Arenberg Doctoral School of Science, Engineering & Technology Faculty of Engineering Department of Mechanical Engineering

Optimal Control Based Eco-Driving Theoretical Approach and Practical Applications Bart SAERENS

Dissertation presented in partial fulfillment of the requirements for the degree of Doctor in Mechanical Engineering

June 2012

Optimal Control Based Eco-Driving Theoretical Approach and Practical Applications

Bart SAERENS

Jury: Prof. dr. ir. Pierre Verbaeten, chair Prof. dr. ir. Eric Van den Bulck, promotor Prof. dr. ir. Moritz Diehl, co-promotor Prof. dr. ir. Jan Swevers Prof. dr. ir. Lieve Helsen Prof. dr. ir. Paul Sas Prof. dr. P.Eng. Hesham Rakha (Virginia Tech Transportation Institute)

June 2012

Dissertation presented in partial fulfillment of the requirements for the degree of Doctor in Mechanical Engineering

© Katholieke Universiteit Leuven – Faculty of Engineering Celestijnenlaan 300A, B-3001 Heverlee (Belgium) Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotocopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher. D/2012/7515/50 ISBN 978-94-6018-514-4

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Be the change that you want to see in the world.

Voorwoord Voilà, dit is het dan: mijn boekske, de kers op de taart van 5 jaar doctoreren. Het begon allemaal in 2006, het laatste jaar van mijn opleiding burgerlijk ingenieur. Samen met Jeroen, en onder toezicht van professor Van den Bulck en professor Swevers, werkte ik een heel jaar aan wat de start van mijn doctoraat zou worden. Als bij toeval startten we het onderzoek rond eco-rijden net toen Al Gore het van de daken schreeuwde en alle auto’s eco moesten zijn. Van timing gesproken. Na enkele maanden noeste arbeid in het motorkot, kwam ik tot het besef dat wetenschappelijk onderzoek wel iets voor mij was. Rond nieuwjaar maakte ik dan ook de beslissing om te doctoreren. In augustus 2007 knutselde ik op een paar weken tijd een dossiertje in elkaar, dat het agentschap voor Innovatie door Wetenschap en Technologie (IWT) genoeg kon bekoren om mij een beurs te geven. Enkelingen vonden het ongelooflijk geweldig dat ik, na het behalen van een diploma burgerlijk ingenieur, nog eens minstens 7 jaar zou gaan studeren voor dokter. De meesten hielden het echter bij het uitspreken van de gevleugelde woorden: “Wat is dat nu eigenlijk doctoreren?” Het korte en simpele antwoord luidt: “Een dikke thesis schrijven.” In de praktijk is het natuurlijk veel meer. Eerst en vooral moet je iets doen waarover je kan schrijven. In mijn geval heb ik mij jaren geamuseerd met het zoeken, verbeteren, en uitvinden van methodes om te berekenen hoe je zuiniger kan rijden. Verder moet je tijdens een doctoraat ook les geven, misschien wel mijn favoriete gedeelte. Naast de P&O over hybride voertuigen en de oefenzitting voertuigpropulsie, was ik 5 jaar lang de man van één van de meest legendarische labo’s: het labo motoren. Ik ben trots dat ik in 2010 het gouden krijtje won voor beste assistent van werktuigkunde. Ik mag uiteraard niet vergeten te vermelden dat dit doctoraat zonder de hulp en inbreng van vele anderen nooit tot stand zou gekomen zijn. In de eerste plaats zijn er mijn ouders, die mij ter wereld hebben gebracht, mij altijd hebben laten kiezen wat ik wou studeren, en dat alles hebben gesponsord. Ik hoop dat zij

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(en mijn grootvader) trots zijn bij het aanschouwen van mijn boekje. Dan zijn er ook de collega’s die voor een leuke werksfeer hebben gezorgd. Ik ga ze niet allemaal bij naam noemen. Met mijn slecht geheugen lijkt het ook niet verstandig om een poging te doen tot. Wie ik wel ga vernoemen, zijn de mannen van bureau 01.058. Danku Filip, Maarten, Maarten, Tom, Vladimir, Asim, Joris, en Lieven. Jeroen, Lieboud, Jan, en Clara verdienen een speciale vermelding. Laten we het even hebben over de mensen met een directe inbreng. In de eerste plaats is er mijn promotor, professor Eric Van den Bulck. Hij wekte mijn interesse voor motoren, en begeleidde mijn eindwerk en mijn doctoraat. Hoewel hij de dingen op z’n eigen eigenaardige manier doet, doet hij ze best erg aardig. Hij is een zeer goede begeleider, en ik waardeer zijn inbreng in mijn onderzoek en de tijd die hij aan mij besteed heeft. Dan is er professor Moritz Diehl, mijn copromotor. Als hij nooit tot in België was gesukkeld, was mijn eindwerk met Jeroen nooit tot een goed eind gekomen. Zijn kennis over optimale controle, eeuwig enthousiasme, en aanzet tot samenwerken en resultaten produceren, hebben een belangrijke inbreng gehad in mijn onderzoek. Professor Hesham Rakha, eigenlijk ook een beetje copromotor, gaf me de kans om 5 maanden onderzoek te doen aan het Virginia Tech Transportation Institute in Blacksburg, USA. Het was daar niet alleen een geweldige persoonlijke ervaring, op het gebied van mijn onderzoek was het zeker ook een verrijking. Danku prof. R. Snoeys fonds, voor de financiering van mijn Amerikaans avontuur. Ik bedank ook de andere mensen van mijn jury: prof. Jan Swevers, prof. Lieve Helsen, en prof. Paul Sas. Hun commentaren en aanbevelingen hebben dit werk zeker beter gemaakt. Verder heb ik ook wat hulp gehad van studenten wiens eindwerk ik begeleidde. Vooral de eindwerken van de laatste jaren hadden een relevante inbreng. Bedankt Jan, Kris en Alexander, Tom, Jurgen, en Michael en Dries. Dan zijn er ook mensen die eerder een ondanks, dan een dankzij rol spelen in het verhaal. En mensen die zich ergens op de vage grens tussen de twee bevinden. Die laatsten zijn vooral de techniekers en de secretaresses. Zonder Ivo, en vooral Hans, was er geen motor om mee te spelen. En wat had ik gedaan zonder Kathleen, Frieda, en Elise? Liggen verrotten achter mijn computer vermoed ik. Langs de andere kant durf ik niet te berekenen hoeveel tijd ik verloren heb door met hen te zitten babbelen en roddelen. Uiteindelijk is een doctoraatsopleiding echter niet enkel puur wetenschappelijk; m’n sociale vaardigheden heb ik alvast goed geoefend. Dan komen we bij het laatste groepje mensen. Mensen ondanks wie ik toch mijn doctoraat heb behaald. Niet dat ik die mensen daarom minder graag heb,

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in tegendeel zelfs. Ik denk hierbij vooral aan Evy. Ondanks haar continue mentale afleiding in het laatste anderhalf jaar van mijn doctoraat, heb je nu toch dit boekje in handen. Kan ik het deze prachtvrouw echter kwalijk nemen dat ze zo ongelooflijk geweldig is, dat het onmogelijk was om nog maar een uur op het werk door te brengen zonder mijn gedachten naar haar te laten afdwalen? Tot slot zijn er nog Kikker, Politiebeer, Memmeke, en Jutteke. Hun nachtelijke aanwezigheid zorgde voor een goede nachtrust die noodzakelijk was om ’s ochtends fris en monter op het werk te zijn. Letterlijk uit het Engels vertaald, dient dit boekje om mijn diploma als dokter in de filosofie te behalen. Enkele filosofische beschouwingen zijn dus zeker niet misplaatst. De reden waarom ik voor dit onderwerp gekozen heb, is puur ideologisch. De mens is doodleuk de wereld aan het kapot maken en ik zie dat niet zo graag gebeuren. Klimaatverandering is niet een mogelijke zorg van de toekomst, het is nu aan het gebeuren. En hoewel de doorsnee westerling in een gematigd klimaat een paar graden extra wel ziet zitten, zijn de globale gevolgen niet zo amusant. Ecosystemen worden bedreigd door meer overstromingen, droogte, bosbranden, en insecten. Een stijging van de gemiddelde temperatuur zorgt voor een verandering in ecosystemen en geografische verspreiding van soorten. Dit met negatieve gevolgen voor de biodiversiteit; plant- en diersoorten worden met uitsterven bedreigd. Wij kunnen mogelijk aan de wieg liggen van de 6e massa-extinctie. De stijging van de temperatuur zorgt voor een daling in de globale voedselproductie. Het zeeniveau zal stijgen. Dichtbevolkte megadeltas in Azië en Afrika, en kleine eilandjes zijn zeer kwetsbaar. De beschikbaarheid van vers water zal stijgen op hogere breedtegraden en in de natte tropen; droge gebieden zullen echter nog droger worden. De opwarming van de aarde brengt ook gezondsheidrisico’s mee door ondervoeding, extreme weersomstandigheden, en een veranderende verspreiding van besmettelijke ziektes. Eco-rijden vermindert het brandstofverbruik en dus de CO2 -uitstoot. Ik ben er van overtuigd dat het gebruik van optimale control om eco-rijden te ondersteunen nog meer brandstof doet besparen. Maar gaan we hiermee de wereld redden? Spijtig genoeg niet. Inderdaad, eco-rijden kan 10 % brandstof besparen. En stel dat iedereen zou gaan eco-rijden. En stel dat we bij alles wat we doen de milieu-impact met 10 % verminderen, dan komen we toch al een heel eind. Niet? Veel te veel min 10 % blijft veel te veel. De levensstijl die wij westerlingen ons toe-eigenen kan niet gedragen worden door onze planeet. Technologie gaat daar niets aan veranderen; er is een gedragswijziging nodig. Simpel voorbeeld: vlees eten. De consumptie van dieren is zowel desastreus voor het dierenwelzijn als voor het milieu. Vlees is niet nodig. Indien je op je voeding let, kan je perfect gezond leven zonder vlees. Als je als vleeseter gezond wil leven, moet je trouwens ook op je voeding letten. Toch eten de

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meeste mensen nog steeds vlees. In de westerse landen zelfs veel te veel, met gezondheidsrisico’s tot gevolg. En waarom? Gewoon omdat het lekker is? Het zit goed fout met onze maatschappij, laat dat duidelijk zijn. We voederen ganzen onder dwang, zodat ze een leverziekte krijgen (waaraan een groot deel sterft voor ze geslacht zouden worden), om met de feestdagen ganzenlever te eten. We sponsoren de burgeroorlog in Congo door het kopen van electronica. Voor de productie is coltan nodig. Een aanzienlijk deel hiervan wordt aangekocht in Congo, omdat het daar goedkoop is. Dat het erts in mensonwaardige omstandigheden wordt ontgonnen (vaak door kinderen) en dat het geld wordt gebruikt om wapens te kopen, nemen we er met plezier bij. We vissen de zeeën leeg, waardoor er van vele vissoorten nog amper exemplaren rondzwemmen. Toch blijven we dit doen en steken we de schuld van de dalende populaties op de dolfijnen. Dit geeft ons meteen een leuk excuus om dolfijnen op rituele manier te gaan slachten en het met zware metalen vervuilde vlees aan onze kinderen te voeden, die daardoor mentaal gehandicapt worden. We kunnen in de rijke landen boven onze stand leven, omdat we andere mensen uitbuiten. Fabrieken sluiten hun deuren, omdat we niet kunnen concurreren met de lage loonlanden. Landen waarin mensen in slechte omstandigheden werken en milieuregels aan de laars worden gelapt. We eten 6 weken oude kuikens die de hoeveelheid vlees hebben van een volwassen kip. Hun poten zijn echter te zwak, waardoor velen breuken hebben. Hun organen zijn niet volgroeid; hun longen draaien overuren. Hun kortademigheid en de amoniakdampen van hun uitwerpselen resulteren in een verbrande luchtpijp. We dragen jeansbroeken die de hele wereld hebben rondgereisd voor ze geassembleerd worden. Het kleuren gebeurt met chemische stoffen, die daarna zonder boe of ba gewoon in de sloot worden geloosd. Via een arbeidsintensief en ongezond proces geven we er een versleten uitzicht aan, omdat we dat nu eenmaal mooi vinden. Dieren worden onverdoofd geslacht uit religieuze overwegingen. Met deze reden wordt dit toegestaan door de Belgische wet. Uit gemakzucht worden alle dieren in veel Belgische slachthuizen halal geslacht. Producten worden opzettelijk ontworpen om relatief snel de geest te geven, zodat de consumptiemolen blijft draaien. Als domme kippen die zich ontlasten waar ze staan, gooien rokers overal hun peuken op de grond. Dit is naar schatting elke dag 2 miljard peuken wereldwijd, goed voor 750.000 ton afval per jaar. Een peuk is niet biologisch afbreekbaar. Na 10 tot 15 jaar is de peuk onder invloed van zonlicht een hoopje plastic poeder geworden. Ondertussen is dat poeder, samen met een hoop giftige stoffen, in het grondwater gelekt. 200 peuken bevatten genoeg nicotine om een volwassen man te doden. Bovendien worden peuken door vogels en vissen vaak aanzien als voedsel. Tegenover 925 miljoen ondervoede mensen, telt onze planeet momenteel 1 miljard bewoners met overgewicht (BMI hoger dan 25). Nog eens 300 miljoen anderen hebben obesitas (BMI hoger dan 30). Concreet sterven er

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zo meer mensen aan typische overgewichtsziekten als hart- en vaatproblemen, dan er mensen overlijden van de honger. De helft van het globaal geproduceerde graan verdwijnt in de magen van de dieren die we opeten. We willen steeds meer en meer, omdat we denken dat dat ons gelukkig maakt. Zelfmoord komt echter veel frequenter voor in rijke landen dan in arme landen. De gemiddelde Belg slikt het meeste antidepressiva van heel de wereld. Onze zogenaamde welvaart blijft groeien, maar mensen worden ongelukkiger. Toch ironisch dat we in het tijdperk van internet en slimme telefoons zo weinig weten van wat er gaande is in de wereld en van wat de impact is van ons alledaags bestaan. De mens is egoïstisch en mentaal lui. We zijn slim genoeg om te weten wat de problemen zijn en hoe we ze kunnen oplossen. In plaats van echter deze intelligentie te gebruiken om de problemen op te lossen, gebruiken we deze intelligentie als een soort van bewijs van onze superioriteit t.o.v. de andere diersoorten. En deze superioriteit gebruiken we als een soort rechtvaardiging van onze daden. Het is hoog tijd dat we nadenken over de consequenties van onze daden en verantwoordelijkheid nemen. Het is hoog tijd voor verandering, maar enkel een verandering van rijstijl is spijtig genoeg niet genoeg. Bart

Preface The only thing that needs to be written in English, is a huge thanks to professor Hesham Rakha. He gave me the opportunity to work on my PhD in the United States for five months. I had a great time in Blacksburg and the Virginia Tech Transportation Institute during the winter of 2009–2010. A lot of people took part in making my USA stay unforgettable, but some took a greater part than others. If they would ever read this, I could never forgive myself not mentioning Marie-Belle, Tom, and Nicolai. Bart

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Abstract Eco-driving is adopting an eco-conscious driving style with the potential to reduce fuel consumption in an easy and cost-effective way. This dissertation develops a methodology to calculate the most fuel-efficient driving behavior. Eco-driving is treated as an optimal control problem. Different low-degree quasistatic polynomial fuel consumption models and optimal control methods are evaluated. The proposed methodology uses quadratic fuel consumption models and Pontryagin’s maximum principle. An explicit and comprehensive expression for the optimal engine control is derived. Gear shifting is introduced by maximizing or minimizing an additional variable. Simulations reveal some interesting properties of optimal driving behavior. These results are used to propose refined eco-drive guidelines.

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Beknopte samenvatting Eco-rijden is het zich eigen maken van een eco-bewuste rijstijl met het potentieel om brandstof te besparen op een eenvoudige en kostefficiënte manier. Dit proefschrift ontwikkelt een methodologie om de meest brandstofefficiënte rijstijl te berekenen. Eco-rijden wordt behandeld als een probleem van optimale controle. Verschillende quasistatische polynomische verbruiksmodellen van lage graad en optimale regelmethodes worden geëvalueerd. De voorgestelde methodologie gebruikt een kwadratisch verbruiksmodel en Pontryagins maximum principe. Een expliciete uitdrukking voor de optimale motorregeling is afgeleid. Schakelen is in rekening gebracht door het maximaliseren of minimaliseren van een extra variabele. Simulaties onthullen enkele interessante eigenschappen van optimaal rijgedrag. Deze resultaten zijn gebruikt om verbeterde ecorijrichtlijnen voor te stellen.

xiii

Contents Abstract

xi

Contents

xv

Nomenclature

xxi

List of Figures

xxvii

List of Tables

xxxi

1 Introduction

1

1.1

Road Transportation and Global Warming . . . . . . . . . . . . .

1

1.2

Reducing Road Transportation Greenhouse Gas Emissions . . .

4

1.3

Eco-Driving: a Definition . . . . . . . . . . . . . . . . . . . . . . 11

1.4

Eco-Driving: How To? . . . . . . . . . . . . . . . . . . . . . . .

13

1.5

Eco-Driving: an Optimal Control Problem . . . . . . . . . . . .

17

1.6

Goals of the Research . . . . . . . . . . . . . . . . . . . . . . .

22

1.7

Outline of the Dissertation and Main Contributions

24

. . . . . .

2 Fuel Consumption Modeling 2.1

27

Quasistatic Polynomial Consumption Models . . . . . . . . . .

xv

27

xvi

CONTENTS

2.2

2.3

2.4

2.1.1

Quasistatic Engine Modeling . . . . . . . . . . . . . . .

28

2.1.2

Fuel Consumption Characteristics . . . . . . . . . . . .

30

2.1.3

Polynomial Fuel Consumption Models . . . . . . . . . . . 31

2.1.4

Emission Modeling . . . . . . . . . . . . . . . . . . . . .

33

2.1.5

Cold Start Modeling . . . . . . . . . . . . . . . . . . . .

34

Assessment of Different Polynomial Models . . . . . . . . . . .

35

2.2.1

Assessment Methodology . . . . . . . . . . . . . . . . .

35

2.2.2

Development of New Consumption Models

. . . . . . .

38

2.2.3

Evaluation of Consumption Models . . . . . . . . . . . .

42

2.2.4

Polynomial Fuel Consumption Models of Diesel Engines and Electric Motors . . . . . . . . . . . . . . . . . . . .

46

Identification of Model Parameters . . . . . . . . . . . . . . . .

49

2.3.1

Identification with OBD Measurements . . . . . . . . .

49

2.3.2

Identification with Publicly Available Data . . . . . . .

50

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3 Theory of Eco-Driving 3.1

3.2

3.3

57

The Minimum-Fuel Vehicle Control Problem

. . . . . . . . . .

57

3.1.1

Mathematical Formulation

. . . . . . . . . . . . . . . .

58

3.1.2

Steady State Analysis . . . . . . . . . . . . . . . . . . .

60

3.1.3

Identification of Subproblems . . . . . . . . . . . . . . .

63

Solution with Euler-Lagrange . . . . . . . . . . . . . . . . . . .

70

3.2.1

Basic Analysis . . . . . . . . . . . . . . . . . . . . . . .

70

3.2.2

Human Factors . . . . . . . . . . . . . . . . . . . . . . .

72

Solution with Pontryagin’s Maximum Principle . . . . . . . . .

74

3.3.1

Basic Analysis . . . . . . . . . . . . . . . . . . . . . . .

74

3.3.2

Accelerations . . . . . . . . . . . . . . . . . . . . . . . .

77

3.3.3

Decelerations . . . . . . . . . . . . . . . . . . . . . . . .

85

CONTENTS

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3.3.4

Driving Between Stop Signs . . . . . . . . . . . . . . . .

88

3.3.5

Approaching Traffic Lights . . . . . . . . . . . . . . . .

89

3.3.6

Eco-Cruise Control . . . . . . . . . . . . . . . . . . . . . . 91

3.3.7

On the Smoothness of the Model . . . . . . . . . . . . .

93

3.3.8

Conclusions on Pontryagin’s Maximum Principle . . . .

95

Solution with Dynamic Programming . . . . . . . . . . . . . . .

96

3.4.1

Discretized Control Problem . . . . . . . . . . . . . . .

96

3.4.2

Eco-Cruise Control . . . . . . . . . . . . . . . . . . . . .

98

3.4.3

Accelerations . . . . . . . . . . . . . . . . . . . . . . . .

99

3.4.4

Decelerations . . . . . . . . . . . . . . . . . . . . . . . .

102

3.4.5

Driving Between Stop Signs . . . . . . . . . . . . . . . .

103

3.4.6

Approaching Traffic Lights . . . . . . . . . . . . . . . .

104

3.4.7

Conclusions on Dynamic Programming . . . . . . . . . .

104

3.5

Solution with Direct Multiple Shooting . . . . . . . . . . . . . .

105

3.6

Application to Other Types of Vehicles . . . . . . . . . . . . . .

107

3.6.1

Heavy-Duty Trucks . . . . . . . . . . . . . . . . . . . . .

107

3.6.2

Continuously Variable Transmission . . . . . . . . . . .

107

3.6.3

Electric Vehicles and Trains . . . . . . . . . . . . . . . .

108

3.6.4

Hybrid Electric Vehicles . . . . . . . . . . . . . . . . . .

108

Implementation in an Eco-Drive Assist System . . . . . . . . .

112

3.7.1

Model Predictive Control . . . . . . . . . . . . . . . . .

112

3.7.2

Parameter Estimation . . . . . . . . . . . . . . . . . . .

112

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

3.4

3.7

3.8

4 Application of the Eco-Drive Theory

117

4.1

Steady State Cruising Velocity . . . . . . . . . . . . . . . . . .

118

4.2

Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

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CONTENTS

4.3

4.4

4.5

4.6

4.2.1

Assessment of the Calculation Method . . . . . . . . . .

122

4.2.2

Human Acceleration Behavior . . . . . . . . . . . . . . .

123

4.2.3

Analysis of Optimal Accelerations . . . . . . . . . . . .

128

4.2.4

Continuously Variable Transmission . . . . . . . . . . .

136

4.2.5

Conclusions on Accelerations . . . . . . . . . . . . . . .

137

Decelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

4.3.1

Assessment of the Calculation Method . . . . . . . . . .

139

4.3.2

Analysis of Optimal Decelerations . . . . . . . . . . . .

140

4.3.3

Possible Fuel Savings . . . . . . . . . . . . . . . . . . . .

146

4.3.4

Conclusions on Decelerations . . . . . . . . . . . . . . .

147

Stop Signs and Traffic Lights . . . . . . . . . . . . . . . . . . .

148

4.4.1

Driving Between Stop Signs . . . . . . . . . . . . . . . .

148

4.4.2

Approaching Traffic Lights . . . . . . . . . . . . . . . .

148

Eco-Cruise Control . . . . . . . . . . . . . . . . . . . . . . . . .

150

4.5.1

Definitions and Assumptions . . . . . . . . . . . . . . .

150

4.5.2

Fuel Consumption Model . . . . . . . . . . . . . . . . .

150

4.5.3

Cost Function . . . . . . . . . . . . . . . . . . . . . . . .

154

4.5.4

Solution Method . . . . . . . . . . . . . . . . . . . . . .

159

4.5.5

Eco-Cruise Control on Steep Slopes . . . . . . . . . . .

163

4.5.6

Conclusions on Eco-Cruise Control . . . . . . . . . . . .

164

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

5 Evaluation of Eco-Drive Guidelines 5.1

Steady State Velocity

167

. . . . . . . . . . . . . . . . . . . . . . .

168

5.1.1

Gear Selection . . . . . . . . . . . . . . . . . . . . . . .

168

5.1.2

Velocity Selection . . . . . . . . . . . . . . . . . . . . . .

168

5.1.3

Pulse and Glide . . . . . . . . . . . . . . . . . . . . . . .

168

CONTENTS

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5.2

Anticipating the Traffic Flow . . . . . . . . . . . . . . . . . . . . 171

5.3

Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

5.3.1

Gear Shifting . . . . . . . . . . . . . . . . . . . . . . . .

172

5.3.2

Acceleration Rate

. . . . . . . . . . . . . . . . . . . . .

172

5.3.3

Automatic Transmission . . . . . . . . . . . . . . . . . .

173

5.4

Decelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

5.5

Driving over Hills . . . . . . . . . . . . . . . . . . . . . . . . . .

176

5.6

Starting and Stopping the Engine . . . . . . . . . . . . . . . . .

178

5.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

6 Conclusions and Recommendations

181

6.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.2

Recommendations for Future Research . . . . . . . . . . . . . .

184

A Eco-Drive Guidelines

187

B Vehicle Modeling

191

B.1 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 191 B.2 Conventional Powertrain . . . . . . . . . . . . . . . . . . . . . .

195

B.3 Hybrid Electric Powertrain . . . . . . . . . . . . . . . . . . . . . 201 C Fuel Consumption Measurements

203

C.1 Engine Dynamometer . . . . . . . . . . . . . . . . . . . . . . .

203

C.2 Chassis Dynamometer . . . . . . . . . . . . . . . . . . . . . . .

206

C.3 On-Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

D Polynomial Fuel Consumption Models

213

D.1 Polynomial Fuel Consumption Models in the Literature . . . .

213

D.2 Engine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

xx

CONTENTS

E Optimal Control

219

E.1 Optimal Control Overview . . . . . . . . . . . . . . . . . . . . .

219

E.2 Discrete Dynamic Programming . . . . . . . . . . . . . . . . . .

223

E.3 Euler-Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . .

226

E.4 Pontryagin’s Maximum Principle . . . . . . . . . . . . . . . . .

227

E.5 Direct Multiple Shooting . . . . . . . . . . . . . . . . . . . . . .

230

Bibliography

231

Curriculum Vitae

247

List of Publications

249

Nomenclature List of Symbols Roman symbols symbol a c0 , c1 , c2 cv E E Fb Ft g H i I J L mf m ˙f m ˙ f,i M n P Q R2 s t

description acceleration vehicle dynamics coefficients viscous friction coefficient Mayer term economy, = (m ˙ f + M)/v brake force traction force gear Hamiltonian total reduction ratio total inertia cost-to-go or total cost integral cost function Lagrangian function fuel consumption fuel mass flow rate idle fuel mass flow rate indirect time constraint scalar rotation speed power lower heating value coefficient of determination distance time

xxi

unit m/s2 N, N·s/m, N·s2 /m2 N·s/m ∼ kg/m N N ∼ 1/m kg ∼ ∼ ∼ kg kg/s kg/s kg/s rpm W J/kg m s

xxii

symbol T ub uc ue v

NOMENCLATURE

description torque brake control clutch control engine control velocity

unit Nm ∼ ∼ m/s or km/h

Greek symbols symbol α β ǫ ε η λ Λ τ θ ω

description

unit

model parameters

∼

small value acceleration coefficient efficiency Lagrangian multiplier adjoint state variable relative AIC engine load (torque fraction) road slope rotation speed

∼ kg/m or l/m ∼ ∼ rad rad/s

Constant parameters symbol g

description gravitational acceleration

value 9.81 m/s2

NOMENCLATURE

xxiii

Subscripts subscript e eco fco lim max min opt ref s ss

description end engine economy line fuel cut-off limit maximum minimum optimal reference start steady state

Operations and notations symbol x˙ x∗ x dx dy ∂x ∂y ∂x ∂y

z

x [u] x [-] x [∼] ∀x ∃x y = f (x)

description time derivative of x optimal value of x average value of x total derivative of x w.r.t. y partial derivative of x w.r.t. y partial derivative of x w.r.t. y, evaluated in z x has unit u x is unitless x has a variable unit for all x there exists an x y is a function of x

xxiv

NOMENCLATURE

List of Acronyms and Initialisms abbreviation AIC ARTA BCPR BMEP BSFC BVP CBG CCC CH4 CNG CO CO2 CONCAWE CVT DMS DP EEA ECC EDAS EE EL EM EMEP EPA EUCAR EUDC EV F-gas FE FWHS GHG GPS H2 HC HEV ICE IEA IPCC ITF LPG MPC

description Akaike’s information criterion arterial LOS A-B brake consumption per rotation brake mean effective pressure brake specific fuel consumption boundary value problem compressed biogas conventional cruise control carbon tetrahydride, methane compressed natural gas carbon monoxide carbon dioxide European oil company organization for environment, health and safety continuously variable transmission direct multiple shooting dynamic programming European Environment Agency eco-cruise control eco-drive assist system electricity economy Euler-Lagrange electric motor European Monitoring and Evaluation Programme Environmental Protection Agency European council for automotive R & D extra urban driving cycle electric vehicle fluorinated greenhouse gas fuel economy freeway high speed greenhouse gas global positioning system hydrogen gas hydrocarbon hybrid electric vehicle internal combustion engine International Energy Agency Intergovernmental Panel on Climate Change International Transport Forum liquified petrol gas model predictive control

NOMENCLATURE

abbreviation mph N2 O NLP NMHC NO NO2 NOx OBD OCP ODE PCC PECC PM PMP RECC RMSE rpm SOC UDDS VSP WOT

xxv

description miles per gallon nitrous oxide, laughing gas nonlinear program non-methane hydrocarbons nitric oxide nitrogen dioxide NO and NO2 on-board diagnostics optimal control problem ordinary differential equation predictive cruise control predictive eco-cruise control particulate matter Pontryagin’s maximum principle reactive eco-cruise control root mean square error rotations per minute state of charge urban dynamometer driving schedule vehicle specific power wide open throttle

xxvi

NOMENCLATURE

Eco-Drive Dictionary coasting Decelerating without propulsion. In the general meaning, this can be with the engine engaged or disengaged. In this dissertation, by coasting is meant decelerating with the engine disengaged. economy line This line in the engine speed-torque plane connects the points that minimize the specific fuel consumption (maximize efficiency) for a given engine output power. eco-cruise control A cruise control system that allows the velocity to vary within given bounds, in order to save fuel. An eco-cruise control system uses the road slope as an input. engine braking Decelerating in gear without touching the accelerator pedal. The internal friction of the engine works as a brake force. engine lugging Vibrating of the engine at low engine speeds. One hears and feels that the engine is struggling to deliver the requested torque. This occurs more easily at high engine torques. Lugging is not very comfortable and can damage the engine. Modern engines are designed better to use low engine speeds. engine stall Phenomenon whereby the engine abruptly ceases operating and stops turning. This usually happens when the engine speed is too low as a result of bad operation of the clutch and gears. fuel cut-off While engine braking, the fuel supply is cut off and there is no fuel consumption. Old engines do not have fuel cut-off. Some engines only cut off fuel above a certain engine speed.

List of Figures 1.1

BSFC map of a gasoline engine . . . . . . . . . . . . . . . . . .

18

1.2

BCPR map of a gasoline engine . . . . . . . . . . . . . . . . . .

19

1.3

The steady state fuel economy of a vehicle . . . . . . . . . . . .

19

1.4

The steady state electricity economy of a Tesla . . . . . . . . .

20

2.1

Comparison between a dynamic and a quasistatic engine model

29

2.2

Fuel consumption map of a gasoline engine . . . . . . . . . . . . 31

2.3

Example of chassis dynamometer measurement data . . . . . .

38

2.4

Measurement data from the engine dynamometer . . . . . . . .

39

2.5

Comparison of real and modeled BSFC and BCPR for model L3

46

2.6

Modeled BSFC of a diesel engine . . . . . . . . . . . . . . . . .

47

2.7

Modeled BCPR of a diesel engine . . . . . . . . . . . . . . . . .

47

2.8

Efficiency map of a synchronous AC motor . . . . . . . . . . .

48

3.1

Steady state fuel economy of a 2004 Toyota Corolla Verso . . . . 61

3.2

Illustration of optimal engine control . . . . . . . . . . . . . . .

79

3.3

Accelerations to vopt = 53.5 km/h for different values of ε . . .

80

3.4

Accelerations to 45 km/h for M = 0 and M < 0 . . . . . . . . . 81

3.5

Illustration of the gear shift strategy for an acceleration to 50 km/h 84

xxvii

xxviii

LIST OF FIGURES

3.6

Illustration of the gear shift strategy for an acceleration to 70 km/h 84

3.7

Illustration of the gear shift strategy for a deceleration from 90 km/h without brakes . . . . . . . . . . . . . . . . . . . . . .

87

Illustration of the gear shift strategy for a deceleration from 90 km/h with brakes . . . . . . . . . . . . . . . . . . . . . . . .

88

Driving between stops signs for different distances . . . . . . .

89

3.10 Illustration of the method to approach traffic lights . . . . . . .

90

3.11 Velocity profile of reactive and predictive ECC . . . . . . . . .

93

3.12 Illustration of the discretization in dynamic programming . . .

98

3.13 Velocity profiles resulting from different integration schemes in DP for ECC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

3.8 3.9

3.14 Velocity profiles of DP accelerations with different grid methods 101 3.15 Comparison of DP accelerations with and without gear shift dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

3.16 Velocity profile of driving between stop signs with DP and different grid types. . . . . . . . . . . . . . . . . . . . . . . . . .

103

4.1

Relative economy for different models with vss = 50 km/h . . .

119

4.2

Relative economy for different models with vss = 70 km/h . . .

119

4.3

Relative economy for different models with vss = 90 km/h . . .

120

4.4

Comparison of optimal accelerations with different solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

4.5

Comparison of optimal accelerations with different fuel consumption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.6

Linear decay versus experimental data for a human acceleration 127

4.7

Optimal acceleration to 70 km/h . . . . . . . . . . . . . . . . .

129

4.8

Engine torque for an acceleration on different road slopes . . .

130

4.9

CVT accelerations to 70 km/h . . . . . . . . . . . . . . . . . . .

138

4.10 Velocity profile of a deceleration from 70 km/h . . . . . . . . .

139

4.11 Influence of fuel cut-off on an optimal deceleration . . . . . . .

142

LIST OF FIGURES

xxix

4.12 Influence of the brake force on an optimal deceleration . . . . .

143

4.13 Influence of the distance on an optimal deceleration . . . . . .

145

4.14 Velocity profiles for approaching traffic lights with DP and PMP 148 4.15 Velocity profiles for approaching traffic lights with an optimal approach and an optimal deceleration . . . . . . . . . . . . . .

149

4.16 Bound between gentle and steep slopes . . . . . . . . . . . . . . . 151 4.17 The optimal steady state velocity vss as a function of the road slope for vref = 90 km/h . . . . . . . . . . . . . . . . . . . . . .

152

4.18 Velocity profiles for different consumption models over a basic road profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

4.19 Altitude profile of a 45 km section of I81 from Roanoke to Blacksburg in Virginia, USA . . . . . . . . . . . . . . . . . . . .

153

4.20 Velocity profiles of CCC and ECC on a 1◦ hill on an engine dynamometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

4.21 Velocity profiles of CCC and ECC on a 2◦ hill on an engine dynamometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

4.22 Velocity profile of ECC with different cost functions, vref = vopt

156

4.23 Velocity profiles of ECC with different cost functions, vref < vopt 157 4.24 Velocity profiles of ECC with different cost functions, vref > vopt 158 4.25 Velocity profiles of ECC with different solution methods . . . .

160

4.26 Velocity profiles of ECC with different solution methods . . . . . 161 4.27 Velocity profiles of ECC with different solution methods . . . . . 161 4.28 Velocity profiles of ECC on a steep slope with different solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

5.1

Illustration of the pulse and glide technique . . . . . . . . . . .

170

6.1

Design scheme of an eco-drive assist system . . . . . . . . . . .

185

B.1 Longitudinal forces acting on a vehicle . . . . . . . . . . . . . .

192

B.2 Scheme of a conventional manual powertrain

195

. . . . . . . . . .

xxx

LIST OF FIGURES

B.3 Comparison of different maximum torque models . . . . . . . .

197

B.4 Scheme of a parallel hybrid powertrain . . . . . . . . . . . . . . . 201 C.1 Universal engine dynamometer . . . . . . . . . . . . . . . . . .

204

C.2 UDDS drive cycle . . . . . . . . . . . . . . . . . . . . . . . . . .

206

C.3 Route in Blacksburg for on-road measurements . . . . . . . . .

210

C.4 Example of an on-road velocity measurement . . . . . . . . . .

210

D.1 Comparison of real and modeled BSFC and BCPR for model P4 215 D.2 Comparison of real and modeled BSFC and BCPR for model P5 215 D.3 Comparison of real and modeled BSFC and BCPR for model P6 216 D.4 Comparison of real and modeled BSFC and BCPR for model T1 216 D.5 Comparison of real and modeled BSFC and BCPR for model T4 216 D.6 Comparison of real and modeled BSFC and BCPR for model T5 217 D.7 Comparison of real and modeled BSFC and BCPR for model T7 217 D.8 Comparison of real and modeled BSFC and BCPR for model T8 217 D.9 Comparison of real and modeled BSFC and BCPR for model L2 218 D.10 Comparison of real and modeled BSFC and BCPR for model L3 218 E.1 Simplified optimal control problem . . . . . . . . . . . . . . . .

220

List of Tables 1.1

Share of different anthropogenic GHGs in total emissions . . .

2

1.2

Share of different sectors in total anthropogenic GHG emissions

2

1.3

Share of different means of transportation in global energy use

3

1.4

Share of different GHGs in total transportation emissions . . .

4

1.5

Comparison of well-to-wheels GHG emissions of different fuels .

6

1.6

European emission standards . . . . . . . . . . . . . . . . . . .

10

2.1

Classification of polynomial fuel consumption models . . . . . .

32

2.2

Polynomial fuel consumption models found in the literature . .

33

2.3

Subsearch for a power-based consumption model . . . . . . . .

40

2.4

Novel polynomial fuel consumption models . . . . . . . . . . .

42

2.5

Evaluation of models based on model structure . . . . . . . . .

43

2.6

Evaluation of models based on the fit quality with engine dynamometer measurements . . . . . . . . . . . . . . . . . . . .

44

Evaluation of models based on the fit quality with chassis dynamometer measurements . . . . . . . . . . . . . . . . . . . .

45

Evaluation of models based on the fit quality with on-road measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Influence of the driving cycle on model quality . . . . . . . . .

50

2.7 2.8 2.9

2.10 Comparison of R2 of models identified with publicly available data 53

xxxi

xxxii

LIST OF TABLES

3.1

Optimal steady state velocity . . . . . . . . . . . . . . . . . . .

62

3.2

Comparison of accelerations for different values of ε and M . . . 81

3.3

Comparison of the travel time when approaching traffic lights .

4.1

Influence of the velocity on the fuel consumption and travel time 120

4.2

Influence of the gear on the fuel consumption . . . . . . . . . . . 121

4.3

Simulation results of accelerations with different fuel consumption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.4

Summary of experimental acceleration data and curve fit results 126

4.5

Analysis of possible fuel savings with an optimal acceleration .

130

4.6

Analysis of possible fuel savings with optimal gear shifting . . .

132

4.7

Optimal gear shift engine speeds for different vss . . . . . . . .

133

4.8

Optimal gear shift engine speeds for different road slopes

. . .

134

4.9

Evaluation of a linear optimal acceleration approximation . . .

135

4.10 Simulation results of optimal decelerations from 70 km/h . . . .

140

4.11 Simulation results for a deceleration with different brake forces

144

4.12 Simulation results for decelerations with different brake forces .

144

4.13 Simulation results for a deceleration with different strategies . .

146

4.14 Fuel consumption for different consumption models over a basic road profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

90

4.15 Evaluation of ECC over a section of I81 with different consumption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.16 Simulation results of ECC with different cost functions . . . . .

157

4.17 Simulation results of ECC with different solution methods . . .

162

5.1

Pulse and glide simulation results . . . . . . . . . . . . . . . . . . 171

5.2

Simulation results for different hill driving strategies . . . . . .

177

B.1 Coefficients of rolling friction . . . . . . . . . . . . . . . . . . .

193

LIST OF TABLES

xxxiii

B.2 Coefficient of road adhesion . . . . . . . . . . . . . . . . . . . .

199

C.1 Specifications of the Toyota 3ZZ-FE engine . . . . . . . . . . .

204

C.2 Specifications of the 2004 Toyota Corolla Verso . . . . . . . . .

205

C.3 Overview of vehicles tested on a chassis dynamometer . . . . .

208

C.4 Overview of vehicles tested on-road . . . . . . . . . . . . . . . . . 211

Chapter 1

Introduction 1.1

Road Transportation and Global Warming

Efficient transportation has always been important for mankind. One of the more important characteristics of our species, walking on two legs, originates from the urge for better mobility. As the environment of our early ancestors became dryer, they needed to cover larger distances to find food. Bipedal walking is potentially more energy efficient than quadrupedal walking, and evolution did the rest [101]. The next big transportation step was taken in central Asia around 3500 BC, when the horse was domesticated. For many ages, the horse was man’s best transportation friend, but by the end of the 19th century the downside of the grown mobility became apparent. The horses needed to eat and this drove up food prices and caused shortages. And what goes in must come out, resulting in massive piles of dung in large cities [45, 140]. In the meanwhile, the Belgian-born engineer Jean Joseph Etienne Lenoir built the first commercial internal combustion engine in 1860. Three years later, he used an improved engine (with petroleum and a carburator) to build the hippomobile, a vehicle with three wheels that managed to complete a fifty-mile trip. The German Nikolaus Otto improved the combustion engine in 1862 with compression and four strokes instead of two. Further improvements were done independently by Gottlieb Daimler and Karl Benz. In 1985, Benz built the first true automobile. In 1903, Henry Ford started the Ford Motor Company and was able to produce cheap cars. Soon, combustion engine propelled vehicles started to replace horses for transportation and the horse-related problems were solved.

1

2

INTRODUCTION

And then there was global warming ... In the last century, the earth’s surface temperature increased by almost 1◦ C. Two thirds of that in the last three decades [42]. The majority of this temperature increase is very likely due to the increase of anthropogenic1 greenhouse gas (GHG) emissions. Table 1.1 shows the share of different anthropogenic GHG emissions in 2004. Carbon dioxide (CO2 ) accounts for more than 3/4 of the effect. Table 1.2 shows the share of different sectors. Transport has a share of 13 % and is the fastest growing source [100]. In the EU the transportation share is 19.3 % and it is the only growing source since 1990 [49]. As a comparison, livestock (the consumption of animal products) contributes 18 %. Table 1.3 shows the share of different means of transportation in the global transportation energy use in 2000 [75]. Road transport accounts for approximately 3/4 of the transportation energy use. Table 1.1: Share of different anthropogenic GHGs in total emissions in 2004 in terms of CO2 -equivalent, IPCC [100]. type

share [%]

CO2 fossil fuel use CO2 deforestation, decay of biomass CH4 (methane) N2 O (nitrous oxide, laughing gas) CO2 other F-gases (fluorinated greenhouse gases)

56.6 17.3 14.3 7.9 2.8 1.1

Table 1.2: Share of different sectors in total anthropogenic GHG emissions in 2004 in terms of CO2 -equivalent, IPCC [100]. type

share [%]

energy supply industry forestry agriculture transport residential and commercial buildings waste and wastewater

25.9 19.4 17.4 13.5 13.1 7.9 2.8

1 anthropogenic

= caused by humans

ROAD TRANSPORTATION AND GLOBAL WARMING

3

Table 1.3: Share of different means of transportation in global energy use in 2000, IEA [75]. type

share [%]

light-duty vehicles heavy freight trucks air ships medium freight trucks buses 2-wheelers rail

44.5 16.2 11.6 9.5 8.8 6.2 1.6 1.5

It is of course too late to prevent global warming. Yet, it is not too late to prevent catastrophic global warming. One can conclude that an increase of the global mean temperature of 2◦ C is the absolute maximum to avoid disastrous consequences. In order to do this, everybody will have to make an effort to reduce GHG emissions. This dissertation is about a simple measure that can be taken in the transportation sector: eco-driving.

4

1.2

INTRODUCTION

Reducing Road Transportation Greenhouse Gas Emissions

Table 1.4 shows that CO2 is the principal transportation GHG. Therefore, the focus in this dissertation will only be on those emissions. CO2 emissions are directly linked to the fuel consumption. What follows is a classification of different methods to reduce road transportation CO2 emissions. This section by no means tries to give a full and complete overview of methods to reduce road transportation GHGs and their qualitative impact. It merely gives an idea of what is possible. Good overviews can be found in e.g. [82, 100, 179, 180, 182]. Table 1.4: Share of different GHGs in total transportation emissions in CO2 equivalent, IPCC [100].

USA Japan EU

CO2 [%]

CH4 [%]

N2 O [%]

F-gases [%]

88.4 96.0 95.3

0.2 0.1 0.3

2.0 2.5 2.8

8.9 1.4 1.7

Reducing Vehicle Loads The work needed to move a vehicle originates from five different vehicle loads: rolling friction, aerodynamic drag, road slope resistance, inertia, and accessory loads. Vehicle weight plays an important role in rolling friction, road slope resistance, and inertia. Vehicle weight can be reduced by using lightweight materials such as aluminum or composite materials. Simpler and cheaper methods are e.g. avoiding unnecessary loads in the vehicle and simply buying a smaller car. As an example: a BMW ActiveHybrid 5 consumes 7 liters per 100 km (l/100km), a Citroën C1 4.6 l/100km. The BMW is a hybrid vehicle, but outweighs the C1 by 1180 kg. Reducing rolling friction can also be achieved by using better tires or maintaining a good tire pressure. Aerodynamic drag can be decreased by lowering the drag coefficient with a more aerodynamic design of the vehicle. Another, simpler measure, is to reduce the frontal surface of the vehicle.

REDUCING ROAD TRANSPORTATION GREENHOUSE GAS EMISSIONS

5

The principal accessory load that can easily be reduced is the air conditioning. This system is powered by a compressor that requires a significant amount of energy. Using the ventilation ducts is the best way to cool the interior of the vehicle, as opening the windows increases the aerodynamic drag.

Improving Powertrain Efficiency The powertrain of a vehicle refers to all the components that generate the traction force that is applied to the road. For a conventional powertrain this is the combustion engine, the clutch or torque converter, the gearbox, the differential, and the wheels. A more efficient powertrain will reduce the fuel consumption and thus CO2 emissions. There are numerous ways to improve engine efficiency, which will not be discussed here. The rest of a conventional powertrain can be improved by reducing friction losses or for example using a gearbox with more gears, such that the engine can be kept in a more efficient operating point. Instead of using a conventional powertrain, one can use a hybrid electric one. A hybrid electric powertrain combines a combustion engine with an electric motor/generator, and a battery or ultracapacitor. The motor/generator can be used to regenerate braking power and can assist the combustion engine while accelerating, thus allowing a smaller engine (downsizing). Furthermore, a hybrid electric powertrain allows for the combustion engine to operate more efficiently.

Using Alternative Fuels The most commonly used fuels for road transportation are fossil gasoline and diesel. Diesel has longer carbon chain lengths, resulting in a higher specific CO2 emission than gasoline: 0.25 versus 0.24 kg CO2 /kWh. However, a diesel engine is more efficient than a gasoline engine. There are other hydrocarbons that have shorter chains and are less carbon-intensive: liquified petrol gas (LPG, 0.22 kg CO2 /kWh) and compressed natural gas (CNG, 0.18 kg CO2 /kWh)2 . The Honda Civic GX, that runs on natural gas, won the 2011 green car of the year award in the USA for the eighth consecutive time. When using hydrocarbons, the origin of the fuel is of major importance. Instead of fossil fuels, one can use biofuels or synthetic fuels. Most common biofuels are ethanol, biodiesel, and compressed biogas (CBG). Biofuels can reduce the net 2 Values

according to the US Energy Information Administration.

6

INTRODUCTION

CO2 emission because they are made from biomass that absorbed CO2 while it was growing. This reduction depends strongly on the fuel, where, and how it was produced. Synthetic diesel is the result of Fischer-Tropsch synthesis from a mixture of carbon monoxide and hydrogen obtained by partial oxidation of e.g. coal or wood, or by steam reforming of natural gas. One can also use noncarbon based fuels like hydrogen gas (H2 ) and last but not least: electricity. When comparing different fuels, the whole well-to-wheels path should be considered. This takes into account both the production of the fuel and driving the vehicle. Table 1.5 shows the well-to-wheels GHG emissions of different fuels [57]. The calculations are based on 2010 technologies. For gasoline, both port injected (PI) and direct injected (DI) engines are considered. Table 1.5: Comparison of well-to-wheels GHG emissions of different fuels, EUCAR/CONCAWE [57]. fuel

GHG emissions [g CO2 -eq./km]

conventional gasoline, PI conventional gasoline, DI conventional gasoline, DI hybrid conventional diesel conventional diesel, hybrid LPG, PI CNG EU mix, PI CBG corn and barley, PI ethanol best, PI ethanol worst, PI biodiesel best biodiesel worst synthetic diesel best synthetic diesel worst H2 wood, fuel cell H2 EU mix, fuel cell electricity EU mix Nissan Leaf, Belgium3

167 166 136 149 120 141 124 49 19 147 27 96 6 335 10 50 68 92

3 Based on 17 kWh/100km for the Nissan Leaf. The base load electricity is nuclear in Belgium. Extra loads, such as charging the vehicle battery, are produced with coal and gas fired power plants (540 g/kWh) [170].

REDUCING ROAD TRANSPORTATION GREENHOUSE GAS EMISSIONS

7

Besides GHG emissions, some fuels also have other negative impacts. Gasoline, LPG, CNG, CBG, ethanol, and H2 used in a spark ignition indirect injection combustion engine with a three-way catalyst, are relatively clean. Diesel is used in a compression ignition engine and emits particulate matter (PM) and NOx (nitric oxide NO and nitrogen dioxide NO2 ). These emissions are lower when using synthetic diesel or biodiesel [13, 98, 114]. The difference in emissions between diesel and gasoline passenger cars is reflected in the European emission standards, shown in table 1.6. Gasoline engines emit more carbon monoxide (CO) and unburned hydrocarbons (HC), mainly as a result of a cold start. The introduction of a PM norm for gasoline cars is for direct injection gasoline engines. Indirect injection gasoline engines hardly emit any PM. Because of NOx and PM emissions, diesel cars are prohibited in California, New York, Massachusetts, Maine, and Vermont. Soot is the result of an incomplete combustion and is a kind of PM. Inhaling PM may cause asthma, lung cancer, cardiovascular issues, birth defects, and premature death [60]. NOx forms acid rain that damages forests, and causes lakes and streams to become acidic and unsuitable for many fish. It also forms nitric acid vapor and related particles. Inhaling these has similar effects as PM. NOx increases nitrogen loading in water bodies and upsets the chemical balance of nutrients. It also leads to oxygen depletion, and reduces fish and shellfish populations. Moreover, NOx can form ozone, which causes lung damage and reduces crop yields. NOx can even react with common organic chemicals to form toxic products that may cause biological mutations [59]. Electric vehicles (EVs), hybrid electric vehicles (HEVs), and fuel cell vehicles (FCVs, running on H2 ) need batteries and electric motors. Rare earth elements are often used to manufacture these batteries and motors. Almost all the rare earth elements in the world come from a Chinese mine north of Baotou. Here, mining and refining rare earths has a serious environmental impact as acids are pumped into streambeds and radioactive waste with toxic metals are injected into an artificial lake [31]. Growing biomass for biofuels might take space and resources away that are normally used for growing food crops. It might even induce deforestation [121].

Improving Infrastructure and Traffic Management The road infrastructure has an important influence on the fuel consumption. For example, improving the condition of the pavement and carpool lanes are simple but effective infrastructure measures. Fuel economy can also be increased with better traffic management. One could e.g. decrease the speed limit or use adaptive traffic light timing [116]. Barth

8

INTRODUCTION

and Boriboonsomsin [17] propose a system that advises vehicles to travel at a specific velocity during congestions. The overall goal is to smooth the traffic flow and thereby decrease fuel consumption. It is found that fuel consumption can be reduced by an order of 10–20 %, without drastically affecting overall travel time. The savings depend on the congestion level.

Changing Driving Behavior It is well known that the way a vehicle is operated has a significant influence on the fuel consumption and thus GHG emissions [37, 38, 65, 67, 135, 189]. ElShawarby, Ahn, and Rakha [58] conducted a series of tests on the Virginia Tech smart road. The findings show that an increase in speed limit from 72 km/h to a speed limit of 88 km/h could result in an increase of 28 % in vehicle fuel consumption, and a 54 % increase with a speed limit increase to 104 km/h, while a decrease in speed limit to 56 km/h results in an increase of 15 % in the vehicle fuel consumption. Three acceleration rates are also tested: 40, 60, and 100 % of the maximum acceleration rate. Aggressive acceleration maneuvers result in higher fuel consumption, if the runs are compared over a fixed distance of travel. De Vlieger [46] concludes that comparable fuel consumption rates are obtained from normal and calm driving. Those from aggressive driving (defined as sudden and high acceleration and heavy braking) are higher, by as much as 40 % in city traffic. Ericsson [61] investigates which properties have main effect on fuel use. Sixteen independent driving pattern factors are calculated for each of 19230 driving patterns collected in real traffic. Nine factors have an important effect on fuel consumption. Four of them relate to acceleration and power demand: factor for acceleration with strong power demand, stop factor, factor for acceleration with moderate power demand, and extreme acceleration factor. Three are related to gear changing: factor for late gear changing from 2nd to 3rd gear, factor for engine speed higher than 3500 rpm, and factor for moderate engine speeds in 2nd and 3rd gear. Two are related to velocity: velocity oscillation factor, and factor for velocity 50–70 km/h. Beckx et al. [19] explore the influence of gear changing behavior on vehicular fuel consumption. A comparison is made between normal and aggressive gear changing. Normal assumes a gear shift when the engine speed exceeds 55 % of the maximum speed, aggressive 80 %. The results indicate that average fuel consumption per trip increases by 30 % when applying aggressive gear changing compared to normal.

REDUCING ROAD TRANSPORTATION GREENHOUSE GAS EMISSIONS

9

Manzie, Watson, and Halgamuge [133] show that a 7 s traffic look-ahead capability can achieve a 33 % fuel economy improvement in urban driving. This is done by smoothing the velocity profile and thus reducing accelerations and decelerations. It can be concluded from the literature that there are four main driving behavior contributors to the fuel consumption: 1. gear shift behavior, 2. vehicle velocity, 3. acceleration and deceleration behavior, 4. amount of accelerations and decelerations, i.e. the smoothness of driving. The route choice can also be seen as part of the driving behavior. It should be mentioned that it significantly influences the fuel consumption as well, see e.g. [4, 62].

Changing Transportation Mode Last but not least, the transportation mode has by far the most influence on the fuel consumption and GHG emissions. Assuming a certain degree of occupancy, collective transport (buses, trains, carpooling, ...) emits less GHGs than private cars. Walking and biking are of course the most eco-friendly ways of transportation.

INTRODUCTION 10

Table 1.6: European emission standards for passenger cars (category M). NMHC = non-methane hydrocarbons. norm

date

Euro 1

July 1992

Euro 2

January 1996

Euro 3

January 2000

Euro 4

January 2005

Euro 5

September 2009

Euro 6

September 2014

fuel

CO [g/km]

HC [g/km]

NMHC [g/km]

NOx [g/km]

HC + NOx [g/km]

PM [g/km]

diesel gasoline diesel gasoline diesel gasoline diesel gasoline diesel gasoline diesel gasoline

2.72 2.72 1.00 2.20 0.64 2.30 0.50 1.00 0.50 1.00 0.50 1.00

0.2 0.1 0.1 0.1

0.068 0.068

0.50 0.15 0.25 0.08 0.18 0.06 0.08 0.06

0.97 0.97 0.70 0.50 0.56 0.30 0.23 0.17 -

0.140 0.080 0.050 0.025 0.005 0.005 0.005 0.005

ECO-DRIVING: A DEFINITION

1.3

11

Eco-Driving: a Definition

The previous section gave an overview of the different methods to reduce road transportation fuel consumption and CO2 emissions. The author distinguishes three different actors in realizing the latter reduction: manufacturers, governments, and individuals. Manufacturers (and also research institutes to some extent) provide technologies and the actual vehicles that are used for transportation. Unfortunately this actor is mainly result driven, aiming for profit and publications, and is highly influenced by individuals (the customer). Companies and research institutes have to operate within the framework that is provided by governments. Governments have a huge influence in the whole global warming story. They decide on taxation, infrastructure, traffic management, vehicle maintenance requirements, traffic restrictions, ... For example reducing the speed limit from 100 to 80 km/h reduces fuel consumption by 15 to 25 % [182]. The influence of the government is illustrated by the share of diesel vehicles in the fleet in Belgium. When buying a new car, 80 % of Belgians choose a diesel car, compared to e.g. 30 % in the Netherlands and 50 % in Germany. This is mainly due to taxing that makes diesel a financially interesting choice. Since the abolition of the eco-premium, sales are shifting more towards gasoline vehicles again. Moreover, only in Europe diesel is regularly used for passenger cars. Driven by fear of not being reelected, politicians rarely take harsh (unpopular) decisions that are necessary if we really want to put a halt to global warming. The last actor is the individual, who has to make personal choices such as: transportation needs, mode of transportation, choice of car to buy, vehicle maintenance (e.g. tire pressure), route choice, climate control (airco, windows, or ventilation), driving behavior, ... The average western individual cares more about personal comfort, consumption, image, power, and money than about the environment and real happiness. This all leads to a complex vicious circle. In its broad definition, eco-driving comprises all of the aforementioned personal choices. It is a behavioral change approach to reducing GHG emissions. In its strict definition, as it is used in this dissertation, it only refers to adopting a fuel-efficient driving behavior. Eco-driving has the potential to reduce the CO2 emissions of road transport and has therefore become a key part of national strategies to reduce CO2 emissions in a number of European countries [71], Belgium not included. Jack Short, secretary-general of the International Transport Forum (ITF), says: “It’s one of the things that is relatively cheap and can be done relatively easily, so it should be part of the package. There’s no silver bullet, and anything that brings 5 –10 % reductions should be grasped.”

12

INTRODUCTION

According to a recent study by Goodyear Dunlop [81], 70 % of European road hauliers have already invested in eco-driving. Since recently not personnel, but fuel is the major expense for these companies, with 30 % of the total costs. Besides reducing GHG emissions, eco-driving has numerous other advantages. It improves road safety and enhances driving skills, and it reduces local air pollutants and noise. Eco-driving also has financial benefits as it saves fuel, costs of accidents, and maintenance costs. Eco-driving is a more responsible way of driving, with less stress and higher comfort for drivers and passengers. Eco-driving is adopting an eco-conscious driving style with the potential to reduce fuel consumption in an easy and cost-effective way.

ECO-DRIVING: HOW TO?

1.4

13

Eco-Driving: How To?

The advantages of eco-driving are clear. Yet, the big question remains: how do we do it? Usually, eco-driving is applied through training courses and workshops. These courses are based on a few guidelines, see appendix A on p. 187. The ones that directly concern the driving behavior are the following: • anticipate traffic flow, • avoid braking, • maintain a steady speed at low rpm, • shift up early, • switch off the engine at short stops. Several reports quantify the fuel savings following an eco-driving course. The international commission for driver testing shows short-term a reduction of fuel consumption between 15 and 25 %. Yet after a while, old driving habits might reemerge and savings are between 5 and 8 % [40]. The same effect is seen by Beusen et al. [23]. Ten drivers were given an eco-drive course and after four months fuel savings were 5.8 % on average. After a longer period of time, some drivers fell back into their old driving habits. The ITF reports short-term savings of 5–15 %, 5 % on average over the mid-term (< 3 years) and 2–3 % in the long-term (> 3 years) [71]. Fuel savings are also researched for buses. For example Wåhlberg reports only a 2 % fuel consumption reduction as a mean over 12 months after training [184], while Rafael-Morales and Gortar [150] mention 16 % for a study in Mexico. In-car equipment can give feedback to the driver on the fuel-efficiency of his/her driving behavior, and thus help to improve long-term fuel economy. By just showing the instant fuel economy to the driver (econometer), a fuel use reduction of 6 % can be attained [29]. There also exist more advanced systems, that are grouped under the name eco-drive assist systems (EDAS). A first type of EDAS analyzes the driving behavior, in real time (during driving) and/or off-line (afterwards). In 2012, the following systems are (commercially) available: EcoGyzerr is an independent eco-driving solution. It logs the velocity and location of the vehicle (using a global positioning system (GPS) and an accelerometer) and analyzes the driving behavior based on heuristics such as: driving time without fuel consumption, hard braking time, soft braking time, velocity, and driver request [54].

14

INTRODUCTION

Renault Optifuel Infomax allows you to track: time spent in the economic zone, fuel consumed in this zone, number of times the brake pedal is depressed, time spent idling, and cruise control usage at full engine load [156]. Fiat Eco:Drive allows you to save telemetric vehicle data on a USB-stick. The driving efficiency is based on four parameters: steady acceleration, steady deceleration, early gear changes, and moderate consistent velocity. Drivers receive a star rating (out of five) for each of these indicators [69]. Ford EcoMode generates a personalized driver operation scorecard by monitoring engine speed, vehicle velocity, accelerator position, clutch position, selected gear, and engine temperature related to three disciplines: gear shifting, anticipation, and velocity. Scores are displayed with a five-leaf icon. While driving, the scoring system also generates hints on how to gain more leaves for each discipline [11]. Honda Eco Assist is found in the 2011 Honda Insight, a hybrid vehicle. It gives feedback on the driving behavior using a colored display behind the speedometer, leaf and flower icons, and an eco-drive bar that the driver needs to keep in the middle. At the end of the trip, the driver receives a leaf-and-flower-score, and can be rewarded with a trophy or punished with a withering plant [12]. Toyota Eco Drive Indicator lights up when the vehicle is being operated in a fuel-efficient manner. It is based on a comprehensive determination that takes into consideration such factors as accelerator use, engine and transmission efficiency, velocity, and level of acceleration [173]. Punch Telematix CarCube driving style assistant displays a number of values for the driver. This includes velocity, engine speed, acceleration, braking behavior, and stationary fuel consumption. These parameters have been set to take into account the type of vehicle and the permitted or fuel-reducing velocity. The driver receives a warning if he/she exceeds the maximum velocity or if his/hers braking or accelerating behavior does not conform to the preset values [51]. ECOdrive is an intelligent driving style stimulator. The ECOdrive unit works together with the electronic gas pedal in the vehicle. Drivers with poor acceleration behavior are actively corrected by the ECOdrive system [52]. Modern Drive Devices (MDD) is a general name for a plug-and-play system that gives real time feedback and also performs an off-line analysis of the driving behavior. van Driel, Tillema, and van der Voort [177] conducted a study on the MDD-concept. Here, the driving style

ECO-DRIVING: HOW TO?

15

analysis is also based on values such as too fast accelerations, too hard decelerations, duration of stop, and duration of idling. FEST (fuel-efficiency support tool) back-calculates the minimal fuel consumption for manoeuvres carried out. If the actual fuel consumption deviates from this optimum, the support tool presents advice to the driver on how to change his or her behavior. Evaluation of the tool by means of a driving simulator experiment revealed that drivers were able to reduce overall fuel consumption by 16 % compared to normal driving. The same drivers were only able to achieve a reduction of 9 % when asked to drive fuel efficiently without support [176]. The aforementioned systems do not explicitly tell the driver what to do in real time. They register and display relevant indicators and possibly give an interpretation. Better systems, the second type, would give the driver direct advice or even automate part of the driving. A gear shift indicator (GSI) is present in a large share of new vehicles. A GSI gives direct advice to the driver by indicating when to shift. A more intervening system is intelligent speed adaptation (ISA). It limits the vehicle velocity to the speed limit. As speeding often increases the fuel consumption, ISA can improve the fuel economy. Microsimulation modeling on four different networks shows fuel savings between 1 and 8 % [34]. The Nissan Eco Pedal [190] informs the driver of excessive acceleration by a counter push-back mechanism in the accelerator pedal. Nissan claims that this can improve fuel efficiency by 5–10 %. The system operation is based on an instantaneous fuel consumption threshold. Upcoming vehicle sensors and telemetrics allow for more functionalities. More and more vehicles are equipped with: a GPS and digital maps; radars that can measure the distance to the preceding vehicle; cameras that see other objects in traffic; ... The future also holds communication possibilities between vehicles and the infrastructure [44]. These technologies can e.g. be used to anticipate traffic light timing. Asadi and Vahidi [14] formulate a control algorithm to schedule an optimum velocity trajectory for the vehicle. The control objectives are: timely arrival at green light with minimal use of braking, maintaining safe distance between vehicles, and cruising at or near set speed. In a specific suburban driving case study, the system reduces fuel consumption by 47 %. A comparable algorithm gives velocity advice to the driver so that the probability of having a green light is maximized. This velocity planning reduces fuel consumption by 12–14 % in a simulation of a 10-intersection signalized corridor [132]. A third type of EDAS makes use of the knowledge of the road slope. Driving on hilly roads can significantly decrease the fuel economy. Park and Rakha [143] show that a 6 % increase of road slope increases the fuel consumption level

16

INTRODUCTION

between 40 and 94 %, while a 1 % road slope increase yields 13 to 18 % extra fuel consumption. Other research [28] shows that the measured fuel economy on a level road is superior compared to hilly roads by 15 to 20 %. This knowledge can be used to lower the fuel consumption by adapting the velocity on hilly roads. This approach is used in heavy-duty truck driving under the name predictive cruise control (PCC). This system uses the downstream road profile to actively change the vehicle velocity through the use of a cruise controller. An experiment on a 120 km segment of a Swedish highway decreases the fuel consumption by 3.5 % without increasing the trip time [88]. Evidently, this approach can also be adopted in driving a passenger vehicle. In this application it is named eco-cruise control (ECC) in accordance with eco-driving. Ahn, Rakha, and Moran [6] show that by simply driving slower uphill and faster downhill within a velocity range of 10 % around the cruise control reference velocity, fuel savings of 10 % are possible. The simulation is based on a 45 km section of I81 in Virginia with 7 different vehicles. Larsson and Ericsson [119] evaluate an acceleration advisory tool. The tool gives resistance in the accelerator pedal when the driver tries to accelerate rapidly. A test carried out in Sweden, shows a significant reduction of aggressive acceleration. Although in general, no significant reduction in fuel consumption is observed. Qian and Chung [149] even report a negative influence of eco-driving on the overall fuel consumption. This is based on acceleration simulations where it is assumed that eco-driving equals moderate and smooth acceleration. Is it a correct assumption that strong accelerations are bad for the fuel economy? Is it good engineering practice to base an EDAS on such a simple heuristic? In fact, as illustrated above, most EDAS are based on similar heuristics. PCC systems are the exception. Common sense indicates that an efficient EDAS and eco-drive guidelines should be based on a good knowledge of the most optimal (fuel-efficient) driving behavior.

ECO-DRIVING: AN OPTIMAL CONTROL PROBLEM

1.5

17

Eco-Driving: an Optimal Control Problem

Transport is “the action of carrying or conveying a thing or person from one place to another”. Thus transport requires mechanical energy that comes from another type of energy. The two main forms of energy that are used for road transport are electric energy stored in a battery, and chemical energy stored in fuel. The work presented in this dissertation mainly focusses on the use of chemical energy, converted with an internal combustion engine (ICE). The techniques are applicable to electric vehicles as well. An important performance indicator is the brake specific fuel consumption BSFC [kg/J]: BSFC =

m ˙f , Pe

(1.1)

where m ˙ f [kg/s] is the fuel mass flow rate and Pe [W] the engine output power, also referred to as engine brake power. Pe = T e · ωe ,

(1.2)

with Te [Nm] the engine output torque (brake torque) and ωe [rad/s] the engine rotation speed. For simplicity and when confusion is not possible, the subscript e will be omitted in the rest of the dissertation. In the automotive world, the rotation speed is often denoted in rpm (rotations per minutes) and then the symbol n is used. The BSFC is linked to the brake efficiency ηb [-] of the engine: BSFC =

1 , ηb Q

(1.3)

with Q [J/kg] the lower heating value of the fuel. Figure 1.1 shows the BSFC map of a 1.6 l gasoline engine. The minimum BSFC (or maximum efficiency) is around 2000 rpm at a torque close to the maximum torque Tmax . Going away from this optimal point will increase the BSFC. The figure also shows the economy line. This line gives the torque Teco that minimizes the BSFC for a given engine power. The economy line and the specific fuel consumption are usually considered very important with regard to efficiency in automotive applications, e.g. [122]. Certain EDAS advise the driver to keep the operating point of the engine as close to the point of minimum BSFC as possible [176]. However, the goal of transportation is not to produce a certain amount of power. The goal is to move a vehicle from one place to another. Thus instead of using the fuel consumption per unit of produced energy as a measure for efficiency, one should consider the ratio between fuel consumption and traveled distance. This is what is called the fuel economy

18

INTRODUCTION

engine torque T [Nm]

Tmax

BSFC [g/kWh]

Teco

100 260 270 280

300

50

350 400 600

1000

1500

2000

2500

3000

3500

engine speed n [rpm]

Figure 1.1: BSFC map of a 1.6 l gasoline engine, including the economy line. FE [kg/m]: FE =

m ˙f , v

(1.4)

with v [m/s] the vehicle velocity. Translated in terms of the engine, this is the brake consumption per rotation of the engine BCPR [kg/rad]: BCPR =

m ˙f . ω

(1.5)

Figure 1.2 shows the BCPR map for the same engine of figure 1.1. The vehicle is a load for the engine and should also be considered when reasoning about efficiency. Therefore, the steady state fuel economy FEss [kg/m] is introduced. This is the fuel that the vehicle consumes when driving at a constant (steady state) velocity and is often expressed in l/100 km. Figure 1.3 shows the steady state fuel economy of a vehicle as a function of the velocity. At the optimal cruising velocity vopt , FEss reaches its minimum value, i.e. usually around 70 km/h. Figure 1.4 shows the equivalent for an electric vehicle, the electricity economy EEss [Wh/km] for a Tesla. The optimal cruising velocity is around 30 km/h.

ECO-DRIVING: AN OPTIMAL CONTROL PROBLEM

19

engine torque T [Nm]

Tmax , Tmin

100

BCPR [mg/rad]

70 60 50

50

40 30 20

0

10

1000

1500

2000

2500

3000

3500

engine speed n [rpm]

fuel economy FEss [l/100km]

Figure 1.2: BCPR map of a 1.6 l gasoline engine.

15

10

5 0

20

40

60

80

100

120

velocity v [km/h]

Figure 1.3: The steady state fuel economy of a vehicle with a small gasoline engine, EMEP/EAA [142].

INTRODUCTION

electricity economy EEss [Wh/km]

20

400

300

200

100 0

50

100

150

200

velocity v [km/h]

Figure 1.4: The steady state electricity economy of a Tesla. The existence of vopt seemingly solves the problem: the optimal driving behavior is cruising at vopt . However, the following questions still remain: • How should we go from standstill to cruising? • How should we stop or decelerate the vehicle? • What if vopt is too fast or too slow? • How should we shift gears? • What about other vehicles in traffic? The first two questions refer to the fact that eco-driving is not a steady state problem, but a dynamic one. Thus, the goal is to control a vehicle, which is a dynamic system, in order to minimize the fuel consumption. This clearly calls for optimal control, also referred to as dynamic optimization. Optimal control finds a control law for a system such that an objective (cost) function is minimized. The system is the vehicle. It has two state variables x: velocity v and traveled distance s [m]; and four control inputs u: engine control ue [-], brake control ub [-], clutch control uc [-], and gear g [-]:   ue    ub  v  x= , u= (1.6)  uc  . s g

ECO-DRIVING: AN OPTIMAL CONTROL PROBLEM

21

The objective function is the total fuel consumption, i.e. the time integral of the fuel mass flow rate m ˙ f . The optimal control problem then looks as follows: min

x(·), u(·)

Zte

m ˙ f (x, u) dt,

(1.7)

0

with te [s] the end time. Besides the objective function, an optimal control problem (OCP) usually also includes constraints. For eco-driving they are the following: • powertrain The powertrain of the vehicle is limited in its operation range. For example the engine has a minimum and maximum rotation speed, and a minimum and maximum torque. • distance In the control problem the vehicle has to travel to a given point, traveling a predefined distance. • travel time The time in which the distance is traveled, can also be limited. • velocity The vehicle has to obey the traffic laws: speed limits, standstill at a red light, ... • traffic The vehicle should keep a safe distance from the preceding vehicle. • system dynamics The solution of the OCP is subject to the dynamic equations of the system.

22

INTRODUCTION

1.6

Goals of the Research

An extensive literature survey shows the potential of eco-driving for fuel consumption reduction. Eco-drive assist systems can save extra fuel compared to just following eco-drive guidelines. These systems are often based on simple heuristics. However, eco-driving is an optimal control problem. It requires optimal control to find the real optimal (most fuel-efficient) driving behavior. The goal of this research is to make existing eco-drive applications more efficient and to contribute to the growth and popularity of eco-driving. To attain this goal, several subgoals can be defined: 1. Analyzing the properties of eco-driving as an optimal control problem. 2. Improving and simplifying existing methods to solve eco-driving as an optimal control problem. 3. Using these methods in eco-drive applications. The core of the research is to develop a simple methodology to calculate the most fuel-efficient driving behavior, one that captures all of the essentials of the optimal control problem. This is done under the following conditions: • The main focus is on passenger cars with a conventional powertrain and a manual transmission. Automatic transmissions are discussed briefly. The methods are equally applicable for heavy-duty trucks and electric vehicles, and partly for hybrid electric vehicles. • Numerical examples are usually given from a European point of view: velocities are in km/h and important speed limits are 30, 50, 70, and 90 km/h. • Validation is mainly done with measurements of gasoline engines. • The constraints of the optimal control problem are basic: given end velocity and distance, travel time, ... Translating real and complex constraints from the surrounding traffic to these basic constraints is out of the scope of the research. • Simple models of longitudinal dynamics and the driveline are used. The fuel consumption is modeled quasistatic. • The way the optimal driving behavior should be implemented or communicated to the driver, is out of the scope of the research.

GOALS OF THE RESEARCH

23

• The methods should remain simple and easily applicable. The research regarding eco-driving at the Katholieke Universiteit Leuven (KU Leuven) started in 2006 with a master’s thesis by Bart Saerens and Jeroen Vandersteen, titled “Minimization of the fuel consumption of a gasoline engine with optimal throttle valve control” [158]. The research was continued in this PhD, and supported by OPTEC (Optimization in Engineering Center, an interdisciplinary research center) and a research stay at the Virginia Tech Transportation Institute (VTTI). The research was funded by a PhD grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).

24

1.7

INTRODUCTION

Outline of the Dissertation and Main Contributions

When using optimal control, there is a need for a system model. This model is split up in a fuel consumption model and a model of the rest of the vehicle. The latter is well covered in the literature and described in appendix B. For this type of application, quasistatic fuel consumption models are used. Low-degree polynomial fuel consumption models are popular because of their simplicity. Chapter 2 assesses different low-degree polynomial fuel consumption models. The contribution of this dissertation is in the way the models are evaluated. Normally this is done based on how well the model can fit the data. The novel approach also takes the implications for eco-driving into account. New polynomial models are found with a subsearch based on Akaike’s information criterion. Two methods to identify the model parameters are presented: based on measurements and based on publicly available data. Chapter 3 describes how an eco-drive control problem can be solved. The control problems are categorized in typical subproblems found in the literature: accelerations, decelerations, driving between stop signs, approaching traffic lights, and eco-cruise control. The subproblems are solved with four different optimal control methods: Euler-Lagrange, Pontryagin’s maximum principle, discrete dynamic programming, and direct multiple shooting. The main contributions of this dissertation are related to the maximum principle. An existing approach is extended and improved. A novel simple formula to calculate the optimal engine control is derived and a new method to calculate the optimal gear and clutch control, based on an adjoint variable, is presented. An important difference between direct and indirect constraints is discussed. In case of direct constraints, this dissertation presents a new method to satisfy the constraints by translating them in understandable parameters. Some contributions are also made in the field of dynamic programming. New algorithmic tricks are presented: an adaptive distance grid, a smart stop for accelerations and decelerations, and a fast algorithm for approaching traffic lights. This chapter also contributes to the general understanding of eco-driving as an optimal control problem. Chapter 4 applies the methods described in chapter 3 and analyzes the solutions. Different fuel consumption models and solution methods are evaluated. The first contribution of this dissertation is that it shows a link between optimal accelerations and typical human accelerations: they both follow a linear decay law. A second contribution is the description of how an optimal deceleration looks like. To the author’s best knowledge, this is not present in the literature. The third contribution concerns eco-cruise control. It is shown that eco-cruise

OUTLINE OF THE DISSERTATION AND MAIN CONTRIBUTIONS

25

control can help to save fuel on gentle slopes. Chapter 5 evaluates existing eco-drive guidelines. The contribution of this dissertation is the proposal of refined eco-drive guidelines, based on the findings in the previous chapters. Chapter 6 concludes the dissertation and gives recommendations for future research.

Chapter 2

Fuel Consumption Modeling The purpose of a fuel consumption model is to have a simulation tool that can predict the amount of fuel that the vehicle will consume when certain controls are applied to it. Such a model is necessary to be able to calculate optimal driving behavior. This chapter discusses the modeling of the fuel consumption. First, the modeling is introduced in section 2.1. This section also gives an overview of different fuel consumption models that are used in the literature for eco-drive purposes. Then, section 2.2 assesses different polynomial fuel consumption models and determines which models are appropriate. Section 2.3 describes two methods to identify the model parameters: based on on-road measurements, and based on publicly available data. Section 2.4 concludes this chapter.

2.1

Quasistatic Polynomial Consumption Models

There are two main groups of fuel consumption models: macroscopic models and microscopic models. Macroscopic fuel consumption models are designed to calculate the average fuel consumption for a large amount of vehicles, on e.g. a national level. These models are not useful for individual vehicle fuel consumption. Microscopic models are capable of predicting the fuel consumption of individual vehicles. Three types are distinguished [18, 96]. Average speed models express fuel consumption for a trip as a function of the average speed, e.g. [64]. This approach is useful in estimating total fuel consumption of vehicles for an entire road network. This type of modeling 27

28

FUEL CONSUMPTION MODELING

approach is not sensitive to changes in vehicle operational modes and is therefore not useful for eco-driving simulations. Speed and acceleration based models use a speed/acceleration matrix, e.g. [113, 195]. Some models use the product of speed and acceleration [109]. Rakha, Ahn, and Trani [152] developed the VT-Micro model based on polynomials of speed and acceleration. Speed and acceleration based models can not capture the influence of gear shifting. Engine model based consumption models represent the fuel consumption based on a model of the engine. A model of the rest of the vehicle is necessary for the link with the driving behavior. These models are the most suitable for eco-driving simulations. The latter model type is used in this dissertation. The total system is a vehicle with a conventional powertrain, it comprises: • combustion engine, • clutch or torque converter, • gearbox, • final drive, • wheels, • longitudinal dynamics. All parts without the longitudinal dynamics, are called the powertrain. The powertrain without the engine is called the driveline. The simulation of the longitudinal dynamics and the driveline is based on relatively simple models that are well known in the literature. They are discussed in appendix B on p. 191.

2.1.1

Quasistatic Engine Modeling

An internal combustion engine (ICE) is controlled by the amount of injected fuel: the fuel mass flow rate m ˙ f [kg/s]. Depending on the engine rotation speed ω [rad/s], it will produce a certain engine torque T [Nm]. In practice, for a gasoline engine with indirect injection, the real control input is the throttle valve angle. This determines the amount of air going into the engine. Based on a measurement of the intake manifold pressure and the engine speed, and a

QUASISTATIC POLYNOMIAL CONSUMPTION MODELS

29

feedback correction with oxygen sensors in the exhaust, the engine control unit (ECU) will inject a certain amount of fuel to have a stoichiometric1 combustion. In a diesel engine, the ECU determines the fuel injection based on the position of the accelerator pedal. The ECU will also control the exhaust gas recirculation valve and possibly the turbocharger. A complete and correct dynamic model of a combustion engine is very complex and difficult to obtain. This is why in practice, for the calculation of the fuel consumption of a vehicle, quasistatic engine models are used [83]. In this type of model, the control input is the engine torque (or a related variable). Depending on the engine speed, a certain amount of fuel is consumed. Figure 2.1 illustrates the difference between a dynamic and a quasistatic engine model.

m ˙f

- ICE 

-T ω

(a) dynamic

m ˙ f

 ICE 

T ω

(b) quasistatic

Figure 2.1: Schematic comparison between a dynamic and a quasistatic engine model [83]. In a dynamic model, the inputs are ω and m ˙ f , and the output is T . In a quasistatic model, the inputs are ω and T , and the output is m ˙ f. A quasistatic model is a simpler representation of the engine than a dynamic model, but does not take into account the dynamics of the engine. These dynamics mainly concern the air flow through the engine. Since these dynamics are much faster than the longitudinal dynamics of a vehicle, it is justified to neglect them. A turbocharger causes slower air flow dynamics. Quasistatic engine models are standardly used in simulations and have been successfully used in practical eco-drive applications, see e.g. [90, 159]. This dissertation will also use quasistatic engine models. 1 A stoichiometric mixture of air and fuel has just enough air to completely oxidize the available fuel.

30

FUEL CONSUMPTION MODELING

A quasistatic engine model consists of different parts: m ˙f =m ˙ f (ω, T ),

fuel mass flow rate,

(2.1a)

ω > ωmin ,

minimum engine speed,

(2.1b)

ω 6 ωmax ,

maximum engine speed,

(2.1c)

T > Tmin (ω),

minimum engine torque,

(2.1d)

T 6 Tmax (ω),

maximum engine torque.

(2.1e)

To model the fuel mass flow rate, a straightforward method would be to use mapped data, or piecewise affine approximations of this data [21, 91, 92, 95, 115, 134, 172]. Although this method is probably the most accurate, it has some disadvantages. First, it takes a lot of time to perform engine tests to obtain detailed consumption maps if they are not provided by the engine manufacturer. Second, some optimal control methods require an analytic expression for the fuel mass flow rate. A good candidate for such a model, that is also widely used, is a polynomial fuel consumption model. This is the type that is mainly used in this dissertation. It should be mentioned that some researchers also use neural network consumption models, e.g. [35, 162, 194]. The models for the minimum and maximum torque are discussed in appendix B.2 on p. 195.

2.1.2

Fuel Consumption Characteristics

Figure 2.2 shows the fuel consumption map of a 1.6 l gasoline engine. The specifications of the engine can be found in appendix C.1 on p. 203. Two obvious observations are that the fuel mass flow rate increases with increasing engine speed and torque: ∂m ˙f > 0, ∂ω

∂m ˙f > 0. ∂T

(2.2)

There is a jump in minimum engine torque Tmin around 3000 rpm, the fuel cutoff speed ωfco . Above this speed, no fuel will be injected at minimum torque. Below ωfco , some fuel is injected. Figures 1.1 on p. 18 and 1.2 on p. 19 show the brake specific fuel consumption BSFC [kg/J] and brake consumption per rotation BCPR [kg/rad] respectively. The BSFC has a minimum value around 2000 rpm at high engine torque, with

QUASISTATIC POLYNOMIAL CONSUMPTION MODELS

31

m ˙ f [g/s]

engine torque T [Nm]

Tmax , Tmin

100

1

50

1. 5 1. 2 5

1. 7 5

2

0. 7 5

0. 5

0

0. 2 5

1000

1500

2000

2500

3000

3500

engine speed n [rpm]

Figure 2.2: Fuel consumption map of a 1.6 l gasoline engine. convex iso-lines below the point of minimum BSFC. The BCPR iso-lines are slightly concave (curving downwards). Indirect injection gasoline engines usually burn a stoichiometric air-fuel mixture. Sometimes, it is possible that the mixture is rich, containing too much fuel. At full throttle, a rich mixture increases the power output of the engine. At high engine power, a rich mixture cools the exhaust gases to prevent the catalytic converter from ageing. The latter can be seen in figure 2.2 in the top right corner where the iso-lines are close together. Since rich burning makes the modeling more complex and is bad for the environment (HC and CO are released in the exhaust), these regions in the engine operation range are excluded for ecodriving.

2.1.3

Polynomial Fuel Consumption Models

A linear quasistatic polynomial fuel consumption model is defined as follows: m ˙f=

M X

αk ω pk uqek ,

p, q ∈ NM ,

α ∈ RM ,

(2.3)

k=1

with ue [∼] the engine control input, α [∼] model parameters, and pk and qk polynomial exponents. A polynomial fuel consumption model can also be

32

FUEL CONSUMPTION MODELING

Table 2.1: Classification of polynomial fuel consumption models. type

consumption model

power-based (PB) linear torque-based (LTB) nonlinear torque-based (NTB) linear load-based (LLB) nonlinear load-based (NLB)

P

k

k P qk , αk ω pP ∀k : pk · qk = 0 pk qk α ω k P k pk PT α ω · l βl T ql k Pk pk qk P k αpkk ω Pτ β τ ql α ω · l l k k

nonlinear in the coefficients: m ˙f=

M X

αk ω

k=1

pk

·

N X

βl uqel ,

p ∈ NM , q ∈ NN ,

α ∈ RM , β ∈ RN , (2.4)

l=1

with β [∼] model parameters. Each nonlinear model has an equivalent linear model, yet the nonlinear form needs less parameters. The control input ue can be the engine torque T [Nm], the engine power P = ωT [W], or the engine load τ [-]. The engine load is defined as the fraction of the maximum torque at a given engine speed: τ (T, ω) =

T Tmax (ω)

.

(2.5)

If ∃k : qk = 2 and ∀k : qk 6 2, then the consumption model is called quadratic. In what follows, extra assumptions for a quadratic fuel consumption model are made: ∃k : qk = 1 and ∀qk > 0 : αk > 0. A fuel consumption model is called affine if ∃k : qk = 1 and ∀k : qk 6 1. One could also refer to this type of model as linear, i.e. linear in the control input (linear ↔ quadratic) as opposed to linear in the model parameters (linear ↔ nonlinear). A classification of polynomial fuel consumption models is given in table 2.1. Power-based, torque-based, and load-based models are defined. Note that in power-based models, ω and P do not occur together in the same term. Because of their simplicity, low-degree polynomial fuel consumption models are commonly used. Table 2.2 gives an overview of polynomial consumption models that are found in the literature. The equations can be found in appendix D.1 on p. 213. Power-based fuel consumption models are often used with the vehicle power as the explanatory variable, instead of engine power (e.g. [5, 106]). The difference between vehicle and engine power are losses in the driveline. Yet, in these simple consumption models, the driveline losses are often modeled by a constant

QUASISTATIC POLYNOMIAL CONSUMPTION MODELS

33

Table 2.2: Polynomial fuel consumption models found in the literature. type model source p q eq. PB

P1 P2 P3

[36] [124, 148] [5, 18, 157]

[0] [0 0] [1 0]

[1] [0 1] [0 1]

(D.1) (D.2) (D.3)

LTB

T1 T2 T3 T4 T5

[90] [169] [169] [185] [145]

[1 2 1] [0 1 1 2] [0 1 2 3 1] [1 2 3 1 2 3 1] [0 1 2 1 0 0]

[0 0 1] [0 0 1 1] [0 0 0 0 1] [0 0 0 1 1 1 2] [0 0 0 1 1 2]

(D.4) (D.5) (D.6) (D.7) (D.8)

NTB

T6

[104, 193]

[1 2 3]

[0 1 2]

(D.9)

NLB

L1

[3, 164]

[0 1 2]

[0 1 2]

(D.10)

driveline efficiency. Thus using vehicle power instead of engine power does not change the model, just changes the parameters slightly. When using vehicle power as a variable, one can define the vehicle specific power (VSP), i.e. the vehicle power divided by the vehicle mass [106]. Instead of using a polynomial model based on VSP, the VSP can be binned and fuel consumption is then defined for each bin [186, 191, 192]. Power-based models often do not take negative engine power into account:  P pk qk Pk αk ω pk P qk, if P > 0, m ˙f= (2.6) k αk ωmin 0 , if P 6 0.

Akçelik and Besley [8] use a special polynomial fuel consumption model that combines engine power, velocity v [m/s], and acceleration a [m/s2 ] as explanatory variables:  α0 + α1 P + α2 ma2 v, if P > 0, m ˙f= (2.7) α0 , if P 6 0,

with m [kg] the vehicle mass.

2.1.4

Emission Modeling

Although this dissertation focusses on fuel consumption and CO2 emissions, they should not be the only concern, as illustrated in section 1.2 on p. 4. Fortunately it is also possible to model emissions such as CO, HC, and NOx

34

FUEL CONSUMPTION MODELING

with polynomial models. Akçelik and Besley [8] model CO, HC, and NOx emissions according to equation (2.7), with different α. Jiménez-Palacios [106] uses the VSP approach for modeling CO, HC, and NOx . Leung and Williams [124] model these emissions with an affine power-based model. The analysis done by Rakha, Ahn, and Trani [152] shows that it is more difficult to model CO, HC, and NOx emissions than CO2 and thus fuel consumption. A data transformation of a polynomial model with a natural logarithm is used to model the mass flow m ˙ of the emissions: ! 3 3 X X
k l . (2.8) m ˙ = exp αk,l · v · a k=0 l=0

This model results in a coefficient of determination R2 of 0.995 for the CO2 emission. For the other emissions the fit is less good: R2 = 0.960 for NOx , R2 = 0.717 for CO, and R2 = 0.689 for HC.

2.1.5

Cold Start Modeling

When the engine is cold, it will consume more fuel compared to a hot-stabilized engine. There exist simple methods for cold start modeling. Rakha, Ahn, and Trani [153] use an extra cold start consumption that decreases with time:    min (t, ths ) ·m ˙ f,hs , (2.9) m ˙ f,cs = 1 + α 1 − ths with m ˙ f,cs [kg/s] the cold start fuel mass flow rate, m ˙ f,hs [kg/s] the hot-stabilized fuel mass flow rate as given by a consumption map or polynomial model, α [-] a cold start fuel consumption parameter, t [s] the time, and ths [s] the time that it takes the engine to warm up. Realistic values are: α = 0.28 and ths = 200 s. Barth et al. [18] use a comparable approach where instead of time, the total amount of consumed fuel mf [kg] is used:    min (mf,tot , mf,hs ) m ˙ f,cs = 1 + α 1 − ·m ˙ f,hs , (2.10) mf,hs with mf,tot [kg] total amount of fuel consumed and mf,hs [kg] the amount of total fuel consumption needed to warm up the engine. A realistic value: mf,hs = 0.1 kg.

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

2.2

35

Assessment of Different Polynomial Models

This section assesses different low-degree quasistatic polynomial fuel consumption models2 . First, the assessment methodology is described. Second, some new polynomial fuel consumption models are developed. Then, the different models are evaluated. Finally, it is illustrated that a polynomial model can also be used to model the fuel consumption of a diesel engine and the efficiency of an electric motor.

2.2.1

Assessment Methodology

The low-degree polynomial fuel consumption models that are considered for the assessment are first of all the models that are found in the literature, see table 2.2. Second, some new fuel consumption models are developed based on a subsearch with Akaike’s information criterion. A priori assessment of the polynomial fuel consumption models is done based on the model structure itself. Three properties are evaluated: 1) existence of a well-defined optimal gear shift behavior; 2) existence of a well-defined optimal velocity control; and 3) the number of model parameters. Properties 1) and 2) are found with vehicles in the real world. Based on these three criteria, a first selection of models is made. 1. Optimal gear shift behavior The overall driveline ratio i [1/m], that depends on the gear, decides on the combination of engine speed ω and torque T to produce a given force F [N] at the wheels for a given velocity v: ω

= iv,

(2.11a)

T

=

F , iη

(2.11b)

with η [-] the efficiency of the transmission. Assuming that η is independent of the gear, gear shifting has no influence on the fuel consumption for a given set of v and F (e.g. given steady state velocity), if: ∀k : pk = 0, ∀k : pk = qk , 2 In

if ue = P, if ue = T.

(2.12)

this context, model means the model structure, not the value of the parameters.

36

FUEL CONSUMPTION MODELING

Even if η depends on the gear, the influence of the gear on the fuel consumption is low in the latter cases. 2. Optimal velocity control Minimum-fuel velocity control of a vehicle minimizes the fuel consumption. Simple fuel consumption models can yield bang-bang control, a result that should preferably be avoided (see section 3.2.1 on p. 70). A polynomial fuel consumption model will imply bang-bang control when it is affine. This means that the fuel mass flow rate should be superlinear in the control variable ue to avoid bang-bang control. 3. Number of parameters A low amount of parameters reduces the model complexity. Fuel consumption models with three or less parameters can be easily calibrated using publicly available data, without the need for measurement data (see section 2.3.2). A posteriori assessment of the polynomial fuel consumption models is done based on the quality of the model fit, which is evaluated based on: 1) the coefficient of determination; 2) the root mean square error; and 3) Akaike’s information criterion. 1. Coefficient of determination The coefficient of determination R2 [-] is the proportion of variability in a data set that is accounted for by the model. It provides a measure of how well future outcomes are likely to be predicted by the model. P 2 [m ˙ f,m (ωi , ue,i ) − m ˙ f,s (ωi , ue,i )] 2 , (2.13) R = 1− iP  2 ˙ f,m (ωi , ue,i ) − m ˙ f,m i m with m ˙ f,m [kg/s] the measured fuel mass flow rate and m ˙ f,s [kg/s] the simulated fuel mass flow rate according to the polynomial fuel consumption model.

2. Root Mean Square Error The root mean square error RMSE [kg/s] is directly linked to R2 , yet gives a more quantitative measure. r RSS RMSE = , (2.14) N with RSS [kg2 /s2 ] the sum of squared residuals, the numerator in equation (2.13), and N the number of data points.

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

37

3. Akaike’s information criterion Akaike’s information criterion AIC [-] is a measure of the goodness of a fit and quantifies the tradeoff between complexity and accuracy of the model [7]. It is a tool for model selection.   RSS AIC = N ln + 2 (K + 1) , (2.15) N with K the number of parameters in the polynomial fuel consumption model. AIC is a measure of accuracy of the model, considering the quality of the model fit and the number of model parameters. This measure is only useful to compare different models whose parameters are identified using the same data set. Three sets of measurement data for the assessment of polynomial fuel consumption models are used: 1) measurements on a universal engine dynamometer; 2) measurements on a chassis dynamometer; and 3) on-road measurements. All measurements are done with gasoline engines. 1. Engine dynamometer Engine speed ω, engine torque T , and the fuel mass flow rate m ˙ f are measured on a universal engine dynamometer. In 500 steady state hotstabilized working points, measurements are taken on a 2004 Toyota 3ZZ-FE 1.6 l port-fuel injected 4 cylinder gasoline engine. The engine speed ranges from 900 to 3800 rpm, and the torque from minimum to maximum torque. The resulting fuel consumption map is shown in figure 2.2. Appendix C.1 on p. 203 discusses the measurements in more detail. 2. Chassis dynamometer Vehicle velocity v and fuel mass flow rate m ˙ f are measured on 11 vehicles. The latter is done by measuring CO2 , CO, and HC in the exhaust. Cold start fuel consumption is converted to hot-stabilized consumption using equation (2.9). Values of the traction force F are calculated with basic longitudinal dynamics. Appendix C.2 on p. 206 discusses the measurements in more detail. 3. On-road tests A global positioning system (GPS) is used to log the velocity v and the road slope θ [rad]. An on-board diagnostics (OBD) reader logs the engine speed ω and the fuel mass flow rate m ˙ f . Values of the traction force F and engine torque T are calculated with basic longitudinal dynamics. Seven vehicles were driven in Blacksburg Virginia, partly on a highway and partly on a signalized arterial through the downtown. Appendix C.3 on p. 209 discusses the measurements in more detail.

38

FUEL CONSUMPTION MODELING

fuel mass flow rate m ˙ f [g/s]

4 data points idle consumption

3

2

1

0

−40

−20

0

20

40

engine power P [kW]

Figure 2.3: Example of chassis dynamometer measurement data. The model parameters of the linear models are identified with a linear least squares fit (Moore-Penrose pseudoinverse), based on the BCPR (this gives the best results). The nonlinear models need a nonlinear regression based on an iteration that requires a starting value. The solution is very sensitive to this starting value, which is obtained by trial and error. The fuel mass flow rate of power-based models and the fuel mass flow rate for vehicles with an automatic transmission is limited to the idle fuel mass flow rate m ˙ f,i [kg/s]. ! X pk qk m ˙ f = max m ˙ f,i , αk ω P . (2.16) k

Figure 2.3 justifies this approach. It shows official EPA fuel consumption measurements from a vehicle with an automatic transmission on a chassis dynamometer.

2.2.2

Development of New Consumption Models

Besides the polynomial fuel consumption models that are found in the literature, several new models are considered in the assessment. These new models are obtained from a base model, using a subsearch regression [33] and the measurements from the engine dynamometer. Six models are developed: 1) a power-based model without engine speed terms; 2) a power-based model with engine speed terms; 3) a linear torque-based model; 4) a nonlinear torque-

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

39

fuel mass flow rate m ˙ f [g/s]

3

2

1

0

0

5

10

15

20

25

30

engine power P [kW]

Figure 2.4: Measurement data from the engine dynamometer. based model; 5) a linear load-based model; and 6) a nonlinear load-based model. The base model has a relatively large amount of parameters and can capture all necessary fuel consumption effects. This model is then simplified using a subsearch based on Akaike’s information criterion. Each term in the base model is discarded and the corresponding AIC is calculated. The simplified model with the lowest AIC is selected. This model can then be used to start a new simplification based on the lowest AIC. The model is simplified as long as the AIC can be decreased by discarding a term. Power-Based Models Figure 2.4 shows the fuel mass flow rate as a function of the engine power as it is measured on the engine dynamometer. Only positive power values are shown. Two important properties are observed: 1) an offset at zero power; and 2) a slight upward curvature. These properties are also observed in figure 2.3. A simple power-based model that can capture these effects is a quadratic model: P4: m ˙ f = α1 + α2 P + α3 P 2 .

(2.17)

Table 2.3 illustrates the model subsearch starting from a base model. For illustration, the base model is chosen to be more complex than the model of equation (2.17). All the possible simplifications are considered by discarding each term. The simplification that has the lowest AIC value is the best simplification. The subsearch yields the following sequence of models:

40

FUEL CONSUMPTION MODELING

Table 2.3: Subsearch for a power-based consumption model. model AIC m ˙ f = α1 + α2 P + α3 P 2 + α4 P 3

−5054.4

m ˙ f = α1 P + α2 P + α3 P m ˙ f = α1 + α2 P 2 + α3 P 3 m ˙ f = α1 + α2 P + α3 P 3 m ˙ f = α1 + α2 P + α3 P 2

−5056.4 −5143.8 −5193.7 −5194.4

m ˙ f = α1 P + α2 P 2 m ˙ f = α1 + α2 P 2 m ˙ f = α1 + α2 P

−5000.9 −4998.6 −5182.6

m ˙ f = α1 P m ˙ f = α1

−4937.4 −4327.6

2

3

1. m ˙ f = α1 + α2 P + α3 P 2 + α4 P 3 , 2. m ˙ f = α1 + α2 P + α3 P 2 , 3. m ˙ f = α1 + α2 P , 4. m ˙ f = α1 P . Only the first two models have an upward curvature. The 4th model in the sequence has no offset at zero power and is therefore not considered. The 2nd model has the lowest AIC of the remaining models and thus is selected as the best model. It is named model P4 in the rest of this section. A power-based model with engine speed terms can also be considered. The proposed base model is the following: m ˙ f = α1 + α2 ω + α3 ω 2 + α4 ω 3 + α5 P + α6 P 2 .

(2.18)

The first four terms take into account the engine friction, the remaining two the efficiency of the production of internal power. Three different sources of friction can be identified: 1) dry (constant or coulomb) friction from certain engine peripherals or external loads (α1 ); 2) viscous friction from engine parts separated by an oil filter, oil pump, and water pump (α2 ω); and 3) aerodynamic friction from gasses flowing through the engine (α3 ω 2 ). Since several existing polynomial fuel consumption models use a cubic friction term (α4 ω 3 ), e.g. models T3–T5 in table 2.2, this is added to the base model too. The second part of the base model is chosen to be quadratic as in model

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

41

P4. A subsearch starting from the presented base model yields the following model based on the lowest AIC: P6: m ˙ f = α1 + α2 ω + α3 ω 2 + α4 P + α5 P 2 .

(2.19)

This model is named model P6. The best model with three or less parameters is the following: P5: m ˙ f = α1 ω + α2 P + α3 P 2 .

(2.20)

It is named model P5 and also considered for further evaluation. Torque- and Load-Based Models Both torque- and load-based models are directly linked to the engine. For these models it is desired that they capture certain observations in the engine map, as indicated before: 1) there is a point of minimum BSFC around 2000 rpm and almost full torque; 2) BSFC iso-lines are convex below this point; and 3) BCPR iso-lines are slightly concave and have a maximum around 2000 rpm. Following the same reasoning used in the proposition of the last power-based model, the following base models are proposed (linear torque-based, nonlinear torque-based, linear load-based, and nonlinear load-based respectively): m ˙f

= α1 + α2 ω + α3 ω 2 + α4 ω 3 + α5 T + α6 ωT + α7 ω 2 T + α8 ω 3 T +α9 T 2 + α10 ωT 2 + α11 ω 2 T 2 + α12 ω 3 T 2 + α13 T 3 + α14 ωT 3 +α15 ω 2 T 3 + α16 ωT 3 + α17 ω 2 T 3 + α18 ω 3 T 3 , m ˙f

m ˙f

=



α1 + α2 ω + α3 ω 2 + α4 ω 3 · β1 + β2 T + β3 T 2 + β4 T 3 ,

= α1 + α2 ω + α3 ω 2 + α4 ω 3 + α5 τ + α6 ωτ + α7 ω 2 τ + α8 ω 3 τ +α9 τ 2 + α10 ωτ 2 + α11 ω 2 τ 2 + α12 ω 3 τ 2 + α13 τ 3 + α14 ωτ 3 +α15 ω 2 τ 3 + α16 ωτ 3 + α17 ω 2 τ 3 + α18 ω 3 τ 3 ,

m ˙ f = α1 + α2 ω + α3 ω 2 + α4 ω 3 · β1 + β2 τ + β3 τ 2 + β4 τ 3 .

(2.21)

(2.22)

(2.23) (2.24)

Starting from the base models and only considering models that can capture the mentioned observations in the maps (visual inspection), the following models are generated with a subsearch: T7: m ˙ f = α1 ω+α2 ω 2 +α3 ω 3 +α4 ωT +α5 ω 2 T +α6 T 2 +α7 ωT 2 +α8 ω 2 T 2 , (2.25)

T9: m ˙ f = α1 + α2 ω 2 · β0 + β1 T + β2 T 2 ,

(2.26)

42

FUEL CONSUMPTION MODELING

Table 2.4: Novel polynomial fuel consumption models. type model p q eq. PB

P4 P5 P6

[0 0 0] [1 0 0] [0 1 2 0 0]

[0 1 2] [0 1 2] [0 0 0 1 2]

(2.17) (2.20) (2.19)

LTB

T7 T8

[1 2 3 1 2 0 1 2] [1 2 3 1 2 1]

[0 0 0 1 1 2 2 2] [0 0 0 1 1 2]

(2.25) (2.29)

NLTB

T9

[0 2]

[0 1 2]

(2.26)

LLB

L2 L3

[1 1 2 3 1 2] [1 2 3 1 2 1]

[0 1 1 1 2 2] [0 0 0 1 1 2]

(2.27) (2.30)

NLLB

L4

[1 2 3]

[0 1 2]

(2.28)

L2: m ˙ f = α1 ω + α2 ωτ + α3 ω 2 τ + α4 ω 3 τ + α5 ωτ 2 + α6 ω 2 τ 2 ,

(2.27)



L4: m ˙ f = α1 + α2 ω 2 + α3 ω 3 · β1 + β2 τ + β3 τ 2 .

(2.28)

T8: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωT + α5 ω 2 T + α6 ωT 2 ,

(2.29)

These models are named T7, T9, L2, and L4 respectively. Note that none of the models have a cubic control term (T 3 or τ 3 ). Cubic terms hardly improve the model fit and cause unnatural iso-lines. Before the presented model search was conducted, the author developed a linear torque-based model:

and a linear load-based model: L3: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωτ + α5 ω 2 τ + α6 ωτ 2 .

(2.30)

These models are named model T8 and L3, and are also considered for further evaluation. An overview of all the new models is given in table 2.4.

2.2.3

Evaluation of Consumption Models

A first selection of consumption models is done based on the model structure as explained in section 2.2.1. Table 2.5 shows the evaluation of the models found in the literature and the models developed in section 2.2.2. A ‘+’ is positive, meaning that the property is well represented. A ‘-’ is negative. The models with two or more positives are selected for further assessment (P4–P6,

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

43

Table 2.5: Evaluation of models based on model structure. Shifting: the model can decently capture the influence of gear shifting. Bang-bang: the model does not result in bang-bang control. Parameters: the model has 3 or less parameters. model eq. shifting bang-bang parameters P1 P2 P3 P4 P5 P6

(D.1) (D.2) (D.3) (2.17) (2.20) (2.19)

+ + +

+ + +

+ + + + + -

T1 T2 T3 T4 T5 T6 T7 T8 T9

(D.4) (D.5) (D.6) (D.7) (D.8) (D.9) (2.25) (2.29) (2.26)

+ + + + + + + + +

+ + + + + +

+ -

L1 L2 L3 L4

(D.10) (2.27) (2.30) (2.30)

+ + + +

+ + + +

-

T1, T4–T9, L1–L4). Model P3 is not considered. Adding a quadratic term to this model results in model P5, which has a lower AIC than P3 and not more than three parameters. To allow for a simple comparison of model fit quality, a relative measure for AIC is defined: Λ=

AICmax − AIC , AICmax − AICmin

(2.31)

with AICmin and AICmax the minimum and maximum AIC value. Λ = 1 for the best model and Λ = 0 for the worst model. Table 2.6 shows the evaluation of the selected models based on the fit quality with the engine dynamometer measurements. Model L2 is the best model. Model P4 is clearly outperformed by the rest. The difference between the

44

FUEL CONSUMPTION MODELING

Table 2.6: Evaluation of models based on the fit quality with engine dynamometer measurements. R2 RMSE Λ model [%] [mg/s] [%] P4 P5 P6

94.22 98.96 99.33

13.93 5.90 4.75

0.0 60.1 74.8

T1 T4 T5 T6 T7 T8 T9

99.06 99.44 99.48 99.43 99.64 99.32 99.43

5.62 4.33 4.18 4.38 3.49 4.77 4.37

63.5 80.8 83.5 80.3 95.6 74.3 80.6

L1 L2 L3 L4

99.01 99.68 99.54 99.54

5.77 3.30 3.92 3.93

61.0 100.0 88.0 87.8

torque-based and load-based models is small. The same can be said for the difference between linear and nonlinear models. Since it is difficult to identify the parameters of nonlinear models, they will not be considered in the further assessment. Table 2.7 shows the evaluation of the selected models based on the fit quality with the chassis dynamometer measurements (average values are shown). Model T1 is the worst, model T4 the best. The difference between models P6, T4–L3 is rather small. The overall fit quality is clearly worse compared to the engine dynamometer measurements. This is due to modeling errors; the engine torque is not directly measured, but estimated based on the vehicle velocity and assumed longitudinal dynamics. Table 2.8 shows the evaluation of the selected models based on the fit quality with the on-road measurements (average values are shown). Models T4–L3 are clearly better than the rest. Model P4 is the worst, model L3 the best. Again, the difference between models T4–L3 is rather small. It is hard to make a good conclusion based solely on the model structure and fit quality. Therefore, a visual evaluation of the engine maps is added to the model assessment: the real engine map is compared to the one resulting from

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

45

Table 2.7: Evaluation of models based on the fit quality with chassis dynamometer measurements, after elimination of nonlinear models. R2 RMSE Λ model [%] [mg/s] [%] P4 P5 P6

90.06 91.07 92.78

15.97 15.44 13.81

25.1 37.3 74.2

T1 T4 T5 T7 T8

89.43 93.43 93.36 93.35 93.32

16.56 13.12 13.18 13.19 13.20

12.8 95.1 93.0 93.1 92.2

L2 L3

92.87 93.29

13.58 13.23

83.6 91.7

Table 2.8: Evaluation of models based on the fit quality with on-road measurements, after elimination of nonlinear models. R2 RMSE Λ model [%] [mg/s] [%] P4 P5 P6

78.87 85.63 84.41

46.18 37.47 37.93

0.0 41.4 49.6

T1 T4 T5 T7 T8

81.18 89.00 88.55 88.88 88.22

36.52 32.55 33.42 32.55 33.82

56.8 85.4 78.8 84.8 74.9

L2 L3

88.91 90.77

32.63 31.74

84.6 86.1

engine torque T [Nm]

FUEL CONSUMPTION MODELING

engine torque T [Nm]

46

100

50

1000

2000

3000

engine speed n [rpm]

100 50 0 1000

2000

3000

engine speed n [rpm]

Figure 2.5: Comparison of the real (solid) and modeled (dashed) BSFC (left) and BCPR (right) for model L3. the model. Only for the engine dynamometer there is a known real map, thus the evaluation is done with those measurements. Figure 2.5 shows the comparison between the real and modeled maps of BSFC and BCPR for model L3. Comparisons for the other models can be found in appendix D.2 on p. 215. Model P4, P5, and T1 do not capture the typical BSFC shape. Models P4, P5, P6, and T1 are outperformed by the rest. The load-based models L2 and L3 are more capable of placing the point of minimum BSFC at the correct engine speed. From all this it can be concluded that models L2 and L3 are the best, closely followed by models T5, T7, and T8. Starting from about 6 parameters, each torque- and load-based quadratic fuel consumption model is appropriate. Furthermore, the most important evaluation criterion is probably how the consumption model affects the fuel savings potential when used in an optimal control calculation. This influence will be studied in chapter 4.

2.2.4

Polynomial Fuel Consumption Models of Diesel Engines and Electric Motors

For now, only indirect injection gasoline engine are considered. The reason is that only data of these engines is available for the author. Direct injection gasoline engines and diesel engines have similar consumption maps and thus it is assumed that the proposed polynomial fuel consumption models are useable. Many people see electric vehicles (EV) and hybrid electric vehicles (HEV) as the future for road transportation. However, it is predicted that it will take until 2030 before they dominate the market [99]. Efficiency maps of electric motors are comparable to BSFC maps and can also be modeled with polynomial

ASSESSMENT OF DIFFERENT POLYNOMIAL MODELS

47

models. Examples of a consumption model of a diesel engine and an efficiency model of an electric motor are given. Measurements on a passive diesel engine dynamometer at the KU Leuven are used. The engine is a Ford/PSA 1.4 l DLD-414 engine. Figures 2.6 and 2.7 show the modeled BSFC and BCPR with model T8 (R2 = 99.23 %). BFSC [g/kWh]

Tmax

engine torque T [Nm]

150

235

100 240 250

50 280 400

0

1500

2000

2500

3000

3500

4000

engine speed n [rpm]

Figure 2.6: Modeled BSFC of a 1.4 l diesel engine. Tmax 100

150 engine torque T [Nm]

BCPR [mg/rad]

80

100

60 40

50 20

0

1500

2000

2500

3000

3500

4000

engine speed n [rpm]

Figure 2.7: Modeled BCPR of a 1.4 l diesel engine.

48

FUEL CONSUMPTION MODELING

motor torque T [Nm]

150

Tmax ηreal [%] ηmodel

100

90

50 70

0

80

2000

85

4000

6000

motor speed n [rpm]

Figure 2.8: Efficiency map of a synchronous AC motor used for EVs and HEVs [174]. Figure 2.8 shows the comparison between the real and modeled efficiency η [-] of the first quadrant of a synchronous motor used for EVs and HEVs. Again, model T8 is used, resulting in R2 = 99.43 %. It should be mentioned that engines with turbocharging have somewhat slower airpath dynamics. The influence on the fuel consumption is negligible [163]. However, the influence can be significant on CO, HC, NOx , and soot emissions during transients. Using dynamic correction factors can take this into account [79].

IDENTIFICATION OF MODEL PARAMETERS

2.3

49

Identification of Model Parameters

This section discusses how the parameters of a polynomial fuel consumption model can be identified for practical applications. Ideally, measurements from an engine dynamometer should be used. Of course in practice, this is not possible for a simple eco-drive application. Two methods are presented here: an identification based on on-board diagnostics measurements (OBD) and an identification based on publicly available data.

2.3.1

Identification with OBD Measurements

The on-board diagnostics system in passenger vehicles allows access to state of health information of various vehicle systems. Since 1996, OBD-II (an improved OBD standard) is mandatory for all cars sold in the US. The European version (EOBD) is mandatory since 2001. Besides standardized diagnostic trouble codes, OBD also provides real-time data. However not all data is mandatory. Engine speed and vehicle velocity are always available. In the US, fuel mass flow rate is normally available too. In Europe, the latter is not always the case. For gasoline engines, the fuel mass flow rate can be calculated based on the mean air flow (MAF) and air-fuel ratio (AFR). Unfortunately, the engine torque is not an output of the OBD system3 and should be calculated based on vehicle dynamics (see appendix B). This can introduce a significant error. Also, the measurements are taken under normal driving conditions and do not spread evenly over the engine operation range. The influence of these effects is studied briefly here. OBD measurements of a vehicle driving over a cycle are simulated. The gasoline engine of the universal engine dynamometer is used in a simulated 2004 Toyota Corolla Verso. The specifications of this vehicle are given in table C.2 on p. 205. Engine speed, vehicle velocity, and fuel mass flow rate are measured every second. All vehicle dynamics are perfectly known and there is no road slope or wind. Load-based model L2 is used. Table 2.9 shows the influence of the driving cycle on the fit quality of the model; R2 and RMSE for a comparison between the model and the real fuel consumption measurements spread evenly in the engine operation range. Only US drive cycles are used because European cycles do not represent realistic driving behavior. City cycles do not yield good results, because they do not cover the engine operation range well. The other cycles yield fairly good results. Other polynomial quadratic fuel consumption models (T5, T7, T8, and L3) yield comparable results. 3 It

is in the OBD version for heavy-duty trucks.

50

FUEL CONSUMPTION MODELING

Table 2.9: Influence of the driving cycle on model quality, using model L2. R2 RMSE cycle [%] [mg/s] EPA75 CITY II HWFET LA92 NYCC SC03 UDDS US06

97.25 67.94 95.78 95.89 75.08 95.19 96.76 94.52

10.30 35.15 12.76 12.58 30.99 13.62 11.18 14.53

In practice, the dynamics of the vehicle are not known perfectly: road slope, wind, rolling friction, and vehicle mass can influence the identification of the model parameters. The influence is quantified based on an average over 20 simulations where the specific value is chosen randomly. Introducing an unknown road slope (based on real roads in Belgium) reduces R2 from 97.25 to 75.96 % on the EPA75 cycle. When the road slope is known and taken into account: R2 = 95.96 %. When a random mass error between −50 and 50 kg on the assumed value is introduced, R2 only reduces to 97.23. When it is assumed that the vehicle drives on good asphalt roads, and in practice it drives over all kinds of pavements (see table B.1 on p. 193), R2 reduces to 95.75 %. Random wind between −5 and 5 m/s yields R2 = 97.21 %. Although this study is far from complete, it shows that it is possible to obtain a decent engine model based on OBD measurements. Preferably, it should not only be based on city driving, but also include high engine speeds and loads. It is also beneficial to drive on level roads or at least know the road slope. It is best to measure when there is no wind, and know the type of road and the total vehicle mass, although their influence is not that high.

2.3.2

Identification with Publicly Available Data

In some cases, needing measurement data can still be an unwanted limitation. Using polynomial fuel consumption models with a low amount of parameters, introduces the possibility to identify these parameters based on publicly available data. A first option is to determine the model parameters with city and highway fuel economy estimates. Decent fuel consumption models need at least three

IDENTIFICATION OF MODEL PARAMETERS

51

parameters. The engine size can be used as a third property. Two models are used4 . The first model is P5, as it appears to be the best model with three parameters:
m ˙ f = max m ˙ f,i , β0 ω + β1 P + β2 P 2 . (2.32)

Preferably, P is used as the engine power. The vehicle power can also be used, but then some accuracy is lost. Appendix B discusses how the engine and vehicle power should be calculated. The engine speed ω occurs in the model and thus needs to be known on the official drive cycles, which is not the case for a passenger vehicle with an automatic gearbox. Model P4 can solve this problem:
m ˙ f = max m ˙ f,i , α0 + α1 P + α2 P 2 . (2.33) Now the engine speed should not be known. This introduces another small loss of accuracy since the model for the rotational inertia of the powertrain contains a part with the gearbox ratio. This part should be dropped, neglecting the rotational inertia of the engine. The latter model (P4) does not reflect the influence of gear shifting as well as model P5, as explained in section 2.2.1.

The parameters are determined as follows:

α0

α2

α1

=

=

=



max m ˙ f,i , 

max ǫ,





   mf,c − mf,h PPhc − ǫ · Pc2 − Ph2 PPhc , tc − th PPhc

   mf,c − mf,h PPhc − α0 · tc − th PPhc , Pc2 − Ph2 PPhc

mf,c −α0 tc −α2 Pc2 Pc

+ 2

mf,h −α0 th −α2 Ph2 Ph

,

(2.34a)

(2.34b)

(2.34c)

4 The models were a joint effort of the author and H. A. Rakha. In the literature they are named Virginia Tech Comprehensive Power-Based Fuel Consumption Model 1 and 2 (VT-CPFM-1 and VT-CPFM-2).

52

FUEL CONSUMPTION MODELING

β0

β2

β1

=

=

=



m ˙ f,i , ωmin





max 

max ǫ,



   mf,c − mf,h PPhc − ǫ · Pc2 − Ph2 PPhc , ωc − ωh PPhc

   mf,c − mf,h PPhc − β0 · ωc − ωh PPhc , Pc2 − Ph2 PPhc

mf,c −β0 ωc −β2 Pc2 Pc

+ 2

mf,h −β0 ωh −β2 Ph2 Ph

,

(2.35a)

(2.35b)

(2.35c)

with mf,c and mf,h [kg] the total amount of fuel used on the city and highway cycle, tc and th [s] the duration of the cycles. ωc , ωh , Pc , Pg , Pc2 , and Ph2 are the sum of the values for each second over the cycle. The limitations on α0 , α2 , β0 , and β2 , together with ǫ are needed to make sure that both α2 and β2 are positive. With ǫ = 1 · 10−6 kg/s, it is usually ensured that the optimal velocity is between 60 and 80 km/h. The idle fuel mass flow rate can be estimated for a gasoline engine as follows [83]: m ˙ f,i =

pf Vd ωmin , QN π

(2.36)

with pf ≈ 4 · 105 Pa the fuel pressure, Vd [m3 ] the engine displacement, Q = 43 · 106 J/kg the lower heating value of gasoline, and N [-] the amount of cylinders. In Europe, the ECE and EUDC cycles are used, and official fuel economy estimates are expressed in l/100km. In the US, the EPA75 and HWFET cycles are used, and fuel economy estimates are expressed in mpg (miles per gallon). The Environmental Protection Agency (EPA) started the use of additional drive cycles in 2008. Until the 2012 model year, the tests are run on the old drive cycles, but the fuel economy estimates are reported for the new cycles using: FEc =

1.18053 · FE′c , 1 − 0.003259 · FE′c

(2.37)

FEh =

1.3466 · FE′h , 1 − 0.001376 · FE′h

(2.38)

with FE [mpg] the fuel economy estimates on the old cycles, and FE′ [mpg] the estimates on the new cycles. Thus for a car built in 2008 or later, the

IDENTIFICATION OF MODEL PARAMETERS

53

fuel economy estimates should be converted to be able to use the EPA75 and HWFET cycles to determine the fuel consumption model parameters. A validation of the identification is done with the chassis dynamometer data described in appendix C.1 on p. 203. Table 2.10 shows the resulting average R2 for both power-based models for different drive cycles. The models have difficulties predicting the fuel consumption on the FWHS cycle. Model P4 results in a better model fit than model P5. A validation is also done with the engine dynamometer data described in appendix C.1. Using an identification with publicly available data instead of the dynamometer measurements, reduces R2 from 94.22 to 90.33 % for model P4. For model P5 it reduces R2 from 98.96 to 89.76 %. It seems that model P4 is more appropriate for use with an identification with publicly available data. Table 2.10: Comparison of R2 of models P4 and P5 identified with publicly available data, for different drive cycles and chassis dynamometer measurements. R2 [%] cycle model P4 model P5 UDDS ARTA LA92 FWHS

85.83 86.04 88.11 65.13

78.32 75.60 81.48 60.16

54

2.4

FUEL CONSUMPTION MODELING

Conclusions

Quasistatic engine models are used in combination with a simple model of the driveline and vehicle dynamics to calculate the fuel consumption of a vehicle. If possible and available, mapped consumption data can be used as a lookup model. Usually this is not the case and then low-degree polynomial fuel consumption models are a good alternative. Polynomial models can also be used with optimal control methods that require an analytical model. Three types of polynomial fuel consumption models are defined, based on the engine control input of the model: 1) power-based, 2) torque-based, and 3) load-based. Torque- and load-based models can be linear or nonlinear. An assessment of different polynomial fuel consumption models is done using three types of data: 1) engine dynamometer measurements; 2) chassis dynamometer measurements; and 3) on-road measurements. Both models found in the literature and new models are evaluated. New models are developed using a subsearch. An evaluation based on the model structure and fit quality shows that torque- and load-based methods outperform power-based models. Quadratic models with less than 10 parameters can predict the fuel consumption to an acceptable degree of accuracy, while capturing the typical iso-line shapes of BSFC and BCPR. Load-based models can locate the point of minimum BSFC better. The actual best model is engine specific, yet different models give good results for most engines. One of them is the following: L8: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωT + α5 ω 2 T + α6 ωT 2 . The parameters of polynomial fuel consumption models can be identified with OBD measurements. It is important that the measurements are obtained in a controlled environment: known pavement, low wind velocity, known vehicle mass, and most important no significant road slopes (or at least known slopes). If measurements are not possible, power-based consumption models with 3 parameters can be identified with publicly available data. The recommended model is the following: P4: m ˙ f = α0 + α1 P + α2 P 2 .

(2.39)

However, this model can not decently capture gear shift behavior. Polynomials can model the fuel consumption of both gasoline and diesel engines, and the efficiency of electric motors. A part of the work on power-based consumption models is published as a paper in Transportation Research Part D: Transport and Environment, titled “Virginia Tech Comprehensive Power-based Fuel Consumption Model:

CONCLUSIONS

55

Model Development and Testing”. Another paper, on different types of fuel consumption models, is submitted to the Journal of Intelligent Transportation Systems (ITS), titled “Assessment of alternative polynomial fuel consumption models for use in ITS applications”. These two papers are the result of a cooperation with the Virginia Tech Transportation Institute.

Chapter 3

Theory of Eco-Driving This chapter comprises the core of this dissertation: the theory of eco-driving. Section 3.1 describes the optimal control problem. This problem is then solved with four different methods: Euler-Lagrange (section 3.2), Pontryagin’s maximum principle (section 3.3), discrete dynamic programming (section 3.4), and direct multiple shooting (section 3.5). In addition, section 3.6 shows how the previous methods can be used for eco-driving applied to other types of vehicles, and section 3.7 discusses how it can be implemented in an eco-drive assist system. Finally, section 3.8 concludes this chapter. This chapter only describes how the optimal driving behavior can be calculated; an analysis of this optimal behavior is given in the next chapter.

3.1

The Minimum-Fuel Vehicle Control Problem

As discussed in section 1.5 on p. 17, eco-driving is an optimal control problem (OCP). In this dissertation, the eco-driving OCP is referred to as the eco-drive control problem or the minimum-fuel vehicle control problem.

57

58

THEORY OF ECO-DRIVING

3.1.1

Mathematical Formulation

The general minimum-fuel vehicle control problem is the following: min

Zte

L (v, ue , g, uc ) dt,

(3.1a)

0


i(g) · T (ue , ω) · uc − Fb (ub ) − cv (g) · v − c0 + c1 v + c2 v 2 dv = , dt I(g, uc ) (3.1b) ds = v, dt

(3.1c)

v(0) = vs ,

(3.1d)

v(te ) = ve ,

(3.1e)

s(0) = 0,

(3.1f)

s(te ) = se ,

(3.1g)

vmin 6 v 6 vmax ,

(3.1h)

ωmin 6 ω 6 ωmax ,

(3.1i)

Tmin (ω) 6 T 6 Tmax (ω),

(3.1j)

0 6 Fb 6 Fmax ,

(3.1k)

g = {1, 2, . . .},

(3.1l)

uc = {0, 1}.

(3.1m)

The control inputs are: engine control ue , which can be torque T [Nm], load τ [-], or power P [W]; brake control ub [-]; gear g [-]; and clutch control uc [-], uc = 1 when the engine is engaged and uc = 0 when the engine is disengaged. When uc = 0, the engine is idling. Controls g and uc are discrete control inputs. The goal is to determine the control inputs that minimize the time integral of the cost function L [kg/s]. The integral cost function L consists of the fuel mass flow rate m ˙ f [kg/s] and possibly mass flow rates of other polluting emissions, as discussed in section

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

59

2.1.4 on p. 33. In the remainder of this dissertation, only the fuel consumption is considered. The cost function can also include an indirect time constraint. This is explained further on. The dynamics of the system are given by equations (3.1b) and (3.1c). These are valid for a vehicle with a manual conventional powertrain, or an automatic conventional powertrain during lockup (always: uc = 1). The origin of equation (3.1b) is given in appendix B.2 on p. 195. As a reminder: v [m/s] is the vehicle velocity, s [m] the traveled distance, i [1/m] the overall driveline ratio (ω = iv), Fb [Nm] the braking force, cv [N·s/m] the viscous friction coefficient, and I [kg] the total inertia. The longitudinal dynamics coefficients c0 [N], c1 [N·s/m], and c2 [N·s2 /m2 ] change with ambient conditions (wind, road slope, road curvature, rolling friction, ...) and are thus possibly a function of time and/or distance. In the dynamics of the control problem, wheel slip and skid, and the maximum traction force are neglected. Equations (3.1d)–(3.1g) give start and end constraints, with te [s] the total travel time and se the total traveled distance. Intermediate constraints are possible too, like the velocity constraint (3.1h). Equations (3.1i) and (3.1j) represent the engine limitations. Equation (3.1k) represents the limitations of the brakes. Using equation (3.1b) and assuming that Fb = 0 and uc = 1, the engine torque can be given as:

T =

I

dv + cv v + c0 + c1 v + c2 v 2 dt . i

(3.2)

This can be split up: T

= Tss + Ti ,

(3.3a)

Tss

=

cv v + c0 + c1 v + c2 v 2 , i

(3.3b)

Ti

=

I dv , i dt

(3.3c)

with Tss the steady state torque to maintain a constant velocity, and Ti the torque to overcome inertia. Now it is investigated whether the eco-drive control problem is convex or not. An optimal control problem is convex if: 1) the objective function is convex, 2) the inequality constraint functions are convex, and 3) the equality constraint

60

THEORY OF ECO-DRIVING

functions are affine [30]. If dv/dt is used as a the control ue , with L = m ˙ f and uc = 1, the optimal control problem is reformulated as follows: min

Zte

m ˙f

0



iv,

Iue + Fb + cv + c0 + c1 v + c2 v 2 i



· dt,

(3.4a)

dv = ue , dt

(3.4b)

ds = v. dt

(3.4c)

The fuel mass flow rate m ˙ f (ω, T ) is a nondecreasing function. Usually it is also a convex function. A function f (x) = h(g(x)) is convex if h is convex and nondecreasing, and g is convex [30]. Thus, m ˙ f (v, ue ) is convex, making the latter control problem convex. Unfortunately, the introduction of gear shifting and clutch control (discrete controls) make the optimal control problem nonconvex, even if m ˙ f is convex. The convexity of the optimal control problem also depends on possible additional constraints.

3.1.2

Steady State Analysis

Recall the definition of the fuel economy FE [kg/m], i.e. the fuel mass flow rate divided by the velocity: FE = m ˙ f /v. For a constant velocity, this is the steady state fuel economy FEss = m ˙ f,ss /v, with m ˙ f,ss the fuel mass flow rate when driving at a constant velocity. Without doing any calculations, it is obvious that the steady state (constant velocity) solution of the eco-drive control problem is to drive at the velocity that minimizes FEss . This velocity is called the optimal (cruising) velocity vopt [m/s]:   ˙ f,ss (v, gmax ) ∂ m = 0. (3.5) ∂v v v=vopt

The highest possible gear gmax is the most fuel-efficient one. Figure 3.1 shows a comparison of the real and modeled steady state fuel economy for different gears as a function of the velocity. The simulated vehicle is a 2004 Toyota Corolla Verso with a 3ZZ-FE engine, whose specifications can be found in appendix C.1 on p. 203. The polynomial fuel consumption model L3 of equation (2.25) is able to model FEss the best. For this combination of engine, vehicle, and ambient conditions, the optimal velocity is attained at the lowest engine speed.

fuel economy FEss [l/100km]

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

61

real model

15

10

5 0

20

40

60

80

100

120

140

velocity v [km/h]

Figure 3.1: Steady state fuel economy of a 2004 Toyota Corolla Verso, comparison between the real values and the values modeled with polynomial fuel consumption model L3. This is sometimes reported, although usually the optimal velocity is around 60–80 km/h in the highest gear [39, 58, 64, 67, 74, 169]. The EMEP/EEA air pollutant emission inventory guidebook proposes the following equation to model the steady state fuel economy [142]: FEss =

a + cv + ev 2 , 1 + bv + dv 2

(3.6)

with a...e parameters that depend on the emission norm, engine size, and fuel of the vehicle. Table 3.1 shows the resulting vopt . It ranges between 66 and 90 km/h. The optimal velocity is of major importance in the eco-drive control problem. Consider the problem where a vehicle has to travel a given distance. The vehicle will start with a transition towards the optimal velocity, and cruise the remaining distance at the optimal velocity. Usually, this is not a satisfactory solution. One will not accept to cruise at 70 km/h on the highway because it will consume the least fuel. Furthermore, the calculated optimal velocity is sensitive to modeling errors. It would be even more unacceptable to drive 70 km/h on the highway, if the actual optimal velocity is 90 km/h. If the traffic allows it, most drivers want to drive at the speed limit. One could think that introducing a lower velocity bound would solve the problem. Yet if the vehicle starts at a velocity lower than this bound, the problem becomes infeasible. Using a velocity constraint ve at the end distance se , does not solve the problem either. The vehicle will just cruise at the optimal velocity and then accelerate at the

62

THEORY OF ECO-DRIVING

Table 3.1: Optimal steady state velocity vopt according to the EMEP/EEA air pollutant emission inventory guidebook [142]. fuel

emission norm Euro 1 Euro 2

gasoline Euro 3 Euro 4 Euro 1 diesel

Euro 2 Euro 3

engine size [cm3 ]

vopt [km/h]

< 1400 1400 . . . 2000 > 2000 < 1400 1400 . . . 2000 > 2000 < 1400 1400 . . . 2000 > 2000 < 1400 1400 . . . 2000 > 2000

70 76 79 70 80 79 76 74 87 66 75 90

6 2000 > 2000 6 2000 > 2000 6 2000 > 2000

67 68 66 68 76 68

end. Several researchers have used a direct time constraint (fixed the end time te ) to solve this problem, see e.g. [92, 120, 172, 178]. Still, this is an unnatural solution; nobody wants to complete a certain trip in an exact period of time. A more elegant solution is to add the travel time to the integral cost function, besides the fuel consumption: Z min (m ˙ f + M) dt, (3.7) with M [kg/s] a scalar. This method is called indirect time constraint. Many researchers determine M iteratively to make the average velocity equal to a given value, e.g. [94, 138]. A better way is to determine M such that the new optimal steady state velocity vss , that minimizes time and fuel combined, equals a desired value: m ˙ f,ss (v, g) + M ∂ v = 0. (3.8) ∂v v=vss

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

63

What the value of vss should be in practice, depends on the situation and is explained further in this chapter. Ivarsson, Åslund, and Nielsen [103] report that some modern heavy-duty diesel engines have nonconvex fuel maps, i.e. the BSFC map has two local minima. A direct time constraint will result in a solution that has two sections with different cruising velocities. When using an indirect time constraint, the solution is cruising at one velocity. Nonconvex fuel consumption results in a velocity range that is unreachable. This means that the shape of FEss (v) is such, that simply adding a constant value to the fuel consumption model can not put the minimum at every velocity.

3.1.3

Identification of Subproblems

Solving the eco-drive control problem of equation (3.1) at once for a complete trip (e.g. driving to work), is almost impossible. First of all because there are no methods that can solve such a big problem fast and efficient. Second, because it is also impossible to simply define all the constraints for that control problem (traffic lights timing, what all other cars in traffic will do, ...). Instead, a trip is split up in small and solvable subproblems. Typical problems are: • acceleration, • deceleration, • driving between stop signs, • approaching traffic lights, • eco-cruise control, • driving in a traffic jam. The details of the subproblems are given further on. Keeping a safe distance from preceding vehicles can always be introduced as extra constraints. Translating the surrounding traffic to specific time, distance, and velocity constraints is out of the scope of this dissertation. Inspiration can be found in research on active cruise control (ACC), see e.g. [43]. In what follows, other vehicles are not taken into account and traffic jam driving is thus not treated. It should be noted that the actual solution of the eco-drive control problem consists of the resulting velocity profile and gears. The control problem is

64

THEORY OF ECO-DRIVING

crafted to the velocity with the indirect time constraint. The engine control and the brake control do not have a practical meaning. Converting them to throttle and brake pedal positions would require extra modeling. Communicating this to the driver wouldn’t be very practical either. Direct and Indirect Constraints Before the different subproblems are elaborated, a definition of time and distance constraints is made. direct time constraint The total time te is fixed. indirect time constraint The total time te is free. However, the time is important, e.g. by means of a desired steady state cruising velocity. direct distance constraint The distance to be traveled se and the end velocity ve are fixed. indirect distance constraint The distance se is free; it does not matter where ve is reached.

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

65

Acceleration The vehicle has to accelerate from the start velocity vs to a given steady state velocity vss and then continues cruising. There is no direct time or distance constraint. The brakes are not used and the engine is engaged. Mathematically, the problem is as follows: min

Zte

(m ˙ f (i(g) · v, ue ) + M) dt,

(3.9a)

0

te free, i(g) · T (ue , ω) − cv (g) · v − c0 + c1 v + c2 v dv = dt I(g)

(3.9b)
2

,

(3.9c)

v(0) = vs ,

(3.9d)

v(te ) = vss ,

(3.9e)

dv (te ) = 0, dt

(3.9f)

ωmin 6 ω 6 ωmax ,

(3.9g)

Tmin (ω) 6 T 6 Tmax (ω),

(3.9h)

g = {1, 2, . . .}.

(3.9i)

66

THEORY OF ECO-DRIVING

Deceleration The vehicle has to decelerate from a steady state velocity vss to a given end velocity ve . There is a direct distance constraint. This problem occurs when the vehicle has to decelerate and then the engine is switched off; the vehicle can immediately continue driving or it can not be predicted when the vehicle continues driving. Example: coming to a stop at the destination or stop sign. Mathematically, the problem is as follows: min

Zte

(m ˙ f (i(g) · v, ue ) · uc + m ˙ f,i (1 − uc ) + M) dt,

(3.10a)

0

te free,

(3.10b)
i(g) · T (ue , ω) · uc − Fb (ub ) − cv (g) · v − c0 + c1 v + c2 v 2 dv = , dt I(g, uc ) (3.10c) ds = v, dt

(3.10d)

v(0) = vss ,

(3.10e)

v(te ) = ve ,

(3.10f)

s(0) = 0,

(3.10g)

s(te ) = se ,

(3.10h)

ωmin 6 ω 6 ωmax ,

(3.10i)

Tmin (ω) 6 T 6 Tmax (ω),

(3.10j)

0 6 Fb 6 Fmax ,

(3.10k)

g = {1, 2, . . .},

(3.10l)

uc = {0, 1}.

(3.10m)

A deceleration can also occur when a vehicle has to slow down to a new steady state velocity. The distance constraint is indirect and equations (3.10e)–(3.10h) are replaced by equations (3.9d)–(3.9f).

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

67

Driving Between Stop Signs The vehicle starts and ends at standstill, traveling a given distance. There is a direct distance constraint. Mathematically, the problem is as follows: min

Zte

(m ˙ f (i(g) · v, ue ) · uc + m ˙ f,i (1 − uc ) + M) dt,

(3.11a)

0

te free,

(3.11b)


i(g) · T (ue , ω) · uc − Fb (ub ) − cv (g) · v − c0 + c1 v + c2 v 2 dv = , dt I(g, uc ) (3.11c) ds = v, dt

(3.11d)

v(0) = 0,

(3.11e)

v(te ) = 0,

(3.11f)

v 6 vss ,

(3.11g)

s(0) = 0,

(3.11h)

s(te ) = se ,

(3.11i)

ωmin 6 ω 6 ωmax ,

(3.11j)

Tmin (ω) 6 T 6 Tmax (ω),

(3.11k)

0 6 Fb 6 Fmax ,

(3.11l)

g = {1, 2, . . .},

(3.11m)

uc = {0, 1}.

(3.11n)

68

THEORY OF ECO-DRIVING

Approaching Traffic Lights The vehicle is cruising at vss at t = 0 and s = 0. Upcoming traffic lights at se are red between te,1 and te,2 . If se /vss < te,1 or se /vss > te,2 , the vehicle can continue to cruise at vss . If that is not the case, the vehicle should accelerate or decelerate to another cruising velocity to pass the traffic lights when they are green. In practice this means right before they turn red (te = te,1 ), or right after they turn green (te = te,2 ). Thus, there is a direct time constraint. Mathematically, the problem is as follows: min

Zte

(m ˙ f (i(g) · v, ue ) · uc + m ˙ f,i (1 − uc ) + M) dt,

(3.12a)

0

te fixed,

(3.12b)


i(g) · T (ue , ω) · uc − Fb (ub ) − cv (g) · v − c0 + c1 v + c2 v 2 dv = , dt I(g, uc ) (3.12c) ds = v, dt

(3.12d)

v(0) = vss ,

(3.12e)

s(0) = 0,

(3.12f)

s(te ) = se ,

(3.12g)

ωmin 6 ω 6 ωmax ,

(3.12h)

Tmin (ω) 6 T 6 Tmax (ω),

(3.12i)

0 6 Fb 6 Fmax ,

(3.12j)

g = {1, 2, . . .},

(3.12k)

uc = {0, 1}.

(3.12l)

THE MINIMUM-FUEL VEHICLE CONTROL PROBLEM

69

Eco-Cruise Control The vehicle is cruising within an allowed velocity band. Knowledge of the ambient conditions and its influence on the fuel economy, can be used to vary the velocity and save fuel. Here, the changing road slope θ(s) is used. The driver can choose a reference velocity vref , a minimum velocity vmin , and a maximum velocity vmax . On a level road, the vehicle is expected to cruise at vref . On slopes, the velocity can be adapted. The goal is that the optimal velocity profile can be fed to a conventional cruise controller. It is therefore assumed that ECC will not disengage the engine from the rest of the powertrain (except for gear shifts) and the brakes are not used. Mathematically, the problem is as follows: min

Zte

(m ˙ f (i(g) · v, ue ) + M) dt,

(3.13a)

0

te free,

(3.13b)


i(g) · T (ue , ω) − cv (g) · v − c0 (s) + c1 (s) · v + c2 (s) · v 2 dv = , dt I(g) (3.13c) ds = v, dt

(3.13d)

vmin 6 v 6 vmax ,

(3.13e)

ωmin 6 ω 6 ωmax ,

(3.13f)

Tmin (ω) 6 T 6 Tmax (ω),

(3.13g)

g = {1, 2, . . .}.

(3.13h)

In contrast with the previous subproblems, the vehicle dynamics coefficients c0 , c1 , and c2 are not constant. They are a function of θ, thus a function of s.

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THEORY OF ECO-DRIVING

3.2

Solution with Euler-Lagrange

This section uses the Euler-Lagrange (EL) differential equation for a first analysis of the eco-drive control problem. A general explanation of this method is given in appendix E.3 on p. 226.

3.2.1

Basic Analysis

Consider the reformulated problem of equation (3.4), with a direct time constraint te . Gear shifting and braking are not considered, the engine is engaged. For simplicity, a torque-based fuel consumption model is used. The application of Euler-Lagrange requires a state end constraint v(te ) = ve . The Lagrangian L [kg/s] includes a distance constraint by using a scalar Lagrangian multiplier λ [kg/m]: L=m ˙ f + λv,

with

Zte

v dt = se .

(3.14)

0

The solution is given by the Euler-Lagrange equation, the fundamental equation of the calculus of variations:   d ∂L ∂L − = 0. (3.15) ∂v dt ∂ v˙ Because ∂L/∂t = 0, this equation reduces to the Beltrami identity [10]: ∂L = C, (3.16) L − v˙ ∂ v˙ v

with C [kg/s] a constant of integration.

The optimal evolution of the vehicle velocity as a function of time is thus given by: dv m ˙ f + λv − C . = ∂L dt ∂ v˙ v

(3.17)

Using equation (3.3), the partial derivative of L with respect to v˙ can be expressed as: ∂L ∂L ∂T I ∂L I ∂m ˙ f = = = , (3.18) ∂ v˙ v ∂T v ∂ v˙ v i ∂T v i ∂T v

SOLUTION WITH EULER-LAGRANGE

71

which yields an expression for the optimal engine torque T ∗ : T ∗ = Tss (v) +

m ˙ f (v, T ) + λv − C . ∂m ˙ f (v, T ) ∂T

(3.19)

v,T

The fuel mass flow rate m ˙ f can be approximated by a Taylor series expansion to the second degree: Ti2 ∂2m ˙ f ∂m ˙ f , (3.20) T + m ˙ f (v, T ) = m ˙ f (v, Tss ) + i ∂T v,Tss ∂T 2 v,Tss 2 with T = Tss + Ti . Furthermore: ∂m ˙ f ∂2m ˙ f ∂m ˙ f + Ti . = ∂T v,T ∂T v,Tss ∂T 2 v,Tss

Inserting equations (3.20) and (3.21) in equation (3.19) yields: ∂2m ˙ f Ti2 ∂m ˙ f T + + λv + C m ˙ f (v, Tss ) + i ∂T v,Tss ∂T 2 v,Tss 2 Ti = . ∂m ˙ f ∂2m ˙ f + T i ∂T v,Tss ∂T 2 v,Tss

(3.21)

(3.22)

This yields a closed algebraic expression for the optimal engine torque in a given gear: v u m ˙ f (v, Tss ) + λv − C ∗ T = Tss (v) ± u . (3.23) u2 ∂2m ˙ f (v, T ) t ∂T 2 v,Tss

The latter is exact if the highest occurring power of T in the fuel consumption model is maximum 2, thus if the fuel consumption model is affine or quadratic. If the model is affine, the denominator under the square root is zero. This results in bang-bang control, i.e. the optimal torque is the maximum or minimum torque. Although some researchers use affine consumption models for eco-driving [36, 127, 169], it is shown that real minimum-fuel control is not bang-bang. Saerens and Vandersteen [158] demonstrate this in their master’s thesis on an engine dynamometer. Furthermore, more complex fuel consumption models (that are more accurate) usually do not yield bang-bang control. For use in eco-drive applications, one should thus discard consumption

72

THEORY OF ECO-DRIVING

models that yield this kind of unrealistic minimum-fuel behavior. Quadratic fuel consumption models have the advantage that they are able to model the fuel mass flow rate quite accurately (see chapter 2) and that the denominator in equation (3.23) is independent of T , which simplifies the expression.

3.2.2

Human Factors

Parameters λ and C in equation (3.23) should be determined such that the end constraints v(te ) = ve and s(te ) = se are met. This is further discussed in section 3.3. However, by incorporating human factors that are displayed during vehicle acceleration, these parameters can be determined as well. The human factors that are to be taken into account are readily recognizable: 1. The acceleration rate is zero at the end of the acceleration course. This can be expressed as: m ˙ f (ive , Tss (ve )) + M + λve = 0.

(3.24)

2. The acceleration is monotonic. This factor can be expressed as: min (m ˙ f (iv, Tss (v)) + M + λv) > 0. v

(3.25)

Given the fact that ∂ m ˙ f /∂v > 0 during the course of the acceleration, the condition above can be written as: dm ˙ f (iv, Tss (v)) λ 6 λmin , with λmin = − (3.26) . dv ve

3. The vehicle acceleration is smooth. The acceleration is more gradual with λ small and because λmin < 0, the equal sign in equation (3.26) generally holds. The steady state fuel mass flow rate m ˙ f,ss (v) = m ˙ f (iv, Tss (v)) is a function of only v and can equally be expanded by a quadratic Taylor series in v, this yields:

m ˙ f (iv, Tss (v)) = m ˙ f (ive , Tss (ve )) + (v − ve )

(v − ve )2 d2 m ˙ f,ss dm ˙ f,ss + . dv ve 2 dv 2 ve (3.27)

SOLUTION WITH EULER-LAGRANGE

73

Taking the human factor conditions into account with λs = λs,min , yields a remarkably simple expression for the optimal acceleration torque Ti∗ : v u 2 ˙ f,ss ud m u u dv 2 v ∗ ′ e. (3.28) Ti = (ve − v)β = (ve − v)u u ∂2m ˙ f,ss t ∂T 2 ve

A linear decay law is readily obtained: a = β(ve − v) =

i ′ β (ve − v). I

(3.29)

This equation is exact for a consumption model that is quadratic. For affine consumption models, a bang-bang control is yielded.

74

THEORY OF ECO-DRIVING

3.3

Solution with Pontryagin’s Maximum Principle

This section uses Pontryagin’s maximum principle (PMP) to solve the ecodrive control problem. To be able to do that, the considered OCP should be simple enough. For difficult OCP’s there is little to no hope that the maximum principle will be able to provide an optimal control law. However for basic subproblems, as explained earlier, PMP can give an exact and elegant solution. First in this section, a basic analysis of the problem is conducted. Then, different kinds of simple eco-drive problems are discussed in detail. Appendix E.4 on p. 227 gives a general explanation of PMP.

3.3.1

Basic Analysis

For simplicity, consider a torque-based fuel consumption model. The brake control ub is the brake force Fb itself. The vehicle dynamics (rolling friction, road slope, road curvature, wind) are constant. According to Pontryagin’s maximum principle, the Hamiltonian H [kg/s] is given by: H

dv + λs v dt = uc · m ˙ f (iv, T ) + (1 − uc ) · m ˙ f,i + M
iT uc − Fb − cv v − c0 + c1 v + c2 v 2 + λs v, +λv I

= m ˙ f + M + λv

(3.30)

with m ˙ f,i the idle fuel mass flow rate, and λv [kg·s/m] and λs [kg/m] adjoint state variables of the velocity and distance respectively. The other parameters are explained in section 3.1.1. In comparison to Euler-Lagrange, there should be no end constraints ve and se . The maximum principle states that the optimal controls T ∗ , Fb∗ , g ∗ , and u∗c minimize the Hamiltonian: H (v, s, λv , λs , T ∗ , Fb∗ , g ∗ , u∗c ) 6 H (v, s, λv , λs , T, Fb , g, uc ) ,

(3.31)

where ∗ denotes optimality. Given the states v, s, λv , and λs , the goal is to find the controls T , Fb , g, and uc that minimize H. The time derivatives of the adjoint states are also given by the maximum principle: dλv ∂H ∂m ˙f cv + c1 + 2c2 v =− = −uc + λv − λs , dt ∂v ∂v I

(3.32a)

dλs ∂H =− = 0. dt ∂s

(3.32b)

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

75

Solving the Boundary Value Problem The normal approach is to solve the boundary value problem (BVP) that results from applying the maximum principle to the OCP, i.e. iteratively determining the initial values of the adjoint state variables λv (0) and λs (0). Kock et al. [115], and Passenberg, Kock, and Stursberg [145] use this approach. They apply a multiple shooting algorithm to solve the BVP. Gear shifting is introduced with given upshift and downshift engine speeds. Abenavoli et al. [3] developed their own special method. It is used to accelerate to vopt , thus the brake force is not used. The iterative procedure is as follows: 1. Start at iteration index j = 0. Choose initial controls T0 (t) and g0 (t), t ∈ [0, tmax ], with tmax the maximum foreseen duration of the acceleration. 2. Integrate the velocity starting at t = 0 until v = vopt . Store the trajectory vj (t) and tj , with vj (tj ) = vopt . 3. Integrate the adjoint state λv,j backwards from t = tj to t = 0 and forwards from t = tj to t = tmax with λs = 0, using Hj (tj ) = 0 to determine λv,j (tj ). 4. Evaluate Hj (t) for t ∈ [0, tj ]. If mint | Hj (t) |< ǫ, then terminate the procedure. 5. For t ∈ [0, tmax ] calculate (Tj+1 (t), gj+1 (t)) = arg minT,g H (t, vj , T, g, λv,j ) and start again at step 2, replacing j by j + 1. Bang-Bang Control Analyzing the problem can give more insight in the solution. It is indicated before in equation (3.23) that affine fuel consumption models result in bangbang control. This case is studied here more in detail. An affine torque-based fuel consumption model is used for illustration: m ˙f = α0 + α1 ωT , where α0 [kg/s] represents the friction losses and α1 [kg/J] the internal engine efficiency. A fixed gear is assumed and the engine is engaged. The part of the Hamiltonian that is affected by the controls is:   λv λv i T− Hu = α1 iv + Fb . (3.33) I I The controls that minimize H, depend on the value of λv . This results in five different control modes:

76

THEORY OF ECO-DRIVING

• λv > 0 According to equation (3.33), T ∗ = Tmin (ω) and Fb∗ = Fb,max minimize H. • λv = 0 T ∗ = Tmin (ω) minimizes H. Fb does not affect the Hamiltonian and thus this is a singular interval for the brake control. Further time differentiation of ∂H/∂Fb does not yield an expression that can determine Fb . The braking force Fb can be chosen freely, but possible end constraints should of course be satisfied. • −α1 vI < λv < 0 T ∗ = Tmin (ω) and Fb∗ = 0 minimize H. Both accelerator and braking pedal are not touched (engine braking). • λv = −α1 vI Fb∗ = 0 minimizes H. T does not affect the Hamiltonian and thus this is a singular interval for the engine control. Stoicescu [169] shows that in this singular interval v = vopt . • λv < −α1 vI T ∗ = Tmax (ω) and Fb∗ = 0 minimize H. The accelerator pedal is pushed pedal to the metal or wide open throttle (WOT). An optimal velocity trajectory consists of a combination of different modes. An optimal control resulting from an affine fuel consumption model is bang-bang control. In fact, here it is bang-off-bang control. The first bang refers to WOT acceleration, the off refers to engine braking, and the second bang to braking at full power. Bang-off-bang control was first used in the literature on eco-driving by Stoicescu [169]. It is later also used by e.g. Liu and Golovitcher [127], and van Keulen et al. [178]. Bang-off-bang control is the normal optimal control law for a minimum-fuel problem according to Kirk [112]. Here, the fuel is actually the propulsive work and not the real fuel. Bang-off-bang control is not an appropriate control law for eco-driving, as explained in section 3.2. The Explicit Approach Solving the boundary value problem has the disadvantage that it always requires an iterative procedure. It also does not provide much insight in the solution. Affine fuel consumption models result in simple solutions that lack the latter disadvantages. The approach that is used in this dissertation is an explicit one. It yields simple analytic solutions and often does not require iterations. This method is explained in the rest of this section.

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

3.3.2

77

Accelerations

The first basic problem is an optimal acceleration. The vehicle accelerates from a start velocity vs to a steady state velocity vss . Afterwards, the vehicle keeps cruising at vss . The time and distance constraint are both indirect. Explicit Control First consider an acceleration towards the optimal velocity vopt and a torquebased fuel consumption model. During an acceleration the brakes will not be used and the engine is engaged. To start with, assume a fixed gear. The engine torque is then the only control. There is no direct time constraint, thus according to the maximum principle: H = 0, see equation (E.23) on p. 228. The optimal control T ∗ should minimize H. Equation (3.30) yields: i ∂H X αk (iv)pk qk T qk −1 + λv = 0, = ∂T I

(3.34a)

k

⇓ λv = −I

X

αk ipk −1 v pk qk T qk −1 .

(3.34b)

k

Inserting the latter equation in (3.30) yields: X k


 αk (iv)pk T qk − ipk −1 v pk qk T qk −1 iT − cv v − c0 − c1 v − c2 v 2 +λs v = 0. (3.35)

This method to eliminate λv is introduced by Schwarzkopf and Leipnik [164]. Further novel analysis reveals some nice properties. If the fuel consumption model is quadratic, an explicit solution for T ∗ can be obtained from the latter equation: v u ˙ f (iv, Tss (v)) + λs v u m , (3.36) T ∗ = Tss (v) + u2 t ∂2m ˙f ∂T 2

with T bounded by Tmin and Tmax . The denominator under the square root is independent of T . This expression is equivalent to equation (3.23), which is derived with Euler-Lagrange. In the acceleration problem, there is no direct

78

THEORY OF ECO-DRIVING

distance constraint. It is the goal to minimize the fuel consumption per traveled distance and to have an acceleration towards vopt . Thus, λs takes into account some sort of indirect distance constraint. The value of λs is found by evaluating the Hamiltonian at v = vopt : X q pk −1 λs = − (3.37) αk ipk vopt (Tss (vopt )) k = −FEss (vopt ). k

Inserting the latter equation in (3.36) yields: v u u FEss (v) − FEss (vopt ) ∗ . T = Tss (v) + u2 t ∂ 2 FE ∂T 2 .

(3.38)

If there is a direct time constraint, then H = 6 0. Moving this value to the right hand side of equation (3.30) and renaming H = −M, yields the Hamiltonian of an eco-drive control with no direct time constraint, but with the time penalized in the objective function (indirect time constraint). Mathematically, these two approaches are the same. The resulting optimal control law is then: v u u E(v) − E(vss ) ∗ T = Tss (v) + u2 , (3.39) t ∂ 2 FE ∂T 2

and

m ˙ f (v, Tss (v)) + M m ˙ f,ss (v) + M = , (3.40) v v with E [kg/m] the economy, a function that combines fuel consumption and an indirect time constraint. E=

Regardless of what control variable ue is used in the consumption model, the following equation holds: v u E(v) − E(v ) u ss ∗ . (3.41) ue = ue,ss (v) + u2 t ∂ 2 FE ∂u2e

This is an important novel equation. First, it gives an explicit and very simple analytic solution. Second, it gives clear insight in the solution. The larger the difference between the current economy and the economy of the aimed at steady state velocity, the larger the acceleration rate towards that velocity. This is illustrated in figure 3.2. The larger the fuel cost for an increase of the acceleration, the lower the acceleration rate towards the steady state velocity. If the fuel consumption model is not affine, equation (3.41) does not result in bang-bang control.

fuel economy FEss and economy E [g/m]

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

79

0.20 FEss E 0.15 E(v) − E(vss )

0.10

0.05

0

0

v 20

40

60 vss 80

100

120

velocity v [km/h]

Figure 3.2: Illustration of optimal engine control. The larger the difference between the current economy and the economy of the aimed at steady state velocity, the larger the acceleration rate towards that velocity. Cost Function and Constraints Consider an acceleration towards a steady state velocity vss . If vss = vopt , then equation (3.38) can be used. In the course of the acceleration, velocities higher than vopt are never attained. Thus if vss > vopt , equation (3.39) should be used, with M according to equation (3.8). For vss < vopt , Bellman’s principle of optimality holds: “A subsolution of an optimal solution of the problem, is itself an optimal solution for the subproblem.” Consider the optimal velocity trajectory from the start velocity vs to vopt . The part of this trajectory from vs to vss is the optimal trajectory for an acceleration from vs to vss . This means that for an acceleration to vss < vopt : M = 0. If the resulting acceleration is too slow to meet e.g. a maximum acceleration distance or time, the value of λs can be adapted: λs = −E(vss ) + ε,

(3.42)

with ε > 0 [kg/m] a distance constraint coefficient. With ε = 0, the acceleration is optimal. Increasing the value of ε increases the value of λs and changes the indirect distance constraint: vss will be attained in a shorter distance. The

80

THEORY OF ECO-DRIVING

introduction of ε results in a new elegant expression for the engine control: v u E(v) − E(v ) + ε u ss . (3.43) u∗e = ue,ss (v) + u2 t ∂ 2 FE ∂u2e In the case that ε = 0 and vss > vopt , the velocity goes asymptotically towards vss , thus taking an infinite time to actually reach vss . To prevent this, one can use v = min (v, (1 − ǫ) · vss ) to calculate the optimal control with equation (3.41), where ǫ [-] is a small value. Figure 3.3 shows accelerations in the highest gear from 35 km/h to vopt = 53.5 km/h. The Toyota Corolla is simulated with a quadratic torque-based model: m ˙ f = α0 + α1 ωT + α2 ω 2 T 2 . Table 3.2 shows the resulting fuel consumption mf , travel time te , and total cost J = mf + M · te . To make a fair comparison, the vehicle cruises at vss after the acceleration, such that the same distance is traveled. Increasing values of ε yield faster accelerations and more fuel consumption. Table 3.2 also shows results for accelerations to 70 km/h, vss > vopt ⇒ M > 0. Increasing values of ε yield faster accelerations, more fuel consumption, and a higher total cost. Figure 3.4 shows accelerations to 45 km/h. In the fuel-optimal case, M = 0 and the velocity is bounded at vss . Table 3.2 shows that if M is determined according to equation (3.8), the fuel consumption increases for vss < vopt .

velocity v [km/h]

55

50

45

εր

40

35

0

200

400

600

distance s [m]

Figure 3.3: Accelerations from 35 km/h to vopt = 53.5 km/h, in the highest gear, for different values of ε, over a distance of 750 m. Increasing ε results in faster accelerations.

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

81

Table 3.2: Comparison of accelerations in the highest gear, for different values of ε. Increasing ε results in a shorter traveling time te , higher fuel consumption mf , and higher total cost J. Bellman’s principle of optimality yields M > 0. vss s M ε mf te J [km/h] [m] [g/s] [l/100km] [g] [s] [g]

53.5 = vopt

750

0

0.0 0.1 0.2 0.3 0.5 ∞

33.07 33.11 33.17 33.19 33.26 33.32

53.84 52.80 52.52 52.36 52.18 52.04

33.07 33.11 33.17 33.19 33.26 33.32

58.50 58.85 59.08 59.23 59.45 59.63

67.73 66.89 66.46 66.24 65.98 65.88

81.76 81.82 81.91 81.96 82.12 82.23

21.51 21.58

40.67 42.06

21.51 17.03

70 > vopt

1200

0.344

0.0 0.1 0.3 0.5 1.0 ∞

45 < vopt

500

0 −0.108

0.0

velocity v [km/h]

45

40 M=0 M 0. With M < 0, the acceleration is slower and yields more fuel consumption compared to M = 0.

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THEORY OF ECO-DRIVING

A distinction should be made between the total distance se and the acceleration distance sa . The latter one being the distance at which the steady state velocity is attained. In the normal case, these distances do not really matter: there is an indirect distance constraint. In some cases though, the acceleration distance might be important. When a vehicle is accelerating on a highway ramp, a certain velocity should be attained within the distance of the ramp. This is a direct distance constraint. The acceleration distance is a function of ε: sa = f (ε). The direct distance constraint can be met by determining ε as the result of an optimization problem: 2

ε = arg min (sa − f (ε)) . ε

(3.44)

Gear Shifting Gear shifting is usually considered a difficult problem to solve because of its discrete control. Therefore, researchers often just neglect gear shifting when calculating minimum-fuel controls with the maximum principle, e.g. [115, 169]. Passenberg, Kock, and Stursberg [145] integrate gear shifting by shifting at fixed engine speeds. Of course, this method does not optimize the gear shift behavior. Hur, Nagata, and Tomizuka [97] use a prediction horizon to decide whether to shift or not. If a gear shift results in a lower fuel consumption in the prediction horizon, the shift is performed. The gear shift behavior depends on the horizon length and the gear is assumed to be constant in the horizon. Gear shifting occurs much more in the eco-drive literature in combination with other solution methods. As shown in the next section, gear shifting is hardly a problem for dynamic programming. Direct methods with gear shifting lead to mixed-integer problems. Fortunately, if the maximum principle is used, gear shifting can be implemented very simply. Optimal control problems with discrete controls are called hybrid optimal control problems. There exists a hybrid maximum principle (HMP) to tackle the problem [167]. HMP requires a specified sequence of discrete controls. For an accelerating vehicle, this is a sequence of increasing gears. The conditions for a switch between two discrete controls, depend on the type of switch: autonomous or controlled. An autonomous switch occurs when the system enters a switching manifold. This could be when the engine speed is higher than a certain value (e.g. upshift speed). This strategy is also used in automatic transmission, where the switching manifold depends on the engine speed (or vehicle velocity) and the engine load. Typically, the more the accelerator pedal is pushed, the higher the upshift speed. A controlled switch does not depend on a switching manifold; the discrete control can actively be chosen, as in a vehicle with a manual transmission. Controlled switches are much easier to handle. The hybrid

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

83

maximum principle requires that adjoint states λ are continuous at a switch, and the Hamiltonian H is continuous at a switch. The adjoint state continuity is a direct consequence from the Weierstrass-Erdmann condition: at every corner of a discontinuous solution the two limiting values of ∂H/∂ v˙ should be equal [27]. Assume that the vehicle is in gear g1 and the following gear in the sequence is gear g2 . An optimal gear shift must satisfy the condition λv (g1 ) = λv (g2 ) at the shift. It is possible that a gear shift that satisfies this condition is not optimal. Schwarzkopf and Leipnik [164] calculate for this purpose λv , assuming that ∂H/∂ue = 0. However, this approach is erroneous if a bound on the control ue is active. Therefore, λv should be calculated with H = 0. According to equation (3.30) with Fb = 0 and uc = 1: λv (g) = − =

m ˙ f + M + λs v

dv dt m ˙ f (i(g)v, T ∗ (g)) + M + (−E(vss ) −I(g) i(g) · T ∗ (g) − cv (g) · v − (c0 + c1 v

+ ε) · v . + c2 v 2 )

(3.45)

Figure 3.5 shows λv as a function of the vehicle velocity for different gears, for an acceleration to 50 km/h. Again, the Toyota Corolla is simulated, but with the 6-parameter model T8 of equation (2.29) on p. 42. This model can decently capture gear shifting. In this example, there is no gear shift for which the values of λv would coincide. This is the result of a lower bound on the engine speed. An upshift should be performed as soon as possible and the optimal gear is the one with the highest value of λv . The adjoint state will be discontinuous at the shift. In figure 3.6, the same is illustrated for an acceleration to 70 km/h. The values do coincide now and gear shifting takes place at higher engine speeds. Again, the optimal gear maximizes λv . Keep in mind that gears with T ∗ 6 Tss can not be selected as they do not result in an acceleration. To validate the presented gear shift method, acceleration simulations were conducted where the time instants to shift were optimized directly. This shift time optimization yields the same result as the maximization of λv . It should be mentioned that for a gear shift problem where the velocity trajectory is given, the solution is very simple. The optimal gear is the one that minimizes the instantaneous fuel mass flow rate. Start Off During the vehicle start off, the OCP changes slightly. For a vehicle with a manual gearbox, it is assumed that ω = ωmin . Thus the clutch is controlled in a

84

THEORY OF ECO-DRIVING

adjoint state λv [kg·s/m]

0

−0.5

−1

gր −1.5

0

10

20

30

40

50

velocity v [km/h]

Figure 3.5: Illustration of the gear shift strategy for an acceleration to 50 km/h. Each line shows λv as a function of v for a fixed gear. The lower left line represents the lowest gear. The optimal gear should maximize λv (solid line parts). In this example, this results in upshifting as soon as possible.

adjoint state λv [kg·s/m]

0

−1

gր −2

0

10

20

30

40

50

60

70

velocity v [km/h]

Figure 3.6: Illustration of the gear shift strategy for an acceleration to 70 km/h. Each line shows λv as a function of v for a fixed gear. The lower left line represents the lowest gear. The optimal gear should maximize λv (solid line parts). In this example, this results in upshifting when the lines of the current gear and the higher gear cross.

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

85

way that the minimum engine speed is attained, until v = ωmin /i(1). Following the same reasoning as before, yet knowing that v = 0 at t = 0, yields: v u v u m ˙ f (i(gss )vss , ue,ss (vss )) +ε u ˙ f (ωmin , ue,ss (v)) − m vss 2 u∗e = ue,ss (v)+ u . (3.46) u ∂2m ˙f t ∂u2e

3.3.3

Decelerations

An optimal deceleration with PMP is somewhat more difficult than an acceleration. In contrast with accelerations, it is not discussed in the literature, except for the trivial case with bang-bang control and no gear shifting, see e.g. [169]. During a deceleration, the brake is a possible control. Moreover, decelerating with a disengaged engine is also a possibility. This introduces a second discrete control. The solution approach is the same as with accelerations. Again considering equation (3.30), only λv has an influence on the minimization of the Hamiltonian. M and λs do not. When the engine is disengaged, the engine torque is not a control any more. In the models for driveline friction and inertia described in section B.2 on p. 195, the gear has no influence when the engine is disengaged. This can simplify the notation: g = 0 for a disengaged engine. Consider a vehicle with a manual gearbox. For notational simplicity, the engine control ue is T again. The optimal controls depend on λv : • λv > 0 Fb∗ = Fb,max minimizes the Hamiltonian. If the engine is engaged, then T ∗ = Tmin and from equation (3.30): λv = −I

m ˙ f (iv, Tmin ) + M + (−E(vss ) + ε) · v . iTmin − Fb,max − (cv v + c0 + c1 v + c2 v 2 )

(3.47)

This mode is called engaged braking. If the engine is disengaged: λv = I

m ˙ f,i + M + (−E(vss ) + ε) · v . Fb,max + cv v + c0 + c1 v + c2 v 2

(3.48)

This mode is called disengaged braking. • λv = 0 This is a singular interval for the brakes. The vehicle stays in the singular

86

THEORY OF ECO-DRIVING

interval if dλv /dt = −∂H/∂v = 0. With uc = 1 this becomes: ∂m ˙ f (iv, Tmin ) − E(vss ) + ε = 0. ∂v

(3.49)

H = 0 yields: m ˙ f (iv, Tmin ) + M + (−E(vss ) + ε) · v = 0.

(3.50)

This is in general only the case if m ˙ f = −M. However M > 0, for the same reason as in the acceleration. Thus if λv = dλv /dt = M = m ˙ f = 0, the brake force Fb can be chosen freely since the velocity trajectory has no influence on the fuel consumption. With λv = 0, the traveled distance does not matter. In fact, the distance is removed from the system and there is no indirect distance constraint. In a real eco-drive problem this is never the case and thus λv = 0 is not a part of the solution, except maybe instantaneous. • λv < 0 Fb∗ = 0 minimizes the Hamiltonian. If the engine is disengaged, then: λv = I

m ˙ f,i + M + (−E(vss ) + ε) · v . cv v + c0 + c1 v + c2 v 2

(3.51)

This mode is called coasting. If the engine is engaged, the case is similar to an acceleration. Yet here the optimal torque is given by: s m ˙ f (iv, Tss (v)) + M + (−E(vss ) + ε) · v ∗ . (3.52) T = Tss (v) − 2 ∂2m ˙ f

∂T 2

If T ∗ = Tmin , the mode is called engine braking. In the case that m ˙ f = 0, it is called engine braking with fuel cut-off. If T ∗ > Tmin , the mode is called normal deceleration. λv is given by equation (3.45). The hybrid maximum principle determines when a switch between gears, or a switch between engagement and disengagement should occur. Since dv/dt < 0, the sign of λv is determined by: m ˙ f + M + (−E(vss ) + ε) · v. It is assumed that the clutch can not slip and that the engine is disengaged as soon as the engine speed would go below the minimum speed in first gear. Figure 3.7 shows λv as a function of the velocity for a deceleration from a steady state velocity of 90 km/h and without brakes (Fb,max = 0 N). It is found that the optimal gear sequence is obtained by minimizing λv (opposed to maximizing λv

adjoint state λv [kg·s/m]

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

2

87

disengaged engaged

gր

0

−2

0

20

40

60

80

velocity v [km/h]

Figure 3.7: Illustration of the gear shift strategy for a deceleration from a steady state velocity of 90 km/h with Fb,max = 0 N. Each line shows λv as a function of v for a fixed gear. The left line represents the first gear. The thick line represents coasting. The optimal gear should minimize λv (solid line parts). In this example, this results in coasting followed by engine braking in second gear, third gear for a split second, second gear, and then first gear. The jumps in the λv -lines occur when the engine goes into fuel cut-off mode. for an acceleration). For this specific example, this results in coasting followed by engine braking in second gear, third gear for a split second, second gear, and then first gear. Figure 3.8 shows λv as a function of the velocity for a deceleration from a steady state velocity of 90 km/h and with brakes (Fb,max = 5 kN). Brakes introduce a bend in the λv -lines and result in a different control sequence. In this example: coasting followed by engine braking in second gear, and then engaged braking in second gear, third, second, first, second, and first. Recall the general equation for λv : λv = I

m ˙ f + M + (−Ess + ε) · v . −iT + Fb + cv v + c0 + c1 v + c2 v 2

(3.53)

The sign depends on the sign of the numerator. When the numerator is positive, the denominator should be large, and engine braking and using the brakes is more likely. With decreasing velocity, the numerator will become positive at some point. This critical velocity is higher when ε is larger. When the velocity is high, the influence of the torque in the denominator is more important than that of the fuel consumption in the numerator and coasting is the optimal mode.

THEORY OF ECO-DRIVING

adjoint state λv [kg·s/m]

88

disengaged engaged

2

0

−2

0

20

40

60

80

velocity v [km/h]

Figure 3.8: Illustration of the gear shift strategy for a deceleration from a steady state velocity of 90 km/h with Fb,max = 5 kN. Each line shows λv as a function of v for a fixed gear. The lower left line represents the first gear. The thick line represents coasting. The optimal gear should minimize λv (solid line parts). In this example, this results in coasting followed by engine braking in second gear, and then engaged braking in second gear, third, second, first, second and first. The jumps in the λv -lines occur when the engine goes into fuel cut-off mode. The bends in the λv -lines for a fixed gear occur when λv = 0, thus when the braking starts. Usually a deceleration comes with a direct distance constraint se , e.g. stopping at an intersection. The total distance is a function of ε: se = f (ε). Comparable to accelerations, the distance constraint can be satisfied as follows: 2

min (se − f (ε)) . ε

(3.54)

Remember that ε > 0. If with ε = 0 the deceleration distance is too short, the deceleration should be preceded by cruising at vss . The actual deceleration is the same as a deceleration with an indirect distance constraint.

3.3.4

Driving Between Stop Signs

The problem of driving between stop signs is fairly easy to solve. If the distance between the stop signs is long enough, the solution is the following: an optimal acceleration from v = 0 km/h to v = vss ; cruising at vss ; an optimal deceleration from v = vss to v = 0 km/h. For short distances, the cruising velocity can not be reached. The solution consists of an acceleration to a

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

89

velocity v [km/h]

60

40

20

0

0

500

1000

1500

2000

distance s [m]

Figure 3.9: Driving between stops sings for different distances, vss = 70 km/h. The start is a part of the optimal acceleration to vss . The start of the deceleration has to be determined such that v = 0 at the desired distance. Only if the distance between the stop signs is long enough, vss can be reached. certain velocity, immediately followed by a deceleration. It follows from the principle of optimality that the acceleration and deceleration are part of the full optimal acceleration and deceleration with ε = 0. This is illustrated in figure 3.9. It shows driving between stops signs for different distances, with vss = 70 km/h and Fb,max = 1 kN.

3.3.5

Approaching Traffic Lights

The vehicle is driving at vss . If maintaining this velocity allows the vehicle to pass a green light, then cruising at vss is the solution. If this is not the case, the vehicle can drive faster to pass the lights right before they turn red, or the vehicle can drive slower to pass the lights right after they turn green. This time constraint will overrule the indirect time constraint; the timing of the traffic lights imposes a direct time constraint. On the other hand, the end velocity ve at which the lights are passed, is not constrained. This can be taken into account by determining a new vss such that the vehicle passes the traffic lights at the required time instant te (i.e. catching the green wave). With te = f (vss ), vss can be determined as follows: 2

vss = arg min (te − f (vss )) . vss

(3.55)

In case of a direct time constraint, M can also be negative. In practice, vss will have bounds. First there is the speed limit vlim . Consider the case where the

90

THEORY OF ECO-DRIVING

vehicle will accelerate to vss = vlim . If this results in a too large (slow) travel time, it can be decreased by increasing ε which wil speed up the acceleration to vss . The second (lower) velocity bound will be a preference of the driver. It is understandable that a driver does not like to approach traffic lights while cruising at a very low velocity. If cruising at this lower velocity bound results in a too large travel time, the vehicle should decelerate right before the traffic lights. The approach is illustrated in figure 3.10. The vehicle is driving at 70 km/h, 500 m before the traffic lights. Different values of vss , ve (the velocity when passing the lights), and ε result in a different travel time. These values are given in table 3.3.

velocity v [km/h]

80

70

60

50

0

100

200

300

400

500

distance s [m]

Figure 3.10: Illustration of the method to approach traffic lights. The timing of the traffic lights imposes a direct constraint on the travel time. The latter is a function of vss , ve , and ε. Table 3.3: Comparison of the travel time te when approaching traffic lights for different values of vss , ve , and ε, with vs = 70 km/h. vss [km/h]

ve [km/h]

ε [l/100km]

te [s]

70 80 80 60 60

70 80 80 60 50

0 0 ∞ 0 0

25.71 23.65 22.83 29.23 30.21

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

3.3.6

91

Eco-Cruise Control

In all the previous problems, the ambient conditions (and thus vehicle dynamics) were constant, i.e. rolling friction, wind drag, and road slope. In practice they do change and this can be used to save fuel. In what follows, the road slope is used as the changing parameter. It can be determined relatively easy (compared to friction and drag) and is often used in this type of application. Eco-cruise control (ECC) uses the property that the cruising velocity vss that minimizes the cost function (fuel and time), depends on the road slope. It will be lower uphill and higher downhill. To start with, M is determined with equation (3.8) for the reference velocity vref on a level road (M > 0). According to equation (3.32b), λs has a constant value. Yet, this is only true if the dynamics are constant. The time derivative of λs is given by: dλs ∂H ∂H ∂θ =− =− . (3.56) dt ∂s ∂θ ∂s The road slope θ changes with the distance s, and so does λs . To determine the optimal velocity profile, two methods can be used: a reactive or a predictive method. Reactive Eco-Cruise Control In the reactive method, no knowledge of the upcoming road slope is used. The reactive eco-cruise control system (RECC) just reacts to the changing road slope. Using ε > 0 does not really make sense here, and thus λs is given by equation (3.30), with dv/dt = 0: λs (θ) = −

m ˙ f,ss (vss (θ)) + M = −E(vss , θ). vss (θ)

The RECC law is then given by:  r ′ ss ,θ)  , if v < vss ,  ue,ss (v ′ ) + 2 E(v ,θ)−E(v ∂ 2 FE   2  ∂ue ue,ss (v), r if v = vss , u∗e =   E(v ′ ,θ)−E(vss ,θ) ′   ue,ss (v ) − 2 , if v > vss ,  ∂ 2 FE 2

(3.57)

(3.58)

∂ue

with:

  max (v, (1 + ǫ) · vss ) , v, v′ =  max (v, (1 − ǫ) · vss ) ,

if v > vss , if v = vss , if v < vss .

(3.59)

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THEORY OF ECO-DRIVING

Here, v ′ is used to prevent that it would take an infinite time to reach vss . Predictive Eco-Cruise Control In the predictive method, knowledge of the upcoming road slope is used. The predictive eco-cruise control system (PECC) can take into account future road slope changes. Continuing equation (3.56):
∂H ∂θ λv ∂ c0 (θ) + c1 (θ) · v + c2 (θ) · v 2 ∂θ dλs =− = . (3.60) dt ∂θ ∂s I ∂θ ∂s

If the road slope changes instantaneously from θ1 to θ2 , the criteria needed to use the maximum principle are not fulfilled. Schwarzkopf and Leipnik [164] consider the limiting case of equation (3.60) for which the time to make the change of road slope goes to zero. Integrating the equation assuming that v and λv are constant during the change, yields: λs (θ2 )−λs (θ1 ) =


λv c0 (θ2 ) + c1 (θ2 )v + c2 (θ2 )v 2 − c0 (θ1 ) − c1 (θ1 )v − c2 (θ1 )v 2 . I (3.61)

Assuming that the length of the road section with the new slope is long enough such that the new cruising velocity vss (θ2 ) can be reached, equation (3.57) holds. The optimal velocity at the change of road grade vθ [m/s] can then be determined by combining equations (3.45), (3.57), and (3.61). The PECC law is the following: 1. Determine vss and λs for the current and next road slope, and vθ for the road slope change. 2. Calculate the distance that would be traveled to reach vθ under the following control law:  r E(v ′ ,θ)−E(vss ,θ) ′  , if v < vθ ,  ∂ 2 FE  ue,ss (v ) + 2 ∂u2 ∗ e r (3.62) ue = E(v ′ ,θ)−E(vss ,θ) ′   , if v > vθ ,  ue,ss (v ) − 2 ∂ 2 FE ∂u2 e

with v ′ according to equation (3.59).

3. If this distance is larger than the distance to the actual road slope change, follow the control law of equation (3.62). Otherwise, follow the control law of equation (3.58) until this distance equals the real distance to the road grade change, then follow (3.62).

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

93

Figure 3.11 shows the resulting velocity profile for the two methods over a simple road profile. RECC simply reacts to a change in road slope by accelerating when vss increases or decelerating when vss decreases. PECC anticipates a road slope change. When vss increases, a deceleration before the road slope change is followed by an acceleration after the change. When vss decreases, an acceleration before the road slope change is followed by a deceleration after the change.

velocity v [km/h]

80

70 RECC PECC vmax vref vmin

60 0

2

4

6

8

10

distance s [km]

Figure 3.11: Velocity profile of reactive and predictive ECC over a simple road profile: 2 km level, 2 km uphill (θ = 3o ), 2 km level, 2 km downhill (θ = −3o ), and 2 km level. In comparison to PECC, RECC does not anticipate a road slope change.

3.3.7

On the Smoothness of the Model

It is important for the vehicle and consumption model to be smooth. Any unsmoothness can result in an unrealistic or unwanted solution of the OCP. It can attract the solution T ∗ to a certain value or repel T ∗ away from a certain value. It therefore is important that an unsmoothness reflects a relevant physical property. This is illustrated using three specific examples: modeling of the driveline efficiency; a piecewise affine fuel consumption model; and modeling of the fuel consumption for negative engine torques. Driveline Efficiency As mentioned in appendix B.2 on p. 200, the driveline efficiency is modeled with a viscous friction coefficient cv . Using this model and equation (3.36) with an

94

THEORY OF ECO-DRIVING

indirect time constraint, the optimal torque during a deceleration is given by: s
l + M + λs v m ˙ f iv, cv v+F c v + F v l ∗ i , (3.63) − 2 T = ∂2m ˙f i ∂T 2 with Fl = c0 + c1 v + c2 v 2 [N] the load force. If the driveline efficiency is modeled with a mechanical efficiency parameter ηm [-], the optimal torque during a deceleration is given by:  s
Fl  +M+λs v m ˙ f iv, iηm  Fl  , if T ∗ > 0,  ∂2 m ˙ f  iηm − 2 ∂T 2 ∗ s T = (3.64)
η F  m ˙ f iv, mi l +M+λs v  ηm Fl ∗   , if T < 0. ∂2 m ˙ f  i − 2 ∂T 2

The latter modeling approach thus can result in a torque jump around T = 0. Piecewise Affine Fuel Consumption Models

As mentioned before, using an affine fuel consumption model results in bangbang control. One way to avoid this, is using a quadratic fuel consumption model. Fröberg and Nielsen [73] avoid this by using a piecewise affine consumption model:  α01 ω + α11 ωT, if T 6 Ts , (3.65) m ˙f= α02 ω + α12 ωT, if T > Ts , with Ts [Nm] the switch torque. This model results in a solution where T ∗ = Ts for a while. Bang-bang control is avoided, but there is a special attraction to Ts . If this switching torque is not wisely chosen, i.e. not reflecting something physical in the engine, this results in unnatural behavior. Fuel Consumption Model for Negative Engine Torques Stoicescu [169] sets a lower torque bound at zero to model the fuel consumption. For a general polynomial fuel consumption model, this is equivalent to: X q (3.66) αk ω pk (max[0, T ]) k . m ˙f= k

This will exclude Tmin < T ∗ < 0 as a solution. Furthermore, this results in a new deceleration mode: T ∗ = 0 with an engaged engine. Yet, T = 0 is

SOLUTION WITH PONTRYAGIN’S MAXIMUM PRINCIPLE

95

irrelevant when it comes down to fuel consumption. The engine brake torque T is the difference between the internal torque and the engine friction torque. The internal torque is directly linked to the fuel consumption, not the engine brake torque. Therefore, T = 0 is an unnatural deceleration mode.

3.3.8

Conclusions on Pontryagin’s Maximum Principle

This section discussed how the minimum-fuel vehicle control problem can be solved with Pontryagin’s maximum principle. An explicit approach yields the following novel equation for the optimal engine control u∗e :

u∗e

v u u E(v) − E(vss ) + ε = ue,ss (v) ± t2 2 ∂ FE ∂u2 e

ue,ss (v) is the engine control to cruise at v and FE = m ˙ f /v is the fuel economy. E(v) = (m ˙ f,ss + M)/v is the economy. The equation holds for a vehicle with a manual transmission when engaged, and for a vehicle with an automatic gearbox during lockup. The steady state velocity vss and parameter ε should be determined to satisfy the end constraints. vss minimizes the economy and is related to M, which determines the total travel time te . Parameter ε is related to λs (the adjoint state variable of the distance): λs = −E(vss ) + ε, which determines the total travel distance se . When there is no direct time constraint, vss can be chosen by the driver. With a direct time constraint, vss should be determined iteratively. The same goes for a distance constraint. With no direct distance constraint, ε = 0. With a direct distance constraint, it should be determined iteratively. Gear shifting can easily be taken into account. During an acceleration, the optimal gear maximizes λv (the adjoint state variable of the velocity). During a deceleration, the optimal gear and clutch control minimize λv . If during a deceleration λv > 0, the brakes should be used. The presented methodology gives a simple and analytic solution. Without direct constraints, only a few function evaluations are needed to calculate the optimal vehicle controls. This allows for a fast and robust implementation in vehicles and can be used for rule-based eco-drive controls.

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THEORY OF ECO-DRIVING

3.4

Solution with Dynamic Programming

This section solves the eco-drive control problem with discrete dynamic programming (DP) [22]. The general method is explained in appendix E.2 on p. 223. Discrete DP is a powerful method that can easily handle difficult constraints, gear shifting, and complicated fuel consumption models. It is easy to understand and to implement. Dynamic programming can also be used in continuous time, leading to the Hamilton-Jacobi-Bellman equation. This will result in the same analytical solution as derived with Euler-Lagrange and Pontryagin’s maximum principle.

3.4.1

Discretized Control Problem

The minimum-fuel vehicle control problem is a continuous problem. Therefore, the continuous control problem of equation (3.1a) is discretized: min

N −1 X

l ∆t,

with l = m ˙ f + M.

(3.67)

i=0

The discretization in time is referred to as a time grid. One can also use a distance grid: min

N −1 X i=0

l ∆s,

with l =

m ˙ f +M . v

(3.68)

To solve this OCP with DP, some algorithmic choices should be made (see appendix E.2 on p. 223): • Time grid or distance grid? • Forward or backward DP? • Interpolation or inverse approach? • Which integration scheme? The algorithmic choice depends on the specific type of problem. Therefore, each problem (acceleration, deceleration, driving between stop signs, approaching traffic lights, and eco-cruise control) is discussed separately again.

SOLUTION WITH DYNAMIC PROGRAMMING

97

A priori, it is decided to use the inverse approach. Equation (3.1b) can easily be inverted to: dv iT − Fb = I + cv v + c0 + c1 v + c2 v 2 = Ft,c , (3.69) dt with Ft,c [N] the controlled part of the traction force at the wheels. For simplicity of notation: g = 0 when the engine is disengaged. Again for simplicity of notation, T is used instead of the general ue . With given values of v, dv/dt, and g the controls are calculated as follows: ( Ft,c and Fb = 0, if Ft,c > iTmin , T = (3.70) i T = Tmin and Fb = iTmin − Ft,c , if Ft,c < iTmin . The cost-to-go function J of equation (E.7) on p. 224 becomes: Ji (vi ) = min [l(vi , vi+1 , g) + Ji+1 (vi+1 )] . vi+1 , g

(3.71)

Note that the distance s is not used explicitly as a state variable. The discretized value of J depends on the discretization scheme and the grid type. Using e.g. a distance grid, the discretization is as follows: m ˙ f (¯ ω , T¯) + M l= ∆s, (3.72a) v¯ ω ¯ = i(gi ) · v¯, a ¯ = v¯

(3.72b)

vi+1 − vi , ∆s

(3.72c)

with ∆s = si+1 − si , a = dv/dt, and T¯ according to equation (3.70) with v¯ and a ¯. In a forward Euler approach: v¯ = vi and s¯ = si , backward Euler: v¯ = vi+1 and s¯ = si+1 , central difference: v¯ = (vi + vi+1 )/2 and s¯ = (si + si+1 )/2. The more complicated trapezoidal rule calculates l as follows: l=

m ˙ f,F +M vi

+ 2

m ˙ f,B +M vi+1

∆s,

(3.73)

with m ˙ f,F and m ˙ f,B the fuel mass flow rate according to forward and backward Euler method respectively. Also an important decision is the grid size. Assume a distance grid. The total distance is se . The choice of ∆s determines the amount of grid points N : se = N ∆s. The velocity should also be discretized: vmax − vmin = Nv ∆v. The complexity of the problem is O(N · Nv2 ). A trade-off between complexity and accuracy should be made. One could be tempted to use a fine grid for s and a rough grid for v. Keep in mind though that the possible transitions of v directly follow from this choice. This is illustrated in figure 3.12.

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      ((      (( ((( ( (   ( ( ( (  (  (  ∆s

6 ∆v ?

Figure 3.12: Illustration of the discretization in dynamic programming. The possible velocity transitions depend on ∆s and ∆v. For example only decreasing ∆s increases the lowest possible acceleration.

3.4.2

Eco-Cruise Control

First, eco-cruise control (ECC) is discussed. Discrete DP for ECC has been used for several years in the department of electrical engineering at the university of Linköping, Sweden. It is applied to heavy-duty trucks. In that application it is named predictive cruise control (PCC). The research was initiated by Hellström [85] and gradually improved by himself and coworkers [86–90, 102]. They use the basic backward algorithm on a distance grid with an inverse approach. As the road slope is a function of the distance, using a distance grid is the most convenient. Figure 3.13 compares the velocity profiles for the different integration schemes. The same basic problem as in figure 3.11 is solved, with ∆s = 50 m and ∆v = 0.5 km/h. The trapezoidal rule normally yields the most accurate result. Yet, it increases the calculation time by more than 50 %. Hellström [86], Hellström, Åslund, and Nielsen [88] show that in some cases a backward Euler approach can yield an unwanted oscillating velocity profile. Therefore the central discretization will be used in this dissertation. Hooker, Rose, and Roberts [92], and Monastyrsky and Golownykh [138] use forward dynamic programming with a distance grid and interpolation for ECC. Park et al. [144] use forward DP with a distance grid and the inverse approach. If the same integration scheme is used, backward and forward DP yield the same result. Computations can be sped up by searching the vi+1 and gi space in a smart way. First of all, only consider gi ∈ {gi+1 − 1, gi+1 , gi+1 + 1}. Second, start the search for vi+1 at vi . Gradually search for higher velocity, as soon as vi+1

SOLUTION WITH DYNAMIC PROGRAMMING

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is infeasible, all higher velocities will be too. Then search for lower velocities in the same way.

velocity v [km/h]

80

70 forward backward central trapezoidal

60 0

1

2

3

4

5

distance s [km]

Figure 3.13: Velocity profiles resulting from different integration schemes in DP for ECC over a simple road profile.

3.4.3

Accelerations

Grid Choice Vehicle accelerations with DP are only discussed in the literature by Hooker, Rose, and Roberts [92], and Monastyrsky and Golownykh [138]. The latter ones stick to a distance grid with forward DP. The problem here is, that at slow velocities, one distance increment takes more time compared to higher velocities. This can give problems for optimizing the gear shift timing. Using a fine grid could solve the problem, however this increases the calculation time. Monastyrsky and Golownykh [138] are aware of this problem, and therefore use ∆s = 10 m compared to ∆s = 20 m for ECC. Taking a look at an optimal acceleration (e.g. figure 3.3), it makes more sense to use a time grid. At low velocities, dv/ds is higher than at high velocities. To capture the low values at high velocities with a distance grid, ∆v should be small. With a large range of v for an acceleration, this results in a high value of Nv , requiring a lot of calculation time. When using a distance grid, the distance is directly constrained. Hooker, Rose, and Roberts [92] prefer a time grid with forward DP for accelerations. In order to obtain a solution that indirectly constraints the traveled distance, they use a transformation of the control problem. Consider the continuous problem with

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M = 0. When v reaches vss , a distance sa is traveled. Including the fuel consumption of the remaining part, cruising at vss until se , yields:  s Za m ˙ (v ) m ˙ f,ss ss f ds + (se − sa ) . (3.74) min  v vss 0

Including the second part in the integrand and using ds = v · dt yields: min

Zte  0

 v m ˙ f,ss (vss ) dt. m ˙f− vss

(3.75)

One can notice the appearance of λs = −m ˙ f,ss (vss )/vss , which shows a link between this approach for DP and PMP. Another option is to use an adaptive distance grid. An a priori assumed velocity profile can determine a variable grid ∆si such that each part takes a prescribed amount of time. Consider an acceleration from vs to vss , traveling the distance se . Assuming a linear acceleration and requiring ti+1 − ti = ∆t yields:   si si+1 = si + vs + (vss − vs ) ∆t. (3.76) se The solution of the DP algorithm can then be used to determine a new variable grid to solve the problem again for a better result. Figure 3.14 shows a comparison of accelerations from 0 to 70 km/h with different methods. A distance grid with ∆s = 12.5 m and a time grid with ∆t = 1 s (more or less the same grid size) both result in a velocity profile that is close to the one obtained with PMP. The adaptive method with ∆t = 1 s needs a second iteration before it gives an acceptable result. Trial and error shows that a time grid usually gives the best result. Backward or Forward DP So far, using backward or forward DP for an acceleration does not make a lot of difference. Remember that when using PMP, an acceleration calculation could be stopped as soon as vss is reached. With backward DP, the total time te or distance se should be taken big enough to allow to reach vss in a minimum-fuel way. Using forward DP, one could actually stop the calculation as soon as this happens (smart stop). This can save a lot of calculation time. Thus the grid length Nv or Nt is not determined a priori; it is increased in each step until: arg min [l(vss , vi−1 , ) + Ji−1 (vi−1 )] = vss . vi−1

(3.77)

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velocity v [km/h]

60

40 PMP distance grid time grid adaptive 1 iteration adaptive 2 iterations

20

0

0

200

400

600

800

1000

distance s [m]

Figure 3.14: different grid grid with ∆t with ∆t = 1 s

Velocity profiles of DP accelerations from 0 to 70 km/u with methods. Both a distance grid with ∆s = 12.5 m and a time = 1 s yield results comparable to PMP. The adaptive method needs two iterations to yield a good result.

If in the previous example a time grid is used with ∆t = 1 s, the normal algorithm needs about 200 s to calculate the velocity profile. Using the smart stop, it takes only 40 s. Using forward DP has another advantage. Suppose there is a combination of distance dependent and time dependent dynamics or constraints. When using backward DP, both time and distance should explicitly be used as variables in the algorithm, immensely increasing the calculation time. When using forward DP, e.g. with a time grid, the traveled distance from the start point can also be stored in a matrix and can then be used to determine the distance dependent dynamics or constraints. Gear Shift Dynamics Up till now, gear shifts where assumed to be instantaneous. In practice they are not. In an automated manual transmission (AMT), i.e. a robotized manual gearbox, it takes about 0.3 s to shift gears [21]. When performed manually, it takes longer. During this gear shift, the engine can not deliver torque to the wheels. Thus during an acceleration, these gear shift dynamics might have a significant influence. DP can easily take this into account. Following the previous analysis, forward DP with a time grid and a central difference is used. If a gear shift occurs, it is assumed to be at the beginning of

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the time increment ∆t. The engine delivers no torque during the shift duration ∆ts and the vehicle will slow down from vi to vi′ . This deceleration is calculated with forward Euler: vi′ = vi −

cv v + c0 + c1 v + c2 v 2 ∆ts . I

(3.78)

After the shift, the vehicle accelerates from vi′ to vi+1 during ∆t − ∆ts . Figure 3.15 shows that the vehicle accelerates slower when gear shift dynamics are taken into account. The example is an acceleration from 0 to 70 km/h with forward DP and a time grid ∆t = 1 s, a velocity discretization ∆v = 1 km/h, and a shift time ∆ts = 0.5 s. Multiple simulations with different ∆t and ∆v show that the influence of shift dynamics is usually rather small. Unfortunately, the solution depends strongly on ∆t and ∆v. Although DP can easily take gear shift dynamics into account, it does not seem to be a good method to study its influence. The mentioned gear shift dynamics refer to the longitudinal dynamics of the vehicle. The engine is still modeled quasistatically.

velocity v [km/h]

60

40

20 without shift dynamics with shift dynamics 0

0

100

200

300

400

500

distance s [m]

Figure 3.15: Comparison of DP accelerations from 0 to 70 km/u with and without gear shift dynamics. Forward DP is used with ∆t = 1 s, ∆v = 1 km/h, and a shift time ∆ts = 0.5 s. Gear shift dynamics result in a slower acceleration.

3.4.4

Decelerations

Decelerations from a steady state velocity as such do not appear in the literature. The different approaches that can be taken to solve the problem, are

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the same as for accelerations. To minimize the calculation time with a smart stop, one should use backward DP. In each step the grid is extended at the beginning. The calculations can be stopped as soon as: arg min [l(vss , v2 ) + J2 (v2 )] = vss .

(3.79)

v2

3.4.5

Driving Between Stop Signs

As the distance between two stop signs is fixed, using a distance grid is the most straightforward approach and is used by Hooker, Rose, and Roberts [92], and Monastyrsky and Golownykh [138]. The solution is a combination of an acceleration and a deceleration, thus it does not matter if one uses forward or backward DP. One could prefer to use a time grid to yield more realistic gear shift behavior. During the forward or backward iteration, the total traveled distance should be stored. If this distance becomes larger than the distance between the stop signs, an infinite cost should be given to the specific transition. Figure 3.16 shows that the grid type can significantly influence the resulting velocity profile. In this example backward DP is used with ∆t = 1 s, ∆s = 12.5 m, and ∆v = 1 km/h.

velocity v [km/h]

30

20

10 distance grid time grid 0

0

50

100

150

200

distance s [m]

Figure 3.16: Velocity profile of driving between stop signs with DP and different grid types. Backward DP is used with ∆t = 1 s, ∆s = 12.5 m, and ∆v = 1 km/h.

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3.4.6

Approaching Traffic Lights

The traffic lights problem with DP is not found in the literature. A smart algorithm is developed. The vehicle is cruising at vss at t = 0 and s = 0. Upcoming traffic lights at se are red between te,1 and te,2 . If se /vss < te,1 or se /vss > te,2 , the vehicle can continue to cruise at vss . If that is not the case, the vehicle should accelerate or decelerate to another cruising velocity to pass the traffic lights when they are green. Assume that a deceleration is the best solution. The optimal velocity profile can be calculated with the following forward DP algorithm. The algorithm is illustrated with a time grid. 1. Choose a grid size ∆t. Discretize the velocity, only considering velocities lower than vss : v = vmin , vmin + ∆v, . . . , vss . 2. Perform a forward iteration and store the traveled distance si . Only consider decelerations: vi < vi−1 . 3. At each step i, start the calculations at vss and work towards vmin . 4. If ti + (se − si )/vi > te,2 , the solution is found.

3.4.7

Conclusions on Dynamic Programming

This section discussed how the minimum-fuel vehicle control problem can be solved with discrete dynamic programming. The ease of implementation and constraint handling makes it a popular method, also for eco-driving calculations. Discrete DP requires the control problem to be discretized. Using a forward approach for an acceleration and a backward approach for a deceleration allows to stop the algorithm as soon as the steady state solution is reached. The choice of discretization has an important influence on the resulting velocity profile. Furthermore, DP is a slow method. This impedes its use in real-time applications.

SOLUTION WITH DIRECT MULTIPLE SHOOTING

3.5

105

Solution with Direct Multiple Shooting

The fourth solution method used in this dissertation is direct multiple shooting (DMS). The general method is explained in appendix E.5 on p. 230. The method is solely used for comparison with other solution methods in the next chapter. This dissertation gives no significant input in the development of a methodology to use DMS for eco-driving. Therefore, this is a short section. DMS for Eco-Driving in the Literature Applications of DMS for eco-driving are limited in the literature. Terwen, Back, and Krebs [172], and Latteman et al. [120] use DMS for ECC for heavy-duty trucks. Huang et al. [94] use another direct approach for ECC for heavy-duty trucks: collocation. Collocation is another method to transform the OCP into a NLP [25]. Gear Shifting The introduction of gear shifting and clutch control, leads to a mixed-integer control problem. There exist different methods to solve this type of problems, e.g. branch & bound, branch & cut, outer approximation, generalized Benders decomposition, rounding strategies, switching time optimization, and penalty term homotopy. A good overview of different methods is given by Sager [160]. Applied to vehicle gear shifting, the rounding strategy is used by Huang et al. [94]. The methodology is simple. First, the control problem is relaxed, i.e. the discrete control is allowed to be continuous. For gear shifting, i ∈ {i(gmax ), ..., i(2), i(1)} becomes i ∈ [i(gmax ), i(1)]. Then, the relaxed control problem is solved and the ratios are rounded to the nearest allowed discrete value. Rounding strategies typically yield solutions that are not optimal. Terwen, Back, and Krebs [172], and Gerdts [78] use a branch & bound method for gear shifting. Branch & bound performs a tree search in the space of the discrete control variables, with an optimal control problem on every node of the search tree. The root consists of the original problem with all control variables relaxed. All nodes in the tree are sons of the father node with an additional control variable fixed. This principle is repeated in every subtree. The full problem is partitioned recursively into small subproblems, based on the fact that the minimum of all solutions of these subproblems is identical to the optimal solution of the full problem. First, the relaxed control problem is solved. Then, one control variable is chosen to be branched, e.g. the gear ratio

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on the first interval of the time grid. A new subproblem is created for each gear. Each subproblem is solved and branched upon by making an extra gear ratio discrete again. When a top of the three is reached, i.e. all gear ratios are discretized, the value of the objective function is stored. There are three cases when a node (subproblem) is not branched on: 1) the relaxed solution is an allowed discrete solution, a feasible solution is found and the objective value is stored; 2) the problem is infeasible, all problems on the subtree will be infeasible too; 3) the objective value is higher than the current upper bound. Using branch & bound requires solving a lot of optimal control problems, which can make the method very slow. Logist et al. [128] use a new methodology for gear shifting. This methodology for solving mixed-integer optimal control problems is developed by Sager [160]. ACADO Toolkit Although it is possible to implement DMS in any programming environment, one can benefit from using specialized software. In this dissertation, the ACADO Toolkit is used. The ACADO Toolkit (acadotoolkit.org) is a software environment and algorithm collection for automatic control and dynamic optimization. It provides a general framework for using a great variety of algorithms for direct optimal control, including model predictive control, state and parameter estimation, and robust optimization. ACADO Toolkit is implemented as selfcontained C++ code and comes along with user-friendly Matlab interfaces. The object-oriented design allows for convenient coupling of existing optimization packages and for extending it with user-written optimization routines. ACADO was mainly developed by B. Houska and H. J. Ferreau, under the supervision of M. Diehl at the Optimization in Engineering Center (OPTEC) at the Katholieke Universiteit Leuven. In its current state, ACADO does not allow for discrete control (gear shifting and clutch control). Implementation Comparable to dynamic programming, the controls can be discretized on a time or distance grid. When a time grid is used without a direct distance constraint, an indirect distance constraint should be introduced as in equation (3.75).

APPLICATION TO OTHER TYPES OF VEHICLES

3.6

107

Application to Other Types of Vehicles

Thus far in this dissertation, optimal control is applied to passenger cars with a conventional powertrain. Although this is the main focus of the dissertation, the presented methods can also be applied to other types of vehicles. What follows is a short discussion on minimum-fuel control applied to: heavyduty trucks, passenger cars with a continuously variable transmission, electric vehicles, hybrid electric vehicles, and electric trains.

3.6.1

Heavy-Duty Trucks

In essence, there is no difference between a passenger car and a heavy-duty truck, thus the same calculation methods can be used. The main difference is the mass-to-power ratio and the number of gears. According to the American Association of State Highway and Transportation Officials (AASHTO) Green Book [2], the design standard for the mass-to-power ratio is 120 kg/kW, almost ten times larger than for a passenger car. Common North American trucks have up to 18 gears, in Europe up to 16.

3.6.2

Continuously Variable Transmission

A continuously variable transmission (CVT) replaces the gearbox in the powertrain. It can change steplessly through an infinite number of gear ratios i, between a minimum and a maximum value. A CVT can improve fuel economy by letting the engine operate at more efficient operating points. Using a continuous instead of a discrete control i, simplifies the control problem when using DMS. For DP, i should be discretized again and introduced in the OCP in the same way as for a normal gearbox. When using i as a continuous control with PMP, the optimal gear should also minimize the Hamiltonian: ∂H/∂i = 0. When using the explicit approach, both T and λv depend on the value of i. Furthermore, T is determined such that H = 0, making H independent of i. Considering a CVT as a gearbox where the amount of discrete gear ratios is infinite, one could use the λv -strategy: maximizing λv during an acceleration and minimizing it during a deceleration.

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3.6.3

Electric Vehicles and Trains

Figures 1.4 on p. 20 and figure 2.8 on p. 48 illustrate that the efficiency of an electric motor is similar to the efficiency of a combustion engine. Taking into account that an electric vehicle (EV) sometimes has no gearbox and usually does not have a clutch, makes minimum-fuel control for an EV simpler than for a conventional passenger car. A vehicle that is very like an EV, is an electric train. Optimal control of trains is a common subject in the literature. The dynamics of a train are very similar to that of a road vehicle, yet slightly more complicated. In a rail bend, the wheels of the train are pushed against the rail, increasing the influence of the curvature on the rolling friction. In a tunnel, the train has to push air out of the tunnel, increasing the aerodynamic drag. Yet just like the road slope, curves and tunnels are a function of the distance and can be treated as the road slope in ECC. The problem is complicated by the fact that the train can not be considered as a point mass because of its length. Assuming an even distribution of the mass over the length, the mass can simply be integrated and converted to a point mass. Optimal train control is in fact a well-defined control problem: the train should drive between two stations, starting and stopping at standstill. The train is not hindered by other trains and often there is a direct time constraint. In train control, the cost function is usually the tractive power. Using PMP, this results in bang-bang control, see e.g. [93, 110, 127]. Franke, Terwiesch, and Meyer [72] use DP, to be able to incorporate a more complex cost function that reflects the setpoint-dependent efficiency of the propulsion system.

3.6.4

Hybrid Electric Vehicles

Introduction A hybrid vehicle has more than one prime mover and power source. A hybrid electric vehicle (HEV) usually has an internal combustion engine (ICE) and an electric motor (EM). In what follows, the combustion engine will be referred to as engine, the electric motor as motor. The motor is a reversible prime mover that can also act as a generator. The electric energy is usually stored in batteries or supercapacitors. The addition of an electric motor to the powertrain yields the following advantages: • The engine can be downsized without sacrificing traction force.

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109

• During a deceleration, the motor can convert kinetic energy to electric energy. In a conventional powertrain, this energy would be converted to heat in the mechanical brakes. • At low velocities and standstill, the engine can be turned off to avoid inefficient operation. • The motor adds an extra degree of freedom that allows the engine to operate in a more fuel-efficient way. A hybrid vehicle can be up to 40 % more fuel-efficient than its conventional brother [74]. HEVs can be classified according to the configuration of the powertrain: • series HEV The powertrain contains two electric motors. One motor drives the vehicle. The electricity is provided by a battery and the other motor that only works as a generator, powered by the engine. There is only an electrical link between the engine and driving motor. • parallel HEV The powertrain is very similar to a conventional one. An electric motor is added to the axle of the engine, creating a mechanical link between them. This is a very basic and popular HEV configuration, adopted by e.g. the Honda Civic. One can also take a conventional front wheel drive vehicle and add a motor to the axle of the rear wheels. The mechanical connection between the motor and engine is through the road, a configuration preferred by PSA. • combined HEV This configuration combines an electrical and a mechanical link between motor and engine. It resembles a series HEV, with a coupling between the two motors. • power-split HEV This is a more complicated configuration that includes an engine, two motors, and a planetary gearbox. It is used in the most popular HEV: the Toyota Prius. • 2-mode HEV Developed by GM, Daimler-Chrysler, and BMW, this complex powertrain includes an engine, two motors, and 2 or 3 planetary gearboxes. It is used in e.g. the BMW X6 Active Hybrid.

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Literature Review Key issue in the fuel economy of an HEV, is the control of the cooperation between the engine and motor(s), called the power management. There is a vast amount of literature covering power management of hybrid vehicles, a considerable amount of them using optimal control. Sciaretta and Guzzella [165] provide a decent overview of the existing literature on power management. Virtually all this research is based on a given velocity trajectory. When both the velocity and the power management are optimized, the fuel savings potential is even larger, however the literature becomes extremely scarce. van Keulen et al. [178] try to exploit this fuel savings potential by first optimizing the velocity trajectory (based on bang-bang control) and then based on this trajectory, the power management of a hybrid truck. Optimality of the combined problem is not discussed. Kim, Manzie, and Sharma [111] do a better job and actually perform a simultaneous optimization of velocity and power management for a parallel HEV. Vehicle telemetrics provide a certain look ahead horizon based on an existing driving cycle. The velocity trajectory can deviate from this part of driving cycle within given bounds. The fuel consumption of the engine and efficiency of the motor are modeled with unspecified polynomials. Information on the method to solve the OCP is scarce, yet it seems that a direct approach is used. They find that with a traffic preview of 15 s, combined optimization saves 6.3 % of fuel compared to an HEV that follows the driving cycle and uses basic rule-based power management. Optimized power management without optimal velocity control saves 2.3 %. The authors admit that the computational burden for solving the nonlinear optimization is a major drawback, and they highlight a need for computationally efficient algorithms. Combined Optimization of Velocity and Power Management This last paragraph is a short discussion on how to perform combined optimal control of the velocity and power management of a parallel HEV. The modeling of this type of powertrain is discussed in section B.3 on p. 201. The cost function L of the OCP should include battery usage: L=m ˙ f + M + ξPb ,

(3.80)

with ξ [kg/J] the equivalence factor and Pb [W] the battery power, Pb > 0 when power is extracted from the battery. The equivalence factor ξ penalizes battery use. The regulatory standard SAE J1711 determines ξ = 0.075 kg/kWh.

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Smart methods determine ξ based on the average efficiencies of EM and battery, knowing that all energy comes from fuel [126]. Assume a hybrid powertrain with a relatively small EM, that is mainly used during acceleration and braking. While cruising, the motor is not used and thus M can still be determined according to equation (3.8). Instead of including the battery power in the cost function, one could also put hard constraints on the battery usage and e.g. require that the state of charge SOC [-] of the battery is the same at the end of the trip as in the beginning. A first possibility to solve the OCP is by using DP. The SOC is also a state variable. The cost-to-go function in a backward approach would then be as follows: Ji (vi , SOCi ) =

min

vi+1 ,SOCi+1 ,g

[l (vi , vi+1 , SOCi , SOCi+1 , g) + Ji+1 (vi+1 , SOCi+1 )] . (3.81)

Adding an extra state variable to the problem, makes the algorithm even slower and inappropriate for real-time implementation. A second possibility is to use PMP. When the engine is engaged, the Hamiltonian looks as follows: H

ds dSOC = L + λv dv dt + λs dt + λb dt 2 = m ˙ f + M + ξPb + λv i(Te +Tm )−Fb −cIv v−c0 −c1 v−c2 v + λs v − λb cb Pb ,

(3.82) with Te [Nm] the engine torque, Tm [Nm] the motor torque, λb [kg] the adjoint variable for the battery SOC, and cb a battery parameter. λs can still be determined by equation (3.42). When the influence of SOC on the internal battery parameters is neglected, dλb /dt = 0. Combining λb and ξ yields: H=m ˙ f +M+ξ ′ Pb +λv

i(Te + Tm ) − Fb − cv v − c0 − c1 v − c2 v 2 +λs v, (3.83) I

with ξ ′ = ξ − λb cb . The occurrence of the extra control Tm impedes the use of an explicit approach. The latter equation shows that if there is an affine relationship between Tm and Pb , the control of the EM will be bang-bang.

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Implementation in an Eco-Drive Assist System

When applying optimal control in an eco-drive assist system (EDAS), modelplant-mismatch will occur: the model of the vehicle is not perfect; information about the surroundings might be wrong (e.g. road slope), traffic might not behave as expected, the driver does not follow the driving advice, ... Thus implementing a complete velocity trajectory and accompanying gears (openloop control), might not give satisfactory results. This problem can be overcome by using model predictive control and parameter estimation.

3.7.1

Model Predictive Control

Model predictive control (MPC) is an advanced control technique that was developed in the sixties, see e.g. [131]. The main idea is the following. The states of the system are measured or estimated, and the optimal control problem is solved over a given prediction horizon. Only the first part of the solution is applied to the system. The next sampling time everything is repeated. Subsequently determining and solving a control problem inherently creates robustness against model-plant-mismatch and introduces feedback. MPC is often used for eco-driving in research and applications, e.g. [88, 89, 94, 95, 108, 115, 120, 145, 172]. When using DP, implementing MPC can be very straightforward. The cost-to-go function J is calculated off-line and stored in a matrix, together with the corresponding optimal controls u∗ . Suppose a distance grid is used for an ECC problem. For the whole trajectory, J is calculated as a function of the traveled distance s and velocity v. For each si and vi , the optimal gear gi and future velocity vi+1 is stored. While driving, the ECC system should only lookup the needed values in a matrix. This approach only works if the control problem (model and constraints) are fixed.

3.7.2

Parameter Estimation

The optimal control problem is based on a model of the vehicle longitudinal dynamics, see equation (3.1b) and (B.10) on p. 194. Important parameters such as the vehicle mass m, rolling friction fr , wind velocity vw , and road slope θ are not fixed and can even change during a trip. To improve the efficiency of an EDAS, these parameters could be estimated online. Johansson [107] and Kozica [117] use standard sensors (GPS, air pressure, and engine torque) to estimate the road slope with an extended Kalman filter and sensor fusion. Eriksson [63] implements an adaptive Kalman

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113

filter to estimate the vehicle mass, based on a multitude of vehicle sensors. Massel, Ding, and Arndt [136] estimate the road slope with the help of two accelerometers and wheel speed sensors. Bae, Ryu, and Gerdes [15] use the measurements of engine torque, road slope, vehicle velocity, and vehicle acceleration to estimate vehicle mass, drag coefficient, and rolling friction. Vahidi, Stefanopoulou, and Peng [175] propose a recursive least squares (RLS) algorithm with multiple exponential forgetting factors to estimate vehicle mass and road slope. RLS is also used by Fathy, Kang, and Stein [68] to estimate the vehicle mass. For the application in an EDAS, one does not need to know e.g. the rolling friction or wind velocity specifically, rather just the overall dynamics. This can be expressed by the following equation: dv = x1 T + x2 + x3 v + x4 v 2 , dt

(3.84)

with x the unknown parameters. Mertens and Vandekeybus [137] select RLS with multiple forgetting parameters as the best estimation algorithm for this problem. The velocity can be obtained through the on-board diagnostics (OBD) of the vehicle, or with a global positioning system (GPS). The acceleration can be obtained via a differentiation of v or with an accelerometer. In contrast to heavy-duty trucks, the engine torque T is not provided by the OBD. It can be calculated using the inverse of the fuel consumption model: m ˙ f = f (ω, T ) → T = f ′ (ω, m ˙ f ). The careful reader notices that the parameter problem is shifted. If a fuel consumption model is obtained through on-road measurements (as in appendix C.3 on p. 209) these parameters should be known. It is assumed that with a good estimate of the parameters and enough measurement data, the errors will cancel each other out. This hypothesis is confirmed by Clockaerts and Proost [41], see also section 2.3.1 on p. 49.

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THEORY OF ECO-DRIVING

Conclusions

This chapter discusses the minimum-fuel vehicle control problem and how it can be solved. The control problem is preferably basic; five different basic subproblems are: 1) acceleration to a steady state velocity, 2) deceleration, 3) driving between stop signs, 4) approaching traffic lights, and 5) eco-cruise control. A distinction is made between direct and indirect constraints. A direct time constraint fixes the total travel time te . An indirect time constraint influences the travel time indirectly by fixing the steady state velocity vss through the parameter M. A direct distance constraint fixes the total traveled distance se . With an indirect distance constraint, the fuel consumption per traveled distance is optimized. This is done using parameter λs . The control problem is based on a vehicle with a manual transmission. It is also valid for an automatic transmission during lockup. Pontryagin’s maximum principle allows for an analytic solution of the control problem and is probably the best way to derive knowledge based robust control strategies. A novel equation is derived that gives the optimal engine control as an explicit function of the current vehicle velocity. This equation is valid when using a quadratic polynomial fuel consumption model. Gear shifting and clutch control can easily be taken into account by evaluating λv , the adjoint state variable of the velocity. During an acceleration, the optimal gear maximizes λv . During a deceleration, the optimal gear and clutch control minimize λv . If λv > 0, the brakes should be used. If there is a direct time or distance constraint in the control problem, M and λs should be determined iteratively. This can be done through the more meaningful parameters vss and ε. Discrete dynamic programming can be used to solve more complex control problems or vehicle models. The dynamics should be discretized, which can have a significant influence on the solution. There are different solution approaches: forward or backward iteration; and time, distance or adaptive distance grid. The best approach depends on the control problem. Forward DP with a time grid for an acceleration and backward DP with a time grid for a deceleration. For these problems, a smart stop criterion can reduce the calculation time. A novel forward algorithm is introduced for approaching traffic lights. For eco-cruise control, a distance grid is the best option. The presented methods can also be used to calculate optimal control of vehicles with a continuously variable transmission, and electric vehicles and trains. To some extent also for hybrid electric vehicles. The next chapter analyzes the solution of the different control problems and compares the different solution methods: Pontryagin’s maximum principle, discrete dynamic programming, and direct multiple shooting. The latter

CONCLUSIONS

115

method is added solely for comparison. A part of the work on the maximum principle and dynamic programming is published in a chapter in the book “Automotive Model Predictive Control: Models, Methods and Applications”, titled “Optimal Control Using Pontryagin’s Maximum Principle and Dynamic Programming”. Earlier work using direct multiple shooting is published in Applied Energy, titled “Minimization of the Fuel Consumption of a Gasoline Engine Using Dynamic Optimization”. The latter work is also published in the proceedings of the IEEE conference on decision and control, under the title “Model Predictive Control of Automotive Powertrains - First Experimental Results”. A paper “Assessment of Eco-Cruise Control Calculation Methods” in the proceedings of the TRB annual meeting discusses different methods to solve the eco-cruise control problem. A more advanced paper on that subject is submitted to Transportation Research Part D, it is titled “Eco-Cruise Control for Passenger Vehicles: Methodology”.

Chapter 4

Application of the Eco-Drive Theory Having a fuel consumption model, an eco-drive control problem, and methods to solve it, this chapter can apply and assess the eco-drive theory. This comprises the assessment of different polynomial fuel consumption models, optimal control methods for efficient calculation of the optimal driving behavior, and the analysis of the optimal driving behavior. This chapter is split up according to the different eco-drive subproblems to allow for a more structured analysis. The first section takes a look at the steady state solution: the cruising velocity. The following sections discuss: accelerations, decelerations, driving between stop signs, approaching traffic lights, and eco-cruise control. Section 4.6 concludes this chapter. All the simulations are done based on the Toyota Corolla Verso of appendix C.1 on p. 203, a vehicle with a manual transmission. A distinction is made between the fuel consumption model for control and for evaluation. The control model is used for the calculation of the optimal control. In line with the conclusions of chapter 2, four different fuel consumption models are used and compared: two power-based polynomial models P4 and P5; a torque-based model T8; and a load-based model L3. Models P4 and P5 only have 3 parameters, which could be identified based on publicly available data. Models T8 and L3 have more

117

118

APPLICATION OF THE ECO-DRIVE THEORY

parameters and are more accurate. Their model equations are: P4: P5: T8: L3:

m ˙f m ˙f m ˙f m ˙f

= α1 + α2 P + α3 P 2 , = α1 ω + α2 P + α3 P 2 , = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωT + α5 ω 2 T + α6 ωT 2 , = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωτ + α5 ω 2 τ + α6 ωτ 2 .

The model for evaluation is used to calculate the fuel consumption. Here, this is a map-based model, an interpolation from the original measurements on the engine dynamometer. Because only one specific vehicle is used in the simulations, the simulation results can not be considered quantitatively representative with respect to specific values such as possible fuel savings, optimal velocities, shift speeds, ...

4.1

Steady State Cruising Velocity

The optimal steady state cruising velocity vopt [m/s], or short optimal velocity, is the cruising velocity that minimizes the fuel consumption per traveled distance, i.e. the steady state fuel economy FEss [kg/m]. Without any constraints, eco-driving implies cruising at this velocity. As mentioned already in section 3.1.2 on p. 60, values of vopt in the literature depend strongly on the source. Simply because they are vehicle specific. Figure 3.1 on p. 61 shows that vopt for the simulated Toyota Corolla Verso is at the lowest possible velocity in the highest gear. Except for fuel consumption model P4, all considered linear fuel consumption models of section 2.2.3 on p. 42 (P5, P6, T1, T4–T8, L2, L3) give the same result. In practice, the economy E = (m ˙ f + M)/v [kg/m], combining fuel consumption and an indirect time constraint, is more important than the fuel economy, see section 3.1.2 on p. 60. M is determined to make a predefined steady state velocity vss the velocity that minimizes the economy. From a fuel consumption point of view, it is unwise to make M negative, or vss < vopt . The problem is that vopt is based on a model, and does not necessarily equal the real value. The real value could be anywhere between the lowest possible velocity in highest gear (around 30 km/h) and 90 km/h. Thus if vopt is uncertain, M should be determined to match the speed limit if the latter one is between 30 and 90 km/h. Above 90 km/h, one should keep in mind that driving slower than the speed limit is probably better. Using very low engine speeds, close to the idle speed, is not always possible as it can cause engine lugging. Thus driving 30 km/h in the highest gear might not be realistic. The fuel consumption model can have an influence on the economy E, which is important for the optimal control of the vehicle. The influence can be seen

STEADY STATE CRUISING VELOCITY

119

relative economy E − Emin [g/km]

in equation (3.43) on p. 80. Figures 4.1–4.3 show the relative economy for different polynomial fuel consumption models for vss = 50, 70, and 90 km/h, three common speed limits in Europe. Models P4, P5, T8, and L3 are shown. These models yield similar results, yet the relative economy E −Emin of model P5 is lower. This results in milder acceleration and deceleration behavior. Overall, the choice of model has little impact on the computed economy around the chosen vss . P4 P5 T8 L3

30

20

10

0

40

60

80

100

120

velocity v [km/h]

relative economy E − Emin [g/km]

Figure 4.1: Relative economy for different polynomial fuel consumption models with vss = 50 km/h.

P4 P5 T8 L3

30

20

10

0

40

60

80

100

120

velocity v [km/h]

Figure 4.2: Relative economy for different polynomial fuel consumption models with vss = 70 km/h.

APPLICATION OF THE ECO-DRIVE THEORY

relative economy E − Emin [g/km]

120

P4 P5 T8 L3

30

20

10

0

40

60

80

100

120

velocity v [km/h]

Figure 4.3: Relative economy for different polynomial fuel consumption models with vss = 90 km/h. The choice of the steady state velocity is an aspect of the driving behavior that probably has the largest influence on the fuel consumption. Table 4.1 shows the relative fuel consumption mf and travel time t for different steady state velocities. At higher velocities, the lower travel time does not compensate for the extra amount of fuel consumption. Lowering the speed limit on highways is a very effective measure to reduce fuel consumption and consequently polluting emissions, e.g. NOx . Also the gear is of major importance. Table 4.2 shows the steady state fuel economy for different gears and different velocities. Table 4.1: Influence of the steady state velocity vss on the fuel consumption mf and travel time t. Lower travel times do not compensate for the extra fuel consumption. vss mf t [km/h] [%] [%] 70 80 90 100 110 120

100 109 120 133 149 166

100 88 78 70 64 59

STEADY STATE CRUISING VELOCITY

121

Table 4.2: Influence of the gear on the steady state fuel economy FEss . vss g FEss [km/h] [-] [%] 50

5 4 3

100 106 122

70

5 4 3

100 107 134

90

5 4 3

100 111 155

122

APPLICATION OF THE ECO-DRIVE THEORY

4.2

Accelerations

This section discusses optimal accelerations. To start, the solution method and fuel consumption model are assessed. Then, human acceleration behavior is studied. This is compared with the optimal acceleration behavior. Last, accelerations with a continuously variable transmission are discussed.

4.2.1

Assessment of the Calculation Method

To start, both a solution method and a fuel consumption method are chosen to use in the rest of this section. Solution Method The assessment is based on: accuracy, ease of implementation, and calculation time. Three different solution methods are considered: 1. Pontryagin’s maximum principle (PMP), see section 3.3.2 on p. 77. 2. Forward discrete dynamic programming (DP) with a time grid and a smart stop, see section 3.4.3 on p. 99. ∆t = 1 s and ∆v = 0.5 km/h. 3. Direct multiple shooting (DMS) with a time grid, see section 3.5 on p. 105. ∆t = 1 s. An example comparison is made between the three solution methods for an acceleration from 35 to 70 km/h over a total distance of 1200 m in the highest gear. Fuel consumption model P4 is used. Figure 4.4 shows the resulting velocity profiles. PMP and DMS yield virtually the same profile, while the discretization is clearly visible in the DP profile. The resulting fuel consumption for DP is about 1 % more. Thus from the point of view of the result, DP is outperformed by PMP and DMS. Another important measure is the calculation time. It must be said that this strongly depends on the software, hardware, and parameters of the algorithm. In this particular case, PMP and DMS each need a few seconds to find the solution. It takes DP more than a half a minute. Other accelerations yield similar results. Taking into account that gear shifting can easily be performed with PMP, it is found to be the best solution method for optimal accelerations. Therefore, only PMP is used in the remainder of this section.

ACCELERATIONS

123

velocity v [km/h]

70

60

50 PMP DP DMS

40 0

200

400

600

800

1000

1200

distance s [m]

Figure 4.4: Comparison of optimal accelerations from 35 to 70 km/h over 1200 m with different solution methods. PMP and DMS yield the same result, while the discretization is clearly visible in the DP velocity profile. Fuel Consumption Model Table 4.3 shows the resulting fuel consumption mf and total cost J = mf +M·te for different fuel consumption models and vss . The accelerations start from standstill. The difference between the different models is quite small. However, there is a clear trend that the more accurate the model, the lower J. The most accurate model, L3, is chosen as the best model and used in the remainder of this section. Figure 4.5 shows the resulting velocity profile for an acceleration to 90 km/h.

4.2.2

Human Acceleration Behavior

In order to do a decent assessment of optimal acceleration behavior, it is important to know what human (typical) acceleration behavior is. Snare [168] reviews extensive studies, some of them rather old, on the acceleration behavior of drivers. He reports a lack of recent measurement data and the focus being mainly on the maximum acceleration rate. In normal traffic situations, drivers almost never use the maximum acceleration capability of the vehicle. While maximum accelerations depend mainly on vehicle characteristics, typical accelerations also depend on human factors. Early researchers conclude that a typical acceleration rate is a fixed percentage of the maximum acceleration rate [1, 130]. Long [129] analyzes old measurement data and recommends linear

124

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.3: Simulation results of optimal accelerations with different polynomial fuel consumption models. vss se mf te J model [km/h] [m] [g] [s] [g] 50

70

90

1600

P4 P5 T8 L3

68.5 68.4 67.4 66.2

122.4 123.2 125.2 131.6

70.0 69.9 68.9 67.8

1700

P4 P5 T8 L3

85.4 86.4 85.4 85.1

99.8 98.1 99.3 99.5

132.4 132.5 132.1 132.0

2000

P4 P5 T8 L3

124.3 125.9 126.5 125.8

93.1 92.2 91.4 91.9

292.5 292.4 291.8 291.9

velocity v [km/h]

80 60 40

P4 P5 T8 L3

20 0

0

500

1000

1500

2000

distance s [m]

Figure 4.5: Comparison of optimal accelerations to 90 km/h over 2000 m with different fuel consumption models.

ACCELERATIONS

125

decay to model human acceleration behavior: a=

dv = α − βv, dt

(4.1)

with a [m/s2 ] the acceleration, and α [m/s2 ] and β [1/s] two parameters. Long [129] recommends α = 2 m/s2 and β = 0.12 1/s for an average acceleration of a passenger car. More recent measurements of human accelerations are conducted by Akçelik and Biggs [9], Searle [166], Snare [168], and Wang et al. [187]. Akçelik and Biggs [9] collected second-by-second velocity-time data using the chase-car method, in urban, suburban, and rural road conditions. They find typical S-shaped velocity profiles that yield a polynomial model. Searle [166] measured accelerating vehicles at an intersection with a radar gun. The drivers were not aware of their participation in the experiment. It is observed that a human acceleration is similar to a maximum acceleration, although less aggressive. Unfortunately, a mathematical expression for the acceleration is not formulated. Snare [168] conducted an acceleration experiment in 2002 at the Virginia Tech Transportation Institute. The test involved accelerating a vehicle from a stop over a distance of approximately 1100 ft (335 m) at a driver’s normal acceleration rate. The test was performed on a relatively flat and straight stretch of roadway controlled by a stop sign and subject to very light traffic volumes that would not interfere with each run. The goal was to simulate the typical acceleration profile of a lead vehicle at a stop line. Twenty different drivers volunteered for the test. The test drivers were random volunteers chosen to reflect the driving population on the road today. Twenty drivers overall were chosen, including eleven men and nine women. The drivers ranged in age from 21 to 45. The test vehicle used in the test was a 1999 Ford Crown Victoria, which had been equipped with a GPS unit that collected velocity data for the vehicle every second. The same vehicle was used for each driver, in order to serve as the control for the experiment. The drivers were each allowed to drive the vehicle for a while to get familiar with the car before starting the testing. The drivers were aware that they were being tested, and were told to accelerate at their normal rate until the end of the test section. To account for variability within the driver, up to twenty-five runs were conducted. From these runs, velocity, time, and distance measurements were recorded. Snare developed a modification of the Rakha model [154] for predicting human acceleration behavior of drivers. This modification applies a reduction factor ξ [-] to the maximum possible acceleration. The average reduction factor is found to be ξ = 0.6. Snare mentions that the linear decay model gives a rather good fit, except at low velocities. It seems from the presented data, that if the data at very low velocities is discarded, the linear decay model can give a better fit.

126

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.4 shows the main statistics of this linear curve fit. The table lists the α and β parameters for each driver. The error margins reflect the 95 % confidence intervals for the two parameters. R2 is the coefficient of determination of the linear decay fit. The average of R2 over the different drivers is 0.84, which is a fairly high value. The distribution of β across the different drivers is almost flat and ranges between 0.041 and 0.147 1/s, with an average value of 0.090 1/s. The first parameter α ranges between 1.13 and 3.42 m/s2 , with an average value of 2.61 m/s2 . Figure 4.6 shows the result for driver 7. Table 4.4: Summary of experimental acceleration data and curve fit results. age te se ve α β R2 # gender [y] [s] [m] [m/s] [m/s2 ] [1/s] [-] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

M M M M M M M M M M M F F F F F F F F F

23 23 27 29 31 31 33 34 35 38 45 21 22 22 22 23 23 24 30 25

21 19 20 20 23 20 20 20 20 23 26 21 21 22 17 23 25 23 20 31

335.2 335.0 315.9 334.6 337.9 302.5 350.6 326.3 311.6 345.3 355.2 333.2 336.4 348.0 318.2 330.3 333.1 321.9 347.1 343.5

26.07 28.99 23.87 24.30 25.62 22.35 29.00 26.09 23.92 25.70 23.11 27.62 28.56 26.25 31.96 23.73 23.12 21.23 29.81 20.68

2.82 3.04 2.90 3.42 2.18 3.41 2.68 2.99 2.93 2.03 1.80 2.66 2.62 2.64 3.06 2.46 1.99 2.56 2.89 1.13

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.07 0.06 0.05 0.10 0.05 0.08 0.05 0.08 0.08 0.05 0.04 0.08 0.05 0.06 0.09 0.08 0.06 0.06 0.08 0.04

0.100 0.085 0.113 0.135 0.070 0.147 0.073 0.105 0.114 0.062 0.068 0.084 0.074 0.093 0.068 0.100 0.076 0.115 0.083 0.041

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.004 0.003 0.003 0.006 0.003 0.005 0.003 0.004 0.005 0.003 0.003 0.005 0.003 0.004 0.004 0.005 0.004 0.004 0.004 0.003

0.82 0.86 0.94 0.88 0.80 0.95 0.86 0.87 0.90 0.76 0.79 0.81 0.85 0.83 0.77 0.87 0.82 0.91 0.84 0.62

Wang et al. [187] collected data of equipped vehicles that were driving in the Atlanta urban area. Only accelerations at an all-way stop sign to a velocity of minimum 32 km/h were taken into account. This way, they have data of 574 accelerations with 76 drivers to built a model. The best fit for the data is given by a nonlinear model: 2

a = (α − βv) ,

(4.2)

ACCELERATIONS

127

acceleration a [m/s2 ]

4 experimental linear decay

3

2

1

0

0

20

40

60

80

100

velocity v [km/h]

Figure 4.6: Linear decay versus experimental data for a human acceleration of driver 7. with α = 1.381 m1/2 /s and β = 0.0396 m−1/2 for straight accelerations. The linear decay model of equation (4.1) is also fitted to the data, with α = 1.883 m/s2 and β = 0.0756 1/s. Although the quadratic model gives the best fit, the difference with the linear decay model is rather small. Other findings are that there is no significant difference between a straight and a turning acceleration, and that the speed limit has no influence on the acceleration behavior. Glauz, Harwood, and John [80] show that the initial acceleration, at v = 0, is less than the maximum attained acceleration. Akçelik and Biggs [9], and Bham and Benekohal [24] indicate an acceleration rate equal to zero at the beginning and end of the acceleration. This implies that the acceleration behavior is dependent on the end velocity and this is opposed to the findings of Wang et al. [187]. Furthermore, Bham and Benekohal [24] compare several acceleration models to experimental data and find the closest fit for models that force the acceleration rate to decrease with velocity, as opposed to models that relate the acceleration rate to time, either in a linear, polynomial, or sinusoidal way. They conclude that velocity is the preferred parameter in validation of acceleration models. Concluding the cited acceleration papers and findings, most models show a good comparison with a linearly decaying acceleration, except for the acceleration model by Akçelik and Biggs [9]. Thus, it can be concluded that a linear decay model is adequate to represent the range of accelerations during the main course of the acceleration event. The reported variations in measured decay rates are due to the differences in driving style that drivers exhibit. However, the linear

128

APPLICATION OF THE ECO-DRIVE THEORY

trend between acceleration rate and velocity seems to be preserved between these different driving styles. An adaptation towards lower accelerations at very low velocities might be desirable. However, this initial acceleration and jerk, which is linked to driving comfort, is minorly contributing to the integrated fuel consumption of the acceleration event.

4.2.3

Analysis of Optimal Accelerations

Figure 4.7 shows an optimal acceleration to 70 km/h. One can observe linear decay in both acceleration and torque. Gear shifting occurs at relatively low engine speeds, around 2000 rpm. What follows is a more detailed study of the engine control, gear shifting, and the linear decay. Engine Control Five different types of acceleration engine control are compared: 1. optimal acceleration, 2. wide-open-throttle (WOT) acceleration, thus at maximum engine torque, 3. an acceleration that follows the economy line, 4. human acceleration at 60 % of the possible acceleration, 5. human acceleration according to a linear decay law with α = 2.6 m/s2 and β = 0.09 1/s. In order to only capture the influence of the torque control, gear shifting is not optimized; upshifts are performed at n = 2000 rpm. Cruising at vss is always in the highest gear. Table 4.5 shows the results of the different types of accelerations to 50, 70, and 90 km/h. The optimal accelerations minimize the total cost J. This is also the case for the fuel consumption for accelerations to 50 and 70 km/h. An acceleration that follows the economy line and a 60 % human acceleration result in a lower fuel consumption than the optimal acceleration for an acceleration to 90 km/h. At this high steady state velocity, the influence of the indirect time constraint is rather high and is reflected in the acceleration time te . Fuel savings are in the order of magnitude of several percent.

129

acceleration a [m/s2 ]

ACCELERATIONS

2

1

0

0

10

20

30

40

50

60

70

velocity v [km/h]

engine torque T [Nm]

120

100

80

60

40

0

10

20

30

40

50

60

70

50

60

70

engine speed n [rpm]

velocity v [km/h]

2000

1000

0

0

10

20

30

40

velocity v [km/h]

Figure 4.7: Optimal acceleration to 70 km/h: acceleration, engine torque, and engine speed as a function of the velocity.

130

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.5: Analysis of possible fuel savings with an optimal acceleration. vss se mf te J model [km/h] [m] [g] [%] [s] [g] [%]

50

70

90

250

optimal WOT economy line 60 % linear decay

19.1 19.8 19.2 19.2 19.8

100.0 103.3 100.2 100.5 103.3

25.6 22.3 23.7 25.1 22.3

19.5 20.1 19.5 19.5 20.1

100.0 103.1 100.0 100.4 103.1

1000

optimal WOT economy line 60 % linear decay

56.6 59.8 57.5 57.3 59.8

100.0 105.6 101.5 101.2 105.6

62.8 58.6 61.5 63.4 58.7

86.1 87.4 86.4 87.1 87.4

100.0 101.4 100.3 101.1 101.4

1100

optimal WOT economy line 60 % linear decay

79.2 83.3 76.5 76.5 83.3

100.0 105.2 96.6 96.6 105.2

56.8 55.1 61.0 62.5 55.1

181.8 182.8 186.7 189.4 182.9

100.0 100.5 102.7 104.2 100.6

torque T [Nm]

100

50

θր 0 0

10

20

30

40

50

60

70

velocity v [km/h]

Figure 4.8: Engine torque for an optimal acceleration to 70 km/h on road slopes between −2 and 2◦ . Figure 4.8 shows the influence of the road slope on the optimal torque. An increasing slope θ results in an increasing engine torque. The linear trend is visible for all slopes.

ACCELERATIONS

131

Gear Shifting Three different types of gear shift strategies are compared: 1. optimal gear shifting that maximizes λv , 2. gear shifting at a given engine speed n, 3. gear shifting that minimizes the brake specific fuel consumption BSFC. Cruising at vss is always in the highest gear. Table 4.6 shows the effect of the different gear shift strategies for accelerations to 50, 70, and 90 km/h. Maximizing λv (optimal gear shifting) minimizes the total cost function J. This is also the case for the fuel consumption for accelerations to 50 km/h. Gear shifting at low engine speeds or according to the minimum BSFC can result in a lower fuel consumption than maximizing λv for an acceleration to 90 km/h. Fuel savings are in the order of magnitude of several percent. Table 4.7 shows the engine speeds at which the optimal shifts occur. The shift speeds increase with increasing vss . Table 4.8 shows that the optimal shift speeds increase with the road slope. Very low shift speeds should possibly be avoided to prevent engine lugging. Normal drivers shift gears at about 50 % of the maximum engine speed [19]. This is at about 3000 rpm with a gasoline engine and about 2500 rpm with a diesel engine. Comparison with Linear Decay As mentioned before, one can observe a linear relation between acceleration and velocity, and engine torque and velocity in figure 4.7. This is in fact a property of all optimal accelerations calculated with a quadratic fuel consumption model. Section 4.2.2 shows that human accelerations also follow the linear decay law. Thus, another advantage of quadratic fuel consumption models is found. Optimal accelerations resemble human accelerations and will be more easily adopted by drivers than acceleration strategies that do not feel natural. Linear decay is also observed in car following behavior [139, 188]. Moreover, when safety in driving is applied as the underlying strategy to determine the optimal velocity control, the outcome is an almost linear relationship between acceleration and velocity [183]. These linear relationships apparently require a small effort for learning how to drive, even in busy traffic. When a linear decay model is fitted to the optimal accelerations and requiring that a(vss ) = 0, the following models are obtained:

132

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.6: Analysis of possible fuel savings with optimal gear shifting. vss se mf te J shift method [km/h] [m] [g] [%] [s] [g] [%]

50

70

90

1000

max λv @ 1500 rpm @ 2000 rpm @ 2500 rpm @ 3000 rpm @ 3500 rpm min BSFC

44.6 45.1 46.5 47.7 48.5 49.7 47.7

100.0 101.2 104.2 107.0 108.8 111.3 106.9

87.4 84.7 79.6 78.5 78.0 77.7 78.5

45.7 46.2 47.5 48.7 49.5 50.6 48.6

100.0 101.1 103.9 106.6 108.3 110.8 106.5

1000

max λv @ 1500 rpm @ 2000 rpm @ 2500 rpm @ 3000 rpm @ 3500 rpm min BSFC

56.4 56.1 56.6 59.4 61.2 63.2 59.1

100.0 99.5 100.4 105.3 108.6 112.0 104.7

63.0 64.8 62.8 60.3 59.3 58.7 60.6

86.1 86.6 86.1 87.7 89.1 90.8 87.6

100.0 100.7 100.1 101.9 103.6 105.5 101.8

1200

max λv @ 1500 rpm @ 2000 rpm @ 2500 rpm @ 3000 rpm @ 3500 rpm min BSFC

85.5 83.2 84.3 85.9 90.0 95.8 84.8

100.0 97.4 98.6 100.5 105.3 112.1 99.2

59.6 63.1 60.8 59.4 58.0 56.8 60.3

193.2 197.3 194.1 193.2 194.8 198.4 193.7

100.0 102.1 100.5 100.0 100.8 102.7 100.3

• acceleration to 50 km/h: a = 1.25 − 0.090 · v, • acceleration to 70 km/h: a = 1.93 − 0.099 · v, • acceleration to 90 km/h: a = 2.21 − 0.088 · v. There is a striking similarity between the values of the second parameter β and the values observed in traffic. When β is approximated by 0.1 1/s, the following simple optimal acceleration law is yielded: a∗ = 0.1 · (ve − v).

(4.3)

Table 4.9 shows that using this simple law for an acceleration yields a significant difference with an optimal acceleration. It does give better results than a human acceleration at 60 % or a WOT acceleration.

ACCELERATIONS

133

Table 4.7: Optimal gear shift engine speeds for different vss . vss n shift [km/h] [rpm] 50

1→2 2→3 3→4 4→5

1450 1608 1488 1176

70

1→2 2→3 3→4 4→5

1819 2052 2027 1871

90

1→2 2→3 3→4 4→5

2329 2524 2434 2266

Comparison with the Literature Optimal accelerations with a strong acceleration at the start, a decreasing acceleration rate, and gear shifting at low engine speeds are also found by Schwarzkopf and Leipnik [164], Hooker, Rose, and Roberts [92], and Monastyrsky and Golownykh [138]. In practical applications, strong accelerations are usually not considered a part of eco-driving. Methods to evaluate the driving behavior often consider strong acceleration as a negative action, e.g. Ericsson [61]. Larsson and Ericsson [119] try to lower the fuel consumption by giving resistance in the accelerator pedal when the driver tries to accelerate rapidly. A test shows no reduction in fuel consumption. Qian and Chung [149] assume that eco-driving equals moderate accelerations. These applications could be improved by the knowledge that an optimal acceleration follows a linear decay law and starts with a rather strong acceleration.

134

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.8: Optimal gear shift engine speeds for different road slopes, with vss = 70 km/h. θ [◦ ]

shift

n [rpm]

−2

1→2 2→3 3→4 4→5

1422 1296 1203 1099

−1

1→2 2→3 3→4 4→5

1583 1792 1707 1480

0

1→2 2→3 3→4 4→5

1819 2052 2027 1871

1

1→2 2→3 3→4 4→5

2079 2211 2217 2093

2

1→2 2→3 3→4 4→5

2200 2374 2344 2247

ACCELERATIONS

135

Table 4.9: Evaluation of a linear optimal acceleration approximation. vss se mf te J method [km/h] [m] [g] [%] [s] [g] [%] 50

70

90

1000

optimal a = 0.1(ve − v) 60 % WOT

44.6 45.6 45.8 46.7

100.0 102.3 102.6 104.7

87.4 81.5 80.6 77.0

45.7 46.6 46.7 47.6

100.0 102.1 102.4 104.3

1000

optimal a = 0.1(ve − v) 60 % WOT

56.4 57.5 57.3 59.7

100.0 101.9 101.7 105.8

63.0 61.1 63.3 58.8

86.1 86.2 87.1 87.3

100.0 100.2 101.3 101.5

1200

optimal a = 0.1(ve − v) 60 % WOT

85.5 87.0 84.6 89.7

100.0 101.8 99.0 105.0

59.6 58.8 63.7 57.8

193.2 193.2 199.7 194.2

100.0 100.0 103.4 100.6

136

APPLICATION OF THE ECO-DRIVE THEORY

4.2.4

Continuously Variable Transmission

There exist different strategies to control a continuously variable transmission (CVT) during an acceleration [146]: • speed envelope strategy This strategy is based on two curves in the engine speed versus vehicle velocity plane. One for 0 % accelerator pedal position, and one for 100 %. The actual position then determines a point in between these two curves. Given the pedal position and the vehicle velocity, the engine speed and thus gear ratio can be determined. • single track strategy This strategy controls the CVT such that the engine is operated at the economy line [76]. The position of the accelerator pedal is interpreted as a power demand, which results in an operating point on the economy line. • off the beaten track strategy This strategy is equivalent to the previous one. Here, the line to be tracked is altered from the economy line to yield better driveability of the vehicle. Pfiffner and Guzzella [146] remark that these strategies are based on heuristics and thus suboptimal. Even the single track (economy line) strategy is suboptimal because it neglects the efficiency of the CVT. They use the direct approach to minimize the fuel consumption during an acceleration while tracking a given velocity profile. The optimal solution is shown to be above the economy line. In the framework of eco-driving, the velocity profile is not given. This adds an extra degree of freedom to the control problem. Assuming that there are no losses in the CVT, one could assume that the optimal strategy is to operate the engine in the overall most efficient point, the point of minimum BSFC. Consider the Toyota Corolla with an ideal CVT that has no losses. The gear ratios are continuous between the values of the first and fifth gear of the manual gearbox. Figure 4.9 shows a comparison between two accelerations to 70 km/h: one that minimizes the BSFC and one that maximizes λv . The acceleration that minimizes the BSFC (economy acceleration) stays in the point of minimum BSFC as long as possible. At low and high velocities, the acceleration should leave this point because of the limitations on the gear ratio. Here, the acceleration follows the economy line. The economy acceleration thus follows the economy line with increasing engine speed and stays as long as possible in the point of minimum BSFC. The optimal acceleration moves along

ACCELERATIONS

137

the economy line with decreasing engine speeds and does not stay in the point of minimum BSFC. The transitions to and from the economy line are done with maximum and minimum gear ratio. Although the optimal acceleration is quite different from the intuitive one that minimizes the BSFC, the fuel saving is only 3 % in this example.

4.2.5

Conclusions on Accelerations

Pontryagin’s maximum principle is the most efficient method to calculate optimal accelerations. It gives fast and accurate results, and can easily take into account gear shifting. An optimal acceleration rate is high at low velocities and decreases for higher velocities. This linear decay behavior corresponds with regular human driving behavior. Optimal gear shifting occurs at relatively low engine speeds. The shift speed increases with road slope and steady state velocity. An optimal acceleration with a CVT follows the economy line with decreasing engine speeds. This is opposed to the intuitive solution that minimizes the BSFC. An optimal acceleration minimizes the total cost J (fuel consumption and travel time). It is thus possible that it does not result in the lowest possible fuel consumption. The effect is larger for accelerations to higher velocities, where the influence of the indirect time constraint is larger. A remark should be made on automatic gearboxes. The implemented control strategy shifts at higher engine speeds when the accelerator pedal is pushed more. Thus at low velocities, this impedes an optimal acceleration.

138

APPLICATION OF THE ECO-DRIVE THEORY

engine torque T [Nm]

120

100

80

60

40 1000

1200

1400

1600

1800

2000

2200

2400

engine speed n [rpm]

engine speed n [rpm]

2000

1500

economy optimal

1000 10

20

30

40

50

60

70

velocity v [km/h]

engine torque T [Nm]

120

100

80

60

40 10

20

30

40

50

60

70

velocity v [km/h]

Figure 4.9: CVT accelerations to 70 km/h. The intuitive economy acceleration minimizes the BSFC and moves along the economy line with increasing engine speed. An optimal acceleration that maximizes λv moves along the economy line with decreasing engine speed.

DECELERATIONS

4.3

139

Decelerations

This section discusses optimal decelerations from a steady state velocity.

4.3.1

Assessment of the Calculation Method

Solution Method Figure 3.7 on p. 87 shows that the optimal gear shift behavior can be rather complicated during an optimal deceleration. Since the gear shifting is essential, only PMP and DP are considered in the evaluation of the solution method. A comparison is done with an example deceleration from 70 km/h over a distance of 600 m, with Fb,max = 1 kN. The backward DP algorithm uses a smart stop and a time grid with ∆t = 1 s and ∆v = 1 km/h. Figure 4.10 shows the resulting velocity profiles. The profiles are similar, yet a clear difference is visible. PMP requires 2 s for the calculation of the velocity profile, DP 1 minute. The previous example uses an indirect distance constraint; the distance to decelerate is long enough and the deceleration is preceded by cruising. When a direct constraint is used, e.g. se = 300 m, the calculation time increases to 20 s for PMP. There is no significant difference for DP. This still makes PMP faster and more accurate than DP. As for accelerations, PMP is the preferred calculation method for decelerations and is used in the rest of this section.

velocity v [km/h]

60

40

20 PMP DP 0

0

100

200

300

400

500

600

distance s [m]

Figure 4.10: Velocity profile of an optimal deceleration from 70 km/h with PMP and DP.

140

APPLICATION OF THE ECO-DRIVE THEORY

Fuel Consumption Model Table 4.10 shows the resulting fuel consumption and total cost for different fuel consumption models and vss , with an indirect time constraint. Overall, model L3 yields the minimum value for J. The difference with models P5 and T8 is rather small. Model P4 results in a significant amount of extra fuel consumption. The reason is that this model can not decently capture gear shift behavior and does not use coasting during the deceleration. It is also beneficial to use the same model for accelerations and decelerations. Therefore, model L3 will be used in the rest of this section. Table 4.10: Simulation results of optimal decelerations from 70 km/h. vss [km/h] 50

70

90

4.3.2

se [m]

model

mf [g]

te [s]

J [g]

400

P4 P5 T8 L3

15.9 10.1 10.1 9.9

40.0 42.9 40.6 41.0

16.3 10.6 10.5 10.4

1200

P4 P5 T8 L3

52.1 35.2 37.6 37.4

78.6 80.2 76.6 75.6

89.1 72.9 73.6 73.0

1600

P4 P5 T8 L3

66.9 53.4 59.0 59.6

81.4 82.6 79.5 78.8

213.9 202.7 202.7 202.0

Analysis of Optimal Decelerations

As mentioned before, optimal decelerations can be complicated. There are two factors that have a large influence on the deceleration: fuel cut-off and brakes. The simulated decelerations are with an indirect time constraint. Fuel Cut-Off Figure 4.11 shows the influence of the fuel cut-off on a deceleration from 70 km/h with no brakes. Three cases are compared: normal fuel cut-off above 3000 rpm (as measured on the engine dynamometer); no fuel cut-off; and always

DECELERATIONS

141

fuel cut-off. In each of these cases, the deceleration starts with coasting (g = 0). This is also observed in decelerations from other vss . More fuel is consumed while coasting compared to engine braking. However, the deceleration is less strong and therefore the deceleration can start sooner. Consequently, the initial cruising part is shorter, consuming less fuel during the initial cruising. At lower velocities, engine braking becomes more interesting. Recall the definition of λv : m ˙ f + M + λs v . (4.4) dv − dt During a deceleration, the optimal gear and clutch control minimizes λv . Usually, λs is negative. At high velocities, the numerator of the latter equation is negative and the deceleration should be as low as possible; coasting is the optimal solution. At low velocities, the numerator is positive and the deceleration should be as high as possible; engine braking is the optimal solution. λv =

With no fuel cut-off, the deceleration starts sooner. With always fuel-cut off, the engine braking starts sooner. Here, engine braking is more interesting since there is no fuel consumption. With normal fuel cut-off, the engine braking part is almost the same as without fuel cut-off. There is an extra part of engine braking in second gear, that results in high engine speeds to have no fuel consumption. For each case, the gear sequence is different, yet never straightforward. Brakes Figure 4.12 shows simulation results of a deceleration from 70 km/h with different brake forces. The brake force changes the gear sequence. Braking only occurs at the end of the deceleration, in two phases and at maximum brake force. When the maximum braking force increases, the braking period decreases. Table 4.11 shows that an increase of braking force, decreases the fuel consumption. This is due to the very last part of the deceleration, where the engine is disengaged to prevent stall. Coasting at very low speeds is bad for both the fuel consumption and the travel time. If this last part is discarded from the deceleration (70 → 8 km/h), increasing the braking force increases the fuel consumption, yet decreases the total cost. This is also shown in table 4.11. Table 4.12 shows more simulation results for decelerations from different vss to 8 km/h. Increasing the brake force decreases the total cost. For the deceleration from 50 km/h this also decreases the fuel consumption.

142

APPLICATION OF THE ECO-DRIVE THEORY

velocity v [km/h]

60

40

normal no fuel cut-off always fuel cut-off

20

0

0

100

200

300

400

500

600

700

500

600

700

distance s [m] 6

gear g [-]

4

2

0 0

100

200

300

400

distance s [m]

engine speed n [rpm]

4000

3000

2000

1000 0

100

200

300

400

500

600

700

distance s [m]

Figure 4.11: Influence of fuel cut-off on an optimal deceleration from 70 km/h with no brakes.

DECELERATIONS

143

velocity v [km/h]

60

40

Fb = 0 kN Fb = 1 kN Fb = 10 kN

20

0

0

100

200

300

400

500

600

400

500

600

400

500

600

distance s [m]

gear g [-]

4

2

0 0

100

200

300 distance s [m]

brake force Fb [kN]

10

5

0 0

100

200

300 distance s [m]

Figure 4.12: Influence of the maximum brake force Fb,max on an optimal deceleration from 70 km/h.

144

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.11: Simulation results for a deceleration from 70 km/h over 600 m with different brake forces. ve Fb,max mf te J [km/h] [kN] [g] [%] [s] [g] [%] 0

0 1 10

15.6 12.6 12.9

100.0 80.7 82.4

73.8 44.7 37.5

50.3 33.7 30.5

100.0 66.9 60.6

8

0 1 10

10.6 12.1 12.8

100.0 113.6 120.3

48.9 42.1 37.2

33.7 31.9 30.3

100.0 94.7 90.0

Table 4.12: Simulation results for decelerations to 8 km/h with different brake forces. vss se Fb,max mf te J [km/h] [m] [kN] [g] [%] [s] [g] [%]

50

70

90

400

0 2 4 6 8 10

9.5 9.2 9.1 9.1 9.1 9.1

100.0 96.8 95.9 95.4 95.4 95.2

43.7 37.0 36.0 35.6 35.5 35.4

10.1 9.7 9.6 9.5 9.5 9.5

100.0 96.2 95.2 94.7 94.6 94.4

600

0 2 4 6 8 10

10.6 12.4 12.6 12.6 12.8 12.8

100.0 116.7 118.3 118.5 120.0 120.3

48.9 39.9 38.3 37.8 37.3 37.2

33.7 31.2 30.6 30.4 30.3 30.3

100.0 92.7 91.0 90.3 90.1 90.0

1100

0 2 4 6 8 10

28.2 35.4 36.7 37.2 37.5 37.7

100.0 125.4 130.1 131.8 132.9 133.6

65.4 52.9 50.5 49.5 49.0 48.7

146.5 131.0 128.0 126.6 126.0 125.7

100.0 89.5 87.3 86.4 86.0 85.8

DECELERATIONS

145

Distance The previous examples use an indirect distance constraint; the deceleration distance is long enough. When this distance is shorter, the deceleration should happen faster and there is a direct distance constraint. Figure 4.13 shows the influence of the distance se on the optimal deceleration. When the deceleration distance decreases, coasting is used less and lower gears are preferred. A lower gear results in a higher engine speed and more friction.

velocity v [km/h]

60

40

20 se = 600 m se = 400 m 0

0

100

200

300

400

500

600

400

500

600

distance s [m] 6

gear g [-]

4

2

0 0

100

200

300 distance s [m]

Figure 4.13: Influence of the distance on an optimal deceleration from 70 km/h to 8 km/h.

146

APPLICATION OF THE ECO-DRIVE THEORY

4.3.3

Possible Fuel Savings

Three different deceleration strategies are compared to assess possible fuel consumption savings: 1. optimal deceleration, 2. coasting deceleration, thus with a disengaged engine, 3. engine braking, with downshifts at a given engine speed. The deceleration stops at 8 km/h and the brakes are not used. Table 4.13 shows the simulation results for these different strategies. For the decelerations from 70 and 90 km/h, coasting results in the lowest fuel consumption. This is in line with the observation that an optimal deceleration uses coasting at higher velocities. The optimal deceleration always minimizes the total cost J. Fuel consumption savings compared to engine braking are around 10 %. Table 4.13: Simulation results for a deceleration with different strategies. vss se mf te J method [km/h] [m] [g] [%] [s] [g] [%]

50

70

90

650

optimal coasting @ 1500 rpm @ 2000 rpm @ 2500 rpm

18.7 21.2 20.2 20.8 20.4

100.0 113.8 108.5 111.4 109.2

61.7 87.5 58.8 57.5 56.8

19.4 22.3 20.9 21.5 21.1

100.0 114.9 107.9 110.7 108.6

1000

optimal coasting @ 1500 rpm @ 2000 rpm @ 2500 rpm

27.2 26.6 29.7 31.2 31.1

100.0 97.8 109.2 114.6 114.6

59.5 107.7 70.6 68.4 66.6

59.9 77.2 62.9 63.3 62.5

100.0 129.0 105.0 105.7 104.3

1600

optimal coasting @ 1500 rpm @ 2000 rpm @ 2500 rpm

53.5 44.3 54.9 56.9 57.9

100.0 82.8 102.6 106.3 108.3

85.6 132.7 89.9 87.0 84.4

208.1 284.1 217.3 214.1 210.5

100.0 136.5 104.4 102.9 101.1

Dornieden et al. [50] compare coasting and engine braking to normal (measured) driving behavior. While engine braking, the engine is always in fuel cut-off

DECELERATIONS

147

mode. A deceleration from 60 km/h over a distance is considered. Coasting increases both the fuel consumption and the travel time. Engine braking decreases the fuel consumption and increases the travel time.

4.3.4

Conclusions on Decelerations

Pontryagin’s maximum principle is the most efficient method to calculate optimal decelerations. It gives fast and accurate results and can easily take into account gear shifting and clutch control. An optimal deceleration usually starts with coasting. A complex gear sequence can occur and is influenced by fuel cut-off and the brake force. The use of brakes at low velocities can actually decrease the fuel consumption.

148

4.4 4.4.1

APPLICATION OF THE ECO-DRIVE THEORY

Stop Signs and Traffic Lights Driving Between Stop Signs

The velocity profile between two stop signs is simply a combination of an acceleration and a deceleration, both without direct distance constraint. Therefore the conclusions on accelerations and decelerations are also valid and this subproblem is not discussed any further.

4.4.2

Approaching Traffic Lights

A comparison is made between PMP and DP. Figure 4.14 shows the velocity profiles for a vehicle approaching traffic lights. The vehicle is driving 70 km/h at 500 m of the lights. The lights turn green in 30 s. There is a significant difference between the velocity profiles for PMP and DP (∆s = 1 s and ∆v = 0.5 km/h). This is due to the discretization in DP; it does not only fix the possible deceleration rates, yet also the possible cruise velocities to approach the lights. A better discretization (finer grid) would improve the DP solution, yet increase the calculation time. PMP results in a fuel consumption of 13.8 g compared to 15.7 g for DP. Each method takes about 20 s of calculation time. There is a direct time constraint. Thus for PMP, vss needs to be determined with an optimization. On the other hand, the novel DP algorithm for approaching traffic lights (see section 3.4.6 on p. 104) is faster than a regular DP algorithm. PMP, mf = 13.8 g DP, mf = 15.7 g

velocity v [km/h]

70

65

60

55

0

100

200

300

400

500

distance s [m]

Figure 4.14: Velocity profiles for approaching traffic lights in 30 s with DP and PMP.

STOP SIGNS AND TRAFFIC LIGHTS

149

To assess the possible fuel savings, it is important to also take into account what happens after the traffic lights are passed. An example is given where the vehicle is driving 70 km/h at 1 km from the traffic lights. The lights should be passed at 80 s. Two cases are considered: optimally approaching the traffic lights, and approaching the lights with an optimal deceleration. In the latter case, the start of decelerating is timed in order to satisfy the time constraint. When the vehicle passes the lights, it should accelerate again to 70 km/h. Figure 4.15 shows the resulting velocity profiles. Using an optimal approach, the vehicle cruises at about 40 km/h to approach the lights. Using an optimal deceleration, the vehicle almost comes to a complete stop at the lights. Over the total length, the optimal approach consumes 73.9 g of fuel, the optimal deceleration 85.1 g (+15 %). optimal approach mf = 73.9g optimal deceleration mf = 85.1 g

velocity v [km/h]

60

40

20

0

0

500

1000

1500

2000

distance s [m]

Figure 4.15: Velocity profiles for approaching traffic lights in 80 s with an optimal approach and an optimal deceleration. PMP is found to be a better method as it yields more accurate results. The latter example shows that there is an important difference between a regular deceleration and a deceleration for approaching traffic lights. The distinction originates from the time constraint: indirect or direct.

150

4.5

APPLICATION OF THE ECO-DRIVE THEORY

Eco-Cruise Control

This section discusses eco-cruise control (ECC). First, some definitions and assumptions are given. Then, different fuel consumption models, cost functions, and solution methods are compared.

4.5.1

Definitions and Assumptions

An eco-cruise control system allows the vehicle to drive in a given velocity band: vmin 6 v 6 vmax . Knowledge of the road slope can be used to save fuel. On a level road, the vehicle is expected to cruise at the reference velocity vref . On a hilly road, the velocity will vary, but the average velocity v¯ is preferably close to vref . A conventional cruise control system (CCC) controls the engine such that it always cruises at vref . In what follows, the velocity band is vref ± 10 km/h, unless specified otherwise. When using discrete dynamic programming: ∆s = 50 m and ∆v = 0.5 km/h. This is a decent trade-off between accuracy and calculation time. When using direct multiple shooting, the distance is discretized with ∆s = 50 m. For simplicity, all simulations are performed in one gear, the highest one. An important difference is made between steep and gentle slopes. On a gentle uphill slope (incline), the vehicle can keep its velocity. On a gentle downhill slope (decline), the vehicle can keep its velocity without using the brakes. On a steep incline, the vehicle will decelerate even with maximum engine torque. On a steep decline, the vehicle can only keep its velocity when it uses the brakes. The difference between gentle and steep depends on the gear and the velocity. When used for ECC, vmin is used for inclines and vmax for declines. Figure 4.16 shows the bounds between steep and gentle slope for the Toyota Corolla Verso. Up to 100 km/h, road slopes between 3 and −3◦ (5 %) are gentle. In the first analysis of ECC, only gentle slopes are considered. As an example road profile, a basic profile is used in the rest of the section. It consists of 5 sections: level, uphill, level, downhill, and level. Unless specified otherwise, each section is 1 km long. It will be referred to as the basic profile.

4.5.2

Fuel Consumption Model

Eco-cruise control is partly based on the property that the optimal steady state velocity vss is a function of the road slope. Some polynomial fuel consumption models can not capture this dependency and lack a possibility to save fuel.

ECO-CRUISE CONTROL

151

uphill bound downhill bound

20 road slope θ [o ]

gր 10

0

−10 20

40

60

80

100

120

velocity v [km/h]

Figure 4.16: Bound between gentle and steep slopes for different gears. Up to 100 km/h, ±3◦ (5 %) grades are gentle in all gears. Consider a torque-based polynomial fuel consumption model and assume that cos(θ) = 1. The dependency of vss with θ can be determined as follows: ∂2E ∂v∂θ

=

∂c0 αk ipk −qk ∂θ h i · (pk − 1)v pk −2 qk Flqk −1 + v pk −1 (c1 + 2c2 v) (qk − 1)Flqk −2 .

P

k

(4.5)

Thus if ∀k : qk = 0 or pq = 1, then vss is independent of θ. This is the case for affine fuel consumption models. As an example, figure 4.17 shows vss as a function of the road slope with vref = 90 km/h for different polynomial fuel consumption models. The two basic power-based models P4 and P5, the linear torque-based model T8, and the linear load-based model L3 are compared. The power-based models result in a linear function. Figure 4.18 shows the resulting velocity profiles for the different consumption models over a the basic road profile with slopes of 2◦ and vref = 90 km/h. The optimal velocity is calculated with DP. Table 4.14 shows that the resulting fuel consumption for the different models is comparable. The table also shows the average velocity v¯ and a comparison with vref = 70 km/h. Models with a slightly higher fuel consumption also have a higher average velocity. A first conclusion is that the difference between the fuel consumption models

APPLICATION OF THE ECO-DRIVE THEORY

steady state velocity vss [km/h]

152

P4 P5 T8 L3

100

90

80 −2

0

−1

1

2

road slope θ [o ]

Figure 4.17: The optimal steady state velocity vss as a function of the road slope, for vref = 90 km/h and different fuel consumption models.

velocity v [km/h]

100

90 P4 P5 T8 L3

80 0

1

2

3

4

5

distance s [km]

Figure 4.18: Velocity profiles for different fuel consumption models over a basic road profile, with vref = 90 km/h.

ECO-CRUISE CONTROL

153

Table 4.14: Fuel consumption for different consumption models over a basic road profile. Model L3 yields a higher fuel consumption, mainly because of a higher average velocity. vref mf v¯ model [km/h] [g] [km/h] 90

P4 P5 T8 L3

254.08 254.99 254.96 255.64

89.18 89.66 89.66 89.75

70

P4 P5 T8 L3

210.09 209.76 210.48 211.06

69.24 69.70 69.72 70.06

is rather small. To further investigate this, a simulation over a more realistic road profile is performed. A 45 km section of Interstate 81 from Roanoke to Blacksburg in the state of Virginia is used, see figure 4.19. The section has slopes between −4 and 4 % (2.3◦ ). The speed limit is 104 km/h (65 mph). An eco-cruise controller is used with vref = 94 km/h, thus the maximum velocity is 104 km/h. Table 4.15 shows that ECC results in a fuel consumption of only 85 % compared to cruising at vref . The difference between the fuel consumption models is small.

altitude h [m]

600

500

400

300 0

10

20

30

40

distance s [km]

Figure 4.19: Altitude profile of a 45 km section of I81 from Roanoke to Blacksburg in Virginia, USA.

154

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.15: Evaluation of ECC over a section of I81 with different fuel consumption models, for vref = 94 km/h. All models result in a fuel consumption of about 85 % compared to cruising at vref (CCC). model CCC P4 P5 T8 L3

[g]

mf

3038.9 2581.3 2589.8 2599.5 2586.0

[%]

v¯ [km/h]

100.0 84.9 85.2 85.5 85.1

94.00 91.65 92.19 92.52 91.88

From the simulations it can be concluded that a simple model, such as the power-based model P4, is appropriate for use in ECC calculations. Only this model is used in the rest of the section on ECC. An extra validation is done on an engine dynamometer. For more information on the engine dyno, see appendix C.1 on p. 203. A simple road profile is considered: 1 km uphill, 1 km downhill, 1 km level. Two velocity profiles with vref = 70 km/h are compared: conventional cruise control and eco-cruise control. The ECC velocity profile is calculated with a reactive PMP algorithm, see section 3.3.6 on p. 91. The velocity profiles are tracked with a PI-controller that controls the throttle valve of the engine. Figure 4.20 shows the result for a 1◦ hill. CCC results in a consumption of 164.3 g and a travel time of 205.6 s. ECC results in a consumption of 162.1 g (−1.3 %) and a travel time of 205.5 s (−0.05 %). Figure 4.21 shows the result for a 2◦ hill. CCC results in a consumption of 172.1 g and a travel time of 205.5 s. ECC results in a consumption of 167.0 g (−3.0 %) and a travel time of 205.4 s (−0.06 %).

4.5.3

Cost Function

For now, only a cost function that includes the fuel consumption and an indirect time constraint is considered. The indirect time constraint M ensures that the average velocity is close to reference velocity. Some researchers use another method to do this: penalizing a deviation from vref . Latteman et al. [120] use the following cost function: L=m ˙ f + φ · (vref − v)2 ,

(4.6)

ECO-CRUISE CONTROL

155

velocity v [km/h]

74

CCC ECC

72 70 68 66 0

500

1000

1500

2000

2500

3000

distance s [m]

Figure 4.20: Velocity profiles of CCC and ECC on a 1◦ hill on an engine dynamometer.

velocity v [km/h]

74

CCC ECC

72 70 68 66 0

500

1000

1500

2000

2500

3000

distance s [m]

Figure 4.21: Velocity profiles of CCC and ECC on a 2◦ hill on an engine dynamometer.

156

APPLICATION OF THE ECO-DRIVE THEORY

φ = 0 kg·s/m2 , M = 0 kg/s φ = 4 · 10−6 kg·s/m2 , M = 0 kg/s

velocity v [km/h]

70

60

50

0

1

2

3

4

5

distance s [m]

Figure 4.22: Velocity profile of ECC with different cost functions, vref = vopt . A deviation penalty keeps the velocity closer to the reference velocity. with φ [kg·s/m2 ] a velocity deviation penalization factor. Hellström, Fröberg, and Nielsen [89] penalize only a negative deviation from the reference velocity:  1, if x > 0, 2 (4.7) L=m ˙ f + φ · (vref − v) · κ(vref − v), κ(x) = 0, if x 6 0. Consider the following cost function that includes fuel consumption m ˙ f, a deviation penalty φ, and an indirect time constraint M: L=m ˙ f + φ · (vref − v)2 + M.

(4.8)

First, the case is considered where vref = vopt = 63 km/h, i.e. optimal for model P4. Figure 4.22 shows a comparison of two velocity profiles (φ = 0 kg·s/m2 and φ = 4 · 10−6 kg·s/m2 ). The optimal velocity profile is calculated with DP for the basic road profile with 3◦ slopes. It is clear that when the deviation from the reference velocity is too large, a deviation penalty can be used. Numerical results are given in table 4.16. Next, the case where vopt > vref = 50 km/h is considered. Figure 4.23 shows a comparison of three cost functions. When no deviation penalty and no time constraint are used, the velocities are relatively high. With a deviation penalty, the velocities will be lower, but are on average still significantly higher than vref . With an indirect time constraint, the velocities will vary around vref . Figure 4.24 shows a comparable case with vopt < vref = 80 km/h.

ECO-CRUISE CONTROL

157

Table 4.16: Simulation results of ECC with different cost functions. vref [km/h] φ [kg·s/m2 ] M [g/s] mf [g] v¯ [km/h] 63

0 · 10−6 4 · 10−6

0 0

222.21 223.34

62.10 62.57

50

0 · 10−6 0 · 10−6 4 · 10−6

−0.1477 0 0

233.04 227.20 228.66

48.66 55.40 53.82

80

0 · 10−6 0 · 10−6 4 · 10−6

0.5428 0 0

235.71 229.68 230.75

79.01 73.07 74.72

φ = 0 kg·s/m2 , M = 0 kg/s φ = 4 · 10−6 kg·s/m2 , M = 0 kg/s φ = 0 kg·s/m2 , M = −0.147 · 10−3 kg/s

velocity v [km/h]

60

50

40 0

1

2

3

4

5

distance s [m]

Figure 4.23: Velocity profiles of ECC with different cost functions, vref < vopt . Without an indirect time constraint, the velocities are relatively high.

158

APPLICATION OF THE ECO-DRIVE THEORY

φ = 0 kg·s/m2 , M = 0 kg/s φ = 4 · 10−6 kg·s/m2 , M = 0 kg/s φ = 0 kg·s/m2 , M = 0.543 · 10−3 kg/s

velocity v [km/h]

90

80

70 0

1

2

3

4

5

distance s [m]

Figure 4.24: Velocity profiles of ECC with different cost functions, vref > vopt . Without an indirect time constraint, the velocities are relatively low. For the case where vref = vopt , having no deviation penalty yields the lowest fuel consumption. If vref 6= vopt , no deviation penalty and no time constraint yield the lowest fuel consumption. No deviation penalty and a time constraint yields the highest fuel consumption. One could advocate that the best cost function to use is the first one. Yet, this one results in an average velocity v¯ that differs the most from the reference velocity vref , and the gain in fuel economy mainly comes from the fact that the average velocity is closer to the optimal velocity. This is illustrated quantitatively on the I81 section. If the section is driven at a constant velocity v = 94 km/h, the fuel consumption is mf = 3038.9 g. Eco-Cruise control with an indirect time constraint on this section with vref = 94 km/h, vmin = 84 km/h and vmax = 104 km/h results in mf = 2581.3 g (−15.1 %) and v¯ = 91.7 km/h. Without a time constraint: mf = 2417.4 g (−20.5 %) and v¯ = 84.8 km/h. Here, most of the fuel savings is a result of the lower velocity, knowing that driving at a constant velocity v = 84 km/h yields a consumption of mf = 2420.2 g. If the driver does not mind driving at a lower average velocity, it would be better to set the cruise controller to vref = 84 km/h, vmin = 74 km/h, and vmax = 94 km/h. These settings result in mf = 2344.5 g (−22.9 %) and v¯ = 82.0 km/h. The conclusion is that an indirect time constraint should be used to have an average velocity that is close to the reference velocity. Yet, keep in mind that a reference velocity closer to the optimal velocity will result in a better fuel

ECO-CRUISE CONTROL

159

economy. If deviations from the reference velocity are considered to be too large, an additional deviation penalty can be used or the allowed velocity band can be narrowed.

4.5.4

Solution Method

Now, the solution method for the optimal control problem is assessed. Following methods are compared: • CCC Perfect conventional cruise control that keeps the vehicle at a constant velocity, the reference case. • RECC Reactive eco-cruise control, calculated with the maximum principle, see section 3.3.6 on p. 91. • PECC-DP Predictive eco-cruise control, calculated with discrete dynamic programming, see section 3.4.2 on p. 98. • PECC-DMS Predictive eco-cruise control, calculated with direct multiple shooting, see section 3.5 on p. 105. • PECC-PMP Predictive eco-cruise control, calculated with the maximum principle, see section 3.3.6 on p. 91. An important distinction between reactive and predictive methods is used. Reactive ECC does not use knowledge of the upcoming road slope. Consider again the basic road profile, yet with variable section length. No velocity constraints are applied and vref = 70 km/h. Figure 4.25 shows a comparison of the velocity profiles for different methods with 3 km sections, figures 4.26 and 4.27 with 2 and 3 km sections respectively. The following observations can be made: • The predictive ECC anticipates a change in road slope. When the road slope increases and vss (θ) decreases, the change will be anticipated with an increase in velocity. The opposite is true for a decrease in road slope.

160

APPLICATION OF THE ECO-DRIVE THEORY

velocity v [km/h]

90

80

70 RECC PECC-DP PECC-DMS PECC-PMP

60

50

0

2

4

6

8

10

12

14

distance s [m]

Figure 4.25: Velocity profiles of ECC with different solution methods, se = 15 km. • PECC-DP and PECC-DMS always yield velocity profiles that are almost the same. They both suffer from the discretization in the method. The DP profile is a combination of straight lines, while the DMS profiles has small wrinkles. • The smaller the section lengths, the larger the differences between the PECC-DMS and the PECC-PMP velocity profiles. PECC, calculated with PMP only gives the correct solution if the road sections with a constant road grade are long enough such that the new optimal steady state velocity can be reached. This assumption becomes less valid with decreasing section lengths. Table 4.17 shows the numerical results. The different methods always yield more or less the same fuel savings. Further evaluation of the different methods can be done based on computation time and ease of implementation. RECC then comes out as the best method. DP is easy to understand and implement, yet it needs a lot of computation time. DMS is rather complicated and prefers specialized software. PECC-PMP is more difficult and slower then its reactive brother.

ECO-CRUISE CONTROL

161

velocity v [km/h]

90

80

70 RECC PECC-DP PECC-DMS PECC-PMP

60

50

0

2

4

6

8

10

distance s [m]

Figure 4.26: Velocity profiles of ECC with different solution methods, se = 10 km.

velocity v [km/h]

90

80

70 RECC PECC-DP PECC-DMS PECC-PMP

60

50

0

1

2

3

4

5

distance s [m]

Figure 4.27: Velocity profiles of ECC with different solution methods, se = 5 km.

162

APPLICATION OF THE ECO-DRIVE THEORY

Table 4.17: Simulation results of ECC with different solution methods. se mf v¯ method [km] [g] [%] [km/h]

15

CCC RECC PECC-DP PECC-DMS PECC-PMP

665.26 653.34 653.78 653.65 652.81

100.0 98.2 98.3 98.3 98.1

70.00 69.61 69.69 69.75 69.50

10

CCC RECC PECC-DP PECC-DMS PECC-PMP

443.51 435.05 435.41 435.48 434.41

100.0 98.1 98.2 98.2 98.0

70.00 69.60 69.71 69.92 69.38

5

CCC RECC PECC-DP PECC-DMS PECC-PMP

221.75 217.09 216.76 216.61 216.91

100.0 97.9 97.4 97.7 97.8

70.00 69.54 69.61 69.64 69.58

ECO-CRUISE CONTROL

4.5.5

163

Eco-Cruise Control on Steep Slopes

So far, the analysis was based on gentle slopes. When driving on steep slopes, prediction of the future road slope becomes more important. On a steep uphill slope, the minimum velocity can not be kept. Accelerating before the slope might be necessary to stay within the velocity bounds. Figure 4.28 shows the velocity profiles when driving a 500 m steep uphill slope, with vref = 70 km/h. RECC does not anticipate the slope at all and the velocity will drop below vmin at the end of the slope. PECC-PMP neither can keep the velocity within the bounds. The algorithm does not take into account the length of the uphill slope and does not really predict the future velocity. If the velocity needs to be kept within the bounds, DP and DMS do a better job.

velocity v [km/h]

80

RECC PECC-PMP PECC-DP PECC-DMS vmin ,vmax

70

60

0

0.5

1

1.5

2

2.5

distance s [km]

Figure 4.28: Velocity profiles of ECC on a steep slope (4◦ ) with different solution methods. The methods based on PMP can not keep the velocity within the bounds. On steep slopes it is also possible to use affine fuel consumption models, see e.g. [36, 89, 93]. Steep slopes are more relevant for heavy-duty trucks than for passenger cars. As mentioned in section 3.6.1 on p. 107, trucks have a massto-power ratio that is ten times higher. Steep slopes for trucks occur at much smaller slopes.

164

4.5.6

APPLICATION OF THE ECO-DRIVE THEORY

Conclusions on Eco-Cruise Control

Eco-Cruise Control can save fuel by varying the velocity over hilly roads. Again, PMP comes out as the best method to calculate the velocity profile. A simple power-based quadratic fuel consumption model is sufficient. Different cost functions are found in the literature. The combination of fuel consumption and an indirect time constraint performs the best. When the vehicle drives over gentle slopes, a quadratic fuel consumption model is necessary to enable fuel savings. Reactive ECC performs almost as good as predictive ECC. When driving over steep slopes, linear consumption models also work. Predictive ECC needs to be used. Reactive ECC has been validated on an engine dynamometer.

CONCLUSIONS

4.6

165

Conclusions

This chapter assesses the eco-drive theory of the previous chapter. Polynomial fuel consumption models and optimal control methods are evaluated, and the optimal driving behavior is analyzed. Pontryagin’s maximum principle is found to be the best method. It gives accurate results and can easily include gear shifting. When only indirect time and distance constraints are used, it is the fastest method. Discrete dynamic programming is also a powerful method that can easily handle different ecodrive problems. Unfortunately, the solution is always strongly influenced by the chosen discretization of the state variables. It is also the slowest method. Direct multiple shooting can also provide fast and accurate results. The disadvantage is that gear shifting is difficult to implement and this method prefers specialized software. Pontryagin’s maximum principle only provides a necessary, and not a sufficient condition for optimality, unlike dynamic programming. It is found that the maximum principle and dynamic programming always find the same solution, within the possible accuracy. Therefore it is concluded that for eco-drive control problems, the maximum principle can find the optimal solution. It is found that more accurate polynomial fuel consumption models result in better solutions. Only for eco-cruise control the difference is negligible and a simple power-based quadratic model is sufficient. All simulations in this chapter have been done with one vehicle. Furthermore, calculated fuel savings strongly depend on the reference case. However, simulations show that eco-driving can save between a few percent, up to 15 %. This is in line with reported fuel savings in the literature. Concerning the analysis of optimal driving behavior, the simulations result in some new findings: • Both optimal and human accelerations follow a linear decay law. • An optimal acceleration with a continuously variable transmission follows the economy line with decreasing engine speeds. • Optimal decelerations can include coasting, engine braking with a complex gear sequence, and braking. Coasting occurs at higher velocities. Using the brakes at low velocities can lower the fuel consumption. • When using a quadratic fuel consumption model, eco-cruise control can save fuel on gentle slopes.

166

APPLICATION OF THE ECO-DRIVE THEORY

It is important to keep in mind that the total cost J is optimized. This is a combination of fuel consumption and travel time. It is thus possible that optimal driving behavior does not result in the lowest possible fuel consumption. The fuel consumption is strongly influenced by the steady state velocity. This velocity should be wisely chosen. The section on eco-cruise control is a part of a paper submitted to Transportation Research Part D, titled “Eco-Cruise Control for Passenger Vehicles: Methodology”.

Chapter 5

Evaluation of Eco-Drive Guidelines Simulations and analytical derivations do not reduce greenhouse gas emissions. Therefore we need practical applications. When it comes down to the application of eco-driving, eco-drive guidelines are by far the most frequently used tool. Dozens of guidelines exist and many sources can easily be found. One can ask the question: How correct are these guidelines? Are they complete? This chapter tries to find an answer. First, existing guidelines are analyzed based on their relevance for the specific eco-drive subproblem. Then, a list of refined guidelines is presented. Only guidelines that concern actual driving behavior are discussed. To most common guidelines are: • anticipate traffic flow, • avoid braking, • maintain a steady speed at low rpm, • shift up early, • switch off the engine at short stops. Appendix A on p. 187 gives more extensive guidelines provided by ECOWILL and EcoDrivingUSA.

167

168

EVALUATION OF ECO-DRIVE GUIDELINES

5.1

Steady State Velocity

Most guidelines agree: • Drive in the lowest possible gear. • Don’t drive too fast. • Maintain a steady velocity. Cruise control can help doing this.

5.1.1

Gear Selection

The ECOWILL guidelines give specific advice on the use of the gears. Assuming a manual transmission with 5 gears, the fifth gear should be used starting from 50 km/h. The Dutch new driving style guidelines (www.hetnieuwrijden.nl) advice the fifth gear starting from 80 km/h. Most modern vehicles can easily handle driving 50 km/h in fifth gear, and there is no reason not to. Vehicles with an automatic transmission always select low engine speeds for steady state driving. The simulations in section 4.1 on p. 118 e.g. show an increase of 34 % fuel consumption when driving 70 km/h in 3rd instead of 5th gear.

5.1.2

Velocity Selection

The simulations in section 4.1 on p. 118 e.g. show an increase of 20 % fuel consumption when driving 90 instead of 70 km/h. The optimal velocity depends on the vehicle and ambient conditions (road condition, slope, wind,...), yet in most conditions it will not be higher than 90 km/h. Use the board computer to track the instantaneous mileage (steady state fuel economy) and learn what the optimal velocity is. Any velocity lower or higher will consume more fuel. Select your velocity wisely. Keep in mind that it is a tradeoff between fuel consumption and travel time.

5.1.3

Pulse and Glide

An analytical analysis of the eco-drive problem shows that the solution is driving at the optimal velocity. It is possible that an oscillating velocity occurs in the solution due to numerical errors. Hellström, Åslund, and Nielsen [88] show that in some cases a backward Euler approach in discrete dynamic programming can yield an unwanted oscillating velocity profile. Vanderbei

STEADY STATE VELOCITY

169

[181] also observes this when using a direct method. He locates the problem in the interior-point methods. Guidelines can be found that actually promote velocity oscillation. The technique is called pulse and glide or burn and glide. Accelerating at maximum torque (pulse) is followed by a deceleration in neutral gear (coasting), over and over again. Switching off the engine during the coasting (glide) is said to work better. Pulse and glide is often used in supermileage competitions. A search on pulse and glide on the internet results mainly in mileage savings reported by people testing their own vehicle. It is agreed that pulse and glide works better for hybrid vehicles. Scientific sources are scarce. The explanation why it should work is that pulse and glide results in a higher average engine efficiency. Lee [123] researches pulse and glide in his PhD thesis. He mentions that it is important that the engine is switched off during the glide. He finds that fuel economy improvement is a little smaller for an HEV than for a conventional vehicle, because the HEV is already well optimized for reducing fuel consumption and using electric energy by preventing the engine from operating in a low efficiency region. An examination of pulse and glide is needed. A first simple analysis is based on the assumption that the pulse and glide period is short. The pulse and glide each take the same amount of time and travel the same distance. The fuel consumption is characterized by the average velocity, which is the steady state velocity vss if no pulse and glide is applied. For steady state driving, the fuel mass flow rate is m ˙ f,ss . Pulse and glide averages the flow rate of pulse and glide. Is is also assumed that the average torque should equal the steady state torque Tss . Figure 5.1 illustrates the pulse and glide. Because the fuel consumption is convex, fuel savings can only occur when the engine is switched off during the glide. When the engine is not switched off, pulse and glide increases the fuel consumption. Unless when an affine consumption model is used. This analysis is based on strong simplifications. Therefore some simulations are performed. Different pulses and glides are considered. Pulse: wide open throttle (WOT) and an acceleration that follows the economy line. Glides: coasting, coasting with the engine switched off, and engine braking. The steady state velocity is 70 km. Table 5.1 confirms the simple analysis. Pulse and glide can only decrease the fuel consumption if the engine is switched off during the glides. Pulse and glide does save fuel, yet every pulse and glide cycle the engine should be switched off and turned on again. When the driver is not careful, this could lock the steering wheel. In a vehicle with an automatic transmissions, the lockup will not be active during strong accelerations. Power losses occur and the pulse and glide will be less effective [123]. Taking everything into account, it is probably not wise to promote pulse and glide.

170

EVALUATION OF ECO-DRIVE GUIDELINES

fuel mass flow rate m ˙f

m ˙ f,max

m ˙ f,ss m ˙ f,i 0

Tmin

0

Tss

Tmax

engine torque T

fuel mass flow rate m ˙f

m ˙ f,max

m ˙ f,ss m ˙ f,i 0

Tmin

0

Tss

Tmax

engine torque T

fuel mass flow rate m ˙f

m ˙ f,max

m ˙ f,ss m ˙ f,i 0

Tmin

0

Tss

Tmax

engine torque T

Figure 5.1: Illustration of the pulse and glide technique. Top figure: pulse and glide with maximum torque and minimum torque (engine braking). Middle figure: pulse and glide with coasting. Bottom figure: pulse and glide with coasting and engine switch off. Because of the convexity of the fuel mass flow rate, pulse and glide can only decrease the fuel consumption when the engine is switched off during coasting.

ANTICIPATING THE TRAFFIC FLOW

171

Table 5.1: Pulse and glide simulation results. The fuel consumption only decreases when the engine is switched off during the glide. v mf pulse glide [km/h] [%] WOT 60–80 economy line

WOT 65–75 economy line

5.2

coasting coasting, engine off engine braking coasting coasting, engine off engine braking

114.9 93.0 137.7 106.1 89.8 120.8

coasting coasting, engine off engine braking coasting coasting, engine off engine braking

113.7 91.7 136.2 104.9 88.4 119.6

Anticipating the Traffic Flow

All guidelines agree that it is important to look ahead as far as possible and to anticipate the traffic flow. Keeping enough distance from the preceding vehicle allows for better anticipation. Most sources mention a 3 s distance. The goal is twofold. First, it allows to drive at a steady velocity, avoiding accelerations and decelerations which increase the fuel consumption. Second, if a deceleration is necessary, it allows the deceleration to start sooner. To illustrate this, a simulation example of the Toyota Corolla Verso is given. An optimal deceleration to a stop over 600 m consumes 12.6 g of fuel. When the traffic was not anticipated well and the driver can only start the deceleration after 300 m, the fuel consumption is 15.0 g (+27 %). In dense traffic, the 3 s distance can decrease the capacity of the traffic flow. This may cause or increase congestion, which of course increases the total fuel consumption.

172

EVALUATION OF ECO-DRIVE GUIDELINES

5.3

Accelerations

5.3.1

Gear Shifting

The guidelines agree that upshifts should be performed as soon as possible, at low engine speeds. Many guidelines specify the engine speed: • ECOWILL: at approximately 2000 rpm, diesel engines at even lower speeds. • www.ecodriving.be: at 2000 rpm for diesel engines and 2500 rpm for gasoline engines. • www.hetnieuwerijden.nl: between 2000 and 2500 rpm. • www.eco-drive.ch: the latest at 2500 rpm and even lower if possible. 1500 rpm in a diesel car. The simulations in section 4.2.3 on p. 128 indeed show that shifting at low engine speeds is the most fuel efficient. Diesel engines have a lower minimum speed and build up torque faster in the low speed region. When accelerating uphill or accelerating to higher velocities, upshifts can be performed at higher engine speeds, yet never higher than 2500 rpm for gasoline engines.

5.3.2

Acceleration Rate

Concerning the acceleration rate, guidelines strongly differ. Some examples: • ECOWILL: avoid full throttle acceleration. • EcoDrivingUSA: avoid rapid starts, ease into accelerations. • www.ecodriving.be: accelerate fast, press down the accelerator pedal almost completely. • eco-drive.ch: accelerate quickly. • www.drivingskillsforlife.com: accelerate smoothly. • www.toyota-global.com: depress accelerator softly for eco-starting.

ACCELERATIONS

173

The guidelines do not seem to agree on the acceleration rate. Several papers promote WOT accelerations (bang-bang) when the research is based on an affine fuel consumption model, e.g. [169]. Measurements show that strong accelerations are not efficient, e.g. [58, 62]. In a vehicle with an automatic transmission, strong accelerations result in high upshift speeds. Drivers that accelerate strongly usually upshift late in a vehicle with a manual transmission. Section 4.2.3 on p. 128 shows that an optimal acceleration starts strong and ends soft (linear decay). The following advice is given by Evans and Takasaki [66]: “Take about 15 to 20 s to accelerate smoothly to a speed of 48 km/h. Use an average acceleration of about 0.8 m/s2 . Use only light pressure on the accelerator pedal while accelerating at speeds greater than 48 km/h, keep acceleration below 0.7 m/s2 .” This guideline is vehicle specific and is far from general, still quite interesting. 48 km/h is the optimal velocity for the researched vehicle; driving faster is bad for the fuel consumption. So if travel time is not taken into account the best thing is basically to accelerate as slow as possible beyond the optimal velocity.

5.3.3

Automatic Transmission

When driving with a vehicle with an automatic transmission, the acceleration rate at the start should be less strong. When the engine delivers a high torque, the lockup is deactivated and significant losses occur in the torque converter. Furthermore, during a strong acceleration the upshifts will occur at high engine speeds. Upshifts can be enforced by shortly releasing the accelerator pedal.

174

EVALUATION OF ECO-DRIVE GUIDELINES

5.4

Decelerations

Let it be clear, first of all one should avoid decelerations. After a deceleration, the vehicle should often accelerate again and consume fuel to overcome inertia. Guidelines refer to this with terms as anticipation, momentum, or kinetic energy. EcoDrivingUSA says: “Slow-and-go is better than stop-and-go”. Optimally approaching traffic lights is a good example of avoiding a deceleration, see section 4.4.2 on p. 148. Again for decelerations, eco-drive guidelines do not seem to agree. guidelines usually agree, engine braking is the way to go:

Brief

• www.ecodriving.be: While decelerating, avoid downshifting and using the brakes. Decelerate while engine braking. • www.hetnieuwerijden.nl: When there is a need to decelerate or stop when approaching traffic lights, release the accelerator pedal and decelerate while engine braking. • www.fahrspartraining.de: Remove foot from the accelerator as early as possible when needing to slow down or stop. • www.mobimix.be: Use engine braking. Use a gear as long as possible and downshift right above 1000 rpm. • Do not shift down to a lower gear too early and keep the car rolling without disengaging the clutch and in as high a gear as possible [180]. Some guidelines give a more complex advice for decelerating. Ecomodder [55] gives the following tip: “When you have to slow down, here’s an approximate hierarchy of methods, from best to worst: coasting, engine off; coasting, engine idling; engine braking with fuel cut-off; conventional friction braking. Choosing the right method depends on traffic conditions and how quickly you need to stop.” ECOWILL identifies the following deceleration techniques: • Engine braking. “This technique is beneficial to saving fuel if the respective engine has a fuel cut-off mode and also while driving at higher speeds.” • Coasting. “This is beneficial for situations like approaching an obstacle or an identified stop (red traffic lights, stop sign). Thus, a relative long distance can be driven at quite constant speed without additional

DECELERATIONS

175

acceleration. Especially for cars without engine fuel cut-off mode this is a good technique. But also for cars with fuel cut-off the option to coast can save fuel at typical low speed driving in cities.” If guidelines mention braking, they agree that braking should be avoided. One can also find e.g.: “avoid sharp braking” [69] and “the level of braking has minor effect on the fuel consumption” [74]. The analysis in section 3.3.3 on p. 85 and the simulations in section 4.3.1 on p. 139 show the following: • Coasting is beneficial at higher velocities. Coasting with the engine switched off is of course better, yet not advised as mentioned before. • At lower velocities, engine braking should be used. Preferably with fuel cut-off, yet it is also beneficial without fuel cut-off. • Using the brakes can actually lower the fuel consumption. This should be done at the end of the deceleration, at very low velocities. The shorter the deceleration distance, the more beneficial engine braking and the use of the brakes become.

176

EVALUATION OF ECO-DRIVE GUIDELINES

5.5

Driving over Hills

Most eco-drive guidelines do not mention driving over hills. If they do, they advice not to use the cruise control on hills. It might react aggressively on a sudden increase of road slope and burn a lot of fuel. There are some specific guidelines: • ECOWILL: When driving uphill, the target is to travel in the highest possible gear with almost full pressure on the accelerator pedal. • Ecomodder: Drive with load, this technique is accomplished by choosing a target rate of fuel consumption. • Ecomodder: Pick a throttle position and hold it. Section 4.5 on p. 150 shows that adapting the velocity on a hilly road is more fuel efficient than keeping a constant velocity. A simulation is done to evaluate the aforementioned guidelines. The first one is not considered. It basically advises driving as fast as possible, yet in the highest gear. Driving with a fixed throttle is simulated as driving with fixed torque. A simulation is performed on the 45 km section of the I81. Table 5.2 shows that when a fixed fuel mass flow rate or torque is used, the fuel consumption decreases compared to reactive ecocruise control (RECC). However, the average velocity v¯ decreases too. Looking at the total cost J, RECC yields the best results. Driving with a fixed fuel flow rate or torque results in large velocity fluctuations. The best advice for driving over hills is probably: “drive slower uphill and faster downhill”.

DRIVING OVER HILLS

177

Table 5.2: Simulation results for different hill driving strategies over I81, with vss = 94 km/h and different velocity bands. v [km/h]

strategy

94

mf [%]

v¯ [km/h]

J [%]

100.0

94.0

100.0

84 < v < 104

RECC m ˙ f fixed T fixed

85.2 85.6 85.8

91.9 92.0 91.4

93.7 94.0 94.0

74 < v < 114

RECC m ˙ f fixed T fixed

84.8 84.4 84.8

91.4 89.2 89.0

93.9 94.9 95.3

64 < v < 124

RECC m ˙ f fixed T fixed

84.8 83.4 83.1

91.4 87.1 85.5

93.9 95.7 96.6

178

5.6

EVALUATION OF ECO-DRIVE GUIDELINES

Starting and Stopping the Engine

All guidelines agree that one should drive off immediately after starting the engine. There is no need to warm the engine while idling. Eco-drive guidelines agree that the engine should be switched off when there is an anticipated stop for a certain period. The duration of this period ranges from 3 s according to the Belgian ecodriving (www.ecodriving.be), to 60 s according to the Dutch new driving style (www.hetnieuwerijden.nl). Although this problem is not discussed in this dissertation yet and is not related to optimal control, it is interesting to do a simplified calculation. When the vehicle is idling during a stop of duration t, the engine consumes the idle fuel mass flow rate m ˙ f,i . The total consumption is: mf = t · m ˙ f,i . When the engine is switched off, it should be switched on again. The starter motor should provide the rotational energy to make the engine turn at the idle speed ωi [rad/s]. The engine has a rotational inertia I [kg·m2 ]. The electric energy comes from the battery, originally produced by the generator. This electric path has an efficiency η [-]. The fuel needed to produce the energy depends on the average brake specific fuel consumption BSFC at which it was produced. Combining this, the duration at which idling and switching off the engine consume the same amount of fuel is given by: Iωi2 BSFC · 2 η t= . m ˙ f,i

(5.1)

Assuming that η = 50 % and BSFC = 500 g/kWh, the duration for the Toyota Corolla Verso is ±3 s. As a round value, 5 s is probably the best guideline. Multiple starting and stopping could drain the battery. This can be a problem in colder weather with an old battery. Modern vehicles often have an automatic start/stop system with an adapted starter motor and battery.

CONCLUSIONS

5.7

179

Conclusions

The conclusion of this section is a refined list of eco-drive guidelines. Maintain a steady velocity and drive in the highest possible gear. Use the board computer and be aware of the optimal velocity (usually around 70 km/h). Keep a distance of 3 s from the preceding vehicle. Drive closer in dense traffic for a fluent traffic flow. Anticipate the traffic. Drive at a constant velocity that guides you through traffic without being hindered. Avoid decelerations if possible. Start an acceleration from a stop strong. Let the acceleration rate drop to zero at the desired cruise velocity. Upshift as soon as possible. When driving uphill or accelerating to high velocities, upshifts can be performed at higher engine speeds. Do not exceed 2500 rpm. When there is a need to decelerate: coast at high velocities; engine brake at moderate velocities, preferably with fuel cut-off; and use the brakes at very low velocities. With a decreasing deceleration distance, engine brake more and make more use of the brakes. Use cruise control only on level roads. Drive slower uphill and faster downhill. Switch off the engine when expected to stop longer than 5 s. Be careful not to drain the battery.

Chapter 6

Conclusions and Recommendations 6.1

Conclusions

Transport is responsible for about 15 % of the anthropogenic greenhouse gas emissions, thus playing a significant role in climate change. To avoid disastrous consequences, the increase of the global mean temperature should stay below 2◦ C. Eco-driving has the potential to reduce fuel consumption and thus CO2 emissions by 10 % in an easy and cost-effective way. There exist different applications of eco-driving: eco-drive guidelines, driving style analyzers, gear shift indicators, intelligent speed adaptation, eco-cruise control, ... The effectiveness of these applications relies on the underlying optimal driving behavior. Eco-driving is an optimal control problem, thus the best way to calculate the optimal driving behavior is using optimal control methods. The main goal of this dissertation is to improve and simplify the existing methods to solve eco-driving as an optimal control problem. Quasistatic low-degree polynomial fuel consumption models are used in combination with a simple model of the driveline and vehicle dynamics to calculate the fuel consumption. A novel assessment approach does not only take into account how well the model can fit the data, but also the implications for eco-driving. Both existing polynomial fuel consumption models, and newly developed ones bases on a subsearch with Akaike’s information criterion, are used in the assessment. It is found that models with less than 10 parameters can still predict the fuel consumption to an acceptable degree of accuracy,

181

182

CONCLUSIONS AND RECOMMENDATIONS

while capturing typical and important properties in the maps of specific fuel consumption and consumption per rotation. It is very important that a quadratic fuel consumption model is used. An example model is the following: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωT + α5 ω 2 T + α6 ωT 2 . If there are no measurements available, consumption models with 3 parameters can be identified with publicly available data. The following model is advised: m ˙ f = α0 + α1 P + α2 P 2 . Polynomials can model the fuel consumption of both gasoline and diesel engines, and the efficiency of electric motors. Four solution methods are considered for solving eco-drive control problems: Euler-Lagrange, Pontryagin’s maximum principle, discrete dynamic programming, and direct multiple shooting. Five subproblems are defined: acceleration, deceleration, driving between stop signs, approaching traffic lights, and ecocruise control. An explicit approach based on Pontryagin’s maximum principle is found to be the best method to solve an eco-drive control problem. It gives a fast and accurate result, and can easily include gear shifting and clutch control. The following simple formula gives the optimal engine control u∗e : v u E(v) − E(v ) + ε u ss ∗ . ue = ue,ss (v) ± u2 t ∂ 2 FE ∂u2e The economy E = (m ˙ f,ss + M)/v is a combination of fuel economy and travel time. The tradeoff between those two is very important and plays a major role in the optimal driving behavior. An important distinction is made between direct and indirect constraints. If the time constraint is indirect, vss equals a desired average or steady state velocity, e.g. the speed limit. If the time constraint is direct, vss should be determined iteratively to satisfy the constraint. If the distance constraint is indirect, ε = 0. If it is direct, ε should be determined iteratively. Optimal gear shifting and (dis)engaging the engine, follows the following rule: maximization of λv during an acceleration, minimization of λv during a deceleration. If λv > 0, the brakes should be used. λv is the adjoint state variable of the velocity. A quadratic fuel consumption model should be used in order to avoid unwanted bang-bang control. New algorithmic tricks for dynamic programming are presented: an adaptive distance grid, a smart stop for accelerations and decelerations, and a fast algorithm for approaching traffic lights. Concerning the analysis of optimal driving behavior, the simulations result in some new findings:

CONCLUSIONS

183

• Both optimal and human accelerations follow a linear decay law. • An optimal acceleration with a continuously variable transmission follows the economy line with decreasing engine speeds. • Optimal decelerations can include coasting, engine braking with a complex gear sequence, and braking. Coasting occurs at higher velocities, using the brakes at low velocities can actually lower the fuel consumption. • When using a quadratic fuel consumption model, eco-cruise control can save fuel on gentle slopes. Eco-drive guidelines are not always correct and often contradict each other. Based on the presented theory and simulations, a refined list of guidelines is proposed: • Maintain a steady velocity and drive in the highest possible gear. Use the board computer and be aware of the optimal velocity (usually around 70 km/h). • Keep a distance of 3 s from the preceding vehicle. Drive closer in dense traffic for a fluent traffic flow. Anticipate the traffic. Drive at a constant velocity that guides you through traffic without being hindered. Avoid decelerations if possible. • Start an acceleration from a stop strong. Let the acceleration rate drop to zero at the desired cruise velocity. Upshift as soon as possible. When driving uphill or accelerating to high velocities, upshifts can be performed at higher engine speeds. Do not exceed 2500 rpm. • When there is a need to decelerate: coast at high velocities; engine brake at moderate velocities, preferably with fuel cut-off; and use the brakes at low velocities. With a decreasing deceleration distance, engine brake more and make more use of the brakes. • Use cruise control only on level roads. Drive slower uphill and faster downhill. • Switch off the engine when expected to stop longer than 5 s. Be careful not to drain the battery.

184

CONCLUSIONS AND RECOMMENDATIONS

6.2

Recommendations for Future Research

Although this dissertation attempts to provide a vast reference on optimal control based eco-driving, it is far from complete. The modeling of the powertrain is based on several assumptions and simplifications. Several extensions can be made to the model, and their influence on the optimal driving behavior can be studied. • Dynamic modeling of the engine and fuel consumption, instead of quasistatic. Engine dynamics might be relevant for engines with a turbocharger. • Automatic transmission without lockup. • Influence of longitudinal dynamics during a gear shift, and dynamic modeling of the clutch and engine during (dis)engagement. • Cold start. • Other emissions: CO, HC, NOx , and soot. It is indicated that the presented methods can also be used for (hybrid) electric vehicles, heavy-duty trucks, and trains. Yet, for these applications the optimal driving behavior is not yet studied. Based on models of different vehicles, one can perform simulations of different eco-drive control problems under different traffic conditions to study the fuel saving potentials. All simulations are focussed on one vehicle. One could also study the system wide impacts of eco-driving vehicles in traffic. The next big step in the research is building an eco-drive assist system. Figure 6.1 shows a possible design scheme of the system. The following modules are present: input This module collects necessary information, divided in three groups: • vehicle states: velocity, engine speed, gear, ... • surroundings: road slope, speed limit, distance to preceding vehicle, location of vehicle, location of stop signs, traffic lights, traffic lights timing, ... • driver preferences: allowed velocity band for ECC, trip destination.

RECOMMENDATIONS FOR FUTURE RESEARCH

185

This information can be collected through the on-board diagnostics, radar, global positioning system, geographic information system, tilt sensor, ... control This module calculates the optimal control, three submodules are: • parameter estimation: estimates the parameters of the longitudinal dynamics, • problem definition: determines the constraints and cost function of the optimal control problem, • optimal control: solves the optimal control problem. The use of Pontryagin’s maximum principle and a quadratic fuel consumption model is advised. output This module communicates the optimal driving behavior to the driver. off-line Additional software on a PC adds to the functionalities of the EDAS. The driver can input vehicle parameters such as vehicle mass, drag coefficient, wheel radius, ... All measurements of the input module are logged, and used to calculate an engine and fuel consumption model. These measurements can also be used to calculate the achieved fuel savings and analyze the driving behavior. input vehicle states surroundings

-

parameter estimation problem definition

driver preferences

optimal control

6 ?

6 ? off-line

log measurements

output

control -

throttle gear brakes

6 ?

vehicle parameters

calculate fuel savings

calculate engine model

analyze driving behavior

Figure 6.1: Design scheme of an eco-drive assist system.

Appendix A

Eco-Drive Guidelines Clear and extensive eco-drive guidelines are given by ECOWILL [56]. The ECOWILL project (eco-driving, widespread implementation for learner drivers and licensed drivers), launched in May 2010, aims at reducing carbon emission by boosting the application of eco-driving across Europe. ECOWILL has been endorsed by the European Commission’s Sustainable Energy Europe Campaign as an official partner. Their guidelines are the following: • Anticipate traffic flow. – Read the road as far ahead as possible and anticipate the flow of traffic. – Act instead of react. – Keep a safety distance of about 3 s to balance speed fluctuations in traffic flow. – Make maximum use of the vehicle’s momentum. • Maintain a steady velocity at low engine speed. – Drive smoothly using the highest possible gear. – Driving at high velocities significantly increases fuel consumption. – Use the cruise control. • Upshift at approximately 2000 rpm. Cars with a diesel engine can be shifted at even lower engine speeds. Rough guidance for shifting and steady driving on a flat road:

187

188

ECO-DRIVE GUIDELINES

– 1st gear: start off only (one vehicle length), – 2nd gear: 20 km/h, – 3rd gear: 30 km/h, – 4th gear: 40 km/h, – 5th gear: 50 km/h, – 6th gear: 60 km/h. • Avoid full throttle acceleration. If strong acceleration is required, use intentionally full throttle acceleration. When accelerating stronger, skipping gears can help to save fuel. • Drive off immediately after starting the engine. Do not warm up the engine. • Switch off the engine when expected to stop longer than 20 s. • When driving uphill: the target is to travel in the highest possible gear with almost full pressure on the accelerator pedal. • Make use of in-car devices, if present, like the tachometers, cruise control, and on-board computer. A good source of eco-drive guidelines in the USA is the eco-driver’s manual by EcoDrivingUSA [53]. EcoDrivingUSA is sponsored by the Alliance of Automobile Manufacturers, which is a trade association representing 10 car manufacturers. Their guidelines are the following: • Avoid rapid starts and stops. Ease into accelerations and brake smoothly. • Slow-and-go is better than stop-and-go. Maintain a constant velocity and look ahead down the road to anticipate stops. • Ride the green wave. Traffic lights are often synchronized so that a motorist driving at a specific speed will pass through a series of green lights without stopping. • Use cruise control. During highway driving, cruise control helps maintain a steady velocity. The benefits come largely from driving on flat terrains. If you are driving on hilly roads, cruise control may cause your engine to speed up on climbing hills and slow down on the other side, reducing mileage, so use cruise control selectively. • Avoid idling. Shut off your engine for any stop anticipated to be longer than 30–60 s.

ECO-DRIVE GUIDELINES

189

• Use the highest gear possible. • Drive your vehicle to warm it up. Other sources on eco-drive guidelines usually list the same, or less, sometimes slightly different guidelines. An extensive list of eco-drive guidelines is given by ecomodder [55].

Appendix B

Vehicle Modeling This appendix describes the vehicle model. The longitudinal dynamics are given in section B.1. Section B.2 presents the model of a conventional powertrain. Section B.3 presents the model of a hybrid electric powertrain.

B.1

Longitudinal Dynamics

This section discusses briefly the modeling of the longitudinal dynamics. More information can be found in e.g. [48, 77, 83, 105, 151]. For fuel consumption calculations, only longitudinal dynamics are considered and the vehicle can be seen as a point mass [83]. The equation of motion is given by: m

dv = Ft − Fa − Fr − Fs , dt

(B.1)

where m [kg] is the total vehicle mass, v [m/s] the vehicle velocity, t [s] the time, Ft [N] the traction force of the wheels, Fa [N] the aerodynamic drag, Fr [N] the rolling friction, and Fs [N] the resistance due to road slope. Figure B.1 shows these longitudinal forces that are acting on the vehicle. The total vehicle mass is the sum of the empty vehicle mass, the mass of the driver and passengers, and the mass of the fuel. The mass of a vehicle can easily be found online. The EU kerb mass includes the vehicle, a full fuel tank, and a 75 kg driver. The US curb mass includes the vehicle, a half full fuel tank, and no driver.

191

192

VEHICLE MODELING

vw Fa mg

v a Fr

Ft

θ

Figure B.1: Longitudinal forces acting on a vehicle. Aerodynamic Drag The aerodynamic drag is given by: Fa =

1 ρa Scd (v − vw )2 , 2

(B.2)

with ρa [kg/m3 ] the air density, S [m2 ] the frontal surface of the vehicle, cd [-] the drag coefficient, and vw [m/s] the wind velocity in the driving direction. The air density at standard conditions (pressure p0 = 101325 Pa and temperature T0 = 15◦ C): ρa = 1.225 kg/m3 . The standard air pressure p [Pa] at altitude h [m] is given by: p

 gM Lh RL = p0 · 1 − T0
5.25588 = 10132500 1 − 2.25577 · 10−5 h , 

(B.3)

with L = 0.0065 K/m the temperature lapse rate, g = 9.81 kg/m2 the gravitational acceleration, M = 0.02896 kg/mol the molar mass of dry air, and R = 8.31447 J/(mol·K) the universal gas constant. The frontal surface is not an easy to find parameter. For a normal vehicle, it can be approximated by: S ≈ 0.85W H,

(B.4)

with W [m] the width and H [m] the height of the vehicle. The drag coefficient of a normal vehicle: cd = 0.3...0.32. The new Toyota Prius is very aerodynamic, with cd = 0.25. A Hummer is the other opposite, with cd = 0.57.

LONGITUDINAL DYNAMICS

193

Rolling Friction The rolling friction is given by:  cr mg cos(θ), if v = 6 0, Fr = 0, if v = 0,

(B.5)

with cr [-] the rolling friction coefficient and θ [rad] the road slope. The rolling friction is complex and depends on the vehicle velocity, the tire pressure, and the conditions of the tires and road. It can be approximated by [70]: cr =

cr,1 (cr,2 v + cr,3 ) , 1000

(B.6)

with coefficients cr,1 , cr,2 , and cr,3 given in table B.1. In a more general format, the rolling friction is modeled as follows: cr = cr,1 + cr,2 v + cr,3 v 2 .

(B.7)

Table B.1: Coefficients of rolling friction [70]. pavement type

pavement condition

cr,1 [-]

concrete

excellent good poor

1.0 1.5 2.0

asphalt

good fair poor

1.25 1.75 2.25

tire type

cr,2 [s/m]

cr,3 [-]

bias ply radial

0.1577 0.1181

6.100 4.575

When the vehicle drives in a curve, the tires provide a lateral force equal to the centrifugal force. A component of that lateral force is in the driving direction, which adds to the rolling friction [74]. The total rolling friction when driving is then given by: Fr = cr mg cos θ + m

v2 sin φ, r

(B.8)

with r [m] the radius of the curve, and φ [rad] the slip angle of the wheels.

194

VEHICLE MODELING

Road Slope Resistance The road slope resistance is given by: Fs = mg sin(θ).

(B.9)

Total Load Force The load force Fℓ [N] is the sum of the aerodynamic drag, rolling friction, and slope resistance. It can be modeled with lumped vehicle dynamics coefficients c [∼]: Fℓ (v, θ, vw )

= Fa + Fr + Fs = c0 + c1 v + c2 v 2 , 1 2 ρa Scd vw + mg (cr,0 cos θ + sin θ) , 2

c0

=

c1

= −ρa Scd vw + cr,1 mg cos θ,

c2

=

1 m ρa Scd + cr,2 mg cos θ + sin φ. 2 r

(B.10a) (B.10b) (B.10c) (B.10d)

CONVENTIONAL POWERTRAIN

B.2

195

Conventional Powertrain

This section discusses briefly the modeling of a conventional powertrain. More information can be found in e.g. [48, 77, 83, 105, 151]. The powertrain of a vehicle is defined as all the components that generate a traction force at the wheels. The powertrain without the internal combustion engine (ICE) is called the driveline. The driveline of a conventional manual vehicle consists of the clutch, gearbox, final drive (differential), wheels, and brakes (manual transmission). Figure B.2 shows the powertrain. In a conventional automatic powertrain, the clutch is replaced by a torque converter (automatic transmission). ω e , Te ω c , Tc ICE

ig

Tb ωw , T'$ w

if

rw,d Ft &% -

Figure B.2: Scheme of a conventional manual powertrain.

Engine A quasistatic engine model consists of different parts: m ˙f =m ˙ f (ωe , Te ),

fuel consumption,

(B.11a)

ωe > ωe,min ,

minimum engine speed,

(B.11b)

ωe 6 ωe,max ,

maximum engine speed,

(B.11c)

Te > Te,min (ωe ),

minimum engine torque,

(B.11d)

Te 6 Te,max (ωe ),

maximum engine torque,

(B.11e)

with m ˙ f [kg/s] the fuel mass flow rate, ωe [rad/s] the engine speed, and Te [Nm] the engine torque. The fuel consumption model is covered in detail in chapter 2 on p. 27. The minimum engine speed ωe,min equals the idle speed. When the accelerator pedal is not pushed, the engine produces the minimum

196

VEHICLE MODELING

torque Te,min . The latter is determined by: engine peripherals; viscous friction from engine parts separated by an oil filter, oil pump, and water pump; and aerodynamic friction from gasses flowing through the engine. Usually, below a certain engine speed, some fuel is still injected when the accelerator pedal is not pushed. Above this fuel cut-off speed ωfco [rad/s], no fuel is injected. The minimum torque can be modeled as:  cT,1 + cT,2 ωe + cT,3 ωe2 , if ωe < ωfco , Te,min = (B.12) cf,1 + cf,2 ωe + cf,3 ωe2 , if ωe > ωfco , with cf [∼] the engine friction coefficients and cT [∼] the minimum engine torque coefficients. One can use the following approximation for minimum engine torque for a gasoline engine: Te,min = 75 · 10−6 · Vd · ωe ,

(B.13)

with Vd [cm3 ] the engine displacement volume. The maximum engine torque Te,max can also be approximated by a polynomial function: X βk ωerk , (B.14) Te,max = k

with β [∼] the maximum torque coefficients, and r [-] the polynomial powers. A common function is [77, 84, 104, 141, 155]: Te,max = β1 + β2 ωe + β3 ωe2 .

(B.15)

Ni [141] suggest to identify the parameters based on three engine specifications. The maximum torque line should go through the points of maximum torque and power, and have a maximum at the maximum torque. Figure B.3 shows that this does not yield a good result (model 1). Adding a cubic term to equation (B.15) doesn’t give a good result either (model 2). Therefore, equation (B.15) is used. Yet, the model is forced to go through the point of maximum torque and 2/3 of this value at minimum engine speed (model 3). The latter identification will be used if no measurements are available to identify the model parameters. Clutch A clutch in a conventional manual powertrain allows the engine to decouple from the rest of the powertrain. This is needed for gear shifting, idling and start off. During start off, the clutch has to slip to ensure that the engine keeps it’s minimum speed to prevent it from stalling.

maximum engine torque Te,max [Nm]

CONVENTIONAL POWERTRAIN

197

400

200

model 1 model 2 model 3

0

1000

2000

3000

4000

5000

6000

engine speed n [rpm]

Figure B.3: Comparison of different maximum torque models. The clutch in a passenger car is usually a dry-plate friction clutch. This type of clutch always transmits the torque without losses. There are three working conditions: Te = Tc , Te = Tc , Te = Tc = 0,

ωe = ω c , ωe = max (ωc , ωe,min ) ωe = ωe,min ,

when engaged, when slipping, when disengaged,

(B.16)

with Tc [Nm] the output torque of the clutch, and ωc [rad/s] the clutch speed. A clutch control variable uc [-] is introduced. When the engine is disengaged: uc = 0, when the engine is engaged or slipping: uc = 1. Large vehicles often use a wet-plate friction clutch. This type of clutch has a better thermal performance. However, it has more losses when engaging. For simplicity, in ωe , Te , and Pe , the subscript when no confusion is possible.

e

is omitted in this dissertation

Torque Converter Vehicles with an automatic powertrain use a torque converter instead of a clutch. A hydrodynamic torque converter consists of an impeller that drives a turbine. It is filled with oil and transmits engine torque by means of the flowing forces of oil. A stator located between impeller and turbine diverts the hydraulic oil back to the input side of the impeller. This means the torque output can be higher than the impeller torque [48]. The torque multiplication µ [-] reaches its

198

VEHICLE MODELING

maximum value when ωc = 0 and µmax = 1.7 . . . 2.5 [48].   ωc Tc = max µmax − , 1 . µ= Te ωe

(B.17)

Modern torque converters have a lockup. This is a system that locks the converter when impeller and turbine are almost rotating at the same speed. The torque converter is not locked during start off and gear shifting. Gearbox and Final Drive The gearbox and the final drive (differential) change the rotation speed of the wheels: ωc = ig (g) · if · ωw ,

(B.18)

with ig [-] and if [-] the reduction ratios of the gearbox and the final drive, g [-] the gear, and ωw [rad/s] the rotation speed of the wheels. This allows the vehicle to have a large velocity range compared to the speed range of the engine. Gear shifting with a manual transmission is modeled as a disengagement of the engine during the shift time tshift [s]. During the gear shift, the engine idles and consumes the idle consumption m ˙ f,i . A realistic shift time is 0.3 s. An automatic transmission uses a planetary gearbox. The torque converter replaces the clutch. Gear shifts are smooth and the engine keeps producing torque during the shift. Wheels The radius of the wheels rw [m] can be derived from the tire type (Xa/bXc): a [mm] is the width of the tire, b [-] is one hundred times the height-width ratio, and c [inch] is the rim diameter. Thus: rw = 0.0127 · c + a · b · 10−5 .

(B.19)

When rolling, the tire is compressed such that the dynamic (actual) wheel radius rw,d [m] is smaller:   rw,d = rw (1 − i), if Ft > 0 (slip), rw rw,d = 1−i , if Ft < 0 (skid), rw,d = (B.20)  rw,d = rw , if Ft = 0,

CONVENTIONAL POWERTRAIN

i=

199

ic Ft , 2Ft,max

(B.21)

with i [-] the wheel slip, ic [-] the linear wheel slip coefficient (±3 %), and Ft,max [N] the maximum traction force. This equation is a linear approximation of Pacejka’s magic formula [16]. The maximum traction force Ft,max [N] is given by: Ft,max = νmta g,

(B.22)

with ν [-] the coefficient of road adhesion and mta [kg] the vehicle mass on the tractive axle. For a front-wheel drive passenger car: mta ∼ 0.6 · m. Values of ν are given in table B.2. Unless mentioned otherwise, the wheel dynamics are neglected and rw,d = rw . Table B.2: Coefficient of road adhesion [70]. pavement

ν [-]

good, dry good, wet poor, dry poor, wet packed snow or ice

1.0 0.9 0.8 0.6 0.25

Brakes The brakes can exert a braking torque Tb [Nm] to the axle of the wheels. In this dissertation this is modeled as a braking force Fb [N]. Powertrain Inertia Besides the vehicle mass, there is also a significant amount of inertia coming from rotating parts in the powertrain. The rotational inertia mainly comes from the engine and the wheels. The total powertrain rotational inertia Ip [kg] is then given by: Ip (g) =



ig (g) · if rw,d

2

Ie +

1 2 Iw , rw,d

(B.23)

200

VEHICLE MODELING

with Ie [kgm2 ] the rotational inertia of the engine, and Iw [kgm2 ] the rotational inertia of all the wheels. Simplified, the total vehicle inertia I [kg] is modeled as [171]:   2 I = m · 1.04 + 0.0025 · (ig if ) . (B.24) Driveline Efficiency Usually, the friction losses in the driveline are modeled with a mechanical efficiency ηm :  ηm Te ωe , if Te > 0, Tw ωw = (B.25) Te ωe if Te 6 0, ηm , with Tw [Nm] the wheel axle torque. However, this approach is not based on physics and may lead to strange optimal control behavior. Therefore in the control applications in this dissertation, the friction losses are modeled with a viscous friction coefficient cv [Ns/m]:   2 2 2  if 1 ig (g) · if cv,1 + cv,2 + cv,3 . (B.26) cv = rw,d rw,d rw,d A method based on heuristics to find realistic values, is to match the friction with a realistic efficiency for average values of engine speed and torque. Total Dynamics The total powertrain is considered stiff and is modeled as: ωe · rw,d ωe v= , = ig (g) · if i(g) dv , dt [1/m] the total driveline ratio.

Ft = i(g) · uc T − Fb − cv v − Ip (g) with i = ig if /rw,d

(B.27) (B.28)

Combining equations (B.1), (B.10) and (B.28) yields: i(g) · uc T − Fb − cv (g) · v − (c0 + c1 v + c2 v 2 ) dv = . dt I(g)

(B.29)

This is the ordinary differential equation (ODE) for the total vehicle dynamics for a conventional manual vehicle, or for a conventional automatic vehicle during torque converter lockup (uc = 1).

HYBRID ELECTRIC POWERTRAIN

B.3

201

Hybrid Electric Powertrain

This section discusses briefly the modeling of a parallel hybrid electric powertrain. Figure B.4 shows the scheme of a possible parallel configuration. An electric motor (EM) is added to a conventional powertrain, connected between the clutch and the gearbox. Thus, the EM is always rotating when the vehicle is driving.

ICE

ig

if

- battery EM 

Tb '$ &%

Figure B.4: Scheme of a parallel hybrid powertrain. The ICE has an output torque Te . The EM has an output torque Tm . Adding the latter torque to equation (B.29) yields: dv i(g) · uc Te + Tm − Fb − cv (g)v − (c0 + c1 v + c2 v 2 ) = . dt I(g)

(B.30)

Compared to a conventional powertrain, the viscous friction coefficient cv will be higher due to friction in the EM. The extra mass of the EM and battery increases the vehicle mass and inertia I. The electric power consumption of the motor Pm,e can be modeled by a polynomial function, comparable to the fuel mass flow rate of an ICE: X pk qk αk ωm Tm , (B.31) Pm,e = k

with ωm [rad/s] the rotation speed of the EM. Different models can be used for positive and negative motor torques. With regards to the battery, the state of charge SOC [-] is defined. The SOC is the relative amount of energy stored in the battery: 0 6 SOC 6 1. For the health of the battery, the operational bounds of the SOC are usually stricter. A proportional relationship exists between the change of SOC and the power

202

VEHICLE MODELING

Pb [W] going in or out the battery: dSOC = −cb Pb , dt

(B.32)

with cb [1/W] a battery parameter. The battery itself also has an efficiency. For a small change in SOC, the relationship between Pm,e and Pb is more or less linear [111]. There exist also more complex efficiency models [83]. Assume that the relationship between Pm and Pb can be modeled by a polynomial in the entire range of SOC. Combining this model with the polynomial model of equation (B.31) yields: X pk qk Pb = αk ωm Tm . (B.33) k

Appendix C

Fuel Consumption Measurements This chapter discusses more in detail the measurements that are used for the assessment of different polynomial fuel consumption models in section 2.2 on p. 35.

C.1

Engine Dynamometer

The universal engine dynamometer is shown in figure C.1. It consists of a gasoline engine and an electric DC motor. The combustion engine is a 2004 Toyota 1.6 l VVT-i 3ZZ-FE gasoline engine, donated by Toyota Belgium. This series is used in the Toyota Corolla in Europe since 2002. Table C.1 shows the specifications of the engine. The following variables are measured steady state and hot-stabilized (warm engine): engine speed The engine speed is measured with a Heidenhain ROD 426 rotary encoder, mounted on the crankshaft. The measurements are logged with a dSpace DS1104 R&D controller board and averaged over 60 s. engine torque The engine torque is measured with a Kistler 4504A torque measuring flange between the combustion engine and the DC motor. The

203

204

FUEL CONSUMPTION MEASUREMENTS

Figure C.1: Universal engine dynamometer.

Table C.1: Specifications of the Toyota 3ZZ-FE engine. cylinders 4, inline 16, DOHC valves chain, VVT-i fuel

electronic indirect injection octane 95 or higher

material weight

aluminum 101 kg

max power max torque max BMEP

81 kW (109 pk) @ 6000 rpm 150 Nm @ 3800 rpm 11.8 bar

displacement bore x stroke compression ratio

1598 cm3 79 x 81.5 mm 10.5

ENGINE DYNAMOMETER

205

measurements are logged with the dSpace controller board and averaged over 60 s. fuel consumption The fuel consumption is measured with a Sartorius Economy EA electronic precision scale, on which the fuel tank is mounted. During 60 s, the scale is read out every 0.5 s through an RS232 serial connection with a PC. These measurements are then used to determine the fuel mass flow rate. air-fuel ratio The air-ruel ratio is measured with a broadband lambda sensor. The measurements are logged with the dSpace controller board and averaged over 60 s. The maximum torque model is the following: Tmax = β1 + β2 ω + β3 ω 2 .

(C.1)

This model yields the best fit with the measurements. The engine is e.g. used in the Toyota Corolla Verso. This vehicle is used for simulation purposes in combination with the 3ZZ-FE engine. The specifications are given in table C.2. Table C.2: Specifications of the 2004 Toyota Corolla Verso. m 1365 kg S 2.87 m2 cd 0.3 rw 0.3159 m if 4.31 ig [3.32, 2.00, 1.36, 1.01, 0.82] cv 0.0017 + 2.77 · 10−4 · i2g

206

C.2

FUEL CONSUMPTION MEASUREMENTS

Chassis Dynamometer

A chassis dynamometer measures power delivered to the surface of a drive roller by the vehicle wheels. This type of dynamometer is used to determine official fuel economy estimates on drive cycles. The tests are done in a controlled environment, with strict procedures. The scheduled velocity profile is displayed on a monitor, while a test driver controls the gas and brake pedals such that the vehicle velocity follows these reference values within pre-specified error bands [83]. The measurements used in this dissertation are obtained from USA test cycles. Unfortunately, the measurements are rather old. Data of 4 different test cycles is available: urban dynamometer driving schedule (UDDS), arterial LOS A-B (ARTA), LA92, and freeway high speed (FWHS). As an example, figure C.2 shows the UDDS cycle. 100

velocity v [km/h]

80 60 40 20 0

0

200

400

600

800

1,000

1,200

1,400

time t [s]

Figure C.2: UDDS drive cycle. The available measurements are the vehicle velocity and the mass flow of CO2 , CO, and HC at the tailpipe, measured each second. Based hereon, engine speed, engine torque, and fuel mass flow rate are calculated. engine speed The gear, final drive ratio, and wheel radius are obtained from car specifications on the internet. Wheel slip and skid is neglected. The gear is determined based on the EPA gear shift pattern: 1st to 2nd at 15 mph, 2nd to 3rd at 25 mph, 3rd to 4th at 40 mph, 4th to 5th at 45 mph. This pattern is used for the official testing of manual vehicles. Since there is no other way to determine the gear, it is also used for automatic vehicles in this dissertation. The minimum engine speed is assumed to be 900 rpm

CHASSIS DYNAMOMETER

207

and slip in the torque converter is only taken into account during start off. engine torque The load force is calculated with equation (B.10). There is no wind, road slope, or road curvature. The density of the air is taken at 20◦ C and sea level. The frontal surface and drag coefficient are taken from car specifications on the internet. The rolling friction is based on equation (B.6) and table B.1 on dry good asphalt pavement. The acceleration is calculated with a backward difference, as this yields the best results. The total inertia is given by equation (B.24). The viscous friction coefficients are based on heuristics. It is assumed that the brakes are not used and the minimum torque is limited using equation (B.13). fuel mass flow rate The fuel mass flow rate is calculated indirectly. The CO2 , CO, and HC flow rates are used to determine the fuel mass flow rate. The delay in the system (from fuel injection to exhaust), is compensated by a constant time shift. This shift is determined by optimizing the coefficient of determination R2 [-] for a fit with a quadratic power-based model. Based on the measurements, it is not possible to identify parameters for a maximum torque model since there is no information on when the torque is actually at its maximum. Therefore, it is based on the engine specifications. Table C.3 shows the 11 vehicles for which measurements are used.

FUEL CONSUMPTION MEASUREMENTS 208

model Honda Accord Toyota Camry Ford Explorer Toyota Corolla Ford F150 Mazda Protegé Ford Ranger Honda Civic Hyundai Elantra Ford Ranger Chrysler LHS

Table C.3: Overview of vehicles tested on a chassis dynamometer. mass engine displacement year cylinders transmission [kg] [cc] UDDS 1996 1996 1996 1996 1996 1992 1996 1996 1994 1994 1994

1325 1470 1838 1050 1962 1070 1365 1110 1198 1365 1579

L4 V6 V6 L4 V6 L4 L4 L4 L4 L4 V6

2163 2999 3998 1590 4916 1865 2294 1590 1835 2294 3523

automatic automatic automatic automatic manual manual manual manual automatic manual automatic

x x x x x x x x x x

available cycles ARTA LA92 FWHS x x

x x

x x

x

x

x

x

x

x

ON-ROAD

C.3

209

On-Road

On-road measurements are taken with 7 vehicles from the Virginia Tech Transportation Institute (VTTI). The followed route goes along the US460 freeway and downtown Blacksburg, see figure C.3. An Auterra Dyno-Scan onboard diagnostics (OBD) reader and a Laipac Tech GD30-L GPS data logger are used to obtain measurements. Wind velocity and direction, temperature, and air pressure are measured at a weather station at the Virginia Tech Montgomery Airport. The measurements are taken at days with calm wind and no precipitation. engine speed The engine speed is directly obtained from the OBD reader. fuel mass flow rate The fuel mass flow rate is directly obtained from the OBD reader. engine torque The load force is calculated with equation (B.10). Dividing the measured wind velocity by two yields the best results. This is due to the fact that the airport is in open field, while big parts of the route are blocked by buildings and trees. The road slope is calculated with the height from the OBD reader and then filtered. The density of the air is calculated with the temperature and pressure from the weather station. The frontal surface and drag coefficient are taken from car specifications on the internet. The rolling friction is based on equation (B.6) and table B.1 on dry good asphalt pavement. The acceleration is calculated with a backward difference, based on the filtered GPS velocity measurements. The total inertia is given by equation (B.24). The vehicle mass takes into account the mass of the passenger and the amount of fuel in the tank. The viscous friction coefficients are based on heuristics. For automatic vehicles, torque multiplication is taken into account based on equation (B.17). It is assumed that the brakes are not used and the minimum torque is limited using equation (B.13). The torque data is interpolated to match the engine speed and fuel mass flow rate measurements. Only hot-stabilized measurements are used (> 80◦ C). Table C.4 shows the 7 vehicles whose measurements are used. Figure C.4 shows an example of the velocity measurement. The measurements do not yield a good maximum torque model. Therefore the identification is based on the engine specifications.

210

FUEL CONSUMPTION MEASUREMENTS

Figure C.3: Route in Blacksburg for on-road measurements.

velocity v [km/h]

100

50

0

0

500

1000

1500

2000

time t [s]

Figure C.4: Example of an on-road velocity measurement.

ON-ROAD

model

Table C.4: Overview of vehicles tested on-road. mass engine displacement year cylinders [kg] [cc]

Oldsmobile Alero GLS Coupé Mazda Protegé sedan Ford Contour LX Ford Explorer XLS 4WD Chevrolet Silverado 2500 ext. cab. Cadillac Escalade AWD Cadillac STS

2002 2003 1999 2000 2000 2002 2006

1399 1653 1256 1870 2389 2565 1790

V6 L4 L4 V6 V8 V8 V8

3350 1191 1988 4015 5967 5967 4565

transmission automatic manual automatic automatic automatic automatic automatic

211

Appendix D

Polynomial Fuel Consumption Models D.1

Polynomial Fuel Consumption Models in the Literature

This section gives a brief overview of low-degree polynomial fuel consumption models that are found in the literature, see table 2.2.

P1: m ˙ f = α1 P.

(D.1)

P2: m ˙ f = α1 + α2 P.

(D.2)

P3: m ˙ f = α1 ω + α2 P.

(D.3)

T1: m ˙ f = α1 ω + α2 ω 2 + α3 ωT.

(D.4)

T2: m ˙ f = α1 + α2 ω + α3 ωT + α4 ω 2 T.

(D.5)

213

214

POLYNOMIAL FUEL CONSUMPTION MODELS

T3: m ˙ f = α1 + α2 ω + α3 ω 2 + α4 ω 3 + α5 ωT.

(D.6)

T4: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 + α4 ωT + α5 ω 2 T + α6 ω 3 T + α7 ωT 2 . (D.7)

T5: m ˙ f = α1 + α2 ω + α3 ω 2 + α4 ωT + α5 T + α6 T 2 .

(D.8)



T6: m ˙ f = α1 ω + α2 ω 2 + α3 ω 3 · β1 + β2 T + β3 T 2 .

(D.9)



L1: m ˙ f = α1 + α2 ω + α3 ω 2 · β1 + β2 τ + β3 τ 2 .

(D.10)

ENGINE MAPS

D.2

215

Engine Maps

engine torque T [Nm]

engine torque T [Nm]

This section shows a comparison between the real and modeled BSFC and BCPR for different polynomial fuel consumption models.

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.1: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model P4.

100

50

1000

2000

3000

engine speed n [rpm]

100 50 0 1000

2000

3000

engine speed n [rpm]

Figure D.2: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model P5.

engine torque T [Nm]

POLYNOMIAL FUEL CONSUMPTION MODELS

engine torque T [Nm]

216

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.3: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model P6.

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.4: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model T1.

100

50

1000

2000

3000

engine speed n [rpm]

100 50 0 1000

2000

3000

engine speed n [rpm]

Figure D.5: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model T4.

217

engine torque T [Nm]

engine torque T [Nm]

ENGINE MAPS

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.6: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model T5.

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.7: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model T7.

100

50

1000

2000

3000

engine speed n [rpm]

100 50 0 1000

2000

3000

engine speed n [rpm]

Figure D.8: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model T8.

engine torque T [Nm]

POLYNOMIAL FUEL CONSUMPTION MODELS

engine torque T [Nm]

218

100

50

1000

2000

3000

100 50 0 1000

2000

3000

engine speed n [rpm]

engine speed n [rpm]

engine torque T [Nm]

engine torque T [Nm]

Figure D.9: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model L2.

100

50

1000

2000

3000

engine speed n [rpm]

100 50 0 1000

2000

3000

engine speed n [rpm]

Figure D.10: Comparison of the real (solid) and modeled (dashed) BSFC and BCPR for model L3.

Appendix E

Optimal Control This chapter gives an overview of the optimal control methods that are used in this dissertation. There are two main classes: discrete and continuous optimal control problems. Eco-driving is a continuous problem. The first section defines what a continuous optimal control is and gives an overview of different methods that can be used to solve this type of problem. Sections E.2–E.5 explain the solution methods that are further used in this dissertation more in detail.

E.1

Optimal Control Overview

Optimal control (also referred to as dynamic optimization) tries to find an optimal control law for a given system or process [47]. This optimal control law minimizes a cost function. Mathematically, a continuous optimal control

219

220

OPTIMAL CONTROL

problem (OCP) looks as follows: Zte

min

x(.), u(.), te

L(x(t), u(t)) dt

+ E(x(te )),

(E.1a)

(initial value)

(E.1b)

0

subject to x(0) − xs = 0, x(t) ˙ − f (x(t), u(t)) = 0,

∀t ∈ [0, te ],

(ODE model)

(E.1c)

h(x(t), u(t)) > 0,

∀t ∈ [0, te ],

(path constraints)

(E.1d)

(end constraints)

(E.1e)

r(x(te )) = 0.

The problem is visualized in figure E.1. The goal is to minimize the objective (cost) function of equation (E.1a). The cost function depends on the state variables x, the controls u, and possibly on the independent variable t (usually the time). It can consist of a Mayer term E and a Lagrange term L. The horizon length te can be fixed or free for optimization. The optimization can be subject to different constraints. Equation (E.1b) gives the initial state values, equation (E.1c) gives the model in ordinary differential equations (ODE), equation (E.1d) gives path constraints, and equation (E.1e) gives the terminal constraints. The above formulation is by far not the most general. Unnecessary notational overhead is avoided by omitting e.g. differential algebraic equations, multi-phase motions, or coupled multipoint constraints, which are, however, all treatable by selected optimal control methods.

6 initial value xs

s

p 0

path constraints h(x, u) > 0 s terminal constraint r(x(te )) = 0

states x(t)

controls u(t) t

pte

Figure E.1: Simplified optimal control problem.

OPTIMAL CONTROL OVERVIEW

221

Generally speaking, there are three basic approaches to address continuous optimal control problems: (a) dynamic programming, (b) indirect methods, and (c) direct methods. (a) Dynamic programming (DP) [20, 22] uses the principle of optimality of subarcs to compute recursively a feedback control for all times t and all x. In the continuous time case, this leads to the Hamilton-JacobiBellman equation, a partial differential equation in state space. Methods to numerically compute solution approximations exist, e.g. [125], yet the approach severely suffers from Bellman’s curse of dimensionality and is restricted to small state dimensions. (b) Indirect methods use the necessary conditions of optimality of the infinite problem to derive a boundary value problem (BVP) in ordinary differential equations, as e.g. described in [32]. This BVP must numerically be solved, and the approach is often sketched as first optimize, then discretize. The class of indirect methods encompasses also the well known calculus of variations and the Euler-Lagrange (EL) differential equations, and Pontryagin’s maximum principle (PMP) [147]. The numerical solution of the BVP is mostly performed by shooting techniques or by collocation. The two major drawbacks are that the underlying differential equations are often difficult to solve due to strong nonlinearity and instability, and that changes in the control structure, i.e. the sequence of arcs where different constraints are active, are difficult to handle: they usually require a completely new problem setup. (c) Direct methods [25, 26, 118, 161] transform the original infinite optimal control problem into a finite dimensional nonlinear programming problem (NLP). This NLP is then solved by variants of state-of-the-art numerical optimization methods, and the approach is therefore often sketched as first discretize, then optimize. One of the most important advantages of direct compared to indirect methods is that they can easily treat inequality constraints, like the inequality path constraints in the formulation above. This is because structural changes in the active constraints during the optimization procedure are treated by well developed NLP methods that can deal with inequality constraints and active set changes. All direct methods are based on a finite dimensional parameterization of the control trajectory, but differ in the way the state trajectory is handled. For solution of constrained optimal control problems in real world applications, direct methods are nowadays by far the most widespread and successfully used techniques. However, for rather simple optimal control problems, indirect methods and dynamic programming still have their value. The analytical

222

OPTIMAL CONTROL

nature of Pontryagin’s maximum principle enables one to get a physical insight in the control problem. Dynamic programming is very easy to implement and can easily handle all sorts of constraints and dynamics. The methods that are further used in this dissertation are: • dynamic programming, • Euler-Lagrange, • Pontryagin’s maximum principle, • direct multiple shooting. These methods are briefly explained in the following sections.

DISCRETE DYNAMIC PROGRAMMING

E.2

223

Discrete Dynamic Programming

Dynamic Programming (DP) is a very strong method to solve optimal control problems. It was developed by Richard Bellman. DP can handle all kinds of constraints, discrete controls, and non-differentiable dynamics. In this dissertation, the continuous control problem is discretized and discrete DP is used. The main disadvantage is that is suffers from the curse of dimensionality. A good textbook on dynamic programming is written by Bertsekas [22]. Basic Algorithm Consider the following discretized optimal control problem: NP −1

min x,u

l(xi , ui )

+ E (xN ) ,

(E.2a)

i=0

subject to x0 − xs = 0, xi+1 − f (xi , ui ) = 0,

i = 0, . . . , N − 1,

hi (xi , ui ) > 0,

i = 0, . . . , N,

r (xN ) = 0,

(initial value)

(E.2b)

(discrete system)

(E.2c)

(path constraints) (E.2d) (end constraints)

(E.2e)

with x ∈ X the discretized state variables and u ∈ U the discretized controls. Dynamic programming can easily get rid of inequality constraints hi and r by giving infinite costs l (x, u) or E(x) to infeasible points (x, u). The basis of dynamic programming is a backward iteration. For i = N − 1, N − 2, . . . , 0: Ji (xi ) = min [l (xi , u) + Ji+1 (f (xi , u))], u | {z }

(E.3)

JN (xN ) = E(xN ),

(E.4)

=J˜i (xi ,u)

starting with:

with J the cost-to-go function. In each step, Ji (x) is obtained for all x ∈ X. Based on J˜i , one can then obtain feedback laws for i = 0, 1, . . . , N − 1: u∗i = arg min J˜i (xi , u), u

(E.5)

224

OPTIMAL CONTROL

where ∗ denotes optimality. For given initial values xs , one can thus obtain the optimal trajectories of xi and ui by the closed-loop system: xi+1 = f (xi , u∗i (xi )) .

(E.6)

This is a forward recursion yielding u0 , . . . , uN −1 . The presented algorithm is the normal backward dynamic programming. DP gives the global optimal solution. Control Space Consider equation (E.3). For all possible discretized controls u ∈ U, J˜ is evaluated. In the normal case, a dicretized control will not make the system evolve to a new state that is exactly a discretized state: f (xi , u) = xi+1 ∈ / X and thus Ji+1 (xi+1 ) can not be evaluated. Therefore it should be interpolated from neighboring grid points. If xi+1 = f (xi , u) can be inverted to u = f ′ (xi , xi+1 ), then a simpler approach can be used: Ji (xi ) = min [l(xi , f ′ (xi , xi+1 )) + Ji+1 (xi+1 )] . xi+1

(E.7)

This is called the inverse approach. Forward Algorithm One can perform the DP algorithm forwardly. A forward iteration is then followed by a backward recursion. This method is called forward dynamic programming. Start with J0 : J0 (x0 ) =



0, if x0 = xs , ∞, if x0 6= xs .

(E.8)

Then perform a forward iteration. For i = 1, 2, . . . , N : Ji (xi ) = min [l(xi−1 , xi ) + Ji−1 (xi−1 )] . xi−1

(E.9)

Forward dynamic programming with the inverse approach is also known as Dijkstra’s algorithm.

DISCRETE DYNAMIC PROGRAMMING

225

Integration Let’s focus on the calculation of l(x, u). When the discrete OCP originates from a continuous OCP, l is a discretization of the Lagrange term L, see e.g. equation (E.1a). Thus: l(xi , u) =

tZi+1

(E.10)

L(x, u) dt.

ti

Depending on the integration scheme, l can have different values. Forward Euler: l(xi , u) = (ti+1 − ti ) · L(xi , u).

(E.11)

Backward Euler: l(xi , u) = (ti+1 − ti ) · L(f (xi , u), u).

(E.12)

Central:

Trapezoidal rule: l(xi , u) = (ti+1 − ti ) ·



 xi + f (xi , u) ,u . 2

(E.13)

L(xi , u) + L(f (xi , u), u) . 2

(E.14)

l(xi , u) = (ti+1 − ti ) · L

The Curse of Dimensionality Dynamic programming is a powerful method. The big disadvantage is the required computation time. The reason is that DP basically tries a lot of different solution, though in a smart way. Assume a time grid of size N . The states x are discretized, with NX possible values. The controls u are discretized with NU possible values. A DP calculation has then a complexity O(N · NX · NU ). When using the inverse approach O(N · NX2 ). Now assume that there is one state variable x, with e.g. NX = 100. The complexity with an inverse approach is O(1002 · N ). If there is a second state, also with 100 possible values, the complexity grows to O(1004 ·N ). This is what is meant with the curse of dimensionality. The complexity (and calculation time) of the problems grows fast with an increasing amount of state and control variables. If one wants to keep the calculation times low, a rough discretization of the continuous problem should be chosen, resulting in a rough and inaccurate result.

226

E.3

OPTIMAL CONTROL

Euler-Lagrange

Consider the following basic control problem: min

Zte

L (x, x, ˙ t) dt,

(E.15)

0

with constraints x(0) = xs and x(te ) = xe . Each function xi that minimizes or maximizes the cost function must satisfy the equation [10, 27]:   d ∂L ∂L − = 0. (E.16) ∂xi dt ∂ x˙ i This is the Euler-Lagrange differential equation and is the fundamental equation of the calculus of variations. It was developed around 1750 by Leonhard Euler and Joseph Louis Lagrange. If the Lagrangian L does not explicitly depend on time, then ∂L/∂t = 0 and the Euler-Lagrange equation simplifies to the Beltrami identity: L−x˙ i

∂L = C, ∂ x˙ i

where C is an integration constant.

(E.17)

PONTRYAGIN’S MAXIMUM PRINCIPLE

E.4

227

Pontryagin’s Maximum Principle

The origins of the maximum principle date back to the calculus of variations and the work by Euler and Lagrange. It was fully developed in the fifties and sixties by Pontryagin and his coworkers [147]. A comprehensive explanation of the maximum principle is written by Kirk [112]. The Basics Regard the following simplified optimal control problem:

min

x(.), u(.), te

Zte

L (x(t), u(t), t)

(E.18a)

dt,

0

subject to x(0) − xs = 0,

(initial value)

(E.18b)

x(t) ˙ − f (x(t), u(t)) = 0,

∀t ∈ [0, te ],

(ODE model)

(E.18c)

h(u(t)) > 0,

∀t ∈ [0, te ],

(control bounds)

(E.18d)

(end constraints)

(E.18e)

x(te ) − xe = 0.

The controls u can be constrained; for simplicity the state variables x are considered unconstrained. To solve the optimal control problem, the Hamiltonian H is defined [32]: H (x(t), u(t), λ(t), t) ≡ L (x(t), u(t), t) + λT [f (x(t), u(t), t)] .

(E.19)

Here, λ are the adjoint state variables or costates. The dynamics of these variables are given by: ∂H (x(t), u(t), λ(t), t) . λ˙ i (t) = − ∂xi

(E.20)

The maximum principle states that an optimal control must minimize the Hamiltonian: H (x(t), u∗ (t), λ(t), t) 6 H (x(t), u(t), λ(t), t) ,

h(u∗ (t)) > 0.

(E.21)

The superscript ∗ denotes optimality. This is a necessary, but not (in general) a sufficient condition for optimality. Possible terminal constraints on the adjoint

228

OPTIMAL CONTROL

variables are given by the following equations: λ(te )T · δxe = 0,

(E.22)

H (x(te ), u(te ), λ(te ), te ) · δte = 0,

(E.23)

and

where δxe,i = 0 if xe,i is constrained and δte = 0 if δte is constrained. Thus when a state variable x has no terminal constraint, the according adjoint state λ has one. The OCP is translated to a boundary value problem (BVP). The actual optimization is done analytically and one has to find the initial values for the adjoint variables λ(0), such that the terminal constraints on states and adjoint states are met. What follows are a few extensions to the basics of the maximum principle. Mayer Term in the Objective A Mayer term E (x(te ), te ) can also be included in the cost function. All the above mentioned equations stay valid, except for equations (E.22) and (E.23) that are replaced by: T  ∂E (E.24) (x(te ), te ) − λ(te ) · δxe = 0, ∂x and   ∂E H (x(te ), u(te ), λ(te ), te ) + (x(te ), te ) · δte = 0. ∂t

(E.25)

Special Conditions for the Hamiltonian Some cases can simplify the maximum principle: • if u(t) is unbounded: ∂H (x(t), u∗ (t), λ(t), t) = 0, ∂u

(E.26)

• if te is fixed and H does not depend explicitly on time, then: dH (x(t), u∗ (t), λ(t), t) = 0, dt

(E.27)

PONTRYAGIN’S MAXIMUM PRINCIPLE

229

• if te is free and H does not depend explicitly on time, then: H (x(t), u(t), λ(t), t) = 0.

(E.28)

Singular Arcs In some cases, the maximum principle of equation (E.21) gives no further information on the optimal controls, i.e. the controls do not affect the Hamiltonian. A time interval in which this occurs is called a singular interval or singular arc. A control that keeps the system on a singular arc is a singular control. The optimal controls can be found by further differentiating equation (E.26) until it explicitly depends on u:   d ∂H (x(t), u(t), λ(t), t) . (E.29) dt ∂u Solution of the Boundary Value Problem A simple method to numerically solve the BVP is single shooting. Start with an initial guess of the values λ(0). The system can then numerically be integrated using equations (E.18c) and (E.20). Finally, check the end constraints. If those are not satisfied, one can iteratively refine the guess of λ(0) with e.g. Newton’s method for root finding.

230

E.5

OPTIMAL CONTROL

Direct Multiple Shooting

Direct methods parameterize the controls u(t), such that the original problem is approximated by a finite dimensional nonlinear program (NLP). This NLP can be solved with specialized methods. Therefore, the approach is often described as first discretize, then optimize. There are three methods to transform the OCP to a NLP: direct single shooting, direct multiple shooting, and direct collocation. Here, only multiple shooting [26] is discussed. The controls are discretized on a time grid t0 , t1 , . . . , tN : u(t) = qi ,

if t ∈ [ti , ti+1 ],

(E.30)

with q the discretized controls. Starting from an artificial initial state value si , the variables in each interval [ti , ti+1 ] can be obtained by a forward integration of the dynamic system, using controls qi : x˙ i (t) xi (ti )

= f (xi , qi ) , = si .

t ∈ [ti , ti+1 ],

(E.31)

In each interval, the cost can be computed: li (si , qi ) =

tZi+1

L (xi (t), qi ) dt.

(E.32)

ti

Then, the following NLP is solved:

min s,q

N −1 X

li (si , qi ) + E(sN ),

(E.33a)

i=0

subject to xs − s0 = 0, xi (ti+1 ) − si+1 = 0, h(si , qi ) 6 0, r(sN ) 6 0.

(E.33b) i = 0, . . . , N − 1,

(E.33c)

i = 0, . . . , N,

(E.33d) (E.33e)

Bibliography [1] [2] [3]

[4]

[5]

[6]

[7] [8]

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Curriculum Vitae Personal Data Bart Saerens born February 22, 1984 in Asse (Belgium) Belgian, male [email protected] Education 1996–2002 Secondary school (wetenschappen-wiskunde) at Sint-Donatus Instituut (Merchtem, Belgium), graduated summa cum laude. 2002–2007 Mechanical-Electrotechnical Engineering, specialization Aerospace Engineering, at Katholieke Universiteit Leuven. Graduated magna cum laude, with a Master’s thesis titled “Minimalisatie van het brandstofverbruik van een benzinemotor door optimale gasklepsturing”. This thesis was rewarded with the third place at the K VIV-Ingenieursprijzen. 2007–2012 PhD programme at Katholieke Universiteit Leuven, funded by a PhD grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen). In 2009–2010, five months were spent at the Virginia Tech Transportation Institute in the USA.

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List of Publications Articles in international journals [1] B. Saerens, J. Vandersteen, T. Persoons, J. Swevers, M. Diehl, and E. Van den Bulck. Minimization of the fuel consumption of a gasoline engine using dynamic optimization. Applied Energy, 86(9):1582–1588, 2009. doi:10.1016/j.apenergy.2008.12.022 [2] H. Rakha, K. Ahn, K. Moran, B. Saerens, and E. Van den Bulck. Virginia Tech Comprehensive Power-based Fuel Consumption Model: Model Development and Testing. Transportation Research Part D: Transport and Environment, 16(7):492–503, 2011. doi:10.1016/j.trd.2011.05.008 [3] B. Saerens, H. A. Rakha, K. Ahn, and E. Van den Bulck. Assessment of alternative polynomial fuel consumption models for use in ITS applications. Submitted to Journal of Intelligent Transportation Systems. [4] B. Saerens, H. A. Rakha, M. Diehl, and E. Van den Bulck. EcoCruise Control for Passenger Vehicles: Methodology. Submitted to Transportation Research Part D: Transport and Environment.

Articles in national journals [1] B. Saerens and J. Vandersteen. Brandstof besparen met intelligente software. Het ingenieursblad, (6):25–35, June 2008.

Book chapters [1] B. Saerens, M. Diehl, and E. Van den Bulck. Optimal Control Using Pontryagin’s Maximum Principle and Dynamic Programming, in 249

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LIST OF PUBLICATIONS

Automotive Model Predicitive Control: Models, Methods and Applications, volume 402/2010 of Lecture Notes in Control and Information Sciences, pp. 119–138. Springer, Berlin/Heidelberg, 2010.

Articles in international conference proceedings [1] B. Saerens, M. Diehl, J. Swevers, and E. Van den Bulck. Model Predictive Control of Automotive Powertrains - First Experimental Results. In Proc. 47th IEEE Conf. on Decision and Control, pp. 5692–5697, Cancun, Mexico, Dec. 9–11, 2008. [2] M. Kopacka, B. Saerens, H. J. Ferreau, B. Houska, M. Diehl and B. RohalIlkiv. Design of a MPC controller running on dSpace hardware using ACADO toolkit. In Proc. 9th Int. Conf. on Process Control, Kouty nad Desnou, Czech Republic, June 7–10, 2010. [3] H. A. Rakha, K. Ahn, K. Moran, B. Saerens, and E. Van den Bulck. Simple Comprehensive Fuel Consumption and CO2 Emission Model based on Intstantaneous Vehicle Power. In Proc. 90th Transportation Research Board Annual Meeting, Washington, D.C., USA, Jan. 23–27, 2011. [4] S. Park, H. A. Rakha, K. Ahn, K. Moran, B. Saerens, and E. Van den Bulck. Predictive Eco-Cruise Control System: Model Logic and Preliminary Testing. In Proc. 91th Transportation Research Board Annual Meeting, Washington, D.C., USA, Jan. 22–26, 2012. [5] B. Saerens, H. A. Rakha, and E. Van den Bulck. Assessment of EcoCruise Control Calculation Methods. In Proc. 91th Transportation Research Board Annual Meeting, Washington, D.C., USA, Jan. 22–26, 2012.

Abstracts in international conference proceedings [1] B. Saerens, J. Vandersteen, J. Swevers, M. Diehl, and E. Van den Bulck. Dynamic optimization for minimizing the fuel consumption of a gasoline engine. In Proc. 27th Benelux Meeting on Systems and Control, pp. 126, Heeze, The Netherlands, March 18–20, 2008. [2] B. Saerens, M. Diehl, J. Swevers, and E. Van den Bulck. Optimizing NMPC of a gasoline engine. In L. Magni, D. M. Raimondo, and R. Allgöwer, editors, Proc. Int. Workshop on Assessment and Future Directions of NMPC, pp. 38, Pavia, Italy, Sept. 5–9, 2008.

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[3] B. Saerens and E. Van den Bulck. Minimum-fuel control of combustion engine powertrains. In Proc. 28th Benelux Meeting on Systems and Control, pp. 136, Spa, Belgium, March 16–18, 2009. [4] B. Saerens, E. Van den Bulck, and M. Diehl. Linear Approximation of Minimum-Fuel Vehicle Accelerations. In Book of Abstracts of the 14th Belgian-French-German Conf. on Optimization, pp. 143, Leuven, Belgium, Sept. 14–18, 2009. [5] B. Saerens and E. Van den Bulck. Analysis of Eco-Driving Using Polynomial Fuel Consumption Models. In Proc. 30th Benelux Meeting on Systems and Control, pp. 102, Lommel, Belgium, March 15–17, 2011.

Abstracts in national conference proceedings [1] B. Saerens, M. Diehl, and E. Van den Bulck. Minimum-fuel vehicle driving with optimal control. In Proc. GRASMECH 2008 poster-day, Royal Military Academy, October 3, 2008.

Arenberg Doctoral School of Science, Engineering & Technology Faculty of Engineering Department of Mechanical Engineering TME Celestijnenlaan 300A B-3001 Heverlee