# Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints
## 1 Introduction  
In light of the ongoing trend towards urbanization and population growth, the integration of electrified public transportation has emerged as a critical component in cities worldwide. Serving as a sustainable mode of transportation, it not only reduces traffic congestion and air pollution but also affords greater convenience for commuters [1]. However, establishing and sustaining a highly efficient and dependable electrified public transportation system is an intricate undertaking that involves multifarious factors such as network design, timetable preparation, vehicle scheduling, personnel allocation, and charging arrangements [2, 3]. Among these factors, vehicle scheduling is an indispensable element that determines the quality of service and cost-effectiveness of the public transportation system [4]. Despite the widespread adoption of traditional fixed-route and fixed-capacity vehicle scheduling methods that allocate a predetermined number of vehicles to fixed routes, such approaches may be insufficiently adaptable to meet the dynamic requirements of travel demands, resulting in suboptimal resource allocation [5, 6]. To optimize the provision of public transportation vehicles, it is imperative to account for the variability in demand across different trips and to leverage resource sharing and reallocation as necessary. MB units in Figure 1, also known as automated modular vehicles, constitute a novel vehicle technology that can dynamically adjust the capacity and size ofplatoons by assembling or separating different modules [7; 8]. Particularly in transportation scheduling systems characterized by multiple trips, MB units can be customized to match passenger numbers and minimize the waste of vehicle resources and road congestion. For instance, modular units can be extracted from trips with lower demand and form large capacity MB platoons for trips with higher passenger demand, leading to cost-effective resource utilization [9; 10]. In recent years, modular bus technology is now being validated in certain real-world applications, such as the United Arab Emirates [11] and Germany [12]. Currently, modular bus-related research has been explored in multiple aspects, including optimization of schedule strategy optimization, passenger transfer plan design [13; 14], specialized station deployment for modular unit reorganization [15], and driving trajectory planning [16]. Various studies collectively indicate that the concept of modular buses holds promise in reducing operating costs, improving service quality, and enabling more flexible vehicle dispatching, thereby potentially serving as an effective solution to address resource allocation challenges in traditional bus systems [17]. With respect to modular bus schedules relevant to the present research, the majority of studies have emphasized the optimization of MB platoon capacity and departure timetables in transit systems. [PERSON] et al. [18; 19], [PERSON] et al. [20], [PERSON] et al. [21], and [PERSON] et al. [22] addressed when and how many modular buses to dispatch on a single bus route to minimize passenger waiting time and vehicle operating costs. [PERSON] et al. [23] extended it to a network encompassing multiple bus stations, and [PERSON] et al. [24] optimized the composition and serve frequency of MB units and conventional buses on multiple lines. However, most of these studies primarily focus on the operation of a single-vehicle platoon composed of different numbers of modular units on the bus route or network, neglecting the more complex coupling/decoupling mechanisms among multiple MB platoons in a bus network scheduling system. Furthermore, [PERSON] et al. [25; 26] and [PERSON] et al. [27] considered the interplay between MB platoons in the scheduling process. The former incorporated passenger arrival times to devise optimal departure times and node separation strategies for module vehicles in Y-shaped shared corridors, minimizing both passengers waiting time and vehicle dispatching costs.