# Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints
## 1 Introduction
The latter has leveraged a modular transit network to expand service coverage, whereby only the coupling/decoupling of main modules, trailer modules, and passengers with impeccably aligned itineraries are deemed valid scheduling operations. Evidently, in scenarios where multiple modular bus platoons coexist in the network, the coupling/decoupling operations can only be effectuated when the MB platoons or units converge at the same location and conform to time constraints, while the uneven demand between trips also emerges as a pivotal factor influencing the coupling/decoupling operations. Figure 1: Modular bus unit (Source: [[http://wwwnext-future-mobility.com/](http://wwwnext-future-mobility.com/)]([http://wwwnext-future-mobility.com/](http://wwwnext-future-mobility.com/)) (accessed on 5 February 2023)). However, the aforementioned studies have largely overlooked the energy limitations and charging arrangement of electric-powered modular buses. It is well known that charging strategies, influenced by factors such as charging time, charging station capacity, and power resources, are crucial for the sustainable operation of electrified transportation systems [28; 29; 30; 31]. In comparison to conventional buses, MB units are characterized by smaller sizes and battery capacities. Consequently, in the midst of demanding trip tasks, scheduling plans must not only account for fluctuations in passenger demand between trips but also ensure that the consumption energy of MBs during their travel sequence does not exceed the battery's capacity while flexibly devising charging plans to avert situations of insufficient power. To bridge the gaps, we propose a modular bus scheduling system under range and charging constraints to reduce system costs while enhancing scheduling flexibility. Given that MB scheduling falls under the category of vehicle scheduling problems, wherein a group of modular buses needs to be efficiently assigned tasks under timetable constraints, we can draw insights from existing research on electric buses. Two main categories of modeling methods can be identified. One is mainly based on a directed network to model the scheduling problem. [PERSON] [32] and [PERSON] et al. [33] developed a spatiotemporal network graph and scheduling model for electric buses, in which charging stations are treated as time nodes. The arc-based model in their study can be solved by commercial solvers (GUROBI or CPLEX), while the path-based model is solved by a branch-and-price framework. [PERSON] et al. [34] also models based on spatiotemporal graphs, except that a genetic algorithm solution method is adopted. In addition, there are some studies that extend the spatiotemporal graph to a complex network covering both vehicle and passenger flow [35]. Another category directly models the schedule regarding electric buses. [PERSON] et al. [36] proposed a mathematical model that simultaneously optimizes the charging station locations and scheduling schemes and solved it using a variable neighborhood search algorithm. [PERSON] et al. [37] considered the impact of vehicle procurement costs, operating costs, and charging facility installation costs to establish a regional schedule model for electric buses. [PERSON] et al. [38; 39] proposed mixed integer linear programming to address the mixed bus fleet scheduling problem, taking into account timetables and energy constraints. The modeling method based on a directed network graph, as a focal point of our study, provides more precise sequential relationships among trips, vehicles, and charging processes. However, it is not entirely applicable to the MB scheduling problem, as a conventional bus can cover one trip with fixed capacity, whereas there is a coupling between MB platoons leading to dynamic capacity. In brief, we establish a modular bus unit scheduling model and method under range and charging constraints to achieve sustainability and economy of the transit system.