# An Automatic Train Operation Based Real-Time Rescheduling Model for High-Speed Railway

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Description

#### 3.1. Notations

#### 3.2. Scenario

#### 3.3. Model Assumption

- The high-speed railway system can achieve real-time and precise perception of the information of disturbance occurrence by implementing sensor networks and information fusion in complex environments.
- At each station, there are designated arrival and departure tracks for up-direction and down-direction trains, respectively. Given the similarity in the rescheduling processes for up-direction and down-direction trains, our study centers on examining the real-time rescheduling problem for a single direction.
- The train speed trajectories with different driving strategies are pre-calculated in this paper.

## 4. Model Formulation

#### 4.1. Objective Function

#### 4.2. Departure Time and Dwell Time Constraints

#### 4.3. Running Time Constraints

#### 4.3.1. Driving Strategy

#### 4.3.2. Train Stop Plan

#### 4.4. Headway Constraints

#### 4.5. Station Capacity Constraints

#### 4.6. Real-Time Rescheduling Model

## 5. Case Study

#### 5.1. Case 1: Impact of Different Disturbance Scenarios

#### 5.2. Case 2: Evaluation of Different Driving Strategies

#### 5.3. Case 3: Comparative Analysis with the Timetable without Optimization

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Corman, F.; Meng, L. A review of online dynamic models and algorithms for railway traffic management. IEEE Trans. Intell. Transp. Syst.
**2014**, 16, 1274–1284. [Google Scholar] [CrossRef] - Cacchiani, V.; Huisman, D.; Kidd, M.; Kroon, L.; Toth, P.; Veelenturf, L.; Wagenaar, J. An overview of recovery models and algorithms for real-time railway rescheduling. Transp. Res. Part B Methodol.
**2014**, 63, 15–37. [Google Scholar] - Fang, W.; Yang, S.; Yao, X. A survey on problem models and solution approaches to rescheduling in railway networks. IEEE Trans. Intell. Transp. Syst.
**2015**, 16, 2997–3016. [Google Scholar] - Dorfman, M.; Medanic, J. Scheduling trains on a railway network using a discrete event model of railway traffic. Transp. Res. Part B Methodol.
**2004**, 38, 81–98. [Google Scholar] [CrossRef] - Törnquist, J.; Persson, J.A. N-tracked railway traffic re-scheduling during disturbances. Transp. Res. Part B Methodol.
**2007**, 41, 342–362. [Google Scholar] [CrossRef] - D’Ariano, A.; Pacciarelli, D.; Pranzo, M. A branch and bound algorithm for scheduling trains in a railway network. Eur. J. Oper. Res.
**2007**, 183, 643–657. [Google Scholar] - Schöbel, A. A model for the delay management problem based on mixed-integer-programming. Electron. Notes Theor. Comput. Sci.
**2001**, 50, 1–10. [Google Scholar] [CrossRef] - Duendar, S.; Sahin, I. Train re-scheduling with genetic algorithms and artificial neural networks for single-track railways. Transp. Res. Part C Emerg. Technol.
**2013**, 27, 1–15. [Google Scholar] [CrossRef] - Corman, F.; D’Ariano, A.; Pacciarelli, D.; Pranzo, M. A tabu search algorithm for rerouting trains during rail operations. Transp. Res. Part B Methodol.
**2010**, 44, 175–192. [Google Scholar] [CrossRef] - Lamorgese, L.; Mannino, C. The track formulation for the train dispatching problem. Electron. Notes Discret. Math.
**2013**, 41, 559–566. [Google Scholar] [CrossRef] - Meng, L.; Zhou, X. Robust single-track train dispatching model under a dynamic and stochastic environment: A scenario-based rolling horizon solution approach. Transp. Res. Part B
**2011**, 45, 1080–1102. [Google Scholar] [CrossRef] - Yang, L.; Zhou, X.; Gao, Z. Credibility-based rescheduling model in a double-track railway network: A fuzzy reliable optimization approach. Omega
**2014**, 48, 75–93. [Google Scholar] [CrossRef] - Pellegrini, P.; Marlière, G.; Rodriguez, J. Optimal train routing and scheduling for managing traffic perturbations in complex junctions. Transp. Res. Part B Methodol.
**2014**, 59, 58–80. [Google Scholar] [CrossRef] - Zhan, S.; Kroon, L.G.; Veelenturf, L.P.; Wagenaar, J.C. Real-time high-speed train rescheduling in case of a complete blockage. Transp. Res. Part B Methodol.
**2015**, 78, 182–201. [Google Scholar] [CrossRef] - Xu, P.; Corman, F.; Peng, Q.; Luan, X. A train rescheduling model integrating speed management during disruptions of high-speed traffic under a quasi-moving block system. Transp. Res. Part B Methodol.
**2017**, 104, 638–666. [Google Scholar] [CrossRef] - Wang, P.; Goverde, R.M. Multi-train trajectory optimization for energy efficiency and delay recovery on single-track railway lines. Transp. Res. Part B Methodol.
**2017**, 105, 340–361. [Google Scholar] [CrossRef] - Wu, W.; Liu, R.; Jin, W. Modelling bus bunching and holding control with vehicle overtaking and distributed passenger boarding behaviour. Transp. Res. Part B Methodol.
**2017**, 104, 175–197. [Google Scholar] [CrossRef] - Luan, X.; Wang, Y.; De Schutter, B.; Meng, L.; Lodewijks, G.; Corman, F. Integration of real-time traffic management and train control for rail networks—Part 1: Optimization problems and solution approaches. Transp. Res. Part B Methodol.
**2018**, 115, 41–71. [Google Scholar] [CrossRef] - Liu, F.; Xun, J.; Liu, R.; Yin, J.; Dong, H. A real-Time rescheduling approach using loop iteration for high-Speed railway traffic. IEEE Intell. Transp. Syst. Mag.
**2023**, 15, 318–332. [Google Scholar] [CrossRef] - Wu, W.; Lin, Y.; Liu, R.; Jin, W. The multi-depot electric vehicle scheduling problem with power grid characteristics. Transp. Res. Part B Methodol.
**2022**, 155, 322–347. [Google Scholar] [CrossRef] - Zhan, S.; Wong, S.; Shang, P.; Lo, S. Train rescheduling in a major disruption on a high-speed railway network with seat reservation. Transp. A Transp. Sci.
**2022**, 18, 532–567. [Google Scholar] [CrossRef] - Yuan, J.; Jones, D.; Nicholson, G. Flexible real-time railway crew rescheduling using depth-first Search. J. Rail Transp. Plan. Manag.
**2022**, 24, 100353. [Google Scholar] [CrossRef] - Liu, F.; Xun, J.; Zhou, M.; He, S.; Dong, H. A driving strategy based integrated rescheduling Model for high-Speed railway by using the parallel intelligent method. In Proceedings of the 2022 IEEE 2nd International Conference on Digital Twins and Parallel Intelligence (DTPI), Boston, MA, USA, 24–28 October 2022. [Google Scholar]
- Zhang, C.; Gao, Y.; Cacchiani, V.; Yang, L.; Gao, Z. Train rescheduling for large-scale disruptions in a large-scale railway network. Transp. Res. Part B Methodol.
**2023**, 174, 102786. [Google Scholar] [CrossRef]

Input Parameters | |
---|---|

Symbol | Description |

$\mathbb{N}$ | A set of stations in the same direction, $\mathbb{N}$ = {1, 2, ⋯, N}. |

n | Station number, $n\in \mathbb{N}$. |

$\mathbb{T}$ | A set of trains, $\mathbb{T}$ = {1, 2, ⋯, T}. |

$i,j$ | Train number, $i,j\in \mathbb{T}$. |

$\mathbb{L}$ | A set of driving strategies, $\mathbb{L}$ = {1, 2, ⋯, L}. |

l | Driving strategy number, $l\in \mathbb{L}$. |

${R}_{i,n}^{l}$ | The inter-station travel time of train i between |

station n and station $n+1$ using driving strategy l. | |

${H}_{i,j,n}^{a}$ | The arrival headway between train i and train j at station n. |

${H}_{i,n}^{a,p}$ | The minimum arrival headway between train i and |

the preceding train at station n when train i does not stop. | |

${H}_{i,n}^{a,s}$ | The minimum arrival headway between train i and |

the preceding train at station n when train i stops. | |

${H}_{i,j,n}^{d}$ | The departure headway between train i and train j at station n. |

${H}_{i,n}^{d,p}$ | The minimum departure headway of train i at station n |

when the preceding train does not stop. | |

${H}_{i,n}^{d,s}$ | The minimum departure headway of train i at station. n |

when the preceding train stops. | |

e | The minimum stopping time. |

$e{p}_{i,n}$ | Time of train i pass station n. |

${O}_{i,n}^{a}$ | The arrival time of train i at station n in the initial timetable. |

${O}_{i,n}^{d}$ | The departure time of train i from station n in the initial timetable. |

Decision Variables | |

Symbol | Description |

${d}_{i,n}$ | Time of train i departure from station n. |

${a}_{i,n}$ | Time of train i arrival at station n. |

${\eta}_{i,n}^{l}$ | A binary variable: If the driving strategy of train i between |

station n and station $n+1$ is l, ${\eta}_{i,n}^{l}=1$, otherwise, ${\eta}_{i,n}^{l}=0$. | |

${\mu}_{i,j,n}$ | A binary variable: The order of train i and train j between |

station n and station $n+1$. if train i is later than train j, | |

${\mu}_{i,j,n}=1$, otherwise, ${\mu}_{i,j,n}=0$. | |

${Q}_{i,n}$ | The number of trains stop at station n when train i |

arrive at station n. |

Index | Station | Distance | Index | Station | Distance |
---|---|---|---|---|---|

1 | Xuzhoudong | - | 8 | Danyangbei | 25 km |

2 | Suzhoubei | 79 km | 9 | Changzhoubei | 32 km |

3 | Bengbunan | 77 km | 10 | Wuxidong | 57 km |

4 | Dingyuan | 53 km | 11 | Suzhounan | 26 km |

5 | Chuzhou | 62 km | 12 | Kunshannan | 32 km |

6 | Nanjingnan | 59 km | 13 | Shanghaihongqiao | 43 km |

7 | Zhenjiangnan | 69 km | - | - | - |

Time Limit | 180 (s) | 360 (s) | ||||
---|---|---|---|---|---|---|

Index | Total | Computing | Gap (%) | Total | Computing | Gap (%) |

Delay (s) | Time (s) | Delay (s) | Time (s) | |||

(10,1) | 0 | 13 | 0 | 0 | 13 | 0 |

(20,1) | 698 | 21 | 0 | 698 | 21 | 0 |

(30,1) | 4632 | 180 | 2.35 | 4535 | 360 | 0.43 |

(45,1) | 17,050 | 180 | 5.49 | 16,032 | 360 | 4.45 |

(60,1) | 28,778 | 180 | 8.52 | 27,393 | 360 | 6.91 |

(10,3) | 24 | 4 | 0 | 24 | 4 | 0 |

(20,3) | 1751 | 6 | 0 | 1751 | 6 | 0 |

(30,3) | 4888 | 37 | 0 | 4888 | 37 | 0 |

(45,3) | 14,497 | 180 | 3.12 | 13,849 | 360 | 2.68 |

(60,3) | 24,343 | 180 | 5.1 | 24,149 | 360 | 4.89 |

(10,5) | 817 | 3 | 0 | 817 | 3 | 0 |

(20,5) | 3125 | 8 | 0 | 3125 | 8 | 0 |

(30,5) | 7137 | 25 | 0 | 7137 | 25 | 0 |

(45,5) | 16,087 | 180 | 3.45 | 15,846 | 360 | 2.87 |

(60,5) | 28,316 | 180 | 5.12 | 27,914 | 360 | 4.66 |

Disturbance Time | 20 min | 30 min | ||||
---|---|---|---|---|---|---|

Speed (Km/h) | 3 | 5 | 7 | 3 | 5 | 7 |

(295,305) | 2260 | 2260 | 2260 | 9222 | 9104 | 8980 |

(294,306) | 2144 | 2144 | 2144 | 8538 | 8476 | 8450 |

(292,308) | 1895 | 1895 | 1895 | 7881 | 7682 | 7675 |

(290,310) | 1667 | 1667 | 1667 | 7492 | 8792 | 8775 |

(288,312) | 1425 | 1425 | 1425 | 6800 | 6712 | 6795 |

(286,314) | 1210 | 1210 | 1210 | 6266 | 6200 | 6193 |

(284,316) | 1026 | 1026 | 1026 | 5704 | 5528 | 5510 |

(282,318) | 884 | 884 | 884 | 5179 | 5089 | 5055 |

(280,320) | 698 | 698 | 698 | 4638 | 4535 | 4530 |

Scenario | Our Model (s) | TWO (s) | Decline Ratio (%) |
---|---|---|---|

(10,1) | 0 | 7200 | 100 |

(20,1) | 698 | 14,400 | 95.1 |

(30,1) | 4632 | 21,600 | 78 |

(45,1) | 17,050 | 32,400 | 47 |

(60,1) | 28,778 | 43,200 | 33 |

(10,3) | 24 | 7200 | 99.8 |

(20,3) | 1751 | 14,400 | 87.8 |

(30,3) | 4888 | 21,600 | 77.3 |

(45,3) | 14,497 | 32,400 | 55.2 |

(60,3) | 24,343 | 43,200 | 43.6 |

(10,5) | 817 | 7200 | 88.6 |

(20,5) | 3125 | 14,400 | 78.2 |

(30,5) | 7137 | 21,600 | 66.7 |

(45,5) | 16,087 | 32,400 | 50.3 |

(60,5) | 28,316 | 43,200 | 34.4 |

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## Share and Cite

**MDPI and ACS Style**

Liu, F.; Xun, J.
An Automatic Train Operation Based Real-Time Rescheduling Model for High-Speed Railway. *Mathematics* **2023**, *11*, 4546.
https://doi.org/10.3390/math11214546

**AMA Style**

Liu F, Xun J.
An Automatic Train Operation Based Real-Time Rescheduling Model for High-Speed Railway. *Mathematics*. 2023; 11(21):4546.
https://doi.org/10.3390/math11214546

**Chicago/Turabian Style**

Liu, Fan, and Jing Xun.
2023. "An Automatic Train Operation Based Real-Time Rescheduling Model for High-Speed Railway" *Mathematics* 11, no. 21: 4546.
https://doi.org/10.3390/math11214546