## 3 Intra-Platoon Potential Field
Therefore, it was necessary to make the platoon lane change model consistent with the previous individual lane change model in the base parameter values of the field. The parameter values are shown in Table 1.  
On the basis of the parameter values that were calibrated using the consistency with the non-platoon vehicle lane-changing model, the remaining parameter \(\gamma\) was calibrated using the optimal platoon space headway. Mature research on the theoretical value of the platoon lane-changing gap is lacking at present. There are essential differences between vehicle lane changing and car following, but the critical state at the beginning and end of \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & \(\lambda\) & \(\alpha\) & \(\beta\) & \(\tau\) & \(\epsilon\) \\ \hline Value & 0.0641 & 0.0738 & 0.2331 & 2.6990 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: The results of parameter calibration. lane changing is the car-following state. Therefore, the desired time headway in CAVs car following could be used to calibrate the optimal distance at the beginning of lane changing. However, due to the oblique movement of vehicles in the process of lane changing, it is difficult to maintain a fixed distance, which cannot be calculated using the headway formula. The distance between the two vehicles would fluctuate up and down at the optimal distance and form a distance interval. Therefore, this optimal distance cannot be the critical minimum collision distance for car following since this would easily allow for collisions in the lane-changing process. In addition, cooperative adaptive cruise control (CACC) is a low-level CAVs system. Some research was done in this regard. [PERSON] (2018) suggested that the desired time headway of a car following in a CACC strings flow was 1.4 s with 0.2 s as the standard deviation. Meanwhile, [PERSON] (2018) claimed that CACC strings can operate with 0.6 s as the minimum inter-vehicle time gap. According to the definition of the optimal distance in this study, 1.4 s was selected as the desired time headway to calibrate the optimal distance at the beginning of the lane-changing process, as shown in Equation (7). It is also worth saying that 1.4 s is just the desired headway time under a low-level CAVs system. With the improvement of the CAVs system level, a closer headway should be adopted. \[D=v_{e}\cdot T \tag{7}\]  
where \(D\) is the optimal distance at the beginning of lane changing, \(v_{e}\) is the velocity of the ego vehicle, and \(T\) is the desired time headway, which was set to 1.4 s.  
In the IPF, there is a minimum ring of the field contour. When the minimum ring of the ego vehicle is tangent to that of the front one, the distance between the two vehicles is the optimal distance in the physical sense. Therefore, this study defined the optimal distance as Equation (8):  
\[D=k_{e}+k_{f} \tag{8}\]  
where \(D\) is the optimal distance, \(k_{e}\) is the distance from the minimum point in the driving direction to the centroid of the ego vehicle, and \(k_{f}\) is the distance between the minimum point in the negative driving direction of the front vehicle and its centroid. They are derived from the IPF, whose derivative is set to zero. The formulas were as follows:  
\[\left\{ \begin{array}{l}k_{e}=\frac{\sqrt{A}e^{\alpha v_{e}+\beta a_{e}}}{ \sqrt{\gamma\cdot T}}\\ k_{f}=\frac{\sqrt{A}e^{\alpha_{f}+\beta a_{f}}}{\sqrt{\gamma\cdot T}}\end{array} \right. \tag{9}\]  
\(\gamma\) was calibrated using simultaneous equations, namely, Equations (7)-(9). The results are shown in Equation (10):  
\[\gamma=\frac{\lambda\left(e^{\alpha v_{e}+\beta a_{e}}+e^{\alpha v_{f}+\beta a _{f}}\right)^{2}}{\left(v_{e}\cdot T\cdot\tau\right)^{2}} \tag{10}\]  
The model can provide dynamic feedback of the gravitational potential field distribution of the platoon vehicles in different motion states. The risk maps are generated, as shown in Figure 3. The green contour represents a horizontal curve that is formed by projecting points with the same potential field into a plane.