## 1 Introduction  
Due to fast progress of urban motorization, today traffic congestion is considered as a common and serious problem. According to the records traffic congestion has caused nearly 7 billion hours of travel delay in 471 urban areas of the United States in 2014, which is more than two times longer than that recorded in 1982 [1; 2; 3]. The frequent deceleration and acceleration of vehicles while driving will not only block the traffic but can also lead to unwanted environmental pollution. Hence, great efforts have been made to alleviate traffic congestion based on car-following models. A car-following model, as a microscopic model [4; 5; 6; 7; 8; 9; 10; 11], describes how a car which follows another car responds to the changes in preceding traffic conditions on a one-way street that limits overtaking. Since the introduction of the car-following model by [PERSON] in 1953 [12], many researchers have studied various factors influencing drivers' car-following behaviors from different points. Based on a driver's stimulus response mechanism, the optimal velocity model (OVM) was put forward by [PERSON] et al. [8]. The OVM can be defined as:  
\[\tilde{x}_{n}=a[V(\Delta x_{n})-\dot{x}_{n}], \tag{1}\]  
where \(a\) describes driver sensitivity; \(\Delta x_{n}=x_{n+1}(t)-x_{n}(t)\) means the headway of two successive cars;  
\(V(\Delta x_{n})\) denotes the optimal velocity function considering preceding headway change; \(\dot{x}_{n}\) is vehicle velocity \(n\); \(\tilde{x}_{n}\) is vehicle acceleration \(n\). Although this model is able to well explain different traffic flow processes like the frequent stopping and going of vehicles and instability of traffic flow, [PERSON] suggested that the OVM produces excessive accelerations and decelerations. Therefore, motivated by pedestrian dynamics, they proposed a generalized force model (GFM) considering the difference of negative velocities [13]. The GFM can be defined as:  
\[\ddot{x}_{n}=a[V(\Delta x_{n}(t))-\dot{x}_{n}(t)]+\lambda H(-\Delta\dot{x}_{n}(t ))\Delta\dot{x}_{n}(t), \tag{2}\]  
where \(H(\cdot)\) denotes the function of Heaviside and \(\lambda\) is relative velocity response intensity. However, the GFM ignores the effect of positive velocity difference, which misleads the vehicle moving delay calculation. To make up for this deficiency of this model, [PERSON] et al. [9] developed the full velocity difference model (FVDM), which considers both negative and positive differences in velocity. The kinematic function of the model can be expressed as:  
\[\ddot{x}_{n}=a[V(\Delta x_{n}(t))-\dot{x}_{n}(t)]+\lambda\Delta\dot{x}_{n}(t), \tag{3}\]  
The numerical simulations confirm that the FVDM can describe vehicle start delays and disturbance propagations. After that, many variants of the FVDM [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24] were developed based on velocity differences. Considering the influence of lateral separation distance between two successive vehicles, [PERSON] et al. established the non-lane-based full velocity difference car-following model (NLBCFM) [14; 15]. [PERSON] et al. explored the relationship between the driver's attribution and driving behavior and redefined a novel car-following model by introducing a driver personality parameter [16]. [PERSON] and [PERSON] found that drivers would adjust their anticipation of optimal velocity beforehand by assessing future traffic conditions [18]. Therefore, an anticipation optimal velocity model (AOVM) was introduced. [PERSON] and [PERSON] investigated the effect of vehicular gap fluctuation on the flow of traffic [19]. By adding the space gap perturbation factor, the modified model further improves traffic flow stability, decreasing the consumption of energy. Numerous researchers have made efforts to consider many properties of traffic flow, and to construct lots of improved models from different points of view.