Available via license: CC BY 4.0
Content may be subject to copyright.
sustainability
Article
Charging Station Allocation for Electric Vehicle Network Using
Stochastic Modeling and Grey Wolf Optimization
Rawan Shabbar 1, Anemone Kasasbeh 1and Mohamed M. Ahmed 2, *
Citation: Shabbar, R.; Kasasbeh, A.;
Ahmed, M.M. Charging Station
Allocation for Electric Vehicle
Network Using Stochastic Modeling
and Grey Wolf Optimization.
Sustainability 2021,13, 3314. https://
doi.org/10.3390/su13063314
Academic Editors:
Karim El-Basyouny and Tae J. Kwon
Received: 8 February 2021
Accepted: 12 March 2021
Published: 17 March 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Department of System Science and Industrial Engineering, State University of New York at Binghamton,
Binghamton, NY 13902, USA; rshabba1@binghamton.edu (R.S.); akasasb1@binghamton.edu (A.K.)
2Department of Civil and Architectural Engineering, University of Wyoming, Laramie, WY 82071, USA
*Correspondence: mahmed@uwyo.edu; Tel.: +1-(307)-766-5550; Fax: +1-(307)-766-2221
Abstract:
Optimal placement of Charging stations (CSs) and infrastructure planning are one of
the most critical challenges that face the Electric Vehicles (EV) industry nowadays. A variety of
approaches have been proposed to address the problem of demand uncertainty versus the optimal
number of CSs required to build the EV infrastructure. In this paper, a Markov-chain network model
is designed to study the estimated demand on a CS by using the birth and death process model.
An investigation on the desired number of electric sockets in each CS and the average number of
electric vehicles in both queue and waiting times is presented. Furthermore, a CS allocation algorithm
based on the Markov-chain model is proposed. Grey Wolf Optimization (GWO) algorithm is used to
select the best CS locations with the objective of maximizing the net profit under both budget and
routing constraints. Additionally, the model was applied to Washington D.C. transportation network.
Experimental results have shown that to achieve the highest net profit, Level 2 chargers need to be
installed in low demand areas of infrastructure implementation. On the other hand, Level 3 chargers
attain higher net profit when the number of EVs increases in the transportation network or/and in
locations with high charging demands.
Keywords:
electric vehicles; charging stations; metaheuristic optimization; GWO algorithm; allocation
1. Introduction
Electric Vehicles (EVs) are a promising solution to resolve the greenhouse emission
issue as well as the fossil fuels scarcity problem in the future. Even though the demand
for EVs is expected to increase in the next few years, it is still constrained by many factors
including gasoline prices, battery costs, and the availability of Charging Station (CS)
infrastructure. On the other hand, investors are willing to invest in CS infrastructure if and
only if enough number of consumers are available in the network.
To address the problem above, also called the “chicken-egg problem”, many design
parameters related to the available CS in the EV transportation network were studied in
the literature [
1
–
3
]. Such parameters need to be involved to determine the best Electric
Vehicle Charging Station (EVCS) infrastructure. These parameters include: location, level,
and size and capacity of the available CSs as discussed below:
•Location:
Several candidate locations for installing CS were suggested in previous
studies which can be categorized into:
I.
The frequent stopping locations, such as refueling stations and parking lots [
4
–
6
].
II.
The EV possible stopping locations based on route-distance tracking and/or
grid network availability [7,8].
•Level:
Three charging levels were investigated and considered in the infrastructure
planning designs and algorithms in a few studies [8]:
I. Level 1 (standard): operating at 120 V, 15 or 20 A, and 1.44 kW.
II. Level 2 (commercial): operating at 240 V, single-phase, 40 A and 6 kW.
Sustainability 2021,13, 3314. https://doi.org/10.3390/su13063314 https://www.mdpi.com/journal/sustainability
Sustainability 2021,13, 3314 2 of 20
III.
Level 3 (Fast Charger): operating at high-voltage, high-current (D.C.) and 90 kW.
•Size and Capacity
: While many papers refer to the size of CS as CS capacity, the
two terms are defined differently in this paper. CS capacity is defined as the number
of EVs the CS can handle at a time including both EVs that are being served at the
moment in addition to the EVs waiting in the queue. Whereas, CS size is determined
by the number of available charging sockets at the station [
9
]. The CS Level affects the
service time, while the capacity and the size affects the queuing time at the CS [
10
].
Many constraints need to be taken into consideration when developing the optimiza-
tion algorithm of allocating EVCSs. Based on related work in the literature, these
constraints can be classified as: (I) Budget-related constraints including demand and
cost constraints, and (II) Route-related constraints including available routes between
candidate locations, the distance EV can go before next charge, traffic, weather, etc.
In this paper, both budget and route-related constraints are considered. A Markov-
chain based model is proposed to study the relation among the demand on CS, the size
and the capacity of the CS, and the quality of service represented by the mean length of the
queue in the CS. In addition, a CS infrastructure design is presented. The infrastructure is
based on the net profit estimated by the proposed model and under both route and budget
constraints using a metaheuristic Grey Wolf Optimization (GWO) algorithm.
The rest of this paper is organized as follows: First, a literature review on EVCSs’
allocation algorithms and infrastructure planning is introduced. Second, the proposed
methodology is presented including the stochastic modeling approach, a brief overview
of the GWO algorithm, and the implementation of EVCSs’ allocation as an optimization
problem using GWO. Third, a simulation on the proposed model is demonstrated and
experimental results are examined. Then, work significance and limitations are discussed.
Finally, conclusions, future work, and recommendations are drawn.
2. Literature Review
Many approaches have been proposed to address the optimization problem of allo-
cating EVCSs. Proposed models in the literature can be categorized into flow-capturing
models and node-serving models.
In flow-capturing models, the demand is represented by vehicle flow in any route.
CSs need to be allocated in the route between the origin and the destination. Mainly, the
optimization problem in flow-capturing is formulated as a Mixed Integer Programming
Problem (MIP), where the objective is to maximize the EV flow in all available routes
within the EV transport network, and the constraints are either route- or demand-related
constraints or both. The model was first proposed by Kuby and Lim [
11
] and named flow-
refueling location model (FRLM) [
12
]. FLRM extends flow-intercepting model [
13
,
14
] which
aims at maximizing the traffic flow given the number of facilities. However, instead of
allocating a single refueling station, FLRM can allocate multiple stations based on demand
and maximum allowable travel distance [
12
,
15
]. Since then, several mathematical models
for FLRM were proposed [
1
,
12
,
15
,
16
]. Nevertheless, flow-capturing models still suffer
from several limitations such as: (I) inflexibility: it is very hard to capture all information
needed to simulate a realistic EV network, (II) high-complexity: the complexity of the
model increases when the size of the EV network, the available tours, and/or the number
of available CSs increase, and (III) centrality: the model does not address the problem of
decentralized networks.
On the other hand, node-serving models aim at maximizing EVCS coverage by either
maximizing the demand or minimizing the travel distance for specific refueling locations
(nodes). Liu [
17
] proposed an algorithm to deploy the three levels of CSs in an EV network.
The algorithm was developed based on demand in each area and aimed at minimizing the
separation distance between CS nodes [
6
]. Shi and Zheng [
18
] proposed another approach
where fuzzy c-mean clustering method was used to cover EV charging demands in each
cluster [
19
]. Node-serving models showed great performance in solving EVCS allocation
in urban areas, where many constraints are needed to be defined and uncertain/different
Sustainability 2021,13, 3314 3 of 20
demand levels exist for each node in the EV network. However, EV network should be
modeled carefully so that all constraints are stated, and a feasible yet realistic solution
is applied.
To solve EVCS allocation, mathematical programming models were primarily used in
the literature. However, for such models, when the number of constraints and/or decision
variables increases, the computational complexity increases. Hence, finding an optimal
solution becomes challenging. On the other hand, metaheuristic algorithms are used to find
global (near-optimal) solution with less computational effort. For this reason, metaheuristic
algorithms show higher flexibility when it comes to simulating realistic scenarios in EV
network. For instance, Vazifeh et al. [
20
] defined the optimization problem of EVCS
allocation as a non-deterministic polynomial-time hard (NP-Hard) problem [
21
]. To solve a
NP-Hard problem, the total area was partitioned into small square cells. The optimization
problem was solved using Genetic Algorithm (GA). In addition to GA, swarm-intelligence
metaheuristic algorithms were also used in a number of studies in the literature such
as the work by Rahman et al. [
22
], and Vasant et al. [
23
] who used Gravitational Search
Algorithm (GSA) and Particle swarm Optimization (PSO) to implement a CS infrastructure
considering energy prices and battery capacity constraints [24,25].
To estimate the demand for EVCS, big data and surveys were mainly used [
4
,
6
,
8
,
21
].
The problem with the existing methods is that demand uncertainty is not captured. For this
reason, Markov-chain theory was applied in a few studies to stochastically model the arrival
rate of EVs at a CS [
9
,
11
,
26
,
27
]. For instance, a death-and-birth process model was proposed
by Farkas and Prikler [
26
]. A mathematical model for the average waiting rate and the
mean queue length was calculated based on the steady-state probabilities. Birth-and-death
rates are represented by the arrival rate of EVs and the charging rate, respectively.
To summarize the above discussion, EVCS allocation is a challenging problem and
is extremely crucial to the EV industry. Although flow-capturing models are the most
popular solutions proposed in literature. These models suffer from inflexibility, high-
complexity, and centrality. On the other hand, node-serving models such as the models
proposed by Vazifeh et al. [
20
], Rahman et al. [
22
], and Vasant et al. [
23
] showed better
flexibility and were able to simulate realistic scenarios for an EV network. However,
none of these models captured the uncertainty of EV demand. Estimating EV demand
has been studied using different methods in the literature. The demand uncertainty is
captured using Markov-chain model. However, none of the proposed models to estimate
demand uncertainty using Markov-chain were utilized in allocating EVCS. In this paper,
we combine the Markov-chain theory with node-serving approach to allocate EVCS.
First, Markov-chain model is proposed to provide a relation between the demand and
the desired number of sockets per CS, as well as the desired number of CSs in an urban
area to maintain the best quality of service. Then, the work of Farkas and Prikler [
26
] was
extended by considering the balking parameters; EVs will enter with a probability
α
in the
case when the served EVs in the CSs equals the total number of charging slots.
Secondly, a node-serving model to allocate the best CS locations was developed. A
Grey Wolf Optimizer (GWO) was utilized to find the best set of CSs from all available
locations. The set of CSs were selected under routing constraints. The Markov-chain model
was used to estimate the net profit of the selected set of CSs at each time iteration and is
applied to the GWO as an objective function. The network was optimized by selecting the
best CS locations for the EVCS infrastructure that provide the maximum estimated profit
under predefined routing constraints.
The proposed algorithm is different from related work in that a Markov-chain model
is combined with a metaheuristic algorithm to capture realistic scenarios of CS demands,
costs, quality of service, and the corresponding profit. GWO is used to solve the EVCS
allocation problem. Compared to other metaheuristic algorithms such as GA and PSO,
GWO is considered as one of the fastest and the most efficient metaheuristic algorithms for
the reason of using three solutions at the convergence to achieve better global solution [
28
].
Finally, both budget and routing constraints are also considered including the cost of
Sustainability 2021,13, 3314 4 of 20
building a CS, waiting time at the CS, available routes between CSs, and the maximum
distance an EV can go after a full charge. The worst-case scenario was considered for EV
distance range with battery capacity set to 20 kWh and adding the consumption caused by
traffic, weather, and using radio.
Contributions are summarized below:
(I)
The proposed algorithm simulates the stochastic behavior of EVCS infrastructure
including demand and cost uncertainty using Markov-chain processes.
(II)
The parameters affecting the CS quality of service and the achieved profit of EV
network were investigated using birth-and-death model and sensitivity analysis.
(III)
GWO is proposed to optimize the NP-hard allocation problem. GWO is known for
lower complexity and higher flexibility compared to other solutions proposed in the
literature to address the EVCS infrastructure problem.
(IV)
Both budget and routing constraints are considered in the optimization model.
(V)
Unlike FLRM, state of charge (SOC) uncertainty is included in the proposed model.
Sustainability can be achieved using the proposed model by gradually implement-
ing the network in multiple phases according to current demand. Therefore, both EV
consumers and CS investors are satisfied.
3. System Model
This section provides a detailed description of the proposed Markov-chain-based
model of EVCS, an overview of GWO algorithm, and EVCS allocation using GWO.
3.1. Markov-Chain of EV Charging Station
Queuing in CS can be modeled as a birth-and-death process, where the birth rate is
the arrival rate of EVs to the CS, the death rate is the charging rate, and the state represents
the number of EVs at the CS at time t.
The arrival time of EV to the CS is assumed to follow a Poisson process with
λ
rate,
and the charging rate is following exponential process with
µ
rate [
17
,
26
]. CS has c sockets
available for EVs, and a maximum capacity of N vehicles. The model is also modified to be
a reneging model in which the EV driver will enter the CS with probability
α
if (s)he found
the CS full at arrival time. Hence, the Markov-chain is modeled as in Figure 1.
Sustainability 2021, 13, x FOR PEER REVIEW 4 of 20
The proposed algorithm is different from related work in that a Markov-chain model
is combined with a metaheuristic algorithm to capture realistic scenarios of CS demands,
costs, quality of service, and the corresponding profit. GWO is used to solve the EVCS
allocation problem. Compared to other metaheuristic algorithms such as GA and PSO,
GWO is considered as one of the fastest and the most efficient metaheuristic algorithms
for the reason of using three solutions at the convergence to achieve better global solution
[28]. Finally, both budget and routing constraints are also considered including the cost
of building a CS, waiting time at the CS, available routes between CSs, and the maximum
distance an EV can go after a full charge. The worst-case scenario was considered for EV
distance range with battery capacity set to 20 kWh and adding the consumption caused
by traffic, weather, and using radio.
Contributions are summarized below:
(I) The proposed algorithm simulates the stochastic behavior of EVCS infrastructure in-
cluding demand and cost uncertainty using Markov-chain processes.
(II) The parameters affecting the CS quality of service and the achieved profit of EV net-
work were investigated using birth-and-death model and sensitivity analysis.
(III) GWO is proposed to optimize the NP-hard allocation problem. GWO is known for
lower complexity and higher flexibility compared to other solutions proposed in the
literature to address the EVCS infrastructure problem.
(IV) Both budget and routing constraints are considered in the optimization model.
(V) Unlike FLRM, state of charge (SOC) uncertainty is included in the proposed model.
Sustainability can be achieved using the proposed model by gradually implementing
the network in multiple phases according to current demand. Therefore, both EV consum-
ers and CS investors are satisfied.
3. System Model
This section provides a detailed description of the proposed Markov-chain-based
model of EVCS, an overview of GWO algorithm, and EVCS allocation using GWO.
3.1. Markov-Chain of EV Charging Station
Queuing in CS can be modeled as a birth-and-death process, where the birth rate is
the arrival rate of EVs to the CS, the death rate is the charging rate, and the state represents
the number of EVs at the CS at time 𝑡.
The arrival time of EV to the CS is assumed to follow a Poisson process with λ rate,
and the charging rate is following exponential process with µ rate [17,26]. CS has c sockets
available for EVs, and a maximum capacity of N vehicles. The model is also modified to
be a reneging model in which the EV driver will enter the CS with probability α if (s)he
found the CS full at arrival time. Hence, the Markov-chain is modeled as in Figure 1.
Figure 1. Markov-chain of EV (Electric Vehicle) Charging Station.
The steady-state probabilities are evaluated as below, where 𝑃,𝑘 ∈0,1,…,𝑁 is
the probability that there will be exactly k EVs at the CS at a time [29]:
Figure 1. Markov-chain of EV (Electric Vehicle) Charging Station.
The steady-state probabilities are evaluated as below, where
Pk
,
k∈{0, 1, . . . , N}
is
the probability that there will be exactly k EVs at the CS at a time [29]:
λP0=µP1(1)
P1=λ
µP0(2)
P2=1
2λ
µ2
P0(3)
P3=1
3! λ
µ3
P0(4)
Sustainability 2021,13, 3314 5 of 20
. . .
Pc=1
c!λ
µc
P0(5)
Pc+1=α
c
1
c!λ
µc+1
P0(6)
Pc+2=α
c21
c!λ
µc+2
P0(7)
It can be concluded that the steady-state probability
Pn
can be obtained from the
following formula:
Pn=
1
n!λ
µnP0f or n ≤c
1
c!λ
µnα
cn−cP0f or n >c
(8)
It is known that: N
∑
k=0
Pk=1 (9)
Hence, the steady-state probability of state 0 can be calculated as follows:
P0=1
1+∑c
k=11
k!λ
µk+∑N
k=c+1α
ck−c1
c!λ
µk(10)
The average number of EV drivers in the CS (L):
L=
N
∑
k=0
k Pk(11)
Average amount of time the EV driver spends in the CS is (
W
) which follows Lit-
tle’s low:
W=L/λa(12)
where
λa
is the average arrival rate of entering EV drivers considering only actual arrivals:
λa=λ(1−PN)(13)
The average number of customers waiting in queue is (Lq):
Lq=
N
∑
k=c
(k−c)Pk(14)
Average amount of time the EV driver spends in the queue is (
Wq
) which follows
Little’s low:
Wq=Lq/λa(15)
Expected net profit (Pro f i t):
Pro f it =λ(1−PN)×€−cu (16)
where cu is the operating cost of the CS per minute and €is the profit per a single EV.
3.2. Grey Wolf Optimization (GWO)
GWO is one of the swarm intelligence optimization algorithms. The basic concept of
metaheuristic algorithms is to generate high-quality solutions based on an initial random
Sustainability 2021,13, 3314 6 of 20
population of possible solutions and a fitness function (or objective function). The genera-
tion is inspired by natural systems behaviors and performed until a termination condition
has been reached.
The basic idea of the metaheuristic swarm intelligence GWO algorithm is to simulate
the hunting procedure done by grey wolves in the wildlife [
30
]. Basically, four types of
wolves are simulated: alpha wolves (
..
α)
, beta wolves (
..
β)
, gamma wolves (
..
δ)
and omega
wolves (
..
ω)
ordered from the strongest to the weakest. To obtain a new solution, the best
three solutions represent the first three types of wolves respectively cooperating in hunting
the prey and achieving a better solution.
The social hierarchy, tracking, encircling, and attacking prey by grey wolf are sim-
ulated and mathematically modeled using GWO. It can be obtained by considering the
best solution as
..
α
, second-best solution as
..
β
, third-best solution as
..
δ
, and the rest of the
solutions as ..
ω.
Encircling prey is modeled as below [30]:
→
D=|C·→
Xp(t)−→
X(t)|(17)
→
X(t+1)=→
Xp(t)−A·→
D(18)
where
t
is the current iteration,
A
and
C
are coefficient vectors,
→
Xp
is the position vector of
the prey, and →
Xis the position vector of a grey wolf.
The coefficient vectors Aand Care obtained from the equations below:
A=2a r1−a(19)
C=2r2(20)
where
r1
and
r2
are random values uniformly distributed between 0 and 1, and
a
is a
linearly decreasing coefficient from 2 to 0.
a=2−t2
Maxiter (21)
For hunting, alpha wolves (
..
α)
are leading the rest of wolves. While beta and sigma
(
..
β
,
..
δ
) wolves are also participating the hunting process. The first three solutions are always
saved and used for forming the new solution.
→
X1=→
X..
α−A1·→
D..
α(22)
→
X2=→
X..
β−A2·→
D..
β(23)
→
X3=→
X..
δ−A3·→
D..
δ(24)
where: →
D..
α=|C1·→
X..
α−→
X|(25)
→
D..
β=|C2·→
X..
β−→
X|(26)
→
D..
δ=|C3·→
X..
δ−→
X|(27)
The new position (solution) is formed by combining the three best positions:
→
X(t+1)=
→
X1+→
X2+→
X3
3(28)
Sustainability 2021,13, 3314 7 of 20
It can be observed that the new solution is positioned in a random place within the
circle of the three top solutions: alpha (
..
α)
, beta (
..
β
), and gamma (
..
δ
). In other words, the
three top solutions determine the new solution position.
3.3. GWO for EV Charging Station Allocation
Assume that the possible CS locations are given with all the needed information about
demand in each area (represented by the arrival rate of EV drivers), the cost for locating a
CS in each location ($ per minute), and a specific available size per each CS. GWO initialize
a population of random solutions. Each solution includes a subset of the candidate CSs
available for the desired infrastructure. First, each solution in the population is evaluated
using a fitness function. The fitness function calculates the achieved profit when using
the selected subset of CSs based on the Markov-chain process with a pre-known arrival
rate and operating cost for each CS. In addition, the total number of selected CSs is added
to the objective function to minimize the number of selected CSs. Then, a new group of
solutions (positions) are generated based on the method explained in the previous section
and then evaluated using the fitness function. The procedure is repeated until meeting
stopping criteria, which, in our case, is the maximum number of iterations. Finally, the best
solution (subset of CSs selected) is studied under routing constraints. In other words, the
EV is guaranteed to accomplish a complete route between selected CSs. This is done by
using different techniques: (I) Calculating the distance between CSs. The distance between
each CS and the closest one should be less than or equal to half the maximum distance an
EV can go with a full charge without recharge. This is a valid assumption in the routing
problem and has been used in FRLM and other routing models to address EV routing
problem in the literature [
12
–
16
,
19
,
22
,
31
]. The assumption is based on allocating CSs in the
middle of the EV origin to distention O-D path. The worst-case scenario was considered
in terms of EV driving distance to address urban area traffic, weather, and using radio;
(II) Capturing the uncertainty of State of Charge (SOC). The uncertainty is added to the
charging rate (µ) when the model is implemented.
The proposed algorithm is shown in Equations (29)–(33); the objective function (fitness)
aims at maximizing the profit while minimizing the total number of selected CSs. Where
Pro f it
is the net profit after implementing the EVCS network,
|I|
is the total number of
selected CSs,
|S|
is the total number of potential CS locations, and
ω1
,
ω2
are weights
(priority ratios). The first constraint evaluates the net profit following Equation (16) where
λi
,
PN,i
, and
cui
are the arrival rate, steady-state probability, and the operating cost
per minute at CS
i
, respectively. Second and third constraints evaluate the steady-state
probability at CS
i
. Where
Ni
and
ci
represent the capacity and size of CS
i
, respectively.
ε
represents the uncertainty of SOC and is a random variable. The fourth constraint ensures
that the driving distance between any two selected CSs in the network (
di,j
) to be less than
half the distance the EV can go (
dEV
). The last two constraints guarantee that the subset
of the selected CSs (
I
) is part of the potential CS location set (
S
), and that both sets are
integers representing the CS id. It is worth mentioning that the set of selected CSs
(I)
at any
time represents the position (solution) formed by GWO
(→
X)
and represents the decision
variable.
min (f=ω1∗1
Pro f i t +ω2×|I|
|S|)(29)
s.t.
Pro f it =
I
∑
i
(λi(1−PN,i)×€−cui)(30)
PN,i=∑N≤ci
n
1
n!λi
µ+εn
P0,i+∑Ni>ci
n
1
ci!λi
µ+εnα
cin−ci
P0,i∀i∈I(31)
Sustainability 2021,13, 3314 8 of 20
P0,i=1
1+∑ci
k=11
k!λi
µ+εk+∑Ni
k=ci+1α
cik−ci1
ci!λi
µ+εk∀i∈I(32)
di,j≤1
2dEV ∀i,j∈I,i6=j(33)
I∈S(34)
I,S∈Z(35)
4. Experimental Results
To simulate the proposed model, python 2.7 [
32
] is used with Bokeh package for
visualization [
18
]. The following parameters are considered in the proposed mathematical
model. These parameters have been used and estimated by Farkas and Prikler [26]:
N
: Number of parking slots (including sockets) = maximum allowable number of EVs
in the CS (Maximum Capacity) = EVs being served + EVs waiting in the queue
c: Number of charging sockets
µ: Charging rate (service rate) (1/min)
λ: EV arrival rate to the CS (1/min)
α: The entering probability when the CS is full (0.3 for all experiments).
It is worth mentioning that the birth-and-death model is following the queuing theory,
the M/M/c (in complete Kendall’s notation) queue [
33
]. This is a queue with Poisson
arrivals, and
c
servers with exponentially distributed service times with
N
places in the
queue. When the customer/EV driver is served by a slot
ci
, the slot
ci
is considered busy,
otherwise, ciis considered idle (waiting for a new customer).
As an illustrative example, the algorithm is performed on Washington D.C. trans-
portation network. The possible locations of CSs were assumed to be the refueling station
locations in the city. The data of refueling station locations were extracted from Socrata [
34
],
while the routing driving distance (in meters) between different locations was evaluated
based on Google API [
35
]. Strictly speaking, the driving distance from station A to station
B is not the same driving distance from station B to A. Finally, costs and rates at each CSs
were estimated based on (District of Columbia Open Data [
9
]; Smith and Castellano [
33
];
Davis et al. [
8
]) as follows. Cost is proportional to Average Standardized Land Price per
Square Foot in D.C. [
23
]. Installation and electricity costs of charging stations were added
to the estimated cost of land based on Smith and Castellano [
33
] who estimated annual
electricity and installation costs to be $686/yr and $1270/yr for Level 2, respectively, and
$1128 and $5100 for Level 3, respectively. Finally, the arrival rate is estimated to be propor-
tional to the Annual Average Daily Traffic (AADT) in D.C. [
36
]. Figure 2illustrates demand
and cost distribution based on the available data. It can be observed that D.C. highway
towards Maryland have the highest traffic compared to other areas. On the other hand,
costs are considered higher around downtown areas.
Two types of experiments are held. First, the algorithm is performed under the
assumption that SOC level is 0.5 (constant). This assumption was considered in most
FRLM-based approaches [
9
,
12
–
16
,
19
,
22
]. Then, the algorithm is performed under SOC
level uncertainty [20].
Parameters used in the experiment are described in Table 1. Table 1a shows GWO
parameters. Solutions of GWO are represented by bit vectors (continuous version). Each
bit in the vector corresponds to a CS in the network. When the bit (rate) is greater than 0.5,
the CS is selected. Otherwise, the CS is neglected.
Sustainability 2021,13, 3314 9 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 10 of 20
Figure 2. Demand and cost by potential CS location.
4.1. Sensitivity Analysis
To study the effect of the assumptions on the network model, the model was tested
while varying the independent variables α, €, c, N, and λ. All statistical analyses have
been performed using Minitab 18 under 95% confidence interval (CI) for difference. In
other words, samples are considered significantly different if the p-value is less than 0.05.
4.1.1. Single Charging Station
Table 1b summarizes the parameter settings for a CS built in a refueling station in an
urban area [9,26,38,39]. The maximum CS capacity of the CS (N) is 10 EVs, the number of
sockets available for charging (c) ranges between 1 and 5, the arrival rate of EVs (λ) equals
one EV every 60 min to one EV every 10 min, and the service rate (µ) is based on charging
levels. For instance, for CS level 1, charging may last for 2 h. Hence, the service rate is
1/120 (1/min). The entering probability (α) is kept to 0.3. This means that the probability
of an EV to enter a CS, if it arrives at the station when it is full (N cars are currently in the
station), is 30%.
Level 1 (service time 120 min), Level 2 (service time 60 min), and Level 3 (service time
30 min) chargers are compared in Figure 3. It can be observed that when the arrival rate
is low, all charging levels will provide similar performance including blocking rate, wait-
ing time, and the number of vehicles in the queue. Whereas, the performance differs when
the arrival rate increase and/or the number of sockets minimized. Additionally, an inves-
tigation on the best pricing is held for two different arrival rate scenarios (Figure 3c,d).
The operating cost per minute is assumed to be 1$/min [39]. On the other hand, to achieve
positive net profit, the gross profit per EV should be greater than $10, and $20 for high
and for low arrival rates, respectively. The effect of number of sockets (c) were studied
(Figure 3e,f). When the number of sockets in the CS increases, the arriving EVs will have
higher possibility to find an available slot to charge. The waiting time decreases. Hence,
the blocking rate decreases. Finally, the entering probability effect was investigated (Fig-
ure 3h,i). The higher the probability the EV enters the CS, the higher the queue length
(Figure 3h). Whereas, since the arrival rate is higher than the charging rate for Levels 1
and 2, and when the entering probability is high, the arriving EVs leave when they find
the CS capacity is full. The possibility of serving new customers reduces. Hence, the net
profit decreases (Figure 3i). It is worth mentioning that, for the sake of comparison be-
tween the three levels, the operating cost was assumed to be equal for all charging levels
Figure 2. Demand and cost by potential CS location.
Table 1.
Experiment parameter settings: (
a
) GWO parameter settings; (
b
) Charging Station parameter settings in an urban
area; (c) Charging Station parameter settings for Washington D.C.
(a) GWO parameter settings
Lower Bound 0
Upper Bound 1
Dimension Number of possible CSs
Population Size 50
Number of Generations 20
(b) Charging Station parameter settings in an urban area
N 10
c 5
µ[1/30, 1/60, 1/120]
λ[1/60, 1/10]
cu ($) 5 (Level 2), 10 (Level 3)
€($) 50 (Level 2), 70 (Level 3)
α0.3
(c) Charging Station parameter settings for Washington D.C.
N 10
c 5
µ[1/30, 1/60]
λProportional to Annual Average Daily Traffic (AADT) in D.C. [36]
Low: [0.002–0.08], High: [0.01–0.4]
Installation & Electricity Cost ($ per min) 0.0057 (Level 2), 0.018 (Level 3) [37]
Operating Cost ($ per min)
Proportional to Average Standardized Land Price per Square Foot in D.C. [
23
]
[0.15–0.5]
cu ($) Operating Cost + Installation & Electricity Cost
€($) 15 (Level 2), 18 (Level 3)
α0.3
Sustainability 2021,13, 3314 10 of 20
There are limited locations to build CSs. Hence, the capacity of the CS and the number
of available sockets will also be limited. Furthermore, the number of EVs will be growing
exponentially and expected to be very small at the first stage of implementing the EVCS
infrastructure [22]. Hence, the arrival rate in the first stages is expected to be low.
4.1. Sensitivity Analysis
To study the effect of the assumptions on the network model, the model was tested
while varying the independent variables
α
,
€
, c, N, and
λ
. All statistical analyses have been
performed using Minitab 18 under 95% confidence interval (CI) for difference. In other
words, samples are considered significantly different if the p-value is less than 0.05.
4.1.1. Single Charging Station
Table 1b summarizes the parameter settings for a CS built in a refueling station in an
urban area [
9
,
26
,
38
,
39
]. The maximum CS capacity of the CS (N) is 10 EVs, the number of
sockets available for charging (c) ranges between 1 and 5, the arrival rate of EVs (
λ
) equals
one EV every 60 min to one EV every 10 min, and the service rate (
µ
) is based on charging
levels. For instance, for CS level 1, charging may last for 2 h. Hence, the service rate is
1/120 (1/min). The entering probability (
α
) is kept to 0.3. This means that the probability
of an EV to enter a CS, if it arrives at the station when it is full (N cars are currently in the
station), is 30%.
Level 1 (service time 120 min), Level 2 (service time 60 min), and Level 3 (service
time 30 min) chargers are compared in Figure 3. It can be observed that when the arrival
rate is low, all charging levels will provide similar performance including blocking rate,
waiting time, and the number of vehicles in the queue. Whereas, the performance differs
when the arrival rate increase and/or the number of sockets minimized. Additionally, an
investigation on the best pricing is held for two different arrival rate scenarios (
Figure 3c,d
).
The operating cost per minute is assumed to be 1$/min [
39
]. On the other hand, to achieve
positive net profit, the gross profit per EV should be greater than $10, and $20 for high
and for low arrival rates, respectively. The effect of number of sockets (c) were studied
(Figure 3e,f). When the number of sockets in the CS increases, the arriving EVs will
have higher possibility to find an available slot to charge. The waiting time decreases.
Hence, the blocking rate decreases. Finally, the entering probability effect was investigated
(
Figure 3h,i
). The higher the probability the EV enters the CS, the higher the queue length
(Figure 3h). Whereas, since the arrival rate is higher than the charging rate for Levels 1
and 2, and when the entering probability is high, the arriving EVs leave when they find
the CS capacity is full. The possibility of serving new customers reduces. Hence, the net
profit decreases (Figure 3i). It is worth mentioning that, for the sake of comparison between
the three levels, the operating cost was assumed to be equal for all charging levels in this
experiment, while the actual operating cost of Level 3 is supposed to be higher than that
for Level 2. This assumption is relaxed for the rest of the paper.
Sustainability 2021,13, 3314 11 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 11 of 20
in this experiment, while the actual operating cost of Level 3 is supposed to be higher than
that for Level 2. This assumption is relaxed for the rest of the paper.
Figure 3. Birth-and-Death Markov-chain sensitivity analysis: (a) Waiting Time (𝑐=5); (b) Queue Length (𝑐=5); (c) Net Profit (𝑐=
5,𝜆=1/20); (d) Net Profit (𝑐=5,𝜆=1/10); (e) Blocking Probability (𝜆=1/20); (f) Waiting Time (𝜆=1/20); (g) Queue Length (c =
5, 𝜆=1/10); (h) Net Profit (c = 5, 𝜆=1/10).
4.1.2. Multiple Charging Stations
In the previous section, the effect of independent variables on the performance of a
single charging station with birth-and-death Markov model was investigated. Similarly,
sensitivity analyses are applied on the proposed network model applied to Washington
D.C. network in high arrival rate (Table 1c). It is worth mentioning that the GWO model
is a metaheuristic algorithm, which means that the solution is a global solution which
might slightly change every time the algorithm is executed depending on the search pop-
ulation and the starting position. For this reason, the model was performed twenty-five
times and statistical analysis was applied to evaluate the significance based on ANOVA
(summarized in Table 2a). Overall results represent the average output of the twenty-five
model performances.
First, the effect of varying the entering probability is studied, statistically. Figure 4a,b
show that varying α does not show significant effect on the net profit nor number of se-
lected charging stations. However, as discussed in Section 4.1.1., the possibility of having
CSs full is higher, and the possibility of serving new customers reduces. Hence, the net
(a) Waiting Time (𝑐=5) (b) Queue Length (𝑐=5)
(f) Waiting Time (𝜆=1/20)
(c) Net Profit (𝑐=5,𝜆=1/20)
(d) Net Profit (𝑐=5, 𝜆=1/10) (e) Blocking Probability (𝜆=1/20)
(g) Queue Len
g
th (c = 5, 𝜆=1/10) (h) Net Profit (c = 5, 𝜆=1/10)
Figure 3.
Birth-and-Death Markov-chain sensitivity analysis: (
a
) Waiting Time (
c=
5); (
b
) Queue Length (
c=
5); (
c
) Net
Profit (
c=
5,
λ=
1
/
20); (
d
) Net Profit (
c=
5,
λ=
1
/
10); (
e
) Blocking Probability (
λ=
1
/
20); (
f
) Waiting Time (
λ=
1
/
20);
(g) Queue Length (c = 5, λ=1/10); (h) Net Profit (c = 5, λ=1/10).
4.1.2. Multiple Charging Stations
In the previous section, the effect of independent variables on the performance of a
single charging station with birth-and-death Markov model was investigated. Similarly,
sensitivity analyses are applied on the proposed network model applied to Washington
D.C. network in high arrival rate (Table 1c). It is worth mentioning that the GWO model
is a metaheuristic algorithm, which means that the solution is a global solution which
might slightly change every time the algorithm is executed depending on the search
population and the starting position. For this reason, the model was performed twenty-five
times and statistical analysis was applied to evaluate the significance based on ANOVA
(summarized in Table 2a). Overall results represent the average output of the twenty-five
model performances.
Sustainability 2021,13, 3314 12 of 20
Table 2. (a) Sensitivity analysis: ANOVA test. (b) Sensitivity analysis: t-test charging levels.
(a)
Parameter Measure Adj SS Adj MS F-Value p-Value (<0.05)
Entering Probability (α) Net Profit 60.76 15.19 1.37 0.247
Entering Probability (α)Number of CS
Selected 253.6 63.39 2.11 0.084
Gross Profit per EV (€) Net Profit 20,959 6986.31 489.88 0.000
Gross Profit per EV (€)Number of CS
Selected 3685 1228.40 41.10 0.000
CS Capacity (N) Net Profit 17.75 5.916 0.72 0.545
CS Capacity (N) Number of CS
Selected 105.7 35.24 1.13 0.339
(b)
Measure CS Level Mean StDev T-Value p-Value (<0.05)
Net Profit—High Rate Level 2 46.58 2.68 −6.81 0.000
Level 3 53.50 4.32
Net Profit—Low Rate Level 2 2.42 1.10 −3.98 0.000
Level 3 3.73 1.22
Number of Selected CS—High Rate
Level 2 29.60 5.35 2.06 0.045
Level 3 26.60 4.94
Number of Selected CS—Low Rate Level 2 7.60 3.33 −3.72 0.001
Level 3 11.56 4.15
Net Profit—High Rate Level 3 53.5 4.32 −1.71 0.100
Level 3 SOC 61 21.6
Net Profit—Low Rate Level 3 3.73 1.22 −0.19 0.850
Level 3 SOC 3.8 1.25
Number of Selected CS—High Rate
Level 3 26.6 4.94 0.03 0.977
Level 3 SOC 26.56 4.93
Number of Selected CS—Low Rate Level 3 11.56 4.15 0.51 0.610
Level 3 SOC 10.96 4.12
First, the effect of varying the entering probability is studied, statistically.
Figure 4a,b
show that varying
α
does not show significant effect on the net profit nor number of
selected charging stations. However, as discussed in Section 4.1.1., the possibility of having
CSs full is higher, and the possibility of serving new customers reduces. Hence, the net
profit decreases. On the other hand, when the entering probability is high, more EVs are
expected to enter the CS.
Figure 4c,d show how gross profit per EV (
€
) significantly affect both the net profit
and the number of selected CSs. When the gross profit per EV increases, the net profit
increases. As a result, the number of charging stations needed to obtain the desired positive
net profit decreases.
Number of slots (c) and CS capacity (N) are studied in Figure 4e–h. Results show
that the model is flexible to be adjusted to overcome the lower capacity in CSs by slightly
increasing the number of charging stations selected.
Finally,
µ
and
λ
effect on the proposed model are examined in Figures 5and 6and
Table 2b. At the first stages of infrastructure planning, the arrival rate is expected to be as
low as one EV per 8 h. In this experiment, the arrival rate was set to be in the range of one
EV per 12.5 min for first stages (low arrival rate), and in the range of one EV per 2.5 min
(high arrival rate) for later stages when the EV number in the network increases. In both
stages, the arrival rate is set to be proportional to AADT as discussed earlier in this section.
Sustainability 2021,13, 3314 13 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 12 of 20
profit decreases. On the other hand, when the entering probability is high, more EVs are
expected to enter the CS.
Figure 4c,d show how gross profit per EV (€) significantly affect both the net profit
and the number of selected CSs. When the gross profit per EV increases, the net profit
increases. As a result, the number of charging stations needed to obtain the desired posi-
tive net profit decreases.
Number of slots (c) and CS capacity (N) are studied in Figure 4e–h. Results show that
the model is flexible to be adjusted to overcome the lower capacity in CSs by slightly in-
creasing the number of charging stations selected.
Finally, µ and λ effect on the proposed model are examined in Figures 5 and 6 and
Table 2b. At the first stages of infrastructure planning, the arrival rate is expected to be as
low as one EV per 8 h. In this experiment, the arrival rate was set to be in the range of one
EV per 12.5 min for first stages (low arrival rate), and in the range of one EV per 2.5 min
(high arrival rate) for later stages when the EV number in the network increases. In both
stages, the arrival rate is set to be proportional to AADT as discussed earlier in this section.
(b) Effect of α on Number of Selected CSs
(a) Effect of α on Net Profit
(d) Effect of € on Number of Selected CSs
(c) Effect of € on Net Profit
(e) Effect of c on Net Profit (f) Effect of 𝑐 on Number of Selected CSs
(g) Effect of 𝑁 on Net Profit (h) Effect of 𝑁 on Number of Selected CSs
Figure 4.
GWO (Grey Wolf Optimization) sensitivity analysis: (
a
) Effect of
α
on Net Profit; (
b
) Effect of
α
on Number of
Selected CSs; (
c
) Effect of
€
on Net Profit; (
d
) Effect of
€
on Number of Selected CSs; (
e
) Effect of
c
on Net Profit; (
f
) Effect of
con Number of Selected CSs; (g) Effect of Non Net Profit; (h) Effect of Non Number of Selected CSs.
At first stages, and assuming SOC level to be half-full, and comparing the implementa-
tion of a network of only Level 2 chargers (
µ
= 1/60), and a network of only Level 3 chargers
(
µ
= 1/30), the average number of selected CSs is 7.6, and 11.56 out of 151 total available
locations at Washington D.C., respectively. The aggregate net profit is $2.43 and $3.73 per
minute for Level 2 and 3, respectively. According to the sensitivity analysis presented in
Table 2b, the effect of charger level is significant (p-value < 0.05). It can be concluded that
slow chargers can achieve better performance with respect to the number of CSs selected
and comparable net profit when installed in the first infrastructure implementation stages
(Figure 5and Table 2b).
Sustainability 2021,13, 3314 14 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 13 of 20
Figure 4. GWO (Grey Wolf Optimization) sensitivity analysis: (a) Effect of α on Net Profit; (b) Effect of α on Number of
Selected CSs; (c) Effect of € on Net Profit; (d) Effect of € on Number of Selected CSs; (e) Effect of c on Net Profit; (f)
Effect of 𝑐 on Number of Selected CSs; (g) Effect of 𝑁 on Net Profit; (h) Effect of 𝑁 on Number of Selected CSs.
At first stages, and assuming SOC level to be half-full, and comparing the implemen-
tation of a network of only Level 2 chargers (µ = 1/60), and a network of only Level 3
chargers (µ = 1/30), the average number of selected CSs is 7.6, and 11.56 out of 151 total
available locations at Washington D.C., respectively. The aggregate net profit is $2.43 and
$3.73 per minute for Level 2 and 3, respectively. According to the sensitivity analysis pre-
sented in Table 2b, the effect of charger level is significant (p-value < 0.05). It can be con-
cluded that slow chargers can achieve better performance with respect to the number of
CSs selected and comparable net profit when installed in the first infrastructure imple-
mentation stages (Figure 5 and Table 2b).
On the other hand, when the network grows and the number of EVs increases in the
network, the fast (Level 3) chargers achieve higher net profit than the slow (Level 2)
chargers with $53.50 compared to $46.58. The average selected number of CSs is 26.6, and
29.6 for Level 3, and Level 2, respectively.
Uncertain SOC level is investigated in Figure 6. For Level 3 chargers and high arrival
rate, the average number of CSs selected is around 26.6 for both models (assuming fixed
SOC and uncertain SOC). However, the net profit achieved by uncertain SOC model is
$61.03 compared to $53.50 achieved by assuming a half-full SOC model. Which, in turn,
shows that uncertain SOC level might affect the selection of the best CS location and the
estimated net profit, while the model and CS level performances remain the same at the
same arrival rate according to t-test shown in Table 2b (p-value > 0.05).
Figure 5. Comparison between Level 2 and Level 3 under low and high arrival rates.
2.42
46.58
3.73
53.50
0
10
20
30
40
50
60
Low Rate High Rate
Aggregate Profit ($ per min)
Level 2 Level 3
7.6
29.6
11.56
26.6
0
5
10
15
20
25
30
35
Low Rate High Rate
Number of CS Selected
Level 2 Level 3
Figure 5. Comparison between Level 2 and Level 3 under low and high arrival rates.
Sustainability 2021, 13, x FOR PEER REVIEW 14 of 20
Figure 6. Comparison between fixed and uncertain SOC Level 3 under low and high arrival rates.
Table 2. (a) Sensitivity analysis: ANOVA test. (b) Sensitivity analysis: t-test charging levels.
(a)
Parameter Measure Adj SS Adj MSF-Value p-Value (<0.05)
Entering Probability (α) Net Profit 60.76 15.19 1.37 0.247
Entering Probability (α) Number of CS Selected 253.6 63.39 2.11 0.084
Gross Profit per EV (€) Net Profit 20,959 6986.31 489.88 0.000
Gross Profit per EV (€) Number of CS Selected 3685 1228.40 41.10 0.000
CS Capacity (N) Net Profit 17.75 5.916 0.72 0.545
CS Capacity (N) Number of CS Selected 105.7 35.24 1.13 0.339
(b)
Measure CS Level Mean StDev T-Value p-Value (<0.05)
Net Profit—High Rate Level 2 46.58 2.68 −6.81 0.000
Level 3 53.50 4.32
Net Profit—Low Rate Level 2 2.42 1.10 −3.98 0.000
Level 3 3.73 1.22
Number of Selected CS—High Rate Level 2 29.60 5.35 2.06 0.045
Level 3 26.60 4.94
Number of Selected CS—Low Rate Level 2 7.60 3.33 −3.72 0.001
Level 3 11.56 4.15
Net Profit—High Rate Level 3 53.5 4.32 −1.71 0.100
Level 3 SOC 61 21.6
Net Profit—Low Rate Level 3 3.73 1.22 −0.19 0.850
Level 3 SOC 3.8 1.25
Number of Selected CS—High Rate Level 3 26.6 4.94 0.03 0.977
Level 3 SOC 26.56 4.93
Number of Selected CS—Low Rate Level 3 11.56 4.15 0.51 0.610
Level 3 SOC 10.96 4.12
4.2. CS Allocation Using GWO
Finally, to validate GWO performance, both selected CS locations and the conver-
gence are illustrated in Figure 7. The convergence of GWO is plotted for both profit and
number of CS selected for all previous scenarios. The proposed GWO is a multi-objective
algorithm that counts for both the number of selected CSs, and the net profit. The fitness
function (Equation (29)) value is a standardized ratio that combines a weighted value of
2.07
53.50
3.73
61.03
0
10
20
30
40
50
60
70
Low Rate High Rate
Aggregate Profit ($ per min)
Level 3 Level 3 - SOC
11.56
26.6
10.96
26.56
0
5
10
15
20
25
30
Low Rate High Rate
Number of Selected CS
Level 3 Level 3 - SOC
Figure 6. Comparison between fixed and uncertain SOC Level 3 under low and high arrival rates.
On the other hand, when the network grows and the number of EVs increases in
the network, the fast (Level 3) chargers achieve higher net profit than the slow (Level 2)
chargers with $53.50 compared to $46.58. The average selected number of CSs is 26.6, and
29.6 for Level 3, and Level 2, respectively.
Uncertain SOC level is investigated in Figure 6. For Level 3 chargers and high arrival
rate, the average number of CSs selected is around 26.6 for both models (assuming fixed
SOC and uncertain SOC). However, the net profit achieved by uncertain SOC model is
$61.03 compared to $53.50 achieved by assuming a half-full SOC model. Which, in turn,
shows that uncertain SOC level might affect the selection of the best CS location and the
estimated net profit, while the model and CS level performances remain the same at the
same arrival rate according to t-test shown in Table 2b (p-value > 0.05).
4.2. CS Allocation Using GWO
Finally, to validate GWO performance, both selected CS locations and the convergence
are illustrated in Figure 7. The convergence of GWO is plotted for both profit and number
of CS selected for all previous scenarios. The proposed GWO is a multi-objective algorithm
that counts for both the number of selected CSs, and the net profit. The fitness function
(Equation (29)) value is a standardized ratio that combines a weighted value of both
objective metrics, where the weights for the number of CS, and profit were 0.1, and
0.9, respectively.
Sustainability 2021,13, 3314 15 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 16 of 20
Level 3—Low
Level 2—Low
Potential Location Selected Location by Level 3
(a) GWO selection of Level 3 CSs in low arrival rate network.
(b) GWO selection of Level 2 CSs in low arrival rate network.
Selected Location by Level 2
Figure 7. Cont.
Sustainability 2021,13, 3314 16 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 17 of 20
Level 3—Hi
g
h
Level 2—Hi
g
h
(d) GWO selection of Level 2 CSs in high arrival rate network.
(c) GWO selection of Level 3 CSs in high arrival rate network.
Potential Location Selected Location by Level 3 Selected Location by Level 2
Figure 7. Cont.
Sustainability 2021,13, 3314 17 of 20
Sustainability 2021, 13, x FOR PEER REVIEW 18 of 20
Figure 7. GWO Performance. EV charging station allocation (left); GWO Convergence (right).
5. Research Significance and Limitations
The proposed algorithm provides a framework to address the EVCS allocation prob-
lem. The model is flexible and shows high convergence. Unlike mathematical program-
ming-based approaches, the proposed algorithm can optimize an EV network with large
number of potential CS locations. Demand uncertainty and routing constraints are also
captured in the proposed model. On the other hand, the model can only be utilized to
allocate a single CS level at a time. The model needs to be modified to be able to select
which level is suitable to which location. Additionally, while it is impossible for mathe-
matical programming models to be utilized to allocate CSs in a large network due to the
extremely high number of combinations, the model needs to be compared with similar
node-serving approaches which utilized metaheuristic algorithms such as GSA and PSO.
Finally, prioritizing between the number of CSs to be located versus the total cost of im-
plementation needs to be investigated.
6. Conclusions
A stochastic model for charging stations of electric vehicles is proposed. The model
is based on birth-and-death Markov-chain process where the arrival rate of EVs is the
birth rate, and the charging rate is the death rate. When the number of EVs in the CS is
greater than the number of sockets available, then the EV drivers enter the CS with prob-
ability α. The Markov-chain model is used to estimate the expected net profit and is ap-
plied as an objective function to the GWO to select the best CSlocations out of all possible
Level 3 – Hi
g
h Rate -
Selected Location by Level 3 Selected Location by Level 2
Potential Location
(e) GWO selection of Level 3 CSs - ROC in high arrival rate network.
Figure 7. GWO Performance. EV charging station allocation (left); GWO Convergence (right).
In the equation above, higher weight is given to net profit because the installation costs
of CS are included. This means that the more the CSs selected, the higher the installation
costs, and the lower the achieved net profit. Hence, net profit also depends on the number
of selected CSs. Nevertheless, these weights can be tuned based on the infrastructure
design priorities. It can be observed from the figure that the number of selected CSs will
be reduced until the achieved net profit is maximized. The best solution is the one that
balances both the number of CSs and the net profit as shown in the figure. Additionally,
it can be shown from Figure 7that mostly CSs at higher demand rates and lower costs
are the ones that will mostly be selected. From the figure, selected CS locations do not
reflect demand. The locations are selected based on the combination of demand, cost, and
travel distance. According to the model, the travel distance constraint is still achieved if
the demand is low and can be covered by a CS that is built away from the downtown area.
The model avoids selecting a location within that area due to high installation costs.
5. Research Significance and Limitations
The proposed algorithm provides a framework to address the EVCS allocation prob-
lem. The model is flexible and shows high convergence. Unlike mathematical programming-
based approaches, the proposed algorithm can optimize an EV network with large number
of potential CS locations. Demand uncertainty and routing constraints are also captured
in the proposed model. On the other hand, the model can only be utilized to allocate
a single CS level at a time. The model needs to be modified to be able to select which
level is suitable to which location. Additionally, while it is impossible for mathematical
programming models to be utilized to allocate CSs in a large network due to the extremely
high number of combinations, the model needs to be compared with similar node-serving
Sustainability 2021,13, 3314 18 of 20
approaches which utilized metaheuristic algorithms such as GSA and PSO. Finally, prior-
itizing between the number of CSs to be located versus the total cost of implementation
needs to be investigated.
6. Conclusions
A stochastic model for charging stations of electric vehicles is proposed. The model is
based on birth-and-death Markov-chain process where the arrival rate of EVs is the birth
rate, and the charging rate is the death rate. When the number of EVs in the CS is greater
than the number of sockets available, then the EV drivers enter the CS with probability
α
.
The Markov-chain model is used to estimate the expected net profit and is applied as an
objective function to the GWO to select the best CSlocations out of all possible locations.
The proposed algorithm was applied to Washington D.C.’s transportation infrastructure,
where fueling stations are suggested to be the candidate locations of CSs, and the distance
between target locations was calculated based on driving distance.
The model can simulate realistic scenarios and can provide accurate estimation of the
net profit. To achieve the best planning for the EVCS infrastructure, slow CSs of Level 2
need to be installed in the areas with low arrival rates of network implementation, and fast
CSs (Level 3) can be used to achieve higher profit in later stages and/or high demand areas
in the early stages.
For future work, the algorithm can be improved by considering other factors such
as the chicken-egg problem, which can be addressed by performing the model in a multi-
period manner by increasing the demand gradually. Further improvement can be achieved
by considering different types and sizes of charging stations including the charging rate for
each type along with SOC level uncertainty. Finally, demand variation effect on pricing can
be investigated to maximize the net profit of EVCS.
Author Contributions:
The authors confirm the contribution to the paper as follows: study concep-
tion and design, R.S., A.K, and M.M.A.; data preparation and reduction, R.S., and A.K.; analysis and
interpretation of results, R.S., A.K. and M.M.A.; draft manuscript preparation, R.S., A.K. and M.M.A.
All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data presented in this study are openly available in District of
Columbia open data. Gas Stations. 2015. Available online: https://opendata.dc.gov/ (accessed on
15 May 2019).
Acknowledgments:
The authors would like to thank Sung Hoon Chung for his generosity in sharing
his knowledge to explain the EVCS chicken-egg problem and Markov-chain process modeling, as
well as for his comments on earlier drafts of the paper. All opinions and results are solely those of
the authors.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Andrenacci, N.; Ragona, R.; Valenti, G. A demand-side approach to the optimal deployment of electric vehicle charging stations
in metropolitan areas. Appl. Energy 2016,182, 39–46. [CrossRef]
2.
Bayram, I.S.; Michailidis, G.; Devetsikiotis, M.; Granelli, F. Electric Power Allocation in a Network of Fast Charging Stations.
IEEE J. Sel. Areas Commun. 2013,31, 1235–1246. [CrossRef]
3.
Berman, O.; Larson, R.C.; Fouska, N. Optimal Location of Discretionary Service Facilities. Transp. Sci.
1992
,26, 201–211. [CrossRef]
4.
Bokeh Development Team. Bokeh: Python Library for Interactive Visualization. 2018. Available online: https://bokeh.pydata.
org/en/latest/ (accessed on 18 April 2018).
5.
Cai, H.; Jia, X.; Chiu, A.S.; Hu, X.; Xu, M. Siting public electric vehicle charging stations in Beijing using big-data informed travel
patterns of the taxi fleet. Transp. Res. Part D Transp. Environ. 2014,33, 39–46. [CrossRef]
Sustainability 2021,13, 3314 19 of 20
6.
Chen, T.D.; Kockelman, K.M.; Khan, M. The Electric Vehicle Charging Station Location Problem: A Parking-Based Assignment
Method for Seattle. In Proceedings of the 92nd Annual Meeting of the Transportation Research Board, Washington, DC, USA,
13–17 January 2013.
7.
Chung, S.H.; Kwon, C. Multi-period planning for electric car charging station locations: A case of Korean Expressways. Eur. J.
Oper. Res. 2015,242, 677–687. [CrossRef]
8.
Davis, M.A.; Oliner, S.D.; Pinto, E.J.; Bokka, S. Residential land values in the Washington, DC metro area: New insights from big
data. Reg. Sci. Urban Econ. 2017,66, 224–246. [CrossRef]
9. District of Columbia Open Data. Gas Stations. 2015. Available online: https://opendata.dc.gov/ (accessed on 15 May 2019).
10.
Dong, J.; Liu, C.; Lin, Z. Charging infrastructure planning for promoting battery electric vehicles: An activity-based approach
using multiday travel data. Transp. Res. Part C Emerg. Technol. 2014,38, 44–55. [CrossRef]
11.
Kuby, M.; Lim, S. The flow-refueling location problem for alternative-fuel vehicles. Socio-Econ. Plan. Sci.
2005
,39, 125–145.
[CrossRef]
12.
Eberle, U.; Von Helmolt, R. Sustainable transportation based on electric vehicle concepts: A brief overview. Energy Environ. Sci.
2010,3, 689–699. [CrossRef]
13.
Emary, E.; Zawbaa, H.M.; Hassanien, A.E. Binary grey wolf optimization approaches for feature selection. Neurocomputing
2016
,
172, 371–381. [CrossRef]
14.
Farkas, C.; Prikler, L. Stochastic modelling of EV charging at charging stations. Renew. Energy Power Qual. J.
2012
, 1046–1051.
[CrossRef]
15.
Franke, T.; Krems, J.F. Interacting with limited mobility resources: Psychological range levels in electric vehicle use. Transp. Res.
Part A Policy Pr. 2013,48, 109–122. [CrossRef]
16.
Google Developers. Getting Started with the Google Places API for Work. 2017. Available online: https://developers.google.com/
(accessed on 27 July 2017).
17.
Liu, J. Electric vehicle charging infrastructure assignment and power grid impacts assessment in Beijing. Energy Policy
2012
,51,
544–557. [CrossRef]
18.
Shi, Q.S.; Zheng, X.Z. Electric Vehicle Charging Stations Optimal Location Based on Fuzzy C-Means Clustering. Appl. Mech.
Mater. 2014,556-562, 3972–3975. [CrossRef]
19. Hodgson, M.J. A Flow-Capturing Location-Allocation Model. Geogr. Anal. 2010,22, 270–279. [CrossRef]
20.
Vazifeh, M.M.; Zhang, H.; Santi, P.; Ratti, C. Optimizing the deployment of electric vehicle charging stations using pervasive
mobility data. Transp. Res. Part A Policy Pr. 2019,121, 75–91. [CrossRef]
21.
Jeff Desjardins. Visualizing the Rise of the Electric Vehicle. 2018. Available online: https://www.visualcapitalist.com/rise-
electric-vehicle/ (accessed on 2 July 2019).
22.
Rahman, I.; Vasant, P.M.; Singh, B.S.M.; Abdullah-Al-Wadud, M. Swarm Intelligence-Based Smart Energy Allocation Strategy for
Charging Stations of Plug-In Hybrid Electric Vehicles. Math. Probl. Eng. 2015,2015, 1–10. [CrossRef]
23.
Vasant, P.M.; Rahman, I.; Singh, B.S.M.; Abdullah-Al-Wadud, M. Optimal power allocation scheme for plug-in hybrid electric
vehicles using swarm intelligence techniques. Cogent Eng. 2016,3. [CrossRef]
24.
Jing, W.; An, K.; Ramezani, M.; Kim, I. Location Design of Electric Vehicle Charging Facilities: A Path-Distance Constrained
Stochastic User Equilibrium Approach. J. Adv. Transp. 2017,2017, 1–15. [CrossRef]
25.
Kendall, D.G. Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov
Chain. Ann. Math. Stat. 1953,24, 338–354. [CrossRef]
26.
Kim, J.; Son, S.-Y.; Lee, J.-M.; Ha, H.-T. Scheduling and performance analysis under a stochastic model for electric vehicle charging
stations. Omega 2017,66, 278–289. [CrossRef]
27.
Kuby, M.; Lim, S. Location of Alternative-Fuel Stations Using the Flow-Refueling Location Model and Dispersion of Candidate
Sites on Arcs. Netw. Spat. Econ. 2006,7, 129–152. [CrossRef]
28.
Lee, Y.-G.; Kim, H.-S.; Kho, S.-Y.; Lee, C. User Equilibrium–Based Location Model of Rapid Charging Stations for Electric Vehicles
with Batteries that have Different States of Charge. Transp. Res. Rec. J. Transp. Res. Board 2014,2454, 97–106. [CrossRef]
29. Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey Wolf Optimizer. Adv. Eng. Softw. 2014,69, 46–61. [CrossRef]
30.
Pinto, F.A.V.; Costa, L.H.M.K.; De Amorini, M.D. Modeling spare capacity reuse in EV charging stations based on the Li-ion
battery profile. In Proceedings of the 2014 International Conference on Connected Vehicles and Expo (ICCVE), Vienna, Austria,
3–7 November 2014; pp. 92–98. [CrossRef]
31.
Python Software Foundation. Python Language Reference, Version 2.7. 2013. Available online: https://www.python.org/
(accessed on 20 July 2017).
32. Ross, S.M. Introduction to Probability Models, 20th ed.; Academic Press: Oxford, UK, 2009. [CrossRef]
33.
Smith, M.; Castellano, J. Costs Associated with Non-Residential Electric Vehicle Supply Equipment: Factors to Consider in the Implementa-
tion of Electric Vehicle Charging Stations; NewWest Technologies: Portland, OR, USA, 2015.
34.
Socrata. About Gas Stations in Washington DC. 2011. Available online: https://opendata.socrata.com/dataset/Gas-Stations-in-
Washington-DC/ (accessed on 27 July 2019).
35.
Tan, X.; Sun, B.; Tsang, D.H.K. Queueing network models for electric vehicle charging station with battery swapping. In
Proceedings of the 2014 IEEE International Conference on Smart Grid Communications (SmartGridComm), Venice, Italy,
3–6 November 2014. [CrossRef]
Sustainability 2021,13, 3314 20 of 20
36.
Tian, Z.; Hou, W.; Gu, X.; Gu, F.; Yao, B. The location optimization of electric vehicle charging stations considering charging
behavior. Simulation 2018,94, 625–636. [CrossRef]
37.
Vardakas, J.S. Electric vehicles charging management in communication controlled fast charging stations. In Proceedings of
the 2014 IEEE 19th International Workshop on Computer Aided Modeling and Design of Communication Links and Networks
(CAMAD), Athens, Greece, 1–3 December 2014. [CrossRef]
38.
Wang, S.; Bi, S.; Zhang, Y.-J.A.; Huang, J. Electrical Vehicle Charging Station Profit Maximization: Admission, Pricing, and Online
Scheduling. IEEE Trans. Sustain. Energy 2018,9, 1722–1731. [CrossRef]
39.
Wu, F.; Sioshansi, R. A stochastic flow-capturing model to optimize the location of fast-charging stations with uncertain electric
vehicle flows. Transp. Res. Part D Transp. Environ. 2017,53, 354–376. [CrossRef]