Satisfaction-Based Optimal Lane Change Modelling of Mixed Traffic Flow and Intersection Vehicle Guidance Control Method in an Intelligent and Connected Environment

Abstract

The information interaction characteristics of connected vehicles are distinct from those of non-connected vehicles, thereby exerting an influence on the conventional traffic flow model. The original lane-changing model for non-connected vehicles is no longer applicable in the context of the new traffic flow environment. The modelling of the new hybrid traffic flow, comprising both connected and ordinary vehicles, is set to be a pivotal research topic in the coming years. The objective of this paper is to present a methodology for optimal mixed traffic flow dynamic modelling and cooperative control in intelligent and connected environments (ICE). The study utilizes the real-time perception and information interaction of connected vehicles for traffic information, taking into account the access characteristics of both connected and non-connected vehicles. The satisfaction-based free lane-changing and mandatory lane-changing models of connected vehicles are designed. Secondly, a mixed traffic flow lane-changing model based on influence characteristics is constructed for the influence area of connected vehicles. This model takes into account the degree of influence that connected vehicles have on non-connected vehicles, with different distances being considered respectively. Subsequently, a vehicle guidance strategy for mixed traffic flows comprising grid-connected and conventional vehicles is proposed. A variety of speed guidance scenarios are considered, with an in-depth analysis of the speed optimization of connected vehicles and the movement law of non-connected vehicles. This comprehensive analysis forms the foundation for the development of a vehicle guidance strategy for mixed traffic flows, with the overarching objective being to minimize the average delay of vehicles. In order to evaluate the effectiveness of the proposed method, the intersection of Gaota Road and Fangshui North Street in Yanqing District, Beijing, has been selected for analysis. The results of the study demonstrate that by modifying the density of the mixed traffic flow, the overall average speed of the mixed traffic flow declines as the density of vehicles increases. The findings reported in this study reflect the role of connected vehicles in enhancing road capacity, maximizing intersection capacity and mitigating the occurrence of queuing phenomena, and improving travel speed through the mixed traffic flow lane-changing model based on impact characteristics. This study also provides some guidance for future control of the mixed traffic flow formed by emergency vehicles and social vehicles and for realizing a smart city.

1. Introduction

The advent of intelligent, connected traffic has prompted a gradual shift in focus towards vehicle networking technology as a potential solution to traffic problems. The real-time acquisition of information about surrounding vehicles and the road traffic environment by connected vehicles in a connected vehicle environment enables the realization of information interaction between vehicles and the surrounding environment. The quantity and scope of information obtained in a connected environment give rise to corresponding alterations in the operational characteristics of traffic flow. To illustrate, a vehicle may utilize the network to ascertain the preceding vehicle’s operational status, thereby anticipating the need for control over its own speed, deceleration, or acceleration. Similarly, a vehicle may utilize the network to ascertain the speed and acceleration spacing of the vehicle ahead, both in the current lane and the intended lane. During lane changes, vehicles obtain information from the surrounding environment regarding the speed and acceleration spacing of the leading vehicle in the current and target lanes. This information impacts the driver’s decision-making process concerning lane changes. The original lane-changing model, which was developed for use in non-connected traffic flow environments, is not applicable to new traffic flow environments. It is therefore evident that further study of the lane-changing model for new mixed traffic flow environments is required in order to support the development of traffic management and control schemes and to alleviate traffic congestion.

The predominant decision-making behavior exhibited by drivers of moving vehicles is that of lane-changing. This behavior is the consequence of a multifaceted interplay between external environmental factors and the driver’s own characteristics, which must be considered in order to predict the potential outcomes of the decision-making process. The vehicle’s lane-changing behavior has the capacity to influence the actual traffic environment. Inappropriate lane-changing behavior has the potential to result in a range of adverse consequences, including traffic congestion, vehicle collisions, and other problems. Yang et al. constructed a forced lane-changing model for vehicles, and an asynchronous updating method was employed to verify the model’s validity [1]. In a further contribution to the field, Li et al. conducted an analysis of the safe time for vehicle lane changing under different traffic states, resulting in the construction of a safe distance model for vehicle lane changing [2]. Wang et al. considered the safe spacing between vehicles and the comfort of passengers during driving, and constructed a lane-changing decision model with the objective of reducing the fluctuation of the traffic flow caused by vehicle changing behavior [3]. Sueyoshi et al. constructed a metacellular automata model based on the modified Nishinari–Fukui–Schadschneider (S-NFS) model to investigate the optimization of traffic speed by vehicles in a mixed traffic flow system [4]. With the progression of technology, scholars have increasingly directed their attention towards lane-changing models in a networked environment, as opposed to solely focusing on traditional models. Wang et al. developed three interaction systems to characterize different intelligent connected vehicles’ environments: the baseline, warning group, and guidance group [5]. Geng et al. proposed an optimal lane-changing and control strategy to solve the collision avoidance of intelligent vehicles in highway driving [6]. An et al. proposed a novel approach to implement both adaptive cruise control and model predictive control for platoon control [7]. Liu et al. presented a cooperative mandatory platoon lane-change strategy for connected and autonomous vehicles [8]. Ali et al. proposed a method that combines the wavelet transform method to pinpoint the correct lane-changing decision point [9]. Liu et al. studied the coordinated lane-changing scheduling problem with the goal of transferring the platoon from an initial state to a target state to minimize a certain cost measurement [10]. Zhang et al. developed the model predictive control (MPC)-based mixed integer nonlinear programming optimizer (MINLP-MPC) model to search optimal lane-change decisions with time window concern [11]. He et al. proposed a novel observation adversarial reinforcement learning approach for the robust lane change decision-making of autonomous vehicles [12]. Du et al. proposed a way to predict lane-change behaviors for intelligent connected vehicles through the driving style-based lane-change environment and driving trajectory-related parameters [13]. Liu et al. proposed a lane selection model based on phase-field coupling and set pair logic [14]. Zheng et al. proposed a cooperative lane changing strategy to improve traffic operation and safety at a diverging area nearby a highway off-ramp [15]. Wang et al. developed a quantitative relationship model based on the artificial potential field (APF) theory with the objective of investigating the dynamic patterns of lane-changing vehicle risks [16]. Peng et al. derived the lane changing optimization method for connected automated vehicles under different penetration rates [17]. Cong et al. presented a modified multi-lane lattice hydrodynamic model considering the passing effect [18]. Cong et al. presented a modified car-following model considering perceived headway errors on gyroidal roads [19]. Cong et al. propose a novel heterogeneous multi-phase traffic flow accounting for human-driven vehicles (HDVs) and connected vehicles (CVs) to close this hole [20]. Tumminello et al. explored the safety and operational performances of traffic-calming measures featuring cooperative driving technologies in urban areas. An urban-scale road network close to a seaside area in the city of Mazara del Vallo, Italy, was properly redesigned and simulated in AIMSUN to assess several design solutions, where connected and automated vehicles (CAVs) have been employed as a more energy-efficient public transportation system [21].

A review of the extant literature reveals that numerous scholars both domestically and internationally have attained noteworthy outcomes in their research endeavors concerning vehicle lane-changing models. These advancements, characterized by sustained and profound research, have contributed to the progressive enhancement and evolution of traffic flow models. However, as connected technology continues to evolve, the traditional lane-changing model is facing challenges in its application to the new mixed traffic flow environment. Further research is necessary to study the intelligent and connected environment of the lane-changing model, providing fundamental theoretical support for managing and controlling traffic issues in this new environment. Nevertheless, there are still areas for improvement in the study of the lane-changing model. In the social vehicle lane-changing model, the existing lane-changing model is predominantly informed by real-time local traffic conditions for decision-making purposes, which can result in low efficiency and frequent lane-changing. In constructing the social vehicle lane-changing model in the intelligent and connected environment, further research is necessary to comprehensively consider the relevant lanes of the vehicle and the historical operating status of other vehicles, as well as changes in the relevant lanes of the vehicles’ operating status in front of the vehicle. In the emergency vehicle lane-changing model, social vehicles must yield to emergency vehicles in order to ensure the priority of emergency vehicles. However, there is a paucity of research based on the intelligent and connected environment in the study of the emergency vehicle lane-changing model. This is because the lane-changing behavior of emergency vehicles and social vehicles under the influence of emergency vehicles will be altered in an intelligent and connected environment. Therefore, it should be combined with information acquisition.

The establishment of an advanced intersection speed guidance system necessitates the utilization of a multitude of wireless communication devices, facilitating the interaction between vehicle and vehicle information, as well as vehicle and road information. Consequently, the high-performance computer acquires the vehicle’s speed, location, and other pertinent data for analysis, the dissemination of traffic information, and the objective of optimizing vehicle speed at intersections. The implementation of such a system is expected to yield several benefits, including enhanced road service quality, reduced congestion, and a decline in traffic accident frequency. Choi et al. present a framework for connected and automated vehicles, including a cloud-based architecture and a deep reinforcement learning system for a sectionalized speed guidance system [22]. Khayyat et al. presented an optimization-free implementation of the overall control architecture based on an unconventional speed profile [23]. Wang et al. developed a driving guidance strategy for an isolated signalized intersection by considering the pre-stop line from the perspective of cooperation driving, speed guidance, and the optimization control method [24]. Wu et al. proposed a two-stage multi-lane unsignalized intersection cooperative control strategy based on mixed platoons in the mixed traffic environment of connected automated vehicles and connected human-driven vehicles [25]. Zhou et al. proposed a cooperative control strategy with three components for platoons consisting of mixed traffic flows [26]. Fu et al. proposed the combination of the Internet of Vehicles and blockchain technologies to form an efficient and accurate autonomous guidance strategy [27]. Ren et al. proposed a boundary guidance strategy for traffic congestion regions, leveraging the real-time traffic information obtained through the Internet of Vehicles [28]. Chen et al. developed a set of cost-minimization paths based on the operation information of connected electric vehicles including both energy consumption and travel time [29]. Wu et al. developed a traffic prediction framework together with a path planning method for connected vehicular networks [30]. Yao et al. proposed a cell transmission model (CTM)-based traffic signal timing model of mixed traffic flow [31]. Yao et al. proposed a decentralized control model for connected automated vehicle trajectory optimization at an isolated signalized intersection with a single-lane road [32]. Huang et al. proposed the guidance strategies based on principles of safety, efficiency, and comfort to help the subject vehicles cross intersections [33]. Bifulco et al. tackled and solved the problem of decentralized crossing at unsignalized intersections for mixed traffic flows [34]. Wu et al. proposed a hierarchical cooperative eco-driving control for a connected autonomous vehicle platoon to enhance traffic mobility efficiency [35]. Coppola et al. proposed an enhanced testing approach of The Green Light Optimal Speed Advisory (GLOSA) services to avoid unnecessary stops at an intersection [36]. Liu et al. presented a joint control approach that simultaneously optimizes traffic signals and the trajectories of cooperative vehicle platooning at urban intersections [37]. Zhao et al. presented an integrated microscopic traffic flow model to describe the relationship with vehicle passing paths and interactions’ control [38]. Zhao et al. established a behavioral simulation model to represent the overall path dispersion at the intersections [39].

A review of the literature reveals a substantial corpus of research conducted by scholars in both domestic and international contexts on the topic of traffic flow and vehicle guidance strategies. However, the majority of these studies are focused on the examination of traffic flow models for connected vehicles in typical traffic conditions or in the context of a fully connected-vehicle environment, with only a limited number of studies examining the mixed traffic flow model, which comprises both connected and ordinary vehicles. The integration of connected vehicles into the existing traffic infrastructure will inevitably lead to the emergence of a mixed traffic pattern comprising vehicles with heterogeneous attributes, thereby rendering the prevailing traffic state an anachronism. Consequently, the characteristics of traffic flow will deviate from those observed in previous studies. The establishment of an efficient hybrid traffic flow model is of great significance for the study of hybrid driving states in incomplete connected vehicle environments. Secondly, traditional guidance methods are predominantly based on historical data, with dynamic traffic parameters obtained in a connected environment not being fully utilized. This has resulted in a failure to adequately reflect the real-time nature of guidance, representing a significant inefficiency in traffic management, as well as a waste of data.

In summary, the present paper addresses the novel challenge of mixed traffic flow modelling, encompassing both connected and conventional vehicles. The paper makes two significant contributions to the field. Firstly, it presents a satisfaction-based model of free and mandatory lane-changing for a mixed traffic flow. For the area affected by connected vehicles, a social vehicle lane-changing model based on influence characteristics is constructed, taking into account the extent of the impact of connected vehicles on social vehicles at different distances. Secondly, an active guidance method for mixed traffic flow at intersections within an intelligent and connected environment is proposed. This method addresses the challenge of coexisting with two distinct types of vehicles, namely, connected and social vehicles, with the aim of enhancing the adaptability of mixed traffic flow. The objective is to facilitate more efficient and flexible navigation through intersections, thereby enhancing the overall efficiency of the traffic system.

The remainder of this paper is organized as follows. Section 2 provides a model for mixed traffic flow lane changing in intelligent and connected environments. Section 3 presents an active guidance method for mixed traffic flow at intersections under the intelligent connected environment. The performance of the proposed model is evaluated and verified through actual cases in Section 4. Conclusions are given in Section 5.

2. Modelling of Mixed Traffic Flow Lane-Changing Based on Global Priority Capacity

In the intelligent and connected environment, two key considerations emerge. Firstly, the operational characteristics of connected vehicles must be taken into account. Secondly, due to the influence of non-connected vehicles, the lane-changing behaviors of social vehicles will change. Consequently, an additional investigation into a novel lane-changing model for mixed traffic flow is imperative to attain a comprehensive representation of traffic operations in a road vehicle priority environment. This will facilitate the development of effective traffic management and control strategies. The research framework is shown in Figure 1.

The proposed model is a satisfaction-based approach to normal and mandatory lane-changing for mixed traffic flows, and it is a social vehicle lane-changing model based on influence characteristics for the areas affected by connected vehicles in intelligent and connected environments (ICE). This takes into account the degree of influence of connected vehicles on social vehicles at different distances.The paper also conducts a scenario analysis for the scenario in which connected vehicles fail to satisfy the stipulated speed guidance, with the objective of enhancing the overall intersection access efficiency. This objective is achieved by integrating the speed guidance and lane-change guidance, thereby establishing an intersection guidance model for mixed traffic flow. The optimization objective is defined as minimizing traffic delay.

2.1. Scenario Description

The model scenario under consideration is a unidirectional two-lane road, and the vehicles in the scenario are classified into the following five categories: emergency vehicles, social vehicles travelling in front of the same lanes, social vehicles travelling behind the same lanes, social vehicles travelling in front of adjacent lanes, and social vehicles travelling behind adjacent lanes. The considered model scenario is shown in Figure 2.

The presence of emergency vehicles on the road invariably leads to disruption to traffic flow. When emergency vehicles enter a driving road that is located in region D (which, for the purposes of this discussion, is assumed to be the length of the road section of the vehicle network impact area), social vehicles in the adjacent lane will generally attempt to avoid the area of influence of the emergency vehicles. The corresponding state of change for social vehicles in the adjacent lane will be acceleration or deceleration in order to maintain their position. The distance between social vehicles and emergency vehicles, coupled with the psychological shifts in the driving process of the driver, results in the classification of the interference into three distinct areas: D₁, D₂, and D₃. These areas are delineated by the varying degrees of anxiety, ease, and urgency experienced by the driver in response to the presence of emergency vehicles.

The D₁ area is in closer proximity to emergency vehicles in the region. The D₁ area of social vehicles that receive emergency vehicles exerts a significant influence on the region of social vehicles, prompting a change in lane positioning. This phenomenon is characterized by the observation that the original travelling lane in front of the car does not necessarily impede acceleration when changing lanes, and that the subsequent speed may not necessarily be improved.

The D₂ area is located at a greater distance from the emergency vehicles in comparison to the D₁ area. The D₂ area, designated for social vehicles, is less affected by the presence of emergency vehicles than the D₁ area. The strategic alteration of lanes in this area is intended to improve the speed of the social vehicles themselves.

The D₃ area is characterized by its considerable distance from the emergency vehicles in the region. In the D₃ area, drivers of social vehicles demonstrate a clear preference for extricating themselves from the vicinity of emergency vehicles. The area of social vehicles changing lanes is characterized by the following: there is space for acceleration in the current driving lane, but if the neighboring lanes offer superior conditions, drivers are likely to opt to change lanes.

2.2. A Study of Lane-Changing Modelling

In the event of an emergency, it is imperative that social vehicles on the road refrain from impeding the progress of emergency vehicles, which are not subject to the same speed restrictions. In the event that a vehicle is situated within the area D, which is designated for intelligent connected environments, it is imperative that the driver does not change lanes in order to avoid interfering with the emergency vehicle’s priority. In accordance with the aforementioned characteristics, it is imperative that vehicles in this lane change lanes. Upon the emergency vehicle’s departure from the area of influence, the social vehicle resumes travel in accordance with the established lane-changing protocols for connected vehicles.

2.2.1. Construction of Mixed Traffic Flow Lane-Changing Model Based on Driver Satisfaction

In contradistinction to non-connected vehicles, connected vehicles are able to interact with surrounding vehicles through the intelligent and connected environments to obtain the movement status of other connected vehicles in a larger range ahead, including the speed and position of the vehicles, etc. According to the content of the zone of concern interval of the connected vehicles, a certain length of the zone ahead of the target vehicle of concern is travelling, and within the defined range, the connected vehicles, through the on-board sensing systems, obtain the concern interval of information of all the connected vehicles within the interval.

Within the zone of concern interval, the target vehicle calculates the maximum speed of the connected vehicles within its vicinity based on the collected information and adopts it as the desired speed. This dynamic process is repeated at regular intervals, with the desired speed being recalculated based on the changing conditions. The speed of the other connected vehicles in the concern range is subject to change in real time as well. The cumulative model for the satisfaction of the connected vehicles travelling in the lane is constructed based on the above analysis, as shown in the following equation.

$$\left\{\begin{array}{c}{W}_a(T)=\kappa (v_a(T)){\displaystyle \frac{v_a(T)-v_{des}(T)}{v_{des}(T)}}+\xi (v_a(T)){\displaystyle \sum _{t=0}^{T-1}\frac{v_a(T)-v_{des}(T)}{v_{des}(T)}}\\ \kappa (v_a(T))+\xi (v_a(T))=1\\ v_{des}(T)=\max (v_i(T))\end{array}\right.$$

(1)

where v_i(t) signifies the velocity of the ith vehicle among all connected vehicles that are ahead of the target connected vehicle within its concern interval range at the specified moment t. In an intelligent and connected environment, vehicle operators are capable of ascertaining the operational speeds of all connected vehicles that are preceding them. Drivers of connected vehicles have expectations regarding the operating speeds of other vehicles in their vicinity. In other words, the expected speed of a driver at any given time is the maximum possible speed of all connected vehicles within their observation range at that particular time.

It is reasonable to hypothesize that emergency vehicles will tend to travel at faster speeds than ordinary social vehicles. Indeed, emergency vehicles may be classified as vehicles with a tendency to change lanes. However, it is important to note that emergency vehicles are different from ordinary social vehicles due to their priority characteristics. As a result, it may not be appropriate to consider the satisfaction accumulation process in the context of emergency vehicles. This is due to the fact that, in order to respond to an emergency, emergency vehicles may need to change lanes during their journey. In such cases, two main situations can be distinguished: normal lane-changing and mandatory lane-changing. The equation of neighboring lane satisfaction of emergency vehicles is as follows.

$$\left\{\begin{array}{c}{W}_b(t)=\delta {\displaystyle \frac{v_b(t)-v_a(t)}{v_a(t)}}+\epsilon {\displaystyle \frac{\overline{V_b}(t)-\overline{V_a}(t)}{\overline{V_a}(t)}}\\ \delta +\epsilon =1\\ \overline{V_a}(t)={\displaystyle \sum _{i=1}^n{v}_{ai}}/n\\ \overline{V_b}(t)={\displaystyle \sum _{i=1}^n{v}_{bi}}/m\end{array}\right.$$

(2)

where $W_b(t)$ and $v_a(t)$ are used to represent the satisfaction of the adjacent lane of the emergency vehicle and the operating speed of the emergency vehicle at the moment of t, respectively. Similarly, $v_b(t)$ is used to denote the speed of the previous vehicle in the adjacent lane at the moment of t, while $v_{ai}$ denotes the speed of the ith vehicle in the current lane. Finally, $v_{bi}$ is used to denote the speed of the ith vehicle in the adjacent lane. The variable $\overline{V_a}(t)$ represents the average speed of the current lane at the moment t. The variable n denotes the number of grid-connected vehicles within the attention range of the current lane of the target vehicle. The variable $\overline{V_b}(t)$ signifies the average speed of the neighboring lane at the moment t. The variable m represents the number of grid-connected vehicles within the attention range of the neighboring lane of the target vehicle. In this paper, the weight coefficients are defined as the variable $\delta$ and $\epsilon$.

In view of the satisfaction calculation previously referenced in relation to the adjacent lanes of emergency vehicles, it can be posited that the lane-changing situation for emergency vehicles can be divided into two distinct cases.

It is only when the adjacent lanes of emergency vehicles satisfy the requisite safety conditions for emergency vehicles changing lanes directly that this is possible.

$$\left\{\begin{array}{ll}{W}_b(t)>0& \hfill \mathrm{(3)}\\ ga{p}_{other}^i(t)>ga{p}_n^i(t)>(1+v_{\max }-\min \left\{v_e(t)+1,v_{e\max }\right\})& \hfill \mathrm{(4)}\\ ga{p}_{back}^i(t)>(1+v_{\max }^i-\min \left\{v_e(t+1)+1,v_{e\max }\right\})& \hfill \mathrm{(5)}\end{array}\right.$$

where $ga{p}_n^i(t)$ is used to denote the distance between the target vehicle and the vehicle in front of the current lane at a given point at time t. Similarly, $ga{p}_{other}^i(t)$ denotes the distance between the target vehicle and the vehicle in front of the neighboring lane at time t. Finally, $ga{p}_{back}^i(t)$ denotes the distance between the target vehicle and the vehicle in the rear of the neighboring lane at time t. v_n is the operating speed of the social vehicle. v_max is the maximum speed of the social vehicle. v_e is the operating speed of the emergency vehicle. v_emax is the maximum operating speed of the emergency vehicle. min{v_e(t) + 1, v_emax} is the minimum value found from the value of v_e over time t plus 1 to the maximum operating speed of the emergency vehicle. min{v_e(t + 1) + 1, v_emax} is the minimum value found from the value of v_e over time t + 1 plus 1 to the maximum operating speed of the emergency vehicle. Equation (3) provides insight into the intention of emergency vehicles to change lanes. In situations where the spacing of neighboring lanes is greater than that of the current lane in front of the vehicle, Equation (4), emergency vehicles are more likely to meet the safety conditions required for a lane change. Equation (5) indicates that the vehicle must meet the same neighboring lanes after the vehicle has changed lanes in order to satisfy the requisite safety conditions.

In the event that the emergency vehicle in question is unable to satisfy the requisite safety conditions for a lane change, it is possible to decelerate to a sufficient degree to enable the emergency vehicle to change lanes in a manner that complies with the requisite safety conditions.

$$\left\{\begin{array}{ll}{W}_b(t)>0& \mathrm{(6)}\\ ga{p}_{back}^i(t)\le (1+v_{nob\max }^i(t)-\min \left\{v_e(t+1),v_{e\max }\right\})& \mathrm{(7)}\\ v_{nob\max }^i(t+1)=v_{nob}^i(t)-1& \mathrm{(8)}\\ rand()<p_1& \mathrm{(9)}\end{array}\right.$$

where $v_{nob}^i(t)$ denotes the speed of the vehicle in front of the adjacent lane at time t. Equation (6) indicates that the emergency vehicle generates the intention to change lanes. min{v_e(t + 1), v_emax} is the minimum value found from the value of v_e over time t + 1 to the maximum operating speed of the emergency vehicle. Equation (7) indicates that the emergency vehicle does not satisfy the safety conditions required for a lane change in the vehicle behind it in the neighboring lane. Equation (8) illustrates that the vehicle situated in the adjacent lane behind the emergency vehicle decelerates in order to facilitate the emergency vehicle’s maneuver. Equation (9) indicates the probability of the vehicle behind the emergency vehicle in the adjacent lane decelerating to yield to the emergency vehicle, represented by the value of the constant p₁.

2.2.2. Construction of the Social Vehicle Lane-Change Type Based on the Influence Characteristics of Emergency Vehicles

In the D₁ area, the distance between social vehicles and emergency vehicles is minimal, and lane changes in this area primarily serve to ensure the priority of emergency vehicles. Consequently, the social vehicle lane-changing behaviour in front of the emergency vehicle’s own lane in this region can be categorized into two distinct cases. The first case pertains to free lane-changing, provided that the requisite conditions for safe lane-changing are satisfied. This enables the potential for speed enhancement to be achieved. The second scenario involves satisfying the conditions for a safe lane change, which necessitates decelerating to a speed that is lower than the original lane speed.

The free lane change behavior of vehicles in the D₁ area is calculated as follows.

$$ga{p}_{other}^i(t)>ga{p}_n^i(t)>(1+v_{n\max }-\min \left\{v(t)+1,v_{\max }\right\})$$

(10)

$$ga{p}_{back}^i(t)>(1+v_{n\max }-\min \left\{v_n(t)+1,v_{n\max }\right\})$$

(11)

where min{v(t), v_max} is the minimum value found from the value of v over time t to the maximum speed of the social vehicle. min{v_n(t) + 1, v_nmax} is the minimum value found from the value of v_n over time t plus 1 to the maximum operating speed of the social vehicle. Equation (10) demonstrates that at time t, the distance between the target vehicle and the vehicle in front of the adjacent lane is greater than the distance between the target vehicle and the vehicle in front of the current lane. Equation (11) indicates that the target vehicle complies with the requisite safety criteria for lane changing, with vehicles positioned behind the adjacent lane.

In order to ensure the safety of emergency vehicles, it is imperative that vehicles approaching these vehicles reduce their speed and decelerate when changing lanes. This protocol is outlined in the D₁ area vehicle deceleration lane-change behavior protocol.

$$ga{p}_e^i(t)<v_{e\max }(t)+1$$

(12)

$$ga{p}_{other}^i(t)=(v_n(t)-1)$$

(13)

$$ga{p}_{back}^i(t)=(1+v_{n\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(14)

where $ga{p}_e^i(t)$ denotes the distance between the target vehicle and the emergency vehicle at a given point in time t. min{v_n(t) + 1, v_max} is the minimum value found from the value of v_n over time t plus 1 to the maximum speed of the social vehicle. The presence of social vehicles in front of the emergency vehicle, as indicated by (12), impedes the rapid movement of the emergency vehicle. Equation (13) indicates that the vehicle decelerates in order to change lanes to the adjacent lane. Equation (14) indicates that the vehicle situated in the adjacent lane maintains the requisite safety distance for a lane change.

The presence of social vehicles in the D₂ area has been shown to result in a reduction in the disturbance caused by emergency vehicles, due to the greater distance between them. The social vehicles situated in this area remain within the D₃ influence radius of the emergency vehicles in front of them, and are unable to escape the influence of the emergency vehicles for a brief period. Consequently, the lane-changing behaviors of vehicles located in this area are primarily driven by the objective of enhancing their own operational speed. In instances where the average speed of vehicles in the target lane exceeds a certain threshold, the vehicle will proceed directly to change lanes. Conversely, if the average speed is lower than that of the current lane, the vehicle will maintain its current position and continue driving.

$$ga{p}_{other}^i(t)>ga{p}_n^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(15)

$$ga{p}_{back}^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(16)

$$\overline{v_b}>\overline{v_a}$$

(17)

$$v_n(t+1)=v_n(t)+1$$

(18)

where $\overline{v_b}$ represents the average speed of the target lane, while $\overline{v_a}$ represents the average speed of the current lane. min{v_n(t) + 1, v_max} is the minimum value found from the value of v_n over time t plus 1 to the maximum speed of the social vehicle. Equation (15) indicates that the distance between the target vehicle and the vehicle in front of the neighboring lane is greater than the distance between the vehicle in front of the current lane. Consequently, the vehicle meets the requisite safety conditions for changing lanes with the vehicle in front of the neighboring lane. Equation (16) indicates that the vehicle meets the requisite safety conditions for changing lanes with the vehicle in the neighboring lane. Equation (17) indicates that the average speed of the adjacent lane is greater than that of the current lane. Equation (18) indicates that the vehicle’s speed increases following the completion of the lane-changing maneuver.

The social vehicles travelling in the D₃ area are least affected by emergency vehicles due to their considerable distance from said vehicles. In instances where a vehicle operator must urgently avoid the influence of an emergency vehicle, the current driving lane must satisfy certain criteria. Firstly, the lane must be able to accelerate to a certain speed. Secondly, the adjacent lanes must satisfy the safety conditions for lane changing. Finally, the adjacent lanes must have an average speed of vehicular operation that is higher than that of the current lane. In this area, the probability of lane changing is increased. The region of social vehicles engaged in lane-changing behavior can be divided into two categories.

Social vehicles have the capacity to accelerate within the confines of the current lane.

$$ga{p}_n^i(t)\ge \min \left\{v_n(t)+1,v_{\max }\right\}$$

(19)

$$ga{p}_{other}^i(t)>ga{p}_n^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(20)

$$ga{p}_{back}^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(21)

$$\overline{v_b}>\overline{v_a}$$

(22)

$$rand()<p_2$$

(23)

where min{v_n(t) + 1, v_max} is the minimum value found from the value of v_n over time t plus 1 to the maximum speed of the social vehicle. Equation (19) indicates that the vehicle has the capacity to accelerate within the confines of this lane. Equation (20) indicates that the neighboring lane offers superior driving space, thereby satisfying the requisite safety conditions for a lane change for the vehicle ahead, which is situated in the same neighboring lane. Equation (21) indicates that the target vehicle meets the requisite safety conditions for a lane change for the vehicle in the neighboring lane that is positioned behind it. Equation (22) indicates that the mean velocity of vehicle operation in the target lane is greater than that of the current lane. Equation (23) indicates that the probability of a lane change is P₂.

In the event that there is no available space for social vehicles to accelerate in the current lane, the following situation is provided.

$$ga{p}_n^i(t)<v_{\max }+1$$

(24)

$$ga{p}_{other}^i(t)>ga{p}_n^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(25)

$$ga{p}_{back}^i(t)>(1+v_{\max }-\min \left\{v_n(t)+1,v_{\max }\right\})$$

(26)

$$\overline{v_b}>\overline{v_a}$$

(27)

$$rand()<p_3$$

(28)

where min{v_n(t) + 1, v_max} is the minimum value found from the value of v_n over time t plus 1 to the maximum speed of the social vehicle. Equation (24) demonstrates that there is no possibility of acceleration within the confines of the current lane. Equation (25) indicates that the distance between the target vehicle and the vehicle in front of the adjacent lane is greater than the distance between the target vehicle and the vehicle in front of the current lane. This satisfies the safety condition for a lane change for the vehicle in front of the same adjacent lane. Equation (26) indicates that the target vehicle meets the requisite safety conditions for a lane change with respect to the vehicle situated behind the neighboring lane. Equation (27) indicates that the mean velocity of vehicle operation in the target lane is greater than that of the current lane. Equation (28) indicates that the probability of a vehicle changing lanes is p₃. In scenarios where lane changes can be executed in a manner that enhances travel speed, the driver will assign priority to lane changes, thereby indicating p₂ > p₃. The probability of emergency vehicles changing lanes is the highest, therefore p₁ > p₂ > p₃.

2.3. Vehicle-Following Lane-Changing Model Implementation

The vehicle-following lane-changing model for meta-cellular automata, as constructed in this paper, is divided into two stages, as follows.

Stage 1: Vehicle Lane-Changing Stage

The emergency vehicle is observed to be operating within the designated emergency vehicle influence area D, and thus is subject to the relevant regulations regarding lane-changing. It is noted that the emergency vehicle is proceeding in the same lane as the social vehicles, in accordance with the established lane-changing rules. The subsequent sequence of events is outlined below.

$$x_n^i(t+1)<x_n^j(t)$$

(29)

$$v_n^i(t+1)<v_n^j(t)$$

(30)

where i and j denote two adjacent lanes.

Stage 2: Single-lane vehicle state evolution-NaSch model

The speed of each vehicle can be taken as 0, 1, 2,..., v_max, where v_max is the maximum speed. During the time period t→t + 1, the vehicles are operated in accordance with the stipulated regulations. The parallel updating method is employed in lieu of the sequential updating method, as it has been demonstrated to be more efficacious in inducing a cascade of transition reactions. In the event of a vehicle decelerating due to random slowing effects, and provided that the density of vehicles on the road is sufficiently high at that particular juncture, it is possible for the following vehicle to also decelerate at the slowing step. If p₀ > 0 is greater than zero, the probability of the subsequent vehicle being decelerated due to random slowing until it comes to a standstill is high, which in turn induces the traffic jam problem. In comparison to the sequential updating method, the spontaneous congestion phenomenon generated by the parallel updating method is more reflective of real-world traffic patterns.

In the context of periodic boundary conditions, the position of the lead vehicle on the road must be identified upon completion of the vehicle position update. In the event that this position exceeds the length of the road system under study (denoted as l_road), the vehicle in question assumes the role of the tail vehicle, entering the system from the end at position x_last = x_lead − l_road, v_last = v_lead. In this equation, x_lead, x_last, v_lead, and v_last represent the position and speed of the lead and tail vehicles on the road, respectively.

The initial conditions are as follows: the leftmost cell of the road is assigned x = 1, and the number of cells contained at the entrance end of the road is v_max cells. This implies that a vehicle can enter the system from any cell. It is necessary to detect the head vehicle as well as the tail vehicle on the road at the moment when the road vehicle completes the update behavior, that is, at time t to time t + 1. The subsequent step involves the determination of the positions of these vehicles, that is, the identification of x_lead and x_last. If x_last is greater than v_max, the vehicle enters the tuple min[x_last − v_max, v_max] at a travelling speed of v_max, with a probability of a. The exit position is then detected, and if x_lead > l_road occurs, the running head vehicle leaves the region with probability b. The rear vehicle then turns to the new head vehicle and evolves according to this law.

The acceleration process of the vehicle is as follows: the driver seeks to obtain the optimal driving environment, i.e., the running speed can reach the desired running speed.

$$b_{n+1}(t)=0, b_n(t)=0$$

(31)

$$v_n(t+1)=\min \left(v_n(t)+1,v_{\max }\right)$$

(32)

where min{v_n(t) + 1, v_max} is the minimum value found from the value of v_n over time t plus 1 to the maximum speed of the social vehicle. The deceleration process of the vehicle, d_safe represents the effective safety distance. This is the distance that the vehicle must decelerate to in order to ensure the safety of the driver.

$$v_n(t+1)=\min \left(v_n(t),d_{safe}\right)$$

(33)

where min{v_n(t), d_safe} is the minimum value found from the value of v_n over time t to the effective safety distance.

If $v_n(t+1)<v_n(t)$, then $b_{n+1}(t)=1$.

$$d_{safe}=x_{n+1}-x_n-l_{veh}$$

(34)

Equation (34) denotes the number of unoccupied metacells between the nth vehicle and the vehicle n + 1 in front of it. l_veh denotes the length of the vehicle for which the d_safe calculates the distance with specific reference to the length of vehicle n + 1.

The probability of random slowing is defined by p₀.

The random slowdown probability is influenced by a multitude of factors, including the driving process encountered in a suboptimal driving environment and the driver’s distinctive personality traits. This ultimately influences the vehicle’s operational process of deceleration, which is characterized by a degree of randomness. It is important to note that this phenomenon is exclusive to social vehicles, as it is not applicable to emergency vehicles.

If $rand()<p_0$, then

$$v_n(t+1)=\max \left\{v_n(t+1)-1,0\right\}, b_n(t)=1$$

(35)

The position update entails the vehicle continuing its forward motion in accordance with the revised speed.

$$x_n(t+1)=x_n(t)+v_n(t+1)$$

(36)

3. An Active Guidance Method for Mixed Traffic Flow

The conventional methodologies for regulating vehicular velocity encompass the following techniques: green wave coordinated control of arterials, single-vehicle speed guidance, and multi-vehicle speed guidance. While single-vehicle speed guidance based on the Advanced Driving Assistance System (ADAS) enhances traffic safety, it is limited in its ability to provide precise speed guidance and lacks the capacity to offer dynamic optimization information. The passive nature of green wave coordinated control and multi-vehicle speed guidance is such that only group-oriented suggestions can be provided. The data transmission method of traditional speed guidance is predominantly wired, which increases the cost of maintenance and operation. Additionally, the processing, analysis, and elimination of erroneous data is challenging, which could potentially impact subsequent speed guidance operations. It is therefore necessary to seek a method of personalized and proactive guidance that differs from the traditional speed guidance and vehicle guidance methods used in the intelligent and connected environment. The utilization of such a system would facilitate the exploitation of the rich data, enhanced accuracy, and expedited time efficiency that characterizes modern information transmission methods. The new system under discussion is designed to satisfy the user’s demand for personalized vehicle guidance services. The subsequent chapter commences with the delineation of a vehicle guidance scenario for mixed traffic flow. This scenario takes into account the disparate conditions encountered by vehicles upon entering the speed guidance zone. It takes into account the varying driving states of connected vehicles and conventional vehicles within the lanes, as well as the presence or absence of vehicles impeding the speed guidance in the two conditions. This analysis leads to the formulation of distinct vehicle guidance strategies and the establishment of a model with parameters such as vehicle delays, driving time, queue length, and other variables as optimization objectives.

3.1. The Theoretical Foundation for Speed Guidance

At the present time, there are a number of mainstream vehicle-following models in existence. These include the stimulus–response-type models, psycho-physiological models, neural network models, and safe distance models.

The psycho-physiological model adjusts the current vehicle’s motion state by considering the stimulus to the driver from the change in the leading vehicle’s motion state. When the speed difference between the leading vehicle and the following vehicle, along with the relative distance, reaches a certain value, the driver will adjust the motion state of their own vehicle. The psycho-physiological follow-up model is a commonly used follow-up model built into the VISSIM 4.30 simulation software.

The classic psycho-physiological model was proposed by Leutzbach [40] in 1986 and is as follows.

$${\displaystyle \frac{d{v}_N\left(T+\tau \right)}{dT}}={\displaystyle \frac{{(\Delta v_N(T))}^2}{2\left(s-\Delta v_N(T)\right)}}+{\displaystyle \frac{d{v}_{N-1}(T)}{dT}}$$

(37)

where $\tau$ is the driver’s reaction time, s is the safety critical gap for following, v_N is the current vehicle speed, and $\Delta v_N(T)$ is the headway between the current vehicle and the immediately preceding vehicle.

The neural network class model employs the potent nonlinear computing capability of the neural network to utilize the values of speed, position, and acceleration of the current vehicle and the leading vehicle as the model’s input parameters, and the acceleration of the vehicle at the subsequent moment as the model’s output parameters. The employment of current coil detectors and video detectors facilitates the acquisition of a substantial volume of following behavior data. Subsequent refinement of the model’s capacity to accurately predict following behavior is enabled by a comprehensive training regimen on the following neural network model. However, the physical meaning of the neural network model remains opaque, and the model parameters are significantly influenced by varying traffic environments. Ensuring the efficacy of the model necessitates the acquisition of a substantial amount of relevant data for training and parameter calibration in various traffic environments.

The safety distance model is a method of avoiding collisions with the leading vehicle by controlling the distance between the front of the current vehicle and the leading vehicle. The more classical approach is the safety distance model proposed by Kometani [41].

$$\Delta v(T)=\alpha v_{N-1}^2(T)+{\beta }_l{v}_N^2(T+\Delta t)+\beta v_N(T+\Delta t)+b_0$$

(38)

where $\Delta t$ is time interval. $\alpha$, ${\beta }_l$, $\beta$, $b_0$ are parameters that require calibration within the model.

Currently, the following classical models are the focus of research:

(1)

Optimal velocity model (OV) model

Bando et al. [42] established the optimal velocity model (OV) model on the basis of the classical following model. The acceleration of the following vehicle is expressed as follows:

$${\displaystyle \frac{d{v}_N\left(T\right)}{dT}}=a\left[V\left(\Delta x_N(T)\right)-\Delta v_N(T)\right]$$

(39)

where a is the sensitivity coefficient to the vehicle in front, $v_N\left(T\right)$ is the velocity of the vehicle N at the moment T, and $V\left(\Delta x_N(T)\right)$ is the headway function of $\Delta x_N(T)=x_{N+1}(T)-x_N(T)$, which is the optimal velocity of the vehicle. There are various types of optimal speed functions $V\left(\Delta x_N(T)\right)$, and one of the more commonly used forms in studies on the car-following model is as follows:

$$V\left(\Delta x_N(T)\right)={\displaystyle \frac{v_{\max }}{2}}\left[\tanh (\Delta x_N-h_c)+\tanh (h_c)\right]$$

(40)

where vmax is the maximum speed and h_c denotes the safe distance between vehicles. Generally, when $\Delta x_N\to 0$, $V\left(\Delta x_N(T)\right)\to 0$ to avoid vehicle collision, i.e., when the vehicles are travelling unimpeded, the interaction between vehicles is negligible.

(2)

Generalized Force (GF) model

Helbing [43] calibrated the optimal velocity model with the actual data obtained, and through parameter identification, the following equation is obtained as the optimal velocity function.

$$V\left(\Delta x\right)=V_1+V_2\tanh \left[C_1(\Delta x-l_c)-C_2\right]$$

(41)

where l_c denotes the length of the vehicle. In the numerical simulation, l_c is set to 5 m.

In the event of the speed of the following vehicle being less than that of the leading vehicle, the Generalized Force (GF) model was established in order to ensure that the model accurately represented the actual traffic situation. This involved the addition of an influence factor for the speed difference. The Generalized Force model is expressed as follows:

$${\displaystyle \frac{d{v}_N\left(T\right)}{dT}}=a\left[V\left(\Delta x_N\right)-v_N(T)\right]+\lambda \Delta v_{N}H(-\Delta v_N)$$

(42)

where $\Delta v_N=v_{N+1}-v_N$ is the speed difference between the following vehicle and the leading vehicle. $H(-\Delta v_N)$ is the Heaviside function. $\lambda$ is the response coefficient of the following vehicle to the speed difference between itself and the leading vehicle, which is the coefficient to be calibrated.

3.2. A Mixed Traffic Flow Guidance Scenario Analysis

In order to facilitate the research and optimization of speed guidance strategies for mixed traffic flows, the following assumptions are made:

• In a connected vehicle with optimal communication conditions, each vehicle is able to communicate with other vehicles and roadside equipment units. Furthermore, the connected vehicles are capable of cooperative control, enabling them to complete assigned driving tasks as directed by the control centre.
• The communication between the vehicle-mounted equipment of the connected vehicles and the roadside equipment is facilitated by a special vehicle networking communication protocol (IEEE 802.11p protocol), which ensures the accuracy and real-time nature of the collected information.
• The influence of external factors, such as non-motorized vehicles and pedestrians, is disregarded.
• The vehicles within the control area are operating in accordance with the relevant regulations.
• All vehicles are travelling at the same desired speed.

A vehicle guidance strategy for mixed traffic flow has been developed, starting from different dimensions of time and space, with the objective of ensuring the maximum efficiency of mixed traffic flow, while guaranteeing the safe passage of vehicles. The specific data required for this analysis are presented in

Table 1. Information pertaining to traffic parameters.

Description	Symbol	Unit
The speed at which a vehicle enters the speed zone	v0	m/s
The time at which a vehicle enters the speed zone	t0	s
The reaction time taken to execute the command	∆t	s
The maximum (low) speed limit of the lane	vmax/vmin	m/s
The comfortable acceleration (or deceleration) of the vehicle	a	m/s2
The color of the light when entering the speed guidance zone	Light	G/R
The remaining time at the end of the green light	tg/tr	s
The guidance speed	vsug	m/s
The length of the speed guidance zone	L	m
The length of the queue at an intersection	l	m
The vehicle safety spacing	ds	m
The speed of smooth traffic	vf	m/s
The speed at which a queue dissipates	vp	m/s

.

3.3. Methodology

It is recommended that the intersection be set upstream of 150 m for the speed guidance area and connected vehicle guidance scenarios. These should be in accordance with the vehicle entering the speed guidance area in front of the interference of ordinary vehicles. Broadly speaking, speed guidance scenarios can be divided into two: those in which there are no vehicles in front of the interference, and those in which there are ordinary vehicles blocking the speed guidance, thereby increasing the difficulty of connected vehicle guidance. In such cases, it is necessary to consider the length of the intersection signal and the interference flow chart of ordinary vehicles, as illustrated in Figure 3.

Upon entering the speed guidance zone, it is observed that other vehicles are present in front of the interference.

(1)

In the event of the light = G, the vehicle enters the speed guidance zone and the intersection signal is green. Despite the optimal guidance speed of vehicle A being calculated, it is possible for it to pass in the green light time, as illustrated in Figure 4a. The presence of the vehicle in front of it constitutes an obstacle, and it is not feasible to proceed according to the guidance speed in order to avoid a potential conflict. Consequently, it is not possible to pass through the intersection within the allotted time frame for the green light.

In instances where traversing the intersection via speed guidance is not a viable option, connected vehicles may be directed to change lanes (see Section 2). Firstly, the neighboring lanes within the speed guidance area must be identified to ascertain whether superior driving conditions are present. In the event that such conditions are present, the vehicle must be guided to the adjacent lanes for the purpose of changing lanes. In the event that vehicle B is also a connected vehicle, it may also be subjected to speed guidance. The lane change at the rear of vehicle B does not impede the smooth passage of the intersection (see Figure 4b).

The objective of vehicle guidance is to ensure the simultaneous operation of two connected vehicles. In order to achieve this, the guidance strategy must satisfy a number of conditions.

When $v_b\Delta t\le d_{other}-d_s$, then

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-\frac{v_{sug}^2-v_0^2}{2a}}{v}}+\Delta t\le t_g\\ v_{\min }\le v_{sug}\le v_{\max }\end{array}\right.$$

(43)

When $v_b\Delta t>d_{other}-d_s$, then

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-v_{sug}^2-v_0^2/2a}{v}}+\Delta t\le t_g\\ {\displaystyle \frac{{v^{\prime }}_{sug}-v_0}{a^{\prime }}}+{\displaystyle \frac{L-{({v^{\prime }}_{sug})}^2-{(v_b)}^2/2a}{v}}+\Delta t\le t_g\\ v_{\min }\le v_{sug}, {v^{\prime }}_{sug}\le v_{\max }\end{array}\right.$$

(44)

where v_b represents the speed of the vehicle situated behind the adjacent lane. d_other denotes the distance between the vehicle in question and the vehicle located behind the adjacent lane at the precise moment of lane change. In the event that the rear vehicle B of the adjacent lane is situated at a greater distance from the lane-changing guiding vehicle A, at this time $v_b\Delta t\le d_{other}-d_s$, the distance traversed by vehicle B within the time frame of vehicle A’s lane change is insufficient to attain the requisite safe distance d. In such instances, the speed guidance of vehicle A, as delineated in Equation (43), is implemented. Subsequently, as the vehicle B in the adjacent lane gradually approaches the vehicle A undertaking a lane change, its distance from vehicle A falls below the requisite safe distance d_s. At this time, $v_b\Delta t>d_{other}-d_s$, the vehicle B must also be provided with a speed guidance signal, which is implemented through the application of Equation (44).

(2)

When light = R, after entering the speed guidance area, the vehicle must adhere to the speed guidance Equation (37) in the event of a red-light phase, the presence of vehicles in front of an obstruction, or a queueing phenomenon, as illustrated in Figure 5.

In order to minimize the delay of vehicles passing through the intersection, it is possible to initially assess the traffic conditions of the two lanes, and subsequently develop speed or lane-changing guidance strategies. In the event that the traffic conditions in the adjacent lanes are superior, vehicle guidance is conducted for connected vehicles, provided that the objective of facilitating a smooth passage through the intersection for all vehicles is met and the safety of the rear vehicles is guaranteed. When l > l′, lane changing is undertaken. When l < l′, lane changing is not performed at this time. In the case where l = l′, it is necessary to ascertain which lane contains a greater number of connected vehicles among those vehicles that have formed a queue in this lane and in adjacent lanes. This is due to the fact that, under the same queue-length dissipation condition, the reaction execution time of the connected vehicles is shorter, and the queue dissipation is faster. (l is the queue length of the lane in which the guided vehicles are located and l′ is the queue length of the neighboring lanes). The conditions that should be satisfied specifically for lane-changing guidance of connected vehicles are as follows:

If $v_b\Delta t\le d_{other}-d_s$, then

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-\frac{v_{sug}^2-v_0^2}{2a}}{v}}+\Delta t\ge t_r+{\displaystyle \frac{l}{v_p}}+\Delta t^{\prime }\\ v_{\min }\le v_{sug}\le v_{\max }\end{array}\right.$$

(45)

If $v_b\Delta t>d_{other}-d_s$, then

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-v_{sug}^2-v_0^2/2a}{v}}+\Delta t\ge t_r+{\displaystyle \frac{l}{v_p}}+\Delta t^{\prime }\\ {\displaystyle \frac{{v^{\prime }}_{sug}-v_{other}}{a^{\prime }}}+{\displaystyle \frac{L-{({v^{\prime }}_{sug})}^2-{(v_{other})}^2/2a}{v}}+\Delta t\ge t_r+{\displaystyle \frac{l}{v_p}}+\Delta t^{\prime }\\ v_{\min }\le v_{sug}, v_{sug}^{\prime }\le v_{\max }\end{array}\right.$$

(46)

where $\Delta t$ represents the vehicle lane-changing execution time, $\Delta t^{\prime }$ denotes the queuing vehicle start-up loss time, $v_{other}$ signifies the speed of the connected vehicle situated in the adjacent lane, $v_{sug}^{\prime }$ is the guidance speed of the connected vehicle situated in the adjacent lane, and $a^{\prime }$ is the guidance acceleration of the connected vehicle situated in the adjacent lane.

The speed guidance zone can be subdivided according to the presence or absence of vehicle interference ahead.

(1)

In the case of light = G (see Figure 6a), upon entering the speed guidance zone, the intersection signal ahead is green and there are no queuing conditions. However, if vehicle A continues to enter the speed guidance zone at its initial speed v forward, it may encounter a red-light phase when reaching the intersection stop line. At this juncture, it is advisable to increase the speed of the vehicles to facilitate the passage of vehicle A through the intersection during the green light interval.

(2)

In the case of light = R (see Figure 6b), the vehicle enters the speed guidance zone with the intersection ahead displaying a red signal. In such instances, the initial speed v is adjusted to prevent the vehicle from reaching the intersection stop line during the red-light phase.

In order to facilitate the expeditious passage of vehicles through the intersection and thereby mitigate delays in the case (1) environment, the guidance speed v_sug should be satisfied.

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-\frac{v_{sug}^2-v_0^2}{2a}}{v}}\le t_g\\ v\le v_{sug}\le v_{\max }\end{array}\right.$$

(47)

In order to minimize the delay of vehicles through the intersection in the context of case (2), it is necessary to ensure that the guidance speed v_sug is satisfied.

$$\left\{\begin{array}{c}{\displaystyle \frac{v_{sug}-v_0}{a}}+{\displaystyle \frac{L-\frac{v_{sug}^2-v_0^2}{2a}}{v}}+\Delta t\ge t_r\\ v_{\min }\le v_{sug}\le v_{\max }\end{array}\right.$$

(48)

The speed limit for road vehicles (defined as v_max) is calculated at 85% of the maximum permitted speed for that particular vehicle, indicating that 85% of all vehicles on the road are travelling at or below the speed limit. This figure is also the speed limit that has been set by the relevant traffic control department. The minimum speed limit, denoted as v_min, is set with the intention of improving driving safety and reducing congestion.

In the event that a vehicle is unable to obtain the requisite vehicle guidance under the aforementioned conditions, it is necessary to abandon the guidance of the vehicle and instead direct one’s attention to the next connected vehicle, with the objective of ensuring that the overall road vehicle delays are minimized and the efficiency of the passage is enhanced.

Concurrently, the total time t_n for vehicle n to enter the speed guidance zone and reach the vehicle stop line at the intersection is calculated, as illustrated in Equation (49).

$$t_n=\Delta t+\left|{\displaystyle \frac{v_{sug}-v_0}{a}}\right|+\left|{\displaystyle \frac{L-\frac{v_{sug}^2-v_0^2}{2a}-v_0\Delta t}{v_{sug}}}\right|$$

(49)

4. Case Study

The present study employs the intersection of Gaota Road and Guishui North Street in Yanqing District, Beijing as a case study for purposes of simulation (see Figure 7). The real connected vehicle data on the mixed traffic flow global optimization lane-changing modelling, as well as the vehicle control interface for controlling the vehicle speed, the signal machine control interface for collecting the corresponding signal timing scheme simulation interface for reading the queue length, the vehicle travel time and other data, and the analysis of the results, are based on this model.

As demonstrated by the field survey and the local traffic police detachment, the channelization and timing information of the intersection was obtained and is presented in

Table 2. The information of channelization.

North		East		South		West
Inlet Road	Exit Road	Inlet Road	Exit Road	Inlet Road	Exit Road	Inlet Road	Exit Road
2 lanes	1 straight left	2 lanes	1 left	2 lanes	1 left	3 lanes	1 left
	1 straight right		1 straight		1 straight		1 straight
			1 straight right		1 right		1 straight right

and

Table 3. Timing information.

Plan	Cycle	Phase 1		Phase 2		Phase 3		Phase 4
1	140	East–west straight ahead at the same time	28	East–west left turn at the same time	26	North–south straight ahead at the same time	32	North left turn and south right turn release	30
2	80	The eastern and western phases are released simultaneously	38	The northern and southern phases are released simultaneously	30
3	60	The eastern and western phases are released simultaneously	26	The northern and southern phases are released simultaneously	22

.

The amber time is 4 s. The all-red time is 2 s and contributes to the overall cycle.

4.1. The Results of Mixed Traffic Flow Lane-Changing Model

This paper utilizes empirical evidence derived from the measurement of vehicle data at the intersection, in conjunction with the assessment of the BeiDou map, to ascertain the total number of single-lane cells within the designated study area. The total number of single-lane cells is determined to be 1000, thereby indicating that the actual length of the lane is 7.5 km. The “Regulations on the Implementation of the Road Traffic Safety Law of the People’s Republic of China” stipulate that the maximum operating speed for social vehicles is 30 km/h, and the maximum speed of vehicles travelling is thus 30 km/h. The maximum operating speed for emergency vehicles is 50 km/h, and the maximum speed of vehicles travelling is thus 150 km/h. At the commencement of the simulation, social vehicles were distributed randomly among the lanes, and the initial speeds were randomly generated by the model. To eliminate the potential for randomness to influence the results, the speed data for vehicles in the final 200 time steps of the simulation is averaged. This includes the average speed, the number of lane changes, and the speed of each vehicle. Furthermore, the positional information of all vehicles, $x_n^j(t)$, is recorded for statistical analysis. In the connected environment, the relationship graphs of running speeds and the number of lane changes are generated for the purpose of comparison of different parameters. The simulation values of each parameter are summarized in

Table 4. Simulation Parameters Selection.

Description	Value
Length of metacells (i)/m	7.5
Number of metacells (N)/grid-lane−1	1000
Social vehicle speed (Vs)/grid-step	0~3
Emergency vehicle speed (Ve)/grid-step	0~5
Traffic density (ρ)	0~1
Time step	10,000
The weight coefficient $\delta$	0.5
The weight coefficient $\epsilon$	0.5

.

The extent of the area of influence of the ICE is a fixed quantity.

$$D_1+D_2+D_3=D$$

(50)

The parameters D, p₁, p₂, and p₃ are defined as follows: D = 200, p₁ = 0.4, p₂ = 0.3, p₃ = 0.2. D₁ and assumes a value range of (0, 100). Similarly, D₂ assumes a value range of (0, 100). The objective of this study is to examine the variation in emergency vehicle travelling time with different values of D₁ and D₂ in three density cases, namely 0.3 (low density), 0.5 (medium density), and 0.7 (high density). When the density is set at 0.3, the emergency vehicle travelling time varies with the values of D₁ and D₂, as illustrated in Figure 8.

As demonstrated in Figure 8, the passage time of emergency vehicles is observed to decrease with the increase in the length of D₁ and D₂, and the decrease is relatively minor. Due to the low density of vehicles at this time, emergency vehicles travelling alongside social vehicles experience a lower degree of interference in the impact area, enabling them to operate smoothly. However, in certain instances, it may be necessary for social vehicles to yield to emergency vehicles. The D₂ area on the emergency vehicle’s passing time is slightly larger than that of D₁, corresponding to the figure. As D₁ increases, the degree of reduction in the emergency vehicle access time is relatively high, due to the D₁ area being the nearest area of the emergency vehicle impact, where social vehicles must yield to emergency vehicles. Given that the D₁ area represents the point of greatest impact for emergency vehicles, it is imperative that social vehicles yield the right-of-way.

When the density is 0.5, the emergency vehicle travelling time varies with the values of D₁ and D₂, as illustrated in Figure 9.

In the scenario of medium density, the passage time of emergency vehicles is subject to a significant decrease in the lengths of D₁ and D₂. The impact of D₂ on emergency vehicle travel time is more pronounced than that of D₁. As D₂ increases, the rate of decrease in emergency vehicle travel time accelerates. When considering the collective influence of the three distances, D₁ emerges as the most immediate area affected by emergency vehicles, necessitating immediate yielding. Consequently, augmenting the length of this area represents a highly effective strategy for reducing the passage time of emergency vehicles.

When the density is 0.7 the emergency vehicle travelling time varies with the values of D₁ and D₂, as illustrated in Figure 10.

As shown in Figure 10, the increasing density of road traffic has resulted in heightened congestion, thereby impeding the ability of social vehicles to ensure the unobstructed passage of emergency vehicles during high-speed driving. The impact of D₁ and D₂ on the passage time of emergency vehicles demonstrates that congestion is at its peak at this juncture, impeding both emergency and social vehicles.

In this paper, the dimensions D₁ (length: 65 m), D₂ (length: 55 m), and D₃ (length: 80 m) are selected for subsequent problems.

As illustrated in Figure 11, further analysis is warranted with respect to the average speed of social vehicles operating under the influence of emergency vehicles, as well as emergency vehicles themselves. The average speed of emergency vehicles is greater than that of social vehicles. As the density of social vehicles increases, the average speed of the two types of vehicles approaches each other. Conversely, as the density of vehicles increases, the average speed of both emergency and social vehicles declines, with this decline becoming more pronounced as the density rises. This phenomenon can be attributed to the fact that as the density of vehicles increases, there is a reduction in the available space for vehicles to change lanes, which in turn limits their ability to maintain high vehicle speeds due to the reduced inter-vehicle spacing. The decline in the average speed of emergency vehicles in comparison to social vehicles is more pronounced. This is due to the necessity for emergency vehicles to maintain high speeds in order to accommodate the movements of social vehicles, as well as to facilitate their own lane changes. The increased congestion resulting from the high density of vehicles poses a significant challenge to the maneuverability of social and emergency vehicles, leading to a substantial decline in their average speeds. At a density of 0.8, the speeds of emergency vehicles and social vehicles are nearly equal, indicating a high degree of road congestion, resulting in both types of vehicles being blocked.

As illustrated in Figure 12, the final average speed is reduced as the density of road vehicles increases, irrespective of whether emergency vehicles adopt a lane-changing strategy. This is due to the fact that as the density of road vehicles rises, the inter-vehicle distance diminishes, rendering it impossible for vehicles to maintain the requisite speed. Consequently, emergency vehicles undertake lane-changing maneuvers, resulting in a slight increase in average speed in comparison with the absence of such maneuvers. When the density is less than 0.2, the performance of the lane-changing operation by emergency vehicles is not significantly affected by the enhancement degree of speed, indicating that the density of road vehicles is relatively low. Consequently, the operation of emergency vehicles in front of social vehicles on emergency vehicles running obstruction is minimal. Furthermore, the enhancement of speed change situation in the performance of the lane-changing operation by emergency vehicles is also insignificant. However, as the density of vehicles increases, the emergency vehicle’s decision to perform a lane-changing operation becomes independent of speed. This suggests that in contexts where the probability of encountering social vehicles is high, emergency vehicles can achieve a more optimal running effect, regardless of whether they perform a lane-changing operation. In the range of vehicle densities between 0.2 and 0.7, the effect of lane-changing behavior on emergency vehicles is particularly pronounced. During this period, the presence of a front vehicle hinders the normal passage of emergency vehicles, thus compromising the safety of emergency vehicle occupants. To mitigate this issue, emergency vehicles must increase their speed by changing lanes, which in turn affects their speed to a greater extent when performing lane-changing maneuvers. As road density increases to a value of 0.7 or above, a number of factors contribute to an increase in congestion. These include a reduction in the availability of operational space for vehicles, a decline in the number of opportunities for emergency vehicles to change lanes, and a corresponding reduction in the overall efficiency of road access. Consequently, the frequency of lane-changing behavior by emergency vehicles, which is designed to enhance speed, also declines. At a road density of 0.8, the speed of emergency vehicles is nearly equal, regardless of whether they perform a lane-changing operation. This finding suggests that the traffic flow state is one of congestion, and that lane-changing behavior at this density is unable to improve the driving speed of emergency vehicles.

As demonstrated in Figure 13, the presence of emergency vehicles has a significant impact on the frequency of social vehicle lane changes at varying densities. In the absence of emergency vehicles, the number of lane changes increases at a slower rate when the density is below or equal to 0.2. At this density, the traffic flow is in a free-flow state, and vehicle operation is smooth. However, as the density increases from 0.2 to 0.55, the number of lane changes made by vehicles rises in line with the growth in density, reaching a peak at a density of 0.55. As the number of vehicles increases, the distance between them decreases, the overall speed declines, and the necessity for vehicle lane changes rises. Vehicles enhance their operational status by performing lane changes, and the number of lane changes gradually increases. As vehicle density exceeds 0.55, the likelihood of road congestion increases in tandem with the vehicle density. This, in turn, leads to a reduction in the available workshop distance for vehicles, despite the necessity for lane changing. The difficulty in meeting the conditions for lane changing results in a decline in the number of lane changes.

In scenarios where the traffic density is less than or equal to 0.2, the vehicle in question is observed to operate in a smooth and uninterrupted manner, with only a small number of vehicles deviating from their original trajectory in order to avoid an emergency vehicle, resulting in a relatively low incidence of lane changes. In the density range of 0.2 to 0.45, the operation of emergency vehicles is undertaken to ensure their priority status and high speed. In such circumstances, social vehicles are required to yield to emergency vehicles, resulting in an increase in the number of lane changes. At a density of 0.45, the number of lane changes reaches its peak. The distinctive characteristics of emergency vehicle operations result in a lower peak number of lane changes at densities of vehicles in the presence of emergency vehicles compared to the absence of such vehicles. In the density range of (0.45, 0.55), the level of congestion on the road increases, and the safety conditions for lane changing become more challenging to meet. When emergency vehicles are present, social vehicles are required to yield to them as much as possible, resulting in a slight reduction in the number of lane changes. As the density exceeds 0.55, the congestion in vehicle operation intensifies, impeding both emergency and social vehicles, and the conditions for vehicle lane-changing become increasingly challenging to satisfy, resulting in a notable decline in the number of lane changes.

In order to gain further insight into the impact of emergency vehicles on the road as a whole, this paper employs a density value of 0.45 and three distinct lane-changing rules within the context of spatio-temporal trajectory class diagrams for analysis. The horizontal coordinate represents the position of the vehicle, with units expressed in grid coordinates, while the vertical coordinate denotes the running time, with units expressed in steps (see Figure 14 and Figure 15). The comparison diagrams demonstrate that the higher constraints on the lane-changing conditions in the model mentioned in [44] result in vehicles failing to utilize road resources fully, thereby causing phase separation and increasing the likelihood of stopping and blocking, with the duration of these blockages being longer. The model proposed in [45] has demonstrably enhanced the resolution of phase separation. A comparison of the proposed model with the aforementioned two vehicle lane-changing models reveals that the phase separation phenomenon has been largely eliminated, suggesting that the number and duration of blockages are significantly reduced, and that the overall operational efficiency of the road is enhanced.

This section explores the consequences of modifying the values proposed in Table 4 on the presented results. Initially, an examination is conducted of the parameters of satisfaction. In this study, the traffic density is set at 0.5 and the proportion of connected vehicles at 0.3. The subsequent analysis employs a simulation-based approach to investigate the impact of these modifications on the satisfaction levels of emergency vehicles in both the current lane and adjacent lanes. The resultant data are presented in Figure 16.

As demonstrated in Figure 16, between the 10th and 15th time steps, the satisfaction value of the emergency vehicle travelling in the adjacent lane is lower, while the satisfaction in this lane is higher before the 10th step. This finding suggests that the fluctuations in satisfaction during this period are not attributable to the vehicle changing lanes. Instead, it is attributed to the change in the speed of the vehicle preceding this lane, which results in the target vehicle’s speed approaching the desired speed. This conclusion is substantiated by comparing the extracted speed and position of the target vehicle with the extracted target vehicle speed and position. In the 30th time step of the further team analysis, the satisfaction of the neighbouring lanes approaches 0, the satisfaction of this lane is larger before this, and then the satisfaction of the neighbouring lanes approaches the satisfaction of the current lane, which indicates that the vehicle has changed lanes at this moment.

Next, an analysis is conducted to ascertain the impact of alterations in the proportional composition of distinct driver categories on the operations of traffic flow. The issue of drivers’ propensity to change lanes is also known to have an effect on the model study (in particular, the weighting coefficients δ = 0.5 and ε = 0.5). Therefore, the proportion of connected vehicles is set to 0.3, and the impact of the proportion of social vehicles on the average speed of emergency vehicles on the discussion road is studied in three density cases: 0.3 (low-density), 0.5 (medium-density), and 0.7 (high-density), as shown in Figure 17.

As shown in Figure 17, the average speed of emergency vehicles in both medium- and high-densities increases continuously with the proportion of social vehicles. This increase is more significant in the medium-density case. When the proportion of social vehicles exceeds 0.7, the increase in the average speed of emergency vehicles becomes more pronounced. Furthermore, when the proportion of social vehicles exceeds 0.8 in the high-density case, the growth in the average speed of emergency vehicles becomes more pronounced, indicating that emergency vehicles in the medium density case have a superior driving space. The lane-changing behaviour in the medium-density case takes into full account the current lane of the vehicle, the driving conditions of the adjacent lanes in the mixed traffic flow, and that lane-changing behaviour can effectively improve the traffic efficiency of overall road sections. Concurrently, as traffic density increases, the utility of non-tendency lane-changing drivers in road traffic is progressively diminished.

This section employs a comparative experimental approach to evaluate the quality of the solutions. Given that different methods are developed based on different constraints, a direct comparison is not meaningful. Instead, the focus is directed towards the identification of common characteristics amongst several extant algorithms, with a subsequent comparison of these with the methods proposed in this paper. The proposed methods are outlined in Algorithms 2 and 3, which are listed in

Table 5. Definitions of comparative algorithms.

Name	Definition
Algorithm 2	As mentioned in [11]
Algorithm 3	As mentioned in [12]

. The ensuing results are presented in Figure 18.

As demonstrated in Figure 18a, the proposed lane-changing control method enhances the smoothness of vehicle lane-changing adaptation. Specifically, the present lane-changing method has been shown to significantly reduce the variability in following spacing, ranging from approximately 35 to 56 metres, and rapidly converge to the desired safe spacing within 15 s. Furthermore, fluctuations in spacing are observed to decrease with increasing traffic density. In contrast, the following spacing of other algorithms varies from 32 to 58.5 metres and exhibits significant fluctuations.

As illustrated in Figure 18b, the stability of various algorithms in lane-changing observation is demonstrated, and the results indicate that the lane-changing control in this paper exhibits superior performance in terms of lane-changing stability under varying traffic densities when compared to Algorithms 2 and 3. Specifically, in the simulation of lane-changing maneuvers in mixed traffic flow under low density conditions (less than 0.4), the number of lane changes in this study reduces by 70.00%, 57.08%, 51.08%, and 42.22%, respectively, compared with Algorithm 2, and reduces 30.00%, 27.50%, 25.24%, and 22.75%, respectively, compared with Algorithm 3. A reduction in the number of lane changes was observed, with decreases of 42.22%, 30.00%, 27.50%, and 25.24%, respectively, compared to the third algorithm. In the mixed traffic flow lane-changing modelling at critical traffic density (0.5–0.7), the method in this paper reduces the number of lane changes by 38.26%, 37.33%, and 40.28% compared to Algorithm 2, and reduces the number of lane changes by 20.82%, 18.60%, and 21.67% compared to Algorithm 3, respectively. In the mixed traffic flow lane-changing modelling at critical traffic density (0.5–0.7), the method in this paper reduces the number of lane changes by 38.26%, 37.33%, and 40.28% compared to Algorithm 2, and reduces the number of lane changes by 20.82%, 18.60%, and 21.67% compared to Algorithm 3, respectively. In the mixed traffic flow lane-changing modelling at saturated traffic density (0.8~1), the method in this paper reduces the number of lane changes by 73.21%, 66.83%, and 58.26% compared to Algorithm 2, and reduces the number of lane changes by 48.11%, 28.29%, and 24.61% compared to Algorithm 3, respectively. As demonstrated in this paper, the proposed method has the capacity to reduce the frequency of lane changes at varying levels of traffic density. This, in turn, has the indirect effect of mitigating the likelihood of vehicle collisions during lane changes, thereby enhancing the operational efficiency of road traffic flow.

As illustrated in

Table 6. Optimal solution results and computational performance of the three methods.

	Algorithm 2	Algorithm 3	The Proposed Method
Optimal solution	80.13%	88.48%	95.65%
Computational performance (s)	12.20	5.74	1.29

, the optimal solution and computational performance for the three methods are summarized. The results demonstrate that all of the aforementioned methods are capable of efficiently achieving optimal results in lane changing, with an average computation time of 1.29 s. In comparison to Algorithm 2 (12.20 s) and Algorithm 3 (5.74 s), the present methods demonstrate a substantial reduction in computation time. Furthermore, the analysis in Table 5 shows that there is an 88.48% probability of attaining a globally optimal solution when employing Algorithm 3, a figure that is further enhanced to 95.65% through the implementation of the proposed method. It is also notable that the computational cost is reduced from 5.74 s to 1.29 s, thereby enhancing the optimality of the solution and reducing the probability of missing the optimal solution, which can be used to explore the optimal lane change decision.

4.2. The Results of Active Guidance for Mixed Traffic Flow

In order to eliminate the effect of transients, the data obtained in the initial 600 s are removed, and the data from 601 to 3600 s are analyzed. It is recommended that the time interval of the study exceed the maximum scenario period (140 s).

The delay time (defined as d_n) for vehicle n to enter the speed guidance zone is calculated using Equation (51), as follows.

$$d_n=t_n-{\displaystyle \frac{L}{v_f}}$$

(51)

The minimum objective function Equation (52) for the vehicle guidance model, which represents the total delay, is presented below.

$$f=\min \left({\displaystyle \sum _{i=1}^n{d}_i}\right)+{\displaystyle \sum _{j=1}^m{d}_j}$$

(52)

where d_i is the total delay time experienced by connected vehicles and d_j is the total delay time encountered by ordinary vehicles.

In this paper, the undersaturated condition is employed as a case study. The vehicle arrival rate, denoted by q, is less than the road capacity, represented by N. Additionally, the intersection queuing dissipation time, represented by τ, is less than the green light time, denoted by t_g. Furthermore, the cumulative number of vehicles arriving in a cycle, represented by q_C, is less than the maximum number of vehicles released at the green light time, denoted by St_g. Here, C represents the signal cycle time and S represents the saturated flow rate. The relationship between these variables is illustrated in Figure 19.

Once the red-light time (t_r) has elapsed, the queue length of the inlet lane reaches its maximum. The maximum queue length is expressed by the equation l_m = qt_r. Subsequently, once the green light time (t_g) commences, the queue departs the intersection with the saturation flow rate (S) simultaneously, and the maximum queue length (l_m) begins to diminish at a gradual rate. The queue is said to dissipate once l_m reaches zero, at which point it has been shown to complete the two line segments in comparison to point B. The dissipation time ($\tau ={\displaystyle \frac{l_m}{S-q}}$) is equal to zero. Subsequent to the dissipation of the green light time (t_g − τ), the queue of vehicles experiences no delay and can traverse the intersection without hindrance.

The total delay time function f of vehicles on the road can be rewritten in order to solve for the area of ∆AOB. The following equation is derived from (53) and is used to calculate the area of ∆AOB.

$$d=\Delta AOB=\Delta B^{\prime }OB-\Delta B^{\prime }AB={\displaystyle \frac{(t_r+\tau )S\tau }{2}}-{\displaystyle \frac{\tau \cdot S\tau }{2}}={\displaystyle \frac{t_r\cdot S\tau }{2}}(veh\cdot s)$$

(53)

Combined with the dissipation time equation, d would be rewritten as Equation (54).

$$d={\displaystyle \frac{Sq{t}_r^2}{2(S-q)}}(veh\cdot s)$$

(54)

Ultimately, the total delay time can be employed to ascertain the average delay time ($\overline{d}$), with the average queue length ($\overline{l}$), as illustrated in Equations (55) and (56).

$$\overline{d}={\displaystyle \frac{d}{qC}}$$

(55)

$$\overline{l}={\displaystyle \frac{d}{C}}=\overline{d}q$$

(56)

It has been demonstrated that there exists a correlation between the average vehicle delay, queue length, and the total delay function. These variables have the capacity to reflect the overall road delay conditions and can be used to assess vehicle travelling time in a more intuitive manner, thereby reflecting the level of service provided by the road. Accordingly, the average vehicle delay, queue length, and vehicle travelling time have been selected as the evaluation indices of the intersection traffic condition. The results obtained from the experiment for the aforementioned parameters are employed to substantiate the advantages and disadvantages of the speed guidance strategy.

(1)

Average delay time $\overline{d}$

In circumstances where traffic density is 0.45 and other driving conditions remain constant, Figure 20 presents a comparison graph of the average delay of a single vehicle. The most crucial metric for assessing the efficacy of the speed guidance strategy is the average vehicle delay. The average vehicle delay in the absence of speed guidance exhibits a fluctuating pattern, ranging from 23 to 27 s. In contrast, the average delay with speed guidance is approximately 21 s, indicating a reduction in average vehicle delay by approximately 16%. Furthermore, the fluctuations in delay with speed guidance are less pronounced, indicating an overall improvement in the equilibrium of the intersection and an enhanced overall traffic efficiency, whilst significantly reducing the travel time of vehicles. The provision of speed and lane change guidance to connected vehicles would result in a reduction in overall delay. This is due to the fact that connected vehicles are able to utilize their own information-gathering capabilities and make driving decisions based on the most accurate information available to them.

Figure 21 illustrates that the average vehicle delay decreases as the penetration of connected vehicles (defined as ղ) increases. This phenomenon can be attributed to the fact that as the penetration of connected vehicles rises, the number of vehicles on the road that are affected by congestion decreases. In contrast to conventional vehicles, which operate as isolated entities, connected vehicles interact with one another, exchanging information. The entry of a connected vehicle into a guidance area enables access to a greater volume of information, thereby facilitating more informed decision-making and reducing the time required for reaction and subsequent action. This phenomenon leads to a decrease in the average delay time.

(2)

Queue length $\overline{l}$

In circumstances where traffic density is 0.45, other driving conditions remain constant. Figure 22 provides a comparison of queue lengths at intersections. The average queue length of intersections under speed guidance is 44.4 m, while the queue length of intersections without speed guidance is 54.6 m. This will increase the waiting time and aggravate the congestion. The queue length in the absence of speed guidance may exhibit fluctuations in response to changes in traffic volume. This is attributed to the inability of conventional vehicles to anticipate queue dynamics, resulting in irregular fluctuations in queue length. Connected vehicles are able to react in advance to the optimal route through the intersection, or to avoid joining the queue by slowing down or changing lanes.

As demonstrated in Figure 23, there is a decline in queue lengths with an increase in the proportion of connected vehicles. This phenomenon occurs because lane change guidance is executed when connected vehicles do not meet the speed guidance criteria. Consequently, connected vehicles can augment the probability of lane changes when confronted with scenarios involving vehicle interference or queuing. Furthermore, vehicles tend to select lanes with the shortest queuing lengths when the red light is in phase.

(3)

Vehicle travel time t

In circumstances where traffic density is 0.45 and other driving conditions remain constant, the comparison graph of vehicle travel time (defined as t) at the intersection is presented in Figure 24. The reduction in vehicle travel time resulting from the implementation of speed guidance can be attributed to a decrease in key parameters, including low average delay rungs and queue length. In the absence of speed guidance, the average vehicle travel time is approximately 30 s. The implementation of speed guidance has been shown to result in a substantial reduction in average travel time, from 30 s to 20.7 s, representing a decrease of approximately 10 s compared to the travel time without speed guidance. This significant reduction in travel time is a notable benefit, and the incorporation of lane-changing guidance for connected vehicles has been demonstrated to simultaneously enhance their operational flexibility and reduce overall congestion, as evidenced by the preceding analysis of delays. Consequently, the time required for vehicle travel is also diminished.

As shown in Figure 25, the travel time from a variety of connected vehicle penetration scenarios demonstrates a decline with an increase in connected vehicle penetration. Moreover, the higher the penetration rate of connected vehicles, the more pronounced the reduction in travel time. Consequently, as the penetration rate of connected vehicles rises, the impact on intersection capacity also rises. It is evident that the advancement of vehicle networking technology will significantly enhance the current stage of traffic congestion.

5. Conclusions

In light of the fact that the original lane-changing model for non-connected traffic flow environments is no longer applicable in new traffic flow environments, further research is required on lane-changing models for new traffic flow environments in order to support the development of traffic management and control schemes. The present paper puts forward a hybrid traffic flow dynamic modelling and cooperative control method in the intelligent and connected environment. The method is based on a satisfaction-based model of free and mandatory lane changing between connected and non-connected vehicles. Secondly, a social vehicle lane-changing model based on influence characteristics is constructed for the influence area of connected vehicles, taking into account the varying degrees of influence exerted by connected vehicles on social vehicles at different distances. Subsequently, a scenario analysis is conducted for the case of connected vehicles in which speed guidance is not satisfied. This is followed by the establishment of a mixed traffic flow intersection guidance model under an intelligent and connected environment, which is based on speed guidance in conjunction with lane-changing guidance. Finally, the model constructed in this paper is verified by using the actual intersection as a case study. The experimental results demonstrate that the implementation of a constructed mixed traffic flow lane-changing model, founded on the influence characteristics of lane-changing behavior, can effectively enhance vehicle operating speed and improve road access efficiency. Conversely, as traffic density increases, the tendency of drivers not to change lanes leads to a gradual decline in road traffic utility. In the density range of (0.45, 0.55), the level of congestion on the road increases, and the safety conditions for lane changing become more challenging to meet. When emergency vehicles are present, social vehicles are required to yield to them as much as possible, resulting in a slight reduction in the number of lane changes. As the density exceeds 0.55, the congestion in vehicle operation intensifies, impeding both emergency and social vehicles, and the conditions for vehicle lane-changing become increasingly challenging to satisfy, resulting in a notable decline in the number of lane changes. A comparison of the optimal solution and computational performance results between this method and different methods demonstrates that the average computation time of this method is 1.29 s, which greatly reduces the computation time. Furthermore, the probability of attaining the global optimal solution is augmented to 95.65% by employing this method. The computational cost is reduced from 5.74 s to 1.29 s, thereby enhancing the optimality of the solution and facilitating the exploration of optimal lane changing decisions. The utilization of the speed guidance method proposed in this paper results in an average delay of approximately 21 s, which is approximately 16% lower than the average delay observed in the absence of speed guidance. In a similar vein, the queue length at the intersection is reduced from 54.6 m to 44.4 m when speed guidance is employed, while the average travelling time is reduced to 20.7 s, which is approximately 10 s shorter than the average travelling time without speed guidance. The combination of speed guidance and a lane-changing strategy enhances the efficiency of intersection access and mitigates the occurrence of queuing. This illustrates the function of connected vehicles in enhancing road accessibility, optimizing intersection access efficiency while simultaneously improving driving speed. This offers valuable insights that can inform future control strategies for mixed traffic flows involving both connected and social vehicles.

There are a number of research topics that warrant further investigation, several of which are discussed in the following sections. Firstly, the impact of pedestrians and non-motorized vehicles on mixed traffic flows was not considered in the construction of the free and mandatory lane-changing models for mixed traffic flows. Secondly, the approach of mixed traffic flows when encountering special circumstances, such as inclement weather or traffic accidents, should be considered.