A New Half-Cycle Adaptive Phasor Estimator Immune to ... - CiteSeerX

A New Half-Cycle Adaptive Phasor Estimator Immune to the Decaying DC Component for Digital Protective Relaying Eugeniusz Rosolowski

Jan Izykowski

Bogdan Kasztenny

Senior Member, IEEE

Member, IEEE

Senior Member, IEEE

Wroclaw University of Technology, Department of Electrical Engineering Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, POLAND

Abstract—The paper presents a new half-cycle adaptive phasor estimation algorithm for digital protective relaying. The algorithm is immune to an exponentially decaying dc component regardless of its initial magnitude and time constant. The algorithm is intended for current signals and improves both the magnitude and impedance measurements. The new measuring technique is based on the recursive half-cycle Fourier filter with a separate adaptive function allowing for thorough rejection of an aperiodic signal component. Analysis of the algorithm’s performance, both analytical and by simulation, is included. The presented algorithm outperforms the classical mimic filtering broadly used in today's relays. Keywords: numerical relaying, dc offset rejection, phasor estimation, half-cycle algorithm.

I. INTRODUCTION As the vast majority of digital protective relays are still found on phasors, the issue of phasor estimation is probably the single most investigated and documented subject in research on digital relaying [1,2]. Digital phasor estimators are designed to meet several contradictory requirements, namely to be fast, immune to frequency excursions, and insensitive to signal pollution including harmonics, subharmonics, high frequency oscillations, noise and decaying dc components [1]. This paper focuses on the last issue. Fault currents can contain the exponentially decaying dc offset with the unpredictable initial magnitude and unknown time constant. The variable nature of the magnitude and time constant is due to both the fault resistance and fault incipience angle being random factors. The decaying dc offset in a fault current can result in near twenty percent overshoot in the estimated magnitude if the full-cycle Fourier algorithm is applied, and in much higher errors if the half-cycle method is used. This would necessitate users to apply comparatively large security margins when setting fast digital relays. Generally, the phasor estimators can be easily designed to remove the pre-defined dc component [1]. The problem arises, however, when the time constant of the dc component

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is not known. Several approaches have been put forward so far to minimize the influence of the decaying dc component on phasor estimation. The most practical solution merges a digital mimic filter with any robust phasor estimator such as the full-cycle Fourier algorithm. The digital mimic filter reproduces numerically its analog counterpart historically applied in line relays [1]. The original mimic circuit is a differentiating element, and therefore, its digital replica is a high-pass digital filter [3]. This, in turn, results in worsened frequency characteristic of the resulting phasor estimator, especially if the half-cycle algorithm is used for phasor estimation. Robustness to the actual (random) value of the time constant is a strong advantage of the digital mimic filter. The other approach is to include the exponentially decaying dc component into the signal model and apply certain curve fitting techniques to design a phasor estimator. Usually, the dc offset is treated as a noise, represented by either deterministic or probabilistic characteristics, rather than parameter to be fitted [4]. Several peculiar estimators originate from this general approach: • Linear estimators when applied to suppress the dc component have to deal with the non-linear function of the signal model of the dc component. This is achieved using first few terms of the Taylor series of the exponential function [2]. • Least Error Squares (LES) technique conceives nonrecursive [5] or recursive [6,7] estimators. The latter can be modified by introducing resetting or forgetting factors [4] for better absorption of non-harmonic frequencies including the decaying dc component. • Application of the state-variable model initiates estimators in the form of a state observer [8,9] or a Kalman filter [10]. The requirement for the a priori noise characteristics is the key weakness of the latter measuring technique. Yet another, numerically simpler, approach to the issue of the decaying dc component is to apply a cosine filter for estimating both the direct (cosine filter) and quadrature (cosine filter delayed by one-fourth of a cycle) orthogonal components [3,11]. This method, however, has poor frequency response when applied in conjunction with a data window shorter than full-cycle. Recently, two new methods for suppressing the dc component have been published [12]. They probably could also be applied with the half-cycle data window, however, the authors of [12] do not publish any practical considerations. The aforementioned methods, although improve the measurement, do not reject the dc component entirely.

The new algorithm is based on the weighting LES technique appropriately applied to the three-state signal model with a half-cycle data window. The new estimator has the form of the regular recursive half-cycle Fourier filter with an additional adaptive correction for the decaying dc component. The fundamental advantage of our algorithm is that it removes the decaying dc offset thoroughly regardless of its time constant and the initial magnitude. The paper begins with traditional LES technique (section II), describes the new algorithm (section III), and finally presents an EMTP-based [13] analysis of the new measuring technique (section IV). II. CLASSICAL LEAST ERROR SQUARES ALGORITHM For all further considerations let us assume the following signal model: y (t ) = Y0 exp(−t / τ ) +

K

∑ Ym cos(mω1t − ϕ m )

(1)

m =1

where: Y0, τ magnitude and time constant of the decaying dc component, respectively, K number of considered harmonics, ω1 =

X (k ) =

1 2

(xc (k ) − jxs (k ))

(5)

The algorithm (3) rejects a dc component perfectly as long as the input signal matches the assumed signal model (1). The standard LES algorithm is, however, very sensitive to the time constant τ , and therefore, cannot be used directly in digital relaying [3]. It is worth noticing that the half-cycle estimator (3) rejects only odd harmonics from the input signal (1). B. The weighting LES algorithm Let us consider a weighting LES algorithm in the following form:

(

2π radial system frequency, T1

)

-1

X(k) = H T(k)G(k)H(k) H T(k)G(k)Y(k)

A. The non-recursive LES algorithm To estimate the phasor out of M consecutive samples of the signal (1) one can use the three-state LES algorithm with the signal model matrix H(k ) composed from the following rows:

h(k − j )= [hc (k − j ), hs ( k − j ), h0 ( M − j − 1) ], j = 0..M − 1 (2a) hc (k − j ) = cos(a(k − j )) , hs (k − j ) = sin (a (k − j ))

(2b)

h ( M − j − 1) = exp(b( M − j − 1) ) (2c) 0 2π T where: a = , b=− 1 , N Nτ N number of samples per cycle, k sample index. Consider the half-cycle (M = N/2) standard LES algorithm based on the signal model matrix (2) with the time constant τ known. The vector of estimates is updated at every sampling instant using the following fundamental equation [1,2]:

X(k) = P(k)H T(k)Y(k)

(

)

where the covariance matrix P = H T(k)H(k)

-1

Traditionally, in order to determine the optimal weighting matrix, the probabilistic characteristics of the measurement errors are required [1]. Those characteristics are seldom available, but the weighting matrix can be considered as a formal tool for arbitrary modification of the standard LES algorithm. For this purpose, (6) needs to be re-written as:

X(k) = PG(k)HTG(k)Y(k) where:

(

(7)

)

PG(k) = HTG(k)H(k)

−1

HTG = HT (k)G(k)

(8a) (8b)

The matrix H G (k ) represents a pattern used in the measurement process for reproducing adequate features inherent in the signal model matrix H (k ) . When using (7)-(8), there is no need to find a physical meaning of the weighting matrix G (k ) because the measurement matrix H G (k ) has its original significance. Thus, the measurement matrix is selected to ensure adequate simplification of the final estimator. Let us consider the measurement matrix H G (k ) formed from the following rows (compare with (2)):

hG (k − j)=[hr (k − j), hi (k − j), h0 ( M − j − 1) ], j = 0..M − 1 (9a)

(3)

hr (k − j ) = cos(a (k − j )) − d c (k )

(9b)

(4a)

hi (k − j ) = sin (a (k − j )) − d s (k )

(9c)

Y(k) = [ y (k − M + 1), y (k − M + 2), .. , y (k )] T (4b) X(k) = [xc (k ), x s (k ), x0 (k )]T

(6)

where G(k ) is a M × M weighting matrix.

T1 period of the fundamental frequency, ϕ m , Y m initial phase angle and magnitude of the m-th harmonic.

and

while the superscript T stands for matrix transposition. Due to the presence of the exponentially decaying dc component in the signal model (2), the matrix P (k ) in (3) is a full matrix with off-diagonal elements depending on the exponential function. As a consequence, the filters of the resulting non-recursive estimator have non-regular coefficients and, unlike the standard full- or half-cycle Fourier algorithms, they cannot be reduced to a simple recursive form. Practically, only the two variables xc and xs are to be determined by (3) as they form the estimated phasor:

(4c)

It is worth emphasizing that the matrix H G (k ) is obtained according to (9) by adequate modification of the original signal model matrix H (k ) (2). For this operation the weighting matrix G (k ) is not necessary. Instead, the adequate values of the coefficients d c and d s need to be determined. These co-

efficients could be chosen based on the assumption that the final estimators for the orthogonal components are not sensitive to the unknown decaying dc component. For the measurement model (9), the estimator is determined as in (7) with the covariance matrix (8a). Note that the functions d c and d s in (9) are fixed in the signal data window. When designing the estimator (7), it is justified to state the following two requirements for these two functions: • the covariance matrix PG(k) of the final algorithm should be maximally reduced in size and simplified in form (preferably diagonal); • the functions d c and d s should be easy to determine from the dc component of the input signal. The first requirement is aimed at achieving the simplest form of the final measuring algorithm. Actually, it is enough to impose this requirement only on the first two rows of the matrix PG (k) related to the estimated orthogonal components. The elements of PG (k) can be easily analyzed from the inverted form:

{ }

PG−1 = qij = H TG (k)H(k) , i, j = 1.. 3

(10)

Imposing the following condition: q1,3 = q 2,3 = 0

d c (k ) = p s sin( ak ) + p c cos(ak ) where: pc = g ⋅ ( r − cos a )

g=

{ }

q13 = q 23 =

M −1

∑ (cos(a (k − j ) ) − d c (k ) )exp(b( M − j − 1) ) = 0 ∑ (sin (a (k − j ) ) − d s (k ) )exp(b( M − j − 1) ) = 0

(13b)

j =0

The functions d c and d s are determined from the conditions (13) as follows:

q12 = q 21 = q 22 =

∑ cos(a ( k − j ) )exp(b( M − j − 1) )

d c (k ) =

M −1

∑ exp(b( M − j − 1) )

(14a)

M −1

∑ sin (a ( k − j ) )exp(b( M − j − 1) )

d s (k ) =

M −1

∑ exp(b( M − j − 1) )

j =0

Simplifying we obtain:

2

(16b) (16c)

M −1

∑ (cos(a (k − j ) ) − d c (k ) ) cos(a (k − j ) )

(17a)

j =0

M −1

∑ (cos(a ( k − j ) ) − d c (k ) )sin (a ( k − j ) )

(17b)

j =0

M −1

∑ (sin (a ( k − j ) ) − d s (k ) ) cos(a ( k − j ) )

(17c)

j =0

M −1

∑ (sin (a ( k − j ) ) − d s ( k ) )sin (a ( k − j ) )

(17d)

j =0

N sin( ak + a / 2) − d c (k ) 4 sin a / 2

(18a)

cos(ak + a / 2) sin a / 2

(18b)

N cos(ak + a / 2) + d s (k ) 4 sin a / 2

(18c)

q12 ( k ) = d c ( k ) q 22 ( k ) =

q 21 (k ) = − d s ( k )

sin( ak + a / 2) sin a / 2

(18d)

From the coefficients (18) we can determine the matrix PS (k) . The final algorithm is in the form of (7) reduced to the two rows (matrix PG (k) is replaced by PS (k) )

j =0

j =0

2

It should be noted that the above coefficients change with time. Assuming the half-cycle window ( M = N / 2 ) we obtain:

M −1 j =0

M

Initially, let us assume that the time constant of the dc component is known. Consequently, the value of r in (16) is also known and it is possible to determine the matrix PS (k) in (12). The four coefficients of the inverted matrix (12) are defined as follows:

q11 ( k ) =

M −1

(1 − r

C. The simplified algorithm

(13a)

j =0

M

From (15)-(16) it is obvious that the obtained functions d c and d s are directly related to the time constant of the dc component. This is in accord with the second postulated condition.

(12)

The elements of PS (k) are independent from the third row of the matrix (10). Therefore, the algorithm (7) under the condition (11) is reduced to estimation only two of the orthogonal components. The conditions (11) define the way for obtaining the coefficients dc and ds, Substituting (2) and (9) into (10) yields the following set of equations:

(1 + r )(1 − r ) )((cos a − r ) + sin a )

r = exp(b )

q11 =

PS−1 = q ij , i, j = 1.. 2

(16a)

p s = g ⋅ sin a

(11)

the first two rows of the matrix (10) can be reduced to the following 2x2 sub-matrix:

(15)

d s (k ) = p c sin( ak ) − p s cos(ak )

(14b)

X S (k) = PS (k)H TS (k)Y(k) where

[

X S (k) = x p (k ), x q ( k )

(19)

]

T

and rows of H S are as in (9) but reduced to first two columns. From (19) we derive the algorithm for phasor estimation as follows: x p ( k ) = p11 ( k ) x r ( k ) + p12 ( k ) x i ( k ) (20) x q ( k ) = p 21 (k ) x r ( k ) + p 22 ( k ) x i (k ) where: x r (k ) = xi (k ) =

j =0

{ } { }

−1

, i, j = 1.. 2

(21b) (21c)

The coefficients p ij can be obtained by matrix inversion as in (21c). However, it is convenient to derive their analytical form. Substituting (18) into (21c) yields: q (k ) − q (k ) p11 ( k ) = 22 p12 (k ) = 12 , , d d q (k ) − q (k ) p 21 ( k ) = 12 p 22 ( k ) = 11 , , (22) d d N N   − g ⋅ ( r + 1)  . 44  The output phasor is now determined by the orthogonal components x p (k ) , x q (k ) . where

d=

III. SELF-TUNING LES ALGORITHM The algorithm (20) fully rejects the dc component from the input signal, but requires the time constant of the decaying dc component (or the parameter r). The parameter r can be estimated on-line as follows. Substituting (9) into (21a-b) yields: x r ( k ) = x c (k ) − δ c ( k ) xi (k ) = x s (k ) − δ s (k ) where

x c (k ) = x s (k ) =

δ s (k ) = d s (k ) x d (k ) x d (k ) =

(24b)

j =0

(25a)

M −1

∑ y (k − j )

(25b)

j =0

It can be seen from (24a,b) that the original orthogonal components xc (k ) , xs (k ) are estimated using the regular half-cycle Fourier algorithm. The new (corrected) values of the orthogonal components xr (k ) and xi (k ) are obtained by applying appropriate correction (23). The correcting functions δc and δs are proportional to the average dc component in the data window x d (25b) and the functions dc and ds (25a).

(26) (27) (28) (29)

The algorithm in its non-recursive (23)-(25) and recursive (26)-(29) forms still requires the time constant of the decaying dc component τ to be known a priori. However, the correlation function is now separated into two operations enabling easy modifications of this estimator. Particularly, this form of the estimator allows for extracting the value of τ in course of the phasor estimation. Taking the above into consideration let us revisit the relations (15) and (16). A. Estimation of the dc component The parameter r can be estimated from the measurements as explained below. Consider the half-cycle Walsh algorithm applied in the recursive form for transformation of the input signal y (k ) : x w (k ) = x w ( k − 1) + wal 2 ( k ) x Σ ( k )

(30)

where wal 2 (k ) is a discrete-time symmetrical square-wave with values of ±1 and the period of the fundamental component. It can be proven that for the input signal y (k ) composed of the fundamental frequency component and the exponentially decaying dc offset the signal abs ( x w (k ) − x w (k − 1)) is proportional to the given dc component. Therefore, the coefficient r as an estimate of the function exp(b) (16c) can be obtained from:

(24a)

M −1

δ c (k ) = d c (k ) x d (k )

x ∆ (k ) = y ( k ) − y ( k − M )

r = r(k ) =

j =0

∑ hs (k − j ) y(k − j)

x Σ (k ) = y ( k ) + y ( k − M ) , and

(23)

M −1

∑ hc ( k − j ) y ( k − j )

where:

where:

j =0

PS = pij = qij

x s ( k ) = x s ( k − 1) + x Σ ( k )h s ( k )

(21a)

M −1

∑ hi ( k − j ) y ( k − j )

x c (k ) = x c ( k − 1) + x Σ (k )h c ( k )

x d ( k ) = x d ( k − 1) + x ∆ ( k )

M −1

∑ hr ( k − j ) y ( k − j )

The estimator (23)-(25) can be re-written in the convenient recursive form:

x w (k ) − x w ( k − 1) x w ( k − 1) − x w ( k − 2)

(31)

The way of determining r (k ) is illustrated in Fig.1. It can be seen that the correct value of r is available after M + 1 samples. Physically, the time constant τ does not change during a given transient. The estimator (31) can produce, however, unrealistic values at first few samples after the fault inception, and therefore, requires some forced stabilization. This can be done by truncating the original estimate as follows: rmin ≤ r ≤ rmax

(32)

The limits are naturally chosen as: rmin = exp

−T1 , rmax = 1 Nτ min

(33)

where τ min is the minimum value of the time constant recommended to be set at around 4 msec.

a)

y(k)

0.65

1.5

-0.1 -0.2

0.63 ps

0.5 0 -0.5 0

10

20

30

40

2

k b)

0.62

-0.3

0.61

-0.4

0.6

-0.5

0.59

-0.6

0.58 0.6

0 -2

-0.7 0.7

Fig.2. Coefficients

-6 -8

xw(k)

0

10

20

30

40

2

k c)

1.5 1 r(k) 0.5

10

20

time

30

40

k

Fig.1. Illustration of estimation of the coefficient r : input signal (a), estimates of x w (k ) (b), dc component and r (c); N = 20 s/c.

Averaging the estimated factor r over a number of samples, say L, can ensure further damping of the unwanted oscillations: r filtered (k ) =

1

pc and p s as functions of r for N = 20 s/c.

1 L −1 roriginal (k − m) L m=0

∑

p c = a1 + r ( a 2 + ra 3 ) p s = b1 + r (b2 + rb3 )

,

(35)

where coefficients a i , bi , i = 1..3 are found using any curvefitting method. The real-time implementation of the new algorithm can be summarized as follows:

|xw(k-1)-xw(k)|

0

0.9

Instead of utilizing (16) for real-time application one can use tabulated representation of these functions. Also, quite convenient form is obtained by applying the third-order polynomial approximation:

-10 -12

0

0.8 expb

wal2(k)

-4

-14

0

pc

0.64

1

-1

ps

pc

(34)

As our simulations show, good results are obtained for L = 2..4. It is worth noticing that, practically, the filter (34) does not delay the phasor estimation. It is so because (34) is applied to the signal parameter that physically does not change, or changes slowly in time. In order to simplify the final equations of the new algorithm, the functions defining the coefficients pc and p s in (16) may be represented in a simpler, less accurate form. For a given value of a (a constant in a given implementation) they are function of r (a variable). Fig.2 presents sample plots of the coefficients pc and p s for M = N / 2 = 10 .

(a) Calculate the orthogonal components x c (k ) , x s (k ) by (26) i.e. using the regular half-cycle Fourier algorithm. (b) Perform the half-cycle Walsh transformation according to (30) for calculation of x w (k ) . (c) If abs(x w (k − 1) − x w ( k − 2) ) < ε a ( ε a - minimum value of the dc component to be considered in the correction), take xc (k ) and xs (k ) as the final results, and skip the steps (d)-(e). Otherwise, (d) Determine the functions d c (k ) and d s (k ) by the following sequence of steps: • estimate the parameter r by (31)-(32), • calculate the functions pc and p s by (35) or use their memory-stored values (Fig.2, for example), • determine d c (k ) and d s (k ) according to (15). (e) Calculate the correcting values δ c (k ) and δ s (k ) (25), x r (k ) , x i (k ) (23), transformation coefficients p11 - p 22 as well as the final values of the orthogonal components (20). The check-point set in step (c) ensures that no division by zero will take place. The condition also by-passes the correction when the dc component is small and the correction is actually unnecessary. Fig.3 presents the signal flow chart of the new algorithm. It should be noted that for real-time application the calculation of the coefficients p11 - p 22 according to (18), (22) can be sufficiently simplified if the trigonometric functions in (18) are stored in the memory.

wal 2(k)

hc(k) + +

y(k) z

−

hs(k)

+

+

+

+

+

+

z-1

z-1

xw(k)

estimation of d c(k), d s(k), p11(k), p 12(k), p 21(k), p 22(k)

xc(k) + δc(k) xs(k)

xr(k)

xi(k)

+

 p11 p12  p p   21 22 

xp(k) xq(k)

- δs(k)

N 2

+

z-1

+

+

z-1

dc(k) xd(k)

ds(k)

Fig.3. Block diagram of the new algorithm.

B. Frequency response Fig.4 presents the frequency response of the new algorithm as compared with the half-cycle Fourier method. The new technique filters out the dc component entirely and significantly damps the subsynchronous frequencies. In contrast, the half-cycle Fourier algorithm exhibits a gain of about 1.3 for the dc component. As seen from the figure, the new algorithm introduces considerable gain for the second harmonic. This is so due to certain similarities in spectrum between the second harmonic and the dc component. C. Numerical stability It is worth emphasizing that the proposed algorithm is not iterative. It uses a feed-forward correction based on the results on certain intermediate calculations. The feedback lines shown in Fig.3 apply to the delayed samples and do not constitute any actual feedback connections or interations.

As a result, the issue of numerical stability and/or convergence is not void. The algorithm is stable and always improves measurement as shown by numerous simulations. D. Illustration example It is interesting to analyze the correcting mechanism of the proposed algorithm. Figs.5–6 present plots of the key signals shown in Fig.3. The algorithm has been fed with the current waveform shown in Fig.1a. The orthogonal components xc (k ) , xs (k ) obtained from the classical half-cycle Fourier algorithm (26) are considerably distorted (Fig.5). The correcting signals δ c (k ) and δ s (k ) result from multiplication of two sinusoidal functions with the period of fundamental frequency (25a) and, therefore, they have double the system frequency (Fig.5). The same applies to the coefficients of the matrix PS (k) (Fig.6). The estimated orthogonal components and the magnitude are controlled very well (Fig.7). The response time is slightly above half a cycle and no transient overshoot occurs even though the dc component is of significant magnitude and duration. IV. EMTP-BASED EVALUATION OF THE NEW ALGORITHM

|H(ω)|

A number of simulation cases reflecting different system conditions have been used in testing. The included example compares the new adaptive algorithm (ADA) against the digital mimic filter (MMC) supported with the half-cycle Fourier algorithm [3]. The MMC algorithm was designed for the fixed value of the time constant of the dc component (25msec).

2

1.5 2

1

A. EMTP model 1

A 330kV, 50Hz system shown in Fig.8 has been simulated using EMTP. The model incorporates the primary system as well as the instrument transformers and analog antialiasing filters set at 375Hz (N = 20 s/c).

0.5

0

B. Simulation example 0

2

4

6

8

10

ω/ω1 Fig.4. Frequency response of the new algorithm (1) as compared with the half-cycle Fourier algorithm (2).

The included example is a LL fault through the fault resistance of 0.1Ω 20km from the substation A (Fig.8). Fig.9 shows the fault-loop current waveforms seen at the substations A and B, respectively.

6

1

xc(k)

X(k)

4 0.5

2 δc(k)

-2 -4

xp(k)

0

0

-0.5 xq(k)

0

10

20

30

40

k

-1

2

0

10

time 30

40

k

Fig.7. Orthogonal components and magnitude of the signal of Fig.1.

0

δs(k)

System A

-2

System B

330 kV

CTs

-4 VTs

xs(k)

-6

Z1=0.8+j17Ω

Z0=1.1+j22Ω

-8 -10

20

0

10

20

time

30

40

k

Line

CTs

180 km R1=0.0292Ω/km L1=1.07mH/km C1=0.012µF/km

R0=0.285Ω/km L0=3.32mH/km C0=0.0082µF/km

VTs Z1=0.6+j10Ω

Z0=0.9+j16Ω

Fig.8. The simulated system.

Fig.5. Operation of the proposed algorithm: output from the half-cycle Fourier ( xc (k ) , x s (k ) ) and the correcting signals ( δ c (k ) and δ s (k ) ).

V. CONCLUSIONS The paper presents a new half-cycle adaptive algorithm for phasor estimation that is immune to the decaying dc component. The algorithm is based on the half-cycle recursive Fourier filters with two correcting functions depending on the time constant of the dc component. The time constant is estimated on-line in a parallel path of the main algorithm. The new algorithm, although based on involved digital signal processing theory, is actually very simple, robust and efficient in implementation. The additional processing power needed to perform the adaptive functions is negligible comparing with the half-cycle Fourier filtering. Both the steady-state and transient performance of the new algorithm are promising.

p11

1

F

p22 0.5

0 p12

VI. ACKNOWLEDGMENTS -0.5

p21 0

10

20

time 30

Fig.6. Coefficients of the matrix

40

k

PS .

The phasor magnitudes as estimated by the MMC and ADA algorithms are shown in Fig.10. When the input current does not contain significant high frequency components (station A), the proposed algorithm gives much more stable result than the MMC method. Because of the half-cycle data window both the algorithms are sensitive to certain extent to high frequency components in the input current (station B).

The authors acknowledge the financial support from the State Committee for Scientific Research (KBN) of Poland (project 8 T10B 009 17). VII. REFERENCES [1]

A.G. Phadke A.G. and J.S. Thorp, Computer relaying for power systems, John Wiley & Sons Inc., 1988.

[2]

M.S. Sachdev (coordinator), "Advancements in microprocessor based protection and communication", IEEE Tutorial, IEEE Publication No. 97TP120-0, 1997.

[3]

G. Benmouyal, "Removal of dc-offset in current waveforms using digital mimic filtering", IEEE Transactions on Power Delivery, Vol.10, No.2, April 1995, pp.621-628.

[4]

A.L. Almeida and A.C. Lima, "Covariance management based RLS algorithm for phasor estimation in severely noisy

3

x 10 4

a)

x 10 4

3

a)

MMC

2 2.5 current, A

current iA, A

1 0 -1

ADA

1 0.5

-2 -3

0

-4

10

1

0 x 10 4

50

100

150

200 b)

0 3 x 10 MMC

50

100

150

200 b)

150

200

current, A

current iB, A

8

0.5 0

ADA 6 2

-0.5 0

-1 -1.5

0

50

100 time, ms

150

200

Fig.9. Fault-loop currents measured at the substation A (a) and B (b) of the system of Fig.8.

systems". IEEE Transactions on Power Delivery, Vol.13, No.4, 1998, pp.1067-1071. [5]

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0

50

100 time, ms

Fig.10. Estimated current magnitudes for the fault-loop currents as seen from the substations A (a) and B (b), respectively.

[12]

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Eugeniusz Rosolowski (SM’97) was born in 1947 in Poland. He received his M.Sc. degree in Electrical Engineering from the Wroclaw University of Technology (WUT) in 1972 where he is presently an Associate Professor. From 1974 to 1977, he studied in Kiev Politechnical Institute from which he received his Ph.D. in 1978. In 1993 he received D.Sc. from the Wroclaw University of Technology. His research interests are in power system analysis and microprocessor application in power systems. Currently he is a Director of the Institute of Electric Power Engineering of WUT. Jan Izykowski (M’97) was born in Poland in 1949. He received his M.Sc. and Ph.D. degrees from the Wroclaw University of Technology in 1973 and in 1976 respectively. In 1973 he joined Institute of Electrical Engineering of the Wroclaw University of Technology where he is presently an Assistant Professor. His research interests are in power system protection, fault locators and transient phenomena of instrument transformers. Bogdan Kasztenny (M'95, SM’98) received his M.Sc. (89) and Ph.D. (92) degrees (both with honors) from the Wroclaw University of Technology (WUT), Poland. In 1989 he joined the Department of Electrical Engineering of WUT. In 1994 he was with Southern Illinois University and during the academic year 1997/98 – with Texas A&M University. Currently, Dr.Kasztenny works for GE Power Management as a Senior Application Engineer. Dr.Kasztenny is a Senior Member of IEEE, holds 2 patents, and has published more than 90 technical papers.