THE COST–TIME CURVE FOR AN OPTIMAL TRAIN JOURNEY ON LEVEL TRACK

ALBRECHT, AMIE; HOWLETT, PHIL; PUDNEY, PETER · 2016 · Crossref

DOI: 10.1017/s1446181116000092

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Summary

This paper establishes the mathematical properties of the cost–time curve for an optimal train journey on level track, demonstrating that the cost function is strictly decreasing and strictly convex with respect to journey time. The research is motivated by the need to design energy-efficient timetables for busy metropolitan rail networks. Specifically, optimizing such networks is often treated as a two-stage process: first, determining optimal transit times for individual journey segments to minimize total energy consumption, and second, synchronizing arrival and departure times to maximize energy recovery from regenerative braking. The precise structure of the cost–time curve is a critical component for the first stage of this optimization, as well as for solving train separation problems where multiple trains share track sections. The authors utilize a point-mass train model where position is the independent variable, and time and speed are dependent state variables. The study focuses on level track, assuming the train starts and finishes at rest. The cost is defined as the net mechanical energy usage per unit mass, accounting for a proportion $\rho$ of energy recovered during regenerative braking. The analysis relies on established optimal train control theory, which identifies four possible control modes: maximum acceleration, speedhold with partial acceleration, coast, and maximum brake. The authors categorize optimal strategies into three cases based on the relationship between the termination speed of the initial acceleration phase and the optimal driving speed $V_\mu$, as well as the level of energy recovery. They derive specific formulas for journey time and distance for each case, utilizing a modified adjoint variable to determine switching points between control phases. The main finding is that for any fixed journey distance, the cost $J$ of the optimal strategy is a strictly decreasing and strictly convex function of the journey time $T$. This relationship is characterized by the formula $dJ/dT = -\psi(V_\mu) < 0$, where $V_\mu$ is the optimal driving speed and $\psi(v) = v^2 r_0(v)$ is a nonnegative, strictly increasing function dependent on resistive acceleration. The paper proves this property for all three identified forms of optimal strategy. The strict convexity implies that there are diminishing returns to increasing journey time for energy savings, while the strictly decreasing nature confirms that longer journeys always consume less energy. These results provide a rigorous theoretical foundation for timetable optimization algorithms. By confirming the convexity and monotonicity of the cost–time curve, the paper supports the use of these curves in optimization models that seek to balance energy efficiency with acceptable journey times. This contributes to solving complex problems in rail operations, such as synchronizing trains to maximize regenerative braking energy transfer and ensuring safe separation between trains on shared tracks. The findings refine previous approaches that relied on linear approximations or empirical data, offering a precise mathematical tool for designing energy-efficient rail networks.

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