A Method for the Assessment of the Optimal Parameter of Discrete-Time Switch Model

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ContentslistsavailableatScienceDirect

Electric

Power

Systems

Research

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

A

method

for

the

assessment

of

the

optimal

parameter

of

discrete-time

switch

model

R.

Razzaghi

∗

,

C.

Foti,

M.

Paolone,

F.

Rachidi

ÉcolePolytechniqueFédéraledeLausanne,1015Lausanne,Switzerland

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received15November2013

Receivedinrevisedform13January2014 Accepted10February2014

Availableonline16March2014

Keywords:

Discrete-timeswitchmodel Modiﬁednodalanalysis

Fixedadmittancematrixnodalmethod Real-timesimulations

a

b

s

t

r

a

c

t

Thispaperproposesanovelmethodfortheoptimalparameterselectionofthediscrete-timeswitch

modelusedincircuitsolversthatadopttheﬁxedadmittancematrixnodalmethod(FAMNM)approach.

Asknown,FAMNM-basedcircuitsolversallowtoreachefﬁcientcomputationtimes,inparticularfor

real-timesimulationapplications,sincetheydonotneedtheinversionofthecircuitnodaladmittancematrix.

However,thesesolversneedtooptimallytunetheso-calleddiscreteswitchconductance,sincethis

parametermightlargelyaffectthesimulationsaccuracy.Withinthiscontext,weproposeamethodfor

thedeterminationofthediscrete-timeswitchconductancewhichisobtainedbyminimizingthedistance

betweentheeigenvaluesoftheoriginalcircuit’snodaladmittancematrixwiththoseassociatedwiththe

circuitincludingthediscrete-timeswitches.Themethodisproventoprovidevaluesofthediscrete-time

switchconductancethatmaximizethesimulationaccuracyandminimizethelossesonthisartiﬁcially

introducedparameter.Additionally,theproposedmethodavoidstheuseoftrial-and-errorprocess

typ-icallyrequiredwhendiscrete-timeswitchconductancesneedtobeaddressedinFAMNMapproach.The

performancesoftheproposedmethodaredemonstratedforcircuitswithsingleandmultipleswitches

inwhichpassiveRLCelementsandtransmissionlinesarebothconsidered.

1. Introduction

Accurateandcomputationallyefﬁcienttime-domainsimulation ofpowersystemsincludingswitches(suchastraditionalcircuit breakersorpowerelectronicdevices),isachallengingsubjectsince thetradeoffbetweenaccuracyandcomputationtimedependson theadoptedmodelsoftheswitchingdevices,especiallywhen real-timeconstraintsneedtobeachieved.

Detailed switch models reproducing their physical proper-ties are used when studying phenomena such as switching losses,arcingtimesandelectromagnetictransientsassociatedwith switching-arcextinction.However,inmanypowersystems appli-cations,thesesophisticatedmodelscannotbeusedbecauseoftheir requiredcomputationaleffortsandcomplexityofimplementation. Asaresult,simpliﬁedswitchmodelshavebeenproposedinthe literature(e.g.[1,2]).

One of the most popular methods consists in representing switchesaslumpedelectricalcomponents.Thesimplestapproach istheso-called two-valuedresistormodel where tworesistors, characterizedbylargedifferencesoftheirresistancevalues, are

∗Correspondingauthor.Tel.:+41216934815;fax:+41216934662. E-mailaddress:reza.razzaghi@epﬂ.ch(R.Razzaghi).

associatedwitheachstateoftheswitch.Thetypical representa-tionconsistsinreplacingtheswitchbyaresistorcharacterizedbya “small”valueofresistanceforthe“closed-state”anda“large”value forthe“open-state”.However,inthiscasethesystem’sadmittance matrixneedstobeupdatedandre-factorizedaftereachswitching statechange(e.g.[3,4]).

Within the context of real-time simulations, updating the admittancematrix imposesadditional computationalburdento thesolutionalgorithmsthatneedtobeexecutedwithina deter-minedtime window. Asa consequence, theadmittance matrix re-factorizationrepresentsamajorobstacletosatisfythe compu-tationalconstraints.

Apossibleapproachtocircumventthisproblemistheuseof modelingtechniquesthatkeepthesystemadmittancematrix con-stant(e.g.,[5,6]).Tothisend,discretecircuitmodelsforswitching deviceswereproposedin[1,7,8].Thebasicideaisthattheswitch couldberepresentedbyarelativelysmallinductancewhenitsstate is‘closed’andbyarelativelysmallcapacitancewhenitsstateis ‘open’.Asaconsequence,thediscrete-timeswitchmodelis repre-sentedbyanequivalentconductance(Gs)inparallelwithacurrent sourcecontrolledbytheso-calledhistoryterm(e.g.,[4]).The con-sequence of such a representation is an approach called ﬁxed admittancematrix nodalmethod(FAMNM)[9].IntheFAMNM, thediscrete-timeswitchconductanceGsiskeptconstantduring http://dx.doi.org/10.1016/j.epsr.2014.02.008

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switchesstate-transitions,thechangeoftheswitchesstateaffects onlythevalueofthecurrentsourcewhichdoesnotappearinthe circuitadmittancematrix.Ontheotherhand,suchaswitch repre-sentationintroducesartiﬁcialtransients[2,5].

Solutionstosolvethisproblemhavebeenproposedinthe liter-ature.Inparticular,adampingresistancecanbeaddedinseriesto thediscrete-timeswitchmodel[10].However,thisapproachposes theproblemoftheoptimalchoiceforthevalueofsuchadamping resistanceaswellastheinclusionofartificiallossesintothesystem. Adifferentapproachtosolvethisproblemistheoptimal selec-tionoftheGsparameter.Onepossibilityistoconsideranapriori valueforGs and,then,findthecorrespondingoptimalvalueby comparingthesimulationresultswithbenchmarkonesinorder tominimizetherelevanterrors[2](i.e.,obtainedbyoff-line sim-ulations where the switches are represented byideal devices). However,suchatrial-and-errorapproachprovidessolutionsthat requirespecific and time consuming assessmentsin which the uniquenessofthesolutionisnotguaranteed.

Theauthorsofthispaperhavepresentedapreliminary analy-sisin[11]onthepossibilityoffindingtheoptimaldiscrete-time switchGsconductancebysolvingasuitableoptimizationproblem. Thecurrentpaperaimsatdiscussingmoreindetailthemethod presentedin[11],andprovidingitsextensiontothecaseof sys-temswithmultipleswitches.Specifically,theproposedmethod isbasedontheminimizationoftheEuclidiandistancebetween theeigenvaluesoftheFAMNM-basednetworkadmittancematrix andthoseassociatedwiththeadmittancematricesofthe refer-encenetworkscorrespondingtoallpossibleswitchespermutations wheretheswitchesarerepresentedbyidealones.Itisworthnoting that,compared totheexistingmethodsthat relyonthe exten-siveandnumeroustrial-and-errorsimulationstudies,theproposed methodallowstoreducethecomputationsrequiredtofindthe optimaldiscrete-timeswitchGsconductance.

Thestructureofthepaperisasfollows:Section2summarizes theformulationoftheFAMNM.Section3describestheproblem definition.Section 4 presentstheproposed methodtofindthe optimumvaluesfortheconductanceofthediscrete-timeswitches generalizedtothecaseofsystemswithmultipleswitches.Section 5presentsthevalidationoftheproposedmethodbymaking refer-encetothreeexamples.Finally,Section6concludesthepaperwith thefinalremarksandpotentialimplementationofwhatdiscussed inthepaper.

2. FAMNMrepresentationoftheswitch

TheideaofFAMNMisthediscrete-timerepresentationofthe switchwithaconstantimpedancemodel[1,7–9].Suchanapproach assumesthat theequivalentmodeloftheidealswitchis piece-wise linearand couldberepresentedby a capacitancewhen it isopenandaninductancewhenitisclosed.Theinductanceand capacitancearerepresented,inadiscreteform,byaconductance inparallelwithacurrentsource.Inordertosetthevalueofthe conductance forboth switchstates, incase thebackwardEuler numerical integrationmethod is used,the following constraint shouldbesatisﬁed:

Gs= Cs

t =

t

Ls (1)

where Cs and Ls are the discrete-time switch capacitance and inductancerespectively,andt isthesimulationtime-step.For othernumericalintegrationmethods,asimilarapproachcouldbe appliedtorelateGstotheswitchcapacitanceandinductance.Itis worthobservingthatEq.(1),withdifferentanalyticalforms,holds alsoincaseothernumericalintegrationtechniquesareused[1].

Asa consequenceof this representation,therelevant model iscomposedofaconstantconductanceinparallelwithacurrent

Fig.1.Idealswitch(a)andcorrespondingdiscrete-timemodel(b). Source:Adaptedfrom[1].

source(seeFig.1).Asafunctionoftheswitchon/offstate,thevalue ofthecurrentsource(jn+1

s inFig.1)isupdatedateachtime-step

basedontheswitchcurrent/voltage.Theadvantageofthismethod isthatthevaluefortheswitchconductanceGsisﬁxedirrespective oftheswitchon/offstate.Asaresult,thenodaladmittancematrix willremainunchangedduringswitchingoperationsastheswitch stateonlyaffectsthevalueoftheshuntcurrentsource.The cur-rentsourceassociatedwiththeswitchatthesimulationstepn+1 isdeﬁnedas[1]:

Jn+1

s

−in

s fortheonstate

Gs

v

ns fortheoffstate

(2) Accordingto[1],oneapproachfordeterminingthevalueofGs istoselectCsandLsequaltothecorrespondingrealswitch param-eters.Then,thevaluesofGsandtcouldbedeterminedusing(1). However,themaindrawbackofthisapproachisthattherequired simulationtimestepmightbecomeextremelysmallresultingin prohibitivecomputationaltimes.

As already stated in the introduction, a different procedure referstotheassessmentofanoptimalGsvaluebymeansofa trial-and-errorprocesswherebenchmarkresultsareobtainedbymeans ofoff-linesimulationscarriedoutbyadoptingidealswitches. How-ever,suchanapproachrequiresnon-negligiblepre-computational efforts.

3. Discrete-timerepresentationofelectricalsystems

Generally,therearetwomaintypesofsolutionmethods cur-rentlyusedintheﬁeldofpowersystem,powerelectronicsand electronic circuit simulations [12]: (i) modiﬁed nodal analysis (MNA)and(ii)state-space(SS)approach.Inthisstudy,inorderto formulatethenetworkequations,MNAhasbeenselectedsimilarly to[1].ThischoiceissupportedbythefactthattheMNAprovides astraightforwardwayofdirectlyintegratingtheaboveillustrated discrete-timeswitchmodel.

Asknown,theMNAformulationisexpressedasfollows:

[An][xn]=[bn] (3)

wherematrix[An],inthediscrete-timedomain,isformedbythe discreterepresentationofthenetworkelements;[xn]isthe vec-torofunknownnetwork’snodevoltagesandbranchcurrents;and [bn]isavectorcomposedoftheindependentsourcesandcurrent historytermsrelatedtothenetworkcomponents.Ateach itera-tion,theunknownvector[xn]iscalculatedand,then,thevector [bn]isupdated.Asalreadymentioned,representingswitcheswith FAMNMallowstokeep[An]ﬁxedduringswitchingtransitions.

InordertosolveEq.(3)indiscrete-time,asuitablenumerical integrationmethodshouldbeused.Amongtheseveralmethods available, backwardEuler and the trapezoidalmethods are the

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Fig.2.Schematicrepresentationofthetestcasecomposedofasingle-conductor transmissionlineandaswitch.

mostcommoninpowersystemapplications[13].Togetherwith thesetraditionalmethods,newnumericalintegrationtechniques basedonmidpointmethodshavebeenproposed.Anexampleof thesetechniquesisthetwo-stagediagonallyimplicitRunge–Kutta methodpresentedin[14].

Inthispaper,thebackwardEulermethodhasbeenadopted. Asitwillbeclariﬁednext,thevalidationoftheproposedmethod hasbeendoneusingtheEMTP-RVsimulationenvironment[15,16] wherethebackwardEulerintegrationmethodwasused.In this respect, it is worth observing that the proposed approach is independentfromtheadoptednumericalintegrationtechnique. Indeed,itreliesontheFAMNMmodelingapproach ofelectrical circuits.Theuseofa differentnumericalintegrationtechniques (i.e.,backwardEulerandthetrapezoidalmethods)willonlychange theelementsthatappearinthediscrete-timenetworkadmittance matrixAn of (3).The backward-Eulerintegrationtechniquehas beenselectedsinceithasbeenlargelyusedintheliterature(e.g., [1,5,9])toformulatetheFAMNMapproach.

Inviewoftheuseof theMNA, allthenetwork components shouldbediscretizedtoformtheso-callednodalequations[17,18]. Lumpedpassiveelements(R,L,C)arerepresentedbytheir dis-cretecompanionmodelscomposedofanequivalentconductance inparallelwitha currentsourcerepresentingthehistory terms [18,19].Thevaluesoftheequivalentconductanceandthecurrent sourcearedeterminedbasedontheappliednumerical integra-tionmethod. Concerningthe case of transmissionlines, one of themost popular modeling approach is the so-called Bergeron

method[20].Itallowsastraightforwardrepresentationofconstant (frequency-independent)transmissionlinemodels[4]and,with someadaptations,itcanalsobeappliedtothecaseof frequency-dependenttransmissionlines[21].Aswellknown,thisapproach isbasedonacircuitrepresentationofthetelegraphers’equations whereeach linetermination isreplaced bymeansofa lumped impedanceinparallelwithacontrolledcurrentorvoltagesource.

3.1. Problemdeﬁnition

Inordertoillustratethisproblem,thetime-domainsimulation oftheinrushofasingle-conductortransmissionlineisconsidered. AsitisshowninFig.2,thenetworkiscomposedofanidealsource, representinganinﬁnitepowerbus,supplyingaHV/MVtransformer thatfeedsa1MW/0.3MVarloadandatransmissionline.The trans-missionlineparametersarethoseofatypical20kVlossyoverhead line,namelya surgeimpedanceof400Ohmsandapropagation timeequalto117␮s.Inordertosimplifytheinterpretationofthe results,asingle-phasemodelhasbeenconsidered.

ThisnetworkhasbeensimulatedinEMTP-RVsimulation envi-ronment where the switch was modeled using either an ideal switch(referencecase),or adiscrete-timemodel withdifferent valuesfortheconductance.Fig.3showstheeffectoftheswitchGs valueontheaccuracyofthecurrentatthefeedingterminalofthe transmissionline.Itcanclearlybeobservedthattheadoptedvalue

0 0.005 0.01 0.015 0.02 -2 -1.5 -1 -0.5 0 0.5 1 time [s] L in e cu rr en t [pu ] Ideal switch FAMNM, G s=0.1 FAMNM, G s=1 FAMNM, G s=10

Fig.3. Linecurrentatthebeginningofthelineforfourswitchrepresentation models(i)idealswitch,(ii)FAMNMrepresentationforGs=0.1,and(iii)FAMNM representationforGs=1,and(iv)FAMNMrepresentationforGs=10.

forGsaffectsdramaticallythesimulationaccuracyoftheFAMNM vs.thereferenceoneinwhichtheidealswitchhasbeenconsidered. 4. Proposedmethodfortheoptimalevaluationof

discrete-timeswitchesconductance

The proposed method is based on the minimization of the Euclidiandistancebetweentheeigenvaluesofthenetwork admit-tancematrix[An]basedonFAMNM,andthoseassociatedwiththe admittancematricesofreferencenetworkscorrespondingtothe allpossibleswitchingpermutations.Inordertoclearlydescribethe proposedmethod,itwillbeﬁrstexplainedforthecaseofanetwork withasingleswitch.Then,itisgeneralizedforthecaseofnetworks withmultipleswitches.

4.1. Proposedmethodfornetworkscomposedofasingleswitch

Theproposedmethodallowsfortheevaluationoftheoptimal valueoftheswitchconductanceGsasthesolutionofan optimiza-tionproblemwheretheobjectivefunctionisassociatedwiththe distancesbetweenthe[An]eigenvaluesobtainedusingtheFAMNM (ofcourse,thissetofeigenvaluesisinherentlyfunctionofGs)and twoothersetsofeigenvaluesofmatrices[An]relatedtothe‘on’ and‘off’statesoftheconsideredswitch.Inthisrespect,itisworth observingthat theproposedapproach is aformal process aim-ingatidentifyingthecharacteristicsofthediscrete-timeswitch conductancethatminimizesasuitablydeﬁnedobjectivefunction thatexhibitsminimumcoincidingwiththoseofobjectivefunctions relatedtoartiﬁciallossesintroducedbytheFAMNMortotheerrors inthereferencesimulatedwaveforms,seeSection4.3forfurther details.

Let[Apn],[Acn],and[Aon]bethenodaladmittancematriceswhen

theswitchisrepresentedbyFAMNM,idealswitchin‘on’state,and idealswitchin‘off’state,respectively.Thethreematricesareall supposedtobeofrankn.Asstatedinafundamentaltheoremof linearalgebra,ifBisabaseofthevectorspaceV(ofdimensionn) andWanothervectorspaceofdimensionn,foreachtransformation ϕ:B→W,thereisoneandonlyonelinearmapT:V→Wsuchthat T

_B=ϕ.Now,since[Apn],[Acn],and[Aon]arerealmatricesofrankn,

wecandeﬁnethreebasesBp,BcandBoofRnassociatedwiththe respectiveeigenvalues:

Bp↔p_i(Gs)=eig{[Apn]}, i=1,2,...,n (4)

Bc↔c_i =eig{[Acn]}, i=1,2,...,n (5)

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Fig.4. DeﬁnitionofEuclidiandistancebetweeneigenvaluesofthenetwork admit-tancematrixbasedonFAMNMandthoseassociatedwiththeadmittancematrices oftworeferencenetworks.

Notethatp_i(Gs),c_i,o_i denoterespectivelythecorresponding

eigenvaluesforeachnodaladmittancematrix.Asalreadyclariﬁed, p_i(Gs)isafunctionofGswhereas,c_iando_i areﬁxed.

Sincetoeachmatrix[Apn],[Anc],and[Aon]wecanassociateaunique

base(Bp,BcandBorespectively),inviewofthefundamental theo-remoflinearalgebra,thesetransformationsareunique.Therefore, theobjectiveistodetermineBpsuchthat T

Bpprovidesresults

thatareascloseaspossibletobothT

_B_candT

_B_o.Apossible met-rictoachievesuchapropertyistominimizethesquaredEuclidian distancesforeacheigenvaluecalculatedas:

c i(Gs)={Re[ p i(Gs)]−Re[ci]} 2 +{Im[p_i(Gs)]−Im[c_i]}2 (7) o i(Gs)={Re[ p i(Gs)]−Re[oi]} 2 +{Im[p_i(Gs)]−Im[o i]} 2 (8) Intheseequations,c

i(Gs)andio(Gs)denote,asafunctionof

Gs,thesquaredEuclidiandistancesbetweentheitheigenvalueof [Apn]andthecorrespondingoneof[Acn]and[Aon],respectively.These

distances,foragenericitheigenvalueareillustratedinFig.4. Then,thefollowingtotaldistance(foragiveneigenvalue)can bedeﬁned:

i(Gs)=_io(Gs)+_ic(Gs) (9)

Itispossibletodeﬁneanobjectivefunctionasthesumofall normalizeddistances: (Gs)= n

i=1

i(Gs) max{i(Gs)}

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Notethatthenormalizationisdoneinordertogiveequalweight toalleigenvaluesdistances.TheoptimumvalueforGsisdeﬁnedas theonethatminimizestheobjectivefunction(10).Inotherwords: G∗

s=arg|Gsmin{(Gs)} (11)

4.2. Extensionfornetworkswithmultipleswitches

ForthecaseofanetworkwithNswitches,thenumberof pos-sibleswitchingpermutationsis2N_._Therefore,_there_are₂N_set_of eigenvaluesofthenodaladmittancematrixassociatedwithideal switchesrepresentations,namely:

x

i=eig{[Axn]},

i=1,2,...,n

x∈2N (12)

wherexisoneofthepossibleswitchespermutations. Addition-ally,theeigenvaluesofnodaladmittancematrixassociatedwith

Fig.5.SchematicrepresentationoftheRLCcasestudyincludingoneswitch.

theFAMNMswitchrepresentationarefunctionofswitches con-ductances,namely:

p_i(Gs1,Gs2,...)=eig{[Apn]}, i=1,2,...,n (13)

where, foreach switch,a differentconductance valuehasbeen considered.

ThesquaredEuclidiandistancesassociatedwiththeith eigen-valuehavetobecalculatedforallpossiblepermutations:

i(Gs1,Gs2,...)=

x ({Re[p_i(Gs1,Gs2,...)]−Re[xi]} 2 +{Im[p_i(Gs1,Gs2,...)]−Im[xi]} 2 ) (14)

Theobjectivefunctionextendedtothegeneralcasereads: (Gs1,Gs2,...)= n

i=1

i(Gs1,Gs2,...) max{i(Gs1,Gs2,...)}

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Theoptimumvaluesfortheswitchesconductancecomefrom thesolutionofthefollowingoptimalproblem:

G∗

s1,G∗s2,...=arg|Gs1,Gs2,...min{(Gs1,Gs2,...)} (16)

Itisworthobservingthatfornetworkswithmultipleswitches, theremightbepreferredswitchingsequences.Thus,allthe permu-tationsarenotusedequally.Therefore,thenumberofthepossible permutationsandnumberoftheswitchconductancesvaluescan besigniﬁcantlyreduced.

4.3. Methodveriﬁcationtowarderrorandlossesfunctions

Inordertoverifythecorrectnessofthesolutionprovidedby (16),wehavefirstcomparedtheobjectivefunctiondefinedby(16) withanerrorfunctioninferredfromthedifferencesbetweenthe voltage/currentwaveformsobtainedforvariousvaluesofGsand reference onesobtainedusing theideal-switch model.Such an errorfunctionincludestime-domainswitchvoltageandcurrent waveformssubsequenttoswitchstatetransitions (inparticular, subsequenttopairsof‘on’–‘off’transitions).Indeed,asitisstatedin [1],switchcurrenterrorin‘off’stateisproportionaltoGs,whereas, switchvoltageerrorin‘on’stateisinverselyproportionaltoGs.This specificpropertyhasbeenexploitedtodefinetheerrorfunction. Specifically,thefollowingprocedurehasbeenadopted:the switch-currenterroriscalculated,foreachoftheN1switchesin‘off’state, asthedifferencebetweentheinstantaneousvaluesoftheswitch currentgivenbytheFAMNMsolverandthecurrentprovidedby areferencesimulationwheretheswitchisconsideredasanideal device.Thesameprocedureisconsideredtocalculatetheswitch voltageerrorforeachoftheN2switcheswhicharein‘on’state.For theithswitch,thecurrentandvoltageerrors,EI

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0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0 Gs 0.6 0. Objectiv Error Fu Losses .8 1 ve Function unction Function

Fig.6.Objective,error,andswitch-lossesfunctionsforthecaseoftheRLCtestcase ofFig.5.

EV

sw_i(Gs1,Gs2,...),arethengivenby: EI ω1(Gs1,Gs2,...)= m

k=0 [iω1(Gs1,Gs2,...)k−ik,ω∗ 1] 2_, _m₌ T t, ω1=1,2,...,N1 (17) EV ω2(Gs1,Gs2,...)= m

k=0 [

v

ω2(Gs1,Gs2,...)k−

v

∗k,ω2] 2_, _m =_tT , ω2=1,2,...,N2 (18)

Notethat in(17)and (18),T isa given timewindow where one possible switching permutation occurs. Discrete variables iω1(Gs1,Gs2,...)k and

v

ω2(Gs1,Gs2,...)k correspond to the

dis-cretizedinstantaneousvaluesofswitchcurrentandvoltagewhen theswitchisrepresentedbyitsapproximatemodeland,thus,they arefunctionofallswitchesconductances.Discretevariablesi∗

k,ω1

and

v

∗_k,ω

2 arethecorrespondingdiscretizedinstantaneousswitch

currentandvoltage,obtainedfromreferencesimulationswhere theswitcharerepresentedasidealdevices.

Todeﬁnetheerrorfunction,theeffectofallNswitchesshould betakenintoaccount.Namely,foreachpossiblepermutation,the sumofthecurrenterrorsforallswitchesin‘off’stateandthesum ofthevoltageerrorsforalltheswitchesin‘on’stateareconsidered as: Ex₍_G s1,Gs2,...)=

ω1

_EI ω1(Gs1,Gs2,...) max(EI ω1(Gs1,Gs2,...)) +

ω2

EV ω2(Gs1,Gs2,...) max(EV ω2(Gs1,Gs2,...)) (19)

Forallpossibleswitchingcases,thesameprocedureisapplied byconsideringthesametimewindowTforthecalculationandby changingtheswitchesstate.

Finally,inordertotakeintoaccountallpossiblepermutations, theoverallerrorfunctionisdeﬁnedasthesumofnormalizederrors calculatedforeachpossiblepermutationin(19),asfollows: E(Gs1,Gs2,...)=

x

_Ex₍_G s1,Gs2,...) max(Ex₍_G s1,Gs2,...))

(20) 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Gs

Objective Function Error Function Losses Function

Fig.7.Objective,error,andswitch-lossesfunctionsforthecaseoftransmissionline testcase(Fig.2)withvariableGs.

Afurtherwaytoverifythecorrectnessofthesolutionprovided by(16)istocompareitwithanotherfunctionthatrepresentsthe switchlosses when thesedevices arerepresented byusingthe FAMNMapproach.Forthecaseofanidealswitch,theswitchlosses arezeroduringthe‘off’and‘on’statessinceswitchcurrent/voltage arenull.

To this end, the switch losses in ‘off’ and ‘on’ states, PO

ω1(Gs1,Gs2,...)andP C

ω2(Gs1,Gs2,...)respectively,canbe

straight-forwardlycalculatedasfollows: PO ω1(Gs1,Gs2,...)= 1 m m

k=0 (

v

oω1(Gs1,Gs2,...)k·i o ω1(Gs1,Gs2,...)k), m=_tT , ω1=1,2,...,N1 (21) PC ω2(Gs1,Gs2,...)= 1 m m

k=0 (

v

cω2(Gs1,Gs2,...)k·i c ω2(Gs1,Gs2,...)k), m=_tT , ω2=1,2,...,N2 (22)

wherediscretevariablesiO

ω1(Gs1,Gs2,...)k,

v

o ω1(Gs1,Gs2,...)k and ic ω2(Gs1,Gs2,...)k,

v

c

ω2(Gs1,Gs2,...)kcorrespondtothediscretized

instantaneousvaluesofswitchcurrentandvoltagein‘off’and‘on’ statesrespectively,whentheswitchisrepresentedbytheFAMNM. Theeffectofallswitchesinapossibleswitchingpermutationis takenintoaccountbyconsideringthesumofallswitcheslossesas:

Px₍_G s1,Gs2,...)=

ω1

PO ω1(Gs1,Gs2,...) max(PO ω1(Gs1,Gs2,...)) +

ω2

PC ω2(Gs1,Gs2,...) max(PC ω2(Gs1,Gs2,...)) (23) wherePx₍_G

s1,Gs2,...)istheoveralllossforone ofthepossible switchingpermutation.

Then,thetotallossesfunctiontakesintoaccountallpossible switchingpermutationsbysummingthenormalizedoveralllosses calculatedin(23): P(Gs1,Gs2,...)=

x

_Px₍_G s1,Gs2,...) max(Px₍_G s1,Gs2,...))

(24)

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5 6 7 8 9 10 11 12 x 10-3 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 time [s] ] u p[ e nil e ht f o d n e e ht t a e g at l o V Ideal Switch FAMNM, Gs*=0.01 FAMNM, Gs=0.5

Fig.8. Time-domainsimulatedwaveformsforvoltageattheendofthetransmission line(secondtestcase–Fig.2)fordifferentvaluesofGs.

Fig.9. SchematicrepresentationoftheRLCcasestudyincludingtwoswitches.

Inthenextsection,wewillshowthattheoptimalGsvalue pro-videdby(16)correspondsalsototheminimumoftheerrorfunction (20)andlossesfunction(24),provingthattheproposedapproach satisﬁesthesetwocriteriaatthesametime.

5. Validationexamples

Inordertovalidateourproposedmethod,twosimulationcases areconsidered.Theﬁrstsimulationreferstoelectricalcircuits com-posedof one switch.The second case studyrefers toelectrical circuitscomposedoftwoswitches.Thesecasestudiesarecarried outbymakingreferencetotwodifferenttypesofelectricalcircuits: (i)circuitscomposedofRLC elementsand (ii)circuitsincluding transmissionlines. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 G_s1 Gs2

Fig.10. ObjectivefunctionusedtoassesstheoptimalGsvaluesforthecaseofRLC circuitwithtwoswitchesshowninFig.9.

2 4 6 8 10 12 14 16 x 10-3 -1 -0.5 0 0.5 1 time [s] S w itc h 2 v o lta g e [ pu ] Ideal Switch FAMNM, Gs1=0.4 , Gs2=0.18 FAMNM, G s1=1 , Gs2=2

Fig.11.Time-domainsimulatedwaveformsforswitch#2voltagefortheoptimal valuesofGs1,Gs2forthecircuitcomposedofRLCelementsandtwoswitches.

5.1. Circuitswithoneswitch

Theﬁrstsimulationcasestudyreferstoanelectricalcircuit com-posedofRLCelementsandoneswitch.Theschematicdiagramof theconsideredcircuitisshownisFig.5.

ThiscircuitissimulatedwithintheEMTP-RVsimulation envi-ronment considering both an ideal model and a discrete-time modelfortheswitch.Aspreviouslymentioned,thebackwardEuler methodwasusedforthenumericalintegrationwithatimestep t=4␮s.

ForthecircuitshowninFig.5,thenodaladmittancematricesare formedforthecaseswheretheswitchisrepresentedby(i)FAMNM, (ii)idealswitchin‘on’case,and(iii)idealswitchin‘off’case.Then, accordingtotheproposedmethod,theobjectivefunction(15)is determined.Inordertocalculatetheerrorandlossesfunctions, thefollowingswitchingtransitionisconsidered:theswitchisin openpositionanditisclosedatt=10ms.Then,itisopenedagain att=20ms.ForallvaluesofGs(i.e.,0≤Gs≤1),Eqs.(15),(20)and

(24)arecalculated.Theobjectivefunctiontogetherwiththe corre-spondingerrorandlossesfunctionsareshowninFig.6.Asitcanbe clearlyobserved,allthethreefunctionshavetheirminimumwhen

Gsisequalto0.11.

Thesecondsimulationexamplereferstoanetworkasshownin Fig.2,whichincludesasingle-conductortransmissionline.

2 4 6 8 10 12 14 16 18 x 10-3 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 time [s] Sw it c h 2 c u rr e n t [ pu ] Ideal Switch FAMNM, Gs1=0.4 , Gs2=0.18 FAMNM, Gs1=1 , Gs2=2

Fig.12.Time-domainsimulatedwaveformsforswitch#2currentfortheoptimal valuesofGs1,Gs2forthecircuitcomposedofRLCelementsandtwoswitches.

(7)

Byapplyingthesameprocedurefor0≤Gs≤1,theobjective,

error,andlossesfunctionsarecalculated(seeFig.7).Asitisshown onFig.7,thesethreefunctionsexhibitthesamebehaviorasforthe previouscase,withacommonminimumoccurringforavalueofGs equalto0.01.

Withreferencetothe secondtest case, Fig.8 illustratesthe time-domainsimulationsofthevoltageattheendofthelinefor dif-ferentvaluesofGsincludingtheoptimalvaluepreviouslyidentiﬁed (G∗

s=0.01).Itcanbeseenthatthesimulationsobtainedusingthe

optimumvaluefortheconductanceareinagreementwiththose obtainedusingEMTP-RV.

Figs.6and7showthattheproposedobjectivefunctioncouldbe utilizedasanefﬁcienttooltoﬁndtheoptimumGsvaluewithout performinganyoff-linebenchmarksimulations,astheeffectofGs ontheadoptedsystemmodelcouldbecorrectlypredicted.

5.2. Circuitswithtwoswitches

Inordertovalidatetheperformanceoftheproposedmethod, additionalinvestigationshavebeendoneforthecaseofnetworks withtwoswitches.Tothisend,thesimulationexamplereferstoan RLCcircuitincludingtwoswitches.Theschematicdiagramofthe consideredcircuitisshownisFig.9.

Accordingtotheproposedmethod,foreachswitch,adedicated

GsvalueisconsideredandEqs.(15),(20),and(24)areformedtoﬁnd theobjective,errorandlossesfunctions.Fig.10showsthe objec-tivefunction.Asitisshownonthisﬁgure,theminimumvalues correspondtoG∗

s1=0.4andGs∗2=0.18.

InFigs.11and12,thetime-domainsimulationsofthevoltage andcurrentofswitch#2ofFig.9showthataverygoodmatchis achievedbetweentheresultsoftheFAMNMrepresentationwith optimalconductancevalueandtheidealrepresentation.

6. Conclusions

Inthispaper,amethodtoﬁndtheoptimumvalueof discrete-timeswitchconductancehasbeenproposed.Asknown,thevalue ofthisparametershouldbechoseninawaytominimizetheerrors introducedbytheapproximaterepresentationofthediscrete-time switchmodel.

The proposed method is based on the minimization of the Euclidiandistancebetweentheeigenvaluesof theFAMNM net-workadmittancematrixandthoseassociatedwiththeadmittance matrices of the reference networks correspondingto all possi-ble switches permutations. Additionally, the proposed method avoidstheuseoftrial-and-errorprocesstypicallyrequiredwhen discrete-timeswitchconductancesneedtobeaddressedinFAMNM approach.

Toprovethecorrectnessoftheproposedmethod,acomparison betweentheproposedmetricandspeciﬁcdeﬁnederrorfunctions hasbeen presentedand discussed.By makingreference totwo differentcategoriesoftest cases,namely(i) circuitswitha sin-gleswitchand(ii)circuitswithmultipleswitches,theproposed methodhasbeenproventobecorrectinidentifyingtheoptimal conductance valueof thediscrete-timeswitch model.It is also shownthattheproposedmethodminimizessimultaneouslythe differenceswithreference-modelcurrent/voltagewaveforms,and lossesonthediscrete-timeswitchconductance.

Withouttheneed of performingoff-linebenchmark simula-tions,theproposedmethodcanrepresentapowerfulsolutionfor therepresentationofswitcheswithinreal-timesimulation plat-formsthatrelyontheuseoftheFAMNM(i.e.,FPGAs).

Acknowledgment

Thepapertakes intoaccountthecommentsreceivedduring thepresentationofapreliminaryversionatthe10thInternational ConferenceonPowerSystemsTransients(IPST2013),Vancouver, Canada,July,2013.ThisworkwassupportedbytheSwissNational ScienceFoundation(ProjectNo.200021-146414,“PowerNetworks FaultLocationbasedonElectronicEmulation”).

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