## 1 Introduction  
The transportation sector constitutes a primary source of greenhouse gas emission and energy consumption. In the United States, the transportation sector is responsible for 30% of the CO\({}_{2}\) emission [1] and one third of the energy consumption [2]. The energy shortage, coupled with the problem of environmental pollution, has brought worldwide attention to electric vehicles (EVs) which brings great environmental and social benefits with zero tailpipe emissions and a lower level of noise pollution. Many countries have already formulated relevant policies to encourage research into EVs and promote their use [3, 4]. For example, countries like France, the Netherlands, and Germany have announced plans to ban the sale of fuel vehicles. [PERSON] et al. [5] proposed a modeling framework to optimize the design of incentive policies. [PERSON] et al. [6] proposed a framework for minimizing the social cost of replacing gas-powered vehicles with battery electric vehicles (BEVs). Additionally, the result shows that it takes 6-12 years for 80% of conventional vehicles to be replaced with EVs. However, the proportion of EVs in all cars still remains low and the major factors that influence the popularization of EVs are the limited travel range and the lack of charging infrastructure. Surveys show that the majority of customers would expect 300 miles of travel range [7] while, for the time being, EVs can only provide a range of 100 miles or so with a full battery. In order to make up for the limited travel range, to mitigate range anxiety, and to promote the penetration rate of EVs, it is of great importance to plan and build up more fast charging stations, taking into account the faint possibility of increasing the travel range in a short time [8]. Recent years have seen a growing number of research on the planning of charging infrastructures. The charging demand constitutes the basis for planning. In [9; 10; 11; 12], emphasis was laid on researching the charging demand and mobility of the EVs according to the uncertainty of traveling. In the classic studies that focus on the planning of charging stations, the charging demand is usually regarded as fixed [13; 14; 15; 16; 17; 18]. In [13], the considered charging demand appears at the destination. [PERSON] et al. [14] assumed that EV would be charged when it was going to run out of power. [PERSON] et al. [16] took centroids of census blocks as charging points, whereas [PERSON] et al. [18] placed charging stations at traffic nodes. This type of planning usually applies a clustering algorithm to the siting of charging stations [13; 14; 15] or minimizes the distance between charging points and charging stations [16; 17]. However, some researchers consider that it is inappropriate to regard charging demands as static and fixed [19], because this type of planning can hardly reflect the mobility of EVs and, actually, the majority of EV users would not make a special visit to charging stations. The flow-capturing location model (FCLM) is widely used to maximize the captured flows based on the assumption that the flow is captured when at least one charging station on the path [20; 21; 22]. The model was used in [23; 24; 25; 19; 26]. Among them, [PERSON] et al. [23] considered the stochastic characteristics of travel patterns based on the classic FCLM, and [PERSON] et al. [24] proposed a multi-objective model to minimize the overall annual cost and simultaneously maximize the captured traffic flow. In [23; 25], an extended version of the FCLM is employed, which is referred to as the flow-refueling location model (FRLM), is employed. The model considers that a combination of two or more facilities may be needed to refuel a flow [27]. However, FCLM and FRLM both ignored the flexibility of users, which indicated that users might accept slightly longer trips to charge their vehicles. [PERSON] and [PERSON] [28] proposed a model called the multipath refueling location model (MPRLM) based on FRLM.