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

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%.