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EE742

Chap.5:ElectromechanicalDynamics

Y. Baghzouz

Fall2011

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introduction Inthischapter,alongertimescaleisconsideredduring

whichtherotorspeedwillvary(orderofseconds 10s

ofseconds transientperiod).

Thechangeinrotorspeedinteractswiththeelectro

magneticchangestoproduceelectromechanical

dynamiceffects.

Someimportantstabilityconceptswillbeintroduced

mathematicallywithphysicalimplications.

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Swingequation

Rotordynamicequation(NewtonsLawonmotion):

Atsteadystate, m= sm,and

Rotorspeed:

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SwingEquation

Aftersubstitution,

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SwingEquation

Inertiaconstant:

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DampingPower

Assumptions:

Generatorequivalentcircuitresemblesthatofaninductionmotor:

Whenignoringrotorsaliency,

where

and

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Dampingpower

Withrotorsaliency,thefollowingformulaisderivedwhenusing dqaxis

decomposition notedependencyonrotorangle.

Forsmallspeeddeviations,theaboveexpressioncanbeapproximatedby

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Dampingpower

Forlargespeeddeviationvalues,itisconvenienttorewritePDas

Criticalspeeddeviationineachaxis:

Criticaldampingpowerineachaxis:

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Equilibriumpoints

Recallthegeneratorpoweranglecharacteristicatsteadystate(chap.3):

Forsalientpole:

Forroundrotor:

Theswingequationcanberewrittenas:

Atequilibrium,

Hence, Usingpoweranglecurveforsimplicity:

NoequilibriumpointwhenPm>criticalpower

oneequilibriumpointwhenPm=criticalpower

twoequilibriumpointswhenPm

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Steadystatestabilityofunregulatedgenerator

Wefirstignorethecontrolsofthegeneratorandturbine(i.e., the

mechanicalpowerandexcitationvoltageareconstant).

Smalldisturbanceorsmallsignalstability:issystemissaidtobe

steadystatestableforaspecificoperatingconditionif,followinga

smalldisturbance,itreachesasteadystateoperatingpointator

closetothepredisturbancecondition.

Herein,thepowersystemmaybelinearizedneartheoperatingpoint

foranalytical

purposes.

Thegeneratorinfinitebusbarsystemisstableonlyinthelefthand

sideofthepoweranglecurve

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Steadystatestabilityofunregulatedgenerator

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Transientpoweranglecharacteristic

Rotoroscillationsoccurinthesametimescaleasthetransient

period generatormodelduringtransientstate: Forgeneratormodelwithconstantfluxlinkage,seeFig.below(Chap.4)

Round

rotorSalient-pole

rotor

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Transientpoweranglecharacteristic

Constantfluxlinkagemodel:

Forageneratorwithasalientpolerotor,

theabovetransientpowerexpressionsimplifiesto

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Transientpoweranglecharacteristic

Classical

generator

model: theconstantfluxlinkagemodelcanbesimplifiedfurtherbyignoringthetransientsaliency,i.e.,assuming

that Thetransientpowerequationbecomesequalto

Notethat

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Examples5.1

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Examples5.2

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Impactofincreaseinload

Theincreaseinload modifiesthetransientcharacteristicasshownbelow(resultsinsmallertransientemfE,hencesmallerpeakvalue

ofthetransientpowercurve).

Thetransientsynchronizingpowercoefficient

issteeperthanitssteadystatecounterpart

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Effectofdamperwinding

Forsmalldeviationsinspeed,thedamperwindingproducesdampingpowerthatisproportionaltospeeddeviationandaddsuptothe airgap

power.ThesignofPDdependsonthesignof .

Therotorwillthereforemovealongamodifiedpowerangletrajectory.

Startingfrompoint2,therotor

beginsto

decelerate

and

reaches

minimumspeedatpoint6.

Pastpoint6,therotoraccelerates

andreachessychronous speed

whenarea246=area635

Therotoroscillationsaredamped

andthesystemquicklyreaches

theequilibriumpoint1.

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Effectofrotorfluxlinkagevariation

Inreality,E isnotconstant (asthearmaturefluxenterstherotorwinding,therotorfluxlinkagechangeswithtime).

Tosimplifytheanalysis,onlythesalientpolemachineisconsidered

(i.e., ))

Linearizethetransientpowerequationattheinitialoperatingpoint:

where Eq canbe inphasewiththespeeddeviation inthiscase,it

inducesanadditionaldampingtorque,(seefirstfigurebelow).

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Effectofrotorfluxlinkagevariation

Updatednecessaryconditionforsteadystatestability(i.e., Eq tobeinphasewiththespeeddeviation ):

Incaseoflargenetworkresistance, Eq canbe180o outofphasewith

thespeeddeviation, generatorinstability canoccurifthisnegative

dampingislargerthatthepositivedampingproducedbythedamper

winding(seesecondfigurebelow).

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Rotorswingsaroundtheequilibriumpoint

Linearizedswingequation:

Rootsofcharacteristicequation:

Threepossiblesolutions:underdamped,criticallydampedandover

damped.

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Mechanicalanaloguesofgeneratorinfinitebus

barsystem Dynamicequationofmassspringdampersystem

Dynamicequationofpendulum(notincludingdamping)

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Steadystatestabilityofregulatedsystem

(whenaction

of

AVR

is

included)

Studyrestrictedtogeneratorwithroundrotorandnegligible

resistance.

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SolveforEqintermsofVs,Vg,and :

Substituteinthepowerequation:

The AVR increases the amplitude

of the power curve significantly

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GeneratorpowerisproportionaltoEqb.

Maximumpoweroccurswhen

Dashed curves 1-6 represent

PEq

() for higher values of Eq.

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TheslowactingAVR(onewithalargetimeconstant)willnotbeable

torespondduringthetransientperiod,hencethestabilitylimitcorrespondsto = /2.

For afastactingAVR(withshorttimeconstant),thestabilitylimit

correspondsto > /2.ThisvaluedependsonthesystemandAVR

parameters(i.e.,conditional

stability).

Theinfluenceoffieldcurrentlimiter isillustratedinthefigurebelow.

Thefieldcurrentlimitisreachedbefore PVgMAX ifXissmall.

Belowthelimitingpoint,thegeneratorsteadystatecharacteristicfollowsPVg.

Abovethelimitingpoint,thegeneratorsteadystatecharacteristicfollowsPEq.

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Transientpoweranglecharacteristicsofregulated

generator

WeconsiderAVRwithlargetimeconstant.Inhere,thetransient

characteristicsisthesameastheunregulatedsystemexcept,

thevalueofEq ishigher(hencehigheramplitudeofPE())sincetheincreased

loadingin

the

regulated

system

causes

an

increase

in

the

steady

state

field

current.

Inaddition,theangle reaches/2beforereacheditscriticalvalue.

Notethat whenthe reacheditscriticalvalue, > /2,henceKE is

negative.SoatwhichpointonthePVg()curvewhenKE=0?

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UsingclassicalmodelandphasordiagraminFig.20b,

RatioofpoweratwhichKE=0tothepoweratwhichKVg=0:

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isstronglydependentonthesystemreactanceX.

Refertothefiguresbelow:

AtoperatingpointA(lightload),0

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Effectorrotorfluxvariation

Determinethephaseshiftbetweenspeedchangeand (seeSection5.5.3fordetails)

ThephasorsbelowshowrepresenttwotypesofAVRs:

ForstaticexciterEfisinphasewithV

Forrotatingmachineexciter,EflagsVby10s ofdegrees. theoutofphasecomponent relativeto reducesdamping(i.e.,

increaseschancesofinstability).

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