Long-term stochastic model predictive control for the energy management of hybrid electric vehicles using Pontryagin's minimum principle and scenario-based optimization

This paper presents a new approach to efficiently integrate long prediction horizons subject to uncertainty into a stochastic model predictive control (MPC) framework for the energy management of hybrid electric vehicles. By exploiting Pontryagin's minimum principle, we show that the energy supply required to obtain a certain change in the state of charge (SOC) of the battery can be approximated using a quadratic equation. The parameters of these mappings depend on the power request imposed by the driving mission and thus allow to compress the time-resolved power profile into only three scalar variables. Having a driving mission divided into several segments of arbitrary length, the corresponding sequence of quadratic approximations allows to reformulate the original energy management problem as a quadratic program, which offers an efficient way to include a large number of future scenarios. The resulting scenario-based stochastic MPC approach prevents SOC boundary violations with a certain probability, which can be controlled by the number of scenarios considered. To validate the quadratic approximation, we study two numerical examples using two different vehicles, a series hybrid electric passenger car and a battery-assisted trolley bus. Finally, a case study based on the operation of the latter is provided, which demonstrates the expected behavior and the real-time capability of the stochastic MPC approach. While the SOC is maintained in predefined boundaries with high probability, the required energy supply is only increased by 1.41% compared to the non-causal optimum.