distributed monitoring and control of load tap changer ... · distributed monitoring and control of...
TRANSCRIPT
-
Distributed Monitoring and Control of Load Tap Changer Dynamics
Bai Cui, Ahmed Zamzam, Guido Cavraro, and Andrey BernsteinNREL
Workshop on Autonomous Energy SystemsAugust 2020
Cui et al. (NREL) LTC dynamics August 2020 1 / 32
-
Introduction
Resilience (GMLC report)
The ability to prepare for and adapt to changingconditions and withstand and recover rapidly fromdisruptions.
Understanding the stability of networked LTCs.
I Stable equilibrium point.I Region of attraction characterization.
Designing algorithm for distributed monitoringand control.
Cui et al. (NREL) LTC dynamics August 2020 2 / 32
-
Load Tap Changer (LTC)
Voltage regulation device that controls the voltage of the Medium Voltage (MV) side bychanging the transformer ratio r .
rk+1 =
rk + ∆r if V2 > V
02 + d and rk < r
max
rk −∆r if V2 < V 02 − d and rk > rmin
rk otherwise
Continuous approximation:
ṙ =1
Tc(V2 − V 02 ) rmin ≤ r ≤ rmax
Cui et al. (NREL) LTC dynamics August 2020 3 / 32
-
Instability Mechanism
Cui et al. (NREL) LTC dynamics August 2020 4 / 32
-
Table of Contents
1 Stability Analysis and ROA Characterization
2 Stability Monitoring and Control
3 Explicit ROA Characterization
4 Simulation Results
5 Conclusions
Cui et al. (NREL) LTC dynamics August 2020 5 / 32
-
Dynamical System Model
The dynamic of networked LTC is governed by the following equation:
ṙi =1
Ti(Vs,i (r)− V0,i ), ∀i ∈ VL
where
Vs = −B−1LL [r ]−1BLGVG .
The set of equilibria of the dynamical system is
M = {r ∈ Rn>0 : ṙi (r) = 0, ∀i ∈ VL} .
Define the set P as
P = {r ∈ Rn>0 : ṙi (r) ≥ 0, ∀i ∈ VL} .
Note that M lies on the boundary of P.
Cui et al. (NREL) LTC dynamics August 2020 6 / 32
-
Dynamical System Model
The dynamic of networked LTC is governed by the following equation:
ṙi =1
Ti(Vs,i (r)− V0,i ), ∀i ∈ VL
where
Vs = −B−1LL [r ]−1BLGVG .
The set of equilibria of the dynamical system is
M = {r ∈ Rn>0 : ṙi (r) = 0, ∀i ∈ VL} .
Define the set P as
P = {r ∈ Rn>0 : ṙi (r) ≥ 0, ∀i ∈ VL} .
Note that M lies on the boundary of P.
Cui et al. (NREL) LTC dynamics August 2020 6 / 32
-
Dynamical System Model
The dynamic of networked LTC is governed by the following equation:
ṙi =1
Ti(Vs,i (r)− V0,i ), ∀i ∈ VL
where
Vs = −B−1LL [r ]−1BLGVG .
The set of equilibria of the dynamical system is
M = {r ∈ Rn>0 : ṙi (r) = 0, ∀i ∈ VL} .
Define the set P as
P = {r ∈ Rn>0 : ṙi (r) ≥ 0, ∀i ∈ VL} .
Note that M lies on the boundary of P.
Cui et al. (NREL) LTC dynamics August 2020 6 / 32
-
Stability Analysis and ROA Characterization
Cui et al. (NREL) LTC dynamics August 2020 7 / 32
-
Stable Equilibrium
Equilibria are the intersection of quadratic hypersurfaces. Two things are known:
A maximum equilibrium exists;
It is asymptotically stable.
r1
r2
Stable Eq.
P
Cui et al. (NREL) LTC dynamics August 2020 8 / 32
-
Region of Attraction
r1
r2
α
e2
e1
e3
P
Theorem (Liu & Vu, 89’)
The set A(r∗) := {r : r ≥ r∗} is a region of attraction of α if r∗ ∈ P and α is the onlyequilibrium point in A(r∗).
Cui et al. (NREL) LTC dynamics August 2020 9 / 32
-
Region of Attraction
r1
r2
α
e2
e1
e3
P
Theorem (Liu & Vu, 89’)
The set A(r∗) := {r : r ≥ r∗} is a region ofattraction of α if r∗ ∈ P and α is the only
equilibrium point in A(r∗) .
Computational considerations
Efficient characterization of P?How to ensure no other equilibria in A(r∗)?
Cui et al. (NREL) LTC dynamics August 2020 10 / 32
-
Region of Attraction
r1
r2
α
e2
e1
e3
P
Observations from the figure
1 The set A(r) contains exactly one equilibrium point for all r ∈ int(P).2 All equilibria other than α are unstable.
3 P is convex.
Cui et al. (NREL) LTC dynamics August 2020 11 / 32
-
Main Results
r1
r2
α
e2
e1
e3
P
Proposition
All equilibria other than α are unstable.
Proposition
There is a unique equilibrium point in A(r∗) = {r : r ≥ r∗} for any r∗ ∈ P \M.
Corollary
The set A(r∗) is a region of attraction of α for any r∗ ∈ P.
Cui et al. (NREL) LTC dynamics August 2020 12 / 32
-
Stability Monitoring and Control
Cui et al. (NREL) LTC dynamics August 2020 13 / 32
-
Stability Assessment
A tap position vector r0 is inside the region of attraction if there is r∗, V ∗ such that
(B̃LL + [bs ][r∗]−2
)V ∗ = h,
[r∗]−1V ∗ ≥ V0,0 ≤ r∗ ≤ r0.
u∗=[r∗]−2V∗−−−−−−−−→B̃LLV ∗ + [bs ]u∗ = h,[V ∗]u∗ ≥ [V0]V0,
[u∗]−1V ∗ ≤ [r0]r0,V ≥ 0.
Convex optimization formulation
minu,V≥0
‖B̃LLV + [bs ]u − h‖2
s.t. [V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0.
Cui et al. (NREL) LTC dynamics August 2020 14 / 32
-
Instability Mitigation
Minimum change in load such that the tap position vector r0 is in P:
r1
r2
P
r0
Minimum effort to guarantee resilience
minV ,r,d
‖d‖2
s.t.(B̃LL + [bs − d ][r ]−2
)V = h
[r ]−1V ≥ V00 ≤ r ≤ r00 ≤ d ≤ bs .
Nonconvex: only admits a convex inner approximation.
Cui et al. (NREL) LTC dynamics August 2020 15 / 32
-
Instability Mitigation: Convex Inner Approximation
minV≥0,u,d
‖d‖22
s.t. B̃LLV + [bs ]u − [d ]u = h[V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0,0 ≤ d ≤ bs .
minV≥0,u,d
‖d‖22
s.t. B̃LLV + [bs ]u − d = h[V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0,0 ≤ d ≤ [bs ]u
Cui et al. (NREL) LTC dynamics August 2020 16 / 32
-
Instability Mitigation: Convex Inner Approximation
minV≥0,u,d
‖d‖22
s.t. B̃LLV + [bs ]u − [d ]u = h[V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0,0 ≤ d ≤ bs .
minV≥0,u,d
‖d‖22
s.t. B̃LLV + [bs ]u − d = h[V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0,0 ≤ d ≤ [bs ]u
Cui et al. (NREL) LTC dynamics August 2020 16 / 32
-
Stability Monitoring and Control
Both monitoring and mitigation problems can be formulated as a single problem:
minu,V≥0
‖B̃LLV + [bs ]u − h‖2
s.t. B̃LLV ≤ h[V ]u ≥ [V0]V0,
[u]−1V ≤ [r0]r0.
Stable if optimal cost = 0.
Can recover needed Q support otherwise.
Cui et al. (NREL) LTC dynamics August 2020 17 / 32
-
ADMM-based Distributed Implementation
minx,z
ns∑i=1
fi (x i )
s.t. x i ∈ Xi , i = 1, . . . , nsW ij = z j ,Vj = z
j , i = 1, . . . , ns , j ∈ N ai
ns : number of connected subgraphs (agents)
Ni : the bus set of the ith agent;N ai : the set of buses adjacent to the ith agent;x i =
({Vj}j∈Ni , {uj}j∈Ni , {W
ij}j∈N ai)
collect the optimization variables of agent i
The corresponding augmented Lagrangian with penalty parameter ρ is
Lρ(x , z ,λ,µ) =ns∑i=1
fi (x i ) +∑i∈B
(λi (Vi − z i ) +
ρ
2(Vi − z i )2
)+
ns∑i=1
∑j∈N ai
(µij(W ij − z j) + ρ
2(W ij − z j)2
).
Cui et al. (NREL) LTC dynamics August 2020 18 / 32
-
ADMM-based Distributed Implementation
ADMM performs the following iterative updates:
x ik+1 = arg minx i∈Xi
{fi (x i ) +
∑j∈B∪Ni
(λjkVj +
ρ
2(Vj − z jk)
2)
+∑j∈N ai
(µijkW
ij +ρ
2(W ij − z jk)
2)}
, ∀i ∈ [ns ]
z ik+1 = arg min
{ ∑j :i∈N aj
(−µjikz
i +ρ
2(W jik+1 − z
i )2)− λikz i +
ρ
2(Vi,k+1 − z i )2
},
λik+1 = λik + ρ
(Vi,k+1 − z ik+1
), ∀i ∈ B
µijk+1 = µijk + ρ
(W ijk+1 − z
jk+1
), ∀i , ∀j ∈ N ai
Cui et al. (NREL) LTC dynamics August 2020 19 / 32
-
ADMM-based Distributed ImplementationInitialization: Multipliers µ(0),λ(0).repeat
[S1] Each agent i receives the multipliers and voltage estimates from its neighbors,update local variables, and broadcasts the resulting voltage to its neighbors.
[S2] Each agent i uses its updated voltages, multipliers, and received bus voltagesand multipliers to compute its voltage estimates and broadcasts them to itsneighbors.
[S3] Each agent i updates its multipliers using its own updated bus voltages,estimated voltages, as well as received voltage estimates.
until Primal & dual residuals are small
0 50 100 150 200
10−7
10−5
10−3
10−1
101
Number of iteration
Rel
ativ
eer
ror
Figure
Cui et al. (NREL) LTC dynamics August 2020 20 / 32
-
Explicit ROA Characterization
Cui et al. (NREL) LTC dynamics August 2020 21 / 32
-
ROA Characterization
Finding minimum tap position in P along certain direction:
minV ,r
c>r
s.t.(B̃LL + [bs ][r ]−2
)V = h (Flow constraint)
V ≥ [r ]V0 (Secondary side voltage requirement)r ≥ 0.
Each direction c determines a (possibly distinct) inner approximation A(r∗(c)) of thetrue ROA, and their union characterizes a maximal inner approximation of the ROA:
A∪ :=⋃c≥0
c>1=1
A(r∗(c)).
Cui et al. (NREL) LTC dynamics August 2020 22 / 32
-
ROA Characterization
r7
r8
0.2 0.5 0.70.2
0.5
0.7
Qpre
Qpost
r∗
Figure: ROA Characterizations for IEEE39-bus system before and after line tripping.
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1Pre-contingency
Post-contingency
Figure: LTC dynamics at bus 8 before andafter line tripping.
Cui et al. (NREL) LTC dynamics August 2020 23 / 32
-
Ellipsoidal Inner Approximation of the ROA
(B̃LL + [bs ][r ]−2
)V = h,
[r ]−1V ≥ V0.
u=[r ]−2V−−−−−−→ B̃LLV + [bs ]u = h,[V ]u ≥ [V0]V0.
The problem of finding the maximum volume inscribed ellipsoid in V -space can be cast as
maxC�0,α
log det C
s.t. [Cξ + α]u(ξ) ≥ [V0]V0 ∀‖ξ‖2 ≤ 1
u(ξ) = [bs ]−1(h − B̃LL(Cξ + α)
)∀‖ξ‖2 ≤ 1
Quite surprisingly, each of the robust SOC constraint above can be reformulated as anSDP with LMIs of dimension 2(n − 1)× 2(n − 1) so the above problem admits atractable reformulation.
Cui et al. (NREL) LTC dynamics August 2020 24 / 32
-
Ellipoidal Inner Approximation of the ROA
Ellipsoid in V -space:
{V = Cξ + α, ‖ξ‖2 ≤ 1}.
With B̃LLV + [bs ]u = h, ellipsoid in u-space is
{u = Dξ + β, ‖ξ‖2 ≤ 1}.
Then the Hadamard division of vectors in the two ellipsoids parametrized by the same ξgives rise to a subset of P2 := {r :
√r ∈ P} that is linear-fractional, which is
C :={r̃ ∈ Rn>0 : r̃i = (c>i ξ + αi )/(d>i ξ + βi ), ‖ξ‖2 ≤ 1, ∀i ∈ VL
}.
We can rewrite C as an SOC set by introducing new variables: let yi = ξ/(d>i ξ + βi ) andti = 1/(d>i ξ + βi ), then C can be rewritten as
C ={r̃ ∈ Rn>0 : r̃i = c>i yi + αi ti , d>i yi + βi ti = 1, ‖yi‖2 ≤ ti ,∀i ∈ VL
}.
Cui et al. (NREL) LTC dynamics August 2020 25 / 32
-
Simulation Results
Cui et al. (NREL) LTC dynamics August 2020 26 / 32
-
Simulation Results: Test System
IEEE 39-bus system (decoupled Q-V model).
r7
r8
0.2 0.5 0.70.2
0.5
0.7
Qpre
Qpost
r∗
Figure: ROA Characterizations for IEEE39-bus system before and after line tripping.
Four scenarios:
1 Steady-state (tap position inQpost) after line (8, 9) outage;
2 Tap position r∗ after line (8, 9)outage;
3 Tap position r∗ after line (8, 9)outage with additional load;
4 Tap position r∗ after line (3, 4)outage with additional load.
Cui et al. (NREL) LTC dynamics August 2020 27 / 32
-
Simulation Results: Convergence Rate
Partition 39-bus system into threesubsystems.
Stopping criterion: relative errorless than 10−4.
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1Without Q support
With Q support
Figure: Dynamics at bus 8 (scenario 3).
ScenarioOptimal # of
Time (sec.)Time per
objective iterations subsystem (sec.)1 0 39 33.01 11.002 4.1870 89 73.52 24.513 12.3824 83 65.35 21.784 20.4829 113 109.72 36.57
Table: Simulation Results on Convergence Rate
Cui et al. (NREL) LTC dynamics August 2020 28 / 32
-
Simulation Results: Reactive Power Support
Scenario Total Load Total support Percentage1 55.10 0 0%2 55.10 1.93 3.50%3 58.004 3.31 5.70%4 58.004 4.68 8.07%
Table: Needed Reactive Power Support.
1 3 4 7 8 9 12 15 16 18 20 21 23 24 25 26 27 28 29
0
2
4
6
8Original Q
Shed Q
Figure: Reactive power load vs support (scenario 3).
Cui et al. (NREL) LTC dynamics August 2020 29 / 32
-
Extension to Full Power Flow Model
2.8 2.9 3 3.1 3.2 3.30.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Figure: PV curve at bus 8.
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
Stable
Unstable
Figure: Dynamics at bus 8.
Stability Monitoring and Control Problem
min ‖Q‖2
s.t. Pi (cii , cij , sij) = −us,igs,i ∀i ∈ N , (i , j) ∈ EQi (cii , cij , sij) = −us,ibs,i + Qi ∀i ∈ N , (i , j) ∈ E
c2ij + s2ij ≤ ciicjj , ∀(i , j) ∈ E
ui ≥ V 20,i , r 2o,iui ≥ cii , ∀i ∈ N .
Cui et al. (NREL) LTC dynamics August 2020 30 / 32
-
Extension to Full Power Flow Model
2.8 2.9 3 3.1 3.2 3.30.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Figure: PV curve at bus 8.
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
Stable
Unstable
Figure: Dynamics at bus 8.
Stability Monitoring and Control Problem
min ‖Q‖2
s.t. Pi (cii , cij , sij) = −us,igs,i ∀i ∈ N , (i , j) ∈ EQi (cii , cij , sij) = −us,ibs,i + Qi ∀i ∈ N , (i , j) ∈ E
c2ij + s2ij ≤ ciicjj , ∀(i , j) ∈ E
ui ≥ V 20,i , r 2o,iui ≥ cii , ∀i ∈ N .
Cui et al. (NREL) LTC dynamics August 2020 30 / 32
-
Conclusions
Summary
New result on stability and ROA characterization of LTC dynamics
Optimization formulations for inner approximation of ROA
Convex formulations for stability monitoring and instability mitigation of LTCdynamics
ADMM-based distributed implementation
Ongoing/future work
Extension to full power flow model
Deal with system uncertainties
Real-time implementation
Cui et al. (NREL) LTC dynamics August 2020 31 / 32
-
Thank you!Questions?
Cui et al. (NREL) LTC dynamics August 2020 32 / 32
Title SlideIntroductionLoad Tap ChangerInstability MechanismTable of ContentsDynamical System Model 1Dynamical System Model 2Dynamical System Model 3Stability Analysis and ROA CharacterizationStable EquilibriumRegion of Attraction 1Region of Attraction 2Region of Attraction 3Main ResultsStability Monitoring and Control 1Stability AssessmentInstability MitigationInstability Mitigation: Convex Inner Approximation 1Instability Mitigation: Convex Inner Approximation 2Stability Monitoring and Control 2ADMM-Based Distributed Implementation 1ADMM-Based Distributed Implementation 2ADMM-Based Distributed Implementation 3Explicit ROA CharacterizationROA Characterization 1ROA Characterization 2Ellipsoidal Inner Approximation of the ROA 1Ellipsoidal Inner Approximation of the ROA 2Simulation ResultsSimulation Results: Test SystemSimulation Results: Convergence RateSimulation Results: Reactive Power SupportExtension to Full Power Flow Model 1Extension to Full Power Flow Model 2ConclusionsThank You