digital control applications in power electronics lez2

8/13/2019 Digital Control Applications in Power Electronics Lez2

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September 1999 1Simone Buso - University of Padova - Lesson 2

LessonLesson 22

DigitalDigital Control of ThreeControl of Three --PhasePhase DC/ACDC/ACConvertersConverters :: SpaceSpace Vector Vector ModulationModulation


Voltage SpaceVoltage Space Vector Vector ModulationModulation

AA threethree --phasephase inverter inverter can generatecan generate threethreeindependent twoindependent two --levellevel phase voltagesphase voltages ..

EightEight different instantaneousdifferent instantaneous inverter inverter configurationsconfigurations ((statesstates )) areare availableavailable ..

By suitablyBy suitably switchingswitching ((modulation strategymodulation strategy ))among these states it is possibleamong these states it is possible toto generategenerateany tripletany triplet ofof averageaverage phase voltagesphase voltages VV1avg1avg ,,VV2avg2avg , V, V3avg3avg ranging fromranging from +E/2+E/2 toto -E/2,-E/2, wherewhereEE is theis the DC linkDC link voltagevoltage ..


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When the load is connected withWhen the load is connected with insulatedinsulatedneutral,neutral, it is only sensitiveit is only sensitive toto lineline -to--to- linelinevoltagesvoltages .. TheThe neutralneutral voltagevoltage doesnt havedoesnt have anyanyeffect on iteffect on it ..

Any voltage tripletAny voltage triplet can be schematicallycan be schematicallyrepresentedrepresented asas a vector a vector laying onlaying on a planea plane ((bibi --dimensionaldimensional representationrepresentation ). In general,). In general, thetheinformation aboutinformation about the value of thethe value of theinstantaneousinstantaneous neutralneutral voltagevoltage ((the thirdthe thirddimensiondimension !)!) cannot be represented andcannot be represented and getsgetslostlost ..



-E

+

v1 v 2 v 3


ThreeThree --phase Voltage Sourcephase Voltage Source Inverter Inverter


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Voltage SpaceVoltage Space Vector Vector ModulationModulationTheThe voltagevoltage vector vector representing the tripletrepresenting the triplet vv11 ,,vv22 , v, v 33 ,, can be drawn bycan be drawn by summingsumming three vectorsthree vectors((lenghtlenght proportionalproportional toto amplitude)amplitude) directeddirected asasthreethree 120120 shiftedshifted axesaxes (a, b, c).(a, b, c).

V1

V2

V3

V

a

c

b

V3

V2V1



We therefore definedWe therefore defined aa

DirectDirect Vector Vector TransformationTransformation

between the tripletbetween the triplet vv11 , v, v22 , v, v 33 ,, andand vectorvector V.V.A similar A similar transformation can be definedtransformation can be defined for for inverter inverter currentscurrents ........


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V

V

V1V2

V3V

( )

V V V V

V V V

=

=

1 2 3

2 3

2 2

3

2

Voltage SpaceVoltage Space Vector Vector ModulationModulationThe direct transformation can beThe direct transformation can be analyticallyanalyticallyformulatedformulated referringreferring to ato a couple of orthogonalcouple of orthogonalaxesaxes andand ( ( usually coincident with axisusually coincident with axis a).a).


V1V2

V3V

a

b

c

2/3 VV V

V V V

V V V

1

2

3

2

32

3

3

2 2

2

3

3

2 2

=

=

=


TheThe reversereverse vector vector transformationtransformation can becan beachieved starting fromachieved starting from aa 2/3 V2/3 V longlong vector vector andandprojecting it on the three axesprojecting it on the three axes a, b, c.a, b, c.


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Voltage SpaceVoltage Space Vector Vector ModulationModulationTheThe inverter inverter statesstates can also be representedcan also be represented asasvoltage vectorsvoltage vectors .. AsAs an examplean example ::

-E

+100V

V1

V2V3

101V

110V010V

011V

001V

StateState 100:100: VV11=E=E VV22=0=0 VV33=0=0



TheThe zero vector zero vector can be generated incan be generated in twotwoequivalent waysequivalent ways ::

-E

+ 111VV1V2V3

but alsobut also ......

StateState 111:111: VV11=E=E VV22=E=E VV33=E=E


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StateState 000:000: VV11=0=0 VV22=0=0 VV33=0=0

-E

+

000V

V1V2

V3



SpaceSpace Vector Vector ModulationModulation (SVM)(SVM) isisperformed byperformed by generatinggenerating ,, within thewithin theswitching periodswitching period ,, aa sequence of differentsequence of differentinverter inverter statesstates ..

The sequence normally consists ofThe sequence normally consists of threethreevectorsvectors ,, one of which is theone of which is the zero vector.zero vector.

TheThe durationsdurations of the threeof the three inverter inverter statesstateshavehave toto satisfy the followingsatisfy the following constraintconstraint ::

11++22++ 33=1=1wherewhere ii is theis the dutyduty --cyclecycle on phaseon phase i.i.



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V100

V110V*

V111


ToTo generategenerate aa triplettripletVV1avg1avg , V, V2avg2avg , V, V3avg3avg((average voltagesaverage voltages )) inin aaswitching periodswitching period ,, vector vector V*,V*, thethe transformationtransformation of of the tripletthe triplet ,, is consideredis considered ..

TheThe two adjacent vectorstwo adjacent vectorsandand aa zero vector zero vector areareapplied successivelyapplied successively ..


V1101

V1002

V*

V100

V110

V1113


TheThe projectionsprojections of of V*V*on the adjacenton the adjacentvectorsvectors determinedetermine thetherespectiverespective dutyduty --cyclescycles ..

TheThe zero vector zero vector dutyduty --cycle iscycle is determineddeterminedfrom the relationfrom the relation ::

11 ++ 22++ 33 =1,=1,if possibleif possible ..


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VectorsVectors V*V* which canwhich canbe generated with thisbe generated with thistechniquetechnique areare thetheonesones included in theincluded in thehexagonhexagon [1].[1].

It is possibleIt is possible toto selectselectdifferent ordersdifferent orders of of applicationapplication for for thethethree vectorsthree vectors ..

V100

V110V*

V111



V100 V110 V111 V 100 V110 V111

V 1

V2

V3

3 T1 T2 TT T

E

E

E

3 T1 T2 T


AA possiblepossible vector vector sequencesequence :: we canwe can dodo betterbetter than thatthan that ......


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2 T111V100 V110 V111 V 100V110 V

V 1

V2

V3

3 T1 T2 TT T

E

E

E

3 T 1 T


This choice reduces theThis choice reduces the number of switchingsnumber of switchings



2TV100 V110 V 111 100V110 V

V1

V2

V3

3 T1 T2 TT T

E

E

E

1 T000V

3 T/2000V

3 T/2

This choiceThis choice minimizesminimizes thethe current ripplescurrent ripples amplitudeamplitude


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V 23 V 12

V10

V 31

V 30

V 20

V N0

V N0

V N0

NeutralNeutral Voltage VariationVoltage VariationVoltage SpaceVoltage Space Vector Vector ModulationModulation


Third Harmonic InjectionThird Harmonic InjectionVoltage SpaceVoltage Space Vector Vector ModulationModulation

V 10

V 23 V 12 V 31

V30

V 20

V N0


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V102E

V10avg

0

Instantaneous and average phase voltage withInstantaneous and average phase voltage withflatflat --top modulationtop modulation .. Each phase switches onlyEach phase switches onlyinin 2/32/3 of theof the fundamentalfundamental periodperiod ..



SummingSumming toto each dutyeach duty --cyclecycle the same commonthe same commoncomponentcomponent ,, constant or variableconstant or variable ::

thethe instantaneousinstantaneous phase voltagesphase voltages changechange ;;

thethe average phaseaverage phase to neutralto neutral voltagesvoltages changechangeaccordinglyaccordingly ;;

the averagethe average phasephase toto phase voltagesphase voltages dodo notnotchangechange ;;

if theif the neutralneutral wire iswire is insulatedinsulated ,, thethe voltage onvoltage onthe loadthe load (Y)(Y) doesdoes notnot changechange ..



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DigitalDigital Implementation of Implementation of SVMSVMSVMSVM is normally theis normally the inner routineinner routine in thein the digitaldigitalcontrol of control of a VSI;a VSI; external currentexternal current loops typicallyloops typicallyprovide theprovide the setset --pointpoint for for the modulator the modulator ::

in thein the a, b, ca, b, c fixed referencefixed reference frameframe ((thethe threethreedutyduty --cyclescycles areare givengiven ););

in thein the ,,,, fixed reference framefixed reference frame ((bibi -dimensional-dimensionalcontrolcontrol :: the averagethe average voltagevoltage vector vector componentscomponentsareare givengiven ););

InIn thethe former former casecase thethe dutyduty --cyclescycles can becan be modifiedmodifiedbyby injectinginjecting aa third harmonicthird harmonic componentcomponent .. TheThe latter latter casecase is suitedis suited for for directdirect SVMSVM implementationimplementation ..


Direct Implementation of Direct Implementation of SVMSVM

Given theGiven the ,, componentscomponents of the setof the set --pointpoint V*,V*,thethe digitaldigital modulator hasmodulator has to computeto compute thetheprojections of the referenceprojections of the reference vector V*vector V* on theon theadjacentadjacent inverter inverter statesstates ..

If If aa floating point processorfloating point processor is availableis available ,, this isthis isnotnot aa problemproblem .. If this is not theIf this is not the case, acase, a lot of lot of different algorithmsdifferent algorithms can be appliedcan be applied . An. Anexampleexample of of SVMSVM algorithmalgorithm is reported inis reported in [3].[3].

Another example is described in the followingAnother example is described in the following ..


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Direct Implementation of Direct Implementation of SVMSVMZZ1y1y

ZZ1x1x

ZZ2x2xZZ2y2y

ZZ3y3y

ZZ3x3x

{{,, }} {{ZZixix ,, ZZiyiy}}

M11

13

023

==

M21

13

113

==

M30

23

113

==



SomeSome regularitiesregularities in the transform matrixesin the transform matrixes MMiican becan be exploitedexploited toto rapidly calculate the Zrapidly calculate the Z ixix ,y,ycomponents of the voltagecomponents of the voltage vector V*:vector V*:

tmptmp = V*= V* / / sqrtsqrt (3);(3);ZZ1x1x = V*= V* -- tmptmp ;;ZZ2y2y = - Z= - Z 1x1x ;;ZZ1y1y = 2= 2 tmptmp ;;ZZ3x3x = Z= Z 1y1y ;;ZZ2x2x = V*= V* ++ tmptmp ;;ZZ3y3y = - Z= - Z 2x2x ;;


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Direct Implementation of Direct Implementation of SVMSVM Once theOnce the ZZixix,y,y components of the voltagecomponents of the voltage vector vector

V* areV* are known it is easyknown it is easy to determineto determine thethe sector sector V*V* lieslies inin ,, e.g.:e.g.:

Z Z1x1x ZZ1y1y < 0 ?< 0 ? yesyes no no

ZZ2x2x ZZ2y2y < 0 ?< 0 ? ZZ1x1x > 0 ?> 0 ? yes yes nono yesyes nono

ZZ3x3x ZZ3y3y < 0 ?< 0 ? ZZ2x2x > 0 ?> 0 ? 1st1st 4th4th ... ... yesyes no no

... ... 2nd2nd 5th 5th



Given the sector Given the sector ,, it is immediateit is immediate to determineto determine whichwhichinverter inverter voltage vectorsvoltage vectors havehave toto be generated andbe generated andconsequently theconsequently the required switching sequencerequired switching sequence ..

The durationsThe durations of the two requiredof the two required inverter inverter statesstates VV11andand VV22 areare proportionalproportional toto thethe ZZixix and Zand Z iyiycomponentscomponents of the averageof the average vector V*vector V* respectivelyrespectively ..

AccordingAccording toto what was previously explainedwhat was previously explained ,, thethezero vector Vzero vector V 00 duration is given by the followingduration is given by the following ::

TT11+T+T 22 +T+T 00=T,=T,unlessunless saturation occurssaturation occurs ..


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Direct Implementation of Direct Implementation of SVMSVM InIn the presencethe presence of saturationof saturation ,, i. e.i. e. when the requiredwhen the required

average voltageaverage voltage vector V* liesvector V* lies outside the hexagonoutside the hexagon ,,different strategiesdifferent strategies can be adoptedcan be adopted ..

AA possibility ispossibility is toto reducereduce the voltagethe voltage vectorvector V*V*amplitude,amplitude, whilewhile keeping its phasekeeping its phase ,, soso as toas to put itput iton theon the hexagon border hexagon border ..

V*V*sat

This well exploits theThis well exploits theinverter inverter capabilitycapability andandis easyis easy toto implementimplement ::

TTisatisat = T= T TTii /(T /(T11+T+T22), i = 1, 2), i = 1, 2

!!



AA roughrough alternative whichalternative which doesdoes notnot requirerequiretroublesome calculationstroublesome calculations ,, isis toto reducereduce the smaller the smaller vector vector componentcomponent enoughenough toto put theput the vector vector on theon thehexagon border hexagon border ::

V*

V*sat

This solution implies anThis solution implies anunavoidableunavoidable error error both inboth inthethe amplitudeamplitude and in theand in thephasephase of the generatedof the generatedvector.vector.


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Direct Implementation of Direct Implementation of SVMSVM IfIf deep saturationdeep saturation occursoccurs , i. e., i. e. at leastat least one of theone of the

two componentstwo components VVii of of vector V*vector V* isis ,, by itself by itself ,,outsideoutside the hexagonthe hexagon ,, another saturation strategyanother saturation strategyis normally adoptedis normally adopted ..

TheThe nearestnearest inverter inverter statestate is steadily generatedis steadily generatedfor for thethe completecomplete switching periodswitching period T.T.

This leads theThis leads the converter toconverter to sixsix --step mode of step mode of operationoperation ..

InIn thethe SVMSVM algorithm the transitionalgorithm the transition from lightfrom lightsaturationsaturation toto deep saturationdeep saturation can be suitablycan be suitablymanagedmanaged



Light saturation areasLight saturation areas

Deep saturationDeep saturationareasareas ,, extending alsoextending alsooutside theoutside the circlecircle


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FinalFinal RemarksRemarks SVMSVM isis very commonlyvery commonly adopted in modernadopted in modern digitaldigital

control of power converterscontrol of power converters ((especially inespecially in drivedriveapplicationsapplications ).).

TheThe implementation of implementation of SVMSVM by means ofby means of CsCs or or DSPsDSPs isis easyeasy toto achieveachieve both directlyboth directly ((if theif therequiredrequired computational power is availablecomputational power is available )) andandindirectlyindirectly ,, byby postpost --processing the phase dutyprocessing the phase duty --cyclescycles withwith aa suitable harmonic injectionsuitable harmonic injection ..

InIn anyany case,case, converter converter saturation must besaturation must beconsidered and suitably dealt withconsidered and suitably dealt with ..


ReferencesReferences

[1][1] J.J. HoltzHoltz , W., W. LotzkatLotzkat , A., A. KhambadkoneKhambadkone ,, On Continuous Control of On Continuous Control of PWMPWM Inverters in the OvermodulationInverters in the Overmodulation RangeRange Including the SixIncluding the Six --StepStepModeMode ,, International Conference onInternational Conference on IndustrialIndustrial Electronics ControlElectronics Control

and Instrumentationand Instrumentation (IECON), 1992,(IECON), 1992, pppp . 307-312.. 307-312.[2] H. W.[2] H. W. VanVan Der Der BroeckBroeck , H. C., H. C. SkudenlySkudenly , G. V., G. V. Stanke Stanke ,, Analysis andAnalysis and

Realization of Realization of aa Pulsewidth Modulator Based on Voltage SpacePulsewidth Modulator Based on Voltage SpaceVectorsVectors ,, IEEEIEEE TransTrans .. on Industry Applicationson Industry Applications ,, VolVol . 24, No. 1,. 24, No. 1,JanJan / /FebFeb , 1988,, 1988, pppp . 142-150.. 142-150.

[3][3] Zhenyu YuZhenyu Yu ,, SpaceSpace -Vector PWM-Vector PWM WithWith TMS320C24x/F24xTMS320C24x/F24x UsingUsing

HardwareHardware andand SoftwareSoftware Determined Switching PatternsDetermined Switching Patterns ,,Application ReportApplication Report SPRA524, TexasSPRA524, Texas InstrumentsInstruments ..

digital control applications in power electronics lez2

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