## 4 Results
### Dynamic Performance Study of the Model  
To examine the dynamic performance of a car-following model involving multiple vehicles with a safety potential field, this study stimulated the movement of a vehicle platoon at a four-way intersection scenario. When the traffic signal was red, all ten vehicles, spaced at 2.5 m apart, remained stationary. At \(t=0\), as the traffic signal changed from red to green, all vehicles began to accelerate and move, aiming to traverse the intersection swiftly while maintaining stability. The parameter settings for both the MIDM-SPF and IDM are detailed in Table 3, with the time delay \(t_{d}\) set at 0.15 s and the maximum velocity at 10 m/s. Figure 5(a) reveals that the MIDM-SPF model achieves higher acceleration than the IDM at the same moment, reaching its maximum acceleration value in a shorter duration. Figure 5(b,c) show that the velocity distribution of the MIDM-SPF model attains its maximum value more rapidly within a brief period. This behaviour aligns with real-world driving at intersections, where vehicles typically strive to accelerate to the maximum speed quickly after a traffic signal change, ensuring safety to expedite passing through the intersection. Moreover, compared to the IDM, the MIDM-SPF covers a greater distance within the same amount of time, suggesting that the car-following scenario at the intersection can facilitate more vehicles passing through in a limited time, thereby significantly enhancing the efficiency of vehicle flow at the intersection. Furthermore, this study created a 400 m circular road scenario to examine the influence of SPFs on traffic flow stability. One hundred vehicles with uniform spacing were tested in this configuration to evaluate their resilience against disturbances and traffic flow oscillations. The sensitivity coefficients, \(\gamma\) and \(\mu\), were varied using a controlled variables method for the experiments. Figure 6(a) depicts the space headway distribution in the MIDM-SPF model for different \(\gamma\) values (0, 0.1, 0.2, and 0.37) when \(\mu\) is set to zero in the circular road scenario. Conversely, Figure 6(b) illustrates the space headway distribution for 100 vehicles in the car-following model when \(\gamma\) is zero and \(\mu\) varies (0, 0.1, 0.2, and 0.29). With increasing \(\gamma\) and \(\mu\), there is a notable decrease in the fluctuation of space headways within the vehicle platoon. This suggests that speed fluctuations caused by initial disturbances in the traffic flow have been effectively absorbed by the system, restoring it to stable traffic conditions. Moreover, as \(\gamma\) and \(\mu\) change, the incidence of vehicles with varying headways due to disturbances diminishes during the traffic flow stabilisation process. This observation implies that the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **ME** & **MAE** & **RMSE** & **R2** \\ \hline IDM & 0.192 & 0.279 & 0.383 & 95.6\% \\ MIDM-SPF & 0.103 & 0.249 & 0.351 & 97.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting error data corresponding to the two models. Figure 5: Front and rear vehicle fitted trajectory data curves: (**a**) front vehicle; (**b**) rear vehicle. Figure 8a clearly demonstrates that, under the influence of disturbances, the space-time evolution of traffic flow, with both \(\gamma=0\) and \(\mu=0\), is characterised by numerous folding bands. These bands, visible as multiple peaks and valleys on the time-space plane, signify that disturbances are continuously propagating and amplifying within the traffic flow. This leads to an unstable state, potentially manifesting as traffic congestion. In contrast, Figure 8b-d show that as the values of \(\gamma\) and \(\mu\) incrementally increase, the undulations in the traffic flow's time-space plot start to decrease, and the folding bands contract. When \(\gamma=0.37\) and \(\mu=0.29\), the folding bands almost vanish, and the overall time-space evolution appears significantly flatter.