A frequency-domain maximum likelihood estimation of ... - IEEE Xplore

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IEEE Transactions on Energy Conversion, Vol. 7.No. 3, September 1992.

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A Frequency-Domain Maximum Likelihood Estimation of Synchronous Machine High-Order Models Using SSFR Test Data I . KAMWA Institut de Recherche d' Hydro-Qdbe'c C.P.1000, Varennes, Canada, J3X IS1

P . VIAROUGE H . LE-HW Electrical Engineering Department Universite'Laval, PQ, Canada GlKIP4

ABSTRACT- This paper proposes a more rigorous and effective numerical scheme for processing noisy signals originating from SSFR tests on synchronous machines. Instead of using an univariate nonlinear least-squares procedure to fit only the weighted sum of magnitude responses, we minimize a multivariate predictionerror criterion based on the determinantof the residuals covariance matrix: such an approach is the only one permitting simultaneous matching of themagnitudeandphaseof ail thestator and Reid harmonic rmponsea with a unique equivalent circuit, of possibly high order. The algorithm pertains to the large class of predktion error methods (PEM) and results in a multiresponse nonlinear regressionprocedurerelated to the maximum likelihood view-point when the residuals distribution is Gaussian.

To demonstrate the efficiencyof the proposed scheme, we first tested our Implementation using noisy Simulated data, based on the Rockport model 3.3. It Is shown, using actual data from the Nanticoke turbogenerator, dating back to the EPRi-project RP9997-2 (l980), that the frequencydomain maximum likelihood approach can be effective for direct estimation of generalized circuits with up to five equivalent windings per axis, providing satisfactorypredictionsof both magnitude and phase as far as the 16th harmonic.

I. Introduction Standstill frequency response (SSFR) testing for synchronous machine modeling presents specific advantages which during recent years have strengthened its importance when compared with a more conventional approach like the unloaded short-circuit test [I],as well as the newly developed timedomain standstill tests with step [2] or pulse-width modulated excitations 131. Besides its inherently strong physical significance, one can credit the following advantages to the SSFR method : It is easy to implement at the factory or during outages for routine maintenancewithout risk to the machine,since the tests involve very little power. The ready availability of modem spectrum analyzers and "powerful" computer tools have eased the data logging and analysis procedures. Unlike the ANSI-standardizedshort-circuit test, the SSFR approach can simultaneously provided the equivalent circuits for both direct and quadrature axes, and at the present time, seems the most appropriate for modeling the machine behaviour at super-synchronous frequencies. One must be careful however not to overestimate the advantages of the SSFR method since standstill time-domain tests involving randomly modulated excitation also exhibit all the aforementionedproperties.

92 WM 299-8 EC A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for presentation at the IEEE/ PES 1992 Winter Meeting, New York, New York, January 2630, 1992. Manuscript submitted August 29, 1991; made available for printing December 18, 1991.

J. DICKINSON Mechanical Engineering Department Universite'hval, PQ, Canada GlKIP4

Furthermore, it has been shown recently that the latter approach fits more closely within the framework of modem system identification theory [3-41,and can therefore take full advantage of all the analytical and numerical tools developed during decades of investigation in this growing field [5-71. For instance, the time-domain perspective easily deals with issues like minimal realizations of the dynamic responses, identifiability of a given model from a given set of input-output observations [7-81 and, through modem filtering theory, can cope more effectively with model and measurement noise [61. A time-domain identification test actually provides several thousand samplesduring one experiment, involving at the same time all the natural frequencies of the underlying system, in terms of both amplitude and phase effects. This results in a high level of inter-frequency redundancy, with a much greater effectivesignal to noise ratio (SNR)than can be obtained from a hundred or so discrete frequency steps involved in a typical SSFR test. Our hope is that, the electric machinery community will fully recognize in the near future, the various merits of stochastic standstill testing for equivalent circuits estimation. Meanwhile,it seems worthwhileto work towards improved methods,to deal in the best possible way with actual SSFR data, since they are increasingly availablefrom joint efforts between utilities, manufacturers and universities [4][9].At the present time, several procedures exist for extracting equivalent circuits with three or less equivalent rotor windings from SSFR data. The most obvious one was fist described in a very lucid manner by Umans et a1 [lo].In their approach, curve fitting techniques are used to derive the d-axis and q-axis transfer functions from test data. The parameters of the assumed equivalent circuit structure are then calculated from a set of nonlinear equations relating these parameters to the time constants corresponding to the derived transfer functions. It turns out that the relevant set of nonlinear equations 1111 are redundant, with the number of unknowns being one less than the number of equations. If these equations are consistent with each other, approximate closed-form solutions can be derived [9-lo], and even optimal equivalent circuit parameters can be obtained through the Broyden method of solving a system of nonlinear equations [12].Unfortunately, as pointed out in [ll],the unavoidable effects of noise on the time constant estimates, render the overdetermined system inconsistent, which adversely affects the accuracy and the uniqueness of the resulting circuit model. To avoid this pitfall, it has been proposedfirstly to construct time responses of the machine using the derived uansfer functions and then to estimate a unique equivalent circuit model by applying the highly efficient maximum likelihood method to this simulated data [4].At fist sight, this approach is attractive, since it effectively deals with the problem of extracting a single set of equivalent circuits from a given set of time constants. However, the simulated time responses used to initiate the maximum likelihood algorithm are obtained from those time constants which are actually suspected to be inaccurate due to noise effects. Therefore it is not really clear that a perfect fit to these questionable simulated responses will lead to an optimal fitting of noisy real life SSFR data. In fact according to Umans, in some cases, a typical curve-fitting routine applied to two separate sets of SSFR data differing by

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less than 2%,can result in time constantsfor the same generator differing by over 100%. This suggests that due to noise effects, the uansfer function estimated by univariate nonlinear leastsquares computer programs [lo-111 suffers from a nonuniqueness problem, and its subsequent use is only justifiable if the resulting circuits provide a satisfactory after-the-fact fitting of actual noisy SSFR data. Faced with this evidence, it appears to us that the method suggested in the appendix of IEEE standard 115-B [13], and briefly illustrated by Dandeno and Service in their discussion of [4] seems simpler and also more direct. This approach now in use at Ontario Hydro, was theoretically justified in [141. Basically, it assumes an equivalent circuit model and predicts its frequencyresponse for a given set of parametervalues, which represent our best knowledge up-to date. The differences between this prediction of (say) xd(jw)and the measured Rd(jw) are evaluated for many values of w and the weighted sum (E) of their amplitude is calculated. If E is not small enough, adjustments are made to the circuit parameters in such a way as to reduce its value. Practically,E is a prediction error criterion stated in an ad-hoc fashion, which measures our level of trust in a given set of parameter values. By iteratively minimizing such a cost function through the use of a specialized computer program, it is possible to achieve a good match between the circuit model and actual SSFR magnitude measurements. The presentpaperfollows this direct approachwhilstaiming at improving its efficiency through the development of a prediction error criterion with a better theoretical foundation. The most obvious advantage of the new scheme stems from the fact that it easily copes with all the amplitude and phase responses involved in a typical SSFR experiment. This better utilization of the available data, added to the inherent ability of the maximum likelihood scheme to deal with random errors, makes it possible to identify equivalent circuits with more than three equivalent rotor windings, which typically achieve better accuracy in representing both the magnitude and phase characteristics of the noisy SSFR data even up to the 16th harmonic (1kHz). Such high order representations are becoming increasingly important in fast transient studies through the EMTP [15], or harmonic prediction using dedicated software U61. In the following section,the PEM for nonlinear multivariate regression is introduced and its essential properties are briefly discussed. Subsequently, its connection with the analysis of synchronous machine SSFR data is established. Section IV deals with implementation issues, while the validation of the scheme using noisy simulated data of the Rockport model 3.3 141 is considered in section V. We particularly pay attention to the effectof varying signal to noise ratios (from 20 to 40db) and starting conditions on the estimates at convergence. In section VI, actual SSFR data from the Nanticoke turbogenerator are thoroughly investigated. It turns out that the new methsd of analysis leads to equivalent circuits which are more accurate than those reported previously. In fact, the newly estimated circuits disclose an extended range of validity and an improved fitting of the various phase measurements.

11. The Frequency-Domain Maximum Likelihood Method A SSFR experiment normally produces for a set 52 = (U,, i=l,...JV) of excitation frequencies, a corresponding set of harmonic responses (zi, i=Z ,...A).In the general case where we are seeking p transfer functions to determine uniquely the system input-output behaviour, z, should be a 2p-dmensional vector, since each response involves both magnitude and phase measurements:

Let us suppose now that the system which generates the actual data (l), can be represented by a mathematical model, denoted by H ( 0 , jw), where 0 is a q-dimensional vector param:terizing the model in an appropriate way. For a given value 0 of 0, the model permits one to evaluate at each frequency step, a value iiof the system frequency response differing from the actual one by an error E ~ which , dependents on both the measurement accuracy and the physical validity of the assumed model structure and parameters values: z, = H ( 6 , COi)

+ E, = ii(6)+ Ei

(2)

For dynamicalidentification experiments, the index i stands for the time instead of the frequency. It is then nztural to call E, the prediction error, since f i looks like a prevision of what the actual data should be, solely on the grounds of our best knowledgeabout the structure and the parameters of the model. However, when we are dealing with a general regression problem, the practice is simply to call &,.theresidual error. If for computational reasons, we collect all the available data in a Nx2p measurement matrix (Z), a correspondingNx2presiduals matrix builds up as follows: E(@=[:\=[

z1

-i@) ... !;Z-z(6)

z',

-

The sample covariance of 1 N

D(6) =- C

Nt=i

(3)

?;(e) ( E ~ )is

given by:

E ~ ( ~ ) E T (=-E(6)TE(6) ~ )N1

(4)

Since the aim of a model H(0jw) is to provide values ii which match the actual data, it seems reasonable to propose that a good model should have small residual errors. This suggests the use of a positive scalar function of D(0) as an estimation criterion. This is the basic consideration involved in the socalled prediction error method for system identification. To proceed further, for mathematical tractability, we will assumed that the assumed model structure H ( 0 j o ) is correct; that is to say, that there existsa (unique) parameter value 0, such thatH(0,jo) is the true mathematical description of the system under study. This hypothesis is very strong, but it permits us to treat the residual errors asessentiallymeasurementerrors. Since such type of errors are basically random, we can make the calculations even easier by further assuming that they are normally distributed,with zero mean and covariance C.Under this Gaussian hypothesis, the likelihood function of the data can be derived from the Bayes rule, and results in the following familiar expression [7][17]:

with m=pN. When the parameter vector 0 is such that p(Z I Q, 8) is maximized,thecorresponding model is very likely,according to actual data. Since the natural logarithm function is monotonic, the maximization of the likelihood function with respect to 8, is equivalent to the minimization of -Log(p(Z I R,8)). After simplification, it is found that for a sufficiently large set of data, the minimization of -Log(p(Z I Q,0)) is in fact equivalent to the minimization of the following cost function [71:

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where the covariance C has been replaced by its sample D(8) as defined in (4). It clearly turns out that the maximum likelihood estimator corresponds to the value of 8 which minimizes a scalar measure V(9) of the sample covariance matrix. Therefore, maximum Uelihood estimation is basically a PEM. Its close relationship with the least squares estimation, which is actually the most frequently used PEM, is well documented [17]. Interestingly,the expression (6) is very close to the cost function used by Keyani et A1[4] for time-domain estimation, the main difference being as to how the covariance matrix of the residuals (or prediction errors) is calculated. In both cases, solving the minimization problem leads to a maximum likelihood estimate of 0. But the above frequency domain criterion is much more attractive when the identification is based by necessity on SSFR experiments.

III. Application to Synchronous Machine

dd(s) = sxpl(s)+ rrrN(s = sxh(s) + r,.rdd(s)

then the field input admittance of the synchronous machine takes the form :

(12b)

r.

x. I

The equivalent circuits derived from the d-q theory constitute the most versatile model of synchronous machines used in computer studies. For a given structure including nd and nq equivalent windings in the rotor representation (Fig. l), the estimation problem consists of detem.lining the following vectors:

(7) which correspond to the d and q axes respectively. When using SSFR to achieve this purpose, the truly measurable machine variables are the transfer (or admittance) functions: y f ( 8 , , s ) = C , [ s-A,(8,)J1B,(8,> l 1 = d,q

(8)

where the state matrices A,, B1,and C, are defined in appendix. If one represents the three classical operational inductances xd(s). G(s) and xq(s) by the following rational fractions :

then, the elements of the matrix Y,,representing the machine admittances at standstill, satisfy the following relationships :

(lob) with dd(s) = Pxd,,(s) + Tax&)

; dq(s)= sxp(s)-r-

(1oC)

At this stage, it is impossible to express the field input admittance immediately in terms of the three standard operational inductances. However, if, on an ad hoc basis, one introduces a new operator xpl x (1 +sTj)(l +STj)...(l + s q d + ' )

?l/(s = - = XH

un(1 +~T;J(~+~T;)...(I+set+')

satisfying the relationship

(1 1)

(1h)

... 1

Figure 1.Generalized Equivalent Circuit Model. (a) = @,Vd; (b)eq = %vq This new operator X ~ S )obviously plays for the field winding, a role similar to the stator operational inductaucexd(s). A more rigorous justification and derivation of this new inductance can be found in [27]. As a matter of fact, xds) does not intervene inthe calculationoftheequivalent circuitsin terms of dynamic time constants, according to methods published up to now [9-111. The difficulty in calculating circuits to fit the field current during rotor perturbation appears as a simple consequence of ignoring this truly informative piece of data during the calculations [3][25]. The test bed and the measurement procedure recommended forthe practical determination of the admittancematrix (8)have already been extensively documented in the literature[13][19]. Theoretically, this involves knowledge of (Ydd(s), yfd(s),yds)) in the d-axis @=3),and that of {yqq(s)}in the q-axis @=I). Since these admittances are linked to operational impedances through one-to-one relationships (cf. equations 10 and 12), we can assume, without loss of generality, the following model for the data: Hd(8d,s ) = [xd(ed,s ) G(ed,s ) x,(ed,s )f

(13~)

H,@,,s)= xq(0,,s)

(136)

According to this model, when the parameter vector 8,is given,avaluei: iscomputedat a frequencyq by firstevaluating the admittance matrix (8) from the state-space description.The numerator and denominator polynomials of the impedances included in the model are then determined using equations (lo)-(12) and the resulting operator evaluated for s = j q . On the experimental side, it is not very stringent to suppose that the measured impedance data are properly per-unitized and converted when necessary in the operational inductance form. This results in an observation vector expressed as follows:

zf = [ X d ( j o i )

G(joi)

x , ( J o ; ) ~ ; Fy = [x,(~O;)]

(14)

where the superscript refers to the axis. From themodel equation (13) and measurement vector (14), it is easy to see that the specific problemof estimating synchronousmachine equivalent circuits using SSFR data has been cast in the general form stated in the previous section. The procedure for evaluating the cost function (6) can now be summarized (f=d,q):

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(1) For each frequency the experimental data are transformed to per-unit inductance values, and collected in a 2pdimensional measurement vector z, consisting of the amplitude and phase of the p elements of T f . (2) Given the parameter vector 61,the modelH,(*, is evaluated at each frequency, and the resulting amplitude and phase information collected in the 2p-dimensional vector z^: . (3) The difference between these two vectors for all (mi, 1 < i < N}, gives the residuals sequence {E, ) , from which the sample covariance D(6,) is determined and subsequently, the cost function (6).

memory resources, and this software, together with the Nelder and Mead computational procedure, has formed the basic numerical tool in the following investigations.

V. Validation with Noisy Simulated data

0 )

IV.Numerical Implementation Issues The attractivenessof the univariate nonlinear least-squares formulation for SSFR data analysis comes from the wide availability of Levenberg-Marquardt [221 and Gauss-Newton [20] computercodes needed to solve the resulting minimization problem. By contrast, we are not aware of a public domain software for determining the minimum of objective functions of the type (6). Basically, this scarcity of proven computer resources can explain why the multiresponse nonlinear regression is so often reduced to an univariate one, by constructing the ith residuals as a weighted sum of the individual responses mismatches [1][141. In order to facilitate the maximum likelihood estimation, we really need to develop a customized program, based on general methods of minimizing a multivariable function such as V ( e ,0).Two basic approaches exist for solving this problem. The first is less efficient,but does notrequire the evaluation of derivatives, whereas the second exhibits fast local convergence properties, but requires fiist and second derivatives, or some corresponding approximation. Among the most widely used methods in the first category, are the Rosenbrock scheme, the Hooke and Jeeves direct search procedure and the Nelder and Mead simplex method [23-241.In the present work, we have made extensive use of the later, mainly because it was available in our computer library. We have found this type of method to be useful when the actual estimate of the parameter is far from the exact solution. Once one is sufficiently close to the minimum, the so-called Newton-type methods based on V(e,R), gradient (G) and Hessian (H) G (e,R) =

awe, R) ~

ae

a2v(e,n) H(e,R)=--;,

ae

Equivalents Circuits SNR

(15b) omitting the division by N . This simplified prediction error criterion has been previously derived through Bayesian argument [211 and is used in the experimental design of nonlinear models for chemical systems, under the name of D-optimal design criterion [81[221. Ignoring the log term makes the estimation less robust, since the new cost function varies more rapidly [171. However a computation scheme for efficiently calculating the gradient and Hessian of (15) proposed by Bates and Watts [221, provides computational advantages from the simplified criterion which more than compensate for the reduced robustness. As a matter of fact, we have successfully implemented their scheme on apersonal computer with limited

2OdB

6OdB

Dynamic Time Constants 2OdB

Tme

xf

2.10560 2.0996 0.02419 r, 2.03000 2.0300 0.00125 XD, 1.93360 1.9276 0.27284 rD, 1.85800 1.8580 0.04035 Xkf, 0.00169 0.00169 0.00000 x D ~ 0.17200 0.1720 0.00000 fD2 0.31580 0.29780 0.00673 Xkn 0.00081 0.00081 0.24058

6OdB

True

1.08290 1.0830 1.08260 0.02367 0.02474 0.02464 0.00761 0.00762 0.00764 4.66610 4.67430 4.67050 0.03093 0.03234 0.03215 0.00767 0.00768 0.00767 TD, 0.01720 0.01793 0.01794 TD2 0.00002 0.00002 0.00000 Tj T,” T: TA Ti TL

Table 2. Estimated and Initial parameters of the Rockport unit. Random initialization (rotor 3.3)[4].

(15~)

are better suited, since then they converge quadratically to the local minimum. Unfortunately,the need to calculate H and G by numerical finite differences [12] adds to the already high computational burden. We have first considered alleviating this burden by simplifying the cost function V(0,R). An obvious way to do this is to ignore the logarithm evaluation,by keeping only the determinant of the sample covariance [ 171: V ( e ,R) = detET(B,R)E(8, R)

In order to validate the frequency-domain maximum likelihood concept and also to test the minimization algorithms involved for numerical accuracy,synthetic frequency response data were generated using model 3.3 parameters derived by Keyani et A1 [4] for the Rockport 1300MW-turbogenerator.The resulting harmonic responses developed for N= 100 frequency steps logarithmically spaced between 0.001Hz and 1000Hz, were then corrupted with uniformly distributed noise { E ~ , i=l,...Ju) of zero mean and varying signal to noise ratios. Both noisy amplitude and phase data where used for the maximum likelihood estimation, which thus implied Nx6 and Nx2 measurement matrices respectively for the d- and q-axes. The correct model structure was retained so that any discrepancies observed in the recalculated values of the machine parameters, could be specifically ascribed to the noise introduced into the data, or to shortcomings in the optimization procedure. For each S N R level,the estimation procedure was initialized in two ways, since we know from previous findings that the solution of the minimization problem often depend on the initial estimates of the parameters [41. Table 1. Estimated and true parameters of the Rockport unit. Biased initialization (rotor 3.3).

Equivalents Circuits SNR 20dB 6OdB

x,

lnitial

0.02450 0.02465 0.00100 r, 0.00125 0.00125 0.00125 XD, 0.24000 0.26732 0.01000 /‘Dl 0.03765 0.03979 O.OIOOO Xkfl 0.01563 0.00248 0.00000 x D ~ 0.00002 O.CO04 0.00100 rD2 0.00678 0.00672 0.00100 Xkp 0.22496 0.23767 0.01000

Dynamic Time Constants 20dB

60dB

lnitial

T j 1.08340 1.08270 0.90252 T,” 0.02492 0.02466 0.00503

Tj 0.00765 0.00764 0.00236 TA 4.67430 4.67220 7.91100 T i 0.03198 0.03211 0.00514 TL 0.00771 0.00767 0.00236 T, 0.01754 0.01782 0.00265 TD2 0.00000 0.00022 0.00265

Tables 1 and 2 show estimated values of the d-axis equivalent circuits, for random and biased initializations. In the random case, it was assumed that no previous knowledge was available (which is really unlikely) whereas in the biased case, an initial error of 80% with random sign was assumed on all

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paranleter values. Null value parameters are initialized at Overall, the results of table 1 show that, even under very adverse conditions, the maximum likelihood estimation can still give realistic results. Figure 2 displays a sample of the validation curves, for a 2OdB SNR with a random initialization similar to that found in 14). The most interesting fact is the robustness of the new scheme under fairly low SNR.It is thus found that from 20dB to 60dB SNR, the method exhibits similar performance. This in fact should not really surprise us, since in this particular case, the Gaussian assumption about the residuals, which was necessary in the derivation of the maximum likelihood estimator, is actually correct, i.e.: (1) there is no modeling error, and (2) the measurement noise is normally distributed.

I

20,

\o -20

"Po-,

VI. Actual Models of the Nanticoke Generator

The EPRI project RP-9997-2 was the first large scale attempt to check the practical usefulness of SSFR testing on large turbogenerators. As a matter of fact, the Ontario Hydro report [lo] which resulted from this effort, contains extensive test data for several units with more than 500MW rating. These data were used to qualify our analysis method and table 3 shows the parameter values obtained for the Nanticoke generator, using various model structures. In figures 3 and 4, we provide graphical validation ofmodels 3.3 and 5.5, and it is obvious that the latter performs much better. As no data was available for x&), only the first two terms of the model (13a) were retained in the estimation. As regards the field-to-stator transfer inductance (sG(s)), the original (AmperelAmpere) data were scaled to have the same per-unit GO value as in [9]. To start the process. we use the equivalent circuits provided in this report as initial values for the model 3.3 parameters esthation. After the model 3.3 was known, we proceeded to model 4.4 using a stage initializationsuggested in [4], which was repeated to pass from model 4.4 to model 5.5. Throughout this process, we observed that it was sometimes suitablefirst to apply the estimation scheme only to magnitude data (truncatedresidual matrix), and then to refine the estimates using both magnitude and phase (full residual matrix). In the lattercase, we usually converted the magnitude to dB,in order .'.-.* * I toachieve amoreuniform scaling for bothmagnitudeandphase. Attempts where also made to check if consistent results could be obtained when Canay reactances of higher order were kept atzerovalues(xw=O,i>l),assuggest by an IEEE workinggroup (Dandenoand Service discussion of [4]).We found in this case Field Opcrationrl lndunrncc rffs) that convergencewas erratic, and the estimates of poor quality. .............................................. "-I Therefore, to achieve maximum flexibility we opted to retain them as unknown and adjustable model parameters. . . . . . . . . +...'. . . . . We consistentlyfound that when morethan three equivalent windings were included in the model, the actual operational '****. ..... inductance magnitudes were fitted with satisfactoryaccuracy. . . . . . . . . . . . . . . . ?.,.. ........ Unfortunately, this was not the case for the phase, which ***.... exhibited p r fitting,even in the critical transientregion, when 10' 10' 100 IO' 102 10' less than five windings were involved in the model.

. -

.%. '

4

the true values of T,' and COare very close, implying a nearcancellation of the corresponding pole and zero pair in the .rd(s)fraction.

'

'.'

0

0'

IO-' Hz

.............................................. 10-2 10' IO IO' 1V Hz

Fi ure 2. Validation of a Circuit Model resulting from noisy data (20dl SNR & Random Initialization). True:Estimated: Inrtial: + However, we observed during these validation tests that the initial conditions do influence the final solution as regards the equivalent circuits. For instance, we found in one case (table 2) a value of xul equal to 0.01563 instead of zero. Yet, no time constant bearing a significant effect on transient dynamics (between 0 and 1OOHz)was in error by more than 5%. This suggests that the actual time constants in the low-frequency range, are "robust"against experimentalerrors in the equivalent circuit parameter values. It also pays to mention here that the Rockport unit model 3.3 used in the above simulations seems numerically illconditioned since from table 1, it appears that

-

....................

~

--

-IO'

-. . . . . . . . . . . ."7 *_.-

_..-I.

Hz

Figure 3. Validation of N a n t i k e q-axis Circuit Models. Model 5.5: -Model 3.3: ---- Test Data: 0.

530

103

10-3

10-2

10-2

10-1

IO-'

100

100

101

10'

102

102

103

103

Hz Figure 4. Validation of Nanticoke d-axis Circuit Models. Model 5.5: __ Model 3.3: ---- Test Data: 0. It has been argued that, si- the RL circuit model is basically a minimum phase system, it is sufficient to fit only the magnitude data, since the phase data are then automatically adjusted [lo][ 141. Although this is theoretically true, under real conditions, the amplitude exhibits less dynamics (i.e. is smoother) than the phase. Therefore, a model can achieve a good graphical fit of the former, yet displaying large inaccuracies in phase which generally exhibits a number of local extrema and inflexion points. In fact, most of the examples reported in the literature show good agreement as regards

magnitude (cf. the Rockport case in the Dandeno and Service discussion of [4]). However in our opinion, they can still suffer from severe phase distortion [9]. which in turn can adversely affect time-domain predictions during fast transients.

VII. Discussion A troublesome fact revealed by the simulations in section V is that the optimization results generally seem sensitive to initial conditions. Actually, no known practical optimization

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scheme can claim global convergence [12]. Therefore, when the cost function is not unimodal due for instance to poor modeling, short data records or excessive measurement noise, it is very likely that the optimization algorithm (especially those of the Newton type) will end on local minima. Even if the statisticalanalysis of the residuals at the convergence point can help assess the significance of a solution, it is the ultimate responsibility of the user to ensure that the final result makes sense physically. If this is not the case, the optimization process should be restarted as many times as necessary with other initial conditions, until a physically satisfactory solution is obtained.

~~~

Table 3. Parameters of the Equivalent Circuits of the Nanticoke Turboalternator (rotor 3.3) I141 . Equivalents Circuits Dynamic Time Constants Model 3.3 4.4 5.5 3.3 4.4 5.5 ~

x, 2.10560 2.0996 2.10200 Tj 2.31900 2.2255 2.03000 '6 2.03000 2.0300 2.03000 T: 0.17790 0.15150 0.18198 Xd,

1.93360 1.9276 1.93000 T j 0.00600 0.00905 0.02017

& 1.85800 1.8580 1.85800 TA 8.09500 8.01440 7.92580 ra 0.00169 0.00169 0.00169 T i 0.29844 0.24003 0.27895 x, x,

0.17200 0.31580 r, 0.00081 xOl 6.31550 fD1 0.06101 411 -1.0781 x, 0.10040 f, 0.01011 x, 1.29770 k, 1.88750 rot 0.00481 k2 0.40580 fo2 0.01645 'k30.09495 ~fo, 0.12210

0.17200 0.31990 0.00083 13.8040 0.13400 -1591 2 0.39540 0.06895 1.80250 3.73670 0.00478 0.80343 0.01132 0.39870 0.05048 xD3 0.14370 fD3 0.21690 4 B 0.00372 0.15390 ~L fa 0.19780 XM 0.00100 0.81760 -0.0113 k5 0.00395 r05 0.98620 6.29040 6.28310 6.15400 X, 2.46900 2.43360 2.45360

~ ,;

The previous data processing difficulties. even if surmountable, help shed light on two fundamental questions: good modeling and the optimal design of experiment [8]. The maximum likelihood estimation is just one useful mathematical tool, and not a panacea for all the technical problems tied to the process of finding a physically sound model for a system. It can never replace good engineering practice during the stages of determining what set of input frequencies best excite the critical modes of the system, or what model structure actually overcomes the identifiability issue. In the presence of a badly conceived test bed,providing insufficientinformation about the system, combined with poor modeling, even the widely acclaimed maximum likelihood formulation leads to a multimodal prediction enor criterion [a]. This in turn gives rise to a nonlinear optimization which is difficult to resolve without strong engineering insight about the system.

0.1720 0.29780 0.00081 9.82720 0.11350 -1.3302 0.11660 0.07295 1.56210 2.49880 0.00478 0.52820 0.01372 0.16980 0.08614 0.00000 0.57010 -0.0237 0.02259 0.69480

TL 0.00975 0.01385 0.02735 T,' 1.13030 1.47630 2.16500 T," 0.08958 0.13175 0.22540 Ti 0.00434 0.00890 0.02799 Tio 2.25360 2.62440 3.32960 Tio 0.21 206 0.30807 0.501 14 T:o 0.00769 0.01572 0.04603 Tf 1.07160 1.00380 1.0352C Tf' 0.15927 0.14044 0.16875 T/' 0.00596 0.00899 0.0199E Tf. 3.43330 3.41130 3.4318C T i 0.23434 0.20771 0.25351 T i 0.00812 0.01258 0.0227C TD, 0.27460 0.22960 0.2732C TD2 0.00263 0.00424 0.01521 0.00020 0.00298 TL 0.00029 0.0037E T," 0.00034 0.0033s T,"o 0.00050 0.0048E TD3 0.00000 0.0017E T: 0.00020 0.00298 T; 0.00036 0.0035 Ti 0.0001E TL 0.00021 Ti 0.0001i Tio 0.00024 TM T:

Ti

.ooooo: 0.0001E 0.0001E

VIII. Conclusion The present paper describes a frequencydomain prediction error method for analyzing efficiently synchronous machine SSFR test data. Under appropriate Gaussian assumptions regarding the residuals, it is shown that the resulting estimator is actually identical to the maximum likelihood's. The method is direct, in that starting from a suitable initial model, using actual SSFR test data, a more accurate equivalent circuit is derived directly, providing an optimum fit to the whole set of magnitude and phase responses. Using noisy simulated data from the Rockport 3.3 model, it isdemonstratedthat satisfactory convergence is achieved with signal to noise ratio as low as 2Odb, or strongly biased initial guesses (eg. +80%on all values). Actual SSFR data of the Nanticoke generating unit was thoroughly analyzed and the graphical "goodness of fit" criterion has shown the newly developed model 5.5 to fit the phase data more closely than the original, tentative Ontario Hydro model 3.3, demonstrating improved performance over the whole input frequency range.

M.References 1. Kamwa, P. Viarouge, E.J. Dickinson, "Direct Estimation of the Generalized Equivalent Circuits of Synchronous Machines from Short-CircuitOscillographs",IEEProc. C,137(6), 1990,p.445451. E.S. Boje, J.C. Balda, R.G. Harley, R.C. Beck, "Time-Domain Identification of S nchronous Machine Parameters from Simple Standstill Test", /&E Trans., EC-4(2), March 1990, pp.164-175. I. Kamwa, P. Viarouge, J. Dickinson, 'Identification of Generalised Models of Synchronous Machines from Time-Domain Tests", IEE R O C . PtC, 138(6), NOV.1991, pp.485-498. A. Keyani, S. Hao, R.P. Schulz, 'Maximum Likelihood Estimation of Generator Stability Constants Using SSFR Test Data', /E€€ Trans., EC-6(1), March 1991, pp.140-154. N.K. Sinha, B. Kuszta, Modeling and ldentification of Dynamic Systems. New-York:Van Nostrand Reinhold,NY, 1983. L. Ljung,SystemIdentification - A 73eow for User. Englewood Cliffs: Prentice-Hall, NJ, 1989. G.C. Goodwin, R.L. Payne, Dynamic System Identification. NewYork: Academic Press, NY, 1977 D.M. Titterington, 'Aspects of Optimal Design in Dynamic Systems", Technometrics, 22(3), August 1980, pp.287-299. "Determination of Synchronous Machine Stability Study Constants",EPRl Report EL-1424, V01.2, Prepared by Ontario Hydro, Dec. 1980. S.D. Umans, J.A. Mallick, G.L. Wilson, 'Modelin of Solid Rotor Turbo-Generators- Parts I 8 11," /€€E Trans.,PA#-97(1), JanlFeb 1978, pp.269-291. A. Keyani, S. Hao, G. Dayal, "The ,Effects of Noise on Fre uenc Domain, .Parameter Estimation of Synchronous Maaine h-odels'. i b d , EC-4(4), Dec. 1989, pp.600-607. J.E. Dennis, Jr, R.B. Schnabel, Numerical Methods for Uncontimization and Nonlinear Equations. EnglewoodCliffs: strained Prentice- all, NJ, 1983.

?

532 (131 'IEEE Standard Procedure for Obtaining Synchronous Machine Parametersby Standstill Frequency ResponseTesting", IEEEStd. 115A-1987. 1141 H.R. Schwenk, 'Deriving Synchronous Machine Models from Fre uenc Res onse Data", in IEEE Symposium Publication 83T80 1 & 6 P h ? . Feb. 1983, pp.31-37. [15] H.K. Lauw, W. S. Meyer, "Universal Machine Modeling for the Representation of Rotating Electric Machinery in an Electroma nebc Transients Program", ibid., PAS-101(5), June 1988; pp.1342-1351. [16] W.W. Xu. H.W. Dommel. J.R. Marti, "A Synchronous Machine Model for Three-phase Harmonic Analysis and EMTP Initialization', lEEEPaper91 WM 210-5PWRS. PresentedattheIEEUPES 1991 Winter Meeting, New-York, NY, February 3-7 1991. 111 K.J. Astrom, "Maximum Likelihoodand Prediction Error Methods", Automatica, 16, 1980, pp.551-574. [18] A. Charnes, E.L. Frome, P.L. Yu, "The E uivalenceofGeneralized Least Squares and Maximum Likelihoo8 Estimation in the Exponential Family', J. Amer. Staris. Assn., 71, 1976, pp.169-171. [19] M.E. Coultes, "Standstill Frequency Response Tests", in /€€E SymposiumPublication83THO 1016PWR, Feb. 1983, pp.26-30. [20] J.E. Denis, D. M. Gay, R. E. Welsh, "An Adaptive Nonlinear Least-Squares Algorithm', ACM Trans. on Math. Software, 7(3), pp.348-368, Sept 1981. [21] G.E.P. Box, N.R. Draper, "The Bayesian Estimation of Common Parameters From Several Responses", Biometrika, 52, 1965, pp.355-365. 1221 D.M. Bates, D.G. Watts, " AGeneralizedGauss-Newton Procedure for Multi-response Parameter Estimation'. SIAM J. Sa. Comp.. 8(1), Jan. 1987, pp.49-55. [23] J.C. Nash, Compact Numerical Methods for Computers, Adam Hilger, New-York, NY, 1990. [24] J.L. Kuster, J.H. Mize, Optimization Techniques with Fortran. New-York: McGraw-Hill, NY, 1973. [25] A.G. Jack, L.G. Men T J Bedford, "Frequency Domain-Based Methods to Predict Furbogenerator Transients with Particular Emphasis on Field Current", I€€ Proc. Parr C, 136(3). 1989, pp.206-214. [26] P.L. Dandeno, A.T. Poray, "Development of Detailed Turbogenerator Equivalent Circuits from Standstill Frequency Responses Measurements", /E€€ Trans., PAS-100(4), pp.16251633, July/Aug, 1981. [27l I. Kamwa, "DBtermination Exp6rimentale des Modeles d'une Machine Synchrone ExcitBe B I'ArrQt par des Tensions en CrBneaux et en Bchelon", IRE0 report #4631. June 1990 (in French).

X. Appendix: State Equations of the Synchronous Machine at Standstill Introducing the two current vectors

3; = [ i d zf 2.1

. .

'* 'iDnA

;3; = ['q

'Q1

*"

and the voltage vectors u,'=[vd vf 0

(la)

01 ;U:' [vq 0 ... 01 (lb) the equations associated with the two equivalent circuits illustrated in figure 1, for zero speed, take the following mamx form [I1 : a . .

1 d3-X.>=-R.~.+U., onJdt I J

j=d,q I

x ~ , , ~'prn,5p2 l

x~.l~J

with U, designating the nominal pulsation in rad/=, whilst the term xu refers to the mutual reactance between windings k and 1. Their numerical value can be obtained using the formulas proposed in [7]. From these basic equations one deduces the following state variable formulation :

y j = C j S j , Bi = o,,X,? ; AJ. = - B IR 1.

The matrix Cj(i=d,q) is determined from the measured currents and must be chosen so that yd = [id ']fi and yq = i,.

Biographies Innocent KAMWA (S'83, M'88) has been with the Hydro-Quebec Research Institute (IREQ) since 1988. He is an associate professor of Electrical Engineering at Laval University in Quebec, Canada. His current interests involve the areas of system identification, real-time control of electric power systems and CASE tools for real-timesystem specification and prototyping. Kamwa received his B.Eng. And Ph.D. Degrees in Electrical Engineering from Laval University in 1984 and 1988 respectively. He is a member of the IEEE Power Engineering and Control System societies. Phili pe VIAROUGE was born inPBrigeux, France, in1954. He received the engineering and Doctor of Engineering degrees from the lnstitut National Polytechnique, Toulouse, France, in 1976 and 1979, res ctively. Since 1979 he has been a professor with the Department of &ctrical Engineering, Laval University, QuBbec, Canada. In this department, he is also working in the Laboratoire dElectrotechnique, d'Electronique de Puissance et de Commande lndustrielle (LEEPCI). His research interests include power electronics, ac drives and the design of electrical machines. Hoan LE-HUY received the B.S. And M.S. Degrees in electrical engineering in 1969 and 1972, respectively, from Laval University, Quebec, Canada, and the Dr.Eng. degree in electrical en ineering in 1980 from the lnstitut National Polytechnique, Grenoble, rance. He is presentlya Professor in the Department of Electrical Engineering at Laval University, where he is en aged in teaching and research in the area of power electronics in e! Laboratoire d'Electrotechnique, d'Electronique de Puissance et de Commande lndustrielle (LEEPCI). Before joining Laval University, he was a Professor of Electrical Engineering, from 1973 to 1987, at the University of Quebec, TroisRividres, Canada, where he worked on microprocessor control of power electronics systems. His research interests include electronically-commutated motors, static converters and digital control systems. Edwin J. DtcKlNSON was born in Carlisle, Cumberland, England, in 1933. He graduatedfrom Jesus College, Cambridge, En land in 1956 and received the Ph.D. degree from Laval University, tuebec, PQ, Canada, in 1964. From 1956 to 1958 he was an Engineer in the Electronics Department of A. V. Roe, Manchester. From 1959 until 1963 he was a Research Assistant in the Aeronautics Laboratory of Lava1 University. For the following year he was an Assistant at the Ecole Normale Sup6rieure en Mecanique et ABronautique, Poitiers, France. He is currently a Professor in the Mechanical Engineering Department, Laval University. His fields of interestare fluid mechanics and instrumentation.

e

533

Discussion A. Keyhani (The Ohio State University, Electrical Engr, Columbus, OH 43210). I would like to commend the authors for a well-written paper and for their efforts to put the experimental determination of synchronous machine parameters in the more general framework of system identification theory.

From Table 3 of the paper, it can be observed that some of the estimated Canay reactances have negative values. Canay indicates [l] that for cylindrical rotor machines, these reactnncos reyrcsent the COT~L‘IIOI-I leakage flux of the damper equivalent windings and field windings yielding positive values for the corresponding reactances. Furthermore, for salient pole machines, the Canay reactances represent the difference flux of the windings, and they normally have negative values [l]. If the Nanticoke generator is a salient pole machine, then the estimated negative values for the Canay reactances are justified, since they confirm the physical distribution of leakage flux within the machine as described by Canay [l]. However, estimation of negative Canay reactances for a round rotor machine needs further elaboration by the authors. In Table 3, the authors have presented several models. It would be of interest to know the values of the cost Since, in general, higher order models function V (@,a). result in a better fit (more degrees of freedom), why didn’t the authors consider higher order models than model 5.5? Was it because the cost function did not change appreciably a s higher order models were considered? Was it because the estimation did not converge? Or did the estimation converge to a seL of parameters which were not physically acceptable? The authors are correct to point out that “the Maximum Likelihood (ML) estimation is just one useful mathematical tool and not a panacea.” However, we need to recognize the power of this approach as compared with other alternatives for modeling test data which are always noisy. From a theoretical viewpoint, when a correct structure and a physically justifiable initial value of parameters are used to estimate the model parameters, the ML estimator will identify the unbiased parameters with the lowest cost function, V(0,Q) [21. That is, any estimate of parameters obtained by any other method will result in a higher cost function. Naturally, the test data used for estimation should provide sufficient information about the system, and the Gaussian assumption of the measurement noise should be valid, which is the case for large data sets [2]. Nevertheless, the global convergence to true parameters cannot be guaranteed because the exact model structure for any real system is never known. For synchronous machine modeling, the circuit model structures used are a t best approximate descriptions of the flux distribution within the machine, and the number of models that can be considered is limited.

Therefore, it is relatively easy t s cornpate the cost function V(0,Q) of all models using different estimation techniques. The discusser believes that the ML estimator will result in the lowest cost function, V(6,Q), when compared with any other estimator; therefore, the ML estimated parameters produce the best “global” convergence that carfibe achieved. The authors have provided the power industry with a

valuable system identification method for estimating synchronous machine high order models using SSFR data. I would appreciate the authors’ comments concerning the questions and issues raised in this discussion.

[l] I.M. Canay, “Extended synchronous-machine model for the calculation of transient processes and stability,” Electric Machines and Electro-mechanics: 1977, Hemisphere Publishing Corporation, pp. 137150.

[2] A. Keyhani and S.I. Moon, “Maximum likelihood estimation of synchronous machine parameters and study of noise effect from DC flux decay data,” IEE Proceedings-Part C, Vol. 139, No. 1 , Jan. 1992. Manuscript received February 10, 1992.

P. L. Dandeno, (University of Toronto): This discusser naturally is pleased by the investigations of the authors into development of generator stability models using frequency response test data. I also note that the authors feel that relatively simple methods, for d and q axes determination of various rotor element values in models, are quite acceptable. These are outlined in the Appendix of IEEE Standard 115A and are expanded on by Mr. Service and myself in a paper presented in 1991. See Reference [l] below. Another recent paper, to be published in the Transactions on Power Systems [2] discusses the application of Model 3.3 (or 3d, 3q, including one differential leakage reactance Lf12,,)to subsynchronous, torsional problems for a large 722 MVa turbogenerator in the 15-50 Hz range. I have recently come to believe that frequency-response-derived models could and should be applied in such studies as well as the usual stability studies. The derivation of the appropriate d and q axes model elements from test standstill frequency response data is probably no more expensive than obtaining calculated data from the generator manufacturer (about $25,000 per machine). I should point out that their nomenclature for time constants (both “short circuit” and “open circuit”) for the turbinegenerator being analyzed is confusing and does not agree with the recent standard on the subject of generator modeling [3]. For example using Td for T:, the short circuit transient time constant, or using Tjo for T:: the open circuit subsubtransient time constant is to be deprecated. Students investigating this subject, as well as others might think that the superscript 2 or 3 associated with these constants might be squared or cubed terms. I imagine their use of such superscripts was based on the need, in models 4.4 or 5.5, to avoid 4 or 5 “primes” as superscripts for any particular open circuit or short circuit constant. A word of caution should be expressed in the interpretation of frequency response data in the ranges from about 500 to 10oO Hz. This data is taken to presumably ensure that the normally required range of (say) 0.001 Hz to 100 Hz or 120 Hz is more than adequately covered. But model fitting, in my estimation should only be attempted up to about 120 Hz. W. Watson, formerly of Ontario Hydro’s Research Division, who pioneered the test procedures and development leading to IEEE Std 115A, warned that capacative effects distorted the phase angle of the operational inductance measurements in the 500-1000 Hz range. As such, the accuracy of any model (such as Model 5.5) might be questioned, using only inductance and resistance elements, when capacative effects might be present. I would be interested in the authors’ reaction to comments in this last paragraph. The application of SSFR testing procedures to salient pole hydraulic generators should be a subject of interest to engineers of Hydro Quebec and I would encourage their engineers, or those at IREQ, to pursue this facet of hydraulic generator stability data determination.

References [l] P. L. Dandeno, J. R. R. Service. “Experience with Translation of Turbogenerator Operational Test Data and Corresponding D and Q Axis Stability Models Into Standard Inductance and Time Constant Forms”, Tenth Power System Computation Conference (PSCC), pp. 371-78, 1990.

534

121 V. Atarod, P. L. Dandeno, M. R. Iravani, “Impact of Synchronous Machine Constants and Models on the Analysis of Torsional Dynamics”, Paper 92 WM 178-4 PWRS. To be published in the IEEE Transactions of Power Systems. 131 IEEE Std 1110 (1991) “Guide for Synchronous Generator Modeling Practices in Stability Analyses”. Manuscript received February 19, 1992.

I. Kamwa, P. Viarouge, H. Le-Huy & E.J.Dickinson: The authors wish to thank the discussers for their interesting comments and questions. We agree with Prof. Dandeno when he states that frequencyresponse-derived models could and should be applied in subsynchronous resonance studies as well as the usual stability studies. In fact, we feel that the accuracy of any EMTP study involving a synchronous machine near the switching area can benefit from the increased frequency resolution provided by such models. Moreover, using the new generation of three-phase harmonic load-flow programs, it now seems feasible to account fully for the frequency conversion phenomenon taking place within a synchronous machine. In spite of the Delta-Wye safeguard, frequency conversion can still have a significant impact on the overall level of harmonic propagation (ref. [16] in the paper). As this effect is most critical at the lower harmonic frequencies (below the loth), models with an actual proven range of accuracy extending up to say 500Hz,could be the most suited for use in harmonic penetration studies. With the present day state of the art, frequency response data, wether calculated or measured, still are the most common but not by any means the sole, nor the most efficient source of such wide-frequency range models, which typically include more than two windings per axis. Prof. Dandeno raises a valid point regarding possible ambiguity in the non-standard nomenclature used for the time constants, particularly for new students investigating this subject. As he suggested, our initial reason for chosing T,’ instead of T”d was in fact that the latter notation becomes rather heavy when generalized up to order 5 or higher. A second more fundamental and less obvious reason, which we realized much latter is that the second time constant (Tj) in a high order model (viz 4.4 and 5.5) is not necessarily the subtransient time-constant (T”d)quoted by the manufacturer, according to the ANSI-I 15 standard. Table 3 illustrates this point quite clearly: T i = 0.010s, from model 3.3 and T i = 0.027s from model 5.5 (3 times more)! Therefore, each model comes out with adifferent value of Ti. The situation is even worse for T j , which should normally correspond to the sub-subtransient. We find that the values of T j calculated from models 3.3 and 5.5 differ by more than 300%.Only in the case of model 3.3, do our T,’ values clearly resemble those we know as T”;. This poses some serious concerns regarding the physical significance of high-order time-constants. We have recently come to believe thattheonly way toenstire aconsistent setofstabilityconstant definitions is to base them on a single model; the most suitable in this respect being the 3.3, since both subtransient and sub-subtransient quantities possess a well understood physical significance [3]. If for convenience, higher order models are used to achieve a better overall fit, we suggest that only reduced-order models be used for determining standard stability constants, in order to avoid unfortunate confusion between the physical {T”d, T”;] and the somewhat mathematical ( T i ,T,’] time constants arising from model 5.5. We are convinced that if the latter are used in a current stability program, it is unlikely that they would provide acorrectprediction of the machine short-circuit oscillograph. It should be pointed out that one of the problems with high-order model fitting stems from the fact that current stsbility programs cannot use models as complex as the 5.5. Due to implementation details, all but few a of them require input data in the form of standard stability constants. As far as we know, only the universal machine (UM) module available within the EMTP can readily implement model 5.5, thus eliminating the computational burden of stability constants.

Regarding Prof. Dandeno’s last remark, we have to pay serious attention to Mr W. Watson’s warning, due to his great experience in implementing frequency response procedures. If the insmmentation accuracy does not permit us to trust data in the ranges from 500 to 1000Hz, there is no good reason to use this data, since it will not convey any relevant information about the magnetic phenomena in the rotor. Nevertheless, the eddy current effect is still significant up to several kHz and the question arises as whether it is important to account for this and in what type of study. We know that the EMTP DCG has been working through Ontario Hydro on improving current transformer models in this respect. Instrumentation should perhaps have improved by the time that accurate high frequency field data become necessary for more sophisticated analysis softwares (cf. the EPRI work in [l]; also, the earlier findings in Ill]) in his first question, Prof. Keyhani points out that, for a so-called Canay reactance (xv,,xv2,...) to be negative, the subject machine should be of a salient pole type. By contrast, a round rotor machine should always display positive Canay reactances. Since the Nanticoke turbogenerator possess a cylindrical rotor (though with some level of unavoidable saliency), the appearance of certain Canay reactances with negative values in the Table 3 (and also in the EPRI- 1424 report), sounds strange according to such a classification. In fact, the argument raised by the discusser was first elaborated by Canay [2] using the physical distribution of flux within a synchronous machine. However, it is our impression that Canay’s original claim was stated for specific models, and seems strictly valid only in the two following circumstances: When the eddy current effects in massive parts are neglected and only the damper and field windings are considered in an approximate equivalent diagram When magnetic phenomena are fully accounted for (exact equivalent diagram) by using an irrational representation of the various actual physical parts, such as the damper bars and wedges, the rotor surface, the field winding and the slots wall. Therefore, one can understand that the Canay statement applies in the first case to an approximate circuit with clearly defined acrual windings, and in the second case, to an exact circuit which accounts for eddy currents through the introduction of impedance branches, proportional to at appropriate nodes of the basic circuit. Both approaches are eminently physical, and lead torepresentations which are most helpful to the designer. By contrast, the windings involved in the generalized circuit model of Fig. 1, convey a limited physical significance, since eddy currents are modelled using a finite expansion of the Canay’s exact irrational representation. Such an approach is the only one permitting us to retain the appealing properties of a linear representation, despite the complexity of the underlying phenomena. As a matter of fact, it is well known that eddy current effects in the machine rotor [4] and transformer core [SI can be accounted for by including a number of equivalent R-L elements (so-called parallel Foster sections) behind the magnetizing branch. The actual number of decoupled R-L sections depends heavily on the required model bandwidth, and can be significant if subsynchronous resonance and harmonic penetration are of concern. For instance, Canay has shown that in a machine with strong dampers, satisfactory prediction of the no-load short-circuit can be achieved without even considering eddy currents in the massive parts 121. However, the sub-subtransient quantities owe their essential physical significance to the skin effect and they influence markedly the switch-on process and the braking torqueof solid-rotor machines 131. A dummy winding in the parallel Foster model of the eddy current [SIcames aclose resemblance with an actual damper winding and the two can hardly be distinguished from each other in the generalized equivalent circuit of Fig. 1. Therefore, only the armature winding, the magnetic branch, and the field winding can be readily tied to physically existing parts of the machine. To illustrate this point, let us consider the reactance matrix associated with the model 5.5 d-axis (table l), calculated from formula(2) in the appendix. It appears that the mutual reactance between the first ammortisseur winding and the remaining rotor windings is surprisingly low. By looking at the corresponding coupling coefficient, one can even conclude that the mutual flux path is stronger

6,

535

Table 2. Cost function value at the final solution

between the first damper and the stator than between the other rotor windings. Since a damper actually behaves like acage in a multi-cage asynchronous motor, such a fact sounds very strange at first sight, if one believes that the r,, -xD, element corresponds to an actual damper winding. However, if it is considered as the outcome of a purely mathematical process of expanding an irrational function in a finite sum of terms, then any physical interpretation of the resulting values is irrelevant, More specifically, it is our opinion that not all of the lumped damping elements in the Table 3 can be matched to physical damper windings. Therefore, the physical meaning of the Canay reactances is lost. In the generalized circuits of Fig. 1, they are to be considered only as useful mathematical entities permitting one to adjust easily the mutual inductance (or the coupling coefficient) between the various rotor elements. From the circuit theory point of view, they are not even necessary for theoretically realizing a d-axis quadripole matching the terminal behavior. However, including a few Canay elements leads to a well-conditioned structure, with sufficient flexibility to cover the most difficult practical case, without excessively increasing the model order. Actually, we are beginning to suspect that some of the problems currently found in modelling the q-axis SSFR phase data could be fixed by introducing a few Canay "stubs" to adjust the coupling coefficient between the various components of this rotor axis. To summarize, only the Canay exact equivalent diagrams 12-31, and the operational impedances lend themselves to physical interpretation. In some sense, the time constants (Tld,T'ld,.. .) and the dynamic reactances ( x ' ~ , x ' ' ~..,.) involved in the x,(s)-x,(s) functions really exist, but can only be realized through an infinite number of linear equivalent circuits (see the discussion of D.R. Brown in [lo]). 'able 1. Mutual reactances and xg xD2

Xa,

xD,

1

1.928 1.928

0.850

K,,,

0.390

1 I1 I

1-

Kdl

1 I

3u

1.928

I K,, I

0.875

1.928

KD,

0.908

x ~ , ~ 0.470 ,

0.090

x ~ , ~ 0.470 ,

0.091

0.470

0.094

(Amplitude in dB and Phase in degrees)

Model

EPRl3.3

New 3.3

New 5.5

New 4.4

available beforehand. Thus, in the case where the errors are serially correlated, the most difficult problem is perhaps to calculate the true likelihood function. For instance, Prof. Keyhani and his co-workers (ref. [4] ofthe paper) resort to a Kalman filter (KF) for achieving this purpose. Even then, they need to assume a covariance matrix Q describing the process noise. When the errors are uncorrelated, this assumption is not critical. However, when the noise is colored (real facts of life), the KF can diverge if the true matrix Q is not used. The problem can only be circumvented by parametrizing the matrix Q itself and estimating its parameters simultaneously with the plant parameters, which is similar to the Recursive Maximum Likelihood approach for ARMAX models [12]. This leads to a huge problem, especially for high dimensional systems. Therefore, alternatives to raw maximum likelihood may be more suitable in the case of colored noise in high-order systems. In our opinion, the iteratively reweighted least-squares method is a good candidate, since it makes the ordinary weighted least-squares method perform as well as the maximum likelihood method, without the burdenofcalculating theexact likelihood function. More specifically, in the case of short datasets with scatteredpointsor grosslyerroneous data (typical of SSFR experiments at low frequencies), the maximum likelihoodestimation can be outperformed by robust variants [6], and even by the least-absolute value 171.In fact, the difficulty ofchoosing the most suitable criterion for the intended goal is often underestimated. We recently learned that in parameter estimation problems, minimizing the trace of the prediction error covariance matrix ieads to more precise values (smaller variance or confidence region) than minimizing its determinant [SI.It thus turns out that the least-squares criterion (trace) is the best for estimation oriented fitting while the maximum likelihood (Log-determinant) is most appropriate for prediction (control) oriented applications.

lin coefficients of model 5.5' I

3

P e .-

E"

xD,04