A Charging Station Service Pricing Method Based on Hierarchical Game

Abstract

The present invention discloses a charging station service pricing method based on hierarchical game theory. First, a game model for electric vehicle charging and charging station pricing is constructed based on hierarchical Stackelberg game, which includes the game characteristics that the pricing of charging stations by charging operation companies affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the pricing of charging stations. Then, the charging cost optimization problem of electric vehicles is defined according to their optimization objectives, and the revenue optimization problem of charging operation companies is defined according to their optimization objectives. After that, the convergence of the problem is analyzed based on optimization theory, and the optimization problems of electric vehicles and operation companies are solved based on optimization algorithms. Finally, the obtained pricing results are used as the pricing strategies of charging operation companies for the charging stations they operate. This method can greatly improve the revenue of charging operation companies by considering the rational decisions of electric vehicle charging.

Description

Technical Field

[0001] The present invention relates to the field of charging station pricing in a smart city, and particularly to a charging station service pricing method based on hierarchical game. Background Art

[0002] In recent years, with the development of fast charging technology, the electric vehicle industry has witnessed rapid growth. As an environmentally friendly alternative to traditional fuel-driven vehicles, electric vehicles have great advantages and potential in reducing carbon dioxide emissions and alleviating the energy crisis. Promoting the use of electric vehicles is regarded as a very promising solution to the global catastrophic energy crisis and environmental pollution. Therefore, the electric vehicle-related industry has attracted increasing attention from the public, policymakers, and academia. With the introduction of some policies by government departments and the demand for sustainable economic and social development, it can be predicted that the number of electric vehicles will continue to grow in the next few years. The popularization of electric vehicles requires the support of public charging infrastructure for electric vehicles. Over the years, as the number of electric vehicles has been increasing, the number of charging stations has also been increasing accordingly. Based on the above background, more and more charging operation companies (such as Teld, Xingxing, State Grid, etc.) have started to deploy charging stations in cities and operate their own charging businesses to achieve commercial profits.

[0003] The participants in the charging operation market include various charging operation companies and electric vehicles in need of charging. For charging operation companies, there is commercial competition among different companies, and each company wants to maximize its own revenue by operating charging stations to provide charging services for electric vehicles. For operating companies, how to price to incentivize electric vehicles to charge at their own charging stations is an important decision-making issue. Price differences will lead to different numbers of vehicles charging at different charging stations, thus affecting the revenue of charging stations. Therefore, there is a mutually restrictive relationship between the prices of charging stations of different companies. Whether it is geographical location or commercial competition, essentially what affects is the number of vehicles charging at the charging station, which is also the most direct factor affecting the pricing of charging stations. For electric vehicles, each vehicle, as a rational and interest-driven individual, is distributed in various regions of the city. When a vehicle has a charging need, how the vehicle owner chooses a charging station to charge so that the total charging cost is the lowest is an important decision-making issue. The factors affecting the choice of charging station by electric vehicles include the pricing of the charging station, the distance to the charging station, and the queuing situation at the charging station. For these three influencing factors, the pricing factor and the distance factor are only determined by the current pricing of the charging station and the current geographical location of the electric vehicle, so they are not affected by other electric vehicles; while the queuing factor is determined by the number of vehicles choosing to charge at the same charging station, so it is affected by the decision-making of other electric vehicles in choosing charging stations.

[0004] The existing research on the charging station service pricing mainly includes two aspects, namely, the pricing strategy research for optimizing the charging station revenue and the pricing strategy research for optimizing the social benefits. However, the existing work mainly processes at the granularity of a single charging station and does not consider whether the total revenue of multiple charging stations under the same company is maximized. Generally, an operating company will operate multiple charging stations, and optimizing the pricing for multiple charging stations simultaneously is more challenging but more practically significant than optimizing the pricing for a single charging station. In addition, although these works consider the strategies of individual electric vehicles, the number of electric vehicles processed is very small, which is inconsistent with the actual urban environment. Therefore, it is crucial to study the revenue optimization of a company operating multiple charging stations in a large-scale urban environment. Summary of the Invention

[0005] In view of the above problems existing in the prior art, the present invention proposes a charging station service pricing method based on hierarchical game. In the smart city environment, on the basis of considering the rational charging decisions of electric vehicles, it focuses on solving the revenue optimization problem of the charging operation company. The present invention introduces the idea of hierarchical game and focuses on maximizing the total revenue of multiple charging stations controlled by a company. The present invention models the electric vehicle charging and charging station pricing problems into a hierarchical Stackelberg game, where the company is the leader and the vehicle flows in different regions are the followers. The present invention combines these two problems into a mathematical programming problem with equilibrium constraints and uses the corresponding optimization algorithm to solve this problem. The obtained pricing result can be used as the pricing strategy of the operating company for the charging station to improve the total revenue of the operating company.

[0006] To achieve the object of the present invention, the technical solution adopted by the present invention is: a charging station service pricing method based on hierarchical game, the method comprising the following steps:

[0007] (1) Establish a charging station service pricing model based on hierarchical game, and the model reflects the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the charging station pricing. The model also depicts the charging cost optimization problem of electric vehicles and the revenue optimization problem of the charging operation company;

[0008] (2) The charging cost optimization problem of electric vehicles depicted in the model is the lower-level optimization problem. Analyze the characteristics of the solution of this problem based on the optimization theory, specifically including whether there is an optimal solution for a single vehicle object, and whether there is an equilibrium solution when multiple vehicle objects compete simultaneously, and whether it is unique;

[0009] (3) The revenue optimization problem of the operating company in the model is the upper-level optimization problem. When solving it, the equilibrium solution of the charging cost optimization problem of electric vehicles needs to be used as its constraint condition, so as to combine the charging cost optimization problem of electric vehicles and the revenue optimization problem of the operating company into an optimization problem aiming at optimizing the revenue of the charging operating company, and analyze the existence of the solution of this optimization problem;

[0010] (4) Design an optimization algorithm to solve the revenue optimization problem of the charging operating company, and use the obtained pricing result as the pricing strategy of the charging stations of the operating company, and apply this pricing strategy to increase the revenue.

[0011] Furthermore, in step (1), the charging station service pricing model based on hierarchical game is described as follows:

[0012] In the model, the city is divided into n regions, and each region has N i electric vehicles that need to be charged, where i ∈ {1, 2,..., n}. In addition, there are m charging stations in the city distributed in different places, and the subscript of the charging station is represented by j, where j ∈ {1, 2,..., m}. These charging stations are operated by l companies, and each company operates H s charging stations, where s ∈ {1, 2,..., l}. The pricing of these charging stations is set by the charging operating company, and the revenue of each company is the sum of the revenues of all the charging stations it operates.

[0013] Furthermore, in step (1), the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the charging station pricing are described as follows:

[0014] In the hierarchical game model, the operating company can dynamically adjust the pricing of the charging stations it operates by predicting the charging behaviors of all electric vehicles in different regions, so as to optimize its own revenue. After knowing the pricing of each charging station, electric vehicles in different regions can adjust their charging decisions and form a traffic flow to different charging stations. The factors affecting the charging decisions of vehicles include the pricing p j of the charging station, the distance d ij from region i to charging station j, and the queuing cost q j of the vehicle at the charging station, where i ∈ {1, 2,..., n}, j ∈ {1, 2,..., m}. The queuing cost q jIt is determined by the number of vehicles charging at the same charging station and is thus affected by the decisions of other electric vehicles to choose charging stations. All electric vehicles want to complete charging at the lowest charging cost. We can consider all the electric vehicles in each area as a whole and optimize the charging efficiency of this whole while ensuring that the charging needs of all electric vehicles are met, thereby reducing the charging cost.

[0015] Furthermore, in step (1), the charging cost optimization problem of the electric vehicle is described as follows:

[0016] The charging costs of all electric vehicles in the same area are affected by the charging station pricing p j , the distance d to the charging station ij and the queuing cost q at the charging station. j Therefore, the charging cost function of the overall vehicle flow from area i to charging station j is as follows:

[0017] C ij (f ij ) = (ω 1 p j + ω 2 q j + ω 3 d ij )f ij

[0018] where f ij represents the number of vehicles from area i to charging station j, 0 ≤ f ij ≤ N i , ω 1 , ω 2 and ω 3 correspond to the weights of the three costs respectively. Considering that the vehicles charging at the charging station change dynamically over time, the number of vehicles at the charging station may change when the vehicle flow is heading towards the charging station. The queuing cost is defined as where f j represents the total number of vehicles from different areas going to charging station j for charging, The charging cost function of the overall vehicle flow in area i is as follows:

[0019]

[0020] For the vehicle flows in different areas, the optimization goal is to minimize the total charging cost. Therefore, the charging cost optimization problem of the total vehicle flow in the area can be defined as follows:

[0021]

[0022]

[0023] Furthermore, in step (1), the revenue optimization problem of the charging operation company is described as follows:

[0024] For each charging station j, its revenue function is as follows:

[0025] V j (p j )=(p j -ε j )f j

[0026] where ε j represents the operation and maintenance cost of charging station j. Additionally, since a company operates H s charging stations, where 0 < H s < m, the revenue function of this company is the sum of the revenues of these H s charging stations, as follows:

[0027]

[0028] The revenue optimization problem of the operation company can be expressed as follows:

[0029]

[0030] s.t. ε j < p j ≤ p max

[0031] where p max represents the highest price that can be set for the charging station. To make the charging station profitable, the price of the charging station should not be less than the maintenance cost. Therefore, ε j < p j .

[0032] Furthermore, in step (2), based on optimization theory, the characteristics of the solution to this problem are analyzed. The equilibrium solution of the lower-layer charging cost optimization problem is specifically described as follows:

[0033] Each area is an entity. Due to the existence of queuing costs, the traffic flow decision in each area is affected by the decisions of other areas. Therefore, the solution to the lower-layer charging cost optimization problem of the total traffic flow in each area can be characterized by the equilibrium state. In the equilibrium state, the traffic flow in no area will reduce the charging cost by changing the strategy of driving towards the charging station, which is specifically expressed as follows:

[0034]

[0035] where represents the optimal charging strategy for electric vehicles in area i, f -iDenote the charging strategy of electric vehicles in other regions except region \(i\), and \(p\) represents the pricing strategy of all charging stations, \(p = [p 1 , p 2 , \cdots, p m T . In the state of the equilibrium solution, if the electric vehicles in other regions do not change their charging strategies, then no electric vehicle in any region can reduce the charging cost by changing the charging strategy.

[0036] Furthermore, in step (2), the existence and uniqueness of the equilibrium solution of the lower-layer charging cost optimization problem are described as follows:

[0037] The objective function of the charging cost optimization problem of the lower-layer vehicle flow is a continuous and second-order differentiable function with respect to the optimization variable \(f i . Its equality constraints and inequality constraint conditions are all convex constraints. Therefore, the feasible region of this problem is a non-empty convex set. It can be proved that the Hessian matrix of the objective function of this problem is a positive definite matrix. Therefore, it is a strictly convex function. Therefore, there exists a unique equilibrium solution for the vehicle flow game between different lower-layer regions.

[0038] Furthermore, in step (3), the equilibrium solution of the electric vehicle charging cost optimization problem is used as the constraint condition for the revenue optimization problem of the upper-layer operation company, which is specifically described as follows:

[0039] Since the solution of the charging cost optimization problem of the lower layer as a whole for each region will affect the pricing problem of the upper-layer charging stations, the optimization problem of the lower layer can be regarded as a sub-problem and substituted into the constraint conditions of the upper-layer problem to form a mathematical programming problem with equilibrium constraints (MPEC), and its specific form is as follows:

[0040]

[0041]

[0042] This problem can be further transformed into the following form based on the KKT conditions:

[0043]

[0044]

[0045] where \(L i \) is the Lagrangian function of the optimization problem of the \(i\)-th region, which is expressed as follows:

[0046]

[0047] where \(v i = [v i1 , v​i2 , …, v im T and λ i are the Lagrange multipliers corresponding to the constraint conditions.

[0048] Furthermore, in step (3), the existence of the solution to the above MPEC problem is analyzed, and the specific description is as follows:

[0049] The feasible region of the decision variables of the upper-layer operating company's revenue optimization problem is a continuous and bounded convex set. The objective function of the operating company is affected by the pricing strategy p and the equilibrium solution f of the lower-layer charging cost optimization problem * For each charging station, when the charging price is very high, electric vehicles may not tend to choose this charging station due to the high price. On the contrary, when the charging price is relatively low, although many electric vehicles may be attracted to charge due to the low price, the total revenue may also be low because of the low unit charge. Therefore, it can be determined that the optimal charging pricing strategy p * always exists. In addition, since there is always a unique equilibrium solution in the lower-layer game, we can infer that there is always an equilibrium in this hierarchical game, that is, there is always an optimal solution to this MPEC problem.

[0050] Furthermore, in step (4), an optimization algorithm is designed to solve the revenue optimization problem of the charging operating company, and the specific description is as follows:

[0051] Since the complementary constraint conditions in the original MPEC problem are difficult to handle, the original MPEC problem can be transformed into a series of continuous and smooth problems, and this series of smooth problems can converge to the solution of the original MPEC problem. Among them, each smooth problem can be expressed as P(μ), as follows:

[0052]

[0053]

[0054] where z ∈ R m*n is an auxiliary variable. It can be seen that the complementary conditions in the original MPEC problem are replaced by the last two constraints in P(μ). When μ = 0, the last two constraints in P(μ) can become two cases: (1) v ij = 0, and (2) f ij = 0, and These two cases exactly correspond to the complementary conditions in the original MPEC problem. P(μ) is a good smooth problem and can be solved using standard optimization tools. When μ approaches 0, the original MPEC problem will converge to a stable result. ​

[0055] Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0056] (1) It is beneficial to expand to large-scale urban environments. Traditional decision-making research for single vehicles often only applies to small-scale situations. Considering that the present invention focuses on the pricing decision of charging stations, it can consider vehicle decisions at a relatively coarse granularity. Therefore, it divides the city into regions, simulates a more realistic urban environment, and speeds up the calculation speed of the model.

[0057] (2) The model is more perfect. Existing models do not consider the competition relationship between electric vehicles or the collaborative cooperation relationship between charging stations under the same company. The present invention simultaneously considers the game process in which the pricing of charging stations of the operating company affects electric vehicle decisions, electric vehicle decisions affect each other, and electric vehicle decisions affect charging station pricing, and also considers the cooperation between charging stations under the same company, which is closer to the actual situation and makes the model more perfect.

[0058] (3) It is beneficial to improve the revenue of charging operation companies. On the one hand, the present invention takes into account the rational decision-making of minimizing charging costs when electric vehicles are charging; on the other hand, the present invention can optimize the charging service revenue of operating companies based on the rational decisions of electric vehicles, which has practical significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Implementation framework of charging station service pricing mechanism;

[0060] Figure 2 Schematic diagram of a model of a charging station service pricing method based on hierarchical game implemented by the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0061] The following further clarifies the present invention in conjunction with the accompanying drawings and specific implementation examples.

[0062] The charging station service pricing method based on hierarchical game described in the present invention simulates the game process in which the pricing of charging stations of the operating company affects electric vehicle decisions, electric vehicle decisions affect each other, and electric vehicle decisions affect charging station pricing by constructing a hierarchical Stackelberg game model. As Figure 1 shown, first, based on the optimization objective of the company, define the revenue function and optimization problem of the company, then based on the optimization objective of the vehicle, define the charging cost function and charging cost optimization problem of the vehicle, and then conduct a convergence analysis on the problem, and design an optimization algorithm to analyze and solve the problem. In addition, the schematic diagram of the model of a charging station service pricing method based on hierarchical game proposed by the present invention is as Figure 2 shown. The specific implementation steps are as follows:

[0063] (1) Establish a charging station service pricing model based on hierarchical game, which reflects the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the charging station pricing. The model also depicts the charging cost optimization problem of electric vehicles and the revenue optimization problem of the charging operation company;

[0064] (2) The charging cost optimization problem of electric vehicles depicted in the model is the lower-level optimization problem. Analyze the characteristics of the solution to this problem based on optimization theory, specifically including whether the optimal solution of a single vehicle object exists, and whether the equilibrium solution exists and is unique when multiple vehicle objects compete simultaneously;

[0065] (3) The revenue optimization problem of the operating company in the model is the upper-level optimization problem. When solving it, the equilibrium solution of the charging cost optimization problem of electric vehicles needs to be used as its constraint condition, so as to combine the charging cost optimization problem of electric vehicles and the revenue optimization problem of the operating company into an optimization problem aiming to optimize the revenue of the charging operation company, and analyze the existence of the solution to this optimization problem;

[0066] (4) Design an optimization algorithm to solve the revenue optimization problem of the charging operation company, and use the obtained pricing result as the pricing strategy of the charging stations of the operating company, and apply this pricing strategy to increase the revenue.

[0067] Furthermore, in step (1), the charging station service pricing model based on hierarchical game is described as follows:

[0068] In the model, the city is divided into n regions, and each region has N i electric vehicles that need to be charged, where i ∈ {1, 2,..., n}. In addition, there are m charging stations in the city distributed in different places, and the subscript of the charging station is represented by j, where j ∈ {1, 2,..., m}. These charging stations are operated by l companies, and each company operates H s charging stations, where s ∈ {1, 2,..., l}. The pricing of these charging stations is set by the charging operation company, and the revenue of each company is the sum of the revenues of all the charging stations it operates.

[0069] Furthermore, in step (1), the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the charging station pricing are described as follows:

[0070] In a hierarchical game model, the operating company can dynamically adjust the pricing of the charging stations it operates by predicting the charging behaviors of all electric vehicles in different regions, so as to optimize its own revenue. After knowing the pricing of each charging station, electric vehicles in different regions can adjust their charging decisions and form a traffic flow to different charging stations. The factors affecting the charging decisions of vehicles include the pricing p j of the charging station, the distance d ij from region i to charging station j, and the queuing cost q j of the vehicle at the charging station, where i ∈ {1, 2, …, n} and j ∈ {1, 2, …, m}. The queuing cost q j is determined by the number of vehicles choosing to charge at the same charging station and is thus affected by the decisions of other electric vehicles to choose charging stations. All electric vehicles want to complete charging with the minimum charging cost. We can regard all the electric vehicles in each region as a whole and optimize the overall charging benefit and reduce the charging cost on the premise that the charging demands of all electric vehicles are met.

[0071] Furthermore, in step (1), the optimization problem of the charging cost of the electric vehicle is described as follows:

[0072] The charging costs of all electric vehicles in the same region are affected by the pricing p j of the charging station, the distance d ij to the charging station, and the queuing cost q j at the charging station. Therefore, the overall charging cost function of the traffic flow from region i to charging station j for charging is as follows:

[0073] C ij (f ij ) = (ω 1 p j + ω 2 q j + ω 3 d ij )f ij

[0074] where f ij represents the number of vehicles from the ith region to the jth charging station, 0 ≤ f ij ≤ N i , ω 1 , ω 2 and ω 3 correspond to the weights of the three costs respectively. Considering that the vehicles charging at the charging station change dynamically over time, the number of vehicles at the charging station may change when the traffic flow is heading towards the charging station. The queuing cost is defined as where f j represents the total number of vehicles from different regions going to charging station j for charging. The charging cost function of the overall vehicle flow in the $i$-th area is as follows:

[0075]

[0076] For the vehicle flows in different areas, the optimization objective is to minimize the overall charging cost. Therefore, the charging cost optimization problem of the total vehicle flow in the area can be defined as follows:

[0077]

[0078]

[0079] Furthermore, in step (1), the revenue optimization problem of the charging operation company is described as follows:

[0080] For each charging station $j$, its revenue function is as follows:

[0081] V j (p j )=(p j -ε j )f j

[0082] where $\varepsilon$ j represents the operation and maintenance cost of charging station $j$. In addition, since a company operates $H$ s charging stations, where $0 < H$ s < m, the revenue function of this company is the sum of the revenues of these $H$ s charging stations, as follows:

[0083]

[0084] The revenue optimization problem of the operation company can be expressed as follows:

[0085]

[0086] s.t. $\varepsilon$ j < p j ≤ p max

[0087] where $p$ max represents the highest price that the charging station can set. To make the charging station profitable, the price of the charging station should not be less than the maintenance cost. Therefore, $\varepsilon$ j < p j .

[0088] Furthermore, in step (2), based on the optimization theory, the characteristics of the solution to this problem are analyzed. The equilibrium solution of the lower-layer charging cost optimization problem is specifically described as follows:

[0089] Each area is an entity. Due to the existence of queuing costs, the traffic flow decision in each area is affected by the decisions of other areas. Therefore, the solution to the charging cost optimization problem of the total traffic flow in the lower-level areas can be characterized by the equilibrium state. In the equilibrium state, the traffic flow in no area will reduce its charging cost by changing the strategy of driving towards the charging station, which is specifically expressed as follows:

[0090]

[0091] where represents the optimal charging strategy of electric vehicles in area i, f -i represents the charging strategies of electric vehicles in other areas except area i, and p represents the pricing strategies of all charging stations, p = [p 1 , p 2 , … p m T . In the state of the equilibrium solution, if the electric vehicles in other areas do not change their charging strategies, then the electric vehicles in no area can reduce their charging costs by changing the charging strategy.

[0092] Furthermore, the existence and uniqueness of the equilibrium solution of the charging cost optimization problem in step (2) are described as follows:

[0093] The objective function of the charging cost optimization problem of the lower-level traffic flow is a continuous and twice differentiable function with respect to the optimization variable f i . Its equality constraint and inequality constraint conditions are all convex constraints. Therefore, the feasible region of this problem is a non-empty convex set. It can be proved that the Hessian matrix of the objective function of this problem is a positive definite matrix. Therefore, it is a strictly convex function. Therefore, there exists a unique equilibrium solution for the traffic flow game among different lower-level areas.

[0094] Furthermore, in step (3), the equilibrium solution of the charging cost optimization problem of electric vehicles is used as the constraint condition for the revenue optimization problem of the upper-level operating company, which is specifically described as follows:

[0095] Since the solution to the charging cost optimization problem of the lower-level areas as a whole will affect the pricing problem of the upper-level charging stations, the lower-level optimization problem can be regarded as a sub-problem and substituted into the constraint conditions of the upper-level problem to form a mathematical programming problem with equilibrium constraints (MPEC), and its specific form is as follows:

[0096]

[0097]

[0098] This problem can be further transformed into the following form based on the KKT conditions:​

[0099]

[0100]

[0101] where L i is the Lagrangian function of the optimization problem in the \(i\)-th region, which is expressed as follows:

[0102]

[0103] where \(v\) i \( = [v\) i1 , \(v\) i2 , \(\cdots\), \(v\) im T and \(\lambda\) i are the Lagrange multipliers corresponding to the constraint conditions.

[0104] Furthermore, in step (3), the existence of the solution to the above MPEC problem is analyzed, which is specifically described as follows:

[0105] The feasible region of the decision variables of the revenue optimization problem of the upper - layer operating company is a continuous and bounded convex set. The objective function of the operating company is affected by the pricing strategy \(p\) and the equilibrium solution \(f\) of the lower - layer charging cost optimization problem * . For each charging station, when the charging price is very high, electric vehicles may not tend to choose this charging station due to the high price. On the contrary, when the charging price is low, although many electric vehicles may be attracted to charge due to the low price, the total revenue may also be low because of the low unit charge. Therefore, it can be determined that the optimal charging pricing strategy \(p\) * always exists. In addition, since there is always a unique equilibrium solution in the lower - layer game, we can infer that there is always an equilibrium in this hierarchical game, that is, there is always an optimal solution to this MPEC problem.

[0106] Furthermore, in step (4), an optimization algorithm is designed to solve the revenue optimization problem of the charging operating company, which is specifically described as follows:

[0107] Since the complementary constraint conditions in the original MPEC problem are difficult to handle, the original MPEC problem can be transformed into a series of continuous and smooth problems, and this series of smooth problems can converge to the solution of the original MPEC problem. Among them, each smooth problem can be expressed as \(P(\mu)\), as follows:

[0108]

[0109]

[0110] where \(z\in R\) m*n ​is a auxiliary variable. It can be seen that the complementarity conditions in the original MPEC problem are replaced by the last two constraints in P(μ). When μ = 0, the last two constraints in P(μ) can become two cases: (1) v ij = 0, and (2) f ij = 0, and These two cases exactly correspond to the complementarity conditions in the original MPEC problem. P(μ) is a well-behaved smooth problem that can be solved using standard optimization tools. When μ approaches 0, the original MPEC problem will converge to a stable result.

Claims

1. A charging station service pricing method based on hierarchical game, characterized in that, the method comprises the following steps: (1) Establish a charging station service pricing model based on hierarchical game, which reflects the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the pricing of charging stations. The model also depicts the charging cost optimization problem of electric vehicles and the revenue optimization problem of the charging operation company; (2) The charging cost optimization problem of electric vehicles depicted in the model is the lower-level optimization problem. Analyze the characteristics of the solution to this problem based on optimization theory, specifically including whether the optimal solution of a single vehicle object exists, and whether the equilibrium solution exists and is unique when multiple vehicle objects compete simultaneously; (3) The revenue optimization problem of the operating company in the model is the upper-level optimization problem. When solving it, the equilibrium solution of the charging cost optimization problem of electric vehicles needs to be used as its constraint condition, so as to combine the charging cost optimization problem of electric vehicles and the revenue optimization problem of the operating company into an optimization problem aiming to optimize the revenue of the charging operation company, and analyze the existence of the solution to this optimization problem; (4) Design an optimization algorithm to solve the revenue optimization problem of the charging operation company, and use the obtained pricing result as the pricing strategy of the charging stations of the operating company, and apply this pricing strategy to increase revenue; The charging station service pricing model based on hierarchical game is described as follows: In the model, the city is divided into n regions, and each region has N i cars that need to be charged, where i ∈ {1, 2, …, n}. In addition, there are m charging stations distributed in different places in the city, and the subscript of the charging station is represented by j, where j ∈ {1, 2, …, m}. These charging stations are operated by l companies, and each company operates H s charging stations, where s ∈ {1, 2, …, l}. The pricing of these charging stations is set by the charging operation companies, and the revenue of each company is the sum of the revenues of all the charging stations it operates; In step (1), the game characteristics that the pricing of the charging stations of the operating company affects the decisions of electric vehicles, the decisions of electric vehicles affect each other, and the decisions of electric vehicles affect the pricing of charging stations are described as follows: In the hierarchical game model, the operating company can dynamically adjust the pricing of the charging stations it operates by predicting the charging behaviors of all electric vehicles in different regions, so as to optimize its own revenue. After knowing the pricing of each charging station, electric vehicles in different regions can adjust their charging decisions, forming a traffic flow to different charging stations. The factors affecting the charging decisions of vehicles include the pricing p j of the charging station, the distance d ij from region i to charging station j, and the queuing cost q j of the vehicle at the charging station, where i ∈ {1, 2, …, n}, j ∈ {1, 2, …, m}, and the queuing cost q j is determined by the number of vehicles choosing to charge at the same charging station and is thus affected by the decisions of other electric vehicles to choose charging stations. All electric vehicles want to complete charging at the lowest charging cost. All electric vehicles in each region can be regarded as a whole. On the premise of ensuring that the charging needs of all electric vehicles are met, optimize the charging benefit of this whole and reduce the charging cost; In step (1), the charging cost optimization problem of electric vehicles is described as follows: The charging costs of all electric vehicles in the same area are affected by the charging station pricing p j , the distance d to the charging station ij , and the queuing cost q at the charging station j . Therefore, the charging cost function of the overall vehicle flow from area i to charging station j is as follows: C ij (f ij ) = (ω 1 p j + ω 2 q j + ω 3 d ij )f ij where f ij represents the number of vehicles from the i-th area to the j-th charging station, 0 ≤ f ij ≤ N i , ω 1 , ω 2 and ω 3 correspond to the weights of three types of costs respectively. Considering that the vehicles charging at the charging station change dynamically over time and the number of vehicles at the charging station may change when the vehicle flow approaches the charging station, the queuing cost is defined as where f j represents the total number of all vehicles going to charging station j for charging from different areas, The charging cost function of the overall vehicle flow in the i-th area is shown as follows: For the vehicle flows in different regions, its optimization goal is to minimize the overall charging cost. Therefore, the charging cost optimization problem of the total vehicle flow in the region can be defined as follows: In step (1), the revenue optimization problem of the charging operation company is described as follows: For each charging station j, its revenue function is as follows: V j (p j ) = (p j -ε j )f j where ε j represents the operation and maintenance cost of charging station j. In addition, since a company operates H s charging stations, where 0 < H s < m, the revenue function of this company is the sum of the revenues of these H s charging stations, as shown below: The revenue optimization problem of the operating company can be expressed as follows: where p max represents the highest price that can be set for the charging station. To make the charging station profitable, the price of the charging station should not be less than the maintenance cost. Therefore, ε j < p j ; In step (2), analyze the characteristics of the solution to this problem based on optimization theory. The equilibrium solution of the lower-level charging cost optimization problem is specifically described as follows: Each region is an entity. Due to the existence of queuing costs, the decision-making of the vehicle flow in each region is affected by the decisions of other regions. Therefore, the solution to the charging cost optimization problem of the total vehicle flow in the lower-level region can be characterized by the equilibrium state. In the equilibrium state, the vehicle flow in no region will reduce the charging cost by changing the strategy of driving towards the charging station, which is specifically expressed as follows: Among them represents the optimal charging strategy for electric vehicles in area i f -i represents the charging strategies for electric vehicles in areas other than area i, and p represents the pricing strategy for all charging stations, p = [p 1 , p 2 , … p m T . In the state of the equilibrium solution, if the electric vehicles in other areas do not change their charging strategies, then the electric vehicles in no area can reduce their charging costs by changing their charging strategies​ In step (2), the existence and uniqueness characteristics of the equilibrium solution of the lower-level charging cost optimization problem are described as follows: The objective function of the charging cost optimization problem for the lower-layer vehicle flow is a continuous and twice differentiable function with respect to the optimization variable f i and its equality constraints and inequality constraints are all convex constraints. Therefore, the feasible region of this problem is a non-empty convex set. It can be proved that the Hessian matrix of the objective function of this problem is a positive definite matrix. Therefore, it is a strictly convex function. Therefore, there exists a unique equilibrium solution for the vehicle flow game between different regions in the lower layer; In step (3), use the equilibrium solution of the charging cost optimization problem of electric vehicles as the constraint condition of the upper-level revenue optimization problem of the operating company, which is specifically described as follows: Since the solution of the charging cost optimization problem for the lower layer as a whole region will affect the pricing problem of the upper layer charging stations, the optimization problem of the lower layer can be regarded as a sub-problem and brought into the constraint conditions of the upper layer problem to form a mathematical programming problem with equilibrium constraints MPEC, and the specific form is as follows: Based on the KKT conditions, this problem can be further transformed into the following form: where L i is the Lagrangian function of the optimization problem in the i-th region, expressed as follows: where ν i = [ν i1 , ν i2 , …, ν im T and λ i are the Lagrange multipliers corresponding to the constraint conditions;​ In step (3), the existence of the solution to the above MPEC problem is analyzed, and the specific description is as follows: The feasible region of the decision variables for the revenue optimization problem of the upper - layer operating company is a continuous and bounded convex set. The objective function of the operating company is affected by the pricing strategy \(p\) and the equilibrium solution \(f\) of the lower - layer charging cost optimization problem. * For each charging station, when the charging price is very high, electric vehicles may not tend to choose this charging station due to the high price. On the contrary, when the charging price is low, although it may attract many electric vehicles to charge because of the low price, the total revenue may also be low due to the low unit charge. Therefore, it can be determined that the optimal charging pricing strategy \(p\) * always exists. In addition, since there is always a unique equilibrium solution in the lower - layer game, we can infer that there is always an equilibrium in this hierarchical game, that is, there is always an optimal solution to this MPEC problem. In step (4), an optimization algorithm is designed to solve the revenue optimization problem of the charging operation company, and the specific description is as follows: Since the complementary constraint conditions in the original MPEC problem are difficult to handle, the original MPEC problem can be transformed into a series of continuous smooth problems, and this series of smooth problems can converge to the solution of the original MPEC problem. Among them, each smooth problem can be expressed as P(μ), as follows: where \(z\in R\) m*n is an auxiliary variable. It can be seen that the complementary conditions in the original MPEC problem are replaced by the last two constraints in \(P(\mu)\). When \(\mu = 0\), the last two constraints in \(P(\mu)\) can become two cases: (1) \(\nu\) ij = 0, and (2) \(f\) ij = 0, and These two cases exactly correspond to the complementary conditions in the original MPEC problem. \(P(\mu)\) is a well - behaved smooth problem and can be solved using standard optimization tools. When \(\mu\) approaches 0, the original MPEC problem will converge to a stable result.

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