Cubic spline approximations of the Dynamic Programming cost-to-go in HEV energy management problems

The energy management problem of a hybrid electric vehicle (HEV) is a non-linear and mixed integer optimization problem. The problem can be solved with Dynamic Programming (DP), but the algorithm requires the problem to be gridded in time, states and control signals. To ensure a high accuracy of the solution the grid must be dense, meaning that the cost-to-go can require several megabytes of memory. The scope of this paper is therefore twofold. The first topic is a sensitivity study, where the effect of a sparsely gridded state is investigated, both for an HEV and a plug-in HEV (PHEV). The study shows that it is possible to use a sparse grid for an HEV, but not for a PHEV. The second topic and the main contribution is a method to approximate the DP cost-to-go with cubic splines. The results indicate that it is possible to use only a few splines, if the knot points are determined based on the characteristics of the cost-to-go. Thereby it is possible to significantly reduce the memory requirements, without any noticeable increase in simulated fuel consumption.