Abstract
Previous work by the authors studied energy optimal control of cam controlled subway trains going from one station to the next on a given schedule. An indirect competing Hamiltonian algorithm was created to handle the associated mixed integer programming problem. A direct approach was also created based on outer convexification, relaxation, and the Krein-Milman theorem. This paper extends the optimization, creating a new form of scheduling that produces a time table for the complete subway line that minimizes total energy consumption from start to end of the line, for a given total transit time. Example computations are given for optimizing the schedule across several stations in the New York subway system. A characteristic of the optimized time table is that the Hamiltonian for the transit from each station to the next must be the same value all over the line. In applications, trains can easily get behind the prescribed schedule, so one must have a method of returning to schedule. The authors previously presented feedback methods to modify the control in a nearly optimal manner en-route to the next station. This is also generalized here to develop feedback to return to schedule at a later station, or by the end of the line. An alternative that could be simpler to implement is also presented, to simply increase the Hamiltonian, which corresponds to requesting an increased average speed, and the motorman could use this until back on schedule.
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Bock, H.: Numerical solution of nonlinear multipoint boundary value problems with applications to optimal control. Zeitschrift für Angewandte Mathematik und Mechanik 58, 407 (1978)
Bock, H.: Numerische Behandlung von zustandsbeschränkten und Chebyshev-Steuerungsproblemen, Technical Report R106/81/11, Carl Cranz Gesellschaft, Heidelberg (1981)
Bock, H., Longman, R.: Optimal control of velocity profiles for minimization of energy consumption in the New York subway system. In: Proceedings of the Second IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Oberpfaffenhofen, pp. 34–43. International Federation of Automatic Control (1980)
Bock, H., Longman, R.: Computation of optimal controls on disjoint control sets for minimum energy subway operation. Adv. Astronaut. Sci. 50, 949–972 (1985)
Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of optimal control problems. In: 9th IFAC World Congress, vol. IX, pp. 242–247. Budapest (1984)
Bryson, A., Ho, Y.C.: Applied Optimal Control. Wiley, New York (1975)
Cadi, A.: Energieoptimale Verteilung von Fahrzeiten auf U-Bahnlinien und optimale Fahrstrategien für Langstrecken, Diplomarbeit, Ruprecht–Karls–Universität Heidelberg (1995)
Franke, R., Meyer, M., Terwiesch, P.: Optimal control of the driving of trains. Automatisierungstechnik 50(12), 606–614 (2002)
Krämer-Eis, P.: Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback–Steuerungen bei beschränkten nichtlinearen Steuerungsproblemen, Volume 166 of Bonner Mathematische Schriften, Universität Bonn (1985)
Leineweber, D., Bauer, I., Bock, H., Schlöder, J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part I: theoretical aspects. Comput. Chem. Eng. 27, 157–166 (2003)
Leineweber, D., Bauer, I., Schäfer, A., Bock, H., Schlöder, J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II). Comput. Chem. Eng. 27, 157–174 (2003)
Sager, S.: Numerical methods for mixed–integer optimal control problems. PhD thesis, Universität Heidelberg (2006)
Viswanathan, C., Longman, R., Domoto, G.: Energy conservation in subway systems by controlled acceleration and deceleration. Int. J. Energy Res. 2, 133–151 (1978)
Acknowledgements
Financial support by the German Federal Ministry of Education and Research program “Mathematics for Innovations in Industry and Service 2013–2016”, grant no 05M2013-GOSSIP, and by the European Union within the 7th Framework Programme under Grant Agreement no 611909 is gratefully acknowledged.
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Bock, H.G., Cadi, A., Longman, R.W., Schlöder, J.P. (2017). Minimum Energy Time Tables for Subway Operation - And Hamiltonian Feedback to Return to Schedule. In: Bock, H., Phu, H., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes HPSC 2015 . Springer, Cham. https://doi.org/10.1007/978-3-319-67168-0_1
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DOI: https://doi.org/10.1007/978-3-319-67168-0_1
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