# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 1 Introduction  
Nowadays, the transfer demand of passengers has increased dramatically due to the rapid development of urban rail transit (URT). Different from single-line operation, cross-line operation (CO) is mainly used to save transfer time and improve the accessibility of travel, which is widely regarded as an effective way to reduce passenger transfer behavior ([PERSON] et al. [1]; [PERSON] et al. [2]). With CO, the spatial distribution of passenger flow has obvious uneven characteristics ([PERSON] [3]), such as a large passenger flow in the collinear zones (the overlapping zones of local lines and cross-lines) and a small passenger flow in the non-collinear zones (the independent zones of local lines). Further, in the collinear zones, cross-line trains should be increased as they can meet passengers' travel demand on both local and cross-lines ([PERSON] et al. [4]). Hence, optimizing train service design for CO has been a hot topic in recent years. As the main method to save transfer time in URT systems, CO has attracted much attention recently. [PERSON] et al. [5] studied the technical solutions for wheel-track design, traction power systems, and communication systems when state-owned railroads cross into tram lines. [PERSON] et al. [6] emphasized the importance of considering networked operations in the planning and construction stage of URT. In addition, [PERSON] et al. [7] considered the use of a jump-stop scheme for cross-line trains and established a capacity allocation model for CO to improve efficiency. To meet the challenge of uneven passenger flow, there has been extensive research on equity in train services. [PERSON] et al. [8] analyzed min-max equity and \(\alpha\)-equity to establish a train schedule optimization scheduling model. [PERSON] et al. [9] considered the equity of waiting times at each transfer station and adopted the min-max equity function to balance URT schedules. [PERSON] et al. [10] established a nonlinear integer programming model to determine the number of reserved trains for each train, thus ensuring the equity of train resources for downstream passengers. [PERSON] et al. [11] proposed a train schedule optimization method that measures equity in terms of differences in passenger travel utility. Based on the concept of \(\alpha\)-equity, [PERSON] et al. [12] constructed a CO train service scheme aiming at minimizing the passenger waiting time, fairness, and operating cost, giving an example to verify that the optimal train service scheme can significantly improve the fairness performance of passenger travel. In order to allocate train capacity properly according to passenger demand, many researchers have attempted to develop train service design using multiple grouping (MG) (see [PERSON] et al. [13], [PERSON] et al. [14], and [PERSON] et al. [15]). However, due to the limitations of the traditional fixed block system, this pattern still leads to serious congestion in some busy zones. In recent years, virtual coupling (VC) technology has attracted the attention of many researchers. Many studies have discussed the basic functions of VC and verified that VC trains can significantly improve the overall line passing capacity (e.g., [PERSON] et al. [16], [PERSON] et al. [17], and [PERSON] et al. [18]). [PERSON] et al. [19] applied VC to full-length and short-turn routes and investigated the efficiency benefit for train operations. [PERSON] et al. [20] further considered the constraints of train operation, rolling stock circulation, etc., in order to achieve the goal of minimizing passenger travel and operating costs. The results showed that VC can effectively adapt to the uneven distribution of passenger flow. [PERSON] et al. [21] verified that VC is one of the most innovative solutions to increase railroad capacity by significantly reducing train headway. [PERSON] et al. [22] established a URT scheduling optimization model based on VC technology.