A2
Power transformers
and reactors
Advances in the interpretation
of transformer Frequency
Response Analysis (FRA)
Reference: 812
September 2020
Advances in the interpretation of
transformer Frequency Response
Analysis (FRA)
WG A2.53
Members
P. PICHER, Convenor
M. LACHMAN, TF2 Leader
P. PATEL, TF4 Leader
B. DIGGIN
R. GRUENSEIS
V. LARIN
M. LOCARNO
S. MIYAZAKI
J. SANCHEZ
B. SZTARI
R.K. TYAGI
Z.D. WANG
P. WERELIUS
CA
US
US
IE
AT
RU
US
JP
FR
HU
IN
GB
SE
S. TENBOHLEN, TF1 Leader
A. SCARDAZZI, TF3 Leader
N. ABEYWICKRAMA
S. GAZIVODA
D. JU
J. LI
H. MARTINS
M. RAEDLER
Y. SHIRASAKA
M. TAHIR
J. VELASQUEZ
M. WEBER
R. ZALESKI
DE
BR
SE
HR
CN
CN
BR
AT
JP
DE
DE
DE
PL
Corresponding Members
R. ALVAREZ
O. AMIROUCHE
H. RAHIMPOUR
J. TUSEK
AR
IT
AU
AU
R. AMINI
J. BORGHETTO
D. SOFIAN
IR
IT
GB
Other contributors
E. ALZIEU
P.A. CROSSLEY
M.H. SAMIMI
FR
GB
DE
B.Z. CHENG
R.S. FERREIRA
GB
CA
Copyright © 2020
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by law”.
ISBN : 978-2-85873-517-4
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Executive summary
The objective of CIGRE Working Group (WG) A2.53 was to develop a guide for the interpretation of
transformer Frequency Response Analysis (FRA).
To achieve this goal, the WG focused on the following tasks:
▪
▪
▪
▪
Understand the frequency response and the factors influencing the measurement
Collect interesting case studies
Review literature on quantitative FRA assessment
Apply numerical indices for FRA interpretation and make recommendations
A selection of 18 case studies from a total of 60 cases collected by the WG is presented in this Technical
Brochure (TB). These examples illustrate the frequency ranges that relate to the different failure modes.
However, it can be inferred from the small number of cases shared at the international level that
experience is very limited in FRA interpretation where there is a clear correlation with a mechanical
failure. This could be due to multiple failure modes playing a role when a winding is damaged. For
instance, if a winding is subjected to mechanical displacement, this is likely followed by a turn-to-turn
failure, which can be detected using conventional electrical tests. Mechanical displacements without
turn-to-turn failure are often investigated by dismantling the active part to identify a mechanically
damaged winding; this is often associated with the winding closest to the core. Since such post-mortem
investigations are costly, they are not common practice and potentially good case studies are being lost.
It is also possible for a transformer to survive the mechanical displacement of a winding and for this
weakened condition to go unnoticed due to lack of testing and inspection.
Because of the limited availability of real case studies, academic institutions have worked intensively in
recent years to generate data that can be used to improve interpretation. This data comes either from
experimental laboratory winding models that can be physically modified to simulate mechanical failure
modes, or from numerical modelling. A considerable amount of literature covers high frequency
transformer modelling, using lumped-element circuits or finite element simulations, numerical indices
for FRA assessment and artificial intelligence. However, the question remains as to the reliability of an
extrapolation from simulated cases to a real-life circumstance, especially when the discrepancy may be
due to external or testing related issues. Therefore, recommendations based on simulations should be
treated with some caution.
This TB contributes to the improvement of FRA interpretation by collecting industry experience and
contributions from academic institutions. Based on academic experimental work and a comparative
analysis of numerical indices as applied to selected case studies, some indices were identified as the
most promising for further investigation into objective FRA interpretation.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Contents
1.
Introduction.............................................................................................................. 11
2.
General understanding of the frequency response............................................... 13
2.1
Introduction .............................................................................................................................................. 13
2.2
Traveling through frequency ranges along the frequency response .................................................. 13
2.2.1
General .............................................................................................................................................. 13
2.2.2
Core ................................................................................................................................................... 13
2.2.3
Winding interaction ............................................................................................................................ 14
2.2.4
Winding structure ............................................................................................................................... 14
2.2.5
Bushings, leads and earth path ......................................................................................................... 14
2.2.6
Leakage channel ............................................................................................................................... 14
2.3
Description of typical frequency responses ......................................................................................... 15
2.3.1
Differences between low-voltage and high-voltage winding responses ............................................. 15
2.3.2
Typical responses of various types of coils ........................................................................................ 16
2.3.3
Frequency response of typical winding configurations ....................................................................... 16
3.
Fundamental understanding of the frequency response through transformer
circuit modelling................................................................................................................ 19
3.1
Introduction .............................................................................................................................................. 19
3.2
Taking a single winding as example ...................................................................................................... 19
3.2.1
Typical frequency responses of single air core winding ..................................................................... 19
3.2.2
Typical frequency responses of single windings with core ................................................................. 21
3.2.3
Modelling a single winding with a full matrix to deal with mutual inductive coupling .......................... 22
3.3
Taking a single-phase two-winding transformer as example .............................................................. 23
3.3.1
General .............................................................................................................................................. 23
3.3.2
Impact of HV winding onto LV winding FRA plot in the low frequency range..................................... 24
3.3.3
Winding interactions in the medium frequency range ........................................................................ 26
3.4
Interactions among phases and three-phase 3-winding transformers ............................................... 26
3.5
Understanding winding natural frequencies ......................................................................................... 26
4.
Factors influencing the measurement ................................................................... 29
4.1
Introduction .............................................................................................................................................. 29
4.2
Measurement setup ................................................................................................................................. 29
4.2.1
Poor earthing ..................................................................................................................................... 29
4.2.2
External busbar connected to bushings ............................................................................................. 30
4.2.3
Poor connections of short-circuiting cables ....................................................................................... 30
4.2.4
Connection between measurement lead screen and bushing flange................................................. 30
4.2.5
Earthing neutral bushing in not-under-test winding ............................................................................ 31
4.2.6
Direction of end-to-end measurement ............................................................................................... 31
4.2.7
Incidence of the selected measurement voltage ................................................................................ 32
4.3
Transformer state .................................................................................................................................... 32
4.3.1
Core earth .......................................................................................................................................... 32
4.3.2
Insulating fluid .................................................................................................................................... 33
4.3.3
Influence of core magnetic state ........................................................................................................ 35
4.3.4
Influence of test bushings .................................................................................................................. 36
4.3.5
Temperature ...................................................................................................................................... 38
4.3.6
Moisture ............................................................................................................................................. 38
4.4
Transformer configuration ...................................................................................................................... 38
4.4.1
Tap position in winding-under-test ..................................................................................................... 38
4.4.2
Tap changer and series/parallel switch in not-under-test winding...................................................... 39
4.4.3
OLTC direction prior to entering neutral position ............................................................................... 40
4.4.4
Opening and earthing/unearthing corner of buried delta winding....................................................... 41
4.5
External factors........................................................................................................................................ 41
4.5.1
External interference .......................................................................................................................... 41
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
4.5.2
Insufficient clearance ......................................................................................................................... 42
4.6
Notes on data analysis ............................................................................................................................ 43
4.6.1
Comparison between phases ............................................................................................................ 43
4.6.2
Comparison of sister units ................................................................................................................. 43
4.6.3
Additional notes ................................................................................................................................. 45
5.
Case studies ............................................................................................................ 46
5.1
Introduction .............................................................................................................................................. 46
5.2
Radial buckling ........................................................................................................................................ 46
5.2.1
Case 1 Buckling of the common and tertiary windings ...................................................................... 46
5.2.2
Case 2 Buckling of the inner LV winding ........................................................................................... 47
5.2.3
Case 3 Buckling of the common winding ........................................................................................... 48
5.2.4
Case 4 Buckling of the inner winding ................................................................................................. 50
5.3
Axial deformation, twisting and tilting ................................................................................................... 51
5.3.1
Case 5 Twisting and loss of clamping ................................................................................................ 51
5.3.2
Case 6 Axial movement of the LV winding ........................................................................................ 53
5.3.3
Case 7 Axial displacement of the stabilising winding ......................................................................... 55
5.3.4
Case 8 Conductors tilting ................................................................................................................... 58
5.4
FRA before and after short-circuit tests ................................................................................................ 59
5.4.1
General .............................................................................................................................................. 59
5.4.2
Case 9 Successful short-circuit test ................................................................................................... 59
5.4.3
Case 10 Successful short-circuit test ................................................................................................. 59
5.4.4
Case 11 Successful short-circuit test ................................................................................................. 60
5.4.5
Case 12 Slightly buckled winding ...................................................................................................... 60
5.4.6
Case 13 Lead movement, axial collapse and spiralling ..................................................................... 62
5.4.7
Case 14 Slight displacement of leads showing possible twisting of the winding ................................ 63
5.5
Electrical failures ..................................................................................................................................... 64
5.5.1
Case 15 Broken earthing connection between core elements of a shunt reactor .............................. 64
5.5.2
Case 16 Short-circuit inside winding .................................................................................................. 64
5.5.3
Case 17 Opening of a parallel winding .............................................................................................. 65
5.5.4
Case 18 Broken bushing connection ................................................................................................. 66
6.
Literature review of quantitative FRA assessment................................................ 68
6.1
Introduction .............................................................................................................................................. 68
6.2
Algorithms based on numerical indices ................................................................................................ 68
6.2.1
Frequency range................................................................................................................................ 68
6.2.2
Definitions .......................................................................................................................................... 68
6.2.3
Numerical indices extracted from frequency response traces ........................................................... 70
6.2.4
Indices based on resonance frequencies .......................................................................................... 73
6.2.5
Numerical indices based on vector fitting of measured traces ........................................................... 75
6.2.6
Vector based method using a sliding window .................................................................................... 75
6.2.7
Summary on numerical indices .......................................................................................................... 77
6.3
Algorithms based on white-box models ................................................................................................ 77
6.3.1
General .............................................................................................................................................. 77
6.3.2
Lumped high-frequency modelling ..................................................................................................... 78
6.3.3
Physical parameters extracted from frequency response measurements ......................................... 80
6.3.4
3D Finite Element Method (FEM) modelling ...................................................................................... 80
6.3.5
Summary on algorithms based on white-box models ........................................................................ 82
6.4
Algorithms based on artificial intelligence ............................................................................................ 82
6.4.1
General .............................................................................................................................................. 82
6.4.2
Decision tree ...................................................................................................................................... 82
6.4.3
Neural networks ................................................................................................................................. 83
6.4.4
Summary on algorithms based on artificial intelligence ..................................................................... 85
7.
Evaluation of selected numerical indices .............................................................. 86
7.1
Introduction .............................................................................................................................................. 86
7.2
Evaluation of the numerical indices....................................................................................................... 86
7.2.1
Experimental studies ......................................................................................................................... 86
7.2.2
Monotonicity....................................................................................................................................... 87
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
7.2.3
7.2.4
7.2.5
Linearity ............................................................................................................................................. 88
Sensitivity .......................................................................................................................................... 89
Dependency on number of data points .............................................................................................. 91
7.3
Discussion ............................................................................................................................................... 91
8.
Comparative analysis of selected numerical indices using case studies ........... 92
8.1
Selection of cases and type of frequency response measurements .................................................. 92
8.2
Selection of indices ................................................................................................................................. 93
8.3
Index analysis results ............................................................................................................................. 93
8.4
Application example: vector based method using a sliding window .................................................. 96
8.5
Discussion ............................................................................................................................................. 102
9.
Conclusion ............................................................................................................. 103
References ....................................................................................................................... 104
6
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figures
Figure 2.1 Main features of a typical frequency response ..................................................................................... 14
Figure 2.2 Frequency response range influenced by leakage channel .................................................................. 15
Figure 2.3 Typical open-circuit frequency response measurements of HV and LV windings ................................ 15
Figure 2.4 Typical responses of common and series windings.............................................................................. 16
Figure 2.5 Typical frequency responses of interleaved vs plain disc windings ...................................................... 16
Figure 2.6 Example of frequency responses of a delta-star transformer with buried tertiary ................................. 17
Figure 2.7 Example of frequency responses of a star-delta transformer with buried tertiary ................................. 17
Figure 2.8 Example of frequency responses of a star-star transformer with buried tertiary ................................... 17
Figure 2.9 Examples of frequency response of autotransformer with accessible tertiary: (a) series winding, (b)
common winding and (c) delta ............................................................................................................................... 18
Figure 3.1: n-stage lumped-element LC ladder network of single air-core winding ............................................... 20
Figure 3.2 Typical FRA responses of single air core windings .............................................................................. 20
Figure 3.3 Typical FRA responses of single windings with core ............................................................................ 21
Figure 3.4 Typical FRA responses of single windings with core (effect of mutual inductance – shift resonant
frequencies up) ...................................................................................................................................................... 23
Figure 3.5 Circuit model for single-phase two winding transformer ....................................................................... 23
Figure 3.6 Typical FRA responses of windings in a transformer ............................................................................ 24
Figure 3.7 Comparison of LV winding FRA plots before and after HV winding is added ....................................... 24
Figure 3.8 Two-stage equivalent network of single-phase two-winding transformer.............................................. 25
Figure 3.9 Comparison of HV winding FRA plots before and after LV winding is added ....................................... 26
Figure 3.10 Frequency response (a) and active admittance of HV winding (b) ..................................................... 27
Figure 3.11 Frequency response (a) and active admittance of HV winding (b) with earthed (1) and unearthed (2, 3)
core and shields (1 and 2 – HVOC; 3 – HVSC) ..................................................................................................... 28
Figure 4.1 Effect of increased earth circuit impedance .......................................................................................... 29
Figure 4.2 External busbar connected to bushings ................................................................................................ 30
Figure 4.3 Poor connections of short-circuit cables ............................................................................................... 30
Figure 4.4 Effect of earthing configuration of screen of measuring lead ................................................................ 31
Figure 4.5 Effect of earthing neutral in not-under-test winding .............................................................................. 31
Figure 4.6 Influence of direction of measurement.................................................................................................. 32
Figure 4.7 Measurements of same 11 kV winding using 3 different voltage levels 3 V (green), 5 V (grey) and 10 V
(red) peak to peak [21] .......................................................................................................................................... 32
Figure 4.8 Influence of core earth (1/2) ................................................................................................................. 33
Figure 4.9 Influence of core earth (2/2) ................................................................................................................. 33
Figure 4.10 Influence of insulating fluid, LV side (1/4) ........................................................................................... 34
Figure 4.11 Influence of insulating fluid, HV side (2/4) .......................................................................................... 34
Figure 4.12 Influence of insulating fluid (3/4) ......................................................................................................... 34
Figure 4.13 Influence of insulating fluid (4/4) ......................................................................................................... 35
Figure 4.14 Effect of core magnetic state (1/3) ...................................................................................................... 36
Figure 4.15 Effect of core magnetic state (2/3) ...................................................................................................... 36
Figure 4.16 Effect of core magnetic state (3/3) ...................................................................................................... 36
Figure 4.17 Influence of test bushings ................................................................................................................... 37
Figure 4.18 Influence of different bushings ............................................................................................................ 37
Figure 4.19 Effect of temperature [23] ................................................................................................................... 38
Figure 4.20 Influence of tap changer position in winding-under-test (1/2) ............................................................. 39
Figure 4.21 Influence of tap changer position in winding-under-test (2/2) ............................................................. 39
Figure 4.22 Effect of various connections of not-under-test winding (1/3) ............................................................. 39
Figure 4.23 Effect of various connections of not-under-test winding (2/3) ............................................................. 40
Figure 4.24 Effect of various connections of not-under-test winding (3/3) ............................................................. 40
Figure 4.25 Influence of tap changer previous to neutral position ......................................................................... 40
Figure 4.26 Effect of buried delta winding configuration (1/2) ................................................................................ 41
Figure 4.27 Effect of buried delta winding configuration (2/2) ................................................................................ 41
Figure 4.28 Influence of external interference ....................................................................................................... 42
Figure 4.29 Frequency response measurements using two devices having different dynamic ranges [29] .......... 42
Figure 4.30 Effect of an insufficient clearance, in this case 20 mm (LV of generator transformer) ........................ 43
Figure 4.31 Comparison between phases ............................................................................................................. 43
Figure 4.32 Comparison of sister units (1/3) .......................................................................................................... 44
Figure 4.33 Comparison of sister units (2/3) .......................................................................................................... 44
Figure 4.34 Comparison of sister units (3/3) .......................................................................................................... 44
Figure 5.1 Case 1 Open-circuit measurements on series windings ....................................................................... 47
Figure 5.2 Case 1 Open-circuit measurements on common windings ................................................................... 47
Figure 5.3 Case 1 Internal inspection .................................................................................................................... 47
Figure 5.4 Case 2 Open-circuit measurements on the LV winding of two identical transformers .......................... 48
Figure 5.5 Case 2 Internal inspection .................................................................................................................... 48
Figure 5.6 Case 3 Series and common windings open-circuit measurements ...................................................... 49
Figure 5.7 Case 3 Tertiary windings open-circuit measurements .......................................................................... 49
Figure 5.8 Case 3 Common windings short-circuit measurements with tertiary windings shorted......................... 49
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 5.9 Case 3 Internal inspection .................................................................................................................... 50
Figure 5.10: Case 4 LV open circuit measurements [30] ....................................................................................... 51
Figure 5.11 Case 4 Internal inspection (left - phase 1 and right - phase 2) [30] .................................................... 51
Figure 5.12 Case 5 Open-circuit measurements on the common windings ........................................................... 52
Figure 5.13 Case 5 Internal inspection [31] ........................................................................................................... 53
Figure 5.14 Case 6 HV open-circuit measurements .............................................................................................. 53
Figure 5.15 Case 6 HV with short-circuit on LV ..................................................................................................... 54
Figure 5.16 Case 6 LV open-circuit measurements ............................................................................................... 54
Figure 5.17 Case 6 LV with short-circuit on HV ..................................................................................................... 54
Figure 5.18 Case 6 Internal inspection .................................................................................................................. 55
Figure 5.19 Case 7 Open-circuit measurements on the series windings ............................................................... 56
Figure 5.20 Case 7 Open-circuit measurements on the common windings ........................................................... 57
Figure 5.21 Case 7 Stabilizing winding open-circuit measurements ...................................................................... 57
Figure 5.22: Case 7 Internal inspection ................................................................................................................. 58
Figure 5.23 Case 8 HV open-circuit measurements .............................................................................................. 58
Figure 5.24 Case 8 Internal inspection .................................................................................................................. 59
Figure 5.25 Case 9 HV and LV open-circuit measurements before and after the short-circuit test ....................... 59
Figure 5.26 Case 10 HV and LV open-circuit measurements on phase A ............................................................. 60
Figure 5.27 Case 11 HV open-circuit measurements on phase 1 (left) and phase 3 (right) .................................. 60
Figure 5.28 Case 12 Measurements before and after three consecutive short-circuit tests .................................. 61
Figure 5.29: Case 12 Internal inspection ............................................................................................................... 62
Figure 5.30 Case 13 Common and tertiary windings measurements before and after the short-circuit test .......... 62
Figure 5.31 Case 13 Internal inspection ................................................................................................................ 63
Figure 5.32 Case 14 HV to neutral open-circuit measurements before and after the short-circuit test .................. 63
Figure 5.33 Case 14 Picture of the internal inspection .......................................................................................... 63
Figure 5.34 Case 15 Open-circuit measurements on failed unit (left) and sister unit (right) .................................. 64
Figure 5.35 Case 16 HV open-circuit measurements ............................................................................................ 64
Figure 5.36 Case 16 LV open-circuit measurements............................................................................................. 65
Figure 5.37 Case 16 HV measurements with LV shorted ...................................................................................... 65
Figure 5.38 Case 17 Open-circuit measurements on the LV side ......................................................................... 66
Figure 5.39 Case 18 Open-circuit measurements on the series+common windings (left) and on the common winding
(right) ..................................................................................................................................................................... 66
Figure 5.40 Case 18 Internal inspection ................................................................................................................ 67
Figure 6.1 Application of a numerical index for FRA interpretation ........................................................................ 70
Figure 6.2 Amplitude variation around resonance points which affects some indices ........................................... 72
Figure 6.3 Resonance and anti-resonance points in an FRA trace ....................................................................... 74
Figure 6.4 Basic principle of winding assessment algorithm.................................................................................. 77
Figure 6.5 Representation of the winding assessment factor SDD for a three-phase transformer ........................ 77
Figure 6.6 Lumped model of a two-winding transformer [60] ................................................................................. 78
Figure 6.7 Method for extraction of main physical indicators [82] .......................................................................... 80
Figure 6.8 FEM model of a two-winding transformer [83] ...................................................................................... 81
Figure 6.9 Components of a decision tree [91] ...................................................................................................... 83
Figure 6.10 Structure of a three-layer ANN system [93] ........................................................................................ 84
Figure 7.1 Experimental setup of 1 MVA distribution transformer windings [103] .................................................. 87
Figure 7.2 Cast-resin transformer [103] ................................................................................................................. 87
Figure 7.3 Monotonic and non-monotonic behaviour of indices against different extents of radial deformation fault
.............................................................................................................................................................................. 88
Figure 7.4 Behaviour of indices with different levels of axial displacement ........................................................... 89
Figure 7.5 Linearity of the indices .......................................................................................................................... 89
Figure 7.6 Large amplitude variation around some anti-resonance points in frequency response of a 34 MVA,
237/5.65 kV transformer ........................................................................................................................................ 89
Figure 7.7 Frequency responses of RLC circuit for discussion of horizontal and vertical sensitivities ................... 90
Figure 7.8 Comparison of CC, CCF and LCC against (a) frequency shifts and (b) amplitude changes ................ 90
Figure 7.9 Comparison of horizontal and vertical sensitivities of indices for two frequency bands: 10–100 kHz and
20–50 kHz ............................................................................................................................................................. 90
Figure 8.1 Index ratio of low frequency band (1–10 kHz) between affected and unaffected phase/winding by the
fault in each case study ......................................................................................................................................... 95
Figure 8.2 Index ratio of mid frequency band (10–500 kHz) between affected and unaffected phase/winding by the
fault in each case study ......................................................................................................................................... 96
Figure 8.3 Index ratio of high frequency band (500 kHz–1 MHz) between the affected and unaffected phase/winding
by the fault in each case study .............................................................................................................................. 96
Figure 8.4 Application of assessment factor SDD in Case 2 open-circuit measurements on the LV winding of two
identical transformers ............................................................................................................................................ 97
Figure 8.5 Application of assessment factor SDD in Case 5 open-circuit measurements on the series windings . 98
Figure 8.6 Application of assessment factor SDD in Case 6 LV with short-circuit on HV ...................................... 98
Figure 8.7 Application of assessment factor SDD in Case 9 HV and LV open-circuit measurements before and after
short-circuit test ..................................................................................................................................................... 99
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 8.8 Application of assessment factor SDD in Case 10 HV and LV open-circuit measurements on phase A
.............................................................................................................................................................................. 99
Figure 8.9 Application of assessment factor SDD in Case 11 HV open-circuit measurements on phase 1 ......... 100
Figure 8.10 Application of assessment factor SDD in Case 12 measurements before and after second and third
short-circuit tests ................................................................................................................................................. 100
Figure 8.11 Application of assessment factor SDD in Case 13 common and tertiary measurements before and after
the short-circuit test ............................................................................................................................................. 101
Figure 8.12 Application of assessment factor SDD in Case 14 HV to neutral open-circuit measurements before and
after the short-circuit test ..................................................................................................................................... 101
Figure 8.13 Minimum value of winding assessment factor SDD (MSDD) in cases 2 to 14 .................................. 102
Tables
Table 6.1 Frequency ranges used in literature for FRA interpretation ................................................................... 68
Table 6.2 Names, abbreviations and main references of the numerical indices ([59]) ........................................... 69
Table 6.3 Level of deformations based on Rxy...................................................................................................... 73
Table 6.4 Description of the circuit elements of the transformer model ................................................................. 78
Table 6.5 Description of the training set [92] ......................................................................................................... 83
Table 7.1 Case studies for index evaluation .......................................................................................................... 86
Table 7.2 Monotonicity of the numerical indices .................................................................................................... 88
Table 7.3 Effect of number of data points .............................................................................................................. 91
Table 8.1 Selected cases for index analysis .......................................................................................................... 92
Table 8.2 Results of the index analysis on the affected phase/winding by the faults presented in six selected case
studies ................................................................................................................................................................... 94
Table 8.3 Results of the index analysis on the unaffected phase/winding by the faults presented in six selected case
studies ................................................................................................................................................................... 95
Table 8.4 Selected cases for application of graphical evaluation method ............................................................. 97
Equations
Equation 3.1 .......................................................................................................................................................... 20
Equation 3.2 .......................................................................................................................................................... 20
Equation 3.3 .......................................................................................................................................................... 21
Equation 3.4 .......................................................................................................................................................... 21
Equation 3.5 .......................................................................................................................................................... 22
Equation 3.6 .......................................................................................................................................................... 25
Equation 3.7 .......................................................................................................................................................... 25
Equation 3.8 .......................................................................................................................................................... 27
Equation 3.9 .......................................................................................................................................................... 28
Equation 6.1 .......................................................................................................................................................... 70
Equation 6.2 .......................................................................................................................................................... 70
Equation 6.3 .......................................................................................................................................................... 70
Equation 6.4 .......................................................................................................................................................... 70
Equation 6.5 .......................................................................................................................................................... 71
Equation 6.6 .......................................................................................................................................................... 71
Equation 6.7 .......................................................................................................................................................... 71
Equation 6.8 .......................................................................................................................................................... 71
Equation 6.9 .......................................................................................................................................................... 71
Equation 6.10 ........................................................................................................................................................ 71
Equation 6.11 ........................................................................................................................................................ 71
Equation 6.12 ........................................................................................................................................................ 71
Equation 6.13 ........................................................................................................................................................ 71
Equation 6.14 ........................................................................................................................................................ 71
Equation 6.15 ........................................................................................................................................................ 71
Equation 6.16 ........................................................................................................................................................ 71
Equation 6.17 ........................................................................................................................................................ 71
Equation 6.18 ........................................................................................................................................................ 71
Equation 6.19 ........................................................................................................................................................ 71
Equation 6.20 ........................................................................................................................................................ 71
Equation 6.21 ........................................................................................................................................................ 71
Equation 6.22 ........................................................................................................................................................ 71
Equation 6.23 ........................................................................................................................................................ 71
Equation 6.24 ........................................................................................................................................................ 74
Equation 6.25 ........................................................................................................................................................ 74
Equation 6.26 ........................................................................................................................................................ 74
Equation 6.27 ........................................................................................................................................................ 74
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Equation 6.28 ........................................................................................................................................................ 74
Equation 6.29 ........................................................................................................................................................ 74
Equation 6.30 ........................................................................................................................................................ 74
Equation 6.31 ........................................................................................................................................................ 74
Equation 6.32 ........................................................................................................................................................ 74
Equation 6.33 ........................................................................................................................................................ 75
Equation 6.34 ........................................................................................................................................................ 75
Equation 6.35 ........................................................................................................................................................ 75
Equation 6.36 ........................................................................................................................................................ 76
10
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
1.
Introduction
Measurement of a frequency response is now commonly used to assess the mechanical integrity of the
active part of power transformers. The analysis of the results, so-called Frequency Response Analysis
(FRA), is based on comparison with a reference measurement which is either a previous measurement
on the same unit, a measurement on an identical transformer or a measurement on another phase of a
three-phase transformer.
In 2008, CIGRE published a Technical Brochure (TB 342) [1] on the assessment of the mechanical
condition of transformer windings using Frequency Response Analysis (FRA). This guide covers the
various measurement techniques available in the industry and makes recommendations on the
standardization of good measurement practices. One chapter is dedicated to FRA interpretation and
several examples of frequency response measurements and diagnostics are reported.
After the CIGRE Working Group had completed their work, an IEC project team was initiated to develop
a standard on the measurement of frequency response. The standard, published in 2012 [2], largely
based on the previously published CIGRE guide, specifies the measurement method (connection and
configuration), the measuring equipment and the measurement records.
In parallel with the CIGRE and IEC initiatives, another working group was active within the IEEE. The
measurement configurations presented in the IEEE guide [3] are similar to those presented by IEC but
there are a few differences in the recommended configurations for a new set of measurements.
Even if the method has recently been studied at the international level in various working groups under
the umbrella of the CIGRE, IEC and IEEE organisations, there was still a need in the industry to obtain
more guidance on the interpretation of the results. In fact, the usual way to interpret the result is to
visually and subjectively compare the frequency response curves and make an interpretation based on
previous experience. The ultimate goal would be to develop an internationally agreed objective
interpretation algorithm that can be applied to condition assessment of transformers after an in-service
event, or as a pass-fail criterion for transformer short-circuit testing.
In this context, CIGRE WG A2.53 was created in September 2015. During the first meeting in June
2016, it was decided to divide the working group into the following four task forces:
▪
▪
▪
▪
Understanding the frequency response and the factors influencing the measurement
Case studies
Literature review of quantitative FRA assessment
Application of indices for FRA interpretation
This Technical Brochure (TB) is structured in the following chapters:
▪
▪
▪
▪
▪
▪
▪
Chapters 2 and 3 – Understanding the frequency response: general overview and
fundamental analysis through circuit modelling.
Chapter 4 – Factors influencing the measurement: the factors that can influence the
measurement and therefore the interpretation. This chapter covers the measurement setup,
the transformer state and configuration, and other external factors.
Chapter 5 – Case studies: a description of case studies showing how FRA can be used to
detect mechanical failure modes and other electrical failures.
Chapter 6 – Literature review of quantitative FRA assessment: a comprehensive review of the
academic state-of-the-art regarding quantitative FRA assessment. Note that an interim report
on this subject was published on behalf of the WG [4].
Chapter 7 – Evaluation of selected numerical indices: the numerical indices are evaluated
based on their performance over key technical parameters.
Chapter 8 – Comparative analysis of selected numerical indices using case studies: a study of
the performance of the numerical indices using selected CIGRE A2.53 case studies datasets.
This chapter concludes with recommendations to investigate promising numerical indices in
the future and to gain more experience by utilizing the indices in several other cases.
Chapter 9 – Conclusion.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Acknowledgments:
The redaction of a TB requires the contribution of all members of the WG. The task force leaders and
some members made extra contributions to put together the text, figures and tables of this document.
To acknowledge this additional contribution, their names are found just under the title of each chapter.
Finally, the authors would like to acknowledge WG members Mario Locarno (US) and Joe Tusek (AU)
for their significant contribution to the editorial and technical review of the whole TB.
Note on terminology:
In this TB, the IEC terminology is generally used, however, the end-to-end measurement, as defined by
IEC [2], is also called open-circuit measurement, and the end-to-end short circuit measurement is also
called short-circuit measurement.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
2.
General understanding of the frequency response
Leader: Mark Lachman (US)
2.1 Introduction
A transformer frequency response is a representation of the interactions between the inductance,
capacitance and resistance elements comprised in the circuit model of the windings. This chapter aims
to provide the reader with a basic understanding of the main features that can be expected in a frequency
response measurement. It presents a description of the physics behind frequency response and a
description of typical responses depending on different winding designs and configurations. In the
following Chapter 3, a more in-depth discussion through transformer circuit modelling is presented.
2.2 Traveling through frequency ranges along the frequency response
2.2.1 General
The analysis of transformer diagnostic data is often aided by having a clear image of the physics behind
the measurement. The tests relying on a single frequency typically allow for an easy grasp of processes
unfolding during the measurement, e.g., the impedance measurement representing the magnetic
coupling between two windings. For FRA, the measurement is performed over a wide range of
frequencies and an insight into the electromagnetic process indicates that it would involve multiple
components of the transformer. Each of these components may manifest itself and become the
dominating factor of various frequency ranges and this process is a challenge to master. Having a visual
insight into the electromagnetic process behind the key resonant points can be instructive for the
frequency response interpretation [5].
In a power transformer, we are dealing with an extremely complicated equivalent network of distributed,
non-linear and frequency-dependent resistance (R), inductance (L) and capacitance (C) elements. As
the frequency of the injected signal changes, various combinations of C and L exchange energy with
each other, resulting in the local extrema of the impedance. Hence, numerous resonant frequencies,
each associated with a different pair of C and L are observed. In frequency ranges between the resonant
points, the circuit will be predominantly inductive or predominantly capacitive. However, even though
the network is seen as inductive, one should recognize that C is always present and makes a
contribution, and, conversely, when the network is capacitive, L plays a role as well.
To interpret the frequency response, there is a need to identify the frequency ranges associated with
the main transformer components. The beginning and the end of each frequency range are influenced
by the transformer characteristics and design, i.e., for different transformers the same component can
manifest itself with different boundary frequencies for these ranges. Therefore, over-relying on fixed
frequency ranges, which were firstly proposed in [6], can lead to erroneous conclusions, as discussed
in Chapters 6 to 8. Despite the complexity of a transformer RLC network, attempts have been made to
suggest the frequency ranges that work as a good rule of thumb for many units [7] as well as develop
algorithms for the automated frequency range identification [8]. Furthermore, experience with identifying
the first natural frequency that manifests in the beginning of the frequency range associated with the
winding is reported in [9, 10].
2.2.2 Core
As the typical amplitude trace is examined in the direction of frequency increase, the first point of interest
(point A in Figure 2.1) is the maximum impedance of the winding-under-test, hence the local minimum
dB value found below a few kHz. To identify components inside the transformer that make-up L and C
in this resonance, we recall that in the low-frequency range under the open-circuit condition, the
magnetic energy is stored in the core. Therefore, here, L represents the ability of the current to create
magnetic flux in the core (magnetization inductance).
This ability can be influenced by the core magnetic state, as well as by any closed circuit acting as an
inductive load (e.g., a shorted turn). The electrical energy in this frequency range, is, for the most part,
confined to the segments of insulation with either a large C or between electrical nodes of the circuit
designed for a higher electrostatic potential difference. These include various capacitive paths to earth,
making up a comprehensive capacitive network coupled to the winding-under-test.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 2.1 Main features of a typical frequency response
2.2.3 Winding interaction
As we travel past the first resonance (400 Hz in Figure 2.1), the next point of interest is the minimum
impedance (point B in Figure 2.1). Once again, we are interested in the transformer components that
make up the C and L that are involved in resonance at this frequency. With respect to L, an increase in
frequency intensifies the impact of the eddy currents in the core laminations. These currents create a
demagnetizing force, which opposes the flux creating them in the first place. The higher the frequency,
the stronger is the effect of pushing the flux out of the core. Hence, part of the flux enters the space
between and within the windings. For this resonance, magnetic energy is stored both in the core and in
the leakage channel, L being a combination of magnetizing and leakage inductance. At the same time,
the storage of electrical energy shifts into C of the inter-winding space. This frequency range exposes
changes in the interaction between the windings.
2.2.4 Winding structure
As we travel further along the frequency response trace, many local extrema points are encountered,
with different types of windings exhibiting a variety of characteristic patterns. These patterns to a large
degree are governed by the relationship between the winding series and shunt capacitances. A
resonance point identified as point C (Figure 2.1) corresponds to one of the winding’s natural
frequencies. It is presumably associated with an oscillation where the electrical energy stored in the
winding turn-to-turn (or layer-to-layer) insulation is exchanged with the magnetic energy stored in the
field created by the leakage flux linked to the corresponding turns.
The natural frequencies (like point C in Figure 2.1) can be identified using the approach described in
[10, 11]. It is based on overlaying the primary winding frequency response data, expressed as real part
of winding admittance, measured with the open and short-circuited secondary winding (presented in
Section 3.5).
2.2.5 Bushings, leads and earth path
The end of the frequency range, influenced by the bulk of the winding, depends on the transformer
design. However, for power transformers, it is generally accepted that in the higher frequency range the
data is influenced by the bushings, leads within the transformer as well as by the earth circuit
encountered by the measurement signal (range D in Figure 2.1). In the case of poor earthing practice,
this can also appear in the other frequency ranges, as discussed in Chapter 4.
2.2.6 Leakage channel
Short-circuiting the secondary winding offers a convenient way to confine the low-frequency range to
being influenced by the space between and within the windings, a space sometimes referred to as the
leakage channel (Figure 2.2). Under short-circuit conditions, the magnetic flux, coupled to the excited
winding, follows a path that includes the segments of the core in series with the leakage channel. Since
14
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
the reluctance of the latter is much higher than that of the core, the changes in the leakage channel
dominate the flux path and, thus, the impedance in the low frequency range. With the geometry of the
leakage channel being almost identical in all three phases, the comparison between phases offers a
convenient way to analyse the short-circuit measurement data.
Figure 2.2 Frequency response range influenced by leakage channel
2.3 Description of typical frequency responses
2.3.1 Differences between low-voltage and high-voltage winding responses
The frequency responses of low-voltage (LV) and high-voltage (HV) transformer windings have some
basic differences [12]:


LV winding responses typically have a higher amplitude (Figure 2.3) than HV winding
responses, especially at low and medium frequencies. This is caused by a smaller number of
turns, lower impedance and electrical length (circumference of the winding multiplied by the
number of turns).
In the example shown, HV windings have a pronounced increasing trend in the 100–1000 kHz
range due to its outer location in the winding assembly, creating a higher series to shunt
capacitance ratio.
Figure 2.3 Typical open-circuit frequency response measurements of HV and LV windings
The data looks different in the case of an autotransformer, where responses of common and series
windings overlap due to the comparable impedances, number of turns, electrical length, etc. (Figure
2.4).
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 2.4 Typical responses of common and series windings
2.3.2 Typical responses of various types of coils
The presence of the interleaved disc winding can be often detected from the frequency response data.
In the frequency range dominated by the winding itself, the presence of the interleaved disc winding
(due to a high series capacitance) is manifested by a significantly smoother trace (i.e., one with fewer
resonance points) than when a plain disc (or helical) winding is present (Figure 2.5).
Plain disc
Interleaved
disc
(series
winding)
Figure 2.5 Typical frequency responses of interleaved vs plain disc windings
2.3.3 Frequency response of typical winding configurations
Transformer winding configuration and design play a very important role in forming the frequency
response. Examples of frequency response for some of the frequently encountered winding
configurations are presented in Figure 2.6 to Figure 2.9. The data are from three phase units with threelimb cores.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
26 MVA, 161/14 kV, Dyn+d
Figure 2.6 Example of frequency responses of a delta-star transformer with buried tertiary
48 MVA, 236/14 kV, YNd+d
Figure 2.7 Example of frequency responses of a star-delta transformer with buried tertiary
36 MVA, 120/25 kV, YNyn+d
Figure 2.8 Example of frequency responses of a star-star transformer with buried tertiary
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
45 MVA, 138/46/13 kV
YNa0d
(a)
45 MVA, 138/46/13 kV
YNa0d
(b)
45 MVA, 138/46/13 kV
YNa0d
(c)
Figure 2.9 Examples of frequency response of autotransformer with accessible tertiary: (a) series
winding, (b) common winding and (c) delta
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
3. Fundamental understanding of the frequency
response through transformer circuit modelling
Leaders: Zhongdong Wang (GB) and Vasily Larin (RU)
3.1 Introduction
A transformer can be modelled as its equivalent circuit network, consisting of a combination of
inductances (L) and capacitances (C) which are geometry dependent electrical parameters. Windings
for the same phase are magnetically coupled, and represented by self and mutual inductances. In
addition to the winding series capacitances and shunt capacitances to earth, a pair of adjacent windings
is also electrostatically coupled, resulting in inter-winding capacitances. Resistive losses can be ignored
as they do not significantly alter the key features of frequency response plots but only produce a
damping effect on the resonances [13]. Consequently, there is a relationship between winding geometry,
equivalent electrical components of the circuit network and the measured frequency responses at the
winding terminals.
The underlying diagnostic principle of FRA is that a physical displacement or deformation of windings
results in changes in the equivalent electrical components of the corresponding winding part. These
changes are reflected in changes or even additions to the observed resonant frequencies in frequency
response curves.
This chapter illustrates with simplified circuit models the electromagnetic circuit theory behind the
frequency response measurement and provides a theoretical basis for understanding of fundamentals
and interpretation. It starts with a review of a single air-core winding with uniform winding structure, and
the complexity is gradually built up by taking single-phase transformer core and winding interactions into
consideration. All the windings modelled and analysed in this chapter are of uniform structure, which
means insulation-graded disc type windings (e.g., an intershielded + plain disc type winding) and
windings consisting of two or more coils (including tap windings) are not included. In addition, winding
terminal conditions, such as the measurement set-up and the bushing effect, are not considered in
modelling.
3.2 Taking a single winding as example
3.2.1 Typical frequency responses of single air core winding
The simplest way to model a transformer winding for high frequencies is to use an n-stage lumpedelement LC ladder network model as shown in Figure 3.1. The winding is assumed to have a uniform
structure, i.e., same conductor, same number of turns per disc, and use of non-graded insulation for all
the discs of the winding. The effect of mutual inductive coupling among the elements is taken into
account by L.
Space coefficient α is a common term used for winding design, as winding structures were evolved due
to the need to pass lightning impulse tests and control the non-linear impulse voltage distribution along
the winding. Space coefficient is defined as 𝛼 = √𝐶𝑔 ⁄𝐶𝑠 and its value is mainly controlled by winding
series capacitance Cs. There is very little room to change the value of Cg, which is dominated by the
main insulation between the winding to earth.
Dependent on the winding structure, whether a single helical winding, a plain, intershielded or
interleaved disc winding, a double helical winding, or even a multi–layer winding, any winding can be
generally categorised into one of two types based on its space coefficient α: high winding series
capacitance (e.g., interleaved disc winding) or low winding series capacitance (e.g., plain disc winding
or single helical winding).
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
L: Total inductance,
Cs: Total winding series
capacitance,
Cg: Total winding shunt
capacitance to ground,
n:
Number of stage.
Figure 3.1: n-stage lumped-element LC ladder network of single air-core winding
Figure 3.2 shows two frequency responses obtained via circuit modelling of different winding types. The
LV winding is for a single helical winding of 13 kV with 74 turns of conductor, and the HV winding is for
an interleaved disc type winding of 144 kV with 140 discs. The LV winding has an α of around 30 and
the HV winding has an α of less than 2. The LV winding circuit model comprises L= 1.581 mH, Cs=
5.67 pF and Cg= 5640 pF; and the HV winding has L= 0.638 H, Cs= 401 pF and Cg= 482 pF. Both
windings are uniformly structured.
Figure 3.2 Typical FRA responses of single air core windings
For the winding with large α, its frequency response is featured with multiple resonances and ‘U-shaped’
pseudo anti-resonances, in between adjacent resonances. For the winding with small α, its frequency
response is featured with one critical anti-resonance and afterwards a smoother rising trend of amplitude
with few or no resonances. Based on the mathematic descriptions of n-stage lumped-element ladder
network model presented in [13], the following equations have been derived to calculate the pseudo
anti-resonant, resonant and critical anti-resonant frequencies [14]. These equations are only valid for
the simplified model of a single air core winding discussed here and therefore are not applicable for a
real transformer involving far more complex electromagnetic interactions. Interested readers may use
the given parameters of the windings above and the equations for an exercise, and compare their own
calculation results with the marked frequencies in Figure 3.2.
𝑓0𝑘 =
1
1
2π√LCs
4α2
√1+
((2k−1)π)2
,
k = 1, 2, 3, … . (n − 1)
Equation 3.1
A pseudo anti-resonance is defined as the local minimum between the adjacent resonances, where the
frequency response amplitude would be concave without the sharp drop into the large negative dB value
as for the normal anti-resonance, hence the frequency response amplitude spectrum has the so-called
U-shape feature around a pseudo anti-resonance.
The frequencies of the resonances, fk, can be calculated as
𝑓𝑘 =
1
2π√𝐿𝐶𝑠
1
α2
(kπ)2
,
𝑘 = 1, 2, 3, … . (𝑛 − 1)
√1+
Equation 3.2
A resonance would display itself as the local maximum in amplitude, and sometimes resonant
frequencies are also called natural frequencies of winding.
The ‘critical’ anti-resonant frequency of the n-stage lumped-element ladder network, fc, can be
calculated as
20
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
𝑓𝑐 =
1
Equation 3.3
2π√𝐿𝐶𝑠
where the frequency response amplitude would drop sharply to a large negative dB value. Hence this
anti-resonant frequency is regarded as ‘normal’. The reason it is also called ‘critical’ anti-resonant
frequency is because it is the highest frequency at which the condition of parallel resonance inside the
winding is still valid, and afterwards the impedance of each parallel branch L/n and nCs will behave
capacitively and the amplitude takes a rising trend with frequency in the rate of
20𝑙𝑜𝑔10 |A| = 20𝑙𝑜𝑔10 (
2𝜋 ∗ 50√𝐶𝑔 𝐶𝑠
𝑒 𝛼 −𝑒 −𝛼
𝑓)
Equation 3.4
2
It can be noticed that α has the dominating effect on the shape of a frequency response. When α is
small enough, there are no multiple resonances expected as the terms
𝛼2
(𝑘𝜋)2
4α2
((2k−1)π)2
in Equation 3.1 and
in Equation 3.2 are too small to distinguish the resonant frequencies, f0k and fk from fc. These
equations explain well why windings with low series capacitance, such as plain disc windings and single
helical windings always have multiple resonances, whereas windings with high series capacitance, such
as interleaved disc type windings, always have less or no resonance.
3.2.2 Typical frequency responses of single windings with core
The next step is to add the core effect by assuming a single winding is now placed onto the transformer
core. Hence the first anti-resonance, one of the low frequency features we normally see on the frequency
response measurements made on a transformer, would appear for the single windings with small α. The
exemplar LV and HV windings described above are now modelled with the core effect added, and their
frequency responses are shown in Figure 3.3. The equivalent magnetizing inductance is Lcore= 0.719 H
for the LV winding, and Lcore= 90 H for the HV winding at power frequency. The equivalent magnetizing
inductance is set as constant in modelling rather than a frequency dependent value, as frequency
dependent magnetizing inductance has shown negligible effect on the corresponding frequency
response curves in [14].
(a) FRA response of HV winding with core
(b) FRA response of LV winding with core
Figure 3.3 Typical FRA responses of single windings with core
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
It can be seen that due to the large value of magnetizing inductance, the amplitude of the HV winding
frequency response at low frequencies has dropped by nearly 40 dB when comparing to that of the aircore configuration. The first low frequency anti-resonance has clearly appeared at the frequency of
930 Hz. This first anti-resonance caused by the core as well as the bulk winding capacitances can be
approximated by Equation 3.5, when considering the magnetic coupling factor β among parts of the
winding through the main flux as well as air flux linkage, in addition to the space coefficient α [14].
2
2
4
𝑎 𝛽 √ 2 𝑎 𝛽 𝑎 2
1−
− 𝛽 −
+ 𝛽
1
1
8
4
64
2
𝑓𝑎𝑟𝑒𝑠1
= ( )2 (
)
2𝜋
𝐶 𝐿
(1−𝛽)
Equation 3.5
𝑠 𝑇
where LT = Lcore+L, and the magnetic coupling factor is defined as 𝛽 = 𝑀12 ⁄𝐿11 , where M12 is the mutualinductance and L11 is the self-inductance of the two units winding model. Interested readers may use
Equation 3.5 with the given parameters of the HV winding, and its magnetic coupling factor β=0.9921
for an exercise and compare their own calculation result with the marked frequencies in Figure 3.3 (a).
As for the LV winding, due to the smaller number of turns linked with the main flux in the core, its
magnetizing inductance is too small and α dominates the frequency response of the LV winding. In
Figure 3.3 (b) the ‘U’ shape in the curve is visible. Comparing with the frequency response of the single
air-core LV winding, it seems that the first pseudo anti-resonant frequency has been shifted to the lower
frequency due to the introduction of the magnetizing inductance, and there is no change for the other
resonances.
It should be also noted that although the transformer core is introduced, the high frequency features on
both the LV and HV winding frequency responses are hardly changed at all.
The impact of transformer core structure on frequency response low frequency characteristics can be
modelled based on the principle of duality [15]. The asymmetry of the three-limb/five-limb core structure
produces dissimilar low frequency characteristics for three phase frequency responses, i.e., there are
often double anti-resonances for the two outer phases frequency responses and the single antiresonance for the middle phase. This can be explained by the different magnetizing inductances seen
by three phase windings, as the main flux path for the middle phase of a three-limb core is different from
those of the other two phases.
3.2.3 Modelling a single winding with a full matrix to deal with mutual inductive
coupling
A transformer is designed to be efficient by arranging windings so they are magnetically closely-coupled.
Hence mutual inductances among windings and among parts of the same winding are substantive. The
windings of the same phase share the same main flux but would experience different leakage fluxes. A
fixed value of transformer impedance can be achieved by carefully designing the winding dimensions
as well as the radial insulation distance between them. Such a magnetic coupling can only be accurately
modelled when we represent it using a full inductive matrix. Note that the number of n*n inductive
matrices can be varied depending on the frequency range of interest. As a rule of thumb, the higher the
frequency, the larger the n.
The next step is to represent mutual inductive coupling effect by modelling the full inductance matrix of
the air-core winding before adding the magnetizing inductance coupling. Figure 3.4 shows the effect of
mutual inductances for the frequency responses of the LV and HV windings.
(a) FRA response of HV winding with mutual fully modelled
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
(b) FRA response of LV winding with mutual fully modelled
Figure 3.4 Typical FRA responses of single windings with core (effect of mutual inductance – shift
resonant frequencies up)
It can be seen as marked in Figure 3.4 (a) and (b) that by taking mutual inductances into consideration,
the resonant, anti-resonant and pseudo anti-resonant frequencies of the FRA are all moved up to higher
frequencies in comparison to the ones produced by the model without mutual inductances which can be
calculated using Equation 3.1 and Equation 3.2. This shift can be quantified as by the ratio of √1 + 𝛽𝐿 ,
where 𝛽𝐿 = 𝑀𝐿12 ⁄𝐿11 , representing the leakage flux coupling factor of the winding. L11 is the selfinductance and ML12 is the mutual-inductance of the two unit air-core winding model. The leakage flux
coupling factor for the LV winding is calculated as βL=0.31 and for the HV winding βL= 0.57, which are
known as dependent to the winding structure.
3.3 Taking a single-phase two-winding transformer as example
3.3.1 General
The next step is to demonstrate the influence of the winding-not-under-test on the frequency response
measurements of a transformer, using a single-phase two-winding transformer as an example [16]. A
single-phase two-winding transformer consists of LV and HV windings which are assembled onto a core
with insulation between the two windings. Its half axis-symmetric cross-section is shown in Figure
3.5 (a), and the equivalent circuit model in Figure 3.5 (b). Here the modelling includes the mutual
magnetic coupling between the two windings, as well as the inter-winding capacitances between the
elements of the windings.
LV
HV
LV
CgL/N
Core
HV
CHL/N
CgH/N
Mcore+Mair
Tank
Lcore+Lair
NCsL
NCsH
CHL/N
CgL/N
(a) Winding arrangement
CgH/N
(b) N-element equivalent circuit
Figure 3.5 Circuit model for single-phase two winding transformer
The modelled frequency responses of both windings are presented in Figure 3.6. In the winding structure
dominated high frequency range, the LV winding frequency response trace shows multiple resonances
at the frequency of 200 kHz and higher, which accords with the frequency response features of single
winding with lower Cs. The HV winding frequency response trace shows a rising capacitive trend with
almost no resonance in the high frequency range because of its higher Cs.
23
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
(a) Amplitude spectra for HV and LV windings in the single-phase transformer
(b) Phase spectra for HV and LV windings in the single-phase transformer
Figure 3.6 Typical FRA responses of windings in a transformer
3.3.2 Impact of HV winding onto LV winding FRA plot in the low frequency range
In the core dominant region which conventionally is regarded as below a few kHz, our general
knowledge is that typical first anti-resonances exist for both winding frequency response traces. This
indicates that the conventional space coefficient α of a single winding is no longer valid to explain the
low frequency characteristic, as shown in Figure 3.7, where is shown the impact of the open HV winding
on the LV winding characteristic.
Figure 3.7 Comparison of LV winding FRA plots before and after HV winding is added
24
TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Indeed, LV winding frequency response measurement in the low frequency range is not only determined
by the magnetizing inductance and its own winding capacitances, but also influenced by the inductive
and capacitive couplings from the HV winding, which is not under test. This can be explained by a twostage equivalent circuit model of the single-phase two-winding transformer as given in Figure 3.8.
E
E
V1
V3
0.5CgL
0.5CHL
0.5CgH
CsL
Core
V1
CsH
LV
0.5CgL
Tank
Core
HV
0.5CgL
V2
0.5CHL
V3
0.5CHL
0.5CgH
CsL
CsH
LV
V4 0.5CgH
0.5CgL
Tank
HV
V2
0.5CHL
V4 0.5CgH
50Ω
50Ω
(a) LV end-to-end FRA measurement
(b) HV end-to-end FRA measurement
Figure 3.8 Two-stage equivalent network of single-phase two-winding transformer
In Figure 3.8, due to the large turns ratio between the HV and LV windings (similar to a generator
transformer), the voltage difference induced at the non-tested winding is determined by the turns ratio
and the potentials of the floating terminals, which can be estimated based on the equivalent capacitive
network. Taking the circuit in Figure 3.8 (a) as example, where V1 is defined as 1 per unit, the voltage
difference V3-V4 is N per units (N being the turns ratio between the HV and LV windings). Note, Figure
3.8 is valid only for the analysis for the low frequency range, as it is a very simple two-stage equivalent
circuit network model. Nevertheless, for the winding-under-test in a transformer, the resulting ‘new’
equivalent shunt capacitance to earth (Ceg) and the equivalent series capacitance (Ces) can be estimated
by Equation 3.6 and Equation 3.7.
𝐶𝑒𝑔 =
𝐶𝐻𝐿 × 𝐶𝑔𝐿
+ 𝐶𝑔𝐻 , 𝐶𝑒𝑠 = 𝐶𝑠𝐻
𝐶𝐻𝐿 + 𝐶𝑔𝐿
𝐶𝑒𝑠 = 𝐶𝑠𝐿 + 𝐶𝑠𝐻 𝑁 2 + (
𝑁2 −4𝑁+4
8
)𝐶𝐻𝐿 ,
𝑓𝑜𝑟 𝐻𝑉 𝑤𝑖𝑛𝑑𝑖𝑛𝑔
𝐶𝑒𝑔 = 𝐶𝑔𝐿 + 𝐶𝑔𝐻
𝑓𝑜𝑟 𝐿𝑉 𝑤𝑖𝑛𝑑𝑖𝑛𝑔
Equation 3.6
Equation 3.7
(N is the turns ratio between the HV/LV windings)
Equation 3.7 helps explain why we can see a large equivalent winding series capacitance for a single
helical winding when it is used for the LV winding in a transformer. It also highlights the magnifying factor
of ‘turns ratio squared’ on the inter-winding capacitance as well as the series capacitance of the windingnot-under-test. With this insight, it is understandable that the LV winding in this transformer has a typical
first anti-resonance as shown in Figure 3.7.
Using Equation 3.6, the equivalent winding capacitances can be estimated for the HV winding, and the
winding series capacitance remains the same before and after it is put near to the LV winding. The shunt
capacitance to earth increases, which slightly increases its space coefficient α; this may help to
understand why changes demonstrated in Figure 3.9 are relatively small.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 3.9 Comparison of HV winding FRA plots before and after LV winding is added
3.3.3 Winding interactions in the medium frequency range
In Figure 3.6 (a), it is clear that the characteristics of HV winding structure, in the frequency range around
10 kHz, have been transferred through winding interaction, i.e., mutual inductive and capacitive
couplings between the two windings, to the LV winding frequency response plot. On the other hand, the
LV winding has fewer turns, low air-core inductances and low winding capacitances, and therefore has
less consequence on the HV winding frequency response plot. However, the impact of the LV’s first
pseudo anti-resonance (just after the 100 kHz) has passed through to the HV frequency response plot
and can be seen as the slight ‘kink’ on the smooth capacitive rising amplitude with frequency.
3.4 Interactions among phases and three-phase 3-winding transformers
Three-phase power transformers may have star and delta connected windings, and sometimes also
have tertiary windings. Tap windings are also available in some transformers for regulating the voltages
and this often means two or more winding coils within a winding, which have different winding structures.
Practical situations like this may present themselves in the frequency response results with much more
complex features.
Although transformer electromagnetic modelling techniques are advanced, it is still challenging to model
the many types of interactions that are present in real three-phase transformers, especially when tertiary
windings are considered and account is taken of the various arrangements and tappings, which leads
to a myriad of cases. Hence there will not be a review of other combinations in this chapter.
3.5 Understanding winding natural frequencies
The interpretation of the FRA measurements with respect to winding type and its design peculiarities
can be made using analysis of winding natural frequencies [10, 17, 18]. Winding natural frequencies are
the fundamental characteristics of the winding and are governed by the electrical length of the winding,
parameters of the winding longitudinal insulation (e.g., dielectric permittivity of turn-to-turn insulation),
arrangement and electromagnetic coupling of different parts of the winding, and constraints on spatial
voltage distribution inside the winding (connection between winding parts, connection of winding
terminals to earth, etc.). The changes in these frequencies are influenced by major faults in the winding
and have different patterns depending on the type and location of the fault.
In general, the natural frequencies of a stand-alone air-core winding can be easily obtained from its
frequency response measurement, since the measured resonant frequencies correspond to the natural
frequencies of the winding. However, if there are several windings on the same magnetic core, as it was
illustrated above using RLC-circuit modelling, the frequency response of a particular winding obtained
by end-to-end measurements has additional resonant frequencies associated with the interaction
between coupled windings.
The natural frequencies can be identified from frequency responses using the approach described in
[17-19]. It is based on overlaying the primary winding frequency response data, expressed as active
admittance, measured with the open and short-circuited secondary winding.
The approach takes advantage of spatial distribution of winding current at the first natural frequencies,
which has nodes where the current changes direction. As a result, the currents in winding sections
connected to these nodes have opposite directions. Then, the electromotive forces, induced in the
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
secondary winding, are mutually compensated, and the magnetic flux generated by the primary winding
penetrates the magnetic core, without encountering reaction from the secondary winding. Thus, at
natural frequencies, the short circuit of the secondary winding has practically no effect on the admittance
of the primary winding and has no influence on the values of the natural frequencies. However, at
frequencies much lower than the first natural frequency of the primary winding, the condition of the
secondary winding, i.e., the presence of a short circuit, has an influence on flux penetration into the core
and on its return path. This, in turn, leads to significant changes in the admittance of the primary winding
and the resonant frequencies related to the interaction between the windings (Figure 3.10 (a)).
(a)
(b)
Figure 3.10 Frequency response (a) and active admittance of HV winding (b)
It is convenient to identify natural frequencies through winding admittance namely by using a local
maxima of active admittance (Figure 3.10 (b)). When source frequency is equal to one of winding natural
frequency, the internal winding resonance occurs and oscillatory currents and active energy
consumption from the power source take place. Thus, the active admittance local maxima can be used
as indicator that source frequency is close to winding natural frequency.
The algorithm for determining the winding natural frequencies [17, 18] contains the following main steps:
1. Using Equation 3.8, approximate evaluation of winding admittance based on measured frequency
responses corresponding to open-circuited and short-circuited secondary winding (A and 𝜑 are the
amplitude and phase values from the frequency response measurement):
−1
1
𝑌12 ≈ [50 (
− 1)]
𝐴∠𝜑
Equation 3.8
2. Approximate evaluation of winding active admittance (conductance) as real part of admittance Y12
or using Equation 3.9:
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
𝐺12 ≈
𝐴 cos 𝜑 − 𝐴2
50(𝐴2 − 2𝐴 cos 𝜑 + 1)
Equation 3.9
3. The natural frequencies are located at the local maxima of the overlaid open and short circuit
frequency responses of the active admittance.
In practice the use of this method makes it possible to identify the natural frequencies of helical- and
disc-type windings, including windings with large series capacitance, for which the resonance peaks at
natural frequencies may not be evidently seen in the original frequency response.
It should be noted that the presence of shunt capacitances in parallel with the 50-Ω input impedance
leads to the error in active admittance evaluated from the frequency response, which increases with
frequency. This is manifested, for example, in an abnormally sharp increase of active admittance
obtained from Equation 3.9 with an increase in frequency (Figure 3.10 (b)). However, as practice shows,
this error begins to strongly affect frequencies closer to 1 MHz and in general does not interfere with
determining the natural frequencies of the windings in the range up to several hundred kHz.
FRA interpretation based on analysis of winding natural frequencies can be used to assess different
types of winding electrical and mechanical faults [12, 17-19].
Application of this approach can be illustrated by detection of unearthed magnetic core and electrostatic
shields. Figure 3.11 shows an example of frequency responses of a two-winding step-up transformer
HV winding having unearthed core and shields [18]. When core was unearthed (red curve), first antiresonance frequency in the range 1–2 kHz became bit lower, new resonance appeared at 400 kHz, and
frequency response in medium and high frequency range shifted up by approximately 7 dB. Meanwhile
the natural frequencies of the HV windings, which are resonance peaks between 8 kHz and 100 kHz
(Figure 3.11, b), remained almost unchanged, indicating that the ‘fault’ is not inside the HV winding.
(a)
(b)
Figure 3.11 Frequency response (a) and active admittance of HV winding (b) with earthed (1) and
unearthed (2, 3) core and shields (1 and 2 – HVOC; 3 – HVSC)
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
4.
Factors influencing the measurement
Leader: Mark Lachman (US)
4.1 Introduction
For frequency response measurements, the likelihood of a transformer having a problem is significantly
lower than the likelihood of the test personnel making a mistake. This chapter reviews the factors
potentially influencing the data. The impact on the frequency response shown in the examples is specific
to units tested and will vary from case to case. However, with these examples, the objective is to raise
awareness about the role these factors may play so that in searching for causes of abnormal data, their
potential contribution is not overlooked. Once these factors are eliminated as suspects or identified as
not relevant, attention can be turned to the transformer.
4.2 Measurement setup
4.2.1 Poor earthing
With transformer earth serving as reference for the signal, the earthing path should not introduce any
tangible impedance into the circuit. Figure 4.1 shows the influence of the earth circuit impedance
accomplished through a progressive increase of the measurement leads shield length. For the unit
tested, the influence becomes apparent above 300 kHz [20].
50 µΩ
100 µΩ
3.5 mΩ
7 mΩ
14 mΩ
As the earthing impedance increases
50 µΩ
100 µΩ
3.5 mΩ
7 mΩ
14 mΩ
Figure 4.1 Effect of increased earth circuit impedance
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4.2.2 External busbar connected to bushings
The open-circuit measurement was performed with cables connected to transformer bushings
connected to external busbars (Figure 4.2). The difference between the traces is clearly visible in the
high-frequency range [20]. The change in the response, obtained when still connected to an external
busbar, results from the mismatch between the characteristic impedance of the measurement cables
and the busbar. While it caused the response to go positive at high frequencies, the rest of the response
remained intact.
Connection to a busbar connected to bushings
60 MVA, 230/34.5 kV, Dyn
Figure 4.2 External busbar connected to bushings
4.2.3 Poor connections of short-circuiting cables
Figure 4.3 shows a short-circuit measurement performed from the high-voltage side. The spread
between the per-phase traces in the low-frequency range is the result of poor connection of shortcircuiting cables on the low-voltage side.
500 MVA, 230/18 kV, YNd
Figure 4.3 Poor connections of short-circuit cables
4.2.4 Connection between measurement lead screen and bushing flange
The connection between the screen of the measuring lead and the bushing flange should be kept as
short and as straight as possible. The shape of that measurement cable segment, having an inductance
of its own, does influence the high-frequency portion of the frequency response trace (Figure 4.4). Hence
the importance of the measurement setup consistency.
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Magnitude (dB)
0
Alu Braids
Cooper wire
-50
-100
2
10
4
6
10
Frequency (Hz)
10
Figure 4.4 Effect of earthing configuration of screen of measuring lead
4.2.5 Earthing neutral bushing in not-under-test winding
Figure 4.5 shows an open-circuit measurement performed on the high-voltage side. The comparison is
made between traces obtained in the presence of earthed and unearthed low-voltage side neutral
bushing.
While in Figure 4.5 the neutral point earths/unearths all three phases of the star-connected winding, in
some units, each phase has its own neutral bushing. During the measurement, these bushings can be
joined together and floating, joined together and earthed, or separated and floating. Each connection
will produce its own unique frequency response signature.
11.2 MVA, 69/13.2 kV, YNyn
LV neutral
earthed
LV neutral
unearthed
Figure 4.5 Effect of earthing neutral in not-under-test winding
4.2.6 Direction of end-to-end measurement
When the transformer design requires a non-uniform insulation distribution along the winding, the
measurement results obtained by applying the voltage to the different ends of the winding may differ.
Figure 4.6 shows the open-circuit measurement performed on a single-phase unit where each end of
the winding was designed for a different insulation level. As a result, the neutral end of the winding had
a smaller bushing and a different capacitance to earth than the one on the line end. This difference
resulted in a deviation between two traces in the medium and high frequency ranges. Different traces
may also be obtained in a three-phase star-connected winding. The signal travelling from the neutral
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
bushing end of the winding under test will be facing a different RLC network than one travelling from the
line bushing. It is therefore important to use the same direction of measurement, as it is specified in the
IEC standard [2].
Figure 4.6 Influence of direction of measurement
4.2.7 Incidence of the selected measurement voltage
The applied voltage may affect the results at low frequencies in configurations where the core is
involved, i.e., in measurement of a winding while other windings are floating (Figure 4.7).
Figure 4.7 Measurements of same 11 kV winding using 3 different voltage levels 3 V (green), 5 V (grey)
and 10 V (red) peak to peak [21]
4.3 Transformer state
4.3.1 Core earth
Figure 4.8 shows an open-circuit measurement performed on the high-voltage side. The comparison is
made between traces obtained in the presence of an earthed and unearthed main core. The unearthed
core resulted (in this case) in changing the low-voltage winding-to-earth capacitance by some 1000 pF.
Figure 4.9 shows the evolution in the frequency response as the core goes from unearthed to earthedthrough-resistors to being solidly earthed.
Sometimes, the unearthed main core (or magnetic shield) may result in the shifting of the HV winding
response in the high-frequency range (e.g., above 100 kHz). This can be explained by applying the
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same rationale as that used to explain the difference between LV and HV frequency responses. When
the main core is unearthed, the coupling between the LV winding and earth is reduced, increasing
potential of the floating LV winding. This additionally increases the capacitive current flowing from LV
winding through the measuring impedance. More discussion can be found in [18].
11.2 MVA, 69/13.2 kV, YNyn
Figure 4.8 Influence of core earth (1/2)
Unearthed
core
Core earthed via
incorrectly connected
resistors
Solidly earthed core
Core earthed
via resistors
Figure 4.9 Influence of core earth (2/2)
4.3.2 Insulating fluid
Figure 4.10 and Figure 4.11 show an open-circuit measurement performed on the low- and high-voltage
sides of a unit with and without oil. The absence of oil decreases the dielectric constant and decreases
capacitance, thus shifting all resonant points of the whole frequency response trace to higher
frequencies (𝑓𝑟𝑒𝑠 = 1⁄
).
√𝐿𝐶
Similarly, an insulating liquid with a dielectric constant higher than that of oil (e.g., natural ester) will shift
the response towards lower frequencies (Figure 4.12 and Figure 4.13).
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15 MVA, 117.87/13.09 kV, Dyn
Figure 4.10 Influence of insulating fluid, LV side (1/4)
15 MVA, 117.87/13.09 kV, Dyn
Figure 4.11 Influence of insulating fluid, HV side (2/4)
Factory with natural ester
Figure 4.12 Influence of insulating fluid (3/4)
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Factory with natural ester
Figure 4.13 Influence of insulating fluid (4/4)
4.3.3 Influence of core magnetic state
The magnetic state of the core changes the magnetizing inductance. These changes become apparent
in the low-frequency range, where they create a characteristic shift of the trace, and, when the core is
sufficiently magnetized, they may also noticeably ‘squash’ the amplitudes of main resonance points
(Figure 4.14).
The magnetic state of the core changes every time the transformer is subjected to (AC or DC) electrical
excitation. When excitation is removed, the core retains some magnetic memory, i.e., a magnetic state
that is unique to the moment of current interruption. However, regardless of whether it was created by
the removal from service, the end of the measurement (e.g., switching surge or winding DC resistance)
or by the process of demagnetization, the magnetic state of the core is a fleeting state. It continues to
change due to the core coupling to a thermal bath—a reservoir of particles in thermal contact with the
core and undergoing Brownian motion. As a result, the small amounts of energy are continuously and
randomly exchanged between the core and the ambient surroundings. It is through these microscopic
interactions that the core will modify its magnetic state while moving towards a thermodynamic
equilibrium. Hence, the state of the core may change in time, even if no external excitation is applied.
Figure 4.15 shows the open-circuit measurement traces obtained on the low-voltage side. The
measurement dates are 9 weeks apart. During that time, the unit, while in transit, saw no electrical
excitation, yet the low-frequency portion of the trace moved.
This phenomenon, sometimes referred to as magnetic viscosity, is demonstrated in more detail in Figure
4.16 [22]. It shows the movement of the trace following the moment when demagnetization ended with
measurements spanning 72 hours.
All in all, the core is constantly moving towards a state of lower energy, i.e., it relaxes, changing its
magnetic state and moving away from the condition created by demagnetization. This affects the lowfrequency range of the trace, leaving the high-frequency range intact. The impact of the magnetic state
is easily recognized as its influence is limited to the low-frequency range. Therefore, two responses,
obtained at different states of the core, will closely overlay in higher frequency ranges where the
influence of the magnetizing inductance disappears (e.g., above 2 kHz).
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
40 MVA, 115/13.2 kV, Dyn
Figure 4.14 Effect of core magnetic state (1/3)
20 MVA, 117/13.2 kV, Dyn
Figure 4.15 Effect of core magnetic state (2/3)
20 MVA, 117.875/13.2 kV, Dyn
Figure 4.16 Effect of core magnetic state (3/3)
4.3.4 Influence of test bushings
There is sometimes a need to compare factory data with field data, especially in the case when regular
bushings are replaced with test bushings. Figure 4.17 shows measurements in both conditions. In the
field test, in addition to having test bushings, the unit is missing oil in the cable boxes. Comparison of
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data shows the expected difference in the unit’s traces in the high-frequency range, which is normally
attributed to the influence of bushings.
Figure 4.18 shows the influence of different bushings on the open-circuit trace on the high-voltage
winding. The difference between the traces is clearly visible in the high-frequency range.
107 MVA, 403/10.3 kV, YNd
Figure 4.17 Influence of test bushings
Figure 4.18 Influence of different bushings
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
4.3.5 Temperature
Large temperature variations can influence the response with the extent depending on the difference in
temperature between the measurements. The impact may be seen in amplitude of the response,
typically noticeable at the resonance points where resistance dominates as well as in the shift of the
trace across the frequency sweep. The latter is attributed to a temperature dependant change in the
dielectric constant.
Usually, factory measurements are performed at room temperature. However, field measurements are
carried out at a variety of temperatures, from ambient to near to service temperature. The extent of
influence depends on the difference in temperature between the measurements. In the example shown
in Figure 4.19, a temperature difference of 40 K influenced the frequency response measurement. The
temperature increase caused a decrease of the resonance frequencies.
-20
-25
Amplitude (dB)
-30
-35
-40
-45
-50
Measurements in repair
shop (red) and in field
(blue), both without oil;
temperature difference
40 K (colder in field)
-55
-60
-65
-70
50000
500000
Frequency (Hz)
4.3.6 Moisture
Figure 4.19 Effect of temperature [23]
T2-A (X1H0X0) on site
T2-A (X1H0X0) in repair shop
Experimental studies showed that transformer moisture diffusion can lead to frequency response
measurement deviations [24-26]. The migration of moisture from paper to oil insulation (when
temperature increases) can make that local resonances in the frequency response change at lower
frequencies, while diffusion of moisture from oil to paper insulation (when temperature decreases) will
shift resonances to higher frequencies [27]. Other techniques, such as dielectric frequency response or
power factor, are often used to assess the level of moisture in transformer insulation and thus can
provide support for interpretation.
4.4 Transformer configuration
4.4.1 Tap position in winding-under-test
Figure 4.20 shows an open-circuit measurement performed on the low-voltage side for the different
inductive-type OLTC positions. The OLTC is located on the low-voltage side. The responses associated
with bridging and non-bridging positions are easily identified in the low-frequency range and, after
zooming in, the different OLTC positions (within each group) are also discerned. The current circulating
through the preventive autotransformer in the bridging positions acts as a load. Resulting increase in
the measured current, i.e., decrease in the measured impedance, shifts the trace amplitude to a higher
value.
Figure 4.21 shows an open-circuit measurement performed on the high-voltage side for the different
DETC positions (each step 2.5%). The DETC is located on the high-voltage side. After zooming in, the
different DETC positions are easily identified.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
12 MVA, 45/13.8 kV, Dyn
Figure 4.20 Influence of tap changer position in winding-under-test (1/2)
12 MVA, 45/13.8 kV, Dyn
Figure 4.21 Influence of tap changer position in winding-under-test (2/2)
4.4.2 Tap changer and series/parallel switch in not-under-test winding
Figure 4.22 to Figure 4.24 show an open-circuit measurement performed on the low-voltage winding for
different positions of switches on the high-voltage winding [28]. The data shows how changing
connections between different points along the not-under-test winding creates characteristic shifts in the
response.
34 MVA, 120-60/13.2, Dyn
Figure 4.22 Effect of various connections of not-under-test winding (1/3)
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
30 MVA, 161-138-115/13.2, Dyn
Figure 4.23 Effect of various connections of not-under-test winding (2/3)
28 MVA, 69/12.47, Dyn
Figure 4.24 Effect of various connections of not-under-test winding (3/3)
4.4.3 OLTC direction prior to entering neutral position
For some transformer designs, the direction from which OLTC returns to the neutral position impacts
the frequency response. Figure 4.25 shows an open-circuit measurement performed on an
autotransformer series winding. The frequency response is obtained twice, both with OLTC in the neutral
position. The difference between the two measurements is in the direction from which the OLTC entered
the neutral position. In these cases, the direction of OLTC return to the neutral should be recorded as
part of the setup. The IEC standard [2] indicates that the direction of movement shall be in the lowering
voltage direction unless otherwise specified.
224 MVA, 230/116/13.2, YNa0d, OLTC: MR, type VR
Direction prior to
entering neutral position
Figure 4.25 Influence of tap changer previous to neutral position
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
4.4.4 Opening and earthing/unearthing corner of buried delta winding
In units with a buried delta winding, one corner is sometimes brought out for external closing of the delta
and earthing. This allows for measurements with the brought-out delta corner open and unearthed,
closed and unearthed or closed and earthed. Figure 4.26 and Figure 4.27 show how different setups
impact the responses of both high- and low-voltage windings [28].
7.5 MVA, 115/4.16 kV, YNyn+d
Figure 4.26 Effect of buried delta winding configuration (1/2)
7.5 MVA, 115/4.16 kV, YNyn+d
Figure 4.27 Effect of buried delta winding configuration (2/2)
4.5 External factors
4.5.1 External interference
In the substation environment, external interference can potentially influence the data. In Figure 4.28,
this impact is observed in the power frequency range. The data can also be impacted by the external
noise when the instrument’s dynamic range is insufficient to handle the existing noise floor level and
therefore doesn’t comply with the IEC standard requirements [2]. The latter is depicted in Figure 4.29
showing measurements using two devices having different dynamic ranges.
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Unit: 300 MVA, 345/138/13 kV, YNad
Figure 4.28 Influence of external interference
H1-H2 (open-circuit and short-circuit)
measurements made with different type
of instrument
Figure 4.29 Frequency response measurements using two devices having different dynamic ranges [29]
4.5.2 Insufficient clearance
An insufficient clearance between transformer terminals and surrounding earthed metallic components
creates additional capacitive coupling that can influence the data as shown in Figure 4.30 [20]. This
example is another manifestation of the importance of consistency in measurement setup.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 4.30 Effect of an insufficient clearance, in this case 20 mm (LV of generator transformer)
4.6 Notes on data analysis
4.6.1 Comparison between phases
Comparison between phases should be performed with extreme care as the expectation of a good
match can lead to erroneous conclusions. Figure 4.31 shows the open-circuit measurement performed
on the HV side of a brand-new unit. The asymmetry of the components surrounding each phase and/or
the difference in routing of the internal leads create(s) a unique benchmark for each phase for a
significant portion of the frequency sweep. In cases like this, analysis should rely on a comparison of
the benchmark of each phase.
15 MVA, 67/34.5 kV, Dyn
Figure 4.31 Comparison between phases
4.6.2 Comparison of sister units
Comparison of sister units should be done while recognizing that not all sisters are created ‘equal’.
Some are ‘identical twins’ and some may have visible differences, with influencing factors including
slight changes in the design/materials, production tolerances and routing of the internal leads. Being
familiar with relevant information helps avoid erroneous conclusions. Figure 4.32 shows the open-circuit
measurement on all phases of series winding in six sister autotransformers. Besides the differences due
to the core magnetic state, which is clearly a normal occurrence, slight deviations are observed in the
higher frequency ranges. Knowing what is normal requires experience with analysing the data for a
particular family of sister units.
Figure 4.33 and Figure 4.34 show the open-circuit measurements performed on the high and low sides
of three sister units. The observed low-frequency deviation is due to the different type of core steel in
one of the units. The impact is like that created by the difference in the core magnetic state, and the true
cause can be identified by demagnetizing the unit, followed by the immediate (to avoid effects of
magnetic viscosity) measurement of the frequency response.
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30 MVA, 115/69/13.2 kV, YNa0d
Figure 4.32 Comparison of sister units (1/3)
10 MVA, 67/12.47 kV, Dyn
Figure 4.33 Comparison of sister units (2/3)
10 MVA, 67/12.47 kV,
Dyn
Figure 4.34 Comparison of sister units (3/3)
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
4.6.3 Additional notes
In addition, the following should be considered:




The preceding discussion shows the importance of documenting details pertaining to
measurement setup.
All factors discussed above should also be considered with respect to the benchmark used for
comparison.
The operational/repair/maintenance history of the unit, e.g., general repairs affecting the inner
geometry, bushing replacement, changing position of internal links, e.g., series versus parallel
winding connection, past through-faults, etc., require new benchmark data.
Ideally, in absence of defects, when all factors are eliminated, the comparison should result in
a near-perfect overlay. However, small ‘natural’ variation between successive measurements
should be expected. Stray impedances at contact points and induced voltages from the
environment will be different. Therefore, the earthing path between measurements should be
reproduced as closely as possible (using photos for instance), even if recreating an identical
setup is often a challenge. Judging the extent of these variations comes with experience in
observing the data.
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5.
Case studies
Leaders: Poorvi Patel (US) and Alaor Scardazzi (BR)
5.1 Introduction
In this chapter, FRA examples are presented and discussed. It is well known that the interpretation of
frequency response measurements is based on a comparison with a reference response (previous
measurement on the same unit, an identical transformer or between the phases of a three-phase
transformer).
As demonstrated in the following case studies, the appearance of new features or major frequency shifts
is cause for concern. Significant changes of the trace amplitude are usually accompanied by significant
phase differences, so phase information is to some extent redundant. In this chapter, only the amplitude
information is used for the visual analysis.
As a good practice, any significant deviation in the comparison with a reference measurement should
be investigated to make sure that the discrepancy is not due to measuring issues as described in
Chapter 4. In this chapter, the measurements are assumed to have been correctly executed using the
best recommended practices.
In the course of this WG, 60 cases were collected and the following 18 cases, covering the main
mechanical and electrical failure modes, are presented in this chapter:


















Case 1 Buckling of the common and tertiary windings
Case 2 Buckling of the inner LV winding
Case 3 Buckling of the common winding
Case 4 Buckling of the inner winding
Case 5 Twisting and loss of clamping
Case 6 Axial movement of the LV winding
Case 7 Axial displacement of the stabilising winding
Case 8 Conductors tilting
Case 9 Successful short-circuit test
Case 10 Successful short-circuit test
Case 11 Successful short-circuit test
Case 12 Slightly buckled winding
Case 13 Lead movement, axial collapse and spiralling
Case 14 Slight displacement of leads showing possible twisting of the winding
Case 15 Broken earthing connection between core elements of a shunt reactor
Case 16 Short-circuit inside winding
Case 17 Opening of a parallel winding
Case 18 Broken bushing connection
5.2 Radial buckling
5.2.1 Case 1 Buckling of the common and tertiary windings
The unit is a three-phase autotransformer rated 750 MVA, 400/275/13 kV.
The analysis is based on a comparison between the three phases because previous measurements
were not available.
Figure 5.1 shows the open-circuit measurements on the series windings. It can be observed that the
resonance frequencies for all three phases line up well in the frequency range above 10 kHz. In the low
frequency range, the two outer phases have two resonances associated with their two different
magnetizing flux paths, and the central phase only shows one resonance due to the core symmetry of
the central phase.
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0
Amplitude (dB)
-10
-20
-30
-40
-50
Phase A
Phase B
Phase C
-60
-70
2.E+1
2.E+2
2.E+3
2.E+4
Frequency (Hz)
2.E+5
2.E+6
Figure 5.1 Case 1 Open-circuit measurements on series windings
0
0
-10
-5
-20
-10
-30
-15
Amplitude (dB)
Amplitude (dB)
Figure 5.2 shows the open-circuit measurements on the common windings. It can be observed that the
phase B response is different from the others in the 20 kHz to 2 MHz range.
-40
-50
-60
-70
Phase A
Phase B
Phase C
-80
-90
-100
2.E+1
2.E+2
2.E+3
2.E+4
Frequency (Hz)
-20
-25
-30
-35
-40
-45
2.E+5
2.E+6
-50
2.E+4
Phase A
Phase B
Phase C
2.E+5
Frequency (Hz)
2.E+6
Figure 5.2 Case 1 Open-circuit measurements on common windings
The unit was inspected internally and then untanked to perform a failure investigation. During the internal
inspection, damaged clamping structure and evidence of significant buckling were found on the phase
B common and tertiary windings (Figure 5.3).
Figure 5.3 Case 1 Internal inspection
5.2.2 Case 2 Buckling of the inner LV winding
The unit is a single-phase generator step-up transformer rated 600 MVA, 22/432 kV.
The assessment is made by comparing measurements on sister units composing the bank of singlephase transformers.
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Figure 5.4 shows the open-circuit measurements on the LV winding of two identical transformers
showing a shift in the resonance frequencies in the 10 kHz to 2 MHz frequency range.
0
0
-5
-5
-15
Amplitude (dB)
Amplitude (dB)
-10
-20
-25
-30
-35
-40
-50
1.E+1
1.E+2
1.E+3
1.E+4
1.E+5
Frequency (Hz)
-15
-20
-25
-30
Phase B
Phase C
-45
-10
1.E+6
Phase B
Phase C
-35
1.E+4
1.E+7
1.E+5
Frequency (Hz)
1.E+6
Figure 5.4 Case 2 Open-circuit measurements on the LV winding of two identical transformers
The internal inspection showed significant buckling of the inner LV winding (Figure 5.5).
Figure 5.5 Case 2 Internal inspection
5.2.3 Case 3 Buckling of the common winding
The unit in this case is a three-phase autotransformer rated 100 MVA, 161/69/13.8 kV.
This unit was in service for many years and was taken out of service for routine maintenance and testing.
As a part of routine testing, frequency response measurements were performed. No previous results
were available for this unit.
Figure 5.6 shows the series and common windings open-circuit measurements.
For the series windings measurements, the resonance frequencies for all three phases line up well in
the above 10 kHz frequency range. The deviation seen in the low frequency range below about 1 kHz
is again due to core influence. Open-circuit measurements on series windings do not indicate any
abnormality.
For the common windings measurements, the resonance frequencies for all three phases show a small
shift in the 50–150 kHz frequency range. The deviation seen in the low frequency range is again due to
the core influence.
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10
0
Serie
s
Amplitude (dB)
-10
Common
-20
-30
-40
-50
H1-X1
H2-X2
H3-X3
-60
-70
-80
1,E+1
1,E+2
1,E+3
1,E+4
1,E+5
Frequency (Hz)
1,E+6
1,E+7
Figure 5.6 Case 3 Series and common windings open-circuit measurements
Figure 5.7 shows the tertiary windings open-circuit measurements. It can be observed that the
resonance frequencies for all three phases have a small shift in the 50–300 kHz range.
Figure 5.7 Case 3 Tertiary windings open-circuit measurements
Figure 5.8 shows the common windings short-circuit measurements with tertiary windings shorted. It
can be observed that the resonance frequencies for all three phases have a small shift in the 50–
300 kHz frequency range. Expanding the view on the low frequency inductive slope to review the
deviation between the three phases showed a deviation of 1.5 dB. The deviation was also greater for
the series windings measurements with common or tertiary windings shorted, but not as large as for
common windings measurements with tertiary shorted.
Figure 5.8 Case 3 Common windings short-circuit measurements with tertiary windings shorted
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After diagnostics such as leakage reactance test and internal inspection, the customer decided to send
the unit to a repair shop for full refurbishment. During untanking, a severely buckled common winding
was found (Figure 5.9).
Figure 5.9 Case 3 Internal inspection
5.2.4 Case 4 Buckling of the inner winding
The unit is a three-phase transformer rated 83.3 MVA, 138/13.8/13.8 kV, Dyn1yn1.
Figure 5.10 illustrates the open-circuit measurements on one of the LV windings in comparison with a
sister unit. For phases 1 and 2, it can be seen that the failed unit resonance and anti-resonance
frequencies between 100 kHz and 400 kHz are significantly different that those of the sister unit while
phase 3 responses are almost identical in this frequency range.
An interesting fact about this case is that the frequency responses shown were made in 2012. A cracked
pressure ring was discovered in 2018 during a tank leak repair. The severity of the mechanical
displacement was later found in the factory when the active part was dismantled. As illustrated in Figure
5.11, the internal inspection showed buckled LV windings in phases 1 and 2. The phase 3 winding (not
shown) was found in good mechanical condition.
(a) Phase 1
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(b) Phase 2
(c) Phase 3
Figure 5.10: Case 4 LV open circuit measurements [30]
Figure 5.11 Case 4 Internal inspection (left - phase 1 and right - phase 2) [30]
5.3 Axial deformation, twisting and tilting
5.3.1 Case 5 Twisting and loss of clamping
In this case, previous measurements before the fault were available. The unit is a three-phase
autotransformer rated 240 MVA, 400/132 kV, YNa0.
Figure 5.12 shows the before and after the fault open-circuit measurements on the common windings.
It can be observed that the traces seem to align well up to 2 MHz. However, the expanded view of the
traces in the 100 kHz to 1 MHz range clearly shows that phase A has a shift in the resonance
frequencies, while phases B and C are perfectly aligned.
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Figure 5.12 Case 5 Open-circuit measurements on the common windings
On the basis of the abnormal results and other indications of failure, the transformer was scrapped. The
investigation showed that phase A was slightly twisted and had lost its clamping (Figure 5.13).
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Figure 5.13 Case 5 Internal inspection [31]
5.3.2 Case 6 Axial movement of the LV winding
The unit is a single-phase transformer rated 21.6 MVA, 166/10 kV. The unit has a two-limb core and the
analysis is done by comparing the traces before and after the fault. The available measurements are:




For limbs 1 and 2: HV open-circuit
For limbs 1 and 2: HV with short-circuit on LV
LV open-circuit
LV with short-circuit on HV
Figure 5.14 shows the HV open-circuit measurements and Figure 5.15 shows the HV short-circuit
measurements. The traces align well. For the open-circuit measurements in Figure 5.14 the deviation
seen in the low frequency range is due to a different core magnetization. These deviations are not visible
in Figure 5.15 because the leakage inductance, instead of the magnetizing inductance, is dominant in
the low frequency range of the short-circuit measurement.
Figure 5.14 Case 6 HV open-circuit measurements
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Figure 5.15 Case 6 HV with short-circuit on LV
Amplitude (dB)
Figure 5.16 and Figure 5.17 show the LV measurements with HV open and short-circuited. The influence
0 range of the open-circuit measurements.
of residual magnetization can be seen in the low frequency
Both the open and short-circuit measurements show a shift in the frequency
Reference resonances, as well as the
appearance of new resonances above 50 kHz. These new resonances are due to the change of the
Failedshipped to a facility for teardown.
natural frequencies of one of the parallel LV windings. The
-5 unit was
During the teardown, evidence was found of damage due to axial movement of a LV winding (Figure
5.18).
0
Amplitude (dB)
-10
Reference
Failed
-20
-30
-40
-10
-15
-20
-50
-60
1.E+1
-25
1.E+2
1.E+3
1.E+4
1.E+5
Frequency (Hz)
1.E+6
1.E+7
-30
-10
1.E+5
Figure 5.16 Case 6 LV open-circuit measurements
0
-12
-20
-30
-40
-50
-60
1.E+1
Frequency (Hz)
-14
Amplitude (dB)
Amplitude (dB)
-10
1.E+6
Reference
Failed
1.E+2
1.E+3
1.E+4
1.E+5
Frequency (Hz)
-16
-18
-20
-22
-24
-26
1.E+7
1.E+6
-28
Figure 5.17 Case 6 LV with short-circuit
on HV
-30
1.E+5
1.E+6
Frequency (Hz)
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Figure 5.18 Case 6 Internal inspection
5.3.3 Case 7 Axial displacement of the stabilising winding
The unit in this case is a three-phase autotransformer rated 150 MVA, 220/115/10.6 kV, YNa0+d.
This unit failed in service due to a lightning strike and it was sent to the factory for investigation and
repair. Measurements were performed after the failure and after repair. Only open-circuit measurements
are available.
Figure 5.19 shows the open-circuit measurements on the series windings. In the frequency range
associated with windings interaction, a small deviation can be noticed on phase W.
Resonance frequency shifts are also observed in the 1 kHz to 10 kHz range for all three phases.
The deviation seen in the low frequency range is due to the different core magnetization.
Figure 5.20 shows the common windings open-circuit measurements. In the frequency range of
windings interaction, a clear deviation can be seen in all three phases. The deviation in the low frequency
range is due to different core magnetization.
Figure 5.21 shows the stabilizing winding open-circuit traces. A clear deviation in all the significant
frequency ranges is observed.
The stabilising winding in phase W was found with an axial displacement (Figure 5.22).
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Phase U
Phase V
Phase W
Figure 5.19 Case 7 Open-circuit measurements on the series windings
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Phase U
Phase V
v
Phase W
v
Figure 5.20 Case 7 Open-circuit measurements on the common windings
Figure 5.21 Case 7 Stabilizing winding open-circuit measurements
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Figure 5.22: Case 7 Internal inspection
5.3.4 Case 8 Conductors tilting
The unit is a three-phase transformer rated 120 MVA, 275/33 kV, YNd1.
The analysis is based on a comparison with an identical sister unit. The sister unit was measured with
oil and the failed unit was measured without oil.
Figure 5.23 demonstrates a good match between the phases of the sister unit. Changes of resonance
frequencies in the range related to winding structure are evidenced on phase C of the failed unit.
The deviations in the low frequency range are attributed to:


External interference causing noise around system frequency and for low amplitude
measurements
Normal asymmetry of the core (lateral vs central phase)
0
-10
Amplitude (dB)
Sister unit
-20
-30
-40
0
-50
-10
Amplitude (dB)
Failed unit
-60
1.E+5
-20
1.E+6
Frequency (Hz)
-30
-40
-50
-60
1.E+5
Figure 5.23 Case 8 HV open-circuit measurements
The tap winding of phase C was found with conductors tilting, as shown in Figure 5.24. Flashover marks
Frequency
(Hz)
in the phase C tap leads and damage in the tap-changer barrier board were also
observed.
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1.E+6
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Figure 5.24 Case 8 Internal inspection
5.4 FRA before and after short-circuit tests
5.4.1 General
In the presented case studies, six FRA were performed after short-circuit tests. Mechanical defects were
found in three of these cases. Any frequency response measurement changes in the traces measured
after short-circuit test should be investigated because the measurement repeatability is expected in a
controlled laboratory environment.
5.4.2 Case 9 Successful short-circuit test
The unit is a three-phase transformer rated 80 MVA, 220/31.5 kV, YNyn0.
Figure 5.25 shows the HV and LV open-circuit measurements for one phase only since the other phases
show the same behaviour. The resonance frequencies line up well.
LV
HV
Figure 5.25 Case 9 HV and LV open-circuit measurements before and after the short-circuit test
The transformer passed the short-circuit test therefore no internal inspection was required.
5.4.3 Case 10 Successful short-circuit test
The unit is a three-phase transformer rated 47 MVA, 120/26.4 kV, YNd1.
Figure 5.26 illustrates the measurements before and after short-circuit tests. The measurements are
essentially identical except for the resonance associated with the core magnetization. Only one phase
is shown, the other phases show similar behaviour.
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0
0
Before
After
-10
Before
After
-20
-30
Amplitude (dB)
Amplitude (dB)
-20
-10
-40
-50
-60
-70
-30
-40
-50
-60
-80
-70
-90
-100
1.E+1
1.E+2
1.E+3 1.E+4 1.E+5
Frequency (Hz)
1.E+6
-80
1.E+1
1.E+7
1.E+2
1.E+3
1.E+4
1.E+5
Frequency (Hz)
1.E+6
1.E+7
Figure 5.26 Case 10 HV and LV open-circuit measurements on phase A
The transformer passed the short-circuit test therefore no internal inspection was required.
5.4.4 Case 11 Successful short-circuit test
The unit is a three-phase autotransformer rated 400 MVA, 400/155 kV, YNa0.
All the measurements gave almost identical traces before and after short-circuit tests similar to the left
graph on Figure 5.27. However, the data records indicate a variation of the response in the 2 kHz to
200 kHz range on one of the phases that was unfortunately left unexplained.
Following the test procedure, the transformer was deemed as a successful short-circuit test therefore
no internal inspection was performed.
0
-20
Before
After
-10
-20
-30
Amplitude (dB)
Amplitude (dB)
0
Before
After
-10
-40
-50
-60
-70
-80
-30
-40
-50
-60
-70
-80
-90
-90
-100
1.E+1
-100
1.E+1
1.E+2
1.E+3 1.E+4 1.E+5
Frequency (Hz)
1.E+6
1.E+7
1.E+2
1.E+3 1.E+4 1.E+5
Frequency (Hz)
1.E+6
1.E+7
Figure 5.27 Case 11 HV open-circuit measurements on phase 1 (left) and phase 3 (right)
5.4.5 Case 12 Slightly buckled winding
The unit is a three-phase transformer rated 3 MVA, 60/6.9 kV, Dd0. Measurements were performed
after each short-circuit test (3 tests). The LV delta windings were open for the test in order to test each
phase separately. The voltage was applied on the LV side with the HV side shorted. More information
on this case can be found in [32].
As the issue was discovered in the w-phase low-voltage winding, only the measurements before and
after failure on this phase are presented in Figure 5.28. The configuration is LV with the HV shorted.
Figure 5.28 shows that the resonance frequencies line up well after the first and second short-circuit
tests, however, after the third test, the unaligned traces indicate a damaged winding.
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After SC 1
After SC 2
After SC 3
Figure 5.28 Case 12 Measurements before and after three consecutive short-circuit tests
The unit was untanked and it was found that the w-phase LV winding had slightly buckled (Figure 5.29).
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Figure 5.29: Case 12 Internal inspection
5.4.6 Case 13 Lead movement, axial collapse and spiralling
This transformer did not pass the short-circuit test. The unit is a three-phase autotransformer rated
450 MVA, 315/120/12.5 kV, YNa0d1.
Open-circuit measurements on all windings were performed before and after the short-circuit test.
The deviations between 100 kHz and 1 MHz were similar in all three phases but more pronounced on
phase C. Only phase C results are shown in Figure 5.30.
Common winding
Tertiary winding
Figure 5.30 Case 13 Common and tertiary windings measurements before and after the short-circuit test
The unit was untanked and damage (lead movement, axial collapse and spiralling) to the phase C
tertiary winding was found (Figure 5.31).
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Figure 5.31 Case 13 Internal inspection
5.4.7 Case 14 Slight displacement of leads showing possible twisting of the winding
This unit is a three-phase autotransformer rated 250 MVA, 400/155 kV, YNa0.
The only measurement available is the open-circuit configuration from the 400 kV terminal to neutral.
The variation observed in Figure 5.32 is very subtle and could be attributed to measurement setup.
A failure occurred during the repeated dielectric tests.
Figure 5.32 Case 14 HV to neutral open-circuit measurements before and after the short-circuit test
As shown in Figure 5.33, internal inspection revealed a slight displacement of the leads indicating a
possible twisting of the phase C tap winding.
Figure 5.33 Case 14 Picture of the internal inspection
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5.5 Electrical failures
5.5.1 Case 15 Broken earthing connection between core elements of a shunt reactor
The unit is a three-phase shunt reactor rated 20 MVA, 66 kV with delta connected windings. More
information on this case can be found in [33]. Measurements are available on a sister unit in good
condition.
Figure 5.34 shows the phase to phase comparison of the failed and sister units. For the sister unit in
good condition, phases U-W and V-U are perfectly superposed. However, the U-W phase of the failed
unit exhibits a change in amplitude and resonance frequencies in the 5–50 kHz range (blue curve).
0
0
Phase U-W
Phase V-U
Phase W-V
Amplitude (dB)
-20
-30
-10
-20
Amplitude (dB)
-10
-40
-50
-60
-30
-40
-50
-60
-70
-70
-80
-80
-90
5.E+3
Phase U-W
Phase V-U
Phase W-V
-90
5.E+3
5.E+4
5.E+4
Frequency (Hz)
Frequency (Hz)
Figure 5.34 Case 15 Open-circuit measurements on failed unit (left) and sister unit (right)
The internal inspection revealed that an earthing connection between two core elements axially
separated by insulators was broken (high resistance). This fault introduced a series capacitance in the
circuit that affected the frequency response.
5.5.2 Case 16 Short-circuit inside winding
In this case, the unit is a three-phase transformer rated 17 MVA, 72/7.2 kV.
This failure mode can be detected by other diagnostic techniques like exciting current or turns ratio. It
can also be easily detected by FRA.
Figure 5.35 shows the HV open-circuit measurements. The resonance frequencies for all three phases
line up well in the frequency range above 10 kHz, however phase C (H3-H2) is deviated similarly to the
result of a short-circuit measurement in the low frequency range.
Figure 5.36 shows the LV open-circuit measurements. It is again clearly observed that phase C is
different (X3-X0).
Figure 5.37 shows the HV with LV shorted, with similar behaviour.
Figure 5.35 Case 16 HV open-circuit measurements
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Figure 5.36 Case 16 LV open-circuit measurements
Figure 5.37 Case 16 HV measurements with LV shorted
The internal inspection revealed a turn-to-turn fault in the HV winding with the other windings in good
condition.
5.5.3 Case 17 Opening of a parallel winding
The unit is a three-phase transformer rated 12 MVA, 110/28 kV, YNyn0. The LV winding of this
transformer can be configured in star of delta using external electrical connections. When the failure
occurred, the LV winding was arranged with two windings connected in parallel in each phase, forming
the star configuration.
Figure 5.38 shows the open-circuit measurements on the LV side. The phase to phase comparison
indicates a significant difference for phase 2.
The internal inspection revealed a broken cable that created an opening of a parallel winding for phase
2. The dc resistance of phase 2 was indeed two times the value of the other phases.
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0
-10
Amplitude (dB)
-20
-30
-40
-50
-60
-70
-80
2.E+1
X1X0
X2X0
X3X0
2.E+2
2.E+3
2.E+4
Frequency (Hz)
2.E+5
2.E+6
Figure 5.38 Case 17 Open-circuit measurements on the LV side
5.5.4 Case 18 Broken bushing connection
The unit is a single-phase autotransformer rated 90 MVA, 525/√3 / 74.5/√3 / 34.5 kV.
Figure 5.39 illustrates the open-circuit measurements on the whole series and common windings and
on the common winding only. It shows that for the common winding open-circuit measurement, the
measurement in the failed condition is under the noise level for the low frequency range indicating an
opening of the circuit. The measurement on the whole winding is showing minor but significant
deviations.
Figure 5.40 shows the opening on the low-voltage terminal of the autotransformer.
This failure mode can be detected by some conventional electrical tests, e.g., DC resistance or
magnetizing current.
0
0
-40
-20
Reference
Failed
Amplitude (dB)
Amplitude (dB)
-20
-60
-80
-100
-120
2.E+1
-40
Reference
Failed
-60
-80
-100
2.E+2
2.E+3
2.E+4
Frequency (Hz)
2.E+5
2.E+6
-120
2.E+1
2.E+2
2.E+3
2.E+4
Frequency (Hz)
2.E+5
2.E+6
Figure 5.39 Case 18 Open-circuit measurements on the series+common windings (left) and on the
common winding (right)
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Figure 5.40 Case 18 Internal inspection
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6.
Literature review of quantitative FRA assessment
Leader: Stefan Tenbohlen (DE)
6.1 Introduction
FRA is a comparative diagnostic method which needs a reference for comparison. As it is detailed in
[1], the reference can be a previous measurement on the same winding, a measurement from another
phase of the same transformer, or a measurement on an identical transformer. After defining the
reference, the new trace is compared with the reference for which different quantitative methods are
described in the literature. These methods can be categorized into three main groups:



algorithms based on numerical indices
algorithms based on a white box model
algorithms based on artificial intelligence (AI)
These three groups will be discussed in this chapter.
6.2 Algorithms based on numerical indices
6.2.1 Frequency range
In the FRA literature regarding the application of indices, it is noteworthy that it is common to find
different frequency ranges used for interpretation. Some researchers use the frequency response data
in the whole frequency range, whereas others divide it into several frequency bands for assessment.
Table 6.1 summarizes the frequency ranges used in FRA interpretation literature.
Table 6.1 Frequency ranges used in literature for FRA interpretation
Reference
Frequency range
Reference
Frequency range
Reference
Frequency range
[39]
[45]
10 Hz–1 MHz
10 kHz–4 MHz
[47]
10 kHz–1 MHz
Single frequency sub-band
[34, 35]
[40-42]
[1, 46]
20 Hz–2 MHz
100 kHz–1 MHz
5 kHz–1 MHz
V < 100 kV
[36-38]
[43, 44]
[1, 46]
20 Hz–1 MHz
10 Hz–3 MHz
5 kHz–2 MHz
V < 100 kV
Three frequency sub-bands
[6, 48]
[51]
0–100 kHz
100–600 kHz
600–1000 kHz
20 Hz–10 kHz
10–100 kHz
100 kHz–1 MHz
[49]
[52]
0–20 kHz
20–400 kHz
400–1 MHz
100 Hz–20 kHz
20–200 kHz
200 kHz–2 MHz
[50]
[53]
300 Hz–50 kHz
50 kHz–1 MHz
1–3 MHz
0–350 kHz
350 kHz–1 MHz
1–2 MHz
More than three frequency sub-bands
[54]
100 Hz–1 kHz
1–10 kHz
10–100 kHz
100 kHz–1 MHz
[3]
[56]
0–2 kHz
2–20 kHz
20–400 kHz
400–1 MHz
[57]
0–2 kHz
2–20 kHz
20 kHz–1 MHz
1–2 MHz
1–10; 10–20 kHz
20–40; 40–100
kHz
100–500 kHz
500–1000 kHz
[55]
10 kHz–1 MHz
1–2 MHz
2–3 MHz
3–5 MHz
[58]
20 Hz–1 MHz
10 ranges
6.2.2 Definitions
The names of the indices along with their main references are shown in Table 6.2.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Table 6.2 Names, abbreviations and main references of the numerical indices ([59])
ABBR.
EQUATION
DEFINITION
REFERENCE
Numerical indices calculated directly using the trace data values
ED
Equation 6.1
Euclidean Distance
[47, 58]
CD
Equation 6.2
Complex Distance
[60]
SD
Equation 6.3
Standard Deviation
[43]
ID
Equation 6.4
Integral Difference
[37, 44]
AID
Equation 6.5
Integral of Absolute difference
[37, 47]
SDA
Equation 6.6
Standardized Difference Area
[48]
ASLE
Equation 6.7
Absolute Sum of Logarithmic Error
[43, 50]
RMSE
Equation 6.8
Root Mean Square Error
[38]
E
Equation 6.9
Expectation
[35]
σe
Equation 6.10
Standard deviation
[35]
SSD
Equation 6.11
Stochastic Spectrum Deviation
[61]
MD
Equation 6.12
Maximum of Difference
[44, 58]
CCF
Equation 6.13
Cross Correlation Factor
[35, 48]
CC
Equation 6.14
Correlation Coefficient
[34, 43]
SSE
Equation 6.15
Sum Squared Error
[50, 58]
SSRE
Equation 6.16
Sum Squared Ratio Error
[47, 58]
SSMMRE
Equation 6.17
Sum Squared Min Max Ratio Error
[50]
CSD
Equation 6.18
Comparative Standard Deviation
[62]
LCC
Equation 6.19
Lin’s Concordance Coefficient
[62]
SE
Equation 6.20
Sum of Error
[62]
LSE
Equation 6.21
Least Squared Error
[54]
MM
Equation 6.22
Minimum Maximum
[57]
JD
Equation 6.23
Jaccard Distance
[63]
Indices calculated based on the resonance frequencies
MDA
Equation 6.24
Mean Deviation of Areas
[35]
MAD
Equation 6.25
Mean Amplitude Deviation
[35]
MFD
Equation 6.26
Mean Frequency Deviation
[35]
IAD
Equation 6.27
Index of Amplitude Deviation
[45, 48]
IFD
Equation 6.28
Index of Frequency Deviation
[45, 48]
Fa
Equation 6.29
Amplitude Function
[36]
Ff
Equation 6.30
Frequency Function
[36]
Wa
Equation 6.31
Weighted Amplitude function
[37, 48]
Wf
Equation 6.32
Weighted Frequency function
[48, 58]
Indices calculated based on vector fitting of the traces
SDP
Equation 6.34
Sum of absolute Displacement of Poles
[45, 64]
FI
Equation 6.35
Faulted-Intact relation
[64]
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6.2.3 Numerical indices extracted from frequency response traces
6.2.3.1 Formulations
Interpretation of frequency response results using numerical indices is based on the fact that the
numerical indices quantify the differences between reference and present frequency response traces.
In this method, a mathematical equation is defined to extract a single value from the present frequency
response trace and its reference in different frequency sub-bands. The equations normally use the
amplitude of the frequency response to calculate the value. The assessment of the transformer is then
carried out based on the final value as depicted schematically in Figure 6.1.
Figure 6.1 Application of a numerical index for FRA interpretation
The frequency response traces are expressed as an amplitude and a phase vector, with discrete
elements in which each element corresponds to one frequency sample. It is noteworthy that the
frequency samples can be distributed linearly or logarithmically through the frequency range, which
should be defined to homogenize the indices. All the indices below are calculated based on the
amplitude vectors except for CD, which implements both the amplitude and the phase response to
calculate the index. These indices are listed in the upper part of Table 6.2.
The equations of these indices are summarized here. In the following equations, Y and X are the
amplitude vectors of the new frequency response trace under investigation and its reference,
respectively, Y(i) and X(i) are the i-th elements of these vectors, f is the vector of frequency samples, N
is the number of samples in a vector, and φX and φY are the phase vectors of the frequency response
traces.
ED  X  Y  (X  Y)T (X  Y)

 Y (i)  X (i) 
N
2
CD 
i 1
( X (i ) cos  X (i )  Y (i ) cos Y (i )) 2 
 i1 ( X (i) sin  (i)  Y (i) sin  (i))2 


X
Y
N
Equation 6.2
Equation 6.1
 Y (i)  X (i)  ,
SD 
N
2
ID 
i 1
N 1
 Y ( f )  X ( f ) df ,
Equation 6.4
Equation 6.3
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
AID 

Y ( f )  X ( f ) df ,
 Y ( f )  X ( f ) df ,
 X ( f ) df
SDA 
Equation 6.5
Equation 6.6
2
ASLE 

N
20 log10 Y (i)  20 log10 X (i)
i 1
N

1
N
RMSE 
,
Equation 6.7


N  Y (i)  X(i) 

 ,
N
i 1 1

X (i) 
i 1
N


Equation 6.8
Y (i)  X (i)
,
N
1
X (i)
i 1
N
N
1
E  
(i),
i 1
N
(i) 


 e  var()  E    E     ,
Equation 6.10
Equation 6.9
𝑁
100
𝑌(𝑖) − 𝑋(𝑖)
𝑆𝑆𝐷 =
∑|
|
𝑁
𝑋(𝑖)
𝑀𝐷 = max(𝑌(𝑖) − 𝑋(𝑖)
𝑖=1
Equation 6.12
Equation 6.11
 ( X (i)  X )(Y (i)  Y )
CCF 
  X (i)  X   Y (i)  Y 
 X (i) Y(i)
CC 
,
  X (i)  Y (i)
N
N
i 1
i 1
N
2
i 1
N
2
N
i 1
Equation 6.13
 i 1Y (i)  X (i) 2
N
,
SSRE 
Equation 6.15

SSMMRE 
S XY 
1
SY2 
N  max( X (i ), Y (i ))
2

 1
i 1  min( X (i ), Y (i ))

 ,
N
CSD 

N i 1
i 1
N 1
Equation 6.18
1 N
S X2   ( X (i )  X ) 2 ,
N i 1
 Y (i)  X (i) 
N
SE 
N
(Y (i ) Y ) ,

 1
i 1  X (i )

 ,
N
[(Y (i)  Y )  ( X (i)  X )]2
2S XY
(Y  X ) 2  SY2  S X2
2

2
N  Y (i )
N
1
 ( X (i)  X )(Y (i) Y )
N i 1
N
2
Equation 6.16
Equation 6.17
LCC 
N
i 1
Equation 6.14
N
SSE 
2
i 1
i 1
N
Equation 6.20
1 N
X   X (i )
N i 1
 ( X (i)  X )(Y (i)  Y )
  X (i)  X 
N
LSE 
i 1
N
2
i 1
1 N
Y   Y (i )
N i 1
Equation 6.21
Equation 6.19
∑𝑁
𝑖=1 min(𝑌(𝑖), 𝑋(𝑖))
𝑀𝑀 = 𝑁
∑𝑖=1 max(𝑌(𝑖), 𝑋(𝑖))
𝐽𝐷 =
Equation 6.22
2
∑𝑁
𝑖=1(𝑋(𝑖) − 𝑌(𝑖))
𝑁
𝑁
2
2
√∑𝑁
𝑖−1 𝑋(𝑖) √∑𝑖=1 𝑌(𝑖) − ∑𝑖=1 𝑋(𝑖)𝑌(𝑖)
Equation 6.23
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
These indices are calculated directly from the frequency response traces without additional calculation
and, therefore, are the easiest ones to implement. The values of the amplitude vectors used for
calculating these indices can be in their original form or on the dB scale. In the literature, a dB scale is
usually used. However, one can also use the original values, which leads to different results than when
using dB scale values. It is noteworthy that there is no report in the literature about the advantages of
each approach.
It is also noteworthy that all of the above indices result in zero for the intact case, except the indices CC
and CCF, which result in a one [48]. CC and CCF can be replaced with 1-CC and 1-CCF to homogenize
all the indices [65]. In this case, a value equal to zero declares a perfect match between the new FRA
trace and its reference.
6.2.3.2 Application examples from the literature
The indices derived directly from the full frequency response traces have different indications for faults.
Authors in [50] employed CC, SSE, ASLE, SSRE and SSMMRE for detection of winding mechanical
deformation. They showed that SSE, which is based on the distance between two traces, is very
sensitive to a large change in amplitude around the peak, which is indicative in frequency response
measurements. Figure 6.2 shows an example of large amplitude variations.
-20
Magnitude (dB)
-30
-40
-50
-60
-70
0
100
200
300 400 500
Frequency (kHz)
600
700
800
Figure 6.2 Amplitude variation around resonance points which affects some indices
Reference [50] also shows some of the limitations of CC. As an example, if Y  cX , in which c is a
constant, CC equals 1, indicating no deviation in the traces although the traces are completely different.
Then, [50] derives SSRE and SSMMRE from SSE. However, these demonstrated weak sensitivity to
variations around the trough points (Figure 6.2). Reference [50] describes that ASLE has the best
performance among the other indices by overcoming the aforementioned defects. It declares that ASLE
has the best relationship with the visual changes in the traces. Furthermore, [50] reports 90% accuracy
when ASLE is used to detect the fault by comparing the traces recorded from different phases of the
same transformer.
Reference [40] uses CC and ED to examine the sensitivity of FRA to different types of mechanical
changes. Authors indicate that CC is insensitive in the case where the frequency response trace only
moves vertically with respect to its reference, without variations in the overall form. In such cases where
CC doesn’t show a fault, some have used two indices, ED and CC, simultaneously. Reference [34]
highlights the inability of CC to respond to a constant difference between frequency responses. As a
result, it seems that using only CC for interpretation is not sufficient. Implementing more than one index
at the same time is also proposed in other studies [48]. Reference [66] also addresses CCF, ASLE and
σ as the mostly used spectral indices and implements them successfully to compare a simulated model
with the measured frequency response trace.
Reference [58] introduces ED and compares it with numbers of other indices. It emphasizes that the
linearity of the indices versus the severity of the fault is important. Reference [58] also calculates ED in
different frequency ranges and implements them for defining the fault type.
Reference [43] extracts three indices from the phase response only and recommends including the
phase response in the interpretation due to its wide variation in case of faults. However, the phase
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
response alone does not always show a regular change with the fault severity. Reference [60] proposes
a new index, CD, which includes both the phase response and the amplitude response. This index
shows a regular change with the fault severity. This paper shows that including the phase response in
the interpretation increases the sensitivity of the index and also decreases the vulnerability of the index
to variations in the measurement setup.
Reference [43] uses CC, ASLE and SD for detecting the axial and radial fault in three large transformers.
It reports that all of three indices can show a fault though CC fails to detect axial displacements less
than 1% of the winding height. Therefore, it is a less sensitive index. This reference employs the indices
for the phase and sister unit comparison and explains that the fault detection is possible only for severe
faults. It also takes 1% of axial displacement to set criterion limits for the indices.
Reference [35] introduces E and σ as an index because E is zero when the traces are identical or when
their difference has an average value of zero. The σ is zero for a constant amplitude difference between
frequency responses. Therefore, these two indices show a good immunity to noise. This reference also
performs a sensitivity check between numbers of indices. The E shows a regular change versus axial
shift, but no clear trend for the radial deformation. The CCF and σ show monotonic behaviour toward
axial and radial shift, except they are reversed. This contribution shows that evaluation of indices for
different frequency ranges can discriminate between the radial and axial fault.
Reference [48] employs a number of indices for defining the type, level and location of a fault. This
reference divides the frequency range into three bands and detects the fault type using the indices
calculated in different frequency bands. It also provides information about the monotonicity of the indices
for various fault severities. It concludes that SDA does not show a regular change versus the fault
severity. Moreover, it indicates that CCF is a less reliable index compared to others. It also proposes to
analyse numbers of indices together, to more precisely determine the state of the winding.
The Chinese standard has defined a criterion to detect deformation inside the transformer based on the
correlation coefficient factor (CCF) [6]. In this standard, three frequency ranges have been described as
follows:
LF: 1-100 kHz
MF: 100-600 kHz
HF: 600-1000 kHz
The interpretation is based on a relative factor, Rxy, which is defined using:
where CCF is calculated using Equation 6.13. Table 6.3 provides the limits based on the RLF, RMF, and
RHF, which are Rxy in the LF, MF, and HF range, respectively.
Table 6.3 Level of deformations based on Rxy
Degree of deformation of
the winding
Severe deformation
RLF < 0.6
Obvious deformation
0.6 ≤ RLF < 1 or RMF < 0.6
Slight deformation
1.0 ≤ RLF < 2.0 or 0.6 ≤ RMF < 1.0
Normal winding
2.0 ≤ RLF, 1.0 ≤ RMF and 0.6 ≤ RHF
Limits for the relative factor, RXY
Reference [35] has used these limits for a 400 kVA transformer and reports that they can detect a radial
deformation and an axial displacement, except for an axial displacement lower than 1 cm. Similarly, [34]
reports on two 100 MVA transformers where the transformers have obvious deformations but the
Chinese standard fails to show deformations, indicating a normal winding in both cases.
6.2.4 Indices based on resonance frequencies
In each FRA trace, there are several peak and trough points which are respectively called resonance
and anti-resonance points in the literature. The following indices are based on the amplitude and
frequency of these points. Correspondingly, it is necessary to find the peak and trough points first.
Afterwards, the following formulas can be used to calculate these indices, in which the subscripts Y and
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
X refer to the new and the reference frequency response trace, respectively, A(i) and f(i) are the
amplitude and the frequency of the i-th resonance or anti-resonance point, and AF is the area below the
frequency response trace between two anti-resonance points, as described in [35].
1
K

mad 
1
K
  A  A ,
Equation 6.25
mfd 
1
K
  f  f ,
Equation 6.26
IAD 
 
IFD 
 
mda 
K
AFY (i)  AFX (i) ,
i 1
Equation 6.24
K
Y ,i
i 1
X ,i
K
Y ,i
i 1
X ,i
N  AY ,i  AX ,i 
i 1
,

AX ,i
Equation 6.27
N  fY , i  f X , i 
i 1
Fa 
Ff 
Equation 6.29

Equation 6.30
AY ,i
N
i 1
N
X ,i
fY ,i
i 1 f
,
X ,i
 A w ,
ai
Equation 6.31

 w fi .
Equation 6.32
AY ,i
i 1
Wf 
Equation 6.28
 A ,
N
Wa 
,

f X ,i
N
X ,i
fY , i
i 1 f
X ,i
*wai and wfi are the weight factors for the i-th resonance point, and their calculation is explained in detail
in [37].
Figure 6.3 illustrates a sample frequency response trace measured from a 6 kV/400 V, 600 kVA
distribution transformer. The resonance and anti-resonance points are shown on the trace. Determining
the peak and trough points may seem simple, but particularly in the case of noise in the measurements,
detecting the true peak and trough points is tricky and may introduce errors to the indices. The inset of
Figure 6.3 magnifies the last trough point in the trace. The question then becomes whether the trough
point is at point 1 or point 2. In the case of small deformations, where peak and trough points do not
change much, a noise induced change from choosing point 2 over point 1 can readily result in ascribing
deformation in the transformer where there was none.
-25
Resonance points
Anti-resonance points
Magnitude (dB)
-30
-35
-40
1
2
-45
-50
0
200
400
600
800
1000
1200
Frequency (kHz)
Figure 6.3 Resonance and anti-resonance points in an FRA trace
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
6.2.5 Numerical indices based on vector fitting of measured traces
It is possible to fit rational functions to a frequency response trace. The main procedure which is most
used for this purpose is called the vector fitting method described in [67-69] and is available online [70].
The derived rational function can also be used for HF power system simulations (Black Box Model).
Equation 6.33
After fitting the rational function, two indices can be calculated as follows:
SDP 

FI  



m
i 1
PY ,i  PX ,i
,
PX ,i

 .


X ,i 

 b
 b
 a
 a
i 1
m
N
i 1
Y ,i
n
i 1 Y ,i
n
i 1 X ,i
Equation 6.34

.



Equation 6.35
where subscripts Y and X refer to the new and the reference trace, respectively, Pi is the i-th pole of the
rational function, a and b are the coefficients of the numerator and denominator of the rational function,
and m and n are the estimation order of the numerator and denominator polynomials, respectively.
It should be noted that the estimation of the rational function is the most difficult task. Furthermore,
rational functions with different orders can be fitted into a single trace, resulting in different values for
the indices. Moreover, [38] reports the sensitivity of this algorithm to the starting pole allocation. This
means that different starting poles for the estimation may lead to dissimilar final poles for one trace,
which means differences in the related indices.
Reference [64] utilises the vector fitting method and defines FI based on the numerator and denominator
coefficients of the fitted rational function. It reports a successful fault severity detection using the FI.
Moreover, it indicates that the FI demonstrates different ranges of values for various faults and,
therefore, it can be used for fault type discrimination. Similarly, [45] employs vector fitting and SDP,
which is based on the frequency response pole location, to prove that SDP shows a regular change
versus three different steps of axial displacement. However, the changes are small compared to the
value of the index.
Despite these successful uses, there are some serious shortcomings in the vector fitting method.
Reference [38] states that pole-zero allocation using the vector fitting method is dependent on the
degree of rational function, the iterations number, and the location of the starting poles. This contribution
segments the frequency response trace between each two local minimums and then, fits second or third
order functions to each segment to estimate the starting poles. It then implements the vector fitting
algorithm using these estimated poles as starting poles. It is shown in [38] that this method demonstrates
better fitting accuracy. However, [38] states nothing about further interpretation experiences.
6.2.6 Vector based method using a sliding window
Several contributions use different frequency ranges for the application of numerical indices. While the
identification of frequency ranges associated with the main transformer components is challenging, it is
evident that there is no generally applicable frequency limit for each range as this depends on the
physical size of the transformer and the ratings of the windings. Therefore, relying on fixed frequency
ranges [5] can lead to erroneous conclusions. Moreover, as described above, numerical indices are
sensitive to the width of frequency sub-bands, consequently, the values of these indicators change with
the change of frequency limits. Thus, it is not possible to set threshold values for numerical indices which
could serve as objective criteria for transformer winding fault detection.
Focusing on these challenges, [71] and [72] use the sliding window approach with two different
numerical indices. In this method, the whole frequency response is scanned in order to cope with the
problem of defining fixed frequency sub-bands. The numerical index is calculated in a window with a
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
specific width and this window is moved from the starting frequency to the ending frequency with a
specific step (Wstep), as shown in Figure 6.4.
In [71], an index called Standard Deviation of Difference (SDD) is introduced (Equation 6.36). Using the
sliding window approach, a vector of SDD, called winding assessment factor, is obtained characterizing
the differences between two frequency response measurements as a function of frequency. Hence, the
winding assessment factor and measured frequency responses can be presented on the same graph.
The winding assessment factor is calculated as


Winding assessment factor  SDD (i )  2 



2 
  Z ( j)  Zw(i)  
WS
j 1
WS  1



Z (i )  X (i )  Y (i )
Zw(i )  Xw(i )  Yw(i )
Equation 6.36
WS

1
Xw(i ) 
X ( j)
WS j 1
WS
Yw(i ) 

1
Y ( j)
WS j 1
i  1, 2,3...N
 f  200 
WS  10  6  res

 200 
Where X and Y are the amplitude vectors of curve 1 and 2, respectively. X(i) and Y(i) are the ith element
of these vectors. Xw(i) and Yw(i) are the means of the ith window. fres is the number of data points per
decade and WS is the window size.
In the international standard IEC 60076-18 [2], the measurement frequency resolution is specified at
200 points per decade minimum. To take into account the effect of different frequency resolution, window
size is made variable and calculated from Equation 6.36. In this way, the effect of different frequency
resolutions is also considered. Figure 6.4 illustrates the basic principle of the presented method. The
representation of the method for identifying the degree of deviation between two frequency response
signatures of a three-phase transformer is shown in Figure 6.5.
For fault diagnosis, the minimum value of the SDD (MSDD) should be assessed. In [71], the applicability
of the method is discussed through six case studies, where the method is applied to frequency response
signatures of real power transformers with and without deformations. The results show the advantage
of this methodology, as it successfully ascertained the deviations in all faulted transformers. Based on
the presented case studies, an objective criterion is also proposed for quantitative assessment of
transformer mechanical faults. As the proposed criterion was only tested on a few transformers, the
method is considered as tentative. The details can be found in [71].
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
Figure 6.4 Basic principle of winding assessment algorithm
Figure 6.5 Representation of the winding assessment factor SDD for a three-phase transformer
6.2.7 Summary on numerical indices
Different frequency ranges can be used to calculate these indices; different frequency bands lead to
different values for the indices. Likewise, different numbers of points can be measured in a defined
frequency range and will have an influence on the calculated value. This disadvantage can be overcome
by using a vector based method with a sliding window.
The important advantage of this approach is its simplicity, especially for indices calculated directly from
the frequency response measurement data points, since the indices convert the differences between
the traces into a single value or a vector of values which can be easily assessed. For instance, a
threshold value can be set to classify the fault severity similar to the Chinese standard [6]. Another
advantage is that the indices are derived only from the measurements. Thus, this group does not need
the details of the internal geometry of the transformer. Correspondingly, it can be easily embedded in
frequency response measurement devices to perform a quick assessment of a transformer’s winding.
Lastly, the indices in this group are repeatable; hence the outputs are unique values, which simplify the
assessment. However, this is not completely true for indices based on vector fitting.
A first limitation is that the results are only numbers, and further interpretation is not possible. The indices
may be unreliable since each transformer has its own signature and special behaviour under different
faults. A second limitation is that the output value, which is the only means for assessment, can be
influenced by measurement uncertainty such as noise and temperature variations.
6.3 Algorithms based on white-box models
6.3.1 General
This section describes two approaches that can be used to model the frequency response. The first
uses a circuit model to explicitly describe the inductance, capacitance, and resistance topology
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representing the windings. The second uses finite element modelling to numerically reproduce the
electromagnetic behaviour of the windings.
6.3.2 Lumped high-frequency modelling
Equivalent electric circuit models of transformer windings have been used by some investigators to
identify faults from the measured frequency responses [73, 74]. The parameters of the models
representing the frequency responses can be calculated using the details of the transformer design [75]
or can be estimated by using terminal measurements [76]. Studying the effects of changes in model
parameters on the frequency responses (as measured at the terminals) helps reveal correlations
between the frequency response variations and faults [77] and identify the nature of the faults. This
method can be used to estimate the location of the faults [76, 77]. It is also used to detect nonmechanical failures in a transformer [78].
There are different approaches for creating a high-frequency lumped model. In this example, the
windings are divided into several sections, and each section is then represented by different circuit
parameters. Figure 6.6 shows the elements normally used for each section [60]. Each element is also
described in Table 6.4.
Figure 6.6 Lumped model of a two-winding transformer [60]
Table 6.4 Description of the circuit elements of the transformer model
Elements
Rs
Series resistance of each section
Description
Ls
Series inductance of each section
Cs
Capacitance between adjacent sections
Cg
Capacitance between each section and the earthed plate near it
Gg, Gs, GHL
Conductance representing the dielectric loss in the insulation system
CHL
Capacitance between sections of LV and HV windings
M
Mutual inductance of each section to the sections of the same or the other winding
There are two approaches for defining the model elements. In the first approach, the elements are
defined based on the transformer geometry and analytical equations. Reference [75] details these
formulas. It is also possible to run an optimisation algorithm after the analytical calculations to reach a
higher agreement between the model output and real measurements. In the second approach, the data
of the internal geometry are not available, and the elements are estimated from the terminal
measurements using optimization algorithms or artificial intelligence methods [79].
It is shown in [74] that in a model winding, if a deviation is observed when subsequently acquired
frequency response data are compared, it is possible to figure out which of the three circuit elements of
the equivalent circuit, namely, series capacitance, shunt capacitance or self-inductance, is responsible
for the observed change. Furthermore, it can also be inferred whether the pertinent circuit element has
increased or decreased in comparison with its initial value. The proposed approach requires frequency
response measurements but does not require reference measurements. In [74], the applicability of the
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method is verified to a limited extent for both disc and interleaved winding designs. Experimental results
are in good agreement with those predicted by simulation and analytical studies.
It is shown in [76] that by using certain properties of driving-point functions and adopting an iterative
circuit synthesis approach, the location, extent, and type of change introduced in a model winding can
be identified based only on frequency response measurements. In this study, a model winding is used,
and based on its measured natural frequencies and pertinent winding data, an equivalent circuit is
synthesized. Next, changes are introduced at different locations in the model winding, and its natural
frequencies are measured. Corresponding to every new set of measured natural frequencies, a new
circuit is synthesised with topology remaining unchanged. A comparison of these circuits with the
reference circuit shows that a mapping could be established between changes introduced in the model
winding and those predicted by the synthesized circuits. Many case studies are presented by
considering continuous-disc and interleaved winding representations. Reasonably good results are
obtained in this paper, and it concludes that localising changes based on frequency response
measurements is feasible.
Based on frequency response measurements on a single continuous-disc winding of a transformer, it is
demonstrated in [77] how faults introduced at different positions along the winding can be localised with
reasonable accuracy. In this paper, discrete changes were simulated, e.g., short-circuiting a few turns
within a disc (i.e., predominantly an inductive change) and/or addition of some tens of pF capacitance
between a disc and earth (i.e., predominantly a capacitive change). Natural frequencies are determined
by open-circuit and short-circuit frequency response measurements, in addition to measuring effective
resistance, shunt capacitance and inductance. The proposed method aims at using the measured data
to iteratively synthesize a lumped-parameter ladder network, corresponding to each set of
measurement. Comparing such synthesized circuits with a reference (or fault-free) circuit reveals the
location and nature of a fault. Results presented in this research demonstrate the potential of this
method.
In [80], a 400 kV disc winding consisting of 86 discs is used to study whether the detailed model is able
to represent the behaviour of large windings in a frequency response form. In addition, disc space
variation is analysed experimentally and mathematically. The accuracy of the model to identify the disc
space variation is verified with the help of the measured frequency responses.
The following results are verified for the test object using measured data:



The detailed RLC model can represent the frequency responses of the test object up to
approximately 400 kHz
The most important finding is that the model parameters calculated by geometrical dimensions
need to be tuned with the help of an optimization method
The optimized model represents the disc space variation effects on frequency responses
accurately
Reference [79] demonstrates the applicability of this modelling approach by simultaneously fitting each
model to the corresponding frequency response data sets without a priori knowledge of the transformer’s
internal dimensions, and then quantitatively assessing the accuracy of key model parameters. This
paper has presented a novel modelling approach which can be used to simulate frequency response
measurements conducted on three-phase power transformers. It is proposed in this paper that the
resulting models can be used as a tool to support FRA interpretation by providing a flexible test bed for
parameter sensitivity analysis. To demonstrate the modelling approach, models for high-voltage end-toend open-circuit, low-voltage end-to-end open-circuit, and capacitive inter-winding measurements were
derived for Dyn-connected transformers. The frequency response measurement procedure resulted in
nine unique frequency responses (three connection permutations for each of the three frequency
response measurement types). A constrained nonlinear optimization algorithm was then applied. This
algorithm simultaneously estimated each of the model frequency responses for the corresponding
frequency response data using a common parameter set. The estimation results are satisfactory for
each frequency response measurement.
To confirm the physically representative nature of the frequency response models, several parameters,
whose values could be accurately determined through internal inspection, were compared against their
estimated counterparts. These parameters are all within a reasonable tolerance of their actual values,
verifying the physically representative nature of the models. The applicability of the proposed method in
the interpretation of FRA is then demonstrated by changing the model parameters to simulate the effect
of winding buckling on the transformer’s frequency response.
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Reference [81] proposes a similar technique to estimate a model that has fewer nodes and can
accurately predict the behaviour of a transformer in a wide range of frequencies. This model is intended
to be used in accurately modelling the transient behaviour of transformer windings. In the proposed
method, based on the frequency response measurements, N dominant resonances are determined, and
it is experimentally shown that the winding has N-1 hidden resonances. Using this idea, the authors
suggest using a 2N-1 section ladder network, which has a minimum number of nodes and can accurately
model the behaviour of the transformer winding. The parameters of this model are determined by
minimizing the error function by using the genetic algorithm. The close agreement between the
simulation and measurement results on the windings of a 20/0.4 kV and 1,600 kVA transformer verifies
the accuracy of the proposed method. However, this paper does not use the model for the FRA
interpretation, and it is not known how the parameters of the limited sections should be modified to
model the mechanical changes.
6.3.3 Physical parameters extracted from frequency response measurements
Rather than modelling the physical components of the transformer using lumped or distributed
parameter models in a more accurate way, a simple and easy approach for extracting the most relevant
physical parameters of the equivalent circuit of a transformer from frequency response measurements
is proposed.
The methodology for the extraction of the physical parameters is illustrated in Figure 6.7. Different
parameters can be extracted from each type of frequency response.




From the end-to-end open-circuit measurement: the magnetizing inductance (Lm), the core
resistance (Rm), the parallel capacitance (Cg), the series capacitance (Cs) and the main mutual
coupling inductance (Lmci). The Lmci is not represented in the equivalent circuit shown in Figure
6.7.
From the short-circuit measurement: winding resistances (R1+R2) and self-inductances (L1+L2)
can be obtained.
From the capacitive inter-winding measurement: the capacitance between windings (C12) can
be extracted.
From the inductive inter-winding measurement: the turns ratio of the transformer (N1/N2) can be
derived.
The complete description of this approach with the detailed equations and explanations can be found in
[82].
End-to-end
FRA Test
Lm
Rm
Cg
End-to-end Short
Circuit FRA Test
Lmci
L12
(L1+L2)
Cs
R12
(R1+R2)
Inter-winding
Capacitive FRA
Test
Inter-winding
Inductive FRA
Test
C12
Cs1
R1
Cs2
L2
L1
Lm
Rm
Cg1
Cg2
N1
C12
R2
N2
N1/N2
Figure 6.7 Method for extraction of main physical indicators [82]
6.3.4 3D Finite Element Method (FEM) modelling
The Finite Element Method (FEM) has many applications in the simulation of the behaviour of power
transformers. These applications include the calculation of energy, leakage fluxes, and electromagnetic
forces. By using FEM and considering the structural details of a transformer, it is possible to build a turn-
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based high frequency FEM model of transformer windings which can calculate the frequency response
of a transformer by considering frequency dependent losses. The main purpose of the high frequency
FEM model is to directly obtain the frequency response traces without solving and evaluating circuit
models. The FEM model also provides an accurate determination of self and mutual inductances
between coils and inter-turn and inter-disc capacitances. Furthermore, it incorporates all frequency
dependent losses such as eddy current effects (skin and proximity effect) in the coils, and dielectric
losses in the insulation structure. Figure 6.8 shows the actual and FEM model of continuous disc
windings [83].
(a) Disc winding model
(b) FEM model
Figure 6.8 FEM model of a two-winding transformer [83]
Reference [83] proposes a high frequency FEM model of transformer windings. It describes that
frequency response simulation is a three-step process. Firstly, a geometric model is created in the
design module. Secondly, a high frequency model is developed which incorporates the frequency
dependent parameters. Lastly, this model is excited with a sinusoidal voltage source to calculate the
frequency response traces for different connection schemes.
It is shown in [83] that in the proposed FEM model of transformer windings, it is possible to implement
a precise fault identical to real situations, which are otherwise difficult to implement, and convert it into
changes to the circuit elements in the circuit model. Furthermore, frequency response traces can be
directly obtained from the 3D FEM model of the windings without solving complex circuit models. In [83],
the applicability of the method is verified for disc type windings. Experimental results are in good
agreement with those predicted by simulation. It is proposed in this paper that the resulting models can
be used as a tool to support FRA interpretation by providing a flexible test bed for sensitivity analysis of
different connection schemes against different mechanical faults. To demonstrate the modelling
approach, axial displacement is analysed experimentally and mathematically. The accuracy of the
model to identify this fault is verified with the help of the measured frequency responses for different
connection schemes, i.e., end-to-end open-circuit, end-to-end short-circuit, capacitive inter-winding and
inductive inter-winding.
In [84], a 1 MVA transformer disc winding consisting of 60 discs is used to study whether the FEM model
is able to represent the behaviour of large windings in a frequency response form. In addition, disc space
variation is analysed experimentally and mathematically. The accuracy of the model to identify the disc
space variation is verified with the help of the measured frequency responses for different connection
schemes.
The following results are verified for the test object using measured data:



The turn-based HF FEM model can represent the frequency responses of the test object up to
approximately 1 MHz.
The proposed model can reliably predict the effect of minor winding deformations.
Numerical indices are evaluated for different levels of axial displacement. The presented results
show that the numerical indices evaluated from simulation agree with the measurements. The
optimized model represents the disc space variation effects on frequency responses accurately.
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
The sensitivity of different connection schemes was analysed, and it was noticed that for the
tested winding, the inter-winding connection schemes have the best sensitivity for detecting a
disc space variation fault.
6.3.5 Summary on algorithms based on white-box models
In general, circuit and FEM models need details of the internal geometry which are usually not available
for all transformers. They are then mostly used by academic institutions to study FRA and by transformer
manufacturers to optimize their winding design for dielectric performance. The 3D FEM approach is
useful to generate data for developing FRA interpretation, for instance simulating the various mechanical
failure modes or studying the effect of the windings’ electrical properties on the frequency response of
transformer.
Some methods estimate the elements from only the terminal measurements. It is possible to extract
physical parameters from the frequency response measurements and compare these numerical values
with a previous reference measurement. This approach does not require a detailed knowledge of the
internal geometry and is thus applicable for general FRA applications.
Circuit models give details about internal physical behaviour which can be used for further interpretation,
namely discriminating the fault and its location. For example, changes in capacitances and mutual
inductances in certain locations can show the fault type and location. The potential of using the output
of such models for cases without reference is shown in some contributions.
Some mechanical changes are difficult to model using circuit models and the precise simulation of the
fault is very important to identify minor winding deformation. The FEM models can overcome this
limitation without solving the complex circuit models, which is the major novelty of this method. However,
they require comparatively significant computation time and memory for 3D calculations.
6.4 Algorithms based on artificial intelligence
6.4.1 General
AI methods include decision trees, genetic algorithms, fuzzy logic and neural networks. Some
researchers use such methods to estimate the parameters of a transformer’s high-frequency model from
data obtained from frequency-response measurements [85]. In other words, AI methods are employed
to build a high-frequency model out of the real measurements. Some other researchers implement AI
techniques for the recognition of frequency-response patterns of the winding to detect any failure of the
internal winding insulation [86]. In this regard, the fault type is also determinable by means of frequency
response classification through AI methods [87-89]. Some other works use AI methods for assessing
the risk of power transformer failures [90].
6.4.2 Decision tree
A decision tree is a decision support tool that uses a tree-like graph or model of decisions and their
possible consequences. One family of decision tree algorithms is called Top-Down Induction on
Decision Trees (TDIDT). It includes algorithms such as CART (Classification and Regression Trees),
CLS (Concept Learning System), ID3 and C4.5.
The components of a decision tree (DT) are shown in Figure 6.9. As can be noted, a DT has a root node
(RN), intermediate nodes, (IN), also called child nodes, splitters and leaves. Any IN can be a RN of a
sub-tree. This leads to the recursive definition of a decision tree. A leaf corresponds to a set of instances
belonging to a single class. The class of the leaf is assigned according to the class corresponding to the
majority of the instances. The leaves represent the automatically extracted concepts.
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Leave
Class j
RN
IN
IN
IN
IN
Leave
Class j
Leave
Class j
Leave
Class j
Figure 6.9 Components of a decision tree [91]
Reference [91, 92] presents a methodology to use decision trees for interpretation of frequency
response traces based on machine learning (ML) algorithms. Table 6.5 presents the classes used in
[92] for training the ML algorithms.
Table 6.5 Description of the training set [92]
Description
Class
N° instances
Healthy winding under the same remanence
condition
A
38
Healthy winding under different remanence
condition
B
75
Short-circuit between turns
C
30
Mechanical deformation
E
31
Different kinds of indicators can be used for condensing information of frequency response data. In this
contribution, the use of correlation coefficients (CC) was considered. Before the calculation of CC
coefficients, the frequency response plots were divided into 5 frequency sub-bands (LF1, LF2, MF, HF1
and HF2), according to the algorithms presented in [92]. Then, in each of these frequency sub-bands,
CC coefficients were calculated for both the amplitude and phase plots. As a result, a total of 10
indicators were extracted. The indicators were numbered from 1 to 10, where 1, 2, 3, 4 and 5 correspond
to the CC coefficients calculated for the plot of the amplitude in the frequency sub-bands LF1, LF2, MF,
HF1 and HF2, respectively. In a similar way, indicators 6, 7, 8, 9 and 10 correspond to the CC
coefficients calculated for the plot of the phase in the five frequency sub-bands.
After some trials with different topologies, two independent classifiers were determined: one for the
classification of low frequency failure modes and the other for the classification of high frequency failure
modes. The performance against cross-validation is good, as 82% of the instances were correctly
classified [92].
6.4.3 Neural networks
AI methods have been of great interest lately for different applications, namely classification problems.
Using these methods, it is possible to classify the changes in the frequency response traces and connect
them to different fault types and even fault locations. Here, a brief explanation is given of artificial neural
networks (ANN), one the most common methods for the classification purpose. As Figure 6.10 shows,
an ANN problem consists of neurons which are divided into three groups: inputs, outputs, and hidden
layers. In an ANN problem, the data are fed to the input neurons. Then, the algorithm calculates the
output of each neuron, which is a function of its inputs and some weight factors:
Output of hidden layers h j  f (
Output layer  f (
W jk hj)
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Figure 6.10 Structure of a three-layer ANN system [93]
The function can be a simple threshold function which has two outputs, but other functions can also be
used. The Sigmoid function is a typical function for this purpose. Finally, the classification is carried out
based on the output neurons. The ANN also uses methods to automatically optimize the weight factors
to minimize the error function between the output and the desired result.
Reference [94] presents a methodology based on ANNs to estimate the parameters of a transformer
high-frequency model using data obtained from frequency-response measurements as one of the ANN
applications. One ANN-based technique is suggested in [86] for the recognition of winding frequencyresponse patterns to detect any failure of the internal winding insulation. Similarly, the type of a fault is
determined in [87] using frequency response analysis by applying the ANN method [48]. Its conclusions,
however, emphasize the necessity for further investigations to classify the severity and location of the
fault. As another example of AI methods, an application of the genetic algorithm related to frequency
response is given in [88]. The genetic algorithm is used for fault diagnosis of power transformers and
also employs methods other than the frequency response method [89].
In [95], an ANN-based method is proposed to detect and locate two types of mechanical defects, i.e.
axial displacement and radial deformation. The investigators create a detailed model of a real
transformer winding that takes into consideration its geometrical characteristics. Thereafter, the
parameters of the detailed model are calculated following different mechanical defects and the
frequency responses are obtained for each case. In the next step, different features are used to train an
ANN as the regression method. It is stated in [95] that it is possible to detect the type, extent and location
of mechanical defects in transformer windings.
In [47], a method for simultaneous detection and location of electrical short-circuit faults and mechanical
defects is presented. Similar to [95], the detailed model of a real transformer winding is employed. Then,
different features are selected from the obtained signals to train an ANN as the classifier. Reference
[47] shows that it is possible to simultaneously detect the defect type and also locate axial displacement,
radial deformation and disc-to-disc short-circuit faults in the transformer winding with a good accuracy.
In [47, 95], cross-correlation and the indices ED, AID, CC, SSE, and SSRE are used as the features to
train the ANN. These references report 98% accuracy for fault type detection and 95% accuracy for
locating the fault. They also indicate that the algorithm performs poorly for radial deformations.
Reference [36] employs Fa and Ff to train a support vector machine for fault classification and shows
that these indices lead to reliable fault type detection. Similarly, this method is used in [36] to distinguish
and classify the faults under one of the axial displacement, radial deformation, disc space variation or
short-circuit types. For this purpose, a support vector machine is used. The results of the proposed
method are verified against the results of past works by application of transformers of different sizes. A
pattern-based method is also suggested in [96] for classification of short-circuit faults in a distribution
transformer using the graphical information of its winding frequency response. Its conclusions, however,
emphasize the necessity for further investigations to classify other winding faults such as axial
displacement and radial deformation.
In [93], an ANN based approach is proposed, that considers the value given by each of nine numerical
indicators and gives an output that would help to identify deviations in frequency response plots. The
indices employed in [93] comprise: correlation coefficient (CC), mean square error (MSE), Sum squared
ratio error (SSRE), Sum squared Min-Max ratio error (SSMMRE), absolute sum of logarithmic error
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(ASLE), absolute difference (DABS), min-max ratio (MM), comparative standard deviation (CSD), root
mean square error (RMSE) which have been introduced in [51, 57, 97-102]. An ANN-based multilayer
feed-forward neural network using a back-propagation algorithm is proposed. The proposed algorithm
was trained and tested with actual field data. The results show that the algorithm can recognize the
different deviations, whether minor, moderate or significant.
6.4.4 Summary on algorithms based on artificial intelligence
There are several AI algorithms, and each one has its own characteristics. This method can use a
combination of several indices for the classification. It can also employ the outputs of the circuit model
for comparison, thereby combining the previous methods.
Because it is by nature an estimate, two runs of this method on the same case can produce different
output. A database, not generally available, is required to train the models. Some contributions propose
using the outputs of circuit models as a database. However, the applicability of these models for a
transformer fleet is itself an issue. Each algorithm can have different numbers of inputs, outputs, middle
layers, and functions, and there is no unique approach for them in the literature.
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7.
Evaluation of selected numerical indices
Leaders: Nilanga Abeywickrama (SE) and Mehran Tahir (DE)
7.1 Introduction
To expand the understanding of numerical indices, this chapter provides a literature review of the
experience with numerical indices and evaluates all the candidate numerical indices on a single platform
to provide a deeper understanding of their characteristics for the assessment of transformer frequency
response signatures. The numerical indices analysed here are calculated directly from the trace data
values (amplitude and phase) as they are the easiest to implement.
In the analysis presented in this chapter, it is assumed that the frequency response measurements are
free from measurement system or other undesirable influences discussed in Chapter 4, which is an
important preceding step to avoid misinterpretation of frequency responses.
7.2 Evaluation of the numerical indices
7.2.1 Experimental studies
Reference [103] provides a comprehensive study to assess the sensitivity of the numerical indices for
different faults. The study collected all the indices and compared and categorised them based on their
characteristics onto a single platform. The authors evaluated the numerical indices using six case
studies from an experimental setup (Figure 7.1) and a moulded transformer (Figure 7.2). They based
their evaluation on the following criteria: monotonicity, linearity and sensitivity. The case studies are
summarized in Table 7.1. The definition of each criteria and the evaluation of all the numerical indices
for different severities of mechanical faults in all cases, are presented in the following sections.
Table 7.1 Case studies for index evaluation
Case
1
2
3
4
5
6
Description
1 MVA distribution transformer windings
10 steps of axial displacements, each step 5 mm
1 MVA distribution transformer windings
5 steps of axial displacement, each step 1 mm
1 MVA distribution transformer windings
5 steps of radial deformation, each step 2.5 mm
1 MVA distribution transformer windings
5 steps of disc space variation, disc 2-3, each step
1 mm
1 MVA distribution transformer windings
5 steps of disc space variation, disc 3-4, each step
1 mm
160kVA, 10.5kV/400V cast-resin transformer
5 steps of axial displacement, each step 2 mm
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(a) Axial displacement
(b) Radial deformation
(c) Disc space variation
Figure 7.1 Experimental setup of 1 MVA distribution transformer windings [103]
Figure 7.2 Cast-resin transformer [103]
7.2.2 Monotonicity
Monotonicity is a property which characterises the increasing and decreasing behaviour of an index.
For a monotonic index, the value of the index always increases with an increasing difference between
two curves. A good numerical index should exhibit monotonic behaviour against increasing severity of
mechanical faults. Usually more severe faults increase the deviations in the frequency response traces
and consequently an index should give higher values for larger deviations. Conversely, if an index is
non-monotonic, it cannot be used to define the extent of a mechanical fault or to quantify the same.
Typical behaviours of some monotonic and non-monotonic indices are shown in Figure 7.3.
The detailed results of monotonicity are presented in Table 7.2. The first six cases are used to check
monotonicity by evaluating indices in the 10 kHz to 1 MHz frequency range, and four connection
schemes (open-circuit, short-circuit, capacitive inter-winging and inductive inter-winding) are
considered. In Table 7.2, a small tick () denotes a monotonic behaviour of an index while a small cross
(x) represents a non-monotonic behaviour. The result shows that different numerical indices have
different behaviours in different cases. A good index should hold monotonic behaviour in all the cases,
even in all connection schemes. Consequently, all the indices that showed non-monotonic behaviour in
one of more cases are discarded from the list of appropriate numerical indices. Under this criterion, most
of the indices including CCF, CC, LCC, SD, CSD, ED, RMSE, ALSE, SE, SSE, and JD are purely
monotonic, and these indices will be evaluated using the next criterion.
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Figure 7.3 Monotonic and non-monotonic behaviour of indices against different extents of radial
deformation fault
Table 7.2 Monotonicity of the numerical indices
7.2.3 Linearity
Linearity is a property of a mathematical relationship that allows it to be graphically represented as a
straight line. To understand the linearity of the numerical indices, the indices are normalized and plotted
in a single graph as shown in Figure 7.4. From a linearity point of view, different numerical indices have
different behaviours against each fault step.
To quantify the linearity of numerical indices, a regression analysis was performed. This statistical
method determines the degree to which two variables are linearly coupled. The fault steps and the
normalized values of the numerical indices are taken as input variables. The output of the regression
analysis is the coefficient of determination that ranges from 0 to 1. The value of coefficient of
determination equal to 1 indicates a complete linear relation. The results of the linearity check are
presented in Figure 7.5. The coefficient of determination reported here is the average of all six cases.
The result shows that indices ASLE, LCC, CCF, ED, SD, CSD, RMSE, SDA, MM, SE and JD exhibit
good linear behaviour while indices CC, SSE, SSD and LSE are weaker in this respect, even if the
linearity coefficient exceeds 0.9. It is important to note that indices such as ED, SD, CSD and SE are
actually the distances between two FRA traces and these have a linear relationship with the fault
severity. However, indices such as SSE and LSE are in fact the square of the distance between FRA
traces and they exhibit a parabolic characteristic as shown in Figure 7.4. This parabolic behaviour makes
them less linear against fault steps.
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Figure 7.4 Behaviour of indices with different levels of axial displacement
Figure 7.5 Linearity of the indices
7.2.4 Sensitivity
According to the principle of frequency response measurement, both frequency and amplitude can
change with the change of the impedance of the transformer. Therefore, an appropriate index should
be sensitive to both the frequency and amplitude shifts. Reference [40] explains that CC has some
deficiencies in this respect, and goes on to derive SSRE and SSMMRE from SSE. However, they
demonstrate weak sensitivity to variations around the trough points as shown in Figure 7.6. To evaluate
the sensitivity of numerical indices against frequency and amplitude shifts, a frequency response is
calculated from a simple RLC series circuit in which horizontal and vertical shifts are applied by varying
the values of circuit elements.
Figure 7.6 Large amplitude variation around some anti-resonance points in frequency response of a
34 MVA, 237/5.65 kV transformer
At first, the resonance frequency is reduced by 1%, 3%, 5%, 10% and 20% by varying the inductance
(L) in the circuit. Secondly, the amplitude of the frequency response is reduced by 1%, 3%, 5%, 10%
and 20% by varying the resistance (R) of the series circuit. The results are demonstrated in Figure 7.7.
To explain the horizontal and vertical sensitivities of the indices, the values of three indices (CC, CCF
and LCC) are compared against different steps of frequency shifts and amplitude changes as shown in
Figure 7.8. It can be seen that each index has a different ability to detect frequency and amplitude
changes. Moreover, it is important to note that indices are less sensitive in detecting amplitude changes.
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To compare and quantify the sensitivities of all the numerical indices, the vertical sensitivity of each
index is plotted against its horizontal sensitivity in one chart as shown in Figure 7.9. Moreover, these
sensitivities also depend on the frequency band. To extend the discussion of sensitivities of indices in
relation to evaluated frequency band, the horizontal and vertical sensitivities are compared for wide (10–
100 kHz) and short (20–50 kHz) frequency bands, as shown in Figure 7.9.
The results show that the numerical indices CSD, ED, LCC, SE, SDA, SD and RMSE possess good
sensitivity to detect both frequency shifts and amplitude changes. In contrast, the indices CCF (used in
Chinese Standard), CC, ASLE and SSE have low sensitivities to detect amplitude changes. It is
important to note that sensitivities of all the indices are improved by narrowing the evaluated frequency
band except ASLE and RMSE.
(a) Percentage shift of resonance frequency
(Horizontal shift)
(b) Percentage change in amplitude (Vertical shift)
Value of index
Value of index
Figure 7.7 Frequency responses of RLC circuit for discussion of horizontal and vertical sensitivities
(a)
(b)
Values of normalized indices
Figure 7.8 Comparison of CC, CCF and LCC against (a) frequency shifts and (b) amplitude changes
Figure 7.9 Comparison of horizontal and vertical sensitivities of indices for two frequency bands: 10–
100 kHz and 20–50 kHz
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7.2.5 Dependency on number of data points
The influence of the number of data points in the measured response is another important aspect of the
application of the numerical indices that should not be overlooked. In [104], authors investigate the effect
of comparing frequency responses obtained with different data size.
In the international standard IEC 60076-18 [2], the data size is not specified in detail, only the minimum
number of points is defined, which is 200 points per decade. In this criterion, the performance of the
indices is evaluated for different data sizes. The data size is changed from 100 to 400 points per decade
and the measurement with 200 points per decade is chosen as the reference. For this purpose, the
frequency response results from a cast resin transformer (case 6) are employed, where response is
measured from 10 Hz to 1 MHz. Initially, the frequency responses are measured for different data size
for the healthy state of the windings. Subsequently, the measurements are repeated when HV winding
is displaced up to 8 mm.
Table 7.3 shows the percentage change of the indices due to different data size. It can be seen that
data size has a huge impact on indices ED, JD, RMSE, and SSE. Consequently, these should not be
used to define the extent of a mechanical fault when measurements are performed with different number
of points. The indices CC, CCF, LCC, SDA and SE have the least influence of the data size. Hence,
these indices can be satisfactorily used to determine the extent of change when measurements are
performed with different data sizes.
Table 7.3 Effect of number of data points
Points per
decade
200
400
300
100
Points per
decade
200
400
300
100
Data size
1000
2000
1500
500
Data size
1000
2000
1500
500
∆CC
(%)
--0.007
-0.008
0.006
∆CCF
(%)
--0.015
-0.016
0.011
∆LCC
(%)
--0.015
-0.016
0.011
∆SD
(%)
-7.66
8.32
-5.91
∆CSD
(%)
-7.65
8.31
-5.86
∆ED
(%)
--30.61
-12.30
25.14
∆JD
(%)
-14.70
15.92
-12.06
∆SDA
(%)
-3.85
3.24
0.90
∆RMSE
(%)
-7.64
8.30
-5.85
∆SE
(%)
-4.83
2.23
-2.56
∆ASLE
(%)
-3.79
1.14
-2.7
∆SSE
(%)
-14.69
15.91
-12.05
7.3 Discussion
The results show that different indices behave differently within each evaluation criterion (monotonicity,
linearity, sensitivity and data size dependency). Looking back at Table 7.2, it is clear that most of the
indices are monotonic, which allows us to draw conclusions on the extent of change between two
frequency response traces. From the linearity perspective, almost all monotonic indices show a good
linearity; only four indices are weaker in this aspect. As for sensitivity, the indices CSD, ED, LCC, SE,
SDA, SD and RMSE stand out from others in terms of showing good sensitivity to detect both frequency
shift and amplitude changes. From a data size dependency viewpoint, the indices CCF, LCC, SDA and
SE are the least influenced by data size.
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8. Comparative analysis of selected numerical
indices using case studies
Leaders: Nilanga Abeywickrama (SE) and Mehran Tahir (DE)
8.1 Selection of cases and type of frequency response measurements
Selecting cases and frequency response types for a fair and comprehensible comparison among the
indices is challenging and hence deserves discussion.
The frequency sub-bands (single or multiple) used by the research community, which are summarized
in Table 6.1 (page 68), cannot be generalized or applied to any type of frequency response. Selecting
the lower frequency band is particularly challenging whenever an open-circuit frequency response is
involved to avoid the influence of the core. This influence may extend up as high as 10 kHz (see Figure
2.6 in page 17) when the high-voltage side open-circuit measurements are performed between the
phase terminals due to an inaccessible neutral terminal or a delta-connected high-voltage winding.
Furthermore, depending on the size of the transformer, the core resonances may extend beyond 1 kHz.
Therefore, a general low frequency sub-band extending below 10 kHz cannot be used for such cases
and, to make matters worse, phase-to-phase frequency responses are often associated with very high
impedance in the low frequency range (as low as −110 dB). These can produce very noisy
measurements as the instruments are at or close to the limits of their dynamic range. The cases provided
with phase-to-phase measurements are therefore discarded for the purposes of using the low frequency
band in the index comparison among the cases.
Comparing the index values of open and short-circuit frequency responses in the low frequency range
is not possible because of quite dissimilar responses caused by influence of the magnetic core on the
open-circuit impedance. It is, however, possible to compare the index values of open-circuit and shortcircuit frequency responses in mid and high frequency bands. For this study, the index analysis was
based on the open-circuit measurements.
The kinds of faults detectable by FRA do not usually change the frequency responses of the primary
and secondary sides of each phase in the same way. For example, the axial deformation of a winding
may affect its open or short-circuit response the most, while buckling on a low-voltage winding could be
seen on the short-circuit response of both low and high-voltage windings. Thus, as it is not possible to
consider one type of open-circuit frequency response across all the selected cases associated with
various kinds of faults, the most affected type of frequency response of each case is used for the
comparison. The output values of each index are neither within a certain range nor possible to normalize,
which is an obstacle for the cross-comparison of the indices. One conceivable way to provide a
reasonable comparison among the indices is to look at the ratio between the index values on a
(relatively) unaffected phase/winding and the most affected phase/winding.
Within the limitations presented above (i.e., no phase-to-phase measurement and at least one
phase/winding unaffected by the fault), the following cases from the WG case studies database are
selected to demonstrate the performance of the numerical indices. The phase/winding most affected by
the fault and the relatively unaffected phase/winding used for the comparison are mentioned in Table
8.1.
Table 8.1 Selected cases for index analysis
Type of frequency response used for index analysis
Description
Affected phase/winding
unaffected phase/winding
Case 2 Buckling of the inner LV winding
LV open-circuit, phase B
LV open-circuit, phase C
Case 5 Twisting and loss of clamping
HV open-circuit, phase A
HV open-circuit, phase B
Case 6 Axial movement of the LV winding
HV open-circuit
LV open-circuit
Case 10 Successful short-circuit test
HV open-circuit, phase 1
HV open-circuit, phase 2
Case 11 Successful short-circuit test
HV open-circuit, phase 3
HV open-circuit, phase 1
Case 14 Slight displacement of leads showing
possible twisting of the winding
HV open-circuit, phase 3
HV open-circuit, phase 1
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All selected cases except Case 2 (associated with a three-phase bank and a spare unit, which can be
considered as sister transformers) were accompanied by reference frequency response measurements
performed on the same transformer at an earlier date. These serve as references for the index analysis.
For Case 2, indices are calculated considering the spare sister unit of the three-phase bank as the
reference response. There is no fault in Cases 10 and 11, and the ratio is merely taken between two
phases; these cases serve as a reference to show how index analysis performs on a healthy
transformer. The frequency response measurements provided with each case were not performed with
the same instrument. Consequently, the frequency resolution and number of measurement points per
decade are not the same among the cases, which may influence the index values even if they are
normalized to the number of points in each frequency band.
8.2 Selection of indices
According to the evaluation of different indices provided in Sections 7.2.2 to 7.2.5 in terms of their
monotonicity, linearity and sensitivity to frequency response changes, some indices (CD, MD, E, etc.)
show very non-monotonic behaviour against most kinds of faults mentioned in Table 7.1. However a
few like SSRE are only non-monotonic in specific frequency response types (e.g., short-circuit
measurement). In terms of linearity criteria, all the indices with square or square root terms, such as
SSRE and SSE, inherently fall into the non-linear category, which makes it difficult to set threshold
values on them. However, their sensitivity to increasing severity of faults is high and therefore can be
an advantage. Traditional correlation-based indices like CCF are insensitive to amplitude changes,
whereas LCC overcomes it quite significantly. According to the analysis presented in Section 7.2.5,
values of some indices (e.g., RMSE) are influenced by the number of data points in the frequency
response.
The following set of indices representing different performance categories are selected to demonstrate
their ability to differentiate affected and unaffected winding/phase by different kinds of faults in each
case study:







SDA – monotonic in all types of simulated faults in Table 7.1, linear and possesses a good
sensitivity to both amplitude and frequency changes
RMSE – monotonic, linear and shows good sensitivity to both amplitude and frequency changes
SE – similar to ASLE, monotonic, linear and shows a good sensitivity to both amplitude and
frequency changes
SSE – monotonic and non-linear due to squaring of SE
SSRE – non-monotonic behaviour only in short-circuit measurement and non-linearity
behaviour similar to SSE
1-CCF – monotonic, linear and exhibits low sensitivity to amplitude changes
1-LCC – monotonic, linear and good sensitivity to both amplitude and frequency changes
8.3 Index analysis results
For the index analysis on the case studies presented in this chapter, 1–10 kHz, 10–500 kHz and
500 kHz–1 MHz sub-bands are used.
Index calculation is performed on affected and unaffected winding/phase frequency response (with
respect to its reference) separately and then the ratio between the two values are defined as the index
ratio. All the analysis results including index values of each frequency band of relevant frequency
responses are summarised in Table 8.2 and Table 8.3. Only the ratio of the same index values between
the affected and unaffected phase/winding is depicted in
Figure 8.1 to Figure 8.3. As already stated, the index ratio is the only conceivable way of comparing the
performance of each index on the same case, as there is no other simple way to normalize them.
The relative change of each index between the affected and unaffected windings indicates that some
are better (higher ratio) than others at identifying different types of faults. SE (ASLE) turns out to be the
weakest compared to others in exposing the frequency response deviations across all the presented
cases. SDA and RMSE exhibit moderate performance. SSRE seems to exaggerate the small deviations
in the 1 – 10 kHz frequency band. Sensitivity of SSE resembles that of SSRE but does not inflate small
deviations in the low frequency band. 1-LCC and 1-CCF perform quite similarly, providing the best
differentiation between affected and unaffected phase/winding, however the ratio can be quite high
compared to other index ratios due to very high correlation in the unaffected phases making the index
value of 1-LCC and 1-CCF closer to zero. That is the reason for very high ratio values in the high
frequency band of Case 5 and the middle frequency band of Case 6. The alleged better performance of
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LCC over CCF cannot be demonstrated, as the faults associated in the selected cases influence both
the resonant frequencies and amplitude.
Table 8.2 Results of the index analysis on the affected phase/winding by the faults presented in six
selected case studies
Low frequency band: 1 - 10 kHz
Case
SSE
SE(ASLE)
1-CCF
1-LCC
RMSE
SSRE
SDA
CSD
Case 2
1.2E+0
1.0E+0
7.1E-3
16.3E-3
110.6E-3
59.3E-3
210.1E-3
808.8E-3
Case 5
8.6E-3
39.9E-3
93.9E-6
119.5E-6
3.0E-3
20.8E-6
2.8E-3
88.6E-3
Case 6
6.9E-3
62.2E-3
107.8E-6
155.6E-6
9.3E-3
135.3E-6
9.6E-3
70.1E-3
Case 10
72.5E-3
218.1E-3
113.2E-6
411.1E-6
5.6E-3
27.3E-6
3.8E-3
170.7E-3
Case 11
844.1E-3
761.7E-3
3.3E-3
3.6E-3
15.5E-3
270.1E-6
12.0E-3
905.7E-3
Case 14
644.9E-3
626.7E-3
1.8E-3
3.2E-3
13.1E-3
216.7E-6
8.2E-3
606.3E-3
Medium frequency band: 10 - 500 kHz
Case 2
19.7E+0
3.6E+0
283.6E-3
289.8E-3
319.6E-3
99.9E-3
302.0E-3
4.4E+0
Case 5
1.0E+0
564.7E-3
11.0E-3
11.1E-3
29.0E-3
767.9E-6
41.8E-3
1.0E+0
Case 6
1.3E+0
817.8E-3
69.2E-3
76.6E-3
62.7E-3
5.7E-3
54.8E-3
1.1E+0
Case 10
24.5E-3
132.6E-3
163.5E-6
221.4E-6
3.5E-3
12.9E-6
3.7E-3
141.9E-3
Case 11
184.8E-3
317.3E-3
1.3E-3
1.4E-3
10.4E-3
111.3E-6
10.0E-3
416.8E-3
Case 14
112.9E-3
310.8E-3
129.2E-6
870.5E-6
8.4E-3
86.4E-6
10.7E-3
132.1E-3
High frequency band: 500 kHz - 1 MHz
Case 2
59.3E+0
5.2E+0
963.8E-3
964.2E-3
880.2E-3
744.4E-3
596.4E-3
7.7E+0
Case 5
17.3E+0
2.6E+0
254.2E-3
258.2E-3
136.4E-3
13.3E-3
84.3E-3
4.2E+0
Case 6
585.0E-3
623.4E-3
57.7E-3
74.8E-3
35.3E-3
1.2E-3
28.0E-3
696.2E-3
Case 10
16.5E-3
115.5E-3
18.6E-3
19.0E-3
4.5E-3
19.6E-6
4.1E-3
129.5E-3
Case 11
2.4E+0
846.1E-3
44.9E-3
45.2E-3
47.7E-3
1.5E-3
28.5E-3
1.6E+0
Case 14
1.1E+0
701.5E-3
24.5E-3
32.0E-3
36.4E-3
894.5E-6
26.5E-3
1.0E+0
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Table 8.3 Results of the index analysis on the unaffected phase/winding by the faults presented in six
selected case studies
Low frequency band: 1 - 10 kHz
Case
Case 2
SSE
1.9E+0
SE(ASLE)
1.3E+0
1-CCF
4.6E-3
1-LCC
RMSE
SSRE
SDA
CSD
27.0E-3
140.3E-3
47.9E-3
201.8E-3
841.3E-3
Case 5
5.6E-3
42.7E-3
59.6E-6
79.6E-6
2.4E-3
5.8E-6
2.9E-3
68.3E-3
Case 6
169.3E-3
321.3E-3
519.3E-6
917.2E-6
8.7E-3
64.5E-6
8.5E-3
316.6E-3
Case 10
24.9E-3
116.2E-3
52.7E-6
147.6E-6
3.3E-3
8.8E-6
2.0E-3
107.7E-3
Case 11
740.5E-3
674.9E-3
3.5E-3
3.7E-3
14.9E-3
275.7E-6
9.3E-3
846.2E-3
Case 14
633.3E-3
579.8E-3
2.9E-3
3.5E-3
13.3E-3
215.3E-6
6.6E-3
732.6E-3
Medium frequency band: 10 - 500 kHz
Case 2
4.3E+0
1.8E+0
57.3E-3
60.2E-3
148.5E-3
23.5E-3
134.2E-3
2.1E+0
Case 5
129.6E-3
258.7E-3
1.5E-3
1.5E-3
10.3E-3
91.2E-6
8.9E-3
358.4E-3
Case 6
244.0E-3
326.1E-3
1.7E-3
1.7E-3
12.3E-3
135.7E-6
7.8E-3
490.0E-3
Case 10
19.1E-3
120.7E-3
110.0E-6
160.8E-6
3.1E-3
10.4E-6
3.4E-3
118.3E-3
Case 11
131.9E-3
250.8E-3
1.3E-3
1.3E-3
9.1E-3
83.9E-6
8.5E-3
360.6E-3
Case 14
36.2E-3
165.4E-3
93.5E-6
306.4E-6
4.9E-3
30.9E-6
6.3E-3
110.0E-3
High frequency band: 500 kHz - 1 MHz
Case 2
16.6E+0
2.7E+0
284.9E-3
289.8E-3
465.8E-3
147.9E-3
310.3E-3
4.1E+0
Case 5
593.4E-3
518.1E-3
7.4E-3
8.3E-3
24.6E-3
463.4E-6
16.0E-3
747.2E-3
Case 6
156.1E-3
372.4E-3
47.2E-6
7.9E-3
18.9E-3
476.5E-6
19.5E-3
132.6E-3
Case 10
103.6E-3
245.4E-3
42.6E-3
93.6E-3
11.6E-3
139.8E-6
10.1E-3
264.0E-3
Case 11
1.3E+0
794.6E-3
57.5E-3
59.2E-3
35.0E-3
1.1E-3
27.4E-3
1.2E+0
Case 14
111.1E-3
301.5E-3
1.1E-3
3.1E-3
12.8E-3
152.5E-6
11.2E-3
302.3E-3
Figure 8.1 Index ratio of low frequency band (1–10 kHz) between affected and unaffected phase/winding
by the fault in each case study
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Figure 8.2 Index ratio of mid frequency band (10–500 kHz) between affected and unaffected
phase/winding by the fault in each case study
Figure 8.3 Index ratio of high frequency band (500 kHz–1 MHz) between the affected and unaffected
phase/winding by the fault in each case study
8.4 Application example: vector based method using a sliding window
To demonstrate the performance of the vector based method using a sliding window presented in
Section 6.2.6, nine cases from Chapter 5 are selected. Table 8.4 summarizes the traces used for this
investigation.
The frequency response measurements of the selected cases along with the calculated SDD (see
Equation 6.36 on page 76) are shown in Figure 8.4 to Figure 8.12. For fault assessment, the minimum
value of the SDD (MSDD) is also displayed in these graphs. It can be seen that the MSDD can identify
the degree of deviation between frequency response traces with an increased sensitivity. However,
some ranges should be disregarded during interpretation. As in open-circuit measurements at low
frequency range, the deviation between frequency responses is due to differences in core
magnetisation. These ranges are indicated by an arrow and the symbol ‘##’.
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Table 8.4 Selected cases for application of graphical evaluation method
Description
Case 2 Buckling of the
inner LV winding
Case 5 Twisting and loss
of clamping
Case 6 Axial movement
of the LV winding
Case 9 Successful shortcircuit test
Case 10 Successful
short-circuit test
Case 11 Successful
short-circuit test
Case 12 Deformation
after 3rd SC test
Case 13 Lead
movement, axial collapse
and spiralling
Case 14 slight
displacement
Type of frequency response used for graphical evaluation method
Reference response
Actual response
LV open-circuit, phase B
LV open-circuit, phase C
(Faulted phase)
HV open-circuit, phase A (before)
HV open-circuit, phase A (After)
(Faulted phase)
HV open-circuit, phase B (before)
HV open-circuit, phase B (After)
(Normal phase)
HV open-circuit, phase C (Before)
HV open-circuit, phase C (After)
(Normal phase)
LV short-circuit (Before)
LV short-circuit (After)
(Faulted phase)
HV open-circuit (Before)
HV open-circuit (After)
(Normal phase)
LV open-circuit (Before)
LV open-circuit (Before)
(Normal phase)
HV open-circuit, phase A (Before)
HV open-circuit, phase A (After)
(Normal phase)
LV open-circuit, phase A (Before)
LV open-circuit, phase A (After)
(Normal phase)
HV open-circuit, phase B (Before)
HV open-circuit, phase B (After)
(Normal phase)
LV short-circuit, phase C (After 2nd
LV short-circuit, phase C (Before)
SC test)
(Normal phase)
LV short-circuit, phase C (After 3rd
LV short-circuit, phase C (Before)
SC test)
(Faulted phase)
Common winding, phase C (before)
Common winding, phase C (After)
(Faulted phase)
Tertiary winding, phase C (before)
Tertiary winding, phase C (After)
(Faulted phase)
HV open-circuit, phase C (Before)
HV open-circuit, phase C (After)
(Faulted phase)
Figure 8.4 Application of assessment factor SDD in Case 2 open-circuit measurements on the LV winding
of two identical transformers
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Figure 8.5 Application of assessment factor SDD in Case 5 open-circuit measurements on the series
windings
Figure 8.6 Application of assessment factor SDD in Case 6 LV with short-circuit on HV
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Figure 8.7 Application of assessment factor SDD in Case 9 HV and LV open-circuit measurements before
and after short-circuit test
Figure 8.8 Application of assessment factor SDD in Case 10 HV and LV open-circuit measurements on
phase A
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Figure 8.9 Application of assessment factor SDD in Case 11 HV open-circuit measurements on phase 1
After SC 2
After SC 3
Figure 8.10 Application of assessment factor SDD in Case 12 measurements before and after second and
third short-circuit tests
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Figure 8.11 Application of assessment factor SDD in Case 13 common and tertiary measurements before
and after the short-circuit test
Figure 8.12 Application of assessment factor SDD in Case 14 HV to neutral open-circuit measurements
before and after the short-circuit test
Figure 8.13 shows the minimum values of the SDD factor for each case. SDD indicator can detect all
the faulted cases with good sensitivity. It is worth noting that MSDD has lower values for the cases with
deformations than for those which are normal. Hence, it is possible to set a threshold to the minimum
values of SDD as a diagnostic criterion, disregarding the frequency ranges. In all presented cases a
value of −5 differentiates the deformed cases from the healthy ones regardless of the size and winding
design of the investigated transformers.
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Figure 8.13 Minimum value of winding assessment factor SDD (MSDD) in cases 2 to 14
8.5 Discussion
The results presented in this chapter provide a good basis and guideline for standardizing processes
for FRA interpretation.
In index ratio analysis based on six selected cases, which can be considered as another criterion that
serves as a way of comparison among indices, the indices LCC (or 1-LCC), CCF (or 1-CCF), SSE and
SSRE show high sensitivity to faults in the case studies, i.e., the relative change in index between
healthy and faulty phase/winding.
There are however a number of other points that must be considered and unified before using numerical
indices as the standard method. First, the definition of a standardized method for selecting frequency
bands in different transformers to unify the implementation of indices. Secondly, the number of points in
a frequency band can affect the value of any index and this should be considered for the establishment
of a threshold. The sliding window method resolves the problem of fixed division of frequency sub-bands
by providing the deviation between frequency response signatures as a function of frequency. An
example of this approach using SDD demonstrated a good sensitivity to detect deformed windings on
nine cases selected from the WG database.
Several complementary indices may also be used to detect faults by harnessing the ability of each index
to detect a particular change in the frequency response. A further step towards objective interpretation
would use values of several indices to train an appropriate machine learning algorithm with supervised
learning.
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9.
Conclusion
In conclusion:




The basis for FRA interpretation is the understanding of the frequency response and the factors
that can influence the measurement.
The selection of case studies presented and analysed in this document can be used as
reference examples to show how a mechanical displacement can be detected using FRA.
Academic contributions help complement the limited number of real case studies data with
laboratory investigations and numerical modelling.
Based on academic contributions and an analysis on selected case studies, some indices were
identified as the most promising for further investigation.
The WG recommends that:



CIGRE continues to offer forums for sharing case studies of FRA interpretation (workshops,
preferential subjects, etc.).
The international transformers community uses the most promising indices for further
development of objective FRA interpretation.
Academic institutions continue to support FRA interpretation related research.
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TB 812 - Advances in the interpretation of transformer Frequency Response Analysis (FRA)
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