Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate

1. Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, UfukTopcu, Steven Low California Institute of Technology

2. Electric Vehicles (EV)are gaining attention • Advantages over internal combust engine vehicles • On lots of R&D agendas

3. Challenges of EV • EV itself • Integration with the power grid • Overload distribution circuit • Increase voltage variation • Amplify peak electricity load Uncoordinated charging demand Coordinated charging Non-EV demand time Coordinate charging to flatten demand.

4. Related works • 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 Continuous charging rate • This work: • Decentralized • Optimally flattened demand • Discrete charging rate

5. Outline • EV model and optimization problem • Continuous charging rate • Discrete charging rate • Results with continuous charging rate [GTL’11] • Results with discrete charging rate

6. EV model withcontinuous charging rate EV n : charging profile of EV n Convex charging rate Area = Energy storage (pre-specified) plug in deadline time

7. EV model withdiscrete charging rate EV n charging rate plug in deadline time Finite

8. Global optimization: flatten demand : charging profile of EV n Utility demand (kW) EV 1 EV N base demand demand time of day Optimal charging profiles = solution to the optimization

9. Continuous / Discrete charging rate Flatten demand: Continuous: convex optimization Discrete: discrete optimization charging rate plug in deadline

10. Outline • EV model and optimization problem • Continuous charging rate • Discrete charging rate • Results with continuous charging rate [GTL’11] • Results with discrete charging rate

11. Distributed algorithm (continuous charging rate) [GTL’11]: L. Gan, U. Topcuand 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.

12. Convergence & Optimality Utility EVs calculate Thm [GTL’11]: The iterations converge to optimal charging profiles:

13. Outline • EV model and optimization problem • Continuous charging rate • Discrete charging rate • Results with continuous charging rate [GTL’11] • Results with discrete charging rate

14. Difficulty with discrete charging rates Utility EVs calculate charging rate Discrete optimization Need stochastic algorithm plug in deadline

15. Stochastic algorithm to rescue 1 1 Discrete optimization over charging rate Convex optimization over plug in deadline Avoid discrete programming

16. Stochastic algorithm to rescue 1 1 Discrete optimization over charging rate Convex optimization over sample plug in deadline Able to spread charging time, even if EVs are identical

17. Challenge with stochastic algorithm • 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. Tool: supermartingale.

18. 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 .

19. Distributed stochastic charging algorithm 1 1 The objective value is a supermartingale.

20. 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.

21. Challenge with stochastic algorithm • 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. Tool: supermartingale.

22. Equilibrium charging profile Def: We call a charging profile equilibrium charging profile, provided that for all k≥1. Genetic algorithm & simulated annealing do 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.

23. Near optimal Thm: Every equilibrium has a uniform sub-optimality ratio bound When the number of EVs is large, very close to optimal.

24. Finite convergence Thm: Algorithm DSC almost surely converges to (one of) its equilibrium charging profiles within finite iterations. Genetic algorithm & simulated annealing never converge in finite steps.

25. Fast convergence Iteration 6~10 Iteration 1~5 base demand demand Iteration 11~15 Iteration 16~20 Stop after 10 iterations time of day

26. Close to optimal Demand (kW/house) Close to flat

27. Theoretical sub-optimality bound Suboptimality ratio # EVs in 100 houses Always below 3% sub-optimality.

28. Summary • 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. suboptimality Thank you!