Arterial Multi-Path Green Wave Control Model Concurrently Considering Motor Vehicles and Electric Bicycles

Abstract

Arterial green wave control can effectively reduce the delay time and number of stops of the coordinated traffic flows. However, existing arterial green wave control methods mostly focus on motor vehicles and provide them with green wave bands, neglecting the electric bicycles that are widespread on the roads. In fact, electric bicycles have become an important tool for short-to-medium trips among urban residents because they are convenient, low-cost, and eco-friendly. To tackle this, an arterial multi-path green wave control model that considers both motor vehicles(cars and buses) and electric bicycles is presented in this paper. The presented model is formulated as a mixed integer linear programming problem. The optimization objective of the model is to maximize the sum of the green wave bandwidths for all coordinated paths of each traffic mode on all road segments. The key constraints of the presented model can be addressed by analyzing the relationships among the green wave bandwidth, coordinated path, common cycle time, offset, phase sequence, etc., to utilize the time–space diagram. The results of the numerical example show that compared with the traditional model for through motor vehicles (cars and buses), the total green wave bandwidths of cars, buses, and electric bicycles generated by the presented model at the entire arterial level has been increased by 36.8%, 47.9%, and 19.3%, respectively.

1. Introduction

Signalized intersections are the bottlenecks of arterials. How design an efficient traffic signal control scheme to reduce the delay time and the number of stops of travelers on arterials is one of the difficult problems faced by traffic authorities. By adjusting the signal control variables such as the offset, phase sequence, and cycle time, arterial green wave control allows for coordinated movements to traverse an arterial with fewer stops. The following is a summary of the research on arterial green wave control methods for single-traffic modes, such as cars or buses. The MAXBAND proposed by Little et al. [1] is a typical arterial green wave control model aiming to maximize the car’s green wave bandwidths. The MULTIBAND developed by Gartner et al. [2] is an enhanced version of the MAXBAND. It can generate green wave bandwidths varying with road segments, addressing the deficiency of the MAXBAND that fails to consider the heterogeneity of road sections. Zhang et al. [3] pointed out that the requirement of left–right symmetry of the green wave band; the symmetrical requirement in the MULTIBAND may cause it to be unable to achieve larger green wave bandwidths, and thus proposed an improved model. In view of the limitation that MAXBAND and MULTIBAND can only provide green wave bands for through movements, Yang et al. [4] proposed a multi-path green wave control model for arterials, but this model failed to consider the heterogeneity of road segments. Arsava et al. [5] proposed an OD-BAND model based on vehicle origin and destination (OD) information, which can provide green wave bands for specified major OD traffic flows on arterials. Afterward, the OD-BAND was improved to generate green wave bands for all potential OD traffic flows [6]. Next, Arsava et al. [7] proposed a network version of the OD-BAND named OD-NETBAND. Following the logic of MAXBAND, Dai et al. [8] proposed an arterial green wave coordination model for buses. However, the authors of [8] did not explore the impact of bus dwell time randomness on green wave bands, possibly resulting in poor green wave coordination. Later, the problem in [8] was better addressed by Kim et al. [9] in their research. Zhang et al. [10] pointed out that when there are a large number of intersections on an arterial, the green wave bandwidth will be very narrow. To solve this problem, they proposed a green wave control model for long-distance arterial, namely MaxBandLA. Wen et al. [11] pointed out that the MaxBandLA did not consider the influence of phase sequence, queue clearing time, traffic volume, etc., on the green wave bandwidth. Moreover, the number of optimal sub-areas in the MaxBandLA was determined by enumeration. Wen et al. corrected the problems existing in the MaxBandLA and proposed an improved model, namely MaxBandLAM. Yao et al. [12] proposed the concept of green wave bandwidth coordination rate to analyze the mapping relationship between adjacent intersections and then established a multi-path green wave control model with the objective of maximizing the total green wave bandwidth coordination rate. Jing et al. [13] proposed an arterial green wave control model named Pband, whose main highlight is the ability to choose the optimal phase scheme from the NEMA phase scheme and the splitting phase scheme. Li et al. [14] combined the advantages of the AM-BAND model [3] and the PBAND model [13] to propose an asymmetric multi-bandwidth model with phase optimization, named the AM-BAND-PBAND model. Jiang et al. [15] conducted an in-depth study on the essential principles of green wave control by proposing a series of new concepts, such as characteristic points, lines, and equations of the traffic signal progression trajectory, and applied them to arterial green wave control. Xu and Tian [16] proposed an arterial partitioning technique based on OD data provided by connected vehicles to improve the green wave control effect of a long-distance arterial. Aiming at the problem that the green wave is easily destroyed under saturated traffic conditions, Bao et al. [17] proposed the method of real-time adjustment of signal timing scheme and speed guidance to improve the arterial green wave control effect based on the vehicle-road cooperative technology. Research on the arterial green wave control for two traffic modes is summarized as follows. Lin et al. [18] developed the INTEBAND, which is an integrated model capable of coordinating cars and buses. Wang et al. [19] proposed an arterial green wave control model for straight and left-turning trams as well as social vehicles. Xu et al. [20] proposed the Lmband, which can solve the inconsistency problem of intersection grouping arising from buses and cars on long-distance arterials. Zhang et al. [21] specifically considered the stochastic nature of bus dwell time and proposed an arterial green wave control model for cars and social vehicles.

Due to its advantages, such as ease of use, flexibility, and low cost, the electric bicycle is one of the most commonly used travel tools for urban residents in China. Therefore, it is necessary to take into account the green wave coordination demand for electric bicycles. However, because there are significant differences in speeds between electric bicycles and motor vehicles, existing arterial green wave control methods, which mainly focus on motor vehicles, cannot solve the green wave control problem of both motor vehicles and electric bicycles. To address the limitations of existing methods, this paper proposes an arterial multi-path green wave control model that considers both motor vehicles and electric bicycles. The main innovations of this paper are summarized as follows. (1) A novel arterial green wave control model for multiple traffic modes is proposed, which can simultaneously generate multi-path green wave bands for motor vehicles (cars, buses) and electric bicycles on an arterial. (2) By introducing 0/1 variables, the connection between the phase sequence and coordination path in the symmetrical phase scheme is established, enabling synchronous optimization of the phase sequence and coordination path. The reason for introducing 0/1 variables to optimize the phase sequence and the coordinated path is that 0/1 variables have clear, logical meanings and are simple to model.

2. Materials and Methods

2.1. Concepts

2.1.1. Outbound and Inbound

As illustrated in Figure 1, the model presented in this paper is built on a two-way arterial with N intersections. The direction from intersection i to intersection i + 1 is defined as the outbound direction, and the opposite direction is defined as the inbound direction.

2.1.2. Coordinated Path

Figure 2 illustrates multiple coordinated paths between two adjacent intersections. As can be seen from Figure 2, there are four coordinated paths for each traffic mode (cars, buses, and electric bicycles) between adjacent intersections in the outbound or inbound directions. The notation $p$ ($\overline{p}$) represents the number of the coordinated paths in the outbound (inbound) direction. When $p$ ($\overline{p}$) is 1, 2, 3, and 4, it represents the outbound (inbound) coordinated paths 1, 2, 3, and 4, respectively.

2.1.3. Phase Scheme

The symmetrical phase scheme is an intersection phase scheme that allows vehicles from opposite directions to go straight simultaneously and also permits vehicles from opposite directions to turn left at the same time. Due to its simple design and implementation, as well as convenient management and maintenance, the symmetrical phase scheme has been widely applied in traffic signal control practice in China. Therefore, a multi-path green wave control model for multiple traffic modes on an arterial based on the symmetrical phase scheme was developed. Figure 3 illustrates six different phase sequences in the symmetrical phase scheme. As shown in Figure 3, the eastbound and westbound through movements occur in Phase 1, the eastbound and westbound left-turn movements occur in Phase 2, the northbound and southbound through movements occur in Phase 3, and the northbound and southbound left-turn movements occur in Phase 4. Taking Phase Sequence 1 as an example, the meaning of phase sequence is explained as follows. In Phase Sequence 1, Phase 1, 2, 3, and 4 obtain the right of way in turn. Other phase sequences can be explained similarly. Among all phase sequences, Phase Sequence 1 is the traditional practice of designing phase sequences. In green wave control, there is often a competitive relationship among different traffic modes. In order to maximize the green wave bandwidths, this paper no longer adheres to the traditional practice of setting the phase sequence as Phase Sequence 1 but regards the phase sequence as a decision variable, thereby expanding the optimization space.

2.2. Objective Function

The objective function of the presented model, which seeks to maximize the sum of green wave bandwidths for all coordinated paths of each traffic mode on all road segments, can be formulated as:

$$\max z={\displaystyle \sum _{i=1}^{N-1}{\displaystyle \sum _{m=1}^3{\displaystyle \sum _{p=1}^4{b}_{i,m,p}}}}+{\displaystyle \sum _{i=1}^{N-1}{\displaystyle \sum _{m=1}^3{\displaystyle \sum _{\overline{p}=1}^4{\overline{b}}_{i,m,\overline{p}}}}},$$

(1)

where $N$ represents the number of intersections on an arterial, $b_{i,m,p}$ represents the green wave bandwidth for the outbound coordinated path $p$ of the traffic mode $m$ between intersections $i$ and $i+1$, ${\overline{b}}_{i,m,\overline{p}}$ represents the green wave bandwidth for the inbound coordinated path $\overline{p}$ of the traffic mode $m$ between intersections $i$ and $i+1$, and when $m$ takes the values of 1, 2, and 3, it indicates the traffic modes as cars, buses, and electric bicycles, respectively.

2.3. Constraints

The following Figure 4 illustrates the relationship between intersection spacing and signal timing and the movement of traffic flow under the green wave control in the form of two-dimensional coordinates. Specifically, Figure 4 shows the spatial–temporal relationships among the green wave bandwidth, coordinated path, offset, phase sequence, red time, green time, etc., between adjacent intersections. Using the time–space diagram illustrated in Figure 4, some key constraints of the proposed model can be derived.

2.3.1. Location of Green Wave Band

According to the definition of the green wave band, it can be seen that the green wave band of each traffic mode must be limited within the green time. Based on Figure 4, the constraint for the location of the green wave band can be expressed as:

$$\left\{\begin{array}{l}{w}_{i,m,p}^{\mathrm{S}}+b_{i,m,p}\le g_{i,p}^{\mathrm{S}}\\ w_{i+1,m,p}^{\mathrm{E}}+b_{i,m,p}\le g_{i+1,p}^{\mathrm{E}}\end{array}\right.i=1,2,\cdots ,N-1;m=1,2,3;p=1,2,3,4,$$

(2)

$$\left\{\begin{array}{l}{\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}+{\overline{b}}_{i,m,\overline{p}}\le {\overline{g}}_{i,\overline{p}}^{\mathrm{E}}\\ {\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}+{\overline{b}}_{i,m,\overline{p}}\le {\overline{g}}_{i+1,\overline{p}}^{\mathrm{S}}\end{array}\right. i=1,2,\cdots ,N-1;m=1,2,3;\overline{p}=1,2,3,4,$$

(3)

where $w_{i,m,p}^{\mathrm{S}}$ ($w_{i+1,m,p}^{\mathrm{E}}$) represents the time from the beginning of the outbound green time at the starting intersection $i$ (ending intersection $i+1$) to the left side of the green wave bandwidth $b_{i,m,p}$, ${\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}$ (${\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}$) represents the time from the end of the inbound green time at the ending intersection $i$ (starting intersection $i+1$) to the right side of the green wave bandwidth ${\overline{b}}_{i,m,\overline{p}}$, $g_{i,p}^{\mathrm{S}}$ ($g_{i+1,p}^{\mathrm{E}}$) represents the outbound green time that the coordinate path $p$ has at the starting intersection $i$ (ending intersection $i+1$), and ${\overline{g}}_{i,\overline{p}}^{\mathrm{E}}$ (${\overline{g}}_{i+1,\overline{p}}^{\mathrm{S}}$) represents the inbound green time that the coordinated path $\overline{p}$ has at the ending intersection $i$ (starting intersection $i+1$).

Because conflicts exist among different paths, it is highly challenging to ensure that every path for each traffic mode can receive a green wave band. In other words, when the path $p$ does not obtain a green wave band, constraint (2) can be violated; when the path $\overline{p}$ does not obtain a green wave band, constraint (3) can be violated. To solve this problem, 0/1 variables $x_{i,m,p}$ and ${\overline{x}}_{i,m,\overline{p}}$, along with a sufficiently large positive number M, are introduced into constraints (2) and (3). As a result, constraints (2) and (3) are transformed into the following constraints (4) and (5).

$$\left\{\begin{array}{l}{w}_{i,m,p}^{\mathrm{S}}+b_{i,m,p}\le g_{i,p}^{\mathrm{S}}+M\left(1-x_{i,m,p}\right)\\ w_{i+1,m,p}^{\mathrm{E}}+b_{i,m,p}\le g_{i+1,p}^{\mathrm{E}}+M\left(1-x_{i,m,p}\right)\end{array}\right.i=1,2,\cdots ,N-1;m=1,2,3;p=1,2,3,4,$$

(4)

$$\left\{\begin{array}{l}{\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}+{\overline{b}}_{i,m,\overline{p}}\le {\overline{g}}_{i,\overline{p}}^{\mathrm{E}}+M\left(1-{\overline{x}}_{i,m,\overline{p}}\right)\\ {\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}+{\overline{b}}_{i,m,\overline{p}}\le {\overline{g}}_{i+1,\overline{p}}^{\mathrm{S}}+M\left(1-{\overline{x}}_{i,m,\overline{p}}\right)\end{array}\right. i=1,2,\cdots ,N-1;m=1,2,3;\overline{p}=1,2,3,4,$$

(5)

where $x_{i,m,p}$ and ${\overline{x}}_{i,m,\overline{p}}$ are 0/1 variables, and M is a sufficiently large positive number.

$x_{i,m,p}=1$ (${\overline{x}}_{i,m,\overline{p}}=1$) indicates that the path $p$ ($\overline{p}$) of traffic mode $m$ between intersections $i$ and $i+1$ is coordinated. In this case, constraints (4) and (5) are transformed into constraints (2) and (3), respectively. $x_{i,m,p}=0$ (${\overline{x}}_{i,m,\overline{p}}=0$) indicates that the path $p$ ($\overline{p}$) of traffic mode $m$ between intersections $i$ and $i+1$ is not coordinated. In this case, constraints (4) and (5) are relaxed.

2.3.2. Green Wave Bandwidths Are Forced to Be 0

When a certain path of traffic mode $m$ between intersections $i$ and $i+1$ is not coordinated, the green wave bandwidth of this path is 0, which can be described using the following constraint. This constraint ensures that the expression of the objective function is correct.

$$\left\{\begin{array}{l}{b}_{i,m,p}-x_{i,m,p}\le 0\\ b_{i,m,p}\ge 0\end{array}\right.i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4,$$

(6)

$$\left\{\begin{array}{l}{\overline{b}}_{i,m,\overline{p}}-{\overline{x}}_{i,m,\overline{p}}\le 0\\ {\overline{b}}_{i,m,\overline{p}}\ge 0\end{array}\right. i=1,2,\cdots ,N-1;m=1,2,3;\overline{p}=1,2,3,4.$$

(7)

2.3.3. Minimum Green Wave Bandwidth

To ensure the effectiveness of a green wave bandwidth, it is necessary to set a minimum for a green wave bandwidth because a green wave bandwidth that is too small is meaningless. The constraint for the minimum green wave bandwidth can be expressed as:

$$\left\{\begin{array}{l}{b}_{i,m,p}\ge b_{\min }z\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4\\ {\overline{b}}_{i,m,\overline{p}}\ge b_{\min }z\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;\overline{p}=1,2,3,4\end{array}\right.,$$

(8)

where $z$ represents the reciprocal of common cycle time and $b_{\min }$ represents the minimum green wave bandwidth.

Similar to the constraint applied to the location of the green wave band, when the 0/1 variables and the large positive number is introduced into constraint (8), it is updated as:

$$\left\{\begin{array}{l}{b}_{i,m,p}\ge b_{\min }z-M\left(1-x_{i,m,p}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4\\ {\overline{b}}_{i,m,\overline{p}}\ge b_{\min }z-M\left(1-{\overline{x}}_{i,m,\overline{p}}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;\overline{p}=1,2,3,4\end{array}\right..$$

(9)

2.3.4. Loop Integer Constraint

The loop integer constraint is used to describe the coordination relationship between two adjacent intersections.

For the outbound direction, according to Figure 4, the loop integer constraint can be formulated as:

$${\theta }_i+r_{i,p}^{\mathrm{S},\mathrm{L}}+w_{i,m,p}^{\mathrm{S}}+t_{i,m,p}z={\theta }_{i+1}+k_{i+1,m,p}+r_{i+1,p}^{\mathrm{E},\mathrm{L}}+w_{i+1,m,p}^{\mathrm{E}}\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4,$$

(10)

where ${\theta }_i$ (${\theta }_{i+1}$) represents the time from the initial time at intersection $i$ ($i+1$) to the beginning of the green time for west–east through movements, $r_{i,p}^{\mathrm{S},\mathrm{L}}$ ($r_{i+1,p}^{\mathrm{E},\mathrm{L}}$) represents the total red time on the left of the green wave bandwidth for coordinated path $p$ at the starting intersection $i$ (ending intersection $i+1$), $t_{i,m,p}$ represents the travel time for the coordinated path $p$ of traffic mode $m$ between intersections $i$ and $i+1$, and $k_{i+1,m,p}$ is an integer variable. Note that if a bus stops on the coordinated path $p$, its travel time includes the dwell time.

Similarly, constraint (10) can be updated as:

$${\theta }_i+r_{i,p}^{\mathrm{S},\mathrm{L}}+w_{i,m,p}^{\mathrm{S}}+t_{i,m,p}z\le {\theta }_{i+1}+k_{i+1,m,p}+r_{i+1,p}^{\mathrm{E},\mathrm{L}}+w_{i+1,m,p}^{\mathrm{E}}+M\left(1-x_{i,m,p}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4,$$

(11)

$${\theta }_i+r_{i,p}^{\mathrm{S},\mathrm{L}}+w_{i,m,p}^{\mathrm{S}}+t_{i,m,p}z\ge {\theta }_{i+1}+k_{i+1,m,p}+r_{i+1,p}^{\mathrm{E},\mathrm{L}}+w_{i+1,m,p}^{\mathrm{E}}-M\left(1-x_{i,m,p}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;p=1,2,3,4.$$

(12)

In Equation (10), the values of $r_{i,p}^{\mathrm{S},\mathrm{L}}$ or $r_{i+1,p}^{\mathrm{E},\mathrm{L}}$ are closely related to the phase sequences (see Figure 3) and the coordinated paths. In this paper, 0/1 variables are introduced to establish a connection between the phase sequences and the coordinated paths, aiming to achieve simultaneous optimization of both. The expressions for $r_{i,p}^{\mathrm{S},\mathrm{L}}$ and $r_{i+1,p}^{\mathrm{E},\mathrm{L}}$ are as follows:

$$r_{i,p}^{\mathrm{S},\mathrm{L}}={\displaystyle \sum _{u=1}^4{y}_{i,u,v}{g}_{i,u}},$$

(13)

$$r_{i+1,p}^{\mathrm{E},\mathrm{L}}={\displaystyle \sum _{u=1}^4{y}_{i+1,u,v}{g}_{i+1,u}},$$

(14)

where $g_{i,u}$ ($g_{i+1,u}$) represents the green time of phase $u$ at intersection $i$ (intersection $i+1$), and $y_{i,u,v}$ ($y_{i+1,u,v}$) represents the sequence between phase $v$ that the coordinated path $p$ passes through and phase $u$ at intersection $i$ (intersection $i+1$). If phase $u$ is before phase $v$ at intersection $i$ (intersection $i+1$), $y_{i,u,v}$ ($y_{i+1,u,v}$) equals 1; otherwise, $y_{i,u,v}$ ($y_{i+1,u,v}$) equals 0.

For the inbound direction, according to Figure 4, the loop integer constraint can be formulated as:

$${\theta }_i+{\overline{n}}_{i,m,\overline{p}}-{\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}-{\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}={\theta }_{i+1}-{\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}-{\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}+{\overline{t}}_{i,m,\overline{p}}z\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;\overline{p}=1,2,3,4,$$

(15)

where ${\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}$ (${\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}$) represents the total red time on the right of the green wave bandwidth for the coordinated path $\overline{p}$ at the ending intersection $i$ (starting intersection $i+1$), ${\overline{t}}_{i,m,\overline{p}}$ represents the travel time for the coordinated path $\overline{p}$ of traffic mode $m$ between intersections $i$ and $i+1$, and ${\overline{n}}_{i,m,\overline{p}}$ is an integer variable. Note that if a bus stops on the coordinated path $\overline{p}$, its travel time includes the dwell time.

Similarly, constraint (15) can be updated as:

$${\theta }_i+{\overline{n}}_{i,m,\overline{p}}-{\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}-{\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}\le {\theta }_{i+1}-{\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}-{\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}+{\overline{t}}_{i,m,\overline{p}}z+M\left(1-{\overline{x}}_{i,m,\overline{p}}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;\overline{p}=1,2,3,4,$$

(16)

$${\theta }_i+{\overline{n}}_{i,m,\overline{p}}-{\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}-{\overline{w}}_{i,m,\overline{p}}^{\mathrm{E}}\ge {\theta }_{i+1}-{\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}-{\overline{w}}_{i+1,m,\overline{p}}^{\mathrm{S}}+{\overline{t}}_{i,m,\overline{p}}z-M\left(1-{\overline{x}}_{i,m,\overline{p}}\right)\hspace{1em}i=1,2,\dots ,N-1;m=1,2,3;\overline{p}=1,2,3,4.$$

(17)

Similarly, the values of ${\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}$ or ${\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}$ are closely related to the phase sequences and the coordinated paths. ${\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}$ and ${\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}$ are expressed by:

$${\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{R}}=1-{\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{L}}-{\overline{g}}_{i,\overline{p}}^{\mathrm{E}}=1-{\displaystyle \sum _{u=1}^4{y}_{i,u,\overline{v}}{g}_{i,u}}-{\overline{g}}_{i,\overline{p}}^{\mathrm{E}},$$

(18)

$${\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{R}}=1-{\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{L}}-{\overline{g}}_{i+1,\overline{p}}^{\mathrm{S}}=1-{\displaystyle \sum _{u=1}^4{y}_{i+1,u,\overline{v}}{g}_{i+1,u}-{\overline{g}}_{i+1,\overline{p}}^{\mathrm{S}}},$$

(19)

where ${\overline{r}}_{i,\overline{p}}^{\mathrm{E},\mathrm{L}}$ (${\overline{r}}_{i+1,\overline{p}}^{\mathrm{S},\mathrm{L}}$) represents the total red time on the left of the green wave bandwidth for coordinated path $\overline{p}$ at the ending intersection $i$ (starting intersection $i+1$), and $y_{i,u,\overline{v}}$ ($y_{i+1,u,\overline{v}}$) represents the sequence between phase $\overline{v}$ that the coordinated path $\overline{p}$ passes through and phase $u$ at intersection $i$ (intersection $i+1$). If phase $u$ is before phase $\overline{v}$ at intersection $i$ (intersection $i+1$), $y_{i,u,\overline{v}}$ ($y_{i+1,u,\overline{v}}$) equals 1; otherwise, $y_{i,u,\overline{v}}$ ($y_{i+1,u,\overline{v}}$) equals 0.

Note that variables $y_{i,u,v}$, $y_{i+1,u,v}$, $y_{i,u,\overline{v}}$, and $y_{i+1,u,\overline{v}}$ must also meet the following additional constraints:

$$y_{k,r,r}=0,$$

(20)

$$y_{k,r,s}+y_{k,s,r}=1\hspace{1em}r\ne s,$$

(21)

$$y_{k,r,t}\le y_{k,r,s}+y_{k,s,t}\hspace{1em}r\ne s\ne t,$$

(22)

where $r$, $s$, and $t$ are the phase numbers at intersection $k$.

Equation (20) indicates that there is no sequence relationship between the same phase. Equation (21) shows that if phase $r$ is before phase $s$, then phase $s$ is after phase $r$; conversely, if phase $r$ is after phase $s$, then phase $s$ is before phase $r$. Equation (22) is used to eliminate unreasonable combinations of 0/1 variables to ensure the optimized phase sequence is correct.

The relationship between the phase sequences and 0/1 variables is shown in

Table 1. The relationship between phase sequences and 0/1 variables.

Phase Sequence	${\mathit{y}}_{\mathit{k},3,2}$	${\mathit{y}}_{\mathit{k},2,4}$	${\mathit{y}}_{\mathit{k},3,4}$
Phase sequence 1	0	1	1
Phase sequence 2	0	1	0
Phase sequence 3	1	0	1
Phase sequence 4	1	0	0
Phase sequence 5	1	1	1
Phase sequence 6	0	0	0

.

As shown in Table 1, the three 0/1 variables $y_{k,3,2}$, $y_{k,2,4}$, and $y_{k,3,4}$ can form eight different 0/1 combinations. However, as can be seen from Figure 3, there are only six types of phase sequences. At this point, among the eight different 0/1 combinations, there are two invalid combinations: 1 and 2. When $y_{k,3,2}=0$, $y_{k,2,4}=0$, and $y_{k,3,4}=1$, it is the invalid combination 1. When $y_{k,3,2}=1$, $y_{k,2,4}=1$, and $y_{k,3,4}=0$, it is the invalid combination 2. The purpose of introducing Equation (22) is to exclude these two invalid combinations.

2.3.5. Common Cycle Time Constraint

The model presented in this paper can optimize the common cycle time, and the corresponding constraint is:

$${\displaystyle \frac{1}{C_{\max }}}\le z\le {\displaystyle \frac{1}{C_{\min }}},$$

(23)

where $C_{\min }$ and $C_{\max }$ are the lower and upper limits on the common cycle time.

Note that the proposed model does not limit the range of the green wave speed but assumes that the green wave speed is a fixed value. If one wants to limit the range of the green wave speed, one can add the corresponding constraints, which can be found in [3].

3. Results

3.1. Geometry of the Test Arterial

As illustrated in Figure 5, the test arterial runs in a west–east direction and consists of five intersections. For convenience, assume that the direction from west to east is the outbound direction and the opposite direction is the inbound direction. Intersections on the test arterial are numbered sequentially from west to east as intersections 1, 2, 3, 4, and 5. Bus stops are located on road segment 1 (between intersections 1 and 2) and road segment 3 (between intersections 3 and 4).

3.2. Traffic Signal Timing of Isolated Intersections

The green time of each phase at each intersection on the test arterial is shown in

Table 2. Phase green time at each intersection.

Intersection	W–E through Phase	W–E Left-Turn Phase	S–N through Phase	S–N Left-Turn Phase
Intersection 1	0.284	0.202	0.282	0.232
Intersection 2	0.241	0.276	0.266	0.217
Intersection 3	0.257	0.212	0.274	0.257
Intersection 4	0.275	0.22	0.261	0.244
Intersection 5	0.252	0.246	0.235	0.267

. The unit of the green time is the common cycle time. The numbering scheme for each phase is as follows. The west–east through phase is numbered 1, the west–east left-turn phase is numbered 2, the north–south through phase is numbered 3, and the north–south left-turn phase is numbered 4. The lower and upper limits for the common cycle time are 80 s and 100 s, respectively.

3.3. Travel Time

The design speeds for green wave bands of cars, buses and electric bicycles are 45 km/h, 36 km/h, and 25 km/h. The average dwell time at the bus stops is shown in

Table 3. Average bus dwell time at each bus stop.

Road Segments	Outbound Dwell Time/s	Inbound Dwell Time/s
Between intersections 1 and 2	19	17
Between intersections 3 and 4	18	17

. The travel time of each traffic mode on each coordinated path can be easily obtained based on the distances between intersections, the design speed for green wave bands, and the bus dwell time.

3.4. Green Wave Bandwidths Generated by the Presented Model

$b_{\min }$ is set to 4 s, and M is set to 20. By inputting all relevant data into the corresponding formulas, a multi-path green wave control model for the test arterial is established. Utilizing the optimization tool Gurobi 11 to solve the model corresponding to the test arterial, the multi-path green wave control scheme for the test arterial generated by the presented model is summarized as follows. The obtained common cycle time is about 91 s. The obtained offsets of intersections 1, 2, 3, 4, and 5 are 40 s, 45 s, 91 s, 45 s, and 1 s, respectively. The obtained phase sequences of intersections 1, 2, 3, 4, and 5 are sequences 4, 4, 1, 1, and 4, respectively. The produced green wave bandwidths are shown in

Table 4. Green wave bandwidths for the test arterial generated by the presented model.

Outbound Bandwidths	Outbound Path 1	Outbound Path 2	Outbound Path 3	Outbound Path 4	Inbound Bandwidths	Inbound Path 1	Inbound Path 2	Inbound Path 3	Inbound Path 4
$b_{1,1,p}$	0.1944	0	0	0	${\overline{b}}_{1,1,\overline{p}}$	0.1653	0	0	0
$b_{1,2,p}$	0	0.232	0.276	0	${\overline{b}}_{1,2,\overline{p}}$	0	0.217	0.195	0.046
$b_{1,3,p}$	0	0.1915	0.2345	0.0495	${\overline{b}}_{1,3,\overline{p}}$	0	0.1746	0.1316	0.1094
$b_{2,1,p}$	0.0687	0.1483	0	0.1087	${\overline{b}}_{2,1,\overline{p}}$	0.1654	0	0.1106	0.1464
$b_{2,2,p}$	0.1633	0.0537	0	0.2033	${\overline{b}}_{2,2,\overline{p}}$	0.257	0	0	0.241
$b_{2,3,p}$	0.0575	0	0.1545	0.0865	${\overline{b}}_{2,3,\overline{p}}$	0.0648	0.1922	0	0.0488
$b_{3,1,p}$	0	0	0	0.1852	${\overline{b}}_{3,1,\overline{p}}$	0	0	0	0.1932
$b_{3,2,p}$	0	0.2312	0.2132	0.0438	${\overline{b}}_{3,2,\overline{p}}$	0	0.2102	0.212	0.0468
$b_{3,3,p}$	0	0.257	0.22	0	${\overline{b}}_{3,3,\overline{p}}$	0	0.244	0.212	0
$b_{4,1,p}$	0	0	0.2285	0.0465	${\overline{b}}_{4,1,\overline{p}}$	0.0438	0.2232	0	0.0518
$b_{4,2,p}$	0.0889	0	0.1571	0.1179	${\overline{b}}_{4,2,\overline{p}}$	0.1152	0.1518	0	0.1232
$b_{4,3,p}$	0.244	0	0	0.252	${\overline{b}}_{4,3,\overline{p}}$	0.2147	0	0	0.2467

.

3.5. Model Comparison

For convenience, the traditional model focusing on the through path for motor vehicles (cars and buses) is referred to as Model 1, while the model presented in this paper is referred to as Model 2.

Note that Model 1 can be obtained by modifying Model 2. The reason is that Model 1 solves the arterial green wave control problem for through cars and buses, while Model 2 considers the green wave demand of both cars and buses as well as electric bicycles and optimizes all paths as decision variables. The key modification of Model 2 is changing the range of the subscripts. $m$ is changed from $m=1,2,3$ to $m=1,2$, $p$ is changed from $p=1,2,3,4$ to $p=4$, and $\overline{p}$ is changed from $\overline{p}=1,2,3,4$ to $\overline{p}=4$. Model 1 was used to generate a green wave control scheme for the test arterial, and the corresponding green wave control problem was solved by the optimization tool Gurobi 11. The green wave bandwidths for the test arterial generated by Model 1 are shown in

Table 5. Green wave bandwidths for the test arterial generated by Model 1.

Outbound Bandwidths	Outbound Path 1	Outbound Path 2	Outbound Path 3	Outbound Path 4	Inbound Bandwidths	Inbound Path 1	Inbound Path 2	Inbound Path 3	Inbound Path 4
$b_{1,1,p}$	0	0	0	0.241	${\overline{b}}_{1,1,\overline{p}}$	0	0	0	0.241
$b_{1,2,p}$	0.126	0.106	0.15	0	${\overline{b}}_{1,2,\overline{p}}$	0.094	0.123	0.108	0
$b_{1,3,p}$	0.1734	0.0586	0.1026	0	${\overline{b}}_{1,3,\overline{p}}$	0.1664	0.0506	0	0
$b_{2,1,p}$	0	0	0	0.213	${\overline{b}}_{2,1,\overline{p}}$	0	0	0	0.149
$b_{2,2,p}$	0	0.08	0.064	0.177	${\overline{b}}_{2,2,\overline{p}}$	0	0	0	0.241
$b_{2,3,p}$	0.0606	0.1564	0.1514	0	${\overline{b}}_{2,3,\overline{p}}$	0	0.2376	0.2536	0
$b_{3,1,p}$	0	0	0	0.241	${\overline{b}}_{3,1,\overline{p}}$	0	0	0	0.257
$b_{3,2,p}$	0.0548	0.2023	0.1653	0	${\overline{b}}_{3,2,\overline{p}}$	0.0583	0.1858	0.1538	0
$b_{3,3,p}$	0.0954	0.1616	0.1246	0	${\overline{b}}_{3,3,\overline{p}}$	0.1114	0.1326	0.1006	0
$b_{4,1,p}$	0	0	0	0.078	${\overline{b}}_{4,1,\overline{p}}$	0	0	0	0.101
$b_{4,2,p}$	0	0	0	0.1595	${\overline{b}}_{4,2,\overline{p}}$	0	0	0	0.1825
$b_{4,3,p}$	0	0.0638	0.0868	0.1882	${\overline{b}}_{4,3,\overline{p}}$	0	0.1098	0.0868	0.1652

. It is worth noting that although the traditional model targets the through paths of motor vehicles, other paths of motor vehicles and electric bicycles may also indirectly benefit from the green wave bands. We can identify these indirectly obtained green wave bands by the time–space diagram generated by Model 1. These indirectly obtained green wave bands are also shown in Table 5. In addition, green wave bandwidths that are too small are considered ineffective. Since the minimum green wave bandwidth is set to 4 s in this study, any green wave band, whether directly or indirectly obtained, with a bandwidth less than $4\times z$ is considered ineffective. Ineffective green wave bandwidths are uniformly treated as zero. This principle applies to both Models 1 and 2.

4. Discussion

The performance of Models 1 and 2 was discussed at both the road segment and arterial levels. According to Table 4 and Table 5, the green wave bandwidths obtained by each traffic mode on each road segment are displayed in

Table 6. The green wave bandwidth of each traffic mode on each road segment.

Road Segments	Cars			Buses			Electric Bicycles
	Bandwidth Generated by Model 1	Bandwidth Generated by Model 2	Improvement	Bandwidth Generated by Model 1	Bandwidth Generated by Model 2	Improvement	Bandwidth Generated by Model 1	Bandwidth Generated by Model 2	Improvement
Between intersections 1 and 2	0.482	0.3597	−25.4%	0.707	0.966	36.6%	0.5516	0.8911	61.5%
Between intersections 2 and 3	0.362	0.7481	106.7%	0.562	0.9183	63.4%	0.8596	0.6043	−29.7%
Between intersections 3 and 4	0.498	0.3784	−24.0%	0.8203	0.9572	16.7%	0.7262	0.933	28.5%
Between intersections 4 and 5	0.179	0.5938	231.7%	0.342	0.7541	120.5%	0.7006	0.9574	36.7%

, and the total green wave bandwidths obtained by the three traffic modes (cars, buses, and electric bicycles) on each road segment are shown in

Table 7. The total green wave bandwidths obtained by the three traffic modes on each road segment.

Road Segments	Bandwidth Generated by Model 1	Bandwidth Generated Model 2	Improvement
Between intersections 1 and 2	1.7406	2.2168	27.4%
Between intersections 2 and 3	1.7836	2.2707	27.3%
Between intersections 3 and 4	2.0445	2.2686	11.0%
Between intersections 4 and 5	1.2216	2.3053	88.7%

. The calculation method for the improvement in Table 6 and Table 7 is ${\displaystyle \frac{B_2-B_1}{B_1}}\times 100\%$. $B_1$ is the bandwidth generated by Model 1 and $B_2$ is the bandwidth generated by Model 2.

According to Table 6, the comparisons between Models 1 and 2 are summarized as follows. (1) For the green wave bandwidths of cars, Model 2 showed a decrease in generation on road segments 1 and 3, but an increase on road segments 2 and 4. (2) For the green wave bandwidths of electric bicycles, Model 2 showed a reduction in generation on road segment 2, while there was an increase on road segments 1, 3, and 4. (3) For the green wave bandwidths of buses, Model 2 achieved an improvement across all road segments. Among the various increases in green wave bandwidth in Table 6, the improvement for cars between Intersections 4 and 5 is the most notable, reaching 231.7%. The possible reason for this is that from Table 4 and Table 5, it can be observed that Model 1 generates a green wave for the car only on the outbound path 4 and the inbound path 4 between intersections 4 and 5, and the bandwidths of the green wave for all other paths are 0, while Model 2 generates a green wave for the car on the outbound paths 3 and 4 and the inbound paths 1, 2, and 4 between intersections 4 and 5.

According to Table 7, it was observed that, compared to Model 1, the total green wave bandwidths generated by Model 2 for cars, buses and electric bicycles were increased on all road segments. Specifically, the smallest increase was on road segment 3, with an increase of 11%, and the most significant increase was on road segment 4, with an improvement of 88.7%.

Based on Table 6 and Table 7, one could observe that although the green wave bandwidths generated by Model 2 for cars or electric bicycles decreased on certain road segments, the presented model (Model 2) achieved an increase in each road segment in terms of the total green wave bandwidths for the three traffic modes.

According to Table 4 and Table 5, in terms of the entire arterial, the total green wave bandwidths obtained by each traffic mode are shown in

Table 8. Comparison of the total green wave bandwidths obtained by each traffic mode between Model 1 and Model 2 at the arterial level.

Traffic Modes	Bandwidth Generated by Model 1	Bandwidth Generated Model 2	Improvement
Cars	1.521	2.08	36.8%
Buses	2.4313	3.5956	47.9%
Electric bicycles	2.838	3.3858	19.3%

. The calculation method for the improvement in Table 8 is the same as the one shown in Table 6.

According to Table 8, the objective function values on the test arterial for Models 1 and 2 are 6.7903 and 9.0614, respectively. Compared to Model 1, the total green wave bandwidths generated by Model 2 for all traffic modes on the test arterial increased by 33.4%. Specifically, compared to Model 1, the total green wave bandwidths of cars, buses and electric bicycles generated by Model 2 at the entire arterial level were increased by 36.8%, 47.9%, and 19.3%, respectively.

Based on the analysis at the road segment and arterial levels, the presented model demonstrates its capability to fully consider the green wave coordination demands of each traffic mode, thereby significantly enhancing the green wave bandwidths of the three traffic modes.

5. Conclusions

Existing arterial green wave control methods primarily focus on motor vehicles, ignoring the important traffic mode of electric bicycles. To address this problem, this paper presents an arterial multi-path green wave control model that considers both motor vehicles and electric bicycles. The conclusions of this study can be summarized as follows.

(1) This paper introduces a model that can simultaneously optimize the paths of multiple traffic modes and coordinated variables, namely offset, phase sequence, and common cycle time.

(2) Traditional arterial green wave control models mainly focus on motor vehicles, ignoring electric bicycles, which results in electric bicycles only occasionally benefiting from the green wave bands. In contrast, the presented model can simultaneously consider the green wave control demands of both motor vehicles and electric bicycles, ensuring that electric bicycles can regularly benefit from the green wave bands.

(3) The proposed model can solve the multi-path green wave control problem of multiple traffic modes at an arterial level, but it cannot solve the network multi-path green wave control problem of multiple traffic modes. In the future, research can be extended from the arterial studied in this paper to a closed traffic network [22] to investigate the green wave control method for multiple traffic modes at a road network level.

(4) The model proposed in this paper assumes that each path for every traffic mode is equally important and fails to fully consider the heterogeneity of different paths. Therefore, how to propose appropriate weight coefficients to assess the importance of different traffic modes on the same path and the importance of different paths for the same traffic mode is another worthwhile research direction.