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Revenue Maximization of Multi-Class Charging Stations with Opportunistic Charger Sharing

Authors:

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 dissatisfaction 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.
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)
ߣ
CS
. . . . . .
ߤ
Fast charging EV
(class-2)
ߣ
ߤ
ݏ
ܵ
Class-iEV
(݅א ͳǡʹ )
EV belongs
to class 1?
YAny idle ݏ
available?
NBlock
Y
N
Any idle ܵെݏ
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 𝜇1
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<𝑠2𝑆when 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),𝑡0}.
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 𝑆𝑠1outlets are occupied by other class-2 EVs
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݅ǡ݆ Non-sharing states (ȳ)
݅ǡ݆ Sharing states (ȳ)
݅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)0𝑖𝑠1,𝑗 𝑆𝑖}, where Ω=𝑠1
𝑘=0(𝑆+1𝑘).
For the corresponding birth-death process shown in Fig. 2, the
transition from some state 𝑥=(𝑖,𝑗)Ωto any adjacent state
𝑦Ωoccurs at the following rates:
𝑞𝑥,𝑦 =
𝜆1,if 𝑦=(𝑖+1,𝑗); 𝑖<𝑠1;𝑗<𝑆1𝑖,
𝜆2,if 𝑦=(𝑖,𝑗+1); 𝑖<𝑠1;𝑗<𝑆1𝑖,
𝑖𝜇1,if 𝑦=(𝑖1,𝑗); 1𝑖𝑠1;𝑗𝑆𝑖,
𝑗𝜇2,if 𝑦=(𝑖,𝑗1); 𝑖𝑠1;1𝑗𝑆𝑖,
0,if otherwise.
(1)
These rates can be re-arranged in matrix form as elements of
the infinitesimal generator matrix 𝑸×Ωsuch that:
𝑸Ω∣×∣Ω=𝑞𝑥,𝑦,if 𝑥=𝑦
𝑦Ω𝑞𝑥,𝑦,if 𝑥=𝑦. (2)
Defining Π=[𝜋𝑖,𝑗 ]1×∣Ωas 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 Π𝑸=0and Π𝒆=1 yields the
following closed-form solution:
𝜋𝑖,𝑗 =1
𝑖!𝑗!𝜆1
𝜇1𝑖𝜆2
𝜇2𝑗
𝜋0,0,(3)
where 0𝑖𝑠1,0𝑗𝑆𝑖, and 𝜋0,0is the normalized
equation given by:
𝜋0,0=
𝑠1
𝑖=0
𝑆𝑖
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 𝑆𝑠1chargers 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=𝑆𝑠1𝑆
𝑗𝜆1
𝜇1𝑆𝑗𝜆2
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 ()is defined using three main cost components;
the mean profit (𝑓𝒫), the mean blocking penalty (𝑓), and
the mean maintenance cost (𝑓). To facilitate the definitions
that follow, we divide the state space into two disjoint sub-
spaces, such that for any given sub-space Ω𝑚,1𝑚𝑀,
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<𝑖𝑚𝑠𝑚+
𝑚1
𝑟=1
Φ𝑟,
0𝑖𝑚+1 𝑠𝑚+1,...,0𝑖𝑀𝑆
𝑀1
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 𝑥Ω,
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
𝑥Ω𝑟
𝑛𝑟
𝑥𝜋𝑥.(9)
B. Derivation of 𝑓
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:
𝑓=
𝑀
𝑟=1 𝑐𝑟
𝑥Ω𝑏
𝑛𝑟
𝑥𝜋𝑥.(10)
C. Derivation of 𝑓
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:
𝑓=
𝑀
𝑟=1 ¯𝑐0
𝑥Ω
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 𝜆𝑚𝜆
𝑚
¯𝑝𝑚𝑝𝑚log 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},
𝑝𝑖𝑝𝑗𝑝𝑘if 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:
=𝑓𝒫𝑓𝑓.(13)
Fig. 3 plots as 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, Λ𝑖𝑁
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)𝑛𝛽Λ
𝑁
𝑛=1
𝜆(𝑛)(15)
subject to 𝑃(𝑛)
0≤𝒩
max
𝑏𝑛∈{1,2,...,𝑁},(16)
𝑁
𝑛=1
𝜆(𝑛)Λ.(17)
The first term in (15) reflects the aggregated network revenue,
where 𝜆(𝑛)𝑛is 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
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௡ୀଵ
Load
balancing
entity
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...
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ሺଵሻ
ߣ
ሺேሻ
ܵ

...
ܵ
...

...
ܵ
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 𝑟/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 (Λ𝑟𝑁
𝑛=1 𝜆(𝑛)
𝑟)then
5: 𝑛𝑒𝑡 Calculate revenue as in Section III
6: if 𝑛𝑒𝑡 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: 𝐵𝐵+𝛽𝑟Λ𝑟𝑁
𝑛=1 𝜆𝑜𝑝𝑡
𝑟,𝑛
13: end for
14: return 𝜆𝑜𝑝𝑡
𝑟,𝑛 ,(𝑛𝑒𝑡 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)
𝑟𝑛
𝑀
𝑟=1
𝛽𝑟Λ𝑟
𝑁
𝑛=1
𝜆(𝑛)
𝑟(18)
subject to
𝑀
𝑟=1
𝑃(𝑛)
𝑟≤𝒩
max
𝑏𝑛∈{1,2,...,𝑁},(19)
𝑁
𝑛=1
𝜆(𝑛)
𝑟Λ𝑟𝑟∈{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 𝜆2500 (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 1,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.
1,Λ2)Type CS1CS2CS33
𝑛=1 𝑛
(𝜆(1)
1,𝜆
(1)
2)1(𝑃1,𝑃
2) (𝜆(2)
1,𝜆
(2)
2)2(𝑃1,𝑃
2) (𝜆(3)
1,𝜆
(3)
2)3(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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Future “smart electric vehicles,” expected to evolve from emerging electric and plug-in hybrid electric vehicles (EV & PHEV) are becoming increasingly attractive. However, the current electric grid is not considered capable of handling the power demand increase required by a large number of charging stations, especially during peak loads. Furthermore, the envisioned critical infrastructure for such vehicles must include the capability for information exchange involving energy availability, distances, congestion levels and possibly, spot prices or priority incentives. In this chapter, we discuss current trends and challenges in this fascinating and rapidly developing area of research. Our emphasis is on topics related to control, demand-response, infrastructure provisioning, and the communications framework necessary to accomplish all of these “smart” features. As a particular application and a form of case study, we zero in on the design and development of charging stations. We describe a candidate PHEV charging station architecture, and a quantitative stochastic model that allows the analysis of its performance, using queuing theory and economics. The architecture we envision has the capability to store excess power obtained from the grid. Our goal is to promote a general architecture able to sustain grid stability, while providing a required level of quality of service; and to further the development of a general methodology to analyze the performance of such stations with respect to traffic characteristics, energy storage size, pricing and cost parameters.
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For dc fast charging stations (FCS), short charging time duration is its main advantage over other ac charging stations. However, due to the characteristics of internal resistance of electric vehicle (EV) batteries, if charging request of an EV is near the full state of charge (SoC), its charging time increases significantly. In this paper, considering the impact of requested SoC on charging time, we propose an operation analysis of FCS, where operators of FCS can set a limit on the EVs’ requested SoC to obtain maximized revenue. We model the FCS operation as an queue and incorporate the dc fast charging model into the queuing analysis, as well as the revenue model of FCS. Simulation results indicate that limiting the requested SoC in an overloaded condition at FCS can increase total charged energy and revenue of FCS, as well as decrease the probability of blocking arriving EVs.