Dynamic Multi-Function Lane Management for Connected and Automated Vehicles Considering Bus Priority

Abstract

Bus lanes are commonly implemented to ensure absolute priority for buses at signalized intersections. However, while prioritizing buses, existing bus lane management strategies often exacerbate traffic demand imbalances among lanes. To address this issue, this paper proposes a dynamic Multi-Function Lane (MFL) management strategy. The proposed strategy transforms traditional bus lanes into Multi-Function Lanes (MFLs) that permit access to Connected and Automated Vehicles (CAVs). By fully utilizing the idle right-of-way of the MFL, the proposed strategy can achieve traffic efficiency improvement. To evaluate the proposed strategy, some experiments are conducted under various demand levels and CAV penetration rates. The results reveal that the proposed strategy (i) improves the traffic intensity balance degree by up to 52.9 under high demand levels; (ii) reduces delay by up to 80.56% and stops by up to 89.35% with the increase in demand level and CAV penetration rate; (iii) guarantees absolute bus priority under various demand levels and CAV penetration rates. The proposed strategy performs well even when CAV penetration is low. This indicates that the proposed strategy has the potential for real-world application.

1. Introduction

In modern urban areas, the increasing traffic demand poses significant challenges to the transportation system. To manage the growing volume of vehicles and ensure smooth traffic flow, prioritizing public transportation, particularly buses, becomes critical [1,2,3]. Buses are a vital component of urban mobility, offering an efficient and environmentally friendly mode of transportation that can carry a large number of passengers [4,5]. Ensuring bus priority not only helps to maintain reliable and timely service for passengers but also encourages a shift from private car use to public transit, thereby reducing overall traffic congestion [6,7,8,9].

To grant transit buses priority, Dedicated Bus Lanes (DBLs) are widely adopted in many cities, such as New York, London, and Shanghai. According to the report by the Ministry of Transport, PRC, China has 20,300 km of DBLs, an increase of 405.3 km by the end of 2023. However, implementing bus priority measures, such as Dedicated Bus Lanes, often leads to the underutilization of road space when buses are not present [10,11]. This underutilization can exacerbate traffic congestion in adjacent lanes, as general vehicles are restricted from using the reserved lanes, leading to an imbalance in traffic intensity among lanes. To address this issue, there is a need for more flexible and dynamic traffic management strategies that can adapt to real-time conditions and optimize the use of available road infrastructure.

With the advent of Connected and Automated Vehicle (CAV) technology, a CAV-based control method has been adopted [12,13,14,15,16,17]. Leveraging CAV technology, some studies focus on controlling CAVs to balance traffic intensity among lanes by utilizing the idle of right-of-way. One innovative approach to achieving this balance is the concept of Intermittent Bus Lanes (IBLs). IBLs are designed to accommodate not only buses but also, under certain conditions, general traffic. By allowing multiple vehicle types to access the same lane, IBLs aim to fully utilize road space and reduce congestion. This strategy balances traffic intensity among lanes by opening IBLs for general vehicles. Such opening can make CAVs bypass too long queues in the General-Purpose Lanes (GPLs). However, this lane management strategy would decrease bus priority levels when general vehicles access the IBL.

To fill the research gap, an innovative lane management strategy is proposed to overcome these shortcomings. This paper proposes a dynamic Multi-Function Lane (MFL) management strategy for CAVs and buses considering bus priority. The proposed strategy makes the following contribution:

➢

Balance traffic intensity between the Multi-Function Lanes (MFLs) and General-Purpose Lanes (GPLs): The proposed strategy converts the traditional bus lane into the MFL for buses and CAVs. It allows CAVs to access the MFL if CAVs have no interference with transit buses. Hence, the proposed strategy can balance traffic intensity among lanes by opening the MFL for CAVs to access when traffic intensity in GPLs is high.

➢

Improve traffic efficiency by dynamically allocating the right-of-way of the MFL for CAVs: The proposed strategy dynamically allocates the idle right-of-way of the MFL for CAVs to utilize no matter if buses are running in the MFL. It enables more CAVs to bypass too long queues in GPLs. Hence, the proposed strategy can effectively improve traffic efficiency at the signalized intersection.

➢

Guarantee absolute bus priority while allowing general vehicles to run in the MFL: The proposed strategy can be applicable even if bus stop times and bus arrivals are stochastic. To cope with the stochasticity, the proposed strategy dynamically reserves a part of the right-of-way for bus priority instead of reserving a whole lane. At the same time, the proposed strategy controls CAVs to access the MFL without interference with bus priority.

➢

The remainder of this paper is structured as follows: Section 2: “Literature review”; Section 3: “Notations and problem statement”, which describes the notations and research problems; Section 4: “Mathematical formulation”, which presents the problem formulation and the associated solution; Section 5: “Evaluation”, which shows the experiment design and associated results; Section 6: “Conclusions”, which details the conclusions of the experiments; Section 7: “Discussion”.

2. Literature Review

Over the past decades, extensive research has been dedicated to addressing the challenges associated with growing urban traffic demands, with a particular emphasis on enhancing public transportation through the implementation of Dedicated Bus Lanes (DBLs). These lanes are pivotal in ensuring the priority necessary for maintaining the reliability and punctuality of urban transit systems [18,19].

To analyze the influence mechanism of DBLs, Ma et al. proposed a novel analytical model of the macroscopic fundamental diagram and passenger macroscopic fundamental diagram [20]. Xu et al. systematically analyzed the impact of curb and median bus-only lane locations on the development and performance of a logic rule-based Bus Rapid Transit Signal Priority (BRTSP) system [21]. These studies emphasized the environmental benefits of buses and their capacity to reduce urban congestion by encouraging a shift from private car use to public transit. Based on the influence mechanism of DBLs, some researchers focused on improving bus operations. Petit et al. proposed an integrated methodological framework to design a spatially heterogeneous bus route network and time-dependent service headways to serve travel demand that varies over time and space [22]. Bayrak et al. proposed a bi-level optimization algorithm to determine Dedicated Bus Lane locations on a network to reduce the total travel time of all network users while considering traffic dynamics [23]. However, the DBL would waste road resources due to its exclusion of general vehicles.

In response to this challenge, several researchers have proposed a more flexible lane management strategy [24,25,26,27]. Zhao et al. presented a dynamic exclusive bus lane design, in which the exclusive bus lane at the exit can be dynamically used for the left-turning buses and the opposing through buses during the various periods of a signal cycle [28]. Qiu et al. proposed two cellular automaton models for a roadway section with two lanes to determine suitable traffic conditions for the IBL strategy implementation [29]. However, while IBLs present a promising solution, they also introduce new challenges, particularly concerning the maintenance of bus priority.

The advent of Connected and Automated Vehicle (CAV) technology has opened new avenues for optimizing lane management strategies [30,31]. In recent studies, Zhang et al. proposed a Trajectory-Based Control (TBC) method for CAVs to access the bus lane without interference with buses [32,33]. Xie et al. explored the use of CAVs to dynamically allocate right-of-way on roads, thereby improving traffic flow and reducing congestion [34]. These studies demonstrated that CAVs could play a crucial role in managing traffic intensity by utilizing underutilized lanes, such as DBLs or IBLs, more effectively. Despite these advancements, the existing research cannot perform well if the bus arrival time and bus stop times are stochastic.

To overcome these shortcomings, this paper proposes a dynamic Multi-Function Lane (MFL) management strategy for CAVs and buses considering bus priority.

3. Notations and Problem Statement

3.1. Notations

The notations are shown in

Table 1. Notations.

General Notations	Explanation
$c$	CAV index
$b$	Bus index
$t$	Time index
$t_s$	Current time step
$C$	Set of CAVs index
$B$	Set of buses index
$T$	Set of times index
$T_{\mathrm{n}\mathrm{o}\mathrm{b}\mathrm{u}\mathrm{s}}$	Set of times index when no bus exists
$i$	Position order in the CAV sequence from front to back
$\mathrm{M}$	A sufficiently big number
$t_{op}^{s,b}$	Start time of the bus b optimization horizon (s)
$t_{op}^{e,b}$	End time of the bus b optimization horizon (s)
$t_{op}^{s,c}$	Start time of the CAV c optimization horizon (s)
$t_{op}^{e,c}$	End time of the CAV c optimization horizon (s)
Geometric parameters	Explanation
$\mathrm{L}$	The length of the control zone, (m).
${\mathrm{x}}^{\mathrm{s}}$	Location of the bus stop
Signal parameters	Explanation
S	Signal cycle (s)
$\mathrm{R}$	Duration of red phase (s)
Gs	Duration of green light for go-through vehicles (s)
Gl	Duration of green light for left-turning vehicles (s)
Traffic parameters	Explanation
${\alpha }_1$	Weight of travel time in the objective
${\alpha }_2$	Weight of comfort level in the objective
${\beta }_1$	Weight of TPT reduction through MFL sharing in the objective
${\beta }_2$	Weight of TPT reduction through accelerating in GPL in the objective
Vehicle parameters	Explanation
$v_{max}^B$	Maximum speed of buses (m/s)
$v_{min}^B$	Minimum speed limit of buses (m/s)
$a_{max}^B$	Maximum acceleration of buses (m/s2)
$a_{min}^B$	Minimum acceleration of buses (m/s2)
$v_{max}^C$	Maximum speed of CAVs (m/s)
$v_{min}^C$	Minimum speed limit of CAVs (m/s)
$a_{max}^C$	Maximum acceleration of CAVs (m/s2)
$a_{min}^C$	Minimum acceleration of CAVs (m/s2)
$l_b$	Length of bus $b$ (m/s)
$l_c$	Length of CAV $c$ (m/s)
$\pi$	The reaction time (s)
$t_h$	Safety headway (s)
$t_{min}$	Minimum bus stopping time
$r$	Turning movement intention
Decision variables	Explanation
$x_t^{b,r}$	Location of bus $b$ (m) with turning movement intention $r$ at time $t$
$v_t^{b,r}$	Instantaneous speed of bus $b$ (m/s) with turning movement intention $r$ at time $t$
$a_t^{b,r}$	Instantaneous acceleration of bus $b$ (m/s2) with turning movement intention $r$ at time $t$
$x_t^{c,r}$	Location of CAV $c$ (m) with turning movement intention $r$ at time $t$
$v_t^{c,r}$	Instantaneous speed of CAV $c$ (m/s) with turning movement intention $r$ at time $t$
$a_t^{c,r}$	Instantaneous acceleration of CAV $c$ (m/s2) with turning movement intention $r$ at time $t$
$F_t^{b,c}$	1, if the preceding vehicle of CAV c in MFL is bus b at time t
$F_t^{c-1,c}$	1, if the preceding vehicle of CAV c in MFL is CAV at time t.
$w_t^{b,c}$	1, if the following vehicle of CAV c in MFL is bus b at time t
$w_t^{c+1,c}$	1, if the following vehicle of CAV c in MFL is CAV at time t
$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}$	Terminal passing time (TPT) of the bus b with turning movement intention $r$ in MFL
$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}$	Terminal passing time (TPT) of the CAV c with turning movement intention $r$ in MFL
$t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}$	Terminal passing time (TPT) of the CAV c with turning movement intention $r$ in GPL

.

3.2. Problem Statement

As shown in Figure 1, the research scenario of this paper is a signalized intersection under a connected and automated traffic environment. The length of the control zone is 480 m, which is divided into an upstream section and a downstream section. In the upstream section, three lanes are adopted. One is the Multi-Function Lane (MFL) and the others are General-Purpose Lanes (GPLs). In the downstream section, four approach lanes are adopted. One is the MFL. The others are, respectively, for left-turning vehicles, go-straight vehicles, and right-turning vehicles. In this scenario, general vehicles are composed of Connected Automated Vehicles (CAVs) and Connected Human-Driven Vehicles (CHVs). The MFL is provided for buses. All CAVs and buses enable real-time communication with other vehicles and the roadside unit via Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I). The roadside units determine when and whether the MFL is open for general vehicles. All CAVs accessing the Multi-Function Lane should pass the intersection without stops during the current signal cycle. To be specific, CAVs with go-forward intentions can only access the Multi-Function Lane when the go-forward green light is on. It is the same with CAVs with left-turning intentions. As for the vehicles with left-turning intentions in the General-Purpose Lanes, they can perform a free lane change to approach the intersection.

This paper proposes a dynamic bus lane management strategy to balance traffic intensity among lanes. The proposed strategy serves three purposes: (1) to balance traffic intensity between the MFL and GPLs; (2) to improve traffic efficiency by dynamically allocating the right-of-way of the MFL for CAVs; and (3) to guarantee absolute bus priority.

4. Mathematical Formulation

The proposed strategy is structured as shown in Figure 2. The proposed strategy has four modules: perception module, right-of-way allocation module, lane change decision-making module, and CAV trajectory planning module. The perception module collects traffic information and vehicle information including signal phase and timing, and turning movement intentions. The right-of-way allocation module is designed to provide a separate right-of-way in the MFL for transit buses and determines the available right-of-way in the MFL for CAVs. The lane change decision-making module determines which and whether CAVs are permitted to access the MFL. The CAV trajectory planning module aims to plan trajectories for CAVs accessing the MFL. Our research focuses on the right-of-way allocation module, the lane change decision-making module, and the CAV trajectory planning module.

4.1. Right-of-Way Allocation Module

The right-of-way allocation module is designed to provide a separate right-of-way in the MFL for transit buses and determines the available right-of-way in the MFL for CAVs. The module first checks for the presence of buses in the control area and operates in different modes based on the results. If a bus is detected, then the module assesses its operational phase: If the bus is starting from a stop, then the module plans an absolutely prioritized trajectory and allocates the corresponding right-of-way to the bus. If the bus has not yet started from the stop, then the module pre-allocates right-of-way based on potential trajectories. In the absence of a bus (anticipating that a bus may enter the MFL at any time), the module assumes a potential bus entry and pre-allocates right-of-way accordingly. After these allocations are completed, the remaining right-of-way in the MFL is made available for CAVs. This module comprises two components: a cost function and bus kinematic constraints, which together form an optimization model for bus trajectories. The detection results of buses in the MFL serve as inputs for this optimization model.

4.1.1. Cost Function of Bus Operation

The cost function to be minimized contains two terms:

$$J_B={\alpha }_1\cdot J_1+{\alpha }_2\cdot J_2$$

(1)

With

$$\begin{array}{cc}{J}_1=\left|v_t^{b,r}-v_{max}^B\right|& \forall b\in B,\forall t\in T,\forall r\in \left\{SD,LT\right\}\end{array}$$

(2)

$$\begin{array}{cc}{J}_2={\displaystyle \sum _t} \left|a_t^{b,r}\right|& \forall b\in B,\forall t\in T,\forall r\in \left\{SD,LT\right\}\end{array}$$

(3)

where ${\alpha }_1$ and ${\alpha }_2$ are weighting factors. $b$ denotes the bus index. $B$ denotes the set of buses index. $t$ denotes the time index. $T$ denotes the set of times index. $v_t^{b,r}$ and $a_t^{b,r}$ are the speed and acceleration of bus $b$ at time t, respectively. $v_{lim}^B$ is the maximum speed limit of buses. $r$ is the turning movement intention. $SD$ and $LT,$ respectively, denote through movement and left-turning movement. Equation (2) denotes that bus $b$ should pass the intersection as fast as possible. Equation (3) denotes the comfort level of bus $b$.

4.1.2. Bus Kinematic Constraints

The kinematic constraints of buses are as described below:

$$\begin{gathered} \begin{array}{c}{v}_{min}^B\le v_t^{b,r}\le v_{max}^B\end{array} \\ \forall b\in B,t_{op}^{b,s}\le t\le t_{op}^{b,e},\forall r\in \left\{SD,LT\right\} \end{gathered}$$

(4)

$$\begin{gathered} \begin{array}{c}{a}_{min}^B\le a_t^{b,r}\le a_{max}^B\end{array} \\ \forall b\in B,t_{op}^{b,s}\le t\le t_{op}^{b,e},\forall r\in \left\{SD,LT\right\} \end{gathered}$$

(5)

$$\begin{gathered} \begin{array}{c}{v}_t^{b,r}=v_{t-1}^{b,r}+a_t^{b,r}\cdot \mathsf{\Delta }t\end{array} \\ \forall b\in B,t_{op}^{b,s}\le t\le t_{op}^{b,e},\forall r\in \left\{SD,LT\right\} \end{gathered}$$

(6)

$$\begin{gathered} \begin{array}{c}{x}_t^{b,r}=x_{t-1}^{b,r}+v_{t-1}^{b,r}\cdot \mathsf{\Delta }t+{\displaystyle \frac{1}{2}}\cdot a_t^{b,r}\cdot \mathsf{\Delta }{t}^2\end{array} \\ \forall b\in B,t_{op}^{b,s}\le t\le t_{op}^{b,e},\forall r\in \left\{SD,LT\right\} \end{gathered}$$

(7)

where $t_{op}^{b,s}$ is the start time of the bus $b$ optimization horizon, which here is equal to the start time of bus $b$. $t_{op}^{b,e}$ is the end time of the bus $b$ optimization horizon, which here is equal to the moment bus $b$ passes through the intersection. $x_t^{b,r}$ is the position of bus $b$ at time t. $v_{min}^B$, $a_{max}^B$, and $a_{min}^B$ are the minimum speed limit, the maximum acceleration, and the minimum acceleration of buses. Equations (4) and (5) means the speed and acceleration of bus $b$ must be limited within feasible ranges. Equations (6) and (7) depict the bus dynamic constraints.

To calculate $t_{op}^{b,e}$ in (4)–(7), a Terminal Passing Time (TPT) prediction method of buses is presented. The Terminal Passing Time (TPT) is defined as the time it takes for a vehicle to pass the intersection from the current moment.

Case 1. Bus $b$ can reach the speed limit $v_{lim}^B$ in the control zone.

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{ll}{\displaystyle \frac{\mathrm{L}-{\mathrm{x}}^{\mathrm{s}}-S^{B,r}}{v_{max}^B}}+{\displaystyle \frac{v_{max}^B}{a_{max}^B}}+t_{{start}^{\prime }}^{b,r}& t_{ar}^{b,r}\le t_s\le t_{stop}^{b,r},\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ {\displaystyle \frac{\mathrm{L}-{\mathrm{x}}^{\mathrm{s}}-S^{B,r}}{v_{max}^B}}+{\displaystyle \frac{v_{max}^B}{a_{max}^B}}+\min \left\{t_{start}^{b,r},t_s+t_{min}\right\}& t_{stop}^{b,r}<t_s\le t_{stop}^{b,r}+t_{min},\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ {\displaystyle \frac{\mathrm{L}-{\mathrm{x}}^{\mathrm{s}}-S^{B,r}}{v_{max}^B}}+{\displaystyle \frac{v_{max}^B}{a_{max}^B}}+t_s& t_{stop}^b+t_{min}<t_s\le t_{start}^{b,r},\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ t_{{start}^{\prime }}^{b,r}=t_s+{\displaystyle \frac{2\left({{\mathrm{x}}^{\mathrm{s}}-x}_{t_s}^{b,r}\right)}{v_{t_s}^{b,r}}}+t_{min}& \forall t_s\in T_{nobus},\\ & \forall r\in \left\{SD,LT\right\},b=-1\end{array}\end{array}\#\end{array}\right.$$

(8)

$$\begin{gathered} \begin{array}{c}{t}_{{start}^{\prime }}^{b,r}=t_s+{\displaystyle \frac{2\left({{\mathrm{x}}^{\mathrm{s}}-x}_{t_s}^{b,r}\right)}{v_{t_s}^{b,r}}}+t_{min}\end{array} \\ \forall r\in \left\{SD,LT\right\},\forall b\in B \end{gathered}$$

(9)

$$\begin{array}{cc}{S}^{B,r}={\displaystyle \frac{{v_{max}^B}^2}{2a_{max}^B}}& \forall r\in \left\{SD,LT\right\},\forall b\in B\end{array}$$

(10)

where $t_s$ denotes the current time step. $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}$ is the TPT when bus $b$ is in the MFL. For each bus $b$ with turning movement intention $r$, $S^{B,r}$ is the distance required to accelerate from 0 to maximum speed, and $x_{t_s}^{b,r}$ indicates the current position. $\mathrm{L}$ represents the length of the control zone. $a_{max}^B$ denotes the maximum acceleration of buses. $t_{min}$ is the minimum bus stopping time. $t_{start}^{b,r}$ is the real starting time for bus $b$. $t_{{start}^{\prime }}^{b,r}$ is the earliest starting time for bus $b$. The bus index $b$ is denoted as −1 to represent a virtual bus entering the MFL at time $t_s$ when no bus exists.

Case 2. Bus $b$ cannot reach the maximum speed $v_{max}^B$ in the control zone.

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}=\left\{\begin{array}{c}\begin{array}{ll}{\displaystyle \frac{v_{max}^{B,r}}{a_{max}^B}}+t_{{start}^{\prime }}^{b,r}& t_{ar}^{b,r}\le t_s\le t_{stop}^{b,r},\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ {\displaystyle \frac{v_{max}^{B,r}}{a_{max}^B}}+\min \left\{t_{start}^{b,r},t_s+t_{min}\right\}& t_{stop}^{b,r}<t_s\le t_{stop}^{b,r}+t_{min},\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ {\displaystyle \frac{v_{max}^{B,r}}{a_{max}^B}}+t_s& t_{stop}^{b,r}+t_{min}<t_s\le t_{start}^b,\\ & \forall r\in \left\{SD,LT\right\},\forall b\in B\\ {\displaystyle \frac{v_{max}^{B,r}}{a_{max}^B}}+t_s+{\displaystyle \frac{2x^s}{v_{ar}}}+t_{min}& \forall t_s\in T_{\mathrm{n}\mathrm{o}\mathrm{b}\mathrm{u}\mathrm{s}},\\ & \forall r\in \left\{SD,LT\right\},b=-1\end{array}\end{array}\right.$$

(11)

$$\begin{array}{cc}{v}_{max}^{B,r}=\sqrt{2a_{max}^B\cdot \left(\mathrm{L}-{\mathrm{x}}^{\mathrm{s}}\right)}& \forall c\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(12)

where $v_{max}^{B,r}$ represents the maximum speed that bus $b$ can reach within the control zone. The TPT is influenced by signal timing as vehicles approach the intersection. Buses with different turning intentions can only pass through the intersection during the corresponding signal phase. Depending on whether bus $b$ can catch the current green light, there are two situations for calculating $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}$:

Situation 1. Bus $b$ can catch the green light in the current signal cycle.

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}=\left\{\begin{array}{c}\begin{array}{lll}{t}_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}& \forall b\in B,r=SD,& \mathrm{R}\le t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\\ t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}& \forall b\in B,r=LT,& \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\le t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+{\mathrm{G}}_{\mathrm{l}}\end{array}\end{array}\right.$$

(13)

where $\mathrm{R}$, ${\mathrm{G}}_{\mathrm{s}}$, and ${\mathrm{G}}_{\mathrm{l}}$ are, respectively, the duration of the red light, through green light, and left-turning green light. S denotes the length of a signal cycle. In this situation, the TPT cannot be affected by multiple phases of signal timing.

Situation 2. Bus $b$ cannot catch the current green light.

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}=\left\{\begin{array}{c}\begin{array}{ll}⌊{\displaystyle \frac{t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+t_s& \forall b\in B,r=LT,0\le t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_s\\ ⌊{\displaystyle \frac{t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+t_s& \forall b\in B,r=SD,0\le t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}\\ ⌊{\displaystyle \frac{t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+t_{\mathrm{s}}& \forall b\in B,r=SD,\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\le t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{S}\end{array}\end{array}\right.$$

(14)

In this scenario, the bus will get through the intersection at the start of the next green light.

4.2. Lane Change Decision-Making Module

The lane change decision-making module is devised to provide a MFL right-of-way allocation scheme that maximizes the benefits for CAVs. This module determines which and whether CAVs are permitted to access MFL. This module contains five submodules: decision-making, solution generation, benefits evaluation, optimal solution updates, and MFL access permission. The decision-making formulation submodule identifies CAV candidates that can access the MFL without interfering with buses. Based on the optimization problem, the solution generation submodule adopts the Monte Carlo Tree Search (MCTS) algorithm to generate a feasible solution in each iteration. For different solutions, the benefit evaluation submodule defines a value function to evaluate the benefit of each solution. After evaluation, the optimal solution update submodule selects the optimal solution. According to the optimal solution, the MFL access permission submodule determines which CAVs are permitted to access the MFL.

4.2.1. Decision-Making Submodule

In this submodule, an optimization problem is formulated to decide whether to permit CAVs to access the MFL. For CAV c with turning movement intention r, the decision $d^{c,r}$ is defined as a binary decision variable:

$$\begin{array}{cc}{d}^{c,r}\in \left\{\mathrm{0,1}\right\}& \forall c\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(15)

where C denotes the set of CAVs index. $d^{c,r}$ = 1 if the decision is to access the MFL; otherwise, 0. A CAV sequence consists of all left-turning and through CAVs in the same lane. The decision vector D of the CAV sequence can be described as:

$$\begin{array}{cc}\mathit{D}={{(d}^{1,r},d^{2,r},\cdots ,d^{i,r})}^T& \left\{\mathrm{1,2},\dots ,i\right\}\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(16)

where $i$ denotes the length of the CAV sequence. The CAV index numbers 1, 2, …, $i$ denote the position order in the CAV sequence from front to back. D is the solution to the optimization problem.

Objective Function: The objective function is designed to make each CAV candidate pass the intersection as fast as possible. Hence, the objective function is formulated as follows:

$$J_d\left(\mathit{D}\right)={\beta }_1\cdot J_3+{\beta }_2\cdot J_4$$

(17)

with

$$\begin{array}{cc}{J}_3={\displaystyle \sum _{c=1}^m} \left(t_{GPL}^{c,r}-t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}\right)\cdot d^{c,r}& \left\{\mathrm{1,2},\dots ,i\right\}\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(18)

$$\begin{array}{cc}{J}_4={\displaystyle \sum _{c=1}^m} \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}-t_{{\mathrm{G}\mathrm{P}\mathrm{L}}^{\prime }}^{c,r}\right)\cdot d^{c,r}& \left\{\mathrm{1,2},\dots ,i\right\}\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(19)

where ${\beta }_1$ and ${\beta }_2$ are weighting factors. $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}$ is the TPT when CAV c is in MFL. $t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}$ is the TPT for CAV c in the GPL when no CAV in the CAV sequence has accessed the MFL. $t_{{\mathrm{G}\mathrm{P}\mathrm{L}}^{\prime }}^{c,r}$ is the TPT for CAV c in the GPL after some preceding CAVs in the sequence have accessed the MFL. Equation (18) indicates the TPT reduction through MFL sharing. MFL sharing maneuvers of preceding CAVs can make space for rear CAVs in GPLs to speed up. Thus, Equation (19) indicates the TPT reduction through accelerating in the GPL. The calculation of TPT is detailed in the evaluation submodule.

MFL sharing possibility constraints: CAVs who want to access the MFL must check if enough space exists in the MFL:

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}\ge F_t^{c-1,c}\cdot t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c-1,r}+F_t^{b,c}\cdot t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}+\left(F_t^{c-1,c}+F_t^{b,c}\right)\cdot t_h$$

(20)

$$\forall c\in \left\{\mathrm{1,2},\dots ,i\right\},\forall t\in T,\forall b\in B,\forall r\in \left\{SD,LT\right\}$$

$$t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}\le w_t^{c+1,c}\cdot t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c+1,r}+w_t^{b,c}\cdot t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{b,r}-\left(w_t^{c+1,c}+w_t^{b,c}\right)\cdot t_h+\left(1-w_t^{c+1,c}-w_t^{b,c}\right)\cdot \mathrm{M}$$

(21)

$$\forall c\in \left\{\mathrm{1,2},...,i\right\},\forall t\in T,\forall b\in B,\forall r\in \{SD,LT\}$$

where M is a large positive real number. b is the bus index and B is the set of buses index. t is the time index and $T$ is the set of times index. $F_t^{b,c}$ = 1 if the preceding vehicle of CAV c in the MFL is bus b at time t; otherwise, 0. $w_t^{b,c}$ = 1 if the following vehicle of CAV c in MFL is bus b at time t; otherwise, 0. $F_t^{c-1,c}$ and $w_t^{c+1,c}$ are for the situation that the preceding or following vehicle is a CAV. $t_{MUL}^{b,r}$ is the TPT of bus b in the MFL. $t_h$ represents the safety headway.

Passing signal cycle constraints: To avoid congestion in the BPL, CAVs who access the BPL must pass the intersection in the current signal cycle:

$$\begin{array}{cc}{t}_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}\le ⌈{\displaystyle \frac{t}{\mathrm{S}}}⌉\cdot \mathrm{S}& \forall c\in \left\{\mathrm{1,2},\dots ,i\right\},\forall t\in T,\forall r\in \left\{SD,LT\right\}\end{array}$$

(22)

4.2.2. Solution Generation Submodule

The solution is the decision vector D. The solution generation can be represented as a tree search problem within the solution space, constrained by the optimization parameters. Unlike constructing the tree structure along the time dimension, this submodule builds the tree in the spatial dimension. According to CAV positions from front to back, each CAV corresponds to one layer of the tree as shown in Figure 3. Starting from the root, the child nodes expand layer by layer until reaching the leaf node. A path from the root to the leaf node can be considered a solution.

The Monte Carlo tree search (MCTS) algorithm is used to generate feasible solutions. In MCTS, each node in the formulated tree is assigned a score to evaluate its potential. Its key idea is to explore the nodes that have more potential to lead to the optimal solution. This algorithm can be summarized in three steps:

Step 1. Selection: Starting at the root node, the most urgent explorable child node $q$ is selected based on the following policy:

$$\underset{q}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}} J_{q,m}=Q_{q,m}+\theta \cdot \sqrt{{\displaystyle \frac{\ln P}{P_{q,m}}}}$$

(23)

where $Q_{q,m}$ is the score of child node q at iteration $m$. P is the number of times the current node has been visited, $P_{q,m}$ is the number of times child node q has been visited before iteration $m$, and θ is a weighting parameter. The first term in (23) encourages the selection of the child node that is currently believed to be optimal, while the second term encourages the exploration of more child nodes. The selection stops when reaching an unexpanded child node or reaching the max selection depth.

Step 2. Expansion: If reaching an unexpanded node in Step 1, then this step randomly selects one unvisited child node in the tree as a new node. If reaching the max selection depth, take the current node as the new node.

Step 3. Simulation: This step runs a Monte Carlo node-selecting simulation from the new node to the leaf node to generate a complete solution. Then, this step checks whether the generated solution is feasible. If the generated solution fulfills the constraints, the generated solution is feasible and then sent to the next submodule to evaluate its benefit.

4.2.3. Benefit Evaluation Submodule

The benefit is the value of the objective function (17) corresponding to the generated feasible solution. To calculate $t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}$, $t_{{\mathrm{G}\mathrm{P}\mathrm{L}}^{\prime }}^{c,r}$, and $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}$ in (17), a Terminal Passing Time (TPT) prediction method is presented. In this method, CAV c needs to select a preceding vehicle (PV) that shares the same turning movement intention. The selected area represents a hypothetical lane in front of CAV c to select a PV. This research team previously developed the selection method of PVs.

TPT in GPLs: For each CAV c with turning movement intention $r$, the $t_{GPL}^{c,r}$ is formulated as follows:

$$\begin{array}{cc}{t}_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}=max\left\{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_h,t_e^{c,r}\right\}& \forall r\in \left\{SD,LT,RT\right\},\forall c\in C\end{array}$$

(24)

where $RT$ denotes right-turning movement. nn denotes the number of vehicles between CAV c and its PV. $t_{GPL}^{c-1,r}$ is the TPT of its PV. $t_e^{c,r}$ is the earliest TPT of CAV c with turning movement intention $r$. Its value depends on whether CAV c can accelerate to the speed limit in the control zone. It is formulated as follows:

Case 1. CAV c can reach the speed limit $v_{lim}^C$ in the control zone.

$$\begin{array}{cc}{t}_e^{c,r}={\displaystyle \frac{\mathrm{L}-x_{t_s}^{c,r}-S_{t_s}^{c,r}}{v_{max}^C}}+{\displaystyle \frac{v_{max}^C-v_{t_s}^{c,r}}{a_{max}^C}}+t_s& \forall r\in \left\{SD,LT,RT\right\},\forall c\in C\end{array}$$

(25)

$$\begin{array}{cc}{S}_{t_s}^{c,r}={\displaystyle \frac{{v_{max}^C}^2-{v_{t_s}^{c,r}}^2}{2a_{max}^C}}& \forall r\in \left\{SD,LT,RT\right\},\forall c\in C\end{array}$$

(26)

where $t_s$ denotes the current time step. For each CAV c with turning movement intention $r$ at time $t_s$, $S_{t_s}^{c,r}$ is the distance required to accelerate to maximum speed, $x_{t_s}^{c,r}$ indicates the current position, and $v_{t_s}^{c,r}$ represents the current speed.

Case 2. CAV c can never reach the speed limit $v_{max}^C$ in the control zone.

$$\begin{array}{cc}{t}_e^{c,r}={\displaystyle \frac{v_{max}^{c,r}-v_{t_s}^{c,r}}{a_{max}^C}}+t_s& \forall r\in \left\{SD,LT,RT\right\},\forall c\in C\end{array}$$

(27)

$$\begin{array}{cc}{v}_{max}^{c,r}=\sqrt{2a_{max}^C\cdot \left(\mathrm{L}-x_{t_s}^{c,r}\right)+v_{t_s}^{c,r^2}}& \forall r\in \left\{SD,LT,RT\right\},\forall c\in C\end{array}$$

(28)

where $v_{max}^{c,r}$ is the maximum speed CAV c can reach in the control zone. The TPT is affected by signal timing when vehicles approach the intersection. Vehicles with different turning movement intentions can only pass the intersection at the corresponding signal phase. According to whether CAV c can catch the current green light, there are two situations related to the calculation of $t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}$:

Situation 1. CAV c can catch the green light in the current signal cycle.

$$t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}=\left\{\begin{array}{c}\begin{array}{ll}{t}_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}& \forall c\in C,r=SD,\mathrm{R}\le t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\\ t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}& \forall c\in C,r=LT,\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\le t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+{\mathrm{G}}_{\mathrm{l}}\end{array}\end{array}\right.$$

(29)

where R, G_s, and G_l are, respectively, the duration of the red light, through green light, and left-turning green light. In this situation, the TPT cannot be affected by multiple phases of signal timing.

Situation 2. CAV c cannot catch the green light in the current signal cycle. In (30) and (31):

$$t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}=\left\{\begin{array}{c}\begin{array}{ll}⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+t_s& 0\le t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}\mathrm{m}\mathrm{o}\mathrm{d} S\le \mathrm{R}+{\mathrm{G}}_s,\\ & \forall c\in C,r=LT,c=1\\ ⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+{\mathrm{G}}_{\mathrm{s}}+t_{\mathrm{s}}& 0\le \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_{\mathrm{h}}\right)\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_{\mathrm{s}},\\ & \forall c\in C,r=LT,c>1\\ t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}& \mathrm{R}+{\mathrm{G}}_s\le \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_h\right)\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{S},\\ & \forall c\in C,r=LT,c>1\end{array}\end{array}\right.$$

(30)

$$t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}=\left\{\begin{array}{c}\begin{array}{ll}⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+{\mathrm{G}}_s+t_{\mathrm{s}}& 0\le t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}\mathrm{m}\mathrm{o}\mathrm{d} S\le \mathrm{R},\\ & \forall c\in C,r=SD,c=1\\ ⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+t_{\mathrm{s}}& \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\le t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{S},\\ & \forall c\in C,r=SD,c=1\\ ⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+t_{\mathrm{s}}& 0\le \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_{\mathrm{h}}\right)\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R},\\ & \forall c\in C,r=SD,c>1\\ ⌊{\displaystyle \frac{t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}}{\mathrm{S}}}⌋\cdot \mathrm{S}+\mathrm{R}+t_{\mathrm{s}}& \mathrm{R}+{\mathrm{G}}_{\mathrm{s}}\le \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_{\mathrm{h}}\right)\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{S},\\ & \forall c\in C,r=SD,c>1\\ t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c,r}& \mathrm{R}\le \left(t_{\mathrm{G}\mathrm{P}\mathrm{L}}^{c-1,r}+\left(n+1\right)\cdot t_h\right)\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{S}\le \mathrm{R}+{\mathrm{G}}_s,\\ & \forall c\in C,r=SD,c>1\end{array}\end{array}\right.$$

(31)

As for through or left-turning CAV c, c = 1 means it is the first vehicle in the selected area. Its TPT is in the next passable duration. c > 1 means there are preceding vehicles. Whether CAV c and its preceding vehicle can enter the intersection in the same green phase is also considered in (30) and (31).

TPT in MFL: For CAV c in the MFL, the TPT can be accurately calculated because the trajectory information of the preceding vehicles and buses in the MFL can be collected. The TPT $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}$ of CAV c in the MFL can be calculated as follows:

$$\begin{array}{cc}{t}_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c-1,r}+t_h,t_e^{c,r}\right\}& \forall c\in C,\forall r\in \left\{SD,LT\right\}\end{array}$$

(32)

where $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c-1,r}$ means the TPT of the preceding CAV in the MFL.

4.2.4. Optimal Solution Updates Submodule

In each iteration, the optimal solution is updated by comparing the benefit of the generated feasible solution with the historical solutions:

$$\underset{m}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}} J_d\left({\mathit{D}}_m\right)$$

(33)

where ${\mathit{D}}_m$ is the generated solution in iteration $m$. Then, the benefit of the generated feasible solution in each iteration is backpropagated to the nodes in its corresponding path.

$$Q_{q,m+1}=Q_{q,m}+J_d\left({\mathit{D}}_m\right)$$

(34)

$$P_{q,m+1}=P_{q,m}+1$$

(35)

where $Q_{q,m+1}$ is the score of child node q at iteration $m$ + 1. $P_{q,m+1}$ is the number of times child node q has been visited before iteration $m$ + 1.

4.2.5. Bus Lane Access Permission

The bus lane access permissions for CAVs are generated based on the optimal solution. They are then sent to the CAV trajectory planning module.

4.3. CAV Trajectory Planning Module

The CAV trajectory planning module can help CAVs sharing the MFL with buses passing the intersection without stops and guarantee absolute bus priority. This module is activated following the generation of the optimal solution by the lane change decision-making module. The module consists of three parts: cost function, vehicle kinematic constraints, and vehicle conflict-free constraints together form an optimization model for CAV trajectories.

4.3.1. Cost Function

The cost function to be minimized contains two terms:

$$J_C={\alpha }_1\cdot J_5+{\alpha }_2\cdot J_6$$

(36)

with

$$\begin{array}{c}{J}_5=\left|x_t^{c,r}-\mathrm{L}\right|+\left|v_t^{c,r}-v_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\end{array}$$

(37)

$$\begin{array}{cc}{J}_6={\displaystyle \sum _t} \left|a_t^{c,r}\right|& \forall c\in C,\forall t\in T,\forall b\in B,\forall r\in \left\{SD,LT\right\}\end{array}$$

(38)

where $x_t^{c,r}$, $v_t^{c,r}$, and $a_t^{c,r}$ are the position, speed, and acceleration of CAV c at time t, respectively. (37) denotes that CAV c should pass the intersection as fast as possible. (38) denotes the comfort level of CAV c.

4.3.2. Vehicle Kinematic Constraints

The movements of CAVs need to be subject to kinematic constraints as described below:

$$\begin{array}{c}\begin{array}{c}{v}_{\mathrm{m}\mathrm{i}\mathrm{n}}^C\le v_t^{c,r}\le v_{\mathrm{m}\mathrm{a}\mathrm{x}}^C\end{array}\\ \forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \left\{SD,LT\right\}\end{array}$$

(39)

$$\begin{array}{c}\begin{array}{c}{a}_{\mathrm{m}\mathrm{i}\mathrm{n}}^C\le a_t^{c,r}\le a_{\mathrm{m}\mathrm{a}\mathrm{x}}^C\end{array}\\ \forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \left\{SD,LT\right\}\end{array}$$

(40)

$$\begin{array}{c}\begin{array}{c}{v}_t^{c,r}=v_{t-1}^{c,r}+a_t^{c,r}\cdot \mathsf{\Delta }t\end{array}\\ \forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \left\{SD,LT\right\}\end{array}$$

(41)

$$\begin{array}{c}\begin{array}{c}{x}_t^{c,r}=x_{t-1}^{c,r}+v_{t-1}^{c,r}\cdot \mathsf{\Delta }t+{\displaystyle \frac{1}{2}}\cdot a_t^{c,r}\cdot \mathsf{\Delta }{t}^2\end{array}\\ \forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \left\{SD,LT\right\}\end{array}$$

(42)

where $t_{op}^{c,s}$ is the start time of the CAV c optimization horizon. $t_{op}^{c,e}$ is the end time of the CAV c optimization horizon and is equal to $t_{\mathrm{M}\mathrm{F}\mathrm{L}}^{c,r}$. Equations (39) and (40) indicate that the speed and acceleration of CAV c must be limited within feasible ranges. Equations (41) and (42) depict the vehicle dynamic constraints.

4.3.3. Vehicle Conflict-Free Constraints

CAVs need to fulfill the vehicle conflict-free constraints in MFLs. Additionally, these constraints guarantee that no interference with buses will be caused.

$$x_t^{c,r}-w_t^{c+1,c}\cdot x_t^{c+1,r}+w_t^{b,c}\cdot \mathrm{M}\ge l_c+\pi \cdot v_t^{c+1,r}$$

(43)

$$\forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \{SD,LT\}$$

$$\mid F_t^{c-1,c}\cdot x_t^{c-1,r}-x_t^{c,r}+F_t^{b,c}\cdot \mathrm{M}\mid \ge l_c+\pi \cdot v_t^{c,r}$$

(44)

$$\forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \{SD,LT\}$$

$$x_t^{c,r}-w_t^{b,c}\cdot x_t^{b,r}+\left(1-w_t^{b,c}\right)\cdot \mathrm{M}\ge l_b+\pi \cdot v_t^{b,r}$$

(45)

$$\forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \{SD,LT\}$$

$$\left|F_t^{b,c}\cdot x_t^{b,r}-x_t^{c,r}+\left(1-F_t^{b,c}\right)\cdot \mathrm{M}\right|\ge l_c+\pi \cdot v_t^{c,r}$$

(46)

$$\forall c\in C,t_{op}^{c,s}\le t\le t_{op}^{c,e},\forall b\in B,\forall r\in \{SD,LT\}$$

where $x_t^{c+1,r}$ and $v_t^{c+1,r}$ are the position and speed of the following CAV behind CAV c at time t, respectively. $x_t^{b,r}$ and $v_t^{b,r}$ are the same circumstances for buses in front of or behind CAV c at time t. $l_b$ and $l_c$ are the length of buses and the length of CAVs, respectively. π represents the average reaction time.

5. Evaluation

In this section, the proposed strategy is evaluated through simulation experiments compared with a baseline of the non-control strategy and a dynamic clear-off lane strategy. The simulation experiments are conducted with a sensitivity analysis under five different congestion levels and five CAV penetration rates. The evaluation is conducted in terms of traffic efficiency improvement validation.

5.1. Experiment Design

The simulation platform is based on PTV-VISSIM [35,36,37,38]. The testbed is shown in Figure 1. Three scenarios with different lane management strategies considering bus priority are adopted to evaluate the proposed strategy: (a) A baseline of non-control strategy: In this scenario, a Dedicated Bus Lane (DBL) is adopted to separate buses from general vehicles. General vehicles are not allowed to access the DBL. Buses as well as general vehicles are not under control; (b) A dynamic clear-off lane strategy: In this scenario, general vehicles are composed of CAVs and CHVs. A dynamic clear-off lane is designed. The dynamic clear-off lane is open for buses and CAVs. Bus stop times are stochastic. All buses are controlled to pass the intersection without stops as early as possible. All CAVs are allowed to access the dynamic clear-off lane only if there is no bus in the control zone; (c) The proposed strategy: In this scenario, general vehicles are composed of CAVs and CHVs. A Multi-Function Lane (MFL) is adopted. The MFL is open for buses and CAVs. Buses stop at the bus station for a stochastic time. All buses are controlled to pass the intersection without stops as early as possible. All CAVs can access the MFL at any time if CAVs have no interference with buses.

To validate the traffic efficiency improvement of the proposed strategy, two Measurements of Effectiveness (MOEs) are selected, including average vehicle delay and average vehicle stops. As for the absolute bus priority validation, the difference between the earliest TPT (accounting for signals) of buses and the actual TPT of buses is selected.

To fairly confirm the validation of the proposed strategy, sensitivity analysis was conducted under three different demand levels (0.8, 1.0, 1.2) and five different CPR (0.2, 0.3, 0.4, 0.5, 0.6). The saturation flow rate has been tested, as shown in

Table 2. Simulation settings.

Parameter	Value
Simulation time horizon (s)	3600
Length of control zone (m)	480
Optimization time interval (s)	1
Cycle of signal time (s)	120
Duration of red light (s)	60
Duration of left-turning green light (s)	30
Duration through green light(s)	30
Saturation flow rate (veh/h)	1200
Departure time interval of buses (s)	300
Bus arrival time error (s)	$U\left(\mathrm{0,60}\right)$ *
Bus stopping time (s)	$U\left(\mathrm{15,25}\right)$ *
Desired speed for buses (m/s)	48
Desired speed for general vehicles (m/s)	60
Maximum speed (m/s)	20
Minimum speed (m/s)	0
Maximum acceleration (m/s2)	3.5
Minimum acceleration (m/s2)	-4
Safe time headway (s)	1.6
Reaction time (s)	0.5
Length of the bus (m)	10
Length of the general vehicle (m)	4
Factor	1
Factor	1
Factor	10
Factor	1

* Uniform distribution.

. Different from the freeway, the factor that vehicles can only get through the signalized intersection during the green time should be taken into consideration. Hence, the demand level of the signalized intersection has to be calibrated by the parameters of the signal plan.

In our experiments, a hypothetical signalized intersection is adopted as the research scenario. In this scenario, some parameters need to be selected and calibrated. The parameters include traffic parameters, vehicle parameters, and some parameters of the optimization model. The traffic parameters and vehicle parameters are based on the real-world application at the signalized intersection. The parameters of the simulation setting are shown in Table 2.

5.2. Results

Compared with the Dedicated Bus Lane management strategy and the dynamic clear-off lane management strategy, the proposed strategy has benefits in traffic intensity imbalance degree reduction and traffic efficiency improvement.

5.2.1. Average Vehicle Delay Reduction Validation

Figure 4, Figure 5 and Figure 6 are the comparison results of average vehicle delay. Compared with the Dedicated Bus Lane management strategy, the proposed strategy has significant benefits in average vehicle delay reduction under various demand levels and CAV penetration rates. Especially when the demand level is high (V/C = 1.2), the proposed strategy can reduce the average delay by between 49.17% and 84.16%. The benefits are more obvious with the increase in CAV penetration rate. The reason is that the proposed strategy can avoid too long queues in GPLs by allowing general vehicles to access the MFL. Compared with the dynamic clear-off lane management strategy, the proposed strategy still has benefits in average vehicle delay reduction. This is because the proposed strategy can allow more CAVs to access the MFL by fully utilizing the idle right-of-way during bus stop times. With the increase in demand level and CAV penetration rate, obvious benefits can be observed.

5.2.2. Average Vehicle Stop Reduction Validation

Figure 7, Figure 8 and Figure 9 are the comparison results of average vehicle stops. Compared with the Dedicated Bus Lane management strategy, the proposed strategy has significant benefits in average vehicle stops under various demand levels and CAV penetration rates. Especially when the demand level is high (V/C = 1.2), the proposed strategy can reduce the average vehicle stops by between 54.77% and 92.87%. It makes sense that the proposed strategy can avoid frequent stop-and-go in GPLs by allowing general vehicles to access the MFL. Compared with the dynamic clear-off lane management strategy, the proposed strategy still has benefits in average vehicle stop reduction. The reasons include that the proposed strategy can allocate more right-of-way of the MFL for allowing general vehicles to access. Such allocation can make more general vehicles pass the intersection during one signal cycle without stops.

5.2.3. Absolute Bus Priority Validation

In our experiment, a fixed signal scheme is adopted. Due to the stochasticity of bus arrival and bus stop times, two cases are considered. One is that the bus can pass the intersection during the current green time. The other is that the bus cannot pass the intersection during the current green time. Figure 10 and Figure 11 show two kinds of trajectory diagrams of buses passing through the intersection during the green light period. In Case 1, the bus can accelerate to its maximum speed and pass through the intersection during the current green light phase, and such a trajectory is generated. In Case 2, the bus is not able to do it, and a trajectory is generated for the bus to pass through the intersection at the start of the next green light phase. In both cases, the bus passes through the intersection at the earliest possible moment, and can be considered as having absolute priority under a fixed signal plan.

6. Conclusions

In this paper, a dynamic Multi-Function Lane management strategy for CAVs and buses considering bus priority is proposed. It converts bus lanes into MFLs to allow CAVs with different turning movement intentions to access. The proposed strategy bears the following features: (i) balance traffic intensity between the Multi-Function Lanes (MFLs) and General-Purpose Lanes (GPLs); (ii) improve traffic efficiency by fully utilizing the idle right-of-way of the mixed-use lane; (iii) guarantee absolute bus priority while allowing general vehicles to run in the MFL. To evaluate the proposed strategy, some experiments are conducted under various demand levels and CAV penetration rates. Some conclusions can be drawn:

• The proposed strategy can achieve average vehicle delay reduction under each CAV penetration rate. It can significantly reduce delay by between 49.17% and 84.16% under high demand levels.
• The proposed strategy can achieve average vehicle stop reduction under each CAV penetration rate. It can significantly reduce stops by between 54.77% and 92.87% under high demand levels.
• The proposed strategy shows more benefits on traffic efficiency improvement with the increase in demand level and CAV penetration rate.
• The proposed strategy performs well even when the CAV penetration is low. This indicates that the proposed strategy has the potential for real-world application.

7. Discussion

In Figure 12, the result shows the vehicle trajectories in the MFL optimized by the proposed strategy. In this experiment, we set the warm-up time to 360 s (three signal cycles). After the warm-up time, we select 1200–1800 s (five signal cycles) to show the trajectory results. The blue lines denote the trajectories of buses. The light blue lines denote CAV trajectories after lane change, and the black dashed lines are the CAV trajectories before lane change. This result shows that CAVs can access the MFL only if they have no interference with buses.

Although the proposed lane management strategy performs better compared with the state-of-the-art strategy, it still has a limitation. The limitation is that transit buses will slow down to approach the intersection if they cannot catch the current green light. Such slowdown would decrease the level of transit bus priority. Hence, future research can combine the proposed strategy with the transit signal priority strategy.