Abstract
The paper is devoted to the solution of the energy minimization problem for a moving train. The train movement is governed by the system of the first order ordinary differential equations where the train speed and the distance along the track are the state variables. The provided locomotive power depends on the control function. The generated traction force is assumed to depend on the velocity of the train and on the control function. Each non-negative value of the control function determines a traction force control while negative values determine a braking force control. The cost functional is defined as the train energy. It is dependent on traction force, speed and control functions. The speed, distance and control functions are assumed bounded. Using the maximum principle and Lagrangian multipliers the system of equations constituting the necessary optimality conditions is formulated. Based on the analysis of the train movement the optimal trajectories in terms of train speed and associated optimal control functions are calculated. A new simplified method is used to calculate the set of the switching times implementing the optimal control function. Numerical examples are provided and discussed.
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References
Albrecht, A., Howlett, P., Pudney, P., Vu, X., 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. Part B 94, 482–508 (2016)
Albrecht, A., Howlett, P., Pudney, P., Vu, X., 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. Part B 94, 509–538 (2016)
Albrecht, A.R., Howlett, P.G., Pudney, P.J., Vu, X.: Energy-efficient train control: From local convexity to global optimization and uniqueness. Automatica 49, 3072–3078 (2013)
Asnis, I.A., Dmitruk, A.V., 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(6), 37–44 (1985)
Bigharaz, M.H., Afshar, A., Suratgar, A., Safaei, F.: Simultaneous optimization of energy consumption and train performances in electric railway systems. In: Preprints of the 19th World Congress of the International Federation of Automatic Control, pp. 6270–6275 (2014)
Burak-Romanowski, R., Woźniak, K.: Energetic aspect of the railway tracks modernization. Tech. Trans. Electr. Eng. 108(13), 13–29 (2011). (in Polish)
Gkortzas, P.: Study on optimal train movement for minimum energy consumption. MSc thesis, Mälardalen University, Sweden (2013). http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-21234
Górecki, H., Fuksa, S., Korytowski, A., Mitkowski, W.: Optimal Control in Linear Systems with the Quadratic Performance Index. Polish Scientific Publisher, Warsaw (1983). (in Polish)
Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)
Howlett, P.: The optimal control of a train. Anna. Oper. Res. 98, 65–87 (2000)
Howlett, P.G., Pudney, P.J., Vu, X.: Local energy minimization in optimal train control. Automatica 45, 2692–2698 (2009)
Miyatake, M., Ko, H.: Optimization of train speed profile for minimum energy consumption. IEEJ Trans. Electr. Electron. Eng. 5, 263–269 (2010)
Montroe, T., Pellegrinii, P., Nobili, P.: Energy consumption minimization problemin a railway network. Transp. Res. Procedia 22, 85–94 (2017)
Myśliński, A., Nahorski, Z., Szulc, K., Radziszewska, W.: Simulation and optimization of the train movement. Research Report, Systems Research Institute, Warsaw, Poland (2017). (in Polish)
Novak, H., Vašak, M., Lešiči, V.: Hierarchical energy management of multi-train railway transport system with energy storages. In: IEEE International Conference on Intelligent Rail Transportation (ICIRT), pp. 130–138 (2016)
Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton (1995)
Rochard, B.P., Schmid, F.: A review of methods to measure and calculate train resistances. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 214(4), 185–199 (2000)
Scheepmaker, G.M., Goverde, R.M.P., Kroon, L.G.: Review of energy-efficient train control and time tabling. Eur. J. Oper. Res. 257, 355–376 (2017)
Vittek, J., Butko, P., Ftorek, B., Makys, P., Gorel, L.: Energy near optimal control strategies for industrial and traction drives with a.c. motors. Math. Probl. Eng. 2017 (2017). https://doi.org/10.1155/2017/1857186, Article ID 1857186, 22 pages
Wang, Y., Ning, B., Cao, F., De Schutter, B., van den Boom, T.J.J.: A survey on optimal trajectory planning for train operations. In: Proceedings of the 2011 IEEE International Conference on Intelligent Rail Transportation (ICIRT 2011), Beijing, China, pp. 589–594 (2011)
Wnuk, M.: The calculation of the optimal velocity of the train under velocity constraints. Technika Transportu Szynowego 4, 54–59 (2012). (in Polish)
Ye, H., Liu, R.: A multiphase optimal control method for multi-train control and scheduling on railway lines. Transp. Res. Part B 93, 377–393 (2016)
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Myśliński, A., Nahorski, Z., Szulc, K., Radziszewska, W. (2018). Energy Consumption Optimal Control of the Train Movement. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2018. AUTOMATION 2018. Advances in Intelligent Systems and Computing, vol 743. Springer, Cham. https://doi.org/10.1007/978-3-319-77179-3_33
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