Jun 10, 1991 - identification experiments in the time-domain. For the identified models, the target application is primarily the solution of power system stability ...
Identification of generalised models of synchronous machines from time-domain tests I. Kamwa, PhD, MIEEE P. Viarouge, Ing.Doc E.J. Dickinson, PhD, MAIAA
lndexing terms: Synchronous motors, Modelling, Circuit theory and design
Abstract: Recent developments have shown that numerical models of synchronous machines are, in many cases, improved by introducing an increased number of dampers or dynamic constants. The major issues affecting the model identification by such an extension, which in fact becomes necessary when the machine possesses a solid rotor are discussed from a time-domain point of view. The first result is that all methods which determine the equivalent circuit from test data in an indirect manner, by using the time constants or operational impedances as intermediate tools, give rise, along the d-axis, to systematic errors in the subtransient and sub-subtransient time constants of the estimated circuits. Unfortunately, when the identification experiment is the standard short circuit, even the direct method suffers from the drawback of weak identifiability of the model, with the result that the uniqueness of the equivalent circuit, directly estimated from test data, is not at all guaranteed. In fact, from an information content point of view, the richer tests are those made at standstill, with a perfect decoupling between the two axes. It is demonstrated by simulation that if the direct approach is applied to such a test, with pulse wave modulated (PWM) voltages as the excitation signals, a numerically cheap and accurate one-step procedure for timedomain identification of circuit models is obtained.
1
Introduction
As the early work of Doherty and Nickle [l], and the even more famous work of Park [2], has shown, from the very early days of electrical energy distribution, grid engineers have realised the crucial importance of a good model of the synchronous machine in simulating the stability or, more generally, the dynamic behaviour of the network subjected to a variety of operational conditions. During the 1970s, the remarkable increase in unit power of turboalternators and the degree of interconnection of multiple distribution grids, especially in North America, Paper X277C (PI), first received 24th July 1990 and in revised form 10th June 1991 P Viarouge is with the Electrical Engineering Department (LEEPCI) and E.J. Dickinson IS with the Mechanical Engineering Department at Universite Laval, Quebec, PQ, Canada, G1K 7P4 I Kamwa is with the Institut de Recherche #Hydro-Queec, CPloOO, Varennes, PQ, Canada, J3X 1st
IEE PROCEEDINGS-C, Vol. 138, No. 6 , NOVEMBER 1991
reinforced the acuity of the stability problem [4-61. This has resulted in a renewed interest in the problem of the modelling and accurate identification of synchronous machines, particularly those with solid iron rotors, which one encounters in increasing numbers in new thermal and nuclear power stations. More recently, the designers of synchronous machines have confirmed the need for improved modelling, when the machines have to be dimensionalised for optimal performance in association with a given static convertor. Often fed by PWM square wave voltages or currents, these machines give rise to a steady state that is essentially a simple succession of transient regimes [23]. The analysis and optimisation of a complete convertormachine unit consequently requires a much more detailed knowledge of the various time constants of the synchronous machine. In addition, it seems somewhat ridiculous to use a sophisticated field calculation to take advantage, during the design process, of the finest structural details of the machine [18], and then, once it has been constructed, to simulate it using programmes based on various parameters which have been estimated graphically (e.g. transient and subtransient quantities)
PI.
In this paper, we propose a new look at the mathematical modelling of synchronous machines, and the numerical estimation of their parameters, from several identification experiments in the time-domain. For the identified models, the target application is primarily the solution of power system stability problems [19, 411, but in some cases, especially when the excitation signal used during the test is relevant, the same models are also well suited for the optimisation and numerical simulation of the operation of the complete convertor-machine units used in power electronic drives [18, 451. The proposed equivalent circuit model constitutes a generalisation of the conventional treatment [9, 191, whereas our approach for estimating the machine parameters [lo, 111 draws its originality from the systematic application of ideas and techniques drawn from system identification theory [12-141. 2
Parametrisation of generalised models
Estimation of the equivalent circuits of Fig. 1 usually consists in determining, for given integers nd and np the following parametric vector of reactances and resistances (per units): [rarJrDi
" '
rDedlrarQi
"'rQ,,~X/XD~
... x D , d I x Q 1 .'. x Q , , I x c l xc
=[xdxx,qxkJ~
"'
xkJndl
(1) 4x5
The latter permits one to reconstitute a satisfactory representation of the machine input-output data which are optimal, in terms of some appropriate criterion. These input-output data typically originate from a dedicated identification experiment, but could also consist of normal operating records, with the machine running in a mode known to be sufficiently informative (e.g. during
the appropriate parameter vector is readily defined by [ll]
e'
=
CT:
'''
Tr+'I
'
'.
Tz:+'
I
' . ' T ; ' I T i 0 . . . ?-ZITD," ' TD"dlxo]
x.
(4)
= [rarfx,x,x,]
'd
Fig. 1 Generalised equivalent circuit model of synchronous machines e, = o.Y, b e, = o,Y,
a
excitation removal, pole slipping [28], loaded short circuit etc.). In the state space representation of the machine, the dependence on the unknown parameter vector 0 E @ can be explicitly expressed in the following manner :
+
+
with q = 3nd 2n, 6. This permits an optimal reconstruction of a given set of machine input-output data, with model dependence on parameters preferably described in terms of transfer functions in the Laplace domain,
y(P) = H(P>04)
m,e) hi[@; x(t, e); u(t)] z 0 i = 1, 2, . . ., r (2) where x E W',U E W",y E W p are, respectively, the state, input and output vectors with q = 3nd 2n, 5 and I = nd nr 3. The state matrices A(@, B(0) and C are given in Appendix 9.1. The hi constitute the r equality and inequality constraints relating a priori x , U and 0. In fact, the parametrisation, eqn. 1 leads directly to equivalent circuit parameters [ l o ] . When the problem posed is more specifically that of estimating the time constants (in seconds) defining the following three operational impedances [38] :
+
+ +
- xx1 w,(l
=
c m - A ~ - mI e )
hi[@;H(p, e); U@)] z 0 i = 1, 2, .. ., r
f i t , e) = c x ( t , e) xo = x(to)
+
+ pTiX1 + pT:) ". (1 + p T r + ' ) + pTioXl + pTZ) .'. (1 + pTX+')
(5)
The closed-form expressions of eight out of nine entries of the basic transfer matrix H(p), are provided in Appendix 9.2, in terms of the time constants found in eqn. 3. The state matrices A,, B, and C , are not to be confused with (A, B, C) in eqn. 2. Starting with the tentative admittance matrix H(p), they define a canonical state space realisation, which, to a large extent, can be chosen arbitrarily. It has been shown elsewhere that such a canonical realisation can constitute a very useful auxiliary tool in the course of the numerical determination of the observed currents (id, if, iq) [ll]. Because the model we really need is that defined by the equivalent circuits illustrated in Fig. 1, the parametrisation (eqn. 4) is not of such immediate interest as eqn. 1 . In fact, when the vector of eqn. 4 is known, we still need supplementary procedures to deduce the vector of eqn. 1. This is why the latter is termed the indirect parameter vector. To understand the behaviour of the system when it is subjected to a forced excitation, one is led to sample the by output y(t) at intervals { t t = to + k T , k = 0, 1, ..., means of a data acquisition system which inevitably introduces a stochastic perturbation { ( ( k ) } , on the measurements { y ( k ) } .It thus appears that the models of eqns. 2 and 4 should be replaced in practice by the following two discrete representations:
y}
+ 1) = F(O)x(k)+ G(O)u(k) y(k, 0) = Cx(k) + t ( k ) ~(0) = x(t0)
x(k x,(l o,(l 486
+ pT:)(l + pTi) ' . ' (1 + pT?+') (1 + pT:+')
+ pT:,Xl + pTiJ
'
"
hi[@;x(k); u(k)] 2 0 i = 1, 2, ...,r
(6)
IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
and ~ ( k=) H(z, @@) + [(k) H(z, 0) = C,[zI - FC(0)]-'G,(0)
h,[O; H(z, e); u(k)] 3 0 i = 1, 2, .. ., r (7) where F and G are related to A and B by the relationships
@(t)= exp (At) is the inverse Laplace transform of H(p, e). Generally, one can assume that the measurement noise { [ ( k ) } , principally due to transducer errors, can be expressed as a sequence of independent random variables, with zero mean and a known variance U calculated from the truncation error of the measurements. But when the noise term is not white, or exhibits a high variance, a more general statespace setup could be necessary to match the observed behaviour of the process [12, 431. Such a model usually involves, in the difference equations, the addition of a supplementary term w(k) known as process noise, requiring the use of a Kalman filter for an optimal one-step ahead prediction of the state [24, 431. Fortunately, under fairly general conditions, { [(k)} tends to behave like white Gaussian noise with a variance largely determined by the resolution of the instrumentation system, which makes the model 6 an adequate state predictor. Therefore a natural predictor of the measured behaviour of the process 6 is given by
j'(k) I 0) = Cx(k, e) = H(z, B)u(k)
(94
with e(k I e) = y(k) - 3 k I @ (94 giving us the prediction error. When a value 0 = 8 renders this sequence {e(k)} white, (i.e. identical in this regard to { [ ( k ) } ) ,there is a high probability that 8 is the parameteric vector of the true model of the synchronous machine. Consequently, with a total of N observations at our disposal, a judicious criterion for determining 8 consists in minimising the scalar function
at least in the scalar output case [13, 391. Nevertheless, this equivalence is no longer valid when m 1 (MIMO system) and the ML criterion becomes, in fact, more effcient in terms of the accuracy achieved than the LS one; although, admittedly, at greater computational cost [13]. Whatever the scalar measure V used, the optimisation problem can be carried out using an iterative GaussNewton or Newton-Raphson procedure, usually requiring the evaluation of the gradient and the Hessian of V [12, 371. A comprehensive code in Fortran, better known as NL2SOL, has been developed by Dennis et al. to solve this optimisation problem in the specific case of the least-squares criterion [36]. Limitations of t h e classical indirect approach
3
In the traditional approach [8, 9, 381, to identify the parameters of the equivalent circuits of a synchronous machine, one firstly estimates the operational impedances from experimental data and hence, by the solution of a system of nonlinear equations [25, 421, deduces algebraically the aforesaid equivalent circuits. Therefore, this approach, which basically relies on the parametrisation 4, is indirect by its very nature. It is the one involved both in the conventional short-circuit treatment [25-271 and the frequency response method [31, 411. It is well known that the central difficulty in this indirect method arises from the necessity of providing a mechanism for deducing the equivalent circuits, which are truly the quantities of interest, from the estimated time constants. The underlying nonlinear problem has been addressed in several publications, with the clearest solutions being those proposed by Umans et al. [31], Ontario Hydro [41], and Canay [27]. It is our aim to quantify the level of inaccuracy generally involved in such a procedure. Let us consider the three rotor windings per axis model (rotor 3.3), of the Nanticoke turbogenerator. Its experimental operational impedances, obtained from standstill frequency response (SSFR) tests by Dandeno and Poray [32] are as follows: x&) = 2.1 16
N
V ( N , q, 0) =
1 trace {eT(k1 O)We(kIO)}
(10)
k=l
where W = diag (wl, . .., w,,). In the interesting case where m = 1 (SIMO system), this cost function, in effect, represents the sum of the quadratic errors through basing our predictions of y(k) on B. The coefficients wi then serve to weight the different outputs in such a manner that they are of comparable order of magnitude. In a more systematic statistical framework, the least-squares (LS) criterion (eqn. IO) can be shown to be closely related to the well known maximum likelihood (ML) criterion [43],
G(p)= 5.477
x,(p) = 2.03
+ 1.788pN1 +O.O493p)(l +0.0065p) ( 1 +7.635p)(l +O.O761p)(l+0.0088p)
(I
+
(1 O.O634p)(1+ 0.00338~) (1 + 7.635p)(1 +O.O761p)(1 +0.0088p)
(11)
(1 + 0.897p)(1+ O.O846p)(1+ 0.0055p) (1 + 1.915p)(l+O.l83p)(l +0.00948p)
In Table 1, we show the equivalent circuits calculated by Dandeno and Poray, using an iterative method similar to that suggested in Reference 34. These results, which can also be found in Reference 32, were first published in the EPRI report EL-1424 by Ontario Hydro. If one applies
Table 1 : Equivalent circuit parameters from experimental operational imoedances Dandeno method ra r, rD, rD2 rn roz ro9 x.,^., -1
xb,-
0.0040 0.000940 0.11420 0.005920 0.005380 0.10810 0.01880 -0.52150 0.89750
Umans method
0.0040 0.000911 0.16259 0.012792 0.005715 0.021250 0.13027 -0.57489 0.69626
IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
Difference
Dandeno Umans method method
%
0 3.09 -42.4 -116 -6.2 80.4 -593 -10.2 22.4
x, xD1 xD2 xo7 xn-z
0.01550 2.7320 0.00753 1.6570 0.11930 x g 2 0.45130 xmd 2.1520 x-,,," 2.0570 x. 0.1720
0.32999 3.8860 0.16299 1.7761 0.52710 0.15794 2.1520 2.0570 0.1720
Difference % -2290 -42.2 -2065 -7.18 -342 65 0 0 0 487
to this same set of operational data, the algorithm ND3 of Umans et al. [31], one obtains an alternative set of equivalent circuits whose parameters are also listed for comparison purposes in Table 1. As can be seen, even though they refer to the same generator, the parameters of the two circuits differ significantly; particularly the field winding reactance which possesses a definite physical meaning and yet displays a variation of more than 2000% between the Dandeno and Umans method! This provokes the question as to which of the two circuits is more representative of the experimental impedances (eqn. 11) actually characterising the turbo-alternator spectral behaviour. The simplest way to answer this question is to calculate the operational impedances associated with each of the equivalent circuits and then to compare the final results with the original impedances given by eqn. 11, which tentatively correspond to the ‘true’ behaviour of the machine. From the circuits of Table 1, we thus deduced the operational impedances of Table 2. These Table 2: Actual operational impedances associated with the calculated eauivalent circuits ~
Experimental
~~
~
Dandeno circuits Actual
r,
0.0065 0.0493 1.788 CO 0.0088 CO 0.0761 7.635 0.0055 T;: 0.0846 0.897 0.009~1 CO 0.183 T& 1.915 T ,, 0.00338 rD2 0.0634
r,
Go
r,
r,,
r.
*
Error, %
0.0087819 35.1 1 0.049329 -0.06 0.36 1.7944 0.84 0.0088736 0.076797 0.92 9.1 8.3293 1 0.0055559 0.26 0.084817 0.08 0.89776 0.77 0.0095527 0.18777 2.6 2.0412 6.6 0.0033740 -0.18 0.063458 0.58 * 0.000940
~~
Umans circuits Actual
Error. %
0.0061456 0.05221 8 1.7854 0.0090997 0.073572 7.6372 0.00 54946 0.084508 0.90014 0.009441s 0.18018 1.9573 0.03380 0.06340 0.000911
-5.45 5.92 -0.1 5 3.4 -3.3 0.03 -0.1 -0.1 1 0.35 -0.4 -1.54 2.2 0.00 0.00
results clearly show that, for the present case, the ND3 algorithm furnishes the best overall results. In practice, owing to underlying assumptions in the procedure for solving the overdetermined system of nonlinear equations [42], nonuniqueness of the equivalent circuits is a common feature; and a method giving good results for one set of operational data, can fail dramatically in another situation (see Canay [27] for an interesting example). With regard to the Umans circuits, we see from Table 2 that the worst discrepancies occur in the hypertransient and subtransient terms along the d-axis (-6%). Should we consider these discrepancies as negligible? In the opinion of the authors, the answer is no; chiefly because these small differences depend solely on the manner chosen to analyse the data, and not on possible deficiencies already existing in the actual data. If one supposes that the estimated time constants which intervene in the experimental operational impedances frequently suffer from errors of the same order of magnitude, one must conclude that the underlying issue of the ND3 algorithm (or its equivalent) can easily double the initial uncertainty regarding the time constants of the machine. Fortunately, in a number of practical situations, the overall error induced by the algorithm is not cumulative, thus leading to an acceptable representativity of the calculated equivalent circuits. For instance, if the shortcircuit operation is considered, the Umans circuits (Table 488
1) lead to the following characteristic polynomial:
+ 0.5601Xp + l.lllOXp+ 11.8347) + 19.1524Xp + 163.0554) x (p + 182.3989Xp + 5.2194 kj376.110)
d(p) = (p
x (p
with zeros agreeing to 6% or better with those derived directly from experimental data (eqn. 11):
d(p) = (p + 0.5593Xp + 1.1148)Xp + 11.8220)
+ 20.2882Wp + 154.0738) x (p + 182.2724Xp + 5.2438 376.140) x (p
It has recently been suggested in Reference 11 that one can improve the conventional time-domain identification of time constants by applying the parameterisation 3 to short-circuit oscillographs. One then undoubtedly obtains the most satisfactory data reduction method for short-circuit analysis. In effect, this leads, in a striking manner, to the operational impedances and not to the poles and zeros of the admittances, in contrast with the ANSI standard [SI. Even though the accompanying optimisation problem is nonlinear, this detracts very little from its merits since, with a code as well conceived as NL2SOL [36], the problems of nonlinearity are easily surmountable, even in the presence of quite poor initial estimates. Unfortunately, the previous validation study demonstrates one point quite clearly: there is no well established method of passing from the time constants to the equivalent circuits that is exempt from certain approximations and subsequent errors. Nevertheless, with a proper choice of data reduction techniques, these inaccuracies can be limited to less than 10%.
4
Limitations of the d i r e c t approach w h e n applied t o short-circuit tests
Faced with the difficulties encountered more specifically in inferring the d-axis circuit from the given operational impedances, it has been proposed in Reference 10 to identify optimally the circuit model by applying the direct parametrisation 1 to short-circuit data. In effect, it seems just as reasonable to determine the circuits first, and subsequently calculate the operational impedances. This inversion allows one to escape the various approximations necessary to implement the ND3 or Ontario Hydro algorithms. However, owing to specific problems related to the very nature of the short-circuit test, this direct method suffers from the drawback of weak identifiability of the model, with the result that the uniqueness of the estimated equivalent circuit is not at all guaranteed. As pointed out in Reference 10 and in a discussion of Reference 16, this nonuniqueness problem is common to all estimation methods that rely on observations collected from experiments not involving a field voltage perturbation. To illustrate the striking duality between the direct and indirect approaches for synchronous machine identification from short-circuit data, we will review briefly the Nanticoke benchmark model which was thoroughly investigated in Reference 10 for direct parametrisation and in Reference 11 for the indirect case. The study can be outlined as follows. We have used the a priori model, termed the true equivalent circuit in the sequel (e*: Dandeno circuit in Table l), to calculate a sudden shortcircuit solution of the generator. The simulated oscilloIEE PROCEEDINGS-C, Vol. 138, No.6, NOVEMBER I991
graphs were covered by N = 2020 points with a sampling rate T = 0.6 ms and initial terminal voltage o&O-) = 0.1 pu. To validate the direct estimation scheme, an attempt is then made to reassemble the equivalent circuit which produced the simulated responses; the latter being considered, for the purpose of our demonstration, to have been measured in a noise free environment. For the validation of the indirect scheme, our aim is to reassemble the operational model, linked to the true circuit. In the analysis, we have assumed that x, is known, which seems reasonably realistic. In addition, in both schemes, we took w = diag(1, 1, 2). For more details, see References 10 and 11. In the direct method, the initial estimate Bo was obtained by forcing a mismatch of & l o % on all the exact parameters defined by 0*, whereas for the indirect scheme a f15% perturbation was used. Figs. 2 and 3
natural modes of the solutions in the two cases are listed along with the exact natural modes of the benchmark model.
04
2
302
3
a
a
0
1
-0.2
0
0.4
02
04
0.2
S
5
a
b
1
2 2
a 3 Q
0
-2 02
0
04
02
5
04
b'
a
II' -4-40
Fig. 3
-1;
I '
0.2 '
S
5
C
d
'
0.4
Assessment ofinitial guessfor direct approach
Field current: initial and - - - - true b Field current initial error c q-axis current: initial and - - - - true
01
(1
2
~
~
d q-axis current initial error Tabla 3: Eigenvalue assessment of the operational impedances estimated from the short-circuit oscillographs5
-2
-
-0 1
-4
0
0.2
I
0.4
0
S C
,
..,.
,
,
0.2
l
0.4
S
d
Fig. 2 Assessment of initial guess for indirect approach P Fjeld current: initial and - -true b Field current initial error c q-axis current: initial and - - - -true d q-axis current initial error ~
~
~
~
reveal the poor quality of these two guesses. In Figs. 4 and 5, one is able to appreciate the good fit of solutions obtained in the two optimal estimation schemes. The estimated and actual waveforms of the field and q-axis currents are shown separately in the Figs. 4 and 5 because they are difficult to identify when superimposed. Equally, the algebraic differences are difficult to identify visually with the waveforms separated, but the computed errors are plotted. Obviously, with an order of magnitude improvement, the indirect method is much more accurate than the direct one, in terms of the prediction error for both the field and q-axis currents. This striking result is better understood by examining Tables 3 and 4 where the I E E PROCEEDINGS-C, Vol. 138, No. 6,NOVEMBER 1991
Initial impedances
Optimal estimate
True impedances
-2.5540 ti376.89 -158.20 -99.890 -23.871 -1 3.910 -1.1691 -0.48634
-4.9172 ii376.40 (2.64%). -182.63 (1.21%) -1 12.88 (0.87%) -20.324 (0.23%) -11.314 (4.05%) -1.1691 (4.95%) -0.55739 (0.01 %)
-5.0508 +i376.37 -1 80.47O -113.87 -20.277 -1 1.792 -1.1139' -0.55731
5 In column 2, the term in parentheses represents the error. * Error on the damping term q-axis natural mode Table 4: Eigenvalue assessment of the equivalent circuit directly estimated from short-circuit oscillographs§ Initial circuit
Estimated circuit
True circuit
-5.7792 ij376.27 -5.0575 ij376.40 (-0.5%)* -5.0583 tj376.40 -162.56 -180.73 (0.15%) -1 80.45O -1 13.55 -1 14.61 (0.65%) -113.87 -20.925 -20.295 (0.09%) -20.275 -12.098 -11.579 (1.8%) -1 1.79' -1.1264 -0.56883 (48.9%) -1.1 139' -0.43228 -0.55607 (0.2%) -0.55731
5 In column 2, the term in parentheses represents the error. * Error on the damping term q-axis natural mode
The modes of the indirect solution are not as inaccurate as one might expect from studying the residual errors pictured in Fig. 4. This reveals the nature of a well-posed 489
identification problem: the solution is well defined and, obviously, the nonuniqueness question is less critical. In contrast, if one looks at the modes associated with the 2 3
a 1
4OO-
a
4,
-4
- 00
02
,yyty 0
,
1
,
lljVI', 4
04
02
b
.
4,
1
04
0.2
.
.
,
,
02
0
, 04
s d
5 C
,
the course of the direct estimation, the ultimate value of the mode (-1.1139) on the q-axis has been roughly halved. In spite of this truncation error, the prediction of the q-axis current is not noticeably affected, as can be seen from the residual plot in Fig. 5. This question, which is one of distinguishability [17] of the current outputs during the short-circuit experiment, has been discussed from a somewhat different point of view in Reference 10. The same issue can also be related to the well known weak observability of certain 4-axis modes during shortcircuit testing. In fact, the q-axis mode (-1.1139), which is obviously irrelevant for the short-circuit modelling, can be completely eliminated by applying a minimal realisation algorithm to the true model [7,40]. This mathematical tool leads to a reduced-order model whose poles and zeros appear in Table 5. The spurious mode (as far as the Table 5: Reduced-order model derived from short-circuit oscillographs d-axis zeros
0
YO,@)
0 -0 005
3 Q
2 Q
- 0 02
-0 04
-0.01
0
02
04
s
02
0
04
s
e f Fig. 4 Assessment of solutionfrom indirect approach a Field current:estimated b Field current: true c q-axis current: estimated d q-axis current: true e Field current h a 1 error 1 q-axis current final error
Field zeros q-axis zeros Yo,@) Ye,@)
Common poles d@)
1 -179.99 -296.39 -113.930 -180.45 2 -112.69 -179.99 -104.680 -113.87 3 -13.021 -15.759 -22.255 -20.275 4 -11.824 -11.824 -5.5681 -11.826 5 -0.12006 -2.6363 -5.0583 j376.40 6 . -0.21 104 -0.55729
*
short circuit is concerned) has disappeared from the eigenvalue list. It is of interest to look at Table 6 where we have recorded the singular values of the Hankel matrix [40] associated with the true equivalent circuit (e*). These values are used to determine the order of the Table 6: Singular value of the Hankel matrix associated with the Nanticoke machine short-circuit oscillographs
1.4708x 1 O4 1.5940x 10' 1.0073x 1 0 . ' ... ...
d, d4 d, 02
04
1.4337x 1 O4 2.3037 x 10' 9.0395x lo-* ... ...
d, d, d,
d, d, d, d,,
3.2595 x 10' 9.1936x lo-' 1.4643x lo-" 1.8490x lo-''
b
-41'
0
'
'
0.2 s
'
I
0.4
-411' 0
,
,
,
1
0.2
0.4
02
04
C
02
20 -0 2
e
0
s f
Fig. 5 Assessment of solutionfrom direct approach Field current:estimated b Field current: true c q-axis current: estimated d q-axis current: true e Field current ha1 error f q-axis current finalerror a
direct estimate of the equivalent circuits, it will be observed that even if the majority of values are accurate, one of them suffers from a large error of 48%. In fact, in 490
minimal realisation, corresponding to the smallest integer n for which the singular value d, is approximately zero [7, 401. Following such a criteria, it is obvious from Table 6 that the minimal realisation of the short-circuit observations is of order 7, which implies one damper less than the full order, or so called exact model. This missing damper corresponds to the q-axis mode (- 1.1139), which thus suffers from weak observability during the shortcircuit test. It is not our intention to assert that the true model of the Nanticoke generator should only include two dampers in the q-axis. We do claim though, as far as the description of its short-circuit oscillographs is concerned that the third damper of the q-axis is irrelevant. Including the latter renders the equivalent circuits nonparsimonious, with regard to short-circuit modelling. However, the third damper is still necessary, for example, in providing an adequate fitting of standstill frequency response data. This is another illustration of the well known detrimental effect of short-circuit testing on q-axis information. This test procedure, (even if it is an old, well established standard), systematically throws away pertinent information about q-axis sub-subtransient quantities which can be easily retained using appropriate standstill tests. IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
5
Justification of direct approach applied t o time-domain standstill tests
Despite the remarkable longevity of short-circuit testing [I, 81, the analyses in the preceding Section prove without doubt that it is far from being the best test procedure strictly from an information content point of view. As a result, specialists have searched for additional methods of obtaining the equivalent circuits. Instead of tests, for example, use has been made of field calculations to predict the parameters numerically [21, 351. However, this approach, which has mainly been restricted to the machine designers, has had little influence on the power system analysts, who generally do not have access to design and construction data. Several authors have considered another type of identification experiment consisting of step tests in both the d and q axes with the machine at standstill [28-301. Despite the difficulties in dealing with the absence of contact pressure and insufficient flux levels, these step response tests possess the advantage of being relatively simple and safe, even with large alternators. However, the most acclaimed standstill approaches, up to now, have relied on harmonic tests [7, 31, 321. Despite a certain amount of controversy [lS, 20,381, several checks, using this procedure, have been published in stability studies [33]. In fact, the success of the method has justified its inclusion in the very recent 1987 ANSI/IEEE No. 115A Standard [34]. 5.1 PWM square wave voltages as excitation signals From the point of view of applications involving chopper-machine units, circuit models inferred from step or harmonic responses are not always easy to justify. When they result from step response tests, they completely neglect hysteresis effects since the direction of the field is never reversed. Hence, they are only suited for the modelling of generators subjected to sudden failures in the lines (step disturbances). On the other hand, it is wellknown that the resistances and inductances take on values which vary slowly with frequency. Equivalent circuits deduced from frequency response tests thus only make sense when covering a sufficiently restricted harmonic range to guarantee more or less constant values of the parameters. Faced with the weaknesses regarding the types of excitation used up to now (sinusoidal or step), it seems appropriate to advocate, in the context of this particular type of application, alternative signals that can be easily generated by the actual choppers feeding the machines when they are in operation. Not only will this ensure the use of a source, which to all intents and purposes is perfect, and whose ideal waveform is known in advance and perfectly reproducible, it will also identify the machine under the same conditions as during normal operation. The hysteresis cycle defined during the identification test will be similar to that occurring under normal operational conditions, for comparable magnetisation levels. A schematic diagram of the experimental setup is shown in Fig. 6 and turns out to be very similar to that used in SSFR tests. The PWM source E, is provided by a chopper (Fig. 7) with the gate T of the MOSFET transistor controlled by a microcomputer also overseeing the data acquisition; this permits a flexible and efficient control of the cyclic ratio U of the excitation. For example, when U = 1, E, is a simple square wave. When the command sequence of the chopper is random, or quasirandom, it is termed simply RBS, or PRBS [U]. This leads to square wave voltages E, with a cyclic ratio IEE PROCEEDINGS-C, Vol. 138, No. 6 , NOVEMBER 1991
v(t) varying continually with time. If one eliminates an
eventual mean value defining the polarisation level of the magnetic circuit E, = (E, + E,)/2, E, is then a signal with a uniform spectrum, whose variance is imposed by the voltage E,.
C
Fig. 6
Experimental setupfor standstill response tests
Fig. 7
P W M voltage source
It would be particularly interesting to repeat the identification of the machine using a number of cyclic ratios, at constant polarisation levels, and, correspondingly, several levels of polarisation with the cyclic ratio constant (E,, U). In this manner, one could construct characteristic families of curves, in the plane, for each parameter of the equivalent circuits xAE,,U), xw(E,. U), rol(E,, U), x,,(E,, U), etc. Besides providing a data bank characterising the machine at different operating points, which would be of great value to the designer of the complete convertor-machine unit, these curves would shed new light on the behaviour of the inductances and resistances of the synchronous machine in terms of cyclic ratio and saturation. In fact, even if the primary justification for using PWM signals was drawn from the modelling of ChopperMachine systems, this type of excitation is equally well suited to identification of general purpose models. Taking advantage of the modulating properties of the chopper, it is possible to generate truly informative test signals with wide band spectra, persistently exciting properties [12], and controllable DC levels. It should be noted that, if one chooses to perform only one such test along the d-axis, this particular test should perturb the field voltage, and not the stator terminal voltage, in contrast to present practices [16, 3G321. By proceeding in this manner, we can log, in one shot, all the information about the standard operational impedances, as well as the required data regarding the poorly known field'input admittance. As a result of this choice, the direct parametrisation method leads, with a minimum of effort, to a well-posed identification problem, whereas the indirect method is obviously ineffective. The d-axis equivalent circuits can then be cheaply and optimally estimated directly from records of two currents (id, ,)i and one voltage (U/). This claim is proved in the following Section.
5.2 Validation by numerical simulation Because of the decoupling arising in the standstill operation of synchronous machines, one must distinguish, in 49 1
the course of the analysis, two constituent, and totally distinct, estimation problems; one exclusively tied to the d-axis network and the other to the q-axis network. Recalling the direct approach outlined in Section 2, the nonlinear least-squares problem is formally split into two separate parts with the following parametrisations:
e%=CrvrlrDi
... X D n d l X m d X k / ~ " ' x k / x d l
"'rD.,Ix/xD~
(124 when the estimation is aimed at the equivalent circuit in the d-axis and
for determining the equivalent circuit in the q-axis. To validate this direct approach to standstill data analysis, we numerically generated data for an hypothetical, static test carried out on the d-axis of the Nanticoke machine, which is used once again as a benchmark model. With the usual notation, the output vector registered was Y
=
[;;I
Two types of excitation were simulated: a step function of amplitude 0.01 pu and a RBS with variance 0.01. These variations can be supposed to apply around a given DC magnetisation level. The other experimental constraints were: T = 3 ms and N = 2020. The results corresponding to each type of excitation are summarised in Tables 7 and 8. In both cases, we imposed an initial Table7: Direct estimation of d-axis circuit from single standstill step response test
eo*
68
Gradient of V ( 6 ) Error, %
-0.08616 0.0048000 0.0040034 -22.90 0.0007520 0.0009359 -48.60 0.43530 0.1370000 0.10391 68 0.0272 9.00455 -1.90920 0.0047360 0.0060330 4.340 24.6060 0.0186000 0.116861 -0.00520 2.1856000 2.0927314 0.116~10-~ 23.3993 -40.6476 0.0090360 0.0105908 -0.0390 1.7216000 2.1559327 -0.1827 0.00820 x k , , -04172000 -0.3768276 -0.0158 27.7416 x , , ~ 0 71 80000 0 7550363 -00234 15 8734 * V(8,)
=
5 07.V ( &
= 0 661 x lo-'
Table 8 . Direct estimation of the d-axis circuit from single stochastic standstill test
en*
a'
Gradient of V ( 6 ) Error. "
0.0048000 0.0040000 0.0007520 0.0009400 rD1 0.1374040 0.1142001 rD2 0.0047360 0.0059200 xI 0.0186000 0.0155000 2.7320011 xD, 2.185600 xD2 0.0090360 0.0075300 xmd 1.7216000 2.1519986 x ~ , , -0.4172000 -0.5215000 x*,. 0.7180000 0.8975000 rB r,
* v(e,) =0.337;~ ( 6=)o m 3 x
-0.213 x -0.735x 0.461 x 0.787x 0.773 x lo-' 0.129 x lo-* 0.241 x lo-' 0.102x -0.109 x 1 0 - 6 -0.127 x lo-'
0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000
1 0 - l ~
error of 20% on all of the precisely known parameters, and assumed a noise-free environment. After 115 iterations, and 250 evaluations of the cost function, using the step response, we stopped the optimisation process as we were still far from the exact circuits. On the other hand the RBS had produced the desired circuits after 25 iterations, in one-fifth of the time. There492
fore it will be noted that, under similar experimental conditions, the RBS leads to a solution much more quickly than the step response approach. The authors of Reference 16 have reported somewhat similar difficulties when using the step voltage, but since they injected their excitation via the stator terminal, the comparison cannot really be pursued much further. It is worthwhile mentioning that a similar study applied to the q-axis network, led to strikingly good results for both excitations. In fact, the Canay mutual reactances intervening in the d-axis network, combined with a greater disparity between rotor and stator time constants, make it more complex (and hence more difficult to identify) than for the q-axis. Therefore the information content of the identification signals used is more critical for the d-axis circuit model than for the q-axis one. From a general point of view, besides possessing the advantages already mentioned, the present strategy is undoubtedly the only one capable of exploiting an important source of information generally ignored in most of the traditional schemes for analysing standstill response tests, i.e. the field current when the excitation is imposed by the rotor (ill). Not only is the latter informative, but above all it is the only one to contain definite information about the field input impedance with the stator open, xlo(p) C11. 20, 311, without which, no complete and unequivocal description of the synchronous machine is possible. Furthermore, as pointed out by Concordia in discussions in References 5 and 31, by putting the excitation on the rotor side, one is able to achieve higher flux levels during the standstill experiment, by means of a proper setting of the DC level of the PWM voltage source. As an additional advantage, a PWM voltage source is known from industrial experience to be more easy to design than its perfectly sinusoidal counterpart, at least for high power ratings. 6
Experimental
results
With regard to the practical implementation of the PWM standstill tests, preliminary results have already been reported in Reference 23. In this Section, we shall first summarise some unpublished findings arising from the application of the methodology described in the previous paper to stochastic data. As we shall see, they provide, through algebraic reduction of the operational impedances, a useful set of initial values for the equivalent circuit parameters. Starting from this realistic guess, we then illustrate the performance of the direct standstill identification scheme described above, based on actual data. 6.1 Standstill operational impedances and equivalent circuit initialisation Extensive tests involving excitations consisting of step and PWM with constant and random cyclic ratio have been performed on an industrial generator rates at 400 V/37 kW. The prime mover was a diesel motor normally used either as a backup power supply or as a basic component of a test bed for studying wind/diesel integration problems. The excitation device was powered by Nickel-Cadmium batteries in a configuration capable of sustaining more than half the nominal current of the diesel generator. Such a design is probably adequate for small and medium size synchronous motors and generators. However, for large utilities and industrial machines, such high levels of magnetising current are not so easily achieved. IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
Data were quantified to an accuracy of 12 bits (1800 sample/s) and recorded using a personal computer, which also controlled the firing sequence of the chopper. Basically, the first stage of the digital processing aimed at estimating the operational inductances from raw timedomain data. This goal was achieved using a black-box modelling approach, very common in system identification theory [12,46]. Basically, the input-output data is modelled by ARMAX type black boxes (autoregressive-moving average with exogenous input [l 2]), whose transfer function coefficients are estimated by the usual methods [12, 131. From the resulting discrete admittances [47], one can calculate the operational impedances of the machine and finally deduce estimates of the equivalent circuits, along both the d- and the q-axis [31]. The ARMAX fitting of the standstill responses to randomly modulated excitations led to the static admittances and operational impedances listed in Table 9 (for the notation, see Appendix 9.2). In fact, these models represent the time-domain records very accurately. In the opinion of the authors, they are at least as accurate, based on the present data, as those derived from standstill frequency response tests. To illustrate this claim, we refer the reader to Fig. 8, where a graphical validation of the field input admittance is provided. For the identification of the models in Table 9, only stochastic data were used, the deterministic data sets being reserved for crossvalidation purposes. Fig. 9 portrays the performance of the two input admittances in the d-axis, during the prediction of a constant cyclic ratio operation at standstill. The estimated models seem very satisfactory for predicting the response of the machine under varied experimental conditions, thus displaying fairly good representativity. In particular, the error in predicting the stator current is systematically less than 0.1% (see Fig. 9). Since we have established evidence for the correctness of the operational models furnished in Table 9, it seems legitimate to use them in calculating initial values of equivalent circuit parameters. For this purpose, our data reduction method parallels that described by Umans et al. in Reference 31, to which we refer the interested reader for the pertinent formulas and implementation details.
With regard to the order of the actual impedances in the two axes, a tentative structure of the equivalent circuit might involve only one damper in both the d- and q-axes (nd = np = 1). Under such circumstances, Table 10 displays the results of the data reduction procedure.
S (I
-6
02
0
04
06
08
10
I 12
b' Validation of thefield input admittance against its generating Fig. 8 data a Field current: model and - - -true* b Field current prediction error The two curves are indistinguishable from one another ~
~
Table 10: Equivalent circuits (pu) calculated from actual omrational imDedances using the Umans f0rmulaS 1311
0.057100 0.24796 x D , 0.25793 x o 7 0.08459 x,. 0.26670 f.. 60 HZ
r,
xa
r,
x,
0.04930 0.0072704 I , , , 0.53789 r0, 0.098594 xmd 0.34070 X. -0.000843
For clarity, it is worth mentioning that the value of the reactance x, comes from the manufacturer and that of the stator resistance r, results from preliminary DC tests. However, it is shown in Reference 23 that, once the test data for the currents {i,,(k), k = 1, 2, ... ,N} and the voltages {U#), k = 1, 2, . .., N} have been recorded, one can immediately calculate the value of the resistance more
Table 9: ARMAX modelling of standstill responses to P W M excitation with random cyclic ratio. Admittances, 0-'
V,,@)
1 + 0.2171 8p' + 0.0003748p2- 6.1307 x 1 O-*p3 1 + 0.2358382p'+ 0.0024421pz+ 2.96956 x 1 O-'p3 + 3.394x 10-'0p4
= 1.8236093
V,,@)
= 0.2110885
1 + 0.03721640' + 0.0000527616p2- 8.7254x 1 O-'p3 1 + 0.2358382p'+ 0.0024421p2+ 2.96956 x 10-'p3 + 3.394 x lO-''p4
p,,@)
= 0.0371822
1 +1.599817p+0.00165513p2+2.6132 x 10-'p3 1 + 0.2358382p'+ 0.0024421p2+ 2.96956 x 10-6p3+ 3.394 x 10-"p4
p,,@)
= 1.7739272
1 + 0.00822326p'- 3.241041 x 1 O-'p2 1 + 0.02396716p' + 5.1551 302 x 10-5pz+ 4.53597x 1 O-qO'
Operational impedances, H L,@) = 0.010098
(1 +0.10804p)(l+O.O0147p) (1 +0.21544p)(l+0.00197p) + 0.00343p) (1 + 0.00934p)
L,@) = 0.0082193('
+ 0.010607~) (1 + 0.001387p) (1 +O.O3573p)(l +O.O01717p)
L,@)
= 0.093978 ( '
G@)
= 0.020388( '
+ 1.5988p)(1 + 0.001272p) (1 +0.21544p)(l+0.00197p)
IEE PROCEEDINGS-C, Vol. 138, No. 6 , NOVEMBER 1991
493
accurately using the following formula:
bations are very poor. This fact constitutes strong experimental evidence for our previous claim that a set of equivalent circuit parameters can match stator transients
N
08,
4,
0.21
I
VI
01
VI
i:m 0
-0 2
0
S
0.8,
a
,
5
0
0
0.5
1.0
5 C
E
0-
5
-
-0.02
0
03
0
z 0.5
D
1.0
--0 2
S
b
T 2 k 0.02.
02
I
5
1.0
03
a
$Lo
0.5
I
02 S
?
k 2 5 0
I
-041 0
01
I
S
b
1 5,
0.5
I I
1.0
S
d
Fig. 9 Crossvalidation oftwo input admittances along d-axis d-axis current
4
b d-axis current error
5 C
c Field current
d Field current error (-model and - - - true) ~
6 2 Direct estimation of d-axis equivalent circuits In this part of our study, we shall focus on the d-axis issue only, because the q-axis equivalent circuit is obvious when n4 = 1. Starting from the initial values in Table 10, we proceed with the estimation using the following observation vector.
Y = [ 3
(12d)
where id, and i f , are the stator and field current, respectively, during a standstill test with excitation on the stator d-axis input and the field input. As in the case of the estimation of the operational impedances, these data originated from stochastic tests, and the two entries of the observation vector were generally chosen to ease the nonlinear least-squares problem. In fact, even if validation of the method (Section 5.2) was first performed using NL2SOL software*, the optimisation algorithm used in the present experimental investigation is a rather simple version of a damped Gauss-Newton algorithm described in more detail in Reference 49. Moreover, this algorithm is cheaply coded with no scaling, using the MATLAB [48] prototyping facilities and from our experience, it does exhibit some difficulties in converging when the problem is ill-posed, e.g. short recording lengths, coloured noise, bad scaling, or lack of parameter identifiability (mainly due to an over parametrisation of the underlying data). We can first evaluate at what level the equivalent circuits of Table 10 can be trusted. Fig. 10 demonstrates that the calculated d-axis circuit is a valuable tool for predicting stator transients, even when the perturbation arises from the field winding. However, it is obvious that its predictions of the field current during rotor pertur-
* While the first author was with the Electrical Engineering Department of the Lava1 University, Quebec, Canada 494
Fig. 10 Assessment of equivalent circuits calculated from actual operational impedance data -( model and - - - -true) 4 Stator excitation b Rotor excitation c Rotor excitation
as well as coupling phenomena between rotor and stator, yet be deficient in characterising the field input admittance. To improve the accuracy of the equivalent circuits (especially for the field input admittance), we have applied the direct identification scheme to raw data, picking a record length of N = 650 samples. This procedure leads us to the parameter values listed in Table 11. Table 11 : Direct estimation of the d-axis circuit parameters (DUI from stochastic standstill tests'
r,
0.057100 0.047211 x, 0.27455 0.011099 x , rD, 0.11228 iD 1.5260 xmd 0.73500 xk,; -0.10630 r.
Initial sum of squares: V(0,) = 1.6066; Final sum of squares: V ( b ) =0.002959(withw,=w,=O.l).
To give the reader more insight and confidence about these results, we provide in Fig. 11, the graphical validation of the model. The waveform predicted by the estimated model is so close to the actual one that it is impossible to distinguish between the two solely by visual inspection. Interestingly enough, the current experienced by the stator when the field winding is excited (Fig. lob), is also accurately predicted, even if the corresponding test data are not used in the fitting process (see eqn. 124. Usually, when a well posed estimation problem is solved, the prediction errors tend to be normally distributed. Looking at Fig. 12,, one sees that this is true in the present case. At the start (Fig. 124, the residual sequence is corrupted by a slow drift, but at the solution point (Fig. 12b), it appears that the residual variance is practically constant. Also, the field current prediction error is less than 0.8% of the true value, in the worst case. IEE PROCEEDINGS-C, Vol. 138, No. 6 , NOVEMBER 1991
Finally, we have applied a cross-check procedure to the estimated equivalent circuit, to verify if it is able to predict equally well more than its generating data. The 081
I
ance can be found in Reference 50. In fact, L&) does not intervene in the calculation of the initial equivalent circuit according to Umans et al. The difficulty in calculating circuits to fit the field current during rotor perturbation, is a simple consequence of ignoring this truly informative piece of datum during the calculations. 02
z- 0 1 -a
S
a I
0
0
0
01
02
03
04
0.5
06
07
04
05
06
07
0.4
0.5
Ob
07
a
2
G-o
+
2
0
0.1
0.2
03
I
5
b
I
15
'-1
'
0
I
01
I 02
03 b
0
0.2
0.1
-a
-
Fig. 11 Assessment of the equivalent circuits directly estimated from standstill responses to a randomly modulated excitation (-model and actual)' The two curves are indistinguishablefrom one another (I Stator excitation b Rotor excitation c Rotor excitation ~
I 0
~
~
1-
zL0.2
0.3
5
C
01
0
I 0.1
0.2
0.2
0.3
0.2
0.3
I
S
a
"._"" 0 2
0.005
0.0101 0
0.1
I
5
b
Fig. 12 Initial andfinal prediction error sequences (I Initial weighted prediction errors b Final prediction errors and field current errors - - - ~
data used for this purpose originated from a pair of tests with a deterministic PWM excitation applied first to the stator and then to the rotor. The results displayed in Fig. 13, indicate that the estimated equivalent circuit is suffciently representative of the actual behaviour under various conditions.
6.3 Discussion In Table 9, we have introduced without further justification, a nonconventional impedance L&). This impedance plays, for the field winding, a role similar to the stator impedance L&). Its calculation from the standstill admittance yf&) (cf. Appendix 9.2) only involves a little algebra and knowledge of the field resistance (rJ. A more rigorous justification and derivation of this new impedIEE PROCEEDINGS-C, Vol 138, No. 6, NOVEMBER 1991
5 C
~
0.1
0.3
Fig. 13 Crossvalidation of the estimated d-axis circuit (-model and - - - -actual) (I Stator excitation b Rotor excitation c Rotor excitation
Even if we claim that it is possible to rely on only one test in the d-axis for the direct estimation of its equivalent circuit, we feel it is necessary to stress that, in practice, it could be useful to make a supplementary test with excitation on the stator side. The first reason for this is that a complete cross-check of the estimated model requires data with excitation on both the stator and rotor side. Secondly, when working with short recording lengths, or noisy data, or high initial uncertainty regarding key parameters (r, and rf), one may need more independent data, to achieve fast convergence to an acceptable solution (see eqn. 1212).Moreover, if one lacks manufacturer's data, or if this is unreliable, knowledge of the operational impedances is mandatory for a realistic initialisation of the direct estimation procedure. As we know, this requires a further experiment with the excitation applied to the stator. 7
Conclusion
The present paper has attempted to put the classical notions of operational impedances and equivalent circuits, into a more general framework. The two complementary parameterisations obtained for synchronous machines are notable extensions of the classical techniques, in that they readily take into account an arbitrary number of dampers or dynamic constants in each axis and are hence particularly adaptable to the digital representation of solid rotor turboaltemators. When we determined the equivalent circuit parameters from test data in an indirect manner, by parametrising our models in terms of time constants (i.e. operational impedances), it was found that two frequently used procedures gave rise 495
along the d-axis to systematic errors, as high as 35%, in the subtransient and sub-subtransient time constants of the resulting circuits. This result of course underlies the basic advantage of methods aimed at determining equivalent circuits from test data directly [lo]. The time-domain test that we first attempted to use, was the standard no-load short-circuit test. This turned out to be a poor candidate because, even the direct method suffered from the drawback of weak identifiability of the model, and there was no guarantee regarding the uniqueness of the equivalent circuit directly estimated from the test data. This difficulty, combined with the need for better probing signals in the case of convertor-machine units, motivated a new procedure for parametric estimation from standstill tests, making use of deterministic or stochastic PWM voltages as the excitation. We feel that the present approach is innovative for two particular reasons: (a) It only requires a single test in the d-axis, consisting of exciting the field winding, with the stator short circuited, and measuring the resulting two currents in the terminal and field windings. In this sense, this test is parsimonious as regards the data required; in addition, it is easy in this manner to achieve higher magnetising flux levels. (b) It also leads straight away to equivalent circuits which are the truly desirable models. In this sense, this approach is direct. 8
References
1 DOHERTY, R.E., NICKLE, C.A.: ‘Three-phase short circuit of synchronous machines’, Trans. AIEE, 1930,49, (V), pp. 7 W 7 1 4 2 PARK, R.H.: ‘Two reaction theory of synchronous machines; generalized method of analysis’, AIEE Trans., 1929,48, (I), pp. 71C728 3 CANAY, 1.M.: ‘Physical significance of sub-subtransient quantities in dynamic behaviour of synchronous machines’, IEE Proc. B, Electr. Power Appl., 1988, 135, (6), pp. 334-340 4 DANDENO, P.L., KUNDUR, P., and SCHULTZ, R.P.: ‘Recent trends and progress in synchronous machine modelling in the electric utility industry’, Proc. IEEE, 1974,62, pp. 941-950 5 IEEE TASK FORCE ON DEFINITION (P.L. DANDENO: CHAIRMAN)’ ‘Current usage and suggested practices in power system stability simulation for synchronous machines’, IEEE Trans., 1986, EC-1, (l), pp. 77-93 6 IEEE TASK FORCE ON DEFINITION (P.L. DANDENO: CHAIRMAN): ‘Supplementary definitions and associated test methods for obtaining parameters for synchronous machine stability study simulations’, IEEE Trans., 1980, PAS-99, pp. 1625-1653 7 KAMWA, I., and VIAROUGE, P.: ‘Estimation structurale et parametrique des turbo-alternateurs a rotor massif par realisation minimale des reponses indicielles‘, Can. J . Electr. Comput. Eng., 1990, 15, (4), pp. 149-157 8 ANSI/IEEE Standard 115-1983, ‘IEEE Guide: test procedures for synchronous machines’ 9 KRAUSE, C.: ‘Analysis of electric machinery’ (McGraw-Hill, New York, 1986) IO KAMWA, I., VIAROUGE, P., and DICKINSON, E.J.: ‘Direct estimation of the generalized equivalent circuits of synchronous machines from short-circuit oscillographs’, IEE Proc. C , 1990, 137, (6), pp. 445451 1 1 KAMWA, I., VIAROUGE, P., and DICKINSON, E.J.: ‘Optimal estimation of the generalized operational impedances of synchronous machines from short-circuit tests’, IEEE Trans., 1990, EC-5, (Z), pp. 4 0 1 4 7 12 LJUNG, L.: ‘System identification: a theory for user’ (Prentice-Hall, 1987) 13 GOODWIN, G.C., and PAYNE, R.L.: ‘Dynamic system identification: experiment design and data analysis’ (Academic Press, New York, 1977) 14 SWIDENBANK, E., and HOGG, B.W.: ‘Application of system identification techniques to modelling a turbogenerator’, IEE Proc. D, Control Theory & Appl., 1989,136, (3), pp. 113121 15 DIGGLE, R., and DINELEY, J.L.: ’Generator works testing sudden-short-circuit or standstill variable-frequency-response method; IEE Proc. C, Gen. Trans. & Distrib., 1981, 128, (4), pp. 177I82 496
16 KEYANI, A., HAO, S., and SCHULZ, R.P.: ‘Maximum likelihood estimation of generator stability constants using SSFR test data’. IEEE Paper 90WM 005-9EC, Presented at the IEEE Power Engineering Society Winter Meeting, 1990, Atlanta, Georgia. 17 DISTEFANO 111, J., and COBELLI, C.: ‘On parameter and structural identifiability: non-unique observahility/reconst~ctibility for identifiable systems, other ambiguities and new definitions’, IEEE Trans., 1980, AC-25, pp. 83C-833 18 DAVAT, B., LAJOIE-MAZENC, M., and HECTOR, J.: ‘Mtthodes de modtlisation des machines tlectriques a rotor massif, alimentks par convertisseur statique’. RGE, 3/85, 1985, pp. 239-245 19 KUNDUR, P., and DANDENO, P.L.: ‘Implementation of advanced generator models into power system stability programs’, IEEE Trans., 1983, PAS-102, pp. 2047-2054 20 JACK, A.G., MENG, L.G., and BEDFORD, T.J.: ‘Frequency domain-based methods to predict turbogenerator transients with particular emphasis on field current’, IEE Proc. C, Gen. Trans. & Distrib., 1989, 136, (3), pp. 2 W 2 1 4 21 KAMABU, T., and MAUN, J.C.: ‘Turbine-generator models by the finite-element method‘, in ROBERT, J., and TRAN, D.K. (Eds.): ‘Modelling and simulation of electrical machines, converters and power systems’ (North-Holland, 1988), pp. 239-246 22 HOEIJMAKERS, M.J.: ‘On the steady-state performance of a synchronous machine with convertor’. Doctoral thesis, Technische Hogeschool Eindhoven, 1984 23 KAMWA, I., VIAROUGE, P., and DICKINSON, E.J.: ‘Experimental modelling of a synchronous machine-chopper system using standstill normal operating records’. Proc. IMACS-TC1 Third Symposium on Modelling and Simulation of Electrical Machines and Power Systems, Nancy 19-24 Sept 1990, France, Vol. 1, pp. 247-253 24 KEYHANI, A., HAO, S., and DAYAL, G.: ‘Maximum likelihood estimation of solid-rotor synchronous machine parameters from SSFR test data’, IEEE Trans., 1989, EC-4, (3), pp. 551-558 25 ALVARADO, F.L., and CANIZARES, C.: ‘Synchronous machine parameters from sudden-short circuit tests by back-solving’, IEEE Trans., 1989, EC-4, (2). pp. 224-236 26 SALVATORE, L., and SAVINO, M.: ‘Experimental determination of synchronous machine parameters’, IEE Proc. B, Electr. Power Appl., 1981, 128, (l), pp. 1-7 27 CANAY, I.M.: ‘Determination of synchronous machine parameters from the characteristic quantities applicable also to sub-subtransient data’, Electr. Mach. Power Syst., 1984,9, pp. 33-47 28 DE MELLO, F.P., HANNET, L.N., SMITH, D., and WETZEL, L.: ‘Derivation of synchronous machine stability parameters from pole slipping conditions’, IEEE Trans., 1982, PAS-101, pp. 3394-3402 29 TURNER, P.J., REECE, A.B.J., and MACDONALD, D.C.: ‘The D.C. decay test for determining synchronous machine parameters: measurement and simulation’, IEEE Trans., 1989, E C 4 (4), pp. 616-623 30 BOJE, E.S., BALDA, J.C., HARLEY, R.G., and BECK, R.C.: ‘Timedomain identification of synchronous machine parameters from simple standstill test’, IEEE Trans., 1990, EC-4, (2), pp. 164175 31 UMANS, S.D., MALLICK, S.A., and WILSON, G.L.: ‘Modelling of solid rotor turbogenerators (in two parts)’. IEEE Trans., 1978, PAS-87, pp. 269-291 32 DANDENO, P.L., and PORAY, A.T.: ‘Development of detailed turbogenerator equivalent circuits from standstill frequency response measurements’, IEEE Trans., 1981, PAS-100, (4). pp. 1 6 4 6 1654 33 DANDENO, P.L., KUNDUR, P., COULTES, M.E., and PORAY, A.T.: ‘Validation of turbogenerator stability models with comparison with power system tests’, IEEE Trans., 1981, PAS-100, (4), pp. 1637-1643 34 ANSIpEEE Standard 115A-1987: ‘Standard procedures for obtaining synchronous machine parameters by standstill frequency response testing’ 35 TURNER, P.J.: ‘Finite-element simulation of turbine generator terminal faults and application to machine parameter prediction’, IEEE Trans., 1987, EC-2, (l), pp. 122-131 36 DENIS, J.E., GRAY, D.M., and WELSH, R.E.: ‘An adaptive nonlinear least-squares algorithm’, ACM Trans. Math. Softw., 1981, 7 , (3). pp. 348-368 37 DENIS, J.E., and SCHNABEL, R.B.: ‘Numerical methods for unconstrained optimization and nonlinear equations’ (Prentice-Hall, 1983) 38 CANAY, I.M.: ‘Identification and determination of synchronous machine parameters’, Brown Boueri Reo., 1984,617, pp. 299-304 39 ASTROM, K.J.: ‘Modelling and identification of power system components’, in HADSCHIN, E. (Ed.): ‘Real-time control of electric power systems’ (Elsevier Publishing Co., 1972), pp. 1-23 40 JUANG, J.N., and PAPPA, R.S.: ‘An eigensystem realization algorithm (ERA) for modal parameter identification and model reduction’, J . Guid. Control & Dyn., 1985,8, pp. 620-627
IEE PROCEEDINGS-C, Vol. 138, No. 6,NOVEMBER 1991
41 EPRI: 'Determination of synchronous machine stability study constants'. EPRI report EL-1424, Vol. 2 (Project 997-2), by Ontario Hydro, Dec. 1980 42 KEYHANI, A., HAO, S., and DAYAL, G.: 'Effects of noise on frequency-domain parameter estimation of synchronous machine models', IEEE Trans., 1989, EC-4, (4), pp. -7 43 ASTROM, K.J.: 'Maximum likelihood and prediction error methods', Automatica, 1980,16, pp. 551-574 44 TULLEKEN, H.J.F.: 'Generalized binary noise test-signal concept for improved identification-experiment design', Automatica ( U K ) , 1990,26, (l), pp. 3 7 4 9 45 LIPO, T.A., and KRAUSE, P.C.: 'Stability analysis of variable frequency operation of synchronous machines', IEEE Trans., 1968, PAS87, (l), pp. 227-234 46 SWIDENBANK, E., and HOGG, B.W.: 'Application of system identification techniques to modelling a turbogenerator', IEE Proc. D, Control Theory & Appl., 1989,136, (3), pp. 113-121 47 KAMWA, I., VIAROUGE, P., and DICKINSON, J.: 'A discrete technique for synchronous machine operational impedances estimation from timedomain standstill tests'. Proceedings of the 12th IMACS World Congress on Scientific Computation, Pt. 3, pp. 24-27,18-22 July 1988, Pans. 48 MOLER, C., LITTLE, I., and BANGERT, S . : PC-MATLAB for MS-DOS personal computers, Ver. 3.1-PC, Feb. 27, 1987. The Math Works, Inc., 20 North Main St., Suite 250, Sherborn, MA 01770. 49 SPATH, H.: 'The damped Taylor's series method for minimizing a sum of squares and for solving systems of nonlinear equations; ACM Algorithm #315', in MILLER, W. (Ed.): 'Collected algorithms from ACM, Vol. I1 (ACM Publications, New York, 1980) 50 KAMWA, 1.: 'Determination expkimentale des modeles d'une machine synchrone excitee a I'arr2t par des tensions en creneaux et en khelon', IREQ report 4631, June 1990 (in French).
Gd
=
Gq
= Cxqq
xdJ
Cxdd
XdD,
xqQl
(144
xtQJ
"'
aijis
the null matrix with dimensions i x j, U, the nominal pulsation in rad/s, and the term xkI refers to the mutual reactance between windings k and I Xdd
+ x,
= xmd
=xd
c Iu
xJJ
+ xmd + i = l ' k l i
= XJ
I=l>...?nd
XD,Dl=XD,+X&++cXkJi #=I
XdJ
= XJd = X
U4f)
d
I = 1, . . ., nd = XDIDl - xDI 1 = 1, .f . nd
xdDl= xDld= x,,, xJDI
= xDIJ
1
= X D ~ D=~X D ~ D , - X
XD,D,
D ~
k, I = 1,
. . ., nd
and
+ x, = xq + xmq+
xqq = xmq XQIQl
= XQI
= 1, ... , nq ..., nd k, I = 1, . . ., nq
I
XQI
xqQl= xQ,q = x,,,,, xaQ, = x ~ , = ~ x,,,,
9
xdDdI
"'
(149)
1 = 1,
From these basic equations one deduces a state variable model which constitutes the most rigorous characterisation of synchronous machines linear regimes
Appendixes
9.1 Equivalent circuits modelling
_ d i -- Ai + BO
If we introduce the two vectors
y = ci
dt
(154
with the basis electrical equations associated with the equivalent circuits in Fig. 1, will take the following matrix form: 1
-X w,
di dt
L
-=
di dt
-=
-{R
+ w,G}i + u
B
=o,X-'
0
1
0
xd
Ond+Z,nq+l
Onq+1.nd+2
x,
with
A
=
-B(R + o,G)
(156)
It follows from these equations that, for a given speed, the state space representation of an autonomous synchronous machine is of an order n = nd +. nq + 3, with m = 3 electrical inputs and p = 3 electrical outputs. The vector of the only measurable currents being y T = [id iJ i q ] , it follows that the output matrix C is given by 0
Ol.nd+2 1
01..-3
1
(1 5 4
Ol.n-nd+3
9.2 Operational impedance modelling By definition, the operational impedances (eqn. 3) satisfy the two following relationships:
1
Cr,
+ ~ x d ( P ) l-i ~m,xq(P)iq = Od - P G @ ) ~ J
(164 + Cr, + px,(P)li, = uq - 0, G(P)uI with o,= w,o,. Ti,T i , and T: are respectively called the transient, subtransient and sub-subtransient (or hypertransient) time constants [3]. Under these conditions, it will be understood that T?+' denotes the (nd)-sub-subtransient.The matrix 0, xd(P)id
xdd
xdJ
xJd
XdDi
x J D ~
xD.dd
xD*dJ
xD~dJi
xqq
xqQi
'qQ,
'Pip
xQn,rl
'QiQi
xQnqQ~
"'
... ... ..'
... ...
x Q ~ Q ~
... xQ~,,Q~
XdDms
'ID.,
X~ld~nd.
xqQnq
x ~ ~ ~ m q
x ~ n , ~ n .
IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
H(P)
=
c(P1 - A ) - l B
=
Ydd
YJd
Yqd
YdJ
YJJ
YqJ
[ydq
YJq
yqql
(1 6 4
contains all the admittances of the machine for a given speed. For linear regimes at fixed speed, this matrix fraction description is equivalent in every respect to the 497
equivalent circuit model of Fig. 1 and as such, has been used in place of the latter for the detailed simulation of various types of linear, constant speed transients [20]. Solving the set of eqns. 16, one obtains the following closed-form relationships between the fractions involved in H(p) and the operational impedances 3:
(174
and
d(P) = ( X d n
X q h 2
+ O:xdn
+ ro(xdd + xqd xdn)P + r.' xdd xqd
Xqn
(17f 1
Putting w, = 0 in eqns. 16-17, one obtains a simplified matrix H(p) characterising the machine at standstill
(184 with
498
IEE PROCEEDINGS-C, Vol. 138, No. 6, NOVEMBER 1991
