## 5 Conclusions  
This paper proposed a car-following model based on the safety potential field within a connected environment. The stability analysis reveals that the stability region of this model is significantly broader than that of the comparative model. However, incorporating a time delay can modify the motion state of vehicles, thereby impacting traffic flow stability. The experiments indicate that accounting for the SPF of preceding and following vehicles, along with the time delay, enhances traffic flow stability and effectively mitigates congestion. The model parameters were calibrated using real-world data from urban roads, and error analysis was conducted to obtain optimal model parameters. A comparison and analysis of the fitting accuracies of the two models show that the DIDM-SPF provides superior fitting accuracy, indicating its enhanced ability to simulate real traffic flows and offer more reliable  
Figure 9: Spatial-temporal trajectories obtained at different MPRs: (**a**) MPR = 20% (**b**) MPR = 40% (**c**) MPR = 60% (**d**) MPR = 80%.  
predictive and evaluative capabilities. These findings offer valuable insights for optimising traffic flow control and enhancing road traffic efficiency.  
Numerical simulations and experimental results indicate that the MIDM-SPF outperforms the IDM in aspects such as vehicle acceleration, speed, and position distribution, aligning more closely with real traffic scenarios. This study also assesses the impact of the MIDM-SPF on the stability of large-scale mixed traffic flow under different MPRs, confirming that the model can effectively stabilise following traffic flow. This model holds significant implications for the development and enhancement of traffic simulation software and algorithms, as well as the design of driving assistance software and platforms. Such advancements can aid drivers in making informed decisions, alleviate traffic congestion, and support the widespread adoption of autonomous driving. Nevertheless, due to the complexity of transportation systems, it is necessary to further expand the calibration samples and conduct more precise parameter calibrations (to verify the model's performance in different weather conditions or various road types, such as urban roads and rural roads). Moreover, the current model does not address the issue of resource consumption in large-scale deployments, which can also impact the stability of traffic flow. Addressing this issue is an important direction for future research and development.  
Conceptualization, Z.W., W.W., K.M. and S.F.; methodology, Z.W., W.W., K.M. and S.F.; software, Z.W., W.W., K.M. and S.F.; validation, Z.W., W.W., K.M. and S.F.; investigation, Z.W., W.W., K.M. and S.F.; resources, Z.W., W.W., K.M. and S.F.; writing--original draft preparation, Z.W., W.W. and K.M.; writing--review and editing, Z.W., W.W. and S.F.; supervision, Z.W., W.W., K.M. and S.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Open Project of Shandong Key Laboratory of Smart Transportation, grant number 2021 SDKLT003.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
This research was supported by Open Project of Shandong Key Laboratory of Smart Transportation.  
The authors declare no conflicts of interest.