# A loss queueing game for electric vehicle charging performance evaluation

Alix Dupont, Yezekael Hayel, Tania Jimenez, Olivier Beaude, Jean-Baptiste Breal

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Alix Dupont, Yezekael Hayel, Tania Jimenez, Olivier Beaude, Jean-Baptiste Breal. A loss queueing game for electric vehicle charging performance evaluation. Performance Evaluation, 2023, 161, pp.102350. 10.1016/j.peva.2023.102350. hal-04552120

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# A loss queueing game for electric vehicle charging performance evaluation

Alix Dupont<sup>a,b</sup>, Yezekael Hayel<sup>a</sup>, Tania Jiménez<sup>a</sup>, Olivier Beaude<sup>b</sup>,
Jean-Baptiste Breal<sup>b</sup>

## Abstract

The demand of electricity at the Charging Stations (CSs) by Electric Vehicle (EV) users is tremendously increasing. However, EV users still face limited resources at the charging stations, both in terms of the number of parking spaces equipped with a charging point, and in terms of available power. This paper deals with the choice of a CS among two CSs by the EV users in a competitive environment. The stochastic nature of arrivals and departures at the CSs is modeled by a queueing system. A queueing game is studied where the EV users are the players and choose the CS that gives the highest expected energy received. An approximation of the expected energy received at the CSs is theoretically provided and the quality of this approximation is numerically illustrated and analyzed through simulations. The existence and uniqueness of the equilibrium of the game is proved, and bounds on the Price of Anarchy (PoA) are also provided. Moreover, the model is simulated using a discrete event framework and a sensitivity analysis of the main metrics of the system with respect to the average parking duration and the power sizing coefficient is provided. The results show that the utility of EV users at equilibrium is close to the optimal utility. This study can help a Charging Point Operator (CPO) to design incentives for EV users in order for instance to limit the parking duration so as to improve the social welfare of the EV users.

**Keywords:** Electric vehicles, charging station, stochastic model, queueing game

<footnote><sup>a</sup>Avignon University</footnote>

<footnote><sup>b</sup>EDF R&D, Paris-Saclay</footnote>
# 1. Introduction

The demand for Electric Vehicles (EVs) is exponentially growing [1, 2]. Between 2018 and 2022, the number of EVs on the road has been multiplied by 4, reaching the number of 16.5 millions of EVs in use in the world [2]. The development of EVs will continue to grow, as major car manufacturers plan to achieve 100% or almost sales of zero emission vehicles in the next two decades [2]. In order to supply in electricity the current and future fleet of EVs, a large deployment of charging points (CPs) is necessary: a maximum ratio of 10 EVs per public CP is recommended by the European commission [3]. However, there still exists many areas with very limited number of CPs which makes difficult to deal with the charging demand of EV users [4], and creates the well-known range anxiety of EV users, i.e. the fear to be out of energy. Moreover, according to a recent french survey [5], a significant proportion of EV users is not satisfied with the current availability of public CPs. However, the construction of CPs and the supply in electrical energy to charge the EVs' batteries represents a considerable cost. As a consequence, supplying the electricity demand is becoming a real challenge for all stakeholders. In this paper, a Charging Station (CS) refers to a set of parking spaces equipped with a CP, for which the maximum power that can be delivered is identical. The main objectives to manage a CS are generally to limit the infrastructure and electricity supply cost, but also to guarantee a high Quality of Charging Service (QoCS) [6] for the EV users, such as the time to fully charge the EV battery or the probability to meet the energy supply demand of EV users at departure. In order to achieve these goals, the Charging Point Operator (CPO) must efficiently forecast the power demand at the CSs, and design management policies that take into account the EVs users behavior. To this end, in this work a model with two CSs at the same site, for instance a shopping center, is presented.

In our model, EV users arrive at the given site equipped with two public CSs. Unlike at gasoline stations, EV users generally have other activities, such as going to the restaurant or the theater, while their EV is charging. Therefore
the parking duration of plugged EVs at parking spaces differs from the time
to complete the charge. Thus a fully charged EV can occupy a parking space
equipped with a CP. As in [7], a queueing system model representing this specific
scenario at a given CS is used: at each time some parked EVs are charging and
some are fully charged but still occupying a CP. Moreover, at each CS, each
EV currently charging has to share the available power within a CS with other
charging EVs. This power depends on some physical local grid constraints or on
the contract between the CPO or the site manager and the electricity supplier.

Upon arrival, EV users choose the CS where they will park and recharge
their battery. This choice naturally depends on EV users previous experiences,
which is often a consequence of the CS characteristics, i.e available instantaneous
power and number of parking spaces equipped with a CP. Based on the modeling
and results obtained in [7], the choice of the CS by the EVs is studied from the
perspective of a queuing game [8]. In our context, EV users see the charging
service as an opportunity: the EV users take advantage of the possibility to
charge at the parking space, but it is not their main reason for coming to this
parking. Because of this, the parking duration of each EV user does not depend
on its energy needed, and the EV users go at a parking space with no charging
spot if the chosen CS is fully occupied. The criterion of choice by the EV users
is based on the expected energy received during their parking duration. The
equilibrium of the game studied is unique, which enables one to anticipate the
EVs distribution to the two CSs.

1.1. *Related work*

A lot of research has been done recently on the optimal design of CS infras-
tructure [9, 10], i.e. number of chargers, as well as on the optimal CS charging
scheduling [6, 11].

In particular, in [12] the minimization of the infrastructure cost at a CS
as a function of the number and types of CPs is studied, while meeting cer-
tain QoCS levels, one of them being a sufficiently high probability that the
EV battery is fully charged at departure. In the optimization problem, dif-
ferent charging scheduling are compared in terms of the infrastructure cost: a
first come first served policy; a simplified policy considering the optimal start
time of the charging period; and a more sophisticated charging policy allowing
interruption during the charging sessions and flexibility on the instantaneous
charging power. It is shown that the latter is more efficient for satisfying the
QoCS and therefore enables to lower the infrastructure costs while meeting the
QoCS conditions.

Nevertheless, only few papers deal with the assignment of EVs to CSs or to
different CPs within a CS. Some studies [13–16] consider that a central scheduler
decides at which location EV users will recharge their battery.

In the context of a residential CS, in [16], a global two-stage optimization
problem is studied with inter-dependent decisions: the optimal number of CPs
and the charging scheduling. An EV to CP assignment rule is taken into ac-
count: an arriving EV user is assigned to the CP with the least number of
connected EVs. The optimization problem enables to reduce the infrastructure
cost. However, no coupling power constraint between the CPs is taken into
account, whereas in most of the cases at residential CSs, CPs are linked with
electrical wires and have to share power between them. In [13] the authors
consider M/M/c queues to derive the expected waiting time of EVs to CSs and
propose a CS assignment algorithm which takes into account the EV location.
However, the assumptions about CP and EV characteristics is unrealistic, for
example the maximal battery storage capacity is 7 kWh. In [14], at each time-
step, a Nash equilibrium of a game where the EV users are players who choose
between several CSs is numerically found. The game takes into account the
interaction between the EVs on the road as well as at the CSs. The solution
equilibrium is then suggested to the EV users which enables them to trust the
central scheduler suggestion because Nash equilibria have the property that no
decision-maker have an interest to deviate from it. In the case studied, the
Nash equilibrium only differ by 15% from the optimal EV to CS assignment,
which shows that the Nash equilibria is an efficient solution for EV users. How-
ever, the charging time is fixed and is supposed the same for all EVs, which is
not the case in reality because the State of Charge (SoC) at arrival can signif-
icantly differ from one EV to another. In [15] the EV assignment problem to
CSs at a highway is studied for long trip context when several charging stops
are necessary. M/M/c queues are considered to find the optimal percentage of
EV choosing each CS along a highway, depending on their location and the EV
battery constraints.

These previous papers consider that a central scheduler suggest or even
decide for the EV users the optimal CS or CP at a CS. However, in reality, EV
users make their own decision on where they will recharge their EV battery. To
this end, [17–20] consider decentralized systems where the EV users make their
own decision. In [17], EV users can choose between two types of CPs: public
charging piles and private charging piles, which are in competition with each
other. This competition between the two types of CPs is modeled through a
2-persons game, in which the decision-makers, i.e. the two types of CPs, set
a charging cost in order to maximize their own profit. The demand response
of the EVs is a probabilistic function which considers the charging cost and
other factors that are input parameters such as the charging duration and the
charging safety. The results shows that at equilibrium of the game, a majority
(about 80%) of EV users choose the public charging mode, which implies that
this charging mode has to be sufficiently deployed. In [18], the optimal EV
assignment to CPs, categorized into two types of charging modes (AC and DC<fnref n="1" />),
at a CS is studied. For each set of CPs with the same charging mode, a waiting
queue is created depending on EV user decisions. The probabilistic choice of
the charging mode is a linear function of the service fee, i.e. the price for
choosing the charging mode. This choice don’t take into account the limited
available power at the CS. The objective is to set the optimal pricing policy
that minimizes the dropping rate, i.e. the proportion of EV users declining
the entrance to the chosen CS because of a too long waiting queue. In [19],
the choice of the CS is not considered but whether or not EV users join the

<footnote n="1">¹AC = Alternative Current; DC = Direct Current.</footnote>
considered fast CS. The result analysis is based on M/M/s queues to derive the waiting time of the EV users. In this use-case, a scheduled pricing policy (as opposed to flat-rate pricing policy is proposed which enables to maximize the CS profit and minimize the EV charging demand during peak hours. [20] considers the EV assignment to public CSs. Each EV users individually minimize the travel time toward the CS and the waiting time upon arrival before a plug is available. The interaction between the EV users is only considered on waiting times at the CSs, and a M/D/c queue is used for modeling the EV arrivals at the CSs. A algorithm that enables to find the Wardrop equilibrium is proposed. However, the charging time is supposed the same for all EV users. Moreover, it is supposed that EV users leave immediately after completing their charge, while they are certainly doing other activities because of the long charging duration. Moreover, when several EVs are simultaneously charging, they may have to draw less power due to limited available power at the CSs. However, in all previous models, the charging time is supposed independent from the number of charging EVs. This is only taken into account in [7], but the EV assignment to CSs is not studied.

## 1.2. Our prior work

In our prior work [21], the decision-making problem by EV users on which CS to choose is investigated. Queueing theory is used to derive the respective probability to leave with a fully charged battery at two CSs [7]. To make the choice of the CS, EV users neglect the possibility that the chosen CS is fully occupied. The expression of the set of equilibria and of the Price of Anarchy (PoA) are found for all parameter values. It shows in particular that the decentralized system, i.e. when all EV users individually make their decision based on their personal interest, is generally efficient in terms of the average QoCS of EV users, except when the two CSs have similar power capacities. In the proposed scenario, the two CSs are not independent in terms of electrical constraints: the sum of the available power at the two CSs is limited, which applies in practice when both CSs are located on the same site. The optimal sizing of the CSs
in terms of power capacity considering a coupled power constraint between the
two CSs is solved analytically in order to maximize the average QoCS for EV
users in the decentralized system.

1.3. *Contributions and organization of the paper*

This paper is an extended version of our prior work [21]. The same situation
is considered in which EV users arrive at a given site where there are two CSs.
In this context, EV users have a charging opportunity, and therefore want to
received as much energy as possible, rather than the binary criteria (used in [21])
of leaving with a fully charged battery or not. In particular, the probability to
leave with a fully charged battery gives less information on the QoCS as it
does not distinguish EVs that received a lot of energy and leave with an almost
fully charged battery and EVs that did not received any energy at all. The
contribution of this paper can be summarized as follows:

1. An approximation of the expected energy received by an EV is proposed and theoretical bounds on this approximation are given.

2. The exact *blocking probability* for an EV to face a full CS is taken into account in the EV users decision.

3. The same static maximum power deliverable by each CP within a CS (because of physical electricity constraints of the chargers) can be different from one CS to another.

4. Simulations using a discrete event framework are provided and enables to study the accuracy of the theoretical results.

This paper is organized as follows. In Section 2, our model is described
and an approximation of the expected number of charging EVs as well as the
expected energy received by an EV are given. In Section 3, a queuing game is
presented and studied, where EV users choose the CS with the highest expected
energy received, which itself depends on the decision of other EV users. Section
4 presents numerical simulations of the model and a quantitative comparison
with theoretical results. Finally, a conclusion is given in Section 5.
<table><caption>Table 1: Table of notations</caption><tr><td>Q<sub>t</sub><sup>i</sup> &isin; &lfloor;0, K<sub>i</sub>&rfloor;</td><td>Number of EVs at CS i at time t.</td></tr><tr><td>Z<sub>t</sub><sup>i</sup> &isin; &lfloor;0, K<sub>i</sub>&rfloor;</td><td>Number of charging EVs at CS i at time t.</td></tr><tr><td>C<sub>t</sub><sup>i</sup> &isin; &lfloor;0, K<sub>i</sub>&rfloor;</td><td>Number of fully charged EVs at CS i at time t.</td></tr><tr><td>D</td><td>Random parking duration (in h).</td></tr><tr><td>E<sub>req</sub></td><td>Random energy needed for a fully charged battery (in kWh).</td></tr><tr><td>B<sub>cap</sub></td><td>Random battery energy storage capacity (in kWh).</td></tr><tr><td>E<sub>i</sub></td><td>Random energy received by an EV at CS i (in kWh).</td></tr><tr><td>E&#x0303;<sub>i</sub></td><td>Approximation of the random energy received at CS i (in kWh).</td></tr><tr><td>SoC<sub>arr</sub></td><td>Random state of charge of the battery when arriving.</td></tr><tr><td>&lambda; (resp. &lambda;<sub>i</sub>) &isin; R<sub>+</sub></td><td>Frequency of EV user arrivals at the site (resp. at CS i) (in h<sup>-1</sup>).</td></tr><tr><td>&nu; &isin; R<sub>+</sub></td><td>Parameter of the random variable D (in h<sup>-1</sup>).</td></tr><tr><td>&mu; &isin; R<sub>+</sub></td><td>Parameter of the random variable E<sub>req</sub> (in kWh<sup>-1</sup>).</td></tr><tr><td>&alpha;<sub>i</sub> &isin; [0, 1]</td><td>Power sizing coefficient.</td></tr><tr><td>K<sub>i</sub> &isin; N<sub>+</sub></td><td>Number of parking spaces at CS i.</td></tr><tr><td>p<sub>max</sub><sup>i</sup> &isin; R<sub>+</sub></td><td>Maximum power each CP can deliver at CS i (in kW).</td></tr><tr><td>z&#x0303;<sub>i</sub> &isin; &lfloor;1, K<sub>i</sub>&rfloor;</td><td>Approximation of the average number of charging EVs at CS i.</td></tr><tr><td>B<sub>i</sub> &isin; [0, 1]</td><td>Blocking probability at CS i.</td></tr><tr><td>&theta; (resp. &theta;<sub>i</sub>) &isin; [0, 1]</td><td>Probability that an EV chooses CS 1 (resp. CS i).</td></tr><tr><td>&theta;<sub>eq</sub> &isin; [0, 1]</td><td>Probability that an EV chooses CS 1 at equilibrium.</td></tr><tr><td>u(.) : [0, 1] &mapsto; R<sub>+</sub></td><td>Utility function of EV users.</td></tr></table>

## 2. Model description with queueing theory

At arrival, an EV user chooses between two CSs according to both the instantaneous expected power and the number of parking spaces at each CS. Indeed, if the chosen CS is fully occupied at time of arrival, the EV user receives no energy as she/he goes to a parking space without a CP; the model is illustrated in Figure 1.
## 2.1. Mathematical modeling of the chosen scenario

Charging behavior of the EV users. The two CSs are supposed to be used by the same population of EVs. The EV arrival time to the CSs follow a common Poisson process with rate $\lambda$. Each EV chooses to join CS i, for $i \in \{1, 2\}$, with a probability denoted by $\theta_i$. This decision is taken before joining a CS, without knowing any information about CSs status, i.e. the number of charging EVs and parked EVs. This assumption is standard, corresponding to the class of unobservable queueing games [8]. The case with information is left for a future extension. Based on preceding assumptions, the arrival process of EVs at CS i follows a Poisson process with rate $\lambda_i := \theta_i\lambda$. Each EV has a battery storage capacity denoted by $B_{cap}$ (in kWh) and a State of Charge (SoC) $SoC_{arr}$ (as a pourcentage of the battery capacity) when arriving at the CSs. $SoC_{arr}$ for each EV mainly depends on the distance traveled in order to reach the parking space since the last charging session, and many other exogeneous parameters like ambient temperature, traffic conditions, etc. [22]. In order to fit with the heterogeneity of the charging demand, the energy $E_{req}$ (in kWh) needed in order to get a fully charged battery of each arriving EV is random and defined by:

$$E_{req} = B_{cap}(1 - SoC_{arr}).$$

EVs have different energy storage capacities and different SoC when arriving at the CSs. As in [9, 10, 18], the energy $E_{req}$ is modeled by an exponential random variable with parameter $\mu$. The EVs parking duration $D$ at the chosen CS is random and follows an exponential distribution with parameter $\nu$, which is in accordance with studies on the parking duration at a mall [23, 24]. In fact, in our model the charging service is an opportunity for the EV customer whose first aim is to go to the shopping center for another kind of service (restaurant, cinema, or other activities). Thus, the parking duration $D$ is supposed to be independent of the energy level required $E_{req}$ in order to get a fully charged battery.

At each instant, among all EVs parked at a CS, some of them are charging and some are not. Indeed, some EVs can stay plugged at the end of their charge.
As the main goal of the EV users is not to charge their EV but to get a service in the commercial center, the redirection of rejected EVs from one CS to the other is not considered.

200

<figure><img src="image_2.png" /><figcaption>Figure 1: Arriving EV users choose between the two CSs. Power is shared among charging EVs at the same CS.</figcaption></figure>

*Charging station characteristics.* CS i ∈ {1, 2} has K_i > 0 parking spaces; each one of them being equipped with one CP. It means that only one EV can charge at one time at each CP. At each CS i and time t, Z_t^i denotes the number of charging EVs and C_t^i the number of EVs that are parked but not charging at this time because their battery is fully charged. Then, the total number of EVs at CS i at time t is Q_t^i := Z_t^i + C_t^i.

205

*Charging station limitations and charging scheduling rule by processor sharing.* Each CP at CS i can deliver the same maximum power $p_{max}^i > 0$, which depends on the physical characteristics of the chargers as stated in [25]. Then the potential global available power at CS i is given by $K_i \times p_{max}^i$. Typically, the global available power is limited and the parameter $\alpha_i \in [0, 1]$ is the proportion of the potential global available power that can be drawn from CS i. This pro-

210
portion determines the number $\lfloor \alpha_i K_i \rfloor$ of CPs that can draw simultaneously the maximum power level $p_{\text{max}}^i$ at CS i.

At each CS, the limited available power is shared among the EVs charging at the CS according to a processor sharing mechanism. This type of charging scheduling policy is already used by some CPOs in practice like Izivia in France<fnref n="2" />. The charging scheduling policy plays a crucial role in the satisfaction of EV users [6] because it determines the instantaneous power received by each EV. When the number of charging EVs $Z_t^i$ at CS i at a given instant t is above $\lfloor \alpha_i K_i \rfloor$, the maximum power $p_{\text{max}}^i$ is equally shared among all EVs within CS i. In turn, at each time t, the instantaneous power $p_i$ delivered to each charging EV at CS i depends on $Z_t^i$ through the following equation:

$$p_i(Z_t^i) = \begin{cases} p_{\text{max}}^i \frac{\min(Z_t^i, \alpha_i K_i)}{Z_t^i} & \text{if } Z_t^i > 0, \\ p_{\text{max}}^i & \text{if } Z_t^i = 0. \end{cases} \quad (1)$$

The instantaneous power received by an EV in the case $Z_t^i = 0$ is mathematically defined although it does not have any physical meaning. It is clear from our context that, for each CS i, the two-dimensional process $(Q_t^i, Z_t^i)$ is a Markov process. EV users determine which CS to join in order to maximize the expected energy in steady state of the Markov process $(Q_t^i, Z_t^i)$ received before leaving. Below, an approximation of the expected energy received is presented.

## 2.2. Approximation of the expected number of charging EVs based on Erlang loss formula

The Poisson Arrivals See Time Averages (PASTA) property [27] applied to our setting states that an EV arriving at CS i at any random instant sees the state $(q, z)$ of the Markov process with probability $\pi_i(q, z)$ where $\pi_i$ is the stationary distribution of the Markov process $(Q_t^i, Z_t^i)$. This stationary distribution $\pi_i$, as well as the expected number $\sum_{q=0}^{K_i} \sum_{z_i=0}^{q} z_i \pi_i(q, z_i)$ of charging EVs, cannot be computed explicitly due to the complexity of the system. In

<footnote n="2">²See [26] for a commercial description of the power scheduling used by Izivia.</footnote>
[7], an approximation $\bar{z}_i$ of the expected number of charging EVs is proposed, which depends in the effective arrival rate $\lambda_i^e$, i.e. the rate at which EVs find an available parking space (EV arriving and not being rejected). In our previous work [21], the effective arrival rate was approximated by $\lambda_i^e \approx \min(\lambda_i, \nu K_i)$ in order to have explicit solutions for the analysis. However, this approximation can be inaccurate for some values of $\lambda_i$. The exact effective arrival rate is given by:

$$ \lambda_i^e = \lambda_i \times (1 - B_i), \qquad (2) $$

where $B_i := \frac{\rho_i^{K_i}}{\sum_{j=0}^{K_i} \frac{\rho_j^i}{j!}}$ is the blocking probability of an Erlang-B queue with load $\rho_i := \theta_i \rho = \theta_i \frac{\lambda_i}{\nu} = \frac{\lambda_i}{\nu}$. Indeed, the number of EVs at CS i corresponds to an M/M/K$_i$/K$_i$ queue. Then, $(1 - B_i)$ is the probability for an EV to be accepted, i.e. to find a available parking space, at CS i.

The approximation $\bar{z}_i$ of the expected number of charging EVs given by [7] is the unique solution of the following equation:

$$ \bar{z}_i = \lambda_i^e \mathbb{E} \left[ \min \left( D, \frac{E_{\text{req}}}{p_i(\bar{z}_i)} \right) \right], \qquad (3) $$

where $p_i$ is the power allocation function for CS i, see (1).

The charging process of an EV is stopped when the maximum capacity of its battery is reached or when the EV leaves the CP (therefore without a fully charged battery). Then the charging duration of an EV is expressed as the minimum between the parking duration and the time to fully charge the EV's battery.

Following the result of Proposition 1 in [21], the explicit expression of the approximation of the expected number of charging EVs at each CS i is obtained by solving the fixed-point equation (3).

**Proposition 2.1.** In stationary regime of the Markov process $(Q_t, Z_t)$, the expected number $\bar{z}_i$ of charging EVs at CS i can be approximated by:

$$ \bar{z}_i = \begin{cases} \frac{\lambda_i^e}{\nu + p_{\text{max}}^i \mu} & \text{if } \frac{\lambda_i^e}{\nu + p_{\text{max}}^i \mu} < \alpha_i K_i, \\ \frac{\lambda_i^e - p_{\text{max}}^i \mu \alpha_i K_i}{\nu} & \text{else.} \end{cases} \qquad (4) $$
Proof of Proposition 2.1. Solving the fixed-point equation (3): $f_i(z) = z$, with $i \in \{1, 2\}$ the CS index, $f_i(z) = \lambda_i^e \mathbb{E} \left[ \min \left( D, \frac{E_{\text{req}}}{p_i(z)} \right) \right]$ and $z \in [0, K_i]$. A more explicit expression of $f_i$ is:

$$f_i(z) = \begin{cases} \frac{1-B_i}{\nu+p_{\text{max}}^i \mu} & \text{if } z \le \alpha_i K_i, \\ \frac{1-B_i}{\nu} \times \frac{z}{z+\frac{p_{\text{max}}^i \mu}{\nu} \alpha_i K_i} & \text{if } z > \alpha_i K_i. \end{cases}$$

The following two cases arise:

* Case $\frac{\lambda_i^e}{\nu+p_{\text{max}}^i \mu} \le \alpha_i K_i$: the unique solution $z^*$ to equation $f(z) = z$ is $z^* = \frac{\lambda_i^e}{\nu+p_{\text{max}}^i \mu}$ (which is smaller than $\alpha_i K_i$).

* Case $\frac{\lambda_i^e}{\nu+p_{\text{max}}^i \mu} > \alpha_i K_i$: the unique solution $z^*$ to equation $f_i(z) = z$ is $z^* = \frac{\lambda_i(1-B_i)-p_{\text{max}}^i \mu \alpha_i K_i}{\nu}$ (which is greater than $\alpha_i K_i$).

■

Proposition 2.1 states that the expression of the average number of charging EVs takes two different forms. The first one, given by the first line of Equation (4), corresponds to the case where the power utilization rate $\frac{\lambda_i^e}{\nu+p_{\text{max}}^i \mu}$ at CS i is sufficiently low compared to the resource $\alpha_i K_i$ available at the CS, such that every charging EV charge at CP maximum power $p_{\text{max}}^i$. The second one, corresponds to the opposite case, where the power utilization rate $\frac{\lambda_i^e}{\nu+p_{\text{max}}^i \mu}$ at CS i is high, so that all EVs charge at a lower power than the maximum one $p_{\text{max}}^i$, see Equation (1) for the power allocation formula. The closed form expression $\bar{z}_i$ allows us to get a closed form expression of an approximation $p_i(\bar{z}_i)$ of the expected power received by an EV at CS i. This enables to express another important metric of the system which is the expected energy received by each EV at CS i.

## 2.3. Approximation of the expected energy received during the parking duration

The real energy received $E$ by an EV which has found an available parking space is determined by the minimum between the need in energy $E_{\text{req}}$ and the potential cumulative power received over the parking duration $D$:

$$E_i = (1 - B_i) \times \min \left( E_{\text{req}}, \int_D p_i(Z_i(t) | Z_i(t) > 0) dt \right). \quad (5)$$
An approximation $\tilde{E}_i(n)$ of the expected energy received at CS i is described. $\tilde{E}_i(n)$ is the energy received at CS i considering a constant number n of charging EVs. In this case, the power received during the charging duration is constant equal to $p_i(n)$, with $p_i(.)$ the instantaneous power received by an EV defined in Equation (1); See Figure 2 for an illustration of this approximation. Therefore, the potential cumulative power $\int_D p_i(Z_i(t)|Z_i(t) > 0)$ received over the parking duration in Equation (5) is simplified to $D \times p_i(n)$, so that:

$$ \tilde{E}_i(n) = (1 - B_i) \times \min(E_{\text{req}}, D \times p_i(n)). $$

$\tilde{E}_i(n)$ then follows an exponential distribution with parameter $\frac{p_i(n) \mu + \nu}{p_i(n) \times (1 - B_i)}$. The following Lemma 2.1 gives an approximation of the expected energy received.

**Lemma 2.1.** The approximated expected energy received $E[\tilde{E}_i(\bar{z}_i)]$ at CS i for any EV, is given by:

$$ \mathbb{E}[\tilde{E}_i(\bar{z}_i)] = \begin{cases} \frac{p_{\max}^i}{p_{\max}^i \mu + \nu} \times (1 - B_i) & \text{if } \lambda_e^i < \alpha_i K_i(\nu + p_{\max}^i \mu), \\ \frac{p_{\max}^i \alpha_i K_i}{\lambda_e^i} & \text{else.} \end{cases} \quad (6) $$

*Proof of Lemma 2.1.* As the approximation of the energy received by an EV follows an exponential distribution with parameter $\frac{p_i(n) \mu + \nu}{p_i(n) \times (1 - B_i)}$, its expectation is:

$$ \begin{aligned} \mathbb{E}[\tilde{E}_i(n)] &= \frac{p_i(n) \times (1 - B_i)}{p_i(n) \mu + \nu} \\ &= \frac{p_{\max}^i \min(n, \alpha_i K_i) \times (1 - B_i)}{p_{\max}^i \min(n, \alpha_i K_i) \mu + n \nu}. \end{aligned} \quad (7) $$
Replacing in Equation (7) n by the expected number of charging EVs $\bar{z}_i$ leads to Equation (6):

$$ 
\begin{aligned}
\mathbb{E}[\tilde{E}_i(\bar{z}_i)] &= \frac{p_{\max}^i \min(\bar{z}_i, \alpha_i K_i) \times (1 - B_i)}{p_{\max}^i \min(\bar{z}_i, \alpha_i K_i) \mu + \bar{z}_i \nu} \\
&= \begin{cases} \frac{p_{\max}^i}{p_{\max}^i \mu + \nu} \times (1 - B_i) & \text{if } \lambda_e^i < \alpha_i K_i (\nu + p_{\max}^i \mu) \\ \frac{p_{\max}^i \alpha_i K_i \times (1 - B_i)}{p_{\max}^i \alpha_i K_i \mu + \frac{\lambda_e^i - p_{\max}^i \mu \alpha_i K_i}{\nu}} & \text{else} \end{cases} \\
&= \begin{cases} \frac{p_{\max}^i}{p_{\max}^i \mu + \nu} \times (1 - B_i) & \text{if } \lambda_e^i < \alpha_i K_i (\nu + p_{\max}^i \mu) \\ \frac{p_{\max}^i \alpha_i K_i \times (1 - B_i)}{\lambda_e^i} & \text{else.} \end{cases} \\
&= \begin{cases} \frac{p_{\max}^i}{p_{\max}^i \mu + \nu} \times (1 - B_i) & \text{if } \lambda_e^i < \alpha_i K_i (\nu + p_{\max}^i \mu) \\ \frac{p_{\max}^i \alpha_i K_i}{\lambda_e^i} & \text{else.} \end{cases}
\end{aligned}
$$

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2: In the approximation, the number of charging EVs is supposed constant equal to the average number of charging EVs.</figcaption></figure>

Section 4 numerically measures the quality of this approximation. The following Lemma 2.2 gives a theoretical bound for the accuracy of the approximation on the expected energy received, given by formula (7). Let $\delta_i$ be the absolute difference between the real energy received given by Equation (5) and the corresponding approximation by an EV at CS i.
**Lemma 2.2.** The error $\delta_i$ at CS i, defined by the absolute value of the difference between the real energy received of an EV during its parking duration and the approximation given by Equation (4) is bounded by:

$$ \delta_i \le \frac{1}{\mu} \left( \frac{\min(1, \alpha_i K_i)}{\min(1, \alpha_i K_i) + \beta_i} - \frac{\alpha_i}{\alpha_i + \beta_i} \right) \times (1 - B_i), $$
with $\beta_i = \frac{\nu}{p_{\max}^i \mu}$. $$

*Proof of Lemma 2.2.* According to Equations (1) and (7), the energy received when being accepted to the chosen CS is upper (resp. lower) bounded by the energy received when there is only 1 (resp. $K_i$) charging EVs. In this case, the power drawn by an EV is $p_{\max}^i \min(1, \alpha_i K_i)$ (resp. $p_{\max}^i \alpha_i$). Therefore,

$$ \begin{aligned} \delta_i &\le (\mathbb{E}[\tilde{E}_i(1)] - \mathbb{E}[\tilde{E}_i(K_i)]) \times (1 - B_i) \\ &= \left( \frac{p_{\max}^i \min(1, \alpha_i K_i)}{p_{\max}^i \min(1, \alpha_i K_i) \mu + \nu} - \frac{p_{\max}^i \alpha_i}{p_{\max}^i \alpha_i \mu + \nu} \right) \times (1 - B_i) \\ &= \frac{1}{\mu} \left( \frac{\min(1, \alpha_i K_i)}{\min(1, \alpha_i K_i) + \frac{\nu}{p_{\max}^i \mu}} - \frac{\alpha_i}{\alpha_i + \frac{\nu}{p_{\max}^i \mu}} \right) \times (1 - B_i). \end{aligned} $$

<figure><img src="image_5.png" /></figure>

The bound on the error $\delta_i$ is obtained by considering the difference between the highest $\bar{\delta}_i$ and the lowest expected energy an EV can draw $\delta_i$, with:

$$ \bar{\delta}_i := \frac{p_{\max}^i \min(1, \alpha_i K_i)}{p_{\max}^i \min(1, \alpha_i K_i) \mu + \nu} \times (1 - B_i), \quad (8) $$

and

$$ \delta_i := \frac{p_{\max}^i \alpha_i}{p_{\max}^i \alpha_i \mu + \nu} \times (1 - B_i). \quad (9) $$

Even if this bound is simple to obtain, as it is based on extreme cases, it gives an explicit expression.

The bound given in Lemma 2.2 is strictly decreasing and convex with respect to the power sizing coefficient $\alpha_i$ for $\alpha_i > \frac{1}{K_i}$. In particular, when $\alpha_i = 1$, i.e. the EVs always get served with the maximal power $p_{\max}^i$, this bound is equal to zero, that means that the expected energy received at CS i given by Equation (7) is the real energy received. When the system is highly congested in terms
of the number of parked EVs, i.e. when the utilization rate $\rho = \frac{\lambda}{\nu}$ is high which implies a blocking probability $B_i$ close to 1, the error is tight. Note also that the error $\delta_i$ is bounded by the expected energy needed $\frac{1}{\mu}$ because the energy received is always lower than the energy needed.

Here the Quality of Charging Service (QoCS) is defined as the expected energy received during the parking duration. Other criteria could be used such as the probability to leave with a fully charged battery [7, 21]. The following Lemma 2.3 gives a relation between the expected energy received during the parking duration and the probability to leave with a fully charged battery, considering a constant number of charging EVs during the parking duration.

**Lemma 2.3.** The probability $\mathbb{P}(F_i)$ to leave CS i with a fully charged battery considering a constant number $n$ of charging EVs during the charging duration is approximated by:

$$ \mathbb{P}(F_i) = \frac{\mathbb{E}[\tilde{E}_i(n)]}{\mathbb{E}[E_{\text{req}}]} \qquad (10) $$

*Proof of Lemma 2.3.* Assuming an EV has been accepted to CS i, its probability to leave with a fully charged battery is the probability that its charging duration $\frac{E_{\text{req}}}{p_i(n)}$ is less than its parking duration D, both following exponential distributions of parameter respectively $p_i(n)$ $\mu$ and $\nu$. Multiplied by the probability to be accepted at the CS, it gives:

$$ \begin{aligned} \mathbb{P}(F_i) &= \mathbb{P}(\frac{E_{\text{req}}}{p_i(n)} < D) \times (1 - B_i) \\ &= \mu \times \frac{p_i(n) \times (1 - B_i)}{p_i(n) \mu + \nu} \\ &= \frac{1}{\mathbb{E}[E_{\text{req}}]} \times \mathbb{E}[\tilde{E}_i(n)]. \end{aligned} $$

The approximated expected energy received $\mathbb{E}[\tilde{E}_i]$, given by Equation (7), at CS i is the average energy needed for a fully charged battery multiplied by the probability to leave with a fully charged battery. Lemma 2.3 therefore states that the EV user satisfaction is very similar whereas she/he wants to leave with
a fully charged battery with the highest probability possible or with the more energy possible.

Assuming that the EV users are rational, they will join the CS that enables to maximize the expected energy received during its parking duration. However, the expected energy received by an EV at CS i depends on the proportion of EV users choosing CS i. This congestion effect, both in terms of the available power, see Equation (1), and in terms of the limited number of parking spaces at the chosen CS, creates an interaction between EV users. This interaction, based on rational decisions made by the EV users, is analyzed in next Section 3, from the perspective of game theory.

### 3. A queueing game framework

In practice, EV users are self-interested and if they have the choice between several CSs, they will choose the one where they can expect maximum energy. In the most simple case, EV users determine which CS to join without any influence from their private information: their SoC upon arrival or parking duration; and without knowing any information about the charging stations occupation level. Their decision is only based on the expectation of the energy received at each CS. The energy received $E_i$ at CS i depends on the stationary distribution of the number of EVs charging which is not explicitly computable. Therefore, the energy received $E_i$ at CS i is approximated by considering that any new arriving EV observes the system in stationary regime (following the PASTA property of queueing systems [27]) and then, in average, $\bar{z}_i$ charging EVs at CS i. Thus, the expectation of the energy received for any new EV at CS i is approximated by $E[\tilde{E}_i(\bar{z}_i)]$. The quality of this approximation is investigated through simulations in the last section of the paper.

As EV users are interacting through a congestion effect (both for the physical access to parking space and power received), they are considered as players in a non-cooperative game setting. Moreover, this scenario is modeled as a non-observable queueing game [28] in normal form, i.e. a queueing game where the
EV users have no information on the CS status, i.e the number of charging and fully charged EVs at the CS, before making their choice.

## 3.1. Equilibria analysis

A possible very large number of EVs interact and there is no specific number of players, then a non-atomic game [29] setting is suitable to model our non-cooperative system. In such game frameworks, the decision of a single EV user is assumed to be negligible, i.e., it has no impact on the decision of the other EVs. This type of game has been widely used in transportation networks and is related to the concept of Wardrop equilibrium (see [30]). This is also typically the case in queueing game models [8]. Let $\tilde{\Gamma}$ denotes the game defined by the 3-tuple $\tilde{\Gamma} = \{\mathcal{N}, \Delta, \tilde{u}(\cdot, \cdot)\}$, where:

* $\mathcal{N} := [0, 1]$ is the set of players, which are the EV users;

* $\Delta := [0, 1]$ is the set of mixed-strategies, which corresponds to the probability to choose CS 1;

* $\tilde{u} : \tilde{\Delta}^2 \to \mathbb{R}_+$ is the utility (payoff) function of an EV that chooses action CS1 with probability $\theta$ and it is defined by:

$$ \forall (\theta, \bar{\theta}) \in \tilde{\Delta}^2, \tilde{u}(\theta, \bar{\theta}) = \theta \mathbb{E}_{\bar{\theta}}[\tilde{E}_1] + (1 - \theta) \mathbb{E}_{1-\bar{\theta}}[\tilde{E}_2], \quad (11) $$

where $\bar{\theta}$ resp. $1-\bar{\theta}$ denotes the mixture probability used to choose CS 1 (resp. CS 2) by any other player in the population assuming that players are indistinguishable. $\mathbb{E}_x[\tilde{E}_i], x \in [0, 1], i \in \{1, 2\}$, is the approximation of the expected energy received by an EV when EV users choose CS i with probability $x$. Note that based on the assumption of an infinite number of players, the approximation of the expected energy $\mathbb{E}_{\bar{\theta}}[\tilde{E}_i]$ received for a particular EV at CS i depends only on the probability $\bar{\theta}$. In realistic scenarios, EV users will choose the CS according to the exact expected energy received, not the approximated one. However, defining the game $\tilde{\Gamma}$ with the approximation enables to find analytic results. The quality of the approximation of some metrics of the game $\tilde{\Gamma}$, in comparison of the game
Γ for which the EV users choose the CS according to the exact energy received, will be analysed through numerical simulations in Section 4.

The solution concept of such non-cooperative games with indistinguishable infinitely many players are equilibria which correspond to strategy profiles for which there is no interest for any player to unilaterally deviate [30]. An equilibrium is therefore a candidate for the EVs distribution, i.e. an allocation of EVs to the CSs, one can empirically observe. As the EVs being indistinguishable, they are supposed to behave equally, therefore symmetric equilibria are studied, i.e. equilibria where all EVs follow the same strategy. A symmetric mixed-strategy equilibrium is formally a constant function from [0, 1] to [0, 1], where the interval [0, 1] corresponds to the mass of players. To simplify notations, it is assimilated to a probability value $\tilde{\theta}_{eq} \in [0, 1]$.

**Lemma 3.1 ([31]).** *the effective arrival rate $\lambda_i^e$, $i \in \{1, 2\}$, increases with respect to the arrival rate $\lambda_i$.*

The result of Lemma (3.1), proven in [31], states that the higher the number of EV users choose CS i, the more EV user will enter CS i. This result is essential for some properties of the game $\tilde{\Gamma}$ as well as of the game $\Gamma$ for which the EV users choose the CS according to the exact expected energy received.

**Definition 1 (Symmetric mixed-strategy equilibrium).** A symmetric mixed-strategy equilibrium is a profile of identical mixed-strategies $\tilde{\theta}_{eq}$ for which no EV user has an incentive to deviate:

$$ \forall \theta \in [0, 1], \tilde{u}(\theta, \tilde{\theta}_{eq}) \leq \tilde{u}(\tilde{\theta}_{eq}, \tilde{\theta}_{eq}). \qquad (12) $$

In some particular configuration settings, the equilibrium is deterministic: all EV users choose the same CS, i.e. $\tilde{\theta}_{eq} \in \{0, 1\}$. In this case, the chosen CS gives a highest expected energy than the other CS, even if all EV users choose it. Proposition 3.1 states that this situation does not happen if the maximum power each CP can deliver at each CS are equal.
**Proposition 3.1.** If the maximum power the CPs can deliver at each CS is equal, i.e. $p_{\text{max}}^{1} = p_{\text{max}}^{2}$, then the equilibrium $\tilde{\theta}_{\text{eq}} \in ]0, 1[$.

*Proof of Proposition 3.1.* Suppose that $p_{\text{max}}^{1} = p_{\text{max}}^{2}$ and $\tilde{\theta}_{\text{eq}} = 0$. Then, because $x \mapsto \mathbb{E}_{x}[\tilde{E}_{2}]$ strictly decreases, $\tilde{u}(\tilde{\theta}_{\text{eq}}, \tilde{\theta}_{\text{eq}}) = \tilde{u}(0, 0) = \mathbb{E}_{1}[\tilde{E}_{2}] < \mathbb{E}_{0}[\tilde{E}_{2}] = \mathbb{E}_{0}[\tilde{E}_{1}] = \tilde{u}(1, 0)$, which contradicts the definition of an equilibrium, given by 1. The reasoning is the same in the opposite case: $\tilde{\theta}_{\text{eq}} = 1$. Therefore $p_{\text{max}}^{1} = p_{\text{max}}^{2} \Rightarrow \tilde{\theta}_{\text{eq}} \in ]0, 1[$. ■

Note that the first principle of Wardrop equilibrium can also be observed in queueing games [8]. Thus, if the equilibrium is inner, i.e. $\tilde{\theta}_{\text{eq}} \in ]0, 1[$, the EV users' utility is the same for both CS: $\mathbb{E}_{\tilde{\theta}_{\text{eq}}}[\tilde{E}_{1}] = \mathbb{E}_{1-\tilde{\theta}_{\text{eq}}} [\tilde{E}_{2}]$.

The proof of the previous proposition relies on the decreasing property of the expected energy received at CS i with respect to the probability that EV users choose CS i. This property is also true if we consider the exact energy received, because the effective arrival rate $\lambda_{i}^{e}$ strictly increases in $\lambda_{i}$. Therefore, the previous proposition is also true for the game $\Gamma$ where the exact energy received is considered in the EV users' utility.

Based on the previous proposition, one way to determine an equilibrium is to look for the distribution of the EV users decision such that the expected energy received by an EV is the same at each CS. Such computations can be difficult in general. Another approach based on the existence of a potential function is described in the next section.

### 3.1.1. A potential function approach

A variant $\tilde{\Gamma}_{\text{pure}} = \{\mathcal{N}, \tilde{\mathcal{A}}, \tilde{u}(\cdot, \cdot)\}$ of the previous game $\tilde{\Gamma}$ is introduced in this section, where $\mathcal{A} := \{\text{CS 1, CS 2}\}$ is the set of actions. In the game $\tilde{\Gamma}_{\text{pure}}$, each EV user chooses a pure action (CS 1 or CS 2), instead of a mixed strategy in the game $\tilde{\Gamma}$. The utility of each EV user depends on her/his pure action and on the proportion of EV users choosing the same action, i.e. the same CS. As described below, the game $\tilde{\Gamma}_{\text{pure}}$ admits a potential function which enables to prove uniqueness of the equilibrium. Several EVs to CSs allocations, i.e. maps
from the mass [0, 1] of EVs to the set of actions {0, 1}, can lead to the same proportion of EV users choosing the first CS, i.e. $\mathcal{L}(S)$ where $\mathcal{L}$ is the Lebesgue measure and S the set of EV users choosing the first CS. Because the utility of EV users consider only the proportion of EVs choosing each CS, by abuse of language, an equilibrium is a proportion of EV users choosing the first CS. An equilibrium in pure strategy of the game $\tilde{\Gamma}_{\text{pure}}$ as follows.

**Definition 2 (Pure-strategy equilibrium).** A pure strategy equilibrium is a proportion $\theta_{\text{eq}}^{\text{pure}}$ of EVs choosing the first CS such that:

$$ \left\{ \begin{array}{ll} \mathbb{E}_{\theta_{\text{eq}}^{\text{pure}}}[\tilde{\text{E}}_1] \geq \mathbb{E}_{1-\theta_{\text{eq}}^{\text{pure}}}[\tilde{\text{E}}_2] & \text{if } \theta_{\text{eq}}^{\text{pure}} > 0, \\ \mathbb{E}_{\theta_{\text{eq}}^{\text{pure}}}[\tilde{\text{E}}_1] \leq \mathbb{E}_{1-\theta_{\text{eq}}^{\text{pure}}}[\tilde{\text{E}}_2] & \text{if } \theta_{\text{eq}}^{\text{pure}} < 1. \end{array} \right. $$

The game $\tilde{\Gamma}_{\text{pure}}$ is equivalent to the game $\tilde{\Gamma}$ in the sense that the equilibria of both games, based on Definitions 1 and 2, are identical, as stated in Proposition 3.2.

**Proposition 3.2.** *The set of equilibria of the game $\tilde{\Gamma}_{\text{pure}}$ and $\tilde{\Gamma}$ are identical.*

*Proof of Proposition 3.2.*

* Assume $\tilde{\theta}_{\text{eq}} > 0$ and $\mathbb{E}_{\theta_{\text{eq}}^{\tilde{\Gamma}}} [\tilde{\text{E}}_1] < \mathbb{E}_{1-\theta_{\text{eq}}^{\tilde{\Gamma}}} [\tilde{\text{E}}_2]$. Therefore, $\tilde{u}(0, \theta_{\text{eq}}^{\tilde{\Gamma}}) > \tilde{u}(\theta_{\text{eq}}^{\tilde{\Gamma}}, \theta_{\text{eq}}^{\tilde{\Gamma}})$, which contradict the equilibrium definition of the game $\tilde{\Gamma}$. Therefore $\mathbb{E}_{\theta_{\text{eq}}^{\tilde{\Gamma}}} [\tilde{\text{E}}_1] \geq \mathbb{E}_{1-\theta_{\text{eq}}^{\tilde{\Gamma}}} [\tilde{\text{E}}_2]$. With the same reasoning, the second condition of Definition 2 holds. It gives that the set of equilibria of the game $\tilde{\Gamma}$ is included in the one of the game $\tilde{\Gamma}_{\text{pure}}$.

* Now let $\theta_{\text{eq}}^{\text{pure}}$ be an equilibrium of the game $\tilde{\Gamma}_{\text{pure}}$, and let $\theta \in [0, 1]$.

– Assume $\theta_{\text{eq}}^{\text{pure}} \in ]0, 1[$. From the definition 2 of an equilibrium of the game $\tilde{\Gamma}_{\text{pure}}$, $\mathbb{E}_{\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_1] = \mathbb{E}_{1-\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_2]$, which gives

$$ \begin{aligned} \tilde{u}(\theta, \theta_{\text{eq}}^{\text{pure}}) &= \theta \mathbb{E}_{\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_1] + (1 - \theta) \mathbb{E}_{1-\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_2] \\ &= \theta_{\text{eq}}^{\text{pure}} \mathbb{E}_{\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_1] + (1 - \theta_{\text{eq}}^{\text{pure}}) \mathbb{E}_{1-\theta_{\text{eq}}^{\text{pure}}} [\tilde{\text{E}}_2] \\ &= \tilde{u}(\theta_{\text{eq}}^{\text{pure}}, \theta_{\text{eq}}^{\text{pure}}). \end{aligned} $$
In particular, $\tilde{u}(\theta, \theta_{eq}^{\text{pure}}) \leq \tilde{u}(\theta_{eq}^{\text{pure}}, \theta_{eq}^{\text{pure}})$, which implies that $\theta_{eq}^{\text{pure}}$ is an equilibrium of the game $\tilde{\Gamma}$.

- Now assume $\theta_{eq}^{\text{pure}} = 0$. From the definition 2 of an equilibrium of the game $\tilde{\Gamma}_{\text{pure}}$, $E_{\theta_{eq}^{\text{pure}}}[\tilde{E}_1] \leq E_1 - \theta_{eq}^{\text{pure}}[\tilde{E}_2]$. Then,

$$ 
\begin{aligned}
\tilde{u}(\theta, \theta_{eq}^{\text{pure}}) &= \tilde{u}(\theta, 0) = \theta E_0[\tilde{E}_1] + (1 - \theta) E_1[\tilde{E}_2] \\
&\leq \theta E_1[\tilde{E}_2] + (1 - \theta) E_1[\tilde{E}_2] \\
&= E_1[\tilde{E}_2] = \tilde{u}(0, 0) = \tilde{u}(\theta_{eq}^{\text{pure}}, \theta_{eq}^{\text{pure}}),
\end{aligned}
$$

which implies that $\theta_{eq}^{\text{pure}}$ is an equilibrium of the game $\tilde{\Gamma}$.

- The same reasoning as for $\theta_{eq}^{\text{pure}} = 0$ holds if $\theta_{eq}^{\text{pure}} = 1$.

It gives that $\forall \theta \in [0, 1]$, $\tilde{u}(\theta, \theta_{eq}^{\text{pure}}) \leq \tilde{u}(\theta_{eq}^{\text{pure}}, \theta_{eq}^{\text{pure}})$ which implies that $\theta_{eq}^{\text{pure}}$ is by definition an equilibrium of the game $\tilde{\Gamma}$. The set of equilibria of the game $\tilde{\Gamma}_{\text{pure}}$ is included in the one of the game $\tilde{\Gamma}$.

415

420

■

Note that the game $\tilde{\Gamma}_{\text{pure}}$ is a particular type of game, it is a congestion game [32]: the utility of the player depends only on the proportion of players choosing the same pure action (CS in our context). It therefore falls within the class of potential games [33] because the utility function is the same for all EVs. As a consequence, first, a pure equilibrium always exists (through the use of the potential function). Second, a potential function $P_o$ associated to the game $\tilde{\Gamma}_{\text{pure}}$ can be defined as follows:

$$ P_o(\theta) = \int_0^\theta E_x[\tilde{E}_1] dx + \int_0^{1-\theta} E_x[\tilde{E}_2] dx. \quad (13) $$

The potential function enables to find pure equilibria of the game $\tilde{\Gamma}_{\text{pure}}$, and then of the game $\tilde{\Gamma}$ (see Proposition 3.2), by solving an optimization problem that is the maximization of the potential function. Moreover, as the potential function $P_o$ is differentiable, a standard algorithm as the gradient-based descent enables to numerically find the pure equilibria.

425
### 3.1.2. Uniqueness and characterization of equilibria

Due to the complicated expression of the utility function, and in particular of the effective arrival rate $\lambda_i^e$ at each CS$_i$, obtaining an explicit form of the equilibria is not possible. However, some properties of the equilibria can be described, in particular due to the increasing effective arrival rate $\lambda_i^e, i \in \{1, 2\}$, with respect to $\lambda_i$, given by Lemma (3.1).

Next Proposition 3.3 proves the uniqueness of the equilibrium, and is a direct consequence of the monotonicity of $\lambda_i^e$ with respect to $\lambda_i$.

**Proposition 3.3 (Uniqueness).** The non-cooperative game $\tilde{\Gamma}$ has a unique equilibrium denoted $\tilde{\theta}_{\text{eq}}$.

*Proof of Proposition 3.3.* As stated in Proposition 3.2, the set of equilibria of the games $\tilde{\Gamma}$ and $\tilde{\Gamma}_{\text{pure}}$ are equivalent. As the set of maximum points of the potential function $P_o$ is identical to the set of equilibria defined previously by Equation (13) of the game $\tilde{\Gamma}_{\text{pure}}$, as shown in [34], it is also identical to the set of equilibria of the game $\tilde{\Gamma}$. $P_o$ is differentiable and its derivative is $P'_o(\theta) = E_\theta[\tilde{E}_1] - E_{1-\theta}[\tilde{E}_2]$. Because $x \mapsto E_x[\tilde{E}_i], i \in \{1, 2\}$ strictly decreases, it is clear that $P'_o$ strictly decreases, which implies that $P_o$ is strictly concave on $[0, 1]$. $P_o$ has therefore a unique maximum point, which concludes the proof. ■

Note that, as in Proposition 3.1, this result is based on the decreasing approximated expected energy received with respect to $\lambda_i$, which is also true for the exact expected energy received. The uniqueness of the equilibrium is also true for the game $\Gamma$ for which the EV users' utility is based on the exact expected energy received.

In many systems in which agents interact, empirical distributions of agents to their possible actions are observed. Uniqueness of the equilibrium enables to predict the empirical configuration of EVs to CSs in a decentralized system, i.e. when the EVs are the decision-maker and individually choose their best strategy.

As seen in Section 3.1.1, the unique equilibrium can be found numerically using standard gradient algorithms. In some special cases, studied below in
details, it is even possible to obtain an explicit formula for the equilibrium. Let us consider the following equation in $\theta_i \in [0, 1]$ for each CS i:

$$ \theta_i \lambda \left( 1 - \frac{\frac{(\theta_i \frac{\lambda}{\nu})^{K_i}}{K_i!}}{\sum_{j=0}^{K_i} \frac{(\theta_i \frac{\lambda}{\nu})^j}{j!}} \right) = (\nu + \mathbf{p}_{\max}^i \mu) \alpha_i K_i. \quad (14) $$

The left term of Equation (14) is exactly the effective arrival rate $\lambda_i^e$ at CS i. The strict monotony of $\lambda_i^e$ with respect to $\lambda_i := \theta_i \lambda$, and therefore with respect to $\theta_i$ also, gives the uniqueness of the solution to Equation (14).

For each CS i $\in \{1, 2\}$, a parameter $\tilde{\theta}_i^{\text{thr}}$ is defined as follows:

$$ \tilde{\theta}_i^{\text{thr}} = \begin{cases} \text{unique solution of Equation (14)} & \text{if } \lambda \left( 1 - \frac{\frac{(\theta_i \frac{\lambda}{\mu})^{K_i}}{K_i!}}{\sum_{j=0}^{K_i} \frac{(\theta_i \frac{\lambda}{\mu})^j}{j!}} \right) > (\nu + \mathbf{p}_{\max}^i \mu) \alpha_i K_i \\ +\infty & \text{otherwise.} \end{cases} \quad (15) $$

$\tilde{\theta}_i^{\text{thr}}$ is an important metric that characterizes the capacity of CS i to offer the maximum instantaneous power $\mathbf{p}_{\max}^i$ to the EVs. When $\tilde{\theta}_i^{\text{thr}} < +\infty$, i.e. there is a solution to Equation (14), the latter $\tilde{\theta}_i^{\text{thr}}$ is a threshold such that if the proportion of EVs is above this threshold, EVs charge at a instantaneous power strictly lower than $\mathbf{p}_{\max}^i$. Observe that this threshold $\tilde{\theta}_i^{\text{thr}}$ increases with $\alpha_i K_i$ (recall that $\lfloor \alpha_i K_i \rfloor$ is the number of parking spaces that can offer simultaneously the maximum power rate $\mathbf{p}_{\max}^i$). Moreover, if the power sizing $\alpha_i$ is sufficiently high ($\alpha_i > \frac{\nu}{\nu+\mathbf{p}_{\max}^i \mu}$), $\tilde{\theta}_i^{\text{thr}} = +\infty$: EVs charge at maximum rate $\mathbf{p}_{\max}^i$ at CS i no matter the probability $\theta_i$ with which EV users choose CS i.

The following Proposition 3.4 and Proposition 3.5 below are particular cases of parameters such that the equilibrium can be determined explicitly. Proposition 3.4 deals with the case when the capacity of both CSs to offer the maximum power $\mathbf{p}_{\max}^i$, i $\in \{1, 2\}$, parameterized by $\tilde{\theta}_i^{\text{thr}}$, i $\in \{1, 2\}$, is low.

**Proposition 3.4 (Low resource in power).** Assume that the following conditions hold:

$$ \tilde{\theta}_1^{\text{thr}} < \frac{\mathbf{p}_{\max}^1 \alpha_1 K_1}{\mathbf{p}_{\max}^1 \alpha_1 K_1 + \mathbf{p}_{\max}^2 \alpha_2 K_2} \quad \text{and} \quad \tilde{\theta}_2^{\text{thr}} < \frac{\mathbf{p}_{\max}^2 \alpha_2 K_2}{\mathbf{p}_{\max}^1 \alpha_1 K_1 + \mathbf{p}_{\max}^2 \alpha_2 K_2}. \quad (16) $$
Then the equilibrium $\tilde{\theta}_{eq}$ is given by:

$$ \tilde{\theta}_{eq} = \frac{p_{\max}^{1} \alpha_{1} K_{1}}{p_{\max}^{1} \alpha_{1} K_{1} + p_{\max}^{2} \alpha_{2} K_{2}}. $$

Moreover, for each CS i, $p_i(\tilde{\theta}_{eq}) < p_{\max}^i$ and the utility $\tilde{U}_{eq}$ at equilibrium for each EV is:

$$ \tilde{U}_{eq} = \frac{p_{\max}^{1} \alpha_{1} K_{1} + p_{\max}^{2} \alpha_{2} K_{2}}{\lambda}. $$

*Proof of Proposition 3.4.* Note that *Conditions (16)* are equivalent to the following system:

$$ \begin{cases} \mathbb{E}_{1-\tilde{\theta}_2^{\text{thr}}}(\tilde{E}_1) \le \mathbb{E}_{\tilde{\theta}_2^{\text{thr}}}(\tilde{E}_2), \\ \mathbb{E}_{1-\tilde{\theta}_1^{\text{thr}}}(\tilde{E}_2) \le \mathbb{E}_{\tilde{\theta}_1^{\text{thr}}}(\tilde{E}_1). \end{cases} $$

Therefore, $\mathbb{E}_{\tilde{\theta}_{eq}}(\tilde{E}_1) = \frac{p_{\max}^1 \alpha_1 K_1}{\tilde{\theta}_{eq} \lambda}$ and $\mathbb{E}_{1-\tilde{\theta}_{eq}}(\tilde{E}_2) = \frac{p_{\max}^2 \alpha_2 K_2}{(1-\tilde{\theta}_{eq}) \lambda}$ and $\tilde{\theta}_{eq}$ verifies the following equation:

$$ \begin{align*} E_{\tilde{\theta}_{eq}}(\tilde{E}_1) = E_{1-\tilde{\theta}_{eq}}(\tilde{E}_2) &\Leftrightarrow \frac{p_{\max}^1 \alpha_1 K_1}{\tilde{\theta}_{eq} \lambda} = \frac{p_{\max}^2 \alpha_2 K_2}{(1-\tilde{\theta}_{eq}) \lambda} \\ &\Leftrightarrow \tilde{\theta}_{eq} = \frac{p_{\max}^1 \alpha_1 K_1}{p_{\max}^1 \alpha_1 K_1 + p_{\max}^2 \alpha_2 K_2}. \end{align*} $$

The value of the utility function $\tilde{U}_{eq}$ at equilibrium follows by applying Equation (11):

$$ \begin{align*} \tilde{U}_{eq} &= \tilde{u}(\tilde{\theta}_{eq}, \tilde{\theta}_{eq}) = \tilde{\theta}_{eq} \mathbb{E}_{\tilde{\theta}_{eq}}[\tilde{E}_1] + (1 - \tilde{\theta}_{eq}) \mathbb{E}_{1-\tilde{\theta}_{eq}}[\tilde{E}_2] \\ &= \tilde{\theta}_{eq} \frac{p_{\max}^1 \alpha_1 K_1}{\tilde{\theta}_{eq} \lambda} + (1 - \tilde{\theta}_{eq}) \frac{p_{\max}^2 \alpha_2 K_2}{(1 - \tilde{\theta}_{eq}) \lambda} \\ &= \frac{p_{\max}^1 \alpha_1 K_1 + p_{\max}^2 \alpha_2 K_2}{\lambda}. \end{align*} $$

470

■

Note that when conditions (16) hold, $\tilde{\theta}_{eq} \in ]0, 1[$, which means that at equilibrium both CSs are chosen by EV users. Therefore, as stated in Proposition 3.1, at equilibrium EV users expect the same approximated energy at both CSs: $\mathbb{E}_{\tilde{\theta}_{eq}}[\tilde{E}_1] = \mathbb{E}_{1-\tilde{\theta}_{eq}}[\tilde{E}_2]$. At equilibrium, the probability with which EV
475 users choose CS i is exactly the proportion of available power $p_{\max}^i \alpha_i K_i$ at CS i in relation to the total available power $p_{\max}^1 \alpha_1 K_1 + p_{\max}^2 \alpha_2 K_2$ at both CSs. Moreover, the utility of EVs users at equilibrium is exactly the ratio between the total available power $p_{\max}^1 \alpha_1 K_1 + p_{\max}^2 \alpha_2 K_2$ and the frequency of arrivals $\lambda$ at both CSs.

480 In the exact opposite case when the capacity of both CSs to offer the maximum power $p_{\max}^i, i \in \{1, 2\}$, parameterized by $\tilde{\theta}_1^{\text{thr}}, i \in \{1, 2\}$, is high, it is also possible to precisely characterize the equilibrium, as stated in the following Proposition 3.5.

**Proposition 3.5 (High resource in power).** Assume that the following condition holds:

$$ \min(\tilde{\theta}_1^{\text{thr}}, \tilde{\theta}_2^{\text{thr}}) = +\infty. \quad (17) $$

Then,

(i) if

$$ \left\{ \begin{array}{l} \frac{\rho_{K_1}^{K_1}}{\sum_{j=0}^{K_1-1} \rho_j^j} \ge 1 - \beta, \\ \\ \frac{\rho_{K_2}^{K_2}}{\sum_{j=0}^{K_2-1} \rho_j^j} \ge 1 - \frac{1}{\beta}, \end{array} \right. \quad (18) $$

with $\beta = \frac{p_{\max}^2(\nu+p_{\max}^1\mu)}{p_{\max}^1(\nu+p_{\max}^2\mu)}$, and $\tilde{\theta}_{eq}$ is the unique solution of:

$$ \beta \sum_{i=0}^{K_1-1} \sum_{j=0}^{K_2} \frac{(1-\theta)^i \theta^j \rho^{i+j}}{i!j!} = \sum_{i=0}^{K_2-1} \sum_{j=0}^{K_1} \frac{\theta^i (1-\theta)^j \rho^{i+j}}{i!j!}. \quad (19) $$

485 Moreover, $\mathbb{E}_{\tilde{\theta}_{eq}}[\tilde{E}_1] = \mathbb{E}_{1-\tilde{\theta}_{eq}}[\tilde{E}_2]$.

(ii) otherwise, $\mathbb{E}_{\tilde{\theta}_{eq}}(\tilde{E}_1) \neq \mathbb{E}_{1-\tilde{\theta}_{eq}}(E_2)$,

$$ \tilde{\theta}_{eq} = \begin{cases} 0 & \text{if } p_{\max}^1 < p_{\max}^2, \\ 1 & \text{if } p_{\max}^1 > p_{\max}^2, \end{cases} $$

and

$$ \tilde{U}_{eq} = \begin{cases} \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu} \times \left(1 - \frac{\rho_{K_2}^{K_2}}{\sum_{j=0}^{K_2-1} \rho_j^j}\right) & \text{if } p_{\max}^1 < p_{\max}^2, \\ \frac{p_{\max}^1}{\nu + p_{\max}^1 \mu} \times \left(1 - \frac{\rho_{K_1}^{K_1}}{\sum_{j=0}^{K_1-1} \rho_j^j}\right) & \text{if } p_{\max}^1 > p_{\max}^2. \end{cases} $$
Proof of Proposition 3.5.

(i) Recall that, by Equation (7),

$$ \forall i \in \{1, 2\}, \forall \theta_i \in [0, 1], \mathbb{E}_{\theta_i}[\tilde{E}_i(\bar{z}_i)] = \frac{p_{\max}^i \min(\bar{z}_i, \alpha_i K_i) \times (1-B_i)}{p_{\max}^i \min(\bar{z}_i, \alpha_i K_i) \mu + n \nu}, \text{ with} $$

$$ B_i := \frac{\frac{\lambda_i^c}{K_i!}}{\sum_{j=0}^{K_i} \frac{\rho_j^i}{j!}} \quad \text{the blocking probability, and that, by Equation (4),} $$

$$ \forall i \in \{1, 2\}, \bar{z}_i = \begin{cases} \frac{\lambda_i^c}{\nu + p_{\max}^i \mu} & \text{if } \frac{\lambda_i^c}{\nu + p_{\max}^i \mu} < \alpha_i K_i, \\ \frac{\lambda_i^c - p_{\max}^i \mu \alpha_i K_i}{\nu} & \text{else,} \end{cases} $$

with $\lambda_i^c = \lambda(1 - B_i)$ the effective arrival rate. As stated in the formula of the threshold $\hat{\theta}_i$ given by Equation (15), $\min(\tilde{\theta}_1^{\text{thr}}, \tilde{\theta}_2^{\text{thr}}) = +\infty$ implies that $\forall i \in \{1, 2\}, \forall \theta_i \in [0, 1], \frac{\lambda_i^c}{\nu + p_{\max}^i \mu} < \alpha_i K_i$, and therefore, $\bar{z}_i < \alpha_i K_i$. It means that the approximation of the expected energy received at CS i is:

$$ \begin{aligned} \forall i \in \{1, 2\}, \forall \theta_i \in [0, 1], \mathbb{E}_{\theta_i}[\tilde{E}_i] &= \frac{p_{\max}^i \bar{z}_i \times (1 - B_i)}{p_{\max}^i \bar{z}_i \mu + \bar{z}_i \nu}, \\ &= \frac{p_{\max}^i}{\nu + p_{\max}^i \mu} \times \left(1 - \frac{\left(\frac{\theta_i \lambda}{\nu}\right)^{K_i}}{K_i!}\right). \end{aligned} $$

Moreover, the equilibrium $\tilde{\theta}_{eq}$ is the solution of

$$ \mathbb{E}_{\theta}[\tilde{E}_1] = \mathbb{E}_{1-\theta}[\tilde{E}_2], \qquad (20) $$

if it exists. The condition for which there is a solution to (20) is:

$$ \begin{cases} \mathbb{E}_1[\tilde{E}_1] \le \mathbb{E}_0[\tilde{E}_2], \\ \mathbb{E}_0[\tilde{E}_1] \ge \mathbb{E}_1[\tilde{E}_2], \end{cases} $$

which is equivalent to (18):

$$ \begin{cases} \mathbb{E}_1[\tilde{E}_1] \le \mathbb{E}_0[\tilde{E}_2], \\ \mathbb{E}_0[\tilde{E}_1] \ge \mathbb{E}_1[\tilde{E}_2], \end{cases} \quad \begin{cases} \frac{p_{\max}^1}{\nu + p_{\max}^1 \mu} \times \left(1 - \frac{\rho_{K_1}^{K_1}}{\sum_{j=0}^{K_1} \rho_j^1}\right) \le \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu}, \\ \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu} \times \left(1 - \frac{\rho_{K_1}^{K_1}}{\sum_{j=0}^{K_1} \rho_j^1}\right) \le \frac{p_{\max}^1}{\nu + p_{\max}^1 \mu}, \end{cases} \\ \begin{cases} \frac{\rho_{K_1}^{K_1}}{\sum_{j=0}^{K_1} \rho_j^1} \ge 1 - \frac{p_{\max}^2 (\nu + p_{\max}^1 \mu)}{p_{\max}^1 (\nu + p_{\max}^2 \mu)}, \\ \frac{\rho_{K_2}^{K_2}}{\sum_{j=0}^{K_2} \rho_j^1} \ge 1 - \frac{p_{\max}^1 (\nu + p_{\max}^2 \mu)}{p_{\max}^2 (\nu + p_{\max}^1 \mu)}. \end{cases} $$
Equation (20), i.e. $E_\theta[\tilde{E}_1] = E_{1-\theta}[\tilde{E}_2]$, equivalent to Equation (19):

$$
\begin{align*}
\mathbb{E}_{\theta}[\tilde{E}_1] &= \mathbb{E}_{1-\theta}[\tilde{E}_2] \\
\Leftrightarrow \frac{p_{\max}^1}{\nu + p_{\max}^1 \mu} \times \left(1 - \frac{\frac{(\theta_1 \lambda)^{K_1}}{\nu}}{\sum_{j=0}^{K_1} \frac{(\theta_1 \lambda)^j}{j!}}\right) &= \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu} \times \left(1 - \frac{\frac{(\theta_2 \lambda)^{K_2}}{\nu}}{\sum_{j=0}^{K_2} \frac{(\theta_2 \lambda)^j}{j!}}\right) \\
\Leftrightarrow \frac{p_{\max}^2(\nu + p_{\max}^1 \mu)}{p_{\max}^1(\nu + p_{\max}^2 \mu)} \times \frac{\sum_{j=0}^{K_1-1} \frac{(\theta_j \lambda)^j}{j!}}{\sum_{j=0}^{K_1} \frac{(\theta_j \lambda)^j}{j!}} &= \frac{\sum_{j=0}^{K_2-1} \frac{(1-\theta_j \lambda)^j}{j!}}{\sum_{j=0}^{K_2} \frac{(1-\theta_j \lambda)^j}{j!}} \\
\Leftrightarrow \beta \times \sum_{j=0}^{K_1-1} \frac{(\theta_j \lambda)^j}{j!} \times \sum_{i=0}^{K_2} \frac{(1-\theta_j \lambda)^i}{i!} &= \sum_{i=0}^{K_2-1} \frac{(1-\theta_j \lambda)^i}{i!} \times \sum_{j=0}^{K_1} \frac{(\theta_j \lambda)^j}{j!}.
\end{align*}
$$

490

This concludes the first part of Proposition 3.5.

(ii) If there is no solution to (20), $\tilde{\theta}_{eq} \in \{0, 1\}$ and $p_{\max}^1 \neq p_{\max}^2$. Suppose $\tilde{\theta}_{eq} = 0$. By definition of an equilibrium,

$$
\begin{align*}
\forall \theta \in [0, 1], \tilde{u}(\theta, 0) &= \theta \mathbb{E}_0[\tilde{E}_1] + (1 - \theta) \mathbb{E}_1[\tilde{E}_2] \\
&\leq \mathbb{E}_1[\tilde{E}_2] = \tilde{u}(0, 0).
\end{align*}
$$

Therefore $\mathbb{E}_0[\tilde{E}_1] \le \mathbb{E}_1[\tilde{E}_2]$. In other words, if all EV users choose CS 2 at equilibrium, the approximation of the expected energy received at CS 2 is greater than the expected energy received at CS 1 at equilibrium. Moreover, as there is no solution to (20), $\mathbb{E}_0[\tilde{E}_1] \ne \mathbb{E}_1[\tilde{E}_2]$ and therefore $\mathbb{E}_0[\tilde{E}_1] < \mathbb{E}_1[\tilde{E}_2]$. Recall that $\forall i \in \{1, 2\}, x \mapsto \mathbb{E}_x[\tilde{E}_i]$ decreases, which implies that $\mathbb{E}_0[\tilde{E}_1] < \mathbb{E}_0[\tilde{E}_2]$. Moreover,

$$
\mathbb{E}_0[\tilde{E}_1] < \mathbb{E}_0[\tilde{E}_2] \Leftrightarrow \frac{p_{\max}^1}{\nu + p_{\max}^1 \mu} < \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu} \Leftrightarrow p_{\max}^1 < p_{\max}^2,
$$

and

$$
\tilde{U}_{eq} = \tilde{u}(\tilde{\theta}_{eq}, \tilde{\theta}_{eq}) = \tilde{u}(0,0) = \mathbb{E}_1[\tilde{E}_2] = \frac{p_{\max}^2}{\nu + p_{\max}^2 \mu} \times \left(1 - \frac{\frac{\rho^{K_2}}{\nu}}{\sum_{j=0}^{K_2} \frac{\rho^j}{j!}}\right).
$$

The same reasoning for $\tilde{\theta}_{eq} = 1$ concludes the second part of Proposition 3.5.

■
In the case of Proposition 3.5, there is enough resource at each CS such that EVs charge at power $p_i = p_{max}^i$ for all EVs to CSs distribution.

When $min(\tilde{\theta}_1^{thr}, \tilde{\theta}_2^{thr}) = +\infty$, for all EVs to CSs distribution. In particular, at equilibrium $\tilde{\theta}_{eq}$, in the approximation, charging EVs benefit with the maximal power at the chosen CS i: $p_i(\bar{z}_i) = p_{max}^i$. In the first case of the previous Proposition 3.5, the two CSs have similar characteristics in terms of the number of parking spaces $Np_i$, or in terms of the maximum instantaneous power $p_{max}^i$ each CP can deliver to EVs, at CS i. The approximation of the expected energy at both CSs is then equal. The second case of Proposition 3.5 corresponds to the opposite case, where the two CSs have very different characteristics. In this case, at equilibrium all EVs choose the same CS because its capacity to offer a high energy is much greater than the other CS.

This section focuses on a decentralized system where EVs individually choose their best strategy (the CS to join) without taking into account the externalities of their actions. For instance, before choosing the CS, an arriving EV user at the site does not consider that she/he will deprive the next arriving EVs of a parking space, or lower the current charging power of the other charging EVs at the chosen CS. This system is not always optimal regarding to the social welfare. A study of the comparison between the decentralized system and the centralized system is shown in next section by studying the Price of Anarchy (PoA) of our system.

## 3.2. Price of Anarchy

In a centralized system where the Charging Point Operator (CPO) has no information on the EVs (their SoC at arrival or parking duration) or the CSs occupancy but can decide for each EV user where she/he will park and recharge her/his EV battery, the CPO may aim at optimizing the satisfaction of an EV users, i.e. their expected energy received. In this case, the optimal utility $\tilde{U}_{opt}$ of an EV user is defined as:

$$ \tilde{U}_{opt} = \max_{\theta \in [0,1]} \tilde{u}(\theta, \theta). \qquad (21) $$
Similarly to the equilibrium, it is difficult to obtain an explicit formula for the optimal utility $\tilde{U}_{opt}$ except in some specific cases. The Price of Anarchy PoA [35] is a well-known metric in game theory that quantifies the inefficiency of a decentralized system in comparison of its centralized counterpart. It is defined as the ratio between the optimal utility $\tilde{U}_{opt}$ and the utility at unique equilibrium $\tilde{U}_{eq}$:

$$ \tilde{P_oA} = \frac{\tilde{U}_{opt}}{\tilde{U}_{eq}}. \qquad (22) $$

Note that the utility at equilibrium is strictly positive so that $\tilde{P_oA}$ is well defined. In general, when there exists possibly several equilibria, the lowest utility among the ones at equilibria is taken in Definition (22) of $\tilde{P_oA}$. Note that $\tilde{P_oA}$ is always greater than 1 because a centralized scheduler can reproduce the equilibrium, i.e. the EVs to CSs distribution in decentralized system. The PoA provides a metric for the quality of the equilibrium. The closer $\tilde{P_oA}$ is to 1, the better the decentralized system is because the corresponding utility is close to the optimal one. The next Proposition 3.6 gives a sufficient condition such that the decentralized system is as efficient as the centralized system, i.e. $\tilde{P_oA} = 1$, as well as upper bounds on the price of anarchy.

**Proposition 3.6.** The PoA verifies the following properties:

(i) Suppose Conditions (16) hold, then: $\tilde{P_oA} = 1$.

(ii) Suppose Conditions (17) hold, then:

$$ \tilde{P_oA} \leq \frac{\min(1, \frac{K_1+K_2}{\rho})}{1 - \frac{\rho^{K_m}}{\sum_{j=0}^{K_m} \frac{\rho^j}{j!}}}, $$

with $m = \argmin(K_1, K_2)$.

**Proof of Proposition 3.6.**

(i) Note that if Conditions (16) hold, an immediate functional analysis shows that utility function $\theta \mapsto \tilde{u}(\theta, \theta)$ strictly increases for $\theta$ in $[0, \tilde{\theta}_1^{thr}[$, is constant for $\theta$ in $[\tilde{\theta}_1^{thr}, 1 - \tilde{\theta}_2^{thr}]$ and then strictly decreases for $\theta$ in $]1 - \tilde{\theta}_2^{thr}, 1]$.
Therefore, $\forall \theta \in [\tilde{\theta}_1^{thr}, 1 - \tilde{\theta}_2^{thr}]$, $\tilde{U}_{\text{opt}} = \tilde{u}(\theta, \theta) = \frac{(p_{\text{max}}^1 \alpha_1 K_1 + p_{\text{max}}^2 \alpha_2 K_2)}{\lambda}$. Using Proposition 3.4 which states that the utility at equilibrium is $\tilde{U}_{\text{eq}} = \frac{(p_{\text{max}}^1 \alpha_1 K_1 + p_{\text{max}}^2 \alpha_2 K_2)}{\lambda}$, it holds that $\tilde{U}_{\text{opt}} = \tilde{U}_{\text{eq}}$, which gives $\tilde{P}_{\text{eq}}A = 1$.

(ii) Firstly,

$$ 
\begin{aligned}
\tilde{U}_{\text{eq}} &\geq \max_{i \in \{1,2\}} \mathbb{E}_1[\tilde{E}_i] \\
&= \max_{i \in \{1,2\}} \left( \frac{p_{\text{max}}^i}{\nu + p_{\text{max}}^i \mu} \right) \times \left( 1 - \frac{\rho^{K_i}}{\sum_{j=0}^{K_i} \rho_j^i} \right) \quad (23) \\
&\geq \max_{i \in \{1,2\}} \left( \frac{p_{\text{max}}^i}{\nu + p_{\text{max}}^i \mu} \right) \times \min_{j \in \{1,2\}} \left( 1 - \frac{\rho^{K_j}}{\sum_{k=0}^{K_j} \rho_k^j} \right).
\end{aligned}
$$

In order to obtain the first inequality, the following fact was used: the utility at equilibrium is greater than the minimal utility for both CSs, and the latter is reached when all EVs go to the same CS.

Secondly,

$$ 
\begin{aligned}
\tilde{U}_{\text{opt}} &= \max_{\theta \in [0,1]} \theta \frac{p_{\text{max}}^1 \mu}{\nu + p_{\text{max}}^1 \mu} \left(1 - \frac{\lambda_i^e(\theta)}{\lambda_1}\right) + \left(1 - \theta\right) \frac{p_{\text{max}}^2 \mu}{\nu + p_{\text{max}}^2 \mu} \left(1 - \frac{\lambda_2^e(1-\theta)}{\lambda_2}\right) \\
&\leq \max_{\theta \in [0,1]} \max_{i \in \{1,2\}} \frac{p_{\text{max}}^i \mu}{\nu + p_{\text{max}}^i \mu} \left(1 - \frac{\lambda_i^e(\theta_i)}{\lambda_i}\right) \\
&= \max_{i \in \{1,2\}} \max_{\theta \in [0,1]} \frac{p_{\text{max}}^i \mu}{\nu + p_{\text{max}}^i \mu} \left(1 - \frac{\lambda_i^e(\theta_i)}{\lambda_i}\right) \\
&\leq \max_{i \in \{1,2\}} \frac{p_{\text{max}}^i \mu}{\nu + p_{\text{max}}^i \mu}.
\end{aligned}
$$

using the hypothesis that $\lambda_e$ increases with $\lambda$, which implies that $\lambda_e \leq \nu K$ (to obtain the first inequality). Therefore,

$$ \tilde{U}_{\text{opt}} \leq \min(1, \frac{K_1 + K_2}{\rho}) \times \max_{i \in \{1,2\}} \left( \frac{p_{\text{max}}^i \mu}{(1 - \epsilon)\nu + p_{\text{max}}^i \mu} \right). \quad (24) $$

(23) with (24) gives:

$$ 
\begin{aligned}
\tilde{P}_{\text{eq}}A &\leq \frac{\min(1, \frac{K_1 + K_2}{\rho}) \times \max_{i \in \{1,2\}} \left( \frac{p_{\text{max}}^i \mu}{(1 - \epsilon)\nu + p_{\text{max}}^i \mu} \right)}{\max_{i \in \{1,2\}} \left( \frac{p_{\text{max}}^i \mu}{(1 - \epsilon)\nu + p_{\text{max}}^i \mu} \right) \times \min_{j \in \{1,2\}} (1 - \mathbb{P}_1(\mathbb{R}_j))} \\
&= \frac{\min(1, \frac{K_1 + K_2}{\rho})}{\min_{j \in \{1,2\}} (1 - \mathbb{P}_1(\mathbb{R}_m))} \\
&= \frac{\min(1, \frac{K_1 + K_2}{\rho})}{1 - \mathbb{P}_1(\mathbb{R}_m)},
\end{aligned}
$$
where $m = \arg\min(K_1, K_2)$.

■

Previous Proposition 3.6 shows that the decentralized system is optimal in terms of social welfare when either the capacity for both CSs to deliver the corresponding maximum instantaneous power $p_{\max}^i$ is poor (part (i) of Proposition 3.6). In the opposite case (part (ii) of Proposition 3.6), EV users tend to overuse the CS that offer the high maximum instantaneous power $p_{\max}^i$, which creates a high blocking probability at this CS. However, when the number of parking spaces at both CSs is high, i.e. the blocking probabilities are low, this overuse does not have a great impact on the utility of EV users. In general, $\tilde{P}_A$ is relatively low, which indicates that in the system studied in this paper, the decentralized system is efficient. This tendency is precised in the next section 4 on a use-case.

Theoretical results on the equilibrium analysis and its comparison with the optimal configuration in a centralized system have been studied through the $\tilde{P}_A$. In order to compare them with an empirical analysis, statistical results from a simulation of the model are shown in Section 4.

## 4. Numerical simulations

The proposed model was simulated using the `Simpy` package, which is a Python discrete-event simulation framework<fnref n="3" />. Inter-arrival duration, parking duration and amount of energy needed for a fully charged battery are randomly generated following exponential distributions as assumed in the model in Section 2. Each EV user chooses a CS to join upon arrival at the given site. If the chosen CS is fully occupied, the EV user leaves without charging. Otherwise, the EV user joins the CS and immediately starts to charge according to the power allocation rule given by Equation 1. After each event at a CS (arrival,

<footnote n="3">³See [36] for the documentation.</footnote>
end of charge or departure of an EV), the current available power, as well as the number of available parking spaces are updated.

Table 2 gives the main parameter values used for the simulations. In [37], the battery storage capacities among the most popular EVs is given, with an average of about 50kWh. In [38], the most frequent charging levels upon arrival is between 50% and 60%, against between 40% and 50% in [39]. Therefore, an initial SoC of 50% is assumed in this scenario, which leads to an average amount of energy needed for a fully charged battery of 25 kWh. The maximum power each CP can deliver to an EV at each CS (11 kW and 22 kW) is in line with the range of typical charging level for EV parking at supermarkets or shopping centers [40]. Note that there is an asymmetry between the two CSs: one has a small number of CPs (equivalently parking spaces) but with a highest power capacity.

<table><caption>Table 2: Parameter values</caption><tr><td>λ = 6</td><td>EVs arrive in average every 10 minutes.</td></tr><tr><td>μ = 1/25</td><td>EVs need in average 25 kWh to have a fully charged battery.</td></tr><tr><td>K<sub>1</sub> = 10<br>K<sub>2</sub> = 4</td><td>Number of parking spaces at each CS.</td></tr><tr><td>p<sup>1</sup><sub>max</sub> = 11<br>p<sup>2</sup><sub>max</sub> = 22</td><td>Maximum power at each charging point (in kW).</td></tr></table>

## 4.1. Performance evaluation for a single charging station

### 4.1.1. Global statistics

The power sizing $\alpha_i$, which depends mainly on the charging infrastructure but also on electrical contracts, is equal to 0.6 at each CS, i.e. 60% of the number of CPs can deliver simultaneously the maximum instantaneous power $p_i^{\text{max}}$, which is a typical power sizing decision for standard charging modes (< 43 kW).

In this section, it is considered that all EVs choose CS 1. Figures 3, 4 and 5 show the histograms of respectively the energy received per EVs among all
<figure><img src="image_6.png" /><figcaption>Figure 3: Histogram of the energy received per EV. *The high proportion (about 30%) of rejected EVs from the CS indicates that the number of parking spaces is not enough compared to the utilization rate, i.e. the frequency of arrivals times the parking duration.*</figcaption></figure>

EVs, the energy received per EV among the accepted EVs, and the average power drawn per EV. The parameters of CS 1 and the EV characteristics are as shown in Table 2. In addition, the average parking duration is 2 hours, which is based on the results from [24]. This value is in accordance with the context of an average parking duration at a shopping center. The power sizing $\alpha_i$, which depends mainly on the charging infrastructure but also on electrical contracts, is equal to 0.6, i.e. 6 CPs can simultaneously deliver the maximum instantaneous power $p_{max}^i$, which is a typical power sizing decision for standard charging modes (< 43 kW). As the previous theoretical results obtained correspond to long-run metrics, about 60000 EVs are simulated. Some EVs stay parked for a small amount of time or lack only little energy to have a fully charged battery, which explain that some accepted EVs receive little energy during their parking
<figure><img src="image_7.png" /><figcaption>Figure 4: Histogram of the energy received per EV among accepted EVs. *The distribution of the energy received among the accepted EVs is very similar to the exponential distribution (the curve in red), which validates the approximation given in the proof of Proposition 2.1.*</figcaption></figure>

duration, as seen in Figures 3 and 4. For example, about 45% of accepted EVs stopped charging because their battery is fully charged. However, even among the EV users that found an available parking space, the average energy received per EV is about 11.5 kWh which is around half of the average energy needed for a fully charged battery which is 25 kWh. This shows that also a high proportion of EVs only received a small proportion of the energy needed for a fully charged battery. Figure 4 shows that among the EV users that found an available parking space, the distribution of the energy received is similar to the exponential distribution, which is our approximation, given in Section 2. Indeed, the average energy received among the accepted EVs (about 11.39 kWh) is very close to the corresponding standard deviation (about 11.46 kWh), which is one of the main characteristics of the exponential distribution. When
<figure><img src="image_8.png" /><figcaption>Figure 5: Histogram of the average power received per EV. The CS is well designed in terms of available power at the CS level, as a majority of EVs received the maximum power of 11 kW or almost.</figcaption></figure>

an EV is rejected, its average power drawn is considered to be 0 kW. Figure 5 therefore highlights the high proportion of EV users rejected from the CS, which is about 30.2% against 30.0% theoretically. This is inline with the fact that the blocking probability is exact considering the stationary regime of the original Markov chain ($Q_t^i, Z_t^i$). The average power received by an EV is about 10.5 kW, which is close to the maximum power level of 11 kW. This illustrates that, in our use-case, most of the time the number of charging EVs is sufficiently low such that all EVs get the maximum power that can be delivered by the CPs.

### 4.1.2. Approximation of energy received per EV: power sizing effect

In the following, each numerical value is an average taken over 50 simulations of 1000 (virtual) hours each, corresponding to about 6000 EV arrivals for each simulation. The average energy and power received by the EVs through
<figure><img src="image_9.png" /><figcaption>Figure 6: Energy received per EV in average with respect to the proportion $\alpha_1$ of the number of parking spaces that can charge simultaneously with maximum power $p_{max}^1 = 11$ kW. Setting $\alpha_1 \approx 0.7$ is sufficient to maximize the expected energy received by the EVs they can charge with the maximum power of 11 kW.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 7: Average power received per EV in average with respect to the proportion of the number of parking spaces that can charge simultaneously with maximum power $p_{max}^1 = 11$ kW. Seeing Figure 6, there is a nonlinear relation between the average energy received and the average power received.</figcaption></figure>

simulation and the corresponding approximated theoretical values determined in Section 2 are illustrated next in Figures 6 and 7. Lower and upper bounds
are also illustrated. These metrics are shown with respect to the power sizing coefficient $\alpha_1$, and when the average parking duration also equals to 2 hours. $\alpha_1$ determines the available instantaneous power at the CS, e.g. there is no power constraint at the level of the CS when $\alpha_i = 1$. Setting this parameter correctly enables a CPO to know the optimal trade-off between the cost of infrastructure and electrical contract and the QoCS for the EV users.

The expected energy received per EV is closely related to the probability for an EV to leave with a fully charged battery (see Lemma 2.3). The latter is plotted in [7] and has a very similar shape as Figure 6. It validates the accuracy of the energy received per EV from the simulations. The average power received per EV is illustrated on Figure 7. The theoretical values of the average power are obtained by using the allocation power formula 1 considering the average number $\bar{z}_1$ of charging EVs given by Equation (4). For $\alpha_1$ sufficiently high (about 0.7), there is enough power at the CS such that the EVs that found an available parking space draw the maximum power of 11 kW. Note that Figure 7 shows the average power for all the EVs, which is strictly lower than 11 kW because of the rejected EV users. For this use-case, we observe that our theoretical expression of the average power is very close to the simulated one.

In our scenario, the error between the theoretical and the simulated average energy received by an EV (as well as the average power), is the highest when $\alpha_1$ is around 0.45. In this case, the difference in terms of energy received is about 10%, which represents about 0.8 kWh. Note that this energy corresponds to about 5 km of autonomy with a standard EV with an average consumption of 0.15 kWh/km for urban trips [41]. This value is high knowing that the average distance when going shopping is around 4 km [42]. However, the power sizing coefficient is usually designed around 0.6. In this case, the theoretical approximations is more accurate: it corresponds to a difference between the theory and the practice of less than 1 km, which is more acceptable.

In both cases, for the energy received and the average power received per EV, the lower and upper bounds get more narrow for high values of $\alpha_1$.
<figure><img src="image_11.png" /><figcaption>Figure 8: Energy received per EV in average with respect to the average parking duration $v^{-1}$. An average parking duration of about 1h45 is the optimal trade-off between having for an EV enough time to fill-in the battery and not too much in order to free a parking space for the next coming EVs.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 9: Average power drawn per EV in average with respect to the average parking duration $v^{-1}$. The decreasing average power received per EV is due to the increasing blocking probability.</figcaption></figure>
4.1.3. *Effect of the parking duration on the approximation of energy received per EV*

Figures 8 and 9 show the average energy and power received by an EV obtained by simulation and by analytical formulae, depending on the average parking duration of an EV in the system. The power sizing $\alpha_1$ equals to 0.6. Note that in our setting the upper bound is exactly equal to the theoretical values. Indeed, in the approximation the average number of charging EVs, given by Equation (4) is sufficiently low such that all charging EVs charge at maximum power $p_{\text{max}}^1$. The average power, and therefore the average energy, received by the EVs is very close to the upper bound.

As seen in Figure 8, the average energy received per EV increases with respect to the average parking duration for low values of the latter and decreases for high average parking duration. As seen in this figure, the average energy received per EV increases with respect to the average parking duration for low values of the latter and decreases for high average parking duration. Considering low average parking duration, the probability for an EV to face a full CS is negligible. However, as the parking duration is low, EVs do not have the time to get a lot of energy. In the opposite case, for high average parking duration, the EVs that entered the CS have more time to fill in their battery. Also, the proportion of fully charged EVs is very high, so only a small proportion of EVs are actually charging, which enables them the charge at maximum power $p_{\text{max}}^1$. However, the proportion of rejected EVs is very high in this case and only a very small proportion of EVs can find an available parking space, which reduces the expected energy received per EV. The unique optimal trade-off between these two situations is an average parking duration of about 1h30 hours, which is inline with average parking duration at a shopping center [24]. Several external factors can influence the parking duration of the EVs. Legal authorities such as a CPO can impose pricing mechanisms that encourage EV users to reduce the time parked in order to improve the quality of service. Recall that in our context the parking duration does not depend on the energy received.
## 4.2. Equilibrium analysis

This section illustrates the scenario in which EV users choose between the two CSs the one with the highest expected energy received after leaving the CS. This scenario involves a non-cooperative game between the EV users. The equilibrium using the exact expected energy received, computed from the simulations, is shown in order to compare it with the approximation proposed in the model. This equilibrium is denoted $\theta_{eq}$. In the following, the power sizing coefficients $\alpha_1$ and $\alpha_2$ equal to 0.6. In order to compute the equilibrium (a probability with which each EV joins each CS) from the simulations, a verification is done on whether the equilibrium is inner ($\theta_{eq} \in ]0, 1[$) or not. If the equilibrium is inner, a dichotomy algorithm is used in order to find the proportion of EV users choosing each CS such that the expected energy received at both CSs are equal. As a stopping criteria of the dichotomy algorithm, a difference of 0.01 kWh between the expected energy received at the CSs is considered. This condition is met in about 10 iterations.

<figure><img src="image_13.png" /><figcaption>Figure 10: Equilibrium with respect to the average parking duration $\nu^{-1}$. A higher proportion of EV users tends to choose CS 1 when the average parking duration increases: the probability to be rejected at CS 2 is then very high due to a lack of parking spaces.</figcaption></figure>

Figure 10 shows the exact equilibrium $\theta_{eq}$ (green curve), and the equilibrium $\tilde{\theta}_{eq}$ (blue curve), see Definition 1, of the game $\tilde{\Gamma}$, i.e considering the theoretical
approximation on the expected energy received.

For low values of the average parking duration, all EV users choose to go to CS 2 because the blocking probability is negligible and they can charge at a highest power than at CS 1. However, when the average parking duration gets higher, the blocking probability at CS 2 gets higher such that some EV users choose CS 1 instead of CS 2.

In Figures 11, 12 and 13, the exact metrics (green curves) computed through the simulations at the exact equilibrium $\theta_{eq}$, as well as the theoretical approximations (blue curves) at the approximated equilibrium $\tilde{\theta}_{eq}$, are shown with respect to the average parking duration $v^{-1}$. According to the figures, the approximations are very accurate. For instance, the approximated energy received at equilibrium $\tilde{\theta}_{eq}$ differs by about 5% from the exact energy received at equilibrium $\theta_{eq}$ and for an average parking duration of 2 hours.

<figure><img src="image_14.png" /><figcaption>Figure 11: Energy received per EV in average at equilibrium with respect to the average parking duration. An optimal trade-off between having a sufficient amount of time to obtain a consequent energy, and a low probability to be rejected can be found (about 1h30 in our use-case).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 12: Average power received per EV in average at equilibrium by an EV with respect to the average parking duration. *With an increasing average parking duration, the average power received by an EV decreases although the average power received by an accepted EVs increases.*</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 13: Price of anarchy with respect to the average parking duration. *When the average parking duration less than 2 hours, in proportion, the EV users tend to overuse CS 2 when they act individually, which lower the quality of charging service.*</figcaption></figure>
### 4.3. Comparison between the centralized and the decentralized system through the price of anarchy

As explained in Section 3.2, the PoA enables to measure the inefficiency of the equilibrium, i.e. when EVs individually choose the best CSs for themselves, in comparison of the optimal EV distribution, i.e. when a centralized scheduler choose the CS for the EV users in order to maximize the average energy received per EV. Recall that the PoA is defined as the ratio between the expected energy received at optimum and at equilibrium. Figure 13 shows the PoA computed with the approximations with respect to the average parking duration $\nu^{-1}$. As seen in Figure 13, the decentralized system is less efficient in terms of the social welfare, i.e. the expected energy received per EV, in particular when the average parking duration is about 1 hour. In this case, the proportion of EV users choosing CS 2 is higher that it is at optimum. Indeed, the maximum power drawn at CS 2 (22 kW) is greater that at CS 1 (11 kW). When choosing CS 2, EV users create negative externalities by increasing the probability for the next arriving EVs to be blocked from CS 2, which has only 4 parking spaces.

## 5. Conclusion

This paper deals with a decision making problem in which EV users choose to join one of two charging stations with finite capacity in terms of number of charging points (CPs). Congestion effects occur at two levels: access to the charging station (CS) and share the limited available power at the CS. Therefore a loss queueing game framework is studied where the players are the EV users. Each EV user chooses the CS to join by maximizing its expected energy received during its parking duration. The latter does not depend on the charging time. In this work, the symmetric mixed-strategy equilibrium is unique. Moreover, characterizations as well as closed form expressions for some parameter values, on the utility at equilibrium are given. Finally, theoretical bounds on the price of anarchy are obtained. The theoretical approximations on the expected energy received and power levels drawn from the CPs are validated
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through numerical simulations for realistic parameter values. This study enables to predict performance of charging service such as the average energy received in a competitive scenario both in terms of parking space and power between EV users at different CSs. The equilibrium analysis can help a charging point operator that aims at optimally designing the CS infrastructure in terms of the maximum power available at the CS level, or at inciting EV users to lower their parking duration for a better EV quality of charging service.

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References

755

[1] BloombergNEF, Electric vehicle outlook 2022, Available: https://about.bnef.com/electric-vehicle-outlook/, [Online; accessed 5 october 2022].

[2] IEA, Global ev outlook 2022, technical report, Available: https://www.iea.org/reports/global-ev-outlook-2020, [Online; accessed 5 october 2022].

760

[3] European Commission, Report from the commission to the european parliament and the council, Available: https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CELEX:52021SC0049&rid=1, [Online; accessed 5 october 2022] (2021).

765

[4] Coda Strategies, Study of the characterization of needs in deployment of charging infrastructure for electric vehicles, Available: https://www.ecologie.gouv.fr/sites/default/files/2019-07-Synth%C3%A8se-IRVE-English.pdf, [Online; accessed 5 october 2022] (2019).

[5] Afirev, Observatoire de la qualité de service de recharge électrique accessibles au public, Available: https://www.observatoire-recharge-afirev.fr/, [Online; accessed 5 october 2022].

770
[6] D. Danner, H. de Meer, Quality of service and fairness for electric vehicle charging as a service, Energy Informatics 4. doi:10.1186/s42162-021-00175-3.

[7] A. Aveklouris, M. Vlasiou, B. Zwart, Bounds and limit theorems for a layered queueing model in electric vehicle charging, Queueing Systems 93 (2019) 83–137. doi:10.1007/s11134-019-09616-z.

[8] R. Hassin, M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, Kluwer Press, 2003.

[9] D. Xiao, S. An, H. Cai, J. Wang, H. Cai, An optimization model for electric vehicle charging infrastructure planning considering queuing behavior with finite queue length, Journal of Energy Storage 29 (2020) 101317. doi: 10.1016/j.est.2020.101317.

[10] Q. Yang, S. Sun, S. Deng, Q. Zhao, M. Zhou, Optimal sizing of pev fast charging stations with markovian demand characterization, IEEE Transactions on Smart Grid 10 (2019) 4457–4466. doi:10.1109/TSG.2018.2860783.

[11] S. Mohammed, F. Titus, B. S. Thanikanti, S. M. Sulaiman, S. Deb, M. N. Kumar, Charge scheduling optimization of plug-in electric vehicle in a pv powered grid-connected charging station based on day-ahead solar energy forecasting in australia, Sustainability 14 (2022) 3498. doi: 10.3390/su14063498.

[12] A. Khaksari, G. Tsaousoglou, P. Makris, K. Steriotis, N. Efthymiopoulos, E. Varvarigos, Sizing of electric vehicle charging stations with smart charging capabilities and quality of service requirements, Sustainable Cities and Society 70 (2021) 102872. doi:10.1016/j.scs.2021.102872.

[13] D. Said, S. Cherkaoui, L. Khoukhi, Queuing model for evs charging at public supply stations, in: 9th International Wireless Communications and
Mobile Computing Conference (IWCMC), IEEE, 2013, pp. 65–70. doi: 10.1109/IWCMC.2013.6583536.

[14] F. Malandrino, C. Casetti, C. Chiasserini, M. Reineri, A game-theory analysis of charging stations selection by ev drivers, Performance Evaluation 83-84 (2015) 16–31. doi:10.1016/j.peva.2014.11.001.

[15] A. Gusrialdi, Z. Qu, M. A. Simaan, Scheduling and cooperative control for charging of electric vehicles’ at highway service stations, in: 53rd IEEE Conference on Decision and Control, IEEE, 2014. doi:10.1109/CDC.2014.7040403.

[16] H. Zhang, Z. Hu, Z. Xu, Y. Song, Optimal planning of pev charging station with single output multiple cables charging spots, IEEE Transactions on Smart Grid 8 (2017) 2119–2128. doi:10.1109/TSG.2016.2517026.

[17] Z. Zhao, L. Zhang, M. Yang, J. Chai, S. Li, Pricing for private charging pile sharing considering ev consumers based on non-cooperative game model, Journal of Cleaner Production 254 (2022) 120039. doi:10.1016/j.jclepro.2020.120039.

[18] Y. Zhang, P. You, L. Cai, Optimal charging scheduling by pricing for ev charging station with dual charging modes, IEEE Transactions on Intelligent Transportation Systems 20 (2018) 3386–3396.

[19] I. Zenginis, J. Vardakas, N. Zorba, C. Verikoukis, Performance evaluation of a multi-standard fast charging station for electric vehicles, IEEE Transaction on smart grid 9 (2017) 4480–4489. doi:10.1109/TSG.2017.2660584.

[20] B. Liu, T. P. Pantelidis, S. Tam, J. Chow, An electric vehicle charging station access equilibrium model with m/d/c queueing, International Journal of Sustainable TransportationarXiv:2102.05851, doi:10.1080/15568318.2022.2029633.

[21] A.Dupont, Y.Hayel, T.Jiménez, O.Beaude, C.Wan, Coupled queueing and charging game model with energy capacity optimization, in:
26th International Conference on Analytical and Stochastic Modelling
Techniques Application, Springer, 2021, pp. 325–344. doi:10.1007/
978-3-030-91825-5_20.

830

[22] Z. Yi, P. Bauer, Effects of environmental factors on electric vehicle energy consumption: A sensitivity analysis, IET Electrical Systems in Transportation 7 (2017) 1–91. doi:10.1049/iet-est.2016.0011.

835

[23] J. Shen, J. Cao, X. Liu, Bag: Behavior-aware group detection in crowded urban spaces using wifi probes, in: WWW’19: The World Wide Web Conference, Association for Computing Machinery, Inc, 2019, pp. 1669–1678. doi:10.1145/3308558.3313590.

840

[24] J. Parmar, D. S. M. Das, Pritikana, A machine learning approach for mod-
elling parking duration in urban land-use, Physica A: Statistical Mechan-
ics and its Applications 572 (2021) 125873. doi:10.1016/j.physa.2021.
125873.

[25] R. Wolbertus, R. van den Hoed, Fast charging systems for passenger electric vehicles, World Electric Vehicle Journal 11 (2020) 1–11. doi:10.3390/wevj11040073.

845

[26] Izivia's site, Available: https://www.izivia.com/bornes-electriques-lieu-de-travail, [Online; accessed 5 october 2022].

[27] R. W. Wolff, Poisson arrivals see time averages, Operations Research 30 (1982) 223–231. doi:10.1287/opre.30.2.223.

850

[28] P. Naor, The regulation of queue size by levying tolls, Econometrica 37 (1969) 15–24.

[29] R. J. Aumann, L. S. Shapley, Value of Non-atomic Games, Princeton University Press, 1974.
[30] D. Schmeidler, Equilibrium points in nonatomic games, Journal of Statistical Physics 7 (1973) 295-300.

[31] Y. David D., S. J. George, Optimal input rates to a system of manufacturing cells, Information Systems and Operational Research (INFOR) 25 (1987) 57-65. doi:10.1080/03155986.1987.11732028.

[32] R. W. Rosenthal, A class of games possessing pure-strategy nash equilibria, International Journal of Game Theory 2 (1973) 65-67.

[33] D. Monderer, L. S. Shapley, Potential games, Games and Economic Behavior 14 (1996) 124-143.

[34] S. William H, Potential games with continuous player sets, Journal of Economic Theory 97 (2001) 81-108. doi:10.1006/jeth.2000.2696.

[35] T. Roughgarden, Selfish Routing and the Price of Anarchy, MIT Press, 2005.

[36] Simpy documentation, Available: https://simpy.readthedocs.io/en/latest/, [Online; accessed 5 october 2022].

[37] F. Gonzalez Venegas, M. Petit, Y. Perez, Plug-in behavior of electric vehicles users: Insights from a large-scale trial and impacts for grid integration studies, eTransportation 10 (2021) 100131. doi:10.1016/j.etran.2021.100131.

[38] J. Smart, S. Schey, Battery electric vehicle driving and charging behavior observed early in the ev project (2012). doi:10.4271/2012-01-0199.

[39] H. Liang, D. Jing, L. Zhenhong, Modeling charging behavior of battery electric vehicle drivers: A cumulative prospect theory based approach, Transportation Research Part C: Emerging Technologies 102 (2019) 474-489. doi:10.1016/j.trc.2019.03.027.
[40] R. Jessica, B. Gary, G. Wendy, J. Chad, E. Larry, Business models for solar powered charging stations to develop infrastructure for electric vehicles, Sustainability 6 (2014) 7358-7387. doi:10.3390/su6107358.

[41] Y. Al-Wreikat, C. Serrano, J. R. Sodré, Driving behaviour and trip condition effects on the energy consumption of an electric vehicle under real-world driving, Applied Energy 297 (2021) 117096. doi:10.1016/j.apenergy.2021.117096.

[42] Travel day vehicle miles data on National Household Travel Survey (NHTS), Available: https://nhts.ornl.gov/vehicle-miles, [Online; accessed 5 october 2022].