Electric vehicle dispatching in smart charging station: a pricing strategy

With the rapid development of electric vehicles (EVs), intelligent charging stations will surely become the focus of development. However, the disordered charging of electric vehicles will cause unbalanced resource utilization. Therefore, it is necessary to study the scheduling problem of electric vehicles in charging stations. In this paper, we propose a new electric vehicle dispatching method to realize balanced utilization of resources and high economic benefits. We model the multi-charging station pricing strategy problem as a non-cooperative continuous strategy game with an integrated objective combining charging cost and charging station capacity. Then the existence of Nash equilibrium is proved, and we combining particle swarm optimization with iterative search method to solve the equilibrium point. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.


Introduction
As the world is facing the problems of environmental pollution [1] and energy crisis [2], the development of green economy is very important. On the one hand, the use of traditional fuel vehicles will produce a large number of harmful exhaust gases and pollute the environment; on the other hand, it intensifies the dependence on non-renewable petroleum resources. Compared with traditional vehicles, electric vehicles cannot only reduce pollutant emissions, but also have better controllability, stability and safety, so electric vehicles have great development value and market potential.
With the development of electric vehicles, it is inevitable to study the charging problem of electric vehicles. Electric vehicle charging has great randomness in time and space, which will have a great impact on the operation management and control scheduling of smart grids. From the perspective of smart grid applications, literature [3][4][5] respectively proposes a new routing protocol for smart grid applications based on wireless sensor networks to improve the reliable, efficient and intelligent operation of the grid. They also propose a new distributed routing protocol based on channel perception in literature [6], which improves the detection reliability and reduces the noise and congested spectrum bands. Literature [7] proposes a multi-objective optimization method to minimize the energy costs and emissions of residential micro-grids. For the proposed optimization method, the augmented  -constraints and fuzzy decision-maker techniques are used to solve the problem.
As the basic charging facility for electric vehicles, electric vehicle charging stations (EVCS) play an important role in the development of electric vehicles. Literature [8] studies the location of charging stations. The authors in [8] establish a location model to minimize the total social cost based on genetic algorithm, and construct an evaluation index system under five location influencing factors.  [9][10][11] study from the perspective of the benefits of charging stations. The authors in [9] design an optimal dynamic resource allocation scheme for electric vehicle parking lots. This scheme optimizes the cost of the parking lot owners' Time of Use Demand Response program, while ensuring the interests of EVs owners. The authors in [10] propose a new economic-based EVCS queuing model. And the charging scheduling strategy that maximizes the long-term profit of charging station owners while minimizing the average delay of electric vehicles is studied. Then they consider the uncertainty of price and number of cars in [11], and establish the required model based on the stochastic optimization method. A long-term sustained profit algorithm for charging station owners is proposed to maximize the long-term profit of charging station owners.
Different from literature [10][11], we consider the real-time benefits of charging station owners, and we take economic benefits and balanced utilization of resources as the integrated optimization goal. In addition, the above literature only consider a single decision maker, and don't consider the competitive relationship among multiple players. We use game theory to study the competitive relationship among multiple charging stations. Besides, we combine iterative search and particle swarm optimization [12] to give a solution algorithm. According to the real-time pricing strategy of charging stations, the charging scheduling of electric vehicles is carried out, and the balanced utilization of charging resources is realized. The main contributions of this paper can be summarized as follows:  A new charging scheduling criterion for electric vehicles based on charging cost is defined.  In this paper, a smart charging station with strong network communication ability, calculation ability and decision-making ability is considered. The game model of real-time competitive pricing strategy for multiple charging stations is constructed, and the solution algorithm is given.
Notations: Z denotes the set of integers and N  the positive integers. R , R  is the set of real numbers and positive real numbers respectively.      represents a downward rounding operation.

Preliminaries
Traditional charging stations rely too much on manpower, and their scheduling is slow and difficult. While the intelligent charging station considered in this paper has strong computing ability and networking communication ability. We study the pricing strategy of charging stations in a certain area to realize the rational utilization of resources.

Charging costs of electric vehicles.
The charging cost of an EV can be divided into time cost and expense cost.
, and the waiting time is the product of the queuing vehicles of each charging pile and the charging time, i.e.

Remark 2.1:
As this paper studies the allocation of electric vehicles in a certain area, the influence of distance travel time on the selection of charging stations can be ignored. Besides, it is assumed that the charging power of all charging piles in each charging station is equal, each car needs to be

Main problem.
In order to describe the main problem, we define an EVs allocation criterion based on expense cost.
denotes the total cost of electric vehicle to charging station i .
For electric vehicles, the total cost needs to be considered when choosing a charging station. However, for charging stations, the real income for charging stations is only the service fee. Assuming that there is a competitive relationship among charging stations in the region, each charging station is rational and selfish. Therefore, each charging station aims to maximize its own interests, that is, to maximize the total service fee. Obviously, if a charging station aims to maximize its total service charge i f W , it needs to choose the unit price as large as possible. However, an increase in the unit price of service fee will lead to a decrease in the number of electric cars it receives. Furthermore, the number of electric vehicles allocated by each charging station is not only affected by the unit price of its own service fee, but also affected by other charging stations' price. So the influence of other charging stations' pricing strategies should be considered when they choose pricing strategies.
From the previous discussion, it can be seen that each smart charging station has a limited number of charging piles. In order to realize the rational utilization of resources, the restriction of charging station capacity should be considered in the decision-making process. Assuming that the ideal capacity of electric vehicles in each charging station is i d n , now this problem can be described as follows.  p . The current problem is how to determine the optimal pricing strategy for each charging station in the interactive decision-making process of multiple players.

Non-cooperative game of smart charging station
According to the previous analysis, there is an interactive decision-making process among multiple charging stations, and the game theory method can well solve this problem. In this section, we will model the pricing decision problem as a multi-player non-cooperative game based on Problem 2.2. The details are discussed as follows.
Define a non-cooperative game which is composed of three tuples: , , G I A P  . 1)Player: is the set of the players, that is, all charging stations. 2)Action:  is a compact set denoting the actions available to player i . At every decision moment, each charging station chooses the price is the payoff of player i under all players' actions A .

Assumption 3.1:
Game G is a static non-cooperative game with complete information, which means that every player knows the triplet G , besides, all players are assumed to be rational and the rationality is common knowledge [13].
In this game, every player will choose the price in set i In order to prove the existence of the pure strategy Nash equilibrium point, it is necessary to explain the relationship between the target revenue of the charging station i P and the pricing of the service fee i f p . We can simply make other irrelevant variables constant, and obtain the functional relationship as shown in Figure 1.  Besides, we know that each charging station has the same mathematical model, and the action set is a compact and convex set. So according to Figure 1 and Theorem 3.1, it can be proved that the noncooperative game between multiple charging stations has pure strategy Nash equilibrium.
According to Table 1, we first use the Algorithm 1 based on particle swarm optimization (PSO) for simulation. After 2.030 seconds and 17 iterations, the equilibrium pricing strategy and the number of cars allocated by each charging station can be obtained. The results are shown in Table 2, and the iterative process is shown in Figure 2. It should be noted that the equilibrium solution is related to the selection of the initial value. Then we use simulated annealing (SA) algorithm [15]  particle swarm algorithm in Algorithm 1 for comparison. After 7.903 seconds and 11 iterations we can get the equilibrium strategy set. The iterative process is shown in Figure 3.
Compare the number of cars allocated by each charging station in Table 2 with the ideal remaining capacity in Table 1, the maximum difference between the two is 0.7175 and the minimum difference is only 0.2706, it is obvious that the two are very close. That is, under the pricing strategy obtained by Algorithm 1, the balanced utilization of the resources of each charging station is realized.   It can be seen from Figure 2 and Figure 3 that the charging station pricing under the two algorithms can quickly converge to the equilibrium pricing policy after a limited number of iterations. This proves the effectiveness of our scheduling strategy. But the simulation time of PSO algorithm is shorter, for example, when we increase the number of charging stations to 10 and the number of cars to be charged to 100, the price can still converge to the equilibrium solution at a fast speed. This is very important in large-scale scenes.

Conclusions
Aiming at the electric vehicle scheduling problem of smart charging stations in an area, this paper simplifies other unnecessary factors, proposes a new optimization problem. We first set up an optimization model with an integrated objective combining charging cost and charging station capacity. Then, the competitive relationship among multiple charging stations is modeled as a non-cooperative game. Next, we present an algorithm combining particle swarm optimization (PSO) with iterative search to solve the equilibrium point. Finally, numerical simulations are used to verify the effectiveness and efficiency of our proposed scheme. Future work includes studying the interactive decision-making between charging stations and users.