# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 1 Introduction
By combining the train schedule with the train scheduling problem, a train service scheme with a minimum passenger waiting time was designed. [PERSON] et al. [23] verified that VC is more suitable for passenger flows with uneven spatial and temporal distributions. VC is a train-coupling technology that enables multiple trains to form a logical whole through vehicle-vehicle communication. In addition, VC allows for more flexible adjustment of coupling schemes to match dynamic passenger demand ([PERSON] et al. [24]; [PERSON] et al. [25]; [PERSON] et al. [26]; [PERSON] et al. [27]). At present, the number of train carriages in CO systems is fixed. With VC, the number of train carriages can be changed during operation. Thus, we attempted to apply VC between cross-line trains and local-line trains, with virtual coupling in the collinear zones to serve passengers in large groups and virtual decoupling in the non-collinear zones to serve passengers in small groups. In this study, we considered train service design in CO during a given period, where the train departure frequency, train composition of each line, and the turn-back stations of the cross-line have to be determined simultaneously. A mathematical model was constructed to minimize the passenger travel cost and the operation cost while ensuring train capacity allocation equity. Meanwhile, using VC, the trains of different lines were coupled in the collinear zone and decoupled in the single-line zone. In this way, it was possible to balance the accumulation of passenger flow in each zone of the CO, reduce the operational risk in some busy zones, and reasonably allocate the train capacity on each line. To sum up, previous studies have the following characteristics: (1) The research on line design mainly focuses on single lines, and trains can only operate on fixed lines, while research on CO is scarce. (2) The research on train-grouping methods in CO mainly focuses on the traditional SG and MG. There is little research on advanced VC. (3) The equity study of URT systems mainly considers passenger travel equity; i.e., the waiting time of passengers is proportional to the time on the train, ignoring the equity of the train capacity. There is no in-depth research on the travel equity gap of train capacity in URT systems. In Table 1, some related published papers are compared. The contributions of this study are summarized as follows. 1. A train service design method for CO to address uneven passenger flow is proposed; in particular, VC is applied. This approach enables trains from different lines to couple in collinear zones and decouple in non-collinear zones. 2. To better measure train capacity equity, this study precisely determines the train capacity utility for all zones by matching passenger demand with allocated train capacity, thereby enhancing the train capacity equity on the whole. 3. An improved adaptive simulated annealing genetic algorithm (ASA-GA) is designed to obtain a near-optimal solution. We test the performance of the proposed scheme using realistic instances from Beijing URT Changping Line and Line 13. The remainder of this paper is organized as follows. In Section 2, we begin by providing a detailed description of the problem and discuss a simple case. In Section 3, a nonlinear integer programming model is built. In Section 4, we design an ASA-GA to solve the proposed problem. After that, a set of numerical experiments with the real-world operation data are implemented in Section 5. In the last section, some conclusions and further research works are presented.