IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 1 January 1982
A TOOL FOR THE COMPREHENSIVE ANALYSISOF POWER SYSTEM DYNAMIC STABILITY
C. F. Imparato
Pacific Gas and Electric CompanySan Francisco, California
Abstract - This is an applications paper reporting onthe methodological basis and the development of aproduction-grade software package for use in the dyn-amic stability (or "small disturbance") analysis oflarge power systems. This package, known as EISEMAN(EIgen_ystem Evaluation - Machine And Network), iscapable of studying a wide range of dynamic stabilityphenomena, such as subsynchronous resonance, inter-machine rotor oscillations, and the effects of excita-tion and turbine-governor systems on stability. All ofthe eigenvalues of the system are calculated, as wellas desired eigenvalue sensitivities. A noteworthy fea-ture of this eigenvalue-based tool is its capabilityof modeling to various degrees of detail the dynamicsof the power system components - the network, synchro-nous machines and control systems. This tool can alsobe used to develop root locus plots and the frequencyresponse characteristics of the power system. Thepresent version of EISEMAN can be used for the analysisof systems containing up to 250 machines, 1,500 buses,2,000 lines, and 500 dynamic states. Several testcases demonstrate the application of this tool.
I. INTRODUCTION
The dynamic stability problem studies the dynamicbehavior of a power system which has been subjected tosmall perturbations. As used in this paper, the term"dynamic" stability is synonymous with the recentlyadopted IEEE definition of "small disturbance" stabil-ity. Typical dynamic stability phenomena are self-excitation, network-torsional interactions, controlsystem-related oscillations, inter-machine (rotorelectromechanical) interactions, turbine-governorrelated oscillations, and monotonic instabilities as-sociated with exceeding the (classical) steady-statepower transmission limits of the system. The potentialfor the occurrence of dynamic instabilities has increa-sed markedly due to recent trends in the design andoperation of power systems, such as the operation ofpower systems closer to their (classical) steady-statestability limits, and the use of series compensation,fast-response excitation systems and machines withsmaller H constants.
A strong need exists for analytical toolscapable of studying the wide range of dynamicstability phenomena. The appropriate level ofdetail of models of the power system components -which include the transmission network, synchronousgenerators, and control equipment - is determinedby the type of phenomena judged to be important ina particular stability study. The information inTable 1 points out the need for a wide range ofmodels in general purpose dynamic stability analysistools.
A second modeling issue focuses on the followingquestion: How much of the system external to thestudy area needs to be represented in order to obtainmeaningful results? This question and related ones,such as what transmission level marks the point belowwhich network, machine and load dynamics can beneglected without significant effects, can at presentonly be dealt with on a trial and error basis. Thequestions themselves underscore the important needfor tools which have the capability of analyzing verylarge power networks, and can also evaluate thesensitivity of stability predictions to the discardedportions of the power system.
Phenomeiaon c o 0n03 C~
of 0 0V ) 4-j 0 0
T <~~ ~~~~C-' I)1. 0
OfH
0 0jinterest tJ a CZ1 U 0
\~~ ~ ~~~\rX r_4'O^ mC,i 0 0 0~uu 00
o I c OH a O
\ \ ;1>xQo.- n o-UcU 0)5 0*H°004 C3-JH-0 i HO J
\-~ 0U
W 0OH U W CaX\ . )Q4¢) 00H 000WX
0aO 00, X0 -405) 4-4 00
Typical c Z H Xo i X o Ha)
Frequency - -Range (Hz)IComponent 10-60 5-60 1-10 0.5-3 <1 0
Dynamics
Machine Stator R RDynamics
Machine Damper R R D DDynamics
Machine FieldDynamics R R R R R R
Machine TorsionalDynamics R
Exciter--Stabilize-Dynamics R R R R
Turbine-GovernorDynamics
NetworkDynamics R R
LoadDDDynamics D D
TABLE 1 - Component Modeling for Dynamic StabilityStudies (R:required; D:desirable)
This paper reviews the statement, component model-ing and analytical formulation of the dynamic stabilityproblem. An efficierit and original approach to theconstruction of the syst-em matrix is presented.Significant computational aspects associated with thestudy of realistically-sized power systems are discus-sed. Finally, the application and versatility ofEISEMAN, the production-grade software package devel-oped at the Pacific Gas and Electric Company, aredemonstrated by several test cases.
II. THE DYNAMIC STABILITY PROBLEM
The dynamic behavior of the power systemoperating in a balanced, three-phase mode may bedescribed by a set of nonlinear differentialequations:
Here, x(t) denotes the p-dimensional state of thesystem and xo is the initial system operating point.
A dynamic stability study investigates thedynamic behaviQr of the power system in the"neighborhood" of a point x, which lies on thetrajectory of (1). Typically, the state x, ofinterest is the operating state of the power system,x.. Because the focus of the study is on the smallexcursions of x about x. in response to a smalldisturbance, it is only necessary to model the power
system accurately in the neighborhood of xs. Inorder to investigate the so-called small-signalbehavior of the power system, the system is "frozen"in the state x_ and the nonlinear differentialequation (1) is linearized about x.. The dynamicstability of the state x, is determined from thestability characteristics of the resulting set oflinear equations
Ax= Ax='AAx7s
(2)
where Ax represents the incremental state vectorabout x,. The Jacobian in (2) is referred to as thesystem matrix. A is a function of all systemparameters and the state x. about which (1) islinearized. The eigenvalues of A (A1 = GI + jwi,i = 1,2,...p) completely specify the nature of themodeled system's response (i.e., frequency and decaycharacteristics) to small disturbances. In the timedomain, the components of the incremental state Axare expressed as linear combinations of the modeseXIt. The oscillatory frequencies of the systemresponse are given by wl/2n, and the al define thedecay rates of the corresponding modes. The systemwhich is modeled by (2) will be stable if all a. are
negative. It will be unstable if any ispositive. If any of the a; are zero, the systemwill either be marginally stable or unstable,depending on the multiplicity of this eigenvalue.
Knowledge of the eigenvalues alone is notsufficient, however, for complete characterizationof stability. The many uncertainties in the power
system data, such as inaccuracy of data, variationsin parameter values and variations in initialconditions, will cause the eigenvalues of A todiffer from those of the actual power system.It is important to know how the excursions inparameter values affect the eigenvalues. For thisreason, information about the sensitivity of theeigenvalues with respect to system parameters, suchas excitation system gains, machine inertias and linereactances, is of importance. This information isobtained from the eigenvectors of A and AT using therelationship (aAX
w: 3A uax1 _ -j \5a -i (3)
WT u.
-j -J
where uj (w1) are the eigenvectors of A (AT)associated with Aj, and a is a system parameter ofinterest [1].
Eigenvalue sensitivity can be used to ascertainwhich power system parameters have a major impact onthe damping of particular modes. Once theseparameters have been identified, the sensitivityinformation can also be used to estimate thechanges in the parameters necessary to assure
adequate damping. This approach can be used toestimate appropriate settings for tunableparameters, such as stabilizer gains. Moreover,knowledge of the eigensystem - the eigenvalues andthe eigenvectors - suffices for the calculation ofthe frequency domain characteristics - gain and phasemargins, Bode and Nyquist plots, etc. - of the powersystem.
Many approaches for dynamical system stabilityinvestigations are based on the determination of thespectrum (i.e., the set of eigenvalues) of thesystem matrix A [2] - [7]. A method for theautomatic formulation of the system matrix forlarge, arbitrary linear systems was presented in[2]. The effects of different levels of modeling ofpower system components were investigated using a
three machine power system in [3].An imaginative frequency domain approach based
on the multidimensional Nyquist criterion isreported in [8]. From the standpoint of computerresource requirements, this approach is more
efficient than eigenvalue-based analysis techniques.However, it is difficult to relate the stabilitymargin information obtained to particular systemparameters when frequency response techniques areused. Eigensystem-based techniques can provideeigenvalue sensitivity information for anyparameters of interest [2].
The dynamic stability tool reported in [51utilizes a transient stability simulation program toconstruct the system matrix via numericaldifferentiation. This approach is attractive andrelatively easy to implement; it may, however, besusceptible to problems of inaccuracy caused by thenumerical instability which is associated in generalwith the numerical differentiation process [9].
A characteristic common to eigenvalue-basedapproaches is the requirement for large amounts ofcomputer storage and time for the study of systemsof high dimensionality. To overcome this problem,approaches that focus on evaluating a subset of thespectrum have been proposed [6], [7]. Thesetechniques may, in certain cases, fail to indicatecritical eigenvalues.
Our work in the dynamic stability area wasundertaken with the aim of developing a tool tohandle effectively the analysis of the wide range ofdynamic stability phenomena for power systems ofpractical interest. We have implemented into aproduction-grade software package an eigenvalue-based dynamic stability analysis tool possessinga large degree of modeling flexibility and capableof studying large-scale systems. The EISEMANpackage can be used to analyze systems with asmany as 250 machines, 1500 buses, 2000 lines, and500 dynamic states. The package has found practicalapplication in the study of subsynchronous resonanceproblems and the planning of series compensationlevels; the identification of possible torsional/water hammer interactions in hydro units; the studyof rotor oscillation damping and inter-machineoscillations; and the study of the effects ofexcitation system and power system stabilizerparameters on system dynamic stability.
III. POWER SYSTEM MODELS
The constituent components of the powersystem - the synchronous machines and associatedequipment, and the transmission network - aredescribed in this section.
Synchronous Machine Electrical Dynamics
A three-phase two-pole synchronous machineconsists of three identical, symmetrically placed,lumped armature windings (a,b,c) in the stator, andup to four lumped rotor windings (F,G,D and Q).The F coil represents the rotor field winding. Thefictitious G coil, whose flux is in the quadratureaxis, is used to represent the effects of eddycurrents which circulate in the solid steel of therotor. The damper windings are represented by the
227
228
fictitious D and Q coils in the direct and quadratureaxes, respectively [10].
The most detailed machine representationcommonly used for the study of dynamic stabilitymodels the dynamics of the stator direct andquadrature axis windings plus the dynamics of thefour rotor windings discussed above.
Simpler machine representations are oftendesirable or necessary for the following reasons:
(i) For the modeling of salient-pole machines,the representation of a fictitious "G" coil andits dynamics is not appropriate.(ii) For studies in which only low-frequency
phenomena are of interest, the modeling of statordynamics may be neglected.(iii) When modeling machines which are electrically"distant" from the study region, the detailedmodeling of these machines may not be necessary.Some or all of the rotor and/or stator dynamics ofthese machines may be neglected.(iv) When approximating the dynamics of "distant"
portions of the power system by dynamic equivalents,a minimum amount of "equivalent synchronous machine"data may be available, necessitating the use ofsimpler machine representations.
Lower-order representations are derived byneglecting the dynamics of appropriate rotor andstator winding fluxes and currents. The number ofdynamical states associated with the differentsynchronous machine models are given in Table 2.
Windings State Variables
Machine Model Rotor With WithoutStator Direct Quadrature Stator Stator
Axis Axis Dynamics Dynamics
Round Rotorwit d,q F,D G,Q 6 4
RoundaRotor - No d,q F G 4 2
Salient Pole with daDamper Windings QF,D 3
Salient Pole - No 3
Damper Windings d,q F 3 1
Classical dqFwtTransient Model d,q constant 2
flux
TABLE 2 - Dynamic Modeling ofSynchronous Machines
The effects of magnetic saturation in thegenerator may also be of interest in some studies.These nonlinear effects are typically accounted forby representing the mutual inductances of themachine as functions of the air-gap fluxes in themachine. Saturation of the leakage inductances isusually neglected [11].
Machine Shaft Dynamics
The machine shaft may be represented by a
multi-mass - spring - dashpot system, using as manyrotating masses as data availability permits [10].The multi-mass model is typically used whensubsynchronous resonance (SSR) and other torsionalsystem-related oscillations are to be analyzed. Forother studies, a single-mass mechanical systemdescription is sufficient. The number of statesassociated with shaft dynamics is equal to twice thenumber of masses represented.
Auxiliary Equipment
The auxiliary equipment of interest in dynamicstability studies consists of excitation systems,power system stabilizers, and turbine-governorsystems. Low frequency mode].s of typical excitation
systems are described in detail in [121. For dynamicstability studies, linearized versions of thesemodels are used. Power system stabilizers arecommonly represented by a signal washout plus twolead-lag pairs, or by a signal washout plus a pairof complex poles and zeros. The input to thestabilizer can be the shaft slip of a machinemass, the machine terminal frequency deviation, orthe machine accelerating power. Turbine-governormodels are described in detail in 113].
Network Representation
Each transmission line is represented as alumped RLC circuit. Lumped parameter representa-tions are also used for series and shunt capacitors,shunt inductors, variable-tap transformers and phase-shifting transformers. When high-frequency dynamicsmay be of interest, the dynamics of the network'sinductive and capacitive elements (i.e., the timederivatives of inductor currents and capacitorvoltages) must be explicitly represented. This levelof modeling is typically used in studies of network-machine interactions such as SSR and self-excitation.The dynamics of the elements in the abc coordinatesystem are transformed via Park's Transformation1141 to a synchronously rotating dqo reference frame,and the zero-sequence dynamics are dropped sincebalanced system operation is assumed. Each capacitoror inductor whose dynamics are explicitly representedgives rise to two dynamical states.
When the focus of a dynamic stability study ison the typically low-frequency inter-machine andinter-area electromechanical oscillations, thedynamics of the network elements may be neglectedand the synchronous frequency, "steady-state" modelof the network elements may be used. This componentmodel is valid for oscillations at or near thesynchronous frequency (ws). Since the "low-frequency"inter-machine modes of interest (WR/2n < 3 Hz) arereferred against a rotating dqo reference frame,then with respect to the stationary abc referenceframe, these modes appear to be oscillations at
frequencies of (Ws ± WR) - Us. Thus, theapproximation that the signals throughout thenetwork are at the synchronous frequency is a
reasonable one when only the "low-frequency"inter-machine phenomena are of interest.
Note that in both models of the network,"infinite buses" may be represented as directconnections to ground since, by definition, thesebuses have no incremental characteristics, and thus,their small-signal dynamic properties are identicalto those of the ground node.
IV. CONSTRUCTION OF THE SYSTEM MATRIX
While conceptually straightforward, the actualautomatic formulation of the system A matrix is quitecomplicated. The approach proposed below accomplishesthe task in a computationally efficient manner. Thenetwork subsystem representAtion is assembled from theRLC element representations in one of two differentways, depending on whether network dynamics are modeledor neglected. When the dynamics of the network are
represented, the capacitor voltages and inductorcurrents chosen as state variables must provide a
minimal complete characterization of the networkdynamics. A linearly independent set of variables isspecified by choosing as state variables a set ofcapacitor voltages possessing the property that theassociated capacitors form no capacitive looops, and a
set of inductor currents possessing the property thatthe associated inductors form no inductive cutsets.The set of state equations which describe the network
dynamics is then systematically formulated using theprocedure developed by Kuh and Rohrer (i5 ].
When the network dynamics are neglected, thesynchronous steady-state description of the networkis used:
_=i- Y v (4)
For a network with b nodes, Y is the 2b x 2b nodaladmittance matrix, and i (v) is the vector of nodalinjections (voltages).
In both network representations, the abcnetwork variables are transformed to thesynchronously rotating reference frame throughPark's Transformation.
The representations for each machine and itsauxiliary equipment are coupled to the networkrepresentation by the machine stator current andvoltage variables, which are related to the nodalinjection and voltage variables at the machinebuses. Because the machine stator currents andvoltages are referenced against the rotatingquadrature axes of the machines, these currents andvoltages must be transformed to the synchronouslyrotating reference frame by a linear transformationof the form r -
Cos 6h -sin hT =
sill 6 hcos 6h(5)
where 6h is the angle between the quadrature axis ofmachine h and the systemwide reference axis.
The equations which describe the network,machine and equipment subsystems are linearizedabout the system state x.s When network dynamicsare represented, the linearized stator currents ofmachine h (Aidh and Aiqh) are components of themachine/equipment state variable (AxM ), and thelinearized terminal voltages of the machine (Avdhand Avqh ) are components of the network state vari-
able N).The system representation is:
-M1 AM M1 ° ...AMAm1[ AXM1
AM Am0M .. A MN Am2 2' 2 ~~2' 2
[ ~ ~ N J [ - . ~~~~~~(6)AXNj A N,M1 AN,M2 AN,N AXN
The matrix AN,NiS usually quite sparse, reflectingthe fact-that a node in a power network is directlyconnected to very few other nodes in the network.Moreover, the coupling matrices AMhN and AN,Mh are
extremely sparse because their non-zero elementsreflect the stator voltage and current inputs to themachine and network subsystems, respectively.Therefore the system matrix A has an "almost" blockdiagonal structure.
When network dynamics are not represented,consistency of modeling assumptions demands that themachine stator dynamics be neglected as well. In thiscase, the linearized equations for the machine/auxili-ary equipment subsystems have the form of (7), in whichthe algebraic equations arise from the assumption that
the incremental stator flux dynamics can be neglected.
h'h h'N h' h
1AX Mh1=h M h'NV Mh' a Mh(7)
~-h
LJL.h,Mh UM,NV_MNI v h(
[At M_
where AiMh = [Ai dh I Aiqh and Av Mh = LAdh, Avqh-
For the network, the algebraic system
[Li [xMMXYMN1R M1
[J Y NM Y NNJL NJ
is used, where
M [aM1' --M2. -Mk
AVM AVMM1 mM2. _M]
229
(8)
and AvN iS the vector of direct and quadrature axisnodal voltages at the non-machine nodes.
The equations in (7) and (8) are combinedinto a linear differential-algebraic system, and thealgebraic equations are eliminated from thesystem formulation using sparsity-oriented Gaussianelimination. This results in the linear differentialsystem^ w _ _ e_
AxMl A MlMAAmMm2 -M1 k AXM1
dt AXZ M21 A m2'1 22 2 k2 (9)dt *-- MJ[M
AX Mk AkKMkMA M2 kMk AX MThe system matrix in (9) is typically of much lowerdimension than the system matrix which representsthe same system with network dynamics (6). This isso because no dynamical states are associated withthe network elements, and because for each machine,the number of states is reduced by two. However,the system matrix (9) is usually much less sparsethan the system matrix (6).
V. EIGENSYSTEM EVALUATION
Once the system matrix A is constructed, theeigensystem of A may be evaluated. Because A isnon-symmetric and has no special exploitablestructure, a general eigenvalue evaluation scheme,based on the conversion of the system matrix toupper Hessenberg form and Francis' QR algorithm, isused. This algorithm is noted for its great accuracy,efficiency, and numerical stability [9]. However,the iterative solution technique which it employsdestroys the sparsity of A in the course of thesolution process. (For this reason, the loss of Amatrix sparsity due to the Gaussian elimination ofthe algebraic network equations in the "low-frequency"formulation (9) is of little consequence.) Sincethe algorithm cannot exploit matrix sparsity, it isnecessary to allocate computer storage for all p2elements of the p-dimensional system matrix A forthe calculation of the eigenvalues.
Constructing-the A matrix from the differentialequations on the subsystem-by-subsystem basis discussedearlier keeps the number of computational operationsassociated with this step - matrix inversions,multiplications, etc. - small, thereby reducing boththe computational resource requirements and numericalaccuracy problems associated with the A matrix forma-tion. Because little numerical error is introducedduring the construction of the system matrix, theaccuracy of the eigenvalues is limited primarily bythe numerical precision of the eigenvalue evaluationsubroutines. The EISPACK general-purpose eigenanalysispackage is used for the calculation of the eigenvalues.For systems with as many as 500 state variables, theeigenvalues calculated have been shown to beaccurate to + 10 sec-
230
For sensitivity analysis of the eigenvalueswith respect to system parameters, eigenvectors of Aand AT must be evaluated. Relationship (3) indicatesthat only those eigenvectors associated with eigen-values of interest need be determined. For thisreason, it is most appropriate to calculate thenecessary eigenvectors individually. An effectivetechnique is the modified inverse iterationalgorithm [16]. Sparsity-oriented techniques can
be employed to great advantage durinig-thesecalculations.
VI. THE EISEMAN PACKAGE
The EISEMAN dynamic stability analysis packageis a computer tool which implements the componentmodeling features and solution methodology discussedabove. The package automatically formulates thesystem A matrix and calculates all of its eigenvaluesand the sensitivity of the eigenvalues to power systemparameters of interest. In addition, EISEMAN can
evaluate the eigenvalues of the machine torsional sub-systems and the eigenvalues of the network subsystem.The package can also be used to produce root loci ofeigenvalues with respect to system parameters.
A wide variety of component models have beenincorporated into the EISEMAN package thus far:
* The synchronous machine rotor can be modeledusing from 0 to 4 rotor circuits, as outlinedin Table 2. The dynamics of the stator windingsmay be modeled or neglected, at the user'sdiscretion.
* Detailed models which account for generator
magnetic saturation are provided.
* The machine shaft may be modeled using as many
masses as the user desires.
*Ten turbine-governor models, with as many as
eight state variables each; nine excitationsystems models, with as many as seven statevariables each; and power system stabilizermodels have been implemented.
*The network model - transmission lines, RLCshunt elements, and transformers - may berepresented with or without dynamics, dependingon the user's needs.
* Power system loads can be represented as havingthe dynamics of equivalent shunt impedanceelements, infinite buses, equivalent synchronousmachines, or they can be neglected.
With the exception of the eigenvalue calculationsubroutines, all of the software for the EISEMAN pack-age was developed at the Pacific Gas and ElectricCompany. The package has been implemented in a modularmanner. Consequently, additional component models can
be easily incorporated.
A most significant feature of the EISEMAN packageis its capability of analyzing large dynamic systems.Power systems with up to 250 machines, 1,500 buses,2,000 lines, and 500 dynamic states can be studiedusing the present version of the program. Insofar as
the authors are aware, no other dynamic stabilityanalysis tool possesses the capability of providing thecomplete spectrum for power systems of this size. ThisFORTRAN package has been implemented using variabledimensioning of arrays, so the capabilities and size ofthe program can be easily increased or decreased toconform to the applications and computer resources athand.
An innovative feature of EISEMAN is the implement-ation of the algorithm discussed. in Section IV for theconstruction of the A matrix. EISEMAN's uniqueapproach to the setup of the network equations and theA matrix is computationally straightforward, and thesparsity-oriented techniques used to form both thesystem A and 3A/3a matrices are highly beneficial fromthe standpoints of preserving accuracy and reducingcomputer CPU time requirements. For large systems inwhich network dynamics are explicitly represented,typically fewer than 5% of the elements of the systemmatrix are non-zero. Very little numerical error isintroduced in the evaluation of the elements of Aand BA/3a.
The amount of computer memory required by thepackage is determined primarily by the dimension pof the state space of the system to be analyzed.The double-precision arithmetic and the QR algorithmemployed for the calculation of the eigenvaluesrequire that the program size be at least 8p2 bytes.The use of overlaying techniques confines thecomputer memory requirements to approximately 8p2 +450 K of memory.
The CPU time used in the calculation ofeigenvalues has been empiricially determinedroughly proportional to p2.83. The CPU timeduring the formation of the system matrix isusually quite small in comparison to theeigenvalue calculation time requirements.
theto beused
VII. APPLICATION EXAMIPLES
The usefulness of a versatile stabilityanalysis tool is illustrated in this section byexamples of EISEMAN's application to several powersystems.
Subsynchronous Resonance Benchmark Test System
This test system was developed as a standardtest case for the purpose of facilitating comparisonsof analytical tools capable of studying SSR. Thelinearized model of the system consists of a singlemachine connected to an infinite bus through a series-compensated transmission line. The machine is modeledas a round rotor machine with damper windings in thedirect and quadrature axes. A six-mass model describesthe machine's torsional dynamics. No auxiliaryequipment is represented. Data for this test case canbe found in [18].
In order to analyze the interaction of thenetwork and the machine torsional subsystems, networkdynamics, stator dynamics, and machine torsionaldynamics must all be explicitly represented. Theeigenvalues of this system and their sensitivitiesto the series capacitance are presented in Table 3.
TABLE 3 - Eigenvalues for SSR BenchmarkTest System (in sec01)
Although each mode of an eigensystem will appear
to a greater or lesser extent in the response of eachOc the state variables of the system (as determined bythe eigenvectors of A), for "weakly coupled" sub-systems, it is often possible to associate the originof certain system modes with one or more subsystems.The sensitivities of the eigenvalues to varioussystem parameters reveal these relationships.Column 5 of Table 3 notes the correspondence betweenthe system eigenvalues and the various subsystems.Columns 3 and 4 present the eigenvalues for thecases in which the damper dynamics and the torsionaldynamics are not modeled.
Figure 1 presents the loci of the real parts ofsome of the eigenvalues as a function of the level ofcapacitive compensation. Note that as the compensationincreases, the subsynchronous electrical frequencydecreases, and the eigenvalues associated with thedifferent torsional modes are destabilized in turn.
Olsec-I
3-
2-
I1I
-1
-2 -
-3
-4
-6 I
COMPLETE MACHINE MODEL MODREiOAMPER WINDINGS NEGLECTED
TORSIONAL TORSIONALMODE 4 MODE 3
TORSIONAL
MODE 2
10 20 30 40 50 6O% COMPEN-
70 80 90 SATION
SUBSYNCHRONOUSNETWORK MODE
-'1 FIG. 1 - Damping of Eigenvalues as aFunction of Compensation
New England Test System
The application of EISEMAN to the study ofinter-machine oscillations is illustrated by this10 machine, 39 bus, 46 line test case. This testsystem was developed as one representative oftransmission systems in the Northeastern UnitedStates. The network diagram and data for this casecan be found in l19].
The dynamic stability phenomena of primaryinterest in this study are the "low-frequency" 1 Hz -
2 Hz oscillations,which are strongly related to themachine swing equations. In the base case studiedhere, network and machine stator dynamics, rotortransient dynamics (F and G windings), electro-mechanical dynamics (single-mass machine models are
used), and excitation system dynamics are modeled..Saturation and turbine-governor dynamics are neglected.Loads throughout the network are modeled dynamically as
equivalent RLC elements.
The eigenvalues calculated by EISEMAN for thiscase are shown in Table 4.
TABLE 4 - Eigenvalues for New EnglandTest System (in secG)
The eigenvalues fall into five distinct groups:
(i) Oscillatory modes at or very near thesynchronous frequency. These modes are closelyassociated with the R-L elements of the transmissionnetwork and loads. The purely real eigenvalues asso-
ciated with R-L networks are transformed to oscilla-tions at the synchronous frequency when viewed fromthe rotating reference frame. The neglect of loaddynamics results in the elimination of all of theeigenvalues in this group for which a > -1, reflectingthe strong association between those system modes andthe dynamics of the network elements used to modelthe loads.
(ii) Other oscillatory modes at frequenciesgreater than 35 Hz. Most of the eigenvalues in thisgroup are closely associated with the representationof line-charging capacitors. The interaction ofthese capacitors with the inductive components ofthe transmission lines results in the creation ofoscillatory modes. These modes occur at frequenciesof 220 Hz and above, and reflect the relativelysmall effects of line-charging in tightlyinterconnected eastern U.S. power systems. Theremaining eigenvalues in this group are associatedwith the capacitive elements which are used tomodels loads for which reactive power consumption isnegative.
(iii) Non-oscillatory modes with -50 < a < -15.The eigenvalues in this group are primarily associatedwith excitation system dynamics.
231
232
(iv) Oscillatory modes with 1 < w < 10. Theeigenvalues in this group correspond to the modesobserved in the responses of the machine rotors tosmall disturbances on the power system. It is thisgroup of eigenvalues which is the primary focus ofthis dynamic stability study.
(v) Lower frequency oscillatory modes andslowly decaying non-oscillatory modes. The eigenvaluesin this group are related to the transient electricaldynamics of. the machines (the field and "G" windings)and to excitation system dynamics.
The neglect of load dynamics and network dynamicshas very little effect on the eigenvalues of primaryinterest, those in group (iv). It is important,however, to note the impacts of different degreesof modeling detail on the computational resourcesrequired for the analysis of system dynamic stability,as illustrated in Table 5.
Computer CPU TimeCase Dynamic Memory (IBM 3033
States (Bytes) OS/MIVS)
New England 28 10K 5. eBase Case 288 l 10 K 55.2 sec
Line Charging 212 810 K 28.6 secNeglected
Load Dynamics 248 940 K 40.4 sec
All NetworkDynamics 66 400 K 2.2 secNeglected
TABLE 5 - New England Test System - Comparisonsof Computer Resource Requirements
Western United States Test System
This study demonstrates the application of EISEMANto the study of potential subsynchronous osciliationson a realistically-sized power system. The system,illustrated in figure 2, is representative of thosein the Western United States. The transmission net-work is composed of a 500 KV transmission system andan underlying 230 KV system. Series compensation isused throughout the 500 KV system in order to increasethe power transmission capability of the transmissionnetwork.
Eight large synchronous machines connected tothe 500 KV system are represented in this studyusing the most detailed synchronous machine model.Because SSR phenomena are the primary focus of thestudy, stator dynamics, network dynamics, andtorsional dynamics must all be explicitly represented.No excitation systems are modeled. Four of the unitsare modeled with ten-mass tors.ional systems. Thetorsional systems of the four remaining machinesare represented with six-mass models. Fifty-sixoscillatory modes, with frequencies from 6 Hz to600 Hz, are associated with these torsional subsystems.
As discussed earlier, several questions of majorimportance in the modeling of large power systemsrevolve around defining the system to be studied.How much of an interconnected power system should bemodeled in order to obtain accurate results from astudy? How should the portions of the system beyondthe chosen study area be represented? In this casestudy, two possible models of the study network areexamined. In the first (System A), only the 500 KVsystem is retained and the 230 KV system is modeledby equivalents at the 8 buses through which the 500 KVand 230 KV systems are interconnected. In the second(System B), both the 500 KV and 230 KV systems are
500 KV SYSTEM
MACHINES MACHINES MACHINES
FIG. 2 - Western United States Test Systemretained, with equivalents at the 15 buses which tiethe 230 KV network to lower voltage networks andmachines or to portions of the 230 KV system definedto be beyond the study area. The equivalents usedhere are calculated from the short circuit duty atthe "boundary" buses using a short circuit analysispackage. Each equivalent impedance developed in thismanner is connected between the appropriate boundarybus and the infinite bus. The System A representationconsists of 79 lines, 68 buses, and 370 statevariables. The System B representation consists of109 lines, 78 buses, and 428 state variables.
As in the New England test case discussed above,the eigenvalues of each of these systems can beclassified into groups which are closely associatedwith the torsional, network, machine transient,machine subtransient, or electromechanical (rotor)dynamics. For either of the two system representa-tions, all but two of the system modes appear to beadequately damped. The positive eigenvalues, whichare closely related to the torsional subsystems ofmachines 1 and 2, are shown below:
System A: +0.17 ±j 126.9+0.08 ±j 114.9
System B: +0.12 ±j 126.8+0.11 ±j 114.9
The sensitivity analysis feature of the EISEMANpackage can be used to determine which series capaci-tors have the greatest impact on the potential SSRproblems, possibly leading to more appropriate choicesfor series compensation. Thit feature may also beused for studying the sensitivities of the eigenvaluesto the system equivalents used. -The sensitivities ofall of the eigenvalues in the region of the twopositively damped modes are presented in Table 6 forSystem A. For this system, the parameters of interestare the inductances of the equivalents which representthe 230 KV transmission system. The data in Table 6shows that the unstable modes are most sensitive to-the equivalent at bus 18.
233
Equivalent(LWq) Eigenvalues and Sensitivities ( kA/)L,g) (in sec-i) l
Is Located +0.17 +j 126.9 -0.09 +j 127.1 +0.09 +j 114.9 -0.08 +j 115.1 -8.9 +j 146.5 -9.2 +j 128.5
TABLE 7 - Sensitivities of Eigenvalues to Equivalents - System B
The sensitivities of the corresponding eigen-values are displayed in Table 7 for System B. Inthis case, the system parameters of interest are theequivalents at buses 69, 79 and 80. (These are the"boundary" buses of the 230 kV system which areclosest to bus 18, the bus at which the greatestsensitivity of the critical eigenvalues to the 230 KVequivalent was noted.) The sensitivity of thecritical eigenvalues to the equivalents is markedlylower in this case. The use of the more detailedsystem representation decreases the sensitivity ofthe study results to the non-modeled portions of thepower system. It must be kept in mind, however,that the more detailed representation increases thesystem dimension from 370 to 428 states, increasingthe CPU time required for the computation of thesystem eigenvalues from 134 seconds to 211 seconlds.
VIII. CONCLUSION
This paper has described in detail an effectiveeigenvalue-based tool, known as the EISEMAN package,which was developed for the comprehensive analysis ofdynamic stability phenomena in power systems. Theefficient manner in which EISEMAN handles the dynamicstability analysis of large power systems, the broadflexibility permitted in the modeling of power systemcomponents, and the original approach to the automatedsetup of the network equations make EISEMAN a uniquetool. EISEMAN has been applied to study subsynchronousresonance phenomena, inter-machine oscillations,control system design and the influence of modelingdetail in system component representation. Extensiveuse of EISEMAN bears out the effectiveness of this toolfor the analysis of power system dynamic stability.
An important feature of the EISEMAN packageis its modular nature. This permits the implemen-tation of additional models in a ratherstraightforward manner. Extension of the workreported here is already underway. The scope of thesystem model is being expanded to incorporate DCtransmission lines, static VAR sources and inductionmachine representations. Dynamic load models, aswell as the application of coherency based dynamicequivalents 117] are being investigated. Results ofwork in these areas will be reported later.
ACKNOWLEDGMENT
The authors wish to thank Mr. J. F. Luini of thePacific Gas and Electric Company for his many fruitfuldiscussions during the development of the EISEMANpackage, and for his assistance in providing some ofthe data used in this paper.
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GEORGE GROSS (M'75) was born in Cluj, Romania.He was graduated from McGill University in Montreal,Quebec, with a B.Eng.(Hons) in Electrical Engineering.He continued his studies at the University ofCalifornia at Berkeley where he was awarded the M.S.and Ph.D. degrees in Electrical Engineering andComputer Sciences. His research work at Berkeleywas in the areas of optimization, nonlinear controland power systems.
Dr. Gross joined the Engineering ComputerApplications Department of Pacific Gas and ElectricCompany in 1974. He is in charge of directing and @
conducting work involving the investigation, evalua-tion and development for computer implementation ofnew mathematical techniques and analytical engineer-ing methodologies for solving problems arising in powersystem planning, operation and control. Dr. Grossis a Supervising Computer Applications Engineer witlthe utility.
Dr. Gross has taught graduate level courses inpower systems planning and operations at the Univer-sity of Santa Clara, California, the Center forProfessional Advaincement, New Jersey, and the Univer-sity Extension, University of California, Berkeley.He is a member of the Computer and Analytical MethodsSubcomnmittee of the Power Systems Engineering Committeeof the IEEE Power Engineering Society.
CARL F. IMPARATO (M'78) received. his B.S. degreein Engineering from the California Institute ofTechnology in 1974 and was awarded the degree ofMaster of Engineering in Electrical Engineering andComputer Sciences by the University of Californiaat Berkeley in 1977. He has been a ComputerApplications Engineer with the Systems EngineeringGroup of the Pacific Gas and Electric Company since1977, working primarily in the areas of dynamicstability and power system equivalencing.
PUI MEE LOOK (M'75) received the B.S. and M.S.degrees in Electrical Engineering and ComputerSciences from the University of California atBerkeley in 1971 and 1972, respectively. Shejoined the Pacific Gas and Electric Company, SanFrancisco, California, in 1972. She is a ComputerApplications Engineer in the Systems EngineeringGroup of the Engineering Computer ApplicationsDepartment. Her work is in the areas of load flow,dynamic stability and optimal power flow.
