stochastic distributed protocol for electric vehicle charging with discrete charging rate lingwen...
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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.
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