, Ivo Ugleˇsi ´c

ContentslistsavailableatScienceDirect

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

Simulation of three-phase transformer inrush currents by using backward and numerical differentiation formulae

Amir Toki ´c

^a

, Viktor Milardi ´c

^b,∗

, Ivo Ugleˇsi ´c

^b

, Admir Jukan

^a

aFacultyofElectricalEngineering,TuzlaUniversity,Tuzla75000,BosniaandHerzegovina

bFacultyofElectricalEngineeringandComputing,UniversityofZagreb,Zagreb10000,Croatia

a r t i c l e i n f o

Articlehistory:

Received28November2014

Receivedinrevisedform31March2015 Accepted24May2015

Keywords:

Three-phasetransformerinrushcurrents Numericaloscillations

L-stability

Extremelystiffsystem Backwarddifferentiationmethod

a b s t r a c t

Thispaperpresentsasimplifiedmodelofathree-phasetransformerdevelopedinthestate-spaceform usingthelineargraphtheory.Thealgorithmforgeneratingthecoefficientmatrixesofthestate-space equationisdescribed.Stiffdetectionproceduresofdifferentialequationsystemsthatdescribethethree- phasetransformerinrushcurrenttransientsareexplained.Itisshownthatthetime-domaintransient responseofthree-phasetransformersmathematicallydescribesextremelystiffsystems.Thenumerical integrationmethodsbasedonstrongstable(AandL)backwarddifferentiationformulaeareusedtosolve extremelystiffdifferentialequationsystemsarisingfromthestate-spaceformulationofthetransformer inrushcurrenttransientequations.Acomparisonofthemeasuredandsimulatedthree-phasetransformer inrushcurrentsshowedverygoodagreement.Theproposedprocedureofmodelingandthesimulation methodareusefultoolsthatcanbeappliedtootherelectricaltransientswhereextremelystiffsystems appear.

©2015ElsevierB.V.Allrightsreserved.

1. Introduction

Thetransformerisoneofthemostimportantelementsofpower systems.Itisimportanttomakeavalidtransformermodelinthe observedtransientelectromagneticphenomena.Therearediffer- entmodelsoftransformersdependingonthefrequency spectra oftransients[1].Accordingto[2],thetransformerinrushcurrent belongstothelow-frequencytransients,frequencyupto1kHz.The transformerinrush currents are low-frequency electromagnetic transientsthatoccurduringenergizationofunloadedtransform- ers.Dependingonthetransformerparameters,residualfluxand themomentofswitchingon,themagneticfluxcanreachatwice highervalueincomparisonwiththeratedoperatingvalue.

Whenthe transformer is switched on and the value of the residualfluxisnearthepointofsaturation,thereductionofthe transformerimpedancetowindingresistanceandlowinductance mayoccurin thesaturationregionofthemagnetizationcurve.

Adirectconsequenceofthisscenarioistheironcoresaturation andtheproductionoftransformerinrushcurrents.Inrushcurrents, whichmaybeseveraltimeshigherthantheratedcurrents,can reducethepowerqualityduetovoltagesag[3],causethefalse

∗Correspondingauthor.Tel.:+38516129976;fax:+38516129890.

E-mailaddresses:amir.tokic@untz.ba(A.Toki ´c),viktor.milardic@fer.hr (V.Milardi ´c),ivo.uglesic@fer.hr(I.Ugleˇsi ´c),admir.jukan@untz.ba(A.Jukan).

operationofprotectiverelaysorfuses[4],damagethetransformer windingsduetodevelopedmechanicalforces[5]andinsomesce- narioscauseharmonicresonanceovervoltages[6,7].

Significantworkhasbeendoneondevelopingthetransformer models for the inrush current analysis. Several approaches to themodelingofthetransformerinthetimedomainsuitablefor inrushcurrentsimulationsaregiveninpapers[8–15].Ingeneral, atransformermodelcanbeseparatedintotwomainparts:the transformerwindingsandthetransformerironcore.Thefirstpart hasa linearand thesecondonehasa nonlinearcharacter.The time-domainmodelingoftransformersispossibleusingthenodal approach(usedinEMTP-basedprograms)orstate-spaceapproach (usedinMATLAB).

Inadditiontotheproblemoftransformermodeling,itisvery important to pay special attention to the choice of simulation algorithm. Thesolutionalgorithm dependsonthechoiceof the appropriatenumericalmethodusedinthesimulationprocedureof themathematicalmodel.Forexample,theEMTP-basedprograms usethecompensationmethodtosolvethesystemsolution[16].

ThesystemisfirstsolvedusingThevenin’sequivalents,ignoringthe nonlinearelements.Apossibleproblemofacompensationmethod isthattheThevenin’sequivalentcannotalwaysbedetermineddue topossiblefloatingnetworkformulations.Inaddition,specialprob- lemsmayariseduringthesimulationofthetransformertransients.

The main purpose of this paper is to present a simplified techniqueofthethree-phasetransformermodelingwithsuitable http://dx.doi.org/10.1016/j.epsr.2015.05.020

0378-7796/©2015ElsevierB.V.Allrightsreserved.

Rp1 Ls1

Rm1 Lh1

Rm2 Rm3

3L0

1:1

Rp2 Ls2 1:1

Rp3 Ls3

Lh2 Lh3

L12

L31

L23

Fig.1.Three-phasetransformercircuitmodel:(a)thestartingmodelwhichincludemutualcouplingbetweenphases,(b)theequivalentcircuitmodel.

solutionalgorithmbasedontheuseofstronglystablenumerical methods.Theproposednumericalmethodwasfoundtobestrongly stableandaccurateforthethree-phasetransformerinrushcurrent simulations.

Thereminderofthispaperisorganizedasfollows:inthefol- lowing section thesimplified circuit model of thethree-phase, three-leg,two-windingtransformerisexplained.Analgorithmfor generatingthestate-spaceequationsdescribingthethree-phase transformerinrushcurrenttransientsisdevelopedinSection3.The characteristicsoftheappropriatedsolutionmethodbasedonback- warddifferentiationformulae(BDF)ornumericaldifferentiation formulae(NDF)aredescribedindetail.Thetransformerinrushcur- rentmeasurementsandnumericalsimulationresultsarecompared inSection4.ThepaperisconcludedinSection5.

2. Simplifiedtransformermodelingprocedure

Thesimplifiedcircuitmodelofthethree-phase,three-leg,and two-windinglaboratorytransformerwithastar-starwindingcon- nectionisshowninFig.1.Theparametersofthecircuitmodelscan beobtainedfromstandardopen,positiveandzerosequencetests.

Theproposedtransformermodelincludesphase-tophasemutual coupling.Thiscircuitmodelwithadditionalzerosequencemag- netizinginductanceisthecommonsimplifiedrepresentationofa three-phasetransformerininrushcurrentsimulations[16,17].

LabelsinFig.1are:R_piwindingresistances,L_siwindingleakage inductances,R_mi corelossresistors,L_hicorehystereticinductors andL0zerosequencemagnetizinginductance,i=1,2,3.

TheselfinductanceLs=L_si andmutual inductanceLm=L_ij are calculatedfrompositive(L_p)andzero(L_z)sequencevalues:

Ls=Lz+2Lp

3 (1)

Lm=Lz−Lp

3 (2)

Thestarting model of thetransformeris shownin Fig.1(a), whiletheequivalentcircuitmodel,whichincludemutualcoupling betweenphases,isshowninFig.1(b).

Inthispartofthepaper,particularattentionwillbedevoted tothe modeling of the nonlinear hysteretic inductor,Fig. 1(b).

Normally, thenonlinearsingle-valuedmagnetization(−im)or nonlinearmulti-valuedhysteresis(−i_h)characteristicsof iron corematerialsaretypically modeledby piece-wiselinearfunc- tionsorsomeothernonlinearanalyticalfunctions.Commonways ofmodelingthesenonlinearcurvesaretouseapiece-wiselinear function[18]orapolynomial[19],arctg[20]orhyperbolicfunc- tion[21].The useof nonlinearanalytical functionsinmodeling nonlinearelectricalcomponentsgenerallyextendsthesimulation timeofdynamicsystemsbecauseoftheNewton–Raphsonitera- tivemethod,comparedtotheuseofapiece-wiselinearmodel.In addition,theuseofcurve-fittingextrapolationtechniquescanlead toproblemsinmodelingthenonlinearinductanceorhysteresisin asaturatedarea.Ontheotherhand,theuseofapiece-wiselinear modeliscloselyrelatedtotheappearanceofunwantedovershoot- ingeffects[22].

Fig.2. (a)Nonlinearhystereticinductor,(b)state-spacemodel.

2.1. Modelingofhystereticironcoreinductor

Inthispaper,amodifiedapproachtohystereticinductormod- elingisused,alreadyshowninthepaper[23].Hystereticinductor isdefinedbyasetofpointsofonebranchofthemajorhysteretic loop:

(ih1,h1), (ih2,h2), ..., (ih_p,h_p)

wherepisthetotalnumberofthesampledpointsof the(half) majorhysteresisloop.Itcanbeseen[24]thatthefinalexpression formagnetizingcurrentofthekthpiece-wiselinearregionofmajor loopintermsoftheactualfluxis:

ih= 1

Lh_k+Sh_k (3)

wherethereare,respectively:

Lm_k=h_k+1−h_k

ih_k+1−ih_k ,Lm_k= sgn ()k

sgn ()k−1Lm_k, Lh_k=Lm_k−Lm_k

Ik= 1

Lm_kk, Sh_k=sgn ()×

Ih_k−Ik

, 1≤k≤p−1

Finally, the nonlinear hysteretic inductor L_h in Fig. 2(a) is modeledthroughlinearinductorLh_k inparallelwithanartificial currentsourceSh_k,Fig.2(b).

Thedevelopedhystereticinductormodeltakesintoaccountthe specialoperatingconditionsoccurringduetothefactthattheoper- atingpointinoneparticularcaseliesoutsidethemajorhysteresis loop.Thedevelopedmodelhasaspecialsubroutineforeliminating thepossibleovershootingeffect[22,23].

3. Solutionprocedureforthethree-phasetransformer inrushcurrentcalculations

Thedevelopedmodelsofnonlinearhystereticinductorsarevery suitableforthedevelopmentofstate-spaceequationsystemsthat describelow-frequencythree-phasetransformertransientssuchas transformerenergization,Fig.3.Anarbitraryintegrationmethod couldbeappliedinastate-spaceform.IntheEMTP-ATPelements arestronglydependentontheintegrationstep;thisfactbecomes apparentwhenatrapezoidalnumericalruleisappliedtotherele- vantbranch.

Itshouldbenotedthatsomeauthorsreportedsomenumeri- calproblems(numericaloscillations)duringthesimulationofthe transformertransients[25–27].Intheseworks,thecauseofthe numericalproblemiscitedas‘stiffsystem’or‘nonlinearityofmag- netizingcurve’.However,theseworksdonotexploreindetailthe causesofunwantednumericaloscillations,whichisoneofthemain goalsofthispaper.

3.1. Modelingofthethree-phasetransformertransients

Thealgorithmprocedureforgeneratingstate-spacematricesis presentedbelow.Thestandardstate-spaceequationthatdescribes transformerinrushcurrentanalysisis:

dX(t)

dt =AX(t)+BU(t)=F(X,t) (4) The input vector contains the system voltages and current sourcesgeneratedfromnonlinearhystereticinductors:

U(t)=

e1 e2 e3 Sh1j Sh2k Sh3l

T

(5) Inordertoreachasolutionofthestandardstate-spaceequation (2),andtogeneratethematrixofAandBcoefficients,thetheory oflineargraphswillbeused[24,28].

First,thepropergraphtreeisdefinedasaseriesofbranches thatconnectallthenodesanditdoesnotcontainanyloop.The remainingbranchesofthegraphmakeacotree,i.e.theconnecting branchesofthegraph.

ForelectricalcircuitmodelinFig.3,anappropriategraphwith aproperlydefinedtreeoracotreecanbeformedasinFig.4.Then, thevariablesofthesystemaredefinedasacurrentthroughthe inductancesthatbelongtothegraph cotree.It shouldbenoted thatthegraphinFig.4containstheinductorcutset(markedinred) whichreducesthedimensionofthestatevectori.e.thedimension ofthewholesystem.

Atthebeginning,thestatevectorisdefinedas:

X˜(t)=

iL1 iL2 iL0 ih1j ih2k ih3l

T

(6) Customizedequationsofthestatespacecanbewrittenas:

L˜d ˜X(t)

dt =A˜X˜(t)+BU˜ (t) (7)

wherecoefficientsofmatrices ˜A, ˜Band ˜Lareunknown.

Theelementsof matrix ˜A areobtainedinstages, columnby column.Hence,allvoltagesources areshortcircuited,while all currentsourcesandinductorsaredisconnected,excludingonlythe inductancethroughwhichthe1Astepsourcecurrentflows.Inthe firstthreecolumns,currentsaretakenthroughordinaryinductors, whileinthelastthreecolumnscurrentsaretakenthroughartificial linearhystereticinductors.

Theanalogousprocedureisusedtodeterminethecoefficientsof matrix ˜B.Inthefirstthreecolumns,allcurrentsourcesandinduc- torsaredisconnected,whileallvoltagesourcesareshortcircuited, excludingonlythevoltagesourceacrosswhichthe1Vstepsource isconnected.

Thelastthreecolumnsaretreatedbyanalogyprovidedthatall inductorsandcurrentsourcesaredisconnected,excludingonlythe currentsourcethroughwhichthe1Astepsourcecurrentflows.

Elementsofthematrices ˜Aand ˜Bare,respectively:

A˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

uL1

i_L1=1 uL1

i_L2=1 uL1

i_L0=1 uL1

i_h

1j=1 uL1

i_h

2k=1 uL1

i_h 3l=1

uL2

iL1=1 uL2

iL2=1 uL2

iL0=1 uL2

ih1j=1 uL2

ih2k=1 uL2

ih3l=1

uL0

iL1=1 uL0

iL2=1 uL0

iL0=1 uL0

i_h1j=1 uL0

i_h2k=1 uL0

i_h3l=1

uh_1j

i_L1=1 uh_1j

i_L2=1 uh_1j

i_L0=1 uh_1j

i_h

1j=1 uh_1j

i_h

2k=1 uh_1j

i_h 3l=1

uh_2k

iL1=1 uh_2k

iL2=1 uh_2k

iL0=1 uh_2k

ih1j=1 uh_2k

ih2k=1 uh_2k

ih3l=1

uh_3l

iL1=1 uh_3l

iL2=1 uh_3l

iL0=1 uh_3l

i_h1j=1 uh_3l

i_h2k=1 uh_3l

i_h3l=1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

Fig.3.Equivalentcircuitmodelfortransformerinrushcurrentsimulations.

B˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

uL1

e1=1 uL1

e2=1 uL1

e3=1 uL1

S_h1j=1 uL1

S_h2k=1 uL1

S_h3l=1

uL2

e1=1 uL2

e2=1 uL2

e3=1 uL2

S_h

1j=1 uL2

S_h

2k=1 uL2

S_h 3l=1

uL0

e1=1 uL0

e2=1 uL0

e3=1 uL0

S_h1j=1 uL0

S_h2k=1 uL0

S_h3l=1

uh_1j

e1=1 uh_1j

e2=1 uh_1j

e3=1 uh_1j

S_h1j=1 uh_1j

S_h2k=1 uh_1j

S_h3l=1

uh_2k

e1=1 uh_2k

e2=1 uh_2k

e3=1 uh_2k

S_h

1j=1 uh_2k

S_h

2k=1 uh_2k

S_h 3l=1

uh3l

e1=1 uh3l

e2=1 uh3l

e3=1 uh3l

S_h1j=1 uh3l

S_h2k=1 uh3l

S_h3l=1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

Sincethelineargraphforthemodelcontainsinductorcutset, theelementsofmatrix ˜Lareobtainedinacompletelydifferentway accordingtotheredefinedandgeneralizedproceduredescribedin thepaper[22].

Theprocedureinthepaper[22]couldnotsolvethescenario oftheappearanceoftheinductorcutset,theproblemhasbeen solved by inserting artificialelements in the original model of thesystem.Thecurrentsourcesareconnectedinseries,inevery branchofthecotreecontainingtheinductances.Inductancesinthe treeremainedintheoriginalpositionswithoutaddingthecurrent sources.Allotherelementsareremovedfromtheelectriccircuit.

Thenmatrix ˜Lisobtainedinstagesbytype,sothattheelementsof thefirsttypeareobtainedwhilemaintainingthecurrentsource ofthefirstinductance ofthecotreeuntil allcurrentsources of theremaininginductancesofthecotreearedisconnectedfromthe electriccircuit.

The coefficients of the first type are obtainedas equivalent inductancethatisseenfromcotreebrancheswherethereisthe firstinductanceofthecotree.Analogously,thecoefficientsofother typesareobtainedasequivalentinductancesthatareseenfrom

othercotreebranchesinwhichthereareresidualinductancesofthe cotree.Thesignsofthecoefficientmatrix ˜Lareobtaineddepend- ingonthecorrelationofthevoltageinductancesinthebranches ofthecotreewiththecorrespondingcurrentthroughtheinduc- tances.Thedetailedprocedureforobtainingthecoefficientsofthe elementsofthementionedmatrixisdescribedinRef.[24].

Whenthematrices ˜A, ˜Band ˜Larecalculatedaccordingtopre- viouslydescribedprocedures,thentheshift ˜X=KX,whereKisa diagonaltransformationmatrix,i.e.

K=

diag(ki,i)

i=1,2,...,6 (8)

withelements:

ki,i=1, i=1,2,3, ki,i= 1 Lh_i(j,k,l)

, i=4,5,6

becomes the state vector containing thecurrents of the linear inductancesandmagneticfluxesonthenonlinearhystereticinduc- tances:

X(t)=

iL1 iL2 iL0 1_j 2_k 3_l

T

(9) Moving to themagnetic fluxes as state variables is suitable because, as integrals of the corresponding voltages, they are changedmoresmoothlythanthecorrespondingcurrents.

NowEq.(7)iswrittenintheform:

L K˜ dX(t)

dt =A K˜ X(t)+B U˜ (t) (10) Thefinalequationobtainedtheformofastandardstate-space equation(2),wherethematrixofthesystemisobtainedas

A=(˜LK)⁻¹A K˜ (11)

Inductor cutset

Ls1−L12−L31

Rm1 Lh1j

Rm2 Rm3

iL1

iL2

ih1j

3L0

Ls2−L23−L12

Ls3−L31−L23

ih2k ih3l

iL0

Sh1j

Lh2k

Sh2k

Lh3l

Sh3l

u2

iL0

iL0 u3

e1(t)

e2(t) e3(t)

Rp1

Rp2

Rp3

L12

L23

L31

iL2

u12

u23

iL3

u31

iL1

Fig.4. Orientedgraphfortransformercircuitmodel.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

B=(˜LK)⁻¹B˜ (12) Thematrices ˜A, ˜B, ˜LandKareobtainedaccordingtotheabove describedprocedure:

A˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

−2(Rp+Rm) −(Rp+Rm) 0 Rm 0 −Rm

−(Rp+Rm) −2(Rp+Rm) 0 0 Rm −Rm

0 0 −3Rm −Rm −Rm −Rm

Rm 0 −Rm −Rm 0 0

0 Rm −Rm 0 −Rm 0

−Rm −Rm −Rm 0 0 −Rm

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

B˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

1 0 −1 Rm 0 −Rm

0 1 −1 0 Rm −Rm

0 0 0 −Rm −Rm −Rm

0 0 0 −Rm 0 0

0 0 0 0 −Rm 0

0 0 0 0 0 −Rm

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

L˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

2(Ls−L13) Ls+L12−L13−L23 0 0 0 0 Ls+L12−L13−L23 2(Ls−L23) 0 0 0 0

0 0 3L0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

,

K=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

1 0 0 Rm 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1

Lh1j

0 0

0 0 0 0 1

Lh2k

0

0 0 0 0 0 1

Lh3l

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

Thepreviouslywrittenmatrix ˜Lisvalidforageneral(unsym- metrical)caseofmutualcouplingbetweenphases.

InthecaseofLm=L_i,j,(i,j=1,2,3,i=/ j)thefollowingrelationis obtained:

LP=Ls−Lm (13)

Nowthematrix ˜Lcanbewritteninthefollowingform:

L˜=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

2Lp Lp 0 0 0 0 Lp 2Lp 0 0 0 0 0 0 3L0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

FinallythematricesAandBare:

A=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

−Rp+Rm

Lp

0 0 2Rm

3LpLh_1j − Rm

3LpLh_2k − Rm

3LpLh_3l

0 −Rp+Rm

Lp 0 − Rm

3LpLh_1j

2Rm

3LpLh_2k

− Rm

3LpLh_3l

0 0 −Rm

L0 − Rm

3L0Lh_1j

− Rm

3L0Lh_2k

− Rm

3L0Lh_3l

Rm 0 −R_m −Rm

Lh_1j

0 0

0 Rm −R_m 0 −Rm

Lh_2k

0

−Rm −Rm −Rm 0 0 −Rm

Lh_3l

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

B=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

2 3Lp − 1

3Lp − 1 3Lp

2Rm

3Lp −Rm

3Lp −Rm

3Lp

− 1 3Lp

2 3Lp − 1

3Lp −Rm

3Lp

2Rm

3Lp −Rm

3Lp

0 0 0 −Rm

3L0 −Rm

3L0 −Rm

3L0

0 0 0 −Rm 0 0

0 0 0 0 −Rm 0

0 0 0 0 0 −Rm

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

Itshouldbenotedthat,ateveryintegrationstep,thematrixA isafunctionoftheoperatingpointpositionwithinallthreemajor hysteresisloops,i.e.:

A=A(j,k,l) (14)

3.2. Backwardandnumericaldifferentiationformulae

Beforetheapplicationofanyappropriatenumericalmethodfor theefficientsimulationofthesystem,itisnecessarytodetermine thecharacteroftheanalyzeddifferentialequationsystem(4).The equationsystem(4)canbenonstiff,stifforanextremelystiffsys- tem.Ingeneral,asystemisstiffiftheeigenvalues(timeconstants) oftheJacobiansystemdiffersignificantlyinmagnitude.

Inthisregard,theconceptofstiffnessisintroduced,whichisa specificcharacteristicofthesystemofdifferentialequationsrelated exclusivelytosolvingthedifferentialequationsystemusingappro- priatenumericalmethods.Thepracticalmeasureofstiffnesscanbe definedviathefollowingquantitativeparameters:stiffnessratio andstiffnessindex:

= max

i

_Re

⁽ⁱ⁾_j,k,l

min

i

_Re

⁽ⁱ⁾_j,k.l

⁽¹⁵⁾

=max

i

Re

⁽ⁱ⁾_j,k,l

₍₁₆₎

where ⁽ⁱ⁾_j,k,l, i=1,2,...,dim(A(j,k,l)) are eigenvalues of state matrices A= A(j,k,l)calculated withineveryintegrationstep.

When1,itisastiffsystemandwhen→∞,itisanextremelyor verystiffsystem.Inothercasesitisanonstiffsystem.Thenumeri- calintegrationofstifforverystiffsystemsbyexplicitnumerical methods shouldbeavoided, becausethesemethods requirean extremelysmallintegrationsteptoensurenumericalstability.

Stiffequationsystemsrepresent problemsforwhich explicit methodsdonotwork[29].RegardingthesecondDahliquistbar- rier,therearenoexplicitA-stablenumericalmethods,andimplicit multistepmethodscanbeA-stableiftheirorderisatmost2[29].

ImplicitA-andL-stablenumericalmethodsarerequiredforthe numericalintegrationofthesekindsofsystems[29,30].

Thenumericalmethodsofthesecondorderofaccuracyandtheir specificcharacteristicswillbepresentedinthefollowingtext.The trapezoidalmethodiscertainlythemostwidespreadoneinthe simulationsoftheelectricalsystems.Thismethodisveryeasyto implementbecauseitissimple,A-stableofthesecondorderand hasthesmallesterrorconstant.Thetrapezoidalmethodappliedto thestateEq.(2)isasfollows:

Xn+1=Xn+t

2 [F(Xn,tn)+F(Xn+1,tn+1)] (17) However,afundamentalweaknessofthetrapezoidalmethodis associatedwiththeoccurrenceofspuriousnumericaloscillations.

Namely,thetrapezoidalmethodisstablebut notstrongstable;

moreprecisely,ithasnocharacteristicsofL-stability,sothatdur- ingthesimulationofextremelystiffsystems,thismethodcangive erroneousresults.Whenusingthetrapezoidalmethod,theampli- tudeandfrequencyofthenumericaloscillationsdependonthe parametersofenergystorageelementsand theintegrationstep size,whichisexploredindetailinthepaper[22].Theconclusion isthatthismethodisadvantageouslyusedforthesimulationof nonstifformoderatelystiffsystems,whileitshouldbeavoidedin thesimulationofextremelystiffsystems.

Toovercometheseproblems,itispossibletouseimplicitback- wardEuler’smethod,sinceitisL-stable.However,thismethodisof thefirstorderanditisinsufficientlyaccurateincomparisonwith thetrapezoidalmethod.Theintegrationstepsizemustbereduced toachieve thesameaccuracy asthetrapezoidalmethod,which increasesthesimulationtime.

Ontheotherhand,thebackwarddifferentiationformulaeofthe pthorder(BDFp)arethefollowing[29–31]:

p m=1

1

m∇^mXn+1=tF(Xn+1,tn+1) (18) BDFparemoreaccurate,andareA(˛)-andL-stable.L-stability propertiesofthesemethods dampouttheresponseof thestiff andextremelystiffcomponents,i.e.BDFsuppressesthenumerical oscillations.

Thenumericaldifferentiationformulae(NDFp)arefine-tuned BDFpwiththefollowingrelations[32,33]:

p m=1

1

m∇^mXn+1=tF(Xn+1,tn+1)+ pp

Xn+1−X^[0]_n+1

(19)

where parameter p=

_p

m=11

m, the starting value X_n+1^[0] =

_p

m=0∇^mXnand aretheoptimallychosenadditionaltermsthat retainmaximumpossiblestability,reducethetruncationerrorand allowlargertimestepsize.NDFparealsoA(˛)-andL-stable.

ThetruncationerrorofBDFpcanbeapproximatedas εBDFp= 1

p+1h^p+1X^(p+1) (20)

whilethetruncationerrorofNDFpcanbeapproximatedas

εNDFp=

_p

m=1

1 m+ 1

p+1

h^p+1X^(p+1) (21)

Itisclearthattheintegrationstepensuresa givenaccuracy.

ForthesamedefinedtoleranceofBDFpandNDFpmethodi.e.from εBDFp=εNDFp,aconnectionbetweentheintegrationstepsusedin thesetwomethods,hBDFpandhNDFp,wasobtained:

1

p+1h^p+1_BDFp=

pp+ 1 p+1

h^p+1_NDFp (22)

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Magnetizing current [A]

Magnetic flux [Vsec]

Fig.5. Transformerhysteresisloop.

Ifthepercentageofthechangeinintegrationstepsofthesetwo methodsismarkedwith:

ıp%= h^p+_NDFp¹ −h^p+_BDFp¹

h^p+_BDFp¹ ·100% (23)

thefollowingvalueisobtained:

ıp%=

⎡

⎣

1/(p+1)

pp+

1/(p+1)

1/(p+1)

−1

⎤

⎦

_·100% (24)

NDF2isL-stablemethodofthesecondorder,soitsrelationship isı2%=26%.ItcanbeconcludedthattheNDF2isabout26%more efficientthantheBDF2,andbecauseofthatNDF2isthepreferred methodinthispaper.

4. Three-phasetransformerinrushcurrents:

Measurementsandsimulations

Thedevelopedthree-phasethree-leggedtransformerequiva- lentcircuitmodelisusedforinrushcurrentanalysis(Fig.4).

Theelectricalsystemparametersare(i=1,2,3):

•sourcevoltages:e_i(t)=311cos(ωt+i120^◦−155^◦)V;

•ratedsystemfrequency:f=50Hz;

•transformerpowerrate:Str=2.4kVA;

•transformerratio:Up/Us=0.38/0.5kV;

•shortcircuitvoltage:u_k%=3.0%;

•windingresistance:R_pi=1.5;

•positivesequenceinductance:L_pi=1.0mH;

•zerosequenceinductance:L_zi=0.9mH;

•corelossresistance:R_mi=4626;

•zerosequencemagnetizinginductance:L₀=5.0mH.

ThemajortransformerhysteresisloopisshowninFig.5.

The presented measurements of three-phase transformer inrushcurrentswereperformedwithMI7111—Power Analyzer (samplerate:256S/period)andwithFLUKE434/5—PowerQuality Analyzer(themaximumsamplingrate200kS/s).

Theaccuracylevelsofthesedevicesare:MI7111—PowerAna- lyzer0.1%plus1digitandFLUKE434/5—PowerQualityAnalyzer 0.1%.

The moment of switching on (T₀=3.10ms) was estimated, althoughtherewasdissipationbetweenphases(3:05ms,3.12ms, 3.15ms).Thesourcevoltagewassinesignaloffundamentalfre- quency (50Hz) although there were distortions of the phase voltages,i.e.THDofphasevoltageswere2.62%,2.80%and2.83%, respectively.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -60

-40 -20 0 20 40 60 80 100

Time [sec]

Current [A]

ph 1 ph 2 ph 3

Fig.6. Inrushcurrentsimulation,trapezoidalmethod,t=80␮s.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -1.5

-1.48 -1.46 -1.44 -1.42 -1.4 -1.38x 10⁷

Time [sec]

Min eigenvalue [1/sec]

Fig.7.Propagationofminimumrealpartofeigenvaluesduringthesimulationtime.

Thesimulationresultsoftransformerinrushcurrentsbyusing theclassicaltrapezoidalmethodwithintegrationstept=80␮s areshowninFig.6.

Theexistenceoftheartificialnumericaloscillationsisevidentby usingthetrapezoidalmethod.Todetectthecauseofthesenumer- icaloscillationsin everyintegrationstep of thesimulation,the maximalandminimaleigenvalues,aswellasthestiffnessratioand stiffnessindexwerecomputed.TheseresultsareshowninFigs.7–9.

Thelimitvaluesofdefinedstiffnessparametersareobtainedby thefollowing:

1.388×10⁷≤j,k≤1.497×10⁷ 0.579×10⁶≤j,k≤6.947×10⁷

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -30

-25 -20 -15 -10 -5 0

Time [sec]

Max eigenvalue [1/sec]

Fig.8. Propagationofmaximumrealpartofeigenvaluesduringthesimulationtime.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0

1 2 3 4 5 6 7x 10⁷

Time [sec]

Eigenvalue ratio

Fig.9. Propagationofeigenvalueratioduringthesimulationtime.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -20

-15 -10 -5 0 5 10

Time [sec]

Current [A]

measured simulated

Fig.10.Measuredandsimulatedtransformerinrushcurrents:phase1.

Itisclearthatthestate-spaceformofthetransformerinrush currenttransientsexhibitsverystiffsystems.Therefore,BDF2and NDF2methodswereusedforfurthersimulationsofthetransformer energization.

Thecomparisonbetweenthemeasuredandsimulatedtrans- formerinrushcurrentsisshowninFigs.10–12.Thesimulations wereperformedbyusingthenumericalmethod(NDF2)withthe integrationstepsizeoft=80␮s.

Theabsolutedifferencebetweenthemeasuredandsimulated transformer inrush currents as a function of time is shown in Figs.13–15accordingtotheformula:

ε(tk)=

_isimul.(tk)−imeas.(tk)

(25)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -20

-15 -10 -5 0 5 10

Time [sec]

Current [A]

measured simulated

Fig.11.Measuredandsimulatedtransformerinrushcurrents:phase2.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -10

0 10 20 30 40

Time [sec]

Current [A]

measured simulated

Fig.12.Measuredandsimulatedtransformerinrushcurrents:phase3.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0

2 4 6 8 10

Time [sec]

Current [A]

Fig.13.Absolutedifferencebetweenthemeasuredandsimulatedtransformer inrushcurrents:phase1.

Thepropagationoftherelativeerrorofthetransformerpeak inrushcurrentsisshowninFig.16.Therelativeerrorofcurrent peakk,1≤k≤8perphasej=1,2,3iscalculatedbyrelation:

ıj(k)=

_isimul.(k)−imeas.(k)

imeas.j ·100% (26)

whereimeasj=max

j

imeas.j(k)

.

Based on the above explanation, it is possible to propose theimplementationofthehybridnumericalmethodwithinthe EMTP-basedprogramsasalinearcombinationofthetraditional trapezoidal method and the proposed BDF2 (NDF2) methods

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0

2 4 6 8 10

Time [sec]

Current [A]

Fig.14.Absolutedifferencethebetweenmeasuredandsimulatedtransformer inrushcurrents:phase2.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0

2 4 6 8 10

Time [sec]

Current [A]

Fig.15.Absolutedifferencebetweenmeasuredandsimulatedtransformerinrush currents:phase3.

1 2 3 4 5 6 7 8

0 0.5 1 1.5 2 2.5 3 3.5

Peak number

Relative error [%]

ph 1 ph 2 ph 3

Fig.16.Relativeerrorofpeakvalueoftransformerinrushcurrents.

0≤≤1:

Xn+1 =

1+ 3

Xn− 3Xn−1+

1+ 3

t

2 F(Xn,tn) +

1−

_t

2 F(Xn+1,tn+1) (27)

Xn+1 =

1+ 5

Xn−3

10Xn−1+

1−

_t

2 F(Xn,tn) +

1+ 5

t

2 F(Xn+1,tn+1)+

10X^[0]_n+1 (28) In expressions (27) and (28), =0 leads to the trapezoidal method,whereas=1leadstotheBDF2(NDF2).Inthisway,itis possibletoovercomesomeproblemsinusingthestandardtrape- zoidalmethod.

5. Conclusions

The simplified three-phase three-legged transformer model basedonlinear graphtheory wasdevelopedin this paper.The modeltakesintoaccountmutualcouplingbetweenthephasesand hystereticcharacterofironcore.

The algorithm for generating system matrices in electrical networksispresentedhere.Theanalyzednetwork maycontain inductorcut-setsand/orcapacitorloops.

Themainfocusofthepaperwastoresearchthespecificchar- acteristicsofthesystemofdifferentialequationsthatdescribethe