Chebyshev–Legendre Collocation for HEV Energy Management: Regularity, Convergence, and Benchmarks

This paper presents a Chebyshev–Legendre (CL) collocation method for the optimal energy management of hybrid electric vehicles (HEVs). The method combines spectral collocation with a regularity analysis. We prove that the collocation error decays algebraically with an asymptotic rate of p=3/2 when at least one kink-type switching time is present. We also prove spectral convergence when no active control constraint is present. We first verify the method using a scalar linear-quadratic regulator (LQR) benchmark. We then apply it to a constrained HEV optimal control problem. The model includes a discontinuous fuel map, but the considered drive cycles do not activate this discontinuity. The optimal solution exhibits bang-bang behavior at the battery power bounds. We compare the CL solution with a basic equivalent consumption minimization strategy (ECMS), an adaptive ECMS (A-ECMS), and a model predictive control strategy (MPC-CL). We also compare the CL method directly with standard Chebyshev and Legendre pseudospectral methods. The results show that the CL method achieves the same accuracy as the Legendre method with reduced computational overhead. Furthermore, we validate the method’s robustness using a new low-load driving cycle that activates the fuel-map jump discontinuity. The offline CL solution and basic ECMS yield statistically indistinguishable costs, with a difference of approximately 0.01%. Basic ECMS is not a real-time strategy. Therefore, we classify it as an offline benchmark alongside CL. Among the evaluated real-time strategies, A-ECMS with Kp=1 achieves the lowest cost increase (+2.53%). MPC-CL follows with a cost increase of +7.47%. More aggressive A-ECMS gains (Kp≥5) produce cost increases between 11.7% and 13.0%. We estimate the numerical self-convergence rate of the control using a least-squares fit. The resulting rate is pL2fit=1.466. This rate agrees with the theoretical prediction p=3/2 to within 2.3%. Finally, we numerically verify each item of the structural assumption on the offline solution.