In Fulfillment of the Requirement for the Degree of Doctor of Philosophy in Science, Engineering and Technology

Distribution and power transformers are essential components of any electrical network, therefore the electrical and mechanical safety of the transformer unit is among the highest concerns of electricity providers. Therefore, for the advancement of the FRA interpretation process, the application of various approaches has been widely discussed including statistical analysis, machine learning, image processing, to name a few.

LIST OF ABBREVIATIONS

FRA ITIC

SD SSE

INTRODUCTION

Problem statement
Motivation
Research objectives
Thesis outline

Apply different severities of the winding short circuit fault across the distribution and current transformers;. The detailed description of the developed physical transformer windings and software models is covered in Chapter 4.

Publications during candidature .1 Journal papers

10.1109/TIA.2021.3118661

10.1109/TPWRD.2019.2909144

Conference papers

10.1109/ICRERA47325.2019.8997076

10.1109/EEEIC.2019.8783252

Complementary journal papers

TRANSFORMER FAULTS AND DIAGNOSTIC TECHNIQUES

Introduction
Transformer fault types and factors
Transformer condition diagnostic techniques

Short circuit impedance (SCI) method
Vibration method
Communication method
Current deviation coefficient (CDC) method

The integrity of the transformer unit may be damaged due to the following internal and external factors. Thermal methods are based on the thermal properties of the active part of the transformer assessed via thermography tests.

Figure 2.1. Transformer faults interconnection

Ultrasonic method
Short circuit impedance and winding stray reactance method
Voltage-current (V-I) locus diagram method
Dissolved Gas Analysis (DGA) method
Transfer function method
FREQUENCY RESPONSE ANALYSIS (FRA)

Introduction
FRA concept

The interpretation of the test results is based on the concentration of the specific gases in ppm (parts per million) in the oil content or the ratio of the different gases (for example ethane/methane) [89]. Based on the type of the injected signal, FRA can be categorized into: (i) Impulse Frequency Response Analysis (IFRA) and (ii) Sweep Frequency Response Analysis (SFRA).

Figure 2.5. Model of the two-port network

FRA interpretation techniques

Visual inspection
Machine learning tools
Statistical analysis
Digital image processing

In [143], learning vector quantization (LVQ) borrowed from artificial neural networks (ANN) is used for transformer insulation damage classification. In addition, in [106] and [150] decision bounds for correlation factor (CCF) and relative factor (RF) for transformer condition classification are defined.

Figure 3.4. FRA signatures of phases A, B, and C with frequency sub-bands illustrated, taken from [133]

FRA Standards .1 DL/T911-2004

CIGRE Working Group A2.26
IEC 60076-18:2012
IEEE C57.149
NCEPRI

The very high frequency subband varies depending on the size of the transformer under test. The practical part of the standard contained actual FRA signatures of the transformers undergoing frequent failures.

Table 1 – Relative factor and the deformation degree Degree of winding deformation Relative factor, R

15) where TF 1i and TF 2i represent transfer function of the fingerprint and the new measurement,

WINDING MODELING

Introduction
Winding RLC model
Self-inductance
Mutual inductance
Turn-to-turn capacitance
Resistance

Disk space variation (DSV)
Disk tilting

SPICE simulation of the mutual coupling between two inductors The complete winding model is shown in Figure 4.6. Disk-to-disk capacitances between turns of the normal and displaced disks during radial deformation faults.

Figure 4.1. Air-core winding with continuously wounded disks Table 4 – Physical parameters of the proposed winding

TRANSFORMER CONDITION CLASSIFICATION METHODOLOGY

Introduction
Statistical analysis

Statistical Indicators (SIs)

Correlation Coefficient (CC)
Euclidean Distance (ED)
Root Mean Square Error (RMSE)
Absolute Sum of Logarithmic Error (ASLE)

Root Mean Square Error (RMSE) is the estimate of the numerical difference between the respective data points of two data sets. The best-case scenario, where two data sets overlap, is represented by the ASLE value equal to 0, while the upper bound for ASLE tends towards +∞, indicating the level of difference between two sets:.

ASLE N

Absolute Difference (DABS)

DABS N

Sum Squared Error (SSE)

SSE N

Comparative standard deviation (CSD)

CSD is an extension of the aforementioned SD, where the deviation of each sample with respect to its mean value is evaluated and used to determine the difference between two sets of data. CSD equal to 0 represents the intact case, where there are no deviations between the two spectra, while the upper bound has no definite limit, as it depends on the level of deviation [173].

CSD Y

Minimum-maximum ratio (MM)
Sum squared ratio error (SSRE)
Sum squared max-min ratio error (SSMMRE)
Normalized correlation coefficient (ρ)

Coherence of SIs
Winding short-circuit emulation

It is understood that when applying the given error condition to FRA data, individual SIs may have conflicting decisions. The transformer winding short circuit scenario is simulated by installing the variable resistance rheostat in parallel with the winding terminals. The worst case scenario corresponds to the complete short circuit of the transformer winding, where all the current coming from the source flows through the rheostat as illustrated in Figure 5.2(b) [110].

Figure 5.1 [180] ©2022 IEEE illustrates the visualized coherence between indices, where

2019 IEEE

Yellow-to-red decision boundary

In principle, the differential relay system of the tested transformer is set to 0.5% of the rated current [176]. Once the measured current value matches the desired value (0.5% of the nominal value), the parallel resistance is recorded and the FRA signature of the given configuration is obtained. Similarly, the decision boundary BY|R is expressed in terms of the critical values of twelve statistical indicators calculated from the FRA data.

SI Green Yellow Red

Classification and Confidence level estimation

Bolstered error estimation

Following the principle of the supported error estimator, the supporting kernel function is centered at each observation and for each working zone the area under the kernel pdf is estimated. Thus, the calculated area under the gain kernel for the given zone represents the confidence level of the test observation belonging to that zone. Consider an example of using the supporting kernels for the classification of the obtained FRA signature.

Figure 5.6. Example of the bolstered resubstitution for linear discriminant analysis, where kernels are expressed with uniform circles, taken from [177]

Green Yellow Red Green Yellow Red

Bootstrap sampling
The FRA measurement of the given scenario is conducted
The 12 SIs are estimated based on the acquired frequency response and the fingerprint signature
The decision boundaries are generated using (63)-(70)
SI-based probabilities C l ( k ) for each label l are estimated using boundaries defined in (69) and (70)
The cumulative decision for each label l is defined via summation of the SI- based confidences and normalization
The working condition of the test object is classified according to the label l with the highest confidence C l

Application of the bootstrapping method for standard deviation estimation was introduced by Efron and Tibshirani [178] and its schematic conceptualization is depicted in Figure 5.9. Consider the application of the bootstrap sampling technique to the FRA data interpretation and classification of the transformer operating condition. The working condition of the test object is classified according to the label l with the highest confidence Cl.

Figure 5.9. Bootstrap sampling scheme taken from [178]

SIMULATION AND PRACTICAL RESULTS, ANALYSIS AND DISCUSSION

Introduction
Case studies: Winding short-circuit

Case 1: 1 kVA distribution transformer

The discussion of the results obtained and comparison with conventional methods is carried out in section 6.4. Transformers are subjected to different levels of winding short-circuit fault by varying the rheostat value according to the size of the test object and its total winding impedance. It is observed that around the first anti-resonance peak at 10 kHz, the frequency response is expected to become flat, which is the result of the winding short circuit.

Figure 6.1. Equipment and test objects: (a) FRA analyzer and 1 kVA open-wounded transformer; (b) 20 kVA dry-type transformer; (c) 40 MVA oil-filled transformer

Green Yellow Red

ASLE 0.2460 0 50 50 DABS 1.0829 0 60 40
SSRE 0.0019 0 0 100 SSMMRE 0.0019 0 0 100
Cumulative 0 52.5 47.5 Assigned Label Yellow

Case 2: 20 kVA distribution transformer

SSE 0.6191 39.18 60.82 0 ASLE 0.1642 0 100 0
SSRE 0.0008 0 60 40 SSMMRE 0.0008 0 70.02 29.98

Case 3: 40 MVA power transformer

ASLE 0.2224 0 70 30 DABS 1.8247 0 0 100
SSRE 0.0018 0 0 100 SSMMRE 0.0018 0 0 100
Cumulative 0 15.0 85.0 Assigned Label Red

Case studies: Mechanical deformation

The visual representation of the predefined boundaries in Figure IEEE and the estimated CC value (0.9996) demonstrate that the given short-circuit scenario does not belong to any boundary, neither green-to-yellow, nor yellow-to-red zone, but instead mostly correspond to the yellow (suspicious) zone. The given CC value is observed to lie in the upper limit of the yellow-to-red boundary, which is numerically reported in 100% confidence level of the given observation being in the red zone. In this section, the classification of the transformer's operating condition and its confidence level is reported using the enhanced error estimation method discussed in Section 5.4.

Figure 6.4. FRA signatures for intact case and short-circuit scenario of 4000 V tap in parallel with 5 kΩ rheostat

Figure 6.8. Illustration of the FRA measurement over the winding software model

Case 1: Disk space variation (DSV) fault

CC-specific confidence levels for Case Study 1 (CC. The joint decision of all 12 indicators should be considered. The SIs evaluated for a given scenario, their corresponding ranking results and confidence levels are summarized in Table 16. Thus, the final assigned label for a given fault condition is green (healthy) operating mode.

Figure 6.9. The FRA signatures of intact and DSV fault scenario

Clearly, the core mass of the current observation, where CC = 0.99745, lies in the red zone, which is why CC reports 100% certainty of belonging to the critical operating state. The CC-specific confidence levels for Case Study 2 (CC The estimated SIs and their respective classification results are reported in Table 17, showing an overall 37.07% and 63.93% confidence to belong to yellow (suspicious) and red ( critical) operating conditions, accordingly Based on the majority voting of the results of the statistical index, the given radial displacement scenario is thus considered to relate to the critical operating condition.

Figure 6.11. The FRA signatures of intact and radial displacement scenario Figure 6.12 displays the Gaussian kernel and corresponding CLs for the CC value estimated from the obtained FRA data

SI SI value Classification result Confidence (%) Green Yellow Red

CONCLUSIONS AND FUTURE WORK

Conclusions
Future work

On-line FRA
Three-phase winding RLC model

The established green-to-yellow (healthy-to-suspicious) and yellow-to-red (suspicious-to-critical) decision limits were applied to different levels of winding short-circuit severity and mechanical deformation. Grid stability and loading must also be considered during FRA line measurements. The detailed RLC model, as the name suggests, was based on distributed winding parameters - resistance, inductance and capacitance.

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APPENDICES

Appendix A. Self- and Mutual inductances calculation

Appendix B. Mutual inductance function