Timetable optimization for a moving block system

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Highlights

  • A general train timetabling problem formulation for fixed and moving block systems with flexible routing.

  • A multi-layer graph approach using velocity expansion to model the problem.

  • A branch-and-cut algorithm to solve the model formulation.

  • A computational study and comparison for realistic data for a corridor in Germany.

Abstract

We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model can solve reasonably sized instances with up to a hundred trains to optimality.

Introduction

Railway is the most environmentally friendly means of transport available. An increased use of railway for both passenger and freight transport is thus highly desirable, but only possible with an increased network and service quality. Digitalization and automation are seen as major technologies to create these conditions without the time and costs required for the building of new tracks. The Digital Rail for Germany sector program is bringing together existing and emerging technologies to develop a digital railway system to its full potential. One important measure is enabling trains to run in absolute braking distance (moving block), rather than obeying fixed-block safety regulations. A second one is the development of a Capacity & Traffic Management System (CTMS) that optimizes the planning and control of rail traffic to fully exploit the features and benefits of digitalized and automated train operation.

Mathematical models for railway timetabling are basically divided into two categories periodic and non-periodic scheduling. The periodic or cyclic railway timetabling problem has been intensively studied, e.g., in Peeters and Kroon (2001) and Liebchen (2008). However, in this paper we focus on non-periodic scheduling which goes back to seminal works by Brännlund et al. (1998) and Caprara et al. (2002). In addition, Lusby et al. (2011), Harrod (2012), and Caimi et al. (2017) provide comprehensive surveys on classical timetabling and train dispatching literature. A huge variety of further aspects like robustness or special applications as freight train routing can be found in Chapters 2, 4, 5, 6, 11, and 12 of Borndörfer et al. (2018).

We will present a model formulation based on a layered graph approach. The model is inspired by the classical work on alternative graph formulations for job-shop scheduling, where orders are modeled by a disjunctive graph, see Mascis and Pacciarelli (2002) and D’Ariano et al. (2007). This methodology has also been successfully applied to train dispatching, see Lamorgese and Mannino (2015). Furthermore, a remarkable non-compact formulation by strengthening and lifting the constraints of a classical Benders’ reformulation, and thus avoiding big-M constraints, was developed by Lamorgese and Mannino (2019).

A similar graph concept and model was investigated by Xu et al. (2017) for a quasi-moving block study on train rescheduling during disruptions. The setting of the problem definition presented in their work differs significantly from the one that we will discuss. In Xu et al. (2017) the application is to reschedule the given high speed trains in a disruption scenario such that the delay for the trains is minimized. Thus, for all trains there is already a timetable trajectory (route in space and time) provided which might be affected by the considered disruption. Therefore some trajectories need to only be adapted in time and not in space in order to minimize the resulting sum of delays. Moreover, the considered scenarios have the topology of a single-line corridor for one high speed train type only.

In contrast, our setting applies to highly heterogeneous mixed traffic, where the capacity depends on several factors, see Harrod (2009).

The contribution of this paper is to enable integrated train timetabling and routing models to handle moving block restrictions. Table 1 briefly classifies our work and the most related literature on timetabling and train dispatching. In more detail, our model has the following features: a flexible routing on a microscopic scale (i.e., on a track-and-switch level), running time lower bounds on arcs, discrete and fixed speed levels at nodes, continuous and flexible velocity functions on arcs, and dynamic headway constraints depending on the braking potential at support points.

Section 2 compares the classical fixed-block signaling system with the moving block safety system for railways. A general problem formulation for timetabling (GTTP) is provided in Section 3, along with running and headway time oracles, and our layered graph constructions. In Section 4, we develop a MIP formulation for the timetable optimization problem for a microscopic railway network and a moving block safety system. Section 5 discusses different algorithmic ingredients towards a branch-and-cut approach. First computational results for a corridor in Germany are presented in Section 6. Finally, we point out some conclusions and an outlook for further developments in Section 7.

Section snippets

Modeling the safety system in railways

The main task of railway signaling systems is to ensure the necessary separation between trains: A moving train must be able to come to a halt at any time without colliding with another train. As braking distances are often longer than sight distances, technical systems are required to guarantee safe railway operations.

The GTTP and a layered graph formulation

In the following section we will define the problem of timetable optimization for a moving block safety system with absolute braking distance. We will consider green field scenarios with the main question: What is the maximal number of trains that can be scheduled, given a set of requested trains (including some stop requirements) within a network? The degree of freedom in the model covers the following decisions:

  • Is a train routed or declined?

  • On which route is a train request routed (sequence

A (disjunctive) MIP model for the GTTP

Having established the layered graphs, it is now possible to present a MIP formulation for GTTP. We introduce a binary decision variable yar for each rR which is one if and only if arc aA is used by request rR, and zero otherwise, see (15). The binary slack variable ur in (17) indicates whether a request is routed or not. Arrival and departure times for each request rR and node vV are formulated by continuous variables t̲vr and  t¯vr in (16).

The most challenging part is the formulation of

A MIP and cut approach

In principle, we are now ready to set up and solve the MIP formulation as described in Section 4. The drawback will be that the number of headway constraints (13) and the number of train ordering variables xer1r2 are typically very large even if most of them will be redundant in an optimal solution. Intuitively, if two trains have a sufficient headway on one (processed layer) edge e, then it is very likely that those trains have a sufficient headway on near-by edges. We therefore use a dynamic

Computational results

This section will discuss the results of applying our model and solution approaches to real-world data.

The focus of the investigation is the automated generation and optimization of conflict-free timetables subject to a moving block regime for the Seelze–Wunstorf–Minden corridor in Germany. Infrastructure and train data (e.g., acceleration etc.) are taken from exports from RailSys with |Iˆ|=9112, |Eˆ|=9391 for an area with about 500 track kilometers in total. The minimum headway time between

Conclusion and outlook

We developed a model formulation which is capable of routing and ordering trains under a moving block regime. We provide a general timetabling definition GTTP which allows to plug-in further details on the calculation of the running or headway time considered in the model. This is done in an arranged way to construct a specially defined layered graph and headway sets which are the basis of the presented MIP formulation.

An intended weakness of the model is clearly that the braking distance is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank our colleagues from the simulation team at the Digital Rail for providing DbbSim and the AI team for being an expert beta user of the developed solver.

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