# An Extended Car-Following Model Considering Generalized Preceding Vehicles in V2X Environment
## 4 Stability Analysis  
To explore the impact of average velocity information of GPV on traffic flow in the V2X environment, linear stability analysis is conducted based on the perturbation method [34; 42; 43]. Assuming that all three lanes in the system are in the same stable state, which means all vehicles maintain the same headway \(h\) and velocity \(V(h)\), at the initial moment, the position of the \(n\) th vehicle can be expressed as follows:  
\[x_{n}^{(0)}(t)=hn+V(h)t \tag{9}\]  
where \(h=L/N\). \(L\) is the length of the road and \(N\) is the total number of vehicles on the road. \(V(h)\) is the optimal velocity. Suppose \(y_{n}(t)\) to be a small deviation from the stable state solution \(x_{n}^{(0)}(t)\)  
\[x_{n}(t)=x_{n}^{(0)}(t)+y_{n}(t) \tag{10}\]  
Substituting Equations (9) and (10) into Equation (6) and linearizing the equation, one can obtain  
\[ \begin{array}{l}\frac{d^{2}y_{n}(t)}{dt^{2}}=p\bigg{\{}a\bigg{[}V^{\prime}(h )\Delta y_{n}(t)-\frac{dy_{n}(t)}{dt}\bigg{]}+\lambda\frac{d\Delta y_{n}(t)}{ dt}\bigg{\}}\\ +(1-p)\bigg{[}\frac{1}{4}\bigg{(}\frac{dy_{n+1}(t)}{dt}+\frac{dy_{n+2}(t)}{dt} +\frac{dy_{n}(t)}{dt}+\frac{dy_{n}(t)}{dt}\bigg{)}-\frac{dy_{n}(t)}{dt}\bigg{]} \end{array}  \tag{11}\]  
where \(\Delta y_{n}(t)=y_{n+1}(t)-y_{n}(t)\) and \(V^{\prime}(h)=\left.dV(\Delta x_{n})\right/d\Delta x_{n}\big{|}_{\Delta x_{n} =h}\). According to vehicles in all three lanes of the road are at the same stable state, one can obtain \(\frac{dy_{n}(t)}{dt}=\frac{dy_{n}(t)}{dt}=\frac{dy_{n+1}(t)}{dt}\). Substituting \(y_{n}(t)=e^{ikn+zt}\) into Equation (11), one can obtain  
\[z^{2}=p\Big{\{}a\big{[}V^{\prime}(h)\Big{(}e^{ik}-1\Big{)}-z\Big{]}+\lambda z \Big{(}e^{ik}-1\Big{)}\Big{\}}+(1-p)\bigg{[}\frac{1}{4}\bigg{(}3 ze^{ik}+ze^{2 ik}\Big{)}-z\bigg{]} \tag{12}\]  
By expanding \(y_{n}(t)\), where \(z=z_{1}(ik)+z_{2}(ik)^{2}+\cdots\), and inserting it into Equation (12), the first- and second-order terms of \(ik\) can be obtained as follows:  
\[z_{1}=V^{\prime}(h) \tag{13}\]  
\[z_{2}=\frac{2 paV^{\prime}(h)-4z_{1}{}^{2}+5(1-p)z_{1}+4p\lambda z_{1}}{4 pa} \tag{14}\]  
For long wavelength modes, the uniformly stable state traffic flow becomes unstable if \(z_{2}<0\), while the uniformly stable state traffic flow remains stable if \(z_{2}>0\). Therefore, the neutral stability condition is given as:  
\[a=\frac{4V^{\prime}(h)-5(1-p)-4p\lambda}{2p} \tag{15}\]  
For small disturbances with long wavelengths, the uniform traffic flow is stable if  
\[a>\frac{4V^{\prime}(h)-5(1-p)-4p\lambda}{2p} \tag{16}\]  
Based on Equation (15) and parameters calibrated in Section 3, the neutral stability curves of the GPV and FVD models in the headway-sensitivity space are as shown in Figure 4. From Figure 4, one can see that the headway-sensitivity phase space is divided into two regions by the neutral stability curves. The first is the stable region, which is above the corresponding neutral stability curve, and the second is the unstable region, which is below the corresponding neutral stability curve. In a stable region,traffic flow is stable, which means that small disturbances will be suppressed and, thus, traffic jams will not occur. In the unstable region, traffic flow is unstable, and density waves emerge. In this region, small disturbances cannot be suppressed effectively, and, on the contrary, it will gradually enlarge with propagation, which could lead to congestion eventually. To explore the impact of the sensitivity parameter \(p\) on the stability of traffic flow, the neutral stability curves of the GPV model with different \(p\), as \(p=~{}0.9,0.8,0.7,0.6,0.5\) respectively, are obtained when \(\lambda\)\(=\) 0.2 as shown in Figure 5. From Figure 5, one can obtain that with the decrease \(p\), the neutral stability curve gradually moves down, and the stable region keeps enlarging. Figure 4: The neutral stability curves of the GPV model and the FVD model with calibrated parameters.