# An Extended Car-Following Model Considering Generalized Preceding Vehicles in V2X Environment
## 2 Model  
Car-following is the behavior, which is ubiquity in the traffic system, that driver manipulates his/her vehicle to follow the vehicles ahead. To study the characteristics of the car-following behavior and traffic flow, multi car-following models [34; 35; 36; 37; 38] were proposed based on various modeling ideas. [PERSON] et al. [34] believe that drivers always attempt to keep a safe velocity depending on the headway between two successive vehicles when following the front vehicle. According to this, [PERSON] proposed a car-following model called the optimal velocity (OV) model, and its motion equation is as follows:  
\[\frac{dv_{n}(t)}{dt}=a[V(\Delta x_{n})-v_{n}(t)] \tag{1}\]  
where \(a\) represents the sensitivity of the driver. \(V(\Delta x_{n})\) is the optimal velocity function, and \(\Delta x_{n}=x_{n+1}(t)-x_{n}(t)\) denotes the headway of the two successive vehicles. \(x_{n}(t)\) and \(v_{n}(t)\) are, respectively, the position and velocity of the \(n\) th vehicle where \(t\) represents time. [PERSON] and [PERSON] [35] found that the OV model would work out excessive acceleration/deceleration during calibration of the optimal velocity function. To improve the OV model, [PERSON] established the generalized force (GF) model by introducing the negative velocity difference. Formulation of the GF model is as follows:  
\[\frac{dv_{n}(t)}{dt}=a[V(\Delta x_{n})-v_{n}(t)]+\lambda H[-\Delta v_{n}(t)] \Delta v_{n}(t) \tag{2}\]  
where \(a\) and \(\lambda\) represent the sensitivity of the driver. \(H\) is the Heaviside function and \(\Delta v_{n}(t)=v_{j+1}-v_{j}\) is the velocity difference between the leading vehicle \(j+1\) and the following vehicle \(j\). The GF model improved the OV model by solving the problem of excessive acceleration/ deceleration. However, there are still some imperfections in the GF model. For instance, the following vehicle will not slow down when the headway is less than the minimum safety headway, and the preceding vehicle is going much faster. Motivated by these, [PERSON] et al. [36] constructed the full velocity difference (FVD) model by further considering the positive velocity difference. Its motion equation is as follows:  
\[\frac{dv_{n}(t)}{dt}=a[V(\Delta x_{n})-v_{n}(t)]+\lambda\Delta v_{n}(t) \tag{3}\]  
Compared with the OV and CF models, the FVD model shows higher performance in simulating traffic flow and, especially, studying the stability of traffic flow. However, the aforementioned models only reflect the interaction between the vehicle and its front vehicle. In a realistic traffic system, drivers not only focus on the vehicle ahead but also pay attention to multi preceding vehicles. Especially in a V2X environment, drivers can obtain massive information (for example, the velocity of an arbitrary number of vehicles ahead). Compared with an arbitrary number of vehicles ahead in the current lane, the driver would pay more attention to nearby vehicles, particularly the GPV, which are in the driver's field of view. Among vehicles that are of GPV, the driver is primarily concerned with the vehicle in front of him/her to maintain a safe distance and avoid a collision. On this basis, the driver would also focus on GPV to optimize car-following behavior. Based on the above contents, an extended car-following model called the GPV model is proposed by introducing the average velocity of GPV, which can reflect the whole traffic situation on the segment [15]. The model's dynamic equation is as follows:  
\[\frac{dv_{n}(t)}{dt}=p[a[V(\Delta x_{n})-v_{n}(t)]+\lambda v_{n}(t)]+(1-p)( \overline{v_{n}}-v_{n}(t)) \tag{4}\]  
where \(a\), \(\lambda\) and \(p\) respectively represent the sensitivity of driver about optimal velocity difference, velocity difference and the difference between GPV's average velocity and self-vehicle velocity, and the drivers are assumed to be ideal and identical, which is expressed by a constant value of sensitivity in the GPV model.