ieee std 1110-1991, ieee guide for synchronous generator modeling practices in stability

IEEE Std 1110-1991

IEEE Guide for Synchronous Generator Modeling Practices in Stability Analyses

SponsorPower System Engineering and Electric Machinery Committeesof theIEEE Power Engineering Society

Approved March 21, 1991

IEEE Standards Board

Abstract: Categorizes three direct-axis and four quadrature-axis models, along with the basic transientreactance model. Discusses some of the assumptions made in using various models and presents thefundamental equations and concepts involved in generator/system interfacing. Covers, generally, thevarious attributes of power system stability, recognizing two basic approaches. The first is categorizedunder large-disturbance nonlinear analysis; the second approach considers small disturbances, where thecorresponding dynamic equations are linearized. Applications of a range of generator models arediscussed and treated. The manner in which generator saturation is treated in stability studies, both in theinitialization process, as well as during large or small disturbance stability analysis procedures isaddressed. Saturation functions that are derived, whether from test data or by the methods, of finiteelements are developed. Different saturation algorithms for calculating values of excitation and internalpower angle, depending upon generator terminal conditions are compared. The question of parameterdetermination is covered. Two approaches in accounting for generator field and excitation system basequantities are identified. Conversion factors are given for transferring field parameters from one base toanother for correct generator/excitation system interface modeling. Suggestions for modeling of negativefield currents and other field circuit discontinuities are included.Keywords: Synchronous generator stability models, modeling practices, saturation practices, stability datadetermination and application

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Foreword

(This Foreword is not a part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in StabilityAnalyses, but is included for information only.)

The Joint Working Group on Determination and Application of Synchronous Machine Models for Stability Studieswas formed in 1973. The scope of the Working Group was updated in 1986 and now reads as follows:

“Define synchronous machine models, particularly for solid iron rotor machines, for use in stability studies, andrecommend standard methods for determining the values of parameters for use in these models by calculation and/ortest. Assess the effect of magnetic saturation on these parameters. Devise and recommend analytical methods forincorporating such machine models, including representation of saturation, into stability programs.”

The working group has been responsible for two particular IEEE Committee Reports on the subject of machinemodeling. The first was published in PA & S in 1980. In 1983, we presented a one-day symposium on the subject ofmachine modeling and generator stability data acquisition at the IEEE PES Winter Power Meeting. Followingpublication of our second IEEE committee report in March 1986 (vol. EC-1), the group and the two committees towhom we report (PSE and Electric Machinery) suggested that application be made to the New Standards Committee(NesCom) of the Standards Board for permission to publish a Guide outlining the work which we had sponsored overthe past fifteen years. A Project Authorization Request was made through the Power System Engineering Committee,which was approved by the IEEE Standards Board on Dec. 12, 1985. This occurred at the December 12, 1985 meetingof NesCom, and the request was issued as PAR No. P1110.

In consulting this document, the reader is advised that it is intended to provide advice of a general nature. The primaryaim of such a guide is to enable an engineer, who is relatively inexperienced in stability analyses, to perceive some ofthe important issues in this branch of power system engineering. As such, any reader, experienced or not, should beable to consult specific chapters of interest, without having to examine the entire contents.

The membership of the Joint Working Group has remained to a large extent unchanged since this the first draft of theguide was published in mid-1986. Significant contributions were also made by P. M. Anderson and F. P. DeMello,Working Group members for several years prior to 1986 and up to 1988. Joint Working Group representation fromoverseas was achieved in the past through Dr. G. Shackshaft. Dr. R. G. Harley joined our Group in 1989. Localassociates of the Chairman who have helped substantially during the past four years are Dr. P. Kundur, G. J. Rogers,and J. R. Service, and Dr. A. Semlyen. D. C. Lee assisted in coordinating the drafting of figures. M.E. Coultes, whopreceded Mr. Lee as a member, was also most helpful.

Since late 1986, the Chair wishes to acknowledge the assistance given through his association with the ElectricalEngineering Department of the University of Toronto. In particular, he appreciates the encouragement and supportgiven by Dr. A. S. Sedra, Dr. G. R. Slemon and Dr. J. D. Lavers of the Electrical Engineering Staff, especially in thelogistics of the Chair's activities through more mundane, but essential support in mailing, office space,communications costs, etc.

Appreciation is also extended to Mrs. Linda Espeut of the Electrical Engineering Department and is especially notedfor her speedy and accurate assembly of the final drafts of this guide.

The Chair of the working group would like to thank the organizations for which those individuals (who contributed tothe development of this document) work.

iii

This document was prepared by the Joint Working Group on Determination and Application of Synchronous MachineModels for Stability Studies. The members of the working group were:

P.L. Dandeno, Chair

B. AgrawalD.H. BakerC. ConcordiaJ.S. Edmonds

L.N. HannettR.G. HarleyD.C. LeeS.H. Minnich

S.J. SalonR.P. SchulzH.R. SchwenkS. Umans

The following persons were on the balloting committee that approved this standard for submission to the IEEEStandards Board:

P. M. AndersonJ. C. AgeeR. A. AldenR. F. BaylessA. BoseC. BowlerV. BrandwajnR. M. BucciR. T. ByerlyA. CalvaerS. ChanM. S. ChenJ. ChowR. ChuC. ConcordiaR. CravenR. CreightonM. DamborgP. L. DandenoF. P. de MelloC. DidriksenH. DommelP. J. DonalekJ. DoudnaR. D. DunlopR. G. FarmerJ. H. FishA. A. Fouad

A. GermondW. B GishH. GlavitschJ. GraingerC. E. GrundA. E. HammadJ. F. HauerR. A. HedinK. HemmaplardhG. T. HeydtD. J. HillJ. HurleyE. KatzR. KumarM. LaubyS. LefebvreC. LuiP. KundurJ. LuiniP. MagnussonA. MahmoudO. MalikY. MansourD. MartinS. MokhtariD. L. OsbornM. A. PaiM. PalM. C. Patel

M. PavellaW. PriceN. D. RaoK. ReichertD. D. RobbR. RobergeL. RodriguezP. A. RuscheN. SainiS. SavulescuR. SchlueterR. P. SchultzG. ScottA. J. SoodK. StantonS. StantonD. A. SwannY. TamuraJ. TangC. TaylorR. J. ThomasJ. Van NessS. VirmaniV. J. VittalO. WasynczukF. F. WuYo N. YuJ. Zaborsky

iv

When the IEEE Standards Board approved this standard on March 21, 1991, it had the following membership:

Marco W. Migliaro, Chair Donald C. Loughry, Vice Chair

Andrew G. Salem, Secretary

Dennis BodsonPaul L. BorrillClyde CampJames M. DalyDonald C. FleckensteinJay Forster*David F. FranklinIngrid Fromm

Thomas L. HannanDonald N. HeirmanKenneth D. HendrixJohn W. HorchBen C. JohnsonIvor N. KnightJoseph L. Koepfinger*Irving KolodnyMichael A. Lawler

John E. May, Jr.Lawrence V. McCallDonald T. Michael*Stig L. NilssonJohn L. RankineRonald H. ReimerGary S. RobinsonTerrance R. Whittemore

* Member Emeritus

Deborah A. CzyzIEEE Standards Project Editor

v

CLAUSE PAGE

1. Introduction .........................................................................................................................................................1

1.1 .................................................................................................................................................................... 11.2 References .................................................................................................................................................. 2

2. Model Classifications or Categories ...................................................................................................................3

2.1 Practical Models Available ........................................................................................................................ 32.2 Nomenclature and Glossary of Frequently used Terms............................................................................. 72.3 References ................................................................................................................................................ 11

3. Classification of Stability Studies .....................................................................................................................12

3.1 Background .............................................................................................................................................. 123.2 Large Disturbances Stability .................................................................................................................... 133.3 Small Disturbance Stability ..................................................................................................................... 143.4 Classification Based on Dominant Modes of System Response.............................................................. 143.5 References ................................................................................................................................................ 143.6 Bibliography............................................................................................................................................. 15

4. Application of Generator Models in Stability Studies ......................................................................................15

4.1 General ..................................................................................................................................................... 154.2 Representation of Generations During Large Disturbances..................................................................... 184.3 Modeling of Machines for Small Disturbance Stability Studies.............................................................. 194.4 References ................................................................................................................................................ 204.5 Bibliography............................................................................................................................................. 20

Annex 4A Calculation of Generator Electrical Torques or Powers (Informative) .......................................................22

5. Representation of Generator Saturation and its Effect on Generator Performance ..........................................24

5.1 General ..................................................................................................................................................... 245.2 Representation of Generator Saturation in the Steady State .................................................................... 245.3 Representation of Saturation Effects During Large Disturbances ........................................................... 265.4 Generator Saturation in Small Disturbance Modeling ............................................................................. 295.5 References ................................................................................................................................................ 315.6 Bibliography..............................................................................................................................................32

Annex 5A Saturation—Past Practices and General Considerations (Informative).......................................................33

Annex 5B Steps Used in a Widely-Used Commercial Stability Program to Account for Saturation During the Step-by-Step Calculations (Informative) ..........................................................................................................36

Annex 5C Procedures in a Second Stability Program to Account for Saturation When Adjusting Mutual Reactances (Informative) ..................................................................................................................................39

Annex 5D Finite-Element-Derived Steady-State Saturation Algorithms (Informative)...............................................44

Annex 5E Comparison of Certain Existing Methods of Accounting for Saturation (Informative) ..............................48

vi

vii

CLAUSE PAGE

6. Determination of Generator Stability Parameters .............................................................................................51

6.1 Stability Parameters Obtained by Testing Generators Under Short-Circuit or Open-Circuit Conditions ................................................................................................................................................ 51

6.2 Frequency Response Testing of Generators............................................................................................. 516.3 Parameters Derived by Two Manufacturers in the Machine Design Stage ............................................. 536.4 Desirability for Uniform Practices in Deriving Machine Stability Parameters ....................................... 536.5 Alternative Forms of Model Representation............................................................................................ 556.6 References .................................................................................................................................................56

Annex 6A Determination of Direct-Axis Parameters from Test Results (Informative) ...............................................58

Annex 6B Alternate or Nonstandard Methods of Obtaining Stability Parameters (Informative).................................63

Annex 6C Generator Stability Data Translations (Informative) ...................................................................................65

7. Field and Excitation Considerations .................................................................................................................71

7.1 Establishing Field-Voltage, Field-Current and Field-Impedance Bases.................................................. 717.2 Calculation of Field Resistance................................................................................................................ 727.3 Field-Circuit Identity................................................................................................................................ 737.4 Special Techniques for Modeling Field-Current Reversal or Field Shorting .......................................... 737.5 References ................................................................................................................................................ 74

Annex 7A Establishing and Comparing Field-Circuit Relationships—Reciprocal vs. Nonreciprocal System(Informative) .................................................................................................................................................................75

Annex 7B Excitation System-Generator Simulation Interfaces (Informative) .............................................................78

IEEE Guide for Synchronous Generator Modeling Practices in Stability Analyses

1. Introduction

1.1

This document has been written to assist the power system analyst in choosing appropriate generator models for powersystem stability studies.

More complex and advanced control concepts have recently been employed to solve a broad range of stability problems.General experience when conducting such studies has indicated that more detailed generator models are necessary,especially when applying excitation system controls for the enhancement of generator dynamic performance.

Discussions were held in 1984 and 1985 to formulate the guide outline, and to decide what it should contain. At thattime, it was felt that computer codes, including generator/network interfacing equations, as well as many otherstability program details, all be included. Space limitations, as well as computer program proprietary restrictions,forced the Joint Working Group to abandon this aspect of generator stability modeling.

Included in this guide, however, is a short section on one approach to generator/network interfacing. In addition,details of excitation system generator interfacing considerations are given, as noted below.

Many improvements, particularly in recent years, have occurred in generator modeling, and the objective of this guideis to set forth much of the current wisdom, and discuss the principal issues that ought to be considered in theapplication of more sophisticated models. These have now become available partly due to better digital simulationcapabilities, and also because of enhanced data acquisition procedures.

In this document, the Working Group has also attempted to touch on some of the fundamental analytical steps in thetreatment of generator stability representation, as proposed by investigators about fifty years ago. References to severalpioneer authors are given in Chapters 2, 4, 5 and 7.

This guide does not attempt to recommend specific procedures for machine representation in non-standard or atypicalcases such as generator tripping and overspeed operation, or the models for harmonics or unbalanced operation. Theparticular modeling requirements for Subsynchronous Resonance studies have been examined by other IEEE-PESgroups, and a second Benchmark Model for SSR Analyses was published in 1985. [3]1

1The numbers in brackets correspond with those listed in each Reference section in each Chapter.

Copyright © 1998 IEEE All Rights Reserved 1

IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR

Three direct-axis and four quadrature-axis models are categorized in Section 2.1 of Chapter 2, along with the basic“transient reactance” model familiar to many. Principal aspects of the various two-axis models relate to their structureand to the parameter values assigned to the elements of individual models. The general concordance between the morefamiliar generator reactances and time constants, and the various; direct and quadrature-axis model structures ispointed out in Section 2.1.

Section 2.1 also discusses some of the assumptions made in using various models, including the basic model. It alsotouches upon the fundamental equations and concepts involved in generator/system interfacing. In Section 2.2, aglossary of terms is given, along with the associated nomenclature used in all the following chapters.

Chapter 3 covers the various attributes of power system stability in a general way, with two basic approaches beingrecognized. The first is categorized under large-disturbance nonlinear analysis. The second approach considers smalldisturbances, where the corresponding dynamic equations are linearized.

In Chapter 4, applications of a range of generator models, documented in Chapter 2, are discussed. These applicationsare also treated in the context of the nature and complexity of large and small power system disturbances that arecovered in Chapter 3.

Chapter 5 covers the manner in which generator saturation is treated in stability studies, both in the initializationprocess, as well as during large disturbance or small disturbance stability analysis procedures. Included in theappendixes of Chapter 5 is the development of saturation functions that are derived either from test data or by themethods of finite elements. The appendixes of Chapter 5 also compare different saturation algorithms for calculatingvalues of excitation and internal power angle, depending upon generator terminal conditions.

After treatment of saturation, the question of parameter determination is thoroughly covered in Chapter 6. Suchparameters are found either by test, as in IEEE Std115-1983 (R1991) [1] or IEEE Std 115A-1987 [2] or are calculatedby manufacturers. The detailed translation of data, from specific direct and quadrature axis model structures, with theirassociated element values, to transient and subtransient reactances and time constants, or to transfer function form ispresented for commonly used models. Some of the assumptions made by manufacturers in producing calculated ordesign parameters are pointed out. In addition, the desirability of a common approach as a basis for such designcalculations is recommended.

In Chapter 7, two approaches in accounting for generator field and excitation system base quantities are identified.Conversion factors are given for transferring field parameters from one base to another for correct generator/excitationsystem interface modeling. The importance and method of correctly determining generator field resistance is stated.Also included are suggestions for the modeling of negative field currents and other field circuit discontinuities.

1.2 References

[1] IEEE Std 115-1983 (R1991), IEEE Test Procedures for Synchronous Machines (ANSI).2

[2] IEEE Std 115A-1987, IEEE Standard Procedures for Obtaining Synchronous Machine Parameters by StandstillFrequency Response Testing.

[3] IEEE Committee Report, Second Benchmark Model for Computer Simulation of Subsynchronous Resonance,IEEE Transactions on PAS, vol. PAS-104, No. 5, May 1985, pp. 1057–1066.

2IEEE publications are available from the Institute of Electrical and electronics Engineers, Inc., Service Center, 445 Hoes Lane, P.O. Box 1331,Piscataway, NJ 08855-1331, USA.

2 Copyright © 1998 IEEE All Rights Reserved

MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991

2. Model Classifications or Categories

2.1 Practical Models Available

The basic approach to generator stability modeling has been to consider an arrangement of three stator windings 120electrical degrees apart, and a rotating (rotor) structure with an excitation or field winding and one or more equivalentrotor body winding(s). The magnetic axis of the field winding is defined as the direct axis, and an orthogonal axis,called the quadrature axis, is located 90 electrical degrees away. The equivalent rotor windings (or circuits) reflectinduced current paths in a round rotor iron body, or in damper bars that are also used in round rotor turbo-generators,as well as in salient pole hydro-generators. One group of equivalent circuits is aligned in the direct axis, and the othergroup is aligned in the quadrature axis.

R. H. Park [7] developed this concept further by mathematically transforming the three-phase stator quantities (such asvoltages, currents and flux linkages) into corresponding two-axis quantities. Park did this by visualizing two fictitiouscircuits—one in the direct axis aligned with the field axis, and the second in the quadrature axis. Under steady stateconditions these fictitious circuits were assumed to rotate in synchronism with the field axis.

The effect of such transformations was to move all the machine time-varying inductance coefficients from the machineflux linkage equations. Time varying three-phase stator voltages, currents, and flux linkages were also transformedinto time invariant direct and quadrature axis quantities under steady-state conditions. His treatment is general enoughthat deviations in speed of the field and rotor structure from steady state could be easily accounted for in the fictitiousstator direct and quadrature axis flux linkage and voltage relationships.

The widespread use of direct axis and quadrature axis equations has developed from these concepts. These can also bevisualized in terms of direct and quadrature axis equivalent circuits. Alternatively, operational inductance expressionsin transfer function form can be used to express the direct and quadrature axis voltage and current relationships.

In considering the equivalent circuits, the model structure for both axes is the basic form or configuration of the model,and its order can be directly observed, ranging usually from first order to third order. Order can be simply defined as thenumber of rotor circuits in either the d or q axis; this concept is expanded upon in a subsequent paragraph in this section.

Model parameter values have historically been given in terms of reactances and time constants for the direct orquadrature axis. Alternatively, resistance and inductance values can be assigned to the elements of direct or quadratureaxis equivalent circuit.

Some of the above points are expanded upon in Section 6.5, where the philosophy behind these modeling approachesis presented. As discussed above, the basic modeling approach usually considered for synchronous machine electricaldynamics is derived from Park's transformations, with corresponding relationships between flux linkages, currents andinductances. This approach is further characterized by a prior knowledge of the stator leakage inductance, and thestator to rotor mutual inductance. To such a fundamental model structure, or configuration in either axis, one mayexplicitly connect an arbitrary number of rotor circuits (including the field winding in the direct axis). For powersystem stability studies, based on experience, judgement and intuition, and often a lack of data, it has been foundsufficient to limit the number of rotor circuits to a small number (usually three as a maximum).

Consider the matrix shown on Table 1, which can be conveniently chosen for describing model structures ranging fromfirst order to third order in each axis, depending upon the number of inductance/resistance series combinationsrepresenting the field and direct axis equivalent rotor circuits,or the number representing quadrature axis equivalentcircuits. There are 12 possible combinations of direct and quadrature axis representations, plus one “constant rotor fluxlinkage” model The most complex (Model 3.3) has a field winding and two equivalent direct axis rotor iron (damper)windings. The quadrature axis structure of Model 3.3 has three equivalent solid rotor iron circuits, or windings. Somecombinations of d and q axis winding configurations are not considered in Table 1, and equivalent circuit structures arenot drawn or discussed. Based on experience, as well as on published material [3,4,5,8] it appears there are sevenmodel structures that could be serious candidates for inclusion in large system stability simulations. Six of these model



structures are drawn in Table 1, and the seventh is the constant rotor flux linkage (or alternatively constant voltagebehind transient reactance) model.

In Table 1, the field winding in each of the six model structures is identified by the initials fd; the direct axis rotor ironbody equivalent circuits are identified as 1d, 2d according to the number of such circuits. In the quadrature axis, theequivalent circuits are identified as 1q, 2q and 3q, depending on the order of the quadrature axis model structure.

Another approach to any of the models would be to consider the roots of the characteristic equation that defines thelinearized model response on open circuit. The roots, for example, in the direct axis for Model 2.2 are usuallyclassified as the inverse of T′do (the direct axis transient open circuit time constant) and the inverse of T″d, (the directaxis subtransient open circuit time constant). If Model 2.2 were used to simulate a terminal fault from an open circuit,the stator currents would be inversely proportional to X″do, (for the subtransient period) and inversely proportional toX′d,(for the transient period). Higher order models than 2.2 such as Model 3.3 would possess 3 time constants and 3values of reactance, and Model 1.1 would only have a single time constant and reactance. Time constant expressionsare discussed in Section 5.2, principally in the context of whether they are field or rotor properties.

In discussing the attributes of particular models, a decision was made by the Working Group to limit the complexity ofvarious models to a third order in both axes, Model 3.3. This decision is based on our experience and that of others [3],[5]. This model, which includes one differential leakage inductance branch, is the most complex that appears to berequired for large size stability programs. There is a fundamental basis [3], [4] for considering this differential leakage,and additional comments on this requirement are offered in Section 4.1.4 of Chapter Auto. Lf12d, as depicted in Fig 1for example, is a leakage inductance proportional to fluxes that link the field winding and the two direct axis equivalentrotor body circuits. These particular fluxes do not link with the stator circuits.

A popular and widely used structure in many current programs can be described, based on Model 2.2, which considerstwo windings in each axis, including the direct axis field winding. Parameter values for this model structure havenormally been supplied by manufacturers of synchronous machines, or have been, for the direct axis, obtained by testsdescribed in IEEE Std 115-1983 (R1991) [1]. Two time constants and two rotor reactances and resistances have beenemployed to describe the response of Model 2.2 in each of the d and q axes, The model is shown here in Fig 2, and itis common practice to use it for representing a majority of machines in stability studies.

In Model 2.2, the differential leakage inductance Lf 1d is proportional to flux that links the field and the one equivalentdirect axis rotor body circuit of Fig 2. This flux does not link the stator circuit. (Thus, by definition Lf1d = Lmf1d -Lad,as per the nomenclature of Section 2.2.3)

Direct axis parameters available from generator manufacturers cover all of the element values required in Fig 2, exceptfor Lf1d the differential leakage inductance [3] [4]. This value cannot be uniquely determined from statormeasurements alone. It should be determined from data based upon both stator and field winding measurements. Thisis discussed further in Section 6.2, as well as in IEEE Std 115A-1987 [2]. If neither field winding measurements nordesign data are available, the value of Lf 1d is typically set to zero, as in Fig 3. However, it should be recognized thatthis choice will affect the armature to field winding transfer relationships. When the field winding identity is ofsignificance in a particular study, it is recommended that field winding measurements be included in the modelderivation. Models of lower complexity than 3.3 and 2.2 have often been used in the past due to unavailability ofhigher-order model data, or because they were not considered vital to the stability assessment process.

Model 2.1 with a single q-axis equivalent damper circuit, has had extensive use for representing hydro-generators.Rotor equivalent circuit parameters for Model 2.1 have, on occasion, had to be estimated for older machines. Use ofsuch estimated values is considered good practice rather than neglecting their presence by reverting, or defaulting toModel 1.0.

Model 1.0 is another structure that represents only the field winding effects in the rotor. It is the simplest model to thatexcitation models can be connected. From this, changes in field flux linkages (E′q) can be determined. Model 1.0 hashad some use in the past, but is not currently recommended for general use.



The simplest model is the classical model that assumes a constant voltage behind transient reactance (E′= constant). Itassumes constant flux linkages in each axis, and also that no transient saliency exists, namely X′d = X′q = X′

Then

Since the voltages and currents of this model are not resolved into direct and quadrature axis components, this modelstructure is placed outside the matrix of Table 5. During a simulation run, the magnitude of the model's internal voltageis kept constant, but the internal angle is changed to reflect the inertial effects for changes in power. An advantage ofthis simple model is that the interfacing of generator and network equations can be accomplished more quickly duringstep-by-step calculations, and the least amount of stability data is needed for it. Thus, the “constant voltage behindtransient reactance” model has virtually replaced Model 1.0 in stability procedures and calculations where simplegenerator models are accepted. Exciter action cannot be represented with this model as it can be with Model 1.0.

Each of the model structures of Table 1 is of a differing order of complexity. Models are constructed by determiningthe parameter values for the model elements (inductances and resistances) based upon measured or analyticallydetermined data that characterize the generator to be modeled (see Section 6.5). The parameter determinationprocedure should begin both with the available generator data and a specification of the model structure. Parameters,for models of less complexity (i.e., fewer equivalent rotor windings), cannot be obtained from models of highercomplexity by simplification (i.e., eliminating rotor winding elements) from the more complex models. As a result,each model should be based upon the direct application of the above procedures to the desired model structure.

2.1.1 Generator—Power System Interfacing

Methods of interfacing generator and network equations are not presented in this guide in any detail. Reference [6]describes one possible approach that permits incorporation of any of the generator stability models, discussed earlierin this section, into network computations. Figure 4 is illustrative of the method employed.

Park's equations forms one building block of this method, where a d-q reference frame for the stator is rotating insynchronism with the rotor axis. This is also discussed in the introduction to Section 2.1.

The network equations are in phasor form, and in the steady-state load flow are positive sequence quantities, also notedin Section Auto.

To solve the network equations in terms of Park's representation, two-axis Real (R) and Imaginary (I) voltage phasorsare chosen. These are synchronously rotating phasors 90° (electrical) apart, with the real-voltage phasors VR chosen tobe in phase with the “swing” bus voltage in the load flow (usually zero degrees). Thus, the real-voltage phasor VR is thereference phasor for all the generator terminal voltages in the network. In Fig 4, δ1′ angle is the electrical displacementbetween system generator #1 rotor position and the reference phasor VR. Angle δ1′ is comprised of the internal angleδ1, of the generator, plus an angle β1. The angle β1 depends on the power system load and generation dispatch. Thus,β2 ..βn will represent the displacement of all the other n-1 generator terminal voltages with respect to phasor VR.

Equations used to transform variables from the d, q reference frame to the power system reference frame include, ingeneral,

(1)

(2)

E ′ E ′q( ) E ′d( )+ 2=

V R Ed sinδ′ Eq cosδ′+=

V I E– d cosδ′ Eq sinδ′+=



Table 1— Selection of Generator Models of Varying Degrees of Complexity



A second basic equation in this process is as follows taken from Eq 15 in [6]:

(3)

This states that the real and imaginary components of machine terminal voltages are equal to the product of an impedancematrix times the real and imaginary components of stator current plus ER and EI. The latter are essentially internalmachine voltages. Both ER and EI and the impedance matrix elements (the latter including relevant stator and rotorinductances) are affected by δ′ , the total electrical angle from the machine rotor position to the system reference phasor.

2.2 Nomenclature and Glossary of Frequently used Terms

2.2.1 Stator Quantities

Ea Synchronous machine stator terminal voltage in p.u.Ed Direct-axis stator voltage in p.u.Eq Quadrature axis stator voltage in p.u.Ia Synchronous machine stator (armature) current in p.u.Id Direct-axis stator current in p.u.Iq Quadrature axis stator current in p.u.

The above capitalized quantities are generally used in equations and phasor diagrams as phasors. (A phasor is acomplex number having the form X = X0ε

jθ, where Xo is the magnitude and θ is the phase angle). This nomenclatureis frequently applied to d-q quantities, where the q-axis is considered the imaginary axis. Where lower case letters (ed,eq, id, iq) appear in stator voltage, current, or flux equations or otherwise, they are considered scalar quantities.

All model parameter symbols listed below are chosen (to the extent possible) to conform with similar modelnomenclature listed in IEEE Std 115A-1987, as well as to the nomenclature of an IEEE Committee Report [3]. Exceptwhere noted, such parameters are in per unit referred to the generator stator.

2.2.2 Field or Rotor Quantifies

Efd Field voltage in the non-reciprocal system (dc volts)efd Filed voltage in p.u. in the reciprocal system.Ifd Field current in the non-reciprocal system (dc amperes).ifd Field current in p.u. in the reciprocal system.Rfd Field resistance in p.u. in the reciprocal system, referred to stator.rfd Fields resistance in the physical ohms.Rkd Direct-axis damper winding or rotor iron equivalent circuit resistances, in p.u., and referred to the stator,

where k=1,2 .. n, the winding number.Rkq Quadrature axis damper winding or rotor iron equivalent circuit resistances in p.u. and referred to the

stator, where k=1,2..n, the winding number.

V R

V I

R– RR XRI

X– IR R– RII

IR

I I

ER

EI

+=



Figure 1—Complete, Third-Order Representation—Both Axes

Figure 2—Complete, Second-Order Representation—Both Axes

Figure 3—“Standard” Second-Order Model—D Axis



Figure 4—Relationship Between d and q axes and Real and Imaginary Axes

2.2.3 Inductance Quantities—Stator and Rotor

The following inductances are usually referred in p.u. to the stator.

Ladu Direct-axis stator to rotor mutual inductance, unsaturated in p.u.Laqu Quadrature-axis stator to rotor mutual inductance, unsaturated in p.u.Ldu Direct axis synchronous inductance (unsaturated) = Ladu +Ll in p.u.Lqu Quadrature axis synchronous inductance (unsaturated) = Laqu +Ll in p.u.Ll Stator leakage inductance (both d and q axis), in p.u.Lfkd Differential leakage inductances in p.u. proportional to fluxes that link one or more damper windings and

the field, but that do not link the stator, k = 1,2,..n, where k is a list of the damper winding to which thesefluxes are mutual.

Lfd Field winding leakage inductance.Lkd Direct-axis equivalent damper winding leakage inductance; k = 1,2,..n.Lkq Quadrature-axis equivalent damper winding leakage inductance; k = 1,2,..n.Lffd Field winding self inductance = Ladu +Lfd +Lf1dLmf1d Mutual inductance between field winding and a direct axis equivalent winding and equals Lf1d + Ladu.

Alternatively, Lf1d = Lmf1d-Ladu, for one chosen differential leakage inductance.L11d Self-inductance of one equivalent rotor winding = Ladu +L 1d +Lf1dL11q Self-inductances of another equivalent rotor winding = Laqu +L1qLd(s) Direct-axis operational inductance, as viewed from the stator terminals.Lq (s) Quadrature-axis operational inductance, as viewed from the stator terminals.G(s) Stator to field operational transfer function.

NOTE — Much of the material in this guide uses reactances when describing the relationships between machine fluxes andcurrents, and this usage arises from the premise that steady -state relationships are being discussed. In generatormodeling, X = ω0 · L, where ω0 = unity or base speed. Thus, L (inductance) and X (reactance) are often usedinterchangeably. It seems difficult to settle on, or recommend a specific usage.

In general, reactances, in p.u. are used to describe steady-state, or initial conditions. Inductances are used, by and large,in places that describe changing or transient conditions.



Usually, inductance or reactance values correspond to the unsaturated value of L or X. In some cases the text includesa subscript “u” or “sat” to clearly distinguish between unsaturated or saturated conditions. The unsaturated conditionis implied in the absence of these 2 subscripts.

2.2.4 Commonly Used Symbols in Stability Programs

Ld,Lq d- and q-axis synchronous inductance, usually quoted unsaturated and in p.u.L′d,L′q d- and q-axis transient inductance, usually quoted in p.u. and saturatedL″d, L″q d- and q-axis subtransient inductance, usually quoted in p.u. and saturated The

above 6 quantities are often replaced by using reactance (X) in p.u. namely X′d,X′q, etc.

T′do, T′qo d-and q-axis transient open-circuit time constants in seconds.T″do, T″qo d- and q-axis subtransient open-circuit time constant in seconds.T′d, T′q d- and q-axis transient short-circuit time constants in seconds.T″d, T″q d- and q-axis subtransient short-circuit time constants in seconds.ψd (or λd) Direct-axis stator flux linkages.ψq (or λq) Quadrature-axis stator flux linkages.ψfd (or λfd) Field flux linkages.ψ1dψ2d (or λ1d, λ2d) Direct-axis damper winding or amortisseur flux linkagesψ1q,ψ2q, ψ3q (or λ1q, λ2q, λ3q) Quadrature axis damper winding or amortisseur flux linkages.

Although flux linkages are denoted in IEEE Std 100-1988 by λ (lambda), common usage in IEEE power literature hasbeen to symbolize flux linkages by ψ (psi), and to use λ for eigenvalue that is defined in Section 2.2.5.

2.2.5 Brief Description of Terms Frequently Used in the Main Body of This Guide

The following terms have been used in this text to describe mathematical procedures or technical concepts. Most of theterms or phrases listed below can be located in IEEE Std 100-1988. A few of the terms listed are not in IEEE Std 100-1988, and can be considered peculiar to the jargon used in this document.

A) Dictionary terms frequently used in this text:

B) Definitions of the familiar time constants and reactances for synchronous machines are given in thedictionary under “direct axis” or “quadrature axis” headings.

C) Non-standard or non-defined terms are given below in the context in which they are used.

characteristic equation: This is the relation found by equating to zero the denominator of a transfer function (such asthe expression for the operational inductance of a generator as viewed from the stator).

state variables: These are the elements of a set of quantities necessary to completely define the dynamic state of asystem, including a power system. The choice of state variables is not necessarily unique for any system. (Alsoreferred to sometimes as dynamic states).

eigenvalue: This is one root of the characteristic equation. The number of eigenvalues in a dynamic system is equal tothe order of the characteristic equation, or the number of state variables. The eigenvalues describe the transientresponse of the system to any disturbance. A transient associated with an eigenvalue takes the exponential form ελitxi,

Air Gap Air Gap Line Permeability, (Incremental)

Armature Exciter

Excitation System Operational Impedance

Operational Inductance Permeability



and this expression is referred to as a mode. The letter xi denotes the eigenvector associated with the eigenvalue λi.Both λi. and xi may be real or complex.

NOTE — The use of λ as a symbol for eigenvalue is consistent with the preferred usage of ψ for flux linkages in Section 2.2.4above.

eigenvector: This vector defines the distribution of the mode, corresponding to a particular eigenvalue, throughout thestate variables of the system.

On a more practical basis, the eigenvector can be thought of as a set of coefficients describing the relative magnitudesof the power system dynamic states when some particular mode is the only one that has been excited by a disturbance.

finite element analysis: This is a numerical procedure used to find the magnetic flux distribution in a generator. In itstwo-dimensional form, the cross section of the generator is divided into a large finite number of geometric elements(frequently triangular). The generator's material properties (rotor steel, stator laminations, air, etc.) are assigned toappropriate elements. Rotor and stator currents are also prescribed. The output of the analysis is the magnetic fluxdistribution (expressed as the vector potential) in the generator cross-section, from which stator and rotor flux linkagesand other quantities of interest can be found. Both nonlinear magnetostatic versions and linear diffusion-equations,which solve for induced currents, are common. Three-dimensional versions have been programmed, but are not incommon use.

rotor body: In a turbo-generator, this is the steel forging into which are machined slots for the field winding,amortisseurs, etc. Because it is solid steel, induced currents can flow in it; these are referred to as rotor body currents.In salient pole machines (hydrogenerators) the rotor body is usually a laminated structure (the pole) encircled by thefield winding. The poles are then attached to a shaft, and in the case of many poles, a central frame is used to supportthem. This frame is then connected to the central shaft. Pole-face damper bars or amortisseurs may be embedded in thelaminations near the pole face.

saliency: This term is now used for all types of synchronous machines (round-rotor and salient-pole) in which theelectromagnetic parameters are significantly different in the direct and quadrature axes. There may be different degreesor amounts of saliency under various conditions, for steady state, transient or subtransient reactances or inductances(calculated or measured) between the direct and quadrature axes. The use of this term arose from consideration ofsalient-pole machines, including hydro-generators, which all have a pronounced physical difference in the magneticflux paths for the direct and quadrature axes.

2.3 References

[1] IEEE Std 115-1983 (R1991), IEEE Test Procedures for Synchronous Machines (ANSI).

[2] IEEE Std 115A-1987, IEEE Standard Procedures for Obtaining Synchronous Machine Parameters by StandstillFrequency Response Testing

[3] IEEE Committee Report: Current Usage and Suggested Practices in Power System Stability Simulations forSynchronous Machines, IEEE Transactions on Energy Conversion, vol. EC-1, March 1986, pp. 77–93.

[4] Canay, I.M., Extended Synchronous Machine Model for Calculation of Transient Processes and Stability.Electrical Machines and Electromechanics - (International Quarterly) no. 1, 1977, pp. 137–150.

[5] Canay, I.M., Identification and Determination of Synchronous Machine Parameters, Brown Boveri Review, June/July 1984.

[6] Kundur, P., and Dandeno, P.L., Implementation of Advanced Generator Models into Power System StabilityPrograms, IEEE Transactions on PAS, vol. PAS-102, July 1983.

[7] Park, R.H.: Two-Reaction Theory of Synchronous Machines, AIEE Transactions, Part I, vol. 48, 1929; Part II, vol.52, 1933.



[8] Schulz, R.P., Jones, W.D., and Ewart, D.N., Dynamic Models of Turbine Generators Derived from Solid Iron RotorEquivalent Circuits, IEEE Transactions on PAS, vol. PAS-92, May–June 1973.

3. Classification of Stability Studies

3.1 Background

The degree of detail and complexity of models required for representing synchronous machines in stability studiesdepends on the nature of the study, as well as the nature of the power system [2]. A striking tendency of the past 25years has been that studies of the dynamic performance of such systems can embrace several or even manyinterconnected power utilities due to the formation of power pools or operating areas. Intra-area, or inter-area studiesinvolving several companies, or several areas or pools, are common, particularly because of changing power transfersarising from the economies of energy interchanges between utilities. As a result, large scale stability programs havebeen developed for such studies, generally embracing the near-term time frame. Studies of this nature are oftenconcerned with power system operation procedures. On the other hand, detailed examinations of system performancecan be concerned with planned additions to power network configurations and with future load and generation patternsexpected several years in the future.

Occasionally, very long-term or “broad brush” type studies are performed. However such studies are more concernedwith generating station location and transmission network configurations; generator detail is secondary. Such studiesare performed to assess the need for future major power system facilities and also fall under the discipline of the powersystem planning function.

Past system performance is sometimes reviewed and simulated in as much detail as possible to determine possiblereasons for misoperation, or for an explanation of past events. However, such a power system analysis forms a smallportion of stability studies.

Until recently, all of the stability studies discussed above analyzed power system responses to faults or other similarcontingencies, and covered several seconds of simulated time. Such time-domain studies still currently embrace moststability analyses. Lately, power system dynamic responses to small perturbations, involving a linearized model ofmajor system components, including generators and their associated controls, have received increased attention.

For convenience in analysis, and to gain a better understanding of the stability problem, it has been the usual practiceto group power system analysis into two general classifications, as noted in Reference [1].

1) Large disturbances, where the equations that describe the dynamics of the power system cannot be linearizedfor the purpose of analysis.

2) Small disturbances, where the equations that describe the dynamics of the power system are linearized for thepurposes of analysis.

From the first of these two general classifications follows a specific definition of the transient, or large disturbancestability of a power system [1]. The second of these two classifications provides a specific definition of steady statestability of a power system [1]. It is further stated in Reference [1] that steady-state stability is also known as smalldisturbance stability of a power system.

As a rule, transient stability simulations are highly nonlinear and are studied using step-by-step integration of thedifferential equations of the dynamic devices that influence system performance. For small disturbance stability, thesame equations are linearized so that the more direct methods of stability prediction based on linear system analysismay be applied.



For stability studies involving faults or other sudden changes to the power system configuration, several factors play asignificant role in determining how the studies are conducted. These include the severity of the disturbance, and theperiod in the time domain to be investigated.

For stability studies concerned with a linearized model of the power system, such factors as the nature of the powersystem responses, including damping of generator oscillations, and the dominant modes or frequencies of power orangular swings, are of prime interest.

The above concepts are discussed accordingly in the following sections.

3.2 Large Disturbances Stability

3.2.1 Short-Term Transient Stability

The traditional transient stability study has been related to the short -term, or transient, response of the power system toa large disturbance such as a fault on the transmission network or sudden loss of generation. It has normally beenconcerned with the behavior of the system up to about 5 seconds following the disturbance, and depending on thedominant modes of oscillation determined in the analysis, can extend to as many as 10 seconds. The instability isnormally due to lack of sufficient generator electrical synchronizing torque, but can also be due to lack of damping torque.

For adequate transient stability performance, the operation of the post-disturbance power system should be stable inthe new steady state, and voltages, power and reactive power flows, and relative phase displacements of generatorsshould have stabilized to acceptable values.

Much of the electric utility industry effort and interest related to large disturbances have concentrated on the short-termtransient response. The system is designed and operated so as to meet a set of reliability criteria concerning transientstability. The criteria specify certain contingencies that the power system should be able to withstand without loss ofsynchronism. These contingencies are selected on the basis that they have a certain level of severity and a significantprobability of occurrence.

Generator electrical stability characteristics have a significant influence on the short-term transient response of the systemand should be adequately represented in transient stability studies. This is discussed in more depth in Chapter 4.

3.2.2 Long-Term Transient Stability

In recent years, the need for studying the response of the system for longer periods has been recognized, and the wordlong-term stability has been introduced. Time frames associated with such analysis extend beyond the usual 5–10seconds. In some cases, such studies can cover several minutes of simulated time. [B1]

Some power system analysts have introduced the phrase “mid-term stability,” which could be considered to be theinterval between transient stability and long-term stability. This concept embraces periods of time up to as many as40 s following a disturbance. The definition of “mid-term stability” is avoided in the stability terms of Reference [1].Comments on such modeling requirements are included in Section Auto under the transient stability headings.

In long-term dynamic response studies, the emphasis is on modeling the sequences of events that follow major systemupsets, with the associated slowly changing phenomena. Prime mover and energy supply system dynamics and theinfluence of automatic generation control, over/under voltage and over/under frequency control and protectionschemes, are likely to be significant. Inter-machine synchronizing power oscillations often may be assumed to havedamped out, resulting in uniform system frequency. In these cases, the generator electrical characteristics usually donot have a significant influence on long-term dynamic response. On the other hand, longer term effects, such ascapacitor switching and restoration of constant mVA loads through tap changer action may render the power systemunstable in such a dynamic state, and in these situations, importance is stressed on generator and excitation systemrepresentation, and appropriate modeling detail.



3.3 Small Disturbance Stability

Small disturbance (or steady-state) stability is concerned with system response to small changes that continually occurin the operation of the system [B2]. The results are usually given in terms of eigenvalues and eigenvectors. Thecalculated system response depends in such situations on the type and modeling of excitation control, as well as on thetype of generator modeling. In the absence of high initial response excitation control, loss of stability may occur dueto lack of sufficient synchronizing torque. This results in monotonic instability, which is also defined in Reference [1].

Generator synchronizing torque is dynamically controlled using high-response, continuously-acting excitationsystems. The stability problem, then, often becomes one of ensuring sufficient damping of system oscillations.Instability is normally characterized by oscillations of increasing amplitude.

The generator electrical dynamic response characteristics have a very significant influence on the small disturbancestability of the system. It is, therefore, important to use accurate generator models in such analyses and particularly indesign studies of excitation control for enhancing stability. (Generator modeling under these conditions is discussedfurther in Chapter 4.)

3.4 Classification Based on Dominant Modes of System Response

In the analysis of power system stability of large systems, usually two distinct types of system oscillations arerecognized. One type is associated with units at a generating station swinging against the rest of the power system.Such oscillations are referred to as “local mode” or “plant mode” oscillations. The oscillations are localized to oneplant or restricted to a small part of the power system. The frequencies of these oscillations are in the range 0.8 to 2.0Hz. For situations where system stability associated with the plant mode is of primary concern, it is necessary torepresent the machines in the local area in detail. Simple models can be used to represent remote machines and systemreduction techniques can be used to reduce the size of representation of remote areas.

The other type of oscillation is associated with swinging of many machines in one part of the system against machinesin other parts. These are referred to as “inter-area mode” oscillations and have frequencies in the range 0.1 Hz to 0.7Hz. They are caused by two or more groups of machines connected by relatively weak ties. Analysis of systemconditions in which the stability of inter-area swing modes is of primary concern requires detailed representation ofthe machines throughout all the areas being simulated. In the initial stages of such a study, it is possible to represent allmachines by a simple classical model, in order to indicate the frequencies of oscillation, and identify the groups ofmachines involved in these dominant modes. If this screening process is resorted to, machines in each individual areashould be modeled consistently. It generally is questionable practice to retain detailed modeling of machines,including their associated controls, in one area, and then resort to modeling approximations in other areas.

In the final analysis of such multi-area power representations, experience and good judgement are helpful indetermining whether the dynamic response of one distinct area can be approximated in the above manner. Thisdecision often can only be taken after initial detailed generator modeling representations of the area under study, andafter examination of critical factors such as inter-area flows, or of voltages at key locations in various areas.

3.5 References

[1] IEEE Committee Report, Proposed Terms and Definitions for Power System Stability. IEEE Transactions onPower Apparatus & Systems, vol., PAS-101, July 1982, pp. 1894–1898.

[2] Concordia, C. and Schulz, R.P., Appropriate Component Representation for the Simulation of Power SystemDynamics, Symposium on Adequacy and Philosophy of Modeling: Dynamic System Performance, IEEE Publication75CHO 970-4-PWR, 1975, pp. 16–23.



3.6 Bibliography

[B1] Dunlop, R.D., Ewart, D.N., and Schulz, R.P., “Use of Digital Computer Simulations to Assess Long-Term PowerSystem Dynamic Response,” IEEE Transactions on Power Apparatus & Systems, vol. PAS-94, 1975, pp. 850–857.

[B2] Kundur, P. and Dandeno, P.L., “Practical Application of Eigenvalue Techniques in the Analysis of Power SystemDynamic Stability Problems,” Proceedings of the 5th PSCC Conference, Cambridge, England, 1975.

4. Application of Generator Models in Stability Studies

4.1 General

As noted previously, system stability studies are generally conducted for one of the following purposes:

1) Power System Planning, to enable decisions to be made on future transmission and generation requirements.2) Power System Operation, to determine operating limits in the light of various system contingencies.3) Post-disturbance Analysis, to simulate past events that have occurred on the system.

In the initial stages of a planning study, many comparisons of different alternatives are required. In such studies, it maybe possible to use simple synchronous machine models, since only a relative measure of the performance of a numberof different system configurations is being sought.

In the latter stages of a system study, the behavior of the chosen system alternatives should be explored in detail. Insome cases, special controls or protection schemes to maintain the stability of the system may be studied. For thesestudies, the synchronous generator modeling should be much more complete.

In operating studies, simplified models may be adequate for real-time determination of operating limits and for somecontingency analysis studies. For the normal determination of operating limits using contingencies included in systemdesign or operating criteria, more accurate modeling is necessary, again requiring more detailed generator models.

It is often in the simulation of past events that have actually occurred on the system that the most demands are madeon the mathematical models used in the simulation.

For planning and operating studies the following seven categories of large disturbances are usually the mostsignificant:

1) Stability of one or more generating stations, following a severe nearby fault.2) Excitation system control behavior and its associated stabilization, under all types of disturbances.3) Power transfer limit determination across one or more inter-area interfaces subsequent to disturbances.4) The sudden loss of one or more large generator(s), or deliberate generation tripping to maintain stability.5) Generator and power system performance with delayed fault clearing and the associated transmission line

relay performance.6) The effects of large industrial loads, or of deliberate load shedding.7) Inter-area tie-line oscillations of a pronounced magnitude.

For generators adjacent to many of the disturbances noted above, important issues in model application include:

• Calculations of generator power or torque during the fault period, or during the initiating situation.• The calculations of post-disturbance generator powers, angles and voltages, for periods of up to several

seconds of simulated time.



In the case of power-system faults in (1) above, in particular, the effects of unbalanced faults cannot be overlooked, butas at general rule in large-scale studies, calculation of generator balanced electrical torque or power developed acrossthe machine air gap is based on the assumption of a balanced or positive-sequence network. The effect of unbalancedfaults on positive-sequence power calculations is obtained by applying an appropriate impedance-to-neutral at thepoint of fault in the positive-sequence network.

The positive sequence system representation used in stability studies does not allow the effects of dc offsets inarmature transients to be modeled. The fundamental frequency rotor losses due to dc offsets and the negative sequence(second harmonic) rotor losses are neglected with the above approach.

Appendix 4A includes a development of the relationships between electrical torque, (Te) and electrical power (Pe), andcomments on neglecting the rate of change of stator flux linkages and/or variations in rotor speed in the torque orpower calculations.

4.1.1 Use of the Simplest Model

The “voltage back of transient reactance” model, discussed in Section 2.1, has been widely applied. A generaltendency is to represent generators in areas remote from disturbances in this way. If in the transient-stability studies,such “remote” machines suffer sudden or sustained voltage changes of only about 0.1 p.u. or less at their statorterminals, then the use of this simple model may be acceptable. Adjacent machines remote from the disturbance canthen be grouped and represented as one machine, as one “voltage behind transient reactance.” In this case, theindividual inertias would be totalled and used in the equivalent generator, along with the Thevenin equivalent of all theinternal voltages, and the parallel value of all the machine transient reactances.

The question of size of machines being simulated in the remote areas should be seriously considered in decisions ontheir degree of modeling. Some engineering judgement needs to be exercised as to which machines can logically beignored. In such cases, they would be balanced against local load. Any generator group consisting of a smallpercentage of the total area load (5% or less) can be balanced out against local load, or at most, represented in all casesby its transient reactance and the corresponding internal voltages. It may be good practice to also retain an appropriatedamping factor for such generator groups.

For simple, yet reasonably accurate reduced order power system analyses, it is important to aggregate machines forwhich the rotor angle excursions are coherent following some initiating disturbance. For short-time-scale studies (1 to1-1/2 s) involving “local” stability problems, this coherency criteria may include all generators electrically remotefrom the disturbance. The reduction of systems for small signal studies is mentioned in Section Auto.

4.1.2 Neglecting Saliency in Second- or Third-Order Models

At one time there was considerable incentive to minimize the number of differential equations required to be solved instability studies. There is sometimes an incentive in reducing the system size to a prescribed level in terms of thenumber of load flow busses to be solved at any “time step”. This follows because the number of algebraic equationsinvolved in the iterative process is a direct function of the number of busses to be represented.

As far as the synchronous machine differential equations are concerned, there would be a very small difference incomputing time per step if all machine equations were written for a third-order model, as opposed, for example, to asecond-order model. With no sub-subtransient saliency in Model 3.3 for example, (that is if ), that modelis computationally more efficient when calculating generator terminal quantities than, for instance, model 2.2 wheresubtransient saliency, was assumed to be present (i.e. ). Simply stated, when and are replacedby a single voltage , the interfacing of generator and network equations can be accomplished more quickly duringstep-by-step stability calculations. Refer also to similar comments (on assuming ) in Section 2.1.

The chief computational limitation on the choice of a Model 3.3 over a Model 2.2 depends on the values of the smallest(effective) time constants associated with Model 3.3. This will likely affect the choice of integration time step.Simulation of high initial response exciters also require small time steps in the 0.005 to 0.01 second ranges. Thus, the

Xq ′″ Xd ′″=

Xq″ Xd″= Eq ′″ Ed ′″E ′″

X ′q X ′d=



application of such excitation systems to certain types of turbo-generators may dictate the stability analysis integrationequations. It should also be noted that elimination of the type of saliency discussed above may slightly distort thevalues of some quadrature axis elements in Model 2.2, and possibly Model 3.3.

4.1.3 Salient Pole Generators—Stability Model Guidelines

Salient pole generators with laminated rotors are usually constructed with copper damper bars located in the polefaces. These damper bars are connected with continuous end-rings and thus, form a squirrel-cage amortisseur circuitthat is effective both in the direct and quadrature axes. Since this amortisseur is the only physical circuit present in theq-axis, a first-order model describes it adequately. Hence, Model 2.1 is recommended for most salient pole generators.

Solid iron pole salient-pole machines may justify a more detailed model structure. Reference [2] discusses variousaspects of salient-pole machine modeling. Some of the above considerations are also touched upon in Reference [1].More information on these particular topics is discussed in [B3].

4.1.4 Round Roter Generators—Stability Model Guidelines

In round-rotor machines, slots are present over part of the circumference to accommodate the field winding. The topsof these slots contain wedges for mechanical retention of the field turns. These wedges are usually made of anonmagnetic metal. In some cases, they are segmented and made of nonmagnetic steel, while in other cases the wedgesare made of aluminum and may be either segmented or full length. In many constructions, a conductive ring under thefield end-winding retaining ring, with fingers extending under the ends of the slot wedges, is used to improveconduction at these connection points.

For infrequent cases where the retaining ring is magnetic, field leakage fluxes and corresponding inductances will beaffected, with noticeable differences existing in frequency response data obtained at or near full load, compared withstand-still data [6].

Copper strips are often inserted under the wedges to provide improved conduction between wedge segments and/or toimprove amortisseur action. In some cases, a complete squirrel-cage winding is formed, while in other cases theconductive paths contribute only marginally to amortisseur action. The effectiveness of the slot-wedge amortisseurcircuit may vary widely depending on the wedge material, segmenting of wedges and the design of the conductivecircuit below the wedge. References [5] and [6] discuss the impact of some of the above details on the direct axis rotormodel. A third-order model for the direct axis definitely gives a better fit to Ld(s) and sG(s) test data than a secondorder one. However, in some cases the smallest time constant ( ) values may be close to the size of the integrationequation time steps. In many cases, a second-order model for the direct axis (Model 2.2 or 2.3) will suffice, along withrecognition of Lf1d, which reflects the differences in mutual coupling between the field, the equivalent rotor wedgeand/or damper circuits and the stator.

For turbo-generators therefore, Model 2.2 can usually describe the operational inductance in both direct andquadrature axes adequately, where no conducting strips are placed under the slot wedges, and where the pole faceregion is without wedges or amortisseur bars.

The pole-face region is often slotted circumferentially, or, on occasion, may be slotted longitudinally for rotor flexingand balancing purposes. In some cases, longitudinal pole face slots may be filled with wedges of either low-conductivity steel or, in the case of machines subject to sub-synchronous resonance (SSR), high-conductivity materialto form a quadrature axis amortisseur. The q-axis equivalent circuit model should account for various current paths inthe rotor iron along the pole faces. The electrical properties of these paths may be affected by circumferential slotting,or by pole face wedges or damper bars.

However, should the pole-face area have conducting wedges or amortisseur bars, Model 2.3 is recommended. It hasalso been found, for a range of rotor damper constructions, that a third-order representation such as Model 2.3 or 3.3gives a better fit between measured and calculated values of Lq(s) including phase angle, and in the 0.1 Hz to 10 Hzrange [4]. Three impedance values are thus available to describe the quadrature axis rotor dynamics over the requisite

T ′″ do



power system frequency range rather than just two. It would appear that electrical damping from currents in the poleface area would more accurately reflect actual conditions when using a third order quadrature axis model, and thisapproach is preferable to the use of damping factors. Concordia treats generator damping torques, and the associateddamping factors that were used in the past, when adequate quadrature axis operational data was unavailable [B1].

Model 3.3 is also suggested [3] for cases where detailed third-order modeling effects of direct and quadrature axisrotor circuits, and generator torques, are considered important in studying rotor “backswings.” This effect is oftencritical in special transient stability power limit calculations.

4.1.5 Application of Excitation Controllers Relative to Generator Models

Direct axis generator models play an important role in determining the effect of excitation system performance ongenerator stability. The largest time constant in the denominator of the transfer function of any direct axis modelreflects the impact of the field time constant. Considering all rotor element values, this effective field time constant isincreased slightly with higher-order models. An accurate knowledge of the magnitude and phase of Ld(s) and sG(s)over a frequency range from .01 Hz to 10 Hz is useful in determining the impact of excitation system voltage changeson calculation of electrical Torque or Power. Various approaches to excitation system applications are discussed in[B2], [B4], and [B5]. Specific examples of the setting of power system stabilizer (PSS) gains and time constants aregiven in Reference [7].

4.2 Representation of Generations During Large Disturbances

4.2.1 Short-term Transient Stability Studies

In transient stability studies involving categories (1) through (6) in Section Auto above, and especially for severe faultconditions or excitation system control design, the responses of the generators, exciters and power system itselfimmediately following the initiating disturbance are of primary interest.

The internal angle and excitation conditions of a generator prior to a disturbance are functions of unit loading andsystem conditions, and are influenced by the magnetic characteristics of the generator. Saturation of the magnetic pathsin generators is discussed more fully in Chapter 5.

For any transient, the depression in system voltages and the corresponding reduction in generator output power due toclose-in faults, results in the acceleration of generator rotors. This is normally accompanied by large increases in fieldvoltage through voltage regulator action to compensate for lost synchronizing torque.

Summary of Recommendations:

The following numbered comments summarize the recommendations in the guide for generator stability modelselection. These comments relate particularly to the seven stability study categories noted in Section Auto.

Basic considerations in generator model selection would suggest that the power system analyst should use, as adesirable minimum, a second-order model for both the direct and quadrature axes, providing reasonably accurateparameters for these chosen models can be either obtained or can be estimated.

1) For turbo-generator representation, a second-order model in both axes (Model 2.2) is nominally required asa minimum. For turbo-generators that also utilize damper bars or conducting wedges in the pole faces, insome cases a second order model may be adequate in the quadrature-axis. A third order model (Model 2.3 or3.3) may be required for the best results, especially to give good representation of generator damping.

2) For excitation system stabilization studies, along with exciter ceiling voltage and response ratio specificationstudies, a second-order model in the direct axis is a minimum requirement. Improved model detail, inrepresentation of the differential leakage reactance (Lf1d), along with representation of the field windingterminals is definitely suggested. This is particularly significant for turbo-generators when damper bars or



other conducting materials are used as part of, or under the field slot wedges, or where magnetic retainingrings are employed [6].

3) Under the conditions of (2) and (3) of Section Auto, a complete third-order model, Model 3.3, should beconsidered, provided that appropriate quantities are available. If this type of information cannot be easilyobtained from the generator manufacturer, it can be derived from frequency-response measurements.

4) For most hydraulic machines, a second-order model for the direct axis is appropriate. For most studies, onlya single-order model is required for the quadrature axis (Model 2.1) [B3]. For most studies, only one q-axistransient reactance is provided for such machines (usually called X″q not X′q ). On occasion, a second-ordermodel (Model 2.2) may be appropriate for the quadrature axis, depending on damper bar, pole-face, andinterpolar space construction details. [2]

5) In many stability studies, a number of tests may be carried out initially to determine a feeling for the overalldynamic response of the power system. These screening studies are often performed using a simple modelsuch as “voltage behind transient reactance” for virtually all machines. More detailed studies based onrecommendations (1) to (4) would then follow. (See also Section 3.4.)

4.2.2 Long-term Transient Stability Studies

Some remarks about the type of investigation as a particular classification of stability have already been made inSection 3.2.2.

As noted in Section 3.2.2, in such a time span, underload tap changing and capacitor switching or line restorationprocedures and switching can come into action. However, prime mover (boiler and reactor) or energy supply dynamicsare also very much in the picture, to change mechanical power inputs. These latter aspects tend to overshadow theelectrical performance of generators, and their associated modeling requirements. In this time frame, they appear to beof secondary importance.

When switching does take place in long-term studies, the program may switch back to a “normal” transient stabilitystudy for a few seconds, in which case, the comments of Section 4.2.1 again apply.

4.3 Modeling of Machines for Small Disturbance Stability Studies

While some studies for examining the small-signal stability of a power systems can be made using time domainsimulation tools, (such as a transient stability program) this is very time consuming and inefficient. For most small-disturbance or small-signal studies, the system equations are linearized about an initial operating point. Eigenvalue/eigenvector or other linear analysis techniques ire then used to assess system stability.

For the generator models, there are two nonlinear functions to account for: magnetic saturation (treated to some extentin Chapter 5) and the torque/power equations. (A third nonlinearity, the “speed voltage” term, has been neglected byusing synchronous speed in conjunction with elimination of stator flux derivatives in the equations of Appendix 4A).

Two methods of obtaining linearized equations for small disturbance stability analysis can be used. In one method, theequations are prelinearized when the computer code is written. In the second method, the computer code describes thenonlinear equations and is used for both time simulation, and small signal analysis. Special provision is made tolinearize about an operating point by repeated calls from a state matrix building subroutine. From an initial steady statecondition, each of the states and inputs is individually perturbed slightly and the effect on the state derivatives andoutputs are measured. The state matrices are built one column at a time. When considering this method, the particularcomputer code being used provides for user selection of the saturation representations. For example, saturation couldbe linearized based upon total, tangential, or minor hysterisis loop relationships between flux and current.

Occasions for which simpler models may be reverted to would involve one distinct or remote area, as mentioned inSection 3.4.



The modeling of generator controls is critical for small signal stability assessment. It is very important that the smallsignal characteristics of excitation systems and power system stabilizers be modeled correctly, and in detail. In manycases, it is just as important to have good models of these control systems as it is to have a good model of thesynchronous machine. The successful tuning of these control systems can be quite dependent on the quality of thesynchronous machine model.

In order to properly account for the generator-excitation system interaction, generator Model 2.2, for turbinegenerators, and Model 2.1, for hydraulic units, are necessary. The initial conditions of generator loading and terminalvoltage, and, hence, field voltage and current initialization algorithms have a significant impact on small signalstability.

Large interconnected power systems often consist of groups of strongly interconnected generators coupled togetherthrough relatively weak transmission systems. Characteristic of such systems are low-frequency inter-area oscillationsthat represent the electromechanical energy interchange between groups of generators. The oscillations are globalrather than local and in consequence, for accurate simulation, the dynamic models used throughout the system shouldhave a consistent degree of detail [B6]. The concept of a study system containing detailed generator models and asimply modeled external system is not valid for inter-area small disturbance stability studies. The same commentsapply to excitation system and governor models for generation control. Reference [8] is an example of a large-sizerecently developed small-disturbance program in which the state-space equations are “prelinearized”.

4.4 References

[1] IEEE Committee Report—Reference [4] of Chapter 2.

[2] Canay, I. M., “Equivalent Circuits of Synchronous Machines for Calculating Quantities of the Rotor DuringTransient Processes and Asynchronous Starting, Part II, Salient Pole Machines,” Brown Boveri Review, vol. 57, March1970.

[3] Canay, I.M., “Physical Significance of Sub-Subtransient Quantities in Dynamic Behavior of SynchronousMachines,”IEE Proceedings, vol 135, Part B, No. 6, Nov. 1988.

[4] C.E.A. Report, “Electrical Parameters of Large Turbine Generators,” Dec. 1986

[5] Doughterty, J.W. and Minnich, S.H., “Operational Inductances for Turbine Generators; Test Data Versus Finite-Element Calculations,” IEEE Transactions on PAS, PAS-102, Oct. 1983, pp. 3393–3404.

[6] Jack, A.G. and Bedford, T.J., “A Study of the Frequency Response of Turbo-Generators with Special Reference toNanticoke GS,” IEEE Transactions, vol. EC-2, no. 3, Sept. 1987, pp. 496–505.

[7] Lee, D.C. and Kundur, P., “Advanced Excitation Controls for Power System Stability Enhancement,” CIGRE Paper38–01, 1986.

[8] Kundur, P., Rogers, G.J., Wong, D.Y., and Lauby, M., “A Comprehensive Computer Program Package for Small-Signal Stability Analyses of Power Systems,” IEEE Transactions on Power Systems, vol. 5, no. 4, Nov. 1990, pp.1076–1083.

4.5 Bibliography

[B1] Concordia, C., “Synchronous Machine Damping and Synchronizing Torques,” AIEE Transactions, vol. 70(1951), pp. 731–737.

[B2] De Mello, F. P., Nolan, P. J., Laskowski, T. F., and Undrill, J. M., “Coordinated Application of Stabilizer inMulti-Machine Power Systems,” IEEE Transactions on PAS, vol. PAS-99, May/June 1980, pp. 892–901.



[B3] Kilgore, L. A., “Calculation of Synchronous Machine Constants—Reactances and Time Constants AffectingTransient Characteristics,” AIEE Transactions, vol. 50, Dec. 1931, pp. 1201–1213.

[B4] Kundur, P., Lee, D.C., Zein-el-Din, H. M., “Power System Stabilizers for Thermal Units: Analytical Techniquesand On-site Validation,” IEEE Transactions on PAS, vol. PAS-100, Jan. 1981.

[B5] Larsen, E. V., and Swann, D. A., “Applying Power System Stabilizers—Part I: General Concepts,” IEEETransactions on PAS, vol. PAS-100, June 1981.

[B6] Stubbe, M., Bihain, A., Deude, J., and Beader, J. C., Stag, “A New Unified Software Program for the Study of theDynamic Behavior of Electrical Power System,” IEEE Transactions on Power Systems, vol. PS-4, Feb. 1989, pp. 129–138.



4A Calculation of Generator Electrical Torques or Powers(Informative)

(This appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in StabilityAnalyses, but is included for information only.)

4A.1

In a majority of stability programs, synchronous machine torque is calculated in terms of the machine's direct andquadrature axis currents and fluxes. Such calculations can be performed either in engineering-units calculations, or inper-unit calculations. As is commonly found in engineering analyses, the choice of per-unit notation results inequations that are more concise. Thus, per-unit equations will be presented in this appendix, although an alternateformulation in engineering units would be equally valid.

Per electrical torque is given by

(1)

in terms of the direct and quadrature axis fluxes, ψd and ψq, and the direct and quadrature axis currents, id and iq.

Accuracy in simulation of synchronous machine performance requires accurate representation of these currents andfluxes. The effects of damper and rotor-body currents should be included in transient simulations, since these currentsplay a significant role both in determining the magnitudes of the transient electrical torque, and in determining thedamping of the electromechanical oscillations associated with the transient.

For Model 2.2, which includes the effects of the field winding, an additional damper winding on the direct axis and twodamper windings on the quadrature axis, the per-unit direct and quadrature axis stator fluxes are given by

(2a)

(2b)

Similar expressions may be derived for higher or lower order models, or for Model 2.2 with the differential leakagereactance Lf1d included, as in Fig 2 of Section 2.1.

The differential equations of the stator winding in Park's form are given below.

(3a)

(3b)

Here, ωr is the rotor speed (in electrical radians per second), ωo is the synchronous electrical frequency (radians persecond), and time (t) is in seconds. The above sign convention for voltages, currents and flux linkages also conformsto the convention that the quadrature axis leads the direct axis, when the field is assumed rotating counter-clockwiserelative to the armature. (See IEEE Std 100–1988 [1] for the definition of “quadrature axis”.) This convention is alsopreferred in the widely used stability programs referred to in Appendixes 5B and 5C.

T e ψd iq ψq id⋅–⋅=

ψd L–=d

id Lad i fq Lad+ i1d⋅ ⋅+⋅

ψq L–=q

iq Laq i1q Laq+ i2q⋅ ⋅+⋅

ed Ra id⋅–1

ωo------

dψd

dt----------

ωr

ωo------ ψq⋅–+=

eq Ra id⋅–1

ωo------

dψd

dt----------

ωr

ωo------ ψd⋅–+=



In large-size transient stability programs, the rate of change of stator flux linkages is usually neglected. This is done

primarily to be consistent with the practice of neglecting power system network transients. Omission of the and

terms implies that stator dc offsets during faults or other unbalances are ignored. This simplification permits a

positive sequence representation of both generator and power network. It also results in a conservative answer, i.e., atransient power limit, for a given disturbance, would be lower with this omission.

The resulting equations, then, would become

(4a)

(4b)

Solving for ψd and ψq, (above) and substituting into (1) for Te, gives

(5a)

or

(5b)

where Pe = electrical power and id 2 + iq2 =Ia

2

It is sometimes customary to ignore changes in electrical machine speed, ωr. Exceptions would occur when sustainedsystem frequency deviates noticeably from the normal 50 Hz or 60 Hz. This is most likely with small isolated powersystems, or where “islanding” conditions suddenly are encountered in a portion of a large power system. Normally ωris very close in value to ωo. Assuming then that ωr/ωo = 1.0, Eq 5b simplifies to:

(6)

Neglecting the instantaneous changes in speed tends to give an optimistic result. References [2] and [3] indicate thatneglecting both d ψ/dt terms compensates for the omission of changes in instantaneous speed. Both of thesesimplifications appear to be justified, not only from a practical consideration of the orders of magnitude of thequantities involved, but also due to the fact that measured test results show reasonable agreement with simplifiedcalculations, as used in Eq 6.

This issue is further discussed in [4]. The authors treat the general case of any machine and then specifically discussPark's equations for a synchronous machine, where their arbitrary reference frame of electrical speed is replaced by therotor reference frame as in Eq 3a and Eq 3b. They finally conclude that neglecting both the rate of change of fluxlinkages in the rotor reference frame, and setting the rotor speed equal to the system electrical speed is required inorder to neglect the electrical transients due to the stator windings of the synchronous machine.

4A2 References

[1] IEEE Std 100–1988, IEEE Standard Dictionary of Electrical and Electronics Terms.

dψq

dt----------

dψd

dt----------

ed Ra id⋅–ωr

ωo------ Ψq⋅–=

eq Ra iq⋅–ωr

ωo------ Ψd⋅+=

T e ed id eq iq Ia2

+ Ra⋅ ⋅+⋅{ }ωo

ωr------=

T e Pe Ia2

Ra⋅+{ }ωo

ωr------=

T e Pe I2

a Ra⋅+=



[2] C. Concordia, “Steady-State Stability of Synchronous Machines as Affected by Voltage -Regulator Characteristics,“AIEE Transactions, vol. 63,, May 1944, pp. 215–220.

[3] D.N. Ewart and F.P. De Mello,” A Digital Computer Program for the Automatic Determination of DynamicStability Limits,” IEEE Transactions on PA&S, Power Apparatus and Systems, vol. 86, July 1967, pp. 867–875.

[4] P.C. Krause, F. Nozari, J.L. Skvarenina and D.W. Olive, “The Theory of Neglecting Stator Transients,” IEEETransactions on PAS, PAS vol. 98, no. 1, Jan/Feb 1979, pp. 141–148.

5. Representation of Generator Saturation and its Effect on Generator Performance

5.1 General

The various implications of synchronous generator saturation have been discussed extensively in the literature formany years. The methods of simulation have been many, and only a limited number of the most frequently usedcomputer simulation procedures will be considered in the following section. In general, saturation effects insynchronous generators should be accounted for in stability programs analyses. However, the effects of the usualsaturation approximations do not have a large impact in most cases on system planning or system operation policies ordecisions. But in special cases, the correct representation of the internal excitation and/or the internal generatorloading angles can be significantly affected by saturation. Correct representation of saturation, therefore, can beimportant for accurate simulation of generator stability performance.

5.2 Representation of Generator Saturation in the Steady State

The effect of saturation on synchronous generator steady-state performance has been recognized for at least sixty yearswhen the initial concern was with accurate calculation of field excitation, for exciter design purposes and sizing.

Standard methods of allowing for saturation, by calculating saturation factors, were codified in 1945 with the firstAIEE test code for synchronous machines. These procedures now appear essentially in the same form in IEEE Std115–1983 (R1991) [1]. Other publications and AIEE papers also treat the subject, and discussed the differences inapproach between salient pole (hydraulic) generators and round-rotor turbo-generators [B1], [B5] and [4]. For the firsthalf of this sixty-year period, emphasis was directed to the effect of saturation on determination of field excitations,while for the past twenty years, the effect of saturation on determining internal angles for turbo-generators hasreceived much attention as well.

Details of these “standard” methods of calculating field excitation by using saturation factors will be given inAppendix 5A. The two following subsections summarize the development of saturation methods leading to presentday stability (computer) analyses. Appendixes 5B through 5D provide more detailed descriptions of present methods.

5.2.1 Salient Pole Machines

In the modeling of salient pole generators, saturation has normally been assumed to occur only in the direct axis. Thegenerator open-circuit saturation curve is used to determine the saturation factors in this axis. An internal voltage“behind some specified reactance” is used to locate the operating point on the open-circuit saturation curve to calculatea saturation factor K (or a saturation correction ∆E). K or ∆E are the ratios (or differences) in excitation between theair gap line and the actual amount of excitation from the curve at the internal voltage operating point.

There are several suggested methods regarding the use of these K or ∆E factors. The K factor is divided into a portion(Xdu − Xp) of the unsaturated synchronous reactance Xdu,and this saturated portion plus Xp constitutes a saturatedsynchronous reactance. Then, after an internal voltage is found, based on stator terminal conditions, this voltage, inturn is multiplied again by the K factor. Justification for this approach is described in Appendix 5A.



Alternatively, another internal voltage, based on unsaturated synchronous reactance was obtained from a phasoraddition of terminal conditions, plus an increment (∆E) added to obtain the correct excitation EI, in per-unitcalculation. Variations in these two basic approaches constituted the methods used for salient pole machines beforecomputer stability programs were in common use. More specifically, a Potier reactance was used to find Ep, for anyterminal operating point, and thus to obtain ∆Ep. This is discussed in IEEE Std 115–1983 (R1991) [1] and is stillreferred to as the IEEE or Standard Method. See also Fig 5A-1 of Appendix 5A. Occasionally, Xl was substituted forXp, and the same process was followed.

Values of Potier reactance suggested by hydraulic machine designers lie between leakage reactance values andtransient reactance values. Typical numbers for Potier reactance, [3] if none are specifically available, can be obtainedfrom the following approximations:

a) Xp = Xl + 0.63(X′d − Xl)b) Xp = 0.8(X′d)

When saliency was first modeled for computer programs for hydraulic machines, a quadrature axis was located withrespect to the terminal voltage, using the unsaturated quadrature axis synchronous reactance. The voltage forsaturation determination was the projection, on the quadrature axis, of the internal voltage behind transient reactance,(E′). This voltage projection (designated E′q) is proportional to field flux linkages.

The incremental saturation correction factor ∆E seems to have had preferred use in many commercial computerstability programs. The internal excitation Xadu··ifd, (or Ladu·ifd) is then the sum of several components. This form ofexcitation determination has had widespread use since the early 1960’s.

Thus, Xad · ifd = EI = E′q + Id(Xd-X′d) + ∆E, here all values are in per unit (Eq 1).

∆E, as noted above, for salient pole machines, is a function of E'q;Xd and X'd are taken as unsaturated and saturatedvalues, respectively.

5.2.2 Solid Rotor Machines

In solid iron rotor turbine generators, saturation can be significant in both the direct and quadrature axes. In past years,various approximations similar to those described in Section 5.2.1, have been used to model saturation, primarily inthe direct axis. One exception is described in [4], which considers saturation at the air gap by calculating an air gapmagnetomotive force (mmf), in space-phase with the air gap flux, and thus with components in both axes. This air gapmmf. is primarily a function of air gap flux, but with a correction for the amount of generator field current. Onoccasion, the effect of the saturation in the quadrature axis was measured, but no design or test data was generallyavailable to produce a “quadrature axis saturation curve.” As is now widely recognized, saturation in the direct axisaffects principally the excitation (EI), while quadrature axis saturation mostly affects Xq, which determines the internaloperating angle between the quadrature axis and the terminal voltage.

Since, in the past, quadrature axis saturation data was not usually obtainable, the generator direct-axis open-circuitsaturation curve was used to determine the operating point for which a saturation factor Kd was calculated. While thePotier reactance proved more than acceptable for hydraulic machines for the purpose of excitation calculation, it wasfound that use of Potier reactance in turbo-generators, using the ∆Ep correction described in the IEEE Standardmethod gave readings of field excitation slightly higher than might have been expected.

An adjusted synchronous reactance method, based on, but not identical to the saturation corrections discussed asReference [4], had also come into use for turbo-generators. The saturation corrections were applied to Xadu, where byaccepted use Xadu = Xdu – Xl. The voltage El, back of leakage reactance Xl., was used to determine a factor Kd, fromthe open circuit saturation curve.

Thus, Xadsat = Xadu/Kd and Xdsat = Xadsat + Xl.



A voltage back of Xdsat located a phasor EQD, which gave a reasonable indication of internal angle, although not aprecise one.

(2)

with all phasors and reactances in per unit. The voltage EQD, multiplied by Kd, then gave the field mmf, EI and aresultant field current, using field base amperes in the nonreciprocal system.

In representing round-rotor effects and saturation at present in one well known commercial stability program,subtransient flux or an internal voltage “behind subtransient reactance” is used as the operating point for calculatingthe direct axis saturation increment factor. The quadrature axis is treated in a somewhat similar manner. Basic detailsof this computer program, for generator stability modeling, are given in Appendix 5B.

In another major program in current use, both direct-axis and quadrature-axis saturation effects are represented byspecifically adjusting Xadu, and Xaqu elements in the stability models. Both axes are involved in the saturationcalculation procedures. In the first option, Kq is assumed equal to Kd, both being based on the open-circuit saturationcurve. Alternatively, in the second option for round-rotor representation, Kd is calculated from a direct-axis saturationfunction. Kq is calculated from a quadrature-axis saturation function to give Xqsat, and a more accurate internal angle.The air gap flux or internal voltage behind leakage reactance is used in all cases to define the operating point on eithersaturation curve. The details of the latter method are given in Appendix 5C, including the development of both a direct-axis and a quadrature-axis saturation function calculated from on-site measurements, which produce internal phaseangle readings, and readings of field current. The reader may obtain more background on this aspect of using on-sitemeasurements by consulting Reference [6] and the discussions of Reference [9].

In the programs discussed in Appendix 5B and 5C, saturation in the quadrature axis, for salient pole machines, isignored.

Several other methods have been developed to allow accurate prediction of both excitation requirements and internalangle simultaneously, but some of these are difficult to implement in stability modeling programs or require longerprogram execution times [10]. For further discussion, see Reference [3] of Chapter 2.

In summary, the methods outlined in Appendix 5B and 5C both give good agreement in calculating field excitations forcomparison with measured results and with each other. Use of a quadrature axis saturation function, as suggested inAppendix 5C shows greater accuracy in predicting generator internal loading electrical angles, than simply using theopen-circuit saturation function as suggested in Appendix 5B.

5.3 Representation of Saturation Effects During Large Disturbances

5.3.1 Current Approaches and Assumptions

This section is intended to cover the issues of note in accounting for saturation during large disturbance (time-domain)studies. By their very nature, such disturbances result in operating conditions that vary significantly from their steady-state values to the extent that a complete nonlinear model is desirable, but only as an ideal.

The current state of the art does not allow for the simple implementation of a complete non-linear, time-dependentstability model. This is true for a number of reasons. First of all, much investigation will be required before time-transient, magnetic field-based or finite element analysis can be developed for large power system representations. Itwould possibly be far too time consuming and costly for all but a few specialized studies.

More specifically, the magnetic saturation of a generator is, in principle, dependent on the electrical current in every“circuit” (as well as the recent magnetic history of the steel). For the three rotor circuit configuration of Model 3.3, oneought to know four currents as inputs, in each axis. Therefore, there may be 8 values in total of circuit parameters orinductances to be adjusted for saturation (4 in each axis). Such an approach appears difficult and impractical to

Thus EQD, Ea jIa Xdsat⋅+=



implement principally due to the lack of input data. As an alternative, emphasis is placed on choosing one or twoimportant inputs or factors. Recognition is given to the fact that current state-of-the-art methods of accounting forsaturation in this latter way are crude. They are, however, reasonably satisfactory and simulation results from transientstability studies, using these models, generally agree reasonably well with on-site measurements recorded duringactual disturbances.

The net result is that, as a practical matter, saturation corrections during the simulation process of large disturbancesare generally limited to adjustment of the magnetizing inductances, Lad and Laq, This is done using essentially thesame techniques as for steady-state saturation discussed in Section 5.2 and Appendix 5A.

It is the opinion of some analysts that during severe system disturbances, when generator currents are usually two orthree times normal, that the stator leakage inductance, Ll, should also be adjusted. Appropriate background material isnoted below [B2, B4], but such procedures are not common practice in current large size stability programs.

During large disturbances, there are major changes in machine stator voltage and stator currents. The manner in whichthese can be accounted for in commercial or widely accepted stability simulations depends to a degree on whether thecomputer codes are based on the “time constant and reactance” approach, or alternatively on the use of d- and q-axismodel structure in which the resistances and reactance values of the structure elements of stator and rotor, includingthe field circuit, are explicitly known. The differences between these two approaches are discussed below in Section5.3.2.

5.3.2 Adjustment of Parameters During Time Domain Calculations

As noted in Section 5.2.1, the adjustment of parameters during time domain calculations, to account for saturation,depends on the organization of the machine flux, current and voltage equations.

Simulation codes are organized in two distinct ways:

1) The machine equations are stated in terms of reactances and time constants (transient and subtransient).Machine parameters are provided in this form according to IEEE Std 115–1983 (R1991) [l]. Historically,these could be measured directly from terminal short-circuit tests, although more recently, they are calculatedby the manufacturer. It is noted that q-axis parameters cannot be obtained from this type of test; however,calculated values of similar q-axis reactances and time constants are frequently furnished, as discussed inSection 6.3.

2) The machine equations are stated in circuit-model form. The parameters required are the stator and fieldleakage inductances, magnetizing inductances, Lad and Laq, and leakage inductances and resistances for eachequivalent rotor body branch. These may be derived from frequency response measurements as specified inIEEE Std 115A–1987 [2], or by analytical procedures. The field resistance rfd in physical ohms should alsobe known.

Additional Comments on the reactances and time constants format noted in (1) above are as follows:

In this approach, the reactances and time constants are furnished as “rated voltage” or “rated current” values. It isgenerally customary to use the rated-voltage values. They reflect test conditions where the terminal short-circuit wasapplied from rated open-circuit voltage. As such, they contain some measure of saturation typical of a large transient,which would occur during normal load. By implication, the constants input to the simulation are, therefore, alreadyadjusted for saturation, although the degree of adjustment to individual circuit constants cannot be identified orquantified. As noted in Section 6.3, it is customary for manufacturers to account for the inductance of generator fluxlevels on both the rotor and stator leakage inductances. In one company, the value of starer leakage inductance Ll thatis nominally quoted is the average of rated-voltage and rated-current conditions.

The adjustment of time constant values during time domain calculations is further described in Appendix 5B. Anexamination of the block diagram of Fig 5B-2 shows that T′do, and T′qo are directly affected in stability computationsby adding a saturation component ∆Lad · ifd to the appropriate feedback loop that mathematically describes the rate of



change of direct axis field flux linkages (E′q) or quadrature axis rotor flux linkages (E′d). Equation 4 of Appendix 5Bis shown for the direct axis field flux linkage changes. These incremental saturation functions do not influencesubtransient time constants as presently utilized and described in that Appendix.

The user also may recognize that a “rated voltage” or “saturated” value of T′do, would be further adjusted based on thesaturation algorithm just discussed above. Users of the “time constant and reactance” approach should also be awarethat the values of open-circuit time constants are usually based on the common acceptance of them being calculated asfollows:

(3)

(4)

Both of the above time constants are in radians, using values of resistance and inductance in per unit.

The above derivations follow from the assumption that Rfd is equal to zero during the subtransient period, and R1d isinfinite during the transient domain. The same comments apply to the quadrature axis, using R1q, and R2q.

Additional Comments on the Equivalent Circuit Format noted in (2) above are as follows:

Where the generator equations are formulated in terms of equivalent circuit constants, customarily only Lad and Laqare adjusted for saturation at each time step. The saturation function used will be the same one that is used for thecalculation of these parameters when the initial values of excitation and load angle are calculated. The values for Ladand Laq that are input to the simulation will be unsaturated values. Various saturation methods that have beendeveloped for adjusting Ladu and Laqu are described in Appendix 5C. In the latest methods, a different saturationfunction is used for each axis.

It is noted that in this approach, the values of the transient and subtransient time constants will be adjusted forsaturation through the adjustment of only one component of Eq 3 and Eq 4 above, namely Ladu. The leakageinductances, Lfd and L1d are not adjusted. A similar comment would apply for the quadrature axis.

In conclusion, and irrespective of the apparent differences, relatively minor variations in results of stability simulationsbetween the two above approaches have been detected for situations where comparisons have been made.

5.3.3 Treatment of Third-Order Models

Higher-order models than those usually furnished by the manufacturer can be derived from frequency response testing,particularly for the q-axis. Models may be derived from frequency response testing at standstill (SSFR), or correctedfrom on-line frequency responses (OLFR). The steady-state values of Xadu (or Ladu) and Xaqu (or Laqu) are arbitrarilyinserted into the model structure in place of the SSFR values Lad(o) or Laq(o). Refer to Section 6.2. This is done inorder that the transient stability modeling process will initialize to the correct steady-state operating point whenconsidering saturation.

Theoretically, the use of small signal circuit constants can be questioned for large disturbance analysis. Theseconstants are governed by the incremental permeability of the rotor iron [11]. Another pertinent factor is possiblevariation in the rotor iron effective skin depth arising from large signal excursions in the flux density, for which theincremental permeability concept might no longer apply. These factors are discussed in more detail in Section 5.4.

Experience shows that small-signal-derived constants can be successfully used in modeling transients involving largedisturbances. The computer results using circuit constants based on “standard” or short-circuit concepts are similar tothose using SSFR-derived constants. In some cases reported on, the models derived from frequency response testing

T ′do

Ladu L fd+

R fd-------------------------- and,=

T ″do1

R1d--------- Lid

Ladu L fd⋅Ladu L fd⋅------------------------+

=



even gave a closer agreement to test results in simulations than did the “standard” models. These simulationsduplicated on-site switching of lines near turbo-generators [5,6,7,8].

5.4 Generator Saturation in Small Disturbance Modeling

5.4.1 General Comments and Theoretical Background

Small disturbance theory involves concepts of linear responses of a generator to perturbations in the generatorarmature and field currents. This means that the permeability describing the behavior of the generator iron can beconsidered constant relative to the magnitude of the small-signal swings in flux density. However, (as a result of thehysteretic nature of iron), the B-H path for small disturbances is different from the path followed for steady-stateoperation in a solid iron rotor, that is, the apparent permeability of the rotor is different. This means that generatorsmall-signal circuit constants are somewhat different from the steady state circuit constants (Ld and Lq). The latterconstants are used solely for small signal study initialization purposes. The rotor permeance is only one component ofthe entire generator magnetic circuit, so that effect of the small-signal permeability on circuit constants can bedetermined only by detailed magnetic analysis or by test. Thus, the small signal operational inductances (or circuitparameters) cannot be deduced in any way from steady-state data.

Figure 5 is schematic representation of the B-H path in rotor iron for both steady-state and small-signal behavior. Thecontinuous curve represents the normal (steady-state) magnetization curve. The small loops show the B-H path forsmall deviations. The normal permeability is defined as the quotient B/H at any point on the normal curve. Theincremental permeability is defined as the slope of the incremental loop. This slope depends on signal amplitude; forsimplicity, the limiting value for zero signal is usually used. The loops depicted are those that would result after manycycles of small signal perturbations. The actual B-H path in arriving at a small signal-steady condition is morecomplicated. For conceptual (and analytical) purposes, it is sufficient to assume that for each value of steady-state fluxdensity in the rotor, there is a corresponding value of incremental permeability. The analysis of small-signal behaviorcan be done by replacing the steady-state permeability by the corresponding value of incremental permeability inevery region of the rotor. The incremental permeability is about 1/10 of the normal (steady-state) permeability forpoints on the “air gap” portion of the normal B-H characteristic. The so-called “initial permeability,” that is measuredaround the toe of the normal B-H curve, is a special case of incremental permeability [11]. This condition applies whenstandstill frequency response measurements are made. Under these test conditions, the operating flux density is zeroover the entire rotor cross section. Because the incremental permeability is smaller at high (and normal) operating fluxdensities, it would be expected that the small signal inductance parameters would be different between standstillconditions and rated operating conditions. This has been confirmed by both analysis and test, as discussed below.

Figure 6 shows the small signal frequency response of the field to armature mutual inductance (both calculated andmeasured) for a turbo-generator. Two pairs of curves are shown; one at zero excitation (standstill), and one for runningon open-circuit under rated-voltage conditions. These latter tests have become known as open-circuit frequencyresponse (OCFR) tests. Since these characteristics were measured for small-signal excitations around their respectivesteady-state operating points, circuit constants derived from them are entirely appropriate for small-signal (eigenvalue)analysis. The zero frequency intercept of the magnitude of Lad is the small-signal value of this mutual inductance. Atthe rated-voltage operating point, the value is about 60% of the steady-state value (Ladu). Other than the low-frequencyresponse below about 1.0 Hz, there is not much difference between the OCFR values and standstill values of Lad forthis particular generator.

Most small signal analysis in power systems is concerned with oscillation modes in the 0.1 Hz to 10 Hz range. Whenconsidering the effects of saturation in general, and based on existing line small signal OCFR test data, it would appearthat the actual small signal value or incremental value of Lad does not affect results significantly in the 0.5 Hz to 10 Hzspectrum. Hence, incremental saturation effects, even if they are modeled differently from the incrementalpermeability concept, should have a relatively minor impact on small signal analyses. The justification for thisstatement is that the rotor values of R1d, L1d, R2d, L2d are considered the important factors in the machinerepresentation under analysis conditions in the frequency range from 0.5 Hz to 10 Hz. This concept has beenconfirmed to some extent by the test data referred to above where one 500 mW, two-pole machine had no amortisseurs,



while the other 500 mW, two-pole machine had amortisseurs in both axes [6]. The machine without the amortisseursis also the same one whose finite element analyses is partly shown in Fig 6.

5.4.2 Guidelines for Practical Analyses

As noted in the previous section, values of magnetizing inductance Lad are depressed significantly to a fraction of theirsteady-state value (Ladu), as verified by OCFR tests [6] and finite element analysis [B3]. For small-signal modelingand eigenvalue analysis, it is theoretically correct to retain the value of magnetizing inductance obtained from the zero-frequency intercept of the small-signal operational inductance. The OCFR test value is the most appropriate value forthis purpose. OCFR test measurements give a value for Xad, but not for Xaq. However, finite-element analyses suggestthat Xaq is depressed to about the same degree as Xad.

Figure 5 —Schematic Examples of Incremental Minor Loops



Figure 6 —Test Data and Finite-Element Calculated Data

The best available rule of thumb for treating saturation in generators for which small-signal data are not availablewould be to take 60% of the manufacturers value for Ladu and Laqu to use in eigenvalue analysis. It is considered likelythat eigenvalue results are not very sensitive to the number used for the magnetizing inductance in any event.

In the last paragraph of Section 5.3.1, it was suggested that values of rotor circuit constants (R1d, L1d etc.) obtainedfrom some type of small signal, or frequency response testing were appropriate to use along with low values of Lad.The latter reflected the effect of incremental permeabilities in the magnitudes of both Ladu and Laqu. Further commentsin determining parameters for this small signal aspect of generator modeling are noted in Appendix 6B1.

5.5 References

[1] IEEE Std 115-1983 (R1991), IEEE Test Procedures for Synchronous Machines (ANSI).


[3] Beckwith, S., “Approximating Potier Reactance,” AIEE Transactions, vol. 56, 1937, pp. 813–818.



[4] Concordia, C., Crary, S. B., and Lyons, J. M., “Stability Characteristics of Turbine Generators, ”AIEETransactions, vol. 57, 1938, pp. 732–744.

[5] Dandeno, P. L., Kundur, P., Poray, A. T. and Coultes, M. E., “Validation of Turbo-generator Stability Models byComparison with Power System Tests,” IEEE Transactions on PA&S, vol. PAS-100, no.4, April 1981, pp. 1637–1645.

[6] “Determination of Synchronous Machine Stability Study Constants,” EPRI Report, EL 1424, vol. 2 (OntarioHydro), Dec., 1980.

[7] “Determination of Synchronous Machine Stability Study Constants,” EPRI Report, EL 1424, vol. 1, Sept. 1980,Westinghouse Electric Corporation.

[8] Hurley, J. D. and Schwenk, H. R. “Standstill Frequency Response Modeling and Evaluation by Field Tests on a 645mVA Turbine Generator”, IEEE Transactions on PA&S, Feb. 1981, vol. PAS-100, no. 2, pp. 828–836.

[9] Minnich, S. H., Schulz, R. P., Baker, D. H., Sharma, D. K., Farmer, R. G., ,and Fish, J. H., “Saturation Functionsof Synchronous Generators from Finite Elements,” IEEE Transactions on Energy Conversion, vol. EC-2, Dec., 1987.

[10] Shackshaft, G. and Henser, P. B., “Model of Generator Saturation for Use in Power System Studies,” IEEProceedings, 1979, 126(8), pp. 759–763.

[11] S. H. Minnich, “Small Signals, Large Signals, and Saturation in Generator Modeling,” IEEE Transactions onEnergy Conversion, vol. EC-1, March 1986.

5.6 Bibliography

[B1] Electrical Transmission and Distribution Reference Book (W.E. Corp., Pittsburgh), 1964, pp. 147–152.

[B2] J. F. Flores, G. W. Buckley and G. McPhereson, Jr., “The Effects of Saturation of the Armature LeakageReactance, of Large Synchronous Machines,” IEEE Transactions on PA&S, PAS-103, 1984, pp. 593–500.

[B3] J.W. Dougherty and S.H. Minnich, “Operational Inductances of Turbine Generators; Test Data vs. Finite-Element Calculations,” IEEE Transactions on PA&S, PAS-102, Oct., 1983, pp. 3393–3404.

[B4] P.J. Turner, “Finite Element Simulation of Turbine Generator Terminal Faults and Application of MachineParameter Prediction,” IEEE Transactions on Energy Conversion, vol. EC-2, no. 1, March 1987, pp. 122–131.

[B5] Power System Stability, Vol. III, (Synchronous Machines), E.W. Kimbark, (1956), pp. 118–128, Library ofCongress Catalog No., 48-6164. John Wiley & Sons, Inc., N.Y., New York, second printing, July, 1962.



Annex 5A Saturation—Past Practices and General Considerations(Informative)

(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in StabilityAnalyses, but is included for information only.)

5A.1

As stability techniques developed over the years, more consideration and emphasis was placed on the effects ofsaturation on the synchronous machine variables, i.e. Xd, X′d, Xq, X′q. With the advent of digital computers, detailedmodels of the excitation system became practical to use. There was a concomitant need to more accurately specifyexcitation system ceiling voltages, in practical terms. This, in turn, necessitated an accurate prediction, primarily offield current, at the commencement of stability time domain studies—the so-called initialization process. Secondly, itwas vital to determine for a specified excitation system ceiling voltage, the actual variations in field current.

The field current or field excitation Lad·Ifd (or EI) during time domain or step-by-step stability calculations is a functionof both direct-axis field flux linkages (E′q) and of armature, load current (id).

As noted in Section 5.2, there were at least two methods of determining the effect of saturation on field excitation thatwere in popular use for many years. The method proposed in IEEE Std 115-1983 (R1991) [1], a saturation functionor increment, ∆E was determined based on a voltage behind Potier reactance, or on a voltage behind leakage reactance.Figure 5A-1A1 shows the calculation procedure in phasor diagram form. Note that the increment ∆Ep, or ∆El, couldbe added directly to the phasor EIU, the voltage behind unsaturated direct axis synchronous reactance, or could beadded to EIU, but as a phasor in phase with Ep. Direct addition to phasor EIU, is shown in Table II calculations.

In the second approach, “Saturated Synchronous Reactance Method,” the unsaturated direct axis synchronousreactance was broken into two components; the larger component was adjusted by a saturation factor Kd or Kp,obtained as discussed in Section 5.2.2.

Thus, , or, , using leakage reactance.

An internal voltage was then determined, where EQD = Ea + jIa ·Xdsat

This is shown in phasor diagram form in Fig 5A-2.

Then Et = Kd · EQD

This operation of multiplying the internal voltage back of saturated synchronous reactance by the factor Kd, (or Kp)can be seen in graphical form where a straight line is drawn from the origin of the open-circuit saturation curve througha point Ep or El on that curve. The extension of the straight line to a value corresponding to a voltage EQD on theordinate of the diagram will determine a point EI which is the excitation on the abscissa. This is shown in Fig 5A-3,and the data are taken for the machine loading indicated for point A in Table I.

A comparison of the method shown in IEEE Std 115-1983 (R1991) and the saturated synchronous reactance method,both using Potier reactance, is shown in Tables II and Table III of this Appendix. For the three test points chosen inTable I, the IEEE method appears to be more accurate in the low excitation range.

The same test points are also compared in Appendix 5E using present-day computer-based algorithms where directand quadrature axes are represented. Note, in particular, the closeness between measured and calculated angles shownin Table III of Appendix 5E. This may be compared to the approximate angles calculated in Table III of this Appendix.

Xdsat

Xdu X p–

K p---------------------- X p+= Xdsat

Xdu Xl–

Kd--------------------- Xl+=



Figure 5A-1 —IEEE Standard Method in Phasor Diagram Form

Figure 5A-2 —Saturated Synchronous Reactance Method in Phasor Diagram Form

Figure 5A-3 —Saturated Synchronous Reactance Method—Graphical Form



Table 5A-I —(Test Results)

Sample Calculations

Table 5A-II —IEEE Standard Method-555 mVA Turbo-24kV (Method 1)

Table 5A-III —Saturated Synchronous Reactance Method 555 mVA Turbo-24kV (Method 2)

Test Point # & EPRI Report #

P mW

Q mvar

Ea kV

Ifd(M) Amps

M = Meas

δ int. angle Degrees (M) M = Meas

A - #623 506.8 +200.4 25.00 3615 36.6°

B - #744 510.0 +11.7 24.13 2911 49.5°

C - #390 306.7 -82.6 23.21 1832 47.4°

Test Point

EI = Ifd(C)

C = CALCIfd(M)

M =MEAS

A 1.175 .500 2.448 2.948 3832 3615 .060

B 1.048 .172 2.088 2.260 2938 2911 .009

C .937 .077 1.310 1.387 1803 1832 –.016

Test Point Kd Xdsat

as a phasor compared to Ea

EI = Ifd(C)

(CALC)Ifd(M)

(MEAS)

A 1.175 1.426 1.478 2.022 /39.8° 2.883 3748 3615 .037

B 1.048 1.164 1.743 1.901 /56.8° 2.213 2877 2911 –.012

C .937 1.078 1.850 1.246 /57.9° 1.344 1747 1832 –.046

E p ∆E p EIUEIU ∆E p+

I fd C( ) I fd M( )–

I fd M( )

------------------------------------

E p

EQDEQD Kd⋅


I fd M( )------------------------------------



Annex 5B

Steps Used in a Widely-Used Commercial Stability Program to Account for Saturation During the Step-by-Step Calculations(Informative)


The equations used in this commercially available transient stability program are shown below for the direct axis. L′dand L″d values are saturated; Ld and Lad are unsaturated.

(4)

(4A)

(4B)

(4C)

Similar equations apply for the quadrature axis. The steady-state relationships may be obtained by setting i1d, i1q, i2qand the rate of change of flux linkages equal to zero. Then Lad·ifd = ψ′d +(Ld - L′d) ·id, in steady state.

Initial values of direct and quadrature axis quantities are found using the initial internal angle, where saturation of Lq is afunction of ψ″ and the open-circuit saturation curve. A phasor representation of this approach is shown in Fig 5B-1.

The manner in which these equations may be solved and the overall calculation process is shown in Fig 5B-2. Thismodel accounts for saturation changes during the step-by-step process by adding an incremental (∆EI or ∆Lad ·ifd) termin both axes with some adjustment to account for saliency between the two axes. The data used for saturation is theopen-circuit saturation curve with values of voltage expressed in per-unit and the field current in per unit as Lad·ifd).The base for Lad·ifd is chosen so that on the air gap line, 1 p.u. voltage or flux is produced by 1 p.u. Lad·ifd.

The following steps account for saturation:

1) Enter the current value of |ψ″| as the voltage value for the saturation curve and determine the correspondingvalue of Lad ·ifd.

2) Subtract from Lad·ifd the air gap value of Lad·ifd for the same value |ψ″| to obtain Lad·∆ifd. With the per-unitsystem so chosen, the air gap value of Lad·ifd will be equal to the current value of |ψ″|.

3) Multiply Lad·∆ifd by to obtain ∆E1d, which is added at the summing junction as shown in Fig 5B-2.

4) Multiply Lad·∆ifd by to obtain ∆EIq, which is added at the lower summing junction as shown in

Fig 5B-2.

ψ″d ψ′dL″d Ll–

L ′d Ll–------------------- ψ1d

L ′d L″d–

L ′d Ll–----------------------+=

dψ1d

dt-------------

1T ″do----------- ψ( ′d ψ1d– id L ′d Ll–( ) )–=

Lad i fd ψ′d Ld L ′d–( )+=⋅ id T ″do

dψ1d

dt-------------

L ′d L″d–( )

L ′d Ll–( )2---------------------------⋅+

dψ′1d

dt--------------

dE ′qdt

-----------1

T ′do---------- E fd Lad i fd⋅–{ }= =

ψ″d

ψ″---------

ψ″q Lq Ll–( )ψ″ Ld Ll–( )--------------------------------



Additional comments on the assumptions implicit in the above methods of representing saturation changes are givenin Section 5.3, which discusses saturation representation during large disturbances.

Figure 5B-1 —Phasor Diagram for Commercial Stability Program #1



Figure 5B-2 —Block Diagram for Generator Model from Current Stability Program



Annex 5C

Procedures in a Second Stability Program to Account for Saturation When Adjusting Mutual Reactances

(Informative)


5C.1 Theoretical and Practical Aspects

This second program is described in Reference [1] of this Appendix. Generator equations are written initially bycalculating direct or quadrature axis flux linkages in terms of direct or quadrature axis circuit element values, anddirect or quadrature axis stator or rotor currents. Xd and Xad when used in Eq 1, Eq 2, and Eq 3 are unsaturated values.

Thus for example, in the general case,

(1)

and the process is continued for ψfd, ψ1d, ψ2d, ψ1q, ψ2q… etc.

The stability program then eliminates stator flux linkages and rotor currents by expressing them in terms ofcomponents of rotor flux linkages and stator currents.

The general form of the initial equations is also given in Fig 10 of Section 6.5. When using this approach, it is alsocustomary to convert ifd and efd per-unit field quantities, which are in the reciprocal system, into nonreciprocal fieldquantities. This is done to permit interfacing with excitation system per-unit relationships. These concepts are noted inAppendix 7B and also completely discussed in [2], which also covers the theory of this second program.

In the steady state, rotor currents i1d, i2d, i1q, i2q … etc., are equal to zero. The direct axis flux linkage equationsimplifies to

(2)

For actual operating conditions, the above equation, (referring to the phasor diagram of Fig 5C-1, and noting that ψd= Eq) can be rewritten, with Id as a phasor:

(3)

ifd will be in per-unit using the reciprocal system. Then with saturation included,

(4)

Xadsat, and (Xaqsat) are adjusted values and depend on the initial operating point.

The steps for specifically adjusting Xadu and Xaqu to obtain Xdsat and Xqsat are given below.

Ψd Xd id⋅– Xad i fd Xad i1d Xad+ i2d⋅ ⋅+⋅+=

ψd Xd id Xad i fd in p.u⋅+⋅–=

Xad i fd Eq Xd jId⋅+=⋅

Xadsat i fd Eq= Xdsat jId⋅+⋅



Step Number

1) Determine a q-axis saturation function Kq, for a calculated air-gap voltage El depending on machine power,reactive power and terminal voltage (q-axis saturation curve is known or see Fig 5C-4 as an example).

2) Determine Xqsat where and Xqsat = Xaqsat + Xl

3) Determine δ the internal machine angle, located by the phasor EQD, where

4) For the same El calculated in step (1) above, find Xdsat, where is determined from a test

saturation curve, or from the open circuit saturation curve.5) Determine ifd, where Xadsat·ifd = Eq + jId·Xdsat, (All values in per-unit reciprocal).

5C.2 Use of On-Site Measurements to Obtain Saturation Factors

One basic approach to determining Kq and Kd factors can be through the use of on-site measurements of generatorquantities. The values of armature voltage, current and power factor (or mW, mvar and kV) are measured. In addition,field current and internal angle are measured. [1, 3]

From the phasor diagram shown in Fig 5C-1, Ed, the projection of the terminal voltage on the direct axis also has themagnitude of a vector corresponding to Iq (Xqsat). Note that Iq is the projection of the armature current Ia on thequadrature axis. The power angle , and the internal angle, have been measured, as have Ea and Ia.

An examination of the phasor diagram in Fig 5C-1 shows that

Ed = Iq·Xqsat and Xqsat = Ed/Iq

An internal voltage back of Xl the leakage reactance, is determined and its magnitude is called El· (El is not shown onFig 5C-1.) The equivalent excitation for this particular operating point is Iexq and, by definition

Iexq = El/Xaqsat

Repeating Eq 2 Xadsat·ifd=Eq+ Id(Xdsat) and noting that Xadsat = Xdsat – Xl we have (Xdsat – Xl) ifd = Eq+Id (Xdsat).

Solving for Xdsat it follows that .

All of the above quantities are per-unit values in the reciprocal system discussed in Reference [6] of Chapter 7.Reciprocal and nonreciprocal field-base concepts are also discussed in the example of Appendix 7A, and shown inFigs 7A-1 and 7A-2.

Figure 5C-2 shows several Iexq and Iexd points obtained from readings for a turbo-generator at power loadings of310 mW and 510 mW at various power factors, overexcited and underexcited. Curve fitting procedures may be used toobtain functions which, when plotted, will fall close to the test points. The shape of the function curves is similar tothat of the open-circuit saturation curve that is also plotted on a reciprocal base.

Figure 5C-3 is an example of how to determine Kd and Kq factors from the function information plotted in Fig 5C-2.

The determination of Kd and Kq factors is based on the arbitrary assumption that the total air-gap flux correspondingto El is presumed to act first in one axis alone, and then in the other axis.The method, while pragmatic, is simple toapply, and provides reasonable agreement between test and calculation.

Xaqsat

Xaqu

Kq------------=

EQD δ∠ Ea 0° Ia jXqsat⋅( )∠=

Xadsat

Xadu

Kd------------=

Xdsat

Eq Xl i fd⋅+

i fd Id–-----------------------------=



Note from Fig 5C-3 or Fig 5C-4, that values of Kd and Kq are unity or greater, depending upon the value of El, whichis a function of Xl, armature voltage current and power factor, and also depending upon where the “break point” fromthe air gap line is assumed. Typical plots of Kd and Kq are also shown in Fig 5E-2 at the end of Appendix 5E and theirvalues are compared with saturation function values obtained from finite element analysis.

5C2.1 References

[1] EPRI Report, “Determination of Synchronous Machine Stability Constants,” EL-1424, vol. 12 (Ontario Hydro)Sept., 1980.

[2] Kundur, P., and Dandeno, P.L., “Implementation of Advanced Generator Models into Power System StabilityPrograms,” IEEE Transactions on PAS, vol. PAS-102, July, 1983, pp. 2047–2054.

[3] Minnich, S. H., Schulz, R. P., Burke, S. H., Sharma, D. K., Farmer, R. G. and Fish, J. H., “Saturation Functions ofSynchronous Generators from Finite Elements,” IEEE Transactions on Energy Conversion, vol. EC-2, Dec. 1987.

Figure 5C-1 —Phasor Diagram for Commercial Stability Program #2



Figure 5C-2 —Typical Excitation Test Points 555 mVA Turbo-generator



Figure 5C-3 —D&Q Axis Saturation Functions

Figure 5C-4 —Typical Q-Axis Saturation Factor (Expanded Scale)



Annex 5D

Finite-Element-Derived Steady-State Saturation Algorithms

(Informative)


5D.1

Under the program described in Reference [1], a study has been made of the use of nonlinear electromagnetic finite-element (FE) methods to represent steady-state saturation. Preliminary studies showed that FE methods could properlypredict steady-state load operating parameters under saturated conditions. Methods were developed for extractingsaturated values for the machine reactances, which could be used in conventional phasor triangle calculations tocorrectly predict excitation current and load angle. Later effort was devoted to the construction of simple algorithms tocorrelate FE-calculated values for saturated reactances. The algorithms are suitable for use in conventional stabilityprograms over a range of overexcited and underexcited conditions.

There are, potentially, three reactances to be fitted, Xq, Xd and Xad. In the per-unit system used, Xd = Xad+Xl, where Xlis the d-axis stator leakage reactance. Preliminary examination of FE results indicated that Xl was reasonably constantover the steady-state operating capability of the generator. On this basis, the fitting was restricted to Xq and Xad anaverage value of Xl being used to construct Xd.

Two generators were used as examples. They will hereafter be referred to as Generator X and Generator Y.

It has been discovered that the reactances can be fitted with a composite function that is the product of two functions.The first is a function of the total voltage (or flux) behind an internal reactance (called for convenience the Potierreactance, although it is not the classical definition). One important conclusion from this decomposition is that Xq,saturation is a strong function of this total voltage . The second saturation function has as its argument only the q-axiscomponent of the Potier flux; it turns out to be a small correction to the saturation correction implied by the total fluxfunction. In equation form, the saturated value for a given reactance becomes:

(1)

(2)

From Eq 2, we note that Ea = Ed + jEq.

(3)

(4)

and either of the above two equations can be used as the argument of F2, since each is the same number in per-unit, andspeed, ω is one per-unit in the steady state.

In Eq 1, Xsat represents either Xad or Xq, and the functions F1 and F2 are numerically different for Xad or Xq.

Xsat Xunsat F1 E p )( F2 ψqp( )⋅⋅=

where E p Ea jIa X p⋅+=

Also, ψqp ψq Iq X p⋅+=

Since Ed ωψq,– Edp Ed– Iq X p,⋅+=–=



Figure 5D-1 shows the total-voltage saturation function, F1 (Ep), for Xq,and Xad for each of the example generators. Inpractice, a table-lookup function was used for these functions. The curves are plots of all the calculated data pointsused to construct the function, and have some lack of smoothness, representing the residual scatter arising from lackof perfect correlation with the Potier voltage.

Figure 5D-2 shows the q-axis flux saturation function, F2 (ψqp), for Xq and Xad for each generator. It was found that aquadratic approximation was sufficient to fit these data. These curves are plots of the quadratic approximation.

As mentioned previously, different values of Potier reactance were needed to fit Xq and Xad and, of course, the valueswere current for the two generators. The values found are shown in Table 5D-I.

Figure 5D-1 —Total Voltage Saturation Functions (F1) for Generator X and Generator Y



Figure 5D-2 —Q-axis Saturation Functions (F2) for Generator X and Generator Y Superposed

Table 5D-I —Values of Potier Reactances Used in Fitting Unsaturated Xq, Xad, and Stator Leakage Reactance

Also given in the above Table are the d-axis stator leakage reactances used to construct Xd from Xad and theunsaturated values of Xq and Xad.

The owners of the generators discussed above measured steady state load angles and field currents over a wide rangeof operating points. The values calculated from the algorithm for each generator were compared with the respectivemeasurements. The details of this comparison can be found below [1, 2]. Generally, the load angles agreed within 2°,and field currents within 2% for both comparisons.

Strictly speaking, the saturation functions, F1, and F2, are unique to each generator in the same sense that the open-circuit-saturation curve is considered unique. Calculations similar to those performed in this project are not currentlyavailable for any machine.

Some approximations suggest themselves as sources of data for arbitrary machines. Since the function F1 for Xad.contains the same data as the open-circuit-saturation curve, that function can be readily obtained from manufacturersdata. Numerical experiments reported in [1] show that a “generic” function, F1 for Xq constructed to lie in the middleof Generator X and Generator Y functions of Fig 5D-1, produces differences of about 2° in load angle when used inplace of the exact Generator X function. The same experiments showed the sensitivity of the calculations to the valueof the Potier reactance, similar in Fig 5D-2 that either could be used. Such approximations, when used for an unknowngenerator, may be a satisfactory improvement over the results of simpler saturation functions. This idea is notrecommended unless the user takes care to validate the procedure against measured values of steady-state loadparameters.

Xp to fit Xq Xp to fit Xad Xl Xq unsaturated Xad unsaturated

Generator X 0.17 0.22 0.150 1.760 1.71

Generator Y 0.17 0.19 0.122 1.492 1.455



5D2 References

[1] EPRI Report, EL3359, “Improvement in Accuracy of Prediction of Electrical Machine Constants, and GeneratorModel for Subsynchronous Resonance Conditions,” (Projects 1288-1 and 1513-1). Vol. 3: Development and BenefitAssessment of a Finite-Element-Based Generator Saturation Model, (General Electric Co.), May 1987.

[2] Minnich, S. H., Schulz, R. P., Baker, D. H., Sharma, D. K., Farmer, R. G., and Fish, J. H., “Saturation Functions ofSynchronous Generators from Finite Elements,” IEEE Transactions on Energy Conversion, vol. EC-2, Dec. 1987.



Annex 5E

Comparison of Certain Existing Methods of Accounting for Saturation

(Informative)


5E.1 Arithmetic Comparison of Methods Described in Appendix 5B and Appendix 5C

An indication of the differences and the similarities in the two computer methods described in the above Appendixescan be obtained by using the measured information from the EPRI Report EL 1424, Volume II. Recordings of powerreactive power, terminal voltage, field current and measurements of internal angle were taken by a public utility on a555.555 mVA unit.

Three representative points were chosen and are shown in Figure 5E-1. Equations chosen to calculate excitationcurrents are given below.

1) As in Appendix 5B, where

2) As in Appendix 5C, where

The appropriate saturation curves of Fig 5E-1 are used to obtain saturation factors, and, subsequently, values of Xdsat,Xqsat. These are expanded portions of Fig 5C-2.

Base mVA=555.555, Base kV = 24.0, Xadu = 1.82; Xaqu = 1.71, Xl = 0.16, X′d (rated volts) = 0.27; X″d (rated volts) =0.215.

Tables 5E-II and 5E-III summarize the results of calculations based on the following readings for three machineloadings:

Table 5E-I —

Test Point # & EPRI Report #

P mW

Q mvar Ea kV

Ifd(M) Amps

M = Meas

δ int. angle Degrees (M) M = MEAS

A - #623 506.8 +200.4 25.00 3615 36.6°

B - #744 510.0 +11.7 24.13 2911 49.5°

C - #390 306.7 -82.6 23.21 1832 47.4°

El Xadu i fd E ′q id Xd X ′d–( ) Xadu i fd (scalar form⋅( )∆+ +=⋅=

Xadsat i fd Eq id Xdsat (scalar form )⋅+=⋅



Table 5E-II —(Table 5E-II gives the results using the methods of Appendix 5B)

Table 5E-III —(Table 5E-III gives the results using the methods of Appendix 5C.)

5E.2 Comparison of Saturation Functions for One Machine—Both Test and Calculated

In Appendix 5C, direct- and quadrature-axis saturation factors were developed, and are shown below in Fig 5E-1, aswell as in Figs 5C-2 and 5C-4. These were based on measurements for a public utility 555.555 mVA turbo-generator.

In order to develop a general sense of the behavior of these functions, obtained from finite element analysis, ascompared to factors developed from test measurements, a series of calculations were made. Several test points forGenerator X taken near full load were used. These were reported on in [1] of Appendix 5D. Kd and Kd factors weredeveloped based on voltages behind leakage reactance in the same manner as described in Appendix 5C.

In order to obtain a general comparison with the calculated results for finite elements for Generator X, the F1 functionsfor that machine were inverted. In addition, factors for the open-circuit saturation curve for Generator X were alsoobtained. The results are shown together in Fig 5E-2

It should be reiterated that the factors Kd and Kq have been developed from on-site readings. They are fitted to readingstaken near full load, and are plotted against El the per unit voltage back of leakage reactance, i.e., the per-unit air gapflux. The F1 and F2 inverted functions are calculated for a much wider range of megawatt loadings. These finiteelement functions are based on using Potier reactance, and hence are plotted against Potier voltage. The open-circuitfactor Kocc is plotted against terminal voltage. Hence, the relative values of the various factors cannot be directlycompared.

The main purpose in showing all the curves together is to demonstrate the fact that the direct axis factors all lie aboveand to the left of the quadrature axis factors; the latter factors, for any particular air gap or Potier voltage are greaterthan the direct axis factors. The break points in the Kd and Kq curves are extrapolated, since limitations in operating theGenerator X unit underexcited did not permit development of test points for values of El below about 0.9 p.u. voltage.

Test Point Xqsat

δ Calc.

δ Meas. -δ Calc.

Ifd(C) A C=Calc

A 1.50 40.1 –3.5 0.31 2.750 3575 –.011

B 1.65 55.4 –5.9 0.16 2.236 2907 –.001

C 1.76 54.4 –7.0 0.07 1.366 1776 –.031

NOTE — 1.0 p.u. EI or Ifd or Xadu · ifd = 1300 A.

Test Point Xqsat δ Calc.

δ Meas. -δ Calc. adsat·ifd (p.u. of 2366)

Ifd(C) A

C=Calc.

A 1.24 36.4° +0.2° 2.014 1.537 3637 +.006

B 1.34 49.8° -0.3° 1.796 1.234 2920 +.003

C 1.43 46.6° +0.8° 1.200 .760 1797 –.019

Summary: Both methods, as calculated, give good checks on field current. Table 5E-II angle calculations are consistently high, since they are based on use of an open-circuit saturation curve to calculate Q-axis saturation.

∆Xad i fd⋅

EI I fd=

= X adu i fd ⋅I fd C( ) I fd M( )–

i fd M( )------------------------------------

X i fd


I fd M( )------------------------------------



The finite element analyses covered values of Potier voltage or Potier flux from around 1.0 p.u. down to virtually zerovoltage.

Figure 5E-1 —Enlarged Portions of Fig 5C-2

Figure 5E-2 —General Comparison—Saturation Functions of Generator X



6. Determination of Generator Stability Parameters

6.1 Stability Parameters Obtained by Testing Generators Under Short-Circuit or Open-Circuit Conditions

In IEEE Std 115–1983 (R1991) [1] various tests are described that define the method of testing for synchronousmachine electrical parameters. The stability constants discussed in previous chapters are the unsaturated synchronousreactances

X

adu

and

X

aqu

and the transient and subtransient short-circuit time constants and reactances

T

′

d

,

T

″

d

,

X

′

d

,

X

″

d

and

T

′

q

,

T

″

q

,

X

′

q

,

X

″

q

.

Stability constants or parameters determined from three-phase short-circuit current tests, or from open-circuit tests, aredirect-axis values only, since, under open-circuit, steady-state conditions, only the field winding is energized andcarries any current. All three-phase short-circuit testing is performed from unloaded open-circuit generator conditions.From values

X

″

d

and

T

″

d

, determined from the initial values of short-circuit current, values

T

″

do

, and

T

′

do

can be found

from the relationships and

Xdu, the unsaturated direct axis synchronous reactance, is found by performing an open circuit test and a sustained orsteady state short circuit current test. Xdu in per-unit is the ratio of the field current required to produce rated armaturecurrent under the above sustained short circuit condition to the field current required to produce rated armature voltageon the air gap line extrapolation of an open circuit saturation curve.

6.2 Frequency Response Testing of Generators

This method of arriving at parameters for generator models has only been developed within the past 15 years. Theactual procedures for testing are described in IEEE Std 115A–1987 [2], but basically involve exciting the stator or fieldof a generator when it is at standstill and is disconnected from the usual generator step-up transformer. Hence, themethod is described as Standstill Frequency Response (SSFR) Testing. All testing work to date has been performed onturbo-generators.

The exciting currents are quite low and the exciting frequencies recommended in the Standard range from .001 Hz upto between 100 Hz and 200 Hz. By positioning one of the stator windings relative to the field winding in one of twoways, either direct-axis or quadrature-axis stator operational impedances (and reactances) may be obtained. The directaxis quantities may be measured with the field open or shorted. Stator to field and field to stator transfer functions orimpedances for the range of exciting frequencies quoted above are also measured. An example of Ld(s), the direct-axisoperational inductance, is shown in Fig 7, and is taken from IEEE Std 115A–1987 [2]. Some brief comments on theinterpretation of the results are given below. The models obtained are small signal models.

Because of the low magnitude of the measuring signals, the magnetic behavior of the generator at standstill is mostnearly described by the incremental permeability of the rotor iron. The values of Lad(0) and Laq(0) obtained from the“zero-frequency” intercepts of the operational inductance curves will be incremental values. For such conditions, theincremental permeability at or very close to zero biasing flux density is substantially coincident with the normalpermeability at the toe of the normal B/H curve. The “patching in” of Ladu from the air gap line in the d-axis modelresults in a relatively minor correction being made to Lad(0). The value of Ladu from the air gap line is substituted forLad(0).

This increase, based on recent test results from about 10 machines, has amounted to somewhere between 8% to 18%and the average is about 12%. The values of Ladu from the air gap line are subsequently corrected for steady-statesaturation in most stability programs in the initialization processes, as discussed in Chapter 5. The same commentsalso apply to the values obtained for Laq(0). It is currently assumed that the correction factor for the quadrature axis isproportional to the Ladu/Lad(0) correction factor. The logic for such corrections, for variations in incrementalpermeability, is also discussed in greater detail in IEEE Std 115A–1987 [2].

T ″do

X ′dX″d--------- T ″d;⋅= T ′do T ′d

X ′du

X ′dv----------- T ′d⋅=



The actual derivation of the model elements values, from frequency response testing results, is also discussed in theAppendix of IEEE Std 115A-1987 [2]. It deals with obtaining parameters for models by using “standstill” test data.The approach used, but not the only one possible, starts with a choice of circuit form for both axes. Figure 1 or 2 inSection 2.1 would be an example of this. Ll is chosen and Lad(0) is then calculated, where Lad(0) = Ld(0)−Ll.

The effective field to armature turns ratio is determined from the standstill frequency response armature to fieldtransfer impedance, along with the measured value of Lad(0). This effective turns ratio also permits a rotor physicalquantity, such as the field resistance, rfd, to be referred to the stator in ohms.

Figure 7 —d-Axis Operational Inductance (Field-Shorted)

Knowing stator leakage, Ll, and Lad(0), both in henrys, and rfd, in ohms, the remaining direct and quadrature axes rotorelements are calculated by assuming some starting value for them, and calculating the error between the frequencyresponse of the resulting equivalent circuit and each measured test point. The value of each undetermined element isthen changed by a small amount and if the error between calculation and test is reduced, the process is continued untilthe error begins to increase again. The process is repeated for all other undetermined elements until the error at eachtest point for both magnitude and phase, between calculation and test, cannot be reduced further, Ladu is subsequentlyused as a saturation correction to reflect an “air gap line” value.

Different weights can be assigned to the amount of error that is allowed at different specific exciting frequencies. Thefrequency range between 0.1 Hz and 10 Hz is often chosen so as to achieve the closest fit between measured data andmodel behavior, so that the calculated values of the equivalent rotor circuit elements, (for example R1d, L1d, or R1q,L1q, etc.) are considered realistic. It should be stressed again that this procedure is just one of several options forparameter fitting.

References below [4, 5, 6] also give additional experimental data using SSFR d- and q-axis models when simulatingon-site test measurements obtained during power system disturbances. Some additional frequency response testing ofmachines when running on open-circuit (OCFR) or operating on-line at load (OLFR) has also been done to investigate



the conditions under which SSFR models ought to be adjusted to compensate for rotational and other on-loadgenerator magnetic characteristics. This is further discussed in Appendix 6B.

6.3 Parameters Derived by Two Manufacturers in the Machine Design Stage

The following discussion relates to the stability constants for large steam turbine generators that are provided by twoU.S. manufacturers. While the detailed calculation procedures used by these companies are of course, different, thereare distinct similarities in the steps used to determine stability constants and the meanings ascribed to them. The intentof this discussion is not to examine detailed calculation methods. Rather, it is hoped that an explanation of themeanings of the stability constants and basic assumptions implicit in their calculation will prove useful to systemsanalysts applying these constants in dynamic synchronous machine models.

Manufacturers provide subtransient, transient and synchronous reactances in the form X″di, X′dv, X′di, X′du and Xdu forthe direct axis, and the quadrature axis quantities have analogous forms. Subscript “i” denotes quantities that would beobtained from short-circuit tests in which the initial voltage is reduced to a value that produces an initial short-circuitcurrent equal to rated armature current. Subscript “v” applies to quantities appropriate for large current variationsassociated with faults from rated terminal voltage. Open-circuit transient time constants for both axes are alsoprovided.

In actual fact, most stability studies are carried out with flux conditions (prefault or pre-disturbance) corresponding torated terminal voltage. It is agreed that prefault flux conditions are generally more important than the magnitudes ofmachine currents (post fault or during disturbances). When X′d is quoted in this guide, rated voltage values of thetransient reactance X′dv are implied, and the subscripts “i” and “v” are employed in this Section solely fordistinguishing subtransient or transient quantities and for accentuating their differences. Calculated values for thedirect axis quantities are intended to match results that would be obtained from the well known three-phase short-circuit tests described in Section . Generally, manufacturers calculate stability constants for the direct axis by firstcalculating parameter values for a circuit structure similar to Model 2.2, and then convert these circuit parameters tosynchronous, transient and subtransient reactances and time-constants. In addition, both manufacturers account for theinfluence of varying machine flux levels only on the leakage reactances (stator and rotor circuits), while the mutualreactances Xadu and Xaqu are assumed to be unchanged when calculating stability constants.

In calculating direct-axis transient reactances the damper path is ignored and parallel combination of the mutual andeffective field leakage reactances is added to the armature leakage. The quadrature axis transient reactances arecalculated using the damper circuit reactances that correspond to particular hunting frequencies. One manufacturerassumes a hunting frequency of 1 Hz, while the second manufacturer assumes 0.6 Hz. Mutual reactances are ignoredin calculating the subtransient reactances.

The direct-axis open-circuit time-constants provided by the second manufacturer are intended to be consistent with

reactances and time-constants that would be obtained from a full voltage short-circuit test., i.e., and

.

As such, T′do is not identically the field constant. The first manufacturer calculates the direct axis transient open-circuittime constant from the equivalent circuit, where the leakage reactances reflect high flux levels associated with a fullvoltage short-circuit. The first calculated value of transient open-circuit time constant ignores the iron (damper)circuit. Both manufacturers consider quadrature axis open circuit time-constants to be poles of an equivalent frequencyresponse interpretation.

6.4 Desirability for Uniform Practices in Deriving Machine Stability Parameters

Section describes methods outlined in IEEE Std 115-1983 (R1991) [1] for obtaining certain synchronous machineparameters. These methods were first documented in an organized way in 1945. For the direct axis, these are Xdu the

T ′do T ′dXdu

X ′dv----------=

T ″do

T ″dX ′dv

X″dv--------------------=



unsaturated synchronous reactance, the direct-axis values of transient and subtransient reactances X′d and X″d, as wellas T′d and T″d, the short-circuit time constants describing the decay of armature currents under three-phase short-circuit test conditions. Section describes a more recent approach in determining parameters for generator stabilitymodels by using frequency response measurements.

In Section 6.3, two slightly different approaches taken by two U.S. manufacturers are presented that are intended toprovide open-circuit direct axis transient (T′do) and subtransient (T″do) time constants. These open-circuit values oftime constants can be obtained from open-circuit tests, or can be calculated from the rate of change of three-phaseshort-circuit current decrements, T′d and T″d, provided by the machine designers. Calculated quantities X′d and X″d arealso available from the same sudden short circuit assumptions.

Based on the above introductory remarks, it would now appear possible for manufacturers to consider supplyingstability data in several alternative forms, noting as well the conditions for which the data applies. These suggestionsare covered under three distinct headings.

1) Calculation of direct-axis quantities using current “standard” procedures. Time constants and reactancesunder prefault flux conditions that yield either “rated voltage” or “rated current” values of direct axis timeconstants and reactances. This would be generally consistent with current practices, which means that directaxis values are based on either assumed or actual test conditions involving a three-phase armature faultapplied from open circuit As noted in Section 6.3, rated voltage values of transient reactance are generally theones that should be used in stability studies.

2) Calculation of quadrature axis quantities using proprietary procedures that may be considered “non-standard”. Information might well be provided as to how the quadrature axis information is obtained,particularly for turbo-generators, since this information is not derived at present in the same way as the directaxis data. For example, an operational expression could be provided on the basis that

and

It should be noted, if appropriate, that the expressions are based on the impedance seen in the rotor solid ironquadrature axis area at some specified per-unit current, and some specified hunting frequency, along with aknowledge of Xq.One of the ambiguities in providing information for turbo-generators in this way is apparent: the direct-axisstability data is based on assumed short-circuit conditions, while the quadrature axis is often based on aspecific value of impedance, as seen at one hunting frequency. The above information for the Xq(s) functionfor solid iron rotors could be expanded upon by noting that Xq(s) normally attenuates between 4 to 6 dB perdecade, in the range from about 0.5 Hz to about 5 Hz. Given a steady state value of Xqu and the impedance ofXq at a specific hunting frequency in the above-mentioned range, Xq(s) can be approximated with abreakpoint and a 4 to 6 db per decade drop off, for frequencies up to about 5 Hz. A transfer function with 2or 3 poles could then be used to approximate this Xq(s) characteristic.In the case of hydrogenerators, the quadrature axis current paths in the damper bars can be well described,and one value of calculated reactance and time constant is usually acceptable to describe sudden flux changesin this axis. (Usually given as X″q and T″qo).

3) Use of recently established calculations or measurement techniques, based on small signal theory.

These above techniques result in direct and quadrature axis equivalent circuit forms with models ranging from first-order model (1.1) to third-order model (3.3), with Model 2.2 as the most common. They have been only applied toturbo-generators and may be obtained in two ways:

a) From operational expressions for both axes based on finite element two-dimensional calculations, asdiscussed in [B3] of Chapter 5.

b) From operational expressions for both axes based on frequency response measurements as described inSection 6.2. Included in the calculations described in (a) and (b) should be advice as to whether the statorleakage (Xl) and the field leakage (Xfd) values were based on flux conditions corresponding to normal orreduced operating voltages.

Xq S( ) Xq

1 sT ′q+( )1 sT ′qo+( )

---------------------------= T ′q T ′qo

X ′qXq--------.⋅=



Concluding Note

In summarizing some of the key issues in Section 6.4, it should be acknowledged that investigations by turbo-generator manufacturers into the dynamic stability properties of turbo-generators in particular go back severaldecades. Reference [3] considers the appropriate expressions for describing direct and quadrature axis operationalinductances (or reactances) of turbo-generators as viewed from the stator terminals. One of the conclusions from theabove reference states, in part, that “by a more detailed consideration of the actual rotor configurations, equationsamenable to attack by frequency response techniques are obtained.” This statement, plus a joint discussion by B.Adkins of this reference and of related papers embrace this same thought. Companion papers and Adkins’ discussionall appear in the same issue of the AIEE Transactions of February 1960, pages 1657–1673. Thus, standstill frequencyresponse measurements were acknowledged as a likely future alternative to the complex magnetic field calculationsconsidered necessary at that time.

6.5 Alternative Forms of Model Representation

In this section, various equivalent forms for the representation of both synchronous machine models and modelparameters are presented and discussed. For the purpose of this section, the following terminology will apply:

model structure: The basic form or configuration of a model constitutes its structure. This structure can be combinedwith model parameters whose values remain to be specified. A model structure is characterized both by its form (e.g.,lumped-parameter equivalent circuit, transfer function, differential equation representation, etc.), as well as its ordernumber of equivalent rotor windings).

model parameter values: Synchronous machine model parameters are derived from characteristics of actual machinebehavior. Although this description may take many forms, two basic categories are evident:

1) Test data obtained from measurements, or2) Analytic “data” obtained from sophisticated analyses that simulate the detailed internal electromagnetic

phenomena of the machine. A common technique for performing such analyses is that of the finite-elementmethod. Using this technique, it is possible to solve for the details of magnetic flux distributions within thesemachines including the nonlinear effects of both magnetic saturation and induced eddy-currents.

model: A complete synchronous machine model consists of a combination of a model structure and a set of parametervalues. Thus, for example, the same model structure in combination with parameter values obtained by different testmethods and under current machine operating conditions could yield different model representations and the machinebehavior predicted by these models may be quite different. This is because the machines themselves are quite complex(nonlinear, high-order, etc.) and the conventional models are relatively simple (lumped-parameter, low-order andlinear with adjustments to handle non-linearities).

Thus, a given model can appear in various equivalent representations, e.g., in the form of an equivalent circuit, or in theform of a transfer function among others. These representations are identical, provided the following conditions aremet:

1) The model parameters for each form of the model have been determined from the same set of test data oranalytically derived data.

2) The order of each equivalent representation should be the same.3) Modifications to the element/parameter values to account for nonlinearities such as saturation effects are

made after the basic parameter values for each of the model elements have been determined. These saturationmodifications should, themselves, be made in a fashion that assures that the behavior of the various forms ofthe model remain consistent.

Figures 8, 9, and 10 illustrate three basic forms that can be used to represent the d-axis of a synchronous machine.These are a lumped-parameter equivalent circuit, a transfer-function representation, and a flux-current/voltagerepresentation. Note that they are each of the same order (i.e. representing the rotor d-axis with two windings).

The equivalent circuit includes an ideal transformer of unknown turns ratio 1:N. This represents the fact that thegenerator consists of 2 coupled windings. Thus, the variables efd and ifd are the field voltage and current, as reflectedto the armature through this turns ratio, and these variables cannot be measured in a physical experiment. On the other



hand, the actual variables that are measured are indicated in the equivalent circuit as voltage e′fd and current i′fd. Thetransfer function representation of Fig 9 is given in terms of measurable variables.

In the equivalent circuit representation of Fig 8 there are eight unknown parameters. On the other hand, in the transferfunction representation of Fig 9, there are 9 measurable parameters; 6 time constants and 3 coefficients. Thus, it wouldappear that by external measurements, one could determine all of the parameters of the equivalent circuit, includingthe exact values of the turns ratio and the armature leakage inductance. However, this is not the case, and in fact, only7 of the 9 parameters of Fig 9 are independent.

The implication of this fact is that for the purposes of making an equivalent circuit, it is not possible to make a set ofmeasurements at the generator terminals that will determine all the values of the equivalent circuit uniquely. One isalways free to pick one parameter value arbitrarily. Note that the resultant model will have uniquely defined terminalcharacteristics, which is essential for modeling purposes [7], [B2].

It is common practice to choose the value of the armature leakage inductance as the free parameter when makingequivalent circuit representations for synchronous machines. Although it is certainly possible to choose the valuetotally arbitrarily without affecting the validity of the resultant model, it is common to choose a value equal to or closeto the manufacturer supplied leakage inductance value which is often based upon an analysis of the flux distributionwithin the machine, under rated conditions.

Of significance for the purposes of this discussion is the fact that the three forms of Figs 8, 9, and 10 of this section areactually equivalent model structures. This can be implied from Appendix 6C which presents some of the equationswhich translate model parameters from one particular form to another.

6.6 References

[1] IEEE Std 115–1983 (R1991), IEEE Test Procedures for Synchronous Machines (ANSI).

[2] IEEE Std 115A–1987, IEEE Standard Procedures for Obtaining Synchronous Machine Parameters by StandstillFrequency Response Testing.

[3] Concordia, C., “Synchronous Machine with Solid Cylindrical Rotor,” AIEE Transactions, Part III-B, Feb. 1960, pp.1650–1657.

[4] Dandeno, P. L., Kundur, P., Poray, A. T. and Coultes, M., “Validation of Turbo-generator Stability Models byComparison with Power System Tests,” IEEE Transactions on PAS, PAS-100, April, 1981, pp. 1637–1645.

[5] EPRI Report , “Determination of Synchronous Machine Stability Study Constants,” EL 1424, vol. 2, Prepared byOntario Hydro, Dec. 1980.

[6] Schwenk, H. R., “Deriving Synchronous Machine Models from Frequency Response Data,” IEEE SymposiumPublication 83THO 101-6-PWR.

[7] Umans, S. D., Mallick, J. A. and Wilson, G. L., “Modeling of Solid Rotor Turbo-Generators Part I Theory andTechniques,” IEEE Transactions on PAS, PAS-97, vol. no. 1, Jan./Feb. 1978.

6.7 Bibliography

[B1] Jackson, W. B, and Winchester, R. L., “Direct and Quadrature Axis Equivalent Circuits for Solid Rotor TurbineGenerators,” IEEE Transactions on PAS, PAS-88, 1969, pp. 1121–1135.

[B2] Umans, S. D., Mallick, J. A. and Wilson, G. L., “Modeling of Solid Rotor Turbo-Generators -Part II - Exampleof Model Derivation and Use in Digital Simulation,” IEEE Transaction on PAS, PAS-97, vol no. 1, Jan./Feb. 1978.



Figure 8 —Equivalent Circuit, d-Axis Model (Second-Order, Model 2.2 of Table I, Section 2.1)

Figure 9 —Transfer Function d-Axis Representation (Second- Order Model)

Figure 10 —Flux/Current Voltage d-Axis Representation (Second-Order Model)



Annex 6A

Determination of Direct-Axis Parameters from Test Results

(Informative)


The most commonly accepted method of obtaining synchronous machine parameters from the short-circuit test ispresented in IEEE Std 115-1983 (R1991) [1]. It should be noted, however, that this method of obtaining parameterspresupposes a model of the synchronous machine. This model is a direct and quadrature axis representation with twotime constants (i.e. a second-order system) in each axis.

The method is best explained by means of an example. Figure 6A-1 shows the phase currents obtained from acomputer simulation of a three-phase short circuit from no-load and rated voltage. It can be seen from Fig 6A-1 thateach phase contains an alternating component and a direct component of current. Since the short circuit is from noload, there will be no initial quadrature axis component of flux, and it has been shown that the symmetrical componentof the current (i.e. the short-circuit current minus the dc offset) obeys the expression:

(1)

E is the prefault voltage, and in this example is assumed to be 1.0 p.u. Xdu is the unsaturated synchronous reactance,and is given as 2.0 per unit.

Figure 10 was then analyzed graphically by first subtracting out the dc offset from each phase and then averaging thesymmetrical currents. The envelope of the result is shown in Fig 6A-2.

Determination of Transient Reactance X′d

The transient reactance may now be found with the aid of Equation (1). T″d is typically small compared to T′d, and thethird term of Equation 1 (the subtransient component) will usually be negligible after a few cycles. If we now :subtractoff the steady-state value, we have the expression

(2)

which is valid a few cycles after the short circuit, i.e. when the subtransient term becomes negligible. Taking thenatural logarithm of Eq 2, we obtain

(3)

which is the equation of a straight line when plotted on semi-log paper. Figure 6A-3 shows a plot for this example. Thetransient component of the current is obtained by extending the straight part of the curve back through the abscissa.

This value is interpreted to be 2.8 p.u. The steady-state component is or . Then

IE

Xdu--------- E

X ′d-------- E

Xdu---------–

εt T ′d⁄– E

X″d---------

EX ′d--------–+ +=

εt T ″d⁄–

I Isteady stateE

X ′d--------

=– EXdu---------–

εt T ′d⁄–

In I( Isteady state ) InE

X ′d--------

=– EXdu---------–

t–T ′d--------⋅

1.02.0------- E

Xdu---------

1X ′d--------- 2.8 0.5 and X ′d 0.303 p.u=+=



Figure 6A-1 —Three-Phase Short-Circuit Currents From No Load



Figure 6A-2 —Envelope of Symmetrical Fault Current

Determination of Subtransient Reactance X′′d

The subtransient reactance is determined in a similar way. Subtract both the steady-state and the transient components ofEq 1 or, in other words, “curve 1” minus “curve 2” in Fig 6A-3. Plotting the results on semi-log paper we obtain from Fig6A-4. This straight line is extended back to the abscissa and gives a value of 0.68 p.u. The initial fault current is:

Io = Isteady state + Itransient + Isubtransient

or Io = 0.5 + 2.8 + 0.68 = 3.98 p.u.

From Eq 1 at t = 0, , since ε0 = 1, so that X″d = 0.251 p. u.= .

Determination of the Direct-Axis Transient Time Constant T′d

The direct axis transient time constant can be found from the slope of Fig 6A-3. Consider curve #2, which representsthe transient component of current. In one time constant period, the current will decrease to 1/ε or .368 of its initialvalue. In the given example, the transient current decreases from 2.8 p.u. to 2.0 p.u. in 0.5 seconds.

Therefore, = 0.714 = ε−0.5/T′d; ln(0.714) = and T′d = 1.486 s.

Determination of Direct-Axis Subtransient Time Constant T″d

The direct axis subtransient time constant is determined in a similar way. Referring to Fig 6A-4, the subtransientcurrent decreases from its initial value of 0.68 to 0.09 in 0.1 s.

IE

X''d--------=

1.03.98----------

2.02.8------- 0.5–

T 'd----------



Therefore, = 0.132 = ε−0.1/T″d; ln(0.132) = and T″d = 0.049 s.

Figure 6A-3 —Symmetrical Fault Current Minus Steady-State Value

0.090.68---------- 0.1–

T ''d----------



Figure 6A-4 —Subtransient Component of Fault Current



Annex 6B

Alternate or Nonstandard Methods of Obtaining Stability Parameters

(Informative)

(This Appendix is not part of IEEE Std 1110–1991, IEEE Guide for Synchronous Generator Modeling Practices in StabilityAnalyses, but is included for information only.)

6B.1 Frequency Response Testing Under Running Conditions

A considerable amount of testing has taken place on a limited number of turbo-generators to ascertain under whatconditions on-line frequency response (OLFR) testing can be found useful. Such tests are necessary to provide a morerealistic source of information on whether SSFR test parameters are acceptable, to reflect actual generator operationalconditions. The main field of the generator is excited over a range of frequencies, when it is fully or partially loaded,and at normal voltage. The range of frequencies that have been found practical are from 0.1 Hz to about 5 Hz. Thereader should consult several appropriate publications on the subject [1], [3].

A simpler form of testing under special operating conditions involves exciting the main field of a generator, overanother limited range of frequencies. The machine is on open circuit, running at rated speed, and with its stator voltageat between 0.5 and 1.0 p.u. of normal voltage. This is reported on in the same publication as cited above [3]. Thisprocedure has been referred to as open-circuit frequency response (OCFR) testing. The range of exciting frequenciesis from about .01 Hz to 10 Hz. This is a broader range than that found practical for OLFR testing. It should berecognized that this latter type of testing provides data only for the direct axis. It has been determined, however, thatmuch useful information about details of rotor direct axis damper construction can be obtained from both OLFR andOCFR testing.

As noted in Section 5.3, the derivation of either SSFR, OCFR or OLFR parameters from test data is influenced by theincremental permeability of the generator iron paths. The small signal operational inductances, especially for Lad (andL aq), are not much different between these three conditions, except at frequencies below about 0.1 Hz. For the rotorequivalent circuit values (R1d,L1d etc.), variations in the rotor noneffective skin depth definitely play an important rolein the generator small-signal model representation. Analytical adjustment of SSFR rotor equivalent circuit values tovalues reflecting on-line conditions is also discussed in another paper [2].

Especially in cases where the rotor has pronounced amortisseur circuits, it has been observed that the running responseis different from the standstill response, and the running, or on-line circuit constants should be used in these cases. Thisphenomenon has been identified with, among other factors, end-ring contacts that are made at running conditions, butwhich may be open at standstill; the phenomenon should not be confused with changes in saturation between standstilland running conditions. The conductivity and length of wedges used to contain the field winding has also been citedas another feature of rotor construction that effects OLFR data. A third and, most likely, very important considerationis whether the end-rings themselves are magnetic or non-magnetic. [4]

6B.2 Stator Decrement Testing with Generators on Load

Short circuit testing is performed at no load, with terminal voltages usually in the range of 0.30 to 0.50 p.u. of normalto avoid excessive stator currents. The transient and subtransient reactances and time constants so obtained arepresumed to reflect unsaturated generator conditions.

Another group of switching tests, recently used, permits the terminal voltage to be in the normal operating range sothat saturation effects can be included in generator response.



Instead of introducing a sudden change in terminal voltage by a three-phase short circuit, the machine is subjected to astep change of armature current by either opening or closing the main breakers. Load rejection tests, i.e., opening thebreaker, produce a change in current that is equal to that at the initial pre-disturbance condition. By selecting the loadingfor the machine before switching, the current can be entirely aligned on the direct axis or on the quadrature axis.Alignment on the direct axis can be easily obtained by keeping the generator power output at zero and raising or loweringthe excitation tc produce or absorb reactive power. To align the stator current in the quadrature axis requires ameasurement of the rotor angle. For this test, the generator power output is normally set at about 0.25 p.u. of the ratedmVA and the excitation is lowered to absorb reactive power. The current is aligned on the quadrature axis by adjusting theexcitation until the power factor angle, with the armature current leading the terminal voltage, is equal to the rotor angle.

Once the current is property aligned and with id = 0, or iq = 0 the breaker is then opened and field-current and stator-voltage quantities are recorded. The change in stator voltage will provide information on the machine's response tochange in current for the direct or quadrature axis,depending on the alignment of the stator current to either axis. Forthe direct axis test, the field current response provides additional information, and it is usually the main source of testdata to determine whether a second- or third-order model is needed. It is difficult to determine the order from just thechanges in stator voltage. For the quadrature axis, the field current ideally will not change, and the recording of thisquantity provides verification whether or not the stator current is fully aligned in the quadrature axis.

The analysis for deriving parameter values as described in Appendix 6A for tests can be similarly used to obtain timeconstants and reactances as initial estimates. However, the final model may be obtained by a series of simulation cases toreproduce the test response, since on-load saturation effects should be included. The process using the simulation runsconsists of a trial-and-error adjustment of parameter values until a match is made with the actual on-site test results.

One problem encountered with the load rejection tests is that, ideally, the field voltage should be kept constant. Forcertain excitation systems this is not possible, in excitation systems for instance whose source is an alternator. The testrequires recording of the field voltage, whose response is included in the simulation. However, some excitationsystems such as a rotating brushless exciter do not have the capability to measure field voltage.

The most recent investigation of stator decrement testing is reported on in the first report listed below [5]. Also noted[6] is earlier work which does not include the refinement process adopted for adjusting for generator saturation underloaded conditions.

6B.3 References

[1] Coultes, M. E., Kundur, P., and Rogers, G.J. “On-Line Frequency Response Tests and Identification of GeneratorModels.” IEEE Symposium Publication 83, THO101-6PWR.

[2] Dandeno, P. L., Kundur, P., Poray, A. T. and Zein-el-Din, H. M. “Adaptation and Validation of Turbo-generatorModel Parameters Through On-Line Frequency Response Measurements,” IEEE Transactions on PAS, PAS-100,April 1981, p. 1656.

[3] EPRI Report, “Determination of Synchronous Machine Stability Study Constants,” EL 1424, vol. 2, Dec. 1980,Ontario Hydro.

[4] Jack, A. G. and Bedford, T.J., “A Study of the Frequency Response of Turbo-generators with Special Reference toNanticoke G.S.” IEEE Transactions on Energy Conversion, EC-2, vol. no. 3, Sept. 1987, p. 456.

[5] “Confirmation of Test Methods for Synchronous Machine Dynamic Performance Models,” EPRI Report , F-L5736, Aug 1988.

[6] “Determination of Synchronous Machine Stability Study Constants,” EPRI Report, EL1424, Vol 3. (PowerTechnologies Inc.), May 1980.



Annex 6C

Generator Stability Data Translations

(Informative)

6C.1 Relationships Between Parameters of the First Two Figures of Section 6.5

The translation of parameters from equivalent circuit form to transfer function form is discussed below in detail sinceis the one situation commonly encountered.

Given Lad,Ll, Lf1d,Lid, Lfd, R1d, Rfd from Fig 8 of Section 6.5. This is the complete d-axis representation of Model 2.2.All L's or inductances should be in henrys and all R's or resistances should be in ohms. Both L and R values should bereferred to the stator.

If the L's and R's are in per unit, instead of physical terms, the a1 and b1 factors should be divided by 2πf to give “T”values below in seconds. Also a2 and b2 values should be divided by (2πf)2 to give “T” values below in seconds

After choosing Ld = Lad + Ll, ,the following factors are calculated:

(1)

From a1, a2, b1 and b2, the following relationships can be developed that define the first two equations of the transferfunction form of Fig 9 of Section 6.5. a1 = T1 + T2; a2 = T1 · T2; b1 = T3 + T4; b2 = T3 · T4

In addition, T5 = L1d/R1d and Go = Lad/Rfd. Rearranging the equations for T1, T2, T3 and T4 gives

(2)

NOTE — Equations 1 and 2 are out of Reference [7] of Chapter 6.

A reverse relationship between the transfer function quantities and the equivalent circuit form can be developed.Reference [7] of Chapter 6 gives steps in the above equation development, as well as steps for proceeding from transferfunction form to equivalent circuit form.

T1, T2, T3 and T4, as calculated from Eq 2 above, correspond respectively in a general sense to T′d, T″d, T′do and T″do,but they are not absolutely equal. The identities given below include given values of Ld.

(3)

a1 Ld R fd L1d L f 1d+( ) R1d L fd L f 1d+( )+{ } Lad Lt R1d R fd+( )⋅+=

Ld R1d R fd⋅ ⋅------------------------------------------------------------------------------------------------------------------------------------------------------------------------

a2 Ld L f 1d L1d L fd+( ) L1d L fd⋅+{ } Lad Ll L1d L fd+( )⋅+=

Ld R1d R fd⋅ ⋅----------------------------------------------------------------------------------------------------------------------------------------------------

b1

Lad L f 1d L fd ) R1d R fd Lad L f 1d L1d+ +( )⋅+⋅+ +(R1d R fd⋅

-------------------------------------------------------------------------------------------------------------------------------

b2

Lad L f 1d ) L1d L fd ) L1d L fd⋅( )+ +(⋅+(R1d R fd⋅

---------------------------------------------------------------------------------------------------=

=

T 1

a1

2-----

12--- a1

24a2– T 3

b1

2-----=

12--- b1

24a2–

T 2

a1

2----- 1

2---– a1

24a2– T 4

b1

2-----=

12--- b1

24b2––=

+ +=

L ′d Ld

T 1

T 3------

Ld T ′dT ′do----------

≡=



(4)

Similar relations hold for the q axis.

See [B1] for a discussion of the basis for the above identities.

6C.2 Use of a Second Detailed Method of Parameter Translation—Equivalent Circuit-To-Time Constant and Inductance Form

The equations of Section 6C1 are completely adequate for translation of model data into transfer function form, andalong with the above identities, are easily programmable.

However, alternate methods of performing these translations are given below, including “closed form” expressions.Also listed are the familiar “approximate” equations. These are all included so that comparisons between detailedexpressions and approximate expressions can be made as a general guide to these translation procedures.

The equations for the second “exact” or detailed method result from assumptions made in the [B2]. Similar equationsfor translation of models up to a third order for both axes are included in Reference [5] of Chapter 6.

In this method, the time constants may be iterated on as indicated in the following equations.

(5)

(6)

In this iterative process, both T′do, and T″do are adjusted following replacement of L11d/R1d in Eq 5 with the value ofTdo in Eq 6. Lffd and L11d are defined in Section 2.2.3 of Chapter 2. A closed form ”exact“ expression for both T′do andT′d, in terms of d- and q-axes model elements can also be derived, thus avoiding any iterations. For example,

(7)

The plus sign before the square root is changed to a minus sign to find T″do.

By definition, Tffd=Lffd/Rfd; T11d=L11d/R1d, while L*fd = Lffd· L11d −L2ad/Rfd · R1d.

Inclusion of Lf1d can be accomplished in Eq 7 by replacing Lad by Lad*= Lad + Lf1d.

The values of L′d and L″d are usually found from an analysis of the response of a second-order d-axis model underconditions where the terminals are short-circuited. Figure 3 of Chapter 2 is an example of such a second-orderequivalent circuit model, where Lf1d is omitted, and Eq 8 and Eq 9 are based on this omission.

(8)

L″d L ′dT 2

T 4------

L ′d T″d

T″do-----------

≡=

T ′do T ″do+L ffd

R fd----------

T 11d

R1d-----------+ T ′do ′≈=

T ″do

L ffd L11d Lad

2–⋅

R fd R1d⋅------------------------------------------ 1

T ′do ′------------⋅=

T ′do12--- T ffd T 11d+( ) T ffd T 11d+( )2

4L∗ fd–+

=

L ′d Ld

L ffd

R fd----------

L11d

R1d-----------

Lad

2

Ld----------

R fd R1d+( )R fd R1d⋅

-----------------------------+–+

L ffd

R fd----------

L11d

R1d-----------+

--------------------------------------------------------------------------------------=



(9)

While the above expression for L″d looks cumbersome, it can be simplified considerably by assuming in the transientperiod that R1d approaches infinity (∞).

In addition, the time constants noted in Eq 5 and Eq 6 can be simplified by assuming that Rfd = ∞ during thesubtransient period and R1 = ∞ during the transient period. The time-constant and inductance expressions thenbecome, under the above conditions, more recognizable, such as the following:

(10)

(11)

Note that the expression for L″d is considerably simplified by the assumptions regarding Rfd = 0 and R1d as are T′do,and T″do. The expression for L″d remains unchanged.

The time constants of Eq 10 and Eq 11 can be modified to include the effect of Lf1d by replacing Lad by Lad *= Lad +Lf1d. As noted previously, time constants will be in radians, when resistances and reactances are in per-unitmeasurement, all values being referred to the stator.

For the quadrature axes equations corresponding to Eq 8, Eq 9, Eq 10, and Eq 11 Laq is substituted for Lad; L1q issubstituted for Lfd; L2q is substituted for L1d; and R1q is substituted for Rfd; R2q is substituted for R1d; Ll remainsunchanged.

Use of these alternative quantities is permissible for any values of quadrature axis elements for the first two methodsdiscussed above. For the approximate method, the smaller of R1q or R2q should be used, along with the correspondinginductance value, to obtain T″qo and subsequently, T′qo.

6C3 Arithmetic Comparison of Methods Described in Sections 6C1. and 6C2

In this section, arithmetic comparisons are made of the time constants and inductances of five machines that have beentested by standstill frequency response methods. Some examples are simple second-order d- and q-axes models, whileother equivalent circuit models derived from the direct axes values of Ld(s) and sG(s) for these machines containvalues of Lf1d, as shown in Fig 2 of Chapter 2, and Fig 8 of Chapter 6.

Example 1 Machine A—4-pole 960 mVA, 24 kV.Given parameters: Ll=.20; Ladu = 1.677; Laqu = 1.644. The following rotor parameter values weredetermined, referred to the stator:

d-axis rotor q-axis rotor

Lfd 0.2434 Rfd 0.00051 L1q 1.0738 R1q .01118

L1d 0.03611 R1d 0.07550 L2q 0.1695 R2q .06063

NOTE — The above data from SSFR testing

L″d Ll1

1Lad-------- 1

L fd-------- 1

L1d--------+ +

---------------------------------------+=

T ′do

L ffd

R fd---------- and L ′d Ll

Lad L fd⋅Lad L fd⋅---------------------+==

T ″do1

R1d--------- Lld

Lad L fd⋅Lad L fd⋅---------------------+

and L″d Ll1

1Lad-------- 1

L fd-------- 1

L1d--------+ +

---------------------------------------+==



Example II Machine B — 4-pole 800 mVA, 26 kV.Given parameters: Ll = .25; Ladu = 1.492; Laqu = 1.511. The following rotor parameter were valuesdetermined, referred to the stator.

Machine Constant (1) Method of App.6C1.(2) Closed Form Method

of App.6C2.(3) Approximate Method

of App.6C2.

T″do 10.04 10.04 10.20

T″do .0201 .0201 .0202

L′d .411 .413 .414

L″d .334 .334 .334

T′qo .691 .691 .645

T″qo .0335 .0335 .036

L″q .796 .795 .849

L′q .334 .334 .341


Lfd 0.04720 Rfd 0.000894 L1q 1.403 R1q 0.01243

L1d 0.0001 R1d 0.00859 L2q 0.3402 R2q 0.04928

L1d 0.2266



of App.6C2.(3) Approximate

Method of App.6C2.

T″do 5.69 5.69 5.16

T″do .0057 .0057 .0062

L′d .459 .461 .461

L″d .447 .447 .446

T′qo .668 .668 .622

T″qo .0535 .0535 .058

L″q .914 .915 .977

L′q .482 .482 .481



Example III Machine B — 2-pole 270 mVA, 18kV.Given parameters: Ll = .16; Ladu = 1.821; Laqu =1.821. The following rotor parameter values weredetermined, referred to the stator.

Example IV Machine D—2-pole 444 mVA, 20 kV.Given parameters: Ll= .162; Ladu = 1.663; Laqu = 1.672. The following rotor parameter values weredetermined, referred to the stator:


Lfd 0.0733 Rfd 0.00105 L1q 0.4465 R1q 0.0055

L1d 0.0001 R1d 0.00713 L2q 0.0838 R2q 0.0267

Lf1d 0.0402




Method of App.6C2.

T″do 5.56 5.56 4.89

T′do .023 .023 .026

L″d .253 .258 .230

L′d .199 .199 .199

T″qo 1.253 1.253 1.103

T′qo .040 .040 .045

L″q .474 .489 .531

L′q .228 .228 .228


Lfd 0.0153 Rfd 0.00127 L1q 0.9144 R1q 0.0108

L1d 0.0058 R1d 0.0065 L2q 0.2328 R2q 0.0366

Lf1d 0.1627




Example V Machine E—2-pole 722 mVA, 26 kV.Given parameters: Ll= 0.19; Ladu = 1.58; Laqu 1.60. Following rotor parameter values weredetermined, referred to the stator:

General Comments on the Calculations (Examples I-V)

For machines C, D and E there is a small difference (2–3%) in values of L″q when calculating this parameter usingEq 1 and Eq 2 of Appendix 6C.1, as opposed to the closed form expressions of Appendix 6C2. In general, theequations of Appendixes 6C.1 and 6C.2 produce results that are almost identical, especially for the direct axis.



Method of App.6C2.

T′do 4.59 4.59 3.85

T″do .0072 .0072 .0086

L′d .319 .322 .323

L″d .314 .314 .314

T′qo .721 .721 .635

T″qo .0526 .0526 .0597

L′q .668 .684 .753

L″q .329 .329 .329


Lfd 0.0153 Rfd 0.00127 L1q 0.9144 R1q 0.0108

L1d 0.0058 R1d 0.0065 L2q 0.2328 R2q 0.0366

Lf1d 0.1627




Method of App.6C2.

T′do 4.15 4.15 3.81

T″do .018 .018 .020

L′d .335 .339 .347

L″d .262 .262 .262

T′qo 1.00 1.00 .92

T″qo .036 .036 .040

L′q .564 .576 .613

L″q .287 .287 .287



The assumptions regarding R1d= ∞ during the transient period and Rfd = 0 during the subtransient period appeargenerally to be quite reasonable for the direct axis for the turbo-generator examples given. However, similar oranalogous assumptions regarding R2q = ∞ during the transient period and R1q =0 during the subtransient period aremore questionable for the same examples. Values of L′q using the approximate formulae range between 7% to 10%higher than when the exact formulae are used, while values of T′qo are all about 9% lower.

The values of d-or q-axis subtransient reactances for the approximate formulae are virtually the same as when usingthe exact equations, as might be expected from examining all relevant equations.

6C4 Bibliography

[B1] Adkins, B. and Harley, R. The General Theory of A.C. Machines, 1975, published by Chapman and Hall(London).

[B2] IEEE Committee Report ,“Supplementary Definitions and Associated Test Methods for Obtaining Parametersfor Synchronous Machine Stability Study Simulations,” IEEE Transactions on PAS, PAS-99, vol. no.4, pp. 1625–1633.

7 Field and Excitation Considerations

7.1 Establishing Field-Voltage, Field-Current and Field-Impedance Bases

In the widely used reciprocal per unit system for synchronous machines, initially advocated in 1945 by Rankin [6], theper unit field current required to produce 1.0 per unit generator terminal voltage open circuit on the air gap line is equalto 1.0/Xadu, where Xadu is in per unit. In other words, base excitation current ifd produces a voltage on the air gap lineof Xadu (Ea base). The nomenclature used here for field quantities is ifd (current), efd (voltage) and Rfd (resistance). Eais the stator terminal voltage in per unit. (See Section 2.2)

Most computer programs use a non-reciprocal system for field current. In the nonreciprocal system a base current (ifd)of 1.0 p.u. is that required to produce Ea base (rated or 1.0 p.u. generator terminal open-circuit voltage) on the d-axisopen-circuit air gap line. Numerically, the nonreciprocal p.u. system is much more convenient to use and to visualize.Thus, the relationship between the values of per-unit field current is as follows:

(1)

The overbars are included here mainly to stress the per-unit concept. Note that upper case nomenclature is used in thenonreciprocal system, for field current and voltage, but not for field resistance. See Section 2.2, and Table 7A-I.

Most excitation system analyses and specifications revolve around the choice of the nonreciprocal systems, since efdin the reciprocal system is small, usually between 0.001 and 0.002 p.u. for rated loading conditions. This concept isexpanded on in Appendix 7A, Table 7A-I.

For convenience, another variable Efd is used to describe field voltage. It is related to efd

by the scaling factor,

so that

I fd (nonreciprocal) Xadu i fd (reciprocal)⋅=

Xadu

R fd------------ so that



(2)

In the steady state, the per unit value of Efd is equal to Xadu·ifd in per-unit. A symbol for Xadu·ifd in common usage isEI, or field excitation, in per-unit.

On the air gap line of the open circuit saturation curve the value of 1.0 p.u. Efd, 1.0 p.u. Xadu·ifd or 1.0 p.u. EIb, allproduce 1.0 p.u. stator voltage.

In Appendix 5B for example, Lad·ifd is used as a base excitation in accounting for saturation in the equations usedduring the step-by-step stability calculations. In this and some other stability programs, changes in speed (ω) areusually accounted for, and in general ω. Ladu=Xadu.

The output voltage of excitation system models should be consistent with the per unit system used with generatormodels. That is, one per unit excitation system voltage is defined as the field voltage, corresponding to the excitationcurrent, which produces one per-unit stator voltage on the air gap line of the generator open-circuit saturation curve.

Although the above relationships might, for the sake of convenience, appear to be arbitrary, there is a sound basis fortheir adoption, as noted in [6], and as discussed in Appendix 7A.

In the above discussions, the two-principal machine constants used in the field excitation equations are Ladu (or Xadu ),and field resistance. In the nonreciprocal system, the field resistance rfd is usually a physical quantity, measured inohms. In synchronous machine calculating procedures, Rfd is usually a per-unit quantity. The next section containssome pertinent guides on obtaining values of field resistance.

Base impedance of the field, referred to the stator is:

Values of base field amperes as used above, are in the reciprocal system and in physical terms.

7.2 Calculation of Field Resistance

One of the important quantities in both the generator representation, as well as in the excitation system modeling is thephysical field resistance rfd. The emphasis for an accurate knowledge of field resistance for the nonreciprocal systemis founded on the need to calculate base field voltage. This is usually obtained by multiplying rfd times the base fieldcurrent from the air gap line (see Appendix 7A and Fig 7A-I).

Should the base field voltage be incorrect, an accurate determination of per-unit ceiling voltages for excitation systemperformance specification is impossible.

It should also be noted that rfd should be corrected to expected operating temperatures of the machine field winding,as specified in ANSI C50.1-1990 [1] and ANSI C50.13-1989 [2]. Another factor to be considered in using correctvalues of field resistance in the reciprocal system is its influence on the value of T′do, which is important in determiningthe rate of change of field flux linkages when assessing excitation system performance on stability improvement.

The value of rfd used should be taken from a physical measure of resistance, and not from the value of open-circuittime constant (T′do). The latter value could be measured from some type of current decrement or short circuit test, andits value would be assumed to be either.

E fd = ecfd

Xadu

R fd------------

Z fdmachine VA

base – field – A(dc) { }( ) 2

--------------------------------------------------------------=

(i) T ′ do X

fd

X

adu +

R

fd

---------------------------=



With reactance and resistance values in per-unit and referred to the stator, time constants are in radians.

It can be seen that, given a value of

T

′

do

, either calculated or measured by test, accurate knowledge of values of statorto rotor mutual reactance or rotor leakage reactances are needed to arrive at a calculation per-unit value of

R

fd

, in thereciprocal system. The physical value of

r

fd

would then be calculated, knowing the base field impedance.

In summary,

r

fd

, in physical terms, should be obtained by direct measurement. Methods described in IEEE Std 115A-1987 [3] also describe an alternative method of arriving at either a physical or a per-unit value of field resistance.Methods of deriving values of

X

fd

are also described in IEEE Std 115A-1987 [3], in order that a two-port circuit for thedirect axis may be determined. Conversion of physical resistance values to per-unit values is given in the Table shownin Appendix 7A.

7.3 Field-Circuit Identity

As a result of calculating

R

fd

, and

X

fd

, for d axis models, based either on measurements, or on procedures as describedin IEEE Std 115A-1987 [3], it is quite practical to thus retain the identity of the field structure of a synchronousgenerator in stability simulations. This retention is particularly desirable in certain types of excitation system andsynchronous machine representations. In such cases, it is a prime requirement that the model should be just as accurateas seen from the field terminals as from the stator terminals. Accordingly, the two-port approach has become anaccepted procedure in excitation system/generator modeling.

An important consideration in excitation system modeling studies is to account for how the excitation system responseis affected by the presence of damper circuits in a turbo-generator rotor, or how it is affected by the so-called “ironshielding” effect and induced currents of the rotor field slots and teeth. This is particularly important where thyristoror other high initial response, high ceiling, excitation systems are being utilized to improve stability limits, along withthe application of power system stabilizers.

Other situations where appropriate field representation is required can occur when the field winding is shorted, or isopen circuited. Although such occasions are relativity infrequent, they should be capable of being convenientlystudied. The need for generator or excitation system designers to accurately establish values of field dischargeresistance is discussed in Reference [4]. The modeling procedures outlined in previous sections, showing a fieldstructure and differential leakage reactance, make such discharge resistance calculations more precise.

The effect of system fault currents on inducing field-current transients is especially important for most rectifier-typeexcitation systems, and retaining the field circuit identity when studying these conditions is advantageous. As acorollary, if the primary interest is only the overall effect of induced field or rotor body currents, as viewed from thestator terminals, and no exciter action is expected, then the two-port approach is not required. Field current identity isalso required when studying diode exciter systems that have an appreciable voltage regulation.

7.4 Special Techniques for Modeling Field-Current Reversal or Field Shorting

With the increasing use in recent years of controlled (or noncontrolled) rectifier type exciters, field current reversal isnot possible. This applies to diode exciters as well as to thyristor (or “static”) exciters, both of which are supplied froman alternating current source.

While the examination of current reversal is not a commonly studied phenomenon, a condition, such as generator poleslipping, will result in a situation where a negative current may be induced in the field circuit. If the negative fieldcurrent is not allowed to flow, a very high voltage may result, depending on the type of rotor construction. In somecases, the damper winding may limit the resulting voltage to acceptable levels, but in other cases, additional protective

(ii) T ′ do T ″ do + X

fd

X

adu +

R

fd

--------------------------- X

1

d

X

adu +

R

1

d

---------------------------+=



circuitry is required. To provide a path for negative generator field currents, “crowbar” circuits (or varistors) arefrequently provided.

In the case of a crowbar, a resistor is inserted across the generator field by thyristors that trigger on over-voltagecreated by the induced negative field current, initially having virtually no path in which to now.

Varistors are non-linear resistors that are inserted across the generator field continuously. During normal conditions,the effective resistance is very high and very little current flows in the varistor. The varistor current increasesexponentially as a function of the applied voltage.

In some instances, no special field-shorting circuit is available and the amortisseurs provide the path for the inducedcurrent in the rotor.

This condition can be simulated by increasing the field leakage inductance to a very large value so as to limit the fieldcurrent to zero, whenever field computations require it to be negative. The field leakage inductance is restored to thenormal value when the field current becomes positive. As the paths for the induced rotor currents are provided only bythe amortisseurs, it is important to ensure that the generator rotor model includes their effects by using second or thirdorder direct-axis models [5].

The above special field modeling provisions are suggested when conditions require them, and are in addition to theusual excitation system modeling details that are not discussed in this document.

7.5 References

[1] ANSI C50.10-1990, Synchronous Machinery Rotating Electrical Machinery

[2] ANSI C50.13-1989, Rotating Electrical Machinery—Cylindrical Rotor Synchronous Generators.


[4] Canay, I. M.: Extended Synchronous Machine Models for Calculation of Transient Processes and Stability.Electrical Machines and Electromechanics—International Quarterly) #1, 1977, pp. 137–150.

[5] Kundur P., and Dandeno, P. L., “Implementation of Advanced Generator Models into Power System StabilityPrograms,”

IEEE Transactions on PAS

, PAS-102, July 1983.

[6] Rankin, A. W., Per-Unit Impedances of Synchronous Machines.

AIEE Transactions

, vol. 64, 1945, (i) pp. 569–572and (ii) pp 839–841.



Annex 7A

Establishing and Comparing Field-Circuit Relationships—Reciprocal vs. Nonreciprocal System

(Informative)


NOTE — again from Eq 1 in Section 7.1 that for values of per unit field current:

or

(1)

Overbars are used here to distinguish per-unit notation between the reciprocal and nonreciprocal systems. The directaxis relationship between terminal voltage and field excitation following from Park's equations (on open circuit) where

fd

= 0,

q

=

d

= 0, is:

(2)

E

I

is a symbol used in place of and also denotes field excitation.

With , and using the value of from Eq 1, it further follows from Eq 2 that

, the excitation system output voltage should equal in the steady state. All three of these symbols possess thesame per-unit quantities in the nonreciprocal system. In addition, 1.0 per-unit excitation required to produce 1.0 per-unit stator voltage on the pen-circuit air-gap line is the same for

E

I

,

I

fd

and

E

fd

.

Typical values for reciprocal and nonreciprocal voltage, current and impedance bases and field quantities for a 555.555mVA, 2-pole turbine generator rated 24 kV are given below in Table 7A-I. Field current (from the open-circuitsaturation curve) required to produce 24 kV on the air gap line, was by extrapolation, 1300 A. Measured fieldresistance

r

fd

was 0.10 ohms (corrected to 100

°

C).

X

adu

calculated = 1.82 p.u. on the 555.555 mVA, 24 kV bases.

The open-circuit saturation curve for this machine is also shown plotted in the conventional (nonreciprocal) per-unitsystem in Fig 7A-I of this Appendix.

In Figs 7A-1 and 7A-2, the two common approaches for establishing base excitation quantities are shown in graphicalform. The open-current saturation curve is identical in each case, and is based on data taken from the 555.555 mVAmachine used as an example in Table 7A-I.

i fd (nonreciprocal)= X adu i fd (reciprocal) ⋅

i fdI fd

Xdv---------=

eq Ψd Xadu i fd⋅= =

Xadu i fd⋅

e fd i fd R fd⋅= i fd

E1 Xadu i fd XaduI fd

Xadu------------ so that E1 I fd=,=⋅=

E fd EI



Table 7A-I —Example of Field Quantity Values for a 555.555 mVA, 24 kV Turbine Generator (Measured Field Values Shown for 0.9 power factor loading (500 mw, 242 mvar, 24.0 kV)

Field Circuit Quantities

Rankin Per Unit Systems Numerical Values

Non-Reciprocal Reciprocal

Base Field Current

I

fd

= 1300 amperes (measured)

i

fd

==

2366 amperes

= 1.0 pu = 1.0 pu

Field Resistance

r

fd

=0.10 ohms @

100

°

C

(measured)

Base Field Voltage E fd = 130 volts

(1300 × 0.10) e fd =234,807 volts

(2366 × 99.242)

Base Field

Z

fd

=

Impedance = 99.242 ohms

pu Field

R

fd

=

Resistance = 0.001010 pu

Field Current rated m V A

I

fd

= 3900 amperes (measured)

i

fd

= 1.648352 pu (3900/2366)

kV, p.f. = 3.0 pu

Corresponding Field Voltage

E

fd

=

390 volts (3900

×

0.10) = 3.0 pu (390/

130)

e

fd

=

0.001661 (390/234.807)

or (1.648352

×

0.001010)

Excitation System Ceiling Specified

E

fd

= 975 volts

e

fd

= 0.004152 p.u.

= 7.5 pu (975/234/807)

XadI fd

555.555x106

2366( )2-------------------------------

0.1099.242----------------



Figure 7A-1 —Non-Reciprocal Excitation Scale

Figure 7A-2 —Reciprocal Excitation Scale



Annex 7B

Excitation System-Generator Simulation Interfaces

(Informative)


7B1

Table 7A-I above shows the principal values associated with field- and rotor-equivalent circuit quantities. It should bereiterated that in some stability programs, the synchronous machine flux linkage, voltage, current and torque equationsare solved using reciprocal system per-unit quantities. On the other hand, it is evident that the excitation systemvariables can be much more conveniently computed and understood using nonreciprocal per-unit values.

Figures 7B-1 and 7B-2 show simple block diagrams based on a commonly used approach that permits the two per-unitsystems to be represented together at the same time in a well known stability program [1].

Figure 7B-1 —Block Diagram Representation of Eq 1, Section 7.1

Figure 7B-2 —Block Diagram Representation of Eq 2, Section 7.1



In the stability program referenced below, rotor flux linkages are chosen as the state variables. The equationsdescribing their solution, for the direct axis, are in per unit:

s(

ψ

rd

) =

R

d

(

M

d

·

ψ

rd

−

C

rd

.I

d

) + (

E

fd

·

Rfd/Xadu)

Rd is a matrix of field and rotor resistances.

Md and Crd are further described in [1], and basically are related to the self and mutual inductances of both stator androtor circuits.

Thus, direct axis flux linkages are functions of field voltage (Efd) and stator current (Id).

7B2 References

[1] Kundur, P., and Dandeno, P. L., “Implementation of Advanced Generator Models into Power System StabilityPrograms,” IEEE Transactions on PAS, PAS-102, July 1983.


ieee std 1110-1991, ieee guide for synchronous generator modeling practices in stability

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