Preface
The basic principle of protective relaying of power systems has not changed for
more than half a century. Almost all power system protective relaying algorithms
are dominated by integral transforms such as the Fourier transform and the wavelet
transform. The integral transform can only provide an average attribute of the signals
or their components. The accuracy of the attribute extraction is significantly
sacrificed by the assumption of periodicity of the signals if the integral transform is
applied to transient signals. It is also well known that the signals are liable to be contaminated
by noise in the formof exponentially decayingDC offsets, high frequency
transients, harmonic distortion, errors caused by non-linearity in the response of the
sensors, and unwanted behaviour of power systems. This contamination is often
provoked by fault conditions, just at the time when the protection relay is required
to respond and trip the circuit breaker to limit damage caused by the fault.
On the other hand, as we know, in most protection relays, complex computation
has to be undertaken within a sampling interval, no matter how small the interval, to
calculate the coefficients relevant to the attributes of the signals by using the integral
transform based on a window of samples, and to calculate the relaying algorithms,
which are derived to represent the relationship between these coefficients and power
system faults. If fast transients and high-order harmonics are to be addressed, extra
computing power and facilities are required. Therefore, it can be seen that the
current power system relaying algorithms suffer from many problems including accuracy,
fast responses, noise, disturbance rejections and reliability.
To tackle the problems of distorted waveforms, disturbances and transient components
of fault voltages and currents, identification of the shapes of complex waveforms
is ideally required instead of the analysis of periodic characteristics, which is
undertaken by the currently used integral transform to obtain the knowledge of distorted
signals indirectly. However, there is currently no generic methodology available
for designing a protection relay that is able to detect the shapes of signals, in
particular for protective relaying purposes.
This book introduces mathematical morphology (MM) for the design and operation
of power system relays. MM has been designated as a new branch of mathematics,
which is totally different from the integral transform-based methods in
basic principles, algorithmic operations and approach. When used for the extraction
of waveform components,MMhas the following merits in comparison with the
integral transform methods: (1) The morphological operators have fast and simple
calculations without using multiplication and division operations. (2) It is applicable
to non-periodic transient signals and not restricted to periodic signals. (3) MM
uses a much smaller sampling window for real-time signal processing, as it does not
require the information of the full signal components. (4) It is able to accurately and
reliably extract the signal componentswithout causing any distortion, as it is a timedomain
signal processing method that does not perform any integral transforms.
We wrote this book in the belief that MM would open an opportunity to develop
more accurate, reliable and faster protective relaying algorithms, leading to a new
generation of power system protection relays. Apart from being an introduction to
basic and advanced MM operators, presented with their pseudo codes, this book
presents a number of MM-based methods developed for power system protection
relays. We hope that this book will be useful for those postgraduates, academic
researchers and engineers working in the area of the design and development of
power system protection relays.
We would like to thank Drs. M.H. Sedaaghi, P. Sun, D.J. Zhang and J.F. Zhang
for their contributions, made during the period of their PhD studies undertaken at
The University of Liverpool, to part of the achievements presented in this book.We
would also like to thank Dr. S. Potts of AREVA for providing the model of current
transformers for the study of inrush current identification and Mr. John Fitch of
National Grid for supporting this work and providing useful discussions.
Special thanks go to Anthony Doyle (the Senior Editor), Nadja Kroke and Sorina
Moosdorf (the Production Editor), and Claire Protherough and Simon Rees (Editorial
Assistants) for their professional and efficient editorial work on this book. Our
thanks are also extended to all colleagues in the Intelligence Engineering and Automation
Research Group, The University of Liverpool, for all assistance provided,
and who have not been specifically mentioned above.
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Historical Background of Digital Protective Relaying Algorithms . . 2
1.3 Development of Protective Relaying Algorithms . . . . . . . . . . . . . . . . 3
1.3.1 SinusoidalWaveform-Based Algorithms . . . . . . . . . . . . . . . . . 3
1.3.2 Digital Filtering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Least Squares-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.4 Differential Equation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.5 Protection Based on Transient Signals . . . . . . . . . . . . . . . . . . . 5
1.3.6 Adaptive Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.7 Artificial Neural Networks for Protective Relaying . . . . . . . . 6
1.3.8 Wavelet TransformMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Introduction of Mathematical Morphology to Protective Relaying
of Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Contents of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 MathematicalMorphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Basic Morphological Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Definitions for Binary Operations . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Set Representations of Functions . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Grey-Scale Dilation and Erosion . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Morphological Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Definitions of Morphological Filters . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Opening and Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Alternating Sequential Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 The Lifting Scheme and MorphologicalWavelets . . . . . . . . . . . . . . . . 25
2.4.1 The Multi-resolution Decomposition Scheme . . . . . . . . . . . . . 25
2.4.2 The Lifting Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 MorphologicalWavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.4 The Multi-resolution Morphological Gradient Algorithm . . . 35
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Phasor Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Phasor MeasurementMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 The Mann and Morrison Algorithm . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 The Rockefeller and Udren Algorithm. . . . . . . . . . . . . . . . . . . 43
3.2.3 The Full-Cycle Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 44
3.2.4 The Half-Cycle Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 45
3.2.5 The Least Squares Error Algorithm . . . . . . . . . . . . . . . . . . . . . 46
3.3 Power System Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Fault Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Influence of Exponentially Decaying DC Offset on FT
Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Morphological Transform for DC Offset Removal . . . . . . . . . . . . . . . 49
3.5 Results of Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Protection of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The Adaptive Distance Relaying Algorithm. . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Fault Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.3 Fault Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Implementation of ADRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Calculation of the Fault Distance by RLS . . . . . . . . . . . . . . . . 67
4.3.2 Settings for the Variable Tripping Zone . . . . . . . . . . . . . . . . . . 69
4.4 Simulation Studies and Results of ADRA . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Ground Faults with Ground-Fault Resistance . . . . . . . . . . . . . 71
4.4.2 Other Types of Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.3 External Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Protection of a Double Circuit Transmission Line . . . . . . . . . . . . . . . . 78
4.6 The Fault Phase Selector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7 Simulation Results and Discussions on Fault Phase Selection . . . . . . 82
4.7.1 Single Line Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7.2 Line-to-Line Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7.3 Simultaneous Faults on Both Line Circuits . . . . . . . . . . . . . . . 87
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Transformer Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Transformer Differential Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 TransformerMagnetising Inrush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 The Magnitude of Magnetising Inrush Current . . . . . . . . . . . . 97
5.3.2 Harmonics of Magnetising Inrush Current . . . . . . . . . . . . . . . 100
5.3.3 The Second-Harmonic Restrained Differential Protection
of Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Morphological Identification of Inrush . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4.1 A Morphological Decomposition Scheme for Inrush
Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4.2 A Multi-resolution Decomposition Scheme for Inrush
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.5 Simulation Studies and Results Analysis . . . . . . . . . . . . . . . . . . . . . . . 109
5.5.1 A TransformerModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5.2 Application of MDS for Inrush Detection . . . . . . . . . . . . . . . . 112
5.5.3 Evaluation of MRDS for Inrush Identification . . . . . . . . . . . . 117
5.6 Further Discussion of the MM-Based Schemes for Inrush
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Bus Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Bus Differential Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Current Transformers for Bus Protection . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Current Transformer Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Saturation of Current Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.6 A Morphological Lifting Scheme for Detection of CT Saturation . . . 130
6.7 A Compensation Algorithm for Distorted Secondary Current . . . . . . 131
6.8 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.8.1 Tests Undertaken at Different Levels of Residual Flux . . . . . 133
6.8.2 Compensation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7 Ultra-High-Speed Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 Principles of UHS Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2.1 Transient-Based Directional Protection . . . . . . . . . . . . . . . . . . 141
7.2.2 Faults in the Forward Direction . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2.3 Faults in the Reverse Direction . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.4 Fault Phase Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3 UHS Directional Protective Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3.1 A Solid Fault in the Forward Direction . . . . . . . . . . . . . . . . . . 147
7.3.2 A High Ground-Fault Resistance Fault in the Forward
Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xii Contents
7.3.3 A Fault with Low Inception Angle in the Reverse Direction . 152
7.3.4 A Solid Fault at the Busbar in the Reverse Direction . . . . . . . 154
7.3.5 Faults of Different Types, Locations, Directions and
Inception Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.6 Phase Selection with Different Fault Types . . . . . . . . . . . . . . . 160
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8 Fault Location on Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.2 Principles of Fault Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.2.1 Transient-Based Positional Protection . . . . . . . . . . . . . . . . . . . 167
8.2.2 Type A: A Passive Method Single-Ended Fault Locator . . . . 168
8.2.3 Type D: A Passive Method Double-Ended Fault Locator . . . 171
8.2.4 Type E: An Active Method Single-Ended Fault Locator . . . . 171
8.2.5 Online Measurement of the Transient Wave Propagation
Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Noise Removal of Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.1 The Edge-Avoiding Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.2 Morphological Edge Reorganisation . . . . . . . . . . . . . . . . . . . . 174
8.3.3 Noise Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4 Accurate Fault Location by Morphological Filters . . . . . . . . . . . . . . . 176
8.4.1 A Solid Phase-A-Ground Fault . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.4.2 A Double-Phase-to-Ground Fault . . . . . . . . . . . . . . . . . . . . . . . 178
8.4.3 A Fault Close to the Busbar . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.4.4 A Fault with a Stable Low Resistance . . . . . . . . . . . . . . . . . . . 180
8.4.5 Switching-In Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.4.6 Calculated Transient Wave Propagation Speed and Fault
Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.5 Morphological Undecimated Wavelet Decomposition for Fault
Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.5.1 Morphological UndecimatedWavelets . . . . . . . . . . . . . . . . . . 184
8.5.2 Fault Location Using MUDW. . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A Electromagnetic Transient Analysis of Transmission Lines . . . . . . . . . . 191
A.1 Distributed Parameter Model of Transmission Lines . . . . . . . . . . . . . 191
A.2 TransientWave Propagation Characteristics . . . . . . . . . . . . . . . . . . . . 193
A.3 Incidence, Reflection and Refraction of Transients . . . . . . . . . . . . . . . 195
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
