Rare event chance-constrained optimal control using polynomial chaos and subset simulation

This study develops a chance-constrained open-loop optimal control (CC-OC) framework capable of handling rare event probabilities. Therefore, the framework uses the generalized polynomial chaos (gPC) method to calculate the probability of fulfilling rare event constraints under uncertainties. Here, the resulting chance constraint (CC) evaluation is based on the efficient sampling provided by the gPC expansion. The subset simulation (SubSim) method is used to estimate the actual probability of the rare event. Additionally, the discontinuous CC is approximated by a differentiable function that is iteratively sharpened using a homotopy strategy. Furthermore, the SubSim problem is also iteratively adapted using another homotopy strategy to improve the convergence of the Newton-type optimization algorithm. The applicability of the framework is shown in case studies regarding battery charging and discharging. The results show that the proposed method is indeed capable of incorporating very general CCs within an open-loop optimal control problem (OCP) at a low computational cost to calculate optimal results with rare failure probability CCs.