# Track Allocation Optimization in Multi-Direction High-Speed Railway Stations

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## Abstract

**:**

## 1. Introduction

- Firstly, to our knowledge, there is litte research that directly pays attention to the track allocation optimization in a multi-direction high-speed railway station. We propose the track allocation problem in the multi-direction high-speed railway station for the first time.
- Secondly, the resources in the throat area are considered. Some researchers assumed that outbound and inbound trains are independent in the single-direction station and separated the track allocation plan into two parts. In this way, headway constraints in the train timetabling problem can make sure that no conflict exists between routes of the trains from the same line in a single-direction station. However, in the multi-direction station, trains’ arrival directions and departure directions are various. Train timetables of different lines are designed separately. The routes of trains from different lines may conflict with each other due to the layout of the throat area. Consequently, we cannot divide the track allocation plan into two independent parts based on their directions. The flexible track utilization rule is applied to this problem and all trains are allocated together. The restrictive resources of one multi-direction station are the switches in the throat area, and the track allocation plan is unfeasible if only the tracks near the platforms are considered. In this paper, the resources in the throat area and arrival-departure tracks are considered simultaneously and outbound and inbound trains are no longer independent in the station, which ensures the feasibility of the result.
- Thirdly, the resources occupation times were not explained exhaustedly and correctly, which lead to a difficult expression of the constraints when solving the track allocation problem of multi-direction stations. We give a detailed analysis of the occupation times of resources in the throat area and arrival-departure tracks in this paper, which facilitates the construction of the complicated conflicting constraints.

## 2. Problem Description

#### 2.1. Multi-Direction Stations in a High-Speed Railway Network

#### 2.2. Occupation Time of Train Routes

#### 2.3. Assumption

- The second type of the station layout mentioned above is discussed.
- Basic information of the station is given, including the minimum preparation time criterion and running time of different routes and the operations each track can provide.
- The speed differences in a station only exist between stop and nonstop trains. Because of the speed limitation in the station, the running time of nonstop trains are the same, regardless of the train types and the speed levels. The same assumption is applied to the stop trains.
- The train timetable and rolling stock plan are given, including the trains’ static operation in this multi-direction station.
- The basic receiving route from one arrival direction to one certain track is unique and fixed. Similarly, the basic departure route is also definite and unique.

## 3. Formulation of Optimization Model

#### 3.1. Notations and Decision Variables

#### 3.2. Objective Function

#### 3.3. Constraints

#### 3.3.1. Time Constraints

#### 3.3.2. Safety Constraints

#### 3.3.3. Operation Constraints

#### 3.3.4. Domain Constraints

## 4. Computational Experiment

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Occupation time of SGs in the station. Blue line: occupation of receiving routes. Black line: occupation of departure routes. Red line: occupation of through routes.

Train ID | Arrival Direction | Departure Direction | Arrival Time | Departure Time | Stop (1) or Nonstop (0) |
---|---|---|---|---|---|

1 | D | A | 12:00:00 | 12:03:20 | 1 |

2 | B | A | 12:02:00 | 12:06:24 | 1 |

3 | B | A | 12:05:10 | 12:11:50 | 1 |

4 | C | D | 12:06:20 | 12:12:20 | 1 |

5 | A | B | 12:09:40 | 12:09:40 | 0 |

6 | B | A | 12:09:50 | 12:18:20 | 1 |

7 | C | D | 12:15:00 | 12:17:00 | 1 |

8 | D | C | 12:16:50 | 12:16:50 | 0 |

9 | B | D | 12:18:10 | 12:24:10 | 1 |

10 | D | C | 12:20:20 | 12:23:20 | 1 |

11 | A | B | 12:22:00 | 12:22:00 | 0 |

12 | C | A | 12:26:10 | 12:30:10 | 1 |

13 | D | C | 12:30:00 | 12:33:00 | 1 |

14 | C | D | 12:30:20 | 12:36:00 | 1 |

15 | B | C | 12:34:30 | 12:36:30 | 1 |

16 | B | A | 12:37:47 | 12:41:00 | 1 |

17 | C | D | 12:38:10 | 12:41:10 | 1 |

18 | D | C | 12:38:30 | 12:42:10 | 1 |

19 | C | B | 12:43:10 | 12:46:10 | 1 |

20 | D | A | 12:44:30 | 12:51:20 | 1 |

21 | B | A | 12:45:46 | 12:45:46 | 0 |

22 | C | D | 12:47:00 | 12:49:00 | 1 |

23 | A | B | 12:47:10 | 12:51:10 | 1 |

24 | C | D | 12:50:30 | 12:52:45 | 1 |

25 | B | C | 12:51:00 | 12:55:00 | 1 |

26 | A | B | 12:52:40 | 12:57:00 | 1 |

27 | C | D | 12:56:00 | 12:58:20 | 1 |

28 | B | A | 12:56:03 | 13:00:03 | 1 |

29 | D | C | 12:57:20 | 13:01:46 | 1 |

30 | A | D | 12:58:00 | 13:02:00 | 1 |

31 | D | A | 13:03:00 | 13:06:20 | 1 |

32 | C | D | 13:07:00 | 13:07:00 | 0 |

33 | D | C | 13:07:00 | 13:10:00 | 1 |

34 | B | A | 13:08:20 | 13:11:20 | 1 |

35 | B | C | 13:13:10 | 13:20:10 | 1 |

36 | D | C | 13:16:00 | 13:16:00 | 0 |

37 | A | D | 13:17:00 | 13:19:00 | 1 |

38 | B | A | 13:18:20 | 13:18:20 | 0 |

39 | D | A | 13:21:00 | 13:26:00 | 1 |

40 | C | B | 13:25:00 | 13:28:00 | 1 |

41 | B | C | 13:25:00 | 13:28:20 | 1 |

42 | B | C | 13:28:40 | 13:31:20 | 1 |

43 | A | A | 13:36:10 | 13:48:00 | 1 |

44 | D | C | 13:40:00 | 13:44:10 | 1 |

45 | A | B | 13:41:00 | 13:43:00 | 1 |

46 | B | A | 13:42:00 | 13:44:00 | 1 |

47 | C | D | 13:45:00 | 13:47:00 | 1 |

48 | D | B | 13:48:40 | 13:53:00 | 1 |

49 | A | C | 13:54:30 | 13:59:00 | 1 |

${\mathit{T}}_{\mathit{min}}^{\mathit{p}\mathit{a}}$ | ${\mathit{T}}_{\mathit{min}}^{\mathit{p}\mathit{d}}$ | ${\mathit{T}}_{\mathit{N}}^{\mathit{a}\mathit{e}}$ | ${\mathit{T}}_{\mathit{S}\mathit{T}}^{\mathit{a}\mathit{e}}$ | ${\mathit{T}}_{\mathit{S}\mathit{T}}^{\mathit{a}\mathit{d}\mathit{e}}$ | ${\mathit{T}}_{\mathit{j}{\mathit{v}}_{\mathit{i}}^{\mathit{d}}}$ | ${\mathit{T}}_{\mathit{b}\mathit{u}\mathit{f}}$ |
---|---|---|---|---|---|---|

180s | 30s | 2s | 4s | 8s | 10s | 10s |

Arrival Direction | Track Number | ${\mathit{T}}_{{\mathit{v}}_{\mathit{i}}^{\mathit{a}}\mathit{j}}\left(\mathbf{s}\right)$ | Track Number | Departure Direction | ${\mathit{T}}_{\mathit{j}{\mathit{v}}_{\mathit{i}}^{\mathit{d}}}(\mathbf{s})$ |
---|---|---|---|---|---|

A | 1/2 | 80 | 1/2 | A | 105 |

A | 3/4 | 70 | 1/2 | B | 85 |

A | 6 | 10 | 1/2 | C | 135 |

A | 9/10 | 95 | 1/2 | D | 50 |

A | 11/12 | 105 | 3/4 | A | 95 |

B | 1/2 | 105 | 3/4 | B | 80 |

B | 3/4 | 95 | 3/4 | C | 125 |

B | 7 | 10 | 3/4 | D | 45 |

B | 9/10 | 70 | 5 | B | 20 |

B | 11/12 | 80 | 5 | D | 10 |

C | 1/2 | 40 | 6 | B | 10 |

C | 3/4 | 35 | 6 | D | 20 |

C | 5 | 10 | 7 | A | 10 |

D | 1/2 | 135 | 7 | C | 20 |

D | 3/4 | 125 | 8 | A | 20 |

D | 8 | 10 | 8 | C | 10 |

D | 9/10 | 35 | 9/10 | A | 80 |

D | 11/12 | 40 | 9/10 | B | 95 |

- | - | - | 9/10 | C | 45 |

- | - | - | 9/10 | D | 125 |

- | - | - | 11/12 | A | 85 |

- | - | - | 11/12 | B | 105 |

- | - | - | 11/12 | C | 50 |

- | - | - | 11/12 | D | 135 |

Publication | Arri./Dep. Route ^{1} | Occupation Time Analysis ^{2} | Pre-acclaim ^{3} | Flexible Rules ^{4} | Multi-direction ^{5} | Rolling Stock Plan ^{6} |
---|---|---|---|---|---|---|

Meng and Zhou (2014) | N | N | N | - | - | N |

Gao et al. (2017) | N | N | N | N | N | N |

Billionnet (2003) | N | N | N | - | - | N |

Sels et al. (2014) | Y | Y | N | - | - | N |

Cardillo et al. (1998) | N | N | N | - | - | N |

Zwaneveld et al. (2001) | Y | N | N | - | - | N |

Qiao (2008) | Y | Y | N | N | N | N |

Wu et al. (2013) | N | N | N | N | N | N |

Zhang (2018) | Y | Y | Y | N | Y | N |

This paper | Y | Y | Y | Y | Y | Y |

^{1}When column ‘Arri./Dep. Route’ contains a ‘Y’, it means that the routes in stations are divided into arrival route, departure routes.

^{2}When column ‘Occupation Time Analysis’ contains ‘Y’, it means that the paper analyzes the occupation time of routes in detail.

^{3}When column ‘Pre-acclaim’ contains ‘Y’, it means that the route occupation time consider the pre-acclaim time.

^{4}When column ‘Flexible Rules’ contains ‘Y’, it means that flexible track utilization rule is adopted in the paper.

^{5}When column ‘Multi-direction’ contains ‘Y’, it means that track allocation problem in multi-direction station is defined.

^{6}When column ‘Rolling Stock Plan’ contains ‘Y’, it means that rolling stock plan is considered in the paper when solving track allocation problem.

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**MDPI and ACS Style**

Zhang, Q.; Zhu, X.; Wang, L.
Track Allocation Optimization in Multi-Direction High-Speed Railway Stations. *Symmetry* **2019**, *11*, 459.
https://doi.org/10.3390/sym11040459

**AMA Style**

Zhang Q, Zhu X, Wang L.
Track Allocation Optimization in Multi-Direction High-Speed Railway Stations. *Symmetry*. 2019; 11(4):459.
https://doi.org/10.3390/sym11040459

**Chicago/Turabian Style**

Zhang, Qin, Xiaoning Zhu, and Li Wang.
2019. "Track Allocation Optimization in Multi-Direction High-Speed Railway Stations" *Symmetry* 11, no. 4: 459.
https://doi.org/10.3390/sym11040459