A remark on energy optimal strategies for a train movement

This article introduces the notion of the critical time in the problem of the energy efficient train control and its calculation in some particular cases. We apply some results of non-linear parametric optimization to show that the number of optimal control levels depends on the relation between the given time of the journey and this critical time. Furthermore, we derive equations for the computation of the switching times. I emphasise exact forms of solutions with a minimal use of numerical mathematics. The results can be used to find the values of the switching times only by solving algebraic equations and to analyse the behaviour of the results with respect to given entry parameters of the problem. References P. Horn. Uber die Anwendung des Maximum-Prinzips von Pontrjagin zur Ermittlung von Algorithmen fur eine energieoptimale Zugsteuerung (About the application of the Pontryagin principle for deriving of the algorithms for an energetically optimal control of a train). Wissensch. Zschr. d. Hochschule fur Verkehrswessen ``Friedrich List'' , 18(4):919--943, 1971. (in German). P. Howlett. An optimal strategy for the control of a train. J. Austral. Math. Soc., Ser. B , 31:454--471, 1990. http://anziamj.austms.org.au/V31/part4/Howlett.html . P. Howlett and J. Cheng. Optimal driving strategies for a train on a track with continuously varying gradient. J. Austral. Math. Soc., Ser. B , 38:388--410, 1997. http://anziamj.austms.org.au/V38/part3/Howlett.html . P. G. Howlett. The optimal control of a train. Annals of OR , 98:65--87, 2000. http://www.ingentaconnect.com/content/klu/anor/2000/00000098/F0040001/0%332912 . P. G. Howlett and J. Cheng. 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(11):1730--1734, 1993. doi:10.1109/9.262051 . P. G. Howlett and A. Leizarowitz. Optimal strategies for vehicle control problems with finite control sets. Dynamics of Continuous, Discrete and Impulsive Systems, B: Applications and Algorithms , 8:41--69, 2001. P. G. Howlett and P. J. Pudney. Energy efficient train control . Springer, London, 1995. E. Khmelnitsky. On an optimal control problem of train operation. IEEE Transactions on Automatic Control , 45:1257--1266, 2000. doi:10.1109/9.867018 . K. P. Li, Z. Y. Gao, and B. H. Mao. Energy-optimal control model for train movements. Chinese Physics , 16(2):359--364, 2007. doi:10.1088/1009-1963/16/2/015 . R. Liu and I. Golovitcher. Energy efficient operation of rail vehicles. Transportation Research Part A: Policy and Practice, Elsevier, United Kingdom , 37:917--932, 2003. doi:10.1109/ICSMC.2001.969927 . B. Bank. Non-Linear Parametric Optimization . Akademie-Verlag, Berlin, 1982. R. Pickhardt. Time- and energy- optimal control of an electric railcar. In Intelligent Transportation Systems, 2000. Proceedings. 2000 IEEE , pages 500--505, 2000. doi:10.1109/ITSC.2000.881120 . P. Pudney and P. Howlett. Optimal driving strategies for a train journey with speed limits. J. Austral. Math. Soc., Ser. B , 36:38--49, 1994. http://anziamj.austms.org.au/V36/part1/Pudney.html . M. F. Bazaraa, H. D. Sherali, and C. M. Shetty. Nonlinear Programming, Theory and Algorithms . John Wiley and Sons, New York, 1993. J. Cheng, Y. Davydova, P. G. Howlett, and P. J. Pudney. Optimal driving strategies for a train journey with non-zero track gradient and speed limits. IMA Journal of Mathematics Applied in Business and Industry , 10:89--115, 1999. doi:10.1093/imaman/10.2.89 . S. H. Han, Y. S. Byen, J. H. Baek, T. K. An, S. G. Lee, and H. J. Park. An optimal automatic train operation (ato) control using genetic algorithms (ga). In TENCON 99. Proceedings of the IEEE Region 10 Conference , volume 1, pages 360--362, 1999. doi:10.1109/TENCON.1999.818425 .