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

- This paper provides a new derivation of a key formula for the rate of energy consumption cost with respect to journey time for an optimal train driving strategy on steep tracks with no speed limits.
- The new derivation shows that as the overall journey time changes then on all suitably chosen track segments the rate of change of segmental energy consumption with respect to segment traversal time must be equal to the rate of change of overall energy consumption with respect to overall journey time.
- The mathematical analysis shows that determination of optimal switching points for traversal of steep sections of track using the local energy minimization principle is closely related to the overall rate of change of energy consumption with respect to journey time.

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.