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THE TWO-TRAIN SEPARATION PROBLEM ON LEVEL TRACK WITH DISCRETE CONTROL

Part of: Optimality conditions Real functions

Published online by Cambridge University Press:  29 October 2018

PHIL HOWLETT
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 phil.howlett@unisa.edu.au
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Abstract

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When two trains travel along the same track in the same direction, it is a common safety requirement that the trains must be separated by at least two signals. This means that there will always be at least one clear section of track between the two trains. If the safe-separation condition is violated, then the driver of the following train must adopt a revised strategy that will enable the train to stop at the next signal if necessary. One simple way to ensure safe separation is to define a prescribed set of latest allowed section exit times for the leading train and a corresponding prescribed set of earliest allowed section entry times for the following train. We will find strategies that minimize the total tractive energy required for both trains to complete their respective journeys within the overall allowed journey times and subject to the additional prescribed section clearance times. We assume that the drivers use a discrete control mechanism and show that the optimal driving strategy for each train is defined by a sequence of approximate speedholding phases at a uniquely defined optimal driving speed on each section and that the sequence of optimal driving speeds is a decreasing sequence for the leading train and an increasing sequence for the following train. We illustrate our results by finding optimal strategies and associated speed profiles for both trains in some elementary but realistic examples.

Keywords

train controlsafe separationoptimal driving strategiesdiscrete controlconstrained optimization

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 60 , Issue 2 , October 2018 , pp. 137 - 174
Copyright
© 2018 Australian Mathematical Society 

Footnotes

*

This is a contribution to the series of invited papers by past ANZIAM medallists (Editorial, Issue 52(1)). Phil Howlett was awarded the 2018 ANZIAM medal

References

Albrecht, A., Howlett, P. and Pudney, P., “The cost–time curve for an optimal train journey on level track”, ANZIAM J. 58 (2016) 10–32; doi:10.1017/S1446181116000092.Google Scholar
Albrecht, A. R., Howlett, P. G., Pudney, P. J. and Vu, X., “Energy-efficient train control: from local convexity to global optimization and uniqueness”, Automatica 49 (2013) 3072–3078; doi:10.1016/j.automatica.2013.07.008.Google Scholar
Albrecht, A., Howlett, P., Pudney, P., Vu, X. and Zhou, P., “Optimal driving strategies for two successive trains on level track subject to a safe separation condition”, in: Proc. American Control Conf., 1–3 July 2015 (ACC, Chicago, 2015) 2924–2929; doi:10.1109/ACC.2015.7171179.Google Scholar
Albrecht, A. R., Howlett, P. G., Pudney, P. J., Vu, X. and Zhou, P., “Energy-efficient train control: the two-train separation problem on level track”, J. Rail Transp. Plann. Manag. 5 (2015) 163–182; doi:10.1016/j.jrtpm.2015.10.002.Google Scholar
Albrecht, A., Howlett, P., Pudney, P., Vu, X. and Zhou, P., “The key principles of optimal train control—Part 1: Formulation of the model, strategies of optimal type, evolutionary lines, location of optimal switching points”, Transp. Res. B 94 (2015) 482–508; doi:10.1016/j.trb.2015.07.023.Google Scholar
Albrecht, A., Howlett, P., Pudney, P., Vu, X. and Zhou, P., “The key principles of optimal train control—Part 2: Existence of an optimal strategy, the local energy minimization principle, uniqueness, computational techniques”, Transp. Res. B 94 (2015) 509–538; doi:10.1016/j.trb.2015.07.024.Google Scholar
Albrecht, A. R., Howlett, P. G., Pudney, P. J., Vu, X. and Zhou, P., “An optimal timetable for the two train separation problem on level track”, Pac. J. Optim. 12 (2016) 327–353; Special Issue to celebrate the 70th birthday of Kok Lay Teo.Google Scholar
Albrecht, A., Howlett, P., Pudney, P., Vu, X. and Zhou, P., “The two-train separation problem on non-level track—driving strategies that minimize total required tractive energy subject to prescribed section clearance times”, Transp. Res. B 111 (2018) 135–167; doi:10.1016/j.trb.2018.03.012.Google Scholar
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”, USSR Comput. Math. Math. Phys. 25 (1985) 37–44; doi:10.1016/0041-5553(85)90006-0.Google Scholar
Baranov, L. A., Erofeyev, E., Golovitcher, I. and Maksimov, V., Automated control for electric locomotives and multiple units, Monograph (Transport, Moscow, 1990).Google Scholar
Baranov, L. A., Meleshin, I. S. and Chin, L. M., “Optimal control of a subway train with regard to the criteria of minimum energy consumption”, Russ. Electr. Engng 82 (2011) 405–410; doi:10.3103/S1068371211080049.Google Scholar
Burdett, R. L. and Kozan, E., “Techniques for inserting additional trains into existing timetables”, Transp. Res. B 43 (2009) 821–836; doi:10.1016/j.trb.2009.02.005.Google Scholar
Burdett, R. L. and Kozan, E., “A sequencing approach for train timetabling”, OR Spectrum 32 (2010) 163–193; doi:10.1007/s00291-008-0143-6.Google Scholar
Caprara, A., Fischetti, M. and Toth, P., “Modeling and solving the train timetabling problem”, Oper. Res. 50 (2002) 851–861; doi:10.1287/opre.50.5.851.362.Google Scholar
Cheng, J. and Howlett, P. G., “Application of critical velocities to the minimisation of fuel consumption in the control of trains”, Automatica 28 (1992) 165–169; doi:10.1016/0005-1098(92)90017-A.Google Scholar
Cordeaux, J. F., Toth, P. and Vigo, D., “A survey of optimization models for train routing and scheduling”, Transp. Sci. 32 (1998) 380–404; doi:10.1287/trsc.32.4.380.Google Scholar
Davis, W. J. Jr, “The tractive resistance of electric locomotives and cars”, Gen. Electr. Rev. 29 (General Electric Review, Schenectady, NY, 1926) 2–24.Google Scholar
Golovitcher, I., “An analytical method for optimum train control computation”, in: Proc. State Universities, Electro-mechanics, Volume 3 Izv. VUZov Ser. Electro-mehan. , (1986) 59–66.Google Scholar
Golovitcher, I., “Control algorithms for automatic operation of rail vehicles”, J. Russian (USSR) Acad. Sci. (Automat. Telemekhan.) 11 (1986) 118–126; doi:10.13140/RG.2.1.4394.1844.Google Scholar
Golovitcher, I., “Optimum control of electric locomotives with regenerative braking”, in: Proc. Moscow Railway Engineering Institute (Trudy MIIT), Volume 811 (Moscow, 1989) 19–24.Google Scholar
Golovitcher, I., “An analytical method for computation of optimum train speed profile considering variable efficiency of locomotive”, in: Proc. State Universities, Electro-mechanics, Volume 2 Izv. VUZov Ser. Electro-mehan. , (1989) 72–81.Google Scholar
Gupta, S. D., Tobin, J. K. and Pavel, L., “A two-step linear programming model for energy-efficient timetables in metro railway networks”, Transp. Res. B 93 (2016) 57–74; doi:10.1016/j.trb.2016.07.003.Google Scholar
Hartl, R. F., Sethi, S. P. and Vickson, R. G., “A survey of the maximum principles for optimal control problems with state constraints”, SIAM Rev. 37 (1995) 181–218; doi:10.1137/1037043.Google Scholar
Higgins, A., Kozan, E. and Ferreira, L., “Optimal scheduling of trains on a single line track”, Transp. Res. B 30 (1996) 147–161; doi:10.1016/0191-2615(95)00022-4.Google Scholar
Howlett, P. G., “An optimal strategy for the control of a train”, J. Aust. Math. Soc. B (now ANZIAM J.) 31 (1990) 454–471; doi:10.1017/S0334270000006780.Google Scholar
Howlett, P., “Optimal strategies for the control of a train”, Automatica 32 (1996) 519–532; doi:10.1016/0005-1098(95)00184-0.Google Scholar
Howlett, P., “The optimal control of a train”, Ann. Oper. Res. 98 (2000) 65–87; doi:10.1023/A:101923581.Google Scholar
Howlett, P., “A new look at the rate of change of energy consumption with respect to journey time on an optimal train journey”, Transp. Res. B 94 (2016) 387–408; doi:10.1016/j.trb.2016.10.004.Google Scholar
Howlett, P. G. and Cheng, J., “A note on the calculation of optimal strategies for the minimisation of fuel consumption in the control of trains”, IEEE Trans. Automat. Control 38 (1993) 1730–1734; doi:10.1109/9.262051.Google Scholar
Howlett, P. and Jiaxing, C., “Optimal driving strategies for a train on a track with continuously varying gradient”, ANZIAM J. (formerly J. Aust. Math. Soc. Ser. B) 38 (1997) 388–410; doi:10.1017/S0334270000000746.Google Scholar
Howlett, P. G. and Leizarowitz, A., “Optimal strategies for vehicle control problems with finite control sets”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 8 (2001) 41–69.Google Scholar
Howlett, P. G. and Pudney, P. J., Energy-efficient train control, Advances in Industrial Control (Springer, London, 1995).Google Scholar
Howlett, P. G., Pudney, P. J. and Milroy, I. P., “Energy-efficient train control”, Control Eng. 2 (1994) 193–200; doi:10.1016/0967-0661(94)90198-8.Google Scholar
Howlett, P., Pudney, P. and Vu, X., “Local energy minimization in optimal train control”, Automatica 45 (2009) 2692–2698; doi:10.1016/j.automatica.2009.07.028.Google Scholar
Ichikawa, K., “Application of optimization theory for bounded state variable problems to the operation of trains”, Bull. JSME Nagoya Univ. 11 (1968) 857–865; doi:10.1299/jsme1958.11.857.Google Scholar
Isayev, I. P. (ed.), Teoriya electricheskoy tyagi [Electric traction theory], 3rd edn (Transport, Moscow, 1987); (in Russian).Google Scholar
Khmelnitsky, E., “On an optimal control problem of train operation”, IEEE Trans. Automat. Control 45 (2000) 1257–1266; doi:10.1109/9.867018.Google Scholar
Kokotovic, P. and Singh, G., “Minimum-energy control of a traction motor”, IEEE Trans. Automat. Control 17 (1972) 92–95; doi:10.1109/TAC.1972.1099870.Google Scholar
Li, X. and Lo, H. K., “An energy-efficient scheduling and speed control approach for metro rail operations”, Transp. Res. B 64 (2014) 73–89; doi:10.1016/j.trb.2014.03.006.Google Scholar
Li, X. and Lo, H. K., “Energy minimization in dynamic train scheduling and control for metro rail operations”, Transp. Res. B 70 (2014) 269–284; doi:10.1016/j.trb.2014.09.009.Google Scholar
Liu, R. and Golovitcher, I. A., “Energy-efficient operation of rail vehicles”, Transp. Res. A 37 (2003) 917–932; doi:10.1016/j.tra.2003.07.001.Google Scholar
Liu, J. and Zhao, N., “Research on energy-saving operation strategy for multiple trains on the urban subway line”, Energies 10(12) (2017) 2156; doi:10.3390/en10122156.Google Scholar
Luenberger, D. G., Optimization by vector space methods (Wiley, New York, 1969).Google Scholar
Milroy, I. P., “Minimum-energy control of rail vehicles”, in: Proc. Railway Engineering Conf. (Institution of Engineers Australia, Sydney, 1981) 103–114.Google Scholar
Scheepmaker, G. M., Goverde, R. M. P. and Kroon, L. G., “Review of energy-efficient train control and timetabling”, European J. Oper. Res. 2017(2) (2017) 355–376; doi:10.1016/j.ejor.2016.09.044.Google Scholar
Strobel, H. and Horn, P., “On energy-optimum control of train movement with phase constraints”, Electr. Inform. Energy Tech. J. 6 (1973) 304–308.Google Scholar
TTG Transportation Technology, http://www.ttgtransportationtechnology.com/energymiser/.Google Scholar
Vu, X., “Analysis of necessary conditions for the optimal control of a train”, Ph. D. Thesis, University of South Australia, 2006. http://search.ror.unisa.edu.au/media/researcharchive/open/9915951966501831 /53111938330001831.Google Scholar
Wang, P. and Goverde, R. M. P., “Two-train trajectory optimization with a green-wave policy”, Transp. Res. Rec.: J. Transp. Res. Board 2546 (2016) 112–120; doi:10.3141/2546-14.Google Scholar
Wang, P. and Goverde, R. M. P., “Multi-train trajectory optimization for energy efficiency and delay recovery on single-track railway lines”, Transp. Res. B 105 (2017) 340–361; doi:10.1016/j.trb.2017.09.012.Google Scholar
Yang, S., Wu, J., Yang, X., Sun, H. and Gao, Z., “Energy-efficient timetable and speed profile optimization with multi-phase speed limits: theoretical analysis and application”, Appl. Math. Model. 56 (2018) 32–50; doi:10.1016/j.apm.2017.11.017.Google Scholar