# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 6 Conclusions and Future Work  
To address non-equilibrium passenger demand more economically, this study investigates the optimization of train service design with CO utilizing VC, whereby the train composition can be changed by coupling/decoupling operations at both ends of the zones. The proposed nonlinear integer programming model aims to minimize passenger travel costs and operation costs while improving train capacity equity. It sets the train departure frequency, train composition, and turn-back stations as decision variables constrained by passenger travel demand, headway, etc. This study proposes an equity index, based on the variance of travel utility for all OD pairs passengers, to measure overall train capacity equity. Furthermore, an improved ASA-GA is proposed to effectively solve the practical case. A set of numerical experiments is conducted on Beijing Changping Line and Line 13 to verify the effectiveness of the proposed model and algorithm. The results demonstrate that the proposed ASA-GA can find a near-optimal solution approximating the exact solution in a shorter computation time, enhancing the algorithm's applicability for model-solving.  
A comparison of the train service scheme under the SG-CO scheme with the SG-No-CO scheme reveals that CO can effectively reduce passenger residence time costs. Furthermore, a comparison of the train service scheme under the VC-CO scheme with that of the MG-CO scheme yields the following results: (1) In terms of passenger travel costs, the average passenger residence time decreases by 16.3%. (2) In terms of equity, the passenger travel utility variance decreases by 11.6%, resulting in a more even distribution of train capacity. (3) In terms of operation costs, the number of carriages used increases by 24, effectively meeting passenger flow demands. Based on these findings, this study concludes that VC effectively balances passenger demand and train capacity, providing an effective method to improve equity performance and reduce passenger travel time.  
This study identifies three directions for future research. (1) VC can be applied to larger network subway networks, such as \"X\"-type lines or systems with three or more lines. In such networks, VC implementation offers greater flexibility. However, this expansion necessitates stricter safety requirements, including the determination of VC locations between lines, ensuring operational safety at transfer stations, and optimizing train turnaround. (2) ASA-GA is a heuristic algorithm that can only obtain approximate solutions, suggesting potential for refinement to improve solution accuracy. (3) This  
Figure 18: The operational cost of different lines.  
study preliminarily explored a train service scheme using VC in CO. In contrast, the train-scheduling problem is more practical and attractive for URT operation, so it is a promising direction to generate feasible timetables based on the obtained train service scheme.  
Conceptualization, L.S., Y.L., Y.X. and H.L. ([PERSON]); Methodology, Y.X.; Software, Y.L. and H.L. ([PERSON]); Validation, D.K. and H.L. ([PERSON]); Resources, L.S. and D.K.; Data curation, H.L. ([PERSON]); Writing--original draft, Y.L.; Writing--review and editing, Y.X. All authors have read and agreed to the published version of the manuscript.  
This study was funded by the Beijing Natural Science Foundation (L231023), the National Natural Science Foundation of China (No. 71901008), and the Beijing Nova Program (20230484443).  
Not applicable.  
The data presented in this study are available on request from the corresponding authors. This data will not be made public due to privacy concerns.  
The authors thank all our team members for their contribution to the experiment. The authors declare no conflicts of interest.