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THE MODELLING OF SYNCHRONOUS MACHINES

A Walton and J S Croft

James Cook University of North Queensland, Australia

INTRODUCTION

The determination of the parameters of synchronous machines has naturally been of interest to both machine designers and plant operators since their introduction as the standard means for generation and distribution of electric power. The ultimate desire would be the prediction of the machine parameters from design information together with the confirmation of the parameters from machine tests. Kilgore (1) and Wright ( 2 ) laid the foundations for such work over fifty years ago but in that time the techniques available for the calculation of the parameters have changed radically and the means of testing have also undergone a re- appraisal.

The use of finite element techniques to determine the flux distribution in the machine is now commonplace and is the accepted means for such analysis, outstripping the limitations of previous methods conceived before the availability of almost unlimited computing power.

The use of frequency response tests to confirm the design predictions is now becoming accepted as an alternative to sudden short circuit tests but there are some problems in the extraction of the machine parameters from the test results and it is this topic which is the substance of this paper.

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 were used exclusively for the specification of machine performance and the analysis of, in particular, the dynamic response.

Initial tests carried out to compare the theoretical predictions with the actual responses of machines produced less than favourable results. Busemann and Casson ( 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 proposed a new approach to the determination of machine parameters with a series of flux decay tests.

These were all examples of step response tests and as such represent only one of the several types of test which can be carried out to identify systems. These will be briefly described in the next section.

TEST METHODS AVAILABLE

Whilst the following methods apply to any controlled system, their application will be

confined to the determination of the time constants and operational inductances of synchronous machines. Five different types of input signal are available for the identification of system parameters:

( f ! Steps (11) Ramps (iii) Impulses

and (iv) (v) Sinusoids Pseudo Random Noise

For second order systems the step response is very widely used in the investigation of system stability and, whilst the parameters of the system are not identified immediately, they can be obtained from further analysis of the time response. This form of test has the advantage that the step change in the input signal is very easily obtained, usually by some form of switching function. When the complexity of the system is greater than second order the parameters cannot be obtained immediately from the step response and, except for information on stability margins, the test is less valuable.

Using a ramp input also enables the plant parameters to be defined for simple systems, the need to generate extremely accurate ramps, however, and the limitations of the results obtained make this form of test less practical than step responses.

The response to an impulse is perhaps the ideal form of testing since it reveals the transfer function of the system immediately. Again, for low order systems, or when the order of the system is known, the extraction of the parameters from the measured response is possible but for high order systems it is once again difficult to extract the parameters. Additionally, the generation of a sufficiently good approximation to an ideal impulse at the power levels required is difficult in practice.

The use of digital signals offers greater possibilities of success than impulses. Either binary or ternary noise signals can be used to excite the system, cross correlation techniques then providing the means for the determination of the impulse response and extraction of the system parameters. When pure white noise signals are used, the cross- correlation of the input and output gives the impulse response of the system. Very often pseudo random M sequences are used (pink noise) as an approximation to white noise and successful results can still be obtained, Lang et. al. (5)

Stimulating the system with sinusoids to determine the frequency response of the plant follows well established concepts which form the kernel of linear control theory, the measurements of gain and phase containing all

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of the information necessar not only to determine the stability of tge system, but also the system parameters. As for the other methods of measurement given above, the system parameters are most easily evaluated for low order systems, increasing difficulty arising as the order rises and smaller variations in gain and phase with change in frequency occur as a result.

In preparation for the description of the means by which the parameters of high order models can be extracted from the results of frequency response tests, the types of models commonly used for synchronous machines will be described in the next section.

GENERATOR MODELS AVAILABLE

Such is the status of machine design that not every machine type built and tested has a pre-determined model. Whilst all of the phenomena associated with each aspect of the various models is well understood, it is not always possible to pre-determine 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 in America (EPRI) initiated a number of contracts for the development of methods for the determination of generator parameters for stability studies which culminated in a workshop on the subject, EPRI ( 6 ) .

The range of models available for generators is quite extensive, each additional feature being represented increasing the order of the model required. In the following section the correspondence between the addition of features and the model representation will be presented.

Steady State Models

Representation of a perfectly cylindrical synchronous machine in the steady state requires only the synchronous reactance, which is simply the sum of the leakage reactance and armature magnetising reactance to be known. When the machine exhibits saliency this model is extended with the introduction of the two reaction theory in which the reactance is separated into components in the direct and quadrature axes.

The d and q axis synchronous reactances then include Xad and Xaq as the corresponding armature magnetising reactances along with the leakage reactance X1.

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 being obtained from the open circuit characteristic.

Transient Response Models

During transients it is also necessary for the equivalent circuit to include the effects of the changing flux linkages across the airgap between the armature winding and the rotor circuits. In the direct axis the field winding must of course be represented along with and damper windings which may be present and also the effects of eddy currents in the rotor body, slot wedges, end

caps etc.

Equivalent circuits for the direct axis are shown in Figure 1 where two damper circuits, are included. Representation of additional rotor circuits requires the addition of more circuits in parallel with the rotor. In the quadrature axis, whilst there is no field winding of course, all of the other induced current may need to be modelled in a similar fashion to those in the direct axis.

In the simplest model, all of the flux passing across the air gap links with all of the rotor circuits but in practice there will be s6me leakage effects and the flux linking each of the rotor circuits is different. This effect is represented by the series reactance between the circuits in question.

Different levels of model complexity are possible by the inclusion or exclusion of circuit elements as necessary. Each of the equivalent circuits will have a unique set of poles and zeros which define the frequency response of the circuit. It is from the measurement of the frequency response that it is hoped that the parameters of the circuit can be determined. For such a range of circuit models available, how can the particular equivalent circuit for a machine be determined. Since all of the equivalent circuits, irrespective of their relative complexities, are combinations of passive resistors and inductors, the plots of the poles and zeros in the complex frequency plane must be a set O E consecutive poles and zeros along the negative real axis. Knowing this it is possible to extract the time constants (and hence the reactances) of the equivalent circuit from the frequency response data.

BODE PLOTS FOR DIFFERENT MACHINE MODELS

Consideration of the frequency response of the operational inductance of the direct axis of a typical machine is sufficient to illustrate the concepts to be considered here.

The Simplest Model

The "classical" model for the direct axis (implicitly assumed when the results of the sudden short circuit tests are used to determine the machine parameters) is that shown in Figure l(a). The transfer function for this four time constant circuit is

The straight line approximation of the magnitude function for this transfer function is shown in Figure 2.

The value of Xd(s) approached asymptotically as the frequency approaches zero is the synchronous reactance in the direct axis (Xd) and the corner frequency of the first lag is that associated with the open circuit transient time constant Tdo'. The value of the inductance on the mid-frequency plateau is that corresponding to the transient reactance Xd' and the lead and lag defining the plateau are the short circuit transient time constant Td' and the open circuit subtransient time constant Tdo". The final asymptotic value of the inductance at high

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frequencies is that corresponding to ,the subtransient reactance Xd" and the final lead time constant defined by the last corner frequency is the short circuit subtransient time constant Td".

In practice, of course, the frequency response is rarely so clearly defined as that shown in Figure 2 with, in particular, no distinguishable plateau being present from which to obtain the transient reactance and the values of Td' and Tdo". It is then necessary to resort to curve fitting techniques to extract values for the time constants and hence component values for the circuit.

Constraining the model of the machine to that of the classical model means that unless the machine does have this simple model, the time constants and parameters can only be a "best fit" to the results. In this case the application of frequency response techniques has not advanced the modeling capabilities for the machine whatever. Even if 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 inserted into the available simulation programs which instigated the new approach to machine testing by Shackshaft.

Higher Order Models

Addition of another parallel R - L branch to the equivalent circuit, for which the time constants Ta and Tb apply, the transfer function then becomes:

Xd(s) = (1 + sTd')(l + sTd")(l t sTa) Xd (1 + s-rao- I [I + s-rao-) (I + STD)

The additional pole-zero pair add another plateau to the frequency response at a position dependant upon the time constants for that circuit.

Extension of the model by the addition of further parallel branches simply adds another pair of time constants to the frequency response, increasing the difficulty in estimating the parameters of the machine. One particular effect which commonly applies is that, with the closer proximity of the poles and zeros, a slope of 2 0 dB per decade never exists and the placing of the poles and zeros becomes more troublesome.

PARAMETER EXTRACTION FROM THE FREQUENCY RESPONSE

It is all very proper of course to extend the detail of the equivalent circuit of the synchronous machine in order to better represent the transient performance. It is quite another to be able to extract, from the measurements made, sufficient detail to enable the parameters for the extended model of the machine to be determined.

The Standard Approach

Faced with the need to fit a frequency response curve with a set of time constants of unknown distribution, the normal approach is to use some standard package utilising some form of least squares, minimax ot some other curve fitting algorithm. 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.

Very often the phase information is given only secondary importance, the majority of the curve fitting and time constant extraction being done on the form of the magnitude variation with frequency. The standard approach is to fit the response to the results which produces the minimum deviation. The phase is sometimes taken into account by equating an equivalent weighting to it compared with the gain e.g. 10 degrees is equivalent to 2 dB but the phase is not used as a prime input for the analysis. It is axiomatic that for this method of analysis, the order of the model must be decided before any curve fitting can begin. It is the intention here to outline a method of fitting parameters to frequency response data which, rather than ignoring the phase information, uses it as the basis for the detection of the positions of the poles and zeros of the transfer function.

The New Method

The procedure proposed here is to dispense with the notion of a pre-determined order for the model and find, using a fairly straightforward algorithm based on well established principles, the best set of time constants from the data for any frequency response. Should it be necessary to sim lify the model for a Darticular amlicaeion, terms can be excluded with' complete knowledge of the degree of error to be expected.

The procedure for the extraction of the parameters for the direct axis is as follows:

The low frequency asymptote of the frequency is usually very well defined in the results and the value of Xd and Tdo' can then be obtained with very good assurance.The term corresponding to this pole can then be used in conjunction with the next zero or subtracted from the results and the residue, which must be made up from a series of pole-zero pairs, is then considered. In this way, the complete frequency range corresponding to the transient response, is covered and finally only the term corresponding to the short circuit subtransient time constant remains.

If the effect of Tdo' is subtracted from the results, the pairs of poles and zeros must always combine to produce a lead function. From classical linear control theory, the effect of adding a single lead to a circuit is very easily defined. From the frequency response of a lead circuit having the first order transfer function

G ( s ) = ( 1 t sT) = a(s + 1/TL (1 + sT/a) ( s + a/T) a > 1

the following can be shown to apply:

(i) the network produces a maximum

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phase shift at a frequency

n = ~/TF (ii) the value of this maximum phase

shift is determined from

and (iii) the overall gain change due to the network 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 lead circuit.

It is here where the phase information is of prime importance since it is very easy to identify a phase shift peak in the frequency response data, determine the frequency at which it occurs and evaluate 'T' and 'a' from (i) and (ii) above. The gain change in (iii) can be used as a cross check or, if the data is fuzzy, the three relationships can be used to provide optimum values for the parameters.

On many occasions the frequency response has such a complex set of terms that a peak in phase is not identifiable. In these circumstances the gain changes with frequency at a rate other than the 20 dB per decade associated with a single term but remains substantially constant over a reasonable frequency range. Differentiating the expression for the gain of the lead circuit shows that at the geometric mean frequency where the maximum phase shift occurs, the slope of the gain function is given by:

(iv) = 20 (a - 1) dB per decade d W (a + 1)

The value of 'a' can then be determined from the slope of the measured frequency response, this relationship replacing that in (ii) above. Used in these ways, most of the practical forms of frequency response can be analysed.

Results of Sample Calculations

The procedures outlined above have been applied to the frequency response measured in the direct axis of a 500 MW generator (7) shown in figure 3.

The asymptotic value of Xd at low frequencies is 7.92 dB and the corner frequency estimated from the -3dB point is 0.0164 i.e. the value of Tdo' is 9.704 secs. Following this pole, the slope of the frequency response gain curve is 13.71 dB/decade which from the previous section gives a value for a of 5.36 for the pole zero combination required for this section. Using the value of Tdo' above indicates a zero at a frequency of 0.0879 Hz.

Subtraction of this pole zero pair from the measured data results in a frequency response having a peak phase shift of -10.1 degrees at a frequency of 0.17 Hz, all deviations in gain error being less than 1dB below this f r e q u e n c y . Applying the derived relationships to these figures produces values for 'a' and 1/T of 1.425 and 0.1424 Hz respectively.

Subtraction of these from the remainder of the frequency response results in a frequency response with a phase peak of 8 degrees at a frequency of 37 Hz and gain errors of less than 0.5 dB up to a frequency of 14 Hz which gives corresponding values for la9 and l/t of 1.323 and 32.2 Hz.

Subtraction of this produces a final residue which in which the gain and phase errors are less than 1.5 dB and 2.3 degrees respectively. The final transfer function is therefore

Xd(s) = 7.92 (1+1.81s)(1+0.784s)(1+0.0374~~ (1+9.704s)(1+1.118s)(1+0.00494s)

The points in figure 3 correspond to the transfer function above. It is evident that because no poles or zeros have time constants close enough to one another to cancel, omission of any pair of time constants to reduce the order of the system would result in quite erroneous simulation of the transient response.

Similar analysis can be applied to results of tests in the quadrature axis and is being applied successfully in work being done on a laboratory machine at James Cook University to determine the optimum parameters for the controllers ( 8 ) .

CONCLUSIONS

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

Existing methods of extracting the parameters 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.

A new procedure for the determination of the time constants required to fit the frequency responses of the measured operational inductances of synchronous machines has been described.

The procedure does not require prior knowledge of the order of the model to be used, is founded on well established principles of linear control theory and has been found to work well for the equivalent circuits used for synchronous machines.

REFERENCES

1 Kilgore, L.A., 1931, "Calculation of Synchronous Machine Constants", Trans.AIEE, 50, 1201-1214.

2 Wright, S . H., 1931 "Determination of Synchronous machine constants by Test", Trans AIEE, so, 1331-1351.

3 Buseman, F.. and Casson, W., 1958 "Results of Full Scale Stability Tests on the British 132kV Grid System". Proc. - IEE, 105, 347-362.

4 Shackshaft, G., and Neilson, R., "Results of Stability Tests on an Underexcited 120 MW Generator", ProcIEE, 119, 175-188

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5 Lang, R.D., Hutchinson, B.E. and Yee, H., 1983, "Microprocessor Based Identification System Applied to Synchronous Generators with Voltage Regulators", ProcIEE, 130, 257-265.

6 Electric Power Research Institute. 1981. . ~~~, Compendium of the EPRI Workshop on Modelling for Stability Calculations.

TL,dl :;dl =,i l_l 4d

'kdl 'kd2

F l g u r e 1 E q u i v a l e n t c i r c u i t s f o r the d i r e c t a x i s

7 Walton, A . , 1981, "Determination of Synchronous Machine Stability Study Constants, A Summary of Work done by NE1 Parsons", EPRI Workshop, St. Louis.

8 Croft, J.S., "Microprocessor Based Control Systems for Synchronous Machines", MEng.Sc. Thesis, James Cook University.

(dBIXd

7 - - I - - I 1

log w 1/Td: l / l d ' l/Td: 1/Td'

F i g u r e 2 Frequency response f o r the d i r e c t a x i s

Phase Xd

(degrees)

10-3 10-2 10- 1 1 10 100

Frequency ( H r )

F i g u r e 3 Frequency response of 5OOW machine I n the D a x i s