Four-Winding Transformer and Autotransformer Modeling
For Load-Flow and Short-Circuit Analysis, Part I: Theory
Carlos A. Muñoz
Universidad Nacional de Ingeniería, Lima, Perú
Alberto Rojas
COVIEM S.A., Lima, Perú
Abstract
Purpose- This paper presents the theory for computing the positive and zero sequence admittance
matrices of three-phase four-winding transformers and autotransformers. These matrices are required
in load-flow and short-circuit calculations.
Methodology- The positive and zero sequence impedances between windings, which are available from
standard transformer test reports, are modeled as mutual impedances between each pair of windings.
These mutual impedances are used to build a 4x4 impedance matrix taking into account the winding
voltage ratios. Also a 4x4 matrix of voltage and current relationship is built. Using these two matrices,
the positive sequence admittance matrix of a transformer or autotransformer is computed. For zero
sequence admittance matrices, the connection group and grounding impedance of each winding is
also taken into account.
Research implications- This theory can be useful to upgrade load-flow and short-circuit software to
handle four-winding transformers and autotransformers, in a reliable and straight-forward manner.
Key words: Four-windings, transformers, autotransformers, sequence networks, load-flow, shortcircuit
I.
Introduction
Four-winding transformers are encountered in high voltage power networks. The Peruvian power
grid has 9 of these transformers. Load-flow and short-circuit software usually has models only for
two-winding and three-winding transformers and autotransformers. The theory for modeling twowinding and three-winding transformers is presented in detail in the classic T&D book of the
Westinghouse [1]. Reference [2] presents the Extended Cantilever Model (ECM) for four-winding
transformers. The ECM model can be used for the positive sequence network. Nevertheless,
reference [2] does not provide enough detailed models for the zero sequence network. A search of
zero sequence models for four-winding transformers and autotransformers carried out in internet
was not successful. Therefore, a research was undertaken to develop complete models of fourwinding transformers and autotransformers to be used for load-flow and short-circuit analysis.
Section II presents the proposed approach. Section III shows the application of the proposed
approach to compute the positive sequence admittance matrix of a four-winding transformer. In
Section IV the proposed approach is used to obtain the zero sequence admittance matrices of
some types of four-winding transformers and autotransformers. Section V presents the validation of
the proposed approach. Finally, in Section VI the main conclusions of this research effort are
outlined.
The proposed approach can be applied also to transformers and autotransformers with more than
four windings. The theory described in this paper can be easily generalized for such a purpose.
II.
Proposed Approach
The standard approach for modeling a two-winding transformer is shown in Figure 1a; Zps is the
short-circuit impedance in ohms measured at the primary winding with the secondary winding shortcircuited, Np is the number of turns of the primary winding and Ns is the number of turns of the
secondary winding. The proposed approach converts Zps to a mutual impedance Mps using
equation (1):
Mps = (½) * Zps * Ns / Np
(1)
Page 1 of 12
Figure 1a
Figure 1b
In the proposed approach, current Ip at the primary winding provokes a voltage drop at the
secondary winding equal to Mps * Ip, and current Is at the secondary winding provokes a voltage
drop at the primary winding equal to Mps * Is. Therefore, the following relationship is satisfied:
Vp + Mps * Is
Ep
Es Vs + Mps * Ip
------------------- = ----- = ----- = -----------------Np
Np
Ns
Ns
(2)
Vp is the voltage at the terminals of the primary winding, Vs is the voltage at the terminals of the
secondary winding, Ip and Is are the currents flowing into the primary and secondary windings
respectively, and finally Ep and Es are voltages applied to an ideal transformer of ratio Np:Ns.
Using equations (1) and (2), it can be shown that if the secondary winding is short-circuited (Vs
equal to zero):
Vp / Np = 2 * Mps * Ip / Ns
(3)
Vp = Zps * Ip
(4)
The proposed mutual impedance Mps allows to reproduce the 60 Hz or 50 Hz behavior of a twowinding transformer under a short-circuit condition. The open-circuit test provides a measured Ym
magnetizing admittance in siemens that can be added to the model connecting it directly to voltage
Ep.
III
Positive Sequence Admittance Matrix – Four-Winding Transformer
Figure 2 shows a four-winding transformer. The windings are named primary, secondary, tertiary
and quaternary. N1 is the secondary to primary voltage ratio, N2 is the tertiary to primary voltage
ratio and N3 is the quaternary to primary voltage ratio. The short-circuit tests of this type of
transformer give the following impedances: Zps, Zpt, Zpq (in ohms measured at the primary
winding), Zst, Zsq (in ohms measured at the secondary winding), and Ztq (in ohms measured at the
tertiary winding). Also, the open-circuit test gives the magnetizing admittance Ym (in siemens
viewed at the primary side).
Figure 2
Page 2 of 12
Using impedances Zps, Zpt, Zpq, Zst, Zsq and Ztq, the following six mutual impedances between
windings can be computed:
Mps = (½) * Zps * N1
Mpt = (½) * Zpt * N2
Mpq = (½) * Zpq * N3
Mst = (½) * Zst * N2 / N1
Msq = (½) * Zsq * N3 / N1
Mtq = (½) * Ztq * N3 / N2
(5)
The following equations can be written relating voltages and currents at each winding:
Vp + Mps*Is + Mpt*It + Mpq*Iq = Ep
(6a)
Vs + Mps*Ip + Mst*It + Msq*Iq = Es = N1*Ep
(6b)
Vt + Mpt*Ip + Mst*Is + Mtq*Iq= Et = N2*Ep
(6c)
Vq + Mpq*Ip + Msq*Is + Mtq*It = Eq = N3 * Ep
(6d)
Ym * (Vp + Mps*Is + Mpt*It + Mpq*Iq) = Ym * Ep = Im
(6e)
Ip + N1 * Is + N2 * It + N3 * Iq = Im
(6f)
Im is the magnetizing current in amperes referred to the primary winding.
Using Vp, Vs/N1, Vt/N2 and Vq/N3 as variables for voltages, and Ip, N1*Is, N2*It and N3*Iq as
variables for currents (all voltages and currents are referred to the primary side), the following 4x4
matrix equation can be written using equations (6a) to (6f):
[VCR] * [V] = [ZMT] * [I]
(7)
Where [V] is a vector of 4 files containing: Vp, Vs/N1, Vt/N2 and Vq/N3, and [I] is a vector of 4 files
containing: Ip, N1*Is, N2*It and N3*Iq. The 4x4 matrix [VCR] (voltage and current relationship) is as
follows:
[VCR]
=
1
1
1
Ym
-1
0
0
0
0
-1
0
0
0
0
-1
0
ZMpt
ZMst
ZMtt
ICpt
ZMpq
ZMsq
ZMtq
ICpq
(8)
The 4x4 matrix [ZMT] is as follows:
[ZMT]
=
ZMpp
ZMsp
ZMtp
1
ZMps
ZMss
ZMts
ICps
(9)
The elements of matrix [ZMT] are:
ZMpp =Mps/N1
ZMsp = Mpt/N2
ZMps = -Mps/N1
ZMss = ((Mst – N2*Mps)/N1)/N2
ZMpt = ((Mst – N1*Mpt)/N2)/N1
ZMst= -Mpt/N2
ZMpq = ((Msq – N1*Mpq)/N3)/N1
ZMsq = ((Mtq – N2*Mpq)/N3)/N2
Page 3 of 12
ZMtp = Mpq/N3
ICps = 1 – (Ym*Mps/N1)
ZMts = ((Msq – N3*Mps)/N1)/N3
ICpt = 1 – (Ym*Mpt/N2)
ZMtt = ((Mtq – N3*Mpt)/N2)/N3
ICpq = 1 – (Ym*Mpq/N3)
ZMtq = -Mpq/N3
(10)
The 4x4 positive sequence admittance matrix of the transformer, called [Ytxf], can be now
computed as:
-1
[Ytxf] = [ZMT] * [VCR]
(11)
[Ytxf] * [V] = [I]
(12)
-1
Matrix [ZMT] is the inverse of matrix [ZMT]. Matrices [ZMT] and [VCR] are non-symmetric, but
matrix [Ytxf] is symmetric; this was verified by performing numerical experiments. Matrix [Ytxf] has
units of siemens (referred to the primary winding) and it can be converted to per unit by multiplying
it by the base impedance of the primary winding.
The equations presented in this section can be used to create MS-Excel spreadsheets, if the
resistive elements of Zps, Zpt, Zpq, Zst, Zsq, Ztq and Ym are neglected. MS-Excel allows inverting
and multiplying real matrices.
IV
Zero Sequence Admittance Matrix
In this section, the proposed approach is applied to several types of four-winding transformers and
autotransformers. A few cases of all possible combinations of winding connections and grounding
are covered. Interested readers can use this methodology to obtain the equations for other cases.
A) Four-Winding Transformer YgYgYgD
Figure 3 shows a four-winding transformer. The first three winding have a start connection and are
grounded with impedances Zgp, Zgs and Zgt. The fourth winding has a delta connection.
Figure 3
A four-winding transformer has six zero sequence impedances: Z0ps, Z0pt, Z0pq, Z0st, Z0sq and
Z0tq. In a similar manner as equation (5), six zero sequence mutual impedances can be defined:
M0ps, M0pt, M0pq, M0st, M0sq and M0tq. The following equations can be written relating zero
sequence voltages and currents:
Page 4 of 12
V0p – 3Zgp*I0p + M0ps*I0s + M0pt*I0t + M0pq*I0qd = E0p
(13a)
V0s – 3Zgs*I0s + M0ps*I0p + M0st*I0t + M0sq*I0qd = E0s = N1*E0p
(13b)
V0t – 3Zgt*I0t + M0pt*I0p + M0st*I0s + M0tq*I0qd = E0t = N2*E0p
(13c)
M0pq*I0p + M0sq*I0s + M0tq*I0t = E0qd = N3*E0p
(13d)
I0p + N1*I0s + N2*I0t + N3*I0qd = I0m = Y0m*E0p
(13e)
I0q is always zero, since the quaternary winding has a delta connection; but there is an I0qd current
that flows in this delta winding. The elements of [Y0txf] allows to compute also I0qd, but the actual
value of this current is I0qd/√3.
Using V0p, V0s/N1, V0t/N2 and V0qd/N3 as voltage variables, and I0p, N1*I0s, N2*I0t and N3*I0qd
as current variables; a 4x4 [ZMT0] matrix and a 4x4 [VCR0] matrix can be defined taking also into
account equations (13a) to (13e):
[VCR0]
=
[ZMT0]
=
1
1
1
Y0m
-1
0
0
0
0
-1
0
0
0
0
0
0
ZM0pp
ZM0sp
ZM0tp
IC0pp
ZM0ps
ZM0ss
ZM0ts
IC0ps
ZM0pt
ZM0st
ZM0tt
IC0pt
ZM0pq
ZM0sq
ZM0tq
IC0pq
(14)
(15)
The elements of matrix [ZMT0] are:
ZM0pp = (M0ps/N1) + 3Zgp
ZM0sp = (M0pt/N2)+3Zgp
ZM0ps = -(M0ps+(3Zgs/N1))/N1
ZM0ss = (-M0ps+(M0st/N2))/N1
ZM0pt = (-M0pt+(M0st/N1))/N2
ZM0st = (M0pt+(3Zgt/N2))/N2
ZM0pq = (-M0pq+(M0sq/N1))/N3
ZM0sq = (-M0pq+(M0tq/N2))/N3
ZM0tp = (M0pq/N3) + 3Zgp
IC0pp = 1 + Y0m*3Zgp
ZM0ts = ((M0sq/N3)-M0ps)/N1
IC0ps = 1 – (Y0m*M0ps/N1)
ZM0tt = ((M0tq/N3)-M0pt)/N2
IC0pt = 1 – (Y0m*M0pt/N2)
ZM0tq = (-M0pq/N3)
IC0pq = 1 – (Y0m*M0pq/N3)
(16)
-1
Now, the zero sequence admittance matrix [Y0txf] can be computed as [ZMT0] * [VCR0]. Matrix
[Y0txf] has units of siemens referred to the primary winding, and it can be converted to per unit
multiplying it by the base impedance of the primary winding.
B) Four-Winding Autotransformer AgAgYgYg
Figure 4 shows a four-winding autotransformer. The first two windings have a start autotransformer
connection and they are grounded with impedance Zgp. The third and fourth windings have a start
transformer connection and they are grounded with impedances Zgt and Zgq.
Page 5 of 12
Figure 4
The primary winding P is also called winding X. A winding Y is modeled as connected between
winding X and the secondary terminals. Therefore, in addition to zero sequence impedances Z0ps,
Z0pt, Z0pq, Z0st, Z0sq and Z0tq, impedances Z0xy, Z0yt and Z0yq can be defined. These
impedances can be computed using the following equations:
2
Z0xy = (N1/(N1-1)) * Z0ps
(17a)
Z0yt = N1*Z0ps + ((N1-1)/N1)*Z0st – (N1-1)*Z0pt
(17b)
Z0yq = N1*Z0ps + ((N1-1)/N1)*Z0sq – (N1-1)*Z0pq
(17c)
We can now compute the following zero sequence mutual impedances:
M0xy = (½) * Z0xy * (N1-1)
M0pt = (½) * Z0pt * N2
M0pq = (½) * Z0pq * N3
M0yt = (½) * Z0yt * N2/(N1-1)
M0yq = (½) * Z0yq * N3/(N1-1)
M0tq = (½) * Z0tq * N3/N2
(18)
Using mutual impedances of equation (18), the following equations relating zero sequence voltages
and currents can be derived:
V0p – 3Zgp*(I0p+I0s) + M0xy*I0s + M0pt*I0t + M0pq*I0q = E0x
(19a)
V0s – 3Zgp*(I0p+I0s) + M0xy*I0p + 2*M0xy*I0s + (M0pt+M0yt)*I0t +
(M0pq+M0yq)*I0q = E0s = N1*E0x
(19b)
V0t - 3Zgt*I0t + M0pt*I0p + (M0pt+M0yt)*I0s + M0tq*I0q = E0t = N2*E0x
(19c)
V0q – 3Zgq*I0q + M0pq*I0p + (M0pq+M0yq)*I0s + M0tq*I0t = E0q = N3*E0x
(19d)
I0p + N1*I0s + N2*I0t + N3*I0q = 0
(19e)
Using V0p, V0s/N1, V0t/N2 and V0q/N3 as voltages variables, and I0p, N1*I0s, N2*I0t and N3*I0q
(all voltages and currents are referred to the primary winding), matrices [ZMT0] and [VCR0] can be
built. The elements of [ZMT0] are:
ZM0pp = (M0xy+(N1-1)*3Zgp)/N1
ZM0sp = (M0pt/N2)+3Zgp
2
ZM0ps = ((2-N1)*M0xy+(N1-1)*3Zgp)/N1
ZM0ss = (((M0pt+M0yt)/N2)-M0xy+3Zgp)/N1
ZM0pt = (M0yt-(N1-1)*M0pt)/N1)/N2
ZM0st = -(M0pt+(3Zgt/N2))/N2
ZM0pq = (M0yq-(N1-1)*M0pq)/N1)/N3
ZM0sq = ((M0tq/N2)-M0pq)/N3
Page 6 of 12
ZM0tp = (M0pq/N3) + 3Zgp
IC0pp = 1
ZM0ts = ((M0pq+M0yq)/N3)-M0xy+3Zgp)/N1
IC0ps = 1
ZM0tt = ((M0tq/N3)-M0pt)/N2
IC0pt = 1
ZM0tq = -(M0pq+(3Zgq/N3))/N3
IC0pq = 1
(20)
Matrix [VCR0] is as follows:
[VCR0]
=
1
1
1
0
-1
0
0
0
0
-1
0
0
0
0
-1
0
(21)
-1
Finally, the zero sequence admittance matrix [Y0txf] can be computed as [ZMT0] * [VCR0].
C) Four-Winding Autotransformer AgAgYgD
Figure 5 shows a four-winding autotransformer. The first two windings have a start autotransformer
connection and are grounded with impedance Zgp. The third winding has a start transformer
connection and is grounded with impedance Zgt. The fourth winding has a delta connection.
Figure 5
Using equations (17) and (18) the mutual impedances M0xy, M0pt, M0pq, M0yt, M0yq and M0tq
are computed. The following equations can be written relating zero sequence voltages and
currents:
V0p – 3Zgp*(I0p+I0s) + M0xy*I0s + M0pt*I0t + M0pq*I0dq = E0x
(22a)
V0s – 3Zgp*(I0p+I0s) + M0xy*I0p + 2*M0xy*I0s + (M0pt+M0yt)*I0t +
(M0pq+M0yq)*I0qd = N1*E0x
(22b)
V0t – 3Zgt*I0t + M0pt*I0p + (M0pt+M0yt)*I0s + M0tq*I0qd = E0t = N2*E0x
(22c)
M0pq*I0p + (M0pq+M0yq)*I0s+ M0tq*I0t = E0qd = N3*E0x
(22d)
I0p + N1*I0s + N2*I0t + N3*I0qd = 0
(22e)
Page 7 of 12
Using V0p, V0s/N1, V0t/N2 and V0qd/N3 as voltages variables, and I0p, N1*I0s, N2*I0t and
N3*I0qd as current variables (all voltages and currents are referred to the primary winding),
matrices [ZMT0] and [VCR0] can be built. The elements of [ZMT0] are:
ZM0pp = (M0xy+(N1-1)*3Zgp)/N1
ZM0sp = (M0pt/N2)+3Zgp
2
ZM0ps = ((2-N1)*M0xy+(N1-1)*3Zgp)/N1
ZM0ss = (((M0pt+M0yt)/N2)-M0xy+3Zgp)/N1
ZM0pt = (M0yt-(N1-1)*M0pt)/N1)/N2
ZM0st = -(M0pt+(3Zgt/N2))/N2
ZM0pq = (M0yq-(N1-1)*M0pq)/N1)/N3
ZM0sq = ((M0tq/N2)-M0pq)/N3
ZM0tp = (M0pq/N3) + 3Zgp
IC0pp = 1
ZM0ts = ((M0pq+M0yq)/N3)-M0xy+3Zgp)/N1
IC0ps = 1
ZM0tt = ((M0tq/N3)-M0pt)/N2
IC0pt = 1
ZM0tq = -(M0pq/N3)
IC0pq = 1
(23)
Matrix [VCR0] is as follows:
[VCR0]
=
1
1
1
0
-1
0
0
0
0
-1
0
0
0
0
0
0
(24)
-1
Finally, the zero sequence admittance matrix [Y0txf] can be computed as [ZMT0] * [VCR0].
D) Four-Winding Autotransformer AAYgD
Figure 6 shows a four-winding autotransformer. The first two windings have a start autotransformer
connection and are ungrounded. The third winding has a start transformer connection and is
grounded with impedance Zgt. The fourth winding has a delta connection.
Figure 6
Using equations (17) and (18) the mutual impedances M0xy, M0pt, M0pq, M0yt, M0yq and M0tq
are computed. The following equations can be written relating zero sequence voltages and
currents:
V0p – V0np + M0xy*I0s + M0pt*I0t + M0pq*I0qd = E0x
(25a)
V0s – V0np + M0xy*I0s + (M0pt+M0yt)*I0t + (M0pq+M0yq)*I0dq = N1*E0x
(25b)
Page 8 of 12
V0t – 3Zgt*I0t + M0yt*I0s + M0tq*I0qd = E0t = N2*E0x
(25c)
M0yq*I0s + M0tq*I0t = E0qd = N3*E0x
(25d)
((N1-1)/N1)*(N1*I0s) + N2*I0t + N3*I0qd = 0
(25e)
Using V0p, V0s/N1, V0t/N2 and V0qd/N3 as voltages variables, and I0p, N1*I0s, N2*I0t and
N3*I0qd as current variables (all voltages and currents are referred to the primary winding);
matrices [ZMT0] and [VCR0] can be built. The elements of [ZMT0] are:
ZM0pp = M0xy
ZM0sp = M0xy
ZM0ps = (M0xy-((N1-1)/N3)*M0yq)/N1
ZM0ss = (M0xy+M0yt/N2-(N1/N3)*M0yq)/N1
ZM0pt = (M0yt-((N1-1)/N3)*M0tq)/N2
ZM0st = (M0yt-((N1/N3)*M0tq)-(3Zgt/N2))/N2
ZM0pq = M0yq/N3
ZM0sq = (M0yq+(M0tq/N2))/N3
ZM0tp = -(M0yq/N3)
IC0pp = 0
ZM0ts = -(M0yt)/N2)/N1
IC0ps = (N1-1)/N1
ZM0tt = ((M0tq/N3)+(3Zgt/N2))/N2
IC0pt = 1
ZM0tq = -(M0tq/N2)/N3
IC0pq = 1
(26)
Matrix [VCR0] is as follows:
[VCR0]
=
1
1
0
0
-N1
-N1
0
0
0
-1
1
0
0
0
0
0
(27)
-1
Finally, the zero sequence admittance matrix [Y0txf] can be computed as [ZMT0] * [VCR0].
V
Validation
A) Software Tools
The equations presented in this paper have been used to create MS-Excel spreadsheets to model
several types of four-winding transformers and autotransformers. Since MS-Excel handles only real
matrices, all the impedances, including grounding impedances, are modeled as reactances; the
resistive elements are neglected.
Another software tool used for validation is WinFdc. WinFdc is a Peruvian load-flow, short-circuit
and harmonics software developed by the authors. WinFdc allows currently modeling balanced
three-phase networks with two-winding and three-winding transformers and autotransformers.
The validation of the proposed approach, as shown in this section, was successful. Therefore,
WinFdc will be upgraded to model four-winding transformers and autotransformers. The results of
this software development effort will be soon presented in a companion paper [3].
Page 9 of 12
B) Four-Winding Transformer Data
The parameters of a four-winding YgYgYgD transformer, installed at 138 kV Juliaca substation in
Peru, are used for the validation process. According to test reports, the parameters are:









Rated voltages: 138/60/22.9/10 kV
Base power for impedances: 30 MVA
Zps = 0.10160 pu
Zpt = 0.24960 pu
Zpq = 0.16348 pu
Zst = 0.13080 pu
Zsq = 0.06360 pu
Ztq = 0.11940 pu
Ym = 0.00110 pu
C) Methodology
The methodology for validation with MS-Excel was to build the positive and zero sequence matrices
[ZMT] and [VCR], and then obtain the positive and zero sequence admittance matrices for the fourwinding transformer or autotransformer. Source impedances are added to build the positive and
zero sequence network admittance matrices [Y1bus] and [Y0bus]. Then, these matrices are
inverted to get [Z1bus] and [Z0bus]. The Thevenin impedances are computed manually and
compared with the diagonal elements of [Z1bus] and [Z0bus].Also, WinFdc is used to model threewinding transformers and autotransformers when one of the four windings has no impact in shortcircuit calculations.
D) Positive Sequence
Using Juliaca’s transformer parameters and assuming the following per unit taps values:




tp = 0.975 (primary tap)
ts = 1.075 (secondary tap)
tt = 0.925 (tertiary tap)
tq = 1.025 (quaternary tap)
Matrix [Ytxf] in pu at 100 MVA is:
P
S
T
Q
P
S
T
3.13008774
-2.89809177
0.27653171
-2.89809177
7.16731813
-1.39510638
0.27653171
-1.39510638
3.25040675
-0.18731366
-3.50129112
-1.73301081
Ytxf pu at 100 MVA including winding taps
Q
-0.18731366
-3.50129112
-1.73301081
5.41426573
Matrix [Ytxf] is symmetric. Connecting the primary winding to an infinite bus (Thevenin impedance
equal to 1.0E-12 pu at 100 MVA), [Z1bus] in pu at 100 MVA is:
P
P
S
T
Q
S
T
1E-12
1.1025E-12
9.48588E-13
1.1025E-12
0.391360732
0.365243968
9.48588E-13
0.365243968
0.711831137
1.05119E-12
0.369992848
0.464040119
Z1bus pu at 100 MVA - Winding P at Infinite Bus
Page 10 of 12
Q
1.05119E-12
0.369992848
0.464040119
0.572494845
Computing the Thevenin impedances at 100 MVA manually, we get:



-1
-1
2
Zps = (((0.10160/2) +(0.0011)) + (0.10160/2))*1.075 *(100/30) = 0.391360732
-1
-1
2
Zpt = (((0.24960/2) +(0.0011)) + (0.24960/2))*0.925 *(100/30) = 0.711831143
-1
-1
2
Zpq = (((0.16348/2) +(0.0011)) + (0.16348/2))*1.025 *(100/30) = 0.572494846
The same values of Thevenin impedances are obtained. Also, this validation process was
performed connecting the secondary, tertiary and quaternary windings to an infinite bus once at the
time; the same values of Thevenin impedances were obtained. This validates the proposed
approach for positive sequence.
E) Zero Sequence – Transformer YgYgYgD
All grounding impedances are zero. Using the same data of Juliaca’s four-winding transformer, and
assuming all taps equal to 1.000 pu, we get [Y0txf] as:
Ytxf0
P
S
T
QD
P
S
T
8.282715198 -3.037522334 -1.387223576
-3.037522334
2.97544401
0.249312769
-1.387223576
0.249312769
2.781055032
-3.857969287 -0.187234445 -1.643144225
Ytxf Autotransformer YgYgYgD in pu at 100 MVA
QD
0
0
0
0
The fourth file of [Y0txf] allows computing the zero sequence current flowing in the delta winding.
We can set to zero all the values of the fourth file and use [Ytxf0] as:
Ytxf0
P
S
T
D
P
S
T
8.282715198 -3.037522334 -1.387223576
-3.037522334
2.97544401
0.249312769
-1.387223576
0.249312769
2.781055032
0
0
0
Ytxf Autotransformer YgYgYgD in pu at 100 MVA
D
0
0
0
0
We connect now the primary winding to a source with impedances Z1 = Z0 = 1.0E-4 at 100 MVA,
and simulate a line to ground short-circuit at the secondary winding. In MS-Excel, using the
proposed approach, we obtain a fault current equal to 1.235029547 kA. Since the tertiary winding is
not connected to other equipment, WinFdc was used to model Juliaca’s transformer as a threewinding transformer YgYgD; the fault current calculated with WinFdc was 1.235 kA. Now, we model
the tertiary winding of Juliaca’s transformer as connected to an external network with Z1 = ∞ pu and
Z0 = 0.1pu at 100 MVA. In this case, using the proposed approach in MS-Excel, we obtain
1.23745026 kA. There is an exact agreement in short-circuit calculations using the proposed
approach and using WinFdc.
F) Zero Sequence – Autotransformer AAYgD
All grounding impedances are zero, unless indicated otherwise. Using the same data of Juliaca’s
four-winding transformer, but modeling it as an autotransformer with the 60 kV winding as the
primary winding, and assuming all taps equal to 1.000 pu, we get [Y0txf] as:
Ytxf0
P
S
T
QD
P
S
T
0.469803866 -1.080548892
0.047893238
-1.080548892
2.485262452 -0.110154448
0.047893238 -0.110154448
2.517445197
0.562851788 -1.294559111 -2.455183987
Ytxf Autotransformer AAYgD in pu at 100 MVA
Page 11 of 12
QD
0
0
0
0
We can set to zero the fourth file of [Y0txf] and it becomes:
Ytxf0
P
S
T
D
P
S
T
0.469803866 -1.080548892
0.047893238
-1.080548892
2.485262452 -0.110154448
0.047893238 -0.110154448
2.517445197
0
0
0
Ytxf Autotransformer in pu at 100 MVA
D
0
0
0
0
Now we connect the primary winding to a source with impedances Z1 = Z0 = 1.0E-4 at 100 MVA,
and simulate a line to ground short-circuit at the secondary winding. In MS-Excel, using the
proposed approach, we obtain a fault current equal to 1.161378905 kA. Since the tertiary winding is
not connected to any other equipment, WinFdc was used to model Juliaca’s transformer as a threewinding AAD autotransformer; the fault current calculated with WinFdc was 1.161 kA. This
autotransformer was also modeled in MS-Excel as AgAgYgD with Zgp equal to 1.0E+12 Ohms, and
the fault current obtained with proposed the approach was 1.161380098 kA. Now, we assume that
the tertiary winding of this autotransformer is connected to and external network with Z1 = ∞ pu and
Z0 = 0.1 pu at 100 MVA. In this case, using the proposed approach, we obtain in MS-Excel
1.162050819 kA. Also the currents in amperes flowing into the primary and secondary terminals of
this AAYgD autotransformer have the same value but with opposite sign; these currents are not
transformed as expected. There is an exact agreement in short-circuit calculations using the
proposed approach and using WinFdc.
VI
Conclusion
A theoretical approach was presented for modeling three-phase four-winding transformers and
autotransformers for load-flow and short-circuit analysis. Equations were presented for several
types of transformers and autotransformers for building the positive and zero sequence admittance
matrices. This approach was successfully validated using it in MS-Excel and comparing it with
short-circuit calculations performed with WinFdc. The transformer or autotransformer data needed
in this approach comes directly from standard transformer test reports. This approach can be used
to upgrade load-flow and short-circuit software packages for modeling four-winding transformers
and autotransformers. This approach can be applied also to transformers with more than four
windings.
References:
[1]
[2]
[3]
[4]
Central Station Engineers of the Westinghouse Electric Corporation, "Electrical Transmission and
Distribution Reference Book", East Pittsburgh, Pennsylvania, USA, September 1950.
Teshmont Consultants LP, "Transformer Modelling Guide”, Calgary, Alberta, Canada, July 08, 2014.
Guide downloaded from internet.
C. A. Muñoz, A. Rojas, "Four-Winding Transformers and Autotransformers Modeling for Load-Flow
and Short-Circuit Analysis, Part II: Application", in preparation and to be soon uploaded to
www.researchgate.net.
A. Rojas, C. A. Muñoz, “WinFdc 2.02 User Manual”, in Spanish, COVIEM S.A., Lima, Peru.
About the authors:
Carlos A. Muñoz is associate professor at Universidad Nacional de Ingeniería at Lima, Peru. He is also the
general manager of COVIEM S.A. COVIEM designs, builds and carries out the commissioning of electrification
projects. His research interests are load-flow, short-circuit, and electrical protections. He can be contacted at
cmunozm@uni.edu.pe.
Alberto Rojas is a consultant engineer. He works at COVIEM S.A. in Lima, Peru, His research interests are
load-flow, short-circuit, harmonics and grounding analysis; and also software development for power systems
studies. He can be contacted at arojas222@yahoo.es.
Manuscript finished on March 18, 2016.
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