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Electric Power Systems Research 79 (2009) 417–425
Contents lists available at ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Parameter estimation of a transformer with saturation using inrush
measurements
S. Bogarra a , A. Font a , I. Candela a , J. Pedra b,∗
a
b
Department of Electrical Engineering, ETSEIT-UPC, Colom 1, 08222 Terrassa, Spain
Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain
a r t i c l e
i n f o
Article history:
Received 29 August 2007
Received in revised form 22 February 2008
Accepted 7 August 2008
Available online 30 September 2008
Keywords:
Transformer modeling
Parameter estimation
Inrush current
a b s t r a c t
This paper presents a method to compute the parameters of a transformer model with saturation using the
voltage and current waveforms of an inrush test and a no-load test. The transformer is modeled with their
electric and magnetic equivalent circuits and a single-valued function that characterizes its non-linear
magnetic behavior. A 3-kVA single-phase transformer and a 5-kVA three-phase three-legged transformer
have been tested in the laboratory. The method to obtain the parameters of the non-linear flux–current
relation that characterize the saturation has been described in the paper. The analytical function used to
adjust the experimental measurements fits them very well.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Transformer modeling is an important subject in transient
studies because an accurate representation of saturation curve is
important for analyzing transformer energization and overvoltages
caused by ferrorresonance [1,2]. The majority of the transformer
models in the bibliography use many parameters to define the nonlinear magnetic circuit [3–5]. A simple model for a three-phase, two
winding transformer with a three-legged iron-core is proposed in
reference [6]. The equivalent circuit used is this paper is similar
to that proposed in [6], but there are some differences between
both circuits: the estimation of transformer saturation characteristics is realized from inrush test and no-load test, and the non-linear
function that represents the iron core saturation is different.
Standard tests usually do not drive transformer cores into deep
saturation and may lead large errors in the estimation of the nonlinear behavior of the saturated reluctances, affecting significantly
the estimation of transformer inrush currents. A measurement
method based on inrush current waveforms has been proposed
in [7] to estimate transformer saturation curves. The proposed
method to estimate the transform non-linear characteristics is
easy-to-use than the presented in [7] and an analytical function is
used to accurately adjust the saturation curve. The measurements
are the inrush test and the no-load test. The non-linear reluctances
which take into account the effect of limb and yoke fluxes in the
three-phase transformer are calculated from single-phase test on
each leg.
2. Transformer model
2.1. Single-phase transformer model
The equivalent electric circuit of single-phase transformer is
shown in Fig. 1, where Rp , Rs , Ldp , Lds are the winding resistances and
constant leakage inductances; the shunt resistance RFe accounts
for the core-losses and the induced voltages due to the core magnetic fluxes across the winding are ep for primary voltage and es for
secondary voltage [6].
Fig. 2 shows the proposed magnetic equivalent circuit of singlephase transformer, where Np ipe and Ns is are the primary/secondary
magnetomotive forces (the primary magnetomotive force depends
on the current ipe = ip − ipR , ipR is the current of the core-losses
resistance, RFe , in the electric equivalent circuit of Fig. 1); is the
non-linear reluctance of the iron, which depends on its own magnetic potential, f = (f).
The equations of the single-phase transformer [6] using the core
fluxes linked by the primary windings, Np = p , are
up =
∗ Corresponding author. Tel.: +34 934016728.
E-mail addresses: bogarra@ee.upc.es (S. Bogarra), fontp@ee.upc.es (A. Font),
candela@ee.upc.es (I. Candela), pedra@ee.upc.es (J. Pedra).
0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2008.08.009
us =
Rp + Ldp
Rs + Lds
d
dt
d
dt
ip +
is +
dp
;
dt
1 dp
;
rt,w dt
ep =
dp
;
dt
es =
1 dp
rt,w dt
(1)
418
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
Fig. 1. Electric equivalent circuit of a single-phase transformer.
where rt,w is the winding turn ratio (rt,w = Np /Ns ), which is measurable in the laboratory.
The magnetic relation from Fig. 2 is
Fig. 3. Magnetic equivalent circuit of a three-phase three-legged transformer.
Np ipe + Ns is = fs
(2)
The winding ratio was measured using a tertiary winding with a
known number of turns (built for this measurement). After applying a voltage to the primary winding and measuring the voltage at
the tertiary winding, the number of turns in the primary winding
was calculated. The turns in the secondary winding were calculated
in a similar way. Finally, the winding ratio thus obtained was tested
with the relation between the voltage in the primary winding and
the voltage induced in the secondary winding.
2.2. Three-phase transformer model
Fig. 3 shows the proposed magnetic equivalent circuit of threephase three-legged transformer. In this case, we need to add the
reluctance of the air path. It can be observed that the yoke reluctances have been added to the outer leg reluctances, and the
reluctance of the air, d , has been considered constant [6]. Fig. 4
shows the electric circuit of a three-phase transformer with a Wye
to ground-Delta connection. The electric relations of the transformer windings are
upk =
usk
epk
Rp + Ldp
d
ipk +
dpk
(3)
(k = a, b, c)
(4)
The electrical connections of the three-phase windings in the
transformer of Fig. 4, imposes the following conditions on the winding voltage of the primary side and the secondary side:
upa = vAN ;
upb = vBN ;
usb = vba ;
usc = vcb
upc = vCN ;
usa = vac ;
Fig. 2. Magnetic equivalent circuit of a single-phase transformer.
In the bibliography, there are many different methods for
approaching the non-linear behavior of the transformer by means
of an anhysteretic magnetization curve (single-valued curve) such
as power series [8], piecewise linear curves [3] or arc tangent function [9].
In this paper, each leg in the three-phase transformer is viewed
as a separate magnetic circuit and the proposed function to represent the core non-linear behavior is a functional relationship
between the magnetic potential in the leg and the flux through
it
f = (f )
(5)
(6)
being the analytical function selected for the non-linear reluctance,
−1
and the magnetic flux and current of Fig. 3 are
Np ipek + Ns isk = fk − fd
a + b + c + d = 0
2.3. Saturation curve
(f )
(k = a, b, c)
dt
dt
d
1 dpk
= Rs + Lds
i +
dt sk rt,w dt
dpk
1 dpk
=
; esk =
rt,w dt
dt
where vAN , vBN and vCN are the phase to ground voltages at the
primary side and vac , vba and vcb are the phase to phase voltages in
the secondary side.
=
K1 − K2
p 1/p
(1 + (|f |/f0 ) )
+ K2
(7)
where K1 , K2 , p and f0 are parameters which allow this single-valued
function to be fitted to the ( − f) transformer saturation curves,
Fig. 5. These four parameters have a clear physical interpretation:
• K1 and K2 are defined by the slope in the linear and the non-linear
zones of the ( − f) curve.
• p influences the shape of the curve.
• f0 is the magnetic potential where saturation begins.
2.4. Linear parameters determination
The short circuit tests can provide fairly accurate estimation
of the transformer winding resistances and leakage inductances.
The magnetizing branch resistance representing losses in the iron
Fig. 4. Electric equivalent circuit of a Wye to ground-Delta transformer.
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
419
Fig. 6. Electric equivalent circuit of a single-phase transformer.
Sb , is defined as
Sb =
Fig. 5. ( − f) characteristic of the proposed saturation curve.
Table 1
Data of the single-phase transformer (3-kVA, 220 + 220/110 + 110 V single-phase
transformer)
Winding resistances and leakage inductances (p.u.)
32.3
Z1b ()
8.1
Z2b ()
3.673E−3
Rp
Ldp
4.193E−6
3.673E−3
Rs
Lds
4.193E−6
(8)
The impedance base is
Zb =
U2
Ub
= N.
Ib
Sb
(9)
In the three-phase transformer, the voltage and current base values are the same as in the single-phase values, but the base power
is defined as
Sb =
3Ub Ib
= 3UN IN ,
2
(10)
and the impedance base is
Non-linear inductance coefficients (p.u.)
M1
37.92
2.217
M2
0.0365
M3
3.02
p1
1.38
p2
i1
0.0225
i2
0.4225
Core-losses resistance (p.u.)
RFe
Ub Ib
= UN IN .
2
Zb =
U2
Ub
= N.
Ib
Sb
(11)
In the single-phase transformer the base values are Sb = 1.5 kVA,
U1b = 311.1 V and in the three-phase transformer Sb = 2.5 kVA,
U1b = 311.1 V.
2.5. Single-phase equivalent circuit
27.37
core of the transformer can be determined through the open circuit
tests. The reluctance of the air path that appears in the magnetic
equivalent circuit of a three-phase three-legged transformer can be
estimated through a zero sequence test. Tables 1 and 2 show the
linear parameters of the 3-kVA single-phase transformer and the
5-kVA three-phase transformer studied in the paper. In the singlephase transformer,
the base values used in
√
√the paper are the peak
voltage, Ub = 2UN and peak current, Ib = 2IN , UN being the phase
to neutral nominal voltage and IN nominal current. The base power,
Fig. 6 shows an electric circuit where the electric circuit of Fig. 1
and the magnetic circuit of Fig. 2 are unified. In this circuit has been
defined the magnetizing current, im , as
Np im = Np ip +
is
(12)
rt,w
where the current of the core-losses resistance, ipR , has been considered negligible. This relation with (2) and (6), produce the
flux–current relation
M(im )im = p
(13)
Table 2
Data of the three-phase transformer (5-kVA, 220 + 220/110 + 110 V three-phase three-leg transformer)
Winding resistances and leakage inductances (p.u.)
Z1b ()
Z2b ()
Rp
Ldp
Rs
Lds
M1
Non-linear inductance coefficients (p.u.)
6.27
Ma (i)
14.59
Mb (i)
7.50
Mc (i)
58.1
14.5
0.0118
0.0118
7.125E−6
7.125E−6
M2
M3
p1
p2
i1
i2
0.6673
1.568
0.7249
2.312E−3
5.651E−3
2.918E−3
18.61
17.45
17.34
1.627
1.657
1.724
0.132
0.0603
0.113
1.357
0.5556
1.290
Air path linear reluctance and core-losses resistance (p.u.)
d
RFe
2190
24.45
420
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
where the non-linear magnetizing inductance is
M(im ) =
M1 − M2
p 1/p
(1 + (|im |/i0 ) )
+ M2
(14)
The relationship between reluctance and magnetizing inductance is
M = Np2 −1 ⇒ M1 = Np2 K1 ;
M2 = Np2 K2
(15)
In the three-phase transformer, only is possible the use of a circuit like Fig. 6 in the transformer bank, where d = 0, which implies
that the magnetic potential fd is null, fd = 0. As can be seen in Fig. 3,
when d = 0 the three-phase three-legged transformer is equivalent to three independent magnetic circuits like the single-phase
transformer of Fig. 2. The consequences of that fd is not null in the
three-phase three-legged transformer (Fig. 3), will be discussed in
Section 5.
Fig. 7. Relation between the primary inrush current and the secondary delta current
in the 5-kVA three-phase three-legged transformer.
3. Single-valued saturation curve determination
The focus of this paper is to determine the non-linear characteristics from the voltage waveform at the transformer secondary
winding and current waveform at the transformer primary winding
produced during inrush-test and no-load test. The flux evaluation
depends on the secondary voltage waveform. The parameter estimation method presented here is easier to use than the presented
in [7], which works with voltage and current waveforms of the
primary winding produced during inrush test and no-load test.
3.1. Single-phase transformer
Flux and current are the waveforms needed to find non-linear
inductances. The main difference of the experimental method proposed here from [7] is that the flux is estimated from the voltage
measured in the non-connected winding, where the current is null
and Eq. (1) results
usk =
1 dpk
rt,w dt
(16)
The measured flux could be evaluated easily trough numerical
integration as
t
p (t) = rt,w
us d + p0
(17)
0
In the single-phase inrush test on the three-phase transformer
the use of the secondary winding voltage and current has the advantage that when the secondary side is connected in delta, the current
idelta is very lower than the inrush current in the primary side.
Therefore, the experimental errors of measuring the parameters
Rs and Ls have a lower influence in (18) than the use of primary side
measurements, because the idelta secondary current is very lower
than the inrush primary current. The relation between the primary
inrush current and the secondary delta current was measured in the
laboratory by ten different tests. The ratio between the maximum
peaks of the secondary delta current in p.u. with the maximum
peaks of the primary inrush current in p.u. was measured in 10 different tests, results being a mean value of 0.105 with a standard
deviation of 0.009. Fig. 7 plots the primary inrush current and the
secondary delta current obtained in a laboratory test performed on
the 5-kVA three-phase transformer.
3.3. Inrush test
Fig. 8 shows the core magnetic flux, p , calculated using (17) in
an inrush test made on the 3-kVA single-phase transformer. Their
linear parameters are shown in Table 1.
Fig. 9 shows the two first cycles of the inrush current obtained in
an inrush test on the 3-kVA single-phase transformer. Fig. 10 shows
eight cycles of the inrush current obtained in a single-phase test
where rt,w is the winding turn ratio, us is the voltage measured in
secondary winding and p0 is the residual flux.
3.2. Three-phase transformer
For simplicity, the three-phase transformer is supposed that is a
transformer bank, where the potential magnetic fd is null. The modifications owed that in the three-legged transformer fd is not null
are studied in Section 5. When three-phase transformer is analysed,
Eq. (1) is applied three times, one to each limb to obtain the three
non-linear core reluctances.
In the three-phase transformers there are two different cases,
when the windings are Wye connected or Delta connected. For the
first case, the flux also can be calculated with (17). In the Delta
connected case, the secondary windings current can be not null,
therefore, Eq. (3) results
pk (t) = rt,w
t
0
usk − Rs + Lds
d
dt
idelta
d + p0k
(18)
Fig. 8. Core magnetic flux linked in the inrush test by the 3-kVA single-phase transformer.
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
Fig. 9. Inrush current of the 3-kVA single-phase transformer.
on the 5-kVA three-phase transformer. Their linear parameters are
shown in Table 2. This test is repeated for each phase in the threephase transformer. In this test, the transformer is without any load,
then the inrush current is the magnetizing current (the current of
the core-losses resistance, ipR , is negligible).
421
Fig. 11. Magnetizing current vs. time curve obtained in the no-load test 3-kVA
single-phase transformer.
4. Residual flux determination
Fig. 11 shows the magnetizing current versus the time in the
no-load test made on the 3-kVA single-phase transformer. To estimate the parameters of the non-linear inductance, it is necessary
previously draw a single-valued (ps − ips ) curve which is averaged
from the measured (pm − ipm ) hysteresis loop. Fig. 12 shows the
measured (pm − ipm ) hysteresis loops (full line) in the single-phase
transformer and the estimated (ps − ips ) single-valued curve (dotted line) for the no-load test.
In Fig. 12, the flux and the current are expressed in p.u.,
being the base values for the 3-kVA single-phase transformer,
Sb = 1.5 kVA, U1b = 311.1 V and U2b = 155.6 V. Therefore, the primary base flux is 1b = 0.99 Wb and the primary base current is
I1b = 9.6 A. The base values for the 5-kVA three-phase transformer
are Sb = 2.5 kVA, U1b = 311.1 V and U2b = 155.6 V. Therefore, the primary base flux is 1b = 0.99 Wb, and the primary base current is
I1b = 5.4 A.
The single-valued saturation curve is a symmetrical function
with respect to the origin as can be observed in Fig. 12. In the
inrush test, the main saturation only is produced in a side of the
saturation curve, therefore, their (pm − ipm ) hysteresis loops does
not have symmetry and it is necessary to make the residual flux
determination because the value of p0 in Eq. (17) is unknown. This
problem is related with the initial flux in the transformer and the
point-of-wave of the voltage in the instant of connexion. The worst
case is usually produced when the voltage is near to the zero value.
The instant of connection is random in every test. In this paper, only
the test that produced a significant inrush current is studied.
Fig. 13 shows the flux–current loops (pm − ipm ) for the no-load
test and the inrush test assuming initial zero flux in (17), p0 = 0. The
flux–current loop of the inrush test has been made using the first
period that include the maximum peak current. It can be observed
that the inrush currents can be three orders of magnitude greater
than the no-load currents.
Graphically, as is shown in Figs. 13 and 14, the residual flux
determination corresponds to vertically shift the flux–current loop
obtained from the inrush test until it overlaps the flux–current loop
from the no-load test. The residual flux is fitted in order to match the
hysteresis loop for inrush test with the hysteresis loop for no-load
test, as can be observed in Fig. 14.
Fig. 10. Inrush current of the 5-kVA three-phase three-legged transformer.
Fig. 12. Flux–current loop obtained from the no-load test (full line) and their singlevalued curve (dotted line) by the 3-kVA single-phase transformer.
3.4. No-load test
422
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
Fig. 13. Estimation of the residual flux from the flux–current loops obtained from
the no-load test and the inrush-test in the 3-kVA single-phase transformer.
This method is repeated independently in each phase in the case
of a three-phase transformer. The residual flux obtained for each
phase of the 5-kVA three-phase transformer is graphically evaluated in Fig. 15, where the single-valued flux–current curve obtained
from the no-load test is represented with a dotted line and the curve
of the inrush test with a full line.
5. Saturation curve parameters estimation
5.1. Single-phase and three-phase transformer bank
The magnetic circuit of the three-phase transformer bank corresponds in Fig. 3 to the case where d = 0, which implies that the
magnetic potential fd is null, fd = 0. This makes that the parameter
estimation in the three-phase transformer bank is identical to the
single-phase transformer.
A close observation of the single-valued saturation curves of
Fig. 15 show that the experimental measurements suggest that
there are a fast saturation, for values of current lower than 0.1 p.u.,
and later an slow saturation until current values of the order of
2 p.u. Therefore, the function used to adjust the experimental data
is the sum of two saturation curves as (7) that can be expressed as
−1
k (f )
=
K1k − K2k
(1 + (|fk |/f1k )
p1k 1/p1k
)
+
K2k − K3k
(1 + (|fk |/f2k )
p2k 1/p2k
)
+ K3k
(19)
Fig. 15. Residual flux estimated with the flux–current curves obtained from no-load
test (dotted line) and inrush test (full line) in the 5-kVA three-phase three-legged
transformer.
In the three-phase transformers, defining the magnetizing currents of each leg as
Np imk = Np ipk +
isk
rt,w
(20)
the non-linear magnetizing inductance of each leg can be expressed
as
M(imk )=
Fig. 14. Inrush flux–current loop obtained considering the calculated residual flux
(broken line) in the 3-kVA single-phase transformer.
M1k − M2k
(1 + (|imk |/i1k )
p1k 1/p1k
)
+
M2k −M3k
(1 + (|imk |/i2k )
p2k 1/p2k
)
+M3k
(21)
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
423
Fig. 17. Measured saturation curve (full line) and adjusted saturation curve (dotted
line) of the 3-kVA single-phase transformer.
formers are showed in Table 1. In order to evaluate the fitted curves,
Fig. 17 shows the measured saturation curve (full line) and the analytical saturation curve (dotted line) for the 3-kVA single-phase
transformer and them are approximately identical.
5.2. Three-legged transformer
Fig. 18 shows the magnetic circuit of the three-phase threelegged transformer in the single-phase inrush test. In this case, the
potential magnetic fd is not null. The relation between the fluxes is
a = b + c + d
(24)
which indicates that although a is saturated, b and c are near
always in the linear zone. Considering that the reluctances b and
c are in the linear zone, their value is much lower than that of the
air reluctance, d . Then, the flux flowing through the air in Eq. (24),
d , is not significant and is considered null.
Therefore, making the hypothesis that in the single-phase inrush
test the two legs that are not excited can be considered linear, the
saturation curve parameters can be calculated in two steps:
• Calculate parameters as a transformer bank.
• Correct the parameters, taking into account the influence of the
two legs that are not excited.
Fig. 15 shows the residual flux calculated for each leg considering
the transformer as a transformer bank. With this information the
non-linear parameters have been calculated. These parameters are
Fig. 16. Curve with fast saturation, MF (im ), curve with the slow saturation MS (im )
and addition curve for the 3-kVA single-phase transformer.
where k = a, b, c. As has been commented before, this non-linear
function can be interpreted as the addition of a fast saturation
MF (imk ) and a slow saturation MS (imk ) defined as
MF (imk ) =
MS (imk ) =
M1k − M2k
(1 + (|imk |/i1k )
(22)
p1k 1/p1k
)
M2k − M3k
(1 + (|imk |/i2k )
p2k 1/p2k
)
+ M3k
(23)
Fig. 16 shows for the 3-kVA single-phase transformer the individual value of each function, MF (im ) and MS (im ), and their addition,
MF (im ) + MS (im ). The non-linear parameters of the analysed trans-
Fig. 18. Magnetic equivalent circuit of a three-phase three-legged transformer in
the single-phase excitation test.
424
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
labelled with the subscript M. The proposed hypothesis implies that
from the calculated parameters, M1kM , M2kM , M3kM , p1kM , p2kM , i1kM
and i2kM , (k = a, b, c), only the coefficients M1k must be modified
(they corresponds to the linear zone, imk = 0).
When the inrush test is made in the leg a, the others legs are in
the linear zone, then
(0) ≈ K1b ;
−1
b
−1
c (0) ≈ K1c
(25)
Fig. 18 indicates that the relation between the three linear reluctances and the reluctance measured is
aM (im ) = a (im ) +
1
(1/b (0)) + (1/c (0))
(26)
Fig. 20. Adjusted saturation curve with a single saturation (dotted line) and adjusted
saturation curve with double saturation (full line) in the 5-kVA three-phase threelegged transformer.
that expressed as magnetizing inductances in the linear zone
(imk = 0) is
1
1
1
=
+
M1aM
M1a
M1b + M1c
(27)
where M1aM is the coefficient calculated as a transformer bank.
Applying this relation for each single-phase test, result the nonlinear equations
1
1
1
=
−
,
M1a
M1aM
M1b + M1c
1
1
1
=
−
,
M1b
M1bM
M1a + M1c
1
1
1
=
−
M1c
M1cM
M1a + M1b
(28)
Those non-linear equations can be solved easily using an iterative algorithm. The final parameters obtained for the three-phase
three-legs transformer are in Table 2. Fig. 19 shows the measured
saturation curves (full line) and the analytical saturation curves
(dotted line) for the 5-kVA three-phase transformer. Also in this
case, both curves are approximately identical.
5.3. Double saturation consequences
Fig. 20 shows the non-linear saturation curve (dotted line)
obtained using only the non-load test for the 5-kVA three-phase
transformer, which clearly is very different to the measured curve
(full line). In Ref. [6] the non-linear characteristic is calculated using
only a no-load test on each leg. This method produces an important error for high excitation currents as can be observed in Fig. 20.
Ref. [10] also proposes the saturation curve determination only
using no-load test measurements. The manufacturer data for calculate the saturation curves are usually a non-load test with the
r.m.s.-voltage as a function of the r.m.s.-current. Therefore, this data
produces the same problem as represented in Fig. 20.
Accurate estimation of the transformer parameters for the low
and high excitation currents is important for studying the impact
of the transformer energization and its inrush current.
6. Conclusions
Fig. 19. Measured saturation curve (full line) and adjusted saturation curve (dotted
line) for the 5-kVA three-phase three-legged transformer.
This paper describes a practical method to adjust the non-linear
saturation curve using an inrush test and a no-load test. The method
proposes to made voltage measurements from the secondary side.
The use of no-load test and inrush test ensures the estimation of
S. Bogarra et al. / Electric Power Systems Research 79 (2009) 417–425
the non-linear characteristics at low and high saturation levels. The
experimental measurements show a fast saturation and a slow saturation in the shape of the non-linear saturation curve. This fact
implies that the use of only the no-load test for the saturation
curve determination can produce an erroneous parameter estimation. The analytical function proposed to adjust the experimental
measurements fits them very well in the case of 3-kVA single-phase
transformer and 5-kVA three-phase transformer.
Acknowledgment
The authors acknowledge the financial support of the Comisión
Interministerial de Ciencia y Tecnología (CICYT) under the project
(DPI2004-00544).
References
[1] R.J. Rusch, M.L. Good, Wyes and wye nots of three-phase distribution transformer connections, Proc. IEEE Conf. Paper, No. 89CH2709-4-C2, 1989.
[2] R.S. Bayless, J.D. Selman, D.E. Truax, W.E. Reid, Capacitor switching and transformer transients, IEEE Trans. Power Deliv. 3 (1) (1988) 349–357.
[3] X. Chen, S. Venkata, A three-phase three-winding core-type transformer model
for low-frequency transient studies, IEEE Trans. Power Deliv. 12 (2) (1997)
775–782.
[4] X. Chen, A three-phase multi-legged transformer model in ATP using the
directly formed inverse inductance matrix, IEEE Trans. Power Deliv. 11 (3)
(1996) 1554–1562.
[5] C.M. Arturi, Transient simulation and analysis of a three-phase five-limb stepup transformer, IEEE Trans. Power Deliv. 6 (1) (1991) 196–207.
[6] J. Pedra, L. Sainz, F. Córcoles, R. López, M. Salichs, PSPICE computer model of
a nonlinear three-phase three-legged transformer, IEEE Trans. Power Deliv. 19
(1) (2004) 200–207.
425
[7] S.G. Abdulsalam, W. Xu, W.L.A. Neves, X. Liu, Estimation of transformer saturation characteristics from inrush current waveforms, IEEE Trans. Power Deliv. 21
(1) (2006) 170–177.
[8] F. Leon, A. Semlyen, Complete transformer model for electromagnetic transients, IEEE Trans. Power Deliv. 9 (1) (1994) 231–239.
[9] C. Pérez-Rojas, Fitting saturation and hysteresis via arctangent functions, IEEE
Power Eng. Rev. 20 (11) (2000) 55–57.
[10] W.L.A. Neves, H.W. Dommel, On modelling iron core nonlinearities, IEEE Trans.
Power Syst. 8 (3) (1993) 417–425.
Santiago Bogarra Rodríguez was born in Gavá, Spain, on 8 May 1966. He received
his Ph.D. in electrical engineering from the Polytechnic University of Catalonia,
Barcelona, Spain, in 2002. He has been associate professor of electrical engineering at the Polytechnic University of Catalonia since 1997. His research interest lies
in the areas of power system transients and insulation coordination.
Antoni Font was born in Spain in 1955. He received his B.S. degree in industrial
engineering from the Universitat Politecnica de Catalunya, Barcelona, Spain. He is
currently pursuing the Ph.D. degree at the Universitat Politecnica de Catalunya. Currently, he is an assistant professor in the electrical engineering Department of the
Universitat Politecnica de Catalunya, where he has been since 1993. His research
interests are electric machines and power system quality.
J. Ignacio Candela was born in Bilbao (Spain) in 1962. He received his B.S. degree in
industrial engineering in Engineering from the Universitat Politècnica de Catalunya,
Barcelona, Spain, in 2000. Since 1991 he has been professor in the electrical engineering Department of the Universitat Politècnica de Catalunya. His main field of
research is power system quality and electrical machines.
Joaquín Pedra was born in Barcelona (Spain) in 1957. He received his B.S. degree
in industrial engineering and his 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 Electrical Engineering Department of the Universitat
Politècnica de Catalunya. His research interest lies in the areas of power system
quality and electrical machines.