CONTROLLER DESIGN FOR A UNIVERSAL POWER INPUT BI-DIRECTIONAL BATTERY CHARGER FOR PLUG-IN ELECTRIC AND HYBRID ELECTRIC VEHICLES EMILIO DAL SANTO DIPARTIMENTO DI INGE

CONTROLLER DESIGN FOR A UNIVERSAL POWER INPUT

BI-DIRECTIONAL BATTERY CHARGER

FOR PLUG-IN ELECTRIC AND HYBRID ELECTRIC VEHICLES

EMILIO DAL SANTO

DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE

Corso di Laurea Magistrale in Ingegneria dell’Automazione

Facolta’ di Ingegneria

Universita’ degli Studi di Padova

Relatore Prof. Silverio Bolognani

iii

Page

ACKNOWLEDGEMENT . . . iii

LIST OFTABLES . . . vi

LIST OFFIGURES . . . xii

ABSTRACT . . . xiii

CHAPTER 1. INTRODUCTION . . . 1

2. AN OVERVIEW ONPHEV POWERELECTRONIC CONVER-TERS . . . 5

2.1. Plug-in Hybrid Ele tri Vehi le . . . 5

2.2. Bi-dire tionalConverters . . . 8

2.3. AC/DC Converters . . . 9

2.4. DC/DC Converters . . . 11

2.5. Battery Model . . . 12

2.6. Design Considerationand Modeling . . . 13

2.7. ControlTe hniques . . . 15

2.8. StabilityIssues . . . 16

3. CONVERTER DESIGN . . . 18

3.1. Boost Converter . . . 19

3.2. Bu k Converter . . . 30

3.3. Bi-dire tionalBu k-Boost Converter . . . 39

3.4. Boost Re tier and PowerFa tor Corre tion . . . 51

3.5. Overall BatteryCharger Model . . . 72

4. DIGITAL CONTROL . . . 94

4.1. Digital Controlfor Swit hing Converters . . . 96

4.2. Sliding Mode Control . . . 97

4.3. True DigitalControl . . . 100

4.4. True DigitalControlfor Swit hing Converters . . . 103

4.5. Comparison WithConventional AnalogController . . . 114

5. STABILITY ANALYSIS AND ISSUES . . . 129

6. CONCLUSION . . . 172

7. FUTURE WORKS . . . 174

APPENDIX . . . 176

A. CIRCUIT ANDCONTROLLER PARAMETERS . . . 176

Table Page

A.1 Values of the parameters for the ir uit omponents. . . 177

Figure Page

2.1 Plug-in HybridEle tri Vehi le s hemati view, pi ture ourtesy of

Argonne NationalLaboratory. . . 6

2.2 Non-lineartime-invariantstate spa e s heme. . . 12

3.1 Basi blo k representation of the ir uits hemati . . . 18

3.2 IdealBoost ir uit representation. . . 21

3.3 Equivalent Boost ir uit representation for

S = ON

,

D = OF F

,

u = 0

. . . 21

3.4 Equivalent Boost ir uit representation for

S = OF F

,

D = ON

,

u = 1

. . . 22

3.5 Non-Ideal Boost ir uit representation. . . 24

3.6 Equivalent Boost ir uit representation for

S = ON

,

D = OF F

,

u = 0

. . . 25

3.7 Equivalent Boost ir uit representation for

S = OF F

,

D = ON

,

u = 1

. . . 26

3.8 IdealBu k ir uit representation. . . 31

3.9 Equivalent Bu k ir uit representation for

S = ON

,

D = OF F

,

u = 0

. . . 31

3.10 Equivalent Bu k ir uit representation for

S = OF F

,

D = ON

,

u = 1

. . . 32

3.11 RealBu k ir uit representation. . . 34

3.12 Equivalent Bu k ir uit representation for

S = ON

,

D = OF F

,

u = 0

. . . 35

3.13 Equivalent Bu k ir uit representation for

S = OF F

,

D = ON

,

u = 1

. . . 35

3.14 Idealbi-dire tional Bu k-Boost ir uit representation. . . 40

3.15 EquivalentBu k ir uit representation forbi-dire tional ir uit. . . 41

3.16 EquivalentBoost ir uitrepresentation for bi-dire tional ir uit. . 42

3.19 EquivalentNon-IdealBoostmode ir uitrepresentationforbi-dire tional

ir uit. . . 44

3.20 Outputvoltageand urrentofBu k-Boost onverter, inBu k mode of operation. . . 49

3.21 Outputvoltageand urrentofBu k-Boost onverter,inBoostmode of operation. . . 50

3.22 FullWave BridgeRe tier ir uit representation. . . 52

3.23 PowerFa tor Corre tion ir uit representation. . . 54

3.24 H BridgeRe tier ir uitrepresentation. . . 56

3.25 Idealbi-dire tional PFC ir uit representation. . . 57

3.26 Idealbi-dire tionalPFC ir uitrepresentation,inAC/DCoperating mode. . . 58

3.27 Idealbi-dire tionalPFC ir uitrepresentation,inDC/ACoperating mode. . . 59

3.28 Bi-dire tionalPFC ir uit representation. . . 60

3.29 Bi-dire tionalPFC ir uitrepresentation,inAC/DC operatingmode. 61 3.30 Bi-dire tionalPFC ir uitrepresentation,inDC/ACoperatingmode. 63 3.31 Outputvoltage ripple of aNon-PowerFa tor regulated Boost Con-verter. . . 66

3.32 Indu tor urrent waveform of a Non-Power Fa tor regulated Boost Converter. . . 66

3.33 Outputvoltage waveformof a Power Fa tor regulated Boost re tier. 69 3.34 Indu tor urrentwaveformofaPowerFa torregulatedBoost re tier. 69 3.35 Outputvoltage waveformof a DC/AC PFCinverter. . . 70

3.36 Indu tor urrent waveformof a DC/AC PFCinverter. . . 71

3.37 EquivalentNon-Ideal ir uit representation for harger. . . 74

3.38 Input urrent and sinusoidal voltagereferen e plots. . . 76

3.41 DC/DC onverter output waveforms plots. . . 78

3.42 Bode plots forthe four transferfun tions orrespondingto

d

AC

= 1

and

d

DC

= 1

. . . 81

3.43 Rootlo usdiagram forthe fourtransfer fun tions orresponding to

d

AC

= 1

and

d

DC

= 1

. . . 81

3.44 Bode plots forthe four transferfun tions orrespondingto

d

AC

= 0

and

d

DC

= 1

. . . 84

3.45 Rootlo usdiagram forthe fourtransfer fun tions orresponding to

d

AC

= 0

and

d

DC

= 1

. . . 85

3.46 Bode plots forthe four transferfun tions orrespondingto

d

AC

= 1

and

d

DC

= 0

. . . 87

3.47 Rootlo usdiagram forthe fourtransfer fun tions orresponding to

d

AC

= 1

and

d

DC

= 0

. . . 87

3.48 Bode plots forthe four transferfun tions orrespondingto

d

AC

= 0

and

d

DC

= 0

. . . 90

3.49 Rootlo usdiagram forthe fourtransfer fun tions orresponding to

d

AC

= 0

and

d

DC

= 0

. . . 90

4.1 S hemati ir uit of proposed true digital ontrol. . . 101

4.2 Simulinkblo k s hemeof True DigitalControl. . . 102

4.3 Regulated DC bus voltage waveform with typi al os illations at twi e the linefrequen y. . . 106

4.4 Powerfa tor orre ted sinusoidalinput urrent. . . 107

4.5 Regulated output urrent and voltage waveforms. . . 110

4.6 Simulinkmodel of xed frequen y True DigitalController. . . 112

4.7 Comparison of input urrents for xed and variable frequen y PWM. 112 4.8 ComparisonofDCbusvoltageforxedand variablefrequen y PWM. 113 4.9 Comparison of the input sinusoidal urrents. . . 115

4.10 Detailof the usp distortioninthe input urrents. . . 116

4.13 Comparison between the two DC bus voltage waveforms and input

voltage. . . 118

4.14 Comparison of the two DC bus voltage waveforms. . . 119

4.15 Comparison of the two output urrent waveforms. . . 119

4.16 Comparison of the two input urrent transients. . . 121

4.17 Comparison of the two DC bus voltage transients. . . 121

4.18 Comparison of the two output urrent transients. . . 122

4.19 Comparisonof the twoinput urrentwaveforms fora hange inthe input voltage. . . 124

4.20 Comparison of the two DC bus voltage waveforms for a hange in the input voltage. . . 125

4.21 Comparison of the twooutput waveforms for a hangein the input voltage. . . 125

4.22 Comparisonofthe input urrent waveforms fora hangeinthe out-put load. . . 126

4.23 Comparison of the two DC bus voltage waveforms for a hange in the output load. . . 127

4.24 Comparisonofthetwooutputwaveforms fora hangeinthe output load. . . 128

5.1 Stable mode of operationseen inthe sinusoidal input urrent. . . 130

5.2 Stable mode of operation seen in the DC bus voltage and output urrent and voltage waveforms. . . 131

5.3 Stable mode of operation seen in the DC bus voltage and output urrent and voltage waveforms. . . 132

5.4 Input urrent waveform and FFTanalysis for

R

out

= 1.2 Ω

. . . 134

5.5 DCbusvoltageandoutput urrentandvoltagewaveformsfor

R

out

=

1.2 Ω

. . . 135

5.6 Phase plane traje tories for

R

out

= 1.2 Ω

.. . . 135

0.5 Ω

. . . 137

5.9 Phase plane traje tories for

R

out

= 0.5 Ω

.. . . 137

5.10 Input urrent waveform and FFTanalysis for

R

out

= 0.3 Ω

. . . 138

5.11 DCbusvoltageandoutput urrentandvoltagewaveformsfor

R

out

=

0.3 Ω

. . . 139

5.12 Phase plane traje tories for

R

out

= 0.3 Ω

.. . . 139

5.13 Input urrent waveform and FFTanalysis for

L

1

= 150 µH

. . . . 141

5.14 DCbus voltageand output urrentand voltage waveformsfor

L

1

=

150 µH

. . . 141

5.15 Phase plane traje tories for

L

1

= 150 µH

. . . 142

5.16 Input urrent waveform and FFTanalysis for

L

1

= 50 µH

. . . 143

5.17 Phase plane traje tories for

L

1

= 50 µH

. . . 143

5.18 Input urrent waveform and FFTanalysis for

L

1

= 500 µH

. . . . 144

5.19 DCbus voltageand output urrentand voltage waveformsfor

L

1

=

500 µH

. . . 145

5.20 Phase plane traje tories for

L

1

= 500 µH

. . . 145

5.21 Input urrent waveform and FFTanalysis for

L

1

= 1000 µH

. . . . 146

5.22 DCbus voltageand output urrentand voltage waveformsfor

L

1

=

1000 µH

. . . 147

5.23 Phase plane traje tories for

L

1

= 1000 µH

. . . 147

5.24 Input urrent waveform and FFTanalysis for

C

1

= 1000 µF

. . . . 149

5.25 DCbus voltageandoutput urrent andvoltagewaveforms for

C

1

=

1000 µF

. . . 149

5.26 Input urrent waveform and FFTanalysis for

C

1

= 500 µF

. . . 150

5.27 DCbus voltageandoutput urrent andvoltagewaveforms for

C

1

=

500 µF

. . . 151

5.28 Phase plane traje tories for

C

1

= 500 µF

. . . 151

5.31 Stable mode of operation seen in the DC bus voltage and output

urrent and voltage waveforms. . . 154

5.32 Input urrent waveform and FFTanalysis for

R

out

= 1 Ω

. . . 155 5.33 DCbusvoltageandoutput urrentandvoltagewaveformsfor

R

out

=

1 Ω

. . . 156 5.34 Phase plane traje tories for

R

out

= 1 Ω

. . . 156 5.35 Input urrent waveform and FFTanalysis for

R

out

= 0.5 Ω

. . . 157 5.36 DCbusvoltageandoutput urrentandvoltagewaveformsfor

R

out

=

0.5 Ω

. . . 158 5.37 Phase plane traje tories for

R

out

= 0.5 Ω

.. . . 158 5.38 Input urrent waveform and FFTanalysis for

R

out

= 0.3 Ω

. . . 159 5.39 DCbusvoltageandoutput urrentandvoltagewaveformsfor

R

out

=

0.3 Ω

. . . 160 5.40 Phase plane traje tories for

R

out

= 0.3 Ω

.. . . 160 5.41 Input urrent waveform and FFTanalysis for

L

1

= 50 µH

. . . 162 5.42 DCbus voltageand output urrentand voltage waveformsfor

L

1

=

50 µH

. . . 162 5.43 Phase plane traje tories for

L

1

= 50 µH

. . . 163 5.44 Input urrent waveform and FFTanalysis for

L

1

= 1000 µH

. . . . 164 5.45 DCbus voltageand output urrentand voltage waveformsfor

L

1

=

1000 µH

. . . 165 5.46 Phase plane traje tories for

L

1

= 1000 µH

. . . 165 5.47 Input urrent waveform and FFTanalysis for

C

1

= 1000 µF

. . . . 167 5.48 DCbus voltageandoutput urrent andvoltagewaveforms for

C

1

=

Power ele troni onverters in Plug-in Hybrid Ele tri Vehi les (PHEV) and

Ele tri Vehi les (EV) require high power and bidire tional power ow apabilities,

with wide input voltage range. This thesis presents a battery harger designed to

operateovera universalinput. The onverterisimplemented usingabasi two-stage

stru ture. The rst stagein ludesa Power Fa torCorre tion (PFC) Boost onverter

to meet power fa tor requirements and improve the e ien y of the system. The

se ond stage is omprised of a bu k onverter whi h is dire tly onne ted to the

batterypa k. Controlofthis onverter hasbeen implementedusingamulti-loopPID

ontroller for the PFC re tier using average urrent ontrol mode. A simple PID

ontroller is implemented in the DC/DC onverter and they both use Pulse Width

Modulation swit hing. In addition, this thesis presents a new digital approa h to

eliminate the feed-forward ontroller from the onventional topology, whi h further

simplies the ontrol strategy. Digital ontrol of a power ele troni system uses a

binary swit hing strategy for the swit hes. High-powersemi ondu tors swit h based

on the sign of the error whi h is used as ontrol signal. A thorough evaluation of

the onverter has been ondu ted to assess the performan e and robustness of the

ontrollerinterms of stability. FFT analysisand phase-planeplots are used inorder

to derive stability maps for the ir uit. Unstable onditions are found and system

CHAPTER 1

INTRODUCTION

In re ent years, Hybrid Ele tri Vehi les (HEV) and Plug-in Hybrid Ele tri

Vehi les (PHEV) have attra ted more and more attention of automotive industry.

Hybrid vehi les have several advantages over onventional ar given their e ien y

and apability of a better fuel e onomy. PHEVs ombine the Internal Combustion

Engine (ICE) with the abilityof harging and dis hargingastorage pa k. It an use

theele tri itystoredwhilethebattery hargeisinahighstate,allowinganall-ele tri

range. At the same time PHEV provides a fuel tank to be used when an extended

driving range isneeded.

Abattery hargerisessentialforthePHEVfun tioning. Thispowerele troni

ir uithas two mainfun tions. It hargesthe battery withaproperStateOfCharge

(SOC) in re harge mode of operation. The other operation mode is alled inverter

mode, whi h means that the battery energy is transferred ba k to the grid. Also

supplyingACele tri ityforon-boardloadsispossible. Thereforethe battery harger

onsists ina multi- onverter system apableof bi-dire tional power ow.

In multi- onverter systems many power ele troni onverters su h as AC/DC

re tiers, DC/DC hoppers, and DC/AC inverters are used as sour es, loads, or

distribution networks to provide power in dierent magnitudes and forms. Re ent

advan ementsinsemi- ondu torte hnologyhaveenhan edtheuseofthese onverters

inPlug-InHybridVehi lesappli ations[57℄. Amulti-stage onversionis onsidereda

ommon hoi eforabattery harger ir uitin[29℄and[22℄. Itin ludesare ti ation

stage and is usually as aded with anoutput regulator.

of the appli ations in terms of e ien y, reliability, ost, volume and weight. Two

main strategies involve an AC/DC bi-dire tional onverter whi h an be separated

from the driving system. The other one ombines the motor driving inverter with

the onverter as anintegrated PHEV motor driving system. Several resear h papers

have been written on the design and the analysis of the onverters, espe ially on a

stand-alone basis [73℄.

Basi stru ture onsists in the as ade of a AC/DC re tier and a DC/DC

onverter pla ed between the battery and the high voltage bus. This thesis in ludes

an example of a bi-dire tional ir uit that an be used with Plug-in Hybrid Ele tri

Vehi les. Thesepowerele troni ir uits anbealsointegratedwithexisting gasoline

orele tri vehi lesto provideplug-in features.

Control ir uitsalsorepresentafundamental omponentof onsideredsystem.

It is responsible to provide a regulated and at urrent at the output to harge the

batterypa k. PowerFa torRegulationalsoneedstobedone forthe input urrentin

order to maximize the e ien y of the system. A new ontrol approa h is analyzed

with the goal to simplify the hardware stru ture. In fa t, lassi al analog ontrol

te hniques need a ompli ated implementation. It usually onsists in a multi-loop

ontroller for the PFC ir uit, in luding a feed-forward ompensator [26℄. DC/DC

onverter stage on the other hand uses only a single PID regulator. Therefore a

new approa h needs to be developed in order to simplify the design of the ontrol

ir uitwhileassuringgoodperforman esandstableoperation. Anoveldigital ontrol

providesa reliableand robust solution forthis appli ation.

Stable behavior of the system has to be assured by the ontroller. Cir uit

designandthe hoi eof riti al omponentsareresponsibleforthegoodperforman es

analysis has to be performed a ordingly tosome pra ti al riteria.

A stable system with desired response is obtained with the use of the novel

digital ontroller. Performed resear h work results to be essential to provide a safe

and reliablesystem. Its stable and sustainable operation are of primary importan e

for riti alpower ele troni ir uits su h asin Plug-inHybrid Ele tri Vehi les.

This thesis has been organized as follows. In the se ond hapter, a brief

in-trodu tionon the appli ation and onits re ent developments has been done. PHEV

on ept is introdu ed and standard ontrol methods are explained. A omplete

lit-erature overview is provided on the most signi ant resear h topi s that have been

ondu ted. Powerele troni onverters designisanalyzedand various ontrol

strate-gies are reviewed, onsidering their performan es and robustness. Importan e of the

bi-dire tionalmulti- onverterbattery harge isunderlinedandvarious ongurations

are presented.

The third hapter des ribes in details the ir uit adopted for the battery

harger appli ation. A omplete analysis is done for the omponents of ea h

on-verter as well as for the overall ir uits. Bu k onverter and Boost onverter are

ombined together inorder to obtain the Bu k-Boost topologyand its bi-dire tional

version. A Power Fa tor Corre tion ir uit is developed to meet stringent

require-ments of this appli ation. Its bi-dire tional version is then ombined together with

theDC/DC onverterintheoverallmodel. Atwo-stagebi-dire tionalbattery harger

is then des ribed and its dierentialequations are derived. Parametri al models are

derived and dierent mathemati al representations of the ir uit are provided. The

state spa e model of ombined ir uit is used in this thesis as the primary analysis

tool toinvestigatethe ontrol designand the stability of the system.

digital design. Digital ontrol te hniques are analyzed and their advantages over

analog regulators are shown. Digital ontrol takes advantage of its exible stru ture

and of its ease of implementation in modern integrated ir uits. A novel approa h

is here presented and its performan es are shown. A detailed omparison with the

lassi alPIDregulatorisalsoperformed. Relatedissuesandpra ti alsolutions,along

with other possible implementationsare presented for the ontroldesign.

In hapter ve, a detailed stability analysis is presented. Classi al tools are

used inorder toinvestigatethe stability of the system through apra ti alapproa h.

Instability onditions are des ribed and dete ted in the operating onditions of the

ir uit. Unstable regions are identied with respe t to ir uits riti al omponents

values. Cir uit design and the hoi e of riti al omponents are thus explained with

the use of stability analysis tools. The performan e of the new ontroller design is

ompared with the lassi al analog ontrol in terms of stability. Robustness of the

ontroller is investigated through the simulation of a variation in the input voltage

orin the load, and its performan es are ommented.

Chapter six dis usses obtained resultsshowing allthe advantagesof proposed

onguration. Analysis results are explained and summarized in this hapter. Need

forfuture worksisdis ussed,in ludingafurthersimpli ationinthe ontroller. Also

a mathemati al onrmation of pra ti al results obtained has to be done. Derived

mathemati al model in all its dierent ongurations an be adopted for a more

CHAPTER 2

AN OVERVIEW ON PHEV POWER ELECTRONIC CONVERTERS

Primarily due to an in reasing environmental ons iousness and a fuel pri e

lift over the last few years, Plug-in Hybrid Ele tri Vehi le market and resear h

interesthas widelygrown. Plug-inHybridEle tri Vehi lesarevehi lesthat ombine

an internal ombustion engine and an ele tri operating energy system, in luding

batteries and power ele troni s ir uits. In parti ular, two spe i and fundamental

ir uits have been analyzed and will be modeled in this thesis, the AC/DC inverter

and the DC/DC onverter, pla edbetween the external universal AC outletand the

battery pa k. Integrated onverters need to be bi-dire tional, in order to let the

energy ow in either dire tion. Ele tri alenergy an beeither stored tothe battery

pa k, from an external power supply or through regenerative braking, or used to

supply powerto anele tri almotor oron board devi es.

Athoroughresear hhasbeen ondu tedonthesetopi s,in ludingdesign

on-siderationsand simulationsof singlephase bi-dire tionalAC/DC inverterwith boost

PowerFa torCorre tionsystem,bi-dire tionalDC/DCswit hing onvertersand

bat-teryevaluation,suitablefor high powerPlug-in HybridEle tri Vehi leappli ations.

2.1 Plug-in Hybrid Ele tri Vehi le

Plug-inhybrid-ele tri vehi leshave re ently emerged asapromising

te hnol-ogy that uses ele tri ity todispla e asigni ant fra tion of petroleum onsumption.

A plug-in hybrid ele tri vehi le (PHEV) is a hybrid vehi le with the ability

to re harge itsenergy storage system with ele tri ity froman o-board sour e, su h

as the ele tri utility grid. Similarly to traditional hybrid ele tri vehi les, it has

Figure 2.1. Plug-in Hybrid Ele tri Vehi le s hemati view, pi ture ourtesy of

Ar-gonne National Laboratory.

harge (SOC), thereby using ele tri ity to displa e liquid fuel that would otherwise

be onsumed. This liquid fuel is typi ally petroleum (gasoline or diesel), although

PHEVs an also use alternatives su h as biofuels or hydrogen [38℄. PHEV batteries

typi ally have larger apa ity than those in HEVs so as toin rease the potential for

petroleum displa ementand all-ele tri range apabilities.

Comparedto onventionalvehi les,PHEVs an redu eairpollution,minimize

dependen e on petroleum and fossil fuels, and lower greenhouse gas emissions that

ontributeto globalwarming. In fa t, PHEVs an avoid use of any fossil fuelduring

theirall-ele tri rangeiftheirbatteriesare hargedfromnu learorrenewablesour es

of energy. In addition to redu ing gasoline onsumption, they have the potential to

also redu e total energy expenses. Existing ommer ial hybrid vehi les have proven

to be su essful omponents of the transportation system in the US and abroad.

Plug-inhybridele tri vehi les(PHEV) an ontributesigni antlytotransportation

system e ien y by introdu ing vehi les that, within a limited range, an operate

entirely in an ele tri mode and be powered by the ele tri ity grid. Conventional

e ient existing hybrids ut gasoline onsumption by around

40

per ent ompared with similar onventional ars. But PHEVs typi ally repla e half of the remaining

gasoline onsumptionwithele tri ity. Thus PHEVs ouldredu e the onsumptionof

liquidfuels by at least

70

per ent ompared with onventional ars.

Oneofthebasi yetimportant omponentsofaPHEVisitsdrivetrain,whi h

in lude the ele tri al motor drive, storage devi e (battery pa k), ontrol ele troni s,

inverter and battery harging ir uit. In parti ular, the power ele troni onverter is

responsible of the power ow from the ele tri al sour e to the load and vi e versa,

allowing the hargingand the dis harging of the battery. This basi omponent will

be here analyzed in details, and an e ient ontrol system will be designed and

dis ussed.

Plug-inHybrid Ele tri Vehi lesand ele tri ars may allowformore e ient

use of existing sour es of ele tri energy, whi h most of the time is unused or is

available asan operating reserve of power in the storage system. This assumes that

vehi les are harged primarily during o peak periods, or equipped with te hnology

toshut o harging duringperiodsof peak demand. Another advantage of aplug-in

vehi le is their potential ability to help the grid during peak loads. This is

a om-plished with vehi le-to-grid te hnology hargers [57℄. Su h vehi les take advantage

of ex ess battery apa ity to send power ba k into the grid and then re harge

dur-ing o peak times using heaper power. Su h vehi les are a tually advantageous to

utilities as well as their owners. Even if su h vehi les mat led to an in rease in the

use ofnighttime ele tri itythey would alsoout ele tri itydemand whi histypi ally

higherinthedaytime. This wouldrepresentagreaterreturnon apitalforele tri ity

2.2 Bi-dire tional Converters

Bidire tional onvertersarenowadayswidelyusedinvariousappli ations,and

are amongthemost studiedPowerEle troni 's ir uits. Appli ationssu hasele tri

vehi les, photovoltai systems, UPS power supplies, general battery based storage

systems and various industrial elds require the development of bi-dire tional

on-verters in order to allow power ow in either dire tion. They are usually employed

asinterfa e ir uitsfor the dierent voltagelevelbuses and have several advantages.

Among all, saving spa e, redu tion of weight and ost of the power systems with

respe ttostandardunidire tional ir uits. Oneofthe most ommonappli ationsisa

battery harger ir uit,inwhi hboth harging(i.e. energy storage), and dis harging

(energy onsumption) methods are implemented, through bi-dire tional power

on-version. Various topologies have been studied, a ordingly to spe i requirements,

based on power apabilities, isolation, input/output relations and onversion type,

number of stages and phases.

Unidire tional onverters anbe lassied intotwobasi ategories, a ording

tovoltage onversion type,DC/DC onverterandtheAC/DC onverter,respe tively

in ludingaDCvoltagepowersour e,oranACvoltageinput,andprovidingadierent

output value of DC voltage. Classi topologies in lude Bu k DC/DC onverters, in

whi h the output voltage is smaller than the input value, boost ir uit in whi h

output voltage is greater than the input, and bu k-boost onverters, in whi h the

output an be either smaller or greater than the input. Similarly AC re ti ation

an be made, through bu k re tiers or boost re tiers. Bi-dire tional power ow

requires those standard ir uitsto be modiedto a ommodate two alternate power

sour es and loads and bi-dire tional operations, thus using DC/DC bi-dire tional

using a as aded onverter system. Whereasunidire tional ir uitshave been widely

dis ussed, bi-dire tional ir uits stillform ana tive resear h eld [30℄.

In the ases where isolation is required, most of the existing bi-dire tional

onverters are of the yba k-forward topologies [11℄. These bu k and boost derived

onverters in ludea transformer in the ir uit whi h provides ele tri alisolation

be-tween the input and the output ports. Built in transformers ele tri ally isolate the

input of the onverter fromitsoutput. Notonly they owe allthe advantages of high

frequen y operation, smallsize and weight of the transformer, but alsoprovidemore

exibilityinaneventualmultipleoutput ontexts[63℄. Infa tisolatedtopologyallows

multi-inputs andmulti-outputs ongurationsusing multi-windingtransformers that

onne t multiplesour eshavingdierentvoltagelevels,orgiveagreaterexibilityin

outputs.

Forhigh-e ien y and high-power appli ationssu h asPHEVs,whi hdo not

require magneti oupling, standard Bi-dire tional swit hing onversion is adopted.

Insu hsystemstheloadisdire tly onne tedtothegridthrougha onversion ir uit.

Resear hers have analyzed several PWM swit hing te hniques, to eliminate

transition responses and to provide soft-swit hing [56℄ using auxiliary and omplex

ir uits,inordertoin reasethesystem'soveralle ien y. Moreover,allthese

swit h-ing onverters an provide voltage regulation as well as prote tion in ase of power

outages. In addition they also show ex ellent performan e in terms of suppressing

in ominglinetransient and harmoni disturban es.

2.3 AC/DC Converters

The rst stage of the power ele troni system for PHEV appli ations

appli ations, to provide highlystable DC voltage at the output while maintaininga

high power fa tor at the input. This onverter is extremely useful in several power

onversiondevi es[55℄andalsomeetsqualityspe i ationsandguidelinestoregulate

power quality. One of main issues of non-power fa tor orre ted ir uits is the

har-moni distortionthat isinje ted ba k into the mainspower line. Furthermorein the

United States and Europe, Federal Communi ations Commission (FCC) and

Euro-pean Asso iationfor Ele tri al,Ele troni ,and InformationTe hnologies, havelately

worked together to introdu e a series of stri t standards to govern ondu ted-noise

emissions and maximum ondu ted noise limits. For this reason an a tive ontrol

ir uit needsto be implemented.

PWM unidire tional inverter in ludes a diode re tier ir uit at the input,

to onvert sinusoidal

90 − 240 V AC

,

50 − 60 Hz

universal voltage input into a DC re tied voltagewaveform. AC/DC ir uit alsorequiresaswit hing boost onverter,

whi h regulates output voltage to an almost onstant value, higher than the input.

Aninverterisadevi ethat onvertsDC urrentfromtheoutputofDC/DC onverter

orthe batteryintoACwhi h an beused forele tri motordrives, andvi eversa. It

istypi ally omprisedofapowermodulein ludinghighpowersemi ondu tor devi es

with high urrent apabilities as BJTs or MOSFETs, sensors, lters and a ontrol

system that regulates the swit hing s heme.

Considering the spe i PHEV requirements,where power is drawn from the

AC side to feed the battery, DC re tied power may also be used for the

ele tri- al motor or the on board ele tri al system, as well for vehi le-to-grid appli ations.

For this spe i use a bi-dire tional ir uit needs to be hosen, and furthermore the

bridge diode re tier is modied into an H-bridge onguration, for power reversal

re-indu tan e needstobeshaped, by onlya s alingfa tor of the ACvoltagewaveform.

A losedloop ontrol[64℄ anbeimplementedas beforeusing avoltage ontrolloop,

withvoltagefeed-forward ompensator. It thusgeneratesthe onstantoutputsignal,

whileaninner urrentshapingregulatorgeneratesanearsinusoidal urrentwaveform.

Cir uit design based on re ent results for High Current Battery Chargers for

PHEVs [44℄ requires a al ulated hoi e of riti al omponents su h asthe indu tor,

based on theoreti al analysis whi h is veries by simulations results of this spe i

ir uit.

Several ontrolstrategies andtopologieshave been studiedin[7℄and [28℄ and

resear hers haveattemptedtoobtain desiredperforman e ofthe system andsuitable

waveforms for input urrent and output voltage in standard operating onditions.

Furthermore a deep analysis is done for the performan e of the onverter,

emphasizing input and output waveforms and improvements due to an appropriate

design or ontrol. Faulty onditions and their ee ts onthe shape of the waveforms

are then studied and on retized in the design of more robust ontrol system and

faulttolerant ir uit.

2.4 DC/DC Converters

DC/DC bi-dire tional onverter forms another basi ir uitfor PHEV

harg-ers. This onverter requireshighpower apabilitiesandworksasaBu k-Boost

topol-ogy, asmentioned. Its bi-dire tionalenergy ow allows alsoboth the harging of the

batterypa k,anditsdis hargethrough theoppositedire tion,supplyingenergyba k

tothe system.

Ithasasimplestru ture thatisderivedfrombasi Bu kandBoosttopologies,

withtheAC/DCre tier explainedabove. Bi-dire tionalbu k-boost onverterworks

as a bu k ir uit, produ ing an almost onstant output voltage a ross the battery,

redu ing high input voltage from the inverter stage. In the opposite dire tion, with

energy owing from the load tothe input side, it operates as aboost onverter thus

in reasing the voltage and allowing batterydis harge.

Modeling di ulties whi h make spa e state dierentialequations derivation

quite ompli ated are related to non-linear and time varying nature of swit hing

onverters. Time-variant matrix stru ture, represented in the s heme of gure 2.2,

may requires the use of Spa e State Averaging te hnique, in order to eliminate the

time dependen e.

Figure 2.2. Non-lineartime-invariant state spa e s heme.

In this thesis, fun tioning of this onverter is analyzed and dierential

equa-tions of its spa e state model are derived in order to simulate its behavior using

MATLAB

r

. As for the other onverters, modeling of the onverter and simulation

are indispensable tools. They are used to investigate further ir uit responses under

spe i onditions.

2.5 Battery Model

Oneofthemostimportant omponentsofthepowerele troni ir uitinPHEV

hani al energy by the ele tri motor. Similarly stored power an be used to feed

on-board ele tri al equipment su h as ele tri power steering, air onditioning

sys-tem,light,pumpset . Therearemanytypesofbatteriesspe i allysuitedforHybrid

Ele tri Vehi le appli ations, in luding Ni kel Iron, Ni kel Cadmium, Ni kel Metal

Hydride, Lithium Polymer and other metal-airbatteries.

Many fa tors hara terize battery quality and spe i performan e riteria,

and form interesting resear h topi s, in luding: energy density, spe i power,

typ-i al voltage, Ampere hour e ien y, energy e ien y, ommer ial availability, ost,

operatingtemperature, self-dis hargerates, life y les and physi al duration. Several

studiesarenowfo usingondevelopingadetailedmodelforabatterypa k[10℄. These

resear hes involve onsideration on the variation of battery load with respe t to its

temperature and its state of harge, realisti harge and dis harge rates, analyzing

battery geometry, optimum temperature of operation, along with suitable harging

methods.

In this thesis a very simple model of battery is used, onsidering an ideal

series of aDC voltage generator and aresistan e. Otherfollowingresear h work will

further investigate the modeling of a battery pa k spe i ally suited for this high

urrent harger.

2.6 Design Consideration and Modeling

Severalpapershaveaddresseddesign onsiderationsforPHEVs'battery

harg-ers, in ludingpra ti al omparative evaluations of dierent as aded ongurations.

In standard literature, the two stages of system have been often studied separately

and have led toa partialanalysis of stability issues. Infa t, many arti leshave

on-sideredthe as adedDC/DC onverterforoutputvoltageregulationasanequivalent

these two ir uits,the dynami behaviorof thisnonlinear system and stability issues

need a thorough investigation.

In terms of stability, the operation of overall ir uit results more restri ted

than that for PFC ir uit with a onstant resistive load, and further onsiderations

are to be done. A new ontroller design has to be developed a ording to analysis

results. Theee tsoftheintera tionbetweenthetwostages,shownin[24℄areveried

forthis model. In fa t,theDC/DC onverterstagerepresentsa onstantpowerload,

equivalent toa negative resistan e seen from the PFC stage onlywhen its output is

perfe tly regulated. In pra ti e, sin e the PFC Boost regulator - is almost always

as aded with a voltage regulator - espe ially for medium to high power ranges,

it is useful onsider the overall as aded stru ture of the power supply. Another

parti ularly important issue is the hoi e of indu tive and apa itive omponents in

the ir uits, in luding the AC side indu tor, bi-dire tional bu k-boost indu tor and

output apa itors.

This thesis onsiders the development of a onverter apable of operating for

universal voltage input. This range is dened nominally from

90 V

to

240 V

and a onstant battery voltage of

48 V

. It must be noted that this hoi e leads to some important onsiderations on the value of the DC bus voltage level. Maximum

in-termediate DC bus voltage level is set to

400 V

between the PFC re tier and the DC/DC onverter. This hoi e along with the modulation index al ulation, whi h

has been widely dis ussed in literature [75℄, maximizesoverall system e ien y.

Furtherdesignspe i ationsalongwiththe ratingsof ir uit omponents

on-sider both mathemati al and a pra ti alapproa h [3℄, based on the investigation of

the performan esof the ir uit. Valuesof riti al omponentshave beenevaluatedin

2.7 Control Te hniques

Oneofthe mostimportanttopi sinthedesignisthe ontrol ir uit,involving

both the bi-dire tional PFC inverter and the DC/DC onverter, to be robust and

fault-tolerant. Asimpleyetee tivetypologyof ontrollersispresented inthisdesign

using a Proportional,Integral and Derivative (PID) regulator. This ontrolstrategy

assures desired response while providing qui k tuning apabilities and a not very

omplex stru ture.

ThePFC onverter ir uitusesamore omplex ontrolalgorithm,in ludinga

few dierent regulators loops. This is due toits need of simultaneously regulate the

outputDCvoltageandshapinginput urrent inorder to orre tthe powerfa tor. In

this ir uit the external ontrolloopis designed for voltage regulation. DC referen e

and a tual sensed signal are used as inputs to produ e a ontrol output su h that

the DCvalue remains onstantregardless variationsof theload, suppliedACvoltage

or output urrent drawn. The voltage ontroller also produ es a referen e signal for

the inner urrent loop, whi h ontrols the shape of input urrent. Outputvoltage is

ontrolledbyasimplePIregulator,inwhi hasignalproportionaltovoltageerrorand

toitsintegralisgenerated. Whentheinputofthevoltageerror ompensatorin reases,

signal generated by PWM ir uit alsoin reases. Therefore for an in rease inoutput

terminalvoltage,theinner urrentregulationloopredu esthe urrentproportionally

to keep the input power onstant. In fa t, this internal urrent error ompensator

uses sensed urrent and regulated voltage signal to determine the referen e. Control

loop error is thus determined by the dieren e between al ulated referen e signal

and a tual indu tor urrent and is pro essed by a PID ontroller. This regulation

system for es the indu tor urrent to followthe referen e sinusoidal waveform.

to orre tly shape the input sinusoidalwaveform and thusrequires a relatively small

bandwidth. The urrent ontrollerprodu esasignalwhi histhenelaboratedthrough

a PWM generator, in order to produ e the appropriate duty y le value for the

gate signal of ir uit MOSFETs. If the voltage de reases, also referen e signal for

input urrent de reases, thus resulting in a lower drawn power. However, in order

to maintain a onstant output power in orresponden e of a redu ed input, urrent

referen e should proportionally de rease [7℄. A voltage feed-forward ompensator

maintains the outputpower onstant and determined onlyby the load, regardless of

inputvariations. Itsoperatingprin ipleaveragesthe input voltage anddivides input

referen e urrent by itssquared value.

Control of bi-dire tional bu k-boost DC/DC onverters result in a simpler

stru ture, as ompared to the multi- ontrol loop for PFC Cir uits, with just one

PI regulator. Proportionaland integral a tionsare used in afeedba k error loopfor

output urrent,whi his omparedwithdesiredbattery urrentreferen e. Theoutput

of the regulator is then used to obtain a PWM signal to generate swit hing signals

for the MOSFETs.

2.8 Stability Issues

Power ele troni ir uitssu h as swit hing onverters, are ommonlyrealized

using a losed loop ontrol system, e.g. PID ontroller in order to minimize the

er-ror between a tual and ommanded response. The Converter an be implemented

SimPower System toolbox in MATLAB

r

Simulink, whi h allows multiple ontrol

strategies, in ludingdire t ir uitapproa hand moreexible digital/dis retedesign.

Behavior of the system an be similarly des ribed through its ir uit-based

repre-sentation as well as itsspa e state model, using dierentialequations. Equivalently,

Stability of a ontrol system is often extremely important and is generally a

primary issue in the engineeringof a system. It isusually relatedto the response of

the system to various inputs or disturban es. Stability analysis of power onverters

is quite di ult due to some intrinsi features of the systems. Variation of model

parameters, su h as input voltage or hange in load resistan e, as well as stru ture

hanges in mode of operations(Continuous mode or Dis ontinuous mode), gives the

system a omplex non-linearmodel.

Stabilityanalysisof thesystem an beperformedbothfor itsopenand losed

loop ongurations. Using various te hniques it is possible to identify instability

is-sues, onstraints in operating onditions, and performan e under faulty onditions

of operation. Frequen y analysis, whi h in ludes methods su h as Routh-Hurwitz

stability riterion or Bode plot and Nyquist diagram have been preferred in many

publi ations[37℄, whileoftenregarding unidire tionalsystems [45℄orlinearized

mod-els [40℄. In parti ular, stability issues for bi-dire tional ir uits, need to be analyzed

inbothpowerows dire tions,sin e ommonwaystoimprovethestabilityusually

af-fe totherdire tion. Similarly,whileea hbi-dire tional onverterinthepowersystem

is designed and optimized separately, when the onverters are as aded, the system

may reveal tobe unstable [68℄.

Otherstabilityanalysiste hniques onsiderforexampleMiddlebrookImpedan e

Criterion [31℄, or linearization of the system around parti ular load onditions, and

Lyapunov theory of the state spa e model using state feedba k ontrol to stabilize

CHAPTER 3

CONVERTER DESIGN

Charging the batteryof aPlug-inHybrid Vehi le fromthe ACoutlet requires

relatively high power apabilities. Allowing a universal input voltage range and

in-verse power owfor dis harging operationof forin-grid appli ations,are also

impor-tant features. This hapter willprovide a detaileddes ription of the ir uitsused in

the battery harger implementation. Their behavior and dynami response is

ana-lyzedthrough thederivationofdierentialequationsand simulatingtheirresponsein

theMATLAB

r

Simulinkenvironment. Figure3.1shows atypi altwo-stage as aded

onverter topology as it is used in [73℄. This is a typi al hoi e for PHEV battery

harger ir uits. The basi s heme of this onguration onsists of two as aded

bi-dire tional power ele troni onverters, an AC/DC re tier/inverter and a DC/DC

onverterrespe tively. The onverteroutputisrequiredtoobtainasmooth ontrolled

urrentinorderto hargethe energystoragedevi e,whi hrequiresstables onditions

and aatwaveform. Given itsrelativelysimplestru ture andwellknowndesign and

ontrolissues, as often des ribed inliterature [7℄, a PFC boost ir uit isis the most

ommonly used ir uit. It is found to have the most suitable onguration for this

appli ation, requiringhigh e ien ies and powerfa tor orre tionat the input. The

se ondstageswit hing onverteristypi allyneeded toeliminatetherippleattheDC

bus voltage, whi his typi allyat twi e the input frequen y and to regulatethe

out-put urrent for the large input voltage range. In this proje t a simple bi-dire tional

bu k-boost ir uithas been hosen.

PFC Boost

(60 KHz)

Step Down DC/DC Converter

(60 KHz)

90 - 240 V 50- 60 Hz AC Input 48V DC Out

This hapter also des ribes the ontroller implementation, using a lassi al

approa h of a simple analog ontrol ir uit, both for the AC/DC and DC/DC

on-verter. A multi-loopregulatorisused to ontroltheDC busvoltageand theshapeof

the inputindu tor urrent,whereas a single ontroller isimplemented forthe se ond

stage output urrent regulation.

Forea h onverter asimpleideal modelisderived and itsmathemati al

equa-tionsare al ulated, onsideringthedynami behaviorof ontinuous ondu tionmode

of operation. Furthermore,a more detailed ir uitdesign in ludesESRresistors and

ON swit hes resistorsand provides more realisti simulationresults.

3.1 Boost Converter

The boost onverter used here is des ribed in ontinuous ondu tionmode of

operation. It is a basi DC/DC onverter that is used to get higher output voltage

than the inputvoltage

V

in

. This highe ien y step-up DC/DCswit hing onverter, onne ted to a DC power sour e is able to hange the output DC value to a higher

voltage level

V

out

. Boost onverter uses a swit h, typi allya BJT or a MOSFET, to modulate the voltage into an indu tor. It has a simple ir uit whi h ontains two

swit hing omponents: a diode and a transistor. The indu tor and the apa itive

lter manage the energy onversion and redu e the ripple inthe output urrent and

voltage. The main operating prin iple an be explained as follows: the swit h is

positionedsu hthat theinputsour e hargesthe indu tor,whilethe apa itoratthe

outputmaintains the outputvoltageusing energy storeda ross itsplates. When the

swit h hanges itsstate, both the DC sour e and the stored energy supply powerto

theload,hen etheoutputvoltageboosts. Whenswit his losedtheindu torabsorbs

energy fromthe input andthe ir uit isseparated intotwoparts. This onguration

ir uit isdes ribed ingreater detail later inthe thesis.

This se tion analyzes ideal boost onverter, its voltage and urrent

relation-ships, and derives a state spa e model for the ir uit. A se ond ir uit onsiders

equivalentseriesresistan es(ESR) ofthe omponents,and isdes ribedthrough state

spa e averaging method. This more pre ise model is then built, making some

im-portant onsiderationspertainingtothe hoi e of omponentvalues anddenition of

the design requirements. In the last part of the se tion its behavior is observed and

simulated with the use of Simulink. Also arst rough ontroller isdesigned.

3.1.1 Boost Converter State-Spa e Model. Analysis of the Boost onverter

needs some general assumptions that will be onsidered also in the next se tion for

the Bu k model. Des ribed ir uit operates in the steady state, and all transients

andimpulses onditionsarenegle ted. Thisimpliesthatallvoltagesand urrentsare

periodi overone swit hingperiod. The ir uitisanalyzedinanequilibriumstate,in

whi htheindu tor urrentneverrea heszero(ContinuousCondu tionMode-CCM).

Swit h

S

has aswit hingfrequen y of

f

s

, and is onsideredtobeopen (swit h OFF) forthetime

t

of f

= (1−D)T

s

,where

T

s

= 1/f

s

istheswit hingperiodand

D

indi ates the Duty Cy le expressed as a per entage of the ommutation period during whi h

the swit h is ON. Besides the ideal swit h remains losed for time

t

on

= DT

s

. Ea h omponent inthis ir uit, as shown ingure 3.2, is onsidered ideal.

Underthese idealassumptions,thesimpleBoost onverter ir uitispresented

for the two possible states of the swit h. In the ON state the swit h is losed and

the sour e input results in an in rease in the indu tor urrent, whereas in the OFF

statethe swit hisopen. In thissituationthe onlypathoered toindu tor urrentis

+

−

V

in

L

C

R

Figure3.2. Ideal Boost ir uitrepresentation.

thusrequirementsontheinputlterarerelaxed. Twosetsofequationsdes ribingthe

dynami s of voltage and urrent relationships are derived for both the losed swit h

ir uit and the open swit h ir uitas follows.

Closed Swit h (

u = 0

)

+

−

V

in

L

C

R

Figure3.3. Equivalent Boost ir uitrepresentation for

S = ON

,

D = OF F

,

u = 0

.

When theswit h

S

is losed,the diode isreverse biasedandequivalent ir uit is shown ingure 3.3.

the followingset of equations:

L

di

L

dt

− V

in

= 0

(3.1)

C

dv

c

dt

+

v

c

R

= 0

(3.2)

These equations, des ribing the ir uit for

u = 0

, an be written in terms of states variables

v

c

and

i

L

as

dv

c

dt

= −

1

RC

v

c

(3.3)

di

L

dt

=

V

in

L

(3.4) Open Swit h (

u = 1

)

+

−

V

in

L

C

R

Figure3.4. Equivalent Boost ir uitrepresentation for

S = OF F

,

D = ON

,

u = 1

.

Whiletheswit hisopen, theindu tor urrent annot hangeinstantaneously,

so the diode

D

be omes forward biased to provide a path for

i

L

. Assumingthat the outputvoltage

V

out

isa onstant,againwith Kir hho'svoltagelawaroundtheouter loop and Kir hho's urrent load inthe same node, following equationsare derived.

V

in

= L

di

L

i

L

= i

c

+ i

R

= C

dv

c

dt

+

v

c

R

(3.6)

Dierential equation (3.5) an be rearranged in terms of apa itor's voltage

andindu tor's urrent,andsubstitutedinequation(3.6),leadingtosystemequations

for

u = 1

.

dv

c

dt

=

i

L

C

−

V

in

RC

−

v

c

RC

(3.7)

di

L

dt

=

V

in

L

−

v

c

L

(3.8)

These two models an be now ombined together with the use of a binary

input variable

u ∈ {0, 1}

asthe value of the swit hing input. It assumes either value

u = 0

when the swit h is losed, or the value

u = 1

when opened, as s hematized in the ir uits. Equations (3.4), (3.7) and (3.5), (3.8) respe tively are ombined to

obtain following dierential globalsystem, written using

v

c

and

i

L

state variables.

dv

c

dt

= −

1

RC

v

c

+

 i

L

C

−

V

in

RC



u

(3.9)

di

L

dt

=

V

in

L

−

v

c

L

u

(3.10)

3.1.2 Boost Converter State-Spa e Model with ESR. In previous analysis

only ideal elements are onsidered, that is input power is transferred to the load

without any losses. In real ir uits, due to intrinsi properties of the materials,

parasiti resistan esare always present. Forthisreason afra tionofthe input power

agooddesign,itisalsoimportanttoanalyzethe ir uit onsideringthemoregeneral

ase of non-ideal omponents. As it has been done for the ideal Boost onverter,

at rst omplete ir uit is presented, analyzing general fun tioning prin iples and

then the two dierent swit h states are dis ussed, deriving dierential equations for

the losed swit h and open swit h ases. The non-ideal behavior of indu tor and

apa itor ishere modeledusing EquivalentSeriesResistan es (ESR):

R

L

as indu tor body resistor and

R

c

as apa itor body resistor. Similarly BJT swit h

S

and diode

D

are modeled through an ON resistor

R

s

and

R

D

. Boost more detailed ir uit is shown in gure 3.5.

+

−

V

in

L

R

L

R

s

R

D

C

R

c

R

Figure3.5. Non-IdealBoost ir uitrepresentation.

Closed Swit h (

u = 0

)

As before, when theswit his ON,diode isreversed biased and theequivalent

ir uit is represented in gure 3.6. Two sets of dierential equations an be written

forthesystem, onsideringthe Kir hho'svoltagelawaroundtheleftmostand

+

−

V

in

L

R

L

R

s

C

R

c

R

Figure3.6. Equivalent Boost ir uitrepresentation for

S = ON

,

D = OF F

,

u = 0

.

V

in

= R

L

i

L

+ R

s

i

L

+ v

L

(3.11)

V

out

= v

c

+ i

c

R

c

(3.12)

These equations, des ribing the ir uit for

u = 0

, an be written in terms of states variables

v

c

and

i

L

as

dv

C

dt

= −

1

C(R + R

c

)

v

c

(3.13)

di

L

dt

= −

(R

L

+ R

s

)

L

i

L

+

V

in

L

(3.14) Open Swit h (

u = 1

)

+

−

V

in

L

R

L

R

D

C

R

c

R

Figure3.7. Equivalent Boost ir uitrepresentation for

S = OF F

,

D = ON

,

u = 1

.

external loop, Kir hho's voltage law for the rightmost loop and from Kir hho's

urrent law onthe upperright node as follows:

V

in

= R

L

i

L

+ v

L

+ R

D

i

L

+ V

out

(3.15)

i

L

= i

c

+ i

out

(3.16)

V

out

= v

c

+ i

c

R

c

(3.17)

Dierentialequation(3.15)and (3.16) anberewritten intermsof indu tor's urrent

and apa itor's voltage. Equation (3.16)is substituted inequation (3.17),leadingto

followingsystem equations for

u = 1

.

dv

C

dt

= −

1

C(R + R

c

)

v

c

+

R

C(R + R

c

)

i

L

(3.18)

di

L

dt

= −

R

L(R + R

c

)

v

c

−

 R

L

+ R

D

L

+

RR

c

L(R + R

c

)



i

L

+

V

in

L

(3.19)

a bilinear behavior due to the nature of the ontrol input, assuming binary values

u ∈ {0, 1}

. Global system an be written using a two set of equations. Values of diode ON resistor and swit h ON resistor are assumed to be

R

s

≃ R

D

with a very reasonable approximation, obtaining

dv

C

dt

= −

1

C(R + R

c

)

v

c

+



R

C(R + R

c

)

i

L



u

(3.20)

di

L

dt

= −

R

L

+ R

s

L

i

L

+

V

in

L

+



RR

c

L(R + R

c

)

i

L

−

R

L(R + R

c

)

v

c



u

(3.21)

The highly non-linear model resulting from the ombination of the two

ir- uits an now be simplied and made suitable for simplied ontrol analysis. The

swit h is repla ed by repla ing a ontinuous element, using the te hnique of system

averaging. In parti ular, infollowingpage average behavioris modeled,su h asonly

informationabout low-frequen y a tion of the onverters is onsidered, ignoring

rip-ple, ommutationsand other fast ee ts. For this averaged model the two swit hes

ongurations an be rearranged, onsidering system equations 3.13, 3.14 and 3.20,

3.21. Here the state is hanginglinearlyfrom itsinitialvalueat the beginningof the

swit hing period, until time instant

t = DT

s

. This approximation onsiders deriv a-tivesto be almost onstant, in the ondition of a triangular ripple waveformor with

a high swit hing frequen y

f

s

, whi h usually holds in reality. The value for the

x

ve tor, representing state variables

x

1

= v

c

and

x

2

= i

L

, an be writtenas

x(DT

s

) ≃ x(0) + ˙x(0)DT

s

≃ x(0) + (A

0

x

+ B

0

u)DT

s

while, attime

t = T

s

, se ond onguration matri esare utilized asfollows

x(T

s

) ≃ x(DT

s

) + ˙x(DT

s

)(1 − D)T

s

≃ x(DT

s

) + (A

1

x

+ B

1

u

)(1 − D)T

s

Therefore, global state evolution be omes

where

D

1

= 1 − D

indi atestime intervalinwhi hthe se ond ongurationisa tive. In equation above, averaged matri es

A = DA

¯

0

+ D

1

A

1

= DA

0

+ (1 − D)A

1

and

¯

B = DB

0

+ D

1

B

1

= DB

0

+ (1 − D)B

1

are dened, asaverages of the ongurations, weighted by the fra tion of the duty y le spent in every onguration. Then it is

possibletosimplifyandprodu ethegeneralformof averagedsystem (equation3.22),

whi h an beadopted alsofor further modelingof other onverters explained later.

˙

x

= ¯

Ax + ¯

Bu

(3.22)

Thisapproximatemodelgivesexa t resultswhenswit hing period

T

s

ismu hshorter thanany other time onstant ofthe ir uit. It has been proved in literature[31℄ that

the new averagedstates dotra kthe average behaviorof

x

. In the innitefrequen y limitthe valuesmat h,avoidingthetime dependen eand thenon-linearitytypi alof

the swit hing systems.

Inordertoobtainasinglestatespa esystemitispossibletodes ribethemodel

through matri es

A

¯

,

B

¯

and torearrange equations using the duty ratio

D

. Here the dynami ofthe system swit hes therefore between

Σ

0

= (A

0

, B

0

, C

0

)

obtained by the value

u = 0

, in the interval

D

1

= (1 − D)T

s

and the system

Σ

1

= (A

1

, B

1

, C

1

)

when the input orresponds to

u = 1

in the interval

D

1

T

s

.

The two spa e state subsystems orrespond to derived dierentialequations.

They an thenberepresentedusing anoni al modelformforStateSpa eLinear

3.2 Bu k Converter

The Bu k DC/DC onverter an be used for step down operation. It redu es

the input voltage

V

in

to the desired outputvoltage level of

V

out

suitable for example for battery harger appli ations. Exa tly like Boost onverter it is a swit hed-mode

powersupply that uses two swit hes (aBJT orMOSFET and a diode), an indu tor

and a apa itor, whereas the load an be assumed simply resistive. Although its

topology is fairly simple, abu k onverter an be highly e ient (easily up to 95%)

and it ispreferred overlinear regulators.

Its operatingprin iple takes advantage of the high ommutationfrequen y of

the swit h thatalternates between onne tingthe indu tor tosour evoltagetostore

energy,anddis hargingtheindu torintotheload. Thankstotheveryshorttransition

time,andapre ise hoi eofindu torand apa itor valuesoutputrippleisminimized

andthedynami transferofpowerfromtheinputtoitsoutputisregulated. Swit hing

frequen yismaintained onstanttothevalue

f

s

,whileitsduty y le isvariedthrough aPulse Width Modulation. In this way the ratio

D = t

on

/T

s

between the time

t

on

in whi hthe swit h is losedand the period

T

s

= 1/f

s

determines the desired DC level at the output. The low-pass stru ture of the onverter, as explainedin next se tion,

guarantees a low frequen y noise spe trum for this modulations heme.

In this hapter Bu k's ideal ir uit is rst presented and then its dynami

equations are derived with the fundamental Kir hho's urrent and voltage laws, in

order to get a state spa e model. Non-ideal omponents are then used to des ribe a

state-spa e model for ontinuous ondu tion mode of operationof the ir uit, whi h

is also implemented in MATLAB

r

Simulink. The end of the hapter analyzes its

ontroldesign,evaluating hoi esof indu torand apa itorvalues andprovidessome

3.2.1 Bu k Converter State Spa e Model. Analysis of the ideal Bu k

on-verterrepresented ingure3.8requiressomeideal assumptions. Asbeforethe ir uit

operates in the steady state, in ontinuous ondu tion mode of operation (CCM),

that is indu tor urrent is always positive and never rea hes zero. Output voltage is

onsideredalmost onstant

V

out

,whileall omponentsare idealand body resistorare negle ted.

+

−

V

in

L

C

R

Figure3.8. IdealBu k ir uit representation.

The swit h

S

is losed inthe interval

DT

s

and open fortime

(1 − D)T

s

. This leads to the following two equivalent ir uits.

Closed Swit h (

u = 0

)

+

−

V

in

L

C

R

Figure3.9 shows equivalent Bu k ir uits when the swit h

S

is losed. Diode

D

isreversed biasedand followingvoltageand urrentrelationships an be obtained using Kir hho's voltage law for inner and outer loops and Kir hho's urrent law

for uppernode

V

in

= L

di

L

dt

+ v

c

(3.25)

v

c

= v

out

(3.26)

i

L

= C

dv

c

dt

+ i

R

(3.27)

These equations des ribe the ir uit for

u = 0

and an be written interms of states variables

v

c

and

i

L

:

dv

c

dt

= −

1

RC

v

c

+

1

C

i

L

(3.28)

di

L

dt

= −

1

L

v

c

+

V

in

L

(3.29) Open Swit h (

u = 1

)

+

−

V

in

L

C

R

Figure3.10. Equivalent Bu k ir uitrepresentation for

S = OF F

,

D = ON

,

u = 1

.

rightmostloops, togetherwith Kir hho's urrentlawonuppernodegivethe

follow-ingdierentialequations.

0 = L

di

L

dt

+ v

c

(3.30)

v

c

= v

out

(3.31)

i

L

= C

dv

c

dt

+ i

R

(3.32)

Equations (3.30) and (3.32) an be rewritten in terms of indu tor's urrent and

a-pa itor'svoltage. Substituting equation(3.31) followingequivalent system for

u = 1

is obtained.

dv

c

dt

= −

1

RC

v

c

+

1

C

i

L

(3.33)

di

L

dt

= −

1

L

v

c

+

V

in

L

u

(3.34)

3.2.2 Bu k Converter State Spa e Model with ESR. Afurther step in

mod-eling the Bu k onverter is assuming non-ideal omponents su h as indu tor and

apa itor, as well as the diode and the swit h. In its ir uit representation parasiti

values are s hematized as equivalent series resistors (ESR):

R

L

,

R

C

,

R

D

and

R

S

, as shown in ir uits hemati ingure 3.11.

Two dierent sets of equations are derived for the ir uit with either open or

losed swit h, based onthe binary values of

u

.

Closed Swit h (

u = 0

)

+

−

V

in

R

s

R

D

L

R

L

C

R

c

R

Figure3.11. RealBu k ir uitrepresentation.

an output voltage a ross the resistor. Figure 3.12 shows equivalent Bu k ir uit for

the losedswit h. WritingKir hho's urrentlawfortheuppernode,andKir hho's

voltagelawin the inner and outer loops we obtain

i

R

= i

L

− i

C

= i

L

− C

dv

C

dt

(3.35)

V

in

= L

di

L

dt

+ (R

S

+ R

L

)i

L

+ Ri

R

(3.36)

V

in

= L

di

L

dt

+ (R

S

+ R

L

)i

L

+ i

C

R

C

+ v

C

(3.37)

Ri

R

= v

C

+ i

C

R

C

(3.38)

Thuswith some simplealgebrai manipulations weget

dv

C

dt

= −

1

(R + R

C

)C

v

C

+

R

(R + R

C

)C

i

L

(3.39)

di

L

dt

= −

R

(R + R

C

)L

v

C

−



RR

C

(R + R

C

)L

+

R

L

L

+

R

S

L



i

L

+

V

in

L

(3.40)

+

−

V

in

R

s

L

R

L

C

R

c

R

Figure3.12. Equivalent Bu k ir uitrepresentation for

S = ON

,

D = OF F

,

u = 0

.

Open Swit h (

u = 1

)

R

D

L

R

L

C

R

c

R

Figure3.13. Equivalent Bu k ir uitrepresentation for

S = OF F

,

D = ON

,

u = 1

.

through the resistor. Bu k onverter equivalent ir uitisshown ingure 3.13. Using

thesamestrategythis onverter anbedes ribedwiththefollowingpairofdierential

equations:

dv

C

dt

= −

1

(R + R

C

)C

v

C

+

R

(R + R

C

)C

i

L

(3.41)

di

L

dt

= −

R

(R + R

C

)L

v

C

−



RR

C

(R + R

C

)L

+

R

L

L

+

R

D

L



i

L

(3.42)

Before ombiningthe two pairs of equations inone single spa e state system,

it is useful to make some onsiderations about real parameters, in order to simplify

the system stru ture. As usually happens in real ir uits, both diode and swit h

resistan es are quitesmall,and an be onsidered as

R

S

= R

D

. Body resistorsof the indu tor and the apa itor may have a signi ant ee t on the output ripple, and

on the e ien y issue for this onverter, therefore they must be onsidered into the

above model.

Using this assumptions equation (3.42) an be simplied and then ombined

withequation(3.37)introdu ingthebinaryinput

u

. Alongwithequations(3.40-3.42) it leads tothe followingoverall swit hing model:

dv

c

dt

= −

1

(R + R

c

)C

v

C

+

R

(R + R

c

)C

i

L

(3.43)

di

L

dt

= −

R

(R + R

C

)L

v

C

−



RR

C

(R + R

C

)L

+

R

L

L

+

R

D

L



i

L

+

V

in

L

u

(3.44)

The swit hing position

u

is taken as input variable, assuming values in the dis rete set

{0, 1}

.

This model an be written in the typi al state spa e system representation.