Stochastic Distributed Protocol for Electric Vehicle
Charging with Discrete Charging Rate
Lingwen Gan, Ufuk Topcu, Steven Low
California 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
demand
Uncoordinated charging
Coordinated charging
Non-EV demand
time
Coordinate charging to flatten demand.
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
This work:
• Decentralized
• Optimally flattened demand
• Discrete charging rate
Continuous
charging rate
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 with
continuous charging rate
EV n
plug in
: charging profile of EV n
deadline
time
Area = Energy storage (pre-specified)
Convex
EV model with
discrete charging rate
EV n
plug in
deadline
Finite
time
Global optimization: flatten demand
1
Utility
: charging profile of EV n
0.9
0.8
EV 1
EV N
0.7
0.6
0.5
demand
base demand
0.4
20:00
0:00
4:00
8:00
12:00
16:00
time of day
Optimal charging profiles = solution to the optimization
20:00
Continuous / Discrete charging rate
Flatten demand:
Continuous: convex optimization
Discrete: discrete optimization
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
Utility
EVs
calculate
Thm [GTL’11]: The iterations converge to optimal charging profiles:
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 optimization
Need stochastic algorithm
plug in
deadline
Stochastic algorithm to rescue
1
plug in
deadline
Discrete optimization
over
Convex optimization
over
Avoid discrete programming
1
Stochastic algorithm to rescue
1
plug in
deadline
1
Discrete optimization
over
Convex optimization
over
sample
Able to spread charging time,
even if EVs are identical
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.
Supermartingale
Def: We call the sequence
,
(a)
(b)
a supermartingale if, for all
Thm: Let
be a supermartingale and suppose that
uniformly bounded from below. Then
For some random variable
.
are
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
• 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.
Equilibrium charging profile
Def: We call a charging profile
charging profile, provided that
equilibrium
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.
Near optimal
Thm: Every equilibrium has a uniform sub-optimality ratio bound
When the number of EVs is large, very close to optimal.
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.
Fast convergence
Iteration 6~10
Iteration 1~5
demand
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
20:00
4:00
12:00
20:00
0.2
20:00
Iteration 11~15
4:00
12:00
20:00
Iteration 16~20
1.2
1.2
1
1
0.8
0.8
Stop after 100.6iterations
0.6
0.4
0.2
20:00
base
demand
0.4
4:00
12:00
20:00
0.2
20:00
time of day
4:00
12:00
20:00
Close to optimal
Demand
(kW/house)
Close to flat
Theoretical sub-optimality bound
0.03
0.025
Suboptimality 0.02
ratio
0.015
0.01
0.005
0
40
80
120
160
200
# EVs in 100 houses
Always below 3% sub-optimality.
240
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
0.03
0.025
0.02
0.015
0.01
0.005
0
40
80
120
160
200
240