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.

1

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

2

nolowvoltagedoesnotnecessarilyimplyvoltagestability voltageinstabilitydoesnecessarilyimplylowvoltage

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

3

Ourtreatmentofvoltagesecuritywillproceedasfollows:

Voltageinstabilityinasimplesystem Voltageinstabilityinalargesystem Brieftreatmentofbifurcationanalysis Continuationpowerflow(pathfollowing)methods Sensitivity methodsSensitivitymethods

4

Voltageinstabilityinasimplesystem

Considertheperphaseequivalentofaverysimplethreephasepowersystemgivenbelow:

V1 V2

Z=R+jX

INode1 Node2

V1 V2

++

1

__

5S12 SD=S12

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

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

+=

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.

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..

9

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

41

212

2 ||||||||

+= VPPVPVV DDD

10

12 ||42||

BBB

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];

11

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)

Plotsofthepreviousequationfordifferentpowerfactors

|V ||V2|

R l l di P

12

Realpowerloading,PD

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

13

theleadingcondition,voltageinstabilityisnotverydetectable.Theleadingconditionoccursduringhightransferconditionswhentheloadislightorwhentheloadishighlycompensated.

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);

14

plot(qd1,v2);

Thecurveonthenextpageillustrates.

QVCurve

|V ||V2|

15QD

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

16

reactance.

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.)

17

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.

18

Implication:TheSYSTEMcanhaveavarsurplusbutexperiencevoltageinstabilityifalocalareahasavardeficiency.

Loadmodeling

Inanalyzingvoltageinstability,itisnecessarytoconsiderthenetworkundervariousvoltageprofiles.

Voltagestabilitydependsonthelevelofcurrentdrawnbytheloads.

Thelevelofcurrentdrawnbytheloadscandependonthevoltageseenbytheloads.

Therefore,voltageinstabilityanalysisrequiresamodelofhowtheloadrespondstoloadvariations.

Thus,loadmodelingisveryinfluentialinvoltageinstabilityanalysis.

19

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

20

Conventionalflorescent 2.07 3.21

PolynomialloadmodelTheZIPorpolynomialmodelisaspecialcaseofthemoregeneral

i l d l i b f 3 i l d l i hexponentialmodel,givenbyasumof3exponentialmodelswithspecifiedsubscripts:

22

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

21

constantimpedancecomponent(p1,q1) lighting constantcurrentcomponent(p2,q2) motor/lighting constantpowercomponent(p3,,q3) loadsservedbyLTCs

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

22

Constantpowerload(p3)isBADsincedemanddoesnotchangeasvoltagedeclines.

The effects of voltage variation on loads and thus of loads on

Someconsiderationsinloadmodeling

Theeffectsofvoltagevariationonloads,andthusofloadsonvoltageinstability,cannotbefullycapturedusingexponentialorpolynomialloadmodelsbecauseofthefollowingthreeaspects.

Thermostaticloadrecovery Inductionmotorstalling/trippingg pp g Loadtapchangers

23

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

24

andittendstoexacerbatevoltageproblemsatthehighvoltagelevel.

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

25

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.

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

27

Underlowvoltageconditionsatthehighside,theLTCwilldecreasetinordertotryandincreaseV1/t.

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.

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

29

andduetostatorendregionheatingfrominducededdycurrents,enforcedbyunderexcitationlimiter(UEL).

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

30

manifestitselfasasharpdiscontinuityinthecurve.

Overloadtime(sec)

Irated1.0

120

10

EffectofOXLactiononPVcurve:

|V|

Onegeneratorhitsreactivelimit

Noreactivelimitsmodeledo

P

(demand)

Note:GeorgiaPowerCo.modelsitsloadabilitylimitatpointx,notpointo.

31

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

32

inthenetwork.Thiseffectiscompoundedbythefactthatlosingacircuitalsomeanslosingitslinechargingcapacitance.

Kundur,onpp.979990,hasanexcellentexamplewhichillustratesmanyoftheaforementionedeffects.Theillustrationwasdoneusingalongtermtimedomainsimulationprogram(Eurostag).

33

Influence of switched shunt capacitorsInfluenceofswitchedshuntcapacitors

I II

P

I

PP P

|V|

WithcapacitorWithoutit

P

capacitor

34

P

(demand)

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.

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

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.

Typesofbifurcations

Thereareatleasttwotypesofbifurcation: Hopf:twoeigenvaluesbecomepurelyimaginary:abirthofoscillatoryorperiodicmotion. Saddlenode:adisappearanceofanequilibriumstate.ThestableoperatingequilibriumcoalesceswithanunstableThe stable operating equilibrium coalesces with an unstableequilibriumanddisappears.Thedynamicconsequenceofagenericsaddlenodebifurcationis:

a monotonic decline in system variablesamonotonicdeclineinsystemvariables.

Sowethinkitisthesaddlenodebifurcationthatcausesvoltage instabilityvoltageinstability.

38

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

39

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 =

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