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A Continuation Method Approach to Finding the Closest Saddle Node Bifurcation Point

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A CONTINUATION METHOD APPROACH TO FINDING THE CLOSEST SADDLE NODE BIFURCATION POINT Yuri V. Makarov Ian A. Hiskens Department of Electrical and Computer Engineering The University of Newcastle, Callaghan, NSW, 2308, Australia Abstract The power ow equations f(x) generally de ne a mapping from state space to a subset of parameter space, i.e., ?f : Rn ! L where L  Rn. The region L de nes the set of parameters for which power ow solutions exist. Therefore the inverse mapping, from parameter
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  ACONTINUATIONMETHODAPPROACHTOFINDINGTHE CLOSESTSADDLENODEBIFURCATIONPOINT  YuriV.MakarovIanA.Hiskens  DepartmentofElectricalandComputerEngineering TheUniversityofNewcastle,Callaghan,NSW,2308,Australia  Abstract  Thepaperconsiderstheproblemofndingsaddlenodebifur-cationpointswhichareclosest(inalocalsense)tothepowersystemoperatingpoint.Thisoptimizationproblemleadstoasetofequationswhichdescribesuch  criticalpoints .Notallsolutionsofthissetofequationsarecriticalpoints.Thepaperthereforeexploresthenatureandcharacteristicsofsolutions.Atwostagealgorithmisproposedforsolvingthecriticalpointproblem.Therststageissimplytondapointonthesingu-larsurface,i.e.,thesurfaceofsaddlenodebifurcationpoints,whichliesisaspecieddirection.Thesecondstageusesacontinuationmethodtomovefromthatinitialpointtothedesiredcriticalpoint.Singularityofthecriticalpointproblem canhaveasignicantinuenceontherobustnessofthecontin-uationmethod.Thepaperinvestigatessingularityconditions.Theproposedalgorithmistestedonaneightbuspowersystem example. Keywords :stabilitymargincalculation,powerowsingular-ities,continuationmethods 1Introduction  Thetrendinmodernpowersystemoperationistoward greaterutilisationofgenerationandtransmissionassets.Thisnecessarilymeansthatsystemsmustoperatemuch closertostabilitylimits.Thereforethereisaneedtobeabletodeterminethoselimitsmoreaccuratelyandreli-ably.Theformofthedesiredstabilitymargininformation varies,dependingonthetypeofpowersysteminvesti-gationbeingundertaken.Investigationsoflargedistur-bancestabilitymaymakeuseofastabilitymarginbased onLyapunovideas,see1]forexample.Inanoperating environmenthowever,itismorecommontousesecurity marginsbasedonquasi-staticpropertiesofthesystem.Thepowerowproblemiscentraltothisformofsecurity margin.Themarginisgenerallybasedonsomemeasureofdistancefromtheoperatingpointtoapointwherethepowerowproblembecomesunsolvable.Thispaperfo-cussesonthislattertypeofsecuritymargin.Apowerowproblemcanbedescribedbythesetof n  nonlinearalgebraicequations y  0 +  f  ( x  )=0(1)where y  0 2  R  n isthevectorofspeciedindependentpa-rameterssuchasactiveandreactivepowersofloadsand generatorsorxedvoltages, x  2  R  n isthestate,consist-ingofnodalvoltages.Thevectorfunction  f  ( x  )denesthesumofpowerowsorcurrentsintoeachbusfromtherestofthenetwork.Ifnodalvoltages x  areexpressedin rectangularcoordinatesthen  f  ( x  )isaquadraticfunction of x  .Thepowerowequations f  ( x  )generallydeneamap-pingfromstatespacetoasubsetofparameterspace,i.e., ?  f  : R  n !L  where L  R  n .Theregion  L  denesthesetofparametersforwhichpowerowsolutionsexist.There-foretheinversemapping,fromparameterspacetostatespaceisonlydenedinside L  .Thisinversemappingisingeneralnotunique,withanumberofsolutionscorre-spondingtoagivenvalueofparameters y  0 2L  .Infact,parameterspacemaybedividedintoregions,witheach regionhavingadierentnumberofsolutionsforagiven valueof y  .Asparameters y  vary,thesolutionsof(1)willalsomoveinstatespace.Parameters y  maymovetoapointwheretwosolutionscoalesce,withfurthervariationof y  result-inginthedisappearanceofthatsolution.Behaviourofthatformisreferredtoasasaddlenodebifurcation.ItfollowsfromtheImplicitFunctionTheoremthatatsuch bifurcationpointsdet D  x f  =det J  ( x  )=0(2)i.e., J  ( x  )istheJacobianmatrixof f  ( x  ).Further,weseethatregionsdenedonthebasisofthenumberofsolutionsforagiven  y  mustbeboundedbysurfacesofpointssatisfying(2).Weshalldenetheboundaryas=  f ( x;y  ): x;y  2  R  n ;y  +  f  ( x  )=0 ; det J  j ( x;y ) =0 g (3)Theprojectionofontostatespaceandparameterspacewillbereferredtoas  x ;   y respectively.Moredetailed analysisofthestructureof L  wasundertakenin11,12].Thecondition(2)meansthatatleastonerealeigen-value   i of J  mustbezero.Undercertainmodelling assumptions,azeroeigenvalueof J  correspondstolossofsmalldisturbancestabilityofthesystem13,14,15].Thereforethe`distance'fromanoperatingpointtopointswhere J  issingular,i.e.,pointsin,providesausefulmeasureofthesecurityofasystem.Asanoperatingpointmovesnearerto,thestabilityregionaroundthatpointreduces37].Animportantpowersystemcontrolproblemthereforeistopreventanoperatingpointfrommovingtoocloseto.Thatis,operatingpointsshouldalwaysbe(atleast)somespecieddistancefrom.Asparameters y  correspondto physicalquantitiesthatcanbemeasuredandcontrolled,itisusefultoconsiderthisdistanceintermsofparameterspace,i.e., d  ( y  )=  k y  ?  y  0 k (4)where y  0 istheoperatingpoint,and  y  2    y ,i.e., y  isa pointonthesolutionboundary16,17,18,19].Theshort-est(orcritical)distancemin  y 2   y d  ( y  )givesameasureofpowersystemsecurityinthemostdangerousdirectionof1  loading19,21,22,23,24,26,27].Inaddition,thecriti-calvector( y  0 ?  y  )denestheoptimalwayofcontrolling thepowersystemtomaximisesecurity.Itslargestcom-ponentsindicateparameterswhichcontributemosttothesecurityconditions17,20,22,26].Inthispaperweaddresstheissueofrobustlyndingtheminimumdistanceto.Thisquestionhasbeeninvesti-gatedbefore19,21,22,23,24,26,27].Weareproposing acontinuationapproachtondingthepointsonwhich areclosest(inalocalsense)totheoperatingpoint.Wecallthesepoints criticalpoints .Singularitiesoftheprob-lemwhichaectsuchmethodsareinvestigated.Thepaperisorganisedasfollows.Section2establishesthemathematicaldescriptionofcriticalpoints.Propertiesofthesolutionsofthecriticalpointproblemarediscussed inSection3.Singularitiesofthatproblemareconsidered inSection4.Section5proposesanalgorithmfornding thecriticalpoints.Aneightbusexampleisconsidered inSection6.AnnumericaltechniquewhichisusefulforthecriticalpointalgorithmofSection5isoutlinedinAp-pendixB. 2Formulationoftheproblem  2.1Parameterweightingfactors  Whenconsideringtheminimumdistancefromanop-eratingpointto,andoptimalcontrolstrategiesforin-creasingthatdistance,itisnecessarytotakeaccountofthefactthatparametersmayhavedierentdimensions,e.g.,buspowersandnodalvoltages.Itmayalsobeneces-sarytoweightparametersofthesametypedierently.Forexampleasmallbutcriticalloadmayneedtobeweighted dierentlytoalarge,butnotsocriticalload.Further,someparametersarexed.Anexampleinthiscasewould bebuspowersatnodeswhichhavenogeneratorsorload connected.Therefore,tomaketheparameterscompatible,`normal-ising'coecientscanbeused19,22,forexample].Thedistancefunctionwouldthenberedenedas d  ( y  )=  k    y  ?  y  0 ] k (5)where   isadiagonalmatrixofweightcoecients,with diagonalelements   i =1 =y  bi (6)Each  y  bi isa`normalising'factorforthe i -thparameter.Thedistance(5)canbeusedasanaperiodicstability(orsecurity)index16,19].Bycomparingthedistance d  ( y  )withaspeciedsafevalueofthestabilityindex  I  s ,itispossibletodecidewhetherthecurrentoperatingstateisdangerousornot.Itshouldbenotedthatthe y  b in(6)shouldbeconstants,andnotdependentontheoperatingpointvalues y  0 .Itwasshownin20]thatdicultiesarisewhen  y  b =  y  0 ,duetotheresultingnonlineardependenceof d  ( y  )on  y  0 . 2.2Optimizationformulation  Oneoftherstthingstoconsiderintheformulationoftheoptimizationproblem min  y 2   y d  ( y  )(7)isthatnotallparametersarefreetovary.Somepa-rameters,suchaspowerinjectedatbuseswhichhaveno loadorgeneration,mustalwaysbexed.Parameterscan beeectivelyheldconstantbeassigningverylargeval-uesofweightcoecients   in(5).Howeverthisleadstoanill-conditionedproblem,andassociatednumericaldiculties.Wethereforeadoptthefollowingpowerow formulation.Let m  equationsin(1)containxedvaluesofparame-ters y  2 =  y  02 =  const .Theother( n  ?  m  )parameters y  1 arefreetovary.Thenthesystem(1)canberewrittenas y  1 +  f  1 ( x  )=0(8) y  02 +  f  2 ( x  )=0(9)Using(8),(9),weseethatthesquareofthedistance d  ( y  )denedat(5)canbewritten  1 d  ( y  ) 2 =  k y  ?  y  0 k 2 (10)=  k y  1 ?  y  01 k 2 (11)=  k y  01 +  f  1 ( x  ) k 2 (12)Considertheoptimizationproblem ext x k y  01 +  f  1 ( x  ) k 2 (13) y  02 +  f  2 ( x  )=0(14)where`ext'denotesextremaofthecostfunction(13).Theoptimizationissubjecttononlinearconstraints(14).From(10)-(12)itcanbeseenthatthecostfunction(13)denesthesquareofthedistancebetweenthepoints y  1 and  y  01 ,withbothpointsbelongingtotheconstrainthy-perplane y  2 =  y  02 =  const ,seeFigure1.IfwedenetheLagrangefunction  l ( x;  )=  k y  01 +  f  1 ( x  ) k 2 +2 y  02 +  f  2 ( x  )] t   (15)thentheconstrainedoptimizationproblem(13),(14)can beformulatedasanunconstrainedproblem ext x; l ( x;  )(16)Solutionsof(16)satisfythenonlinearsystem  J  t 1 ( x  ) y  01 +  f  1 ( x  )]+  J  t 2 ( x  )   =0(17) y  02 +  f  2 ( x  )=0(18)where J  1 =  @f 1 @x and  J  2 =  @f 2 @x .Thissystemofequationscanbewritteninmoregeneralformas   ( x;  )=0(19)Usingthesubstitution  s =  y  01 +  f  1 ( x  )(20)in(17)allowsustorepresentthesystem(17),(18)as ?  s +  y  01 +  f  1 ( x  )=0 y  02 +  f  2 ( x  )=0(21) J  t 1 ( x  ) s +  J  t 2 ( x  )   =0 1 Tosimplifyexpressionswetake   =  I ,where I istheidentitymatrix.Thecasewhen   6 =  I requirestrivialtransformationsofallthefollowingequations. 2  2 = const  200 2  y y y y y y 1 Figure1:Theconstrainthyperplane(14).Itisclearthatthesystem(21)hasthesamesolutionsasthesystem(17),(18),so    ( x;  )=0 ,    ( x;s;  )=0(22)where   ( x;s;  )=0representsthesystem(21).Thelastequationin(21)canberewrittenas J  t ( x  ) s  =0(23)where J  t = J  t 1 J  t 2 ](24) s  = s t   t ] t (25)If s  6 =0,theJacobianmatrix  J  ( x  )issingular,and thevector s  isalefteigenvectorcorrespondingtoazero eigenvalue.Therefore,consideringtheoriginaloptimiza-tionproblem(13),(14),andthecondition(23)for s  6 =0,wecanconcludethatcriticalpoints,i.e.,pointsonthatareminimaldistance(locally)fromtheoperatingpoint y  0 ,satisfythesystem(21). 3Solutionsofthecriticalpoint problem  Thesystem(21)canbeconsideredasanextendedpowerowproblemwheretheusualpowerowequations(thersttwoequationsin(21))aresupplementedbythesin-gularitycondition(23).Statespace,i.e.,thespaceofun-knownvariables,isnowextendedtoincludetheadditionalvariables s  .Sothevariablesare( x;s  ) 2  R  2 n . 3.1Trivialandnontrivialsolutions  Therearetwokindsofsolutionstothecriticalpointproblem(21):  Trivialsolutionscorrespondingtothecondition  s  = 0.Thosesolutionsareactuallythesolutionsoftheusualpowerowproblem(1).Theyareglobalmin-ima(zeros)ofthedistancefunction(4).Alltrivialsolutionscoincideinparameterspace.  Nontrivialsolutionsconformingtothecondition  s  6 = 0.Thosesolutionsbelongtothesingularmargin givenby(3).Solutionof(21)canresultineithertrivialornontrivialsolutions,dependingoninitialestimatesofthevariablesandthenumericalsolutiontechniqueusedforsolvingtheproblem.AtechniquewhichproducesnontrivialsolutionsisproposedinSection5. 3.2Distanceandthelefteigenvector  Letusanalyzethenontrivialsolutionsofthesystem (21).Itisknown,thatthevector s  isanormalvectorto thesingularhypersurface  y .However,weseefromtherstequationof(21),and(8)thatatcriticalpoints,i.e.,solutionsof(21), y  1 ?  y  01 =  y  1 ?  s +  f  1 ( x  )=  ?  s (26)So,because y  2 =  y  02 ,thecompontent ?  s ofthevector ?  s  isthedistancevector y  ?  y  0 .Sincethedistancevectorisanorthogonalvectortothesingularhypersurface  y ,nontrivialsolutionscorrespond tolocalminimaormaximaofthedistancefromthepoint y  0 tothesingularmargin.Theminimumofthedistancesassociatedwiththenontrivialsolutionpointscharacterizesthe\level ofpowersystemsecurity. 3.3Agraphicalillustration  Thegraphicalillustrationofanontrivialsolutionpointof(21)isgivenbyFigure2.Thesolutionpointmustliesomewhereontheintersectionofthesingularmargin   y andconstrainthyperplane y  2 =  y  02 =  const .Thelefteigenvector s  atthenontrivialsolutionpoint y  isper-pendiculartothesingularboundary,anditscomponent s coincideswiththevectorfromthesingularpoint y  to theoperatingpoint y  0 .Thecomponent   (theLagrangemultipliervector)of s  isorthogonaltotheconstrainthy-perplane. 3.4Constraintimpactuponthecriticaldistance  Therelativelengthofthevector   , l  =  k   kk s  k (27)indicatesthesignicanceoftheconstraintset(14)ontheoptimization(13).Considerthefollowingcases,3  = 0 y=  const y y λ 22 λ  y o2  y o - --  ss J(x)  det Figure2:Graphicalillustrationofanontrivialsolution point.1. l  =0,( s 6 =0 ;  =0).Thevector s  liesonthecon-strainthyperplane y  2 =  y  02 =  const ,andthecriticalpointcorrespondstoasolutionoftheunrestricted optimizationproblem ext x k y  0 +  f  ( x  ) k 2 (28)Hencetheconstraintshavenoinuenceonthesolu-tion.2. l  =1,( s =0 ;  6 =0).Thevector s  isorthogo-naltotheconstrainthyperplane.Thismeansthattheconstrainthyperplaneistangenttothesingularmargin  y atthecriticalpoint.Itwillbeshownin Section4.2thatthecorrespondingpointisasingu-larpointof(21).Traditionalnumericaltechniquesencounterdicultiesatsuchpointsinthesameway thatpowerowtechniquesexhibitpoorconvergencenearsingularpoints.Theconstraintsareparticularly signicantinthiscase.3.0 <l  <  1,( s 6 =0 ;  6 =0).Thecorrespondingnon-trivialsolutionisaectedbytheconstraints(14).Thevalueof l  indicatestheextenttowhichthecon-straintsinuencetheminimumdistancefromtheop-eratingpointtothesingularmargin.Asmallvalueof l  indicatesmodestinuenceofconstraints. 3.5Minima,maximaandsaddlepoints  TheJacobianmatrix  J   of(21), J   =  2 6 6 4  J  1 ( x  ) ?  I  0 J  2 ( x  )00 D  ( s  ) J  t 1 ( x  ) J  t 2 ( x  ) 3 7 7 5  (29)  j1j1 j1P3V = 1V = 1V = 1 δδδ = var= var= 012 11133222 = 0.5P = -0.5 Figure3:Simple3buspowersystem.with  D  ( s  )=  @ @x   J  t ( x  ) s  ](30)iseectivelyaHessianmatrixoftheLagrangiancostfunc-tion(15).Itisknown28,forexample]thatthesolutionpointisaminimumwhenthematrix  J   ispositivedenite,anda maximumwhenitisnegativedenite.IftheHessianma-trixisindenite,thecorrespondingsolutionisasaddle.Allthosecasescanbeencounteredinthisoptimization problem.Toillustratethis,weconsiderthefollowingsim-ple3-buspowersystemexample. Example(3bus) PowerbalanceequationsforthesystemshowninFig-ure3canbewrittenas P  1 +sin(  2 ?   1 ) ?  sin   1 =0 P  2 +sin(  1 ?   2 ) ?  sin   2 =0(31)Thevalues P  1 , P  2 areconsideredasfreeparameters y  1 in(8).Thevoltagemagnitudes V  1 =1puand  V  2 =1pu aretakenasxedparameters y  02 in(9).Theoptimization problem(13)thereforetransformsto ext  1 ; 2 ( P  1 ?  0 : 5) 2 +( P  2 +0 : 5) 2 ](32)Figure4showstheplaneofstatevariables  1 ; 2 ,and Figure5theplaneoffreeparameters P  1 ;P  2 ,withparam-eters V  1 ;V  2 xed.Singularmargins  x forthesystem (31)(dashedlines)andcontoursofthecostfunction(32)areplottedontheplane  1 ; 2 inFigure4.Solutionsoftheoptimizationproblem(32)arealsoshown.Points A1 , A2  areminima.Theycorrespondtotrivialsolutionsofthecriticalpointproblem(21). A1 isthe`normal'operating point.Pointsmarked  B  and  C  arenon-trivialsolutionsof(21),with  B1  B5  beingsaddlepointsof(32),and  C1  C3  maxima.Allnon-trivialsolutionsof(21)lieonthesingu-larmargin  x .Notethatthesectionof  x whichsurroundstheop-eratingpoint A1 ,i.e.,theovalthatcontainspoints B1 , B2  , C1 , C2  ,isofprimaryinterest.Ifparameterswerevar-iedcontinuouslyfromtheiroperatingpointvalues,then 4 

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Jul 25, 2017
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