Approximate Pontryagin’s minimum principle applied to the energy management of plug-in hybrid electric vehicles

This paper proposes an optimal energy management strategy based on the approximate Pontryagin’s Minimum Principle (A-PMP) algorithm for parallel plug-in hybrid electric vehicles (HEVs). When the driving cycles are known in advance, the Pontryagin’s Minimum Principle (PMP) can help to achieve the best fuel economy, but real-time control has been unavailable due to the massive computational load required by instantaneous Hamiltonian optimization. After observing some regular patterns in numeric PMP results, we were inspired to apply a novel piecewise linear approximation strategy by specifying the turning point of the engine fuel rate for the Hamiltonian optimization. As a result, the instantaneous Hamiltonian optimization becomes convex. Considering the engine state, there are only five candidate solutions for the optimization. For the engine off state, only one of the available torque split ratios (TSR) is one of these five candidates. The other four TSR candidates are for the engine on state, including the TSR when the engine operates at the best efficiency point for the current speed, the TSR when the engine delivers all the required torque and two terminal TSRs. The optimal TSR is the one with the smallest Hamiltonian of the current engine state. The engine state with the smallest Hamiltonian will be requested for the next time step. The results show that the A-PMP strategy reduced fuel consumption by 6.96% compared with the conventional “All-Electric, Charge-Sustaining” (AE–CS) strategy. In addition, the A-PMP shortened the simulation time from 6 h to only 4 min, when compared with the numeric PMP method. Unlike other approximation methods, the proposed novel piecewise linear approximation caused no severe distortion to the engine map model. The engine state switching frequency is also reduced by 43.40% via both the filter and the corresponding engine on/off optimal control strategy.