david fournier , phd cifre inria /ge transporation , 3rd year
DESCRIPTION
Metro Energy Optimization through Rescheduling: Mathematical Model and Heuristics Compared to MILP and CMA-ES. David FOURNIER , PhD CIFRE Inria /GE Transporation , 3rd year. CWM3EO BUDAPEST. SEPTEMBER 26TH 2014. Context. Use of regenerative braking - PowerPoint PPT PresentationTRANSCRIPT
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Metro Energy Optimization through Rescheduling: Mathematical Model and Heuristics Compared to MILP and CMA-ESDavid FOURNIER, PhD CIFRE Inria/GE Transporation, 3rd yearCWM3EO BUDAPEST SEPTEMBER 26TH 2014
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Context
• Use of regenerative braking
• Metro lines without energy storage
devices
• Key industrial problem and differentiator
for GE
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Energy Saving Techniques
Gonzales et al. « A systems approach to reduce urban rail energy consumption », Energy Consumption and Management vol. 80 2014
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Energy Saving Techniques
Best investment cost / energy saving potential ratio
Gonzales et al. « A systems approach to reduce urban rail energy consumption », Energy Consumption and Management vol. 80 2014
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• Powerful accelerations
after departing a
station
• Powerful braking
before arriving to a
station
Braking and Acceleration Phases
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 610
10
20
30
40
50
60
70
speed(km...
time(s)
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61
-6000
-4000
-2000
0
2000
4000
6000
8000
power(kW)
time(s)
Acc
eler
atin
g Coasting
Station A
Braking
Traction En-ergy Regenerative
Energy
Time slot 1
Time slot 2
Time slot 3
Time slot 4
Station B
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A B
A B
time
Power
Trains synchronization
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Our contribution
1. Classification of energy optimization
timetabling problems
2. Fast and accurate approximation of the
instant power demand
3. Conception, implementation and
comparison of a dedicated heuristic for
the problem7
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Timetable Classification
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{Objective function,
Timetable problems classification
Global consumption G
Power peaks PP
Departure times dep
Dwell times dwe
Interstation times int
Decision variables,
Instant powerEvaluation}
Simulator sim
Linear approx. lin
Non linear approx. nonlin9
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Timetable problems classification
Authors Class ArticleAlbrecht (PP, dwe-int,
sim)Reducing power peaks and energy consumption in rail transit systems by simultaneous metro running time control
Sanso et al. (PP, dwe, sim) Trains scheduling desynchronization and power peak optimization in a subway system
Kim et al. (PP, dep, lin) A model and approaches for synchronized energy saving in timetabling
Miyatake and Ko
(G, dep-int, sim)
Numerical analyses of minimum energy operation of multiple trains under DC power feeding circuit
Peña et al. (G, dep-dwe, lin)
Optimal underground timetable design based on power flow for maximizing the use of regenerative-braking energy
Nasri et al. (G, dwe, sim) Timetable optimization for maximum usage of regenerative energy of braking in electrical railways systems
Fournier et al.
(G, dwe, nonlin)
A Greedy Heuristic for Optimizing Metro Regenerative Energy Usage
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Tackling (G,dwe,nonlin)
• G: Global energy consumption minimization• dwe: Dwell times modifications, rest fixed• nonlin: Power flow based objective function
• +10000 variables
• Non-linear objective function
• No dedicated algorithm11
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Instant Power Demand
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• Bottleneck of the optimization• Arithmetic sum of the instant power demands computed with a
power flow at each time interval
The objective function G
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Timetable instant power demands on each time interval
Time intervals (second)
Pow
er
(kW
)
𝐺= ∑𝑖
𝑇𝑖𝑚𝑒 𝑠𝑡𝑒𝑝𝑠
𝑃 𝑖
: instant power demand of time step 20
- 13
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315
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How to evaluate the transfer of energy
between metros and the total amount of
energy flowing through the electric
substations ?
Instant Power Demand
• Joule effects in the third rail
• Non linear electricity equations
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S1 S2 S3 S4 S5 S6 S7
Metro Line Model
Electric points being:• Metro platforms• Electric substations
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S1 S2 S3 S4 S5 S6 S7
R1 R2 R3 R4 R5 R6 R7 R8 R9
1500V 1500V 1500V 1500V 1500V 1500V
ESS1 ESS2 ESS3 ESS4 ESS5 ESS6
Electric substations are represented by a voltage source. Metro platforms by an electrical potential point.
Metro Line Model
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S1 S2 S3 S4 S5 S6 S7
R1 R2 R3 R4 R5 R6 R7 R8 R9
1500V 1500V 1500V 1500V 1500V 1500V
Pacc=1.5MW Pbrk=0.5MW
Trains accelerate or brake near a metro platform.They consume or produce power.
ESS1 ESS2 ESS3 ESS4 ESS5 ESS6
Each time step, a set of braking and accelerating metros run on the line.
Metro Line Model
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• Voltages in Electric Substations• Power produced/consumed by trains• Resistances between points in line
Known Parameters
Unknown Variables
• Voltages in every electric points• Intensity flowing through the circuit
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Solving electricity equations gives voltages and
intensities
Non revertible electric substations
Total power demand:
S1 S2 S3 S4 S5 S6 S7
R1 R2 R3 R4 R5 R6 R7 R8 R9
1500V 1500V 1500V 1500V 1500V 1500V
ESS1 ESS2 ESS3 ESS4 ESS5 ESS6
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Instant power demand
Pacc=1.5MW Pbrk=0.5MW
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• Electric simulation accurate but slow.• Computation of the power ratio a braking metro will transfer to
an accelerating metro between each pair of stations of the line.
S1 S2 S3 S4 S5 S6 S7
R1 R2 R3 R4 R5 R6 R7 R8 R9
1500V 1500V 1500V 1500V 1500V 1500V
Pacc=1,5MW Pbrk=0,5MWTransfer 85% = 0,425MW
ESS1 ESS2 ESS3 ESS4 ESS5 ESS6
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Distribution matrix
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S1 S2 S3 S4 S5 S6 S7
R1 R2 R3 R4 R5 R6 R7 R8 R9
1500V 1500V 1500V 1500V 1500V 1500V
Transfer 80% = 0,4MW
ESS1 ESS2 ESS3 ESS4 ESS5 ESS6
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Pacc=1,5MW Pbrk=0,5MW
• Electric simulation accurate but slow.• Computation of the power ratio a braking metro will transfer to
an accelerating metro between each pair of stations of the line.
Distribution matrix
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Power flow model
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• Each braking metro can transfer its power to
all accelerating metros• The transfer is attenuated by the distribution
factor• Modelled as a generalized flow
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Power flow model
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BrakingTrains
AcceleratingTrains
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Power flow model
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BrakingTrains
AcceleratingTrains
Metro 1 is braking, regenerating a power
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Power flow model
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BrakingTrains
AcceleratingTrains
It transfers its power to Metro 6.The power transferred is
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Power flow model
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BrakingTrains
AcceleratingTrains
Continue with all braking metros until demand is fulfilled or all braking metros transfer their energy
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• Bottleneck of the optimization• Arithmetic sum of the instant power demands computed with
the power flow at each time interval
The objective function G
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Timetable instant power demands on each time interval
Time intervals (second)
Pow
er
(kW
)
𝐺= ∑𝑖
𝑇𝑖𝑚𝑒 𝑠𝑡𝑒𝑝𝑠
𝑃 𝑖
: instant power demand of time step 20
- 27
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315
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Optimization Methods
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1. Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
implementation (Hansen, 2006) on a vector of continuous variables.
2. MILP model based on the maximization of overlapping
braking/accelerating phases
3. Greedy heuristics. Synchronization for all braking phases of the
accelerating phase that minimizes the local energy consumption.
Optimization methods
• Use of the customer timetable• Modification of the dwell times length by {-3,…,9} seconds.• Slight modification to ensure feasibility.• Time interval of 1s for energy computation.
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• Evolutionary algorithm for non-linear continuous
optimization• New candidates are sampled according to a
multivariate normal distribution• Objective function arbitrarily complex.• Quasi parameter-free
CMA-ES
Wikipedia
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• A candidate solution is a vector of deltas on dwell times .• Initial point is the original timetable .• Bound handling using a penalty function added to the fitness
function
• Before computing the fitness function, vector of deltas is
discretized by rounding float numbers to their closest integer.
CMA-ES
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• Mixed Integer Linear Programming
• Widely used and studied in OR
• CPLEX as a very powerful solver
• Need to linearize all equations
MILP
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• One braking phase can transfer its energy to at most one
acceleration phase• One acceleration phase can benefit from at most one braking
phase• Each potential transfer is attenuated by a distribution factor• Maximization of the overlapping times between braking and
acceleration phases, multiplied by their distribution factor
MILP Linearization
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Greedy Heuristics
• Shift acceleration phases to
synchronize with braking phases
• Shifting acceleration phases are
equivalent to modify dwell times
• Improvements computed with
electric-based equations
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Algorithm
B A
A
A
B A
A
Train 1
Train 2
Train 3
Train 4
Train 5time
B
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Algorithm
B A
A
A
B A
A
1. Choose the first active braking interval of the time horizonTrain 1
Train 2
Train 3
Train 4
Train 5time
B
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Algorithm
A
A
A
B A
AMax shift
B
2. Shift neighbour acceleration phases and compute energy savingsTrain 1
Train 2
Train 3
Train 4
Train 5
3. Shift best neighbour and fix both braking and acceleration phases
Savings maximized
timeB
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Algorithm
A
A
A
B A
A
B
3. Shift best neighbour and fix both braking and acceleration phasesTrain 1
Train 2
Train 3
Train 4
Train 5time
B
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Algorithm
A
A
A
B A
A
B
1. Choose the first active braking interval of the time horizonTrain 1
Train 2
Train 3
Train 4
Train 5time
B
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Algorithm
A
A
A
A
A
B
Max shift
B
2. Shift neighbour acceleration phases and compute energy savingsTrain 1
Train 2
Train 3
Train 4
Train 5
Savings maximized
3. Shift best neighbour and fix both braking and acceleration phases
timeB
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Algorithm
A
A
A
A
A
B
B
3. Shift best neighbour and fix both braking and acceleration phases
time
Train 1
Train 2
Train 3
Train 4
Train 5 B
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Algorithm example
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Results
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Benchmark instances
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• 6 small sample timetables• 15 minutes or 60 minutes• Peak or off-peak hour• 31 stations, 15~30 metros• Available at
http://lifeware.inria.fr/wiki/COR14/Bench
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CMA-ES vs. Heuristics
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CMA-ES : Covariance Matrix Adaptation Evolution Strategy, Hansen 2006
The greedy heuristics outperforms CMA-ES in terms of computation time and objective function on 5 of the 6 benchmark instances
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CMA-ES vs. Heuristics
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MILP linear objective
On 4 instances, CPLEX finds the optimum.On all 6 instances, CPLEX outperforms the greedy heuristics, on the overlapping times objective function
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MILP vs. Heuristics
But comparing the real objective function, the heuristics does better on 5 instances.
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Robustness
Adding noise on variables show that output solutions still save energy even with 3 seconds noises
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Full TimetableHighest gainson peak hours
5.1 % savings
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Take Home Messages
• Classification of timetable energy
optimization problems• Fast and accurate evaluation of the
timetable energy consumption• Greedy dedicated heuristic fast and
effective for offline and online
optimization
Current work on timetable creation
from scratch51