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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 No part of this publication may be reproduced in any form, in an electronic retrieval system of otherwise, without the prior written permission of the publisher. IEEE Standards documents are developed within the Technical Committees of the IEEE Societies and the Standards Coordinating Committees of the IEEE Standards Board. Members of the committees serve voluntarily and without compensation. They are not necessarily members of the Institute. 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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 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 iv When the IEEE Standards Board approved this standard on March 21, 1991, it had the following membership: Marco W. Migliaro, Chair Donald C. Loughry, Vice Chair Andrew G. Salem, Secretary Dennis 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 Background .............................................................................................................................................. 12 Large Disturbances Stability .................................................................................................................... 13 Small Disturbance Stability ..................................................................................................................... 14 Classification Based on Dominant Modes of System Response.............................................................. 14 References ................................................................................................................................................ 14 Bibliography............................................................................................................................................. 15 4. 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 3 Copyright © 1998 IEEE All Rights Reserved 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) Copyright © 1998 IEEE All Rights Reserved 5 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Table 1— Selection of Generator Models of Varying Degrees of Complexity 6 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 A second basic equation in this process is as follows taken from Eq 15 in [6]: VR VI – R RR X RI – X IR – R RII IR 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. Copyright © 1998 IEEE All Rights Reserved 7 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Figure 1—Complete, Third-Order Representation—Both Axes Figure 2—Complete, Second-Order Representation—Both Axes Figure 3—“Standard” Second-Order Model—D Axis 8 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 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. Copyright © 1998 IEEE All Rights Reserved 9 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR 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: Air Gap Armature Excitation System Operational Inductance Air Gap Line Exciter Operational Impedance Permeability Permeability, (Incremental) B) C) 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, 10 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 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. Copyright © 1998 IEEE All Rights Reserved 11 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR [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. 12 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 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. Copyright © 1998 IEEE All Rights Reserved 13 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR 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. 14 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 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. Copyright © 1998 IEEE All Rights Reserved 15 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR 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) 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 2) 18 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 3) 4) 5) 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 ψ q = – L q ⋅ i q + L aq ⋅ i 1q + L aq ⋅ i 2q (2a) (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 q primarily to be consistent with the practice of neglecting power system network transients. Omission of the --------- and dψ dt 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. 24 Copyright © 1998 IEEE All Rights Reserved 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 25 IEEE Std 1110-1991 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) 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. 2) 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 L adu ⋅ L fd 1 = -------- L id + ----------------------- R 1d L adu ⋅ L fd (3) (4) Both of the above time constants are in radians, using values of resistance and inductance in per unit. The above derivations follow from the assumption that Rfd is equal to zero during the subtransient period, and R1d 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. du p du Thus, X dsat = --------------------- + X p , or, X dsat = --------------------l + X l , using leakage reactance. - X –X Kp X –X Kd 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) Ifd(M) Amps M = Meas 3615 2911 1832 δ int. angle Degrees (M) M = Meas 36.6° 49.5° 47.4° Test Point # & EPRI Report # A - #623 B - #744 C - #390 P mW 506.8 510.0 306.7 Q mvar +200.4 +11.7 -82.6 Ea kV 25.00 24.13 23.21 Sample Calculations Table 5A-II —IEEE Standard Method-555 mVA Turbo-24kV (Method 1) Test Point A B C EI = Ep 1.175 1.048 .937 ∆E p .500 .172 .077 E IU 2.448 2.088 1.310 E IU + ∆E p 2.948 2.260 1.387 Ifd(C) C = CALC 3832 2938 1803 Ifd(M) M =MEAS 3615 2911 1832 I fd ( C ) – I fd ( M -----------------------------------) I fd (M) .060 .009 –.016 Table 5A-III —Saturated Synchronous Reactance Method 555 mVA Turbo-24kV (Method 2) EI = E QD ⋅ K d 2.883 2.213 1.344 I fd ( C ) – I fd ( M -----------------------------------) I fd ( M ) .037 –.012 –.046 Test Point A B C Ep 1.175 1.048 .937 Kd 1.426 1.164 1.078 Xdsat 1.478 1.743 1.850 E QD as a phasor compared to Ea 2.022 /39.8° 1.901 /56.8° 1.246 /57.9° Ifd(C) (CALC) 3748 2877 1747 Ifd(M) (MEAS) 3615 2911 1832 Copyright © 1998 IEEE All Rights Reserved 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′ q dψ′ 1d 1 -------------- = ---------- = ---------- { 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) 3) 4) 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″|. d Multiply Lad·∆ifd by --------- to obtain ∆E1d, which is added at the summing junction as shown in Fig 5B-2. q q l Multiply Lad·∆ifd by ------------------------------- to obtain ∆EIq, which is added at the lower summing junction as shown in 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 39 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Step Number 1) 2) 3) 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). aqu Determine Xqsat where X aqsat = ----------- and Xqsat = Xaqsat + Xl - X Kq Determine δ the internal machine angle, located by the phasor EQD, where E QD ∠δ = E a ∠0° ( I a ⋅ jX qsat ) 4) 5) adu For the same El calculated in step (1) above, ﬁnd Xdsat, where X adsat = ----------- is determined from a test X Kd 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). q l fd Solving for Xdsat it follows that X dsat = ---------------------------- . E +X ⋅i i fd – I d 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 41 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Figure 5C-2 —Typical Excitation Test Points 555 mVA Turbo-generator 42 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES 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 43 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 ) E p = E a + jI a ⋅ X p (1) where (2) From Eq 2, we note that Ea = Ed + jEq. Also, ψ qp = ψ q + I q ⋅ X p E d = – ωψ q, – E dp = – E d + I q ⋅ X p , (3) Since (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 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 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 45 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 Generator X Generator Y 0.17 0.17 Xp to fit Xad 0.22 0.19 Xl 0.150 0.122 Xq unsaturated 1.760 1.492 Xad unsaturated 1.71 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 47 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) 2) 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 ) 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 — Ifd(M) Amps M = Meas 3615 2911 1832 δ int. angle Degrees (M) M = MEAS 36.6° 49.5° 47.4° Test Point # & EPRI Report # A - #623 B - #744 C - #390 P mW 506.8 510.0 306.7 Q mvar +200.4 +11.7 -82.6 Ea kV 25.00 24.13 23.21 48 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 ∆X ad ⋅ i fd 0.31 0.16 0.07 = X adu ⋅ i fd 2.750 2.236 1.366 I fd ( C ) – I fd ( M -----------------------------------) i fd ( M ) –.011 –.001 –.031 Test Point A B C Xqsat 1.50 1.65 1.76 δ Calc. 40.1 55.4 54.4 δ Meas. δ Calc. –3.5 –5.9 –7.0 Ifd(C) A C=Calc 3575 2907 1776 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.) Test Point A B C Xqsat 1.24 1.34 1.43 δ Calc. 36.4° 49.8° 46.6° δ Meas. -δ Calc. +0.2° -0.3° +0.8° X adsat·ifd 2.014 1.796 1.200 i fd (p.u. of 2366) 1.537 1.234 .760 Ifd(C) A C=Calc. 3637 2920 1797 I fd ( C ) – I fd ( M -----------------------------------) I fd ( M ) +.006 +.003 –.019 Summary: Both methods, as calculated, give good checks on ﬁeld current. Table 5E-II angle calculations are consistently high, since they are based on use of an open-circuit saturation curve to calculate Q-axis saturation. 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 49 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 Copyright © 1998 IEEE All Rights Reserved 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 d du from the relationships T ″ do = -------- ⋅ T ″ d ; and T ′ do = T ′ d ---------- ⋅ T ′ d X′ X ″d X′ X ′ dv 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]. Copyright © 1998 IEEE All Rights Reserved 51 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR 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 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES 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 du reactances and time-constants that would be obtained from a full voltage short-circuit test., i.e., T ′ do = T ′ d ---------- and X X ′ dv 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 53 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) 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 2) 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) 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. 54 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 55 IEEE Std 1110-1991 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) Copyright © 1998 IEEE All Rights Reserved 57 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 Copyright © 1998 IEEE All Rights Reserved 59 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.= --------- . X'' d 3.98 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 Copyright © 1998 IEEE All Rights Reserved 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 Copyright © 1998 IEEE All Rights Reserved 61 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Figure 6A-4 —Subtransient Component of Fault Current 62 Copyright © 1998 IEEE All Rights Reserved 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. Copyright © 1998 IEEE All Rights Reserved 63 IEEE Std 1110-1991 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 a1 1 2 b1 1 2 - - T 1 = ---- + -- a 1 – 4a 2 T 3 = ---- + -- b 1 – 4a 2 2 2 2 2 a1 1 2 b1 1 2 - - T 2 = ---- – -- a 1 – 4a 2 T 4 = ---- – -- b 1 – 4b 2 2 2 2 2 NOTE — Equations 1 and 2 are out of Reference [7] of Chapter 6. (2) 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 (3) Copyright © 1998 IEEE All Rights Reserved 65 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR T2 L″ d = L′ d ----- ≡ L′ d T 4 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 L ffd ⋅ L 11d – L 2 1 ad = ----------------------------------------- ⋅ ----------T ′ do′ R fd ⋅ R 1d (5) (6) In this iterative process, both T′do, and T″do are adjusted following replacement of L11d/R1d in Eq 5 with the value 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 (8) Copyright © 1998 IEEE All Rights Reserved 66 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 L ad ⋅ L fd 1 1 = -------- L ld + -------------------- and L″ d = L l + ------------------------------------- R 1d L ad ⋅ L fd 1 1 1 ------- + ------- + ------L ad L fd L 1d (10) (11) Note that the expression for L″d is considerably simpliﬁed by the assumptions regarding Rfd = 0 and R1d as are T′do, and T″do. The expression for L″d remains unchanged. The time constants of Eq 10 and Eq 11 can be modiﬁed to include the effect of Lf1d by replacing Lad by Lad *= Lad + Lf1d. As noted previously, time constants will be in radians, when resistances and reactances are in per-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 L1q L2q q-axis rotor 1.0738 R1q 0.1695 R2q .01118 .06063 Copyright © 1998 IEEE All Rights Reserved 67 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Machine Constant T″do T″do L′d L″d T′qo T″qo L″q L′q (1) Method of App.6C1. 10.04 .0201 .411 .334 .691 .0335 .796 .334 (2) Closed Form Method of App.6C2. 10.04 .0201 .413 .334 .691 .0335 .795 .334 (3) Approximate Method of App.6C2. 10.20 .0202 .414 .334 .645 .036 .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 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 L′d L″d T′qo T″qo L″q L′q (1) Method of App.6C1. 5.69 .0057 .459 .447 .668 .0535 .914 .482 (2) Closed Form Method of App.6C2. 5.69 .0057 .461 .447 .668 .0535 .915 .482 (3) Approximate Method of App.6C2. 5.16 .0062 .461 .446 .622 .058 .977 .481 68 Copyright © 1998 IEEE All Rights Reserved MODELING PRACTICE IN STABILITY ANALYSES IEEE Std 1110-1991 Example III 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 L″d L′d T″qo T′qo L″q L′q (1) Method of App.6C1. 5.56 .023 .253 .199 1.253 .040 .474 .228 (2) Closed Form Method of App.6C2. 5.56 .023 .258 .199 1.253 .040 .489 .228 (3) Approximate Method of App.6C2. 4.89 .026 .230 .199 1.103 .045 .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 L1q L2q q-axis rotor 0.9144 R1q 0.2328 R2q 0.0108 0.0366 Copyright © 1998 IEEE All Rights Reserved 69 IEEE Std 1110-1991 IEEE GUIDE FOR SYNCHRONOUS GENERATOR Machine Constant T′do T″do L′d L″d T′qo T″qo L′q L″q (1) Method of App.6C1. 4.59 .0072 .319 .314 .721 .0526 .668 .329 (2) Closed Form Method of App.6C2. 4.59 .0072 .322 .314 .721 .0526 .684 .329 (3) Approximate Method of App.6C2. 3.85 .0086 .323 .314 .635 .0597 .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 L′d L″d T′qo T″qo L′q L″q (1) Method of App.6C1. 4.15 .018 .335 .262 1.00 .036 .564 .287 (2) Closed Form Method of App.6C2. 4.15 .018 .339 .262 1.00 .036 .576 .287 (3) Approximate Method of App.6C2. 3.81 .020 .347 .262 .92 .040 .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 Rankin Per Unit Systems Numerical Values Non-Reciprocal Reciprocal 2366 amperes 1300 amperes = = ifd X ad I fd (measured) = = Field Resistance Base Field Voltage Base Field Impedance pu Field Resistance Field Current rated m V A kV, p.f. Corresponding Field Voltage Excitation System Ceiling Specified Rfd rfd Efd = = 1.0 pu 0.10 ohms @ 100° C (measured) 130 volts (1300×0.10) = 1.0 pu Ifd efd Zfd = = = = = 234,807 volts (2366 × 99.242) 555.555x10 -----------------------------2 ( 2366 ) 99.242 ohms 0.10 --------------99.242 0.001010 pu 1.648352 pu (3900/2366) 0.001661 (390/234.807) or (1.648352 × 0.001010) 0.004152 p.u. (975/234/807) 6 Ifd = = Efd = 3900 amperes (measured) 3.0 pu 390 volts (3900× 0.10) = 3.0 pu (390/ 130) 975 volts 7.5 pu ifd = efd = Efd = = efd = 76 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. Copyright © 1998 IEEE All Rights Reserved 79

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