Calculation and harmonic analysis of transient inrush currents in three-phase transformers

ELSEVIER

Electric Power Systems Research 30 (1994) 93-102

ELECTRIC POWBR 8WSTEI8 RBSBflRgH

Calculation and harmonic analysis of transient inrush currents in three-phase transformers J.C. Y e h , a C.E. Lin, a C.L. H u a n g , a C.L. C h e n g b aDepartment of Electrical Engineering, Cheng Kung University, Tainan, Taiwan bDepartment of Electrical Engineering, Yunlin Polytechnic Institute, Yunlin, Taiwan Received 7 June 1993; in revised form 6 December 1993

Abstract

Measurements of transient inrush currents in three-phase transformers offer important data for power system operation and protection. Because three-phase transformers are more widely used in industrial applications than single-phase transformers, the inrush current of the three-phase transformer is worthy of investigation. This paper proposes a simple method, extended from that for single-phase transformers, to investigate three-phase transformer inrush currents. Harmonic analysis of the inrush currents is carried out. Various structures of three-phase transformers, including winding connections and loading conditions, are discussed. The simulation results are compared with experimental results. The proposed method is simple and effective for transformer banks and five-limb transformers. The harmonic analysis of inrush currents is useful for protective relay design.

Keywords: Transient currents; Harmonic analysis; Transformers

I. Introduction

Since three-phase transformers have a multiple structure compared with that of single-phase transformers, the magnetic loop analysis of a three-phase transformer is more complicated and difficult to study. Analysis of the saturation characteristics is a necessary procedure in studying the inrush current in three-phase transformers for many research and design works. Regarding the inrush current in three-phase transformers, several methods [1-4] have been used to derive an analytical solution or perform a numerical simulation by applying various techniques. The main drawbacks to these methods are the mathematical complexity of the analysis and the procedural difficulties in parametric data acquisition. Furthermore, because these methods cannot cover actual loading conditions accurately, they are not adequate to represent the general operating conditions of power systems. Only special cases have been studied. An improved model [5] to deduce the single-phase inrush current from a finite-difference form using digital data acquisition was developed in our laboratory. This model is simple and accurate for inrush current analysis 0378-7796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0 3 7 8 - 7 7 9 6 ( 9 4 ) 0 0 8 2 3 - M

as well as for other applications concerning single-phase transformers. Based on a general extension of this model, a new method of inrush current study for threephase transformers is proposed. The proposed method can also provide effective analytical data by simple modelling with straightforward procedures. Various structures of three-phase transformers, including electrical winding connections using different three-phase structures, are investigated by considering various operating conditions, such as switch-on angles and loading power factors. Nevertheless, the support of an accurate simulation of the inrush current is not always enough to solve all problems. For example, in protective relays, it is difficult to distinguish fault currents from transient currents in time domain analysis. However, it is always easier to separate the fault currents from transient currents from a frequency domain viewpoint. Under such circumstances, the investigation of harmonic variation can provide strong support for protective relay designs. Harmonic analysis of inrush currents from an energy viewpoint [6] is carried out in this paper. The systematic procedures and consequent results will be helpful to protective relay designers.

94

J.C. Yeh et al./Electric Power Systems Research 30 (1994) 93-102

2. The proposed method 2.1. Analysis of saturation characteristics in three-phase transformers The saturation characteristics of a transformer must be known in order to calculate the transformer inrush current by simulation. M a n y methods to obtain the saturation characteristics of a transformer have been developed in the technical literature [7-11]. Defects may be found in these methods either in their complicated procedures or in the precision of measurements from field test data. A method using a polynomial form to determine transformer saturation characteristics without field test data [ 12] was developed in our laboratory. This method formulated a simple but accurate saturation representation using PC-AT real-time data acquisition. It does not require field test data to represent the saturation curve. Based on this technique, the transformer saturation characteristics can be obtained for further study [5, 6]. In a single-phase transformer, the calculation of the inrush current is simple owing to the independent magnetic circuit; in a three-phase transformer, this calculation becomes much more complicated because of interaction between the magnetic circuits. For different types of three-phase transformer, such as transformer banks, five-limb transformers and three-limb transformers, there are different mechanical structures and magnetic circuits. In addition, electrical connections, such as d e l t a - w y e and delta-delta, will have a strong influence on the saturation characteristics. The saturation characteristics of three-phase transformers are described further in the following sections.

Transformer banks Delta-wye connection. A transformer bank is essentially three separate single-phase transformers. The magnetic circuit of the iron core in each phase of the transformer is independent. Fig. 1 shows the hysteresis loop in phase A at 110 V when three-phase power is supplied to a three-phase transformer bank with d e l t a wye connection. The same hysteresis loop is also obtained in phase A at 110 V when single-phase power is supplied only to the independent phase A. The other phases display the same behavior. This proves that delta wye connected transformer banks can be treated as single-phase transformers Delta-delta connection. Fig. 2 shows a hysteresis loop in phase A at 110 V when a delta-delta connected transformer bank is energized by a three-phase power source. Comparing Fig. 2 with Fig. 1, obviously the peak value of the magnetizing current with the d e l t a delta connection is smaller than that with the d e l t a wye connection. A circular current exists in the delta-delta connected transformer secondary because,

57 Fig. 1. Hysteresis loop in phase A in delta-wye connected threephase transformer banks, measured at the three-phase voltage of IlOV. The same loop is obtained in phase A of the three-phase transfomer bank when it is measured independently at the singlephase voltage of 110 V.

Fig. 2. Hysteresis loop of phase A in delta-delta connected threephase transformer banks measured at the three-phase voltage of ll0V. in general, the three-phase secondary load cannot be exactly balanced. The circular current diminishes the main flux and reduces the risk of iron core saturation. The peak value of the magnetizing current in the saturated region becomes smaller and results in a smaller inrush current, as described in a later section.

Five-limb transformers Different types of three-phase transformer core structure are shown in Fig. 3. The mechanical structure of three-phase transformers has essentially evolved from three single-phase transformers connected by a c o m m o n limb, as in Fig. 3(b). When a three-phase circuit is balanced, the net flux of the c o m m o n limb is zero. For economy, the c o m m o n limb can be removed to form the three-limb three-phase transformer in Fig. 3(d). Alternatively, when the c o m m o n limb is replaced by two side limbs, the five-limb three-phase transformer (Fig. 3(c)) is formed. Delta-wye connection. The magnetic circuits in each phase of the five-limb transformer of Fig. 3(c) can be considered as approximately independent. The flux

J.C. Yeh et al. / Electric Power Systems Research 30 (1994) 93-102

IAI L

95

(a)

•~

~

~

~

B

Fig. 4. Hysteresis loop of phase A in a delta-wye connected five-limb three-phase transformer, measured at the three-phase voltage of 140 V.

A

common i imb

C

(b)

edge

A

limb

(c) Fig. 5. Hysteresis loop of phase A in a five-limb three-phase transformer, measured independently at the single-phase voltage of 140 V.

B

I

(d)

Fig. 3. Three-phase transformer structures.

interaction between phases is not significant. The peak magnetizing current in the saturated region is not affected by the other phases. Fig. 4 shows the hysteresis loop of a three-phase delta-wye connected transformer in phase A at 140V. Fig. 5 shows the hysteresis loop of phase A supplied by a single-phase power source at 140 V. Comparing Figs. 4 and 5, it can be seen that the hysteresis loop in the three-phase delta-wye connected transformer is narrower than in the independent single-phase transformer. However, the peak values of the magnetizing current in both hysteresis loops are nearly the same. The same results apply for different levels of excited voltages. This proves the previous assumption that a delta-wye

connected magnetic current can be approximated by an independent circuit. Delta-delta connection. As with delta-delta connection of transformer banks, the secondary circular current diminishes the main flux and reduces the risk of iron core saturation. The peak magnetizing current in the saturated region indeed becomes smaller. This statement can be proved by observing the experimental results.

Three-limb transformers Delta-wye connection. The mechanical structure of a three-limb transformer is shown in Fig. 3(d). Because the flux in the virtual common limb is not exactly null in practical operation, the flux interaction in each phase becomes strong, making the magnetic circuit difficult to analyze. The magnetic circuit cannot be considered independently. Fig. 6 shows the hysteresis loop of a three-phase delta-wye connected transformer in phase A at 110 V with a three-phase power source. Fig. 7 shows the hysteresis loop in the independent phase A at 110 V with a single-phase power source. Comparing Fig. 6 with Fig. 7, it can be seen that the

96

J.C. Yeh et al./ Electric Power Systems Research 30 (1994) 93-102

Table 1 Data for the transformers used in the example

Primary/secondary rating voltage Primary/secondary rating current Primary/secondary winding resistance Primary winding leakage inductance

Bank

Five-limb

Three-limb

110 V/220 V

110 V/220 V

110 V/220 V

10 A/5 A

10 A/5 A

10 A/5 A

0.10Q/0.35 f2

0.12D/0.65 ~)

0.08D/0.28 D

0.16 mH

0.19 mH

0.07 mH

Fig. 6. Hysteresisloop of phase A in a delta wye connected three-limb three-phase transformer, measured at the three-phase voltage of 110 V. current can be carried out by applying the fast Fourier transform ( F F T ) from an energy viewpoint [6].

3. Example and experiment

_f Fig. 7. Hysteresis loop of phase A in a three-limb three-phase transformer, measured independently at the single-phasevoltage of 110 V. three-phase d e l t a - w y e connected hysteresis loop is strongly influenced by the other phases. The peak magnetizing current in the saturation region is also affected by the other phases and becomes smaller. The complicated flux interaction causes larger errors in the simulation results. However, it is still important that the inrush current in three-limb transformers should be properly investigated. This knowledge can be used to estimate the exact inrush current and to avoid the damage resulting from inrush current. D e l t a - d e l t a connection. In experiments, a smaller peak magnetizing current is observed in a delta-delta connected saturation region. The reason is the same for transformer banks and five-limb transformers.

Three 3 kVA 110 V/220 V three-phase transformers, one transformer bank, one five-limb transformer, and one three-limb transformer, were used for on-site measurement of the inrush current in our laboratory. The fundamental data are indicated in Table 1. The resistance and flux leakage of the secondary winding were neglected. The resistive part of the load impedance, Z 2, was assumed to be very high and regarded as an open circuit; it was fixed at 44 D for all loading tests. The power factors for all the different loading conditions were controlled by changing the series inductor or the parallel capacitor. An experimental circuit for switch-on angle control in three-phase transformers is shown in Fig. 8. The Transformer (Secondary) (PrimarY)R /

Source

t

SW

.

7

ad

1 xt (a)

(or A

connection)

TRIAC

2.2. The numerical simulation

F r o m the above analysis of the saturation characteristics of different types of three-phase transformer, transformer banks and d e l t a - w y e connected five-limb transformers can be considered as three independent magnetic circuits, approximately similar to those of single-phase transformers. The numerical method [ 5, 12] for single-phase transformers, as described in the Appendix using the finite-difference method, can be applied to the above cases. The harmonic analysis of the inrush

l ~ ia

5V

SCR

j ,r'o, f 000 I I

I

(hi: .....................................

'

, ;

Fig. 8. Experiment circuit for switch-on angle control in three-phase transformers: (a) transformer and switches; (b) complete circuit controlling the switches.

J.C. Yeh et al./ Electric Power Systems Research 30 (1994) 93-102 switches ( S W ) in Fig. 8(a) were c o n t r o l l e d b y a real-time c o m p u t e r , the circuit o f which is s h o w n in detail in Fig. 8(b). T h e p h a s e c u r r e n t s ia, ib, a n d ic were s a m p l e d a n d r e a d into a P C - A T t h r o u g h a real-time m e a s u r e m e n t system. Line currents can be c a l c u l a t e d f r o m the threep h a s e currents. A n R S T sequence p o w e r source s u p p l i e d the t h r e e - p h a s e t r a n s f o r m e r s , a n d VRS, the v o l t a g e o f p h a s e A , is used as the reference in the discussion o f the switch-on angle. T o m a k e sure t h a t the r e m a n e n t flux was zero, a d e m a g n e t i z a t i o n process was p e r f o r m e d b e f o r e each e x p e r i m e n t as follows [5, 12]: (1) the threep h a s e t r a n s f o r m e r ' s s e c o n d a r y was k e p t o p e n w i t h o u t a n y load; (2) t h r e e - p h a s e v o l t a g e was a p p l i e d to the p r i m a r y w i n d i n g a n d increased until the core r e a c h e d s a t u r a t i o n ; (3) the t h r e e - p h a s e v o l t a g e was d e c r e a s e d slowly d o w n to zero; (4) the r e m a n e n t flux d r o p s to zero.

Table 2 Effect of switch-on angle ct on peak inrush current (A) in five-limb three-phase transformers ct

Phase

4.1. Inrush current of three-phase transformer banks and five-limb transformers

0° A

S* E+

B

C 45 ° A B

B

Delta-wye connection Switch-on angle. W h e n the t r a n s f o r m e r s e c o n d a r y was o p e n e d a n d u n d e r zero r e m a n e n t flux c o n d i t i o n s , the inrush c u r r e n t s with respect to different s w i t c h - o n angles were investigated. T a b l e 2 c o m p a r e s the simulated a n d m e a s u r e d i n r u s h c u r r e n t s u n d e r s w i t c h - o n angles ~ = 0 °, 45 °, a n d 90 °. Since the possibility o f flux s a t u r a t i o n in p h a s e A at switch-on angles o f 45 ° a n d 90 ° is small, the resulting inrush c u r r e n t s are relatively small. Conversely, the m a x i m u m inrush c u r r e n t in p h a s e A c a n be o b t a i n e d when the switch-on angle is 0 ° . C o n s e q u e n t l y , the inrush c u r r e n t in p h a s e A decreases as the switch-on angle increases. H o w e v e r , for a t h r e e - p h a s e circuit, the p h a s e angle o f each p h a s e is shifted b y 120 ° . W h e n the smaller c u r r e n t is o b t a i n e d in p h a s e A , larger c u r r e n t s c o u l d a p p e a r in phases B o r C simultaneously. T h e influence o f the s w i t c h - o n angle o n inrush c u r r e n t s a p p e a r s to be i r r e g u l a r a n d insignificant. Loading conditions. I n o r d e r to investigate v a r i o u s relations between the inrush c u r r e n t a n d l o a d i n g c o n d i tions, a specific o p e r a t i n g c o n d i t i o n is r e q u i r e d for c o m p a r i s o n : the s w i t c h - o n angle e was c o n t r o l l e d at 0 ° a n d the r e m a n e n t flux l i n k a g e 2r was also zero. U n d e r resistive loading conditions, the inrush c u r r e n t varies very sensitively with l o a d changes. A t y p i c a l inrush c u r r e n t m e a s u r e d for the resistive l o a d o f

114.3 110.6 -14.5

E+ S* E+ S* E+

-13.9 -9.0 -8.5 55.5 54.1 4.1 3.9 -91.4 -89.1 1.4 1.4 78.5 76.1 -74.1 -72.4

E+ S* E+ S* E+ S* E+ S* E+

90° A

2

S*

S*

C T h e inrush currents o f t h r e e - p h a s e t r a n s f o r m e r b a n k s a n d five-limb t r a n s f o r m e r s were investigated. Since the m a g n e t i c circuits in each type o f t r a n s f o r m e r are similar, the results are similar, so we shall present Figures, T a b l e s a n d the discussion m a i n l y for five-limb t r a n s f o r m e r s .

Cycle 1

C 4. R e s u l t s

97

3

42.0 40.1 -4.0

4

5

19.2 11.5 18.9 11.0 -2.8

-3.8 -2.7 -3.7 -2.9 -3.5 -2.8 16.5 9.2 15.9 9.0 2.7 2.5 2.5 2.5 -32.0 -10.1 -30.5 -9.6 1.3 1.3 1.4 1.4 30.1 14.0 28.7 13.1 -16.1 -5.3 --15.7 -5.1

6

7.3 7.1

7

5.5 5.3

4.5 4.3

-2.5

--2.2

-1.9

-1.8

-2.3 -2.5 -2.5 6.4 6.1 2.4 2.4 -5.2 -5.0 1.3 1.4 7.8 7.5 -4.0 -3.9

-2.1 -2.2 -2.1 5.0 4.9 2.3 2.3 -3.8 -3.7 1.3 1.4 5.6 5.5 -3.3 -3.2

-1.9 -2.0 -2.0 4.3 4.2 2.2 2.2 -3.3 -3.2 1.3 1.4 4.8 4.8 -2.7 --2.6

-1.7 -1.9 -1.8 3.8 3.7 2.1 2.1 -2.6 -2.5 1.3 1.4 4.1 4.0 -2.4 -2.4

S*= simulation result; E+ =experimental result.

120 90 A <.,~60 I 30.

I66.7

~ -30

t(msec)

Fig. 9. Inrush current in five-limb three-phase transformers by experiment under resistive load conditions at ct = 0°, 2r = 0 Wb turns,

R2 = 44 f~. R 2 = 44 f~ is s h o w n in Fig. 9. T h e m a g n e t i z i n g inrush c u r r e n t decays in an o s c i l l a t o r y p a t t e r n to the n o r m a l value. Fig. 10 shows the a p p a r e n t v a r i a t i o n o f the p e a k inrush c u r r e n t with l o a d resistance in the transf o r m e r secondary. T h e m a i n flux interacts with the reverse l i n k a g e flux resulting f r o m s e c o n d a r y currents. T h e larger the a p p a r e n t s e c o n d a r y currents, the smaller the inrush c u r r e n t due to the lower risk o f i r o n core saturation. U n d e r inductive loading conditions, the inrush c u r r e n t is sensitive to l o a d p o w e r factors. Fig. 11 shows the v a r i a t i o n o f p e a k inrush c u r r e n t with l o a d p o w e r

J.C. Yeh et al./Electric Power Systems Research 30 (1994) 93-102

98 120"

I00

•

A

Experiment --n-

80. 80 --Experiment --- S i m u l a t i o n

60' < 40 40

20.

0.

0

c l IA

-20 0A

I I 2A 3A Load C u r r e n t

I 4A

BP

-20

=~

0.6

5A

T

0.8

Power Factor

Fig. 10. Effect of resistive load on p e a k inrush current in five-limb three-phase t r a n s f o r m e r s at c~ = 0 °, "~r = 0 W b turns.

I

|

0.7

0.9

1.0

(leading)

Fig. 12. Effect of capacitive load on p e a k i n r u s h current in five-limb three-phase t r a n s f o r m e r s at ~ = 0 °, 2r = 0 W b turns.

I00. A --

~

~

120'

I

80" --Experiment --- S i m u l a t i o n

60"

<

80

40.

< 40

20.

0"

0

B C -20

H 0

I 0.6

I

|

|

0.7 0.8 0.9 Power F a c t o r (lagging)

i.

-20 0.8

Fig. 11. Effect of inductive load on p e a k inrush current in five-limb three-phase t r a n s f o r m e r s at = = 0 °, "~r = 0 W b turns.

!

!

0.9

1.0 Applied Voltage

! i.i

1.2

(p.u.)

Fig. 13. Effect o f voltage on p e a k inrush current in three-phase t r a n s f o r m e r b a n k s at c~ = 0 °, 2 r = 0 W b turns.

factor. The main flux strongly interacts with the real part of the reverse linkage flux resulting from secondary currents. The higher the power factor, the more the peak inrush current due to iron core saturation decreases. Under capacitive loading conditions, the inrush current also has a close relationship with load power factors. Fig. 12 shows the variation of the peak inrush current with load power factor. When the power factor is close to 1.0, the peak inrush current is smaller. The reason for this effect is similar to that for the inductive load. Power source voltage. In Fig. 13, the simulation result shows the influence of the applied voltage in the transformer primary on the resulting inrush current. The approximately linear relationship of the applied voltage to the peak inrush current in phase A is an important and impressive phenomenon in this study.

Delta-delta connection When the secondary windings are delta-connected, a circular current will exist. The magnitude of this circular current is sensitive to the load imbalance. The larger the circular current, the more the main flux is diminished by the reverse linkage flux resulting from this circular current. Therefore, a smaller inrush current is expected. The hysteresis loops with delta-delta and delta-wye connections are shown in Figs. 2 and 1, respectively. Comparing Fig. 2 with Fig. 1, the peak value of the inrush current corresponding to the delta-delta connection is obviously smaller than that corresponding to the delta-wye connection. The inrush currents in the delta-delta and delta-wye connections under various resistive loads are compared in Fig. 14. The corresponding results are listed in Table 3. Fig. 14 and

J.C. Yeh et al. /Electric Power Systems Research 30 (1994) 93 102 120

•

4.3. Harmonic analysis of inrush currents

A-Y --- A-A

A

90.

60"

,(

30

r

C

-30

I

0A

2.5A Load Current

99

5A

A detailed harmonic analysis of the magnetizing inrush current in a three-phase transformer has been completed. A guideline for harmonic suppression and protective relay design has been established. In this paper, only the harmonic suppression for three-phase transformers is discussed. The harmonic variation in the time domain analysis is neglected. The influences of switch-on angle and various loading conditions on harmonic suppression will be discussed in detail. For simplicity, only the results of transformer banks are presented because the same conclusions apply to fivelimb and three-limb three-phase transformers.

Fig. 14. Comparison of delta-delta and delta-wye peak inrush currents in five-limb three-phase transformers. .

"'°-"

Table 3 Comparison of delta delta and delta wye peak inrush currents in five-limb three-phase transformers

R2

oo (0 A) 88 f2 (2.5 A) 44 ~ (5 A)

Phase

lp (A) A-Y

A-A

110.6 - 13.9 -8.5

93.6 - 19.2 -19.3

A

97.4

B

- 8.1

C

-7.7

83.0 --9.0 -21.8

A B C

87.3 - 15.0 - 14.5

78.1 - 14.9 -27.1

A B C

DC

1

100.

--

2

,

_'

:

:

,

3

4

5

6

7

A

8

9

Line S

40"

Table 3 prove that the delta-delta connection leads to a smaller inrush current than the delta-wye connection. e

4.2. Inrush current of three-limb three-phase transformers Based on the previous discussion of saturation characteristics in transformers, strong flux interactions exist in three-limb three-phase transformers. Larger simulation errors occur with three-limb three-phase transformers. Fortunately, the actual peak inrush current is smaller than the simulated value. The peak inrush current can be estimated by simulation. This is helpful in designing protective relay systems. The influence of various operating conditions on the peak inrush current in three-limb three-phase transformers has been thoroughly investigated. The results are similar to those for transformer banks and five-limb three-phase transformers.

i

3

,

5

6

7

8

'

&

10O

80

/ / ~~

Line T

#@

±'°I I

\oA

o.I 1

o

DC

:

1

~

3 4 Harmonic

I

5

I

6

7

8

9

Component

Fig. 15. Effect of resistive load on harmonics in three-phase transformer banks.

100

J.C. Yeh et al./Electric Power Systems Research 30 (1994) 93 102

I n a single-phase transformer, only the line inrush current, that is, the phase current, is selected for h a r m o n i c analysis. However, for three-phase transformers, the line inrush c u r r e n t or the phase inrush c u r r e n t can be selected for analysis. The h a r m o n i c c o n t e n t of the phase inrush current is strongly affected by the t r a n s f o r m e r itself. The influence of the phase n o n s i n u s o i d a l current o n the w i n d i n g a n d eddy c u r r e n t losses has been presented in the literature [ 13-15]. The h a r m o n i c c o n t e n t of the line i n r u s h current is strongly influenced by other devices in the power system. The latter is more i m p o r t a n t from the power system viewpoint. Therefore, the h a r m o n i c analysis o f the line inrush c u r r e n t in three-phase t r a n s f o r m e r s is presented in the following discussion. T a b l e 4 shows the h a r m o n i c c o n t e n t c o n t a i n e d in the first cycle o f the inrush c u r r e n t u n d e r various switch-on angles a n d l o a d i n g conditions. The m e a s u r e m e n t s are carried out independently, one cycle after another. It can be seen that the h a r m o n i c c o n t e n t is rich in the D C

c o m p o n e n t , second, a n d third harmonics. The m a i n results are as follows. (1) T a b l e 4 shows the relationship between the harm o n i c c o n t e n t a n d switch-on angle for each o f the lines R, S, a n d T. The influence of the switch-on angle on the h a r m o n i c c o n t e n t is n o t identical for each line, b u t appears to be irregular a n d insignificant. (2) C o m p a r i n g the results u n d e r resistive loads o f R2 = 44 f~, 88 f~, a n d open circuit, it can be seen that the higher load currents result in smaller h a r m o n i c c o m p o n e n t s . Fig. 15 illustrates that a heavier load can effectively reduce the h a r m o n i c c o n t e n t of all lines in three-phase transformers. (3) C o m p a r i n g the results for inductive loads u n d e r power factors of 1.0, 0.8, a n d 0.6 lagging, there is n o a p p a r e n t h a r m o n i c variation. (4) C o m p a r i n g the results for capacitive loads u n d e r power factors of 1.0, - 0 . 8 , a n d - 0 . 6 leading, the h a r m o n i c s are seen to be sensitive to power factor

Table 4 Comparison of harmonic analyses in three-phase transformers for 2r = 0 Wb turns, various loads power factors and switch-on angle conditions (all harmonics are shown as a percentage of the fundamental current for the first cycle) Line

Harm.

Magnetizing inrush current (%) variation

R variation

L variation

C variation

Z 2 = o(3

Z 2 = 00

Z 2 = (x)

Z 2 = 88

Z 2 = 44

Z 2 = 44

Z 2 = 44

Z2 = 44

ct = 0 °

ct = 4 5 °

ct = 9 0 °

~ =0 °

ct = 0 °

c~ = 0 °

c~ = 0 °

ct = 0 °

Z2 = 44 ct = 0 °

pf = 1

pf = 1

pf = 0.8

pf = 0.6

pf = 0.8

pf = 0.6

R

Dc 2nd 3rd 4th 5th 6th 7th 8th 9th

58.9 68.7 45.9 30.4 18.7 9.5 3.2 1.8 2.5

58.7 64.1 39.9 20.8 11.3 3.5 1.1 4.5 2.9

55.4 80.9 54.6 37.8 21.9 11.9 5.4 2.2 1.5

31.3 34.1 21.9 15.1 9.6 6.9 4.0 2.0 1.3

18.7 18.8 11.5 8.2 4.0 4.2 2.4 1.3 1.3

24.2 21.4 11.6 6.8 7.5 3.2 4.9 2.0 3.4

32.4 21.2 13.9 6.3 8.5 2.8 5.4 2.1 3.7

19.4 19.5 28.3 9.6 18.5 12.3 10.9 12.5 6.5

25.4 27.9 42.2 19.1 26.9 22.2 9.3 15.2 7.6

S

DC 2nd 3rd 4th 5th 6th 7th 8th 9th

58.1 70.4 45.3 29.3 18.1 9.3 3.4 1.8 2.5

35.1 68.4 23.9 16.9 7.3 5.0 2.8 1.5 0.4

55.1 73.3 49.1 33.2 21.0 11.0 4.6 2.0 2.2

43.0 48.9 31.3 20.5 14.3 9.4 5.8 3.3 1.3

21.2 22.8 15.8 9.0 5.9 4.4 3.5 1.4 1.2

16.5 16.5 9.3 5.9 4.8 2.7 2.3 1.0 1.1

20.5 15.4 9.8 5.3 5.6 1.9 3.1 0.3 1.8

39.2 52.3 23.4 31.4 18.5 20.0 21.0 10.7 14.9

64.4 90.0 44.1 52.3 40.3 21.3 32.0 18.5 15.5

T

DC 2nd 3rd 4th 5th 6th 7th 8th 9th

5.3 70.8 4.1 15.7 6.6 2.9 1.5 1.4 0.2

56.1 69.0 47.9 25.3 12.5 4.4 1.7 5.0 3.0

65.5 46.1 5.3 20.2 21.4 13.5 5.3 1.7 0.8

2.0 10.0 0.9 1.5 0.6 1.1 0.8 1.0 0.8

2.3 5.1 1.5 1.3 0.8 1.4 1.3 1.2 1.1

11.7 16.7 8.8 5.9 5.7 4.6 4.1 3.6 3.2

15.0 15.0 7.0 4.4 4.4 3.5 3.1 2.8 2.4

6.8 15.5 15.7 19.4 23.7 23.4 21.3 19.7 15.6

10.4 22.2 24.5 29.2 35.9 31.9 24.8 20.6 15.4

J.C. Yeh et al./Electric Power Systems Research 30 (1994) 93 102

changes: the lower the power factor, the larger the increase in harmonic content.

il

~

5. Conclusions

RI

_

•

•

i2

h +

.~e(t)

This paper demonstrates a simple and effective method, extended from the method for single-phase transformers [5], for predicting the magnetizing inrush current in three-phase transformers. Various investigations are demonstrated related to transformer loading conditions, switch-on angles, remanent flux, and winding connections of three-phase transformers of different types. Harmonic analysis of inrush current from an energy viewpoint has been carried out. The harmonic characteristics of inrush currents are studied in detail by analyzing the simulated results. Furthermore, based on the results obtained, it is concluded that the main harmonics can be suppressed by increasing the loads and power factors. Consequently, more detailed knowledge of inrush currents in three-phase transformers has been gained from this study. By using the proposed methods, preparation time and field tests can be reduced. The proposed methods can also be applied to investigate the problems caused by inrush currents in three-phase transformers.

101

)) _,% )+

v1

7>_

_

0

i

L2 %1

!

Fig. A1. A c t u a l t r a n s f o r m e r inductive loading.

L,

primary circuit inductance, equal to primary winding leakage inductance if source impedance is neglected Emax maximum of e(t) ~o frequency of exciting voltage (rad s -~) c~ switch-on angle of voltage waveform (rad) No load, resistive load, or inductive load The differential equation of the secondary circuit in Fig. A1 is written as dXz/dt = R2i2 + L2 diz/dt

(A3)

where 22

Acknowledgement This work is supported by the National Science Council under research project NSC81-0404-E006-04.

instantaneous mutual flux linkage (Wb turns) in secondary winding i2 instantaneous secondary current R2 secondary circuit series resistance L 2 secondary circuit series inductance When the resistance increases, the load of the transformer secondary circuit drops until it is open-circuited.

Appendix

Numerical method

The proposed method [5, 12] A power transformer is assumed to be connected to an infinite bus. By Kirchhoff's voltage law, the differential equation of the primary circuit in Fig. A1 can be written as e(t) = d 2 l / d t + R1 il + L1 dil Idt

(A1)

and (A2)

with the following nomenclature:

il R1

e k

=

Ema x sin(~ot~ +

instantaneous exciting voltage time measured at switch-on instantaneous mutual flux linkage (Wb turns) in primary winding instantaneous primary current primary circuit resistance, equal to primary winding resistance if source impedance is neglected

c~)

(a21)k/At = ek - R1 (il)k -- Ll (Aii)k/At

(A4) (A5)

where tk = kAt, and k represents the sampling steps in the simulation: (AX,),<

e(t) = Emax sin@ot + ~).

e(t) t 21

Eqs. (A1) and (A2) can be expressed in finite-difference form to get

(Ail)k

= (~1),<+, =

- (Xl),<

(i,)k -- (i,)k_ 1

Similarly, i2 of Eq. (A3) for different load characteristics can be written in finite-difference form as (i2)k = [(AJ[2)k~at -- Le(AiE)k/AtI/R2

(A6)

where (a&)~ = (v2/Vl)(aXl)k

V, and V2 are the primary and secondary rating

102

J.C. Yeh et al./ Electric Power Systems Research 30 (1994) 93-102

Then, (im)k+l corresponding to (21)k+l is calculated.

voltages, respectively, and

For k = k + 1,

( A i 2 ) ~ = (i2)k - ( i ~ ) ~ _ 1

After (iz)k is calculated from Eq. (A6), the new (i1)~ and (Ail)k can be obtained from

(il)k+ 1 = (im)k+ 1 4- ( i 2 ) k ( V 2 / V , ) and

(il)~ = (im)~ 4- ( i 2 ) k ( V 2 / V , )

(A7)

(Ail)~+, = (i,)k+ l -- (il)k

(Ail)k = (il)k - (il)k -1

(A8)

are calculated and substituted into Eq. (A5) together with e~+l from Eq. (A4) to get

where (im)k is the magnetizing current with respect to

(21)k.

(A21)~ + 1/At = e~ + j - R 1 ( il )k + 1 - L1 (Ail)k + ~/At

Using Eqs. ( A 5 ) - ( A 8 ) as one iteration routine, a new (il)k is obtained for the sample k. If the difference between (il)k from Eq. (A7) and the starting value in Eq. (A5) is larger than a specified tolerance, the average of these two values is used as (il)~ for the next iteration routine starting from Eq. (A5) again. A similar process is used for (Ail)k. This routine from Eq. (A5) to (A8) is repeated until both specified error values are achieved. Then, one simulated sample of inrush current i~ at t = k A t , that is, (il)k, is obtained from the iteration process as shown in Fig. A2. The values of (A21)k and (i:)k can be calculated easily from both Eqs. (A5) and (A6) in the last iteration of k. When t = (k + 1)At, (21)~+1 can be calculated from (~l)k+ 1 = (~'l)k 4- (A~l)k

(i9)

Read Emax, a, At, RI, L I, R2, L2, C2" ~i' £~" M, At l

[

I

I (il)0 0, (i2)0=0, (AAI)0=0, I (nil)0=0, k=0 i

1:

Calculate ek using Eq. A-4 ] n=0, q=0

(XI )k

corresponding

'[

,

[il)q= [(il)q+ (i)q-[l/2

q=q+l [Calculate ( i l i a . l

+ •

tail) =t(

z K

•

~

I

q

.

y

l

I

(i2)kq, [

q-I yes

write tk"( i(i 2 )l)k' q qk = k + l ~

n

i

o

Fig. A2. Flowchart for numerical simulation: s l and e2 = tolerances; M =maximum number of simulation samples; n =number of sampled points in the hysteresis loop; q = iteration number.

(A10) By the same iterative procedure, (il)k for all the k can be obtained to get a simulation result of the magnetizing inrush current. The overall simulation procedure is shown in Fig. A2. References [1] W.K. Macfadyen, R.R.S. Simpson, R.D. Slater and W.S. Wood, Method of predicting transient current patterns in transformers, Proc. Inst. Electr. Eng., 120 (1973) 1393-1396. [2] H.L. Nakra and T.H. Barton, Three phase transformer transients, IEEE Trans. Power Appar. Syst., PAS-93 (1974) 1810-1819. [3] R. Yacamini and A. Abu Nasser, The calculation of inrush current in three-phase transformers, IEE Proc. B, 133 (1986) 31-40. [4] M.A. Rahman and A. Gangopadhyay, Digital simulation of magnetizing inrush currents in three-phase transformers, IEEE Trans. Power Delivery, PWRD-1 (1986) 235-242• [5] C.E. Lin, C.L. Cheng, C.L. Huang and J.C. Yeh, Investigation of magnetizing inrush current in transformers, Part I. Numerical simulation, IEEE Trans. Power Delivery, 8 (1993) 246 254. [6] C.E. Lin, C.L. Cheng, C.L. Huang and J.C. Yeh, Investigation of magnetizing inrush current in transformers, Part II. Harmonic analysis, IEEE Trans. Power Delivery, 8 (1993) 255-263. [7] G.W. Swift, Power transformer core behaviour under transient condition, IEEE Trans. Power Appar. Syst., PAS-90 (1971) 2206-2210. [8] J. Rivas, J.M. Zamarro, E. Martin and C. Pereira, Simple approximation for magnetization curves and hysteresis loops, IEEE Trans. Magn., MAG-17 (1981) 1498-1502. [9] E.H. Badawy and R.D. Youssef, Representation of transformer saturation, Electr. Power Syst. Res., 6 (1983) 301-304. [ 10] S. Prusty and M.V.S. Rao, A novel approach for predetermination of magnetization characteristics of transformers including hysteresis, IEEE Trans. Magn., MAG-20 (1984) 607-612. [11] A. Keyhani and S.M. Miri, Nonlinear modeling of magnetic saturation and hysteresis in an electromagnetic device, Electr. Power Syst. Res., 15 (1988) 15 24. [12] C.E. Lin, J.B. Wei, C.L. Huang and C.J. Huang, A new model for transformer saturation characteristics by including hysteresis loops, IEEE Trans• Magn., 25 (1989) 2706 2712. [13] M.S. Hwang, W.M• Grady and H.W. Sanders, Distribution transformer winding losses due to nonsinusoidal currents, IEEE Trans. Power Delivery, PWRD-2 (1987) 140-146. [14] M.D. Hwang, W.M. Grady and H.W. Sanders, Calculation of winding temperatures in distribution transformers subjected to harmonic currents, IEEE Trans. Power Delivery, 3 (1988) 10741079. [15] A.E. Emanuel, The effect of nonsinusoidal excitation on eddy current losses in saturated iron, IEEE Trans. Power Delivery, 3 (1988) 662-671.