# Towards railway traffic management using switching Max-plus-linear systems

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## Abstract

In this paper we present a railway traffic model and a model predictive controller for online railway traffic management of railway networks with a periodic timetable. The main aim of the controller is to recover from delays in an optimal way by changing the departure of trains, by breaking connections, by splitting joined trains, and - in the case of multiple tracks between two stations - by redistributing the trains over the tracks. The railway system is described by a switching max-plus-linear model. We assume that measurements of current running and dwell times and estimates of future running times and dwell times are continuously available so that they can be taken into account in the optimization of the system’s control variables. The switching max-plus-linear model railway model is used to determine optimal dispatching actions, based on the prediction of the future arrival and departure times of the trains, by recasting the dispatching problem as a Mixed Integer Linear Programming (MILP) problem and solving it. Moreover, we use properties from max-plus algebra to rewrite and reduce the model such that the MILP problem can be solved in less time. We also apply the algorithm to a model of the Dutch railway network.

## Keywords

Max-plus algebra Railway networks Rescheduling Model reduction MILP## Notes

### Acknowledgements

This research is supported by the Dutch Technology Foundation STW, project 11025 “Model-Predictive Railway Traffic Management; A Framework for Closed-Loop Control of Large-Scale Railway Systems”. STW is part of the Netherlands Organisation for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs. This research is also funded by TÁMOP-4.2.1./B-11/2/KMR-2011-002 and TÁMOP-4.2.2./B-10/1-2010-0014.

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