Design of CVT-Based Hybrid Passenger Cars IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009 Design of CVT-Based Hybrid Passenger Cars Theo Hofman, Maarten Steinbuch,

572 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009

Design of CVT-Based Hybrid Passenger CarsTheo Hofman, Maarten Steinbuch, Roëll van Druten, and Alex F. A. Serrarens

Abstract—In this paper, the hybridization of a small passengercar equipped with a continuously variable transmission (CVT) isinvestigated. Designing a hybrid drive train is a multiobjectivedesign problem. The main design objectives are fuel consumption,emissions, and performance. However, it is difficult to find a globaloptimal integral design solution due to the interdependence ofdesign choices (parameters) regarding the drive-train topology,component sizes, component technologies, and control strategy,as well as the unknown sensitivity of the design objectives tothe design parameters. In this paper, a parametric optimizationprocedure is presented to solve the design problem, where the maindesign objective is fuel consumption. The effects of parametervariation on fuel consumption have been investigated. Further-more, a reduced hybrid drive-train model is introduced, withwhich the effects of design parameter variation is very quicklystudied with an average error of less than 1.6%.

Index Terms—Continuously variable transmission (CVT), en-ergy source, hybrid electric vehicles (HEVs), vehicle simulation.

NOMENCLATURE

List of acronyms

Symbol DescriptionADVISOR Advanced Vehicle Simulator.AMT Automated manual transmission.AT Automatic transmission.BER Brake-energy recovery.CL Wet-plate clutch.CLC Clutch closed.CLO Clutch open.CVT Continuously variable transmission.CS Control strategy.EM Electric machine.EMA Electromechanical actuation.EMS Energy management strategy.

Manuscript received September 4, 2007; revised January 28, 2008 andApril 20, 2008. First published May 28, 2008; current version publishedFebruary 17, 2009. This work was supported by The Netherlands Organisationfor Scientific Research (NWO) Technology Foundation under the InnovationalResearch Incentives Scheme 2000/2001. The review of this paper was coordi-nated by D. Diallo.

T. Hofman is with the Control Systems Technology Group, Departmentof Mechanical Engineering, Eindhoven University of Technology, 5600 MBEindhoven, The Netherlands (e-mail: [email protected]).

M. Steinbuch is with the Control Systems Technology Group, Departmentof Mechanical Engineering, Eindhoven University of Technology, 5600 MBEindhoven, The Netherlands, and also with 3TU Centre of Competence HighTech Systems, The Netherlands.

R. van Druten and A. F. A. Serrarens are with Drivetrain Innovations B.V.(DTI), 5653 LD Eindhoven, The Netherlands.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2008.926217

EON Engine on.EOFF Engine off.IMA Integrated motor assist.HDA Hydraulic actuation.JP10–15 Japanese 10–15 mode (drive cycle).M1, M2 Simulation models 1 and 2.NEDC New European Drive Cycle.NiMH Nickel metal hydride.OOL Optimal operation line.O Observation.P Primary power source (engine).PRI Primary.S Secondary power source (battery/electric

machine).SI Spark ignition.SEC Secondary.T Transmission technology.TC Torque converter.WOT Wide-open throttle.V Vehicle/drive cycle.

List of Symbols Description (Unit).

c0, c1 Fit coefficient. Characteristic param-eter (-).

ffb Brake force ratio between the front andrear wheels (-).

g Gravity (in meters per second squared).gi Inequality constraints (in watts).hi Equality constraints (in joules).i Iteration step (-).mf Fuel mass flow (in grams per second).mv Total vehicle mass (in kilograms).rCVT CVT speed ratio (-).rd Final drive ratio (-).rt Overall transmission speed ratio (-).rot Optimal overall transmission speed

ratio (-).rud, rod Under- and overdrive speed ratios (-).rw Wheel radius (in meters).t Time (in seconds).tf Final time (in seconds).vM Vehicle speed electric-only threshold

value (in meters per second).vv Vehicle speed (in meters per second).vv Time derivative of vv (in meters per

second squared).xi Design parameter (-).xo

i Optimal design parameter (-).Ef Energy content of used fuel (in joules).Es Energy storage level of battery (in joules).

0018-9545/$25.00 © 2009 IEEE

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HOFMAN et al.: DESIGN OF CVT-BASED HYBRID PASSENGER CARS 573

�Es Battery energy storage level difference(in joules).

Ev Energy vehicle demand for driving over adrive cycle (in joules).

G Matrix with inequality constraints (-).Jw Inertia of rotating parts, including wheels

(in kilogram meter squared).Pe Engine power (in watts).Pb Battery output power (in watts).Pb,min, Pb,max Minimum and maximum battery output

powers (in watts).Pem Electric machine power (in watts).Pf Fuel power (in watts).Ps Battery input power (in watts).Ps,min, Ps,max Minimum and maximum battery input

powers (in watts).Pin, Pout Input and output powers (in watts).Pmax Maximum output power. Characteristic

parameter (in watts).PBER Regenerative brake power (in watts).Pv Vehicle drive power demand (in watts).R Set of rational numbers (-).Te Engine crankshaft torque (in newton

meter).Te,min, Te,max Minimum and maximum engine output

torques (in newton meter).T o

e Optimal engine crankshaft torque (innewton meter).

Tem Electric machine output torque (in newtonmeter).

Tem.min, Tem.max Minimum and maximum electric machinetorques (in newton meter).

Tt,p Transmission input torque (in newtonmeter).

Trl Road load torque (in newton meter).ε Error iteration step (in watts).γ Convergence ratio (-).ηb Battery efficiency (-).ηCVT CVT efficiency (-).ηCVT Average CVT efficiency (-).ηe Engine efficiency (-).ηem Electric machine efficiency (-).ηfd Final drive set efficiency (-).ηs Secondary source efficiency (-).ηsys System efficiency (-).ηt Transmission efficiency (-).ωe Angular engine speed (in radians per

second).ωem Angular electric machine speed (in

radians per second).ωem.min, ωem.max Minimum and maximum angular electric

machine speeds (in radians per second).ωt,p Angular transmission input speed (in

radians per second).ωv Angular vehicle wheel speed (in radians

per second).Φf Objective function or total fuel

consumption (in grams).

I. INTRODUCTION

SALES of hybrid vehicles are becoming more significantevery year. The introduction of the Toyota Prius almost ten

years ago significantly changed the spectrum of the automotivepower train market. The Toyota Prius operates using an electricCVT principle [1], whereas the Honda Civic IMA [2] uses amechanical belt-driven CVT in parallel with an electric motor.The CVT principle in the Prius excels in driving agility and fuelconsumption. However, the CVT is relatively costly, in partic-ular, due to the dual electric machines, power electronics, andbattery pack. This is one of the reasons why this hybrid systemis now introduced in higher class vehicles (e.g., large passengercars and large off-roaders). The Honda IMA system uses matureand low-cost belt-driven CVT technology, in combination witha much smaller electric machine. This saves cost; however,overall performance is penalized due to the limited electricalassist, in particular if the relatively small electric machine iscombined with a down-sized engine.

There are three main functional differences between a hybridvehicle equipped with a CVT and a transmission with a discreteset of gear ratios (e.g., an AMT or an AT).

1) The engine or the electric machine (i.e., only if the elec-tric machine is precoupled to the CVT) can be operatedat the optimal operation points.

2) The electric machines are part of an electrical orelectromechanical CVT (i.e., in case of a series or aseries–parallel hybrid configuration, respectively).

3) A CVT with infinite ratio coverage (e.g., Toyota Prius)allows smooth and quick engine restart.

In this paper, the hybridization of a small passenger carequipped with a CVT is analyzed, and the component sizes(power specifications) of such a car are optimized. The baselinevehicle is a Toyota Yaris equipped with a four-cylinder 1.3-Lgasoline engine, a TC, and a conventional push-belt CVT.A short overview of different CVT solutions is given in the nextsection.

A. Development of CVT Solutions

A pure electrical CVT consists of two electric machines [seeFig. 1 (top)]. Both electric machines can operate as motor orgenerator. If one electric machine is directly coupled to theengine and the other is directly coupled to the driven wheels, thesize (maximum continuous power) of both machines has to beequal to the maximum power of the engine. The correspondingweight and cost would be too high for passenger car applica-tions. Furthermore, the power conversion from the mechanicaldomain to the electrical domain and vice versa is not highlyefficient [3]. To reduce the size of the electric machines andimprove the efficiency of an electrical CVT, a power-splitdevice (planetary gear set), which splits the engine power intoa mechanical and an electrical path, can be used [see Fig. 1(bottom)]. However, due to, e.g., the kinematics of the planetarygear or the constraints on the ratio coverage of the transmission,the reduction of the electric machine sizes is limited yet stillsignificant, compared with a pure electrical CVT.



Fig. 1. Pure electrical, push-belt (discussed in this paper), and electricalpower-split CVTs (Toyota Prius).

In summary, the advantages of a power-split CVT, comparedwith an electrical CVT, are given here.

1) The transmission efficiency is higher since most of thepower is transmitted over the mechanical branch consist-ing of planetary gears.

2) A lower maximum power flow through the electricalbranch is required, which results in smaller electricmachines.

Developments to further reduce the size of the electric ma-chines and further increase the transmission efficiency are doneby Villeneuve [4], Schulz [5], and Kriegler et al. [6]. However,these developments mostly lead to a design that is close tothat of existing or automated manuals, in which the clutchesare replaced by electric machines to enable continuously vari-able shifts. This always leads to a transmission that is muchmore complex than a conventional transmission. Moreover,electronically controlled clutches for clutching shifts are so welldesigned nowadays that it is hard to feel that a shift took place.

The mechanical CVT, which is currently produced by Jatco,Aisin, ZF, Toyota, Subaru, Daimler Chrysler, Audi, etc., isconsidered to be a benchmark for all transmissions [7]. It out-performs the manual transmission regarding fuel consumption.Moreover, the mechanical CVT outperforms current automatedmanuals regarding cost and comfort [8]. In particular, due to theefforts of Japanese transmission manufacturers, the mechanicalCVT will have high potential on a large scale. The potential forfurther improvement of the CVT efficiency and the reduction

of costs is high with new upcoming technologies, such as slipcontrol [9], on-demand actuation systems [10], and involutechain technology [11].

II. PROBLEM DEFINITION

This paper presents a modeling and design approach infinding the optimal values for the following design parametersdenoted as xi: the drive-train topology, component technology,component size (in kilowatts), and control strategy. This isperformed for a parallel hybrid configuration equipped witha push-belt CVT for a small passenger car (Toyota Yaris).The topology determines how the power sources (engine andelectric machine/battery) are connected to the drive train.

The main design criterion, which is used to find the optimalparametric values, is the minimization of the overall fuel con-sumption, which is denoted as Φf , on a defined drive cycle.In addition, constraint functions G(xi) can be defined to keepthe system properties and variables within certain bounds. Thedesign optimization problem is formulated as [12]

minxi

Φf (xi) subject to G(xi) ≤ 0, xi ∈ X ⊆ Rn (1)

where the feasible design space is assumed to be embeddedin set X with dimension n. From a hybrid vehicle propulsiondesign viewpoint, the control design is seen as a subproblem.Therefore, the control model uses a local design variable, suchas the control power flow of the energy storage system (battery/electric machine).

III. CONTRIBUTION AND OUTLINE OF THIS PAPER

Finding the optimal parametric values is a complex taskdue to the strong interdependence of the design parameters.Furthermore, calculation of the optimal control strategy can bea computational burden, particularly if the control strategy isoptimized with the control freedom of the component operationpoint on top of the power distribution problem between themain power sources. In the last part of this paper, to alleviatethe complex design problem, the effects of employing a high-level modeling framework in determining the optimal designparameters are investigated. Initially, detailed component mod-els are used for analyzing the sensitivity of the design para-meters to the design objective. Moreover, the effects of controlfreedom in the operation point on the total fuel consumption areinvestigated. Within this context, one of the main underlyingquestions in this work is given here.

• Can a reduced hybrid drive train model with sufficientaccuracy for design be employed?

This question is answered at the end of this paper.The outline of this paper is given as follows: In Section IV,

the baseline vehicle, engine, and push-belt CVT model usedas a reference are discussed. Accordingly, the hybrid vehi-cle drive train and electrical storage model are discussed.The simulation methods and assumptions are discussed inSection V. In Section VI, the effects of parameter variationon the main design objective fuel consumption for a parallelhybrid vehicle are discussed. A method for the calculation



of the component efficiencies used with the simplified hybriddrive-train model is presented in Section VII. The componentefficiencies are described using a few characteristic param-eters. The fuel consumption using the reduced hybrid drive-train model is calculated and compared with that of a detailedhybrid drive train model. Finally, the conclusions are describedin Section VIII.

IV. MODELING OF THE DRIVE TRAIN COMPONENTS

In this section, the used component models for the engine,transmission, and battery-storage system are discussed. Someof the used models, i.e., the electric machine and battery data,are obtained from ADVISOR [13]. The static-efficiency mapsof the electric machines are (linearly) scaled up or down fortheir properties to match the desired values. Similar scaling ap-proaches have been utilized by others [14]–[17]. Nevertheless,some of the used static-efficiency models, i.e., for the engine,push-belt CVT, and TC, have been derived based on the per-formed measurements. The control design method, as describedin [18], is used for the determination of the optimal controlstrategy. The control method is based on the combination ofrule-based and equivalent consumption minimization strategieshaving sufficient accuracy, with which fuel consumption canquickly be calculated. Next, the reference vehicle model and,accordingly, the hybrid vehicle drive trains, including the drivetrain components, are discussed.

A. Baseline Vehicle Model

The baseline vehicle is a Toyota Yaris (2007) equipped witha 1.3-L SI 64-kW engine, a TC, and a hydraulic actuated push-belt CVT. The fuel consumption of this vehicle was measuredon a dynamo test bench for the NEDC (cold and hot enginestarts), JP10–15 (only hot engine) cycles, additional differ-ent continuous driving speeds, and WOT measurement tests.Thereby, the injected fuel mass flow (in grams per second),engine crankshaft speed, drive shaft speed, and torque weremeasured. Since the engine crankshaft torque and the efficiencyof the actual TC and CVT were not measured, available exper-imental validated models [19] for the TC and belt-driven CVTwere used to calculate backward the required engine crankshafttorque. The collected measurement data and the reconstructedengine torque assumed to mimic the actual engine torque wereused to reconstruct the (hot) engine fuel map. A forward-facingcontrol model consisting of a driver model and a dynamicdrive train model was used to validate the used componentmodels with the measured data. The results are not discussedhere and are beyond the scope of this paper. Instead, onlythe fuel consumption results of the baseline vehicle are given(see Table I). The simulation with a CL shows a relatively smallfuel consumption improvement of 2%. The reconstructed fuelefficiency map as a function of engine torque Te and speed ωe

is shown in Fig. 2. In the figure, the fitted or “smoothed”OOL is also shown. The OOL collects the set of engine op-eration points with maximum efficiency over the engine outputpower range.

TABLE IREFERENCE VALUES: BASE LINE VEHICLE

(FORWARDS-FACING CONTROL MODEL)

Fig. 2. Reconstructed 1.3-L VVTi 64-kW engine efficiency map and“smoothed” OOL.

B. Hybrid Drive Train Model

For the parallel hybrid configuration equipped with a push-belt CVT [Fig. 1 (middle)], the electric machine is coupled atthe engine side of the transmission between the clutch and thetransmission. This allows the engine to be turned off during thepropulsion phases [20] since the vehicle can be propelled onlyby the electric machine. Smooth engine restart during standstillwith this configuration cannot be done without an additionalstarter–alternator. This is however different for the Honda IMA,where the electric machine is directly connected to the enginecrankshaft. The Honda IMA is able to restart the engine duringstandstill with the same electric machine that is used, e.g., forbrake energy recuperation or motor assistance. However, theHonda IMA is not able to efficiently propel the vehicle solelyby the electric machine due to the relatively large engine draglosses.

An electromechanical power-split CVT is used in the ToyotaPrius [Fig. 1 (bottom)], which has only “one mode,” implyingthat the CVT has neither clutches nor brakes. Furthermore,engine restart can be done at vehicle standstill without usinga starter–alternator. The depicted power-split CVT is character-ized by the possible occurrence of recirculation power, whichdiminishes the overall transmission efficiency. The main power-flow cases are indicated by the arrow directions in the figure.In [4], [5], and [21], more details on this subject are given.In Fig. 3, the efficiency map at a transmission input speed of209 rad/s (2000 r/min) as a function of the overall speed ratio



Fig. 3. Efficiency map for the belt-driven CVT at transmission inputspeed ωt,p = 209 rad/s (2000 rad/min).

TABLE IIBASE COMPONENT CHARACTERISTICS

and input torque is shown. The push-belt CVT efficiency isstrongly influenced by the input torque for a fixed input speed.The efficiency increases with an increase of input torque andwith an increase of speed ratio up to the “neutral” speed ratiorCVT = 1. The maximum efficiency for the push-belt CVT is atneutral speed ratio rCVT = 1 (which is indicated by the dashedline in the figure).

An overview of the used model data for the main power sour-ces and the specific electric machine data is presented in Table II.

C. Battery Model

The maximum battery pack power is of the same size as themaximum possible output power specifications of the electricmachine for the parallel hybrid drive train equipped with thepush-belt CVT, i.e., 15 kW. Thereby, a minimum number of40 battery modules for the NiMH with a nominal battery systemvoltage of 288 V are needed to meet the estimated minimumvoltage requirement of 120 V and the maximum current of125 A allowed by the motor controller/electric machine. Thesevalues are kept constant (see Table II).

Fig. 4. Power flow in the hybrid drive train (backward-facing control model).

TABLE IIIVEHICLE MODEL PARAMETERS

V. SIMULATION METHOD

In Fig. 4, the power flows in the hybrid drive trainare schematically visualized, which yields a backward-facingmodel implying that the vehicle speed vv(t) (in meters persecond) prescribed by the drive cycle is assumed to be exactlytracked. This model is used to compute the optimal controlsignals that minimize the fuel consumption.

The inputs are battery input power Ps(t) (in watts), velocityvv(t) described by the given drive cycle, and vehicle drivepower demand Pv(t) (in watts). The outputs are engine speedωe(t) (in radians per second) and torque Te(t) (in newtonmeters), which are used to compute fuel mass flow mf (t) (ingrams per second). The vehicle parameters of the Toyota Yarisdetermine vehicle road load torque Trl(t) (in newton meter).The vehicle model parameters are listed in Table III. Notethat the battery power can be affected by additional electrical(auxiliary) loads, which is not shown in the figure. The vehicleroad load depends on vehicle speed vv(t) (assuming no wheelslip) and is the sum of the roll resistance and the air-dragtorque, i.e.,

Trl(t) = 36.8 + 0.416 vv(t) + 0.113 vv(t)2 (2)

where the vehicle power demand as a function of time Pv(t)becomes

Pv(t) = Trl(t)/rw · vv(t) +(mv + Jw/r2

w

)· vv(t). (3)

The vehicle mass (in kilograms), the inertia of the wheels (inkilogram meter squared), and wheel radius (in meters) aredescribed by parameters mv , Jw, and rw, respectively. Theenergy levels (in joules) in the battery and in the fuel tank arerepresented by variables Es and Ef , respectively.

The simulation procedure, which is described by sequentialcomputation steps, to calculate the fuel mass-flow rate mf (t) as



a function of battery input power Ps(t), given vv(t) and Pv(t),is discussed here.

Step 1: In the first step, transmission efficiencies ηCVT arecomputed for all admissible battery powers Pb, transmissioninput torques Tt,p, speeds ωt,p, and speed ratios rt = rCVT · rd

and are stored in multidimensional lookup tables. The transmis-sion efficiency can be expressed as

ηCVT = ηCVT(ωt,p, Tt,p, rt, Pb). (4)

Step 2: In the second step, fuel mass flows mf (in grams persecond) are computed for all admissible battery output powersPb, given drive power demand Pv(t) and angular vehicle wheelspeed ωv(t) = vv(t)/rw. The fuel mass flows are computed fortwo different control strategies focusing on maximizing engineηe or system efficiency ηsys at each time instant. These twocontrol strategies, which determine the engine operation point,are referred to as CS1 and CS2, respectively.

System efficiency (see Fig. 4) ηsys is defined as the productof engine ηe and transmission efficiency (without the battery)ηCVT and is written as

ηsys = ηe(ωe, Te) · ηCVT(ωt,p, Tt,p, rt, Pb) (5)

with the transmission input torque as the sum of the engineand the electric machine output torque Tt,p = Te + Tem andωt,p = ωe = ωem, respectively, since the input shaft of theCVT is directly connected to the output shaft of the engine.Accordingly, the optimal speed ratio and engine torque areselected as

(rot (Pb, t), T o

e (Pb, t))=arg max(rt,Te)∈R

(η∗|Pb, Pv(t), ωv(t)) (6)

for ∗ ∈ {e, sys} (i.e., CS1 or CS2, respectively), where set Rcovers the feasible solutions that satisfy the constraints

R ={(rt, Te)|ω∗∗,min ≤ ω∗∗ ≤ ω∗∗,max

∧ T∗∗,min(ω∗∗) ≤ T∗∗ ≤ T∗∗,max(ω∗∗)

∧ rud ≤ rCVT ≤ rod ∧ Pb,min ≤ Pb ≤ Pb,max

}(7)

for ∗∗ ∈ {e, em}, and rCVT is limited by the under- and over-drive ratios rud and rod, respectively, for the push-belt CVT(see Table II).

The optimal gear ratios and engine torques for all admissi-ble battery powers for both control strategies (CS1 and CS2)are stored in multidimensional lookup tables and are used tocompute the fuel mass flows using the engine fuel map. If theoptimal gear ratios and engine torques at each time instant areknown for different values of Pb, given Pv(t) and ωv(t), thenthe fuel mass flow can be expressed only as a function of thebattery power, i.e.,

mf = mf (Pb, t|Pv(t), ωv(t)) (8)

for CS1 and CS2.Note that, in case of engine optimal operation (CS1), the

transmission efficiency ηCVT is determined by speed ratiorot (Pb, t) and engine torque T o

e (Pb, t) (prescribed by the OOL),

given a certain vehicle speed ωv(t). However, in a backward-facing control model, the required ro

t (Pb, t) and T oe (Pb, t),

given a certain speed ωv(t), are determined by ηCVT and thegiven Pv(t). Due to this causality conflict, it is impossibleto determine ro

t (Pb, t) and T oe (Pb, t), given a certain speed

ωv(t) exactly. In this paper, for CS1, the transmission lossesare iteratively estimated and are compensated for engine powerPe(t). The iteration error, which is denoted as ε, at time instantt in each iteration step i is defined as

ε(i, t) = |Pe(i + 1, t) − Pe(i, t)| . (9)

The iteration is repeated until the error ε(i, t) between the itera-tion steps at a certain time instant becomes sufficiently small(i.e., ε(i, t) < 10 W). A mathematical condition to preventthe iteration loop from instabilities, in terms of the estimationPe(i, t), is

|Pe(i + 1, t) − Pe(i, t)| < γ |Pe(i, t) − Pe(i − 1, t)|0 < γ < 1. (10)

This condition implies that the error ε of each estimate de-creases in each iteration step i. If the ratio of the error betweenthe previous and subsequent steps becomes equal to 1, theestimates will not further improve. This corresponds to thetransition between stable and unstable iteration loops. At latertime instants, the required Pe is calculated using the knownvalues for the efficiencies at the previous time instant. Thus,the requested Pv(t) is divided by the computed ηCVT.

Step 3: In the third step, the optimal control strategy isdetermined using the control design model described in [18],where the engine efficiency (CS1), or the system efficiency(CS2) at each time instant t is maximized, and the total fuelconsumption Φf is minimized. The optimization problem isfinding control power flow Ps(t), given a certain power demandat the wheels Pv(t) and wheel speed ωv(t), while Φf over adrive cycle with time length tf is minimized, i.e.,

Φf = minPs(t)

tf∫

0

mf (Ps(t), t|Pv(t), ωv(t)) dt (11)

subject to certain component constraints �h = 0, �g ≤ 0, whichare described here. The state is equal to the stored energy Es inthe battery in joules, and the control input is equal to the batteryinput power Ps in watts (see Fig. 4), i.e.,

Ps(t) = min (Pb(t)/ηb (Pb(t)) , Pb(t)ηb (Pb(t))) . (12)

Battery efficiency ηb is only a function of battery power Pb

since the influence of the relatively small variation in the batteryenergy level on the battery efficiency is small and is, therefore,neglected. The energy level in the battery is a simple integrationof the power and is calculated as follows:

Es(t) = Es(0) +

t∫

0

Ps(τ)dτ. (13)



The main constraints on the battery are the energy balanceconservation of Es over the drive cycle and constraints onpower Ps, i.e.,

h1 := Es(tf ) − Es(0) = 0

g1,2 := Ps,min ≤ Ps(t) ≤ Ps,max. (14)

Computation steps 1–3 are performed for different electricmachine sizes (1–15 kW), in combination with other differentdesign parameters. In the next section, the modeling assump-tions, descriptions, and symbols of the investigated designparameters are discussed.

Assumptions and Descriptions of the Design Parameters xi:Initially, for the hybrid vehicle model, nine assumptions aremade.

A.1) The drive-train components (engine, transmission, elec-tric machine, and battery) are modeled as quasi-static-efficiency models.

A.2) The electric machine characteristics (maximum torquecurve and static-efficiency map) are linearly scaled asneeded. The effect of the vehicle mass increase byupscaling of the electric machine is neglected.

A.3) For the whole drive cycle, the electrical auxiliary loadsare assumed to be constant at 200 W. For the baselinevehicle, the electrical auxiliary loads increase the fuelconsumption with 0.13 L/100 km for the JP10–15.

A.4) The minimum engine speed during operation is980 rad/min, which is equal to that of the referencevehicle with a conventional drive train.

A.5) The braking forces are evenly distributed between therear and front wheels. The brake force distribution ratiobetween the front and rear wheels is denoted as ffb. Tomaximize the vehicle stability, an adaptive brake forcedistribution should be considered [22]. However, in thispaper, ideal braking is not considered. Instead, the influ-ence of different constant values for ffb, which is alsoreferred to as regenerative brake fraction, is investigated.

A.6) The maximum electromechanical braking power is lim-ited by the torque, speed, and power constraints ofthe electric machine and battery, respectively. Brakingpowers that are larger than the maximum electrical re-generative braking power are assumed to be dissipatedin the front-wheel brake discs.

A.7) The engine has no drag or idle losses at vehicle standstill.This start–stop function decreases fuel consumptionwith 0.66 L/100 km for the JP10–15. Fuel savings withthe start–stop function during propulsion depends on thedegree of hybridization.

A.8) The fuel use for engine cranking is neglected since theengine can be started with the electric machine in a veryshort time period (typically 100–300 ms). The requiredelectrical energy for engine start (usually, the engine isstarted when the vehicle is already driving) is thereforelimited and very small. However, the power for enginestart is relatively high.

A.9) During braking, the clutch is open (CLO), and theengine is assumed to be off (EOFF) (i.e., inducingno additional drag torques and no idle fuel consump-

Fig. 5. EON/EOFF strategy.

tion). This assumption decreases fuel consumption with0.45 L/100 km.

If, during braking, in a real vehicle, the engine is completelyshut off and disengaged, then quick engine restart duringdriving can be uncomfortable and, therefore, unacceptable.Unless, for example, the electric machine is sufficiently largeto deliver the relatively large engine start-up torques, which aresuperposed on the drive torque demand. Therefore, two otherbraking strategies are investigated where the engine remains on(EON) during braking, in combination with the electric-onlyvehicle speed threshold value, which is denoted as vM . Theengine is always on for vehicle speeds above this value, andvice versa. This will be discussed by the descriptions of thedesign parameters in more detail.

A.10) The electric-only vehicle speed threshold value vM is20 km/h, which is equal to the measured value for theToyota Prius on the JP10–15 cycle [23].

A.11) During the electric-only mode (propulsion and braking),the CVT speed ratio is assumed to be controlled, suchthat the electric machine is operated at the highest ef-ficiency points. In [24], it is shown that this brakingstrategy provides increased recuperated energy.

Next, the descriptions and symbols of the design (xi fori ∈ {1, . . . , 7}) parameters are given.x1 Regenerative brake fraction: This determines the distribu-

tion fraction ffb between the front and rear wheels, wherethe regenerative braking power becomes PBER(t) = ffb ·Pv(t|Pv(t) < 0), with ffb ∈ {0%, 50%, 100%}.

x2 Drive cycle: The vehicle load is influenced by using theNEDC or the Japanese drive cycle (JP10–15) [see (2)].

x3 Component size: The component efficiencies, power-flowcontrol strategy, and potential fuel savings are affected bythe size of the electric machine. The base electric machinesize is linearly scaled as needed (see Table II).

x4 Topology: This determines how the electric machine iscoupled to the transmission, i.e., at the primary side(engine side) (PRI) or the secondary side (wheel side)(SEC) of the transmission. When it is coupled at thesecondary side, a reduction set of gearwheels with a fixedspeed ratio is selected, such that the electric machine isable to reach the maximum drive cycle speed (70 km/hfor the JP10–15). The efficiency of the reduction set isassumed to be constant at 98%. Note that the start–stopfunction is still possible with this configuration since theengine is, in principle, restarted with the starter–alternator.Without a starter–alternator, engine restart with an



Fig. 6. Fuel consumption as a function of electric machine size x3 for different design parameters (x1, x2, and x4 − x7).

electric machine coupled at the wheel side requires arelatively large and more costly electric machine due tothe transmission losses and relatively low torque multiplierfactor [25].

x5 Actuation technology for the push-belt CVT: One of themajor power losses of the conventional push-belt CVT(which are on the order of 50% of the total power losses[26]) is caused by the hydraulic pump. Therefore, theinfluence of applying an EMA system, instead of an HDAsystem, on fuel consumption is investigated. The EMApower, which is only required during the operation of thepush-belt CVT, is assumed to be constant at 90 W. Theefficiency of the EMA system is assumed to be constant at90%, which results in a battery output power demand of100 W.

x6 Regenerative brake strategy (combined with an electric-only vehicle speed threshold value): The influence of theelectric-only vehicle speed threshold value vM on the fuelconsumption is investigated, in combination with theclutch state (i.e., closed or open) during braking. Depend-

ing on the clutch state, the engine applies “fuel cutoff” (nofuel injection) and has engine drag losses or idle fuel massflow during regenerative braking.a) CLO/EON: The engine is on at an idle speed of

610 rad/min and disengaged. The engine has an idle fuelmass flow until vv(t) = vM . However, the engine has no draglosses. The electric machine is operated at the OOL duringbraking.

b) CLC/EON: The engine is on and engaged. The enginehas no fuel mass flow (i.e., fuel cutoff) until vv(t) = vM .However, the engine has drag losses. The drag losses causereduction in the recuperative brake energy. Notice that the push-belt CVT is controlled, such that the engine output shaft isrotating at the minimum speed (610 rad/min), while the enginedrag losses are minimized.

For both cases, the engine is switched off and disengagedduring braking only if vv(t) < vM . The strategies are schemat-ically visualized in Fig. 5. At certain time instants, drive powerdemand Pv is still positive while the vehicle decelerates (due torelatively large road load losses at relatively high speeds). Brake



Fig. 7. Fuel consumption as a function of the technologies, control strategies, topologies, component sizes, and functions.

energy storage can be done at moments where the road loadlosses (air drag and roll resistance losses) are not sufficientlyenough to decelerate the vehicle as desired.x7 Speed ratio control strategy: The CVT speed ratio is

controlled based on an engine optimal operation (CS1) orsystem optimal operation (CS2).

In the next section, the effects of parameter variation(x1 − x7) on the total fuel consumption over a drive cycle Φf

are investigated.

VI. RESULT

A. Effects of Parameter Variation (x1 − x7) onDesign Objective Φf

The responses of design objective (fuel consumption) Φf asa function of electric machine size x3 for the different designparameters x1, x2, and x4 − x7 are shown in Fig. 6. Althoughthe response functions show small local oscillations (localminima), the global trend of the response functions shows aconvex curvature. The optimal electric machine sizes are easilydetermined and are depicted by the white circular marks. Therelatively fuel consumption values are shown in the figure at theright side. A bar diagram with the fuel consumption at optimalelectric machine size xo

3 as a function of the different designparameters is shown in Fig. 7. Observations O, following theparameter variation study, are discussed here.

O.1) A 5-kW electric machine size is sufficient for the EMApush-belt CVT to minimize fuel consumption on theJP10–15. The optimal size is independent of topologychoice x4, the regenerative braking strategy x6, and theCVT speed ratio control strategy x7.

Fig. 8. Influence of the electric machine size on the efficiency.

In Fig. 8, the static-efficiency maps of a small anda large electric machine is schematically visualized.The styled OOLs for both electric machines are alsodepicted. If the electric machine is operated duringthe electric-only modes at the OOL, then, for a largerelectric machine, the desired optimal speed is lowerthan that for a smaller electric machine, given the sameoutput power. However, the electric machine outputtorque is higher, which increases the push-belt CVTefficiency.

O.2) The optimal electric machine size is a tradeoff betweenthe average electric machine and the CVT efficiency,which decrease and increase, respectively, with anincrease in electric machine size.

The optimal electric machine size for the HDApush-belt CVT is approximately 10 kW. The optimalelectric machine size is larger than that for the EMApush-belt CVT due to the higher static losses of theHDA push-belt CVT. For the NEDC, the optimal



Fig. 9. Overview of the relative effects of parameter variation on the fuel consumption.

electric machine size is also 10 kW due to the higherbraking powers, compared with JP10–15. For ffb =100% (100% BER), the optimal electric machine sizeis 6 kW due to the increase in maximum braking powerand regenerative brake energy on the JP10–15.

O.3) The start–stop function at vehicle standstill relativelyreduces the fuel consumption by approximately 13.8%or 14.3%, which depends on the type of actuationtechnology (compare test 4 with test 6 and test 5 withtest 7 in Fig. 7).

O.4) The fuel saving increases with increase in the brakedistribution fraction. It is observed that, if ffb is in-creased from 50% to 100% BER at the front wheels,the fuel consumption is relatively decreased by 12.8%(compare test 14 with test 19 in Fig. 7).

O.5) The vehicle load prescribed by the NEDC increases thefuel consumption by 10%, compared with the JP10–15(compare test 8 with test 14 of Fig. 7).

O.6) Hybridization of the drive train with a 5-kW electricmachine, which is connected at the engine side ofthe EMA push-belt CVT with 50% BER, reduces thefuel consumption by approximately 3.5% or 11.3%,depending on the regenerative brake strategy (exclud-ing the start–stop function at vehicle standstill; com-pare test 7 with test 10 and test 7 with test 14 inFig. 7).

O.7) Connecting the electric machine at the wheel sideof the push-belt CVT reduces the fuel consumptionby 5.1% due to higher transmission efficiency duringregenerative braking (compare test 14 with test 17 inFig. 7).

However, due to the fixed gear ratio between theelectric machine and wheels, the electric machine op-eration points are prescribed by the drive cycle and thevehicle power demand. The operation points are there-fore not optimally selected. In addition, the averageelectric machine efficiency decreases with increase inelectric machine size, which again causes a decreasein fuel savings.

O.8) The EMA push-belt CVT, compared with the HDApush-belt CVT, reduces fuel consumption by 3.1% and5.4% for the baseline and hybrid vehicles, respectively(compare test 4 with test 5 and test 11 with test 14 inFig. 7).

O.9) Fuel cutoff (CLC/EON) during regenerative braking ismore beneficial than idling of the engine (CLO/EON).The relative fuel consumption with CLC/EON, com-pared with that with CLO/EON, is 4.7% lower (com-pare test 10 with test 13 in Fig. 7).

O.10) The influence of system optimal operation CS2, com-pared to engine optimal operation CS1, on fuel savingfor the hybrid vehicle is relatively small. Fuel con-sumption is reduced by 1.3% or 1.1%, which dependson the type of actuation technology (compare test 11with test 12 and test 14 with test 15 in Fig. 7).

In Fig. 9, an overview of the relative effects of the parametervariation on fuel consumption change is shown. Some ofthe bars, which are indicated with an (lower)upper boundaryvalue, represent the average values of the performed tests.The upper and lower boundary values represent the maximumand minimum values, respectively. Since the cross correlation



Fig. 10. Battery storage power and battery energy-level difference.

of parameter variation is not investigated, the relative effectsthat decrease or increase the fuel consumption cannot besummed up. Nonetheless, it can be concluded that potentialfuel savings is strongly affected (5%–13%) by the followingdesign parameters (x1 − x4): the regenerative brake fraction,drive cycle (vehicle load), component size (and start–stopfunction), and topology.

In Fig. 10, the power-flow control strategies Ps(t), includ-ing the battery energy-level difference over time �Es(t), areshown for the comparison of the HDA–PRI, EMA–PRI, andEMA–SEC push-belt CVTs. The other design parameters,which are kept constant, are shown in the figure. In the diagramsat the bottom of the figure, the strategies for the different regen-erative brake fractions are shown. The transmission efficienciesduring the electric-only modes are larger for the EMA–PRIand EMA–SEC CVTs, in comparison with the HDA–PRI CVT.Because of this, effectively more brake energy can be used for

electric driving. Furthermore, if brake fraction ffb increases,then the electric machine is more intensively used during theelectric-only modes.

In this section, the electric-only vehicle speed threshold valuevM was kept constant at 20 km/h for the different regenerativebrake strategies. In the next section, the effects of changing vM

on the fuel consumption for the different brake strategies areinvestigated.

B. Effects of the Electric-Only Vehicle Speed Threshold ValuevM on Design Objective Φf for Different Brake Strategies x7

In Fig. 11, the contour plots of the fuel consumption forthe CS1–EMA–50% BER–PRI–JP1015 as a function of theelectric machine size (in kilowatts) and electric-only vehiclespeed threshold value vM (in kilometers per hour) for theCLO/EOFF, CLO/EON, and CLC/EOFF braking strategies are



Fig. 11. Contour plots of the fuel consumption as a function of vM and the electric machine size for different brake strategies.

shown, respectively. Two observations are made based on theseresults.

O.11) The CLO/EOFF strategy has a negligible effect on thefuel consumption improvement, with increase in vM

of larger than 20 km/h. From the optimized controlstrategy, it followed that propulsion only by the electricmachine during acceleration up to vehicle speeds ofhigher than 20 km/h does not occur.

O.12) The CLC/EON strategy is preferable over the CLO/EON strategy for electric-only vehicle speed thresholdvalues of lower than 40 km/h. The fuel savings byfuel cutoff and the recuperation of brake energy areeffectively larger than the fuel costs of the engineidling during braking, although, with the CLC/EONstrategy, the recuperative brake power is reduced.

C. Effects of Changing Topology x4 and ActuationTechnology x5 on Design Objective Φf andComponent Efficiencies

Table IV shows the fuel consumption results for the baselinevehicle and the hybrid vehicle equipped with the different CVT

TABLE IVFUEL-CONSUMPTION RESULTS WITH CS1, 50% BER, AND vM = 20 km/h

actuation technologies. For the baseline vehicle, the batteryis not used, i.e., Ps(t) = 0. For the hybrid vehicle, the fuel-consumption results are similar to the results shown in Fig. 7.The lowest fuel consumption is achieved with EMA–SEC (seeobservation O.7).

For the hybrid vehicle, the average component efficienciesand the relative fuel savings as a function of the transmissiontechnologies and topologies are shown in Fig. 12. The sec-ondary power source efficiency, which is denoted as ηs, consistsof the battery, electric machine, and transmission efficiency(e.g., the push-belt CVT or the reduction set, including the finaldrive set) from the battery to the wheels during the electric-onlymodes (i.e., regenerative braking and propulsion). The decreasein fuel consumption is in correspondence with increase in the



Fig. 12. Effects of the transmission technology and topology on the relative fuel saving, transmission, engine, and secondary source efficiency.

average component efficiencies. Fuel consumption is stronglycorrelated with secondary power source ηs and transmissionefficiency ηt since engine efficiency ηe remains approximatelyconstant.

VII. REDUCED HYBRID DRIVE-TRAIN MODEL

The computation steps (1–3), as discussed in Section V,are quite tedious to carry out. In this section, a reduced hy-brid drive train model describing the main component modelsand the topology is discussed, which can be used for designanalysis decoupled from the choice of specific components,hybrid drive train configurations, and control strategy. Theidea of a high-level modeling and design framework, as pre-sented in [3], is further investigated. First, the assumptionsfor simplified modeling of the components and topology arediscussed.

• The component efficiencies are modeled as simplifiedparametric power-based fit functions, where, e.g., an affinerelationship between the input and output powers is as-sumed: Pin = c1 · min(Pmax, Pout) + c0.

Parameter Pmax represents the output power limitationof the component. Parameters c1 and c0 represent the fitcoefficients and correspond to the reciprocal of the inter-nal efficiency and the static losses, respectively. The engine(Primary power source = P) and the electric machine/battery(Secondary power source = S) efficiency, which are similarto the efficiency of CVT (Transmission technology = T), arewell described by an affine relationship. The operation pointsdetermine the values for the fit coefficients (c1, c0), which arealso referred to as the characteristic component parameters. The

operations points, given a drive cycle and vehicle parameters,are prescribed by three assumptions.

1) The engine is operated over a whole drive cycle atthe maximum efficiency points (speed ratio controlstrategy CS1).

2) The electric-only modes (regenerative braking andpropulsion only by secondary power source S) over awhole drive cycle are used for S, whereas S is operatedat the OOL (i.e., only if S is precoupled to T), and P isshut off (CLO/EOFF).

3) The engine-only mode over a whole drive cycle is usedfor transmission T, whereas engine P is operated at theOOL, and secondary power source S is shut off (assumingno drag or idle losses).

These assumptions are sufficiently accurate for estimating thecomponent efficiencies since 1) the influence of system optimaloperation, compared with engine optimal operation, on the fuel-consumption reduction for a hybrid vehicle, is relatively small(1%–2%), and 2) the influence of power exchange betweenS and P during driving (motor assisting and charging) on theefficiency of S and T is relatively small, as discussed here.

The battery power influences the CVT efficiency. For thehybrid vehicle, the CVT efficiency is decreased, compared withthe baseline vehicle due to charging and motor assisting duringdriving with S. If the transmission input power is correctedwith the battery power during the hybrid driving modes (i.e., inthis case for charging and motor assisting during driving), thenthe CVT efficiency for the hybrid vehicle is slightly increased,compared with the CVT efficiency for the baseline vehicle.However, the difference is relatively small. This is observed bycomparing the results for T, as listed in Table V.



TABLE VTRANSMISSION EFFICIENCY ηCV T (CS1, JP10-15)

Fig. 13. Simplified hybrid drive train model. Component technology (c0, c1),size (Pmax), and topology models (location of S and P).

In Fig. 13, the reduced hybrid drive-train model for the dif-ferent component technologies and the power-based fit functionis shown schematically.

The effects of parameter variation on the fuel economy andoptimal electric machine size x3 using the reduced modelare investigated next. This is performed to investigate if theassumptions, as posed in the beginning of this section, aresufficiently accurate for design optimization. The varied designparameters are regenerative brake fraction x1, drive cycle x2,and topology x4. For different sizes of S (x3), the character-istic parameters (c1, c0) during the electric-only modes and,for T, during the engine-only mode are calculated for theEMA CVT.

The fuel consumption results for both cycles of the detailedhybrid drive train model, which is denoted as M1, and thereduced model, which is denoted as M2, are both shown inFig. 14.

The fuel consumption of the detailed model M1 is higherthan that calculated with the reduced hybrid drive-train modelM2. The differences tend to increase with increase in the elec-tric machine size. The average relative errors and the maximumand minimum error values are shown in Fig. 15. It shouldbe noted that the vehicle mass increases with increase in theelectric machine size. This has a relatively small effect on thefuel consumption and is therefore neglected. Moreover, it wasfound that the function becomes more convex, yet there is nochange in the observed optimal electric machine size.

For the detailed model M1, the engine is assumed to beoperated at the OOL and therefore determines the opera-tion points of the electric machine during the hybrid driving

Fig. 14. Fuel consumption as a function of the electric machine size. Fuelconsumption is calculated using a detailed hybrid drive train model M1 and areduced hybrid drive train model M2.

modes (i.e., charging and motor assisting during driving). Thiscauses that the efficiency of S to decrease with increase inthe electric machine size. This relatively small effect has notbeen taken into account using the reduced hybrid drive trainmodel M2, which causes the objective function to sometimesnot be convex; rather, a boundary optimum is determined at15 kW. However, in this case, an additional design constraint,which is put on the sensitivity of the objective function to theelectric machine size, can be used to find the optimal electricmachine size.

A different transmission technology (see [3]) or actuationtechnology x5 simply implies a different set of characteristicparameters for T. The same holds for power-split CVTs. Forexample, if the electric machine size at the wheel side of T issufficiently large to maintain the transmission ratio coverage yettoo small to fulfill the required functions of S, then the electricmachine size at the wheel side of T needs to be increased.

For the electrical power-split CVT, the electric machinescoupled at the wheel side of T, which perform the electric-only modes, are defined to be functionally part of S. For thepower-split CVT, changing the size of S indirectly affects thetransmission efficiency. This way, the characteristic parametersfor T, which are determined by the engine-only mode, alsobecome a function of the size of S. However, looking atthe development of the transmission by Toyota, the increasedsize of S is mainly determined by performance and not by areduction in fuel consumption or ratio coverage constraints.

Nevertheless, it can be concluded that the effects of com-ponent sizing on fuel consumption can be investigated usingthe reduced hybrid drive-train model with the assumptionsposed at the beginning of this section very quickly and withsufficient accuracy. The maximum average relative error is1.6%, which is shown in Fig. 15 for a configuration of 100%BER–JP1015–PRI.



Fig. 15. Average relative error and the maximum and minimum error values between the detailed model M1 and the reduced model M2 for the different designparameters.

VIII. CONCLUSION

The optimal electric machine size for a hybrid vehicleequipped with a CVT regarding the minimum fuel consumptionwas determined. Thereby, the effects of parameter variation,e.g., for the CVT technology and the topology, on fuelconsumption were investigated. Other parameters thatstrongly influence the optimal electric machine size arerelated to the vehicle load (vehicle parameters and drivecycle) and the regenerative brake fraction. Finally, areduced hybrid drive-train model with sufficient accuracy(maximum average error < 1.6%) is introduced, with whichthe influence of component technologies, sizes, and topologychoice on the fuel consumption can be studied very quickly.

ACKNOWLEDGMENT

This study is part of “Impulse Drive,” which is a researchproject at the Control Systems Technology Section, De-partment of Mechanical Engineering, Technische UniversiteitEindhoven, Eindhoven, The Netherlands.

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Theo Hofman received the M.Sc. (with honors)and Ph.D. degrees in mechanical engineering fromEindhoven University of Technology, Eindhoven,The Netherlands, in 1999 and 2007, respectively.

From 1999 to 2003, he was a Researcherand Project Leader with Thales-Cryogenics. SinceSeptember 2007, he has been a Postdoctoral Re-searcher with the Control Systems TechnologyGroup, Department of Mechanical Engineering,Eindhoven University of Technology. His researchinterests include the modeling, design, and control

of hybrid propulsion systems.

Maarten Steinbuch received the M.Sc. and Ph.D.degrees from Delft University of Technology, Delft,The Netherlands, in 1984 and 1989, respectively.

From 1987 to 1999, he was a Researcher andGroup Leader with Philips Electronics. Since 1999,he has been a Full Professor with the Control Sys-tems Technology Group, Department of MechanicalEngineering, Eindhoven University of Technology,Eindhoven, The Netherlands. Since 2006, he has alsobeen the Scientific Director of the 3TU Centre ofCompetence High Tech Systems, The Netherlands.

Roëll van Druten received the Master’s and Ph.D.degrees from Technische Universiteit Eindhoven,Eindhoven, The Netherlands, in 1996 and 2001,respectively.

He is currently the Chief Executive Officerof Drivetrain Innovations B.V. (DTI), Eindhoven,which is a licensing and contract-research center thatdeal with automotive power trains, transmissions,and components.

Alex F. A. Serrarens was born in Hulst,The Netherlands, on September 6, 1973. Hereceived the M.Sc. degree in mechanical engineeringand the Ph.D. degree in the field of powertrain control of passenger cars with CVT fromTechnische Universiteit Eindhoven, Eindhoven, TheNetherlands, in 1997 and 2001, respectively.

He is currently a business partner with Drive-train Innovations B.V. (DTI), Eindhoven, which isa licensing and contract-research center that dealswith automotive power trains, transmissions, and

components.


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