POWER SYSTEM DYNAMICS AND STABLITY CHAP1

Dynamics and Stability

1. Power System Dynamics and Stability
Ch. 1-Introduction:
Synchronous Machines Rotor Dynamics
Swing Equation
Power Flow /Transfer and Rotor angle
by
Dr. Wondwossen Astatike Haile
BSc (BEET), MSc (Power Engineering), PhD (Electrical and Electronics Engineering)
Department of Electrical Power Engineering
College of Engineering, Defense University
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2. Chapter1. Introduction to power system stability problem
Powersystemdynamicsisalsocalledpowersystemstabilityproblem.
Modernpowersystemsareverywidelyinterconnected.
Interconnectionresultsinoperatingeconomy,increasedreliability,and
mutualassistanceofdifferentsystems.
Inthemeantime,interconnectionwillalsocontributetothestabilityproblem.
Duetothisstabilityproblembecomeanimportantconcernforpowersystem
engineersinaninterconnectedsystem.
Methodologyforpowersystemstabilityproblemanalysisaremodellingofthe
system,andoncethemathematicalmodelofapowersystemisdeveloped,oneis
toobtainthesolutionthroughnumericaltechniques.
Developmentofmathematicalmodelofpowersystemincludesmathematical
modelforsynchronousmachines,excitationsystems,voltageregulator,governor
andloads.
Powersystemstabilitymaybedefinedasthatpropertyofapowersystemthat
enablesittoremaininstateofoperatingequilibriumundernormal
conditionandtoregainanacceptablestateofequilibriumafterbeing
subjectedtoadisturbance.Thatis,
Abilitytoremaininoperatingequilibrium
Equilibriumbetweenopposingforces
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3. Chapter1. Introduction to power system stability problem
Classificationofpowersystemstabilityare:
1.Anglestability
2.Voltagestability
1.Anglestability:furtherclassifiedinto
Smallsignalstability
Transientstability
Mid-termstability
Long-termstability
2.Voltagestability:alsoclassifiedinto
Largedisturbancevoltagestability
Smalldisturbancevoltagestability
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4. Chapter1. Introduction to power system stability problem
Rotoranglestability
Itistheabilityofinterconnectedsynchronousmachinesofapowersystem
toremaininsynchronism.i.e.abilitytomaintainsynchronismandtorque
balanceofsynchronousmachines.
Toanalysepowersystemstabilitywehavetounderstandthedynamicsofthe
rotoranddevelopamathematicalequationstodescribethedynamicsofthe
rotor.
SynchronousMachinesRotordynamicsandtheswingequation
Theequationgoverningthemotionoftherotorofasynchronousmachineis
basedontheelementaryprincipleindynamics.
Itstatesthatanacceleratingtorqueistheproductofthemomentofinertiaand
angularacceleration.
Thisisafundamentallawonwhichtheswingequationisbasedon.
Asynchronousmachinemayoperateeitherasasynchronousgeneratororasa
synchronousmotor.
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5. Chapter1. Introduction to power system stability problem
Fig. (a) and (b)representation of a machine rotor comparing direction of rotation of
mechanical and electrical torques
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6. Chapter1. Introduction to power system stability problem
Where,
Tm and Teoperate on the rotor in opposite direction
The mechanical torque, Tm is provided by the prime mover
The electrical torque, Teis developed by the interaction of magnetic field and
stator currents.
From the above diagrams,
Therotorrotatesinthedirectionofthemechanicaltorqueinthecaseof
generatorandinthedirectionofelectricaltorqueinthecaseofmotor.
Understeadyoperatingconditionthesetwotorquesareequalandtherotorofthe
synchronousmachinerotateswithsynchronousspeed.
However,whendisturbancesoccurthereexistsunequilibriumbetweenthe
twotorquesandthetwotorquesarenotequalandthedifferenceiscalled
acceleratingtorque
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7. Chapter1. Introduction to power system stability problem
The swing equation
A differential equation can be written relating the accelerating torque, moment of
inertia and acceleration. That is,
6
6
•In mkssystem of units,
•J= the total moment of inertia in Kg-m
2
•m= angular displacement of rotor with respect to a stationary axis in mechanical
radians
•t=time in seconds
•Tm= mechanical or shaft torque supplied by prime mover less retarding torque
due to rotational losses in N-m
•Te= the net electrical torque or electromagnetic torque in N-m
•Ta= the net accelerating torque in N-m
It is convenient to measure the rotor angular position with respect to reference
axis which rotates at synchronous speed. Therefore, we define,
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8. Chapter1. Introduction to power system stability problem
(2)
•sm= synchronous speed of the machine in mechanical radians/sec
•m= the angular displacement of rotor in mechanical radians from the
synchronously rotating reference axis.
The derivatives of eq.2 with respect to time are
(3)
And taking the second derivatives of eq.3 gives us,
.
.
.
.
(4)
Substituting eq.4 into eq.1, we obtain,
??
.
?
??
.
(5)
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9. Chapter1. Introduction to power system stability problem
Inpowersystemstudieswearemorecomfortablewiththetermsinpowerlike
watts,kilowattsandmegawatts.
Inthisregardmultiplyingeq.5bymwhichisdenotedbyeq.6yieldseq.7,
??
??
(6)
?
.
?
??
.
(7)
•Where,
•mTm=Pm
•mTe=Pe
•Pm= shaft power input to the machine less rotational losses
•Pe= the electrical power crossing the air gap
Eq.7 may be written as
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10. Chapter1. Introduction to power system stability problem
?
.
?
??
.
(8)
•Where,
•J= moment of inertia in Kg/m
2
•m= speed in rad/s
•Jm= is called angular momentum
Inpracticalcondition,therotorspeedmisnearlyequaltothesynchronousspeed.
Thedifferencebecomelargeonlywhenthemachinelossessynchronism.
Forthepurposeofsimplicitym=sm.Then,thecoefficientJmistheangular
momentumoftherotor,atsynchronousspeedsmitisdenotedbyManditis
calledinertiaconstantofthemachine.
?
.
?
??
.
(9)
•Where,
M=Jm
The M term varies over a wide range depending on the type of machine. That is
whether a synchronous generator or turbo generator. In that case it demands to
define another Inertia constant, H as follows:
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11. Chapter1. Introduction to power system stability problem
Inertia constant, H
H is defined by
6
-
.
? ??
?????
{MJ/MVA} (10)
Where,
•Smach= the three phase rating of the machine in MVA
Solving for M, from equation 10 , we get,
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12. Chapter1. Introduction to power system stability problem
6?
??
(11)
Substituting for M in eq.9, we find
6?
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?
.
?
??
.
????
?????
(12)
OR
6?
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?
.
?
??
.
(13)
Where,
2I
5I=?D
−LANQJEP IA?D=JE?=H LKSAN
2A
5I=?D
−LANQJEP AHA?PNE?=H LKSAN
Theperunitsystemofcalculationisveryconvenientinpowersystemanalysis.
Generally,Pm(pu)&Pe(pu)arerepresentedasPm&Peonlyforsimplicity.
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13. Chapter1. Introduction to power system stability problem
So that Eq.13 becomes,
6?
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?
.
?
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.
(14)
Finally, eq.14 is rewritten as,
6?
?
?
.

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.
(15)
Eq.15iscalledtheswingequationforsynchronousmachine.
Itisapplicablebothforgeneratorandmotor.
Theonlydifferenceisincaseofmotor
Pmbecomenegative
and
Pe
becomenegative
andtheequationbecome
6?
?
?
.

??
.
(16)
ForasystemwithanelectricalfrequencyfHz,eq.15isrewrittenas
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14. Chapter1. Introduction to power system stability problem
?
?
?
.

??
.
(17)
Where,
•isinelectricalradians
Forinelectricaldegreeseq.17willberewrittenas
?
5<4?
?
.

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.
(18)
Theswingequationiscalledsecondordernon-lineardifferentialequation.
Thesolutionoftheswingequationiscalledaswingcurve.
Inanalysis,thesolutionofthesecondorderdifferentialequationcanbe
obtainedbywritingitintotwofirstorderdifferentialequationsrepresentedby
thefollowingtwoequationsin(19)&(20):
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15. Chapter1. Introduction to power system stability problem
6?
?
?
??
(19)
?
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(20)
Whentheswingequationissolvedweobtaintheexpressionforasafunction
oftime.
Thegraphofthesolutioniscalledswingcurveofthemachineandinspectionof
theswingcurvesofallthemachinesofthesystemwillshowwhetherthe
machinesremaininsynchronismafteradisturbanceornot.
Inamultimachinesystem,theoutputandhencetheacceleratingpowerofeach
machinedependupontheangularposition–and,tobemorerigorous,alsoupon
theangularspeeds-ofallthemachineofthesystem.
Thus,fora3-machinesystemtherearethreesimultaneousdifferentialequations.
Thatis,
?5?
.
5
??
.
?5
??
?6
??
?7
??
(21)
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16. Chapter1. Introduction to power system stability problem
?6?
.
6
??
.
?5
??
?6
??
?7
??
(22)
?7?
.
7
??
.
?5
??
?6
??
?7
??
(23)
Thesystemconsideredinequations21to23isindynamiccondition.
Whenthesystemrotorisindynamiccondition,itdevelopssome damping
torque.
Thedampingtorqueisproportionaltothespeeddeviationwithrespecttothe
synchronouslyrotatingfield.
Tosimplifytheanalysis,usuallythedampingtorqueisignoredandthefinal
equationswilltakethefollowingfinalforms:
?5?
.
5
??
.
(24)
?6?
.
6
??
.
(25)
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17. Chapter1. Introduction to power system stability problem
?7?
.
7
??
.
(26)
Power versus angle relationships
Animportantcharacteristicofpowersystemstabilityistherelationshipbetween
interchangepowerandpositionsofrotorsofsynchronousmachines.
Thisrelationshipishighlynonlinear.
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Th is imag e c an n o t c u r r en tly b e d isp lay ed .

18. Chapter1. Introduction to power system stability problem
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19. Chapter1. Introduction to power system stability problem
IfweplotpowerangleandpowerPorelectricalpowerPeitlookslike:
Fig. power versus angle curve
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?
(27)
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20. Chapter1. Introduction to power system stability problem
Onsamepoweranglecurvediagramoffigureabove,ifwedrawthe
mechanicalinputline,themechanicalinputisnotfunctionof.
Therefore,itcomesouttobealineparalleltoline/axis.
Forthesystemoperatingatpoint“a”andifitisperturb,thenit
developtheforcesandreturnbacktotheoperatingpoint“a”.
However,ifthesystemismadetooperateatpoint“b”,whichisalso
anequilibriumpoint,andperturb,thesystemwillloseitsstability.
Itwillnotdeveloprestoringforcestoreturnitbacktopoint“b”.
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21. Seminar Topics/Titles
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Group 1
LightningPhenomena,its
impactonstableoperationof
powersystemandcontrol
mechanisms
Group 2
Role/functionsofFACTSdevicesand
differentcompensationdevicesfor
stableoperationofpowersystem
Group 3
Modellingaspectsofsomeof
thecommonlyused
compensationdevices
Group 4
Latestdevelopmentinpowersystem
dynamicsandstability.
TrytoseetheimpactofDistributed
Generationsystemsinthestable
operationofpowersystemand
possibleremediesforensuring
stability.

22. END OF
Ch. 1-Introduction to power system stability
problem

Thank you
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