## 2 Mixed Traffic Flow Car-Following Model  
CAVs and HDVs exhibit different characteristics due to their distinct driving entities, rendering traditional car-following models unsuitable for the new traffic environment. To analyze the impact of CAV integration on traffic flow and understand the characteristics of mixed traffic flow, it is necessary to develop a car-following model tailored for mixed traffic conditions. The following sections consider the impacts of CAV integration and propose improvements to existing car-following models. To represent a more realistic mixed traffic environment, this study incorporates reaction time and the psychological factors of different human drivers when following CAVs, improving existing car-following models to establish a mixed-traffic-flow car-following model. This research categorizes the four cases shown in the figure as follows. In urban road mixed traffic flow, CAVs and HDVs coexist with random distribution, forming four car-following scenarios (Figure 1). When an HDV is the following vehicle, two scenarios may occur: 1 the HDV follows another HDV, or 2 the HDV follows a CAV. In both scenarios, regardless of whether the leading vehicle is an HDV or a CAV, the following HDV lacks onboard equipment and vehicle-to-vehicle (V2V) communication capabilities. Therefore, the following vehicle relies solely on the driver's physiological perception to obtain information about the leading vehicle. Consequently, in this study, these two scenarios are combined into one and referred to as the scenario where an HDV is the following vehicle. This study categorizes these into three types: CAV following CAV, CAV following HDV, and HDV following HDV. When a CAV follows an HDV, it cannot use vehicle-to-vehicle communication and will rely on onboard equipment to obtain information about the preceding vehicle and optimize its driving state. This scenario refers to a Degraded Connected and Automated Vehicle (DCAV). Figure 1: Vehicle following situation in mixed traffic flow. (CAVs: connected and autonomous vehicles; HDVs: human-driven vehicles). 1. CAV Following CAV The Cooperative Adaptive Cruise Control (CACC) model, proposed by the PATH laboratory at the University of California at Berkeley [27] based on a constant time headway strategy, is a popular model for CAV car-following research. The modified CACC model considering reaction time is expressed as: \[a_{c}=\frac{k_{p}(x_{c}-s_{c}-l-(t_{c}+\tau_{c})v)+k_{d}\Delta v}{\Delta t+k_{d}t _{c}}\] (1) In the above equation, \(a_{c}\) is the acceleration of CAV. \(x_{c}\) is the headway of the CAV. \(s_{c}\) is minimum safe distance for the CAV, which set to 3 m. \(l\) is the length of the vehicle, which is set at 5 m. \(v\) is the current speed. \(\Delta v\) is the speed difference between the two adjacent vehicles. \(t_{c}\) is the desired time headway for CAV, set to the highest acceptable value of 0.6 s; \(\tau_{c}\) is the reaction time for CAV, set to 0 s; \(\Delta t\) is the update interval for speed, set to 0.01 s based on the PATH real-vehicle experiments; \(k_{p}\) and \(k_{d}\) are control coefficients, set to 0.45 and 0.25, respectively, based on the calibration results from real-vehicle experiments for the CACC model. (2) CAV Following HDV The Adaptive Cruise Control (ACC) model, based on a constant desired time headway, can obtain surrounding vehicle information through onboard equipment, aligning with the characteristics of DCAV. The modified ACC model considering reaction time is expressed as: \[a_{d}=k_{1}[x_{d}-s_{d}-l-(t_{d}+\tau_{d})v]+k_{2}\Delta v\] (2) In the above equation, \(a_{d}\) is the acceleration of the DCAV. \(x_{d}\) is the headway of DCAV. \(s_{d}\) is minimum safe distance for DCAV, which is set to 2 m. \(t_{d}\) is the desired time headway for DCAV, set to the highest acceptable value of 1.1 s for drivers. \(\tau_{d}\) is the reaction time for DCAV, set to 0.2 s [28].