Identification of synchronous generator saturation models based on on-line digital measurements

J.-C. Wang H.-D. Chiang C.-T. Huang Y.-T. Chen

Indexing terms: Saturated machine modelling, Measurement appronch, Stochastic approximation, Nonlinear models

Abstract: Accurate stability analysis requires the precise modelling of machine saturation pheno- mena. The paper presents a technique suitable for identifying several typical machine saturation models. It is shown that the technique can be used to identify nonlinearities associated with satura- tion and cross-magnetising effects. An algorithm is designed to estimate all parameters of the machine models and the associated saturation functions. Numerical studies on a synchronous generator are included, using online measured data of the Tai- power system recorded by the plant transient recording and analysis system.

1 Introduction

Accurate synchronous generator models allow for the precise calculation of power system stability limits. However, some of the models, such as linear models in databases, may not be reasonable approximations of real equipment. Due to economic considerations, machines may be required to be operated in the magnetically satu- rated region. It has been long recognised that stability calculation requires properly accounting for the iron saturation of the synchronous machines. Various models accounting for the nonlinear effects of machine magnetic saturation have been proposed [l-111. In recent studies of machine saturation models, the saturation effect is usually represented by modifying machine reactances using saturation factors as a function of d and q magnetic fluxes and/or accounting for magnetic coupling between the d and q axis (cross-magnetising effect).

Up to the present, despite its importance, little effort has been made to apply online measured data to identify these models currently available in literature. In this paper, the task of identifying machine saturation models based on online measured data is presented. A sixth- order machine saturation model structure is identified using online neasurements.

In References 12 and 13 an identification technique is developed to estimate the parameters of a sixth-order synchronous generator model based on online measure-

0 IEE, 1995 Paper 1745C (Pl), first received 8th February 1993 and in revised form 20th September 1994 J.-C. Wang and H.-D. Chiang are with the School of Electrical Engin- eering, Cornell University, Ithaca, NY 14853, USA C.-T Huang and Y.-T. Chen are with the System Planning Department, Taiwan Power Company, Taipei, Taiwan, Republic of China

1EE Prof.-Gener. Transm. Distrib., Vol. 142, No. 3, May 1995

ments. In particular, the conjugate gradient based identi- fication technique was shown to be capable of identifying the linear generator model. Since this technique requires the exact gradient of a performance objective, which is very difticult to derive for nonlinear models, this method needs to be modified to cope with the identification of nonlinear models.

Due to a number of technical reasons associated with the problem under study, machine saturation models are identified with a technique based on the stochastic approximation method. Stochastic approximation is a well known recursive procedure for finding extrema of functions. Several merits prompt the application of sto- chastic approximation to the problem of synchronous generator model identification : (i) stochastic approx- imation has strong theoretical support and the recursive procedure provides a consistent estimate of parameters, thus the resulting parameter estimates possess some nice statistical properties such as unbiasedness and asymp- totic normality; (ii) stochastic approximation does not require exact gradient information and it can provide insight into the shape of the performance measure without requiring exact derivatives or a large number of function evaluations. This technique can be applied to finding extrema of non-differentiable functions, and thus can be applied to overcome the nonlinearities of machine models.

2 General model structure for saturated machines

We consider the following nonlinear state-space model which describes synchronous generator dynamical behav- iour :

-- dh(x) - fp (x , U) + W) dt

y = c x + qt) (1) Suppose in addition that the disturbance term Yt) and C(t) are white and small, and the nominal disturbance-free system trajectory is x*(t) when the system input is u*(t). Let Ax@) = x(t) - x*(t). Ay(t),= y(t) - y*(t) and Au(t) = u(t) - u*(t). Then, the linearised system is expressed as

i = A,,(t)x(t) + B,,(t)u(t) + t+(t)

y = Cx(t) + u(t) (2) where x(t) = [Aid Aifd Aild Aiq $il! Ai2,]' is a state vector. u(t) = [AV, Aefd Au, Am] is an input vector. f i t ) = [Aid Aifd AiJT is an output vector.

225

A,(t) = cawx1 - ' (a /ax ) f (x , 4 lx.(t). @(t) and Bdt) = Cah/axl - '(a/au)f(x> U) I=(,), U'( [ ) . Both A,@) and Bp(4 explicitly depend on the parameter vector, p , of the satu- ration models.

The above procedure for he linarisation of nonlinear machine models can be applied to general sixth-order machine model structures, for example, those given in the Appendix. In order to obtain the detailed expressions of A,(t) and B,(t) for the given models, the flux/current relationship should be specifically modelled.

3

Various saturation represents in conjunction with differ- ent model structures [l, 141 have been previously pro- posed. In this Section, a popular sixth-order model structure along with the two saturation representations is employed. Details of the development for the linearised models are contained in the Appendix.

Specific saturated machine model structures

3.1 Model I The voltage equations for a sixth-order model structure and flux/current relationship for Model I are deferred to the Appendix. For modelling the iron saturation effect, X, , and X, , are used to represent the saturated reac- tances, i.e. X, , = cad X , , and X , = cq X,, . c, and coq can be, respectively, approximated by some functions of +,,d and described by

where $ad = Xadu(ifd + i,, - id) and JI, = Xa,(il, + i,, - i,). Xadu and X , , stand for unsaturated rectances.

It can be seen that the fluxes depend nonlinearly on the machine currents due to the fact that X, , and X, , are implicit nonlinear functions of JIod and JI,,,, and thus the fluxes are in turn nonlinear functions of the d and q axis currents.

32 Model11 A different flux/current relationship from the first model is employed in this study, but the voltage equations are no different from before. Both the iron saturation and cross magnetising effects are considered in the model representation.

Let us first consider the saturation effect in machine modelling. In order to accurately represent the saturation effect, X, , and X, , are now modelled as variables that can be properly modified to denote the saturated reac- tances and related to the unsaturated reactances, X,, and X.,", according to X,, = cad X,, and X , = cq X,,, . The variables cad and coq are expressed in terms of machine ampere turns as follows:

where AT, = i,, - i ld - id and AT, = i l , + it,,- i,. Next, the modelling of the cross-magnetising effect is

studied. The effect of the cross-magnetising is to decrease the magnitudes of the magnetic flux components in the d

226

(4)

Linearised saturated machine models around the oper- ating points are then established from the voltage equa- tions and the flux/current relationships (see Appendix). The nonlinear models are identified via the linearised models which can be readily used in problem formula- tion.

4 Problem formulation

In this Section, we present a problem formulation for identifying the saturated generator using online measure- ments. The problem formulation turns out to be a non- linear optimisation problem.

In general, the linearised model based on a series of state-space equations for the original saturation models can be described by the following compact form

i = A,(t)x + Bp(t)u

y = cx

Since input and output signals are sampled at discrete instants, it is necessary to convert the state-space models from continuous- to discrete-time descriptions. Assuming a zero-order hold on the inputs and a given sampling time T, the discrete-time state-space system can be written as

x(n + 1) = mp(n)x(n) + r,(n)u(n) y = cx

where the system matrices are obtained by

By the nature of nonlinearity arising in the derivations, exact computations of @ and r are impossible, one must resort to approximations. Since the function matrix exponentials are involved, Pade approximation can be conveniently used [lS].

Based on the discrete-time state-space form, we can proceed to calculate the transfer function H(z), which takes the form:

Each entry of H(z), Hij(z) = Nij(z)/D(z), i = 1, 2, 3, j = 1, 2, 3, 4, has in general a numerator Nij(z) often coprime with a common nondegenerate denominator D(z) of sixth-order polynomials in the forward shift operator z. If the estimates for the machine model parameter vector, p , is available, those coefficients can be easily computed.

I E E hoc.-Gener. Trarum. Distrib., Vol. 142, No. 3, May I995

According to the above equations, the relations

D(z)AiAz) = N , ,(z)AoAz) + N,,(z)Ae,&)

+ Ni&)A~,(z) + NiAZ)AW

D(z)Ai,Az) = N21(Z)AUAZ) + N22(Z)Aefd(Z)

between inputs and outputs of the generator model are

+ N23(z)Auq(z) + N24(z)Aw

D(z)Aiq(z) = N31(z)Audz) + N32(z)Ae/Az)

+ N 3 3 ( Z ) 4 ( 4 + N 3 4 W w (6) The expressions for D(z) and Ni j ( z ) for the two models are contained in the Appendix.

From the mathematical expression, H(z) is param- eterised by a 90-dimensional coefficient vector 8, which for the two models under study can be found in the Appendix. One can derive the kth predicted value of the direct axis current, field current and quadrature axis current from previous observations of outputs and present in addition to the previous observations of inputs. Let A$, Ai{ and A 3 be the predictions at the kth sample point. The instantaneous errors can be expressed as linear functions of 0 (thus can be regarded as nonlinear functions of p) since the predictions can be given by linear regressions of the following three equations con- sisting of a prediction error model representation:

(7) The weighted least mean square (WLMS) is then taken as a criterion for the model fitness. At the present parameter estimate point, say p. WLMS is

where j = d, f and q. = Ai; - Ai$p), &{@) = Ai{ - A$(p) and dt(p) = Ai8 - A&).

The problem of identifying the saturation machine models then boils down to a nonlinear least squares minimisation problem of finding a parameter vector such that the WLMS error is minimised.

Remarks From the above formulation, an exact explicit formula for gradient vector 8(p) is rather difficult to establish, if not impossible. As a result, neither the standard least squares method nor the maximum likelihood method can be directly applied to the minimisation problem. To resolve the difficulty, a finite difference approximation to the gradient vector (or Hessian matrix) is used, making stochastic approximation suitable for identifying the non- linear machine models.

5 Stochastic approximation

For model parameter estimation, stochastic approx- imation methods create stochastic equivalents to the clas- sical (conjugate) gradient methods. In this section, we discuss asymptotically consistent and efficient stochastic approximation procedures. The task here is to find an estimate for the local minimum of the function in eqn. 8. We briefly describe both gradient type stochastic approx- imation and conjugate gradient type stochastic approx- imation methods for machine saturation model identification.

IEE hoc.-Gener. Transm. Distrib., Vol. 142, No. 3, May 1995

There are two major classes of stochastic approx- imation (SA) methods, i.e. simultaneous perturbation SA and nonsimultaneous perturbation SA [16-181. For the problem at hand, it appears that the nonsimultaneous stochastic perturbation technique is more suitable due to the obvious numerical reason that widespread eigen- values of an associated autocorrelation matrix in the regression model exist.

Given an objective function E@), the nonsimultaneous perturbation stochastic approximation method starts with an estimate po for p , and proceeds to update the estimate by means of the technique of nonsimultaneous perturbation. In contrast to simultaneous perturbations, nonsimultaneous perturbation provides a better approx- imation to the gradient vector and is easier to manipulate for generator parameter estimation based on online digital measurements.

Consider the problem formulated in the preceding Section: we now desire to select a parameter vector, p, which minimises the weighted average error function, 8@). If this function is known and smooth, then the basic Newton procedure can be used to solve for p

Pk = p k - l - ,Pp'(p'- ')Bp(p'-')

where JPp(;) is the Hessian of 4.) and is invertible. However, If &(.) cannot be exactly computed and/or Sp( .) is computationally burdensome, but the system inputs and outputs can be observed and their measure- ments mixed with noise can be taken, then we might try a Monte-Carlo method which is based on a 'noisy' finite difference form for the gradient in the above equation.

To set this up, we need some additional notations and definitions. Let {ck} be a sequence of positive finite differ- ence intervals, approach zero as k + CO and let ei stand for the unit vector in the ith coordinate direction. Also, let p' be the kth estimate of the optimal parameter. Define the two-sided finite difference vectors as D8(pk, ck) and the observation noise 5' by

ith component of Db(p', c') is

[8@' + c'ei) - &(p' - ckei)] 2c'

Dd'(p', c') =

where ck is proportional to (c'), and we have assumed that d have uniformly bounded third derivatives.

The parameters in a finite difference stochastic approximation algorithm is updated according to

a'$?' - ' p k = p k - l -

where 3' = Dd(pk, ck) = (ad//ap) + c'. Of course, one- sided difference and other expressions are also applicable in this formula. The finite difference iteration makes sense even if B,,(.) does not exist at all p, and it has been shown to converge to the desired value under conditions which do not require the existence of the derivative.

The conjugate type SA has the potential of superior performance to the gradient type SA. In the conjugate type SA, the direction vector is produced according to the usual conjugate gradient method, except that the exact gradient vector is replaced with an approximated gradient mixed with the noise process. The conjugate type SA moves along a direction which produces conju- gacy to the previous direction. Although the direction vector is corrupted by an unknown noise process, the iterations in the long run converge to the optimal param-

221

eter estimate. The conjugate gradient type SA algorithm for an unconstrained problem consists of parameter and search direction updating as follows

Pk = p k - 1 - aksk

sk = -$ + 8-1

where U' is a smoothing sequence. In order to achieve good convergence, the smoothing

sequence a' and the perturbation sequence ck are chosen to meet the following conditions [16-181:

m m m 1 a* = a2 1 (sky < CO 1 ukck < m ck - 0 k = 1 k = l k = l

f ($), < a2 k = I

From a practical implementation viewpoint, the optimum choice of {ak} and { ck } are the form uk = a/k" and ck = c/ky, where 0.75 < a < 1 and 1 - a < y < U - 0.5. Both algorithms stop whenever the [,-norm of the gradient vector at the current iterate is less than 0.1% and the I,-norm of the initial gradient vector.

6 Numerical results

At 10:36 a.m., September 25, 1991, a 345 kV transmission line between Hsinta G/S and Lungchi E/S was subject to a single phase-to-ground fault caused by lightning. The fault lasted for 5 cycles and was cleared properly and the circuit was automatically restored. Instantaneous ter- minal voltages, armature currents, speed, power angle, field voltage and current were captured during the occurrence by a plant transient recorder and analysis (PTRA) system [19] . These data are used for the sub- sequent model parameter estimation for the synchronous generator rated at 666.75 MVA, 22 kV, 3600 RPM and 0.85 PF.

The algorithm, developed in Section 5, has been applied to synchronous machine modelling. Using the conjugate gradient SA algorithm with the smoothing sequence 0.98/ko.' and the perturbation sequence 0.01 3/ k0.35, two typical saturation machine models were derived based on online recorded data. Parameters of Model I are identified using data set. From a typical run of the proposed algorithm, the resulting model param- eters are given in Table 1. Several graphs describing this implementation results are shown in Figs. 1-6, where the

Table 1 : The parameters of Model I

Machine Values Machine values parameters (P.u.) parameters (P.u.)

X' 0.120 a, 0.993 x e d " 1.3041 a, -0.01 54 ' t d 0.1035 a2 0.1 49 x,,, 0.00001 a3 0.0598 XI d 0.096 a4 -0.0596 X,," 1.2563 bo 0.987

x*, 0.0834 b, 0.3741 Re 0.0032 b, 0.485 Rtd 0.001 b, 0.1 89 R ? d 0.01 69 X,, 1.290 R I D 0.0111 x,, 1.1 59 4. 0.0641 Ed 0.024 558% E'@ 0.005 61 % E' 0.02240% I 0.009 166

XI s 0.3213 b, 0.0145

Weighting factors w,+ = 1 .E, w, = 0.1. w.. = 0.6

model responses are compared with the measured outputs. These figures show the fitness of simulation to the measurement and the low amplitudes of the residual signals. For this identified model, the misadjustments E ~ ,

0 63 0 62 L A

,O 61

' 0 60 0 59 0 58

0 100 200 300 400 500 600 a samples

2 2 , I

1 4 1 I 0 100 200 300 400 500 600

samples b

Fig. 1 a Direct-axis voltage b Field voltage

Direct axis voltage andfield voltage

0721 I

0 70

0 68

0 66 U

0 641 0 100 200 300 400 500 600 a samples

1 0 100 200 300 400 500 600

samples b

Quadrature axis uoltage andfrequency deuiation Fig. 2 Quadature-axis voltage Angular frequency Ructualions

v.21 I

0 8 0 7 0 6 0 5

-G

0 100 200 300 400 500 600 a samples

A " , V J

0 8 0 7

' 0 6 0 5

0 100 200 300 400 500 600 b samples

Fig. 3 P Simulated direct axis currcnt b Measured direct axis current

Simulated and measured direct axis currents

IEE Proc.-Gener. Tranrm. Distrib., Vol. 142, No. 3, May 1995 228

E, and venience, their values are given in Table 1.

are calculated according to eqn. 9, and for con-

Simulations are also performed for Model 11. The conju- gate gradient SA algoirithm is implemented with the

O 100 200 300 400 500 600 a samples

1 L , 1 1 3 5 1 3

'1 25

1 2

U

115 1 0 100 200 300 400 500 EO0 b somples Direct axis current estimation error and simultatedfield current Fig. 4

a Direct-axis current estimation error b Simulated field current

1 L ,

1 1 -

11 0 100 200 300 400 500 600 a samples

4 . \I

'0 2 v = " O a -2

0 100 200 300 400 500 600 b mmples

Fig. 5 a Measured field current b Field current estimation error

Measuredfield current andfield current estimation error

0 100 200 300 400 500 600 a samples

-4 I 0 100 200 300 400 500 600

samples Quadrature axis current by estimation and measurement Fig. 6

a Measured field current b Field current

IEE Proc.-Gener. Transm. Distrib., Vol. 142, No. 3, May 1995

smoothing sequence ak = 0.95/k0.', and the perturbation sequence cx = 0.01/k0.37. The parameters of this model are identified. It can be seen that the model currents are close to the recorded machine output signals. Evidently, the model outputs are similar to those presented earlier and for simplicity are not presented. For comparison purposes, the misadjustments, e d , E, and c q , are displayed together with the identified model parameters in Table 2.

Table 2: The Darameten of Model II

Machine Values Machine Values parameters (P.u.) parameters (P.u.)

X. 0.120 cod 0.374 X,," 1.3041 coq 0.293 x,, 0.1035 zd 0.259

x, d 0.096 yd 0.259 x*,, 1.2563 y q 0.163 X. - 0.3213 AT-. 0.623

x#kd 0.00001 aq 0.133

0.0834 AT:: 0.561 0.0032 X,, 1.286

x;: R* R,d 0.001 x,, 1.1 533 R . , 0.01 69 Rt.a 0.01 1 1 R20 0.0641 Ed 0.023 77% E'd 0.005 448% E" 0.021 30% 8 0.008 459

Weighting factors wd = 1.8, w, = 0.1, w , = 0.6

It has been additionally verified that, for both of the test cases, saturation models slightly better represent, in terms of WMSE, the machine dynamic behaviour than the unsaturated models.

A comparison between the above two saturation rep- resentations is performed based on the calculated weighted average of the instantaneous errors. As shown in both Table 1 and Table 2, the second saturation model yields lower model errors than the first model. The differ- ence might be attributed to the machine cross- magnetising effects of the observed dynamic behaviours.

Extensive computer simulations have been carried out in order to evaluate the convergence behaviour and to assess the overall adaptive performance of the algorithm. In a typical run, the weighted mean square error in dB [-20 log (41 at each iteration is stored. Fig. 3 depicts a learning curve in which WMSE was plotted against the number of iterations. Fig. 3 also shows the behaviour of the algorithm when the second machine model is identi- fied.

From the above simulation studies, the following observations are made:

(i) Accurate saturated machine models for the Taipo- wer system based on on-line digital measurement were developed.

(ii) Quantitative residual assessments to determine the model fitness and to evaluate the accuracy of the derived model were performed.

(iii) The usefulness of the direct online measurements of the machine transient behaviours for model identifica- tion is illustrated.

(iv) An identification technique suitable for online measurement-based parameter estimation for saturated models of the synchonous machines is developed.

(v) A suitable algorithm for deriving machine model parameters using online data recorded during a fault occurrence is implemented.

229

7 Conclusions

100

90

80

70

6 0 -

50

40

3 0

Online direct measurement of machine transient behav- iour has been proved to be useful for the accurate deriva- tion of saturated machine model parameters. The

-

-

-

-

-

-

“ I I

-8 1 I 0 100 200 300 400 500 600

samples

Fig. 7 Quadrature current estimation error

- _ 0 2 4 6 8 10 1 2 1 4

iterat ions

Fig. 8 Convergence of the proposed algorithm

identification of various typical generator models from online measured data has been considered and formu- lated as optimisation problems of minimising the weighted least mean squares of the instantaneous errors between observation and prediction responses. A tech- nique suitable for nonlinear model identification has been proposed, analysed and implemented. Numerical studies have been performed to verify the utility of the developed technique to effectively identify the nonlinear machine models in the Taipower system. Applying the online mea- sured data from the plant transient recorder and analysis system, parameters associated with two typical saturation generator models have been estimated to accurately rep- resent the synchronous generators in the Taipower systems. In addition, it has been verified via numerical studies that the saturated machine models can better rep- resent the machine dynamic behaviour than unsaturated machine models.

8 References

1 EL-SERAFI, A.M., ABDALLAH, AS., EL-SHERBINY, M.K., and BADAWY, E.H.: ‘Experimental study of the saturation and the crossmagnetizing phenomenon in saturated synchronous machines’, IEEE Trans. on Energy Conversion, 1988,3, (4), pp. 815-823

2 EL-SERAFI, A.M., and ABDALLAH, AS.: ‘Saturated synchronous reactances of synchronous machines’, IEEE Trans. on Energy Con- uersion, 1992, 7, (3), pp. 570-579

230

3 VAS, P., HALLENIUS, K.E., and BROWN, J.E.: ‘Cross-saturation in smooth-air-gap electrical machines’, IEEE Trans., 1986, EC-1, (l), pp. 103-112

4 XIE, G., and RAMSHAW, R.S.: ‘Nonlinear model of synchronous machines with saliency’, IEEE Trans., 1986, EC-1, (3), pp. 198-203

5 SHACKSHAFT, G., and HENSER, P.B.: ‘Model of generator satu- ration for use in power system studies’, IEE Proc., 1979, 126, (E), pp. 759-163

6 ABDEL-HALIM, M.A., and MANNING, C.D.: ‘Modeling satura- tion of laminated salient-pole synchronous machines’, IEE Proc. B, 1987,134, (4), pp. 215-223

7 SLEMON, G.R.: ‘Analytical models for saturated synchronous machines’, IEEE Trans., 1971, PASW, (2), pp. 409-417

8 RAMSHAW, R.S., and XIE, G.: ‘Nonlinear model of nonsalient synchronous machines’, IEEE Trans., 1984, PAS-103, (7), pp. 1809- 1815

9 BRANDWAJN, V.: ‘Representation of magnetic saturation in the synchronous machine model in an electromagnetic transients program’, IEEE Trans., 1980, PAS-99, (S), pp. 1996-2001

10 SAUER, P.W.: ‘Constraints on saturation modelling in AC machines’, IEEE Trans. on Energy Conversion, 1992, 7 , (l) , pp. 161-167

1 1 HARLEY, R.G., LIMEBEER, D.J.N., and CHIRRICOZZI, E.: ‘Comnarative studv of saturation methods in synchronous machine modeis’, I E E Proc.-B, 1980, 127, (1). 1-7

12 HUANG, C.T., CHEN, Y.T., CHANG, C.L., HUANG, C.Y., CHIANG. H.D.. and WANG. J.C. : ‘On-line measurement based model parameter estimation for synchronous generators: model development and identificaton schemes’, IEEE Trans. on Energy Conversion, 1994, 9, (2). pp. 330-336

13 WANG, J.C., CHIANG, H.D., HUANG, C.T., CHEN, Y.T., CHANG, C.L., and CHIOU, C.Y.: ‘On-line measurement based model parameter estimation for synchronous generators: numerical studies’, IEEE Trans. on Energy Conversion, 1994.9, (2). pp. 337-343

14 STYBLINSKL M.A., and TANG, T.S.: ‘Experiments in nonconvex optimization: stochastic approximation with function smoothing and simulated annealing’, Neural Networks, 1990,3, pp. 467-483

15 GOLUB, G.H., and VAN LOAN, C.F.: ‘Matrix computations’(The Johns Hopkins University Press, 1989)

16 RUSZCZYNSKI, A., and SYSKI, W.: ‘Stochastic approximation method with gradient averaging for unconstrained problems’, IEEE Trans., 1983, AC-28, (12), pp. 1097-1105

17 FABIAN, V.: ‘On asymptotic normality in stochastic approx- imation’, Annals Math. Stat., 1968.39, (4), pp. 1327-1332

18 DUPUIS, P., and SIMHA, R.: ‘On sampling controlled stochastic approximation’, IEEE Trans. on Automatic Control, 1991, 36, (8), pp. 915-924

19 CHIOU, C.T., HAUNG, C.H., CHEN, Y.T., HUANG, C.T., and LIN, C.J.: ‘A powerful personal computer-based plant transient recording and analysis system’, IEEE Trans. on Power Systems, 1993,8, (3), pp. 849-857

9 Appendix

9.1 A model structure for synchronous generators The synchronous machine model is usually represented by a R-L lumped parameter equivalent circuit. For one damper winding on the d axis, two dampers on the q axis and the d-q axis fluxes coupled, the model differential equations for synchronous machines are assumed described by a set of differential equations [ 11 :

I*= w, dt - R f d i f d + e f d

--= “Id -R, , i ld + uld 0, dt

IEE Proc.-Genm. Transm. Distrib., Vol. 142, No. 3, May 1995

where w and w, are, respectively, rotor angular speed and base angular frequency in radians per second. All other quantities are in the per unit system. The reciprocal system has been adopted in converting nonrated quan- tities to rated quantities.

TO incorporate the saturation and cross-magnetising

follows:

*d = Xdd(-id) + x4di/d + Xadi ld

$,.d = Xa,,( - id) + Xfdd if, + cd X,, ild

$ id = xad(-id) + cdxadi/d + x l i d i l d

M =

effects, the -relationship between flux and current expressed as follows:

$, = xqq(- i,) + x,, i,, + x,, i,, $1, = x a q ( - i q ) + x l l q i l q + X a q i Z q

(13)

Linearised flux/current formula is given by AY = SAI.

$d = xdd(-id) f xadi/d + xodi ld - SAY)

$fd = xad(-id) + X f d d i/d + cdxad i l d - $2 , = Xaq( - in) + x,, 4, + xzz, iz,

where x , d = Cad x,,du and x,, = Cas fYoqL!. $ i d = x a k i d ) + cdXodi /d + x i l d i l d -

- R, 0 0

0 - RJd 0

- xddl -- xi -- Xi

0 0 0

0 0 -Rid W 0 U

ws ws U,

- 0 0 0

S is expressed as follows: $, = Xqq( - i,) + X., il, + X,, izq - S,(Y)

0 0 -x,,1 xz xz 0 0 -xz Xll,, xz 0 0 -xz xz XZZ,,

where = Lid, $/d, ,$ld, iq, $ I , ? $ZqlY. xdd, x/,dd3 x i i d , X,, , Xi,, and X Z 2 , in the above equations are given by

where xdd = xL + xad

Xfdd = x/d f Xfkd + xad x l = xodO

= x l d + Xfkd + xad

x,, = XL + x,, Xll, = xi, + x,, x z z , = xz, + x,,

Given a flux current relationship, the small signal behav- iour between flux and current can be established. To this end, let us define the following variables: AY = [AGd

Ail9 Aizq]. Then, in general, we can write AY = SAI. Now we are in a position to obtain the overall d-q

axis state-space representation for a balanced symmetri- cal three-phase, synchronous machine with a field winding and three damper windings on the rotor in matrix/vector form. This representation is a combination of the dynamic eqn. 10 and the flux linkages/current relationships (expr. 11). The whole system equation can be written in compact form as:

A$/d A $ l d A$, A$Lq A$ZqlT and A I = [Aid Ai,.,, Ai,

f = A,x + B,u

y = c x

where x E R6 = AI is a state vector. U = [AV, he,., AV, AwlT is an input vector. y = [Aid Aifd AiqIr is an output vector. A , = w,S- 'M and B, = w,S-'B.

B =

In the above equations, p is a parameter vector, which along with matrices, S and M , vary with the extent of machine saturation and should be specified later after the saturation representations are given.

9.2 A linearised fluxlcurrent relationship for Model I For this model, the flux/current relationship is given as

I E E Proc.-Gener. Transm. Distrib., Vol. 142, No. 3, May 1995

9.3 A linearised fluxlcurrent relationship for Model 11 For this second model, the fluxlcurrent relationship is described as follows:

$d = x d d ( - id) + xad

$fd = xad(-id) + X J d d i f d + C d X a d i l d + $dq

$Id = X a A - i d ) + CdXmfiJd + X l l d i l d + $dq

$q = X ~ q ( - ~ q ) + x q i l q + x q i Z q - $46

$ i q = X a q ( - i q ) + x l l q i ~ q

+ i l d + $dq

xaqi2q - $qd

$2q = Xoq(-iq) + X q i l q + X 2 2 q i z q - $qd (14)

Linearised flux/current formula is then given by AY = SAI. S takes on the form:

- x d d l XI x1 0 0 0 xf fd l cdxl

-xl c d x l x l l d l 0 %, j s = [ 0 0 0 -Xqq1 x2

0 0 0 -x2 X 1 1 , l x2

0 0 0 -x, x2 X 2 Z q l

where

= XodO - adXadu(iJdO + j l d 0 - idO)

+ YqA-iqo + i iqo + i zqo + cog)

Xz = X,,, - aSX.,.(ilq~ + ~Z,O - i d

- Y d q ( t - ifdo - j l d 0 - cod)

XI2 = YqdifdO + i l d 0 - id01

XZl = -ydq(-iqo + i,,, + i,,,) (15)

M can be derived based on the right-hand side of eqn. 14 and the expressions for l(ld and $q in terms of d-q axis

currents and field current.

0 0

M =

0 0 0 0

0 0 0 0 0 0

R o 0 0

xddl, xfddl, x l l d l , xqql , x l l q l and x 2 Z q l in the above equations are defined by the following:

Xddl = xL + xl

xfddl = xJd + xJkd + x l l d l = x l d + X f k d + x,,, = X L + x, xllql = Xl, + x, X 2 2 , l = x 2 , + x,

For this model, the parameter vector to be identified is P = x d u , x f d , x f W , xaqt,, x 2 q , Ro? RJd,

R i d , RI,, Rz,, a,, al, a2, a,, a4, bo, b,, b z , b,, b,, Xed, X.ql T.

232 IEE Proc.-Genm. Transm. Disrrib., Vol. 142, No. 3, May 1995