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The optimal operation of railway systems minimizing total energy consumption is discussed in this paper. Firstly, some measures of finding energy-saving train speed profiles are outlined. After the characteristics that should be considered in optimizing train operation are clarified, complete optimization based on optimal control theory is reviewed. Their basic formulations are summarized taking into account most of the difficult characteristics peculiar to railway systems. Three methods of solving the formulation, dynamic programming (DP), gradient method, and sequential quadratic programming (SQP), are introduced. The last two methods can also control the state of charge (SOC) of the energy storage devices. By showing some numerical results of simulations, the significance of solving not only optimal speed profiles but also optimal SOC profiles of energy storage are emphasized, because the numerical results are beyond the conventional qualitative studies. Future scope for applying the methods to real-time optimal control is also mentioned. 

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An algorithm optimizing train running profile with Bellman's Dynamic Programming (DP) is investigated in this paper. Optimal running trajectory of a train which minimizes amount of total energy consumption has been produced under fixed origin and destination, stipulated running time and various track profile. Many previous works on this area adopt the numerical techniques of calculus of variations, Pontryagin's maximum principle, 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 the state variable as speed limitations, nonlinear running resistance and variable grade profiles. Basic numerical DP algorithm can cope with such comlicated conditions and give the globally optimal solution. But this method consumes too large computation time for practical uses. We have made the improvements for shorter calculation time of whole optimization process and reducing the numerical error. The confined state space and irregular lattice play most important role for them. Dynamic meshing and effective utilization of system memory also realize shorter computation time. The effectiveness of the proposed method is demonstrated using various complicated running conditions.