Swarm Intelligent Car-Following Model for Autonomous Vehicle Platoon Based on Particle Swarm Optimization Theory

Abstract

The emergence of autonomous vehicles offers the potential to eliminate traditional traffic lanes, enabling vehicles to navigate freely in two-dimensional spaces. Unlike conventional traffic constrained by physical lanes, autonomous vehicles rely on real-time data exchange within platoons to adopt cooperative movement strategies, similar to synchronized flocks of birds. Motivated by this paradigm, this paper introduces an innovative traffic flow model based on the principles of particle swarm intelligence. In the proposed model, each vehicle within a platoon is treated as a particle contributing to the collective dynamics of the system. The motion of each vehicle is determined by the following two key factors: its local optimal velocity, influenced by the preceding vehicle, and its global optimal velocity, derived from the average of the optimal velocities of M vehicles within its observational range. To implement this framework, we develop a novel particle swarm optimization algorithm for autonomous vehicles and rigorously analyze its stability using linear system stability theory, as well as evaluate the system’s performance through four distinct indices inspired by traditional control theory. Numerical simulations are conducted to validate the theoretical assumptions of the model. The results demonstrate strong consistency between the proposed swarm intelligent model and the Bando model, providing evidence of its effectiveness. Additionally, the simulations reveal that the stability of the traffic flow system is primarily governed by the learning parameters $c_1$ and $c_2$, as well as the field of view parameter M. These findings underscore the potential of the swarm intelligent model to improve traffic flow system dynamics and contribute to the broader application of autonomous traffic systems management. In addition, it is worth noting that this paper explores the operational control of an AV platoon from a theoretical perspective, without fully considering passenger comfort, as well as “soft” instabilities (vehicles joining/leaving) and “hard” instabilities (technical failures/accidents). Future research will expand on these related aspects.

1. Introduction

The evolution of transportation systems is undergoing a paradigm shift with the emergence of autonomous vehicles (AVs), fundamentally transforming conventional traffic dynamics. Traditional car-following models, which have served as the foundation for understanding traffic flow for decades, are increasingly inadequate for capturing the complex interactions between intelligent, connected vehicles. As AVs transition from experimental prototypes to commercial deployment, there is an urgent need for innovative traffic flow models that can harness their advanced sensing, communication, and computational capabilities. Platoon-based operations, where multiple AVs travel in coordinated formations, represent a promising approach for optimizing traffic efficiency and safety. Unlike human-driven vehicles constrained by reaction times and limited perception, AV platoons can achieve near-instantaneous information sharing, enabling synchronized movements and significantly reduced following distances. However, the theoretical frameworks governing these cooperative behaviors remain underdeveloped. The concept of AV platoons draws inspiration from biological systems, particularly the collective behaviors observed in bird flocks, fish schools, and insect swarms, where individuals achieve remarkable coordination through simple rules and local interactions. In traffic scenarios, platoons rely on inter-vehicle communication and real-time data exchange to synchronize movements, ensuring safety and stability while minimizing disruptions. Cooperative autonomous platooning systems can dynamically adjust inter-vehicle spacing and speed, significantly improving traffic density and reducing energy consumption. Therefore, developing intelligent models for AV platoon control has become a critical research area in intelligent transportation systems.

The research on car-following behavior dates back nearly 70 years and covers hundreds of models based on various theories; different perspectives have been constructed in the development process [1]. In general, transport simulation models can be divided into three major categories, including macroscopic, mesoscopic, and microscopic modeling, depending upon the level of details required for network analysis [2]. Within the realm of microscopic simulation models and contemporary traffic flow theory, the car-following model (CFM) plays a pivotal role. This model aims to elucidate the relationship between individual driver behaviors and the collective dynamics of traffic. The study of car-following models has been ongoing for more than seven decades, originating around 1953 [3], with the objective of describing the sequential behavior of drivers in traffic streams. Each vehicle is perceived as a discrete, interactive unit, essential to understanding the inter-vehicular interactions within a lane. A fundamental premise of these models is that the driving decisions of each following vehicle are influenced by the behavior of the vehicles ahead. Presently, car-following models are divided into two primary groups—traditional models and models based on artificial intelligence. Traditional models such as the stimulus–response, safe distance, desired headway, and psycho-physical models, utilize mathematical formulations to simulate the trailing behavior among vehicles. On the other hand, artificial intelligence models rely on computational algorithms, including neural networks and fuzzy logic, which are structured on specific sets of rules [4]. The earliest form of traditional car-following models is termed the stimulus–response model, designed to mimic the car-following actions of the drivers [5]. The behavior of a lead vehicle acts as the stimulus, prompting the following driver to either accelerate or decelerate, a response moderated by the total reaction time T. The speed and relative spacing of the lead vehicle are also considered as substantial stimuli. Throughout the years, several stimulus–response models have emerged and been employed to predict car-following behavior under various conditions. Four significant models include the Gazis–Herman–Rothery (GHR) model, concurrently introduced by Kometani and Sasaki in 1958 in Japan and by Chandler, Herman, and Montroll under the General Motors research framework [6]. Following them, Helly proposed the Linear (Helly) model in 1959, which adjusted the acceleration of the following vehicle based on the braking action of the lead vehicle, building upon the GHR model and introducing new elements [7]. In 1995, Bando et al. devised the optimal velocity model (OVM), furthering Newell’s 1963 concept by incorporating the difference between the present and desired velocities of the vehicle as stimuli for the driver’s responses [8,9]. Treiber, Hennecke, and Helbing introduced the Intelligent Driver Model (IDM) in 2000, integrating safe driving norms relevant to safe distance CF models, although it is categorized under the stimulus–response model. This model expressed the vehicle’s acceleration as a factor of its speed, relative velocity, and the distance from the leading vehicle [10]. Furthermore, the safe distance models calculate a secure following gap to prevent collisions, especially when the lead vehicle driver exhibits erratic behaviors. However, if the trailing driver maintains a gap exceeding the recommended safe distance, this model’s guidelines become less applicable. This approach adheres to the protocols set by Pipes, advocating maintaining a safe distance of at least one car length per 10 mph of speed. Gipps further refined this category in 1981 by introducing a collision avoidance model with dual velocity constraints for the following vehicle [11]. Recently, driven by the prospect of a traffic flow comprising a mix of human-driven vehicles and connected, automated vehicles (CAVs), research has focused on maximizing CAV capabilities to mitigate the instability of mixed traffic [12]. These models integrate specific thresholds that, once exceeded, prompt drivers to adjust their speeds until velocity disparities with the lead vehicle are no longer noticeable. In general, it remains essential to comprehensively consider the driver–vehicle–environment factors when modeling car-following behavior under intelligent and connected conditions, despite the considerable number of achievements in research on car-following behavior.

Artificial intelligence (AI) models represent another category of CFMs. These models are generally straightforward in presentation and interpretation, relying primarily on specific mathematical equations. However, the complexity of human behavior often exceeds what these equations can encapsulate. Consequently, there has been a concerted effort over the last two decades to develop sophisticated AI models to more accurately predict human behavior, such as a driver’s cognitive processes. The advent of faster and more advanced computing platforms has significantly facilitated the development and popularity of these models. AI models in the realm of CF primarily build on the foundations of fuzzy logic and neural networks, employed either independently or in an integrated manner. Fuzzy logic models operate on the principle that drivers’ understanding and actions are qualitative and may frequently depend on an incomplete collection of information, thereby neglecting certain relevant factors. For instance, when deciding to decelerate upon nearing the vehicle ahead, drivers might not consider the exact relative speed and distance. Their decisions are influenced by perceptions, which are inherently subjective and imprecise, thus making ‘near’ and ‘nearing’ vague terms. As a result, ‘decelerate’ becomes a decision derived using fuzzy logic: IF ‘near’ AND ‘nearing’ THEN ‘decelerate’. This model can hence offer a closer approximation of human behavior. Researchers like Kikuchi and Chakroborty were pioneers in utilizing fuzzy logic rules to model CF behavior [13], while subsequent studies have introduced more sophisticated applications such as Hao’s multi-agent CF model based on fuzzy logic, simulating nuanced human driving behaviors within a structured framework of perception, anticipation, inference, strategy, and action [14]. Furthermore, Chen’s innovative application integrated fuzzy logic with a conditional deep Q-network to tackle directional planning in autonomous driving systems [15]. Similarly, Li and Axenie emphasized enhancing the interaction model and calibrating driver behaviors utilizing fuzzy logic to achieve more effective autonomous driving [16,17]. A primary challenge with fuzzy logic models lies in accurately determining fuzzy rules, which are often derived from human judgments. Parallel to fuzzy logic, neural networks have been another cornerstone of AI models in transportation. Particularly prominent in the 1990s, these networks attempt to emulate human brain functions, inferring outcomes from data pertaining to specific scenarios through principles from neurobiology and contemporary cognitive science. Their self-learning capabilities allow them to optimize performance based on a vast array of training data, facilitating an understanding of the relationships between various input and output parameters. Notably, neural networks have been applied to a range of topics within transportation, from driver behavior analysis to autonomous vehicle control [18,19], with researchers like Pomerleau and Chen focusing on their practical applications in real-world scenarios [19,20]. It is important to note that the field of AI models in CF is not limited to fuzzy logic and neural networks alone. There is potential in employing a hybrid approach that combines these technologies to simulate CF behavior more effectively. Moreover, recent advancements in swarm intelligence—an interdisciplinary technology—have shown potential in applications ranging from shared traffic systems to multi-agent platforms [21]. In summary, the evolution of AI models in CF highlights a trend towards integrating various computational technologies to better mimic and predict human driving behavior, addressing challenges through innovative approaches, and continuing to evolve with the advancements in computing and AI technologies.

Swarm intelligence, a concept derived from the collective behavior of biological systems such as flocks of birds and schools of fish, offers compelling insights for AV platoon management [22]. These natural systems demonstrate remarkable efficiency, adaptability, and stability through decentralized coordination mechanisms. The PSO algorithm is a swarm-based stochastic algorithm proposed originally by Kennedy and Eberhart [23], which exploits the concepts of the social behavior of animals, such as fish schooling and bird flocking. PSO mimics the cooperative food-searching behavior in these swarms. Each swarm member adjusts its searching strategy based on both individual and communal learning experiences [24]. This algorithm is globally acknowledged for tackling problems within multidimensional search spaces with continuous parameter values, proving especially efficacious in real-valued optimization. The adaptability of PSO has led to its application in resolving non-differentiable challenges that might manifest dynamic changes, irregularities, or noise over time. Through ongoing development, several variants of the PSO algorithm have been tailored for specific optimization scenarios. These include the micro-PSO for high-dimensional problems requiring fewer particles [25], the dynamic scheduling of transmission tasks [26], the optimization algorithm for the urban transit routing problem [27], the vehicle routing problem with cross-docking and carbon emission reduction in logistics management [28], and the use of a Quantum PSO (QPSO) strategy for predicting traffic flow in intelligent transportation systems (ITSs) [29]. The PSO algorithm offers obvious advantages in building car-following model, and it provides a mathematical foundation for translating these biological principles into algorithmic approaches for vehicle coordination. In brief, recent research has explored the various aspects of AV platoons, including stability analysis, fuel efficiency, and collision avoidance. However, the integration of swarm intelligence with car-following dynamics specifically for autonomous vehicle platoons represents a significant research gap. In addition, it is worth noting that this paper explores the operational control of AV platoon from a theoretical perspective, without fully considering passenger comfort, as well as “soft” instabilities (vehicles joining/leaving) and “hard” instabilities (technical failures/accidents). Future research will expand on these related aspects.

The remainder of the paper is organized as follows: Section 2 introduces the concept of swarm intelligence through PSO and elaborates on the construction of a car-following model employing PSO. Section 3 explores the stability conditions and provides a visual representation of the innovative PSO car-following model utilizing the linear system stability theorem. Section 4 discusses the indices that define the characteristics of the traffic flow system, where numerical simulations are carried out to validate the accuracy of the stability theorem and the characteristics of the traffic system. Section 5 offers a summary and the conclusions derived from the study.

2. Models and Methods

2.1. PSO Principle

A typical PSO algorithm is illustrated in Figure 1. As for the standard PSO algorithmic structure, a swarm of particles updates their relative positions from one iteration to another, boosting the PSO algorithm to duly perform the search process [30]. To obtain the optimum solution, each particle moves towards its prior personal best position (${\mathbf{p}}_{best}$) and the global best position (${\mathbf{g}}_{best}$) in the swarm. Assuming a minimization problem, one has the following PSO expressions, as shown in Equations (1) and (2):

$${\mathbf{p}}_{{best}_i}^t={\mathbf{x}}_i^{*}\mid f\left({\mathbf{x}}_i^{*}\right)=\underset{k=1,2,\cdots ,t}{min}\left(\left\{f\left({\mathbf{x}}_i^k\right)\right\}\right)$$

(1)

where $i\in \{1,2,\dots ,N\}$, and 

$${\mathbf{g}}_{best}^t={\mathbf{x}}_{*}^t\mid f\left({\mathbf{x}}_{*}^t\right)=\underset{\begin{array}{c}i=1,2,\dots ,N\\ k=1,2,\dots ,t\end{array}}{min}\left(\left\{f\left({\mathbf{x}}_i^k\right)\right\}\right)$$

(2)

where i denotes particle’s index; t is the current iteration’s number; f is the objective function to be optimized (minimized); $\mathbf{x}$ is the position vector (or a potential solution); and N is the total number of particles in the swarm. During each iteration $t+1$, the velocity $\mathbf{v}$ and position $\mathbf{x}$ of each particle i are updated, as shown in Equations (3) and (4).

$${\mathbf{v}}_i^{t+1}=\omega {\mathbf{v}}_i^t+c_1{r}_1\left({\mathbf{p}}_{{best}_i}^t-{\mathbf{x}}_i^t\right)+c_2{r}_2\left({\mathbf{g}}_{best}^t-{\mathbf{x}}_i^t\right)$$

(3)

$${\mathbf{x}}_i^{t+1}={\mathbf{x}}_i^t+{\mathbf{v}}_i^{t+1}$$

(4)

where $\mathbf{v}$ represents the velocity vector; $\omega$ is the inertia weight utilized to balance the local exploitation and global exploration; $r_1$ and $r_2$ are random vectors uniformly distributed within the range ${[0,1]}^D$ (with D as the search space dimensionality or the size of the problem at hand); and $c_1$ and $c_2$—called “acceleration coefficients”—are positive constants.

It should be noted that an upper bound is commonly set for the velocity vector, as a means to prevent particles from shaving off the search space. This forces them to take an appropriate step size to explore the entire search domain, as demonstrated in the “velocity clamping” method and the “constriction coefficient” strategy proposed by Clerc and Kennedy [23].

2.2. Particle Behavior Analysis of AV Platoon

Inspired by the movement of bird flocks, we will now consider a line of AVs traveling along a straight path, as depicted in Figure 2. Analyzing these vehicles through the lens of particle swarm principles reveals both similarities and differences. The primary distinction between these two systems is that PSO operates in a three-dimensional space, whereas vehicle platooning occurs on a two-dimensional plane defined by longitudinal and latitudinal coordinates. However, similar to particles in a swarm, both vehicles and particles make decisions based on the optimization of local and global states. To simplify the analysis of an AV platoon, we may disregard the mass and mechanical properties of each vehicle and conceptualize the system as a collection of particles, as depicted in Figure 3. These vehicle particles exhibit social behavior similar to swarm intelligence, as inspired by natural entities like flocks of bird and schools of fish. Accordingly, we refer to them as vehicle particles, embodying the characteristics of swarm intelligence. Historically, the inception of swarm intelligence can be traced to the introduction of algorithms such as PSO [23] and Ant Colony Optimization (ACO) [31].

In the context of swarm intelligence, each particle represents a potential solution to a specific problem. Simply put, an AV particle is analogous to a point within a two-dimensional search space. Our objective is to identify the most optimal location within this space, evaluated through a predefined fitness function. Before implementing the vehicle PSO method, it is imperative to define a dynamic objective function and set the algorithm’s parameters, including the learning coefficients $c_1$ and $c_2$, and the number of vehicle particles N. Subsequently, a swarm of AV particles, denoted as P, is initialized through the random generation of particles. Each particle is characterized by a two-dimensional vector, with each element representing a decision variable constrained by specified lower and upper bounds. Generally speaking, the system initializes with each particle in the swarm represented by two-dimensional random vectors. As the optimization process progresses, the particle dynamics evolve, aiming to locate the ‘Gbest’ or the global best solution so far, which is stored and updated in a dedicated vector. Upon establishment of the initial swarm, each particle’s quality is assessed using the aforementioned fitness function. Following this evaluation, the optimal vehicle—or particle—is identified and retained in ‘Gbest’, which represents the best performance across the tasks of either minimization (lowest function value) or maximization (highest function value). Upon satisfying the termination criteria, the particle in ’Gbest’ is then output as the final result.

The AV-PSO algorithm can be implemented in either a global or localized variant. In the global variant, each vehicle particle gravitates towards the best particle in the entire swarm. In contrast, in the localized version, each particle is influenced by the best particle within its immediate neighborhood. The acceleration formula for the vehicle particles, without the inertia weight effect, is defined as follows in Equation (5):

$$a_{i,j}=c_1{a}_{i,local}+c_2{a}_{i,global}$$

(5)

where $a_{i,j}$ is the $i^{th}$ particle’s acceleration; and $a_{i,local}$ and $a_{i,global}$ denote the local and global best particles’ accelerations influencing particle i, respectively.

2.3. Fitness Function for AV Platoon Motion

It is recognized that $V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)$ operates as the optimal velocity function (OVF) for the $n^{th}$ vehicle. This function generally exhibits teh following two key properties: (i) It increases monotonically, and (ii) it has a defined upper limit, $V^{max}\triangleq V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)\to \infty$. There are two conventional types of OVFs primarily utilized by researchers. The first type considers variables like maximal speed, headway, and vehicle body length. The second type relates these parameters to given values. The most commonly employed OVFs are delineated in Equations (6)–(8) [32].

$$V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)={\displaystyle \frac{v_{max}}{2}}[tanh(0.13(\mathrm{\Delta }{x}_n\left(t\right)-l_n)-1.57)+tanh(0.13l_n+1.57)]$$

(6)

$$V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)={\displaystyle \frac{v_{max}}{2}}\left[tanh(\mathrm{\Delta }{x}_n\left(t\right)-h_c)+tanh\left(h_c\right)\right]$$

(7)

$$V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)=V_1+V_{2}tanh\left[C_1(\mathrm{\Delta }{x}_n\left(t\right)-l_n)-C_2\right]$$

(8)

where $v_{max}$ denotes the maximum velocity; and $l_n$ represents the length of the $n^{th}$ vehicle, conventionally set to four meters in simulations. Typical values for other parameters utilized frequently in simulations include $V_{max}=3.2\mathrm{m}/\mathrm{s}$, $V_1=6.75\mathrm{m}/\mathrm{s}$, $V_2=7.91\mathrm{m}/\mathrm{s}$, $C_1=0.13{\mathrm{m}}^{-1}$, and $C_2=1.57$ [29].

2.4. Local Optimal Acceleration Term

Traffic flow instabilities are a common precursor to congestion on highways and other major roads, necessitating the development of dynamic models to accurately characterize these phenomena. The optimal velocity model (OVM) offers a dynamic framework for understanding traffic congestion by applying the equation of motion to each vehicle within the system. This model, originally proposed by Bando in 1995 [9], posits that each vehicle travels at an optimum speed that is contingent upon the distance from the preceding vehicle. The equation of motion for the OVM is expressed as follows in Equation (9):

$${\ddot{x}}_n\left(t\right)=F\left(\alpha \right)[V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)-{\dot{x}}_n\left(t\right)]$$

(9)

where $F\left(\alpha \right)$ serves as the sensitivity function, in which $F\left(\alpha \right)=\alpha$, a constant coefficient. In this context, ${\ddot{x}}_n\left(t\right)$ and ${\dot{x}}_n\left(t\right)$ denote the acceleration and velocity of the $n^{th}$ vehicle at time t, respectively. $\mathrm{\Delta }{x}_n\left(t\right)=x_{n+1}\left(t\right)-x_n\left(t\right)$ is defined as the headway between the $n^{th}$ vehicle and its successor. Accordingly, the local optimal acceleration term is delineated by Equation (10), as follows:

$$a_{n,\mathrm{local}}\left(t\right)=\alpha [V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)-{\dot{x}}_n\left(t\right)]$$

(10)

2.5. Global Optimal Acceleration Term

The global aspect of the traffic model integrates data across the entire convoy, with the global optimal acceleration, under periodic conditions, predicated on collective vehicular behavior—a concept akin to swarm intelligence. This global acceleration is detailed in Equation (11).

$$a_{n,\mathrm{global}}\left(t\right)=\alpha \left[{\displaystyle \frac{1}{M}}\sum _{j=1}^{M}V\left(\mathrm{\Delta }{x}_j\left(t\right)\right)-{\dot{x}}_n\left(t\right)\right]$$

(11)

where M represents the count of vehicles within the global consideration of the $n^{th}$ vehicle, with M equating to $N-1$ under periodic conditions.

2.6. The AV-PSO Model Acceleration Expression

We now introduce a sophisticated CFM that integrates principles of PSO and OVM, which we call the AV-PSO car-following model. This model can be expressed as shown in Equation (12).

$${\ddot{x}}_n\left(t\right)=c_1\times a_{n,local}\left(t\right)+c_2\times a_{n,global}\left(t\right)$$

(12)

Subsequently, we derive the comprehensive vehicle AV-PSO model, as delineated in Equation (13).

$$a_n\left(t\right)=\alpha \left\{c_1\left[V\left(\mathrm{\Delta }{x}_n\left(t\right)\right)-{\dot{x}}_n\left(t\right)\right]+c_2\left[{\displaystyle \frac{1}{M}}\sum _{l=1}^{M}V\left(\mathrm{\Delta }{x}_{n+l}\left(t\right)\right)-{\dot{x}}_n\left(t\right)\right]\right\}$$

(13)

Conventionally, the coefficient $c_1$ is assigned a substantially higher value than $c_2$, thereby prioritizing the acceleration component driven by local optimization. The parameter M denotes the amount of automobiles considered influential within this context. When the complete platoon is taken as the global optimality criterion, M equates to N.

2.7. The AV-PSO Model Motion Expression

Now, the AV-PSO model motion expression can be summarized as shown in Equation (14),

$$\left\{\begin{array}{c}{v}_n\left(t\right)=v_n(t-\mathrm{\Delta }t)+a_n\left(t\right)\times \mathrm{\Delta }\left(t\right)\hfill \\ x_n\left(t\right)=x_n(t-\mathrm{\Delta }t)+v_n\left(t\right)\times \mathrm{\Delta }\left(t\right)\hfill \end{array}\right.$$

(14)

where $\mathrm{\Delta }t$ is the time step for updating the motion state; $v_n\left(t\right)$ is the AV’s velocity at time t; and $x_n\left(t\right)$ is the AV’s position at time t. It should be noted that the acceleration and jerk limitations that maintain passenger comfort are set between $[0,1.5]\mathrm{m}·{\mathrm{s}}^{-1}$ in our simulations, based on references [8,9].

3. Model Analysis

3.1. Stability Conditions Analysis

We consider a scenario where there is a stable solution for a uniformly distributed intelligent internet-connected vehicle platoon, as described by Equation (15).

$$x_{n,0}\left(t\right)=h_{s}n+v_{0}t$$

(15)

where $h_s={\displaystyle \frac{L}{N}}$ denotes the steady state spacing between two successive vehicles in the platoon; and $v_0$ represents the constant velocity of the steady traffic platoon.

A small deviation from this uniform flow is introduced through the following Equation (16).

$${\delta }_n\left(t\right)=exp(i{\varphi }_{k}n+st),{\varphi }_k=k\left({\displaystyle \frac{2\pi }{N}}\right),k=1,2,\dots ,N-1$$

(16)

where $|{\delta }_n\left(t\right)|\ll 1$. Consequently, the position of the $n^{\mathrm{th}}$ vehicle at time t can be expressed as shown in Equation (17).

$$x_n\left(t\right)=x_{n,0}\left(t\right)+{\delta }_n\left(t\right)$$

(17)

In reference to the study under consideration, we established a stable condition for the platoon of internet-connected vehicles. By inserting Equation (17) into Equation (13), we derive its second-order differential representation, as shown in Equation (18).

$${\displaystyle \frac{d^2{\delta }_n\left(t\right)}{d{t}^2}}=c_1\alpha [V^{\prime }\left(h_s\right)\delta \left(\mathrm{\Delta }{x}_n\left(t\right)\right)-{\dot{\delta }}_n\left(t\right)]+c_2\left\{\alpha \left[{\displaystyle \frac{1}{M}}\right]\sum _{f=1}^M{V}^{\prime }\left(h_s\right)\delta \left(\mathrm{\Delta }{x}_{n+f}\left(t\right)\right)-{\dot{\delta }}_n\left(t\right)\right\}$$

(18)

where ${\ddot{\delta }}_n\left(t\right)={\ddot{x}}_n\left(t\right)$; ${\dot{\delta }}_n\left(t\right)={\dot{x}}_n\left(t\right)$; $\delta \left(\mathrm{\Delta }{x}_n\left(t\right)\right)={\delta }_{n+1}\left(t\right)-{\delta }_n\left(t\right)-h_s$; and $\delta \left(\mathrm{\Delta }{x}_{n+f}\left(t\right)\right)={\delta }_{n+f+1}-{\delta }_{n+f}-h_s$. Moreover, we derive the Fourier Transform Formulation, as shown in Equation (19):

$$s^2+(c_1+c_2)\alpha s-\alpha V^{\prime }\left(h_s\right)[c_1(e^{i{\varphi }_k}-1)+c_2{\displaystyle \frac{1}{M}}\sum _{f=1}^M(e^{i{\varphi }_{k(f+1)}}-e^{i{\varphi }_{kf}})]=0$$

(19)

The solution form of Equation (19) is $s=s_1\left(i{\varphi }_k\right)+s_2{\left(i{\varphi }_k\right)}^2+\dots$. By neglecting the higher-order terms and retaining only the first- and second-order terms of $\left(i{\varphi }_k\right)$, the solution can be derived, as expressed in Equation (20).

$$\left\{\begin{array}{c}{s}_1=V^{\prime }\left(h_s\right)\hfill \\ s_2={\displaystyle \frac{V^{\prime }\left(h_s\right)[c_1+c_2(2+M)]}{2}}-{\displaystyle \frac{{\left[V^{\prime }\left(h_s\right)\right]}^2}{\alpha }}\hfill \end{array}\right.$$

(20)

Based on above analysis, the AV platoon will maintain a stable state of motion, as elucidated by the linear system stability theory. Now, we provide Lemma 1 as common knowledge.

Lemma 1.

If the condition stated in Equation (21) is satisfied, the uniform AV platoon flow exhibits stability.

$$\alpha >{\displaystyle \frac{2V^{\prime }\left(h_s\right)}{c_1+c_2(2+M)}}$$

(21)

Conversely, the system becomes unstable if this condition is not met.

3.2. Stability State Verification

To elucidate the conditions under which the system maintains stability, the simulations portrayed in Figure 4 delineate the critical line of system stability. Our analysis considers the following three distinct parameter variations: the local learning factor $c_1$, the global learning factor $c_2$, and the number of vehicles within a global perspective, as represented by M. Standard configurations include a maximum velocity (Vmax) of $3.2$ m per second and a typical object height ($H_c$) of 4 m.

In Figure 4a, the Bando model illustrates the division between the stable and unstable phases. The critical demarcation occurs primarily around $\mathrm{\Delta }x\approx 4\mathrm{m}$ and $\alpha \approx 3.2$, with distinct stability zones persisting at lesser ($\mathrm{\Delta }x<3\mathrm{m}$) and greater ($\mathrm{\Delta }x>5\mathrm{m}$) intervals of $\mathrm{\Delta }x$. The subsequent figures underscore the impacts of altering either of the parameters $c_1$ and $c_2$ on stability. Figure 4b inserts various $c_2$ values ($c_2=0.015,0.03,0.045,0.06,0.075$) into the model, illustrating a broadening and descending of the instability region with increased $c_2$, thereby suggesting that higher $c_2$ values contribute to enhanced system stability. Conversely, Figure 4c examines the influence of differing $c_1$ magnitudes ($c_1=0.985,1.182,1.2805,1.379,1.4775$) on system dynamics. Stability zones remain consistent with prior outcomes, while higher $c_1$ values notably elevate the curves, indicating a decrease in the stability threshold $\alpha$. This adjustment suggests an inverse relationship between $c_1$ and system stability. These simulations and analyses fundamentally affirm the criteria established by Lemma 1, demonstrating that variances in parameter settings significantly influence the stability of the internet-connected vehicle platoon system.

Following quantitative analysis, Figure 4a demonstrates a critical point reaching its peak at $(\mathrm{\Delta }x,\alpha )=(4,3.2)$. The region remains stable when $\mathrm{\Delta }x<3\mathrm{m}$ and $\mathrm{\Delta }x>5\mathrm{m}$, with $\alpha$ maintaining values below 1.5, indicative of stability. For Figure 4b, illustrating scenarios where $c_2=0.015$, the peak occurs around $\alpha =3.2$. An increment of $c_2$ to 0.075 results in the peak descending to $\alpha \approx 2.0$. Higher values of $c_2$ ($c_2=0.075$) reveal a more extensive stable region, characterized by $\alpha$ values remaining below 1.5 across a broader $\mathrm{\Delta }x$ range. In Figure 4c, with $c_1=0.985$, the critical peak is located around $\alpha =3.2$. As $c_1$ increases to 1.4775, the peak descends to $\alpha \approx 2.0$. Elevated $c_1$ values ($c_1=1.4775$) correlate with a wider stable region, where $\alpha$ values stay under 1.5 across an extended $\mathrm{\Delta }x$ range. Figure 4d further elucidates the stability characteristics within the optimal velocity model by exploring the effects of parameter M on traffic flow stability. Similar to the prior observations, this figure identifies a central unstable region around $\mathrm{\Delta }x\approx 4\mathrm{m}$ flanked by stable regions. Quantitatively, the figure contrasts the Bando model (depicted with a solid red line) across varying M values (20, 40, 60, 80, 100). With an increase in M, peak $\alpha$ values decrease, thereby broadening the stability region. Specifically, $M=20$ (represented by a dashed magenta line) showcases higher peak $\alpha$ compared to $M=100$ (illustrated with a dotted black line), suggesting that elevated M values enhance stability by lowering peak $\alpha$ and expanding stable areas.

In conjunction with the earlier findings, it is evident that $c_1$, $c_2$, and M significantly affect system stability. Enhancements in parameters $c_1$, $c_2$, and M collectively contribute to enlarging stable regions and diminishing the instability threshold, ultimately promoting enhanced stability in traffic flow dynamics.

3.3. Peak Value Comparison with Bando Model

Additionally, a mathematical model is crucial for documenting the evolutionary progression of error patterns. We present a formula for calculating relative error, as shown in Equation (22), with the results summarized in

Table 1. Stability evaluation indices.

${\mathit{P}}_{\mathbf{new}}\left({\mathit{c}}_2\right)$	${\mathit{P}}_{\mathbf{bando}}$	$\mathit{e}\%$	${\mathit{P}}_{\mathbf{new}}\left({\mathit{c}}_1\right)$	${\mathit{P}}_{\mathbf{bando}}$	$\mathit{e}\%$	${\mathit{P}}_{\mathbf{new}}\left(\mathit{M}\right)$	${\mathit{P}}_{\mathbf{bando}}$	$\mathit{e}\%$
0.015	2.4335	23.95	0.9850	2.4335	23.95	20	2.4335	23.95
0.030	1.9453	39.21	1.1820	2.1164	33.85	40	1.9814	38.08
0.045	1.6203	49.37	1.2805	1.9870	37.91	60	1.6710	47.78
0.060	1.3883	56.62	1.3790	1.8724	41.49	80	1.4447	54.85
0.075	1.2144	62.05	1.4775	1.7704	44.68	100	1.2724	60.24

.

$$e={\displaystyle \frac{P_{Bando}-P_{new}}{P_{Bando}}}\times 100$$

(22)

where $P_{new}$ denotes the peak values derived from Equation (22) using different parameters $c_1$, $c_2$, and M; $P_{Bando}$ represents the peak of the Bando model’s critical line, set at $P_{Bando}=3.2$ s⁻¹. Table 1 lists the stability evaluation indices, facilitating a comparative analysis.

As evidenced in Table 1, the modifications to the learning factor lead to a considerable expansion in the stable zone of the traffic flow system. Specifically, the system’s stability range extends drastically from $23.95\%$ to $62.05\%$, with an increase in the learning factor from $0.015$ to $0.075$. Furthermore, adjustments to the learning factor from $0.985$ to $1.4775$ lead to an expansion of the stability zone from $23.95\%$ to $44.68\%$, and alterations of the variable from 20 to 100 elevate the zone from $23.95\%$ to $60.24\%$. This significant improvement in the system’s stability underscores the effectiveness of the AV-PSO model in improving traffic flow stability.

4. Simulation

4.1. Simulation Setup

In our study, the parameters selected for simulations are as follows: length $L=400$ mm, maximum velocity $v_{max}=3.2$ m/s, comfortable headway $H_c=4$ m, and acceleration constant $\alpha =0.5$ m/s². We conduct all simulations under an open boundary condition, wherein the AV platoon travels along a predefined straight route without executing overtakes. At the onset, the initial vehicle operates at a pre-determined velocity, and a deviation of 1 m from the stable position is introduced to the first leading vehicle’s initial location to test the traffic flow system’s resilience and recovery capability. Three distinct simulation scenarios are executed to evaluate and affirm the traffic flow system’s characteristics influenced by learning factors $c_1$ and $c_2$, along with the visual field range M.

4.2. Simulation Control Flow Chart

The simulation control flow includes three steps. The first step is the Initialization as depicted as follows. The second step is to Create two-dimensional Gbest vector as depicted in alg1. The last step is to Assign the best particle(AV) $P_i$ to the Gbest as depicted in alg2:

(1)

Initialization

Randomly initialize AV swarm P:

Scale: N particles(AVs) in the swarm

Optimal Goal: 2^th dimensional objective function $V(•)$ for the $i^{th}$ AV.

Velocity: Range of velocity for $j^{th}$ dimension $[V_{j,min},V_{j,max}]$ of the $i^{th}$ AV.

Learning Factors: $c_1$, $c_2$.

(2)

Create two-dimensional Gbest vector:

Algorithm 1 Create two-dimensional Gbest vector the $i^{th}$ AV

1: for each $i^{th}$ particle(AV) $P_i$ in AV swarm P do 2:     Assign the current $i^{th}$ particle(AV) $P_i$ to $Pbes{t}_i$ 3:     Assign initial velocity $V_i$ for each particle(AV) $P_i$ 4:     Evaluate the particle(AV) $P_i$ using the objective function $of(•)$ 5: end for

(3)

Assign the best particle(AV) $P_i$ to the Gbest:

While the termination criterion is not met, use the following algorithm for each $i^{th}$ AV particle in the swarm P.

Algorithm 2 AV-PSO Motion Update

1: for each dimension j do 2:     for each AV(particle) i do 3:         Update the acceleration $a_{i,j}$ using the formula including both local and global information: 4:         $a_{i,j}\leftarrow c_1·a_{i,\mathrm{local}}+c_2·a_{i,\mathrm{global}}$ 5:         Update the AV(particle) velocity $v_{i,j}$ using the formula: 6:         $v_{i,j}\leftarrow v_{i,j}+a_{i,j}$ 7:         if $v_{i,j}<v_{j,\min }$ then 8:            $v_{i,j}\leftarrow v_{j,\min }$ 9:         end if 10:         if $v_{i,j}>v_{j,\max }$ then 11:            $v_{i,j}\leftarrow v_{j,\max }$ 12:         end if 13:         Update position of AV(particle) i 14:         Evaluate the particle $P_i$ using the objective function $V(•)$; 15:         $i\leftarrow i+1$ 16:     end for 17:     $j\leftarrow j+1$ 18: end for

4.3. Car-Following Behavior and Characteristics

Car-following behavior involves the adjustment of a following vehicle’s motion to align with that of the leading vehicle. The parameters defined for this behavior include $c_1=0.65$, $c_2=0.35$, and $M=5$. The vehicles within a platoon are sequentially labeled from the lead vehicle as #100 to the last vehicle as #1. To analyze the car-following dynamics, three distinct groups of vehicles (leading, intermediate, and tail) are examined through their velocity profiles as depicted in Figure 5, Figure 6 and Figure 7, respectively.

Figure 5a displays the velocity profiles of vehicles #100 (dashed red line), #99 (solid green line), and #98 (dotted blue line), emphasizing their initial responses and the ongoing adjustments in velocity. Vehicle #100 maintains a relatively steady pace post an initial adjustment period, while Vehicles #99 and #98 exhibit more significant oscillations before reaching stability, indicating their response latency and adjustment dynamics to the leader vehicle’s pace. Quantitatively, in the initial phase (0–100 s), Vehicle #100 accelerates to approximately $2.2\mathrm{m}/\mathrm{s}$ and then levels off. Vehicles #99 and #98 initially peak at higher velocities of around $2.5\mathrm{m}/\mathrm{s}$ and $2.4\mathrm{m}/\mathrm{s}$, respectively, before decelerating. In the oscillation phase (100–300 s), both Vehicles #99 and #98 display diminishing oscillations in velocity, suggesting a damping response as they adjust to the pace of Vehicle #100. Around 300 s, their velocities begin to stabilize at around $2\mathrm{m}/\mathrm{s}$. In the steady-state phase (300–500 s), all vehicles consistently maintain a velocity close to $2\mathrm{m}/\mathrm{s}$, exemplifying well-adjusted car-following behavior. Figure 5b provides a detailed examination of the extreme points in the velocity profiles for vehicles #99 and #98. The designated points ($p_{98}\left(n\right)$ and $p_{99}\left(n\right)$) correspond to the peak and trough values illustrated in the primary graph. Initially, both vehicles peak in velocity at around 200 s; car #99 reaches approximately $2.5\mathrm{m}/\mathrm{s}$ and car #98 approximately $2.4\mathrm{m}/\mathrm{s}$. Subsequent extrema ($p_{98}\left(2\right)$ and $p_{99}\left(2\right)$, et al.) display a diminishing trend in oscillation amplitude. By the fifth extreme point ($p_{98}\left(5\right)$ and $p_{99}\left(5\right)$), the velocities have stabilized near $2\mathrm{m}/\mathrm{s}$.

An analysis of these graphs suggests classic car-following dynamics, where the vehicles initially experience oscillations due to speed adjustments, followed by a phase of stabilization in which a uniform velocity of approximately $2\mathrm{m}/\mathrm{s}$ is achieved. The parameters $c_1=0.65$, $c_2=0.35$, and $M=5$ seem to support this stabilization, indicating effective car-following dynamics under the specified conditions.

4.3.1. Group II: Middle AV

The two graphs illustrate the car-following behavior of vehicles #50 and #51 over a prolonged period. Figure 6a depicts the velocity profiles of both cars, while Figure 6b highlights the extreme values (peaks and troughs) of these profiles over time.

In Figure 6, both cars initially exhibit minor oscillations, which escalate before eventually diminishing and stabilizing at around $2\mathrm{m}/\mathrm{s}$. The dotted magenta line represents car #50, while the solid blue line represents car #51. Quantitatively, the initial phase (0–500 s) sees both cars starting with a velocity near $2\mathrm{m}/\mathrm{s}$ accompanied by minor oscillations, indicative of the initial adjustment phase. During the mid-phase (500–3000 s), significant oscillations occur with increasing amplitude, then gradual reductions are observed. The peak velocities reached during this phase are approximately $2.6\mathrm{m}/\mathrm{s}$, and the troughs descend towards about $1.6\mathrm{m}/\mathrm{s}$. This responsive adjustment leads to periodic accelerations and decelerations as the vehicles align their speeds. Finally, in the stabilization phase (3000–4000 s), the oscillations dampen, and the velocities of both cars converge to around $2\mathrm{m}/\mathrm{s}$, signaling synchronized speed maintenance and the establishment of a steady driving state. Figure 6a also details the extremum points of the velocity profiles, illustrating an evolution through the phases. The initial peaks ($p_{50}\left(1\right)$ and $p_{51}\left(1\right)$) see both cars reaching about $2.4\mathrm{m}/\mathrm{s}$ at approximately 500 s. As the phase progresses, the amplitudes of the subsequent peaks ($p_{50}\left(2\right)$, $p_{51}\left(2\right)$, et al.) increase up to around 2000 s before commencing a downward trend. By the penultimate peak ($p_{50}\left(4\right)$ and $p_{51}\left(4\right)$), the velocities start stabilizing around $2.2\mathrm{m}/\mathrm{s}$. The culmination of this phase is marked by the final peaks ($p_{50}\left(5\right)$ and $p_{51}\left(5\right)$ around 3000 s), where the velocities converge to approximately $2\mathrm{m}/\mathrm{s}$, symbolizing the end of the oscillation phase and the onset of a stable traffic flow.

An analysis of the car-following behavior, as depicted in the graphs, illustrates a characteristic trajectory of initial fluctuations, followed by a progression towards stability. The parameters $c_1=0.65$ and $c_2=0.35$ induce pronounced initial oscillations, typical of the adaptive stages in which vehicles adjust their speeds in response to the dynamics of adjacent ones. As time progresses, these oscillations diminish, culminating in a stable car-following behavior where cars sustain a steady speed of approximately $2\mathrm{m}/\mathrm{s}$. This behavior demonstrates efficient vehicle dynamics adaptation, which facilitates stable traffic flow.

4.3.2. Group III: Tail AV

Figure 7a,b present the velocity profiles and peak values of car #2 (leading, shown with a solid magenta line) and car #1 (following, shown with a dotted blue line). Initially characterized by minor fluctuations, both vehicles later exhibit significant oscillations in velocity, which escalate in amplitude prior to subsiding and stabilizing around $2\mathrm{m}/\mathrm{s}$. From a quantitative perspective, during the initial phase (0–1000 s), both vehicles begin with velocities near 2 m/s, experiencing slight oscillations that mark the early adjustment period. In the oscillation phase (1000–6000 s), the velocity of each car displays considerable swings; amplitude initially swells and then systematically wanes. Peak velocities during this time rise to approximately $2.6\mathrm{m}/\mathrm{s}$, with troughs dropping to about $1.6\mathrm{m}/\mathrm{s}$, emphasizing the ongoing adaptation period. During the stabilization phase (6000–7000 s), the oscillations gradually diminish, and the velocities of the vehicles converge towards a stable state of approximately $2\mathrm{m}/\mathrm{s}$, reflective of effective speed synchronization. Figure 7b provides further insights into the dynamics of these extreme values. Initially, both vehicles reach peak velocities around $2.4\mathrm{m}/\mathrm{s}$ at roughly 1000 s. These peaks show an increasing trend up to 2000 s, after which they begin to decline. By the fourth peak, the velocity moderates to about $2.2\mathrm{m}/\mathrm{s}$, and by the final peaks at around 6000 s, the velocities converge to around $2\mathrm{m}/\mathrm{s}$, signaling a transition to a steady-state phase.

Collectively, the examination of these graphical trends confirms that the setting of parameters $c_1=0.65$, $c_2=0.35$, and $M=5$ are foundational in shaping the initial stage adjustments and stabilization, demonstrating robust car-following dynamics adapted for cohesive traffic flow.

4.4. Exploration of Influence of Learning Factors $c_1,c_2$ on Traffic System Stability

4.4.1. Typical Velocity Profiles

Figure 8 provides a detailed examination of the velocity dynamics of car #50 as influenced by different combinations of learning factors, $c_1$ and $c_2$. Each graph offers insights into the relationship between the adjustment of these factors and the resulting changes in velocity oscillation patterns, convergence times, and the occurrence of oscillations.

In Figure 8a, the learning factors are set as $c_1=0.95$ and $c_2=0.05$. This configuration results in oscillations with relatively high amplitudes, fluctuating between approximately $1\mathrm{m}/\mathrm{s}$ and $3\mathrm{m}/\mathrm{s}$. The velocity stabilizes around $2\mathrm{m}/\mathrm{s}$ after about 4000 s. Notably, prominent oscillations are observed during the initial 3000 s, which gradually diminish by 4000 s. In Figure 8b, the learning factors are set as $c_1=0.75$ and $c_2=0.25$. Here, the amplitude of the oscillations is moderate, peaking at around $2.5\mathrm{m}/\mathrm{s}$ and dropping to about $1\mathrm{m}/\mathrm{s}$. The convergence time sees the velocity stabilizing at approximately $2\mathrm{m}/\mathrm{s}$ after about 3000 s. This graph shows a slight reduction in the number of oscillations and a more rapid damping compared to the first scenario. In Figure 8c, the learning factors are set as $c_1=0.65$ and $c_2=0.35$. In this case, the oscillations manifest a lower amplitude, ranging up to around $2\mathrm{m}/\mathrm{s}$ and descending to approximately $1\mathrm{m}/\mathrm{s}$. The convergence toward stability occurs at about $1.5\mathrm{m}/\mathrm{s}$ after some 2500 s. Further reduction in the number of oscillations is observed, indicating a quicker stabilization of the system. In Figure 8d, the learning factors are set as $c_1=0.55$ and $c_2=0.45$. The oscillations show a lower amplitude, peaking around $2\mathrm{m}/\mathrm{s}$ and dropping to about $1\mathrm{m}/\mathrm{s}$. Velocity convergence happens around $1.5\mathrm{m}/\mathrm{s}$ after approximately 2000 s, with an even further diminished number of oscillations, leading to rapid stabilization of the system.

This systematic variation in dynamics under different learning parameters offers valuable insights into optimizing control settings for desired vehicular behavior. With the coefficient set $(c_1=0.55,c_2=0.45)$, the system exhibits the lowest amplitude oscillations ranging from 1 m/s to 2 m/s, with a convergence time of approximately 2000 s. This configuration demonstrates the fewest oscillations, indicative of rapid stabilization. Generally, as the value of $c_2$ increases, the following trends are observed:

(1)

Amplitude of Oscillations: There is a decrease in the amplitude of oscillations, which contributes to a more stable velocity profile characterized by diminished extreme fluctuations.

(2)

Convergence Time: The convergence time decreases, implying that the system attains a stable velocity more expeditiously.

(3)

Number of Oscillations: A reduction in the number of oscillations is noted, signaling a smoother transition to a stable state with fewer velocity fluctuations.

Overall, enhancing $c_2$ improves the stability and efficiency of the car-following behavior. Higher values result in lower amplitude oscillations, quicker convergence to a steady state, and reduced oscillations, thereby fostering better overall performance of the traffic flow system.

4.4.2. Time-Spatial Graph

A detailed analysis of the spatiotemporal diagrams reveals the following insights:

Figure 9a ($c_1=0.95,c_2=0.05$) exhibits dense and irregular patterns indicating significant fluctuations in vehicle spacing, large amplitude fluctuations, a longer convergence time around 1500 s, and frequent oscillations, indicating a less stable traffic flow. Figure 9b ($c_1=0.75,c_2=0.25$) shows more uniform patterns with reduced fluctuations, notably smaller amplitude fluctuations, a quicker convergence around 1200 s, and fewer oscillations, suggesting a more stable traffic flow. Figure 9c ($c_1=0.65,c_2=0.35$) showcases even more uniform patterns, further reduced fluctuations, smaller amplitude fluctuations compared to the previous diagrams, a faster convergence by around 1000 s, and even fewer oscillations, indicating a more stable and efficient traffic flow. Figure 9d ($c_1=0.55,c_2=0.45$) presents the most uniform patterns indicating minimal fluctuations, the smallest amplitude fluctuations among all diagrams, the quickest convergence at approximately 900 s, and the fewest oscillations, which suggest the most stable and efficient traffic flow. A comparative analysis of all four spatiotemporal diagrams elucidates the following benefits as $c_2$ increases: a decrease in the amplitude of fluctuations, fostering more consistent and stable distances between vehicles; a reduction in convergence time, demonstrating quicker stabilization of the traffic flow; and a decline in oscillation frequency, indicating fewer adjustments and smoother traffic operation.

Overall, elevating the parameter $c_2$ notably augments the stability and efficiency of traffic flow dynamics. An increase in $c_2$ correlates with more consistent spacing between vehicles, the accelerated attainment of a stable traffic state, and a reduction in the oscillatory variation of vehicle distances. These observations suggest that higher $c_2$ values foster more stable and efficient car-following behaviors in traffic systems.

4.5. Influence of Viewing Field M on Stability of AV Platoon System

The parameter M is widely recognized as defining the optimal velocity influence range universally applicable across a vehicle platoon. A series of simulations were executed to thoroughly assess the effects of M on system characteristics, both quantitatively and qualitatively, as depicted in Figure 10. For these simulations, the learning factors were set to $c_1=0.9$ and $c_2=0.1$. The viewing fields investigated were $M=10,40,60$, and 80, listed in Figure 10a, Figure 10b, Figure 10c, and Figure 10d, sequentially increasing to observe progressive impacts.

4.5.1. Trends in AV Acceleration

In the realm of traffic flow modeling, an exhaustive analysis was conducted to assess the performance of three distinct acceleration strategies—local optimal acceleration, global optimal acceleration, and weighted acceleration—across various viewing field horizons. The purpose was to delineate the dynamic behavior and stability imparted by each approach.

(1)

Local Optimal Acceleration (Red Dashed Line): This strategy manifests minimal oscillatory behavior across all the evaluated horizons ($M=10$, $M=40$, $M=60$, $M=80$), symbolizing a conservative tactic focusing on stability during vehicle‘s speed adjustments. The near-zero acceleration values signify a steady adherence to uniform velocity, underscoring a cautious operational mode irrespective of horizon length.

(2)

Global Optimal Acceleration (Blue Dotted Line): Exhibiting significant oscillations across all horizons, this strategy reflects an assertive optimization approach aimed at swift vehicular speed adjustments for traffic flow optimization. The oscillatory nature varies with horizon length; oscillations are more frequent in shorter horizons and less frequent but larger in longer horizons, reaching up to ±1.5 m/s². This dynamic nature seeks to rapidly enhance traffic flow but may occasionally compromise stability, particularly in scenarios with extended horizons.

(3)

Weighted Acceleration (Green Solid Line): This strategy presents a judicious balance between conservative and assertive approaches, demonstrating moderate oscillations conducive to both responsiveness and stability. Through integrating elements of both the local and global optimal accelerations, it aims to offer a pragmatic solution for dynamic traffic management. The consistent oscillatory behavior across various horizons, maintained within a range of ±1 m/s², underscores its adaptability and robustness in varying traffic conditions.

In essence, the Local Optimal Acceleration strategy is delineated by its reserved and consistent nature, which achieves stability, albeit possibly at the expense of quick adaptability. In contrast, the global optimal acceleration strategy employs a vigorous approach that focuses on swift optimization, but could potentially compromise stability due to its substantial and horizon-sensitive oscillations. Meanwhile, the weighted acceleration strategy strikes a balance, maintaining moderate oscillations and serving as a practical median between stability and responsiveness. Analyzing the performance across various horizons ($M=10$, $M=40$, $M=60$, $M=80$) indicates that while the local and weighted strategies exhibit consistent behavior, the oscillations associated with the global strategy decrease in frequency but increase in magnitude as the horizon extends, illustrating the dynamic interaction between optimization and stability in traffic management. The thorough assessment across various horizons ($M=10$, $M=40$, $M=60$, and $M=80$) delineates distinct characteristics for each strategy. Local optimal acceleration upholds the most stable performance with minimal oscillation, ideal for fostering consistent traffic flow. Global optimal acceleration, high in responsiveness, tends to destabilize as the oscillation amplitude grows with increasing horizon length, aligning with scenarios that demand rapid modulation but risk flow instability. Weighted acceleration offers a harmonious blend, with its moderate oscillations optimally balancing both stability and responsiveness attributes.

The influence of horizon length is notably pronounced in global optimal acceleration, where shorter spans necessitate frequent adjustments, while longer durations prompt fewer, yet critical, modifications. This detailed investigation accentuates the trade-offs between stability and responsiveness in car-following behaviors, underlining the importance of choosing suitable horizon lengths tailored to the desired equilibrium between these factors, contributing beneficial insights towards optimizing traffic management and bolstering transportation system efficiencies.

4.5.2. Time-Spatial Spectrum Graph

This section delves into the impact of different prediction horizons ($M=10$, $M=40$, $M=60$, and $M=80$) on AV motion state, as illustrated in Figure 11. We employ spectral density analysis as a quantitative tool to discern and evaluate the periodic patterns and interaction intensities in car-following dynamics. Such an analysis is instrumental in optimizing traffic flow and enhancing the stability and efficiency of transportation systems.

In Figure 11a ($M=10$), the spectral plot for a 10-view horizon delineates a distinct periodic pattern, characterized by alternating bands of high (red) and low (blue) spectral densities. These regions indicate, respectively, significant vehicle interactions and more stable car-following distances. The recurring nature of these patterns suggests regular oscillations in vehicle proximity. Spectral density values predominantly fluctuate between 180 and 220, peaking at around 1500 s. The observed periodicity of approximately 500 s signifies a consistent pattern of vehicle interaction, with stable amplitude oscillations throughout the analysis period. In Figure 11b ($M=40$), at a 40-view horizon, the spectral plot maintains a periodic pattern, albeit with more pronounced and frequent high-density (red) regions relative to the 10-view horizon, indicating increased vehicle interactions. This pattern exhibits slight irregularities, pointing to variable periodicities in vehicle spacing. The spectral density ranges are similar to the 10-view horizon, approximately between 180 and 210, yet with a peak at around 1000 s. The periodicity shortens to about 400 s, suggesting heightened vehicle interaction frequency. Furthermore, the amplitude of these oscillations illustrates more notable fluctuations in car-following distances over time. In Figure 11c ($M=60$), the plot for a 60-view horizon portrays a similar periodic pattern to the 10-second horizon, with alternating high-density (red) and low-density (blue) bands, indicative of fluctuations between intense and stable vehicle interactions. This regular pattern across the observation period suggests persistent oscillations in vehicle spacing. Spectral density values range from approximately 180 to 220, with the most significant density noted around 1500 s. The periodicity remains around 500 s, evidencing a consistent interval of vehicle interaction, with the amplitude of these oscillations maintaining relative stability over the period studied. In Figure 11d ($M=80$), at an 80-view horizon, the plot displays a periodic pattern akin to the 60-view horizon, but with more frequent and pronounced high-density (red) regions. This suggests increased and more intense vehicle interactions, with minor irregularities indicating variations in the periodicity of vehicle distances. While the spectral density values and periodicity trends mirror those observed in shorter horizons, the increased amplitude and irregularity in the pattern highlight more dynamic interactions in car-following behavior at this extended horizon.

This comprehensive spectral density analysis underlines the critical influence of prediction horizons on the dynamics of car-following behavior, offering valuable insights for traffic management and system optimization. The analysis of spectral density values, which range from approximately 180 to 210, reveals a peak density at approximately 1000 s. The periodicity of oscillations, noted at around 400 s, signifies a higher frequency of vehicle interactions compared to the 60-view horizon. Additionally, the amplitude of these oscillations is observed to be higher, suggesting more pronounced fluctuations in the distances between vehicles over time. The spectral density analysis, conducted across various horizons ($M=10$, $M=40$, $M=60$, $M=80$), offers valuable insights into the periodic patterns and the intensity of vehicle interactions in traffic flows. At a shorter horizon ($M=10$), the analysis displays a consistent periodic pattern with stable oscillations, indicative of regular vehicle interactions at intervals of approximately 500 s. As the horizon extends to 40, the frequency of vehicle interactions escalates, with a decrease in periodicity to about 400 s and an increase in the amplitude of oscillations, reflecting heightened fluctuations in vehicle distances. For intermediate horizons ($M=60$), the interactive pattern remains consistent with that observed at the 10-view horizon, albeit with increased intensity and frequency of interactions. The periodicity is maintained at approximately 500 s with stable oscillation amplitude, suggesting consistent car-following behavior. At the longest assessed horizon ($M=80$), the results indicate the most frequent and intense vehicle interactions characterized by prominent high-density regions, a 400 s periodicity, and greater amplitude in oscillations, highlighting notable fluctuations in vehicle distances over time.

This complete analysis underscores the influence of horizonal length on both the periodic patterns and the intensity of vehicle interactions in traffic flows. Shorter horizons foster stable and regular interactions, while longer horizons lead to increased frequency and intensity of interactions, with significant fluctuations in vehicle distances. These observations underscore the necessity of selecting an appropriate horizon length in car-following models to optimize traffic flow and enhance the efficiency of transportation systems.

5. Discussion & Conclusions

The rise of AVs has redefined traditional traffic systems, rendering fixed road lanes less relevant as AVs can dynamically organize into cooperative platoons driven by specialized algorithms. Inspired by the natural coordination observed in bird flocks within two-dimensional spaces, this study introduces a novel car-following model founded on the principles of PSO. The proposed model treats each AV as an independent agent operating within a two-dimensional plane, driven by local and global optimal velocity factors. Local optimal velocity is influenced by the nearest preceding vehicle, while global optimal velocity aggregates the influence of vehicles within a defined observational range. The stability of the AV-PSO model is assessed using linear system theory, with learning parameters ($c_1$, $c_2$) and the field of view (M) playing a pivotal role in ensuring traffic system stability. In addition, it is worth noting that this paper explores the operational control of AV platoon from a theoretical perspective, without fully considering passenger comfort, as well as “soft” instabilities (vehicles joining/leaving) and “hard” instabilities (technical failures/accidents). Future research will expand on these related aspects.

Extensive simulation studies have yielded several significant quantitative findings, particularly the following: (1) The AV-PSO model effectively captures car-following behaviors, aligning with empirical observations and the existing models. (2) The learning parameters $c_1$ and $c_2$ are critical for achieving and maintaining system stability. Notably, a quantitative comparison of the amplitude variations reveals significant differences between the two parameter configurations. With $c_1=0.95$ and $c_2=0.05$, the velocity oscillations reach a maximum amplitude of approximately 3.0 m/s and drop to a minimum of nearly 0 m/s, resulting in a total velocity range of 3.0 m/s over an extended oscillation period of approximately 3000 s (from 1000 s to 4000 s). In contrast, when the parameters are set to $c_1=0.55$ and $c_2=0.45$, the maximum amplitude decreases to about $2.5$ m/s, with a minimum of $0.9$ m/s, yielding a significantly reduced velocity range of $1.6$ m/s and a shorter oscillation duration of approximately 2000 s (from 1000 s to 3000 s). These measurements demonstrate that the balanced parameter configuration reduces amplitude variation by approximately $47\%$ (from $3.0$ m/s to $1.6$ m/s) and shortens convergence time by about 33%, clearly indicating superior damping properties and system stability with more balanced learning parameters. (3) The field of vision parameter (M) significantly impacts system stability, as evidenced by the acceleration profiles in the provided figures. When comparing $M=80$ with $M=10$ while maintaining identical learning parameters $(c_1=0.9,c_2=0.1)$, the smaller field of vision demonstrates markedly different oscillation patterns. With $M=10$, acceleration oscillations exhibit a more consistent frequency and reach peak amplitudes of approximately ±1 m/s², while the larger field of vision ($M=80$) produces more irregular oscillations with slightly lower peak amplitudes around ±0.9 m/s². Notably, the system with $M=10$ shows faster convergence, stabilizing completely by 4200 s, whereas the $M=80$ system still displays residual oscillations at the same time point. This counter-intuitive result suggests that an intermediate field of vision, rather than maximizing observational range, optimizes platoon stability, with quantitative analysis showing an approximately $25\%$ improvement in convergence time when using a more focused field of vision tailored to immediate traffic conditions.

Compared to traditional car-following models, the proposed AV-PSO model offers significant advantages, including its ability to dynamically adapt to complex traffic environments and optimize vehicle behavior both locally and globally. However, it has some limitations. For instance, its reliance on predefined learning parameters may reduce adaptability in highly heterogeneous traffic scenarios, and additional computational overhead may be required for large-scale implementations. Future studies should explore these issues in greater depth and broaden the model’s scope by investigating its performance under varying conditions, such as different traffic densities, vehicle types, and speed limits. Furthermore, potential improvements could include integrating real-world data to refine parameter settings, applying dynamic learning mechanisms to adapt in real time, and incorporating environmental factors such as weather conditions and road gradients.