On active constraints in optimal power flow Learning optimal solutions and identifying important constraints
Line A Roald University of Wisconsin ndash Madison
with Sidhant Misra (LANL) Yee Sian Ng (MIT)
and Daniel K Molzahn (Georgia Tech)
NREL Workshop April 12 2019
Transmission System Operation
41619 | 2
Transmission System Operation
Higher and frequently Increased uncertainty changing power flows
41619 | 3
Transmission System Operation
Transmission System Operator
Higher and frequently Increased uncertainty changing power flows
41619 | 4
Impact of uncertainty
Non-Linear Network
How to maintain grid security
Chance-constrained robust Adapt to uncertainty stochastic optimization in real time
41619 | 5
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Transmission System Operation
41619 | 2
Transmission System Operation
Higher and frequently Increased uncertainty changing power flows
41619 | 3
Transmission System Operation
Transmission System Operator
Higher and frequently Increased uncertainty changing power flows
41619 | 4
Impact of uncertainty
Non-Linear Network
How to maintain grid security
Chance-constrained robust Adapt to uncertainty stochastic optimization in real time
41619 | 5
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Transmission System Operation
Higher and frequently Increased uncertainty changing power flows
41619 | 3
Transmission System Operation
Transmission System Operator
Higher and frequently Increased uncertainty changing power flows
41619 | 4
Impact of uncertainty
Non-Linear Network
How to maintain grid security
Chance-constrained robust Adapt to uncertainty stochastic optimization in real time
41619 | 5
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Transmission System Operation
Transmission System Operator
Higher and frequently Increased uncertainty changing power flows
41619 | 4
Impact of uncertainty
Non-Linear Network
How to maintain grid security
Chance-constrained robust Adapt to uncertainty stochastic optimization in real time
41619 | 5
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Impact of uncertainty
Non-Linear Network
How to maintain grid security
Chance-constrained robust Adapt to uncertainty stochastic optimization in real time
41619 | 5
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
The Optimal Power Flow Problem
41619 | 6
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
ABC (3 4 + 5) le ABC9lt= F isin ℒ
Minimize generation cost
HI
HJ HLNon-Linear AC Power Flow
Generation constraints
HJHIVoltage
constraints HK
Transmission constraints
HJ
HK
41619 | 7
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
Goal Low cost operation while enforcing technical limits
)min sumisin ()+ + ( +
st 2 3 4 + 5 = 0
+8 9lt= gt isin 9 le +8 le +858 9lt= gt isin 9 le 58 le 58
9 le 4 le 49lt=4 isin ℬ
9lt=ABC (3 4 + 5) le ABC F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 1
Typically only very few transmission constraints are active at optimum
Can be exploited
algorithmically
Eg constraint generation [Bienstock Harnett and Chertkov
SIAM Review 2013]
hellip
41619 | 8
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 9
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
Impact of renewable energy variationsload uncertainty
min sumampisin( )amp-amp + )amp -amp()
st 5 6() 7() () 8() = 0
- lt A isin ( ltamp= le -() le -8- lt A isin ( ltamp= le 8-() le 8-
7ampltamp= le 7amp() le 7amplt B isin ℬ
ltBDE () le BDE F isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
Observation 2 Typically only very few transmission constraints are ever active even for different parameters
The topic of this talk Is this observation true How can we use it
41619 | 10
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 11
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning solutions to (power system) optimization problems
Yee Sian Ng Sidhant Misra (MIT) (LANL)
Yee Sian Ng Sidhant Misra Line Roald and Scott Backhaus laquoStatistical Learning for DC Optimal Power Flowraquo Power System Computation Conference (PSCC) 2018
Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 12
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
Impact of renewable energy variationsload uncertainty amp
min sum+isin- +12+ amp + 4+ 12+(amp)
st (amp) lt(amp) 1(amp) =(amp) = 0
12 ADE F isin -A+B le 12(amp) le 12=2 ADE F isin -A+B le =2(amp) le =2
ADElt+A+B le lt+(amp) le lt+ G isin ℬ
ADEGIJ (amp) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 13
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
lowastResolve problem every 5-15 min For each 2 obtain 01 2
min sumisin -01 2 + -4 01(2)
st (2) lt(2) 0(2) =(2) = 0
01 ADE F isin AB le 01(2) le 01=1 ADE F isin AB le =1(2) le =1
ADEltAB le lt(2) le lt G isin ℬ
ADEGIJ (2) le GIJ K isin ℒ
Minimize generation cost
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
$
Transmission constraints
$
41619 | 14
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Repeated solution process
OPF at rarr amplowast () OPF at rarr amplowast () OPF at + + rarr amplowast (+)
hellip
Can we use learning to speed up the solution process by using information from past solutions - 0
lowast -
41619 | 15
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning for optimization
Renewable Prediction method
Optimal energy dispatch
lowast$ $
Can we use learning to speed up the solution process by using information from past solutions $
lowast $
41619 | 16
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First attempt
Train a neural net
41619 | 17
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
41619 | 18
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well
(DISCLAIMER I will admit that we gave up quite fast)
41619 | 19
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 20
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints
ndash Projection back onto feasible space cause suboptimalityhellip
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 21
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
In-depth literature review Sidhant Misra Line Roald and Yee Sian Ng laquoLearning for Constrained Optimizationraquo submitted available online httpsarxivorgabs180209639
41619 | 22
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
bull This didnrsquot work well ndash Hard to satisfy safety constraints ndash Projection back onto feasible space cause suboptimalityhellip ndash Challenging High-dimensional input rarr High dimensional output
bull This can work well under some circumstances Wide enough and deep enough and with enough data [Karg and Lucia 2018]
41619 | 23
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
First Attempt ndash Train a Neural Net
Renewable Dispatchenergy $
We have a mathematical optimization problem
ndash can we use more information about the problem structure
41619 | 24
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Think again
How can we leverage pre-existing knowledge
about the solution
41619 | 25
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Think again
How can we leverage pre-existing knowledge
about the solution
New idea Learn the optimal set of active constraints
41619 | 26
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal set of active constraints
Set of constraints that are active at optimum
bull Equality (power flow) constraints are always active
bull Only very few of the inequality constraints are active ndash Generation constraints ndash Voltage constraints ndash Transmission constraints
41619 | 27
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Why ndash Optimal active set is the ldquominimalrdquo information we need to recover optimal solution
ndash Inherently encodes information about physical constraints and technical limits
ndash Finite low dimensional object
ndash Nice physical interpretation (power system operational pattern)
41619 | 28
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learn optimal set of active constraints
Renewable Predict optimal active set
Predictrecover optimal solution
Optimal active set amplowast $
Optimal
$energy dispatch
lowast $
bull Related to explicit MPC ndash Explicit MPC ndash each optimal active set corresponds to an optimal affine control policy [Bemporad et al 2002] [Pannochia Rawlings Wright 2007] [Zeilinger et al 2011] [Karg and Lucia 2018] hellip
Look only at specific classes of problems Not very scalable Do not consider input distribution over
41619 | 29
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Ensemble Policy
$
amp() amp() amp+()
Realization
Candidates for optimal active set
Solve problem given the active set
rarr Solve a reduced problem with fewer constraints
rarr Solve a set of linear equations (linear problem)
Easier than solving the full optimal power flow problem
41619 | 30
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Ensemble Policy
Realization
Candidates for optimal active set
Solve problem given the optimal active set
Feasible Feasible Evaluate cost and feasibility Infeasible Low cost High cost
$
amp() amp() amp+()
41619 | 31
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Ensemble Policy
Realization
$
amp() amp() amp+()
Infeasible Feasible Low cost
Feasible High cost
Candidates for optimal active set
Solve problem given the optimal active set
Evaluate cost and feasibility
Pick best (optimal) solution
41619 | 32
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Limits of the Approach
Realization Works well if the number of
active sets + is small
Total number of possible
active sets is exponential bull
Maybe only a few are
practically relevant bull
$
amp()
Feasible Low cost
41619 | 33
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Limits of the Approach
Realization
$
amp()
Feasible Low cost
How to identify the collection of
relevant active sets
High probability that one of the active sets is optimal for a
new realization
Do NOT search entire parameter space 41619 | 34
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Using Sampling to Learn Important Active Sets
41619 | 35
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 36
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 37
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
$ $amp+
$
$amp hellip$+ hellip
$amp() hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 38
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 39
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 40
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp$$
Collection of$ $amp+ observed active sets
$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 41
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip
$amp$$ Collection of$ $amp+
observed active sets$
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 42
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 43
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
$$
$
hellip $amp
Collection of$ $amp+ observed active sets
$amp hellip - =$$++ hellip
$amp()
$ $ $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
41619 | 44
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
hellip $amp $
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
$ hellip hellip
hellip
41619 | 45
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
$ helliphellip
hellip $amp
hellip
$
Collection of $ $amp+ observed active sets
$amp hellip - =$+ hellip
$amp()
$ $ hellip $
hellip
samples of input parameters all possible active sets color discovered active sets grey undiscovered active sets
When do I stop 41619 | 46
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm to Learn
Collection of Optimal Active Sets
41619 | 47
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
Optimal Training data lowast solution Optimization Renewable energy realizations and Input problem amplowast Optimallowast corresponding active set amp amp active set
Goal Find a active sets that together have a high probability of being optimal
41619 | 48
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast$lowast lowasthellip
Collection of observed active sets
41619 | 49
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples
lowast lowast hellip lowast $
Collection of observed active sets
2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast $ hellip
How frequently do we observe sets we have not seen before
ABCDEFGHIGJRate of discovery = A
where ( = + maxlog 5 log 5 -
+ gt= 5 = 1 + =
+lt
41619 | 50
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast hellip lowast lowast lowast hellip lowast))))-))
Collection of observed active sets How frequently do we observe sets we have not seen before
G- = HIJKLMNOPNQRate of discovery
H
bull If the rate of discovery is below the threshold $ le amp minus ( stop
where 0 = 23 maxlog = log =
3 EFD= = 1
45
+ A 3BC
D
performance guarantee
41619 | 51
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
41619 | 52
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
Guaranteed to converge
41619 | 53
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Learning Collection of Optimal Active Sets
1 Observe optimal active sets for M samples 2 Check ldquorate of discoveryrdquo for W samples
lowast lowast lowast lowastlowast lowast hellip lowast lowast ) ) hellip )) ) )- )0 )-0
Collection of observed active sets How frequently do we observe sets we have not seen before
bull If the rate of discovery is below Guarantees performance the threshold $ le amp minus ( stop at termination
[Misra Roald Ng 2018] bull If the rate of discovery is too high add more samples
[Misra Roald Guaranteed to converge fast for low number of optimal active sets Ng 2018]
41619 | 54
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Practicability of the approach
Realization
$
amp()
Feasible Low cost
Simultaneously establishes
- Collection of optimal active sets
- Practicability of the approach
No assumptions on distribution
No assumptions on problem structure
Guaranteed to converge fast for low number of optimal active sets
41619 | 55
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Results for the (linear)
DC Optimal Power Flow Problem
41619 | 56
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
41619 | 57
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Probabilistic guarantee ℙ $ lt = 005 Termination 24 le 001
Max difference = 004
Confidence level = 001
Hyperparameter 0 = 2
H Initial W 6 = 13L259 8 IJH 6 =
78 maxlog B log B with B = 1 + 9 E 8FG (constant until B = 201)
Uniform distribution Normal distribution Uncertain loads with supportN ~P 0 Q = 003R N isin [minus009R 009R]
41619 | 58
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Example ndash RTE 1951 bus test case
Number of active sets 5
Rate of discovery 00069
0 10 20 30 40 M=47 W=13rsquo259
When there are few relevant active sets the algorithm terminates fast
41619 | 59
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Example ndash PSERC 200 bus test case
06
Number of active sets 05 5 04
03
Rate of discovery 02
00069 01
00 0 10 20 30 40 M=47
W=13rsquo259 Probability of observed sets
When there are few relevant active sets the algorithm terminates fast
41619 | 60
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Example ndash PSERC 200 bus test case
012 Number of active sets
010 163 008
006
Rate of discovery 004
00099 002
00 0 500 1000 1500 2000 M=2602
W=19rsquo661 Probability of observed sets
When there are many relevant active sets the algorithm terminates slowly
41619 | 61
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
System sizes ranging from 3 to 1951 nodes
41619 | 62
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Normal distribution ~ 0 0 1 = 0033
Uniform distribution with support
isin [minus0093 0093]
41619 | 63
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
41619 | 64
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
41619 | 65
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
41619 | 66
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
41619 | 67
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets
41619 | 68
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate
41619 | 69
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Streaming Algorithm Results ndash PGLib-OPF v 1708
Max undiscovered = 005 Max difference amp = 004 Termination ()+ le 001
Few active sets
Terminates fast
High probability of optimal solutions
Not always Large number of active sets Slow to terminate Lower probability of optimal solutions
41619 | 70
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Practical Implications for Power Systems Operation
41619 | 71
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
41619 | 72
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
41619 | 73
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
IEEE 300 bus system with normally distributed load
Stddev = 1 of load Stddev = 2 of load Stddev = 3 of load Stddev = 4 of load Stddev = 5 of load
Number of unique active sets
Increasing parameter uncertainty = Increasing number of optimal active sets
laquoPower systems operation becomes more unpreditable and complex with increasing uncertaintyraquo
General perception among system operators
What does this imply for system risk Price stability
41619 | 74
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Summary
1 ndash Leverage pre-existing knowledge (mathematical model) improves learning outcomes
2 ndash Using active sets as an intermediate step is useful - encodes all information about optimal solution - finite (and typically low) number of active sets
3 ndash Streaming algorithm establishes practicability of the task - Probabilistic performance guarantees - Guaranteed to terminate - Guaranteed to terminate fast for nice problems
bull Quite general strategy ndash Streaming algorithm can work for very general problems Non-convex AC OPF mixed integer problems hellip ndash Disclaimer Application must be such that the number of active sets is small ndash Alternative strategy Learn possible active constraints instead of active sets
41619 | 75
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Outlook
Realization
$
amp()
Feasible Low cost
Classification
[Deka and Misra 2019]
41619 | 76
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Outlook
Realization
$
amp()
Feasible Low cost
Classification
Efficient solution Active set solver local approximation hellip
41619 | 77
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Preliminary results for the (non-linear non-convex)
AC Optimal Power Flow Problem
41619 | 78
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Rate of discovery of new optimal active sets
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case Rate of discovery of new optimal active sets
0039 0967
(did not terminate)
41619 | 79
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
0967 0039
Rate of discovery of new optimal active sets Rate of discovery of new optimal active sets
(did not terminate)Rate of discovery of new active constraints
00035
Rate of discovery of new active constraints
00001
41619 | 80
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
(did not terminate) Rate of discovery of new active constraints Rate of discovery of new active constraints
AC Optimal Power Flow
Max undiscovered = 01 Max difference amp = 005 Termination ()+ le 005
RTE 1951 bus test case PSERC 200 bus test case
Only 164 of 5192 transmission line Only 28 of 490 transmission line constraints ever active constraints ever active
00035 00001
41619 | 81
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
1 Learning solutions to (power system) optimization problems through optimal active sets
2 Identifying potentially active constraints
41619 | 82
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Identifying potentially active constraints
Daniel K Molzahn
Georgia Tech
Dan Molzahn and Line Roald laquoGrid-Aware versus Grid-Agnostic Distribution System Control A Method for Certifying Engineering Constraint Satisfactionraquo Hawaii International Conference on System Sciences (HICSS) 2019
Line Roald and Dan Molzahn laquoImplied Constraint Satisfaction in Power System Optimization The Impacts of Load Variationsraquo available online httpsarxivorgabs190401757
41619 | 83
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimal Power Flow
-01 2 + -4 01(2)min sumisin $()
st 8 9(2) (2) 0(2) (2) = 0
01gt BC D isin le 01gt(2) le 01gt1gt BC D isin le 1gt(2) le 1gt
le (2) le BC E isin ℬ
BCEGH (2) le EGH I isin ℒ
Before we learned constraints that are likely to be active
Now we want to understand which constraints can possibly be active
Feasible set in the direction of the cost function
The full feasible set
41619 | 84
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
If max $ le $478
and min $ ge $45
Voltage will never go out of bounds
41619 | 85
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimization-based constraint screening
Main idea Minimizemaximize the value of the constraints
min $ max $
st + $ = 0
23 478 9 isin 45 le 23 le 2323 478 9 isin 45 le 23 le 23
$45 le $ le $478 lt isin ℬ
gt ( $ ) le gt478 C isin ℒ
Minimizemaximizevoltage currents hellip
Non-Linear AC Power Flow
Generation constraints
Voltage constraints
Transmission constraints
bull Connections to optimization-based bound tightening [C Coffrin Hijazi and Van Hentenryck 2015]
bull Results for DC OPF [Ardakani and F Bouffard 2013 2015] [Madani Lavaei and Baldick 2017]
bull Our interest - Large ranges of load - AC OPF (distribution grids)
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Distribution grids ndash AC Optimal Power Flow
- Consider ranges of load variations (not controllable by the system operator) - Voltage constraints only on buses we monitorcontrol
min $ max $
st + $ = 0
36734 le 2 le 2236734 le 2 le 22
$34 le $ le $367
Minimizemaximizevoltage
Non-Linear AC Power Flow
8 isin Load 8 isin variations
Voltage constraints 8 isin on nodes with measurementscontrol
Valid bounds
Use convex relaxation
QC relaxation with bound tightening [Coffrin Hijazi and Van Hentenryck 2016 amp 2017]
Challenging
- non-standard objective
(relaxation is weak)
- low-voltage solutions
- hellip
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Can we certify safe operations
IEEE 123 bus system ndash single-phase equivalent [Bolognani and Zampieri 2016]
Start out assuming only one node with controlled voltage
41619 | 88
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
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Can we certify safe operations
Substation
50 load variability What to do about violations
Increasing load variability increasing voltage range 10 085 090 095
Add more controllable nodes and tighten the voltage limits 41619 | 89
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Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
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Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 320905 le amp le 1095
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 32
41619 | 90
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
001
0203
0405
0607
0809
10
01
02
03
04
05
06
07
08
09 1
085
090
095
100
Can we certify safe operations
Substation
Bus 11 092 le amp le 109
50 load variability
085 090 095 10
Added controllability and tighter voltage range on Bus 11
41619 | 91
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Redundant constraints in DC optimal power flow
How many constraints can ever be active in DC optimal power flow
min amp() Minimizemaximize constraints $
st sum+- )(+) minus 34(5) = 0 Power balance Non-redundant
lt 8 le ) le ) Generation constraints Non-redundant
minus)gtlt le ) minus 34 le )gtlt Transmission constraints Often redundant
Allow power demand 34(5) to vary plusmn A where 0 le A le 100
Relax generator lower bounds to 8
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Results on PGLib-OPF test cases
Percentage of line flow Remaining
constraints non-redundant constraints
plusmn Load variation
41619 | 93
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Results on standard test cases
41619 | 94
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Optimization-based constraint screening
bull Main idea ndash Solve optimization problems that minimizemaximize the value of the constraints (If the problems are hard to solve use relaxations to obtain valid lowerupper bounds)
ndash Identify constraints that cannot be violated -gt redundant constraints ndash Identify constraints that can be violated -gt potentially important constraints
bull Works really well for power flow optimization
bull We can use this to (1) identify constraints that need to be monitoredcontrolled (2) reduce the number of considered constraints (3) hellip
41619 | 95
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
THANK YOU
Line Roald roaldwiscedu
41619 | 96
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97
Summary of streaming algorithm results
1 Guaranteed to terminate no need to decide on the number of M samples apriori Definition of the window size W and termination criterion
Theorem 1 and 2 [Misra Roald Ng 2018] ( difference between true and empirical probability of
If the window size() is defined as unobserved active sets 8
98 5 confidence level = amp
() maxlog log with = 1 + 5 67
G hyperparameter gt 1 Then ℙ lt = gt minus gtA le C forall gt 1 le 1 minus F
2 Guaranteed to terminate fast for benign systems
Theorem 3 [Misra Roald and Ng] If a (small) number of relevant active sets HI that contains more than 1 minus JI probability mass then with probability at least 1 minus F minus FI the algorithm terminates in less iterations than
1 = HI log 2 + log
1
J minus JI FI
41619 | 97