The CSSS is closely related to its comprehensive strength. The evaluation index system for the comprehensive strength of CSs is shown in Figure 1, mainly including the contract capacity of CSs, the number of fast/slow charging piles, service fees, and charging prices.

In Figure 1, the contract capacity and the number of fast/slow charging piles are positive factors. The higher the value of these factors, the stronger the comprehensive strength C _n of CSs. It also indicates a stronger attraction to the surrounding demand area, where more users are willing to go to CS _n for charging, resulting in a larger service scope S _n . Conversely, the charging price and service fees are negative factors. The higher the value, the smaller the CSSS.

The evaluation indicators are standardized by the extreme value method as follows. The positive and negative indicators are standardized, as shown in Eqs 1, 2, respectively: T n , i = T n , i − T min ⁡ , i T max ⁡ , i − T min ⁡ , i , (1) T n , i = T min ⁡ , i − T n , i T max ⁡ , i − T min ⁡ , i , (2) where T _n.i is the original value of index i of CS _n and T _max.i and T _min.i represent the maximum and minimum values of index i, respectively.

The contribution of CS _n is calculated with index i, as shown in Eq. 3: p n , i = T n , i ∑ i = 1 N T n , i , (3) where p _n.i is the contribution of CS _n with index i and N is the number of CSs.

The entropy of index i is calculated as shown in Eq. 4: e i = − 1 ln ⁡ N ∑ i = 1 N p n , i ⁡ ln p n , i . (4)

The evaluation index weight of CSs could be obtained as shown in Eq. 5: γ i = g i ∑ i = 1 N g i , (5) g i = 1 − e i , (6) where γ _i is the weight of index i and g _i and e _i are the coefficients of difference and entropy, respectively.

Then, the comprehensive strength of CS _n could be quantified as shown in Eq. 7: C n = ∑ i = 1 5 c i γ i T n . i T max ⁡ . i , (7) where C _n represents the comprehensive strength of CS _n and c _i indicates the correlation coefficient. c _i = 1 when γ _i is positive, and c _i = −1 if γ _i is negative.

The formula of the point charge field strength is obtained by Eq. 8: E e = k e q e r e 2 , (8) where E _e represents the point charging field strength, k _e is the electrostatic force constant with the unit N·m²/C², q _e is the charge electric quantity, and r _e indicates the distance between the center and charge.

Analogous to the electrostatic field generated by a point charge, the attraction of CSs to the surrounding area will not change abruptly, in accordance with the characteristics of the typical non-rotating scattered field. According to the Helmholtz theorem, the function of the non-rotating scattered field is always inversely proportional to the distance vector R ². Therefore, the SFS of the CSSS, akin to the electric field strength of a point charge, is defined to quantify the attraction strength of CSs to the surrounding charging demand areas (CDAs).

It is assumed that a certain area is divided into K small grid areas with a certain precision, which is defined as CDAs. Taking CS _n as the center, SFS E _n.k of CS _n for CDA _k is defined considerating the average distance and the comprehensive strength, which is obtained by Eq. 9: E n . k = k c C n r n . k 2 , (9) where k _c is the service constant and r _n.k represents the distance between CS _n and CAD _k , which can be calculated by the longitude and latitude.

The Wilson model is appropriate for studying spatial interactions (Zhang et al., 2010). Thus, the charging service capacity ratio of CS _n to CDA _k can be described as shown in Eq. 10: T n . k = K C n D k e − β r n . k , (10) where K represents the normalization factor, describing the regional difference of the CSs and the demand area. K = 1 when the regional difference is ignored. D _k indicates the total charging demand in CDA _k . β is the attenuation factor, determining the speed of service attenuation.

Combined with the electrostatic force constant k _e , and assuming each CS has the same service constant, k _c is derived as follows: k c = K C a v e D a v e e − β r a v e r a v e 2 C a v e 2 , (11) β = 2 N t max S a v e _ n , (12) where C _ave is the average comprehensive strength of all CSs, D _ave represents the average demand of all demand areas, and r _ave is the average distance between the CSs and demand areas. Calculated using Statistical Product and Service Solutions (SPSS) software, t _max represents the number of CSs with a comprehensive score greater than 0. S _{ave_n} represents the average area of each CS served.

In the two-dimensional space, a facility or an enterprise is attractive to the users within a certain range of its surroundings, catering to their needs and preferences. Similarly, as the charging service provider, the CSs act as the central hub, offering charging service to electric vehicle users in their vicinity. Then, CSSS, defined as the geographical spatial distribution range of electric vehicle users who receive charging services from that CS, is usually of an irregular shape, as shown in Figure 2. Notably, the dotted line represents the SFS emitted by CSs, and the CSs are attractive to EVs at the same time.

Any CS _n will attract the CDAs, resulting in a CSSS that is constrained by the parameters of the surrounding CS and itself. According to the strongest occupation method proposed by Isard, if SFS E _n.k of CS _n to the demand area k is larger than SFS E _m.k of other CS _m , the demand area k is identified as the CSSS of CS _n , as shown in Eq. 13: S n = S E n . k ≥ E m . k , m = 1 , . . . , N m ≠ n , (13) where S _n represents the CSSS of CS _n and the demand area k is defined as the market demand area. Eq. 13 indicates that among all CSs, CS _n has the greatest attraction to the demand area k, and the demand area k belongs to the CSSS of CS _n .

In a scenario of a perfectly competitive market, it is assumed there are N CSs with different sizes, which belong to H different operators, H ≤ N. All charging loads are evenly distributed throughout the area; users will only choose one of the N CSs to charge when needed.

A multi-objective real-time pricing model is established considering the profit rate and power deviation rate. The objective is optimized by maximizing the total profit rate of all CSs and minimizing the power deviation rate, as shown in Eq. 17. max ⁡ E = ⁡ max ω n E p _ l r + ω b c E p _ p c , (17) E p _ l r = ∑ t = 1 T ∑ n = 1 N P n . t − P e + P n . w P r . t , (18) E p _ p c = ∑ t = 1 T Δ P t − ∑ n = 1 N q n . t − q e v . t Δ P t , (19a) Δ P t = P W . t + P L . t − P w . t − P l . t , (20a) where E _{p_lr} and E _{p_pc} represent the profit rates of the CS and the reduced power deviation rate, respectively. ω _n and ω _ab represent the weights of the above two, respectively. P _e and P _n.w are the electricity purchasing cost and the operation and maintenance cost of the CS, respectively. ΔP _t is the power deviation between the real-time output of renewable energy and the day-ahead forecast. P _W.t and P _w.t are the real-time and day-ahead wind power output, respectively. P _L.t and P _l.t are the real-time and day-ahead photovoltaic outputs, respectively. If ΔP _t >0, indicating that the real-time output of wind power and photovoltaic at time t is greater than the predicted output, it is necessary to guide users to charge and increase the charging load to absorb the excess wind and photovoltaic power. Otherwise, if ΔP _t <0, indicating that the real-time output of wind and photovoltaic power at time t is less than the day-ahead predicted output, users need to be encouraged to reduce the charging load.

The constraints of optimization model are as follows:

Market demand constraint 0 < q n . t < q e v . t . (21)

Eq. 21 implies that the charging demand of the CS does not exceed the predicted charging load.

Charging load constraint ξ l q e v . t ≤ ∑ n = 1 N q n . t ≤ ξ h q e v . t , (19b) where ξ _l and ξ _h represent the lower and upper limit coefficients, respectively, which means that the sum of the charging demand should not exceed a certain range of the day-ahead charging load.

Charging price constraint P r . n 1 − φ l ≤ P n . t ≤ P r . n 1 + φ h , (20b) where φ _l and φ _h represent lower and upper limits of the charging price, respectively.

The basic idea of a real-time pricing optimization model based on CSSS is shown in Figure 3. First, the initial CSSS was obtained by the strongest occupation method. Then, based on the pricing optimization model, the charging station operator obtains the real-time charging price of charging stations, which affects the initial CSSS, and further develops into the real-time CSSS.

The detailed steps of real-time pricing optimization based on CSSS are designed as follows:

Step 1: The CSSS is calculated

Furong District in Changsha is taken as an example for simulation, and the data on CSs are all obtained from Hunan Economic and Technological Research Institute of the State Grid. Notably, the output data of wind power and photovoltaic are for the total province. First, the map of Furong District is divided into 16,288 market areas with latitude and longitude accuracy of 0.0005 using ArcGIS drawing software. Then, the first three representative CSs are selected from Furong District for calculation as examples. The basic data of these three CSs are shown in Table 1. Based on the evaluation index system, the weight of each index of the CS is obtained by the entropy method. The weight of the contract capacity, the number of fast/slow-charging piles, the service fee, and the charging price are 0.221, 0.486, 0.201, 0.051, and 0.041 yuan/kWh, respectively. Afterward, SPSS is employed for the factor analysis. A total of 143 CSs have a comprehensive score greater than 0.

For the load of the three charging stations q _1,t , q _2,t , and q _3,t , its boundary is given by setting ξ _l = 0.8 and ξ _h = 1.2 in Eq. 19a and φ _l = φ _h = 0.1 in Eq. 20a. In addition, according to the related policy issued by the Comprehensive Department of the National Energy Administration, the electricity purchasing price in Hunan Province is 0.46244 yuan/kWh. The daily maintenance cost of each CS is P _1.w = 600 yuan, P _2.w = 550 yuan, and P _3.w = 500 yuan. Based on the project data, the day-ahead wind power and photovoltaic output data of a typical day are shown in Supplementary Figure AB1. It is assumed that the wind power and photovoltaic prediction error is 10%, and the real-time output meets the normal distribution.

Without considering the interaction of the charging price, the initial CSSS and occupancy of each CS are calculated based on the strongest occupation method, as shown in Figure 4.

As shown in Figure 4, the CSSS of CS₂ is the largest without considering the influence of the charging price. It is noted that CS₂ has the largest number of fast charging piles, and also, no other competing CS is present in the surrounding area. Thus, its CSSS occupies half of Furong District. As shown in Table 2, the comprehensive strength of CS₁ is the strongest. However, due to the remote location and the competition from CS₃, the CSSS of CS₁ only occupied 17% of the market share and is smallest among the three CSs. In addition, from the shape of the CSSS of CS₁ and CS₃, the boundary presents an inward curved arc, which indicates that the further away from the two CSs, the more area belongs to CS₁. That is, compared to CS₃, the distance from one point to CS₁ is larger, but it is still attracted by CS₁. Therefore, it can be concluded that the SFS of CS₁ is greater than that of CS₃. The average peak–valley–flat charging price of CSs is then calculated by the real-time pricing optimization model mentioned in Section 4, as shown in Table 2.

Table 2 shows that the charging prices of CS₁ and CS₂ are lower than the policy price, while the charging price of CS₃ is higher than the policy price. Due to the higher comprehensive strength of CS₁ and CS₂, the electric vehicle users will pay more attention and be more sensitive to price changes. Accordingly, the influence parameter of the policy price difference k _p.n in Eq. 16 is also larger, indicating that charging prices can increase the charging demand and profit if they are lower than the policy price. In comparison, the comprehensive strength of CS₃ is the smallest, and users pay the least attention to its price change. Thus, CS₃ will maintain its income by setting a higher price than the policy price, increasing by 0.0137 yuan/kWh during the peak period and 0.0235 yuan/kWh during the flat period.

Then, the change in CDAs is analyzed for the peak, valley, and flat periods, and three time points of 6:00, 10:00, and 18:00 are selected. The number of CDAs occupied by each CS at these time points is shown in Supplementary Figure AB2. It is observed that there is no significant change in the CSSS of CS₂ due to the small competitive pressure. On the contrary, the CSSS changed with the charging price mainly for CS₁ and CS₃. Therefore, the CSSS changes for CS₁ and CS₃ are discussed as primary. The number of CDAs occupied by CS₁ and CS₃ and their CP changes are shown in Figure 5.

Afterward, the real-time charging price is taken into consideration. Since charge station 1 has a higher comprehensive strength and a small charging price difference of 0.02 yuan, the market share increases from 17% in Figure 4 to 20% (the number of CDAs is 3,212 at 6:00). Correspondingly, CS₃, which is closer to CS₁, lost part of its CSSS due to its smaller comprehensive strength, and its market share decreased from 33% to 29%.

With the increase in the charging price, the number of CDAs occupied by CS₁ decreased by 245 from 6:00 to 18:00. At the same time, the number of CDAs occupied by CS₃ increased by 253. This can be attributed to the fact that CS₁ has stronger comprehensive strength, and users are more responsive to its charging price changes. Thus, as the charging price of charge station 1 increases, the number of users of CS₁ decreases faster than those of CS₃, resulting in a more significant change in the CSSS of CS₁.

It can be seen that the charging price and CSSS are affected by many factors, such as comprehensive strength, geographical location, and number of competitors around.

The impacts of the number of CSs on the CSSS, power deviation rate, and profit margin are analyzed to verify the relevant conclusions given in Section 5.2. Taking five CSs and eight CSs as examples, the data of CSs are shown in Table 1.