Optimal Speed Profile Determination with Fixed Trip Time in the Electric Train Operation of the Cat Linh-Ha Dong Metro Line based on Pontryagin's Maximum Principle

  • T. T. T. A. Anh Department of Electrical Engineering, University of Transport and Communications, Vietnam
  • N. V. Quyen Department of Applied Mechanics, Hanoi University of Science and Technology, Vietnam https://orcid.org/0000-0001-9486-1431
Volume: 10 | Issue: 6 | Pages: 6488-6493 | December 2020 | https://doi.org/10.48084/etasr.3856


The significant energy consumption for railway electric transportation operation poses a great challenge in outlining saving energy solutions. Speed profile optimization based on optimal control theory is one of the most common methods to improve energy efficiency without the railway infrastructure investment costs. The paper proposes an optimization method based on Pontryagin's Maximum Principle (PMP), not only to find optimal switching points in three operation phases: accelerating, coasting, braking, and from these switching points being able to determine the optimal speed profile, but also to ensure fixed-trip time. In order to determine trip time abiding by the scheduled timetables by applying nonlinear programming puts the Lagrange multiplier λ in the objective function regarded as a time constraint condition. The correctness and energy effectiveness of this method have been verified by the simulation results with data collected from the electrified trains of the Cat Linh-Ha Dong metro line in Vietnam. The saving energy levels are compared in three scenarios: electrified train operation tracking the original speed profile (energy consumption of the route: 144.64kWh), train operation tracking the optimal speed profile without fixed-trip time (energy consumption of the route: 129.18kWh), and train operation tracking the optimal speed profile and fixed trip time (energy consumption of the route: 132.99kWh) in an effort to give some useful choices for operating metro lines.

Keywords: energy-saving, energy-efficient operation methodology, timetable optimization, metro system


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G. R. Kazemiyan, A. Rasooli, and S. Rafipoor, "The advantages of rail transport compared to road within the city, based on a sustainable development approach, case study Tehran Metro Line 4," Research and Urban Planning, vol. 6, no. 23, pp. 77-94, Jan. 2016.

V. R. Vuchic, Urban Transit Systems and Technology, 1st ed. Hoboken, NJ, USA: Wiley, 2007. DOI: https://doi.org/10.1002/9780470168066

S. Su, T. Tang, and Y. Wang, "Evaluation of Strategies to Reducing Traction Energy Consumption of Metro Systems Using an Optimal Train Control Simulation Model," Energies, vol. 9, no. 2, p. 105, Feb. 2016. DOI: https://doi.org/10.3390/en9020105

J. Yang, L. Jia, S. Lu, Y. Fu, and J. Ge, "Energy-Efficient Speed Profile Approximation: An Optimal Switching Region-Based Approach with Adaptive Resolution," Energies, vol. 9, no. 10, pp. 1-27, 2016. DOI: https://doi.org/10.3390/en9100762

P. Howlett, "The Optimal Control of a Train," Annals of Operations Research, vol. 98, no. 1, pp. 65-87, Dec. 2000. DOI: https://doi.org/10.1023/A:1019235819716

P. G. Howlett and P. J. Pudney, Energy-Efficient Train Control. London, UK: Springer-Verlag, 1995. DOI: https://doi.org/10.1007/978-1-4471-3084-0

X. Vu, Analysis of necessary conditions for the optimal control of a train: New neccessary conditions for energy-efficient train control. VDM Verlag Dr. Müller, 2009.

A. Albrecht, P. Howlett, P. Pudney, X. Vu, and P. Zhou, "The key principles of optimal train control-Part 1: Formulation of the model, strategies of optimal type, evolutionary lines, location of optimal switching points," Transportation Research Part B: Methodological, vol. 94, pp. 482-508, Dec. 2016. DOI: https://doi.org/10.1016/j.trb.2015.07.023

A. Albrecht, P. Howlett, P. Pudney, X. Vu, and P. Zhou, "The key principles of optimal train control-Part 2: Existence of an optimal strategy, the local energy minimization principle, uniqueness, computational techniques," Transportation Research Part B: Methodological, vol. 94, pp. 509-538, Dec. 2016. DOI: https://doi.org/10.1016/j.trb.2015.07.024

E. Khmelnitsky, "On an optimal control problem of train operation," IEEE Transactions on Automatic Control, vol. 45, no. 7, pp. 1257-1266, Jul. 2000. DOI: https://doi.org/10.1109/9.867018

R. Liu and I. M. Golovitcher, "Energy-efficient operation of rail vehicles," Transportation Research Part A: Policy and Practice, vol. 37, no. 10, pp. 917-932, Dec. 2003. DOI: https://doi.org/10.1016/j.tra.2003.07.001

S. Lu, S. Hillmansen, T. K. Ho, and C. Roberts, "Single-Train Trajectory Optimization," IEEE Transactions on Intelligent Transportation Systems, vol. 14, no. 2, pp. 743-750, Jun. 2013. DOI: https://doi.org/10.1109/TITS.2012.2234118

H. Ehteshami, S. Javadi, and S. M. Shariatmadar, "Improving the Power Quality in Tehran Metro Line-Two Using the Ant Colony Algorithm," Engineering, Technology & Applied Science Research, vol. 7, no. 6, pp. 2256-2259, Dec. 2017. DOI: https://doi.org/10.48084/etasr.1551

M. Domínguez, A. Fernández, A. P. Cucala, and L. P. Cayuela, "Computer-aided design of ATO speed commands according to energy consumption criteria," presented at the COMPRAIL 2008, Toledo, Spain, Aug. 2008, pp. 183-192. DOI: https://doi.org/10.2495/CR080191

M. Domínguez, A. Fernández, A. P. Cucala, and P. Lukaszewicz, "Optimal design of metro automatic train operation speed profiles for reducing energy consumption," Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Aug. 2011. DOI: https://doi.org/10.1177/09544097JRRT420

T. Kara and M. C. Savas, "Design and Simulation of a Decentralized Railway Traffic Control System," Engineering, Technology & Applied Science Research, vol. 6, no. 2, pp. 945-951, Apr. 2016. DOI: https://doi.org/10.48084/etasr.631

K. K. Wong and T. K. Ho, "Dynamic coast control of train movement with genetic algorithm," International Journal of Systems Science, vol. 35, no. 13-14, pp. 835-846, Oct. 2004. DOI: https://doi.org/10.1080/00207720412331203633

S. Acikbas and M. T. Soylemez, "Coasting point optimisation for mass rail transit lines using artificial neural networks and genetic algorithms," IET Electric Power Applications, vol. 2, no. 3, pp. 172-182, May 2008. DOI: https://doi.org/10.1049/iet-epa:20070381

A. T. H. T. Anh, N. V. Quyen, N. T. Hai, N. V. Lien, and V. H. Phuong, "Speed profile optimization of an electrified train in Cat Linh-Ha Dong metro line based on pontryagin's maximum principle," International Journal of Electrical and Computer Engineering (IJECE), vol. 10, no. 1, pp. 233-242, Feb. 2020. DOI: https://doi.org/10.11591/ijece.v10i1.pp233-242

X. Sun, H. Lu, and H. Dong, "Energy-Efficient Train Control by Multi-Train Dynamic Cooperation," IEEE Transactions on Intelligent Transportation Systems, vol. 18, no. 11, pp. 3114-3121, Nov. 2017. DOI: https://doi.org/10.1109/TITS.2017.2682270

N. T. M. Chau, Hanoi Urban Railway Project Cat Linh-Ha Dong Line. Package 1: EPC Contact. Technical Design, Book 2: Traffic organization and operation management. Hanoi, Vietnam: Railway Project Management Unit, Vietnam Railway Administration, 2014.

W. Davis, The tractive resistance of electric locomotives and cars. General Electric, 1926.

N. T. Hai, "Evaluation of effect Pontryagin's Maximum Principle for optimal control train by criteria of energy save," in 2010 International Symposium on Computer, Communication, Control and Automation (3CA), Tainan, Taiwan, May 2010, vol. 1, pp. 363-366. DOI: https://doi.org/10.1109/3CA.2010.5533807


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