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New algorithms for microprocessor-based distance relaying
Electric Power Systems Research, 15 (1988) 233 - 238
233
New Algorithms for Microprocessor-Based Distance Relaying G. C. KAKOTI and H. K. VERMA
Department of Electrical Engineering, University of Roorkee, Roorkee 247 667 (India) (Received June 11, 19SS)
ABSTRACT
System impedance calculation for distance relaying requires extraction o f fundamental frequency components from highly distorted post-fault relaying signals. Many algorithms are available for extracting the fundamental frequency component, but most o f them involve a large amount o f computation, whilst others have low accuracy, or poor convergence, or both. This paper describes two algorithms based on the Hartley transform. They are computationally simple, have good frequency response and give fast convergence in the calculation o f resistance and reactance. Their efficacy has been examined using representative voltage and current relaying signals. The results obtained and presented here are gratifying.
INTRODUCTION
Use of computers for transmission line protection has been the subject o f research in many countries since the late 1960s [1 - 4]. The suitability of the microprocessor, which has the distinct advantages over the minicomputer of low cost, low component count and small size, has been studied for the protection of transmission lines. The basic requirement of distance protection with the microprocessor is the availability of computationally simple algorithms for signal processing. The post-fault relaying signals (voltage and current) contain harmonics eind a DC component which affect relaying accuracy adversely. These signals are filtered to extract their fundamental frequency components. Subsequent computations are carried out with the filtered signals to find the impedance (resistance and reactance) of the fault loop which is then compared with a 0378-7796/88/$3.50
preset value to determine if a trip signal should be issued. Some of the algorithms developed so far for extraction of the fundamental frequency component are based on the Fourier series and Walsh and Haar functions [4- 7]. These algorithms require extraction of real and imaginary components to compute the magnitude and phase. The amount of computation being large and the inter-sample period being small (typically 1.25 ms for 50 Hz signals with 16 samples per cycle), the distance relaying of a single line needs a minicomputer or a multi-microprocessor system [8]. In this paper a new algorithm based on the Hartley transform is proposed by which the magnitude and phase angles of the relaying signals can be computed with relatively fewer calculations. A simplified version of the same algorithm eliminates multiplications altogether, making it more suitable for implementation on the microprocessor. The algorithms have been tested for typical relaying signals on a main-frame computer and their convergence has been found to be even better than those of Fourier transform algorithms.
ALGORITHM FOR EXTRACTION OF THE FUNDAMENTAL FREQUENCY COMPONENT
The Hartley transform of a continuous time-varying real signal h(t) is given by H(co) -
1
~ Jh(t) cas(--cot) dt
(1)
where cas(--cot) = cos(cot) -- sin(cot) To extract a frequency component present in a time-varying signal using the Hartley © Elsevier Sequoia/Printed in The Netherlands
234 1.2 I.
transform [9, 10] on a digital computer, t h e signal must be digitized. If it is sampled at N equidistant instants over a period of one fundamental frequency cycle and these samples are denoted b y h0, hl . . . . , hN-1, the discrete Hartley transform can be written
1.0(
0.75
z
as
0.~
0.2S
H(m) -
~ hn cas ~//N
mn
(2) cto ( a ) °'°
n = 0
1,0
2.0
3.0
NORMA/ISEO
where m is the frequency c o m p o n e n t to be extracted (equal to one for the fundamental frequency), and 2rr/N is the sampling interval in radians. With N = 16, the computed o u t p u t Yk at the kth instant of computation, using the present sample hk and the immediate past 15 samples, hk-1, hk-2, ..., h k - l s , is given by 1 Yk - X/-ff~[hk + h k - 4 - - h ~ - s - - h k - 1 2
6.O
7-O
6.0
T.o
$.0
1,2~.
0./5
z
0,so o
0.3
0.0
o.o
+ 1.307(h~-1 + h~-3 -- hA-9 -- h k - 11) + 1.414(hk-2 -- h~- 10) + 0.541(hk-s -- hk-? -- h~-la + hA-ls)]
(3) Using this relation, the fundamental component can then be extracted with three multiplications and thirteen additions/subtractions. Multiplication by 1 / ~ f 1 6 is ignored here as the expressions for both the voltage and current signals will include this factor which will be cancelled o u t in the impedance calculation. The frequency response of the algorithm given by expression (3), and hereafter called the Hartley transform algorithm, has been c o m p u t e d and is shown in Fig. l(a). The curve shows a 100% gain for the fundamental positive frequency and complete rejection of DC and harmonics. The minor lobes present between the harmonics are attributed to the limited sampling window.
ALGORITHM
5,0
4.0
FREQUENCY* F(N )
SIMPLIFICATION
The above algorithm involves at least three multiplications for extracting the fundamental frequency component. However, by making suitable approximations, these
to
2.0
3.0
~.o
s.o
6.0
(b) NOR.AUSE O FREQUE.CV, F ( . Fig. 1. Frequency response for a full cycle window with N = 16: (a) Hartley transform algorithm; (b) simplified Hartley transform algorithm.
multiplications can be replaced b y a few binary shifts and additions and thus reduce computations considerably. It may also be noted that once multiplications are eliminated, no storage of weighting factors will be required. Accordingly, the multipliers 1.307, 1.414 and 0.541 are replaced by 1.25, 1.5 and 0.5, respectively, so that eqn. (3) reduces to 1 y~ -
X/r~. [h~ + h k _ 4 -
h~ -.s - - h k - 1 2
+ 1.25(hk_1 + hk-3 -- hk-9 -- hk-11) + 1.5(hk-2 -- hA-10) + 0.5(h~- s -- hA-~ -- hA-13 + hA-ls)]
(4) The frequency response of this simplified Hfirtley transform algorithm has been evaluated and is shown in Fig. l(b). As can be seen from the curve, the algorithm rejects DC and even harmonics completely, and shows a marginal response to odd harmonics. The response to the fundamental is slightly higher than that of the unmodified algorithm b u t it will pose no problem in distance relaying as
235
the fundamental c o m p o n e n t s of b o t h voltage and current will have the identical factor which will disappear in the impedance calculation. The computational requirement of the simplified algorithm for extracting the fundamental frequency c o m p o n e n t of one signal is 19 binary shifts/additions/subtractions. A benchmark program written for the Intel 8086 microprocessor to extract one fundamental frequency c o m p o n e n t shows that the proposed unmodified version of the algorithm needs 943 T states, whereas the requirement of the simplified version is only 339 T states. Thus, the modification leads to a reduction of the processing time to almost one-third of its value.
DISTANCE RELAYING WITH THE PROPOSED ALGORITHM
YP/ ¢v- yp
I p / ~i
Ip / ¢v - - dPi
(5)
va = Vp cos q~k
(6)
i~ = Ip cos ~bs~
(7)
where ¢~k and ¢i~ are the angular displacements of the voltage and current samples from the peaks of the respective waves. At the ( k - 2)th instant, the value of the voltage (for N = 16) is given b y
cos ¢~,k cos -4 + sin ¢vk sin
vh
(8)
Ip sin ¢tk = 1.414 ik-2 -- ih
(9)
Noting that ~b~-- ¢i = Cvk -- ¢~k, expression (5) can be written as
vp s,>-
z = _--/¢o,
¢,,
-
[(cos Cvk cos ~bik + sin Cvk sin ¢ ~ )
-
ip
+ j(sin ~vh cos ~b~k-- cos ¢vk sin ~b~k)] Since the real and imaginary parts of impedance Z are the resistance R and reactance X, respectively, the above equation yields + sin ~bv~ sin ~b~k)
and X = VP (sin Cvk cos ¢ik -- cos ~bvk sin ¢~k)
where Vp and Ip are the peak values and ~bv and q~l are the phase angles (with respect to a c o m m o n reference) of the fundamental frequency c o m p o n e n t s of the voltage and current signals applied to the relay. The fundamental frequency c o m p o n e n t s can be obtained b y using either the proposed unmodified Hartley transform algorithm (3) or the simplified Hartley transform algorithm (4). The fundamental frequency c o m p o n e n t s of voltage and current at any sampling instant k can be expressed,as
=
--
similarly,
R = Vp(cos ¢~h cos ~
The impedance seen b y the relay at any instant is
z-
Vp sin ¢ ~ = 1.414 vk-2
Substituting the values o f Vp cos¢~k, Vp sin ~b~k, etc. from eqns. (6) - (9), ? .?
v k ik .-I- Pk lk
R
i~ 2 + i~ 2
(10)
and t.
X-
.?
Uk lk ~ Vk lk ik 2 + i~ 2
(11)
where t
vk = Vp s m
and .f
lk -- Ip sin ~bik
The multiplier 1.414 appearing in eqns. (10) and (11) can be replaced b y 1.407 to eliminate the multiplication operation on the microprocessor (as multiplication b y 1.407 can be achieved b y 1 + 1 / 4 + 1 / 8 + 1 / 3 2 which only requires binary shifts/additions). No noticeable change is observed in the c o m p u t e d values of the resistance and reactance. The microprocessor will c o m p u t e the resistance and reactance at every sampling instant from the filtered voltage and current signals using eqns. (10) and (11) and compare them with preset operating values. A trip signal
236
will be generated if and when the calculated values are found to be lower than the operating values at two sampling instants in succession.
PERFORMANCE EVALUATION
The performance of a distance relay using the proposed algorithms has been studied on a DEC-2050 computer for typical relaying signals. Under steady-state conditions the current and voltage on a transmission line are sinusoidal in nature and differ in phase by a small angle. The post-fault signals may contain varying degrees of harmonics and a decaying DC component depending upon the instant of occurrence of a fault. The occurrence of a fault when the voltage is passing through its peak value causes maximum harmonics in the post-fault voltage signal. On the other hand, the value of the DC offset in the current is maximum for faults initiated near the voltage zero. The harmonics and DC offset present in current and voltage signals are of decaying nature and after a few cycles from the occurrence o f a fault both the signals attain steady-state post-fault values. With the above considerations in mind, the following typical voltage and current signals have been used to evaluate the performance of the relay. (1) Prefault v(t) = sin(cot) (1 p.u. magnitude)
The prefault resistance and reactance are 0.940 and 0.342 p.u. respectively. The postfault steady-state values are 0.0868 and 0.4924 p.u., respectively, which is thus the theoretical operating threshold of the relay. The resistance and reactance computed for the above relaying signals under the two cases mentioned using the unmodified Hartley transform algorithm are shown in Figs. 2(a) and (b). In the same Figures the values computed using the Fourier transform algorithm are also shown for comparison. With the Hartley transform algorithm for the fault occurring at the voltage peak, the convergence towards post-fault steady-state values is somewhat better and there is practically no fluctuation after 20 ms (1 cycle) from the occurrence of the fault. But, for the fault occurring at voltage zero, though the convergence occurs after 20 ms, wide fluctuations are observed around the steady state after 20 ms. This is attributed to the high content of DC present in the current signal. 1.0
voltage is passing
v ' ( t ) = sin(cot) + [0.3 sin(3cot)
0
0
0
FOURIER
O~
0
~
REACTANCE
0.~ o
0.2
RESt STANCE o
o.o
'
-10.0
-$-T$
'
2. S0
.~ls
(a)
ls.'o TIME
21'2s
2TISO ,,'?s
- ~
G
I ~0
IN M I L L I S E C O N D S
1.0, :) 0.8,
when
0
l
i(t) = sin(cot -- 20 °) (1 p.u. magnitude) (2) Postfault (a) for fault through zero
HARTLEY
O
O.g
0
0
0
HARTLEY 0 FOURIER
¢g 0.$
REACTANCE
z
~ 0-~
0
•
+ 0.2 s i n ( 5 c o t ) ] e -t/°'°2s 0.;
i'(t) = 2 sin(cot -- 80 °) + 1.6 e -t/°'°2° + 0.1 sin(3cot -- 26.67 °) e -t/°'°=s (b) for fault when voltage is passing through the maximum value
(b)
00,0
RE S[STANCE
-3.7S
Z.SO
8~3
15.0
21.25
27.50
332/S
40.0
TIME ,N M,LLISECOND$
Fig. 2. R and X computed with the Hartley transform algorithm: (a) fault at voltage peak; (b) fault at voltage zero.
v " ( t ) = sin(cot) + 0.4 sin(3cot) + 0.2 sin(5cot) i " ( t ) = 2 sin(cot -- 80 °) + 0.08 e-t/°'°3° + 0.4 sin(3cot -- 26.6 °) + 0.2 sin(5cot -- 16 °)
The performance of the simplified Hartley transform algorithm is not significantly different from that of the unmodified algorithm, as revealed by Figs. 2 and 3. Thus the
237 1. O - q
i
A ratio error of +5% in the CT and +1% in the VT were assumed in the relaying signals. A m a x i m u m of +3.85% variation in the c o m p u t e d values o f the resistance and reactance was observed in the post-fault steadystate regions. The effects of quantization (limited resolution) and error in the conversion of an analogue/digital converter (ADC) on the performance of the proposed simplified Hartley algorithm have been studied on a mainframe computer. The digitized data were multiplied with appropriate scaling factors limiting the m a x i m u m digitized value to (a) 8, (b) 10, (c) 12, and (d) 14 bits o f binary data. In addition, a conversion error of one least significant bit was assumed. The computed resistance and reactance show m a x i m u m errors o f 2.2%, 0.32%, 0.062% and 0.016% respectively, for the 8, 10, 12 and 14 bit ADCs with respect to the values obtained for no quantization and conversion errors.
SIMPLIFIE D HARTLEY
• O.e
O
O
O
O FOURIER
O.G
REACTANCE
O.d 0-i O O )-10.0
I -3.'S
I
I 2,SO
~
,.?S
(. a _
T,ME
I 21,2S
1.0 [N
~ -~
-
O
I. 2? SO
[ .-7S
40.0
MILL, S E C O N D S
1.0-CL ~
R E Sl STA RCE
Q
O O
SIMPLIFIED
HARTLEY
O FOURIER
o. G ( z
REACTANCE
K O z 0.4
O O
O•] O -10.0
(b)
R E r " STA NC [
-3-75
2. SO
S. TS TIME
15.0
21.25
2?.SO
"33-?S
/*GO0
IN M I L L I S E C O N ~
Fig. 3. R and X computed with the simplified Hartley transform algorithm: (a) fault at voltage peak; (b) fault at voltage zero.
CONCLUSION
approximation made in the development of the simplified algorithm does n o t adversely influence the performance of the distance relay and is therefore valid. On the other hand, the simplification reduces the computations considerably. A study has been made to test the performance of the simplified Hartley algorithm in the presence of errors in a current transformer (CT) and a voltage transformer (VT).
A comparison of the computational requirement of the proposed algorithm has been made with those of the Fourier transform, Walsh and Haar function algorithms with N = 16 samples and is shown in Table 1. The proposed algorithms require less computation than the other algorithms. The proposed simplified version of the Hartley transform algorithm requires far less computation than does the unmodified algorithm, whereas their performances in distance
TABLE 1 Computational requirements of full-cycle-window algorithms (using N = 16 for resistance and reactance computation Algorithm
Fundamental frequency tomponent extraction (voltage and current) Binary shifts/ Multiplications/ additions/ divisions subtractions
Fourier Walsh Haar Hartley Simplified Hartley
56 60 60 26 38
12 16 16 6 0
Resistance and reactance computation
Binary shifts/ Multiplications/ additions/ divisions
Total
Binary shifts/ Multiplications/ additions/ divisions subtractions
subtractions
3 3 3 5 21
8 8 8 10 8
59 63 63 31 59
20 24 24 16 8
238 r e l a y i n g are similar. Oscillations a f t e r I cycle in t h e case o f faults o c c u r r i n g at t h e p e a k o f t h e voltage w a v e are v e r y small. B u t in t h e case o f faults o c c u r r i n g at t h e v o l t a g e z e r o , oscillations ( a r o u n d t h e s t e a d y - s t a t e value) b e y o n d 1 cycle are o b s e r v e d . F r e q u e n c y responses of the two algorithms show c o m p l e t e r e j e c t i o n o f DC a n d even h a r m o n i c s . I n t h e case o f t h e simplified a l g o r i t h m , o d d h a r m o n i c s , albeit in v e r y small f r a c t i o n s , w o u l d b e passed t h r o u g h . This, h o w e v e r , d o e s n o t a f f e c t t h e results adversely, as revealed by the study of the relay performance. The p r o p o s e d s i m p l i f i c a t i o n is, t h e r e f o r e , justified.
3 G. B. Gilchrist, G. D. RockfeUer and E. A. Undren, High-speed distance relaying using a digital computer, Part-I. System description, IEEE Trans., PAS-91 (1972) 1235 - 1243. 4 M. Ramamoorthy, Application of digital computers to power system protection, J. Inst. Eng. (India), 52 (1972) 235- 238. 5 P. G. McLaren and M. A. Redfern, Fourier series techniques applied to distance protection, Proc. Inst. Electr. Eng., 122 (1975) 1301 - 1305. 6 J. W. Horton, The use of Walsh functions for high speed digital relaying, IEEE PES S u m m e r Meeting, San Francisco, 1975, Paper No. A75582-7, pp. 1- 9. 7 D. B. Fakruddin and K. Parthasarathy, Simplified algorithms based on Haar transforms for signal recognition in protective relays, Proc. IEEE, 73 (1985) 40- 42.
REFERENCES
8 Computer
1 G. D. Rockfeller, Fault protection with digital computer, IEEE Trans., PAS-88 (1969) 438464.
2 B. J. Mann and J. F. Morrison, Relaying a three phase line with a digital computer, IEEE Trans., PAS-90 (1971) 742 - 750.
Relaying -- IEEE
Tutorial
Course,
IEEE, New York, Pubn. 79 EHO 148-7 PWR. 9 R. V. L. Hartley, A more symmetrical Fourier analysis applied to transmission problems, Proc. IRE, (Mar.) (1942) 144- 150. 10 H. J. Meckelburg and D. Lipka, Fast Hartley transform algorithm, Electron. Lett., 21 (1985) 341 - 343.
233
New Algorithms for Microprocessor-Based Distance Relaying G. C. KAKOTI and H. K. VERMA
Department of Electrical Engineering, University of Roorkee, Roorkee 247 667 (India) (Received June 11, 19SS)
ABSTRACT
System impedance calculation for distance relaying requires extraction o f fundamental frequency components from highly distorted post-fault relaying signals. Many algorithms are available for extracting the fundamental frequency component, but most o f them involve a large amount o f computation, whilst others have low accuracy, or poor convergence, or both. This paper describes two algorithms based on the Hartley transform. They are computationally simple, have good frequency response and give fast convergence in the calculation o f resistance and reactance. Their efficacy has been examined using representative voltage and current relaying signals. The results obtained and presented here are gratifying.
INTRODUCTION
Use of computers for transmission line protection has been the subject o f research in many countries since the late 1960s [1 - 4]. The suitability of the microprocessor, which has the distinct advantages over the minicomputer of low cost, low component count and small size, has been studied for the protection of transmission lines. The basic requirement of distance protection with the microprocessor is the availability of computationally simple algorithms for signal processing. The post-fault relaying signals (voltage and current) contain harmonics eind a DC component which affect relaying accuracy adversely. These signals are filtered to extract their fundamental frequency components. Subsequent computations are carried out with the filtered signals to find the impedance (resistance and reactance) of the fault loop which is then compared with a 0378-7796/88/$3.50
preset value to determine if a trip signal should be issued. Some of the algorithms developed so far for extraction of the fundamental frequency component are based on the Fourier series and Walsh and Haar functions [4- 7]. These algorithms require extraction of real and imaginary components to compute the magnitude and phase. The amount of computation being large and the inter-sample period being small (typically 1.25 ms for 50 Hz signals with 16 samples per cycle), the distance relaying of a single line needs a minicomputer or a multi-microprocessor system [8]. In this paper a new algorithm based on the Hartley transform is proposed by which the magnitude and phase angles of the relaying signals can be computed with relatively fewer calculations. A simplified version of the same algorithm eliminates multiplications altogether, making it more suitable for implementation on the microprocessor. The algorithms have been tested for typical relaying signals on a main-frame computer and their convergence has been found to be even better than those of Fourier transform algorithms.
ALGORITHM FOR EXTRACTION OF THE FUNDAMENTAL FREQUENCY COMPONENT
The Hartley transform of a continuous time-varying real signal h(t) is given by H(co) -
1
~ Jh(t) cas(--cot) dt
(1)
where cas(--cot) = cos(cot) -- sin(cot) To extract a frequency component present in a time-varying signal using the Hartley © Elsevier Sequoia/Printed in The Netherlands
234 1.2 I.
transform [9, 10] on a digital computer, t h e signal must be digitized. If it is sampled at N equidistant instants over a period of one fundamental frequency cycle and these samples are denoted b y h0, hl . . . . , hN-1, the discrete Hartley transform can be written
1.0(
0.75
z
as
0.~
0.2S
H(m) -
~ hn cas ~//N
mn
(2) cto ( a ) °'°
n = 0
1,0
2.0
3.0
NORMA/ISEO
where m is the frequency c o m p o n e n t to be extracted (equal to one for the fundamental frequency), and 2rr/N is the sampling interval in radians. With N = 16, the computed o u t p u t Yk at the kth instant of computation, using the present sample hk and the immediate past 15 samples, hk-1, hk-2, ..., h k - l s , is given by 1 Yk - X/-ff~[hk + h k - 4 - - h ~ - s - - h k - 1 2
6.O
7-O
6.0
T.o
$.0
1,2~.
0./5
z
0,so o
0.3
0.0
o.o
+ 1.307(h~-1 + h~-3 -- hA-9 -- h k - 11) + 1.414(hk-2 -- h~- 10) + 0.541(hk-s -- hk-? -- h~-la + hA-ls)]
(3) Using this relation, the fundamental component can then be extracted with three multiplications and thirteen additions/subtractions. Multiplication by 1 / ~ f 1 6 is ignored here as the expressions for both the voltage and current signals will include this factor which will be cancelled o u t in the impedance calculation. The frequency response of the algorithm given by expression (3), and hereafter called the Hartley transform algorithm, has been c o m p u t e d and is shown in Fig. l(a). The curve shows a 100% gain for the fundamental positive frequency and complete rejection of DC and harmonics. The minor lobes present between the harmonics are attributed to the limited sampling window.
ALGORITHM
5,0
4.0
FREQUENCY* F(N )
SIMPLIFICATION
The above algorithm involves at least three multiplications for extracting the fundamental frequency component. However, by making suitable approximations, these
to
2.0
3.0
~.o
s.o
6.0
(b) NOR.AUSE O FREQUE.CV, F ( . Fig. 1. Frequency response for a full cycle window with N = 16: (a) Hartley transform algorithm; (b) simplified Hartley transform algorithm.
multiplications can be replaced b y a few binary shifts and additions and thus reduce computations considerably. It may also be noted that once multiplications are eliminated, no storage of weighting factors will be required. Accordingly, the multipliers 1.307, 1.414 and 0.541 are replaced by 1.25, 1.5 and 0.5, respectively, so that eqn. (3) reduces to 1 y~ -
X/r~. [h~ + h k _ 4 -
h~ -.s - - h k - 1 2
+ 1.25(hk_1 + hk-3 -- hk-9 -- hk-11) + 1.5(hk-2 -- hA-10) + 0.5(h~- s -- hA-~ -- hA-13 + hA-ls)]
(4) The frequency response of this simplified Hfirtley transform algorithm has been evaluated and is shown in Fig. l(b). As can be seen from the curve, the algorithm rejects DC and even harmonics completely, and shows a marginal response to odd harmonics. The response to the fundamental is slightly higher than that of the unmodified algorithm b u t it will pose no problem in distance relaying as
235
the fundamental c o m p o n e n t s of b o t h voltage and current will have the identical factor which will disappear in the impedance calculation. The computational requirement of the simplified algorithm for extracting the fundamental frequency c o m p o n e n t of one signal is 19 binary shifts/additions/subtractions. A benchmark program written for the Intel 8086 microprocessor to extract one fundamental frequency c o m p o n e n t shows that the proposed unmodified version of the algorithm needs 943 T states, whereas the requirement of the simplified version is only 339 T states. Thus, the modification leads to a reduction of the processing time to almost one-third of its value.
DISTANCE RELAYING WITH THE PROPOSED ALGORITHM
YP/ ¢v- yp
I p / ~i
Ip / ¢v - - dPi
(5)
va = Vp cos q~k
(6)
i~ = Ip cos ~bs~
(7)
where ¢~k and ¢i~ are the angular displacements of the voltage and current samples from the peaks of the respective waves. At the ( k - 2)th instant, the value of the voltage (for N = 16) is given b y
cos ¢~,k cos -4 + sin ¢vk sin
vh
(8)
Ip sin ¢tk = 1.414 ik-2 -- ih
(9)
Noting that ~b~-- ¢i = Cvk -- ¢~k, expression (5) can be written as
vp s,>-
z = _--/¢o,
¢,,
-
[(cos Cvk cos ~bik + sin Cvk sin ¢ ~ )
-
ip
+ j(sin ~vh cos ~b~k-- cos ¢vk sin ~b~k)] Since the real and imaginary parts of impedance Z are the resistance R and reactance X, respectively, the above equation yields + sin ~bv~ sin ~b~k)
and X = VP (sin Cvk cos ¢ik -- cos ~bvk sin ¢~k)
where Vp and Ip are the peak values and ~bv and q~l are the phase angles (with respect to a c o m m o n reference) of the fundamental frequency c o m p o n e n t s of the voltage and current signals applied to the relay. The fundamental frequency c o m p o n e n t s can be obtained b y using either the proposed unmodified Hartley transform algorithm (3) or the simplified Hartley transform algorithm (4). The fundamental frequency c o m p o n e n t s of voltage and current at any sampling instant k can be expressed,as
=
--
similarly,
R = Vp(cos ¢~h cos ~
The impedance seen b y the relay at any instant is
z-
Vp sin ¢ ~ = 1.414 vk-2
Substituting the values o f Vp cos¢~k, Vp sin ~b~k, etc. from eqns. (6) - (9), ? .?
v k ik .-I- Pk lk
R
i~ 2 + i~ 2
(10)
and t.
X-
.?
Uk lk ~ Vk lk ik 2 + i~ 2
(11)
where t
vk = Vp s m
and .f
lk -- Ip sin ~bik
The multiplier 1.414 appearing in eqns. (10) and (11) can be replaced b y 1.407 to eliminate the multiplication operation on the microprocessor (as multiplication b y 1.407 can be achieved b y 1 + 1 / 4 + 1 / 8 + 1 / 3 2 which only requires binary shifts/additions). No noticeable change is observed in the c o m p u t e d values of the resistance and reactance. The microprocessor will c o m p u t e the resistance and reactance at every sampling instant from the filtered voltage and current signals using eqns. (10) and (11) and compare them with preset operating values. A trip signal
236
will be generated if and when the calculated values are found to be lower than the operating values at two sampling instants in succession.
PERFORMANCE EVALUATION
The performance of a distance relay using the proposed algorithms has been studied on a DEC-2050 computer for typical relaying signals. Under steady-state conditions the current and voltage on a transmission line are sinusoidal in nature and differ in phase by a small angle. The post-fault signals may contain varying degrees of harmonics and a decaying DC component depending upon the instant of occurrence of a fault. The occurrence of a fault when the voltage is passing through its peak value causes maximum harmonics in the post-fault voltage signal. On the other hand, the value of the DC offset in the current is maximum for faults initiated near the voltage zero. The harmonics and DC offset present in current and voltage signals are of decaying nature and after a few cycles from the occurrence o f a fault both the signals attain steady-state post-fault values. With the above considerations in mind, the following typical voltage and current signals have been used to evaluate the performance of the relay. (1) Prefault v(t) = sin(cot) (1 p.u. magnitude)
The prefault resistance and reactance are 0.940 and 0.342 p.u. respectively. The postfault steady-state values are 0.0868 and 0.4924 p.u., respectively, which is thus the theoretical operating threshold of the relay. The resistance and reactance computed for the above relaying signals under the two cases mentioned using the unmodified Hartley transform algorithm are shown in Figs. 2(a) and (b). In the same Figures the values computed using the Fourier transform algorithm are also shown for comparison. With the Hartley transform algorithm for the fault occurring at the voltage peak, the convergence towards post-fault steady-state values is somewhat better and there is practically no fluctuation after 20 ms (1 cycle) from the occurrence of the fault. But, for the fault occurring at voltage zero, though the convergence occurs after 20 ms, wide fluctuations are observed around the steady state after 20 ms. This is attributed to the high content of DC present in the current signal. 1.0
voltage is passing
v ' ( t ) = sin(cot) + [0.3 sin(3cot)
0
0
0
FOURIER
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0
~
REACTANCE
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0.2
RESt STANCE o
o.o
'
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IN M I L L I S E C O N D S
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when
0
l
i(t) = sin(cot -- 20 °) (1 p.u. magnitude) (2) Postfault (a) for fault through zero
HARTLEY
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0
0
HARTLEY 0 FOURIER
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REACTANCE
z
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0
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+ 0.2 s i n ( 5 c o t ) ] e -t/°'°2s 0.;
i'(t) = 2 sin(cot -- 80 °) + 1.6 e -t/°'°2° + 0.1 sin(3cot -- 26.67 °) e -t/°'°=s (b) for fault when voltage is passing through the maximum value
(b)
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Fig. 2. R and X computed with the Hartley transform algorithm: (a) fault at voltage peak; (b) fault at voltage zero.
v " ( t ) = sin(cot) + 0.4 sin(3cot) + 0.2 sin(5cot) i " ( t ) = 2 sin(cot -- 80 °) + 0.08 e-t/°'°3° + 0.4 sin(3cot -- 26.6 °) + 0.2 sin(5cot -- 16 °)
The performance of the simplified Hartley transform algorithm is not significantly different from that of the unmodified algorithm, as revealed by Figs. 2 and 3. Thus the
237 1. O - q
i
A ratio error of +5% in the CT and +1% in the VT were assumed in the relaying signals. A m a x i m u m of +3.85% variation in the c o m p u t e d values o f the resistance and reactance was observed in the post-fault steadystate regions. The effects of quantization (limited resolution) and error in the conversion of an analogue/digital converter (ADC) on the performance of the proposed simplified Hartley algorithm have been studied on a mainframe computer. The digitized data were multiplied with appropriate scaling factors limiting the m a x i m u m digitized value to (a) 8, (b) 10, (c) 12, and (d) 14 bits o f binary data. In addition, a conversion error of one least significant bit was assumed. The computed resistance and reactance show m a x i m u m errors o f 2.2%, 0.32%, 0.062% and 0.016% respectively, for the 8, 10, 12 and 14 bit ADCs with respect to the values obtained for no quantization and conversion errors.
SIMPLIFIE D HARTLEY
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O
O
O
O FOURIER
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-
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Fig. 3. R and X computed with the simplified Hartley transform algorithm: (a) fault at voltage peak; (b) fault at voltage zero.
CONCLUSION
approximation made in the development of the simplified algorithm does n o t adversely influence the performance of the distance relay and is therefore valid. On the other hand, the simplification reduces the computations considerably. A study has been made to test the performance of the simplified Hartley algorithm in the presence of errors in a current transformer (CT) and a voltage transformer (VT).
A comparison of the computational requirement of the proposed algorithm has been made with those of the Fourier transform, Walsh and Haar function algorithms with N = 16 samples and is shown in Table 1. The proposed algorithms require less computation than the other algorithms. The proposed simplified version of the Hartley transform algorithm requires far less computation than does the unmodified algorithm, whereas their performances in distance
TABLE 1 Computational requirements of full-cycle-window algorithms (using N = 16 for resistance and reactance computation Algorithm
Fundamental frequency tomponent extraction (voltage and current) Binary shifts/ Multiplications/ additions/ divisions subtractions
Fourier Walsh Haar Hartley Simplified Hartley
56 60 60 26 38
12 16 16 6 0
Resistance and reactance computation
Binary shifts/ Multiplications/ additions/ divisions
Total
Binary shifts/ Multiplications/ additions/ divisions subtractions
subtractions
3 3 3 5 21
8 8 8 10 8
59 63 63 31 59
20 24 24 16 8
238 r e l a y i n g are similar. Oscillations a f t e r I cycle in t h e case o f faults o c c u r r i n g at t h e p e a k o f t h e voltage w a v e are v e r y small. B u t in t h e case o f faults o c c u r r i n g at t h e v o l t a g e z e r o , oscillations ( a r o u n d t h e s t e a d y - s t a t e value) b e y o n d 1 cycle are o b s e r v e d . F r e q u e n c y responses of the two algorithms show c o m p l e t e r e j e c t i o n o f DC a n d even h a r m o n i c s . I n t h e case o f t h e simplified a l g o r i t h m , o d d h a r m o n i c s , albeit in v e r y small f r a c t i o n s , w o u l d b e passed t h r o u g h . This, h o w e v e r , d o e s n o t a f f e c t t h e results adversely, as revealed by the study of the relay performance. The p r o p o s e d s i m p l i f i c a t i o n is, t h e r e f o r e , justified.
3 G. B. Gilchrist, G. D. RockfeUer and E. A. Undren, High-speed distance relaying using a digital computer, Part-I. System description, IEEE Trans., PAS-91 (1972) 1235 - 1243. 4 M. Ramamoorthy, Application of digital computers to power system protection, J. Inst. Eng. (India), 52 (1972) 235- 238. 5 P. G. McLaren and M. A. Redfern, Fourier series techniques applied to distance protection, Proc. Inst. Electr. Eng., 122 (1975) 1301 - 1305. 6 J. W. Horton, The use of Walsh functions for high speed digital relaying, IEEE PES S u m m e r Meeting, San Francisco, 1975, Paper No. A75582-7, pp. 1- 9. 7 D. B. Fakruddin and K. Parthasarathy, Simplified algorithms based on Haar transforms for signal recognition in protective relays, Proc. IEEE, 73 (1985) 40- 42.
REFERENCES
8 Computer
1 G. D. Rockfeller, Fault protection with digital computer, IEEE Trans., PAS-88 (1969) 438464.
2 B. J. Mann and J. F. Morrison, Relaying a three phase line with a digital computer, IEEE Trans., PAS-90 (1971) 742 - 750.
Relaying -- IEEE
Tutorial
Course,
IEEE, New York, Pubn. 79 EHO 148-7 PWR. 9 R. V. L. Hartley, A more symmetrical Fourier analysis applied to transmission problems, Proc. IRE, (Mar.) (1942) 144- 150. 10 H. J. Meckelburg and D. Lipka, Fast Hartley transform algorithm, Electron. Lett., 21 (1985) 341 - 343.