A Comparison Study Between Power-Split Cvts and a Push-Belt CVT

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A comparison study between power-split CVTs and a push-belt CVT

Pablo Noben DCT 2007.100

Master’s thesis

Coaches: ir. T. Hofman dr. P.A. Veenhuizen

Supervisor: prof. dr. ir. M. Steinbuch

Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group Automotive Engineering Science Group

Eindhoven, August, 2007

Abstract

Since the ongoing discussion about the global warming of the earth, this is mainly attributed to the greenhouse gasses. Which on their part are produced by vehicles, by internal combustion engines to be more accurate. More and more car manufacturers and research institutes are investigating alterna- tives for internal combustion engines. However changing the fuel for vehicles is very diﬃcult because this is a complex collaboration of governments, oil companies and car manufacturers. On top of this years and years of development has resulted in the current infrastructure of the gas stations.

A ﬁrst solution to decrease the fuel consumption of vehicles is to develop hybrid power trains. With a hybrid power train, a power train consisting of two power sources is meant here. Normally this is an internal combustion engine combined with one or more electric machines. The fuel consumption of a hybrid vehicle is lower compared to a conventional vehicle because electric machines can recover brake energy which can be stored in a battery or a similar electrical storage system. Later this electric energy can be used to power the electric machines and assist the internal combustion engine to power the vehicle and herewith decreasing the fuel consumption. To realize a hybrid power train the most commercially successful hybrid vehicle makes use of a power split transmission. This power split transmission contains of a kinematic chain and an electric variator. The electric variator controls the overall transmission ratio and herewith the working point of the internal combustion engine.

In literature different methods can be found to investigate power split transmissions. In this thesis some methods are treated. Schulz and Villeneuve investigate both a specific power split transmission. Mattsson on the other hand investigates several power split transmissions. Eventually the method by Mattsson is used to investigate the Toyota Hybrid System and the Renault IVT concept power train. Both transmissions are alike, however the Toyota hybrid system is a single mode transmission and the Renault IVT is a dual mode transmission. To compare these transmission considering efficiencies, a simplified loss model is used.

Finally the Toyota hybrid system, as hybrid power train, is compared to a push-belt CVT, as conventional power train. Hereto three cases are studied, first the working point efficiency of the ICE is considered only. Followed by the implementation of constant efficiencies for the power train components. At the end working point dependent efficiencies are implemented for the electric machines in case of the Toyota hybrid system and for the push-belt CVT in case of the push-belt CVT. Herewith the overall efficiency of both power trains are compared.

iii iv Samenvatting

Door de voortdurende discussie over de opwarming van de aarde, die voor het merendeel aan de broei- kasgassen wordt toegeschreven. Die op hun beurt worden veroorzaakt door voertuigen, meer precies door verbrandingsmotoren. Steeds meer automobielfabrikanten en onderzoeksinstellingen zoeken naar alternatieven voor verbrandingsmotoren. Hoewel het erg moeilijk is om de brandstof van voertuigen te veranderen omdat dit wordt beïnvloed door de complexe samenwerking van regeringen, olie maatschappijen en automobielfabrikanten. Daarnaast is de huidige infrastructuur van tankstations door de jaren heen uitgegroeid tot een goed georganiseerd netwerk.

Een eerste oplossing om het verbruik van voertuigen te verlagen is om een hybride aandrijflijn te ontwikkelen. Met een hybride aandrijflijn wordt hier een aandrijflijn bedoeld die door twee vermo- gensbronnen aangedreven wordt. Normaal gesproken is dit een verbrandingsmotor, gecombineerd met een of meerdere elektrische machines. Het brandstofverbruik van een hybride voertuig is lager ten opzichte van een conventioneel voertuig doordat de elektrische machines remenergie kunnen te- rugwinnen, wat opgeslagen kan worden in batterijen of een soortgelijk elektrisch opslag systeem. Na- derhand kan deze elektrische energie gebruikt worden om de elektrische machines aan te drijven en de verbrandingsmotor assisteren, hierdoor daalt het brandstofverbruik. Het meest commercieel suc- cesvolle hybride voertuig maakt gebruik van een vermogenssplit transmissie. Deze vermogenssplit transmissie bestaat uit een mechanisch gekoppelde keten van tandwielen en een elektrische variator. De elektrische variator regelt de transmissie ratio en hiermee ook het werkpunt van de verbrandingsmotor.

In de literatuur kunnen verschillende onderzoeken gevonden worden met betrekking tot vermogenssplit transmissies. In dit verslag worden verschillende methodes behandeld. Schulz en Villeneuve onderzoeken allebei een speciﬁeke vermogenssplit transmissie. Mattsson echter onderzoekt verschillende vermogenssplit transmissies. Uiteindelijk is de benadering van Mattsson gebruikt om het hybride systeem van Toyota en de Renault IVT te onderzoeken. Beide transmissies lijken op elkaar hoewel het hybride systeem van Toyota maar een mode heeft terwijl de Renault IVT twee modes heeft. Met een simpel rendementsmodel worden deze transmissies vergeleken met elkaar.

Uiteindelijk is het hybride system van Toyota vergeleken met een duwband CVT. Hiervoor worden drie situaties bestudeerd, allereerst worden alleen de werkpunts afhankelijke rendementen van de verbrandingsmotor beschouwd. Gevolgd door de toevoeging van constante rendementen voor de transmissie componenten. En uiteindelijk worden de werkpunts afhankelijke rendementen van elektrische machines voor het hybride systeem van Toyota en van de duwband CVT voor de transmissie met duwband CVT. Hiermee wordt het totale rendement van beide transmissies met elkaar vergeleken.

v vi Contents

Abstract iii

Samenvatting v

Nomenclature ix

1 Introduction 1

2 Methodologies 7 2.1 Method by Schulz ...... 8 2.2 Method by Villeneuve ...... 10 2.3 Method by Mattsson ...... 13 2.4 Discussion methodologies ...... 14

3 Analyzing IVTs 15 3.1 Kinematics ...... 15 3.1.1 Toyota hybrid system ...... 15 3.1.2 Renault IVT ...... 20 3.1.3 Comparison THS & Renault IVT ...... 22 3.2 Mattsson ...... 23 3.2.1 Toyota hybrid system ...... 23 3.2.2 Renault IVT ...... 23 3.2.3 Eﬃciencies ...... 24

4 Detailed efficiency analysis 27 4.1 Internal combustion engine ...... 28 4.2 Toyota hybrid system ...... 29 4.2.1 Case A - ICE efficiency ...... 30 4.2.2 Case B - Constant transmission component efficiencies ...... 31 4.2.3 Case C - EM working point efficiencies ...... 32 4.3 Push-belt CVT ...... 33 4.3.1 Case A - ICE efficiency ...... 34 4.3.2 Case B - Constant transmission component efficiencies ...... 36 4.3.3 Case C - CVT working point efficiencies ...... 37 4.4 Discussion ...... 39 4.4.1 ICE strategy ...... 39 4.4.2 Comparison case A, B & C ...... 41 4.4.3 Comparison THS & push-belt CVT ...... 42

5 Conclusion & recommendations 45 5.1 Conclusion ...... 45 5.2 Recommendations ...... 46

vii viii CONTENTS

Bibliography 47

A Kinematics planetary gear 49

B Clariﬁcation of the method by Schulz 51 B.1 M-ﬁle for the method by Schulz ...... 52

C Clariﬁcation of the method by Villeneuve 55 C.1 M-ﬁle for the method by Villeneuve ...... 56

D Clariﬁcation of the method by Mattsson 59

E Kinematics THS 61

F Kinematics push-belt CVT 65

G Toyota Prius speciﬁcations 67 Nomenclature

Abbreviations

2WD / 4WD 2 / 4 Wheel Drive AMT Automated Manual Transmission AT automatic transmission BER Brake Energy Recovery CO2 Carbon dioxide CVT Continu Variable Transmission DNR Drive-Neutral-Reverse ECU Engine Control Unit EM Electrical Machine EMPAct CVT Electro-Mechanical Pulley Actuation CVT GNR Geared Neutral Ratio HEV Hybrid Electric Vehicle ICE Internal Combustion Engine IMA Integrated Motor Assist IVT Inﬁnitely Variable Transmission OD Overdrive SOC State Of Charge SOOL System Optimal Operation Line SUV Sport Utility Vehicle THS Toyota Hybrid System e-line economy line rpm revolutions per minute

Symbols

Symbol Deﬁnition Unit

P power [kW ] T torque [Nm] loh level of hybridization [−] r radius [m] r ratio [−] v velocity [m/s] z number of teeth [−]

Φ power split ratio [−]

ix x NOMENCLATURE

η eﬃciency [−] λE power ratio [−] ω rotational velocity [rad/s]

Subscripts

Subscript Deﬁnition

A1 planetary gear A1 A2 planetary gear A2 B planetary gear B C planetary gear C C1 clutch 1 C2 clutch 2 CVT push-belt CVT EM Electrical Machine F 1 brake 1 F 2 brake 2 H higher shaft ICE Internal Combustion Engine L lower shaft MG1 motor / generator 1 MG2 motor / generator 2 c carrier ev electrical variator f fuel fd ﬁnal drive gn geared neutral pg planetary gear prim primary pulley r ring gear s sun gear sec secondary pulley tr transmission v vehicle vin variator in vout variator out wls wheels Chapter 1

Introduction

Every year more discussions arise about the global warming of the earth. This again got an extra boost since the documentary An Inconvenient Truth is released. Many scientists dedicate the global warming to the production of to much carbon dioxide (CO2) emissions. More voices say this is caused for a large part by the growing transportation sector. Passenger cars are a main part of this sector. People are urged to help solving this problem by using less fuel, e.g. drive more fuel economic cars. This is also encouraged by the rising oil prices. Which is caused by the oil getting scarce on one hand. On the other hand, events which attributed to previous spikes are the North Korea’s missile launches, the crisis between Israel and Lebanon, the Iranian nuclear brinksmanship and the Iraq war. People especially notice these rising oil prices by the increasing gas prices. The gas price over the past years can be seen in Fig. 1.1. The gas price of the year 2007 is slightly lower compared to the year 2006, however the year 2007 is only covering the ﬁrst quarter.

Governments stimulate more and more to awaken the people to use less oil. With respect to vehicles, fuel consumption guides are composed, e.g. Brandstof Verbruiks Boekje [2]. Next to stimulating the people to drive more fuel efficient with tips like keep the revolutions of the engine under 2500 rpm, drive constant velocities (use the cruise control) and check the tyre pressure every month, e.g. Het Nieuwe Rijden [14]. Understandably small vehicles are more fuel efficient, however people are used to luxurious and spacious vehicles. With a hybrid power train, mid-sized vehicles can reach the fuel efficiency of conventional smaller vehicles. However the social acceptance of hybrid vehicles is still low because hybrid vehicles are assumed to be slow, to have high maintenance costs and to be dan- gerous because of the high voltages present. Nevertheless the sales figures of the hybrid vehicles are rising. Nowadays more and more car manufacturers are producing hybrid vehicles. Next to even more concept hybrid vehicles which are developed by car manufacturers. In this section first the different classifications for hybrid vehicles are discussed. Next the different configurations for hybrid power trains are described.

150

140 ] l 130 100 / 120

110

100 gas price [euro

80 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

Figure 1.1: Development of the gas price (euro95) over the past years, data obtained from CBS [4]

1 2 CHAPTER 1. INTRODUCTION

Classiﬁcations

Hybrid vehicles can be divided into different classifications, ranging from conventional vehicle to plug- in hybrid vehicle. Normally a vehicle has no electric driving power, the higher the hybridisation, the more electric driving power the vehicle has. Like a start-stop system is not typically for only hybrid vehicles, because this can be realized by an alternation in the engine control unit (ECU) of mostly every conventional car. On the other hand, hybrid vehicles are generally spoken equipped with this feature, an example of a conventional vehicle with a start-stop system is the Citroën C3 Stop & Start [10]. In contrast to a plug-in hybrid vehicle which tends to a complete electric vehicle. Basically it should be possible to make a vehicle which is only powered by electric motors and ‘refuels’ by plugging-in. The classification which ranges from conventional to plug-in hybrid vehicle can be divided in a conventional vehicle, muscle-, mild-, full- and a plug-in-hybrid vehicle. This classification is listed in Table 1.1 and is further described by D. Friedman [5].

A muscle hybrid vehicle is not designed to improve the fuel consumption a lot however has extra power which is generated with the electric machines. Compared to equally sized conventional vehicles it uses less fuel. A mild hybrid vehicle is designed to improve the fuel consumption however has a small electric machine. It is not able to run on electrical power only. A full hybrid vehicle is like a mild hybrid vehicle with bigger electric machines, it is possible to drive on electrical power only. The last category, and the highest level of hybridisation, is the plug-in hybrid vehicle. This is the closest to the full electric vehicle, it can be plugged in to recharge the batteries and has a large range to drive on electric power only.

Table 1.1: Hybrid checklist: Is this vehicle a hybrid? From A New Road, The Technology and Potential of Hybrid Vehicles [5] conventional muscle mild full plug-in Does this vehicle... vehicle hybrid hybrid hybrid hybrid shut off the engine at traffic lights X X X X X and in stop-and-go traffic use regenerative braking and X X X X operates above 60 V use a smaller engine than a X X X conventional version with the same performance drive using only electric power X X recharge batteries from the wall X plug and have a range of at least 30 km on electricity alone

Conﬁgurations

The power trains of hybrid electric vehicles can be divided into three different configuration types. These configurations are series, parallel and series/parallel power train. In Fig. 1.2 these configurations are graphically presented together with a comparison of the gained effects. Typical hybrid functions are electric drive off, full electric driving, recharging of the secondary power source (e.g. battery) while driving, power assist, brake energy recovery (BER) and engine shut-off when no driving power is required.

Series hybrids

A series hybrid system consists of an Internal Combustion Engine (ICE) which drives a generator. The electric power generated can be stored in a battery or can directly be used to power an electric 3 motor. The vehicle is powered by an electric motor which uses this generated electricity to drive the wheels. This is called a series hybrid system because the power flows to the wheels in series, i.e. the engine power and the motor power are in series. A series hybrid system can run a small ICE in the efficient operating region relatively steadily, generate and supply electricity to the electric motor and efficiently charge the battery. It has two electric motors, one functions as a generator and the other as an electric motor. The advantage of a series hybrid power train is the ICE can operate at a constant speed so the fuel consumption is low. This system is very flexible as well, because only the ICE and generator are mechanical coupled, the motors can be placed in the wheels or wherever necessary. Disadvantages are the power from the ICE always goes over two electric machines, which both have a certain efficiency and during highway driving this can result in a lower efficiency compared to a conventional, mechanical transmission. This system is used in the TNO Hybrid Carlab.

Parallel hybrids

In a parallel hybrid system, both the ICE and the electric machine drive the wheels, and the drive power from these two sources can be utilized according to the prevailing conditions. This is called a parallel hybrid system because the power ﬂows to the wheels in parallel. In this system, the battery is charged by switching the electric machine to operate as a generator. To drive the vehicle with the electric power

Battery

Inverter

Engine Motor Generator

Reduction Drivewheels gear

Battery

Inverter

Transmission Engine Motor/ generator

Reduction Drivewheels gear

Battery

Generator Inverter Powersplitdevice

Engine Motor

Reduction Drivewheels gear

Figure 1.2: Three major types of hybrid configurations used in hybrid vehicles currently on the market, from The Fifth Toyota Environmental Forum [15] 4 CHAPTER 1. INTRODUCTION stored in the battery, the electric machine operates as motor. Although it has a simple structure, the parallel hybrid system cannot drive the wheels from the electric machine while simultaneously charging the battery since the system has only one electric machine. When the battery needs to be recharged while driving, an amount of the power delivered by the ICE goes to the electric machine. This configuration is, among other configurations, applied by Honda in the integrated motor assist (IMA) system and by Torotrak in their hybrid infinitely variable transmission (IVT) system.

Series/parallel hybrids A series/parallel hybrid system combines the series hybrid system with the parallel hybrid system in order to maximize the beneﬁts of both systems. It has two electric machines, and depending on the driving conditions, uses only one electric machine or the driving power from both the electric machines and the ICE, in order to achieve the highest eﬃciency level. Furthermore, when necessary, the system drives the wheels while simultaneously generating electricity using one electric machine as generator. A disadvantage of this system is its complexity, because the power train needs to combine the power from the ICE and both the electric machines. The advantage of this system is it can cover all hybrid functionalities depending on the sizes of the electrical machines and the electrical storage system. This is the system used in a lot of hybrid systems nowadays, among with the Toyota Prius, Lexus GS450h, Ford Escape Hybrid and the Mercedes-Benz S-Classe Hybrid.

Commercial hybrid vehicles More and more car manufacturers have hybrid vehicles in their range of products. Audi started in 1994 with the Audi 80 Duo, because the price was to high it could not be sold, some years later Audi introduced the A4 Duo. After selling 90 items for still a high price, Audi decided to stop production and to explore the possibilities of direct diesel injection to improve fuel consumption. The ﬁrst commercially successful hybrid vehicles are the Toyota Prius and Honda Insight.

The Toyota Prius, a four-door sedan, is equipped with the Toyota Hybrid System (THS), a series/parallel hybrid power train. Toyota also developed the THS-C (THS with a continue variable transmission (CVT)) for the Estima and Alphard minivan. Next to the THS-M (mild THS) in the Crown sedan. Over the years the THS is improved considering fuel consumption and driving performance which resulted in the THS II. This hybrid power train is on its turn adjusted to ﬁt into larger vehicles. In several sport utility vehicles (SUVs) of Toyota and Lexus the THS II for SUVs is available now. These vehicles are available in a 2WD and 4WD model, the 4WD model is realized with an extra electric machine on the rear wheels. Lexus adapted the THS II for a front engine, rear-wheel driven vehicle, the GS450h.

The Honda Insight, a small two-seater, is equipped with the IMA system, a parallel hybrid power train developed by Honda. The IMA is enlarged to reach the required driving performance for the Civic Hybrid and the Accord hybrid.

The hybrid model by Ford is the Escape Hybrid. This has a similar power train as the THS II for SUVs however is designed by Ford itself. The Escape Hybrid is available in a 2WD and 4WD model. With the Mercury brand, Ford brought the Mariner Hybrid on the market which is a sibling of the Escape Hybrid, however only in a 4WD model available.

Remarkable is the point of view from which a hybrid vehicle is developed, like Toyota and Honda developed a completely new vehicle. Toyota developed a mid-sized hybrid vehicle to reduce emissions, especially in urban driving. On the other hand, Honda developed a small hybrid vehicle to reach the best fuel economy as possible. Both Toyota and Honda optimized the power train for a new model and later implemented this power train into new versions of existing models. Other car manufacturers develop concept vehicles based on existing models equipped with hybrid power train concepts. 5

Another interesting point is the level of hybridisation (loh) of a hybrid vehicle. In Fig. 1.3 the level of hybridisation is shown for the commercially available hybrid vehicles. The level of hybridisation is deﬁned as in Eq. (1.1). P loh = EM (1.1) PICE

Where PEM stands for the maximum power delivered by the electric machine, when the power train consists of two electric machines the electric machine is depicted which mainly operates as motor. PICE is the maximum power delivered by the ICE. Clearly Honda has a low level of hybridisation, combined with relatively low vehicle weight. The second generation Toyota Prius has the highest level of hybridisation and still low vehicle weight, especially compared to the vehicles with just a little lower level of hybridisation. These all have a higher vehicle weight. The most desirable region is the top left corner, here the vehicle weight is low and the level of hybridisation is high. Low vehicle weight is an important factor because the fuel consumption increases drastically with increasing vehicle weight. In the contrary a hybrid vehicle is heavier compared to a conventional vehicle. This is caused by the electrical storage system and the electric machines. However a hybrid vehicle gains more fuel eﬃciency compared to the costs of fuel economy caused by the extra vehicle weight.

0.9 Toyota Prius II ↑ Toyota Highlander4WD ↓ 0.8 ← Lexus RX400h Toyota Highlander 2WD ↑ Ford Escape 2WD ↓ 0.7 Toyota Prius → ← Ford Escape 4WD Mercury Mariner ↑ ← Lexus GS450h

0.6 Toyota Kluger → ← Toyota Harrier

0.5 Toyota Estima →

loh [−] 0.4 ← 0.3 Audi A4 Duo

← Honda Civic 0.2 ← Honda Insight ← Suzuki Twin 0.1 Toyota Alphard → Hond Accord → 0 800 1000 1200 1400 1600 1800 2000 2200 2400 Vehicle weight [kg]

Figure 1.3: Level of hybridisation for commercially available hybrid vehicles

Outline of the report Almost all car manufacturers have carried out studies or are investigating the possibilities of diﬀer- ent hybrid power trains. Research institutes and automotive suppliers are investigating hybrid power trains as well. Some interesting power trains are the Bosch Dual-E transmission, the Renault IVT, the TNO Hybrid Carlab and the LuK power split CVT for instance.

The Bosch Dual-E transmission is a six speed automated manual transmission (AMT) equipped with two electric machines to create a continuously ratio change between the gears, on top of this hybrid functions are gained. The Renault IVT is a dual mode IVT realized with a combination of planetary gears and two electric machines. Depending on the electrical energy storage capacity this power train ranges from pure transmission to a full hybrid power train. TNO realized a series hybrid vehicle based on a Volkswagen Beetle, herein a diesel ICE and two electric machines result in the power train. With this vehicle the influence of different control strategies and the size of different power train components is investigated. LuK on the other hand developed a power split transmission based on a chain CVT, a planetary gear and two fixed gear ratios. Next to an electric machine which can fulfill most hybrid functionalities. 6 CHAPTER 1. INTRODUCTION

Since the oil is getting more scarce every year and the oil price is rising, it becomes more important to improve the fuel efficiency of vehicles. A hybrid power train is a viable solution to improve the fuel consumption of vehicles. To realize a hybrid power train different configurations are possible. The most used configuration is the series/parallel hybrid power train, which is realized by means of power split transmissions.

The main goal of this thesis is to investigate the construction and functionality of different power split transmissions. And the emphasis lays on the influence of the power train components on the overall power train efficiency of a power split transmission compared to a conventional transmission.

To achieve this goal first three methods to investigate power split transmissions are discussed. The transmission developed by Bosch is investigated by M. Schulz [13]. Especially he is interested in circulating power flow and how to determine the region where no circulating power flow occurs. This in order to develop a strategy leading to a fuel-efficient vehicle operation. The power split transmission developed by Renault is investigated by A. Villeneuve [17]. Who looks at the ratio between the electrical variator power and the ICE power to investigate the transmission. Which also gives an indication for the circulating power flow in the transmission, however Villeneuve tolerates some circulating power flow. In contrast to Schulz and Villeneuve, P. Mattsson [8] derived a method to determine suitable values of the basic speed ratios for a general CVT. In order to derive this method Mattsson first investigates several power split transmissions. In Chapter 2 these different methods are discussed.

Next a closer look is taken at two power split transmissions, the THS and the Renault IVT. First the kinematics are derived for these transmission and secondly these transmission are described by the method derived by Mattsson. This is discussed in Chapter 3. At the end a detailed efficiency analysis is made. Hereto the THS and the push-belt CVT are investigated, especially the power train component efficiencies and their influence on the overall power train efficiency. This is discussed in Chapter 4. Finally, in Chapter 5 the conclusions and recommendations are given. Chapter 2

Methodologies

The most commercially successful hybrid vehicles are equipped with a power split transmission. Power split transmissions like the name suggests split the power in the transmission. This is applied to achieve high eﬃciencies combined with outstanding driving comfort. The eﬃciency is high because the mainstream of the power is transmitted through a kinematic chain, normally consisting of one or more planetary gears combined with some gear stages. The rest of the input power passes through a variator which controls the transmission ratio and herewith the working point of the ICE. In case of an electric variator, hybrid functionalities are easy to implement. In case of a mechanical variator, hybridisation is somewhat more complicated however not impossible. The driving comfort is characterized by the transmission ratio which is continuously variable and herewith considered to be very comfortable.

In literature various power split transmissions are investigated, in this chapter three publications are discussed. First the hybrid power split transmission developed by Bosch is discussed, M. Schulz [13] investigates this transmission by means of power flow and especially circulating power flow. Next the hybrid power split transmission developed by Renault is discussed, A. Villeneuve [17] investigates this transmission by means of power flow as well, however in a different manner. Finally the publication of P. Mattsson [8] is discussed. This publication reports a general description of power split transmissions and how to quantify them.

First some general notations used in this thesis are presented. The overall speed ratio used in this thesis is deﬁned as the output speed over the input speed, Eq. (2.1). Where the wheels are the output of the transmission and the ICE functions as input.

ωwls rtr = (2.1) ωICE

Where rtr stands for the overall speed ratio. The rotational speed of the wheels and ICE are represented by ωwls and ωICE respectively. In power split transmissions normally the power is split over a kinematic chain and a variator path. An interesting parameter is the ratio between the variator input speed and the variator output speed. This is shown in Eq. (2.2).

ωvout rev = (2.2) ωvin

Where rev is the electric variator speed ratio. The rotational input and output speeds of the electric variator are represented by ωvin and ωvout respectively. Next to these speed ratios, power ratios are useful as well to indicate the power split transmissions. The power split ratio is deﬁned as the electric variator input power over the ICE input power, Eq. (2.3). P Φ = vin (2.3) PICE

7 8 CHAPTER 2. METHODOLOGIES

Where Φ stand for the power split ratio and the power delivered by the ICE is represented by PICE.

The power which goes into the electric variator is indicated with Pvin .

This chapter reviews the literature concerning the usefulness of using the methods derived by M. Schulz, A. Villeneuve and P. Mattsson.

2.1 Method by Schulz

The Robert Bosch GmbH company developed the Dual-E transmission, a hybrid power train based on a six speed automated manual transmission (AMT). This power train is investigated by M. Schulz [13]. In Fig. 2.1(a) the schematic representation of the Dual-E transmission is shown. Where the crankshaft is connected to the ICE and the output shaft is connected to the driven wheels. The loop-like arrange-

rMG2 MG2

rA2

MG1

PMG1 C1 P1 L

rA1 PC1 PL

ICE wls MG1 P P ICE P P wls rMG1 C2 H C2P2 H

P MG2 MG2 (a) Schematic representation of the Dual-E transmis- (b) Internal and external power ﬂow of the sion Dual-E transmission

Figure 2.1: Bosch Dual-E transmission, [13] ment of the shafts give the supposition circulating power may occur in this system. The knowledge of the region where circulating power occurs is essential for developing a fuel efficient strategy. Hereto first the power flow in the transmission is investigated and followed by the kinematics. In Fig. 2.1(b) the internal and external power flows are depicted. The energy balance can be formulated by the linear algebraic Eq. (2.4), herein the inertia effects, compliance of the members and the mechanical power losses are neglected. Ax = b (2.4) With  1 1 0 0   0 0 1 1  A =   (2.5)  −1 0 1 0  0 −1 0 1

T x = PC1 PC2 PL PH (2.6)

T b = PICE Pwls PMG1 PMG2 (2.7) 2.1. METHOD BY SCHULZ 9

Where x is the internal power flow and b is the external power flow. The ICE and the two electric machines provide the power input to the transmission, PICE, PMG1 and PMG2 respectively. Together with the output power, Pwls, they represent the external power flow. The power of the ICE is split into PC1 and PC2 driving the carriers of the two planetary gears P1 and P2 respectively. The outputs PL and PH from the ring gears of the planetary gears drive the two countershafts L and H. The coefficient matrix A is singular.

An analysis shows that each solution can be expressed as a linear combination of three basic solutions and one solution which represents cyclic power flow. The internal power flow is not uniquely defined by Eqs. (2.4) to (2.7), a detailed investigation of the kinematics and dynamics of the transmission is necessary. The kinematics for a planetary gear are derived in App. A. Combining these equations with the constraints imposed by the meshing gears result in the equations for the rotational speeds of the EMs, which depend on the rotational speed of the ICE and the output shaft. This results in Eq. (2.8), an elaborate clarification can be found in App. B. ω −a b ω MG1 = (L) ICE (2.8) ωMG2 −c d(H) ωwls

The derivation of the constants a, b(L), c and d(H) can be found in App. B. Neglecting the mechanical losses and the inertia of the power train components, the relations for the torques result into Eq. (2.9).

T a c T ICE = MG1 (2.9) Twls −b(L) −d(H) TMG2

Where the constants a and c are independent of the engaged gear. In contrast to the constants b(L) st nd and d(H) which are dependent of the engaged gear. When the 1 and 2 gear are engaged this results for the torque relations into Eq. (2.10). T 7.2 7.1 T ICE = MG1 (2.10) Twls −25.8 −17.6 TMG2 For a graphical representation of the operating range free from circulating power, the ratio of the total mechanical power of the EMs to the output power is introduced, Eq. (2.11).

PMG1 + PMG2 λE = (2.11) Pwls

1.5 1st gear 2nd gear 1 3rd gear 4th gear th 0.5 5 gear 6th gear ] − [ 0 E λ

-0.5

-1

-1.5 0 1 2 3 4 5 6 7 8

1/rtr [−]

Figure 2.2: Overview of the power ratio for each discrete transmission gear 10 CHAPTER 2. METHODOLOGIES

Where the power ratio is represented with λE. The power from the electric machines is represented with PMG1 and PMG2, where the output power is given by Pwls. In Fig. 2.2 the power ratio is plotted against the inverse of the overall speed ratio. The inverse of the overall speed ratio is taken because Schulz opts for this variable to indicate the circulating-power-free region. Each line represents one discreet ratio of the six speed automated manual transmission used in the Bosch Dual-E transmission.

The region where no circulating power ﬂow occurs is in between two half lines, e.g. in between the lines representing the 1st and the 2nd gear (the two top-right lines). When the transmission is in the st st 1 gear, TMG2 is zero. So when the transmission is purely in the 1 gear the electric power is only delivered or consumed by MG1. In contrast to the line which represents purely the 2nd gear where TMG1 is zero. In between the the two half lines both the EMs are operating. The half line representing rd the 3 gear, again TMG2 is zero, analogously for the rest of the discreet gear ratios. A more elaborate clariﬁcation about determining the circulating-power-free operating range can be found in Schulz [13].

When λE is zero, the output power is completely delivered by the ICE. If the half line of a gear intersects with the line where λE is zero, both the EMs do not cooperate. In the other hand, in between two half lines and on the line where λE is zero, the sum of the powers generated by the EMs is zero so the power generated by one EM is completely consumed by the other EM. On the basis of this knowledge, a fuel-eﬃcient operating strategy can be developed, taking into account that circulating power leads to high mesh losses.

2.2 Method by Villeneuve

Renault studied the possibilities of power split automatic transmission architectures in order to se- lect the most flexible and efficient variator. Their conclusion is an electric variator based upon two electric machines is one of the best solutions considering the existing technologies. At first a single mode transmission was developed, however the ratio range is limited and the EMs have to be large to cope with the power passing through the variator. The drawback of large EMs is they are heavy and expensive. Hereto a dual mode power split transmission is designed, one mode is dedicated to the low vehicle speed range and the second mode to the high vehicle speed range. This reduces the power passing through the variator and herewith the size of the EMs. The mode change has to be

Electric Machine1

r2 F1 ICE F2

C B A2 A1 r1 Damper

rfd

Differential wls wls Electric Machine2

Figure 2.3: Schematic representation of the Renault IVT 2.2. METHOD BY VILLENEUVE 11 transparent and seamless to the driver because it is not really a gear change and is not made at the driver’s request. Hereto a set of rules and conditions are developed to optimise the mode change. In Dual mode electric inﬁnitely variable transmission [17] A. Villeneuve describes the development of the dual mode Renault IVT. The resulting design is shown in Fig. 2.3.

The transmission can be divided in a kinematic chain and an electric variator. The kinematic chain consists of gear stages and planetary gear sets. This kinematic chain has two inputs and two outputs, when two input speeds are set the output speeds can be calculated according to Eq. (2.12).

ω a b ω ω ICE = MG1 = M MG1 (2.12) ωwls c d ωMG2 ωMG2

When the inertia eﬀects, compliance of the members and the mechanical power losses are neglected, the torques can be calculated according to Eq. (2.13).

T T MG1 = −MT ICE (2.13) TMG2 Twls

Because the Renault IVT is a dual mode transmission, the matrix M needs to be deﬁned for each mode. The elaborate clariﬁcation can be seen in App. C. For mode 1 this results in Eq. (2.14) and for mode 2 this results in Eq. (2.15).

0.4424 −0.4310 M = (2.14) 1 0.0231 −0.1083

−0.3119 −0.4310 M = (2.15) 2 −0.1664 −0.1083

As can be seen, the constants b and d are the same for each mode. This corresponds to the rule which ensures the power through the electric variator is zero during mode change. Since ωMG1 = ωMG2 = 0. Other rules are the overall in- and output speed should continue during mode change which implies the mode change should occur at a given transmission ratio. The overall transmission ratio is the same at this point for mode 1 and mode 2. Next to the rule of continuity of the in- and output speed of the electrical variator during mode change.

Table 2.1: Power circulation in the Renault IVT

−1 < Φ < 0 Φ < 0 negative power recirculation

Φ < −1

0 < Φ < 0.5 true power split Φ > 0

0.5 < Φ < 1

Φ > 1 positive power recirculation 12 CHAPTER 2. METHODOLOGIES

To analyze the Renault IVT, the power split ratio defined in Eq. (2.3) is used. When the variator input is working as a motor, Pvin is negative. When the variator input is working as a generator, Pvin is positive. The possible power flows are listed in Table 2.1. Negative circulating power flow is undesirable because this implies larger EMs and a lower efficiency of the power train. The desired operating range is true power split to keep the size of the power train components relatively small and ensure the transmission efficiency. However little negative power recirculation is tolerated in order to obtain a sufficient ratio range.

When the Renault IVT purely operates as transmission the electrical storage system is omitted. This is investigated with Fig. 2.4. As can be seen, the power split ratio does not exceed the value ± 0.5. This means the power through the electric variator is less then half the power delivered by the ICE. Only at low transmission ratios the power split ratio tends to inﬁnity. However no losses are taken into account here. If losses will be implemented, the power split ratio will no longer tend to inﬁnity. This is caused by the ICE power which is no longer equal to zero. The advantage is the ICE can not deliver zero power when rotating, driving the power output to zero is possible by organising the losses in the transmission to exactly compensate the ICE power. Logically the ICE power is as low as possible in this situation.

At low transmission ratios it can also be seen the power split ratio is negative which implies negative power recirculation. However the ﬁrst node, where the power ratio is equal to zero, is designed to be as close as possible to a transmission ratio of zero. Herewith the transmission will spend only limited time in negative power recirculation mode.

With increasing transmission ratio the power split ratio increases as well. The power split ratio increases up to nearly 0.5, which means almost half the power delivered by the ICE is transmitted through the electrical variator. When the transmission ratio increases further, the power split ratio decreases and becomes even zero. In this point the transmission switches from mode 1 to mode 2.

In mode 2 the power split ratio is negative and reaches nearly a value of −0.25, which means almost one quarter of the power delivered by the ICE recirculates via the electrical variator in negative sense. At increasing transmission ratio, the power split ratio increases and becomes positive again. This means the transmission reaches true power split mode.

1.5

0.5 ]

− 0 Φ [ -0.5

-1

-1.5

-2 0 0.1 0.2 0.3 0.4 0.5

rtr [−]

Figure 2.4: The power split ratio versus the overall transmission ratio for the Renault IVT 2.3. METHOD BY MATTSSON 13

v v in variator out

in out kinematicchain

Figure 2.5: General power split transmission

2.3 Method by Mattsson

In order to model continuously variable power split transmissions, P. Mattsson analysed the speed and torque relations in a general matter for several power split transmissions [8]. In Fig. 2.5 the schematic representation of a power split transmission is shown. The transmission is assumed to be a loss-free transmission with linear speed and torque relations. It has two speed degrees of freedom and two torque degrees of freedom. The speed relations can be written according Eq. (2.16). ωvin a b ωin ωin = · = Iω · (2.16) ωvout c d ωout ωout

The torque relations can be written as Eq. (2.17). Tin t Tvin −a −c Tvin = −Iω · = · (2.17) Tout Tvout −b −d Tvout

The overall transmission ratio is defined as in Eq. (2.1). With the speed relations defined as in Eq. (2.16) and the variator ratio defined as in Eq. (2.2) this results in Eq. (2.18).

a · rev − c rtr = − (2.18) b · rev − d The power split ratio can also be described with the relations in Eqs. (2.16) and (2.17), this results in Eq. (2.19). The elaborate clariﬁcation can be found in App. D. P −T · ω r a · d − b · c Φ = vin = vin vin = − ev · (2.19) Pin Tin · ωin a · rev − c b · rev − d Assume that a power split transmission is designed so that the amount of power transmitted through the variator is always less than the input power. It can be shown that the overall speed ratio range of the transmission will be smaller than the speed ratio range of the variator. Hereto it is important to investigate the variator transmission ratio and the power split ratio. An elaborate derivation is given in App. D. 14 CHAPTER 2. METHODOLOGIES

2.4 Discussion methodologies

In this section three methods are discussed. Schulz and Villeneuve are each subjected to one specific power split transmission. On the other hand Mattsson uses his method in a much more general manner, this is applicable for any kind of power split transmission in a straight forward manner. On top of this Mattsson evaluated his method with a simplified loss model as well. When the general equations for a power train are derived the efficiencies are easily to implement. Mainly therefore this method is used in the next section to investigate the THS and the Renault IVT. Because for both power trains the same method is adopted, the power trains can be compared with each other.

All three methodologies discuss the importance of circulating power flow. The method by Schulz is to determine the region where no circulating power flow occurs. In his opinion circulating power highly influences the overall transmission efficiency and is therefor undesirable. The method by Villeneuve on the other hand investigates the amount of circulating power. The transmission is designed in such a way this circulating power is within proportion and the overall speed ratio is reasonable. The main difference between the Bosch Dual-E transmission and the Renault IVT is, Bosch has five driving ranges and Renault has only two driving modes. The method by Mattsson is comparable to the method used by Villeneuve. However the variator used by Mattsson is not an electric variator necessarily, this can also be a mechanical variator like a push-belt CVT. The main goal of Mattsson is to determine the dimensions of the transmission considering the power split ratio combined with the overall speed ratio and the variator speed ratio. Chapter 3

Analyzing IVTs

In the previous chapter each method was used to analyze a diﬀerent power split transmission. In this chapter two transmissions are investigated by the method derived by Mattsson. The two transmissions are the Toyota hybrid system (THS) and the Renault IVT. The method by Mattsson is chosen because herewith a simple analysis can be made without losses and on top of this eﬃciencies can be implemented in a very straightforward manner. However to have a benchmark, initially the kinematics of the two transmissions are derived and analyzed.

3.1 Kinematics

In order to analyze the THS and the Renault IVT the kinematics are derived. The dynamics are neglected, thus no inertias and spring-damper eﬀects are considered. This means considering the in- and output power these are equal to each other. First the THS is analyzed using the kinematics, followed by the Renault IVT.

3.1.1 Toyota hybrid system The THS is shown in Fig. 3.1(a) with the planetary gear enlarged. The electrical system, like battery, supercapacitors and inverter, is not shown here. In this thesis the THS is investigated at constant vehicle speeds only. When the THS operates at a constant vehicle speed, in a steady state point, the battery is not discharged or recharged. When the battery would be recharged in a steady state operating point, the battery would be recharged completely. As a consequence the operating point of the power train

c pg r s

MG1 in out batt rfd MG2

electricpath mechanicalpath (a) Schematic representation of the THS, [16] (b) Simpliﬁed representation of the THS

Figure 3.1: Toyota hybrid system

15 16 CHAPTER 3. ANALYZING IVTS would change because the battery can not obtain more electric power. Correspondingly is the situation when the battery would be discharged in a steady state operating point. The control strategy would notice the battery depleting and at a certain state of charge (SOC) level the working point would be changed to recharge the battery.

The electrical energy storage system (e.g. battery) is not taken into account because only steady state operation points are considered. This means the power generated by one EM is completely consumed by the other EM, considering the eﬃciencies to be 100 %. As one EM operates as motor, the other operates as generator. If losses are taken into account the generator has to compensate the losses for both the EMs. The generator in Fig. 3.1(a) is referred to as MG1 and the motor is referred to as MG2 because both the EMs can operate as motor and generator.

The planetary gear is often referred to as the power split device. By deﬁnition the sum of the powers acting on the planetary gear is always equal to zero. Hereto the power is split into two paths or brought together from two paths. In Fig. 3.1(b) a simpliﬁed representation of the THS is shown. The kinematics of a planetary gear are derived in App. A, in Eq. (3.1) these are listed to recapitulate.

zr rpg = zs ωs + rpgωr − (1 + rpg) ωc = 0 rpgTc + (1 + rpg) Tr = 0 (3.1) Tc + (1 + rpg) Ts = 0 Tr − rpgTs = 0

Where rpg is the planetary gear ratio, with z the number of teeth of a gear, ω stands for the rotational speed. The subscripts refer to the diﬀerent parts of the planetary gear, with r the ring gear, s the sun gear and c the planet carrier.

With MG1 the working point of the ICE is inﬂuenced via the planetary gear. In Fig. 3.2 two engine operating points at identical vehicle speed are schematically represented in a nomograph. In Fig. 3.2(a) the sun gear has positive rotational speed, hence MG1 operates as generator. In Fig. 3.2(b) the working point of the ICE (connected to the carrier) is inﬂuenced by lowering the rotational speed of MG1 (connected to the sun gear) compared to Fig. 3.2(a). Because the electrical energy storage system is not taken into account and no losses considered, the ICE provides the road load power. The power on the carrier is for both situations the same. Because the rotational speed of the carrier is lowered, the torque on the carrier increases. With Eq. (3.1) the torques acting on the rest of the planetary gear are calculated. For both the sun gear and the ring gear the torques increase. The situation in Fig. 3.2(a) corresponds with the situation shown in Fig. 3.4. The situation in Fig. 3.2(b) corresponds with the situation shown in Fig. 3.5(b). To drive in reverse with the ICE turned on, the carrier of the planetary gear has a positive rotational

sun carrier ring sun carrier ring

gear gear gear Tr gear w w w w w w s c r s c r T s Tr

Tc Ts 0 + 0 + torque, torque, speed speed

rpg rpg 1 Tc 1

(a) Positive speed of the sun gear (b) Negative speed of the sun gear

Figure 3.2: Nomographs of the planetary gear during forward driving with the ICE on 3.1. KINEMATICS 17

sun carrier ring sun carrier ring gear gear gear gear w w w w w w s c r s c r

Ts Tr

+ + T torque, T c torque, T speed s speed 0 r 0

rpg 1 rpg 1 (a) Planetary gear relations during reverse (b) Planetary gear relations in the geared neu- driving tral ratio

Figure 3.3: Nomographs of the planetary gear during reverse driving and in geared neutral ratio speed. The ring gear needs a negative rotational speed to drive in reverse, so the generator needs to compensate the velocity diﬀerence, thus always has a positive rotational speed and operates as a generator. This is shown in Fig. 3.3(a). The speed of the sun gear increases very fast when the carrier speed only increases a little. In Fig. 3.5(a) the situation is shown, corresponding to the situation in Fig. 3.3(a). The THS is designed to drive reverse on electric power only, normally the ICE will not be started. Another interesting situation is the geared neutral ratio (GNR). This is the transmission ratio for which the power only is transmitted via the mechanical path. For the planetary gear this means no power goes via the sun gear. This is realized by putting the rotational speed of the sun gear to zero. This is shown in Fig. 3.3(b). However the torque acting on the sun gear is not zero. The GNR for the complete power train of the THS is given by Eq. (3.2).

(1 + rpg) rgn = rfd (3.2) rpg

Where rgn stands for the geared neutral ratio and rfd stands for the ﬁnal drive ratio. The power ﬂow in the THS can be divided into several situations. During normal driving, the engine delivers power and is assisted by the electric motor as shown in Fig. 3.4. In this situation the battery can be charged or discharged, whatever necessary. When cruising, driving a constant speed, the ICE delivers the power to drive the wheels. The power generated by MG1 is consumed by MG2 to assist the ICE. To recharge the battery when cruising the ICE delivers more power then necessary and the extra power is converted into electric energy by MG1 which is stored in the battery.

Another situation can be seen in Fig. 3.5(b), here circulating power flow occurs. This means power circulates inside the transmission. Circulating power flow is undesired because the overall transmission efficiency will decrease drastically. Therefore this situation is avoided in the THS.

c pg r s

MG1 in batt fd out MG2

electricpath mechanicalpath power

Figure 3.4: Hybrid vehicle propulsion 18 CHAPTER 3. ANALYZING IVTS

c pg r c pg r s s

MG1 MG1 in batt fd out in batt fd out MG2 MG2

electricpath electricpath mechanicalpath mechanicalpath power power (a) Reverse driving with the ICE turned on (b) Forward driving and MG1 operates as a motor

Figure 3.5: Circulating power ﬂow

In Fig. 3.6 two situations for electric driving are shown, these contain speciﬁcally BER, electric forward and reverse driving. When braking or releasing the throttle, the THS recovers the kinetic energy of the vehicle. This is realized by means of MG2 which operates as a generator in this situation. The rest of the power train is not operating. Full electric driving is also realized with the MG2, the possibility to drive electric is highly dependent on the SOC of the battery. When the SOC is low the ICE will be started to drive the vehicle and to recharge the battery.

c pg r c pg r s s

MG1 MG1 in batt fd out in batt fd out MG2 MG2

electricpath electricpath mechanicalpath mechanicalpath power power (a) BER (b) Electric driving

Figure 3.6: Power ﬂow with the ICE turned oﬀ

In Fig. 3.7 the power ﬂow can be seen which occurs in engine only mode. This situation is like in GNR, however in GNR the MG2 can generate or consume electrical power in hybrid mode. In hybrid mode a battery is present, however this will not be further discussed in this thesis. In App. E the kinematics for the THS, pure functioning as transmission, are derived. The electrical storage system is neglected and herewith the hybrid functions are not investigated. The relations between the rotational speeds

c pg r s

MG1 in batt fd out MG2

electricpath mechanicalpath power

Figure 3.7: Engine only propulsion 3.1. KINEMATICS 19

Table 3.1: Gear ratios in the THS

rpg rfd 78 36 44 75 30 39 30 26 are shown in Eq. (3.3).

vv ωwls = rwls

ωfdout = ωwls ωfdout ωfd = in rfd (3.3) ωr = ωMG2 = ωfdin ωs = (1 + rpg) ωc − rpgωr = ωMG1 ωc = ωICE

The relations between the torques are derived with the help of power conservation and the results are shown in Eq. (3.4).

TICE + Tc = 0 Tc + (1 + rpg) Ts = 0 Tr − rpgTs = 0 Ts + TMG1 = 0 (3.4) Tfdin + TMG2 + Tr = 0

sign(Pfdin ) Tfdin ηfd + rfdTfdout = 0

Tfdout + Twls = 0 Twls + Trl = 0

The gear ratios stated in the equations above are listed in Table 3.1. To analyze the THS, the dimensionless power ratio of the electric variator over the ICE, versus the overall transmission ratio is plotted in Fig. 3.8(a). This power ratio is a measure for the proportion of mechanical power which is converted into electric power. Because the THS is considered in transmission mode only, circulating power ﬂow occurs when this value is negative. In case the power ratio is equal to zero, the speed ratio is equal to the GNR. In Fig. 3.8(b) the speed ratio between the EMs (rev) is shown. When the speed of MG1 goes

2 5 4 1.5 3 1 2 0.5 1 ] ] − [ − 0 0 ev Φ [ r -0.5 -1 -2 -1 -3 -1.5 -4 -2 -5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

rtr [−] rtr [−] (a) (b)

Figure 3.8: The power split ratio and the electric variator ratio versus the overall transmission ratio for the THS power train 20 CHAPTER 3. ANALYZING IVTS to zero the ratio goes to infinity, for MG1 this means it is going to switch from generator to motor. In case this ratio is positive, positive power split occurs in the planetary gear. Positive power split means the power from the ICE is split to the MG1 and to the final drive. In case the ratio is negative, circulating power flow occurs. The overall speed ratio is shown for positive values only, this corresponds with forward driving. The rotational speed of MG2 is positive as well.

Electric Machine1

r2 F1 ICE F2

C B A2 A1 r1 Damper

rfd

Differential wls wls Electric Machine2

Figure 3.9: Schematic representation of the Renault IVT

3.1.2 Renault IVT

The Renault IVT is a dual mode power split transmission which is brieﬂy discussed in Section 2.2. In Fig. 3.9 the schematic representation of the Renault IVT is shown. Here this transmission is further investigated by means of the kinematics. To derive the kinematics of this system it is divided into two modes. In mode 1, brake 1 is closed so the carrier of the planetary gear C is locked. In mode 2, brake 2 is closed so the ring of the planetary gear B and the sun of the planetary gear C are locked. The switch between the modes occurs when the rotational speeds of the clutches are equal to zero. Therefore the dynamics of the clutches are not taken into account in this analysis.

cz1 rz1 A1 sz1 c B r B B

r3 Brake2 MG2 in out r1 rfd MG1

Brake1 r2 sC c C C rC

sZ2 electricpath rZ2 A2 cZ2 mechanicalpath

Figure 3.10: Simpliﬁed representation of the Renault IVT 3.1. KINEMATICS 21

c1 r1 c1 r1 A1 A1

s1 s1 c c B r B r s s

r3 Brake2 r3 Brake2 MG2 MG2 in out in out r1 rfd r1 rfd MG1 MG1

Brake1 Brake1 r2 s r2 s c c r C r C

s2 s2 A2 electricpath A2 electricpath r2 c2 mechanicalpath r2 c2 mechanicalpath power power (a) Mode 1 (b) Mode 2

Figure 3.11: Power ﬂow in the Renault IVT

In Fig. 3.10 the schematic representation of the Renault IVT is shown. The resemblance with the THS can be seen, however the Renault IVT is more complex. It has more planetary gears, reduction gears and even two clutches. The EMs in the Renault IVT always operate in opposite function, when one EM operates as generator the other EM functions as motor. In Fig. 3.11 the possible power ﬂow in the Renault IVT is shown. In mode 1, a part of the power splits into one part via the electrical variator path and the rest via the mechanical variator path. In mode 2 all the power is transferred via the electrical variator. With Fig. 3.9 the kinematics can be derived. In general, when no clutches are closed, the speed relations which hold are stated in Eq. (3.5).

ωcA1 ωrA2 ωICE = = r1 r1 ωsA2 ωrC ωMG1 = = r2 r2 ωsB ωMG2 = r3 ωwls ωr = = ωc A1 rfd A2

ωsA1 = ωcB (3.5) ωF 2 = ωrB = ωsC

ωF 1 = ωcC

ωsA1 = (1 + rA1) ωcA1 − rA1ωrA1

ωsA2 = (1 + rA2) ωcA2 − rA2ωrA2

ωsB = (1 + rB) ωcB − rBωrB

ωsC = (1 + rC ) ωcC − rC ωrC Secondly the torque relations which hold for the Renault IVT transmission are listed in Eq. (3.6).

TICE + r1TcA1 + r1TrA2 = 0

TMG1 + r2TsA2 + r2TrC = 0

TMG2 + r3TsB = 0

TrA1 − rA1TsA1 = 0

TcA1 + (1 + rA1) TsA1 = 0

TrA2 − rA2TsA2 = 0

TcA2 + (1 + rA2) TsA2 = 0 (3.6)

TrB − rBTsB = 0

TcB + (1 + rB) TsB = 0

TrC − rC TsC = 0

TcC + (1 + rC ) TsC = 0

TrB + TsC + TF 2 = 0

rfdTwls + TrA1 + TcA2 = 0 The gear ratios stated in the equations above are listed in Table 3.2. With these relations the kinematics for each mode are calculated, this results in Fig. 3.12. As can be seen, mode 1 is used in low ratios 22 CHAPTER 3. ANALYZING IVTS

Table 3.2: Gear ratios in the Renault IVT

r1 r2 r3 rA1 rA2 rB rC rfd 31 27 30 75 67 63 79 41 − − 55 51 26 41 25 29 45 67 and hereby also known as the low velocity mode. Mode 2 is used in higher ratios and corresponds to the high velocity mode. The switch from one mode to the other always occurs in the point where the overall transmission ratio is the same for mode 1 and mode 2. In this point the power split ratio is for both modes equal to zero. Hence the power through the clutches is equal to zero, which is realized by zero rotational speed of the clutches.

2 5

1.5 4 3 1 2 0.5 1 ] ] − − 0 [ 0 ev Φ [ r -0.5 -1 -2 -1 -3 -1.5 mode 1 mode 1 -4 mode 2 mode 2 -2 -5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

rtr [−] rtr [−] (a) (b)

Figure 3.12: Analysis Renault IVT power train

3.1.3 Comparison THS & Renault IVT With the above derived equations the transmissions now are compared with each other. The overall speed ratio is deﬁned as in Eq. (2.1). Together with the power ratio, deﬁned as in Eq. (2.3), the transmissions can be represented as in Fig. 3.13(a). Villeneuve states the Renault IVT transmission is designed to maintain the power through the electrical variator under 50% of the power delivered by the ICE. With this taken into consideration, it can be seen the THS stays under 100% for the same region where the Renault IVT stays under 50%. An extra remark has to be made because Villeneuve also states the hybrid functionality depends on the amount of energy storage capacity. In this thesis the Renault IVT is analyzed in pure transmission mode, i.e. all the power generated by one EM is used by the other EM so no power is stored into or used from a battery or any kind of secondary power source. Therefore the THS is also investigated in pure transmission mode. When the electric variator ratio versus the overall ratio is plotted for both transmissions, it is remarkable how both transmissions start around the same value for the electric variator ratio and when the overall ratio is over 1, again both transmissions go to about the same value for the electric variator ratio. 3.2. MATTSSON 23

2 5

1.5 4 3 1 2 0.5 1 ] ] − − 0 [ 0 ev Φ [ r -0.5 -1 -2 -1 -3 -1.5 THS THS -4 Renault IVT Renault IVT -2 -5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

rtr [−] rtr [−] (a) (b)

Figure 3.13: The THS and Renault IVT compared by means of power split ratio and variator ratio

3.2 Mattsson

As discussed in Section 2.3 Mattsson derived a general manner to investigate power split transmission. In this section the THS and Renault IVT will be described by this method. Hereto the Eqs. (2.16) to (2.19) are used.

3.2.1 Toyota hybrid system In this section the THS is analyzed according to the method of Mattsson. With the kinematics and the general equations which hold according to Mattsson. The characteristic power train matrix results into Eq. (3.7). " # 1 1 3.6 −10.15 I = (1+rpg ) rpg rfd = (3.7) ω 0 1 0 3.9 rfd

With these deﬁnitions the same results are obtained as in Section 3.1.

3.2.2 Renault IVT To analyze the Renault IVT with the method derived by Mattsson it has to be divided into two parts, one for each mode. In mode 1, brake 1 is closed and brake 2 is open, so the carrier of planetary gear C stands still, herewith the planetary gear C functions as a simple gear ratio. For mode 1 the matrix Iω results in Eq. (3.8). " # − r1rA2 (1+rA2) I = r2 r2rfd (3.8) ωI r1 1 ((1 + rA1) (1 + rB) − rA2rBrC ) (rBrC (1 + rA2) − rA1 (1 + rB)) r3 r3rfd

When the power train characteristics for the Renault IVT are used, the parameters for mode 1 result in Eq. (3.9).

2.85 −11.36 I = (3.9) ωI 0.61 −11.66

In mode 2, brake 1 is open and brake 2 is closed, in this situation the ring of the planetary gear B is standing still, all the power goes over the electric variator. For mode 2 the matrix Iω results in 24 CHAPTER 3. ANALYZING IVTS

Eq. (3.10). " # − r1rA2 (1+rA2) 2.85 −11.36 I = r2 r2rfd = (3.10) ωII r1 rA1 (1 + rA1) (1 + rB) − (1 + rB) −4.38 8.22 r3 r3rfd

For the Renault IVT these deﬁnitions result into the same ﬁgures as in Section 3.1 as well.

3.2.3 Efficiencies With the relations derived by Mattsson a simplified loss model can be determined. In Fig. 3.14 the power flow affected by losses for each direction of variator power is shown. A total variator efficiency ηv is used, without considering how these losses are distributed. For the planetary gear arrangement a generalised efficiency is introduced, ηp. The gear stage efficiencies ηrv and ηr are products of the efficiencies of the physical gear stages involved. For the two cases in Fig. 3.14 the overall efficiencies can be written as in Eqs. (3.11) and (3.12).

ηtot = (ηp (1 − Φ) + ηvηrvΦ) ηr (3.11)

ηtot = (ηp (1 + ηvηrvΦ) − Φ) ηr (3.12)

Where ηtot is the overall power train eﬃciency.

rv rv Pv hv h rvP v hv h rvP v Pv P P P+hv h rv Pv P-Pv h (P-Pv ) p r r h (P+h h P ) p v rv v h (P-P )+h hP) h h (P+h h P )-P) h ( p v v rv v r ( p v rvv v r (a) Positive variator power ﬂow (b) Negative variator power ﬂow

Figure 3.14: Simpliﬁed loss model

The constant efficiencies used in both models for the THS and Renault IVT are listed in Table 3.3. In Fig. 3.15 the overall power train efficiency is plotted versus the overall transmission ratio. The maximum efficiency for the THS occurs in the GNR, this can easily be explained because in this situation all the power goes over the mechanical path. The EMs require no power and hereby no losses occur in the EMs. The three maximum efficiencies for the Renault IVT occur at those speed ratios where one of the rotational speeds of the EMs is equal to zero. In the Renault IVT all the power goes over the mechanical path when maximum efficiency is reached, in the same way as in the THS. It is remarkable the Renault IVT has almost over the complete speed ratio better transmission efficiency

Table 3.3: Constant efficiencies used in the Mattsson model Part η [-] Planetary gear 0.98 Final drive 0.90 Generator (MG1) 0.75 Motor (MG2) 0.75 3.2. MATTSSON 25 compared to the THS. However this is a simplified loss model and for both power trains the same efficiencies are used. Hereto both transmissions should be investigated further in order to determine the efficiencies of each power train component. In this thesis the THS is further investigated and compared with a conventional transmission. Unfortunately not a lot information of the Renault IVT is available and is hereto not further investigated.

1 0.9 0.8 0.7 0.6 ] − [ 0.5 tot η 0.4 0.3 0.2 THS 0.1 Renault IVT 0 0 0.2 0.4 0.6 0.8 1

rtr [−]

Figure 3.15: Overall power train eﬃciency of the THS and Renault IVT with the method by Mattsson 26 CHAPTER 3. ANALYZING IVTS Chapter 4

Detailed eﬃciency analysis

In Chapter 3 a dimensionless analysis of two power split transmissions is described. A start was made to include efficiencies with the simplified method derived by Mattsson. This gives an indication of where the maximum efficiency possibly is reached, concerning the transmission. The ICE efficiency is left out of this consideration. Herewith still no conclusions can be made concerning the influence of working point efficiencies on the overall efficiencies of the power train. In this chapter this will be further investigated with the THS (a power split transmission) and a push-belt CVT (a conventional transmission).

To investigate these two power trains three different cases are used. In the first case the ICE efficiency is depicted from an efficiency map generated by a computer model, calibrated using test data from a dyno, ADVISOR [3]. The rest of the power train components efficiencies are assumed to be 100 %. Secondly, as an extension of the first case, the efficiencies of the power train components are considered to be constant however selected more realistically compared to the first case. In the third case, as an extension of the second case, the efficiency of the variator is depicted from an efficiency map. For the THS this means the efficiencies of the EMs are depicted from efficiency maps, ADVISOR [3]. For the push-belt CVT this means the efficiency is calculated with the model derived by P. Albers [1]. In Table 4.1 an overview of the three cases is shown.

Table 4.1: The three research cases for the THS and push-belt CVT Case A Case B Case C ηICE efficiency map efficiency map efficiency map ηtr 100 % 56 % / 80 % efficiency map ηpg 100 % 98 % 98 % ηfd 100 % 90 % / 98 % 90 % / 98 %

Where η stand for efficiency. The subscripts ICE, pg and fd stand for ICE, planetary gear and final drive respectively. The subscript tr stands for transmission, for the THS this means the combination of the two EMs. For the CVT this comes down to the efficiency of the push belt CVT. In the event of case B this comes down to an efficiency for the combination of two EMs of 56 % and for the push-belt CVT to 80 %. The THS can be found in the Toyota Prius. For this reason the vehicle characteristics of

Table 4.2: Vehicle characteristics of the Toyota Prius (ﬁrst generation) Vehicle weight 1275 [kg] Drag coeﬃcient 0.26 [-] Projected frontal surface 1.746 [m2] Tyre geometry P165/65R15

27 28 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS

120

100

80 ] kW

[ 60 rl P 40

0 0 20 40 60 80 100 120 140 160 180

vv [km/h]

Figure 4.1: Road load power as function of the vehicle speed for the Toyota Prius for diﬀerent road gradients, 0 % (—), 5 % (− ·), 10 % (− −) and 15 % (···). the Toyota Prius are used and the main characteristics are listed in Table 4.2. In App. G an extensive list of speciﬁcations can be found.

With these vehicle characteristics, the road load power as a function of the vehicle speed can be calculated. The result can be seen in Fig. 4.1. In this thesis the road gradient is presumed to be equal to zero, thus only considering ﬂat roads. The power trains are only investigated in steady state operations, i.e. driving a constant vehicle speed. Hereto three constant vehicle speeds are depicted, namely 50 km/h, 80 km/h and 120 km/h. These speciﬁc speeds are chosen because these are the maximum velocities for urban, rural and motorway driving in the Netherlands since the Wegenverkeerswet 1994 [9]. The characteristic values when driving these three constant vehicle speeds with the Toyota Prius are listed in Table 4.3.

Table 4.3: Road load for diﬀerent requested vehicle speeds

vv [km/h] Prl [kW] Trl [Nm] ωwls [rad/s] 50 2.47 55.1 44.7 80 5.77 80.6 71.6 120 14.25 132.7 107.4

4.1 Internal combustion engine

In the Toyota Prius a specially designed ICE is installed. It makes use of the Miller cycle, which is a modern modification of the Atkinson cycle. This so called Atkinson cycle has an expansion ratio greater than the compression ratio. To realise the different compression ratios during intake and expansion stroke the Atkinson cycle uses a complex linkage system. Unlike the Miller cycle, which uses unique valve timing to obtain the same desired results, [11]. The disadvantage of using the Miller cycle is fact that the ICE delivers less power, though in the THS the EMs compensate this difference in power. Besides this, during normal driving the ICE delivers sufficient power.

A comparison between a Miller cycle ICE and a conventional Otto cycle ICE is shown in Fig. 4.2. The brake speciﬁc fuel consumption map of the Prius ICE is shown in Fig. 4.2(a). It can be seen that the proﬁle is very smooth and the fuel consumption is lower compared to the conventional Otto cycle ICE 4.2. TOYOTA HYBRID SYSTEM 29

110 160 100 230 140 90 220 120 80 230 230 230 240 260 240 70 100 240 250 250 300 260 260 250 240 240 [Nm]

[Nm] 60 80 250 275 275 250 250 275 50 300 ICE ICE 275 60 260

T 300 T 40 275 275 300 275 300 30 300 300 40 300 350 350 350 20 400 350 350 350 400 400 500 400 400 20 400 10 500 500 600 600 600 0 0 150 200 250 300 350 400 100 200 300 400 500 600 700 ωICE [rad/s] ωICE [rad/s] (a) Fuel consumption map of a 1.5 l Atkinson cycle (b) Fuel consumption map of a conventional 1.6 l ICE in [g/kWh] Otto cycle ICE in [g/kWh]

Figure 4.2: Fuel consumption maps with e-lines of two spark ignition ICEs, ADVISOR [3] which is shown in Fig. 4.2(b). The Otto cycle ICE can deliver higher torques and the applicable speed range is broader. Because the Prius ICE has overall better fuel consumption compared to the Otto cycle ICE, the Prius ICE is used to investigate the THS and the push-belt CVT for the diﬀerent cases.

4.2 Toyota hybrid system

The different functionalities of the THS have been described in Section 3.1.1. In this section this power split transmission is further investigated, with special focus on the overall efficiency. To investigate the overall efficiency the three above described cases are studied. When driving a constant speed the requested road load power is constant if there are no natural influences taken into account. The ICE needs to deliver the power required by the road load and has to compensate the losses in the power train. However the ICE has a set of working points which deliver the same power. The working point of the ICE is determined by the overall speed ratio. With Fig. 4.3 the relations for the rotational speeds are derived. vv ωwls = rwls ωwls ωfd = rfd (4.1) ωr = ωMG2 = ωfd ωs = (1 + rpg) ωc − rpgωr = ωMG1 ωc = ωICE From the above stated equations it is clear the speed of the ICE or MG1 can be chosen freely for a specific vehicle speed. When one of these speeds is chosen, the rest of the rotational speeds are deter-

PMG1el PMG2el PMG1 PMG2 MG1 MG2

Ps ICE pg fd wls Pf PICE Pc Pr Pfdin Pfdout Pwls Prl

Figure 4.3: Schematic view of the THS 30 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS mined. It has to be kept in mind that the rotational speeds of the ICE and EMs are however restricted within their boundaries. The torques in the THS are derived with the law of power conservation. For a first impression the transmission efficiencies (η) in Eqs. (4.2) can be set equal to 1. In App. E an elaborate clarification of the kinematics of the THS is described.

Twls + Trl = 0 sign(Pfd) Tfdηfd − rfdTwls = 0 Tfd + TMG2 + Tr = 0 sign(PMG1) sign(PMG2) TMG1ηMG1 + TMG2ηMG2 = 0 (4.2) Ts + TMG1 = 0 Tr − rpgTs = 0 Tc + (1 + rpg) Ts = 0 TICE + Tc = 0

4.2.1 Case A - ICE eﬃciency

In this section the THS is investigated when driving a constant speed and only the efficiency of the ICE is taken into account, case A. In Fig. 4.4(a) the working points at different vehicle speeds are calculated, the circle shows where the overall efficiency of the power train reaches its maximum value. When only the efficiency of the ICE is taken into account it is clear the maximum efficiency is at the e-line of the ICE. For the ICE in the THS the e-line is the same as the maximum torque line. In Fig. 4.4(b) the overall efficiency is plotted versus the speed ratio. In this figure the maximum efficiency occurs at maximum speed ratio, which is equivalent to the minimum ICE speed. The maximum values for the overall efficiencies per vehicle speed are listed in Table 4.4. The efficiency is the best at high vehicle speed, this is caused because only the ICE efficiency is considered.

100 0.35 90 0.375 80 0.35 0.35 0.3 70 0.35

] 0.325 60 ] 0.25 − 0.325 [

Nm 50 0.325 [ 0.3 tot

η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.4: Working points and eﬃciencies for the THS for case A at vehicle speeds 50 km/h (− −), 80 km/h (···) and 120 km/h (− ·)

Table 4.4: Overall eﬃciency for the THS in case A

vv [km/h] ηtot [-] 50 0.24 80 0.32 120 0.37 4.2. TOYOTA HYBRID SYSTEM 31

4.2.2 Case B - Constant transmission component eﬃciencies

Next the THS is investigated with constant efficiencies adopted for the power train components, case B. To determine the overall efficiency of the THS with constant power train component efficiencies, the efficiencies first have to be determined for each component separately. The final drive efficiency is based on the combination of the efficiencies of a chain (92 %) and two gear pairs (each 99 %). The planetary gear efficiency is based on a combination of two gear pair efficiencies (each 99 %). From the efficiency maps of the EMs in Fig. 4.7 it can be seen the efficiencies reach from 50 % to 85 %. With the method used to determine the working point dependent efficiencies, an average efficiency for the EMs is estimated. When for the EM efficiency 75 % is chosen, the resulting efficiencies are similar to the working point dependent efficiencies. The values for the power train component efficiencies are listed in Table 4.5.

Table 4.5: Constant eﬃciencies used for the THS Part η [-] Planetary gear 0.98 Final drive 0.90 Generator (MG1) 0.75 Motor (MG2) 0.75

The results of the simulations for the three vehicle speeds are shown in Fig. 4.5(a), the circles indicate the maximum overall efficiency for a vehicle speed. When the efficiencies of each part of the power train are taken into account, the overall efficiency does not reaches its maximum at the e-line. In Fig. 4.5(b) it is clear the maximum overall efficiency does not occur at the maximum speed ratio. The maximum overall efficiency occurs at the GNR of the THS. The GNR means the power is transmitted via the mechanical path only. In this situation the efficiencies of the planetary gear and final drive need to be compensated. The theoretical representation of the power flow when the THS is in the geared neutral ratio is shown in Fig. 4.6. In this case MG1 stands still, while still consuming some electric power to support the torque on the planetary gear. MG2 still rotates proportional to the vehicle speed and generates some electric power to keep MG1 at rest, because in this model no electrical storage system is considered. The specific maximum overall efficiency values are listed in Table 4.6.

100 0.35 90 0.375 80 0.35 0.35 0.3 70 0.35 ]

60 0.325 ] 0.25 −

0.325 [

Nm 50 0.325 [

0.3 tot η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.5: Working points and eﬃciencies for the THS for case B at the vehicle speeds 50 km/h (− −), 80 km/h (···) and 120 km/h (− ·) 32 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS

MG1 MG2 ICE pg fd wls

Figure 4.6: Power ﬂow in the THS when operating in geared neutral ratio

Table 4.6: Overall eﬃciency for the THS in case B

vv [km/h] ηtot [-] 50 0.21 80 0.25 120 0.29

4.2.3 Case C - EM working point efficiencies To investigate the influence of working point dependent efficiencies, the choice has been made to solely take the working point efficiencies of the electric machines into account. Next to the working point dependent efficiencies of the ICE. This approach is also named case C. The efficiencies of the planetary gear and final drive are assumed to be constant because these do not differ a lot when the working point changes. To implement the working point dependent efficiencies for the electric machines, the data from ADVISOR is used like the ICE data. The efficiency maps of the electric machines are shown in Fig. 4.7, here the efficiencies for positive speeds and torques are plotted only, the first quadrant. To calculate the efficiencies for negative speeds and torques, the power losses are calculated. First the power losses are calculated in the first quadrant, these are then mirrored to the second quadrant. The result of this mathematical equation results in Eq. (4.3). This is further explained by L. Guzzella et al. [6].

1 η(−) = 2 − (4.3) EM (+) ηEM

(+) Where ηEM represents the efficiency of the EM for positive speed and torque, the motor efficiency. (−) Analogously for ηEM , this represents the efficiency of the EM in the second quadrant, the generator efficiency.

60 300

0.5 0.6 50 0.6 250

0.8 0.7 40 0.75 200 0.75 0.75

0.8

0.65 [Nm] [Nm] 0.85 0.8 2

1 30 150 0.85 MG MG T T 20 0.7 100 0.85 0.85 10 0.8 0.8 50 0.9 0.9 0.9 0.85 0.85 0.85 0.85 0.8 0.8 0.8 0 0 0 100 200 300 400 500 0 100 200 300 400 500 600 ωMG1 [rad/s] ωMG2 [rad/s] (a) MG1 (b) MG2

Figure 4.7: Eﬃciency maps of MG1 and MG2, ADVISOR [3] 4.3. PUSH-BELT CVT 33

Like in the previous section, the maximum overall efficiency with working point dependent efficiencies also occurs at the geared neutral ratio, which can be seen in Fig. 4.8(b). In Fig. 4.8(a) the circles show the working point of the ICE where the overall efficiency reaches its maximum value for a certain vehicle speed. The specific values for the maximum overall efficiencies are listed in Table 4.7.

100 0.35 90 0.375 0.35 80 0.35 0.3 70 0.35 ]

60 0.325 ] 0.25 −

0.325 [

Nm 50 0.325 [

0.3 tot η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.8: Working points and eﬃciencies for the THS for case C at the vehicle speeds 50 km/h (− −), 80 km/h (···) and 120 km/h (− ·)

Table 4.7: Overall eﬃciency for the THS in case C

vv [km/h] ηtot [-] 50 0.21 80 0.25 120 0.29

4.3 Push-belt CVT

A push-belt CVT can be compared with the THS since its transmission ratio can be varied continuously as well. The THS has however an inﬁnite ratio coverage in contrast to the push-belt CVT which has a limited ratio coverage between low and overdrive (OD) ratio. The push-belt CVT is in low ratio when the input speed is higher then the output speed. The push-belt CVT is in OD ratio when the input speed is lower then the output speed. This is graphically represented in Fig. 4.9. Additionally

primary pulley

secondary pulley

LOWratio ODratio

Figure 4.9: A push-belt CVT in LOW and OD ratio 34 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS

ICE CVT fd wls Pf PICE Pprim Psec Pfdin Pfdout Pwls Prl

Figure 4.10: Schematic view of the power train with push-belt CVT the push-belt CVT needs a torque converter to drive off and a Drive-Neutral-Reverse-set (DNR) for full vehicle functionality. However in this section the clutch and DNR-set are neglected because only driving at constant vehicle speeds is dealt with. Because the push-belt CVT has limited ratio coverage it is important to make a decision about the final drive ratio. This influences also the driving comfort, e.g. maximum vehicle speed and drive off power (acceleration). To compare the push-belt CVT with the THS, the vehicle characteristics are taken from the Toyota Prius as are the maximum vehicle speed (170 km/h) and drive off power (1 200 Nm), App. G. With Fig. 4.10 the kinematics are determined. For the rotational speeds Eq. (4.4) holds.

vv ωwls = rwls ωwls ωsec = rfd (4.4) ωsec ωprim = rCVT ωICE = ωprim

Where the subscripts prim and sec stand for the primary and secondary pulley. The subscript CVT stands for the push-belt CVT. From the above equations it is clear the ICE speed can be influenced by the push-belt CVT ratio. The torques in the push-belt CVT are derived with the law of power conservation. For a first impression the transmission efficiencies in Eq. (4.5) can be set equal to 1, now only the efficiency of the ICE is considered. In App. F an elaborate clarification of the kinematics of the push-belt CVT is described.

TICE + Tprim = 0 sign(Pprim) TprimηCVT + rCVT Tsec = 0 T + T = 0 sec fdin (4.5) sign(Pfdin ) Tfdin ηfd + rfdTfdout = 0

Tfdout + Twls = 0 Twls + Trl = 0

4.3.1 Case A - ICE efficiency First the push-belt CVT is investigated with ICE efficiency only, case A. Hereto first the final drive has to be determined. In this case is chosen to first match the maximum vehicle speed, which is 170 km/h. The maximum ICE speed is 418 rad/s and with the push-belt CVT in OD this results in a secondary pulley speed of 880 rad/s. With the maximum vehicle speed, the maximum ICE speed and the overdrive ratio, the final drive ratio can be calculated, Eq. (4.6). Assumably no slip occurs between belt and pulleys in the push-belt CVT.

vv = 170 km/h = 47 m/s

ωwls = 152 rad/s

ωsec = ωICE · rCVT = 418 · 2.1 = 880 rad/s ωwls 152 rfd = = = 0.17 (4.6) ωsec 880 Using a final drive ratio of 0.17, the maximum torque of the ICE is not reached to drive top speed. This is shown with the dashed line (− −) in Fig. 4.11. Hereto the final drive ratio is increased to 0.2 so the ICE can be used in a more efficient region. This is shown with the solid line (—) in Fig. 4.11, the 4.3. PUSH-BELT CVT 35

100 90 0.375 0.35 80 0.35 70 0.35 ] 60 0.325

Nm 0.325 [ 50 0.325

T 0.3 40 0.3 0.3 30 0.25 0.25 20 0.25 0.2 0.2 0.2 10 150 200 250 300 350 400 ω [rad/s]

Figure 4.11: Working range of the ICE in combination with the transmission grey region is the working range of the push-belt CVT. Next it will be checked whether the push-belt CVT with the determined final drive ratio can reach the same drive off power as the THS. With the THS, drive off is preformed full electric with MG2 which can deliver 305 Nm. Combined with the final drive ratio in the THS this results in 1191 Nm at the wheels. With the calculated final drive and the ICE, this torque can not be reached at the wheels with the push-belt CVT. Hereto an EM is added between the ICE and push-belt CVT. Now the torque delivered by the ICE and EM is enforced by the push-belt CVT and final drive. The size of the EM is calculated in Eq. (4.7).

Twls = 1 200 Nm

Tsec = rfd · Twls = 0.2 · 1 200 = 240 Nm

TICE + TEM = rCVT · Tsec = 0.45 · 240 = 108 Nm (4.7) When driving off, the ICE can deliver 90 Nm so the EM has to deliver about 20 Nm. However this transmission is investigated when driving constant speeds. This means the EM will not assist. In the first place this transmission will therefore be analysed without the EM. The overall ratio of this transmission reaches from 0.09 till 0.42. In Fig. 4.11 the usable area of the ICE is shown, the grey area under the OD line. The maximum torque clearly can not be reached for most ICE speeds, accordingly the most efficient region of the ICE can not be reached. In Fig. 4.12(a) the working points of the ICE

100 0.35 90 0.375 80 0.35 0.35 0.3 70 0.35

] 0.325 60 ] 0.25 −

0.325 [

Nm 50 0.325 [

0.3 tot η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.25 0.15 20 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.12: Working points and efficiencies for the power train with a push-belt CVT for case A at the vehicle speeds 50 km/h (- -), 80 km/h (···) and 120 km/h (− ·) 36 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS are shown for three vehicle speeds. For each speed the maximum overall efficiency is indicated with a circle, clearly this occurs for all speeds in OD ratio. In Fig. 4.12(b) the overall efficiency is plotted versus the overall speed ratio, the maximum overall efficiency occurs at the maximum overall speed ratio. The specific values for the maximum overall efficiencies are listed in Table 4.8. In this case the overall efficiency is equal to the ICE efficiency.

Table 4.8: Overall eﬃciency of the push-belt CVT in case A

vv [km/h] ηtot [-] 50 0.24 80 0.29 120 0.33

4.3.2 Case B - Constant transmission component eﬃciencies

In case B the transmission components are modelled with a constant efficiency. Here the transmission components are the push-belt CVT and the final drive. The constant efficiencies of the power train components are listed in Table 4.9. The final drive efficiency is in contrast to the THS higher, because in a push-belt CVT the secondary axle is direct connected with the final drive via a single gear pair. Accordingly the final drive efficiency is based on a single gear pair (98 %). The constant push-belt CVT efficiency is based on the working point dependent efficiencies calculated with the model derived by Albers, [1].

Table 4.9: Constant eﬃciencies of the power train with push-belt CVT Part η [-] push-belt CVT 0.80 Final drive 0.98

Again first the final drive ratio has to be determined. The assumption of no slip between the belt and the pulleys still holds. Therefore Eq. (4.6) to calculate the final drive ratio holds as well. Because constant efficiencies are implemented for the push-belt CVT and final drive, the power required from the ICE is significantly higher as in the previous section, case A. Hereto the final drive ratio does not need to be adjusted and the final drive ratio of 0.17 is maintained. However the required drive off power is higher because of the losses in the power train. With this final drive ratio the OD line is plotted in the

100 90 0.375 0.35 80 0.35 70 0.35 ] 60 0.325 0.325 Nm [ 50 0.325 T 0.3 0.3 40 0.3 30 0.25 0.25 20 0.25 0.2 0.2 0.2 10 150 200 250 300 350 400 ω [rad/s]

Figure 4.13: Overdrive of the push-belt CVT in the eﬃciency map of the ICE 4.3. PUSH-BELT CVT 37

ICE eﬃciency map, Fig. 4.13.

The specifications to drive off are determined according to the THS electric drive off which results in 1 200 Nm at the wheels. Which is the same as in the previous section. The only difference is the EM has to be larger because the losses in the power train now need to be compensated. The new size of the EM is calculated in Eq. (4.8).

Twls = 1 200 Nm rfd · Twls 0.17 · 1 200 Tsec = = = 211.7 Nm ηfd 0.98 rCVT · Tsec 0.45 · 211.7 TICE + TEM = = = 119.1 Nm (4.8) ηCVT 0.8 The EM needs to deliver about 30 Nm when constant efficiencies are used for the power train components. Again the power train will be calculated without EM because only constant driving is considered. The speed ratio of the power train now reaches from 0.08 till 0.36. In Fig. 4.14(a) the working points of the ICE are shown for three vehicle speeds. For each speed the maximum overall efficiency is indicated with a circle, clearly this occurs for all speeds in OD ratio. In Fig. 4.14(b) the overall efficiency is plotted versus the speed ratio, the maximum overall efficiency occurs at the maximum speed ratio. The specific values for the maximum overall efficiencies are listed in Table 4.10.

100 0.35 90 0.375 0.35 80 0.35 0.3 70 0.35 ]

60 0.325 ] 0.25 −

0.325 [

Nm 50 0.325 [

0.3 tot η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.14: Working points and eﬃciencies for the power train with a push-belt CVT for case B at the vehicle speeds 50 km/h (- -), 80 km/h (···) and 120 km/h (− ·)

Table 4.10: Overall eﬃciency of the push-belt CVT in case B

vv [km/h] ηtot [-] 50 0.20 80 0.23 120 0.27

4.3.3 Case C - CVT working point efficiencies In case C the efficiency of the push-belt CVT is calculated in each working point. The final drive is modelled with a constant efficiency. To calculate the working point efficiency of the push-belt CVT, the model derived by Albers [1] is used. A drawback of this model is the covered speed range is limited between 50 and 300 rad/s for the primary pulley. While the ICE has a range from 104 rad/s 38 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS till 418 rad/s. Consequently the speed range is reduced to 104 rad/s till 300 rad/s. The final drive ratio is selected to be the same as calculated for constant efficiencies, since the average efficiency of the push-belt CVT is a well representation of the working point dependent efficiency. The resulting working points of the ICE are shown in Fig. 4.15(a). The circles indicate the maximum efficiency for each vehicle speed. The overall efficiencies for each vehicle speed are plotted in Fig. 4.15(b). In Table 4.11 the values of the maximum overall efficiencies are listed.

100 0.35 90 0.375 0.35 80 0.35 0.3 70 0.35 ]

60 0.325 ] 0.25 −

0.325 [

Nm 50 0.325 [ tot

0.3 η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.15: Working points and eﬃciencies for the power train with a push-belt CVT for case C at the vehicle speeds 50 km/h (- -), 80 km/h (···) and 120 km/h (− ·)

Table 4.11: Overall eﬃciency of the push-belt CVT for case C

vv [km/h] ηtot [-] 50 0.19 80 0.23 120 0.27

The overall efficiency of the power train with push-belt CVT is lower compared to the overall efficiency of the THS, Table 4.7. This is partly caused by the final drive ratio. To enlarge the practicable working area of the ICE, the final drive ratio can be enlarged. To estimate the new value for the final drive ratio, the drive off torque relations are used. Herein the size of the EM is enlarged to 50 Nm, which is about the same size as the EM used in the Honda Civic Hybrid. The Honda Civic Hybrid is about the same size as the Toyota Prius and is equipped with the EM in between the ICE and the push-belt CVT. Herewith is the EM torque enforced by the The final drive ratio is calculated in Eq. (4.9).

(TICE + TEM ) (90 + 50) Tsec = · ηCVT = · 0.80 = 249 Nm rCVT 0.45 Tsec · ηfd 249 · 0.98 rfd = = = 0.20 (4.9) Twls 1200 With this final drive ratio the new overall efficiencies can be calculated. The working points of the ICE are shown in Fig. 4.16(a) and the overall efficiency in Fig. 4.16(b). The values for the maximum overall efficiency are listed in Table 4.12. The overall efficiencies are the same compared to the THS, except for 50 km/h the difference is 1 % lower in contrast to the THS. 4.4. DISCUSSION 39

100 0.35 90 0.375 0.35 80 0.3 0.35 70 0.35 ]

60 ] 0.25

0.325 0.325 − [

Nm 50 0.325 [ tot

0.3 η 0.2 T 0.3 40 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.16: Working points and eﬃciencies for the power train with a push-belt CVT and improved ﬁnal drive ratio for case C at the vehicle speeds 50 km/h (- -), 80 km/h (···) and 120 km/h (− ·)

Table 4.12: Overall eﬃciency of the push-belt CVT in case C, with improved ﬁnal drive ratio

vv [km/h] ηtot [-] 50 0.20 80 0.25 120 0.29

4.4 Discussion

In the previous sections two power trains are investigated by means of three research cases. In this section these two power trains are compared with the emphasis on the overall power train efficiency. In the first place a closer look will be taken to the efficiency of the ICE. Followed by a comparison of the different cases for each power train. Finally the power trains are compared for one particular research case.

4.4.1 ICE strategy Each ICE has a specific efficiency map, equivalent to the fuel consumption map. With the transmission the working point of the ICE can be influenced along a hyperbola for equal ICE power. Two main strategies can be distinguished, engine optimal control and system optimal control. In this section these control strategies are discussed.

Engine optimal operation Engine optimal operation means the ICE operates in its most optimal working point considering fuel eﬃciency. The eﬃciency of other power train components are therefore not taken into account. At each level of power demanded from the ICE the engine operating point can be calculated. This results in the economy line (e-line) of the ICE. In automated transmissions (e.g. AT, AMT or CVT) this is normally endeavoured.

For the Atkinson cycle ICE used in the THS the e-line is the same as the maximum torque line. This can also be seen in Fig. 4.17(a). For a conventional Otto cycle ICE this is however diﬀerent. In Fig. 4.17(b) an example is given for a conventional 1.6 l petrol ICE. With these two fuel consumption 40 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS

110 160

100 230 140 90 240 220 120 80 230 230 240 240 260 240 300 250 70 100 260 260 250 275 260

240 240 275

250 [Nm]

[Nm] 60 250 250 80 250 50 275 275

ICE 260 ICE 275 60 T T 40 275 275 275 300 300 300 30 300 300 40 300 350 350 20 350 350 350 350 400 400 400 400 400 500 500 20 400 10 500 600 600 600 0 0 150 200 250 300 350 400 100 200 300 400 500 600 700 ωICE [rad/s] ωICE [rad/s] (a) Fuel consumption of a 1.5 l Atkinson cycle ICE (b) Fuel consumption of a conventional 1.6 l Otto cycle ICE

Figure 4.17: Fuel consumption maps and e-lines of two spark ignition ICEs, ADVISOR [3] maps the overall efficiency is calculated in combination with the THS. The efficiencies for the three different research cases, at the three considered vehicles speeds and for the two different ICEs are listed in Table 4.13. Where the efficiencies have the subscript for the corresponding research case. The two ICEs are indicated by ‘Atkinson’ for the Atkinson cycle ICE and ‘Otto’ for the Otto cycle ICE.

Table 4.13: Overall eﬃciencies for e-line operation η [-] η [-] η [-] v [km/h] A B C v Atkinson Otto Atkinson Otto Atkinson Otto 50 0.24 0.23 0.20 0.11 0.20 0.15 80 0.32 0.29 0.19 0.17 0.23 0.18 120 0.37 0.33 0.26 0.21 0.28 0.22

As can be seen both ICEs have better overall efficiency when working point dependent efficiencies are used for the EMs (case C). Apparently the EMs have better efficiency in the working point compared to the constant efficiencies used in case B. The difference between the Atkinson and Otto cycle ICE can already be seen in Fig. 4.2. The Atkinson cycle ICE has over the complete range better fuel economy and herewith better efficiency compared to the Otto cycle ICE.

System optimal operation

System optimal operation means the efficiencies of the complete power train is taken into account together with the ICE efficiency. The difference between e-line operation and system optimal operation is the ICE is possibly operated at a working point with lower efficiency compared to e-line operation. However the efficiency of the power train components is higher and hereto results in a better overall efficiency. The fuel consumption is lower because less power is demanded from the ICE and the power losses in the power train are lower. As the e-line, for system optimal operation a system optimal operation line (SOOL) can be calculated as well. However the SOOL is dependent on the power train components used and their working point efficiencies. The efficiencies for the different research cases, at the three considered vehicles speeds are listed in Table 4.14.

When the ICE efficiencies are taken into account only, the overall efficiency is equal to the ICE efficiency. It goes without saying the overall efficiency is considerably lower when constant or working 4.4. DISCUSSION 41

Table 4.14: Overall efficiencies for system optimal operation η [-] η [-] η [-] v [km/h] A B C v Atkinson Otto Atkinson Otto Atkinson Otto 50 0.24 0.23 0.21 0.18 0.21 0.18 80 0.32 0.29 0.25 0.22 0.25 0.22 120 0.37 0.33 0.29 0.26 0.29 0.26 point dependent efficiencies are used for the power train components instead of 100 % efficiency. The differences between the maximum overall efficiencies for constant efficiencies (case B) and working point dependent efficiencies (case C) is nil. This is expected for the THS because the maximum efficiency occurs when all the power is transmitted through the mechanical path, the GNR. In the GNR the efficiencies of the planetary gear and the final drive are of significance only and these are the same for both ICEs. The Atkinson cycle ICE results in better overall efficiency, this is expected because the Atkinson cycle ICE has better fuel economy overall and herewith better efficiency.

4.4.2 Comparison case A, B & C

In the Sections 4.2 and 4.3 three cases are studied to investigate the influence of component efficiencies on the overall efficiency. To compare the different cases, these are plotted together for one vehicle speed (80 km/h) in Fig. 4.18. The diamonds indicate the working point where maximum overall efficiency is reached for the THS. In case of the push-belt CVT this is indicated with circles. In Fig. 4.18(a) the working points of the ICE in combination with the THS are plotted. In Fig. 4.18(b) analogously the working points of the ICE in combination with the push-belt CVT are plotted. Clearly for both power trains the required power from the ICE is the lowest when the efficiencies of the power train components are considered to be 100 %. For the push-belt CVT the optimal working point occurs at minimum ICE speed, however the usable working area is smaller compared to the THS. This is caused by the OD ratio of the push-belt CVT in combination with the final drive ratio.

The THS reaches its optimal working point at the GNR, this can clearly be seen in Fig. 4.19(a). In Fig. 4.19 the overall efficiency versus the speed ratio is shown. When constant losses are taken into account, the efficiency is significantly lower and a maximum can be found at the geared neutral ratio

100 100 90 0.375 90 0.375 80 80 0.35 0.35 0.35 0.35 70 0.35 70 0.35

] 60 0.325 ] 60 0.325 0.325 0.325

Nm 50 0.325 Nm 50 0.325 [ 0.3 [ 0.3 T 40 0.3 0.3 T 40 0.3 0.3 30 30 0.25 0.25 0.25 0.25 20 0.25 20 0.25 0.2 0.2 0.2 0.2 0.2 0.2 10 10 0 0 150 200 250 300 350 400 150 200 250 300 350 400 ω [rad/s] ω [rad/s] (a) ICE working points for the THS (b) ICE working points for the push-belt CVT

Figure 4.18: ICE working points in combination with the THS and the push-belt CVT at 80 km/h for case A (—), case B (···), case C (− ·) and for the push-belt CVT, case C with improved ﬁnal drive ratio (− −) 42 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS

0.35 0.35

0.3 0.3 ]

0.25 ] 0.25 − [ − [ tot tot η

η 0.2 0.2

0.15 0.15

0.1 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 − rtr [ ] rtr [−] (a) Overall eﬃciency for the THS (b) Overall eﬃciency for the push-belt CVT

Figure 4.19: Overall efficiencies for the THS and the push-belt CVT at 80 km/h for case A (—), case B (···), case C (− ·) and for the push-belt CVT, case C with improved final drive ratio (− −) for the THS. For lower speed ratios the results with constant efficiency are similar to the results with working point dependent efficiencies. However for higher speed ratios the differences are significant. For higher speed ratios the efficiency of the transmission is higher using working point dependent efficiencies compared to constant efficiencies. Obviously working point dependent efficiencies are more realistic. To calculate the maximum efficiency of the complete power train, constant efficiencies for the components are sufficient, however an accurate estimate for the EM efficiencies is necessary. For the push-belt CVT the maximum overall efficiency occurs for every method at the maximum speed ratio, or lowest ICE speed. As can be seen in Fig. 4.19(b).

To calculate the working point efficiency of the push-belt CVT, the model derived by Albers is used. However in this model the push-belt CVT is actuated hydraulically, hereto an oil pump is necessary which has a negative influence on the fuel economy, especially in partial load. When an EM is added to the power train, higher voltages are available and herewith it is interesting to actuate the push- belt CVT electrically, like the Electro-Mechanical Pulley Actuation CVT (EMPAct CVT) [7]. With the EMPact CVT efficiency can be gained compared to the conventional push-belt CVT because the oil pump is not used to change the ratio, this is realized by an EM.

4.4.3 Comparison THS & push-belt CVT In Case C the THS and the push-belt CVT are modelled with working point dependent efficiencies. In this section both the power trains are compared for one vehicle speed (80 km/h). In Fig. 4.20 the working points of the ICE and the overall efficiency are shown for the THS and the push-belt CVT. The THS reaches its maximum overall efficiency in the geared neutral ratio, while the push-belt CVT reaches its maximum overall efficiency in the overdrive ratio. These ratios differ from one another, however result in the same overall efficiency.

For driveability the ICE speed is desired to be as low as possible, because this is normally experienced as comfortable. For the push-belt CVT the ICE speed is about 30 rad/s lower compared to the ICE speed with the THS. In case the ICE speed is higher, the required torque from the ICE is lower and herewith the torque reserve which is available for acceleration is higher, the performance increases. When a fuel economical power train is designed it is clear some concessions have to be made with respect to performance.

As mentioned before, these simulations are all evaluated in transmission function only. This means no hybrid functionality is considered. When the power train with push-belt CVT is hybridised, this 4.4. DISCUSSION 43

100 0.35 90 0.375 0.35 80 0.35 0.3 70 0.35 ]

60 0.325 ] 0.25 −

0.325 [

Nm 50 0.325 [

0.3 tot η 0.2 T 40 0.3 0.3 30 0.25 0.25 0.15 20 0.25 0.2 0.2 0.2 10 0.1 0 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6

ω [rad/s] rtr [−] (a) ICE working points (b) Overall eﬃciency

Figure 4.20: Working points of the ICE and overall efficiencies in case C at 80 km/h for the THS (···) and the push-belt CVT (− −) can be done at reasonable low costs, however the disadvantage is it will not be able to drive off full electrically. A hybrid push-belt CVT power train has also the disadvantage the brake energy recovery is low because the EM is small and the power has to go via the push-belt CVT, thus the push-belt CVT efficiency has to be taken into account. 44 CHAPTER 4. DETAILED EFFICIENCY ANALYSIS Chapter 5

Conclusion & recommendations

5.1 Conclusion

In Chapter 2 different methods are described to investigate power split transmissions. Basically these methods are based on the same principal, the main differences are the investigated power ratios. Schulz investigates the power ratio of the electrical machines over the output power in order to determine the operating points free from recirculating power flow. Villeneuve and Mattsson investigate the power split ratio in order to keep this within proportion. On the other hand Villeneuve and Mattsson also investigate the variator ratio and how this develops over the overall transmission ratio.

In Chapter 3 the method by Mattsson is used to investigate the Toyota hybrid system and the Renault IVT. Hereto first the kinematics are derived followed by the matrices which are used by Mattsson. Herewith a simplified loss model can be implemented for both transmissions. Here can be seen the Toyota hybrid system has only a small region where the efficiency is better compared to the Renault IVT. However both transmissions are investigated in pure transmission mode and the Toyota hybrid system is designed to operate as a hybrid power train. The Renault IVT on the other hand is designed to operate purely as transmission, however hybrid functionalities can be gained by enlarging the electrical storage system.

In Chapter 4 the Toyota hybrid system and the push-belt CVT are investigated. First only the efficiency of the internal combustion engine is taken into account. Obviously both transmissions reach maximum efficiency at the lowest speed of the internal combustion engine. Next constant values for the power train components are implemented. The push-belt CVT again reaches maximum efficiency at the lowest speed of the internal combustion engine. The Toyota hybrid system on the other hand reaches maximum efficiency in the geared neutral ratio, which means the power is only transmitted through the mechanical path. When the working points efficiencies are implemented for the electric machines in case of the Toyota hybrid system, not much changes considering the overall efficiencies. In case of working points efficiencies for the push-belt CVT the overall efficiency do not change much as well. However when the final drive ratio is changed slightly the overall efficiency changes more. It can be concluded the overall construction of the power train is very important. All the components work together and herewith the gear ratios and the components used, influence each other strongly. Both the transmissions reach about the same overall efficiency, however they are only investigated in pure transmission mode only. Hereto the Toyota hybrid system is presumed not to be operated in the driving conditions where it gains most of its fuel consumption. At constant vehicle speeds both transmissions reach the same overall efficiency in case the Atkinson cycle internal combustion engine is used.

45 46 CHAPTER 5. CONCLUSION & RECOMMENDATIONS

5.2 Recommendations

• Take the hybrid functionalities of the Toyota Prius into account and compare this with a power train with push-belt CVT, equipped with an electric machine. • Investigate stop-and-start for the THS and the hybrid push-belt CVT. In this mode the most fuel is saved because the vehicle is driven electrically and the battery is constantly recharged when decelerating. • Purchase a Toyota Prius, or another hybrid vehicle, to investigate the different driving situations and how the power train operates in these situations. • If a Toyota Prius is purchased, all the components can be tested and efficiency maps can be measured. Herewith a more detailed and up-to-date model can be developed. • The second generation of the Toyota Prius is currently available and the third generation will be introduced soon. It would be interesting how the Toyota Prius is developed over the years and how the efficiency is improved. Bibliography

[1] P.H.W.M. Albers. Fuel economy beneﬁts of advanced techniques for cvt actuation and control. Master’s thesis, TU/e, February 2005. [2] ANWB. Brandstof verbruiks boekje. RDW, January 2007.

[3] AVL. ADVISOR 2004 SP1. http://www.avl.com, 2004.

[4] Centraal Bureau voor de Statistiek. Prijzen motorbrandstoﬀen. http://statline.cbs.nl/ StatWeb/Table.asp?D2=78-83,88,93,98,103&LA=nl&DM=SLNL&PA=7521&D1=2-5, 2007. [5] D. Friedman. A New Road, The Technology and Potential of Hybrid Vehicles. UCS Publications, January 2003. http://www.ucusa.org. [6] L. Guzzella and A. Sciarretta. Vehicle Propulsion Systems: Introduction to Modeling and Optimization. Springer-Verlag, Berlin Heidelberg, 2005. [7] T.W.G.L. Klaassen. The empact CVT: dynamics and control of an electromechanically actuated CVT. Technische Universiteit Eindhoven, 2007. [8] P. Mattsson. Continuously Variable Split-Power Transmission with Several Modes. Chalmers university of technology, Göteborg, Sweden, 1996.

[9] Minesterie van Verkeer en Waterstaat. Wegenverkeerswet 1994. www.wetten.nl/ Wegenverkeerswet%201994, 1994.

[10] PSA Peugeot Citroën. Stop & Start media presentation. http://www.psa-peugeot-citroen.com/ document/presse_dossier/dp_stop-and-start_en1094545369.pdf, September 2004. [11] W.W. Pulkrabek. Engineering Fundamentals of the Internal Combustion Engine. Pearson Edu- cation, Inc., Upper Saddle River, New Jersey, second edition, 2003. [12] D.J. Sanger. Matrix methods in the analysis and synthesis of coupled diﬀerentials and diﬀerential mechanisms. In Fourth World Congress on the Theory of Machines and Mechanisms, Newcastle, UK, pages 27–31, 1975. [13] M. Schulz. Circulating mechanical power in a power-split hybrid electric vehicle transmission. In Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, volume 218, pages 1419–1425. Professional Engineering Publishing, December 2004.

[14] SenterNovem. Het nieuwe rijden. http://www.hetnieuwerijden.nl/english.html.

[15] Toyota. The Fifth Toyota Environmental Forum. http://www.toyota.co.jp/en/k_forum/, June 2003. [16] Toyota. Toyota Hybrid System II. Toyota Motor Corporation, Public Aﬀairs Division, May 2003. [17] A. Villeneuve. Dual mode electric inﬁnitely variable transmission. SAE, 2004.

47 48 BIBLIOGRAPHY Appendix A

Kinematics planetary gear

sun carrier ring gear gear w w w s c r

Ts Tc 0 + torque, speed

rpg 1

Figure A.1: Torque and speed relations in a planetary gear

The nomographic of a simple planetary gear can be seen in Figure A.1. The deﬁnition for the planetary gear ratio can be found in Eq. (A.1) where zr and zs are the number of teeth of the ring and sun gear respectively, this relation can also be seen in Figure A.1. The relation between the rotational speeds can be found in Eq. (A.2).

zr ωc − ωs rpg = = (A.1) zs ωr − ωc rpg (ωr − ωc) = ωc − ωs

(1 + rpg) ωc = rpgωr + ωs (A.2)

On ﬁrst hand it is assumed there are no losses in the planetary gear, so there are no torque or power losses. These relations can also be written as in Eqs. (A.3) and (A.5).

X T = 0

Tc + Tr + Ts = 0 (A.3)

X P = 0

Pc + Pr + Ps = 0 (A.4)

ωcTc + ωrTr + ωsTs = 0 (A.5)

The above equations can be combined, which results in relations for the torques working on the

49 50 APPENDIX A. KINEMATICS PLANETARY GEAR

planetary gear. First the relation between Tc and Tr can be written as:

ωcTc + ωrTr + ωsTs = 0 ⇒

ωcTc + ωrTr + ((1 + rpg) ωc − rpgωr)(−Tc − Tr) = 0 ⇒

(ωc − (1 + rpg) ωc + rpgωr) Tc + (ωr − (1 + rpg) ωc + rpgωr) Tr = 0 ⇒

rpgTc + (1 + rpg) Tr = 0 (A.6)

For a relation between Tc and Ts the following holds:

ωcTc + ωrTr + ωsTs = 0 ⇒ 1 ωcTc + ((1 + rpg) ωc − ωs)(−Tc − Ts) + ωsTs = 0 ⇒ rpg 1 + rpg 1 1 + rpg 1 ωc − ωc + ωs Tc + ωs − ωc + ωs Ts = 0 ⇒ rpg rpg rpg rpg

Tc + (1 + rpg) Ts = 0 (A.7)

And ﬁnally the relation between Tr and Ts can be obtained:

ωcTc + ωrTr + ωsTs = 0 ⇒ 1 (rpgωr + ωs)(−Tr − Ts) + ωrTr + ωsTs = 0 ⇒ 1 + rpg rpg 1 rpg 1 ωr − ωr − ωs Tr + ωs − ωr − ωs Ts = 0 ⇒ 1 + rpg 1 + rpg 1 + rpg 1 + rpg

Tr − rpgTs = 0 (A.8) So to recapitulate, the kinematics of a simple planetary gear are:

zr rpg = zs ωs + rpgωr − (1 + rpg) ωc = 0

rpgTc + (1 + rpg) Tr = 0

Tc + (1 + rpg) Ts = 0

Tr − rpgTs = 0 These equations apply for all simple planetary gears used in this thesis, when the losses and the dynamics are not considered. Appendix B

Clariﬁcation of the method by Schulz

In this section the derivation of the constants used in the M-ﬁle are given. Hereto the relations between the rotational speeds are used. The ratios concerning the lower gear train are deﬁned as in Eq. (B.1).

ωc1 rA1 = ωICE ωs1 rMG1 = (B.1) ωMG1 ωwls rL = ωr1

For the higher gear train the ratios are deﬁned as in Eq. (B.2).

ωc2 rA2 = ωICE ωs2 rMG2 = (B.2) ωMG2 ωwls rH = ωr2

With these ratios and the relation which holds for a planetary gear, Eq.(B.3) holds for the Bosch Dual-E transmission.

ωwls (1 + rpg) rA1ωICE = rpg + rMG1ωMG1 rL (B.3) ωwls (1 + rpg) rA2ωICE = rpg + rMG2ωMG2 rH

These equations can be written into Eq. (B.4) with the constants a, b(L), c and d(H).

ωMG1 = −aωICE + b ωwls (L) (B.4) ωMG2 = −cωICE + d(H)ωwls

Where the constants are listed in Eq. (B.5).

a = − (1+rpg )rA1 rMG1 rpg b(L) = − rMG1rL (B.5) c = − (1+rpg )rA2 rMG2 d = − rpg (H) rMG2rH

The gear ratios which hold for the Bosch Dual-E transmission are listed in Table B.1. The torque relations can also be written by means of these constants. The more detailed derivation can be found in Schulz [13], here only the results are given in Eq. (B.6).

TICE = aTMG1 + cTMG2 (B.6) Twls = −b(L)TMG1 − d(H)TMG2

51 52 APPENDIX B. CLARIFICATION OF THE METHOD BY SCHULZ

Table B.1: Gear ratios of the Bosch Dual-E transmission

rL rH

rpg rA1 rA2 rMG1 rMG2 r1 r3 r5 r2 r4 r6 42 18 23 11 11 19 27 42 − − − − − = r = r = r 12 45 40 44 30 35 24 18 1 3 5

The Eqs. (B.4) and (B.6) can be written in matrix notation according to Eq. (B.7).

ω −a b ω MG1 = (L) ICE ωMG2 −c d(H) ωwls (B.7) T a c T ICE = MG1 Twls −b(L) −d(H) TMG2

According to the analysis of circulating power by Schulz [13] and the power ratio deﬁned as in Eq. (B.8) the relation between the power ratio and the overall transmission ratio can be determined.

PMG1 + PMG2 λE = (B.8) Pwls The boundary of the circulating-power-free operating range is given by the limiting case as deﬁned in Eq. (B.9).

TMG1TMG2 = 0 (B.9)

In this case either TMG1 = 0 or TMG2 = 0 holds. For TMG1 = 0, this results into Eq. (B.10). For TMG2 = 0, this results into Eq. (B.11).

c ωICE λE = 1 − (B.10) d(H) ωwls

a ωICE λE = 1 − (B.11) b(L) ωwls

B.1 M-ﬁle for the method by Schulz

This ﬁle is used to calculate the power ratio of the total mechanical power of the EMs over the total output power for the Bosch Dual-E transmission. For each driving range the power ratio has to be calculated, this can be divided in two power ratios. One for the lower shaft and one for the higher shaft. For the lower shaft this is “labda_EL” and for the higher shaft this is “labda_EH”. In total six power ratios are calculated, one for each gear ratio. %------% Gear ratios Bosch Dual-E transmission: r_pg = 42/12; r_A1 = -18/45; r_A2 = -23/40; r_MG1 = 11/44; r_MG2 = 11/30; r_1 = -19/35; r_3 = -27/24; r_5 = -42/18; % r_L, lower shaft r_2 = r_1; r_4 = r_3; r_6 = r_5; % r_H, higher shaft r_L = [r_1 r_3 r_5]; r_H = [r_2 r_4 r_6]; r_all = [0,8]; % overall transmission ratio B.1. M-FILE FOR THE METHOD BY SCHULZ 53

%------% Determine the power ratio for each gear ratio: a = -r_A1*(1+r_pg)/r_MG1; c = -r_A2*(1+r_pg)/r_MG2; for j = 1:3 b(j) = -r_pg/r_MG1/r_L(j); d(j) = -r_pg/r_MG2/r_H(j); labda_EL(j,:) = 1-(c./d(j))./r_all; labda_EH(j,:) = 1-(a./b(j))./r_all; end %------

For each gear the power ratio can be calculated. This is plotted in Fig. B.1 with on the horizontal axis the inverse of the overall transmission ratio. On the vertical axis the power ratio deﬁned as in Eq. (B.8).

1.5 1st gear 2nd gear 1 3rd gear 4th gear th 0.5 5 gear 6th gear ] − [ 0 E λ

-0.5

-1

-1.5 0 1 2 3 4 5 6 7 8

1/rtr [−]

Figure B.1: Overview of the circulating-power-free operating ranges 54 APPENDIX B. CLARIFICATION OF THE METHOD BY SCHULZ Appendix C

Clariﬁcation of the method by Villeneuve

In this section the derivation of the constants used in the M-file is given. Hereto the relations between the rotational speeds are used. The relation between the overall in and output speed and the electric variator in and output speed is defined as in Eq. (C.1). ω a b ω ω ICE = · MG1 = M · MG1 (C.1) ωwls c d ωMG2 ωMG2 To calculate the coefficients a and b the following equations are used:

ωcA1 ωICE = r1

ωrA2 = ωcA1 1 ωc = (ωs + rA1ωr ) A1 (1+rA1) A1 A1

ωsA1 = ωcB

ωrA1 = ωcA2 1 ωc = (ωs + rA2ωr ) A2 (1+rA2) A2 A2 ωr = r1ωICE A2 (C.2) ωsA2 = r2ωMG1 1 ωc = (ωs + rBωr ) B (1+rB ) B B

ωrB = ωF 2

ωsB = r3ωMG2

ωF 2 = ωsC

ωsC = (1 + rC ) ωcC − rC ωrC

ωrC = r2ωMG1

ωcC = ωF 1

In mode 1 this results in Eq. (C.3) for the coeﬃcients a1 and b1. r2 (1 + rA2) rA1 rBrC r3 ωICE = − ωMG1 + ωMG2 (C.3) r1 (1 + rA1 + rA2) (1 + rA2) (1 + rB) r2 (1 + rB)

To calculate the coeﬃcients c1 and d1 the following equations are needed on top of the above stated equations: ω = r ω wls fd rA1 (C.4) ωwls = rfdωcA2

So now for the coeﬃcients C1 and D1 this results in Eq. (C.5): r2rA2rfd (1 + rA1) rBrC r3 ωout = − ωMG1 + ωMG2 (C.5) (1 + rA1 + rA2) rA2 (1 + rB) r2 (1 + rB)

55 56 APPENDIX C. CLARIFICATION OF THE METHOD BY VILLENEUVE

For mode 1 this can be recapitulated into Eq. (C.6).   r2rA1(1+rB )−r2rB rC (1+rA2) r3(1+rA2) a1 b1 r1(1+rA1+rA2)(1+rB ) r1(1+rA1+rA2)(1+rB ) M1 = =   (C.6) c d (1+rA1)(1+rB )−rA2rB rC r3rA2rfd 1 1 r2rfd (1+rA1+rA2)(1+rB ) (1+rA1+rA2)(1+rB )

When the values for the gear ratios are substituted in matrix M1 this results in Eq. (C.7).

0.4424 −0.4310 M = (C.7) 1 0.0231 −0.1083

In mode 2 this results in: (1 + rA2) r2rA1 r3 ωICE = ωMG1 + ωMG2 (C.8) r1 (1 + rA1 + rA2) (1 + rA2) (1 + rB) And rfd (1 + rA1) r3rA2 ωwls = r2ωMG1 + ωMG2 (C.9) (1 + rA1 + rA2) (1 + rA1) (1 + rB)

For mode 2 this can be recapitulated into Eq. (C.10).   r2rA1 r3(1+rA2) a2 b2 r1(1+rA1+rA2) r1(1+rA1+rA2)(1+rB ) M2 = =   (C.10) c2 d2 r2rfd(1+rA1) r3rA2rfd (1+rA1+rA2) (1+rA1+rA2)(1+rB )

When the values for the gear ratios are substituted in matrix M2 this results in Eq. (C.11).

−0.3119 −0.4310 M = (C.11) 2 −0.1664 −0.1083

As can be seen, the constants b1 and b2 are equal, as are d1 and d2. This corresponds to the rule which ensures the power through the electric variator is zero during mode change. Since ωMG1 = ωMG2 = 0.

Table C.1: Gear ratios in the Renault IVT planetary gears

r1 r2 r3 rfd rA1 rA2 rB rC 31 27 30 41 75 67 63 79 − − 55 51 26 67 41 25 29 45

C.1 M-ﬁle for the method by Villeneuve

This ﬁle is used to calculate the power ratio of the variator input power over the ICE power of the Renault IVT. For each mode the constants of the matrix M are diﬀerent. These constants are calculated with the kinematics derived for the Renault IVT.

%------% Gear ratios Renault IVT: ra1 = 75/41; % Planetary gear A1 ra2 = 67/25; % Planetary gear A2 rb = 63/29; % Planetary gear B rc = 79/45; % Planetary gear C C.1. M-FILE FOR THE METHOD BY VILLENEUVE 57 r1 = -31/55; r2 = 27/51; r3 = 30/26; rfd = -41/67; r = [0:0.01:2]; % overall transmission ratio (w_wls/w_ICE) %------% Mode 1: mode1.a = (r2*ra1*(1+rb)-r2*rb*rc*(1+ra2))/r1/(1+ra1+ra2)/(1+rb); mode1.b = r3*(1+ra2)/r1/(1+ra1+ra2)/(1+rb); mode1.c = r2*rfd*(((1+ra1)*(1+rb)-ra2*rb*rc)/(1+ra1+ra2)/(1+rb)); mode1.d = r3*ra2*rfd/(1+ra1+ra2)/(1+rb); mode1.M = [mode1.a mode1.b ; mode1.c mode1.d]; mode1.rev = -(mode1.a.*r-mode1.c)./(mode1.b.*r-mode1.d); mode1.Phi = ((mode1.c-mode1.a.*r).*(mode1.a.*mode1.d-mode1.b.*mode1.c))./... ((mode1.b.*r-mode1.d).*(mode1.c+mode1.d.*mode1.rev).*... (mode1.b.*mode1.rev+mode1.a)); %------% Mode 2: mode2.a = r2*ra1/r1/(1+ra1+ra2); mode2.b = r3*(1+ra2)/r1/(1+ra1+ra2)/(1+rb); mode2.c = r2*rfd*(1+ra1)/(1+ra1+ra2); mode2.d = r3*ra2*rfd/(1+ra1+ra2)/(1+rb); mode2.M = [mode2.a mode2.b ; mode2.c mode2.d]; mode2.rev = -(mode2.a.*r-mode2.c)./(mode2.b.*r-mode2.d); mode2.Phi = ((mode2.c-mode2.a.*r).*(mode2.a.*mode2.d-mode2.b.*mode2.c))./... ((mode2.b.*r-mode2.d).*(mode2.c+mode2.d.*mode2.rev).*... (mode2.b.*mode2.rev+mode2.a)); %------

10 2 8 1.5 6 1 4

0.5 2 ] − ] [

− 0 0 ev r Φ [ -2 -0.5 -4 -1 -6 -1.5 -8

-2 -10 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

rtr [−] rtr [−] (a) (b)

Figure C.1: The power split ratio and the variator ratio versus the overall transmission ratio for the Renault IVT 58 APPENDIX C. CLARIFICATION OF THE METHOD BY VILLENEUVE Appendix D

Clariﬁcation of the method by Mattsson

The method derived by Mattsson [8] is stated here.

ωvin a b ωin ωin = · = Iω · (D.1) ωvout c d ωout ωout

According to [12], the torque relationships can then be written as:

Tin t Tvin −a −c Tvin = −Iω · = · (D.2) Tout Tvout −b −d Tvout

Equilibrium of power between the CVT input and output shafts gives:

ω T out = − in = i (D.3) ωin Tout Similarly, equilibrium of power between the variator input and output shafts gives:

ωvout Tvin = − = iv (D.4) ωvin Tvout Eq. (D.1) can be written:

1 a b 1 in 1 a + b · i · ωvin = · · ω ⇐⇒ · ωvin = · ωin (D.5) iv c d i iv c + d · i

Eq. (D.5) can be expressed in matrix form:

1 −a − b · i ω 0 · vin = (D.6) iv −c − d · i ωin 0

Non-trivial solution for:

1 −a − b · i = 0 ⇐⇒ −c − d · i + a · iv + b · i · iv = 0 (D.7) iv −c − d · i

59 60 APPENDIX D. CLARIFICATION OF THE METHOD BY MATTSSON

The relationship between the overall speed ratio, i, and the variator speed ratio, iv, can then be written as:

a · i − c i = − v (D.8) b · iv − d This function has the derivative:

di a · d − b · c = 2 (D.9) div (b · iv − d) Eq. (D.2) can be written as:

1 −a −c 1 · Tin = · · Tvin (D.10) −1/i −b −d −1/iv ⇐⇒ 1 −a + c/iv · Tin = · Tvin (D.11) −1/i −b + d/iv Eq. (D.11) includes two equivalent relationships between the torques. Any of the two relationships then gives the ratio between the variator input torque and the CVT input torque as:

T i vin = − v (D.12) Tin a · iv − c The ratio between the variator output torque and the CVT input torque becomes:

T 1 T 1 vout = − · vin = (D.13) Tin iv Tin a · iv − c Furthermore, the ratio between the variator input torque and the CVT output torque read as:

T T i vin = −i · vin = − v (D.14) Tout Tin b · iv − d Finally, the ratio between the variator output torque and the CVT output torque can be written:

T i T 1 vout = · vin = (D.15) Tout iv Tin b · iv − d Any of the two relationships in Eq. (D.5), together with Eq. (D.8), gives the quotient between the variator input speed and the CVT input speed as:

ω a · d − b · c vin = − (D.16) ωin b · iv − d The variator power ratio in the loss-free case can then be written:

P −T · ω i a · d − b · c v = vin vin = − v · (D.17) P Tin · ωin a · iv − c b · iv − d Identiﬁcation of the Eq. (D.8) and (D.9) gives: P i di v = v (D.18) P i div cf. Eq. (1.1) Appendix E

Kinematics THS

PMG1el PMG2el PMG1 PMG2 MG1 MG2

Ps ICE pg fd wls Pf PICE Pc Pr Pfdin Pfdout Pwls Prl

Figure E.1: Schematic view of the THS

With Fig. E.1 the kinematics can be derived, a gear ratio is deﬁned as in Eq. (E.1). ω r = out (E.1) ωin

For the rotational speeds the following holds.

vv ωwls = rwls

ωfdout = ωwls

ωfdout ωfdin = rfd

ωr = ωMG2 = ωfdin

ωs = (1 + rpg) ωc − rpgωr = ωMG1

ωc = ωICE

From the equations above it can be seen the rotational speeds are not distinct when only the vehicle speed (vv) is given. A combination of rotational speeds of the ICE and MG1 is still possible, however when one of these speeds is chosen, the other speed is distinct. When no losses are taken into account the complete THS power train can be seen as in Fig. E.2 and each part in the power train can be seen as in Fig. E.3. For the powers this results in Eq. (E.2).

MG1 MG2 P P f ICE pg fd wls rl

Figure E.2: The THS considered as one systeem

61 62 APPENDIX E. KINEMATICS THS

Pin plant Pout

Figure E.3: Deﬁnitions of power on a plant

sign(Pin) Pinηplant + Pout = 0 (E.2)

For Fig. E.2, Eq. (E.3) holds. This can be done for each part of the power train.

Pf ηTHS + Prl = 0 (E.3)

Pf ηICE + PICE = 0

Pc + Ps + Pr = 0 sign(PMG1) PMG1ηMG1 + PMG1el = 0

sign(PMG2el ) PMG2el ηMG2 + PMG2 = 0

sign(Pfdin ) Pfdin ηfd + Pfdout = 0 Pwls + Prl = 0

Because Pf is deﬁned positive, both PICE and Prl are negative when driving. However when the vehicle is coasting, no combustion occurs in the engine, the engine delivers negative power (Pf < 0) because of the pumping eﬀects. With the above equations the torque relations can be derived. For the planetary gear the relations are derived in App. A. For the interaction between the power train parts an example is given for the relation between the ICE and the carrier.

PICE + Pc = 0

ωICETICE + ωcTc = 0

TICE + Tc = 0

An extra relation is used to connect the two EMs with each other, when no battery and electric system is taken into account, Eq. (E.4) holds.

PMG1el + PMG2el = 0 (E.4)

P pg P Pf ICE PICE c r (a) ICE (b) Planetary gear P P P P MG1 MG1 MG1el MG2el MG2 MG2 (c) MG1 (d) MG2 P P P P fdin fd fdout wls wls rl (e) Final drive (f) Wheels

Figure E.4: Power relations deﬁned for each part of the power train 63

TICE + Tc = 0

Tc + (1 + rpg) Ts = 0

Tr − rpgTs = 0

Ts + TMG1 = 0

Tfdin + TMG2 + Tr = 0

sign(Pfdin ) Tfdin ηfd + rfdTfdout = 0

Tfdout + Twls = 0

Twls + Trl = 0 64 APPENDIX E. KINEMATICS THS Appendix F

Kinematics push-belt CVT

ICE CVT fd wls Pf PICE Pprim Psec Pfdin Pfdout Pwls Prl

Figure F.1: Schematic view of the CVT

With Fig. F.1 the kinematics can be derived, a gear ratio is deﬁned as in Eq. (E.1). For the rotational speeds the following holds.

vv ωwls = rwls

ωfdout = ωwls

ωfdout ωfdin = rfd

ωsec = ωfdin ωsec ωprim = rCVT ωICE = ωprim

For Fig. F.2, Eq. (E.3) holds. This can be done for each part of the power train.

Pf ηICE + PICE = 0

sign(Pprim) PprimηCVT + Psec = 0

sign(Pfdin ) Pfdin ηfd + Pfdout = 0 Pwls + Prl = 0

Because Pf is deﬁned positive, both PICE and Prl are negative when driving. However when the vehicle is coasting, no combustion occurs in the engine, the engine delivers negative power (Pf < 0) because of the pumping eﬀects. With the above equations the torque relations can be derived. For the interaction between the power train parts an example is given for the relation between the ICE and the

Pf ICE CVT fd wls Prl

Figure F.2: The power train with the CVT considered as one systeem

65 66 APPENDIX F. KINEMATICS PUSH-BELT CVT

CVT.

PICE + Pprim = 0

ωICETICE + ωprimTprim = 0

TICE + Tprim = 0

Pf ICE PICE Pprim CVT Psec (a) ICE (b) Planetary gear P P P P fdin fd fdout wls wls rl (c) Final drive (d) Wheels

Figure F.3: Power relations deﬁned for each part of the power train

TICE + Tprim = 0

sign(Pprim) TprimηCVT + rCVT Tsec = 0

Tsec + Tfdin = 0

sign(Pfdin ) Tfdin ηfd + rfdTfdout = 0

Tfdout + Twls = 0

Twls + Trl = 0 Appendix G

Toyota Prius speciﬁcations

Table G.1: Speciﬁcations of the Toyota Prius

Hybrid Synergy Driver ICE Type Aluminum DOHC 16-valve VVT-i 4-cylinder Displacement 1 497 cm3 Bore × stroke 75.0 mm × 84.7 mm Compression ratio 13.0 : 1 Maximum power 43 kW (58 hp) at 4 000 rpm Maximum torque 102 Nm at 4 000 rpm Induction system Multi-point with ETCS-i Electric generator (MG1) Type Permanent magnet AC synchronous motor Maximum rotational speed 5 500 rpm Maximum power 15 kW (20 hp) at 2 600 − 5 500 rpm Maximum torque 55 Nm at 0 − 2 600 rpm Maximum voltage 288 V Electric motor (MG2) Type Permanent magnet AC synchronous motor Maximum power 30 kW (45 hp) at 1 040 − 5 600 rpm Maximum torque 305 Nm at 0 − 400 rpm Maximum voltage 288 V HSDr system Combined power 74 kW (101 hp) Combined torque 421 Nm Performance Top speed 170 km/h Acceleration 0 − 100 km/h 14.1 s

Fuel economy and CO2-emissions (1999/100 EEC) City 20 km/l Highway 23.8 km/l Combined 23.3 km/l Continued on next page

67 68 APPENDIX G. TOYOTA PRIUS SPECIFICATIONS

Table G.1 – continued from previous page Hybrid Synergy Driver

CO2-emissions, combined 104 g/km Tyres Tyre size 165/65R15 Weight Empty weight 1 240 kg Loading capacity (legal) 450 kg Coeﬃcient of drag 0.30 Projected frontal surface 1.746 m2 78 Planetary gear ratio 30 36 44 75 Final drive ratio 39 30 26