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Research Papers

Optimal Gear Ratio Planning for Flywheel-Based Kinetic Energy Recovery Systems in Motor Vehicles

[+] Author and Article Information
J. F. Dunne

School of Engineering and Informatics,
The University of Sussex,
Falmer, Brighton BN1 9QT, UK
e-mail: j.f.dunne@sussex.ac.uk

L. A. Ponce Cuspinera

School of Engineering and Informatics,
The University of Sussex,
Falmer, Brighton BN1 9QT, UK
e-mail: L.Ponce-Cuspinera@sussex.ac.uk

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 1, 2014; final manuscript received February 17, 2015; published online March 26, 2015. Assoc. Editor: Junmin Wang.

J. Dyn. Sys., Meas., Control 137(7), 071012 (Jul 01, 2015) (13 pages) Paper No: DS-14-1357; doi: 10.1115/1.4029929 History: Received September 01, 2014; Revised February 17, 2015; Online March 26, 2015

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.

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Figures

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Fig. 1

(a) Representation of the KERS with a super CVT, step-up gear box, and bearings. (b) Schematic diagram of the KERS, CVT, step-up gear box, and bearings.

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Fig. 2

Optimal least effort input wheel speed ω∧w for the frictionless KERS via the single transition approach (“•”) and via Pontryagin's maximum principle (solid line)

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Fig. 3

Optimal least effort CVT gear ratio G for the frictionless KERS via the single transition approach (•) and via Pontryagin's maximum principle (solid line)

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Fig. 4

Optimal least effort flywheel speed for the frictionless KERS via the single transition approach (•) and via Pontryagin's maximum principle (solid line)

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Fig. 5

Optimal least effort CVT gear ratio G for the frictionless KERS via the multiple transition approach: first pass: discrete time (•); second pass: continuous-time (•); Pontryagin's maximum principle (solid line). Dynamic programming uses a mesh of 6 × 6 discrete kinematics and six discrete gear ratios corresponding to each of the four transitions each of 2.5 s intervals.

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Fig. 6

Optimal least effort CVT gear ratio G for the frictionless KERS via the multiple transition approach (•) and via Pontryagin's maximum principle (solid line) but where dynamic programming uses exact discrete kinematics and exact discrete gear ratios corresponding to each of the four transitions each of 2.5 s duration

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Fig. 7

Optimal least effort CVT gear ratio G for the KERS with bearing friction but no CVT power losses via the single transition approach (•) and via Pontryagin's maximum principle (solid line)

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Fig. 8

Optimal least effort flywheel speed for the KERS with bearing friction but no CVT power losses via the single transition approach (•) and via Pontryagin's maximum principle (solid line)

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Fig. 9

Empirical CVT efficiency model ηcvt as a function of input torque Tw with input wheel speed ωw as a parameter, for minimum input wheel speed ωwmn = 40 rad/s and the maximum input wheel speed ωwmx = 400 rad/s

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Fig. 10

Optimal least effort CVT gear ratio G for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and CVT gear ratio assuming constant acceleration input wheel velocity (solid line)

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Fig. 11

Optimal least effort flywheel speed for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and flywheel speed assuming constant acceleration input wheel velocity (solid line)

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Fig. 12

Optimal least effort cost J for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and cost J assuming constant acceleration input wheel velocity (solid line)

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Fig. 13

Optimal minimum energy loss CVT gear ratio G for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and CVT gear ratio assuming constant acceleration input wheel velocity (solid line)

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Fig. 14

Optimal minimum energy loss flywheel speed for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and flywheel speed assuming constant acceleration input wheel velocity (solid line)

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Fig. 15

Optimal minimum energy loss cost J for the KERS with bearing friction, plus CVT, and step-up gear box power losses via the single transition approach (•); and cost J assuming constant acceleration input wheel velocity (sold line)

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