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A multi-layer neural network approach to identification of mechanical damage in power transformer windings Singh, Arvind 2009

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A Multi-Layer Neural Network Approach to Identification of Mechanical Damage in Power Transformer Windings  by ARVIND SINGH B.Sc., The University of The West Indies, 2003 Ma.Sc., The University of British Columbia, 2006  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  March 2009 c Arvind Singh, 2009  Abstract Power transformers are among the most critical of assets for electric utilities in the financial impact that their failure can bring. Asset Managers need to be able to determine the right time for replacement, refurbishment or relocation of these devices, with an increasing degree of confidence, in order to minimize the total cost of operation over the equipments’ life. This has brought a change from scheduled maintenance to condition based monitoring (CBM), where the state of the transformer is continuously monitored to evaluate its working condition. A key method of transformer CBM, which effectively detects mechanical damage to the structure of the transformer windings, is Frequency Response Analysis (FRA). FRA relies on comparison of electrical admittance signatures to determine if the winding has become deformed. One of the major problems it still faces is the interpretation of differences in the signatures. To date, experts are needed to analyse graphs, drawing from experience in order to produce educated guesses as to what the differences in admittance functions denote. However, in the recent past, there has been some headway in programming computer based solutions for the problem of interpretation. The use of Artificial Neural Networks (ANNs) has perhaps been the most promising in this respect. ANNs perform in the same way that human experts do, drawing upon experience to map a change in shape of a signature to a physical change in the winding system. However, one of the major drawbacks of these methods is the large training data-sets required for the neural network to learn. The work reported in this thesis seeks to address this problem by generating training datasets from analytical models of the transformer. Due to the large number of simulations that need to be performed a customized solution method was developed to speed up computations. A combination of back propagation and radial basis function networks were then used to classify the type, location and severity of winding movement. The results showed that the neural network approach was not only accurate but tolerant to high noise levels.  ii  Contents Abstract . . . . . Table of Contents List of Tables . . List of Figures . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  ii iii v vi  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Chapter Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 3  2 Asset Management . . . . . . . . . . . 2.1 Components of an Asset Management 2.2 Major Asset Management Roles . . . 2.3 Asset Management Competencies . . 2.4 The Asset Manager . . . . . . . . . . 3 Power Transformer Failure . . . . . 3.1 Failure Definitions . . . . . . . . . 3.2 Overview of Failure Mechanisms . . 3.3 Major Causes of Power Transformer 3.4 Winding Failure . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . System . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . Failure . . . . .  . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  5 7 8 10 11  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . 18 . 19 . 20 . 22 . 24  4 Transformer Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 Traditional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Frequency Response Analysis (FRA) . . . . . . . . . . . . . . . . . . . . . . 35 5 Overview of Investigation . . . . . . . . . . . . 5.1 Development of Online Monitoring Techniques 5.2 Time Domain Reflectometry (TDR) . . . . . . 5.3 Signature Interpretation . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . 42 . 43 . 43 . 44  6 Transformer modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1 Overview of High Frequency Transformer Models . . . . . . . . . . . . . . . 49 6.2 Modelling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iii  7 Fast Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . 7.1 Traditional EMTP Simulation . . . . . . . . . . . . . . . . . . . . 7.2 Eigen-decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Method 1: Simulation using Direct Capacitance Calculation . . . 7.4 Method 2: Simulation using Direct Inductance Calculation . . . . 7.5 Method 3: Direct Calculation of both Capacitance and Inductance 7.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . 66 . . . . . . 66 . . . . . . 70 . . . . . . 72 . . . . . . 75 Matrices 76 . . . . . . 82 . . . . . . 88  8 Signature Interpretation . . 8.1 Simulation Details . . . . 8.2 Neural Network Classifier 8.3 Results . . . . . . . . . . . 8.4 Summary . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  9 Conclusions and Future Work . . . . . . . . . . 9.1 Contributions to Condition Monitoring of Power 9.2 Other contributions . . . . . . . . . . . . . . . . 9.3 Future Work . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . 90 . 92 . 96 . 104 . 108  . . . . . . . . Transformers . . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  109 109 110 111  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10 APPENDIX A: Parameter Calculation 10.1 Capacitance . . . . . . . . . . . . . . . 10.2 Inductance . . . . . . . . . . . . . . . 10.3 Velocity of Propagation . . . . . . . . 11 APPENDIX B : Reduction Methods 11.1 Reduction to Capacitive Network . . 11.2 Single Transmission Line (STL) and (MSTL) . . . . . . . . . . . . . . . . 11.3 Hybrid Transmission Line (HTL) . . 11.4 Summary . . . . . . . . . . . . . . .  for Overhead Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . .  122 122 126 126  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphase Single Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  127 127  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  12 APPENDIX C: Isolating Transformer Response from External System  iv  132 138 144 145  List of Tables 2.1  Health index and expected transformer lifetime . . . . . . . . . . . . . . . .  16  3.1 3.2  Primary causes of transformer failure . . . . . . . . . . . . . . . . . . . . . . Transformer component failure . . . . . . . . . . . . . . . . . . . . . . . . .  23 25  4.1 4.2 4.3 4.4 4.5  Sensors in Siemens’s power transformer monitoring system . . . Common faults and associated gasses . . . . . . . . . . . . . . . Comparison of SFRA and FRA-LVI according to Tenbohlen and Comparison of SFRA and FRA-LVI according to Britton . . . . Faults detectable by FRA . . . . . . . . . . . . . . . . . . . . .  . . . . .  28 34 37 38 39  6.1  Drawing board data for simulated winding of fig. 6.10 . . . . . . . . . . . . .  56  7.1 7.2 7.3  Overhead for various simulation methods . . . . . . . . . . . . . . . . . . . 83 Important computations inside frequency loop for various simulation methods 83 Time taken to perform simulation in tens of thousands of clock tics . . . . . 84  8.1  Base transformer parameters  . . . . . . . . Ryder . . . . . . . .  . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . .  v  94  List of Figures 2.1 2.2 2.3  Three basic functions of asset management . . . . . . . . . . . . . . . . . . . Three pillars of asset management competency . . . . . . . . . . . . . . . . . Operational role of Asset Manager . . . . . . . . . . . . . . . . . . . . . . . .  9 10 14  3.1 3.2 3.3 3.4 3.5  Overview of physical fault processes . . . . . . . . Effect of severe fault conditions on withstand ability Radial forces due to current surge . . . . . . . . . . Steady state magnetization . . . . . . . . . . . . . . Transient magnetization of winding . . . . . . . . .  20 21 26 27 27  5.1 5.2 5.3 5.4  Acquiring training data for neural network classifier . . . . . . . . . . . . . . Role of trained classifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mismatch between baseline signatures for actual transformer and circuit model Example of trending movement index to compensate for natural variations in frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13  . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  Top view of disk winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . Side view of disk winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-section of disk type winding . . . . . . . . . . . . . . . . . . . . . . . Visualization of capacitance and inductances forming winding structure . . LC mesh formed from multiple turns and layers . . . . . . . . . . . . . . . . Purely capacitive network . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible component network comparison using terminal models. . . . . . . . Synthesized network from admittance matrix fitting . . . . . . . . . . . . . Multiphase transmission line representation of transformer . . . . . . . . . . Drawing board data for simple three turn winding . . . . . . . . . . . . . . Concentric cylinder turn model . . . . . . . . . . . . . . . . . . . . . . . . . Ring windings used for direct inductance calculation . . . . . . . . . . . . . Integral region for inductance calculation ignoring effect of intermediate conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Integral region for inductance calculation with intermediate conductors . . .  vi  45 46 47 48 49 50 50 51 52 52 53 54 55 56 59 60 63 64  7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18  Multi-phase representation of transformer winding . . . . . . . . . . . . . . Hyperbolic-pi representation of multi-phase system . . . . . . . . . . . . . . Matrix representation of trasmission line . . . . . . . . . . . . . . . . . . . . Algorithm for solution using traditional EMTP . . . . . . . . . . . . . . . . Turn arrangement and numbering for 3-disc, 4-turn transformer . . . . . . . Optimized algorithm for solution method using direct capacitance calculation only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized algorithm for solution method using direct inductance calculation only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformer circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized algorithm for method using direct calculation of both inductance and capacitance matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trends in performance for varying number of frequency points and system sizes Speed increases as system size varies . . . . . . . . . . . . . . . . . . . . . . Speed increase as number of frequency points varies . . . . . . . . . . . . . . Trend in average speed increase as number of frequency points varies . . . .  67 68 69 70 72 74 75 77 82 85 86 87 88  Simple LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 System admittance matrix form for 4-layer tranformer . . . . . . . . . . . . 92 Sectionalized transformer winding with important dimensions . . . . . . . . 93 Cluster configuration for parallel simulation . . . . . . . . . . . . . . . . . . 95 Characteristic impedance signatures for radial movement at different winding positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Transadmittance signatures for radial movement at different winding positions 96 Perceptron layers in backpropagation network . . . . . . . . . . . . . . . . . 97 Neuron model for RBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Activation function for radial basis function . . . . . . . . . . . . . . . . . . 99 RBN network architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Network inputs based on difference plot . . . . . . . . . . . . . . . . . . . . . 100 Network inputs based on percentage change plot . . . . . . . . . . . . . . . 101 Multi-layer classfication scheme . . . . . . . . . . . . . . . . . . . . . . . . . 102 Admittance matrix visualization of network responsibilities for a system 4 layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Errors in layer one classification of validation data . . . . . . . . . . . . . . . 105 Error levels for noisy inputs to layer 1 . . . . . . . . . . . . . . . . . . . . . . 105 Errors in layer 2, movement localization . . . . . . . . . . . . . . . . . . . . 106 Error surface for layer 2 with noisy input . . . . . . . . . . . . . . . . . . . 107  vii  8.19 Layer 3 network errors due to noisy inputs . . . . . . . . . . . . . . . . . . . 108 10.1 Conductor over perfectly conducting ground plane . . . . . . . . . . . . . . 122 10.2 Replacement of ground plane with image conductor . . . . . . . . . . . . . . 123 10.3 Equipotential lines around line charge . . . . . . . . . . . . . . . . . . . . . 124 11.1 C network reduction (3 pancakes, 2 turns per pancake) . 11.2 C network reduction (3 pancakes, 7 turns per pancake) . 11.3 C network reduction (3 pancakes, 15 turns per pancake) 11.4 C network reduction (10 pancakes, 2 turns per pancake) 11.5 C network reduction (10 pancakes, 7turns per pancake) 11.6 C network reduction (10 pancakes, 15 turns per pancake) 11.7 C network reduction 20 pancakes, 2 turns per pancake) . 11.8 C network reduction (20 pancakes, 7 turns per pancake) 11.9 C network reduction (20 pancakes, 15 turns per pancake) 11.10STL reduction (3 pancakes, 2 turns per pancake) . . . . 11.11STL reduction (3 pancakes, 7 turns per pancake) . . . . 11.12STL reduction (3 pancakes, 15 turns per pancake) . . . 11.13STL reduction (10pancakes, 2 turns per pancake) . . . . 11.14STL reduction (10 pancakes, 7 turns per pancake) . . . 11.15STL reduction (10 pancakes, 15 turns per pancake) . . . 11.16STL reduction (20pancakes, 2 turns per pancake) . . . . 11.17STL reduction (20 pancakes, 7 turns per pancake) . . . 11.18STL reduction (20pancakes, 15 turns per pancake) . . . 11.19Single winding layer representation using HTL reduction 11.20HTL 1 reduction (3 pancakes, 2 turns per pancake) . . . 11.21HTL 1 reduction (3pancakes, 7 turns per pancake) . . . 11.22HTL 1 reduction (3pancakes, 15 turns per pancake) . . 11.23HTL 1 reduction (10pancakes, 2 turns per pancake) . . 11.24HTL 1 reduction (10pancakes, 7 turns per pancake) . . 11.25HTL 1 reduction (10 pancakes, 15 turns per pancake) . 11.26HTL 1 reduction (20 pancakes, 2 turns per pancake) . . 11.27HTL 1 reduction (20 pancakes, 7 turns per pancake) . . 11.28HTL 1 reduction (20 pancakes, 15 turns per pancake) .  . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  128 128 129 129 130 130 131 131 132 133 134 134 135 135 136 136 137 137 139 139 140 140 141 141 142 142 143 143  12.1 System impedance reflected between measurement points . . . . . . . . . . . 145 12.2 Impedance stabilizing filter on transformer output . . . . . . . . . . . . . . . 146  viii  Chapter 1 Introduction Traditionally, electric utilities were publicly regulated as vertically integrated monopolies responsible for generation, transmission and distribution. Prior to the 1970s, load growth was doubling more than once per decade and equipment was mostly new. Since earnings were growing, utilities could spend liberally to achieve high levels of system performance while minimizing most aspects of risk. Today, the industry is vertically unbundling so that generation, transmission and distribution can be operated as separate businesses. With slow load growth, ageing equipment, declining rate bases, rate freezes and regulatory uncertainty, electric utilities are looking for ways to increase earnings, credit ratings and stock price. Since the transmission and distribution business is asset intensive there has been re-newed interest in applying asset management principles to effectively running utilities. On the technical side of asset management schemes, the monitoring, through sophisticated sensor networks, of critical-asset conditions is of vital importance. This thesis focuses on one such critical asset of the power system, namely the power transformer, more specifically EHV and UHV transformers in the transmission system. These transformers pose perhaps the most difficult life management problem aside from generators. Their large size, expensive cost and long life-times mean that they are not stocked by grid operators. Unexpected failure, however, can severely affect the system operation. The extra burden placed on the remainder of the system can leave the grid susceptible to contingency conditions that would be covered in normal operation. Considering the long lead times involved in the tendering, manufacturing and testing processes, it is extremely important that optimal decisions are made in a timely manner concerning the repair or replacement of damaged units. In addition to making maintenace decisions, condition monitoring allows Asset Managers to making economic decisions concerning the degree to which the transformers can be loaded. By 1  contrast, the unexpected failure of distribution transformers, does not have as severe an impact in terms of service interruption, replacement cost and grid security. The tracking of the condition of these transformers is, therefore, not of major importance. Several diagnostic methods have developed over time as a response to the need for condition assessment. Among these, Dissolved Gas Analysis (DGA) and Frequency Response Analysis (FRA) have emerged as the industry standard tests for assessing the condition of the transformer insulation / oil and the integrity of the winding structure, respectively. This thesis focuses on Transmission Line Diagnostics (TLD), a transfer function method similar to FRA, which has recently been developed at the University of British Columbia, [1]. Both FRA and TLD represent the winding as a transfer function characteristic. Changes in the measured transfer function indicate changes in the winding structure. However, a reliable method to relate measured changes in the transfer function to actual physical changes in the winding is still a major problem. The solution proposed entails building a library of known deformation cases based upon analytical models of the transformer. Neural networks are then employed to extract information pertaining to the location, direction and severity of winding movement. The project therefore can be seen in two distinct parts: 1. Accurate modelling of the transformer. 2. Development of classification schemes. Each of these by itself is inadequate if the other cannot be accomplished. That is, if an accurate model cannot be developed, the classification scheme becomes sub-optimal and if the data features do not allow classification then there is no point to developing an accurate model. That being said, for accurate models to be developed, a test transformer of realistic size, along with facilities to house it would be needed. This cost can run even into the hundreds of thousands of dollars. It would have been unwise, therefore, to embark upon this phase of the project first only to find that the classification stage (which does not require any extra facilities) fails. In addition, accurate models have already been developed in areas such as fast transient modelling and vector fitting analysis. Therefore, the development of classification schemes was undertaken first, and forms the work presented in this thesis while the design, and custom manufacturing of a modular test transformer and associated facilities is planned for future work. Although the thesis does not deal with practical experimentation, it is backed up by four years of experimental work, by the author, in other aspects of the winding diagnostic problem. These experiments have been performed both in  2  the University’s high voltage laboratory and at PowerTech Labs, a subsidiary of BC Hydro [2]. It should be noted that the report does not consider sources of characteristic-signature corruption such as partial discharges, or outlier effects such as electrohydrodynamic movement of oil.  1.1 1.1.1  Chapter Summaries Chapter 2: Asset Management  This chapter locates the present research in the landscape of modern power system operation. An overview of the roles and competencies of an effective asset management scheme is covered. Specific attention is given to the role of the Asset Manager since it is the Asset Manager who is directly responsible for evaluating the condition of critical assets and making timely replacement or maintenance decisions as well as economic asset utilization.  1.1.2  Chapter 3: Power Transformer Failure  Definitions of failure under operational and preventative paradigms are outlined. The economic impact of losing a transformer is illustrated with data from a case study and statistics for the failures by cause and component are given. An overview of failure mechanisms is also provided.  1.1.3  Chapter 4: Transformer Diagnostics  Following from the previous chapter, diagnostic methods which have emerged over time for diagnosing problems in power transformers are outlined with particular attention being paid to FRA. An overview of baseline measurement techniques is given as well as advantages and disadvantages of the impulse and swept frequency methods of measurement.  1.1.4  Chapter 5: Overview of Investigation  An overview of completed and ongoing related work in the field by the author is provided. A description of how the completed system is expected to work in practice as well as forseeable technical challenges are outlined. 3  1.1.5  Chapter 6: Transformer Modelling  This chapter provides a summary of existing high frequency transformer models. In addition, the calculation of the inductance and capacitance matrices for the case of transmission line models of transformers is discussed. Anomalies arising out of modelling assumptions are exposed using a simple example.  1.1.6  Chapter 7: Fast Simulation Methods  Generic circuit simulation software is efficient, however, does not take into consideration the unique characteristics of a transformer winding system. This chapter develops a number of algorithms developed specifically to exploit the special properties of transformer system. Algorithms are developed for each approach to parameter calculation given in Chapter 6.  1.1.7  Chapter 8: Signature Interpretation  This chapter discusses the multi-layer neural network approach used for matching changes in measured impedance characterstics to changes in the physical structure of the tranformer. The separability of the problem, which is an important property of the data for neural networks is illustrated via a simple example. Pre-conditioning of the input data is discussed as well as the choice of networks for each stage. The network performance is tested with and without random changes corrupting the characteristics, which represent both electrical noise and model mismatch.  1.1.8  Chapter 9: Conclusions and Future Work  A review of contributions made in the thesis is presented as well as recommendations for future work in the area.  4  Chapter 2 Asset Management Asset management of power systems has become very much a hot topic over the past few years. This is undoubtedly due to the criticality of physical assets to the operation of the system. However, a survey of literature reveals that there is still wide variation in the understanding of what is meant by the term ”Asset Management”. The following are some quotations related to asset management extracted from recent literature: “It is the ability to understand, and manage the trade-offs between risk, cost and performance; to optimize the financial and commercial performance of an asset base. The key lies in optimizing and balancing the trade-offs across a variety of key financial and non-financial metrics rather than simply attempting to minimize whole life cost or risk... It is a philosophy that must encompass the organization at every level and be accepted within the organization’s mindset... Success in asset management is achieved when a company can successfully integrate the various components and manage the trade-offs between operational performance, risk exposure and financial performance. It takes a combination of people, processes and tools to successfully pull off an asset management transformation.” [3]. “Asset Management’s objective is to make all infrastructure related decisions based on a single set of stakeholder-driven criteria. The payoff is a set of spending decisions capable of delivering the greatest stakeholder value from the investment dollars available... Fundamental to asset management is the trade-off between risk and return. Investors identify acceptable risk and asset management techniques are used to achieve this level of risk for the highest possible return.”[4]. “Asset Management can be defined as a systematic process of, cost effectively, 5  operating, maintaining and upgrading electrical assets by combining engineering practices and economic analysis with sound business practices.”[5] “Asset Management is the process of guiding the acquisition, use and disposal of assets to make the most of their future economic benefit and manage the related risks and cost over their entire lifetime. ”[6]. “Asset Management is the balancing act in which asset performance is balanced against stakeholders’ expectations” [7]. It is also telling that a number of papers ([4] and [8]) make it a point to state, in addition to what asset management comprises, what it does not comprise. Generally, these are aimed at systems or functions that are often touted as Asset management systems but are just portions of the broad scope of true Asset Management. The following list gives some examples of what experts try to distinguish Asset Management from: • Reliability Centred Maintenance (RCM) • Equipment Condition Monitoring (ECM) • Pushing equipment closer to operating limits for longer periods of time • Risk reviews for cancelled projects • Tracking of assets and prioritizion of spending requests • Maintenance Management Systems (MMS) • Work Management Systems (WMS) • Automated Mapping and Facilities Management and Geographic Information Systems (AM/FM/GIS) The major disparity can be explained by considering the two possible meanings of the term ”Asset Management”. The first meaning can be understood by rewording the term to be ”Management based on Assets” and refers to a corporate method of aligning infrastructure decisions with stakeholder-driven criteria (most often based on profit maximization). This requires a corporate culture, business processes and information systems designed to make rigorous and consistent financial decisions based on asset level data, resulting in a mid to long- term investment plan aimed at maximizing shareholder value under the constraints of performance and risk, [4]. The second meaning becomes clear by re-wording the term as ”Management of an Asset”. In this context it often relates to the technical processes of condition assessment and remaining 6  lifetime estimation of specific components in the system. The reason for the dual meaning of the word will become clearer as the components of corporate Asset Management are outlined.  2.1  Components of an Asset Management System  Mohseni, [3], gives the following components as being critical to Asset Management: • An asset investment planning system to optimize asset investment across capital and O&M investment opportunities. • An asset repository that can link the operational information about assets with their financial information so that a true life cycle cost of various assets can be tracked and leveraged. • A network and asset database that will enable better and more efficient systems planning and reliability analysis. • A focused maintenance and inspection system to track asset condition, performance and cost of M&I at the asset level so that not only ”rogue” assets will be identified but also asset conditions are better monitored to maintain the desired reliability level. Considerations in developing such a strategy include: • Business strategy. • Network impact. • Customer impacts. • Environmental influences. • Criticality of the assets in the context of the network. Mohseni further cites the following as being critical infrastructures for achieveing effective asset management. • Asset repository containing asset attributes, asset reliability and unit cost of investment, repair, maintenance and inspection. 7  • A Geographical Information System (GIS) that denotes the geographical location of assets. • An investment planning model that determines the optimal strategy for acquiring an asset to meet load growth and service reliability. • Maintenance and Inspection (M&I) planning model for determining the optimal schedule for M&I work. • The WMS for managing the M&I work if the utility conducts its own M&I work. • The systems which manage and monitor the performance of contractors if the utility decides to outsource the M&I work. • A material management system. • A supply chain management system.  2.2  Major Asset Management Roles  Experts have defined three distinct roles (or functions) that need to be implemented for effective asset management, ([3], [4]): 1. Asset Owner 2. Asset Manager 3. Service Provider While these may seem to form a hierarchal relationship, with Asset Owners at the top making decisions that trickle down to the Service Providers, this is not the case. Brown and Humphrey, [4], provide the following illustration (fig. 2.1) of the interlinked relationship between these three roles.  8  Figure 2.1: Three basic functions of asset management Although fig. 2.1 provides a direct flow from Service Provider to Asset Owner, it is more likely that this interaction is routed through the Asset Managers who can put the raw asset level data in a corporate context.  2.2.1  Asset Owner  Asset Owners are responsible for setting corporate goals of performance, risk and cost for the company. They set the operating context for the Asset Manager. Asset Owners are the regulatory licence holders and focus on corporate governance. They are linked to the Asset Managers through contractual relationships and track performance through Service Level Agreements (SLAs).  2.2.2  Asset Managers  Asset Managers occupy the middle ground between the Asset Owners and Service Providers. They are responsible for translating the corporate goals of the Asset Owners into policies and procedures for the Service Providers as well as to transform the data gathered from the service providers into a form useable by Asset Owners for shaping the future course of the company. Today, they are faced with investment decisions but are also responsible for the much wider issue of reducing the operational cost of ageing assets. 9  2.2.3  Service Providers  Service Providers carry out the procedures laid out by asset managers and provide feedback on actual cost and performance of assets.  2.3  Asset Management Competencies  The business structures supporting asset management have been visualised in different ways. Brown and Humphrey, outline the following as the three pillars of a robust asset management structure (illustrated in fig. 2.2, [4]): 1. Management 2. Engineering 3. Information  Figure 2.2: Three pillars of asset management competency It should be noted that these do not translate directly into the functions of Asset Owner, Manager and Service Provider. This poses a major difficulty as developing expertise in all areas is needed for optimal asset management. For instance, many times, procurement contracts are handled by financial departments of the utilities which result in neglect of the technology life-cycle cost, which can severely impact on operations. 10  2.4  The Asset Manager  From the breakdowns given above it is easy to see how the disparity in the asset management outlooks arise. Many papers which qualify the term ’Asset Management’ with a device, or system, for example, ”Asset Management of Power Transformers” are talking about the job of the Asset Manager in making decisions regarding the course of action that should be followed for those assets, especially when approaching their end of life. In this regard, there is a wide consensus on the job of the Asset Manager as one which revolves around proper maintenance of equipment, [5]. “It is crucial to provide a realistic assessment of consequence of failure for major assets as the effect on the network can be significantly more expensive than the cost of replacement of the element itself ” - [9] “One of the most important questions (of Asset Management), if not the most important is that regarding the replacement or delay thereof, of a piece of equipment.” [9] Asset Managers generally have three courses of action with respect to critical assets. Aside from continuing normal operation, they can choose, [5]: 1. Maintenance • The restoration of equipment to its intended condition and performance through corrective action. 2. Refurbishment • This is a special subset of maintenance which involves the replacement of equipment to comply with current technical practice, safety standards and required operating performance. 3. Asset Strengthening • The expansion or upgrading of plant to improve the capacity or quality (or both) of supply. It is achieved by upgrading or uprating existing equipment to introduce a new stream of income. Maintenance can be further analysed in the following three categories, [6]: 11  1. Corrective and Emergency: • This is applied after a failure has occured or when an imminent failure is detected. It is carried out as soon as possible after the condition is detected. 2. Preventive Maintenance: • This consists in carrying out predefined activities at fixed time intervals on the basis of experience or manufacturer recommendations. 3. Predictive Maintenance: • This provides a quantitative connection between maintenance and reliability. It usually incorporates some type of analysis and assessment and is performed based on predictive estimates of the equipment condition. Data is analyzed from inspections, diagnostic tests or other means of condition monitoring in forming predictions. The type of maintenance strategy employed depends on a number of factors, relating to cost including: 1. Consequence and cost of failure. 2. Cost of replacement / repair. 3. Cost of monitoring and analysis. At first, it may seem like Predictive Maintenance is the most favourable since it has the greatest potential to minimize unscheduled down time. However, this is not the case. In some cases, where failure does not have severe repercussions, it may be more feasible to wait till a device has failed and then replace it, while in other cases it may be of paramount importance to prevent failure and unscheduled downtime as much as possible.  2.4.1  Methodology  There is no set procedure for an asset manager to follow in developing an asset management scheme. However, a very robust methodology that can be smoothly translated to an asset management framework is the disaster mitigation and response framework proposed by the Infrastructure Interdependency Simulator (I2Sim) team at UBC, [10]. The framework has 12  been developed to model multiple infrastructures and their interdependencies for simulation of disaster crises. The solution to the system, as a whole, yields vital information relating to system deficiencies, which could then be strengthened ahead of time. It would also theoretically provide resource dispatchers with a set of near optimal dispatch decisions in a dynamically changing environment. The asset management problem is similar, in that the interdependencies of multiple assets must be considered in order to make maximally profitable decisions. Using this methodology the Asset Manager would need to take the following approach: • Decompose the system into subsystems based on functionality and location. • Identify interdependencies among subsystems. This would allow the Asset Manager to determine how system changes in one component affect the entire system, and more specifically how the profitability of the company is affected. • Identify operational modes for each subsystem. This enables the Asset Manager to determine how much the system can be burdened based on the current state of the subsystems. This is even more crucial for utilities which may be critical to survival in disaster conditions. • Identify known failure modes for each subsystem. This is strongly related to the operational modes stated previously. The operational mode of a subsystem is dependent on its physical state and knowledge of its failure modes enables proper monitoring schemes to be developed to track these states. • Monitoring the condition of subsystems involves the design of sensor networks to detect incipient and evolving degradation conditions. These sensors must be able to survive in harsh industrial environments. This is the true backbone of the asset management scheme which by definition is the decision making process based on asset level data. In this manner the Asset Manager’s job can be visualized in fig.2.3.  13  Figure 2.3: Operational role of Asset Manager  2.4.2  Decision Making  The Asset Manager’s job is not as simple as monitoring the equipment and replacing them when their condition demands it. They often have to ask the following questions, [9]. “Are there any future network changes, which can result in replacement or elimination of the component?” “Can collective replacements be made, due to connected components also reaching their end of life, which would allow cost saving or technical benefit?” “ Are more modern system layouts possible by making collective replacement of multiple components?” These considerations have to be analysed case by case. Uncontrollable forces, such as network changes, environmental policies, etc., may demand that equipment be changed at 14  certain times. In addition, governments may regulate the restrictions applied to reliability and cost of service. These may vary depending on the type of customer, whether industrial or domestic, or depending on the location of the customer, for example, power being exported across regional or national borders may be regulated differently from power meant to be sold locally. In the absence of such constraints, however, replacement is a complex optimization problem that can be stated as: “ To determine the time of replacement that will result in the minimal overall cost and deterioration over the lifetime of the asset.”. This is difficult because cost and deterioration do not share a fixed relationship, although they are often described using a common set of parameters such as: • Energy not supplied due to interruptions. • Need for corrective maintenance. • Need for preventive maintenance. • Operating costs. • Energy losses. • Non-ideal conditions of the asset. • Non-ideal constructions of the asset. • Replacement costs. Regardless of these extra considerations though, maintenance is still the key focus for the Asset Manager. In recent times, the need to exert greater levels of control over the operating condition of primary plant assets, in order to be consistent with greater corporate goals, has led to a shift from traditional, scheduled maintenance schemes, to condition based monitoring. This allows Asset Managers to track the state of equipment and make more informed maintenance decisions on critical assets. It also allows more economic utilisation of the asset by allowing the Asset Manager to determine what are the safest levels to which the devices can be loaded. A major implication of this is that the evolution of CBM technologies, to more closely track the critical states of an asset can give greater insight into the operational health of that asset. This would in turn allow the Asset Manager to take decisions pertaining to asset operation and maintenance in a more timely manner with a greater degree of confidence. The assets that an asset manager is responsible for can be grouped into two classes, [5]. 15  1. Primary Plant Assets • These make up the direct energy delivery system. Their failure results in the loss of supply to the consumer. Overhead lines, power and instrument transformers, high and low voltage switchgear and cables are examples of primary plant assets. 2. Secondary Plant Assets • These consist chiefly of indoor equipment that serve to monitor equipment, and provide operators with a degree of confidence in the state of the primary network. Examples include: telecommunications, protection relays, metering and monitoring equipment. Among primary assets, the power transformer is of special interest because of its long lead time in replacement and cost. Asset Managers gather information on these devices from various condition monitoring systems and develop health indices for the transformers. Table 2.1, [11], illustrates a typical table that Asset Managers might use for assessing the health of transformers. Health Index  Condition  85-100  Very good  70-85  Good  50-70  Fair  30-50  Poor  0-30  Very poor  Expected Lifetime More than 15 years More than 10 years From 3-10 years Less than 3 years Near to end of life  Requirements Normal Maintenance Normal Maintenance Increase diagnostic testing, possible remedial work or replacement needed depending on criticality Start planning process to replace or rebuild considering risk and consequences of failure Immediately assess risk: Replace or rebuild based on assessment  Table 2.1: Health index and expected transformer lifetime In addition to these indices, Asset Managers must take into consideration the following: importance, age, loading and environment [16]. The greater the impact on the system a transformer has, the higher its importance should be rated. This is especially true when there is no back up for it, which is common in the case of large power transformers which are expensive to stock and which deteriorate due to natural ageing, even if not in service. Transformer age is also important in that it gives an indication of the deterioration of components and hence the probability of failure. It should be noted that the time in years 16  since commissioning is not always the best indication of age. Instead, age should be thought of as time till end of life, an indication of which would be given by tables such as table 2.1. Load profiles determine the mechanical stresses experienced by the transformer coil assembly. Profiles which are relatively constant are easier on the mechanical structures compared to profiles with drastic fluctuations such as those for arc furnaces. Environment refers generally to whether the transformer is housed inside a building or is exposed to the elements. Transformers that are inside tend to suffer more from over-heating due to inhibited air flow. The work presented here seeks to aid the Asset Manager to protect and to make best economic use of one of the most expensive pieces of equipment in a high voltage power network by extracting meaningful data relating to the physical state of the power transformer winding from measured transfer function characteristics at high frequencies. It is also of use in performing past history assessments when such assets are being bought or sold. That is, in competitive markets companies may engage in buying and selling used assets to reduce capital costs of new projects or maintenance costs when replacing assets. In such cases, knowledge of the condition of the transformer with respect to winding damage would be useful in negotiating cost of purchase or sale.  17  Chapter 3 Power Transformer Failure Power transformers carry large amounts of power in the transmission system. They are among the most crucial physical assets in a power system in terms of their capital cost, network impact and cost due to unexpected failure and thus warrant close monitoring by Asset Managers. Large power transformers are complex systems, usually constructed from a hundred or more layers of disk-wound copper conductor on a solid ferro-magnetic core. The insulating fluid used to surround the core and winding structure fall into three categories: 1. Dry Type • Air, AramidTM paper, silicone coated fiberglass, NomexTM sleeving and porcelain. These are usually untanked to allow better air-flow. 2. Oil Filled • High-dielectric cooling oil, kraft paper, wood and porcelain. 3. Fluid Filled • SiliconeTM , R-Temp concern.  TM  , AskeralTM , InerteenTM . Used in areas where fire is a  Outages due to power transformer failure cost the company money not only in replacement or repair, but also in buying power from other companies to supply their customers, in environmental clean up, customer and collateral damage and increased insurance premiums. These costs can quickly run into millions of dollars in the space of just a few days. A case study carried out by Pacificorp, [12], estimated the failure of a 520MVA transformer to reach approximately US$18 Million in just 8 days. The following break down was given: Equipment costs: $3.5 Million 18  • Transformers: 3 Million Collateral Equipment damage • Environmental clean up: $0.5 Million Company Losses • Self-insured deductible: $1 Million • Environmental clean up $0.5 Million Replacement Power on Spot market $1.5 Million per day • $100 per MWh spot market price • $30 per MWh continuing production cost • 500MW purchased on the spot market • Total cost to purchase power: 500x(100+30)x24hrs = $1.56 Million per day  3.1 3.1.1  Failure Definitions Operational Failure  A transformer is said to have failed when it is forced out of service by a specific event that results in the inability of the transformer to operate properly under nominal system conditions. Only the present state of the transformer is considered in deciding whether to keep the transformer in service or not. That is, the transformer is considered to be operational as long as it can operate under normal operating conditions.  3.1.2  Preventive Failure  A transformer is said to have failed when it can no longer withstand the fault conditions for which it was originally designed. This means that the transformer may be fine in the operational sense but could be classified as having failed because the next fault condition would cause it to fail. It entails assessing how ”fit” the transformer is and making a decision as to whether it is safe to operate based not only on its present condition but on the probability of catastrophic damage occurring by abnormalities that are likely to happen in the normal course of operation. 19  3.2  Overview of Failure Mechanisms  The behaviour of a transformer depends on the physical state it is in, that is, the arrangement (shape, symmetry etc) of the windings, and the condition of the insulation (both paper and oil). When transformer ratings are specified, they relate to the voltages that the dielectrics can withstand before breakdown and the short circuit forces that can be endured by the structures that hold the windings in place. It follows, therefore, that transformer failures stem from exceeding of the withstand capabilities in these categories. The transformer can be looked at as a combination of three systems: electrical, mechanical and chemical. Figure 3.1 gives an overview of how these interact during over-current conditions.  Figure 3.1: Overview of physical fault processes These processes operate to lower the withstand capability of the transformer, increasing its probability of failure as shown in fig.3.2  20  Figure 3.2: Effect of severe fault conditions on withstand ability In the normal course of operation, the physical state of the transformer is stable. This means that the elastic limits of the clamps and windings have not been exceeded and the dielectrics have not broken down in any area. When a severe fault occurs which causes these 21  elastic limits to be exceeded or the dielectrics to break down, then plastic deformation of windings or chemical changes in the insulation take place respectively. The transformer does not return to its pre-fault physical state, but moves to a new state instead. For example, clamps may be slightly stretched thus reducing the clamping pressure after the fault is cleared or gasses may be liberated in the oil. This new state that the transformer moves to has an altogether different set of ratings associated with it. As long as the new set of ratings is still acceptable then the transformer can be kept in service. If not taken out of service, the transformer can reach an unstable state where new dynamics have to be taken into consideration (e.g. vibrations of the winding if the clamp has been sufficiently loosened). In the normal course of operation, the state may continue to degrade till the transformer completely collapses. Traditionally, as long as the transformer was stable (up to ’n’ faults on fig. 3.2) then it would be kept in service till the next scheduled maintenance time. However, in order to avoid critical failures the transformer has to be taken out of service at the ’nth’ over-current condition, at the latest, in order to prevent the system going to an unstable state. Since the withstand capabilities decrease with the number of severe over-current conditions encountered then the probability of the new withstand being exceeded in a given time increases correspondingly and the time between severe over current events decreases.  3.3  Major Causes of Power Transformer Failure “All transformers fail the same way, all the time” [13].  Tanguay succinctly describes all transformer failures as stemming from insulation breakdown. When the insulation can no longer contain the high-voltage it is supposed to, the transformer fails. The root causes for failure, however, have a bit more variety. He gives table 3.1, [13], detailing primary causes of transformer failure as well as components of the transformer which most often fail.  22  % failure for year Lightning Surges Line Surges / External Short Circuits Poor Workmanship Deterioration of Insulation Overloading Moisture Inadequate Maintenance Sabotage, Malicious Mischief Loose connections Other  1975 32.3 13.6 10.6 10.4 7.7 7.2 6.6 2.6 2.1 6.9  1983 30.2 18.6 7.2 8.7 3.2 6.9 13.1 1.7 2.0 8.4  1998 12.4 21.5 2.9 13 2.4 6.3 11.3 0 6 24.2  Winding Movement Evident YES YES YES NO NO NO YES NO YES —  Table 3.1: Primary causes of transformer failure  3.3.1  Voltage Transients  Voltage waves from lightning and switching surges are both beyond the control of the operator. They have the ability to severely stress the insulation system, leading to insulation breakdown or weakening that may lead to eventual failure.  3.3.2  External Short Circuits  Short circuit currents, or lightning surge currents, cause correspondingly large magnetic fields inside the winding structure of the transformer. These exert massive electromechanical forces on the windings, and the clamps holding them in place. They pose considerable danger as they can result in the complete mechanical collapse of the coil arrangement.  3.3.3  Poor Workmanship  Lax quality control in the manufacturing process can lead to a number of undesirable conditions such as loose brackets which cause winding vibration, non-uniform surfaces which contribute to accelerated insulation breakdown and may lead to internal corona and partial discharge, oil impurity, and short circuits between turns. During maintenance, it is not unheard of for tools to be mistakenly left in the transformer tank. These items may get pulled around the tank by the strong magnetic fields, and oil movement in the transformer under normal operation. When these collide with the windings they can quickly destroy insulation and cause short circuits. This is largely responsible for the infant mortality rate of transformers.  23  3.3.4  Heat and Overloading  These processes act gradually by causing accelerated insulation degradation over a long period of time. They can be thought of as increasing the rate of ageing of the transformer, eventually leading to the cracking and flaking of the insulation. This is exacerbated by the natural expansion and contraction of the structure due to magnetization and demagnetization. Once energized conductor is exposed, the likelihood of arcs and consequently winding failure increases.  3.3.5  Moisture  Moisture enters the transformer through its natural breathing process. This contaminates the cooling fluid and eventually makes its way to the insulation. There it tends to sit and not move back into the cooling fluid, lowering the effective di-electric strength of the insulation. In extreme cases, over-contamination can make the insulation conductive. Failed transformers may sometimes contain gallons of water in the insulation paper.  3.3.6  Insulating Fluid Deterioration  Insulating fluid serves two purposes: firstly as an insulating material between the high voltage carrying conductors comprising the windings and the grounded tank and core, and secondly as a coolant, conducting heat away from the windings which carry considerable current. During normal operation, the insulating fluid is exposed to oxygen, which causes oxidation of the hydrocarbonic fluid. The process is catalyzed by heat, moisture, electrical stress and contact with the copper and aluminium from the windings and tank respectively. Oxidation results in fatty-acid deposits which cause sludge like build up on the windings. This in turn traps heat close to the windings and weakens the tensile strength of the insulating paper.  3.4  Winding Failure  The primary causes of transformer failure have been mentioned. However, they do not necessarily give direct insight into the physical state of the transformer. Although some causes are measureable physical quantities, others such as voltage and current transients are not. Because of this, it is useful to also look at the components associated with failure. Table 3.1 shows that the average failures exhibiting winding movement is 63.5 % over the three 24  years, with 54.1% in the latest year. This is consistent with separate findings by Tanguay shown in table 3.2, [13], which reveals that some 71% of failures result from winding problems. These results, however, are for transformers without on-load tap changers (OLTCs) which account for approximately 41% of failures where they are installed. OLTCs suffer predominantly from sparking and erosion of contacts, which cause sticking of the moving mechanisms. Silver coated contacts in older transformers may react with sulphur contained in transformer oil leading to high contact resistance from aluminium sulphide compounds, [14]. However, this situation may change as new technologies such as arc-free tap changers emerge. Component Failure HV Windings LV Windings Bushings Leads Tap Changers Gaskets Other  Percentage / % 48 23 2 6 0 2 19  Table 3.2: Transformer component failure One of the predominant causes of winding collapse is high currents through the winding. Under external short-circuit conditions, the transformer can undergo stresses over 100 times the forces it was designed to withstand. Figure 3.3 illustrates how radial forces develop in the windings.  25  Figure 3.3: Radial forces due to current surge These transient forces on the winding can exceed 100,000 lbs in transformers as small as 10MVA, [15], and are most dangerous because they have the ability to loosen and deform clamping structures. If this is left unchecked, it can result in the entire winding structure coming apart with explosive results. High forces also stretch the insulation and may cause cracks or brittleness and speed the rate at which it naturally deteriorates or even crush it completely. In an infinitely long solenoid, there would be no radial fluxes if the entire solenoid was energized uniformly. However, when the solenoid is of finite length, as in the case of realworld structures such as transformer windings, radial fluxes develop at the top and bottom of the coil. These give rise to axial forces which can lead to winding collapse when clamps, which usually exert great compression forces on the winding to keep it in place, are damaged. Strains caused by axial forces can be limited by design changes, [16], however radial forces are more difficult to deal with as they rise with the capacity of the transformer. In addition, the combination of radial and axial forces can produce twisting of windings which severely stress insulation and even fracture it where it has become brittle. Axial forces may also arise due to high frequency inter-turn current resonances. During normal operation, the entire winding is energized by the same current and so flux at the centre of the coil arrangement cancels out for the most part, as in fig. 3.4.  26  Figure 3.4: Steady state magnetization However when a fast surge propagates through the winding, the current is confined to sections at a time as shown in fig. 3.5  Figure 3.5: Transient magnetization of winding  27  Chapter 4 Transformer Diagnostics The previous chapter outlined several causes of transformer failure. Due to this variety of failure modes, there is no single test that can be designed to accurately determine the condition of a transformer. As a result, several monitoring techniques have evolved to give operators an indication of the transformer state. The basic idea in all of these is the comparison of measured parameters of the transformer, to a previous record which represents the transformer in good health. To achieve this goal, transformers may be outfitted with networks of sensors. Liebfried, [17], provides table 4.1 outlining as many as 45 sensors that may be placed on the transformer from the Siemens Power Transformer Monitoring System. Measured Quantity Voltage Current Tap changer Position Temperature Gas in Oil Oil Moisture Compensator Moisture Pump and Fan Condition Air Velocity Oil Velocity Compensator Oil Level Tap changer tank oil level HV bushing oil pressure Total  Number of Sensors 3 5 1 9 4 1 1 8 4 4 1 1 3 45  Table 4.1: Sensors in Siemens’s power transformer monitoring system An extensive bibliography of condition monitoring techniques has been compiled by Singh, [18]. Wang, [19], has provided a comprehensive summary of traditional condition monitoring 28  methods as well as detailed bibliography for further reference. Some common methods found in those works and others are given in the following sub-sections.  4.1 4.1.1  Traditional Methods Visual Inspection  Visual inspection is the most reliable method to determine the winding condition. The transformer has to be taken out of service, drained and opened up to be inspected, the condition of clamps, windings and insulation can then be inspected to determine if there are any noticeable problems. It requires trained and experienced personnel to carry out inspections and can lead to long out of service times for the transformer, which is undesirable. This method is likely to be retained only as a final verification when a less invasive method detects the presence of critical damage.  4.1.2  Short Circuit Impedance( [20], [21], [22])  The low voltage winding terminals are shorted to each other and the input current voltage and power are measured. A deviation of 2% or greater is considered to be indicative of significant winding movement. This technique has to be performed offline in a test lab and is only effective for significant winding distortion.  4.1.3  Leakage Reactance Test [23]  The short circuit impedance test set-up can also be used to calculate the leakage reactance of the transformer. If the winding has expanded, the leakage reactance would increase as a consequence. This method is sensitive to certain types of distortion only, namely distortion that results in increased distance between the primary and secondary coil. It does not pick up distortions such as twisting of windings and is ineffective at high frequencies due to the skin effect.  4.1.4  Winding Ratio Test [23]  The winding ratio test is another offline test that can be used to detect faulty winding conditions. The transformer’s voltage ratio is tested to ensure that the proper turns-ratio is present. This can be used to detect short circuit or open circuit conditions. 29  4.1.5  Winding Resistance Test [24]  The winding resistance test is also an offline method. It operates on the principle that any change in the geometry of the conductor would show up as a change in the winding resistance. For example, if the winding expands then the length of the winding would increase while the cross sectional area would decrease. This would cause an increase in the resistance of the winding. The technique requires highly sensitive equipment to detect fraction of an ohm changes. In addition, since the temperature at which the experiments are carried out would influence the quality of the readings, temperature information has to be recorded to ensure repeatability when conducting future experiments. Generally, variations of more than 5% are considered indicative of damage.  4.1.6  Vibration Test [23], [20]  Vibration testing involves the mounting of acoustic sensors on the tank wall of the transformer to sense vibrations caused by the continuous magnetization and demagnetization of the core and windings. These acoustic signals form the signature for the winding. Abnormal vibrations may be caused by loose core and coil segments, shield problems, deteriorated bearings etc. This method has the advantage of being an online method. However, the externally mounted sensors are highly susceptible to vibration noise from the external environment.  4.1.7  Bushing Power Factor [25]  Many HV bushings are fitted with contacts which permit measurement of the capacitive current flowing from the live conductor to earth through the bushing insulation. If the bushing and core are in good condition, the power factor should correspond to the rated value. Significant deviations indicate that the insulation is deteriorating, which may in turn mean that more heat than usual is being dissipated within the bushing. This can lead to explosive failures.  4.1.8  Furanic Analysis  Furans are oxidized-hydrocarbonic compounds (chemical structure C4 − H4 − O ). The most stable of these compounds, 2-furfuraldehyde or 2-FAL, is released in the oil as a result of cellulosic paper degradation brought on by exposure to prolonged periods of overheating and oxygen. 30  4.1.9  Degree of Polymerization  Degree of polymerization measures the strength of cellulose paper. It measures the average number of glucose molecules making up cellulose chains in the paper. Insulating paper consists of large linear polymeric molecules comprising hundreds of glucose units. When these molecules fracture, the degree of polymerization decreases. Like furanic generation, the process is catalyzed by excessive heat and oxygen. A value of 200 or less generally indicates complete loss of mechanical strength.  4.1.10  Infrared Thermography  This is a non-invasive method of identifying areas of excessive heat loss. It is useful in detecting cooling system problems, problems with electrical contacts and hot spot location. Thermal imagers, however, are not able to give a picture of the internal structures of the tank. Temperature rises of over 20 degrees are considered serious and over 50 degrees indicative of critical conditions demanding immediate attention.  4.1.11  Core-to-Ground Test  This is a simple test used to check for un-intentional ground paths to the tank. The designed ground between core and tank is removed and the resistance between the two is measured. A low resistance indicates the presence of unwanted ground paths. Conversely, a high impedance while the designed ground is in place may indicate a problem with the connection.  4.1.12  Excitation Current  This is used to test electrical faults such as short-circuits between turns and poor connections. It is also used to detect core problems such as core delaminations, and core lamination shorts. Other problems, such as online tap changer (OLTC) problems also show up. The method aims to measure the magnetizing current of the transformer. It is carried out on individual phases and the results of each compared.  4.1.13  Tap Changer Torque Measurement  The tap changer torque gives an indication of contact damage and sludge build up which directly relate to the state of the tap changer. In practice, the current drawn by the tap 31  changer drive motor is monitored. Problems of this nature can cause catastrophic damage to transformers resulting in outages lasting for months. In extreme cases, it can also result in the permanent decommissioning of the transformer.  4.1.14  Oil Temperature Measurement  Top and bottom oil temperatures are measured to estimate hot spot temperatures. In addition, oil temperatures in the diverter switch compartment can give indications of problems in those compartments. However, this method is not very sensitive and unless measured values are extremely high, do not indicate immediate threats to the device.  4.1.15  Partial Discharge Monitoring  Partial discharge monitoring is one of three areas of monitoring that demand special attention (the other two being Dissolved Gas Analysis, DGA, and Frequency Response Analysis, FRA ). It is one of the most discussed topics in the area of condition monitoring. Partial discharges have severe effects on the transformer insulation system, which is the only part of the transformer which cannot be re-instated to original condition. Though discharges may not result in immediate failure they drastically increase the rate of insulation ageing which eventually leads to complete failure in the mid to long term. Boggs, [26], gives a good overview of partial discharge processes. In his paper he describes partial discharge as “... some form of electrical activity within a system which results in a rapid change of the electric field configuration which causes a current to flow in a conductor connected to the external world.” They arise from three main sources: 1. Floating Components. 2. Corona. 3. Insulation voids. Floating metallic components within a system, especially if held in place by insulating materials which may have permittivities different from that of the surrounding oil or gas insulation may cause considerable field distortion. When very strong electric fields are present these field distortions may cause dielectric break down and hence partial discharge. Corona takes place when the insulation in the vicinity of the conductor begins to break down and becomes an ionic conductor. These types of discharges are easy to to detect and even locate, using acoustic and optical sensors. 32  Void discharge, however, is the most technologically challenging to detect. Unlike Corona, which can be detected, and also located, using acoustic or optical sensors quite easily, void discharges buried deep within the insulation system cannot be seen. Ionization of fluids within insulation voids in effect reduce the spacing provided by the insulation material and also distort the field distribution within the insulating material. This is similar in principle to the case of floating components. Discharges of this nature follow no distinct pattern, they vary widely in magnitude and scale inversely with system size, making detection in large systems difficult. Over time, several techniques have been developed to try and localize void discharges, but a definitive approach still eludes the engineering community at large. Acoustic methods, which are successful in locating discharges in the oil surrounding the windings are useless when these processes operate within the dense winding structure. Promising work in this area includes use of the continuous wavelet transform (CWT) to provide both time and frequency discrimination, the use of bayesian classifier based neural networks, heteroscedastic probabalistic neural networks, [27, 28], time domain reflectometry, [29], and incremental based knowledge approaches, [30].  4.1.16  Dissolved Gas Analysis (DGA)  Dissolved Gas Analysis is perhaps the most popular method of transformer diagnostics. As the name implies, the method relies on tracking levels of gasses disolved in the transformer oil. These gasses can be related to the presence of electrical faults within the transformer. DiGiorgio, [31], gives an excellent overview of the chief methods used in analyzing DGA results. The gasses that are produced in oil fall into three general categories: 1. Hydrogen and Hydrocarbons eg. Methane, Ethane, Ethylene, Acetylene, Hydrogen. 2. Carbon oxides i.e. Carbon Monoxide and Carbon Dioxide. 3. Non fault Gasses such as Nitrogen and Oxygen. These gasses remain dissolved in the oil or get released into the headspace in transformers which have a gas blanket. Specific gasses can be related to specific types of faults as summarized in table 4.2, [31].  33  Fault type Oil corona Cellulose corona Low temperature oil pyrolysis High temperature oil pyrolysis Low temperature cellulose pyrolysis High temperature cellulose pyrolysis Arcing  Gas liberated hydrogen hydrogen carbon monoxide / dioxide methane, ethane ethylene, hydrogen, methane, ethane carbon dioxide / monoxide carbon monoxide / dioxide hydrogen, acetylene, methane, ethane ethylene  Table 4.2: Common faults and associated gasses The levels of these gasses are interpreted under four main schemes: 1. Total Combustible Gas (TCG). 2. Dornenburg Ratios (DR). 3. Rogers’ Ratios (RR). 4. Logarithmic Nomograph. TCG is a popular method because it is fast and easy to implement. It measures combustible gases in the gas blanket above the oil. Consequently, it only applies to those transformers which have a gas blanket and not to those with conservators. Because of this, faults are not immediately obvious as gasses must first saturate the oil before they diffuse into the gas blanket. DR tests involve measuring the two separate gas ratios, C2 H2 /C2 H4 and CH4 /H2 . The combination of gas ratios determine the type of fault that may be extant. Practically this is visualised in the form of a graph with the first ratio on the abscissa and the second on the ordinate axis. The plot region is divided into regions which correspond to different fault types, such as thermal, arcing and corona. RR tests utilise four ratios, C2 H6 /CH4 and C2 H4 /C2 H6 in addition to the two used in the DR tests. The ratio levels for each gas are divided into ranges, each of which is assigned a number. The four number combination for a particular case is a unique code that corresponds to a specific diagnosis. Practically, this is carried out by a simple look up table. The method can detect various degrees of heating, partial discharge, circulating currents in the winding, core and tank and insulation temperature increase. The logarithmic nomograph is a set of variable linear log scales on which a number of gas levels are plotted.The points are then connected. If a certain point lies above its threshold 34  value the slope of the lines connected to it are analyzed to give an indication of the type of fault present.  4.2  Frequency Response Analysis (FRA)  FRA involves measuring the trans-impedance or trans-admittance of the winding. It is given special treatment here because of its similarity to the TLD method focused on later in this work. This method has come to be somewhat of an industry standard, because of its high sensitivity to a number of different types of winding distortions. It is based on the fact that a change in winding geometry results in a change of the RLC parameters of the winding. Usually, it is thought of as a means to measure the capacitive leakage between the transformer windings. Practically, the admittance of the winding may be compared with that of a healthy winding in one of three ways: 1. Time based comparison. 2. Type based Comparison. 3. Construction based comparison. Time based signatures are simply the signature of the transformer obtained at an earlier date. This is the most reliable signature that can be used. When the transformer is new, the signature is recorded and this serves as the reference for all future measurements. This data is usually not available though, since transformers have been in service for decades while condition monitoring techniques are relatively new. This means that alternative methods for obtaining baseline signatures usually have to be employed. Type based signatures involve obtaining a signature from an identically constructed transformer that is known to be in good condition. This may be a relatively new transformer that has been installed at another substation or one that services a low fault area so that the only change in transfer function would arise from natural ageing. The main problem with this method is that even for identically specified transformers, winding designs over time may have changed, causing slightly different transfer functions, in addition, designs are made within a certain tolerance level so that there would be slight variations from one transformer to the next even if the winding designs used are identical. To solve this problem Christian and Feser, [32], have proposed the statistical calculation of tolerance bands using transfer functions from a large group of same-type transformers to distinguish differences arising from winding distortions to differences arising from different manufacturing processes. 35  Construction based signatures are used on multi-leg transformers where windings are not zigzag connected. The process entails using the windings on different legs as mutual references. Each winding is tested separately and then transfer functions compared. Christian and Feser, find that the geometrical properties of the core-and-coil assembly as well as the type of vector group have noticeable effects on the comparability of the results of different legs. The approach is not constrained to three-phase transformers. Where three single phase transformers are used the technique is equally viable. The technique of using windings as mutual references has the advantage of requiring no past or external data for comparisons. Measurements, [32], show that the frequency responses of three windings are near identical. The technique has the following advantages: the problem of different manufacturing techniques does not affect it since the windings are part of the same unit. Past data is not needed since asymmetries are evident from comparison of present data. Unless a three phase fault that symmetrically distorts all the windings occurs, which is almost never the case, winding movement and distortion can always be ascertained. The disadvantage of the technique is that if the three windings undergo change in their frequency response characteristics then the asymmetry would give misleading results, making the winding look like it has moved more or less than it actually has. Two methods are typically used to carry out the FRA measurements: 1. Swept Frequency. 2. Low Voltage impulse. Swept Frequency Response Analysis (SFRA) involves injecting a sinusoidal voltage into one end of the winding and then measuring the output current at the other end. The transadmittance is then calculated from the phasor quantities. This is repeated for a wide range of frequencies to obtain the transadmittance signature. White noise is also sometimes used as an injected quantity since it contains all frequencies at equal power levels. The Fourier transform is then applied to the input and output measurements to obtain their phasor representations and finally the transadmittance is calculated according to equation 4.1.  Ytrans =  Iout (ω) [S] Vin (ω)  (4.1)  The Low Voltage Impulse method or (FRA-LVI) involves injecting a low voltage pulse at the input and recording the current at the output terminal. The input and output waveforms are then processed in the same way as is done with white noise in SFRA measurements. 36  There has been much debate about which of these methods is actually better. Tenbohlen and Ryder, [33], have explored the relative advantages and disadvantages of the methods, which are summarized in tables 4.3 and 4.4, [34]. SFRA advantages High signal to noise ratio  LVI disadvantages Signal to noise ratio decreases with frequency as higher frequency components have less energy. Frequency resolution is fixed and poor at low frequencies Several pieces of equipment needed (function generator, rogowski coil, digital oscilloscope) Difficult to filter out broad band noise  Wide frequency range Adaptable frequency increments (better resolution at low frequencies) Only one piece of measuring equipment needed (Network analyzer) SFRA disadvantages measurement is relatively long(several minutes)  LVI advantages measurement is short (typically less than one minute).  Table 4.3: Comparison of SFRA and FRA-LVI according to Tenbohlen and Ryder  37  SFRA advantages Intuitive and straightforward  LVI disadvantages Difficult to filter out broad band noise Repeatability not of a high standard without the use of very expensive equipment LVI advantages Voltage and currents measured at the transformer, thus minimizing the effect of the external circuit setup.  SFRA disadvantages Network analyzer traditionally used which does not have sufficient power to appreciably excite windings at high frequencies due to large inductive load. Insufficient power to excite windings at very high frequencies due to high capacitive load of insulation system.  Low inductance shunts may be used to reduce damping compared to 50 ohm internal impedance of network analyzer in SFRA.  Low level signals result in high measurement errors and poor repeatability especially on high current windings. Extremely sensitive to measurement set up since cable lengths may exceed 50 feet. Table 4.4: Comparison of SFRA and FRA-LVI according to Britton Ryder, proponent of SFRA, describes the key indicators of damage when comparing SFRA signatures as: • Changes to the overall shape of the graph. • The creation of new resonant frequencies or the elimination of existing resonant frequencies. • Large shifts in existing resonant frequencies. Through a series of case studies he puts forth the information in table 4.5, [35], regarding the ability of FRA to detect fault conditions.  38  Type of Fault No core earth Multiple core earths Foreign object Additional turns on yoke Additional turns on limbs Short-circuited turns Mechanical damage to windings Mechanical damage to core Windings unclamped Loose turns "Normal" ageing  Detectable Probably not detectable except under laboratory conditions. Usually not detectable Not detectable. Detectable. Detectable. Detectable. Detectable. Detectable if very severe. Probably not detectable except under laboratory conditions. Detectable. Detectable if very severe.  Table 4.5: Faults detectable by FRA The main difficulty of FRA is repeatability of results. Homagk and Liebfried, [36], have suggested methods for decreasing the error in reproducibility by paying attention to bushing, terminal and ground connections.  4.2.1  Online FRA  Thus far, FRA is still predominantly an offline technique. However, work based on online FRA is not unheard of. Two known methods are: 1. System transient measurement. 2. Injection through bushing tap. System transients form a natural source of high frequency energy that impact transformers. By measuring these input transient voltages and the output transient currents, the need for separate high-frequency injection instrumentation is eliminated. Wimmer, [37], has examined the use of stochastic system transients and transients from the switch on event of the transformer. The major difficulty of this type of method is very poor reproducibility due to the transfer function’s dependence on the system state. The transformer switch on event is better in this respect since the effect of the system is reduced. However, if the transformer is being switched on, this implies that the transformer is originally off, which raises the question of performing an on-site offline test instead, which is more reliable. Injection through the transformer bushing tap is a means of performing SFRA online. Setayeshmehr, [38], illustrates the effectiveness of the method for winding short circuits of 39  faults using a model transformer and bushing. The experiment was carried out with a network analyzer for signal injection, made possible by the single capacitance bushing model used. However, where bushings are centre tapped capacitors, the division ratio often makes low power injection unfeasible. To remedy this, DeRybel and Singh, [2, 39], at the University of British Columbia, have developed a high power, high frequency injection and measurement system, capable of injecting pulse trains up to 2MHz frequency at a power of 1kW.  4.2.2  Interpretation Techniques  The most difficult problem encountered in FRA, is the one of interpretation. When differences in winding signatures show up, experts, drawing upon long experience, are usually required to decipher them. However, there have been a few attempts at forming more rigid methods for interpreting the meaning of signature changes. One such method known as the “Chinese Method” (due to its being developed by the Electric Power industry of China, [40]) splits the frequency range (up to 1MHz) into three sections and calculates correlation factors for each range. Bounds are set for the corelation factors of each range relating to the severity of winding movement that is likely to have caused the change. This is a similar approach to the ratio methods used in DGA. Wimmer, [41], gives a good evaluation of the method and concludes that it offers good support to FRA measurements but is not mature enough to replace expert analysis. Another slowly emerging field in this area is the use of Artificial Neural Networks (ANNs). The basic task given to the network is to extract from the differences in the plots the nature of the change in the system. Preliminary work has been done on this by Doble, [42], as well Nirgude, [43]. Both methods utilize feed forward back propagation neural networks with great success. Doble uses a large number of actual test cases (thousands of transformers) grouped by type of failure and uses their entire frequency spectrum as the source of data, while Nirgude performs transfer function fitting to estimate the poles and zeros of the admittance function to use as inputs. The major downfall of these methods though is the large number of test cases required. Ragavan and Satish, [44], have developed a method of network synthesis. A ladder network is synthesized from the transformer frequency response. When a change is noticed in the signature of a transformer, a new ladder network is sythesized. By comparison to the ladder network for the baseline signature, deformations that may have occurred in the transformer can be determined. The attraction of this method lies in its ability to not only determine the existence of faults but also to localize and quantify them. However, it suffers from poor 40  algorithmic performance, requiring billions or even trillions of iterations for convergence, resulting in excessively long computation times (days) for even small systems ( 5 inductors and capacitors). It should be noted that circuit synthesis is a non-unique operation. However, by constraining the circuit topology to a particular form a unique solution can be obtained but the resultant circuit is still an approximation to the complex three-dimensional electromagnetic structure of the transformer. Ragavan and Satish have performed validations on table-top sized coil models and shown that inductances and capacitances added to the physical model show up in the correct location with correct value in the synthesized circuit model. However, validation on a full-scale model to show whether these findings hold on larger systems has not been performed. This is because the construction of a full scale test model is extremely expensive, with the cost possibly running into hundreds of thousands of dollars. In addition, the deformation of the transformer would in effect render the model useless for future studies.  41  Chapter 5 Overview of Investigation Research into diagnostics of power transformers, using transfer function methods to detect winding displacements, has been an on-going exercise for the Power Systems Group at the University of British Columbia over the last few years. The initial work began with exploration of extending the frequency range of FRA tests beyond 1MHz which was common practice in industry, [45]. After this, came the development of the TLD project, [1, 46], which was based on the visualization of the transformer winding as a very long transmission line and sought to characterize it by calculating the characteristic impedance of the winding, which could be used as the signature of the transformer. In practice, the input and output voltages and currents are measured, and the surge impedance calculated through equation 5.1. Simulations with detailed models, [1, 47], have shown it to have greater sensitivity than FRA for various types of winding movement. Prototypic evaluations of the method were carried out in the UBC HV lab while field validation of the method was carried out at Powertech Labs, a subsidiary of BC Hydro, and confirmed the method’s usefulness as an alternative, or complement to FRA, [2].  Zc =  2 Vin2 − Vout [Ω] 2 2 Iin − Iout  (5.1)  Following the development of this method, the scope of research was extended to the development of an online measurement system to circumvent the obvious restrictions of offline testing. In addition to this, preliminary work into alternative methods of detecting faults such as Time Domain Reflectometry (TDR) has been done. An interpretation methodology was also deemed necessary to aid the Asset Manager in making more informed decsions based on available CBM data. It should be noted once more that the transformers of concern for these studies are EHV 42  and UHV transformers found in the transmission system. Distribution transformers are not of major concern as they are more easily replaceable and their loss does not have as severe an impact on the electrical system as does the loss of EHV and UHV transformers.  5.1  Development of Online Monitoring Techniques  A reliable solution to the problem of online measurement still eludes the engineering community at large. The following areas were identified as major set-backs in the field: • Signal injection of sufficient power and known frequency content. • A method of reducing the influence of the external power network on the measured signatures. A high power (1kW), high frequency (up to 2.5MHz) square wave injection system, [48] was designed. This system could be used to inject high-frequency test signals into the bushing tap of a transformer during online tests (or into the terminal directly during offline tests). The extra power is required to have good signal to noise ratio, especially given the shunting effect of the centre tapped bushing capacitance which diverts a considerable part of the signal energy to ground. The repeatability of online tests are also influenced by system changes external to the transformer. A method to reduce this effect is currently under investigation. Appendix C discusses the general approach currently being followed, however, this work is still in the development phase and completion of a low voltage prototype is expected soon.  5.2  Time Domain Reflectometry (TDR)  TDR is a well known method of locating faults in cables and transmission lines. Discontinuties in a transmission line cause reflections which can be sensed. Based on the time between the injected and reflected pulse the location of the discontinuity can be detected. In the case of transformers, however, it is not a simple matter. The major difficulty lies in multiple transmission modes arising from the strong capacitive couplings between the turns on the transformer winding. Essentially, these couple the signal energy to the end of the transformer very quickly. The dominant signal propagation path is therefore the short top-bottom path of the coil through the insulation and not the 43  long path along the conductor winding. This means that signals show up at the end of the winding almost instantaneously. To gain any useable discrimination, sub-nanosecond pulses would have to be used on relatively large transformer models (scaled down models would decrease the upper limit of the pulse width required). This equipment limitation has delayed explorations in this avenue till suitable equipment is acquired. Simulation work in this area is also hindered due to lack of a suitable line model, which takes into account propagation through insulation. Multiphase transmission line models assume instaneous signal transmission through insulative coupling branches owing to their being orders of magnitude shorter than the length of the conductor. This makes them useless for reflectometry work of this nature, in which the insulation path is comparable in length to the conductive path.  5.3  Signature Interpretation  Signature interpretation, more specifically the interpretation of characteristic impedance signatures, (TLD method), is the major focus of the work presented in this thesis. The use of neural networks seems to be one of the most promising areas in this regard. Their use, however, is hampered by the need for large training sets. The building of such training sets from actual transformers is often not feasible due to a sheer lack of transformers available to acquire data from. This is especially true of large power transformers which are not as prevalent in the system as distribution transformers. To address this, training the networks on analytical models is proposed. The major setback of this method however, is the ability to very accurately model the transformer. Ragavan and Satish’s work on network synthesis by iteration, [44], shows that it is possible to get within 2% of the transformer characteristic with a lumped network. Other authors have developed transmission line models which also closely follow the measured frequency response of the transformer up to megahertz frequencies using drawing board data of the physical dimensions and layout of the transformer windings. These suggest that the method of analytically trained networks can in fact be used to produce accurate models required to train damage classifiers. The overall methodology illustrated in fig. 5.1.  44  Figure 5.1: Acquiring training data for neural network classifier Using either network synthesis or drawing board data, the base circuit model is extracted from the transformer. Changes are then made to the parameters of this circuit model to represent incipient winding movements in the form of axial compression or radial expansion of winding sections. These can give early indications of problems developing within the transformer. Simulations are then carried out to calculate the frequency responses of each new case. The magnitude spectra from these cases are used as inputs to train the neural networks. Each input spectrum is mapped to an output data set representing the type, location and severity of winding movement. In many cases, where field engineers may not have the internal details of the transformer available, and are not able to get the data from the transformer manufacturer, network synthesis is the only choice for extracting a circuit model. Furthermore, using a network synthesis approach would allow the details of simulation to be hidden from the user. The user would only need to give the software package a baseline signature from which a network could automatically be synthesized, modified, simulated and used to train the classifier networks. 45  Once the training is complete, CBM data (frequency responses) from the transformer can be input directly to the classifier to determine if any movement has taken place as shown in fig. 5.2.  Figure 5.2: Role of trained classifier  5.3.1  Practical Considerations  The practical implementation of the sysem has three principal challenges: • Resolution of the instrumentation system. • Mismatch between the measured signature and modelled signature. • Natural variations in the transformer response. The resolution of the instrumentation system limits the level of change that can be detected. To ensure that this situation is not exacerbated, the resolution of the classifier should be 46  much higher than the resolution of the instrumentation system. That is, the classifier should be trained to detect much smaller changes than the instrumentation can detect so that whatever changes the instrumentation does record are seen by the user. The mismatch between the circuit model response and the actual response of the transformer will show up as a change in signature, as shown in fig. 5.3, if compared to the response of the circuit model.  Figure 5.3: Mismatch between baseline signatures for actual transformer and circuit model This is mitigated, however, by using difference curves to train the networks. A measured response is not compared to the response of the circuit model but to the measured baseline response from the actual transformer. The difference between these two measured curves forms the input to the classifier. There is still expected to be minor differences in the measured difference curves from the actual transformer and the computed difference curves. To compensate for this when a difference curve is computed from the circuit model, random changes of bounded magnitude should be made to it, giving rise to a number of different responses corresponding to a single case. Training the network with these extra cases 47  would increase its immunity to the mismatch between the circuit model response and actual transformer response. Natural variations in the transformer response may also be interpreted as winding movement. This can be reduced through averaging several measured signatures to yield a more repeatable signature. In spite of this, there is still expected to be slight variation in measured signatures that may be possibly classified wrongly. However, by trending the classifier output, as shown in fig. 5.4, insight into the state of the transformer can still be gained.  Figure 5.4: Example of trending movement index to compensate for natural variations in frequency response  48  Chapter 6 Transformer modelling 6.1  Overview of High Frequency Transformer Models  High voltage power transformers are generally constructed with rectangular conductors in a circular disk type arrangement as shown in figs. 6.1 and 6.2, [49]. Figure 6.3, [50], illustrates the winding cross section.  Figure 6.1: Top view of disk winding  49  Figure 6.2: Side view of disk winding  Figure 6.3: Cross-section of disk type winding The type of model extracted from the physical system, largely depends on the type of study being carried out. In particular, the frequencies of interest determine the level of detail that must be modelled. As with overhead transmission line modelling, models usually fall into one of three categories. 1. Lumped models. 2. Terminal models. 3. Transmission line models.  6.1.1  Lumped Network models  Lumped networks are developed by considering each turn as a section. Each section contains a series inductance representing the conductor, a series capacitance representing the 50  capacitance between adjacent turns and a parallel capacitance representing the capacitance between layers. Figure 6.4, [50], illustrates how the capacitances are visualised for three turns on a core. The inductance (shaded region) lies along the direction of the coil.  Figure 6.4: Visualization of capacitance and inductances forming winding structure When multiple turns and layers are involved, as is always the case, the resultant network is a dense mesh of capacitances and inductances as shown in fig.6.5, [50]. Following the full lumped description, which would contain a large number of turns, circuit reductions would normally be made in order to provide faster simulation and meet memory requirements. In studies concerned with the initial distribution of the voltages the inductances would often be left out altogether and a purely capacitive circuit, illustrated in fig. 6.6, [50], would be used under the assumption that at high frequencies the inductive reactance becomes infinite when compared to the capacitive reactance.  51  Figure 6.5: LC mesh formed from multiple turns and layers  Figure 6.6: Purely capacitive network  52  These methods are quite accurate up to frequencies of about 20kHz. Beyond this sections must be subdivided in order to obtain accurate results.  6.1.2  Terminal Models  Terminal models involve approximating the transformer admittance characteristics at the terminals of the transformer, [51]. The elements of the nodal admittance matrix are approximated with rational functions consisting of real as well as complex conjugate poles and zeros. This can be represented in EMTP simulations as an RLC network. This method has been shown to produce accurate terminal results for frequencies in the megahertz range, [52]. There is potential for this type of method to replace the iteration technique proposed in [44], since they also extract physical circuit models from measured data. Such a system would allow direct comparison of component networks as shown in fig. 6.7.  Figure 6.7: Possible component network comparison using terminal models. At the moment, however, this method models the system by combinations of parallel RL, RC and RLC branches for the low, high and mid-frequency range as shown in fig. 6.8.  53  Figure 6.8: Synthesized network from admittance matrix fitting These circuits do not resemble drawing board circuit models such as fig.6.5. This means changes in component values do not map directly to physical locations on a transformer, and so, much work needs to be done if they are to become useful in transformer diagnostics.  6.1.3  Transmission Line Models  As mentioned previously, to simulate frequencies higher than approximately 20kHz, winding sections, modelled as lumped elements must be subdivided. A natural extension of this is the subdivision of the turn into incremental pi-circuits resulting in a transmission line representation of each turn. The entire transformer can then be modelled as a multiphase transmission system as shown in fig. 6.9.  54  Figure 6.9: Multiphase transmission line representation of transformer Such a model has fully distributed capacitances, inductances as well as losses. Transmission line models like their lumped counterparts use data that can be extracted by examining a cross-section of the winding. These models are typically used in very fast transient overvoltage studies, [53, 54, 55, 56, 57, 58, 59, 60, 61], where frequencies can be over 500kHz. Due to the accuracy of these models at higher frequencies, they are the most suitable for developing training sets required by the neural network classification system.  6.2  Modelling Approaches  One of the major problems faced in the application of a transmission line model to a transformer, is the calculation of the inductance and capacitance matrices which characterize the system. Unlike overhead transmission lines which are circular and spaced far apart, transformer windings are rectangular and packed tightly together. This feature of the winding arrangement casts doubt over the validity of some of the governing equations, in terms of the manner in which they are used to calculate parameters for transformers, particularly the assumed field distributions and the relationship between the inductance, capacitance and velocity of propagation. Since the present day modelling techniques have borrowed largely from transmission line modelling, a brief review of the parameter calculation as found in that field is given in Appendix A. Due to the relationship between velocity of propagation capacitance and inductance (equation 6.1) three approaches for calculating the inductance and capacitance parameters can 55  be theorized:  1. Calculate capacitances directly and inductances through velocity of propagation. 2. Calculate inductances directly and capacitances through velocity of propagation. 3. Calculate both capacitances and inductances separately.  v=  1 LC  [ms−1 ]  (6.1)  Where the inductance , L, has units of Hm−1 and the capacitance, C, has units of F m−1 All of these methods can be found in current literature dealing with high frequency transformer modelling. Ideally, these methods should all produce the same results, however, due to some extra considerations made in the modelling of transformer windings, this is not evident. The computational anomalies are illustrated using the simple system of fig. 6.10 and table 6.1.  Figure 6.10: Drawing board data for simple three turn winding  parameter Parameter value permittivity permeability insulation thickness conductor height conductor width radius of inner turn height above ground velocity of propagation  value 8.85x10^-12 4pix10^-7 1mm 1cm 1cm 1m 10 cm 3x10^8ms^-1  Table 6.1: Drawing board data for simulated winding of fig. 6.10  56  6.2.1  Calculation of Capacitances Directly and Inductances through Velocity of Propagation  The direct calculation of the capacitances relies on the tightly packed structure of the windings and their rectangular cross section. Two methods for calculating capacitance between windings are commonly found: 1. Rectilinear calculations. 2. Concentric cylinder calculations. For rectilinear calculations the usual assumptions made include the following: 1. The curvature of the winding can be ignored. 2. The ratio of the distance between adjacent conductor surfaces and the height of the conductor is small enough so that the capacitance between adjacent turns can be modelled as parallel plate capacitors. 3. Intermediate conductors shield electric flux such that capacitance only exists between neighbouring conductors. The last point is especially important as it indicates that the capacitance matrix is not full. In fact, since a conductor is represented as having couplings only to adjacent conductors, the maximum number of coupled phases is five (a conductor bounded by four others). This places an upper limit on the number of non-zero elements in each column of the capacitance matrix, an important result in improving simulation time as will be shown in subsequent chapters. For the three-turn coil being simulated the capacitances are calculated using the well known equation for parallel plate capacitors, which can be found in common electromagnetic theory text books such as [62]:  C=  A [F ] d  Where A is the cross-sectional area, d is the distance between the plates, and  is the di-electric constant. 57  (6.2)  The following is obtained for the capacitance matrix:  −10  C = 1 × 10  0.4513 −0.4425 0 −0.4425 0.8939 −0.4425 0 −0.4425 0.4513  [F ]  (6.3)  The inductance can then be calculated through equation 6.1 as:  −5  L = 1 × 10  0.4522 0.4157 0.4076 0.4157 0.4240 0.4157 0.4076 0.4157 0.4322  [H]  (6.4)  While there is nothing immediately surprising about the form of the matrices, the full inductance matrix implies there is coupling between non-adjacent conductors while the capacitance matrix implies there is not. The argument for the validity of this is usually that magnetic flux is not shielded by the presence of conductors so that a full inductance matrix is feasible. However, the series impedance jωL, (calculated from the above inductance matrix) when inverted to find the series admittance matrix, would give a matrix of the same form as the sparse capacitance matrix, since L−1 = v 2 C, from equation 6.1 . This would indicate that the same system has finite non-zero impedances between nodes, however, would also have zero admittances between those same nodes. The concentric cylinder model has only come into use in the recent past. It involves visualizing adjacent turns as very short concentric co-axial cylinders as shown in fig. 6.11.  58  Figure 6.11: Concentric cylinder turn model In this case, the capacitance to ground is still calculated using equation 6.2 but the interturn capacitances are calculated through equation 6.5 which can also be found in common electromagnetic theory text books such as [62].  C=  h ln  r2 r1  [F ]  (6.5)  This is a more accurate representation as it preserves the winding geometry. However, the same problems associated with the sparse and full forms of the capacitance and inductance matrices are present.  6.2.2  Calculating Inductances Directly  For this method the rectangular cross-section of the conductor is ignored and it is considered to be a current filament. However, the circular winding geometry is preserved in finding self and mutual inductances as shown in fig. 6.12.  59  Figure 6.12: Ring windings used for direct inductance calculation For this system, the equations governing the inductance are not as simple as that for rectilinear lines and the solution must employ elliptic integrals, K and E. For turns of radius r1 and r2 metres, separated by a distance of h metres, equations 6.6 and 6.7 from [63] give the self and mutual inductances respectively.  Lself = µr log( √ Lmutual = µ r1 r2  8∗r )−2 GM D  [H]  2 2 − k K(k) − E(k) k k  (6.6)  [H]  (6.7)  where GM D = geometric mean distance of conductor to itself  µ = µr µ0 = permeability of the material surrounding the turns and  k=  4r1 r2 (r1 + r2 )2 + h2  (6.8)  Once the inductance matrix is calculated, the capacitance matrix is calculated through equation 10.14. Using this method the matrices for the same system are: 60  −0.7206 0.2101 0.0421 0.2101 0.0149 −0.0504 0.0421 −0.0504 0.0315  −8  C = 1 × 10  −7  L = 1 × 10  0.0282 0.1189 0.1527 0.1189 0.3330 0.3740 0.1527 0.3740 0.7476  [F ]  [H]  (6.9)  At first glance, it is apparent that the capacitance matrix is not in admittance form. One diagonal element is negative and four off diagonals are positive, which correspond to the presence of negative capacitances. The matrices are also orders of magnitude different from the calculations starting from direct calculation of the capacitance matrix presented before, however, other authors studying different systems have found differences as low as 40%, [56]. The capacitance matrix is also full, indicating that couplings between non-adjacent turns are present. This method assumes current filaments and so mutual impedances become increasingly inaccurate for turns that are close together, [64]. This can be overcome by subdividing the turn cross-section to better model the rectangular construction. A popular method of subdivision is Lyle’s method, [65], which splits rectangular conductors into two filaments. However, for square conductors as in the example system used, the two filaments overlap so that no subdivision is necessary.  6.2.3  Calculation of both L and C separately  The last method possible is the calculation of the capacitance and inductance matrices separately. Using the direct calculations for inductance, 6.9, and capacitance, 6.3, from the previous sections the following velocity matrix results:  10  v = 1 × 10  1.2947i 1.2502i 1.2345i 0.7869 0.7674 0.7532 0.2857 0.2733 0.2820  [ms−1 ]  As can be seen, the velocity matrix is complex and has magnitudes that exceed the speed of light by more than an order of magnitude. The matrix is also a full asymmetrical matrix whereas in the modelling of overhead lines the matrix is diagonal.  61  6.2.4  Observations  There are two main observations to be made: 1. The conductor to insulation volume ratio in transformers is high, whereas in overhead transmission lines it is very low. 2. Static field assumptions may not hold for high frequencies as well as they do for low frequencies. 6.2.4.1  Presence of Conductors  The presence of conductors distort both the electric and magnetic field distributions. Considering an electrostatic situation for the case studied, the electric flux will leave and terminate on conductors normal to their surfaces. For the magneto-static situation the magnetic field will fully penetrate adjacent conductors. On the other hand electric fields would be absent within the conductor . This means that the integral of the magnetic field within the conductor is non-zero whereas the integral of the electric field would be zero. The presence of conductors effectively shortens the distance between non-adjacent conductors for electric field calculations but not for magnetic field calculations, in the static case. 6.2.4.2  Effect of High Frequencies  Static field approximations are traditionally used (and are suffcient) in parameter calculations for low frequency simulations. In fact, in the case of filamentary conductors, the static approximations are sufficient for any frequency range, given there is a sufficient degree of spatial discretization. That is, the higher the frequency the smaller the discretized line segments need to be. However, for the case of real conductors closely packed together there are additional concerns relating to the internal flux distributions of the conductors and the effect of the conductor geometry on the external field distributions. The absence of electric field within the conductors for static fields arises from the fact that conductors have free moving charges. Any applied electric field will cause the electrons to re-orient themselves such that the field in the conductor is zero. This process takes place not at the speed of light but at the drift velocity of the electrons which is very slow (in the order of millimetres per second). That being said, the distances the electrons may have to move would be significantly smaller than the width of the conductor. This inertia associated with the re-orientation of the electrons would act as a high pass filter to the electric field. As the frequency of the electric field increases the electrons would not be able to move fast 62  enough to neutralize the field within the conductor. In such a case, flux would no longer terminate normal to the surface of the conductors and the fields even in densely packed structures would become more radial. The conductors in effect become transparent to the flux at high frequencies just as they are transparent to steady magnetic flux. However, this phenomena becomes apparent only in the gigahertz frequency range, after which it is even possible to measure the permittivity of the conductor. The opposite effect is noted in the case of magnetic fields. This is a major drawback of the current method of direct inductance calculation. Mutual inductances are calculated by ignoring the effect of intermediate conductors on the flux distribution. There are two reasons behind this simplification. The first is that conductors do not contain magnetic monopoles so that the conductors are transparent to the magnetic flux lines. This, however, is a static field assumption. For low frequencies skin effect is negligible and consequently so is the field distortion. However, as frequency increases, the skin effect becomes stronger and the conductors become magnetic-flux devoid regions. The second reason is that traditionally overhead transmission lines are spaced relatively far apart so that the density of conductor to insulation in the system is very low. For transformers, however, the ratio of conductor density to insulation density is much higher. This affects the calculation of mutual inductance as it alters the regions for which the flux integral is non-zero as shown in fig. 6.13 and 6.14.  Figure 6.13: Integral region for inductance calculation ignoring effect of intermediate conductors  63  Figure 6.14: Integral region for inductance calculation with intermediate conductors The increase in frequency also changes the flux distribution since high frequencies cutting conductor surfaces at an angle induce eddy currents which negate those fluxes. The result is that the magnetic field streamlines run parallel to the surface of the conductors, in their vicinity. Once again, because of the closely packed nature of the winding, insulation only exists in the vicinity of the conductor surface. The implications of this are that, unlike overhead transmission line modelling where it is the far field approximation of the field distribution that is used, in transformer modelling it is the near field distribution that must be used. It is likely that the form of the equations for direct inductance calculation would then change from that of equations 6.6 and 6.7 to one similar to that of the concentric cylinder capacitance equation (equation 6.5) since that is built on an electric field distribution that is normal to the conductor surfaces. The phenomena is mildly evident in frequencies as low as 60Hz but at the megahertz frequencies of interest for diagnostic tests, it effectively makes the entire conductor a flux devoid region. The skin depth for good conductors can be approximated by equation 6.10.  δ=  1 πf µσ  [m]  (6.10)  where µ = 4π10−7 N A−2 For copper which has an electrical conductivity, σ, of 5.967 Sm−1 , the skin depth at 1MHz it 64  has a value of approximately 6.52×10−5 m or 0.0652mm. Considering an insulation thickness of about 1mm this is more than 30 times smaller than the distance between conductors.  65  Chapter 7 Fast Simulation Methods Due to the inability of reduction methods (see Appendix B) to accurately model the system at high frequencies, customized algorithms were developed. These were catered to the specific circuit topology of transformer windings, particularly sparsity of the capacitive system. Due to the variety in modelling techniques viz. the calculation of the inductance and capacitance matrices, which are all currently used in high frequency simulation of transformer windings, algorithms for each case will be developed. To review, these methods are: 1. Direct capacitance calculation. 2. Direct inductance calculation. 3. Direct calculation of inductance and capacitance. Before looking at these, however, an overview of the traditional EMTP style simulation is given and computational bottlenecks identified.  7.1 7.1.1  Traditional EMTP Simulation Overall Solution Technique  The traditional EMTP solution, [66], is centred around building a nodal-admittance-matrixrepresentation of the system and then solving for the unknown voltages at the nodes as in equation 7.1.  66            I Y Yab   Vk   k  =  aa Iu Yba Ybb Vu  (7.1)  where Ik and Vk are the currents and voltages at known-voltage nodes, Iu and Vu are the currents and votages at unknown-voltage nodes. The sub-matrices Yaa , Yab , Yba , Ybb represent the system admittance matrix. Sub-matrix Yaa , which represents the admittance connections of the known-voltage nodes, cannot be defined at this stage, due to ideal voltage sources having zero impedance and so cannot be used in the solution. The vector of unknown voltages is therefore solved by solving equation 7.2 for Vu .  [Iu ] = [Yba ] [Vk ] + [Ybb ] [Vu ]  7.1.2  (7.2)  Building System Matrices  The solution illustrated by equations 7.1 and 7.2 hinges on the construction of the Y matrix extracted from the circuit model. For the case of the transformer winding problem, the winding is visualized as a multi-phase transmission system as shown in fig.7.1.  Figure 7.1: Multi-phase representation of transformer winding  67  Such a transmission system can be represented by an exact pi-circuit, that is, a pi-circuit with hyperbolic corrections, as shown in fig. 7.2.  Figure 7.2: Hyperbolic-pi representation of multi-phase system Equations 7.3 and 7.4 give the hyperbolic correction factors calculated from the nominal impedance and admittance parameters.  [Zhyperbolic ] = [Znominal ] ×  [Y hyperbolic ] = [Ynominal ] ×  sinh([γ]) [γ] tanh  [γ] 2  [γ] 2  [Ω]  (7.3)  [S]  (7.4)  where γ, the propagation function, is given by equation 7.5:  γ=  [Znominal ] ∗ [Ynominal ] =  [Rnominal + jωLnominal ] ∗ [Gnominal + jωCnominal ]  (7.5)  Since these equations are in matrix form, they must be converted to the modal domain in order to calculate the square roots and hyperbolic-trigonometric functions. The diagonal matrix of eigen-values, λ, and matrix of eigen-vectors, Λ, of γ 2 are found. The diagonalized system can then be used to calculate the corrected series-impedance and shunt-admittance using equations 7.6 and 7.7.  68  [Zseries ] = [Znominal ] × [Λ] ×  [Yshunt ] = [Ynominal ] × [Λ] ×  sinh([λ]) × Λ−1 [λ]  tanh(  λ 2  λ 2  )  × Λ−1  [Ω]  (7.6)  [S]  (7.7)  The Zseries matrix must then be inverted to bring it to admittance form, Yseries , so that it can be added to the system admittance matrix. For a set of sending-end nodes, k, and receiving-end nodes, m, the shunt and series matrices are added to the system matrix as illustrated in fig. 7.3.  Figure 7.3: Matrix representation of trasmission line The overall solution technique is illustrated in fig. 7.4.  69  Figure 7.4: Algorithm for solution using traditional EMTP  7.1.3  Computational Bottlenecks  The critical (time-consuming) aspects of the simulation process from the preceding subsections are: 1. Calculation of eigen-values and eigen-vectors. 2. Inversion of eigen-vector matrix. 3. Dense multiplication of nominal impedance matrix to correction matrix. 4. Inversion of Zseries matrix.  7.2  Eigen-decomposition  In classical EMTP methodology, due to the form of the propagation function, equation 7.5, where Rnominal is a diagonal matrix, the eigen-decomposition changes with each frequency 70  and these computations must be performed for every single frequency point. Where large multi-phase systems are involved, this severely affects the simulation time. However, for the problem of transformer windings being simulated at high frequencies, skin effect and more importantly, proximity effect come into play. When these are taken into consideration the losses become dependent on the inductance and capacitance matrices. Equations 7.8 and 7.9 show the new form of the nominal impedance and admittance equations, [58, 67]. 2ω [L] σµd2  [Ω]  (7.8)  Ynominal = (jω + ωtan(δ)) [C]  [S]  (7.9)  Znominal = jω +  where ω is the frequency in radians per second, rad s−1 . d is the distance between the turns. δ is the di-electric loss angle. µ = 4π10−7 N A−2 The propagation function γ can now be written as equation 7.10.  γ=  Znominal Ynominal =  jω +  2ω (jω + ωtan(δ)) ∗ σµd2  [L][C]  (7.10)  This is an extremely important result since the frequency dependent portion of γ is a onedimensional complex multiplier. The system to be diagonalized is now [L][C], which is independent of frequency. This means that the eigen-vectors of a given network topology only need to be found once, as opposed to at every frequency point. The eigen-values for different frequencies can then be found through a simple scaling. In the methods which involve direct calculation of either [L] or [C] (but not both), there is an additional simplification that can be made Since the per unit length inductance and capacitance are related through the velocity of propagation as shown in equation 7.11;  v=  1 LC  [ms−1 ]  The propagation function can then further be re-written as equation 7.12.  71  (7.11)  γ=  jω +  2ω 1 (jω + ωtan(δ)) ∗ [I] 2 σµd v  (7.12)  In these cases, since the system is already diagonal, the need to compute eigen-values and eigen-vectors is eliminated altogether. Since the [L][C] product is a diagonal matrix with equal entries, the eigen-vector matrix is the identity matrix and the eigen-values, λ, are all equal to the complex pre-multiplier of the identity matrix in equation 7.12, 2ω jω + σµd (jω + ωtan(δ)) ∗ v1 . The eigen-values can thus be explicitly calculated at ev2 ery frequency while the eigen-vector matrix is constant as the identity matrix, significantly speeding up the computation process.  7.3  Method 1: Simulation using Direct Capacitance Calculation  Figure 7.5 shows a cross section of a transformer windings with 3 winding layers and 4 turns per layer.  Figure 7.5: Turn arrangement and numbering for 3-disc, 4-turn transformer Considering capacitances to adjacent turns only, the capacitance matrix takes the form of equation 7.13.  72      ∗ ∗ ∗       ∗ ∗ ∗       ∗ ∗ ∗ ∗       ∗ ∗ ∗       ∗ ∗ ∗ ∗       ∗ ∗ ∗ ∗ ∗   [C] =     ∗ ∗ ∗ ∗ ∗       ∗ ∗ ∗       ∗ ∗ ∗       ∗ ∗ ∗ ∗       ∗ ∗ ∗ ∗   ∗ ∗ ∗  (7.13)  This is a sparse matrix with at most 5 non-zero elements per column. The inductance matrix is then calculated through equation 7.14.  [L] =  1 [C]−1 2 v  [Hm−1 ]  (7.14)  where v is the speed of light A full inductance matrix yields a full series impedance matrix. This would usually indicate that the series admittance matrix is also full. However, since the inductance matrix was calculated using the sparse capacitance matrix, the series admittance matrix is also sparse and can be calculated directly from the capacitance matrix. Section 7.2 described how the eigen-vector matrix of the propagation function is equal to the identity matrix and the eigen-values are all equal. This means that the hyperbolic correction matrix of equation 7.6, [Λ] × sinh([λ]) × [Λ−1 ], is a diagonal matrix with all elements equal, [λ] k[I], where the hyperbolic correction factor, k, must be evaluated at every frequency step. The hyperbolically corrected impedance matrix can then be written as:  Zseries = k × Znominal  Zseries = k × jω +  73  2ω [L] σµd2  (7.15)  (7.16)  Zseries = k × jω +  2ω σµd2  1 [C]−1 = K[C]−1 2 v  Yseries = [Zseries ]−1 =  1 [C] K  (7.17)  (7.18)  The series admittance matrix is, therefore, sparse and has the same form as the capacitance matrix. This method is very fast as it does not contain any of the computational bottlenecks mentioned previously. In addition, the inductance matrix does not even need to be calculated as the [Yseries ] matrix can be obtained directly from the capacitance matrix. The nodal voltages can then be solved for using the traditional EMTP approach. The overall algorithm for this method is given in fig. 7.6.  Figure 7.6: Optimized algorithm for solution method using direct capacitance calculation only  74  7.4  Method 2: Simulation using Direct Inductance Calculation  This method starts off with the calculation of a full inductance matrix based on Maxwell’s equations for circular filaments. The capacitance matrix can then be obtained through the velocity of propagation using equation 7.19.  [C] =  1 [L]−1 v2  [F m−1 ]  (7.19)  where v is the speed of light in ms−1 . Once the capacitance matrix is found, the solution method is identical to the method for direct capacitance calculation, with the exception that the system admittance matrix is now a full matrix and the solution requires a dense linear solver. The overall algorithm is given in fig. 7.7.  Figure 7.7: Optimized algorithm for solution method using direct inductance calculation only  75  7.5  Method 3: Direct Calculation of both Capacitance and Inductance Matrices  When both the capacitance and inductance matrices are calculated separately, using the concentric cylinder equations for capacitance and the concentric ring equations for inductance, the [L][C] product was shown to be a full asymmetrical matrix, unlike the diagonal velocity matrices obtained for overhead transmission lines. As a result, the elimination of the eigen-decomposition can no longer be achieved. In addition to this, the [L][C] product is a full matrix, which gives rise to a full hyperbolic correction matrix. This is important to note, especially in the case of the admittance matrix. When the sparse admittance matrix is hyperbolically corrected it becomes full, implying that the correction introduces new admittances into sections of the system where none existed before. All sparsity in the system is thus lost. However, since the losses are still expressed as complex scalings of the inductance and capacitance matrices, (equations 7.8 and 7.9), the eigen-decomposition only needs to be performed once, and scaling factors for the eigen-values can be computed at every frequency step. Further to this, since the eigen-vectors for different frequencies differ only by scaling then they can be held constant. The series impedance matrix can be written as equation 7.20. [Zseries ] = k(ω) × [Znominal ] × [Λ] ×  sinh([λ]) × Λ−1 [λ]  (7.20)  × [Λ−1 ], only needs to be calculated once, and The hyperbolic correction term, [Λ] × sinh([λ]) [λ] the series impedance matrix, [Zseries ], at new frequencies can be obtained through scaling. A similar analysis applies for the shunt admittance system. Once this is done the simulation proceeds in the same manner as the algorithm for Method 2 (direct inductance calculation). The peculiarity of this method, in that the sparsity of the admittance system is lost due to hyperbolic correction can be overcome by simply ignoring the hyperbolic equations, resulting in a nominal pi circuit line representation instead of an exact pi circuit representation. Such an approximation gets better when the line length is short compared to the wavelength. At one megahertz frequency a transformer winding of diameter 1m would be approximately 1-3% of the wavelength making the approximation very feasible. Even though the shunt admittance matrix is sparse, the series admittance matrix is full. This means that the system admittance matrix would also be full and the sparsity of the shunt admittance system cannot be exploited using a traditional EMTP, nodal admittance approach. To circumvent this problem, a diakoptic approach is proposed. 76  7.5.1  Diakoptic Simulation  Since inversion is a costly computational process, the defining equations of the system have been re-written so that the series-impedances are left in impedance form and do not need to be inverted to admittance form. Consider the specific circuit to be simulated, shown in fig. 7.8.  Figure 7.8: Transformer circuit model All currents are defined as flowing from node k to node k + 1 and denoted by Ik−k+1 . The current injections to each node can then be written as: I1 = −Isrc−1 + I1−2 + (V1 )Y11 + (V1 − V2 )Y12 + (V1 − V3 )Y13 + .... + (V1 − Vn+1 )Y1,n+1  I2 = −I1−2 + I2−3 + (V2 − V1 )Y21 + (V2 )Y22 + (V2 − V3 )Y23 + .... + (V2 − Vn+1 )Y2,n+1  I3 = −I2−3 + I3−4 + (V3 − V1 )Y31 + (V3 − V2 )Y32 + (V3 )Y33 + .... + (V3 − Vn+1 )Y3,n+1  In+1 = −In−n+1 +In+1−g +(Vn+1 −V1 )Yn+1,1 +(Vn+1 −V2 )Yn+1,2 +(Vn+1 −V3 )Yn+1,3 +....+(Vn+1 )Yn+1,n+1 (7.21) It should be noted that these equations do not contain information relating to the series impedances of the transmission lines. The node voltages are next described by the following link equations:  Vsrc = Vsrc (known)  77  V1 = Vsrc − Isrc−1 Rin  V2 = V1 − (I12 Z11 + I23 Z12 + I34 Z13 + ...... + In,n+1 Z1,n )  V3 = V2 − (I12 Z21 + I23 Z22 + I34 Z23 + ...... + Inn+1 Z2,n )  V4 = V3 − (I12 Z31 + I23 Z32 + I34 Z33 + ...... + Inn+1 Z3,n )  Vn+1 = Vn − (I12 Zn1 + I23 Zn2 + I34 Zn3 + ...... + In,n+1 Zn,n )  (7.22)  Since there are no current sources in the circuit, the L.H.S. currents, Ik , in equation 7.21 are all zero. Equation 7.23 shows the matrix form of the system of equations.  78    79                                      Isrc 0 0 0 .. . 0 0 0 0 0 .. . 0 0                                                                              =  ∗ 0 0 0 .. .  ∗ Y11 Y21 Y31 .. .  ∗ Y12 Y22 Y32 .. .  ∗ Y13 Y23 Y33 .. .  ∗ ... ... ... ...  ∗ Y1,n+1 Y2,n+1 Y3,n+1 .. .  ∗ −1  0 Yn+1,1 Yn+1,2 Yn+1,3 . . . Yn+1,n+1 0 1 −1 0 −Rin 0 1 −1 0 0 1 −1 0 .. . 0 0 1 .. .. . 0 . 0 −1 0 0 0 1 0  ∗ 1 −1  ∗ 1 −1  ∗  ∗  ∗  ∗ 0  1 ...  0 0 0 −Z11 −Z12 −Z13 −Z21 −Z22 −Z23  ...  −1 0 0 . . . −Z1,n . . . −Z2,n  1 0 0 0  −Z31 −Z32 −Z33 . . . −Z3n 0 .. .. .. .. .. . . . . . 0 −Zn1 −Zn2 −Zn3 . . . −Znn 0 0 0 0 0 0 −Rout                                       Vsrc V1 V2 V3 .. .                  Vn+1    Isrc−1    I1−2    I2−3    I3−4    ..   .   In−n+1    In+1−gnd  (7.23)  This matrix can be partitioned into 9 submatrices as shown in equation 7.24.       0 0 0            =     Yaa Yab Pua Yba Ybb Pub Pla Plb −Zseries        Vk Vu Ilink        (7.24)  Where Yaa , Yab , and Pua cannot be defined at this stage due to the zero impedance seen at the voltage source node. The P (peeking) matrices represent the connections between the admittance (Y ) and impedance (Z) subsystems and are very sparse (maximum of 2 entries per column). The Y subsystem is built from the capacitance matrix of the winding structure as shown in fig. 7.3, resulting in a sparse matrix. The Z subsystem represents the series impedance of the transmission system. This is obtained through direct calculation of the inductance matrix. The Zseries and Ybb matrices are calculated from the inductance and capacitance matrices for the first frequency point and scaled for subsequent frequency points. Once the submatrices of equation 7.24 are set up. The sparsity of the Y and P subsystems can exploited using UMFPACK, [68, 69, 70, 71], for fast solution as follows:  0 = [Yba ][Vk ] + [Ybb ][Vu ] + [Pub ][Ilink ]  (7.25)  0 = [Pla ][Vk ] + [Plb ][Vu ] − [Zseries ][Ilink ]  (7.26)  However, Yba is 0 because the only element connecting the source to the rest of the system is Rin which is accounted for in Zseries matrix in the link equations. Equation 7.25 then becomes:  0 = [Ybb ][Vu ] + [Pub ][Ilink ]  (7.27)  [Vu ] = −[Ybb ]−1 [Pub ][Ilink ]  (7.28)  and Vu can be written as:  Substituting for Vu in equation 7.26 : 0 = [Pla ][Vk ] − [Plb ][Ybb ]−1 [Pub ] + [Zseries ] [Ilink ] 80  (7.29)  Let Zp = [Ybb ]−1 [Pub ]  [Ybb ][Zp ] = [Pub ]  (7.30)  Equation 7.30 represents the first computation step of the solution. Since Ybb is sparse, UMFPACK is again used to solve for Zp . Equation 7.29 becomes: 0 = [Pla ][Vk ] − ([Plb ][Zp ] + [Zseries ]) [Ilink ]  (7.31)  The second step of computation is to evaluate [Zt ] = [Plb ][Zp ]+[Zseries ]. Since Plb is a peeking matrix with just a single one and a single negative one per column, a custom routine is used to carry out the matrix multiplication to avoid wasting time in multplications by ones and zeros that would take place in generic multiplication routines. The vector Pla is zero except for the first element which is one. This allows the product [Pla ][Vk ] to be built explicitly. Equation 7.31 is now reduced to its final form:  [Zt ][Ilink ] = [Pla ][Vk ]  (7.32)  Since the Zt matrix is dense, LAPACK, [72], compiled against Goto’s BLAS, [73, 74] is used for fast linear solution. This method results in a solution for the branch currents. The input and output voltages can then be obtained by calculating the voltage drop across the voltage source and ground impedances. The overall algorithm is illustrated in fig. 7.9.  81  Figure 7.9: Optimized algorithm for method using direct calculation of both inductance and capacitance matrices  7.6 7.6.1  Results Algorithm Comparison  The following important features of the new methods can be noted: 1. Moving the building of the Z & Y matrices outside the frequency loop. 2. Replacing costly eigen-decomposition with a simple scaling to find hyperbolic correction factors for methods 1 and 2. 3. Eliminating dense matrix multiplication in applying hyperbolic corrections for methods 1 and 2. 4. Replacing dense matrix inversion of Zseries with sparse linear solution of Ybb for method 3.  82  Tables 7.1 and 7.2 provide comparisons of the computational complexity of each method in terms of computational processes inside and outside the frequency loop. EMTP  direct C calculation  direct L calculation  Build dense L matrix  Build sparse C matrix  Build dense L matrix  direct L and C calculation Build dense L matrix  dense inversion to get C matrix  Build sparse C matrix  Build dense C matrix  Table 7.1: Overhead for various simulation methods EMTP  direct C calculation  direct L calculation  direct L and C calculation 1 dense scaling  Dense eigen-decomposition 6 dense multiplications (hyperbolic corrections) Dense inversion of series impedance matrix Dense Linear solution of system admittance matrix  2 sparse scalings  2 dense scalings  build sparse admittance matrix  build dense admittance matrix  1 sparse scaling  sparse linear solution  dense linear solution  1 sparse linear solution 1 gathering function  1 dense linear solution Table 7.2: Important computations inside frequency loop for various simulation methods  7.6.2  Performance Comparison  Of the algorithms developed, the diakoptic method presented in method 3 would set the lower limit of performance increase that could be gained in using the optimized algorithms. The separate calculation of inductance and capacitance matrices has also become the most common choice of modelling method in recent times. Table 7.3 shows the number of clock ticks vs number of frequency points for systems of varying sizes. In practice, the inductance matrix was estimated through the capacitance matrix as in method 1, but left in series impedance form. The simulations were carried out on an Athlon 64, 2GHz, 1GB RAM machine running the Kubuntu 8.04 Linux operating system. Systems with varying numbers of phases were tested 83  and the number of clock ticks (in tens of thousands) taken to simulate varying numbers of frequency points were recorded in table 7.3. Num freq pts. Sim type System size / phases 10 50 100 200 500 1000  1  1  Diakoptic EMTP  0 0 2 9 61 296  0 1 5 26 324 2,913  10  10  Diakoptic EMTP  0 3 13 57 493 2,371  0 9 40 245 3,229 29,367  100  100  Diakoptic EMTP  1 28 135 612 4,935 23,805  3 72 390 2,315 31,165 255,920  Table 7.3: Time taken to perform simulation in tens of thousands of clock tics Figure 7.10 illustrates the raw trends evident from table 7.3. The legend uses the notation D for the Diakoptic simulation method and E for conventional EMTP.  84  Figure 7.10: Trends in performance for varying number of frequency points and system sizes The computational speed increase gained from using the diakoptic method can be estimated by dividing the number of clock ticks for the conventional EMTP approach by the number obtained for the diakoptic approach. Figures 7.11 and 7.12 show the speed increases obtained against the system size (number of phases in transformer model) and against number of frequency points calculated. Further, fig. 7.13 trends the average speed increase as the system size increases.  85  Figure 7.11: Speed increases as system size varies  86  Figure 7.12: Speed increase as number of frequency points varies  87  Figure 7.13: Trend in average speed increase as number of frequency points varies The performance trends show not only that significant speed increases (2-10 times) can be gained, but that as the number of phases modelled increases, the performance gap between both methods also increases.  7.7  Summary  Efficient methods of simulating transformer windings, represented by multi-phase transmission systems, were developed and shown to give significant performance benefits especially for larger systems where an order of magnitude speed increase was recorded. In addition to UMFPACK which was used to handle sparse matrix computations, the software developed made use of the state-of-the-art linear algebra computational packages, GotoBLAS and LAPACK, to speed up matrix operations. Since the conventional EMTP method was also coded using these packages, even greater performance gains can be expected if compared 88  to commercial EMTP packages which may not make use of these optimized libraries. For example, [75], quotes 8 hours required to simulate 1000 frequency points of a system of 236 sections. Using the simulation method developed, along with optimized linear algebra packages, this could be done in under 2 minutes. Machines with multi-processor architectures can give further peformance benefits by simply compiling GotoBLAS for a specified number of threads. The performance gains make the generation of large training sets required for neural network training very feasible, even where transformers with a large number of turns need to be simulated.  89  Chapter 8 Signature Interpretation Interpretation of differences observed in signatures is the main problem facing transfer function diagnostic methods such as FRA and TLD. The complex nature of the transformer winding causes shifts in resonant frequencies, addition or subtraction of new resonances, etc. There is no straightforward way to relate these changes to actual physical changes inside the transformer. It is quite possible that a number of network topologies may have very similar frequency responses. For example, the LC-mesh that represents the actual electrical network of the transformer can be approximated by a set of admittance branches as in vector fitting solutions, [51], [76] . However, for a given network topology, specific changes in that network may cause unique changes in the shape of the impedance and admittance functions. This is the basic premise of the work being presented. Consider the simple network of fig. 8.1:  Figure 8.1: Simple LC circuit The input impedance can be written as: 90  Zinput  XC1 XL2 + XC1 Rload = jωL1 + = XL1 + XC1 + XL2 + Rload  1 1 jωL2 + jωC Rload jωC1 1 1 + jωL2 + RLoad jωC1  (8.1)  A change in the inductance L1 , ∆L1 , will result in the modification of the input impedance:  Zinput = jω(L1 + ∆L1 ) +  1 1 jωL2 + jωC Rload jωC1 1 1 + jωL2 + RLoad jωC1  (8.2)  Similarly a change in inductance L2 will result in:  Zinput = jωL1 +  1 1 jω(L2 + ∆L2 ) + jωC Rload jωC1 1 1 + jω(L2 + ∆L2 ) + RLoad jωC1  (8.3)  If the modified impedance function due to a change in L2 is not unique, then it should be possible to compute a value of L1 which would yield the same impedance function.  jω(L1 + ∆L1 ) +  1 1 jωL2 + jωC Rload jωC1 1 1 + jωL2 + RLoad jωC1  = jωL1 +  1 1 jω(L2 + ∆L2 ) + jωC Rload jωC1 1 1 + jω(L2 + ∆L2 ) + RLoad jωC1  (8.4)  The equivalent change in inductance, ∆L1 , is related to the change, ∆L2 , through equation 8.5  ∆L1 =  (L2 + ∆L2 ) + 1 − ω 2 L2 C1 + jωRC1  Rload jωC1 + ω 2 ∆L  2 C1  −  L2 +  Rload jω  1 − ω 2 L2 C1 + jωRload C1  (8.5)  From equation 8.5, it can be seen that an inductance in a different location would have to be frequency dependent in order to give rise to the same impedance function. Since, at high frequencies, inductances are dependent solely on the geometry of the winding, they can be considered constant over the frequency range. The example thus illustrates that changes in the shape of the impedance or admittance functions contain the information about the specific location of winding which has undergone change. Another way of thinking of the uniqueness problem arises out of the transmission line representation of the transformer. An impulse applied to the winding would produce a reflection pattern that is dependent on the location of discontinuities in the winding. Deformations occurring in different locations would cause different reflection patterns in the response of the winding. Since these patterns directly relate to the frequency response of the winding it can be assumed that the location of a deformation has a unique effect on the frequency response. 91  This property of the winding signatures indicates that the problem of localization of winding movement would be separable in a virtual problem space, making neural network approaches a viable classification strategy. A convenient way of visualizing this separability is with the system admittance matrix. Consider a system of 4 winding layers, with couplings to adjacent layers. The form of the system admittance matrix would resemble that of fig. 8.2.  Figure 8.2: System admittance matrix form for 4-layer tranformer Radial distortions of the winding, for instance, would show up primarily in the the diagonal blocks, since they contain the couplings between turns within a layer. Axial distortions, on the other hand, would show up primarily in the off-diagonal blocks which contain couplings between one layer and another.  8.1  Simulation Details  To simplify the localization problem the transformer was sectionalized, with each section containing a number of winding layers, as illustrated in fig. 8.3. The important dimensions of the transformer are also illustrated.  92  Figure 8.3: Sectionalized transformer winding with important dimensions The winding sections specify portions of the windings which have the same dimensional data, (dip , dit ). It is not necessary that sections all have the same number of layers as the software developed reads vectors of values for the dimensional data shown in fig. 8.3. This allows winding movement to be applied to arbitrary sections of the winding.  8.1.1  Transformer Parameters  8.1.1.1  Base Model  The drawing board data for the transformer to be modelled as the base case is given in table 8.1. The parameters of the model were estimated to reflect the conductor size and geometrical arrangement typical of power transformers in the 500kV range.  93  Parameter Number of pancakes Number of turns per pancake Number of Sections Inter-pancake distance inter-turn distance Inter-section distance radius of innermost winding core radius tank radius conductor height conductor width relative permeability relative permittivity  Value 50 10 10 5mm 2mm 5mm 0.5m 0.45m 0.75m 1cm 0.5cm 1 2.3  Table 8.1: Base transformer parameters  8.1.1.2  Deformation Models  Deformations were modelled by changing the drawing board data of table 8.1. For example, radial deformations would show up as changes in the inter-turn distances while axial deformations would show up as changes in the inter-pancake and inter-section distances. These changes in physical distances are then used to calculate new inductance and capacitance matrices associated with the deformed transformer which are then used for simulation. It should be noted that, due to the strong electromagnetic forces expected during short circuit conditions, the transformer windings are kept under considerable compression forces. These forces prevent the winding from moving to any significant degree. As a result, expansions or compressions in the winding, even in the order of millimetres represent significant deformation. It should also be noted that although the change in separation between two turns may be fractions of a millimetre this represents a significant change in capacitance between the turns since the normal separation between turns is in the order of 1mm. As mentioned before, the winding was divided into 10 sections, each comprising 5 winding layers. For each section the training data was obtained by making changes to the inter-turn distance in that layer for radial deformations and changes in the inter-pancake distance for the axial deformations. For radial deformations, 50 increments were used equally spaced starting at at 2mm (no distortion, just the insulation thickness) to 1cm between turns (10 cm total expansion in radius of the entire winding). For the case of axial compressions, 50 increments were also used, starting at 5mm spacing for no distortion and ending at 2mm spacing (representing just the insulation thickness). 94  The validation data for each case were chosen to be the midpoints of the spacing increments for the training data. For example, if the training cases represented were distances of 1mm, 2mm, 3mm etc., then the validation cases would be chosen as 1.5mm, 2.5mm etc. This data set should represent the worst case for the classifier as the patterns are as far as possible from the trained patterns. If the network could accurately classify them, it would mean it generalizes well for all other cases.  8.1.2  Cluster Simulations  In addition to the customized algorithms, and optimized linear algebra routines used for solving the system, parallelization was also used to decrease total simulation time. Unlike transient simulations where the solution at a given time is dependent on the solution at past time points, steady state simulations at a given frequency are completely independent of simulations at other frequencies. This property makes the simulations inherently parallelizable. The Large Area Real-time Appication (LARA) cluster at the UBC Power Lab, [77], was used to run simulations in parallel. Figure 8.4 illustrates the cluster configuration.  Figure 8.4: Cluster configuration for parallel simulation The simulation set, comprised of 500,000 frequency points with the following breakdown; 500 points per characteristic, 100 characteristics per section, 10 sections per transformer. Ten nodes of the 16 available were used, one for each section, resulting in a computational load of 50,000 simulations per CPU. The total time taken for all simulations was approximately 7.5 hours. Figures 8.5 and 8.6 show examples of the characteristics obtained for 1cm radial winding movement at different sections of the winding. 95  Figure 8.5: Characteristic impedance signatures for radial movement at different winding positions  Figure 8.6: Transadmittance signatures for radial movement at different winding positions  8.2  Neural Network Classifier  A neural network classifier was decided upon because of the following properties of the system:  96  1. The problem is believed to be separable. That is, the shape of the characteristics contain information pertaining to the location and type of change that has occurred. 2. The relations and patterns, although present, are too complex to humanly discern so that a rule base cannot be built manually. Two types of networks were identified as being suitable to the problem: 1. Backpropagation (BP) networks. 2. Radial Basis Networks (RBNs).  8.2.1  Back Propagation (BP)  Back-propagation perceptron networks are usually the choice where neural network based classifiers are used in transformer diagnostic procedures such as DGA. It has also been used in the very few cases of neural network classifiers used in interpretation of FRA characteristics. Back Propagation networks utilize a perceptron network to draw a non-linear decision boundary between data sets on a virtual problem space. The networks usually consist of three perceptron layers; an input layer, output layer and intermediate or hidden layer as shown in fig. 8.7.  Figure 8.7: Perceptron layers in backpropagation network 97  The size of the input and output layer, that is, the number of perceptrons in those layers, correspond to the dimensions of the input and output data sets. However, the size of the hidden layer is left to the designer of the network. If too few perceptrons are used, the network may not converge to an acceptable level of error. If, on the other hand, too many perceptrons are used, the network overfits the data yielding a network which can recognize familiar patterns but does not have the ability to generalize what it has learned to new data. The problem of choosing the optimal number of perceptrons for the hidden layer is not a simple one and trial and error remains the most reliable means. In addition to this, the network requires repetitive training. That is, the inputs and outputs are applied to the network over and over again (epochs) until the the weights in the network converge. The number of epochs required for the error to be acceptable may run into the tens of thousands for complex mappings. This can become very time consuming especially for networks with a large number of perceptrons. Regardless of these shortcomings, BP networks are still the most widely used for classification problems of this type because, as long as the problem is separable, a properly designed network is guaranteed to find a solution. They are also preferred because of familiarity, as they have been in existence for a relatively long period of time and are amongst the more popular and easy to understand neural networks. This network was found to give the best performance for the classification problems of deciding type and location of the change in the winding, however, it performed very poorly as a quantifier.  8.2.2  Radial Basis Networks (RBNs)  RBNs are another type of neural network that are gaining popularity for classification problems. These networks were found to outperform the BP networks for the problem of quantification, however, they did not perform the task of classification well. These networks typically require more neurons than BP networks, but they can be designed in much shorter time because they do not rely tens of thousands of epochs of training. The neuron model as illustrated in Matlab R is shown in fig. 8.8, [78].  98  Figure 8.8: Neuron model for RBN The neuron model is quite different from the well known perceptron model. Whereas the perceptron uses weights to scale its inputs, the RBN neuron sees the inputs and weights as two vectors and computes the distance between them via the dot product. The ouput from 2 this stage is then operated on by the activation function e−n , illustrated in fig. 8.9, [78].  Figure 8.9: Activation function for radial basis function A special property of this network is that it can remember input patterns perfectly. The ability to interpolate between inputs for new data, however, must be tweaked by adjusting the spread factor. Another peculiarity is that the network architecture is not built solely on the RBN neurons. The network is usually built with one RBN layer and a linear layer as shown in fig. 8.10, [78].  99  Figure 8.10: RBN network architecture  8.2.3  Pre-conditioning of Input Data  An important facet of neural network classification is the preconditioning of the input data. The frequency response of a transformer is essentially a series of values corresponding to the characteristic impedance (or transadmittance) of the winding. These values can be used directly as inputs to the classifier. However, since it is the change in shape that contains the useful information, the difference between a plot and the baseline signature would be more useful. Figure 8.11 illustrates a difference plot for a radial deformation in the 5th section of the winding.  Figure 8.11: Network inputs based on difference plot 100  Immediately evident is the numerical dominance of the lower frequency inputs (lower numbered inputs). As a result of this, the features of the data at the higher end of the system may be drowned out. There is need, therefore, for some type of normalization. This is achieved by using percentage change plots which in effect scale the change noticed at a particular frequency by the value of the baseline plot at that frequency, as shown in fig. 8.12.  Figure 8.12: Network inputs based on percentage change plot The higher frequency inputs are still considerably lower in value than the low frequency inputs. However, this now gives a better picture of how the data features change, which can be useful in reducing the size of the network, and also the number of simulations that need to be performed. If the most useful data can be shown to lie in a particular frequency band, then the number of simulated frequency points can be restricted to that band.  8.2.4  Multi-layer Classification  Three pieces of information are desired from the classification scheme; type of movement, location of movement and degree of movement. The features of the data associated with each of these may be different. Using the admittance matrix visualization, the type is reflected in the change being in the diagonal or off diagonal blocks, the location of the fault further specifies a sub-section of either the diagonal or off diagonal blocks and the degree of movement the magnitude of change in the values of that block. In practice, using a single network to perform all classifications simultaneously did not produce good results,  101  therefore, separate networks geared toward the specific functions were designed. Figure 8.13 illustrates the multi-layer scheme used for classification.  Figure 8.13: Multi-layer classfication scheme Each box in fig. 8.13 represents a separate network. The topmost layer uses the largest network and decides whether the movement is radial or axial. This network is trained with the complete set of input training data available. The result determines whether the radial or axial localization network of the second layer would be used. The radial and axial localization networks are trained only on cases of radial or axial deformation respectively. The output from the localization layer is a section number. The section number determines 102  which network in the third layer will be used. These third layer networks are trained only on deformations of a certain type and location and output an index corresponding to a certain degree of movement (DoM). Figure 8.14 illustrates the decisions the network has to make using the admittance matrix visualization.  Figure 8.14: Admittance matrix visualization of network responsibilities for a system 4 layers The first layer determines if the change has occurred on the diagonal or off diagonal of the admittance matrix. The second, looks only at that region and further determines which subsection has changed. The final network determines the degree of change that has occurred in that specific block. As an example, the outputs from the layers may be: • Layer 1: Radial • Layer 2: Section 8  103  • Layer 3: 0.5 - 0.8 mm expansion By trending the movement over time, an operator would be able to tell the rate at which the winding structure is moving, and can better determine the proper time for performing maintenance, retiring the unit as well as determining how much units can be safely overloaded.  8.3  Results  Signature corruption can arise out of noisy measurements as well as the mismatch between the physical transformer and circuit model responses. Both of these change the shape of the difference curves which are used to train the networks. Since the shape of these curves are the basis on which classification is made, it is critical that the networks be able to create mappings that are able to accurately classify data corrupted by noise as well as differences arising out of model mismatch. The networks’ performance in the face of both of these is tested by making random changes to the input curves fed to the classfier. The following subsections detail the results of presenting the networks with such data.  8.3.1  Layer 1: Radial / Axial Type Classification  Type classification involved training a back propagation network with 1000 input patterns; 500 radial deformation patterns and 500 axial deformation patterns. The network output ”1” for radial deformation and ”-1” for axial deformation. Figure 8.15 showing the errors in classificaton, illustrates that the network was able to achieve near perfect classification of the validation data, with the sum squared error of the output data being in the order of 10−9 .  104  Figure 8.15: Errors in layer one classification of validation data Figure 8.16 shows that this layer is very resilient to noise, even for a noise level which causes 50% change in the input data. This indicates that the data features associated with the movement type are very distinct and well separated on the problem space, a feature that is especially important in compensating for model mismatches.  Figure 8.16: Error levels for noisy inputs to layer 1  105  8.3.2  Layer 2: Localization Layer  The localization layer comprises two networks, for radial and axial movements separately. Here again back propagation networks were chosen. Each network was trained only with the data pertinent to it. That is, the radial network was trained with 500 input patterns containing radial deformations and the axial network trained with 500 patterns containing axial deformations. Each network contained ten outputs corresponding to the ten sections of the transformer winding. A ”1” was output at the changed section and ”-1” for the others. The networks for both deformation types perform near identically. The error surface given in fig. 8.17 shows that the radial localization network fails to classify properly for the points which represent the least movement at each section. Typically, the initial 1 to 3 cases of a section (ridges at regular intervals) give errors. Beyond those, the network error is very small (within 2% of the target value).  Figure 8.17: Errors in layer 2, movement localization Figure 8.18 shows that this layer also has a high tolerance to noise, indicating once more that the data features are well separated. Since the separation on the problem space is similar to the layer 1 case, in that both can be visualized as different sections of the system admittance matrix, it is expected that the data features share similar characteristics in terms of noise tolerance.  106  Figure 8.18: Error surface for layer 2 with noisy input  8.3.3  Layer 3: Quantification Layer  The quantification layer comprises 20 networks (10 each for radial and axial). Each network comprises a Radial Basis Network, trained with 50 input patterns corresponding to the incremental changes in that section. These networks are not as robust in dealing with noise influences as the upper layers of the scheme. Figure 8.19 shows the estimated displacement (target) vs the actual displacement (validation set). Although the network shows some inaccuracy as the noise level increases, the movement trend is still followed well enough for an operator to make a decision based on the state of the winding as the max error is less than 5% even for 20% signal corruption.  107  Figure 8.19: Layer 3 network errors due to noisy inputs  8.4  Summary  As previously stated, the combination of these networks would give the Asset Manager information relating to the type, location and severity of winding movement represented by a change in characteristic impedance (or transadmittance). This can be integrated into a visualisation tool that would present the Asset Manager with a 2D or 3D map of the movement in the transformer sections, allowing them to perform a virtual inspection of the winding structure.  108  Chapter 9 Conclusions and Future Work 9.1  Contributions to Condition Monitoring of Power Transformers  This thesis makes the following contributions to the area of condition based monitoring of power transformers:  9.1.1  A Classification Scheme for Winding Movement  A multi-layer neural network approach based on analytical training sets was developed to localize deformations in EHV and UHV power transformers as well as quantify the extent of deformation. Such a system would be a useful tool for Asset Managers of power systems, enabling them to identify units which are in danger of failing, or allowing them to make informed decisions on whether transformers can be temporarily overloaded for economic spot-market strategies for energy sales. The system comprised two backpropagation layers and a single radial basis layer. The first stage classfication was used to identify whether the winding movement was radial or axial in nature. The second layer determined the location of the movement and the third quantified the degree of movement. The networks were shown to extract data features very well and also had very high tolerance to signature corruption which could arise out of noisy measurements or model mismatches between the real transformer response and the circuit model response.  109  9.1.2  Fast Simulation Methods  The neural network approach developed requires a large number of training sets to learn the problem properly. Since this is not available from actual field data, established analytical models, which are accurate over the frequency range of interest, were used to simulate winding movements. Traditional simulation techniques for generic circuit simulation are not optimized for solving the particular problem at hand. Therefore, customized algorithms were developed to take advantage of the special properties of the matrix representation of the sysem. This allowed hundreds of thousands of frequency points to be simulated in a matter of hours, whereas it would normally take months. These methods would also be of use to the study of fast overvoltages and high-frequency interwinding resonances where the current solutions have a great deal of redundancy that considerably lengthens the simulation times required.  9.2 9.2.1  Other contributions Anomalies in Simulation Techniques  The analysis of simulation techniques used in multiphase transmission line modelling of transformers has led to the exposure of several anomalies that do not exist in classical modelling of overhead transmission lines. Assumptions made with regard to the form of the capacitance matrix, the relationship between the impedance and admittance matrices and the geometry of the windings, give rise to dense complex velocity matrices which sometimes exceed the speed of light, negative capacitances and also dense impedance matrices which have sparse admittance forms.  9.2.2  Improvements to reduction methods  A summary of reduction techniques has been provided. In addition to this, several improvements were made to existing reduction methods: • A method for distributing the capacitances in a capactive network was developed. • A simple multiphase extension of the STL method was developed by encorporating capacitances to adjacent winding layers. • A hybrid transmission line was developed which combined transmission line models with parallel capacitance branches. 110  9.3 9.3.1  Future Work Physical Validation  The next stage of this project should be the development of a modular test transformer of realistic size. This would enable more rigorous model validation. It would also faciltate the development of a more detailed and practical procedure for extracting an accurate and precise circuit model from a given transformer which could then be used as the basis for the analytical training set. Without this step, further investigation into software methods of classification etc. would not be very useful for practical applications.  9.3.2  Improving Winding Damage Models  The current winding damage models reflect incipient movements in the windings. They allow for detailed changes but are none-the-less very simple. The spacing between winding layers or turns are changed to reflect winding movement. Since the changes may also take place in a combination of ways, alternative approaches to generating training sets may be required. One possibility for this is manipulating the system admittance matrix directly instead of the physical dimensions of the system. For instance, making random changes to known rows and columns of the admittance matrix ( and upating the diagonal ) may allow the network to be more robust in terms of the complexity of the deformations that it can successfully locate.  9.3.3  Hardware development  If the method is to be made into a fully online technique, proper instrumentation and signal processing needs to be developed that would allow a high degree of repeatability and low signal to noise ratio. A technique for eliminating the effect of network changes is also required. These elements of the online system are currently under development at the University of British Columbia by Singh and De Rybel.  9.3.4  Asset Management Scheme  The work presented here is an interpretation scheme for one condition monitoring method on a single piece of equipment. An asset manager, however, would have to deal with multiple sets of such data for multiple assets. To enable this, a number of infrastructures need to be 111  developed: A database of critical assets with information linking the operation of each asset to its financial information (including cost of maintenance); a focussed maintenance scheme involving extensive networks of sensors to track asset condition in terms of reliability and performance capability; knowledge of the inter-dependencies between critical assets. Once these facilities are in place a reliable method of simulating the asset inter-dependencies to determine the most cost effective time for maintenance, replacement or asset strengthening needs to be developed. Such a method would draw information from extensive sensor networks to determine the present state of operation of the system. This would be combined with a system model interconnecting the system components to each other to allow the asset manager to simulate the effect of decisions on the system. 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The Mathworks, 3 Apple Hill Drive 3 Applehill Drive, Natick, MA 01760-2098, U.S.A., 10 ed., March 2008.  119  Glossary AM - Asset Management AM/FM/GIS - Automated Mapping and Facilities Management and Geographic Information Systems ANN - Artificial Neural Network BLAS - Basic Linear Algebra Subprograms CBM - Condition Based Monitoring DGA - Dissolved Gas Analysis DR - Dornenburg Ratios ECM - Equipment Condition Monitoring EHV - Extra High Voltage EMTP - Electro-Magnetic Transients Program FRA - Frequency Response Analysis GIS - Geographic Information System LAPACK - Linar Algebra PACKage M&I - Maintenance and Inspection MMS - Maintenance Management System O&M - Operation and Maintenance OM - Operational Mode RCM - Reliability Centred Maintenance RR - Rogers Ratios SLA - Service Level Agreements TCG - Total Combustible Gas 120  TDR - Time Domain Reflectometry TLD - Transmission Line Diagnostics UHV - Ultra High Voltage UMFPACK - Unsymmetric Multi-Frontal PACKage WMS - Work Management System  121  Chapter 10 APPENDIX A: Parameter Calculation for Overhead Lines 10.1  Capacitance  Consider a single overhead transmission line of radius r over a perfectly conducting horizontal ground plane as shown in fig. 10.1  Figure 10.1: Conductor over perfectly conducting ground plane The problem that needs to be solved is finding the capacitance between the line and ground. There is no well known way to calculate the capacitance between a point and plane directly. 122  Through the method of images, however, the ground plane, which is of uniform potential, can be replaced by an image of the transmission line below it and of opposite potential to the actual line charge. This is illustrated in fig. 10.2.  Figure 10.2: Replacement of ground plane with image conductor The capacitance can now be calculated according to the equations relating systems of charge distributions. The approach to be taken can be generalized as: 1. Assume a charge distribution. 2. Calculate the electric vector-field distribution. 3. Integrate all non-zero field vectors between the two required points to get the potentials at those points and the potential coefficients. 4. Invert the potential coefficients to get obtain the capacitance. For the case of an infinite line (in the absence of a ground) , a cylindrical coordinate system is used and the electric field resolved into radial and axial components. Since the line is infinite the axial components cancel. The resulting radial field distrbution is quantified by equation 10.1.  Eradial =  123  ρl 2π R  (10.1)  Where R is the perpendicular distance from the line charge to the external point, and ρl is the line charge density. Finding the inductance follows the same technique, but starts with assuming a current distribution. For two external points R = a and R = b, where b > a the potential difference can be calculated by integrating the electric field, between them. It does not matter if the points are in different directions since no work is done in travelling along the equipotential circumference of a given radius, as illustrated in fig. 10.3  Figure 10.3: Equipotential lines around line charge The angular displacement between the points can always be brought to zero by travelling along the lines of equipotential, since no work (against the electric field) is required to travel along these lines. Integrating equation 10.1, the potential difference between the two points is obtained as equation 10.2.  Vab =  pl b ln 2π a  [V ]  (10.2)  An absolute potential can be derived by considering one point to be at zero potential at an infinite distance away. Interestingly, this cannot be directly substituted for in equation 10.2 but must be substiuted for in equation 10.1 prior to integration of the electric field. The absolute potential at a radial distance , R , then becomes equation 10.3. 124  Vabsolute =  ρl ln(R) 2π  [V ]  (10.3)  The potential at a point due to a di-polar line charge arrangement can be found by the superposition of the potentials due to each line charge independently. This yields equation 10.4.  Vabsolute,dipole =  Dp ρl ln 2π dp  [V ]  (10.4)  Where Dp is the distance to the image conductor and dp is the distance to the actual conductor from an arbitrary external point. The aforementioned approach is the method traditionally found in textbooks on electromagnetics. However, the same result for absolute potential can be gained, without having to consider points at infinity by considering the potential at the radius of the conductor and travelling from that point to the external point. In this case there is no need to initially revert to the electric field equation as is necessary if considering a theoretical ground at infinity. Potential difference from actual conductor to arbitrary point a:  Vpa =  ρl dp ln 2π r  [V ]  (10.5)  Potential difference from image conductor to arbitrary point a: Vpi = −  Dp ρl ln 2π r  [V ]  (10.6)  Absolute potential due to actual conductor to arbitrary point a: Vp1 = Va − Vpa  [V ]  (10.7)  Absolute potential due to image conductor to arbitrary point a: Vp2 = −Va − Vpi  [V ]  (10.8)  Total absolute potential:  Vp = Vp1 + Vp2 =  ρl 2π  ln  Dp dp − ln r r  125  [V ]  (10.9)  Vabsolute,dipole =  ρl Dp ln 2π dp  [V ]  (10.10)  The potential coefficients are defined as the voltage divided by the charge density and is equal to the inverse of the capacitance as shown in equation 10.11. 1 1 Dp =P = ln C 2π dp  10.2  (10.11)  Inductance  The magnetic field resulting from a theoretical differential current element has the same form, with the exception direction, as the electric field resulting from a differential charge element. As a result the solutions of integrating the electric and magnetic fields are largely similar. The inductance is calculated by considering the field associated with a steady current along an infinite line. The final equation used in calculation is gven in equation 10.12  L=  10.3  Dp µ ln 2π dp  (10.12)  Velocity of Propagation  The rectilinear velocity of propagation of the electromagnetic fields in a dielectric medium is determined by the permeability and permittivity of the medium as shown in equation 10.13.  v=  1 µ  [ms−1 ]  (10.13)  The µ product can also be calculated by the product of equations 10.12 and 10.11. Giving rise to the relationship between velocity, capacitance and inductance:  v=  1 LC  126  [ms−1 ]  (10.14)  Chapter 11 APPENDIX B : Reduction Methods The software required to carry out the simulations was written in Matlab R . Initial estimates on the computation time required to compute an estimated 500,000 frequency points ran to months of continuous computing (Athlon 64 2GHz, 1GB RAM) . This represents a serious obstacle to the use of analytical models in building training sets for neural based classifiers. One method commonly used to deal with this problem of lengthy simulation times is circuit reduction. This section examines a number of existing circuit reduction techniques and suggests simple improvements that can be made to them. The performance of each method was observed for different size systems. In total over 500 different scenarios were examined and a select few are reproduced in here.  11.1  Reduction to Capacitive Network  This simplification follows the logic that at high frequencies the series impedance of the transformer behaves increasingly like an open circuit while the shunt admittance behaves increasingly like a short circuit. The series impedance is thus neglected and the transformer is modelled using the shunt parameters of the system. It is important to note that this does not result in reduction of the system size. However, the shunt matrix is very sparse since only admittances to adjacent turns are modelled. This sparsity can be exploited for fast solution of the system. Typically the absence of the series impedance would yield a lumped capacitive system. However, the simulations here achieve distributedness by initially taking into consideration the series impedance to calculate hyperbolic correction factors which are then used to correct the lumped admittances.  127  Figures 11.1 to 11.9 show that this method performs poorly over the selected range of frequencies regardless of the system configuration.  Figure 11.1: C network reduction (3 pancakes, 2 turns per pancake)  Figure 11.2: C network reduction (3 pancakes, 7 turns per pancake)  128  Figure 11.3: C network reduction (3 pancakes, 15 turns per pancake)  Figure 11.4: C network reduction (10 pancakes, 2 turns per pancake)  129  Figure 11.5: C network reduction (10 pancakes, 7turns per pancake)  Figure 11.6: C network reduction (10 pancakes, 15 turns per pancake)  130  Figure 11.7: C network reduction 20 pancakes, 2 turns per pancake)  Figure 11.8: C network reduction (20 pancakes, 7 turns per pancake)  131  Figure 11.9: C network reduction (20 pancakes, 15 turns per pancake)  The capacitive reduction does not show any resonances in the frequency characteristics and is thus unsuitable for use in both fast overvoltage estimation as well as condition monitoring studies.  11.2  Single Transmission Line (STL) and Multiphase Single Transmission Line (MSTL)  The STL model first described in [58] is one of the most popular methods of circuit reduction used in the fast overvoltage estimation. This method maps the multiphase representation of each layer of the transformer onto a single phase line. The STL tries finds an equivalent frequency dependent characteristic impedance that would allow the end-to-end connected arrangement in the multiphase transmission system describing the transformer to be merged into a single homogenous line of length equal to the sum of the individual line lengths, while preserving the input-output characteristics. For a layer with capacitance to ground of Cg , inter-turn capacitance of Cit , a velocity of propagation v and average turn length of l , the equivalent characteristic impedance is given by equation 11.1.  ZC,equivalent =  1 v Cg + 2Cit 1 − cos( ωl v  132  (11.1)  Assuming lossless TEM propagation the equivalent inductance can be estimated through equation 11.2.  ZC =  L C  (11.2)  Each winding layer is considered to be an independent system (capacitances between pancakes are ignored) therefore the reduced system is a series connection of single phase lines. The shunt admittance matrix is thus diagonal, allowing for very fast solution, even with the re-calculation of the capacitance and inductance matrices for every frequency. It is interesting to note that the the inter-turn capacitance in the multi-phase system is mapped to a capacitance between the turn and ground in this reduced single phase system. The MSTL is an incremental addition to the STL. In this method the equivalent L and C are found just as in the STL case and just as before each pancake is reduced to a single transmission line. However, in the MSTL approach the inter-pancake capacitances are included in the system yielding a sparse, but not diagonal, capacitance matrix. The inductance matrix derived from the capacitance matrix is full. Figures 11.10 to 11.18 show how both methods compare with the full multiphase representation.  Figure 11.10: STL reduction (3 pancakes, 2 turns per pancake)  133  Figure 11.11: STL reduction (3 pancakes, 7 turns per pancake)  Figure 11.12: STL reduction (3 pancakes, 15 turns per pancake)  134  Figure 11.13: STL reduction (10pancakes, 2 turns per pancake)  Figure 11.14: STL reduction (10 pancakes, 7 turns per pancake)  135  Figure 11.15: STL reduction (10 pancakes, 15 turns per pancake)  Figure 11.16: STL reduction (20pancakes, 2 turns per pancake)  136  Figure 11.17: STL reduction (20 pancakes, 7 turns per pancake)  Figure 11.18: STL reduction (20pancakes, 15 turns per pancake)  The frequency response characteristics of both the STL and the MSTL follow the general shape of the full multiphase systems they model. However, the resonances evident in the multiphase systems are largely absent in the plots for both reduction methods. The impedance function is shifted downward. This shift is seen to increase as the number of turns per winding pancake, i.e. the number of turns mapped to a single turn, increases. It 137  can also be seen that when the number of turns per pancake is small, the STL and MSTL yield near identical results. As the number of turns per pancake increases the loss of the inter-pancake capacitances become more evident and the STL increasingly diverges from plots for both the MSTL and the full turn-by-turn system.  11.3  Hybrid Transmission Line (HTL)  The HTL model is a semi-distributed model, mixing both transmission line models and lumped parameters. Whereas the inter-turn capacitance in the STL methods outlined in the previous section are incorporated in the capacitance to ground, the HTL lumps the capacitance across the line representing the reduces the order of the system by bundling the turns in a winding layer together. The first step in reduction is the reduction of the capacitance matrix. This is done by summing the relevant elements of the capacitance matrix together. Equation 11.3 shows the reduction for a system of of 3 winding layers and 3 turns per layer. The summation term ΣC is the sum of the ground capaciance and the other capacitances in the row or column.                         ΣC −Cit −Cip −Cit ΣC −Cit −Cip −Cit ΣC −Cip −Cip ΣC −Cit −Cip −Cip −Cit ΣC −Cit −Cip −Cip −Cit ΣC −Cip −Cip ΣC −Cit −Cip −Cit ΣC −Cit −Cip −Cit ΣC                          (11.3)  ⇓        3Cg + 3Cip −3Cip −3Cip 3Cg + 3Cip −3Cip −3Cip 3Cg + 3Cip        The 9 phase system is thus reduced to a 3 phase system. It should be noted that the interturn capacitances Cit are absent from the reduced system. To compensate for this, these 138  capacitances are summed as in equation 11.4 and added in parallel across each equivalent transmission line as in fig. 11.19. 1 Cit,total  =  1 Cit,1−2  +  1 Cit2−3  + ...... +  1 Cit,(n−1)−n  (11.4)  Figure 11.19: Single winding layer representation using HTL reduction The inductance can be estimated in two ways: 1. Reduction of inductance matrix from full multphase system. 2. Inversion of reduced capacitance matrix. Reduction of the inductance matrix is carried out in exactly the same way as for the capacitance matrix in eq.11.3 and preserves the total inductance in the system. Method 2 preserves the TEM mode of propagation in the reduced system. Figures 11.20 to 11.28 show how both hybrid methods compare to the full multiphase system.  Figure 11.20: HTL 1 reduction (3 pancakes, 2 turns per pancake) 139  Figure 11.21: HTL 1 reduction (3pancakes, 7 turns per pancake)  Figure 11.22: HTL 1 reduction (3pancakes, 15 turns per pancake)  140  Figure 11.23: HTL 1 reduction (10pancakes, 2 turns per pancake)  Figure 11.24: HTL 1 reduction (10pancakes, 7 turns per pancake)  141  Figure 11.25: HTL 1 reduction (10 pancakes, 15 turns per pancake)  Figure 11.26: HTL 1 reduction (20 pancakes, 2 turns per pancake)  142  Figure 11.27: HTL 1 reduction (20 pancakes, 7 turns per pancake)  Figure 11.28: HTL 1 reduction (20 pancakes, 15 turns per pancake)  The results show an interesting trend for Hybrid Method 1, with the approximations getting better as the number of turns per pancake increases. The approximation also seems to get better with increasing frequency. Unlike the STL methods, resonances show up in the reduced system. However, the resonant peaks are higher than their counterparts in the full multiphase system. 143  With respect to Hybrid Method 2 (which preserves the TEM mode of propagation), the results show that the difference in inductance calculation significantly affects the accuracy of the approximation. Unlike method 1, which was the most accurate of the reduction methods presented, this variation of the method yields worse approximations than the STL methods, giving reasonable approximations in isolated cases, with no discernable trend in performance.  11.4  Summary  The Hybrid Transmission Line (HTL), preserving the total inductance of the system was shown to be the most accurate reduction method, giving good reproduction of resonances especially at higher frequencies, and performing relatively well even when many turns were grouped together. The commonly used STL method was also improved upon by introducing multiphase couplings into the system giving rise to the MSTL system which was shown to be more accurate than the STL method as the number of turns increases. While some of these methods, particularly the (M)STL and HTL models may be of use for calculation of transient over-voltages, they are not very useful for diagnostic purposes. This is because diagnostics rely on the comparison of frequency response characteristics. The reduced circuits would need to follow the response of the unreduced circuit very closely if they are to be useful, otherwise they would give results that may seem to indicate drastic changes in the transformer structure when in fact there are none.  144  Chapter 12 APPENDIX C: Isolating Transformer Response from External System The varying impedance of the external system, due to changes in system state, as seen by a transformer can have an effect on the measured impedance of the transformer itself. This is because the impedance of the external system is reflected within the transformer as shown in fig. 12.1.  Figure 12.1: System impedance reflected between measurement points The method being explored to overcome this problem is the insertion of low pass filters at 145  the transformer terminals as shown in fig. 12.2.  Figure 12.2: Impedance stabilizing filter on transformer output The general assumption is that filter will be transparent to the system frequency, but present a very high impedance to the high-frequency measurement signals being used. The filter would in effect dominate the system impedance at high frequencies so that changes in the system state have a negligible effect on the value of the reflected impedance. Two methods are being explored to achieve this: • Shunt filters. • Series filters. Line traps, designed for broadband over power line (BPL) applications, in essence, are shunt filters which divert the high frequency carrier signals from the system into signal receivers. A similar system can be used for shunting the high frequency test signals on the output of the transformer, in effect creating a constant, known termination, in place of the variable system impedance at high frequencies. This method is currently being simulated, with promising results, and plans to test a low voltage prototype are in the foreseeable future. A series filter design would involve an ungrounded current transformer on the transformer terminal with a high order filter on the secondary side reflecting a very low 60Hz impedance 146  and an extremely high impedance for the frequencies of the test signal. This design is still in the conceptual stage of development.  147  

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