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Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate Lingwen Gan, Ufuk Topcu, Steven Low California Institute of Technology

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

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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.

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Related works Continuous charging 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

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

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EV model with continuous charging rate EV n time plug indeadline Convex Area = Energy storage (pre-specified) : charging profile of EV n

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EV model with discrete charging rate time plug indeadline Finite EV n

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

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Continuous / Discrete charging rate Discrete: discrete optimization Continuous: convex optimization Flatten demand: plug indeadline

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

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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. UtilityEVs “cost” penalty Both the utility and the Evs only needs local information.

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Convergence & Optimality Thm [GTL’11]: The iterations converge to optimal charging profiles: UtilityEVs calculate

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

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Difficulty with discrete charging rates UtilityEVs calculate Discrete optimization Need stochastic algorithm plug indeadline

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Stochastic algorithm to rescue Discrete optimization over plug indeadline Convex optimization over Avoid discrete programming 1 1

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Stochastic algorithm to rescue Discrete optimization over plug indeadline Convex optimization over sample Able to spread charging time, even if EVs are identical 1 1

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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.

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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.

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

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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.

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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.

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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.

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

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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.

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

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

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

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Summary 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.

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