## 7 Case Study
### Simulation Results Considering Only Fast Charging  
Based on the travel pattern model, we obtain 113 sets of traveling data of users whose travel range exceeds the maximum travel range of EVs. In the upper model, the population size of the algorithm is set to be 100, and the iteration times is 50. The algorithm is run independently 10 times and the set with the highest success ratio is the feasible solution. Accordingly, we obtain nine sets of feasible solutions with a success ratio of 85.84%. Later, we determine the optimal routs and optimal charging strategy for each user according to Sections 3.3 and 5.1. Moreover, we adopt the hybrid method to determine the capacity of the charging stations. The charging stations are rated according to the sizing results. The nine planning schemes are shown in Table 5, and the locations of these charging stations in the corresponding power grid are shown in Figure 14. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Station Level** & **Fixed Investment Cost/\(\times\)10\({}^{3}\)\$ & **Minimum Number of Chargers** \\ \hline
1 & 1061 & 45 \\
2 & 800 & 30 \\
3 & 477 & 15 \\
4 & 323 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Land rental cost and time cost of users in different areas. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(p\) & 4 & - \\ \(n_{\text{cmax}}\) & 2 & - \\ \(P_{eq}\) & 15 [25] & kWh \\ \(P_{\text{slow}}\) & 3.5 [23] & kW \\ \(\omega\) & 15 [23] & kWh/100 km \\ \(\eta\) & 0.9 [46] & - \\ \(R\) & 100 & km \\ \(\lambda\) & 0.1 & - \\ \(d_{min}\) & 10 & km \\ \(P_{ch}\) & 96 [47] & kW \\ \(s\) & 30 [48] & m\({}^{2}\) \\ \(C_{ap}\) & 100 & - \\ \(C_{com}\) & 208.33 [17] & \$/kW \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The study parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Area Type** & **Residential Area** & **Industrial Area** & **Commercial Area** \\ \hline Land rental cost (\(\$/\text{m}^{2}\)) [40] & 330 & 109 & 1070 \\ Time cost (\(\$/\text{h}\)) & 5.76 [49] & 5.25 & 3.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The siting results of the lower model. Based on the locations in the power grid and the capacity of charging stations, we check the grid constraints and find that only solutions 3-7 can meet the power grid constraints. After the constraint test, we are able to work out the total social cost \(C\) and the satisfaction index \(r\) of the five feasible solutions via Equations (20)-(23), The result is shown in Figure 15, and the detailed total social cost is shown in Figure 16.  
Figure 14: The locations of potential charging stations in the power grid. Figure 15: The total social cost and satisfaction index. From Figure 15, we can see that the fifth solution has an overwhelming economic benefit and social benefit. Accordingly, the fifth planning scheme is chosen as the final planning scheme. It is worth noting that there is no Pareto curve in Figure 15, and we regard the fact that input solutions were screened in the upper model and the number of input solutions is limited as the primary reasons. Additionally, the two indicators have no stationary relationship and a multi-objective model is necessary to obtain the optimal solution. Furthermore, the situation demonstrates the good performance of the fifth solution. Although the contradiction between users and charging stations has been considered in the form of the waiting cost in the total social cost, from Figure 16, we notice that the waiting cost of users has little influence on the total social cost while the land rental cost is an essential part of the total social cost. The siting result is shown in Figure 17, and we can know that the charging stations are all placed in the residential area considering the cheap land rental cost. The locations of charging stations in the power grid is shown in Figure 18.  
Figure 16: The detailed total social cost for each solution. Figure 17: The siting results.