Stochastic Distributed Protocol for Electric Vehicle Charging with Discrete Charging Rate

Lingwen Gan, Ufuk Topcu, Steven LowCalifornia Institute of Technology

Electric Vehicles (EV)are gaining attention

• Advantages over internal combust engine vehicles• On lots of R&D agendas

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.

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

Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate

• Results with continuous charging rate [GTL’11]• Results with discrete charging rate

EV model withcontinuous charging rate

EV n

time

char

ging

rate

plug in deadline

ConvexArea = Energy storage (pre-specified)

: charging profile of EV n

EV model withdiscrete charging rate

time

char

ging

rate

plug in deadline

Finite

EV n

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

Continuous / Discrete charging rate

Discrete: discrete optimization

Continuous: convex optimization

Flatten demand:ch

argi

ng ra

te

plug in deadline

Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate

• Results with continuous charging rate [GTL’11]• Results with discrete charging rate

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.

Convergence & Optimality

Thm [GTL’11]: The iterations converge to optimal charging profiles:

Utility EVs

calculate

Outline• EV model and optimization problem– Continuous charging rate– Discrete charging rate

• Results with continuous charging rate [GTL’11]• Results with discrete charging rate

Difficulty with discrete charging rates

Utility EVs

calculate

Discrete optimizationNeed stochastic algorithmch

argi

ng ra

te

plug in deadline

Stochastic algorithm to rescue

Discrete optimizationover

char

ging

rate

plug in deadline

Convex optimizationover

Avoid discrete programming

1

1

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

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.

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 .

Distributed stochastic charging algorithm

1

1

The objective value is a supermartingale.

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.

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.

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.

Near optimal

When the number of EVs is large, very close to optimal.

Thm: Every equilibrium has a uniform sub-optimality ratio bound

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.

Fast convergence

time of day

demand

basedemand

Stop after 10 iterations

Iteration 1~5 Iteration 6~10

Iteration 11~15 Iteration 16~20

Close to optimal

Demand(kW/house)

Close to flat

Theoretical sub-optimality bound

Suboptimalityratio

# EVs in 100 housesAlways below 3% sub-optimality.

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.