*5.1. Scenario Description*

The simulation examples for a time-varying road network are designed to explore the performance of the SDD and CSB strategies. The structure of the road network is designed based on the Sioux Falls network, which is often adopted to simulate travel optimization problems [54–56]. The network consists of 24 nodes and 76 links, as shown in Figure 5. The road network comprises eight nodes with charging stations, which are marked as CS 1 to CS 8. The other nodes, numbered 1–16, are the normal ones without charging stations, which may generate charging demands in every time slot.

**Figure 5.** Sioux Falls road network for the simulation example.

Each charging station node in the road network has parameter μ*<sup>j</sup>* to reflect the charging levels of the charging station, where *j* = {1,2, ... ,7,8}. For the normal nodes, each one has parameter λ*<sup>i</sup>* to reflect the stochastic characteristic of charging demand generation, where *I* = {1,2, ... ,15,16}. Tables A1 and A2 (Appendix A) list the values of μ*<sup>j</sup>* and λ*i*, respectively. The simulation example assumes that the initial number of EVs at each charging station is equal to zero, that is, ϕ*<sup>j</sup>* = 0, for each charging station *j* in the road network.

The information for each charging demand *Ct <sup>i</sup>* includes the travel destination *<sup>d</sup><sup>t</sup> <sup>i</sup>* and remaining energy *et i* , which are randomly generated in the simulation example. In every time slot, the travel destination *d<sup>t</sup> <sup>i</sup>* is randomly selected from other normal nodes in the road network if a charging demand occurs in a specific normal node. Moreover, the remaining energy *e<sup>t</sup> <sup>i</sup>* varies within a given interval, and we suppose that its value ranges from 7.2 kWh to 16.8 kWh by referring to the battery capacity of EVs. In general, EVs with a 24-kWh battery are widely used in unban transportation systems [33]. The energy consumption on each link in a time-varying road network varies as time slot passes. The simulation example randomly determines parameter *Et <sup>a</sup>* from given intervals for link *a* at time slot *t* to reflect such a characteristic. The value intervals of the energy consumption *Et <sup>a</sup>* on each link *a* are listed in Table A3 (Appendix A).

Similar to the energy consumption *E<sup>t</sup> <sup>a</sup>*, the driving time on each link *a* also has a time-varying characteristic. Similar to parameter *Et <sup>a</sup>*, the values of parameter τ*<sup>t</sup> <sup>a</sup>* in every time slot are randomly determined based on given intervals for each link *a*. The value intervals of driving time τ*<sup>t</sup> <sup>a</sup>* on each link *a* are listed in Table A4 (Appendix A). The number of time slots represents the driving time on each link given that the time is slotted into the time slots with identical duration. Without loss of generality, the duration for each time slot is not constrained in the simulation example. In the real-world situation, the duration for time slots could be valued according to actual requirements.

Table A5 (Appendix A) lists the length of each link *a*, which is denoted by *la* (km), in the road network. Considering the structure characteristic of road networks, the links with a symmetric relation have the same length.

### *5.2. Simulation Results and Analysis*

On the basis of the example scenario, the SDD and CSB strategies are applied in the route guidance problem with stochastic charging demands in a time-varying road network. The total number of time slots is set as *T* = 102, *T* = 103, *T* = 104, *T* = 105, and *T* = 10<sup>6</sup> to analyze the performance during different time horizons. The SDD and CSB strategies can ensure the reachability of selected charging stations for the charging demands in every time slot, as mentioned in Section 4. The charging demands of EV drivers can be satisfied by both strategies. Therefore, the simulation example focuses on the effects of the proposed strategies on the operation efficiency of charging stations. The number of EVs in a charging station is a critical factor that reflects the operation state of the charging station. Figure 6 presents the average number of EVs at each charging station during different time horizons *T* under the proposed strategies, which is computed by averaging over all time slots over the entire number of EVs.

In Figure 6, cases (a)–(h) respectively show the average number of EVs in CS 1–CS 8 during different time horizons. The change trends of the average EV number during different time horizons reflect the stability of charging stations under specific scenarios. Stability is an important criterion for guaranteeing the operation efficiency of charging stations. If the average number of EVs in a charging station has a flat change trend as the time horizon increases, then the charging station would operate stably for the given scenarios [57]; otherwise, the average number of EVs in the charging station would increase rapidly as the time horizon increases. The figure depicts that the SDD and CSB strategies can stabilize the operation states for CS 1–CS 8 under the example scenario, because the number of EVs in all charging stations has flat change trends as time horizon *T* varies. Although fluctuation trends exist when the time horizon ranges from *T* = 10<sup>2</sup> to *T* = 104 for several charging stations under specific strategies, such as CS 2 under the SDD strategy, CS 3 under the CSB strategy, and CS 5 under both strategies, all charging stations could reach a stable state after the time horizon *T* = 104. A comparison of the average EV number in CS 1–CS 8 with a stable state shows a difference between the SDD and CSB strategies. For the SDD strategy, the average number of EVs in CS 5 is greater than that in other charging stations because the vehicle balance of charging stations is not considered. The average EV number under the CSB strategy has a similar trend in all charging stations.

**Figure 6.** *Cont*.

**Figure 6.** Average electric vehicle (EV) number at each charging station during different time horizons *T*.

Although the average EV number is a critical reflection of the stability of each charging station, it cannot perfectly represent the actual number of EVs in every time slot. The EV number in charging stations at different time slots may vary during the time horizon. As time slots pass, there exists the obvious difference between the maximum and minimum numbers of EVs in a charging station. If the EV number in a selected charging station is relatively large, then the drivers would be reluctant to charge their vehicles by using it at the corresponding time slot. This condition would negatively influence the implementation efficiency of the route guidance service. Therefore, during the time horizon *T*, the maximum number of EVs at each charging station is often regarded as the bottleneck in the application of route guidance strategies under real-world situations. Figure 7 presents the maximum number of EVs at each charging station during different time horizons *T* based on the simulation example to compare the performances of the SDD and CSB strategies.

**Figure 7.** Maximum EV number at each charging station during different time horizons *T*.

The maximum number of EVs at each charging station under the SDD and CSB strategies is depicted in Figure 7, where cases (a)–(e) illustrate the results during different time horizons, ranging from *T* = 102 to *T* = 106, respectively. In case (a), the maximum number of EVs in CS 1–CS 8 under the SDD strategy is less than that under the CSB strategy. However, when the time horizon *T* = 103, as shown in case (b), the EV number in CS 5 under the SDD strategy is larger than that under the CSB strategy. In case (c), the SDD strategy enlarges the maximum number of EVs in most charging stations, especially in CS 5, compared with case (b). The extreme gap of maximum EV number among the charging stations is equal to 32. By contrast, the maximum number of EVs under the CSB strategy has a moderate degree of change for all charging stations. The maximum number of EVs in CS 3 and CS 5 in particular has a decreasing trend unlike case (b). The extreme gap of maximum EV number among the charging stations is equal to 7. When the time horizon *T* = 104, as shown in case (d), the maximum number of EVs under the SDD strategy increases for all charging stations, and the maximum EV number in CS 5 and CS 7 is larger than that under the CSB strategy. The extreme gap in the maximum EV number among the charging stations under the SDD and CSB strategies is equal to 41 and 7, respectively, thereby indicating a visible difference in vehicle balance among different charging stations between the two strategies. In case (e), the maximum EV number in CS 5, CS 6, and CS 7 under

the SDD strategy is larger than that under the CSB strategy. The extreme gap of maximum EV number among the charging stations reaches 48 under the SDD strategy. By contrast, the maximum number of EVs presents a balanced state for different charging stations under the CSB strategy. The extreme gap of maximum EV number among the charging stations is equal to 7.

A comparison of the performances of the SDD and CSB strategies based on the simulation example shows that the CSB strategy has an advantage in vehicle balance among different charging stations, especially in the situation with a long time horizon. Thus, using the CSB strategy would avoid the negative influence of the large number of EVs in a charging station. Unlike the CSB strategy, the SDD strategy would enlarge the gap of the number of EVs at different charging stations as the time horizon increases, thereby affecting the operation efficiency of the charging stations that have relatively more vehicles. However, in the situation with a short time horizon, the SDD strategy, which considers the travel cost of EV drivers, could address stochastic charging demands because of the unobvious difference in the performance of the two strategies in such a situation. In summary, the CSB strategy is suitable to be applied in long-term transportation scenario due to its ability to stabilize the charging service system. By contrast, the SDD strategy fits the short-term transportation scenario to reduce the travel cost of individual drivers.

### *5.3. Parameter Analysis for Scenario Characteristics*

When discussing the route guidance problem for stochastic charging demands, in addition to time horizon, the scenario characteristics have significant effects on the performance of the proposed strategies. For problem formulation, parameter λ*<sup>i</sup>* and μ*<sup>j</sup>* are used to present the stochastic characteristics of charging demands and processes, respectively, as mentioned in Section 3. Such parameters can also reflect the scenario characteristics in terms of the EV scale and charging level. For instance, a large parameter λ*<sup>i</sup>* represents a large EV scale in node *i*. A large parameter μ*<sup>j</sup>* illustrates a high charging level of the charging station in node *j*. Parameters λ*<sup>i</sup>* and μ*<sup>j</sup>* are set as different values to explore the performance of route guidance strategies under different scenario parameters. The values of parameter λ*<sup>i</sup>* for all normal nodes *i* are set as identical value λ to highlight the effects of parameter values on the simulation results. The values of parameter μ*<sup>j</sup>* for all charging station nodes *j* are also set as identical value μ. The time horizon is set as *T* = 10<sup>6</sup> for all parameter scenarios. Figure 8 presents the maximum number of EVs at each charging station under the SDD strategy as parameters μ and λ vary. The value of λ is set as 0.1, 0.2, 0.3, 0.4, and 0.5. The value of μ is set as 0.6, 0.7, 0.8, 0.9, and 1.0. A parameter scenario consists of a pair of parameters λ and μ. Thus, a total of 25 parameter scenarios are considered.

In Figure 8, cases (a)–(h) respectively illustrate the maximum EV number in CS 1–CS 8 under the SDD strategy for the different parameter scenarios. In several parameter scenarios, the maximum EV number in a specific charging station may exceed its sustainable limit, which indicates that the charging station is unstable. For such a scenario, we let the maximum EV number be equal to zero in the figure. The threshold of sustainable number of EVs for all charging stations is set as 120. In case (a), as parameter μ increases, the maximum EV number in CS 1 presents a decreasing trend. This phenomenon indicates that the maximum EV number reduces as the charging level of the charging station increases. By contrast, as parameter λ increases, the maximum number of EVs in CS 1 has an increasing trend, which indicates that the maximum EV number increases as the EV scale increases in the road network. Among all parameter scenarios, the peak and lowest values of the maximum EV number are equal to 2 and 29, respectively. In case (b), as the scenario parameters change, the change trend of the maximum EV number in CS 2 is similar to that in CS 1. However, unlike case (a), unstable parameter scenarios exist in case (b). Among all stable parameter scenarios, the peak and lowest values of the maximum EV number equal 3 and 82, respectively. In cases (c)–(h), the change trend of the maximum number of EVs in CS 3–CS 8 is also similar to that in CS 1. Similar to CS 2, the unstable state would exist in CS 3–CS 8 under specific parameter scenarios. Among all stable parameter scenarios, the lowest values of the maximum EV number in CS 3–CS 8 are all equal to 4. Comparatively, the peak values of the maximum EV number in CS 3–CS8 are respectively equal to 103, 41, 95, 58, 69, and 67. For a transportation system, the charging

service is unstable until all charging stations can reach stability. Therefore, if at least one unstable charging station exists in the road network under a parameter scenario, then the SDD strategy cannot be applied in the parameter scenario. The parameter scenarios that cannot support the SDD strategy can be determined based on such a criterion. The parameter pairs of the unstable scenarios include (λ = 0.3, μ = 0.6), (λ = 0.3, μ = 0.7), (λ = 0.4, μ = 0.6), (λ = 0.4, μ = 0.7), (λ = 0.4, μ = 0.8), (λ = 0.4, μ = 0.9), (λ = 0.5, μ = 0.6),(λ = 0.5, μ = 0.7), (λ = 0.5, μ = 0.8), (λ = 0.5, μ = 0.9), and (λ = 0.5, μ = 1.0). For all the stable parameter scenarios in each case, a significant change trend can be observed as the parameters vary, which indicates that the SDD strategy is sensitive to the change in scenarios.

**Figure 8.** Change trends of the maximum EV number at each charging station under the SDD strategy, where λ is the possibility of charging demand occurrence in normal nodes at each time slot, and μ is the possibility of an EV leaving the charging station after completing charging at each time slot.

On the basis of the parameter scenarios and the time horizon mentioned above, the CSB strategy is further applied in the route guidance problem with stochastic charging demands. As parameters μ and λ vary, the maximum EV number at each charging station is obtained, as shown in Figure 9.

**Figure 9.** Change trends of the maximum EV number in each charging station under the CSB strategy, where λ is the possibility of charging demand occurrence in normal nodes at each time slot, and μ is the possibility of an EV leaving the charging station after completing charging at each time slot.

In Figure 9, cases (a)–(h) respectively present the maximum number of EVs in CS 1–CS 8 under the CSB strategies for the different parameter scenarios. In case (a), the maximum number of EVs in CS 1 broadly presents a flat increasing trend as parameter μ decreases and parameter λ increases. For several individual scenarios, a moderate fluctuation occurs as the parameters vary. The results indicate that a change in scenarios has relatively limited effects on the CSB strategy compared with that on the SDD strategy. Among all parameter scenarios, the peak and lowest values of the maximum EV number in CS 1 are equal to 7 and 20, respectively. In cases (b)–(h), the maximum number of EVs in CS 2–CS 8 shows a similar change trend to that in CS 1. Among all parameter scenarios, the lowest values of the maximum EV number in CS 2–CS 8 are respectively equal to 9, 10, 11, 10, 9, 11, and 10. Comparatively, the peak values of the maximum EV number in CS 2–CS 8 are equal to 26, 30, 32, 30, 28, 29, and 27, respectively. Unlike the SDD strategy, the CSB strategy can stabilize the state of CS 1–CS 8 for all parameter scenarios, thus indicating its advantage in charging station stability.

A comparison of the simulation results in Figures 8 and 9 indicates that, for the SDD and CSB strategies, the maximum EV number increases as parameter μ decreases and parameter λ increases, with different change trends. Given the implication of parameters μ and λ, the simulation results conform to the operation state of charging stations in real-world situations.
