EXAMPLE OF APPLICATION OF THE PONTRYAGIN’S MINIMUM PRINCIPLE “PMP EXTENSION”: ZERMELO PROBLEM (WITH CURRENT SPEED MORE THAN BOAT SPEED HYPOTHESIS)

We present an example of application covering several cases using the extension of the Pontryaguine minimum principle (PMP) in the case where we add a constraint on reaching a target variety at the final time: the Zermelo problem with current speed more than Boat speed hypothesis, where we consider a boat crossing a channel under a strong current and where we try to reach the opposite bank by minimizing the lateral offset or by minimizing the crossing time.

Online optimization of Pontryagin’s minimum principle for a series hydraulic hybrid wheel loader

The hydraulic hybrid powertrain has great potential for reducing fuel consumption and emission of off-road vehicles. The energy management strategy is the key to hybrid powertrain and currently there are many well-developed strategies. Of which the Pontryagin’s minimum principle is of research interest since it is a global optimization method while less computational burden than dynamic programming. However, it requires full cycle information to calculate co-state value in the principle, making it not implementable. Therefore in this study an implementable Pontryagin’s minimum principle is proposed for a series hybrid wheel loader, where the optimal co-state value in the principle is trained through repetitive wheel loader duty cycle. The Pontryagin’s minimum principle formulations of hybrid wheel loader are developed. The online co-state training algorithm is presented. A dynamic simulation model of hybrid wheel loader is developed. The fuel consumption of hybrid wheel loader with proposed strategy is compared with dynamic programming strategy and rule-based strategy in wheel loader long and short loading cycles. Results show the fuel consumption with proposed strategy is close to dynamic programming result and is lower than rule-based strategy. Finally, the influence of pressure level of hybrid powertrain on vehicle fuel consumption is studied.

Adaptive energy control strategy of PHEV based on the Pontryagin’s minimum principle algorithm

The optimization of energy control strategy is one of the key technologies of plug-in hybrid vehicles (PHEVs) to improve the capabilities of energy saving and emission reduction. In order to improve fuel economy of PHEV, adaptive equivalent minimum fuel consumption strategy (A-ECMS) is proposed. Firstly, optimization methods of different energy control strategies are analyzed, and the Pontryagin’s Minimum Principle (PMP) and the equivalent fuel consumption theory are selected to optimize energy control strategy of the PHEV. Secondly, the configuration of PHEV and research objectives of the power control system are determined. Thirdly, the energy control problem is analyzed by the PMP theory, and the improvement measures for the energy control problem are put forward by the equivalent minimum fuel consumption strategy (ECMS). Fourthly, after analyzing the relationship between the equivalent factor and reference SOC, adaptive equivalent minimum fuel consumption strategy (A-ECMS) model is established by MATLAB/Simulink. Finally, combined with Cruise software, the PHEV simulation model is simulated, and the simulation results are analyzed. The results show that compared with the CD/CS energy control strategy, the A-ECMS energy control strategy reduced the 100 km fuel consumption of the vehicle by 7% under three times WLTC driving condition.

Computationally Efficient Energy Management in Hybrid Electric Vehicles Based on Approximate Pontryagin’s Minimum Principle

This article presents an energy management method for a parallel hybrid electric vehicle (HEV) based on approximate Pontryagin’s Minimum Principle (A-PMP). The A-PMP optimizes gearshift commands and torque distribution for overall energy efficiency. As a practical numerical solution in PMP, the proposed methodology utilizes a piecewise linear approximation of the engine fuel rate and state of charge (SOC) derivative by considering drivability and fuel economy simultaneously. Moreover, battery aging is explicitly studied by introducing a control-oriented model, which aims to investigate the effect of battery aging on the optimization performance in the development of the HEVs. An approximate energy management strategy with piecewise linear models is then formulated by the A-PMP, which targets a better performance for the Hamiltonian optimization. The gearshift map is extracted from the optimal results in the standard PMP to hinder frequent gearshift by considering both drivability and fuel economy. Utilizing an approximated Hamilton function, the torque distribution, gearshift command, and the battery aging degradation are jointly optimized under a unified framework. Simulations are performed for dynamic programming (DP), PMP, and A-PMP to validate the effectiveness of the proposed approach. The results indicate that the proposed methodology achieves a close fuel economy compared with the DP-based optimal solution. Moreover, it improves the computation efficiency by 50% and energy saving by 3.5%, compared with the PMP, while ensuring good drivability and fuel efficiency.