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.

References

1.
Pfiffner
,
R.
,
Guzzella
,
L.
, and
Onder
,
C. H.
,
2003
, “
Fuel-Optimal Control of CVT Powertrains
,”
Control Eng. Pract.
,
11
(
3
), pp.
329
336
.10.1016/S0967-0661(02)00219-8
2.
Setlur
,
P.
,
Wagner
,
J. R.
,
Dawson
,
D. M.
, and
Samuels
,
B.
,
2003
, “
Nonlinear Control of a Continuously Variable Transmission (CVT)
,”
IEEE Trans. Control Syst. Technol.
,
11
(
1
), pp.
101
108
.10.1109/TCST.2002.806434
3.
Shen
,
S.
, and
Veldpaus
,
F. E.
,
2004
, “
Analysis and Control of a Flywheel Hybrid Vehicular Powertrain
,”
IEEE Trans. Control Syst. Technol.
,
12
(
5
), pp.
645
660
.10.1109/TCST.2004.824792
4.
Liu
,
J.
,
Zhou
,
Y.
,
Cai
,
Y.
, and
Su
,
J.
,
2007
, “
The Application of Generalized Predictive Control in CVT Speed Ratio Control
,”
IEEE
International Conference on Automation and Logistics,
Jinan, China
, Aug. 18–21, pp.
649
654
.10.1109/ICAL.2007.4338644
5.
Youmin
,
W.
, and
Penghuang
,
C.
,
2009
, “
The Optimal Test PID Control for CVT Control System
,”
IEEE International Conference on Intelligence and Intelligent Systems
,
Shanghai, China
, Nov. 20–22, pp.
1
5
.
6.
Tanaka
,
H.
,
2002
, “
Speed Ratio Control of a Parallel Layout Double Cavity Half-Toroidal CVT for Four-Wheel Drive
,”
JSAE Rev.
,
23
(
2
), pp.
213
217
.10.1016/S0389-4304(02)00163-7
7.
Adachi
,
K.
,
Ochi
,
Y.
, and
Kanai
,
K.
,
2006
, “
Development of CVT Control System and Its Use for Fuel-Efficient Operation of Engine
,”
Asian J. Control
,
8
(
3
), pp.
219
226
.10.1111/j.1934-6093.2006.tb00273.x
8.
Delkhosh
,
M.
, and
Foumani
,
M. S.
,
2013
, “
Optimisation of Full-Toroidal Continuously Variable Transmission in Conjunction With Fixed Ratio Mechanism Using Particle Swarm Optimisation
,”
J. Veh. Syst. Dyn.
,
51
(
5
), pp.
671
683
.10.1080/00423114.2013.765588
9.
Druzhinina
,
M.
,
Stefanopoulou
,
A. G.
, and
Moklegaard
,
L.
,
2002
, “
Speed Gradient Approach to Longitudinal Control of Heavy-Duty Vehicles Equipped With Variable Compression Brake
,”
IEEE Trans. Control Syst. Technol.
,
10
(
2
), pp.
209
220
.10.1109/87.987066
10.
Wu
,
B.
,
Lin
,
C.-C.
,
Filipi
,
Z.
, and
Peng
,
H.
,
2004
, “
Optimal Power Management for a Hydraulic Hybrid Delivery Truck
,”
Veh. Syst. Dyn.
,
42
(
1-2
), pp.
23
40
.10.1080/00423110412331291562
11.
Yeo
,
H.
,
Hwang
,
S.
, and
Kim
,
H.
,
2006
, “
Regenerative Braking Algorithm for a Hybrid Electric Vehicle With CVT Ratio Control
,”
Proc. Inst. Mech. Eng. Part D
,
220
(
11
), pp.
1589
1600
.10.1243/09544070JAUTO304
12.
Mukhitdinov
,
A. A.
,
Ruzimov
,
S. K.
, and
Eshkabilov
,
S. L.
,
2006
, “
Optimal Control Strategies for CVT of the HEV During Regenerative Process
,”
IEEE
Conference on Electric and Hybrid vehicles (ICEHV), pp.
1
12
.10.1109/ICEHV.2006.352278
13.
Carbone
,
G.
,
Mangialardi
,
L.
,
Bonsen
,
B.
,
Tursi
,
C.
, and
Veenhuizen
,
P. A.
,
2007
, “
CVT Dynamics: Theory and Experiments
,”
Mech. Mach. Theory
,
42
(
4
), pp.
409
428
.10.1016/j.mechmachtheory.2006.04.012
14.
Srivastava
,
N.
, and
Haque
,
I.
,
2009
, “
A Review on Belt and Chain Continuously Variable Transmissions (CVT): Dynamics and Control
,”
Mech. Mach. Theory
,
44
(
1
), pp.
19
41
.10.1016/j.mechmachtheory.2008.06.007
15.
Bayindir
,
K. C.
,
Gözüküçük
,
M. A.
, and
Teke
,
A.
,
2011
, “
A Comprehensive Overview of Hybrid Electric Vehicle: Powertrain Configurations, Powertrain Control Techniques and Electronic Control Units
,”
Energy Convers. Manage.
,
52
(
2
), pp.
1305
1313
.10.1016/j.enconman.2010.09.028
16.
Tie
,
S. F.
, and
Tan
,
C. W.
,
2013
, “
A Review of Energy Sources and Energy Management System in Electric Vehicles
,”
Renewable Sustainable Energy Rev.
,
20
, pp.
82
102
.10.1016/j.rser.2012.11.077
17.
Cross
,
D.
, and
Brockbank
,
C.
,
2009
, “
Mechanical Hybrid System Comprising a Flywheel and CVT for Motorsport and Mainstream Automotive Applications
,”
SAE
Technical Paper No. 2009-01-1312.10.4271/2009-01-1312
18.
Hua
,
L.
,
Jian
,
Z.
,
Da
,
X.
, and
Xiaojun
,
M.
,
2009
, “
Design for Hybrid Electric Drive System of Armoured Vehicle With Two Energy Storage Devices
,”
International Conference on Sustainable Power Generation and Supply
, Vol.
1–4
, pp.
1747
1750
.
19.
Bischof
,
G.
,
Reisinger
,
K.
,
Singraber
,
T.
, and
Summer
,
A.
,
2014
, “
Investigation of a Passenger Car's Dynamic Response Due to a Flywheel-Based Kinetic Energy Recovery System
,”
Veh. Syst. Dyn.
,
52
(
2
), pp.
201
217
.10.1080/00423114.2013.869609
20.
Boretti
,
A.
,
2010
, “
Comparison of Fuel Economies of High Efficiency Diesel and Hydrogen Engines Powering a Compact Car With a Flywheel Based Kinetic Energy Recovery Systems
,”
Int. J. Hydrogen Energy
,
35
(
16
), pp.
8417
8424
.10.1016/j.ijhydene.2010.05.031
21.
Brogan
,
W.
,
1991
,
Modern Control Theory
, 3rd ed.,
Prentice Hall
, Upper Saddle River, NJ.
22.
Pérez
,
L. V.
,
Bossio
,
G. R.
,
Moitre
,
D.
, and
García
,
G. O.
,
2006
, “
Optimization of Power Management in an Hybrid Electric Vehicle Using Dynamic Programming
,”
Math. Comput. Simul.
,
73
(
1–4
), pp.
244
254
.10.1016/j.matcom.2006.06.016
23.
Johannesson
,
L.
,
Pettersson
,
S.
, and
Egardt
,
B.
,
2009
, “
Predictive Energy Management of a 4QT Series–Parallel Hybrid Electric Bus
,”
Control Eng. Pract.
,
17
(
12
), pp.
1440
1453
.10.1016/j.conengprac.2009.07.004
24.
Song
,
X.
,
Zulkefli
,
M.
,
Sun
,
Z.
, and
Miao
,
H.
,
2011
, “
Automotive Transmission Clutch Fill Control Using a Customized Dynamic Programming Method
,”
ASME J. Dyn. Syst., Meas. Control
,
133
(
5
), p.
054503
.10.1115/1.4003797
25.
van Berkel
,
K.
,
Hofman
,
T.
,
Vroemen
,
B.
, and
Steinbuch
,
M.
,
2012
, “
Optimal Control of a Mechanical Hybrid Powertrain
,”
IEEE Trans. Veh. Technol.
,
61
(
2
), pp.
485
497
.10.1109/TVT.2011.2178869
26.
Pérez
,
L.
, and
Pilotta
,
E. A.
,
2009
, “
Optimal Power Split in a Hybrid Electric Vehicle Using Direct Transcription of an Optimal Control Problem
,”
Math. Comput. Simul.
,
79
(
6
), pp.
1959
1970
.10.1016/j.matcom.2007.03.006
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