Characteristics of equivalent circuits of synchronous machines

A. Wa Ito n

Indexing terms: Equivalent circuits, Synchronous machines

Abstract: Frequency response tests have now become accepted as a sound alternative to sudden short-circuit tests for the determination of parameters for synchronous machines for transient studies. The standard approach to the extraction of machine parameters from the results of frequency response tests generally concentrate on curve-fitting techniques to match the measured magnitude and phase with a set of time constants. These processes are fraught with difficulty in respect of needing to: first, define the order of the model before the analysis can commence; and secondly, initiate the curve fitting with initial estimates of the parameters. Unfortunately, there is not a unique set of time constants which produce a frequency response which fits the measured data, and a better method than a blind numerical method is therefore needed. The author presents the application of standard linear systems theory to predict the positions of the poles and zeros in the frequency response and to determine the order of the equivalent circuit required to model the machine accurately. The process breaks quite naturally into two parts; the extraction of the time constants from the frequency response, and the determination of the parameters of the equivalent circuit from those time constants. The basis for the measurement technique is reviewed and the effects of different levels of complexity of the equivalent circuit are considered with respect to the increased difficulty normally experienced in extracting the parameters. The ease with which this method copes with the higher-order models and the sequential nature of the process, working from the lowest frequency to the highest frequency in the frequency response, justify accepting the procedure. Results obtained from tests on production machines are used to illustrate the procedures for both time constant extraction and equivalent circuit parameter determination to confirm the capabilities of the methods.

1 Introduction

The determination of the parameters of synchronous 0 IEE, 1996 IEE Proceedings online no. 19960003 Paper received 31st January 1995 The author is with the Department of Electrical & Computer Engineering, James Cook University of North Queensland, Townsville, Q4811, Australia

IEE Proc.-Electr. Power AppL, Vol. 143, No. 1. Januury 1996

machines has naturally been of interest to both machine designers and plant operators since their intro- duction as the standard means for the supply of electric power. The ultimate aim of the machine designer must be the prediction of the machine performance and therefore necessarily the machine parameters from design information, confirmed by the results of machine tests.

The initial need was for the determination (or predic- tion) of the steady-state power-producing capability of the machine and Kilgore [l] and Wright [2] laid the foundations for such work over 50 years ago. Since then, radical improvements have taken place in the techniques available for the calculation of the parame- ters. The use of finite element analysis to determine the flux distribution in the machine and hence the induct- ances is now commonplace and is the accepted means for such analysis, outstripping the constraints of previ- ous methods conceived before the availability of almost unlimited computing power. In addition, the control- lers have also undergone a revolution, the means of testing have been re-appraised and various aspects of the response of the machine and the power system have come to the fore.

The reactances and time constants obtained from the time-honoured sudden short circuit tests have been the foundation of the model of the synchronous machine for transient studies and the results from these tests have been used almost exclusively for the specification of machine performance and, in particular, for the analysis of the dynamic response.

Initial tests carried out to compare the theoretical predictions with the actual responses of machines pro- duced less than favourable results. Busemann and Cas- son [ 3 ] reported on tests carried out at Cliff Quays power station in 1958 and Shackshaft and Neilson [4] on the more significant tests at Northfleet in 1972. As a result of these tests, Shackshaft [5] proposed a new approach to the determination of machine parameters with a series of flux decay tests.

In their various guises these are all examples of step response tests. The use of frequency response tests, car- ried out at both standstill and synchronous speed, is now becoming widely accepted as an alternative to sud- den short-circuit tests for the determination of the machine parameters but there are some problems with the identification of the order of model required for a given machine and the extraction of the machine parameters from the test results.

It is this topic which IS the substance of this paper, which separates naturally into two sections: the first on the extraction of time constants for the machine from the results of frequency response tests, and the second

31

on the determination of the machine parameters from those time constants. The process developed does not depend on any numerical technique for either the deter- mination of the order of the system, nor the placement of the poles and zeros, but applies straightforward lin- ear circuit analysis and reduction techniques for their determination.

2 Generator modelling

2. I Steady-state models Representation of a perfectly cylindrical synchronous machine in the steady state requires knowledge of only the synchronous reactance (Xs), which is simply the sum of the leakage reactance (Xa) and armature mag- netising reactance (Xm). When the machine exhibits saliency this model is extended with the introduction of the two reaction theory in which the reactance is sepa- rated into components in the direct and quadrature axes. The d and 4 axis synchronous reactances then include Xmd and Xmq as the corresponding armature magnetising reactances along with the leakage reac- tance Xa.

In this steady-state representation of the machine it is not necessary to include the rotor circuit, the steady- state EMF induced in the armature winding by the field flux being the only data required and this is obtained from the open circuit characteristic.

2.2 Transient response models During transients it is necessary for the equivalent cir- cuit to include the effects of the rate of change of flux linkages crossing the air gap. In the direct axis the field winding must of course be represented along with damper windings and also the effects of eddy currents in the rotor body, slot wedges, end caps etc. which may be present. Whilst there is no field winding in the quad- rature axis, all of the other induced current effects need to be modelled in a similar fashion to those in the direct axis.

Such is the status of machine design that not every machine type built and tested has a detailed predeter- mined model. Whilst all of the phenomena associated with each aspect of the various models are well under- stood, it is not always possible to know in advance the relative levels of each effect to enable an a priori model to be assumed. It was with this background that the Electric Power Research Institute (EPRI) in America initiated a number of contracts for the development of methods for the determination of generator parameters for stability studies which culminated in an EPRI workshop on the subject [6,7].

I

I rotor 0

L _ _ _ _ _ _ _ _ _ _ _ _ i stator L

1

a b C Fig. 1 Equivalent circuits f o r synchronous machines

The range of models available is quite extensive, as indicated by the selection shown in Fig. 1. No distinc-

32

tion is made between circuits which model the direct and quadrature axes in this figure as it is the number of parallel paths on the rotor and hence the order of the model for a given equivalent circuit which is being emphasised here. If the direct axis is being modelled, the field winding is identified by the appropriate sub- script for that branch in the equivalent circuit. As the analysis in this paper is centred on the direct axis, the equations derived in this paper will be assumed to be for the direct axis only. With appropriate annotation, the analysis of the results of tests carried out in the quadrature axis follows exactly the same form as that described here for the direct axis.

The results of the sudden short-circuit test are nor- mally only relevant for models with two rotor circuits in the direct axis, that shown in Fig. lb. It is virtually impossible to obtain the parameters for a machine model with an equivalent circuit more complicated than this from the results of a sudden short-circuit test which highlights the need for some other test method to be used for the more complex models. Whilst the complete range of test signals for the identification of linear systems is available, the discussion here will be concerned only with the results of frequency response tests.

3 Frequency response tests

Frequency response tests have become the accepted means for obtaining information on the parameters for higher-order models of machines. Whilst the test proce- dures are well established there is no universally accepted method for the extraction of the parameters for the machine from the magnitude and phase infor- mation obtained from the tests.

The concept of the operational inductance of the equivalent circuit which is derived from the impedance measured at the stator terminals for either the direct or quadrature axis is an attractive means of presenting and comparing frequency response test results. Meas- urement of the rotorhtator transfer functions provides alternative data from which values for the parameters can also be obtained. Each set of test results in its own right defines a set of parameters and judgement must be used in combining the results from each test in order to detennine the best set of parameters for a machine. Two cases will be considered in detail.

3.7 Case I: One rotor circuit

3. I . I Operational inductance: The most basic of equivalent circuits is that for the quadrature axis with just one damper winding or the direct axis with only the field circuit represented, as shown in Fig. la. It can be seen, almost by inspection, that the operational inductance (Ld(s)} of this circuit is given by

R f ( L a + L m ) + s(LmLf + LaLm + LaLf) R f + s(Lm + L f ) Ld(s) =

(1)

R f ( L a + L m ) + s(LmLf + LaLm + LaLf) R f + s(Lm + L f ) Ld(s) =

(1) which, written in the standard form, becomes

(1 + sTd) (La + Lm) Ld(s) =

1 + sTdo where

L m + L f Rf

Tdo =

IEE Proc-Electr Power Appl., Vol. 143, No. I , January 1996

(3)

LaLm La + L m Lam =

Tdo' and Td' are seen to be the traditional open-circuit and short-circuit transient time constants of the machine, respectively.

3.1.2 Stator/rotor transfer functions: The results from these tests enable the open-circuit and short-circuit time constants to be identified separately and can therefore be very useful for parameter extrac- tion or comparison. The statorhotor voltage transfer function (VsVsIVJ) for the circuit of Fig. la with the field winding open circuited can easily be shown to be

1 sLm v s V f 1 + sTdo' R f (4)

- - _

When the test is conducted with the field winding short circuited the transfer function relating stator voltage to field current is

vs R f ( L a + Lm) I f L m - = (1 + sTd') (5)

In this simple case the parameters of the equivalent cir- cuit are exactly related to the principal time constants of the circuit, the values of the leakage and magnetising inductances and the value of the field resistance. Rarely in machine design does such a simple model hold; it is therefore necessary to consider higher order models for the representation of machine characteristics.

3.2 Case 2: Two rotor circuits The addition of a damper winding to the rotor circuit produces the 'standard' or 'classical' second order model for the machine, the circuit shown in Fig. lb.

3.2. I Operational inductance: It is convenient to first present the relationships which, although approximate, have been previously accepted as ade- quate for this equivalent circuit. The exact equations derived in the next paragraph can then be compared with these. Using conventional terminology the 'stand- ard' equation for the direct axis operational inductance is:

( 1 + sTd') ( 1 + sTd") Ld(s) = (La+ Lm) (6)

(1 + sTdo ' ) ( l + sTdo") where the subtransient open-circuit and short-circuit time constants are defined as

in which Lmf is the parallel combination of Lm and Lf, and Lmfa is the parallel combination of Lm, Lf and La.

The operational inductance of the equivalent circuit shown in Fig. l b can be shown to be

(La + Lm)(R f + sLf ) (Rk + SLk) + sLaLm(R f + sL f + Rk + sLk)

( R f + s L f ) ( R k + sLk) + sLm(Rf + sL f + Rk + sLk)

{ {

Ld(s) =

The coefficients of the terms in the numerator and

IEE Proc-Electr. Power Appl., Vol. 143, No. I , January 1996

denominator can be rearranged to simplify the above equation to:

1 1

1 + s{(Lk + Lam)/Rk + ( L f + Lam) /R f } + s 2 ( L f + Lam)(Lk + L a m f ) / R f R k

1 + s { ( L f + L m ) / R f + (Lk + L m ) / R k } + s 2 ( L m + L f ) ( L k + L m f ) / R f R k

(9)

[ [

Ld(s) =

x (La + Lm)

Recognising that all of the terms have the dimension of time and that some of the them are indeed the time constants presented in the previous section, the equa- tion can be further reduced to:

(La+Lm) (10) 1 + s(Td' + T A ) + ~ ~ T ~ ' T ~ ' ' 1 + s(Tdo' + T~)+s2Tdo'Tdo'' Ld(s) =

where Lk + Lam

Rk Llc + L m

R k and TB = TA =

The exact time constants of the poles and zeros of eqn. 10 are, of course, determined by extracting the roots of the quadratic equations making up the numer- ator and denominator. For the numerator, the roots are

giving values T1 and T3 and for the denominator

(1'4 4 -("do' + TB) 1 Tdo' + TB -

2Tdo'Tdo" -J 2 Tdo'Tdo" Tdo'Tdo" S = giving values T2 and T4 The operational inductance can then be written in the standard form of a two-over-two transfer function

If in the exact solution T A is assumed to be equal to Td" and TB equal to Tdo" the exact solution then degenerates to the approximate solution. Since the per unit value of the resistance of the damper winding is normally much greater than that of the field winding, this approximation holds true for superficial analysis but is a fundamental source of error in more compre- hensive studies.

3.2.2 Rotor/stator transfer functions: Once again these transfer functions can be used to provide frequency responses which to some extent separate the open-circuit and short-circuit time constants. The standard stator voltage/rotor voltage transfer function with the field open circuit is

(14) 1 + sTlc sLm

( 1 + sTdo') (1 + sTdo") R f and the transfer function with the field winding short circuited is V s - ( 1 + s T d ' ) ( l + sTd") Rk(La + Lm) I f - 1 + sTk L m

vs - - - V f

- (15)

where Lk T k = - R k

33

The exact solution for the transfer functions with the field winding open circuited and short circuited can be shown to be v s V f - (16)

1 + sTk s Lm - - 1 + ~ ( T d d + TB) + s2TdoTdo Rf

and

(17) VS - 1 + S(TA + Td) + s2TdTd R k ( h + Lm) - I f - 1 + sTk Lm

where TA and TB are the time constants defined in eqn. 10 and the same roots again apply, i.e. T1 to T4.

As was the case for the simpler circuit, the rotorlsta- tor tests enable some degree of separation of the open- circuit and short-circuit time constants and provide additional data for parameter extraction and compari- son.

3.3 More complex models As the machine swings asynchronously in the rotating field, the complete rotor experiences the same changing field. Any paths which allow induced currents to flow in the body of the rotor must be represented in the equivalent circuit, in addition to the currents which flow in defined windings. As more parallel branches are added to the rotor circuit, the analysis becomes more cumbersome, exacerbating the identification problem.

It should be noted that in both the approximate and the exact analysis of the first- and second-order models, the presence of the additional parallel path on the rotor in the higher-order model was simply to add another pole-zero pair to the transfer functions. It is precisely this feature for this type of circuit which is the basis for the parameter extraction method to be outlined in the next section.

The effect of extending the model to include further parallel branches on the rotor for other eddy current effects etc. will therefore add other pole-zero pairs in the frequency response, necessitating the identification of additional pairs of time constants. It will be seen that this increase in complexity, which causes definitive problems with numerical extraction methods, offers no such difficulty whatever to the algorithm to be described here.

When a third-order circuit is included on the rotor, as shown in Fig. IC, the equation for the exact opera- tional induclance is naturally a ratio of third-order pol- ynomials for which the numerator is given by

[ 1 + s { ( L f + L a m ) / R f + (Lk + Lam)/Rk + ( L j + Lam) / R j }

+ s 2 { ( L f * L j + Lj * Lam + Lam * L f ) / R f * Rj + (Lj * Lk + Lk * Lam + Lam * Lj)/Rj * Rk + (Lf * Lk + L f * Lam + Lam * Lk)/Rk * R f }

+ 2 ( L f * Lj + Lk + Lf * Lk * Lam + L j * Lk * Lam + Lj * Lf * Lam)/Rf * Rk * Rj](La+ L m )

(18) The denominator is of exactly the same form as the numerator but with Lam replaced by Lm in each of the coefficients of s, s2 and s3.

These then reduce to the standard transfer function form of

(1 + sT1) (1 + sT3) (1 + sT5) (1 + sT2) (1 + sT4) (1 + sT6) Ld(s) = (La + L m ) (19)

The task of determining the additional time constants is again simply the determination of another pair of pole-zero time constants, exactly the same as for the extension from a first-order model to a second-order model.

4 l i m e constant extraction from frequency response data

It is all very proper, of course, to extend the detail of the equivalent circuit of the synchronous machine to better represent the transient performance. It is quite another to be able to extract sufficient detail from the measurements made to enable the parameters of the extended model of the machine to be determined. To differentiate between the conventional and new meth- ods of analysis the terms numerical and analytical, respectively, will be used in these discussions.

4.7 Numerical approach Faced with the need to fit a frequency response curve with a set of time constants of unknown number and distribution, the normal approach is to use a standard numerical analysis package utilising some form of curve-fitting algorithm with an appropriate error func- tion. In the case of a synchronous machine which is often tested over a frequency range of five decades, weighting functions can also be used to provide the emphasis required for the extraction of time constants over the full frequency range. These can be applied to either the magnitude information, phase information, or both, according to the wishes of the investigator. It is axiomatic that for this method of analysis, the order of the model must be decided before the curve-fitting process can begin.

Very often the phase information is either ignored or given only secondary importance, the majority of the curve-fitting and time-constant extraction being done using the variation of the magnitude with frequency. The phase is sometimes taken into account by giving it a comparative weighting compared with the magnitude but the phase is not used as a prime input for the anal- ysis. It is the intention here to outline a method of fit- ting parameters to frequency response data which, rather than ignoring the phase information, uses it as the prime basis for the detection of the positions of the poles and zeros of the transfer function. Additionally, the order of the model is not specified as an input but is provided as an output from the parameter extraction process.

Constraining the model of the machine to that of the classical second-order model means that unless the machine does have this simple model, the time con- stants and parameters can only be a best fit to the results. If this is the case, the application of frequency response techniques has not advanced the modelling capabilities for the machine. When a higher-order model is required, the choice is wide and the order required is not known. It was just this lack of precision which was responsible for the poor results obtained in the simulation programs then available which insti- gated the new approach to machine testing developed by Shackshaft [5] .

4.2 Analytical approach The crux of the extraction method offered here, first reported at conferences [8,9], is that, since all of the equivalent circuits irrespective of their relative com-

IEE Proc.-Electr. Power Appl., Vol. 143, No. 1, January 1996 34

plexities are combinations of passive resistors and inductors, the circuit must always be represented by a set of consecutive poles and zeros along the negative real axis in the complex frequency plane, starting with a pole, which is shown in any respectable text on net- work analysis [lo]. Knowing this, it is possible to develop a strategy for the extraction of the time con- stants for the equivalent circuit from the frequency response test results.

The procedure proposed here is to dispense with the notion of a predetermined order for the model and find, using a fairly straightforward algorithm based on well-established principles, the best set of time con- stants from the data for any frequency response. Should it be necessary to simplify the model for a par- ticular application, terms can be excluded with com- plete knowledge of the type and degree of error being introduced.

From linear systems theory, the frequency response {G(s)} of a pole-zero pair forming a first-order lag cir- cuit having a time constant for the pole of T s and that for the zero of Tla s (a > l), is

80

6 0 - .c

U c cl 40

: 20-

(20) l + s T / a - s + a / T G ( s ) = - 1 + sT a ( s + 1 / T )

~

-

The following general relationships can be shown to apply for this circuit. (i) The network produces a maximum phase shift at the geometric mean or centre frequency (wm), given by

(ii) The value of this maximum phase shift (@m) is determined from

w m = &IT (21)

(iii) The overall gain change due to the pole-zero pair is given by 20 log a From (i) and either one of the other relationships it is possible to uniquely define the transfer function of the lag circuit. In practice it is (i) and (ii) which combine to provide the most useful method of parameter extrac- tion and it is here where the phase information is of prime importance. Since it is very easy to identify a phase peak in the frequency response data and the fre- quency at which it occurs, the evaluation of 'T and 'a' from (i) and (ii) above is also very easy. The magnitude change in (iii) is rarely sufficiently well defined that it can be used effectively for parameter extraction, but it must, of course, be taken into account in the overall transfer function. Used in this way, all of the practical forms of frequency responses for synchronous machines can be analysed.

4.3 Results of sample calculations The procedures outlined above have been applied to the direct axis impedance measured on a 500 MW gen- erator [6,7]. To confirm the capability of the new method, the extraction of the parameters for the direct axis impedance of this machine using the method pre- sented here will be processed in full and compared with parameters obtained using numerical procedures. The complete set of results for this machine (and others) will be presented in subsequent papers. The input to the process is the measured impedance data presented as magnitude (dB) and phase (deg.), shown in Fig. 2 . The results of the numerical analysis will be presented first, followed by the analytical results.

IEE Proc.-Elech. Power Appl., Vol. 143, No. I , JQnUQry 1996

The armature resistance is the low frequency asymp- totic value of the real part of the impedance of the machine which gives a value of 0.0019098Q. Subtrac- tion of this from the impedance data shown in Fig. 2 and dividing by jw gives the operational inductance shown in Fig. 3. It is this frequency response which is the starting point for both the numerical and analytical method of parameter extraction. The low frequency asymptote of the magnitude gives a value for Ld of 0.005003H. This will be subtracted from the magnitude of the impedance in the following analysis to provide a convenient zero dB datum for the results.

100, , 1 1 , - I ! , . , . " 8 " ' ' ' ' 1 ' ' """" ' """" '"''"1 .. . .___ .- .

__.....---

E -20

-4 0

-6 0 10-3 lo2 IO-l loo io

frequency Fig.2 Input impedance datu

._--

5

0

-5

$ -10

5 -20 II Q -1 5 U

a, 73 -25

$, -30 0 E -35

-40

4.3.1 Numerical method: This data has been ana- lysed by Harris and Prashad [Ill producing the set of time constants in the transfer fijnction shown below. Only the values of the time constants of the poles and zeros are given, the (1 + ST) format being assumed.

(23) (0.746) (0.0289) (0.001 69)

X d ( s ) = 0.005003 (4.967)(0.0297)(0.00172) It is necessary to compare the frequency response of this transfer function with that obtained from the test results to determine the accuracy of the numerical method. Whilst this could be done by simply compar- ing the total frequency response, it is illustrative to extract the frequency responses of the pole-zero pairs in sequence, starting at the low frequency end of the spec- trum. In this way the effect of each pair of pole-zero time constants can be seen individually. The validity of a time constant pair will be evident from the residual

35

magnitudes and phase left after their subtraction from the original data. Ideally the subtraction of the effects of all of the poles and zeros would result in a final residual which would have zero phase and magnitude error over the complete range of the frequency response.

Subtracting the frequency response of the first pole- zero pair (0.746/4.967) from the measured frequency response gives the residues shown in Fig. 4. The maxi- mum magnitude and phase errors are +1.9 and -2.4dB and +5.6 and -6.3deg.

frequency Fig. 4 Residuals after subtraction of one polelzero pail

frequency Fig. 5 Residuals after subtraction of two pole-zero pairs

6

4

$

5

.c 0 .2 U

;0

2 -2 t m

-4

10-3 102 10-1 100 101 102 103 frequency

Fig. 6 Residuals after three pole-zero pair

Repeating the above procedure for the second pair of time constants (0.0289l0.0297) gives the residues shown in Fig. 5. The maximum magnitude and phase errors are now +1.9 and -2.ldB and +5.5 and -6.2deg., respectively.

Repeating the above procedure for the third pair of time constants gives the residue shown in Fig. 6. The maximum phase and gain errors are now +1.9 and - 1.9dB and +5.5 and -5.7deg., respectively.

4.3.2 Analy-tical method: A computer program has been written [I21 in a WINDOWS environment which provides a user-friendly interactive process for the pole-zero extractions from the frequency response of the operational inductance as outlined in Section 4.2.

The first phase maximum in the operational induct- ance data has a value of -42.91 deg. and occurs at a fre- quency of 0.08Hz. The values of the two time constants required to produce this maximum phase, calculated using the relationships given in Section 4.2 are 4.565 and 0.8669s. As they are determined from the individual input data points, these time constants are not necessarily the best possible ones to use. The pro- gram therefore plots the frequency response of this pole-zero pair on the screen along with the measured data and the residue left after subtraction of the pole and zero from the operational inductance data. The values of the centre frequency and the time constant ratio are then allowed to be varied online to enable the best pair of time constants (i.e. those which produce the smallest total residues in phase and magnitude over the frequency range affected by that pole-zero pair) to be selected. In this way the operator has an immediate indication of the quality of the pole-zero fit and the error remaining when it is extracted from the measured data.

For this data, the final time constants are 4.3844 and 0.9047s producing the results shown in Fig. 7. The residual frequency response after subtraction of the pole-zero pair is plotted on the graph. For a perfect match, the residual magnitude and phase would sit on the zero axes over the frequency range sensitive to the given pole-zero pair. It should be noted that there are only small variations around zero in both phase and magnitude up to a frequency of 0.2Hz, the maximum errors in magnitude and phase being +0.5, 4 . 7 d B and +0.7, -6.3deg., respectively.

frequency Fig. 7 Residuals after subtraction of one pole-zero pair

36 IEE Proc.-Electr. Power Appl., Vol. 143, No. I , January 1996

The frequency response which remains after subtrac- tion of the frequency response of the first pole-zero pair has a phase maximum of -3.9deg. at a frequency of 0.7Hz. Time constants of 0.2392 and 0.2061 s for the pole and zero, respectively, are finally extracted by the program to give the residuals shown in Fig. 8. It is clear that the successive subtraction of these pole-zero pairs has catered for all that is needed in the frequency response from DC up to approximately 1 Hz with max- imum errors of +0.4, -3.4dB and +0.9, -6.2deg.

\ I

frequency

Fig. 10 Residuals after subtraction of four pole-zero pairs

10-3 10-2 10-1 i o 0 io1 102 103 frequency

Fig. 8 Residuals after subtraction of two pole-zero pairs

L

1

% O

-0 -1 L Q

0 G -2 0 J

E-3 0

E -L

-5

-6 10-3 10-2 10-1 io0 101 i o 2 103

frequency Fig. 9 Residuals after subtraction of three pole-zero pairs

The maximum residual phase error is now -6.2deg. at a frequency of 96Hz. Two further sets of pole-zero pairs are required to reduce the remaining frequency response to zero dB and zero phase.

The third pair of time constants are 0.004925 and 0.003991 which when subtracted extend the model to a frequency of about 20Hz as shown in Fig. 9 and the fourth pair are 0.002321 and 0.0001688s. The final transfer function is then

(0.9047) (0.1669) (0.003991) (0 .OOO 1688) (4.3844) (0.1962) (0.004925)(0.0002321) X d ( s ) = 0.005003

(24) The final residual frequency response then has magni- tude errors between +0.8 and -0.8dB and phase errors of +0.9 and -0.7deg. and over the whole frequency range as shown in Fig. 10. The constancy of fit over the whole frequency range is evident. The final compar- ison of Figs. 6 and 10 finally show the excellent results achieved with the new method.

4.4 Summary It is clear that a fourth-order model is needed to fit the frequency response test data and that the analytical method produces a better fit than the numerical method over the full range of frequency. Not only has a better fit been produced by the analytical method, but the sequential determination of the pole-zero pairs from the phase information clearly best represents the physical nature of the circuit being modelled.

5 Determination of the machine parameters

The methods available for the determination of the machine parameters for a known set of time constants and the corresponding equivalent circuit will now be discussed, building upon the process for the extraction of time constants from the frequency response data presented previously. Again, for brevity, only the proc- ess for data obtained for tests in the direct axis will be examined. Once again, either numerical optimisation processes or an analytical approach can be used.

5. I Extraction process Having determined the values of the time constants of the poles and zeros in the transfer function it is now necessary to reverse the process and determine the val- ues of resistance and inductance of the branches of the equivalent circuit from the time constants. Knowing the number of time constants required to adequately model the operation inductance defines the order of that circuit and it would be possible to iterate the val- ues of the parameters of the rotor branches to find the best fit. Once again, this breaks the physical link between the time constants extracted from the fre- quency response and the equivalent circuit, More cor- rectly, the analysis of the equivalent circuits described in Section 3 should be used as the basis for the param- eter extraction.

To enable comparisons between the numerical and analytical approach to be made, the analysis presented here will be for a rotor circuit with three branches shown in Fig. IC using the notation developed in Sec- tion 3.3. The value of the synchronous reactance (Xd) has already been determined and it is next necessary to determine or assign a value to the leakage reactance (Xa) which also then defines the value of the magnetis- ing reactance (Xm). The remaining unknown parame- ters are then only those of the rotor branches of the equivalent circuit. Using a purely numerical approach,

31 IEE Proc-Electr. Power Appl., Vol. 143, No. I , January 1996

values of resistance and inductance of the rotor circuits can be iterated to produce a frequency response which matches the measured data. More properly, however, it will be shown that it is possible to extend the analytical approach one stage further. The two approaches will now be considered in more detail.

5.1.1 Numerical methods: Using the set of time constants given in Section 4.3, Harris and Prashad [Ill applied a variety of numerical optimisation strategies to determine the best parameters of the third order equivalent circuit. The parameters determined by Har- ris and Prashad are shown in Table 1. It should be stressed that these were obtained by taking the values of the time constants extracted from the frequency responses and iterating the values of the branch com- ponents to produce the best fit.

Rather than taking this totally numerical approach it is possible to set up relationships between the extracted time constants and the equations derived in Section 3. The process for any form of solution must be equiva- lent to equating the time constants extracted from the transfer function to those derived for the equivalent cir- cuit. The process for doing this is set out below using eqn. 18 and 19 which were derived in Section 3.3 for the third-order model.

For the denominator

Ti * T3 + T3 * T5 + T5 * Ti L k * Lf + Lj + L m + Lf * L m

Rf * R j Lj * Lk+ L j * L m -k Lk * Lm

Rj * Rh Lf * Lk + Lf * Lm + Lk * Lm

Rk*Rf

L f * Lj * Lk + Lj * Lk * Lm

- -

+ +

TI * T3 * T5 R f * R j * R k

(25) and for the numerator

Lf +Lam L k t L a m L j + L a m + R f + Rk R3

T2 + T4 + T6 = T2 * T4 + T4 * Tt3 + T6 * T2

- Lk * Lf + Lj *Lam + Lf * Lam Rf * R j

Lj * Lk + Lk * Lam + Lj * Lam Rf *Rj

Lf * L k + L f * L a m + L k * L a m R k * Rf

-

+ + { Lf * L j * L k + L j * L k * L a m

+ Lf * Lk * Lam+ Lf * Lj *Lam T2 * T4 * T6 =

Rf * Rj * R k (26)

This reveals a set of six nonlinear equations which can be solved numerically using, for example, the fsolve function in MATLAB. As with all numerical forms of solution, initial estimates are required for the process and this can once again lead to the final values for the parameters being dependent on those initial estimates,

which often leads to negative values of resistance and inductance being found for some branches. A system- atic procedure which attempts to overcome this short- coming is to use the time constants for the lowest frequency pole-zero pair in conjunction with the equa- tions for the first-order model to define starting values to be used in the second-order model. The solutions to the second-order modulation give initial values for the solution of the third-order model and so on. Even then, negative values for some components often occur.

Table 1: Comparison of direct axis parameters obtained by analytical and numerical methods

Parameter Analytical Numerical

Rf

Lf

Tf = LVRf

Rj

Li Ti= LjlRj Rk

Lk Tk = LWRk

0.1378 1.08

0.8425

6.1 14

9.4761

0.2123

0.0224

21 0.59

1.5272

0.00725

0.413

0.382

205

1.68

0.0082

495

0.588

0.00119

Values in mQ and mH as appropriate

5.1.2 Analytical method: Since the operational inductance (Ld(s)} is simply the leakage reactance in series with the parallel combination of the magnetising reactance and the rotor impedance, it is a simple task to determine the frequency response of the equivalent rotor impedance Zv(s).

(27) sLm{Ld(s) - La}

Zr(s) = Lm + La - Ld(s)

which, of course, gives another frequency response data set. This process mirrors that of subtracting the arma- ture resistance from the measured impedance to pro- duce the frequency response data for the operational inductance. Assuming a value of 8.2% (0.000456Q) for the leakage reactance for the machine (to be consistent with Harris and Prashad) produces the frequency response plotted in Fig. 12. It would be possible to invoke another numerical curve fitting technique to this frequency response but there is no need to since the actual impedance of the rotor (Zv) for the third-order model can easily be shown to be

(28) R p ( 1 f s T f ) ( l + sTlc)(l+ s T j )

Zr = (1 + sTv) (1 + s T w )

where TA lk, r j , Tv and Tw are related to the time constants of Ld(s) and the values of La and Lm by a set of linear equations. In fact, the time constants of the numerator are the individual time constants of each of the rotor branches,

T f = L f / R f , Tlc = Llc/Rk and T j = L j / R j (29) and Rp is the parallel combination of resistances of the three rotor branches:

(30) Rf * R k * R j

Rf * R k + R k * Rj + Rj * Rf R p = The low frequency asymptotic value of the magnitude of the frequency response Zv(s) shown in Fig. 12 is Rp which is therefore known and has a value of

38 IEE Proc.-Electr. Power Appl., Vol. 143, No. I , January 1996

0.00019876 for this machine. It should be noted that the value of this asymptote is also related to the circuit parameters and time constants by the equation

Q 6 0 - 8

c3 2 0 -

2 0 -

L

U Q 40

U

U1

z-20

(31) Lm2 R p =

Ld(T4+T5 + T6 - T1- T2 - T3)

-

-

The relationships between the unknown rotor circuit parameters and the known time constants can then be written as a simple linear matrix equation.

Tj+Tk T f + T k T f + T j ] [i] = & I u + T w ] T j*Tk T f * T k T f * T j Rk Tu * Til:

Inversion of this matrix then gives the values of Rf, Rj and Rk, from which, since the time constants are known, the inductances are simply obtained from eqn. 30. The values of the parameters obtained are given in Table 1.

1 __ 1 1 Rf

( 3 2 )

-45 -40 ! i I 0-3

frequency Fig. 1 1 culated

Operational inductances, comparison of input and numerical cal-

l o o 80 r--- ' ' ' ' " ' ' r ' ' ' -7

-40 1

...

/ /

-GO/- ' " " "" ' ""- l i 1 -U 10-3 10-2 10-1 100 I O ~ io2 lo3

frequency Fig. 12 Derived rotor impedance

There are other versions of third-order models which can be used in which additional leakage inductances are included between the branches of the rotor circuit. Whilst these require the determination of additional rotor parameters, since the frequency response of the rotor must still be constrained to be equal to Zr(s), the additional parameters cannot, and do not, change the order of the system and simply add an additional arbi-

IEE Proc.-Electr. Power Appl., Vol. 143, No. 1. January 1996

trary component to the inductances which make up the time constants of the circuit. They do not add to the complexity of the solution process. Fig. 13 shows the operational inductance obtained using this process with the time constants for the third-order model deter- mined in Section 4.3.2.

.. _.

1

-45 , , , , I ,,,, , , i ,,,,,, , , 1 , 1 1 , , , ~ I , / 1 # 1 < 1 , 1 1 1 1 1 1 1 1 , , 10-3 10-2 10-1 100 I O ' 102 103

frequency Fig. 13 culated

Operational inductances, comparison of input and analytical cal-

5.2 Comparison of results Comparison of the graphs shown in Figs. 11 and 13 show conclusively that the results for the analytical method are far superior to those for the numerical approach .

Whilst there is no means for determining which of these are the most appropriate set of values, compari- son of the time constants of the three rotor circuits (Tf, T j and Tk) is quite illuminating. Those obtained using the analytical approach have a very systematically ordered set of values in the expected range whilst those obtained using the numerical approach are quite arbi- trary and do not follow such a systematic trend. This difference is very much in line with what would be expected when comparing the results of two systems one based on fundamental principles of circuit theory, and the other on the blind application of numerical methods.

It must be said that the results produced by Harris and Prashad [ 1 I ] were compromises, attempting to obtain the best fit for both the direct axis impedance and the rotorhtator transfer function for the machine. In addition to the better degree of fit, however, it is the mechanism by which the time constants for the equiva- lent circuit are obtained by a sequential process which sweeps through the frequency response from the lowest to the highest, systematically modelling the features of additional rotor circuits until those required have been identified. which has been confirmed.

6 Conclusions

Frequency response methods are becoming the stand- ard means for the determination of the parameters of synchronous machines for transient stability studies.

Existing numerical methods for extracting the param- eters for the equivalent circuits from the measured data tend to be rather arbitrary and have not produced the level of agreement with measured responses necessary for validation purposes.

From the basic properties of cascaded L-R circuits and the fundamental characteristics of their frequency

39

responses, a new set of analytical relationships for the equivalent circuit parameters has been established. A new procedure for the determination of the time con- stants required to fit the frequency responses of the measured operational inductances of synchronous machines has then been described.

The procedure does not require prior knowledge of the order of the model to be used, it is founded on well-established principles of linear systems theory and has been found to work well for the equivalent circuits used for synchronous machines. Whilst the final level of agreement between the measured and predicted fre- quency responses using both methods are similar in the case studied, it is evident from the results of the inter- mediate stages shown, that the method described here is modelling the effects of the individual rotor circuits much more completely.

Methods for the extraction of the time constants for a synchronous machine and then the determination of the equivalent circuit parameters have been shown to produce very good agreement between the initial meas- ured data and the final results. The final errors in the gain and phase were within +I- one dB and one deg., respectively, over the full frequency range.

Further work is being carried out using data for the quadrature axis impedance and the statorhotor transfer functions for comparison with those for the direct axis. In addition, work is being carried out on laboratory-

size machines to enable the results of standstill and online frequency response tests to be compared.

7

1

2

3

4

5

6

7

8

9

References

KILGORE, L.A.: Calculation of synchronous machine con- stants, Truns. A IEE, 1931, 50, pp. 1201-1214 WRIGHT. S.H.: Determination of Synchronous machine con- stants by test, Truns. AIEE, 1931, 50, pp. 1331-1351 BUSEMAN, F., and CASSON, W.: Results of full scale stability tests on the British 132 kV grid system, Proc. IEE, 198?, 10, pp. 347-362 SHACKSHAFT, G., and NEILSON, R.: Results of stability tests on an underexcited 120 MW generator, Proc. IEE, 119, pp. 17-188 SHACKSHAFT, G.: New approach to determination of syn- chronous machine parameters from tests, Proc. IEE, 121, pp. 1385-1391 Electric Power Research Institute, Compendium of the EPRI Workshop on modelling for stability calculations (EPRI, 198 1) WALTON, A.: Determination of synchronous machine stability study constants, a summary of work done by N.E.I. Parsons, EPRI Workshop, St. Louis, USA, 1981 WALTON, A., and CROFT, J.S.: The modelling of synchronous machines, Fourth Int. Conf. on Electrical Machines and Drives, London. 1989 WALTON, A.: The extraction of parameters for synchronous machines from the results of frequency response tests, Int. Conf. on Electrical Machines and Drives, 1993, pp. 480-484

10 Van Valkenberg: Network analysis (Prentice Hall, 3rd edn.) 11 HARRIS, M.R., and PRASHAD, F.R.: Improved methods for

inter-relating circuits and frequency-response data for synchro- nous machines, Int. Conf. on Electrical Machines and Drives, 1989, pp. 192-197

12 CROFT, J.S.: Microprocessor based control systems for synchro- nous machines, MEng Sc thesis, James Cook University, North Queensland, Australia

40 IEE Proc -Electr Power Appl , Vol 143, No 1, January 1996