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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007
Theoretical Calculation of Inrush Currents in
Three- and Five-Legged Core Transformers
Luis Sáinz, Felipe Córcoles, Joaquín Pedra, Member, IEEE, and Luis Guasch
Abstract—In this paper, a theoretical study of the three-phase
transformer behavior in the presence of a sag is presented by considering that the fault clearing is produced simultaneously in all
phases. Analytical expressions of the magnetic flux and the inrush
current after voltage recovery are obtained, and the results presented in the literature are analytically justified. The influence of
depth, duration, and the initial point-on-wave on the peak value of
the inrush current are studied for all sag types. Simple expressions
to obtain the value of the current peaks are presented.
Index Terms—Inrush current, transformer model, voltage sag.
Fig. 1. Electric equivalent circuit of a Wye G-Wye G three-phase transformer.
I. INTRODUCTION
A
VOLTAGE SAG is a short-duration reduction in
root-mean-square (rms) voltage. The most severe voltage
sags are produced by faults in power systems, and it can lead
to malfunctioning equipment [1]. In the case of power transformers, the sudden voltage recovery, when a fault is cleared,
can saturate the core transformer. This transformer saturation
can involve high inrush currents [2], which are very sensitive
to the voltage recovery instant. Therefore, a precise study of
the problem must take into account that this instant can only
have discrete values, since fault clearing is produced in the
natural current zeros [3], [4]. Nevertheless, as in the previous
study of [5], the consideration that fault clearing is produced
simultaneously in all phases simplifies the study and provides
a preliminary estimation of the inrush currents. The current
peaks calculated by assuming that voltage recovery can only
be produced in the current zeros are lower than those obtained
when this assumption is not made. In the studies of [4] and
[5], the three-phase transformer model is implemented and
simulated in PSpice.
In this paper, the inrush currents caused by symmetrical and
unsymmetrical voltage sags are analytically studied in detail by
considering that fault clearing is produced simultaneously in all
of the phases. This simplification allows analytical expressions
of the magnetic flux and inrush current after a voltage sag to be
Manuscript received February 23, 2006; revised May 5, 2006. This work was
supported by Grant DPI2000-0994. Paper no. TPWRD-00081-2006.
L. Sáinz, F. Córcoles, and J. Pedra are with the Department of Electrical Engineering, Universitat Politècnica de Catalunya, Barcelona 08028, Spain (e-mail:
sainz@ee.upc.es; corcoles@ee.upc.es; pedra@ee.upc.es).
L. Guasch is with the Department of Electronic, Electrical, and Automatic
Engineering, Universitat Rovira i Virgili, Tarragona 43007, Spain (e-mail:
llguasch@etse.urv.es).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2006.881428
obtained. Inrush current dependence on sag type, depth, duration, and the initial point on wave is characterized. These expressions allow the transformer behavior presented in [5] to be
explained and an approximate value for the peak of the inrush
current to be analytically calculated. This can be useful in power
system studies, such as relay coordination, transformer stress,
etc.
II. TRANSFORMER MODEL
Two types of three-phase transformers are studied in this
paper, namely three- and five-legged core transformers. The
model proposed in [4] and [5] is used to study the three-legged
transformer, whereas a particular case of this model is used to
study the five-legged core transformer.
A. Electric- and Magnetic-Circuit Models
The electric and magnetic relations of the Wye G-Wye G
three-phase transformer are shown in Figs. 1 and 2
(1)
In this model, each leg is viewed as a separate magnetic element, and the expression used to represent the core nonlinear
that relates the magnetic potential
behavior is a function
in the leg and the flux through it
0885-8977/$25.00 © 2007 IEEE
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SÁINZ et al.: THEORETICAL CALCULATION OF INRUSH CURRENTS IN CORE TRANSFORMERS
987
Fig. 2. Magnetic equivalent circuit of a three-legged three-phase transformer.
where
,
, , and
are experimental parameters
which allow this single-valued function to be fitted to the
transformer saturation curve [5].
has been
In Fig. 2, it can be observed that the reluctance
considered constant because it represents the air path in a threelegged transformer.
In order to eliminate the winding turns, the above relations
are rewritten according to [4] as follows:
Fig. 3. Magnetic fluxes in a symmetrical voltage sag of characteristics
.
0.4, t
4.375T , and
1 =
(3)
is the winding turn ratio and
where
is the core magnetic flux linked by the primary windings. The
relation between the magnetic potential in the leg and the flux
through it is rewritten as
In this paper, only the primary voltages, currents, and total
fluxes are of interest. Thus, these primary variables are renamed
as follows for clarity purposes:
(4)
=0
h
=
The above considerations are also true for the transformer
bank. Thus, the study of the five-legged transformer can be extended to the transformer bank.
In all of the cases studied in this paper, the secondary supplies
a rated resistive load of 0.8 per phase. In fact, load magnitude
and load character (resistive or inductive) have no influence on
the transformer behavior [5].
III. ANALYTICAL CHARACTERIZATION OF THE CURRENT PEAKS
References [4] and [5] show that the main reason why transformer current peaks appear is found in the dc component of
with
the total magnetic flux after the sag; for example,
in Fig. 3. Transformer saturation occurs when this
dc component is not null, causing the current to be high. This
section presents an analytical study of the total magnetic flux
and
and the inrush current when the voltage sag ends
with
. This study is conducted in a Wye G-Wye G
three-phase five-legged transformer (or a transformer bank).
B. Transformer Data
A. Total Flux Peak Determination
The study of the three-legged model has been performed with
a 60-kVA, 380/220-V Wye G-Wye G three-phase transformer
whose linear and nonlinear parameters have been obtained from
experimental measurements [4].
The study of the three-phase five-legged transformer is also
performed with the same data as the three-legged model but
considering:
• reluctance values of legs and are forced to be equal to
the values of leg ;
has been considered null be• value of the reluctance
cause, during transformer saturation, it will not be significant in comparison with the leg reluctances.
Considering the transformer model (3), the total magnetic
flux linked by the primary windings can be calculated as
(5)
where the primary total magnetic flux is
the voltage drop in the primary winding resistances (
with
) has been neglected.
The total magnetic flux is derived from (5) and gives
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, and
(6)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007
TABLE I
MAGNITUDE AND ANGLE OF VOLTAGE SAG PHASORS
TABLE II
AND t TO AVOID OBTAINING FLUX AND CURRENT PEAKS.
VALUES OF
VALUES OF
TO OBTAIN THE MAXIMUM FLUX AND CURRENT PEAKS
WHEN t IS FIXED, AND i
1
1
Taking into account that
When the supply voltage is
with
, and the transformer is in steady state, the
magnetic flux is expressed as
(7)
As the mean value of the flux is zero in steady state, the next
condition must be true for the initial flux
(8)
(12)
the postsag flux (11) can be rewritten as
and the final expression of the flux is
(13)
(9)
The supply voltages assumed when a sag is produced are
where the two first terms are identical to the terms of the case
without a sag (9).
Then, the final expression of the flux when voltage is recovered is
.
(10)
,
The pre and postsag voltage is defined as
,
.
The magnitude and angle values of the voltage phasors during
and
) are shown in Table I. They are determined
the sag (
from the sag expressions of Table II in [4], which were obtained
from the sag classification of [1].
Flux continuity implies that flux can be calculated at any instant by means of (6), even if the voltage changes during the sag.
The flux after the sag (Fig. 3) can be calculated as
(11)
where the sag is produced between times
and
.
(14)
where
(15)
This term is constant and its physical meaning is a dc
magnetic flux. The Appendix shows the expressions of
obtained from (15) for the different sag types.
The current peak is produced when there is a flux peak bemagnetic curve (3). Therecause they are related by the
fore, the peak values of the total magnetic fluxes must be calculated to determine the current peaks.
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SÁINZ et al.: THEORETICAL CALCULATION OF INRUSH CURRENTS IN CORE TRANSFORMERS
The flux peak produced by the sag depends on the depth ,
the duration , and the initial point-on-wave . Thus, the following function must be studied:
989
In this case, the magnetic equation of (3) gives
(20)
This expression allows the relation
to be calculated
(21)
and the use of the nonlinear expression
in (3) yields
(16)
(22)
(17)
Fig. 4(a) shows the influence of the nonlinear curve parameters
on the shape of the nonlinear saturation
curve. The above nonlinear relation can be approximated by a
piecewise linear saturation curve. If only two pieces are used
Fig. 4(a), their parameters can be related easily to the parameters of the nonlinear function (22). The two-piece linear saturation curve is defined as
where
In the case shown in Fig. 3, the flux peak is produced in phase
, and its value is
.
must be analyzed to study the influence
Then,
of sag type, depth, duration, and the initial point-on-wave on the
maximum and minimum values of the postsag magnetic flux.
is obtained from the analytical exThe flux
in the Appendix. For example, the dc compressions of
ponent of the total magnetic fluxes in a type B sag (Appendix)
is
(23)
where
is the saturation flux. The
current peak can be easily determined from (23).
Fig. 4(b) shows the current peak determination from the sat,
, and
uration curve for a type B sag of
ranging from 0 to . Fig. 4(b) shows that the difference between the use of the nonlinear saturation curve (continuous line)
and the two-piece approximation (dashed line) is small.
The expression for calculating the current peak when
is derived from (23)
(24)
(18)
where
and, according to (16), the flux peak gives
is the phase with the greatest flux peak.
IV. BEHAVIOR STUDY OF THE THREE-PHASE TRANSFORMER
(19)
Fig. 4(b) shows the curve
0.73 and
characteristics
for a type B sag of
5.5 obtained from (19).
B. Current Peak Determination
The transformer total flux and the primary currents can be related if some simplifications are made in the transformer model.
• Transformer is at no load (or the secondary currents are of
less importance than the primary currents when the trans(or
).
former is saturated)
is null, as the five-legged transformer is
• Reluctance
considered (the magnetic potential is also null).
• Core-loss resistances are neglected
.
The expressions of the previous section allow the transformer
current peak to be analytically calculated. Thus, the influence of
sag type, duration , depth , and the initial point-on-wave on
the current peaks can be studied. It must be noted that in the study
of the previous section, the source has no internal impedance.
This leads to an overestimation of the transformer current peaks
because the source impedance damps these peaks [5].
All of the analytical calculations in this paper have been
obtained by programming the analytical expressions from
Section III and the Appendix in MATLAB [6].
To analyze the duration, initial point-on-wave and depth influence on the current peaks, the following series of sags have
been studied for the seven sag types, A to G.
• Duration influence: Four series of sags of depth
have been studied
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.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007
Fig. 4. (a) Influence of the parameters on the shape of the saturation curve. (b) Current peak determination from the saturation curve.
• Initial point-on-wave influence: Four series of sags of
depth
have been studied
(25)
.
• Depth influence: One series of sags with the most unfavorable duration and the most unfavorable initial point-onand , obtained from Table II, have been
wave
studied
(26)
A. Three-Phase Five-Legged Transformer
In order to study this transformer analytically, the data of [4]
have been considered imposing that the three legs are identical
(the reluctance values of legs and are set equal to the values
of leg ) and the air-path linear reluctance
is null. Thus, this
study can also be extended to the transformer bank.
Figs. 5 and 6 show the sag duration and the initial point-onwave influence on the current peak, respectively (series to
and
to ).
By comparing these figures with the corresponding ones
in [5], it can be noted that the analytical study predicts the
five-legged transformer behavior (and the transformer bank
behavior) accurately. Nevertheless, the analytical results are
slightly lower than the simulation results in [5] because the
resistance voltage drop has been neglected.
1) Duration and Initial Point-on-Wave Influence: The value
of
and
required to avoid obtaining current peaks (which
corresponds to minimum flux values) when voltage is recovered
can be calculated by imposing
in the expressions of
the Appendix. These values are shown in Table II.
The most unfavorable situation is produced when the maximum flux peak (which corresponds to the maximum current
peak) is obtained. The maximum flux peak can be calculated
analytically. At this point, the difference between the maximum
flux peak (which takes into account all current peaks corresponding to all values of
and ), and the maximum flux
peak when
is fixed (which takes into account only the current peaks corresponding to all values of , but only one value
of ) must be considered
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991
Fig. 5. Three-phase five-legged transformer: analytical results of the sag duration influence on the current peak for the seven sag types.
Fig. 6. Three-phase five-legged transformer: analytical results of the initial
point-on-wave influence on the current peak for the seven sag types.
The value of
required to obtain
(for example, in
series , , , or ) is obtained by imposing
in the expressions of Appendix A. These values
are shown in Table II for the relation
.
shown in Table II, only
Among all of the values of
the highest values produce the most unfavorable situations, that
. According to the Appendix,
is, the maximum flux peaks
these most unfavorable situations are produced for specific
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007
Fig. 7. Three-phase five-legged transformer: analytical results of the depth influence on the most unfavorable current peak for the seven sag types.
values of
with
(when
i.e., series 7) and for the corresponding
relations of Table II.
The maximum flux peak can be calculated as
(28)
where
for type A,B,D,E, and F sags
for type C sag
for type G sag.
(29)
The maximum current peak
can be calculated by the
substitution of (28) in (24). It can be noted that type A, B, D, E,
and F sags have the highest values.
2) Depth Influence: According to series , Fig. 7 shows the
depth influence on the maximum current peak.
For a given
and , the expressions of the Appendix show
that the relation between
and the depth is a straight line
with negative slope (i.e., there is a linear relation). Then
(30)
where
does not depend on the sag depth .
Considering (23), the relation between the current peaks and
the depth is also linear with two possible negative slopes depending on whether the total magnetic flux is above or below
the saturation flux
or
.
B. Three-Phase Three-Legged Transformer
In order to study this transformer analytically, the data of [4]
for the three legs have been considered.
According to series to and to , Figs. 8 and 9 show
the sag duration and the initial point-on-wave influence on the
current peaks, respectively.
By comparing these figures with the corresponding ones in
[5], it can be observed that although the analytical model conis null (
in
siders that the value of the reluctance
Section III-B), the study predicts the three-legged transformer
behavior accurately. This good agreement is achieved because
(the magnetic potenthe magnetic potential in the reluctance
tial , (3)) is not significant in comparison with the remaining
magnetic potentials when the transformer is saturated [
, which is equivalent to considering
, as in (20)]. By
comparing the five- and three-legged transformer results (Figs. 5
Fig. 8. Three-phase three-legged transformer: analytical results of the sag duration influence on the current peak for the seven sag types.
and 6 with Figs. 8 and 9), it can be observed that the main difference is in the symmetry of the waveforms whereas the current
peaks do not essentially change.
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SÁINZ et al.: THEORETICAL CALCULATION OF INRUSH CURRENTS IN CORE TRANSFORMERS
993
The procedure to calculate the maximum current peaks is the
same as the procedure of the previous section. However, the
three-legged transformer has fewer most unfavorable situations
than the five-legged transformer for type A, E, and G sags because the central leg is shorter than the outer legs.
2) Depth Influence: The study of the depth influence leads
to the same conclusions as those obtained for the five-legged
transformer.
V. CONCLUSION
The effects on the current peaks of symmetrical and unsymmetrical voltage sags in Wye G-Wye G three-phase transformers
have been analytically studied by considering that fault clearing
is produced simultaneously in all of the phases. Analytical expressions for the total magnetic flux and the inrush current when
the transformer is saturated have been obtained. This saturation
is produced when voltage is recovered. This simple model allows the behavior of the current of the five- and the three-legged
transformers to be explained when a sag is produced. These analytical expressions also allow the current peak value to be analytically estimated. It can be observed that the obtained current
peaks are higher than those obtained when it is considered that
voltage recovery can only be produced in the current zeros [4].
Thus, following the analytical process indicated in this paper,
this consideration must be taken into account in future studies
to obtain more precise results.
APPENDIX
DC COMPONENT OF THE TRANSFORMER
TOTAL MAGNETIC FLUXES
The expression of the dc component of the transformer total
magnetic fluxes when voltage is recovered can be obtained from
and
(15) by considering
with
(31)
which can be rewritten as follows:
(32)
Fig. 9. Three-phase three-legged transformer: analytical results of the initial
point-on-wave influence on the current peak for the seven sag types.
1) Duration and Initial Point-on-Wave Influence: As in the
case of the five-legged transformer, Table II shows the values
and required to avoid obtaining current peaks, and the
of
to obtain the maximum current peaks when
is
values of
.
fixed
Equation (32) can be rewritten by using duration and the initial point-on-wave as variables
(33)
Thus, for the different sag types, the dc component of the total
magnetic flux can be obtained from expression (33) and by con-
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007
sidering
and the values of
• Type A
and
and
,
of Table I.
,
Using trigonometric relations, the above expression can be
rewritten as
(34)
(39)
• Type B
• Type E
(35)
• Type C
(40)
• Type F
(36)
Using trigonometric relations, the above expression can be
rewritten as
(41)
(37)
Using trigonometric relations, the above expression can be
rewritten as
(38)
(42)
• Type D
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SÁINZ et al.: THEORETICAL CALCULATION OF INRUSH CURRENTS IN CORE TRANSFORMERS
• Type G
995
[5] L. Guasch, F. Córcoles, J. Pedra, and L. Sáinz, “Effects of symmetrical
voltage sags on three-phase three-legged transformers,” IEEE Trans.
Power Del., vol. 19, no. 2, pp. 875–883, Apr. 2004.
[6] The MathWorks, MATLAB 5.3 and Simulink 3.0. Natick, MA, 1999.
Luis Sáinz was born in Barcelona, Spain, in 1965. He
received the B.S. degree in industrial engineering and
the Ph.D. degree in engineering from the Universitat
Politècnica de Catalunya, Barcelona, Spain, in 1990
and 1995, respectively.
Since 1991, he has been Professor in the Department of Electrical Engineering, Universitat Politècnica de Catalunya. His main field of research is power
system quality.
(43)
Using trigonometric relations, the above expression can be
rewritten as
(44)
Felipe Córcoles was born in Almansa, Spain, in
1964. He received the B.S. degree in industrial engineering and the Ph.D. degree in engineering from
the Universitat Politècnica de Catalunya, Barcelona,
Spain, in 1990 and 1998, respectively.
Currently, he is a Professor in the Department
of Electrical Engineering, Universitat Politècnica
de Catalunya, where he has been since 1992. His
research interests are electric machines and power
system quality.
Joaquín Pedra (S’85–M’88) was born in Barcelona,
Spain, in 1957. He received the B.S. degree in industrial engineering and the Ph.D. degree in engineering
from the Universitat Politècnica de Catalunya,
Barcelona, Spain, in 1979 and 1986, respectively.
Since 1985, he has been Professor in the Department of Electrical Engineering, Universitat Politècnica de Catalunya. His research interest lies in the
areas of power system quality and electric machines.
REFERENCES
[1] M. H. J. Bollen, Understanding Power Quality Problems: Voltage Sags
and Interruptions. Piscataway, NJ: IEEE Press, 2000.
[2] E. Styvaktakis, M. H. J. Bollen, and I. Y. H. Gu, “Transformer saturation after a voltage dip,” IEEE Power Eng. Rev., vol. 20, no. 4, pp.
62–64, Apr. 2000.
[3] M. H. J. Bollen, “Voltage recovery after unbalanced and balanced
voltage dips in three-phase systems,” IEEE Trans. Power Del., vol. 18,
no. 4, pp. 1376–1381, Oct. 2003.
[4] J. Pedra, L. Sáinz, F. Córcoles, and L. Guasch, “Symmetrical and unsymmetrical voltage sag effects on three-phase transformers,” IEEE
Trans. Power Del., vol. 20, no. 2, pt. 2, pp. 1683–1691, Apr. 2005.
Luis Guasch was born in Tarragona, Spain, in 1964.
He received the B.S. degree in industrial engineering
and the Ph.D. degree in engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in
1996 and 2006, respectively.
Currently, he is a Professor in the Department of
Electronic, Electrical, and Automatic Engineering,
Universitat Rovira i Virgili, Tarragona, where he has
been since 1990. His research interests are electric
machines and power system quality.
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