# A Convex Optimization Algorithm for Electricity Pricing of Charging Stations
## 1 Introduction  
With the popularization of traffic electrification, the number of electric vehicles (EVs) is increasing year by year. It is estimated that the number of EVs will reach 11 million in 2025 and 30 million in 2030 [1]. In order to satisfy the charging demand of EVs, a large number of charging stations (CSs) have been built in the distribution network. Considering the charging behavior of EV closely related to people's daily work and rest, EV users usually charge after work in the evening. In addition, the operation of the distribution network will be severely impacted by a large number of charging loads [2]. Therefore, the electric energy plan of CSs must be optimized to ensure the orderly charging of EVs [3]. Electric vehicle users are guided to change charging time by CSs setting charging prices [4]. The relationship among electric vehicle users, charging station operators, and distribution network operators (DSO), needs to be taken into consideration. However, existing algorithms are inefficient in solving this problem [5]. Therefore, it is very meaningful to develop the optimization algorithm for electric power planning of charging stations.  
For the time being, the research on charging price of EVs mainly focuses on time-of-use price [6]. In [7; 8; 9], the distribution locational marginal pricing (DLMP) method is applied to optimize the charging price of EVs to reduce the impact of charging load on distribution network. In [10; 11; 12], the electricity demand and travel characteristics of EV users are considered, and charging price is optimized by theprice demand elasticity coefficient. In [13], a Stackelberg leadership game model is applied to optimize the electricity prices for encouraging EV users to participate in the electricity market, improving their own interests, and promoting the development of smart energy systems. However, the problem of power planning for charging stations is a multi-objective mixed integer nonlinear programming (MINLP) and some constraints are nonconvex. Heuristic algorithms are usually used to solve this problem such as genetic algorithm (GA) in [14] and particle swarm optimization (PSO) in [15], which take a long time to solve and easily fall into local optimum.  
Aiming at getting the global optimal solution quickly, the main contribution of this paper is to propose a convex optimization algorithm for electricity pricing of charging stations. Firstly, the model is transformed into a convex optimization problem by second-order conic relaxation and Karush-Kuhn-Tucker optimality conditions. Secondly, a polyhedral approximation method is applied to construct a mixed integer linear programming, which can be solved quickly by branch and bound method. Finally, the model is solved iteratively to obtain the Pareto front according to the scalarization basic theorem.  
On the one hand, the optimal solution can be obtained in the strict sense provided with a convex optimization problem. The conic problem is approximated to a simple type based on polyhedral approximation in order to improve the speed of solution. In addition, the tripartite game problem is decomposed into a Stackelberg leadership game and a cooperative game. Besides, the multi-solution property of the multi-objective optimization problem is preserved under the scaling method.  
The remainder of the paper is organized as follows. Section 2 introduces the model of charging station electric power planning. Section 3 proposes the convex optimization algorithm based on second order conic relaxation (SOCR) and polyhedral approximation method. Simulations and analysis are carried out in Section 4. Finally, a conclusion is drawn in Section 5.