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Charging of Electric Vehicles Utilizing RandomWind: A Stochastic Optimization Approach

Mathew Goonewardena and Long Bao Le

Abstract—In this paper, we present a general stochastic frame-work for the optimization of charging of electric vehicles (EVs)utilizing wind energy. The framework considers different keycomponents of the Smart Grid including stochastic wind energy,bulk and real-time purchase of energy from the grid operator,penalties or sell back of unutilized committed energy and flexibledemand of the EV users. We formulate the joint vehicle chargingand power purchase problem as a stochastic optimization pro-gram. Then we describe how to obtain its solution numerically.We also present a widely used alternative problem formulationusing expected values of the random wind and real-time prices.Numerical results are presented to demonstrate the efficacy ofthe proposed framework and the significant performance gaincompared to an expected-value approach.

Index Terms—smart grid, wind power, power scheduling,electric vehicle, stochastic optimization.

I. INTRODUCTION

Recent years have witnessed two major trends in the energy

sector. One is the transformation of the current power grid

to a Smart Grid (SG) and the other is the wide penetration

of renewable energy sources (RESs) such as wind and solar

energy. Electric vehicles (EVs), SG and RESs have been

recognized as important ingredients of the future power system

[1]. In fact, EVs are expected to become a major component

of the power grid due to their environmental, economic and

strategic advantages [2]. However, if not properly managed,

the increasing penetration of EVs may result in various adverse

factors on the operation and stability of the power grid [3]. In

addition to EVs, there are strong interests in implementing

several dynamic and flexible pricing schemes [4] and building

reliable communications infrastructure for two-way informa-

tion exchange between end users and the utility [5].

There is a rich collection of literature that studies various

research issues related to EVs, SG and RESs, and their

interactions. In [6], intelligent SG based charging and legacy

charging are compared for economic benefits where various

simulation results are presented. This paper, however, does

not consider the renewable energy or flexible demand of EVs.

In [7], day ahead pricing is assumed for the scheduling of

EVs under the control of an aggregator. In addition, dispatch

is considered to be done separately; day ahead predicted price

of energy is assumed to be error free; and charging rates of the

batteries are constant at either zero or the maximum rate. A

matching-based algorithm is proposed to solve the scheduling

problem.

The authors are with INRS-EMT, University of Quebec, Montreal, QC,Canada. Emails: {goonewardena, long.le}@emt.inrs.ca.

In [8], the EV scheduling problem utilizing wind power

is considered assuming average wind power whose solution

is developed using the game theory approach. However, the

Nash equilibrium achieved by the proposed algorithm may

not be efficient in general. In [9], a dynamic potential game

theory based approach is employed to maximally utilize wind

power. Here, the objective function is based on expectation

of difference of wind and conventional power. In [10], the

authors investigate the stochastic demand and generation in

the unit commitment problem and only heuristic algorithms

are developed.In this paper, we attempt to develop a general model

for optimization of charging of EVs considering different

design and modeling aspects, namely, SG, RESs and financial

planning. Our framework unifies the problem of procuring

of energy (from the SG and/or RESs), and the problem

of energy scheduling among EVs. Furthermore, our model

considers flexible demand of EVs, continuous charging rates

and penalties for over committed energy. Thus, the proposed

model is quite general and extensive. We formulate the joint

EV charging control and power purchase as a stochastic

optimization problem. Then, we present how to represent it

in an appropriate form so that the optimal solution can be

obtained by a suitable numerical technique. Numerical results

are then presented to illustrate the efficacy of the proposed

framework and the significant performance gain compared to

an alternative formulation based on expected values of random

wind and real-time price.The remaining of this paper is organized as follows. Sec-

tion II presents the system model and problem formulation.

In Section III, we propose solutions for the stochastic and

expected-value formulations of the joint charge control and

energy purchase optimization problem. Numerical results are

presented in Section IV followed by conclusion in Section V

II. SYSTEM MODEL

We consider a scenario where the charging of a number of

EVs is coordinated by an aggregator, which is illustrated in Fig

(1). An aggregator is a hub that brings together a collection of

EVs and interfaces them to the power sources in a controlled

manner [11]. In our model the aggregator has a dedicated

renewable energy source, which is considered to be a wind

turbine. The arrows in Fig 1 indicate the possible direction of

energy exchange. The aggregator is assumed to be interfaced

with the power grid to receive additional energy whenever the

wind generator cannot meet the demand requested by all EVs.We consider two pricing schemes for energy purchased

through the SG, namely bulk pricing and real-time pricing.

GC'12 Workshop: Smart Grid Communications: Design for Performance

978-1-4673-4941-3/12/$31.00 ©2012 Crown 1520

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Figure 1. System model

In particular, the aggregator is assumed to purchase energy

from the SG through bilateral contract under the bulk pricing

scheme. The aggregator is informed of the hourly prices of

bulk energy by the SG a few hours before the charging period

and the aggregator purchases energy at that price [7] for the

complete charging period at one time.

Wind power generated in each time slot of the charging

duration is random, thus the aggregator would not know

exactly how much energy to purchase from the bulk market for

each hour to match the total demand. Therefore, if the wind

energy and the purchased bulk energy is not sufficient to meet

the demand in any particular hour, the aggregator is assumed

to purchase energy from the real-time market. We assume that

the energy price from the real time market is necessarily higher

than the bulk energy price for any hour [12], [13].

We further assume that the purchased bulk energy is bind-

ing. That means if the wind energy and the bulk energy, which

is purchased in advance, is in excess of the total demand for

one particular hour, then the excess energy is retained by or

purchased back by the SG but at a lower price than that the

aggregator pays. This is the penalty imposed by the SG for

having to commit equipment to generate energy.

EV users are assumed to inform their energy requirement

or the desired state of charge (SOC [7] also referred to as

state of energy in other literature [14]) to the aggregator. We

allow flexible demand which means the EV users can request

the final SOC to be within a range. Our design objective is to

allocate energy among users and to manage the purchase of

energy from the different markets to maximize profit subject

to charging constraints imposed by the EV users and the SG

operator.

The charging profiles of the involved EVs depend on the

price of energy and available wind energy in each hour. We

assume that each EV can start their charging from any time

slot during the optimization period, which is determined by

the solution of the underlying optimization problem [7], [15].

Both bulk energy and real-time energy prices are assumed to

be fixed within each time unit, which is chosen to be one hour

in this paper for simplicity.

A. Wind and Real-time Price Models

Stochastic modeling for wind energy has been an active

research topic (e.g., see [16], [17] and the references therein).

In this paper, we adopt the model proposed in [8], [18].

Specifically, we assume that wind energy in hour h, denoted

as ωh, is modeled as the sum of an accurately predicted mean

value ωh, and a zero-mean random variable errw, i.e.,

ωh = ωh + errw.

Our formulation allows errw to obey any distribution. For the

numerical results presented in Section IV, we, however, choose

the Gaussian error, i.e., errw ∼ N(0, σ2

w

)where σw represents

the standard deviation of the underlying random variable.

The real-time price of electricity can be forecasted using

certain price forecast techniques [19]. Similar to the wind

model, we assume that the real-time price of energy of the hour

h, denoted as ψh, is the sum of an accurately predicted mean

value ψh and an associated error errp with known distribution,

i.e.,

ψh = ψh + errp.

In fact, it has been verified in [20] that the forecasted error

can be modeled as either a zero-mean Gaussian distribu-

tion errp ∼ N(0, σ2

p

)or a zero mean uniform distribution

errp ∼ Uniform[−α, α]. We adopt uniform errors to obtain

numerical results in Section IV. It should be noted that the

proposed model can work with other models of hourly random

wind and price models as long as they obey well-defined

distributions and ωh, ψh R≥0 with probability near 1.

B. Problem Formulation

Let xhi denote the allocation of energy to EV i for hour hand it is in units of energy. The set of EVs is denoted by N and

the number of EVs is equal to N (i.e., |N| = N ). We assume

that there are H time slots (hours) in the changing period.

We represent the energy profiles of all EVs in a vector, x as

follows:

x �[x11, x

12, . . . , x

1N , x

21, . . . , x

HN−1, x

HN

].

Let yh denote the amount of purchased bulk energy for

hour h. We present all bulk energy purchased for all hours

of the changing period in a vector y �[y1, . . . , yH

]. Let ph

denote the price per energy unit at which energy is charged

to the EVs during hour h by the aggregator. In addition, let

ch be the price of bulk energy for hour h as informed by the

SG. Let [Di,min, Di,max] denote the flexible range of energy

demand of the EV i where we have Di,min ≤ Di,max, ∀i.Also, let Ei,max denote the maximum amount of energy that

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can be transferred to the battery of EV i in one hour, which

indeed is the maximum charger rate [21]. Let Uh denote the

limit imposed by the SG on the maximum amount of bulk

energy that the aggregator can purchase for hour h. This limit

is mostly due to generation capacity but can be omited by

seting a high value to Uh.Our design objectives are to maximize the utilization of

wind energy and to minimize the cost of purchased energy

from the SG. Thus, we can formulate the charging control

and energy purchase optimization problem to maximize the

profit of the aggregator. To this end, let us denote a vector

u � [x|y] =[x11, . . . , x

HN , y

1, . . . , yH], which concatenates

two vectors x and y representing our optimization variables.

Expressing the objective function explicitly for this profit

maximization problem turns out to be not trivial. This is

because the random wind prevents the aggregator from de-

termining exactly the amount of energy to be purchased from

the bulk energy market to match the total demand. In fact,

the aggregator may need to purchase additional energy from

the real-time market or sell back excess energy later. We can

define the objective function describing the profit that can be

earned by the aggregator as follows:

f(u) �∑h

ph∑i

xhi −∑h

ch · yh −Q(u) (1)

where the first term in this function is the revenue earned by

the aggregator for charging EVs and the second term is the

cost of bulk energy purchased for the charging period. The

final term Q(u) accounts for the additional “cost” due to the

fact that the aggregator may purchase more energy to meet the

total demand. We derive this term in the next section when we

present the solution approach. With this function, we can write

down the deterministic equivalent version of the underlying

stochastic optimization problem as follows:

minimizeu

− f(u) (2)

subject to

0 ≤ xhi , yh, ∀i, h (3)

yh ≤ Uh, ∀h (4)

xhi ≤ Ei,max, ∀i, h (5)

Di,min ≤∑h

xhi ≤ Di,max, ∀i (6)

where recall that u = [x|y] describes our optimization

variables; (3) presents standard non-negative constraints for

scheduled and purchased energy in each hour; (4) captures

the constraints on the total bulk energy that the aggregator can

purchase in each hour; (5) describes the maximum scheduled

energy for individual EVs in each hour; and (6) describes the

feasible demand range for individual EV users.

III. JOINT CHARGE CONTROL AND ENERGY PURCHASE

OPTIMIZATION

A. Transformation for the Stochastic Formulation

An important step toward solving the underlying problem

(2)-(6) is to derive the final term Q(u) in its objective function.

To this end, let us define function Q(u, ξ) as follows:

Q(u, ξ) � minimizez,v

−∑h

{−rh (ξ) · zh + qh · vh} (7)

subject to∑i

xhi = wh (ξ) + yh + zh − vh, ∀h (8)

0 ≤ vh, zh, ∀h (9)

where rh (ξ) denotes the price of energy in the real-time

market; qh represent the price at which excess energy is

purchased back by the SG; zh denotes the real-time energy

purchased by the aggregator; vh is the excess energy sold

back to the SG; and wh (ξ) denotes the wind energy which

in our model is equal to ωh (see Section II-A). All these

quantities correspond to hour h of the charging period. We

assume that ch ≤ rh (ξ) and qh ≤ ch. In addition, we stack

the optimization variables zh and vh to form the corresponding

vectors z �[z1 · · · zH]

and v �[v1 · · · vH]

.

In (7), the term rh (ξ) · zh represents the total cost due

to additional energy purchase to meet the total demand and

qh · vh is the revenue that the aggregator may receive in

selling back excess energy to the SG. The constraint (8)

captures the balance between energy supply and demand in

each hour h. Therefore, Q(u, ξ) represents the minimum

additional cost that occurs since the aggregator may need to

purchase additional energy to match the actual demand or sell

back excess energy. Since the optimization problem (7)-(9)

is separable in different hours h, we can solve H per hour

optimization problems separately. To this end, let us define

Qh(u, ξ) � minimizezh,vh

− {−rh (ξ) · zh + qh · vh} (10)

subject to∑i

xhi = wh (ξ) + yh + zh − vh (11)

0 ≤ yh, zh (12)

which is the minimum additional cost for hour h. Then, we

have the following relation

Q(u, ξ) =∑h

Qh(u, ξ). (13)

From this, we can define Q(u) in (2) as follows:

Q(u) � Eξ {Q(u, ξ)} (14)

where the expectation is taken over the distribution of the ran-

dom variables ξ = [ω1, . . . , ωH , ψ1, . . . , ψH ], which consists

of the random variables describing wind energy (i.e., ωh) and

real-time price of electricity (i.e., ψh) at each time slot h. Let

vh∗ denote the optimal value of energy that is sold back to

the SG. Then, it can be written as

vh∗ = minvh

{vh|vh =

(yh + wh (ξ)

)−∑i

xhi , vh ≥ 0

}(15)

=

⎧⎪⎨⎪⎩0 if∑

i xhi −

(yh + wh (ξ)

) ≥ 0(yh + wh (ξ)

)−∑i x

hi otherwise,

(16)

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This is because vh∗ > 0 only if there is excess energy (from

wind energy plus bulk energy purchase) over the demand for

that hour. In addition, we can write the optimal value of real-

time energy purchased zh∗ as

zh∗ = minzh

{zh|zh =

∑i

xhi −(yh + wh (ξ)

), zh ≥ 0

}(17)

=

⎧⎪⎨⎪⎩0 if∑

i xhi −

(yh + wh (ξ)

) ≤ 0∑i x

hi −

(yh + wh (ξ)

)otherwise.

(18)

Here, if the energy of wind and bulk purchase is insufficient

to fulfill the demand for that hour then the deficiency zh∗ is

greater than zero. Using the results in (16) and (18), we can

express Qh(x, ξ) in (10) as

Qh(u, ξ) = rh (ξ) · zh∗ − qh · vh∗. (19)

Substitute this result into (13) and (14), we can rewrite

the optimization problem (2)-(6) in terms of zh∗ and vh∗ as

follows:

minimizeu

−{∑

h

ph∑i

xhi −∑h

ch · yh (20)

−Eξ

∑h

{rh (ξ) · zh∗ (ξ)− qh · vh∗ (ξ)}}

subject to∑i

xhi = yh + wh (ξ) + zh∗ − vh∗, ∀h (21)

constraints (3), (4), (5), and (6) . (22)

This equivalent optimization problem now depends only on

u = [x|y] (vh∗and zh∗are functions of u from 16 and 18).

However, we need to exploit the statistical knowledge about

the random variable ξ to transform it into another solvable

deterministic program. This is explored in the following.

B. Numerical Solution for the Stochastic Formulation

The probability density functions we consider in the for-

mulation are continuous and thus it involves multidimensional

integrations to evaluate Q(u) in (14) analytically. This is

usually a challenging task. One typical method to resolve

this issue is to discretize the probability distribution to obtain

a discrete distribution on a finite support. The lower and

upper bounds for such an approximation can be derived by

using the generalized Jensen’s and Edmundson & Madansky

inequalities, respectively [22].

For the purpose of obtaining numerical results we fol-

low linearization method presented in [23]. In particular, the

deterministic equivalent form of (20) is transformed into a

large scale linear program the size of which depends on

the cardinality of the discrete distribution (the number of

discrete points). Let us consider the evaluation of the following

expectation from (20).

Eξ

{∑h

{rh (ξ) · zh (ξ)− qh · vh (ξ)}} .

By the modeling assumptions presented in Section II-A, we

have the distribution for real-time price and wind energy in

each hour. This expectation is separable and can be written as∑h

Eωh,ψh

{rh

(ψh

) · zh (ωh)− qh · vh (ωh)} .Assume that we consider the three-sigma range of the

normal distributions of wind energy and real-time price for

hour h and discretize that into a number of regions. We then

identify each region by its conditional expected value. Let Ωhwbe the set of discrete points of the wind energy distribution

of hour h and Ωhr be the set of discrete points of the real-

time price distribution of hour h. Then, we have |Ωhw| × |Ωhr |realization points of the joint wind and real-time price discrete

distribution. We denote the resulting discrete distributions for

random wind and real-time price by ωhdw and ωhdr random

variables, respectively.

The stochastic constraints∑i x

hi = yh + wh (ξ) + zh −

vh, ∀h can now be replaced with a finite number of determin-

istic constrains one for each realization of the joint random

variable ξhd =[ωhdw, ψ

hdr

]. The joint probability mass of this

random vector is the product of the two probability masses

as the price and the winder energy are independent where

we denote it by pk,hwr (k ∈ Ωhw × Ωhr ). From the obtained

discrete distributions, we can write the extensive form [22]

of the program (20)-(22) as

minimizeu,z,v

−{∑

h

ph∑i

xhi −∑h

ch · yh

−∑h

∑k∈Ωh

w×Ωhr

pk,hwr

{rhk z

hk − qhvhk

}⎫⎬⎭ (23)

subject to∑i

xhi = yh + whk + zhk − vhk , k ∈ Ωhw × Ωhr , ∀h

0 ≤ xhi , yh, zhk , v

hk , k ∈ Ωhw × Ωhr , ∀i, h

constraints (4), (5), and (6) .

where whk , zhk , and vhk denote wind energy, excess energy, and

real-time energy for a particular realization k ∈ Ωhw × Ωhrin hour h; and z, v denote the corresponding vectors. The

optimization problem (23) is a deterministic program, which

can be solved by any available solver. We use the CVX solver

to obtain numerical results presented in the next section [24].

C. Expected-Value Formulation

We consider an alternative formulation based on average

values of the underlying random variables. In particular, the

random wind energy and real-time price values for each hour

are replaced by their average values wh = ωh and rh = ψh

in (20), respectively. Then the bulk energy purchase and EV

charging profiles can be obtained by solving the following1523

5

optimization problem

minimizeu,z,v

−{∑

h

ph∑i

xhi −∑h

ch · yh

−∑h

rh · zh +∑

qh · vh}

(24)

subject to∑i

xhi = yh + wh + zh − vh, ∀h (25)

constraints (3), (4), (5), and (6) . (26)

Let u∗EV = [x∗EV|y∗EV] be the optimal solution of this problem.

Then, we can substitute u = u∗EV into (20)-(22) to obtain the

average cost under this expected-value formulation. Then, we

can calculate the performance gain in terms of optimal cost

between the two formulations as

Gain =CSF − CAVF

CAVF× 100 (27)

where CSF and CAVF are the optimal costs due to the stochastic

formulation (20)-(22) and expected-valued formulation (24)-

(26), respectively.

IV. NUMERICAL RESULTS

We present numerical results to demonstrate the perfor-

mance achieved by the proposed framework. Simulation is

carried out for the following parameters unless stated other-

wise. The real-time price is assumed to vary between 1.5 to 2times that of the bulk energy price for any hour and the selling

price to the EVs is 1.5 times the average bulk energy price

for any hour in the charging period. Wind energy estimation

error distribution of N(0, (0.4)2

)and 50 numbers of EVs are

assumed with Di,min = 10 kWh and Di,max = 30 kWh. The

maximum bulk energy purchase per hour (i.e., Uh) is 50 kWh.

The bulk energy prices for different hours are presented in

Fig. 2.

We illustrate the total energy allocation for EVs due to

the stochastic and expected-value formulations in Fig. 2. This

figure shows that the stochastic formulation results in a wide

variations in the amount of allocated energy, which better

adapts to the price of the bulk energy market. In particular,

when the bulk energy is cheap from midnight to 3am, the

maximum possible level of energy is allocated. In contrast,

when the energy is expensive as in the initial hours and some

last hours of the charging period, a far less quantity of energy

is allocated to EVs. It can also be observed that the expected-

value formulation does not account for the price variations in

different hours very well and it results in nearly same levels

of allocated energy in all hours.

In Fig. 3, we illustrate the performance gain between the

optimal costs due to stochastic and expected-value solutions.

Here, we fix maximum total demand of all EV users while

varying the average amount of wind energy. This figure shows

that significant gains can actually be achieved by the proposed

stochastic formulation over the expected-value counterpart. In

addition, larger gains can be achieved when the ratio between

the expected wind energy and the maximum total demand is

21−22 22−23 23−24 00−1 1−2 2−3 3−4 4−5 5−6 6−750

60

70

80

90

100

Charging Hour

En

ergy

(k

Wh

)

Stochastic Solution

Expected Value Solution

.25 .25.20.15.15.15.20.25Cost of Bulk Energy

Figure 2. Total energy allocation

13 18 24 29 35 40 −−− −−− 804

6

8

10

12

14

16

18

20

Total Expected Wind Energy to Total Demand Ratio (%)

Gai

n (%

)

σw

= 0.5

σw

= 1

σw

= 2

Figure 3. Gains versus total expected wind energy to total demand ratio fordifferent values of wind energy prediction error variance σ2

w

smaller. This is intuitive since the stochastic formulation is

able to exploit better the random wind when there is relatively

less wind energy compared to the total demand.

Next we consider the effect of the limitation on bulk energy

purchase (i.e., Uh) on the gain. We explore the gain against the

ratio Uh/ωh as shown in Fig. 4. As the limitations imposed on

bulk energy purchase by the SG are relaxed, this figure shows

that the gain initially increases and then slightly decreases.

The initial increase in gain is due to the reduction in real-time

energy purchase as more bulk energy is purchased. As we have

kept the same values of maximum and minimum demands (i.e.,

1 2 3 4 5 6 70

5

10

15

20

25

Maximum Bulk Energy Purchase to Total Expected Wind Energy Ratio

Gai

n (%

)

σw

= 1

σw

= 1.5

σw

= 2

Figure 4. Gains versus total max. bulk energy to total wind energy ratio forvarying values of wind energy prediction error varianceσ2

w

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6

20 25 30 35 40 45 50 55 6010

12

14

16

18

20

22

24

26

28

Maximum Demand per EV (kWh)

Gai

n (

%)

E[ωh] = 10kWh

E[ωh] = 20kWh

E[ωh] = 40kWh

Figure 5. Gain vs Maximum Demand for different values of mean windenergy per hour E

[ωh

]

20 40 60 80 100 120 140 160 180 20010

12

14

16

18

20

22

24

26

28

30

Hourly Expected Wind Energy (kWh)

Gai

n (

%)

N = 100

N = 500

N = 1000

N = 10000

Figure 6. Gain vs. expected wind energy per hour for different values ofnumber of EVs N

Di,max, Di,min), further bulk energy purchase does not provide

much gain, as we cannot sell all that to the users.

In Fig. 5, we study the influence of maximum demand (i.e.,

Di,max) and hourly expected wind energy (i.e., ωh) on the gain

of the stochastic solution. The figure implies that as Di,max

increases, the stochastic formulation results in increasing gain.

In Fig. 6, we evaluate the variation of performance gain as the

number of EVs vary. It can be deduced from the figure that the

performance improves with the increase of number of EVs and

also the dependance of gain on the expected mean wind energy

reduces drastically as the number of EVs increases. Both the

properties are highly desirable for scalable and predictable

reasons.

V. CONCLUSION

We have proposed a stochastic optimization based frame-

work for optimal charging control of EVs and energy purchase

planning from the SG. In addition, we have also presented

an alternative formulation, which is widely used to handle

similar cases, based on expected values of random wind

and real-time price to demonstrate the potential performance

gain of the proposed stochastic framework. Finally, we have

presented numerical results to demonstrate the efficacy and the

significant performance gain due to the proposed framework.

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