EFFECTS OF SYNCHRONOUS MACHINE MODELING IN LARGE SCALE SYSTEM STUDIES

Pkul L. DandenoOntario Hydro, Toronto, Ontario

and Chairman, Northeast Power CoordinatingCouncil Task Force on System Studies

Abstract-A study was made of the accuracy of utility power sys-tem stability simulations, as affected by the complexity of the syn-chronous generator models and the data used with the models. Theinvestigation was performed by simulating the operation of a repre-sentative multimachine power system with many combinations of dis-turbance, initial load, system stiffness, model detail and model data.The findings from this study provide guidelines for the selection of thesimplest computational machine models for use in large scale stabilityanalyses, with the goal of minimizing the cost of computer usage whileassuring sufficient accuracy of the results. Recommendations are madefor selecting machine data more suitable for use in stability studies thanthe standard machine data.

INTRODUCTION

An important step in power system planning is the examination ofdynamic and transient stability characteristics of alternative systemdesigns. This examination generally involves the time simulation of thebehavior of many generators and their controls using a digital computerstability program. The computation cost of this process is a function ofthe complexity with which the power systern elements are modeled.This paper discusses how the accuracy obtained in large scale systemstability studies is related to the complexity of models used forsynchronous machines in the system and the data used with thosemodels.

The generator data presently used in stability studies are based onANSI Standards Section C42. 10. The transient and subtransient directand quadrature axis reactances and time constants in the ANSI standardsare based on assumed terminal short circuit conditions. The standardsimplicitly assume that there are two rotor circuits in each axis of themachine; the field, one d-axis amortisseur, and two q-axis amortisseurs.Recently a very complex two-axis set of equivalent circuits of a solidiron turbine generator was presented [ I] . The complexity of this modelis judged too great to allow its practical use in multimachine simulationstudies of large power systems. A model of the solid iron rotor generatorhas been derived [21 from the complex Jackson and Winchester model.The derived model, which is more complex than the models designedto use the ANSI data, has 7 rotor circuits: three q-axis amortisseurs,two d-axis amortisseurs, the field and a mutual impedance between thefield and d-axis amortisseurs. This derived model provides dynamicresponses equivalent to the Jackson-Winchester equivalent circuitmodels for time simulations by accounting for the effects of rotor bodycurrents implied in the equations relating terminal (stator, field) quan-tities.

This paper is based upon a study of the effects of machine model-ing [3] sponsored by a Working Group of the Northeast Power Coordi-nating Council (NPCC). The NPCC Working Group's interest in this areawas prompted by a prior examination of these problems by the IEEEComputer and Analytical Methods Subcommittee and its CAPS WorkingGroup. The NPCC study used the derived 7-rotor circuit model as abench mark to determine the accuracy of several machine models, lesscomplex than the bench mark. The less complex models used two sets.of generator data. One set was the ANSI standard data and the secondset was derived from the frequency characteristics of the Jackson and

Paper T 72 514-8, recommended and approved by the Power System Engi-neering Committee of the IEEE Power Engineering Society for presentation at theIEEE PES Summer Meeting, San Francisco, Calif., July 9-14, 1972. Manuscriptsubmitted February 15,1972; made available for printing May 4,1972.

Ronald L. Hauth Richard P. SchulzGeneral Electric Co.

Schenectady, New York

Winchester model. The study included 79 simulation tests of transientstability and dynamic stability in multimachine and single machine con-figurations. The paper also discusses the influence of other factors onsimulation accuracy, including the equivalent system stiffness, the mag-nitude of the disturbance, the initial power factor of the generator, andwhether the excitation system has high initial response.

SCOPE OF THE STUDY

The influence of generator modeling complexity on the accuracyof stability study results varies with many factors. The dynamic be-havior of a real generator varies in a nonlinear way with the electricalload on the generator. Therefore, a model chosen to represent thegenerator must be accurate over a wide range of operating conditions,e.g., real power output and excitation. A real generator's dynamic per-formance also varies with the transmission system to which it is con-nected and the electrical proximity between it and others of comparablesize in the system. If it is closely coupled to other generators of nearequal size, its observed dynamic behavior when subjected to a testdisturbance will be different from its behavior when electrically remotefrom other generators. These considerations, plus others such as theseverity of disturbance (fault, loss of line or generation) and the type ofexcitation system (conventional-rotating machine or high initial re-sponse) assumed with the generator, influence the accuracy require-ments of a given generator model.

Transmission Systems Modeled

Two transmission system models were used; a detailed 39-bus,46-line, 10-machine model and a single machine connected to an infinitebus through a series impedance line. The second model was derivedfrom the detailed model. The first was the model of a 345 KV trans-mission system typical of the New England area shown in Figure 1a. Itis the same system used by several investigators on the ERC-RP-90 re-search project. [4] All system loads were assumed to be constant im-pedance loads for this study. One of the ten machines was an equivalentpower source representing parts of the U.S.-Canada interconnectedsystem. That equivalent machine was called USCAN and its rotor axiswas chosen as the reference axis for the angles of the nine other ma-chines.

The test unit, which was modeled in nine ways and to whichdisturbances were applied, was assumed to occupy either of two sitesin the system of Figure la: a "remote generation" site or a "closelycoupled" site. The site at Bus 329 was electrically remote from thenine remaining generators, i.e., the impedance "looking into" bus 329was about 0.5 p.u. on the test generator's 800 MVA base. The"closely coupled" site was selected as bus 322 since there the test unitwas electrically closely coupled with the nontest unit at bus 323 withthe impedance "looking into" bus 322 about 0.3 p.u. on the testgenerator's MVA base. The intent of studying these two sites was toshow the effects of differing system stiffness on the accuracy obtainedwith the simpler models. The combination of low natural system damp-ing and the multifrequency behavior of the 10-machine system simula-tions, however, made it difficult to measure the small differences indamping between the results obtained using the different machinemodels. Therefore, a second system configuration was devised which didnot possess a multifrequency behavior.

574

Z"'12 r(a

39-Bus, 46 line, 10 Machine SystemFigure la.

The second system configuration studied was the one-machine-infinite bus system model shown in Figure lb. Three versions of theone-machine system were studied: Nominal line, short line and longline as characterized by their electrical stiffness. In order to relate thissystem model to the 39 bus system, the nominal line version was madeelectrically equivalent to the 39-bus system when looking into that sys-tem from bus 329. The transfer impedance (Z'1'2 as defined in Figurelb) across the single machine system was varied over approximately anine to one range to determine the effect of natural frequency or sys-tem stiffness on the accuracy of the simpler models. This change wasmade by varying Xe and Re from a range of 0 (short line) to .014 +j.0556 (nominal line) to .161 + j.332 (long line) on a 100 MVA base.The load YL was adjusted to match power factor of the 39 bus systemand to allow stable power transfer over the long line.

The one-machine-infinite bus system was also used for studyingthe influence of modeling complexity and data differences on computedcritical clearing times.

Types of Analyses - Disturbances

The study considered numerous multimachine simulations withvarious combinations of generator models, generator data sets, initialloadings, excitation systems and system disturbances made using the10 machine and one machine transmission systems discussed above.Most of these simulations were performed by imposing a stub faultapplied on the high voltage side of the test generator's station trans-former. This stub fault was cleared approximately one-half cycle beforethe critical clearing time and the system response was computed for athree-second period after the application of the fault. The swing curvesobtained using different generator models were then compared and thedifferences due to modeling complexity and/or data differences werenoted.

Small-disturbance time simulations were performed to observewhether inclusion of more or less complex models of the solid rotoreddy current (iron) effects made a significant difference in those cases.The small-disturbance studies were performed using the 39 bus systemwherein the line between buses 16 and 21 was tripped out, without afault, to perturb the system. Swing curves were computed using dif-ferent models and the results were compared.

The effect of modeling complexity and data differences on com-puted values of critical clearing times was also investigated using the

- 8int'o Q0

et M

rT +jXT

e HIGH

e b LOOne-machine-infinite-bus system

Figure lb.

simple one-machine-infinite bus system in Figure 1 b. The system con-figuration assumed was the "nominal line" version of the one machinesystem, which was the electrical equivalent of the "remote" site of the10-machine system, and generator power factor was shifted by modify-ing the initial voltages at the machine terminals and infinite bus.

The time domain simulations described above were performedusing the FACE multimachine program described in Reference 5 andthe multimachine simulation program in Reference 6.

To confirm the damping performance observed in the time domainsimulations, several dynamic stability studies were performed using theprogram described in Reference 7. That program determined the powerlimits of dynamic stability, defined in the P, Q plane; i.e. real powerversus reactive voltampere outputs of test generator. The power systemmodel used in those cases was the nominal line version of the system inFigure lb. The Nyquist criterion was used to detect the onset of instabilityassuming infinitesimal perturbations of the test machine's internal angle.

Prime Mover and Excitation System Models

No prime mover dynamics were considered in any of the simula-tion studies performed. All the generators, test and nontest units, wereassumed to have a constant input torque.

Two excitation systems were considered in the time-domainsimulations performed. The test unit was supplied with either a con-ventional rotating exciter system or a high initial response thyristor-typeexciter. The excitation systems on the nontest units of the 10-machinesystem were the same as those of the ERC study. [41 All of the one-machine infinite bus simulation runs were made with the conventionalexciter on the test unit since early 10-machine runs showed the highinitial response excitation system partly masked the differences be-tween the results obtained with different machine models. Power Sys-tem Stabilizers were not modeled with either system.

Generator Models

The machine models studied were from the most complex to thesimplest:

The bench mark (Model 4) [2] This model included the fieldcircuit and six rotor body circuits, three in each axis. The performancewith this model was used as a reference to asses the accuracy of theother models.

575

High order model (Model 3) This model included two rotor circuitsin each axis; it is the most complex model now available in large scalestability programs, and uses all the presently defined data. It consistedof the field circuit plus one amortisseur in the d-axis and twoamortisseurs in the q-axis.

Intermediate order models (Models IJ2 and 2) These had a singlequadrature axis rotor circuit and a field circuit. They differed by in-clusion of a second d-axis rotor body circuit in Model 2.

Simple model (Model 1) This model had no q-axis rotor circuitsand represented only the field circuit.

TAB

Order in d-axis--.Order in q-axisI

3LE I - MODELS STUDIED

1 2 _ t 3

f f cdl1 d2f, d1d2 ` fkd

0 0 Model 1.1 q, Model 121 Model 22 qlq? 2 Model 33 ql,3q2,'q3 Model 4

f = field circuit; d1, d2 = d-axis amortisseurs;ql.q2nq3= q-axis amortisseurs; fkd = mutual between

field and d-axis amortisseurs

MODEL 11/2 MODEL 3

XRf ..

~~~~~~~RidRfIdi Xad f~ 'd Xad Xld Xf

efd efd

Xt R Rlq RNq R~~~~Iq iqXiq XIq X2q

Data requiredxQ, xd, xd, Td,

x x' T'xq5 q qo

Data requiredx , T' . T"xqxd,xd, xd d do

x~x' q T' qo

Figure 2

The relative differences in order of complexity between themodels are illustrated in Table I. As indicated in Figure 2, implementa-tion of Model 1 /2 requires two principal time constants T'0 and T'qo.In contrast, Model 2 requires the use of TU0 in addition to T'0 andTo due to its added complexity in the d-axis. One goal of this studywas to find a model which gave reasonably accurate results withoutincurring excessive computing costs. Potential computational costsavings were anticipated with use of Model 1%2 since it could be imple-mented without the need for very small integration time steps whichare often required when small time constants such as T"j and T"o aremodeled. Using Model I /2, the computation may be three to five timesfaster if not limited by models of other equipment. Model 1 alsopromised to be inexpensive to use for the same reasons.

The data required for implementing Model 3 is indicated in Figure2. Notice that subtransient time constants and reactances are neededwith that model. Since the numerical values of X' and Xj are nearlyequal for round solid rotor synchronous machines subtransient saliencyis often neglected by setting X" equal to X' for use in stability studies.This assumption affords a computational simplification in the stabilitystudy of multiple machine power systems if system loads can be re'

presented by constant impedances to ground. Models 11/2 and 1, how-ever, do not afford this simplification since saliency effects generallycannot be assumed negligible when either of these models are used. Theresult is that for multiple machine stability studies, the machine equa-tions have to be solved iteratively every time step. At first this appearsto negate the advantage that Models 1,/2 and 1 have over other modelswhich require use of small time steps to insure numerical stability. How-ever, since most stability programs in use today provide for treatmentof nonlinear loads (not constant impedance), iterative solutions arerequired at each time step anyway, even when saliency can be neglected.Therefore, for many stability studies performed the use of Models 1 or1 /2 instead of Models 2 or 3 could result in a net saving in computer-use costs for large scale system studies.

The bench mark Model 4 was derived from design data and adetailed equivalent circuit model. The derivation of this model, itssimulation form and the results of comparisons of it to test results aregiven in Reference 2.

Saturation was represented in Models 11/2, 2, 3 and Model 4 byadjusting the mutual reactances in both axes as a function of the virtualvoltage behind leakage reactance. The method used accounts for satura-tion in both d and q axes, and is described in Reference 2. Since Model1 lacked q-axis amortisseurs, saturation effects for it were assumed as afunction of E', proportional to field flux linkages. The different treat-ments of saturation caused the initial steady-state rotor angle of theModel 1 simulation runs to differ slightly from the steady state anglefor similar conditions with other models.

Generator Data

Two representative generators were chosen for which compre-hensive solid rotor equivalent circuit data was available. One was a two-pole 3600 rpm generator, typical of fossil fuel thermal units. Thesecond was a four-pole 1800 rpm generator typical of nuclear poweredunits. Both units had an 800 MVA rating; the data for the 3600 rpmunit data is given in Table II.

TABLE IIGENERATOR DATA FOR 3600 RPM UNIT

Direct AxisMODEL X X' X" T'1dd d do3

13A2A

IAlB

1. 751.751.751.751.751.751.751. 75

.285

.285

.285

.285

.285

.275

.240

.275

.24 4.0

T"do.029

--- 4.0 ---- 1.68--- 4.0 ---- jl.68.24 5. 2 .011 1l.68.24 5.2 .011 1.68--- 5.2 ---- 11.68--- 5.2 ---- 1.68--- 5.2 ---- 1.68

Quadrature AxisX X' X" T' T"q q q qo qo1.68 .47 .24 .540 .Q53

.47 --- .540 ----

.47 .24 1.96

.24 --- .540.053_ _ _

.47 --- 1.96 ----

Two sets of numerical data were used for each test unit. The firstset was the standard data as defined by ANSI standard C42. 10 andnormally supplied by generator manufacturers. This set includes fourtime constants and seven reactance values: T' O, T' T" T"o and

I it I it~ ~ ~ ~ o'do qo

xd, xd, Xd, xq, xq, xq, xQ. The second or modified set of data was ob-tained by adjusting one or more of the eleven standard constants tomore closely agree with the Model 4 data.

Model numbers shown with the letter A or B appended to themindicate the use of modified values of reactances and time constants;otherwise standard data was used. For example, Model 3A was identicalin form (Figure 2) to Model 3. However, whereas standard data values,T' , T O, Tqo xd, etc. were used with Model 3, some of these datawere modified numerically for use with Model 3A. The data were modi-fied to force the dynamic terminal characteristics Ld(ico) and Lq(ji),

576

of the equivalent circuits for the simpler models to approximate thedynamic terminal characteristics of the bench mark Model 4. Thereasons for considering modified data are considered next.

The terminal characteristics of a synchronous machine can bedescribed in terms of the d-axis and q-axis impedances, as viewed look-ing into terminals of the two circuits.

Zd(s) = Rd(s) + S Ld(s) . . . (1)Z (s) =R (s) + sL (s) .... . . . (2)q q q

These equations are operational expressions of the complex impedancesand s is the familiar Laplace operator. If s is set equal to jco, the rela-tionships describe the variation of impedances Zd and Zq with slip fre-quency ji for sinusoidal slip frequency currents flowing in the equiva-lent circuits. The inductive terms Ld(s) and Lq(s) relate the flux link-ages Qd and 4/q to the currents id and iq' respectively. These relation-ships account for the effects of field and rotor body currents on theterminal (stator) characteristics of the generator.

The operational expressions for Ld(s) and Lq(s) are of importancein describing why the modified data was required. The operationalexpression for Ld(s) can be generally expressed as a ratio of poly-nomials in s. The behavior of the solid iron rotor generator was rep-resented by a solid iron rotor model in the FACE program, Ref. 5.The d-axis inductance of the model may be represented by a ratio offactored polynomials

L (s) X (1+As)(1Bs)(1+Cs)(1-Ds) (3)d s d (1+Tls)(lT2s)(1+T3S)(1T4s)The order of numerator and denominator polynomials determines thed-axis order of Model 4. Model 3 has second order inductances in bothaxes so an approximate second order expression for its d-axis opera-tional inductance, Ld(s) would be:

L (s) = (1+Es)3(1+Fs)approx d(1+T5s).lT6T(4)It is clear that Ld(s) in (4) can never equal Ld(s) in (3) unless, bychance, two of the numerator coefficients equal two of the denominatorcoefficients. However, E, F, T5 and T6 in (4) can be chosen so that a"best fit" of magnitude and phase of Ld(jcO) is achieved over a fre-quency range of interest.

Model 3, using standard data, results in an approximate expressionfor Ld(s) as:

Ld(s)MOD3 Xd (1+Gs)(1+H"s) (5)d OD d(1TT s)(1+T" s .....

where G T and H Txd do xd do

In general the "best fit" coefficients do not all equal the correspondingstandard (Model 3) coefficients. Thus Model 3 (standard) data isgenerally not a "best fit" to Model 4 data. The "best fit" data in Equa-tion 4 was determined from a study of the Model 4 data. That "bestfit" data (E, F, T5 and T6) was called the "modified" d-axis data foruse with a second order d-axis circuit. By following a similar "best fit"approach for arriving at the coefficients of Lq(s) using a second orderq-axis circuit, Model 3A was defined. Similar reasoning was used toestablish modified data for the other models.

Generally, the reactances of the modified data closely match thereactances of the standard data. The differences between some of thetime constants and their "best fit" counterparts were significant. Forexample, it was found that the principal d-axis time constant valuederived from the solid rotor equivalent circuit data was nearly thirty

percent higher than T'0 for both sample generators. The principalq-axis time constant, so derived, was more than double the value ofTo for both sample generators. The results obtained in the stabilitystudies were found to be quite sensitive to these differences in data.

RESULTS

The results presented in this paper are representative of the totalbody of results of the study. [3] For brevity figures are given for the10-machine large disturbance multiswing stability studies and thecritical clearing time studies. Brief descriptions of results are given forthe three other types of analyses described above.

Large Disturbance Multiswing Studies-Ten Machine System

Qualitative comparisons of the 10-machine system results arepresented here with the aid of Figures 3 and 4. Figure 3 shows a com-parison between results obtained with Models 1, 1IV2, and 3 (usingstandard data) and the bench-mark results obtained using Model 4.These results were obtained with the 3600 rpm test unit located at the"remote" site and initially operating at I p.u. power (800 MW). Theconventional excitation system was assumed in use on the test unit.The voltage profile on the system was tilted to obtain the indicatedrange of initial reactive output and consequent range of excitation.Power output was maintained at 800 MW from the test unit. The faultclearing time, shown for each of the operating power factor conditions,was chosen as approximately 0.5 cycles less than the critical clearingtime obtained using Model 4 for each power factor condition con-sidered.

The excitation level on the test generator increases, moving fromleft to right in Figure 3. The accuracy of results obtained using Models1/2 and 3, relative to the Model 4 results, increase as excitation levelincreases. Model 1 also becomes more accurate but its ranking in Figure3c became unclear, because of the different initial angle.

Figure 3 also illustrates that increased model complexity did notalways result in increased accuracy of results when standard data wasused.

Figure 3 illustrates the difference between Model 4 results andthose obtained using the standard machine data in the simpler models.Figure 4 shows results from the same simulation studies except modi-fied values of time constants and reactances, derived from the solidrotor equivalent circuit, were used in place of the standard quantities.

In general, the results obtained using the modified data were incloser agreement to the bench mark Model 4 results than the resultsobtained using standard machine data. Model 3A consistently gaveresults in close agreement with Model 4. Model l1V2A was also a goodapproximation to Model 4 but was more sensitive to changes in excita-tion level than was Model 3A. Model lB was also quite faithful to theModel 4 results except its lack of amortisseur circuit representationcauses it to be less reliable in predicting damping performance. Theevidence for that point is most clear in Figure 4c. No apparent improve-ment was found in damping performance of Model 1 B over that forModel 1. Similar results were observed when the 1800 rpm unit wasstudied with Models 3, 3A and the bench mark Model 4.

Using a high initial response exciter with the test unit had theeffect of masking differences between the results obtained using the dif-ferent machine models. This may be illustrated by comparing Figures 5and 6. In those figures the swing curves (angles) obtained using ModelsIA, 2A, 3A and bench mark Model 4 are shown. The swing curves forthe various models are more closely grouped when a high initial re-sponse exciter was used (Figure 5) than they were with a conventionalexciter (Figure 6). The masking effect is most evident in those figuresfor Model 2A. The large difference between Model 2A results andModel 4 results can be explained in terms of the value of the opera-tional inductances at the swing frequency.

Figures 5 and 6 also show how the computed terminal voltage isaffected by differences in model complexity.

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Large Disturbance Multiswing Studies-One Machine System

To better quantify the effects of machine model on damping ofrotor angle swings, several fault cases were run with the one machine toinfinite bus system. This system has only one mode of rotor angleoscillations and show the damping differences more clearly than themultimodal swings of the multimachine system.

Models 1B, I1/2A, and 4 were studied together with three systemstiffness levels and three test unit initial power factor levels, as discussedin the Scope. The "short line" and "long line" cases were run at unitypower factor, and the nominal reactance line was used at 0.98 p.f.underexcited, unity p.f., and 0.95 p.f. overexcited machine conditions.The "short line" represented a system reactance of about twelve per-cent on the machine's MVA base, which corresponds to a practicallower limit on system reactance. The "long line" case constituted the"weakest" system considered with a tie line reactance of about 2.7 p.u.on the machine's MVA base. Two significant observations were made

from these results. First, the differences in damping performance weresmall, as in the 10-machine simulations. Model 3A was not even con-sidered since the differences between Model 1'/2A and the bench-markmodel were so small. The second point is that these differences changedsignificantly when the system stiffness was varied. In all but one casethe results using Model l1/2A exhibited less damping than Model 4.That one exception was in the case of the "long line" where ModelI1/2A exhibited more damping than Model 4. This apparent reversal inthe damping error exhibited by Model lV2A can be explained in termsof the variation of swing frequency from case to case, and the models'operational inductances.

Critical Clearing Time Studies

The effects of model complexity and data on computed values ofcritical clearing time for fault conditions were studied by applying athree phase stub fault on the high side bus of the test generator's trans-

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former using the single machine infinite bus system with the "nominalline" configuration. The results are illustrated in Figure 7. The ordinatein that figure is a measure of the complexity of the models and corre-sponds to the sum of d-axis and q-axis orders. For each model con-sidered, a vertical bar was drawn with its horizontal distance from theorigin corresponding to the critical clearing time computed using thatmodel. The width of each bar is 0.2 cycles since the resolution of esti-mating critical clearing time was dictated by the integration time step,which was 0.2 cycles.

For each of the three power factors considered, the critical clear-ing time computed using Model 4 was the reference value in the follow-ing discussion. The switching time computed using the simpler models,when standard data was used, were consistently less than the Model 4results. That is, the results using standard data in all simpler modelswere pessimistic. Generally, when the modified data was used the resultswere optimistic, i.e., greater than Model 4 switching time. The singleexception was for Model 1B, which was consistently pessimistic. For agiven model complexity, the errors were smaller with the modified datathan with the standard data.

The relative errors evident when using the simpler models weresmaller for higher levels of excitation. That fact is apparent in Figure 7where the vertical bars are more closely grouped for the overexcitedcases than they are for the unity power factor and underexcited cases.

Small Disturbance, Multiswing Stability Studies

Multiswing time simulations were performed on the 39 bus, 10 ma-chine system where the disturbance was caused by tripping out the linebetween buses 16 and 21 identified in Figure 1 a. In those cases the testunit was assumed to occupy the "closely coupled" site. Only Models 3,3A, 1 1/2A and the bench mark Model 4 were considered in these studies.The purpose for the small disturbance runs was to provide simulationsof 5 seconds in length for analyzing the difference in damping suppliedby the various models.

In these tests, the results with Models 3, 3A and I1/2A were veryaccurate and not significantly different one from another. The resultswere only slightly dependent on the choice of exciter (conventional orhigh initial response) on the test unit.

Damping Studies - Frequency Domain

A quantitative measure of the damping performance, for smalldisturbance analyses, was also sought using the dynamic stability com-puter program described in Reference 7 to find the dynamic stabilitylimits of the one machine infinite bus system and to evaluate thedamping obtained with several models at one realistic operating condi-tion. The absolute stability limit and a family of relative stability limitswas found for each of Models 3, 3A, and I1/2A. Using the "nominal

line" transmission system, the results for the models at unity p.f.were:Model 3A, 2.1 p.u.power; Model 3, 2.05 p.u.; and Model I1/2A, 2.14 p.u.power. The relative stability limits confirmed the relative damping ratiocalculated from the time domain swing curves, 0.04.

CONCLUSIONS

This study has shown that the accuracy of the results obtainedfrom stability analyses depends on both the complexity of the machinemodels used and the selection of the machine data used in these models.The conclusions first discuss the effect of modeling complexity on ac-curacy, assuming standard data (based on ANSI stan'dards) is used withthe models. Secondly, the effect of modifying the machine data for themodel in use is discussed.

Conclusions With Standard Data: When using the standard ma-chine data, a more complex model of a synchronous machine did notalways give more accurate results in the stability studies performed. Forexample, Model 1, the simplest model studied, was shown to be quiteadequate for predicting first swing stability of a system in some cases.In contrast, the high order model (Model 3) was less accurate in thesame first swing study. Model 12 with intermediate complexity gavereasonable accuracy, better than Model 3 and comparable to Model 1.

The study showed that the relative accuracy of the models wassignificantly affected by the initial power factor of the machine. Also,the accuracy of Model 1 in predicting damping in multswing stabilitystudies was inferior to that of the other models.

Conclusions With Modified Data: The accuracy of all modelsstudied was greatly improved when the standard machine data wasmodified to match the results of a frequency response analysis of asolid iron rotor equivalent circuit. The dependence of the accuracy onthe initial loading was also reduced greatly. With the modified data, therelative accuracy of the models increased when model complexity wasincreased. That is, the best results were obtained with Model 3 for bothfirst swing (critical clearing) and multiple swing stability analyses; andthe accuracy of Model 3 was nearly unaffected by initial power factor.

Model 1/2 was found to be very accurate for large disturbance(fault) stability studies and virtually exact for small disturbances (linetrip, without fault). Model 1, even with the modified data, was inferiorto the other models in predicting the long-term damping performanceof the machine being modeled.

RECOMMENDATIONS

When used with the standard data, the machine model calledModel 1/2 in this study appears to be sufficiently accurate for use inmultiswing transient stability studies. Because it requires use of only thelarger time constants T'0 and T'0, Model 1 /2 may allow the use oflarger computational time steps than those needed with models re-quiring the use of Tq0and Tqo. Therefore, Model 11/2 is recommendedfor use in large scale stability studies where significant computer-usagecost savings may be realizable without serious sacrifices in accuracy.

The use of machine data modified to suit the machine modelchosen results in a significant improvement in accuracy of all modelsconsidered in the study. The modified data was derived from frequen' yresponse characteristics of the detailed reference model which are notgenerally available. This suggests that further work is required to definenew data for use in stability studies and to devise general methods forcomputing these data for a specific generator. This effort must beundertaken on an industry-wide basis resulting in a revision of thestandard definitions for data used in power system stability studies.

When used with the appropriate modified data, Model 3 was foundto be an excellent model for the stability analyses performed in thestudy. Model 3 is therefore recommended for use in the more criticalstability studies performed by the industry after the new stabilitystudy data has been defined and made available. These more criticalstability studies might include situations with one or more of theseconditions: a machine operating underexcited or in some other weakly

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coupled situation or, a fault occurring very close to machine understudy or, the machine being modeled is considered to be a significantelement of the system.

ACKNOWLEDGEMENTS

Many people from several organizations contributed to the workfrom which this paper is drawn. The authors gratefully acknowledge theparticular contributions of: the Northeast Power Coordinating Counciland its Task Force on System Studies, which was chaired by Halbert E.Pierce, Jr. during the time the work was started and accomplished; tothe NPCC Review Committee with which the authors worked: ErnestG. Neudorf, Edward M. Gulachenski and George C. Loehr; and to engi-neers of the General Electric Company for their advice. Dr. PrabhaKundur of Ontario Hydro also assisted the review committee byproviding results for several of the tests.

REFERENCES

[1] W. B. Jackson, and R. L. Winchester, "Direct and Quadrature-AxisEquivalent Circuits For Solid-Rotor Turbine Generators," IEEETransactions on Power Apparatus & System, vol. PAS-88, No. 7,July, 1969, pp. 1121-1136.

[2] R. P. Schulz, W. D. Jones and D. N. Ewart "Dynamic Models ofTurbine Generators Derived from Solid Rotor Equivalent Circuits,"paper submitted for the 1972 IEEE Summer Power Meeting.

[3] R. L. Hauth and R. P. Schulz, "Effects of Synchronous MachineModeling in Large Scale System Studies," final report for North-east Power Coordinating Council (NPCC) Task Force on SystemStudies, System Dynamic Simulation Techniques Working Group,Dec. 1, 1971.

[4] J. M. Undrill, and A. E. Turner, "Construction of Power SystemElectromechanical Equivalents by Modal Analysis," IEEE Transac-tions on Power Apparatus and Systems, vol. PAS-90, no. 5, Sept./Oct. 1971, pp. 2049-2059.

[5] D. N. Ewart, R. P. Schulz, "FACE Multimachine Power SystemSimulator Program." PICA Conference May 18-21, 1969. Confer-ence Record No. 69 C1-PWR, pp. 133-153.

[6] Prabhashankar Kundur, "Digital Simulation and Analysis of PowerSystem Dynamic Performance," Ph.D Thesis, University ofToronto; May, 1967.

[7] D. N. Ewart, F. P. deMello, "A Digital Computer Program For TheAutomatic Determination of Dynamic Stability Limits," IEEETransactions on Power Apparatus Systems, vol. PAS-86, No. 7,pp. 867-875, July, 1967.

Discussion

A. M. El-Serafl and M. A. Badr (University of Saskatchewan, Sask.,Canada): The Authors are to be commended for their interesting study.

The use of the conventional equivalent circuit representation, asshown in Fig. 2 of the paper, does not accurately represent the syn-chronous machine as far as rotor quantities are concerned. Themeasured variations of the rotor currents following a three phase suddenshort circuit of a turbo-alternator were found to differ greatly fromthose calculated using this equivalent circuit [ 1] . This is due to the factthat such an equivalent circuit is based on an approximate applicationof the per-unit system, in which the couplings between the rotorcircuits on one axis are assumed equal. A more accurate representationof the synchronous machine has been found by modifying this conven-tional equivalent circuit [ 1] .

Do the Authors think that taking this point into consideration willappreciably affect the accuracy of the machine models used for theirstudies?

REFERENCE

[1] I. M. Canay, "Causes of discrepancies on calculation of rotorquantities and exact equivalent diagrams of the synchronous ma-chine", IEEE Transactions, PAS-88, No. 7, pp. 1114-1120, July1969.

Manuscript received July 31, 1972.

Swarn S. Kalsi (General Electric Company, Philadelphia, Pa. 19124):The authors should be commended for presenting a paper examiningthe suitability of different models of a synchronous machine for multi-machine studies. Models 1, 11/2, and 2 are well known and Model 2 iswidely accepted [Ref. 1, 2] as an accurate model of a synchronous ma-chine. However, due to the complex phenomena of the quadrature axis,a solid rotor synchronous machine may be adequately represented bythe Model 3. Before attempting to complicate the machine models anyfurther, I feel that Models 1 to 3 should first be comprehensively testedagainst test results of a single machine system. Theoretically, to achieveexact duplication of conditions inside a machine, an infinite number ofcircuits must be used in each axis of the equivalent machine. Additionof more circuits increases the complexity of the model and the Model 4is an example of this type. Moreover, the most crucial aspect of ma-chine modeling is to devise methods for evaluating parameters whichare good for calculating the machine performance for all operating con-ditions.

The discusser developed a model [Ref. 3] which is equivalent topaper's Model 3. Few methods for calculating the machine parameterswere suggested, but the two important test methods are:

Method A: Short Circuit Test at Machine Terminals - equivalentto standard parameters as defined in ANSI Standard C42-10.

Method B: Static Impedance Test - based on an operational func-tion similar to Eqn. (3) of the paper.

Transient and sub-transient reactances calculated by above meth-ods agreed, but transient and sub-transient short-circuit time constantsdid not. Ratio of sub-transient short-circuit time constant measured byMethod-A to that measured by Method-B, was about 3. It must benoted that the Jackson-Winchester Model (Ref. 1 of the paper) isequivalent to Method-B and parameters determined by this methodexhibit more damping than those determined by Method-A. Dependingupon the type of disturbance studied, two sets of parameters give dif-ferent results. Using parameters determined by the terminal short-circuitmethod in the Model-2, good agreement was observed between test andcalculated swing curves [Ref. 2]. Referring to Fig. (3) of the paper,I presume that swing curve 4 was calculated using parameters obtainedfrom the Jackson-Winchester Model and the other three curves arecalculated with standard parameters. Since a sudden short circuit gener-ates a fast transient, the higher damping effect incorporated in parame-ters calculated by the Jackson-Winchester Model, severely damped swingcurve 4. I feel that swing curve 3 (calculated with the Model-3 usingstandard parameters) should be more accurate. This difference ofopinion can be resolved by comparing calculated result against testresult, preferably on a single machine system because in a multi-ma-chine system, the swing curve of each machine is affected by energytransferred to and from other machines.

While making calculations for curves in Fig. 3 to 6, I wonder if thefollowing voltage components were considered.

i) Stator Transients - P4d and P4'q terms in the primary voltageequations.

ii) Voltage drop in system reactance (X/wo pl) due to rate ofchange of current.

Omission of these voltages caused a considerable error in the calcula-tions of a three-machine system [Ref. 1]. On the other hand, if thestator transients and sub-transient saliency are neglected in the Model-2,it is possible to write machine and system equations together in termsof phasors [Refs. 1, 2]. This arrangement saves considerable com-puting time, since an integrating time step as large as 0.5 cycle can beused without causing numerical instability, and the results are stillwithin 1% of those obtained with integrating time step of 0.1 cycle.

REFERENCES

[1] S. S. Kalsi and B. Adkins, "Transient Stability of Power Systemscontaining both Synchronous and Induction Machines", Proc.IEE, Vol. 11 8, No. 10, October 197 1, pp. 1467-1474.

[21 R. G. Harley and B. Adkins, "Calculation of the Angular BackSwing Following a Short Circuit of a loaded Alternator", Proc.IEE, Vol. 117, No. 2, February 1970, pp. 377-386.

[3] S. S. Kalsi, D. D. Stephen, and B. Adkins, "Calculation of SystemFault Current Due to Induction Motors", Proc. IEE, Vol. 118,No. 1, January 1971, pp. 201-215.

Manuscript received July 26, 1972.

G. Shackshaft (Central Electricity Generating Board, London, England):The authors present a fascinating account of studies to assess the ac-curacy of several machine models. Having been concerned with stability

Manuscript received August 7, 1972

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analysis at the Central Electricity Generating Board in England for anumber of years, I am particularly interested to see that the authors'recommendations with regard to Model 3 line up exactly with the multi-machine program which we are using at present [ 1], and which we findto be satisfactory.

Until recently we used Model 2, but as a result of comparisonswith system tests carried out over many years and particularly thosecarried out at Northfleet Power Station [2] recently, we have nowabandoned this model. Its main deficiency is in its inadequate represen-tation of the quadrature axis, which is particularly important at leadingpower factor operation.

The authors will be aware that we have laid great stress on theimportance of validating our models by carrying out large scale systemfault-throwing tests; very complex models, like the Jackson andWinchester model, have never been used in the development of ouranalytical methods.

At present, we also have the problem which the authors discusswith reference to Model 3, i.e. that of obtaining adequate data; how-ever, I do not expect that this will be difficult to overcome. Data isclearly crucial in such advanced models and I feel that I must expressmy unease about the use of the Jackson and Winchester model to modi-fy manufacturers data. In my view, one may be deluding oneself on theabsolute accuracy of the models by matching operational impedanceswith the Jackson and Winchester model, whilst ignoring importantpractical factors such as the saturation of machine leakage reactancesunder high current conditions. The following table illustrates this effectas measured on a 120 MW turbo-generator at Northfleet Power Sta-tion [21.

Type of Test VAr Rejection Short Circuit Short CircuitTest at 50% Volts at 100% VoltsApprox. statorcurrent change 0.5 2.0 5.0(p.u.)X'd (p.u.) 0.478 0.27 0.245X"'d (p.u.) 0.288 0.19 0.17T'd (seconds) 1.7 0.97 0.63T"d (seconds) 0.083 0.046 0.026

I conclude that saturation of leakage flux paths is an importantconsideration in stability studies such as those described in the paper,in which the current variations are very large, and should be obliged ifthe authors could tell me whether or not the accuracy of the Jacksonand Winchester model has been tested under such current conditionsand if so, with what results.

REFERENCES

[1 ] P. Humphreys and M. R. Payne, "Development of Digital Programfor the Simulation of Power-Angle Disturbances by Comparisonwith System Tests" paper to Fourth Power Systems ComputationConference at Grenoble, 1972.

[2] G. Shackshaft and R. Neilson "Results of Stability Tests on anUnderexcited 120 MW Generator" Proceedings I.E.E., February1972, Vol. 119, pp. 175-188.

Paul L. Dandeno, Ronald L. Hauth, and Richard P. Schulz: We wel-come the discussors' comments and are pleased to continue the discus-sion with them.

Manuscript received September 18, 1972.

There seems to be some misunderstanding about our intent inincluding model 4 within the study reported in this paper. Model 4,which is exactly the Solid Iron Rotor Model reported in reference 2,was used as a standard of comparison for the lower order models. Weare not proposing that its structure be implemented in large scalestability programs, but rather we are advocating that models 1 /2 and 3be used in large scale stability programs, depending principally upon thedata available. The data used for the lower order models was either thatavailable directly from the manufacturer as calculated according to thepresent standards, or that which was determined through the use of thefull Jackson-Winchester model.

Dr. El-Serafi and Mr. Badr note that neither models 1/2 nor 3accurately represent the synchronous machine as far as rotor circuitcurrents are concerned. We concur; our intent was to evaluate theadequacy of lower order models for representing terminal character-istics for those machines as elements of a large scale power system. Inthe course of the study, the variations of field current were examinedfor the several models for several of the disturbances; these comparisonswere not made for all the disturbances since the fundamental intent ofthe study was the representation of the generator as it affected the over-all system dynamics. Those comparisons of field current which weremade, showed significant differences between field current waveformsobtained with the various models. The discussors point out that Dr.Canay presented a modified conventional equivalent circuit whichallowed the accurate determination of rotor circuit currents. The formof the Solid Rotor Iron Model used as model 4 is a similar equivalentcircuit, and provides a realistic determination of rotor circuit currents.

In reply to Dr. Kalsi, we note- that the simulation programs used(Reference 5, 6) did not include the P4d, PPq terms in the stator volt-age equations nor did the programs include the rate of change of energystorage in the transmission system elements' equations. These inclusionswould have significantly increased the cost of the simulations, particu-larly when studying the 39 bus, 46 line, 10 machine system shown inFig. 1A. There have been recent recommendations [Ref. 11] for theinclusion of the p P terms for machines and transmission networks inpower system simulation. A recent paper [12] shows the effect ofignoring these terms to be very small, as Dr. Kalsi notes.

We are pleased that Dr. Shackshaft's experience in his choice ofmodel 3 over model 2 confirms that which we recommend. Dr.Shackshaft has been instrumental in the pursuit by CEGB of large scalestability testing. We have no direct experience similar to those testsreported by Dr. Shackshaft; i.e., 50%/ var rejection, and short circuitsfrom 50% and 100% initial voltage, on large generators. The extent andnature of the various test confirmations we have of Solid Iron RotorModel are indicated in the latter part of a companion paper, reference2, which was given IEEE no. T 72 515-5. We are interested in theimplications of saturation on machine leakage reactances that he hasreported. The method used for representing saturation effects in theSolid Iron Rotor Model used for Model 4 is shown in Fig. 8 of reference2. It can be appreciated that this means of including saturation doeshave an effect upon transient and subtransient reactances and timeconstants, although they may not be of the order that Dr. Shackshaftreported.

The discussors and we concur in the need to obtain appropriatedata for power system studies. A working group of the P.E.S. Com-mittees on Power System Engineering and Rotating Machinery hasbeen recently formed called the "Working Group on Determination ofSolid Rotor Synchronous Machine Stability Study Constants". Its goalis to review the determination of reactances and time constants (orinductances and resistances) and saturation data of generators and,ultimately, to prepare standards for determining the data.

REFERENCES

[1 1] R. P. Schulz, "Synchronous Machine Modeling", IEEE paperC 72 616-1, a talk prepared for the Symposium "Adequacy andPhilosophy of Modeling: System Dynamic Performance" SanFrancisco, July 9-14, 1972.

[12] J. L. Dineley, and A. J. Morris "Synchronous GeneratorTransient Control: Part I - Theory and Evaluation of AlternativeMathematical Models" IEEE PICA 1971 Conference, pp. 182-190.

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