## 2 Methods
### Model Building
#### 2.2.2 Lower-Layer Planning Model  
The lower-layer planning model is chosen to construct a generalized minimum cost model from the personal perspective of electric vehicle users. Its goal is to minimize the cost of user travel. The generalized minimum travel cost from the perspective of electric vehicle users includes driving time, charging waiting time, charging time, and the electricity cost consumed to reach the destination. The objective function is  
\[\text{min}N=\sum\limits_{a\in A}\left[\left(t_{a}+\left(T_{c}+W_{qj}\right) \cdot\psi_{j}\cdot\varphi_{j}\right)\cdot U+P_{E}t_{a}\cdot\eta\right] \tag{4}\]  
Based on existing literature research, lower-level planning is approached from the user's perspective. When an electric vehicle travels at a constant speed, its energy consumption is the lowest, corresponding to the lowest electricity cost. When an electric vehicle is moving at a constant speed on a flat road, it travels at a constant speed on a level road. The two forces acting in the horizontal direction, namely, the traction force and the friction force, are balanced with each other; that is, the traction force is equal to the friction force. When the traction force remains constant, the power of the engine and the work carried out by the electric vehicle at the destination can be calculated based on the driving speed of the electric vehicle. The constraint conditions are as follows. 1 User arrival rate  
Due to insufficient charging station resources and long charging times for individual vehicles, electric vehicles may face queuing problems when arriving at charging stations, directly affecting service efficiency. Therefore, when establishing a charging station location model, it is advisable to avoid excessively long queuing times as much as possible. Assuming there is a charging station on section \(a\), and there are \(y\) independent paths from starting point \(O\) to point \(j\), and each path consists of g sections connected in series, the arrival probability calculation rule is as follows:  
\[\lambda_{j}=\left\{ \begin{array}{l}\frac{d_{1}\cdot\mathrm{H}}{C_{0}}\cdot x _{1},g=1\\ \left[\begin{array}{l}\sum\limits_{j=1}^{g}\frac{d_{g}\cdot\mathrm{H}}{C_{0} }+\left(1-\frac{d_{1}\cdot\mathrm{H}}{C_{0}}\right)\cdot\left(\frac{d_{1} \cdot\mathrm{H}}{C_{0}}+\frac{d_{2}\cdot\mathrm{H}}{C_{0}}\right)+\ldots\\ \sum\limits_{i=1}^{g}\left.+\left(1-\frac{d_{1}\cdot\mathrm{H}}{C_{0}}\right) \cdot\left(\frac{d_{1}\cdot\mathrm{H}}{C_{0}}+\frac{d_{2}\cdot\mathrm{H}}{C_{ 0}}+\ldots+\frac{d_{g}}{C_{0}}\right)\cdot x_{g}\end{array}\right],g\geq 2 \end{array} \right. \tag{5}\]  
In the queuing system formed by electric vehicles arriving at charging stations for charging, the pattern of electric vehicles arriving at charging stations can be represented by the Poisson distribution, generally following the principle of first come, first served. The queuing rule is a multi-service desk waiting system; service providers generally provide serial services. In summary, the electric vehicle charging queue system can be seen as a limited service desk queue system, namely, system \(M/M/b/K\). Among them, \(b\) is the number of charging stations within the charging station (due to the constraint of charging station definition, each station requires at least three charging devices). \(K\) indicates the maximum car capacity that the charging station space can accept. When the number of cars reaches \(K\), the system will shut down and vehicles that need to be charged will directly move to the next charging station. Therefore, if there is a true customer arrival rate \(\lambda_{ej}\) that is less than \(\lambda_{j}\), the effective arrival rate is  
\[\lambda_{ej}=\lambda_{j}(1-P_{K}) \tag{6}\]  
2 The average service capacity of charging stations per unit time  
\[\mu_{j}=\left\{ \begin{array}{l}\frac{n}{t_{j}},0\leq n<b_{j}\\ \frac{b_{j}}{t_{j}},b_{j}\leq n<K\end{array} \right. \tag{7}\]  
3 Service intensity of charging station queuing system  
\[