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Capacitor voltage transformer induced ferroresonance—causes, effects and design considerations
Electric Power Systems Research, 21 (1991) 23 - 31
23
Capacitor Voltage Transformer Induced Ferroresonance--Causes, Effects and Design Considerations B. S. A S H O K K U M A R a n d S U A T E R T E M
Department of Electrical and Computer Engineering, University of Southwestern Louisiana, Lafayette, LA 70504 (U.S.A.) (Received J u l y 25, 1990)
ABSTRACT
The combination of the equivalent coupling capacitance and the nonlinear magnetizing inductance of the isolating transformer of a capacitor voltage transformer (CVT) can cause a variety of nonlinear oscillations. In such networks, odd-order subharmonics can appear as stable oscillations, whereas subharmonics of any order can appear as transient oscillations. Such oscillations are known to interfere with the operation of high speed relays. This paper discusses the different types of oscillations that are possible in ferroresonant circuits, the conditions under which they appear, their effect on high speed relays, and the methods for minimizing their influence. It is shown that for a C V T the fundamental frequency voltage signal retains the discriminative features of protection, even in the presence of subharmonic resonance.
1. I N T R O D U C T I O N
The problem of nonlinear voltage oscillations in power networks has gained importance due to a number of failures of transformers and protective systems. The elements which contribute to the initiation of such oscillations are the iron cores of the power transformers and the capacitor voltage transformers (CVTs). When the nonlinear magnetizing reactance of a power or instrument transformer appears in series with an effective capacitance, a series ferroresonant circuit is formed. Such a circuit can exhibit a variety of nonlinear phenomena, such as: (a) jump resonance, (b) subharmonic resonance of even and odd o r d e r - - e i t h e r in a stable mode or in a transient mode, and 0378-7796/91/$3.50
(c) amplitude-modulated almost-periodic oscillation. Several factors, such as driving voltage, extent of loading, type of disturbance, asymmetry due to the disturbance, resistance, hysteresis, etc., influence the nat ure of such oscillations. These oscillations are known to produce elevated voltages and currents which result in failure of equipment and protective systems [1, 2]. The elevated voltages and currents are the result of the resonant nat ure of the oscillations. Further, the nonlinear nat ure of the oscillations substantially affects the magnitude and phase of the fundamental frequency signals, thereby influencing the operation of the high speed protective devices. Even in the transient mode, these oscillations can persist for several cycles and cause serious malfunctioning of the relays. It is therefore necessary t hat the general features of protection such as speed and discrimination be reexamined in the presence of nonlinear oscillations. The different types of oscillations are exhibited under varying conditions, and their areas of existence are known to overlap each other. The subsequent sections analyze and examine the various types of nonlinear oscillations, effect of different parameters, their influence on high speed protection, and possible design considerations.
2. C A P A C I T O R V O L T A G E T R A N S F O R M E R (CVT)
The advantages of cost, size and simplicity has made the CVT a very valuable device for metering and protection. The CVT and its equivalent circuit, referred to the low voltage side (115 V), are shown in Fig. l(a) and (b). In the equivalent circuit, NL represents the nonlinearity due to the magnetizing impedance of ~:~ E l s e v i e r S e q u o i a / P r i n t e d in T h e N e t h e r l a n d s
24
ship of the n o n l i n e a r i t y is s h o w n in Fig 2. It is r e p r e s e n t e d by the fifth-degree p o l y n o m i a l I = al)~ + a:~)5~+ as)~ '~
I
C1 t
= 0.016057. - 0.0168)~ 3 + 0.052)o '~
R
(1)
T V~
R
C
_[ N:I
"
(a)
C
3. JUMP RESONANCE
I:~jrO~n
L
NiILt (b) Fig. 1. (a) The CVT and (b) its equivalent circuit parameters. the i s o l a t i n g t r a n s f o r m e r . Le includes the tuning i n d u c t a n c e and the l e a k a g e i n d u c t a n c e of the t r a n s f o r m e r . The e q u i v a l e n t c a p a c i t a n c e Ce is t u n e d w i t h the i n d u c t a n c e Le. V a r i o u s b u r d e n i m p e d a n c e s are c o n s i d e r e d to represent v a r y i n g b u r d e n s up to 250 VA at p o w e r f a c t o r s of u n i t y and 0.8 lagging. All r e s u l t s a r e p r e s e n t e d on a b a s e of l 1 5 V , 2 0 0 V A and 377 rad/s. T h e c u r r e n t - f l u x l i n k a g e relation-
1.5
r~
U n d e r c e r t a i n c o m b i n a t i o n s of n e t w o r k p a r a m e t e r s the v o l t a g e and c u r r e n t Jn a ferr o r e s o n a n t circuit s u d d e n l y j u m p s to h i g h e r levels [3, 4]. This c o n d i t i o n is g e n e r a l l y associated w i t h s u b s t a n t i a l odd-order h a r m o n i c s and a f u n d a m e n t a l at the d r i v i n g frequency. A f e r r o r e s o n a n t circuit goes into the j u m p r e s o n a n c e m o d e at h i g h levels of d r i v i n g voltages, or due to s w i t c h i n g o p e r a t i o n s at low levels of d r i v i n g voltages. W h e n such a c o n d i t i o n prevails, the r e s u l t i n g o v e r v o l t a g e s and h i g h c u r r e n t s c a n d a m a g e the e q u i p m e n t . M a n y f a c t o r s d e t e r m i n e the c o n d i t i o n s u n d e r w h i c h j u m p r e s o n a n c e c a n occur. In a series f e r r o r e s o n a n t circuit the j u m p r e s o n a n c e c a n o c c u r w h e n the effective series c a p a c i t i v e rea c t a n c e is m o r e t h a n the i n d u c t i v e r e a c t a n c e of the n e t w o r k at a d r i v i n g frequency. Ano t h e r f a c t o r w h i c h c a n c o n t r i b u t e to the j u m p r e s o n a n c e is a low p o w e r f a c t o r c a p a c i t i v e load. T h e s e c o n d i t i o n s are g e n e r a l l y not satisfied by the CVT. T h e r e f o r e , the possibility of the CVT c a u s i n g the j u m p r e s o n a n c e is quite remote. On the o t h e r hand, u n l o a d e d series compens a t e d t r a n s m i s s i o n lines with i n t e r m e d i a t e load tapping, and u n b a l a n c e d a n d u n g r o u n d e d d i s t r i b u t i o n s y s t e m s satisfy the c o n d i t i o n s for jump resonance.
4. SUBHARMONIC OSCILLATIONS
1.0
c~
0.5
I 0.5
I 1.0 I
in
1 1.5
pu
Fig. 2. M a g n e t i z i n g c h a r a c t e r i s t i c s of t h e i s o l a t i n g transformer.
T h e series f e r r o r e s o n a n t circuit c a n go into a m o d e of s u b h a r m o n i c o s c i l l a t i o n s o v e r a v e r y wide r a n g e of n e t w o r k p a r a m e t e r s . D u r i n g s u c h oscillations, the v o l t a g e a n d c u r r e n t s will h a v e l a r g e c o m p o n e n t s of f r e q u e n c i e s below the f u n d a m e n t a l frequency. The s u b h a r m o n i c freq u e n c i e s are g e n e r a l l y 1/3, 1/5, 1/7, etc., of the d r i v i n g frequency. T h e s e odd-order s u b h a r m o n ics a p p e a r e i t h e r as stable o s c i l l a t i o n s or as t r a n s i e n t s . T h e following Sections detail the different f a c t o r s w h i c h influence the subharmonic resonance.
25
4.1. Mathematical analysis The block diagram c o r r e s p o n d i n g to the series f e r r o r e s o n a n t circuit of Fig. l(b) is given in Fig. 3, w h e r e
of the o u t p u t s u b h a r m o n i c to the input subh a r m o n i c gives the didf [6], as
N = al + 3a3(P 2 + 2)~m)/4 -1-5a5(~t 4 -4- 6~t2)~m2 -f- 3)~m4)/8
G(S) = R3(S2LeC~ + S R e C e -t- ] ) ( R 2 + S L 2) --p[5as).m(2p e + 3).m2)/8
× [SC~R.3(R2 + SL2) A- S(S2LeC~ +SReCe+I)(R3+R2+SL2)I
1
(2)
+ 3aa).m/4] exp[ --j(30 -- ~)] --5a5~3~.m exp[j(30 - ~)]
and the n o n l i n e a r i t y NL is defined by (1). The m e t h o d adopted for a n a l y z i n g the subharmonic oscillation utilizes the dual-input describing f u n c t i o n (didf) as suggested by West and D o u c e [5]. Since the s u b h a r m o n i c oscillations h a v e p r e d o m i n a n t l y two f r e q u e n c y components, it is n e c e s s a r y to d e t e r m i n e the response of the n o n l i n e a r element to two s i m u l t a n e o u s inputs of different frequencies. The effective gain of the n o n l i n e a r i t y to one f r e q u e n c y in the presence of the o t h e r is referred to as the didf. In the i n v e s t i g a t i o n of a s u b h a r m o n i c whose f r e q u e n c y is of order l/n, a signal )'m sin(~t) is r e g a r d e d as predetermined, and the didf to a second input p sin(cgt/n + 0) is superposed on the N y q u i s t locus of the l i n e a r part G(ju~/n) of the system. The v a l u e of p at which the n e g a t i v e r e c i p r o c a l of the didf intersects the N y q u i s t locus gives the amplitude c o r r e s p o n d i n g to an overall loop gain of u n i t y in the feedback system. A c o m b i n a t i o n of 2~m and p c o r r e s p o n d i n g to the point of intersection gives the possible steady-state oscillation. Let the driving voltage be E = Em sin(t~t) If the input to the n o n l i n e a r i t y is assumed to be ).(t) = p sin(~ot/3 + 0) + Amsin(~ot + ~)
(3)
t h e n the didf for the s u b h a r m o n i c of order 1/3 can be d e t e r m i n e d by s u b s t i t u t i n g (3) in (1). N e g l e c t i n g h i g h e r o r d e r harmonics, the r a t i o
G(s)
Fig. 3. Block diagram representation of the CVT.
(4)
The c o n d i t i o n for the existence of stable subh a r m o n i c oscillations is G(j~o/3) = - 1IN
(5)
The values of g, )°m, 0 and ~ at which the c o n d i t i o n (5) is satisfied defines the subharmonic response. At the j u n c t i o n of the block diagram, the r e l a t i o n s h i p for the signals of f u n d a m e n t a l f r e q u e n c y may be established as follows: a l + a3 {3()~m2 A- 2tt2)/4
_ p 3 exp[j(30 - a)]/(4).m) } -- 5).mp 3 e x p [ - - j ( 3 0 -- ~)]/8 -4-a5 [5().m 4 -~- 61/2)~m2 -~- 3~t4)/8 -- 5~ 3(~ 2 _~ 4)~m2)/( 162m )]
× exp[j(30 -- ~)] = - 1/G(j(9) + (Wm/2m)lGo(je))/G(j~o)] x exp[j(0 - a)]
(6)
w h e r e 0 is the angle of Go(j(o)/G(jo)). The s u b h a r m o n i c response can be o b t a i n e d a n a l y t i c a l l y by solving eqns. (5) and (6) by any i t e r a t i v e method.
4.2. Region of stable subharmonics The locus of - 1 / N on the complex plane for )'m = 1.0 p.u., which corresponds approxim a t e l y to a driving voltage of 1.0p.u., is shown in Fig. 4. The area w i t h i n the envelope gives the region of existence of stable subharmonics. F r o m a study of several such diagrams it was found t h a t for a specific G(j~/3) = G(jl/3), the amplitude of subharmonic oscillation changes only w i t h i n a narrow r a n g e for different 2m. Therefore, even at low driving voltages, the response will h a v e large s u b h a r m o n i c amplitudes. In all cases, the locus of - 1 / N shifts to the r i g h t in the second and third q u a d r a n t s , and c o n v e r g e s at the origin for large values of g. If the t r a n s f e r
26
z
I
•
~eal
G(s)
-2
--4
Fig. 4. R e g i o n of 1/3 f r e q u e n c y s u b h a r m o n i c s for t he CVT a t )-m -- 1.0 p.u.
function G(jl/3) of the linear part of the system is very close to the origin in the third quadrant, the circuit can oscillate with high values of #. As the operating point G(jl/3) moves away from the origin in the third quadrant, the amplitude of the subharmonic oscillation reduces and ultimately the oscillations disappear. When G(jl/3) is outside the envelope, there is no possibility of stable subharmonics of order 1/3. The region of subharmonics spreads out on the negative side of the complex plane within an angle of approximately (180 _+ 27) degrees. In addition, the regions of higher order subharmonics overlap, and some of them are slightly wider. For any one set of parameters of the CVT, several subharmonic responses of different frequencies are therefore possible. So far, it has not been possible to predict which responses would prevail for a given set of parameters. The particular response depends not only on the parameters, but also on the conditions at the time of switching and the magnitude of the driving voltage. The locus of G(jl/3) for the CVT is shown in Fig. 5. The loci are drawn for different values of load burden and equivalent capacitance Ce. Superposed on the same diagram are the envelopes of the didf for different 2m. The most favorable parameter conditions for the appearance of subharmonic resonance are large values of C~ and low burden. Increased values of the equivalent resistance Re slightly shift the G(jl/3) point downwards, and hence have a marginal tendency to reduce the susceptibility to resonance. The area covered by - 1 / N is by far the largest for moderate levels of 2m. Therefore a remote fault on the power network is the most favorable operating con-
dition for the CVT to cause subharmonic resonance. In addition, a minimum permanent load can greatly assist in the prevention of resonance. The ferroresonance suppression circuit providing effective loading at subharmonic frequencies is a good method available for avoiding the subharmonics. Design of a CVT with lower equivalent capacitance can further ease the problem.
4.3. Transient subharmonics If the operating point G(j 1/3) of a ferroresonant circuit is outside the region of subharmonics, then stable oscillations cannot appear. However, this is not an adequate guarantee that components of subfrequencies do not appear during transient conditions. This is particularly true when G(jl/3) is in the third quadrant--outside, but in the vicinity of the region of stable subharmonics. Factors like hysteresis, asymmetrical operation of the nonlinearity due to DC components, parameter dependence on frequency and temperatures, changes in equivalent capacitance, etc., are responsible for causing substantial subharmonics in a transient mode. A ferroresonant suppression circuit may apparently solve the problem of stable subharmonics, but still may not be able to prevent the failure of high speed protection due to transient subharmonics. A previously published test report on a CVT with a ferroresonant suppression circuit appears to show subharmonics during the transients [7]. To illustrate the above point, the results from a series ferroresonant circuit are presented in Figs. 6 and 7. The oscillograms of Figs. 6(a) and 7(a) show 1/3 frequency
27 Im -3.0
-2.0
!
!
-1.0 '-'
v~
G(s)
/
/ i ),=
/
/
/
/
/ J
/ -1.0
O. &
I I
u~
/.?"
,E) _
t',-
I I I I I I
I
-2.0
\ \ \
\
\
\ \
Fig. 5. L o c u s of G(jl/3) for different C e a n d burden: C , , ; - - , e n v e l o p e s of -1/N.
-3.0
, c o n s t a n t b u r d e n at 0.8 l a g g i n g power factor;
, constant
%(a)
(b)
(c)
(d)
Fig. 6. T r a n s i e n t s u b h a r m o n i c s at m o d e r a t e voltages.
(a)
(b)
(c)
(d)
Fig. 7. T r a n s i e n t s u b h a r m o n i c s at low voltages.
2~
monics in the steady state. After initiation of the oscillations, the circuit parameters and the driving voltage were sufficiently changed, beyond the point of disappearance of stable resonance. The several other transients presented are taken under these conditions. The following points are based on the observation of a number of such transients. (a) The decaying rate of the transient subharmonics does not remain constant and varies considerably for the same set of network parameters. (b) For the same network condition, harmonics of order 1/3, 1/5, 1/7, etc., or even a combination, may appear as transients. In such cases, the higher frequency components decay faster. (c) Transient subharmonics are possible even at very high values of series resistance. (d) Substantially large levels of subharmonics may appear even at low driving voltages.
4.5. Even-order transient subharmonics The literature documents that even-order subharmonics cannot exist in ferroresonant circuits with symmetrical saturation types of nonlinearities. When the saturation is asymmetrical, stable subharmonics of even order are possible [8]. However, McCrumm has reported the laboratory simulation of stable even-order subharmonics in a circuit consisting of a capacitor, an inductor and the magnetizing impedance of a transformer [2]. It is therefore
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~/
2/
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1
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\
~.,. _ d ~
\
~
-.4
.I
II/ .sll/
"~ ~,
,,
;/
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,~
\
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h
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\
\
.4
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I
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3,,,/ /,~ / 4/ # /
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11 %.
l/
Fig. 8. R e g i o n of e v e n - o r d e r (1/2) s u b h a r m o n i c s a t )~m= 1.0 p.u.: , symmetrical saturation; , asymmetric a l s a t u r a t i o n w i t h a n offset of ) . o - 0.5 p.u.: 1, y - 0 . 2 ; 2, p - 0.4; 3, I~ = 0.6; 4, ~ = 0.8 p.u.
reasonable to assume that hysteresis and such other factors may be responsible for the initiation of even-order subharmonics, since it makes the nonlinearity asymmetrical. The analytical development of the didf for a nonlinearity with hysteresis is quite complex. Numerical methods are best suited for nonlinearities with hysteresis, minor loops and discontinuities. The region of the subharmonic of order 1/2 for a symmetrical saturation nonlinearity is shown in Fig. 8. As can be expected the area of existence is very small, and therefore its initiation is difficult. When hysteresis and minor loops are considered, the region of subharmonics of order 1/2 slightly enlarges and shifts downwards. If the nonlinearity is rendered asymmetrical by a DC offset of 0.5 p.u., then the order 1/2 subharmonic region expands many times, as shown by the broken lines of Fig. 8. In fact, subharmonics of even order are easily initiated when the nonlinearity is operated in an asymmetrical mode. An asymmetrical operation of this type is possible during switching conditions due to DC components, residual magnetization, etc. Therefore, subharmonics of even order can be expected to appear in a transient mode in switching and fault signals. Reference 9 gives more information on the various factors affecting the even-order subharmonic resonance.
4.6. Ferroresonant suppression circuit design The ferroresonant suppression circuits provide an effective loading at subharmonic frequencies, and thereby reduce the possibility of resonance. A resistive loading shifts the point G(jl/3) mostly in the downward direction in the third quadrant. To eliminate both the stable and the transient subharmonics, the ferroresonant circuit should move the point as far away from the negative real axis as possible. This requires a tuned parallel circuit with a sufficiently low loading resistance in series. An inductive loading instead of a resistive loading moves the point G(jl/3) deep into the negative side of the complex plane, and shifts over to the positive side. Two simplified series ferroresonant circuits with resistive and inductive burdens are shown in Fig. 9. The series parameters are chosen such that L~ = 1 / C e , which gives G(jl/3) = 0.89R2/( - Gel9 + j0.3/R2)
29 Le
Ce
Le
Ce
(a)-
(b) Fig. 9. Simplified CVT circuits with (a) resistive and (b) inductive burdens. for resistive burden, and G(jl/3) = 0.89L2/(0.89 - GEL2) for inductive burden. The above equations are derived for a highly simplified version of the CVT, and therefore any conclusion derived should be treated with caution. Moving the operating point G(jl/3) to the right side of the complex plane appears to be easier with inductive loading. Therefore inductive burden or suppression circuits designed with inductance can be equally effective in avoiding both transient and stable subharmonic resonance. The locus of G(jl/3) for a burden of 0.65 p.u. at a lagging power factor of 0.707 is also shown in Fig. 5. Relative to a power factor of 0.8 this locus always stays below, indicating the effectiveness of inductive loading. 5. EFFECT ON HIGH SPEED RELAYING The present-day high speed relays, particularly the solid-state relays, are designed to identify and locate the faults in periods ranging from 5 to 20 ms. From the foregoing discussion, it is evident that components of subharmonics should be expected in the fault signals, particularly on systems adopting CVTs and series capacitors. When subharmonics appear during the first cycle, the net voltage and c ur r e nt will be completely different from normal, and any relay which operates based on either the magnitude or phase of the voltage and c ur r e nt would give a wrong
operation. There are a number of reported cases of relay maloperation due to this reason. Two different methods are available for solving this problem. (a) The first approach is to ensure t hat the system is not susceptible to the generation of subharmonics. This should be realized by suitable designs for the parameters of the CVTs, suppression circuits, and loading. The ferroresonant suppression circuit is probably the best method available for its reduction. However, the suppression circuits should be designed for moving the operating point G(jl/3) to the right side of the complex plane. Suggestions for the use of switched suppression circuits and saturable reactors are somewhat questionable, because subharmonic components can still appear during the first cycle after the fault. (b) The second approach to the problem is to remove the subharmonic components from the fault signals by using a band-pass filter. The filtering of the subharmonics would be feasible only if the filtered fundamental retains the principal features of protection, such as sensitivity and selectivity. However, filters designed with a sharp cutoff for low frequencies introduce substantial time delays.
5.1. Filtering compatibility As stated in the previous Section, a suitably designed band-pass filter can eliminate the subharmonics of any type, and give only the fundamental signal to the relay and measuring circuits. If the discriminative feature is to be retained, then the magnitude and phase of the fundamental voltage and current should not undergo unpredictable changes in the presence of subharmonics. In other words, if the magnitude and phase of the fundamental frequency voltage and current signals do not change in the presence of subharmonics, then the filtered signals can be used as relaying signals. If the fundamental signals undergo substantial changes, and the changes are predictable, then the signals would still be useful as relay signals. The variation of )~rn and ~, that is, the magnitude and angle of the filtered output voltage for varying loads, is shown in Fig. 10, with and without subharmonics. The Figure shows practically no change in magnitude or phase of the output voltage. Several results similar to Fig. 10 indicated a maximum magnitude deviation of about 3.0%, and a
30
0.~
I00
..~ 90
0. (
SO
I
1
0.25
J
0.5
loading
al
0.8
I
0.75
1.0
I>t ]tl~;
F i g . 10. V a r i a t i o n o f 2 m a n d ~ w i t h t h e l o a d i n g : -, without subharraonics; with subharmonics. ( E = 0.5 p . u . , C~. = 5.0 p . u . )
phase deviation of around 2 ~. Therefore, the filtered fundamental signal of the CVT should be able to give reliable discrimination. In fact, this is a natural outcome of the resonant nature of the CVT. Since L~ and Ce are tuned at the fundamental frequency, the voltage drop across them should be that due to the resistance only. The above observation does not hold for subharmonics generated from the system side. The variation of p and 0 for varying load conditions is shown in Fig. 11. The subharmonic flux linkage changes substantially with E, Ce and the loading. The corresponding subharmonic cur r ent as obtained from (1) does not behave in any predictable manner. This information has no direct bearing on the usefulness of the CVT output for use in relaying. When subharmonics are generated on the
9Q
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G
"
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~
~
7s
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region of unstable\
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F i g . 11. V a r i a t i o n , 0.
I 0.25 Loading
I 0.5 at 0.8
pf
I 0.75 lag,
of p and 0 with the loading: --
I 1.0
,P;
power network (not due to the CVT) the phase and magnitude of the subharmonic currents become unpredictable, and therefore the illtered fundamental voltage and currents would be practically useless. From a CVT, only the fundamental output voltage is needed. Its phase and magnitude are not affected when the initiation of subharmonics is by the CVT. Even though the fundamental voltage signal in the presence of subharmonic resonance retains the discriminative features of protection, the filtering requirements are quite severe. The relative magnitude of p as compared with 2m is quite large, particularly at low driving voltages corresponding to fault conditions on the system. Therefore, the filter should provide an at t enuat i on of around 40 to 50 dB at 20 Hz. Any band-pass filter designed for this requirement would give a delay of approximately 10- 15 ms. In addition, the possibility of even-order subharmonics in the transient mode makes the problem more acute, demanding the same at t enuat i on at 30 Hz, with a correspondingly higher delay.
6. C O N C L U S I O N
The coupling capacitance of the CVT is known to cause subharmonic oscillations. The tuned condition of the capacitance and the equivalent inductance makes it free from jump resonance. While the most common resonance is at 1/3 frequency, other lower order resonant frequencies are also possible. A ferroresonant suppression circuit can eliminate stable subharmonics, but an inadequate design may give subharmonics in a transient mode, with the transient persisting for considerable periods of time. A ferroresonant circuit properly designed with inductance can be equally effective in avoiding the resonance. The output of a CVT may also have even-order subharmonics in a transient mode. If a filter is adopted, the filtered fundamental output retains sufficient discriminative protective features. In a laboratory simulation of the series ferroresonant circuit on an analog computer, an amplitude-modulated almost-periodic oscillation was encountered several times. So far, the authors have not been able to establish its modes and boundaries of existence analytically.
31 ACKNOWLEDGEMENT
The authors wish to express their appreciation of the encouragement provided by Dr William A. Klos of the University of Southwestern Louisiana. NOMENCLATURE
R2, R~, L 2
=(C1 + C2)N 2, equivalent capacitance of CVT stray capacitance =Em sin(oJt), driving voltage gains of linear part of system magnetizing current of transformer didf of nonlinearity NL equivalent resistance and inductance of CVT burden parameters
0
phase of )~ with respect to E initial phase of p with respect to
Ce C4 E
G(s), Go(s) I N Re, Le
E
;.(t) /~rn P
net flux linkage at any instant amplitude of normal frequency flux linkage in iron core amplitude o f 1/3 f r e q u e n c y flux linkage in iron core
c)
frequency, rad/s
REFERENCES 1 E. J. Dolan, D. A. Gillies and E. W. Kimbark, Ferroresonance in a transformer switched with an EHV line, IEEE Trans., PAS-91 (1972) 1273 - 1280. 2 J. D. McCrumm, An experimental investigation of subharmonic currents, Trans. AIEE, 60 (1941) 533540. 3 B. S. Ashok Kumar, A. K. Tripathi, K. P a r t h a s a r a t h y and G. C. Kothari, Approach to the problem of ferroresonance in EHV systems, Proc. Inst. Electr. Eng., 119 (1972) 672 - 676. 4 G. W. Swift, An analytical approach to ferroresonance, IEEE Trans., PAS-87 (1968) 42-47. 5 J. C. West and J. L. Douce, Dual input describing function and its use in the analysis of nonlinear feed back systems, Proc. Inst. Electr. Eng., Part B, 103 (1956) 463- 473. 6 M. S. Hughes, Distance relay performance as affected by CVTs, Proc. Inst. Electr. Eng., 121 (1974) 1557 - 1566.
7 ASEA Pamphlet--RT09-119 E, Edition 1, Technical Information, Micafil Ltd., CH-8048 Zurich. 8 J. C. West, Analytical Techniques for Nonlinear Control Systems, English Universities Press, 1960. 9 B. S. Ashok Kumar and C. L. Blanchfield, Nonlinear oscillations in power networks, Proc. Int. Conf. Computers, Systems and Signal Processing, 1984, Inst. Eng. India, Bangalore, pp. 1233-1236. 10 J. W. Teape, R. D. Slater R. R. S. Simpson and W. S. Wood, Hysteresis effects in transformers including ferroresonance, Proc. Inst. Electr. Eng., 123 (1976) 153 - 157.
23
Capacitor Voltage Transformer Induced Ferroresonance--Causes, Effects and Design Considerations B. S. A S H O K K U M A R a n d S U A T E R T E M
Department of Electrical and Computer Engineering, University of Southwestern Louisiana, Lafayette, LA 70504 (U.S.A.) (Received J u l y 25, 1990)
ABSTRACT
The combination of the equivalent coupling capacitance and the nonlinear magnetizing inductance of the isolating transformer of a capacitor voltage transformer (CVT) can cause a variety of nonlinear oscillations. In such networks, odd-order subharmonics can appear as stable oscillations, whereas subharmonics of any order can appear as transient oscillations. Such oscillations are known to interfere with the operation of high speed relays. This paper discusses the different types of oscillations that are possible in ferroresonant circuits, the conditions under which they appear, their effect on high speed relays, and the methods for minimizing their influence. It is shown that for a C V T the fundamental frequency voltage signal retains the discriminative features of protection, even in the presence of subharmonic resonance.
1. I N T R O D U C T I O N
The problem of nonlinear voltage oscillations in power networks has gained importance due to a number of failures of transformers and protective systems. The elements which contribute to the initiation of such oscillations are the iron cores of the power transformers and the capacitor voltage transformers (CVTs). When the nonlinear magnetizing reactance of a power or instrument transformer appears in series with an effective capacitance, a series ferroresonant circuit is formed. Such a circuit can exhibit a variety of nonlinear phenomena, such as: (a) jump resonance, (b) subharmonic resonance of even and odd o r d e r - - e i t h e r in a stable mode or in a transient mode, and 0378-7796/91/$3.50
(c) amplitude-modulated almost-periodic oscillation. Several factors, such as driving voltage, extent of loading, type of disturbance, asymmetry due to the disturbance, resistance, hysteresis, etc., influence the nat ure of such oscillations. These oscillations are known to produce elevated voltages and currents which result in failure of equipment and protective systems [1, 2]. The elevated voltages and currents are the result of the resonant nat ure of the oscillations. Further, the nonlinear nat ure of the oscillations substantially affects the magnitude and phase of the fundamental frequency signals, thereby influencing the operation of the high speed protective devices. Even in the transient mode, these oscillations can persist for several cycles and cause serious malfunctioning of the relays. It is therefore necessary t hat the general features of protection such as speed and discrimination be reexamined in the presence of nonlinear oscillations. The different types of oscillations are exhibited under varying conditions, and their areas of existence are known to overlap each other. The subsequent sections analyze and examine the various types of nonlinear oscillations, effect of different parameters, their influence on high speed protection, and possible design considerations.
2. C A P A C I T O R V O L T A G E T R A N S F O R M E R (CVT)
The advantages of cost, size and simplicity has made the CVT a very valuable device for metering and protection. The CVT and its equivalent circuit, referred to the low voltage side (115 V), are shown in Fig. l(a) and (b). In the equivalent circuit, NL represents the nonlinearity due to the magnetizing impedance of ~:~ E l s e v i e r S e q u o i a / P r i n t e d in T h e N e t h e r l a n d s
24
ship of the n o n l i n e a r i t y is s h o w n in Fig 2. It is r e p r e s e n t e d by the fifth-degree p o l y n o m i a l I = al)~ + a:~)5~+ as)~ '~
I
C1 t
= 0.016057. - 0.0168)~ 3 + 0.052)o '~
R
(1)
T V~
R
C
_[ N:I
"
(a)
C
3. JUMP RESONANCE
I:~jrO~n
L
NiILt (b) Fig. 1. (a) The CVT and (b) its equivalent circuit parameters. the i s o l a t i n g t r a n s f o r m e r . Le includes the tuning i n d u c t a n c e and the l e a k a g e i n d u c t a n c e of the t r a n s f o r m e r . The e q u i v a l e n t c a p a c i t a n c e Ce is t u n e d w i t h the i n d u c t a n c e Le. V a r i o u s b u r d e n i m p e d a n c e s are c o n s i d e r e d to represent v a r y i n g b u r d e n s up to 250 VA at p o w e r f a c t o r s of u n i t y and 0.8 lagging. All r e s u l t s a r e p r e s e n t e d on a b a s e of l 1 5 V , 2 0 0 V A and 377 rad/s. T h e c u r r e n t - f l u x l i n k a g e relation-
1.5
r~
U n d e r c e r t a i n c o m b i n a t i o n s of n e t w o r k p a r a m e t e r s the v o l t a g e and c u r r e n t Jn a ferr o r e s o n a n t circuit s u d d e n l y j u m p s to h i g h e r levels [3, 4]. This c o n d i t i o n is g e n e r a l l y associated w i t h s u b s t a n t i a l odd-order h a r m o n i c s and a f u n d a m e n t a l at the d r i v i n g frequency. A f e r r o r e s o n a n t circuit goes into the j u m p r e s o n a n c e m o d e at h i g h levels of d r i v i n g voltages, or due to s w i t c h i n g o p e r a t i o n s at low levels of d r i v i n g voltages. W h e n such a c o n d i t i o n prevails, the r e s u l t i n g o v e r v o l t a g e s and h i g h c u r r e n t s c a n d a m a g e the e q u i p m e n t . M a n y f a c t o r s d e t e r m i n e the c o n d i t i o n s u n d e r w h i c h j u m p r e s o n a n c e c a n occur. In a series f e r r o r e s o n a n t circuit the j u m p r e s o n a n c e c a n o c c u r w h e n the effective series c a p a c i t i v e rea c t a n c e is m o r e t h a n the i n d u c t i v e r e a c t a n c e of the n e t w o r k at a d r i v i n g frequency. Ano t h e r f a c t o r w h i c h c a n c o n t r i b u t e to the j u m p r e s o n a n c e is a low p o w e r f a c t o r c a p a c i t i v e load. T h e s e c o n d i t i o n s are g e n e r a l l y not satisfied by the CVT. T h e r e f o r e , the possibility of the CVT c a u s i n g the j u m p r e s o n a n c e is quite remote. On the o t h e r hand, u n l o a d e d series compens a t e d t r a n s m i s s i o n lines with i n t e r m e d i a t e load tapping, and u n b a l a n c e d a n d u n g r o u n d e d d i s t r i b u t i o n s y s t e m s satisfy the c o n d i t i o n s for jump resonance.
4. SUBHARMONIC OSCILLATIONS
1.0
c~
0.5
I 0.5
I 1.0 I
in
1 1.5
pu
Fig. 2. M a g n e t i z i n g c h a r a c t e r i s t i c s of t h e i s o l a t i n g transformer.
T h e series f e r r o r e s o n a n t circuit c a n go into a m o d e of s u b h a r m o n i c o s c i l l a t i o n s o v e r a v e r y wide r a n g e of n e t w o r k p a r a m e t e r s . D u r i n g s u c h oscillations, the v o l t a g e a n d c u r r e n t s will h a v e l a r g e c o m p o n e n t s of f r e q u e n c i e s below the f u n d a m e n t a l frequency. The s u b h a r m o n i c freq u e n c i e s are g e n e r a l l y 1/3, 1/5, 1/7, etc., of the d r i v i n g frequency. T h e s e odd-order s u b h a r m o n ics a p p e a r e i t h e r as stable o s c i l l a t i o n s or as t r a n s i e n t s . T h e following Sections detail the different f a c t o r s w h i c h influence the subharmonic resonance.
25
4.1. Mathematical analysis The block diagram c o r r e s p o n d i n g to the series f e r r o r e s o n a n t circuit of Fig. l(b) is given in Fig. 3, w h e r e
of the o u t p u t s u b h a r m o n i c to the input subh a r m o n i c gives the didf [6], as
N = al + 3a3(P 2 + 2)~m)/4 -1-5a5(~t 4 -4- 6~t2)~m2 -f- 3)~m4)/8
G(S) = R3(S2LeC~ + S R e C e -t- ] ) ( R 2 + S L 2) --p[5as).m(2p e + 3).m2)/8
× [SC~R.3(R2 + SL2) A- S(S2LeC~ +SReCe+I)(R3+R2+SL2)I
1
(2)
+ 3aa).m/4] exp[ --j(30 -- ~)] --5a5~3~.m exp[j(30 - ~)]
and the n o n l i n e a r i t y NL is defined by (1). The m e t h o d adopted for a n a l y z i n g the subharmonic oscillation utilizes the dual-input describing f u n c t i o n (didf) as suggested by West and D o u c e [5]. Since the s u b h a r m o n i c oscillations h a v e p r e d o m i n a n t l y two f r e q u e n c y components, it is n e c e s s a r y to d e t e r m i n e the response of the n o n l i n e a r element to two s i m u l t a n e o u s inputs of different frequencies. The effective gain of the n o n l i n e a r i t y to one f r e q u e n c y in the presence of the o t h e r is referred to as the didf. In the i n v e s t i g a t i o n of a s u b h a r m o n i c whose f r e q u e n c y is of order l/n, a signal )'m sin(~t) is r e g a r d e d as predetermined, and the didf to a second input p sin(cgt/n + 0) is superposed on the N y q u i s t locus of the l i n e a r part G(ju~/n) of the system. The v a l u e of p at which the n e g a t i v e r e c i p r o c a l of the didf intersects the N y q u i s t locus gives the amplitude c o r r e s p o n d i n g to an overall loop gain of u n i t y in the feedback system. A c o m b i n a t i o n of 2~m and p c o r r e s p o n d i n g to the point of intersection gives the possible steady-state oscillation. Let the driving voltage be E = Em sin(t~t) If the input to the n o n l i n e a r i t y is assumed to be ).(t) = p sin(~ot/3 + 0) + Amsin(~ot + ~)
(3)
t h e n the didf for the s u b h a r m o n i c of order 1/3 can be d e t e r m i n e d by s u b s t i t u t i n g (3) in (1). N e g l e c t i n g h i g h e r o r d e r harmonics, the r a t i o
G(s)
Fig. 3. Block diagram representation of the CVT.
(4)
The c o n d i t i o n for the existence of stable subh a r m o n i c oscillations is G(j~o/3) = - 1IN
(5)
The values of g, )°m, 0 and ~ at which the c o n d i t i o n (5) is satisfied defines the subharmonic response. At the j u n c t i o n of the block diagram, the r e l a t i o n s h i p for the signals of f u n d a m e n t a l f r e q u e n c y may be established as follows: a l + a3 {3()~m2 A- 2tt2)/4
_ p 3 exp[j(30 - a)]/(4).m) } -- 5).mp 3 e x p [ - - j ( 3 0 -- ~)]/8 -4-a5 [5().m 4 -~- 61/2)~m2 -~- 3~t4)/8 -- 5~ 3(~ 2 _~ 4)~m2)/( 162m )]
× exp[j(30 -- ~)] = - 1/G(j(9) + (Wm/2m)lGo(je))/G(j~o)] x exp[j(0 - a)]
(6)
w h e r e 0 is the angle of Go(j(o)/G(jo)). The s u b h a r m o n i c response can be o b t a i n e d a n a l y t i c a l l y by solving eqns. (5) and (6) by any i t e r a t i v e method.
4.2. Region of stable subharmonics The locus of - 1 / N on the complex plane for )'m = 1.0 p.u., which corresponds approxim a t e l y to a driving voltage of 1.0p.u., is shown in Fig. 4. The area w i t h i n the envelope gives the region of existence of stable subharmonics. F r o m a study of several such diagrams it was found t h a t for a specific G(j~/3) = G(jl/3), the amplitude of subharmonic oscillation changes only w i t h i n a narrow r a n g e for different 2m. Therefore, even at low driving voltages, the response will h a v e large s u b h a r m o n i c amplitudes. In all cases, the locus of - 1 / N shifts to the r i g h t in the second and third q u a d r a n t s , and c o n v e r g e s at the origin for large values of g. If the t r a n s f e r
26
z
I
•
~eal
G(s)
-2
--4
Fig. 4. R e g i o n of 1/3 f r e q u e n c y s u b h a r m o n i c s for t he CVT a t )-m -- 1.0 p.u.
function G(jl/3) of the linear part of the system is very close to the origin in the third quadrant, the circuit can oscillate with high values of #. As the operating point G(jl/3) moves away from the origin in the third quadrant, the amplitude of the subharmonic oscillation reduces and ultimately the oscillations disappear. When G(jl/3) is outside the envelope, there is no possibility of stable subharmonics of order 1/3. The region of subharmonics spreads out on the negative side of the complex plane within an angle of approximately (180 _+ 27) degrees. In addition, the regions of higher order subharmonics overlap, and some of them are slightly wider. For any one set of parameters of the CVT, several subharmonic responses of different frequencies are therefore possible. So far, it has not been possible to predict which responses would prevail for a given set of parameters. The particular response depends not only on the parameters, but also on the conditions at the time of switching and the magnitude of the driving voltage. The locus of G(jl/3) for the CVT is shown in Fig. 5. The loci are drawn for different values of load burden and equivalent capacitance Ce. Superposed on the same diagram are the envelopes of the didf for different 2m. The most favorable parameter conditions for the appearance of subharmonic resonance are large values of C~ and low burden. Increased values of the equivalent resistance Re slightly shift the G(jl/3) point downwards, and hence have a marginal tendency to reduce the susceptibility to resonance. The area covered by - 1 / N is by far the largest for moderate levels of 2m. Therefore a remote fault on the power network is the most favorable operating con-
dition for the CVT to cause subharmonic resonance. In addition, a minimum permanent load can greatly assist in the prevention of resonance. The ferroresonance suppression circuit providing effective loading at subharmonic frequencies is a good method available for avoiding the subharmonics. Design of a CVT with lower equivalent capacitance can further ease the problem.
4.3. Transient subharmonics If the operating point G(j 1/3) of a ferroresonant circuit is outside the region of subharmonics, then stable oscillations cannot appear. However, this is not an adequate guarantee that components of subfrequencies do not appear during transient conditions. This is particularly true when G(jl/3) is in the third quadrant--outside, but in the vicinity of the region of stable subharmonics. Factors like hysteresis, asymmetrical operation of the nonlinearity due to DC components, parameter dependence on frequency and temperatures, changes in equivalent capacitance, etc., are responsible for causing substantial subharmonics in a transient mode. A ferroresonant suppression circuit may apparently solve the problem of stable subharmonics, but still may not be able to prevent the failure of high speed protection due to transient subharmonics. A previously published test report on a CVT with a ferroresonant suppression circuit appears to show subharmonics during the transients [7]. To illustrate the above point, the results from a series ferroresonant circuit are presented in Figs. 6 and 7. The oscillograms of Figs. 6(a) and 7(a) show 1/3 frequency
27 Im -3.0
-2.0
!
!
-1.0 '-'
v~
G(s)
/
/ i ),=
/
/
/
/
/ J
/ -1.0
O. &
I I
u~
/.?"
,E) _
t',-
I I I I I I
I
-2.0
\ \ \
\
\
\ \
Fig. 5. L o c u s of G(jl/3) for different C e a n d burden: C , , ; - - , e n v e l o p e s of -1/N.
-3.0
, c o n s t a n t b u r d e n at 0.8 l a g g i n g power factor;
, constant
%(a)
(b)
(c)
(d)
Fig. 6. T r a n s i e n t s u b h a r m o n i c s at m o d e r a t e voltages.
(a)
(b)
(c)
(d)
Fig. 7. T r a n s i e n t s u b h a r m o n i c s at low voltages.
2~
monics in the steady state. After initiation of the oscillations, the circuit parameters and the driving voltage were sufficiently changed, beyond the point of disappearance of stable resonance. The several other transients presented are taken under these conditions. The following points are based on the observation of a number of such transients. (a) The decaying rate of the transient subharmonics does not remain constant and varies considerably for the same set of network parameters. (b) For the same network condition, harmonics of order 1/3, 1/5, 1/7, etc., or even a combination, may appear as transients. In such cases, the higher frequency components decay faster. (c) Transient subharmonics are possible even at very high values of series resistance. (d) Substantially large levels of subharmonics may appear even at low driving voltages.
4.5. Even-order transient subharmonics The literature documents that even-order subharmonics cannot exist in ferroresonant circuits with symmetrical saturation types of nonlinearities. When the saturation is asymmetrical, stable subharmonics of even order are possible [8]. However, McCrumm has reported the laboratory simulation of stable even-order subharmonics in a circuit consisting of a capacitor, an inductor and the magnetizing impedance of a transformer [2]. It is therefore
/
//
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J
~/
2/
/ / /
/
/
I
/
/
/
/
\ \
\
/
1
\ \
\
\
\
\
\
~.,. _ d ~
\
~
-.4
.I
II/ .sll/
"~ ~,
,,
;/
\
\
~'~
,~
\
%
4
h
\
\~.
"2 3
?'0
\
\
.4
\
\
\
\ \
~,\
I
/ '
X
\ \
I
/ Z
\
3,,,/ /,~ / 4/ # /
/
11 %.
l/
Fig. 8. R e g i o n of e v e n - o r d e r (1/2) s u b h a r m o n i c s a t )~m= 1.0 p.u.: , symmetrical saturation; , asymmetric a l s a t u r a t i o n w i t h a n offset of ) . o - 0.5 p.u.: 1, y - 0 . 2 ; 2, p - 0.4; 3, I~ = 0.6; 4, ~ = 0.8 p.u.
reasonable to assume that hysteresis and such other factors may be responsible for the initiation of even-order subharmonics, since it makes the nonlinearity asymmetrical. The analytical development of the didf for a nonlinearity with hysteresis is quite complex. Numerical methods are best suited for nonlinearities with hysteresis, minor loops and discontinuities. The region of the subharmonic of order 1/2 for a symmetrical saturation nonlinearity is shown in Fig. 8. As can be expected the area of existence is very small, and therefore its initiation is difficult. When hysteresis and minor loops are considered, the region of subharmonics of order 1/2 slightly enlarges and shifts downwards. If the nonlinearity is rendered asymmetrical by a DC offset of 0.5 p.u., then the order 1/2 subharmonic region expands many times, as shown by the broken lines of Fig. 8. In fact, subharmonics of even order are easily initiated when the nonlinearity is operated in an asymmetrical mode. An asymmetrical operation of this type is possible during switching conditions due to DC components, residual magnetization, etc. Therefore, subharmonics of even order can be expected to appear in a transient mode in switching and fault signals. Reference 9 gives more information on the various factors affecting the even-order subharmonic resonance.
4.6. Ferroresonant suppression circuit design The ferroresonant suppression circuits provide an effective loading at subharmonic frequencies, and thereby reduce the possibility of resonance. A resistive loading shifts the point G(jl/3) mostly in the downward direction in the third quadrant. To eliminate both the stable and the transient subharmonics, the ferroresonant circuit should move the point as far away from the negative real axis as possible. This requires a tuned parallel circuit with a sufficiently low loading resistance in series. An inductive loading instead of a resistive loading moves the point G(jl/3) deep into the negative side of the complex plane, and shifts over to the positive side. Two simplified series ferroresonant circuits with resistive and inductive burdens are shown in Fig. 9. The series parameters are chosen such that L~ = 1 / C e , which gives G(jl/3) = 0.89R2/( - Gel9 + j0.3/R2)
29 Le
Ce
Le
Ce
(a)-
(b) Fig. 9. Simplified CVT circuits with (a) resistive and (b) inductive burdens. for resistive burden, and G(jl/3) = 0.89L2/(0.89 - GEL2) for inductive burden. The above equations are derived for a highly simplified version of the CVT, and therefore any conclusion derived should be treated with caution. Moving the operating point G(jl/3) to the right side of the complex plane appears to be easier with inductive loading. Therefore inductive burden or suppression circuits designed with inductance can be equally effective in avoiding both transient and stable subharmonic resonance. The locus of G(jl/3) for a burden of 0.65 p.u. at a lagging power factor of 0.707 is also shown in Fig. 5. Relative to a power factor of 0.8 this locus always stays below, indicating the effectiveness of inductive loading. 5. EFFECT ON HIGH SPEED RELAYING The present-day high speed relays, particularly the solid-state relays, are designed to identify and locate the faults in periods ranging from 5 to 20 ms. From the foregoing discussion, it is evident that components of subharmonics should be expected in the fault signals, particularly on systems adopting CVTs and series capacitors. When subharmonics appear during the first cycle, the net voltage and c ur r e nt will be completely different from normal, and any relay which operates based on either the magnitude or phase of the voltage and c ur r e nt would give a wrong
operation. There are a number of reported cases of relay maloperation due to this reason. Two different methods are available for solving this problem. (a) The first approach is to ensure t hat the system is not susceptible to the generation of subharmonics. This should be realized by suitable designs for the parameters of the CVTs, suppression circuits, and loading. The ferroresonant suppression circuit is probably the best method available for its reduction. However, the suppression circuits should be designed for moving the operating point G(jl/3) to the right side of the complex plane. Suggestions for the use of switched suppression circuits and saturable reactors are somewhat questionable, because subharmonic components can still appear during the first cycle after the fault. (b) The second approach to the problem is to remove the subharmonic components from the fault signals by using a band-pass filter. The filtering of the subharmonics would be feasible only if the filtered fundamental retains the principal features of protection, such as sensitivity and selectivity. However, filters designed with a sharp cutoff for low frequencies introduce substantial time delays.
5.1. Filtering compatibility As stated in the previous Section, a suitably designed band-pass filter can eliminate the subharmonics of any type, and give only the fundamental signal to the relay and measuring circuits. If the discriminative feature is to be retained, then the magnitude and phase of the fundamental voltage and current should not undergo unpredictable changes in the presence of subharmonics. In other words, if the magnitude and phase of the fundamental frequency voltage and current signals do not change in the presence of subharmonics, then the filtered signals can be used as relaying signals. If the fundamental signals undergo substantial changes, and the changes are predictable, then the signals would still be useful as relay signals. The variation of )~rn and ~, that is, the magnitude and angle of the filtered output voltage for varying loads, is shown in Fig. 10, with and without subharmonics. The Figure shows practically no change in magnitude or phase of the output voltage. Several results similar to Fig. 10 indicated a maximum magnitude deviation of about 3.0%, and a
30
0.~
I00
..~ 90
0. (
SO
I
1
0.25
J
0.5
loading
al
0.8
I
0.75
1.0
I>t ]tl~;
F i g . 10. V a r i a t i o n o f 2 m a n d ~ w i t h t h e l o a d i n g : -, without subharraonics; with subharmonics. ( E = 0.5 p . u . , C~. = 5.0 p . u . )
phase deviation of around 2 ~. Therefore, the filtered fundamental signal of the CVT should be able to give reliable discrimination. In fact, this is a natural outcome of the resonant nature of the CVT. Since L~ and Ce are tuned at the fundamental frequency, the voltage drop across them should be that due to the resistance only. The above observation does not hold for subharmonics generated from the system side. The variation of p and 0 for varying load conditions is shown in Fig. 11. The subharmonic flux linkage changes substantially with E, Ce and the loading. The corresponding subharmonic cur r ent as obtained from (1) does not behave in any predictable manner. This information has no direct bearing on the usefulness of the CVT output for use in relaying. When subharmonics are generated on the
9Q
4.0
"
G
"
\
\-
\
\ a
~
~
7s
I.o
~
/
X
region of unstable\
\ I/3rd ~bha=~nio
\ %
70,
O. 0
F i g . 11. V a r i a t i o n , 0.
I 0.25 Loading
I 0.5 at 0.8
pf
I 0.75 lag,
of p and 0 with the loading: --
I 1.0
,P;
power network (not due to the CVT) the phase and magnitude of the subharmonic currents become unpredictable, and therefore the illtered fundamental voltage and currents would be practically useless. From a CVT, only the fundamental output voltage is needed. Its phase and magnitude are not affected when the initiation of subharmonics is by the CVT. Even though the fundamental voltage signal in the presence of subharmonic resonance retains the discriminative features of protection, the filtering requirements are quite severe. The relative magnitude of p as compared with 2m is quite large, particularly at low driving voltages corresponding to fault conditions on the system. Therefore, the filter should provide an at t enuat i on of around 40 to 50 dB at 20 Hz. Any band-pass filter designed for this requirement would give a delay of approximately 10- 15 ms. In addition, the possibility of even-order subharmonics in the transient mode makes the problem more acute, demanding the same at t enuat i on at 30 Hz, with a correspondingly higher delay.
6. C O N C L U S I O N
The coupling capacitance of the CVT is known to cause subharmonic oscillations. The tuned condition of the capacitance and the equivalent inductance makes it free from jump resonance. While the most common resonance is at 1/3 frequency, other lower order resonant frequencies are also possible. A ferroresonant suppression circuit can eliminate stable subharmonics, but an inadequate design may give subharmonics in a transient mode, with the transient persisting for considerable periods of time. A ferroresonant circuit properly designed with inductance can be equally effective in avoiding the resonance. The output of a CVT may also have even-order subharmonics in a transient mode. If a filter is adopted, the filtered fundamental output retains sufficient discriminative protective features. In a laboratory simulation of the series ferroresonant circuit on an analog computer, an amplitude-modulated almost-periodic oscillation was encountered several times. So far, the authors have not been able to establish its modes and boundaries of existence analytically.
31 ACKNOWLEDGEMENT
The authors wish to express their appreciation of the encouragement provided by Dr William A. Klos of the University of Southwestern Louisiana. NOMENCLATURE
R2, R~, L 2
=(C1 + C2)N 2, equivalent capacitance of CVT stray capacitance =Em sin(oJt), driving voltage gains of linear part of system magnetizing current of transformer didf of nonlinearity NL equivalent resistance and inductance of CVT burden parameters
0
phase of )~ with respect to E initial phase of p with respect to
Ce C4 E
G(s), Go(s) I N Re, Le
E
;.(t) /~rn P
net flux linkage at any instant amplitude of normal frequency flux linkage in iron core amplitude o f 1/3 f r e q u e n c y flux linkage in iron core
c)
frequency, rad/s
REFERENCES 1 E. J. Dolan, D. A. Gillies and E. W. Kimbark, Ferroresonance in a transformer switched with an EHV line, IEEE Trans., PAS-91 (1972) 1273 - 1280. 2 J. D. McCrumm, An experimental investigation of subharmonic currents, Trans. AIEE, 60 (1941) 533540. 3 B. S. Ashok Kumar, A. K. Tripathi, K. P a r t h a s a r a t h y and G. C. Kothari, Approach to the problem of ferroresonance in EHV systems, Proc. Inst. Electr. Eng., 119 (1972) 672 - 676. 4 G. W. Swift, An analytical approach to ferroresonance, IEEE Trans., PAS-87 (1968) 42-47. 5 J. C. West and J. L. Douce, Dual input describing function and its use in the analysis of nonlinear feed back systems, Proc. Inst. Electr. Eng., Part B, 103 (1956) 463- 473. 6 M. S. Hughes, Distance relay performance as affected by CVTs, Proc. Inst. Electr. Eng., 121 (1974) 1557 - 1566.
7 ASEA Pamphlet--RT09-119 E, Edition 1, Technical Information, Micafil Ltd., CH-8048 Zurich. 8 J. C. West, Analytical Techniques for Nonlinear Control Systems, English Universities Press, 1960. 9 B. S. Ashok Kumar and C. L. Blanchfield, Nonlinear oscillations in power networks, Proc. Int. Conf. Computers, Systems and Signal Processing, 1984, Inst. Eng. India, Bangalore, pp. 1233-1236. 10 J. W. Teape, R. D. Slater R. R. S. Simpson and W. S. Wood, Hysteresis effects in transformers including ferroresonance, Proc. Inst. Electr. Eng., 123 (1976) 153 - 157.