Advances in Industrial Control

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Neuro-Control and its Applications Sigeru Omatu, Marzuki Khalid and Rubiyah Yusof

P.G. Howlett and P.J. Pudney

Energy-Efficient Train Control

Wilh 76 Figures

, Springer

Philip G. Howlett Peter J. Pudney

Scheduling and Control Group School of Mathematics University of South Australia The Levels Campus Pooraka South Australia 5095 Australia

British Library Cataloguing in Publication Data Howlett, Philip G.

Energy Efficient Train Control. - (Advances in Industrial Control) I. Title II. Pudney, Peter J. III. Series 385.1 ISBN-13:978-1-4471-3086-4

Library of Congress Cataloging-in-Publication Data Howlett, Philip, 1944-

Energy efficient train control I Philip Howlett and Peter Pudney. p. cm. - - (Advances in industrial control)

Includes bibliographical references (p. ) and index. ISBN-13:978-1-4471-3086-4 e-ISBN-13:978-1-4471-3084-0 DOl: 10.1007/978-1-4471-3084-0

1. Railroads - - Automatic train control. TF638.H69 1995 625.1 - - dc20

I. Pudney, Peter, 1961-. II. Title. III. Series. 95-31200

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Typesetting: Camera ready by authors

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SERIES EDITORS' FOREWORD

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology impacts all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, ..... , new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination.

The number of texts reporting an in-depth study of the control of transportation systems is not so many. In this contribution to the Advances in Industrial Control Series, Phil Howlett and Peter Pudney report on well over a decade of research into the energy efficient aspects of rail transportation. The work described is characterised by the development of a sound theoretical framework for the optimal control problems to be solved.

The presentation of modelling, and optimisation problems forms the main thrust of the text after the introductory chapters. These first chapters describe the background to the train efficiency research work and introduce the reader to the theoretical concepts required later in the text. Also of considerable interest to practitioners is the description of the in-cab technology, the METROMISER, which resulted from some of the research work.

Not surprisingly, the pioneering theoretical and technological research reported in this volume has now won several awards and we are delighted that a full presentation of this long programme of research is available for wider dissemination as an Ale volume.

MJ. Grimble and M.A. Johnson Industrial Control Centre Glasga»:. Scotland, U.K.

PREFACE

Rail is potentially a very efficient form of transport, but must deliver attractive alternatives to road and air transport if it is to be competitive. The challenge for rail is to meet the needs of potential users. For heavy-haul freight, railways will need to be reliable and cost-effective. For intercity passengers, rail must provide a fast and comfortable alternative to air travel, or a relaxed and luxurious holiday service.

Suburban and mass transit systems will need to overcome the problems of peak congestion and match the perceived convenience of the car. In theory, rail can provide energy-efficient transport on a dedicated corridor free of external interference. Can this theory be turned into effective practice?

For railway operators there are many factors to consider. The customers want

prompt and reliable service that must also be cost-effective. The railways must make a profit. On congested lines, scheduling problems are seen as dominant. For the long haul across continents, arrival on time and reduction of fuel consumption are more

important. There are of course other concerns. Safe operation is a critical factor and the adoption of new signalling technology is high on the agenda. In this book, however, we restrict our attention to the problem of energy-efficient control.

Beginnings

In 1980 Ian Milroy returned from Loughborough University in the UK to the School

of Electrical Engineering at the South Australian Institute of Technology, now the

University of South Australia. He had just completed his PhD thesis, Aspects of Automatic Train Control, which included some novel work on driving strategies [1, 2]. In particular he showed that for short journeys an energy-efficient driving strategy has three control phases: maximum acceleration, coast, and maximum brake.

The work on driving strategies was continued by a post-graduate Electrical

Engineering student, Kim Tyler, supervised by Ian Milroy and by David Lee from the

School of Mathematics and Computer Studies. They discovered a fourth optimal control mode, speed-hold, applicable on longer journeys [3] and used an analogue

computer to simulate efficient speed profiles for journeys of varying distance and duration.

The Transport Control Group

In 1982 Ian Milroy formed the Transport Control Group at the South Australian Institute of Technology to work on a project funded by the South Australian Department of Transport. The aim of the project was to determine whether the suggested driving strategies were effective in practice and, if so, to develop a system for achieving fuel savings on suburban trains in Adelaide.

The first part of the project involved calculating efficient speed profiles for various sections of track on the Adelaide rail network. This was done by Basil Benjamin from the School of Mathematics and Computer Studies, Jerry Kautsky from the Flinders University of South Australia and Nancy Nichols from the University of Reading. The project manager, Andy Long, then boarded a train with a stopwatch and a pre-computed speed profile and advised the driver when to change control. On each trip the train completed the section within a few seconds of the desired time, and the time spent accelerating was much less than normal.

Metromiser

The next stage of the project was to design and build a system that could compute an efficient driving strategy in real time and display appropriate driving advice to the driver. Basil Benjamin and a post-graduate Computer Studies student, Peter Pudney, developed algorithms and computer software for the system. Computer hardware was designed and built by Tony Gelonese from the School of Electrical Engineering. The resulting system, now known as Metromiser, monitored the state of a journey and, using stored timetable and route information, advised the driver when to coast and brake so that the train arrived at each stop on time and consumed as little energy as possible.

Metromiser was first evaluated in Adelaide in February 1985. The unit was fitted to a diesel-hydraulic railcar running in normal service on an 80-minute round trip. Twenty trips were evaluated-ten trips with driving advice and ten trips without advice. Metromiser achieved a fuel saving of 15% and significant improvements in timekeeping [4].

Since then Metromiser has been evaluated on suburban trains in Adelaide, Melbourne, Brisbane and Toronto. For each trial a Metromiser unit was installed on a test train and operated in normal service. Fuel consumption and timekeeping data

were collected over a three week period. All trials showed significantly improved timekeeping and substantial fuel savings.

Theoretical Developments

During the period 1982-85 the theoretical basis for the work was also extended. After early discussions with Ian Milroy it was decided by the Group that Phil Howlett, from the School of Mathematics and Computer Studies, should seek a more rigorous mathematical justification for the strategies suggested by Milroy. Following discussions with Jim Michael at Adelaide University and Bruce Craven at Melbourne University in 1984, Howlett did indeed give a rigorous justification of the work and at the same time produced the first theoretical confmnation that speed-holding should be used if the journey time is relatively large, and that an optimal driving strategy used an accelerate-hold-coast-brake control sequence. He also found a special relationship between the holding speed and the speed at which braking should begin [5,6]. This relationship was found independently and at much the same time by Asnis et al in the Soviet Union [7].

Long-Haul Trains

Although much had been achieved with the Metromiser project, the strategy and algorithms used were only effective when stops were less than about 10 km apart. The group began planning a second project to develop a fuel conservation system for long-haul freight trains. In 1986 the Transport Control Group was awarded a National Energy Research, Development and Demonstration Council (NERDDC) contract to develop such a system.

Long-haul freight trains presented several new problems:

• the performance of a train is initially unknown, and changes with the direction of the wind and when wagons are loaded or unloaded

• the trains have a low power to mass ratio, and so gradients have a greater effect

• there can be many different speed limits between stops

A different train model was now required. The original model had assumed continuous control of acceleration. This is not the case for the diesel-electric locomotives used by Australian National, where each locomotive has eight discrete control notches, and each notch determines a constant rate of fuel supply to the diesel generator. An appropriate model was developed by Benjamin et al [8]. Benjamin also directed work to develop parameter estimation techniques for the new model.

The Australian long-haul rail network has long sections of single-line track and only occasional crossing loops. This means that design of appropriate crossing schedules is critical in improving the performance of a long-haul system. Graham Mills and Sonya Perkins joined the group to work on the scheduling aspects of the project. This work is described in an article in the Asia-Pacific Journey o/Operational Research [9].

Further NERDDC grants were awarded to the Group in 1987-88 and in 1989-90. Development of the long-haul system was subsequently completed and commercial production begun. In 1989 the Transport Control Group received a National Energy Innovation Award for Outstanding Achievement in Research and Development into, and the Effective Management of, Transport Energy Use.

In 1989 Howlett and Benjamin began a systematic consideration of their new model for the train control problem. They assumed a fixed number of discrete control settings with each setting corresponding to a constant rate of fuel supply. Total fuel consumption was used as the cost functional. The first serious theoretical work with this model began with the arrival of Cheng Jiaxing, a visiting scholar from Anhui University in the Peoples Republic of China. By December 1989 preliminary results were presented by Cheng and Howlett at the Conference of Australian Institutes of Transport Research at the Flinders University of South Australia. Even with the new model, the research confirmed the fundamental optimality of the power-hold-coast­brake strategy for level track.

Scheduling and Control of Trains on a Network

By early 1990 it was clear that the heuristic algorithms used to compute efficient long-haul speed profiles could be improved by incorporating the most recent theoretical work. Further investigations by Howlett and Cheng had suggested that the effects of gradient and speed limits could be incorporated into the new model. At the same time Graham Mills believed that much more could be done to improve long-haul scheduling.

In 1991 the South Australian Institute of Technology was incorporated into the new University of South Australia. In the same year the Group, now renamed the Scheduling and Control Group, and again lead by the redoubtabl~ Ian Milroy, received a Generic and Industrial Research and Development (GIRD) grant. The project proposed two major areas of investigation. The first, under the direction of Graham Mills, would develop algorithms for scheduling trains on a long-haul network. The second, directed by Phil Howlett, would develop energy-efficient control strategies and devise numerical algorithms for calculating these strategies.

The original commercial collaborator ceased operation in 1991, soon after the project began, and it was necessary to find another partner to provide technical support. Fortunately the Group had done some work on automatic braking systems for another railway systems supply company, who were keen to develop stronger ties with the University, and they became the new partner for the GIRD project early in 1992. The research program and specifications for the long-haul system were completed in 1993, but commercial development was hampered by licence negotiations with the original partner. Nevertheless, the Group received another National Energy Innovation Award and a survey paper by Howlett, Milroy and Pudney [10] was selected as a finalist for the applications prize at the Twelfth World Congress of the International Federation of Automatic Control (IFAC) in Sydney, Australia, 1993.

Solar Cars

More recently the Scheduling and Control Group, now led by Phil Howlett, has worked with the Aurora Vehicles Association to determine energy-efficient driving strategies for a solar car. Aurora Vehicles are the leading Australian developers of solar car technology. After some preliminary work in cooperation with the Commonwealth Scientific and Industrial Research Organisation (CSIRO) in 1993, the Group obtained a three-year Australian Research Council Collaborative grant with Aurora to develop energy management strategies for solar and electric cars. The Aurora Q 1 solar car finished the 1993 World Solar Challenge in fifth position overall, and was the leading Australian entry. The first five of the 52 entries averaged more than 70 km/h for the 3000 km journey, breaking the record 67 km/h set by the GM Sunraycer in 1987. In November 1994 the QI was driven 4000 km across Australia from Perth to Sydney in eight days, beating the previous best time of 20 days set by Hans Tholstrup and Larry Perkins in 1982.

ACKNOWLEDGEMENTS

We would like to thank present and past members and associates of the Scheduling and Control Group for their contributions to the solution of the train control problem. They are: Ian Milroy, Kim Tyler, David Lee, Basil Benjamin, Bob Northcote, Doug Seeley, Andy Long, Tony Gelonese, Jerry Kautsky, Nancy Nichols, Bruce Craven, Jim Michael, Andrew Skinner, Peter McWhirr, Bob Payne, Sally Rice, Graham Mills, Sonya Perkins, Bernie Chandler, Cheng Jiaxing and Tania Tamopolskaya. In particular Ian Milroy, Cheng Jiaxing and Basil Benjamin have co-authored some of the original papers and assisted with proof-reading. We would also like to thank our colleagues in Trans Adelaide, Australian National, The Canadian Institute of Guided Ground Transport, Teknis Systems, Queensland Rail, National Rail Corporation, and Westinghouse Brake and Signal Australia.

Parts of this book are based on previously published material. Chapters 7, 11 and 13 are rewritten from papers [6, 11, 12] published in the Journal of the Australian Mathematical Society, Series B, and some of the material in Chapter 12 appeared in a paper [13] published in Computational Techniques and Applications, CTAC-9J; this material appears with the kind permission of the Australian Mathematical Society, Australian National University, Canberra ACT 0200. Chapter 8 is rewritten from a paper [14] published in Automatica, and Chapters 1,2,3 and 12 contain material from a paper [10] publis~ed in Control Engineering Practice; this material appears with the kind permission of Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB UK. Chapter 10 contains material from a paper [15] published in Transactions on Automatic Control, and appears with permission of IEEE Publishing Services, 445 Hoes Lane, Piscataway NJ 08855-1331.

CONTENTS

Section A: Introduction

1 THE TRAIN CONTROL PROBLEM ........................................................... 1

1.1 The Original Formulation ................................................................................. 1 1.2 Solution of the Original Problem ...................................................................... 3 1.3 Initial Results .................................................................................................... 6 1.4 Further Early Results ........................................................................................ 7 1.5 Metromiser ........................................................................................................ 7 1.6 Long-Haul Operations .................................................................................... 14

2 MODELLING THE TRAIN CONTROL PROBLEM ............................... 17

2.1 The Mechanical Energy Model ...................................................................... 17 2.2 The Fuel Consumption Model ........................................................................ 20

2.2.1 Traction Characteristics for a Diesel-Electric Locomotive ............... 20 2.2.2 Braking Characteristics for a Diesel-Electric Locomotive ................ 20 2.2.3 Modelling the Control Mechanism .................................................... 21 2.2.4 The Equations of Motion ................................................................... 22

2.3 Trains with Distributed Mass ......................................................................... 23

3 PRACTICAL DRIVING STRATEGIES ..................................................... 25

3.1 Approximation of Measurable Control.. ......................................................... 25 3.2 Speed-Holding ................................................................................................ 30

4 CONSTRAINED OPTIMISATION-AN INTUITIVE VIEW ................. 33

4.1 Constrained Optimisation ............................................................................... 33 4.2 Linearlsation Techniques ................................................................................ 36 4.3 Hyperplanes and Half-Spaces ......................................................................... 38

4.3.1 Hyperplanes in Hilbert Space ............................................................ 38 4.3.2 Hyperplanes in the Dual Space ......................................................... .40

4.3.3 Half-Spaces ....................................................................................... 40 4.4 Linear Mappings ............................................................................................ 41 4.5 The Adjoint Mapping ..................................................................................... 45 4.6 Convex Cones ................................................................................................ 47

4.6.1 Polar Cones ....................................................................................... 48 4.6.2 Positivity ........................................................................................... 51 4.6.3 Cones in the Dual Space ................................................................... 51 4.6.4 The Cone Separation Theorem .......................................................... 51 4.6.5 The Cone Inclusion Theorem ............................................................ 52 4.6.6 Farkas Theorem for Convex Cones ................................................... 53

4.7 The Optimisation Theorem ............................................................................ 55 4.8 Linear Constrained Minimisation .................................................................. 56 4.9 The Kuhn-Tucker Conditions ........................................................................ 58 4.10 The Pontryagin Principle ................................................................................ 60

Section B: Analysis of the Mechanical Energy Model

5 EXISTENCE OF AN OPTIMAL STRATEGy .......................................... 65

5.1 Introduction .................................................................................................... 65 5.2 Precise Formulation of the Train Control Problem ........................................ 65 5.3 Sufficient Conditions for a Feasible Strategy ................................................ 69 5.4 Existence of an Optimal Strategy ................................................................... 71 5.5 Conclusions .................................................................................................... 76 5.6 Appendix-Convergence in Banach Spaces .................................................. 76

6 NECESSARY CONDITIONS FOR AN OPTIMAL STRATEGy ........... 81

6.1 An Equivalent Formulation of the Train Control Problem ............................ 81 6.2 Necessary Conditions for an Optimal Strategy .............................................. 84 6.3 The Adjoint Differential Equation and the Pontryagin Principle ................... 98 6.4 Conclusions .................................................................................................. 105 6.5 Appendix-The Radon Integral ................................................................... 105

7 DETERMINATION OF OPTIMAL DRIVING STRATEGIES ............ 111

7.1 A Special Case of the Train Control Problem ............................... : .............. 111 7.2 The Nature of the Optimal Strategy ............................................................. 112 7.3 The Complete Solution ................................................................................ 119 7.4 Examples ........................................................................... : .......................... 130 7.5 Summary ...................................................................................................... 132 7.6 Limitations of the Model .............................................................................. 133

Section C: Analysis of the Fuel Consumption Model

8 CRITICAL SPEEDS AND STRATEGIES OF OPTIMAL TYPE .....•... 137

8.1 Introduction .................................................................................................. 137 8.1.1 A Driver Perspective on the Problem .............................................. 137 8.1.2 A Well-Posed Problem .................................................................... 138

8.2 Formulation of the Train Control Problem ................................................... 138 8.2.1 The Control Strategies ..................................................................... 138 8.2.2 The Equations of Motion ................................................................. 139 8.2.3 A Precise Statement of the Problem ................................................ 140

8.3 The Nature of the Resistive Acceleration ..................................................... 140 8.4 The Fundamental Speed Profiles .................................................................. 141 8.5 Necessary Conditions for a Strategy of Optimal Type ................................. 143 8.6 Calculating a Feasible Strategy of Optimal Type ......................................... 147 8.7 Numerical Examples ..................................................................................... 151 8.8 Conclusions and Future Developments ........................................................ 156

9 MINIMISATION OF FUEL CONSUMPTION •••••..........••.•..••.••..•........... 157

9.1 Additional Notation ...................................................................................... 157 9.2 Approximating the Minimum-Cost Strategy ................................................ 158 9.3 Approximate Speed-Holding Strategies ....................................................... 161 9.4 The Structure of an Optimal Strategy ........................................................... 170

9.4.1 The Power Phase ............................................................................. 170 9.4.2 The Transition from Power to Speed-Hold ..................................... 172 9.4.3 The Transition from Speed-Hold to Coast ...................................... 175 9.4.4 The Coast Phase .............................................................................. 176

9.5 A Speed-Holding Strategy of Optimal Type ................................................ 178

10 A MORE GENERAL MODEL ................................................................... 183

10.1 The Equations of Motion .............................................................................. 183 10.2 Statement of the Train Control Problem ....................................................... 184 10.3 The Nature of the Model .............................................................................. 185 10.4 The Main Results .......................................................................................... 186 10.5 The Fundamental Speed Proflles .................................................................. 187 10.6 Necessary Conditions for a Strategy of Optimal Type ................................. 188 10.7 The Critical Speeds ................................................................... : ................... 190 10.8 Examples ...................................................................................................... 192 10.9 Approximating the Minimum-Cost Strategy ................................................ 196 10.10 Approximate Speed-Holding Strategies ....................................................... 198 10.11 The Structure of an Optimal Strategy ........................................................... 204

10.11.1 The Power Phase ............................................................................. 204 10.11.2 The Transition from Power to Speed-Hold ..................................... 206 10.11.3 The Transition from Speed-Hold to Coast ...................................... 209 10.11.4 The Coast Phase .............................................................................. 209

10.12 A Speed-Holding Strategy of Optimal Type ................................................ 211

11 SPEED LIMITS ........................................................................................... 215

11.1 Vehicle Model .............................................................................................. 215 11.2 Journey Model. ............................................................................................. 216 11.3 Fundamental Speed Profiles ......................................................................... 216 11.4 Strategies of Optimal Type .......................................................................... 217 11.5 Critical Speeds ............................................................................................. 221 11.6 Examples ...................................................................................................... 225

12 NON-CONSTANT GRADIENT ................................................................. 229

12.1 Notation ........................................................................................................ 229 12.2 Solution of the Equations of Motion ............................................................ 230 12.3 Track Gradient Analysis and Terminology .................................................. 233 12.4 The Speed Profiles ....................................................................................... 233 12.5 The Constraints ............................................................................................ 235 12.6 Necessary Conditions for a Strategy of Optimal Type ................................ 237

12.6.1 The Lagrangean Function ............................................................... 237 12.6.2 The Key Equations .......................................................................... 237 12.6.3 The Kuhn-Tucker Conditions ......................................................... 239

12.7 Derivation of the Key Equations .................................................................. 241 12.7.1 Calculation of some Useful Derivatives ......................................... 241 12.7.2 The Kuhn-Tucker Equations ........................................................... 244

12.8 An Alternative Form for the Key Equations ................................................ 247 12.9 The Strategies of Optimal Type ................................................................... 248

12.9.1 The Properties of the Effective Energy Density Function .............. 248 12.9.2 The Structure of a Strategy of Optimal Type .................................. 250

12.10 An Algorithm for Solving the Key Equations .............................................. 251 12.11 Examples for Non-Steep Track .................................................................... 256

12.11.1 Level Track Strategies Applied to Track with Small Gradients ..... 257 12.11.2 Strategies of Optimal Type on Track with Small Gradients ........... 259

12.12 Examples for Steep Track ............................................................................ 263

13 CONTINUOUSLY VARYING GRADIENT •.••••••.••••••.•.••.•••••••••••••.••••••.•.• 267

13.1 Some Additional Notation ............................................................................ 267 13.2 A General Form for the Key Equations ........................................................ 269 13.3 Necessary Conditions for a Strategy of Optimal Type ................................. 270

13.3.1 An Intuitive Derivation of the Key Equations ................................. 270 13.3.2 Lagrangean Function and Kuhn-Tucker Equations ......................... 272 13.3.3 Some Results from Perturbation Theory ......................................... 274 13.3.4 Calculation of some Useful Derivatives .......................................... 278

13.4 An Algorithm for Solving the Key Equations .............................................. 282

14 PRACTICAL STRATEGY OPTIMISATION .......................................... 285

14.1 A Simple Journey ......................................................................................... 285 14.2 Undulating Track .......................................................................................... 287 14.3 Speed-Holding on Steep Track ..................................................................... 287 14.4 Overlapping Control Intervals ...................................................................... 290

14.4.1 Overlapping Control Intervals for Steep Sections ........................... 291 14.4.2 Other Overlaps ................................................................................. 292

14.5 Initial and Final Speeds ................................................................................ 292 14.6 Speed Limits ................................................................................................. 293 14.7 A Practical Algorithm for Energy-Efficient Strategies ................................ 293

14.7.1 Overview ......................................................................................... 293 14.7.2 Speed-Holding ................................................................................. 295 14.7.3 Speed-Holding with Speed Limits ................................................... 295 14.7.4 Initial and Final Phases .................................................................... 296 14.7.5 Calculating Journey Time ................................................................ 297 14.7.6 Finding the Correct Hold Speed ...................................................... 297

REFEREN CES _ .......................... _ ........................................................... _ .......... 299

IND EX ................................................................................................................ _ 303