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Revenue Maximization of Multi-class Charging

Stations with Opportunistic Charger Sharing

Kihong Ahn∗, Aresh Dadlani†, Kiseon Kim∗, and Walid Saad‡

∗School of Electrical Engineering and Computer Science, GIST, Gwangju, South Korea

†Department of Electrical and Electronic Engineering, Nazarbayev University, Astana, Kazakhstan

‡Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA

Email: {gandio, kskim}@gist.ac.kr, aresh.dadlani@nu.edu.kz, walids@vt.edu

Abstract—Distribution of limited smart grid resources among

electric vehicles (EVs) with diverse service demands in an

unfavorable manner can potentially degrade the overall proﬁt

achievable by the operating charging station (CS). In fact,

inefﬁcient resource management can lead to customer dissatis-

faction arising due to prolonged queueing and blockage of EVs

arriving at the CS for service. In this paper, a dynamic electric

power allocation scheme for a charging facility is proposed and

modeled as a bi-variate continuous-time Markovian process, with

exclusive charging outlets being allotted to EVs of different

classes in real-time. The presented mechanism enables the CS to

guarantee the quality-of-service expected by customers in terms

of blocking probability, while also maximizing its own overall

revenue. By adopting a practical congestion pricing model within

the deﬁned proﬁt function, the revenue optimization framework

for a single CS is further extended to a load-balanced network

of CSs. Simulation results for the single CS and networked

models reveal considerably higher satisfaction levels for congested

fast charging EV customers and improved attainable system

revenue as compared to a baseline scenario which assumes no

classiﬁcation based on EV service preferences.

Index Terms—Electric vehicle charging model, continuous-time

Markov chain, revenue maximization, shared chargers.

I. INTRODUCTION

Perceptible advancement of battery and converter technolo-

gies over recent years has stimulated the large-scale penetra-

tion of plug-in electric vehicles (EVs) as eco-friendly and cost-

efﬁcient substitutes in the transportation sector. With threefold

increase in global EV sales since 2013, a soaring 42% (i.e.

over 773,600 units) was recorded in 2016 alone [1]. Amid

the growth of EV market shares and energy policy regulations

worldwide, EV charging operations are foreseen to pose new

challenges for demand response management in smart grids

[2]–[4]. A key issue in this line concerns devising optimal

energy management paradigms that meet the expectations of

customers with different service preferences without over-

exploiting the limited resources and thus, jeopardizing the

stability of power grids. To avert potential supply-demand

imbalances arising due to the growing engagement of EVs,

it is thus crucial to address and control factors impacting the

performance limits of power supply infrastructures.

Customer satisfaction, in terms of waiting time and blocking

probability, is the foremost performance measure mostly re-

garded in the EV charging decision process [5]. Depending

on the queueing discipline and service distribution, a few

number of seminal studies aim at maximizing the proﬁt of

a charging station (CS) by minimizing the penalty associated

with delayed and evicted EVs [6]–[8]. Despite the efforts

made to meet service-level agreements (SLAs), the works

referred above mandate all EVs arriving at a CS to enter a

waiting space prior to commencing service. Nonetheless, as

usually witnessed in conventional gas stations, an impatient

EV customer may prefer to be blocked on arrival (ﬁnding no

idle charging outlet) rather than to be delayed in queue for

service [9]. Early blockage of such EV owners permits them to

either retry after some random time period or visit another CS

in local proximity thus, eliminating the waiting cost incurred.

Intertwined with customer satisfaction is the strategy various

CS operators undertake to allocate scarce grid resources. The

non-trivial revenue maximization problem becomes even more

challenging when accommodating EVs with different battery

charging speciﬁcations and proﬁles (AC Level-1, AC Level-

2, DC Fast) [10], [11]. While more recent works consider

either dedicated or shared resource pools in determining the

utility of the charging network [12]–[14], theoretical aspects

of an opportunistic sharing-based charger provisioning strategy

under energy constraints have yet to be scrutinized.

Building on the premise of immediate resource allocation

to EVs upon arrival, this paper introduces a novel real-time

scheme for allocating shareable chargers to EVs subject to

distinct service needs by taking their incoming trafﬁc rates into

account. To our best knowledge, all existing works allocate CS

chargers to EVs based on the simple ﬁrst-come-ﬁrst-served

sharing strategy. In contrast, here, we propose an efﬁcient,

opportunistic charger allocation strategy in a bufferless CS

model that favors fast charging over slow charging requests

during congestion in both, single and networked CS settings.

Our results reveal higher attainable system revenue as com-

pared to the baseline counterpart which does not classify EV

service requests. In that course, our main contributions are:

∙Continuous-time Markov chain (CTMC) characterization

of a dynamic resource allocation scheme for a single CS

with two EV service classes (i.e. slow and fast).

∙Optimal revenue framework formulation for the multi-

class CS model using an amended pricing function to

alleviate congestion and meet quality-of-service (QoS)

expectations of EV customers.

∙Network-level performance analysis of multiple CSs me-

Slow charging EV

(class-1)

ߣଵ

CS

. . . . . .

ߤଶ

Fast charging EV

(class-2)

ߣଶ

ߤଵ

ݏଵ

ܵ

Class-iEV

(݅א ͳǡʹ )

EV belongs

to class 1?

YAny idle ݏଵ

available?

NBlock

Y

N

Any idle ܵെݏ

ଵ

available?

Allocate

Y

N

Fig. 1: Schematic design and charger allocation strategy of a

two-class charging model.

diated by a centralized load dispatching entity and sim-

ulation comparison with an undifferentiated single-class

baseline model under varying trafﬁc intensities.

The rest of this paper is organized as follows. The proposed

performance model for a single CS is presented in Section II,

followed by details on the revenue maximization formulation

with an effective pricing mechanism in Section III. Section IV

discusses the optimal resource allocation scheme for a network

of multiple CSs. Numerical simulations and discussions follow

in Section V. Finally, Section VI concludes the paper.

II. SINGLE CS SYSTEM MODEL

We consider a CS facility equipped with 𝑆charging outlets

capable of charging 𝑀=2service-differentiated classes of

EVs. The CS is inﬂexible in meeting additional demands as

it is assumed to draw 𝑆discretized units of constant power

from the grid. In other words, the CS reaches its maximum

capacity when all the charging outlets are busy. For 𝑖∈{1,2},

EVs of class 𝑖are assumed to follow a Poisson arrival with

rate 𝜆𝑖>0and exponentially distributed customer service time

with mean 𝜇−1

𝑖>0. To account for the diversity range in slow

and fast charging requests, EVs belonging to classes 1and 2

are allocated 𝑠1and 𝑠2chargers at any given time, respectively,

such that 𝑠1<𝑠2≤𝑆when class-2has non-preemptive priority

in using the charger outlets of class-1. Given these notations

and assumptions, the single CS model can thus, be expressed

as a ﬁnite-space CTMC {𝑋(𝑡),𝑡≥0}.

To capture the state of the system for 𝑀=2 at time 𝑡,we

deﬁne 𝑁1(𝑡)and 𝑁2(𝑡)as the number of class-1 and class-2

EVs being served by the system at time 𝑡, respectively. Without

loss of generality, we consider class-2 to have service priority

over class-1. An EV of class 2 is allocated one of the 𝑠1outlets

only if (i) all 𝑆−𝑠1outlets are occupied by other class-2 EVs

1,SͲݏଵ+10,SͲݏଵ+1 ݏଵͲ2,

SͲݏଵ+1

ߣଵߣଵ

ߤଵʹߤଵ

ߣଵ

...

ߣଶ

(SͲݏଵ+2ሻߤଶߣଶ

(SͲݏଵ+2ሻߤଶߣଶ

(SͲݏଵ+2ሻߤଶ

...

...

...

0,SͲ1ߣଵ

ߤଵ

ߣଶ

(SͲ1)ߤଶߣଶ

(SͲ1)ߤଶ

1,SͲ1

(ݏଵ-2)ߤଵ

0,S

ߣଶ

Sߤଶ

1,00,0 2,0 ݏଵͲ1,0

ߣଵߣଵߣଵߣଵ

...

ߤଵʹߤଵ͵ߤଵ(ݏଵ-1)ߤଵ

1,10,1 2,1 ݏଵͲ1,1

ߣଵߣଵߣଵߣଵ

...

ߤଵʹߤଵ͵ߤଵ

ݏଵ,0

ߣଵ

ݏଵߤଵ

(ݏଵ-1)ߤଵ

ߣଶ

ߤଶߣଶ

ߤଶߣଶ

ߤଶߣଶ

ߤଶ

ߣଶ

ʹߤଶߣଶ

ʹߤଶߣଶ

ʹߤଶ

ߣଶ

ߤଶ

ߣଶ

ʹߤଶ

ݏଵ,1

ߣଵ

ݏଵߤଵ

ߣଶ

ʹߤଶ

ߣଶ

ߤଶ

ߣଶ

ʹߤଶ

...

...

...

...

...

...

ߣଵߣଵߣଵߣଵߣଵ

1,SͲݏଵ

0,SͲݏଵ2,SͲݏଵݏଵͲ1,SͲݏଵ

...

ߤଵʹߤଵ͵ߤଵ(ݏଵ-1)ߤଵ

ߣଶ

(SͲݏଵ+1ሻߤଶߣଶ

(SͲݏଵ+1ሻߤଶߣଶ

(SͲݏଵ+1ሻߤଶߣଶ

(SͲݏଵ+1ሻߤଶ

ݏଵ,SͲݏଵ

ݏଵߤଵ

ߣଶ

(SͲݏଵ)ߤଶߣଶ

ߣଶߣଶߣଶߣଶ

(SͲݏଵ)ߤଶ(SͲݏଵ)ߤଶ(SͲݏଵ)ߤଶ(SͲݏଵ)ߤଶ(SͲݏଵ)ߤଶ

ݏଵͲ1,

SͲݏଵ+1

ߣଵ

(ݏଵ-1)ߤଵ

ߣଶ

(SͲݏଵ+1ሻߤଶ

ߣଶ

(SͲݏଵ+2ሻߤଶ

...

݅ǡ݆ Non-sharing states (ȳଵ)

݅ǡ݆ Sharing states (ȳଶ)

݅ǡ݆ Blocking states (ȳ)

Fig. 2: Markovian two-class charging model representation.

and (ii) at least one of the 𝑠1outlets is idle. Subsequently, the

system in Fig. 1 can be characterized by a bi-variate process

𝑋(𝑡)=𝑁1(𝑡),𝑁

2(𝑡)taking values in the state space Ω=

{(𝑖, 𝑗)∣0≤𝑖≤𝑠1,𝑗 ≤𝑆−𝑖}, where ∣Ω∣=𝑠1

𝑘=0(𝑆+1−𝑘).

For the corresponding birth-death process shown in Fig. 2, the

transition from some state 𝑥=(𝑖′,𝑗′)∈Ωto any adjacent state

𝑦∈Ωoccurs at the following rates:

𝑞𝑥,𝑦 =

𝜆1,if 𝑦=(𝑖′+1,𝑗′); 𝑖′<𝑠1;𝑗′<𝑆−1−𝑖′,

𝜆2,if 𝑦=(𝑖′,𝑗′+1); 𝑖′<𝑠1;𝑗′<𝑆−1−𝑖′,

𝑖′𝜇1,if 𝑦=(𝑖′−1,𝑗′); 1≤𝑖′≤𝑠1;𝑗′≤𝑆−𝑖′,

𝑗′𝜇2,if 𝑦=(𝑖′,𝑗′−1); 𝑖′≤𝑠1;1≤𝑗′≤𝑆−𝑖′,

0,if otherwise.

(1)

These rates can be re-arranged in matrix form as elements of

the inﬁnitesimal generator matrix 𝑸:Ω×Ω→ℝsuch that:

𝑸∣Ω∣×∣Ω∣=𝑞𝑥,𝑦,if 𝑥∕=𝑦

−𝑦∈Ω𝑞𝑥,𝑦,if 𝑥=𝑦. (2)

Deﬁning Π=[𝜋𝑖,𝑗 ]1×∣Ω∣as the steady-state probability vector,

with each element 𝜋𝑖,𝑗 denoting the probability of having 𝑖

number of class-1 and 𝑗number of class-2 vehicles in service

at the CS, the linear system Π⋅𝑸=0and Π⋅𝒆=1 yields the

following closed-form solution:

𝜋𝑖,𝑗 =1

𝑖!𝑗!𝜆1

𝜇1𝑖𝜆2

𝜇2𝑗

𝜋0,0,(3)

where 0≤𝑖≤𝑠1,0≤𝑗≤𝑆−𝑖, and 𝜋0,0is the normalized

equation given by:

𝜋0,0=

𝑠1

𝑖=0

𝑆−𝑖

𝑗=0

1

𝑖!𝑗!𝜆1

𝜇1𝑖𝜆2

𝜇2𝑗

−1

.(4)

The blocking probability for each class can now be derived

from the stationary probability distribution. An incoming EV

of class-2 is blocked only if (i) all 𝑆−𝑠1chargers are occupied

by other EVs of class-2 and (ii) the remaining 𝑠1chargers are

busy. Denoted by 𝑃2, the blocking probability of class-2 EVs

can be calculated as below:

𝑃2=

𝑖+𝑗=𝑆

𝜋𝑖,𝑗 =𝜋0,0

𝑆!

𝑆

𝑗=𝑆−𝑠1𝑆

𝑗𝜆1

𝜇1𝑆−𝑗𝜆2

𝜇2𝑗

.(5)

Similar to (5), the blocking probability of a class-1 EV, 𝑃1,

includes an additional term associated with threshold 𝑠1<𝑆:

𝑃1=𝑃2+

𝑆−(𝑠1+1)

𝑗=0

𝜋𝑠1,𝑗 .(6)

As a result, the proportion of vehicles denied service due to

blockage is denoted by 𝒩𝑏and can be obtained as follows:

𝒩𝑏=𝜆1

𝜆1+𝜆2

𝑃1+𝜆2

𝜆1+𝜆2

𝑃2.(7)

III. SINGLE CS REVENUE FORMULATION

The revenue earned by the operating CS can be quantiﬁed

in terms of the charging model dynamics. As the power grid

resources are shared instantaneously among EV owners willing

to pay for services in accordance to their needs, the revenue

function (ℛ)is deﬁned using three main cost components;

the mean proﬁt (𝑓𝒫), the mean blocking penalty (𝑓ℬ), and

the mean maintenance cost (𝑓ℳ). To facilitate the deﬁnitions

that follow, we divide the state space into two disjoint sub-

spaces, such that for any given sub-space Ω𝑚,1≤𝑚≤𝑀,

the constituent system states correspond to cases when class-𝑚

EVs are assigned idle chargers primarily reserved for EVs with

lower service demands rather than being blocked from service.

Let Φ𝑚represent the set of class-𝑚idle chargers allocable to

faster charging EVs that ﬁnd all dedicated chargers occupied

upon arrival. Thus, Ω𝑚can be formally generalized as:

Ω𝑚=(𝑖1,...,𝑖

𝑚,...,𝑖

𝑀)𝑠𝑚<𝑖𝑚≤𝑠𝑚+

𝑚−1

𝑟=1

∣Φ𝑟∣,

0≤𝑖𝑚+1 ≤𝑠𝑚+1,...,0≤𝑖𝑀≤𝑆−

𝑀−1

𝑟=1

𝑠𝑟.

(8)

A. Derivation of 𝑓𝒫

The overall proﬁt made by the CS is differentiated according

to the service grades of incoming EVs. For any state 𝑥∈Ω,

let 𝑛𝑚

𝑥and 𝜋𝑥denote the number of class-𝑚EVs in state

𝑥and the stationary probability of being in 𝑥, respectively.

Also, let 𝑝𝑚be the price paid by a class-𝑚EV for service; fast

charging customers are required to pay more amount of money

as compared to low charging customers [8], [9]. Subsequently,

the average proﬁt achievable for any 𝑀number of EV service

classes is computed as:

𝑓𝒫=

𝑀

𝑟=1 𝑝𝑟

𝑥∈Ω𝑟

𝑛𝑟

𝑥𝜋𝑥.(9)

B. Derivation of 𝑓ℬ

The average penalty incurred due to the blockage of a class-

𝑚EV from service is given below, where 𝑐𝑚symbolizes the

compensation cost reimbursed to the blocked EV customer and

Ω𝑏is the set of all blocking states exempliﬁed in Fig. 2:

𝑓ℬ=

𝑀

𝑟=1 𝑐𝑟

𝑥∈Ω𝑏

𝑛𝑟

𝑥𝜋𝑥.(10)

C. Derivation of 𝑓ℳ

Similar to [9], the ﬁnal component in our revenue model is

related to the subsidiary maintenance fees covering ancillary

expenses (installation, labor, acquisition, etc.), with ¯𝑐0and ¯

𝑑0

taken to be positive constants:

𝑓ℳ=

𝑀

𝑟=1 ¯𝑐0

𝑥∈Ω

𝑛𝑟

𝑥𝜋𝑥+¯

𝑑0𝑆. (11)

D. Pricing Policies for 𝑝𝑘and 𝑐𝑘

To comply with the customer QoS satisfaction level speci-

ﬁed in the SLA, an adjustable pricing mechanism is needed to

mitigate congestion of EV service requests at the CS without

compromising the net revenue substantially. By extending the

myopic pricing policy of [15] to our multi-class setting, we

adopt the following principle in steady-state:

𝑝𝑚=

¯𝑝𝑚,if 𝜆𝑚≤𝜆∗

𝑚

¯𝑝𝑚+¯𝑝𝑚−log 𝜆∗

𝑚

𝜆𝑚,if 𝜆𝑚>𝜆

∗

𝑚,(12)

where 𝜆∗

𝑚∈[0,𝜆

max

𝑚]is the maximum arrival rate satisfying

the QoS target denoted as 𝒩max

𝑏and ¯𝑝𝑚is the normal charging

price ﬁxed by the CS operator. It is obvious that an arriving

fast charging EV ﬁnding all chargers dedicated to its class busy

is required to pay a relatively higher price to utilize an idle

charger of a slower charging class, i.e. ∀𝑖, 𝑗, 𝑘∈{1,2,...,𝑀},

𝑝𝑖≤𝑝𝑗≤𝑝𝑘if 𝑖<𝑗<𝑘. Likewise, since idle chargers of

low charging classes are shared with EVs from higher classes

opportunistically, we deﬁne the compensation cost to be 𝑐𝑚=

𝛼𝑝𝑚where 𝛼∈[0,1] is set by the CS operator.

Consequently, the total revenue of a single CS with multiple

service classes is calculated as below:

ℛ=𝑓𝒫−𝑓ℬ−𝑓ℳ.(13)

Fig. 3 plots ℛas function of 𝑆for three different (𝑠1,𝑠

2)

ratios in a single CS with 𝑀=2service classes. The revenue

model is evaluated for parameters (𝜆1,𝜆

2)=(6,6),(𝜇1,𝜇

2)=

(1,3),(¯𝑝1,¯𝑝2)=(3,4),𝑁max

𝑏=5%,(𝜆max

1,𝜆

max

2)=(10,10),

𝛼=0.7,¯𝑐0=0.1, and ¯

𝑑0=0.02. The baseline is deﬁned to be

a single-class system with trafﬁc intensity equal to the sum of

the trafﬁc intensities of customer classes 1 and 2, and 𝑠1=𝑆.

The blocking probability for such a system is thus, given as:

𝑃0=𝜋0,0

𝑆!𝜆1

𝜇1

+𝜆2

𝜇2𝑆

.(14)

Fig. 3: Net revenue proﬁle of a single CS in terms of 𝑆for a

two-class charging scenario.

The existence of an optimal 𝑆value for which the net revenue

peaks is evident in this ﬁgure. Moreover, the overall revenue

for all three ratios converges as 𝑆increases. This is because

abundance of chargers reduces the EV blocking probability.

IV. NETWORK-LEVEL RESOURCE ALLOCATION

In this section, we analyze the revenue maximization prob-

lem for a network of 𝑁closely-located CSs as in Fig. 4. We

assume a central load dispatcher that distributes the incoming

trafﬁc to the most appropriate CS in a probabilistic manner.

Higher resource utilization and real-time decision-making are

the main advantages of a centrally operated load dispatcher

[15]–[17]. Also, we use 𝑆𝑛to denote the number of chargers

installed at station CS𝑛, where 𝑛∈{1,2,...,𝑁}. Based on

the decomposition property of Poisson processes, Λ𝑖denotes

a Poisson process with aggregated rate 𝑁

𝑛=1 𝜆(𝑛)

𝑖, where the

service requests arriving at CS𝑛are independent and follow a

Poisson process with rate 𝜆(𝑛)

𝑖. As a result, Λ𝑖≥𝑁

𝑛=1 𝜆(𝑛)

𝑖.

A. Undifferentiated Charging Requests

Serving as the baseline for our performance comparison, all

EV service requests are considered to be equally prioritized in

this case. In other words, chargers are allocated to EVs on a

ﬁrst-come-ﬁrst-served basis irrespective of their service class

thus, resulting in the following revenue maximization problem

for the classless scenario:

maximize

{𝜆(𝑛)}

𝑁

𝑛=1

𝜆(𝑛)ℛ𝑛−𝛽Λ−

𝑁

𝑛=1

𝜆(𝑛)(15)

subject to 𝑃(𝑛)

0≤𝒩

max

𝑏∀𝑛∈{1,2,...,𝑁},(16)

𝑁

𝑛=1

𝜆(𝑛)≤Λ.(17)

The ﬁrst term in (15) reﬂects the aggregated network revenue,

where 𝜆(𝑛)ℛ𝑛is the fraction contributed by CS𝑛. The second

term calculates the penalty associated with EVs blocked by the

central dispatcher when overwhelmed by service requests. The

fee 𝛽∈[0,1] is decided by the central dispatcher authorized

to decline service demands under high request rates. For the

classless scenario, where 𝑠1=𝑆,𝑃(𝑛)

0denotes the blocking

probability of EVs arriving at CS𝑛.

ߣ

ሺଶሻ

class-iEV

Ȧ ߣ

ሺሻ

ே

ୀଵ

Load

balancing

entity

ଵ

...

ߣ

ሺଵሻ

ߣ

ሺேሻ

ܵଵ

ଶ

...

ܵଶ

...

ே

...

ܵே

Fig. 4: Network of CSs governed by a central load dispatcher.

Algorithm 1 Centralized pricing-based load balancing

Input: 𝑁,Λ𝑟,Δ𝜆,𝛽𝑟,𝑁max

𝑏,¯𝑝𝑟, where 𝑟∈{1,2,...,𝑀}

Output: Optimal 𝜆(𝑛)

𝑟values, corresponding net revenue

1: Find all (Λ𝑟/Δ𝜆+1)

𝑁permutations with step size Δ𝜆.

2: Discard invalid permutations that add up to more than Λ𝑟.

3: for each valid permutation of class 𝑟do

4: if (𝑀

𝑟=1 𝑃(𝑛)

𝑟≤𝒩max

𝑏) and (Λ𝑟≥𝑁

𝑛=1 𝜆(𝑛)

𝑟)then

5: ℛ𝑛𝑒𝑡 ←Calculate revenue as in Section III

6: if ℛ𝑛𝑒𝑡 is the maximum so far then

7: 𝜆𝑜𝑝𝑡

𝑟,𝑛 ←Save selected rate permutation.

8: end if

9: end if

10: end for

11: for each class 𝑟do

12: 𝐵←𝐵+𝛽𝑟Λ𝑟−𝑁

𝑛=1 𝜆𝑜𝑝𝑡

𝑟,𝑛

13: end for

14: return 𝜆𝑜𝑝𝑡

𝑟,𝑛 ,(ℛ𝑛𝑒𝑡 −𝐵)

B. Differentiated Charging Requests

We now account for charger allocation in a setting where

the central entity optimally distributes the service trafﬁc of

each class among the CSs so as to maximize the overall

revenue. Unlike the baseline, the computational complexity of

revenue optimization increases exponentially with the number

of EV classes. The central entity executes Algorithm 1 to

distribute the service requests to each CS in the network. The

algorithm returns the optimal 𝜆(𝑛)

𝑖values that generates the

maximum revenue. The revenue maximization framework for

a network of 𝑁CSsisasfollows,where𝛽𝑖and 𝑃(𝑛)

𝑖denote

respectively, the eviction fee for class-𝑖(𝑖∈{1,2,...,𝑀})

and the blocking probability of class-𝑖EVs at CS𝑛:

maximize

{𝜆(𝑛)

𝑟}

𝑁

𝑛=1

𝑀

𝑟=1

𝜆(𝑛)

𝑟ℛ𝑛−

𝑀

𝑟=1

𝛽𝑟Λ𝑟−

𝑁

𝑛=1

𝜆(𝑛)

𝑟(18)

subject to

𝑀

𝑟=1

𝑃(𝑛)

𝑟≤𝒩

max

𝑏∀𝑛∈{1,2,...,𝑁},(19)

𝑁

𝑛=1

𝜆(𝑛)

𝑟≤Λ𝑟∀𝑟∈{1,2,...,𝑀}.(20)

V. P ERFORMANCE EVA L U AT I O N A N D DISCUSSIONS

The proposed CS performance model is evaluated in terms

of the charging blocking probabilities of each customer class

(a) 𝑆=10,𝜆1=12,(𝜇1,𝜇

2)=(1,3).(b) 𝑠1=4,(𝜇1,𝜇

2)=(1,3).(c) 𝑆=10,𝜆1= 12,(𝜇1,𝜇

2)=(1,3).

Fig. 5: Single CS performance evaluation: (a) charging blocking probability versus 𝜆2, (b) charging blocking probability versus

𝑆, and (c) net revenue versus 𝜆2.

and the net system revenue. For sake of better demonstration,

we adopt the two-class model throughout this section. Unless

stated otherwise, the parameters are set to be as in Fig. 3.

A. Single CS Performance Analysis

Fig. 5 illustrates the impact of various control parameters

on the performance indicators in a single CS setting. In

particular, Fig. 5a shows the blocking probabilities of each

service class in terms of class-2 EV arrival rate. For lower

values of 𝜆2, class-2 EVs experience a much lesser chance

of service denial than the slow charging EVs of class-1. The

blocking probabilities of the two classes however, converge

towards the baseline mark as 𝜆2increases. Moreover, for

smaller 𝑠1values, newly arriving class-2 EVs are probable

to ﬁnd an idle charger from a larger set of dedicated chargers

thus, yielding lower dissatisfaction of class-2 customers. For

instance, the blocking probability of fast charging EVs rises by

approximately 142% as 𝑠1increases from 3to 7at 𝜆2=10.In

contrast, 𝑃1is reduced by 28.5% for the same 𝜆2value. Here,

even though the satisfaction of class-1 customers are slightly

compromised over class-2 customers, the overall satisfaction

level determined as 𝑁max

𝑏is still guaranteed. Nonetheless,

such favoring behavior gradually fades away as 𝜆2increases.

In Fig. 5b, we observe the inﬂuence of the charger set size

on the system blocking probability for varying EV arrival rates.

Note that 𝑃2reaches zero for a smaller set size of 𝑆=10

chargers when 𝜆1>𝜆2, while 𝑃1stabilizes at 0.69. This ﬁgure

reveals that the proposed charging model prioritizes congested

fast-charging EVs over slow charging EVs as the number of

installed chargers at the CS increases. In comparison to the

baseline, our proposed strategy facilitates high charging EVs

(who pay more) opportunistically by allocating them any idle

𝑠1. Nonetheless, if a class-1 EV ﬁnds all 𝑠1chargers occupied

on arrival, it is unwillingly evicted for a compensation fee.

The net revenue earned by the CS under varying 𝜆2values

is shown in Fig. 5c. Note that unlike the case of 𝑠1=7,the

revenue for 𝑠1=3 slightly decreases at 𝜆2=12 before rising

up again. Such behavior for 𝜆2∈[12,40] can be explained

based on the high 𝑃1value used in Fig. 5a; the corresponding

penalty paid for evicting class-1 EVs is higher than the proﬁt

made by accommodating congested class-2 EV customers. The

total revenue however, beats the baseline scenario and peaks

at an extreme value of 𝜆2≈500 (not shown in the ﬁgure).

B. Network-level Performance Analysis

We now evaluate the performance of our strategy in a net-

worked environment comprising of 𝑁=3 identical charging

stations with 𝑆=5 chargers each, 𝒩𝑚𝑎𝑥

𝑏=0.25%,Δ𝜆=1,

𝛽𝑟=0.9,(𝜆max

1,𝜆

𝑚𝑎𝑥

2)=(7,7), and (𝜇1,𝜇

2)=(1,2). Table I

summarizes the optimal EV arrival rates distributed among the

three CSs by the central entity for which the maximum revenue

is obtained considering the undifferentiated service class (Type

A) and the differentiated service classes with 𝑠1=2 (Type B)

and 𝑠1=4 (Type C).

The impact of dedicating chargers to speciﬁc service classes

on the overall network revenue is manifold. Unlike the un-

differentiated baseline model, Type B and Type C scenarios

reveal reduction in the blocked fraction of fast charging EVs.

In addition, while the baseline case outperforms its Type B

counterpart in maximizing the net revenue, it is dominated

by Type C where slow charging EVs are primarily allocated

a larger number of installed chargers for service. On the

other hand, Type B demonstrates a more evenly balanced

service request load at the expense of evicting a fraction of

the low charging EV arrivals to meet the SLA requirements.

The revenue gap between Type A and Type B reduces as Λ2

becomes greater than Λ1, which is in complete agreement with

the results of Fig. 5. For instance, as Λ2increases from 3 to 5,

Type C yields almost 80% increase in revenue as compared to

the baseline case. For (Λ1,Λ2)=(3,5), the system witnesses a

revenue increase of 104.7%. This is merely due to the fact that

congested fast charging EVs intrinsically tend to pay more to

utilize a lower class charger rather than be declined service.

Therefore, by dedicating a larger fraction of the charger set to

TABLE I: System performance and revenue comparison of Type A (undifferentiated service), Type B (differentiated service,

𝑠1=2), and Type C (differentiated service, 𝑠1=4) network scenarios.

(Λ1,Λ2)Type CS1CS2CS3∑3

𝑛=1 ℛ𝑛

(𝜆(1)

1,𝜆

(1)

2)ℛ1(𝑃1,𝑃

2) (𝜆(2)

1,𝜆

(2)

2)ℛ2(𝑃1,𝑃

2) (𝜆(3)

1,𝜆

(3)

2)ℛ3(𝑃1,𝑃

2)

(5,3)

A(2,0) 2.4046 (0.0367,0.0367) (1,2) 2.4046 (0.0367,0.0367) (2,1) 2.7899 (0.0697,0.0697) 7.599

B(1,1) 1.7084 (0.2004,0.0032) (1,1) 1.7084 (0.2004,0.0032) (1,1) 1.7084 (0.2004,0.0032) 3.3252

C(3,0) 2.2687 (0.2061,0) (0,3) 7.5737 (0.0142,0.0142) (2,0) 1.9333 (0.0952,0) 11.7758

(5,5)

A(3,0) 3.06 (0.1101,0.1101) (0,5) 4.9487 (0.0697,0.0697) (2,0) 2.4046 (0.0367,0.0367) 10.4133

B(1,2) 2.7867 (0.2047,0.0197) (1,2) 2.7867 (0.2047,0.0197) (1,1) 1.7084 (0.2004,0.0032) 5.4819

C(3,0) 2.2687 (0.2061,0) (0,5) 14.6296 (0.0697,0.0697) (2,0) 1.9333 (0.0952,0) 18.8316

(3,5)

A(1,0) 1.2796 (0.0031,0.0031) (2,0) 2.4046 (0.0367,0.0367) (0,5) 4.9487 (0.0697,0.0697) 8.6329

B(1,2) 2.7867 (0.2047,0.0197) (1,1) 1.7084 (0.2004,0.2004) (1,2) 2.7867 (0.2047,0.2047) 7.2819

C(1,0) 1.1138 (0.0154,0) (2,0) 1.9333 (0.0952,0) (0,5) 14.6296 (0.0697,0.0697) 17.6768

slow charging EV customers, the proposed allocation strategy

generates more revenue while guaranteeing the satisfaction

expectation of customers belonging to both service classes.

Imposing a stricter QoS level would distribute the incoming

service load more evenly, while generating lower revenue due

to EVs blocked by the load balancer.

VI. CONCLUSION

In this paper, we have proposed a novel congestion pricing-

based resource allocation model for electric vehicle charging

stations accommodating customers with distinct service pref-

erences in real-time. The presented charger sharing strategy

was formulated using a multi-dimensional Markovian chain

to attain the maximum possible revenue for a single as well

as network of charging stations. In comparison to the single-

class undifferentiated system set-up, the out-performance of

the proposed strategy in terms of blocking probability and

system revenue maximization was conﬁrmed via simulation

results. Service classes permitted to non-preemptively utilize

chargers primarily reserved for slower EV charging classes

experience lower service eviction as compared to the classless

baseline. An interesting follow-up on this work would involve

analysis of the intrinsic retrial behavior of blocked EV owners

who randomly attempt to access grid resources and its impact

on various performance indicators under several system con-

ﬁgurations. In addition, ﬁnding the optimal number of chargers

allocable to each class under different EV trafﬁc intensities

would provide better insights on the system model behavior

and deployment cost estimate.

ACKNOWLEDGMENT

This work was supported by the GIST Research Institute

(GRI), South Korea, in 2017 and, in part, by the US National

Science Foundation (NSF) under Grant ECCS-1549894.

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