**1. Introduction**

The market share of new energy vehicles without fossil fuels has been increasing rapidly in recent years, especially electric vehicles (EVs), which provide better performance, higher efficiency, and zero emissions [1,2]. To promote the usage of EVs, governments around the world have successively issued a series of active policies to provide subsidies for EV purchases and to deploy public charging infrastructures at convenient locations [2,3]. However, the market acceptance of EVs seems to fluctuate with the decreasing subsidy. On the one hand, limited driving range, insufficient public charging infrastructures, and longer charging time are the main reasons for at-home charging. On the other hand, the deployment of EV charging stations may induce more trips [4,5], that is, demands between each origin to destination (OD) pair that respond to the available fast charging are elastic, so does the route choice. Therefore, it is especially important to study how to effectively and reasonably deploy EV public charging stations within the hybrid network of rapidly increasing EVs and dominated gasoline vehicles (GVs), which should help reduce the range anxiety of EV users and maximize the coverage rate of EVs.

According to previous studies, a variety of factors affect the location of charging stations, including, among others, preference and travel choice behavior of users [6,7], travel demand of users [8], information of the en-route energy consumption of EVs [9–12], information of the remained range of EVs [13,14], charging speed and demand of battery [7,15], road environment [5,16], land

supply [17], elastic charging demands [18]. However, it is not practical to build a complicated location model and to solve the location problem by incorporating so many macro and micro factors into the planning and decision making of EV charging stations. This paper mainly studies the uncertainties due to the choice behavior of EV and GV users under elastic demands and uncertain path constraints. A bilevel model is proposed in this paper, where the upper model deals with a maximum flow-covering (MFC) problem, while the lower model is a stochastic user equilibrium model with elastic demands (SUE-ED). It is worth noting that factors, such as road congestion, charging speed, range limit of EVs, and capacity of charging stations, are already considered in users' travel choices. The lower model considers the elastic demands and the distance constraints of EVs, which have a significant influence on route choice and the distribution of link flow. Finally, a generalized Lagrangian function is constructed to prove the equivalence between the stochastic user equilibrium model and the elastic demand model.

The remainder of this paper is structured as follows: Section 2 reviews the relevant literature. Section 3 affirms the problem hypothesis, analyzes three charging paths of the EVs, and gives the symbolic description used in the article. Section 4 establishes a bilevel model. Sections 5 and 6 introduce the algorithms used to solve the model and numerical analysis applying to the Nguyen–Dupuis network, respectively. Finally, the research results and future research directions are discussed in Section 7.

## **2. Literature Review**

The conventional and dominant three types of location optimization models include the point demand model [19,20], the flow demand model [20–23], and the multi-objective optimization model [24–27]. The P-Median model assumes that "charging demand is generated in the road network node" and is widely used as one of the point demand models. The flow demand model is formulated on the basis of the Flow Capturing Location Model (FCLM). For the first time, some researches proposed the Flow Refueling Location Model (FRLM), where mileage limitations were explicitly considered in facility location issues [28–31]. One branch of FRLM aims to maximize the demand coverage by locating a fixed number of charging facilities, which is called the maximum coverage location problem. Although the multi-objective optimization models have the advantage of addressing more complicated experimental requirements, they are not good at dealing with the uncertainty planning problem [32].

In a hybrid network with both EVs and GVs, the layout of EV charging stations and the distribution of EV flow affect mutually. Elastic demands in a hybrid network result in many more uncertainties. Most of the previous studies used the User Equilibrium Model (UE) to locate charging facilities. In these studies, Xu et al. dealt with the user equilibrium problem in a hybrid transport network with battery switching stations and road grade constraints [33]. Jing et al. gave a comprehensive discussion of the equilibrium network model [34]. Jiang et al. introduced the path distance constraints into the UE model [13]. Zheng et al. proposed a bilevel model where the upper layer minimized travel costs, and the lower layer aimed to find the path-constrained EVs equalization flow [35]. The bounded rationality of EV users led to much more complicated energy consumption [7,36,37], and, therefore, route choice behavior reflected more unobserved heterogeneity, which resulted in various elastic demands.

The network design problem with elastic demands, where the induced or transferred OD demand is the subject of responses in traveler itinerary choices to enjoy the improvement of new infrastructure, have several formulations with various motivations. Ge et al. [38] is one of the early attempts to consider both the proportion of EVs and the charge rate of EVs when determining the elastic charging demands from the total number of vehicles on road connections. The elastic demand was formulated based either on feedback of congested travel and congested station on route choices [39] or on the assumption that charge demand between OD pairs follows a nonlinear inverse cost function without considering the pre-generating paths and charging combinations [40].

It should be noted that it is hard to consider all these constraints simultaneously, say, the elastic demand of the road network, the capacity of the charging stations, and the range limit of EVs by using the SUE-ED model in location problems of public EV charging stations. In this study, a novel

bilevel public charging station location model that combines a flow-capturing location model and a multi-objective optimization model is proposed.
