Short-circuit test based maximum likelihood estimation ... - IEEE Xplore

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IEEE Transactions on Energy Conversion, Vol. 14, No. 2, June 1999

Line-to-line Short-circuit Test Based Maximum Likelihood Estimation of Stability Model of Large Generators R. Wamkeue, (StM) kcole Polytechnique, Montrt!al, Qukbec, Canada

I. Kamafa, (M) Canada

Abstrut-This paper presents a time-domain statistical identification method for synchronous-machine linear parameters from the standard line-to-line short-circuit test. The measurements are recorded on a 13.75-MVA hydrogenerator at Hydro-Qukbec's Rapide-des-Quinze generating station. A complete mathematical model for synchronousmachine asymmetrical test analysis is proposed. An efficient algorithm is built to accurately calculate the standard equivalent circuit from time-constants and operational inductances. The maximum likelihood estimator derived from the generalized least-squares method is then used for ' parameter identification. Validation of the estimated model response against the measured running-time domain data confirms the effectiveness of the proposed estimation technique.

X. DS-DO,(SM) Ecok Polytechnique, Montrkal, Qukbec, Canada

A serious difficulty in using the generalized least-squares and maximum likelihood estimators is estimation of the unknown covariance matrix of innovation sequence. In this sense, these two estimators are said to be equivalent [51. The purpose of this paper is to use an alternative procedure derived from the generalized least-squares method to implement maximum-likelihood identification of machine parameters from the line-to-line short-circuit test. The main objectives of the work are listed below: Develop an appropriate state space model for the synchronous-machine asymmetrical short-circuit test analysis. Present a suitable maximum likelihood identification procedure for synchronous-machinemodeling.

I. Introduction Many papers have been published on synchronous machine modeling and parameter estimation using on-line test data [l]. These tests generally take into account the saturation effects, centrifugal force effects on the damper windings and rotational effects of the coupling axes. Although the three phase sudden short-circuit test is not a typical formal running time-domain response testing (RTDR), it has the merit of presenting the same advantages in addition to be more informative and easier to implement. More recently, problems in the resulting estimated parameters using this test have shaken users' confidence, however. Among other things, the analysis of the no-load threephase short-circuit test as proposed by authors does not give any information on the quadrature-axis components or the field current; morever, it is always difficult to explain the harmonic effects of sequential circuit-breaker closing on the estimated parameters during the test. Consequently, there appears to be an advantage in performing more informative alternative testing such as line-to-line short-circuit tests, which do not cause the problem just mentioned [2]. The maximum-likelihood estimator has been extensively used for synchronous machine parameter identification [3,4]. The likelihood function remains an unbiased, consistent estimator for stochastic system modeling.

Build the measurement setup for the line-to-line short-circuit test together.with the analysis principle using the proposed model structure. Provide a suitable algorithm for computing initial parameter values of the standard circuit from the machine short-circuit time constants. Derive an accurate syncronous-machine model for Rapidedes-Quinze generator based on line-to-line faults.

11. Generalized synchronous-machine models and

parametrization It is established in [2] that the equations of the accurate synchronous machine model including the neutral connection for both symmetrical and asymmetrical test analysis, can be described by equations (1-5) where E = -1 for the generator and E = 1 for the motor behavior of the machine. In addition to being in a simple compact form, this formulation could also allow the sign convention problem to be avoided when organizing the machine equations, and to derive a machine state space model with currents or flux as state variables.

PE-193-EC-0-2-1998 A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Energy Conversion. Manuscript submitted July 1, 1997; made available for printing March 2, 1998.

0885-8969/99/$10.00 0 1998 IEEE

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n = n + n +4 is the order of the system; nd and n are the d 4 4 total damper windings in d- and 4- axis respectively, d 1 = 1,2-..n ,I' = 1,2.-.n and p = - I is the n order identity d 4 dt'n matrix and w is the rotor speed in (pu). These equations rn suggest the equivalent circuits shown in Fig.1. where w and m are respectively the process and measurementnoise sequence with the following covariance matrices [3,4]: ~ [ w=]o ;

Q = E[ WwT] ;

R = E[ mm'];

E[x(O)]= 0 ;

0

Fig.1: Generalized equivalent circuit model of synchronous machines. (a) e =-a y, ; (b) ed = w m y d 4 m 4 In the previous equations, h denotes zero-sequence axis components. In a noisy environment, when state and measurement errors are taken into account, the general form of a deterministidstochastic discrete linear state-space model is given by: xgo = xg (0) xg(k +i,e) = A g ( k , e ) X g ( k , e ) + B g ( k , B ) u g ( k ) + F ( k ) w ( k )

~ [ r n=] o P0 = E [ m T ]

(16) (17)

The algebraic equivalence (18) relates the deterministic current and flux models. All the state-space matrices of current mode1 in (18) and the transformation matrix T are defined in the Appendix. The matrices A , E , C ,D , F and G are functions of the g

g

s

g

unknown parameter 8 . Q is the process noise covariance matrix. The b equations in (6) are 1 + 3 constraints relating x , U , and 8 , a priori, which may ensure system stability and accelerate algorithm convergence. n . ( A ) denotes the jth eigenvalue of J g A . g

111. Maximum likelihood identification method

yg(k,8) = C g x g ( k , @ + D g u g ( k ) + G ( k h ( k )

A. The maximum likelihood estimator

fi[s:x,(k,e);u,(k)]r0,1=1,2.--y; 8 =i,w(6) )) c 0 , j = l..-n b Re(n.(A I g Q(k,e> 1 0 R(k,O)> 0

The maximum likelihood estimate of parameter vector is the value of8 that maximizes the joint probability that y(k,8)is equal to the actual measurements y(k) in hand [5, 81. This is equivalent to minimizing the negative logarithm of (19) to conform with optimization conventions.

with

g' =

[Ps

e=[eT

T

T g,]

;y g

=[is

if]

T ; u g =[vs

8, = diug(Q)

v,]

(7)

(8)

1 N V ( 8 )=-2 k Z = 1 ( & ( k ) T R ( 8 ) - 1 & ( k ) ) + ~ N l o g ( d e t ( R ( 8 ) (19) ) E=y-Yp;

R(8) = E["]

(20)

169 where E is the innovation sequence and R(6) the corresponding covariance matrix. y is the predicted noisy observations. Since P

the generalized least-squares estimate BG of 8 is equal to its maximum likelihood estimate eM , tho following t h r m p procedw of the generalized least-squares method is used to compute OM [5].

The minimization of (19) is a nonlinear optimization problem, which can be solved by means of Newton-type iterative algorithms using finite difference computation. Therefore, starting with an initial guess obtained from the method developed in section V, the maximum-likelihood estimate of parameter vector 8 is evaluated by choosing a suitable constrained optimization algorithm such as developed in reference [9].

IV. Test setup 1.

Set R(6) = I and minimize (19) with respect to 8

2.

Calculate R(8) = -!g ~ ( k ) e ( k using ) ~ the residuals from Nk=l step 1

3.

Form the cost function (19) and solve the minimization problem.

Steps 2 and 3 are repeated until convergence is attained. At each iteration i , Rice) is updated at step 2 ; thus (19) behaves like

The experimental arrangement of the test is shown in Fig. 2. The machine under test is a wye-connected three-phase turbinegenerator at Hydro-Qu6bec's Rapide-des-Quinze generating station, rated at 13.75 MVA, 13.2 KV, 0.8 power factor, 60Hz, and 180 rpm. A 10% line-to-line short-circuit is obtained by suddenly applying a single-phase short-circuit between the two phases a and b of the unloaded generator running at synchronous speed using switch k as shown in Fig. 2. 0

I

the weighted least squares that will be reweighted during future iterations. Therefore, the procedure is also called the iteratively reweighted least-squares estimator.

B. The predictor-correctorand optimization algorithm The predicted state and output variables are computed using the discrete form (21) obtained by combining the discrete deterministic model of the system-machine with the Kalman prediction-correctionformulation [7] x

P

(k +1,6)= [A(k,6)-K C E (k,6) P I

Data Acquisition System

Fig. 2: Line-to-line short-circuit test experimentation where the index p denotes the predicted variable values, and Ol,z is 1x z order zero matrix; K is the limiting (steady-state) Kalman gain matrix defined in (22) and P a solution of Ricatti matrix equation (23)

K = PCT(R+CPCT)-l

(22)

P = A(k ,Os)[ P - KCP + Q]AT (k,e,)

(23)

z Q = jexp-Aa F F T e ~ p - A T a d ~

The testing is done according to the standard procedure described in EEE Standard 115 [lo] and confirmed by IEC Standard [ll]. The technical equipment used for data acquisition is fully explained in the test report [12]. The recorded rotor speed variation is illustrated in Fig. 3.

V. Initial conditions and short-circuit test modeling A. Control voltages

(24)

0

The covariance matrix of the discretized process noice Q is

In order to compute d-q-h armature voltages, it is important to evaluate , the angle by which the d-axis leads the center line of the phase a winding in the direction of rotation. Its expression is given by (25) [13], where Po = P(t = 0) :

obtained from (24). In practice, Q = zFFT [8]. SinceF is unknown, an adequate value Qo of Q is chosen to initialize the Kalman Filter. Further estimates of Q are computed together with the system parameter vector Os of (8).

The rotor angle in steady state is then:

n

= -- (a-

2

6)(25b)

170

where a is the phase angle of armature voltage of phase U at t = 0 [ 1 3 ] . Fig. 3 shows that the rotor speed variation in the interval chosen [ I S 5 1.721, is less than 0.1%; f l = 155s is the instant of fault application. Therefore, it can be assumed that the machine is rotating at synchronous speed ( Q)m = Q)s ). If we set p = 0 as the zero-reference angle [13],equation (25) becomes 0

(a) measured wm

1,

short-circuit. Since the armature voltages are not symmetrical as illustrated in Fig. 4 and Fig. 5 (a), the input armature voltages in Park reference frame are not step voltages as in case of the three phase short-circuit test (see Figs. 6-7). Their sinusoidal characteristic which includes harmonics, contributes more to exciting the machine modes strongly. In fact, these input voltages have a large spectral density making the line-to-line short-circuit test a goad choice for synchronous-machine parameter identification.

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