IEEE Std 1110-1991
IEEE Guide for Synchronous Generator
Modeling Practices in Stability Analyses
Sponsor
Power System Engineering and Electric Machinery Committees
of the
IEEE Power Engineering Society
Approved March 21, 1991
IEEE Standards Board
Abstract: Categorizes three direct-axis and four quadrature-axis models, along with the basic transient
reactance model. Discusses some of the assumptions made in using various models and presents the
fundamental equations and concepts involved in generator/system interfacing. Covers, generally, the
various attributes of power system stability, recognizing two basic approaches. The first is categorized
under large-disturbance nonlinear analysis; the second approach considers small disturbances, where the
corresponding dynamic equations are linearized. Applications of a range of generator models are
discussed and treated. The manner in which generator saturation is treated in stability studies, both in the
initialization process, as well as during large or small disturbance stability analysis procedures is
addressed. Saturation functions that are derived, whether from test data or by the methods, of finite
elements are developed. Different saturation algorithms for calculating values of excitation and internal
power angle, depending upon generator terminal conditions are compared. The question of parameter
determination is covered. 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/excitation system interface modeling. Suggestions for modeling of negative
field currents and other field circuit discontinuities are included.
Keywords: Synchronous generator stability models, modeling practices, saturation practices, stability data
determination and application
The Institute of Electrical and Electronics Engineers, Inc.
345 East 47th Street, New York, NY 10017, USA
Coprigth © 1991 by the
Institute of Electrical and Electronic Engineeers, Inc.
All rights reserved. Published 1991
Printed in the United States of America
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prior written permission of the publisher.
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Foreword
(This Foreword is not a part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
The Joint Working Group on Determination and Application of Synchronous Machine Models for Stability Studies
was 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, and
recommend standard methods for determining the values of parameters for use in these models by calculation and/or
test. Assess the effect of magnetic saturation on these parameters. Devise and recommend analytical methods for
incorporating 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 machine
modeling. The first was published in PA & S in 1980. In 1983, we presented a one-day symposium on the subject of
machine modeling and generator stability data acquisition at the IEEE PES Winter Power Meeting. Following
publication of our second IEEE committee report in March 1986 (vol. EC-1), the group and the two committees to
whom 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 over
the 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 meeting
of 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 primary
aim of such a guide is to enable an engineer, who is relatively inexperienced in stability analyses, to perceive some of
the important issues in this branch of power system engineering. As such, any reader, experienced or not, should be
able 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 the
guide 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 from
overseas was achieved in the past through Dr. G. Shackshaft. Dr. R. G. Harley joined our Group in 1989. Local
associates 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, who
preceded 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 Electrical
Engineering Department of the University of Toronto. In particular, he appreciates the encouragement and support
given by Dr. A. S. Sedra, Dr. G. R. Slemon and Dr. J. D. Lavers of the Electrical Engineering Staff, especially in the
logistics 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 noted
for 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 to
the development of this document) work.
iii
This document was prepared by the Joint Working Group on Determination and Application of Synchronous Machine
Models for Stability Studies. The members of the working group were:
P.L. Dandeno, Chair
B. Agrawal
D.H. Baker
C. Concordia
J.S. Edmonds
L.N. Hannett
R.G. Harley
D.C. Lee
S.H. Minnich
S.J. Salon
R.P. Schulz
H.R. Schwenk
S. Umans
The following persons were on the balloting committee that approved this standard for submission to the IEEE
Standards Board:
P. M. Anderson
J. C. Agee
R. A. Alden
R. F. Bayless
A. Bose
C. Bowler
V. Brandwajn
R. M. Bucci
R. T. Byerly
A. Calvaer
S. Chan
M. S. Chen
J. Chow
R. Chu
C. Concordia
R. Craven
R. Creighton
M. Damborg
P. L. Dandeno
F. P. de Mello
C. Didriksen
H. Dommel
P. J. Donalek
J. Doudna
R. D. Dunlop
R. G. Farmer
J. H. Fish
A. A. Fouad
iv
A. Germond
W. B Gish
H. Glavitsch
J. Grainger
C. E. Grund
A. E. Hammad
J. F. Hauer
R. A. Hedin
K. Hemmaplardh
G. T. Heydt
D. J. Hill
J. Hurley
E. Katz
R. Kumar
M. Lauby
S. Lefebvre
C. Lui
P. Kundur
J. Luini
P. Magnusson
A. Mahmoud
O. Malik
Y. Mansour
D. Martin
S. Mokhtari
D. L. Osborn
M. A. Pai
M. Pal
M. C. Patel
M. Pavella
W. Price
N. D. Rao
K. Reichert
D. D. Robb
R. Roberge
L. Rodriguez
P. A. Rusche
N. Saini
S. Savulescu
R. Schlueter
R. P. Schultz
G. Scott
A. J. Sood
K. Stanton
S. Stanton
D. A. Swann
Y. Tamura
J. Tang
C. Taylor
R. J. Thomas
J. Van Ness
S. Virmani
V. J. Vittal
O. Wasynczuk
F. F. Wu
Yo N. Yu
J. Zaborsky
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 Bodson
Paul L. Borrill
Clyde Camp
James M. Daly
Donald C. Fleckenstein
Jay Forster*
David F. Franklin
Ingrid Fromm
Thomas L. Hannan
Donald N. Heirman
Kenneth D. Hendrix
John W. Horch
Ben C. Johnson
Ivor N. Knight
Joseph L. Koepfinger*
Irving Kolodny
Michael A. Lawler
John E. May, Jr.
Lawrence V. McCall
Donald T. Michael*
Stig L. Nilsson
John L. Rankine
Ronald H. Reimer
Gary S. Robinson
Terrance R. Whittemore
* Member Emeritus
Deborah A. Czyz
IEEE Standards Project Editor
v
CLAUSE
1.
PAGE
Introduction .........................................................................................................................................................1
1.1 .................................................................................................................................................................... 1
1.2 References .................................................................................................................................................. 2
2.
Model Classifications or Categories ...................................................................................................................3
2.1 Practical Models Available ........................................................................................................................ 3
2.2 Nomenclature and Glossary of Frequently used Terms............................................................................. 7
2.3 References ................................................................................................................................................ 11
3.
Classification of Stability Studies .....................................................................................................................12
3.1
3.2
3.3
3.4
3.5
3.6
4.
Background .............................................................................................................................................. 12
Large Disturbances Stability .................................................................................................................... 13
Small Disturbance Stability ..................................................................................................................... 14
Classification Based on Dominant Modes of System Response.............................................................. 14
References ................................................................................................................................................ 14
Bibliography............................................................................................................................................. 15
Application of Generator Models in Stability Studies ......................................................................................15
4.1
4.2
4.3
4.4
4.5
General ..................................................................................................................................................... 15
Representation of Generations During Large Disturbances..................................................................... 18
Modeling of Machines for Small Disturbance Stability Studies.............................................................. 19
References ................................................................................................................................................ 20
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
5.2
5.3
5.4
5.5
5.6
General ..................................................................................................................................................... 24
Representation of Generator Saturation in the Steady State .................................................................... 24
Representation of Saturation Effects During Large Disturbances ........................................................... 26
Generator Saturation in Small Disturbance Modeling ............................................................................. 29
References ................................................................................................................................................ 31
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
CLAUSE
6.
PAGE
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............................................................................................. 51
6.3 Parameters Derived by Two Manufacturers in the Machine Design Stage ............................................. 53
6.4 Desirability for Uniform Practices in Deriving Machine Stability Parameters ....................................... 53
6.5 Alternative Forms of Model Representation............................................................................................ 55
6.6 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
7.2
7.3
7.4
7.5
Establishing Field-Voltage, Field-Current and Field-Impedance Bases.................................................. 71
Calculation of Field Resistance................................................................................................................ 72
Field-Circuit Identity................................................................................................................................ 73
Special Techniques for Modeling Field-Current Reversal or Field Shorting .......................................... 73
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
vii
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 power
system 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 that
time, it was felt that computer codes, including generator/network interfacing equations, as well as many other
stability 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 guide
is to set forth much of the current wisdom, and discuss the principal issues that ought to be considered in the
application of more sophisticated models. These have now become available partly due to better digital simulation
capabilities, 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 the
treatment of generator stability representation, as proposed by investigators about fifty years ago. References to several
pioneer 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 atypical
cases such as generator tripping and overspeed operation, or the models for harmonics or unbalanced operation. The
particular modeling requirements for Subsynchronous Resonance studies have been examined by other IEEE-PES
groups, 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 structure
and to the parameter values assigned to the elements of individual models. The general concordance between the more
familiar generator reactances and time constants, and the various; direct and quadrature-axis model structures is
pointed out in Section 2.1.
Section 2.1 also discusses some of the assumptions made in using various models, including the basic model. It also
touches upon the fundamental equations and concepts involved in generator/system interfacing. In Section 2.2, a
glossary 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 being
recognized. The first is categorized under large-disturbance nonlinear analysis. The second approach considers small
disturbances, where the corresponding dynamic equations are linearized.
In Chapter 4, applications of a range of generator models, documented in Chapter 2, are discussed. These applications
are also treated in the context of the nature and complexity of large and small power system disturbances that are
covered in Chapter 3.
Chapter 5 covers the manner in which generator saturation is treated in stability studies, both in the initialization
process, as well as during large disturbance or small disturbance stability analysis procedures. Included in the
appendixes of Chapter 5 is the development of saturation functions that are derived either from test data or by the
methods of finite elements. The appendixes of Chapter 5 also compare different saturation algorithms for calculating
values 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. Such
parameters are found either by test, as in IEEE Std115-1983 (R1991) [1] or IEEE Std 115A-1987 [2] or are calculated
by manufacturers. The detailed translation of data, from specific direct and quadrature axis model structures, with their
associated element values, to transient and subtransient reactances and time constants, or to transfer function form is
presented for commonly used models. Some of the assumptions made by manufacturers in producing calculated or
design parameters are pointed out. In addition, the desirability of a common approach as a basis for such design
calculations 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/excitation
system 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 Standstill
Frequency 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 120
electrical degrees apart, and a rotating (rotor) structure with an excitation or field winding and one or more equivalent
rotor 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) reflect
induced 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 other
group is aligned in the quadrature axis.
R. H. Park [7] developed this concept further by mathematically transforming the three-phase stator quantities (such as
voltages, currents and flux linkages) into corresponding two-axis quantities. Park did this by visualizing two fictitious
circuits—one in the direct axis aligned with the field axis, and the second in the quadrature axis. Under steady state
conditions 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 machine
flux linkage equations. Time varying three-phase stator voltages, currents, and flux linkages were also transformed
into time invariant direct and quadrature axis quantities under steady-state conditions. His treatment is general enough
that deviations in speed of the field and rotor structure from steady state could be easily accounted for in the fictitious
stator 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 be
visualized in terms of direct and quadrature axis equivalent circuits. Alternatively, operational inductance expressions
in 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 the
number 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 or
quadrature axis. Alternatively, resistance and inductance values can be assigned to the elements of direct or quadrature
axis equivalent circuit.
Some of the above points are expanded upon in Section 6.5, where the philosophy behind these modeling approaches
is presented. As discussed above, the basic modeling approach usually considered for synchronous machine electrical
dynamics is derived from Park's transformations, with corresponding relationships between flux linkages, currents and
inductances. This approach is further characterized by a prior knowledge of the stator leakage inductance, and the
stator to rotor mutual inductance. To such a fundamental model structure, or configuration in either axis, one may
explicitly connect an arbitrary number of rotor circuits (including the field winding in the direct axis). For power
system stability studies, based on experience, judgement and intuition, and often a lack of data, it has been found
sufficient 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 from
first order to third order in each axis, depending upon the number of inductance/resistance series combinations
representing the field and direct axis equivalent rotor circuits,or the number representing quadrature axis equivalent
circuits. There are 12 possible combinations of direct and quadrature axis representations, plus one “constant rotor flux
linkage” 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. Some
combinations of d and q axis winding configurations are not considered in Table 1, and equivalent circuit structures are
not drawn or discussed. Based on experience, as well as on published material [3,4,5,8] it appears there are seven
model structures that could be serious candidates for inclusion in large system stability simulations. Six of these model
Copyright © 1998 IEEE All Rights Reserved
3
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
structures are drawn in Table 1, and the seventh is the constant rotor flux linkage (or alternatively constant voltage
behind 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 iron
body equivalent circuits are identified as 1d, 2d according to the number of such circuits. In the quadrature axis, the
equivalent 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 the
linearized model response on open circuit. The roots, for example, in the direct axis for Model 2.2 are usually
classified as the inverse of T′do (the direct axis transient open circuit time constant) and the inverse of T″d, (the direct
axis 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 to
X′d,(for the transient period). Higher order models than 2.2 such as Model 3.3 would possess 3 time constants and 3
values of reactance, and Model 1.1 would only have a single time constant and reactance. Time constant expressions
are 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 of
various 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 be
required 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 1
for example, is a leakage inductance proportional to fluxes that link the field winding and the two direct axis equivalent
rotor 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 considers
two windings in each axis, including the direct axis field winding. Parameter values for this model structure have
normally been supplied by manufacturers of synchronous machines, or have been, for the direct axis, obtained by tests
described in IEEE Std 115-1983 (R1991) [1]. Two time constants and two rotor reactances and resistances have been
employed 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 it
is 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 equivalent
direct 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, except
for Lf1d the differential leakage inductance [3] [4]. This value cannot be uniquely determined from stator
measurements alone. It should be determined from data based upon both stator and field winding measurements. This
is discussed further in Section 6.2, as well as in IEEE Std 115A-1987 [2]. If neither field winding measurements nor
design data are available, the value of Lf 1d is typically set to zero, as in Fig 3. However, it should be recognized that
this choice will affect the armature to field winding transfer relationships. When the field winding identity is of
significance in a particular study, it is recommended that field winding measurements be included in the model
derivation. Models of lower complexity than 3.3 and 2.2 have often been used in the past due to unavailability of
higher-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 of
such estimated values is considered good practice rather than neglecting their presence by reverting, or defaulting to
Model 1.0.
Model 1.0 is another structure that represents only the field winding effects in the rotor. It is the simplest model to that
excitation models can be connected. From this, changes in field flux linkages (E′q) can be determined. Model 1.0 has
had some use in the past, but is not currently recommended for general use.
4
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
The simplest model is the classical model that assumes a constant voltage behind transient reactance (E′= constant). It
assumes constant flux linkages in each axis, and also that no transient saliency exists, namely X′d = X′q = X′
Then E′ =
( E′ q ) + ( E′ d ) 2
Since the voltages and currents of this model are not resolved into direct and quadrature axis components, this model
structure is placed outside the matrix of Table 5. During a simulation run, the magnitude of the model's internal voltage
is kept constant, but the internal angle is changed to reflect the inertial effects for changes in power. An advantage of
this simple model is that the interfacing of generator and network equations can be accomplished more quickly during
step-by-step calculations, and the least amount of stability data is needed for it. Thus, the “constant voltage behind
transient reactance” model has virtually replaced Model 1.0 in stability procedures and calculations where simple
generator 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 determining
the parameter values for the model elements (inductances and resistances) based upon measured or analytically
determined data that characterize the generator to be modeled (see Section 6.5). The parameter determination
procedure 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 higher
complexity 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 earlier
in 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 in
synchronism 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 noted
in Section Auto.
To solve the network equations in terms of Park's representation, two-axis Real (R) and Imaginary (I) voltage phasors
are chosen. These are synchronously rotating phasors 90° (electrical) apart, with the real-voltage phasors VR chosen to
be in phase with the “swing” bus voltage in the load flow (usually zero degrees). Thus, the real-voltage phasor VR is the
reference phasor for all the generator terminal voltages in the network. In Fig 4, δ1′ angle is the electrical displacement
between system generator #1 rotor position and the reference phasor VR. Angle δ1′ is comprised of the internal angle
δ1, 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, in
general,
V R = E d sinδ′ + E q cosδ′
(1)
V I = – E d cosδ′ + E q sinδ′
(2)
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IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Table 1— Selection of Generator Models of Varying Degrees of Complexity
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A second basic equation in this process is as follows taken from Eq 15 in [6]:
VR
VI
=
– R RR X RI
IR
– X IR – R RII
II
+
ER
EI
(3)
This states that the real and imaginary components of machine terminal voltages are equal to the product of an impedance
matrix times the real and imaginary components of stator current plus ER and EI. The latter are essentially internal
machine voltages. Both ER and EI and the impedance matrix elements (the latter including relevant stator and rotor
inductances) 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
Ed
Eq
Ia
Id
Iq
Synchronous machine stator terminal voltage in p.u.
Direct-axis stator voltage in p.u.
Quadrature axis stator voltage in p.u.
Synchronous machine stator (armature) current in p.u.
Direct-axis stator current in p.u.
Quadrature axis stator current in p.u.
The above capitalized quantities are generally used in equations and phasor diagrams as phasors. (A phasor is a
complex number having the form X = X0εjθ, where Xo is the magnitude and θ is the phase angle). This nomenclature
is 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 model
nomenclature listed in IEEE Std 115A-1987, as well as to the nomenclature of an IEEE Committee Report [3]. Except
where noted, such parameters are in per unit referred to the generator stator.
2.2.2 Field or Rotor Quantifies
Efd
efd
Ifd
ifd
Rfd
rfd
Rkd
Rkq
Field voltage in the non-reciprocal system (dc volts)
Filed voltage in p.u. in the reciprocal system.
Field current in the non-reciprocal system (dc amperes).
Field current in p.u. in the reciprocal system.
Field resistance in p.u. in the reciprocal system, referred to stator.
Fields resistance in the physical ohms.
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.
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.
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Figure 1—Complete, Third-Order Representation—Both Axes
Figure 2—Complete, Second-Order Representation—Both Axes
Figure 3—“Standard” Second-Order Model—D Axis
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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
Laqu
Ldu
Lqu
Ll
Lfkd
Lfd
Lkd
Lkq
Lffd
Lmf1d
L11d
L11q
Ld(s)
Lq (s)
G(s)
Direct-axis stator to rotor mutual inductance, unsaturated in p.u.
Quadrature-axis stator to rotor mutual inductance, unsaturated in p.u.
Direct axis synchronous inductance (unsaturated) = Ladu +Ll in p.u.
Quadrature axis synchronous inductance (unsaturated) = Laqu +Ll in p.u.
Stator leakage inductance (both d and q axis), in p.u.
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 these
fluxes are mutual.
Field winding leakage inductance.
Direct-axis equivalent damper winding leakage inductance; k = 1,2,..n.
Quadrature-axis equivalent damper winding leakage inductance; k = 1,2,..n.
Field winding self inductance = Ladu +Lfd +Lf1d
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.
Self-inductance of one equivalent rotor winding = Ladu +L 1d +Lf1d
Self-inductances of another equivalent rotor winding = Laqu +L1q
Direct-axis operational inductance, as viewed from the stator terminals.
Quadrature-axis operational inductance, as viewed from the stator terminals.
Stator to field operational transfer function.
NOTE — Much of the material in this guide uses reactances when describing the relationships between machine fluxes and
currents, and this usage arises from the premise that steady -state relationships are being discussed. In generator
modeling, X = ω0 · L, where ω0 = unity or base speed. Thus, L (inductance) and X (reactance) are often used
interchangeably. 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.
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Usually, inductance or reactance values correspond to the unsaturated value of L or X. In some cases the text includes
a subscript “u” or “sat” to clearly distinguish between unsaturated or saturated conditions. The unsaturated condition
is implied in the absence of these 2 subscripts.
2.2.4 Commonly Used Symbols in Stability Programs
Ld,Lq
L′d,L′q
L″d, L″q
T′do, T′qo
T″do, T″qo
T′d, T′q
T″d, T″q
ψd (or λd)
ψq (or λq)
ψfd (or λfd)
ψ1dψ2d (or λ1d, λ2d)
ψ1q,ψ2q, ψ3q (or λ1q, λ2q, λ3q)
d- and q-axis synchronous inductance, usually quoted unsaturated and in p.u.
d- and q-axis transient inductance, usually quoted in p.u. and saturated
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.
d-and q-axis transient open-circuit time constants in seconds.
d- and q-axis subtransient open-circuit time constant in seconds.
d- and q-axis transient short-circuit time constants in seconds.
d- and q-axis subtransient short-circuit time constants in seconds.
Direct-axis stator flux linkages.
Quadrature-axis stator flux linkages.
Field flux linkages.
Direct-axis damper winding or amortisseur flux linkages
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 has
been 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 the
terms or phrases listed below can be located in IEEE Std 100-1988. A few of the terms listed are not in IEEE Std 1001988, and can be considered peculiar to the jargon used in this document.
A) Dictionary terms frequently used in this text:
B)
C)
Air Gap
Air Gap Line
Armature
Exciter
Excitation System
Operational Impedance
Operational Inductance
Permeability
Permeability, (Incremental)
Definitions of the familiar time constants and reactances for synchronous machines are given in the
dictionary under “direct axis” or “quadrature axis” headings.
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 as
the 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 a
system, including a power system. The choice of state variables is not necessarily unique for any system. (Also
referred to sometimes as dynamic states).
eigenvalue: This is one root of the characteristic equation. The number of eigenvalues in a dynamic system is equal to
the order of the characteristic equation, or the number of state variables. The eigenvalues describe the transient
response of the system to any disturbance. A transient associated with an eigenvalue takes the exponential form ελitxi,
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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 2.2.4
above.
eigenvector: This vector defines the distribution of the mode, corresponding to a particular eigenvalue, throughout the
state variables of the system.
On a more practical basis, the eigenvector can be thought of as a set of coefficients describing the relative magnitudes
of 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 its
two-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 to
appropriate elements. Rotor and stator currents are also prescribed. The output of the analysis is the magnetic flux
distribution (expressed as the vector potential) in the generator cross-section, from which stator and rotor flux linkages
and 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 in
common 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 the
field winding. The poles are then attached to a shaft, and in the case of many poles, a central frame is used to support
them. This frame is then connected to the central shaft. Pole-face damper bars or amortisseurs may be embedded in the
laminations near the pole face.
saliency: This term is now used for all types of synchronous machines (round-rotor and salient-pole) in which the
electromagnetic parameters are significantly different in the direct and quadrature axes. There may be different degrees
or 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 of
salient-pole machines, including hydro-generators, which all have a pronounced physical difference in the magnetic
flux 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 Standstill
Frequency Response Testing
[3] IEEE Committee Report: Current Usage and Suggested Practices in Power System Stability Simulations for
Synchronous 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 Stability
Programs, 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.
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[8] Schulz, R.P., Jones, W.D., and Ewart, D.N., Dynamic Models of Turbine Generators Derived from Solid Iron Rotor
Equivalent 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 studies
depends on the nature of the study, as well as the nature of the power system [2]. A striking tendency of the past 25
years has been that studies of the dynamic performance of such systems can embrace several or even many
interconnected power utilities due to the formation of power pools or operating areas. Intra-area, or inter-area studies
involving several companies, or several areas or pools, are common, particularly because of changing power transfers
arising from the economies of energy interchanges between utilities. As a result, large scale stability programs have
been developed for such studies, generally embracing the near-term time frame. Studies of this nature are often
concerned with power system operation procedures. On the other hand, detailed examinations of system performance
can be concerned with planned additions to power network configurations and with future load and generation patterns
expected several years in the future.
Occasionally, very long-term or “broad brush” type studies are performed. However such studies are more concerned
with generating station location and transmission network configurations; generator detail is secondary. Such studies
are performed to assess the need for future major power system facilities and also fall under the discipline of the power
system planning function.
Past system performance is sometimes reviewed and simulated in as much detail as possible to determine possible
reasons for misoperation, or for an explanation of past events. However, such a power system analysis forms a small
portion of stability studies.
Until recently, all of the stability studies discussed above analyzed power system responses to faults or other similar
contingencies, and covered several seconds of simulated time. Such time-domain studies still currently embrace most
stability analyses. Lately, power system dynamic responses to small perturbations, involving a linearized model of
major 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 practice
to group power system analysis into two general classifications, as noted in Reference [1].
1)
2)
Large disturbances, where the equations that describe the dynamics of the power system cannot be linearized
for the purpose of analysis.
Small disturbances, where the equations that describe the dynamics of the power system are linearized for the
purposes of analysis.
From the first of these two general classifications follows a specific definition of the transient, or large disturbance
stability of a power system [1]. The second of these two classifications provides a specific definition of steady state
stability of a power system [1]. It is further stated in Reference [1] that steady-state stability is also known as small
disturbance stability of a power system.
As a rule, transient stability simulations are highly nonlinear and are studied using step-by-step integration of the
differential equations of the dynamic devices that influence system performance. For small disturbance stability, the
same equations are linearized so that the more direct methods of stability prediction based on linear system analysis
may be applied.
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For stability studies involving faults or other sudden changes to the power system configuration, several factors play a
significant role in determining how the studies are conducted. These include the severity of the disturbance, and the
period 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 power
system responses, including damping of generator oscillations, and the dominant modes or frequencies of power or
angular 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 to
a large disturbance such as a fault on the transmission network or sudden loss of generation. It has normally been
concerned with the behavior of the system up to about 5 seconds following the disturbance, and depending on the
dominant modes of oscillation determined in the analysis, can extend to as many as 10 seconds. The instability is
normally 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 in
the new steady state, and voltages, power and reactive power flows, and relative phase displacements of generators
should have stabilized to acceptable values.
Much of the electric utility industry effort and interest related to large disturbances have concentrated on the short-term
transient response. The system is designed and operated so as to meet a set of reliability criteria concerning transient
stability. The criteria specify certain contingencies that the power system should be able to withstand without loss of
synchronism. These contingencies are selected on the basis that they have a certain level of severity and a significant
probability of occurrence.
Generator electrical stability characteristics have a significant influence on the short-term transient response of the system
and 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 word
long-term stability has been introduced. Time frames associated with such analysis extend beyond the usual 5–10
seconds. 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 the
interval between transient stability and long-term stability. This concept embraces periods of time up to as many as
40 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 system
upsets, with the associated slowly changing phenomena. Prime mover and energy supply system dynamics and the
influence of automatic generation control, over/under voltage and over/under frequency control and protection
schemes, are likely to be significant. Inter-machine synchronizing power oscillations often may be assumed to have
damped out, resulting in uniform system frequency. In these cases, the generator electrical characteristics usually do
not have a significant influence on long-term dynamic response. On the other hand, longer term effects, such as
capacitor switching and restoration of constant mVA loads through tap changer action may render the power system
unstable in such a dynamic state, and in these situations, importance is stressed on generator and excitation system
representation, and appropriate modeling detail.
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3.3 Small Disturbance Stability
Small disturbance (or steady-state) stability is concerned with system response to small changes that continually occur
in the operation of the system [B2]. The results are usually given in terms of eigenvalues and eigenvectors. The
calculated system response depends in such situations on the type and modeling of excitation control, as well as on the
type of generator modeling. In the absence of high initial response excitation control, loss of stability may occur due
to 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 excitation
systems. 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 disturbance
stability of the system. It is, therefore, important to use accurate generator models in such analyses and particularly in
design studies of excitation control for enhancing stability. (Generator modeling under these conditions is discussed
further 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 are
recognized. 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 one
plant or restricted to a small part of the power system. The frequencies of these oscillations are in the range 0.8 to 2.0
Hz. For situations where system stability associated with the plant mode is of primary concern, it is necessary to
represent the machines in the local area in detail. Simple models can be used to represent remote machines and system
reduction 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 machines
in other parts. These are referred to as “inter-area mode” oscillations and have frequencies in the range 0.1 Hz to 0.7
Hz. They are caused by two or more groups of machines connected by relatively weak ties. Analysis of system
conditions in which the stability of inter-area swing modes is of primary concern requires detailed representation of
the machines throughout all the areas being simulated. In the initial stages of such a study, it is possible to represent all
machines by a simple classical model, in order to indicate the frequencies of oscillation, and identify the groups of
machines involved in these dominant modes. If this screening process is resorted to, machines in each individual area
should 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 in
determining whether the dynamic response of one distinct area can be approximated in the above manner. This
decision often can only be taken after initial detailed generator modeling representations of the area under study, and
after 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 on
Power 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 System
Dynamics, Symposium on Adequacy and Philosophy of Modeling: Dynamic System Performance, IEEE Publication
75CHO 970-4-PWR, 1975, pp. 16–23.
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3.6 Bibliography
[B1] Dunlop, R.D., Ewart, D.N., and Schulz, R.P., “Use of Digital Computer Simulations to Assess Long-Term Power
System 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 System
Dynamic 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)
2)
3)
Power System Planning, to enable decisions to be made on future transmission and generation requirements.
Power System Operation, to determine operating limits in the light of various system contingencies.
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 may
be possible to use simple synchronous machine models, since only a relative measure of the performance of a number
of 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. In
some cases, special controls or protection schemes to maintain the stability of the system may be studied. For these
studies, 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 some
contingency analysis studies. For the normal determination of operating limits using contingencies included in system
design 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 made
on the mathematical models used in the simulation.
For planning and operating studies the following seven categories of large disturbances are usually the most
significant:
1)
2)
3)
4)
5)
6)
7)
Stability of one or more generating stations, following a severe nearby fault.
Excitation system control behavior and its associated stabilization, under all types of disturbances.
Power transfer limit determination across one or more inter-area interfaces subsequent to disturbances.
The sudden loss of one or more large generator(s), or deliberate generation tripping to maintain stability.
Generator and power system performance with delayed fault clearing and the associated transmission line
relay performance.
The effects of large industrial loads, or of deliberate load shedding.
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.
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In the case of power-system faults in (1) above, in particular, the effects of unbalanced faults cannot be overlooked, but
as at general rule in large-scale studies, calculation of generator balanced electrical torque or power developed across
the machine air gap is based on the assumption of a balanced or positive-sequence network. The effect of unbalanced
faults on positive-sequence power calculations is obtained by applying an appropriate impedance-to-neutral at the
point of fault in the positive-sequence network.
The positive sequence system representation used in stability studies does not allow the effects of dc offsets in
armature 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), and
comments on neglecting the rate of change of stator flux linkages and/or variations in rotor speed in the torque or
power 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 general
tendency 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 stator
terminals, then the use of this simple model may be acceptable. Adjacent machines remote from the disturbance can
then be grouped and represented as one machine, as one “voltage behind transient reactance.” In this case, the
individual inertias would be totalled and used in the equivalent generator, along with the Thevenin equivalent of all the
internal 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 on
their degree of modeling. Some engineering judgement needs to be exercised as to which machines can logically be
ignored. In such cases, they would be balanced against local load. Any generator group consisting of a small
percentage of the total area load (5% or less) can be balanced out against local load, or at most, represented in all cases
by its transient reactance and the corresponding internal voltages. It may be good practice to also retain an appropriate
damping factor for such generator groups.
For simple, yet reasonably accurate reduced order power system analyses, it is important to aggregate machines for
which the rotor angle excursions are coherent following some initiating disturbance. For short-time-scale studies (1 to
1-1/2 s) involving “local” stability problems, this coherency criteria may include all generators electrically remote
from 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 in
stability studies. There is sometimes an incentive in reducing the system size to a prescribed level in terms of the
number of load flow busses to be solved at any “time step”. This follows because the number of algebraic equations
involved 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 in
computing time per step if all machine equations were written for a third-order model, as opposed, for example, to a
second-order model. With no sub-subtransient saliency in Model 3.3 for example, (that is if X q ′″ = X d ′″ ), that model
is computationally more efficient when calculating generator terminal quantities than, for instance, model 2.2 where
subtransient saliency, was assumed to be present (i.e. X q ″ = X d ″ ). Simply stated, when E q ′″ and E d ′″ are replaced
by a single voltage E′″ , the interfacing of generator and network equations can be accomplished more quickly during
step-by-step stability calculations. Refer also to similar comments (on assuming X′ q = X ′ d ) 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
16
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
application of such excitation systems to certain types of turbo-generators may dictate the stability analysis integration
equations. It should also be noted that elimination of the type of saliency discussed above may slightly distort the
values 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 pole
faces. These damper bars are connected with continuous end-rings and thus, form a squirrel-cage amortisseur circuit
that is effective both in the direct and quadrature axes. Since this amortisseur is the only physical circuit present in the
q-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 various
aspects 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 tops
of these slots contain wedges for mechanical retention of the field turns. These wedges are usually made of a
nonmagnetic metal. In some cases, they are segmented and made of nonmagnetic steel, while in other cases the wedges
are made of aluminum and may be either segmented or full length. In many constructions, a conductive ring under the
field end-winding retaining ring, with fingers extending under the ends of the slot wedges, is used to improve
conduction at these connection points.
For infrequent cases where the retaining ring is magnetic, field leakage fluxes and corresponding inductances will be
affected, with noticeable differences existing in frequency response data obtained at or near full load, compared with
stand-still data [6].
Copper strips are often inserted under the wedges to provide improved conduction between wedge segments and/or to
improve amortisseur action. In some cases, a complete squirrel-cage winding is formed, while in other cases the
conductive paths contribute only marginally to amortisseur action. The effectiveness of the slot-wedge amortisseur
circuit may vary widely depending on the wedge material, segmenting of wedges and the design of the conductive
circuit below the wedge. References [5] and [6] discuss the impact of some of the above details on the direct axis rotor
model. A third-order model for the direct axis definitely gives a better fit to Ld(s) and sG(s) test data than a second
order one. However, in some cases the smallest time constant ( T′″ do ) values may be close to the size of the integration
equation time steps. In many cases, a second-order model for the direct axis (Model 2.2 or 2.3) will suffice, along with
recognition of Lf1d, which reflects the differences in mutual coupling between the field, the equivalent rotor wedge
and/or damper circuits and the stator.
For turbo-generators therefore, Model 2.2 can usually describe the operational inductance in both direct and
quadrature axes adequately, where no conducting strips are placed under the slot wedges, and where the pole face
region is without wedges or amortisseur bars.
The pole-face region is often slotted circumferentially, or, on occasion, may be slotted longitudinally for rotor flexing
and balancing purposes. In some cases, longitudinal pole face slots may be filled with wedges of either lowconductivity steel or, in the case of machines subject to sub-synchronous resonance (SSR), high-conductivity material
to form a quadrature axis amortisseur. The q-axis equivalent circuit model should account for various current paths in
the 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 has
also been found, for a range of rotor damper constructions, that a third-order representation such as Model 2.3 or 3.3
gives a better fit between measured and calculated values of Lq(s) including phase angle, and in the 0.1 Hz to 10 Hz
range [4]. Three impedance values are thus available to describe the quadrature axis rotor dynamics over the requisite
Copyright © 1998 IEEE All Rights Reserved
17
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
power system frequency range rather than just two. It would appear that electrical damping from currents in the pole
face area would more accurately reflect actual conditions when using a third order quadrature axis model, and this
approach is preferable to the use of damping factors. Concordia treats generator damping torques, and the associated
damping 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 axis
rotor circuits, and generator torques, are considered important in studying rotor “backswings.” This effect is often
critical 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 on
generator stability. The largest time constant in the denominator of the transfer function of any direct axis model
reflects the impact of the field time constant. Considering all rotor element values, this effective field time constant is
increased 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 changes
on 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 are
given 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 fault
conditions or excitation system control design, the responses of the generators, exciters and power system itself
immediately 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 and
system conditions, and are influenced by the magnetic characteristics of the generator. Saturation of the magnetic paths
in 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 to
close-in faults, results in the acceleration of generator rotors. This is normally accompanied by large increases in field
voltage 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 model
selection. 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 a
desirable minimum, a second-order model for both the direct and quadrature axes, providing reasonably accurate
parameters for these chosen models can be either obtained or can be estimated.
1)
2)
18
For turbo-generator representation, a second-order model in both axes (Model 2.2) is nominally required as
a minimum. For turbo-generators that also utilize damper bars or conducting wedges in the pole faces, in
some cases a second order model may be adequate in the quadrature-axis. A third order model (Model 2.3 or
3.3) may be required for the best results, especially to give good representation of generator damping.
For excitation system stabilization studies, along with exciter ceiling voltage and response ratio specification
studies, a second-order model in the direct axis is a minimum requirement. Improved model detail, in
representation of the differential leakage reactance (Lf1d), along with representation of the field winding
terminals is definitely suggested. This is particularly significant for turbo-generators when damper bars or
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
3)
4)
5)
IEEE Std 1110-1991
other conducting materials are used as part of, or under the field slot wedges, or where magnetic retaining
rings are employed [6].
Under the conditions of (2) and (3) of Section Auto, a complete third-order model, Model 3.3, should be
considered, provided that appropriate quantities are available. If this type of information cannot be easily
obtained from the generator manufacturer, it can be derived from frequency-response measurements.
For most hydraulic machines, a second-order model for the direct axis is appropriate. For most studies, only
a single-order model is required for the quadrature axis (Model 2.1) [B3]. For most studies, only one q-axis
transient reactance is provided for such machines (usually called X″q not X′q ). On occasion, a second-order
model (Model 2.2) may be appropriate for the quadrature axis, depending on damper bar, pole-face, and
interpolar space construction details. [2]
In many stability studies, a number of tests may be carried out initially to determine a feeling for the overall
dynamic response of the power system. These screening studies are often performed using a simple model
such as “voltage behind transient reactance” for virtually all machines. More detailed studies based on
recommendations (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 in
Section 3.2.2.
As noted in Section 3.2.2, in such a time span, underload tap changing and capacitor switching or line restoration
procedures and switching can come into action. However, prime mover (boiler and reactor) or energy supply dynamics
are also very much in the picture, to change mechanical power inputs. These latter aspects tend to overshadow the
electrical performance of generators, and their associated modeling requirements. In this time frame, they appear to be
of secondary importance.
When switching does take place in long-term studies, the program may switch back to a “normal” transient stability
study 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 domain
simulation tools, (such as a transient stability program) this is very time consuming and inefficient. For most smalldisturbance 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 extent
in Chapter 5) and the torque/power equations. (A third nonlinearity, the “speed voltage” term, has been neglected by
using 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, the
equations are prelinearized when the computer code is written. In the second method, the computer code describes the
nonlinear equations and is used for both time simulation, and small signal analysis. Special provision is made to
linearize about an operating point by repeated calls from a state matrix building subroutine. From an initial steady state
condition, each of the states and inputs is individually perturbed slightly and the effect on the state derivatives and
outputs are measured. The state matrices are built one column at a time. When considering this method, the particular
computer code being used provides for user selection of the saturation representations. For example, saturation could
be 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 in
Section 3.4.
Copyright © 1998 IEEE All Rights Reserved
19
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
The modeling of generator controls is critical for small signal stability assessment. It is very important that the small
signal characteristics of excitation systems and power system stabilizers be modeled correctly, and in detail. In many
cases, it is just as important to have good models of these control systems as it is to have a good model of the
synchronous machine. The successful tuning of these control systems can be quite dependent on the quality of the
synchronous machine model.
In order to properly account for the generator-excitation system interaction, generator Model 2.2, for turbine
generators, and Model 2.1, for hydraulic units, are necessary. The initial conditions of generator loading and terminal
voltage, and, hence, field voltage and current initialization algorithms have a significant impact on small signal
stability.
Large interconnected power systems often consist of groups of strongly interconnected generators coupled together
through relatively weak transmission systems. Characteristic of such systems are low-frequency inter-area oscillations
that represent the electromechanical energy interchange between groups of generators. The oscillations are global
rather than local and in consequence, for accurate simulation, the dynamic models used throughout the system should
have a consistent degree of detail [B6]. The concept of a study system containing detailed generator models and a
simply modeled external system is not valid for inter-area small disturbance stability studies. The same comments
apply to excitation system and governor models for generation control. Reference [8] is an example of a large-size
recently 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 During
Transient Processes and Asynchronous Starting, Part II, Salient Pole Machines,” Brown Boveri Review, vol. 57, March
1970.
[3] Canay, I.M., “Physical Significance of Sub-Subtransient Quantities in Dynamic Behavior of Synchronous
Machines,”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 FiniteElement 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 to
Nanticoke 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 Paper
38–01, 1986.
[8] Kundur, P., Rogers, G.J., Wong, D.Y., and Lauby, M., “A Comprehensive Computer Program Package for SmallSignal 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 in
Multi-Machine Power Systems,” IEEE Transactions on PAS, vol. PAS-99, May/June 1980, pp. 892–901.
20
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
[B3] Kilgore, L. A., “Calculation of Synchronous Machine Constants—Reactances and Time Constants Affecting
Transient 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 Techniques
and 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,” IEEE
Transactions 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 the
Dynamic Behavior of Electrical Power System,” IEEE Transactions on Power Systems, vol. PS-4, Feb. 1989, pp. 129–
138.
Copyright © 1998 IEEE All Rights Reserved
21
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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 Stability
Analyses, 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 and
quadrature axis currents and fluxes. Such calculations can be performed either in engineering-units calculations, or in
per-unit calculations. As is commonly found in engineering analyses, the choice of per-unit notation results in
equations that are more concise. Thus, per-unit equations will be presented in this appendix, although an alternate
formulation in engineering units would be equally valid.
Per electrical torque is given by
T e = ψ d ⋅ iq – ψ q ⋅ id
(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 and
fluxes. The effects of damper and rotor-body currents should be included in transient simulations, since these currents
play a significant role both in determining the magnitudes of the transient electrical torque, and in determining the
damping 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 two
damper windings on the quadrature axis, the per-unit direct and quadrature axis stator fluxes are given by
ψ d = – L d ⋅ i d + L ad ⋅ i fq + L ad ⋅ i 1d
(2a)
ψ q = – L q ⋅ i q + L aq ⋅ i 1q + L aq ⋅ i 2q
(2b)
Similar expressions may be derived for higher or lower order models, or for Model 2.2 with the differential leakage
reactance Lf1d included, as in Fig 2 of Section 2.1.
The differential equations of the stator winding in Park's form are given below.
1 dψ d ω r
e d = – R a ⋅ i d + ------ ---------- – ------ ⋅ ψ q
ω o dt ω o
1 dψ d ω r
e q = – R a ⋅ i d + ------ ---------- – ------ ⋅ ψ d
ω o dt ω o
(3a)
(3b)
Here, ωr is the rotor speed (in electrical radians per second), ωo is the synchronous electrical frequency (radians per
second), and time (t) is in seconds. The above sign convention for voltages, currents and flux linkages also conforms
to the convention that the quadrature axis leads the direct axis, when the field is assumed rotating counter-clockwise
relative to the armature. (See IEEE Std 100–1988 [1] for the definition of “quadrature axis”.) This convention is also
preferred in the widely used stability programs referred to in Appendixes 5B and 5C.
22
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
In large-size transient stability programs, the rate of change of stator flux linkages is usually neglected. This is done
dψ
dt
primarily to be consistent with the practice of neglecting power system network transients. Omission of the ---------q- and
dψ d
---------- terms implies that stator dc offsets during faults or other unbalances are ignored. This simplification permits a
dt
positive sequence representation of both generator and power network. It also results in a conservative answer, i.e., a
transient power limit, for a given disturbance, would be lower with this omission.
The resulting equations, then, would become
ωr
e d = – R a ⋅ i d – ------ ⋅ Ψ q
ωo
ωr
e q = – R a ⋅ i q + ------ ⋅ Ψ d
ωo
(4a)
(4b)
Solving for ψd and ψq, (above) and substituting into (1) for Te, gives
ωo
2
T e = { e d ⋅ i d + e q ⋅ i q + I a ⋅ R a } -----ωr
(5a)
or
ωo
2
T e = { P e + I a ⋅ R a } -----ωr
(5b)
where Pe = electrical power and id 2 + iq2 =Ia2
It is sometimes customary to ignore changes in electrical machine speed, ωr. Exceptions would occur when sustained
system frequency deviates noticeably from the normal 50 Hz or 60 Hz. This is most likely with small isolated power
systems, or where “islanding” conditions suddenly are encountered in a portion of a large power system. Normally ωr
is very close in value to ωo. Assuming then that ωr/ωo = 1.0, Eq 5b simplifies to:
T e = Pe + I
2
a
⋅ Ra
(6)
Neglecting the instantaneous changes in speed tends to give an optimistic result. References [2] and [3] indicate that
neglecting both d ψ/dt terms compensates for the omission of changes in instantaneous speed. Both of these
simplifications appear to be justified, not only from a practical consideration of the orders of magnitude of the
quantities involved, but also due to the fact that measured test results show reasonable agreement with simplified
calculations, as used in Eq 6.
This issue is further discussed in [4]. The authors treat the general case of any machine and then specifically discuss
Park's equations for a synchronous machine, where their arbitrary reference frame of electrical speed is replaced by the
rotor reference frame as in Eq 3a and Eq 3b. They finally conclude that neglecting both the rate of change of flux
linkages in the rotor reference frame, and setting the rotor speed equal to the system electrical speed is required in
order 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.
Copyright © 1998 IEEE All Rights Reserved
23
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
[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 Dynamic
Stability 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,” IEEE
Transactions 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 for
many years. The methods of simulation have been many, and only a limited number of the most frequently used
computer simulation procedures will be considered in the following section. In general, saturation effects in
synchronous generators should be accounted for in stability programs analyses. However, the effects of the usual
saturation approximations do not have a large impact in most cases on system planning or system operation policies or
decisions. But in special cases, the correct representation of the internal excitation and/or the internal generator
loading angles can be significantly affected by saturation. Correct representation of saturation, therefore, can be
important 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 years
when 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 first
AIEE test code for synchronous machines. These procedures now appear essentially in the same form in IEEE Std
115–1983 (R1991) [1]. Other publications and AIEE papers also treat the subject, and discussed the differences in
approach between salient pole (hydraulic) generators and round-rotor turbo-generators [B1], [B5] and [4]. For the first
half 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 has
received much attention as well.
Details of these “standard” methods of calculating field excitation by using saturation factors will be given in
Appendix 5A. The two following subsections summarize the development of saturation methods leading to present
day 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. The
generator 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 calculate
a saturation factor K (or a saturation correction ∆E). K or ∆E are the ratios (or differences) in excitation between the
air 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 saturated
synchronous reactance. Then, after an internal voltage is found, based on stator terminal conditions, this voltage, in
turn is multiplied again by the K factor. Justification for this approach is described in Appendix 5A.
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MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Alternatively, another internal voltage, based on unsaturated synchronous reactance was obtained from a phasor
addition of terminal conditions, plus an increment (∆E) added to obtain the correct excitation EI, in per-unit
calculation. Variations in these two basic approaches constituted the methods used for salient pole machines before
computer stability programs were in common use. More specifically, a Potier reactance was used to find Ep, for any
terminal operating point, and thus to obtain ∆Ep. This is discussed in IEEE Std 115–1983 (R1991) [1] and is still
referred to as the IEEE or Standard Method. See also Fig 5A-1 of Appendix 5A. Occasionally, Xl was substituted for
Xp, and the same process was followed.
Values of Potier reactance suggested by hydraulic machine designers lie between leakage reactance values and
transient reactance values. Typical numbers for Potier reactance, [3] if none are specifically available, can be obtained
from the following approximations:
a)
b)
Xp = Xl + 0.63(X′d − Xl)
Xp = 0.8(X′d)
When saliency was first modeled for computer programs for hydraulic machines, a quadrature axis was located with
respect to the terminal voltage, using the unsaturated quadrature axis synchronous reactance. The voltage for
saturation 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 computer
stability programs. The internal excitation Xadu··ifd, (or Ladu·ifd) is then the sum of several components. This form of
excitation 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 saturated
values, 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 in
the direct axis. One exception is described in [4], which considers saturation at the air gap by calculating an air gap
magnetomotive force (mmf), in space-phase with the air gap flux, and thus with components in both axes. This air gap
mmf. is primarily a function of air gap flux, but with a correction for the amount of generator field current. On
occasion, the effect of the saturation in the quadrature axis was measured, but no design or test data was generally
available to produce a “quadrature axis saturation curve.” As is now widely recognized, saturation in the direct axis
affects principally the excitation (EI), while quadrature axis saturation mostly affects Xq, which determines the internal
operating 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-circuit
saturation curve was used to determine the operating point for which a saturation factor Kd was calculated. While the
Potier reactance proved more than acceptable for hydraulic machines for the purpose of excitation calculation, it was
found that use of Potier reactance in turbo-generators, using the ∆Ep correction described in the IEEE Standard
method 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 as
Reference [4], had also come into use for turbo-generators. The saturation corrections were applied to Xadu, where by
accepted use Xadu = Xdu – Xl. The voltage El, back of leakage reactance Xl., was used to determine a factor Kd, from
the open circuit saturation curve.
Thus, Xadsat = Xadu/Kd and Xdsat = Xadsat + Xl.
Copyright © 1998 IEEE All Rights Reserved
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IEEE GUIDE FOR SYNCHRONOUS GENERATOR
A voltage back of Xdsat located a phasor EQD, which gave a reasonable indication of internal angle, although not a
precise one.
Thus, E QD = E a + jI a ⋅ X dsat
(2)
with all phasors and reactances in per unit. The voltage EQD, multiplied by Kd, then gave the field mmf, EI and a
resultant 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 calculating
the direct axis saturation increment factor. The quadrature axis is treated in a somewhat similar manner. Basic details
of 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 by
specifically adjusting Xadu, and Xaqu elements in the stability models. Both axes are involved in the saturation
calculation procedures. In the first option, Kq is assumed equal to Kd, both being based on the open-circuit saturation
curve. Alternatively, in the second option for round-rotor representation, Kd is calculated from a direct-axis saturation
function. 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 either
saturation curve. The details of the latter method are given in Appendix 5C, including the development of both a directaxis and a quadrature-axis saturation function calculated from on-site measurements, which produce internal phase
angle readings, and readings of field current. The reader may obtain more background on this aspect of using on-site
measurements 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, is
ignored.
Several other methods have been developed to allow accurate prediction of both excitation requirements and internal
angle simultaneously, but some of these are difficult to implement in stability modeling programs or require longer
program 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 for
comparison with measured results and with each other. Use of a quadrature axis saturation function, as suggested in
Appendix 5C shows greater accuracy in predicting generator internal loading electrical angles, than simply using the
open-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 steadystate 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-dependent
stability model. This is true for a number of reasons. First of all, much investigation will be required before timetransient, magnetic field-based or finite element analysis can be developed for large power system representations. It
would 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, one
ought to know four currents as inputs, in each axis. Therefore, there may be 8 values in total of circuit parameters or
inductances to be adjusted for saturation (4 in each axis). Such an approach appears difficult and impractical to
26
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
implement principally due to the lack of input data. As an alternative, emphasis is placed on choosing one or two
important inputs or factors. Recognition is given to the fact that current state-of-the-art methods of accounting for
saturation in this latter way are crude. They are, however, reasonably satisfactory and simulation results from transient
stability studies, using these models, generally agree reasonably well with on-site measurements recorded during
actual disturbances.
The net result is that, as a practical matter, saturation corrections during the simulation process of large disturbances
are generally limited to adjustment of the magnetizing inductances, Lad and Laq, This is done using essentially the
same 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 or
three times normal, that the stator leakage inductance, Ll, should also be adjusted. Appropriate background material is
noted 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 which
these can be accounted for in commercial or widely accepted stability simulations depends to a degree on whether the
computer codes are based on the “time constant and reactance” approach, or alternatively on the use of d- and q-axis
model structure in which the resistances and reactance values of the structure elements of stator and rotor, including
the field circuit, are explicitly known. The differences between these two approaches are discussed below in Section
5.3.2.
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)
2)
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 calculated
by 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 in
Section 6.3.
The machine equations are stated in circuit-model form. The parameters required are the stator and field
leakage inductances, magnetizing inductances, Lad and Laq, and leakage inductances and resistances for each
equivalent rotor body branch. These may be derived from frequency response measurements as specified in
IEEE Std 115A–1987 [2], or by analytical procedures. The field resistance rfd in physical ohms should also
be 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 is
generally customary to use the rated-voltage values. They reflect test conditions where the terminal short-circuit was
applied 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, already
adjusted for saturation, although the degree of adjustment to individual circuit constants cannot be identified or
quantified. As noted in Section 6.3, it is customary for manufacturers to account for the inductance of generator flux
levels on both the rotor and stator leakage inductances. In one company, the value of starer leakage inductance Ll that
is 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. An
examination of the block diagram of Fig 5B-2 shows that T′do, and T′qo are directly affected in stability computations
by adding a saturation component ∆Lad · ifd to the appropriate feedback loop that mathematically describes the rate of
Copyright © 1998 IEEE All Rights Reserved
27
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
change of direct axis field flux linkages (E′q) or quadrature axis rotor flux linkages (E′d). Equation 4 of Appendix 5B
is shown for the direct axis field flux linkage changes. These incremental saturation functions do not influence
subtransient 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 the
saturation algorithm just discussed above. Users of the “time constant and reactance” approach should also be aware
that the values of open-circuit time constants are usually based on the common acceptance of them being calculated as
follows:
L adu + L fd
T ′ do = --------------------------, and
R fd
T ″ do
(3)
L adu ⋅ L fd 
1 
= ---------  L id + ----------------------R 1d 
L adu ⋅ L fd 
(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 is
infinite 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 Laq
are adjusted for saturation at each time step. The saturation function used will be the same one that is used for the
calculation of these parameters when the initial values of excitation and load angle are calculated. The values for Lad
and Laq that are input to the simulation will be unsaturated values. Various saturation methods that have been
developed for adjusting Ladu and Laqu are described in Appendix 5C. In the latest methods, a different saturation
function is used for each axis.
It is noted that in this approach, the values of the transient and subtransient time constants will be adjusted for
saturation through the adjustment of only one component of Eq 3 and Eq 4 above, namely Ladu. The leakage
inductances, 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 simulations
between 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 corrected
from on-line frequency responses (OLFR). The steady-state values of Xadu (or Ladu) and Xaqu (or Laqu) are arbitrarily
inserted into the model structure in place of the SSFR values Lad(o) or Laq(o). Refer to Section 6.2. This is done in
order that the transient stability modeling process will initialize to the correct steady-state operating point when
considering saturation.
Theoretically, the use of small signal circuit constants can be questioned for large disturbance analysis. These
constants are governed by the incremental permeability of the rotor iron [11]. Another pertinent factor is possible
variation in the rotor iron effective skin depth arising from large signal excursions in the flux density, for which the
incremental 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 large
disturbances. The computer results using circuit constants based on “standard” or short-circuit concepts are similar to
those using SSFR-derived constants. In some cases reported on, the models derived from frequency response testing
28
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
even gave a closer agreement to test results in simulations than did the “standard” models. These simulations
duplicated 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 generator
armature and field currents. This means that the permeability describing the behavior of the generator iron can be
considered constant relative to the magnitude of the small-signal swings in flux density. However, (as a result of the
hysteretic nature of iron), the B-H path for small disturbances is different from the path followed for steady-state
operation in a solid iron rotor, that is, the apparent permeability of the rotor is different. This means that generator
small-signal circuit constants are somewhat different from the steady state circuit constants (Ld and Lq). The latter
constants are used solely for small signal study initialization purposes. The rotor permeance is only one component of
the entire generator magnetic circuit, so that effect of the small-signal permeability on circuit constants can be
determined only by detailed magnetic analysis or by test. Thus, the small signal operational inductances (or circuit
parameters) 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. The
continuous curve represents the normal (steady-state) magnetization curve. The small loops show the B-H path for
small deviations. The normal permeability is defined as the quotient B/H at any point on the normal curve. The
incremental permeability is defined as the slope of the incremental loop. This slope depends on signal amplitude; for
simplicity, the limiting value for zero signal is usually used. The loops depicted are those that would result after many
cycles of small signal perturbations. The actual B-H path in arriving at a small signal-steady condition is more
complicated. For conceptual (and analytical) purposes, it is sufficient to assume that for each value of steady-state flux
density in the rotor, there is a corresponding value of incremental permeability. The analysis of small-signal behavior
can be done by replacing the steady-state permeability by the corresponding value of incremental permeability in
every region of the rotor. The incremental permeability is about 1/10 of the normal (steady-state) permeability for
points on the “air gap” portion of the normal B-H characteristic. The so-called “initial permeability,” that is measured
around the toe of the normal B-H curve, is a special case of incremental permeability [11]. This condition applies when
standstill frequency response measurements are made. Under these test conditions, the operating flux density is zero
over the entire rotor cross section. Because the incremental permeability is smaller at high (and normal) operating flux
densities, it would be expected that the small signal inductance parameters would be different between standstill
conditions 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 and
measured) for a turbo-generator. Two pairs of curves are shown; one at zero excitation (standstill), and one for running
on open-circuit under rated-voltage conditions. These latter tests have become known as open-circuit frequency
response (OCFR) tests. Since these characteristics were measured for small-signal excitations around their respective
steady-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. At
the rated-voltage operating point, the value is about 60% of the steady-state value (Ladu). Other than the low-frequency
response below about 1.0 Hz, there is not much difference between the OCFR values and standstill values of Lad for
this particular generator.
Most small signal analysis in power systems is concerned with oscillation modes in the 0.1 Hz to 10 Hz range. When
considering the effects of saturation in general, and based on existing line small signal OCFR test data, it would appear
that the actual small signal value or incremental value of Lad does not affect results significantly in the 0.5 Hz to 10 Hz
spectrum. Hence, incremental saturation effects, even if they are modeled differently from the incremental
permeability concept, should have a relatively minor impact on small signal analyses. The justification for this
statement is that the rotor values of R1d, L1d, R2d, L2d are considered the important factors in the machine
representation under analysis conditions in the frequency range from 0.5 Hz to 10 Hz. This concept has been
confirmed to some extent by the test data referred to above where one 500 mW, two-pole machine had no amortisseurs,
Copyright © 1998 IEEE All Rights Reserved
29
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
while the other 500 mW, two-pole machine had amortisseurs in both axes [6]. The machine without the amortisseurs
is 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 their
steady-state value (Ladu), as verified by OCFR tests [6] and finite element analysis [B3]. For small-signal modeling
and eigenvalue analysis, it is theoretically correct to retain the value of magnetizing inductance obtained from the zerofrequency intercept of the small-signal operational inductance. The OCFR test value is the most appropriate value for
this purpose. OCFR test measurements give a value for Xad, but not for Xaq. However, finite-element analyses suggest
that Xaq is depressed to about the same degree as Xad.
Figure 5 —Schematic Examples of Incremental Minor Loops
30
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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 available
would be to take 60% of the manufacturers value for Ladu and Laqu to use in eigenvalue analysis. It is considered likely
that 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.) obtained
from 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 comments
in 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).
[2] IEEE Std 115A-1987, IEEE Standard Procedures for Obtaining Synchronous Machine Parameters by Standstill
Frequency Response Testing.
[3] Beckwith, S., “Approximating Potier Reactance,” AIEE Transactions, vol. 56, 1937, pp. 813–818.
Copyright © 1998 IEEE All Rights Reserved
31
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
[4] Concordia, C., Crary, S. B., and Lyons, J. M., “Stability Characteristics of Turbine Generators, ”AIEE
Transactions, 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 by
Comparison 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 (Ontario
Hydro), 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 645
mVA 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 Functions
of 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,” IEE
Proceedings, 1979, 126(8), pp. 759–763.
[11] S. H. Minnich, “Small Signals, Large Signals, and Saturation in Generator Modeling,” IEEE Transactions on
Energy 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 Leakage
Reactance, 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. FiniteElement 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 Machine
Parameter 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 of
Congress Catalog No., 48-6164. John Wiley & Sons, Inc., N.Y., New York, second printing, July, 1962.
32
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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 Stability
Analyses, but is included for information only.)
5A.1
As stability techniques developed over the years, more consideration and emphasis was placed on the effects of
saturation on the synchronous machine variables, i.e. Xd, X′d, Xq, X′q. With the advent of digital computers, detailed
models of the excitation system became practical to use. There was a concomitant need to more accurately specify
excitation system ceiling voltages, in practical terms. This, in turn, necessitated an accurate prediction, primarily of
field current, at the commencement of stability time domain studies—the so-called initialization process. Secondly, it
was 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 function
of 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 that
were in popular use for many years. The method proposed in IEEE Std 115-1983 (R1991) [1], a saturation function
or 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, could
be added directly to the phasor EIU, the voltage behind unsaturated direct axis synchronous reactance, or could be
added 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 synchronous
reactance 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.
X
–X
Kp
X
–X
Kd
du
p
du
Thus, X dsat = --------------------- + X p , or, X dsat = --------------------l + X l , 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 through
a point Ep or El on that curve. The extension of the straight line to a value corresponding to a voltage EQD on the
ordinate 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 in
Table 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 direct
and quadrature axes are represented. Note, in particular, the closeness between measured and calculated angles shown
in Table III of Appendix 5E. This may be compared to the approximate angles calculated in Table III of this Appendix.
Copyright © 1998 IEEE All Rights Reserved
33
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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
34
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Table 5A-I —(Test Results)
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°
306.7
-82.6
23.21
1832
47.4°
Test Point #
& EPRI Report #
C - #390
Sample Calculations
Table 5A-II —IEEE Standard Method-555 mVA Turbo-24kV
(Method 1)
EI =
Test
Point
Ifd(M)
M =MEAS
I fd ( C ) – I fd ( M )
----------------------------------I fd
Ep
∆E p
E IU
E IU + ∆E p
Ifd(C)
C = CALC
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
(M)
Table 5A-III —Saturated Synchronous Reactance Method 555 mVA Turbo-24kV
(Method 2)
Test
Point
EI =
Ifd(C)
(CALC)
Ifd(M)
(MEAS)
I fd ( C ) – I fd ( M )
----------------------------------I fd ( M )
Ep
Kd
Xdsat
E QD as a phasor
compared to Ea
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
Copyright © 1998 IEEE All Rights Reserved
E QD ⋅ K d
35
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Annex 5B
Steps Used in a Widely-Used Commercial Stability Program to Account for
Saturation During the Step-by-Step Calculations
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
The equations used in this commercially available transient stability program are shown below for the direct axis. L′d
and L″d values are saturated; Ld and Lad are unsaturated.
L″ d – L l
L′ d – L″ d
ψ″ d = ψ′ d ------------------- + ψ 1d ---------------------L′ d – L l
L′ d – L l
dψ 1d
1
------------- = ----------- ( ψ ′ d – ψ 1d – i d ( L′ d – L l ) )
T ″ do
dt
dψ 1d ( L′ d – L″ d ) 

L ad ⋅ i fd = ψ′ d + ( L d – L′ d )  i d + T ″ do ------------- ⋅ --------------------------2- 
dt

( L′ d – L l ) 
dE′
dψ′ 1d
1
-------------- = ----------q- = ---------- { E fd – L ad ⋅ i fd }
T ′ do
dt
dt
(4)
(4A)
(4B)
(4C)
Similar equations apply for the quadrature axis. The steady-state relationships may be obtained by setting i1d, i1q, i2q
and 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 a
function 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. This
model accounts for saturation changes during the step-by-step process by adding an incremental (∆EI or ∆Lad ·ifd) term
in both axes with some adjustment to account for saliency between the two axes. The data used for saturation is the
open-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)
2)
Enter the current value of |ψ″| as the voltage value for the saturation curve and determine the corresponding
value of Lad ·ifd.
Subtract from Lad·ifd the air gap value of Lad·ifd for the same value |ψ″| to obtain Lad·∆ifd. With the per-unit
system so chosen, the air gap value of Lad·ifd will be equal to the current value of |ψ″|.
3)
Multiply Lad·∆ifd by ---------d to obtain ∆E1d, which is added at the summing junction as shown in Fig 5B-2.
4)
q q
l
- to obtain ∆EIq, which is added at the lower summing junction as shown in
Multiply Lad·∆ifd by -------------------------------
ψ″
ψ″
ψ″ ( L – L )
ψ″ ( L d – L l )
Fig 5B-2.
36
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Additional comments on the assumptions implicit in the above methods of representing saturation changes are given
in Section 5.3, which discusses saturation representation during large disturbances.
Figure 5B-1 —Phasor Diagram for Commercial Stability Program #1
Copyright © 1998 IEEE All Rights Reserved
37
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Figure 5B-2 —Block Diagram for Generator Model from Current Stability Program
38
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Annex 5C
Procedures in a Second Stability Program to Account for Saturation When
Adjusting Mutual Reactances
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
5C.1 Theoretical and Practical Aspects
This second program is described in Reference [1] of this Appendix. Generator equations are written initially by
calculating direct or quadrature axis flux linkages in terms of direct or quadrature axis circuit element values, and
direct 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,
Ψ d = – X d ⋅ i d + X ad ⋅ i fd + X ad ⋅ i 1d + X ad ⋅ i 2d
(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 of
components 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 also
customary to convert ifd and efd per-unit field quantities, which are in the reciprocal system, into nonreciprocal field
quantities. This is done to permit interfacing with excitation system per-unit relationships. These concepts are noted in
Appendix 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 equation
simplifies to
ψ d = – X d ⋅ i d + X ad ⋅ i fd in p.u
(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:
X ad ⋅ i fd = E q + X d ⋅ jI d
(3)
ifd will be in per-unit using the reciprocal system. Then with saturation included,
X adsat ⋅ i fd = E q + X dsat ⋅ jI d
(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.
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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)
aqu
Determine Xqsat where X aqsat = ----------- and Xqsat = Xaqsat + Xl
3)
Determine δ the internal machine angle, located by the phasor EQD, where
X
Kq
E QD ∠δ = E a ∠0° ( I a ⋅ jX qsat )
X
Kd
4)
adu
- is determined from a test
For the same El calculated in step (1) above, find Xdsat, where X adsat = -----------
5)
saturation curve, or from the open circuit saturation curve.
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 generator
quantities. 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 the
magnitude of a vector corresponding to Iq (Xqsat). Note that Iq is the projection of the armature current Ia on the
quadrature 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 on
Fig 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).
E +X ⋅i
i fd – I d
q
l
fd
-.
Solving for Xdsat it follows that X dsat = ----------------------------
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 in
Figs 7A-1 and 7A-2.
Figure 5C-2 shows several Iexq and Iexd points obtained from readings for a turbo-generator at power loadings of
310 mW and 510 mW at various power factors, overexcited and underexcited. Curve fitting procedures may be used to
obtain functions which, when plotted, will fall close to the test points. The shape of the function curves is similar to
that 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 corresponding
to El is presumed to act first in one axis alone, and then in the other axis.The method, while pragmatic, is simple to
apply, and provides reasonable agreement between test and calculation.
40
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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, which
is a function of Xl, armature voltage current and power factor, and also depending upon where the “break point” from
the 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 their
values 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 Stability
Programs,” 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 of
Synchronous Generators from Finite Elements,” IEEE Transactions on Energy Conversion, vol. EC-2, Dec. 1987.
Figure 5C-1 —Phasor Diagram for Commercial Stability Program #2
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Figure 5C-2 —Typical Excitation Test Points 555 mVA Turbo-generator
42
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IEEE Std 1110-1991
Figure 5C-3 —D&Q Axis Saturation Functions
Figure 5C-4 —Typical Q-Axis Saturation Factor
(Expanded Scale)
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Annex 5D
Finite-Element-Derived Steady-State Saturation Algorithms
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
5D.1
Under the program described in Reference [1], a study has been made of the use of nonlinear electromagnetic finiteelement (FE) methods to represent steady-state saturation. Preliminary studies showed that FE methods could properly
predict steady-state load operating parameters under saturated conditions. Methods were developed for extracting
saturated values for the machine reactances, which could be used in conventional phasor triangle calculations to
correctly predict excitation current and load angle. Later effort was devoted to the construction of simple algorithms to
correlate FE-calculated values for saturated reactances. The algorithms are suitable for use in conventional stability
programs 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 Xl
is the d-axis stator leakage reactance. Preliminary examination of FE results indicated that Xl was reasonably constant
over the steady-state operating capability of the generator. On this basis, the fitting was restricted to Xq and Xad an
average 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 Potier
reactance, 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-axis
component of the Potier flux; it turns out to be a small correction to the saturation correction implied by the total flux
function. In equation form, the saturated value for a given reactance becomes:
X sat = X unsat ⋅ F 1 ( E p ) ⋅ F 2 ( ψ qp )
where
(1)
E p = E a + jI a ⋅ X p
(2)
From Eq 2, we note that Ea = Ed + jEq.
Also,
Since
ψ qp = ψ q + I q ⋅ X p
(3)
E d = – ωψ q, – E dp = – E d + I q ⋅ X p ,
(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, and
speed, 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.
44
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MODELING PRACTICE IN STABILITY ANALYSES
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Figure 5D-1 shows the total-voltage saturation function, F1 (Ep), for Xq,and Xad for each of the example generators. In
practice, a table-lookup function was used for these functions. The curves are plots of all the calculated data points
used to construct the function, and have some lack of smoothness, representing the residual scatter arising from lack
of 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 a
quadratic 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 values
were 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
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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
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
Also given in the above Table are the d-axis stator leakage reactances used to construct Xd from Xad and the
unsaturated values of Xq and Xad.
The owners of the generators discussed above measured steady state load angles and field currents over a wide range
of operating points. The values calculated from the algorithm for each generator were compared with the respective
measurements. 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 opencircuit-saturation curve is considered unique. Calculations similar to those performed in this project are not currently
available 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 manufacturers
data. Numerical experiments reported in [1] show that a “generic” function, F1 for Xq constructed to lie in the middle
of Generator X and Generator Y functions of Fig 5D-1, produces differences of about 2° in load angle when used in
place of the exact Generator X function. The same experiments showed the sensitivity of the calculations to the value
of the Potier reactance, similar in Fig 5D-2 that either could be used. Such approximations, when used for an unknown
generator, may be a satisfactory improvement over the results of simpler saturation functions. This idea is not
recommended unless the user takes care to validate the procedure against measured values of steady-state load
parameters.
46
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
5D2 References
[1] EPRI Report, EL3359, “Improvement in Accuracy of Prediction of Electrical Machine Constants, and Generator
Model for Subsynchronous Resonance Conditions,” (Projects 1288-1 and 1513-1). Vol. 3: Development and Benefit
Assessment 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 of
Synchronous Generators from Finite Elements,” IEEE Transactions on Energy Conversion, vol. EC-2, Dec. 1987.
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Annex 5E
Comparison of Certain Existing Methods of Accounting for Saturation
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
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 Appendixes
can be obtained by using the measured information from the EPRI Report EL 1424, Volume II. Recordings of power
reactive power, terminal voltage, field current and measurements of internal angle were taken by a public utility on a
555.555 mVA unit.
Three representative points were chosen and are shown in Figure 5E-1. Equations chosen to calculate excitation
currents are given below.
1)
As in Appendix 5B, where E l = X adu ⋅ i fd = E′ q + i d ( X d – X ′ d ) + ∆( X adu ⋅ i fd (scalar form )
2)
As in Appendix 5C, where X adsat ⋅ i fd = E q + i d ⋅ X dsat (scalar form )
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 machine
loadings:
Table 5E-I —
Test Point #
& EPRI Report #
48
Ea kV
Ifd(M)
Amps
M = Meas
δ int. angle
Degrees (M)
M = MEAS
+200.4
25.00
3615
36.6°
510.0
+11.7
24.13
2911
49.5°
306.7
-82.6
23.21
1832
47.4°
P
mW
Q
mvar
A - #623
506.8
B - #744
C - #390
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Table 5E-II —(Table 5E-II gives the results using the methods of Appendix 5B)
E I = I fd
Ifd(C) A
C=Calc
I fd ( C ) – I fd ( M )
----------------------------------i fd ( M )
2.750
3575
–.011
0.16
2.236
2907
–.001
0.07
1.366
1776
–.031
Xqsat
δ
Calc.
δ Meas. δ Calc.
∆X ad ⋅ i fd
A
1.50
40.1
–3.5
0.31
B
1.65
55.4
–5.9
C
1.76
54.4
–7.0
Test
Point
= X adu ⋅ i fd
NOTE — 1.0 p.u. EI or Ifd or Xadu · ifd = 1300 A.
Table 5E-III —(Table 5E-III gives the results using the methods of Appendix 5C.)
i fd (p.u. of 2366)
Ifd(C) A
C=Calc.
I fd ( C ) – I fd ( M )
----------------------------------I fd ( M )
2.014
1.537
3637
+.006
-0.3°
1.796
1.234
2920
+.003
+0.8°
1.200
.760
1797
–.019
δ Calc.
δ Meas. -δ
Calc.
X adsat·ifd
1.24
36.4°
+0.2°
B
1.34
49.8°
C
1.43
46.6°
Test
Point
Xqsat
A
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.
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, as
well 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, as
compared to factors developed from test measurements, a series of calculations were made. Several test points for
Generator X taken near full load were used. These were reported on in [1] of Appendix 5D. Kd and Kd factors were
developed 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 functions
for that machine were inverted. In addition, factors for the open-circuit saturation curve for Generator X were also
obtained. 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 readings
taken near full load, and are plotted against El the per unit voltage back of leakage reactance, i.e., the per-unit air gap
flux. The F1 and F2 inverted functions are calculated for a much wider range of megawatt loadings. These finite
element functions are based on using Potier reactance, and hence are plotted against Potier voltage. The open-circuit
factor Kocc is plotted against terminal voltage. Hence, the relative values of the various factors cannot be directly
compared.
The main purpose in showing all the curves together is to demonstrate the fact that the direct axis factors all lie above
and to the left of the quadrature axis factors; the latter factors, for any particular air gap or Potier voltage are greater
than the direct axis factors. The break points in the Kd and Kq curves are extrapolated, since limitations in operating the
Generator X unit underexcited did not permit development of test points for values of El below about 0.9 p.u. voltage.
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
The finite element analyses covered values of Potier voltage or Potier flux from around 1.0 p.u. down to virtually zero
voltage.
Figure 5E-1 —Enlarged Portions of Fig 5C-2
Figure 5E-2 —General Comparison—Saturation Functions of Generator X
50
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MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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 synchronous
machine electrical parameters. The stability constants discussed in previous chapters are the unsaturated synchronous
reactances Xadu and Xaqu 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, are
direct-axis values only, since, under open-circuit, steady-state conditions, only the field winding is energized and
carries 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
X′
X ″d
X′
X ′ dv
du
- ⋅ T ′d
from the relationships T ″ do = --------d- ⋅ T ″ d ; and T ′ do = T ′ d ----------
Xdu, the unsaturated direct axis synchronous reactance, is found by performing an open circuit test and a sustained or
steady state short circuit current test. Xdu in per-unit is the ratio of the field current required to produce rated armature
current under the above sustained short circuit condition to the field current required to produce rated armature voltage
on 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. The
actual procedures for testing are described in IEEE Std 115A–1987 [2], but basically involve exciting the stator or field
of a generator when it is at standstill and is disconnected from the usual generator step-up transformer. Hence, the
method is described as Standstill Frequency Response (SSFR) Testing. All testing work to date has been performed on
turbo-generators.
The exciting currents are quite low and the exciting frequencies recommended in the Standard range from .001 Hz up
to between 100 Hz and 200 Hz. By positioning one of the stator windings relative to the field winding in one of two
ways, either direct-axis or quadrature-axis stator operational impedances (and reactances) may be obtained. The direct
axis quantities may be measured with the field open or shorted. Stator to field and field to stator transfer functions or
impedances for the range of exciting frequencies quoted above are also measured. An example of Ld(s), the direct-axis
operational inductance, is shown in Fig 7, and is taken from IEEE Std 115A–1987 [2]. Some brief comments on the
interpretation 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 most
nearly 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, the
incremental permeability at or very close to zero biasing flux density is substantially coincident with the normal
permeability at the toe of the normal B/H curve. The “patching in” of Ladu from the air gap line in the d-axis model
results in a relatively minor correction being made to Lad(0). The value of Ladu from the air gap line is substituted for
Lad(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-state
saturation in most stability programs in the initialization processes, as discussed in Chapter 5. The same comments
also apply to the values obtained for Laq(0). It is currently assumed that the correction factor for the quadrature axis is
proportional to the Ladu/Lad(0) correction factor. The logic for such corrections, for variations in incremental
permeability, is also discussed in greater detail in IEEE Std 115A–1987 [2].
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The actual derivation of the model elements values, from frequency response testing results, is also discussed in the
Appendix 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 in
Section 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 field
transfer impedance, along with the measured value of Lad(0). This effective turns ratio also permits a rotor physical
quantity, 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 rotor
elements are calculated by assuming some starting value for them, and calculating the error between the frequency
response of the resulting equivalent circuit and each measured test point. The value of each undetermined element is
then changed by a small amount and if the error between calculation and test is reduced, the process is continued until
the error begins to increase again. The process is repeated for all other undetermined elements until the error at each
test point for both magnitude and phase, between calculation and test, cannot be reduced further, Ladu is subsequently
used 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. The
frequency range between 0.1 Hz and 10 Hz is often chosen so as to achieve the closest fit between measured data and
model 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 for
parameter fitting.
References below [4, 5, 6] also give additional experimental data using SSFR d- and q-axis models when simulating
on-site test measurements obtained during power system disturbances. Some additional frequency response testing of
machines when running on open-circuit (OCFR) or operating on-line at load (OLFR) has also been done to investigate
52
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IEEE Std 1110-1991
the conditions under which SSFR models ought to be adjusted to compensate for rotational and other on-load
generator 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 two
U.S. manufacturers. While the detailed calculation procedures used by these companies are of course, different, there
are distinct similarities in the steps used to determine stability constants and the meanings ascribed to them. The intent
of this discussion is not to examine detailed calculation methods. Rather, it is hoped that an explanation of the
meanings of the stability constants and basic assumptions implicit in their calculation will prove useful to systems
analysts 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 for
the direct axis, and the quadrature axis quantities have analogous forms. Subscript “i” denotes quantities that would be
obtained from short-circuit tests in which the initial voltage is reduced to a value that produces an initial short-circuit
current equal to rated armature current. Subscript “v” applies to quantities appropriate for large current variations
associated with faults from rated terminal voltage. Open-circuit transient time constants for both axes are also
provided.
In actual fact, most stability studies are carried out with flux conditions (prefault or pre-disturbance) corresponding to
rated terminal voltage. It is agreed that prefault flux conditions are generally more important than the magnitudes of
machine currents (post fault or during disturbances). When X′d is quoted in this guide, rated voltage values of the
transient reactance X′dv are implied, and the subscripts “i” and “v” are employed in this Section solely for
distinguishing subtransient or transient quantities and for accentuating their differences. Calculated values for the
direct axis quantities are intended to match results that would be obtained from the well known three-phase shortcircuit tests described in Section . Generally, manufacturers calculate stability constants for the direct axis by first
calculating parameter values for a circuit structure similar to Model 2.2, and then convert these circuit parameters to
synchronous, transient and subtransient reactances and time-constants. In addition, both manufacturers account for the
influence of varying machine flux levels only on the leakage reactances (stator and rotor circuits), while the mutual
reactances 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 and
effective field leakage reactances is added to the armature leakage. The quadrature axis transient reactances are
calculated using the damper circuit reactances that correspond to particular hunting frequencies. One manufacturer
assumes a hunting frequency of 1 Hz, while the second manufacturer assumes 0.6 Hz. Mutual reactances are ignored
in calculating the subtransient reactances.
The direct-axis open-circuit time-constants provided by the second manufacturer are intended to be consistent with
X
X ′ dv
du
reactances and time-constants that would be obtained from a full voltage short-circuit test., i.e., T ′ do = T ′ d ---------and
T ″ d X ′ dv
T ″ do = -------------------.
X ″ dv
As such, T′do is not identically the field constant. The first manufacturer calculates the direct axis transient open-circuit
time constant from the equivalent circuit, where the leakage reactances reflect high flux levels associated with a full
voltage 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 frequency
response 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 machine
parameters. These methods were first documented in an organized way in 1945. For the direct axis, these are Xdu the
Copyright © 1998 IEEE All Rights Reserved
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
unsaturated synchronous reactance, the direct-axis values of transient and subtransient reactances X′d and X″d, as well
as T′d and T″d, the short-circuit time constants describing the decay of armature currents under three-phase shortcircuit test conditions. Section describes a more recent approach in determining parameters for generator stability
models by using frequency response measurements.
In Section 6.3, two slightly different approaches taken by two U.S. manufacturers are presented that are intended to
provide open-circuit direct axis transient (T′do) and subtransient (T″do) time constants. These open-circuit values of
time constants can be obtained from open-circuit tests, or can be calculated from the rate of change of three-phase
short-circuit current decrements, T′d and T″d, provided by the machine designers. Calculated quantities X′d and X″d are
also available from the same sudden short circuit assumptions.
Based on the above introductory remarks, it would now appear possible for manufacturers to consider supplying
stability data in several alternative forms, noting as well the conditions for which the data applies. These suggestions
are covered under three distinct headings.
1)
2)
Calculation of direct-axis quantities using current “standard” procedures. Time constants and reactances
under prefault flux conditions that yield either “rated voltage” or “rated current” values of direct axis time
constants and reactances. This would be generally consistent with current practices, which means that direct
axis values are based on either assumed or actual test conditions involving a three-phase armature fault
applied from open circuit As noted in Section 6.3, rated voltage values of transient reactance are generally the
ones that should be used in stability studies.
Calculation of quadrature axis quantities using proprietary procedures that may be considered “nonstandard”. 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 direct
axis data. For example, an operational expression could be provided on the basis that
( 1 + sT ′ q )
X ′q
X q ( S ) = X q --------------------------- and T ′ q = T ′ qo ⋅ -------- .
( 1 + sT ′ qo )
Xq
3)
It should be noted, if appropriate, that the expressions are based on the impedance seen in the rotor solid iron
quadrature axis area at some specified per-unit current, and some specified hunting frequency, along with a
knowledge of Xq.
One of the ambiguities in providing information for turbo-generators in this way is apparent: the direct-axis
stability data is based on assumed short-circuit conditions, while the quadrature axis is often based on a
specific value of impedance, as seen at one hunting frequency. The above information for the Xq(s) function
for solid iron rotors could be expanded upon by noting that Xq(s) normally attenuates between 4 to 6 dB per
decade, in the range from about 0.5 Hz to about 5 Hz. Given a steady state value of Xqu and the impedance of
Xq at a specific hunting frequency in the above-mentioned range, Xq(s) can be approximated with a
breakpoint and a 4 to 6 db per decade drop off, for frequencies up to about 5 Hz. A transfer function with 2
or 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 changes
in this axis. (Usually given as X″q and T″qo).
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 firstorder model (1.1) to third-order model (3.3), with Model 2.2 as the most common. They have been only applied to
turbo-generators and may be obtained in two ways:
a)
b)
54
From operational expressions for both axes based on finite element two-dimensional calculations, as
discussed in [B3] of Chapter 5.
From operational expressions for both axes based on frequency response measurements as described in
Section 6.2. Included in the calculations described in (a) and (b) should be advice as to whether the stator
leakage (Xl) and the field leakage (Xfd) values were based on flux conditions corresponding to normal or
reduced operating voltages.
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Concluding Note
In summarizing some of the key issues in Section 6.4, it should be acknowledged that investigations by turbogenerator manufacturers into the dynamic stability properties of turbo-generators in particular go back several
decades. Reference [3] considers the appropriate expressions for describing direct and quadrature axis operational
inductances (or reactances) of turbo-generators as viewed from the stator terminals. One of the conclusions from the
above reference states, in part, that “by a more detailed consideration of the actual rotor configurations, equations
amenable 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’ discussion
all appear in the same issue of the AIEE Transactions of February 1960, pages 1657–1673. Thus, standstill frequency
response measurements were acknowledged as a likely future alternative to the complex magnetic field calculations
considered 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 model
parameters 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 combined
with 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 order
number of equivalent rotor windings).
model parameter values: Synchronous machine model parameters are derived from characteristics of actual machine
behavior. Although this description may take many forms, two basic categories are evident:
1) Test data obtained from measurements, or
2) 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-element
method. Using this technique, it is possible to solve for the details of magnetic flux distributions within these
machines 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 parameter
values. Thus, for example, the same model structure in combination with parameter values obtained by different test
methods and under current machine operating conditions could yield different model representations and the machine
behavior 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 and
linear 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 the
form of a transfer function among others. These representations are identical, provided the following conditions are
met:
1) The model parameters for each form of the model have been determined from the same set of test data or
analytically 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 saturation
modifications should, themselves, be made in a fashion that assures that the behavior of the various forms of
the 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/voltage
representation. 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 the
generator consists of 2 coupled windings. Thus, the variables efd and ifd are the field voltage and current, as reflected
to the armature through this turns ratio, and these variables cannot be measured in a physical experiment. On the other
Copyright © 1998 IEEE All Rights Reserved
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IEEE GUIDE FOR SYNCHRONOUS GENERATOR
hand, the actual variables that are measured are indicated in the equivalent circuit as voltage e′fd and current i′fd. The
transfer 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 transfer
function representation of Fig 9, there are 9 measurable parameters; 6 time constants and 3 coefficients. Thus, it would
appear that by external measurements, one could determine all of the parameters of the equivalent circuit, including
the exact values of the turns ratio and the armature leakage inductance. However, this is not the case, and in fact, only
7 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 of
measurements at the generator terminals that will determine all the values of the equivalent circuit uniquely. One is
always free to pick one parameter value arbitrarily. Note that the resultant model will have uniquely defined terminal
characteristics, 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 making
equivalent circuit representations for synchronous machines. Although it is certainly possible to choose the value
totally arbitrarily without affecting the validity of the resultant model, it is common to choose a value equal to or close
to the manufacturer supplied leakage inductance value which is often based upon an analysis of the flux distribution
within 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 are
actually equivalent model structures. This can be implied from Appendix 6C which presents some of the equations
which 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 Standstill
Frequency 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 by
Comparison 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 by
Ontario Hydro, Dec. 1980.
[6] Schwenk, H. R., “Deriving Synchronous Machine Models from Frequency Response Data,” IEEE Symposium
Publication 83THO 101-6-PWR.
[7] Umans, S. D., Mallick, J. A. and Wilson, G. L., “Modeling of Solid Rotor Turbo-Generators Part I Theory and
Techniques,” 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 Turbine
Generators,” 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 - Example
of Model Derivation and Use in Digital Simulation,” IEEE Transaction on PAS, PAS-97, vol no. 1, Jan./Feb. 1978.
56
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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)
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Annex 6A
Determination of Direct-Axis Parameters from Test Results
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
The most commonly accepted method of obtaining synchronous machine parameters from the short-circuit test is
presented in IEEE Std 115-1983 (R1991) [1]. It should be noted, however, that this method of obtaining parameters
presupposes a model of the synchronous machine. This model is a direct and quadrature axis representation with two
time 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 a
computer simulation of a three-phase short circuit from no-load and rated voltage. It can be seen from Fig 6A-1 that
each phase contains an alternating component and a direct component of current. Since the short circuit is from no
load, there will be no initial quadrature axis component of flux, and it has been shown that the symmetrical component
of the current (i.e. the short-circuit current minus the dc offset) obeys the expression:
E –t ⁄ T ″d
E
E
E –t ⁄ T ′d  E
I = --------- +  -------- – --------- ε
+ --------- – --------  ε
 X ″ d X′ d 
X du  X ′ d X du
(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 the
symmetrical 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 the
third term of Equation 1 (the subtransient component) will usually be negligible after a few cycles. If we now :subtract
off the steady-state value, we have the expression
–t ⁄ T ′d
E
E
I – I steady state =  -------- – ---------  ε
 X ′ d X du 
(2)
which is valid a few cycles after the short circuit, i.e. when the subtransient term becomes negligible. Taking the
natural logarithm of Eq 2, we obtain
E
–t
E
In ( I – I steady state ) = In  -------- – ---------  ⋅ ------- X ′ d X du  T ′ d
(3)
which is the equation of a straight line when plotted on semi-log paper. Figure 6A-3 shows a plot for this example. The
transient 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
1.0
------2.0
or
E
--------- .
X du
Then
1
--------- = 2.8 + 0.5 and X ′ d = 0.303 p.u
X′d
58
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Figure 6A-1 —Three-Phase Short-Circuit Currents From No Load
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IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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 of
Eq 1 or, in other words, “curve 1” minus “curve 2” in Fig 6A-3. Plotting the results on semi-log paper we obtain from Fig
6A-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.
1.0
E
From Eq 1 at t = 0, I = -------, since ε0 = 1, so that X″d = 0.251 p. u.= ---------- .
3.98
X'' d
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 represents
the transient component of current. In one time constant period, the current will decrease to 1/ε or .368 of its initial
value. In the given example, the transient current decreases from 2.8 p.u. to 2.0 p.u. in 0.5 seconds.
2.0
– 0.5
Therefore, ------- = 0.714 = ε−0.5/T′d; ln(0.714) = ---------- and T′d = 1.486 s.
2.8
T' d
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 subtransient
current decreases from its initial value of 0.68 to 0.09 in 0.1 s.
60
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MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
0.09
– 0.1
Therefore, ---------- = 0.132 = ε−0.1/T″d; ln(0.132) = ---------- and T″d = 0.049 s.
0.68
T'' d
Figure 6A-3 —Symmetrical Fault Current Minus Steady-State Value
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61
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Figure 6A-4 —Subtransient Component of Fault Current
62
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MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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 Stability
Analyses, 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 what
conditions on-line frequency response (OLFR) testing can be found useful. Such tests are necessary to provide a more
realistic source of information on whether SSFR test parameters are acceptable, to reflect actual generator operational
conditions. 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. The
reader 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, over
another limited range of frequencies. The machine is on open circuit, running at rated speed, and with its stator voltage
at between 0.5 and 1.0 p.u. of normal voltage. This is reported on in the same publication as cited above [3]. This
procedure has been referred to as open-circuit frequency response (OCFR) testing. The range of exciting frequencies
is from about .01 Hz to 10 Hz. This is a broader range than that found practical for OLFR testing. It should be
recognized that this latter type of testing provides data only for the direct axis. It has been determined, however, that
much useful information about details of rotor direct axis damper construction can be obtained from both OLFR and
OCFR testing.
As noted in Section 5.3, the derivation of either SSFR, OCFR or OLFR parameters from test data is influenced by the
incremental permeability of the generator iron paths. The small signal operational inductances, especially for Lad (and
L aq), are not much different between these three conditions, except at frequencies below about 0.1 Hz. For the rotor
equivalent circuit values (R1d,L1d etc.), variations in the rotor noneffective skin depth definitely play an important role
in the generator small-signal model representation. Analytical adjustment of SSFR rotor equivalent circuit values to
values 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 response
is different from the standstill response, and the running, or on-line circuit constants should be used in these cases. This
phenomenon has been identified with, among other factors, end-ring contacts that are made at running conditions, but
which may be open at standstill; the phenomenon should not be confused with changes in saturation between standstill
and running conditions. The conductivity and length of wedges used to contain the field winding has also been cited
as another feature of rotor construction that effects OLFR data. A third and, most likely, very important consideration
is 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 normal
to avoid excessive stator currents. The transient and subtransient reactances and time constants so obtained are
presumed to reflect unsaturated generator conditions.
Another group of switching tests, recently used, permits the terminal voltage to be in the normal operating range so
that saturation effects can be included in generator response.
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IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Instead of introducing a sudden change in terminal voltage by a three-phase short circuit, the machine is subjected to a
step change of armature current by either opening or closing the main breakers. Load rejection tests, i.e., opening the
breaker, produce a change in current that is equal to that at the initial pre-disturbance condition. By selecting the loading
for 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 lowering
the excitation tc produce or absorb reactive power. To align the stator current in the quadrature axis requires a
measurement of the rotor angle. For this test, the generator power output is normally set at about 0.25 p.u. of the rated
mVA and the excitation is lowered to absorb reactive power. The current is aligned on the quadrature axis by adjusting the
excitation 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 statorvoltage quantities are recorded. The change in stator voltage will provide information on the machine's response to
change in current for the direct or quadrature axis,depending on the alignment of the stator current to either axis. For
the direct axis test, the field current response provides additional information, and it is usually the main source of test
data to determine whether a second- or third-order model is needed. It is difficult to determine the order from just the
changes in stator voltage. For the quadrature axis, the field current ideally will not change, and the recording of this
quantity 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 time
constants and reactances as initial estimates. However, the final model may be obtained by a series of simulation cases to
reproduce the test response, since on-load saturation effects should be included. The process using the simulation runs
consists 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. For
certain excitation systems this is not possible, in excitation systems for instance whose source is an alternator. The test
requires recording of the field voltage, whose response is included in the simulation. However, some excitation
systems 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 under
loaded conditions.
6B.3 References
[1] Coultes, M. E., Kundur, P., and Rogers, G.J. “On-Line Frequency Response Tests and Identification of Generator
Models.” 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-generator
Model 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 to
Nanticoke 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-L
5736, Aug 1988.
[6] “Determination of Synchronous Machine Stability Study Constants,” EPRI Report, EL1424, Vol 3. (Power
Technologies Inc.), May 1980.
64
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
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 since
is 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 be
referred 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:
a 1 = L d { R fd ( L 1d + L f 1d ) + R 1d ( L fd + L f 1d ) } + L ad ⋅ L t ( R 1d + R fd )
----------------------------------------------------------------------------------------------------------------------------------------------------------------------L d ⋅ R 1d ⋅ R fd
a 2 = L d { L f 1d ( L 1d + L fd ) + L 1d ⋅ L fd } + L ad ⋅ L l ( L 1d + L fd )
--------------------------------------------------------------------------------------------------------------------------------------------------L d ⋅ R 1d ⋅ R fd
( L ad + L f 1d + L fd ) ⋅ R 1d + R fd ⋅ ( L ad + L f 1d + L 1d )
b 1 = ------------------------------------------------------------------------------------------------------------------------------R 1d ⋅ R fd
( L ad + L f 1d ) ⋅ ( L 1d + L fd ) + ( L 1d ⋅ L fd )
b 2 = --------------------------------------------------------------------------------------------------R 1d ⋅ R fd
(1)
From a1, a2, b1 and b2, the following relationships can be developed that define the first two equations of the transfer
function 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
a
b
1 2
1 2
T 1 = ----1- + --- a 1 – 4a 2 T 3 = ----1- + --- b 1 – 4a 2
2 2
2 2
a1 1 2
b 1 2
T 2 = ----- – --- a 1 – 4a 2 T 4 = ----1- – --- b 1 – 4b 2
2 2
2 2
(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 transfer
function 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.
T1
L′ d = L d  ------ ≡ L d
 T 3
T′ d 
 -------- T′ do
Copyright © 1998 IEEE All Rights Reserved
(3)
65
IEEE Std 1110-1991
T2
L″ d = L′ d  ------ ≡ L′ d
 T 4
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
T″ d 
 --------- T″ 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-ToTime Constant and Inductance Form
The equations of Section 6C1 are completely adequate for translation of model data into transfer function form, and
along 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 detailed
expressions 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 equations
for 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.
L ffd T 11d
T ′ do + T ″ do = --------- + ----------- ≈ T ′ do′
R fd R 1d
T ″ do
(5)
L ffd ⋅ L 11d – L 2
1
ad
= ------------------------------------------ ⋅ -----------T ′ do′
R fd ⋅ R 1d
(6)
In this iterative process, both T′do, and T″do are adjusted following replacement of L11d/R1d in Eq 5 with the value of
Tdo 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 and
T′d, in terms of d- and q-axes model elements can also be derived, thus avoiding any iterations. For example,

1
2
T ′ do = ---  ( T ffd + T 11d ) + ( T ffd + T 11d ) – 4L∗ fd 
2

(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 under
conditions where the terminals are short-circuited. Figure 3 of Chapter 2 is an example of such a second-order
equivalent circuit model, where Lf1d is omitted, and Eq 8 and Eq 9 are based on this omission.
L ffd L 11d L ad 2 ( R fd + R 1d )
--------- + ----------- – ---------- + ----------------------------R fd ⋅ R 1d
R fd R 1d
Ld
L′ d = L d -------------------------------------------------------------------------------------L ffd L 11d
--------- + ----------R fd R 1d
66
(8)
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
1
L″ d = L l + --------------------------------------1
1
1
-------- + -------- + -------L ad L fd L 1d
(9)
While the above expression for L″d looks cumbersome, it can be simplified considerably by assuming in the transient
period that R1d approaches infinity (∞).
In addition, the time constants noted in Eq 5 and Eq 6 can be simplified by assuming that Rfd = ∞ during the
subtransient period and R1 = ∞ during the transient period. The time-constant and inductance expressions then
become, under the above conditions, more recognizable, such as the following:
L ffd
L ad ⋅ L fd
T ′ do = ---------- and L′ d = L l + -------------------R fd
L ad ⋅ L fd
T ″ do
(10)
L ad ⋅ L fd
1
1
- and L″ d = L l + --------------------------------------= ---------  L ld + -------------------
R 1d
L ad ⋅ L fd
1
1
1
-------- + -------- + -------L ad L fd L 1d
(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-unit
measurement, 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 is
substituted for Lfd; L2q is substituted for L1d; and R1q is substituted for Rfd; R2q is substituted for R1d; Ll remains
unchanged.
Use of these alternative quantities is permissible for any values of quadrature axis elements for the first two methods
discussed above. For the approximate method, the smaller of R1q or R2q should be used, along with the corresponding
inductance 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 been
tested by standstill frequency response methods. Some examples are simple second-order d- and q-axes models, while
other equivalent circuit models derived from the direct axes values of Ld(s) and sG(s) for these machines contain
values 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 were
determined, referred to the stator:
d-axis rotor
Lfd
0.2434
Rfd
0.00051
L1d
0.03611
R1d
0.07550
NOTE — The above data from SSFR testing
Copyright © 1998 IEEE All Rights Reserved
L1q
L2q
q-axis rotor
1.0738
R1q
0.1695
R2q
.01118
.06063
67
IEEE Std 1110-1991
Machine Constant
T″do
T″do
(1) Method of App.6C1.
10.04
.0201
(2) Closed Form Method
of App.6C2.
10.04
.0201
(3) Approximate Method
of App.6C2.
10.20
.0202
L′d
L″d
.411
.334
.413
.334
.414
.334
T′qo
T″qo
.691
.0335
.691
.0335
.645
.036
L″q
L′q
.796
.334
.795
.334
.849
.341
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 values
determined, referred to the stator.
Lfd
L1d
L1d
NOTE
68
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
d-axis rotor
0.04720
Rfd
0.000894
0.0001
R1d
0.00859
0.2266
— The above data from SSFR testing
L1q
L2q
1.403
0.3402
q-axis rotor
R1q
R2q
0.01243
0.04928
Machine Constant
T″do
T″do
(1) Method of App.6C1.
5.69
.0057
(2) Closed Form Method
of App.6C2.
5.69
.0057
(3) Approximate
Method of App.6C2.
5.16
.0062
L′d
L″d
.459
.447
.461
.447
.461
.446
T′qo
T″qo
.668
.0535
.668
.0535
.622
.058
L″q
L′q
.914
.482
.915
.482
.977
.481
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
Example III
IEEE Std 1110-1991
Machine B — 2-pole 270 mVA, 18kV.
Given parameters: Ll = .16; Ladu = 1.821; Laqu =1.821. The following rotor parameter values were
determined, referred to the stator.
Lfd
L1d
Lf1d
NOTE
d-axis rotor
0.0733
Rfd
0.00105
0.0001
R1d
0.00713
0.0402
— The above data from SSFR testing
L1q
L2q
q-axis rotor
0.4465
R1q
0.0838
R2q
0.0055
0.0267
Machine Constant
T″do
T′do
(1) Method of App.6C1.
5.56
.023
(2) Closed Form Method
of App.6C2.
5.56
.023
(3) Approximate
Method of App.6C2.
4.89
.026
L″d
L′d
.253
.199
.258
.199
.230
.199
T″qo
T′qo
1.253
.040
1.253
.040
1.103
.045
L″q
L′q
.474
.228
.489
.228
.531
.228
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 were
determined, referred to the stator:
Lfd
L1d
Lf1d
NOTE
d-axis rotor
0.0153
Rfd
0.00127
0.0058
R1d
0.0065
0.1627
— The above data from SSFR testing
Copyright © 1998 IEEE All Rights Reserved
L1q
L2q
q-axis rotor
0.9144
R1q
0.2328
R2q
0.0108
0.0366
69
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
(2) Closed Form Method
of App.6C2.
4.59
.0072
(3) Approximate
Method of App.6C2.
3.85
.0086
Machine Constant
T′do
T″do
(1) Method of App.6C1.
4.59
.0072
L′d
L″d
T′qo
T″qo
.319
.314
.721
.0526
.322
.314
.721
.0526
.323
.314
.635
.0597
L′q
L″q
.668
.329
.684
.329
.753
.329
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 were
determined, referred to the stator:
Lfd
L1d
Lf1d
NOTE
d-axis rotor
0.0153
Rfd
0.00127
0.0058
R1d
0.0065
0.1627
— The above data from SSFR testing
L1q
L2q
q-axis rotor
0.9144
R1q
0.2328
R2q
0.0108
0.0366
Machine Constant
T′do
T″do
(1) Method of App.6C1.
4.15
.018
(2) Closed Form Method
of App.6C2.
4.15
.018
(3) Approximate
Method of App.6C2.
3.81
.020
L′d
L″d
.335
.262
.339
.262
.347
.262
T′qo
T″qo
1.00
.036
1.00
.036
.92
.040
L′q
L″q
.564
.287
.576
.287
.613
.287
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 using
Eq 1 and Eq 2 of Appendix 6C.1, as opposed to the closed form expressions of Appendix 6C2. In general, the
equations of Appendixes 6C.1 and 6C.2 produce results that are almost identical, especially for the direct axis.
70
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
The assumptions regarding R1d= ∞ during the transient period and Rfd = 0 during the subtransient period appear
generally to be quite reasonable for the direct axis for the turbo-generator examples given. However, similar or
analogous assumptions regarding R2q = ∞ during the transient period and R1q =0 during the subtransient period are
more 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 using
the 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 Parameters
for 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], the
per unit field current required to produce 1.0 per unit generator terminal voltage open circuit on the air gap line is equal
to 1.0/Xadu, where Xadu is in per unit. In other words, base excitation current ifd produces a voltage on the air gap line
of Xadu (Ea base). The nomenclature used here for field quantities is ifd (current), efd (voltage) and Rfd (resistance). Ea
is 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-axis
open-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:
I fd (nonreciprocal) = X adu ⋅ i fd (reciprocal)
(1)
The overbars are included here mainly to stress the per-unit concept. Note that upper case nomenclature is used in the
nonreciprocal 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 efd
in the reciprocal system is small, usually between 0.001 and 0.002 p.u. for rated loading conditions. This concept is
expanded 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,
X adu
----------- so that
R fd
so that
Copyright © 1998 IEEE All Rights Reserved
71
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
 X adu
E fd = e cfd  ------------
 R fd 
(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 is
EI, 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, all
produce 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 used
during the step-by-step stability calculations. In this and some other stability programs, changes in speed (ω) are
usually 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 generator
models. That is, one per unit excitation system voltage is defined as the field voltage, corresponding to the excitation
current, 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 for
their 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 in
ohms. In synchronous machine calculating procedures, Rfd is usually a per-unit quantity. The next section contains
some pertinent guides on obtaining values of field resistance.
Base impedance of the field, referred to the stator is:
machine VA
Z fd = -------------------------------------------------------------2
( { base – field – A(dc) } )
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 the
physical field resistance rfd. The emphasis for an accurate knowledge of field resistance for the nonreciprocal system
is founded on the need to calculate base field voltage. This is usually obtained by multiplying rfd times the base field
current 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 system
performance 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 correct
values of field resistance in the reciprocal system is its influence on the value of T′do, which is important in determining
the 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-circuit
time constant (T′do). The latter value could be measured from some type of current decrement or short circuit test, and
its value would be assumed to be either.
X fd + X adu
(i) T ′ do = -------------------------R fd
72
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
X fd + X adu X 1 d + X adu
- + --------------------------(ii) T ′ do + T ″ do = -------------------------R fd
R1 d
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 stator
to rotor mutual reactance or rotor leakage reactances are needed to arrive at a calculation per-unit value of Rfd, in the
reciprocal system. The physical value of rfd would then be calculated, knowing the base field impedance.
In summary, rfd, in physical terms, should be obtained by direct measurement. Methods described in IEEE Std 115A1987 [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 Xfd are also described in IEEE Std 115A-1987 [3], in order that a two-port circuit for the
direct axis may be determined. Conversion of physical resistance values to per-unit values is given in the Table shown
in Appendix 7A.
7.3 Field-Circuit Identity
As a result of calculating Rfd, and Xfd, for d axis models, based either on measurements, or on procedures as described
in IEEE Std 115A-1987 [3], it is quite practical to thus retain the identity of the field structure of a synchronous
generator in stability simulations. This retention is particularly desirable in certain types of excitation system and
synchronous machine representations. In such cases, it is a prime requirement that the model should be just as accurate
as seen from the field terminals as from the stator terminals. Accordingly, the two-port approach has become an
accepted procedure in excitation system/generator modeling.
An important consideration in excitation system modeling studies is to account for how the excitation system response
is affected by the presence of damper circuits in a turbo-generator rotor, or how it is affected by the so-called “iron
shielding” effect and induced currents of the rotor field slots and teeth. This is particularly important where thyristor
or other high initial response, high ceiling, excitation systems are being utilized to improve stability limits, along with
the application of power system stabilizers.
Other situations where appropriate field representation is required can occur when the field winding is shorted, or is
open circuited. Although such occasions are relativity infrequent, they should be capable of being conveniently
studied. The need for generator or excitation system designers to accurately establish values of field discharge
resistance is discussed in Reference [4]. The modeling procedures outlined in previous sections, showing a field
structure 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-type
excitation systems, and retaining the field circuit identity when studying these conditions is advantageous. As a
corollary, if the primary interest is only the overall effect of induced field or rotor body currents, as viewed from the
stator terminals, and no exciter action is expected, then the two-port approach is not required. Field current identity is
also 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 is
not possible. This applies to diode exciters as well as to thyristor (or “static”) exciters, both of which are supplied from
an alternating current source.
While the examination of current reversal is not a commonly studied phenomenon, a condition, such as generator pole
slipping, will result in a situation where a negative current may be induced in the field circuit. If the negative field
current is not allowed to flow, a very high voltage may result, depending on the type of rotor construction. In some
cases, the damper winding may limit the resulting voltage to acceptable levels, but in other cases, additional protective
Copyright © 1998 IEEE All Rights Reserved
73
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
circuitry is required. To provide a path for negative generator field currents, “crowbar” circuits (or varistors) are
frequently provided.
In the case of a crowbar, a resistor is inserted across the generator field by thyristors that trigger on over-voltage
created 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 increases
exponentially 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 induced
current in the rotor.
This condition can be simulated by increasing the field leakage inductance to a very large value so as to limit the field
current to zero, whenever field computations require it to be negative. The field leakage inductance is restored to the
normal value when the field current becomes positive. As the paths for the induced rotor currents are provided only by
the amortisseurs, it is important to ensure that the generator rotor model includes their effects by using second or third
order direct-axis models [5].
The above special field modeling provisions are suggested when conditions require them, and are in addition to the
usual 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.
[3] IEEE Std 115A-1987, IEEE Standard Procedures for Obtaining Synchronous Machine Parameters by Standstill
Frequency Response Testing.
[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 Stability
Programs,” 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. 569–572
and (ii) pp 839–841.
74
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Annex 7A
Establishing and Comparing Field-Circuit Relationships—Reciprocal vs.
Nonreciprocal System
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
NOTE — again from Eq 1 in Section 7.1 that for values of per unit field current:
i fd (nonreciprocal)= X adu ⋅ i fd (reciprocal)
or
I fd
i fd = -------X dv
(1)
Overbars are used here to distinguish per-unit notation between the reciprocal and nonreciprocal systems. The direct
axis relationship between terminal voltage and field excitation following from Park's equations (on open circuit) where
fd = 0, q = d = 0, is:
e q = Ψ d = X adu ⋅ i fd
(2)
EI is a symbol used in place of X adu ⋅ i fd and also denotes field excitation.
With e fd = i fd ⋅ R fd , and using the value of i fd from Eq 1, it further follows from Eq 2 that
I fd
E 1 = X adu ⋅ i fd = X adu ------------ , so that E 1 = I fd
X adu
E fd , the excitation system output voltage should equal E I in the steady state. All three of these symbols possess the
same per-unit quantities in the nonreciprocal system. In addition, 1.0 per-unit excitation required to produce 1.0 perunit stator voltage on the pen-circuit air-gap line is the same for EI, Ifd and Efd.
Typical values for reciprocal and nonreciprocal voltage, current and impedance bases and field quantities for a 555.555
mVA, 2-pole turbine generator rated 24 kV are given below in Table 7A-I. Field current (from the open-circuit
saturation curve) required to produce 24 kV on the air gap line, was by extrapolation, 1300 A. Measured field
resistance rfd was 0.10 ohms (corrected to 100 °C).
Xadu 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-unit
system 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 graphical
form. The open-current saturation curve is identical in each case, and is based on data taken from the 555.555 mVA
machine used as an example in Table 7A-I.
Copyright © 1998 IEEE All Rights Reserved
75
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
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
Base Field
Current
Ifd
Rankin Per Unit Systems
Numerical Values
Non-Reciprocal
Reciprocal
2366 amperes
1300 amperes
=
=
ifd
X ad I fd
(measured)
=
=
Field
Resistance
rfd
=
Base Field
Voltage
Efd
=
1.0 pu
0.10 ohms @
100° C
(measured)
130 volts
(1300×0.10)
=
1.0 pu
efd
=
234,807 volts
(2366 × 99.242)
Zfd
=
555.555x10
------------------------------2
( 2366 )
=
99.242 ohms
=
0.10
---------------99.242
6
Base Field
Impedance
pu Field
Resistance
Field Current
rated m V A
kV, p.f.
76
Rfd
=
Ifd
=
=
3900 amperes
(measured)
3.0 pu
390 volts
(3900× 0.10) =
3.0 pu (390/
130)
Corresponding
Field Voltage
Efd
=
Excitation
System Ceiling
Specified
Efd
=
975 volts
=
7.5 pu
ifd
=
0.001010 pu
1.648352 pu
(3900/2366)
efd
=
0.001661
(390/234.807)
or
(1.648352 × 0.001010)
efd
=
0.004152 p.u.
(975/234/807)
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
Figure 7A-1 —Non-Reciprocal Excitation Scale
Figure 7A-2 —Reciprocal Excitation Scale
Copyright © 1998 IEEE All Rights Reserved
77
IEEE Std 1110-1991
IEEE GUIDE FOR SYNCHRONOUS GENERATOR
Annex 7B
Excitation System-Generator Simulation Interfaces
(Informative)
(This Appendix is not part of IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in Stability
Analyses, but is included for information only.)
7B1
Table 7A-I above shows the principal values associated with field- and rotor-equivalent circuit quantities. It should be
reiterated that in some stability programs, the synchronous machine flux linkage, voltage, current and torque equations
are solved using reciprocal system per-unit quantities. On the other hand, it is evident that the excitation system
variables 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-unit
systems 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
78
Copyright © 1998 IEEE All Rights Reserved
MODELING PRACTICE IN STABILITY ANALYSES
IEEE Std 1110-1991
In the stability program referenced below, rotor flux linkages are chosen as the state variables. The equations
describing their solution, for the direct axis, are in per unit:
s(ψrd) = Rd (Md · ψrd − Crd .Id) + (Efd · 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 and
rotor 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 Stability
Programs,” IEEE Transactions on PAS, PAS-102, July 1983.
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