# On the Local and String Stability Analysis of Traffic Collision Risk
###### Abstract  
Conventional traffic stability studies primarily concentrate on the evolution of disturbances in vehicle motion but seldom consider how collision risk changes spatially and temporally. This study bridges the gap by extending the principles of traffic stability analysis to the field of traffic safety, focusing specifically on the temporal and spatial dynamics of collision risk. Leveraging the concepts of local and string stability, we formulate conditions under which collision risk behaves in a stable manner over time and space through the transfer function approach. A comparative analysis between conventional traffic stability and the newly introduced concept of collision risk stability reveals that while conditions for local stability are largely aligned in both domains, the criteria for string stability differ. These theoretical insights are substantiated through microscopic simulations using a variety of car-following models. The simulations also indicate that the consistency between theoretical and simulation outcomes diminishes as the disturbance magnitude increases, which is attributed to the linearization errors inherent in applying the transfer function in the theoretical derivations. traffic safety; traffic stability; collision risk stability; surrogate safety measurement +
Footnote †: journal: Applied  
0000-0002-4690-33  
and CACC on traffic safety. Comparable approaches have been adopted in other studies [PERSON] et al. [19; 20]. Traffic stability analysis investigates how the disturbance of a leading vehicle evolves over time and space. Generally, traffic stability can be categorized into (1) local stability, which refers to the stability of a vehicle's movement over time under the influence of a small disturbance, i.e., the vehicle is locally stable if the disturbance diminishes over time; (2) string stability, which refers to the stability of a platoon of vehicles over space under the influence of a small disturbance on the leading vehicle, i.e., the platoon is string-stable if the disturbance strictly diminishes along the platoon. Numerous studies in the literature have been conducted on applying traffic stability analysis in different scenarios, to name a few [21; 22; 23; 24; 25; 26]. While a substantial body of research has been devoted to the analysis of traffic stability, the predominant focus has been on understanding how disturbances in vehicle motion, such as speed, position, and spacing, evolve over time and space. However, the conventional traffic stability analysis may not be readily applicable to evaluating traffic collision risk. This leads us to the central research question of our study: Can the principles of conventional traffic stability be effectively applied to the analysis of collision risk in traffic systems? Additionally, we hypothesize that applying the methodologies of traffic stability analysis will yield significant insights into the dynamics of collision risk, both temporally and spatially. The question of whether the principles underlying conventional traffic stability can be seamlessly applied to analyze collision risk remains largely unexplored, leaving a critical research gap yet to be filled [16; 27]. This study aims to bridge this research gap by extending the principles and methodologies of conventional traffic stability analysis to the field of traffic safety, with a specific focus on understanding how collision risk evolves both temporally and spatially. The contributions of our work are summarized as follows. 1. The scope of conventional traffic stability analysis is broadened by introducing a specialized framework for evaluating collision risk stability. This framework provides insights into how collision risk changes dynamically across both temporal and spatial dimensions. Furthermore, the framework is compatible with safety surrogate measures (SSMs) for comprehensive risk assessment in various traffic safety scenarios. 2. Both local and string stability criteria concerning traffic collision risk are theoretically derived.