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Contributions to modeling and control of power converters for renewable energy applications

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Title:
Contributions to modeling and control of power converters for renewable energy applications
Creator:
Soriano-Rangel, Carlos A.
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
College of Engineering and Applied Sciences, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Radenkovic, Miloje
Committee Members:
Mancilla-David, Fernando
Golkowski, Mark
Altman, Tom
Angulo, Alejandro

Notes

Abstract:
Electric power obtained from renewable energy sources such as photovoltaic panels, wind turbines and fuel cells comes in different varieties and needs to be processed to meet the needs of the final user. This processing task is realized with different types power electronic converters such as DC–DC and AC–AC. This has caused converters to become an important enabling technology throughout the whole power system and are expected to play an even bigger role in the future power grid. The ultimate goal of the proposed research is to model and control power converters in a variety of applications related to renewable energy sources. Specifically, this work focuses on (i ) implementation of a new switching strategy for a state–of–the–art DC–DC converter intended for renewable energy applications, (ii ) development of a new DC–DC converter topology tailored for renewable energy applications, (iii ) experimental validation of nonlinear controls for converters feeding constant power loads, and (iv) proposing a low frequency AC transmission based on AC–AC converters.

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University of Colorado Denver
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Auraria Library
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Copyright Carlos A. Soriano-Rangel. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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CONTRIBUTIONSTOMODELINGANDCONTROLOFPOWERCONVERTERS FORRENEWABLEENERGYAPPLICATIONS by CARLOSASORIANO{RANGEL B.S.,InstitutoTecnologicodeCiudadMadero,2010 M.Eng.,InstitutoTecnologicodeCiudadMadero,2012 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy EngineeringandAppliedSciences 2018

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ThisthesisfortheDoctorofPhilosophydegreeby CarlosASoriano{Rangel hasbeenapprovedforthe EngineeringandAppliedScienceProgram by MilojeRadenkovic,Chair FernandoMancilla{David,Advisor MarkGolkowski TomAltman AlejandoAngulo Date,December15,2018 ii

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Soriano{Rangel,CarlosA(Ph.D.,EngineeringandAppliedSciences)ContributionstoModelingandControlofPowerConvertersforRe newableEnergyApplicationsThesisdirectedbyAssociateProfessorFernandoMancilla{David ABSTRACT Electricpowerobtainedfromrenewableenergysourcessuchasph otovoltaicpanels, windturbinesandfuelcellscomesindierentvarietiesandneedstob eprocessedtomeet theneedsofthenaluser.Thisprocessingtaskisrealizedwithdie renttypespowerelectronicconverterssuchasDC{DCandAC{AC.Thishascausedconv erterstobecomean importantenablingtechnologythroughoutthewholepowersystem andareexpectedto playanevenbiggerroleinthefuturepowergrid. Theultimategoaloftheproposedresearchistomodelandcontro lpowerconverters inavarietyofapplicationsrelatedtorenewableenergysources.Sp ecically,thisworkfocuseson( i )implementationofanewswitchingstrategyforastate{of{the{a rtDC{DC converterintendedforrenewableenergyapplications,( ii )developmentofanewDC{DC convertertopologytailoredforrenewableenergyapplications,( iii )experimentalvalidation ofnonlinearcontrolsforconvertersfeedingconstantpowerload s,and( iv )proposingalow frequencyACtransmissionbasedonAC{ACconverters. Theformandcontentofthisabstractareapproved.Irecommen ditspublication. Approved:FernandoMancilla{David iii

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TABLEOFCONTENTS CHAPTERI.BACKGROUND....................................1 Introduction......................................1Objectives.......................................2Impact.........................................2 II.OPTIMIZEDSWITCHINGSTRATEGYFORADC{DCCONVERTER.... 4 Motivation&PresentStateofKnowledge.................... ..4 Complementaryswitchingstrategyof[42]................... 6 TechnicalApproach..................................7 Proposedproportionalswitchingstrategy................. ..7 Voltagegainanalysis...............................8Inputcurrentrippleanalysis...........................9ExperimentalResults...............................13 Contributions&Impact................................1 3 III.DC{DCCONVERTERWITHQUADRATICGAIN................ 15 Motivation&PresentStateofKnowledge.................... ..15 TechnicalApproach..................................1 6 Converteranalysis................................18Designconsiderations..............................22 Contributions&Impact................................2 7 IV.NON{LINEARCONTROLOFBUCK{BOOSTCONVERTERSFEEDINGCP LS28 Motivation&PresentStateofKnowledge.................... ..28 TechnicalApproach..................................3 0 ControlLawwithKnownLoadPower D ....................31 AdaptiveControlUsingaPowerEstimator..................3 5 Simulationandexperimentalresults......................36 iv

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Contributions&Impact................................4 3 V.NON{LINEARCONTROLOFBUCKCONVERTERSFEEDINGCPLS....4 4 Motivation&PresentStateofKnowledge.................... ..44 TechnicalApproach..................................4 5 ControlLawwithKnownLoadPowerP....................45AdaptivecontrolusinganI&Ipowerestimator............... .47 Computersimulationsandexperiments...................... .47 Averagedsimulations...............................47 Gainsensitivityanalysis..........................47 Switchedsimulationsandexperiments.....................49 Contributions&Impact................................5 0 VI.HEXVETER{BASEDLOWFREQUENCYACTRANSMISSION........ 51 Motivation&PresentStateofKnowledge.................... ..51 LowFrequencyACSystem..............................5 2 Transmissionlineparameters............................. 52 Temperaturedependance............................53Frequencydependance..............................53 Hexverter........................................55Optimaltransmissionfrequency.......................... .57 Results.........................................60Conclusions......................................61Contributions&Impact................................6 2 VII.CONCLUDINGREMARKS.............................6 3 REFERENCES.......................................64 v

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CHAPTERI BACKGROUND Introduction Electricpowerobtainedfromrenewableenergysourcessuchasph otovoltaicpanels, windturbinesandfuelcellscomesindierentvarietiesandneedstob eprocessedtomeet theneedsofthenaluser.Thisprocessingtaskisrealizedwithdie renttypespowerelectronicconverterssuchasDC{DCandAC{AC.Thishascausedconv erterstobecomean importantenablingtechnologythroughoutthewholepowersystem andareexpectedto playanevenbiggerroleinthefuturepowergrid. DC{DCconvertersareoftenusedtointerphaserenewableenerg ysourcessuchasfuel cellswiththegrid.Thesesourcesrequirespecicneedssuchaslowin putcurrentripple andlargevoltagegaintoallowoptimaloperationandtomaximizethede vice'slifecycle. Thisposesaneedtoproposenewconvertertopologieswiththesef eaturesandalsotoinvestigateimprovedoperationofexistingones.Theseproblemsare addressedinChapters IIandIII.Additionally,giventheincreasedpenetrationofrenewa bleenergysources,it hasbecomefrequenttondconvertersinteractingwithotherco nverters.However,convertersarecommonlyoperatedinaclosedloopfashion,whichcause sthemtobehaveas constantpowerloadswhenconnectedincascade.ThusinChapter sIVandV,non{linear controlsforabuck{boostandabuckconverterfeedingaconsta ntpowerloadareproposed,testedwithsimulationsandvalidatedthroughexperiments. Finally,anexampleapplicationofAC{ACconvertersistointerphasew indturbines whichgeneratepoweratavariablefrequencywiththepowergridwh ichoperatesataconstantfrequency.Thisabilityofconverterstogenerateanoutpu tpowerofafrequencydifferenttotheinputfrequencywillbeappliedtoaLowFrequencyAC{ ACtransmissionsysteminChapterVI.Giventhattheseriesimpedanceoftransmissionlin esisproportional totheoperatingfrequency,ifthefrequencyisreduceditispossib letoincreasethemaximumpowerrowofacongestedtransmissionline. 1

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Objectives Thegoaloftheproposedresearchistomodelandcontrolpower convertersinavarietyofapplicationsrelatedtorenewableenergysources.Specica lly,thisworkfocuseson ( i )implementationofanewswitchingstrategyforastate{of{the{a rtDC{DCconverter intendedforrenewableenergyapplications,( ii )developmentofanewDC{DCconverter topologytailoredforrenewableenergyapplications,( iii )experimentalvalidationofnonlinearcontrolsforconvertersfeedingconstantpowerloads,and ( iv )proposingalowfrequencyACtransmissionbasedonAC{ACconverters.Thus,thefo llowingobjectivesare set: Applyandvalidateexperimentallyanalternativecommutationstrate gyforastate{ of{the{artDC{DCconverter. Design,analysisandexperimentalvalidationofanovelDC{DCconve rtertopology forrenewableenergyapplications. Performexperimentalvalidationofnon{linearcontrolsforcascad e{connectedconverters. DesignalowfrequencyACtransmissionsystembasedonAC{ACconv ertersand testthroughsimulations. Impact Thetechnicalsolutionsdevelopedinthisworkwillallowabetterunder standingofapplicationsofpowerelectronicconvertersinpowersystemsinarene wableenergycontext. Theancillaryservicesthatconverterscanprovideinthetransmiss ionsystemcouldhelp betterutilizethepoweravailablefromrenewableenergysourcesto improvethestabilityof system. Furthermore,lowfrequencyACtransmissioncouldimprovethecap acityofexisting transmissionlinesandhelpdecongestthetransmissionsystem.Add itionally,converters havebecomeanenablingtechnologytointegraterenewableenergy sourcestothegrid, 2

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suchasfuelcellsandphotovoltaicpanels.Theserenewableenergy sourceshavespecic needsandrequirespecializedconvertertopologiesforoptimalope ration.Therefore,developingnewtopologiesandimprovingtheoperationofexistingonesw illfurtherincrease thepenetrationofconvertersandrenewableenergysources.F inally,newsolutionstothe problemofvoltageregulationofcascadeconnectedconvertersn eedtobedevelopedinordertoguaranteestableoperationofthepowersystem. 3

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CHAPTERII OPTIMIZEDSWITCHINGSTRATEGYFORADC{DCCONVERTER Motivation&PresentStateofKnowledge Thevoltageprovidedbyanumberofsmallpower{generatingsourc essuchasrenewablesisusuallylowinamplitude.Asaresult,aboost{typearchitectur ewithalargevoltagegainisrequiredtolinkthisvoltagetoaninverter[61,2,14].Anoth erimportantrequirementforaconverterinrenewableenergyapplications|e.g.,fu elcells|istodraina continuouscurrentwithminimumripple[8,55,30].Therefore,conve rterscombiningthese twofeaturesareexpectedtondmanyapplicationswithintherene wableenergyeld.In aneorttofullltheserequirements,ahybridswitched{capacito r(SC)/interleavedboost convertertopologyfeaturingresonantswitchingamongthecons tituentSCswaspresented in[42].Inthistopology,theSCstructuregivestheconverteralarg evoltagegain,while twointerleavedinductorscanceltheinputcurrentrippleatapres electeddutycycle. Otherconverterswithoutmagneticcouplingwhichareabletoprovid earelatively largevoltagegainwhiledrawingcurrentwithminimumripplehavebeenpr oposed.In [9]and[16],adualboosttopologycancelingtheinputcurrentripplea t50%dutycycleis proposed.Theauthorsin[47]augmentaboostconverterwithanin putlterforcanceling theinputcurrentripple,butatacostofincreasedtopologycomple xity. ThehybridSC/interleavedconverterof[42]isworthfurtheratte ntionbecauseitis abletoprovidealargervoltagegainthanitscounterparts,andallow sonetoselectanarbitrarydutycycletodrawripple{freecurrentfromthesource.A sdescribedin[42],the converterismodulatedthroughcomplementaryswitchingamongth econstituenttransistors.Herein,theworkof[42]isextendedbyproposingamodulation strategybasedon proportionalswitching. Thetopologyproposedin[42]isillustratedinFig.II.1(a).Asthegur esuggests,the topologycontainstwotransistors( S 1 , S 2 ),threediodes( D 1 , D 2 , D 3 ),threecapacitors( C 1 , C 2 , C 3 ),twoinductorsforenergystorage( L 1 , L 2 )andasmallinductor( L 3 )forcurrent 4

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S 1 S 2 R C 1 C 2 C 3 D 1 D 2 D 3 L 1 L 2 + ++ + + + L 3 (a) R D 1 D 2 D 3 v 0 v in L 1 L 2 + ++ + + + L 3 R D 1 D 2 D 3 L 1 L 2 + ++ + + + L 3 (b)(c) S 1 S 2 C 1 C 2 C 3 S 1 S 2 C 1 C 2 C 3 R D 1 D 2 D 3 L 1 L 2 + ++ + + + L 3 R D 1 D 2 D 3 L 1 L 2 + ++ + + + L 3 (d)(e) S 1 S 2 C 1 C 2 C 3 S 1 S 2 C 1 C 2 C 3 i in i in i in i in i in v 0 v in v 0 v in v 0 v in v 0 v in FigureII.1:CircuitschematicofthehybridSC/interleavedboostco nverter. limitingthrough D 3 duringthechargeinterchangeamongtheSCs.Itcorrespondsto the hybridizationofSCandinterleavedconverters,providingthebene tsofboth.TheSC structureattheoutputsideprovidesahighvoltagegain,whilethet wointerleavedinductorsattheinputsideoftheconvertercanceltheinputcurrentr ippleatapreselectedduty cycle( d ? ). Dependingontheswitchingstrategy,thetransistorsmayormayn otswitchinacomplementarymanner.Ingeneral,thetopologycommutesamongfou rdierentequivalent circuits.Transistors'states( S 1 =ON, S 2 =OFF),( S 1 =OFF, S 2 =ON),( S 1 =OFF, S 2 =OFF) and( S 1 =ON, S 2 =ON)leadtotheequivalentcircuitsillustratedinFigs.II.1(b)throug h (e). Boththecomplementarystrategyof[42]andtheproportionalst rategyproposed hereincanbefullycharacterizedbydeningareferencedutycycle .Forconvenience,the 5

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dutycycleof S 2 isselectedasthereferencedutycycle,andistermed d throughoutthe restofthesection, d ( t ):= d 2 ( t )= 1 T S Z t + T S t q 2 ( ) d: (II.1) In(II.1), q 2 istheswitchingfunctionoftheswitch S 2 and T S istheswitchingperiod. Thedynamicsof i L 1 , i L 2 and v C 1 maybeconvenientlyanalyzedconsideringtheiraveragebehavior,astheyfeaturetriangularwaveformssimilartothos eintraditionalDC{DC converters[12].Ontheotherhand, C 2 , C 3 ,and L 3 formaresonantSCcircuitandthereforetheirdynamicbehaviorhastobeformulatedwithadditionalcon siderations[42].The eectoftheproposedswitchingstrategyonthevoltagegainandin putcurrentripplecan befullycharacterizedfocusingonlyon L 1 and L 2 ,wheredynamicaveragingapplies.Underthisassumption,switchingfunctionsmaybereadilyreplacedbyt heircorresponding dutycycles.Complementaryswitchingstrategyof[42]Inthisstrategythetransistorsswitchcomplementarily,i.e.,when S 1 isclosed, S 2 isopen andviceversa.Becauseofthecomplementaryswitching, S 1 'sdutycycleissynthesizedas d 1 =1 d andtheconvertercommutesbetweentheequivalentcircuitsofFig s.II.1(b)and (c)only. Akeyfeatureofthetopologyisthattheconverter'sinputcurren tcorrespondstothe sumofthecurrentsthrough L 1 and L 2 .Sincetheycharge/dischargeinacomplementary manner,theripplestendtocanceleachother.Byproperlysizing L 1 and L 2 ,theripples fullycanceleachotherat d ? ,drawingripple{freecurrentfromthesource.Itwasshownin [42] d ? = L 2 = ( L 1 + L 2 ).AccordingtotheequivalentcircuitsinFigs.II.1(b)and(c),the averagedynamicsof i L 1 and i L 2 ischaracterizedby(II.2)and(II.3), di L 1 dt = d v in v C 1 L 1 +(1 d ) v in L 1 ; (II.2) di L 2 dt = d v in L 2 +(1 d ) v in v C 2 L 2 : (II.3) 6

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TechnicalApproachProposedproportionalswitchingstrategy TheequivalentcircuitsofFig.II.1suggesttheconvertercanalsoo peratewhileboth transistorsaresimultaneouslyeitherOFForON.Fig.II.1(c)indicat esthatwhileboth transistorsareOFFtheinductorsdischargethroughcapacitors thatcanhavedierent voltages.WhenbothtransistorsareON,theinductorschargeth roughindependentloops. Therefore, S 1 and S 2 neednotswitchcomplementarilyandcanhavearbitrarydutycycles . Thestrategyproposedhereintakesadvantageofthisfactands ynthesizesthedutycycle oftheswitch S 1 as d 1 = k d ,thatis, S 1 'sdutycycleisselectedtobeproportionalto d , where k = L 1 =L 2 .Furthermore,thecarriersignalutilizedtogenerate S 1 'sswitchingfunctionisphaseshifted180 withrespectto S 2 'scarriersignal.Boththecomplementaryand proportionalswitchingstrategiesyieldthesameswitchingsequenc ewhenoperatingat d ? : if d = d ? , d 1 = k d ? =1 d ? . Inapracticalsetting,theconverterwilloperateat d ? mostofthetime.However, disturbancesintheinputvoltageorloadcurrentmaycausetheclos ed{loopregulatorto changethedutycycleinordertoachievethedesiredoutputvoltag e.Underthisscenario eachstrategyyieldsadierentswitchingfunctionfor S 1 ,allowingtheconvertertocommutethroughtheequivalentcircuitsofFig.II.1(d)and(e)aswell.I tisshowninthesequelproportionalswitchingyieldssuperiorperformanceintermso fthetwokeyfeatures oftheconverter:voltagegainandinputcurrentripple.Whiletheav eragedynamicsfor i L 2 remainsunchanged,thedynamicsfor i L 1 becomesasin(II.4).Thus,(II.3)and(II.4) describethecurrents'dynamicsunderproportionalswitching. di L 1 dt =(1 kd ) v in v C 1 L 1 + kd v in L 1 : (II.4) Since k isutilizedtosynthesizethevalueof d 1 ,itfollowsthattheinductors'sizeoughtto beselectedthroughthedesignprocesssuchthat L 2 >L 1 .Theanalysisbelowisperformed for L 2 =2 L 1 asanexample,whichleadsto d ? =66 : 67%and k =0 : 5. 7

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Voltagegainanalysis 2 4 6 8 10 V 0 V in 0 0.5 1 " i inN 0 20 40 60 80 100 0 10 20 d (%)THD(%) Complementarystrategy Proportionalstrategy d ? =66.67% Complementarystrategy Proportionalstrategy d ? =66.67% Complementarystrategy Proportionalstrategy d ? =66.67% FigureII.2:Comparativeplotsofvoltagegain(top),normalizedinpu tcurrentripples(middle)andTHDs(bottom). ItmaybeobservedfromFig.II.1thattheconverter'soutputvolt agecorrespondsto V 0 = V C 1 + V C 3 .Thevalueof V C 1 readilyfollowsfromzeroingouttheleftsideof(II.2)or (II.4)forcomplementaryorproportionalswitching,respectively .Thevoltageacross C 3 is computedfromtheSCstructure:inaverage, V C 2 = V C 3 ,and V C 2 isevaluatedbyzeroing outtheleftsideof(II.3).Makingthenecessarysubstitutions,th econverter'sgainiscomputedasin(II.5)and(II.6)forcomplementaryandproportionals witching,respectively. V 0 V in = 1 d d 2 ; (II.5) V 0 V in = 2 d kd d kd + kd 2 : (II.6) ThetopplotofFig.II.2showsthevoltagegainasafunctionofthedu tycyclefor bothswitchingstrategies. Theadvantageoftheproposedproportionalstrategyistwofold .First,thevoltagegain underproportionalswitchingismoresensitivetochangesinthedut ycyclewhendeparting 8

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from d ? .Thismeanstheconvertercanoperateoverawidervoltagerange forthesame changeinthedutycycle.Inputcurrentrippleanalysis i in ( t ) i L 2 ( t ) i L 1 ( t ) T S (1 d ) T S 1 d ? d Carrier q 2 ( t ) q 1 ( t ) q 2 ( t ) q 1 ( t ) 0 q 2 ( t ) 0 tt i L 1 i in i L 2 dT S t 0 m L 1 m L 2 m +L 2 m +L 1 FigureII.3:Illustrationofcomplementaryswitchingfor dd ? .Illustrativewaveformsfor d = 60%and d =80%arepresentedinthebottomplotsofFigs.II.4andII.5,respe ctively. Inallthreeguresitisshownthattheinputcurrentrippleisnotzer o,whichisexpected as d 6 = d ? .Althoughtheaverageinputcurrentcanbereadilycomputedasth esumof thecurrentsthroughinductors L 1 and L 2 ,thequanticationofitsripplerequirestotake intoaccounttheswitchingfunctionsofeachstrategy.Ripplescan becomputedas i = m t ,where m representsthecurrent'srising(orfalling)slope,and t isthetimeofa particularswitchingstate.Theslopesfor i L 1 and i L 2 arereadilyobtainedfromtheaverage dynamicsdescribedby(II.2)or(II.4),and(II.3).TableII.1summa rizestheresults.The superindexindicateswhetherthesloperises(+)orfalls( ). Itisnoteworthythatslopesdependsolelyonthestateoftheircor respondingswitch. When S 1 ison(o), i L 1 rises(falls).Similarly,when S 2 ison(o), i L 2 rises(falls).Furthermore,risingslopesareconstant,andthereforeindependen tofthedutycycleandthe 9

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TableII.1:Risingandfallingslopesfor i L 1 and i L 2 . m +L 1 m L 1 m +L 2 m L 2 V in L 1 V in V C 1 L 1 V in L 2 V in V C 2 L 2 switchingstrategy.Giventhedenitionof V C 1 and V C 2 ,itfollowsthat m L 2 dependsonthe dutycyclebutisindependentoftheswitchingstrategy,while m L 1 dependsonboth. 1 d ? d Carrier S 2 q 2 ( t ) q 1 ( t ) q 2 ( t ) q 1 ( t ) 0 i L 1 i L 2 i in i L 1 ( t ) i L 2 ( t ) kdT S q 2 ( t ) 0 dT S T S 0 i in ( t ) 10 Carrier S 1 tttt kd (1 kd d ) T S 2 d ? m +L 1 m +L 2 m L 2 m L 1 FigureII.4:Illustrationofproportionalswitchingfor d
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i inN forcomplementaryswitchingiscomputedas, i inN = i in i L 2 = 1 d kd kd : (II.7) Underproportionalswitching,theinputcurrentfeaturesdiere ntripplecharacteristicsdependingonwhether dd ? .Itfollowsfromthedenitionof d ? thatif dd ? ( d =80%). Carriersignals(top),switchingfunctions(middle)andinductorcur rents(bottom). thecommutationstate( S 1 =OFF, S 2 =OFF)asthethirdswitchingstatefortheconverter. Thus,when dd ? leadsto kd> (1 d ).Thismeansboth switchesareonduringsometime,adding( S 1 =ON, S 2 =ON)asthethirdswitchingstate fortheconverter.Theconverterthuscommutesamongtheequ ivalentcircuitsofFigs. II.1(b),(c)and(e)when d>d ? .Theboundarybetweenthetwocases,thatis, d = d ? reducestocomplementaryswitching.Illustrativewaveformsfors witchingfunctionsare presentedinthetopandmiddleplotsofFigs.II.4andII.5for dd ? ,respectively.Thebottomplotsoftheguresalsorevealthattheinp utcurrentrisesandfalls 11

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twiceperswitchingcycle,whichisindeedaconsequenceofthethirdc ommutationstate. Asinthecaseofcomplementaryswitching,thecurrentthroughth einductorsrisesand fallsaccordingtotheswitches'state.However,thethirdcommut ationstateallowsforthe slopestoaddorcanceltwiceperswitchingcycle,givingtheinputcur rentitsripplecharacteristic.Alengthybutstraightforwardwaveformanalysisallows toelaborateTablesII.2 andII.3for dd ? ,respectively,wheretheinputcurrentslopesandassociated timesareidentied.Each m t productinthetablesgivesthetotal i theinputcurTableII.2:Inputcurrentrippleslopeandconductingtimesfor dd ? . t 1+ kd + d 2 T S (1 kd ) T S 1+ kd + d 2 T S (1 d ) T S m m +L 1 + m +L 2 m L 1 + m +L 2 m +L 1 + m +L 2 m +L 1 + m L 2 rentrisesorfalls.Thepeakinputcurrentripple i in correspondsthentothelargest i . Itisstraightforwardtoshowthatforbothcasesthelargest i occurswhentheinputcurrent'sslopeisgivenby m +L 1 + m L 2 .Therefore,for dd ? .Makingthenecessarysubstitutions,thenormalizedinputcurren trippleisthus calculatedasin(II.8). i inN = 8>>><>>>: 1 d kd 1 d ; if dd ? : (II.8) 12

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Currentripplesforeachswitchingstrategyarecomparativelyeva luatedbyplotting(II.7) and(II.8)againstthedutycycle.Resultsarepresentedinthemidd leplotofFig.II.2. Itmaybeobservedthatundertheproportionalstrategythepe akcurrentrippleiseither smallerthanorequaltothatofthecomplementarystrategy,and thereforeitmaybeconsideredsuperior.Theinputcurrent'stotalharmonicdistortion(T HD)wasalsoevaluated (numerically)asafunctionofthedutycycle.Resultsarepresente dinthebottomplotof Fig.II.2.Theinputcurrentunderproportionalswitchingfeature sasmallerTHDwithin thefeasibleoperatingrangeforcomplementaryswitching( d> 50%).Anotheradvantage ofproportionalswitchingisthatthedominantharmonicstakeplace atahigherfrequency. Because i in rises/fallstwiceperswitchingperiod,thedominantharmonicsbegint oappear attwicetheswitchingfrequency.Undercomplementaryswitching, thedominantharmonic isattheswitchingfrequency.ExperimentalResults Theconverterwasprototypedinthelaboratory.Fig.II.6showsa photooftheexperimentalsetup.Giventheinductancevalueoftheinterleavedind uctorsutilizedinthe experiment,theripple{cancelingdutycycleis d ? =66 : 67%.Experimentally,ripple{freeoperationwasobtainedat d ? =64%.Thismaybecausedbythetoleranceontheinductors' inductancevalueandaslightcoresaturation. Theexperimentfocusesonvalidatingtheproportionalswitchingst rategyintermsof itsinputcurrentripple.Figs.II.7(a)and(b)presentillustrativewa veformswhenoperatingat d ? =64%andatasmallerdutycycle( d =60%),respectively.Bothguresillustrate currentsthrough L 1 and L 2 ,andtheconverter'sinputcurrent.Fig.II.7(a)validateszero inputcurrentrippleoperation.Fig.II.7(b)validatesthetheoretic alconsiderations,asthe experimentalwaveformsareconsistentwiththoseofFig.II.4.Contributions&Impact Aproportionalswitchingstrategyfortheconverterintroduced in[42]wasproposed, testedandcomparedwiththecomplementaryswitchingstrategy. 13

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C 1 = C 2 = C 3 =20 F L 3 =1 H Gatedrivers From microcontroller L 2 =160 H L 1 =80 H D 1 S 1 S 2 D 2 D 3 FigureII.6:Hardwarephoto.Theswitchingfrequencywassetat5 0kHz. ATexasInstrumentsmicrocontrollermodelMSP430G2231wasutiliz edtoprovidethegatedriversignals.MOSFETsIRFP4668PBFanddiodesMBR4 0250G wereselectedfortherealization.Valuesofinductanceandcapacit anceareprovidedinthephoto. i in ( t ) i L 2 ( t ) 3A2A1A i in ( t ) i L 2 ( t ) 0.5A/div,5 s/div i L 1 i L 2 (a) i in ( t ) i L 2 ( t ) 3A2A1A i in ( t ) i L 2 ( t ) i L 1 i L 2 (b) 0.5A/div,5 s/div 0 0 i in =0 i in m +L 1 m L 2 m L 1 m L 2 m +L 2 m +L 1 m +L 2 m L 1 i L 1 ( t ) i L 1 ( t ) i L 1 ( t ) i L 1 ( t ) FigureII.7:Experimentalwaveformsfor(a) d =64%and(b) d =60% underproportionalswitching. Itwasdemonstratedthroughsimulationsandexperimentsthatth eproportional switchingstrategyoutperformsthecomplementaryswitchingstr ategyintermsof voltagegainandinputcurrentripple 14

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CHAPTERIII DC{DCCONVERTERWITHQUADRATICGAIN Motivation&PresentStateofKnowledge ConvertersforDC{DCpowerprocessingfeaturinghighvoltagega inplayanimportantroleintheinterconnectionoflowvoltagepowergenerationsou rcestoinverters.They arebecomingparticularlypopularwithintherenewableenergysecto r,wheresometechnologiessuchasphotovoltaicpanelsandfuelcellsgeneratepowera tlowvoltage[51,39]. ItiswelldocumentedintheliteraturethattraditionalboostDC{DC powerconvertersare notwellsuitedforhigh{gainapplications,duetotheparasiticcircuit resistanceand/or thenitecommutationtimeoftheconstituentsemiconductors[42] [34].Inaddition,convertersintendedforrenewableenergyapplicationsoughttoalsod rawalmostcontinuous currentfromtheDCsource.Largecurrentripplewouldresultinah ighRMScurrentthat producesextralossesandhighertemperaturesthatmayacceler atedegradationofthe renewablesource[21].Furthermore,acontinuousinputcurrenta llowsextractionofthe maximumpoweravailablefromtherenewablesource[57].Converters featuringbothhigh gainandripple{freeinputcurrentarethusexpectedtondnumer ousapplicationsasmarketpenetrationofDCsourcebasedrenewableenergiescontinues toincrease.Topologies withthesetwofeaturesfoundintheliteraturemaybeclassiedasm agnetic{coupledand nonmagnetic{coupled. Mostmagnetic{coupledapproaches[31,24,13,1,18,17]takead vantageoftransformersandcoupledinductorswithlargeturnratiostoachievehighvolta gegainwhiletheinputcurrentrippleisreducedthroughinterleavedinductorsorded icatedripplecanceling networks.Sometopologiesoeradditionalfeaturessuchaszero voltageswitchingorelectricalisolation.[31,24].Otherapproaches[13,1]usecoupledinduct orstoachievenear{ zeroinputandoutputcurrentripplesbuttheinductorturnratiois reservedfortheripple cancelingprocess.However,magneticcouplingintroducescomplex ityinthedesignand operationoftheconverter[24].Specically,reducedeciencydue towindings'equivalent 15

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seriesresistance(ESR)isreportedin[18].In[17],thecoupledinduct orsutilizedtoachieve highgainincreasetheinputcurrentripple,which,inturn,hastobee liminatedusingan additionalnetwork. Nonmagnetic{coupledsolutionssuchas[42,52,15,54,48]repres entanattractivealternativesincetheyusuallyyieldsimplertopologicalrealizationsandc ommutationstrategies.Thesetopologiesfeaturenocoupledinductorsandarealsotr ansformerless.In[42], theauthorspresentatopologythatutilizesswitchedcapacitorst oprovidehighgainand isabletocanceltheinputcurrentrippleataselectabledutycycleus ingtwointerleaved inductors.Moreover,aswitchingstrategyproposedlaterin[52]f urtherreducesthecurrentrippleandimprovesthevoltagegainof[42]forawideroperating range.Thedouble dualboosttopologydiscussedin[15]alsoreliesoninterleavedinduct orsforripplereduction,andobtainshighvoltagegainaddingtheoutputsoftheconstit uentboostconverters. Asimilartopologywithlowinputcurrentripplebutasmallervoltagegainis presentedin [54].Thetopologypresentedin[48]achieveshighvoltagegainandlowin putcurrentripplebyusingacascadedstructurecomposedofaninterleavedboos tconverterandathree{ levelboostconverter. Severaltopologiesfeaturingaquadraticgainhavebeenproposed andanalyzedinthe literature[26,20,38,25].Theseconvertersareabletosynthesiz ethedesiredoutputvoltageforawiderangeofinputvoltages,whilemaintainingthedutycycle withinreasonable values.However,topologieswithquadraticgainavailableintheliterat uredonotaddress inputcurrentripplecancelingtechniques.TechnicalApproach TheschematicinFig.III.1(a)illustratestheproposedtopology.Tr ansistors S 1 and S 2 areoperatedsynchronouslyutilizingthesameswitchingfunction.B ecauseofthis,the converterhastwoequivalentcircuitsresultingfromtheswitchinga ction:( S 1 = S 2 =ON) and( S 1 = S 2 =OFF).Theequivalentcircuitsfortheon{ando{statearedepict edin Figs.III.1(b)and(c),respectively. 16

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(a)(b)(c) L 2 v in i L 2 + + v 0 R S 2 C 2 + D 2 S 1 L 1 i L 1 i in C 1 D 1 v C 1 v C 2 + L 2 v in i L 2 + + v 0 R C 2 + D 2 L 1 i L 1 i in C 1 D 1 v C 1 v C 2 + L 2 v in i L 2 + + v 0 R S 2 C 2 + D 2 S 1 L 1 i L 1 i in C 1 D 1 v C 1 v C 2 + S 2 S 1 FigureIII.1:Circuitschematicoftheproposedconverter:(top) topological realization,(middle)equivalentcircuitfortheon{state,and(bott om)equivalentcircuitfortheo{state. Fromtheconverter'sgainpointofview,thearchitecturecorresp ondstothecascaded connectionofamodiedversionofabuck{boostconverter( S 1 , D 1 , L 1 , C 1 )andaboost converter( S 2 , D 2 , L 2 , C 2 ).Thisgivesthetopologyaquadraticgain,providingawide rangeofoperationfortypicalvaluesofdutycycle.Thistopologica lrealizationyieldsa converterwiththe keyfeature offorcingthecurrentdrawnfromthesourcetobecomputedasthedierencebetweenthecurrentsthroughthetwoco nstituentinductors, L 1 and L 2 ,whichinturnallowstosizetheseinductorstodrawripple{freecurr entfromthe sourceataselectabledutycycle.Theproposedtopologyhasanon {roatingoutput,pro17

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vidingacommonpointfortheloadandpowersource. Itisnoteworthythatthemodiedbuck{boostconverterutilizedin theproposed topologyhasbeenstudiedinotherworkssuchas[43,58].Hereinthe buck{boosttopologyisutilizedasabuildingblocktorealizea noveltopology withfeaturesnotaddressedin [43,58].Converteranalysis Whileoperatingundercontinuousconductionmode(CCM),tradition alDC{DCconvertersfeaturetriangularwaveformsforinductorcurrentsan dcapacitorvoltages,which allowanalysisoftheconverterutilizingthesmall{rippleapproximation[1 2].Theproposed topologybelongstothesameclassofconverters,andtherefore thesmall{rippleapproximationapplies.AsdiscussedinthedesignconsiderationsofSectionI II.2.2,CCMisassuredbyproperselectionofpassivecomponentsandswitchingfre quency.Thedynamics of i L 1 , i L 2 , v C 1 ,and v C 2 maythusbeanalyzedconsideringtheiraveragebehaviorandthe switchingfunctionsof S 1 and S 2 maybereadilyreplacedbytheircorrespondingdutycycles.Since S 1 and S 2 areswitchedsynchronously,auniquedutycycleoftheconverter d ( t ) isdened.Itcorrespondstothepercentageoftimeovertheswit chingperiod T s thatboth transistorsareonaccordingtotheirswitchingfunction q ( t ),as(III.1)indicates. d ( t )= 1 T s Z t + T s t q ( ) d: (III.1) Assumingalosslesssystemfornow,theequationsthatrepresent theaveragedynamicsfor thecurrentthroughinductors L 1 and L 2 are, L 1 di L 1 dt = d ( v in )+(1 d )( v in v C 1 ) ; L 2 di L 2 dt = d ( v C 1 v in )+(1 d )( v C 1 v in v C 2 ) : (III.2) Insteadystate,theaveragevoltageacrossinductorsmustbee qualtozero.Thus,byzeroingtheleft{hand{sideof(III.2)thesteadystatevoltageacro ss C 1 and C 2 isreadilyobtainedasin(III.3). V C 1 = 1 1 D V in and V C 2 = D (1 D ) 2 V in : (III.3) 18

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In(III.3)(andthroughtherestofthesection)capitallettersd enotesteadystatequantities.ItisevidentfromFig.III.1that V 0 = V C 2 ,andthereforetheconverterfeaturesa quadraticvoltagegaingivenbytheproductbetweengainsofabuck {boostandboostconverter, G = V 0 V in = D (1 D ) 1 (1 D ) = D (1 D ) 2 : (III.4) Acomparisonof(III.4)againstthequadraticboostconverterwh osegainis1 = (1 D ) 2 [26]showsthatthetopologypresentedhereinfeaturesalowervo ltagegain,but,asdiscussedbelow,hasthenovelcapabilityofachievingzeroinputcurre ntrippleataselectable dutycycle.Furthermore,(III.4)showsthat G canbelessthanunityandhencethetopologyisalsoabletobuckthevoltage,whichmaybeconsideredanadvan tageinsomeapplications[59].Giventhattheproposedtopologyisintendedforrenew ableenergyapplicationswheretechnologiessuchasphotovoltaicpanelsandfuelcellsg eneratepoweratlow voltage,thefocushereinisonboostoperation. Theconverterstatespacemodeliscompletebystatingthedynam icsfor v C 1 and v C 2 , C 1 dv C 1 dt = d ( i L 2 )+(1 d )( i L 1 i L 2 ) ; C 2 dv C 2 dt = d v C 2 R +(1 d ) i L 2 v C 2 R : (III.5) Insteadystate,theaveragecurrentthroughcapacitorsmust beequaltozero,which leadstothefollowingexpressionsforaveragevaluesoftheinducto rcurrents, I L 1 = 1 (1 D ) 2 V 0 R and I L 2 = 1 1 D V 0 R : (III.6) Comparisonwithotherconverters Fig.III.2showsacomparativeevaluationfortheidealvoltagegainp rovidedbythe proposedconverteragainstacoupleofstate{of{the{artcomp etitivetopologies,thatis, transformerlessconvertersabletoprovide bothahighvoltagegainandinputcurrentripple cancelation .Theripple{cancelingboostconverterfrom[42,52]andthedouble dualboost converterproposedin[15]areselectedforthecomparison.Theg ainoftheinterleavedrealizationofaconventionalboostconverterisalsoincludedforrefe rence. 19

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n r r !"# $ r r%! &rr%! 'r#"%! FigureIII.2:Idealvoltagegainoftheproposedtopologycompare dagainst othercompetitivetopologies. Astheguresuggests,theproposedtopologyisabletoprovideala rgergainanda wideroperatingrangefortypicalvaluesofdutycycles,say20to8 0%.Thismakesthe topologycompetitiveagainstothersolutionsavailableintheliteratur e. TableIII.1:Comparisonoftheproposedtopologyagainstcompetit ive converters. LCSDG ^ V T Proposedconverter2222 d (1 d ) 2 dV in (1 d ) 2 = V 0 Ripple{cancelingboost3323 2 d kd 1 d kd + kd 2 V in 1 d
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isthemaximumvoltageblockedbyatransistor(forthecaseofther ipple{cancelingboost converter k isadesignparameterrelatedtotheinductorssize[42,52]). Itisreadilyobservedfromthetablethattheproposedtopologyha sthesameora lowercomponentcountascomparedtotheotherconverters.Th etablealsosuggeststhat theproposedtopologyhasaswitchthatwillblocktheoutputvoltag eoftheconverter, whiletheswitchoftheripple{cancelingboostanddoubledualboostc onverterblocks alowervoltage.Thesetwoconvertersarethusabletoprovidethe sameoutputvoltage withlessstressintheirconstituentswitches,butatapriceofamor ecomplexrealization. Forexample,theripple{cancelingboostconverterreliesonareson antswitchedcapacitor structuretoachieveahighgain.Finally,itisnoteworthytomentiont hat,unliketheother competitivetopologies,theproposedconverterallowsbuckopera tionaswell. Analysisunderrealconditions Theconverter'sstatespacemodelgivenby(III.2)and(III.5)ca nbereadilyaugmentedtoincludetheparasiticeectofESRforthevariouscompon ents.Dening R in , R Lj , R Sj astheESRoftheinputvoltagesource,theinductors L j and S j for j =1 ; 2,respectively,itispossibletocomputetheconvertergainundermorer ealisticconditions, G = D D 2 + R L 2 + DR S 2 R + R L 1 + DR S 1 D 2 R + D 2 D 2 +2 D + D R in D 2 R (III.7) where D =1 D hasbeenintroducedtoavoidnotationcluttering.Fig.III.3illustratesthesensitivityoftheconverter'sgainwithrespecttovaria tionsintheinductors ESR.Asiscustomary,sensitivitiesaregivenastheratiobetweenth eloadresistance andtheinductor'sESR.Foreachinductor,theratioisvariedaccor dingto R L =R = 0 : 0005 ; 0 : 0010 ; 0 : 0015 ; 0 : 0020 ; 0 : 0025whiletheotheriskeptconstantat R L =R =0 : 0015. Theplotalsoshowsthemaximumgains b G L 1 and b G L 2 ,obtainedfromvaryingthecorrespondinginductor.Itmaybeobservedthattheconverter'sga inismuchmoresensitive tovariationsin R L 1 ,whichisexpectedsincetheinductor L 1 carriesalargercurrent.ESR 21

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nrr FigureIII.3:Sensitivitiesoftheconverter'sgaintovariationsinthe inductorsESR. ofsemiconductorsalsolimitthegainoftheconverter,butthiscanb eovercomebyselectingappropriatedevices[62].Theconvertermaythusoeramaximum gainofaround15. Designconsiderations Sizingofreactivecomponentsandtheprincipleofripplecancelationf ortheinputcurrentarediscussedinthissection.Inductorssizingforripplecancelation Thedynamicsof i L 1 and i L 2 givenby(III.2)suggeststhat,duringon{state, i L 1 rises withaslopeof v in =L 1 whereas i L 2 riseswithaslopeof( v C 1 v in ) =L 2 .Fortheo{state, dischargingslopesfor i L 1 and i L 2 are( v in v C 1 ) =L 1 and( v C 1 v in v C 2 ) =L 2 ,respectively. Consideringalsothaton{ando{statelast DT s and(1 D ) T s seconds,respectively,the ripplefor i L 1 and i L 2 maybequantiedas, I L 1 = V in L 1 DT s and I L 2 = V in L 2 D 2 1 D T s : (III.8) ItisalsoobservedinFig.III.1thattheinputcurrentoftheconver tercorrespondstothe dierencebetween i L 1 and i L 2 .Sincebothinductorschargeanddischargeatthesame time,theinputcurrentripplecanbecomputedas I in = I L 1 I L 2 .Making I in =0 andusing(III.8),theinputcurrentwillberipple{freewhen L 2 L 1 = D ? 1 D ? : (III.9) 22

PAGE 28

Thisisakeyfeatureoftheconverteroperation.Itallowstoselect L 1 and L 2 duringthe designprocesssuchthattheconverterdrawscontinuouscurre ntfromthesourceataselectable( nominal )valueofthedutycycle, D ? .Furthermore,theprincipleofoperationof theconvertersuggeststhatbothinductorsmay(ormaynot)be coupledforimprovingthe design,butcouplingisnotrequiredforenergytransferasitisintop ologiessuchasthe rybackconverter.Therefore,theproposedconvertermaybe consideredtransformerless. Astheoperatingdutycycledepartsfrom D ? ,rippleintheinputcurrentwillappear accordingto I in = D L 1 D 2 L 2 (1 D ) V in T s : (III.10) Itisnoteworthythatthisisasharedfeatureamongthetopologies usedforcomparisoninSectionIII.2.1.However,thisrippleisstillsmallascomparedto aconverterwith noripplecancelation. v in L 1 v C 1 v in L 2 i L 1 ( t ) v in v C 1 L 1 i L 1 v C 1 v in v C 2 L 2 t 0 i L 2 ( t ) i in ( t ) i L 2 T s DT s FigureIII.4:Instantaneousinductorsandinputcurrentsobtain edduring theon{stateando{stateoftheproposedconverterforaspe cicdutyratio ( D =66 : 7%). Asanexample,consideradesirednominalvoltagegainof G ? =12.Using(III.4),the dutycycleisfoundtobe75%.Theinductorscanbesizedas L 2 =3 L 1 utilizing(III.9) toachieveripple{freeoperationat D ? =75%.Fig.III.4illustratestheripplecancelation 23

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processfor D ? =75%.Thegureshowstheinstantaneouscurrentthrough L 1 and L 2 and theripplefreeinputcurrentcomputedas i in = i L 1 i L 2 . Also,(III.4),(III.6),and(III.8)allowcomputationoftheminimumv alueof L 2 that guaranteesCCM, L 2 RD ? (1 D ? ) 2 T s 2 ; (III.11) whichprovidesfurtherdesignguidelinesfor L 2 . Capacitorssizing Capacitorsmaybeselectedtomeetvoltageripplespecications.Vo ltageripplemay bequantiedfollowingaprocedureanalogoustothatusedinthesizin gofinductors,but consideringthedynamicsforcapacitorsinsteadofthatforinduct ors.From(III.5)and (III.6)ripplesacross C 1 and C 2 arereadilycomputed, V C 1 = V 0 RC 1 D 1 D T s and V C 2 = V 0 RC 2 DT s : (III.12) Inordertovalidatetheapproach,theconverterillustratedinFig. III.1wassimulated intheMatlab'sSimPowerSystemssoftwareandprototypedinthelab oratory.Fig.III.5 showsaphotooftheexperiment,highlightingthevariouscomponen tsinrelationshipto Fig.III.1.Forbothsimulationsandexperiments,a90Wconverterw ithaninputvoltage of V in =12Vwasimplemented.Theconverter'snominalgainwasselectedba sedonthe inductorsavailableatthetimeoftheexperiments .Inductorswithinductance L 1 =18 H and L 2 =39 Hyieldfrom(III.9)aripple{freedutycycleof D ? =68 : 42%.Utilizing (III.4),theconverter'snominalgainisfoundtobe G ? =6 : 86.Basedofthediscussionin SubsectionsIII.2.1andIII.2.2,itisnoteworthythatnominalgainso f G ? upto15maybe obtainedmodifyingtheselectionofinductors. Theloadresistanceis R =60nandcapacitors C 1 and C 2 weresizedaccordingto (III.12)andselectedat C 1 = C 2 =40 F.Theconverterisoperatedataswitchingfrequencyof f s =50kHz,utilizingoptocoupledgatedriverstocontrolthemosfets . Fig.III.6illustrateswaveformsundertheaforementionedoperat ingconditions.Simu24

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L 2 L 1 i L 2 i in i L 1 S 2 D 2 C 2 C 1 S 1 D 1 Frommicrocontroller Resistiveload Driver circuits FigureIII.5:Prototypephoto.Switches S 1 and S 2 arerealizedwithpower mosfetsIPP075N15N3withlowESRof7.5mn.Theinternaldiodesofm osfets IRF3710wereusedas D 1 and D 2 .Gatesignalsaresynthesizedthroughan8{ bitmicrocontrollermodelATmega328p,andappliedtothemosfetst hrough gatedriversmodelHCPL{3120. xxx i L 1 xxxxx i L 2 xxx i in xxxx v C 1 xxxxx v in xxxx v C 2 xxx q s Time(5 ¹ s/div) 0 20 40 60 80 xxx v C 1 xxx v C 2 xxx v in (V) 0 10 20 Time(5 ¹ s/div)(V) xxx q s 0 10 20 xxx i L 1 xxxxxxx i L 2 xxxxxxx i in (A) FigureIII.6:Simulated(left)andexperimental(right)waveforms :(top) inductorandinputcurrents( i L 1 ;i L 2 ; and i in ),(middle)capacitor,outputand inputvoltages( v C 1 ;v C 2 = v o ; and V in ),and(bottom)drivingswitchingfunction for D ? =68 : 42%. latedandexperimentalwaveformsarepresentedsidebysideforc omparison.Thetopplots showtheripplecancelationprocess;middleplotsillustratethevoltag egain;andthedrivingswitchingfunctionat D ? ispresentedonthebottomplots.Itmaybeobservedthein25

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nrn n !n FigureIII.7:Gainoftheproposedconverterasafunctionofthed utycycle. putcurrentrippleisindeedcanceledaspredictedby(III.9).Theme asuredoutputvoltage isaround75V,whichtranslatesintoagainofabout6.22.Thisvalue,s lightlylessthan G ? ,isconsistentwiththepredictionforthenonidealgainmadeby(III .7).Theexcellent matchbetweensimulatedandexperimentalwaveformswiththethe oreticalconsiderations discussedintheprevioussectionsvalidatetheconverter'sprinciple ofoperation.Theconvertergainobtainedfromcomputersimulationsalongwithexperimen talmeasurementsfor dierentoperatingpointsisshowninFig.III.7.Itmaybeobservedt hatboththesimulatedandexperimentalgainsfeaturethesameshapeasthatofth etheoreticalgainillustratedinFig.III.3. nrr ! n" FigureIII.8:Eciencyoftheproposedconverterfordierentva luesofthe outputpower. Finally,Fig.III.8showstheeciencyoftheconverteratdierentt hroughputpower levels.Thevaluesarecomputedinthesimulationandmeasuredinthee xperimentalsetup 26

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byvaryingtheloadresistorwhilekeepingallotherparametersunch anged.Ascanbeseen, theeciencyremainsabove90%fortheconverter'soperatingran ge,andmaybefurther improvedinacommercialimplementation.Theeciencyvaluesarealso similartothose reportedbythecompetitivetopologiesdiscussedinSubsectionIII .2.1[42,52,15],aswell astothosereportedforotherconverterswithquadraticgains[2 0,38,25]. Contributions&Impact Aconvertertopologywithhighgainandinputcurrentripplecancellat ionataselectabledutycyclewasdeveloped,prototypedandtested. Theproposedtopologyfeatureshighvoltagegainandgoodinputcu rrentripplecancellationwhencomparedtosimilartopologiesintheliterature. 27

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CHAPTERIV NON{LINEARCONTROLOFBUCK{BOOSTCONVERTERSFEEDING CPLS Motivation&PresentStateofKnowledge DC{DCpowerconvertersarewidelyemployedinpowerdistributionsy stemsto achievetheregulationofvoltagebetweentheDCsourceandtheloa d[63].Generally speaking,theDC{DCbuck{boostpowerconverterismoreadvant ageousduetothefact thatitpossessestheabilityofstep{upandstep{downmodes.Alth oughthecontrolof theseconvertersinthefaceofclassicalloadsiswell{understood, insomemodernapplicationstheloadsdonotbehavelikestandardpassiveimpedances,in steadtheyaremore accuratelyrepresentedasconstantpowerloads(CPLs),whichc orrespondtorst{third quadranthyperbolasintheloadsvoltage{currentplane.Thisscen ariosignicantlydiers fromtheclassicaloneandposesanewchallengetocontroltheoris t,see[4,28,11,33,56] forfurtherdiscussiononthetopicand[49]forarecentreviewoft heliterature.Itshould beunderscoredthatthetypicalapplicationofthisdevicerequires largevariationsofthe operatingpoint|therefore,thedynamicdescriptionofitsbehavio rcannotbecapturedby alinearizedmodel,requiringinsteadanonlinearone. Severaltechniqueshavebeenproposedforthevoltageregulatio nofthebuck{boost converterwithaCPLinthepowerelectronicsliterature.However, tothebestoftheauthors'knowledge,noneofthemprovidesarigorousstabilityanalys isforthenonlinear model.In[40],theactive{dampingapproachisutilizedtoaddressthe negativeimpedance instabilityproblemraisedbytheCPL.Themainideaofthismethodistha tavirtualresistanceisconsideredintheoriginalcircuittoincreasethesystemd amping.However,the stabilityresultisobtainedbyapplyingsmall{signalanalysis,whichisvalid onlyinasmall neighbourhoodoftheoperatingpoint.Anewnonlinearfeedbackco ntroller,whichiscalled \LoopCancellation",hasbeenproposedtostabilizethebuck{boos tconverterby\cancellingthedestabilizingbehaviourcausedbyCP"[41].Thecontrolpro blemturnsinto 28

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thedesignofacontrollerforthelinearsystembyusingloopcancellat ionmethod.However,theconstructionisbasedonfeedbacklinearizationthat,as iswell{known,ishighly non{robust.Aslidingmodecontrollerisdesignedin[50]forthisproble m.However,for theconsiderednonlinearsystem,thestabilityresultisobtainedbya doptinglinearsystemtheory.Inaddition,asitiswidelyacknowledged,thedrawbacks ofthismethodare thattheproposedcontrollawsuersfromchatteringanditsrela yactioninjectsahigh switchinggain.Thedeleteriouseectofthesefactorsisclearlyillust ratedinexperiments shownin[50],whichexhibitaverypoorperformance.In[19]anadapt iveinterconnection anddampingassignmentpassivity{basedcontroller(IDA{PBC)[37, 36],withguaranteed stabilityproperties,wasproposed.Unfortunately,theresulting controllawisrelatively complexstymingitspotentialapplicationinapracticalscenario.In[6 ]anapproximationofthecontrollawwasimplementedinanexperimentalbenchmar k.Aconsequenceof thecontrollerreductionwasthatthetuningprocedureforitsgain sturnedouttobequite critical|yieldingabelowparperformance. Thetopologyofabuck{boostconverterfeedingaCPL,isshowninF ig.IV.1.Under thestandardassumptionthatitoperatesincontinuousconductio nmode(CCM),theaveragemodelisgivenby L di dt = (1 u ) v + uE; C dv dt =(1 u ) i P v ; (IV.1) where i 2 R > 0 istheinductorcurrent, v 2 R > 0 theoutputvoltage, P 2 R > 0 thepower extractedbytheCPL, E 2 R > 0 istheinputvoltageand u 2 [0 ; 1]isthedutyratio,which isthecontrolsignal. Considerthesystem(IV.1)verifyingthefollowingconditions. Thepowerload P is unknown buttheparameters L;C and E are known Thestates i;v are measurable 29

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FigureIV.1:CircuitrepresentationoftheDC{DCbuck{boostconv erter withaCPL Fixa desiredoutputvoltage v ? 2 R > 0 andcomputetheassociatedassignableequilibriumpoint( i ? ;v ? ) 2E .Designastaticstate{feedbackcontrollawwiththefollowing features. ( i ? ;v ? )isanasymptoticallystableequilibriumoftheclosed{loopwithawell{de ned domainofattraction. Itispossibletodeneasetn R 2> 0 whichis invariant andinsidethedomainof attractionoftheequilibrium.Thatis,asetinsidetherstquadrant verifying ( i (0) ;v (0)) 2 n ) ( i ( t ) ;v ( t )) 2 n ; 8 t 0(IV.2) lim t !1 ( i ( t ) ;v ( t ))=( i ? ;v ? ) : (IV.3) TechnicalApproach Tosimplifythenotation,andwithoutlossofgenerality,inthesequel weconsiderthe normalizedmodelofthesystem,whichisobtainedusingthechangeo fcoordinates x 1 := 1 E r L C i x 2 := 1 E v; (IV.4) anddoingthe timescale change = t p LC thatyieldsthemodel _ x 1 = (1 u ) x 2 + u _ x 2 =(1 u ) x 1 D x 2 (IV.5) 30

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where D := P E 2 r L C ; _ ( )denotes d d ( )andallsignalsareexpressedinthenewtimescale . Thesystem(IV.5)canbeexpressedinthefollowingform _ x = f ( x )+ g ( x ) u; (IV.6) where f ( x )= 264 x 2 x 1 D x 2 375 ;g ( x )= 264 1+ x 2 x 1 375 : (IV.7) Theassignableequilibriumset E inthecoordinates x isgivenby E x := x 2 R 2> 0 j x 1 D x 2 D =0 (IV.8) Noticethatthecontrolproblemistranslatedintothedesignofast atefeedbackfor thesystem(IV.5)suchthatagiven x ? 2E x isasymptoticallystable. Itisimportanttorecallthatthesignalofinterestistheoutputvo ltage v ,therefore, forthexed x 2 ? 2 R > 0 ,the x 1 ? 2 R > 0 isdenedvia x 1 ? = D x 2 ? + D: (IV.9) ControlLawwithKnownLoadPower D Thecontrollerdesignproceedsinthreesteps. (i)Performachangeofcoordinatesandapartialfeedbacklineariz ationtotransform (IV.5)intoasimplecascadedsystem. (ii)ApplytheIDA{PBCtechniquetothissystemtoshapeitsenergyf unctionandensurelocalstabilityofthedesiredoperatingpoint. (iii)AddaPIcontrolleraroundthenewpassiveoutputforasymptot icstability. 31

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Proposition1 Thesystem(IV.5)withthestatecoordinatechange z =( x )= 264 1 2 j x j 2 + x 2 x 2 375 ; (IV.10) where j x j 2 := x > x ,andtheinput u =1 1 x 1 ( D x 2 + w ) ; (IV.11) takesthecascadedform _ z 1 = q 2 z 1 z 2 2 2 z 2 D (1+ 1 z 2 ) ; _ z 2 = w: (IV.12) Beforeclosingthissubsectionnoticethattheassignableequilibriums et E inthecoordinates z isgivenby E z := z 2 R 2> 0 j q 2 z 1 z 2 2 2 z 2 D (1+ 1 z 2 )=0 : (IV.13) Therefore,foraxed z 2 ? 2 R > 0 ,thecorresponding z 1 ? 2 R > 0 isdenedvia z 1 ? = D 2 2 1+ 1 z 2 ? 2 + z 2 2 ? 2 + z 2 ? : (IV.14) Proposition2 Considerthesystem(IV.12)with D known inclosed{loopwiththeIDA{ PBC w = ak 1 ( k 2 + z 1 ) arctan " 1+ z 2 p 2 z 1 z 2 2 2 z 2 # + w PI ; (IV.15) where k 1 ;k 2 and a arenon{zeroconstantsand w PI isanexternalsignal. (P1)Theclosed{loopsystemtakesthesimpleport{Hamiltonianform _ z = 264 0 a a 0 375 r H d ( z )+ 264 01 375 w PI (IV.16) 32

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where H d ( z )= 1 2 a 2 Dz 2 (1+ z 2 ) q 2 z 1 z 2 2 2 z 2 (1+2 z 1 )arctan " 1+ z 2 p 2 z 1 z 2 2 2 z 2 # +2 D ln z 2 + k 1 2 ( z 1 + k 2 ) 2 ; (IV.17) and r denotesthetransposedgradient. (P2)Fix z 2 ? 2 R > 0 andcompute z 1 ? 2 R > 0 via(IV.14).Select k 1 and a tosatisfy a z 4 2 ? aD (1+ z 2 ? ) 2 ( z 3 2 ? D 2 ) z 2 ? aD (1+2 z 1 ? ) : (IV.18) andselecttheconstant k 2 as k 2 := 1 ak 1 arctan 1+ z 2 ? p 2 z 1 ? z 2 2 ? 2 z 2 ? z 1 ? : (IV.19) If w PI =0theoperatingpoint z ? isalocallystableequilibriumwithLyapunovfunction H d ( z ). From(IV.16)with w PI =0,onegets _ H d ( z )= r T H d ( z )_ z = 1 2 r H T d ( z ) 0B@ 264 0 a a 0 375 + 264 0 a a 0 375 1CA r H d ( z )=0 : Hence,thetrajectoriescannotconvergetotheequilibrium.Toma ketheequilibrium asymptoticallystable,wefollowthetechniqueproposedin[36]andad daPIcontroller aroundthenewpassiveoutputgivenas y = h ( z;D ):= 01 > r H d ( z ) = 1 a D 1+ 1 z 2 q 2 z 1 z 2 2 2 z 2 : (IV.20) 33

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Proposition3 Fix z 2 ? 2 R > 0 andcompute z 1 ? 2 R > 0 via(IV.14).Considerthesystem (IV.12)with D known inclosed{loopwiththeIDA{PBCproposedinProposition2with theouter{loopPI _ = y w PI = k P y k I (IV.21) where k P > 0 ;k I > 0and y givenin(IV.20). (R1)( z ? ; 0)isalocallyasymptoticallystableequilibriumwiththeLyapunovfunctio n W ( z; )= H d ( z )+ k I 2 2 : (IV.22) (R2)Thereexistsapositiveconstant c a suchthattheset n a := f ( z; ) 2 R 2> 0 R j W ( z; ) c a g ; (IV.23) veries ( z (0) ; (0)) 2 n a ) ( z ( t ) ; ( t )) 2 n a ; 8 t 0 lim t !1 ( z ( t ) ; ( t ))=( z ? ; 0) : (R3)Thereexistsapositiveconstant c suchthattheset n:= f z 2 R 2> 0 j H d ( z ) c g ; (IV.24) veries z (0) 2 nand (0)=0 ) z ( t ) 2 n ; 8 t 0 lim t !1 ( z ( t ) ; ( t ))=( z ? ; 0) : 34

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AdaptiveControlUsingaPowerEstimator Inthissection,weproposeanadaptiveversionofthepreviouslypr esentedIDA{PBC forthecaseofunknownpower D .Theproposedadaptivecontrolleris,then,obtainedreplacing D by ^ D inthecontrollerofProposition3. TheestimatorofthisparameterisdesignedfollowingtheI&Itechniq uepresentedin [3].Inthepropositionbelowweprovethattheparameterestimation error ~ D ( t ):= ^ D ( t ) D; tendstozeroexponentiallyfastforallinitialconditions.Proposition4 ConsidertheaveragemodeloftheDC{DCbuck{boostconverter with CPL(IV.12).Deneanon{lineestimateof D generatedwiththeI&Iestimator ^ D = D I r z 1 z 1 z 2 +1 ; (IV.25) _ D I = rD I + r 2 z 1 z 1 z 2 +1 + r z 1 w ( z 2 +1) 2 + z 2 1+ z 2 q 2 z 1 z 2 2 2 z 2 ! ; (IV.26) where r> 0isafreegain. Forall initialconditionsofthesystemandall D I (0),wehave ~ D ( t )= e rt ~ D (0) : (IV.27) Toclarifytheformoftheadaptivecontrollerwedenetheestimate softhetermsappearinginthecontrolthatdependontheconstant D as: ^ u :=1 1 x 1 ( ^ D x 2 +^ w ) ^ z 1 ? := ^ D 2 2 1+ 1 z 2 ? 2 + z 2 2 ? 2 + z 2 ? ^ k 2 := 1 ak 1 arctan 1+ z 2 ? p 2^ z 1 ? z 2 2 ? 2 z 2 ? ^ z 1 ? ^ w := ak 1 ( ^ k 2 + z 1 ) arctan " 1+ z 2 p 2 z 1 z 2 2 2 z 2 # +^ w PI ^ y := 1 a ^ D 1+ 1 z 2 q 2 z 1 z 2 2 2 z 2 : 35

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where ^ D ( t )isanon{lineestimateof D generatedwithanI&Iestimatorbelow.TheadaptivePItakestheform _ =^ y; ^ w PI = k I k P ^ y: (IV.28) Noticethatallthe\hat"functionscanbewrittenintheform ^ ( )=( D; )+ " ( ~ D; ) ; where " ( ^ D; )issomesuitablydenedfunction,whichsatises " (0 ; )=0.Therefore,the closed{loopdynamicscanbewrittenastheasymptoticallystablesys temofProposition3 perturbedbyanadditivetermthatvanishesatzeroandconverge sexponentiallyfasttoit. Thus,wecaninvokewell{knownresultsofasymptoticstabilityofcas cadedsystems, e.g. , Proposition4.1of[46]toestablishASoftheadaptivescheme.Thede tailsofthisproof areomittedforbrevity|theinterestedreaderisreferredto[19] forafullderivation. Simulationandexperimentalresults Simulationsaremadetocomparetheperformanceoftheproposed controllerwiththe onedesignedin[19],whichisgivenas u ( x ):= 1 x 21 +( x 2 +1) 2 x 2 ( x 2 +1)+ x 1 ( x 1 D x 2 ) x 2 ( x 2 +1) x 1 + 2 x 1 x 2 x 2 +1 k 1 x 1 2( k 2 + x 21 )+ x 22 D (1+ x 2 ) 2 x 21 + x 22 + p 2 Dx 1 arctan x 1 r x 21 + x 22 2 (2 x 21 + x 22 ) 3 2 + 1 2 x 2 (2 x 21 + x 22 ) 3 2 2 x 21 ( x 2 +1) 2 2 x 2 q 2 x 21 + x 22 2 Dx 1 (1+ x 2 )+ x 2 (2 x 21 + x 22 )( 1+2 k 1 x 2 ( k 2 + x 21 )+ k 1 x 32 ) + p 2 Dx 22 arctan x 1 q x 21 + x 22 2 ! ; (IV.29) where k 1 isatuninggainsatisfying k 1 > max f k 0 1 ;k 00 1 g ; 36

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00.511.52 Time(s) 15 20 25Output voltage Vo/V set 5 set 7 set 9 00.050.10.150.2 Time(s) 15 20 25Output voltage Vo/V set 5 set 7 set 9 0.50.550.60.65 Time(s) 23 24 25 26Output voltage Vo/V set 5 set 7 set 9 00.511.52 Time(s) 1 2 3 4 5Inductor current i/A set 5 set 7 set 9 FigureIV.2:ResponsescurvesforproposedIDA{PBCwith r =1to changesinthepower D .(a)outputvoltage{(b)and(c)zoomsforit,(d)inductorcurrent. 00.511.52 Time(s) 15 20 25 30 35P/ W Actual 2 5 10 0.480.50.520.540.560.58 Time(s) 20 22 24 26 28 30 32P/ W Actual 2 5 10 FigureIV.3:Transientsperformanceoftheestimator ^ D understepchanges oftheparameter D fordierentthegain r andazoomoftherststep. 37

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where k 0 1 ;k 00 1 aretwoconstantsandtheconstant k 2 isdenedas k 2 := 1 k 1 0BB@ D (1+ x 2 ? ) 2 x 1 ? (2 x 21 ? + x 22 ? ) p 2 Dx 1 ? arctan x 1 ? r x 21 ? + x 22 ? 2 2 x 1 ? (2 x 21 ? + x 22 ? ) 3 2 1CCA x 22 ? 2 x 21 ? : Thecontroller(IV.29)is,clearly,muchmorecomplexthantheonepr oposedherein, e.g. ,(IV.11),(IV.15)and(IV.21). Asstatedin[6],forthecomplicatedcontroller(IV.29)thepowerful softwarepackages suchas Least-Squares-Fitting fromMatlabor Nonlinear-Model-Fit fromWolframMathematica,provideanecientwaytoobtainanapproximatedcontroller .Whenthevariable intervals,whichincludestheequilibrium,arechosenas x 1 2 [0 : 001 ; 0 : 2] ;x 2 2 [1 ; 3] ;D 2 [0 : 0001 ; 1],theapproximatedcontrollerisgivenby: u = 0 : 1702+1 : 005 D x 1 x 2 0 : 0143 x 2 ( D x 2 ? + D )+0 : 2412 x 2 ? : Itisobviousthattheaboveapproximatedcontrolleriseasytoimplem entinpractical scenario.However,aconsequenceofthecontrollerreductionist hatthetuningprocedure foritsgainsturnedouttobequitecritical. Inallsimulations,wehavechosenthesystemparametersshowninT ableIV.1and thenxedthedesiredequilibriumas x ? =(0 : 0893 ; 5 3 ) : Forsimplicity,wesimulatethe scaledsystem(IV.5),forwhichwehave D =0 : 0558.Dependingonthecontext,theplots areshowneitherfor x orfor( i;v )|thatwerecallaresimplyrelatedbythescalingfactors givenin(IV.4). Fig.IV.2revealstheresponsescurvesoftheoutputvoltage v anddutyratio u undertheproposedadaptiveIDA{PBCwithdierentgainsetsand r =1.Thedierent setsofgainsareshowninTableIV.2.Itisclearlyseenthatincreasing k 1 improvesthe convergencespeedoftheoutputvoltage.TogetherwithFig.6,itis concludedthatthere isatrade{obetweentherisetimeandthesizeofthedomainofattr action.Therefore, 38

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TableIV.1:ParametersofDC{DCbuck{boostconverterwithaCPL ParametersSymbolsValues InputVoltage E 15 V ReferenceOutputVoltage v ? 25 V Inductance L 107.5 H Capacitance C 1380 F NominalExtractedPower P 20 W 00.010.020.030.040.05 Time(s) 10 15 20 25Output voltage Vo/V New IDA-PBC IDA-PBC 00.010.020.030.040.05 Time(s) 0.4 0.5 0.6 0.7 0.8 0.9u New IDA-PBC IDA-PBC FigureIV.4:Transientresponsesforthetwocontrollers.(a)out putvoltage and(b)dutyratio. theparameters k 1 ;k p arechosensuchthatreasonablerisetimeandadequatesizeofthe domainofattractionareobtained.Furthermore,inFig.IV.3wesho wtheroleoftheobservertuninggain r .Asitcanbeseen,forlarger r ,thespeedofconvergenceoftheestimatorisfaster|aspredictedbythetheory.Butitisnoticedthatt hereisatradeobetweenconvergencespeedandnoisesensitivity. InFig.IV.4wepresentthetransientresponsesoftheoutputvolt ageanddutyratio forthetwocontrollers.Itisobservedthattheproposedcontro llerhasabettertransient response.Moreover,thecomplexityoftheproposedcontrolleris signicantlylowerthan 39

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(a) (b) FigureIV.5:Phaseportraitoftheclosed{loopsystemwithpropose dcontroller,threesublevelsetsn(blacklineshows u =1,orangelinefor u =0 : 5, greenlinefor u =0)andtrajectories(red)fordierentinitialconditions,level curvesoftheproposedcontrollerandazoom. thepreviouscontroller(IV.29). Thephaseportraitsoftheclosed{loopsystemwiththeproposedc ontrollerareshown 40

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TableIV.2:Gainsets Previouscontroller[19]Proposedcontroller(IV.15) Gain k 1 Gainsa k 1 k P k I set410.0010.30.5 set10.001set510.0010.70.5set20.01set610.010.31set30.1set710.010.71 set810.10.31.3set910.10.71.3 inFigs.IV.5.Similarlytothepreviouscontroller(IV.29),itisseeninFig. IV.5(b)that theclosed{loopvectoreldhasanotherequilibriumin R 2> 0 thatcorrespondstoasaddlepoint.InFig.IV.5,theveinitialconditionsareconsidered,whos estatetrajectories areshowninred.ItisobservedfromFig.IV.5(a)thatthreesublev elsetsn a |dened in(IV.24)|ensurethetrajectoriesremainin R 2> 0 .Moreover,theothercolorlinesarethe levelcurvesoftheproposedcontroller,whereblacklinesshows u =1,orangelinesshows u =1,greenlinesshows u =1.Itmeansthatwhentheinitialconditionisputtheregion from u =0(blackline)to u =1(orangeline),thecontrolsignalalwayssatisesthepracticalconstraint u 2 [0 ; 1].Thisistosaythatwhenweputtheinitialconditionintothe intersectionofthedomainofattractionandtheregionfrom u =0to u =1,itisensured thatthestatetrajectorieswillconvergetothedesiredequilibrium pointandthecontrol signalbelongstotheset[0 ; 1]. 41

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Experimentalresults InordertodemonstratetheperformanceoftheproposedIDA{ PBCanexperiment withabuck{boostconverterconnectedtoaCPLwasrealized.The powercircuitconsists oftwoo{the{shelfconverterboardsVishayDale MPCA 75136connectedincascade.One oftheboardsisconguredtoworkasavoltage{controlledbuckco nverter,whichactsas theCPL.Theotherboardisconguredasabuck{boostconverte randcontrolledusing theproposedIDA{PBC.Thepictureofthesetupusedforexperim entalimplementation areshowninFig.IV.6. FigureIV.6:ExperimentalsetupofDC{DCbuck{boostconverter witha CPL Tovalidatetheeectivenessoftheproposedapproach,weprese ntinthefollowingfour practicalcaseswhichrepresentdierentscenariosthatareofin terestinpracticalapplications. TherstexperimentisconcernedwiththestudyofDC{DCbuck{bo ostconverter withaCPLoperatinginboostmode.Theinputvoltage E anddesiredoutputvoltage v 2 ? aresetto15 V; 25 V ,respectively.Fortherstscenario,theloadpower P ischanged 42

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15 16 24 25 26 00.20.40.60.811.21.41.61.82 2 3 4 25 26 15 16 00.20.40.60.811.21.41.61.82 0 0.5 1 FigureIV.7:ResponsecurvesofDC{DCbuck{boostconverterwit haCPL for:(a)boostmode( E =15 V;v ? =25 V ),with P steppedfrom20 W to30 W and(b)buckmode( E =25 V;v ? =15 V ),with P steppedfrom5 W to7 : 5 W from20 W to30 W .Asecondexperimentiscarriedouttoexaminetheoutputvoltager egulationunderbuckmodeinthepresenceofsteptypechangeofload power.Forthiscase, theinputvoltage E anddesiredoutputvoltage v 2 ? aresetto25 V; 15 V ,respectively.The loadpowerisinitially P =5 W andwasincreasedto7 : 5 W .Theresultingoutputvoltage andinductorcurrentareshowninFigs.IV.7(a)and(b).Overall,th eproposedcontroller successfullymaintainstheoutputvoltageatthedesiredvaluerega rdlessofloadpower changes.Contributions&Impact Asolutiontotheproblemofvoltageregulationofabuck{boostconv erterfeedinga CPLwasgivenandthenvalidatedthroughexperiments. AgloballyexponentiallyconvergentI&Iestimatorwasproposedand implementedin anexperimentalsetup. 43

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CHAPTERV NON{LINEARCONTROLOFBUCKCONVERTERSFEEDINGCPLS Motivation&PresentStateofKnowledge Aswasstatedinthepreviouschapter,thecontrolofconverter sfeedingCPLsposesa problemtocontroltheorists.Herein,anIDA{PBCisappliedtoabuc kconverterfeeding aCPLanditistestedthroughsimulationsandexperiments.Assuming CCM,theaverage modelofabuckconverterfeedingaCPLisgivenby: _ x 1 = 1 L x 2 + E L u; _ x 2 = 1 C x 1 P Cx 2 ; (V.1) where x 1 2 R > 0 istheinductorcurrent, x 2 2 R > 0 theoutputvoltage, P 2 R > 0 the powerextractedbytheCPL, E 2 R > 0 istheinputvoltageand u 2 [0 ; 1]isthedutyratio, whichisthecontrolsignal. Considertheprevioussystemverifyingthefollowingconditions: Thepowerload P is unknown buttheparameters L;C and E are known Thestates x 1 ;x 2 are measurable Fixa desiredoutputvoltage x 2 ? 2 R > 0 andcomputetheassociatedassignableequilibriumpoint( x 1 ? ;x 2 ? ) 2E .Designastaticstate{feedbackcontrollawwiththefollowing features. ( x 1 ? ;x 2 ? )isanasymptoticallystableequilibriumoftheclosed{loopwithawell{ deneddomainofattraction. Itispossibletodeneasetn R 2> 0 whichis invariant andinsidethedomainof attractionoftheequilibrium.Thatis,asetinsidetherstquadrant verifying ( x 1 (0) ;x 2 (0)) 2 n ) ( x 1 ( t ) ;x 2 ( t )) 2 n ; 8 t 0(V.2) lim t !1 ( x 1 ( t ) ;x 2 ( t ))=( x 1 ? ;x 2 ? ) : (V.3) 44

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TechnicalApproach ThesystemofabuckconverterfeedingaCPLcanbeexpressedint heform _ x = f ( x )+ g ( x ) u; (V.4) where f ( x )= 264 1 L x 2 1 C x 1 P Cx 2 375 ;g ( x )= 264 E L 0 375 : (V.5) Theequilibriumforthissystemisgivenas: E := f ( x 1 ;x 2 ) 2 R 2> 0 j x 1 P x 2 =0 g (V.6) ControlLawwithKnownLoadPowerP Now,wewillapplytheIDA{PBCtechniquetothissystemconsideringt heload P isknow,toshapeitsenergyfunctionandensurelocalstabilityofth edesiredoperating point. From(V.5),wecandene g ? ( x )=[01],whichisthefull{rankleftannihilatorof g ( x ). Thematchingequationforthesystemisgivenby g ? [ F d r H d ( x ) f ( x )]=0 : (V.7) Now,deningtheinterconnectionmatrix F d as F d = 264 0 1 C 1 C 0 375 ; (V.8) thegradientoftheenergyfunction r H d ( x )canbecomputed.Substituting(V.8)in(V.7) weobtain: [01] 0B@ 264 0 1 C 1 C 0 375 264 r x 1 H d r x 2 H d 375 264 1 L x 2 1 C x 1 P Cx 2 375 1CA =0 : (V.9) 45

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Now r x 1 H d canbeimmediatelyobtainedfrom(V.9)as r x 1 H d = x 1 P x 2 : Theenergyfunctionisgenericallydenedas: H d ( x )=( x 1 )+( x 2 ) ; where( x 1 )canbeobtainedfrom r x 1 H d as ( x 1 )= 1 2 x 21 Px 1 x 2 : Now,wewillchoose( x 2 ),whichisafreefunctionas ( x 2 )= k 1 2 ( x 2 + k 2 ) 2 : Therefore, H d ( x )canbeexpressedas H d ( x )= 1 2 x 21 Px 1 x 2 + k 1 2 ( x 2 + k 2 ) 2 : Now,bycomputing r x 2 H d wecancompletethegradientvector r H d ( x )as: r H d ( x )= 264 x 1 P x 2 Px 1 x 22 + k 1 ( x 2 + k 2 ) 375 : Then,energyshapingcontrollerforabuckconverterfeedingaco nstantpowerisgiven as: u ( x )=( g T g ) 1 g T ( F d r H d ( x ) f ( x )) = L CE Px 1 x 22 + k 1 ( x 2 + k 2 ) + 1 E x 2 : (V.10) Finally,thePBCwithdampinginjectionisdenedas: u ( x )= L CE Px 1 x 22 + k 1 ( x 2 + k 2 ) + 1 E x 2 k p E L x 1 P x 2 : (V.11) Now,wewilllookatbothstabilityandconvergencecriteriatodenet hecontrolgains k 1 and k 2 . 46

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Inordertoguarantee r H d ( x ) j x = x ? =0, k 2 needstobedenedas: k 2 = Px 1 ? k 1 x 22 ? x 2 ? : Inordertoguarantee r 2 H d ( x ) j x = x ? > 0, k 1 needstobedenedas: k 1 > P 2 x 42 ? + 2 Px 1 ? x 32 ? : Usingtheseequationstocalculatethecontrolgains k 1 and k 2 willguaranteethatthe systemisbestableandconvergestotheequilibrium.iAdaptivecontrolusinganI&Ipowerestimator Inordertorendertheclosedloopsystemadaptive,apowerestima torisdesignedusingtheI&Itechniqueas ^ P = ^ P I 1 2 rCx 22 (V.12) _ ^ P I = rx 1 x 2 + 1 2 r 2 Cx 22 r ^ P I : (V.13) Thisestimatorwillprovidethecontroltheunknownpowerandwillallo wvoltageregulationunderchangesintheload.ComputersimulationsandexperimentsAveragedsimulations AbuckconverterfeedingaCPLinclosedloopwiththeproposedIDA{ PBCofand theestimatorof(V.12){(V.13)wasimplementedinMatlabusingblockd iagrams. Gainsensitivityanalysis Inthissubsection,theeectofthetuninggains k 1 and k p isanalyzedthroughMatlab simulations.Theparametersforthesimulationareconcentratedin TableV.1.Inthissimulations,astepwillbeappliedtothepower P andthetransientresponseofbothstates x 1 and x 2 willbeanalyzed. 47

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TableV.1:Simulationandexperimentalparameters ParameterCase1 E30V x 2 ? 20V L106.5 H C1380 F P20,40 FigureV.1:Simulatedcurrent(top),voltage(middle)andpower(bo ttom) waveformsfortheadaptiveIDA{PBCwith r =60, k p =1 e -7anddierent valuesof k 1 . ItcanbeseenfromFig.V.1thatthegain k 1 controlstheresponsespeed.However,it canalsobeseenthatfor k 1 =30theresponseseemstohaveanovershootandnooscillations,whilefor k 1 =50and k 1 =100,boththeovershootandoscillationsincrease.In Fig.V.2itcanbeseenthatthevalueof k p regulatestheresponsetime.Largervaluesof k p implyalongerresponsetime.Finally,theresponseoftheestimatorf ordierentvalues 48

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FigureV.2:Simulatedcurrent(top),voltage(middle)andpower(bo ttom) waveformsfortheadaptiveIDA{PBCwith r =60, k 1 =100anddierent valuesof k p . FigureV.3:Simulatedtransientperformanceoftheestimate ^ P understep changesoftheparameter P with k 1 =100, k p =1 e -7andvariousestimator gains r . of r showninV.3showsthatlargervaluesof r yieldshorterresponsetimesforthepower estimator.Switchedsimulationsandexperiments Inthissection,switchedsimulationandexperimentalresultsofthe proposedIDA{ 49

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0.460.4650.470.4750.480.4850.490.4950.50.5050.51 0.5 1 1.5 2 2.5 Experimental filtered Simulated filtered 0.460.4650.470.4750.480.4850.490.4950.50.5050.51 19 19.5 20 20.5 21 Experimental filtered Simulated filtered FigureV.4:Simulatedandexperimentalwaveformsfor x 1 and x 2 with P steppedfrom20Wto40W. PBCforabuckconverterfeedingaCPLarepresented.Theparam etersforbothexperimentsandsimulationsareconcentratedinTableV.1.Theestimatorg ainwasselectedas r =60andthecontrollergainswerechosenas k 1 =100and k p =1e-7. ItcanbeseeninFig.V.4thattheexperimentalresultsareclosetot hesimulation results.Additionally,itcanbeseenthatinbothsimulationsandexper iments,thereisa steadystateerrorofaround2%.Thisisexpectedduetoresistive lossesnotincludedin themodelsuchasinductor'sandcapacitor'sESRsandothersensin gresistorsinstalledin theboards.Contributions&Impact Asolutiontotheproblemofvoltageregulationofabuckconverterf eedingaCPL wasgivenandthenvalidatedthroughexperiments AgloballyexponentiallyconvergentI&Iestimatorwasproposedand implementedin anexperimentalsetup 50

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CHAPTERVI HEXVETER{BASEDLOWFREQUENCYACTRANSMISSION Motivation&PresentStateofKnowledge Thetransmissionsystem,whichisakeypartofthepowergrid,iseve r{growingand needstobeupgradedconstantlytomatchthepowerneedsofthe growingpopulation. However,transmissionlinesnowadayshavebeenpushedclosertot heiroperationallimits duetotheslowupgradeofthetransmissioninfrastructure[60].AC Transmissionsystems havebeencommonlyusedforpowertransmissionovershortdistan ces,howeverincertain applicationswherelongtransmissiondistancesareinvolved,DCsyst emshavebeenchosenoverACsystems.Currently,powerelectronicconvertersma kepossibletoimplement alternativestoHVDCsystemssuchasLowFrequencyAC(LFAC)Tr ansmissionSystems, alsotermedFractionalFrequencyTransmissionSystems.LFACsy stemsareapromising solutionforpowertransmissionoverlargedistances[23]andtheya lsohavethepotential ofincreasingthepowertransfercapacityofanoverheadtransm issionline[7,32]. LFACtransmissionsystemshavebeenproposedintheliterature,m ostlyforintegrationofoshorewindfarmstothegrid[22,27].In[44],theauthorsst udyaLFACtransmissionsystemoperatingat16.7Hz,whichisrealizedwithtwoback{to {backtwo{level VoltageSourceConverters(VSC)ateachendoftheline.Acycloco nverter{basedLFAC transmissionsystemisproposedandcomparedwithaVSCHVDCrealiz ation.ItisconcludedinthispaperthatthistheproposedLFACsystemhasalowerc ostthanHVDC, howevernotbymuch.Aback{to{backModularMultilevelConverte r(MMC){based LFACsystemisproposedin[53].Theproposedtransmissionsystemis comparedwitha cyclconverter{basedrealizationandwithanHVDCapproach.Thea forementionedimplementationsdonotdiscusstheselectionofthetransmissionfreque ncy.Theeectofnon{ standardtransmissionsystemsisdiscussedin[29]andin[10],theaut horsdiscusstheoptimaltransmissionfrequencyfortheintegrationofano{shorewin dfarmtothegrid.The problemfocusedmostlyonminimizingthecostofreactors,transfo rmersandtheo{shore 51

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infrastructure,whichledtoaminimumwhenthefrequencyisapprox imately90Hz. Herein,theLFACsystemisconsideredtoberealizedwithtwoHexver ters[5],aVSC memberoftheMMCfamily[45].Anunconstrainedoptimizationproblemw illbeformulatedtondtheoptimalfrequencyfortheLFACsystembasedonm inimizingtheenergy variationsinsidetheHexverter.LowFrequencyACSystem V 1 6 1 V 2 6 2 X =2 fL V g1 V g2 FigureVI.1:TransmissionlineinterconnectingtwoACsystems. Alosslesstransmissionlineusedtotransferpowerbetweentwosys temsisshownin Fig.VI.1.Thepowerrowingthroughthislineneglectingshuntelement sandtheseries resistancecanbeexpressedas: P = V 1 V 2 X sin( 1 2 ) ; (VI.1) where V 1 , V 2 , 1 and 2 aretheLine{to{NeutralRMSvoltagesandanglesateachendof theline,and X isthelinereactance,whichisdeterminedbythelineinductanceandth e transmissionfrequency( f ).Herein,aLFACsystemisproposedwhichwillberealizedby placing3{ /3{ AC/ACpowerconvertersattheendsofatransmissionlinetoreduc ethe transmissionfrequency.Thiswillinturnreducethelinereactance, increasingthemaximumthroughputpoweroftheline.Transmissionlineparameters Themainparametersthatdescribeatransmissionlineareitsseriesr esistance( r ac ), reactance( x ac )andshuntadmittance( y )perunitoflength.Theseparametersdependon 52

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thegeometricaldistributionofthephasewires,thematerialsused tobuildthelineand thesystemfrequency. Inordertocalculate r ac and x ac ,theDCresistanceofthewiremustbeobtained. WhenDCcurrentisrowingthroughawire,itsresistancecanbecalcu latedas: r dc = 1 a 2 ; (VI.2) where istheresistivityofthewireinnmand a istheconductorradiusinmeters. Temperaturedependance TheDCresistanceofawirewillchangeiftheambienttemperaturech anges.The changeinresistanceduetothetemperaturecanbeconsideredby calculatinganewdc resistance r 0 dc as r 0 dc = r dc (1+ ( T )) where isthetemperaturecoecient, r dc isthedcresistanceofthewire,and T isthe temperaturechange.Frequencydependance WhenACcurrentisrowing,thenon{uniformityofthecurrentdistr ibutioninthewire increasesas f increases.Thisisknownastheskineect.Duetothiscurrentdistr ibution,linesofmagneticruxwillexistinsidetheconductor.Sectionsof thewirecloserto thesurfacearenotlinkedtotheseinteriorlinesofmagneticrux,th us,sectionscloserto thecenterofthewirewillhavemoreruxlinkagethanthosecloserto thesurface.This meansthatmorevoltagewillbeinducedclosertothecenterofthew irethanonthesurface.Giventhatthetotalvoltagegradientmustbethesameinthe conductor,thecurrentwillnotbeuniformlydistributedandwillincreaseasitapproache sthesurfaceofthe 53

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wire,compensatingfortheoppositevariationoftheinducedvoltag e.Itisnoteworthythat notonlythemagnitudeofthecurrentdensitywillvaryoverthecro sssectionbutalsoits phaseangle. Theimpedanceoftheaconductorofradius a [35]duetotheskineectconsideringa temperaturechangeis: z i = r 0 dc ma 2 M 0 ( ma ) M 1 ( ma ) \ 0 ( ma ) 1 ( ma )+ 3 4 : (VI.3) where ma = p 2 0 r f r dc (VI.4) J 0 ( ka )=ber 0 ( ma )+ j bei 0 ( ma )= M 0 ( ma )e j 0 ( ma ) (VI.5) J 1 ( ka )=ber 1 ( ma )+ j bei 1 ( ma )= M 1 ( ma )e j 1 ( ma ) : (VI.6) J 0 ( ka )and J 1 ( ka )areBesselfunctionsofzeroandrstorder,while k 2 = j ! 0 Then,the seriesresistance r ac andinternalreactance x i canbeobtainedfrom(VI.3)as r ac = r 0 dc ma 2 M 0 ( ma ) M 1 ( ma ) cos 0 ( ma ) 1 ( ma )+ 3 4 (VI.7) x i = r 0 dc ma 2 M 0 ( ma ) M 1 ( ma ) sin 0 ( ma ) 1 ( ma )+ 3 4 : (VI.8) TheACreactance x ac oftheconductoristhenobtainedas x ac = x i +2 10 7 2 f ln 1 a +ln D ab +ln D bc ; (VI.9) where D ab and D bc arethedistancebetweeneachofthephasewires.Theseriesimped ance perunitoflengthisdenedas z = r ac + jx ac : (VI.10) 54

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Finally,therelationshipbetweenvoltageandcurrentatbothendso fadistributedline atdistance d isdescribedas: 264 V S ( d ) I S ( d ) 375 = 264 cosh( rd ) Z c sinh( rd ) sinh( rd ) =Z c cosh( rd ) 375 264 V R I R 375 (VI.11) where r = p zy isthepropagationconstantand Z c = p z=y thecharacteristicimpedance. Asitwasdiscussedin[35],theaforementionedmodelisrathercomple xandthus,anestimatedmodelwhichyieldsnearlyidenticalparametersforthefreq uencyrangeofinterestwasproposed.Thismodel,whichcapturestheskineectbysca lingthefrequency{ dependentparameterswiththeratiooffrequencies,willbeusedf ortheproposedoptimizationproblem.Hexverter AsimpliedaveragedcircuitofaHexverterisshowninFig.VI.2.Letth einputand outputvoltagesandcurrentsbedenedas: v i ; o m ( t )= ^ V i ; o cos 2 f i ; o t 2( m 1) 3 (VI.12) i i ; o m ( t )= ^ I i ; o cos 2 f i ; o t 2( m 1) 3 + i ; o ; (VI.13) assumingnophaseshiftbetween v i 1 ( t )and v o 1 ( t )at t =0.Thepeakvoltagesandcurrentsfortheinputandoutputsystemsare ^ V i ; o and ^ I i ; o ,respectively.Eachofthevoltage anglesisassumedtobetrackedand i ; o isthephaseshiftbetweeneachofthesystem's currentandvoltage.Itisnoteworthytoremarkthateachofthe systemsoperatesatadifferentfrequency,i.e. f i 6 = f o .ThecaseoftheHexverteroperatingatthesameinputand outputfrequenciesisnotconsideredherein.Analyzingthetermina lquantitiesforeachof theHexverterbranchesandassumingthelossesinthebranchimpe dances( Z b )negligible yieldsthefollowingexpressionsforthevoltageandcurrentofbran ch k [5]: 55

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v b k ( t )= ^ V i cos(2 f i t k +1+2( 1) k 3 )+ ^ V o cos(2 f o t k 2( 1) k 3 ) v go ( t )( 1) k (VI.14) i b k ( t )= ^ I i cos(2 f i t i 2 k +2 ( 1) k 6 )+ ^ I o cos(2 f o t o 2 k +( 1) k 6 )+ i cir ( t ) ; (VI.15) where v go ( t )isthevoltagedierencebetweentheinputandoutputreference pointsand i cir ( t )iscurrentcirculatinginsidetheHexverter. i b 1 ( t ) Z b v b 1 ( t ) Hexverter v i 1 ( t ) v i 2 ( t ) v i 3 ( t ) v o 1 ( t ) v o 2 ( t ) v o 3 ( t ) i i 1 ( t ) i i 2 ( t ) i i 3 ( t ) i o 1 ( t ) i o 2 ( t ) i o 3 ( t ) Z b Z b Z b Z b Z b i b 2 ( t ) v b 2 ( t ) i b 3 ( t ) v b 3 ( t ) i b 4 ( t ) v b 4 ( t ) i b 5 ( t ) v b 5 ( t ) i b 6 ( t ) v b 6 ( t ) FigureVI.2:Hexverteraveragedcircuitmodel. Additionally,thefollowingrelationshipsbetweentheHexverterpeak inputandoutput 56

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voltagesandcurrentsaredenedas: 264 ^ V o ^ I o 375 = 264 c 0 01 =c 375 264 ^ V i ^ I i 375 ; (VI.16) where c isaconstant.Thus,theHexverterisconsideredasalosslessfreq uencychanger. Thebranchpower p bk addingtotheenergyofbranch k canbecalculatedastheproductofthearmvoltageandcurrent.Assumingthat v go = V go =0andthatthecirculating currentisconstant( i cir = I cir ),thebranchpowercontainsatotalofsevenfrequencies, whoseamplitudescanbefoundinTableVI.1.Giventhatbothsystems arealsoconsidered balance,allsixbranchpowerscanbecalculatedbyusingonlythevolt ageandcurrentof onephase.Then,thebranchenergyvariationcausedbyanyofth espectralcomponentsat frequency f n isgivenby: e f n bk = 1 f n ^ p f n bk ; (VI.17) whichimpliesthatbranchpowersatlowerfrequencieswillhaveahighe rimpactontheenergyvariation.InthenextSection,thesumofallthebranchener gyvariationswillbethe objectiveofaminimizationproblemwhichwillselecttheoptimaltransm issionfrequency. Optimaltransmissionfrequency TheproposedLFACtransmissionsystemisdepictedinFig.VI.3.Given thattheenergystoragerequirementsoftheconverterarecloselyrelatedt otheoverallcost,itwillbe theobjectiveoftheproposedunconstrainedoptimizationproblem tominimizethebranch energyvariationsoftheHexvertersusedtorealizetheLFACtran smissionsystem,while alsoconsideringskineectonthetransmissionlineparameters.Itis noteworthytomentionthatthelossesintheHexverterareconsiderednegligible,andt hatthereceivingend ofthetransmissionlineisoperatingatunitypowerfactor. 57

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TableVI.1:Branchpowercomponents Frequency Resultingcomponentinbranchpower f 1 = f i ^ p f i bk =[ I 2 cir ^ v 2 i ] 1 2 f 2 = f o ^ p f o bk =[ I 2 cir ^ v 2 o ] 1 2 f 3 = f i + f o ^ p f i + f o bk = 1 6 [3^ v 2 o ^ i 2i +3^ v 2 i ^ i 2o + 4 3 P i P o + 4 3 Q i Q o + 4( 1) k p 3 ( P i Q o P o Q i )] 1 2 f 4 = f i f o ^ p f i f o bk = 1 6 [3^ v 2 o ^ i 2i +3^ v 2 i ^ i 2o + 8 3 P i P o 8 3 Q i Q o ] 1 2 f 5 =2 f i ^ p 2 f i bk = p 3 6 ^ v i ^ i i f 6 =2 f o ^ p 2 f o bk = p 3 6 ^ v o ^ i o f 7 =0 ^ p constbk = 1 6 P i 1 6 P o +( 1) k ( p 3 18 ( Q i Q o )) PCC2 PCC1 Hex 2 Hex 1 I S V S I R V R I 1 V 1 I 2 V 2 DistributedTL V g1 V g2 FigureVI.3:ProposedLFACTransmissionSystem. 0102030405060 20 30 40 50 FigureVI.4:Totalenergyvariationsasafunctionofthetransmiss ionfrequency( f ). GiventhecomplexpoweratPCC2and V 2 , I 2 iscalculated.Then(VI.16)isusedassuming c =1toobtainthevoltage( V R )andcurrent( I R )atthereceivingendofthetrans58

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missionline. Theparametersofthetransmissionlineconsideredin[35]willbeused ,whichwere reportedtobeobtainedfromanexistinglineoperatingat345kV,60 Hz.Next, V S and I S arecalculatedusing(VI.11)andgiventoHexverter1asoutputqua ntities.Finally, I 1 is calculatedassumingthat V 1 isknownandthat S S = S 1 ,albeitatdierentfrequencies. RecallingthattheHexverterswereassumedlossless,i.e. P i = P o andthattheyoperatewith Q i = Q o , I cir isconsidered0andthus,thebranchpowersforbothconverters canbecalculated.Finally,theenergyvariationsforallthebranche sofbothHexverters arecalculatedandconstitutethecostfunctionfortheproposed optimizationmodel: C ( f )= 6 X k =1 7 X n =1 1 f n ^ p f n 1 b k 1 !! | {z } Hexverter1 + 6 X k =1 7 X n =1 1 f n ^ p f n 2 b k 2 !! | {z } Hexverter2 : (VI.18) Sevenenergyvariationsarecalculatedforeachofthesixbranche softhetwoHexvertersateachendofthetransmissionlineandaddedtogether.Theen ergyvariationsfor theproposedLFACtransmissionsystemfordierentvaluesof f with S R =400MVA, V 1 = V 2 =345 = p 2kVandatransmissiondistanceof400kmisshowninFig.VI.4. 21222324252627 24.2 24.4 24.6 24.8 25 FigureVI.5:Totalenergyvariationsfordierentchangesintempe rature. Itcanbeseenthatthefunctionisconvexandthatithastwoasymp totes,oneat f =0andanotheroneat f = f g .Thisisexpectedgiventhatsomeenergyvariationsare inverselyproportionaltoboth f and f g f .Thus,wecanguaranteethatthefrequency 59

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foundwillbetheglobalminimumevenwhenusinganunconstrainedopt imizationsolver bygivinganinitialguessas0 f 0 f g . Thecostfunctionfordierentambienttemperaturesisdepictedin Fig.VI.5.Itcan beseenthatalthoughahighertemperatureyieldsahigheroptimalf requency,thevariationsarelessthan1Hz. 0102030405060 90 92 94 96 FigureVI.6:Transmissionlineeciencyasafunctionofthetransmiss ion frequencyfordierentpowerlevels. Finally,theeciencyofthetransmissionlinefordierentthroughpu tpowersasa functionofthetransmissionfrequencyisshowninFig.VI.6,wheret hedistanceoftheline wasconsidered400kmlong.Astheresultssuggest,theeciencyo fthelineishigherat higherfrequencieswithintherangeofinterest.Results Theoptimaltransmissionfrequencywasfoundforthreetestcas eswhilevaryingthe throughputpowerandtheambienttemperature.Forthepresen tanalysis,thelinewas consideredtobe400kmlongandthevoltagesoutsidetheLFACtran smissionsystemare 345kVLine{to{lineRMS.Fortherstcase,thefrequencyforbot hsystemswasconsideredtobe60Hz,thesecondcaseconsidered60Hzatoneendand5 0Hzattheotherand nally,thethirdscenarioassumesbothfrequenciestobe50Hz. Fig.VI.7showstheoptimaltransmissionfrequencyfoundwhenvar yingthepower atthereceivingend.Itcanbeseenthatfor S R > 400MVAalargerthroughputpower 60

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20 22 24 19 20 21 22 2004006008001000 18 19 20 21 FigureVI.7:Optimalfrequencyfoundfordierentreceivingendpo wers. (Top)60Hz/60Hz,(mid)60Hz/50Hz,(bottom)50Hz/50Hz. yieldsaloweroptimaltransmissionfrequency.However,thevariat ionsofthefrequency forawiderangeofthroughputpowersisaround4Hzmaximum.Addit ionally,theresults showthatlowerexternalfrequenciesalsoyieldaloweroptimaltran smissionfrequency. Theoptimalfrequencyasafunctionoftheambienttemperatureis showninFig. VI.8.Evenlargevariationsoftemperatureyieldsmalldierencesint heoptimalfrequency found.However,theexternalfrequencieshavealargerimpacto nthetransmissionfrequencyfound.Conclusions AssumingtheproposedLFACTransmissionsystemisimplementedona nexisting transmissionline,onlytemperatureandthroughputpowerwillchan geduringnormaloperation.Thus,theproposedapproachcanbeusedtoobtainanda djustthetransmission frequencyasafunctionofthetemperatureorthethroughputp owerdesiredthroughthe line.Finally,itisimportanttoremarkthattheresultsobtainedarepa rticularto:( i )the convertertopologychosen,( ii )thenatureoftheobjectivefunction,and( iii )thefactthat itisanunconstrainedoptimizationproblem.Addingconstraintstoth eproblemsuchas linecapacityiscurrentlybeinginvestigated. 61

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23.8375 23.83775 23.838 21.535 21.54 21.545 21.55 020406080100 20.0995 20.1 20.1005 20.101 FigureVI.8:Optimalfrequencyfoundfordierentambienttemper atures. (Top)60Hz/60Hz,(mid)60Hz/50Hz,(bottom)50Hz/50Hz. Contributions&Impact AmethodtodeterminetheoptimaltransmissionfrequencyforaLF actransmission systemwasproposed Theoptimalfrequencywasfoundforseveralambienttemperatu resandthroughput powers Itwasfoundthatevenlargevariationsintheambienttemperature resultedinsmall dierencesintheoptimalfrequency 62

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CHAPTERVII CONCLUDINGREMARKS Powerconvertersarealreadypresentthroughoutthewholepow ersystem.Theyplaya vitalroleinrenewableenergyapplicationsandareakeyenablingtech nologyforthefuture ofthegrid.Inthiswork,severalcontributionstoconvertersina renewableenergycontext wereincluded.DC{DCconvertertopologieswithfeaturessuchash ighvoltagegainand inputcurrentripplecancellationwerestudied,buildandtestedexte nsively.Additionally, theinteractionofcascade{connectedconverters,whereoneo fthembehavesasaCPL,was investigated.Non{linearcontrolstrategiestoaddresstheprob lemofvoltageregulationin suchscenarioswereimplementedandtestedinanexperimentalset up.Whilethesecontrollersstillhavesomedisadvantagessuchassteadystateerror andarelativelydicult gain{tuningprocess,theypresentaviableapproachtothisproble m.Finally,guidelines todeterminetheoptimaltransmissionfrequencyforaLowFreque ncyACtransmission systemwerefound.Althoughtheseguidelinesapplyonlywhentheco nverterchosenfor AC{ACconversionisaHexverter,theycanreadilybeextendedtor ealizationswithother topologies. 63

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