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.

1. Introduction

Traffic safety remains a critical focus for both the academic and the industrial sectors. According to [1], road traffic accidents claim approximately 1.19 million lives annually worldwide. To mitigate the frequency of traffic accidents, academic and industrial communities are increasingly investing in the analysis of accident-related factors using real-world data [2,3,4], predicting hazardous conditions in advance [5], and employing vehicle connectivity and automation technologies for real-time vehicle motion optimization [6].

In traffic safety analysis, the evaluation of the collision risk plays a fundamental role [7,8,9,10]. A significant area of research focuses on the evolution of traffic safety conditions over time and space [11], which is conducive to reducing the likelihood of safety-critical events and providing early warnings. In particular, with the recent innovations in vehicle automation technologies, such as adaptive cruise control (ACC) and cooperative adaptive cruise control (CACC), the study of the evolution of traffic safety conditions may be critical to the decision-making process of when and where the take-over maneuver should happen [12].

Stability analysis has been demonstrated to be highly effective in examining the evolution of disturbances over time and space, as well as providing qualitative insights [13,14]. Recent research has extended analytical approaches to the field of traffic safety, suggesting that traffic stability is a key attribute for enhancing traffic safety where high traffic stability could bring significant benefits to traffic safety [15,16]. Several studies have used theoretical stability analysis to evaluate traffic safety. For example, stability analysis was employed in Wu et al. [17] to formulate traffic safety conditions. Both theoretical linear stability analysis and simulations were utilized in Dong et al. [18] to evaluate the impact of ACC and CACC on traffic safety. Comparable approaches have been adopted in other studies Li et al. [19], Li et al. [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.

• 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.
• Both local and string stability criteria concerning traffic collision risk are theoretically derived. The reciprocal of gap time, a widely used SSM, serves as the safety indicator for traffic in risk analysis. Based on the derived criterion, the similarities and differences between the newly proposed risk stability and conventional traffic stability are revealed.
• The theoretical analysis is verified through microscopic traffic simulations that employ multiple car-following models. Additionally, the simulations reveal a decline in the consistency between theoretical and simulation outcomes as the magnitude of disturbances increases, likely due to the linearization errors inherent in car-following models.

The rest of this paper is structured as follows. Section 2 introduces the fundamentals of conventional traffic local and string stability. In Section 3, the modeling of collision risk is presented, along with definitions for risk local and string stability. Section 4 sets forth the conditions for local and string stability concerning collision risk and compares them with the conventional stability analysis. Microscopic traffic simulations applying different car-following models are conducted in Section 5 to verify the theoretical insights. Finally, Section 6 summarizes the work with conclusions, and its limitations and future directions are discussed.

2. Preliminary on Conventional Traffic Stability Analysis

This section offers an overview of the fundamentals of conventional traffic stability analysis, serving as a concise primer to facilitate the understanding of the proposed risk stability concepts in the following section. For a more detailed and comprehensive review of conventional traffic stability analysis, readers may refer to Sun et al. [28] and Montanino et al. [29].

In the microscopic modeling of traffic flow, the longitudinal driving behavior of individual vehicles is captured using car-following models. A representative form of a general car-following model is as follows:

$${\ddot{x}}_n\left(t\right)=f(\Delta x_n\left(t\right),\Delta {\dot{x}}_n\left(t\right),{\dot{x}}_n\left(t\right))$$

(1)

where $x_n\left(t\right)$, ${\dot{x}}_n\left(t\right)$, and ${\ddot{x}}_n\left(t\right)$ denote the position, speed, and acceleration of vehicle n at time t, respectively; $\Delta x_n\left(t\right)$ and $\Delta {\dot{x}}_n\left(t\right)$ denote the spacing difference and speed difference between the following vehicle n and the leading vehicle $n-1$ (as demonstrated in Figure 1), i.e., $\Delta x_n\left(t\right)=x_{n-1}\left(t\right)-x_n\left(t\right)$, $\Delta {\dot{x}}_n\left(t\right)={\dot{x}}_{n-1}\left(t\right)-{\dot{x}}_n\left(t\right)$; and $f(·)$ denotes a general form of a car-following model.

Nonlinear car-following models are usually linearized via Taylor expansion at the equilibrium state in traffic stability analysis. The linearized form of Equation (1) can be written as

$${\ddot{x}}_n\left(t\right)=f_{\Delta x}\Delta x_n\left(t\right)+f_{\Delta \dot{x}}\Delta {\dot{x}}_n\left(t\right)+f_{\dot{x}}{\dot{x}}_n\left(t\right)$$

(2)

where $f_{\Delta x}=\frac{\partial f}{\partial \Delta x}{|}^{eq},f_{\Delta \dot{x}}=\frac{\partial f}{\partial \Delta \dot{x}}{{|}^{eq},f_{\dot{x}}=\frac{\partial f}{\partial \dot{x}}|}^{eq}$ are the first-order Taylor expansion coefficients at equilibrium state. In the frequency domain, the transfer function is formulated based on the car-following model linearized above as

$$Q_n\left(s\right)=\frac{f_{\Delta x}+f_{\Delta \dot{x}}s}{s^2+(f_{\Delta \dot{x}}-f_{\dot{x}})s+f_{\Delta x}}$$

(3)

which reflects the propagation of disturbance from vehicle $n-1$ to vehicle n in the frequency domain.

2.1. Definition of Traffic Local Stability

The traffic local stability is defined in the same way as bounded-input–bounded-output (BIBO) stability in system engineering [30]. Traffic local stability focuses on the disturbance evolution over time, indicating whether the vehicle’s state can recover to the equilibrium state in a sufficiently long period of time. Specifically, after a minor disturbance (e.g., a sudden deceleration) is forced on vehicle n at time t, if the disturbance decays over time and the vehicle recovers to its initial state $x_n^{eq}$, eventually, the vehicle is considered traffic local stable. The mathematical expression of local stability in the time domain is written as

$$\begin{array}{cc}\hfill x_n\left(t\right)-x_n^{eq}\left(t\right)& =y_n\left(t\right),\underset{t\to \infty }{lim}{y}_n\left(t\right)\to 0\hfill \\ \hfill {\dot{x}}_n\left(t\right)-{\dot{x}}_n^{eq}\left(t\right)& ={\dot{y}}_n\left(t\right),\underset{t\to \infty }{lim}{\dot{y}}_n\left(t\right)\to 0\hfill \end{array}$$

(4)

where $x_n^{eq}\left(t\right)$ and ${\dot{x}}_n^{eq}\left(t\right)$ denote the equilibrium state of $x_n\left(t\right)$ and ${\dot{x}}_n\left(t\right)$, respectively; $y_n\left(t\right)$ represents the disturbance, which is assumed to be differentiable; and ${\dot{y}}_n\left(t\right)$ is its first derivative.

With the linearized car-following model in Equation (2), the local stability criterion is derived as follows:

$$f_{\Delta \dot{x}}>f_{\dot{x}}\mathrm{and}{f}_{\Delta x}>0$$

(5)

2.2. Definition of Traffic String Stability

Unlike traffic local stability, string stability focuses on the evolution of disturbance of a vehicle platoon [31]. The traffic local stability and string stability should be distinguished. Local stability investigates whether the subject vehicle will return to the equilibrium state. In contrast, string stability focuses on the disturbance propagation along the vehicle platoon, i.e., if the disturbance will be amplified or attenuated along the vehicle platoon.

A sketch of a vehicle platoon is demonstrated in Figure 1. After a small disturbance $y_n\left(t\right)$ is introduced to $x_n\left(t\right)$, vehicle n will respond to the disturbance, and the disturbance will propagate to vehicle $n+1$ as $y_{n+1}\left(t\right)$, where $y_{n+1}\left(t\right)=x_{n+1}\left(t\right)-x_{n+1}^{eq}\left(t\right)$. If the disturbance decays over the following vehicles and the vehicle platoon recovers to its initial state, eventually, the vehicle platoon is considered traffic-string-stable. The criterion for string stability is formulated as

$$\frac{\left|\right|y_n\left(t\right){\left|\right|}_p}{\left|\right|y_{n+1}\left(t\right){\left|\right|}_p}>1$$

(6)

where $\left|\right|·{\left|\right|}_p$ represents the p-norm ($p=1,2,\dots ,\infty$), and $p=2$ is normally adopted to analyze the string stability [32], which is also known as $L_2$ string stability. With $p=2$, Equation (6) is reformulated as

$$\frac{\left|\right|Y_n\left(s\right){\left|\right|}_2}{\left|\right|Y_{n+1}\left(s\right){\left|\right|}_2}>1$$

(7)

where $Y_n\left(s\right)=\mathcal{L}\left(y_n\left(t\right)\right)$ and $Y_{n+1}\left(s\right)=\mathcal{L}\left(y_{n+1}\left(t\right)\right)$ denote the Laplace transform of $y_n\left(t\right)$ and $y_{n+1}\left(t\right)$, respectively.

By substituting the linearized car-following model (Equation (2)) into Equation (7), the string stable criterion is derived as follows. For a detailed derivation, refer to [28].

$$f_{\dot{x}}^2-2f_{\Delta x}-2f_{\Delta \dot{x}}{f}_{\dot{x}}>0$$

(8)

3. Introduction of Traffic Collision Risk Stability

Conventional traffic stability analysis focuses on the stability of vehicle motion, e.g., speed, acceleration, and position. In this section, we extend the analysis of traffic stability to safety and introduce collision risk local and string stability.

In traffic safety research, to assess the rear-end collision risk of vehicles on the road, safety surrogate measures (SSMs) are widely adopted [33]. SSMs are usually derived from the currently available measures, such as the position, velocity, and acceleration of the subject vehicle and the preceding vehicle [34]. There are various SSMs, such as the gap time (GP), which characterize the time to collision by assuming the leading vehicle suddenly stops while its follower keeps traveling at the current speed [35], and the crash index (CI), which goes further to cover the momentum information of the vehicles to characterize the severity of the accident [36].

In this study, we define the collision risk, $r_n\left(t\right)$, referenced from a general SSM, as follows:

$$r_n\left(t\right)=f_{SSM}(\Delta x_n\left(t\right),{\dot{x}}_n\left(t\right),{\dot{x}}_{n-1}\left(t\right),{\ddot{x}}_n\left(t\right),{\ddot{x}}_{n-1}\left(t\right))$$

(9)

It is worth noting that we would further characterize a positive relationship between $r_n\left(t\right)$ and the collision risk; i.e., a higher value of $r_n\left(t\right)$ signifies higher collision risk. The variable $r_n\left(t\right)$ can be based on any SSM. If a chosen SSM is inversely correlated with collision risk (e.g., GP), the reciprocal can be applied to ensure that the measure aligns with the notion that a higher value correlates with a greater collision risk.

Stability analysis regards the equilibrium state as the baseline and investigates the variation between the current state and the equilibrium state. The variation between the current risk state and the equilibrium risk is defined as the risk disturbance, denoted by $\Delta r_n\left(t\right)$, which is expressed as

$$\Delta r_n\left(t\right)=r_n\left(t\right)-r_n^{eq}\left(t\right)$$

(10)

where $r_n^{eq}\left(t\right)$ represents the collision risk at an equilibrium state, where $x_n-x_{n+1}=d^{eq}$, ${\dot{x}}_n={\dot{x}}_{n+1}=v^{eq}$ and ${\ddot{x}}_n={\ddot{x}}_{n+1}=0$, with $v^{eq}$ and $d^{eq}$ denoting the velocity and relative space between two neighbor vehicles in the equilibrium state. In other words, $r_n^{eq}\left(t\right)=f_{SSM}(d^{eq},v^{eq},v^{eq},0,0)$.

3.1. Definition of Collision Risk Local Stability

Consider vehicle n to be in equilibrium until a disturbance affecs the vehicle’s velocity. Vehicle n is risk-local-stable if the risk disturbance is suppressed over time and the collision risk $r_n\left(t\right)$ can eventually return to the equilibrium state; otherwise, the vehicle is risk-local-unstable. The mathematical expression of risk local stability is formulated as

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{lim}\Delta r_n\left(t\right)\to 0\hfill \end{array}$$

(11)

In the frequency domain, the difference between traffic local stability and the proposed risk local stability can be viewed from the block diagram in Figure 2. The green blocks and orange blocks represent the input and output of the system, respectively. The input is the Laplace transform of the disturbance from vehicle $n-1$, denoted by $Y_{n-1}\left(s\right)$. $\Delta X_n\left(s\right)=X_n\left(s\right)-X_n^e\left(s\right)$ and $\Delta R_n\left(s\right)=R_n\left(s\right)-R_n^{eq}\left(s\right)$ are the outputs, which denote the space variation and risk variation from the equilibrium state in the frequency domain, respectively. The yellow block denotes the initial state, which is also the equilibrium state. The blue block is used to indicate the frequency-domain operators. $Q_n\left(s\right)$ is the transfer function of vehicle n, which is derived from the car-following model (see Equation (3) for an example). the $R\left(s\right)$ calculator is a manifestation of $f(·)$ in Equation (9) in the frequency domain. s is the frequency domain operation that corresponds to the differential operator in the time domain. $X_n\left(s\right)$, $V_n\left(s\right)$, and acceleration $A_n\left(s\right)$ represent the position, velocity, and acceleration state of vehicle n, respectively.

In traffic local stability analysis, in light of the input disturbance $Y_{n-1}\left(s\right)$ and the equilibrium state of vehicle n, $x_n^{eq}\left(t\right)$, vehicle n generates the response as $X_n\left(s\right)$. Then, the deviation from the state $X_n\left(s\right)$ to the equilibrium state $X_n^{eq}\left(s\right)$ is obtained as $\Delta X_n\left(s\right)$. Traffic local stability investigates if the output disturbance $\Delta X_n\left(s\right)$ will be bounded with a bounded input disturbance $Y_{n-1}\left(s\right)$. Similarly, the deviation of $R_n\left(s\right)$ to the equilibrium risk state $R_n^{eq}\left(s\right)$ is obtained as $\Delta R_n\left(s\right)$. Unlike traffic local stability, risk local stability includes an extra calculation (i.e., $R\left(s\right)$) for the collision risk (i.e., Equation (9)) and investigates if $\Delta R_n\left(s\right)$ is bounded with a bounded input disturbance $Y_{n-1}\left(s\right)$.

3.2. Definition of Collision Risk String Stability

Traffic string stability studies the disturbance propagation along the vehicle fleet or platoon. Inspired by the definition and criterion of traffic string stability, the criterion of risk string stability in the time domain can be formulated as

$$\frac{\left|\right|\Delta r_n\left(t\right){\left|\right|}_p}{\left|\right|\Delta r_{n+1}\left(t\right){\left|\right|}_p}>1$$

(12)

which implies the p-norm of risk disturbance of vehicle $n+1$ is smaller than that of vehicle n, and risk disturbance is compressed along the vehicle fleet. Regularly, 2-norm or ∞-norm is adopted for exploration, where 2-norm indicates the disturbance energy and ∞-norm implies the maximum disturbance. In this study, similar to the traffic string stability analysis, we focus on the 2-norm string stability.

In the frequency domain, the block diagram of traffic and risk string stability are shown in Figure 3. The legend is the same as Figure 2. Away from the equilibrium states, $X_n^{eq}\left(s\right)$ and $X_{n+1}^{eq}\left(s\right)$, the biases of the position for vehicle n and $n+1$ are derived as $\Delta X_n\left(s\right)$ and $\Delta X_{n+1}\left(s\right)$, respectively. The evaluation of traffic string stability aims to compare the p-norm of $\Delta X_n\left(s\right)$ and $\Delta X_{n+1}\left(s\right)$. If the p-norm of $\Delta X_{n+1}\left(s\right)$ is smaller than that of $\Delta X_n\left(s\right)$, the platoon is considered traffic-string-stable. To demonstrate risk string stability, $R\left(s\right)$ calculators are introduced to the diagram. The inputs of $R\left(s\right)$ calculator for vehicle n are the position, velocity, acceleration of vehicle n and vehicle $n+1$, according to Equation (9). The velocity and acceleration are obtained from $X_n\left(s\right)$ through the differentiation operator s. After the initial state, the deviation of collision risk for vehicle n and vehicle $n+1$ are $\Delta R_n\left(s\right)$ and $\Delta R_{n+1}\left(s\right)$.

In contrast to traffic string stability, which examines the relationship between $\Delta X_n\left(s\right)$ and $\Delta X_{n+1}\left(s\right)$, the focus of risk string stability is on the relationship between $\Delta R_n\left(s\right)$ and $\Delta R_{n+1}\left(s\right)$.

4. Stability Analysis on Traffic Collision Risk

For the collision risk indicator, a commonly used SSM, gap time (GP) [35], is selected for the sake of simplicity. Note that the definition of the risk indicator can be based on any form of SSM. In other words, any SSM can be adopted to quantify the collision risk $R\left(s\right)$. With a different $R\left(s\right)$ calculator (or $f_{SSM}$), the risk disturbance $\Delta R_n\left(s\right)$ will be different. In the following analysis, we shall adopt GP as an example of SSM for the sake of simplicity.

GP is defined as the time required to collide if the current vehicle remains at the current velocity and the preceding vehicle suddenly stops:

$$G{P}_n\left(t\right)=\frac{x_{n-1}\left(t\right)-x_n\left(t\right)}{{\dot{x}}_n\left(t\right)}$$

(13)

To make the interpretation more straightforward, we modify the original definition of GP. In the original definition, a lower GP value is indicative of a higher collision risk. In this study, we opt for the reciprocal of GP (denoted by RGP) as the risk indicator. In the revised schema, a larger $r_n\left(t\right)$ directly signifies a higher collision risk:

$$r_n\left(t\right)=RGP=\frac{{\dot{x}}_n\left(t\right)}{x_{n-1}\left(t\right)-x_n\left(t\right)}$$

(14)

4.1. Risk Disturbance Modeling

In the initial conditions, traffic is assumed to be in an equilibrium state, which in turn implies that $r_n$ starts from an equilibrium state as well:

$$r_n^{eq}=\frac{v^{eq}}{d^{eq}}$$

(15)

where $v^{eq}={\dot{x}}_n^{eq}$ and $d^{eq}=x_{n-1}^{eq}-x_n^{eq}$ denote the velocity and desired spacing in the equilibrium state.

At a certain point, the equilibrium state of traffic is disrupted by thedisturbance. The disturbance of vehicle $n-1$ is denoted by $y_{n-1}\left(t\right)$, and the disturbance of vehicle n is $y_n\left(t\right)$. Then, the collision risk for vehicle n can be rewritten as

$$r_n\left(t\right)=\frac{v^{eq}+{\dot{y}}_n\left(t\right)}{d^{eq}+y_{n-1}\left(t\right)-y_n\left(t\right)}$$

(16)

Since the relationship between $Y_n\left(s\right)$ and $Y_{n-1}\left(s\right)$ can be obtained from the transfer function in Equation (3), the time-domain $y_n\left(t\right)$ can be obtained through convolution:

$$y_n={\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n\left(s\right)\right)$$

(17)

where ${\mathcal{L}}^{-1}(·)$ denotes the inverse Laplace transform operator.

Similarly, ${\dot{y}}_n\left(t\right)$ can be calculated as

$${\dot{y}}_n={\mathcal{L}}^{-1}\left(s{Y}_{n-1}\left(s\right)Q_n\left(s\right)\right)$$

(18)

Substituting Equations (17) and (18) into Equation (16), $r_n\left(t\right)$ is formulated as:

$$r_n\left(t\right)=\frac{v^{eq}+{\mathcal{L}}^{-1}\left(s{Y}_{n-1}\left(s\right)Q_n\left(s\right)\right)}{d^{eq}+y_{n-1}-{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n\left(s\right)\right)}$$

(19)

Accordingly, the collision risk for vehicle $n+1$ is represented by

$$r_{n+1}\left(t\right)=\frac{v^{eq}+{\mathcal{L}}^{-1}\left(s{Y}_{n-1}\left(s\right)Q_n^2\left(s\right)\right)}{d^{eq}+{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n\left(s\right)\right)-{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n^2\left(s\right)\right)}$$

(20)

The risk disturbances for vehicle n (denoted by $\Delta r_n\left(t\right)$) and vehicle $n+1$ (denoted by $\Delta r_{n+1}\left(t\right)$), deviated from equilibrium risk ($r_n^{eq}$), can be formulated as

$$\Delta r_n\left(t\right)=\frac{v^{eq}+{\mathcal{L}}^{-1}\left(s{Y}_{n-1}\left(s\right)Q_n\left(s\right)\right)}{d^{eq}+y_{n-1}-{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n\left(s\right)\right)}-\frac{v^{eq}}{d^{eq}}$$

(21)

$$\Delta r_{n+1}\left(t\right)=\frac{v^{eq}+{\mathcal{L}}^{-1}\left(s{Y}_{n-1}\left(s\right)Q_n^2\left(s\right)\right)}{d^{eq}+{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n\left(s\right)\right)-{\mathcal{L}}^{-1}\left(Y_{n-1}\left(s\right)Q_n^2\left(s\right)\right)}-\frac{v^{eq}}{d^{eq}}$$

(22)

4.2. Risk Local Stability Analysis

As per Section 3.1, the criterion of risk local stability should meet the following criterion:

$${\int }_{-\infty }^{+\infty }\left|{\mathcal{L}}^{-1}\left(\frac{\mathcal{L}(\Delta r_n\left(t\right))}{Y_{n-1}\left(s\right)}\right)e^{-st}\right|\mathrm{d}t<\infty$$

(23)

which leads to:

$$Re\left({\delta }_i\right)<0,\forall \mathrm{poles}{\delta }_i\mathrm{of}\frac{\mathcal{L}(\Delta r_n\left(t\right))}{Y_{n-1}\left(s\right)},s=j\omega ,i=1,2,3...$$

(24)

where $Re(.)$ calculates the real part of the poles.

Remark 1.

If a vehicle is traffic-local-stable, as governed by the car-following model, it will also be locally stable with respect to collision risk, and vice versa.

Proof. 

Firstly, we show that being traffic-local-stable is sufficient for being risk-local-stable. Based on the definition of traffic local stability (Equation (4)), if the traffic is local-stable, we will have:

$$y_{n-1}\left(t\right)\to 0,{\dot{y}}_n\left(t\right)\to 0\mathrm{and}{y}_n\left(t\right)\to 0,\mathrm{when}t\to \infty$$

(25)

According to Equation (16), when Equation (25) holds, the following equation is satisfied.

$$\underset{t\to \infty }{lim}{r}_n\left(t\right)=\frac{v^{eq}}{d^{eq}}=r_n^{eq}$$

(26)

Hence, we have

$$r_n\left(t\right)-r_n^{eq}\to 0,\mathrm{when}t\to \infty$$

(27)

which implies the risk is local-stable.

Then, we show that being risk local-stable is sufficient for being traffic-local-stable. If risk is local-stable, we have Equation (27). Combining Equation (27) and Equation (16), we have

$$v^{eq}{y}_n\left(t\right)+d^{eq}{\dot{y}}_n\left(t\right)-v^{eq}{y}_{n-1}\left(t\right)\to 0,\mathrm{when}t\to \infty$$

(28)

Further, for the risk local stability, the leading vehicle could be in the equilibrium state, i.e., $r_{n-1}\left(t\right)=r^{eq}$ and $y_{n-1}\left(t\right)=0$, while risk local stability only focuses on the evolution of the perturbation for vehicle n over time. Then, Equation (28) is further simplified to

$$v^{eq}{y}_n\left(t\right)+d^{eq}{\dot{y}}_n\left(t\right)\to 0,\mathrm{when}t\to \infty .$$

(29)

Consider a large enough t, we have the following ordinary differential equation (ODE):

$$v^{eq}{y}_n\left(t\right)+d^{eq}{\dot{y}}_n\left(t\right)=0$$

(30)

The solution to the equation is either $y_n\left(t\right)=0$ or $y_n\left(t\right)=C{e}^{-\frac{v^{eq}}{d^{eq}}t}$, where the constant C equals $y_n\left(0\right)$. Both solutions imply traffic local stability when t goes to infinity.

The proof is complete.    □

4.3. Collision Risk String Stability Analysis

For risk string stability, the criterion is written as

$$\frac{\int |\Delta r_{n+1}{\left(t\right)|}^2\mathrm{d}t}{\int |\Delta r_n{\left(t\right)|}^2\mathrm{d}t}<1$$

(31)

Based on Equations (21) and (22), the risk stability criterion is related to the configuration of the car-following model, the equilibrium-state velocity and spacing, and the input disturbance.

To further the analysis, this study employs a sine-form disturbance, a waveform commonly favored in stability studies. Sine-form disturbances are known for their capability to effectively capture the oscillatory behavior in traffic flows, allowing for a comprehensive evaluation of a system’s resilience against disturbance [37,38]. In this study, the sine-form disturbance is given in a general form as

$$y_{n-1}\left(t\right)=Asin\left(\omega t\right)\mathsf{\Theta }\left(t\right)$$

(32)

where A indicates the magnitude of the disturbance and $\omega$ denotes the frequency of the disturbance. $\mathsf{\Theta }\left(t\right)$ is the Heaviside step function, which is adopted to ensure the causality of the perturbation. The Laplace transform of Equation (32) is

$$Y_{n-1}\left(s\right)=\frac{A\omega }{s^2+{\omega }^2}$$

(33)

Substituting Equation (33), Equation (21) and Equation (22) into Equation (31), the criterion is reformulated as:

$$\int |\Delta r_n{\left(t\right)|}^2\mathrm{d}t<\int {|\Delta r_{n+1}\left(t\right)|}^2\mathrm{d}t$$

(34)

where

$\int |\Delta r_n{\left(t\right)|}^2\mathrm{d}t=$

$$\int |\frac{v^{eq}+{\mathcal{L}}^{-1}\left(\frac{A{f}_{\Delta \dot{x}}\omega s^2+A{f}_{\Delta x}\omega s}{s^4+(f_{\Delta \dot{x}}-f_{\dot{x}})s^3+(f_{\Delta x}+{\omega }^2)s^2+(f_{\Delta \dot{x}}-f_{\dot{x}}){\omega }^{2}s+f_{\Delta x}{\omega }^2}\right)}{d^{eq}+Asin\left(\omega t\right)\mu \left(t\right)-{\mathcal{L}}^{-1}\left(\frac{A{f}_{\Delta \dot{x}}\omega s+A{f}_{\Delta x}\omega }{s^4+(f_{\Delta \dot{x}}-f_{\dot{x}})s^3+(f_{\Delta x}+{\omega }^2)s^2+(f_{\Delta \dot{x}}-f_{\dot{x}}){\omega }^{2}s+f_{\Delta x}{\omega }^2}\right)}-\frac{v^{eq}}{d^{eq}}{|}^2\mathrm{d}t$$

(35)

$\int |\Delta r_{n+1}{\left(t\right)|}^2\mathrm{d}t=$

$$\int |\frac{v^{eq}+{\mathcal{L}}^{-1}\left({\dot{Y}}_{n+1}\left(s\right)\right)}{d^{eq}+{\mathcal{L}}^{-1}\left(\frac{A{f}_{\Delta \dot{x}}\omega s+A{f}_{\Delta x}\omega }{s^4+(f_{\Delta \dot{x}}-f_{\dot{x}})s^3+(f_{\Delta x}+{\omega }^2)s^2+(f_{\Delta \dot{x}}-f_{\dot{x}}){\omega }^{2}s+f_{\Delta x}{\omega }^2}\right)-{\mathcal{L}}^{-1}\left(Y_{n+1}\left(s\right)\right)}-\frac{v^{eq}}{d^{eq}}{|}^2\mathrm{d}t$$

(36)

${\dot{Y}}_{n+1}\left(s\right)=$

$$\frac{A{f}_{\Delta \dot{x}}\omega s^3+2f_{\Delta x}{f}_{\Delta \dot{x}}A\omega s^2+A{f}_{\Delta x}^2\omega s}{(s^6+2(f_{\Delta \dot{x}}-f_{\dot{x}})s^5+({(f_{\Delta \dot{x}}-f_{\dot{x}})}^2+2f_{\Delta x}+{\omega }^2)s^4+\left(2(f_{\Delta x}+{\omega }^2)(f_{\Delta \dot{x}}-f_{\dot{x}})\right)s^3+({(f_{\Delta \dot{x}}-f_{\dot{x}})}^2+2f_{\Delta x}){\omega }^2+f_{\Delta x}^2)s^2+2f_{\Delta x}(f_{\Delta \dot{x}}-f_{\dot{x}}){\omega }^{2}s+f_{\Delta x}^2{\omega }^2}$$

(37)

$$Y_{n+1}\left(s\right)=\frac{1}{s}{\dot{Y}}_{n+1}\left(s\right)$$

(38)

Remark 2.

The criterion for risk string stability, specified in Equations (34)–(38), poses a particular challenge for analytical exploration due to its complexity and structure. Given these complications, it becomes difficult to proceed with further analytical deductions or insights. Therefore, to gain a more in-depth understanding of the implications of the criterion, we will employ numerical solutions in the following section for a more comprehensive analysis.

4.3.1. Numerical Solutions’ Settings

There are three factors for the traffic stability and risk stability criteria, i.e., $f_{{\Delta }_x},f_{\Delta \dot{x}}$, and $f_{\dot{x}}$, capturing the sensitivity to the change in relative position, relative velocity, and current velocity. The higher absolute value of all these three parameters indicates a stronger sensitivity of the car-following model to the changes. In addition, the setting of the car-following model should satisfy the rational driving constraints defined in Wilson and Ward [39], which suggest that vehicles tend to have large acceleration when the relative position/speed is large or when the following vehicle speed is low. The rational driving constraints can be further simplified as follows:

$$f_{{\Delta }_x}>0,f_{\Delta \dot{x}}>0,f_{\dot{x}}<0$$

(39)

Moreover, the values of these three parameters cannot be infinite due to physical conditions. For instance, if $f_{\Delta x}$ is too large, the following vehicle will have a huge deceleration even if the relative position only slightly changes. Each parameter should be limited to a rational interval. In this study, the specific ranges for numerical solutions of $f_{\Delta x},f_{\Delta \dot{x}}$, and $f_{\dot{x}}$ are $[0,5]$, $[0,5]$, and $[-2.5,0]$. Each parameter is sampled at an interval of 0.01. In total, $500\times 500\times 250$ solutions are presented.

4.3.2. Numerical Solutions’ Results

The numerical results are shown in Figure 4, where the difference between the criterion of traffic string stability and risk string stability is visualized. In Figure 4a, the purple dots represent the numerical solutions satisfying both th traffic string stability criterion (Equation (8)) and the risk string stability criterion (Equation (31)). Meanwhile, in Figure 4b, the red dots represent the numerical solutions satisfying only the traffic string stability criterion (Equation (8)) but not the risk string stability criterion. The orange dots denote both risk- and traffic-unstable conditions.

To better unveil the similarities and differences between traffic string stability and risk stability, several 2-D diagrams are provided in Figure 5, which clearly s shows that when the vehicle platoon is risk-string-stable, it is also traffic-string-stable. Moreover, Figure 5a,b reflect that the difference between risk and traffic string stability is rising with the growth of $f_{\Delta x}$.

Figure 4 and Figure 5 show that if the platoon is traffic-risk-string-stable, it is also traffic-string-stable. However, if the platoon is traffic-string-stable, it is not necessarily risk-string-stable.

It is important to note that the points illustrated in Figure 4 are not suggestive of distinct states ($f_{\Delta x},f_{\Delta \dot{x}},f_{\dot{x}}$) within the traffic dynamics. Rather, they are illustrative examples designed to demonstrate the potential behavior of the traffic dynamics under varying circumstances. Due to the complex nature of the system, correlations among the variables and their derivatives exist. Consequently, the diversity of points in Figure 4 serves to encapsulate a range of possibilities and to demonstrate the distinction between traffic flow stability and risk stability.

5. Simulation Verification

In this section, we employ microscopic traffic simulations as a means to validate the insights gained from the stability analysis on traffic collision risk. Comparing simulation results with theoretical derivation is well established in the literature as a robust approach to confirm both the consistency and the applicability of stability analyses [22,40]. Specifically, we are interested in evaluating how well the simulation results align with the theoretical analysis concerning risk stability.

5.1. Simulation Setup

Consider a platoon of vehicles traveling along a straight road with a single lane in the simulation; at a certain point $t_0$, a disturbance causes the acceleration of the leading vehicle. The disturbance is a sine function as $Asin\left({\omega }_{0}t\right)$, where A is the amplitude of the disturbance and ${\omega }_0$ is the disturbance frequency. The velocity increases from $v^{eq}$ to $v^{eq}+\frac{2A}{\omega }$ in time $t_0$ to $t_0+\frac{\pi }{\omega }$ and decreases from $v^{eq}+\frac{2A}{\omega }$ to $v^{eq}$ in $t_0+\frac{\pi }{\omega }$ to $t_0+\frac{2\pi }{\omega }$. Figure 6 shows an example of the acceleration and velocity profiles of the leading vehicle.

The Intelligent Driver Model (IDM) [41] is selected as the car-following model in the demonstration of Figure 7. The IDM acceleration $a_i\left(t\right)$ for vehicle i at time t is formulated as

$${\ddot{x}}_n\left(t\right)=\alpha \left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4-{\left({\displaystyle \frac{S_0+max\{0,v_n\left(t\right)T+{\displaystyle \frac{v_n\left(t\right)\Delta v_n\left(t\right)}{2\sqrt{\alpha \beta }}}\}}{S_i\left(t\right)}}\right)}^2\right]$$

(40)

where $\alpha$ and $\beta$ denote the maximal acceleration and deceleration and $S_0$ represents the minimum gap. The IDM parameter settings are the same as those in Kesting et al. [42]: $v_0=120$ km/h, $S_0=2$ m.

The Taylor expansion coefficients as follows of the IDM model are calculated as follows

$$f_{\Delta x}=\frac{2\alpha (s_0+v^{eq}T)}{{\left(s^{eq}\right)}^3},f_{\Delta \dot{x}}=\frac{\sqrt{\alpha }{v}^{eq}(s_0+v^{eq}T)}{\sqrt{\beta }{\left(s^{eq}\right)}^2},f_{\dot{x}}=-\alpha \left[\frac{4{\left(v^{eq}\right)}^3}{v_0^4}+\frac{2T(s_0+v^{eq}T)}{{\left(s^{eq}\right)}^2}\right]$$

where $s^{eq}=\frac{s_0+v^{eq}T}{\sqrt{1-{\left(\frac{v^{eq}}{v_0}\right)}^4}}$.

Figure 7 shows the speed profile and the risk profile for three cases: (a) both traffic- and risk-string-stable; (b) risk-string-unstable but traffic-string-stable; (c) both traffic- and risk-string-unstable. It can be seen that for Figure 7a,c, the traffic string stability and risk string stability are the same (i.e., both stable or both unstable), while for Figure 7b, it is shown that even the traffic is string-stable, the risk can be string-unstable, which further verifies the insight in Section 4.3.2.

5.2. Consistency Results

The simulation verification procedure is presented in Algorithm 1. The inputs of the process include the equilibrium velocity $v^{eq}$, the standstill space $s_0$, the number of vehicles involved N, the disturbance amplitude A, and the disturbance duration time $t_0$. Further, $\delta t$ is the time slot to update the vehicles’ states, and i and j are the countable numbers to calculate the consistency of theoretical derivation and simulation results. For each time step for each vehicle, $r_n\left(t\right)$ is calculated from the simulation and then compared with the theoretical value to check whether the theoretical result is consistent with the simulation result regarding stability. Finally, the consistency is calculated. As shown in Algorithm 1, the specific range selected for $\alpha$, $\beta$, and T is $[0.1,5](\mathrm{m}/{\mathrm{s}}^2)$, $[0.1,5](\mathrm{m}/{\mathrm{s}}^2)$ and $[0.1,5]\left(\mathrm{s}\right)$.

Algorithm 1: Simulation verification

input: $v^{eq}$, $s_0$, A, N, $t_0$, $\omega$, $\Delta t$, $i=j=0$

output: Consistency

Besides IDM, two kinds of widely used car-following models, optimal velocity model (OVM) [40] and ACC control model [43], are applied to simulate the movements of vehicles. The consistency results are presented in

Table 1. Consistency between theoretical derivation and simulation results.

Disturbance ($\frac{2\mathit{A}}{{\mathit{\omega }}_0}$)	Model	Traffic (Local)	Traffic (String)	Risk (Local)	Risk (String)
	IDM	98.6%	91.9%	98.4%	90.9%
0.3	OVM	98.7%	89.6%	97.6%	89.7%
	ACC	99.8%	98.9%	98.5%	95.9%
	IDM	98.1%	89.2%	97.9%	88.1%
0.8	OVM	98.3%	88.2%	96.5%	86.6%
	ACC	98.5%	97.9%	98.3%	95.8%
	IDM	96.9%	88.9%	96.1%	87.0%
1.5	OVM	95.8%	87.1%	95.6%	84.2%
	ACC	98.3%	97.6%	97.2%	94.8%
	IDM	95.5%	86.8%	95.3%	85.6%
3	OVM	95.6%	81.6%	95.1%	80.7%
	ACC	98.0%	96.4%	96.9%	93.3%

. We see that high consistency is obtained between theoretical values and simulation results for both traffic and risk stability analysis.

In addition, when the disturbance magnitude is large, the consistency between the theoretical value and simulation results decreases. This could be due to the linearization nature of applying the transfer function approach in the theoretical stability analysis, whereby the car-following model linearization is conditioned on the disturbance magnitude being small. If the disturbance is large, to a certain extent, the error between the linearized model and the original model may grow and lead to inconsistent results.

Further, as can be seen from Table 1, the ACC model exhibits the highest consistency owing to the fact that the adopted ACC model is linear. In addition, note that even with a linear car-following model, it is difficult to achieve 100% consistency for either traffic or risk stability since the continuous-time CF model is discretized in the simulation.

6. Conclusions

Conventional traffic stability analysis focuses on the evolution of the disturbance of vehicle motions, while it is still unknown how the safety condition (collision risk) evolves over time and space. In this study, the traffic stability analysis is extended to the collision risk analysis. A commonly used surrogate safety measure (SSM), gap time (GP), is adopted to quantify the collision risk. The linear analysis technique is employed to investigate small disturbances in the context of collision risk, specifically around an equilibrium state. Then, inspired by the conventional traffic stability analysis, risk local and string stability are defined, and the boundary conditions for risk local and string stability are formulated for a general car-following model. The study shows that risk local stability exhibits consistency with traffic local stability within a rational driving constraint. However, the risk string stability differs from the traffic string stability in terms of boundary conditions. In the specific rational range, the traffic string stability criterion is a sufficient but unnecessary condition for the risk string stability criterion. Numerical solutions are found and microscopic simulations with various car-following models are performed to validate the theoretical derivation and the applicability of the proposed risk stability analysis. Results show that both traffic stability analysis and risk stability demonstrate high consistency between theoretical derivation and simulation when the disturbance magnitude is small.

This study has substantial implications for the enhancement of transportation safety. For example, by integrating the identified risk stability conditions into traffic management systems, a more adaptive response to risk disturbances could be developed. And understanding of risk propagation dynamics may engender safer driving behaviors, which could be further promoted in driver education initiatives. This work can be expanded in several directions. Firstly, to advance our understanding of traffic safety dynamics, future research should consider incorporating nonlinear methods for a more comprehensive understanding of car-following behavior under a wider range of conditions. Secondly, it would be interesting to incorporate vehicle heterogeneity, along with considering cases where $\omega >1$, aligning with the driver characteristics outlined in [29,32], to provide a more robust and comprehensive traffic safety analysis. Furthermore, this paper investigates safety from a theoretical perspective. In future work, applying real-world empirical data to validate the proposed risk stability analysis is an interesting direction. Such an application would not only enhance the practical relevance of the theoretical findings but also provide a deeper understanding of real-world traffic safety conditions and lead to more accurate prediction of collision risks.