Optimal Charging Scheduling by Pricing for EV …cai/its18-ev-dual.pdfThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1

Optimal Charging Scheduling by Pricing for EVCharging Station With Dual Charging Modes

Yongmin Zhang , Member, IEEE, Pengcheng You , Student Member, IEEE,

and Lin Cai , Senior Member, IEEE

Abstract— With the increasing penetration of electric vehicles(EVs) and various user preferences, charging stations oftenprovide several different charging modes to satisfy the variousrequirements of EVs. How to effectively utilize the chargingcapacity to minimize the service dropping rate is a pressingand open issue for charging stations. Given that EV ownersare price-sensitive to the charging modes, we intend to designan optimal pricing scheme to minimize the service droppingrate of the charging station. First, we formulate the operationof a dual-mode charging station as a queuing network withmultiple servers and heterogeneous service rates, and analyze therelationship between the service dropping rate of the chargingstation and the selections of EVs. Then, we formulate a customerattrition minimization problem to minimize the number of EVsthat leave the charging station without being charged and proposean optimal pricing approach to guide and coordinate the chargingprocesses of EVs in the charging station. The simulation has beenconducted to evaluate the performance of the proposed chargingscheduling scheme and show the efficiency of the proposed pricingscheme.

Index Terms— Charging stations, charging modes, queuingtheory, charging scheduling, pricing.

I. INTRODUCTION

ELECTRIC vehicles (EVs) have been considered to be akey technology to cut down the massive greenhouse gas

emissions from the transportation sector, and they are alsoexpected to mitigate the fossil fuels scarcity problem [1].Thanks to the policies and plans for promoting EVs fromthe regions and countries worldwide (e.g., the sales of EVsincluding PHEVs in US will reach 50% of total sales of mobilevehicles by 2030, and Europe has the similar targets [2]),the amount of EVs is expected to reach a sizable market sharein the next decade.

Manuscript received October 19, 2017; revised August 13, 2018; acceptedOctober 5, 2018. This work was supported in part by the Natural Sciencesand Engineering Research Council of Canada (NSERC) and in part by NSFCunder Grant 61702450 and Grant 61629302. The Associate Editor for thispaper was E. Kosmatopoulos. (Corresponding author: Lin Cai.)

Y. Zhang is with the State Key Laboratory of Industrial Control Technologyand Innovation Joint Research Center for Industrial Cyber Physical Systems,Zhejiang University, Hangzhou 310027, China, and also with the Departmentof Electrical and Computer Engineering, University of Victoria, Victoria,BC V8P 5C2, Canada (e-mail: [email protected]).

P. You is with the Department of Mechanical Engineering, Johns HopkinsUniversity, Baltimore, MD 21218 USA (e-mail: [email protected]).

L. Cai is with the Department of Electrical and Computer Engineering,University of Victoria, Victoria, BC V8P 5C2, Canada (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TITS.2018.2876287

However, due to limited cruising range, EVs may requirefrequent recharging when they travel to a faraway destination,and hence charging convenience is one of the most impor-tant concerns for EV owners. In addition, due to the highcost of battery replacement, battery lifetime-related cost isanother important concern for EV owners. Given the slowevolution of battery technologies, the key issue to is howto address these challenges without waiting for new batterytechnologies.

Charging stations play an important role in providingcharging services to EVs. Typically, there exist two dif-ferent charging modes for conventional public chargers:i) AC Level II (charging power typically is 10-22 kW); andii) Direct Current Quick Charging (DCQC) (charging powertypically is 50-120 kW). Given the EV charging requirement,AC Level II has a longer charging duration while DCQChas a shorter charging duration, which may further reducebattery lifetime. Because of deployment cost concerns, boththe number of charging stations and the number of chargersin charging stations are limited. How to use limited chargingstation resources to satisfy the various EV charging require-ments has attracted attention, such as developing intelligentcharging station architectures [3]–[6], and optimizing the loca-tion and sizing of charging stations [7]–[11]. Nevertheless,the non-cooperative charging behaviors of EVs will lead tocongestion at charging stations and reduce their operationalefficiency.

To guide and coordinate the charging behaviors of EVs,there have been extensive researches on developing chargingscheduling schemes, for EV charging stations [12]–[17], forbattery swapping stations [18]–[20], and for charging sta-tions with renewable energy [21]–[24]. However, these worksassumed that all the chargers in the charging station are usingthe similar charging mode, such as AC Level II or DCQC.Since different EVs may have different charging behaviorsand charging service requirements, i.e., short charging dura-tion or long battery lifetime or both of them, the chargingstation with single charging mode has a low flexibility andadaptability to satisfy the requirements of multi-class cus-tomers, and thus limits the service quality.

To deal with the various EV charging requirements,the charging station can install two types of chargers, suchas part of them with the AC Level II mode and part ofthem with the DCQC mode. Consequently, the charging stationcan provide different charging services to different classes of

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2 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS

customers based on their behaviors. Furthermore, guiding theEV owners to select adequate charging modes can reduce thecongestion and improve the service quality of the chargingstation. This motives us to design an optimal schedulingscheme for the charging station with dual charging modesto minimize the service dropping rate. To the best of ourknowledge, this is the first paper addressing the chargingscheduling problem for the charging station with dual chargingmodes by designing an optimal pricing scheme.

Generally, the selections of EV owners depend on severalfactors, i.e., service fee, charging duration, waiting time,etc [12]. Given that EV owners are price sensitive, we analyzethe relationship between the service dropping rate and theservice rate of the charging station and then design an optimalpricing scheme to guide and coordinate the charging processesof EVs, such that the number of EVs that leave the chargingstation without being charged can be minimized and theoperation efficiency of the charging station can be improved.The contributions of this papers can be summarized as follows:

• We model the operation processes of the charging stationwith dual charging modes as a queuing network withmultiple servers and heterogeneous service rates.

• Based on the queuing theory, we analyze the relationshipbetween the selections of EVs and the service droppingrate of the charging station, and prove that the servicedropping rate is a convex function of the service rates.

• We formulate a customer attrition minimization problemfor the charging station and propose an optimal pricingscheme to guide and coordinate the charging processesof EVs to minimize the service dropping rate.

• Simulation results show that the proposed pricing schemecan reduce the service dropping rate of the chargingstation with dual charging modes significantly comparingto the charging station with a single charging mode.

The rest of the paper is organized as follows: Section IIpresents the operation models for the charging station, andformulates the charging scheduling problem as a customerattrition minimization problem based on queuing theory. Therelationship between the selections of EVs and the servicedropping rate of the charging station is analyzed, and anoptimal pricing scheme is designed by utilizing the price sensi-tiveness of EV owners in Section III. Section IV demonstratesthe operational performance analysis based on simulationresults. Finally, Section V concludes our work.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Considering a charging station, there are N1 AC chargersand N2 DC chargers to provide charging services for connectedEVs.1 Given the charging demand of one EV, both the servicefee and the battery lifetime-related cost of the AC chargersare much lower than the DC chargers. But, the chargingduration for the AC chargers is much longer. As shown in[12] and [25], the dominant factors that affect the selections

1For simplicity, we use the AC charger and the DC charger to denote thecharger with the AC Level II charging mode and the charger with the DCQCcharging mode, and the AC mode and the DC mode instead of the AC LevelII charging mode and the DCQC charging mode in this paper, respectively.

Fig. 1. Operation model and queuing model for the charging station. (a)Operation model. (b) Queuing model.

of EVs are service fee, waiting time, battery lifetime-relatedcost, etc. In this paper, we assume that the service fee andthe battery lifetime-related cost will affect the EVs’ selectionswhile the waiting time will affect the service dropping rateof the charging station. When the corresponding queue lengthis too long, EV will leave the charging station without beingcharged. Otherwise, EV will be charged by one charger withthe selected charging mode immediately or late.

For each EV, it needs to select one charging mode when itarrives at the charging station. Then, it will be charged whenthere is at least one available charger with the selected charg-ing mode; otherwise, it needs to wait in the correspondingqueue until one charger with the selected charging mode isavailable, or leaves the charging station directly without beingcharged if the corresponding queue length is too long. Here,the percentage of EVs that leave the charging station directlydenotes the service dropping rate, which is one of the mainfactors related to the Quality of Service (QoS) of the chargingstation. Our objective is to design an optimal pricing schemebased on EV owners’ preferences to guide them to selectadequate charging modes, such that the service dropping rateof the charging station can be minimized. The operation modeland the queuing model for the charging station are shownin Fig. 1. A summary of notations is given in Table I.

Note that, given the ubiquitous communication networks,in the future, EVs and charging stations can exchange pricingand waiting time information remotely. Such that, an EV candecide which charging mode it prefers and whether go tothe charging station for charging service or not based on itsQoS. In this scenario, the queuing model and the proposed


ZHANG et al.: OPTIMAL CHARGING SCHEDULING BY PRICING FOR EV CHARGING STATION WITH DUAL CHARGING MODES 3

TABLE I

NOTATION DEFINITIONS

pricing algorithm in this paper can still be applicable, usingthe concept of virtual queues.

A. Operation Model of Charging Stations

Considering the realistic scenarios that the average EVarrival rate during peak vs. off-peak hours may be different,we divide the whole day into T time slots (i.e., an hour isone time slot), and assume that the average EV arrival ratewithin the time slot remains the same, while that for differenttime slots may change. Let t , t = 1, 2, . . . , T , denote t-th timeslot. The arrival of EVs at each time slot follows a Poissonprocess with the average rate of λt during time slot t [26]. Theprobability for n EVs arriving at the charging station duringtime slot t is given by

P{n} = eλt (λt )n

n! , n = 0, 1, 2, · · · .

When EVs arrive at the charging station, they will selecttheir charging modes based on their preferences and thecorresponding service fee. Let pA

t and pDt respectively denote

the probability for one EV selecting the AC mode and the DCmode during time slot t , respectively. The values of pA

t andpD

t satisfy

pAt + pD

t = 1, (1)

0 ≤ pAt , pD

t ≤ 1. (2)

Let λAt and λD

t denote the expected number of EVs that selectthe AC mode and the DC mode during time slot t , respectively.We have

λAt = λt pA

t and λDt = λt pD

t . (3)

Generally, based on the existing data analysis in [27],the EV charging requirements follow a lognormal distributionwith a mean value E and a standard deviation of Ed . Thus,we assume that the expected charging requirement of each EVis E . Let T A and T D denote the expected service time for oneAC charger and one DC charger to fully charge a battery withthe capacity of BC , respectively. T A > T D always holds. Themean service times for one EV selecting the AC mode andthe DC mode are ET A/BC and ET D/BC , respectively.

Based on the selections of EVs, they enter two separatequeues: 1) one queue for EVs that select the AC mode, denotedby Q A

t ; 2) another queue for EVs that select the DC mode,denoted by QD

t . The queue lengths of Q At and QD

t at timeslot t can be given by the following equations:

Q At = max{0, Q A

t−1 + λAt − V A

t − O At − L A

t }, (4)

QDt = max{0, QD

t−1 + λDt − V D

t − O Dt − L D

t }, (5)

where V At and V D

t respectively denote the total number ofthe available AC chargers and DC chargers at slot t , O A

t andO D

t respectively denote the number of EVs that have beenfully recharged by the AC chargers and the DC chargers andleave the charging station during time slot t , and L A

t and L Dt

respectively denote the number of EVs that select the AC modeand the DC mode and leave the charging station without beingcharged due to the long corresponding queue length.

In this paper, the maximum queue length is used to denotethe maximal tolerable queue length of the EV owners, denotedby Q A

t and QDt respectively. If Q A

t ≥ Q At (QD

t ≥ QDt ),

the coming EVs that select the AC (DC) mode will leave thecharging station without being charged; otherwise, they willbe charged immediately or wait in the corresponding queue.Let Lt denote the service dropping rate of the charging stationduring time slot t . The expected value of Lt can be given by

Lt = L At + L D

t , (6)

where

L At = λA

t Pr{Q At |Q A

t ≥ Q At }, (7)

L Dt = λD

t Pr{QDt |QD

t ≥ QDt }. (8)

Obviously, the service dropping rate Lt depends on theselections of EVs λA

t and λDt , and the queue lengths Q A

t andQD

t , as well as the maximum queue lengths Q At and QD

t . Sincethe values of Q A

t and QDt are determined by EV owners, it is

difficult for the charging station to change them. According to(4) and (5), Q A

t and QDt mainly depend on the selections of

EVs λAt and λD

t , respectively, which can be tuned to reducethe service dropping rate Lt .

B. Selection Model of EVs

Generally, due to the high construction cost of the DC mode,the service fee of the DC mode is much higher than that for



the AC mode. Furthermore, the battery lifetime-related cost forthe AC mode is much lower than that for the DC mode [28].Different EVs may have the different preferences, but are pricesensitive [16]. Since different charging stations always havethe similar service fee of the AC mode, we set the service feeof the AC mode as a constant and adjust the service fee ofthe DC mode.

Let C At and C D

t denote the service fee per kWh during timeslot t , and C A

B and C DB denote the battery lifetime-related cost

per kWh, with the AC mode and the DC mode, respectively.Here, C A

t < C Dt and C A

B < C DB always hold. Generally,

the service fee of the DC mode can only be adjusted in agiven range. Let C D and C

Ddenote the lower bound and upper

bound on the service fee of the DC charger. Thus, the servicefee C D

t should always satisfy

C D ≤ C Dt ≤ C

D. (9)

As mentioned above, the selections of EVs mainly dependon the service fees C A

t and C Dt and the battery lifetime-related

costs C AB and C D

B . Specifically, in this paper, we define pAt and

pDt as follows:

pDt = −aC D

t + b, (10)

pAt = 1 − pD

t , (11)

where a = 1C A

t +β−(C DB −C A

B ), b = aC

D, and β denotes the

service time reduction cost of the DC mode comparing withthe AC mode. It can be seen that a is a price-dependent termand b is a price-independent term.

To ensure 0 ≤ pAt , pD

t ≤ 1, we set C D = CD − (C A

t + β −(C D

B − C AB )), which usually is larger than 0. Such that, when

C Dt equals its lower bound C D , all the coming EVs will select

the DC mode, i.e., pDt = 1 and pA

t = 0; when C Dt = C

D,

all the coming EVs will select the AC mode, i.e., pDt = 0

and pAt = 1. Note that, our proposed algorithm also applies

to other probability models, such as linear or concave ones.

C. Service Dropping Rate of Charging Station

Let pLt denote the probability for EVs leaving the charging

station without being charged during time slot t . The expectedvalues of pL

t can be given by

pLt = Pr{Q A

t |Q At ≥ Q A

t } pAt + Pr{QD

t |QDt ≥ QD

t } pDt . (12)

Thus, the expected value of the service dropping rate Lt canbe rewritten as

Lt = λt pLt , (13)

due to λAt = λt pA

t and λDt = λt pD

t . It can be found thatthe service dropping rate Lt can be reduced by adjusting theservice fee C D

t , which affects the values of pAt and pD

t .

D. Customer Attrition Minimization Problem

The service dropping rate is not only an important parameterfor the loss of benefit, but also one of the major factors for thesatisfaction of EV owners. To minimize the service droppingrate of the charging station, we formulate a customer attrition

minimization problem. Since EV arrives at different time slotfollow a Poisson process, which has independent increments,we just need to minimize the service dropping rate of thecharging station during each time slot, such that the totalservice dropping rate can be minimized. By now, the customerattrition minimization problem can be formulated as follows:

P0: minC D

t

Lt = λt pLt ,

s.t . C D ≤ C Dt ≤ C

D,

pDt = −aC D

t + b, (14)

pAt + pD

t = 1, (15)

1 ≤ pAt , pD

t ≤ 0, (16)

(4), (5), and (12). (17)

The objective is to minimize the service dropping rate Lt byadjusting the service fee C D

t . The first constraint defines theavailable range of C D

t . The constraints (14)-(16) describe therelationship between the service fee C D

t and the selections ofEVs, pA

t and pDt . The other constraints show relationships

among pLt , Q A

t , QDt , pA

t , and pDt .

III. OPTIMAL EV CHARGING SCHEDULING SCHEME

To solve the customer attrition minimization problem,we first analyze the performance measure of the queue systemto explore the possible range of optimal price and the rela-tionship between the service dropping rate Lt and the servicefee C D

t . Then, we transform the primal problem into a convexoptimization problem. At last, we propose an optimal pricingscheme to guide the EVs to select adequate charging modes,such that the total service dropping rate of the charging stationcan be minimized.

According to the definitions of pAt and pD

t , it can be foundthat the relationships among λA

t , λDt and C D

t are linear. Forany given service fee C D

t and arrival rate λt , λAt and λD

t canbe calculated according to (3), (10) and (11). Also, C D

t andλA

t can be obtained when the value of λDt is given. Hence,

in the following sections, we can replace the variable C Dt by

λDt for easy computation.

A. Performance Measures Based on Queue Theory

Let μA and μD denote the average service rate for one ACcharger and that for one DC charger, respectively. Based on thesystem model, the values of μA and μD , and their relationshipcan be given by

μA = BC

ET A, (18)

μD = BC

ET D, (19)

μA = T D

T AμD. (20)

Obviously, μA < μD since T D < T A.The charging processes of EVs at the charging station

can be formulated as two independent queuing networks,i.e., M/M/N1 with ρA

t = λAt /(N1μ

A) and M/M/N2 with



ρDt = λD

t /(N2μD), respectively. Here, ρA

t and ρDt are the

corresponding utilization factors for different charging modes.For the optimal service fee C D

t , we have the followingTheorem:

Theorem 1: For any given arrival rate λt , the availablerange for the optimal service fee C D

t can be given by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

N1μA + bλt − λt

aλt

≤ C Dt ≤ bλt − N2μ

D

aλt, if λt ≥ N1μ

A + N2μD;

bλt − N2μD

aλt

< C Dt <


aλt, otherwise.

Proof: According to the queuing theory, if with infinitebuffer, the necessary conditions for these two queue systemsto be stable are ρA

t < 1 and ρDt < 1. If the mean arrival

rate is greater than the mean service rate, the necessaryconditions for the stable queue systems cannot be satisfied,which makes the server keeping busy and the queue growingwithout bound2 [29]. Thus, based on the arrival rate λt andthe service rate, we classify the available range of C D

t into thefollowing two cases:

CASE I: If λt ≥ N1μA + N2μ

D , there is no available

service fee satisfying λt p At

N1μA < 1 and λt pDt

N2μD < 1 simultaneously.

If ρAt ≥ 1, the chargers with the AC mode will keep

busy since their queuing system becomes overloaded andthe service dropping rate will increase with respect to λA

t ,and L A

t ≈ λAt − N1μ

A; otherwise, the service dropping rateL A

t ≈ λAt Pr{Q A

t |Q At ≥ Q A

t } = λAt (1 − Pr{Q A

t |Q At < Q A

t }).The parameters for L D

t are similar to those for L At . Hence,

the service dropping rate Lt can be given by the followingcases:

• If λAt > N1μ

A and λDt < N2μ

D , we have

Lt ≈ λt − N1μA − λD

t Pr{QDt |QD

t < QDt }; (21)

• If λAt ≥ N1μ

A and λDt ≥ N2μ

D , we have

Lt ≈ λt − N1μA − N2μ

D; (22)

• If λAt < N1μ

A and λDt > N2μ

D , we have

Lt ≈ λt − N2μD − λA

t Pr{Q At |Q A

t < Q At }. (23)

Since Pr{Q At |Q A

t < Q At } < 1 and Pr{QD

t |QDt < QD

t } < 1,(22)<(21) and (22)<(23) always hold. To minimize the servicedropping rate Lt , ρA

t ≥ 1 and ρDt ≥ 1 should be satisfied

simultaneously. Thus, the available range for the optimalservice fee C D

t is

λt pAt ≥ N1μ

A ⇒ C Dt ≥ N1μ

A + bλt − λt

aλt,

λt pDt ≥ N2μ

D ⇒ C Dt ≤ bλt − N2μ

D

aλt.

2In a realistic queuing system, the buffer size is limited, so the queue lengthwill be limited, while a portion of the arrivals will be blocked when the bufferis full.

CASE II: If λt < N1μA + N2μ

D , it means that there existsan optimal service fee C D

t that can satisfy ρAt < 1 and ρD

t < 1simultaneously. If ρA

t < 1 and ρDt < 1, it means that λA

t <N1μ

A and λDt < N2μ

d always hold, and the service droppingrate Lt can be calculated by

Lt = λAt Pr{Q A

t |Q At ≥ Q A

t } + λDt Pr{QD

t |QDt ≥ QD

t }= λt − λA

t Pr{Q At |Q A

t < Q At } − λD

t Pr{QDt |QD

t < QDt }.(24)

It can be found that (24)<(21) and (24)<(23) always hold.Hence, the optimal service fee C D

t should satisfy

λt pAt < N1μ

A ⇒ C Dt <


aλt,

λt pDt < N2μ

D ⇒ C Dt >

bλt − N2μD

aλt.

The available range for the optimal service fee C Dt is obtained.

According to the available range for the optimal servicefee C D

t given by Theorem 1, if the arrival rate λt satisfiesλt ≥ N1μ

A + N2μD , we have the following Lemma for the

optimal service fee C Dt :

Lemma 1: The optimal service fee C Dt can be any value

in the available range [ N1μA+bλt−λtaλt

, bλt−N2μD

aλt] when λt ≥

N1μA + N2μ

D .Proof: For the service dropping rate Lt given by (22),

it can be found that the minimal value of the service droppingrate Lt is a constant when λt ≥ N1μ

A + N2μD , and the only

condition for the optimal service fee C Dt is to make sure of

that ρAt ≥ 1 and ρD

t ≥ 1 are satisfied simultaneously. Thus,the optimal service fee C D

t can be any value in the available

range [ N1μA+bλt−λtaλt

, bλt−N2μD

aλt].

Therefore, we only need to design the optimal pricing schemefor the problem when λt < N1μ

A + N2μD .

Let n1 and n2 denote the steady state of the chargingprocesses of the AC mode and the DC mode, in which there aren1 and n2 customers in their corresponding systems, includethe customers in service, respectively. Let pA

t (n1) denote theprobability that in steady state the number of customers presentin the AC mode is n1 during time slot t and pD

t (n2) denotethe probability that in steady state the number of customerspresent in the DC mode is n2 during time slot t . Then, we have

pAt (n) =

⎧⎪⎪⎨

⎪⎪⎩

pAt (0)

(N1ρAt )n

n! , if n ≤ N1;

pAt (0)

(ρAt )

nN N1

1

N1! , if n ≥ N1;(25)

pDt (n) =

⎧⎪⎪⎨

⎪⎪⎩

pDt (0)

(N2ρDt )n

n! , if n ≤ N2;

pDt (0)

(ρDt )

nN N2

2

N2! , if n ≥ N2;(26)

where

pAt (0) = [

N1−1∑

n=0

(N1ρAt )n

n! + (N1ρAt )N1

N1!1

1 − ρAt

]−1, (27)



pDt (0) = [

N2−1∑

n=0

(N2ρDt )n

n! + (N2ρDt )N2

N2!1

1 − ρDt

]−1. (28)

In this paper, we use the steady-state distribution of EVs tomodel the service processes of the charging station and solvethe customer attrition minimization problem.

B. Service Dropping Rate of Charging Station

Since the maximal queue lengths Q At and QD

t for EVsduring time slot t are constant, the service dropping rate Lt

can be given by

Lt =∞∑

n1=N1+Q At

λAt pA

t (n1) +∞∑

n2=N2+Q Dt

λDt pD

t (n2) (29)

where∑∞

n1=N1+Q At

λAt pA

t (n1) = λAt (p A

t (N1+Q At ))(N1+Q A

t )

1−ρ At

denotes the blocking traffic for the queue of the AC mode

and∑∞

n2=N2+Q Dt

λDt pD

t (n2) = λDt (pD

t (N2+Q Dt ))(N2+QD

t )

1−ρDt

denotesthe blocking traffic for the queue of the DC mode. Accordingto the definitions of pA

t (n) and pDt (n) given by (25) and (26),

it can be found that pAt (n) and pD

t (n) also depend on thevalues of λA

t and λDt . Hence, the service dropping rate Lt

mainly depends on the values of λAt and λD

t .

C. Problem Transformation

According to Theorem 1 and the relationship among λAt ,

λDt , pA

t , pDt and C D

t given by (3), (10) and (11), the optimalλD

t should satisfy

λt − N1μA < λD

t < N2μD . (30)

By now, the primal Problem P0 can be transformed to thefollowing problem P1, in which the variable is λD

t , i.e.,

P1: minλD

t

Lt , (31)

s.t . λt − N1μA < λD

t < N2μD. (32)

In this problem, the goal is to find the optimal λDt to minimize

the service dropping rate Lt , and the available range for λDt

is given by constraint (32). If the objective function Lt isa convex function of λD

t , Problem P1 can be transformedinto a typical convex optimization problem. Thus, we firstestablish that, if the problem is a convex optimization problem,the solution of this problem is indeed toward the globaloptimum [30]. To solve this problem, we first prove thatconvexity of the service dropping rate Lt with respect to λD

t ,and then propose an optimal pricing scheme to obtain theoptimal price C D

t .

D. Proof of Convexity

According to (29), the service dropping rate Lt is

Lt = λAt pA

t (n′1)

1

1 − ρAt

+ λDt pD

t (n′2)

1

1 − ρDt

,

where n′1 = N1 + Q A

t and n′2 = N2 + QD

t . Due to λAt =

N1μAρA

t and λDt = N2μ

DρDt , we have

Lt = L At + L D

t ,

where

L At = N1μ

AρAt pA

t (n′1)

1

1 − ρAt

= N1μA pA

t (0)N N1

1 (ρAt )n′

1+1

N1!1

1 − ρAt

, (33)

L Dt = N2μ

DρDt pD

t (n′2)

1

1 − ρDt

= N2μD pD

t (0)N N2

2 (ρDt )n′

2+1

N2!1

1 − ρDt

. (34)

It can be found that L At and L D

t have the similar structure.Thus, we first prove that the second part L D

t is an increasingand convex function of λD

t , and then we can derive that thefirst part L A

t is a decreasing and convex function of λDt due

to λAt = λt − λD

t . Because ρAt is a linear function of λA

t andsuch a variable substitution will not change the convexity ofthe objective function, we take L D

t as the objective functionand ρD

t as the variable in the following part. For the secondpart L D

t , we have the following theorem:Theorem 2: The service dropping rate L D

t is an increasingand convex function of the service rate ρD

t .Proof: The proof can be found in Appendix A.

Since L At and L D

t have the similar structure and λAt = λt −

λDt , we have the following Lemma for the relationship between

L At and ρD

t :Lemma 2: The service dropping rate L A

t is a decreasing andconvex function of the service rate ρD

t .Proof: The proof can be found in Appendix B.

Theorem 3: The service dropping rate Lt is a convex func-tion of λD

t .

Proof: Since Lt = L At + L D

t , ∂2 L At

∂(ρDt )2 > 0 and ∂2 L D

t

∂(ρDt )2 > 0,

we have ∂2 Lt

∂(ρDt )2 > 0. Since λD

t = ρDt N2μ

D , ∂λDt

∂ρDt

= N2μD > 0

always holds. Hence, ∂2 Lt

∂(λDt )2 = (N2μ

D)2 ∂2 Lt

∂(ρDt )2 > 0. Thus,

the service dropping rate Lt is a convex function of λDt .

Because the relationship between λDt and C D

t is linear, the ser-vice dropping rate Lt also is a convex function of C D

t .

E. Optimal Pricing Scheme

Since the objective function Lt is a convex function ofλD

t and the constraint (32) is a linear constraint for λDt ,

the transformed Problem P1 is a convex optimization problem.Since this optimization problem will be solved by the chargingstation, it can be solved using existing centralized tools, suchas fmincon function [31] or CVX toolbox [32] in Matlab.We omit the details of how to solve this problem.

By solving Problem P1, the optimal λDt can be obtained,

then the optimal pricing C Dt can be calculated by

C Dt = b

a− λD

t

aλt. (35)

The process for obtaining the optimal price C Dt can be

sketched as Algorithm 1.Theorem 4: The minimal Lt can be achieved by the optimal

pricing scheme shown in Algorithm 1.



Algorithm 1 Optimal Pricing Scheme for the Charging Station

Initialization λt , N1, μA, N2, μ

D, Q At , QD

t

• If λt ≥ N1μA + N2μ

D

Chooses λDt randomly in [N2μ

D, λt − N1μA];

• Else

Obtains the optimal λDt by solving Problem P1;

• end

Calculates the optimal price C Dt by (35);

return C Dt

Fig. 2. The EV arrivals of the charging station.

Proof: According to Theorem 1 and Lemma 1, if λt ≥N1μ

A + N2μD , the minimal service dropping rate Lt ≈ λt −

N1μA − N2μ

D can be obtained by any value in the availablerange, i.e., C D

t ∈ [ N1μA+bλt−λtaλt

, bλt−N2μD

aλt]. If λt < N1μ

A +N2μ

D , since the customer attrition minimization problem is aconvex optimization problem according to Theorem 3, thereexists an unique optimal solution, which can be obtained bysolving Problem P1 [30]. Thus, the proposed optimal pricingscheme in Algorithm 1 can minimize the service droppingrate Lt .

IV. PERFORMANCE EVALUATION

In order to demonstrate the performance of the proposedalgorithm, we take one charging station with dual chargingmodes and time-varying arrivals of EVs as a case study,and then analyze the effects of the number of chargers,the maximal waiting queue length, and the arrival rate on theoptimal pricing of the charging station.

A. Case Study

Consider a charging station with N1 = 15 AC chargers andN2 = 8 DC chargers. The average service rates for each ACcharger and DC charger are μA = 2

5 per hour and μD = 125

per hour, respectively. According to [33], the cost for replacinga 24kWh battery is about $5500, and the lifetime with theAC mode and the DC mode are about 650 and 500 cycles,respectively. Thus, we can derived that C A

B ≈ $0.35/kWh andC D

B ≈ $0.46/kWh. The maximal queue lengths are Q At = 10

and QDt = 8. We set C A

t = $0.15/kWh, CD = $0.8/kWh,

β = $0.2/kWh, and E = 16kWh. The arrivals of EVs duringdifferent time slot can be found in Fig. 2. Assume that thereexists another two construction plan: i) charging station withthe AC mode, in which all the DC chargers (including thewaiting space) are replaced by the AC chargers, and all the

Fig. 3. Optimal price C Dt where C A

t = $0.15/kWh and selections of EVsat different time. (a) Optimal price C D

t . (b) Selection of EVs.

Fig. 4. The numbers of EVs that leave the charging station without beingcharged and the corresponding expected queue lengths at different times.(a) L A

t and L Dt . (b) Q A

t and QDt .

EVs can only select the AC mode, i.e., N1 = 31 and Q At = 10;

ii) charging station with the DC mode, in which all the ACchargers are replaced by the DC chargers, i.e., N2 = 23and QD

t = 18, while the service fee of the DC mode is$0.63/kWh.

The optimal price C Dt of the charging station with dual

charging modes and the expected selections of EVs can befound in Fig. 3. It can be found that the available range for theoptimal price changes and the optimal price of the DC modeis time-varying due to time-varying EV arrival rate. With theincrease of the arrival rate λt , the available range for optimalprice will be narrowed, which means that the selections forthe optimal price become less, and the optimal price C D

t forthe DC mode will be decreased, such that more EVs willselect the DC mode. However, since the service fees in thecharging stations with single mode are constants, all the EVsin i) charging station with the AC mode can only select theAC mode while only part of EVs in ii) charging station withthe DC mode will select the DC mode due to its high batterylifetime-related cost.

The total service dropping rates for different chargingstations and the expected queue lengths can be found in Fig. 4.It can be found that the optimal pricing scheme can minimizethe service dropping rate since it can guide the EVs selectingthe suitable charging mode to improve the utilization ofchargers in the charging station. For i) the charging stationwith the AC mode, part of EVs leaves due to its limited serviceability and the long waiting queue, while for ii) the chargingstation with the DC mode, part of EVs will not select thischarging station due to its high battery lifetime-related cost andhigh charging service fee. For the charging station with dualcharging modes under optimal pricing scheme, by adjusting



Fig. 5. Optimal solution with the increase of arrival rate λAt : a) the optimal price C D

t ; b) the selections of EVs λAt and λD

t ; c) the service dropping ratesL A

t and L Dt ; and d) the expected queue lengths Q A

t and QDt .

TABLE II

THE SERVICE DROPPING RATE (UNIT: EVS)

the service fee of the DC mode, both the number of EVsthat leave the charging station without being charged and theexpected queue lengths can be minimized, especially when thearrival rate is high.

With same arrival rates and construction space, i) chargingstation with the AC mode has the smallest service ability andlongest waiting queue, ii) charging station with the DC modehas the largest service ability and the smallest waiting queue,and charging station with dual charging modes under optimalpricing scheme has a middle service ability and waiting queue.In addition, i) charging station with the AC mode has thelargest service dropping rate due to limited service ability, andii) charging station with the DC mode has a higher servicedropping rate due to its high battery lifetime-related cost, andthe charging station with dual charging modes under optimalpricing scheme can minimize the service dropping rate, whichhas a higher flexibility and adaptability to deal with the time-varying arrival rate of EVs.

The expected service dropping rate for different chargingstations can be found in Table II. It can be found that thecharging station with dual charging modes under the proposedpricing scheme can minimize the total service dropping rate ofthe charging station. The charging station with the AC modehas the highest service dropping rate due to its limited chargingservice rate even when it has 31 chargers. The charging stationwith the DC mode has a middle service dropping rate due to itshigh service fee and high battery lifetime-related cost. As weknow, since the charging station with the DC mode has enoughcharging service ability to service more EVs, reduce its servicefee can cut down the service dropping rate. However, we foundthat only when the charging station with the DC mode reduceits the service fee to $0.59/kWh, which is much lower thanthe service fee in charging station with dual charging modes,it has the similar service dropping rate of the charging stationwith dual charging modes.

B. Relationship Between Arrival Rate and SystemPerformance

The maximal service ability of the charging station withdual charging modes is 25.2 EVs/hour. In order to explore theperformance of our proposed pricing scheme, we set the arrivalrate from [1, 35] and respectively show the optimal price C D

tand its available range, the minimal service dropping rates L A

tand L D

t , the selections of EVs λAt and λD

t , and the expectedqueue lengths for the AC mode and the DC mode in Fig. 5.

From the simulation results in Fig. 5(a), it can be foundthat, when the arrival rate of EVs is smaller than the maximalservice ability of the charging station, with the increase ofthe arrival rate λt , the available range for the optimal pricebecomes narrow to ensure that both of the utilization factorsρA

t and ρDt are smaller than 1, and the optimal price C D

tdecreases; otherwise, the available range for the optimal pricebecomes wider to ensure that both of the utilization factors ρA

tand ρD

t are larger than 1, and the optimal price C Dt can be any

value in the available range. Specially, when the arrival rate ofEVs is smaller than the maximal service ability of the chargingstation, the charging station needs to select an optimal pricingby solving Problem P1 to minimize the total service droppingrate for both the AC mode and the DC mode; and when thearrival rate of EVs is larger than the maximal service ability ofthe charging station, the charging station just needs to selectone available value in the available range since any value inthe available range will obtain the same service dropping rate.

Figs. 5(b)-5(d) show the selections of EVs, the servicedropping rates and the expected queue lengths, respectively.With the increase of EV arrival rate, the selections of EVsfor both the AC mode and the DC mode, the service droppingrate and the expected queue lengths will be increased. In orderto minimize the service dropping rate, it can be found thatthe selections of EVs for both the AC mode and the DCmode increase smoothly. Also, the service dropping rate growswhen the arrival rate of EVs is larger than a certain thresholdand the expected queue lengths reach their upper bounds ofthe charging station. Since the service rate of the AC modeis much lower than the service rate of the DC mode, moreEVs are assigned to select the DC mode to minimize the totalservice dropping rate.

C. The Effects of N1 and N2 on Service Dropping Rate Lt

To explore the effect of the charging facilities in thecharging station on the service dropping rate Lt , we fix the



Fig. 6. The minimal service dropping rate Lt affected by the values of N1and N2: (a) N1 = 15 and N2 ∈ [1, 20]; (b) N2 = 8 and N1 ∈ [1, 20].

Fig. 7. The minimal service dropping rate Lt affected by the values of Q At

and QDt : (a) Q A

t = 10 and QDt ∈ [1, 20]; (b) QD

t = 8 and Q At ∈ [1, 20].

EV arrival by λt = 22, and perform the following simula-tions: 1) fixing N1 = 15 while adjusting N2 from 1 to 20,the minimal service dropping rate Lt is shown in Fig. 6(a);2) fixing N2 = 8 while adjusting N1 from 1 to 20, the minimalservice dropping rate Lt is shown in Fig. 6(b). It can befound that the minimal service dropping rate Lt in the firstsimulation is decreasing much faster than that in the secondone. That is because the service rate of each DC charger ismuch higher than that of each AC charger. For a chargingstation with limited space, more DC chargers can reduce theservice dropping rate, but may decrease the number of EVsdue to its high battery lifetime-related cost.

D. The Effects of Q At and QD

t on Service Dropping Rate Lt

Since the maximal queue length affects the minimal servicedropping rate Lt , we conduct several simulations to demon-strate the effects of Q A

t and QDt on the minimal service

dropping rate Lt . First, we fix Q At = 10 and adjust QD

tfrom 1 to 20, whose minimal service dropping rate Lt isshown in Fig. 7(a). Then, we fix QD

t = 8 and adjust Q At

from 1 to 20, whose minimal service dropping rate Lt isillustrated in Fig. 7(b). Obviously, the service dropping rateLt in the first simulation will decrease more quickly than thatin the second simulation. That is because the service rate forthe DC chargers is much higher than that of the AC charger.However, increasing the waiting space of the charging stationmay not improve the service dropping rate significantly, sincethe service dropping rate mainly depends on the service abilityof the charging station.

V. CONCLUSION

In this paper, we modeled the operation of the chargingstation with dual charging modes as a queuing network with

the multiple servers and heterogeneous service rates, andanalyzed the relationship between the service dropping rateand the selections of EVs. Then, by making use of pricesensitiveness of EV owners, we designed an optimal pricingscheme to guide and coordinate the charging processes of EVsto minimize the service dropping rate of the charging station.Simulation results are provided to demonstrate the efficiencyof the proposed charging scheduling scheme.

In our future work, we will consider the optimal pricingscheme for the charging station, where the arrival rate of thecharging station depends on the service fee and the EVs canchange their selections when the selected queue length is toolong. Also, we intend to design an algorithm to determines theoptimal number of chargers with dual charging modes basedon the distribution of EVs, which can maximize the total profitof the charging stations and improve the service quality of thecharging station.

APPENDIX A

Proof: For the special case when QDt = 0, according

to the Erlang’s C formula [34], the loss probability forM/M/N/N can be given by

B(N2, N2ρDt ) = pD

t (0)N N2

2 (ρDt )N2

N2!1

1 − ρDt

=(N2ρD

t )N2

N2!(1−ρDt )

∑N2−1n=0

(N2ρDt )n

n! + (N2ρDt )N2

N2 !(1−ρDt )

=( N2−1∑

n=0

N2!(1 − ρDt )

n!(N2ρDt )N2−n

+ 1)−1

. (36)

The existing works [35], [36] have proved that B(N2, N2ρDt )

is an increasing and convex function of ρDt when

N2 is given. Thus, both of the first and the sec-ond derivatives B(N2, N2ρ

Dt ) with respect to ρD

t arelarger than zero, i.e., ∂ B(N2, N2ρ

Dt )/∂ρD

t > 0 and∂2 B(N2, N2ρ

Dt )/∂(ρD

t )2 > 0.According to (34), L D

t can be rewritten as

L Dt = N2μ

D(ρDt )Q D

t +1 B(N2, N2ρDt ). (37)

The first derivative L Dt with respect to ρD

t is

∂L Dt

∂ρDt

= N2μD(∂(ρD

t )Q Dt +1

∂ρDt

B(N2, N2ρDt )

+ (ρDt )Q D

t +1 ∂ B(N2, N2ρDt )

∂ρDt

).

Since N2μD > 0, ∂(ρD

t )QDt +1

∂ρDt

= (QDt + 1)(ρD

t )Q Dt > 0,

B(N2, N2ρDt ) > 0, (ρD

t )Q Dt +1 > 0, and ∂ B(N2,N2ρD

t )

∂ρDt

> 0,

we have ∂L Dt

∂ρDt

> 0.



The second derivative L Dt with respect to ρD

t is

∂2 L Dt

∂(ρDt )2

= N2μD(∂2(ρD

t )Q Dt +1

∂(ρDt )2

B(N2, N2ρDt )

+ 2∂(ρD

t )Q Dt +1

∂ρDt

B(N2, N2ρDt )

∂ρDt

+ (ρDt )Q D

t +1 ∂2 B(N2, N2ρDt )

∂(ρDt )2

). (38)

Since ∂2(ρDt )QD

t +1

∂(ρDt )2 = QD

t (QDt +1)(ρD

t )Q Dt −1 > 0, ∂(ρD

t )QDt +1

∂ρDt

>

0, ∂ B(N2,N2ρDt )

∂ρDt

> 0, and ∂2 B(N2,N2ρDt )

∂(ρDt )2 > 0, ∂2 L D

t

∂(ρDt )2 > 0 always

holds.Since ∂L D

t

∂ρDt

> 0 and ∂2 L Dt

∂(ρDt )2 > 0 always hold, the service

dropping rate L Dt is an increasing and convex function of the

service rate ρDt .

APPENDIX B

Proof: Since λAt = Lt − λD

t , ρAt = λA

t /N1μA and ρD

t =λD

t /N2μD , we have

ρAt = Lt − N2μ

DρDt

N1μA, (39)

and the first derivation of ρAt with respect to ρD

t is

∂ρAt

∂ρDt

= − N2μD

N1μA. (40)

Since L Dt has the same structure with L A

t , it can be easily

proved that ∂L Dt

∂ρDt

> 0 and ∂2 L Dt

∂(ρDt )2 > 0. Due to ∂ρD

t

∂ρ At

< 0,

we have ∂L Dt

∂ρ At

< 0 and ∂2 L Dt

∂(ρ At )2 > 0. Thus, the service dropping

rate L Dt is a decreasing and convex function of ρA

t .

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Yongmin Zhang (S’12–M’15) received the Ph.D.degree in control science and engineering fromZhejiang University, Hangzhou, China, in 2015. Hewas a Visiting Student with the California Insti-tute of Technology. He is currently a Post-DoctoralResearch Fellow with the Department of Electricaland Computer Engineering, University of Victoria,BC, Canada. His research interests include resourcemanagement and optimization in wireless networks,smart grid, and mobile computing. He received theBest Paper Award from the IEEE PIMRC 2012 and

the IEEE Asia–Pacific Outstanding Paper Award in 2018.

Pengcheng You (S’14) received the B.S. degree(Hons.) in electrical engineering and the Ph.D.degree in control science and engineering fromZhejiang University, China, in 2013 and 2018,respectively. He was a Visiting Student with theSingapore University of Technology and Designand the California Institute of Technology and aResearch Intern with the Pacific Northwest NationalLaboratory. He is currently a Post-Doctoral Fellowwith the Department of Mechanical Engineering,Johns Hopkins University. His research focuses on

smart grid, especially electric vehicle, and power market.

Lin Cai (S’00–M’06–SM’10) received the M.A.Sc.and Ph.D. degrees in electrical and computer engi-neering from the University of Waterloo, Waterloo,Canada, in 2002 and 2005, respectively. Since 2005,she has been with the Department of Electricaland Computer Engineering, University of Victoria,where she is currently a Professor. Her researchinterests span several areas in communications andnetworking, with a focus on network protocol andarchitecture design supporting emerging multimediatraffic over wireless, mobile, ad hoc, and sensor net-

works. She was a recipient of the NSERC Discovery Accelerator SupplementGrants in 2010 and 2015 and the Best Paper Award at the IEEE ICC 2008 andthe IEEE WCNC 2011. She has served as a TPC Symposium Co-Chairfor the IEEE Globecom 2010 and Globecom 2013, as an Associate Editorfor the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, the EURASIP Journal onWireless Communications and Networking, the International Journal of SensorNetworks, and the Journal of Communications and Networks, and as aDistinguished Lecturer of the IEEE VTS Society.

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