Optimal gear ratio planning for flywheel-based kinetic energy recovery systems in motor vehicles

An efficient computational methodology is proposed for optimal gear ratio planning in motor vehicle kinetic energy recovery systems (KERS) using a flywheel and continuously variable transmission (CVT). Initial modeling of a clutch-less KERS, comprising an input wheel, CVT, flywheel, and bearings, shows that the “least effort” or “minimum energy loss” optimal control problem can be formulated in two ways: one being a conventional two-state formulation involving input wheel angular velocity and CVT gear ratio, for which least effort control can be solved in simple cases with Pontryagin's maximum principle. The second formulation involves a single-state CVT gear ratio equation for which the input wheel angular velocity and acceleration appear as unknown time-dependent parameters. A novel multiparameter optimization methodology is proposed using the single-state formulation to find optimal CVT gear ratios by adopting two discrete time scales: one being a small time scale for numerical integration of the model, and the second involving discrete transitions, hundreds of times larger. Using Chebyshev polynomial expansions (CPEs) to initially generate sets of zero-energy-loss least effort kinematics for use as the time-dependent parameters in the CVT gear ratio equation, two solution approaches are developed. The first involves a single large discrete time transition, which only requires discretization of the input wheel angular acceleration at the start and end-of-transition. The second approach involves multiple large-scale discrete time transitions as a generalization of the first, but additionally needing discretization of the input wheel angular velocity, and the CVT gear ratio, plus dynamic programming to find the optimum. Both approaches are tested using the clutchless KERS model by assuming a “super CVT” gear ratio range (but with no restrictions for use with slipping clutches). Comparison with least effort control via Pontryagin's maximum principle shows that the single transition approach is in practice far superior. The single transition approach is then used to compare a minimum energy loss clutchless KERS gear ratio plan, with one obtained using constant input wheel angular acceleration as a benchmark. This comparison, involving power losses throughout the KERS, shows the very clear benefits of adopting an optimal gear ratio plan.