## 7 Conclusions
Additionally, multiple train services will also impact regular passengers, because medium- and long-distance passengers will purposefully take express trains and adjust their arrival times. Therefore, in future research, passenger flow prediction and control will be considered in the ESTNDM to build a more efficient and robust line plan and timetable. (2) The ESTNDM solving algorithm took a long time and was not easy to apply to large-scale problems, such as all-day timetables and long lines. Although the actual application scenarios will not require the high solving efficiency for the model in this study, the computing efficiency for large-scale problems must be improved; otherwise, it will be difficult to ensure the optimality of the solution in a limited solving time. Therefore, in future research, algorithms, such as the column generation algorithm (CG), or software, such as the general algebraic modeling system (GAMS), which are suitable for solving large-scale problems, and linearization methods that approximate and simplify the nonlinear model in this paper will be considered. (3) This study focused on a single urban rail line, which would not be very applicable to some cities where urban rail systems are operated as a network, because it ignores the interactions of passengers between lines. Under the network operation, the imbalance in the temporal and spatial distributions of passenger flow is obvious due to more demanding het erogeneity factors, such as the convenience of transfers. Therefore, our future research will consider transfer costs and passenger path choice behavior in urban rail transit networks and extend the model proposed in this paper to a three-dimensional space-time network design, which will be an integrated optimization model of line planning and timetabling under network operation to provide multi-train services adapting the imbalance in the temporal and spatial distributions of passenger flow. Investigation, Y.G. and C.J. ; conceptualization, Y.G. and C.J. ; methodology, Y.G., Z.W., and C.J. ; supervision, C.J. and Z.H. ; validation, Y.G. and Z.H. ; data curation, Y.G. and Z.H. ; writing--original draft preparation, Y.G. and Z.W. ; writing--review and editing, C.J., Z.W. and Z.H. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (NSFC) under Grant 72371018, 62132003, and the Science and Technology Research and Development Plan Project of China State Railway Group Co., Ltd. under Grant N2022G025. Not applicable. The data presented in this study are available in the article. Special thanks are given to the Center for High Performance Computing at Beijing Jiaotong University ([[http://hpc.bjtu.edu.cn](http://hpc.bjtu.edu.cn)]([http://hpc.bjtu.edu.cn](http://hpc.bjtu.edu.cn)) (accessed on 11 June 2024)) for computing their resources and support. Author [PERSON] was employed by the company Line Branch Company Affiliated with Beijing Mass Transit Railway Operation Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Notations. **Table A1. **_Cont._  
\begin{tabular}{l l} \hline \hline \multicolumn{2}{l}{**Parameters and Symbols**} \\ \hline \(t_{si}^{(\mathrm{ss})}\), \(t_{si}^{(\mathrm{s})}\), \(t_{si}^{(\mathrm{e})}\) & Dwell times of train \(i\in\mathcal{I}\) at station \(s\in\mathcal{S}\). Superscripts \((s*)\), \((s)\) and \((e)\) indicate the minimum dwell time, actual dwell time, and additional dwell time due to overtaking, respectively. \\ \(t_{si}^{(\mathrm{rs})}\), \(t_{si}^{(\mathrm{r})}\) & Running times of train \(i\in\mathcal{I}\) at section \([s,s+1]\), \(\forall s,s+1\in\mathcal{S}\). Superscripts \((r*)\) and \((r)\) indicate the pure running time and the actual run time, respectively. \\ \(\mathbf{Q}\) & The set of OD flows with timestamp. \\ \(\mathcal{W}\) & The index set of OD pairs with timestamp.