loads Voltage of a regulation matrix converter with balanced and unbalancedthree-phase Technology Journal of Applied Researchand

Availableonlineatwww.sciencedirect.com

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology13(2015)510–522

Original

Voltage regulation of a matrix converter with balanced and unbalanced three-phase loads ^夽

E. López-Robles

, J.J. Rodríguez-Rivas

, E. Peralta-Sánchez

, O. Carranza-Castillo

aInstitutoPolitécnicoNacional,CiudaddeMéxico,Mexico

bUniversidadPopularAutónomadelEstadodePuebla,Puebla,Mexico

Received28January2015;accepted5June2015 Availableonline23October2015

Abstract

Thispaperaddressesthedesign,simulationandexperimentalvalidationofavoltagecontrolforathree-phasetothree-phaseMatrixConverter workingunderbalancedandunbalancedresistiveloads.Theconverterisbasedonbidirectionalswitchesandspacevectorpulsewidthmodulation isusedtocontroltheirturn-onandturn-offtimes.Trackingandrepetitivecontrolsaredesignedandimplemented;havingthislastoneamajor impactontheperformanceoftheoutputvoltageregulationforbalancedorunbalancedloads.Theexperimentalcontrolsetupiscomprisedofa fieldprogrammablegatearrayboard,adigitalsignalprocessorandagraphicsinterfaceboard.

AllRightsReserved©2015UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisan openaccessitemdistributedundertheCreativeCommonsCCLicenseBY-NC-ND4.0.

Keywords: Matrixconverter;Balancedandunbalancedload;Outputvoltageregulation;Trackingandrepetitivecontrol

1. Introduction

CA/CApowerelectronicconvertersarehighlydemandedon theindustry,transportandservicesectors. Usuallythesecon- vertershavebeenimplementedbyusingatwostageconversion process:aCA/CDconverterfollowedbyanotherCD/CAresult- inginaDCbuslinkwhichneedsstorageenergyelementssuch asaninductorwhethertheconverterworksasacurrentsource converter,oracapacitorifitworksasavoltagesourceconverter.

AnotherwayofCA/CAconversionistheMatrixConverter (MC)whichiscomprisedbybidirectionalswitches(Wheeler, Clare,Empringham,Apap,&Bland,2003).Theconversionin thisconverterisstraightforwardwithoutaDClink.Pulsewidth modulation(PWM)tocontroltheswitchingsequenceisusedfor theconversionprocess.OneofthemainadvantagesoftheMCis thatasithasnoDClinkthevolumeandweightoftheconverter

夽 PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.

∗Correspondingauthor.

E-mailaddress:[email protected](J.J.Rodríguez-Rivas).

arereducedduetotheremovalofelectrolyticcapacitorswhich contributeswith30–50%ofthetotalvolumeinaconventional converter(Wheeleretal.,2003).

TheuseofMCsiscurrentlylimited.Themainreasonsare:(1) limitedavailabilityofthebidirectionalpowerelectronicdevices workingathighfrequencies,(2)controlimplementationiscom- plex,(3)theoutputvoltageisonly86.6%oftheinputvoltage,(4) thecostoftheprotectionsystemsforthebidirectionalswitches is high(Podlesaketal., 2005).Howeverwiththe technology progress itis expectedthat the MC will become areal alter- native to the traditional voltage source converter (VSC). An experimental 150kVA matrixconverterhas beenreportedin Podlesaketal.(2005).TheMCsarecurrentlyusedinapplica- tionssuchaswindpowergenerators(Cha&Enjeti,2003),no breakenergysources(Ratanapanachote,Cha,&Enjeti,2004), motordrives(Chekhet,Sobolev,&Shapoval,2000)andmarine propulsiondrives(Dalal,Syam,&Chattopadhyay,2006).

Simulationandexperimentalresultsofa3×3MCwithinput andoutputfiltersarepresentedanddiscussedinthispaper.A particular four-stepspace vectormodulation(SVM)sequence isused.Tracking(TC)andrepetitivecontrols(RC)areusedto regulatetheoutputvoltageunderbalancedandunbalancedload conditions.

http://dx.doi.org/10.1016/j.jart.2015.10.003

1665-6423/AllRightsReserved©2015UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopen accessitemdistributedundertheCreativeCommonsCCLicenseBY-NC-ND4.0.

Electric load Output filter

ia

SCa1 SCb1 SCc1

SCc2 SCb2

SCa2 SBa1

SBa2

SAa1 SAb1 SAc1

SAa2 SAb2 SAc2

Bidirectional switch Input filter

VA

VB

VC SBb1

SBb2

SBc1 SBc2

Output phase a. Output phase b. Output phase c.

Fig.1.3×3matrixconverter.

2. 3×3Matrixconverter

Fig.1showsa3×3MCcomprisedbya9bidirectionalswitch arraywhichconnectstheoutputphasesa,b,orcwiththeinput phasesA,BorC.Inputandoutputfiltersareillustratedaswell.

Bidirectional switches are controlled under the following constraints:

1. Astheconverterisfedbyathree-phasevoltagesource,only abidirectionalswitchperoutputphasecanbeturnedonto avoidashortcircuitbetweentwoinputphases.

2. Astheloadsareusuallyinductiveitmustbeguaranteedthat anoutputphasemustbeconnectedtoaninputphaseatany timeandabruptchangesofcurrentmustbeavoidedfornot generatinghighdv/dtthatmaydamagetheswitches.

Bearinginmindthesetwoconstraintsanopen-loopcontrol basedonSVMandafour-stepswitchingstrategywasimple- mented.Inputandoutput filters were used toavoidinjecting harmoniccurrentsintothemainsandharmonicvoltagesintothe load.TheSVM,four-stepswitchingstrategy,inputandoutput filterdesignsaredetailednext.

2.1. SpacevectorPWM

SpacevectorPWMisusedinthisworkbecauseithassome advantagestoreduceharmoniccontent(THD)ontheinputcur- rentandoutputvoltage(Casadei,Serra,&Tani,1998;Casadei, Serra,Tani,&Zarri,2009),beingfeasibletocompensateunbal- anced voltages by means of applying instantaneous voltage vectors which are calculated from the states of the switches (Casadei, Serra, Tani, &Zarri,2002; Grzegorz, 2009;Huber

&Borojevic,1995).

2.2. Four-stepcurrentcommutation

Constraints statedabove mustbe accomplishedduring the commutationofoneoftheinputphasestoanotherinputphase having the sameoutput phase. By means of externalcurrent sensorsthesenseoftheoutput-phasecurrentoftheconverteris deduced,thatcurrentsenseisneededtoimplementthecommu- tationstrategy.ThecommutationstrategyisillustratedinFig.2 (Deihimi&Khoshnevis,2009;Empringham,Wheeler,&Clare, 1998).

The procedure to disconnect input-phase A and connect input-phaseBtotheoutput-phaseaaccountingforthecurrent senseillustratedinFig.2(io>0)isasfollows:

SAa1 SAa2 – SBa1 SBa2 - SCa1 SCa2 11 – 00 - 00

01 – 00 - 00

01 – 01 - 00

00 – 01 - 00

10 – 00 - 00

10 – 00 - 00

10 – 00 - 10

00 – 00 - 10 10 – 10 - 00

00 – 10 - 00

00 – 10 - 00

00 – 10 - 10

00 – 00 - 10 00 – 00 - 01

00 – 00 - 11 00 – 00 - 01

ia(t)

ia(t)<0 ⁰ ia(t)>0

Output voltage a Input

voltage C Input voltage B

Input voltage A

SCa1 SBa1 SCa2 SAa1

SAa2

SBa2

01 – 00 - 01 01 – 00 - 00

00 – 01 - 01 00 – 01 - 00

00 – 11 - 00

Fig.2.Four-stepcurrentcommutationinmatrixconverter.

1. DisconnectSAa2.Theloadcurrentisnotinterrupted.

2. ConnectSBa1.Allowingthecurrenttoflowfrominput-phase Btooutput-phasea.

3. DisconnectSAa1.The switchSBa1isclosedallowingthe flowoftheoutputcurrent.

4. ConnectSBa2.PhaseBisconnectedtophasea.

Delaytimes(t1,t2andt3)areincludedbetweenstepandstep (seeFig.3).

Asimilaranalysiscanbecarriedoutfortheconditionio<0.

Theswitchingsequenceofthebidirectionalswitchesisillus- tratedontherightsideofFig.2.Thestateofeachoneofthe6 transistorsisrepresentedby“1”whenthetransistorisonand

“0”whenthetransistorisoff.

2.3. Inputfilter

In ordertominimize thecurrent harmonicsinjected tothe mains an input filter is used (see Fig. 1). Fig. 4 illustrates

SAa1 SAa2

SBa2

t1 t2 t3

SBa1

Fig.3.Blankingtimesdiagramfor4-stepcurrentcommutation.

the equivalentfilter circuit perphase, whereLand Care the inductance and capacitance of the filter, RL is the internal resistance of the inductor and R is a damping resistance (Wheeler&Grant,1997).

The values of the components in the input filter are:

L=0.7mH,C=26␮F,RL=50m,andR=56.Thetransfer function(TF)ofthefilteris:

H(s)= 686.8s+5.449×10⁷

s²+758.2s+5.449×10⁷ (1)

Fig.5illustratestheBodeplotofthefilterusingthreedifferent dampingresistances.

ForadampingresistanceofR=56theresonancefrequency isfr=1.18kHzandthegainisof20dB.Takingintoaccountthat theswitchingfrequencyoftheMCis12.8kHzanattenuation of−40dBisachieved.ThevalueofRchosenresultsinagood

Vin

RL L

R

i_L (t ) i_R (t )

i_c (t ) C

Fig.4.Equivalentcircuitperphasefortheinputfilter.

0 20

–20

–40

–60

–80 40

10² 10³ 10⁴ 10⁵

Frecuency (Hz) Magnitude (dB)

System: Hs Frequency (Hz): 1.84e+003 Magnitude (dB): –3 System: Hs

Frequency (Hz): 1.81e+003 Magnitude (dB): 19.9

R=56 Ohms.

R=20 Ohms.

R=200 Ohms

Fig.5.Bodediagramoftheinputfilter.

Vin

L RL

C Vout

Fig.6.Equivalentcircuitperphasefortheoutputfilter.

balancebetween theamplification atfrequenciescloseto the resonancefrequencyandattenuationathighfrequencies.

2.4. Outputfilter

Theoutputfilterisusedtoreducethevoltageharmonicsin theloadduetothecommutationofthebidirectionalswitches.

Fig.6illustratestheequivalentcircuitperphasefortheoutput filter(Dewan&Ziogas,1979;Kim,Choi,&Hong,2000).

–40 –30 –20 –10 0 10 20 30

10² 10³ 10⁴

Frecuencia (Hz) Magnitude (dB)

C=68 e-6F C=100 e-6F

System: Hs Frequency (Hz): 1.71 e+003 Magnitude (dB): 28.8

System: Hs Frequency (Hz): 2.65 e+003 Magnitude (dB): –3 System: Hs

Frequency (Hz): 1.41 e+003 Magnitude (dB): 27.1

Fig.7.Bodediagramofoutputfilter.

Setpoint F

3.5 + 1.1

–

k Controller

Z²–1.319z+0.9699

Z²+0.422z+0.402 Z²–1.319z+0.9699 0.3273z+0.3239

Plant (output filter)

Output voltage 1

Fig.8.Blockdiagramofthetrackingcontrol.

20

10

–10

–20

–30

–40 180

90

–90 0 0

Fase (deg)Magnitud (dB)

Frecuencia (Hz)

1.71 kHz Diagrama de bode

10¹ 10² 10³ 10⁴

Fig.9.Bodediagram.

The values of the components in the output filter are:

L=0.128mH,C=68␮FandRL=50m.TheTFofthefilter is:

H(s)= 1.149×10⁸

s²+390.6s+1.149×10⁸ (2) Fig.7illustratestheBodeplotoftheTF,itcanbeseenthat thecutofffrequencyis2.65kHzforthecapacitorvalueproposed andtheresonantfrequencyis1.71kHzwithagainof28.5dB.

IfthecapacitorvalueisincreasedasshowninFig.7,thewidth bandisreducedbeingthatthereasontochooseacapacitorof 68␮F.

3. Controlofthematrixconverter

InordertocontroltheoutputvoltageoftheMCandtocom- pensate theunbalancedoutput voltagesforunbalanced loads, two differenttypesofcontrollers:trackingandrepetitivecon- trollersareimplemented.

r(z)

e(z) rc(z)

C(z)

S(z) Z^–k K_r P(z) Y(z)

+ +

+

+ – +

Q(z) Z^–N

Fig.10.Blockdiagramoftherepetitivecontrol.

20 0 –20 –40 –60 –80

–100 0 –45 –90 –135 –180

–225

10¹ 10² 10³ 10⁴

Fase (deg)Magnitud (dB)

Frecuencia (Hz) FC=(400 Hz, –3db)

Diagrama de bode

Fig.11.BodediagramoffilterS(z).

Fig.12.BlockdiagramoftheRC.

1 0.5

ia>0

ia<0

SAa1 (blue) & SAa2 (red)

SBa1 (blue) & SBa2 (red)

SCa1 (blue) & SCa2 (red)

Time (s) –0.5

0

1.5 1 0.5 0

1.5 1 0.5 0

1.5 1 0.5 0

4.03 4.04 4.05 4.06 4.07 4.08 4.09

x 10^–2

Fig.13.4-Stepcommutation.

a

0 20

–20 –40 –60 –80 40 60 80

0.13 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.14

Va Vb Vc

Time (s)

0.13 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.14

Va Vb Vc

Time (s)

0.13 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.14

Va Vb Vc

Time (s)

0.105

0.1 0.11 0.115 0.12 0.125 0.13 0.135 0.14

Ia Ib Ic

Time (s)

Voltage (V)

b

0 20

–20 –40 –60 –80 40 60 80

Voltage (V)

c

0 20

–20 –40 –60 –80 40 60 80

Voltage (V)

d

0 5 10 15

–5

–10

–15

Current (A)

Fig.14.Openloopoutputvoltageswithunbalancedloadandfilters(a),addingTC(b),addingTC+RC(c);outputcurrentswithTC+RC(d).

30

20

10

0

–10

–20

–30

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14

TC TC + RC

Time (s)

Error (V)

Fig.15.Voltageerror.

3.1. Trackingcontrol

Theoutputvoltageisthevariabletobecontrolledwiththe trackingcontrol(TC)itsvalueismeasuredattheterminalsof theoutput-filtercapacitoranditisthefeedbacksignal.TheTF oftheoutputfilter(Eq.(2))isconsideredastheplantfortheTC.

Applyingz-transformtoEq.(2)resultsasfollows:

H(z)_plant = 0.3273z+0.3239

z²−1.319z+0.9699 (3)

FromEq.(3)thecharacteristicequationcanbeobtainedbeing theirs poles p1,2=0.4949±1.0237j which are located out of theunitcircle.Inordertoimprovethecontrolsystemstability thesepoleshavetobecanceledwiththezerosofthecontroller resulting ina TF for the controller withnew stablepoles as follows:

H(z)_controller =z²−1.319z+0.9699

z²+0.422z+0.402 (4) SisotoolsofMatlabv7.6wasusedfortheTCdesign.Fig.8 illustratestheblockdiagramoftheTC.

Fig.9showstheBodeplotofthecontroller.Itcanbeseenthat thecontrollerattenuatesthepeakvalueofthegain(seeFig.7)at theresonancefrequencyof1.71kHzoftheoutputfilter(plant).

From Eq. (4) andthe block diagramof Fig. 8 arecursive equation(Eq.(5))canbededuced.TheMCcontrolisbasedon Eq.(5)whichisprogrammedinaDSP.

Y(k)=−0.402Y(k−2)−0.422Y(k−1)

+1.1∗ [0.9699U(k−2)−1.319U(k−1)+U(k)]

(5) where

U(k)=3.5∗V(k)_ref−V(k)_fb (6) k–isanintegervalue(numberofsample),V(k)ref,V(k)fb–are thereferenceandoutputvaluesrespectivelyforaksample,U(k) –isthesubtractionofV(k)ref,fromV(k)fb,Y(k)–istheoutput variableofthecontroller.

3.2. Repetitivecontrol

Repetitivecontrol(RC)isusedtochangethereferencevalue appliedtotheTCtoreducethenoiseattheoutputanddiminish theperiodicalerrorsthatcanappearinthecontrolloop(Inoue, 1990;Yeh&Tzou,1995).Fig.10showsablockdiagramofthe RCwhere:r(z)isthereferencesignal,e(z)istheerrorsignal,Z^−N isthedelayinthefeedbackloop,Nisthenumberofsamplesina periodofthefundamentalfrequency,Q(z)isaconstant,S(z)isa low-passfilterthatworksascompensatorprovidingrobustness tothecontrolsystemandimprovingthestabilitymargin,Z^−kis adelaytomaketheoutputsignaloftheblockC(z)tobein-phase withtheinputsignalr(z),KristhegainoftheRC,rc(z)isthe modifiedreference,P(z)istheTFoftheTCanditisconsidered astheplant(outputfilter)ofthecontrol,C(z)isthepolynomial transferfunctionfortheRCandY(z)istheoutput.

ThepolynomialC(z)isasfollows:

C(z)=Z^−N∗S(z)∗Z^−k∗K_r

1−[Z^−N∗Q(z)] (7)

ThevalueofQ(z)inFig.10isaconstantequalto0.95(Zhang, Kang,Xiong,&Chen,2003).ThecompensatorS(z)isasecond order IIR filter (García-Cerrada, Pinzón-Ardila, Feliu-Batlle, Roncero-Sánchez, & García-González, 2007) with a cutoff

Output filter Input filter

MC

Load

DSP, FPGA, HPI

Fig.16.Experimentalsetup.

frequencyof400Hz(fundamentalfrequencyoftheMCoutput) anda sampling frequencyof 12.8kHz which isequal tothe switchingfrequencyoftheMC.TheMatlabtoolbox“fdatool”

isusedforthedesignofthefilter.TheTFofthefilteris:

S(z)=0.0084z²+0.0169z+0.0084

z²−1.724z+0.7575 (8)

Fig. 11 shows its Bode plot. The value of N is the ratio between the switching and fundamental frequencies.

N=12,800/400=32. The value of Kr is 0.6 (García-Cerrada etal.,2007;Zhangetal.,2003).Theprocedureforcomputing thevalueofkisbasedonasimulationwheretheoutputofC(z) andthereferencesignalr(z)arecompared(Fig.10),kisadjusted untilthemismatchbetweenC(z)andr(z)iscompensated.The valueofkfoundisequalto23thereforeZ^−k=Z⁻²³.

ThefinaldiagramoftheRCisshowninFig.12.AdelayZ⁻¹ isaddedtothereferencesignaltoguaranteethatthissignalis in-phasewiththefeedbacksignal.

BysplittingFig.12inthreesectionstherecursiveequations arededucedasfollows:

1.

e(z) +

0.95 +

Z^-32

MI(z)

MI(k)=e(k)+0.95∗MI(k−32) (9)

LeCroy

LeCroy Measure

value status

P1: max(C2) 80.8 V

P2: --- C2

C2

P3: --- P4: --- P5: --- P6: ---

Measure value status

P1: max(C2) 22.16 A

P2: freg(C2) 403.27415 Hz

P4: --- P5: --- P6: ---

P3: ---

a

b

vb_salida

Vb_salida

Fig.17.Outputsignalswithoutfilters:phasevoltage(a),currents(b),linevoltage(c),inputcurrent(d).

LeCroy

LeCroy Measure

value status

P1: max(C1) 136.1 V

P2: freq(C1) 12.1140420 kHz C1

C4

LA

P3: --- P4: --- P5: --- P6: ---

Measure value status

P1: max(C4) 25.13 A

P2: freq(C4) 12.37451 kHz

P3: --- P4: --- P5: --- P6: ---

c

d

Fig.17. (Continued)

2. MI_ 2( )k =0.6*MI k( −23)

0.6

Z^-21 Z^-2 MI_2(z) )

z ( I

MI 2(k)=0.6∗MI(k−23) (10)

3.

2 2

0.0084 0.0169 0.0084 1.724 0.7575

z z

z z

+ +

− +

MI_2(z) e2_S(z)

e2_S(k)=−0.7575∗e2_S(k−2)+1.724∗e2_S(k−1) +

0.0084∗MI2(k−2)+0.0169∗MI2(k−1)

+0.0084∗MI2(k)

(11)

4. Simulationresults

The 3×3 MC was simulated in Matlab/Simulink v.7.6.

Fig.13illustratesthea-phasecurrentandtheswitchstatesfor a4-stepcommutation(seeFig.1).Fig.14showstheMCout- putvoltagesperphase,athree-phaseunbalancedstarconnected resistiveloadof4,8and10wasused.

Fig. 14a shows the output voltagesper phase of the MC inopen loop,the outputandinputfilters are connected.This figureshows theunbalancedshape ofthe voltagesduetothe unbalancedload.

Fig.14b showstheresultsafteraddingaTCtocontrol the output voltage. The stability was improvedas it was pointed outabovehoweverthewaveformsarestillunbalanced.Finally Fig.14cillustratestheresults ofaddinganRCtocontrol the

LeCroy LeCroy

Measure value status

P1: max(C3) 81 V

P2: freg(C3) 400.886 Hz

P4: freg(C4) 401.633 Hz

P5: --- P6: ---

P3: max(C4) 18.4 A

Measure value status

P1: max(C4) 17.7 A

P2: freg(C4) 60.0118 Hz

P4: --- P5: --- P6: ---

P3: ---

v out

C0

C4

i out

i in

a

b

Fig.18.Outputphasevoltageandcurrent(a),inputcurrent(b)intheMC.

outputvoltage. Abalancedoutputvoltageisachieveddespite theunbalancedloadused.TheoutputcurrentswithTC+RCare showninFig.14d.Inadditiontogetbalancedoutput-voltages whenRCisaddedtothecontrolsystemareductionontheerror oftheoutputvoltageoftheMCisachieved.Fig.15illustrates the outputvoltageerror underTC andTC+RC.The error is calculatedasthesubtractionoftheoutputvoltagemeasuredat theoutputcapacitorfromthereferencevoltage.

5. Hardwareimplementation

WiththeaimsofvalidatingthecontroloftheMCalaboratory prototypewasbuilt.Fig.16showsthelaboratorysetup.Itcanbe

seentheinputandoutputfilters,theMCandthecontrolboards whicharecomprisedbyadevelopmentkitDSPTMS320C6713, communicationboardHPI(HostPortInterface)andanFPGA board. The printed circuit board of the MC andFPGA were designed by The Nottingham University inthe UK (Buso&

Mattave, 2006)and thesewereacquired for thisproject. The communicationboardHPIallowsthedatatransmissionbetween PCandDSPusinganUSBportbeingpossiblethemonitoring ofdifferentvariablesindeDSPbyusinganinterfacedeveloped inMatlab. Theratedoutputpowerof theMCisof 7.5kW.It iscomprisedby12bidirectionalswitchesinamatrixarrayof threerowsbyfourcolumns(onlythreecolumnswereusedin thiswork).EachbidirectionalswitchiscomprisedbytwoIGBT

Phase A

Phase B

Phase C v out

Measure value status

P1: max(C1) 80 V

P2: freg(C1) 400.909 Hz

P4: freg(C4) 400.961 Hz

P5: --- P6: ---

P3: max(C4) 13.9 A

Measure value status

P1: max(C2) 80 V

P2: freg(C2) 400.936 Hz

P4: freg(C4) 399.931 Hz

P5: --- P6: ---

P3: max(C4) 10.6 A

Measure value status

P1: max(C3) 80 V

P2: freg(C3) 400.052 Hz

P4: freg(C4) 400.007 Hz

P5: --- P6: ---

P3: max(C4) 9.3 A C4

C3

C3

i out

v out

i out

v out

i out

LeCroy

LeCroy

LeCroy

Fig.19.PhaseoutputvoltagesandcurrentswithTC+RCandunbalancedload.

transistors and isolated gate drivers. Each output phase has attacheda current sensor used toimplement the 4-stepcom- mutation.

6. Experimentalresults

Fig.17showstheexperimentalresultsfortheMCwithout inputandoutput filters. Itcan beseena highfrequencyhar- monic content dueto the switching devices. Fig. 18a shows thephasevoltageandoutputline-currentswithaY-connected resistiveload.Fig.18billustratesaphaseoftheinputcurrent.

InputandoutputfiltersareconnectedtotheMCforresultsof Figs.18and19showoutputvoltagesandcurrentsper-phasefor anunbalanced3-phaseY-connectedresistiveloadof 4,8 and10.Byanalyzingthisfigureitcanbeseenthatthevoltage peakvaluesareof80Vbeingofthesamevalueasthereference valuevalidatingthesimulatingresultsshowninFig.14c.

Theoutputcurrentshavedifferentmaximumvalues:13.9A, 10.6Aand9.3A duetoan unbalancedload andtheir values correlatewellwiththoseshowninFig.14d.

7. Conclusions

The AC/AC power conversionin aMC is straightforward without needing to carryout a two-stage conversion process resulting inaconverterwithout aDCbus link andin conse- quenceastorageenergyelementisneedless.MCiscomprised byabidirectionalswitch-array;thenumberof switchesrelies onthetypeof matrix.Thebidirectionalswitches needahigh performanceswitchingcontroltoavoidoperationalfails.Inthis workthesimulationandexperimentalvalidationofa3×3MC werecarriedoutassessingtheoutputvoltageregulationforbal- ancedandunbalancedloads.A4-stepcommutationwasassessed resultinginasafeoperationintheMC.Inputandoutputfilters wereusedtoavoidinjectingharmoniccurrentsintothemains andharmonicvoltagesintotheload.Thecompensationofthe voltagecontrolloopwascarriedoutbyaTCwherethepolesof theTFoftheoutputfilterwerecanceledwiththezerosofthe controllerincreasingthesystemstability.BymeansoftheRC thereferencesignalappliedtotheTCcanbevariedtoreducethe noiseattheoutputanddiminishtheperiodicalerrorsthatcan appearinthecontrol loop. Theexperimentalresultsillustrate thattheoutput voltageamplitudescorrelatewellwiththeref- erencesignalwhenTC+RCareusedeventhoughunbalanced loadsareconnected.

Conflictofinterest

Theauthorshavenoconflictsofinteresttodeclare.

Acknowledgement

ThisworkwassupportedbytheNationalPolytechnicInsti- tuteinMexicoCity.

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