## 5 Stability Analysis of Mixed Traffic Flow Based on PS-IDM
### Stability Analysis  
#### 5.2.1 Homogeneous Traffic Flow  
In Equation (7), when \(p=0\), it degenerates to a homogeneous HDV traffic flow; when \(p=1\), it degenerates to a homogeneous CACC vehicle traffic flow stability condition. Based on Equation (7) and \(\forall w>0\), the stability conditions of the homogeneous HDV traffic flow are given by Equation (10) below. \[ \begin{split} f_{1}=\frac{1}{2}\left(-\frac{4 av_{\text{c}}^{3}}{v_ {f}^{4}}-\frac{2 aT\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}{s_{0}+v _{\text{c}}T}+\lambda_{1}\frac{\text{d}c_{i}}{\text{d}v_{\text{e}}}\right)^{2} -\left(\sqrt{\frac{a}{b}}\frac{v_{\text{c}}\left[1-\left(v_{\text{c}}/v_{f} \right)^{4}\right]}{s_{0}+v_{\text{c}}T}\right)\left(-\frac{4 av_{\text{c}}^{3} }{v_{f}^{4}}-\frac{2 aT\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}{s_{ 0}+v_{\text{c}}T}+\lambda_{1}\frac{\text{d}c_{i}}{\text{d}v_{\text{e}}}\right) -\\ 2a\frac{\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]\sqrt{ \left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}}{s_{0}+v_{\text{c}}T}\geq 0 \end{split}  \tag{10}\]  
When the free flow velocity is 0-33.3 m/s, the stability of HDV at different equilibrium velocities at \(T=1.1\) s, \(T=1.6\) s, and \(T=2.2\) s is analyzed by using Equation (10), as shown in Figure 6.  
It can be observed from Figure 6 that homogeneous HDV traffic flow is unstable at 0-22.3 m/s when the value of is 1.1 s; the homogeneous HDV traffic flow is unstable at 1.2-21.3 m/s when the value of \(T\) is 1.6 s; the homogeneous HDV traffic flow is unstable at 4.4-20 m/s when \(T\) is 2.2 s. With an increase in \(T\) value, the unstable velocity range of the homogeneous HDV traffic flow is gradually reduced. Furthermore, traffic flow tends to be stable, regardless of the value of \(T\). However, when \(T\) is 2.2 s, the traffic flow transforms into a stable state the fastest. Similarly, the stability condition of the traffic flow of homogeneous CACC vehicles should satisfy Equation (11). \[ \begin{split} f_{2}=\frac{1}{2}\left(-\frac{4 av_{\text{c}}^{3}}{ v_{f}^{4}}-\frac{2 aT\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}{s_{0}+v _{\text{c}}T}+\lambda_{2}\frac{\text{d}c_{i}}{\text{d}v_{\text{e}}}\right)^{ 2}-\left(\sqrt{\frac{a}{b}}\frac{v_{\text{c}}\left[1-\left(v_{\text{c}}/v_{f} \right)^{4}\right]}{s_{0}+v_{\text{c}}T}\right)\left(-\frac{4 av_{\text{c}}^{3} }{v_{f}^{4}}-\frac{2 aT\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}{s_{0} +v_{\text{c}}T}+\lambda_{2}\frac{\text{d}c_{i}}{\text{d}v_{\text{e}}}\right) -\\ (1-r)2a\frac{\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right] \sqrt{\left[1-\left(v_{\text{c}}/v_{f}\right)^{4}\right]}}{s_{0}+v_{\text{c}} T}\geq 0\end{split}  \tag{11}\]  
Figure 6: Analytical diagram of the stability of homogeneous HD trafficThe sensitivity analysis of CACC vehicle acceleration feedback coefficients \(r=0.1\), \(r=0.2\), and \(r=0.3\) is shown in Figure 7. Figure 7a-c show the stability at \(T=1.1\) s, \(T=1.6\) s, and \(T=2.2\) s, respectively. Figure 7a shows that the unstable velocity range gradually decreases in the case of \(T=1.1\) s with an increase in \(r\). When \(r=0.1\), the stability of the homogeneous CACC vehicle traffic flow increases with the velocity, reaching the zero point at \(23\) m/s. The unstable state turns to a stable state, and then it is stable within the free flow velocity. When \(r=0.2\), the stable zero point of the homogeneous CACC vehicle traffic flow shifts to the left, which is unstable at \(0\)-\(21\) m/s, and stabilizes after the traffic velocity exceeds \(21\) m/s. When \(r=0.3\), the unstable velocity boundary continues to shrink, and the zero-boundary point moves to the right to \(17.9\) m/s. Figure 7b shows that the traffic flow of homogeneous CACC vehicles is stable in the initial stage and then gradually decreases, when \(r=0.1\), in the case of \(T=1.6\) s. It reaches the first zero boundary point at \(1.6\) m/s and becomes unstable.