A Novel Mean Field Game-Based Strategy for Charging Electric Vehicles in Solar Powered Parking Lots

Abstract

We develop a strategy, with concepts from Mean Field Games (MFG), to coordinate the charging of a large population of battery electric vehicles (BEVs) in a parking lot powered by solar energy and managed by an aggregator. A yearly parking fee is charged for each BEV irrespective of the amount of energy extracted. The goal is to share the energy available so as to minimize the standard deviation (STD) of the state of charge (SOC) of batteries when the BEVs are leaving the parking lot, while maintaining some fairness and decentralization criteria. The MFG charging laws correspond to the Nash equilibrium induced by quadratic cost functions based on an inverse Nash equilibrium concept and designed to favor the batteries with the lower SOCs upon arrival. While the MFG charging laws are strictly decentralized, they guarantee that a mean of instantaneous charging powers to the BEVs follows a trajectory based on the solar energy forecast for the day. That day ahead forecast is broadcasted to the BEVs which then gauge the necessary SOC upon leaving their home. We illustrate the advantages of the MFG strategy for the case of a typical sunny day and a typical cloudy day when compared to more straightforward strategies: first come first full/serve and equal sharing. The behavior of the charging strategies is contrasted under conditions of random arrivals and random departures of the BEVs in the parking lot.

1. Introduction

The massive introduction of battery electric vehicles (BEVs) [1,2,3] in modern power systems is bound to have important impacts, positive or negative, depending on the way this novel situation is managed [4]. There will be a great pressure to introduce numerous charging stations where the need is anticipated, but if too many high speed charging BEVs are connected at any one time (for example upon departure from work towards residence place), that may create both local transformer and eventually system wide overloads [5]. Many works in the literature [6,7,8,9,10] present algorithms for an optimal scheduling of vehicle-to-grid (V2G). The authors in [6] propose a centralized algorithm based on reinforcement learning which reduces the total power grid load variance by $65\%$ in a test scenario of 300 consecutive days by charging 50 homogeneous BEVs in each hourly time slot in a neighborhood of 250 households. The authors in [7] propose a centralized weighted fair queuing (WFQ) algorithm with a 5 min time slot control switch in each smart charger to charge 300 homogeneous BEVs, by favoring those arriving with less charge. The algorithm selects a subset of BEVs to charge in each interval during peak demand when there is not enough energy. They compare the results with a first come first serve (FCFS) algorithm. They show that when the supply demand ratio (SDR) is equal to 1, there is $5\%$ of BEVs which cannot leave their homes on time while it is $7\%$ for FCFS. Recently, the authors in [10] study the case where the maximum charging power depends on the state of charge of the BEV’s battery. They propose a centralized mixed integer linear programming (MILP) algorithm to charge a fluctuating heterogeneous population of BEVs at a single station considering availability of each BEV in order to minimize time-dependent electricity costs.

On the other hand, adequate management of the battery storage associated with an aggregate of BEVs can turn such an aggregate into a virtual power plant. Thus, in a context of integration with clean sources of energy, such as photovoltaics, despite other energy harvesting and storage techniques in the literature [11,12], BEVs’ batteries could be storing the solar energy available during the day when BEVs are parked [13,14,15,16]. As is well known in the photovoltaics rich state of California for example, a so-called power demand duck curve [17] is observed: the peak demand occurs at the end of the day, upon return of working people to their homes. At that point, available solar radiation has all but disappeared and while solar energy may have been used by consumers during the day, there is a need for a high electric power ramp at dusk followed by several hours of sustained high power consumption. The latter power demand will be most likely met by fossil based energy sources, unless some other mitigating actions are taken. In [18,19], the authors show in their geographic context, that if the electric energy storage contained in a large number of BEVs is properly utilized, this could help significantly reduce the power needed from fossil sources during the evening peak.

The authors in [20], whose objective is close to ours in this paper, propose a centralized linear programming (LP) algorithm, in a solar powered parking lot of a car-share service to fairly distribute the available solar energy amongst 97 heterogeneous BEVs by favoring those arriving with less charge. They study the case where the SDR is strictly inferior to 1, and that all BEVs are available during the daily charging session of 5 h in the parking lot. They demonstrate, by charging a subset of 5 BEVs during each time slot, a reduction of $60\%$ of yearly average standard deviation in the battery charge levels at the end of recharging compared to the equal sharing (ES) approach. The authors in [6,7,10,20] do not propose a decentralized algorithm. A decentralized algorithm scheme allows individual BEVs to determine their own charging pattern. Their decisions could, for example, be made on the basis of time-of-day, electricity price or battery state of health [21,22].

Table 1. Classification of related work.

Aggregators		Potential Goal
	User Satisfaction	Monetary Benefits	Grid Impact
PLOs	our work, [20]	[18,21]	[9]
DSOs	[7]	[10,22]	[6]

below places our work in the BEVs charging optimization when the aggregators are parking lot operators (PLOs) or distribution system operators (DSOs).

Our objective in this paper is to propose an algorithm for sharing solar photovoltaic (PV) power amongst homogeneous BEVs parked in a parking lot, or a collection of federated parking lots. The BEVs belong to commuters working in the neighborhoods of these parking lots and could recharge at least partially depending on sunshine availability, their batteries at the parking lot charging stations. One particular business model is that the parking lots aggregator would charge a yearly fee for use of a parking space and the associated charging station. In a potential extension of the business model, if the BEVs’ owners wish to recuperate part of their parking costs, they could choose to participate in a financially compensated grid support operation coordinated by the aggregator of the parking lots. In that case, it would be in the interest of the aggregator to equalize charging of BEVs to maximize the probability of vehicle-to-grid charging participation.

We suggest relying on an adequately tailored variation of a Mean Field Game-based algorithm scheme [23,24] which, while it fills all BEVs simultaneously, tends to provide more instantaneous charging to the BEVs with the lowest current fill levels. The problem is formulated as large population game on a finite charging interval. In [21], the authors study the existence, uniqueness and optimality of the Nash equilibrium of the charging problems to minimize local electricity costs and to fully charge. In a decentralized computational mechanism, they show in a deterministic case that the large population charging games will converge to a unique Nash equilibrium which is globally optimal for a homogeneous population.

In what follows we shall present a novel Mean Field Game-based charging algorithm to calculate the operator broadcasted decentralized algorithm laws according to the potential solar energy available. Subsequently, the performance of these laws will be compared to that of two common algorithms used in the literature. The first algorithm, first come first full (which, in a dynamic context, will be upgraded to first come first serve), consists of recharging maximally (or up to an adequately updated SOC in real time) the BEVs in order of arrivals at the parking lot. The second algorithm, equal sharing, consists of sharing equally at all times the available solar power amongst battery BEVs still not fully charged. All algorithms make full use of the available daily energy. Furthermore, for the purpose of meaningfully comparing the performance of the different algorithms in our case studies, we assume that the SDR is less than one.

In order to implement the charging algorithms, we make the following assumptions:

• There exists a communication infrastructure to coordinate BEVs charging in the parking lot.
• The BEVs are equipped with microprocessors in the chargers allowing them to locally compute and implement a local feedback-based charging algorithm.

The rest of this paper is organized as follows. In Section 2, we present the theoretical underpinnings and details of the MFG-based algorithm in the case of homogeneous BEVs. In Section 3, we present the numerical results assuming a fixed population of BEVs which are charged in the parking lot simultaneously with common characteristics of batteries. In Section 4, we present the algorithmic modifications and the numerical results in a more realistic situation where BEVs arrive and depart randomly in the parking lot. Finally, in Section 5, we conclude and give an outlook on future research.

2. Mean Field Game-Based Control of a Large Population of BEVs

2.1. Battery Model

We consider a population of n homogeneous BEVs in a parking lot. The assumption of a large population is needed only if, as we do in (1) below, we assume randomness in the dynamics of battery charging and later on in our analysis we will assimilate the empirical mean of SOCs with its mathematical expectation (a predictable deterministic quantity) by virtue of the law of large numbers. Because of the linearity of the model, the analysis will be perfectly exact for arbitrarily small numbers of BEVs if the battery charging processes remain deterministic. Each BEV i, $i=1,\cdots ,n$, has a SOC $x_{i,arr}$ upon arrival which results of a daily traffic pattern from home to parking lot. We can then write the SOC stochastic dynamics for BEV i as follows [21]:

$$\begin{array}{c}\hfill d{x}_{i,t}=b{u}_{i,t}dt+\nu d{\omega }_i,\end{array}$$

(1)

where $t\in [t_0,T]$ is time in hours (h), $x_{i,t}$ is the SOC in per unit (pu) of capacity, $b=\frac{\alpha }{\beta }$, $\alpha \in (0,1]$ is the charger efficiency in pu$/h$, $\beta$ is the battery capacity in $kWh$, $u_{i,t}\in {\mathbb{R}}^{+}$ is the charging rate in $kW$, ${\omega }_i$ is a normalized Brownian process, $\nu$ is the intensity of that Brownian noise and ${\omega }_i$ is assumed independent of ${\omega }_j$ for $i\ne j$. The term $\nu d{\omega }_i$ defines the stochasticity of the SOC which can result physically from fluctuations in the charging and losses of the battery. We first present the algorithm in the simple case where all BEVs are started charging at the same time and depart at the end of control horizon, i.e., $t_0=t_{begin}$ and $T=t_{end}$. Then the algorithm will be updated in a dynamic context, where $[t_0,T]$ will be made up of successive fixed short control horizons representing random arrivals and random departures of BEVs between $t_{begin}$ and $t_{end}$.

2.2. Considerations

The charging algorithms tested in this paper will be compared with respect to the following requirements:

2.2.1. Fairness

We develop here our notion of fairness. While it is very hard to realize in practice, we wish that BEVs have near identical SOCs at the time they leave the parking lot irrespective of the time they arrived or their SOC at arrival. Furthermore, for BEVs that are simultaneously present in the parking lot, at no time should the BEV that arrived with a lower SOC be allowed to have its SOC exceed that of the other BEV. Such a criterion will lead us to the definition of a fairness coefficient by means of which we shall contrast the performance of various charging algorithms. We aim at:

(a)

BEVs having relatively close SOCs at the time of their departure (under an assumption that they will spend statistically equivalent amounts of time in the parking lot).

(b)

Pairs of BEVs present in the parking lot at the same time must maintain a fixed relative ordering of their SOC values (SOC of BEV $i>$ SOC of BEV j) throughout their simultaneous stay, and until one of them leaves.

This leads us to the definition of a fairness coefficient (FC) that will help compare the fairness of the various charging strategies.

• Given (a) above, we consider that FC should be proportional to the the reduction in standard deviation of the SOCs’ BEVs at departure time relative to the SOC standard deviation at arrival time.
• Given (b) above, we should count for all BEV pairs that qualify, the number of violations of the fixed SOC relative value ordering criterion during their simultaneous stay. Let $N_O$ be the number of such violations, and let $\left(\genfrac{}{}{0pt}{}{N}{2}\right)$ be the number of BEV pairs in a population of N BEVs. Let $\eta$ be the ratio of the two. We consider that FC must decrease exponentially with the value of $\eta$.

We suggest using the following expression:

$$\begin{array}{c}\hfill FC=\frac{{\sigma }_{x_{i,arr}}-{\sigma }_{x_{i,dep}}}{{\sigma }_{x_{i,arr}}}{e}^{-\eta },\mathrm{where} \eta =\frac{N_O}{\left(\genfrac{}{}{0pt}{}{N}{2}\right)}=\frac{2N_O(N-2)!}{N!}.\end{array}$$

(2)

• The degree of fairness of the charging strategy is measured by FC if $FC>0$.
• The charging strategy is considered neutral if $FC=0$.
• The charging strategy is rejected if $FC<0$.

2.2.2. Decentralization

From the point of view of the parking lot operator, decentralized charging algorithm laws are quite desirable because they minimize the need to observe the state of charge of individual batteries, a process which is both complex and invasive. Furthermore, a local algorithm allows a user to interrupt the process at any time, particularly if the parking operator has designed a charging scheme based on a poor model of the battery.

We wish to address the decentralized control of battery recharging of a set of BEVs as part of a so-called Mean Field Game (MFG). The control will be of linear-quadratic (LQ) type [25,26]. The parking lot operator broadcasts an average SOC target trajectory (${\overline{x}}_t^{target}$) based on the solar energy forecast for the current day. The goal is that the BEVs store up as much of the solar energy available as possible and yet share the energy in a manner which will be deemed as fair. The proposed algorithm requires that the parking lot operator know the average SOC of the BEVs upon arrival (${\overline{x}}_0$). This can be achieved by recording initial SOCs as BEVs enter the parking lot.

The key point of the approach is the prescription of a daily quadratic cost $J_i$ for each BEV i to ensure that the BEVs that are initially fuller recharge less quickly than those that are less full, so that final SOC standard deviation of departing BEVs is reduced; while still maintaining the goal of using up the available solar power at all instants. More precisely the cost functions are designed so that by optimizing the individual BEV costs $J_i$, one achieves the aggregator’s goals (fairness and decentralization) while using all available solar energy in the parking lot.

2.3. Establishment of Individual Battery Cost Function

The battery cost function is a mathematical expectation. It is designed by the aggregator and defined for a BEV as follows:

$$\begin{array}{c}\hfill J_i(x_{i,0},u_{i,t})=\mathbb{E}\left[{\int }_{t_0}^T{e}^{-\delta t}[\frac{q_t^y}{2}{(x_{i,t}-y)}^2+\frac{r}{2}{u}_{i,t}^2+\frac{q_{x_0}}{2}{(x_{i,t}-x_{i,0})}^2]dt|x_{i,0}\right],\end{array}$$

(3)

where $\mathbb{E}$ is the expectation operator, $t_0$ is the charging starting time, T is the charging stopping time, $\delta$ is a discount coefficient to ensure convergence of the cost, y is the collective direction of the BEVs’ SOCs which is equal to 1 in our case (it serves as a direction signal to all BEVs, such that all BEVs should move toward y but not beyond), r is a coefficient which penalizes the level for charging rate, $q_{x_0}$ is a pressure coefficient aimed at limiting the distance from the SOC $x_{i,0}$ (for the state of health of the user’s battery and fairness to others) and $q_t^y$ is the pressure field trajectory. The latter is common to all BEVs and is numerically obtained as the solution of a system of differential equations. It is the key quantity which will drive all SOCs towards a full state of charge while sharing instantaneously available solar energy in a way that reduces the standard deviation of SOCs. Its computation is further detailed below and it is at the heart of our inverse Nash equilibrium procedure. Note that the class of quadratic cost functions has been frequently used in the MFG literature [23,25,27].

The BEVs will collectively recharge their batteries with a time dependent coefficient ($q_t^y$) penalizing the gap between the current SOC ($x_{i,t}$) and the ultimate destination direction of SOCs defined by the value of y. Once the target for steady-state mean SOC of BEVs (${\overline{x}}_T^{target}$) is reached, $q_t^y$ will settle to a constant value, allowing the reaching of a mean SOC steady-state that meets the constraints set by the parking lot operator.

2.4. Optimal Control Problem and Solutions

We begin by making the following variable changes to simplify the expression of cost $J_i$ in Equation (3):

$$\begin{array}{cc}\hfill X_{i,t}& =(x_{i,t}-y)e^{-\delta t/2},U_{i,t}=u_{i,t}{e}^{-\delta t/2},\hfill \\ \hfill V_t& =\nu e^{-\delta t/2} \mathrm{and} Z_{i,t}=(x_{i,t}-x_{i,0})e^{-\delta t/2}.\hfill \end{array}$$

(4)

Then:

$$\begin{array}{cc}\hfill d{X}_{i,t}& =d{x}_{i,t}{e}^{-\delta t/2}-\frac{\delta }{2}{x}_{i,t}{e}^{-\delta t/2}dt+\frac{\delta }{2}y{e}^{-\delta t/2}dt\hfill \\ \hfill & =(b{u}_{i,t}dt+\nu d{\omega }_i)e^{-\delta t/2}-\frac{\delta }{2}(x_{i,t}-y)e^{-\delta t/2}dt\hfill \\ \hfill & =-\frac{\delta }{2}{X}_{i,t}dt+b{U}_{i,t}dt+V_{t}d{\omega }_i.\hfill \end{array}$$

(5)

The solution approach is based on assuming a quadratic form of the optimal cost function [26] (with coefficients $\pi ,s$ and $\gamma$):

$$\begin{array}{c}\hfill J_i^{*}\left(X_{i,t}\right)=\frac{1}{2}{\pi }_{i,t}{X}_{i,t}^2+s_{i,t}{X}_{i,t}{e}^{-\delta t/2}+{\gamma }_{i,t}.\end{array}$$

(6)

We then write the dynamic programming equation corresponding to this guess [28].

$$\begin{array}{c}\hfill \frac{\partial J_i^{*}}{\partial t}+\underset{U}{min}\left\{\frac{q_t^y}{2}{X}_{i,t}^2+\frac{q_{x_0}}{2}{Z}_{i,t}^2+\frac{r}{2}{U}_{i,t}^2+\frac{\partial J_i^{*}}{\partial X}(-\frac{\delta }{2}{X}_{i,t}+b{U}_{i,t})+\frac{{\partial }^2{J}_i^{*}}{\partial X^2}\frac{V_t^2}{2}\right\}=0.\end{array}$$

(7)

Differentiating with respect to $U_{i,t}$ yields:

$$\begin{array}{cc}\hfill & {\nabla }_U\{\frac{q_t^y}{2}{X}_{i,t}^2+\frac{q_{x_0}}{2}{Z}_{i,t}^2+\frac{r}{2}{U}_{i,t}^2\}+{\nabla }_U\left\{\frac{\partial J_i^{*}}{\partial X}(-\frac{\delta }{2}{X}_{i,t}+b{U}_{i,t})+\frac{{\partial }^2{J}_i^{*}}{\partial X^2}\frac{V_t^2}{2}\right\}=0\hfill \\ \hfill & ⟶U_{i,t}^{*}=-\frac{b}{r}\frac{\partial J_i^{*}}{\partial X}.\hfill \end{array}$$

(8)

The second derivative with respect to $U_{i,t}$ is $r>0$, so the value found will indeed correspond to a minimum:

$$\begin{array}{c}\hfill U_{i,t}^{*}=-\frac{b}{r}\left[{\pi }_{i,t}{X}_{i,t}+s_{i,t}{e}^{-\delta t/2}\right].\end{array}$$

(9)

The optimal control therefore depends on the values ${\pi }_{i,t}$ and $s_{i,t}$. The expressions of ${\pi }_{i,t}$ and $s_{i,t}$ are then determined by identification:

$$\begin{array}{cc}\hfill & \frac{1}{2}{X}_{i,t}^2\frac{d{\pi }_{i,t}}{dt}+X_{i,t}{e}^{-\delta t/2}(\frac{d{s}_{i,t}}{dt}-\frac{\delta }{2}{s}_{i,t})+\frac{d{\gamma }_{i,t}}{dt}\hfill \\ \hfill & =-\frac{q_t^y}{2}{X}_{i,t}^2-\frac{q_{x_0}}{2}{Z}_{i,t}^2+\frac{b^2}{2r}({\pi }_{i,t}^2{X}_{i,t}^2+2X_{i,t}{\pi }_{i,t}{s}_{i,t}{e}^{-\delta t/2}+s_{i,t}^2{e}^{-\delta t})\hfill \\ \hfill & +\frac{\delta }{2}{X}_{i,t}({\pi }_{i,t}{X}_{i,t}+s_{i,t}{e}^{-\delta t/2})-\frac{V_t^2}{2}{\pi }_{i,t}\hfill \\ \hfill & =X_{i,t}^2(\frac{b^2}{2r}{\pi }_{i,t}^2+\frac{\delta }{2}{\pi }_{i,t}-\frac{q_t^y}{2}-\frac{q_{x_0}}{2})+X_{i,t}{e}^{-\delta t/2}(\frac{b^2}{r}{\pi }_{i,t}{s}_{i,t}+\frac{\delta }{2}{s}_{i,t}-y{q}_{x_0}+x_{i,0}{q}_{x_0})+\dots \hfill \end{array}$$

(10)

The analysis results in the following system of differential equations that must be solved backwards:

$$\begin{array}{cc}\hfill \frac{d{\pi }_{i,t}}{dt}& =\frac{b^2}{r}{\pi }_{i,t}^2+\delta {\pi }_{i,t}-q_t^y-q_{x_0},\hfill \\ \hfill \frac{d{s}_{i,t}}{dt}& =(\delta +\frac{b^2}{r}{\pi }_{i,t})s_{i,t}+q_{x_0}(x_{i,0}-y),\hfill \end{array}$$

(11)

and the optimal control law is given by:

$$\begin{array}{c}\hfill u_{i,t}^{*}=-\frac{b}{r}\left[{\pi }_{i,t}(x_{i,t}-y)+s_{i,t}\right].\end{array}$$

(12)

The coefficient $q_t^y$ appearing in the differential equation of $\pi$ is unknown at this stage. Nonetheless, it must respect the fact that when the BEVs use the optimal control $u_{i,t}^{*}$, their empirical average trajectory (${\overline{x}}_t$), assimilated thanks to the law of large numbers to the mathematical expectation of the SOC of a generic BEV $\left(\mathbb{E}\left[x_{i,t}\right]\right)$, will correspond to the average target trajectory (${\overline{x}}_t^{target}$) imposed by the parking lot operator. The charging strategy that is developed, relies on knowing the anticipated solar energy during the day. Based on the dynamics of the SOC $x_{i,t}$ in Equation (1), the average target ${\overline{x}}_t^{target}$ needed is determined by integrating the curve of the forecast solar power ($u_{W_t}$) over the time $[t_0,T]$ of interest and dividing, for a case of homogeneous battery capacities, by the total number of BEVs present in the parking lot for recharging. The total forecast solar energy W available in the parking lot is assumed less than the total energy that all BEVs would need to fully recharge their batteries.

2.5. Calculation of $q_t^y$ by Nash Equilibrium Inversion

We calculate the pressure field $q_t^y$ directly by numerical resolution of differential equations. The system of differential equations to be solved is obtained by imposing that under the action of $q_t^y$, and the associated optimal control law in Equation (12), the average trajectory ${\overline{x}}_t$ of the BEVs follows the average target trajectory ${\overline{x}}_t^{target}$ broadcasted by the parking lot operator. This restriction allows us to write the Mean Field equations based on taking the mathematical expectation of the SOC of a generic battery in the population subject to decentralized control law (Equation (12)).

$$\begin{array}{cc}\hfill \frac{d{\overline{x}}_t^{target}}{dt}& =b{\overline{u}}_t^{*}=-\frac{b^2}{r}[{\pi }_t({\overline{x}}_t^{target}-y)+{\overline{s}}_t],\hfill \\ \hfill \frac{d{\pi }_t}{dt}& =\frac{b^2}{r}{\pi }_t^2+\delta {\pi }_t-q_t^y-q_{x_0},\hfill \\ \hfill \frac{d{\overline{s}}_t}{dt}& =(\delta +\frac{b^2}{r}{\pi }_t){\overline{s}}_t+q_{x_0}({\overline{x}}_0-y),\hfill \end{array}$$

(13)

where ${\pi }_t={\pi }_{i,t}$ and ${\overline{s}}_t=\frac{{\sum }_{i=1}^n{s}_{i,t}}{n}$. With this approach, the goal is to obtain a mathematical relationship between $\frac{d{\overline{x}}_t^{target}}{dt}$ and $q_t^y$. This is the so-called inverse Nash algorithm, its first steps were developed in the control of electric space heaters [27].

With the first equation of (13), we can write the relation between ${\pi }_t$ and ${\overline{s}}_t$:

$$\begin{array}{c}\hfill {\pi }_t=-\frac{{\overline{s}}_t}{{\overline{x}}_t^{target}-y}-\frac{r\frac{d{\overline{x}}_t^{target}}{dt}}{b^2({\overline{x}}_t^{target}-y)}.\end{array}$$

(14)

The differential equation governing the dynamics of ${\overline{s}}_t$ is then written as follows:

$$\begin{array}{cc}\hfill \frac{d{\overline{s}}_t}{dt}& =-\frac{{\overline{s}}_t^2{b}^2+r{\overline{s}}_t\frac{d{\overline{x}}_t^{target}}{dt}}{r({\overline{x}}_t^{target}-y)}+({\overline{x}}_0-y)[q_{x_0}+\frac{\frac{d{\overline{x}}_t^{target}}{dt}}{b^2({\overline{x}}_t^{target}-y)}].\hfill \end{array}$$

(15)

To solve this differential equation numerically in the interval of time $[t_0,T]$, we need to specify a terminal condition at T. Since this equation is solved backwards in time, this is equivalent to determining ${\overline{s}}_T$. By choosing a time horizon T such that the solar power curve has already fallen to zero at T, ${\overline{x}}_t^{target}$ will settle at ${\overline{x}}_T^{target}$. Thus, $\frac{d{\overline{x}}_t^{target}}{dt}=0,t\ge T$, and this is consistent with imposing $\frac{d{\pi }_t}{dt}=\frac{d{\overline{s}}_t}{dt}=0,t\ge T$. We will then assimilate $q_T^y,{\pi }_T,{\overline{s}}_T$ to their steady-state values. This allows us to write:

$$\begin{array}{cc}\hfill 0& =-\frac{b^2}{r}[{\pi }_T({\overline{x}}_T^{target}-y)+{\overline{s}}_T],\hfill \\ \hfill 0& =\frac{b^2}{r}{\pi }_T^2-q_T^y-q_{x_0},\hfill \\ \hfill 0& =\frac{b^2}{r}{\pi }_T{\overline{s}}_T+q_{x_0}({\overline{x}}_0-y),\hfill \end{array}$$

(16)

and yields:

$$\begin{array}{cc}\hfill q_T^y& =q_{x_0}\left(\frac{{\overline{x}}_0-{\overline{x}}_T^{target}}{{\overline{x}}_T^{target}-y}\right),\hfill \\ \hfill {\pi }_T& =\sqrt{\frac{r}{b^2}(q_{x_0}+q_T^y)},\hfill \\ \hfill {\overline{s}}_T& ={\pi }_T(y-{\overline{x}}_T^{target})\mathrm{or}{\overline{s}}_T=\frac{r{q}_{x_0}(y-{\overline{x}}_0)}{{\pi }_T{b}^2}.\hfill \end{array}$$

(17)

Thereafter, we solve numerically and backwards the differential equation of $\frac{d{\overline{s}}_t}{dt}$, which yields the trajectory of ${\overline{s}}_t$. The latter is re-injected into the equation of ${\pi }_t$. Finally, we have all the necessary ingredients to calculate $q_t^y$ from:

$$\begin{array}{c}\hfill q_t^y=\frac{b^2}{r}{\pi }_t^2+\delta {\pi }_t-\frac{d{\pi }_t}{dt}-q_{x_0}\end{array}$$

(18)

3. Numerical Results in the Case of a Fixed Population of BEVs

3.1. Required Data

• Homogeneous population of BEVs: we consider an average BEV with $\alpha =0.85$ and $\beta =23kWh$ representing the average values of a realistic population of BEVs.
• Simulation parameters: we consider installing 100 solar panels in the parking lot for recharging 400 BEVs between $t_{begin}=6h$ and $t_{end}=18h$ in the sunny day case or the cloudy day case with a random normal distribution of SOCs upon arrival (with a mean of $0.15$ and a standard deviation of $0.10$). Here the charging of all BEVs starts at $t_{begin}$ and stops at $t_{end}$ (i.e., $t_0=6h$ and $T=18h$), $dt=0.01h,\nu =0.01,\delta =0$ (we set $\delta$ to zero to work on a finite control horizon), $q_{x_0}=1000,r=0.001,y=1$.

3.2. MFG Inverse Nash Algorithm of Charging BEVs

Figure 1 shows the outline of the operation of the algorithm.

Below is the detailed Algorithm 1.

Algorithm 1 MFG inverse Nash

Require:$t_0$ (charging starting time), T (charging stopping time), $u_{W_t}$ (forecast solar power  at time t), n (number of BEVs present in the parking lot, i.e., at time $t_0$), $x_{i,0}$ (BEVs’ SOCs  at time $t_0$), ${\overline{x}}_0=\frac{{\sum }_{i=1}^n{x}_{i,0}}{n}$, $\alpha ,\beta ,b=\frac{\alpha }{\beta },dt,\nu ,q_{x_0},r,y,\delta$. Ensure:The parking lot operator computes the pressure field ($q_t^y$) of n BEVs using the steps:   1. Solve $\frac{d{\overline{x}}_t^{target}}{dt}=\frac{1}{n}b{u}_{W_t}$ and note the mean target SOC (${\overline{x}}_T^{target}$) of n BEVs.   2. Calculate $q_T^y=q_{x_0}\left(\frac{{\overline{x}}_0-{\overline{x}}_T^{target}}{{\overline{x}}_T^{target}-y}\right),{\pi }_T=\sqrt{\frac{r}{b^2}(q_{x_0}+q_T^y)}$ and ${\overline{s}}_T={\pi }_T(y-{\overline{x}}_T^{target})$.   3. Solve $\frac{d{\overline{s}}_t}{dt}=-\frac{{\overline{s}}_t^2{b}^2+r{\overline{s}}_t\frac{d{\overline{x}}_t^{target}}{dt}}{r({\overline{x}}_t^{target}-y)}$$+({\overline{x}}_0-y)\left[q_{x_0}+\frac{\frac{d{\overline{x}}_t^{target}}{dt}}{b^2({\overline{x}}_t^{target}-y)}\right]$ backwards.   4. Calculate ${\pi }_t=-\frac{{\overline{s}}_t}{{\overline{x}}_t^{target}-y}-\frac{r\frac{d{\overline{x}}_t^{target}}{dt}}{b^2({\overline{x}}_t^{target}-y)}$ and determine $\frac{d{\pi }_t}{dt}$ (by mean value theorem).   5. Calculate $q_t^y=\frac{b^2}{r}{\pi }_t^2+\delta {\pi }_t-\frac{d{\pi }_t}{dt}-q_{x_0}$. Ensure: Each BEV $i,i=1,2,3,\cdots ,n$, computes its local feedback strategy using the steps:   1. Solve $\frac{d{s}_{i,t}}{dt}=(\delta +\frac{b^2}{r}{\pi }_t)s_{i,t}+q_{x_0}(x_{i,0}-y)$ backwards with $s_{i,T}=\frac{r{q}_{x_0}(y-x_{i,0})}{{\pi }_T{b}^2}$.   2. Solve $d{x}_{i,t}=b{u}_{i,t}^{*}dt+\nu d{\omega }_i=-\frac{b^2}{r}\left[{\pi }_t(x_{i,t}-y)+s_{i,t}\right]dt+\nu d{\omega }_i$.

3.3. Obtaining the Average Target SOC by Using Daily Solar Energy in the Parking Lot

Realistic generation curves based on historical meteorological data are used (Figure 2), assuming that similar generation curves can be predicted using for example, a machine learning based model. The meteorological data is obtained from photovoltaic geographical information System (PVGIS) made available by the European Commission. A typical meteorological year in the city of Montreal (45.50 North, 73.58 West) is used with a data resolution of one hour. The very same data can also be found in Canadian Weather Energy and Engineering Datasets (CWEEDS). The solar photovoltaic (PV) power output is then modeled with the simulation software TRNSYS using type 103 appropriate for modeling the electrical performance of mono and polycrystaline PV panels.

We determine two different real solar power curves (sunny day and cloudy day cases) in order to compare the influence of the difference in generation on the behaviour of the BEVs charging. Starting with the sunny day case, and 400 BEVs, we get an average target curve that saturates at the end of the horizon (${\overline{x}}_T^{target}=0.90$). Looking at the cloudy day case and as a result of that a lower energy output at the end of the horizon, charging the same number of BEVs with the same distribution SOCs upon arrival (${\overline{x}}_0=0.15$) would result in an average target that is low (${\overline{x}}_T^{target}=0.50$). The parking lot operator would then announce the situation the day before, so that the BEVs arrive next day more full in the parking lot. However here, for comparison purposes, we shall work with the same distribution SOCs upon arrival as for the sunny day case.

3.4. Pressure Field, Empirical per BEV Average SOC and Individual SOCs of BEVs Using MFG Inverse Nash Algorithm

In Figure 3, Figure 4 and Figure 5, first we see that the trajectories of the empirical per BEV average SOC and the mean target SOC broadcasted by the parking lot operator are quite the same in both days, thanks to the law of large numbers. And as expected in steady-state we have constant values, the proposed MFG algorithm makes full use of the available daily energy between 6 a.m. and 6 p.m. The results of individual SOCs of BEVs in the sunny day case, in Figure 6, show a strong reduction in standard deviation ${\sigma }_{x_{i,T}}$ while in the cloudy day case, in Figure 7, we have a slight reduction in standard deviation. Additionally, all the curves’ behaviour in the cloudy day case well reflects the characteristic of solar fluctuations, and more importantly the results are less desirable in the cloudy day case relative to the sunny day case because the charging rate needed to achieve full solar utilization is lower in the cloudy day case. Furthermore, we confirm the main features of the proposed MFG algorithm, that of filling more batteries that were emptier to start with while bringing all batteries close to a predefined mean target.

3.5. Different Charging Strategies

• First come first full (FCFF), which fills up to maximum capacity each BEV in the order of their arrival in the parking lot.
• Equal sharing (ES), which fills all the BEVs in the parking lot at equal rates until BEVs quit charging when they are full.
• Mean Field Game (MFG), which fills all the BEVs according to the MFG inverse Nash algorithm (see Algorithm 1 in Section 3.2).

3.6. Comparison of Charging Strategies

In the following, we present the results for three charging strategies in Figure 8 and Figure 9.

3.6.1. FCFF Strategy

It is the easiest strategy to implement. A centralized signal is sent to fill maximally the BEVs in the parking lot regardless of their SOCs upon arrival, and only depending on the ordering of their arrival times. This approach is not suitable at all for our charging objective. It does not meet any of the requirements. First, it is a centralized charging strategy since the parking lot operator must not only note the order of arrivals of BEVs in the parking lot, but also when they finish filling up. Secondly, at the end of the day, there will generally be users who have not recharged their batteries at all ($12\%$ of BEVs in the sunny day case and $59\%$ of BEVs in the cloudy day case). Thirdly, we have the highest standard deviations of SOCs upon departure: ${\sigma }_{x_{i,T}}=0.280$ (an increase of $175\%$) in the sunny day case and ${\sigma }_{x_{i,T}}=0.426$ (an increase of $318\%$) in the cloudy day case.

3.6.2. ES Strategy

This is a straightforward charging scheme. Using the forecast solar power available throughout the day, the parking lot operator estimates at every instant an average level per BEV available for charging and thus, the average charging rate. As BEVs get charged, the number of BEVs still in demand must be updated. Thus, this scheme, although superior to the previous one, is not decentralized. By giving the same amount to everyone, one does not significantly reduce ${\sigma }_{x_{i,T}}$ relative to ${\sigma }_{x_{i,0}}$. We have ${\sigma }_{x_{i,T}}=0.082$ (a reduction of $20\%$) in the sunny day case and ${\sigma }_{x_{i,T}}=0.102$ in the cloudy day case. Indeed, the SOCs’ standard deviation remains close to what it was at the beginning, except for a slight reduction due to some BEVs completely filling up in the sunny day case.

3.6.3. MFG Strategy

This is our novel charging strategy scheme based on the idea of inverse Nash control. The parking lot operator, after broadcasting ${\overline{x}}_T^{target}$, prescribes a decentralized strategy via an inverse Nash algorithm in the smart charger in the parking lot. Each user applies their optimal control locally so that, the average trajectory (${\overline{x}}_t$) of all users corresponds to the average target trajectory (${\overline{x}}_t^{target}$) broadcasted by the parking lot operator. The standard deviation at the end of recharging has been significantly reduced. We have ${\sigma }_{x_{i,T}}=0.013$ (a reduction of $87\%$) in the sunny day case and ${\sigma }_{x_{i,T}}=0.061$ (a reduction of $40\%$) in the cloudy day case. Furthermore, except for the broadcasting of some initialization data by the parking lot operator, the charging scheme is decentralized. This strategy meets all the requirements of our charging objective.

In

Table 2. Comparison of fairness coefficient and departing SOCs ($min{x}_{i,0}=0$ and $max{x}_{i,0}=0.45$) for a fixed population of 400 BEVs.

		Sunny Day			Cloudy Day
	$min{\mathit{x}}_{\mathit{i},\mathit{T}}$	$max{\mathit{x}}_{\mathit{i},\mathit{T}}$	FC	$min{\mathit{x}}_{\mathit{i},\mathit{T}}$	$max{\mathit{x}}_{\mathit{i},\mathit{T}}$	FC
ES	$0.76$	1	$0.19$	$0.35$	$0.79$	0
FCFF	0	1	$-0.72$	0	1	$-2.13$
MFG	$0.88$	$0.93$	$0.86$	$0.40$	$0.67$	$0.40$

the fairness coefficient (FC), as defined previously in Section 2.2.1, is computed for each charging strategy.

The results in both days show that the MFG strategy is the fairest strategy while the FCFF strategy is rejected.

4. Numerical Results in the Case of a fluctuating Population of BEVs

4.1. Considerations

• In the parking lot, the BEVs start arriving before 6 a.m., stop arriving at 12 p.m., start departing after 3 p.m. until after 6 p.m. (with $t_{begin}$ = 6 h, $t_{end}$ = 18 h).
• We consider a Poisson distribution for the arrivals (Figure 10). A Poisson distribution allows us to realistically fill parking spaces in the parking lot, where the number of arrivals at each time interval decreases exponentially until some fixed stopping time. Each 15 min arriving time interval ${\phi }_a,a=1,2,3,\cdots ,24$, a subpopulation of $n_a$ BEVs is connected to the charging stations at random times $t_j^{\left(a\right)}<6+0.25(a-1)$ a.m., $j=1,\cdots ,n_a$. We then charge, between $6+0.25(a-1)$ a.m. and $6+0.25a$ a.m., all ${\sum }_{k=1}^a{n}_k$ BEVs present in the parking lot. For example, the first $n_1$ BEVs connected to their charging stations before 6 a.m. will start charging at 6 a.m., the next $n_2$ BEVs connected between 6 a.m. and 6:15 a.m. will start charging at 6:15 a.m., and so on. The last $n_{25}$ BEVs, arrive between 11:45 a.m. and 12 p.m., are connected at random times $t_j^{\left(25\right)}<12$ p.m., $j=1,\cdots ,n_{25}$. We then charge, between 12 p.m. and 3 p.m., the total expected number of ${\sum }_{k=1}^{25}{n}_k=400$ BEVs in the parking lot.
• We also consider a Poisson distribution for the departures (Figure 10). Here, a Poisson distribution is used for the same reason as in the case of arrivals but will be defined to empty the parking lot so that on average the departing BEVs spend the same parking time. Each 15 min departing time interval ${\phi }_d,d=26,27,28,\cdots ,38$, a subpopulation of $n_d$ BEVs is disconnected from the charging stations at random times $t_j^{\left(d\right)}>3+0.25(d-26)$ p.m., $j=1,\cdots ,n_d$. We then charge, between $3+0.25(d-26)$ p.m. and $3+0.25(d-25)$ p.m., the remaining $\left(400-{\sum }_{k=26}^d{n}_k\right)$ BEVs in the parking lot. Here, we cannot delay recharging the BEVs within a given departing time interval ${\phi }_d$ as the charging remains continuous despite the random departures of the BEVs. It is only at the end of ${\phi }_d$ that we can record the exact number of vehicles that have disconnected from their charging stations. We then consider a small-size battery in the parking lot to store the unused energy $W_{{\phi }_d}^{*}$ in a given departing time interval ${\phi }_d$, i.e., not really distributed to the BEVs already left the parking lot. $W_{{\phi }_d}^{*}$ is therefore added to the forecast solar energy $W_{{\phi }_{d+1}}$ acquired in the next departing time interval ${\phi }_{d+1}$.
• For all other parameters, we use the same values as in the case of a fixed population of BEVs (both in the sunny day case and in the cloudy day case).
• To summarize here, BEVs recharging is updated in 15-min cycles between 6 a.m. and 6 p.m. except between 12 p.m. and 3 p.m. when the cycle lasts 3 h.

4.2. Updated MFG Inverse Nash Algorithm

Each charging time interval ${\phi }_c$ (considering both arrival time interval ${\phi }_a$ and departing time interval ${\phi }_d$), i.e., $c=1,2,\cdots ,25,26,27,\cdots ,38$ (where ${\phi }_1=[6,6.25[h,{\phi }_2=[6.25,6.5[h,\cdots ,{\phi }_{25}=[12,15]h,{\phi }_{26}=]15,15.25]h,{\phi }_{27}=]15.25,15.5]h,\cdots ,{\phi }_{38}=]17.75,18]h$), the parking lot operator calculates the mean target SOC trajectory of all BEVs present in the parking lot. Unlike in the case of a fixed population of BEVs in the parking lot where we worked with a solar curve over the entire control horizon between $t_{begin}$ and $t_{end}$, here we work with successive fixed short control horizons at the start of which the number of BEVs present in the parking lot is recorded. In order to enforce the Riccati steady-state conditions (illustrated in Section 2), in each charging time interval ${\phi }_c$ we add to the solar power curve $u_{W_{{\phi }_c}}$ a fictitious extension ${\tilde{u}}_{W_{{\phi }_c}}$ into ${\phi }_{c+1}$ with the solar power falling smoothly to zero at the end of ${\phi }_{c+1}$. Below, in Figure 11, is the example of a piece of solar power curve between 8 a.m. and 10 a.m. in the sunny day case with fictitious added parts.

The MFG inverse Nash algorithm (see Algorithm 1 in Section 3.2) will be applied here in each charging time interval ${\phi }_c$ on short control horizons (i.e., here $t_0$ is the beginning of ${\phi }_c$ and T is the end of ${\phi }_c$) with the resulting mean target SOC trajectory ${\overline{x}}_t^{target}$, $t\in \{{\phi }_c\cup {\phi }_{c+1}\}$, knowing that resulting solar power curve is equal to $\{u_{W_{{\phi }_c}}\cup {\tilde{u}}_{W_{{\phi }_c}}\}$ (Figure 11).

4.3. Adjustment of Different Charging Strategies

• First come first full (FCFF), which fills up to maximum capacity each BEV among the n BEVs present in the parking lot in the order of their arrival in the parking lot in each charging time interval ${\phi }_c$. We propagate this order of arrivals to the next charging time interval, ${\phi }_{c+1}$.
• First come first serve (FCFS), is a refined version of FCFF, i.e., has the same characteristics as the FCFF except that instead of filling up to maximum capacity the n BEVs present in the parking lot, they are filled in each charging time interval ${\phi }_c$ up to

  $$\frac{{\sum }_{i=1}^n{x}_{i,0}}{n}+\frac{1}{n}{\int }_{t_0}^T{u}_{W_t}dt.$$
• Equal sharing (ES), which fills the n BEVs present in the parking lot at equal rates (until BEVs quit charging when they are full) over each charging time interval ${\phi }_c$.
• Mean Field Game (MFG), which fills the n BEVs present in the parking lot by using the inverse Nash algorithm in each charging time interval ${\phi }_c$ (see Algorithm 1 in Section 3.2).

4.4. Comparison of Charging Strategies Considering SOCs’ BEVs at Departure Times

First, we present in Figure 12 the results of the SOCs’ averages and standard deviations for the remaining BEVs in the parking lot knowing that the SoCs’ averages before the departures (i.e., before 3 p.m.) are all the same for all charging strategies in both days, as they use all solar energy in the parking lot. As expected, the FCFS strategy comes first with the highest SOCs’ averages and the lowest SOCs’ standard deviations for the remaining BEVs as its perfectly equalizes the BEVs’ SOCs regardless of their SOCs at the beginning of recharging, while the MFG strategy comes second as its also equalizes the BEVs’ SOCs by maintaining some fairness criterion (with regard to their SOCs at the beginning of recharging). The ES strategy, considered as the base case, comes third, while the FCFF is clearly the worst strategy with the lowest SOCs’ averages and the highest SOCs’ standard deviations as its fills up to maximum capacity the BEVs’ SOCs.

Then, we present in Figure 13 and Figure 14 the results of SOCs for the departing BEVs knowing that the average parking time is 8 h.

In both days, we note that the MFG strategy remains the best between the four options insofar as the resulting standard deviation for the departed BEVs are concerned.

In the sunny day case, the MFG strategy reduces significantly the standard deviation (${\sigma }_{x_{i,dep}}=0.026$, a reduction of $75\%$). The FCFF strategy gives the worst standard deviation (${\sigma }_{x_{i,dep}}=0.296$, an increase of $190\%$) with $12\%$ of users who have not recharged their batteries at all. The FCFS strategy is the improved version of the FCFF, as we can see that the BEVs are recharged with a better standard deviation (${\sigma }_{x_{i,dep}}=0.062$, a reduction of $39\%$). The ES strategy results in a slight increase of $16\%$ of the standard deviation (${\sigma }_{x_{i,dep}}=0.118$) due to the fluctuating population of BEVs and to some BEVs completely filling up.

In the cloudy day case, the MFG strategy also reduces significantly the standard deviation (${\sigma }_{x_{i,dep}}=0.024$, a reduction of $76\%$) due to the fluctuating population of BEVs. The FCFF strategy gives the worst standard deviation (${\sigma }_{x_{i,dep}}=0.457$, an increase of $348\%$) with $54\%$ of users who have not recharged their batteries at all. The FCFS strategy is the improved version of the FCFF, as we can see that the BEVs are recharged with a better standard deviation (${\sigma }_{x_{i,dep}}=0.053$, a reduction of $48\%$). The ES strategy results in a slight reduction of $21\%$ of the standard deviation (${\sigma }_{x_{i,dep}}=0.081$) due to the fluctuating population of BEVs.

In

Table 3. Comparison of fairness coefficient and departing SOCs ($min{x}_{i,arr}=0$, $max{x}_{i,arr}=0.45$) for a fluctuating population of 400 BEVs.

		Sunny Day			Cloudy Day
	$min{\mathit{x}}_{\mathit{i},\mathit{dep}}$	$max{\mathit{x}}_{\mathit{i},\mathit{dep}}$	FC	$min{\mathit{x}}_{\mathit{i},\mathit{dep}}$	$max{\mathit{x}}_{\mathit{i},\mathit{dep}}$	FC
ES	$0.50$	1	$-0.12$	$0.32$	$0.77$	$0.19$
FCFF	0	1	$-0.83$	0	1	$-2.78$
FCFS	$0.79$	$0.98$	$0.18$	$0.42$	$0.58$	$0.19$
MFG	$0.85$	$0.94$	$0.37$	$0.41$	$0.62$	$0.61$

the fairness coefficient (FC), as defined previously in Section 2.2.1, is evaluated for each charging strategy.

The results show that the MFG strategy remains the fairest strategy in both days while the FCFF strategy is rejected in both days and the ES strategy is rejected in the sunny day case.

5. Conclusions and Future Research

We have considered the situation of a large daytime work parking lot with homogeneous battery electric vehicles (BEVs) for simplicity, and solar sources based electricity charging. We have used realistic data to implement deterministic daily solar power curves with photovoltaic panels in a parking lot for a typical sunny day and a typical cloudy day. One should note that a large heterogeneous population of BEVs can be analyzed by assuming that it is possible to group the BEVs into classes considered homogeneous. Thus, all the BEVs of a class share the same physical parameters and, in order to better redistribute energy according to individual BEV needs, the forecast solar energy can be distributed by favoring a class with more BEVs, a larger-size battery and a lower charger efficiency.

In Section 3, a fair, and decentralized MFG strategy, for recharging a large fixed population of BEVs, has been developed. The goal was to reduce significantly the SOCs’ standard deviation while elevating the SOCs of BEVs to a satisfactory level regardless of their SOCs upon arrival. A comparison was carried out with an equal sharing (ES) strategy and a first come first full (FCFF) strategy which we saw could result in some unsatisfied individual users with little SOCs at the end of recharging. In Section 4, we considered a large fluctuating population of homogeneous BEVs. This new situation allowed us to improve the FCFF strategy into first come first serve (FCFS) strategy. The results showed that the MFG strategy remains the most desirable charging strategy with regards to the standard deviation of SOCs upon departure and fairness criterion. Finally, we did much better than the literature [20] ($60\%$ as maximum reduction of SOCs’ standard deviation in the case of a fluctuating population of BEVs) as we illustrated in the summary

Table 4. Standard deviation increase/reduction of SOCs’ BEVs.

	400  Fixed	BEVs  in a	400  Fluctuating	BEVs  in a
	Sunny Day	Cloudy Day	Sunny Day	Cloudy Day
FCFF	$\uparrow 241\%$	$\uparrow 318\%$	$\uparrow 151\%$	$\uparrow 464\%$
FCFS	$\uparrow 0\%$	$\uparrow 0\%$	$\downarrow 47\%$	$\downarrow 35\%$
MFG	$\downarrow 84\%$	$\downarrow 40\%$	$\downarrow 78\%$	$\downarrow 70\%$

below when we compared our results to the base case which is here the ES strategy.

In future research, we shall extend this work by considering stochastic solar acquisition in the parking lot and explore MFG based algorithms this time for potential partial restitution of battery energy from solar charged BEVs to the grid during evening peak hours.