An algorithm optimizing train running profile with Bellman’s Dynamic programming (DP) is investigated in this paper. Optimal train running trajectory which minimizes amount of total consumed energy has been produced under fixed origin and destination, stipulated running time, limited electric motive force and electric brake by VVVF controlled induction motor/generator blended with mechanical brake, several local speed constraints and local inclines. Many previous works on this area adopt the numerical techniques of calculus of variations, Pontryagin’s maximum principle, incremental method, and so on. But these methods often meet some difficulties accounting for complicated actual train running preconditions, e.g. complicated functions which describe electrical motive/brake torque, local constraints of state variables as speed limits, nonlinear running resistance and variable grade profile. DP can cope with such complicated conditions. It can directly deal with such difficult constraints of an optimal control problem, except for terminal boundary condition. At DP process of a former research, position and velocity of train and total running time are divided into nonuniform lattice and the numerical algorithm solves state equations partially and calculates interpolated local valuation. The authors have made the improvements for reducing calculation time of optimization process and improving accuracy of solution. Optimal run-curve can be obtained in practically acceptable computational time even when it is applied to actual complicated running conditions. The error of distance and speed at destination is required less than 0.6m and 0.1m/sec respectively. The authors have concluded that the small error guarantees the reliability of the results.
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Ko et al. (2004) studied this question.