Discrete Mixed-Integer Shooting (DMIS): Algorithm and Application to Plug-In Hybrid Electric Vehicle Energy Management Accounting for Fuel Cranking and Actual Powertrain Efficiency Maps

This article presents an algorithm for mixed-integer nonlinear optimal control problems (OCPs) formulated directly in discrete time. In our current application, we address the energy management of a plug-in hybrid electric vehicle (PHEV) seeking to achieve minimum trip fuel consumption under a terminal state-of-charge (SOC) equality constraint while avoiding powertrain efficiency map approximations in order to maximize its fuel economy (FE). An additional engine cranking state with a nonsmooth state and control transition cost is then incorporated to prevent undesired engine on/off behaviors. We pursue a single shooting-based numerical algorithm, motivated by the success of the computationally efficient Pontryagin’s Minimum Principle (PMP)-based single shooting in achieving similar FE results as dynamic programming (DP) for the power-split optimization problem without the additional engine cranking state shown in prior work. The Discrete Mixed-Integer Shooting (DMIS) algorithm, with costate backward-in-time propagation, is presented here, avoiding unstable shooting iterations that are typically encountered in PMP-based single shooting. When applying DMIS to solve the power-split problem with cranking fuel consideration, we show through simulation that it leads to better FE results than the PMP-based single shooting without the cranking state, thus demonstrating the effectiveness of the DMIS algorithm. Finally, experimental dynamometer test results with the model predictive powertrain controller solved by DMIS for minimum trip fuel consumption are presented and discussed, demonstrating its real-time execution and the cranking fuel reduction capability.