Nose and Inrush
description
Transcript of Nose and Inrush
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Summary of Papers
1.P.SauerandM.Pai,PowerSystemSteadyStateStabilityandtheLoadFlowJacobian,IEEETransactionsonPowerSystems,Vol.5,No.4,Nov.1990
2.V.AjjarapuandC.Christy,TheContinuationPowerFlow:AToolforSteadyStateVoltageStabilityAnalysis,IEEETransactionsonPowerSystems,Vol.7,No.1,Feb.,1992.
3.S.Greene,I.Dobson,andF.Alvarado,SensitivityoftheLoadingMargintoVoltageCollapsewithRespecttoArbitraryParameters,IEEETransactionsonPowerSystems,Vol.12,No.1,Feb.1997,pp.232240.
4.S.Greene,I.Dobson,andF.Alvarado,ContingencyRankingforVoltageCollapseviaSensitivitiesfromaSingleNoseCurve,IEEETransactionsonPowerSystems,Vol.14,No.1,Feb.1999,pp.262272.
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Voltage Security
Voltagesecurityistheabilityofthesystemtomaintainadequateandcontrollablevoltagelevelsatallsystemloadbuses.Themainconcernisthatvoltagelevelsoutsideofaspecifiedrange can affect the operation of the customers loadsrangecanaffecttheoperationofthecustomer sloads.
Voltagesecuritymaybedividedintotwomainproblems:1 L lt lt l l i t id f d fi d1.Lowvoltage:voltagelevelisoutsideofpredefinedrange.2.Voltageinstability:anuncontrolledvoltagedecline.
Youshouldknowthat lowvoltagedoesnotnecessarilyimplyvoltageinstability no low voltage does not necessarily imply voltage stability
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nolowvoltagedoesnotnecessarilyimplyvoltagestability voltageinstabilitydoesnecessarilyimplylowvoltage
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Th h b l i di id l th t h i ifi tl
Resources
Therehavebeenseveralindividualsthathavesignificantlyprogressedthefieldofvoltagesecurity.Theseinclude:
AjjarapufromISU
Van Cutsem: See the book by Van Cutsem and Vournas.VanCutsem:SeethebookbyVanCutsemandVournas.
Alvarado,Dobson,Canizares,&Greene:
Thereareacoupleothertextsthatprovidegoodtreatmentsofthe subject:thesubject: CarsonTaylor:PowerSystemVoltageStability PrabhaKundur:PowerSystemStability&Control
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Ourtreatmentofvoltagesecuritywillproceedasfollows:
Voltageinstabilityinasimplesystem Voltageinstabilityinalargesystem Brieftreatmentofbifurcationanalysis Continuationpowerflow(pathfollowing)methods Sensitivity methodsSensitivitymethods
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Voltageinstabilityinasimplesystem
Considertheperphaseequivalentofaverysimplethreephasepowersystemgivenbelow:
V1 V2
Z=R+jX
INode1 Node2
V1 V2
++
1
__
5S12 SD=S12
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jBGYjXRZ =+= NoteB>0jj
121212 jQPS +=
)sin(||||)cos(|||||| 212121212
112 += BVVGVVGVP
121212 jQPS +
)sin(||||)cos(||||||
)(||||)(||||||
212121212
112
21212121112
= GVVBVVBVQ )s (||||)cos(|||||| 21212121112 GVVVVVQ
LetG=0.Then.
)sin(|||| 212112 = BVVP
6)cos(|||||| 2121
2112 = BVVBVQ
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NowwecangetSD=PD+jQD=(P21+jQ21)by
exchangingthe1and2subscriptsinthepreviousequations. negatingg g
)sin(||||)sin(|||| 122121
===
BVVBVVPPD
)cos(||||||
)sin(||||
2
2121
+==
=
BVVBVQQ
BVV
)cos(||||||
)cos(||||||
21212
2
1221221
+=
+==
BVVBV
BVVBVQQD
Define12 =1 2
1221 sin|||| BVVPD =
712212
2
1221
cos||||||
||||
BVVBVQD
D
+=
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Define: isthepowerfactorangleoftheload,i.e., p g
IV = 2
ThenwecanalsoexpressSD as:|||| 2*
2eIVIVS jD ==
)sin1(||||
)sin(cos|||| 2
jIV
jIV +=
)tan1(
)cos
1(cos|||| 2
jP
jIV
+=
+=
)tan1( jPD +=
Define=tan.Then
Notethatphi,andthereforebeta,ispositiveforlagging,
ti f l di
8)1( jPjQPS DDDD +=+=
negativeforleading.
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Sowehavedevelopedthefollowingequations.
12212
2
1221
cos||||||
sin||||
BVVBVQ
BVVP
D
D
+=
=
12212 cos|||||| BVVBVQD +=
)1( jPjQPS DDDD +=+=
EquatingtheexpressionsforPD andforQD,wehave:
2 |||||| BVVBVPQ
1221 sin|||| BVVPD = 12212
2
12212
2
cos||||||
cos||||||
BVVBVP
BVVBVPQ
D
DD
=+
+==
Squarebothequationsandaddthemtoget..
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122
12222
22
122
22 )cos(sin||||)||( BVVBVPP DD +=++
222
21
222
21212212
||||)||(
)(||||)||(
BVVBVPP DD
DD
=++
Manipulationyields:
2
( ) [ ] 01||||2|| 222
22
21
222 =++
+
BPVV
BPV DD
Notethatthisisaquadraticin|V2|2.Assuch,ithasthesolution:
2/12
1
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212
2 ||||||||
+= VPPVPVV DDD
10
12 ||42||
BBB
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Letsassumethatthesendingendvoltageis|V1|=1.0pu1andB=2pu.Thenourpreviousequationbecomes:
[ ])2(11 2/12 + DDD PPP [ ]2
)2(11|| 22 +
= DDDPPPV
%pf=0.97laggingbeta=0 25
You can make
beta=0.25pdn=[00.10.20.30.40.50.60.70.78];v2n=sqrt((1beta.*pdn sqrt(1pdn.*(pdn+2*beta)))/2);pdp=[0.780.70.60.50.40.30.20.10];v2p=sqrt((1beta.*pdp+sqrt(1pdp.*(pdp+2*beta)))/2);pd1=[pdnpdp];v21=[v2nv2p];Youcanmake
thePVplotusingthefollowing
tl b d
[ p]%pf=1.0beta=0pdn=[00.10.20.30.40.50.60.70.80.90.99];v2n=sqrt((1beta.*pdn sqrt(1pdn.*(pdn+2*beta)))/2);pdp=[0.990.90.70.60.50.40.30.20.10];v2p=sqrt((1beta.*pdp+sqrt(1pdp.*(pdp+2*beta)))/2);d2 [ d d ]matlabcode. pd2=[pdnpdp];v22=[v2nv2p];
%pf=.97leadingbeta=0.25pdn=[00.10.20.30.40.50.60.70.80.91.01.11.21.3];v2n=sqrt((1beta.*pdn sqrt(1pdn.*(pdn+2*beta)))/2);pdp=[1.3 1.2 1.1 1.0 0.9 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0];
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pdp [1.31.21.11.00.90.70.60.50.40.30.20.10];v2p=sqrt((1beta.*pdp+sqrt(1pdp.*(pdp+2*beta)))/2);pd3=[pdnpdp];v23=[v2nv2p];
plot(pd1,v21,pd2,v22,pd3,v23)
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Plotsofthepreviousequationfordifferentpowerfactors
|V ||V2|
R l l di P
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Realpowerloading,PD
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SomecommentsregardingthePVcurves:1.Eachcurvehasamaximumload.Thisvalueistypicallycalledthemaximumsystemloadorthesystemloadability.2.Iftheloadisincreasedbeyondtheloadability,thevoltageswilldeclineuncontrollably.3.Foravalueofloadbelowtheloadability,therearetwovoltagesolutions.Theupperonecorrespondstoonethatcanbereachedinpractice.Theloweroneiscorrectmathematically,butIp y,donotknowofawaytoreachthesepointsinpractice.4.Inthelaggingorunitypowerfactorcondition,itisclearthatthevoltagedecreasesastheloadpowerincreasesuntiltheloadability.g p yInthiscase,thevoltageinstabilityphenomenaisdetectable,i.e.,operatorwillbeawarethatvoltagesaredecliningbeforetheloadability is exceeded.loadabilityisexceeded.5.Intheleadingcase,oneobservesthatthevoltageisflat,orperhapsevenincreasingalittle,untiljustbeforetheloadability.Thus,inthe leading condition voltage instability is not very detectable
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theleadingcondition,voltageinstabilityisnotverydetectable.Theleadingconditionoccursduringhightransferconditionswhentheloadislightorwhentheloadishighlycompensated.
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QVCurvesWeconsideroursimple(lossless)systemagain,withtheequations
12212
2
1221
cos||||||
sin||||
BVVBVQ
BVVP
D
D
+=
=
Now,againassumethatV1=1.0,andforagivenvalueofPDandV2,compute12 fromthefirstequation,andthenQfromthe
secondequation.RepeatforvariousvaluesofV2 toobtainaQVcurveforthespecifiedrealloadPD.
v1=1.0;b=1.0;
YoucanmakethePVplotusingthefollowingmatlabcode.
pd1=0.1v2=[1.1,1.05,1.0,.95,.90,.85,.80,.75,.70,.65,.60,.55,.50,.45,.40,.35,.30,.25,.20,.15];sintheta=pd1./(b*v1.*v2);theta=asin(sintheta);qd1=v2.^2*b+v1*b*v2.*cos(theta);
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plot(qd1,v2);
Thecurveonthenextpageillustrates.
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QVCurve
|V ||V2|
15QD
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Homework
1.DrawthePVcurveforthefollowingcases,andforeach,determinetheloadability.
a.B=2,|V1|=1.0,pf=0.97lagging
b B=2 |V1|=1 0 pf=0 95 laggingb.B=2,|V1|=1.0,pf=0.95lagging
c.B=2,|V1|=1.06,pf=0.97lagging
d.B=10,|V1|=1.0,pf=0.97lagging
Identifytheeffectonloadabilityofpowerfactor,sendingendvoltage,andlinereactance.
2.DrawtheQVcurvesforthefollowingcases,andforeach,determinethemaximumQD.
a.B=1,|V1|=1.0,PD=0.1
b.B=1,|V1|=1.0,PD=0.2
c. B=1, |V1|=1.06, PD=0.1c.B 1,|V1| 1.06,PD 0.1
d.B=2,|V1|=1.0,PD=0.1
IdentifytheeffectonmaximumQD ofrealpowerdemand,sendingendvoltage,andlinet
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reactance.
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SomecommentsregardingtheQVCurves
Inpractice,thesecurvesmaybedrawnwithapowerflowprogramby
1.modelingatthetargetbusasynchronouscondenser(ag g y (generatorwithP=0)havingverywidereactivelimits
2.Setting|V|toadesiredvalue3 Solving the power flow3.Solvingthepowerflow.4.ReadingtheQofthegenerator.5.Repeat24forarangeofvoltages.
QVcurveshaveoneadvantageoverPVcurves:Theyareeasiertoobtainifyouonlyhaveapowerflow(standard
powerflowswillnotsolvenearorbelowthenoseofPVcurvespbuttheywillsolvecompletelyaroundthenoseofQVcurves.)
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Voltageinstabilityinalargesystem:Influentialfactors:
Loadmodeling Reactivepowerlimitsongenerators Loss of a circuit Lossofacircuit Availabilityofswitchableshuntdevices
Twoimportantideasonwhichunderstandingoftheabovei fl1. Voltageinstabilityoccurswhenthereactivepowersupply
cannotmeetthereactivepowerdemandofthenetwork.
influencesrest:
Transmissionlineloadingistoohigh Reactivesources(generators)aretoofarfromloadcenters Generatorterminalvoltagesaretoolow.
I ffi i l d i i Insufficientloadreactivecompensation2.Reactivepowercannotbemovedveryfarinanetwork(varsdonottravel),sinceI2Xislarge.
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Implication:TheSYSTEMcanhaveavarsurplusbutexperiencevoltageinstabilityifalocalareahasavardeficiency.
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Loadmodeling
Inanalyzingvoltageinstability,itisnecessarytoconsiderthenetworkundervariousvoltageprofiles.
Voltagestabilitydependsonthelevelofcurrentdrawnbytheloads.
Thelevelofcurrentdrawnbytheloadscandependonthevoltageseenbytheloads.
Therefore,voltageinstabilityanalysisrequiresamodelofhowtheloadrespondstoloadvariations.
Thus,loadmodelingisveryinfluentialinvoltageinstabilityanalysis.
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Exponentialloadmodel
Atypicalloadmodelforaloadatabusistheexponentialmodel:
=
= 00VQQVPP
0
00
0 VQQ
VPP
wherethesubscript0indicatestheinitialoperatingconditions.Theexponents and arespecifictothetypeofload,e.g.,
Incandescentlamps 1.54 Roomairconditioner 0.50 2.5F f 0 08 1 6Furnacefan 0.08 1.6Batterycharger 2.59 4.06Electroniccompactflorescent1.0 0.40
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Conventionalflorescent 2.07 3.21
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PolynomialloadmodelTheZIPorpolynomialmodelisaspecialcaseofthemoregeneral
i l d l i b f 3 i l d l i hexponentialmodel,givenbyasumof3exponentialmodelswithspecifiedsubscripts:
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VVVV
++
=
++
= 3
02
0103
02
010 qV
VqVVqQQp
VVp
VVpPP
0.1321 =++ ppp 0.1321 =++ qqq
whereagainthesubscript0indicatestheinitialoperatingconditions.
Usually,valuesp2 andq2 arethelargest.
Sothismodeliscomposedofthreecomponents:
constant impedance component (p1 q1) lighting
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constantimpedancecomponent(p1,q1) lighting constantcurrentcomponent(p2,q2) motor/lighting constantpowercomponent(p3,,q3) loadsservedbyLTCs
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Understanding the effect of each component on voltage instability
EffectofLoadmodeling
Understandingtheeffectofeachcomponentonvoltageinstabilitydependsonunderstandingtwoideas:
l i bili i ll i d h h d d d hi1.Voltageinstabilityisalleviatedwhenthedemandreduces.ThisisbecauseIreducesandI2Xreactivelossesinthecircuitsreduce.
2.Sincevoltageinstabilitycausesvoltagedecline,alleviationofvoltageinstabilityresultsifdemandreduceswithvoltagedecline.This gives the key to understanding the effect of load modelingThisgivesthekeytounderstandingtheeffectofloadmodeling.
constantimpedanceload(p1)isGOODsincedemandreduces with square of voltagereduceswithsquareofvoltage. constantcurrentload(p2)isOKsincedemandreduceswithvoltage.C l d ( ) i BAD i d d d
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Constantpowerload(p3)isBADsincedemanddoesnotchangeasvoltagedeclines.
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The effects of voltage variation on loads and thus of loads on
Someconsiderationsinloadmodeling
Theeffectsofvoltagevariationonloads,andthusofloadsonvoltageinstability,cannotbefullycapturedusingexponentialorpolynomialloadmodelsbecauseofthefollowingthreeaspects.
Thermostaticloadrecovery Inductionmotorstalling/trippingg pp g Loadtapchangers
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Heatingloadisthemostcommontypeofthermostaticload,anditThermostaticloadrecovery
g yp ,isoneforwhichweareallquitefamiliar.Althoughmuchheatingisdonewithnaturalgasastheprimaryfuel,someheatingisdoneelectrically and even gas heating systems always contain someelectrically,andevengasheatingsystemsalwayscontainsomeelectriccomponentsaswell,e.g.,thefans.
h h i l d i l d h / lOtherthermostaticloadsincludespaceheaters/coolers,waterheaters,andrefrigerators.
Whenvoltagedrops,thermostaticloadsinitiallydecreaseinpowerconsumption.Butaftervoltagesremainlowforafewminutes,theload regulation devices (thermostats) will start the loads or willloadregulationdevices(thermostats)willstarttheloadsorwillmaintainthemforlongerperiodssothatmoreofthemareonatthesametime.Thisisreferredtoasthermostaticloadrecovery,d i d b l bl h hi h l
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andittendstoexacerbatevoltageproblemsatthehighvoltagelevel.
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Three phase induction motors comprise a significant portion of
Inductionmotorstalling/tripping
Threephaseinductionmotorscompriseasignificantportionofthetotalloadandsoitsresponsetovoltagevariationisimportant,especiallysinceithasaratheruniqueresponse.
Considerthesteadystateinductionmotorperphaseequivalentmodel.
Za=R1+jX1 X2 I2
Zb=Rc//jXm
R2+R2(1-s)/s =R2 / s
V1c//j / s
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The(referredtostator)rotor current is given by:
Inductionmotorstalling/tripping
2 ')/'('
jXsRZVI th
++=
rotorcurrentisgivenby: 22 ')/'( jXsRZth ++
bZVV = baZZZZZ //where ba
th ZZVV
+= 1
andba
babath ZZ
ZZZ+
== //
Under normal conditions the slip s is typically very small less than 0 05Undernormalconditions,theslipsistypicallyverysmall,lessthan0.05(5%).Inthiscase,R2/s>>R2,andI2 issmall.
B t lt V d th l t ti t d l dButasvoltageV1 decreases,theelectromagnetictorquedevelopeddecreasesaswell,themotorslowsdown.Ultimately,themotormaystall.Inthiscase,s=1,causingR2/s=R2.Thus,oneseesthatthecurrentI2 is
h l f ll d di i h f l di i B f Xmuchlargerforstalledconditionsthanfornormalconditions.BecauseofX1andX2 oftheinductionmotor,thelargestallcurrentrepresentsalargereactiveload.
26Largemotorshaveundervoltagetrippingtoguardagainstthis,butsmallermotors(refrigerators/airconditioners)maynot.
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Tapchangers:
L d h (LTC OLTC ULTC TCUL) f hLoadtapchangers(LTC,OLTC,ULTC,TCUL)aretransformersthatconnectthetransmissionorsubtransmissionsystemstothedistributionsystems.Theyaretypicallyequippedwithregulationcapabilitythatallowthemtocontrolthevoltageonthelowsidesothatvoltagedeviationonthehighsideisnotseenonthelowside.
t:1
V1 V1/t
t:1HVside
LVside
V1 andtaregiveninpu.
Inperunit,wesaythatthetapist:1,where tmayrangefrom0.851.15pu a single step may be about 0 005 pu (5/8%=0 00625 is very common)asinglestepmaybeabout0.005pu(5/8%=0.00625isverycommon) achangeofonesteptypicallyrequiresabout5seconds. thereisadeadbandof23timesthetapsteptopreventexcessivetapchange
U d l l di i h hi h id h LTC ill d
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Underlowvoltageconditionsatthehighside,theLTCwilldecreasetinordertotryandincreaseV1/t.
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Tapchangers:
Thus,aslongastheLTCisregulating(notatalimit),avoltagedeclineonthehighsidedoesnotresultinvoltagedeclineattheload in the steady state so that even if the load is constant Zload,inthesteadystate,sothateveniftheloadisconstantZ,itappearstothehighsideasifitisconstantpower.Soasimpleloadmodelforvoltageinstabilityanalysis,forsystemsusingLTC,isconstantpower!
There are 2 qualifications to using such a simple model (constant power):Thereare2qualificationstousingsuchasimplemodel(constantpower):1.Fastvoltagedipsareseenatthelowside(sinceLTCactiontypicallyrequiresminutes),andifthedipislowenough,induction motors may trip resulting in an immediate decrease ininductionmotorsmaytrip,resultinginanimmediatedecreaseinloadpower.2.OncetheLTChitsitslimit(minimumt),thenthelowside
28voltagebeginstodecline,anditbecomesnecessarytomodeltheloadvoltagesensitivity.
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Generatorcapabilitycurve:
Fieldcurrentlimitduetofieldheating,enforced by overexcitation limiter on I
QenforcedbyoverexcitationlimiteronIf.
Armature current limit due toQmax
Armaturecurrentlimitduetoarmatureheating,enforcedbyoperatorcontrolofPandIf.
Typicalapproximation
d iP
usedinpowerflowprograms.
Limitduetosteadystateinstability(smallinternalvoltageEgivessmall|E||V|Bsin),
Qmin
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andduetostatorendregionheatingfrominducededdycurrents,enforcedbyunderexcitationlimiter(UEL).
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Effectofgeneratorreactivepowerlimits:
1.Voltageinstabilityistypicallyprecededbygeneratorshittingtheirupperreactivelimit,somodelingQmax isveryimportanttoanalysisofvoltageinstability.
2.MostpowerflowprogramsrepresentgeneratorQmax asfixed.However,thisi i i d h h ld b i d I li Q i fi disanapproximation,andonethatshouldberecognized.Inreality,Qmax isnotfixed.ThereactivecapabilitydiagramshowsquiteclearlythatQmax isafunctionofPandbecomesmorerestrictiveasPincreases.AfirstorderimprovementtofixedQmaxis to model Q as a function of PistomodelQmax asafunctionofP.
3.Qmax issetaccordingtotheOvereXcitationLimiter(OXL).ThefieldcircuithasaratedsteadystatefieldcurrentIfmax,setbyfieldcircuitheatinglimitations.Sinceheating is proportional to , we see that smaller overloads can be tolerated 2dtI fheatingisproportionalto,weseethatsmalleroverloadscanbetoleratedforlongertimes.Therefore,mostmodernOXLsaresetwithatimeinversecharacteristic:
4.AssoonastheOXLactstolimitIf,thenno
timeoverload
f
ffurtherincreaseinreactivepowerispossible.WhendrawingPVorQVcurves,theactionofageneratorhittingQmax,will
IfI
2.0OXLcharacteristic
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manifestitselfasasharpdiscontinuityinthecurve.
Overloadtime(sec)
Irated1.0
120
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EffectofOXLactiononPVcurve:
|V|
Onegeneratorhitsreactivelimit
Noreactivelimitsmodeledo
P
(demand)
Note:GeorgiaPowerCo.modelsitsloadabilitylimitatpointx,notpointo.
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L f i itLossofacircuit
Comparereactivelosseswithandwithoutsecondcircuit
/
AssumebothcircuitshavereactanceofX.
I/2
I/2
I
P
X
XI/2 P P
Qloss=(I/2)2X+ (I/2)2X=I2X/2 Qloss=I2XQloss (I/2) X (I/2) X I X/2 Qloss I X
Implication:Lossofacircuitwillalwaysincreasereactivelossesin the network. This effect is compounded by the fact
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inthenetwork.Thiseffectiscompoundedbythefactthatlosingacircuitalsomeanslosingitslinechargingcapacitance.
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Kundur,onpp.979990,hasanexcellentexamplewhichillustratesmanyoftheaforementionedeffects.Theillustrationwasdoneusingalongtermtimedomainsimulationprogram(Eurostag).
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Influence of switched shunt capacitorsInfluenceofswitchedshuntcapacitors
I II
P
I
PP P
|V|
WithcapacitorWithoutit
P
capacitor
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P
(demand)
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But,shuntcompensationhassomedrawbacks:
Itproducesreactivepowerinproportiontothesquareofthethe voltage therefore when voltages drop so does the reactivethevoltage,thereforewhenvoltagesdrop,sodoesthereactivepowersuppliedbythecapacitor.
It h i ti l l b d hi h t bl Ithasamaximumcompensationlevelbeyondwhichstableoperationisnotpossible(Seepg.972ofKundur,andnextslide).
(A h d d SVC d t h th 2 d b k )(AsynchronouscondenserandanSVCdonothavethese2drawbacks)
ItresultsinaflatterPVcurveandthereforemakesvoltageinstabilit less detectable Therefore as the load gro s in areasinstabilitylessdetectable.Therefore,astheloadgrowsinareaslackinggeneration,moreandmoreshuntcompensationisusedtokeepvoltagesinnormaloperatingranges.Bysodoing,normal
35operatingpointsprogressivelyapproachloadability.
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V1=1.0 V2
PL
EachQVcurve/Capacitorcharacteristicintersectionshowstheoperatingpoint.Notethatforthefirstthreeoperatingpoints,asmall increase in Q comp (indicated by
|V |
PLQL=0
smallincreaseinQcomp(indicatedbyarrows)resultsinvoltageincrease,butforthelastoperatingpoint(950),moreQcomp(say960)resultsinavoltagedecrease.S=|V2|2B*Sbase
1.2
|V2|300Mvar450Mvar675Mvar
950Mvar
with|V2|=1.0
1.0
0.8QV-curves drawnusing synchronouscondensor approach.
0.6
362004006008001000120014001600
CapacitiveMvars
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Bifurcationanalysis(ref:A.GaponovGrekhov,NonlinearitiesinactionandalsoVanCutsem&Vournas,Voltagestabilityofelectricpowersystems.)
Abifurcation,foradynamicsystem,isanacquisitionofanewqualitybythemotionthedynamicsystem,causedbysmallchangesin its parameters. A power system that has experienced a bifurcation
Considerrepresentingthedynamicsofthepowersystemas:
initsparameters.Apowersystemthathasexperiencedabifurcationwillgenerallyhavecorrespondingmotionthatisundesirable.
)(0
),,(
pyxG
pyxFx
=
=&Eqts.1
),,(0 pyxG=Adifferentialalgebraicsystem(DAS):H i bl f h ( lHerex representsstatevariablesofthesystem(e.g.,rotorangles,rotorspeed,etc),y representsthealgebraicvariables(busvoltagemagnitudes&voltageangles),andp representstherealandreactivepowerinjections
37ateachbus.ThefunctionF representsthedifferentialequationsforthegenerators,andthefunctionG representsthepowerflowequations.
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Typesofbifurcations
Thereareatleasttwotypesofbifurcation: Hopf:twoeigenvaluesbecomepurelyimaginary:abirthofoscillatoryorperiodicmotion. Saddlenode:adisappearanceofanequilibriumstate.ThestableoperatingequilibriumcoalesceswithanunstableThe stable operating equilibrium coalesces with an unstableequilibriumanddisappears.Thedynamicconsequenceofagenericsaddlenodebifurcationis:
a monotonic decline in system variablesamonotonicdeclineinsystemvariables.
Sowethinkitisthesaddlenodebifurcationthatcausesvoltage instabilityvoltageinstability.
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The Jacobian matrix of eqts 1 is
TheunreducedJacobian:
TheJacobianmatrixofeqts.1is
YX FFJ
=YX GG
J
anditisreferredtoastheunreducedJacobianoftheDAS,where
x
Jx&
Eqt 2
=
y
J0
Eqt.2
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Wemayreduceeq.2byeliminatingthevariable y
ThereducedJacobian:
=
yx
GF
GFx
Y
Y
X
X
0&
yGG YX0Thismeansweneedtoforcethetoprighthandsubmatrixto0,whichwecandobymultiplyingthebottomrowbyFYGY1 andthenaddingtothetoprow.
=
yx
GGGGFFx XYYX 0
0
1&
yGG YX0Thisresultsin: [ ] xGGFFx XYYX = 1& [ ]SothatthereducedJacobianmatrixisaSchurscomplement:
GGFFA 140
XYYX GGFFA =
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Stability:
Fact1:Theconditionsforasaddlenodebifurcationare)(F&1. Equilibrium:
2. Singularity of the unreduced Jacobian),,(0
),,(
pyxG
pyxFx
=
=
=
Y
Y
X
X
GF
GF
J2. SingularityoftheunreducedJacobian
det(J)=0(a0eigenvalue,J noninvertible). YX
Implication1:ThestabilityofanequilibriumpointoftheDASdependson
Fact2:ThedeterminantofaSchurscomplementtimesthedeterminantof
theeigenvaluesoftheunreducedJacobianJ.ThesystemwillexperienceaSNBasparameterp increaseswhenJ hasazeroeigenvalue.
Implications 2:
pGY givesthedeterminantoftheoriginalmatrix:det(J)=det(A)*det(GY)ifGY isnonsingular.
Implications2:1. IfGY isnonsingular,thensingularityofA impliessingularityofJ so
thatwemayanalyzeeigenvaluesofA toascertainstability.h f h b l l h
41
2. ThefactthatGY maybenonsingular,yetA singular,meansthatloadflowconvergenceisnotasufficientconditionforvoltagestability.
-
SingularityofloadflowJacobian:
I li i 2Implications2:1. IfGY isnonsingular,thensingularityofA impliessingularityofJ so
thatwemayanalyzeeigenvaluesofA toascertainstability.2. ThefactthatGY maybenonsingular,yetA singular,meansthat
loadflowconvergenceisnotasufficientconditionforvoltagestability
Singular SingularSingular(unstable)
stability.
NonsingularAGY J
NonsingularNonsingular(stable)
Y
42
-
SingularityofloadflowJacobian:
S l i bili l i i l l d fl J bi i ldSovoltageinstabilityanalysisusingonlyaloadflowJacobianmayyieldoptimistic resultswhencomparedtoresultsfromanalysisofA,thatis,stablepoints(basedonGy)maynotbereallystable.y=>However,IbelieveitistruethatpointsidentifiedasunstableusingtheloadflowJacobianwillbereallyunstable(Schurscomplementdoes not support that singularity of G implies singularity of JdoesnotsupportthatsingularityofGY impliessingularityofJ,however,becauseitisonlyvalidifGY isnonsingular).
SingularSingular(unstable)Nonsingular
Singular(unstable)Nonsingular
Singular(unstable)Nonsingular
Note: Sauer and Pai 1990 provide an indepth analysis of the relation
Nonsingular(stable) AGY J
g(stable) (stable)
43
Note:SauerandPai,1990,provideanin depthanalysisoftherelationbetweensingularityofGY andsingularityofJ,andshowsomespecialcasesforwhichsingularityofGY impliessingularityofJ.
-
SingularityofloadflowJacobian:
S h l d fl J bi l i idSo,weassume thatloadflowJacobiananalysisprovidesanupperboundonstability.
Fact:Thebifurcation(zeroeigenvalueofGY)oftheloadflowJacobian corresponds to the turnaround point (i e the noseJacobiancorrespondstothe turn aroundpoint (i.e.,the nose point)ofaPVorQVcurvedrawnusingapowerflowprogram.
Thiscanbeprovenusinganoptimizationapproach.Seepp.218220ofthetextbyVanCutsemandVournas.
WehavepreviouslydenotedthepowerflowequationsasG(x,y,p)=0,butnowwedenotethemasG(y,p)=0,withoutthedependenceonthe
44
statevariablesx (whichrelatetothemachinemodelingandinclude,minimally, and ofeachmachine).
-
Soweturnourefforttoidentifyingthesaddlenodebifurcation(SNB)forthepowerflowJacobianmatrix.
TheJacobiancanreachaSNBinmanyways.Forexample, increasetheimpedanceinakeytielinep y increasethegenerationlevelatageneratorwithweaktransmission,whiledecreasinggenerationatallothergenerators.
increasetheloadatasinglebus |V|g increasetheloadatallbuses.Inallcases,wearelookingforthenosepointoftheV curve,where istheparameterthatisbeingincreased.)
| |
Mostapplicationsfocusonthelastmethod(increaseloadatallbuses).Key questions here are:Keyquestionshereare: directionofincrease:arebusloadsincreasedproportionally,orinsomeotherway? dispatchpolicy:howdothegeneratorspickuptheloadincrease?
45Wewillassumeproportionalloadincreasewithgovernorloadflow(generatorspickupinproportiontotheirrating)
-
D fi iti l i t th ti diti h t i dDefine:criticalpoint theoperatingconditions,characterizedbyacertainvalueof,beyondwhichoperationisnot
acceptable.
|V|Question1:Whatcancausethecriticalpointtodiffer
p
fromtheSNBpoint?
Question 2:Question2:Howcanknowledgeofthecriticalpointprovideasecuritymeasure?
Question3:DoesthePVcurveprovideaforecast ofthesystemtrajectory?
46
-
Solutionapproachestofinding*, thevalueof correspondingtoSNB.
Approach1:Searchfor* usingsomeiterativesearchprocedure.
1.i=12. Using (i), solve power flow using NewtonRaphson.2.Using ,solvepowerflowusingNewton Raphson.
Here,weiterativelysolveG(y,p)=0.Ateachstep,wemustsolvefory intheeqt:GYy =p
3 If solved3.Ifsolved,(i+1)=(i)+ .
i=i+1goto2g
elseifnotsolved,*=(i+1)
endif4.End
Butbigproblem:as getscloseto*,GY becomesillconditioned
(close to singular) This means that at some point before the critical
47
(closetosingular).Thismeansthatatsomepointbeforethecriticalpoint,step2willnolongerbefeasible.
-
Approach2:Usethecontinuationpowerflow(CPF).
Predictor stepPredictorstep
Correctorstepp
S lNoPass*? Selectcontinuationparameter
No.
Yes.
48
Stop
-
Thepredictorstep:
)(0 pyG=
Thepowerflowequationsarefunctionsofthebusvoltagesandbusanglesandthebusinjections:
),(0 pyG=Augmentthepowerflowequationssothattheyarefunctionsof(dependenceonp iscarriedthroughthedependenceon).
),(0 yG=
N i th t )(0 VG
pp0
Nowrecognizethat
=
Vy
sothat ),,(0 VG=
If we want to compute the change in the power flow equations dGIfwewanttocomputethechangeinthepowerflowequationsdGduetosmallchangesinthevariables,V,and,
thatmoveusclosertotheloadabilitypoint as we move from one solution i to another close solution i+1 then
49
aswemovefromonesolutionitoanotherclosesolutioni+1,thendG=G((i),V(i),(i)) G((i+1),V(i+1),(i+1))=0 0=0
-
ddGdVd
VdGdd
dGdGd ++=
dVddHere,eachsetofpartialderivativesareevaluatedattheoperatingconditionscorrespondingtotheoldsolution.Ifthepowerflowequationsarelinearwiththe3sets of variables in the region between the old solution and the (close) new one the
0=++= dGdVdGddGdGd
setsofvariablesintheregionbetweentheoldsolutionandthe(close)newone,thefollowingissatisfied:
0++
dd
VdVd
dd
Gd
Eq.3 [ ] 0=
Vdd
GGG V
BUT h dd d k t th fl bl ith t ddi
q [ ] 0
dVdGGG V
BUT,wehaveaddedoneunknown,, tothepowerflowproblemwithoutaddingacorrespondingequation,i.e.,inG(,V,)=0,thereareareNequationsbutN+1variables,sothatineq.3,thematrix[G GV,G],hasNrows(thenumberofeqts
50
beingdifferentiated)andN+1columns(thenumberofvariablesforwhicheacheqtisdifferentiated). Soweneedanotherequationinordertosolvethis.Whattodo?
-
Theanswertothiscanbefoundbyidentifyinghowwewillbeusingusingthesolution to eqt. 3. Note the solution corresponding to the new point is:solutiontoeqt.3.Notethesolutioncorrespondingtothe new pointis:
+ ')(),1( dipi Here the p indicates
+
=
+
+
''
)(
)(
),1(
),1(
dVdVV
i
i
pi
piHerethe p indicatesthatthisisthepredictedpoint.
d
Ifwedefine tobethestepsize,thenwecanrewritethisas
+
=
+
+
Vd
dVV i
i
pi
pi
)(
)(
),1(
),1(
dd 'where
+
+
d
VdVVipi )()1(
=
dVd
dVd
''
51
dd
-
Wecalltheupdatevector(withthedifferentials)thetangentvector,denotedbyt.
=
dVd
t d
Thisvectorprovidesthedirection tomoveinordertofindanewsolution(i+1,p)fromtheoldone(i).We can think of this in terms of the following pictureWecanthinkofthisintermsofthefollowingpicture..
52
-
Tangentvector
|V|
53
-
Note:Inspecifyingadirectionusinganndimensionalvector,onlyn1oftheelementsareconstrained oneelementcanbechosentobeanyvaluewe
Forexample,considera2dimensionalvector.
ylike.
x2
x2=x1tan(30)so:
thedirectionisspecifiedby
Direction=30o
x
selectingx1=1,x2=0.5774, thedirectionisspecifiedbyselectingx1=0.5,x2=0.2246.
Sowecansetoneofthetangentvectorelementstol lik h h h l
x1g 1 , 2
54
anyvaluewelike,thencomputetheotherelements.Thisprovidesuswithourotherequation.
-
Supposethatwesetthekthparameterinthetangentvectortobe1.0.Thenourequationgivenaseq.3canbeaugmentedtobecome:
0
dGGG V
=
10
ddV
GeGG
k
V
where
]0...010...00[=ke
k
]0...010...00[
ke
Toselect,wewouldhave:k
55
]1...000...00[=keWhichwouldforced=1.
-
Theparameterforwhichweselectkiscalledthecontinuationparameter,anditcanbeanyloadlevel(orgroupofloadlevels),oritcanbeavoltagemagnitude.Initially,whenthesolutionisfarfromthenose,thecontinuationparameteristypically.
+
+ dd
i
i
pi
pi
)(
)(
)1(
),1(
+
=
+
+
ddVVV
i
i
pi
pi
)(
)(
),1(
),1(
tyy ipi )(),1( +=+
Theparameter iscalledthestepsize,anditcanbeselected
usingvarioustechniques.Thesimplestoftheseistojust
56
setittoaconstant.Letstrythisonoursimpleproblemformulatedatthebeginningoftheseslides.
-
HOMEWORK#2,DueMonday,Jan26.1.Usingtheequationsatthebottomofslide7,withthelefthandside(PDandQD)andalsoV1givenbytheproblemstatement,weknoweverythingexcept V2 and theta.
12212
2
1221
cos||||||
sin||||
BVVBVQ
BVVP
D
D
+=
=
exceptV2andtheta.2.Now,justbringtherighthandsideofthese2equationsovertothelefthandside,andyouhavethe2equationsthatcorrespondtoG(y,p)=0.
3.Solvetheseequationstogetthecorrespondingpowerflowsolution(butyoudonotneedNewtonRaphson todothis youcanjustusetheequationt th b tt f lid 10)atthebottomofslide10).
4.NowyouneedtoreplacethevaluespecifiedintheequationsforPD(whichis0.4accordingtotheproblemstatement)with0.4*lambda.Thisgivesyoutheequationsintheformofslide49:0=G(theta,V,lambda).Note,however,thatGisreallytwoequations:G1andG2.y q
5.Nowyouneedtoformulatetheequationsontheslide55.Thisisamatteroftakingderivativesandthenevaluatingthosederivativesatthesolutionthatyouobtainedabove.Note,however,theeachelementinthematrixofslide55actuallyrepresents2elements.Thatis:
|dG1/dtheta dG1/dV dG1/dlambda||dG2/dtheta dG2/dV dG2/dlambda||0 0 1 |
6.Evaluateeachoftheabovematrixelementsatthesolutionobtainedin#9and#10willbeexplainedinnext
step3.7.Thensolvetheseequationsforthetangentvector.Youcandothisbyinvertingtheabovematrix(usematlab oracalculatortodothis)andthenmultiplytherighthandsidebythisinvertedmatrix.
8 Then take a step using an appropriately chosen step size per the
pfewslides.
57
8.Thentakea step usinganappropriatelychosenstepsizepertheequationonslide56.
9.Beginningfromyourpredictedpointthatyouidentifiedinstep8of#2a,developequationsforapproacha,solvethem,andidentifytheresultingcorrectedpointintermsofvoltageandpower.10.Repeat#9exceptimplementapproachb.
-
Correctorstep
Note,however,thatthepredictedpointwillsatisfythepowerflowequationsonlyifthepowerflowequationsarelinear, which they are not.linear,whichtheyarenot.
Soourpointneedscorrection.Thisleadstothecorrectorstep.
Therearetwodifferentapproachesforperformingthecorrectorstep.
Approacha: Perpendicularintersectionmethod.
Approachb: Parameterizationmethod
58
-
Approacha:perpendicularintersection
Here,wefindtheintersectionbetweenthepowerflowequations(thePVcurve)andaplanethatisperpendiculartothe tangent vectorthetangentvector.
|V| ty(i)
)(0 )1( +iG (i+1)y(i+1,p)
ySolvesimultaneously,fory(i+1)
{ } 0),1()1( = ++ tyy pii),(0 )1( += iyG y(i+1)
The last equation says the inner
Thelastequationsaystheinner(dot)productof2 vectorsiszero.
59
UseNewtonRaphsontosolvetheabove(requiresonly13iterationssincewehavegoodstartingpoint).Ifnoconvergence,cutstepsize()byhalfandrepeat.
-
Approachb:ParameterizationThecorrectorstepisperformedbyidentifyingacontinuationparameter (seeslide62) canbe fixingitatthevaluefoundinthepredictorstep; thensolvingthepowerflowequations.g p q
|V| t
(i 1 )
y(i)
Solve simultaneously
y(i+1)
y(i+1,p)
0),( )1(
+ iyG
Solvesimultaneously,fory(i+1)
Verticalcorrectionscorrespondtoafixedloadcontinuationparameter,horizontal
0)1(
=
+ ikycorrectionstoafixedvoltagecontinuationparameter.
Here,yk(i+1) isthecontinuationparameter;itisthevariableyk(i+1) thatcorrespondstothekthelementdyk(i+1) inthetangentvectorandisusually atfirstbutoftenbecomessomethingelse
60
yk g y gasthenosepointisneared.Theparameter isthevaluetowhichyk isset,whichwouldbethe
valuefoundinthepredictorstep. Asinapproacha,wecansolvethisusingNewtonRaphson.Ifnoconvergence,cutstepsize()byhalfandrepeat.
-
Detectionofcriticalpoint:
Wewillknowthatwehavesurpassedthecriticalpointwhenthesignofd inthetangentvectorbecomes
negative,becauseitisatthispointwheretheloadingreachesamaximumpointandbeginstodecrease.
|V| increasing
x decreasing
61
-
Selectionofcontinuationparameter:
The continuation parameter is selected from among Thecontinuationparameterisselectedfromamong
andthestatevariablesiny accordingtotheonethatischangingthemostwith.Thiswillbetheparameterthat
Theonechangingthemostwith ismost sensitive and
hasthelargestelementinthetangentvector. relativelyunstressedconditions(farfromnose):generally
relatively stressed conditions (close to nose): generally the
mostsensitiveandrepresentsavariablethatwewanttobecarefulwithaswelookfor
relativelystressedconditions(closetonose):generallythevoltagemagnitudeoftheweakestbus,asitchangesagreatdealas ischanged,whenwearecloseto*.
anothersolution,soitmakessensetokeepitconstant.
+ dipi
)(
)(
)1(
),1(
+
=
+
+
ddVVV
i
i
pi
pi
)(
)(
),1(
),1(Typically,ykisgoingtobeoneofthese.
62
-
Selectionofcontinuationparameter(unstressedcondition):
The continuation parameter is selected from among Thecontinuationparameterisselectedfromamong
andthestatevariablesiny accordingtotheonethatischangingthemostwith.Thiswillbetheparameterthat
hasthelargestelementinthetangentvector. relativelyunstressedconditions(farfromnose):generally.
> This looks like below
|V| y(i)
=>Thislookslikebelow.
y(i+1)
y(i+1,p)
yHere, isfixed.
63
-
relativelystressedconditions(closetonose):generallythe
Selectionofcontinuationparameter(stressedcondition):
y ( ) g yvoltagemagnitudeoftheweakestbus.Here,thevoltagebeingplottedischosenasthecontinuationparameter.
|V|
(i+1 p)
y(i)
y(i+1)
y(i+1,p)
Here,|V|isfixed.
Essentially,avariableisfixedasaparameter(thevoltage),andtheparameter()istreatedasavariable.Thisprocessofselecting
64
p ( ) p g
avariabletofixissometimescalledtheparameterizationstep.ScottGreene,Ph.D.dissertation,1998.
-
Acentralquestion:
Howdoesthecontinuationtechniquealleviatetheillconditioning problem experienced by a regular power flow ?conditioningproblemexperiencedbyaregularpowerflow?
Refer to the solutions procedures for the two corrector approaches.Refertothesolutionsproceduresforthetwocorrectorapproaches.PerpendicularinteresectionSolvesimultaneously,for y(i+1)
ParameterizationSolvesimultaneously,for y(i+1)
),(0 )1( += iyG0
)( )1(
+ iyG
fory fory
{ } 0),1()1( = ++ tyy pii 0)1( =
+ iky
I b th N t R h t l d t bt i th
65
Inbothcases,weuseNewtonRaphsontosolve,soweneedtoobtaintheJacobian.ButtheJacobianisslightlydifferentthaninnormalpowerflow.
-
TheJacobianofthepowerflowequationsisjustGy,buttheJacobian of the equations in the two corrector approachesJacobianoftheequationsinthetwocorrectorapproacheswillhaveanextrarowandcolumn.
GG
k
k
x
x
y
y
C
G
C
G
Here,Cistheadditionalequation,andxk istheselectedcontinuationparameter.
ThisadditionofarowandcolumntotheJacobianhastheff f i i h di i i h h i leffectofimprovingtheconditioningsothatthepreviouslysingularpointscaninfactbeobtained.Inotherwords,theadditionalrowandcolumnprovidesthatthisJacobianis
i l * h h d d J bi i i l
66
nonsingularat* wherethestandardJacobianissingular.
-
Knowncodesforcontinuationmethods:
1. ClaudioCanizarresatUniversityofWaterloo:CcodeSeehttp://www.power.uwaterloo.ca/~claudio/claudio.htmlUWPFLOWisaresearchtoolthathasbeendesignedtocalculatelocalbifurcationsrelatedtosystemlimitsorsingularitiesinthesystemJacobian.Theprogramalsogeneratesaseriesofoutputfilesthatallowfurtheranalyses,suchastangentvectors,leftandrighteigenvectorsatasingularbifurcationpoint,Jacobians,powerflowsolutionsatdifferentloadinglevels,voltagestabilityindices,etc
2. IhaveMatlabcodethatdoesit fromScottGreene.3. VenkataramanaAjjarapu(ISU):Fortrancode4. Powertechhasaprogram
67
-
Calculationofsensitivitiesforvoltageinstabilityanalysis
Whatisasensitivity?
Itisthederivativeofanequationwithrespecttoavariable.Itshowshowparameter1changeswithparameter2.
Itis:exactwhenparameter2dependslinearlyonparameter1.Itisapproximatewhenparameter2dependsnonlinearlyonparameter1,
butitisquiteaccurateifitisonlyusedclosetowhereitiscalculated.
68
-
ConsiderthesystemcharacterizedbyG(y).Thenis the sensitivity of the equation G with respect to y
*yyG
isthesensitivityoftheequationGwithrespecttoy,evaluatedaty*.
G(y) SlopeisG/yevaluatedaty*.
y
y*yy
yy
Itsusefulnessisthatonceitiscalculated,itcanbeusedtoQUICKLYevaluatef(y)fromG(y)G(y*)+(G/y|y*)y,
69
BUTONLYASLONGASyISCLOSETOy*.
-
Consider parameter p: we desire to obtain the sensitivity ofConsiderparameterp:wedesiretoobtainthesensitivityofG(y,p)top.Typicalparameterspwouldbeabusload,abuspowerfactor,oragenerationlevel.
Veryimportanttodistinguishbetween voltagesensitivities
voltageinstabilitysensitivities
Whatisthedifferencebetweenthemintermsof whattheymean? how to compute them ? howtocomputethem?
70
-
Sensitivities for bus voltageSensitivitiesforbusvoltage
Thesewecomputeatthecurrentoperatingcondition.
Foragivencontinuationparameter,theycanbeobtainedfrom the first predictor step in the continuation power flowfromthefirst predictorstepinthecontinuationpowerflow.
|V|
dRecallthatthisprovidesuswith
=d
dVtthetangentvector,givenby:
Th t t t i th t f
Thetangentvectoristhevectorofsensitivitieswithrespecttoasmallchangein,sotheportionofthevector
71
Currentoperatingpoint
designatedasdV isexactlythevoltagesensitivities.
-
Sensitivitiesforvoltageinstability
Here it is important to realize that the measure of voltage instabilityHere,itisimportanttorealizethatthemeasureofvoltageinstability,theloadingmargin,dependsonanoperatingcondition
differentfromthepresentoperatingcondition.
Theimplicationisthatwemustlookatsensitivitiesof the loading margin not of the voltageoftheloadingmargin,notofthevoltage.
|V|
Sowewantthesensitivitiesevaluated at this point, i.e.,
Loadingmargin
evaluatedatthispoint,i.e.,theSNBpoint.
72
Currentoperatingpoint
-
DerivationofloadingmarginsensitivitiesatSNBpoint.
LetS bethevectorofrealandreactiveloadpowers,d k b h di i f l d iandk bethedirectionofloadincrease.
kSS 0 +=Also,defineLastheloadingmargin(ascalar),sothattheloadpowersresultingintheSNBpointaregivenby:
kSS L0 +=
WedesiretofindthesensitivityoftheloadingmarginLtoah i th t W d t thi iti it b L
0
73
changeintheparameterp.WedenotethissensitivitybyLp.
-
ConsiderthesystemcharacterizedbyWewantthesensitivityofthe loading margin to p.
G(y,S,p)=0
Assumption:thesystemhasaSNBat(y*,S*,p*),i.e.,:
theloadingmargintop.
p y (y , , p ), ,
1.G(y*,S*,p*)=0 (anequilibriumpoint)
2.Gy(y*,S*,p*)issingular(zeroeigenvalue),andw isalefteigenvectorofGy(y*,S*,p*),correspondingtothezeroeigenvaluesothat(bydefinitionofthelefteigenvector)
wT Gy(y*,S*,p*)=0wT=0y(y , , p )NotethatGy(y*,S*,p*),beingsingular,cannotbeinverted,butwecancomputeit(thatis,Gy(y*,S*,p*)),anditseigenvectors.
74
3.wT GS(y*,S*,p*) 0
-
Thepoints(y,S,p)satisfyingnumbers1and2correspondtoSNBpoints,
andwecanobtainacurveofsuchpointsbyvaryingpaboutitsnominalvaluep*.
LinearizationofthiscurveabouttheSNBpointresultsin
0++ GSGG 0***
=++ pGSGyG pSywherethenotation|* indicatesthederivativesareevaluatedattheSNBpoint.
P lti li ti b th l ft i t lt iPremultiplicationbythelefteigenvectorw resultsin:
0***
=++ pGwSGwyGw pT
ST
yT
75By#2onthepreviousslide,thefirsttermintheaboveiszero.So...
-
0=+ pGwSGw TST Eqt.*0
**+ pGwSGw pS
Nowrecalltherelationoftheloadpowerstotheloadingmargin.
kLSkSS
=+= L0
kLS =Substitutingthisexpressionfortheloadpowersintoeqt.*,
pGwkGwLpGwkLGw pTT
pTT ==+
****0
And the loading margin sensitivity to parameter p is:
GwLL pT
*
=
=
Andtheloadingmarginsensitivitytoparameterpis:Sopmaybe,forexample,realpowerloadatabus(todetect the most effective load
76
kGwpL
STp
*
*=
= detectthemosteffectiveload
shedding)orreactivepoweratabus(todeterminewheretositeashuntcap).
-
SomecommentsaboutcomputingLpGw
LT
pT
p*
*
=
ThelefteigenvectorwmustbecomputedfortheJacobian G evaluated at the SNB point
kGw STp
*
*
JacobianGyevaluatedattheSNBpoint.
Youonlyneedtocomputew andGS once,independentofhow many sensitivities you need. Methods to compute the left eigenvectorhowmanysensitivitiesyouneed.Methodstocomputethelefteigenvectorw includeQRorinverseiteration.
Thevectorofderivativeswithrespecttotheparameterp,whichisGp, isp,typicallysparse.Forexample,ifyouwanttocomputethesensitivitytoabuspower,thentherewouldbeonly1nonzeroentryinGp.
Thematrixofderivativeswithrespecttotheloadpowers,GS,usingconstantpowerloadmodels,isadiagonalmatrixwithonesintherowscorrespondingtoloadbuses.ThisisbecauseaparticularloadvariablewouldONLYoccuri th ti di t th b h it i l t d d f th
77
intheequationcorrespondingtothebuswhereitislocated,andfortheseequations,thesevariablesappearlinearlywith1ascoefficient.
-
Some comments about extensionsSomecommentsaboutextensions
MultiplesensitivitiesmaybecomputedusingGp (amatrix)insteadofGp (avector).Inthiscase,theresultisavector.
kGw
GwL T
pT
p*
*
=
kGw S * Gettingmultiplesensitivitiescanbeespeciallyattractivewhenwewanttofindthesensitivitytoseveralsimultaneouschanges.Onegoodexampleistofindthesensitivitytochangesinmultipleloads.
AspecialcaseofthisistofindthesensitivitytochangesatALLloads,whichisl l l k h
A iti it t li t b bt i d b l tti t i l t
verytypical,givenaparticularloadingdirectionk.Then
=i
pi iLkL
*loads all
78
Asensitivitytoalineoutagemaybeobtainedbylettingp containelementscorrespondingtotheoutagedlineparameters.
-
Somecommentsaboutextensions
A iti it t li t b bt i d b l tti t i l t Asensitivitytoalineoutagemaybeobtainedbylettingp containelementscorrespondingtotheoutagedlineparameters:R(seriesconductance),X(seriesreactance),andB(linecharging).Thenusethemultipleparameterapproach.
kGw
GwL
T
pT
p*
*
=
Zpq=R+jX
kGw S *
pLL p = *
p q
jB jBpp * Here,p =[RXB]T.
Notethatp isNOTSMALL!ThereforeLmayhaveconsiderableerror.
Forthatreason,thisoneneedstobecarefulaboutusingthisapproachtocomputetheactualloadingmarginsfollowingcontingencies.
79
However,itcertainlycanbeusedforRANKINGcontingencies.Onemightconsiderhavingaquickapproximationandalongexactriskcalculation.
-
Some comments about alternatives
Greene,etal.,alsoproposeaquadraticsensitivity whichrequires calculation of a second order term L This is used
Somecommentsaboutalternatives
requirescalculationofasecondordertermLpp .Thisisusedtogetherwiththelinearsensitivityaccordingto
2)(1 LLL 2*
*)(
2pLpLL ppp +=
Itrequiressignificantlymorecomputationbutcanprovidegreateraccuracy over a larger range of paccuracyoveralargerrangeofp.
InvariantSubspaceParameticSensitivity(ISPS)byAjjarapu.Advantages:
basedondifferentialalgebraicmodel
80
g providessensitivitiesatANYpointonthePVcurve
- voltageInrush