# A Time-Space Network Model Based on a Train Diagram for Predicting and Controlling the Traffic Congestion in a Station Caused by an Emergency

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## Abstract

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## 1. Introduction

#### 1.1. The Description of the Subject of Research and the Motives to Take

- Strong timeliness. Because of the transmissibility of congestion, it is of great significance to predict congestion in time for maintaining and improving the capacity of the transport network in an emergency.
- Time-space. The final output of the model should be which station will become congested first and when the station begins to become congested. These output results will become the basis of the decision of the railway transportation managers.
- Strongly controllable. Rail transportation is a strongly controlled mode of transport, which means that the turnover process of railcar flow and the distribution of railcar flow in the transport network are subject to schedules and plans.
- Multiple highly dynamic constraints. Congestion prediction and control in an emergency is a complicated system involving many components. First, the capacity in the network will be reduced in an emergency. Second, the state of railcar flow will change dynamically with the operation process and the state of some railcar flows changes more than twice within the decision-making period horizon. Third, the completion rate of the transportation plan and scheduled train departure will be affected by the emergency.

#### 1.2. Literature Review

- Based on the train diagram, we constructed a time-space network for the first time by considering the transition of the railcar state.
- The problem is split into two sub-problems: feasible path set generation and railcar flow distribution. An improved A* algorithm based on railcar flow route was used to generate a feasible path set of any pair of transportation demands. A dynamic railcar flow distribution model in an emergency which can be solved by mathematical programming software was established to simulate the process of railcar flow operation in emergency, and the railcar flow distribution in the future period was estimated based on the distribution results of railcar flow in the time-space network, so as to predict where there will be traffic congestion and when the traffic congestion will begin in the station.

## 2. Problem Description

#### 2.1. The Process Resulting in Traffic Congestion due to the Emergency

#### 2.2. The Turnover Process of Railcar and Transition of Railcar State

- Train arrival. The activity begins when the inbound train carrying railcars arrives at the station’s receiving yard and ends when railcars detached from the inbound train are pushed over the top of the station’s hump.
- Train break-up. The activity begins when the railcars starts to slide off the hump and ends when all railcars detached from the inbound train roll down to the appropriate track in the station’s switchyard.
- Loaded railcar delivery. The activity begins when all loaded railcars detached from the inbound train roll down to the station’s switchyard and ends when the loaded railcars are pulled to the freight yard by the local locomotive.
- Empty railcar delivery. The activity begins when all empty railcars detached from the inbound train roll down to the station’s switchyard and ends when the empty railcars are pulled to the freight yard by the local locomotive.
- Railcar loading. The activity begins when all empty railcars detached from the inbound train start to load in the freight yard and ends when the empty railcars have finished loading the goods.
- Railcar unloading. The activity begins when all loaded railcars detached from the inbound train start to unload in the freight yard and ends when the loaded railcars have finished unloading the goods.
- Loaded railcar pickup. The activity begins when the loaded railcars detached from the inbound train start to hang to the local locomotive and ends when all the loaded railcars are pulled back to the station’s switchyard from the freight yard by the local locomotive.
- Empty railcar pickup. The activity begins when the empty railcars detached from the inbound train start to hang to the local locomotive and ends when all the empty railcars are pulled back to the station’s switchyard from the freight yard by the local locomotive.
- Railcar assembly. The activity begins when all railcars detached from the inbound train are pulled back to the station’s switchyard from the freight yard and ends when the railcars are able to form an outbound train.
- Train formation. The activity begins when the railcars detached from many inbound trains are able to form an outbound train and ends when the formation of an outbound train is completed.
- Train departure. The activity begins when the formation of an outbound train is completed and ends when the outbound train carrying railcars starts from the departure yard.

- Origin station. The operation processes of a railcar flow mainly include the initialization process of loaded railcar flow and the initialization process of empty railcar flow. The initialization process of empty railcar flow consists of railcar unloading, empty railcar pickup, railcar assembly, train formation, and train departure. The process realizes the transition of loaded railcar flow into empty railcar flow. The initialization process of loaded railcar flow consists of railcar loading, loaded railcar pickup, railcar assembly, train formation, and train departure. The process realizes the transition of loaded railcar flow into empty railcar flow.
- Transfer station. The main operation process is the transfer process of railcar flow. The transfer process of railcar flow consists of train arrival, train break-up, railcar assembly, train formation, and train departure.
- Destination station. The operation processes of a railcar flow mainly include the local operation process of loaded railcar flow, the secondary operation process, the local operation process of empty railcar flow, the empty railcar allocation process, and the formation and departure process. The local operation process of loaded railcar flow consists of train arrival, train break-up, loaded railcar delivery, and railcar unloading. The secondary operation process is the process of reloading for the loaded railcar after unloading. The local operation process of empty railcar flow consists of train arrival, train break-up, empty railcar delivery, and railcar loading. The empty railcar allocation process consists of empty railcar pickup, railcar assembly, train formation, and train departure. The formation and departure process consists of loaded railcar pickup, railcar assembly, train formation, and train departure. The local operation process of loaded railcar flow realizes the transition of loaded railcar flow into empty railcar flow. The secondary operation process and the local operation process of empty railcar flow realize the transition of loaded railcar flow into empty railcar flow.

#### 2.3. Problem Abstraction

## 3. Methods

#### 3.1. Construction of time-space Network Based on the Train Diagram

#### 3.2. Feasible Path Set Generation Algorithm Based on Improved A* Algorithm

#### 3.3. The Dynamic Railcar Flow Distribution Model in an Emergency

#### 3.3.1. Set Parameter and Variable Definitions

#### 3.3.2. Objective Function

#### 3.3.3. Constraint Conditions

#### 3.3.4. Model Synthesis

#### 3.3.5. Solution Method

#### 3.4. The Method of Calculating the Real-Time Number of Railcars at Each Station in Each Period

## 4. Numerical Studies

#### 4.1. Data Input

#### 4.2. Results and Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AN | Arrival Node |

DN | Departure Node |

LRSEN | Loaded Railcar State End Node |

ERSEN | Empty Railcar State End Node |

SOCN | Secondary Operation Completion Node |

IN | Initial Node |

EN | End Node |

TA | Train Arc |

LOA | Local Operation Arc |

SOA | Secondary Operation Arc |

FADA | Formation And Departure Arc |

ERAA | Empty Railcar Allocation Arc |

TOA | Transfer Operation Arc |

SA | Stay Arc |

IA | Initial Arc |

EA | End Arc |

DA | Detention Arc |

LRL | Loaded Railcar Layer |

ERL | Empty Railcar Layer |

PL | Public Layer |

IP | Initial Path |

SOP | Secondary Operation Path |

ETLP | Empty To Loaded Path |

DP | Detention Path |

DRFDMUE | Dynamic Railcar Flow Distribution Model Under Emergency |

DRFDMUN | Dynamic Railcar Flow Distribution Model Under Normality |

SCU | Station Capacity Utilization |

TL | Total Loss |

ALPCR | Average Loading Plan Complete Rate |

AEPCR | Average Emptying Plan Complete Rate |

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**Figure 7.**Station Capacity Utilization (SCU) time series curves corresponding to different stations in an emergency.

Studies | Holmberg et al. (1998) | Narisetty et al. (2008) | Heydari et al. (2017) | This Model |
---|---|---|---|---|

Combines railcar routing and distribution? | Yes | No | No | Yes |

Formulation | Capacitated network design | Transportation problem | Path-based capacitated network | Multi-commodity flow problem |

Considers transition of railcar state? | No | No | No | Yes |

Capacitated? | Yes | No | Yes | Yes |

Solution method | Tabu Search | LP | LP | Heuristic decomposition |

Time-Space Node Type | Type Index |
---|---|

AN | 1 |

DN | 2 |

LRSEN | 3 |

ERSEN | 4 |

SOCN | 5 |

IN | 6 |

EN | 7 |

time-space Arc Type | Type Index |
---|---|

TA | 1 |

LOA | 2 |

SOA | 3 |

FADA | 4 |

ERAA | 5 |

TOA | 6 |

SA | 7 |

IA | 8 |

EA | 9 |

DA | 10 |

Hierarchy | Hierarchy Index |
---|---|

LRL | 1 |

ERL | 2 |

PL | 3 |

time-space Path Type | Type Index |
---|---|

IP | 1 |

SOP | 2 |

ETLP | 3 |

DP | 4 |

Sets | Descriptions |
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$H$ | Set of hierarchy index; |

$N$ | Set of time-space node; |

${B}_{N}$ | Set of type indexes of time-space nodes; |

$A$ | Set of time-space arc; |

${A}_{ns}$ | Set of time-space arcs starting from time-space node $n,n\in N$; |

${A}_{ne}$ | Set of time-space arcs ending in time-space node $n,n\in N$; |

${B}_{A}$ | Set of type indexes of time-space arcs; |

$P$ | Set of time-space path; |

${P}_{ns}$ | Set of time-space paths starting from time-space node $n,n\in N$; |

${P}_{ne}$ | Set of time-space paths ending in time-space node $n,n\in N$; |

${B}_{P}$ | Set of type indexes of time-space paths; |

${F}_{1}$ | Set of initial loaded railcar flow; |

${P}_{f1}$ | Set of feasible paths of initial loaded railcar flow ${f}_{1},{f}_{1}\in {F}_{1}$; |

${F}_{2}$ | Set of initial empty railcar flow; |

${A}_{f2}$ | Set of feasible arcs of initial empty railcar flow ${f}_{2},{f}_{2}\in {F}_{2}$; |

$L$ | Set of loading plan; |

${P}_{l}$ | Set of time-space$\text{}\mathrm{path}\text{}\mathrm{in}\text{}\mathrm{line}\text{}\mathrm{with}\text{}\mathrm{loading}\text{}\mathrm{plan}\text{}l,l\in L$; |

$E$ | Set of emptying plan set; |

${A}_{e}$ | Set of time-space$\text{}\mathrm{arc}\text{}\mathrm{in}\text{}\mathrm{line}\text{}\mathrm{with}\text{}\mathrm{emptying}\text{}\mathrm{plan}\text{}e,e\in E$; |

${A}_{T}$ | Set of train arc affected in an emergency; |

$G$ | Set of period; |

$S$ | Set of station; |

${A}_{g}^{s}$ | Set of time-space arcs used to calculate the number of railcars at station $s$ in period $g$ |

Parameters | Descriptions |

${b}_{n}$ | Type index of time-space node $n,{b}_{n}\in {B}_{N},n\in N$; |

${h}_{n}$ | Hierarchy index of time-space node $n,h\in H,n\in N$; |

${b}_{a}$ | Type index of time-space arc $a,{b}_{a}\in {B}_{A},a\in A$; |

${h}_{a}$ | Hierarchy index of time-space arc $a,h\in H,a\in A$; |

${t}_{a}$ | Time of time-space arc $a,a\in A$; |

${c}_{a}$ | Capacity of time-space arc $a,a\in A$; |

${b}_{p}$ | Type index of time-space path $p,{b}_{p}\in {B}_{P},p\in P$; |

${h}_{p}$ | Hierarchy index of time-space path $p,h\in H,p\in P$; |

${t}_{p}$ | Time of time-space path $p,p\in P$; |

${A}_{p}$ | time-space arc set of time-space path $p,p\in P$; |

${q}_{f1}$ | Number of initial loaded railcar flow ${f}_{1},{f}_{1}\in {F}_{1}$; |

${q}_{f2}$ | Number of initial empty railcar flow ${f}_{2},{f}_{2}\in {F}_{2}$; |

${q}_{l}$ | $\mathrm{Planned}\text{}\mathrm{loading}\text{}\mathrm{number}\text{}\mathrm{for}\text{}\mathrm{one}\text{}\mathrm{day}\text{}\mathrm{of}\text{}\mathrm{loading}\text{}\mathrm{plan}\text{}l,l\in L$; |

${r}_{l}$ | $\mathrm{Completed}$ when emergency occurs, $l\in L$; |

${q}_{e}$ | $\mathrm{Planned}\text{}\mathrm{emptying}\text{}\mathrm{number}\text{}\mathrm{for}\text{}\mathrm{one}\text{}\mathrm{day}\text{}\mathrm{of}\text{}\mathrm{emptying}\text{}\mathrm{plan}\text{}e,e\in E$; |

${r}_{e}$ | $\mathrm{Completed}$$\mathrm{when}\text{}\mathrm{emergency}\text{}\mathrm{occurs},e\in E$; |

$m$ | Maximum number of marshaling railcars in a train; |

$s{t}_{a}$ | Start moment in minutes of time-space arc $a,a\in A$; |

$e{t}_{a}$ | End moment in minutes of time-space arc $a,a\in A$; |

$e{t}_{g}$ | End moment in minutes of period $g,g\in G$; |

${s}_{n}$ | Station of time-space node $n,{s}_{n}\in S,n\in N$; |

${n}_{as}$ | Start node of time-space arc $a,a\in A$; |

${n}_{ae}$ | End node of time-space arc $a,a\in A$; |

Decision variables | Descriptions |

${x}_{a}$ | Empty railcar flow number of time-space arc $a$; |

${y}_{p}$ | Loaded railcar flow number of time-space path $p$; |

${z}_{p}^{f1}$ | Loaded railcar flow number of feasible path $p$ of initial loaded railcar flow $f1$; |

${v}_{a}^{f2}$ | Empty railcar flow number of feasible arc $a$ of initial empty railcar flow $f2$. |

${d}_{s}^{g}$ | Number of railcars at station$p$ in period$g$; |

O | D | Railcar Flow Number (Generation Moment in minutes) | Type | Railcar Flow Route |
---|---|---|---|---|

A | B | 100(0), 200(180), 150(250) | Loaded | A–B |

A | C | 50(0), 60(120), 90(160) | Loaded | A–C |

A | D | 30(0), 30(160), 40(200) | Loaded | A–C–D |

B | C | 20(100), 50(300) | Loaded | B–C |

B | D | 150(0) | Loaded | B–D |

C | D | 60(0), 50(100), 80(160), 100(230), 50(350) | Loaded | C–D |

A | 200(100), 20(170), 100(230), 80(310), 30(350) | Empty | ||

B | 60(0), 100(160), 60(200), 100(270), 60(360) | Empty | ||

C | 100(120), 80(250), 160(300), 150(360), 150(450) | Empty |

Loading Station | Destination Station | Planned Loading Number | Completed Loading Number |
---|---|---|---|

A | B | 120 | 0 |

A | C | 160 | 0 |

B | C | 100 | 0 |

B | D | 100 | 0 |

C | D | 240 | 0 |

Boundary Station | Planned Emptying Number | Completed Emptying Number |
---|---|---|

D | 300 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Qu, Z.; He, S.
A Time-Space Network Model Based on a Train Diagram for Predicting and Controlling the Traffic Congestion in a Station Caused by an Emergency. *Symmetry* **2019**, *11*, 780.
https://doi.org/10.3390/sym11060780

**AMA Style**

Qu Z, He S.
A Time-Space Network Model Based on a Train Diagram for Predicting and Controlling the Traffic Congestion in a Station Caused by an Emergency. *Symmetry*. 2019; 11(6):780.
https://doi.org/10.3390/sym11060780

**Chicago/Turabian Style**

Qu, Zihan, and Shiwei He.
2019. "A Time-Space Network Model Based on a Train Diagram for Predicting and Controlling the Traffic Congestion in a Station Caused by an Emergency" *Symmetry* 11, no. 6: 780.
https://doi.org/10.3390/sym11060780