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THE TWO-TRAIN SEPARATION PROBLEM ON LEVEL TRACK WITH DISCRETE CONTROL

Part of: Real functions Optimality conditions

Published online by Cambridge University Press:  29 October 2018

PHIL HOWLETT
PHIL HOWLETT*
Affiliation:
Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), School of Information Technology and Mathematical Sciences, University of South Australia, South Australia 5095, Australia email [email protected]
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Abstract

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When two trains travel along the same track in the same direction, it is a common safety requirement that the trains must be separated by at least two signals. This means that there will always be at least one clear section of track between the two trains. If the safe-separation condition is violated, then the driver of the following train must adopt a revised strategy that will enable the train to stop at the next signal if necessary. One simple way to ensure safe separation is to define a prescribed set of latest allowed section exit times for the leading train and a corresponding prescribed set of earliest allowed section entry times for the following train. We will find strategies that minimize the total tractive energy required for both trains to complete their respective journeys within the overall allowed journey times and subject to the additional prescribed section clearance times. We assume that the drivers use a discrete control mechanism and show that the optimal driving strategy for each train is defined by a sequence of approximate speedholding phases at a uniquely defined optimal driving speed on each section and that the sequence of optimal driving speeds is a decreasing sequence for the leading train and an increasing sequence for the following train. We illustrate our results by finding optimal strategies and associated speed profiles for both trains in some elementary but realistic examples.

Keywords

train controlsafe separationoptimal driving strategiesdiscrete controlconstrained optimization

MSC classification

Primary: 49K15: Problems involving ordinary differential equations
Secondary: 26A48: Monotonic functions, generalizations 26A51: Convexity, generalizations
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 2 , October 2018 , pp. 137 - 174
Copyright
© 2018 Australian Mathematical Society 

Footnotes

*

This is a contribution to the series of invited papers by past ANZIAM medallists (Editorial, Issue 52(1)). Phil Howlett was awarded the 2018 ANZIAM medal

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