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
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
of BEVs which cannot leave their homes on time while it is
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
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 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.
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] (
as maximum reduction of SOCs’ standard deviation in the case of a fluctuating population of BEVs) as we illustrated in the summary
Table 4 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.