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![Page 1: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/1.jpg)
Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate
Lingwen Gan, Ufuk Topcu, Steven LowCalifornia Institute of Technology
![Page 2: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/2.jpg)
Electric Vehicles (EV)are gaining attention
• Advantages over internal combust engine vehicles• On lots of R&D agendas
![Page 3: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/3.jpg)
Challenges of EV• EV itself• Integration with the power grid– Overload distribution circuit– Increase voltage variation– Amplify peak electricity load
time
demand
Non-EV demand
Uncoordinated charging
Coordinated charging
Coordinate charging to flatten demand.
![Page 4: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/4.jpg)
Related works
Continuouscharging rate
This work:• Decentralized• Optimally flattened demand• Discrete charging rate
• Centralized charging control– [Clement’09], [Lopes’09], [Sortomme’11]– Easy to obtain global optimum– Difficult to scale
• Decentralized charging control– [Ma’10], [GTL’11]– Easy to scale– Difficult to obtain global optimum
![Page 5: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/5.jpg)
Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate
• Results with continuous charging rate [GTL’11]• Results with discrete charging rate
![Page 6: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/6.jpg)
EV model withcontinuous charging rate
EV n
time
char
ging
rate
plug in deadline
ConvexArea = Energy storage (pre-specified)
: charging profile of EV n
![Page 7: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/7.jpg)
EV model withdiscrete charging rate
time
char
ging
rate
plug in deadline
Finite
EV n
![Page 8: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/8.jpg)
Global optimization: flatten demand
Utility
EV NEV 1
time of day
dem
and
(kW
)
: charging profile of EV n
base demanddemand
Optimal charging profiles = solution to the optimization
![Page 9: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/9.jpg)
Continuous / Discrete charging rate
Discrete: discrete optimization
Continuous: convex optimization
Flatten demand:ch
argi
ng ra
te
plug in deadline
![Page 10: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/10.jpg)
Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate
• Results with continuous charging rate [GTL’11]• Results with discrete charging rate
![Page 11: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/11.jpg)
Distributed algorithm (continuous charging rate)[GTL’11]: L. Gan, U. Topcu and S. H. Low, “Optimal decentralized protocols for electric vehicle charging,” in Proceeding of Conference of Decision and Control, 2011.
Utility EVs
“cost” penalty
Both the utility and the Evs only needs local information.
![Page 12: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/12.jpg)
Convergence & Optimality
Thm [GTL’11]: The iterations converge to optimal charging profiles:
Utility EVs
calculate
![Page 13: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/13.jpg)
Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate
• Results with continuous charging rate [GTL’11]• Results with discrete charging rate
![Page 14: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/14.jpg)
Difficulty with discrete charging rates
Utility EVs
calculate
Discrete optimizationNeed stochastic algorithmch
argi
ng ra
te
plug in deadline
![Page 15: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/15.jpg)
Stochastic algorithm to rescue
Discrete optimizationover
char
ging
rate
plug in deadline
Convex optimizationover
Avoid discrete programming
1
1
![Page 16: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/16.jpg)
Stochastic algorithm to rescue
Discrete optimizationover
char
ging
rate
plug in deadline
Convex optimizationover
sample
Able to spread charging time,even if EVs are identical
1
1
![Page 17: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/17.jpg)
Challenge with stochastic algorithm
Tool: supermartingale.
• Examples of stochastic algorithm– Genetic algorithm, simulated annealing– Converge almost surely (with probability 1)– Converge very slowly• In order to obtain global optima• Do not have equilibrium points
• What we do?– Develop stochastic algorithms with equilibrium points.– Guarantee these equilibrium points are “good”.– Guarantee convergence to equilibrium points.
![Page 18: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/18.jpg)
Supermartingale
Def: We call the sequence a supermartingale if, for all ,(a)(b)
Thm: Let be a supermartingale and suppose that are uniformly bounded from below. Then
For some random variable .
![Page 19: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/19.jpg)
Distributed stochastic charging algorithm
1
1
The objective value is a supermartingale.
![Page 20: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/20.jpg)
Interpretation of the minimization
To find the distribution, we minimize
Average load of others Direction to shift
Shift in the direction to flatten the average load of others.
![Page 21: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/21.jpg)
Challenge with stochastic algorithm
Tool: supermartingale.
• Examples of stochastic algorithm– Genetic algorithm, simulated annealing– Converge almost surely (with probability 1)– Converge very slowly• In order to obtain global optima• Do not have equilibrium points
• What we do?– Develop stochastic algorithms with equilibrium points.– Guarantee these equilibrium points are “good”.– Guarantee convergence to equilibrium points.
![Page 22: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/22.jpg)
Equilibrium charging profile
Def: We call a charging profile equilibrium charging profile, provided that
for all k≥1.
Genetic algorithm & simulated annealingdo not have equilibrium charging profiles.
Thm: (i) Algorithm DSC has equilibrium charging profiles; (ii) A charging profile is equilibrium, iff it is Nash equilibrium of a game; (iii) Optimal charging profile is one of the equilibriums.
![Page 23: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/23.jpg)
Near optimal
When the number of EVs is large, very close to optimal.
Thm: Every equilibrium has a uniform sub-optimality ratio bound
![Page 24: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/24.jpg)
Finite convergence
Thm: Algorithm DSC almost surely converges to (one of) its equilibrium charging profiles within finite iterations.
Genetic algorithm & simulated annealingnever converge in finite steps.
![Page 25: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/25.jpg)
Fast convergence
time of day
demand
basedemand
Stop after 10 iterations
Iteration 1~5 Iteration 6~10
Iteration 11~15 Iteration 16~20
![Page 26: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/26.jpg)
Close to optimal
Demand(kW/house)
Close to flat
![Page 27: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/27.jpg)
Theoretical sub-optimality bound
Suboptimalityratio
# EVs in 100 housesAlways below 3% sub-optimality.
![Page 28: Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062515/56649c815503460f94939d48/html5/thumbnails/28.jpg)
Summary
Thank you!
suboptimality
• Propose a distributed EV charging algorithm.– Flatten total demand– Discrete charging rates– Stochastic algorithm
• Provide theoretical performance guarantees– Converge in finite iterations– Small sub-optimality at convergence
• Verification by simulations.– Fast convergence– Close to optimal.