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

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.