HLCG: A Hierarchical Lane-Changing Gaming Decision Model for Heterogeneous Traffic Flow on Two-Lane Highways

Abstract

Discretionary lane-changing behavior is one of the most common highway operations, which seriously affects traffic efficiency and safety. Nowadays, connected and automated vehicles (CAVs) are advancing rapidly, though not yet fully widespread. As a result, a mixed traffic environment with traditional human-driven vehicles (HDVs) and CAVs will persist for the foreseeable future. To achieve effective automatic lane-changing maneuvers, it’s necessary to propose a lane-changing decision model for heterogeneous traffic flow on two-lane highways. This paper firstly extends longitudinal car-following models based on the intelligent driver model and lateral lane-changing models using quintic polynomial curves to accommodate heterogeneous traffic flow, and introduces hyperbolic tangent transition models as well as modified virtual vehicle models to enable HDVs and CAVs to navigate the hybrid conditions of car-following and lane-changing, respectively. Then, this paper designs a hierarchical lane-changing gaming (HLCG) framework based on Stackelberg game theory and Harsanyi transformation theory, including a CAV decision model integrating dynamic safety domains and optimal lane-changing trajectories, and an HDV decision model considering different human driving styles, with a comprehensive payoff function concerning collision safety, traffic efficiency, and ride comfort. Finally, to verify the overall performance of the proposed HLCG decision model, simulations under different traffic densities and CAV penetration rates are carried out. The results show that the cooperation between CAVs can improve collision safety, stability and ride comfort during lane changes. Moreover, the proposed HLCG framework outperforms traditional game theory in terms of traffic efficiency, safety, ride comfort and shockwave generation across different traffic conditions.

Discretionary lane change on highways is a common maneuver to obtain potential driving space for the subject vehicle (SV), but often has negative impacts on traffic efficiency and safety. Thus, coordination among connected and automated vehicles (CAVs) represents a cutting-edge solution for alleviating traffic accidents and congestion ( 1 ). However, because of current technological and economic constraints, CAVs cannot completely replace traditional human-driven vehicles (HDVs) in the near term. Consequently, a traffic environment comprising HDVs and CAVs will persist for the next 20−30 years ( 2 ). Therefore, how to mitigate the effects of discretionary lane-changing behaviors on heterogeneous traffic in the decision-making and trajectory-planning stage of CAVs becomes a key challenge to tackle.

Early studies focused on developing lane-changing decision models that mimic natural driving characteristics, with a particular emphasis on recognizing human decision-making and interactions. These approaches include rule-based models ( 3 ), discrete choice-based models ( 4 ), incentive-based models ( 5 ), and artificial intelligence-based models ( 6 ). Additionally, emerging driving environments like the connected environment, as well as the automated environment are gradually influencing lane-changing dynamics ( 7 ). However, in mixed traffic environments involving both HDVs and CAVs, the intricate interactions and limited accurate real-time information among vehicles may affect traffic efficiency and safety ( 8 ). Therefore, the decision-making and trajectory-planning of CAVs not only need to optimize local performance during lane changes, but also need to account for uncertain interactions with other traffic participants to achieve overall optimality.

Among classic lane-changing decision models, game theory-based models offer a nuanced perspective on interactions between HDVs and CAVs, revealing a more realistic image of driving behaviors compared with other traditional models ( 9 ). Yu et al. ( 10 ) proposed a game theory-based lane-changing model considering the interactions between CAVs and HDVs in mixed traffic scenarios, focusing on estimating the aggressiveness or cooperation intentions of the surrounding vehicles. Zhang et al. ( 11 ) improved the aggressiveness estimation method in Yu et al. ( 10 ), and developed a game-theoretic model predictive controller to address mandatory lane changes in mixed-driving environments. Yan et al. ( 12 ) proposed a multi-vehicle game-theoretic trajectory-planning framework for both HDVs and CAVs, incorporating non-cooperative interactions between CAVs and HDVs, as well as partial cooperative interactions between CAVs. Yu et al. ( 13 ) developed a multi-player lane-changing game theory model for CAVs to facilitate simultaneous lane-changing decision-making and trajectory-planning based on the status of the surrounding vehicles. In these papers, trajectories of the following vehicles on the adjacent lane were predicted through estimating their driving styles, and then CAVs interacted with them during lane changes. However, these studies often neglected the inaccurate motion predictions of the surrounding vehicles as a result of incomplete information of the mixed traffic, and also neglected the potential cooperation between CAVs when HDVs executed unexpected lane changes.

In discretionary lane changes, the decision-makers are the lane changers themselves, serving as the central unit of analysis for these maneuvers. Typically, interactions between vehicles on two-lane highways are modeled using a two-player Stackelberg game ( 14 ). Yu et al. ( 10 ) expanded on this by employing two-player Stackelberg game and three-player Stackelberg game. Wang et al. ( 15 ) created a highway-based two-vehicle Stackelberg game model for discretionary lane-changing decision in mixed traffic involving autonomous cars and trucks. Ji et al. ( 16 ) developed a decision-making strategy for self-driving vehicles merging into another lane in dense traffic, ensuring the manageable computation through comprehensive vehicle interactions. Given the complex characteristics of heterogeneous traffic during discretionary lane changes, Harsanyi theory is utilized to transform a static game with incomplete information into a game with complete but imperfect information ( 17 ), which can better capture the complex human behaviors and enhance the cooperation between CAVs and HDVs. Ji et al. ( 9 ) provided an overview of lane-changing behaviors in both traditional and connected environments, presenting a comprehensive game theory-based lane-changing decision model that considered all related factors (i.e., location prediction and time prediction error, false alarm rate, and detection rate). Talebpour et al. ( 18 ) differentiated between mandatory and discretionary lane changes, applying Harsanyi transformation theory to address games of incomplete information, thereby improving the reliability and safety of lane-changing maneuvers. While existing studies often focused on multi-vehicle cooperation and trajectory-planning in either purely traditional or connected environments, they frequently overlooked interactions between CAVs and HDVs. In addition, although global traffic efficiency is the ultimate optimization goal, meeting local performance remains fundamental to lane-changing behaviors. Besides, hardly any existing research on discretionary lane-changing decision modeling simultaneously took both local and global performance of heterogeneous traffic flow into consideration.

According to the analysis above, this paper makes three contributions to the existing research. Firstly, this paper modifies the longitudinal car-following models and lateral lane-changing models to accurately capture the distinct driving characteristics of both HDVs and CAVs. Then, this paper introduces transition models as well as modified virtual vehicle models to ensure HDVs and CAVs applicable for hybrid conditions during lane changes, while comprehensively considering human factors and cooperative adaptive cruise control (CACC) technologies. Secondly, this paper proposes a hierarchical lane-changing gaming (HLCG) framework based on Stackelberg game theory and Harsanyi transformation theory, with a comprehensive payoff function concerning collision safety, traffic efficiency, and ride comfort. It includes a CAV gaming decision model combining dynamic safety domains and optimal trajectories, and an HDV gaming decision model considering different human driving styles. Finally, this paper conducts extensive simulations, where it evaluates local performance of five-key-vehicle scenarios involving different vehicle types and extreme conditions, as well as global performance of mixed traffic flow across varying CAV penetration rates and traffic densities. These simulations assess the positive impacts of CAVs on overall performance of the mixed traffic flow on two-lane highways and verify the advantages of the proposed HLCG decision model over traditional game theory with regard to traffic efficiency, safety, ride comfort, and shockwave generation.

The remainder of this paper is organized as follows: the second section develops the microscopic heterogeneous traffic flow models. The third section details the design of the proposed HLCG decision model. The fourth section presents comprehensive local and global simulation analyses. Finally, the fifth section offers a concise summary and outlines potential research prospects.

Heterogeneous Traffic Flow

This paper focuses on the discretionary lane-changing maneuvers of HDVs and CAVs in the two-lane highway scenario (Figure 1). The traffic conditions consist of SV, the vehicle in the front of SV in the same lane (VFS), the vehicle in the front of SV in the target lane (VFT), the vehicle in the rear of SV in the same lane (VRS), and the vehicle in the rear of SV in the target lane (VRT). Driving behaviors on two-lane highways generally consist of two maneuvers: longitudinal car-following maneuvers and lateral lane-changing maneuvers.

Figure 1.

Discretionary lane-changing maneuvers in the two-lane highway scenario.

Longitudinal Car-Following Model

Based on previous simulation analysis ( 19 ), the intelligent driver model (IDM) demonstrates better performance in the two-lane highway scenario. Therefore, in this paper, the longitudinal car-following models of HDVs and CAVs are both derived from IDM. The expression equation of IDM is presented in Equation 1.

$$\{\begin{array}{c}\multicolumn{1}{c}{a_n(s_n,v_n,\mathrm{\Delta }{v}_n)=a(1-{\left(\frac{v_n}{v_0}\right)}^{\delta }-{\left(\frac{s^{*}(v_n,\mathrm{\Delta }{v}_n)}{v_0}\right)}^2)}\\ \multicolumn{1}{c}{s^{*}(v_n,\mathrm{\Delta }{v}_n)=s_0+\max (0,v_n·T_s+\frac{v_n·\Delta v_n}{2\sqrt{\mathrm{ab}}})}\end{array}$$

(1)

where $a_n$ is the desired acceleration of SV; $s_n$ is the actual car-following distance between SV and its preceding vehicle; $v_n$ is the actual speed of SV; $\mathrm{\Delta }{v}_n$ means the actual speed difference between SV and its preceding vehicle; $a$ means the maximum desired acceleration; $v_0$ represents the desired speed of SV; $\delta$ is the acceleration index; $s^{*}(v_n,\mathrm{\Delta }{v}_n)$ is the desired car-following distance; $s_0$ is the static safety distance; $T_s$ is the safety time headway of SV; and $b$ is the comfort deceleration.

Human-Driven Vehicle

In previous studies ( 20 – 25 ), IDM was utilized to describe the longitudinal car-following behaviors of HDVs. This paper introduces driver’s reaction time and estimation error to the HDV longitudinal car-following model to account for the longer reaction delay and larger perception error of HDVs compared with CAVs. The model equation incorporating the reaction delay time is expressed in Equation 2.

$$a_n=f(s_n(t-{\tau }_{s_n}),v_n(t-{\tau }_{v_n}),\mathrm{\Delta }{v}_n(t-{\tau }_{\mathrm{\Delta }{v}_n}))$$

(2)

where ${\tau }_{s_n}$ , ${\tau }_{v_n}$ , and ${\tau }_{\mathrm{\Delta }{v}_n}$ are the reaction time with respect to the actual car-following distance, the actual speed of SV and the actual speed difference between SV and its preceding vehicle, respectively.

The estimation error is calculated by a Wiener process. The estimated car-following distance and the estimated speed difference between SV and its preceding vehicle are expressed in Equation 3.

$$\{\begin{array}{c}\multicolumn{1}{c}{s_n^{\mathrm{est}}\left(t\right)=s_n\left(t\right)e^{v_s{w}_s\left(t\right)}}\\ \multicolumn{1}{c}{\Delta v_n^{\mathrm{est}}\left(t\right)=\Delta v(t)+s_n(t){\sigma }_r{w}_{\Delta v_n}(t)}\end{array}$$

(3)

where $s_n^{\mathrm{est}}$ is the estimated car-following distance; $v_s$ is the relative standard deviation of the estimated car-following distance from the actual car-following distance; $\mathrm{\Delta }{v}_n^{\mathrm{est}}$ is the estimated speed difference between SV and its preceding vehicle; ${\sigma }_r$ means the constant standard deviation of relative approach rate; and $w_{s_n}(t)$ and $w_{\mathrm{\Delta }{v}_n}(t)$ are both random variables normally distributed at (0, 1) to depict the variation over time of the estimation error of the car-following distance and the speed, respectively.

In summary, the HDV longitudinal car-following model considering driver’s reaction time and estimation error is expressed in Equation 4.

$$a_n^{\mathrm{HDV}}=f(s_n^{\mathrm{est}}(t-{\tau }_{s_n}),v_n(t-{\tau }_{v_n}),\mathrm{\Delta }{v}_n^{\mathrm{est}}(t-\Delta {\tau }_{v_n}))$$

(4)

Connected and Automated Vehicle

This paper defines CAVs as adaptive cruise control (ACC) vehicles, integrating the constant acceleration heuristic into IDM to formulate the CAV longitudinal car-following model ( 26 ).

The maximum acceleration leading to no crashes is given by Equation 5.

$$a_{\mathrm{CAH}}(s_n,v_n,\mathrm{\Delta }{v}_n,a_{n-1})=\{\begin{array}{c}\multicolumn{1}{c}{\frac{v_n^2{\tilde{a}}_{n-1}}{{(v_n-\mathrm{\Delta }{v}_n)}^2-2s_n{\tilde{a}}_{n-1}},(v_n-\mathrm{\Delta }{v}_n)\mathrm{\Delta }{v}_n\le 2s_n{\tilde{a}}_{n-1}}\\ \multicolumn{1}{c}{{\tilde{a}}_{n-1}-\frac{\mathrm{\Delta }{v}_n^2\Theta (\mathrm{\Delta }{v}_n)}{2s_n},(v_n-\mathrm{\Delta }{v}_n)\mathrm{\Delta }{v}_n>2s_n{\tilde{a}}_{n-1}}\end{array}$$

(5)

where $a_{\mathrm{CAH}}$ is the maximum acceleration leading to no crashes; $a_{n-1}$ is the acceleration of the preceding vehicle; ${\tilde{a}}_{n-1}$ satisfies ${\tilde{a}}_{n-1}=\min (a_{n-1},a)$ ; and $\Theta$ means the Hoviside function, eliminating the effects of negative approach rate on $a_{\mathrm{CAH}}$ .

In summary, the CAV longitudinal car-following model with ACC function is expressed in Equation 6.

$$a_n^{\mathrm{CAV}}=a_{\mathrm{ACC}}=\{\begin{array}{cc}\multicolumn{1}{c}{a_{\mathrm{IDM}},}& \multicolumn{1}{c}{a_{\mathrm{CAH}}\le a_{\mathrm{IDM}}}\\ \multicolumn{1}{c}{(1-c)a_{\mathrm{IDM}}+c(a_{\mathrm{CAH}}+b\tanh \left(\frac{a_{\mathrm{IDM}}-a_{\mathrm{CAH}}}{b}\right)),}& \multicolumn{1}{c}{a_{\mathrm{CAH}}>a_{\mathrm{IDM}}}\end{array}$$

(6)

where $a_{\mathrm{ACC}}$ is the acceleration of the ACC model; $a_{\mathrm{IDM}}$ is the acceleration of the traditional IDM; and c is the cooling factor.

Lateral Lane-Changing Model

Drawing from prior research ( 27 ), the quintic polynomial curve can satisfy the constraint conditions of vehicle motions at the initial and final states of lane change. Suppose the expression of lane-changing trajectory using the quintic polynomial curve shown in Equation 7.

$$y\left(x\right)={\omega }_0+{\omega }_{1}x+{\omega }_2{x}^2+{\omega }_3{x}^3+{\omega }_4{x}^4+{\omega }_5{x}^5,x\in [x_0,x_f]$$

(7)

where $y$ is the lateral displacement of SV; ${\omega }_i$ is the polynomial corresponding coefficient; $x$ is the longitudinal displacement of SV; $x_0$ is the longitudinal displacement at the beginning of lane change; and $x_f$ is the longitudinal displacement at the end of lane change.

The boundary conditions of the quintic polynomial trajectory are set in Equation 8.

$$\{\begin{array}{c}\multicolumn{1}{c}{y\left(x_0\right)=0,y\left(x_f\right)=y_f}\\ \multicolumn{1}{c}{y^{\prime }\left(x_0\right)=u\tan {\theta }_0,y^{\prime }\left(x_f\right)=0}\\ \multicolumn{1}{c}{y^{″}\left(x_0\right)=a_{y_0},y^{″}\left(x_f\right)=0}\end{array}$$

(8)

where $y_f$ is the lateral displacement at the end of lane change; $u$ is the longitudinal speed of SV at the beginning of lane change; ${\theta }_0$ is the yaw of the vehicle at the beginning of lane change; and $a_{y_0}$ is the lateral acceleration of the vehicle at the beginning of lane change.

Substituting Equation 8 into Equation 7 yields the coefficients of quintic polynomial lane-changing trajectory function, as shown in Equation 9.

$$\{\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}\multicolumn{1}{c}{{\omega }_0=0}\\ \multicolumn{1}{c}{{\omega }_1=\tan {\theta }_0}\end{array}}\\ \multicolumn{1}{c}{{\omega }_2=\frac{a_{y_0}}{2u^2}}\end{array}}\\ \multicolumn{1}{c}{{\omega }_3=\frac{10y_f}{x_f^3}-\frac{6\tan {\theta }_0}{x_f^2}-\frac{3a_{y_0}}{2u^2{x}_f}}\\ \multicolumn{1}{c}{{\omega }_4=-\frac{15y_f}{x_f^4}+\frac{8\tan {\theta }_0}{x_f^3}+\frac{3a_{y_0}}{2u^2{x}_f^2}}\\ \multicolumn{1}{c}{{\omega }_5=\frac{6y_f}{x_f^5}-\frac{3\tan {\theta }_0}{x_f^4}-\frac{a_{y_0}}{2u^2{x}_f^3}}\end{array}$$

(9)

Human-Driven Vehicle

This paper introduces the HDV longitudinal car-following model in Equation 4 to the HDV lateral lane-changing model, as is expressed in Equation 10.

$$\{\begin{array}{c}\multicolumn{1}{c}{x^{\mathrm{HDV}}=x_0+u(t-t_0)+{\int }_{t_0}^{t_f}({\int }_{t_0}^{t_f}{a}_n^{\mathrm{HDV}}(t)\mathrm{dt})\mathrm{dt}}\\ \multicolumn{1}{c}{y^{\mathrm{HDV}}=y\left(x^{\mathrm{HDV}}\right)}\end{array}$$

(10)

where $x^{\mathrm{HDV}}$ and $y^{\mathrm{HDV}}$ are the longitudinal and lateral displacement of HDV, respectively; $t_0$ is the beginning moment of lane change; and $t_f$ is the end moment of lane change.

Referring to the lane-changing time on the common high-speed condition ( 28 ), this paper sets the HDV lane-changing time (t_f - t₀) as a constant, which is 4 s.

Connected and Automated Vehicle

This paper introduces the CAV longitudinal car-following model in Equation 6 to the CAV lateral lane-changing model in Equation 11.

$$\{\begin{array}{c}\multicolumn{1}{c}{x^{\mathrm{CAV}}=x_0+u(t-t_0)+{\int }_{t_0}^{t_f}({\int }_{t_0}^{t_f}{a}_n^{\mathrm{CAV}}(t)\mathrm{dt})\mathrm{dt}}\\ \multicolumn{1}{c}{y^{\mathrm{CAV}}=y\left(x^{\mathrm{CAV}}\right)}\end{array}$$

(11)

where $x^{\mathrm{CAV}}$ and $y^{\mathrm{CAV}}$ represent the longitudinal and lateral displacement of CAV, respectively.

In this paper, it is assumed that CAVs can calculate the dynamic safety domains, consisting of the lower boundary for lateral stability and the upper boundary for collision safety, and follow the ideal optimal trajectory exactly for lane changes. Therefore, the CAV lane-changing time is dynamic, which can be obtained by the proposed ideal optimal trajectory introduced in later subsection.

• (a) Lower boundary for lateral stability. According to Equation 7, the expression of lateral acceleration with respect to the longitudinal displacement can be obtained by Equation 12:

$$a_y\left(x\right)=2{\omega }_2{u}^2+6{\omega }_3{u}^{2}x+12{\omega }_4{u}^2{x}^2+20{\omega }_5{u}^2{x}^3,x\in [x_0,x_f]$$

(12)

when ${a_y}^{\prime }\left(x\right)=0$ , the maximum lateral acceleration during lane change can be obtained as shown in Equation 13.

$$a_{\mathrm{ymax}}(x)=\{\begin{array}{c}\multicolumn{1}{c}{\left|a_{y0}\right|,\begin{array}{c}{\omega }_5=0\mathrm{or}{\omega }_5\ne 0,\Delta <0\hfill \end{array}}\\ \multicolumn{1}{c}{\max \{\left|a_y\left(x_d\right)\right|,\left|a_{y0}\right|\},{\omega }_5\ne 0,\Delta \ge 0}\end{array}$$

(13)

where $x_d$ is the solution to the equation ${a_y}^{\prime }\left(x\right)=0$ ; and $\mathrm{\Delta }$ is the discriminant.

To prevent lane-changing instability, the conditions in Equation 14 must be met.

$$a_{y_{\max }}\le a_y^{\max }$$

(14)

Based on the boundary conditions set in the existing results ( 27 ), let $a_y^{\max }=2m/s^2$ . By solving this inequality, the lower boundary for lateral stability $x_f^{\mathrm{lower}}=x_{f_{\min }}$ can be obtained.

• (b) Upper boundary for collision safety. A rectangular vehicle model is used to represent the vehicle to assess potential collisions between SV and its surrounding vehicles (Figure 2). The discretionary lane-changing behavior in this paper occurs under stable car-following conditions, so there is no need to consider VRS−SV rear-end scenario, which is eliminated from the scenario.

Figure 2.

Discretionary lane-changing maneuvers using rectangular vehicle models.

The length of the vehicle is represented by $L_i$ , the width of the vehicle is represented by $W_i$ , and $i$ represents different vehicles. The origin of the vehicle coordinate system for VFS, VFT, and VRT is located near the midpoint of the side that may collide with SV, while SV’s centroid G is taken as the origin of the vehicle coordinate system. The centroid position of SV is measurable, and for this study, the distance in x-direction between the centroid G of SV and the front end is $g$ and the distance in y-direction between the centroid G and the body is $W_{\mathrm{SV}}/2$ . $\theta$ is the heading angle during lane change. $D$ is the lane centerline distance. Considering the space limitation, the collision conditions are not derived in this paper, and the final safety conditions are shown in Equation 15.

$$\{\begin{array}{c}\multicolumn{1}{c}{(x_{\mathrm{SV}}+\sqrt{g^2+{\left(\frac{W_{\mathrm{SV}}}{2}\right)}^2}\cos (\theta -\frac{\arctan W_{\mathrm{SV}}}{2g})-x_{\mathrm{VFS}})\tan \theta <y_{\mathrm{SV}}+\sqrt{g^2+{\left(\frac{W_{\mathrm{SV}}}{2}\right)}^2}\sin (\theta -\frac{\arctan W_{\mathrm{SV}}}{2g})-(y_{\mathrm{VFS}}+\frac{W_{\mathrm{VFS}}}{2})}\\ \multicolumn{1}{c}{x_{\mathrm{SV}}+g<x_{\mathrm{VFT}}}\\ \multicolumn{1}{c}{x_{\mathrm{SV}}-(L_{\mathrm{SV}}-g)>x_{\mathrm{VRT}}}\end{array}$$

(15)

Therefore, there exists a critical $x_f$ such that the lane-changing process of SV satisfies the collision safety conditions, and $x_f$ is regarded as the upper boundary for collision safety, that is, $x_f^{\mathrm{upper}}=\min (x_{f_{\max }}^{\mathrm{VFS}},x_{f_{\max }}^{\mathrm{VFT}},x_{f_{\max }}^{\mathrm{VRT}})$ .

In summary, the intermediate region between the lower boundary for lateral stability and the upper boundary for collision safety forms the dynamic safety domains of the CAV lateral lane-changing model, that is, $S=\{x_f|x_f\in (x_f^{\mathrm{lower}},x_f^{\mathrm{upper}})\}$ .

• (c) Optimal lane-changing trajectory. An optimization objective function is established to balance the trade-off between comfort and efficiency, as shown in Equation 16.

$$J=\mathrm{\alpha }(\frac{a_{y_{\max }}}{a_{y_{\max }}^{\lim }}{)}^2+\mathrm{\beta }(\frac{a_{y_{\max }}^{\prime }}{a_{y_{\max }}^{\prime \lim }}{)}^2+(1-\mathrm{\alpha }-\mathrm{\beta })(\frac{x_f}{x_{f_{\max }}^{\lim }}{)}^2$$

(16)

where $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the weights of lateral acceleration and jerk, respectively. Based on the boundary conditions set in the existing results ( 27 ), let $a_{y_{\max }}^{\lim }=0.2g,a_{y_{\max }}^{\prime \lim }=0.24g/s^3,x_{f_{\max }}^{\lim }=u{t}_{f_{\max }}^{\lim }$ and $t_{f_{\max }}^{\lim }=7s$ .

Transition Model

Based on our previous work ( 29 ), it is known that in the discretionary lane-changing scenario (Figure 1), when SV changes from lane 2 to lane 1, its lead vehicle shifts from VFS to VFT. This state difference between VFS and VFT triggers a sudden jump in the output of the IDM-based car-following model. Similarly, the lane-changing behavior will cause a similar parameter jump for VRS and VRT, resulting in unreasonable outputs and fluctuations to the model. Therefore, to achieve a smooth transition and reflect the realistic lane-changing maneuvers of HDVs, the transition function is introduced between the original acceleration and the new acceleration as shown in Equation 17.

$$a_{\mathrm{tar}}=\mathrm{\Psi }\left(\tau \right)a_{\mathrm{new}}+(1-\mathrm{\Psi }\left(\tau \right))a_{\mathrm{ori}}$$

(17)

where $a_{\mathrm{tar}}$ is the desired acceleration of SV with transition function; $a_{\mathrm{new}}$ and $a_{\mathrm{tar}}$ are the desired acceleration of SV after and before lane change, respectively; $\mathrm{\Psi }\left(\tau \right)$ is the transition function, which satisfies $\mathrm{\Psi }\left(t_0\right)=0$ and $\mathrm{\Psi }\left(t_f\right)=1$ ; and $\tau$ is the timing of lane change.

In comparison with different transition functions to modify the longitudinal car-following model in the lane-changing process, the hyperbolic tangent function can achieve the best ride comfort and system stability ( 27 ). Therefore, this paper introduces the hyperbolic tangent function to realize the smooth transition as shown in Equation 18.

$$\{\begin{array}{c}\multicolumn{1}{c}{\Psi \left(\tau \right)=\frac{1}{2}(\tanh (\lambda \tau -\gamma )+1)}\\ \multicolumn{1}{c}{\gamma =\arctan h(0.98)\approx 2.3}\\ \multicolumn{1}{c}{\lambda =\frac{\mathrm{ar}\tan h(0.98)+\gamma }{t_f}\approx \frac{4.6}{t_f}}\end{array}$$

(18)

where $\gamma$ and $\lambda$ are the parameters reflecting the phase and speed of the transition function, respectively.

Modified Virtual Vehicle Model

The integration of CAVs has led to the development of cooperative operations, which enables direct control of CAVs’ motion states, thereby representing a cutting-edge solution for alleviating traffic accidents and congestion during discretionary lane change. One prominent method is the virtual vehicle model initially introduced by Uno et al. ( 30 ), which involves mapping lane 1 vehicles to lane 2 based on their distance from a pre-determined lane-changing point. Then, speed control strategies were implemented based on the car-following relationship between the virtual vehicle and its preceding vehicle in the adjacent lane. While the virtual vehicle theory can ensure the safety of the lane-changing process by establishing a virtual queue, it may not align with actual traffic characteristics without considering the speed differences between the vehicles across lanes as analyzed in the previous subsection. In this paper, a modified virtual vehicle model based on motion boundaries is proposed for CAV cooperative motion planning. And the motion boundary conditions of the initial and final motion states of the virtual vehicles are determined by the input as shown in Equation 19.

$$\{\begin{array}{c}\multicolumn{1}{c}{x_{\mathrm{VV}}\left(t_0\right)=x_{\mathrm{VFT}}\left(t_0\right)}\\ \multicolumn{1}{c}{v_{\mathrm{VV}}\left(t_0\right)=v_{\mathrm{VFT}}\left(t_0\right)}\\ \multicolumn{1}{c}{x_{\mathrm{VV}}\left(t_f\right)=x_{\mathrm{SV}}\left(t_f\right)}\\ \multicolumn{1}{c}{v_{\mathrm{VV}}\left(t_f\right)=v_{\mathrm{SV}}\left(t_f\right)}\end{array}$$

(19)

where $x_{\mathrm{VV}}$ and $v_{\mathrm{VV}}$ represent the displacement and speed of the virtual vehicles, respectively.

In this paper, we use cubic polynomial curves to realize the longitudinal motion planning for the modified virtual vehicle model, achieving a smooth transition between the motion boundary conditions ( 31 ). The longitudinal motion trajectory equation and the speed equation are shown in Equation 20.

$$\left(\begin{array}{cc}{x}_{\mathrm{VFT}}\left(t_0\right)\hfill & x_{\mathrm{SV}}\left(t_f\right)\hfill \\ \multicolumn{1}{c}{v_{\mathrm{VFT}}\left(t_0\right)}& v_{\mathrm{SV}}\left(t_f\right)\hfill \end{array}\right)=\left(\begin{array}{cc}{b}_{11}\hfill & b_{12}\hfill \\ \multicolumn{1}{c}{b_{12}}& 2b_{13}\hfill \end{array}\begin{array}{cc}{b}_{13}\hfill & b_{14}\hfill \\ \multicolumn{1}{c}{3b_{14}}& 0\hfill \end{array}\right)\left(\begin{array}{c}\begin{array}{cc}1\hfill & 1\hfill \\ \multicolumn{1}{c}{t_0}& t_f\hfill \\ \multicolumn{1}{c}{t_0^2}& t_f^2\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}{t}_0^3\hfill & t_f^3\hfill \end{array}}\end{array}\right)$$

(20)

The undetermined coefficients $b_{\mathrm{ij}}$ in the state equation can be obtained by solving Equation 21.

$$\left(\begin{array}{c}\begin{array}{c}{b}_{11}\hfill \\ \multicolumn{1}{c}{b_{12}}\\ \multicolumn{1}{c}{b_{13}}\end{array}\hfill \\ \multicolumn{1}{c}{b_{14}}\end{array}\right)={\left(\begin{array}{c}\begin{array}{cc}1\hfill & t_0\hfill \\ \multicolumn{1}{c}{1}& t_f\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & 1\hfill \end{array}}\end{array}\begin{array}{c}\begin{array}{cc}{t}_0^2\hfill & t_0^3\hfill \\ \multicolumn{1}{c}{t_f^2}& t_f^3\hfill \\ \multicolumn{1}{c}{2t_0}& 3t_0^2\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}2t_f\hfill & 3t_f^2\hfill \end{array}}\end{array}\right)}^{-1}\left(\begin{array}{c}\begin{array}{c}{x}_{\mathrm{VFT}}\left(t_0\right)\hfill \\ \multicolumn{1}{c}{x_{\mathrm{SV}}\left(t_f\right)}\\ \multicolumn{1}{c}{v_{\mathrm{VFT}}\left(t_0\right)}\end{array}\hfill \\ \multicolumn{1}{c}{v_{\mathrm{SV}}\left(t_f\right)}\end{array}\right)$$

(21)

Then, the states of VRT can realize smooth transition when VRT follows the virtual vehicle, and the computational burden can be reduced simultaneously.

Hierarchical Lane-Changing Gaming Decision Framework

Discretionary lane change aims to improve driving efficiency under the premise of safety. We neglect the subjective factor of returning to the original lane after lane-changing (i.e., keep right/left principle) to simply the process of lane-changing. To ensure that SV’s lane-changing behaviors will not cause excessive negative impacts on the surrounding vehicles, the decision-making process of SV’s lane change is modeled as a gaming process between SV and VRT as shown in Equation 22.

$$\mathit{G}=\{\mathit{NV},\mathit{A},\mathit{U}\}$$

(22)

where $\mathit{G}$ denotes the two-player Stackelberg game; $\mathit{NV}$ represents the set of participants in the game, that is, SV and VRT; $\mathit{A}$ is the set of actions; and $\mathit{U}$ indicates the set of payoff functions.

Generally, SV has priority in making driving decisions as it encounters traffic situations earlier than VRT, and its decisions are rarely restrained by VRT. Therefore, the interactions between SV and VRT can be modeled as a Stackelberg game with SV as the leader ( 14 ).

For SVs with lane-changing intentions, their action set includes changing lanes and keeping car-following. VRTs, according to varying driving styles and situations, have actions including changing lane and keeping car-following (i.e., at constant speed, accelerating and decelerating) ( 32 ).

Later, a probability set of actions in the game and a comprehensive set of payoff functions addressing collision safety, traffic efficiency, and ride comfort will be introduced.

Based on the gaming decision process, the HLCG decision model for heterogeneous traffic is proposed (Figure 3). Initially, assuming SV as a CAV, it gathers essential information (e.g., whether VRT or VRS is a CAV or HDV, etc.) from its surrounding environment. Subsequently, SV and VRT engage in a strategic game. Applying Stackelberg game principles and Harsanyi transformation theory, the payoff functions are computed across various action combinations, aiming to identify an optimal action for both SV and VRT. In scenarios where SV conducts lane changes, dynamic safety domains and optimal lane-changing trajectories are calculated to enhance collision safety and lateral stability. Simultaneously, if VRT is also a CAV, modified virtual vehicle models are adopted, otherwise transition models are accepted, to improve cooperation between CAVs, thereby enhancing traffic efficiency and ride comfort.

Figure 3.

Hierarchical lane-changing gaming (HLCG) framework.

State Prediction

As the influence of lane-changing decisions always lasts for some time ( 33 ), the evaluation of the payoff functions must account for the delayed impacts of lane-changing behaviors. To accurately assess the payoff functions’ magnitude, it is crucial to predict how the strategies affect the state changes of SV and its surrounding vehicles. A state vector $\mathit{z}=(v,x{)}^T$ is defined to describe the vehicle’s state in Equation 23.

$$\frac{d}{\mathrm{dt}}\mathit{z}=\frac{d}{\mathrm{dt}}\left(\begin{array}{c}v\hfill \\ \multicolumn{1}{c}{x}\end{array}\right)=\left(\begin{array}{c}\stackrel{·}{v}\hfill \\ \multicolumn{1}{c}{v}\end{array}\right)=\mathit{F}({\mathit{z}}_{{\mathit{t}}_{\mathit{c}}},\mathit{A})$$

(23)

where $t_c$ denotes the current moment, and the state of the system at the current moment is ${\mathit{z}}_{{\mathit{t}}_{\mathit{c}}}=(v_c,x_c{)}^T$ ; and $\mathit{A}$ is the actions adopted by the vehicle.

The prediction step is $\Delta t$ , then the predicted state of the vehicle can be calculated in Equation 24.

$${\hat{\mathit{z}}}_{{\mathit{t}}_{\mathit{c}}}=\mathit{z}\left(t_c\right)+{\int }_{{\mathit{t}}_c}^{{\mathit{t}}_c+\Delta \mathit{t}}\mathit{F}({\mathit{z}}_{{\mathit{t}}_{\mathit{c}}},\mathit{A})\mathrm{dt}$$

(24)

As shown in the discretionary lane-changing scenario (Figure 1), SV, VRS and VRT are affected by SV’s potential lane-changing behaviors. Different strategy combinations lead to distinct state changes among these three vehicles, necessitating varied methods for predicting their states. These predictions illustrate how different strategies affect SV and its surrounding vehicles. According to the longitudinal car-following models of CAVs and HDVs in the second section, the expected speed and displacement of SV after a prediction step $\Delta t$ can be calculated in Equation 25 and Equation 26, respectively.

$${\hat{v}}_{t_c+\Delta t}=v_c+{\int }_{t_c}^{t_c+\Delta t}{\stackrel{·}{v}}_{\mathrm{tar}}\mathrm{dt}=v_c+{\int }_{t_c}^{t_c+\Delta t}(\mathrm{\Psi }\left(\tau \right){\stackrel{.}{v}}_{\mathrm{new}}+(1-\mathrm{\Psi }\left(\tau \right)){\stackrel{·}{v}}_{\mathrm{ori}})\mathrm{dt}$$

(25)

$${\hat{x}}_{t_c+\Delta t}=x_c+{\int }_{t_c}^{t_c+\Delta t}v(t)\mathrm{dt}=x_c+{\int }_{t_c}^{t_c+\Delta t}(v_c+{\int }_{t_c}^{t_c+\Delta t}(\mathrm{\Psi }\left(\tau \right){\stackrel{·}{v}}_{\mathrm{new}}+(1-\mathrm{\Psi }\left(\tau \right)){\stackrel{·}{v}}_{\mathrm{ori}})\mathrm{dt})\mathrm{dt}$$

(26)

Within a prediction step, it is assumed that the leading vehicles VFS and VFT, unaffected by SV’s potential lane-changing behaviors, will maintain constant acceleration. The vehicle’s state after $\Delta t$ can be calculated using Newton’s laws of motion:

$${\hat{\mathit{z}}}_{{\mathit{t}}_{\mathit{c}}}(t_c+\Delta t)=\left(\begin{array}{c}{v}_{\mathrm{ic}}\hfill \\ \multicolumn{1}{c}{x_{\mathrm{ic}}}\end{array}\right)+\left(\begin{array}{cc}\Delta t\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{1}{2}\Delta t^2}& \Delta t\hfill \end{array}\right)·\left(\begin{array}{c}{\stackrel{·}{v}}_{\mathrm{ic}}\hfill \\ \multicolumn{1}{c}{v_{\mathrm{ic}}}\end{array}\right)$$

(27)

Payoff Function

The lateral motions of SV during lane change are strictly constrained by the lateral lane-changing model proposed in this paper, so the payoff functions only take the longitudinal motion characteristics of SV into account ( 15 ).

Collision Safety

On the basis of no collision with surrounding vehicles, if there is more travelling space and longer time headway, the collision safety payment should be lower. The collision safety payoff function $U_{\mathrm{safe}}$ is defined as follows in Equation 28.

$$U_{\mathrm{safe}}=\{\begin{array}{c}\multicolumn{1}{c}{\frac{J}{\mathrm{ga}{p}_{\mathrm{cri}}-s_0},\mathrm{ga}{p}_{\mathrm{cri}}\ge s_0+0.2}\\ \multicolumn{1}{c}{5J,\mathrm{ga}{p}_{\mathrm{cri}}<s_0+0.2}\end{array}$$

(28)

where $\mathrm{ga}{p}_{\mathrm{cri}}$ is the most dangerous car-following distance between the SV and its preceding vehicle, which satisfies $\mathrm{ga}{p}_{\mathrm{cri}}=\min (\mathrm{ga}{p}_{\mathrm{SV}},\mathrm{ga}{p}_{\mathrm{VRS}},\mathrm{ga}{p}_{\mathrm{VRT}})$ ; and $J$ is a constant, indicating the collision safety payoff function gain factor.

Traffic Efficiency

This paper refers to the acceleration gain model in the minimizing overall braking induced by lane change (MOBIL) model ( 34 ), consisting of SV’s acceleration gain and its surrounding vehicle’s acceleration gain. The traffic efficiency payoff function $U_{\mathrm{eff}}$ is defined as follows.

$$U_{\mathrm{eff}}=M\sum _i{\mathrm{\beta }}_i(a_{\left(i\right)\mathrm{new}}-a_{\left(i\right)\mathrm{ori}})$$

(29)

where $M$ is a constant coefficient; $i$ represents different vehicles, that is, SV, VRS and VRT; $a_{\left(i\right)\mathrm{ori}}$ and $a_{\left(i\right)\mathrm{new}}$ are the acceleration of different vehicles before and after lane change, respectively; and ${\mathrm{\beta }}_i$ is the weight coefficient of different vehicles.

Ride Comfort

The ride comfort issue caused by longitudinal motions includes longitudinal acceleration and jerk. Both of them can directly affect ride comfort, so the ride comfort payoff function $U_{\mathrm{com}}$ is defined as follows.

$$U_{\mathrm{com}}=C(\left|a_{\mathrm{cri}}\right|+j_{\mathrm{cri}})$$

(30)

where $C$ is a constant coefficient; $a_{\mathrm{cri}}$ and $j_{\mathrm{cri}}$ are critical acceleration and jerk, respectively; $a_{\mathrm{cri}}=\min (a_{\mathrm{SV}},a_{\mathrm{VRT}},a_{\mathrm{VRS}},0)$ and $j_{\mathrm{cri}}=\max (\left|j_{\mathrm{SV}}\right|,\left|j_{\mathrm{VRT}}\right|,\left|j_{\mathrm{VRS}}\right|)$ . As a result of the acceleration limitation of longitudinal car-following models, there is no excessive acceleration so that the maximum deceleration of the affected vehicles is regarded as the $a_{\mathrm{cri}}$ .

Solution Method

By predicting the vehicles’ states and calculating the payoff functions, the safety, efficiency and comfort payments for each strategy in the vehicle strategy space can be obtained. The integrated payoff function in the SV decision-making stage is defined as follows:

$$U_{\mathrm{SV}}={\mathrm{\rho }}_{\mathrm{safe}}{U}_{\mathrm{safe}}+{\mathrm{\rho }}_{\mathrm{eff}}{U}_{\mathrm{eff}}+{\mathrm{\rho }}_{\mathrm{com}}{U}_{\mathrm{com}}$$

(31)

where ${\mathrm{\rho }}_{\mathrm{safe}}$ , ${\mathrm{\rho }}_{\mathrm{eff}}$ , and ${\mathrm{\rho }}_{\mathrm{com}}$ are the weighting coefficients for safety, efficiency, and comfort payments, respectively.

Finding the optimal strategy for SV involves minimizing $U_{\mathrm{SV}}$ , which is influenced by the coping strategies of VRT. SV lacks real-time information about VRT’s surrounding vehicles during the game, preventing accurate assessment of VRT’s payoff under different strategies. Consequently, SV struggles to determine the best coping strategy choice, rendering the decision-making game in regard to incomplete information. However, with the assistance of the Harsanyi transformation theory, it is possible to transform the incomplete information static game into a complete but imperfect information game ( 17 ). This assumes knowledge of the probabilities associated with VRT’s potential action choices.

In this paper, $p$ and $q$ represent probabilities of different actions taken by VRT when SV changes lane and keeps car-following, respectively, satisfying $p_{\mathrm{Dec}}+p_{\mathrm{Acc}}+p_{\mathrm{CL}}=1$ and $q_{\mathrm{Kee}}+q_{\mathrm{CL}}=1$ . Where Kee = keeping the original lane and car-following behavior based on its original car-following model; Dec = decelerating; Acc = accelerating; and CL = changing lane. In the human driving phase, this probability distribution reflects human driver’s driving style; while in the fully automatic driving phase, it is a design parameter of the self-driving system.

According to the idea of mixed-strategy Nash equilibrium ( 35 ), SV takes the minimization of the expectation of its own payoff functions as the optimal strategy, and the combination of the strategy space and the mixed-strategy probability distribution of VRT can be used to compute the expected payoff functions under different strategies (i.e., changing lane and not changing lane) of SV:

$$E(U_{\mathrm{SV}}|\mathrm{CL})=\sum _{n=1,2,3,4}{p}_n{U}_{\mathrm{SV}}^{1n}$$

(32)

$$E(U_{\mathrm{SV}}|\mathrm{NCL})=\sum _{n=1,2,3,4}{q}_n{U}_{\mathrm{SV}}^{2n}$$

(33)

The optimal action for SV can be expressed as:

$$A_{\mathrm{SV}}^{*}=\underset{A_{\mathrm{SV}}}{\arg \min }E(U_{\mathrm{SV}})$$

(34)

Simulation and Analysis

In this section, simulations are conducted at both local and global levels. Local simulations demonstrate distinctions in car-following and lane-changing performance between CAVs and HDVs and interactions between these vehicle types. Meanwhile, global simulations assess the impacts of the proposed HLCG decision model on heterogeneous traffic, in comparison with traditional game theory.

Local Simulation

There are four different combinations of vehicle types for SV and VRT during lane change: SV_HDV-VRT_HDV, SV_HDV-VRT_CAV, SV_CAV-VRT_HDV, and SV_CAV-VRT_CAV. Local simulations are conducted using Carla and OpenCDA as platform ( 36 ). Three working conditions are designed to comprehensively simulate vehicle performance during lane change across these different vehicle type combinations. The microscopic simulation parameters are shown (Table 1).

Table 1.

Basic Parameters in Local Simulation

Symbol	Description	Value
Basic parameters of car-following model
$a$	maximum desired acceleration (m/s2)	1.4
$b$	comfort deceleration (m/s2)	2
$s_0$	static safety distance (m)	2
$T_{\mathrm{s}}$	safety time headway of SV (s)	1.5
$\delta$	acceleration index	4
$v_0$	desired speed (m/s)	33.3
$\mathrm{c}$	cooling factor	0.99
$\tau$	reaction time of HDV driver (s)	0.4
$v_s$	relative standard deviation of car-following distances	0.1
${\sigma }_r$	constant standard deviation of relative approach rate (/s)	0.01
Basic parameters of lane-changing model
$t_{f_{\mathrm{HDV}}}$	desired duration of lane change for HDV (s)	4
$\mathrm{\alpha }$	weight of the lateral acceleration target term	0.3
$\mathrm{\beta }$	weight of the lateral jerk target term	0.2
$\mathrm{D}$	lane centerline distance (m)	3.5
$\mathrm{L}$	length of the vehicle (m)	4.9
$\mathrm{W}$	width of the vehicle (m)	2.1
$\mathrm{g}$	distance between the centroid G and the front end (m)	2.45
Parameters of condition 1
$x_{\mathrm{VFS}-\mathrm{SV}}^1$	initial distance between VFS and SV (m)	50
$x_{\mathrm{VFT}-\mathrm{SV}}^1$	initial distance between VFT and SV (m)	25
$x_{\mathrm{SV}-\mathrm{VRT}}^1$	initial distance between SV and VRT (m)	25
/	desired speed of VFS (m/s)	33.3
/	desired speed of VFT (m/s)	33.3
$b_{\mathrm{VFS}}$	deceleration of VFS (m/s2)	2
Parameters of condition 2
$x_{\mathrm{VFS}-\mathrm{SV}}^2$	initial distance between VFS and SV (m)	50
$x_{\mathrm{SV}-\mathrm{VFT}}^2$	initial distance between SV and VFT (m)	100
$x_{\mathrm{SV}-\mathrm{VRT}}^2$	initial distance between SV and VRT (m)	200
/	desired speed of VFS (m/s)	22.2
/	desired speed of VFT (m/s)	33.3
Parameters of condition 3
$x_{\mathrm{VFS}-\mathrm{SV}}^3$	initial distance between VFS and SV (m)	25
$x_{\mathrm{SV}-\mathrm{VFT}}^3$	initial distance between SV and VFT (m)	25
$x_{\mathrm{SV}-\mathrm{VRT}}^3$	initial distance between SV and VRT (m)	75
/	desired speed of VFS (m/s)	22.2
/	desired speed of VFT (m/s)	33.3

Note: HDV = human-driven vehicles; SV = subject vehicle; VFS = the vehicle in the front of subject vehicle in the same lane; VFT = the vehicle in the front of subject vehicle in the target lane.

Local simulation takes different combinations of vehicle types, varying traffic densities, varying desired speed and extreme traffic conditions into account. Condition 1 describes the discretionary lane change caused by the sudden deceleration of VFS in congested condition; condition 2 and condition 3 consider the discretionary lane change in equilibrium in moderate and congested condition, respectively. The simulation results for the three conditions are shown as follows (Table 2). As a result of space constraints, condition 1 is used as an example to analyze the performance of SV and VRT during lane change.

Table 2.

Local Performance of Different Conditions

SV−VRT type	SV−lateral		VRT−longitudinal
	Max. acceleration (m/s2)	Max. jerk (m/s3)	Max. acceleration (m/s2)	Max. jerk (m/s3)
Condition 1
HDV-HDV	1.604	19.440	5.250	14.760
HDV-CAV			3.157	6.687
CAV-HDV	0.817	9.284	2.046	3.706
CAV-CAV			1.048	1.371
Condition 2
HDV-HDV	1.496	60.640	7.965	43.710
HDV-CAV			7.510	40.710
CAV-HDV	0.865	29.850	7.929	23.760
CAV-CAV			1.959	2.527
Condition 3
HDV-HDV	1.840	16.600	1.707	3.138
HDV-CAV			1.159	1.129
CAV-HDV	0.978	7.927	1.188	1.809
CAV-CAV			0.603	0.812

Note: CAV = connected and automated vehicles; HDV = human-driven vehicles; SV = subject vehicle; VRT = the vehicle in the rear of subject vehicle in the target lane; max. = maximum.

Performance of SV

Since SV is the leader in the two-player Stackelberg game, the lane-changing performance of SV is solely influenced by its own vehicle type. When SV is an HDV, lane changes are executed based on the desired lane-changing duration (i.e., $t_{f_{\mathrm{HDV}}}=4s$ ). Otherwise, when SV is a CAV, it leverages the information of its surrounding vehicles to analyze the dynamic safety domains and then determine the optimal lane-changing trajectory within safety domains through multi-objective optimization considering lateral acceleration, jerk, and lane-changing efficiency (Figure 4).

Figure 4.

Connected and automated vehicles (CAV) lateral lane-changing model: (a) dynamic safety domain and (b) optimal lane-changing trajectory.

According to the simulation results (Table 2), CAVs demonstrate better lateral performances thanks to the proposed dynamic safety domains and optimal lane-changing trajectory. To be more specific, in condition 1, CAVs exhibit a maximum lateral acceleration of 0.817 m/s² and a maximum lateral jerk of 9.284 m/s³, which are 49.1% and 52.2% lower than that of HDVs, respectively (Figure 5). This gentler lane-changing behavior of CAVs enhances both ride comfort and lateral stability throughout the maneuver.

Figure 5.

Local performance of human-driven vehicles (HDV) and connected and automated vehicles (CAV) during lane change in condition 1: (a) lateral position, (b) lateral acceleration, and (c) lateral jerk.

Performance of VRT

During the lane-changing process of SV, VRT’s leading vehicle transitions gradually from VFT to SV, thereby affected by both VRT and SV vehicle types. When VRT is an HDV, VRT calculates the desired acceleration by hyperbolic tangent transition function. Only when both VRT and SV are CAVs does VRT utilize the optimal lane-changing trajectory of SV for generating a modified virtual vehicle model as its leading vehicle. Under this circumstance, the role of the modified virtual vehicle model is to realize smooth coordination among CAVs and to improve the performance of the key vehicles considering the information sharing and computational capabilities of CAVs.

According to the simulation results (Table 2), when both SV and VRT are CAVs, both ride comfort and stability are improved throughout the lane-changing process, indicating the superior performance of the modified virtual vehicle model. As can be clearly depicted in the simulation results (Figure 6), When VRT is an HDV and SV is a CAV, the maximum deceleration and jerk during car-following process decrease by 3.204 m/s² and 11.054 m/s³, respectively, compared with when VRT and SV are both HDVs. This improvement indicates CAVs’ ability to enhance the ride comfort for VRT. When VRT and SV are both CAVs, employing a virtual vehicle model for the hybrid conditions of car-following and lane-changing results in a maximum deceleration of 1.048 m/s² and a maximum jerk of 1.371 m/s³; while using hyperbolic tangent transition function when VRT is a CAV and SV is an HDV results in a maximum deceleration of 3.157 m/s² and a maximum jerk of 6.687 m/s³. Among the four vehicle type combinations, both VRT and SV being CAVs demonstrate superior performance in ride comfort and longitudinal stability.

Figure 6.

Local performance of different vehicle type combinations in condition 1: (a) longitudinal velocity, (b) longitudinal acceleration, and (c) longitudinal jerk.

Global Simulation

A two-lane highway traffic flow simulation system is established using MATLAB, incorporating both HDVs and CAVs. The CAV penetration rate is introduced to represent the proportion of CAVs in the mixed traffic environment. CAVs maintain a desired speed of 33.3 m/s, while HDVs have a uniformly random desired speed from 27.8 m/s to 33.3 m/s. Nine working conditions are designed to comprehensively simulate vehicle performance during lane change across different traffic densities and CAV penetration rates. The proposed HLCG decision model is evaluated for its advantages over traditional cooperative and non-cooperative game theory method ( 35 ) in overall traffic efficiency and safety. The global simulation parameters are shown (Table 3).

Table 3.

Basic Parameters in Global Simulation

Symbol	Description	Value
Basic parameters of mixed traffic flow
$V_{\mathrm{CAV}}$	desired speed of CAVs (m/s)	33.3
$V_{\mathrm{HDV}}$	desired speed of HDVs (m/s)	27.8–33.3
$V_{\mathrm{CA}\mathrm{V}0}$	average initial speed of CAVs (m/s)	30.6
$V_{\mathrm{fm}0}$	average initial speed of HDVs (m/s)	27.8
$N_1$	initial number of vehicles on lane 1	50
$N_2$	initial number of vehicles on lane 2	50
Basic parameters of game-theoretic decision model
${\mathrm{\rho }}_{\mathrm{safe}}$	secure payment weight factor	0.2
${\mathrm{\rho }}_{\mathrm{eff}}$	efficiency payment weight factor	0.6
${\mathrm{\rho }}_{\mathrm{com}}$	comfort payment weight factor	0.2
$J$	safety payment function gain factor	1
$M$	efficiency payment function gain factor	4
$C$	comfort payment function gain factor	2
${\mathrm{\beta }}_{\mathrm{SV}}$	efficiency payment function SV weight	0.6
${\mathrm{\beta }}_{\mathrm{VRT}}$	efficiency payment function VRT weight	0.2
${\mathrm{\beta }}_{\mathrm{VRS}}$	efficiency payment function VRS weight	0.2
$[p_{\mathrm{Dec}},p_{\mathrm{Acc}},p_{\mathrm{CL}}{]}_{\mathrm{CAV}}$	CAV driving style	[1, 0, 0]
$[q_{\mathrm{Kee}},q_{\mathrm{CL}}{]}_{\mathrm{CAV}}$		[1, 0]
$[p_{\mathrm{Dec}},p_{\mathrm{Acc}},p_{\mathrm{CL}}{]}_{\mathrm{HDV}}$	HDV driving style	[0.8, 0.15, 0.05]
$[q_{\mathrm{Kee}},q_{\mathrm{CL}}{]}_{\mathrm{HDV}}$		[0.9, 0.1]
Parameters of condition 4
$T_{\mathrm{ah}1}^4$	average initial headway in lane 1 (s)	3
$T_{\mathrm{ah}2}^4$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	90%
Parameters of condition 5
$T_{\mathrm{ah}1}^5$	average initial headway in lane 1 (s)	3
$T_{\mathrm{ah}2}^5$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	60%
Parameters of condition 6
$T_{\mathrm{ah}1}^6$	average initial headway in lane 1 (s)	3
$T_{\mathrm{ah}2}^6$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	30%
Parameters of condition 7
$T_{\mathrm{ah}1}^7$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^7$	average initial headway in lane 2 (s)	2
$P_{\mathrm{CAV}}$	CAV penetration rate	90%
Parameters of condition 8
$T_{\mathrm{ah}1}^8$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^8$	average initial headway in lane 2 (s)	2
$P_{\mathrm{CAV}}$	CAV penetration rate	60%
Parameters of condition 9
$T_{\mathrm{ah}1}^9$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^9$	average initial headway in lane 2 (s)	2
$P_{\mathrm{CAV}}$	CAV penetration rate	30%
Parameters of condition 10
$T_{\mathrm{ah}1}^{10}$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^{10}$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	90%
Parameters of condition 11
$T_{\mathrm{ah}1}^{11}$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^{11}$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	60%
Parameters of condition 12
$T_{\mathrm{ah}1}^{12}$	average initial headway in lane 1 (s)	2
$T_{\mathrm{ah}2}^{12}$	average initial headway in lane 2 (s)	3
$P_{\mathrm{CAV}}$	CAV penetration rate	30%

Note: CAV = connected and automated vehicles; HDV = human-driven vehicles; SV = subject vehicle; VRS = the vehicle in the rear of SV in the same lane; VRT = the vehicle in the rear of subject vehicle in the target lane.

Evaluation Index

An optimal lane-changing decision model should aim to increase overall speed of the traffic flow while minimizing lane-changing frequency and impact of lane change on surrounding traffic conditions. Therefore, traffic efficiency, safety, ride comfort, stability and shockwave generation should be considered. Based on this analysis, the following metrics are chosen as criteria for evaluating the performance of lane-changing decision models.

• (a) Average lane-changing benefit. The primary objective of discretionary lane change is to improve driving efficiency. Therefore, assessing how lane-changing behavior impacts on vehicle speed is crucial in determining the reasonableness of lane-changing decisions. To quantify this indicator, the average lane-changing benefit is defined as follows:

$$\epsilon =\frac{\overline{\Delta v}}{n}$$

(35)

where $\epsilon$ is the average lane-changing benefit; $\overline{\Delta v}$ is the average speed increase; and $n$ is the total times of lane change.

• (b) Average longitudinal jerk. The lateral ride comfort is controlled by lane-changing models and is analyzed in the “Local Simulation” subsection. In this “Global Simulation” subsection, we focus on the longitudinal ride comfort of traffic flow. Given that fluctuations in longitudinal acceleration can reflect both the stability and ride comfort of the whole system, the average longitudinal jerk is selected as the evaluation index of the lane-changing decision model.

• (c) Deceleration rate to avoid the crash. To further evaluate the safety performance of the lane-changing decision models, the deceleration rate to avoid the crash (DRAC) is chosen as a key measure ( 37 ).

$$\mathrm{DRAC}=\frac{{(V_{\mathrm{front}}-V_{\mathrm{rear}})}^2}{X_{\mathrm{front}}-X_{\mathrm{rear}}}$$

(36)

where $V_{\mathrm{front}}$ and $V_{\mathrm{rear}}$ represent the velocities of the vehicle in front and behind, respectively; and $X_{\mathrm{front}}\mathrm{and}{X}_{\mathrm{rear}}$ and denote the positions of the vehicle in front and behind, respectively.

• (d) Average speed and low-speed region volume. This paper uses both average speed and low-speed region volume as the evaluation index of traffic efficiency ( 38 ). The low-speed region volume is computed by integrating the product of velocity, longitudinal displacement, and time, as depicted in Equation 37.

$$V_{\mathrm{ls}}=\sum _{i=1}^n[v_{\mathrm{is}}-v_i(x,t)]\Theta [v_{\mathrm{is}}-v_i(x,t)]\mathrm{dx}·\mathrm{dt}D:\{\begin{array}{c}x\in [x_0,x_f]\hfill \\ \multicolumn{1}{c}{t\in [t_0,t_f]}\end{array}$$

(37)

where $V_{\mathrm{ls}}$ is the low-speed region volume; $\mathrm{dx}·\mathrm{dt}$ represents the area of a single differential element within the space−time domain $D$ ; and $v_i(x,t)$ represents the speed of the vehicle i within the specific differential element, while $v_{\mathrm{is}}$ represents the expected speed of vehicle i.

• (e) Speed standard deviation. The impact of lane changes on surrounding traffic conditions (e.g., shockwave generation) is evaluated based on the speed standard deviation (SSD). The value of SSD serves as an indicator for identifying shockwave formation and tracking its propagation ( 39 ).

Simulation Results

Global Simulation takes different traffic densities and CAV penetration rates into account. Condition 4−condition 6 describe the discretionary lane change in moderate condition under different CAV penetration rates; condition 7−condition 9 describe the discretionary lane change in congested condition under different CAV penetration rates; and condition 10−condition 12 describe the discretionary lane change in mixed condition under different CAV penetration rates. The simulation results for the nine conditions are as follows (Table 4).

• (a) Average lane-changing benefit. In the simulation results of average lane-changing benefit in different conditions (Figure 7), the error bars (one standard deviation) indicate the result variability. As shown in the figure, the variance is increased when the CAV penetration rate grows, or the congestion level increases. The solid line indicates the most significant improvement made by the proposed HLCG model, compared with the baseline (i.e., 30% CAV penetration rate). It can be observed that the largest improvement of average lane-changing benefit is gained in the congested traffic condition (condition 7−condition 9), that is, 322% compared with the baseline. When the CAV penetration rate grows, the average lane-changing benefit increases significantly because the proposed algorithm for each CAV can coordinate the lane-changing maneuvers implicitly.

• (b) Average longitudinal jerk. In the simulation results of average longitudinal jerk in different conditions (Figure 8), the largest improvement is gained in the moderate traffic condition (condition 4–condition 6), that is, 57.15% in moderate traffic, while 52.17% in congested traffic. When the CAV penetration rate grows, the average longitudinal jerk decreases significantly because the proposed algorithm for each CAV can implement the optimal lane-changing trajectory in the dynamic safety domains, which improves the ride comfort.

• (c) Speed standard deviation. In the simulation results of SSD in different conditions (Figure 9), the largest improvement is gained in the congested traffic condition (condition 7–condition 9), that is, 31.97% compared with the baseline. After CAVs are introduced to the market, their influences on SSD are not obvious at low CAV penetration rates. When the CAV penetration rate grows and reaches a high degree (i.e., 90%), the value of SSD decreases significantly, indicating that CAVs can mitigate the negative impact of lane changes on surrounding traffic conditions. It can be derived that the cooperation among CAVs during lane change can alleviate shockwave formation and propagation.

• (d) Deceleration rate to avoid the crash. In the simulation results of DRAC in different conditions (Figure 10), the largest improvement is gained in the moderate traffic condition (condition 4–condition 6), that is, 57.15% in moderate traffic, while 52.17% in congested traffic. When the CAV penetration rate grows, the average longitudinal jerk decreases significantly because the proposed algorithm for each CAV can implement the optimal lane-changing trajectory in the proposed dynamic safety domains.

Table 4.

Global Performance of Different Conditions

Condition	Average lane-changing benefit (10−3 m/s)	Average longitudinal jerk (10−2 m/s3)	DRAC (10-3 m/s2)	Average speed (m/s)	SSD (m/s)
Condition 4	2.1181	0.5893	3.0947	31.4818	1.5797
Condition 5	0.8956	0.8448	4.1036	30.5118	1.8984
Condition 6	0.6118	1.3755	5.3837	29.8670	1.9056
Condition 7	3.0130	0.7279	3.6969	32.1021	1.3237
Condition 8	1.3015	1.0647	4.9887	30.8369	1.9457
Condition 9	0.7132	1.5219	5.7489	30.0430	1.9459
Condition 10	2.4454	0.6656	4.0511	31.8772	1.3841
Condition 11	0.8354	0.9921	4.8095	30.6054	2.0580
Condition 12	0.6087	1.3998	5.8581	29.9066	1.9326

Note: DRAC = deceleration rate to avoid the crash; SSD = speed standard deviation.

Figure 7.

Average lane-changing benefit in different conditions.

Figure 8.

Average longitudinal jerk in different conditions.

Figure 9.

Speed standard deviation in different conditions.

Figure 10.

Deceleration rate to avoid the crash in different conditions.

As traffic safety is the prerequisite, the proposed HLCG model is evaluated for its advantages over the traditional game theory method in congested conditions (i.e., condition 7–condition 9). According to the simulation results (Figure 11), when the CAV penetration rate grows, the value of DRAC decreases significantly, showing that the game theory-based methods are adaptable to the mixed traffic environment. To be more specific, our HLCG model improves DRAC 13.2% and 35.5% in condition 8 and condition 7 compared with the baseline (condition 9), while the traditional game theory improves 5.0% and 27.6%, respectively. Despite that when the CAV penetration rate is low, the safety index of our HLCG is slightly worse than that of the traditional game theory, when the CAV penetration rate grows, our HLCG can improve the traffic safety compared with the traditional game theory, thereby enhancing 5.67% and 7.94% in condition 8 and condition 7, respectively. Results in congested conditions show that our HLCG model outperforms the traditional game theory especially when CAV penetration rates grow. Simultaneously, the increases in the number of CAVs lead to lower DRAC in all conditions, which indicates the significant role of CAVs in improving the safety of mixed traffic.

• (e) Average speed and low-speed region volume. In the simulation results of average speed in different conditions (Table 4), the largest improvement is gained in the congested traffic condition (condition 7–condition 9), that is, 6.85% compared with the baseline. When the CAV penetration rate grows, the average speed increases in all the conditions.

Figure 11.

Deceleration rate to avoid the crash (DRAC) of different models in congested conditions.

To visualize traffic efficiency more intuitively, two-dimensional contour maps of the two main-lane traffic flow based on the HLCG model in congested conditions are shown as follows (Figure 12). With the increase of CAV penetration rates, the high-speed area (yellow) increases, and the average speed of traffic flow increases, indicating that CAVs can improve the efficiency of traffic flow. Additionally, the scattered color blocks in the figure decrease, indicating that the consistency of vehicle speed in the traffic flow is improved. To summarize, our HLCG can significantly improve overall traffic efficiency as CAV penetration rates increase.

Figure 12.

Contour maps of traffic flow based on the hierarchical lane-changing gaming (HLCG) model in congested conditions: (a) 30% connected and automated vehicles (CAV) penetration rates, (b) 60% CAV penetration rates, and (c) 90% CAV penetration rates.

As traffic efficiency is the objective of discretionary lane change, the proposed game theory-based HLCG model is also evaluated for its advantages over the traditional game theory method in congested conditions (i.e., condition 7–condition 9). According to the simulation results (Figure 13), when the CAV penetration rate grows, the average speed of traffic flow increases, and our HLCG outperforms the traditional game theory with higher values in all the working conditions. Moreover, the value of low-speed region volume of our HLCG is always lower than that of the traditional game theory, showing that our HLCG enables the heterogeneous traffic flow to approach the desired speed in less time and mileage, therefore improving overall traffic efficiency. Simultaneously, the significant role of CAVs in improving the efficiency of mixed traffic is more obvious at high CAV penetration rates.

Figure 13.

Average speed and low-speed region volume of different models in congested conditions: (a) average speed and (b) low-speed region volume.

Conclusions

Discretionary lane-changing maneuvers are common on highways but can significantly impact traffic efficiency and safety. Meanwhile, the traffic environment comprising both CAVs and HDVs will likely persist for the foreseeable future. To achieve effective automatic lane-changing maneuvers in this mixed traffic environment is a meaningful research direction. This paper modifies the longitudinal car-following models and lateral lane-changing models, to better reflect the characteristics and differences between CAVs and HDVs, and introduces transition models as well as modified virtual vehicle models to ensure HDVs and CAVs applicable for the hybrid conditions during lane change. Furthermore, this paper designs an HLCG framework based on Stackelberg game and Harsanyi theory, including a CAV decision model integrating dynamic safety domains and optimal trajectories, and a HDV decision model considering different human driving styles, with a comprehensive payoff function concerning collision safety, traffic efficiency, and ride comfort. Local simulation results show that compared with HDVs, CAVs can reduce collision risk through dynamic safety domain analysis, while the optimal trajectory planning can reduce the lateral acceleration and jerk by 49.1% and 52.2%, thereby improving lateral stability and ride comfort during lane change. Global simulation demonstrates that the proposed HLCG framework has great potential in mitigating the effects of discretionary lane-changing behaviors on heterogeneous traffic. The overall performance of traffic flow increases when the CAV penetration rate grows, or the congestion level increases. In regard to traffic efficiency and safety, our HLCG outperforms the traditional game theory across different CAV penetration rates and traffic densities. The HLCG framework shows great adaptability to the mixed traffic flow of HDVs and CAVs.

As one of the few discretionary lane-changing algorithms developed for mixed traffic, some challenges need to be addressed along its future development pathway: Firstly, human–machine interactions in game theory-based algorithms need further investigation, since the modeling of human behaviors is still challenging. Secondly, human-in-the-loop simulation tests can be carried out to obtain valid data for more in-depth human behavior research. Thirdly, multi-lane highways can be considered in the design of discretionary lane-changing model to improve the system integrity.