The key principles of optimal train control—Part 1: Formulation of the model,strategies of optimal type,evolutionary lines,location of optimal switching points

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. We review the state-of-the-art and establish the key principles of optimal train control for a general model with continuous control. The model with discrete control is not considered. We assume only that the tractive and braking control forces are bounded by non-increasing speed-dependent magnitude constraints and that the rate of energy dissipation from frictional resistance is given by a non-negative strictly convex function of speed. Partial cost recovery from regenerative braking is allowed. The cost of the strategy is the mechanical energy required to drive the train. Minimising the mechanical energy is an effective way of reducing the fuel or electrical energy used by the traction system. The paper is presented in two parts. In Part 1 we discuss formulation of the model, determine the characteristic optimal control modes, study allowable control transitions, establish the existence of optimal switching points and consider optimal strategies with speed limits. We find algebraic formulae for the adjoint variables in terms of speed on track with piecewise-constant gradient and draw phase plots of the associated optimal evolutionary lines for the state and adjoint variables. In Part 2 we will establish important integral forms of the necessary conditions for optimal switching, find general bounds on the positions of the optimal switching points, justify the local energy minimization principle and show how these ideas are used to calculate optimal switching points. We will prove that an optimal strategy always exists and use a perturbation analysis to show the strategy is unique. Finally we will discuss computational techniques in realistic examples with steep gradients and describe typical optimal strategies for a complete journey.