Phil Howlett

Brief paper: Local energy minimization in optimal train control

The calculation of optimal driving strategies for on-board control of freight trains is a challenging task. In this paper we calculate the critical switching points for a globally optimal strategy on a track with steep gradients using a new local energy minimization principle. The method has been used successfully in Australia to calculate optimal switching points and hence provide in-cab advice to train drivers on long-haul freight trains.

Optimal strategies for the control of a train

This paper describes a method for the calculation of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with non-constant gradient. The journey must be completed within a given time, and it is desirable to minimise the fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomotive. For each fixed finite sequence of control settings, we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. Several realistic examples are given with the results of associated numerical calculations. The examples demonstrate the profound effect of even a small gradient on a strategy of optimal type. We show that a strategy of optimal type with alternate phases of coast and maximum power can be used to approximate the idealised minimum cost strategy.

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.

Energy-Efficient Train Control

Abstract Over the past decade, the Scheduling and Control Group has conducted an extensive program of research into the theory and practice of energy-efficient train control. Two distinct systems have been developed for providing train drivers with advice on energy-efficient driving strategies. In normal operation, the Metromiser system for suburban railways is achieving fuel savings in excess of 13% and dramatic improvements in timekeeping. The Long-Haul Fuel Conservation System provides driving advice and dynamic rescheduling on long-haul rail networks. The paper outlines the theoretical basis for our work, and illustrates results with selected examples.

A note on the calculation of optimal strategies for the minimization of fuel consumption in the control of trains

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimize fuel consumption. It is assumed that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. During each phase of the journey the power developed by the locomotive is determined by the rate of fuel supply. For each given sequence of throttle settings it has been shown that fuel consumption is minimized by finding an optimal duration for each phase. This suboptimal strategy has been called a strategy of optimal type. In this note we will show that a comparison of suboptimal strategies allows us to find an idealized strategy of optimal type that minimizes fuel consumption. >

An optimal strategy for the control of a train

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimise the energy required to drive the train. It has been shown with an appropriate formulation of the problem that an optimal strategy exists and that this strategy must satisfy a Pontryagin type criterion. In this paper the Pontryagin principle will be used to find the nature of the optimal strategy and this information will then be used to determine the precise optimal strategy.

Energy-efficient train control

Abstract Over the past decade, the Scheduling and Control Group has conducted an extensive program of research into the theory and practice of energy-efficient train control. Two distinct systems have been developed for providing train drivers with advice on energy-efficient driving strategies. In normal operation, the Metromiser system for suburban railways is achieving fuel savings in excess of 13% and dramatic improvements in timekeeping. The Long-Haul Fuel Conservation System provides driving advice and dynamic rescheduling on long-haul rail networks. The paper outlines the theoretical basis for the work, and illustrates results with selected examples.

Analytic Perturbation Theory and Its Applications

Mathematical models are often used to describe complex phenomena such as climate change dynamics, stock market fluctuations, and the Internet. These models typically depend on estimated values of key parameters that determine system behavior. Hence it is important to know what happens when these values are changed. The study of single-parameter deviations provides a natural starting point for this analysis in many special settings in the sciences, engineering, and economics. The difference between the actual and nominal values of the perturbation parameter is small but unknown, and it is important to understand the asymptotic behavior of the system as the perturbation tends to zero. This is particularly true in applications with an apparent discontinuity in the limiting behavior - the so-called singularly perturbed problems. Analytic Perturbation Theory and Its Applications includes a comprehensive treatment of analytic perturbations of matrices, linear operators, and polynomial systems, particularly the singular perturbation of inverses and generalized inverses. It also offers original applications in Markov chains, Markov decision processes, optimization, and applications to Google PageRank and the Hamiltonian cycle problem as well as input retrieval in linear control systems and a problem section in every chapter to aid in course preparation. Audience: This text is appropriate for mathematicians and engineers interested in systems and control. It is also suitable for advanced undergraduate, first-year graduate, and advanced, one-semester, graduate classes covering perturbation theory in various mathematical areas. Contents: Chapter 1: Introduction and Motivation; Part I: Finite Dimensional Perturbations; Chapter 2: Inversion of Analytically Perturbed Matrices; Chapter 3: Perturbation of Null Spaces, Eigenvectors, and Generalized Inverses; Chapter 4: Polynomial Perturbation of Algebraic Nonlinear Systems; Part II: Applications to Optimization and Markov Process; Chapter 5: Applications to Optimization; Chapter 6: Applications to Markov Chains; Chapter 7: Applications to Markov Decision Processes; Part III: Infinite Dimensional Perturbations; Chapter 8: Analytic Perturbation of Linear Operators; Chapter 9: Background on Hilbert Spaces and Fourier Analysis; Bibliography; Index

Application of critical velocities to the minimisation of fuel consumption in the control of trains

Abstract This paper introduces a method for the determination of optimal control strategies in an important engineering application. A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimise fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. In each case the power developed by the locomotive is directly proportional to the rate of fuel supply. For a given sequence of throttle settings we show that fuel consumption is minimised if the settings are changed only when the velocity reaches one of the critical values. We call this strategy a strategy of optimal type. It is now possible to determine the strategy which minimises fuel consumption by considering only strategies of optimal type. Several realistic examples are given and we discuss systematic numerical calculations procedures.

Optimal driving strategies for a train on a track with continuously varying gradient

This paper derives key equations for the determination of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with continuously varying gradient. The journey must be completed within a given time and it is desirable to minimise fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomotive. For each setting the power developed by the locomotive is directly proportional to the rate of fuel supply. We assume a single level of braking acceleration. For each fixed finite sequence of control settings we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. We show that the equations can be derived using an intuitive limit procedure applied to corresponding equations obtained by Howlett [9, 10] in the case of a piecewise constant gradient. The equations will also be derived directly by extending the methods used by Howlett. We discuss a basic solution procedure for the key equations and apply the procedure to find a strategy of optimal type in appropriate specific examples.