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Electric Power Systems Research, 17 (1989) 109 - 117

109

Digital Differential Relaying for Generator Protection: Development of Algorithm and Off-line Evaluation H. K. VERMA and K. SOUNDARARAJAN

Department of Electrical Engineering, University of Roorkee, Roorkee-24 7667 (India) (Received February 21, 1989)

ABSTRACT

The paper proposes the use o f real parts, instead o f the R M S values, o f the fundamental frequency c o m p o n e n t s o f the differential and sum currents for the purpose o f digital differential relaying for generator protection. For the extraction o f the real parts o f the fundamental frequency components o f current signals, a simple filter based on cross-correlation o f the distorted signal with an even heptagonal wave is suggested. The frequency response o f this filter is almost identical to that o f the filter based on the discrete Fourier transform, while its computational requirement is far less. The paper also examines filters based on cross-correlation with square and trapezoidal waves and discards them on the grounds o f p o o r rejection o f odd harmonics. The suggested use o f real parts in place o f R M S values and a heptagonal wave correlation filter instead o f a Fourier transform based filter for extracting them reduce the computational needs o f the differential relaying algorithm so drastically that a complete three-phase differential relay can be implemented on a 16-bit microprocessor w i t h o u t supplementing it with a coprocessor or multiplier. The efficacy o f these measures has been evaluated off-line on a mainframe computer and the test results are presented in the paper.

I. INTRODUCTION

Biased differential relaying has been universally applied for the protection of large generators on stator winding short-circuits. It uses the ratio of the differential current to the sum current in a stator coil as the fault discriminant. A trip signal is generated by the relay when this fault discriminant exceeds a 0378-7796/89/$3.50

preset threshold, called the sensitivity factor or the bias setting of the relay, and the differential current exceeds a minimum operating current setting. Analog (passive} filters have been used in the conventional generator differential relays to extract the fundamental frequency components of the distorted current inputs and thereby ensure a good relay accuracy in the presence of DC and harmonics. Only a few works on digital differential relaying for generators have been reported in the literature [1, 2]. The digital filter for extraction of the fundamental frequency components of the distorted relaying signals, which has been used extensively in digital relaying, is based generally on the discrete Fourier transform, where the cosine and sine waves of the fundamental frequency are cross-correlated with a distorted signal to get expressions for the real and imaginary parts of the fundamental frequency component of the signal [1, 3, 4]. As the filter involves several multiplications and the calculation of RMS values requires square and square-root operations, differential relaying needs a fairly fast minicomputer [1]. The authors of this paper propose the use of only one (real) component (instead of both the real and imaginary components) of the fundamental frequency in defining the fault discriminant and the trip criterion. This reduces the amount of computation drastically. For a further saving in the data processing time, cross-correlation of the distorted signal wave with other than sinusoidal waves is investigated. Finally, a differential relaying algorithm involving a fundamental frequency filter based on cross-correlation with heptagonal waves and a fault discriminant defined in terms of the running averages of the real parts of the fundamental frequency components of the differential and © Elsevier Sequoia/Printed in The Netherlands

110

sum currents is developed. The efficacy of this algorithm has been evaluated off-line on a mainframe c o m p u t e r and the test results are reported in the paper.

2. REVISED D E F I N I T I O N OF THE F A U L T DISCRIMINANT

Adhering to the trip criterion used in biased generator differential relays of electromechanical and static (hardwired electronic) types, and granting that only fundamental frequency c o m p o n e n t s of currents are to be used in checking the trip condition in the digital (computer based) relay, the following will be the algorithm for the digital differential relaying. S t e p 1. C o m p u t e the fundamental frequency components of the differential and sum currents, Idl and Is1, from the incoming and outgoing winding currents of one phase. S t e p 2. Check if the following trip criterion is satisfied: Idi ~>I0

and

D>~S

(1)

where D is the fault discriminant defined conventionally as D = Id,/Isl

(2)

I0 is the minimum operating current of the relay, and S the sensitivity factor of the relay. If the criterion is satisfied, issue a trip signal. S t e p 3. Repeat steps (1) and (2) for the other two phases. S t e p 4. Repeat steps 1 - 3. Reference 1 uses cross-correlation of an input current signal with cosine and sine waves to filter (extract) the real and imaginary parts of the fundamental frequency c o m p o n e n t of the signal. From the real and imaginary parts of the fundamental frequency of the incoming and outgoing winding currents so obtained, the magnitudes of the fundamental frequency c o m p o n e n t s of the differential and sum currents are c o m p u t e d . The definition of the fault discriminant given by relation (2) is then used to check the trip condition. This processing involves multiplication, square and square-root operations, which are all time-consuming, and

hence an HP2100 minicomputer was used for real time implementation of the algorithm. The first and major step in the direction of simplifying and reducing the computations is to redefine the fault discriminant in terms of real components alone. Denoting this fault discriminant by D', it may be computed from

D,=

average value of real part of fundamental frequency c o m p o n e n t of Id average value of real part of fundamental frequency c o m p o n e n t of I~

(3) The computation involved is now less than half of that needed for definition (2) of the fault discriminant because imaginary components are n o t to be extracted and no square and square-root operations are needed.

3. F I L T E R ALGORITHM

The second measure for reducing the computation is to use such reference waves as would lead to a computationally simpler filter algorithm for the extraction of the fundamental frequency component. 3.1. cross-correlation w i t h sinusoidal waves (Fourier t r a n s f o r m )

With a sampling rate of N samples per fundamental time period and an observation window of one full cycle, the cross-correlation of a digitized signal {x(n)} with cosine and sine waves is expressed as [3] 2N-1

Yri(k) = -:-:~ x ( k N n-0 2

Yil(k) -

- - n) cos(k -- n)

(4)

N-1

- ~ x ( k -- n) sin(k -- n) N ,~=o

(5)

where c o s ( k - - n ) and s i n ( k - - n ) are the digital values of the cosine and sine functions of the fundamental frequency at the (k -- n)th sampling instant, and the filter outputs Yrl(k) and yil(k) are the real and imaginary components of the fundamental frequency at the kth sampling instant. For N = 16, these expressions become

111 signal { x ( n ) ) can also be o b t a i n e d t h r o u g h its cross-section w i t h t h e even and o d d square waves o f a m p l i t u d e s +1 and f u n d a m e n t a l f r e q u e n c y , illustrated in Fig. 1. Restricting o u r i n t e r e s t t o t h e real c o m p o n e n t alone, the cross-correlation is w r i t t e n as

y r l ( k ) = ( 1 / 8 ) [ x k -- x k - 8 + 0.9239(xk _ 1 + Xk -

--

xk

-

7 --

-

9

15)

+ 0 . 7 0 7 1 ( x k _ : - - xk - 6 - "1- X k _

xk

Xk--

10

14)

2N--1

y,x(k) = ~ ~

+ 0 . 3 8 2 7 ( x k - 3 - - x k - s - - xk-11

(8)

x ( k - - n ) E V E N ( k - - n)

n=0 + x k - 13)]

(6)

w h e r e E V E N is the even r e f e r e n c e f u n c t i o n . F o r an even square wave it is given b y

Yil(k) = ( 1 / 8 ) [ x k -4 - - x k - l ~ + 0.9239(xk-3 + xk-s--xk-ll X k --

E V E N ( k - - n) =

13)

+1

forn =0toNI4-1 and n = 3N/4 to N -- 1

--

for n = N / 4 to 3N]4 -- 1

+ 0.7071(xk-2 + x~-6--xk-lo -- Xk -

Solving f o r N = 16, we get

14)

+ 0.3827(xk_ 1 + xk-7-

xk-9

x~ _ is)]

Y~l(k) = ( l / 8 ) [ ( x k - is +

-

OBSERVATION WINDOW +1

~o

(EVEN)

b-

O ,< Z-1

÷1

T (ODD)

X- 1 . k-3 SAMPLING

k-lk INSTANTS

(9)

The frequency response, obtained by using the z - t r a n s f o r m , f o r t h e real c o m p o n e n t filter a l g o r i t h m given b y eqn. (9) is p l o t t e d in Fig. 2 ( f r e q u e n c i e s have b e e n n o r m a l i z e d b y taking t h e f u n d a m e n t a l f r e q u e n c y as u n i t y ) .

T h e real and imaginary parts o f t h e f u n d a m e n t a l f r e q u e n c y c o m p o n e n t o f the digitized

k-8

+ Xk--12

(xk-11 + xk-lo + x~-9 + x~-8

+ x~,-~ + x k - 6 + x k - s + x k - 4 ) ]

3.2. C r o s s - c o r r e l a t i o n w i t h s q u a r e w a v e s

k-12

q" X k - 1 3

+ X h - 3 + X k - ~ + X k - l + Xk)

(7)

w h e r e x ~ _ , = x ( k - - n ) a n d xk = x ( k ). T h e filter a l g o r i t h m r e p r e s e n t e d b y eqns. (6) and (7) will, h e r e a f t e r , be r e f e r r e d t o as the F o u r i e r t r a n s f o r m filter algorithm.

k-16

Xk-14

k+4

k+$

k+12

~'

Fig. 1. Even and o d d square waves of the f u n d a m e n t a l frequency (N

=

16).

k+16

112 0.80-

3.3. Cross-correlation w i t h t r a p e z o i d a l w a v e s

I 0.60. / - -

~

The even and odd trapezoidal waves, shown in Fig. 4, may be used as the reference waves for cross-correlation. Noting the values of the even reference function from the Figure and expanding expression (8) for N = 16, we obtain

~

0.40-

~ 0.200.00~" o

~

~

~

~.

NORMALISED

,~

'

~

Yr](k) = (1/8)[(xk-

~

FREQUENCY,fn

Is --

xk-9 -- x k - s

--xk-7+Xk-l+ xk)

Fig. 2. F r e q u e n c y r e s p o n s e o f a filter based o n crossc o r r e l a t i o n w i t h a s q u a r e wave (eqn. (9)).

_

+ (2/3)(x*- 14-- X k - l o - - Xk--6 + X k - 2 )

1.00-

+ (l/3)(xk-

1o. L 0.25 .

1

2 3 /. 5 6 NORMALISEDFREQUENCY,fn

8

7

1.00-

Fig. 3. F r e q u e n c y r e s p o n s e o f a filter based o n crossc o r r e l a t i o n w i t h a sine wave (eqn. (6)).

i" 0.75-

The response of the filter algorithm based on the Fourier transform (eqn. (6)) is plotted in Fig. 3. A comparison clearly indicates that the filter derived from cross-correlation with the square wave fails to eliminate odd harmonics. Consequently, if this filter is used then the relay may have a poor accuracy in the event of odd harmonics being present in the current signals.

050-

o.25-

0.00. 0.00

1

2 3 4 5 6 7 NORMALISED FREQUENCY,fn "

(EVEN)

0

-I,

I

I I I

,I-1

1

uJ 0 I-;Z 0

,,X

\:/

/

i

',

\!

\

/

J,

(ODD)

I k-lq

k-12

k--8

k-4

k

k+4

8

Fig. 5. F r e q u e n c y r e s p o n s e o f a filter based o n crossc o r r e l a t i o n w i t h a t r a p e z o i d a l wave (eqn. (10)).

4-1

,< I

(10)

The frequency response of this filter is plotted in Fig. 5. It gives a better rejection of the odd harmonics than does the filter based on cross-correlation with a square wave.

0

I

11 - - x k - s

+ xk-3)]

~ 0.50-

0-00

13 - - x ~ _

k-~-8

k+12

k+16

Fig. 4. E v e n a n d o d d t r a p e z o i d a l waves o f t h e f u n d a m e n t a l f r e q u e n c y ( N = 16).

113 3.4. Cross-correlation with heptagonal waves For a better frequency response of the filter, the even and odd heptagonal waves shown in Fig. 6 can be used for cross-correlation. The number of samples per cycle is taken as 12 so that the sampling instants coincide with the corners of the heptagon. Taking N = 12 and proceeding as before, the following expression is obtained for the real component: yrl(k)

=

( l / 6 ) [ ( x k -- xk -6) + 0 . 8 7 5 ( x k - l i -- xk-7 -- x k - s + xh-1) + 0.5(xk_

10 -

xk-

s -

xk-4

+ xk- 2)]

(11) The frequency response of this filter is plotted in Fig. 7. The rejection of odd harmonics here is almost as good as that of the filter based on cross-correlation with a sinusoidal wave. But the a m o u n t of computation is much less and, most significantly, no multiplication is involved (the factor 1/6 is ignored since this disappears in the calculation

I

-,

I

i

l/ /V /

I

oI

I

-1-1

,

k-12

k-9

i 1

I

\

(EVEN)

'

~--S I

1 I

\/i\/