## 3 Impedance Function Model of Mixed Traffic Flows
### Improvement of the Travel Time Model  
The travel times are based in part on the US Federal Highway Administration's (FHWA) BPR resistance function [9]. The BPR function is widely used in traffic engineering to estimate the impact of road congestion on travel time. In this model, the consumption time of different types of vehicles at recharge stations is considered, especially the charging time of EVs at charging stations. \[\text{T}=\text{T}_{\text{BPR}}+\text{T}_{\text{W}} \tag{2}\]  
In Formula (2), \(\text{T}_{\text{BPR}}\) is the time spent by a vehicle driving on a road section, which is included in the BPR function proposed by the US FHWA, and \(\text{T}_{\text{W}}\) is the time spent by the vehicle at a recharge station, that is, during a recharge operation at a gas station or charging station. It is assumed that the traffic on the selected road is composed of EVs and traditional vehicles, regardless of the impact of other types of vehicles. In the BPR base function, a fixed value \(\text{C}_{0}\) in Formula (3) is used to represent the actual capacity of the road section. However, in practice, different percentages of EVs will result in different impacts on the capacity of the road. Through the simulation of data [33], it was found that there is a certain relationship between the penetration rate of EVs and C\({}_{0}\). The simulation results show that as the penetration rate of EVs increases, the road capacity also increases accordingly. We fit this relationship based on a function diagram, as shown in Figure 2, where the red dotted line represents the fitted trend line, and the Bev-C\({}_{0}\) function is shown in Formula (3). \[\text{C}=\text{C}_{0}\times\left(1+0.0017\text{B}_{\text{EV}}{}^{2}+0.0049\text {B}_{\text{EV}}-0.0037\right) \tag{3}\]  
According to the functional relationship between the proportion of EVs and the road capacity, the C parameter in the BPR function is improved to obtain T\({}_{\text{BPR}}\), as expressed in Formula (7). \[\text{Q}=\text{Q}_{\text{EV}}+\text{Q}_{\text{GV}} \tag{4}\]  
\[\text{B}_{\text{EV}}=\frac{\text{Q}_{\text{EV}}}{\text{Q}} \tag{5}\]  
\[\text{B}_{\text{GV}}=\frac{\text{Q}_{\text{GV}}}{\text{Q}}=1-\frac{\text{Q}_{ \text{EV}}}{\text{Q}} \tag{6}\]  
\[\text{T}_{\text{BPR}}=\text{T}_{0}\times(1+\alpha\bigg{(}\frac{\text{Q}_{ \text{EV}}+\text{Q}_{\text{GV}}}{\text{C}}\bigg{)}^{\beta}) \tag{7}\]  
where B\({}_{\text{EV}}\) is the proportion of EVs in the traffic flow, B\({}_{\text{GV}}\) is the proportion of GVs in the traffic flow, Q is the mixed traffic volume on the road, Q\({}_{\text{EV}}\) is the traffic volume of EVs on the road, and Q\({}_{\text{GV}}\) is the traffic volume of fuel-consuming vehicles on the road. Queuing theory provides a useful framework for considering the effect of the charging time of EVs and the refueling time of traditional vehicles on the impedance of nodes in traffic networks. Queueing theory is a branch of mathematics in which queueing systems are studied, and this theory can be applied to analyze and optimize systems with random arrival rates and service rates. In this context, we consider a charging station or gas station as a queuing system with several key parameters: the arrival rate, service rate, queue length, and average waiting time. 1. The arrival rate \(\lambda\) refers to the rate at which vehicles arrive at the charging station or gas station, depending on the traffic flow. 2. The service rate \(\mu\) refers to the rate at which the station provides services to vehicles, which is affected by the performance and efficiency of the charging or refueling equipment. Figure 2: Effect of the electric vehicle blending ratio on the road capacity. 3. The queue length \(L_{q}\) refers to the number of vehicles waiting for service at a station, which depends on the difference between the arrival rate and the service rate. 4. The average waiting time \(W_{q}\) refers to the average time vehicles spend in the queue waiting for service at a station, which is jointly affected by the arrival rate, service rate, and queue length.