Electric Power Systems Research 144 (2017) 32–40
Contents lists available at ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Influence of tertiary stabilizing windings on zero-sequence
performance of three-phase three-legged YNynd transformers. Part I:
Equivalent circuit models
Angel Ramos a,∗ , Juan Carlos Burgos b
a
b
Gas Natural Fenosa, Madrid, Spain
Universidad Carlos III de Madrid, Leganés, Madrid, Spain
a r t i c l e
i n f o
Article history:
Received 4 October 2015
Received in revised form 27 July 2016
Accepted 29 October 2016
Available online 23 November 2016
Keywords:
Three-winding transformers
Tertiary stabilizing windings
Transformer zero-sequence performance
Onsite zero-sequence impedance
measurements
Transformer equivalent circuits
a b s t r a c t
The presence of a stabilizing winding (or tertiary stabilizing winding when is used to for auxiliary
applications) in three-phase three-legged YNynd transformers remarkably affects the zero-sequence
performance of both the transformer and the network. This paper presents a detailed analysis of the
influence of the stabilizing winding on the zero-sequence behavior of three-phase three-legged YNynd
transformers. Based on a complete set of onsite zero-sequence measurements taken in three power
transformers, transformer zero-sequence performance is analyzed in relation to internal design features
such as stabilizing winding position relative to high-voltage and low-voltage windings or the presence
of magnetic shunts in the tank. Based on these measurements, this paper assesses the ability of various
equivalent circuit models to reproduce zero-sequence performance accurately. A companion paper that
complements this study evaluates the influence of stabilizing windings on tank overheating hazard and
short-circuit duty in the event of asymmetrical faults.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction
Delta stabilizing windings have been used since early electric
power system development to avoid some of the network operation
drawbacks of using wye connections in power transformers and
autotransformers [1–3]. Utilization of stabilizing windings even
became mandatory under some local regulations, and other external uses were also found for these windings [4,5].
Although the need for “critical” analysis of the decision to
include a stabilizing winding in Yy-connected transformers [6–8],
particularly in relation to three-phase three-legged transformers,
was pointed out many years ago, many utilities still maintain this
practice for reasons of “tradition” without analyzing the decision.
Making an appropriate decision about the need for stabilizing
windings requires deep understanding of the transformer’s zerosequence performance. Although many authors have studied zerosequence behavior and transformer modeling in detail [9–15], there
is little information in technical literature providing an overview
∗ Corresponding author at: Architecture and Network Design Manager, Electrical
Distribution Division, Gas Natural Fenosa, Avda. San Luis 77, Madrid 28033, Spain.
E-mail addresses:
aramosg@gasnatural.com, aramosgomez@gmail.com (A. Ramos)
http://dx.doi.org/10.1016/j.epsr.2016.10.065
0378-7796/© 2016 Elsevier B.V. All rights reserved.
of three-phase three-legged YNynd power transformer behavior
depending on whether the stabilizing winding is closed or open.
For all these reasons, in this paper (Part I) a complete set of zerosequence measurements was performed on three YNynd power
transformers of different internal design under different operating
conditions (with and without stabilizing winding) in order to evaluate the accuracy of several equivalent circuit models in predicting
impedances and currents. In the second part of this study (Part II),
presented in a companion paper, these results will be used to analyze tank overheating hazard and short-circuit duty in three-phase
three-legged transformers. This information will be useful to design
and planning engineers when conducting thorough assessment of
the need for stabilizing windings.
2. Onsite zero-sequence measurements
To analyze stabilizing windings’ influence on zero-sequence
performance and its dependence on the windings’ relative position
and on the presence of magnetic shunts, a complete set of tests
was performed on three three-phase three-legged YNynd power
transformers (see Table 1 and Fig. 1 for nameplate data and internal characteristics). These internal designs cover the vast majority
of core-form construction types of power transformers.
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
33
Fig. 1. Schematic description of the internal construction design of the transformers measured.
Table 1
Characteristics of transformers subject to zero-sequence measurements.
Table 4
Results and calculations of zero-sequence measurements taken in transformer #2.
Data
Transformer
#1
#2
#3
MVA
kV
Vector group
Short-circuit impedance
Inner winding
Intermediate winding
Outer winding
Magnetic shunts
25/25/8.33
45/16.05/10
YNyn0 + d11
10.8%
Stabilizing
Low-voltage
High-voltage
NO
75/75/25
220/71/10
YNyn0 + d11
14.0%
Stabilizing
Low-voltage
High-voltage
YES
150/150/50
230/71/20
YNyn0 + d11
14.1%
Low-voltage
High-voltage
Stabilizing
NO
Test code
V (%)
I1 (%)
I2 (%)
I3 (%)
Z0 (%)
HOO
HOC
HSO
HSC
LOO
LOC
LSO
LSC
(In Table 6)
0.136
0.171
0.110
(In Table 6)
0.055
0.115
0.039
(In Table 6)
0.699
1.258
0.821
–
–
0.766
0.322
–
–
1.223
0.931
(In Table 6)
1.171
0.839
1.218
–
0.695
–
0.164
–
1.160
–
0.893
135.29
19.41
13.63
13.45
142.30
4.68
13.76
3.24
Table 5
Results and calculations of zero-sequence measurements taken in transformer #3.
Table 2
Set of zero-sequence measurements to be performed.
Test number
Test code
HV winding
LV winding
Stabilizing winding
1
2
3
4
5
6
7
8
HOO
HOC
HSO
HSC
LOO
LOC
LSO
LSC
Energized
Energized
Energized
Energized
Open circuit
Open circuit
Short circuit
Short circuit
Open circuit
Open circuit
Short circuit
Short circuit
Energized
Energized
Energized
Energized
Delta open
Delta close
Delta open
Delta close
Delta open
Delta close
Delta open
Delta close
Table 3
Results and calculations of zero-sequence measurements taken in transformer #1.
Test code
V (%)
I1 (%)
I2 (%)
I3 (%)
Z0 (%)
HOO
HOC
HSO
HSC
LOO
LOC
LSO
LSC
(In Table 6)
0.249
0.150
0.145
(In Table 6)
0.033
0.054
0.019
(In Table 6)
1.581
1.590
1.555
–
–
0.503
0.225
–
–
1.386
1.659
(In Table 6)
0.599
0.507
0.587
–
1.243
–
0.387
–
0.545
–
0.360
80.14
15.76
9.43
9.35
86.09
5.58
10.76
3.29
The internal design of transformer #1 (with inner stabilizing
winding and no magnetic shunts) is typically employed in low- and
medium-power YNynd transformers. When rated power is above
30–40 MVA, transformer designers include magnetic shields, as in
transformer #2. For high-power transformers (above 100 MVA),
three-winding YNynd transformers may include outer stabilizing
windings, as in transformer #3 (with or without magnetic shields).
As demonstrated in Ref. [16], accurate representation of the
zero-sequence performance of three-phase three-legged YNynd
transformers can be achieved by taking onsite low-voltage measurements, requiring 8 tests as indicated in Table 2.
The results and calculations deriving from the tests performed
on the three transformers in Table 1 are shown in Tables 3–5, which
present measurements of voltage in the energized winding and of
currents in the HV side (I1 ), LV side (I2 ) and stabilizing winding
Test code
V (%)
I1 (%)
I2 (%)
I3 (%)
Z0 (%)
HOO
HOC
HSO
HSC
LOO
LOC
LSO
LSC
(In Table 6)
0.045
0.109
0.062
(In Table 6)
0.123
0.053
0.065
(In Table 6)
0.265
0.896
0.897
–
–
0.366
0.502
–
–
0.782
0.487
(In Table 6)
0.359
0.375
0.470
–
0.257
–
0.396
–
0.324
–
0.061
87.98
16.85
12.17
6.90
104.27
34.20
14.06
13.88
(I3 ). Voltage and current values are shown in percentage of base
quantities (rated voltage and rated apparent power). Calculation of
zero-sequence impedance is indicated in Eq. (1),
Z0(ABC) =
V
I/3
(1)
where (ABC) is the test code (e.g., HOO in test 1 in Table 2) and V
and I are the measurements taken in the energized winding.
Zero-sequence impedance modules are presented in percentages in Tables 3–5, calculated as indicated in Eq. (2),
Z0(ABC) (%) =
Z0(ABC)
Zbase
=
V (%)
I (%)
(2)
Relation between Eqs. (1) and (2) could be easily derived from
well-known relations between rated and base quantities.
As illustrated in Ref. [16], non-linear behavior of HOO and
LOO measurements does not produce a major problem of model
accuracy when predicting short-circuit currents. Consequently, to
facilitate understanding of Tables 3–5 these tests consider average
impedance values. All values measured in these tests are shown in
Table 6. Some additional considerations about “no-load” tests (i.e.
HOO and LOO measurements) can be found after Table 6.
As indicated previously, an accurate representation of the
zero-sequence performance of three-phase three-legged YNynd
transformers can be achieved by means of low-voltage onsite
measurements. With the exception of no-load tests, the rest of zerosequence impedances (i.e. HOC, HSO, HSC, LOC, LSO and LSC) show
a good consistency between low-voltage onsite measurements and
high-voltage factory tests [16,17]. This is an expected result as these
34
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
Fig. 2. Nonlinear performance of ferromagnetic materials [19].
Table 6
Results and calculations of zero-sequence measurements taken in no-load tests.
Transformer
Test code
V (%)
I (%)
#1
HOO
0.397
0.801
0.950
0.149
0.649
0.805
0.526
0.980
1.141
0.186
0.737
0.893
75.45
81.74
83.28
80.09
88.11
90.17
0.100
0.161
0.154
0.255
0.314
0.522
0.075
0.117
0.110
0.180
0.220
0.360
133.62
136.96
139.62
141.70
142.64
145.23
0.041
0.075
0.102
0.122
0.108
0.167
0.219
0.274
0.048
0.085
0.115
0.137
0.105
0.160
0.209
0.261
85.89
88.04
88.80
89.18
103.11
104.18
104.53
105.27
LOO
#2
HOO
LOO
#3
HOO
LOO
Z0 (%)
impedances are largely dominated by leakage fluxes with linear
behavior [18].
In the case of “no-load” tests (i.e. HOO and LOO) of three-legged
transformers, zero-sequence flux closes its path from the magnetic
core to the tank through oil gaps and structural parts. This flux
circulation path is significantly linearized due to the presence of
non-ferromagnetic elements, but the non-linear effect of the tank
steel magnetic permeability is still present as can be deduced from
the dependence of impedance with voltage as shown in Table 6.
Under no-load conditions (in the absence of delta winding), the
transformer zero-sequence response is markedly influenced by the
magnetic permeability of the tank steel. Actually, the measurements in Table 6 are fragments of a continuous dependence of
no-load zero-sequence impedances versus applied voltage, which
is qualitatively similar to that in Fig. 2 [19].
As this slightly non-linear behavior of the transformer suppose a source of error in HOO and LOO measurements, it must be
checked if this error is a problem or not in order to obtain a sufficiently accurate representation of zero-sequence performance of
the transformers in the calculation of short circuit currents (such as
those in the companion paper [20]). These measurements mainly
affect zero-sequence magnetizing impedance, whose importance
when analyzing unbalanced loads and faults is expected to be much
lower than that of the other zero-sequence impedances. In Ref.
[17], a complete sensitivity analysis is carried out in order to evaluate actual implications of the uncertainty of the zero-sequence
magnetizing impedance, concluding that on-site low-voltage measurements can be used to obtain a faithful representation of
transformer zero-sequence behavior for steady-state short-circuit
calculations.
In some other calculations (i.e. tank losses due to unbalanced
currents or inrush current calculations) a more detailed model taking into account non-linear behavior should be used [21], as linear
equivalent circuits (as those proposed in Section 3) could not provide enough accuracy.
3. Calculation of zero-sequence equivalent circuit model
parameters
Once a complete set of measurements is available, the next
step is to ascertain whether these measurements are suitable
for establishing zero-sequence equivalent circuits that represent transformer performance when delta stabilizing windings
are closed or open. Two approaches were considered: classic Ttype equivalent circuits whose simplicity usually makes this the
option selected by engineers, and the more complex six-impedance
equivalent circuit described in Ref. [10] as necessary for accurate
determination of internal currents in three-winding YNynd transformers.
3.1. Three-impedance (T-type) equivalent circuit models
Utilization of T-type equivalent circuits in transformers is
mandatory in commercial power system analysis tools used by protection, operation and planning engineers. For this reason, several
types of T-type equivalent circuits for YNynd transformers will be
analyzed in order to ascertain the best alternative.
The first alternative to be considered for the zero-sequence
equivalent circuit of the transformer (model A) is based on the
model proposed by IEC standard 60076-8 [22], and considers the
equivalent circuit shown in Fig. 3. This circuit has a branch in parallel with the magnetizing branch, so when the delta winding is
closed, current flows in both impedances. If the delta winding is
open, current flows through the magnetizing impedance only.
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
35
Table 7
Zero-sequence impedance (T-type) calculation results for transformer #1.
Zero-sequence equiv. circuit (model A)
Zero-sequence equiv. circuit (model B1)
Zero-sequence equiv. circuit (model B2)
Impedance
Z0 (%)
Impedance
Z0 (%)
Impedance
Z0 (%)
Z01
Z02
Z03
Z0M
QME = 8.82%
10.08
−0.20
6.07
78.38
Z 01
Z’02
Z’03
–
QME = 0.16%
9.79
−0.42
5.99
–
Z”01
Z”02
–
Z”0M
QME = 1.51%
1.08
9.53
–
77.84
Fig. 3. T-type zero-sequence equivalent circuit model A (switch S is open when
delta is open).
Fig. 5. Six-impedance zero-sequence separate equivalent circuit (model C).
Fig. 4. T-type zero-sequence separate equivalent circuits (model B).
The second alternative (model B) is based on IEEE standard
C57.12.90 [23] and considers two different T-type equivalent circuits for the YNynd transformer depending on whether the delta
stabilizing winding is closed or open (Fig. 4).
The impedances of the zero-sequence equivalent circuits in
Figs. 3 and 4 were calculated from the zero-sequence impedance
measurements presented in Tables 3–5. To obtain the circuit
parameters in model B1 (Fig. 4—B1), measurements HOC, HSC, LOC
and LSC were used, while measurements HOO, HSO, LOO and LSO
were used for the circuit in model B2 (Fig. 4—B2).
As the number of measurements is higher than that
of the impedances, an optimization process based on
the Levenberg–Marquardt algorithm [24] was performed.
Levenberg–Marquardt algorithm mixes the Gauss–Newton
method and the gradient descendent method to improve the convergence of the problem. In the optimization process, the objective
was to minimize the quadratic mean error (QME) between the
measured impedances (shown in Tables 3–5) and those obtained
from the equivalent circuit. The equations used in the optimization
process for each model are detailed in the Appendix A. The results
of calculation of the zero-sequence impedances of the equivalent
circuits are shown in Tables 7–9. The QME is also indicated in the
tables. Resistive components of the measurements were ignored,
as their influence on the results is not relevant [16].
3.2. Six-impedance equivalent circuit model
As indicated previously, Garin states [10] that an abbreviated
equivalent circuit (T-type) is sufficient for system calculations, but
that the completed equivalent circuit (six branches) is required to
calculate the current circulating in the delta stabilizing winding.
To verify that statement and ascertain the veracity of the currents predicted by the T-type and six-branch equivalent circuit
models, a third alternative is proposed, as indicated in Fig. 5. In
this equivalent circuit, the stabilizing winding is represented by a
branch with a switch S that is closed when the delta winding is
closed.
In the same way as stated above for T-type equivalent circuits,
the impedances of the zero-sequence equivalent circuit shown in
Fig. 5 were calculated from zero-sequence impedance measurements. Optimization based on the Levenberg–Marquardt algorithm
[24] was applied to Eqs. (A.17)–(A.24) shown in the Appendix A.
Table 10 shows the results of the zero-sequence impedance calculations and the QME.
4. Analysis of results
This section compares the ability of the different zero-sequence
equivalent circuits to reproduce transformer behavior accurately.
For this purpose, and for each test configuration, the predicted
impedances and currents obtained from the various equivalent circuit models will be compared with real measurements. It should
be noted in Tables 7–10 that some impedances could be negative
as a result of the mathematical transformations used to obtain the
models, without representing capacitive effects.
4.1. Models’ accuracy in predicting impedance
In Section 3, the overall appropriateness of the different
equivalent circuits was assessed by means of the QME obtained
when performing the optimization process to obtain the models’
impedances. Table 11 summarizes the QME values. For model B,
the average value of models B1 and B2 is shown.
The results summarized in Table 11 show that model B offers
greater accuracy (QME below 1%). Model C also presents quite low
QME values (in the 1–3% range). Conversely, model A presents
36
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
Table 8
Zero-sequence impedance (T-type) calculation results for transformer #2.
Zero-sequence equiv. circuit (model A)
Zero-sequence equiv. circuit (model B1)
Zero-sequence equiv. circuit (model B2)
Impedance
Z0 (%)
Impedance
Z0 (%)
Impedance
Z0 (%)
Z01
Z02
Z03
Z0M
QME = 5.32%
14.39
−0.45
5.26
131.87
Z’01
Z’02
Z’03
–
QME = 0.15%
14.13
−0.61
5.28
–
Z”01
Z”02
–
Z”0M
QME = 1.03%
4.96
9.12
–
131.70
Table 9
Zero-sequence impedance (T-type) calculation results for transformer #3.
Zero-sequence equiv. circuit (model A)
Zero-sequence equiv. circuit (model B1)
Zero-sequence equiv. circuit (model B2)
Impedance
Z0 (%)
Impedance
Z0 (%)
Impedance
Z0 (%)
Z01
Z02
Z03
Z0M
QME = 1.15%
−1.54
15.73
23.12
89.34
Z’01
Z’02
Z’03
–
QME = 0.22%
−1.59
15.65
18.48
–
Z”01
Z”02
–
Z”0M
QME = 0.64%
−0.46
14.61
–
88.99
Table 10
Zero-sequence impedance (6-impedances) calculation results for transformers #1,
#2 and #3.
Zero-sequence equiv. circuit (6-impedances)
Impedance
Transformer #1
Z0 (%)
Transformer #2
Z0 (%)
Transformer #3
Z0 (%)
Z01
Z02
Z03
Z012
Z013
Z023
QME
73.35
−204.26
229.94
8.35
−39.07
5.19
2.54%
177.07
−82.58
67.49
11.90
−67.78
4.31
1.17%
834.42
579.08
102.77
12.64
15.09
−107.95
1.26%
Table 11
Comparison of the different models’ quadratic mean error.
Model
A
B
C
Quadratic mean error (QME)
Transformer #1
Transformer #2
Transformer #3
8.82%
0.83%
2.54%
5.32%
0.59%
1.17%
1.15%
0.43%
1.26%
greater dispersion in the different transformers, producing values
above 5% in transformer #1 and #2.
Despite being topologically quite similar, the accuracy of models
A and B are very different. As it was indicated, the main difference between models A and B is that while in model A the same
impedances Z01 and Z02 represent the behavior of a YNynd transformer in all conditions, model B uses different impedances in the
series branches (Z 01 different to Z 01 and Z 02 different to Z 02 )
to represent the behavior of a YNynd transformer whether stabilizing winding is open or closed. The underlying physical reason
behind the difference in accuracy of models A and B is that in
a three-legged core-type wye–wye transformer, when calculating
open-circuit zero-sequence impedances, the tank acts as if it were
a high-impedance delta-connected outermost winding [10]. On the
contrary, in a three-winding YNynd transformer this effect is not
present due to the delta stabilizing winding.
In the case of transformers #1 and #2, model B1 represents the
zero-sequence equivalent circuit of a wye-wye transformer with
an innermost delta (real) winding and model B2 represents the
zero-sequence equivalent circuit of a wye-wye transformer with
an outermost delta (virtual) winding. In this case, leakage fluxes
trajectories are quite different and this is the reason why Z 01 is so
different to Z 01 and Z 02 is so different to Z 02 . As model A is not
capable to represent accurately both situations, QME in model A
is significantly higher than in model B. On the other hand, in the
case of transformer #3, due to its internal design, both the stabilizing winding and the tank when acts as a virtual delta winding are
positioned as the outermost winding. In this case, leakage fluxes
trajectories when stabilizing winding is closed or open are more
similar and the differences in QME of model A and B are not too
high.
To provide more detailed information about impedance prediction errors in each test configuration, Table 12 shows the
impedance values obtained in each measurement and the relative
error of the predicted impedances of the different equivalent circuit models. As revealed in Ref. [16], the errors are normally higher
in no-load configurations due to transformers’ non-linear behavior,
without significantly affecting zero-sequence performance analysis. The conclusions drawn about the different models from Table 11
can be extrapolated to the results in Table 12.
4.2. Accuracy of the different models’ current prediction
As explained previously, a T-type equivalent circuit model is
expected to be sufficiently accurate for system calculations but not
for internal current prediction. However, a more complex model
(with at least six branches) is expected to offer more accurate current prediction.
The aim of this section is to verify the current prediction accuracy of the different equivalent circuits for both external (primary
and secondary windings) and internal (stabilizing winding) currents.
Table 13 shows each model’s average error in primary and
secondary current prediction for the different test configurations,
and Table 14 shows the mean error in stabilizing winding current prediction. Tables 15–17 show detailed winding current values
obtained in each measurement and the relative error (in relation to
measurement) of the predicted currents for the different equivalent
circuit models.
As expected, in model C current prediction accuracy remains
good (error less than 3–5%) for both internal and external currents.
Conversely, equivalent circuit A made poor predictions not only for
internal but also for external currents, especially in one of the transformers. Meanwhile, model B offers good predictions for external
currents and good predictions for delta winding currents in about
75% of cases. The reason why model B has less accuracy in predicting delta winding currents than model C is related with tank
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
37
Table 12
Comparison of the different models’ impedance prediction.
Test code
Model
Transformer #1
Transformer #2
Transformer #3
Z0 (%)
Error (%)
Z0 (%)
Error (%)
HOO
Measured
A
B
C
80.14
88.46
78.92
77.25
–
10.38
1.52
3.60
135.29
146.26
136.66
136.85
–
8.11
1.01
1.15
87.98
87.80
88.53
89.01
–
0.20
0.63
1.17
HOC
Measured
A
B
C
15.76
15.71
15.78
16.36
–
0.29
0.13
3.80
19.41
19.45
19.41
19.34
–
0.20
0.00
0.34
16.85
16.83
16.89
16.91
–
0.14
0.24
0.35
HSO
Measured
A
B
C
9.43
9.88
9.57
9.62
–
4.77
1.49
2.02
13.63
13.94
13.49
13.52
–
2.26
1.03
0.84
12.17
11.84
12.09
12.38
–
2.75
0.66
1.72
HSC
Measured
A
B
C
9.35
9.87
9.34
9.28
–
5.59
0.12
0.76
13.45
13.90
13.44
13.35
–
3.32
0.07
0.77
6.90
6.93
6.88
6.82
–
0.48
0.23
1.13
LOO
Measured
A
B
C
86.09
78.18
87.37
88.26
–
9.19
1.49
2.52
142.30
131.42
140.82
140.65
–
7.65
1.04
1.16
104.27
105.07
103.60
102.28
–
0.77
0.64
1.90
LOC
Measured
A
B
C
5.58
5.43
5.57
5.73
–
2.62
0.18
2.73
4.68
4.61
4.67
4.77
–
1.53
0.21
1.90
34.20
34.10
34.13
34.63
–
0.30
0.20
1.26
LSO
Measured
A
B
C
10.76
8.73
10.60
10.99
13.76
12.52
13.90
13.89
–
8.98
1.02
0.95
14.06
14.16
14.15
14.23
–
0.73
0.62
1.18
LSC
Measured
A
B
C
3.29
3.41
3.30
3.25
3.24
3.29
3.23
3.29
–
1.63
0.19
1.55
13.88
14.05
13.91
13.97
–
1.22
0.22
0.66
–
18.85
1.53
2.14
–
3.77
0.19
1.17
Table 13
Comparison of the different models’ mean errors in primary and secondary current
prediction.
Model
A
B
C
Quadratic mean error (QME)
Transformer #1
Transformer #2
Transformer #3
7.7%
0.8%
2.3%
3.8%
1.3%
1.7%
1.2%
0.8%
1.3%
Table 14
Comparison of the different models’ mean errors in stabilizing winding current
prediction.
Model
A
B
C
Quadratic mean error (QME)
Transformer #1
Transformer #2
Transformer #3
29.2%
27.0%
2.3%
16.0%
9.3%
2.2%
23.0%
11.4%
2.0%
current circulation when zero-sequence voltages are applied to the
transformer [10]. For example, in case of zero-sequence voltages
applied to HV winding, HV ampere-turns should be balanced with
LV ampere-turns, stabilizing winding ampere-turns and ampereturns from tank currents. This last term (ampere-turns from tank
currents) is not taken into account by model B, as this is a T-type
model (see Fig. 4—B1).
Z0 (%)
Error (%)
5. Conclusions
Differences in the zero-sequence performance of three-phase
three-legged YNynd power transformers when the stabilizing
winding is closed and when it is open is a key factor in assessing the need for this winding. This performance is determined by
zero-sequence flux circulation inside the transformer under different network operating conditions, meaning that internal design
features such as the relative position of the windings (HV, LV and
stabilizing) or the presence of magnetic shields in the tank are of
great importance.
This paper proposed taking a complete set of onsite low-voltage
zero-sequence measurements to obtain overall information about
the zero-sequence performance of the three most frequent design
types found in core-form power transformers, complementing the
results published by the authors in Ref. [16].
Based on these measurements, the parameters of different
equivalent circuits were calculated using a set of equations and an
optimization process. These circuit models comprised two T-type
equivalent circuits (the simpler option commonly used in representation of transformers in power system analysis tools) and a more
complex six-branch equivalent circuit.
For the T-type models, the model whose parameters are different depending on whether the stabilizing winding is closed or open
was more accurate than the alternative with common parameters
in all cases. Separate T-type zero-sequence equivalent circuits usually produce a complete and sufficiently accurate description of the
zero-sequence behavior of YNynd transformers for all stabilizing
38
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
Table 15
Comparison of the different models’ current prediction for transformer #1.
Test Code
Model
Current prediction for transformer #1
I1 (%)
Error (%)
I2 (%)
Error (%)
I3 (%)
Error (%)
HOC
Measured
A
B
C
1.581
1.586
1.579
1.523
–
0.33
0.09
3.62
–
–
–
–
–
–
–
–
1.243
1.472
1.579
1.232
–
18.39
27.02
0.91
HSO
Measured
A
B
C
1.590
1.517
1.566
1.558
–
4.58
1.49
2.01
1.386
1.521
1.395
1.364
–
9.74
0.67
1.60
–
–
–
–
–
–
–
–
HSC
Measured
A
B
C
1.555
1.473
1.557
1.567
–
5.27
0.15
0.79
1.659
1.527
1.674
1.741
–
7.93
0.96
4.99
0.387
0.050
0.117
0.372
–
87.00
69.66
3.84
LOC
Measured
A
B
C
–
–
–
–
–
–
–
–
0.599
0.617
0.601
0.584
–
2.86
0.35
2.49
0.545
0.572
0.601
0.550
–
4.99
10.36
0.95
LSO
Measured
A
B
C
0.503
0.553
0.507
0.496
–
10.06
0.96
1.31
0.507
0.624
0.514
0.496
–
23.19
1.52
2.14
–
–
–
–
–
–
–
–
LSC
Measured
A
B
C
0.225
0.203
0.222
0.231
–
9.92
1.22
2.73
0.587
0.565
0.586
0.594
–
3.65
0.21
1.16
0.360
0.337
0.363
0.372
–
6.45
0.96
3.35
Table 16
Comparison of the different models’ current prediction for transformer #2.
Test code
Model
Current prediction for transformer #2
I1 (%)
Error (%)
I2 (%)
Error (%)
I3 (%)
Error (%)
HOC
Measured
A
B
C
0.699
0.698
0.699
0.701
–
0.19
0.00
0.34
–
–
–
–
–
–
–
–
0.695
0.671
0.699
0.670
–
3.48
0.56
3.58
HSO
Measured
A
B
C
1.258
1.230
1.271
1.269
–
2.23
1.02
0.83
1.223
1.235
1.189
1.188
–
0.96
2.77
2.84
–
–
–
–
–
–
–
–
HSC
Measured
A
B
C
0.821
0.794
0.821
0.827
–
3.24
0.05
0.75
0.931
0.872
0.929
0.928
–
6.32
0.24
0.34
0.164
0.075
0.107
0.163
–
54.56
34.66
0.80
LOC
Measured
A
B
C
–
–
–
–
–
–
–
–
1.171
1.189
1.173
1.149
–
1.58
0.23
1.84
1.160
1.144
1.173
1.181
–
1.38
1.19
1.83
LSO
Measured
A
B
C
0.766
0.831
0.800
0.800
–
8.48
4.48
4.41
0.839
0.921
0.830
0.831
–
9.83
1.04
0.98
–
–
–
–
–
–
–
–
LSC
Measured
A
B
C
0.322
0.311
0.331
0.331
–
3.47
2.80
2.70
1.218
1.197
1.218
1.198
–
1.78
0.01
1.70
0.893
0.851
0.887
0.914
–
4.60
0.62
2.43
winding operation options. Nevertheless, in some cases, stabilizing
winding current prediction showed errors, though of slight significance. It should be noted that in case of no load condition and
stabilizing winding open, non-linearity of magnetizing impedances
can introduce a certain error in the use of the models.
The six-impedance equivalent circuit supposes greater complexity both in obtaining the parameters and in calculating circuit
currents. Nevertheless, this option offers greater accuracy when
a precise evaluation of internal transformer currents (especially
circulating current inside the delta stabilizing winding) is required.
These experimental data and results will be of great use in analysis of tank overheating hazard and short-circuit duty in three-phase
three-legged YNynd power transformers, which will be addressed
in the companion paper (Part II) to this study.
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
39
Table 17
Comparison of the different models’ current prediction for transformer #3.
Test code
Model
Current prediction for transformer #3
I1 (%)
Error (%)
I2 (%)
Error (%)
I3 (%)
Error (%)
HOC
Measured
A
B
C
0.265
0.265
0.264
0.264
–
0.13
0.24
0.36
–
–
–
–
–
–
–
–
0.257
0.211
0.264
0.250
–
18.07
2.75
2.76
HSO
Measured
A
B
C
0.896
0.921
0.902
0.880
–
2.80
0.64
1.72
0.782
0.783
0.774
0.762
–
0.18
0.93
2.51
–
–
–
–
–
–
–
–
HSC
Measured
A
B
C
0.897
0.893
0.899
0.907
–
0.44
0.27
1.18
0.487
0.481
0.487
0.490
–
1.21
0.02
0.60
0.396
0.327
0.412
0.410
–
17.39
4.10
3.56
LOC
Measured
A
B
C
–
–
–
–
–
–
–
–
0.359
0.360
0.359
0.354
–
0.28
0.19
1.26
0.324
0.286
0.359
0.325
–
11.76
10.97
0.38
LSO
Measured
A
B
C
0.366
0.379
0.375
0.369
–
3.70
2.56
0.92
0.375
0.373
0.373
0.371
–
0.64
0.53
1.09
–
–
–
–
–
–
–
–
LSC
Measured
A
B
C
0.502
0.507
0.513
0.516
–
1.01
2.26
2.86
0.470
0.464
0.469
0.467
–
1.21
0.22
0.66
0.061
0.034
0.044
0.060
–
44.87
27.90
1.31
Z0(HOO) = Z 01 + Z 0M
Acknowledgement
The authors would like to thank Miguel Angel Cardiel, from
Universidad Carlos III de Madrid, for performing equivalent circuit
calculations.
Z0(HSO) = Z
Z0(LOO) = Z
Z0(LSO) = Z
01
02
02
+Z
+Z
+Z
(A.13)
02 //Z 0M
(A.14)
0M
(A.15)
01 //Z 0M
(A.16)
Appendix A.
- Alternative C equivalent circuit (Fig. 5):
In this section, the equations that allow transition from measured quantities to equivalent circuit impedances are presented for
each model. For simplicity, the development of shunt impedances
is not shown.
Z0(HOO) = Z01 // Z01Y + (Z02Y + Z02 ) // (Z03Y )
Z0(HSC) = Z01 // Z01Y + (Z02Y ) // (Z03Y + Z03 )
(A.1)
Z0(HOC) = Z01 + Z0M //Z03
(A.2)
Z0(LOO) = Z02 // Z02Y + (Z01Y + Z01 ) // (Z03Y )
Z0(HSO) = Z01 + Z0M //Z02
(A.3)
Z0(LOC) = Z02 // Z02Y + (Z01Y + Z01 ) // (Z03Y + Z03 )
Z0(HSC) = Z01 + Z0M //Z02 //Z03
(A.4)
Z0(LOO) = Z02 + Z0M
(A.5)
Z0(LSO) = Z02 + Z0M //Z01
(A.6)
Z0(LSO) = Z02 + Z0M //Z01
(A.7)
Z0(LSC) = Z02 + Z0M //Z01 //Z03
(A.8)
Z0(LSO) = Z02 // Z02Y + (Z01Y ) // (Z03Y )
Z0(LOC) = Z02 + Z03
(A.11)
Z0(LSC) = Z02 + Z01 //Z03
(A.12)
- Alternative B2 equivalent circuit (Fig. 4—B2):
(A.21)
(A.22)
(A.23)
(A.24)
Z01Y =
Z012 · Z013
Z012 + Z013 + Z023
(A.25)
Z02Y =
Z012 · Z023
Z012 + Z013 + Z023
(A.26)
Z03Y =
Z013 · Z023
Z012 + Z013 + Z023
(A.27)
(A.9)
(A.10)
(A.20)
where impedances Z01Y − Z02Y − Z03Y are corresponding startriangle transformation from Z012 − Z013 − Z023 :
- Alternative B1 equivalent circuit (Fig. 4—B1):
Z0(HSC) = Z01 + Z02 //Z03
Z0(LSC) = Z02 // Z02Y + (Z01Y ) // (Z03Y + Z03 )
(A.18)
(A.19)
Z0(HOO) = Z01 + Z0M
Z0(HOC) = Z01 + Z03
(A.17)
Z0(HOC) = Z01 // Z01Y + (Z02Y + Z02 ) // (Z03Y + Z03 )
Z0(HSO) = Z01 // Z01Y + (Z02Y ) // (Z03Y )
- Alternative A equivalent circuit (Fig. 3):
References
[1] L.F. Blume, Influence of transformers connections on operation, AIEE Trans. 33
(May) (1914) 753–770.
40
A. Ramos, J.C. Burgos / Electric Power Systems Research 144 (2017) 32–40
[2] J.F. Peters, Harmonics in transformer magnetizing currents, AIEE Trans. 34
(September) (1915) 2157–2182.
[3] L.N. Robinson, Phenomena accompanying transmission with some types of
star transformer connections, AIEE Trans. 34 (September) (1915) 2183–2195.
[4] J.F. Peters, M.E. Skinner, Transformers for interconnecting high-voltage
transmission systems for feeding synchronous condensers from a tertiary
winding, AIEE Trans. 40 (June) (1921) 1181–1199.
[5] J. Mini Jr, L.J. Moore, R. Wilkins, Performance of auto transformers with
tertiaries under short-circuit conditions, AIEE Trans. 42 (October) (1923)
1060–1068.
[6] R.O. Kapp, A.R. Pearson, The performance of star–star transformers, J. Inst.
Electr. Eng. 1 (January (6)) (1955) 9–12.
[7] B.A. Cogbill, Are stabilizing windings necessary in all Y-connected
transformers? AIEE Trans. Power Appar. Syst. 78 (October (3)) (1959)
963–970.
[8] O.T. Farry, Tertiary windings in autotransformers, AIEE Trans. Power Appar.
Syst. 80 (April (3)) (1961) 78–82.
[9] A.N. Garin, Zero-phase sequence characteristics of transformers. Part I:
sequence impedances of a static symmetrical three-phase circuit and of
transformers, Gen. Electr. Rev. 43 (March (3)) (1940) 131–136.
[10] A.N. Garin, Zero-phase sequence characteristics of transformers. Part II:
equivalent circuits for transformers, Gen. Electr. Rev. 43 (April (4)) (1940)
174–179.
[11] K. Schlosser, Die Nullimpedanzen des Voll- und des Spartransformatoren
(Zero-sequence impedance in transformers and autotransformers),
Brown-Boveri-Nachr. 44 (February) (1962) 78–83 (in German).
[12] M. Christoffel, Zero-sequence reactances of transformers and reactors, Brown
Boveri Rev. 52 (November/December (11/12)) (1965) 837–842.
[13] K.P. Oels, Ersatzschaltungen des Transformators für das Nullsystem
(Zero-Sequence Equivalent Circuits for Transformers) ETZ-A Bd, 89, 1968, pp.
59–62 (in German).
[14] R. Allcock, S. Holland, L. Haydock, Calculation of zero-phase sequence
impedance for power transformers using numerical methods, IEEE Trans.
Magn. 31 (May (3)) (1995) 2048–2051.
[15] T. Ngnegueu, M. Mailhot, A. Munar, Zero-phase impedance and tank heating
model for three-phase three-leg core type transformers coupling magnetic
field and electric circuit equations in a finite element software, IEEE Trans.
Magn. 31 (May (3)) (1995) 2068–2071.
[16] A. Ramos, J.C. Burgos, et al., Determination of parameters of zero-sequence
equivalent circuits for three-phase three-legged YNynd transformers based
on onsite low-voltage tests, IEEE Trans. Power Deliv. 28 (July (3)) (2013)
1618–1625.
[17] A. Ramos, Consideraciones acerca de la Utilización de Arrollamientos de
Estabilización en Transformadores Estrella-Estrella (in Spanish). Pd. D.
Dissertation, Universidad Carlos III de Madrid, 2016.
[18] M.F. Lachman, Y.N. Shafir, Influence of single-phase excitation and
magnetizing reactance on transformer leakage reactance measurement, IEEE
Trans. Power Deliv. 12 (October (4)) (1997) 1538–1546.
[19] S.V. Kulkarni, S.A. Khaparde, Transformer engineering, in: Design and Practice,
Marcel Dekker, 2004.
[20] A. Ramos, J.C. Burgos, Influence of stabilizing windings on zero-sequence
performance of three-phase three-legged YNynd transformers. Part II: tank
overheating hazard and short-circuit duty, Electr. Power Syst. Res. XXXX. (in
revision).
[21] N. Chiesa, B.A. Mork, H.K. Høidalen, Transformer model for inrush current
calculations: simulations, measurements and sensitivity analysis, IEEE Trans.
Power Deliv. 25 (October (4)) (2010) 2599–2608.
[22] IEC Std. 60076-8, Power Transformers, Part 8: Application Guide. 1997.
[23] IEEE Std. C57.12.90, IEEE Standard Test Code for Liquid-Immersed
Distribution, Power and Regulating Transformers, 2010.
[24] D. Marquardt, An algorithm for least-squares estimation of nonlinear
parameters, J. Soc. Ind. Appl. Math. 11 (2) (1963) 431–441.