Green Wave Control Model Simultaneously Considering Passenger Cars and Buses in Closed Road Networks

Abstract

Existing green wave control methods for passenger cars and buses mainly focus on maximizing bandwidths at the arterial level. There is little research on green wave control for both at the closed road network level, which makes it difficult to improve the efficiency of the entire area. To address this, a green wave control model that considers both passenger cars and buses in closed road networks is presented in this paper. The objective function of the model is to maximize the sum of the weighted bandwidths of passenger cars and buses on each segment of the road network. The relationships between car green bands, bus green bands, offsets, phase sequences, red time, green time, etc. are analyzed on the level of arterials and road networks, respectively, using time–space diagrams. Based on these analyses, the key constraints of the model are constructed accordingly. In addition, 0/1 variables and a sufficiently large positive number M are introduced to relax some of the constraints to ensure that the presented model has feasible solutions. The results of the numerical example demonstrate that compared with the fixed phase sequence schemes 1, 2, and 3, the total weighted bandwidth generated by the presented model increased by 9.5%, 16.4%, and 17%, respectively. Compared with the model without constraint relaxation, the presented model can still find a global, optimal solution when the common cycle time is fixed, while the model without constraint relaxation has no feasible solution.

1. Introduction

Currently, many cities are facing traffic congestion problems. As a cost-effective way to reduce traffic congestion, green wave control enables vehicles to pass through signalized intersections without stopping as much as possible by collaboratively designing signal timing schemes for each intersection. Many green wave control models have been proposed by researchers in past years. Little et al. [1] conducted an in-depth analysis of the green wave control problem and proposed the well-known green wave control model for arterials, namely MAXBAND. MAXBAND fails to take into account the differences in road segments and generates the same bandwidth on all segments, which may not achieve a good green wave control effect. To address this issue, Gartner et al. [2] proposed an arterial green wave control model called MULTIBAND, where the bandwidths vary with the road sections. Yang et al. [3] proposed a two-stage traffic signal optimization model for diverging diamond intersections, where the first stage is to maximize the capacity, and the second is to maximize the weighted green wave bandwidth for critical paths. In MULTIBAND, the green wave band is required to be symmetrical along the progression line. Zhang et al. [4] developed a modified MULTIBAND to relax the requirement to yield larger bandwidths. Kim et al. [5] proposed a method to generate dynamic green wave bandwidth using closed-loop signal data. Cheng et al. [6] proposed a model that can simultaneously optimize green wave bandwidth and crossover spacing for diverging diamond interchanges. Ji and Cheng [7] presented a two-way arterial green wave control method under asymmetric release mode. Jing et al. [8] proposed a variable green wave bandwidth maximization model for arterials that can simultaneously optimize phase plans (NEMA phasing and split phasing), offsets, and phase sequences. Addressing the limitations of the traditional algebraic methods that are mainly suitable for single-cycle control, Lu et al. [9] proposed an algebraic method for arterial green wave control in unequal double-cycle scenarios. Typical research for green wave control in road networks is summarized as follows. Chang et al. [10] extended the MAXBAND and proposed a model that can address the green wave control problem in closed road networks, namely MAXBAND-86. Based on MULTIBAND, Stamatiadis and Gartner [11] proposed a road network version of the variable green wave bandwidth model called MULTIBAND-96. Zhang et al. [12] proposed two green wave bandwidth optimization models to generate traffic signal coordination schemes for long arterials and grid road networks. Lu et al. [13], taking one-way road networks as the research subject and considering the impact of pedestrian-exclusive phases, proposed a green wave coordination control method for one-way road networks. Based on MAXBAND [1] and [14], Li and Wang [15] proposed a multi-path bandwidth optimization model for traffic networks considering variable phase structures. Jing et al. [16] considered coordinated paths as decision variables and incorporated them into bandwidth optimization, proposing a collaborative optimization method of path selection and green wave control for unclosed road networks.

The aforementioned green wave control models for arterials or road networks are based on a single traffic mode—passenger cars—while ignoring another common motorized traffic mode for urban commuting: buses. In fact, buses, as an indispensable means of urban commuting, have advantages such as high capacity, efficiency, low carbon emissions, and environmental friendliness. The green wave control demand for buses cannot be ignored. Consequently, it is necessary to consider the green wave control demand of both passenger cars and buses concurrently. Dai et al. [17] analyzed the constraint relationship between green wave bands for passenger cars and buses on arterials and proposed a green wave bandwidth optimization model with the minimization of the weighted sum of bus travel time as the optimization objective. Ma et al. [18] proposed a multi-mode (passenger cars and transit vehicles) arterial green wave control model with the function of an arterial system partition. Aiming at the green wave control problem of passenger cars and buses, Florek [19] proposed the BUS-MULTIBAND and the BUS-AM-BAND based on the MULTIBAND and the AM-BAND [4]. Xu et al. [20] proposed an arterial bus signal priority method considering multi-path coordination of social vehicles, addressing the issue where implementing bus priority can reduce the green wave control optimization space for social vehicles.

In summary, many scholars have studied the green wave control of passenger cars and buses at the arterial level. However, there is little research on the green wave control of the two at the level of closed road networks. The existing arterial green wave control methods for passenger cars and buses cannot address the green wave control problem in closed road networks. The reason is that the green wave control for passenger cars and buses in closed road networks is different from that on arterials. A closed road network consists of multiple intersecting arterials, and these arterials are not independent of each other within the network. Moreover, the arterial green wave control can only improve the efficiency of one single road, while the green wave control for closed road networks can simultaneously coordinate the green wave bands of multiple intersecting arterials within the network, which is more conducive to achieving efficient regional control effects. Therefore, this paper takes passenger cars and buses as the research objects, analyses the spatio-temporal relationships between their driving characteristics and green wave control using time–space diagrams, and then presents a green wave control model for cars and buses in closed road networks. The highlights of the presented model are summarized as follows. (1) Phase sequences under the symmetric phase scheme are employed as a decision variable, which breaks the traditional practice that phase sequence is fixed under the symmetric phase scheme and improves the applicability of the model. (2) A sufficiently large positive number M and 0/1 variables are introduced to relax certain constraints, which can prevent infeasible solution scenarios when solving the model.

2. Materials and Methods

2.1. Concepts

The topology of the closed road network studied in this paper is shown in Figure 1. The road network is closed; that is, each intersection is located in a closed loop consisting of multiple road segments. The numbers in parentheses shown in Figure 1 represent intersection numbers. The notation (i, j) represents the jth intersection on the ith arterial. In addition, when an intersection is located in two different arterials where green wave control will be implemented, it will have two different numbers, but the two numbers indicate the same intersection. It is assumed that the direction from west to east (the arterial runs west-east) or south to north (the arterial runs north-south) is defined as the outbound, and the opposite direction is defined as the inbound.

2.2. Model Formulation

2.2.1. Objective Function

Taking the sum of the weighted bandwidths on each road segment as the optimization goal, the objective function of the green wave control model for passenger cars and buses in closed road networks can be formulated as

$$\max {\displaystyle \sum _{i=1}^A{\displaystyle \sum _{j=1}^{N_i-1}\left({\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)}},$$

(1)

where A is the number of arterials in the road network, N_i is the number of intersections on the ith arterial, $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$] is the outbound [inbound] car bandwidth between (i, j) and (i, j + 1), $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$] is the outbound [inbound] bus bandwidth between (i, j) and (i, j + 1), ${\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$] is the weighting factor for $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$], and ${\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$] is the weighting factor for $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$].

The number of passengers per hour in the outbound [inbound] through cars at (i, j + 1) [(i, j)] can be used as the weighting factor ${\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$]. The number of passengers per hour in the outbound [inbound] through buses at (i, j + 1) [(i, j)] can be used as the weighting factor ${\alpha }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{\alpha }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$]. In addition, the weighting factors can also be specified by traffic engineers according to the control needs.

2.2.2. Constraints

The relationship between the driving characteristics of passenger cars and buses and green wave control at the arterial level can be described in Figure 2 and Figure 3.

• Constraints on the location of green wave bands

As can be easily seen from Figure 2, in order to limit the green wave bands of cars within the green time, the following constraints need to be satisfied:

$$\{\begin{array}{l}{w}_{\left(i,j\right)}^{\mathrm{car}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j\right)}^D\\ w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j+1\right)}^D\\ {\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j\right)}^D-{\overline{\epsilon }}_{\left(i,j\right)}^{\mathrm{car}}\\ {\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j+1\right)}^D\end{array},$$

(2)

where $w_{\left(i,j\right)}^{\mathrm{car}}$ [${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$] is the time from the beginning [end] of the green time at (i, j) to the left [right] side of the car outbound [inbound] bandwidth, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}$] is the time from the beginning [end] of the green time at (i, j + 1) to the left [right] side of the car outbound [inbound] bandwidth, $g_{\left(i,j\right)}^D$ [$g_{\left(i,j+1\right)}^D$] is the green time for the coordinated movement D at (i, j) [(i, j + 1)], D = {WET, SNT}, WET denotes the west–east through movement, SNT denotes the south–north through movement, and ${\overline{\epsilon }}_{\left(i,j\right)}^{\mathrm{car}}$ denotes the initial queue clearance time corresponding to the car inbound bandwidth at (i, j).

Due to the difference in speeds between cars and buses, it may result in a situation where cars and buses cannot simultaneously obtain green wave bands on certain road segments. To ensure that there is a feasible solution, the 0/1 variables $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ and a sufficiently large positive number M are introduced to relax some of the constraints in the presented model.

With the introduction of $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and M, Equation (2) can be updated as

$$\{\begin{array}{l}{w}_{\left(i,j\right)}^{\mathrm{car}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j+1\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ {\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j\right)}^D-{\overline{\epsilon }}_{\left(i,j\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ {\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le g_{\left(i,j+1\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\end{array}.$$

(3)

When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ equals to 0, it indicates that there are car green wave bands between (i, j) and (i, j + 1), and Equation (3) will transform into Equation (2). When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ equals to 1, it indicates that there are no car green wave bands between (i, j) and (i, j + 1), and Equation (2) is relaxed.

Similarly, the constraints on the location of the bus green wave bands are formulated as

$$\{\begin{array}{l}{w}_{\left(i,j\right)}^{\mathrm{bus}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le g_{\left(i,j\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\\ w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le g_{\left(i,j+1\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\\ {\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le g_{\left(i,j\right)}^D-{\overline{\epsilon }}_{\left(i,j\right)}^{\mathrm{bus}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\\ {\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le g_{\left(i,j+1\right)}^D+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\end{array},$$

(4)

where $w_{\left(i,j\right)}^{\mathrm{bus}}$ [${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$] is the time from the beginning [end] of the green time at (i, j) to the left [right] side of the bus outbound [inbound] bandwidth, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}$] is the time from the beginning [end] of the green time at (i, j + 1) to the left [right] side of the bus outbound [inbound] bandwidth, and ${\overline{\epsilon }}_{\left(i,j\right)}^{\mathrm{bus}}$ denotes the initial queue clearance time corresponding to the bus inbound bandwidth at (i, j).

When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ equals to 0, it indicates that there are bus green wave bands between (i, j) and (i, j + 1). When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ equals to 1, it indicates that there are no bus green wave bands between (i, j) and (i, j + 1).

• Arterial-level loop integer constraints

Since all intersections are operated with the same cycle time (common cycle time), there is an arterial-level loop integer constraint for cars, according to Figure 2, and it can be expressed as

$${\overline{\varphi }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\varphi }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}=m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}},$$

(5)

where ${\varphi }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{\varphi }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$] denotes the time from the center of an outbound [inbound] red time at (i, j) to the center of the nearest outbound [inbound] red time at (i, j + 1), the centers of an outbound [inbound] red time at (i, j) and (i, j + 1) are situated closest to the left [right] side of an outbound [inbound] green wave band for cars, and $m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ is an integer variable, representing an integer multiple of the common cycle time.

As shown in Figure 2, ${\varphi }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and ${\overline{\varphi }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ can be further formulated by

$${\varphi }_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}=0.5r_{\left(i,j\right)}^D+w_{\left(i,j\right)}^{\mathrm{car}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}-0.5r_{\left(i,j+1\right)}^D,$$

(6)

$${\overline{\varphi }}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}=0.5r_{\left(i,j\right)}^D+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}z-{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}-0.5r_{\left(i,j+1\right)}^D,$$

(7)

where $r_{\left(i,j\right)}^D$ [$r_{\left(i,j+1\right)}^D$] is the red time for the coordinated movement D at (i, j) [(i, j + 1)], $t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [${\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$] is the car travel time between (i, j) and (i, j + 1) in the outbound [inbound] direction, and $z$ is the reciprocal of the common cycle time.

Combining Equations (5)–(7), Equation (5) can be updated as

$$\left(r_{\left(i,j\right)}^D-r_{\left(i,j+1\right)}^D\right)+\left(w_{\left(i,j\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+\left(t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)z-\left(w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}\right)=m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}.$$

(8)

By relaxing Equation (8), the equality constraint shown in Equation (8) can be equivalently transformed into two inequality constraints, which are Equations (9) and (10).

$$\left(r_{\left(i,j\right)}^D-r_{\left(i,j+1\right)}^D\right)+\left(w_{\left(i,j\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+\left(t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)z-\left(w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}\right)\le m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M.$$

(9)

$$\left(r_{\left(i,j\right)}^D-r_{\left(i,j+1\right)}^D\right)+\left(w_{\left(i,j\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+\left(t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)z-\left(w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}\right)\ge m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M.$$

(10)

When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ equals to 0, Equations (9) and (10) will transform into Equation (8). When $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ equals to 1, Equation (8) is relaxed.

Similarly, the arterial-level loop integer constraint with relaxation for buses is expressed as

$$\left(r_{\left(i,j\right)}^D-r_{\left(i,j+1\right)}^D\right)+\left(w_{\left(i,j\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)+\left(t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)z-\left(w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}\right)\le m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M,$$

(11)

$$\left(r_{\left(i,j\right)}^D-r_{\left(i,j+1\right)}^D\right)+\left(w_{\left(i,j\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)+\left(t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)z-\left(w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}\right)\ge m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M,$$

(12)

where $t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ [${\overline{t}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$] is the bus travel time between (i, j) and (i, j + 1) in the outbound [inbound] direction and $m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ is an integer variable representing an integer multiple of the common cycle time.

• Network-level loop integer constraints

Figure 4 illustrates a closed loop consisting of four road links. Once the offsets of road links 1, 2, and 3 are determined, the offset of road link 4 will be determined accordingly. In other words, the offset of road link 4 cannot be independently determined. This is the reason for the formation of the network-level loop integer constraints.

The network-level loop integer constraints are described in Figure 5.

The network-level loop integer constraint for cars is formulated as

$$\begin{array}{l}{\varphi }_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}-{\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}+{\varphi }_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}+\\ {\overline{\varphi }}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}-{\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}+{\overline{\varphi }}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}=n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}\end{array},$$

(13)

where ${\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}$, ${\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}$, ${\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}$, and ${\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}$ are intranode offset variables and they can be explained uniformly using the symbol ${\psi }_{\left(a,b\right);\left(c,d\right)}^{F_1{F}_2}$, and $n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}$ is an integer variable representing an integer multiple of the common cycle time.

${\psi }_{\left(a,b\right);\left(c,d\right)}^{F_1{F}_2}$ denotes the time difference between the center of the red time for movement F₁ and the nearest center of the red time for movement F₂ at (a, b) or (c, d). It is positive if the center of the red time for movement F₁ is to the right of the center of the red time for movement F₂. Otherwise, ${\psi }_{\left(a,b\right);\left(c,d\right)}^{F_1{F}_2}$ is negative. F₁ = {S, W, N, E} and F₂ = {W, N, E, S}. S, W, N, and E denote the through movement at the south, west, north, and east approaches, respectively.

Based on the definitions of ${\varphi }_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}$, ${\varphi }_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}$, ${\overline{\varphi }}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}$, and ${\overline{\varphi }}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}$, Equation (13) can be updated as

$$\begin{array}{l}0.5\left(r_{\left(m,n\right)}^{\mathrm{SNT}}+r_{\left(e+1,h\right)}^{\mathrm{WET}}+r_{\left(m+1,k\right)}^{\mathrm{SNT}}+r_{\left(e,f\right)}^{\mathrm{WET}}\right)-0.5\left(r_{\left(m,n+1\right)}^{\mathrm{SNT}}+r_{\left(e+1,h+1\right)}^{\mathrm{WET}}+r_{\left(m+1,k+1\right)}^{\mathrm{SNT}}+r_{\left(e,f+1\right)}^{\mathrm{WET}}\right)+\\ \left(w_{\left(m,n\right)}^{\mathrm{car}}+w_{\left(e+1,h\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)-\left(w_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+w_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f+1\right)}^{\mathrm{car}}\right)+\\ \left(t_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+t_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)z-\\ {\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}-{\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}=n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}\end{array}.$$

(14)

By relaxing Equation (14), the equality constraint shown in Equation (14) can be equivalently transformed into two inequality constraints, which are Equations (15) and (16).

$$\begin{array}{l}0.5\left(r_{\left(m,n\right)}^{\mathrm{SNT}}+r_{\left(e+1,h\right)}^{\mathrm{WET}}+r_{\left(m+1,k\right)}^{\mathrm{SNT}}+r_{\left(e,f\right)}^{\mathrm{WET}}\right)-0.5\left(r_{\left(m,n+1\right)}^{\mathrm{SNT}}+r_{\left(e+1,h+1\right)}^{\mathrm{WET}}+r_{\left(m+1,k+1\right)}^{\mathrm{SNT}}+r_{\left(e,f+1\right)}^{\mathrm{WET}}\right)+\\ \left(w_{\left(m,n\right)}^{\mathrm{car}}+w_{\left(e+1,h\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)-\left(w_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+w_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f+1\right)}^{\mathrm{car}}\right)+\\ \left(t_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+t_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)z-{\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}-\\ {\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}\le n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}+\left(x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)M\end{array},$$

(15)

$$\begin{array}{l}0.5\left(r_{\left(m,n\right)}^{\mathrm{SNT}}+r_{\left(e+1,h\right)}^{\mathrm{WET}}+r_{\left(m+1,k\right)}^{\mathrm{SNT}}+r_{\left(e,f\right)}^{\mathrm{WET}}\right)-0.5\left(r_{\left(m,n+1\right)}^{\mathrm{SNT}}+r_{\left(e+1,h+1\right)}^{\mathrm{WET}}+r_{\left(m+1,k+1\right)}^{\mathrm{SNT}}+r_{\left(e,f+1\right)}^{\mathrm{WET}}\right)+\\ \left(w_{\left(m,n\right)}^{\mathrm{car}}+w_{\left(e+1,h\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)-\left(w_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+w_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{w}}_{\left(e,f+1\right)}^{\mathrm{car}}\right)+\\ \left(t_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+t_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+{\overline{t}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)z-{\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}-\\ {\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}\ge n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}-\left(x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}+x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}+x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}+x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}\right)M\end{array},$$

(16)

where $x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{car}}$, $x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{car}}$, $x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{car}}$, and $x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{car}}$ are 0/1 variables.

When all these four binary variables equal to 0, Equations (15) and (16) will transform into Equation (14). Otherwise, Equation (14) is relaxed.

Similarly, the network-level loop integer constraint with relaxation for buses is expressed as

$$\begin{array}{l}0.5\left(r_{\left(m,n\right)}^{\mathrm{SNT}}+r_{\left(e+1,h\right)}^{\mathrm{WET}}+r_{\left(m+1,k\right)}^{\mathrm{SNT}}+r_{\left(e,f\right)}^{\mathrm{WET}}\right)-0.5\left(r_{\left(m,n+1\right)}^{\mathrm{SNT}}+r_{\left(e+1,h+1\right)}^{\mathrm{WET}}+r_{\left(m+1,k+1\right)}^{\mathrm{SNT}}+r_{\left(e,f+1\right)}^{\mathrm{WET}}\right)+\\ \left(w_{\left(m,n\right)}^{\mathrm{bus}}+w_{\left(e+1,h\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)-\left(w_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+w_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(e,f+1\right)}^{\mathrm{bus}}\right)+\\ \left(t_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+t_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)z-{\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}-\\ {\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}\le n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+\left(x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)M\end{array},$$

(17)

$$\begin{array}{l}0.5\left(r_{\left(m,n\right)}^{\mathrm{SNT}}+r_{\left(e+1,h\right)}^{\mathrm{WET}}+r_{\left(m+1,k\right)}^{\mathrm{SNT}}+r_{\left(e,f\right)}^{\mathrm{WET}}\right)-0.5\left(r_{\left(m,n+1\right)}^{\mathrm{SNT}}+r_{\left(e+1,h+1\right)}^{\mathrm{WET}}+r_{\left(m+1,k+1\right)}^{\mathrm{SNT}}+r_{\left(e,f+1\right)}^{\mathrm{WET}}\right)+\\ \left(w_{\left(m,n\right)}^{\mathrm{bus}}+w_{\left(e+1,h\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)-\left(w_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+w_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{w}}_{\left(e,f+1\right)}^{\mathrm{bus}}\right)+\\ \left(t_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+t_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+{\overline{t}}_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)z-{\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}-{\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}-\\ {\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}-{\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}\ge n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{bus}}-\left(x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}+x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}+x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}+x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}\right)M\end{array},$$

(18)

where $x_{\left(m,n\right),\left(m,n+1\right)}^{\mathrm{bus}}$, $x_{\left(e+1,h\right),\left(e+1,h+1\right)}^{\mathrm{bus}}$, $x_{\left(m+1,k\right),\left(m+1,k+1\right)}^{\mathrm{bus}}$, and $x_{\left(e,f\right),\left(e,f+1\right)}^{\mathrm{bus}}$ are 0/1 variables, and $n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{bus}}$ is an integer variable representing an integer multiple of the common cycle time.

Based on the definitions of ${\psi }_{\left(e+1,h\right);\left(m,n+1\right)}^{\mathrm{SW}}$, ${\psi }_{\left(e+1,h+1\right);\left(m+1,k+1\right)}^{\mathrm{WN}}$, ${\psi }_{\left(e,f+1\right);\left(m+1,k\right)}^{\mathrm{NE}}$, and ${\psi }_{\left(e,f\right);\left(m,n\right)}^{\mathrm{ES}}$, the expressions of these intranode offset variables are related to the phase scheme and phase sequence adopted at an intersection. The symmetrical phase scheme (see Figure 6) has the advantages of simple implementation and low management cost and is widely used in traffic signal optimization practices in China. Based on this, the expressions of these intranode offset variables under the symmetrical phase scheme are given in the following discussion. Meanwhile, in view of the fact that a conventional phase sequence (sequence 1 in Figure 6) is not always adopted in traffic signal optimization practices, 0/1 variables are introduced to optimize the phase sequences, which contributes to better control effects and improves the applicability of the presented model.

Suppose an intersection is numbered (p, q) and (r, s) in the W-E and S-N arterials, respectively. The expressions of ${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{SW}}$, ${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{WN}}$, ${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{NE}}$, and ${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{ES}}$ are as follows:

$${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{SW}}=\{\begin{array}{l}\left(2y_{\left(p,q\right);\left(r,s\right)}-u_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}-0.5r_{\left(p,q\right)}^{\mathrm{WET}}+g_{\left(r,s\right)}^{\mathrm{SNT}}\right)+\\ \left(2z_{\left(p,q\right);\left(r,s\right)}-v_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}+0.5r_{\left(p,q\right)}^{\mathrm{WET}}-g_{\left(p,q\right)}^{\mathrm{WEL}}\right),g_{\left(r,s\right)}^{\mathrm{SNL}}\le g_{\left(p,q\right)}^{\mathrm{WEL}}\\ \left(2y_{\left(p,q\right);\left(r,s\right)}-u_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}-0.5r_{\left(p,q\right)}^{\mathrm{WET}}+g_{\left(r,s\right)}^{\mathrm{SNT}}\right)+\\ \left(v_{\left(p,q\right);\left(r,s\right)}-2z_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}+0.5r_{\left(p,q\right)}^{\mathrm{WET}}-g_{\left(r,s\right)}^{\mathrm{SNL}}\right),g_{\left(r,s\right)}^{\mathrm{SNL}}>g_{\left(p,q\right)}^{\mathrm{WEL}}\end{array},$$

(19)

$${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{WN}}=\{\begin{array}{l}\left(u_{\left(p,q\right);\left(r,s\right)}-2y_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}-0.5r_{\left(p,q\right)}^{\mathrm{WET}}+g_{\left(r,s\right)}^{\mathrm{SNT}}\right)+\\ \left(v_{\left(p,q\right);\left(r,s\right)}-2z_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}+0.5r_{\left(p,q\right)}^{\mathrm{WET}}-g_{\left(p,q\right)}^{\mathrm{WEL}}\right),g_{\left(r,s\right)}^{\mathrm{SNL}}\le g_{\left(p,q\right)}^{\mathrm{WEL}}\\ \left(u_{\left(p,q\right);\left(r,s\right)}-2y_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}-0.5r_{\left(p,q\right)}^{\mathrm{WET}}+g_{\left(r,s\right)}^{\mathrm{SNT}}\right)+\\ \left(2z_{\left(p,q\right);\left(r,s\right)}-v_{\left(p,q\right);\left(r,s\right)}\right)\left(0.5r_{\left(r,s\right)}^{\mathrm{SNT}}+0.5r_{\left(p,q\right)}^{\mathrm{WET}}-g_{\left(r,s\right)}^{\mathrm{SNL}}\right),g_{\left(r,s\right)}^{\mathrm{SNL}}>g_{\left(p,q\right)}^{\mathrm{WEL}}\end{array},$$

(20)

$${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{NE}}={\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{SW}},$$

(21)

$${\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{ES}}={\psi }_{\left(p,q\right);\left(r,s\right)}^{\mathrm{WN}},$$

(22)

where $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ are 0/1 variables, which are employed to optimize the phase sequences.

$u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ should satisfy the following constraints:

$$\{\begin{array}{l}{u}_{\left(p,q\right);\left(r,s\right)}+v_{\left(p,q\right);\left(r,s\right)}=1\\ u_{\left(p,q\right);\left(r,s\right)}-y_{\left(p,q\right);\left(r,s\right)}\ge 0\\ v_{\left(p,q\right);\left(r,s\right)}-z_{\left(p,q\right);\left(r,s\right)}\ge 0\end{array}.$$

(23)

The relationship between $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ and phase sequences are shown in

Table 1. Relationship between $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ and phase sequences.

Combination of 0/1 Variables	Phase Sequences
$u_{\left(p,q\right);\left(r,s\right)}=1$ and $y_{\left(p,q\right);\left(r,s\right)}=0$	sequences 2 or 5
$u_{\left(p,q\right);\left(r,s\right)}=1$ and $y_{\left(p,q\right);\left(r,s\right)}=1$	sequences 3 or 4
$v_{\left(p,q\right);\left(r,s\right)}=1$ and $z_{\left(p,q\right);\left(r,s\right)}=0$	sequence 1
$v_{\left(p,q\right);\left(r,s\right)}=1$ and $z_{\left(p,q\right);\left(r,s\right)}=1$	sequence 6

.

• Bandwidth ratio constraints

To balance the outbound and inbound bandwidth, the bandwidth ratio constraints for cars and buses are expressed as

$$\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right){\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}-\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\ge 0,$$

(24)

$$\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right){\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}-\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\ge 0,$$

(25)

where $k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [$k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$] is the car [bus] ratio of inbound to outbound bandwidth between (i, j) and (i, j + 1).

The ratio of passengers per hour in the inbound through cars [buses] at (i, j) to passengers per hour in the outbound through cars [buses] at (i, j + 1) can be used as the value of $k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ [$k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$]. Additionally, the values of $k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and $k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$ can be specified by traffic engineers according to the control needs.

Equations (24) and (25) can be updated to the following constraints by relaxing them:

$$\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right){\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}-\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\ge 0,$$

(26)

$$\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right){\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}-\left(1-k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)k_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\ge 0.$$

(27)

• Mutual offset constraints

The mutual offset constraint is due to the fact that cars and buses are controlled by the same traffic signal system. Consequently, car bands and bus bands should have an identical offset. The mutual offset constraints are formulated as

$$\left(w_{\left(i,j\right)}^{\mathrm{car}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+m_{\left(i,j\right),\left(i,j+1\right)}-\left(w_{\left(i,j\right)}^{\mathrm{bus}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)=0,$$

(28)

where $m_{\left(i,j\right),\left(i,j+1\right)}$ is an integer variable representing an integer multiple of the common cycle time.

Equation (28) can be equivalently transformed into two inequality constraints by relaxing it, as shown in Equations (29) and (30).

$$\left(w_{\left(i,j\right)}^{\mathrm{car}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+m_{\left(i,j\right),\left(i,j+1\right)}-\left(w_{\left(i,j\right)}^{\mathrm{bus}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)\le \left(x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)M,$$

(29)

$$\left(w_{\left(i,j\right)}^{\mathrm{car}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\right)+m_{\left(i,j\right),\left(i,j+1\right)}-\left(w_{\left(i,j\right)}^{\mathrm{bus}}+t_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}z-w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)\ge -\left(x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\right)M.$$

(30)

• Minimum bandwidth constraints

If the bandwidth is too small (for example, the bandwidth is 1 s), vehicles cannot actually get the green wave experience. Therefore, a green wave bandwidth that is too small is meaningless. To avoid the above situation, it is necessary to meet the following minimum bandwidth constraints.

$$\{\begin{array}{l}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\ge b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car},\min }z-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\ge {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car},\min }z-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\ge b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus},\min }z-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\\ {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\ge {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus},\min }z-x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\end{array},$$

(31)

where $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car},\min }$, ${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car},\min }$, $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus},\min }$, and ${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus},\min }$ are the minimum for the corresponding bandwidths.

• Constraints that bandwidths are forced to be 0

When there is no green wave band on a road segment, the bandwidth is 0, and the following constraints should be satisfied:

$$\{\begin{array}{l}{b}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le 1\\ {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}\le 1\\ b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le 1\\ {\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}+x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\le 1\\ b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}},{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}},b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}},{\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}\ge 0\end{array}.$$

(32)

When there is no green wave band on a road segment, Equation (32) can force the corresponding bandwidth to be 0, which ensures that the objective optimization function expression is correct.

• Common cycle time constraints

To optimize the common signal cycle, the following constraints should be satisfied:

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

(33)

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

• Additional constraints on interference variables

It is obvious that the interference variables $w_{\left(i,j\right)}^{\mathrm{car}}$, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, ${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, ${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}$, $w_{\left(i,j\right)}^{\mathrm{bus}}$, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$, ${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$, and ${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}$ also need to satisfy the following constraints:

$$\{\begin{array}{l}{w}_{\left(i,j\right)}^{\mathrm{car}},w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}},{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}},{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}\ge 0\\ w_{\left(i,j\right)}^{\mathrm{bus}},w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}},{\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}},{\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}\ge 0\\ w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}-{\epsilon }_{\left(i,j+1\right)}^{\mathrm{car}}\ge -x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}M\\ w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}-{\epsilon }_{\left(i,j+1\right)}^{\mathrm{bus}}\ge -x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}M\end{array},$$

(34)

where ${\epsilon }_{\left(i,j+1\right)}^{\mathrm{car}}$ [${\epsilon }_{\left(i,j+1\right)}^{\mathrm{bus}}$] denotes the initial queue clearance time corresponding to the car [bus] outbound bandwidth at (i, j + 1).

Assume that a closed road network consists of $m_{\mathrm{WE}}$ W-E arterials and $n_{\mathrm{SN}}$ S-N arterials. In this road network, the number of road segments is $\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$, the number of intersections is $m_{\mathrm{WE}}\times n_{\mathrm{SN}}$, and the number of closed loops is $\left(n_{\mathrm{SN}}-1\right)\times \left(m_{\mathrm{WE}}-1\right)$. The number of constraints for the proposed model is shown in

Table 2. The number of constraints for the proposed model.

Type of Constraints	Number of Constraints
Constraints on the location of green wave bands	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 8$
Arterial-level loop integer constraints	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 4$
Network-level loop integer constraints	$\left(n_{\mathrm{SN}}-1\right)\times \left(m_{\mathrm{WE}}-1\right)\times 4$
Bandwidth ratio constraints	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 2$
Mutual offset constraints	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 2$
Minimum bandwidth constraints	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 4$
Constraints that bandwidths are forced to be 0	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 8$
Common cycle time constraints	1
Additional constraints on interference variables	$\left[\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}\right]\times 10$
Constraints about $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ (Constraint 23)	$m_{\mathrm{WE}}\times n_{\mathrm{SN}}\times 3$

. The number of variables for the proposed model is shown in

Table 3. The number of variables for the proposed model.

Type of Variables	Number of Variables
Bandwidth $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, ${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, $b_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$, and ${\overline{b}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$	$\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$
Interference variables $w_{\left(i,j\right)}^{\mathrm{car}}$, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, ${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$, ${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{car}}$, $w_{\left(i,j\right)}^{\mathrm{bus}}$, $w_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$, ${\overline{w}}_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$, and ${\overline{w}}_{\left(i,j+1\right)}^{\mathrm{bus}}$	$\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$
0/1 variables $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and $x_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$	$\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$
Integer variables $m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{car}}$ and $m_{\left(i,j\right),\left(i,j+1\right)}^{\mathrm{bus}}$	$\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$
Integer variables $n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{car}}$ and $n_{\left(e,f\right),\left(m+1,k+1\right)}^{\mathrm{bus}}$	$\left(n_{\mathrm{SN}}-1\right)\times \left(m_{\mathrm{WE}}-1\right)$
0/1 variables $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$	$m_{\mathrm{WE}}\times n_{\mathrm{SN}}$
Integer variables $m_{\left(i,j\right);\left(i,j+1\right)}$	$\left(n_{\mathrm{SN}}-1\right)\times m_{\mathrm{WE}}+\left(m_{\mathrm{WE}}-1\right)\times n_{\mathrm{SN}}$
$z$	1

.

2.3. Model Solution

The objective function and constraints of the presented model are linear, and some of the variables are integers, which indicates that the presented model is a mixed integer linear programming (MILP) model. The branch-and-bound method can be used to solve the MILP problem. Many optimization software, such as LINGO 20.0, GUROBI 11.0, and CPLEX 22.1.1, have a built-in algorithm to solve the MILP problem.

3. Results

3.1. Basic Parameters of the Test Road Network

Figure 7 illustrates the distances between adjacent intersections and locations of bus stops in a 3 × 3 closed road network. The green splits at each intersection are shown in

Table 4. Green splits at each intersection.

Intersections	S-N through Movements	S-N Left-Turn Movements	W-E through Movements	W-E Left-Turn Movements
(1, 1)/(4, 1)	0.29	0.22	0.27	0.22
(1, 2)/(5, 1)	0.28	0.22	0.26	0.24
(1, 3)/(6, 1)	0.27	0.23	0.26	0.24
(2, 1)/(4, 2)	0.27	0.23	0.27	0.23
(2, 2)/(5, 2)	0.26	0.24	0.27	0.23
(2, 3)/(6, 2)	0.27	0.23	0.26	0.24
(3, 1)/(4, 3)	0.29	0.22	0.25	0.24
(3, 2)/(5, 3)	0.29	0.22	0.27	0.22
(3, 3)/(6, 3)	0.27	0.23	0.27	0.23

. The range of the common cycle time is between 100 and 120 s. The lower limit of green wave bands for cars and buses on all road segments is 4 s. The green wave speeds of cars and buses on each road segment are set to be 45 km/h and 36 km/h, respectively. The weighting factors for car and bus bandwidths on each road segment are shown in

Table 5. Weighting factors of bandwidths.

Road Segments	Weighting Factors for Outbound Car Bandwidth	Weighting Factors for Inbound Car Bandwidth	Weighting Factors for Outbound Bus Bandwidth	Weighting Factors for Inbound Bus Bandwidth
Between (1, 1) and (1, 2)	274	291	361	306
Between (1, 2) and (1, 3)	288	280	306	380
Between (2, 1) and (2, 2)	296	280	255	380
Between (2, 2) and (2, 3)	301	314	210	324
Between (3, 1) and (3, 2)	291	278	342	255
Between (3, 2) and (3, 3)	289	310	256	210
Between (4, 1) and (4, 2)	314	319	210	272
Between (4, 2) and (4, 3)	339	297	256	306
Between (5, 1) and (5, 2)	278	306	225	380
Between (5, 2) and (5, 3)	338	306	272	144
Between (6, 1) and (6, 2)	287	318	289	225
Between (6, 2) and (6, 3)	308	314	224	180

. The bandwidth ratios of cars and buses on each road segment are shown in

Table 6. Bandwidth ratios.

Road Segments	Car Bandwidth Ratios	Bus Bandwidth Ratios
Between (1, 1) and (1, 2)	1.062	0.848
Between (1, 2) and (1, 3)	0.972	1.242
Between (2, 1) and (2, 2)	0.946	1.490
Between (2, 2) and (2, 3)	1.043	1.543
Between (3, 1) and (3, 2)	0.955	0.746
Between (3, 2) and (3, 3)	1.073	0.820
Between (4, 1) and (4, 2)	1.016	1.295
Between (4, 2) and (4, 3)	0.876	1.195
Between (5, 1) and (5, 2)	1.101	1.689
Between (5, 2) and (5, 3)	0.905	0.529
Between (6, 1)and (6, 2)	1.108	0.779
Between (6, 2) and (6, 3)	1.019	0.804

. When there are bus stops on road segments, the average bus dwell time at each stop is shown in

Table 7. Average dwell time at each bus stop.

Road Segments	Outbound Dwell Time/s	Inbound Dwell Time/s
Between (1, 1) and (1, 2)	19	18
Between (2, 2) and (2, 3)	15	18
Between (4, 2) and (4, 3)	17	17
Between (6, 1) and (6, 2)	18	16

. The initial queue clearance time on each road segment is assumed to be 0. In addition, the sufficiently large positive number M is set to be 100 when generating the green wave control schemes for the test road network.

3.2. Green Wave Control Scheme Generation

The relevant parameters of the test road network, such as green splits, travel time, weighting factors of bandwidth, etc., were input into the presented model, and the corresponding MILP problem was solved by the optimization software LINGO 20.0. According to the LINGO solution report, the state of the solutions is globally optimum. The optimum common cycle time is about 100.39 s, which can be rounded to 100 s in the traffic signal optimization practice. The optimum bandwidths and relative offsets are shown in

Table 8. Optimum bandwidths and relative offsets on each road segment in the test road network.

Road Segments	Car Outbound Bandwidths/s	Car Inbound Bandwidths/s	Bus Outbound Bandwidths/s	Bus Inbound Bandwidths/s	Relative Offsets/s
Between (1, 1) and (1, 2)	12.6	13.4	12.3	12.6	51
Between (1, 2) and (1, 3)	7.4	11.1	15.7	19.4	52
Between (2, 1) and (2, 2)	15.9	17.1	25.8	27.0	51
Between (2, 2) and (2, 3)	22.9	19.9	4.0	4.0	49
Between (3, 1) and (3, 2)	9.1	8.7	17.3	16.9	49
Between (3, 2) and (3, 3)	19.2	20.2	24.1	23.1	51
Between (4, 1) and (4, 2)	16.0	16.2	25.5	25.7	51
Between (4, 2) and (4, 3)	23.2	22.2	4.7	5.6	49
Between (5, 1) and (5, 2)	22.7	18.3	20.4	24.8	49
Between (5, 2) and (5, 3)	18.7	17.2	26.0	26.0	48
Between (6, 1) and (6, 2)	27.0	21.0	0	0	47
Between (6, 2) and (6, 3)	23.9	24.4	18.3	17.8	50

. The relative offset is the time difference between the beginning of the outbound green time at (i, j) and that at (i, j + 1). The optimum phase sequences are shown in

Table 9. Optimum phase sequences at each intersection in the test road network.

Intersections	Optimum Phase Sequences
(1, 1)/(4, 1)	sequences 3 or 4
(1, 2)/(5, 1)	sequence 6
(1, 3)/(6, 1)	sequences 3 or 4
(2, 1)/(4, 2)	sequences 3 or 4
(2, 2)/(5, 2)	sequence 1
(2, 3)/(6, 2)	sequences 3 or 4
(3, 1)/(4, 3)	sequence 6
(3, 2)/(5, 3)	sequences 2 or 5
(3, 3)/(6, 3)	sequence 1

.

3.3. Comparison

To verify the validity of the proposed model, two comparison studies were conducted. The first comparison is between optimizable phase sequence and fixed phase sequence schemes. The second comparison is between models with relaxation constraints and models without relaxation constraints under a fixed common cycle time.

In the first comparative study, the optimizable phase sequence refers to those phase sequences that can be optimized at each intersection. That is, phase sequences are decision variables. Fixed phase sequence means that the phase sequence at each intersection is predetermined. We have designed three different fixed phase sequence schemes, and the specific phase sequences for each intersection are detailed in

Table 10. Three different fixed phase sequence schemes.

Intersections	Fixed Phase Sequence Scheme 1	Fixed Phase Sequence Scheme 2	Fixed Phase Sequence Scheme 3
(1, 1)/(4, 1)	sequences 2 or 5	sequence 6	sequence 1
(1, 2)/(5, 1)	sequence 1	sequence 1	sequence 1
(1, 3)/(6, 1)	sequences 3 or 4	sequences 3 or 4	sequences 2 or 5
(2, 1)/(4, 2)	sequence 6	sequences 3 or 4	sequences 3 or 4
(2, 2)/(5, 2)	sequences 3 or 4	sequence 1	sequence 6
(2, 3)/(6, 2)	sequences 2 or 5	sequences 2 or 5	sequences 3 or 4
(3, 1)/(4, 3)	sequence 6	sequences 2 or 5	sequences 2 or 5
(3, 2)/(5, 3)	sequence 1	sequence 1	sequence 6
(3, 3)/(6, 3)	sequences 2 or 5	sequence 1	sequence 1

.

When generating green wave schemes for different fixed phase sequence schemes, the 0/1 variable $u_{\left(p,q\right);\left(r,s\right)}$, $v_{\left(p,q\right);\left(r,s\right)}$, $y_{\left(p,q\right);\left(r,s\right)}$, and $z_{\left(p,q\right);\left(r,s\right)}$ in the proposed model should be set to the corresponding value. In this way, four kinds of bandwidth optimization models were obtained: the proposed model and the models corresponding to fixed phase sequence schemes 1, 2, and 3, respectively.

In the second comparative study, green wave control schemes for the test road network were generated by the model with constraint relaxation (the proposed model) and the model without constraint relaxation. The objective function values of these two models when the common cycle time is fixed are presented in

Table 11. Objective function values for the models with and without constraint relaxation.

Common Cycle Time/s	Objective Function Values
	Model with Constraint Relaxation	Model without Constraint Relaxation
100	2367.249	no feasible solution
101	2366.957	no feasible solution
102	2348.060	no feasible solution
103	2328.882	no feasible solution
104	2328.562	2328.562
105	2313.568	2313.568
106	2298.305	2298.305
107	2282.394	2282.394
108	2261.124	2261.124
109	2239.132	2239.132
110	2216.762	2216.762
111	2194.422	2194.422
112	2172.057	2172.057
113	2149.771	2149.771
114	2126.787	2126.787
115	2103.997	2103.997
116	2081.600	2081.600
117	2059.586	2059.586
118	2037.945	2037.945
119	2014.743	2014.743
120	1990.558	1990.558

.

4. Discussion

In the LINGO solution report presenting the control scheme shown in Table 8, only the 0/1 variable $x_{\left(6,1\right),\left(6,2\right)}^{\mathrm{bus}}$ equaled to 1, which indicates that there were car green wave bands on each road segment in the test road network, and so were bus green wave bands except the road segment between (6, 1) and (6, 2). This was also consistent with the bandwidth results shown in Table 8. The reasons for the bus bandwidth on the road section between (6, 1) and (6, 2) being 0 are as follows. Due to constraints of factors such as road segment length, vehicle travel time, bus dwell time, traffic light time, etc., cars and buses cannot simultaneously obtain green wave bands that meet the limit requirements on the road segment between (6, 1) and (6, 2). At the same time, since the weighting coefficients of the car bandwidths on this road segment are relatively large, the green wave band was allocated to cars instead of buses.

According to the LINGO solution reports for the first comparative study, the objective function values (the sum of weighted bandwidths) under optimizable phase sequence and fixed phase sequence schemes 1, 2, and 3 are 2378.660, 2171.864, 2044.018, and 2032.889, respectively. Compared with the fixed phase sequence schemes 1, 2, and 3, the total weighted bandwidth generated by the presented model increased by 9.5%, 16.4%, and 17%, respectively. It can be observed that the optimizable phase sequence contributes to obtaining larger bandwidths.

As can be seen from Table 11, regardless of which fixed value of the common cycle time was taken, the model with constraint relaxation could always achieve the global optimal solutions, while the model without constraint relaxation failed to obtain a feasible solution when the common cycle time was fixed at 100, 101, 102, or 103 s.

5. Conclusions

Passenger cars and buses are common trip modes for city residents. To improve the traffic efficiency of passenger cars and buses in closed road networks, this paper presents a network-level green wave control model that considers both passenger cars and buses. The presented model is formulated as a mixed integer linear programming problem, and its global optimal solution can be obtained using the branch-and-bound algorithm. Based on the analysis results of the test road network, the following main conclusions can be drawn.

(1) The model proposed in this paper is able to simultaneously coordinate the green wave for cars and buses on each arterial road in the closed road network. The average car outbound (inbound) bandwidths and the average bus outbound (inbound) bandwidths are 18.2 s (17.5 s) and 16.2 s (16.9 s), respectively, on each road segment in the test road network.

(2) The model proposed in this paper can not only optimize the phase sequences to obtain the optimal phase sequence scheme but also is applicable to the case of fixed phase sequence scheme. Compared with the fixed phase sequence scheme, the optimizable intersection phase sequence contributes to obtaining larger bandwidths.

(3) When the common cycle time is fixed at certain values, the model without constraint relaxation may have no feasible solution, while the proposed model can avoid this situation and still obtain the global optimal solution. Therefore, it is very necessary to relax some constraints.

(4) Generally, through-traffic flows have larger volumes on road segments. Therefore, the proposed model takes the through passenger cars and buses as the research objects and provides green wave bands for them. However, when the left-turning volumes are higher, there is a need to provide green bands for the left-turning traffic flows, which cannot be achieved by the proposed model. Thus, the proposed model should be extended in the future so that it can provide green bands for multiple path flows.

(5) The proposed model focuses on providing green wave bands for passenger cars and buses, potentially neglecting the green wave control demand of electric bicycles. In fact, electric bicycles are commonly used for commuting short and medium distances in China due to their convenience and time-saving benefits. Therefore, another important research direction is to consider not only the green wave control needs of motorized vehicles but also of electric bicycles.