Available online at www.sciencedirect.com
Electric Power Systems Research 78 (2008) 1119–1128
Short survey
Sweep frequency response analysis (SFRA) for the assessment of
winding displacements and deformation in power transformers
J.R. Secue ∗ , E. Mombello
Instituto de Energı́a Eléctrica (IEE), Universidad Nacional de San Juan, Av. Lib. Gral. San Martı́n 1109 Oeste, 5400 San Juan, Argentina
Received 18 November 2006; received in revised form 18 July 2007; accepted 4 August 2007
Available online 27 September 2007
Abstract
The sweep frequency response analysis (SFRA) is an analysis technique for detecting winding displacement and deformation (among other
mechanical and electrical failures) on power and distribution transformers. Nowadays, there is an increasing interest in SFRA method because of its
sensibility in detecting mechanical faults without opening the unit. SFRA as a diagnostic technique must integrate both the off-line measurements
and the interpretation of the data in order to provide an assessment of the condition of the windings. However, guidelines for the measurement
and record interpretation are not available. The evaluation is presently done by experts in the topic through the visual inspection or with the help
of statistical parameters such as the correlation coefficient and the standard deviation. However, criteria like the limits of normal variation of the
parameters, and the features observed in the records in the presence of a determined type of fault could not to coincide. Although, there are some
proposals for making the interpretation more objective, neither of them integrate human expertise along with the different kind of parameters
obtained from the evaluation of the records in a diagnostic model. This paper presents a survey on the alternatives in the measurement techniques
and interpretation of SFRA measurements, describing some sources of uncertainty in applying this methodology.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Transformer; Sweep frequency response analysis (SFRA); Winding displacement; Winding deformation
1. Introduction
A reliable detection of mechanical failures in power transformers due to winding displacement and deformations requires
the implementation of a sensitive technique for the detection of
this type of damage. Some of the diagnosis techniques used for
this purpose are:
1. Measurement of short-circuit impedance (reactance) [1,2].
2. Vibro-acoustic method [3,4].
3. Frequency response analysis—FRA, obtained by two methods [4–47].
a. Low-voltage impulse—LVI.
b. Sweep frequency response analysis—SFRA.
4. Measurement of frequency response of stray losses—FRSL
[5,6].
∗
Corresponding author. Tel.: +54 2644226444; fax: +54 2644210299.
E-mail addresses: janneth.secue@gmail.com (J.R. Secue),
mombello@iee.unsj.edu.ar (E. Mombello).
0378-7796/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2007.08.005
FRA technique is widely used because of its high sensitivity and is based on the concept that changes in the windings
due to deformation and displacements cause a change in the
impedances of the transformer and consequently a modification
of its frequency response.
Frequency response analysis includes SFRA and LVI. Most
of the literature about this topic indicates that the term FRA is
understood as it was introduced by Dick and Erven [7]: “The
FRA method uses a sweep generator to apply sinusoidal voltages at different frequencies to one terminal of a transformer
winding. Amplitude and phase of signals obtained from selected
terminals of the transformers are plotted directly as a function of
frequency”. This definition coincides with the current definition
of SFRA method used by several authors, and is the one used in
this paper.
The research works carried out in previous works [5,7–16]
allows to establish the characteristics of LVI and SFRA and to
determine some advantages that SFRA has over LVI as: higher
signal to noise ratio, bigger repeatability and reproducibility and
less requirements of measurement equipment.
This paper presents a review of the SFRA methodology.
Section 2 focuses on the different aspects to be considered
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for the implementation of this metrology: frequency range and
connection of the non-tested terminals. Section 3 presents a
description of the SFRA’s sensitivity, and several aspects related
to the diagnosis. Several sources of uncertainty and inaccuracies
influencing the results of the measurement and the subsequent
diagnosis are showed in Section 4.
2. SFRA metrology
In order to analyze the SFRA metrology some general concepts of metrology must be given:
Measurand: Particular quantity subject to measurement.
Method of measurement: A logical sequence of operations,
described generically, used in the performance of measurements and based in a principle of measurement or scientific
base.
Measurement procedure: Set of operations, described specifically, used in the performance of measurements according to a
given method.
2.1. SFRA measurand
There exist two possibilities of measurand in applying the
SFRA method: Transfer Function (Vout /Vin ) and Impedance
(Vin /Iout ). As explained in the literature [8], the transfer function obtained from voltage ratio Vout /Vin has no direct relation
with the impedance measurement. The voltage ratio (Vout /Vin )
is most frequently used as transfer function to be measured. The
reasons for using impedance or voltage ratio transfer functions
are not clearly established in the literature.
Reference [17] indicates that admittance measurement is usually less sensitive to small geometric changes than voltage ratio
measurement. In [7,18] both types of measurement are made in
order to obtain diagnosis criteria. In [7] these measurements are
used also to calculate parameters of an equivalent circuit of the
transformer winding.
The main characteristics of the SFRA measurand are twofold: frequency range and number of frequencies, which have
been not clearly defined.
2.1.1. Frequency range
Table 1 contains a list of references reporting SFRA measurements. The third column indicates whether the measured
magnitude was an impedance (Z) or a transfer function (H) and
second column the frequency range, which varies from 10 Hz to
10 MHz.
2.1.2. Set of frequencies to be used during the test
The necessary time for performing an SFRA test (typically
several minutes) is related to the bandwidth and number of spot
frequencies, which are not universally defined. The number of
spot frequencies used or recommended by the different authors
is different. For example, 1000 spot frequencies are used in [7],
2000 in [8] and 3000 in [18].
Table 1
Frequency ranges used for SFRA measurements
Reference
Frequency
Measurement
[4,19]
[6,9]
[7]a
[8,17]
[11]
[12]
[13,20–22]
[14]b
[16]
[18]
[23]
[24]
[25]
[26,27]
[28,29]
[30]
[31]
[32,33]
[34,35]
[36]c
[37]
[38]
100 Hz to 1 MHz
Up to 2 MHz
1 kHz to 10 MHz
20 Hz to 2 MHz
10 kHz to 1 MHz
10 Hz to 2 MHz
Up to 1 MHz
10 Hz to 1 MHz
5 Hz to 2 MHz
100 Hz to 3 MHz
100 Hz to 1 MHz
Up to 10 MHz
Up to 2 MHz
50 Hz to 1 MHz
50 Hz to 200 KHz
10 Hz to 1 MHz
Up to 10 MHz
10 Hz to 10 MHz
Up to 200 kHz
1 kHz to 1 MHz
1 kHz to 450 kHz
10 kHz to 3 MHz
Z
H
H–Z
H
H
H
H
H
H
H–Z
H
Z–H
Z–H
H
H
H
H
H
Z
H
H
H
a
Measurements up to 10 MHz were carried out, but it is concluded that the
upper limit of the useful frequency range is 1 MHz.
b It is mentioned that the upper limit of the reproducible range is probably at
least 1 MHz.
c Several measurements have been performed in order to define the frequency
range for the test, it has been established that the upper limit for the reproducible
range is 1 MHz.
According [8], the relative spacing between adjacent spot
frequencies must be always less than 2%.
2.2. SFRA measurement procedure
There are three important aspects involved in the measurement procedure: terminals connection of tested and
non-tested terminals, types of measurement (transferred and
non-transferred) and set of measurements to be performed.
2.2.1. Terminals connection
a. Non-tested terminals grounded through a damping resistor
of 1 k. In [7] a damping resistor of 1 k is connected from
each non-tested terminal of the transformer to the grounded
tank. These resistors help to damp out secondary oscillations
in non-excited windings and to minimize stray capacitance
at the bushing terminals. Similarly in [36] resistors are connected to all tested and non-tested terminals.
b. All non-tested terminals open (floating). This configuration
is used in [17,24,29,39,40]. In [29] non-tested terminals
are left floating and the HV/LV neutral is grounded. Reference [39] states that earthing or short-circuiting non-tested
terminals constrains the flux in the transformer to follow
certain paths what results in the loss of potentially useful
data.
c. Measurements using non-tested terminals short-circuited.
Reference [21] indicates that short-circuiting the non-tested
J.R. Secue, E. Mombello / Electric Power Systems Research 78 (2008) 1119–1128
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2.2.3. Quantity of measurements (tests)
The number of measurements to be performed depends on
factors such as:
a. Set of tested terminals and connection form.
b. Set of non-tested terminals and connection form.
c. Type of measurement considered.
Fig. 1. Non-transferred measurement of a transfer function.
windings helps to remove the core effect at lower frequencies
below 200 kHz. This effect is identified in [8] as the cause of
the variation of the frequency response in the 2 kHz range,
which becomes more noticeable when non-tested terminals
are floating.
The authors of [4] perform tests using grounded (shortcircuited) non-tested terminals as well as using open non-tested
terminals in order to obtain complementary information from
the measurements. In a similar way, in [41] measurements
using ungrounded (short-circuited) non-tested terminals and
open non-tested terminals are proposed, too.
2.2.2. Types of measurement
a. Non-transferred measurements: The terminals used in the test
belong to the same voltage level (see Fig. 1).
b. Transferred measurements: The terminals used in the test
belong to the different voltage levels (see Fig. 2).
These types of measurement can be performed either for
the case of transfer function or impedance measurement.
For example, in [18] the input voltage and the input current of the low-voltage winding are measured to estimate the
input impedance; this is a non-transferred measurement of the
impedance. On the other hand, the output voltage of the highvoltage winding is measured to estimate a transfer function. This
is transferred measurement.
Fig. 2. Transferred measurement of a transfer function.
Comparing proposals [4,7,41] for a two-winding three-phase
wye–wye connection transformer, it can be seen that the different
proposals define the set of non-tested terminals and the number
of tests to be carried out in a different way.
In reference [4] 15 tests are suggested, with both open and
grounded (short-circuited) non-tested terminals:
a. Non-transferred measurements:
- Measurements performed having each terminal of the HV
windings as input and the respective neutral terminal as
output. Similarly, also the terminals of the LV were used as
inputs and the LV neutral terminal as output. All measurements performed using open non-tested terminals (HV and
LV).
- Measurements performed having each terminal of the HV
windings as input and the respective neutral terminal as
output, keeping HV non-tested terminals open and shortcircuited LV terminals. Similarly, also the terminals of the
LV winding were measured. These measurements characterize the HV/LV windings and the leakage impedances
between primary and secondary winding.
b. Transferred measurements:
- Measurements using the terminals to each phase of the
HV winding and the corresponding terminals (same phase)
of the LV winding, being non-tested terminals (HV and
LV) open and neutral terminals grounded. These measurements describe the leakage impedances between primary
and secondary winding.
Reference [41] reduces the number of tests from 15 tests to 12
since measurements between each LV terminal and LV neutral
terminal keeping non-tested LV terminals open and non-tested
HV terminals short-circuited are not proposed.
In reference [7] a set of 24 tests using grounded non-tested
terminals is proposed. The test to be performed are:
a. Non-transferred measurements:
- Measurements performed having each terminal of the HV
windings as input and the respective neutral terminal as
output. Similarly, also the terminals of the LV were used
as inputs and the LV neutral terminal as output.
- Supplementary measurements using both HV and LV neutral terminals.
- Supplementary measurements inverting the role input
–output of each pair of terminals.
b. Transferred measurements:
- Measurements using the terminals to each phase of the
HV winding and the corresponding terminals (same phase)
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of the LV winding, for highlighting capacitive coupling
between HV and LV windings.
These quantities do not include additional tests for different
tap positions.
acteristics or an old one in good condition. This is the most
difficult alternative, because it is not easy to find transformers of the same constructive characteristics, whose
operation conditions has been similar.
3.2. Features extracted from the frequency response
3. Fault diagnosis using SFRA
The sensitivity of SFRA has been extensively tested in several works [4,7,8,14,16,18,20,22–24,27,28,30–34,36–43,45] by
means of faults simulations in laboratory and of real cases studies
of transformers in service. Several types of faults can be detected
by SFRA, such as winding movements, winding deformation,
unclamped screws at the ends of a winding (loss of clamping
pressure), inter-turn faults, loosening of connecting leads of HV
and LV windings to the bushings, poor tank grounding, multiple
core grounding, among others.
There is linguistic agreement between some experts
[8,14,24,31], that major faults (caused by large movements of
the core or windings) are identified in the low frequency range
whereas minor faults (interturn faults, connection leads and
small displacements) are identified in the high frequency range.
However, there is no agreement regarding the frequency range
to be used during the tests.
A drawback of SFRA as diagnostic tool is that there are no
standard procedures to analyze and interpret the measurement
records yet. The diagnosis task is performed by experts through
the visual inspection of the records or with the help of some
statistical and mathematical parameters. The analysis depends
on factors as the type of recordings used for the comparison,
the features extracted from the frequency response, etc. These
factors are described below.
3.1. Type of recordings
SFRA method is based on the analysis of frequency response
recordings taken during the lifetime of the transformer. There
are two possibilities:
a. Analysis of recordings taken on windings having reference recordings. It is assumed that a set of historical
records representing a healthy state of the transformer is
available.
b. Analysis without reference recordings. If there are not historical recordings of the transformer there are two possibilities:
- Analysis using recordings which belong to different phases
of the same transformer. Due to the asymmetries inherent
in the transformer design, there are differences between
phases that must be considered. It has been reported in
[15] that normal difference inter-phase are not comparable
with the difference present in the case of significant winding displacements or deformations. This analysis has the
advantage that the measurements are made under the same
conditions.
- Analysis using recordings of twin transformers. The comparison is made on the basis of recordings from a twin
transformer, either a new transformer with the same char-
When reference recordings exist, the following features are
analyzed [39]:
- Changes in the shape of the curve.
- Appearance of new resonant frequencies or disappearance of
existing ones.
- Large shifts in existing resonant frequencies.
In order to establish the differences between recordings some
parameters have been defined, namely: correlation coefficient
(CC), standard deviation (S.D.) and maximum absolute difference (DABS). An analysis of the sensibility for detecting faults
of CC and S.D. for different frequency ranges is done in [42,43].
The conclusion was that CC is a useful statistical parameter,
while S.D. is an unreliable comparison parameter.
In [18] the disadvantages of CC and S.D. are analyzed. The
authors of [18] state that CC is not sensitive for detecting changes
in the frequency response characterized by a similar shape but
having a constant difference in magnitude, and that an undesirable overestimation of the parameter S.D. takes place when the
order of magnitude of the two responses analyzed differs not as a
consequence of any fault but as a consequence of the slight shift
of a peak, which is normal in this type of measurement. Other
parameters such as: sum of squares error (SSE), sum squared
ratio error (SSRE), sum square max–min error (SSMMRE), and
absolute sum of logarithmic error (ASLE) were proposed by the
authors in order to correct these undesirable characteristics of
the CC and S.D. However, most of them, excepting ASLE, have
undesirable numerical disadvantages. ASLE was presented as
the most reliable parameter which was designed to make the fully
log-scaled comparison in the magnitude frequency response; its
application considers a previous process of interpolation proposed by the authors. The normal range of variation for these
parameters has not been set yet.
On the other hand, some proposals have been made in order
for the interpretation of SFRA measurements to be more objective:
Ryder’s proposal [39]. This includes the calculation of CC
by ranges of frequencies, relative change in first resonant frequency, relative chance in minimum low frequency amplitude
and relative change in number of high frequency resonances.
The criteria to be applied to determine whether a particular
behavior constitutes a normal variation or not is not included
as a part of the method.
Frequency response modeling using an equivalent circuit
[19,27,28,30]. These models consider the behavior of the
core and windings as a function of the frequency. In [19]
the equivalent circuit uses sections having different topology,
representing a particular bandwidth defined on the basis of
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1123
Table 2
Characteristics of the proposals for the interpretation of SFRA measurements
resonant or antiresonant frequencies. The complete model size
depends on the quantity of the resonant and antiresonant points
which are identified in each particular case. In [27,28,30] the
initial equivalent circuit is simplified to three circuits for low,
medium and high frequency. In [27] the equivalent transfer
functions have low order (second and third), which do not
fit the frequency response measured; the parameters of the
functions are obtained using the invfreqs MATLAB command
(Signal Processing Toolbox). In these proposals the variations
of the parameters of these circuits are used for comparison
purposes. The function invfreqs finds a continuous-time transfer function that corresponds to a given complex frequency
response. From a laboratory analysis standpoint, invfreqs is
useful in converting magnitude and phase data into transfer
functions [48].
Modeling of transformers based on the internal geometry
and material properties [32,37,38,40,44–47]. These models
are a theoretical approach based on numerical simulation,
their importance is related with the possibility of evaluating the sensitivity of the method for different kinds of
faults.
Mathematical models. In these proposals the frequency
response is modeled as a rational function with real coefficients.
In [25] the rational function is solved through the already mentioned MATLAB function invfreqs. In [26,29] the problem of
finding the polynomial coefficients is solved using invfreqs and
non-iterative frequency-domain subspace-based identification
algorithms. The parameters of the rational model are proposed
for comparison purposes, but their sensibility to different kind
of faults is not reported.
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Artificial neural network (ANN) for fault diagnosis [22,25,34].
The use of ANNs for failure identification is proposed in these
research works. These ANNs are trained by simulating faults in
a specific test transformer. In [22] the CC and SD are calculated
for low, medium, high frequency ranges and also for whole
frequency range; then they are used as inputs of the ANN. The
output is a number, 0 indicates a normal state of the windings,
and 1 indicates abnormal state. The procedure is similar in [34]
but only the CC is used as inputs of the ANN and the output is
a number, if the output value exceeds a limit (the unity) then
the diagnosis is a fault in the transformer windings. In [25]
several networks are trained using the absolute frequency from
poles and zeros, natural frequency, and damping coefficient as
inputs; this ANN is able to identify and classify the type of
fault.
Table 3
Parameters used for frequency response analysis
ASLE(x, y) =
i=1
N
N
DABS(x, y) =
(1)
|yi − xi |
i=1
(2)
N
N
CC(x, y) =
N
N
xi2 yi2
i=1
MM =
xi yi
i=1
N
Table 2 summarizes the main characteristics of the proposals.
4. Uncertainty and inaccuracies using SFRA
N
20 log10 yi − 20 log10 xi
i=1
min(xi , yi )
i=1
N
(3)
(4)
max(xi , yi )
i=1
In practice there are several sources of uncertainty and inaccuracies that can influence the measurement results.
A linear single-input single-output system can be formally
characterized by means of its impulse response h(t) or by its
frequency response, H(jω). The frequency response is a representation of the system’s response to sinusoidal inputs at varying
frequencies. The response of a linear system to a sinusoidal
input is a sinusoid having the same frequency but a different magnitude and phase. The frequency response is defined
as the magnitude and phase differences of the used transfer
function.
It means that in order to perform a frequency response analysis in transformers there are two factors that must be taken into
account:
- The analysis must be performed using a frequency range in
which the system can be assumed to be linear.
- Not only the magnitude but also the phase response should be
analyzed.
The first factor is associated with the effects of the iron core
nonlinearity, which depends on the frequency range. The authors
of [10] propose to perform the measurements at frequencies
greater than 1 kHz for which it is supposed that the transformer
behaves linearly and the iron core does not play a significant role.
In [8] the effect of the core becomes significant for frequencies
lower than 2 kHz; in [8,21] it is stated that the effects of the
core are reduced if non-tested windings are short-circuited. In
[46,47] the nonlinear effects of the core has been assumed to
be negligible above 10 kHz, it is considered that the penetration depth of the magnetic field decreases with an increase of
frequency. However, in [49] it was shown that there is a considerable inductance above 1 MHz. So the interaction between
core and windings exists in the whole frequency range, even at
higher frequencies.
The second factor refers to the fact that the phase response
must be taken into account.
Except for CC, xi and yi are the ith elements
of the frequency responses to be
compared. N is the number of samples
For CC : xi = xn − µx
(5)
and yi = yn − µy
(6)
where µx and µy are the arithmetic average for {xn }, {yn }n=1,. . .,N
Usually, it is only considered the magnitude response for
the diagnosis, if the phase response is used, then it must be
correctly represented. When the phase is shown from −180 to
+180◦ (−π, π rad) there are some jumps when the angle exceeds
one of the limits. This wrapped phase may be corrected by certain algorithms that unwrap the phase, as the unwrap function
(MATLAB’s Signal Processing Toolbox).
Another kind of error of the frequency response measurement
is the presence of outliers in the records. These normally affect
the magnitude and the phase and cause distortions in any statistical test based on sample means and variances; for example
the parameter SSE is very sensitive to outliers. The detection of
outliers in the measured frequency response is possible through
visual inspection or techniques for discrete signals which use
forward differences. Outliers can be suppressed by means of an
interpolation process as proposed in [18], which considers the
non-equidistance of the used spot frequencies (logarithmically
distributed).
5. Case study showing the sensitivity of the
mathematical and statistical parameters
As an example, the parameters DABS, CC and MM
(Min–Max relation) are calculated to show their sensitivity to
variations in the frequency response and how they could be
used for diagnostic purposes. Table 3 shows the corresponding
mathematical expressions of these parameters. The parameter
DABS has the same form as ASLE excepting the factor 20 log10
and both parameters give the same results if the input data for
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Fig. 3. Transfer function amplitude.
Fig. 4. Transfer function phase.
DABS is given in decibels for the magnitude response. Min–Max
(MM) allows comparing the similarity of a data set. The parameter CC is sensible to resonance frequency displacements and
to differences in the amplitude provided that they are not constant.
In analyzing similarity by means of CC, MM and DABS, the
highest possible similarity level is CC = 1, MM = 1 or DABS = 0.
The tests carried out on a three-phase transformer (132
/69/13.86 kV–60/60/50 MVA wye–wye–delta) at factory were
used to evaluate the sensitivity of the parameters. The transfer
function was measured using an HP 4192A Impedance/Transfer
Function Analyzer. The measurements were made at 1000
logarithmically spaced spot frequencies in a frequency range
from 1 kHz up to 1 MHz. The analysis was performed using
records belonging to different phases of the primary winding.
To evaluate the parameters, the measurements were processed in order to suppress outliers and to correct wrapped
phase. Figs. 3 and 4 show the amplitude and phase response,
respectively.
Table 4 shows the values of the different parameters applied
to magnitude and phase response for defined frequency ranges,
which were selected in order to highlight the sensitivity of the
parameters as a consequence of variations between phases.
Analysis of the results:
• Range 1: 1–10 kHz
The parameters DABS and Min–Max are sensitive to slight
differences in amplitude between the responses, and CC to the
similarity in the shape of the responses, magnitude and phase.
• Range 2: 10–20 kHz
The parameter DABS assesses the differences in amplitude
between phases as a result of slight shifts in the resonant
frequency at 13 kHz corresponding to phase B with respect
to the other phases. The parameters CC and Min–Max also
Table 4
Parameters evaluated for a primary winding
Frequency
range (Hz)
Phases
Magnitude
response
Phase
response
DABS
CC
Min–Max
DABS
CC
Min–Max
1 kHz to 10 kHz
AN-BN
BN-CN
CN-AN
2.1E−02
8.6E−02
6.5E−02
1.000
1.000
1.000
0.999
0.994
0.996
1.0E−03
2.7E−03
1.7E−03
1.000
1.000
1.000
0.999
0.996
0.998
10 kHz to 20 kHz
AN-BN
BN-CN
CN-AN
1.2E+00
1.7E+00
7.0E−01
0.980
0.962
0.995
0.955
0.937
0.973
1.7E−01
2.4E−01
7.4E−02
0.979
0.958
0.994
0.803
0.737
0.897
20 kHz to 40 kHz
AN-BN
BN-CN
CN-AN
2.5E−01
1.2E+00
1.2E+00
0.999
0.981
0.980
0.988
0.944
0.942
2.9E−02
3.7E−02
3.9E−02
1.000
1.000
1.000
0.989
0.987
0.986
40 kHz to 100 kHz
AN-BN
BN-CN
CN-AN
4.2E−01
8.9E−01
1.2E+00
0.977
0.833
0.786
0.982
0.962
0.951
3.8E−02
6.3E−02
8.6E−02
0.943
0.770
0.677
0.993
0.989
0.985
100 kHz to 500 kHz
AN-BN
BN-CN
CN-AN
2.2E−01
4.6E−01
5.8E−01
0.982
0.922
0.919
0.989
0.978
0.973
2.1E−02
3.1E−02
3.4E−02
0.986
0.939
0.937
0.997
0.995
0.994
500 kHz to 1 MHz
AN-BN
BN-CN
CN-AN
1.1E−01
1.5E−01
2.3E−01
0.983
0.969
0.936
0.994
0.993
0.989
1.2E−02
1.5E−02
2.1E−02
0.913
0.825
0.625
0.998
0.998
0.997
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Table 5
Characteristic of the parameters CC, MM, DABS
No sensitivity to
Sensitivity to
CC
Changes in the shape of the responses
characterized by a constant difference in the
amplitude
Changes to the shape of the curve as a consequence of:
• The creation of new resonant frequencies or the elimination of existing
resonant frequencies
• Shifts in existing resonant frequencies
• Non-constant amplitude differences
MM
Changes in the shape of the responses, which
do not involve changes in amplitude
Changes of the shape especially related to amplitude variations in the responses as
a consequence of:
• The creation of new resonant frequencies or the elimination of existing
resonant frequencies
• Shifts in existing resonant frequencies
• Changes in which the responses may be similar in shape but have a constant
difference in amplitude
DABS
Changes in the measurements which do not
involve changes in amplitude
Changes of the amplitude in the curves as a consequence of:
• The creation of new resonant frequencies or the elimination of existing
resonant frequencies
• Shifts in existing resonant frequencies
• Non-constant amplitude differences
identify the shift at this resonant frequency (reduced values
compared with 1).
• Range 3: 20–40 kHz
The vertical variations for the phases A and B with respect
to the phase C observed in the magnitude response are identified by DABS. The slight shifts of the resonant frequencies,
which are not easily visualized, and the non-constant vertical changes between the responses are identified by CC and
Min–Max. The parameter Min–Max quantifies the amplitude
variations better than CC.
In this case the phase response has no detectable variations
and CC indicates that the shapes are similar for the three
responses, and the values of DABS and Min–Max lead to
conclude that there are no important variations in amplitude.
• Range 4: 40–100 kHz
The differences between the magnitude response of phase
C with respect to those of phases A and B, which evidence
not only shifts but non-constant variation of the amplitude, are
identified for all parameters. However, a higher sensitivity of
the CC can be appreciated. If the diagnosis be based on the
analysis of CC, these low values could be interpreted as a bad
condition for the phase C, but at the same time the values of
Min–Max for magnitude and phase responses indicate that the
differences are even smaller than those of the preceding range.
The CC is highly sensitive to shape variations and Min–Max
quantifies amplitude variations better, hence these parameters
complement each other.
• Ranges 5 and 6: 1000–500 kHz/500 kHz to 1 MHz
As in the previous range, CC is sensitive to the presence of some local minima and maxima, their displacements
and slight non-constant amplitude differences. DABS and
Min–Max evidence that the amplitude differences are not
significant.
Table 5 summarizes the characteristics observed in the parameters.
The analysis using several parameters is most reliable since
it is possible to confirm or weaken the diagnosis. As it has
been shown in this example, the use of only one parameter
could lead to an overestimation or underestimation of specific and isolated variations present in the frequency response.
Besides, if the sensitivity of several parameters is integrated,
it is possible to construct an automatic diagnostic system.
For example, if an axial displacement causes only amplitude
changes in the transfer function (TF) in a specify frequency
range as described in [38], DABS and Min–Max will vary
and if the differences are not uniform (which is the most
possible situation) CC will be considerably sensitive to this
change as well, allowing to confirm the diagnosis. In order
to identify the type of fault, a certain knowledge of the corresponding typical values of the used diagnostic parameters is
necessary.
The use of statistical or other parameters obtained from
the frequency response model as a rational function (poles,
zeros, residues, etc.) in applying SFRA methodology entails
the processing of measured data. The parameters of the rational function obtained from the frequency response must be
found by means of a fitting process. A very efficient and
new method for doing this is Vector Fitting [50], a robust
and public domain software whose approach is given in the
form of partial fractions. The method presents satisfactory
numerical stability in the case of high order approximations
and wide frequency ranges, which is necessary since the frequency response of power transformers normally has several
resonance frequencies in the frequency range from 1 kHz to
1 MHz.
A diagnostic technique based on multiple parameters of different kind (e.g., CC, ASLE, poles and residues) in order to
take advantage of the different sensitivity of each parameter
in the identification of features characterizing specific faults,
which also takes the uncertainty, the imprecision and the experts’
knowledge into account has not been proposed yet.
J.R. Secue, E. Mombello / Electric Power Systems Research 78 (2008) 1119–1128
1127
6. Conclusions
Acknowledgement
Nowadays, there is great interest in SFRA because of its sensitivity in detecting mechanical failures without opening the unit.
If SFRA is to be used as a diagnostic technique, it must integrate
the off-line measurements and the interpretation of the data in
order to provide assessment to the mechanical condition of the
windings and core.
The survey presented above shows that there are no guidelines
for the measurement and although there are several proposals
for the interpretation of the recordings, neither of them integrate
human expertise and represent it by means of an expert system
which require a knowledge structure.
The following characteristics must be taken into account in
order to obtain a suitable diagnostic tool based on SFRA:
The co-author J.R. Secue would like to thank the German
Academic Exchange Service (DAAD) for the financial support
of this work.
• The complex nature of the frequency response. Magnitude and
phase responses must be taken into account for the analysis.
• There are several physical interactions present during the measurement, such as the interactions between windings, core
and tank, depend on the type of winding excited, secondary
winding type, terminal configuration, non-tested terminals
connection, etc. An equivalent circuit could integrate all these
physical phenomenon, but it requires the information of geometrical and physical properties of the materials which is not
easily available.
• The measurements are affected by certain errors, thus any
result is only an approximation or estimation of the real value
of the measurand. Consequently, the parameters calculated
from the measurements are also affected by errors.
• A diagnostic methodology which makes use of parameters of different kind, e.g., statistical and those obtained
from rational functions, had not been yet proposed. In
order to integrate all parameters, it is necessary to consider
that those coming from rational functions contain amplitude and phase information. On the contrary, in the case of
statistical parameters it is necessary to calculate them separately, that is, one for magnitude and another one for phase
response.
• The knowledge of the experts in the topic is valuable but it
has not been integrated in a systematic way. This knowledge
can be illustrated, for example, by the fact that there is a
linguistic agreement between some experts with regard to the
relation between the severity of the fault and the frequency
range analyzed.
All these factors indicate that the diagnosis based on SFRA
is not straightforward, however, fuzzy causal diagnosis can be
applied in order to obtain a diagnostic methodology. The consistency fuzzy approach and abduction fuzzy approach described
in [51] could be adopted for the solution due to its effectiveness for dealing with several sources of uncertainty, such as
those described for SFRA and those obtained in the process
of feature extraction, i.e., the characteristic parameters. Furthermore, the experts’ knowledge can be integrated with this
approach.
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