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THE COST–TIME CURVE FOR AN OPTIMAL TRAIN JOURNEY ON LEVEL TRACK

Part of: Optimality conditions Real functions

Published online by Cambridge University Press:  01 July 2016

AMIE ALBRECHT Open the ORCID record for AMIE ALBRECHT [Opens in a new window] ,
PHIL HOWLETT Open the ORCID record for PHIL HOWLETT [Opens in a new window]  and
PETER PUDNEY Open the ORCID record for PETER PUDNEY [Opens in a new window]
AMIE ALBRECHT
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 amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au, peter.pudney@unisa.edu.au
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 amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au, peter.pudney@unisa.edu.au
PETER PUDNEY
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 amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au, peter.pudney@unisa.edu.au
*
✉phil.howlett@unisa.edu.au
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Abstract

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In this paper, we show that the cost of an optimal train journey on level track over a fixed distance is a strictly decreasing and strictly convex function of journey time. The precise structure of the cost–time curves for individual trains is an important consideration in the design of energy-efficient timetables on complex rail networks. The development of optimal timetables for busy metropolitan lines can be considered as a two-stage process. The first stage seeks to find optimal transit times for each individual journey segment subject to the usual trip-time, dwell-time, headway and connection constraints in such a way that the total energy consumption over all proposed journeys is minimized. The second stage adjusts the arrival and departure times for each journey while preserving the individual segment times and the overall journey times, in order to best synchronize the collective movement of trains through the network and thereby maximize recovery of energy from regenerative braking. The precise nature of the cost–time curve is a critical component in the first stage of the optimization.

Keywords

optimal train controlcost–time curvesenergy efficiency

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 58 , Issue 1 , July 2016 , pp. 10 - 32
Copyright
© 2016 Australian Mathematical Society 

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