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A new look at the rate of change of energy consumption with respect to journey time on an optimal train journey

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  • Howlett, Phil

Abstract

We present a new derivation of a key formula for the rate of change of energy consumption with respect to journey time on an optimal train journey. We use a standard mathematical model (Albrecht et al., 2015b; Howlett, 2000; Howlett et al., 2009; Khmelnitsky, 2000; Liu and Golovitcher, 2003) to define the problem and show by explicit calculation of switching points that the formula also applies for all basic control subsequences within the optimal strategy on appropriately chosen fixed track segments. The rate of change was initially derived as a known strictly decreasing function of the optimal driving speed in a text edited by Isayev (1987, Section 14.2, pp 259–260) using an empirical resistance function. An elegant derivation by Liu and Golovitcher (2003, Section 3) with a general resistance function required an underlying assumption that the optimal strategy is unique and that the associated optimal driving speed is a strictly decreasing and continuous function of journey time. An earlier proof of uniqueness (Khmelnitsky, 2000) showed that the optimal driving speed decreases when journey time increases. A subsequent constructive proof (Albrecht et al., 2013a, 2015c) used a local energy minimization principle to find optimal switching points and show explicitly that the optimal driving speed is a strictly decreasing and continuous function of journey time. Our new derivation of the key formula also uses the local energy minimization principle and depends on the following observations. If no speed limits are imposed the optimal strategy consists of a finite sequence of phases with only five permissible control modes. By considering all basic control subsequences and subdividing the track into suitably chosen fixed segments we show that the key formula is valid on each individual segment. The formula is extended to the entire journey by summation. The veracity of the formula is demonstrated with an elementary but realistic example.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:transb:v:94:y:2016:i:c:p:387-408
    DOI: 10.1016/j.trb.2016.10.004
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    References listed on IDEAS

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    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.
    3. 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.
    4. Li, Xiang & Lo, Hong K., 2014. "Energy minimization in dynamic train scheduling and control for metro rail operations," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 269-284.
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    Cited by:

    1. Canca, David & Andrade-Pineda, José Luis & De los Santos, Alicia & Calle, Marcos, 2018. "The Railway Rapid Transit frequency setting problem with speed-dependent operation costs," Transportation Research Part B: Methodological, Elsevier, vol. 117(PA), pages 494-519.
    2. 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.
    3. Yang, Songpo & Chen, Yanyan & Dong, Zhurong & Wu, Jianjun, 2023. "A collaborative operation mode of energy storage system and train operation system in power supply network," Energy, Elsevier, vol. 276(C).
    4. 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.
    5. 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.
    6. 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.

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