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

Published online by Cambridge University Press:  17 February 2009

P. G. Howlett  and
J. Cheng
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Abstract

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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.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 3 , January 1997 , pp. 388 - 410
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Asnis, I. A., Dmitruk, A. V. and Osmolovskii, N. P., “Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle”, U. S. S. R. Comput. Maths. Math. Phys. 25 (1985) 37–44.CrossRefGoogle Scholar
[2]Benjamin, B. R., Long, A. M., Milroy, I. P., Payne, R. L. and Pudney, P. J., “Control of railway vehicles for energy conservation and improved timekeeping”, Proc. IEA Conference on Railway Engineering, (Perth, Western Australia, 1987) 41–47.Google Scholar
[3]Benjamin, B. R., Milroy, I. P. and Pudney, P. J., “Energy-efficient operation of long-haul trains”, Proceedings of the Fourth International Heavy Haul Railway Conference, Institution of Engineers Australia, (Brisbane, 1989) 369–372.Google Scholar
[4]Birkhoff, G. and Rota, G., Ordinary Differential Equations, 3rd ed. (Wiley, 1978).Google Scholar
[5]Howlett, P. G., “The optimal control of a train”, Study Leave Report School of Mathematics, (University of South Australia, 1984).Google Scholar
[6]Howlett, P. G., “Existence of an optimal strategy for the control of a train”, School of Mathematics Report No. 3, (University of South Australia, 1988).Google Scholar
[7]Howlett, P. G., “Necessary condition on an optimal strategy for the control of a train”, School of Mathematics Report No. 4, (University of South Australia, 1988).Google Scholar
[8]Howlett, P. G., “An optimal strategy for the control of a train”, J. Austral. Math. Soc. Ser. B 31 (1990) 454–471.CrossRefGoogle Scholar
[9]Howlett, P. G., “Optimal strategies for the control of a train on a track with non-constant gradient” school of Mathematics Report No. 6 (University of South Australia, 1992).Google Scholar
[10]Howlett, P. G., “Optimal strategies for the control of a train”, Automatica 32 (1993) 519–532.CrossRefGoogle Scholar
[11]Howlett, P. G., Cheng, J. and Pudney, P. J., “Optimal strategies for energy-efficient train control”, Paper presented at the SIAM Symposium on Control Problems in Industry, (San Diego, 1994).Google Scholar
[12]Howlett, P. G., Milroy, I. P. and Pudney, P. J., “Energy-efficient train control”, Control Engineering Practice 2 (1994) 193–200.CrossRefGoogle Scholar
[13]Howlett, P. G., Milroy, I. P. and Pudney, P. J., “Scheduling and control of trains beyond 2000”, Invited paper, Special Issue, Journal of Advanced Transportation (1994), to appear.Google Scholar
[14]Howlett, P. G., Pudney, Peter and Benjamin, Basil, “Determination of optimal driving strategies for the control of a train”, Proceedings CTAC 91, (Adelaide, 1992) 241–248.Google Scholar
[15]Jiaxin, Cheng and Howlett, P. G., “Critical velocities for the minimisation of fuel consumption in the control of trains”, School of Mathematics, Report No. 1 (University of South Australia, 1990).Google Scholar
[16]Jiaxin, Cheng and Howlett, P. G., “Optimal strategies for the minimisation of fuel consumption in the control of trains”, School of Mathematics, Report No. 3 (University of South Australia, 1990).Google Scholar
[17]Jiaxin, Cheng and Howlett, P. G., “Application of critical velocities to the minimisation of fuel consumption in the control of trains”, Automatica 28 (1992) 165–169.CrossRefGoogle Scholar
[ 18]Jiaxin, Cheng and Howlett, P. G., “A note on the calculation of optimal strategies for the minimisation of fuel consumption in the control of trains”, IEEE Transactions on Automatic Control 38 (1993) 1730–1734.CrossRefGoogle Scholar
[19]Milroy, I. P., “Aspects of automatic train control”, Ph. D. Thesis, Loughborough University, 1980.Google Scholar