IEEE Transactions on Power Delivery, Vol. 7, No. 2, April 1992 789

Fast Adaptive Schemes for Tracking Voltage Phasor and Local Frequency in Power Transmission and Distribution Systems

1. W w A , MEMBER, IEEE R. GRoNDIN, MEMBER, IEEE

Instituf de Recherche d Hydro-Qdbec (IREQ) 1800, monte'e Ste-Julie, Varennes, Qukbec, Canada, J3X IS1

ABsrmcT - Real-time measurements of voltage phasor and local frequency deviation find applications in computer-based relaying, static state estimation, disturbance monitoring and control. Thls paper proposes two learnin schemes for fast esti- mation of these basic quantities. We attacaed the problem from a system identification pers ectlve In o osition to the well-establlshed Extended Kahan Fliterlng (I!&) technlque. It is shown that, from a simple non-linear model of the s stem voltage which Involves only two parameters, the Recursive L a s t Squares (RLS) and the Least Means Squares (WS) algorithms can each provide dynamic estimates of the volta e phasor. The finite derivative of the phase deviation, fol lowd by a moving-average filter, then leads to the local frequency deviation. A constant forgetting factor included in these algorithms rovides both fast adaptation in time-varying situations and good)smoothing of the estlmates when necessary.

I. Introduction With the introduction of microcomputers in substations,

digital monitoring of the characteristics of the power system voltage has become feasible, According to Thorp and his co-workers [5][17], real-time static state estimation is an obvious application of such measurements. They have derived a means of improving accuracy of the state estimation processes by including phasor measurements from a few buses. Their approach relies on the output of a computer-based transmission line protection system, known as the Symmetrical Component Distance Relay (SCDR), which uses a recursive method for phasor calculation [4]. The computational cost of this DFT- based algorithm is very low, but its performance can be adversely affected by decaying DC components or low SNR. Besides static state estimation, voltage phasor tracking in real-time has found applications in Power System Disturbance Recorders (PSDR) [3], static load modeling 1211, flicker monitoring [18] and modem relaying 1201.

As a matter of fact, devices specifically dedicated to highly accurate measurement of frequency have been described for specific applications like power system stabilizers [81[141. However3 most situations the digitalestimateof the frequency deviation is needed concurrently with other decision quantities [MI. Therefore, its value is usually obtained as a by-product of a more general-purpose algorithm, based for instance on the extended Kalman filtering [20] or the recursive least error squares techniques [ 161. Basically, Girgis and his co-workers [1][2][19] referred to a three-state Kalman filter, with time- varying transition and observation matrices.

"his paper was presented at the Transmission

and Distribution Conference in Dallas, Texas

from September 22-27,1991at the Dallas Con-

vention Center. Sponsored by the IEEE Power

Engineering Society.

Their approach is to be compared with that of [6] where the authors devised an EKF technique for combined estimation of positive sequence phasor and frequency deviation. Unfor- tunately, a common problem with these Kalman filters is the high computational requirements, due to transcendental functions evaluation in real-time.

Using a time invariant state transition matrix, Kamwa and Srinivasan E181 have recently devised a filter structure with improved computational speed and tracking performance. But their EKF displayed limited adaptation during jumps in the signals or data reset, even when supplemented with some sort of forgetting factor. In a dynamic environment such as a power system, this poor adaptability constitutes a flaw common to all methods which explicitly or implicitly hypothesized the sta- tionarity of signal statistics [2][16]. Therefore, the need still exists for more clever implementations of the various real-time algorithms, which could alleviate the computational burden and enhance the adaptation speed during transients [ 191 .

To fulfill this need to some extent, this paper describes two new methods suitable for fast estimation of voltage phasor and frequency deviation. They arose from the application of the RLS [12] and LMS [ll] algorithms to the identification of a simple non-linear phase voltage model. This identification view point departs significantly from that of the state estimation which is typical of EKF approaches. A forgetting or adaptation factor allows for limiting the working memory of the extended algorithms, by emphasizing the most recent data located in a sliding window. These concepts of adaptation and tracking are mainly borrowed from [9] and [13] where mathematical background on the identification of time-varying systems can be found.

Section I1 of the paper first refers to a basic non-linear and time-varying model of the system voltage. It then shows how to band-pass filter the observations in order to alleviate the effects of harmonics aliasing, subsynchronous resonance and offset. Sections I11 and IV describe the two learning schemes fortrackingthemodelparametersinreal-time.Next, we validate the algorithms on simulated data and then evaluate their performance on actual data. After discussing various tuning details, we compare in the last section the performances of the new adaptive filters with a standard EKF, from both compu- tational, accuracy and tracking perspectives.

II. Model of the System Voltage The time variations of the phase voltage at a given power

system node are basically described by the following obser- vation model:

in which v(t) is 8 zero mean white noise, with varianceR(t) and 0 (t)isasuitabletime-varyingparameter vector. Since theoutput function h(.) can be readily written as

088S-8977/91/$3.0001992 IEEE

-2- 790 for a grid with a nominal frequencyf, (Hz), it appears that a suitable parameter vector for the generic model (la) is given by

Therefore, to model a sequence of voltage-sampled values {e,(k), k=1,2 ,...}, it is possible to use the following discrete representation:

where vk, 8, and t k refer to the value of v(t) , 9(t) and t at the discrete time index k. In terms of parameter vector 0 k , the voltage phasor is given by [17]:

- E ( k ) =Akej" =Es@) + J E c ( k ) (3)

with j = m. At this stage, we shall make two useful obser- vations about the stated model.

(i) The white noise hypothesis in (la) is only a matter of convenience. It is required to ensure the optimality of the least squares-based identification scheme, but is unnecessary for the algorithm to work in practical situations where the noise could be colored (by harmonics or trends). In a later section, this point will be further illustrated through an example.

(ii) If the sampling is uniform with rate F,=lIT(Hz), the value of t at a discrete time index is given by tk=k*T. Thus, the phase shift in the model (2) due to the carrier frequencyf,, is expressed as follows:

(4) = (2nf , ) tk = (2nf> (kT)

Because of the periodicity (modulo n2n) of the sine func- tion, it is sufficient to know the value of 9, only at p=q(F,lf,) discrete points, where q is an appropriately picked integer. Therefore, if we choose for the sake of convenience to sample the voltage at a rate of N points per cycle of the fundamental frequency, the knowledge of the discrete model (2) from now to infinity ( t E (O,=[ ), requires the storage of only the fiiite sequence (9,~>,j=O,l,..N-l) (p=N,q=l ) .

Since the instantaneous frequency deviationf(t) (in Hz) is defined by [4]

its digital estimate can be derived for the sequence [ & k=0,1, ...), through the forward finite derivative approximation:

In practice, the straightforward use of the above formula leads to very noisy estimates off,. To alleviate this defect, we can prefilter the sequence [ek, k=0,1 ,...) through a simple moving average filter with memory length h 1221:

(7)

The use of 6 k instead of e, in (7) provides smoother frequency estimates fk, whose noise level can be furrher reduced by another low-pass filtering stage:

Therefore, once the instantaneous voltage phasor (E(@, k=0,1,..) is identified ftom the sample values of the phase voltage, the estimator defined by (8) provides the accurate values of the frequency deviation I f , , k=0,1,..) at very low computational cost.

III. Dealing with Harmonics and Trends The hypothesis of a pure sinusoidal voltage in white noise

assumed by model (2) is a mathematical idealization which probably never occurs in practice, especially in modem distri- bution power systems were power electronic devices have notably increased the level of harmonics. Furthermore, it is well known that acquisition systems can introduce significant offset (or bias) on the voltage sample values after a long period of normal operation. This phenomenon results from the drift in the electronic amplifier, which slowly decalibrates the instrumen- tation system. In a K h a n filtering context, a straightforward account of the bias is achieved by including it in the state vector.

In a series-compensated transmission network, subsyn- chronous resonances (at 6Hz or more in the future Hydro- Qubbec transmission system) represent another significant phenomenon which acts as asupplemental nuisance on the ideal model (2). To account for these various nuisance signals, an ad hoc procedure could involve collecting their influence in the noise term v (k). In fact, this is precisely why it is advisable to work in a stochastic framework, even when the underlying process is deterministic. From a practical perspective, this blind approach would work transparently as long as the unavoidable nuisance terms are small compared to the useful signal (high SNR).

A solution to the problem could be to improve the basic discrete model, by including auxiliary terms which account for all physical phenomena, known to bear a significant effect. In the context discussed above, such a model can take the following form:

e$) =Aksin(27if, tk+~,)+eo(k)+e,(k)+e,(k)+v(k) (8)

where e&), ekk) and e,@) represent the trend (decaying DC component or varying offset), the subsynchronous and the harmonic contributions at time k, respectively. An accurate model of the two latter terms should involve an extensive number of parameters, since the power system can simulta- neously disclose several harmonics and subsynchronous reso- nance frequencies. Such a high order model with a great number of parameters obviously involves a more tedious identification procedure. Therefore, our requirement for areal-time algorithm with a very low computational burden forces us to reject this extended model approach, even if it is defensible. By opposi- tion, we have chosen the more conservative solution of prefil- tering out the nuisance terms, so as to keep only the fundamental term modeled by (2).

A useful band-pass digital filter, very effective in rejecting DC offset and harmonics, has been proposed in [231. Its general form derived from the Fourier Transform is as follows:

In fact, this is an FIR filter, which can be recasted in a more familiar form:

- z 4 + z - ~ - z") + Z-' - 2' e,(k) (9b) 1

~

3-

1

-30 _ . ..... . . . . ..,.........; .... .....,,... ...,.. i .................... . _

-40 0 100 200 300

I I I

Hz Figure 1. Frequency response of the band-pass filter.

IV. Algorithm Derivation Recursive identification of linear systems has become a very

well established theory [7][11]. For convenient reference, the most basic algorithm for solving this problem in real-time is summarized in the appendix. It arose from the Kalman filtering theory and, with a proper tuning of its design parameters it can efficiently track time variations occumng in the system parameters.

Since our model (2) is not a linear one, we cannot use the algorithm in the appendix as such. Instead, we shall proceed as in the extended Kalman fdter methodology [7], by approxi- mating the nonlinear model with its fFst order Taylor series around the current parameter estimate B(k):

e,,(k) y(k I e) = h + cpT(k)$(k) + v(k) ( I O U )

where A&) = [O(k) - @)I is the deviation of 8 from its most recently estimated value and

is the gradient of the observation function W), evaluated at &). If we assume, as in the appendix, that the small variations of this parameter vector evolve according to a random walk model, the linearized process can then be described by the following linear state equations:

$ ( k + I ) = $ ( k ) + w ( k ) (1OC)

Ay ( k ) = q T ( k )$(k ) + ~ ( k ) (10d)

where {wfk) ) is supposed Gaussian with covariance Q(k). Therefore, the standard Kalman filter (appendix A) can be used with the linearized model (10) to compute the gain matrix L(k) at each iteration. The recursive estimation of the parameter vector is performed using the gain so obtained, and the actual prediction error given by (1 1).

The basic recursive algorithm in appendix A is quite powerful. It can deal with time varying parameters by assigning proper values to covariance matrix Q(k) and variance R(k). The two most popular choices for these design parameters give rise to the RLS and LMS schemes. They are both based on the concept of a forgetting (or adaptation) factor, which can be in a general manner assumed to be a sequence {k(k); k > 0) where IA(k)l c I . In the case of a constantforgeitingfucror A, the filter forgets past samples more slowly as the value of 1 increases.

According to results established by Ljung and Soderstrbm (121, the RLS algorithm with adaptation facility is obtained from the basic filter by picking for Q(k) and R(k) the following time-varying expressions:

R ( k ) = k(k) (12)

while the normalized version of Widrow's LMS algorithm results from

In the latter method, p(k) = 1 - k ( k ) refers to theforgerring rare [13]. After small algebra, the parameter updates in both algorithms can be recasted in more conventional forms:

8(k) = 8(k - I)+ L ( k M k ) ( 1 4 ~ )

with

for the RLS; and

for the LMS. The asymptotic behaviour of the LMS algorithm has been extensively studied by Widrow and his co-workers [lo-111, but rigorous analysis of the convergence properties of the finite-memory RLS is still in progress [13][24]. In fact, both algorithms perform very well in practice, and are extensively used in adaptive control and adaptive signal processing [7].

V, Performance evaluation through simulation The extended RLS and LMS algorithms as described in this

paper were coded in FORTRAN and tested off-line on an IBMcompatible personal computer. In the first study, we assume the following form for the phase voltage:

ea(r) = (1 +a(r))sin(wcr + d , sin(2xfr)+d_sin(2~f,t))

792

0.00 0 . 2 5 0 .50 0.75

s e c s e c

Figure 2. RLS and LMS algorithms on simulated data. True and Estimated (---).

Thus, the amplitude is sinusoidally modulated atf,=lHz andf,=20Hz (d,=d,=I%). The carrier frequency is also mod- ulated by sinusoidal waveforms at f,=lHz and fi=5Hz (d,=0.005, d2=0.002). Finite word length digital measurement introduces unavoidable quantization errors. The equation (16) simulates this effect:

y ( k ) = A D C r o u n d ( e , ( k T ) / ~ C ) ,& =0,1,2, ... N (16)

where T=11720 Hz (N=12 points per cycles) and the quantizer resolution is ADC=111500 per unit (PA) voltage. This reflects a 12-bit analog-to-digital converter set up to allow a maximum reading of about 204811500-1.33 p.u. voltage.

With h=0.8 and p=O.8 chosen to be constants, both the RLS and LMS filters perform very well, and produce nearly similar estimates. The results displayed at figure 2 show a good tracking capability when the amplitude or phase deviation is both small and rapidly varying. This good tracking property is especially useful in short term-flicker monitoring [MI. Also, the low predxtion error demonstrates that the filter is able to cope with a reasonable quantizing noise level.

In the field of communications, the LMS algorithm is well known for its low computational requirements, while the RLS is most appreciated for its fast adaptation properties. In par- ticular, our studies show the latter adapts to fast varying signal more quickly than any Kalman filter we have tested. But the LMS in this paper does not display such a good transient capacity. Considering more specifically the filter reported in (181, we have made a simple comparison using the responses

to simultaneous steps of one radian in the phase and 10% in the amplitude. As it appears in figure 3, this discontinuous beha- viour strongly affects the LMS tracking capability, which reaches the steady state only after a large overshoot in the opposite direction (amplitude). By opposition, the Kalman filter of [ 181 reacts more smoothly and promptly, thus demonstrating a larger bandwidth.

In terms of computational speed, lunited tests have been made in our laboratory on three candidate algorithms: the LMS, the EKF in [IS] and the EKF in [ 11. As a matter of fact, they rank in this order, the Girgis EKF being roughly four times slower than the LMS. We should however, acknowledge that in our implementation ofthe Girgis algorithm, we have included a step for estimating the amplitude from the in-phase and in-quadrature components. In this context, we were not sur- prised to find that, the Kamwa EKF was two tines faster than the Girgis. Perhaps it should also be mentioned that the benchmark on an Intel-386 CPU running at 3 3 W with a coprocessor, reported for the LMS algorithm a mean execution time of 0.6ms per iteration. On an industrial STD-type machine with an Intel-286 CPU at 16MHz speed, the processing time rose to 2ms.

To gain more confidence in the tracking capability of the new adaptive filters, a stability program has been used to generate a data set, more representative of an actual situation in the transmission system. The benchmark power system consists of a single machine linked to an infinite bus, through a transmission line.

-5- (a 1 51

- - -7.5

793

- - I I l I ] I l I l -7.5 I I I 1 ~ 1 1 1 1

-20 1, 0.00 0.02 0.04 0 2 4 6

sec sec Figure 3. Step responses of the LMS and Kalman filters. Figure 4. RLS performance on stability data.

True (J and Estimated (---). LMS and EKF in [ 181 (---).

2.57 (a) LMS: Amplitude (%I o-, ( c ) RLS: Amplitude (%I

-7.5 1 - 6 1 (b) LMS: Residuals (%I (d) RLS: Residuals (%I

0 . o

-2.5

-5.0

Figure 5. LMS and RLS filters applied on raw data.

194

The generator, operating at unit active power and 0.85 p.u. reactive power, is subjected to a -IO%change in its mechanical power set-point, followed by another change of +IO%. The phase-a voltage during the transient is recorded a t a sampling rate of 2rns and the values are rounded to integer digits, using (16) withADC=I/1500. The performance of the RLS filter with k=0.8 is shown in figure 4. The amplitude time variations are very well tracked. But the phase estimates disclose a small overshoot during fast decaying transient.

~

-6-

VI. Experimental results

We have obtained data records (960 sampleskycle) from a load monitoring system installed in the Hydro-Quebec trans- mission system. The extended LMS and RLS algorithms were first applied on the raw data. The results shown in figure 5 are unsatisfactory, since they disclose, for both algorithms, a high variance of residuals, as well as harmonic interference on the estimated amplitude. From the prediction errors waveform, it is obvious that the recorded phase voltage contains significant offset. Furthermore, its spectral analysis has also revealed the presence of second and third harmonics.

- 1 . 5 ~ (a) LMS: A m p l i t u d e (%I

-2 .5

- * . ~ -3 . o a 1

(b l LMS: Residuals (%I .03

-1.0 , I , I I I l l 1

0 . 0 0.5 I . o sec

Figure 6. LMS algorithm applied on filtered data.

To alleviate the detrimental effect of the offset and har- monics, the filter devised in section I1 has been applied to the data. Then we proceeded again to the estimation of the phasors, now using the LMS and the RLS algorithms with the filtered data. The results presented in figure 6 (LMS only) have improved dramatically. More specifically, the harmonic inter- ference on the estimates has been completely eliminated, as has the offset in the prediction errors. It is now possible to recognize and more accurately characterize a significant short-term flicker.

VII. Conclusion This paper describes two original adaptive schemes for the

combined estimation of the voltage phasor and frequency. Basically, both methods refer to a simple non-linear model of the system phase voltage involving only two parameters. Once the model is appropriately linearized, the Recursive Least Squares (RLS) and the Least Means Squares (LMS) algorithms (supplemented with forgetting and adaptation factors) are first used to provide dynamic estimates of the voltage phasor. A finite derivative of the phase deviation, followed by a moving-average filter, then leads to the local frequency deviation. Numerical tests on simulated data dsclosed a satisfactory and similar performance for both IUS- and LMS-based learning schemes. However, extensive experiments on actual data from the Hydro-Quebec transmission system clearly demonstrate that to improve the accuracy, a pre-filtering of the raw data is called for, in order to eliminate the unavoidable effects of harmonics and offset.

VIII. References [l] A. A. Girgis, J. Qiu, "Measurement of the Parameters of Slowly Time

Varying High Frequency Transients", /€E€ Transadions on lnsfrumentation and Measuremenf, lM-38(6), Dec. 1989,

[2] A. A. Girgis, T. L. D. Hwang, "Optimal Estimation of Voltage and Frequency Deviation Using Linear and Non-Linear Kalman Filtering: Theory and Limitations", /€€E Transactions on Power Apparatus and Sysfems, PAS-103(10), October 1984, pp.2943-2949.

[3] T.A. Gough and G.R. Berub6, "Power System Disturbance Recorder", Paper presented to Canadian Electrical Association, Transmission and Station Planning and Operations Subsection, Toronto, Ontario, March 1989.

[4] A.G. Phadke, J.S. Thorp, M.G. Adamiak, "A New Measurement Technique for Tracking Voltage Phasors, Local System Frequency and Rate of Change of Frequen , /E€€ Transadions on Power Apparatus and Systems, PAS-1%;5). May 1983, pp.1025-1038.

K.J. Karimi, "State Estimation with Phasor Measurements", /€€,?'Transactions on Power Systems, PWRS- l ( l ) , February 1986, pp.233-238.

[6] G. Andria. L. Salvatore, "Inverter Drive Si nal Processing via DFT and EKF, /E€ Proc. €?,137(2), March 1980, pp.111-119.

m G. C. Goodwin, K. S. Sin, Ad five FiHering, Predidion and Contrd. Englewood Cliffs: PrentiidDall, NJ, 1984.

[SI V. Eckhart, P. Hippe, G. Hoseman, "Dynamic Measuring of Fre- quency and Frequency Oscillations in Multiphase Power Systems", IEEE Transactions on Power Delivery, PWRD4(1), Jan. 1989.

[Q] L. Ljung, S. Gunnarsson, "Adaptation and Tracking in System Identification-A Survey", Aufomatica, 26(1), pp.7-21, January 1990.

[ lo] B. Widrow, J. McCool, M. C. Larimore, C. R: Johnson, JR., "Stationa and Nonstationary Learning Characteristics of the LMS Adaptive%lter", Proceedings of the /€€E, 64(8), August 1976,

[ll] B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1985.

[I21 L Ljung, T. Soderstrom, Theory and Pracfice of Recursive E d - rnafion, MIT Press, Cambridge, MA, 1983.

[13] E. Ewada. 0. Macchi, "Convergence of the RLS and LMS Adaptive Fitters", /€€E Transactions on Circuits and Sysfems, CAS-Wq, July 1087, pp.799-803.

[14] F.W. Kea and W.H. South, "Design of a Power System Stabilizer Sensing Frequency Deviation", /€€E Transacffons on Power ApperafusandSysfems, PAS-90(2), MarcMAprill971, pp.707-713.

[15] Y . Bar-Shalom, T.E. Fortman, Trackin and Data Association, Academic Press Inc., New York, NY, 1988.

[le] M.S., Sachdev, M. Nagpal, "A Recursive Least Error quares Algorithm for Power System Relaying and Measurement Applica- tions", IEEE Pa er 9OSM 473-9 PWRD, Presented at the IEEE/PES 1990 Summer Reeting, Minneapolls, Minnesota, July 1519, 1890.

pp.1057-1063.

[SI A.G. Phadke, J.S. Tho

pp.95102.

pp.1151-1162.

-7- 195

In the linear case, the determination of the increment p L ( e )

chiefly depends on the regressor covariance matrix H = E (cpcpT), also called Hessian mamx. We shall suppose now that the parameter vector varies with time, according to the following law:

9(k) =w - u,w(k)) (22)

Here [w(k)) would be a sequence of random vectors, presumably white and Gaussian. Most commonly, where no specific information on the parameter variations is on hand, one can assume that f(.,e,-) takes the form of a random walk. Therefore, a convenient model for the combined description of parameter and observation dynamics is as follows:

6 ( k ) = 6(k - 1) + w(k)

Y (k> = cpT(k)9(k> + v(k) (23)

If we show Q(k) to be the covariance of w(k) and R(k) the variance of v(k), the well known Kalman filter provides the optimal estimates of { B(k)) given past observations:

[17] J.S. Thorp, A.G. Phadke, K.J. Karimi, "Real Time Voltage-Phasor Measurements for Static State Estimation", /E€€ Transactions on Power Apparatus and Systems, PAS-104(1 l), November 1985, pp.3098-3106.

[18] I . Kamwa, K. Srinivasan, 'A Kalman Filter-based Technique for Combined Digital Estimation of Voltage Flicker and Phase in Power Distribution Systems", submitted for presentation at the IEEVPES 1992 Winter Meeting, New York, NY.

[19] A.A. Girgis, W. L. Peterson, "Adaptive Estimation of Power System Frequency Deviation and its Rate of Change for Calculabng Sudden Power System Overloads", I€€€ Transactions on Power Delivew, PWRD-5 (2), April 1990, pp.585-594.

[ZO] M.S. Sachdev, H.C. Wood, N.G. Jonhson, "Kalman Filtering Techni ue Applied to Power System Measurements for Relaying", /E€€ Yransactions on Power Apparatus and Systems, PAS- 104( 12), December 1985, pp.35653573.

[21] D.C. Lee, K. Srinivasan. "Specification of Data Acquisition System for Load Model Parameter Estimation", Canadian Electrical Asso- ciafion, Report 113 T 372, April 1986.

[22] C.C. Foster, Real lime Programming - Neglected Topics, Addison-Wesley, Reading, MA, 1981.

[23] A. Papoulis, Signal Analysis, McGraw-Hill, New York, NY, 1977.

[24] M. Niedzwiecki, L. Guo, "Nonasymptotic Results for Finite-Memory WLS Filters", /E€€ Transactions on Automatic Control, AC36 (2), February 1991, pp.198-206.

M. Appendix: Recursive Identification of a Time-Varying Linear System

Let us assume that the observed signal y Q is the output of a system, parametrized by an unknown vector The basic purpose of the description is to predict the future outputs of the system y^(k I e). The underlying system is linear when the prediction y^(k I e) is a linear function of 9:

y^@ I 9) = CpT(W(k> +v(k) (17)

where (v(k)) is a Gaussian observation noise. This predictor represents the so-called linear regression model. When the system does not have any known exogenous input, the regressor can consist of lagged outputs like in the linear prediction theory U]:

~ p ~ ( k ) = [ - y ( k - l ) - y ( k - 2 ) ... - y ( k - n ) ] (18)

System identification deals with the problem of finding the parameter vector that gives the best description of the observed data (y(k) ,k>O]. This problem can be in turn expressed as finding the parameter vector that minimizes the least squares criterion:

where the prediction error is given by

Ljung and Soderstrom [12] have shown that the minimum @)of the cost function (19) can be determined on-line, through the so-called recursive prediction error mFthod (FWEM). At time k, the algorithm recursively provides 9(k) through

(21) 6 ( k ) = 6(k - 1) + p,((60'), cpo)),J I k)E(k)

6(k) = 6(t - 1) +L(k)E(k), e(0) = e, E(k) = y ( k ) - cpT(k)6(k - 1)

Clearly, the above Kalman filter (or RPEM algorithm) provides an optimal solution to the problem of minimizing (19) only if the actual statistics Q(k) and R(k) of the hypothetically Gaussian sequences [w(k)] and {v(k)) are perfectly known. Since such an hypothesis is very unlikely, especially in a the-varyipg context, we shall deal in practice with estimates &k) and R(k) that obviously lead to sub-optimal estimates of B(k). It thus appears that Q(k) and R(k) are merely design parameters whose ultimate values (Q(k) and k(k)) are to bc chosen by the Kalman filter user, for purposes such as: fast tracking of abrupt changes, smooth estimates of B(k), and accurate predictions of y(k).

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-time system 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.

ROBERT GRONDIN (S'77, M'80) received his B.A.Sc. (Electrical Engi- neering, University of Sherbrooke, Quebec, Canada) in 1976 and his M.Sc. (INRS Energie, Varennes, Quebec, Canada) in 1979. He then joined the Hydro-Quebec Research Institute (IREQ) where he is involved in the development of real-time systems applied to the automatic control and protection of power systems. He is a member of IEEE Power Engineering, Automatic Control, and Industrial Elec- tronics societies.