## 4 Numerical Analysis
### Analysis of Travel Time Characteristics  
To intuitively show the impacts of Q, L, and B\({}_{\text{EV}}\) on T, we apply the 3D mapping function in Python to produce a heatmap. The \(x\)-axis of the graph represents the traffic flow Q, the \(y\)-axis represents the mileage L, the \(z\)-axis represents the penetration rate of electric vehicles B\({}_{\text{EV}}\), the color of the data points represents the travel time T, and a warmer (colder) color represents a longer (shorter) travel time, as shown in Figure 3.  
As illustrated in Figure 3, as Q increases, T increases nonlinearly. This indicates that the impact of traffic congestion on travel time is most significant for high traffic flows. The impact of B\({}_{\text{EV}}\) on T depends on Q. At low traffic volumes, the penetration rate of EVs has little influence on travel time. However, under high traffic conditions, an increase in the penetration rate of EVs leads to an increase in the charging time, thus extending the travel time. The influence of L on travel time is linear, and the travel time increases linearly with increasing mileage. This comprehensive analysis method provides a new perspective for understanding the impedance characteristics in different travel situations, overcomes the shortcomings of existing models by utilizing a mixed traffic impedance function, and provides accurate and comprehensive decision support for traffic planning and management. 1. The time spent by a vehicle in the replenishment queuing system t\({}_{\text{q}}\)  
t\({}_{\text{q}}\) represents the time spent at a recharge node, which is calculated according to the queuing system Formula (16) in Section 3.1, and the results for 10 kinds of EVs with different penetration rates are calculated, as shown in Figure 4.  
According to the analysis of Figure 4, the relationship between traffic volume (Q) and recharge time (t\({}_{\text{q}}\)) under different B\({}_{\text{EV}}\) values can be described as follows. Figure 3: Travel time T.  
The vertical axis (t\({}_{\rm q}\)) represents the time spent at a recharge node, and the horizontal axis (Q) represents the traffic volume. The curves in different colors show the penetration rates (B\({}_{\rm EV}\)) of different electric vehicles, ranging from 0 to 1 with an interval of 0.1. (1) Low EV penetration (B\({}_{\rm EV}\) = 0 to 0.3)  
When B\({}_{\rm EV}\) is low (e.g., B\({}_{\rm EV}\) = 0 to 0.3), the replenishment time shows a relatively slow upward trend with increasing traffic volume. In the small range of Q, the replenishment time slowly increases. For example, when Q < 500, t\({}_{\rm q}\) increases slowly. This shows that for a low penetration rate of EVs, the number of EVs on the road is small, the vast majority of vehicles are still fuel-powered vehicles, the number of charging piles is sufficient relative to the number of EVs, the usage rate of charging piles is low, and a vehicle does not need to wait for a long time to complete the charging process; therefore, the time spent at a recharge node is relatively short. (2) Moderate EV penetration (B\({}_{\rm EV}\) = 0.4 to 0.7)  
As B\({}_{\rm EV}\) increases to moderate levels (B\({}_{\rm EV}\) = 0.4 to 0.7), the increase in the replenishment time becomes increasingly significant. When Q < 500, t\({}_{\rm q}\) increases rapidly, indicating that the use frequency of charging piles increases at this time, resulting in an increase in the average waiting time. When Q > 500, the increase in the number of EVs leads to a significant increase in charging demand, and more vehicles need to be replenished at charging piles, leading to an increase in the frequency of use of charging piles at recharge nodes; additionally, the average waiting time also increases. When Q > 1000, charging piles begin to approach their maximum service capacity. Although the traffic volume continues to increase, the use of the charging pile approaches saturation; consequently, the replenishing time tends to stabilize, and the upward trend slows.