ISSN1100-1607
Control Lyapunov Functions:
A Control Strategy for Damping of Power
Oscillations in Large Power Systems
Mehrdad Ghandhari
Stockholm2000
DoctoralDissertation
Royal Institute of Technology
Dept. of Electric PowerEngineering
Electric Power Systems
In thepresent climate of deregulation and privatisation, the utilitiesare
oftenseparatedintogeneration, transmissionanddistributioncompanies
so as to help promote economic eÆciency and encourage competition.
Also, environmental concerns, right-of-way and cost problems have de-
layedtheconstruction of bothgenerationfacilitiesand new transmission
lines whilethe demand for electric power has continued to grow, which
mustbemetbyincreasedloadingofavailablelines. Aconsequenceisthat
power system damping is often reduced which leads to a poordamping
of electromechanical power oscillations and/or impairment of transient
stability.
The aim of this thesis is to examine the ability of Controllable Series
Devices (CSDs),such as
Uni edPower FlowController(UPFC)
ControllableSeries Capacitor(CSC)
QuadratureBoosting Transformer (QBT)
forimprovingtransient stabilityand dampingofelectromechanical oscil-
lationsina powersystem.
Forthesedevices, ageneralmodelisusedinpowersystemanalysis. This
model is referred to as injection model which is valid for load ow and
angle stabilityanalysis. The model isalso helpfulfor understandingthe
impact ofthe CSDson powersystem stability.
A control strategy for damping of electromechanical power oscillations
is also derived based on Lyapunov theory. Lyapunov theory deals with
dynamical systems without input. For this reason, it has traditionally
been applied only to closed{loop control systems, that is, systems for
which the input has been eliminated through the substitution of a pre-
determined feedbackcontrol. However, inthisthesis,Lyapunov function
candidates are used in feedback design itself by making the Lyapunov
derivative negative when choosing the control. This control strategy is
calledControlLyapunovFunction(CLF) forsystems withcontrolinput.
Keywords: Controllable Series Devices (CSDs), Uni ed Power Flow
Controller (UPFC), Quadrature Boosting Transformer (QBT), Control-
lableSeriesCapacitor(CSC),Lyapunovfunction,ControlLyapunovFunc-
tion (CLF),SIngle Machine Equivalent(SIME), VariableStructureCon-
trol(VSC).
TRITA-EES{0004ISSN1100-1607
First of all, I would like to express my deepest gratitude and apprecia-
tion to my supervisor, Professor Goran Andersson, for his support and
guidancethroughout thisproject.
I would like to extend my warmest thanks to Dr. Ian A. Hiskens for
his constant support, inspiringdiscussions and valuable suggestions, es-
peciallyduringmyvisitat theUniversityof Newcastle, Australia.
I gratefully acknowledge numerous useful comments by the members of
the project steering committee, namely, Mojtaba Noroozian, Lennart
Angquist, Bertil Berggren of ABB and Magnus Danielsson of Svenska
Kraftnat. Also, nancialsupportfromthesecompaniesthroughtheElek-
traprogram is gratefullyacknowledged.
Manythanks to thesta of ElectricPowerSystems forproviding stimu-
latingandfriendlyatmosphereforstudyandresearchandhelpindi erent
aspects. Myspecialthanksto Mrs. LillemorHyllengrenforallher assis-
tance.
A specialthanksto Lars Lindkvistforhis assistance withSIMPOW.
Many thanks to Professor Mania Pavella and Damien Ernstfor helping
mewith SIMEduringmy visitat theUniversityof Liege, Belgium.
Finally,Iwouldliketo extendmydeepestgratitudeand personalthanks
to those closest to me. In particular, I would like to thank my dear
mother for teaching me the value of education and my lovely Karin for
hersupportand encouragementduringthisperiodoflate workinghours.
Mehrdad Ghandhari
Stockholm
September2000
Acronym Description
AC Alternating Current
AVR Automatic Voltage Regulator
BT Boosting Transformer
CLF ControlLyapunov Function
CSC ControllableSeries Capacitor
CSDs ControllableSeries Devices
DAE Di erential{Algebraic Equations
DC Direct Current
ET Excitation Transformer
FACTS FlexibleAC TransmissionSystems
GOMIB Generalized One{Machine In niteBus
OMIB One{Machine In niteBus
PSS PowerSystem Stabilizer
QBT Quadrature BoostingTransformer
RNM Reduced Network Model
s.e.p StableEquilibriumPoint
SIME SIngle MachineEquivalent
SPM Structure PreservingModel
TCSC ThyristorControlledSeriesCapacitors
TSSC ThyristorSwitchedSeriesCapacitors
UPFC Uni edPowerFlowController
VSC Variable Structure Control
Abstract iii
Acknowledgments v
1 Introduction 1
1.1 Backgroundand Motivationof Project . . . 1
1.2 Aims ofthePerformed Work . . . 2
1.3 Outlineof theThesis . . . 5
1.4 Listof Publications. . . 6
2 Power SystemOscillations 9 2.1 Sourcesof Mitigating PowerSystem Oscillations . . . 10
2.2 Summary . . . 12
3 Modeling of Power Systems 13 3.1 Reduced Network Model . . . 15
3.2 StructurePreservingModel . . . 19
3.3 Summary . . . 21
4 Modeling of Controllable Series Devices 23
4.1 OperatingPrinciple of Controllable SeriesDevices . . . . 23
4.1.1 Uni edPowerFlowController . . . 23
4.1.2 QuadratureBoosting Transformer . . . 24
4.1.3 ControllableSeries Capacitor . . . 24
4.2 InjectionModel . . . 25
4.2.1 InjectionModelof UPFC . . . 26
4.2.2 InjectionModelof QBT . . . 29
4.2.3 InjectionModelof CSC . . . 30
4.3 Summary . . . 31
5 Lyapunov Stability 33 5.1 Mathematical Preliminaries . . . 33
5.2 LyapunovFunction . . . 38
5.3 Total Stability . . . 46
5.4 Applicationof LyapunovFunctionto PowerSystems . . . 48
5.4.1 EnergyFunction forReducedNetwork Model . . . 48
5.4.2 EnergyFunction forStructure PreservingModel . 49 5.5 Summary . . . 54
6 Control Lyapunov Function 55 6.1 GeneralFramework. . . 55
6.2 Applicationof CLFto theStructure PreservingModel . . 68
6.3 Summary . . . 71
7 Numerical Example 73 7.1 Two{Area Test System. . . 74
7.2 IEEE 9-BusTest System . . . 79
7.3 Nordic32A Test System . . . 81
8 Single Machine Equivalent 85
8.1 Foundations . . . 85
8.2 ControlLawBased on SIME . . . 87
8.3 Numerical Examples . . . 88
8.4 Selectionof theGains ofControlLaws . . . 102
8.5 Summary . . . 103
9 Variable Structure Control with Sliding Modes 105 9.1 Background . . . 105
9.2 Methodof EquivalentControl . . . 110
9.3 Summary . . . 117
10 Closure 119 10.1 ContributionsoftheThesis . . . 119
10.2 Conclusions . . . 120
10.3 Discussions and Future Work . . . 121
3.1 A multi{machinepowersystem. . . 13
4.1 Basic circuitarrangement of aUPFC. . . 24
4.2 Basic circuitarrangement of aQBT. . . 25
4.3 Basic circuitarrangement of aCSC. . . 25
4.4 Equivalent circuitdiagramof a CSD. . . 26
4.5 Vector diagram ofthe equivalentcircuitdiagram. . . 26
4.6 Representationof theseriesconnectedvoltage source. . . 27
4.7 Replacementoftheseriesvoltagesourcebyacurrentsource. 27 4.8 Injection modelof theseriespartof theUPFC. . . 28
4.9 Injection modelof theUPFC. . . 28
4.10 CSClocatedina losslesstransmissionline. . . 30
5.1 Stabilityboundary(dottedlines)and stabilityregionof x s . 38 5.2 Estimate ofthe stabilityregionof x s .. . . 42
5.3 The OMIB system. . . 43
5.4 Phase portraitof theOMIB system. . . 44
6.1 The OMIB systemwitha CSD. . . 60
6.2 Phase portraitof theOMIB systemduringthefault. . . . 63
6.3 Phase portraitof theOMIB systemafter thefault. . . 64
6.4 The2{machine in nitebustest system. . . 65
6.5 Variationoftherotor angles. . . 67
6.6 Variationoftheenergy function. . . 68
7.1 Thetwo{area testsystem. . . 74
7.2 VariationofP vs timeforthe systemmodel1. . . 75
7.3 VariationofP vs timeforthe systemmodel2. . . 76
7.4 VariationofP vs timeforthe systemmodel3. . . 77
7.5 VariationofP vs timeforthe systemmodel4. . . 78
7.6 TheIEEE 9{bus test system. . . 79
7.7 VariationofP vs timeinthe IEEE9{bus test system. . . 80
7.8 VariationofP vstimewithCSDsintheIEEE9{bussystem. 81 7.9 TheNordic32A test systemproposedbyCIGRE. . . 82
7.10 VariationofP vstimeintheNordic32Atestsystem,LF32{ 028. . . 83
7.11 VariationofP vstimeintheNordic32Atestsystem,LF32{ 029. . . 84
8.1 Two{area powersystem. . . 87
8.2 Case1: VariationofP vs. timeinthetwo{areatestsystem and phaseportrait ofthecorresponding GOMIBsystem.. 90
8.3 Case2: VariationofP vs. timeinthetwo{areatestsystem and phaseportrait ofthecorresponding GOMIBsystem.. 91
8.4 Case3: VariationofP vs. timeinthetwo{area testsystem. 92 8.5 VariationofP vs timeinthe IEEE9{bus system. . . 93
8.6 VariationofP vs timeinthe Nordic32Atest system. . . . 94
8.7 TheBrazilianNorth{South interconnection system.. . . . 95
8.8 Case 1: Variation of P vs. time in the Brazilian North{
Southinterconnectionsystemandphaseportraitofthecor-
8.9 Case 2: Variation of P vs. time in the Brazilian North{
Southinterconnectionsystemandphaseportraitofthecor-
responding GOMIBsystem. . . 98
8.10 Case 3: Variation of P vs. time in the Brazilian North{
Southinterconnectionsystemandphaseportraitofthecor-
responding GOMIBsystem. . . 99
8.11 Case 4: Variation of P vs. time in the Brazilian North{
Southinterconnectionsystemandphaseportraitofthecor-
responding GOMIBsystem. . . 100
8.12 Case 5: Variation of P vs. time in the Brazilian North{
South interconnection system. . . 101
8.13 Phase portraitof theGOMIB systemof thetest system. . 102
9.1 Phase portrait ofthe systemfork = 3(dotted line)and
k = 2 (dashed line, and also solid lines which are indeed
the eigenvectors). . . 106
9.2 Phase portraitof thesystemcontrolledbyVSC,c
1
=
1
. 107
9.3 Phase portrait of the system when g
1
<
1 and g
1
>
1 ,
respectively. . . 108
9.4 Phase portrait of the OMIB system after the fault, when
CSC is controlled by CLF and VSC with sliding mode,
respectively. . . 114
9.5 Phase portrait of the OMIB system after the fault, when
an energy function(dottedline) and a Lyapunov function
(solidline)areusedforderivingthecontrollaw,respectively. 116
Introduction
1.1 Background and Motivation of Project
Historically,power systems were designed and operated with large mar-
gins. Itwascomparativelyeasytomatchloadgrowthwithnewgeneration
and transmission equipment. So, systems normallyoperated ina region
where behavior was fairly linear. Only occasionally would systems be
forced to extremes wherenonlinearitiescould begin to have some signi -
cant e ect. However, because of politicaland environmentalissues, such
as thebuilding and the locations of new generation and impedimentsof
thebuildingtransmissionfacilities,there isagreaterneedtomake maxi-
mumuseof existingfacilities. Asaconsequence, some transmissionlines
becomemoreloadedthanwasplanned(whentheywerebuilt)whichleads
toreducedpowersystemdampingofoscillationsandtodecreasedsystem
stabilitymargins. Also,astheelectricityindustrymovestowardanopen
access market, operating strategies will become much less predictable.
Hence,therelianceonnearly linearbehavior(whichwasadequateinthe
past)mustgivewaytoanacceptancethatnonlinearitiesaregoingtoplay
an increasinglyimportant role inpower systemoperation. Itis therefore
vital that analysis tools perform accurately and reliablyin the presence
of nonlinearities[1 ].
Development of devicesforincreasing the transmissioncapacity of lines,
and controlling thepower owin transmissionsystemgoeson presently.
Manyofthesenewapparatusescan bematerializedonlydueto thelatest
development in high{power electronics to be used in the main circuits 1
combined withcontrolstrategies thatrelyon themoderncontrolsystem
software and hardware.
ByusingpowerelectronicscontrollersaFlexibleACTransmissionSystem
(FACTS) can be produced which o ers greater control of power ow,
secure loading and dampingof powersystem oscillations[2 ]. The device
conceptscanbeclassi edintothoseoperatinginshuntwiththepowerline
in which cases the injected currents are controlled, and those operating
in series with the power line in which cases the inserted voltages are
controlled. The rst category includes system components, such as the
Static Var Compensator (SVC),and thelatter category includessystem
components, such as
Uni edPowerFlowController(UPFC)
Controllable SeriesCapacitor (CSC)
Quadrature BoostingTransformer(QBT)
which all henceforth will be called Controllable Series Devices (CSDs).
Application of these devicesto power ow control and damping control
in electricpowersystems isdescribedin[3 ].
Generally, in the modeling of such devices for studies of power system
behavior,thefastswitchingactioninherentinpowerelectronicsisignored.
Instead,thedevicesarerepresentedbyapproximatemodelswhichexhibit
continuousbehavior. Theaimistoensurethattheexactandapproximate
representations have asimilar\average" e ect on thesystem. Ofcourse,
anyphysicallimitationsintheactualdevicemustbeaccurately re ected
in theapproximatemodel[1 ].
1.2 Aims of the Performed Work
Modern power systems are large scale and complex. Disturbances typ-
ically change the network topology and result in nonlinear system re-
sponse. Also, because of deregulation the con guration of the intercon-
nected grid will routinely be in a state of change. Therefore, the tradi-
tionalcontrollawsbasedonlinearizedsystemmodelsareoftenof limited
1
Thecircuitsofthedevicewherethepoweris owingareusuallyreferredtoasmain
value. Thus, a control strategy that will counteract a wide variety of
disturbancesthatmayoccur inthepowersystemis attractive.
The aim of this project is to investigate and evaluate the enhancement
of the performance of the control laws which are derived for nonlinear
systems. Also,aquestionofgreatimportanceistheselectionoftheinput
signalsfortheCSDsinordertodamppoweroscillationsinane ectiveand
robust manner. For a CSD controller sited in the transmission system,
it is attractive to extract an input signal from the locally measurable
quantitiesat thecontrollerlocation.
In the rstpart of theproject,two controlstrategies, namely:
Variable StructureControl
EnergyFunction Method
were studied and the results were reported in [4]{[6]. It was concluded
that the Energy Function Method was more suitable than the Variable
Structure Control for controlling CSDs in a multi{machine power sys-
tem. Therefore, furtherresearch regarding EnergyFunctionMethodwas
motivated.
It shouldbe noted that Energy Function Method will henceforth be re-
named to ControlLyapunovFunction(CLF).
The overall aim of the research of this part of the project is to try to
resolve some issuesregardingCLFand verifyits applicabilityto realistic
powersystems. The followingtopicsareplannedto be addressed:
In uence oflosses.
In uence ofmore detailedmodels.
Use oflocal inputsignalsand coordinationof di erentcontrollers.
These itemswillbeelaborated below.
SofarControlLyapunovFunction(CLF) isprovento workinpowersys-
temswithoutlosses. Oneissue istheunavailabilityof CLFto e ectively
handle power system losses, where the losses are either from transmis-
sion systems or from the transfer conductances in the reduced system
theoretical problem. Thisproblemisalso validwhenvoltage dependence
of realloadsandmore detailedmodelsofsynchronousmachinesincluded
AutomaticVoltageregulator(AVR)andturbineregulatorareconsidered.
Oneoftheaimsoftheproposedprojectistostudythee ectsthatcanbe
expectedwhencontrollawsfortheCSDs(whicharebaseduponsimpli ed
system models)are appliedto realisticsystems.
In the somewhat simpli edmodelused to derive theCLF based control
law, it can be veri edthat localinputsignals, e.g. power ows on lines,
can successfully be usedto damp power oscillations. However, there are
twoissuesthatwillbeinvestigatedfurtherinthisprojecttogainabetter
basis andunderstanding,namely:
ArelocalsignalssuÆcientalsowhenmorecomplicatedandrealistic
modelsofthepowersystemcomponentsare used?
Even ifit can be proven thatlocalsignalscan stabilizethesystem,
a remote input signal may be more e ective for this purpose. A
pertinent questionis forwhich power systemconditions thisis the
case.
Arelatedquestion(atleastfromatheoreticalpointofview)concernsthe
coordination ofseveral CSD controller. A relevant questionis then:
Do CSDswithCLFcontroladverselya ect each other?
Theaimoftheprojectistoanswertheabovequestionsthroughanalytical
work andsimulationsof realisticpowersystems.
The project can be seen asa very natural extensionand continuationof
theworkdoneatthedepartmentandreportedin[7 ]. Theemphasisin[7]
wason (steady{state)power owcontrolandonlinearanalysisofpower
systems with CSDs, butsome possibilitiesof nonlinearcontrolwere also
brie y investigated.
TheprojectwasalsocoordinatedwithaprojectbytheNUTEKREGINA
projectonCoordinatedandRobust ControlofPowerSystemswhichwas
reportedin[8 ]. Themainissuesof[8 ]involvethedesignofcontrolstrate-
gies ofpowersystemsforthecasewhenseveralinteractingcontrollersare
the project reported in [8 ] have some points of interactions which were
coordinated,anditisbelievedthatthesetwoprojectshavebene tedfrom
each other infruitfulway.
Another project was dealing with damping of power oscillations by use
of High Voltage Direct Current (HVDC) systems and was reported in
[9]. Many of the questions and problems of the proposed project are
similarto thoseof[9 ],butthestudiedsolutionsareofcoursedi erent. A
fruitfulinteractiontookalsoplaceinthiscase. Allthedescribedprojects
together are part of long term plan of the department to develop and
investigate the possibilities and virtues of controllable devices in power
systems. Thisplanincludesalsothedevelopmentofrelevantanalysisand
simulation tools.
1.3 Outline of the Thesis
Chapter 2 brie y explains the e ects and consequence of power system
oscillationsinapowersystem. Thischapter alsooutlineshowthese oscil-
lationsaremitigatedinapowersystem. Discussioninthischapterlargely
follows thatin[10 ] and referencestherein.
Chapter3presentsthemathematicalmodelsforapowersystemrequired
in formulating the stability problem. Both Reduced Network Model
(RNM) and Structure Preserving Model (SPM) are presented in this
chapter. Discussion inthis chapter largely follows that in [11 ], [12 ] and
referencestherein.
Chapter 4 explains the operating principles of the Uni ed Power Flow
Controller (UPFC), the Quadrature Boosting Transformer (QBT) and
ControllableSeries Capacitor(CSC). A generalmodelisalso derived for
thesedevices. Thismodelwhichisreferredtoasinjectionmodel,ishelpful
forunderstandingthe impactof these componentson power systems.
Chapter 5starts byreviewingsome relevant conceptsfromnonlineardy-
namicalsystems theory. Then,this chapter analyzesstabilityof equilib-
riumpointsbyapplyingLyapunovtheorems. Formechanicalandelectri-
calsystems,thephysicalenergy(orenergy{like)functionsareoftenused
as Lyapunov function candidates. The time derivatives of these energy
functionsarehowevernegativesemide nite,andtherefore,thesefunctions
applyingtheLaSalle'sinvarianceprincipleandthetheoremofBarbashin
and Krasovskii,theasymptoticstabilityofan equilibriumpointcan also
be justi ed by the energy functions. Discussion in this chapter largely
follows that in[13 ]{[16].
Chapter 6 introducesthe concept ofControlLyapunovFunctionfor sys-
tems withcontrolinput. Theso{called aÆnesystems arestudied inthis
chapter. Discussion inthischapter largely follows thatin[17 ] and refer-
ences therein.
Chapter 7 provides the results of numerical examples. In this chapter,
thecontrollaws derivedinChapter6areappliedto varioustest systems.
Chapter 8 introduces theconcept of SIngle Machine Equivalent (SIME).
SIMEisahybriddirect-temporaltransientstabilitymethod,whichtrans-
formsthetrajectoriesofamulti{machinepowersystemintothetrajectory
of a Generalized One{Machine In niteBus (GOMIB)system. Basically,
SIMEdealswiththepost-faultcon gurationofapowersystemsubjected
to a disturbance which presumably drives it to instability. Under such
condition,SIMEusesatime{domainsimulationprograminordertoiden-
tifythemode ofseparationofits machinesintotwo groups,namely,crit-
ical and non-critical machines which are replaced by successivelya two{
machine equivalent. Then,this two{machine equivalent is replaced bya
GOMIB system. Discussioninthischapter largely followsthat in[12 ].
Chapter 9 introduces the concept of Variable Structure Control (VSC)
and VSC with sliding mode. With VSC, dynamical systems are con-
trolledwithdiscontinuousfeedbackcontrollers. VSC hasbeendeveloped
duringthelastfourdecades,andischaracterizedbyacontrollawwhichis
designed todrivethesystemtrajectoriesonto aspeci edline(orsurface)
in the state space. The slidingmode describesthe particular case when
the system trajectories are constrained to lie upon a line (or surface).
Discussion inthischapter largelyfollowsthat in[44 ] and[48 ].
Finally,inChapter 10,weprovidethe conclusionsand alsosome sugges-
tions forfutureworkare given.
1.4 List of Publications
Work performed duringthis projecthas beenpublished inthe following
1. M. Norrozian,L.
Angquist,M.GhandhariandG. Andersson, \Use
of UPFC for Optimal Power Flow Control", Proceedings of Stock-
holm Power Tech., pp. 506{511, June1995.
2. M.Norrozian,L.
Angquist,M.GhandhariandG.Andersson,\Series{
Connected FACTS Devices Control Strategy for Damping of Elec-
tromechanical Oscillations", Proceedings of 12 th
PSCC, pp. 1090{
1096, August 1996.
3. M. Norrozian,L.
Angquist,M. GhandhariandG. Andersson, \Use
ofUPFCforOptimalPowerFlowControl", IEEETrans. onPower
Delivery, Vol. 12, No. 4,pp. 1629{1635, October1997.
4. M. Norrozian,L.
Angquist,M. GhandhariandG.Andersson, \Im-
provingPowerSystem Dynamics bySeries{Connected FACTS De-
vices", IEEE Trans. on Power Delivery, Vol. 12, No. 4, pp.
1636{1642, October1997.
5. M.Ghandhari,G.Andersson,M.NorrozianandL.
Angquist,\Non-
linearControlofControllableSeriesDevices(CSD)",Proceedingsof
the 29 th
North American Power Symposium (NAPS), pp. 398{403,
October1997.
6. M. Ghandhari, Control ofPowerOscillations inTransmission Sys-
tems Using Controllable Series Devices, Licentiate Thesis, Royal
Institute of Technology, TRITA{EES{9705, ISSN1100{1607, 1997.
7. M. Ghandhariand G. Andersson, \Two Various ControlLaws for
Controllable Series Capacitor (CSC)", Power Tech. Budapest 99,
September1999.
8. M. Ghandhari and G. Andersson, \A Damping Control Strategy
for Controllable Series Capacitor (CSC)", Proceedings of the 31 th
NorthAmerican PowerSymposium (NAPS), pp. 398{403,October
1999.
9. M.Ghandhari,G.AnderssonandI.A.Hiskens, \ControlLyapunov
Functions for Controllable Series Devices", SEPOPE, Brazil, (in-
vited paper), May2000.
10. M. Ghandhari, G. Andersson and I. A. Hiskens, \Control Lya-
punov Functions for Controllable Series Devices", Submitted to
11. M.Ghandhari,G.Andersson,D.ErnstandM. Pavella, \AControl
Strategy for Controllable Series Capacitor in Electric power Sys-
tems", Submitted to Automatica.
Power System Oscillations
Anelectricalpowersystemconsistsofmanyindividualelementsconnected
together to form a large, complex system capable of generating, trans-
mittingand distributing electricalenergy over a largegeographical area.
Because of this interconnection of elements, a large variety of dynamic
interactionsarepossible,someof whichwillonlya ectsome ofelements,
otherswilla ect partsof thesystem, whileothers maya ect thesystem
asa whole.
Ingeneral, powersystemstabilitycanbedividedinto(rotor)anglestabil-
ityand voltage stability. In thisthesis, the angle stabilityis considered.
Power system stability is a term applied to alternating current electric
power systems, denoting a condition in which the various synchronous
machinesofthe systemremain \insynchronism", or\instep" with each
other. Conversely,instabilitydenotes a conditioninvolving \lossof syn-
chronism", or falling \out of step" [19 ]. The stability problem involves
thestudyoftheelectromechanicaloscillationsinherentinpowersystems.
Power systems exhibit various modes of oscillation due to interactions
among system components. Many of the oscillations are due to syn-
chronousgeneratorrotorsswingingrelativetoeachother. Theelectrome-
chanical modes involving these masses usually occur in the frequency
range of 0.1 to 2 Hz. Particularly troublesome are the interarea oscil-
lations, which typically are in the frequency range of 0.1 to 1 Hz. The
interarea modesareusuallyassociatedwithgroupsof machinesswinging
relative to other groupsacross a relativelyweak transmissionpath. The
higher frequency electromechanical modes (1 to 2 Hz) typically involve
one or two generators swinging against the rest of the power system or
electrically close machinesswinging against each other (called also local
modes). Inmanysystems,thedampingofthese electromechanical swing
modesis acritical factor foroperating ina securemanner.
Becauseofpoliticalandenvironmentalissues,suchasthebuildingandthe
locationsofnewgenerationandimpedimentsofthebuildingtransmission
facilities,there is agreater needto make maximumuseof existingfacili-
ties. Asaconsequence,sometransmissionlinesbecomemoreloadedthan
was plannedwhen they were built. In particular, heavy power transfers
can create interareadamping problemsthat constrain systemoperation.
The oscillationsthemselves maybe triggeredthroughsome event ordis-
turbance on the power system orbyshiftingthe system operatingpoint
across some steady-state stability boundary where oscillations may be
spontaneously created. Controller proliferation makes such boundaries
increasingly diÆcult to anticipate. Once started, undamped oscillations
oftengrowinmagnitudeoverthespanofmanyseconds. Theseoscillations
maypersistformanyminutesandbelimitedinamplitudeonlybysystem
nonlinearities. In some cases, large generator groups loose synchronism
and part orall of the electrical network is lost. The same e ect can be
reachedthrough slow cascading outages when theoscillationsare strong
and persistent enough to cause uncoordinated automatic disconnection
of key generators or loads. Sustained oscillationscan disrupt the power
systeminotherways,evenwhentheydonotproducenetworkseparation
orlossof resources. Forexample, powerswingsthatarenottroublesome
inthemselvesmayhaveassociatedvoltageorfrequencyswings,whichare
unacceptable. Such considerations can limit power transfers even when
stabilityis notadirect concern.
2.1 Sources of Mitigating Power System
Oscillations
The torqueswhichin uencethemachineoscillationscanbeconceptually
split into synchronizing and damping components of torque. The syn-
chronizingcomponent\holds"themachinestogetherandisimportantfor
system transient stabilityfollowing largedisturbances. For smalldistur-
bances, thesynchronizingcomponent oftorquedeterminesthefrequency
The damping component determinesthe decay of oscillationsand isim-
portant for system stability following recovery from the initial swing.
Damping is in uenced by many system parameters. It is usually small
and can sometimes become negative in the presence of controls, which
arepracticallytheonly\source"ofnegative damping. Negativedamping
can lead to spontaneous growth of oscillations until relays begin to trip
systemelements.
Much history existsin thepowersystem literatureon the application of
supplemental modulation controls to existing regulators in order to aid
dampingofpowerswings. Whenadevice,itsregulatorandsupplemental
controlareaddedtothepowersystem,theymustoperatesatisfactorilyin
thepresenceofmultiplepowerswingmodesoverawiderangeofoperating
conditions.
Conventionally,thedampingofpowersystemoscillationsisperformedby
Power System Stabilizer (PSS) which is an added device to Automatic
VoltageRegulator(AVR)ofthegenerator. ThebasicfunctionofthePSS
isto extendstabilitylimitsbymodulatinggenerator voltage throughthe
exciter to provide positive damping torque to power swing modes. By
modulatingtheterminalvoltage thePSSa ectsthepower owfromthe
generator,whicheÆcientlydampslocalmodes. PSShasthedisadvantage
of working through the same element that had resulted in the negative
damping originally. Also, the achievable damping of interarea modes is
lessthanthatoflocalmodes. Sincesystemdampingissmallatbest,itis
reasonable to use new devicesfor more damping. For e ective damping
withoutdisturbingthenetworksynchronizingtorques,itisessentialthat
the damping device generate a torque whose phase is precisely de ned
andcanoperatecontinuously. Thesesrequirementsseembestsatis edby
thefast responseand staticcharacter of powerelectronicsdevices.
In recent years, the fast progress in the eld of power electronics has
opened new opportunities for the power industry via utilization of the
FACTS deviceswhich o er an alternative means to mitigate power sys-
tem oscillations. Theyareoperated synchronouslywiththetransmission
lineand maybeconnectedeitherinparallelproducingcontrollableshunt
reactive current forvoltage regulation,or inserieswith theline forcon-
trollingpower owon thetransmissionline.
Unlike PSS control at a generator location, the speed deviations of the
machinesofinterestusedasinputsignals(measurements)arenotreadily
since the usualintent is to damp interarea modes, which involve a large
number of generators, speed signals themselves are not necessarily the
best choice for an input signal for devices in the transmission line. For
the FACTS devices, it is typically desirable to extract an input signal
from locally measurable quantities. Selectingappropriate measurements
is usuallya very mostimportant aspect ofcontroldesign.
2.2 Summary
Power systems exhibit various modes of oscillation due to interactions
among system components. Particularly troublesome are the interarea
oscillations, which typically are in the frequency range of 0.1 to 1 Hz.
Conventionally,thedampingofpowersystemoscillationsisperformedby
Power System Stabilizer(PSS).However, dueto the fastprogress inthe
eld of power electronics, the FACTS deviceso er an alternative means
to mitigatepowersystemoscillations.
Modeling of Power Systems
In thischapter, themathematical modelsforapowersystemrequired in
formulating thestabilityproblem willbe presented. Both Reduced Net-
workModel(RNM)andStructurePreservingModel(SPM)arepresented
inthischapter.
Figure3.1showsamulti{machinepowersystemwhichhasatotalofn+N
nodesof whichthe rst nare internalmachine nodes and theremaining
N areloadbuses, thatis,networknodes.
Transmission Network GEN 1
1 1
n n
V + Ð G +
2 2
n n
V + Ð G +
2 n 2 n
V Ð G
1 1
E @ ¢Ð
2 2
E @ ¢Ð
n n
E @ ¢Ð
1
jx¢ d
2
jx¢ d
jx¢ dn
2 1 n 2 1 n
V + Ð G +
2 2 n 2 2 n
V + Ð G +
n N n N
V + Ð G +
GEN 2
GEN n
Figure3.1. Amulti{machinepowersystem.
In Figure 3.1,
E 0
k
=E 0
k
6
Æ
k
(k =1n) is the internal machine voltage
phasorbehind the transient reactance x 0
dk
which includesthe reactance
of transformer. E 0
k
is themagnitudeof theinternal machine voltage and
Æ
k
is the internal machine angle of the k{th machine. All the angles
are measured with respect to a synchronously rotating reference in the
system.
V
k
=V
k
6
k
(k=n+1n+N) is theload busvoltage phasor
with magnitudeV
k
and phase angle
k .
Historically, loads are presented by the following three types or models
in terms of their load voltage characteristics (also called \static loads"),
namely:
Constant power
Constant current
Constant impedance
They form fundamental basis in modeling a majority of loads with the
exceptionofsomemotorloadsrequiringspecialconsiderationduringlarge
disturbances. Astatic load isdescribedby
P
L
=P
Lo (
V
V
o )
mp
Q
L
=Q
Lo (
V
V
o )
mq
(3.1)
where P
Lo
and Q
Lo
are the active and reactive powers at the nominal
voltage V
o
, respectively. mp and mq are the voltage exponents of the
activepowerandthereactive powerwhichcanassumeanyvalueranging
from 0to 3 basedon thenature ofthecompositeloadcharacteristic ata
given bus. V is thecurrent voltage.
Having mp =mq =0, the active and reactive components of the static
load have constant power characteristics. For mp = mq = 1 and mp =
mq =2,the active and reactive componentsof the static load have con-
stant current and constant impedancecharacteristics, respectively.
Afrequentlyusedrepresentationofthestaticloadsasfunctionsofvoltage
and frequencydeviationsmay be writtenas(also calledZIPmodel)
P
L
=P
Lo (a
0 V
0
+a
1 V
1
+a
2 V
2
)(1+k
P
f)
Q
L
=Q
Lo (b
0 V
0
+b
1 V
1
+b
2 V
2
)(1+k
Q
f)
(3.2)
wherea
i ,b
i ,k
P andk
Q
aretherespectivevoltageandfrequencysensitivity
3.1 Reduced Network Model
The Reduced Network Model (RNM) is based on the following assump-
tions:
The various network components are assumed to be insensitive to
changes infrequency.
Each synchronousmachine isrepresented bya voltage phasorwith
constant magnitudeE 0
behind itstransientreactance.
Themechanicalangle ofthesynchronousmachinerotor isassumed
to coincide with the electrical phase angle of the voltage phasor
behindthetransient reactance.
Loads are represented asconstant impedances, i.e. mp= mq = 2
in(3.1).
Mechanical powerinputto generatorsis assumedconstant.
Saliencyis neglected,i.e. x 0
d
=x 0
q .
Stator resistance isneglected.
This is the simplest power system model used in stabilitystudies. It is
usuallylimited to analysisof rst{swingtransients.
Power systems are most naturally described by Di erential{Algebraic
Equations(DAE).Anadvantageoftheassumptionofconstantimpedance
loads is that it is possible to eliminate the network nodes to obtain an
equivalent systemwhichonlyconsistsof nonlineardi erentialequations.
Thisis achieved bythefollowingsteps:
1. Performa pre{fault load owcalculation. Calculate theequivalent
steady{state impedance loadsin theform ofadmittances as
y
Lk
= P
Lk jQ
Lk
V 2
k
; k =n+1n+N
foreach loadbus, and add theseelementsto the
Y matrix.
2. Calculatetheinternalmachinevoltagesbehindtransientreactances
as
E 0
k
=
V
n+k +jx
0
dk P
Gk jQ
Gk
V
n+k
; k=1n
foreach of thenmachines.
3. Augmentthe
Y
bus
matrixbyadmittances corresponding tothema-
chine transient reactances as
y
k
= 1
jx 0
dk
; k=1n
to create theninternal machine nodes.
4. Thisaugmentedadmittancematrixcansymbolicallybepartitioned
as
n N
^
Y
BUS
=
Y
A
Y
B
Y
C
Y
D
n
N
The relation between injected currents and node voltages is now
given by
I
G
0
=
Y
A
Y
B
Y
C
Y
D
E
G
V
L
where
E
G
isthevector of theinternalmachinevoltages behindthetran-
sient reactances and
V
L
isthe vector of loadbus voltages. Since there is
no injectedcurrentsinthenetwork nodes,thissystemcan bereducedto
internalmachinenodesas
I
G
=(
Y
A
Y
B
Y 1
D
Y
C )
E
G
=
Y
int
E
G
The powerinjectedintheinternalmachinenodek can nowbecalculated
by
P
Gk
=R ef
E 0
k
I
Gk g
=E 0
k 2
G
kk +
n
X
l=1
l6=k E
0
k E
0
l (B
kl sin(Æ
kl )+G
kl cos(Æ
kl ))
(3.3)
Æ
kl
=Æ
k Æ
l
G
kk
isthe short-circuitconductanceof thek{thmachine.
G
kl
isthe transferconductance in
Y
int
,k6=l.
B
kl
isthetransfer susceptancein
Y
int
,k 6=l.
Let
C
kl
=E 0
k E
0
l B
kl
F
kl
=E 0
k E
0
l G
kl
(3.4)
Now, themotion ofthek{thmachine isgiven by (k =1n)
_
Æ
k
=!
k
M
k _
!
k
=P
mk D
k
!
k P
Gk
(3.5)
or
_
Æ
k
=!
k
M
k _
!
k
=P
k D
k
!
k n
X
l=1
l6=k (C
kl sin(Æ
kl )+F
kl cos (Æ
kl ))
(3.6)
where
P
k
=P
mk E
0
k 2
G
kk
P
mk
isthe mechanical power inputto thek{thmachine.
D
k
>0 isthedamping constant ofthe k{thmachine.
M
k
>0is themoment of inertiaconstant of thek{thmachine.
!
k
is the rotor speed deviation of the machine k with respect to a syn-
chronouslyrotating reference.
Intheanalysisofanglestability,thefocusofattention isonthebehavior
of the machine angles with respect to each other. In order to clearly
distinguishbetweentheforcesthatacceleratethewholesystemandthose
are transformed into the Center Of Inertia (COI) reference frame. The
positionofthe COIisde nedby
Æ
COI
= 1
M
T n
X
k=1 M
k Æ
k
; M
T
= n
X
k=1 M
k
(3.7)
Next, thestate variablesÆ
k and !
k
aretransformedto theCOIvariables
as
~
Æ
k
= Æ
k Æ
COI
~
!
k
= !
k
!
COI
These COIvariablesareconstrainedby
n
X
k=1 M
k
~
Æ
k
=0
n
X
k=1 M
k
~
!
k
=0
(3.8)
Swing equations (3.6) can now be rewritten in the COIreference frame
as (k=1n)
_
~
Æ
k
=!~
k
_
~
!
k
= 1
M
k [P
k n
X
l=1
l6=k C
kl sin(Æ
kl )+
M
k
M
T P
COI D
k
~
!
k ]
1
M
k n
X
l=1
l6=k F
kl cos (Æ
kl )
=f
k +p
k
(3.9)
where
P
COI
= n
X
k=1 (P
mk P
Gk )
f
k
= 1
M
k [P
k n
X
l=1
l6=k C
kl sin(Æ
kl )+
M
k
M
T P
COI D
k
~
!
k ]
p
k
= 1
M
k n
X
l=1 F
kl cos (Æ
kl )
(3.10)
Note that
~
Æ
kl
=
~
Æ
k
~
Æ
l
=Æ
kl
and uniformdampingis considered.
System (3.9) can indeedbe consideredasanOrdinaryDi erentialEqua-
tion (ODE)of theform
_
x=F(x)=f
o
(x)+p(x) (3.11)
wherex=[
~
Æ !~] T
is thevector of thestate variables. In (3.11)
f
o
(x)=[!~ [f
1
f
n ]]
T
; p(x)=[0 [p
1
p
n ]]
T
and
~
Æ=[
~
Æ
1
~
Æ
n
] ; !~ =[!~
1
!~
n ]
3.2 Structure Preserving Model
It is known that load characteristics have a signi cant e ect on system
dynamics. Inaccurateloadmodelingmayleadto apowersystemoperat-
ing in modes that result inactual system collapse or separation. In the
Reduced Network Model (RNM), impedance loads are assumed. Hence,
inthe context of systemmodeling,RNM precludesconsiderationof load
behaviors (i.e. voltage and frequencyvariations) at load buses. Further-
more, in the context of physical explanation of results, reduction of the
transmissionnetworkleadsto lossof network topology.
Structure Preserving Model (SPM) have beenproposed( rst in [20 ]) to
overcomesomeoftheshortcomingsoftheRNM,andtoimprovethemod-
elingofgeneratorsandloadrepresentations. An advantageof usingSPM
isthatfromamodelingviewpoint,itallowsmorerealisticrepresentations
of powersystem components,especiallyload behaviors.
Consider again themulti{machine powersystem shownin Figure3.1. It
is assumed that the mechanical power input is constant and the stator
resistance is neglected. The one{axis generator model is used for the
generators. This model includesone circuit for the eld winding of the
rotor, i.e. thismodel considers thee ects of eld ux decay. Note that
in theone{axis generator model, thevoltage behindthe direct transient
reactance is no longer a constant. The loads are modeled by equation
(3.1) withmp=0 andarbitrary mq. Thetransmissionlinesaregiven by
loads and the d{axis transient reactances x 0
d
. The kl{th element of the
admittance matrixis de ned by
Y
kl
=G
kl +jB
kl
, whereG
kl
represents
solely the resistances of the respective transmission lines. In general,
because ofthehigh ratioof reactance toresistance, thetransmissionline
resistances can be neglected. Thus,
Y
kl
=jB
kl .
The dynamics of the k{th generator are described by the following dif-
ferentialequationswithrespectto theCOIreference frame. Notethatin
the followingequations
~
Æ
k
~
l
=Æ
k
l and
~
k
~
l
=
k
l
. Thus, for
k =1n
_
~
Æ
k
=!~
k
M
k _
~
!
k
=P
mk P
Gk D
k
~
!
k M
k
M
T P
COI
T 0
dok _
E 0
qk
= x
dk x
0
dk
x 0
dk V
n+k cos (Æ
k
n+k )
+E
fdk x
dk
x 0
dk E
0
qk
(3.12)
where
P
Gk
= 1
x 0
dk E
0
qk V
n+k sin(Æ
k
n+k )
x 0
dk x
qk
2x 0
dk x
qk V
2
n+k
sin(2(Æ
k
n+k ))
(3.13)
P
COI
isgiven by (3.10).
x
dk ,x
qk
arethed{axisandtheq{axissynchronousreactances ofthek{th
machine.
E 0
qk
istheq-axisvoltage behindtransientreactanceof thek{thmachine.
T 0
dok
is the d{axis transient open{circuit time constant of the k{th ma-
chine.
E
fdk
istheexcitervoltageofthek{thmachinewhichisassumedconstant.
E
fd
can be either constant ( xed excitation) or can vary due to Auto-
matic Voltage Regulator (AVR) action. When theexciter controlaction
is included in the generator model, due to AVR modeling, at least one
For the lossless system the following equations can be written at bus k
whereP
k
istherealpowerandQ
k
isthereactive powerinjected intothe
systemfrom busk.
For k=(n+1)2n
P
k
= n+N
X
l =n+1 B
kl V
k V
l sin(
k
l )+
E 0
q(k n) V
k sin(
k Æ
k n )
x 0
d(k n)
+ x
0
d(k n) x
q(k n)
2x 0
d(k n) x
q(k n) V
2
k
sin(2(
k Æ
k n ))
Q
k
=
n+N
X
l =n+1 B
kl V
k V
l cos(
k
l )+
V 2
k E
0
q(k n) V
k cos (
k Æ
k n )
x 0
d(k n)
x 0
d(k n) x
q(k n)
2x 0
d(k n) x
q(k n) V
2
k
[cos (2(
k Æ
k n )) 1]
and fork =(2n+1)(n+N)
P
k
=
n+N
X
l =n+1 B
kl V
k V
l sin(
k
l )
Q
k
=
n+N
X
l =n+1 B
kl V
k V
l cos (
k
l )
Therefore, for k = (n+1)(n+N) the power ow equations can be
writtenas
P
k +P
Lk
= 0
Q
k +Q
Lk
= 0
(3.14)
3.3 Summary
Dynamicsofmulti{machinepowersystems aredescribedbytheReduced
Network Model ortheStructure PreservingModel. IntheReduced Net-
workModelimpedanceloadsareassumed. Thus,itispossibletoeliminate
nonlinear di erential equations. However, in thecontext of physicalex-
planation of results, reduction of the transmission network leads to loss
of networktopology.
In the Structure Preserving Model, dynamics of multi{machine power
systems are described by Di erential{Algebraic Equations. Thus, from
a modeling viewpoint, it allows more realistic representations of power
system components,especiallyload behaviors.
Modeling of Controllable
Series Devices
In this chapter, the operating principles of a Uni ed Power Flow Con-
troller (UPFC), a Quadrature Boosting Transformer (QBT) and a Con-
trollable Series Capacitor (CSC) are described. Also, a general model
is derived for these devices. The models are derived in a single{phase
positive{sequencephasorframe. Thismodelwhichisreferredto asinjec-
tion model,is helpfulfor understandingthe impact ofthese components
on power systems. Furthermore, this model can easily be implemented
intoexistent powersystemanalysis programs.
4.1 Operating Principle of Controllable Series
Devices
4.1.1 Uni ed Power Flow Controller
A uni edpower ow controller consists of two voltage source converters
[21 ]. Theseconverters areoperated froma commonDC linkprovidedby
a DC capacitor, see Figure 4.1. Converter 2 provides the main function
oftheUPFCbyinjectinganACvoltagewithcontrollablemagnitudeand
phase angle in serieswith the transmissionline via a seriestransformer.
The basic function of converter 1 is to supply or absorb the real power
demand by converter 2 at the common DC link. Converter 1 can also
generate or absorb controllable reactive power ifit is desired. This con-
vertercan thereby provideindependent shunt reactive compensation for
the line. Converter 2 supplies or absorbs locally the required reactive
power, and exchanges the active power asa result of the seriesinjection
voltage.
Converter 1
Converter 2
Series transformer Shunt
transformer
V i V j
ij ij
P + jQ
ji ji
P + jQ V se
Figure4.1. BasiccircuitarrangementofaUPFC.
4.1.2 Quadrature Boosting Transformer
Based on feasible semiconductor switches and converter topologies for
high{powerapplications,di erentPhase ShiftingTransformer(PST)cir-
cuitcon gurationsareidenti ed. Inthisthesis,theso{calledQuadrature
Boosting Transformer (QBT) (i.e. the injected voltage is perpendicular
to the inputterminal voltage) is considered. Figure 4.2 shows the basic
circuit of a Quadrature Boosting Transformer (QBT). The phase angle
di erencebetweentheQBTterminalvoltages isachievedbyseriallycon-
necting a Boosting Transformer (BT) into the transmission line. The
powerwhichisinjectedinto thetransmissionlinebythisboostingtrans-
former must be taken from the network by the Excitation Transformer
(ET). Theconvertercontrolsthemagnitudeandthephase angleof
V
se .
4.1.3 Controllable Series Capacitor
A Controllable SeriesCapacitor (CSC)can bematerialized byThyristor
Converter
BT ET
V i V j
ij ij
P + jQ
ji ji
P + jQ V se
Figure4.2. BasiccircuitarrangementofaQBT.
pacitors (TSSC), as shown in Figure 4.3. In a simpli ed study, a CSC
can be consideredasa continuouslycontrollable reactance (normallyca-
pacitive) whichis connectedinserieswiththetransmissionline.
TSSC TCSC
Figure4.3. Basiccircuitarrangement ofaCSC.
4.2 Injection Model
Figure4.4showstheequivalentcircuitdiagramofaCSDwhichislocated
between buses i and j in a transmission system. A UPFC and a QBT
inject a voltage
V
se
in serieswith the transmission line througha series
transformer, see Section4.1. The active power P
se
involved inthe series
injection is taken from the transmission line (i.e. P ) through a shunt
transformer. TheUPFC generates orabsorbs theneeded reactive power
(i.e. Q
se
and Q
sh
) locally by the switching operation of its converters,
whilethereactive powerQ
se
injectedinserieswiththetransmissionline
by the QBT, is taken from the transmission line (i.e. Q
sh
). In Figure
4.4, x
s
isthee ectivereactanceoftheUPFC(ortheQBT)seenfromthe
transmissionlinesideof theseriestransformer.
V i V j
jx s
V '
V se
se , se
P Q
sh sh
P Q I sh
I se
Figure 4.4. EquivalentcircuitdiagramofaCSD.
Figure 4.5 shows thevector diagramof the equivalentcircuit diagramof
a CSD.
γ
I se
V '
V se
V i
β
Figure 4.5. Vectordiagramoftheequivalentcircuitdiagram.
4.2.1 Injection Model of UPFC
ToobtainaninjectionmodelforaUPFC,we rstconsidertheseriespart
V se
V i V j
jx s
I se
V '
Figure4.6. Representationoftheseriesconnectedvoltagesource.
The seriesconnected voltage sourceismodeledbyanideal seriesvoltage
V
se
which is controllable in magnitude and phase, that is,
V
se
= r
V
i e
j
where0r r
max
and 0 2.
Theinjectionmodelisobtained(asshowninFigure4.7)byreplacingthe
voltage source
V
se
bya currentsource
I
inj
= jb
s
V
se
inparallelwith x
s .
Note thatb
s
=1=x
s .
i i i
V V G = Ð V V G j = Ð j j
jx s
I inj
Figure4.7. Replacementoftheseriesvoltagesourcebyacurrentsource.
The current source
I
inj
correspondsto injectionpowers
S
i and
S
j which
arede ned by
S
i
=
V
i (
I
inj )
= rb
s V
2
i
sin( ) jrb
s V
2
i
cos ( )
S
j
=
V
j (
I
inj )
=rb
s V
i V
j sin(
ij
)+jrb
s V
i V
j cos (
ij )
where
ij
=
i
j .
Figure4.8showstheinjectionmodeloftheseriespartoftheUPFC,where
P
i
= real(
S
i
) ; Q
i
= imag(
S
i )
P = real(
S ) ; Q = imag(
S )
(4.1)
i i i
V V G = Ð V V G j = Ð j j
jx s
i i
P jQ + P j + jQ j
Figure 4.8. InjectionmodeloftheseriespartoftheUPFC.
The apparent power supplied by the series voltage source is calculated
from
S
se
=
V
se
I
se
=re j
V
i (
V 0
V
j
jx
s )
Active and reactive powers suppliedbytheseriesvoltage sourceare dis-
tinguished as:
P
se
= rb
s V
i V
j sin(
ij
+ ) rb
s V
2
i
sin( )
Q
se
= rb
s V
i V
j cos (
ij
+ )+rb
s V
2
i
cos ( )+r 2
b
s V
2
i
AssuminganidealUPFC(i.e. lossesareneglectedintheUPFC),wehave
then P
sh
= P
se
. For the UPFC, Q
sh
is independently controllable, and
we assumethat Q
sh
=0. Note thatQ
sh
can alsohave a nonzero value.
TheinjectionmodeloftheUPFCisconstructedfromtheseriesconnected
voltage sourcemodelshown inFigure4.8byaddingP
sh +jQ
sh
to busi.
Figure 4.9 shows theinjectionmodelof theUPFC.
i i i
V V G = Ð V V G j = Ð j j
jx s
si si
P + jQ P sj + jQ sj
Figure4.9. InjectionmodeloftheUPFC.
In Figure4.9
P
si
=rb
s V
i V
j sin(
ij + )
P
sj
= P
si
Q
si
=rb
s V
2
i
cos ( )
Q
sj
= rb
s V
i V
j cos (
ij + )
(4.2)
wherer and are thecontrolvariablesofthe UPFC.
ForthepurposeofdevelopingacontrolstrategyfortheUPFC,itisuseful
to applythefollowingcontrolvariables.
Since
rsin(
ij
+ )=rcos( )sin(
ij
)+rsin( )cos(
ij )
rcos (
ij
+ )=rcos( )cos(
ij
) rsin( )sin(
ij )
(4.3)
let
u
up1
=rcos( ) ; u
up2
=rsin( ) (4.4)
Substituting(4.3) and (4.4) into (4.2),thefollowingis obtained.
P
si
=b
s V
i V
j (u
up1 sin(
ij )+u
up2 cos(
ij ))
P
sj
= P
si
Q
si
=u
up1 b
s V
2
i
Q
sj
= b
s V
i V
j (u
up1 cos(
ij ) u
up2 sin(
ij ))
(4.5)
Note that
r = q
u 2
up1 +u
2
up2
; =arctan(
u
up2
u
up1 )
4.2.2 Injection Model of QBT
The argument given in Subsection4.2.1 is also validfor constructing an
injection model for a QBT. For this device, the injected voltage
V
se is
perpendicular to the input terminal voltage
V
i
. Thus, = =2, see
Figure 4.5.
Assuming an ideal QBT (i.e. losses are neglected in theQBT), we have
then P =P
se
. Forthe QBT,the reactive powerinjected inserieswith
thetransmissionlineistakenfromtheshuntpartoftheQBT.Therefore,
Q
sh
=Q
se .
The injectionmodeloftheQBTisconstructedfromtheseriesconnected
voltage sourcemodelshown inFigure4.8byaddingP
sh +jQ
sh
to busi.
Thus, Figure4.9also showstheinjectionmodelof theQBT,where
P
si
=u
q b
s V
i V
j cos (
ij )
P
sj
= P
si
Q
si
=u
q b
s V
2
i +u
q b
s V
i V
j sin(
ij )
Q
sj
=u
q b
s V
i V
j sin(
ij )
(4.6)
In (4.6),u
q
=rsin( ) and r
max
u
q
r
max
since ==2.
Note that
r=ju
q
j ; =sgn(u
q )
2
where sgn(:)is thesign function.
4.2.3 Injection Model of CSC
SupposeaCSCisislocatedbetweenbusesiandjinalosslesstransmission
lineas showninFigure 4.10.
i i i
V V G = Ð V V G j = Ð j j
jx L - jx c I se
Figure4.10. CSClocatedinalosslesstransmissionline.
For studies involving load ow and angle stability analysis, the CSC is
modeled as a variable reactance, i.e x
c
in Figure 4.10. However, for the
purposeofdevelopinga controlstrategyandhavingsame modelsforthe
Figure 4.4 is also valid for the CSC if
I
sh
is set to zero and x
s is the
transmissionlinereactance, i.e. x
s
=x
L
. Furthermore, ==2inFigure
4.5. Thus,Figure 4.10 can bereplaced by Figure4.4 wherex
s
=x
L and
I
sh
=0. From Figure 4.10, we have
I
se
=
V
i
V
j
j(x
L x
c )
In Figure4.4,
V
se
= jx
c
I
se
and inFigure 4.7, we have
I
inj
=
V
se
jx
L
= jx
c
I
se
jx
L
= x
c
x
L
I
se
TheinjectionmodeloftheCSCisthenobtainedby(4.1). Notethatsince
I
sh
= 0, we have P
sh
=Q
sh
= 0. Thus, Figure 4.9 is also validfor the
CSC, where
P
si
=u
c b
s V
i V
j sin(
ij )
P
sj
= P
si
Q
si
=u
c b
s (V
2
i V
i V
j cos(
ij ))
Q
sj
=u
c b
s (V
2
j V
i V
j cos(
ij ))
(4.7)
and
u
c
= x
c
x
L x
c
(4.8)
4.3 Summary
The injection models of the Controllable Series Devices are derived in
a single{phase positive{sequence phasor frame. The injection model is
helpful for understanding the impact of the Controllable Series Devices
on power systems. This model can easily be used for the purpose of
developing control laws. Furthermore, this model can be implemented
intoexistent powersystemanalysis programs.
Lyapunov Stability
This chapter starts by reviewing some relevant concepts from nonlinear
dynamical systems theory. Stabilityofequilibriumpointsin thesenseof
Lyapunov is also presented. Lyapunov stabilitytheorems give suÆcient
conditionsforstability. Theydonotsaywhether thegivenconditionsare
necessary. There arehowevertheorems which establish(at least concep-
tually)thatformanyofLyapunovstabilitytheoremsthegivenconditions
areindeednecessary. SuchtheoremsareusuallycalledconverseLyapunov
theorems. Thesetheoremshavebeenabasisfortheintroductionof Con-
trolLyapunovFunctionforsystems withcontrolinput.
5.1 Mathematical Preliminaries
Considerthe nonlinearsystem
_ x=f
o
(x) (5.1)
wherexisthen{dimensionalstate vectorwhichbelongstotheEuclidean
spaceR n
. Thesystemisspeci edbythevector eldfunctionf
o
:D!R n
whichiscontinuousandhascontinuous rst-orderpartialderivativeswith
respect to x on a domain D R n
into R n
. System (5.1) is also called
autonomous since timedoesnotappearexplicitlyinf
o
. Letthesolution
to (5.1) be given by
x(t)= (t t
o
;x
o )
where x
o
is the initial conditions and t
o
is the initial time which is ar-
bitrary. Since the dependence of the solution on the initial time is not
essential, wecan assume withoutlossof generalitythat t
o
=0. Thus,
x(t)= (t;x
o )
Since f
o
(x) is continuous, and has continuous rst-order partial deriva-
tiveswithrespecttox,asolutionto(5.1)(satisfyingtheinitialconditions
x(0) = (0;x
o ) =x
o
) exists on some time interval a<t <b containing
0, and that the time interval can be extended at both ends as long as
kf
o
[x(t)]kremainsbounded. Furthermore,thesolutionisuniqueand dif-
ferentiable inbotht and x
o .
One of themost important geometric properties of autonomous systems
of theform (5.1)is thatthere isonlyone solutionx(t)= (t;x
o
) passing
throughanygiveninitialstatex(0)= (0;x
o )=x
o
. Hence,trajectoriesin
statespacecanneverintersecteachother. Incontrast,fornonautonomous
systems, thesolution x(t)= (t;t
o
;x
o
) depends on the startingtime t
o ,
so that the solution from x
o at t
o
=0 will generally not be thesame as
the solutionstartingfromx
o
at some other initialtimet
o 6=0.
An equilibriumpoint x
e
for(5.1) is de ned byf
o (x
e
)=0. The point x
e
is Lyapunovstable (or stablein thesenseof Lyapunov) ifsolutions that
start nearx
e
remain nearx
e
forall t0. More precisely,anequilibrium
pointx
e
isLyapunovstableif forany >0 (nomatter howsmall)there
exists aÆ =Æ()>0 such thatforeveryx(0) inwhich
kx(0) x
e k<Æ
the solutionx(t)satis es
kx(t) x
e
k< ; 8t>0
If, inaddition,
kx(t) x
e
k!0 as t!1
then x
e
is asymptotically stable. An equilibrium point x
e
that is not
Lyapunov stability of a solution to a system of nonlinear equations (at
least locally) can be de ned by examining the linearized equations of
motion. ThisapproachisknownasLyapunov's rst(orindirect)method.
Linearizingthenonlinearsystem(5.1)aroundanequilibriumpointx
e ,we
obtain
x_ =Ax
where
A=
@f
o (x)
@x
x=x
e
= 2
6
6
4
@f
1 (x)
@x
1
@f
1 (x)
@xn
.
.
. .
.
. .
.
.
@f
n (x)
@x
1
@f
n (x)
@x
n 3
7
7
5
x=x
e
whichisalsocalledtheJacobianmatrixat x
e
. We sayx
e
ishyperbolicif
Ahasnoeigenvalues()withzerorealpart. MatrixAiscalledastability
matrixora Hurwitzmatrixif alleigenvaluesof AsatisfyR e
i
<0.
Theorem 5.1. Let x
e
be an equilibrium point for the nonlinear system
(5.1) . Then,
1. x
e
isasymptotically stableif A is a Hurwitz matrix.
2. x
e
isexponentially stable if andonly if A is a Hurwitzmatrix.
3. x
e
isunstable if R e
i
>0 for one or more of the eigenvalues of A.
The proof can be foundin[13 ].
Anasymptoticallyorexponentiallystableequilibriumpointishenceforth
denoted by x
s
. A few topological concepts of R n
are reviewed in the
following. Thestabilityregionofx
s
(denotedbyS(x
s
))isaregioninthe
state space from whichall trajectoriesconverge to x
s
. Moreprecisely,
S(x
s
)=fx: lim
t!1
(t;x
o )=x
s g
A subset S R n
is said to be open, if for every vector x 2 S, one can
ndan {neighborhoodof x
N(x;)=fz2R n
:kz xk<g
such that N(x;) 2 S. A set S is bounded if there is r > 0 such that
