## 2 The Extended Model  
In the V2V communication environment, the external stimulus information that affects car-following behavior mainly includes two aspects. One is the direct stimulation, which comes from the preceding vehicle; the driver reacts directly according to his observation of their distance from and the acceleration of the preceding car. The other is the indirect stimulation, which represents the traffic density that is able to reflect the traffic congestion level; when a driver perceives traffic flow density in front, he will make some adjustments in advance, enabling a reduction in the frequency of acceleration and deceleration and thus maintaining a relatively stable driving status. The response to this stimulus depends on the perception of state information of leading vehicles. However, as an important macro parameter, traffic flow density is difficult to obtain beforehand. This is because the acquisition of traffic density requires a real-time counting of the vehicles distributed on each lane per kilometer. In fact, it is difficult to guarantee a precise performance through traditional survey methods, including access methods, remote sensing images and geomagnetic vehicle detectors. Fortunately, communication between vehicles favors the calculation of the average of the headway, which is inversely proportional to the traffic flow density. Therefore, to enhance the stability of driving, a modified model for car-following, called the density and acceleration velocity difference (DAVD) model, is proposed through adding traffic flow density and acceleration difference feedback parameters into the FVD model. The kinematic equation of the model is expressed as:  
\[\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[(1-p)V[\Delta x_{n}(t)]+pV(1/k)-v_{n}(t)]+ \beta a_{n+1}(t)+\lambda\Delta v_{n}(t), \tag{4}\]  
where \(\alpha\) is driver sensitivity to headway; \(\beta\) denotes response intensity of the acceleration of the preceding vehicle; \(\lambda\) is relative velocity sensitive coefficient o; \(x_{n}(t)\) and \(v_{n}(t)\) are the location and velocity of car \(n\) at time \(t\), respectively; \(a_{n+1}(t)\) is leading vehicle \(n+1\) acceleration; \(\Delta x=x_{n+1}(t)-x_{n}(t)\) is space headway from car \(n\) to car \(n+1\); \(\Delta v_{n}=v_{n+1}(t)-v_{n}(t)\) is the difference of velocity between two successive vehicles; \(k\) is traffic flow density, so, \(1/k=\sum_{j=0}^{m-1}\Delta x_{n+j}(t)/m\) is described as the average of the headway; \(p\) is the influence coefficient of traffic density, and its value determines the extent to which the driver perceives the traffic state ahead. So, the extended model may be rearranged as:  
\[\frac{d^{2}x_{n}(t)}{dt^{2}}=\alpha[(1-p)V[\Delta x_{n}(t)]+pV[\frac{1}{m} \sum_{j=0}^{m-1}\Delta x_{n+j}(t)]-v_{n}(t)]+\beta a_{n+1}(t)+\lambda\Delta v _{n}(t), \tag{5}\]  
When the value of \(p\) is 0 or \(m\) is 1, the state of car-following is only under the effect of headway, acceleration and the velocity difference of the preceding vehicle. In this case, as \(\beta\) is 0, this model is transformed into the FVD model. \(V(\cdot)\) is the function of optimal velocity, which adopts the same equation that was proposed and calibrated by [PERSON] and [PERSON] [PERSON] and [PERSON] (2013). The OV formula is defined as:  
\[V(\Delta x_{n}(t))=V_{1}+V_{2}\text{tanh}(C_{1}(\Delta x_{n}(t)-l_{c})-C_{2}), \tag{6}\]where \(V_{1}\) = 6.75 m/s, \(V_{2}\) = 7.91 m/s, \(C_{1}\) = 0.13 m\({}^{-1}\), \(C_{2}\) = 1.57, \(l_{c}\) means vehicle length, which is taken as 5 m in this model.