^{1}

^{1}

^{2}

^{2}

^{2}

^{1}

^{2}

Train timetable stability is the possibility to recover the status of the trains to serve as arranged according to the original timetable when the trains are disturbed. To improve the train timetable stability from the network perspective, the bilevel programming model is constructed, in which the upper level programming is to optimize the timetable stability on the network level and the lower is to improve the timetable stability on the dispatching railway segments. Timetable stability on the network level is defined with the variances of the utilization coefficients of the section capacity and station capacity. Weights of stations and sections are decided by the capacity index number and the degrees. The lower level programming focuses on the buffer time distribution plan of the trains operating on the sections and stations, taking the operating rules of the trains as constraints. A novel particle swarm algorithm is proposed and designed for the bilevel programming model. The computing case proves the feasibility of the model and the efficiency of the algorithm. The method outlined in this paper can be embedded in the networked train operation dispatching system.

Train timetable is the fundamental file for organizing the railway traffic, which determines the inbound and outbound time of trains. Railways are typically operated according to a planned (predetermined) timetable, and the quality of the timetable determines the quality of the railway service. So it is most important to map a high quality timetable for all kinds of trains.

But there is a dilemma that we place as much as possible trains on the timetable chart, and simultaneously we should enhance the possibility to adjust the timetable when disruptions occur. The randomly occurring disturbances may cause train delays and even disrupt the entire train operation plan. In the railway network, every station and section are planned to serve the trains according to the schedule, often compactly. So a slightly delayed train may cause a domino effect of secondary delays over the thorough network. Although the buffer times added to the minimum running time in the sections and minimum dwell at stations in scheduled timetables may absorb some train delays and assure some degree of timetable stability, the large buffer time will reduce the capacity of the railway.

Therefore, to ensure both the capacity and the order of the train operation, a reliable, stable, robust timetable, and the feasible efficient rescheduling of the planned timetable must be worked out. A superior quality timetable cannot only decide the inbound and outbound time at stations, and the more important, can offer the possibility to recover the operation according to the planned timetable when the trains are disturbed by accidents randomly. Timetable stability is the index to measure the

So when assigning the trains paths and mapping out the train schedule, the train number assigned to the sections and buffer time distribution should be designed carefully, not only considering the section capacity and station capacity, but also the minimal running time at each section and the minimal dwell on the station.

We define the networked timetable stability quantitatively, considering that the railway network with the goal is to optimize the timetable stability to offer more possibilities to reschedule the trains on the railway network to deal with disturbs in the train operation process.

The outline of the paper is as follows. First, Section

It is a hot topic now to assure the reliability, safety, and stability of the traffic control system as discussed in [

The research experienced two developing periods. In the earliest period, the focus was the timetable on the dispatching section, according to the operation mode of the railway and the basis to study the train timetable stability is formed. A discrete dynamic system model was built to describe the timetable with the max-plus algebra based on the discussion of the timetable periodicity and analyzed the timetable stability, as proposed by Goverde [

Research on timetable stability progressively expanded to the railway network, for the study focusing on the timetable stability of dispatching cannot suit the networked timetable design and optimization. Engelhardt-Funke and Kolonko considered a network of periodically running railway lines. They built a model to analyze stability and investments in railway networks and designed an innovative evolutionary algorithm to solve the problem in [

And we can see that the networked timetable stability is related to not only time but also the utilization coefficient capacity of the railway network, as discussed in [

Networked timetable stability must be studied from two levels. The upper level is to study the relation between the trains flow and the capacity of the sections and stations and the ability to recover the timetable when an emergency occurs determined by the relation. The lower level is to study the distribution plan of the buffer time for each train in the sections running process and the stations dwelling, to eliminate the negative effects of the disturbs.

The goal of the upper programming is to decide the number of trains assigned on each railway section and at the stations. The fundamental restriction is that the number of trains assigned to the sections and stations must not exceed the capacity of the sections and the stations. And the number of trains received by the stations must be equal to the total number of the trains running through the sections which are connected to the relative station.

The lower programming is to determine the buffer time distribution plan. The running time through a whole section planned in the timetable is more than that it requires if it runs at its highest speed. So there is a period of time called buffer time that can be distributed for the sections running and stations dwelling to absorb the delay caused by the random disturbances. The restriction is that buffer time allocated to each station and section must be longer than or equal to zero.

To define the timetable stability on the network level, the load on the sections and stations is the key factor. So the load index numbers must be defined first.

The load index number of a station on the railway network is

The index number of the capacity of a station is

Then the station weight is

The load index number of a section on the railway network is

The index number of the capacity of a section is

Then the weight of the section is

Then with the load index numbers of the stations and sections, the timetable stability on the network level is defined as

The goal of the upper programming is to optimize the timetable stability on the network level, so

Restrictions require that the number of the trains running through a section cannot be greater than the number of the trains that the section capacity allows. Likewise, the total trains number going through a station cannot exceed the station capacity of receiving and sending off trains.

And the total numbers of the trains distributed on the sections connected to the station must be equal to the number of arriving trains at the station:

Take it for granted that there are

The smaller the value of the

Likewise, take it for granted that there are

The smaller the value of the

On the basis of considering of the running adjustability dispersion and the dwelling adjustability dispersion, the timetable stability on the dispatching section level is defined as

Then we take the timetable stability on the network level as the optimizing goal of the upper programming:

When rescheduling the trains on the sections, the minimum running time and the minimum dwelling time must be considered. The rescheduled running time and dwelling time must be longer than the minimum time, which is described in (

Depending on the analysis in Sections

We can see that the networked timetable stability is directly related to timetable stability on the network level and the dispatching section level. The programming in Sections

Considerable attention has been paid to fuzzy particle swarm optimization (FPSO) recently. Abdelbar et al. proposed the FPSO [

Computation in the PSO paradigm is based on a collection (called a swarm) of fairly primitive processing elements (called particles). The neighborhood of each particle is the set of particles with which it is adjacent. The two most common neighborhood structures are

PSO can be used to solve a discrete combinatorial optimization problem whose candidate solutions can be represented as vectors of bits;

Let

Fuzzy PSO differs from standard PSO in only one respect: in each neighborhood, instead of only the best particle in the neighborhood being allowed to influence its neighbors, several particles in each neighborhood can be allowed to influence others to a degree that depends on their degree of charisma, where charisma is a fuzzy variable. Before building a model, there are two essential questions that should be answered. The first question is how many particles in each neighborhood have nonzero charisma. The second is what membership function (MF) will be utilized to determine level of charisma for each of the

The answer to the first question is that the

The answer to the other question is that there are numerous possible functions for charisma MF. Popular MF choices include triangle, trapezoidal, Gaussian, Bell, and Sigmoid MFs; see [

Let

The charisma

The charisma

The charisma

The charisma

Because

In Fuzzy PSO, velocity equation is

Hybrid rule requires selecting two particles from the alternative particles at a certain rate. Then the intersecting operation work needs to be done to generate the descendant particles. The positions and velocities of the descendant particles are as follows: according to the intersecting rule, inheriting from the FPSO, see [

Thus, the HFPSO is built. In the model,

It can be seen that the network timetable stability optimizing model is a nonlinear one and it is an NP-hard problem. Generally, it is very difficult to solve the problem with mathematical approaches. Evolutionary algorithms are often hired to solve the problem for their characteristics. First, its rule of the algorithm is easy to apply. Second, the particles have the memory ability which results in convergent speed and there are various methods to avoid the local optimum. Thirdly, the parameters which need to select are fewer, and there is considerable research work on the parameters selecting.

In addition, the HFPSO hires the fuzzy theory and the hybrid handling method when designing the algorithm. Thus it has the ability to improve the computing precision when solving the optimization problem. And it utilizes intersecting tactics to generate the new generation of particles to avoid the precipitate of the solution. It is adaptive to solve the timetable stability optimization. And its easy computing rule determines the applicability in the solving of timetable stability optimization problem.

The size of the particle swarm is set to be 30 to give consideration to both the calculating degree of accuracy and computational efficiency. For

The size of the particle swarm is also set to be 30.

There are 22 stations and 10 sections in the network, as indicated in Figure

The railway network in the computing case and the original distribution plan of the trains.

And the planned operation diagram on path 1-2–5–7 is shown in Figure

Planned operation diagram on path 1-2–5–7.

Planned operation diagram on path 1–3–6-7.

According to the networked timetable stability definition in Section

Computing results of the planned timetable stability.

Related computing results of

Key nodes | Trains number through station | Nodes capacity | Load | Capacity index | Nodes degree | Degree index | Stations weight |
---|---|---|---|---|---|---|---|

1 | 44 | 66.0 | 0.6667 | 0.3099 | 2.0000 | 0.1111 | 0.2245 |

2 | 23 | 34.5 | 0.6667 | 0.1620 | 3.0000 | 0.1667 | 0.1760 |

3 | 21 | 31.5 | 0.6667 | 0.1479 | 3.0000 | 0.1667 | 0.1607 |

4 | 0 | 15.0 | 0.0000 | 0.0704 | 4.0000 | 0.2222 | 0.1020 |

5 | 23 | 36.0 | 0.6389 | 0.1690 | 3.0000 | 0.1667 | 0.1837 |

6 | 21 | 30.0 | 0.7000 | 0.1408 | 3.0000 | 0.1667 | 0.1531 |

Related computing results of

Section | Trains number through section | Sections capacity | Load | Capacity index | Sections weight |
---|---|---|---|---|---|

1-2 | 23 | 30.0 | 0.7667 | 0.1622 | 0.1622 |

1–3 | 21 | 27.5 | 0.7636 | 0.1486 | 0.1486 |

2–4 | 0 | 6.5 | 0.0000 | 0.0351 | 0.0351 |

2–5 | 23 | 23.5 | 0.9787 | 0.1270 | 0.1270 |

3-4 | 0 | 6.5 | 0.0000 | 0.0351 | 0.0351 |

3–6 | 21 | 21.0 | 1.0000 | 0.1135 | 0.1135 |

4-5 | 0 | 8.0 | 0.0000 | 0.0432 | 0.0432 |

4–6 | 0 | 5.0 | 0.0000 | 0.0270 | 0.0270 |

5–7 | 23 | 31.0 | 0.7419 | 0.1676 | 0.1676 |

6-7 | 21 | 26.0 | 0.8077 | 0.1405 | 0.1405 |

According to Table

We reallocate all the 44 trains on the modest railway network, as shown in Figure

Computing results of rescheduled timetable stability.

Related computing results of

Key nodes | Trains number through station | Nodes capacity | Load | Capacity index | Nodes degree | Degree index | Stations weight |
---|---|---|---|---|---|---|---|

1 | 44 | 66.0 | 0.6667 | 0.3099 | 2 | 0.1111 | 0.2245 |

2 | 23 | 34.5 | 0.6667 | 0.1620 | 3 | 0.1667 | 0.1760 |

3 | 21 | 31.5 | 0.6667 | 0.1479 | 3 | 0.1667 | 0.1607 |

4 | 10 | 15.0 | 0.6667 | 0.0704 | 4 | 0.2222 | 0.1020 |

5 | 24 | 36.0 | 0.6667 | 0.1690 | 3 | 0.1667 | 0.1837 |

6 | 20 | 30.0 | 0.6667 | 0.1408 | 3 | 0.1667 | 0.1531 |

Related computing results of

Section | Trains number through section | Sections capacity | Load | Capacity index | Sections weight |
---|---|---|---|---|---|

1-2 | 23 | 30.0 | 0.7667 | 0.1622 | 0.1622 |

1–3 | 21 | 27.5 | 0.7636 | 0.1486 | 0.1486 |

2–4 | 5 | 6.5 | 0.7692 | 0.0351 | 0.0351 |

2–5 | 18 | 23.5 | 0.7660 | 0.1270 | 0.1270 |

3-4 | 5 | 6.5 | 0.7692 | 0.0351 | 0.0351 |

3–6 | 16 | 21.0 | 0.7619 | 0.1135 | 0.1135 |

4-5 | 6 | 8.0 | 0.7500 | 0.0432 | 0.0432 |

4–6 | 4 | 5.0 | 0.8000 | 0.0270 | 0.0270 |

5–7 | 24 | 31.0 | 0.7742 | 0.1676 | 0.1676 |

6-7 | 20 | 26.0 | 0.7692 | 0.1405 | 0.1405 |

Distribution plan of the trains according to the computing results.

According to Table

According to the computing results of the lower level programming, we adjust the timetable, moving some of the running lines of the trains. The two-dot chain line is the newly planned running trajectory and the dotted line is the previously planned trajectory. The timetable on path 1-2-5–7 is shown in Figure

Rescheduled timetable of path 1-2–5–7 according to the computing results of the lower level programming.

Rescheduled timetable of path 2–4-5 according to the computing results of the lower level programming.

Rescheduled timetable of path 1–3–6-7 according to the computing results of the lower level programming.

Rescheduled timetable of path 3-4–6 according to the computing results of the lower level programming.

From Figures

From Figures

The timetable stability on the dispatching section level is

The bilevel programming model is appropriate for the networked timetable stability optimizing. It comprises the timetable stability of the network level and the dispatching section level. Better solution can be attained via hybrid fuzzy particle swarm algorithm in networked timetable optimizing. The timetable is more stable, which means that it is more feasible for rescheduling in the case of disruption, when it is optimized by hybrid fuzzy particle swarm algorithm. The timetable rearranged based on the timetable stability with bilevel networked programming model can make the real train movements very close to, if not the same with, the planned schedule, which is very practical in the daily dispatching work.

The results also show that hybrid fuzzy particle algorithm has significant global searching ability and high speed and it is very effective to solve the problems of timetable stability optimizing. The novel method described in this paper can be embedded in the decision support tool for timetable designers and train dispatchers.

We can do some microcosmic research work on the timetable optimizing based on the railway network in the future based on the method set out in the present paper, enlarging the research field, adding the inbound time, and outbound time of the trains at stations.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is financially supported by National Natural Science Foundation of China (Grant 61263027), Fundamental Research Funds of Gansu Province (Grant 620030), and New Teacher Project of Research Fund for the Doctoral Program of Higher Education of China (20126204120002). The authors wish to thank anonymous referees and the editor for their comments and suggestions.