Control of Delay Propagation in Railway Networks Using Max-Plus Algebra

Abstract

In this thesis max-plus algebra is introduced and applied to the problem of controlling train delays. Two control strategies for the propagation of delays are discussed. The first is by letting certain trains run faster when a delay is detected, the second is by breaking connections between trains that have to wait for each other in order to enable passengers to changeover from one train to the other. The goal is to understand how to model the propagation of delays when different control strategies are applied, in order to provide train operators tools for making quick decisions on how to intervene when a delay is detected.
The models provided in the report are in the form of max-plus-linear systems and switching max-plus linear systems. These can be programmed in Python to automate the decision making. The report starts with providing a basic understanding of max-plus algebra, where also max-plus linear systems and switching max-plus linear systems are explained. Subsequently, a railway network is designed that serves as an example during this thesis. This railway network is modelled into a max-plus linear system and, additionally, a desirable train timetable for passengers is designed for this network by means of the power algorithm. The main results are two switching max-plus linear systems that model the propagation of delays when the two control strategies are applied. The report ends with a larger railway network at which all acquired knowledge is applied. It can be concluded that the models in this thesis provide methods to calculate exactly how the delay propagates through the network when certain control strategies are applied and, based on that, decisions can be made quicker. Moreover, it is possible to calculate the consecutive departure times. As a result the passengers can be informed quickly about the new departure times as a consequence of the delay and how long it will take for the trains to run according to timetable again. This thesis adds the modelling of faster running trains to existing literature. We have seen that speeding up trains is also a control strategy to solve delays and can be modelled systematically.