## 5 Stability Analysis of Mixed Traffic Flow Based on PS-IDM
### Model Parameter Calibration  
In the model, the following parameter values were assigned according to [PERSON] and [PERSON] [11]: \(a=1\) m/s\({}^{2}\), \(v_{f}=33.3\) m/s, \(l=5\) m, and \(S_{0}=2\) m. The value of \(b\) for the HD and CACC vehicles were 2 and 2.8 m/s, respectively. [19]. The time headway correction value \(\Delta T\) and lateral distance utility parameters \(\lambda\) were calibrated using actual survey data. When the time headway was greater than 3 s, the front and rear cars were less restrictive. After screening, 292 groups of valid data from the test analysis were extracted, with the time headway being between 0 and 3 s. To obtain the time headway correction value of different driver personalities, the data were clustered and analyzed based on the \(k\)-means algorithm. The driving styles of drivers are usually classified into three categories: aggressive, ordinary, and conservative [20]. Therefore, a clustering number of 3 was used. The results for clusters 1, 2, and 3 were 2.2, 1.6, and 1.1, respectively, and are shown in Figure 5. \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Definition** \\ \hline \(S_{i}(t)\) & PS boundary of vehicle \(i\) at time \(t\) \\ \(x_{i}(t)\) & Horizontal coordinates of vehicle \(i\) at time \(t\) \\ \(y_{i}(t)\) & Horizontal coordinates of vehicle \(i\) at time \(t\) \\ \(x\) & Points on the PS boundary \\ \(y\) & Points on the PS boundary \\ \(A_{i}\) & The forward distance of vehicle \(i\) \\ \(B_{i}\) & The lateral distance of vehicle \(i\) \\ \(\dot{v}\) & Vehicle acceleration \\ \(v_{0}\) & The current velocity of the vehicle \\ \(a\) & Maximum acceleration \\ \(v_{f}\) & Free-flow velocity \\ \(s_{1}\) & Desired vehicle distance \\ \(h\) & The current space headway of the vehicle \\ \(l\) & Vehicle length \\ \(s_{0}\) & Minimum parking distance \\ \(T_{s}\) & Safe time headway \\ \(\Delta v\) & Velocity difference \\ \(b\) & Comfortable deceleration \\ \(\Delta T\) & Time headway correction value \\ \(T\) & Corrected time headway \\ \(c_{i}\) & Lateral intrusion distance \\ \(\lambda\) & Lateral distance utility parameter \\ \(F(S)\) & Disturbance transfer function \\ \(S\) & Laplace domain \\ \(f_{\Delta v}\) & Partial differentials of \(\Delta v\) \\ \(f_{h}\) & Partial differentials of \(h\) \\ \(f_{v_{0}}\) & Partial differentials of \(v_{0}\) \\ \(v_{e}\) & Equilibrium velocity \\ \(h_{e}\) & Equilibrium space headway \\ \(F(S_{1})\) & Disturbance transfer function of HD \\ \(F(S_{2})\) & Disturbance transfer function of CACC \\ \(r\) & Acceleration feedback coefficient \\ \(j\) & Imaginary number \\ \(w\) & Frequency of frequency domain \\ \(p\) & Penetration rate of CACC vehicle \\ \hline \end{tabular}
\end{table} 
Table 2: Notations Summary Table. Based on the above analysis, the longitudinal stability study was conducted to explore individual differentiation at the values of 2.2, 1.6, and 1.1 s.  
The difference between the maximum abscissas under free flow conditions was screened, and it was assumed that the current driver had obtained enough personal space, and the difference between the vehicle velocity and the abscissa was fitted to obtain the mathematical relationship between \(v_{0}\) and \(2B_{i}\). The lateral direction of PS was obtained through further processing. The relationship between distance \(B_{i}\) and vehicle velocity \(v_{0}\) is shown in Equation (8). \[B_{i}=0.029v_{0}+0.7364 \tag{8}\]  
where \(B_{i}\) is the lateral distance of the vehicle \(i\), \(i\) is the vehicle number, and \(v_{0}\) is the current velocity of the vehicle. \(B_{i}\) was substituted into the second item \(\frac{(x-x_{i}(t))^{2}+(y-y_{i}(t))^{2}}{B_{i}(t)^{2}}\leq 1\) of Equation (1) to obtain the partial PS boundary of vehicle \(i\) at \(x<x_{i}(t)\), which was used to quantify \(c_{i}\), the lateral intrusion distance of vehicle \(i\), as expressed in Equation (9).