Cities identify urban rail transit as a service complementary to other sustainable forms of urban public transit systems. More generally, designing low-carbon urban rail transit systems is a critical component of reducing emissions and addressing climate change at the urban scale [1]. Daily operations account for more than 90% of total greenhouse gas (GHG) emissions throughout the entire life cycle of a metro [2]. Major energy-saving enhancements for urban rail transit thus aim to improve the operational energy performance. Theoretical investigations primarily focus on operational energy-saving approaches, such as trajectory control strategies (e.g., optimizing the operation time between stations [3], departure/arrival times [4], dwell and running times of a train [5,6], and speed profiles [7]) and energy storage systems (e.g., the utilization of regenerative braking technology [8]). The efficiency of trajectory control strategies is mostly restricted by the safety distance between trains and the stability of the power grid. Regenerative braking technology can save an appropriate percentage of energy consumption (around 33% in the traction energy flow [9]). The energy transfer [10,11] and utilization efficiency [12] of regenerative braking are widely considered to minimize energy loss during braking. Meanwhile, some studies have explored the voltage-stabilizing effects and aim to enhance the efficiency of energy storage systems directly [13]. The increasing number of urban rail transit systems under development have made it even more essential to design measures for systematically reducing operational energy consumption levels.

In urban rail transit systems, energy usage is typically classified as traction or non-traction consumption activities. The energy required to operate rolling stock (i.e., trains) within the system is referred to as traction energy consumption [9], and accounts for more than 50% of the energy consumption of metro operations [11]. Previous studies primarily focused on different energy-saving strategies targeted at traction consumption by optimizing train speed profiles or timetables, both with and without regenerative braking functions. With regenerative braking, the energy produced during the braking process can be recovered and reused, resulting in a reduction in the overall energy consumption [14]. In addition, different types of trains have adopted distinct control conditions in different environments (e.g., the optimal utilization of the gravitational potential energy generated by a train when moving downhill [15]) to further reduce energy consumption levels.

The optimization of train timetables, meanwhile, mainly considers the overlap between the traction and braking phases and the duration of the running time [16,17]. The existing research also focuses on energy-saving operational strategies and timetable optimization methods based on parallel operation diagrams. The rapid development of urban rail transit systems makes train operations more complex and the utilization of non-parallel operation diagrams more frequent. Previous studies have focused on energy-saving operational strategies and timetable optimization based on parallel train operation diagrams but have overlooked the further utilization of non-parallel operation diagrams in the development of urban rail transit systems. Under non-parallel operation diagrams, the complexity of passenger flows and non-parallel train running features create significant challenges for train scheduling.

Energy-saving operational strategies have been investigated in two major ways: the utilization of the potential gravitational energy of a train [15] and the flexibility of control conditions [18]. These investigations were conducted both with and without considering the implementation of regenerative braking technology. Prior to the implementation of regenerative braking technology, previous conventional studies focused on optimizing operations and reducing energy consumption levels based on the characteristics of individual track sections. He et al. (2015) optimized the train speed profiles of each inter-station line segment through a section separation analysis by incorporating both running line-related features [19]. Deng et al. (2021) considered the predetermined running times for each inter-station track segment and optimized the control condition sequences and durations along each section by regulating the multi-speed parameters [18]. Meanwhile, some studies explore the design of timetables that optimize the overlap times of control conditions between adjacent trains, thereby enhancing the utilization efficiency of regenerative braking energy. Sun et al. (2017) optimized the sequence of control conditions for adjacent trains [20]; Bai et al. (2020) investigated the application of the secondary traction of the train when optimizing the control conditions of a single train in the section [21,22].

An alternate approach to energy saving is schedule optimization, particularly prior to the implementation of regenerative braking technology. He et al. (2021) proposed a two-stage energy-saving calculation method to globally optimize the allocation of total train running times between different sections [23]. Gao et al. (2020) optimized the allocation of inter-station running times by calculating the optimal control strategy for inter-station train operations along a Pareto frontier [24]. When regenerative braking technology was implemented, many previous studies focused on optimizing the temporal and spatial elements of parallel operation diagrams to minimize energy consumption levels and design optimal timetables. Peng et al. (2017) used the controllability of train headway and dwell times to minimize the total energy consumption [10]. Ran et al. (2020) assumed that train headway and turnaround times were fixed, and then moderately optimized the dwell time, inter-station running time, and turnaround time so as to minimize the net energy consumption [25]. Wei et al. (2020) considered the overlapping time distribution and distance between the front and rear trains, and then optimized the arrival, departure, and dwell times of trains traveling in the same direction [12].

While reducing operational energy consumption levels, it is also critical to consider other factors, especially passenger demands. He et al. (2020) developed an integrated optimization method to minimize both the energy consumption and transfer waiting time cost for transfer passengers [3]. Wu et al. (2020) designed a multi-objective timetabling optimization and incorporated the consideration of the crowdedness of passengers [26]. Xie et al. (2021) proposed an energy-saving timetable for a high-speed railway line, meanwhile avoiding time delays for passengers [5].

In light of the more and more complex train speed profiles and increasing passenger demands, non-parallel train operation diagrams are crucial for address rapid urban rail system developments. But non-parallel operation diagrams introduce increased complexity in terms of the running features of trains and the characteristics of passenger flows.