**1. Introduction**

Dependence on petroleum contributes to a serious environmental and energy problem. The transportation sector is one of the major economic industries that contribute to energy consumption and greenhouse gas emissions. The International Energy Agency found that the transportation sector contributes 28% of global energy consumption and 23% of global greenhouse gas emissions [1]. Electric vehicles (EVs), which are highly energy-efficient, are recognized as a promising solution to alleviate the problem of fossil fuel dependency and increasing greenhouse gas emissions, especially if the energy used for their charging is obtained from a renewable energy source [2]. However, unlike conventional internal combustion engine vehicles, EVs have a relatively short driving range because of their limited battery capacity, thus requiring drivers to recharge their vehicles often to reach their destinations. Insufficient charging infrastructure often causes difficulties in finding charging stations, thus resulting in the range anxiety of EV drivers [3]. Range anxiety can be effectively alleviated by providing guidance information for EV charging based on specific service platforms, such as a smart

charging service. Drivers could file charging demands to the charging service provider and receive recommended charging stations through their mobile devices [4]. Guidance strategies should be developed in consideration of charging demand information to determine the recommended charging stations and corresponding travel routes. The traffic condition on a road network also affects the route choice of EVs from departure points to charging stations, because the traffic volume would vary as time progresses in real-world situations, due to the factors such as rush hour, which is regarded as the time-varying characteristic for a road network [5]. Such a characteristic would influence vehicle driving state and should be considered in the route guidance strategies. The stochastic characteristic intrinsic to the charging demands substantially affects the strategies, thus further increasing the difficulty of dealing with charging demands. Large-scale charging behaviors with stochastic characteristics considerably affect the operation efficiency of charging stations. Therefore, given the widespread adoption of EVs, solving stochastic charging demands in complex real-world situations is a critical issue for the current and future global transportation system.

Directing EVs to suitable charging stations is an important and fundamental problem for the adoption of EVs in urban transportation systems. As introduced in Section 2, the traditional methods of route guidance for EVs mainly focus on the problems with deterministic charging demands in a static road network. Although several studies considered the impacts of a time-varying road network on the driving state of vehicles, less attention was paid to the stochastic characteristics intrinsic to charging demands. Therefore, a route guidance method that can be used to deal with stochastic charging demands and that takes into account the time-varying road network is expected. To fill the gap, this study develops route guidance strategies for stochastic charging demands in a time-varying road network from two different perspectives. The performance of the route guidance strategies is explored by considering their effects on the operation efficiency of charging stations. Both strategies can direct EVs with stochastic charging demands to reachable charging stations by considering the time-varying traffic conditions on the routes.

This study makes the following unique contributions: firstly, the stochastic characteristic of charging demands is investigated given the situation with large-scale adoption of EVs. A route guidance problem with stochastic charging demands is formulated by combining the time-varying road network. A dynamic recursive equation is developed to obtain the EV number in charging stations, which varies as time progresses. Secondly, to address stochastic charging demands, two route guidance strategies are established, and the operation efficiency of charging stations and travel cost of individual drivers are considered in the strategies. In actual situations, the operation efficiency of charging stations could be affected by the number of EVs in them, because the sustainable number of EVs for a charging station is limited and the queuing time is increased as vehicle number increases. Therefore, we use the number of EVs in charging stations as the matric to reflect the operation efficiency of charging stations. In addition, the travel cost of individual drivers is generally composed of travel time, energy consumption, and charging cost. These travel cost components are closely correlated with driving distance. Thus, the driving distance is employed to reflect the integration of travel cost components. The reachability of the selected charging stations can be ensured by both strategies in a time-varying road network. Lastly, the proposed strategies are applied in simulation examples to provide guidance for stochastic charging demands in a time-varying road network. The performance of the two strategies is compared in different simulation scenarios, and application recommendations in terms of the strategies are presented based on the simulation results.

The remaining portions of this paper are arranged as follows: in Section 2, the literature review is presented. In Section 3, the route guidance problem is formulated by considering the stochastic charging demands in a time-varying road network. In Section 4, the metrics regarding the charging station selection are analyzed from two different perspectives, and the route guidance strategies are presented. In Section 5, the simulated results are presented to compare the performance of the proposed strategies. Lastly, the conclusions and future studies are discussed in Section 6.

### **2. Literature Review**

EVs are a potential solution to environmental and energy problems because they have high energy efficiency and can be charged by using renewable energy source. Thus, the traveling and charging problems of EVs attracted increasing interest from the scientific community. Given the limited driving range of EVs, several studies attempted to find optimal routes for EVs on the basis of the framework of the constrained shortest path problem [6–8]. However, charging behavior was not involved in the methods. Considering the charging demands incurred by EV travels, Wang et al. used geometric approaches and designed an algorithm for route guidance by considering the charging demand information from drivers [4]. Driving direction and distance were used as choice indicators for optimal charging stations. Sweda et al. proposed two heuristic methods for making adaptive routing and recharging decisions for EVs [9]. Charging costs were involved in the solution. In addition to charging processes, Qin and Zhang and Said et al. considered the effects of queuing time on charging station selection [10,11]. Queuing theory was used to optimize the route guidance. Several studies combined driving time, charging time, and queuing time to discuss charging and route optimization for EVs [12–14]. Wang et al. incorporated energy constraints during travel and proposed an energy-aware routing model for EVs [15]. Cao et al. and Liu et al. considered the effects of charging costs on charging station selection to investigate EV charging problems [16,17]. Yagcitekin and Uzunoglu developed a smart route guidance strategy based on double-layer optimization theory [18]. Sun and Zhou compared the effects of different factors on route selection of EVs by using a cost-optimal algorithm [19]. The trade-off between traveling cost and time consumed was obtained to guide drivers in traveling and charging. Wang et al. integrated drivers' intended traveling and charging choices [20]. A multiobjective model was established to provide guidance for EV charging; the objectives include minimized traveling time, charging costs, and energy consumption. In view of the environmental effects for EV adoption, many studies aimed to search for energy-efficient routes for EVs under different situations [21–25]. However, the aforementioned methods for route guidance are mainly based on problems in a static road network, in which the time or energy consumed in each link is constant. Consequently, the effects of traffic condition on driving state are ignored, which makes the solution unrealistic in complex situations with respect to urban road networks.

In view of this, Alizadeh et al. incorporated time-varying traffic conditions in the traveling and charging problems for EVs to improve the accuracy of route guidance schemes [26]. An extended transportation graph was used to find the optimal routes. Yi and Bauer proposed a model to investigate the effects of traffic condition on energy cost of EVs [27]. The primal–dual interior point algorithm was used to construct the optimal paths. Zhang et al. proposed a multiobjective routing model for EV travel, in which the effects of traffic condition on travel time, driving distance, and energy consumption were considered [28]. The ant colony optimization algorithm was employed to search for optimal routes. Jafari and Boyles incorporated route reliability in the solution for an EV traveling problem under a road network with time-varying traffic condition [29]. Daina et al. explored the EV charging problem by considering uncertain traffic conditions on the basis of random utility theory [30]. The trade-off among driving distance, charging time, and costs for charging selection was analyzed. Huber and Bogenberger utilized real-time traffic information to investigate the time-varying characteristic of traffic conditions and their effects on the EV driving state [31]. Several works introduced network equilibrium theory to explore the optimization models for EV charging and traveling [32–35]. They modeled changing traffic conditions by changing the number of vehicles in each link. However, most existing methods assume that the charging demands of EV drivers are predetermined and overlook their stochastic characteristics. In real-world situations, charging demands with variable information may be made at different periods, and the charging service providers are unable to know the information before they receive the charging demands. Therefore, the previous methods are unable to solve stochastic charging demands in complex real-world situations.

Furthermore, to address stochastic charging demands, Hung and Michailidis proposed a route guidance strategy based on the queuing modeling framework, in which charging demands occur in accordance with a general process during a time period [36]. Nevertheless, the study did not consider the time-varying characteristic of road networks. EVs were assumed to operate with a constant speed in the road network. Energy consumption, which considerably influences the reachability of charging stations, was also ignored in the method. To our knowledge, few studies investigated the route guidance methods of EVs by comprehensively considering stochastic charging demands and a time-varying road network.

Overall, even though the previous studies made achievements in route guidance for EVs, there are still some limitations, as mentioned above. To further clarify the existing studies, we summarize the aforementioned references with respect to their considerations, as listed in Table 1 (considered factors are marked as "√"; otherwise, unconsidered factors are marked as "×"). In view of the limitations, two heuristic-based strategies for route guidance are proposed and introduced in the following sections, which aim to deal with stochastic charging demands in a time-varying road network.


**Table 1.** Previous studies with respect to their considerations.

### **3. Problem Description**

During trips, EV drivers often need to recharge their vehicles to reach their destinations, thus resulting in charging demands in a road network with EVs. The charging demand information is assumed to include drivers' travel destinations and the remaining energy of vehicles. Travel destination is one of the critical factors in drivers' travel demand. The remaining energy of EVs can be directly obtained through built-in vehicle dashboards. When EV drivers notice that the remaining energy of their vehicle may be insufficient to reach their destinations, they send the information of charging demands to the charging service provider by using their mobile devices. When receiving charging demands from EV drivers, the charging service provider needs to determine the suitable charging station for each charging demand according to the given information. Note that charging demands in real-world situations have uncertainty and variability by time. That is, the charging demands received in different periods may contain different information about travel destinations of drivers and the remaining energy of EVs. Specifically, both travel destinations and remaining energy have uncertainty from the perspective of the charging service provider, because they derive from the individual travel demands of drivers and operation state of EVs, respectively, thus preventing the charging service provider from predicting the detailed information of charging demands in advance. In situations with large-scale charging demands, multiple charging demands during identical periods often have different information on travel destinations and remaining energy. In actual situations, the charging service provider can obtain the information on EV and charging station locations by using positioning devices. EV drivers do not need to send the location information to the charging service provider, whether the charging demands occur or their detailed information is not predetermined at different periods. Thus, the charging demands in a road network have a significant stochastic characteristic as time progresses. To realize problem formulation, the time is discretized into finite time slots normalized to integral units. Let {1, ... , *t*, ... , *T*} denote the set of time slots, where *T* is the total number of the time slots. With the identical duration for each time slot, the time horizon increases as *T* increases. The charging demand that occurs in node *i* and at time slot *t* is denoted by *Ct i* , which is a binary variable in the problem formulation. Its value is equal to 1 if there exists a charging demand that occurs in normal node *i* (*i* = 1, ... , *m*) at time slot *t*; otherwise, it is equal to 0. For each charging demand

(*Ct <sup>i</sup>* = 1), its information includes the travel destination and remaining energy, which are denoted as *dt <sup>i</sup>* and *et i* , respectively. The travel destination *d<sup>t</sup> <sup>i</sup>* and remaining energy *<sup>e</sup><sup>t</sup> <sup>i</sup>* from different charging demands may vary. The charging service provider cannot understand or predict *C<sup>t</sup> <sup>i</sup>* before time slots *t*. Thus, the decision-making for all the charging demands needs to be determined based on the traffic condition at corresponding time slots.

The stochastic characteristic of charging demands is one of the challenges in situations with large-scale adoption of EVs. Traffic conditions in a road network also affect the driving speed and energy consumption of EVs, thus affecting their traveling and charging process [37]. Traffic conditions often have time-varying characteristic in real-world situations because of environmental factors [38]. Consequently, the energy and time that are spent while traversing the same links may vary at different time slots. The time-varying characteristic of a road network should be considered to improve the effectiveness of route guidance schemes. The road network structure and EV operating characteristics are combined, and the time-varying road network is defined as *G* = (*V*, *A*, τ*<sup>t</sup> <sup>a</sup>*, *Et <sup>a</sup>*), where *V* and *A* denote the sets of nodes and links, respectively. In set *V*, two types of nodes, namely, normal and charging station nodes, exist. The latter has the ability to charge EVs. For problem formulation, set *V* is assumed to consist of *m* normal nodes and *n* charging station nodes. τ*<sup>t</sup> <sup>a</sup>* and *E<sup>t</sup> <sup>a</sup>* in *G* denote the driving time and energy consumption on link *a* at time slot *t*, respectively, where *a* ∈ *A*. In every time slot, the values of τ*t <sup>a</sup>* and *E<sup>t</sup> <sup>a</sup>* randomly change within a reasonable range, which reflects the time-varying characteristic of the road network. The conventional optimization problems with time-varying networks often assume that the links remain stable for the duration of a time slot [39]. For the problem of route guidance for EV charging, the assumption signifies the constant driving time and energy consumption during a separate time slot. The assumption conforms to the traffic condition characteristic in an actual road network if the duration for each time slot is relatively short. Therefore, we follow such an assumption in the route guidance problem for EV charging.

The route guidance problem for EV charging is formulated by combining stochastic charging demands and time-varying road networks. The charging demands are assumed to occur in the normal nodes only. The charging station nodes could not generate charging demands. Such an assumption is reasonable, because drivers seek help from the charging service provider only when they have difficulty finding nearby charging stations. The guidance strategies aim to help EV drivers from normal nodes select heuristic suggested charging station nodes based on specific objectives. The binary decision variable in the problem formulation is denoted by *xt ij*, which is equal to 1 if the charging demand generated in normal node *i* (*i* = 1, ... , *m*) at time slot *t* is assigned to charging station node *j* (*j* = 1, ... , *n*); otherwise, this variable is 0. In every time slot, all the normal nodes in a road network have the potential to generate charging demands. However, whether the charging demands occur or not in every time slot is an uncertain event. In order to reflect such a characteristic of charging demand occurrence, the possibility of the charging demand occurring in node *i* at each time slot is denoted as λ*i*(0 ≤ λ*<sup>i</sup>* ≤ 1), as shown in Equation (1).

$$\begin{cases} \Pr(\mathcal{C}\_i^t = 1) = \lambda\_i\\ \Pr(\mathcal{C}\_i^t = 0) = 1 - \lambda\_i \end{cases}, t \in \{1, \dots, T\}, j \in \{1, \dots, m\}, \tag{1}$$

where Pr (Λ) represents the possibility of the occurrence of event Λ.

Note that, given the definition of charging demand occurrence as shown in Equation (1), we assume that each normal node is able to generate at most one charging demand following a specific possibility during a time slot. The assumption conforms to the characteristic of charging demand occurrence in actual situations if the duration for each time slot is relatively short. Moreover, to reduce the complexity of problem formulation, we assume that the charging demand occurrence for each node follows a uniform distribution. For example, the possibility of charging demand occurrence for node *i* has a uniform value λ*<sup>i</sup>* as time progresses, which does not have a time-varying characteristic and is influenced by the node location. Although node *i* has a constant possibility λ*<sup>i</sup>* for every time slot, the travel destination and remaining energy from the charging demands may vary at different time slots.

When addressing the charging demands at each time slot, the first step is to ensure that the remaining energy can enable the EVs to reach the target charging stations. In a time-varying road network, the energy consumption between normal and charging station nodes may vary at different time slots. Thus, before charging stations are selected, the energy consumption on the routes should be observed, and only the reachable charging stations can be considered as candidates, as shown in Figure 1.

**Figure 1.** Reachable and unreachable charging stations (CS).

In Figure 1, the energy consumptions between node 1 and CS 1 at time slots *t*<sup>1</sup> and *t*<sup>2</sup> are different. The green check mark indicates that the EV can traverse the route, and the red cross indicates that the EV cannot do so because it has insufficient remaining energy. The figure indicates that the reachability of the same charging station may vary at different time slots because of the time-varying traffic conditions on the road network. The driving time on the routes may also change at different time slots, thus determining the suitable time slots as the EVs reach charging stations. The charging service provider is assumed to know the information about the traffic conditions on all the links at the beginning of each time slot, acquiring such information through either real-time traffic information from the transport sector or short-term traffic flow prediction [40]. The basic framework for the stochastic route guidance problem in a time-varying road network is presented in Figure 2.

**Figure 2.** Stochastic route guidance problem in a time-varying road network.

In Figure 2, τ*<sup>t</sup>* (1,CS1) is the driving time from node 1 to charging station node CS 1 under the traffic condition at time slot *t*. The energy consumption between node 2 and charging station node CS 1 at time slot *t* is denoted by *Et* (2,CS1) . The charging demands that occur in nodes 1, 3, and *m* at time slot *t* are denoted as *Ct* <sup>1</sup>, *<sup>C</sup><sup>t</sup>* <sup>3</sup>, and *<sup>C</sup><sup>t</sup> <sup>m</sup>*, respectively. The objective of the problem is to provide guidance for every charging demand by considering the traffic condition at time slot *t*. The recommended

charging station nodes would be selected for the charging demands based on specific route guidance strategies. In the figure, the decision for charging station selection is denoted as *x<sup>t</sup>* (1,1) <sup>=</sup> 1, *xt* (3,2) = 1, and *x<sup>t</sup>* (*m*,n) <sup>=</sup> 1. For instance, *xt* (1,1) = 1 indicates that the charging demand that occurs in node 1 is assigned to the charging station node CS 1. How to determine the value of *xt ij* at each time slot *t* is the critical issue to solve the route guidance problem for EV charging in a time-varying road network. This issue should be considered from two aspects. Firstly, the route guidance strategies satisfy the charging demands of EV drivers; that is, an EV should be able to reach the selected charging station under its current remaining energy. For this reason, the relationship between remaining energy and traffic condition needs to be considered. Secondly, the charging behavior has significant effects on the operating state of charging stations, especially in a situation with large-scale adoption of EVs. In every time slot, multiple charging demands may occur in a road network, and the charging stations may have to accept multiple EVs. Given the limited charging rate, the number of EVs in a charging station increases as the time slots pass. However, mass EV charging significantly affects the operating state of charging stations; that is, it may prolong the queuing time and even present a potential burden on local power systems [41–43]. Therefore, aside from drivers' charging demands, the number of EVs at each charging station is another important factor that needs to be considered by route guidance strategies.

We attempt to develop a dynamic recursive equation based on the operation characteristics of charging stations to explore the change trend of EV number in charging stations under the situation with large-scale stochastic charging demands. *St <sup>j</sup>* denotes the number of EVs that complete charging and leave charging station *j* at time slot *t*. Without loss of generality, the problem assumes that at most one EV can leave a charging station after completing charging at each time slot. The assumption conforms to the actual operating situation in charging stations if the duration for each time slot is relatively short. The possibility of the event that an EV leaves the charging station *j* after completing charging at each time slot is denoted as μ*j*(0 ≤ μ*<sup>j</sup>* ≤ 1), as shown in Equation (2).

$$\begin{cases} \Pr(S\_j^t = 1) = \mu\_j\\ \Pr(S\_j^t = 0) = 1 - \mu\_j \end{cases}, t \in \{1, \dots, T\}, j \in \{1, \dots, n\}. \tag{2}$$

The parameter μ*<sup>j</sup>* can reflect the charging levels of the chargers in charging station *j*. During the actual charging processes, the chargers with different charging levels have different charging rates for EVs [44]. Under the definition of *S<sup>t</sup> <sup>j</sup>* and μ*j*, the duration between two adjacent events of an EV leaving charging station *j* follows a geometric distribution [45]. The EV number in charging station *j* at time slot *t* is denoted by *U<sup>t</sup> j* . The dynamic recursive equation for *U<sup>t</sup> j* is

$$\mathsf{U}\_{j}^{t} = \begin{cases} \mathsf{q}\_{j\prime} & t = 1, j \in \{1, \ldots, n\} \\ \max\{\mathsf{U}\_{j}^{t-1} + \sum\_{i=1}^{m} \sum\_{t''=1}^{t-1} \mathsf{C}\_{i}^{t''} \mathsf{x}\_{ij}^{t''} - \mathrm{S}\_{j}^{t-1}, \, \mathrm{0}\}, \quad t \in \{2, \ldots, T\}, j \in \{1, \ldots, n\} \end{cases} \tag{3}$$

where ϕ*<sup>j</sup>* represents the initial EV number in charging station *j* within a specific time horizon, and *t* is the time slot when the charging demand from node *i* occurs. In the equation, the time slots *t* and *t* satisfy the following relationship:

$$t = t'' + \tau\_{(i,j)}^{t''}.\tag{4}$$

Equation (4) indicates that the EV with charging demand *C<sup>t</sup> <sup>i</sup>* can reach charging station *j* after the driving time τ*<sup>t</sup>* (*i*,*j*) . Note that, in the equation, the driving time τ*<sup>t</sup>* (*i*,*j*) is defined as the number of time slots with identical duration.

For the problem formulation, the probability variables λ*<sup>i</sup>* and μ*<sup>j</sup>* are introduced to simulate the events of charging demand occurrence and EVs leaving charging stations during the time horizon. However, in real-world situations, the charging service provider could receive the charging demand information and know the number of vehicles that are leaving charging stations at the beginning of

each time slot. Therefore, the probability variables λ*<sup>i</sup>* and μ*<sup>j</sup>* do not appear in the dynamic recursive equation. Without loss of generality, the problem assumes that the routes with minimum energy consumption are selected as the travel routes between departure points and charging station nodes. Another assumption is that EV drivers can reach their destinations by charging their vehicles only once, because EVs with a single charge often have sufficient energy to reach their destinations in an urban road network [46].
