## 5 Stability Analysis of Mixed Traffic Flow Based on PS-IDM
### Stability Analysis
#### 5.2.2 Heterogeneous Traffic Flow  
According to the discrimination conditions in Section 4, the discrimination conditions for the stability of heterogeneous traffic flow can be generalized using Equations (10) and (11) to obtain Equation (12) as follows.  
\[f=(1-p)f_{1}+pf_{2}\geq 0 \tag{12}\]  
where \(f\) is the stability of heterogeneous traffic flow, \(f_{1}\) and \(f_{2}\) are each the stability of homogeneous traffic flow, and \(p\) is the penetration rate of the CACC vehicle.  
Using a homogeneous traffic flow analysis method to adjust the CACC vehicle penetration rate, the mixed traffic flow stability region under different CACC vehicle penetration rates and different equilibrium velocities can be calculated. As shown in Figure 8, the stable region conditions for four types of \(r\) values and three types of \(T\) values are given. The black area in the figure is the unstable area, and the white area outside the black area is the stable area. In Figure 8, when the CACC vehicle penetration rate is 0, the boundary value of the black unstable area and the white stable area is the velocity boundary value of the homogeneous HDV traffic flow. It can be obtained from the analysis results in Figure 6.  
As shown in Figure 8, when the CACC vehicle penetration rate is 1, the velocity boundary and the values of \(r\) are negatively correlated, i.e., the larger the value of \(r\), the smaller the unstable velocity range of the CACC vehicle. When \(r\) = 0.1, the velocity  
Figure 8: Stability region of heterogeneous traffic flow.  
boundaries are 0\(\sim\)23 m/s, 1.6\(\sim\)22 m/s, and 4.6\(\sim\)20.8 m/s, corresponding to \(T\) = 1.1 s, 1.6 s, and 2.2 s, respectively. When \(r\) = 0.2, the velocity boundaries are 0\(\sim\)21 m/s, 3\(\sim\)19.6 m/s, and 7.5\(\sim\)17.5 m/s, corresponding to \(T\) = 1.1 s, 1.6 s, and 2.2 s, respectively. When \(r\) = 0.3, \(T\) = 1.1 s and 1.6 s correspond to the velocity boundaries 0\(\sim\)17.9 m/s and 7.5\(\sim\)14.5 m/s, respectively. When the value of \(T\) is 2.2 s, the traffic flow of homogeneous CACC vehicles is stable in the free-flow velocity range. When \(r\) = 1, the unstable region is the smallest.  
Therefore, under the same value of \(T\), as the feedback coefficient \(r\) increases, the unstable region of heterogeneous traffic flows shrinks. Furthermore, under the same value of \(r\), with an increase in time headway \(T\), the unstable region of heterogeneous traffic flow is significantly reduced.  
For each individual unstable domain, when \(r\) \(\geq\) 0.2, the traffic flow instability region of homogeneous CACC vehicles is within the homogeneous HDV traffic flow instability region, i.e., in the homogeneous HDV traffic flow stability region, and the heterogeneous traffic flow of any CACC vehicle permeability is stable. Especially, when \(r\) = 0.1, the unstable velocity range of the homogeneous CACC vehicle traffic flow is slightly larger than that of the homogeneous HD traffic flow.  
Until \(r\) = 0.3, \(T\) = 2.2, \(p\)\(\geq\) 0.72, a constant stability region appears. When \(r\) = 1, the CACC vehicle penetration rate reaches 0.395, 0.24, and 0.16, and the heterogeneous traffic flow is stable within the free flow velocity range of \(T\) = 1.1 s, 1.6 s, and 2.2 s, respectively.