Computerised evaluation of the magnetising inrush current in transformers

Electric Power Systems Research, 2 ( 1 9 7 9 ) 179 - 182 © Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s

179

Computerised Evaluation of the Magnetising Inrush Current in Transformers

M O H I B U L L A H B A S U a n d K. P. B A S U

Department of Electrical Engineering, Zakir Husain Engineering College, Aligarh Muslim University, Aligarh (India) ( R e c e i v e d N o v e m b e r 13, 1 9 7 8 )

SUMMARY

Design of protective systems in large p o w e r transformers necessitates a detailed knowledge of the magnitude and shape of the magnetising inrush current during the switching operation. It is very difficult to evaluate the inrush current analytically owing to nonlinearity in the magnetisation (B-H) curve. This paper suggests a m e t h o d for the computerised evaluation of the magnetising inrush current in transformers without any drastic linearising assumption for the B - H curve.

INTRODUCTION

When a transformer is suddenly energised a transient current, known as the magnetising inrush current, flows for a short period of time until normal flux conditions are established. Often the magnitude of this current reaches 5 - 10 times the full-load current. Inrush current depends u p o n various operating conditions like voltage magnitude, switching angle, the B - H characteristic of the core, residual magnetism, resistance in the primary circuit, etc. Some of the harmful effects of the inrush current are voltage dips at the consumer terminals, blowing of fuses, excessive unbalanced magnetic pull, and inadvertent operation of protective relays. To eliminate, or at least to reduce, these harmful effects a thorough knowledge of the inrush current is a necessity. Various investigators [1 - 3] have studied the nature of inrush currents experimentally. Analytical evaluation of this current is difficult owing to the nonlinear nature of the B - H curve. Approximate values may be

obtained by linearising the magnetising curve [4] with two straight lines or by approximating it with a hyperbolic curve [5]. For an accurate evaluation of this current a drastic linearising assumption of the magnetisation curve is not acceptable. Moreover, detailed study of the governing factors of the inrush current and the control of these factors necessitates a computerised m e t h o d for the evaluation of the inrush current.

ANALYSIS

For the purpose of the analysis, leakage between primary and secondary flux is neglected. Hysteresis and eddy currents in the core are also neglected. The differential equation of the primary and secondary circuit of a transformer may be written as Ema x

de sin(co t + ~ ) = e = nl ~ + rl il

d¢

n2 -dr = R2i2

+L2 di2 dt

(1)

(2)

n2 im = i I - - ~ Z2 nl

(3)

~b = f(im)

(4)

.

Equation (4) gives the relation between im and ¢ or, in other words, between H and B of the core. The variation of the primary current il with time has to be calculated. When the secondary is on open circuit il becomes equal to ira, the magnetising current. The non-linear relation between ¢ and im suggests a piecewise solution to the problem. Writing down the differential equations in

180

finite d i f f e r e n c e f o r m , a f t e r simplification we have

A¢ = ( e - - r l i l ) -

At

(5)

nl

[ i;=oli'~=o, t%o ]

1 Ai2 = (n2A¢ -- R 2 i 2 A t ) L2

(6)

[ sEr . = o

I

T h e c o m p u t e r algorithm f o r the c o m p u t a t i o n o f i~, i~., etc. m a y n o w be w r i t t e n as

t K+I = t g + A t

(7)

~K+I

= c K 4- A(b K

(8)

i~ ÷1

=i~

(9)

+Ai~

A¢ K and Ai2K are d e t e r m i n e d f r o m eqns. (5) and (6) and il is d e t e r m i n e d f r o m eqn. (3) after finding i 2 a n d im. im should be determ i n e d f r o m eqn. (4) a f t e r ~ is obtained. N values o f @d and imd c o m p u t e d f r o m the B - H curve d a t a o f t h e t r a n s f o r m e r core are used as input: ( ~ < ~ K + I < ~b/+l

l = {1, ( N -

CAICULAT5 i~ U$/~'G t

CAJ-cuIA'rE lKm

,

~

,;A/.CU/.ATK

L

t

J

il~ fArOtv/

,.U,*T, ON (=~,

]

1)}

Assuming a linear relationship b e t w e e n ~ and im in the small interval o f ~d, i/+l .l • K+I = i/md + md - - /md (~)K+I __~h) lm

(10)

The magnetisation curve in a region above ~ m ay be assumed to be linear,having a slope •N "N--1 N (lind - - l i n d ) / ( ~ d - - ~ b N - 1 )

Emax = 3 2 5 . 0 V

If ¢K+I e x c e e d s ~N, • K+I

'm

n 1 = 135 turns

•N "N-1 "N 4- / m d - - / m d (~bK+l

='me @aN @~_~

Fig. 1. F l o w chart.

--¢~)

(11)

nl/n2 = 2 r 1 = 0 . 3 6 9 $2

Based o n the above e q u a t i o n s , a flow c h a r t has been p r e p a r e d (Fig. 1).

INPUT D A T A A N D C O M P U T E R

RESULTS

Based o n t h e f l o w c h a r t a c o m p u t e r p r o g r a m has been p r e p a r e d f o r evaluation o f the magnetising inrush c u r r e n t o f single-phase t r a n s f o r m e r s . T h e p r o g r a m was fed with the d a t a f r o m a 4 k V A , 230 V / l 1 5 , 50 cycle/s t r a n s f o r m e r . T h e values o f @a and imd a r e o b t a i n e d f r o m the @-i characteristic o f Fig. 2. F o r a c c u r a t e results a large n u m b e r o f values o f @d and /d is fed t o the c o m p u t e r . O t h e r d a t a are:

t = 0.0005 s R 2 is assumed to be very high for the evaluation o f inrush c u r r e n t u n d e r no-load c o n d i t i o n s , i °, i ° are assumed t o be zero. ~o s h o u l d be equal to the residual m a g n e t i s m , b u t in the cases c o n s i d e r e d it is also assumed t o be zero. Results were o b t a i n e d u n d e r various o p e r a t i n g c o n d i t i o n s , n a m e l y : (a) no-load at zero degree switching with rl = 0 . 3 6 9 $2 ; (b) as f o r (a) with r 1 = 2 . 8 6 9 ~2 ; (c) full load (R2 = 3 . 2 6 7 0 4 ~2, L 2 = 0) at zero degree switching with r 1 = 0 . 3 6 9 $2 ; (d) no-load at 5 = 0 °, 30 °, 6 0 °, 85.4 °, 90 °, 120 ° and 150 ° with r I = 0 . 3 6 9 ~Z.

181 11ol

the magnetisation curve. The program may also be used in the evaluation of inrush currents in 3-phase transformers having three single-phase units connected in a star. The program can easily handle changes in (i) primary resistance, (ii) load impedance, (iii) switching angle and (iv) residual magnetism.

~.07Z tO~t$

98Z 938 892

J

848 ( 804

76O i770

j

715

ACKNOWLEDGEMENT

167o

625

The authors are grateful to Professor Jalaluddin, Head of the Department of Electrical Engineering, Aligarh Muslim University, for providing all the facilities of the c o m p u t e r centre.

~ 58o Z 535 ~o492 .J

~ 447

4o2 "~

57.8

NOMENCLATURE t~ 26S 223.4

E rll a x

1~'8.5

CD

~34

t

80.98 44.5

~tJ~RENT

,Yl A~pe~,es

o 8,5 t'T0

|.414 9.2 18"4

2.1~28 11.32 19.8P-

4.25 12.72

&1.6~

~

QPeak Values) 5.1;6 14,14~.2.1~Z

h i , n2

7.07' 8.5 15.55 - - 2 ~T.o- 24'8

Fig. 2. ¢ - i characteristic.

rl

Rz, L2 All the results are presented graphically in Figs. 3 and 4. From the results obtained it m a y be easily concluded that, with no residual flux, inrush current becomes maximum at zero degree switching. With increased primary circuit resistance its valtie decreases and with secondary load its value increases. As the switching angle increases, the inrush current decreases and becomes a minimum at an angle very near to 90 ° , depending u p o n the primary resistance. In the cases considered, it is a minimum at 5 = 85.4 ° with r z = 0.369 ~2. In the first few cycles, one-half of the current wave almost vanishes and a considerable period elapses before the current is symmetrical around the time axis.

CONCLUSION

A m e t h o d has been proposed and a c o m p u t e r program prepared for the evaluation of the magnetising inrush current in single-phase transformers. Results obtained from the program are fairly accurate because of the reasonably accurate representation of

i l , i2 im

B,H imd, ~d

peak value of supply voltage angular frequency time, s switching angle number of primary and secondary turns, respectively flux in the transformer core primary circuit resistance secondary circuit resistance and inductance primary and secondary current, respectively magnetising current flux density and magnetic field in the core data of im and ¢ obtained from the magnetisation (B-H) curve of the core

REFERENCES 1 L. F. Kennedy and C. D. Hayward, Harmonic current restrained relays for differential protection. Trans. A m . Inst. Electr. Eng., 57 (1938) 262 - 271. 2 L. F. Blume, G. Camilli, S. B. Farnham and H. A. Peterson, Transformer magnetising inrush currents and influence on system operation. Trans. A m . Inst. Electr. Eng., 63 (1944) 366 - 375. 3 W. K. Sonnemann, C. L. Wanger and G. D. Rockefelar, Magnetising inrush phenomenon in transformer banks. Trans. A m . Inst. Electr. Eng., Part 3, 77 (1958) 884 - 889. 4 V. I. Malyshev, Determining the maximum inrush current in the no-load closure of a transformer. Electr. Technol. USSR, 2 (1968) 136 - 142. 5 A. D. Drozdov and V. A. Borisov, Methods of calculating the kick of magnetising current of large power transformers in power systems. Electr. Technol. USSR, 4 (1968) 28.

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