## 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. [PERSON] 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, [PERSON] et al. [2] proposed an arterial green wave control model called MULTIBAND, where the bandwidths vary with the road sections. [PERSON] 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. [PERSON] et al. [4] developed a modified MULTIBAND to relax the requirement to yield larger bandwidths. [PERSON] et al. [5] proposed a method to generate dynamic green wave bandwidth using closed-loop signal data. [PERSON] et al. [6] proposed a model that can simultaneously optimize green wave bandwidth and crossover spacing for diverging diamond interchanges. [PERSON] and [PERSON] [7] presented a two-way arterial green wave control method under asymmetric release mode. [PERSON] 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, [PERSON] 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. [PERSON] 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, [PERSON] and [PERSON] [11] proposed a road network version of the variable green wave bandwidth model called MULTIBAND-96. [PERSON] et al. [12] proposed two green wave bandwidth optimization models to generate traffic signal coordination schemes for long arterials and grid road networks. [PERSON] 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], [PERSON] and [PERSON] [15] proposed a multi-path bandwidth optimization model for traffic networks considering variable phase structures. [PERSON] 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. [PERSON] et al.