Research on Speed Guidance Strategies for Mixed Traffic Flow Considering Uncertainty of Leading Vehicles at Signalized Intersections

Abstract

In the context of intelligent connected environments, this study explores methods to guide the speed of mixed traffic flow to improve intersection efficiency. First, the composition of traffic flow is analyzed, and a car-following model for mixed traffic flow is established, considering reaction time and the psychology of human drivers. Secondly, considering the uncertainty factors of the leading vehicle, we establish a speed guidance model for mixed traffic flow platoons. Finally, a simulation environment is built using Python and SUMO, evaluating the speed guidance effect from the perspectives of different traffic volumes and CAV penetration rates based on average stop times and average delays. The research findings indicate that the speed guidance algorithm proposed in this paper can reduce the number of parking times and delays at intersections. When the mixed traffic flow remains constant, the higher the penetration rate of CAV, the more effective the guidance becomes. However, when the traffic flow reaches a certain level, congestion intensifies, and the effectiveness of the guidance gradually diminishes. Therefore, this study is more applicable to long-distance intersections or key intersections on interconnected roads outside urban areas.

1. Introduction

As vehicular network technology matures, connected and automated vehicles (CAVs) (CAVs are equipped with advanced onboard sensors, controllers, actuators, and other devices. They integrate modern communication and network technologies, enabling intelligent information exchange and sharing between vehicles, humans, infrastructure, and backend systems) are poised to integrate into the broader traffic environment. The networked and autonomous nature of CAV resolves the coupling issues between control methods and human drivers, providing new solutions to existing traffic problems.

Before the complete implementation of fully CAVs, the phenomenon of mixed traffic flow with CAVs and traditional manually driven vehicles (HDVs) will become the norm. Due to the differences in driving subjects, CAVs and HDVs exhibit different characteristics, and the mixed traffic flow composed of both increases the complexity of the traffic environment. Therefore, it is necessary to conduct forward-looking research to reasonably analyze new problems that may exist in mixed traffic flow and to anticipate and predict them. This will provide a theoretical foundation for the future planning, design, and traffic management of corresponding road networks.

Therefore, analyzing the car-following characteristics of mixed traffic flow and leveraging the advantages of CAVs in mixed traffic scenarios to enhance traffic control efficiency and safety is a current research focus. Many scholars have explored this area, improving existing models like the IDM [1,2,3], CACC [4,5], and Gipps models [6,7] to suit various mixed traffic scenarios. For instance, Xinjuan Ma [8] et al. enhanced car-following models using driver memory, while Zongfang [9] et al. considered multiple leading vehicles in a full-velocity difference model.

In speed guidance, scholars have incorporated signal timing to plan CAV trajectories for eco-friendly intersection crossing. For example, Kamal et al. [10] proposed an eco-driving vehicle control system for intersection vehicle collaboration. Zhou [11] aimed to minimize delay time, fuel consumption, and safety risk, finding the optimal speed trajectory scheme for intersection vehicles under fixed signal timing conditions. Xu et al. [12] aiming to minimize vehicle travel time, proposed a speed control strategy to reduce fuel consumption during trips. The results demonstrated that this speed control strategy could achieve a 23.7% reduction in fuel consumption. Faraj M. [13] focused on minimizing idling time and the number of stops at signalized intersections by introducing a cooperative speed optimization framework based on game theory. Wu [14] utilized the entropy weight method to determine the weights for a comprehensive optimization objective, including vehicle delay, CO₂ emissions, and fuel consumption, and employed a particle swarm optimization algorithm to solve for the optimal speed. Some scholars have also focused on the efficiency of intersection traffic flow. Yi [15] considered the impact of weather factors in the optimal speed planning of vehicles and established a minimum transportation energy optimization model with travel time constraints and a minimum travel time optimization model with transportation energy constraints. Shuai L [16] et al. proposed two speed guidance algorithms based on single-vehicle and multi-vehicle travel time optimization. Xu L [17] et al. proposed corresponding speed guidance strategy models based on the different situations of guided vehicles arriving at intersections. Ma W [18] et al. proposed a hierarchical multi-objective optimization framework based on sampling vehicle trajectories at isolated signalized intersections to increase vehicle throughput, reduce queue lengths, and traffic delays. Yu and Long [19] established an optimal speed recommendation curve based on the predictive behavior of leading vehicles in the foresight vision and traffic signal timing. Some scholars have focused on unsignalized intersections. M. Amirgholy [20] developed a stochastic traffic model to optimize the headway of connected and automated vehicles (CAVs), allowing them to pass through unsignalized intersections continuously. Zohdy et al. [21] designed an adaptive cruise control (ACC) system for autonomous vehicles that can effectively reduce vehicle collisions and delays at intersections. Pei [22] proposed a speed control method for vehicles at unsignalized intersections in a connected vehicle environment to improve traffic efficiency.

In summary, there is extensive research on speed guidance and trajectory optimization for purely CAVs, but there is relatively little research on speed guidance strategies for mixed traffic flow. Yao H [23] et al. proposed a joint optimization framework at intersections for mixed traffic flows with the goal of reducing traffic emissions. Baby [24] et al. proposed a new robust advisory control framework based on model predictive control (MPC) to optimize the fuel efficiency of CAVs in heterogeneous urban traffic flows. In this control framework, CAVs are considered to provide advisory commands to HDV to follow, in order to improve their own speed and overall fuel economy. Xiao [25] et al. proposed a two-layer hierarchical longitudinal control method for travel time and trajectory along multiple intersections on arterial roads under mixed traffic of CAVs and HDVs. Wang [26] et al. proposed a trajectory prediction-based DRL method for mixed traffic environments.

Based on existing research results, this paper combines factors such as vehicle reaction time and the driving style of HDV when following CAVs to establish a car-following model for mixed traffic flow considering multiple factors. Secondly, considering that most existing research on speed guidance strategies focuses on single-lane driving environments under fully CAV conditions, this study constructs a passage model for mixed traffic flow in a two-lane driving environment. Additionally, recognizing that current research on platoon speed guidance strategies often involves excessive assumptions, resulting in an overly idealized research environment, this study considers the uncertainty interference of leading vehicles during the driving process. Various interference factors are studied, ultimately establishing a speed guidance model for mixed traffic flow platoons. The specific arrangement of the paper is as follows: The second section presents the establishment of the car-following model for mixed traffic flow. The third section presents the passage model at signalized intersections. The fourth section provides the speed guidance strategy considering the uncertainty interference of leading vehicles during the driving process. The fifth section includes the experimental design and result analysis to verify the effectiveness of the model guidance.

2. Mixed Traffic Flow Car-Following Model

CAVs and HDVs exhibit different characteristics due to their distinct driving entities, rendering traditional car-following models unsuitable for the new traffic environment. To analyze the impact of CAV integration on traffic flow and understand the characteristics of mixed traffic flow, it is necessary to develop a car-following model tailored for mixed traffic conditions. The following sections consider the impacts of CAV integration and propose improvements to existing car-following models.

To represent a more realistic mixed traffic environment, this study incorporates reaction time and the psychological factors of different human drivers when following CAVs, improving existing car-following models to establish a mixed-traffic-flow car-following model. This research categorizes the four cases shown in the figure as follows.

In urban road mixed traffic flow, CAVs and HDVs coexist with random distribution, forming four car-following scenarios (Figure 1). When an HDV is the following vehicle, two scenarios may occur: ① the HDV follows another HDV, or ② the HDV follows a CAV. In both scenarios, regardless of whether the leading vehicle is an HDV or a CAV, the following HDV lacks onboard equipment and vehicle-to-vehicle (V2V) communication capabilities. Therefore, the following vehicle relies solely on the driver’s physiological perception to obtain information about the leading vehicle. Consequently, in this study, these two scenarios are combined into one and referred to as the scenario where an HDV is the following vehicle. This study categorizes these into three types: CAV following CAV, CAV following HDV, and HDV following HDV. When a CAV follows an HDV, it cannot use vehicle-to-vehicle communication and will rely on onboard equipment to obtain information about the preceding vehicle and optimize its driving state. This scenario refers to a Degraded Connected and Automated Vehicle (DCAV).

(1)

CAV Following CAV

The Cooperative Adaptive Cruise Control (CACC) model, proposed by the PATH laboratory at the University of California at Berkeley [27] based on a constant time headway strategy, is a popular model for CAV car-following research. The modified CACC model considering reaction time is expressed as:

$$a_c={\displaystyle \frac{k_p\left(x_c-s_c-l-\left(t_c+{\tau }_c\right)v\right)+k_d\Delta v}{\Delta t+k_d{t}_c}}$$

(1)

In the above equation, $a_c$ is the acceleration of CAV.

• $x_c$ is the headway of the CAV.
• $s_c$ is minimum safe distance for the CAV, which set to 3 m.
• $l$ is the length of the vehicle, which is set at 5 m.
• $v$ is the current speed.
• $\Delta v$ is the speed difference between the two adjacent vehicles.
• $t_c$ is the desired time headway for CAV, set to the highest acceptable value of 0.6 $\mathrm{s}$;
• ${\tau }_c$ is the reaction time for CAV, set to 0 s;
• $\Delta t$ is the update interval for speed, set to 0.01 s based on the PATH real-vehicle experiments;
• $k_p$ and $k_d$ are control coefficients, set to 0.45 and 0.25, respectively, based on the calibration results from real-vehicle experiments for the CACC model.

(2)

CAV Following HDV

The Adaptive Cruise Control (ACC) model, based on a constant desired time headway, can obtain surrounding vehicle information through onboard equipment, aligning with the characteristics of DCAV. The modified ACC model considering reaction time is expressed as:

$$a_d=k_1\left[x_d-s_d-l-\left(t_d+{\tau }_d\right)v\right]+k_2\Delta v$$

(2)

In the above equation, $a_d$ is the acceleration of the DCAV.

• $x_d$ is the headway of DCAV.
• $s_d$ is minimum safe distance for DCAV, which is set to 2 m.
• $t_d$ is the desired time headway for DCAV, set to the highest acceptable value of 1.1 s for drivers.
• ${\tau }_d$ is the reaction time for DCAV, set to 0.2 s [28].
• $k_1$ and $k_2$ are control coefficients, set to 0.23–2 and 0.07–2, respectively, based on the calibration results from real-vehicle experiments for the CACC model.

The parameters $l$, $v$ and$\Delta v$ have the same meanings as in Equation (1).

(3)

HDV as the following vehicle

The Intelligent Driver Model (IDM), proposed by Treiber [29] et al., has the advantages of fewer parameters and clear physical meaning, better reflecting the car-following characteristics of HDV. Considering the varying levels of trust that human drivers have towards CAV, drivers in ‘Scenario (4)’ in Figure 1 are categorized into three car-following styles: hesitant, steady, and trustful. Based on this, we introduce a maximum acceleration sensitivity coefficient ${\lambda }_n$ and a safe headway sensitivity coefficient ${\omega }_n$ into the IDM model. In contrast, ‘Scenario (3)’ in Figure 1 represents a human-driven vehicle following another human-driven vehicle. The car-following style in this scenario is defined as ‘normal’, which is not affected by the two sensitivity coefficients, both of which are set to a value of 1. The modified IDM model considering reaction time and driver characteristics is expressed as:

$$a_h={\lambda }_n{a}_{\max }\left\{1-{\left({\displaystyle \frac{v}{v_f}}\right)}^4-{\left[{\displaystyle \frac{s_h+v\left({\omega }_n{t}_h+{\tau }_h\right)+v\Delta v{\left(2\sqrt{a_{\max }b}\right)}^{-1}}{x_h-l}}\right]}^2\right\}$$

(3)

In the above equation, $a_h$ is the acceleration of the HDV.

• $a_{\max }$ is the maximum acceleration, set to 1 $\mathrm{m}/{\mathrm{s}}^2$;
• $v_f$ is the free-flow velocity, set to 11.1 $\mathrm{m}/\mathrm{s}$;
• $s_h$ is minimum safe distance for HDV, set to 2 $\mathrm{m}$;
• $t_h$ is the desired time headway for HDV, set to 1.5 $\mathrm{s}$;
• ${\tau }_h$ is the reaction time for DCAV, set to 0.4 $\mathrm{s}$ [30];
• $b$ is the desired comfortable deceleration, set to 2.8 $\mathrm{m}/{\mathrm{s}}^2$;
• $x_h$ is the headway of HDV.

The parameters $l$, $v$ and $\Delta v$ have the same meanings as in Equation (1).

The values of the sensitivity ${\lambda }_n$ and ${\omega }_n$ coefficients [31] are shown in

Table 1. Table of sensitive coefficients ${\lambda }_n$ and ${\omega }_n$ with values and following styles.

Style of Car-Following	$$\mathbf{Maximum}\mathbf{Acceleration}\mathbf{Sensitivity}\mathbf{Factor}{\mathit{\lambda }}_{\mathit{n}}$$	$$\mathbf{Safety}\mathbf{Locomotive}\mathbf{Time}\mathbf{Distance}\mathbf{Sensitivity}\mathbf{Factor}{\mathit{\omega }}_{\mathit{n}}$$
Hesitant	0.97	1.91
Steady	1.31	1.30
Trust	1.70	0.65
Normal	1.00	1.00

:

3. Signalized Intersection Traffic Flow Model for Intelligent Connected Environment

Building upon the previously established multi-factor car-following model for mixed traffic flow, this study aims to analyze the impact of CAV on intersection efficiency within mixed traffic. Considering the characteristics of HDV when passing through intersections and the real-time traffic information reception capabilities of CAV, a decision-making model for intersection passage in mixed traffic flow is developed.

The study scenario is subject to the following constraints:

(1)

The research scope is limited to straight-through vehicles at a single intersection with fixed signal timing;

(2)

The effects of adjacent intersections, roadside parking, pedestrians, and bicycles are not considered;

(3)

CAV can obtain real-time road traffic information;

(4)

The wireless communication between vehicles and between vehicles and traffic signals is assumed to be reliable, with no delay in traffic information transmission.

3.1. Decision Area Division

(1)

For HDVs, based on their characteristics at intersections, the decision area is divided into free-driving and passing decision areas, constructing a model to predict HDV passage at intersections.

(2)

For CAVs, leveraging their ability to receive real-time traffic information, the decision area is divided into free-driving, platoon formation, and speed guidance areas, constructing a passage model for CAV at intersections. The decision zones for CAVs and HDVs are shown in Figure 2. And the roadside units depicted in the figure are capable of real-time information sharing with the intelligent networked system.

3.2. HDV Passing Model

(1)

Free-Driving Area Evolution Rules:

When the leading vehicle makes an emergency stop, the following vehicle updates its speed based on a minimum safe distance to avoid collision.

(2)

Passing Decision Area Evolution Rules:

Based on the vehicle’s status and traffic environment, we determined the maximum travel distance and classified the following scenarios: green light with long remaining time, green light with short remaining time, red light with long remaining time, and red light with short remaining time.

①

HDV as Lead Vehicle

Scenario a: The vehicle enters the intersection when the traffic light is green, and the remaining time is relatively long. During this green light period, it can pass the stop line, and a certain probability is given for accelerating or maintaining speed to pass the stop line.

Scenario b: The vehicle enters the intersection when the traffic light is green, and the remaining time is relatively short. During this green light period, it cannot pass the stop line. The vehicle accelerates or maintains speed until the next green light period to pass the stop line, usually choosing to accelerate.

Scenario c: The vehicle enters the intersection when the traffic light is red, and the remaining time is relatively long. It can reach the stop line when the red light ends and pass the stop line during the green light period.

Scenario d: The vehicle enters the intersection when the traffic light is red, and the remaining time is relatively short. It cannot reach the stop line when the red light ends and accelerates or maintains speed until the stop line.

②

HDV as the following vehicle

Introduce the HDV as the following vehicle’s driving style, and the driver follows the same pattern as the leading vehicle for decision making. The evolution rules of the HDV decision zones are illustrated in Figure 3.

3.3. CAV Passing Decision Model

(1)

Free-driving area evolution rules: CAV in the free-driving area is not influenced by traffic signals, applying only relevant car-following models.

(2)

Passing decision area evolution rules: Based on the vehicle’s status and interaction with the driving environment, we determined scenarios such as green light with long remaining time, green light with short remaining time, red light, and consider headway constraint to optimize speed guidance.

①

CAV as the leading vehicle

Scenario e: The vehicle enters the intersection when the traffic light is green and the remaining time is relatively long. During this green light phase, it can pass the stop line without exceeding the speed limit, thereby accelerating to provide the maximum passing time for subsequent vehicles.

Scenario f: The vehicle enters the intersection when the traffic light is green, and the remaining time is relatively short. During this green light phase, it cannot pass the stop line. If accelerating allows it to pass during this green light phase, it accelerates; if accelerating does not allow it to pass during this green light phase, it decelerates to pass the stop line just as the next green light phase starts.

Scenario g: The vehicle enters the intersection when the traffic light is red. If it can reach the stop line by the end of this phase, it decelerates to pass the stop line just as the next green light phase starts; if it cannot reach the stop line by the end of this phase, it accelerates.

②

CAV as the following vehicle

Scenario h: The vehicle enters the intersection when the traffic light is green and can pass the stop line during this green light phase. It determines whether accelerating is influenced by the leading vehicle; if not influenced, it accelerates; if influenced, it closely follows the leading vehicle.

Scenario i: The vehicle enters the intersection when the traffic light is red, introducing the queue dissipation time, and decelerates to pass the stop line just as the next green light phase starts.

The evolution rules of the decision zones for CAV intelligent traffic are illustrated in Figure 4.

All other situations must comply with the following two constraints:

①

Solve for the recommended acceleration based on the control strategy when the CAV is the leading vehicle.

②

To ensure driving safety, the solved acceleration must meet the constraints of the improved CACC and ACC car-following models on speed.

4. Speed Guidance Strategy Considering the Uncertainty of Leading Vehicles

During the speed guidance process, uncertainties caused by HDVs must be explicitly considered, including intersection queuing and disturbances from preceding HDV. Merely predicting their passage decisions at intersections is insufficient to determine the optimal speed for guiding the corresponding connected CAVs. The subsequent sections utilize optimal control theory and rolling optimization models to optimize and derive the target speeds under special conditions.

4.1. Guidance Mechanism

Based on vehicle speed, signal phase, signal cycle, and surrounding traffic environment information, recommended speeds or acceleration are provided to achieve speed guidance. The principles for improving traffic efficiency in this study are as follows:

(1)

If the guided vehicle cannot pass through the intersection at its current speed (Figure 5, Principle 1), it should be accelerated if it can pass the intersection within the current green light phase by accelerating. Conversely, if the vehicle cannot pass through the intersection even at the maximum speed limit, it should be guided to decelerate and wait for the next green light phase. The vehicle should decelerate to an appropriate speed to ensure it can pass the intersection at the beginning of the next green light phase.

(2)

If the guided vehicle can pass through the intersection at its current speed (Figure 5, Principle 2), it should be accelerated to pass through the intersection at the maximum speed, ensuring driving safety. This acceleration guidance aims to maximize the intersection’s throughput and allocate as much passing time as possible for subsequent vehicles.

4.2. Speed Guidance Model for Mixed Traffic Flow

To guide the CAV platoon in mixed traffic flow, the platoon is divided, and the headway is used to determine the platoon’s boundary. If the headway between adjacent vehicles is less than a standard value, the vehicles are classified as the same platoon; otherwise, they form different platoons.

When the vehicle platoon size is 1, the traffic strategy and expected acceleration are consistent with the algorithm of the CAV traffic model constructed in the previous chapter. When the vehicle platoon size is greater than 1, the entire platoon is regarded as a whole entity, and the strategy of leading one vehicle is used to lead multiple vehicles, reducing the complexity of the algorithm and improving the traffic efficiency. Using the car-following model constructed for CAV above, the target acceleration for the leading vehicle passing through the stop line is calculated. Subsequently, the speeds of the preceding vehicle and the improved CACC model are combined to keep a tight car-following distance, ensuring that as many vehicles in the platoon as possible pass the stop line. If a vehicle in the platoon cannot pass the stop line within the current green light phase, the platoon needs to be disbanded.

When the leading vehicle of the target platoon moves into the speed guidance area, the driving state of the vehicle platoon is represented as $P_{ij}\left(v_j,x_{ij},t_{r,g},t_{j-1,f}\right)$. Here, $v_{ij}$ $\left(j=1,2,3,\cdots Q\right)$ is the current speed of the vehicle in the $j$th platoon in the $i$th queue; $x_{ij}$ is the position of the vehicle in the $j$th platoon in the $i$th queue; “j” represents the number of vehicle platoons within a green light phase, with a minimum platoon size of 1. Therefore, the upper bound of “j” is the maximum number of vehicles, Q, that can pass through during a complete green light phase; $t_{r,g}$ is the remaining time of the current green light phase or the duration of the red light phase; $t_{j-1,f}$ $\left(j=1,2,3,\cdots Q\right)$ is the time required for the last vehicle platoon to pass completely. According to the traffic situation of the CAV at the signalized intersection discussed in the previous chapter, the traffic strategy of the target platoon passing through the intersection is divided into deceleration, acceleration, deceleration to a stop, and acceleration guidance.

(1)

Acceleration guidance strategy

When the target platoon is guided to accelerate and pass through the intersection, if the remaining time of the current green light phase is relatively long or the duration of the red light phase is relatively long, and there is no leading vehicle, or the leading vehicle does not hinder the acceleration of the target platoon, the maximum acceleration limit $v_{\max }$ of the following vehicle to the leading vehicle is calculated according to Formula (1), and the acceleration of the following vehicle is $a_{c,ij}\left(i=2,3,\cdots ,W_n;j=1,2,3,\cdots Q\right)$:

$$a_{c,ij}={\displaystyle \frac{k_p\left(x_{c,ij}\left(t\right)-s_c-l-\left(t_c+{\tau }_c\right)v_{ij}\left(t\right)\right)+k_d\left(v_{\max }-v_{ij}\left(t\right)\right)}{\Delta t+k_d{t}_c}}$$

(4)

Let the platoon size be denoted as $W_n$, where the minimum platoon size is 1 and its unit is a car. And the maximum platoon size is the maximum number of vehicles passing through within one complete green light phase, denoted as Q. Therefore, the upper bound of j is Q. $x_{c,ij}\left(t\right)$ is the position of the vehicle in the $j$th platoon in the $i$th queue at the time of $t$; $v_{ij}(t)$ is the current speed of the vehicle in the $j$th platoon in the $i$th queue at the time of $t$; all other parameters, $k_p,k_d,t_c,{\tau }_c,s_c,l,v_{\max },\Delta t$, can be found in Equations (1)–(3).

The best acceleration of the rear car is:

$$a_{c,ij}^{*}=\min \left[a_{c,ij},a_{\max }\right]>0$$

(5)

$a_{c,ij}^{*}$ in represents the best acceleration between the acceleration of the rear vehicle and the maximum acceleration as the best acceleration of the rear vehicle.

The time $t_{1j}\left(j=1,2,3,\cdots Q\right)$ for the leading vehicle to pass the stop line shall meet the following constraints:

$$t_{1j}=\{\begin{array}{l}{\tau }_d+{\displaystyle \frac{v_{\max }-v_{1j}}{a_{\max }}}+{\displaystyle \frac{L_g-v_{1j}{\tau }_d-\frac{v_{\max }^2-{\left(v_{1j}\right)}^2}{2a_{\max }}}{v_{\max }}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{HDV}\\ {\tau }_c+{\displaystyle \frac{v_{\max }-v_{1j}}{a_{\max }}}+{\displaystyle \frac{L_g-v_{1j}{\tau }_c-\frac{v_{\max }^2-{\left(v_{1j}\right)}^2}{2a_{\max }}}{v_{\max }}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{CAV}\end{array}$$

(6)

$$t_{1j}\in \{\begin{array}{ll}(0,t_{r,g}]\hfill & \mathrm{Green}\mathrm{light},\mathrm{no}\mathrm{traffic}\mathrm{at}\mathrm{signal}\hfill \\ \left[t_{j-1}+h_s,t_{r,g}\right]\hfill & \mathrm{Green}\mathrm{light},\mathrm{vehicles}\mathrm{present}\mathrm{at}\mathrm{signal}\hfill \\ \left[t_{r,g},t_{r,g}+T_{green}\right]\hfill & \mathrm{Red}\mathrm{light},\mathrm{no}\mathrm{traffic}\mathrm{at}\mathrm{signal}\hfill \\ \left[t_{j-1}+h_s,t_{r,g}+T_{green}\right]\hfill & \mathrm{Red}\mathrm{light},\mathrm{vehicles}\mathrm{present}\mathrm{at}\mathrm{signal}\hfill \end{array}$$

(7)

$v_{\max }$ is the maximum speed limit; ${\tau }_d$ and ${\tau }_c$ are the reaction times for DCAV and CAV, respectively; $t_{r,g}$ is the remaining time of the current green light phase or the duration of the red light phase; $L_g$ is the control range of speed guidance areas; $T_{green}$ is the total duration of the green light phase for the intersection’s traffic signal; and $h_s$ is the standard headway. $t_{j-1}$ $\left(j=1,2,3,\cdots Q\right)$ is the time required for the last vehicle platoon to pass completely. All other parameters, $v_{\max },a_{\max },{\tau }_d,{\tau }_c$, can be found in Equations (1)–(4).

The time $t_{2j}\left(j=1,2,3,\cdots Q\right)$ for the following vehicles in the platoon to cross the stop line should satisfy the following constraints:

$$t_{2j}\in \{\begin{array}{ll}(0,t_{r,g}-h_s]\hfill & \mathrm{Green}\mathrm{light}\mathrm{phase}\hfill \\ \left[t_{r,g},t_{r,g}+T_{green}-h_s\right]\hfill & \mathrm{Red}\mathrm{light}\mathrm{phase}\hfill \end{array}$$

(8)

$$t_{2j}={\tau }_c+{\displaystyle \frac{v_{\max }-v_{ij}}{a_{c,ij}^{*}}}+{\displaystyle \frac{L_g+x_{ij}-v_{ij}{\tau }_c-\frac{v_{\max }^2-{\left(v_{ij}\right)}^2}{2a_{c,ij}^{*}}}{v_{\max }}}$$

(9)

All parameters can be found in Equations in the preceding text.

(2)

The deceleration guidance strategy

When the target platoon enters the speed guidance area, when the remaining time of the green light phase is shorter or the remaining time of the red light phase is longer, guide the platoon leader to slow down to the target speed $v_{1j}^{*}\left(v_{\min }<v_{1j}^{*}<v_{\max },j=1,2,3,\cdots Q\right)$ to achieve the purpose of the green light phase just starting when the leading vehicle drives to the stop line, then the time $t_{1j}\left(j=1,2,3,\cdots Q\right)$ through the stop line shall meet the following constraints:

$$t_{1j}=\{\begin{array}{l}{\tau }_d+{\displaystyle \frac{v_{1j}^{*}-v_{1j}}{b}}+{\displaystyle \frac{L_g-v_{1j}{\tau }_d-\frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{1j}\right)}^2}{2b}}{v_{1j}^{*}}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{HDV}\\ {\tau }_c+{\displaystyle \frac{v_{1j}^{*}-v_{1j}}{b}}+{\displaystyle \frac{L_g-v_{1j}{\tau }_c-\frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{1j}\right)}^2}{2b}}{v_{1j}^{*}}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{CAV}\end{array}$$

(10)

$$t_{1j}\in \{\begin{array}{ll}(t_{r,g}+T_{red},t_{r,g}+T_{red}+T_{green}]& \mathrm{Green}\mathrm{light},\mathrm{no}\mathrm{traffic}\mathrm{at}\mathrm{signal}\\ \left[t_{j-1}+h_s,t_{r,g}+T_{red}+T_{green}\right]& \mathrm{Green}\mathrm{light},\mathrm{vehicles}\mathrm{present}\mathrm{at}\mathrm{signal}\\ \left[t_{r,g},t_{r,g}+T_{green}\right]& \mathrm{Red}\mathrm{light},\mathrm{no}\mathrm{traffic}\mathrm{at}\mathrm{signal}\\ \left[t_{j-1}+h_s,t_{r,g}+T_{green}\right]& \mathrm{Red}\mathrm{light},\mathrm{vehicles}\mathrm{present}\mathrm{at}\mathrm{signal}\end{array}$$

(11)

In these formulas, $T_{red}$ is the total duration of the red light phase at the intersection signal (including the yellow light’s duration), and all the other parameters, $v_{\max },a_{\max },{\tau }_d,{\tau }_c,\mathrm{b}$, can be found in Equations in the preceding text.

From Formula (4), the following deceleration of the car is $b_{c,ij}$ $\left(j=1,2,3,\cdots Q\right)$, and the best deceleration $b_{c,ij}^{*}$ of the car is:

$$b_{c,ij}^{*}=\min \left[\left|b_{c,ij}\right|,\left|b\right|\right]$$

(12)

The min represents that the best deceleration of the rear car is the minimum value between the two.

The time $t_{2j}\left(j=1,2,3,\cdots Q\right)$ through the rear line in the platoon shall meet the following constraints:

$$t_{2j}={\tau }_c+{\displaystyle \frac{v_{1j}^{*}-v_{ij}}{b_{c,ij}^{*}}}+{\displaystyle \frac{L_g+x_{ij}-v_{ij}{\tau }_c-\frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{1j}\right)}^2}{2b_{c,ij}^{*}}}{v_{1j}^{*}}}$$

(13)

$$t_{2j}\in \{\begin{array}{ll}(0,t_{r,g}+T_{red}+T_{green}-h_s]& \mathrm{green}\mathrm{phase}\\ \left[t_{r,g},t_{r,g}+T_{green}-h_s\right]& \mathrm{red}\mathrm{phase}\end{array}$$

(14)

(3)

Acceleration and deceleration guidance strategy

When vehicle $k$ in platoon $j$ cannot keep up with the front portion of the platoon through the intersection, the platoon is divided into two segments, with vehicle $k-1$ marking the end of the front segment and vehicle $k$ leading the rear segment. Acceleration or deceleration guidance is applied to the front segment vehicles, whereas the rear segment vehicles undergo deceleration guidance. This approach constitutes a variable acceleration–deceleration guidance strategy.

The variable acceleration–deceleration guidance strategy is applicable in the following two scenarios:

When the target platoon enters the speed guidance zone and the signal light is green, with a preceding platoon $j-1$ moving steadily in front of the signal, the acceleration process of some vehicles in the target platoon is affected during their attempt to proceed.

Upon the target platoon’s entry into the speed guidance zone under a red signal light, there is a relatively short remaining duration and a queued platoon $j-1$ waiting ahead of the signal.

In these scenarios, the guidance strategy involves the front portion of the platoon either accelerating or decelerating to align with the tail of platoon $j-1$, following it through the green light phase to cross the stop line. Meanwhile, the remaining vehicles decelerate towards the stop line, preparing to proceed during the subsequent green light phase. The target speed for the lead vehicle of the preceding platoon $j-1$ is set in accordance with the speed $v_{j-1}\left(j=1,2,3,\cdots Q\right)$ of platoon $j-1$, so:

$$\{\begin{array}{l}{v}_{j-1}{\tau }_d+{\displaystyle \frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{ij}\right)}^2}{2d_{1j}}}+v_{j-1}{t}_f+s_d\ge L_g+x_s,\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{HDV}\\ v_{j-1}{\tau }_c+{\displaystyle \frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{ij}\right)}^2}{2d_{1j}}}+v_{j-1}{t}_f+s_c\ge L_g+x_s,\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{CAV}\end{array}$$

(15)

In the formula: $x_s$ is the standard front spacing, take 15 $\mathrm{m}$; $d_{ij}$ represents the acceleration (or deceleration) of the ith vehicle in the jth platoon; $t_f$ is the average dissipation time for vehicles, and all the other parameters can be found in Equations in the preceding text.

From Formula (4), the best additional (decreased) speed of the rear car in the previous platoon is $d_{ij}^{*}$ $\left(j=1,2,3,\cdots Q\right)$, so:

$$\begin{array}{l}{\displaystyle \frac{{\left(v_{ij}\right)}^2}{2x_c}}\le d_{ij}^{*}\le \left|d_{ij}^{*}\right|\le a_{\max }\end{array}$$

(16)

①

Guide the front part of the platoon of vehicles to accelerate

When guiding the leading vehicle of the front part of the target platoon to accelerate, the time $t_{1j}\left(j=1,2,3,\cdots Q\right)$ through the stop line shall meet the following constraints:

$$t_{1j}=\{\begin{array}{l}{\tau }_d+{\displaystyle \frac{v_{j-1}-v_{1j}}{d_{1j}^{*}}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{HDV}\\ {\tau }_c+{\displaystyle \frac{v_{j-1}-v_{1j}}{d_{1j}^{*}}},\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{CAV}\end{array}$$

(17)

$$t_{1j}\in \{\begin{array}{ll}(t_f+h_s,t_{r,g}]& \mathrm{green}\mathrm{phase}\\ \left[t_f+h_s,t_{r,g}+T_{green}\right]& \mathrm{red}\mathrm{phase}\end{array}$$

(18)

The time $t_{2j}\left(j=1,2,3,\cdots Q\right)$ for the rear car in the platoon to pass through the stop line shall meet the following constraints:

$$t_{2j}={\tau }_c+{\displaystyle \frac{v_{j-1}-v_{ij}}{d_{ij}^{*}}}+{\displaystyle \frac{L_g+x_{ij}-v_{ij}{\tau }_c-\frac{{\left(v_{j-1}\right)}^2-{\left(v_{1j}\right)}^2}{2d_{ij}^{*}}}{v_{j-1}}}$$

(19)

$$t_{2j}\in \{\begin{array}{ll}(0,t_{r,g}-h_s]& \mathrm{green}\mathrm{phase}\\ \left[t_{r,g},t_{r,g}+T_{green}-h_s\right]& \mathrm{red}\mathrm{phase}\end{array}$$

(20)

The maximum number of vehicles, $N_{k-1}$, in the target platoon that can accelerate through the stop line is:

$$N_{k-1}=\lfloor {\displaystyle \frac{t_{r,g}-t_f-\left(k-2\right){\tau }_c-{\tau }_d}{h_s}}\rfloor$$

(21)

②

After guiding part of the platoon of vehicles to slow down

When guiding part of the target platoon to slow down, guide the vehicles to slow down to the target speed $v_{1j}^{*}\left(v_{\min }<v_{1j}^{*}<v_{\max },j=1,2,3,\cdots Q\right)$ to achieve the green light phase, which just begins when the vehicle starts at the stop line. Then, the time of some vehicles in the target platoon $t_{ij}\left(i=k,\cdots ,W_n,j=1,2,3,\cdots Q\right)$ passing the stop line should meet the following constraints:

$$t_{ij}={\tau }_c+{\displaystyle \frac{v_{1j}^{*}-v_{ij}}{d_{ij}^{*}}}+{\displaystyle \frac{L_g+x_{ij}-v_{ij}{\tau }_c-\frac{{\left(v_{1j}^{*}\right)}^2-{\left(v_{1j}\right)}^2}{2d_{ij}^{*}}}{v_{1j}^{*}}}+h_s$$

(22)

$$t_{ij}\in \{\begin{array}{l}(t_{r,g}+T_{red},t_{r,g}+T_{green}+T_{red}]\hspace{1em}\hspace{1em}\mathrm{Green}\mathrm{phase}\\ \left[{\tau }_{r,g}+{{\rm T}}_{green}+{{\rm T}}_{red},{\tau }_{r,g}+2{{\rm T}}_{green}+{{\rm T}}_{red}\right]\mathrm{Red}\mathrm{phase}\end{array}$$

(23)

(4)

Slow down and stop guidance strategy

When the target platoon enters the speed guidance area, the signal light is red or green. Because there are many vehicles queuing in front of the signal light, no vehicles in the platoon can idle at the parking line. Currently, to avoid emergency braking of the vehicle, the optimal reduction in speed of the first car $b_{1j}\left(j=1,2,3,\cdots Q\right)$ is:

$$b_{1j}=\{\begin{array}{l}{\left(v_{1j}\right)}^2/2\left(L_g-v_{1j}{\tau }_d\right),\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{HDV}\\ {\left(v_{1j}\right)}^2/2\left(L_g-v_{1j}{\tau }_c\right),\mathrm{Front}\mathrm{vehicle}\mathrm{type}\mathrm{is}\mathrm{CAV}\end{array}$$

(24)

The optimal reduction in speed of the rear car $b_{ij}\left(i=2,3,\cdots W_n,j=1,2,3,\cdots Q\right)$ in the platoon meets the following constraints:

$$b_{ij}={\displaystyle \frac{{\left(v_{1j}\right)}^2}{2\left(L_g-x_{ij}-\left(i-1\right)x_s-v_{1j}{\tau }_c\right)}}$$

(25)

$$\{\begin{array}{ll}{t}_{r,g}<{\tau }_c-{\displaystyle \frac{v_{ij}}{b_{ij}}}+\left(i-1\right)h_s<t_{r,g}+T_{red}& \mathrm{green}\mathrm{phase}\\ t_f+h_s\le {\tau }_c-{\displaystyle \frac{v_{ij}}{b_{ij}}}+\left(i-1\right)h_s<t_{r,g}+T_{red}+T_{green}& \mathrm{red}\mathrm{phase}\\ {\left(v_{ij}\right)}^2/2x_c\le b_{ij}& \end{array}$$

(26)

Then, the optimal speed reduction rate $b_{ij}^{*}$$\left(j=1,2,3,\cdots Q\right)$ is:

$$\left|b_{ij}^{*}\right|=\min \left[\left|b_{ij}\right|,a_{\max }\right]$$

(27)

The min represents that the optimal speed reduction is the minimum value between the two.

4.3. Optimal Speed Adjustment Method under Uncertain Disturbances

During the speed guidance process, uncertainties introduced by HDVs necessitate explicit consideration of queuing at intersections and disturbances caused by the lead HDV. Relying solely on predictions of their crossing decisions at intersections is insufficient to derive the optimal guided speeds for the corresponding CAVs. Hence, this study employs optimal control theory and a rolling optimization model to refine the process, yielding target speeds tailored to these exceptional circumstances.

Considering the impact of HDV on speed guidance, the sources of uncertainty are categorized into two driving scenarios: intersection queuing constraints (Figure 6) and HDV-following constraints (Figure 7), as depicted in the following figures.

(1)

Optimal Control under Queue Constraints

When CAVs approach a signalized intersection with queued traffic (depicted in Figure 6), their progression through the intersection is contingent upon the dissipation of the queue ahead, waiting for the signal to permit passage. Consequently, in this driving scenario, the guidance of CAVs is influenced by the dynamics of the downstream queue dispersion. Apart from the constraints outlined previously, an additional path constraint comes into play in this context, which can be stated as follows:

$$x_i(\mathrm{t})\le s_{i,p}(\mathrm{t})$$

(28)

where $s_{i,p}(\mathrm{t})$ is the path constraint caused by the CAV $i$ to avoid the collision between the CAV and the queuing vehicles in front.

It can be seen from Figure 8 that, as long as the CAV $i$ does not cross the driving track $s_{i,p}(\mathrm{t})$ of the virtual vehicle, it cannot collide with the vehicle 3, so as to ensure the safe driving of the traffic flow. In order to determine the $s_{i,p}(\mathrm{t}\mathrm{t})$ curve part, the cumulative data of the queuing from the process of finite vehicles at the intersection to dissipation were collected, the trajectory data set corresponding to different vehicles $d+x_d$ passing through the intersection was established, and the polynomial function of N order was used to fit the following analytical formula:

$$s_{i,p}(\mathrm{t})=\max \left\{x_f-d-x_d,{\displaystyle {\displaystyle \sum _{j=0}^N}{k}_j{t}^{N-j}}\right\}$$

(29)

where: $k_j$ is the polynomial coefficient and N is the highest order of the polynomial.

After the curve $s_{i,p}(\mathrm{t})$ is determined, the time $t_r$ and speed $v_r$ of the virtual vehicle leaving the stop line can be introduced. In this study, the end speed $v_i$ (the speed when passing the stop line) under the queuing constraint is set to $v_r$ the smaller speed in the free flow speed $v_f$, and the end time $t_i$ (the time when passing the stop line) is set to $t_r$. At this point, the CAV $i$ can obtain the optimal driving trajectory according to the beginning and end conditions.

(2)

Velocity Rolling Optimization under Leading Vehicle Disturbances Ahead

When a CAV approaches an intersection and there is an HDV ahead of it (as depicted in Figure 7), uncertainties inherent in the behavior of the HDV prevent the CAV from accurately predicting the HDV trajectory or the time it will take to clear the intersection. Consequently, the CAV movement is constrained by the actions of the leading vehicle. In this scenario, the CAV must overcome the dynamic disturbances caused by the preceding HDV and adopt a path constraint that differs from those applied under queuing conditions. Specifically, the path constraint in this context is dynamically time-varying, implying:

$$x_i\left(t\right)\le x_{i-1}\left(t\right)+x_d$$

(30)

where: $x_i\left(t\right)$ is the real-time location of the current CAV and $x_{i-1}\left(t\right)$ is the real-time position of the front vehicle of the current CAV $i-1$ (the vehicle $i-1$ is a driving vehicle).

Since some of the dynamic path constraints $x_{i-1}\left(t\right)+x_d$ cannot be learned in advance, to determine the optimal speed trajectory of the CAV in this driving scenario, the control strategy was optimized through the speed rolling time domain in this study, and the principle is shown in Figure 9.

In this study, the idea of rolling time domain optimization for dynamic planning of the target vehicle speed trajectory, as shown in Figure 9, is used. Each moment $t_i^n$ is determined for a long time period $\left[t_i^n,t_r^n\right]$ as a time window, each short time period $t_s^n$ is determined as a sub-time window, the optimal control strategy at this time $t_s^n$ is determined and executed, and the calculation process is repeated ($t_i^{n+1}=t_i^n+t_s^n$) in the next moment, executed in the corresponding range until the CAV passes through the parking line.

In order to ensure the safety of the driving scenario, assuming that there is a CAV $i$ and artificial driving of an HDV in the moment $t_i^n$ and CAV $i$, and that they have the same speed and position $\left(v_i^n,x_i^n\right)$, by the second chapter, the ACC is improved using a car-following model to determine its speed track $A_{\mathrm{i}}^n\left(t\right)$. If the trajectory $A_{\mathrm{i}}^n\left(t\right)$ above the trajectory $R_{\mathrm{i}}^n\left(t\right)$ is set $R_{\mathrm{i}}^n\left(t\right)$ to the CAV speed trajectory limit, on the contrary, the trajectory $R_{\mathrm{i}}^n\left(t\right)$ above the trajectory $A_{\mathrm{i}}^n\left(t\right)$ will be set to the intelligently made vehicle speed trajectory limit, as shown in Figure 10. For CAV, the actual execution speed track is $E_{A,\mathrm{R}}^n\left(t\right)$:

$$E_{A,\mathrm{R}}^n\left(t\right)=\min \left[A_{\mathrm{i}}^n\left(t\right),R_{\mathrm{i}}^n\left(t\right)\right]$$

(31)

The min represents that the CAV’s actual execution speed track has the minimum value between the two.

The velocity rolling horizon optimization strategy is continually executed through iterative processes until the target CAV crosses the stop line or, during execution, it is detected that the preceding HDV has joined the queue in front of the stop line; in this next instance, the velocity control strategy is determined using the optimization control method tailored for queuing constraints.

5. Simulation Experiments and Numerical Results Analysis

To evaluate the effectiveness of the proposed speed guidance model, a simulation experiment was designed using Python and SUMO to construct the simulation scenario. By comparing parameters such as platoon delays and the number of stops before and after guidance, the impact of the speed guidance strategy under different traffic volumes and penetration rates was analyzed.

5.1. Experiment Setup

To evaluate the efficiency of urban signalized intersections, this study selected two indicators: average vehicle delay and average number of stops. These indicators were used to analyze the effectiveness of the speed guidance strategy proposed in this study.

(1)

Simulation Scenario Construction

Road Network: Utilizing the “netedit” tool, a city road network with a two-lane intersection was created. The road lengths for the east, south, west, and north approaches to the intersection were set to 300 m, 50 m, 50 m, and 50 m, respectively, with traffic signals installed at the intersection.

Vehicle Definition and Generation: Three types of vehicles were defined: CAV, DCAV, and HDV, each with a length of 5 m. At the beginning of the simulation, the generation probabilities of CAV and HDV were controlled to obtain traffic flows with different CAV penetration rates, ensuring a random distribution of CAV and HDV.

Simulation Process: The simulation step length was set to 1, updating vehicle positions and speeds within each time step.

(2)

Traffic Parameter Settings

Road Parameters: The maximum speed for all roads in the simulation was set to 11.1 m/s, with each lane width set to 3.5 m, the standard width for urban roads. To simulate realistic conditions at urban road intersections, white solid lines were placed within 60 m of the intersection to prohibit lane changes in this area.

Vehicle Behavior Model: Vehicles entered the road at random speeds with their departure positions set to zero. The car-following model used was the mixed traffic flow car-following model established in Section 2 Formulas (1)–(3),while the lane-changing model employed was the Krauss model.

Traffic Signal Configuration: Traffic signals in the simulation had fixed phases, with red light time, all-red time, green light time, and yellow light time set to 14 s, 3 s, 30 s, and 3 s, respectively. The simulation scenario is depicted as illustrated in Figure 11.

5.2. Results Analysis

Vehicles entered the road with random velocities in the experiments. To mitigate the effects of random errors, 20 consecutive experiments were conducted for each combination of traffic flow rate and CAV penetration level, with the data from these simulations being compiled for analysis.

(1)

Impact of Traffic Flow

With a CAV penetration rate of 40%, the lane traffic volume was set from 1000 veh/h to 2200 veh/h.

①

Average Number of Stops

②

Average delay

As shown in

Table 2. Comparison of the number of stops before and after guidance under varying traffic volumes.

Parking Times
	1000 veh/h		1200 veh/h		1400 veh/h		1600 veh/h		1800 veh/h		2000 veh/h		2200 veh/h
NO.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.
1	21	5	31	12	46	16	69	19	58	20	72	16	50	31
2	22	6	32	15	39	19	65	19	68	19	64	26	79	29
3	19	7	36	11	52	21	69	22	85	28	83	31	75	33
4	21	7	34	18	41	14	59	27	82	19	56	28	64	37
5	21	5	37	15	45	24	53	22	58	19	66	35	70	26
6	21	6	32	10	62	16	42	21	72	12	83	32	85	33
7	21	6	32	13	51	25	51	28	68	32	75	34	79	38
8	21	5	33	14	47	20	62	15	68	30	67	27	51	46
9	21	6	32	12	59	18	55	23	68	31	57	35	56	19
10	21	8	35	15	60	26	48	27	75	31	56	34	62	19
11	21	6	32	11	39	12	71	17	77	17	67	32	54	17
12	21	6	32	18	40	15	52	21	63	27	72	26	67	21
13	21	6	34	15	45	25	58	18	63	15	67	30	50	23
14	23	6	31	10	48	29	43	28	63	23	83	29	46	29
15	21	5	33	13	39	17	52	21	72	25	68	33	61	21
16	21	6	31	14	43	23	63	28	73	22	60	29	70	47
17	22	7	31	16	50	24	57	33	65	27	60	34	84	19
18	21	9	34	12	42	19	65	26	67	18	84	35	82	29
19	19	8	34	17	41	20	61	22	83	22	80	34	93	15
20	20	6	39	14	42	25	53	15	72	22	74	33	71	31
A.S	0.209	0.070	0.326	0.140	0.453	0.204	0.556	0.225	0.676	0.228	0.673	0.301	0.652	0.278

Note: NO. represents the number of simulations, N. represents before guidance, G. represents after guidance, and A.S represents the average number of stops.

and

Table 3. Comparison of delay before and after guidance under different flows.

Delay(s)
	1000 veh/h		1200 veh/h		1400 veh/h		1600 veh/h		1800 veh/h		2000 veh/h		2200 veh/h
NO.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.	N.	G.
1	205	54	278	119	361	86	459	85	385	85	507	133	325	136
2	206	53	283	103	352	92	490	83	452	84	419	159	474	134
3	193	69	311	86	403	109	430	88	483	148	486	189	418	227
4	207	66	305	119	292	93	411	149	523	75	420	195	394	201
5	206	39	343	139	359	105	391	119	371	73	454	154	412	160
6	205	50	282	87	392	142	282	73	457	80	484	224	459	166
7	204	63	283	95	421	99	334	86	372	182	445	194	479	166
8	197	50	295	115	317	106	423	68	452	126	443	182	380	141
9	183	49	276	119	309	92	336	109	355	96	441	191	410	134
10	206	79	316	103	372	123	299	157	477	143	441	232	418	130
11	205	43	285	86	398	107	453	90	523	78	452	234	390	116
12	207	54	288	119	342	121	338	70	370	115	506	161	437	141
13	204	46	292	139	363	118	397	81	380	81	450	198	323	127
14	219	53	270	87	299	104	265	88	325	96	475	218	321	146
15	186	59	296	95	424	92	373	99	383	129	387	219	418	159
16	186	52	272	115	378	98	361	102	476	106	448	214	351	211
17	208	75	277	114	319	103	376	150	470	130	448	232	457	160
18	204	74	311	114	420	99	432	96	394	80	568	154	507	135
19	193	67	302	99	326	106	379	127	413	108	441	191	455	111
20	184	53	340	120	354	121	332	69	486	69	460	185	433	149
A.D (s)	1.918	0.556	2.821	1.044	3.439	1.017	3.610	0.957	4.080	1.001	4.379	1.847	3.943	1.462

Note: NO. represents the number of simulations, N. represents before guidance, G. represents after guidance, and A.D represents the average delay.

, when the penetration rate was 40% and the traffic flow rates were 1000 veh/h, 1200 veh/h, 1400 veh/h, 1600 veh/h, 1800 veh/h, 2000 veh/h, and 2200 veh/h, the average numbers of stops before and after guidance decreased by 0.130, 0.180, 0.249, 0.34, 0.331, 0.396, and 0.374, respectively. The average delays decreased by 1.362 s, 1.777 s, 2.422 s, 2.653 s, 3.079 s, 2.532 s, and 2.481 s, respectively. Additionally, at a constant penetration rate, as the traffic volume increased, the reductions in average stops and average stop delay were more significant. However, when the volume reached a certain level, congestion intensified, and the effectiveness of the guidance gradually diminished.

(2)

Penetration Rate Impact Analysis

At a traffic flow of 1800 veh/h, the penetration rates of CAV varied, reaching 20%, 40%, 60%, and 80%.

①

Average Number of Stops

②

Average delay

From

Table 4. A comparison of the average number of stops before and after guidance across different penetration rates.

	Average Number of Stops
	20%		40%		60%		80%
NO.	N.	G.	N.	G.	N.	G.	N.	G.
1	69	12	58	20	59	12	43	5
2	82	44	68	19	62	22	52	1
3	84	34	85	28	63	20	59	6
4	89	35	82	19	60	13	52	2
5	69	28	58	19	54	13	58	4
6	58	45	72	12	55	5	51	0
7	78	51	68	32	65	14	46	5
8	67	22	68	30	49	8	55	3
9	95	14	68	31	53	20	58	7
10	86	15	75	31	57	17	51	4
11	65	15	77	17	61	9	45	2
12	91	54	63	27	56	14	52	5
13	84	61	63	15	61	21	50	9
14	76	21	63	23	56	14	52	6
15	81	26	72	25	51	4	50	2
16	81	31	73	22	69	14	56	11
17	69	33	65	27	59	5	56	6
18	75	41	67	18	64	17	62	3
19	91	36	83	22	51	14	47	6
20	96	31	72	22	58	8	46	1
A.S	0.793	0.324	0.700	0.229	0.581	0.132	0.520	0.044

Note: NO. represents the number of simulations, N. represents before guidance, G. represents after guidance, and A.S represents the average number of stops.

and

Table 5. Comparison of delay before and after guidance under different permeability rates.

	Delay(s)
	20%		40%		60%		80%
NO.	N.	G.	N.	G.	N.	G.	N.	G.
1	355	22	385	85	484	12	281	10
2	645	178	452	84	406	63	397	1
3	481	133	483	148	464	33	577	22
4	518	145	523	75	469	13	419	4
5	468	91	371	73	337	52	353	12
6	423	196	457	80	395	7	360	0
7	401	240	372	182	476	26	337	15
8	453	113	452	126	383	8	452	5
9	469	43	355	96	297	22	477	9
10	520	46	477	143	478	33	423	28
11	314	56	523	78	521	11	294	2
12	609	155	370	115	538	20	402	11
13	522	221	380	56	408	21	434	36
14	497	85	325	96	326	27	381	13
15	558	128	383	129	363	7	383	20
16	524	164	476	106	525	16	470	16
17	559	177	470	130	331	9	472	10
18	551	171	394	80	313	47	552	3
19	556	156	413	108	389	51	350	40
20	486	155	486	69	383	8	348	4
A.D (s)	4.954	1.337	4.273	1.029	4.143	0.243	4.081	0.130

Note: NO. represents the number of simulations, N. represents before guidance, G. represents after guidance, and A.D represents the average delay.

, it is evident that with the penetration rates of connected and autonomous vehicles (CAVs) set at 20%, 40%, 60%, and 80%, the average numbers of stops decreased by 0.469, 0.471, 0.449, and 0.476 times, respectively, after the implementation of guidance measures. Simultaneously, the average stopping delays were reduced by 3.617 s, 3.244 s, 3.900 s, and 3.951 s accordingly. Furthermore, when the traffic volume remained constant, higher penetration rates led to more significant reductions in both the average number of stops and the average stopping delay.

(3)

Visualization of Results

To illustrate these findings graphically, a set of simulation data closely approximating the mean stop counts was selected. Trajectory plots were then generated for differing traffic volumes and penetration rates, depicting the empty paths of each vehicle according to their respective positions over time. This visual representation offers a clear insight into how varying levels of CAV penetration, under consistent traffic conditions, impact vehicle movement patterns and efficiency.

As can be seen from Figure 12, when the penetration rate of connected autonomous vehicles was 40%, the spatio-temporal chart before and after the guidance was compared with that before the guidance. The trajectory trend after the guidance was relatively gentle, indicating that connected autonomous vehicles slow down in advance when approaching the intersection after being guided, which is most likely to avoid stopping and waiting in front of the signal light. Meanwhile, as for HDVs in the mixed traffic flow, under the influence of CAVs, some HDVs passed the intersection without stopping. Through the longitudinal comparison of the spatio-temporal trajectory diagram of different traffic flows, the position of the trajectory fluctuation before guidance gradually moved forward with the increase in traffic flow, which was due to the continuous increase in its queue length and slow walking length. Meanwhile, the trajectory fluctuation to the horizontal after guidance gradually decreased, which was due to the increasing number of HDVs being indirectly affected by the guided CAVs. Thus, the number of vehicle stops at the entire intersection was reduced. In Figure 12 and Figure 13 Lane “E0-1” represent lane 1; Lane “E0-0” represent lane 0.

As can be seen from Figure 13, when the flow rate was 1800 veh/h, through horizontal comparison of the spatio-temporal graph and trajectory diagram before and after guidance, the trajectory trend after guidance was obviously flat compared with that before guidance, for the same reasons as those shown Figure 10. Through the longitudinal comparison of the spatio-temporal trajectory diagram of different traffic flows, it was found that with the increase in the permeability of connected autonomous vehicles, the trajectory fluctuation to the horizontal before guidance was significantly reduced, while after guidance, the trajectory fluctuation trend was larger and gradually became flat. At this time, connected autonomous vehicles affect almost all HDVs.

To sum up, after guidance, the CAV slowed down near the intersection in advance to achieve the purpose of passing the intersection without stopping, and this indirectly affected the manual driving vehicles so that some HDVs also passed the intersection without stopping. In addition, with the increase in traffic volume, the number of artificially driven vehicles indirectly affected by connected autonomous vehicles increased, and with the increase in the penetration rate of connected autonomous vehicles, connected autonomous vehicles gradually occupied a dominant position, and the impact caused by connected autonomous vehicles in both cases also gradually increased. Currently, the effect of applying the speed guidance strategy in this study is better.

6. Discussion

Experiments have shown that the proposed speed guidance algorithm can improve intersection efficiency to a certain extent, exhibiting the following characteristics under different traffic volumes and penetration rates: when the CAV penetration rate is constant, as traffic volume increases and the average number of stops and the average stop delay decrease significantly, but as the volume reaches a certain level, the guidance effect gradually diminishes. When the mixed traffic flow volume is constant, the higher the CAV penetration rate, the better the guidance effect. The speed guidance scheme proposed in this study can be applied to each approach lane of the intersection to enhance the overall efficiency of the intersection, and its optimization effects can also be extended to a broader area to improve regional traffic efficiency.

The speed guidance algorithm proposed in this study guides the speed of CAV and indirectly affects HDV. When the CAV penetration rate is constant, as traffic volume increases, the number of guided CAV increases, indirectly affecting more HDV, which leads to a greater reduction in the average number of stops and average stop delay. However, when the traffic volume reaches a certain level, the degree of congestion increases. Although the number of guided CAV is high, the dense traffic flow on the road may prevent CAV from executing the guidance strategy (for example, if the guidance strategy is to accelerate, but due to vehicle queuing and small inter-vehicle distances, the safety distance constraint in the car-following model prevents the CAV from executing the acceleration command). When traffic volume suddenly decreases, the guidance strategy remains effective, but its impact may be diminished due to the sudden reduction in the number of CAVs. Conversely, when traffic volume suddenly increases to the point of congestion, vehicles queue and move slowly or even stop, rendering the speed guidance algorithm ineffective. When the mixed traffic flow volume is constant, a higher penetration rate of CAVs means that more vehicles in the mixed traffic flow are capable of providing guidance, resulting in a better guidance effect.

Although this study considers the uncertainty of leading vehicles and conducts extensive simulation experiments on the proposed guidance algorithm, there are currently no practical experiments, and the real-world applicability of the proposed speed guidance method has not been assessed. Additionally, this research is based on certain assumptions, whereas actual urban road traffic scenarios (including pedestrians, bicycles, and buses) are more complex. Furthermore, this study only addresses speed guidance, without involving issues related to vehicle route optimization and traffic signal timing adjustments.

Considering that the current research focuses on improving the traffic efficiency of urban road intersections and has not addressed the rapid transit needs of priority vehicles (e.g., ambulances, fire trucks, police cars), future research will analyze the impact of speed guidance measures on the environment, such as vehicle energy consumption and emissions. Additionally, further studies will be conducted on more traffic scenarios, such as rural roads, highways, and unsignalized intersections.

7. Conclusions

To leverage the advantages of CAV in mixed traffic flow and enhance intersection efficiency, this study thoroughly analyzes the car-following characteristics of mixed traffic and develops a speed guidance strategy. The main contributions and conclusions of this study are as follows. First, considering the different driving characteristics of CAV and HDV, the study incorporates reaction time and driver car-following behavior to construct a mixed-traffic-flow car-following model. This model improves the applicability of various car-following models in mixed traffic flow, aligning better with future realistic mixed traffic environments.

Additionally, based on real-time information such as vehicle speed, position, and traffic signal status, and considering the uncertainty factors of leading vehicles, a speed guidance algorithm is constructed. This algorithm enables vehicles to pass through intersections without stopping or with minimal stopping, thereby enhancing intersection efficiency. While the manuscript uses a crossroad as an example, this model is applicable to all signal-controlled intersections, whether they are crossroads, roundabouts, or “T” intersections. Furthermore, simulation experiments were conducted where detectors were set up to measure the delay time and the number of stops for vehicles at intersections under both guided and unguided conditions. These experiments verify the effectiveness of the proposed speed guidance algorithm. The results indicate that the proposed speed guidance algorithm can reduce the number of stops and delays for vehicles at intersections to a certain extent. Moreover, when the mixed traffic flow is constant, a higher penetration rate of CAVs leads to a better guidance effect.

Lastly, the research focuses on intersections outside urban areas, where the algorithm is applicable under free-flow and stable-flow conditions. However, it should be noted that when traffic volume increases to the point of causing congestion, the effectiveness of the guidance is significantly reduced.