## 3 Coupling Relationship between Charging Stations and Travel Route
### Feasibility Analysis of Trip Chains  
OD is a potential trip chain with a distance of \(d\) and \(R\) is the maximum travel range of EVs. It can be divided into two possible scenarios:  
1. When \(d\leq R\) which indicates that the traveling distance is smaller than the maximum travel range of EVs, charging is not necessary, thus the trip is a success; and
2. When \(d>R\) which indicates that the traveling distance exceeds the maximum travel range of EVs, charging is necessary and we need to the judge whether the trip is a success and analyze which charging station(s) the user needs to charge at. Here we analyze Scenario 2 while considering the range anxiety problem. When the SOC is lower than 20%, the user will be imperative to charge. It is assumed that a linear relationship exists between the maximum travel range and SOC [43; 44]. The mileage when anxiety begins is set as \(R^{\prime}\) which could be calculated via Equation (6):  
\[R^{\prime}=(1-20\%)R, \tag{6}\]  
We obtain D\({}_{i}\), which satisfies the condition \(d_{OD_{i}}>R\), which means that when the EV cannot reach D\({}_{i}\) within its travel range, it shall be charged during the range of OD\({}_{i}\). The principle of simulating the process of users heading for charging stations to charge is as follows:  
Figure 6: A journey with four potential routes. Figure 7: A set of potential trip chains. 1. If there is no charging station found in the range of (0,\(R\)), the user cannot reach the next destination. The trip chain is deemed a failure. 2. If there are any charging stations found in the range of (\(R^{\prime}\),\(R\)), we will opt for the charging station closer to the start point. As the battery runs at a lower level, the user will become anxious and choose to charge immediately. 3. If there are any charging stations found in the range of (0,\(R^{\prime}\)), we will choose the charging station farthest from the starting point. As the vehicle could still provide sufficient power, the charging demand is not that urgent. The time cost in the charging station would be lower for the user and she/he could charge more if she/he charges at a low-power state, thus, leading to a higher probability of a successful trip. We assume that the user has successfully charged at charging station C\({}_{k}\) within the range of OD\({}_{i}\), then charging station C\({}_{k}\) is set as the start point O\({}^{\prime}\) and we continue to judge the feasibility of the remaining route; if that trip chain fails, we continue with the next one until all are judged. If all the potential trip chains cannot meet the demand of the journey, the journey is deemed as a failure, which indicates that the siting of the charging stations cannot satisfy the demand of the EV user. Figure 8 shows a closed-loop trip chain with two destinations. In this study, we convert the set of tortuous routes in the urban traffic network into straight ones for analysis. In the figure, the black dot represents the destination while the red dot indicates the charging station. The solid arc is the distance boundary corresponding to \(R\) with O/O\({}^{\prime}\) as the starting point, and the dotted arc is the distance boundary corresponding to \(R^{\prime}\) with O/O\({}^{\prime}\) as the starting point. As shown in Figure 8a, we search for the node which satisfies the condition \(d_{OD_{m}}>R\) and find node D\({}_{2}\), which means that the user needs fast charging in the range of OD\({}_{2}\). Then we search for a charging station and find \(R^{\prime}\leq d_{OC_{2}}\leq R\), which indicates that the user should charge the vehicle at the charging station C\({}_{2}\). Thus, C\({}_{2}\) is set as the new starting point O\({}^{\prime}\) as shown in Figure 8b. Repeat the procedures and we find that another fast charging is needed in O\({}^{\prime}\)D\({}_{3}\)(C\({}_{2}\)D\({}_{3}\)). Following the simulating principle, the user will charge at C\({}_{4}\), thus, turning C\({}_{4}\) into a new starting point O\({}^{\prime}\) as is shown in Figure 8c.