## 3 Methodology  
### Car-Following Model Based on Safety Potential Field  
From the perspective of physics, a field can be defined as a space where objects in specific states exert interaction forces on other objects without physical contact over a certain spatial distance. The varying relative positions of these objects lead to differences in the interaction of forces, generating potential energy related to attributes such as distance, size, and weight. In this way, fields and potentials with different properties combine throughout the physical space, displaying the capacity for objects to interact with one another. Similarly, in traffic scenarios, there is a correspondence between vehicle driving behaviour and potential field theory. In car-following scenarios, sudden disturbances can prompt notable oscillations in vehicle speed due to the requirement for maintaining safe driving distances. Vehicles constantly adjust their speeds to preserve a safe distance from the vehicle ahead, a dynamic adjustment strategy akin to the influence of force in a potential field, which ultimately contributes to traffic flow stability. This also reflects the driving risks that emerge as vehicles respond to changes in scenario elements during motion. In mixed traffic environments featuring CAVs and human-driven vehicles (HDVs), CAVs can obtain extensive state information about the surrounding environment, including their own locations, lane lines, and information about surrounding vehicles. In these traffic scenarios, any element impacting a vehicle's driving state can be considered a field source in the physical space. The safety potential field characterises driving safety in car-following scenarios based on the interaction forces of these elements. To analyse the amalgamation of vehicles, roads, and environmental elements in coupled scenarios in a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Authors**} & **Information** & \multirow{2}{*}{**Time**} & **Stability** & **Calibration** & \multirow{2}{*}{**Mixed Traffic**} \\  & **from Multiple** & **Delay** & **Analysis** & **of Relevant** & \\  & **Vehicles** & & & **Parameters** & \\ \hline Li et al. [35] & X & X & X & & X \\ [PERSON] et al. [36] & & X & X & X & X \\ [PERSON] et al. [37] & X & X & X & & X \\ [PERSON] et al. [38] & & X & X & & X \\ [PERSON] et al. [39] & X & X & X & X & \\ [PERSON] et al. [40] & & X & X & & \\ This study & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of the car-following behaviour modelling works related to SPF. layered manner, this study adopts an adaptive layered model for scenario representation, as per [PERSON] et al. [41]. Figure 1a,b illustrates a typical car-following scenario, comprising three key elements: lane-marking lines, road boundary lines, and vehicles. Consequently, in the traffic environment, CAVs are influenced by the potential fields of lane-marking lines, road boundary lines, and other vehicles, denoted as \(U_{L}\), \(U_{S}\), and \(U_{v}\), respectively. The safety potential field is calculated using (1). \[|U_{total}|=w_{L}|U_{L}|+w_{s}|U_{s}|+w_{v}|U_{v}|, \tag{1}\]  
where \(U_{L}\) represents the lane-marking field, \(U_{S}\) denotes the road boundary field, \(U_{v}\) is the vehicle field, and \(w_{L}\), \(w_{s}\), and \(w_{v}\) are their respective weight coefficients. For the SPF model, derivation, and simulation concerning lane-marking lines, road boundary lines and other vehicles, please refer to Appendix A, specifically Appendix A.1. Assuming all vehicles travel along the road's centreline, the lane-marking and road boundary potential fields become irrelevant. Consequently, there is no need to consider lane-changing behaviour or vehicle deviation angles in this scenario. The intensity of the vehicle potential field exerted by the \((n-i)\)th and \((n+j)\)th vehicles on the nth vehicle can be expressed as follows. Figure 2 illustrates a multi-vehicle car-following scenario involving \(n+j\) vehicles (where both \(n\) and \(j>0\)).