## 4 Conditions for Stability Judgment  
The equilibrium velocity of the traffic flow is defined as \(v_{e}\), and the space headway of the traffic flow is defined as equilibrium \(h_{e}\). By considering only the velocity disturbance \(v_{n}(t)-v_{e}\) and the inter-vehicle distance disturbance \(h_{n}(t)-h_{e}\) in the traffic flow, the first-order Taylor expansion of the model at \(f(h_{e},v_{e},0)\) was obtained. Subsequently, the IDM traffic flow disturbance transfer function was derived from the Laplace transform of the velocity input signal and the Laplace transform of the velocity output signal.  
The disturbance transfer function in the equilibrium state of traffic flow is expressed in Equation (4) as follows.  
\[F(S)=\frac{f_{\Delta v}S+f_{h}}{S^{2}+(f_{\Delta v}-f_{v_{0}})S+f_{h}} \tag{4}\]  
where \(S\) is the Laplace domain, and \(f_{\Delta v}\), \(f_{h}\), and \(f_{v_{0}}\) are the partial differentials of \(\Delta v\), \(h\) and \(v_{0}\), respectively, in Equation (3).  
Subsequently,\(f_{\Delta v}\), \(f_{h}\), and \(f_{v_{0}}\) are substituted to obtain the disturbance transfer function \(F(S_{1})\) of the car-following model of the HD.  
\[F(S_{1})=\frac{\sqrt{\frac{g}{b}\frac{v_{\text{e}}\left[1-\left(v_{\text{e}}/ v_{f}\right)^{4}\right]}{s_{0}+v_{\text{e}}T}}S+2a\frac{\left[1-\left(v_{ \text{e}}/v_{f}\right)^{4}\right]\sqrt{\left[1-\left(v_{\text{e}}/v_{f} \right)^{4}\right]}}{s_{0}+v_{\text{e}}T}} \tag{5}\]  
The feedback information of the preceding vehicle is added based on the CACC car-following model, and the acceleration of the preceding vehicle is often ignored in the car-following model. In this study, the acceleration feedback coefficient \(r(0\leq r\leq 1)\) of the preceding vehicle was added to derive the disturbance transfer of the CACC car-following model. The function is given by Equation (6).  
\[F(S_{2})=\frac{rS^{2}+\sqrt{\frac{g}{b}\frac{v_{\text{e}}\left[1-\left(v_{ \text{e}}/v_{f}\right)^{4}\right]}{s_{0}+v_{\text{e}}T}}S+2a\frac{\left[1- \left(v_{\text{e}}/v_{f}\right)^{4}\right]\sqrt{\left[1-\left(v_{\text{e}}/v _{f}\right)^{4}\right]}}{s_{0}+v_{\text{e}}T}} \tag{6}\]  
where \(r\) is acceleration feedback coefficient of CACC vehicles.  
In the mixed traffic flow, \(F(S_{1})\) and \(F(S_{2})\) act together; let \(S=jw\) according to the following system stability conditions [7]:  
\[\begin{split}&\left|F(S)\right|_{\infty}=\left|F(jw)\right|_{ \infty}=\left|F_{1}(jw)^{(1-p)}F_{2}(jw)^{p}\right|_{\infty}\\ &=\left(\frac{f_{\Delta v}jw+f_{h}}{-w^{2}+(f_{\Delta v}-f_{v_{0 }})jw+f_{h}}\right)^{(1-p)}\left(\frac{-rw^{2}+f_{\Delta v}jw+f_{h}}{-w^{2}+( f_{\Delta v}-f_{v_{0}})jw+f_{h}}\right)^{p}\\ &=\left(\frac{\sqrt{\left(f_{\Delta v}w\right)^{2}+\left(f_{h} \right)^{2}}}{\sqrt{\left(f_{\Delta v}-f_{v_{0}}\right)^{2}w^{2}+\left(f_{h}-w ^{2}\right)^{2}}}\right)^{(1-p)}\left(\frac{\sqrt{\left(f_{\Delta v}w\right)^ {2}+\left(f_{h}-rw^{2}\right)^{2}}}{\sqrt{\left(f_{\Delta v}-f_{v_{0}}\right)^ {2}w^{2}+\left(f_{h}-w^{2}\right)^{2}}}\right)^{p}\leq 1\end{split} \tag{7}\]  
where \(j\) and \(w\) are imaginary numbers and frequencies of frequency domains, \(\left|\cdot\right|_{\infty}\) is the maximum amplitude of transfer function frequency domain, \(p\) is the penetration rate of the CACC vehicle.  
The notations involved in the model are shown in Table 2.