Electric Power Systems Research

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Electric Power Systems Research

jou rn al h om e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r

Modeling power networks using dynamic phasors in the dq0 reference frame

Yoash Levron

^a,∗

, Juri Belikov

^b

aTheAndrewandErnaViterbiFacultyofElectricalEngineering,Technion—IsraelInstituteofTechnology,Haifa3200003,Israel

bFacultyofMechanicalEngineering,Technion—IsraelInstituteofTechnology,Haifa3200003,Israel

a r t i c l e i n f o

Articlehistory:

Received26July2016 Receivedinrevisedform 20November2016 Accepted28November2016

Keywords:

Quasistatic Averagesignals Dynamicphasors Inter-areaoscillations Multi-machine Stability

a b s t r a c t

Thedynamicbehavioroflargepowersystemshasbeentraditionallystudiedbymeansoftime-varying phasors,undertheassumptionthatthesystemisquasi-static.However,withincreasingintegrationoffast renewableanddistributedsourcesintopowergrids,thisassumptionisbecomingincreasinglyinaccurate.

Inthispaper,wepresentadynamicmodelofgeneraltransmissionanddistributionnetworksthatuses dynamicphasorsinthedq0referenceframe.Themodelisformulatedinthefrequencydomain,andis basedonthenetworkfrequencydependentadmittancematrix.Wealsopresentasimplifiedversionof thismodelthatisobtainedbyafirst-orderTaylorapproximationofthedynamicequations.Theproposed modelsextendthequasi-staticmodeltohigherfrequencies,whileemployingdq0signalsthatarestatic atsteady-state,andthereforecombinetheadvantagesofhighbandwidthandawell-definedoperating point.Themodelsareverifiednumericallyusingthe9-,30-,and118-bustest-casenetworks.Simulations showthatfrequencyresponsesofallmodelscoincideatlowfrequenciesanddivergeathighfrequencies.

Inaddition,responsesofthedq0modelinthetimedomainandintheabcreferenceframeareveryclose tothoseofthetransientmodel.

©2016ElsevierB.V.Allrightsreserved.

1. Introduction

Dynamicprocessesoccurringinlargeelectricpowersystemsaremodeledatvaryinglevelsofabstraction[1,2].Insimplestaticmodels, voltagesandcurrentsareassumedtobesinusoidal,andarerepresentedbyphasors.Thenetworkistypicallydescribedinthiscaseby thepowerflowequations.Ontheotherextreme,transientmodelsdescribethesystemusingnonlineardifferentialequations,whichare solvednumericallyinthetimedomain[3–8].Athirdtypeofmodel,basedontime-varyingphasors,iscalledtheaverage-value,dynamic orquasi-staticmodel[2].Thismodelissufficientlyaccurateaslongasphasorsareslowlychangingincomparisontothesystemfrequency [9,10].Akeyadvantageofthequasi-staticmodelisthatitenableslongnumericsteptimes,andhencecanacceleratethesimulationprocess.

Inaddition,sincethequasi-staticmodelemploysphasorsinsteadofsinusoidalACsignals,thesystemoperatingpointiswell-defined,a propertywhichconsiderablysimplifiesstabilitystudies.Duetotheseproperties,quasi-staticmodelshavebeenusedextensivelyinthe analysisofdynamicinteractionsthatoccurintimeframesofsecondstominutes,andhavehistoricallyenabledstudiesofmachinesstability, inter-areaoscillation,andslowdynamicphenomena[1–3,11].

Inrecentyears,withtheemergenceofsmalldistributedgeneratorsandfastpowerelectronicsbaseddevices,theassumptionofquasi- staticphasorsisbecomingincreasinglyinaccurate.Duetotheseemergingtechnologies,voltageandcurrentsignalscancontainhigh harmoniccomponents,andcanexhibitfastamplitudeandphasevariations[9].Thesedevelopmentshaveledtoanewclassofmodels basedondynamicphasors.Theconceptofdynamicphasorshasemergedfromaveragingtechniquesemployedinpowerelectronics,and hasbeenextendedtothree-phasesystems[12].Dynamicphasorsgeneralizetheideaofquasi-staticphasors,andrepresentvoltageand currentsignalsbyFourierseriesexpansionsinwhichtheharmoniccomponentsareevaluatedoveramovingtimewindow[13].This approachoffersthebenefitsofaphasorbasedanalysis,highaccuracy,andfastsimulations[14,15].

Duetotheseproperties,dynamicphasorshavebeenusedintheanalysisofsynchronousandinductionmachines[12,16],HVDCsys- tems[14,17,18],FACTSdevices[10],sub-synchronousresonance[19,20],asymmetricsystems[12,21],andasymmetricfaults[16,22].For

∗Correspondingauthor.

E-mailaddresses:yoashl@ee.technion.ac.il(Y.Levron),juri.belikov@ttu.ee(J.Belikov).

http://dx.doi.org/10.1016/j.epsr.2016.11.024 0378-7796/©2016ElsevierB.V.Allrightsreserved.

Fig.1.Aunitinthenetwork,showingsignalsintheabcreferenceframe.

instance,adynamicphasormodelofalinecommutatedHVDCconverterispresentedin[14].Themodelrepresentsthelow-frequency dynamicsoftheconverter,andhaslowercomputationalrequirementsthanaconventionaltransientmodel.Adynamicphasorrepresen- tationofanunbalancedradialdistributionsystemispresentedin[21],wherevariouscomponentsincludingaphotovoltaicsourceandan inductionmachinearemodeled,anddynamicinteractionofthesecomponentsisshown.Asimulatorbasedondynamicphasorshasbeen developedin[15,23],whichusesdifferentialswitched-algebraicstate-resetequationstodescribethesystemcomponents.Thesimulation toolcombinesadvantagesoftransientstabilityandelectro-magneticsimulationprograms,andisdemonstratedontheIEEE39bustest- casenetwork.Work[24]extendsthedynamicphasorconcepttomulti-generator,multi-frequencysystems.Thetheorypresentedin[24]

enablesstudyofelectricpowersystemswithoutassumingasinglefrequency.Theproposedapproachisvalidatedonatwin-generator system,andcanbeextendedtolargernetworks.Inaddition,dynamicphasorsareutilizedinstateestimation[25–29],insystemswith varyingfrequencies[30],andalsoinmicrogrids[31].

Theserecentworksaremainlyfocusedonmodelingspecificgeneratorsand loads,and donotprovidea completemodeloflarge transmissionnetworks.Tobridgethisgap,thispaperpresentsadynamicmodelofgeneraltransmissionanddistributionnetworksusing dynamicphasorsinthedq0referenceframe,andshowshowthismodelcanbeimplementedinpracticeusingtime-domainstateequations.

Thefirstmodelpresentedinthisworkisformulatedinthefrequencydomain,andisbasedonthenetworkfrequencydependentadmittance matrix.Sinceadirectnumericimplementationofthismodelischallengingingeneral,wepresentasimplifiedversionofitcalledthefirst- orderdq0model,whichisobtainedbyafirst-orderTaylorapproximationofthedynamicequations.Theproposedfrequency-domainmodel extendsthequasi-staticmodeltohigherfrequencies,whileemployingdq0signalsthatarestaticatsteady-state,andthereforecombines theadvantagesofhighbandwidthandawell-definedoperatingpoint.Thestandardquasi-staticmodelfollowsasaspecialcasefromthe modelatlowfrequencies.

Thepapercontinuesasfollows.Section2recallsbasicconceptsofdynamicphasorsandexplainshowtoformulatequasi-staticmodelsin termsofdq0quantities.Section3presentstheproposeddq0model.Thefirst-orderdq0modelispresentedinSection4.Section5provides simpleillustrativeexamples,followedbyvariousnumericalresultsprovidedinSection6.ConcludingremarksaredrawninSection7.

2. Dynamicphasorsinthedq0referenceframe

Forsystemsinsteady-state,thedq0transformationmapssinusoidalsignalstoconstantquantities.Thistransformationiscompatible withstandardelectricmachinemodels,andwithemergingmodelsofrenewableanddistributedgenerators[10,15,32,33].

AssumeageneralpowernetworkcontainingNthree-phaseunits,withvoltagesandcurrentsaspresentedinFig.1.

Thedq0transformation(asdefinedin[26])isgivenby

⎡

⎢ ⎣

x^d(t) x^q(t) x⁰(t)

⎤

⎥ ⎦

⁼²₃

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

cos(ωst) cos

ωst−2

3 cos

ωst+2 3

−sin(ωst) −sin

ωst−2

3 −sin

ωst+2 3 1

2

1 2

1 2

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

⎡

⎢ ⎣

x^a(t) x^b(t) x^c(t)

⎤

⎥ ⎦

^{, (1)}

whereωsisthenominalsystemfrequency,andxisreplacedwitheitheravoltage(vn)oracurrent(in)signal.Aprominentfeatureof thedq0transformationisthatforbalancedandstaticsystems,thedirectandquadraturecomponentsx^d,x^qareconstants,andx⁰=0.Ina quasi-staticsystemx^a,x^b,x^carenearlysinusoidal,withslowlyvaryingamplitudesandphases.Inthiscase,thesystemcanbemodeledin termsoftime-varyingphasors,whicharedefinedintermsofthedq0componentsas

V_n(t)= 1

√2

v^d_n(t)+jv^q_n(t)

, I_n(t)= 1

√2

i^d_n(t)+ji^q_n(t)

. (2)

Underthequasi-staticapproximationthesedqcomponentsareidenticaltothexycomponents,asdescribedin[34].Thefollowinganalysis definesthestandardquasi-staticmodelandpowerflowequationsintermsofdq0signals.Quasi-staticnetworksareusuallydescribedby theadmittancematrixY^bus.Definethevectors:

V^d(t)=

v^d₁(t),...,v^d_N(t)

T

, I^d(t)=

i^d₁(t),...,i^d_N(t)

T

, V^q(t)=

v^q₁(t),...,v^q_N(t)

T

, I^q(t)=

i^q₁(t),...,i^q_N(t)

T

, V⁰(t)=

v⁰₁(t),...,v⁰_N(t)

T

, I⁰(t)=

i⁰₁(t),...,i⁰_N(t)

T

. (3)

Thenthecomplexvoltageandcurrentphasorsrelateby I^d(t)+jI^q(t)=Y^bus

V^d(t)+jV^q(t)

(4)

andtherealandimaginarypartsofthisequationsare I^d(t)=Re

Y^bus

V^d(t)−Im

Y^bus

V^q(t), I^q(t)=Im

Y^bus

V^d(t)+Re

Y^bus

V^q(t). (5)

Theseequationsconstituteaquasi-staticmodel.Equivalently,quasi-staticnetworksarealsodescribedbythepowerflowequations[33]

Pn(t)=|Vn(t)|

N k=1

|y_nk||V_k(t)|cos

∠y_nk+ı_k−ın

, Qn(t)=−|Vn(t)|

N k=1

|y_nk||V_k(t)|sin

∠y_nk+ı_k−ın

. (6)

Inwhichtheconstantsy_nkareelementsofthenetworkadmittancematrixY^bus,andpowers,amplitudesandphasesareformulatedin termsofdq0signalsas

Pn(t)=Re

Vn(t)In(t)^∗

= 1 2

v^d_n(t)i^d_n(t)+v^qn(t)i^q_n(t)

, Qn(t)=Im

Vn(t)In(t)^∗

= 1 2

v^qn(t)i^d_n(t)−v^d_n(t)i^q_n(t)

,

Vn(t)

_=√¹

2

v^d_n(t)

2

+

v^q_n(t)

2

, ın(t)=∠Vn(t)=atan2

v^q_n(t),v^d_n(t)

. (7)

Ifthesystemisnotquasi-static,thephasorsabovecanbegeneralizedbydynamicphasors.ThesecanbedefinedbyaFourierseries expansion,asin[16].However,tosimplifynotations,wehavechoseninthispapertodefinedynamicphasorswithrespecttotheFourier transformas

⎡

⎢ ⎣

V_n^d(ω) V_n^q(ω) V_n⁰(ω)

⎤

⎥ ⎦

⁼

_∞

−∞

⎡

⎢ ⎣

v^d_n(t) v^qn(t) v⁰_n(t)

⎤

⎥ ⎦

^e−jωtdt,

⎡

⎢ ⎣

I_n^d(ω) I_n^q(ω) I_n⁰(ω)

⎤

⎥ ⎦

⁼

_∞

−∞

⎡

⎢ ⎣

i^d_n(t) i^q_n(t) i⁰_n(t)

⎤

⎥ ⎦

^{e−jωtdt. (8)}

3. Largenetworkdynamicsinthedq0referenceframe

Thepowerflowequationsandtheequivalentdq0modelin(5),(6)areonlyvalidunderthequasi-staticapproximation,assumingslow variationsinamplitudesandphases[34].Thefollowingtheoremextendsthequasi-staticmodeltohigherfrequencies,anddescribesthe systemdynamicsforgeneraldq0signals.

Theorem1. Insymmetricpowernetworks,adynamicmodelbasedondq0dynamicphasorscanbedescribedas

⎡

⎢ ⎣

I^d(ω) I^q(ω) I⁰(ω)

⎤

⎥ ⎦

⁼

⎡

⎣

^U(ω)₋jR(ω) ^jR(ω)U(ω) ⁰0

0 0 Y^bus(ω)

⎤

⎦

⎡

⎢ ⎣

V^d(ω) V^q(ω) V⁰(ω)

⎤

⎥ ⎦

^{. (9)}

Thefollowingdefinitionsareusedin(9):

U(ω):=1 2

Y^bus(ω+ω_s)+Y^bus(ω−ω_s)

, R(ω):=1 2

Y^bus(ω+ω_s)−Y^bus(ω−ω_s)

, (10)

and

V^d(ω)=

V₁^d(ω),...,V_N^d(ω)

T

,I^d(ω)=

I^d₁(ω),...,I_N^d(ω)

T

, V^q(ω)=

V₁^q(ω),...,V_N^q(ω)

T

,I^q(ω)=

I^q₁(ω),...,I_N^q(ω)

T

, V⁰(ω)=

V₁⁰(ω),...,V_N⁰(ω)

T

,I⁰(ω)=

I₁⁰(ω),...,I⁰_N(ω)

T

. (11)

Proof. Followingthedefinitionofthedq0transformationin(1),voltagesandcurrentscanbeexpressedassumsofthreeelements V^a(t)=V^d(t)cos(ωst)−V^q(t)sin(ωst)+V⁰(t), I^a(t)=I^d(t)cos(ωst)−I^q(t)sin(ωst)+I⁰(t). (12) Similarexpressionsapplytophasesb,cwithaproperphase-shiftof±2/3.Transformationtothefrequencydomainusingthemodulation propertyyields

V^a(ω)= 1 2

V^d(ω−ωs)+V^d(ω+ωs)+jV^q(ω−ωs)−jV^q(ω+ωs)

+V⁰(ω),

I^a(ω)=1 2

I^d(ω−ω_s)+I^d(ω+ω_s)+jI^q(ω−ω_s)−jI^q(ω+ω_s)

+I⁰(ω). (13)

Accordingtodefinition

I₁^a(ω) ··· I^a_N(ω)

T

=Y^bus(ω)

V₁^a(ω) ··· V_N^a(ω)

T

. (14)

Substituting(14)in(13)yields

I^d(ω−ωs)+I^d(ω+ωs)+jI^q(ω−ωs)−jI^q(ω+ωs)+2I⁰(ω)

=Y^bus(ω)

V^d(ω−ωs)+V^d(ω+ωs)+jV^q(ω−ωs)−jV^q(ω+ωs)+2V⁰(ω)

. (15)

Usingrelation(10),equation(15)canberewrittenas

I^d(ω−ωs)+I^d(ω+ωs)+jI^q(ω−ωs)−jI^q(ω+ωs)+2I⁰(ω)=2Y^bus(ω)V⁰(ω)+U(ω−ωs)V^d(ω−ωs)+jR(ω−ωs)V^q(ω−ωs) +U(ω+ωs)V^d(ω+ωs)+jR(ω+ωs)V^q(ω+ωs)+j(−jR(ω−ωs)V^d(ω−ωs)+U(ω−ωs)V^q(ω−ωs))−j(−jR(ω+ωs)V^d(ω+ωs)

+U(ω+ωs)V^q(ω+ωs)), (16)

whichaftercomparisonofcoefficientsresultsin(9).䊐

NotethatTheorem1holdsforsymmetricnetworks,inwhichthematrixY^bus(ω)equallyappliestoeachofthethreephases.However, thegeneratorsandloadsconnectedtothenetworkarenotnecessarilylinearorsymmetric.

Thefollowingcorollaryrelatestheproposeddq0modeltothestandardquasi-staticmodel.

Corollary1. WhenY^bus(ω+ω_s)=Y^busandY^bus(ω−ω_s)=(Y^bus)^∗,equation(9)reducestothequasi-staticexpressionsin(5).

ThiscorollarystatesthatifthegeneralY^bus(ω+ωs)canbeapproximatedbyaconstantmatrixwhenω→0,thenthedynamicmodelis quasi-static.

4. Thefirst-orderdq0model

Sincethefulldq0modelin(9)is,ingeneral,hardtoimplementnumerically,thissectionpresentsasimplifiedversionofitcalledthe first-orderdq0model,whichisobtainedbyafirst-orderTaylorapproximationofthedynamicequations.Inthiscaseitisassumedthatthe dq0signalsarebandlimited,andhavesignificantenergyonlyatlowfrequencies.Thisenablesrepresentationoftheadmittancematrixby aTaylorseriesaroundω=0as

Y^bus(ω+ωs) ≈

M k=0

ω^k k!

∂Y^bus(ω+ω_s)

∂ω^k |ω=0, Y^bus(ω−ωs) ≈

M k=0

ω^k k!

∂Y^bus(ω−ωs)

∂ω^k |ω=0,=

M k=0

(jω)^kj^k k!

∂Y^bus(ω+ωs)

∂ω^k

∗

|ω=0,

Y^bus(ω) ≈

M k=0

ω^k k!

∂Y^bus(ω)

∂ω^k |ω=0,

(17)

whereintheexpressionforY^bus(ω−ωs)wehaveutilizedY^bus(ω)=(Y^bus(−ω))^∗andemployednestedfunctionsderivatives.Thestandard quasi-staticapproximationisobtainedbyazero-orderTaylorserieswithM=0suchthat

Y^bus(ω+ωs)≈Y^bus(ωs), Y^bus(ω−ωs)≈

Y^bus(ωs)

_∗

. (18)

Iftheseexpressionsaresubstitutedintotheoriginalmodel(9),thenthemodelreducestothequasi-staticequationsin(5).Amoreaccurate dynamicmodelisobtainedbythefirst-orderTaylorseriesapproximation(M=1)as

Y^bus(ω+ωs)≈Y^bus(ωs)+jωF, Y^bus(ω−ωs)≈(Y^bus(ωs))^∗+jωF^∗, Y^bus(ω)≈Y^bus(0)+jωF₀, F=−j∂Y^bus(ω+ωs)

∂ω |ω=0, F₀=−j∂Y^bus(ω)

∂ω |ω=0. (19)

Substitutionoftheseexpressionsin(9)yields

⎡

⎢ ⎣

I^d(ω) I^q(ω) I⁰(ω)

⎤

⎥ ⎦

^=(M2+M1jω)

⎡

⎢ ⎣

V^d(ω) V^q(ω) V⁰(ω)

⎤

⎥ ⎦

⁽²⁰⁾

with

M₁=

⎡

⎣

^ReIm^{{^FF^}} ⁻Re^Im{F^{F}^{} 0}0

0 0 F₀

⎤

⎦

, M₂=

⎡

⎢ ⎣

Re

Y^bus(ω_s)

−Im

Y^bus(ω_s)

0 Im

Y^bus(ωs)

Re

Y^bus(ωs)

0

0 0 Y^bus(0)

⎤

⎥ ⎦

^{. (21)}

Thismodelcanbeimplementedinpracticeusingalinearstate-spacemodel dx

dt =Ax+Bu y =Cx+Du,

(22)

Fig.2.Example1:anetworkwithasingleinductor.

Fig.3. Example2:longtransmissionlinefeedingamatchedload.

whereu=(V^d(t),V^q(t),V⁰(t))^T,y=(I^d(t),I^q(t),I⁰(t))^T.Thematrices A,B,C,Darechosensuchthatatlowfrequenciesthefrequency responseofthestate-spacemodel(22)isequalto(20).Thegeneralfrequencyresponseisgivenbyy(ω)=(C(jωI−A)⁻¹B+D)u(ω),andis approximatedatω→0by

y(ω)≈

C

−A⁻¹

I+jωA⁻¹

B+D

u(ω)

=

−CA⁻¹B+D−jωCA⁻²B

u(ω).

(23)

Comparisonof(23)and(20)provides

−CA⁻²B=M1, −CA⁻¹B+D=M2. (24)

From(24),C,Dcanbecomputedforanyfull-rankmatricesA,Bas

C=−M1B⁻¹A², D=M2−M1B⁻¹AB. (25)

ThematricesA,Bcanbearbitrarilychosen.ItistypicallybeneficialtoselectA,BsuchthatD=0,toobtainzerogainatω→∞.Thiscriterion resultsin

D=M2−M1B⁻¹AB⇒A=BM⁻¹₁ M2B⁻¹. (26)

Ingeneral,ifthematrixM₁isnotfull-rank,thenAcanbecomputedbysolvingastandardleast-squaresminimizationproblem A=BPB⁻¹, P=argmin

X M1X−M2²2. (27)

ThematrixBcanbearbitrarilychosen,forinstanceB=I.

5. Dynamicdq0models:simpleexamples

ThesimplenetworkinFig.2demonstratesthedifferencebetweenthequasi-staticmodelandthefulldq0model.

Inthisexample,thequasi-staticmodel(5)provides

V^d(ω)=−ωsLI^q(ω), V^q(ω)=ωsLI^d(ω). (28)

Incomparison,thedq0modelcanbecomputedby(9),andresultsin

V^d(ω)=jωLI^d(ω)−ωsLI^q(ω), V^q(ω)=ωsLI^d(ω)+jωLI^q(ω), V⁰(ω)=jωLI⁰(ω). (29) Notethatwhenω→0thequasi-staticandthedq0modelsareequivalent.

Anotherexample(Fig.3)isthelongtransmissionline.Inthisexamplethedq0dynamicphasorsareshowntorelatethroughtimedelays.

Assumealongpowerlineoflengthl.Thelineislossless,withinductanceLxandcapacitanceCxperunitlength.Itfeedsamatchedloadwith animpedanceequaltothecharacteristicimpedanceZ_c=

L_x/C_x.Theresultingloadcurrentisdelayedinrespecttothesourcevoltage [33]as

i^abc₂ (t)= 1

Zcv^abc₁ (t−d), (30)

wherethetimedelayisgivenbyd=l

LxCx.

Fig.4.Comparisonoffrequencyresponsesinthe9-busnetworkfordqquantitiesfrominput1tooutput4.

Fig.5. Comparisonoffrequencyresponsesinthe118-busnetworkfordqquantitiesfrominput76tooutput77.

Thecorrespondingdq0modelcanbecomputedby(9).DefiningY^bus(ω)=(1/Zc)e^−jωdandapplyingtheinverseFouriertransform,the resultis

i^d₂(t)= 1 Z_c

cos(ωsd)v^d₁(t−d)+sin(ωsd)v^q₁(t−d)

, i^q₂(t)= 1 Z_c

−sin(ωsd)v^d₁(t−d)+cos(ωsd)v^q₁(t−d)

, i⁰₂(t)= 1

Z_cv⁰₁(t−d). (31) Thedirectandquadraturecurrentcomponentsaredelayedinrespecttothesourcevoltage.

6. Numericalresults

Inthissectionthevariousmodelsarecomparedintermsoffrequencyandtransientresponses.Threemodelsaretested:thefulldq0 model,thefirst-ordermodel,andthequasi-staticmodel.Thesearecomparedtothefulltransientmodel,whichisconstructedinstate- spaceusingMatlab’sSimscapePowerSystemssoftwarepackage.The9-,30-,and118-busnetworksfromtheMatPowerdatabase[35]are used.

Fig.6. Comparisonofthefirst100largestHankelsingularvaluesofthedq0modelandthefirst-ordermodelinthe118-busnetwork.

Fig.7.Comparisonofstep-responsesinthe9-bussystem.Theinputs(bottomplot)arestepsinphasesa,b,catbus1.Theoutputisthecurrentofphaseaatbus4.

AnarrayofBodeplotsrepresentingthedq0,first-order,andquasi-staticmodelsisgraphicallyillustratedinFigs.4and5for9-and 118-busnetworks.Theplotsdepictthemagnitude(indB)andphase(indegrees)forseveralinput/outputpairs.Thestepinputisapplied toageneratorlocatedatbus1andtheoutputismeasuredatbus4forthecaseof9-busnetwork.Similarly,for118-busnetworkthestep inputisappliedtoageneratorlocatedatbus76andtheoutputismeasuredatbus77.Thefiguresshowthatthefrequencyresponses coincideatlowfrequencies(ω→0)anddivergeathighfrequencies.Specifically,thequasi-staticmodelisrepresentedbyaconstantgain, andhasaconstantgainandphaseatallfrequencies.Thefirst-ordermodelresponsecoincideswiththedq0modelresponseoverawider frequencyrangeanddeviatesonlyathighfrequencies.ThesefiguresillustratethemainresultpresentedinTheorem1thatdq0model extendsbothquasi-staticandfirst-ordermodelsbytakinghigh-frequencydynamicphenomenaintoaccount.

Fig.6comparesthelargestsingularvaluesofthedq0modelandthefirst-ordermodelforthe118-busnetwork.Itcanbeseenthat themostenergetic(largest)valuesaresimilar,meaningthatthemajorityofenergyispreserved.Thisinturnverifiesthequalityofthe first-ordermodel.

Figs.7–11comparethetransientresponseofthevariousmodelsforthe9-,30-and118-busnetworks.Allinputsandoutputsarethree phasesignals.Thetransientmodelissimulateddirectlyintheabcreferenceframe.Thedq0model,quasi-staticmodel,andthefirst-order modelarerepresentedinthedq0referenceframe,soinputandoutputsignalsareconvertedaccordinglyusing(1).Tothisend,thedq0 modelisrepresentedinthestate-spaceformusingimplementationavailablein[36].Thefirst-ordermodelisimplementedusingequations (22)–(27).

Figs.7and8providesimulationresultsforthecaseof9-busnetwork.InFig.7thestepinputsignals(shiftedintimeforeachofthephases) areappliedtothegenerator1atbus1andtheoutputsaremeasuredatbus4.InFig.8amorecomplicatedscenarioisconsidered.Before thetransient,theinputsarebalancedthreephasesinusoidalsignalsu^a₁=sin(250t)atbus1.Then,thetransientistriggeredbyshorting phaseatoground,i.e.,u^a₁=0.Notethatu^c,d₁ =sin(250t±2/3).Theoutputisthecurrentofphaseameasuredatbus4.Thesimulation resultsconfirmtheabovefrequencydomainanalysispresentedinFig.4andthefactthatthequasi-staticmodelbecomesinaccurateat highfrequencies.Thesimilarsettingswereusedtoobtainsimulationresultsforthe30-and118-busnetworks(seeFigs.9–11).Thefigures showthattransientresponsesofthedq0modelsaresimilartothoseofthetransientmodels.Thequasi-staticmodelisimplementedby constantgains,andisinaccurate,exceptforbalancedsinusoidalinputs.Thefirst-ordermodelismoderatelyaccurate.

Fig.8.Comparisonoftransientsinthe9-bussystem.Beforethetransient,theinputsarebalancedthreephasesinusoidalsignalsatbus1.Thetransientistriggeredby shortingphaseatoground.Theoutputisthecurrentofphaseaatbus4.

Fig.9.Comparisonofstepresponsesinthe30-busnetwork.Theinputsarestepsatt=0inphasesa,b,c.(a)inputatbus1,outputatbus2,(b)inputatbus5,outputatbus 7,(c)inputatbus8,outputatbus28,(d)inputatbus11,outputatbus9.

Table1

Errorsintransientresponses.

Fig.7 Fig.8(a) Fig.10 Fig.11(a)

MSE

dq0 0.9595×10⁻⁸ 0.9572×10⁻⁹ 0.6557×10⁻⁸ 0.9340×10⁻⁷

fo 0.0035 3.4425×10⁻⁵ 4.6116×10⁻⁵ 4.9407×10⁻⁴

qs 0.1929 0.0020 0.0039 0.0170

ISE

dq0 0.7922×10⁻⁸ 0.9706×10⁻⁹ 0.3570×10⁻⁸ 0.8491×10⁻⁷

fo 0.0035 3.4446×10⁻⁵ 1.1498×10⁻⁵ 1.2363×10⁻⁴

qs 0.1913 0.0020 9.6554×10⁻⁴ 0.0042

Fig.10.Comparisonofstep-responsesinthe118-bussystem.Theinputs(bottomplot)arestepsinphasesa,b,cofbus76.Theoutputisthecurrentofphaseaatbus77.

Fig.11.Comparisonoftransientsinthe118-bussystem.Beforethetransient,theinputsarebalancedthreephasesinusoidalsignalsatbus76.Thetransientistriggeredby shortingphaseatoground.Theoutputisthecurrentofphaseaatbus77.

ErrorsinthemodelstransientresponsesarepresentedinTable1,wheretwotypicalmetrics,themeansquarederror(MSE)andthe integralsquarederror(ISE),areused.Theerrorsarecalculatedincomparisontothetransientmodel,whichservesasareference.Itcanbe seenthatthemagnitudesoferrorsforthedq0modelaresignificantlylowerthanthoseoffirst-orderandquasi-staticmodels.

7. Conclusion

Dynamicbehaviorandstabilityoflarge-scalepowersystemshavebeentraditionallystudiedbymeansoftime-varyingphasors,under theassumptionthatthesystemisquasi-static.However,withincreasingintegrationoffastrenewableanddistributedsourcesintopower grids,thisassumptionisbecomingincreasinglyinaccurate.Thispaperusesdynamicphasorsinthedq0referenceframetodescribethe dynamicsoflargetransmissionanddistributionnetworks.Theproposedmodelsdonotemploytheassumptionofquasi-staticphasors, andthereforeremainaccurateoverawiderangeoffrequencies.Thefulldq0modelisbasedonthefrequencydependentadmittance matrix,andisapproximatedusingtheTaylorseriesexpansiontogeneratemodelsoflowercomplexity:theclassicquasi-staticmodel isobtainedbyazero-orderapproximation,whilethemoreaccuratefirst-ordermodelisobtainedbyafirst-orderapproximation.The developedmodelscombinetwopropertiesofinterest.Ononehand,theyprovidetheadvantagesofthedq0referenceframe.Specifically,

themodelsemploynonrotatingdq0signalsthatarestaticatsteady-state,andthusenableawell-definedoperatingpoint.Ontheother hand,theproposedmodelsimprovetheaccuracyofthequasi-staticmodelathighfrequencies,enablingrepresentationoffastdynamic phenomenainnetworksthatincludesmalldistributedgeneratorsandfastpowerelectronicsbaseddevices.Inparticular,theseproperties cancontributetoaccurateandlowcomplexitystabilityanalysisinsuchnetworks.Simulationsshowthatthefrequencyresponsesofall modelscoincideatlowfrequenciesanddivergeathighfrequencies.Inaddition,responsesofthedq0modelinthetimedomainandinthe abcreferenceframeareveryclosetothoseofthetransientmodel.

Acknowledgment

TheworkwaspartlysupportedbyGrandTechnionEnergyProgram(GTEP)andaTechnionfellowship.

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