Charging with Discrete Charging Rate

bankpottstownAI and Robotics

Oct 23, 2013 (3 years and 7 months ago)

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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

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.