IDEAS home Printed from https://ideas.repec.org/a/eee/transb/v94y2016icp509-538.html
   My bibliography  Save this article

The key principles of optimal train control—Part 2: Existence of an optimal strategy, the local energy minimization principle, uniqueness, computational techniques

Author

Listed:
  • Albrecht, Amie
  • Howlett, Phil
  • Pudney, Peter
  • Vu, Xuan
  • Zhou, Peng

Abstract

We discuss the problem of finding an energy-efficient driving strategy for a train journey on an undulating track with steep grades subject to a maximum prescribed journey time. In Part 1 of this paper we reviewed the state-of-the-art and established the key principles of optimal train control for a general model with continuous control. We assumed only that the tractive and braking control forces were bounded by non-increasing speed-dependent magnitude constraints and that the rate of energy dissipation from frictional resistance was given by a non-negative strictly convex function of speed. Partial cost recovery from regenerative braking was allowed. Our aim was to minimize the mechanical energy required to drive the train. We examined the characteristic optimal control modes, studied allowable control transitions and established the existence of optimal switching points. We found algebraic formulae for the adjoint variables in terms of speed on track with piecewise-constant gradient and drew phase plots of the associated optimal evolutionary lines for the state and adjoint variables. In Part 2 we will establish integral forms of the necessary conditions for optimal switching, find general bounds on the positions of the optimal switching points, justify an extended local energy minimization principle and show how these ideas can be used to calculate the optimal strategy. We prove that an optimal strategy always exists and use a perturbation analysis to show that the optimal strategy is unique. Finally we discuss computation of optimal switching points in two realistic examples with steep grades and describe the optimal control strategies and corresponding speed profiles for a complete journey with several different allowed journey times. In practice the strategies described here have been shown to reduce the costs of energy used by as much as 20%.

Suggested Citation

  • Albrecht, Amie & Howlett, Phil & Pudney, Peter & Vu, Xuan & Zhou, Peng, 2016. "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, Elsevier, vol. 94(C), pages 509-538.
    • Handle: RePEc:eee:transb:v:94:y:2016:i:c:p:509-538
    DOI: 10.1016/j.trb.2015.07.024
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0191261515002076
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.trb.2015.07.024?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Phil Howlett, 2000. "The Optimal Control of a Train," Annals of Operations Research, Springer, vol. 98(1), pages 65-87, December.
    2. Liu, Rongfang (Rachel) & Golovitcher, Iakov M., 2003. "Energy-efficient operation of rail vehicles," Transportation Research Part A: Policy and Practice, Elsevier, vol. 37(10), pages 917-932, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. He, Deqiang & Yang, Yanjie & Chen, Yanjun & Deng, Jianxin & Shan, Sheng & Liu, Jianren & Li, Xianwang, 2020. "An integrated optimization model of metro energy consumption based on regenerative energy and passenger transfer," Applied Energy, Elsevier, vol. 264(C).
    2. Zhang, Lang & He, Deqiang & He, Yan & Liu, Bin & Chen, Yanjun & Shan, Sheng, 2022. "Real-time energy saving optimization method for urban rail transit train timetable under delay condition," Energy, Elsevier, vol. 258(C).
    3. Luan, Xiaojie & Wang, Yihui & De Schutter, Bart & Meng, Lingyun & Lodewijks, Gabriel & Corman, Francesco, 2018. "Integration of real-time traffic management and train control for rail networks - Part 2: Extensions towards energy-efficient train operations," Transportation Research Part B: Methodological, Elsevier, vol. 115(C), pages 72-94.
    4. Lai, Qingying & Liu, Jun & Haghani, Ali & Meng, Lingyun & Wang, Yihui, 2020. "Energy-efficient speed profile optimization for medium-speed maglev trains," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 141(C).
    5. Xuan Lin & Qingyuan Wang & Pengling Wang & Pengfei Sun & Xiaoyun Feng, 2017. "The Energy-Efficient Operation Problem of a Freight Train Considering Long-Distance Steep Downhill Sections," Energies, MDPI, vol. 10(6), pages 1-26, June.
    6. Luijt, Ralph S. & van den Berge, Maarten P.F. & Willeboordse, Helen Y. & Hoogenraad, Jan H., 2017. "5years of Dutch eco-driving: Managing behavioural change," Transportation Research Part A: Policy and Practice, Elsevier, vol. 98(C), pages 46-63.
    7. Zhaoxiang Tan & Shaofeng Lu & Kai Bao & Shaoning Zhang & Chaoxian Wu & Jie Yang & Fei Xue, 2018. "Adaptive Partial Train Speed Trajectory Optimization," Energies, MDPI, vol. 11(12), pages 1-33, November.
    8. Huanhuan Lv & Yuzhao Zhang & Kang Huang & Xiaotong Yu & Jianjun Wu, 2019. "An Energy-Efficient Timetable Optimization Approach in a Bi-DirectionUrban Rail Transit Line: A Mixed-Integer Linear Programming Model," Energies, MDPI, vol. 12(14), pages 1-24, July.
    9. Wang, Pengling & Goverde, Rob M.P., 2017. "Multi-train trajectory optimization for energy efficiency and delay recovery on single-track railway lines," Transportation Research Part B: Methodological, Elsevier, vol. 105(C), pages 340-361.
    10. Guangzhao Meng & Chaoxian Wu & Bolun Zhang & Fei Xue & Shaofeng Lu, 2022. "Net Hydrogen Consumption Minimization of Fuel Cell Hybrid Trains Using a Time-Based Co-Optimization Model," Energies, MDPI, vol. 15(8), pages 1-21, April.
    11. Canca, David & Zarzo, Alejandro, 2017. "Design of energy-Efficient timetables in two-way railway rapid transit lines," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 142-161.
    12. Wang, Xuekai & Tang, Tao & Su, Shuai & Yin, Jiateng & Gao, Ziyou & Lv, Nan, 2021. "An integrated energy-efficient train operation approach based on the space-time-speed network methodology," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 150(C).
    13. Zhan, Shuguang & Wang, Pengling & Wong, S.C. & Lo, S.M., 2022. "Energy-efficient high-speed train rescheduling during a major disruption," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 157(C).
    14. Ning, Jingjie & Zhou, Yonghua & Long, Fengchu & Tao, Xin, 2018. "A synergistic energy-efficient planning approach for urban rail transit operations," Energy, Elsevier, vol. 151(C), pages 854-863.
    15. Lian, Deheng & Mo, Pengli & D’Ariano, Andrea & Gao, Ziyou & Yang, Lixing, 2024. "Energy-saving time allocation strategy with uncertain dwell times in urban rail transit: Two-stage stochastic model and nested dynamic programming framework," European Journal of Operational Research, Elsevier, vol. 317(1), pages 219-242.
    16. Wu, Fuliang & Bektaş, Tolga & Dong, Ming & Ye, Hongbo & Zhang, Dali, 2021. "Optimal driving for vehicle fuel economy under traffic speed uncertainty," Transportation Research Part B: Methodological, Elsevier, vol. 154(C), pages 175-206.
    17. Yang, Songpo & Liao, Feixiong & Wu, Jianjun & Timmermans, Harry J.P. & Sun, Huijun & Gao, Ziyou, 2020. "A bi-objective timetable optimization model incorporating energy allocation and passenger assignment in an energy-regenerative metro system," Transportation Research Part B: Methodological, Elsevier, vol. 133(C), pages 85-113.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cheng Gong & Shiwen Zhang & Feng Zhang & Jianguo Jiang & Xinheng Wang, 2014. "An Integrated Energy-Efficient Operation Methodology for Metro Systems Based on a Real Case of Shanghai Metro Line One," Energies, MDPI, vol. 7(11), pages 1-25, November.
    2. Luan, Xiaojie & Wang, Yihui & De Schutter, Bart & Meng, Lingyun & Lodewijks, Gabriel & Corman, Francesco, 2018. "Integration of real-time traffic management and train control for rail networks - Part 2: Extensions towards energy-efficient train operations," Transportation Research Part B: Methodological, Elsevier, vol. 115(C), pages 72-94.
    3. Wang, Xuekai & Tang, Tao & Su, Shuai & Yin, Jiateng & Gao, Ziyou & Lv, Nan, 2021. "An integrated energy-efficient train operation approach based on the space-time-speed network methodology," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 150(C).
    4. Pier Giuseppe Sessa & Valerio Martinis & Axel Bomhauer-Beins & Ulrich Alois Weidmann & Francesco Corman, 2021. "A hybrid stochastic approach for offline train trajectory reconstruction," Public Transport, Springer, vol. 13(3), pages 675-698, October.
    5. Mariano Gallo & Marilisa Botte & Antonio Ruggiero & Luca D’Acierno, 2020. "A Simulation Approach for Optimising Energy-Efficient Driving Speed Profiles in Metro Lines," Energies, MDPI, vol. 13(22), pages 1-17, November.
    6. Albrecht, Amie & Howlett, Phil & Pudney, Peter & Vu, Xuan & Zhou, Peng, 2018. "The two-train separation problem on non-level track—driving strategies that minimize total required tractive energy subject to prescribed section clearance times," Transportation Research Part B: Methodological, Elsevier, vol. 111(C), pages 135-167.
    7. Jie Yang & Limin Jia & Shaofeng Lu & Yunxiao Fu & Ji Ge, 2016. "Energy-Efficient Speed Profile Approximation: An Optimal Switching Region-Based Approach with Adaptive Resolution," Energies, MDPI, vol. 9(10), pages 1-27, September.
    8. Yang, Xin & Chen, Anthony & Ning, Bin & Tang, Tao, 2016. "A stochastic model for the integrated optimization on metro timetable and speed profile with uncertain train mass," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 424-445.
    9. De Martinis, Valerio & Weidmann, Ulrich A., 2015. "Definition of energy-efficient speed profiles within rail traffic by means of supply design models," Research in Transportation Economics, Elsevier, vol. 54(C), pages 41-50.
    10. Yang, Lixing & Li, Keping & Gao, Ziyou & Li, Xiang, 2012. "Optimizing trains movement on a railway network," Omega, Elsevier, vol. 40(5), pages 619-633.
    11. Kang, Liujiang & Sun, Huijun & Wu, Jianjun & Gao, Ziyou, 2020. "Last train station-skipping, transfer-accessible and energy-efficient scheduling in subway networks," Energy, Elsevier, vol. 206(C).
    12. Howlett, Phil, 2016. "A new look at the rate of change of energy consumption with respect to journey time on an optimal train journey," Transportation Research Part B: Methodological, Elsevier, vol. 94(C), pages 387-408.
    13. Albrecht, Amie & Howlett, Phil & Pudney, Peter & Vu, Xuan & Zhou, Peng, 2016. "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, Elsevier, vol. 94(C), pages 482-508.
    14. Canca, David & Zarzo, Alejandro, 2017. "Design of energy-Efficient timetables in two-way railway rapid transit lines," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 142-161.
    15. Luijt, Ralph S. & van den Berge, Maarten P.F. & Willeboordse, Helen Y. & Hoogenraad, Jan H., 2017. "5years of Dutch eco-driving: Managing behavioural change," Transportation Research Part A: Policy and Practice, Elsevier, vol. 98(C), pages 46-63.
    16. Li, Xiang & Lo, Hong K., 2014. "An energy-efficient scheduling and speed control approach for metro rail operations," Transportation Research Part B: Methodological, Elsevier, vol. 64(C), pages 73-89.
    17. Ye, Hongbo & Liu, Ronghui, 2016. "A multiphase optimal control method for multi-train control and scheduling on railway lines," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 377-393.
    18. Gupta, Shuvomoy Das & Tobin, J. Kevin & Pavel, Lacra, 2016. "A two-step linear programming model for energy-efficient timetables in metro railway networks," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 57-74.
    19. Wang, Chao & Meng, Xin & Guo, Mingxue & Li, Hao & Hou, Zhiqiang, 2022. "An integrated energy-efficient and transfer-accessible model for the last train timetabling problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    20. Xuesong Feng & Hanxiao Zhang & Yong Ding & Zhili Liu & Hongqin Peng & Bin Xu, 2013. "A Review Study on Traction Energy Saving of Rail Transport," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-9, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:transb:v:94:y:2016:i:c:p:509-538. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/548/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.