Electric Power Systems Research, 7 (1984) 271 - 277 271

D a m p i n g E f f e c t o f L o a d o n t h e F e r r o r e s o n a n c e P h e n o m e n o n in P o w e r

N e t w o r k s

M. M. SAIED*, H. M. ABDALLAH and A. S. ABDALLAH

Electrical Engineering Department, Faculty of Engineering, Assiut University, Assiut (Egypt)

(Received December 5, 1983)

SUMMARY

A generalized approach to the ferroreso- nance phenomenon, occasionally resulting from power transformers connected to transmission systems, is given. The effect of the transformer load current on the ferroreso- nance phenomenon at different load power factors is studied. A second-order equation determining the value of the critical load impedance at any load power factor is de- rived. Also, the effect of the transmission line parameters and the saturation degree of the transformer magnetization characteristic on the value of the critical load current and hence on the ferroresonance phenomenon is shown.

1. INTRODUCTION

As power transmission distances have increased, new problems, both technical and economic, have been encountered. As trans- mission distances continue to increase, it is expected that, in addition to problems which have already been exposed, new ones may arise which previously have not required consideration.

One of the most serious problems in electrical power systems is the possibility of overvoltages resulting from a ferroresonant condition between the system capacitances and the nonlinear inductance of the trans- former magnetization characteristic. This phenomenon has been studied extensively by many investigators in the last 50 years [1 - 6]. From these papers, the ferroresonance phe- nomenon can be defined as a special case of jump resonance in which the nonlinearity is the transformer iron core magnetization

*Present address: Department of Electrical and Computer Engineering, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Kuwait.

curve. Jump resonance refers to circuits comprising a nonlinear inductor and a linear capacitor with or without other elements present. Investigators have used different approaches in studying the ferroresonance phenomenon, such as the incremental describ- ing function (IDF) method, which is used by Maklad and Zaky [3], or by studying the phenomenon in the G_-l(jco) plane, which was also clone by Pmsty and Sanayal [4 - 6].

In all the previous papers [1 6], the authors studied the ferroresonance on a general nonlinear reactor [ 1] or an unloaded transformer [ 2 - 6]. As in the actual power systems, the power transformers are loaded most of the time. So, it is important to study the effect of the load on this phenomenon; this is the main object of this work. Here, the load has been considered with regard to predicting the possibility of the ferroreso- nance phenomenon in the power system under load conditions. The load is considered by studying the circuit of Fig. l(a). It illus- trates a transmission line represented by its exact equivalent 1r-section connected to a loaded power transformer, with all param- eters referred to its high voltage side. Also, a new generalized second-order equation giving the critical load at any power factor for any power system circuit parameters is derived. The effect of the line parameters and the

R t

%It , ~ t cl

: !

. • L ! it I1 i2 R2 L2

i,-Td ,oT J fl t i l t i~

| . ! I i3

R 3

L 3

Vol tage j T .L . ' _ .j_ Transformer _L Load Sourc. l - T T

Fig. l(a). Equivalent circuit of system considered, with CI = C~ + C i and R e = rc/(Ggr c + 1).

0378-7796/84/$3.00 © Elsevier Sequoia/Printed in The Netherlands

272

degree of transformer core saturation on the critical load current are also studied.

2. METHOD OF ANALYSIS

The block diagram of the suggested system, which is derived in Appendix A, is shown in Fig. l (b) , where:

/ f

Go(s)

+ ! : , ( :a) X ( 8C1 Re + R, + sL ) ]1-1

G(s) = (R 1 +sL1)Is[I+(R 1 + sLI) X

( : :)II-' X s C I + ~ + R , Rc + sL' . ( lb)

and N(s) = Im(S)/F(s) is the nonlinear relation between the transformer core flux linkage F(s) and the magnetization current Ira(s). E°(~ Fig. l(b). Block diagram of system.

2.1. Analysis on the G_-lOco) plane The nonlinearity of the transformer charac-

teristic is represented by [ 2 ]

im(t ) = Clof(t) + C,fn(t) n = 3, 5, 7 . . . .

The envelope equation is given by

y = --(n -- 1)(x + Clo)/2~/n

Putting

2~/n C N -

n - - 1

then

y = --(x + Clo)/CN (2)

where

co2L1 co2L' X ---- - -co2C 1 + +

R12 + co2L12 R '2 + co2Lr2

coR 1 co coR' + - - + (3b)

Y - R12 + co2LI2 Rc R '2 + co2L'2

L' = L2 + L3

and

R ' = R 2 + R a

By letting L' = R ' = 0% the values of x and y are found to be the same as those of Prusty and Sanayal [6] for the special case of no load.

From Fig. l (b) ,

Im(S) = N'F(s) (4)

and

IEo/Fml = I[1 + N'G_(jco)]/G_o(jco)I (5)

where

N ' = Clo + (5/8)CsFm 4 (6)

From eqns. (1), (4), (5) and (6), it follows that

64Eo 2

25Cs2(R12 + c02L12)

16 = Fm 10 + - - C5(x + Clo)Fm 6 +

5

64 + -- Cs2[(x + Clo) 2 + y2]Fm2

25

This equation, which relates the supply volt- age E0 to the maximum flux linkage Fro, is similar to that derived by Prusty and Sanayal [5, 6] for no-load conditions but with differ- ent values of x and y. The roots of this equation, and hence the jump points Fro1, Fm2 , Fm3 , Fro4 , E01 , and E02 , c a n be obtained in the same way as in refs. 5 and 6.

2.2. Critical parameters of the transmission line

The critical parameters of the transmission line can be obtained by substituting the values of x and y from eqns. (3) in eqn. (2) which gives:

CNR1 + coL1 C1° + A 1 = 0 (7) --C1 + co(R12 + co2L12 ) + co2

where

CN CNR' + coL' - + ( 8 )

(3a) AI coRc co(R, 2 + co2L,2)

The critical value Ce of the capacitance C1 is obtained b y solving eqn. (7) for C1, i.e.

CI0 CNR 1 + coL1 C c = A I + +

co: co(R12 + 032L12)

Equation (7) can be writ ten as

A 2 R I 2 + (CN/CO)R 1 + L 1 + c02A2L12 = 0 (9)

where

A 2 = A 1 - - C 1 + ClO/CO 2

By solving eqn. (9) for R1 (orLl), the critical values of the transmission line resistance (or inductance) can be obtained.

2.3. Critical load current ILc The value of the critical load impedance

ZLc can be obtained by solving the equation

KZL 2 + ZL[(2KR2 -- CN) cos ~b +

+ (2KX2 -- 1) sin ~] + K(R2 2 + X2 2) +

- - CNR2 --X2 = 0 (10)

where

Clo CN CNR l + coL1 K = o~C I

Re RI 2 + co2LI 2

Equation (10) is derived in Appendix B, and, knowing the value of ZLo the value of the critical load current as a percentage of the full load current IFL can be obtained:

ILc/IFL ~ (ZFL/ZLc) X 100% (11)

Equations (10) and (11) give the value of the critical load current at any power factor in terms of the circuit parameters.

3. APPLICATION AND RESULTS

3.1. System data As an example showing the effect o f load

current on the ferroresonance phenomenon, a system composed of a loaded transformer connected to a long transmission line is considered. The equivalent circuit o f this system is shown in Fig. l (a) , and the values of the system parameters referred to the high voltage side are given in Table 1.

3.2. Ef fect o f load current at cos ¢J = 0.8 lag F i g u r e s 2 ( a ) a n d (b ) s h o w t h a t , as I L in-

creases from 0 to 100% of IFL, the possibility of the occurrence of the ferroresonance

273

TABLE 1

System parameters referred to high voltage side, ¢o = 1 p.u.

System parameters Values (p.u.)

R l 0.0008836 L1 0.0099278 R2 0.0 L2 0.0015075 R3 0.00804 L3 0.00603 Ce 139.46 G~ 21.7253 r c 1.3397 C i 0.12252 C~ 139.58312 R e 0.0445 ZFL 0.01005 C1o 1 Cs 4.3417892 n 5

The load parameters are given for a lagging power factor of 0.8.

Fm P.U. 2 t N o - L ~

°'IIi~ ~ ~) 1.2 i,c i,6 i.li 2. 0/- I

Eo P.U. Fig. 2(a). Relation between F m and E0 at cos ~ = 0.8 lag.

~ Fm P.U.

2.01

1.2 I L / I F , L.

0 I 2 3 t S t~ 9%

Fig. 2(b). Relat ion between Fro, Fro2, Fm3, Fro4 and I L at cos ~b ffi 0.8 lag.

phenomenon has already disappeared at a value of load current equal to ILc = 5.77% of IFL, and no jump can occur at higher values.

274

3.3. Effect o f load current at different power factors

Figure 3 can be obtained by varying the values of the load parameters due to different power factors and drawing the variation of the jump points with the load current in each case. In this Figure, it is clear that the value of IL¢, and hence the possibility of the occur- rence of the ferroresonance phenomenon, depends mainly on the power factor. The locus of ILc can be predicted, as shown by the dashed curve.

. . . P ' U "

/ U . e - L o g

2.1.82 ~ ~ . .

"°t " i i L I 1 l I J 2 t,'/o ' 0"60 4 8 1~2 t' 6 2~0 I L/I F.~.

Fig. 3. Effect of load current on the jump points at d i f f e r en t p o w e r factors .

3.4. Effect o f load current on jump severities and percentage unstable zone

In Ref. 5, it is defined that:

the percentage unstable zone (PUZ)

Fm2-- Fml - × 100%

Fm,

the jump-up severity - Fro3 - - F m l

× 100%

F m 2 - - F m 4 the jump-down severity =

Fro2 X 100%

Figure 4 shows that increasing the load current IL will decrease the PUZ at all load power factors except in the case of zero power factor loading, at which the PUZ will

I P .UZ. ° /o

32 -0 - L e a d

0 m i i ~. i , i i I " ! , i i i ,~ • ' 0 l, 8 12 16 20 2 ; 2 6 %

Fig. 4. Ef fec t o f load c u r r e n t on the pe rcen tage u n s t a b l e zone at d i f f e ren t p o w e r factors .

increase. For each power factor, the PUZ equals zero for the critical values of the load current ILc given in Table 2. The above observations are also found to be valid for the jump-up and jump-down severities.

3.5. Determination o f ILc at any load power factor

From eqns. (10) and (11), Fig. 5 can be obtained, which gives the value of ILc at any load power factor. For zero lead power factor, it gives a value for ILc equal to 667% of/FL"

.jotlLC / IF.L. 190~00 h Lead. • Log

H P.F. P.F.

70

60

so i

- 6 0 o 5 0 -30 -10 10 30 5 0 70 9 0 Load impedance

a n g t e

Fig. 5. Relation between ILc and the load impedance angle ~b.

T A B L E 2

Values o f ILc for d i f f e ren t power fac to rs

Power factor 0 - lag 0.8 - lag Unity 0.8 - lead 0 - lead ILc/IFL 8.7% 5.77% 7.6% 24.8% > 100%

3.6. Effect of degree of transformer core saturation, n

As expected, Fig. 6 shows that increasing n will increase the value of ILc. This means that increasing the degree of saturation, n, of the transformer magnetization characteristic will increase the jump region and hence the possibility of occurrence of the ferroreso- nance phenomenon.

I c'rFL '°1/t 80 L ead',,-~. La P,F. R~,

70

60 50 I / / No .Tump

t.O

30

IC Iump ~ n=o

-6o-~o-';o-~ i'o 3'o s'o ;o 9'o:,, Fig. 6. Effect of the degree of saturation, n.

3.7. Effect of system parameters R], L1 and C1 on ILc

The results obtained are summarized in Table 3, where RI , LI , C1 have the values given in Table 1. Table 3 shows that increas- ing R1 will generally reduce the value of ILc, while increasing L1 or C1 will increase it.

4. CONCLUSIONS

(1) The load current damps the tendency of the ferroresonance phenomenon to occur.

275

At a certain critical value it will prevent the ferroresonance phenomenon. This value of ILc depends mainly upon the load power factor. The value of ILc is very large for zero leading power factor because it means the addition of capacitances to the system, which, of course, increases the possibility of the jump phenomenon.

(2) Increasing R1, which represents the series resistance of the transmission line, will damp the jump phenomenon because it will absorb the jump energy.

(3) Increasing the transmission line param- eters L1 or C1 will increase the possibility of the jump phenomenon because it will increase the interaction between the sys tem induc- tances and capacitances.

(4) Increasing the transformer core satura- tion degree, n, will increase the possibility of the jump phenomenon.

NOMENCLATURE

Clo, C5

Ci

eo(t)

Eo

Eo(s) Eol , Eo2

f(t)

F(s)

coefficients of equation repre- senting transformer magnetiza- tion curve effective transformer internal ca- pacitance, p.u. instantaneous value of input voltage at line sending end, p.u. value of input voltage at sending end, p.u. Laplace transform of eo(t) values of input voltage at sending end corresponding to jump-up and jump<lown phenomena, re- spectively, p.u. instantaneous value of flux link- age, p.u. Laplace transform of f(t)

TABLE 3

Effect of system parameters on value of ILc

System parameters ILc/IFL (%) (p.u.)

cos ¢ ffi 0 cos ¢ ffi 0.8 (lag) (lag)

cos ~b = 1.0 cos ~b = 0 .8 cos ~ = 0 ( l ead) ( l ead)

RI , L1, C1 8.7 5.8 7.6 24.8 667 1.5R1, Lz, Cl 6.7 4.4 5.9 19.7 667 RI , 1.5L1, C1 47.9 30.6 38.1 89.4 667 RI , LI , 1.25C1 46.7 29.8 73.3 88 667

276

Fm peak value of flux linkage, p.u. Fml,Fm3 peak values of flux linkage

corresponding to jump-from and jump-to points, respectively, for jump-up phenomena, p.u.

Fm2,Fm4 peak values of flux linkage corresponding to jump-from and jump-to points, respectively, for jump<lown phenomena, p.u.

G~, C~ half transmission line equivalent shunt conductance and capaci- tance, respectively, p.u.

ira(t) instantaneous value of trans- former magnetizing current, p.u.

Im(S) Laplace transform of ira(t) IFL transformer full load current,

p.u. ILc value of critical load current as

percentage of full load current n saturation degree of transformer

magnetization curve N symbol representing nonlinearity

of transformer magnetization c u r v e

N' describing function of trans- former nonlinear magnetization curve

r e shunt resistance representing transformer core losses, p.u.

R 1, L 1 transmission line equivalent series resistance and inductance, re- spectively, p.u.

R2, L2 transformer equivalent series re- sistance and inductance, respec- tively, p.u.

R3, L3 values of load resistance and inductance, respectively, p.u.

x real part of G_-l(jw) XI, X2, X3 reactances of inductances LI , L2,

L3, respectively, at power fre- quency

y imaginary part of G_-l(j¢0) ZLe magnitude of critical load imped-

ance, p.u. ZFL magnitude of load impedance

corresponding to IFL, p.u. ZL magnitude of load impedance

referred to high voltage side, p.u. angle of load impedance Z_ L

REFERENCES

1 C h a u n c e y Guy , Suits , S tudies in non- l inea r c i rcui ts , AIEE Trans., 50 ( 1 9 3 1 ) 724 - 732.

2 G. W. Swif t , A n ana ly t ica l a p p r o a c h to fer roreso- nance , IEEE Trans., PAS-88 ( 1 9 6 9 ) 42 - 47.

3 M. S. Maklad and A. A. Zaky , M u l t i m o d a l opera- t i on of a f e r r o r e s o n a n t c i rcui t w i th qu in t i c non l inea r i t y , IEEE Trans., MA G-12 ( 1 9 7 6 ) 380 - 384 .

4 S. P rus ty and S. K. Sanayal , New a p p r o a c h for s t u d y of f e r ro re sonance , Proc. Inst. Electr. Eng., 123 ( 1 9 7 6 ) 916 - 918.

5 S. P rus ty and S. K. Sanayal , S o m e new so lu t ions to f e r ro r e sonance p r o b l e m in p o w e r sys tem, Proc. Inst. Electr. Eng., 124 ( 1 9 7 7 ) 1207 - 1211 .

6 S. P rus ty and S. K. Sanaya l , E f fec t o f core loss on m u l t i m o d a l o p e r a t i o n of a paral lel fer roreso- n a n t c i rcui t : some general ised conc lus ions , Proc. Inst. Electr. Eng., 126 ( 1 9 7 9 ) 826 - 832.

A P P E N D I X A

D e v e l o p m e n t o f the sys t em block diagram From Fig. l(a), the following equations can

be written

dr(t) v l d t ) -

dt

1 dr(t) i4(t) -

Re dt

df(t) _ i3( t )R' + L ' di3(t) dt dt

d2f(t) ie(t) = C1 dt 2

ira(t) = C 1 0 f ( t ) + C s f S ( t )

i l ( t ) = ie( t ) + ira(t) + i3(t) + i4(t)

eo(t) = R i l l ( t ) + L1 - - dil(t)

+ v12(t) dt

After Laplace transformation and simplifica- tion:

F ( s ) = Go(s )Eo( s ) - - G ( s ) I m ( s )

where

o0,8,:ls[l Rl sL,x 1 )]I

× sC1 + ~ + R ' Rc + s L '

G(s) fRI+sL1)s[I+(RI+sL1)X

× (sCl + Rc + R, l s L , ) ] f

The block diagram is shown in Fig. l(b).

APPENDIX B

Determination o f the critical load impedance Substituting from eqns. (3) in (2) and

simplifying, it follows that

CNR' + eL'

R '2 + ~2L~2

CIo C N CNR 1 + coL1 = coCl

o~ Rc R12 + ~2L,2

Putting

C10 CN CNR1 + c~L1 K = coC1

Rc R12 + ~2L12

(B-l)

277

SO, also,

CNR' + o~L' K - (B-2)

a'2 + o~2L '2

As

R'= R2 + R3

eL' = X' = X2 + X3

then

R' =R2 + ZLCOS ¢

X' ---- X 2 4" Z L sin 4)

Substituting these relations in eqn. (B-l), and rearranging, gives:

KZL 2 + ZL[(2KR2 -- CN) cos 4) +

+ ( 2 K X 2 - - 1) sin 4)] + K(R2 2 + X2 2) +

-- CNR2 -- X2 = 0 (B-3)

The solution of eqn. (B-3) gives the critical value ZLc of the load impedance ZL.