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
time
demand
Non

EV demand
Uncoordinated charging
Coordinated charging
Coordinate charging to flatten demand.
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
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
time
p
lug in
deadline
Convex
Area = Energy storage (pre

specified)
: charging profile of EV
n
EV model with
discrete charging rate
time
p
lug in
deadline
Finite
EV
n
Global optimization: flatten demand
Utility
EV
N
EV
1
t
ime of day
: charging profile of EV
n
b
ase demand
demand
Optimal charging profiles = solution to the optimization
Continuous / Discrete
c
harging
r
ate
Discrete: discrete optimization
Continuous: convex optimization
Flatten demand:
p
lug 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 optimization
Need stochastic algorithm
p
lug in
deadline
S
tochastic algorithm to rescue
Discrete
optimization
o
ver
p
lug in
deadline
Convex
optimization
o
ver
Avoid discrete programming
1
1
Stochastic algorithm to rescue
Discrete optimization
o
ver
p
lug in
deadline
Convex
optimization
o
ver
sample
Able to spread charging time,
e
ven 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.
S
upermartingale
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
f
or 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
When the number of EVs is large, very close to optimal.
Thm
:
E
very 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 annealing
n
ever
converge in finite steps.
Fast convergence
t
ime of day
demand
b
ase
demand
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
Suboptimality
ratio
# EVs in 100 houses
Always below 3% sub

optimality.
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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