RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING.

Paulo Fernando Ribeiro, Technological University of Eindhoven, The Netherlands Carlos Augusto Duque, Federal University of Juiz de Fora, Brazil Paulo Márcio da Silveira, Federal University of Itajubá, Brazil Augusto Santiago Cerqueira, Federal University of Juiz de Fora, Brazil With special relation to smart grids, this book provides a clear and comprehensive explanation of how Digital Signal Processing (DSP) and Computational Intelligence (CI) techniques can be applied to solve problems in the power system. Its unique coverage bridges the gap between DSP, electrical power and energy engineering systems, demonstrating the application of many different DSP and IC techniques to typical and expected system conditions with practical power system examples. Discussing many recent advances on DSP for power systems, this book enables engineers and researchers to understand the current state of the art and to develop new tools. It presents: • an overview of the power system and electric signals, with a description of the basic concepts of DSP commonly found in power system problems; • the application of several signal processing tools to problems, looking at power signal estimation and decomposition, pattern recognition techniques and detection of the power system signal variations; • a description of DSP in relation to measurements, power quality, monitoring, protection and control and wide area monitoring; ® • a companion website with real signal data and several examples of MATLAB code, DSP algorithms and samples of signals for further processing, understanding and analysis. Power Systems Signal Processing for Smart Grids can be a helpful guide for utilities engineers as well as researchers and postgraduate students investigating, designing and operating the intelligent grid of the future. It is intended to facilitate the learning and application of signal processing analysis and the understanding of power quality, protection and control of energy systems in general.

www.wiley.com/go/signal_processing

Power Systems Signal Processing for Smart Grids

Power Systems Signal Processing for Smart Grids

Ribeiro Duque da Silveira Cerqueira

Power Systems Signal Processing for Smart Grids

Paulo Fernando Ribeiro Carlos Augusto Duque Paulo Márcio da Silveira Augusto Santiago Cerqueira

POWER SYSTEMS SIGNAL PROCESSING FOR SMART GRIDS

POWER SYSTEMS SIGNAL PROCESSING FOR SMART GRIDS Paulo Fernando Ribeiro Technological University of Eindhoven, The Netherlands

Carlos Augusto Duque Federal University of Juiz de Fora, Brazil

Paulo M arcio da Silveira Federal University of Itajub a, Brazil

Augusto Santiago Cerqueira Federal University of Juiz de Fora, Brazil

This edition first published 2014 # 2014 John Wiley and Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. MATLAB1 is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB1 software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB1 software.

Library of Congress Cataloging-in-Publication Data Ribeiro, Paulo F. Power systems signal processing for smart grids / Paulo F. Ribeiro, Paulo Marcio da Silveira, Carlos Augusto Duque, Augusto Santiago Cerqueira. 1 online resource. Includes bibliographical references and index. Description based on print version record and CIP data provided by publisher; resource not viewed. ISBN 978-1-118-63921-4 (MobiPocket) – ISBN 978-1-118-63923-8 (ePub) – ISBN 978-1-11863926-9 (Adobe PDF) – ISBN 978-1-119-99150-2 (hardback) 1. Electric power systems. 2. Signal processing–Digital techniques. 3. Smart power grids. I. Title. TK1005 621.310 7–dc23 2013023846 A catalogue record for this book is available from the British Library. ISBN: 978-1-119-99150-2 Set in 10/12 pt Times by Thomson Digital, Noida, India 1 2014

Contents About the Authors

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Preface

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Accompanying Websites

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Acknowledgments

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1 Introduction 1.1 Introduction 1.2 The Future Grid 1.3 Motivation and Objectives 1.4 Signal Processing Framework 1.5 Conclusions References

1 1 2 3 4 8 10

2 Power Systems and Signal Processing 2.1 Introduction 2.2 Dynamic Overvoltage 2.2.1 Sustained Overvoltage 2.2.2 Lightning Surge 2.2.3 Switching Surges 2.2.4 Switching of Capacitor Banks 2.3 Fault Current and DC Component 2.4 Voltage Sags and Voltage Swells 2.5 Voltage Fluctuations 2.6 Voltage and Current Imbalance 2.7 Harmonics and Interharmonics 2.8 Inrush Current in Power Transformers 2.9 Over-Excitation of Transformers 2.10 Transients in Instrument Transformers 2.10.1 Current Transformer (CT) Saturation (Protection Services) 2.10.2 Capacitive Voltage Transformer (CVT) Transients 2.11 Ferroresonance 2.12 Frequency Variation

11 11 12 12 13 15 17 21 25 27 29 29 42 45 47 47 54 55 56

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2.13 Other Kinds of Phenomena and their Signals 2.14 Conclusions References

56 57 58

3 Transducers and Acquisition Systems 3.1 Introduction 3.2 Voltage Transformers (VTs) 3.3 Capacitor Voltage Transformers 3.4 Current Transformers 3.5 Non-Conventional Transducers 3.5.1 Resistive Voltage Divider 3.5.2 Optical Voltage Transducer 3.5.3 Rogowski Coil 3.5.4 Optical Current Transducer 3.6 Analog-to-Digital Conversion Processing 3.6.1 Supervision and Control 3.6.2 Protection 3.6.3 Power Quality 3.7 Mathematical Model for Noise 3.8 Sampling and the Anti-Aliasing Filtering 3.9 Sampling Rate for Power System Application 3.10 Smart-Grid Context and Conclusions References

59 59 60 64 67 71 71 72 73 74 75 78 79 79 80 81 84 84 85

4 Discrete Transforms 4.1 Introduction 4.2 Representation of Periodic Signals using Fourier Series 4.2.1 Computation of Series Coefficients 4.2.2 The Exponential Fourier Series 4.2.3 Relationship between the Exponential and Trigonometric Coefficients 4.2.4 Harmonics in Power Systems 4.2.5 Proprieties of a Fourier Series 4.3 A Fourier Transform 4.3.1 Introduction and Examples 4.3.2 Fourier Transform Properties 4.4 The Sampling Theorem 4.5 The Discrete-Time Fourier Transform 4.5.1 DTFT Pairs 4.5.2 Properties of DTFT 4.6 The Discrete Fourier Transform (DFT) 4.6.1 Sampling the Fourier Transform 4.6.2 Discrete Fourier Transform Theorems 4.7 Recursive DFT 4.8 Filtering Interpretation of DFT

87 87 87 90 92 93 95 97 98 98 103 104 108 109 110 110 116 116 117 120

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4.8.1 Frequency Response of DFT Filter 4.8.2 Asynchronous Sampling 4.9 The z-Transform 4.9.1 Rational z-Transforms 4.9.2 Stability of Rational Transfer Function 4.9.3 Some Common z-Transform Pairs 4.9.4 z-Transform Properties 4.10 Conclusions References

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123 124 126 128 131 131 133 133 133

5 Basic Power Systems Signal Processing 5.1 Introduction 5.2 Linear and Time-Invariant Systems 5.2.1 Frequency Response of LTI System 5.2.2 Linear Phase FIR Filter 5.3 Basic Digital System and Power System Applications 5.3.1 Moving Average Systems: Application 5.3.2 RMS Estimation 5.3.3 Trapezoidal Integration and Bilinear Transform 5.3.4 Differentiators Filters: Application 5.3.5 Simple Differentiator 5.4 Parametric Filters in Power System Applications 5.4.1 Filter Specification 5.4.2 First-Order Low-Pass Filter 5.4.3 First-Order High-Pass Filter 5.4.4 Bandstop IIR Digital Filter (The Notch Filter) 5.4.5 Total Harmonic Distortion in Time Domain (THD) 5.4.6 Signal Decomposition using a Notch Filter 5.5 Parametric Notch FIR Filters 5.6 Filter Design using MATLAB1 (FIR and IIR) 5.7 Sine and Cosine FIR Filters 5.8 Smart-Grid Context and Conclusions References

135 135 135 138 140 142 142 144 146 148 151 153 154 155 155 156 159 161 161 163 163 165 166

6 Multirate Systems and Sampling Alterations 6.1 Introduction 6.2 Basic Blocks for Sampling Rate Alteration 6.2.1 Frequency Domain Interpretation 6.2.2 Up-Sampling in Frequency Domain 6.2.3 Down-Sampling in Frequency Domain 6.3 The Interpolator 6.3.1 The Input–Output Relation for the Interpolator 6.3.2 Multirate System as a Time-Varying System and Nobles Identities 6.4 The Decimator

167 167 167 168 169 169 170 172 172 174

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6.4.1 Introduction 6.4.2 The Input–Output Relation for the Decimator 6.5 Fractional Sampling Rate Alteration 6.5.1 Resampling Using MATLAB1 6.6 Real-Time Sampling Rate Alteration 6.6.1 Spline Interpolation 6.6.2 Cubic B-Spline Interpolation 6.7 Conclusions References

174 174 175 175 176 177 180 184 184

7 Estimation of Electrical Parameters 7.1 Introduction 7.2 Estimation Theory 7.3 Least-Squares Estimator (LSE) 7.3.1 Linear Least-Squares 7.4 Frequency Estimation 7.4.1 Frequency Estimation Based on Zero Crossing (IEC61000-4-30) 7.4.2 Short-Term Frequency Estimator Based on Zero Crossing 7.4.3 Frequency Estimation Based on Phasor Rotation 7.4.4 Varying the DFT Window Size 7.4.5 Frequency Estimation Based on LSE 7.4.6 IIR Notch Filter 7.4.7 Small Coefficient and/or Small Arithmetic Errors 7.5 Phasor Estimation 7.5.1 Introduction 7.5.2 The PLL Structure 7.5.3 Kalman Filter Estimation 7.5.4 Example of Phasor Estimation using Kalman Filter 7.6 Phasor Estimation in Presence of DC Component 7.6.1 Mathematical Model for the Signal in Presence of DC Decaying 7.6.2 Mimic Method 7.6.3 Least-Squares Estimator 7.6.4 Improved DTFT Estimation Method 7.7 Conclusions References

185 185 185 187 188 191

8 Spectral Estimation 8.1 Introduction 8.2 Spectrum Estimation 8.2.1 Understanding Spectral Leakage 8.2.2 Interpolation in Frequency Domain: Single-Tone Signal 8.3 Windows 8.3.1 Frequency-Domain Windowing 8.4 Interpolation in Frequency Domain: Multitone Signal

227 227 227 229 232 236 236 240

192 195 198 200 201 203 203 205 205 207 209 211 212 213 214 215 216 224 224

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8.5 Interharmonics 8.5.1 Typical Interhamonic Sources 8.5.2 The IEC Standard 61000-4-7 8.6 Interharmonic Detection and Estimation Based on IEC Standard 8.7 Parametric Methods for Spectral Estimation 8.7.1 Prony Method 8.7.2 Signal and Noise Subspace Techniques 8.8 Conclusions References 9 Time-Frequency Signal Decomposition 9.1 Introduction 9.2 Short-Time Fourier Transform 9.2.1 Filter Banks Interpretation 9.2.2 Choosing the Window: Uncertainty Principle 9.2.3 The Time-Frequency Grid 9.3 Sliding Window DFT 9.3.1 Sliding Window DFT: Modified Structure 9.3.2 Power System Application 9.4 Filter Banks 9.4.1 Two-Channel Quadrature-Mirror Filter Bank 9.4.2 An Alias-Free Realization 9.4.3 A PR Condition 9.4.4 Finding the Filters from P(z) 9.4.5 General Filter Banks 9.4.6 Harmonic Decomposition Using PR Filter Banks 9.4.7 The Sampling Frequency 9.4.8 Extracting Even Harmonics 9.4.9 The Synthesis Filter Banks 9.5 Wavelet 9.5.1 Continuous Wavelet Transform 9.5.2 The Inverse Continuous Wavelet Transform 9.5.3 Discrete Wavelet Transform (DWT) 9.5.4 The Inverse Discrete Wavelet Transform 9.5.5 Discrete-Time Wavelet Transform 9.5.6 Design Issues in Wavelet Transform 9.5.7 Power System Application of Wavelet Transform 9.5.8 Real-Time Wavelet Implementation 9.6 Conclusions References 10 Pattern Recognition 10.1 Introduction 10.2 The Basics of Pattern Recognition 10.2.1 Datasets 10.2.2 Supervised and Unsupervised Learning

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243 246 247 250 254 254 262 269 270 271 271 274 274 276 279 280 282 282 284 288 290 290 292 294 295 298 298 300 300 301 305 305 308 308 313 316 318 319 319 321 321 322 323 323

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10.3 Bayes Decision Theory 10.4 Feature Extraction on the Power Signal 10.4.1 Effective Value (RMS) 10.4.2 Discrete Fourier Transform 10.4.3 Wavelet Transform 10.4.4 Cumulants of Higher-Order Statistics 10.4.5 Principal Component Analysis 10.4.6 Normalization 10.4.7 Feature Selection 10.5 Classifiers 10.5.1 Minimum Distance Classifiers 10.5.2 Nearest Neighbor Classifier 10.5.3 The Perceptron 10.5.4 Least-Squares Methods 10.5.5 Multilayer Perceptron 10.5.6 Support Vector Machines 10.6 System Evaluation 10.6.1 Estimation of the Classification Error Probability 10.6.2 Limited-Size Dataset 10.7 Pattern Recognition Examples in Power Systems 10.7.1 Power Quality Disturbance Classification 10.7.2 Load Forecasting in Electric Power Systems 10.7.3 Power System Security Assessment 10.8 Conclusions References 11 Detection 11.1 Introduction 11.2 Why Signal Detection for Electric Power Systems? 11.3 Detection Theory Basics 11.3.1 Detection on the Bayesian Framework 11.3.2 Newman-Pearson Criterion 11.3.3 Receiving Operating Characteristics 11.3.4 Deterministic Signal Detection in White Gaussian Noise 11.3.5 Deterministic Signals with Unknown Parameters 11.4 Detection of Disturbances in Power Systems 11.4.1 The Power System Signal 11.4.2 Optimal Detection 11.4.3 Feature Extraction 11.4.4 Commonly Used Detection Algorithms 11.5 Examples 11.5.1 Transmission Lines Protection 11.5.2 Detection Algorithms Based on Estimation 11.5.3 Saturation Detection in Current Transformers 11.6 Smart-Grid Context and Conclusions References

323 324 324 325 325 325 326 327 328 329 329 329 330 334 337 342 348 349 350 350 350 351 353 353 353 355 355 355 356 356 357 358 358 363 368 368 369 370 370 371 371 373 377 380 381

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12 Wavelets Applied to Power Fluctuations 12.1 Introduction 12.2 Basic Theory 12.3 Application of Wavelets for Time-Varying Generation and Load Profiles 12.3.1 Fluctuation Analyses with FFT 12.3.2 Methodology 12.3.3 Load Fluctuations 12.3.4 Wind Farm Generation Fluctuations 12.3.5 Smart Microgrid 12.4 Conclusions References

383 383 384 385 385 386 387 389 390 392 392

13 Time-Varying Harmonic and Asymmetry Unbalances 13.1 Introduction 13.2 Sequence Component Computation 13.3 Time-Varying Unbalance and Harmonic Frequencies 13.4 Computation of Time-Varying Unbalances and Asymmetries at Harmonic Frequencies 13.5 Examples 13.5.1 Inrush Current 13.5.2 Voltage Sag 13.5.3 Unbalance in Converters 13.6 Conclusions References

395 395 396 397 398 401 401 404 407 410 411

Index

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About the Authors Paulo Fernando Ribeiro achieved a PhD in Electrical Engineering from the University of Manchester and has worked in academia, industrial management, electric companies and research institutes in the fields of power systems, power electronics and power quality engineering, transmission system planning, strategic studies for power utilities, transmission and distribution system modeling, space power systems, power electronics for renewable generation, flexible AC transmission systems, signal processing applied to power systems, superconducting magnetic energy storage systems and smart grids. His professional experience includes teaching at US, European and Brazilian universities, and he has held research positions with the Center for Advanced Power Systems at Florida State University, EPRI and NASA. He is a Distinguished Lecturer and Fellow of the IEEE and IET and has written over 200 peer-reviewed papers, chapters and technical books. He is an active member of IEC, CIGRE and IEEE technical committees, including the chair of the IEEE Task Force on Probabilistic and Time-Varying Aspects of Harmonics and membership of the IEC 77A Working Group 9 (Power Quality Measurement Methods) and the CIGRE C4.112 (Guidelines for Power Quality Monitoring: Measurement Locations, Processing and Presentation of Data). Carlos Augusto Duque achieved a BS degree in Electrical Engineering from the Federal University of Juiz de Fora, Brazil in 1986, and a MSc and PhD degree from the Catholic University of Rio de Janeiro in 1990 and 1997, respectively, in Electrical Engineering. Since 1989 he has been a Professor in the Electrical Engineering Faculty at Federal University of Juiz de Fora (UFJF), Brazil. During 2007 and 2008 he joined the Center for Advanced Power Systems (CAPS) at Florida State University as a visiting researcher. His major research works are in the area of signal processing for power systems including the development of a power quality co-processor, the time-varying harmonic analyzer and signal processing for synchophasor estimation. He is currently the head of the Research Group of Signal Processing Applied to Power Systems, UFJF and associated researcher of the Brazil National Institute of Energy. He has written over 120 peer-reviewed papers and chapters of technical books, and is the author of several patents.

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About the Authors

Paulo M arcio da Silveira achieved a DSc degree in Electrical Engineering from the Federal University of Santa Catarina, Brazil in 2001. He has industrial design, academic and research experience in power system equipment, substation, protection and power quality issues, operation of power systems studies and development of protective devices and power quality monitoring algorithms for power utilities applications. He has conducted research on transmission and distribution system modeling, monitoring, measurement and signal processing for fault identification, fault location, protective relays, power quality and energy metering. He has worked as a consultant on power quality and power system protection, conducting research for different Brazilian utilities through the Brazilian Electricity Regulatory Agency (ANEEL). Dr Silveira was a visiting researcher at the Center for Advanced Power System at the Florida State University in Tallahassee, US in 2007, when he worked with real-time digital simulations. He is an associate professor at the Itajuba´ Federal University (UNIFEI) in Brazil, where currently he is also the coordinator of a post-graduate course on Power System Protection, the coordinator of the Electrical Compatibility for Smart Grid Study Center (CERIn), and the head of the Electrical and Energy System Institute of the UNIFEI. Augusto Santiago Cerqueira achieved a DSc degree in Electrical Engineering at the Federal University of Rio de Janeiro, Brazil in 2002. In 2004, he began his academic and research activities at the Federal University of Juiz de Fora (UFJF), where he is currently an associate professor. His academic and research activities mainly involve electronic instrumentation, digital signal processing, computational intelligence for power systems and experimental high-energy physics. He has participated in and coordinated research projects related to power quality issues, applying signal processing and computational intelligence techniques for power quality monitoring and diagnosis. He is coordinator of the UFJF group at the Large Hadron Collider at CERN (European Organization for Nuclear Research), which conduct research into experimental high-energy physics instrumentation, signal processing and computational intelligence mainly for signal detection and estimation.

Preface This book has grown out of a cooperation between friends who have a common interest, expertise and passion for power systems (PS) and signal processing (SP). It has evolved as a consequence of SP projects applied to power quality (PQ) and power systems in general. The rapid growth of computational power associated with the cross-fertilization of applications and use of SP for analysis and diagnosis of system performance has led to unprecedented development of new methods, theories and models. The authors have come to appreciate the potential for much wider applications of SP, prompted in particular by the modernization of electric power systems via the current and comprehensive developments associated with the implementation of smart grid (SG) technologies. The increasing complexity of the electric grid requires intensive and comprehensive signal monitoring followed by the necessary signal processing for characterizing, identifying, diagnosing and protecting and also for a more accurate investigation of the nature of certain phenomena and events. SP can also be used for predicting and anticipating system behavior. For electrical engineering SP is a vital tool for clarifying, separating, decomposing and revealing different aspects and dimensions of the complex physical reality of the operation of electrical systems, in which different phenomena are usually intricately and intrinsically aggregated and not trivially resolved. SP can be qualified by the analytical aspects of the electrical systems, and can help to expose and characterize the diversity, unity, meaning and intrinsic purpose of electrical parameters, system phenomena and events. As the electric grid becomes more complex, modeling and simulation become less capable of capturing the influence of the multitude of independent and intertwined components within the network. SP deals with the actual system and not with modeling abstraction or reduction (although it may be used in connection with simulations), so may clarify aspects of the whole through a multiplicity of analytical tools. Consequently, SP allows the engineer to detect and measure the behavior and true nature of the electric grid. Today, the vast majority of analog signals are converted to digital signals. In the context of electrical systems, this conversion is carried out by numerous secondary smart digital devices that perform the tasks of controlling, metering, protecting, supervising or communicating with other components of the system. Moreover, the quality of such smart devices is enhanced by their ability to perform digital signal processing (DSP). The term DSP is used to describe the mathematics, algorithms and techniques used to manipulate signals after they have been converted into a convenient digital form in order to

Preface

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Figure 1 Power systems signals in the context of smart grids.

address a wide variety of needs such as the enhancement of visual images, recognition and generation of speech and compression of data for storage and transmission [1]. The aim of this book is to further promote the use of DSP within power systems, and to expand its application in the context of smart grids. Various techniques are presented, discussed and applied to typical and expected system conditions. Figure 1 illustrates a sample of the gamma of waveforms of typical power systems signals in a context of traditional and smart-grid power system environments. Chapter 1 describes the motivation for the use of signal processing in different applications of power systems in the context of the smart grids of the future. A wide variety of digital measurements and data analysis techniques required to deliver diagnostic solutions and correlations is provided. Chapter 2 provides a comprehensive list of power system events and phenomena in terms of time-varying voltage and current signals, characterizing these in terms of magnitude, phase

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and waveform. It will become apparent that many signals can be represented by a mathematical expression (e.g. exponential DC, faults, waveform distortions). Chapter 3 describes the different aspects as related to voltage transformers, current transformers, analog filters and analog to digital converters. These components are sources of noise and errors, and impose speed constraints. Due to the lack of information about acquisition systems for electric power signals, this chapter addresses a few of the important demands that are generally neglected in common signal processing literature. Chapter 4 covers discrete transforms essential in the analysis and synthesis of power systems signal processing. The chapter describes the discrete-time Fourier transform (DTFT), discrete Fourier transform (DFT) and z-transform, as well as a summary of the continuous transforms. Although these transforms are widely treated in several textbooks, the focus of the authors is on specific and common power systems applications. Chapter 5 covers basic aspects of power system signal processing. These include digital signal operators (delay, adders, multipliers), digital signal operations (modulation, filtering, correlation and convolution), finite impulse response filters and infinite impulse response filters. Several power systems applications are used to illustrate these concepts. Chapter 6 covers the multirate and sampling frequency alterations, a common time-variant method used in power systems to change the sampling frequency or to analyze a signal. Such an example is using filter banks or wavelet transform. (Filter banks and wavelet transform are covered in Chapter 9, but the digital principles for the implementation of these structures are presented in Chapter 6.) Offline and real-time frequency alterations for power systems application are also discussed. In Chapter 7 the focus is on algorithms that are capable of estimating parameters such as phasor, frequency, RMS (root mean square), harmonics and transients (decaying exponential) for real-time and offline applications. The basic concepts of estimation theory are presented, including the Cramer–Rao lower bond (CRLB), the MVU estimator, BLUE and LSE estimators. The smart-grid environment is one of higher-complexity electrical signals, which need to be properly and accurately measured. Chapter 8 covers the basic concepts of spectrum analysis and parametric and nonparametric spectrum estimations. Common errors in parametric estimation are covered, including aliasing, scalloping loss and spectrum leakage. Among the parametric methods discussed are the Prony, Pisarenko, MUSIC and ESPRIT methods. Chapter 9 introduces a unified view of time-frequency decomposition based on filter banks and wavelet transforms for power system applications. The short-time Fourier transform (STFT) is presented, and the basic principle of filter banks theory and its connection with wavelets is discussed. The basic theory of the wavelet and relevant signal processing techniques are described. Guidance on how to choose the mother wavelet for power system applications is provided. Chapter 10 covers pattern recognition as an essential enabling tool for the operation and control of the upcoming electric smart-grid environment. The chapter highlights the main aspects and necessary steps required for providing necessary tools to operate the grid of the future. Chapter 11 presents the basic aspects of detection theory using the Bayesian framework and discusses the deterministic signal detection for white Gaussian noise. Chapter 12 discusses the application of wavelet analysis to determine fluctuation patterns in generation and load profiles. This is achieved by the filtering of its wavelet components based

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Figure 2 Structure of the book.

on their RMS values, from which it is possible to identify the most-relevant scaling factors. The procedure reveals fluctuation patterns which cannot be visualized via frequency decomposition methods. Chapter 13 describes an application in which the evaluation of unbalances and asymmetries in power systems can be facilitated by the use of a time-varying decomposition method based on SW-DFT. The time-varying harmonics and their positive-, negative- and zero-sequence components are calculated for each frequency. Figure 2 depicts the structure of the book. Finally, some philosophical considerations with regards to the utilization and reception of this book (or any other book) is adapted below from the writings of British author C. S. Lewis:

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‘A scientific or engineering work such as this can be either received or used. When we receive it, we exercise our senses and imagination and various other powers according to a pattern suggested by the authors. When we use it we treat it as an assistance for our own activities. . . . Using is inferior to receiving because, in science and engineering, using merely facilitates, relieves or palliates our research/applications; it does not add to it.’ [2] The authors hope that the reader will both use and receive this book as a valuable and thought-provoking guide and tool.

References 1. Smith, S.W. (1997) The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing. 2. Lewis, C.S. (1961) An Experiment in Criticism, Cambridge University Press.

Accompanying Websites To accompany this book, two websites have been set up containing MATLAB1 files for additional waveforms of typical non-linear loads; these can be signal-processed by different techniques for further understanding. Two MATLAB1-based time-varying harmonic decomposition techniques are also available on site for waveform processing. Please visit http://www.ufjf.br/pscope-eng/digital-signal-processing-to-smart-grids/ Password: dspsgrid Or http://www.wiley.com/go/signal_processing Readers are welcome to send additional waveforms for signals and MATLAB1 scripts to be included in the database to Professor Paulo Fernando Ribeiro at [email protected].

Acknowledgments The authors would like to thank PhD students Tulio Carvalho, Mauro Prates, Leandro Manso,  and Pedro Machado for their valuable support and for Ballard Asare-Bediako, Vladimir Cuk comments, suggestions and assistance in preparing simulations, illustrations and experiments used in this text. Thanks are also due to Dr Jan Meyer from the University of Dresden for his suggestions and contributions to Chapter 3, Dr Jasper Frunt for his contributions to Chapter 12 and Tulio Carvalho and Totis Karaliolios for their contributions to Chapter 13. Thanks also to Mrs Adriana S. Ribeiro for her proofreading of all chapters and helpful editorial suggestions. The authors are especially grateful to The INERGE - Brazilian Institute of Electric Energy Science and Technology, Brazil, for the sponsoring Prof. Paulo Ribeiro as a visiting research professor during the preparation of this manuscript. The authors are also grateful to the Federal University of Juiz de Fora, Federal University of Itajuba, Technical University of Eindhoven, Netherlands, CNPq, and FAPEMIG, Brazil. The authors would like to thank their wives and families for their support during the last couple of years of persistent and unrelenting production process, in which new ideas, concepts and experiments have been developed, updated and refined.

1 Introduction 1.1 Introduction A power system is one of the most complex systems that have been made by man. It is an interconnected system consisting of generation units, substations, transmission, distribution lines and loads (consumers). Additionally, these encompass a vast array of other equipment such as synchronous machines, power transformers, instrument transformers, capacitor banks, power electronic devices, induction motors and so on. In this context the smart grid has contributed even further to this complex situation, of which a better understanding is required. Given these conditions, signal processing is becoming an essential assessment tool to enable the engineer and researcher to understand, plan, design and operate the complex and smart electronic grid of the future. Signal processing is used in many different applications and is becoming an important class of tools for electric power system analysis. This is partly due to a readily available vast arsenal of digital measurements that are needed for the understanding, correlation, diagnosis and development of key solutions to this complex context of smart grids. Measurements retrieved from numerous locations can be used for data analysis and can be applied to a variety of issues such as:     

voltage control power quality and reliability power system and equipment diagnostics power system control power system protection.

This book focuses on electrical signals associated with power system analysis in terms of characterization and diagnostics, or where signal-processing techniques can be useful such as for the analysis of possible concerns about individual loads and/or state of the system. A large variety of equipment can be used to capture and characterize system variations. These include monitors, digital fault recorders, digital relays, various power system controllers and other intelligent electronic devices (IEDs). Furthermore, power system conditions and events require signal processing techniques for the analysis of its recorded signals. This book Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo Marcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

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Power Systems Signal Processing for Smart Grids

promotes attentiveness to issues in the signal processing community. It will provide an overview of these techniques for the understanding and promotion of solutions to its concerns.

1.2 The Future Grid The future of the developed, developing and emerging countries in a global economy will rely even more on the availability and transport of electrical energy. It is believed that in the near future the global consumption of electrical energy will grow to unprecedented levels. Additionally, security and sustainability have become major priorities both for industry and society. The deployment of sustainable/renewable energy sources is crucial for a healthy relationship between man and his environment. These changes are driven by a number of developments in society, where the transition to a more sustainable society is a priority. Moreover, the availability of various new technologies and the deregulation of the electric industry may have an additional impact on future developments. The sustainable and low-carbon imprinting of a society and problematical energy storage requires an integrated power grid which will play a central role in the achievement of energyefficiency targets and savings. However, the large-scale incorporation of renewable energy production and novel forms of consumption will substantially increase the complexity of its electricity distribution system. The urgency requirement of this complex smart energy grid is evident from the extensive research and development in this area. An overall picture of this new complex infrastructure is shown in Figure 1.1, where the smart grid of the future can be seen as a merging of the power system and control information technologies.

Figure 1.1 The grid of the future.

Introduction

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Figure 1.2 The complexity of the smart grid: technologies, stakeholders, dimensions.

The complexity of a smart grid (illustrated in Figure 1.2) might be classified as:  dimensional complexity  technological complexity  stakeholder complexity.

The science and art of designing technological systems within a complex societal environment is a challenging job. In order to produce systems that are synchronized with all the different normative moments of each complexity, new projects must take into account the abovementioned evolving reality. In philosophical terms, a simultaneous realization of different laws and norms is required, where dimensional, technological and stakeholder issues with conflicting objectives and interests need to be accommodated in a well-integrated manner. In this context, signal processing emerges as one of the most important and effective tools for investigating the operation of such a system.

1.3 Motivation and Objectives In topics such as power quality, research has traditionally been motivated by the need to supply acceptable voltage quality to end-user loads where voltage, current and frequency deviations in the power system are normal concerns of a systems operator. The characterization of the incompatibilities caused by these deviations requires an understanding of the phenomena themselves. Listed among the possible aspects to be

4

Power Systems Signal Processing for Smart Grids

investigated are the need for efficient representation of the voltage and current variations and the signal processing to understand how equipment behaves. There is also a need for continuous monitoring that can capture deviations, events and variations and the correlation with equipment performance, decomposition, modeling, parametric estimation and identification algorithms. This book aims at utilizing more widely and effectively the signal processing tools for electrical power and energy engineering systems analysis. The text uses an integrated approach to the application of signal processing in power systems by means of the critical analysis of the methodologies recently developed or in innovative ways. The main techniques are critically illustrated, compared and applied to a variety of power systems signals. Both traditional and advanced signal processing tools for monitoring and control of power systems are considered. To meet future requirements, methods and techniques shall be engaged to explore the full range of signals that derive from the complex interaction between suppliers, consumers and network operators. The book is not only intended to convey the theoretical concepts, but also to demonstrate the application. How do engineers in the research and development of electrical grids cope with this increased complexity? It is impossible for an engineer to take the full complexity of these systems into account? During the design process, focus is generally on one or two aspects, one or two components or systems or the perspective of one or two parties involved. In other words, the complex system is reduced to a simplified, neatly arranged subsystem in order to design a new component, to study its performance and to optimize its stability. Through the years this has proven to be a very practical approach as long as the system does not experience major changes, allowing engineering judgment to be used in the simplification process. Unfortunately, a direct consequence of this is that it is not the whole system that is considered: only a reduced system. In research and development, reduction is unavoidable. Engineers and researchers therefore have to be aware that they study and design in the context of reduced realities. As a consequence, they have to question themselves continuously whether they are missing any relevant dimensions. In practice, engineers cannot easily handle all the technical and non-technical dimensions of an electrical system due to the enormous complexity of smart grids and the requirements of all parties involved, including the requirements of governments and powerful stakeholders. As a consequence it is easy to miss relevant dimensions, to overlook important interactions between technical systems, to neglect the interests of certain parties and to lose a great amount of information. The interaction between multitudes of participants produces very complex signals that must be monitored and processed in order to determine the state of and developments around devices and systems, as depicted in Figure 1.3.

1.4 Signal Processing Framework The condition of the grid can be fully assessed through the measurement and analysis of signals at different points in the system. Figure 1.4 illustrates the basic concept of signals and parameters that can be processed and derived in steps. First, three-phase signals are decomposed into time-varying harmonics and these are then processed by symmetrical components. The result provides the engineer with a unique tool to visualize the nature of time-varying imbalances and asymmetries in power systems.

Introduction

5

Figure 1.3 Signals, technologies and interactions.

Figure 1.5 further summarizes the signal processing that includes the measurement, monitoring and processing sequences from acquisition, analysis, detection, extraction and classification of the waveforms which might carry useful information for identification of system events, phenomena and load characteristics. As new signal processing tools are developed to deal with the smart grid developments, it useful to remember that the development of signal processing began in the late 1970s. Figure 1.6 shows the progression of these developments starting with the Fourier series and progressing to time-frequency decompositions, analyzers and advanced signal processing for smart grids. Figure 1.7 shows a summary of these signal processing aspects in the context of smart girds, emphasizing applications, techniques and specifications. In Figure 1.8 a comprehensive approach to the use of signal processing is illustrated. Here it can be seen that voltage and current signals at a specific point (even in a remote location) can be used to determine impedance, power factors, power flow, stability and so on, where such information can be used by the system operator for more efficient control of the electric grid.

Power Systems Signal Processing for Smart Grids

6

Voltage and Current Signals Analog Conditioning (transducers, low-pass filters)-ADC

SP 1st step

Results

CurveFitting Fittingtechniques techniques Curve 1 Sum, LSM, Sum, LSM,derivatives derivatives Fourierseries series Fourier FourierTransform Transform 2 Fourier STFT (DFT, SWDFT, FFT) Model system Model system Differential equation Differential equation

Travelling Waves Travelling Waves Techniques Techniques

3

4

Time-Frequency Time-Frequency techniques 5 techniques (Wavelet…) (Wavelet, …) Other filter banks

Filter banks

Special Specialfilters filters Denoising, Denoising,Notch, Notch, Kalman…

SP 2nd step

1 2 3 5 6 7

RMS phase phasor frequency

1 2 5 6

harmonics orders THD

2

2

Frequency Spectrum

3

1 2 7

2 5 7

1

Time-Varying Components

7

Fast Transients

1

Probabilistic parameters

Symmetrical Components

1

Histogram

4 5 6

Scalegram

2 3

Impedance

Unbalances, Asymmetries

2

1 2 3 5

Pattern Recognition

3

4 5

Wavelet Transform

4

2

4 5

Other transformations (Walsh, Hilbert…)

5

2

Time-Varying Phasors

1 4 5

Other time-frequency techniques

6

2 4 7 8

Power, power factor, energy

4 5

Collective RMS

7

2 3

PQ identification and classification

2 4 6

Normalization

8

2 3 4 5 8

Fault location, incipient defects

4

5

6 2 4 5 6

1

1

1

Steady-State Components

Probabilistic Analysis

Results

6

Kalman…

Making decision: Protection, Control, Supervision, Planning.

Making decision: Protection, Control, Supervision, Planning.

Figure 1.4 Basic concept of signals and parameters that can be processed and derived.

Finally, Figure 1.9 illustrates the perspective of a system, highlighting where signal processing can take place at different points within the network and providing crucial information to system operators. Finally, the use of a phasor measurement unit (PMU), wide area networks (WANs), home area networks (HANs) and local area networks (LANs), together with developments in information and communications technology (ICT), can be integrated with power quality and energy measurements. Signal processing techniques can then be utilized to facilitate the control, protection and diagnosis of performance of the complex transmission and distribution of the micro cyber-physical smart grid of the future (see Figure 1.10). Excellent literature has been published [1–7] describing the types of measurements and their technical specifications for power quality and other power systems operation performance requirements.

Introduction

7

Figure 1.5 Measurement, monitoring and signal processing sequence.

DFT

Fourier

FFT

Modified DFT and Filter Banks

Wavelets

Wavelets , Modified DFT and Filter Banks

STFT

Time-varying Harmonic Analyzer

Advanced Signal Processing in Smart-grid

Figure 1.6 Summary of signal processing development.

Signal Processing for Smart Grids Characterization

Processing Techniques Time

Identification

Frequency Time-Frequency

Analysis

Specifications Sampling

Segmentation

Artificial Intelligence

Figure 1.7 Signal processing techniques process.

Power Systems Signal Processing for Smart Grids

8

Figure 1.8 A comprehensive system-wide signal processing analysis.

1.5 Conclusions A broad perspective of the material covered by the book is given in this chapter. We also expand on how to apply in an integrated fashion both traditional and advanced signal processing techniques for monitoring and control of power systems, particularly in the context of future complex smart grids. The methods and techniques explore the full range of signals that account for the interaction of a greater number of generation sources and active consumers with non-linear time-varying loads. The increased complexity of the electric grid, prompted by the development and implementation of smart grid technology and systems, requires a higher level of signal processing techniques. The authors hope this book will increase this awareness and assist with the visualization of solutions and applications.

Introduction

9

Figure 1.9 System perspective of signal processing.

µ-Grid Generation

Transmission

LAN

LAN

Distribution

LAN

Generation

Customer

HAN FAN

MAN WAN

HAN NAN/EAN

Generation Control Transmission Control

Distribution Control Market Signals

AMI

HAN

TMN Telecommunication Management System

Figure 1.10 The complex transmission and distribution of the micro cyber-physical smart grid of the future.

10

Power Systems Signal Processing for Smart Grids

References 1. European Standard EN50160 (1999) Voltage characteristics of electricity supplied by public distribution system, CENELEC, Brussels, Belgium. 2. CIGRE WG C4.07 (October, 2004) Power quality indices and objectives. Technical Report No 261. CIGRE/ CIRED Working Group C4.07, Power Quality Indices and Objectives, CIGRE Technical Brochure TB 261, Paris. 3. European Standard EN50160 (2007) Voltage characteristics of electricity supplied by public distribution system, CENELEC, Brussels, Belgium. 4. ERGEG (December, 2006) Towards voltage quality regulation in Europe. ERGEG public consultation paper E06-EQS-09-03. 5. Council of European Energy Regulators (December, 2012) Guidelines of good practice on the implementation and use of voltage quality monitoring systems for regulatory purposes. Council of European Energy Regulators ASBL Energy Community Regulatory Board. 6. IEC 61000-4-30 (2003) Testing and measurement techniques – power quality measurement methods. International Electrotechnical Commission, Geneva, Switzerland. 7. IEEE PC37.242/D11 (Oct, 2012) IEEE Draft Guide for Synchronization, Calibration, Testing, and Installation of Phasor Measurement Units (PMU) for Power System Protection and Control. Institute of Electrical and Electronics Engineers.

2 Power Systems and Signal Processing 2.1 Introduction A key aspect of signal processing in power systems is determining which parameters should be measured and to what accuracy, as well as which signal processing methods provide the best characterization and analysis of the signals to be investigated. For example, in many types of studies only the voltage measurements are necessary for an adequate evaluation. However, there are many reasons to measure the current, frequency and active and reactive power of a power system. The study and application of digital signal processing techniques for the control, protection, supervision and monitoring of smart grids requires an understanding of the electrical system behavior under both normal and unusual or uncharacteristic situations. For any reading the basic sinusoidal signal (voltage and current) may be modified for different reasons, and as such will present distinguishing features in its waveforms. In this chapter we describe the main phenomena in power systems of time-varying and/or steady-state conditions, in terms of voltage and current. The aim is to characterize each of those taking into consideration its magnitude, phase and waveforms. Furthermore, evidence will be provided showing that many of these signals may be represented by a mathematical expression, such as exponential DCs, faults, harmonics and others. This is not however the case for completely chaotic signals, such as ferroresonance, subsynchronous oscillations and voltage fluctuations. The processes that generate such events involve highly non-linear elements such as arc resistances, steel core and so on. Taking the above into account it can be said that many of the phenomena mentioned can be recreated in simulation models, but others can only be represented by their specific measurements. Finally, this chapter will demonstrate the importance of knowledge of the phenomena that occur in power systems in order that the correct tools for signal processing can be properly applied. This is especially true in the context of increased system complexity due to the advent of smart grids. Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo M
arcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

Power Systems Signal Processing for Smart Grids

12

The following sections describe the different types of signals in electrical systems under different conditions. Due to the wide range of possible waveforms, only the most common waveforms are mentioned here. The representation of the electrical waveforms of the electric grid can be compared to the representation of the electrocardiogram (ECG) representing the electric function of a human heart, giving insight into its health and function. In the same manner, an evaluation of the electrical signals of a power grid can give the electrical engineer the ability to diagnose and predict possible malfunctions of the electric system.

2.2 Dynamic Overvoltage 2.2.1 Sustained Overvoltage Sustained overvoltage means an increased voltage of an industrial frequency (50–60 Hz) above the rated values. This overvoltage can appear in different regions of the power system such as a generator output or a load terminal. Figure 2.1 is an illustration of a sustained overvoltage. In general, the excess of reactive power is the primary cause of the overvoltage in an electrical power system. Over a given time period, the reactive power consumed by inductive loads is no longer consumed due to an abnormal occurrence. The immediate effect of this excess is the increase in voltage in different parts and components of the system. Specifically, in a transmission line the overvoltage can occur in its receiving terminal, either by load rejection or during an energization, when the terminal is opened. This effect is known as the Ferranti effect and is due to the voltage drop across the line impedance and by the absorption of a capacitance-charging current. This can happen when the line is energized with an open-ended terminal impedance and results in a voltage rise at the terminal. Both the Dynamic Overvoltage

1

Voltage (pu)

0.5

0

–0.5

–1

0

0.05

0.1 Time (s)

Figure 2.1 An example of an overvoltage.

0.15

Power Systems and Signal Processing

13

Figure 2.2 Voltage profile in an open-ended transmission line.

inductance and capacitance are therefore responsible for the production of this phenomenon. This will be more pronounced the longer the line and the higher the service voltage. In reality, the resistance, inductance and capacitance values of a long transmission line must be considered as distributed parameters. Figure 2.2 illustrates the effect of reactive power flowing (not active power) in the direction of the voltage source when the voltage at the receiving terminal (R) is higher than its sending terminal (S). Depending on its intensity and duration, a sustained overvoltage causes the deterioration of the insulation characteristics of the power equipment. For transformers and shunt reactors, for example, the overvoltage can result in:
excessive current due to a saturation of the core; such a current will be distorted with harmonics and consequently cause unwanted interference in the rest of the system;
local damage due to overheating, since the magnetic field during the saturation is sustained at a high level; and
premature aging (loss of insulation characteristics). Adequate surge protection is therefore necessary to disconnect the equipment and/or the transmission lines.

2.2.2 Lightning Surge Lightning can be a source of significant voltage surges. It can hit anywhere in an electric system, and affects the equipment and connected loads. This is true for both high-voltage (HV) and low-voltage (LV) devices. Electric charges build up in thunderclouds to such an extent that they can break through the atmospheric insulation. This may result in an electric discharge from cloud to ground, and such a current can reach 20–200 kA. If a lightning discharge occurs directly or in the vicinity of a transmission line, it will cause the movement of electrical charges (outbreaks). These discharges travel close to the speed of

14

Power Systems Signal Processing for Smart Grids

light and move through the power conductors reaching substations. Furthermore, the substation itself might be subject to lightning. When these high currents are discharged through the earthing structure of the system, these can cause significant voltage surges. The lightning surges can also be induced. Such is the case of a voltage produced by electrostatic or electromagnetic induction on cables located in close proximity to the point where the lightning hits, such as in a shield wire. A lightning surge has duration of the order micro–milliseconds. When these transients reach equipment (a transformer, a reactor or an insulator chain) through their terminals, the surge can cause cracks in the insulation and start the process of short-circuiting. The main protection in a substation is provided by power surge arresters. These are installed at the entry point of lines and power transformers. Transformers and other equipment can also be protected by the so-called overvoltage ‘spark gaps’. It is important to mention that any transients in the primary circuit (high voltage) caused by lightning also affect its secondary circuits through capacitor voltage transformers (CVTs), voltage (potential) transformers (VTs), current transformers (CT) or through electrostatic and/or electromagnetic induction. The standard lightning current has a fast rise followed by a slow decay. The typical waveform has a rise of 1.2 ms and decay of 50 ms. Such a waveform can be generated in a HV laboratory in order to test equipment for lightning currents. The waveform, shown in Figure 2.3, can be described by the equation:
 vðtÞ ¼ V 0 et=tb  et=ta

(2.1)

where V0 is initial voltage, t is time and ta and tb are time to reach 30% and 90% of the peak value, respectively. In this case, ta ¼ 71 ms and tb ¼ 0.2 ms. Not all instruments are equipped to receive or capture a lightning strike on a voltage signal, due to the short duration of the phenomenon. Figure 2.4 depicts an example of such a voltage signal.

Figure 2.3 Standard lightning voltage.

Power Systems and Signal Processing

15 Voltage Signal

1

Voltage (pu)

0.5

0

–0.5

–1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (s)

Figure 2.4 Voltage spike from lightning surge.

2.2.3 Switching Surges Switching in power systems (close and re-closing operations) causes switching surges in highvoltage systems and their auxiliary control circuits. The switching surge occurs in many different forms and has many different sources. Normally these are associated with a change in the operating state of the system, which means a switching involving trapped energy and its release. Transients in power circuits are caused by the transition from one state to another as, for example, when a circuit breaker opens. In this transitory period the energy accumulated in the electromagnetic fields is redistributed causing transients, as shown in Figure 2.5. The exchange of energy between electric and magnetic fields occurs not only in the fundamental frequency (50 or 60 Hz); there will also be oscillations between the related fields at other frequencies. These will depend on the involved inductances and capacitances of the circuit. The radian frequency of the transient component is given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r 2 1  vS ¼ LC 2L

(2.2)

where vS is the natural angular frequency (rad/s); r is damping resistance; L is inductance; and rffiffiffiffiffiffi 1 C is capacitance. If r ¼ 0, vs ¼ v0 ¼ . LC

Power Systems Signal Processing for Smart Grids

16 2.5

Va Vb Vc

2 1.5

Voltage (pu)

1 0.5 0 –0.5 –1 –1.5 –2 –2.5

0

0.02

0.04

0.06

0.08

0.1 0.12 Time (s)

0.14

0.16

0.18

0.2

Figure 2.5 Energization of a 400 kV transmission line indicating the three phases Va, Vb, and Vc. Vd r

jx Ic I cc

V(t) C

Figure 2.6 Cleaning a short circuit.

For the circuit shown in Figure 2.6 it can be demonstrated that the terminal voltage at the opened circuit-breaker (Vd) overlaps the fundamental signal. This is approximately defined: h i (2.3) V d ¼ V c ¼ V M 1  et=t cosðv0 tÞ where Vc is the terminal voltage at the shunt capacitance and VM is the fundamental amplitude of the source, VðtÞ ¼ V M cosðv0 tÞ

(2.4)

and t is the time constant of the circuit: 2L : (2.5) r The high-frequency damped transient term V M ¼ et=t cosðv0 tÞ is normally referred to as transient recovery voltage (TRV). Its maximum value (Vd,max) depends on r (damping t¼

Power Systems and Signal Processing

17

Figure 2.7 A transient recovery voltage at the breaker terminals. When r 6¼ 0.

resistance) and on frequency (v0), and its value is within the range: V M < V d;max < 2V M :

(2.6)

If r is considered equal to zero, we have a non-damped transient with frequency fS, defined: 1 fS ¼ 2p

rffiffiffiffiffiffi 1 : LC

(2.7)

In this case, V d;max ¼ 2V M . The characteristic of the transient recovery voltage Vd is shown in Figure 2.7. Circuit breakers are designed with special devices to minimize the magnitude and the impact of these transients. If the rate of rising of the TRV (RRRV) in kV s1 exceeds the rate of recovery characteristic of the dielectric strength or the voltage-withstanding capability between the contacts, the breaker will be unable to hold off the voltage and a re-strike will occur. Re-strike is defined as the breaker conduction of a current half-cycle after the successful interruption at a zero-crossing current. Sometimes the circuit breakers are specified for this special switching of critical circuits.

2.2.4 Switching of Capacitor Banks It is common practice to install a shunt capacitor in order to improve the power factor and the voltage profile at all voltage levels of an electric power system. These shunt capacitor banks are switched on and off as necessary. The switching operations include energizing, de-energizing, fault clearing, reclosing and energization of a capacitor bank when another bank is already in operation. The latter operation is known as ‘back-to-back switching’.

Power Systems Signal Processing for Smart Grids

18

If involving a capacitor bank, all the actions previously mentioned may cause huge transients in both voltage and current signals. Transient characteristics depend on the combination of the initiating mechanism and the electric circuit characteristics at the source of the transient. Circuit inductance and capacitance are responsible for the oscillatory nature of transients. 2.2.4.1 Energization The energization of a shunt capacitor bank through a predominantly inductive source results in an oscillatory transient that can reach twice the normal system peak voltage (Vpk). Figure 2.8 illustrates the simplified equivalent system for the energizing transient. The characteristic frequency fS of this transient is given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 1 XC MVAsc p ffiffiffiffiffiffiffiffi ffi fS ¼ : (2.8)  f0  f0 XS MVAr 2p LS C The peak inrush current (Ipk) is determined by: I pk ¼

V pk ZS

;

ZS ¼

rffiffiffiffiffi LS C

(2.9)

where fS is the characteristic frequency (Hz); LS is the positive sequence source inductance (H); C is the capacitance of the capacitor bank (F); f0 is the system frequency (50 or 60 Hz); XS is the positive sequence source impedance (V); XC is the capacitive reactance of capacitor bank (V); MVAsc is the three-phase short circuit capacity (MVA); MVAr is the three-phase capacitor bank rating; Vpk is the peak line-to-ground bus voltage (V); and ZS is the surge impedance (V). Because the capacitor voltage cannot change instantaneously, energizing a capacitor bank results in an immediate system voltage drop toward zero. This is followed by an oscillating transient voltage superimposed on the fundamental power frequency waveform. The peak voltage magnitude depends on the point-on-wave (i.e. the instant of energization), and can reach twice the normal system voltage (Vpk in per-unit) under worst-case scenarios.

MVAsc Zf Ipk

MVAr Vpk

Ls

Capacitor bank

Rs Ipk

Figure 2.8 System representation of a capacitor bank.

Power Systems and Signal Processing

19

Figure 2.9 Voltage transient during capacitor energizing.

Nevertheless, in a real system the transient magnitude is less than the theoretical 2.0 perunit energization when considering the system losses, loads and other damped elements. Typically the real magnitude levels range from 1.2 to 1.8 per-unit and the transient frequencies generally fall in the range of 300–1000 Hz. Figure 2.9 illustrates the voltage oscillation during the energization of a capacitor bank, obtained by simulation. Figure 2.10 illustrates the waveforms from a digital fault recorder (DFR). Note that the frequency of the voltage transient is the same as for the capacitor inrush current.

Figure 2.10 Real oscillography of a capacitor bank energization (source www.pqview.com).

Power Systems Signal Processing for Smart Grids

20

Normally, utilities are not very concerned about the transient overvoltage. This is due to the fact that surges generally lie below the level of insulation coordination. Due to the frequency band however, these transients may pass through step-down transformers directly into the industrial and commercial loads. As such, they may cause problems or damage equipment. Normally the secondary capacitor-switching transients are a function of the turns ratio of the step-down transformer. If the customer uses capacitors for power factor corrections on the low-voltage side, a severe overvoltage situation can result in the switching of highvoltage capacitors. This will be due to voltage magnification at the low-voltage capacitors in remote locations. Typically, these overvoltages might simply damage low-energy surgeprotective devices or cause a trip of power electronic equipment. Nevertheless, some cases have been reported of complete failure of end-user equipment. For major details of voltage magnification issues see reference [1]. 2.2.4.2 De-Energization When a circuit breaker opens at a specific time and the current goes to zero, the voltage wave at the open end (where the capacitor bank is installed) goes into a DC mode and the charge remains. This is due to the presence of the shunt capacitance at the end of the circuit, as shown in Figure 2.11a. It is very important to understand the transient recovery voltage (TRV) across the circuit breaker. Figure 2.11b represents these waveforms. Note that the TRV can reach 2 pu. If the TRV magnitude exceeds the allowed ratings of the breaker, a re-strike may occur. 2.2.4.3 Back-to-Back Switching of Capacitor Banks This event is normally associated with the high-intensity inrush of currents, with high frequencies overlapping the fundamental component. Nowadays, the most common practice to limit the current magnitude and frequency is to use series reactors with individual capacitor banks. Pre-insertion resistors or inductors can also be used with some types of switches or circuit breakers. The techniques of synchronized switching may be possible for certain types of circuit breakers; this is currently an important approach for smart grids. These are attempts to close each phase of a three-phase system towards its ideal condition, that is, when the voltage is passing through zero.

Figure 2.11 Capacitor bank de-energization: (a) voltages across the capacitor; (b) TRV.

Power Systems and Signal Processing

21

Figure 2.12 Capacitor inrush current during a back-to-back energization.

The frequency and magnitude of the inrush current during back-to-back switching depends on the size of the existing capacitor in service, the impedance of the discharging loop and the instantaneous voltage at the terminals of the capacitor bank at the time of the energization (point-on-wave). Normally, the impedance of the loop is much lower than the system impedance. This causes a higher inrush current if compared to the current during the energization of an isolated bank. Figure 2.12 illustrates the inrush current during back-to-back energization, characterized by the presence of high-frequency components. Although the high frequency lasts only a few milliseconds, this may exceed the momentary transient-frequency capability of the switching device. It can also cause fuse blows, false operation of protective relays and excessive errors in current transformers located at the feeder cables of grounded-wye capacitor banks.

2.3 Fault Current and DC Component Transmission and distribution lines are the components of a power system most exposed to environmental conditions. Rain, wind, lightning, fire, objects carried by the wind, birds and airplanes are among events that may affect the operation of a distribution system or a transmission line. When there is a breakdown of the insulation between conductors, this can be a path to an abnormal flow of current. The current tends to flow through the area of low resistance, bypassing the rest of the circuit. In such an event, the current can be very large or almost negligible, depending on configuration of the system and mainly of the grounding. In any case, this is known as a short circuit or simply a ‘fault’. In power plants and substations, short-circuits occur involving buses, connections, switchgear, transformers, reactors, capacitor banks, power electronic circuits and other equipment. Among the abovementioned events, the most common cause of a short circuit in a transmission line is lightning. A lightning strike can directly reach an energized cable or

22

Power Systems Signal Processing for Smart Grids

a cable guard of a transmission line. In some cases there will be a trigger of a potential that will open arcs between live parts of the line and its grounding. This then results in a short circuit of industrial frequency. When a short circuit occurs in close proximity to a generator terminal (synchronous machine), a decaying AC current can be observed. This decaying pattern is due to the fact that the magnetic flux through the windings of the rotating machine cannot change instantaneously. This is due to the magnetic nature of its inner circuit. Considering a three-phase source at the instant the fault occurs, each phase current has a value that depends on the generator load at that instant. Following the incidence of a fault the current in each of the phases changes rapidly to very high values. This is dependent on the type of fault (three-phase, phase-tophase, single-phase), its location and the generator sub-transient reactance x00 d . This very high current immediately begins to decay, first at a rapid rate determined by the transient time constant t00 d and then at a slower rate as defined by the transient time constant t0 d . When presented to the generator such a physical situation makes calculations quite difficult, and can be interpreted as a reactance that varies over time. Figure 2.13 shows this phenomenon at three non-discrete levels of current as established in one phase of a three-phase generator. The subtransient period (2 or 3 cycles) is characterized by x00 d and t 00 d . The transient period (several cycles) is characterized by x0 d and t0 d , and the current in steady-state or synchronous period. This is imposed by the direct axis reactance (xd). It is important to note that if a short circuit occurs far from a generator, this AC decaying does not exist practically. Furthermore, all short circuits in an AC circuit are also associated with the appearance of a DC component. Figure 2.14 illustrates the concept involved in a switching inductive or a capacitive circuit. For the circuit of Figure 2.14 consider the voltage source E: E ¼ EM sinðvt þ cÞ;

Figure 2.13 Short-circuit waveform with AC decaying.

(2.10)

Power Systems and Signal Processing

23

Figure 2.14 Equivalent circuit.

the impedance to the point of fault: Z1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ R21 ;

(2.11)

and the electrical angle of the circuit: x1 : R1

(2.12)

d icc ¼ E sinðvt þ cÞ dt

(2.13)

u ¼ arctan The differential equation for this circuit is: R1 icc þ L1

where the angle c characterizes the instant in which the circuit is closed or short circuited. The solution of this equation shows that the short-circuit current icc has two terms: icc ¼ I sinðvt þ dÞ  I sind et=T 1 :

(2.14)

The first term of the equation is a sinusoidal term, where d ¼ c  u. The second is a nonperiodic term that decays exponentially over time and is dependent on the primary time constant T1, defined: T 1 ¼ L1 =R1 :

(2.15)

This second term is normally referred as the DC component or DC offset of the fault current. The intensity of the DC component depends on the point-on-wave. Equation (2.14) shows that depending on the time instant in which the switching occurs, the value of the DC component magnitude can be higher or lower (even zero, if c ¼ u). On the other hand the duration of the DC component depends on the time constant T1. In a distribution system the normal value for L1/R1 is lower than the L1/R1 relation in a transmission system. As a consequence, the DC component disappears very quickly as shown in Figure 2.15b, compared to a DC component in transmission lines occurrences as in Figure 2.15a. Figure 2.16 shows a real waveform of a current during a phase-to-ground fault. Note the presence of the DC offset that causes an asymmetry in relation to the time axis. In some cases the transient overvoltage at high frequencies may be marked with the DC component, considering the voltage drop on the impedance of the source.

Power Systems Signal Processing for Smart Grids

24

Figure 2.15 Short-circuit waveform with c ¼ 90 and (a) T1 ¼ 0.1; and (b) T1 ¼ 0.02.

Figure 2.16 A phase-to-ground fault with DC offset.

The three-phase waveforms during a fault close to a generator can be more complex. Since the DC component is mixed with the AC current decay, the short-circuit current can be approximated [2]: iccðtÞ ¼

pffiffiffi 2







1  100   10 I 00k  I 0k etd þ I 0k  I k etd þ I k sinðvt  dÞ þ I 00k eT 1 sind

(2.16)

where I 00k , I 0k and Ik are the RMS currents in transient and sub-transient periods and in a steady state, respectively. Figure 2.17 illustrates the typical case of a three-phase fault close to a generator. Finally, it is important to remember that there are different types of faults. They may occur as a result of the abovementioned situations, involving grounding or not:





single-phase fault (phase-to-ground); phase-to-phase fault; phase-to-phase-to-ground; and three-phase fault.

Power Systems and Signal Processing

25

Figure 2.17 Three-phase short-circuit waveforms.

It is observed that the incidence of a single-phase fault is always higher than other disturbances. The statistics published in literature show that around 85–90% of all short-circuits in transmission lines are single-phased.

2.4 Voltage Sags and Voltage Swells Voltage sags (dips) are defined as an attenuation voltage signal with a duration of approximately 0.5 cycles up to 1 min. The typical magnitude is 0.1–0.9 pu. This phenomenon is often associated with short circuits; however, it can also be caused by switching transients of heavy loads such as large motors. The standards [3,4] define undervoltage as a similar phenomenon; its duration is longer than 1 min however. The phenomenon ‘voltage sag’ can be completely defined by measuring its magnitude (RMS value) and its duration. The magnitude depends on the fault conditions: location, resistance and the electrical system conditions such as as topology, source impedance ratio (SIR), short-circuit level, grounding system and so on. The duration of the voltage sag is directly related to the timing of the power system protection. This includes fault detection, fault magnitude measurements, trip command by the relay or the time of fault clearing by the circuit-breaker. In a three-phase system the voltage magnitude and duration is however also dependent on the kind of fault and the connection of the transformer(s) windings between the fault location and the point of the interest. This is normally the point of common coupling (PCC), where the sensitive load is installed. Such an example is that of a step-down Dy transformer. A single phase-to-ground fault on the primary side (D) will result in voltage sags on the secondary side (y) in two phases.

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Power Systems Signal Processing for Smart Grids

It is important to emphasize that sags are one of the most important phenomena studied in power quality, considering its harmful effects on loads such as information technology (IT) equipment, electronic control systems, rotating machines (mainly induction motors), variable speed drivers, contactors and relays. It is however highly problematic to establish the indices to be used as a reference when assessing the performance of the whole system. This is due to the number of points to be monitored and the random nature of the phenomenon, which calls for a long period of measurements. Figure 2.18 represents a record from a DFR showing (a) a phase-to-ground fault current and (b) the respective voltage sag of this phase in a 230 kV system. In this case, the propagation of the voltage sag through a Dy transformer is also measured in two phases on 69 kV subtransmission system (c).

Figure 2.18 Voltage sags caused by short circuit: (a) phase-to-ground fault current in 230 kV system; (b) the voltage sag measured at the 230 kV busbar; and (c) the voltage sage measured at the 69 kV system.

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Voltage swells are defined as an increase of the voltage signal with a duration of 0.5 cycles up to 1 minute, of typical magnitude 1.1–1.8 pu. This phenomenon can also be associated with short circuits. It can cause malfunctioning or damage of electronic equipment, controllers, computers and so on. In isolated systems or power sources (generators, transformers) with neutral earth through high impedance, normally the three-phase voltage profile during a phase-to-ground fault will present an increase of the voltages in at least two phases. The magnitude of the overvoltage (t > 1 min) or swell (0.5 cycle < t < 1 min) depends on the earthing level or on the relation between zero sequence impedance and positive sequence impedance (Z0=Z1). Equation (2.17) gives an idea of the voltage increase during a phase-to-ground fault: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2 1 X 0 =X 1 þ1 (2.17) e¼ 3 ðX 0 =X 1 Þ þ 2 2 where e is the factor of increase of the single-phase voltage, defined: V phase-to-ground ¼ eV phase-phase :

(2.18)

Consider a system with a high-impedance earthing and X0/X1 ¼ 10. In this case, during a solid phase-to-ground fault with voltage going to zero the sound lines (single-phase voltage) will be 0.88 of the phase-phase voltage. This means an increase of 1.53 pu or 53% on the single-phase voltages of the system. In the case where X0/X1 ¼ 1 (theoretical isolated system), this results in V phase-to-ground ¼ V phase-phase since e ¼ 1 (i.e. an increase of 73% or 1.73 pu on the sound phase voltage). On the other hand, in a solid grounding system where X0=X1 ¼ 1, there will be no swells (or overvoltage). A swell occurs less frequently compared to voltage sags, as the great majority of transmission and distribution systems are not isolated. A complementary study of shortduration voltage variations (sags and swells), including interruptions, can be read in [5,6].

2.5 Voltage Fluctuations Voltage fluctuations can be described as random or cyclical voltage waveforms. These can be observed when a load requires abrupt variations of current, especially when reactive components are present. The characteristics of voltage fluctuations depend on the load type and the power system capacity. Basically, there are two important parameters for voltage fluctuations: (1) the frequency; and (2) the magnitude. The best-known load that causes this voltage variation is an arc furnace, where the voltage waveform varies in magnitude due to the fluctuating nature or intermittent operation of connected loads. Typically the magnitudes of these variations do not exceed 10% of the rated voltage. This variation in magnitude is usually much lower than the sensitivity threshold of most equipment. Consequently, operational problems are experienced only on rare occasions. The main disturbing effect of these voltage fluctuations are changes in the illumination intensity of light sources, commonly called scintillation or flicker. ‘Flicker’ is the term used to refer to the subjective sensation that is experienced by the human eye when changes occur in the illumination intensity of some kinds of lamps, mainly incandescent. Technically, a voltage fluctuation is an electromagnetic phenomenon. Flicker is an undesirable result of the voltage fluctuation manifested in the luminosity emitted by incandescent lamps. Unfortunately,

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Power Systems Signal Processing for Smart Grids

Figure 2.19 A typical voltage fluctuation.

these two terms are often used synonymously in the literature. Figure 2.19 illustrates an example of a fluctuating voltage waveform. The flicker signal is defined by its RMS magnitude expressed as a percentage of the fundamental; IEC 61000-4-15 [7] describes the methodology as well as the specifications in terms of instruments for flicker measurement. An example of RMS voltage fluctuation caused by arc furnace operation is depicted in Figure 2.20. It is important to mention that, over the last 10 years, worldwide renewable energy capacity for many technologies has increased at rates of 10–60% annually. Wind power and many other

Figure 2.20 RMS voltage fluctuation.

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renewable technologies such as photovoltaic cells have experienced accelerated growth. Typically, 10–100 MW are produced in many different areas and connected to the local grid. Many challenges arise from this new scene in terms of operation, especially considering that wind and photovoltaic (PV) power systems are often sources of voltage fluctuations. A reasonable system to control voltage fluctuations should therefore exist.

2.6 Voltage and Current Imbalance Synchronous generators are three-phase voltage sources in power systems. The voltages should be of the same magnitude in each of the three terminals, positioned at 120 in relation to each other, for a practically balanced and symmetrical system. In practice however it is impossible to obtain a 100% balanced system since the many loads are primarily single-phase loads, especially when in a low- or medium-voltage network. As a consequence, currents that flow through these systems are no longer balanced. Special three-phase loads can also cause voltage unbalances such as arc furnaces. These may have different impedances in the path of the high current during the melting process [8]. Unbalanced voltages may appear for various other reasons, for example, the self-impedance and mutual impedance between different phases in a non-transposed transmission line. One of the main tools to assess the imbalance of current and voltage signals is the method of symmetrical components. Unbalanced currents flow through the system and can be decomposed into positive, negative and zero sequences, depending on the electric network and the transformers connections. These currents will produce additional losses of power (energy). Furthermore, they may cause undesirable heating at some points (wiring connections), and different voltage drops in each phase of a three-phase network. These cause unbalanced voltages at the point of common coupling. In turn, the unbalanced voltages may seriously affect three-phase loads. This is particularly true for motors and asynchronous rectifiers. For example, a negative sequence voltage will induce double-frequency currents at the rotor of a machine, producing additional heating and pulsating torques. Figure 2.21 shows the three-phase voltages measured in a PCC coupled to a weak system (low level of short circuit). Finally, it is important to mention that most international standards present limits and indices of voltage imbalance; these are outwith the scope of this book, however. For more details of unbalanced and asymmetrical voltages and currents see [5].

2.7 Harmonics and Interharmonics Harmonics have always been present in electric power systems. However, due to the widespread use of power electronics during the last decades, these have increased in magnitude. As such, current and voltage harmonic studies have become very important issues in all kind of installations. The typical definition of a harmonic is ‘a sinusoidal component of a periodic wave or quantity having a frequency that is an integral multiple of the fundamental frequency’. [9]. When a harmonic or harmonics are added in a periodic and sinusoidal signal its waveform becomes distorted, representing a non-ideal condition for an electrical grid as shown in Figure 2.22. Harmonics can be treated by their numbers or orders, taking into account the fundamental frequency as reference. For a fundamental frequency of 60 Hz, the fifth harmonic means

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Power Systems Signal Processing for Smart Grids

Figure 2.21 Unbalanced voltage (VT secondary) in a PCC.

5  60 Hz or 300 Hz and the eleventh means 11  60 or 660 Hz. Figure 2.23 illustrates some harmonics components. A periodic distorted or non-sinusoidal waveform can be obtained by adding different sine and cosines, each one a frequency multiple of the fundamental. A rectangular waveform can be composed in principle by the sum of all odd-order harmonics (sine) whose amplitudes are equal to the inverse of their order. In practice this particular case of the rectangular waveform does not converge uniformly, even considering the sum of a large number of harmonic components. This is known as the Gibbs phenomenon [10].

Figure 2.22 Distorted waveform by harmonics.

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Figure 2.23 Examples of harmonic orders.

Figure 2.24 depicts the sum: 41 X 1 n¼1

n

sinðn2pf 0 tÞ:

Note that the maximum value of the oscillation does not reduce, even if more terms are added to the series. The Gibbs phenomenon occurs for every signal that presents discontinuity.

Figure 2.24 The composition of a function and the Gibbs phenomenon.

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32

To be consistent, all periodic signals can be represented by a sum of sine and cosine where the mean-square convergence is defined [10]. The rectangular signal is one that converges in the mean-square sense. As a waveform can be created, the reverse operation is also true. A distorted, but periodic, waveform can be decomposed into the sum of several sine and cosines, each with its own amplitude or coefficient. This technique established by Jean B. Joseph Fourier (1768–1830) is well known and has been constantly used in order to obtain the harmonic spectrum. This concept can be summarized by Equations (2.19–2.22): xðtÞ ¼ a0 þ a1 cos v0 t þ a2 cos 2v0 t þ . . . þ an cos nv0 t þ b1 sin v0 t þ b2 sin 2v0 t þ . . . þ bn sin nv0 t; xðtÞ ¼ a0 þ

1 X n¼1

2 a0 ¼ T0

ðT 0

an cos nv0 t þ

1 X n¼1

bn sin nv0 t ¼ a0 þ

1 X

cn cosðnv0 t  fn Þ;

(2.19)

(2.20)

n¼1

ð ð 2 T 2 T xðtÞdt; an ¼ xðtÞcos nv0 t dt; bn ¼ xðtÞsin nv0 t dt; T0 0 T0 0  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn : cn ¼ a2n þ b2n ; fn ¼ arctan  an

(2.21) (2.22)

The harmonic spectrum (as shown in Figure 2.25) is obtained by this technique, referred to as the Fourier series. The theoretical treatment of the Fourier series will be presented in Chapter 4.

Figure 2.25 A decomposed waveform using Fourier series.

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Figure 2.26 The influence of the harmonics phase.

The magnitude of a certain harmonic order depends on the type of the non-linear load (or loads), as well as on the actual conditions of the electrical system. As shown in Equation (2.22) the phase of each harmonic (phase spectrum) can also be obtained. It should be noted that the phase of a harmonic changes the final waveform, as shown in Figure 2.26, although the harmonic content remains the same. The phase of a harmonic in relation to its fundamental frequency can assume any value between 0 and 2p. In a three-phase system the harmonics flowing through the phases of a system in a balanced way can be decomposed in terms of its symmetrical components. In other words, each order has a sequence that can be positive, negative or zero. Figures 2.27–2.29 illustrate this point, presenting an example of each sequence. Table 2.1 illustrates how harmonic sequence corresponds to the RMS value of the harmonic h: all triple (or multiple) harmonics are of zero sequence. These are also called homopolar harmonics. Among the sources of harmonics in power systems, three groups of equipment can be distinguished: (1) with a magnetic core (transformers, reactors, rotating machines); (2) special loads (arc furnaces, arc welders, discharge lamps); and (3) electronic and power electronic equipment. Figures 2.30–2.33 introduce some waveforms, their sources (compact fluorescent lamp, television, notebook and six-pulse rectifier, respectively) and their harmonic spectrum. Table 2.2 summarizes the current waveform for several devices. Although it is not the focus of this chapter, the harmonic levels in a grid must be supervised and monitored. For this, it is necessary to design instruments that quantify the overall harmonic content, both for current and voltage. Among several factors, one of the most

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Power Systems Signal Processing for Smart Grids

Figure 2.27 Negative sequence harmonic.

Figure 2.28 Positive sequence harmonic.

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Figure 2.29 Zero sequence (or homopolar) harmonic.

Table 2.1 Harmonic sequence. Order h Sequence

0 0

1 þ

2 

3 0

4 þ

5 

6 0

7 þ

8 

9 0

10 þ

11 

12 0

13 þ

Figure 2.30 Compact fluorescent lamp: current waveform and harmonic spectrum.

... ...

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Power Systems Signal Processing for Smart Grids

Figure 2.31 Television: current waveform and harmonic spectrum.

Figure 2.32 Notebook: current waveform and harmonic spectrum.

Figure 2.33 Current waveform of a six-pulse rectifier.

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Power Systems Signal Processing for Smart Grids

common is the total harmonic distortion (THD) factor. This is used for voltage or current, according to: sffiffiffiffiffiffiffiffiffiffiffiffi Nh P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2n 2 2 2 2 n¼2 h 2 þ h 3 þ h 4 þ . . . þ hn THDf ¼ ¼ (2.23) h1 h1 where h represents the RMS value of the harmonic and n is the harmonic order. International Standards always require some limits of THD for voltage and current, as in relation to the rated voltage of the system. However, there is some controversy about what should be the best values or limits to be adopted, the best measurement protocol to be used by instrumentation and the best procedure for establishing a measurement campaign. The presence of high levels of harmonic currents in an electrical network can in fact jeopardize the system components and loads. This is mainly if the point of common coupling (PCC) is linked to weak sources. In this case, the voltage THD can reach levels above those established by standards. These voltage harmonic distortions are caused by voltage drops on the impedances between the source and the PCC when the harmonic currents flow through them. Considering the presence of both non-sinusoidal voltage and current, the effects can be harmful. They can cause additional losses of the phase and neutral conductors, appearance of non-active distortion power, deterioration of the power factor of the plant, additional torque in rotating machines, additional losses of iron and copper in transformers, the burning of capacitors caused by resonance overvoltages or improper operations of protective relays. Other problems include measurements errors and telecommunications interference. A distorted waveform is not always and only composed of harmonics that are integer multiples of the fundamental. These can also be composed of non-multiples, commonly called interharmonics. Interharmonics are signals with a frequency that is a non-integer multiple of a fundamental frequency. Studies of electrical events associated with interharmonics are still in progress, but there is currently a great deal of interest in this phenomenon. Interharmonics have recently become more significant since the many types of power electronic systems, cycle converters and similar have led to an increase in their magnitude. According to the IEC 61000-2-1 standard [11] the interharmonics (voltages or currents) are further frequencies which can be observed between the harmonics of the power frequency voltage and current which are not an integer of the fundamental: ‘they can appear as discrete frequencies or as a wide-band spectrum’. By analogy to the order of a harmonic, the order of interharmonic is given by the ratio of the interharmonic frequency to the fundamental frequency (f i =f 1 ). The frequency of 92 Hz represents an interharmonic of 1.533 ‘order’. Also according to the IEC recommendation, the order of an interharmonic is denoted m [12]. If this ratio is less than unity the frequency is also referred to as a subharmonic frequency. The term ‘subharmonic’ does not yet have any official definition; it is simply a particular case of an interharmonic and its frequency is less than the fundamental frequency. However, the term has been used in many references. It is not straightforward to deal with interharmonics because their presence in a signal makes it a non-periodic signal within a given observation window, causing the spread or spillover to the next frequency domain. It can be observed in Figure 2.34 that considering the actual data window, the waveform is non-periodic. In fact, the waveform content is composed

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Figure 2.34 A waveform with interharmonic.

by 1 pu of the fundamental, 0.3 pu of the third harmonic and 0.11 pu of a non-integer multiple, in this case 92 Hz. As consequence, the conventional spectral analysis will cause false frequencies if an incorrect data window is utilized, as shown in Figure 2.35a. The basic sources of interharmonics include variable-load electric drives, arcing loads, static converters (in particular with direct and indirect frequency converters or inverters) and ripple controls. Interharmonics can also be caused by oscillations in systems comprising series or parallel capacitors, during switching processes or where transformers are subject to saturation.

Figure 2.35 From the waveform with interharmonics: (a) a ‘false’ harmonic spectrum; and (b) the true components.

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42

Spectrum 1 0.9 0.8

Magnitude

0.7 0.6 0.5 interharmonic frequency - 92 Hz

0.4 0.3 0.2 0.1 0

0

30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

Frequency [Hz] (b)

Figure 2.35 (Continued )

The most common effects of interharmonics are thermal effects, low-frequency oscillations in mechanical systems, disturbances in fluorescent lamps and electronic equipment operation. Others are interference with control and protection signals in power supply lines, overloading passive parallel filters for high-order harmonics, telecommunication interference and acoustic disturbances. More about harmonics and interharmonics see [22].

2.8 Inrush Current in Power Transformers A power transformer works by transferring apparent power from one side to another with an efficiency of 95–98%. In order to do this, an electromagnetic transformation with a magnetizing current is necessary. Under normal steady-state conditions the transformer magnetizing current associated with the operating flux level is relatively small, with a value that varies between 0.5% and 2% of its rated current. In order to minimize the costs, weight and size, transformers are designed to operate near the knee point of the exiting curve. This exciting current is normally non-sinusoidal, as shown in Figure 2.36. This non-sinusoidal 4

1.5

3

1

2

Iexc%

flux (pu)

0.5 0

1 0

–1

–0.5

–2 –1 –1.5 –3000

–3 –2000

–1000

0

Iexc (mA)

1000

2000

3000

–4

0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time (s)

Figure 2.36 Hysteresis cycle and the exciting current.

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voltage flux

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time (s)

Figure 2.37 A graphical relationship between voltage and flux.

current is due to the magnetization curve and the hysteresis cycle, illustrated in the figure. In other words, on a minor scale, a transformer is a source of harmonic currents during its normal operation. On the other hand, when a transformer is initially connected to a source of AC voltage, there may be a substantial surge of current through the primary winding of the transformer. This inrush current is required to establish its magnetic field. It is known that the rate of change of the instantaneous flux in a transformer core is proportional to the instantaneous voltage drop across its primary winding. In a continuously operating transformer, flux and voltage are phase-shifted by 90 since the flux is the integral of the voltage: f¼

ð 1 1 sin vt dt ¼  cos vt Nt vN t

(2.24)

where Nt is the number of turns. Figure 2.37 illustrates the relationship between voltage and flux. If the winding inductances were linear, the current would have exactly the same waveform as the flux. Both would be lagging the voltage waveform by 90 . Let us assume that the primary winding of a transformer is suddenly connected to an AC voltage source at the exact moment in time that the instantaneous voltage is at its maximum positive value. Under these circumstances, both core flux and coil current start from zero and build up to the same peak values experienced during continuous operation. There is theoretically no inrush current in this scenario; however, the inductance is not linear and saturation can be expected to occur, especially since transformers are usually working near the knee point. Taking the flux to twice its normal maximum will cause hard saturation, requiring a very large exciting current. However, this is not the worst-case scenario. If a transformer

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Power Systems Signal Processing for Smart Grids

Figure 2.38 A real inrush current in a three-phase bank transformer.

is energized at the zero point of the voltage wave with a maximum residual flux in the same direction, the saturation may be greater and the inrush current will be established reaching 5–20 times the rated current. Moreover, the current waveform will be non-sinusoidal and fully offset from the time axis as sharp pulses, as depicted in Figure 2.38. For three-phase transformers, as can be seen in Figure 2.38, each phase will have a different exciting current since the point-on-wave at which excitation begins is different for each phase voltage. Thus, even if one phase experiences non-saturation (non-residual flux, point-on-wave at its peak), the other two phases will necessarily have a distorted waveform with great magnitude. In brief, the magnitude of the inrush current strongly depends on the exact time of the connection. If the transformer has some residual magnetism in its core at the moment of the connection to a source, the inrush could be even more severe.

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The decay of the inrush current is fast for the first few cycles, but after this it decays slowly and it will take several seconds for the current to reach normal levels. The time constant depends on: (1) the transformer size:10 cycles for small transformers to 1 min for a large one [13]; (2) the resistance of the system; (c) the variable inductance of the transformer. In some cases the current waveform will still be distorted for a relatively long period such as 30 min after the energization point [14]. Other elements that influence the inrush current are the core design, the type of three-phase connection and the sort of steel used for its construction. It is important to emphasize that the inrush current is rich in harmonics. It contains all orders of harmonics. However, excluding the significant DC offset also present, the second- and the third-order harmonics are by far the greatest in magnitude. The second harmonic will be present in all inrush waveforms of all three phases. Its proportion will vary with the degree of saturation and the presence of DC offset in the core flux. The second harmonic magnitude has been reported as being about 20–60% of the transformer’s rated current value. Higher harmonics are also present, but their proportion is much smaller than that of the second and third. Other factors that control the magnitude and duration of an inrush current are the conditions surrounding the energization of a transformer or a bank transformer. These include: (1) initial energization, normally producing the maximum value; (2) recovery inrush from a fault next to the transformer, when the voltage returns to normal value after the action of the protection system; and (3) sympathetic inrush. The latter is when a current occurs in an energized transformer when a nearby transformer (paralleling a second transformer) is energized. This last case is due to the fact that the flowing inrush current finds a parallel path in the previously energized bank. The DC offset flowing may actually saturate the core of the previously energized bank, causing a kind of inrush as shown in Figure 2.39. It is important to emphasize that the inrush current is a very interesting time-varying harmonics process, since each harmonic can be seen changing its magnitude and phase during the entire process. However, the main concern regarding an inrush current is the transformer protection. Under normal conditions, the inrush cannot activate the protection system. For a transformer to operate correctly however, the protective relay needs to distinguish between an internal fault and an inrush current. Nowadays, with more intelligent numerical relays, this task is relatively simple. In order to minimize the magnitude and the distortion of the energized current, a new system of controls for smart grids must be developed so that accurate synchronization can lead to the closing instant of a transformer’s circuit breakers.

2.9 Over-Excitation of Transformers The magnetic flux inside a transformer core is directly proportional to the applied voltage and inversely proportional to the power frequency. Overvoltage and/or under-frequency conditions can cause over-excitation conditions that can saturate the transformer core. This over-excitation causes heating and increases the exciting current, as well as noise and vibration in a transformer. A typical waveform can be seen in Figure 2.40. The most significant harmonics of this waveform are the third, with about 40% of the fundamental and the fifth with 20%. Normally the fifth harmonic is used to detect over-excitation, since the third can be filtered by a delta connection of the transformer or the delta connection of the CTs.

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Power Systems Signal Processing for Smart Grids

Figure 2.39 An example of a sympathetic inrush.

Figure 2.40 Exiting current of an overxcited transformer.

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2.10 Transients in Instrument Transformers Instrument transformers are used for measurement, control and protective applications together with equipment such as meters, relays and other devices of control. Their objective in electric systems is of great importance, considering that these devices are the means by which the values of high current and high voltage are reduced to values such as 1–5 A or 100–120 V, enabling them to be handled by the secondary instruments. This offers the advantage that measurements and protective equipment can be standardized on a few values of current and voltage. Nowadays there is an entire range of conventional instrument transformers that cover lowvoltage 600 V to extra-high voltage of 1000 kV, such as current transformers (CTs), voltage (potential) transformers (VTs) and capacitor voltage transformers (CVTs). Instrument transformers are designed to present good accuracy in steady-state or during faults in the electrical systems for which they are specified. However, certain events can extrapolate the magnitude levels for which they are prepared. This can result in a distorted waveform of the secondary signal and can be caused by the performance of its own instrument transformers. These secondary transients may affect the performance of the protection and control systems, endangering the reliability of the entire system. Of the three devices mentioned above, CTs and CVTs are of most interest in terms of transients on their secondary terminals. The main issues are discussed in the following sections.

2.10.1 Current Transformer (CT) Saturation (Protection Services) The performance of a CT can be assessed by considering its ability to accurately reproduce the primary current waveform at its secondary. Normally, two different aspects must be considered: symmetrical and asymmetrical fault current. 2.10.1.1 Symmetrical Fault Current AC saturation occurs when the current magnitude is higher than expected during a symmetrical fault for which the CT was designed. Usually the accuracy of a CT (typically 5–10% for protective devices) must be confirmed with a standard burden connected in its secondary from the rated current up to 20 times the rated current. If the magnitude of a fault current exceeds 20 times the nominal current, the CT enters into a saturation regimen. It is worth mentioning that every CT has its magnetization curve. Its flux density b is defined by Equation (2.25): b¼k

E N2 f A

(2.25)

where E is electromotive force; N2 is the number of secondary turns; f is the system frequency; A is the cross-section of the core; and k is a manufacturing constant. The secondary rated voltage of a CT which will be available to the relay is calculated based on this factor of 20. Suppose a CT with rated current equal to 500-5A and a nominal standard burden of 25 VA or 1 V of impedance (VA ¼ 1  52). For this example, when the current reaches 20 times the rated current, the terminal voltage will be 100 V (20  5 A  1 V ¼ 100 V). This is actually very close to the knee point of the excitation curve. Any current magnitude higher than a factor of 20 greater will cause an increase in the excitation current, which may be above 10% of the nominal value; this is an unacceptable error. In addition to the error itself, such a current

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Power Systems Signal Processing for Smart Grids

Figure 2.41 Secondary current of a CT during AC saturation (actual and expected).

waveform is distorted, consequently distorting the secondary current flowing to the secondary devices (relays, meters, electronic devices, etc.). Figure 2.41 shows a secondary current waveform of a CT 100-5A, C100 (Zburden ¼ 0.5 þ j0.866 V) with a symmetrical fault current equal 4000 Arms (40  100 A). Figure 2.42 is another example with a burden equal to Zburden ¼ 1 þ j0 V.

Figure 2.42 Secondary current of a CT during AC saturation (actual and expected).

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2.10.1.2 Asymmetrical Fault Current From the definition of short-circuit current (Equation (2.14)), we assume a short-circuit current that is completely asymmetrical i.e. d ¼ 90 . This gives: h i i1 ¼ icc ¼ I cosðvtÞ  et=T 1

(2.26)

where T1 is the time constant of the system, given by Equation (2.13), or T 1 ¼ X 1 =vR1 where X1 and R1 are the reactance and the resistance of the primary system up to the fault point. Considering a lossless CT with resistive burdens, the following expression for the flux inside the CT core can be derived: 2

3

6egt  eat sinðvt þ wÞ 7 eat sin w 6 7  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ¼ R2T I 6 7 a2 þ v2 5 4 ag a2 þ v 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

(2.27)

f alternated

f DC

where w ¼ arctan

R2T ; vLm

(2.28)

where Lm is the magnetizing inductance and R2T is the total resistance of the secondary side (burden plus secondary coil). Further, g ¼ 1=T 1 and a ¼ 1=T 2 , with T2 being the time constant of the CT, given by T2 ¼

Lm : R2T

(2.29)

If we consider the case Lm ! 1, Equation (2.27) reduces to h i R I 2T sinðvtÞ f ¼ R2T IT 1 et=T  1 þ v

(2.30)

which comprises another exponential term added to an alternated term and whose evolution with time is illustrated by Figure 2.43. It is important to note that the relationship between the maximum exponential flux and alternated flux is given by fexp;max L1 x1 ¼ vT 1 ¼ v ¼ : falt;max R1 R1

(2.31)

Equation (2.31) highlights the importance of the X/R ratio of the primary system when analyzing CT transient performance e.g. if considering that this ratio will directly influence the saturation time.

Power Systems Signal Processing for Smart Grids

50 25

Normalized Flux

20 15

Total Total flux

10

Exponential Exponential flux

5

Alternated Alternated flux

0 –5

0

0.1

0.2

0.3 Time (s)

0.4

0.5

0.6

Figure 2.43 Theoretical flux evolution.

The previous consideration is of a theoretical nature since Lm may assume a high value but not infinity (1). Thus, when Lm >> R2T and hence w ffi 0, Equation (2.27) can be simplified as: f¼

R2T I gt R2T I ½e  eat  þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðvtÞ: ag a2 þ v 2

(2.32)

In this case, the core flux also consists of an AC and a DC component. The DC component is composed of two different exponentials: (1) g, defined by the time constant of the primary system (i.e. related to the X1/R1 ratio) and (2) a, defined by the time constant of the CT. It is well known that the DC component reduces proportionally with the reduction of the X/R ratio of the complete circuit, from generator to short-circuit point. In generators the ratio of sub-transient reactance to resistance may reach values as high as 70 [15]. On the other hand, in circuits far away from generators (e.g. utility power distribution systems and industrial power systems), the X/R ratio is lower and the DC component decays quickly. Figures 2.44 and 2.45 depict the calculated flux, assuming no saturation. The first case is for a system with T1 ¼ 0.053 s (X1/R1 ¼ 20, typical for power transmission systems). The second case is for a system with T1 ¼ 0.013 s (X1/R1 ¼ 5, typical for power distribution systems). For both examples, T2 ¼ 0.663 s. Each design CT core has a saturation level or factor (according to IEEE Std C37.110 [16] guidelines), denoted Ks. This depends mainly on core section and type of lamination material (stalloy, mumetal, etc.), and is calculated from the expression: K s ¼ V x =V s

(2.33)

where Vx is the saturation voltage and Vs is the secondary voltage for a symmetrical fault. Considering Figures 2.44 and Figure 2.45, if the core has a saturation level of 6 pu for the first case of Figure 2.44 the CT will saturate as soon as the first cycles occur. On the other hand, the CT for the second case (Figure 2.45) will not saturate. The saturation therefore also

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25 Exponential definedby byTT2 2 Exponential defined

Normalized Flux

20

Total flux Total flux

15 10 Exponential definedby byTT1 1 Exponential defined

5 Alternated Alternatedflux flux

0 –5

0

0.1

0.2

0.3 Time (s)

0.4

0.5

0.6

Figure 2.44 Flux evolution (T1 ¼ 0.053 s and T2 ¼ 0.663 s).

6

Exponential definedbyby Exponential defined T2T2

5 Normalized Flux

Total Total flux flux

4 3 2

Exponential definedbyby Exponential defined T1T1 Alternated Alternated flux flux

1 0 –1 0

0.1

0.2

0.3 Time (s)

0.4

0.5

0.6

Figure 2.45 Flux evolution (T1 ¼ 0.013 s and T2 ¼ 0.663 s).

depends on two important external factors: the level of fault and the X/R ratio. Figure 2.46 shows in detail a comparison of a 20% increase in magnitude of the fault current and a change in X/R ratio from 5 to 7. For this case the CT saturates in approximately 0.02 s; the greater the fault current and the X/R ratio, the smaller the time to saturation. The question now is: what happens to the secondary current when the core saturates? From the theoretical model of saturation depicted in Figure 2.47, we can attempt to answer this question. The model shows that when the core tends towards saturation, the primary current becomes equal to the magnetizing current i0 and the secondary current i2 tends to zero. Furthermore, when the flux falls below the saturation level the secondary current i2 follows the primary current i1, as illustrated in Figure 2.48. Observe that when the transient flux decreases

52

Power Systems Signal Processing for Smart Grids

Figure 2.46 Comparison: a flux without saturation and with saturation.

Figure 2.47 A theoretical model of saturation curve.

below the saturation level (the bold line), the waveform of the secondary current is no longer distorted. In practice, the model shown in Figure 2.48 does not occur. The current does not tend to zero instantaneously, but there is a smoothed change. The data presented in Figure 2.49 are the result of a simulation, representing the secondary current during a CT saturation considering CT 100-5A, C100 (Zburden ¼ 0.5 þ j0.866 V). It was submitted to an asymmetrical fault of 1000 Arms with maximum DC offset. Note that the current in this case does not reach 20 times the rated current; instead, the saturation occurs due to the DC component of the fault.

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Figure 2.48 Theoretical model of secondary current of a saturated CT.

Figure 2.49 Secondary current of a CT during saturation caused by DC offset.

Figure 2.50 depicts a real waveform; during a three-phase fault two CTs go into saturation. Finally, it is important to observe that CT saturation time is normally a function of current fault, core flux density, CT parameters, CT burden including wire impedance and the DC time constant. Sometimes the time to saturation is long enough for high-speed relaying devices to

54

Power Systems Signal Processing for Smart Grids

Figure 2.50 Real secondary current of CTs during saturation.

detect faults prior to its CT collapse. In many cases however, the CT saturation may cause improper operation of protective relays, jeopardizing the system performance. Specific CT performance parameters, including CT type TPX, TPY and TPZ, can be found in IEC 61869-6 [17].

2.10.2 Capacitive Voltage Transformer (CVT) Transients Capacitive voltage transformers (CVTs) convert transmission class voltages to standardized low and easily measurable values. These are used for metering, protection and control of a high-voltage system. They are the predominant source of voltage signals for impedance relays in HV and extra-high voltage (EHV) systems and provide a costefficient way of obtaining secondary voltages for EHV systems. Additionally, CVTs serve as coupling capacitors for coupling high-frequency power line carrier signals to the transmission line. The non-linear nature of the elements that constitute the primary circuit of the CVT gives rise to electromagnetic phenomena that may affect the secondary signal. The most common events are the transient voltage drop and ferroresonance. During line faults, when the primary voltage collapses and the energy stored in the stack capacitors and the tuning reactor of a CVT needs to be dissipated, the CVT generates severe transients that affect the performance of protective relays. The higher the system impedance ratio, the worse the CVT transient will be. Figure 2.51 presents an example of CVT operation during a fault occurring at the zero crossing of the primary voltage. As seen from the figures, the CVT transients can last for up to two cycles and reach a magnitude of up to 40% of the nominal voltage. This small signal seems of no importance; however, it can cause great difficulties for a protective relay to distinguish quickly between faults at the reach point and faults within the protection zone. More details about this kind of problem can be found in reference [18].

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Figure 2.51 Transient voltage drop in the secondary of a CVT [18].

2.11 Ferroresonance Ferroresonance is a very complex phenomenon. The book which covers it thoroughly has yet to be written and few papers on the topic have been published. According to reference [19], the following categories or classes of ferroresonant circuits have been reported: 1. transformer supplied accidently in one or two phases; 2. transformer energized through the grading capacitance of one or more open circuit breakers; 3. transformer connected to a series-compensated transmission line; 4. voltage transformer connected to an isolated neutral system; 5. capacitor voltage transformer; 6. transformer connected to a de-energized transmission line running in parallel with one or more energized lines; 7. transformer supplied through a long transmission line or a cable with a low short-circuit power. Ferroresonance is a type of resonance. It can suddenly change from a steady-state response (sinusoidal frequency) to another with a chaotic behavioral response. It is characterized by a high-current fundamental-frequency state. However, it is also characterized for subharmonic, quasi-periodic and even chaotic waveforms and a random time duration in any circuit containing a non-linear inductor. The associated high overvoltage can cause dielectric and thermal problems to the transmission and distribution systems, as well as to secondary devices in CVTs or VTs. It may exhibit different modes of operation that are not experienced in linear

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Figure 2.52 Ferroresonance at the secondary of a CVT.

systems. Finally, the frequency of the voltage and current waveforms may be different from the sinusoidal voltage source. This phenomenon is not predictable using linear theories. Unusual solutions are necessary to detect, identify and prevent equipment damage through ferroresonance. To illustrate, Figure 2.52 is a kind of ferroresonance signal recorded in a CVT secondary after a circuit-breaker has cleared a fault. For a better understanding of the phenomenon, including ferroresonance modes, methods of identifying, modeling and preventing ferroresonance, see reference [20].

2.12 Frequency Variation In normal operating conditions the system frequency experiences very small variations. In abnormal situations however, such as imbalance between generation and load, the frequency can experience large variations. All these events are supposed to be stabilized by the control and protection systems, leading the frequency back towards its tolerable limits of variation. In critical situations the protection and control systems can however fail. Under such circumstances the power system experiences losses of synchronism and stability. This phenomenon constitutes a vicious circle. Furthermore, because a large number of algorithms are used in control and protection, it is assumed that the frequency is constant. If there is a large deviation in frequency the algorithms cannot work properly. Figure 2.53 shows the frequency variation when the interlinked Brazilian system suffered a major system event. The plot shows the frequency in several sites spread along the Brazilian grid. The large variation of the frequency at certain sites after islanding two main regions can clearly be seen.

2.13 Other Kinds of Phenomena and their Signals There are many others types of disturbances that can appear in a power system, such as: secondary arcs during a single-pole auto-reclosing power system oscillation or stable and unstable power swings; the loss of excitation in synchronous machines; the appearance of third harmonic voltage at generator neutral during stator winding phase-to-ground fault; and the non-zero current crossing during faults. Problems are also experienced with circuit break re-strike and re-ignition, reversal current, reversal voltage and sub-synchronous oscillations in compensated transmission lines.

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Figure 2.53 Frequency measured in several sites during a strong event in the interlinked Brazilian Power System.

Nowadays, all of the above and their signals (voltage and current) can be recorded using digital fault recorders (DFRs); numerical relays, phase measurement units (PMUs) and other intelligent electronic devices (IEDs) can also be used. Analysis of system disturbance is very important since it can provide feedback regarding the integrity of the equipment and the overall system, answering the basic questions: What happened? Why did it happen? What is going to be done about it? For more about these questions and answers, see reference [21].

2.14 Conclusions This chapter gives a comprehensive but concise overview of the most common power system phenomena (time-varying or not). These are discussed in terms of voltage, current signals and waveforms. The idea is to characterize many of them by considering the magnitude, phase and waveforms and to show that many signals can be represented by a mathematical expression (exponential DC, faults, harmonics in general, etc.). The chapter also highlights some of the challenges in applying signal processing techniques to electrical power signal analysis. Although voltage and current quality are one of the most common applications for these techniques, the approach and tools can be used by the entire power engineering community. This chapter gives an extensive list of other issues for which advanced signal processing tools may provide solutions.

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Power Systems Signal Processing for Smart Grids

It is expected that a greater integration and utilization of signal processing by the power engineering community will be necessary in order to cope with the higher complexity of future intelligent networks or smart grids.

References 1. Natarajan, R. (2005) Power System Capacitors, CRC Press, Boca Raton. 2. Reimert, D. (2006) Protective Relaying for Power Generation Systems, CRC Press, Taylor & Francis Group. 3. IEC61000-4-30 (2003) Electromagnetic compatibility (EMC), Part 4, Section 30: Power quality measurement. International Electrotechnical Commission, Geneva, Switzerland. 4. IEEE-1159 (1995) Recommended practice for monitoring electric power quality, IEEE Std. 1159–1995. IEEE Standards Board, Piscataway, USA. 5. Bollen, M.H.J. (2000) Understanding Power Quality Problems: Voltage Sags and InterrruptionsWi, Wiley– IEEE Press. 6. Dugan, R.C., McGranaghan, M.F., Santoso, S. and Beaty, H.W. (2003) Electrical Power Systems Quality, 2nd edn, McGraw-Hill Companies, Inc. 7. IEC61100-4-15 (1997) IEC 61000-4-15, Flickermeter: Functional and design specifications. IEC, Geneva, Switzerland. 8. Baggini, A. (2008) Handbook of Power Quality, John Wiley & Sons, Chichester, England. 9. IEEE-519. (1993) Recommended practices and requirements for harmonic control in electric power systems. IEEE Industry Applications Society, IEEE Standards Board, Piscataway, USA. 10. Mitra, S.K. (2006) Digital Signal Processing: A Computer Approach, McGraw–Hill Companies, Inc. 11. IEC61000-2-1 (1990) Electromagnetic compatibility (EMC) - Part 2: Environment - Section 1: Description of electromagnetic environment for low-frequency conducted disturbances and signalling in public power supply systems. International Electrotechnical Commission. 12. IEC 61000-2-2 (2002) Electromagnetic compatibility (EMC) - Part 2-2: Environment-compatibility levels for low-frequency conducted disturbances and signalling in public low-voltage power supply systems. International Electrotechnical Commission. 13. Blackburn, J.L. (2007) Protective Relaying: Principles and Application, 3rd edn, CRC Press. 14. Anderson, P.M. (1999) Power System Protection, IEEE Press. 15. GE Power. Dimensioning of current transformers for protection application. Application note, GER3973. GE Power. 16. IEEE Std. C37.110 (2007) IEEE guide for the application of current transformers used for protective relaying purposes. IEEE Power Engineering Society. 17. IEC61869-6 (2012) Current transformer for transient performance. Part 6. International Electrotechnical Commission. 18. Kasztenny, B., Sharples, D., Asaro, V. and Pozzuoli, M. (April 2000) Digital relays and capacitive voltage transformers: balancing speed and transient overreach. 3rd Annual Conference for Protective Relay Engineers, GE Power. 19. Jacobson, D.A.N. (2003) Examples of ferroresonance in a high voltage power system. In Proceedings of Power Engineering Society General Meeting, IEEE Vol. 2, 13–17 July 2003. 20. Ang, S.P. (2010) Ferroressonance simulation studies of transmission systems. Thesis. University of Manchester. 21. Ibrahin, M.A. (2012) Disturbance Analysis for Power Systems, Wiley & Sons. 22. Acha, E. and Madrigal, M. (2001) Power Systems Harmonics: Computer Modelling and Analysis, Wiley & Sons. 23. Arrilaga, J. and Watson, N.R. (2003) Power System Harmonics, Wiley & Sons.

3 Transducers and Acquisition Systems 3.1 Introduction Intelligent electronic devices (IEDs) such as protective relays, digital fault recorders (DFRs), energy meters, power quality monitors, signal analyzers or other secondary devices are required to have reasonably accurate reproduction under the conditions of their power system, whether normal or unusual operational conditions. Between that power system and the core of a processing device a chain of elements however exists, as shown in Figure 3.1. The first components in connection with the power grid are instrument transformers. Instrument transformers are used for measurement, control and protective applications, together with a diverse array of equipment such as meters, relays and other devices of control. In electrical systems their role is of primary importance as they are the means of ‘stepping down’ the current or voltage of the system to measurable values, such as 1–5 A in the case of current transformer or 100–120 V in case of a voltage transformer. This offers the advantage that measurements and protective equipment can be standardized on only a few values of current and voltage. Another important function of instrument transformers is to decouple (isolate) the primary circuit from the secondary. This means that there is no electrical connection between the primary and secondary circuits. The transfer of information between the primary and the secondary circuits is achieved through an electromagnetic transformation. There are basically three types of conventional instrument transformers: voltage transformer (VT), capacitive voltage transformer (CVT) and current transformer (CT). Recent technology allows the replacement of a conventional instrument transformer by a non-conventional transducer, for example an optical current transformer (OCT), optical voltage transformer (OVT), Rogowski Coil and others. The objective of this chapter is to present the basic concepts and types of transducers currently used as well as the recently introduced technologies.

Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo M
arcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

Power Systems Signal Processing for Smart Grids

60 Current and Voltage inputs

Instrument Transformers

Surge protection and filters

Signal conditioning

Sample/hold A/D

IED V, I Microprocessor GPS

RAM, ROM, PROM, EPROM, communication channels, digital inputs....

Figure 3.1 Block diagram of an IED.

3.2 Voltage Transformers (VTs) As for any kind of transformer, VTs (formally potential transformers or PTs) obey the law of electromagnetic transformation as depicted in Figure 3.2. The nominal rate is given by V1/V2 ¼ N1/N2 ¼ RVT. This class of conventional transformers has both primary and secondary windings. The primary winding is connected directly to the power circuit, either between two phases or between one phase to ground depending on the rating of the transformer and on the requirements of the application. These are specially designed to accurately reflect the primary voltage signal of the power system, at a rated frequency. In a low voltage this is a secondary signal; these may therefore be used in any kind of low-voltage measurement or protective devices. The flux density b in the magnetic circuit is given by Equation (2.23) (see Chapter 2), and the relation between flux density and current intensity H is not linear, as depicted in Figure 3.3. This curve represents the saturation curve of the transformer core. The VTs can also be represented by their equivalent circuit as shown in Figure 3.4, where R1, X1, R2, X2, Rm and Xm are the resistance and the reactance of the primary circuit, secondary circuit and the fictitious magnetizing shunt element, respectively. When analyzing the equivalent circuit (Figure 3.4) the same characteristic curve that is shown in Figure 3.3 can be obtained in laboratory, relating the internal voltage E2 and the magnetizing current I0.

Figure 3.2 Voltage transformer.

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61

Figure 3.3 Magnetizing curve.

E3

X1

E4

R1

Io

X2

I1

I2 Ix

V1

R2

E2

Ir Xm

Rm

V2

Zb

Figure 3.4 Equivalent circuit of VTs.

Equation (2.23) can also be written as b¼k

E2 ½ðR2 þ jX 2 Þ þ Z B I 2 ¼k ; N2f A N2f A

(3.1)

where ZB is the burden impedance which represents the consumption of the secondary instruments. It is important to emphasize that any VT introduces an error to the process, even in steady state. The main cause of this error is the voltage drop on the primary and on the secondary circuits, defined: ðK n  V 2  V 1 Þ  100 DV ¼  100 V1 V1 where as Kn is the VT nominal rate.

(3.2)

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62

V1 Ir Ix

E2

g V2

Io

E3

E3

E4

R2I2

X1I1

X2I2 I2 Io

I1

R1I1

Figure 3.5 Phasorial diagram.

The secondary current I2 and the magnetizing current I0 are responsible for the voltage drops. In other words: the greater the burden, the greater the secondary current. This implies an increase of the internal voltage E2, causing an increase of the magnetization current according to the saturation curve. Note also that I1 ¼ I0 þ I2 (phasorial sum). As such, the errors imposed by a VT are partly caused by the burden current (error due to ZB); the other share is I0. Figure 3.5 depicts the phasor diagram which shows that V2 is different from V1 in both magnitude and phase. The VTs are therefore specified according to the service (for metering or relaying) in different accuracy classes. Table 3.1 lists the accuracy classes for measurement according to IEC 60044-2 [1]. For example, a VT class A must have a magnitude error between 0.5% and 0.5% and a phase displacement of 2 to 2 minutes. For protection services, see the data listed in Table 3.2 [2]. According to the IEEE Standard C57.13 [3], the accuracy classes for metering are 0.3%, 0.6% and 1.2%. These are related to a composite error (e% and g (min)), graphically represented by

Table 3.1 Accuracy classes for metering (IEC 60044-2). V ¼ (0.9  1.1) Vpn (rated voltage)

Accuracy class

A B C

eV %

g (min)

0.5 1.0 2.0

2 30 60

Table 3.2 Accuracy classes for relaying (IEC60044-2) (k depends on earthing/grounding mode; a value of 1.1 or 1.5 or 1.9 may be assumed) V ¼ (0.25  0.9)Vpn

ccuracy class

e (%) E F

3 5

g (min) 120 250

V ¼ (1.1  k) Vpn e (%)

g (min)

3 10

120 300

Note: additional requirements about VTs were recently introduced by IEC 61869: Part 1–9 [2].

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Magnitude Error (ε%) –1.40

(1-ε%) 1.014 1.2%

–1.20 –1.00

1.012 1.010

0.6%

–0.80

1.008 0.3%

–0.60

1.006

–0.40

1.004

–0.20

1.002

0

1.000

+0.20

0.998

+0.40

0.996

+0.60

0.994

+0.80

0.992

+1.00

0.990

+1.20

0.988

+1.40 70 60 50 40 30 20 10 0 10 20 30 40 50 60 – Phase displacement (δ) in minutes

0.986 +

Figure 3.6 Parallegram of accuracy according to IEEE Standard C57.13 [3].

the accuracy classes parallelogram (Figure 3.6). The accuracy classes are linked to the standard burdens as follows:

Burden VA

W 12.5

X 25

M 35

Y 75

Z 200

ZZ 400

For example, 0.3 W means that for a maximum burden of 12.5 V the accuracy class 0.3% is guaranteed. Normally for protective relays, a class 1.2 is sufficient. More details on the subject can be found in the IEEE Standard C57.13 [3]. In terms of rated voltage, VTs are designed from a low voltagepto an extra high voltage. Normally the secondary windings are specified from 57.73 (100/ 3) to 120 V. Finally, it is important to mention that conventional (inductive) voltage transformers are usually used for harmonic measurements. However, the frequency responses of these instruments limit the accuracy of the measurements. As a result, they may not be suitable for high-order harmonics. An example of this limitation is illustrated in Figure 3.7, where the measured frequency response and transformation of a square wave are shown. Further information on this can be found in reference [4]. In the context of new technologies and smart grids, higher-frequency loads are becoming more common.

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Figure 3.7 (a) Frequency response of a VT and (b) signal transformation of a square wave (measurements). Measurements taken at University of Dresden laboratory, Germany [4].

3.3 Capacitor Voltage Transformers A high-voltage VT must have a large number of turns in the primary circuit as well as a lower primary current. For a VT in 138 kV the number of turns can be greater than 100 000 and the primary current lower than 2 mA. Thus, the design of a high-voltage VT is very complex considering the large number of turns of very thin wire, and can experience problems of fixation, isolation, resistance to breakage and so on. This makes such equipment very expensive. Due to the issues above, it is common that in high- and extra-high-voltage systems (above 145 kV) a voltage transformer operating through a capacitive voltage divider is used. This equipment is referred to as a capacitive voltage transformer (CVT). In other words, a CVT is basically a capacitive voltage divider with a voltage transformer (inductive) connected to a point of medium voltage, normally 10–25 kV. Furthermore, a secondary divider is used at the lowvoltage (100–120 V) level and is used for metering, protection and control as shown in Figure 3.8. The capacitive divider represents an equivalent source of capacitive impedance and can therefore be compensated by a reactor (tuning inductance L) connected in series with the tapping point. An ideal reactor should allow for minimum error in the process.

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65

A B C

I + I1 C1 Vc1 L

Vc2

C2

I1

I

filter

I2 V2

Conventional VT

Burden

Figure 3.8 Capacitor voltage transformer.

The reactor L is normally adjustable in order to form a tuned circuit with a group of capacitors C1 þ C2. This is achieved in such a fashion that the current load I2 does not influence the accuracy of the secondary voltage V2. Figure 3.9 shows the equivalent circuit of a CVT, emphasizing the compensation reactor, the VT equivalent circuit and the burden. An equivalent circuit when seen from the perspective of the burden terminals (i.e. with zero source impedance when the excitation circuit and the resistance of the primary and secondary winding of the VT are neglected) is shown in Figure 3.10, where V TH ¼

XC2 Vf XC1 þ XC2

(3.3)

I + I1 XC1

R1

XL

Vf I

R2

XL1 I0

I1 XC2

V1

Rm

Xm

XL2 I2 Rb

E2

V2 Xb

Figure 3.9 Capacitor voltage transformer equivalent model.

Power Systems Signal Processing for Smart Grids

66 XC1 XL

XL1

XL2

I1 = I2

XC2

Rb V2

Vth

Xb

Figure 3.10 The Thevenin equivalent circuit.

and Z TH

  XC 1 XC 2 : ¼ j XL1 þ XL2 þ XL  XC 1 þ XC 2

(3.4)

Considering Figure 3.10, if the total inductive reactance is exactly equal to its capacitive reactance (C1/C2), the circuit will be tuned. This means that the resulting impedance will be equal to zero and no voltage drop is considered. This is true when XL ¼

XC 1 XC 2  ðXL1 þ XL2 Þ: XC 1 þ XC 2

(3.5)

The secondary terminal voltage is therefore equal to the Thevenin voltage (i.e. V2 ¼ VTH), and depends only on XC1 and XC2. A CVT normally contains a particular ferroresonance suppression circuit, as shown in Figure 3.11. The ferroresonance suppression circuit does not however adversely affect the transient response. The analysis is similar to other types of ferroresonance circuits that were discussed in Chapter 2.

Anti-ferroresonant filter C

L Rf

V1

Zb Cf

Lf

Figure 3.11 Anti-ferroresonant filter.

V2

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67

For information on the accuracy class and standard burden for metering and protective relays, consult the IEEE Standard 57.13 [3] and IEC 60044-5 [5]. It is important to remember that additional requirements about CVTs were recently introduced by IEC 61869, Part 1–9 [2]. The drawback of the CVT is that its accuracy is dependent on the harmonic content of the primary voltage. This may not be a problem since in HV and EHV the harmonic distortions are very small. In a future grid this may become more problematic, however. Finally, it is important to mention that a capacitor voltage transformer serves as a coupling capacitor for power line carrier signals. These are normally of high frequency (30–500 kHz) and are conducted/dispersed through the transmission line (end-to-end) in order to have a unity protection via teleprotection schemes [6].

3.4 Current Transformers The main purpose of a CT is to reduce the primary current from the power system to a measurable and standardized secondary current. A CT has the primary winding in series with the power circuit, at which it is necessary to make measurements, protection and monitoring. CTs obey the same principle of electromagnetic transformation as for VTs. However, two special operating conditions exist where the primary current is absolutely independent of the transformer itself and works practically in short circuit, considering that the burden has very small impedance. Figure 3.12 illustrates the CT windings, and the transformation rate is given by I1/I2 ¼ N2/N1 ¼ RCT. Similarly to VTs, CTs can be represented by their equivalent circuit as shown in Figure 3.13; simplifications such as no primary impedance and discounting the secondary reactance need

Figure 3.12 Current transformer.

R2

Io

I1

I2

Ir

Ix Xm

Rm

E2

Figure 3.13 Current transformer equivalent circuit.

V2

Zb

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68

I2 g I1 I0

Figure 3.14 Phasorial diagram of currents.

to be taken into account. This second consideration is valid since the current flowing in a conductor through the hole in a toroidal core gives rise to a magnetic flux in the space surrounding the conductor including the core. If the core material is of high permeability, almost all of the created flux will be localized in the core material and the dispersion flux can be neglected. Even in a steady state, a CT can introduce errors in the transformation. Its causes are related only to the magnetizing current I0: I_ 1 ¼ I_ 2 þ I_ 0 :

(3.6)

Figure 3.14 shows the phasor diagram and, in a hyperbolic way, I2 is different from I1 in magnitude and phase. The magnitude of the error as related to the secondary current can be calculated as: jej% ¼

I0 jI 1  I 2 j 100 ¼ 100%: I2 I2

(3.7)

CTs are also specified according to their service in different accuracy classes, such as for metering or relaying. However, it must be emphasized that a CT core measurement is different from the CT for protection. Table 3.3 lists the accuracy classes for CT measurements [7]. Other classes yet to be considered are 3% and 5% without limits for phase errors. For protection services, the data listed in Table 3.4 must be taken into account. According to the IEEE Standard [2], the accuracy classes for metering are 0.3%, 0.6% and 1.2%. These classes are related to a composite error (e% and g (min)), graphically represented by the accuracy classes parallelogram depicted in Figure 3.15. These accuracy classes are linked to the standard burdens such as B0.1, B0.2, B0.5, B1.0, B2.0, B4.0 and B8.0. For example, 0.3 B 1.0 means that if a burden is used up to or equal to 1 V, the accuracy class of 0.3% is guaranteed. Table 3.3 Accuracy classes for metering (IEC 60044-1). Accuracy class

Percentage current (ratio) error

% rated current

5

20

100

120

5

20

100

200

0.4 0.75 1.5 3

0.2 0.35 0.75 1.5

0.1 0.2 0.5 1.0

0.1 0.2 0.5 1.0

15 30 90 180

8 15 45 90

5 10 30 60

5 10 30 60

0.1 0.2 0.5 1

Phase displacement (minutes)

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69

Table 3.4 Accuracy classes for relaying (IEC 60044-1). Class 5P 10P

Current error at rated primary current (%)

Phase displacement at rated current (minutes)

Composite error at rated accuracy limit primary current (%)

1 3

60

5 10

Note: Additional requirements about CTs were recently introduced by IEC 61869: Part 1–9 [2].

The IEEE Standard C57.13 also describes how the CT for protection services must be specified. Firstly, all protection CTs must have a maximum relative error equal to 10%, up to 20 times the rated current. In other words, the CT should not be saturated within this limit. Considering this statement, the accuracy classes for a protection CT are designated by two symbols that effectively describe the performance for the permanent state: C and T. Class C encompasses CTs where the leakage flux in the core has no appreciable effect on the transformation ratio within the limits of current (1–20 times Irated) and has a specified standard burden. The ratio can be calculated through the excitation curves and equivalent circuits. Class T includes CTs where the core leakage flux has an appreciable effect on the transformation ratio within the limits of current (1–20 times Irated) and has a specified standard burden. The appreciable effect is defined as 1% of the difference between the current Magnitude Error (ε%) 1.2%

–1.20 –1.00

1.012 1.010

0.6% –0.80 –0.60

(1-ε%) 1.014

–1.40

0.3%

1.008 1.006

–0.40

1.004

–0.20

1.002

0

1.000

+0.20

0.998

+0.40

0.996

+0.60

0.994

+0.80

0.992

+1.00

0.990

+1.20

0.988

+1.40

0.986 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 – + Phase displacement (δ) in minutes

Figure 3.15 Accuracy classes parallelograms.

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and calculated ratio corrections using the excitation curves. This transformation ratio should be tested. Finally, the nominal secondary voltage must be specified. This voltage can be calculated when the secondary current is less than or equal to 20 times the rated current passing through the burden. For example, consider a CT with the following data: 1200 – 5 – 5 A; 0.3 B 1.0; C100. This describes a CT with a primary current rating equal to 1200 A and a secondary current rating equal to 5 A with two cores: one for measurement and the other for protection. The protection CT (core) is specified for a standard burden equal to 1.0 V. An example may be a short circuit: if the secondary current is equal to 100 A (20  5 A) the voltage at the burden terminals will be 100 V. This voltage is the approximated excitation curve knee-point. It is important to mention that, for an error of 10%, the limit admitted for protection CTs of 5 A secondary is its own exciting current of 10 A, where the exciting voltage E2 reaches a value above the knee-point curve; see the following equation and Figure 3.16 for further explanation: e20 % ¼

I 0;20 100% ¼ I 0;20 %: 20  5

(3.8)

Chapter 2, Section 2.10, demonstrates what happens to the secondary current when the excitation voltage reaches such this point or a point above it.

Figure 3.16 Excitation curves of a multi-ratio CT (IEEE Standard 57.13).

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3.5 Non-Conventional Transducers For decades, certain voltage and current transducers (the so-called unconventional transducers) were confined to the literature or else had very limited practical applications. Although they can be accurate, one of their major drawbacks is the poor ability to supply secondary power to feed instrumentation basically composed of voltage and current coils. With the advent of digital instrumentation the inconvenience of a high secondary burden was eliminated, increasing the future possibilities for the use of these efficient transducers. The latest research in the field of voltage and current monitoring in high voltage has focused on obtaining, manufacturing and installing new transducers which are safer, more economical and technically advantageous. These new solutions proposed for measurement and protection of electrical systems promise to bring great benefits in both performance and applications in the near future, combined with lower costs. These sensors are not exactly new; they operate on the basis of principles that have been known of since the beginning of the 20th century. However, it is only now that they are in great demand. This is mainly due to the proliferation and installation of intelligent electronic devices (microprocessor relays, meters, etc.) that require only the voltage and/or current with almost no secondary power. To cater for these new technological tendencies, the most promising transduction devices for voltage and current are: (1) resistive voltage divider; (2) optical voltage transducer; (3) the Rogowski coil; and (4) optical current transducer.

3.5.1 Resistive Voltage Divider The principle of the resistive voltage divider is depicted in Figure 3.17. It is used for the sensing of voltage and presents some advantages when compared to voltage transformers (VTs), such as: not saturable; linear; small; lightweight; and does not cause ferroresonance. Due to its high linearity, one of the great benefits of a resistive voltage divider is the possibility of its use at several voltage levels. For example, the same sensor applied at 69 kV can also be used at 138 kV since it has sufficient isolation. The transducers do not need to be replaced. For this purpose several techniques can be employed such as derivation, using secondary voltage tapes along the resistive divider, the use of auxiliary resistive dividers (comparable to auxiliary VTs) or a simple adjustment of the dynamic range software of the

U1

Z1 U2

Z2 Z1

Z2

U1

Z2

Figure 3.17 Voltage resistive divider.

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analog/digital converters at the input of the instrumentation. As a result, this reduces the diversity of sensors needed for very different applications. As opposed to conventional VTs, resistive voltage dividers do not cause ferroresonance and are not destroyed by this phenomenon. They can even be used in abnormal conditions of operation. Resistive voltage dividers experience several major disadvantages however, listed below.  Losses by Joule effect. Since the losses are proportional to the square of the voltage this effect becomes significant for very high voltage. High values of resistance should be used to minimize such losses.  Parasitic capacitances play an important role in determining the accuracy of the divider. The higher the voltage level involved, the higher are the values of the resistors and the most significant are the values of these capacitances. As a consequence, there is a reduction in bandwidth of the voltage signal, which will limit the harmonic frequencies to the order of a few kiloHertz. This range is however sufficient for most power system applications, including harmonic analysis and measures of power quality.  No galvanic decoupling. Due to the inherent coupling between primary and secondary current, special care must be taken to overcome this deficiency.  Low output capability (small burden). Actually, this fact is guaranteed by the new generation of intelligent electronic devices. To ensure high accuracy of the resistive dividers, the resistors must have the same coefficient of drift with temperature. It is possible to reach values of 0.2–0.5% of accuracy classes for the long-term stability, the effect of parasitic capacitances and drift with temperature.

3.5.2 Optical Voltage Transducer This kind of transducer is better referred to as a voltage sensor through the piezo-optical effect. The operating principle of these sensors is based on the phenomenon of change in the physical size and shape of piezo-electric crystals when subjected to electric fields. These changes are detected by the rotation of polarized light through an optical fiber wrapped around a crystal. This phenomenon is known as the Pockel effect, illustrated by Figure 3.18. Since this effect is directly proportional to the electric field applied to the crystal, the applied voltage can be accurately measured. Polarizer Photo diode

Electro-optic Crystal Polarizer

Delay Filter Light Source Voltage Source

Figure 3.18 Optical voltage transducer.

Polarizer

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73

Figure 3.19 Rogowski coil.

Unlike a resistive voltage divider, an optical voltage transducer has the advantage of being galvanically decoupled. Other advantages include no losses by the Joule effect and a negligible effect of parasitic capacitances. However, disadvantages are the high cost associated with its production, complex technology and 90 of phase rotation on the output signal.

3.5.3 Rogowski Coil The measurement principle of current through the Rogowski coil has been known since 1912. This coil consists of a winding, uniformly distributed in a core of non-magnetic material. The simplest possible arrangement consists of a toroid air core, where a great number of turns are wound around the coil and one turn comes back inside the toroid, as shown in Figure 3.19. A Rogowski coil provides measurements that are galvanically decoupled. Furthermore, it operates over extremely large bands compared to the frequencies concerned in power systems. The frequency response of the coil can reach the order of megaHertz, as illustrated by Figure 3.20.

Figure 3.20 Frequency response of a Rogowski coil [8].

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Additionally, these advantages affect the weight and size of the circuit sensor. One of the disadvantages of the Rogowski coil is that its output is a voltage signal. To obtain the current signal proportional to the primary signal, the voltage must be integrated. In the past, because of the low accuracy due to the use of analog integrators, this method was inadequate. With today’s advanced techniques of digital signal processing, the integration can be numerically performed. The error of a Rogowski coil is typically around 0.5%. However, in addition to the load, the influence of the frequency, temperature, current in the adjacent phases and the accuracy of its mechanical construction must also be taken into account. Much work has been done to reduce the errors imposed by the Rogowski coil. One of the more encouraging results is shown by Ramboz [9], where a Rogowski coil was constructed using toroid porcelain with a metallized surface and turns performed with a high-precision laser beam. For the two prototypes prepared, errors were found to be 0.05% and 0.26%. This may be considered excellent when compared to conventional coils.

3.5.4 Optical Current Transducer The operation of an optical current transducer (OCT; see also magneto-optical current transducer or MOCT) is based on the Faraday effect, in which polarized light suffers a phase rotation in the presence of a magnetic field. Figure 3.21 illustrates the operating principle of an OCT. It is important to note that these devices are able to provide the necessary power to modern IEDs, since microprocessor devices only represent a small burden to the transducers. The key element in the system is a Faraday-effect sensor composed of an element called a rotor, which is a block of a special glass or crystal. This rotor has an opening for the passage of the conductor in which current is being monitored. The fiber-optic cables are connected to the block through a set of lenses that guide and polarize the light beams. This block is passive and is the only component of the transducer that is installed at a high voltage level.

Figure 3.21 Optical current transducer.

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75

A polarized light beam surrounds the rotor, which is also subjected to a magnetic field. A second polarizing filter captures the light beam. The interaction between light and magnetic field (proportional to the current) establishes a phase modulation to the light, captured by a photodetector diode that feeds an electronic amplifier. This amplifier produces a low voltage proportional to the instantaneous value of the current in the primary conductor. It can be shown that, for an arbitrary closed path around a current conductor, the phase modulation depends only on the collected current, that is, the sensor does not respond to external fields. A pair of fiber-optic cables is connected to each Faraday sensor. These cables carry the light between the sensor and the electronic module in the control room, sometimes hundreds of meters away. All these components are passive and stable over time. The active electronics system, that is, the light source and the signal processing circuit, is fully installed in the control room on a rack which easily accessible and not harmful to the environment. The expression that relates the measured current and the angular variation is u ¼ 2yi, where y is the Verdet constant and i the instantaneous current. The selection of the crystal is always a compromise involving its optical characteristics, operating range and thermal stability. The combination of the sensor and electronic amplifier should maintain an accuracy of 0.2–0.5% in a range of 0.01–2 pu current levels. In this case, the transfer function approximates a linear function for angles of rotation in the range of 25 since it uses a material of low y to avoid amplification of noise. On the other hand, low values of y can lead to large instantaneous errors; these are still much lower than those provided by conventional high-quality CTs, however. Finally, since modern instrumentation represent negligible secondary loads (burdens), the application of unconventional transducers becomes increasingly attractive, either for cost/ benefit or performance reasons. When the performance of conventional CTs is compared to special CTs and other unconventional current transducers, the first impressive characteristic is the non-saturation capability of the Rogowski coil and OCT. Although the secondary voltage provided by these devices is small (low burden), the linearity is maintained as can seen in Figure 3.22. In other words, the secondary signal is a quasi-perfect copy of the primary signal for a great range of magnitude, in this case primary current. It must be emphasized that this ‘new’ technology is increasingly used in electrical systems. There is no doubt that this is a path of no return. On the other hand, conventional VTs and CTs will still find their place in the real world for a long time.

3.6 Analog-to-Digital Conversion Processing This chapter has so far been focused on instrument transformers. The next sections however focus on other issues related to signal processing, such as the analog-to-digital (A/D) conversion and the anti-aliasing filter. The electronic design and analysis are beyond the scope of this book; only the signal processing aspects are addressed here. From an economical perspective, the price of an analog-to-digital converter has been continually decreasing as resolution and the conversion speed increase. However, an important point regarding IEDs is the choice of analog-to-digital converter (ADC). The engineer has to make decisions such as the converter rate, resolution and conversion time. Figure 3.23 represents the tradeoff between conversion rate (Hz) versus the number of bits. The right scale shows the percentage of active ADCs with a specific number of bits. For example, 34.88% of

76

Power Systems Signal Processing for Smart Grids

Figure 3.22 Voltage versus current characteristics of transducers (adapted from [8]).

the converters investigated had 12-bit resolution. The left scale shows the speed in gigaHertz of the ADC. It can be noticed that as the converter resolution increases, the speed decreases. This figure shows that for ADCs with the same number of bits, there is at least one that reaches the maximum frequency operation (right scale).

Figure 3.23 ADC resolution versus conversion rate.

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77

The resolution of a converter is related to its number of bits B and can be calculated using the expression: Dxð%Þ ¼

1  100 2B  1

(3.9)

or, in terms of the full-scale (FS) range, Dx ¼

1  xFS 2B  1

(3.10)

where x represents voltage or current. However, if the dynamic range of x is xdr and the conditioner circuit is appropriately designed to conform to the signal presented in the ADC, the resolution can be written: Dxdr ¼

1  xdr : 2 1

(3.11)

B

Figure 3.24 illustrates how the dynamic range can be converted to use the full resolution of the converter. The first step is subtract the input signal from the small value of the range va. The new interval of variation becomes the range from 0 up to vb  va. The next step is to multiply the resultant signal in order that kðvb  va Þ ¼ vFS , where vFS corresponds to the maximum binary code 1111 . . . 12. A better resolution of the converter is obtained in this way. The choice of resolution for the converter depends on the variation range of the signals. Additionally, the conversion rate will depend on the dynamics (or frequency content) of the corresponding signal. Generally, power system applications can be divided into three major areas with regards to data acquisition requirements: (1) supervision and control; (2) protection; and (3) power quality and diagnosis.

ν FS

νb ν dr

Range of variation

Range of variation after scaling and offset circuit

νa

0

Figure 3.24 Dynamic range conversion.

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Table 3.5 typical range of variation in control and supervision application. Signal Voltage (V) Current (A)

Variation range

Minimum resolution (#bits)

90 jV j 130; V N ¼ 100 V 0 jI j 10; I N ¼ 5 A

10 12

3.6.1 Supervision and Control A typical range for current and voltage variations in the control/supervision application is listed in Table 3.5. Despite the fact that the dynamic range of the voltage signal is 2  40 ¼ 80 V, the common design for the voltage conditioner and converter uses the dynamic range 2  130 V. This is because a simplification was obtained by the conditioner and, in abnormal situations, the voltage can drop below 90 V. If a 10-bit converter is used, the quantization error is given by Equation (3.9), i.e. Dv ¼

2  130 0:25 V: 210  1

Di ¼

2  10 20 mA: 210  1

For current, the resolution is

For these resolutions the error introduced by the quantization processing assumes that the converter is ideal. In practice however, the converters are not ideal and an error of 1/2 LSB (low significant bit) is commonly found in real converters. As it only has 9 bits, a 10-bit converter must be used in order to compensate for the internal errors of the converter. Resolution is therefore recalculated using 9 bits length: Dv ¼

2  130 0:5 V; 29  1

Di ¼

2  10 40 mA: 29  1

The percent relative errors (Dxr ð%Þ) are given taking into account the nominal value xN: Dxr ð%Þ ¼

Dx  100: xN

(3.12)

For the previous examples, Dvrc. 0.5% and Dirc. 0.8%. According to Table 3.6, a 10-bit converter can be used in the voltage acquisition system even when using a class A VT. For current conversion however, 10 bits cannot be used in CT

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Table 3.6 Typical range of variation in control and supervision. Signal

Variation range

Voltage (V)

0 jV j 150;

Current (A)

0 jI j 200;

V N ¼ 100 V I N ¼ 5A

Minimum resolution (#bits) 10 bits ! Dvr 0:6% 12 bits ! Dvr 0:15% 14 bits ! Dvr 0:04% 12 bits ! Dir 4% 14 bits ! Dir 1% 16 bits ! Dir 0:05%

classes 0.1 and 0.2 as more bits are needed for the current acquisition system. For example, if a 12-bit converter is used, the percent relative error is: Dir ð%Þ ¼

20 100 0:2%:  5 2 1 11

12-bit converters then match the current error for all CT classes. The last column of Table 3.6 summarizes the converter resolution for control and its supervision applications.

3.6.2 Protection In protection applications, the voltage can reach up to 1.5 VN with VN ¼ 100 V, and the current can reach up to 40 IN with IN ¼ 5 A. These limits require an ADC with higher resolution than those used for control application. Typically 14- and 16-bit resolutions are used in a numerical relay. Table 3.6 summarizes the typical range variation and resolution for protection applications. The final results are obtained using equations (3.9) and (3.11). Note that, to compensate for internal errors, the number of bits used in these equations was B  1. Comparing Table 3.6 with Table 3.2, it becomes clear that a 10-bit converter leads to smaller errors of the VT class E; however, 14 bits or more are needed for current.

3.6.3 Power Quality Power quality applications are mainly concerned with voltage quality. There are several parameters that need to be calculated directly from the voltage measured, such as sag, swell, transient or flicker. Of these, the flicker measurement demands a higher precision in voltage acquisition systems and requires a high resolution in the ADC. The standard [10] specifies a resolution 0.1% over the range 10–150% of declared input voltage Udin. For B ¼ 12, then Dvr ð%Þ ¼

3U din 100 ¼ 0:3%:  11 2  1 U din

If this is true, then the converter must have 14 bits or more to reach the desired resolution. In the case of harmonics measurements, the standards state that the error of 5% is the limit for each harmonic. For example, in distribution systems where the limit of the odd harmonics is 3% of the fundamental component, the final resolution is: 3%  5% ¼ 0:15% which requires a converter of 11 bits or more.

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Table 3.7 Accuracy and minimum resolution for the ADC (odd harmonics). System

Limit (odd harmonics) (%)

Accuracy (%)

Resolution # bits

3.0 1.5 1.0

0.150 0.075 0.050

11 bits 12 bits 12 bits

Distribution Sub-transmission Transmission

Table 3.7 summarizes the minimum resolution required for the voltage measurement of odd harmonics with a 5% uncertainty [11]. Note that: (1) the limit for even harmonics is 25% less than for the odd harmonics; and (2) measuring the current harmonics requires a higherresolution converter, typically >15 bits.

3.7 Mathematical Model for Noise Voltage and current signals in a power system are not pure sinusoidal functions. There are several distortions, such as those described in Chapter 2. Some can be modeled as additive white noise, with a normal distribution of zero mean and variance equal to s 2e or, mathematically: y½n ¼ x½n þ e½n

(3.13)

where y½n is the observed discrete-time signal, x½n is the desired signal and e½n is the Gaussian white noise. Figure 3.25 illustrates the mathematical model commonly used in signal processing. There is only a small control regarding the error e1 ðtÞ. This is the function of a transducer or background error present in the system and other less significant errors. Typical values of signal-to-noise ratio SNRA are >27 dB; 40 dB is the most widely used in practice. On the other hand, e2 ½n is the error due the quantization process and is well known in literature. For example, the signal to noise ratio due to the quantization processing SNRADC is defined: SNRADC ¼ 6:02B þ 1:76 dB

(3.14)

A 10-bit converter therefore introduces an SNR of 61.96 dB. The number of bits of a converter may not be defined exclusively by SNRA. For example, if the analog SNR, SNRA ¼ 40 dB, a 7-bit converter leads to the SNR introduced by the ADC, SNRADC ¼ 44 dB. A larger resolution may be needed for the following reasons. If the converter resolution is higher, it may be possible to utilize signal processing tools in order to attenuate the noise ε 1(t )

ε 2[n] ADC y [n]

x(t ) SNRA Analog

SNRADC Digital

Figure 3.25 Mathematical model including noise.

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process e1 ðtÞ, such as a moving average filter that can attenuate the noise from the process if e1 ðtÞ has a zero mean. The converter must be chosen with a quantization error smaller than the measurement error.

3.8 Sampling and the Anti-Aliasing Filtering This section addresses the questions of the anti-aliasing filter and sampling principles (for A/D converter resolution, see Section 3.7). The sampling theorem is discussed here in a conceptual way; its mathematical analysis is developed in the next chapter. The sampling theorem states that the continuous-time signal xðtÞ can be represented by its samples (or by discrete-time signal x½n) in such a way that x½n ¼ xðtÞjt¼nT s , where Ts is the sampling time if the sampling frequency is F s > 2 f max , where fmax is the maximum frequency in the input signal.

Example 3.1 Determine the minimum sampling frequency to analyze a voltage signal with a spectrum that contains up to the 15th harmonic. The maximum frequency in this signal is f max ¼ 15  60 Hz, and the system frequency is assumed to be 60 Hz. The sampling frequency or sampling rate is given by F s > 30  60 Hz: In general, if the IED needs to perform analysis up to harmonic h, the theoretical limit for the sampling rate is F s > 2  h  f 1 Hz

(3.15)

where f 1 is the fundamental frequency in Hz. Some textbooks establish the sampling theorem as F s > 2 f max . The equality is in fact the mathematical limit if the signal has low energy in that frequency and the ideal low-pass filter can be built. In practice however, an ideal filter cannot be built and the sampling rate limit should be much higher than the equality. For example, using a sample three times higher than the theoretical limit relaxes the low-pass filter design, that is, F s  3  2  h  f 1 Hz:

(3.16)

Since the harmonic content is not known in advance, it is necessary to guarantee that aliasing will not occur. Aliasing destroys the information in low frequency, because the highfrequency spectrum corrupts the low-frequency spectrum. Aliasing is avoided by prefiltering the analog signal through a low-pass filter. This analog low-pass filter is known as an antialiasing filter or guard-filter. An illustrative example of aliasing in the time domain is presented in Figure 3.26. The input signal is composed of the fundamental and the 15th harmonic. The sampling rate used was F s ¼ 16  60 Hz, which evidently does not fulfill the sampling theorem. The samples of this signal are represented by crosses and correspond to the fundamental frequency, but of smaller

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Figure 3.26 Example of aliasing in time domain.

amplitude then the original frequency. In this case it is easy to mathematically verify the aliasing error: xðtÞ ¼ A sinð2pf 1 tÞ þ m sinð15  2pf 1 tÞ x½n ¼ xðtÞjt¼1=16f 1 ¼ A sinð2pf 1 n=16f 1 Þ þ m sinð15  2pf 1 n=16f 1 Þ ¼ sinð2pn=16Þ þ m sinð2pn=16 þ 32pn=16Þ ¼ ðA  mÞsinð2pn=16Þ: Note that a single discrete-time component appears after discretization. The frequency is the normalized fundamental frequency, now in radians (2p=16), and the amplitude was reduced. This means that information was lost when aliasing was not avoided. The correct design of an anti-aliasing filter is crucial to maintain the correct information. Figure 3.27 shows the parameters that must be used in the filter design. In the figure below, fc

Magnituede (dB)

0 aliasing

Harmonics

A dB

fc

Fs/2

Fs/2 - fc

Freq (Hz )

Figure 3.27 Specification of the anti-aliasing filter.

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83

is the cutoff frequency and A is the minimum attenuation in decibels. The region between fc and Fs/2 is not used to carry information, and aliasing can occur in this region. The harmonic of interest must fall in the harmonics region. Since the frequency response at the cutoff frequency has an attenuation of 3 dB, the cutoff frequency must match: fc ¼ k  H  f1

(3.17)

where H is the higher harmonic, f 1 the fundamental frequency and k is a gain factor that guarantees a minimum distortion for harmonic H (typically k ¼ 1.2). The minimum attenuation A must be higher than the SNRADC, that is, A  6:02B þ 1:76:

(3.18)

Example 3.2 Specify the Butterworth filter that allows analysis up the 15th harmonic, assuming an ADC resolution of 14 bits. The cutoff frequency must be f c  1:2  15  60 ¼ 1080 A 86:04dB: The attenuation expression (in dB) for a Butterworth filter of order n in the stopband region is given by: G ¼ 20  n  log V

(3.19)

where n is the filter order, V is the normalized frequency and G is the attenuation in decibels. For an attenuation of 86.04 dB and using a third-order Butterworth filter, Equation (3.18) yields: 86:04 ¼ 60  log V

) V ¼ 29 rad=s:

The normalized frequency indicates that the cutoff frequency is equal to 1 rad/s. The last calculation shows that the attenuation reaches 86.04 for V ¼ 29 rad=s. This point corresponds to point F s  f c in Figure 3.27. To denormalize the previous frequency, the results are multiplied by the true cutoff frequency which, in the present example, is f c 1080 Hz. We then have: V0 ¼ 29  2p  1080 or

f 0 ¼ 29  1080 ¼ 31:32 kHz:

The sampling frequency can then be determined as f s  f 0 þ f c ¼ 32:4 kHz

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which corresponds to 540 samples per cycle: a very high sampling rate for the low frequency of interest. If the order of the Butterworth filter was changed to a 5th order, than the necessary sampling rate to keep the SNRADC is: f s ¼ 8:9 kHz; or 148 samples per cycle: The design can be relaxed because the ADC is not ideal and cannot give ideal SNRADC. Remember that a B-bit ADC is effectively used as B–1 bits. When this is taken into account, the sampling rate can be reduced. For example, it is possible to use a sampling rate of 128 samples per cycle. The sampling rate can be reduced if the magnitude characteristic of the filter is known and an internal compensation can be applied to the obtained results from the harmonics of high order. For this situation, 64 samples per cycle can be used.

3.9 Sampling Rate for Power System Application Protection and control applications usually use signal information in low frequency. Traditionally a relay algorithm works with 16 samples per cycle, utilizing only the fundamental component and, in some applications, the second and third harmonic. New protection algorithms require higher frequencies however, and it is a common finding in technical protection literature that the algorithm may run with more than 64 samples per cycle; such devices have not yet been commercialized. Use of the high-frequency information of the signal is common in power-quality and datalogger applications. Such an example is an impulsive phenomena that may last only a few nanoseconds (high-frequency spectrum) and demands a high-speed ADC. In general, impulsive phenomena are not captured directly by sampling the analog signal but may be noticed when an auxiliary analog filter is used to enlarge the duration of the pulse, generally a high-pass filter. Transient phenomena can last from micro-seconds to milliseconds, as listed in Table 3.8. High-frequency phenomena require a high-speed converter which can address actual speeds of 1024 samples per cycle; this cannot be found in commercial power-quality equipment. Measuring these high-frequency events requires special equipment or circuits such as a high-pass filter.

3.10 Smart-Grid Context and Conclusions Many system events may demand equipment other than those of power frequencies that exist in the electric grid. Some of the consequent waveforms are time-varying and contain highfrequency signals that are conducted through the power lines; these need to be properly measured and, if necessary, their negative impacts compensated for. The increasing Table 3.8 Transients phenomena. Frequency band Low frequency ( V0

Definition equation/tips for solution Definition No Time shifting property (Table 4.4) Frequency shifting property (Table 4.4) Euler relationship and properties Euler relationship and properties Definition Euler relationship and properties Definition Definition

The reader however needs to be aware that, if attempting to find a Fourier transform (or its inverse), some integrals do not converge, this does not necessarily means that the Fourier pair does not exist. A more in-depth mathematical analysis is needed to solve this. The approach here will be to present a Fourier pair without this verification. Table 4.3 lists the most common Fourier transform pair. The last column in the table indicates whether the pair obey Equations (4.25) and (4.26). If yes, the reader will be able to verify the pair through the definition equations. If not, the reader has to use the concept of generalized function, not presented in this book.

Example 4.2 Find the Fourier transform of the gate function:  gt0 ðtÞ ¼

1 jtj  t0 0 jtj > t0 :

Figure 4.7a depicts the gate function. The Fourier transform can be obtained directly from its definition:

Fð jVÞ ¼

ð t0 t0

ejVt dt ¼

 2 sin Vt0 2  jVt0 : e þ e jVt0 ¼ V 2jV

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Magnitude spectrum (b)

gt 0(t )

(a)

2 1.8 1.6

1 Amplitude

1.4 1.2 1 0.8 0.6 0.4 0.2 –t0

t0

0 –20

t

–15

–10

0 5 –5 Frequency (rad/s)

10

15

20

Figure 4.7 (a) Function gate and (b) magnitude spectrum.

The magnitude spectrum AðVÞ of Fð jVÞ is presented in Figure 4.7b for t0 ¼ 1. The gate function is widely used in signal processing and its transform is referred to as a sinc function; note that the sinc function has 0s spread periodically.

Example 4.3 Find the Fourier transform of the function:  f ðtÞ ¼

eat

for t  0

0

for t < 0

where a is a positive and a real constant. This function can be written as f ðtÞ ¼ eat uðtÞ, and the Fourier transform is Fð jVÞ ¼

ð1

eðaþjVÞt dt ¼

0

1 : a þ jV

The pair can then be written as f ðtÞ ¼ eat uðtÞ $

1 : a þ jV

The time function f ðtÞ and the magnitude spectrum of Fð jVÞ are represented in Figure 4.8a and b respectively for a ¼ 5. Note that the magnitude spectrum has energy over a wide range of the frequency axis. An interesting fact is that the higher the a value (the exponential vanishes quicker), the wider the magnitude spectrum (the spectrum vanishes slower). This type of signal is very common during a fault in a transmission line. It is known as the exponential

Discrete Transforms

103

Exponential function

Magnitude spectrum (b)

2

9

1.8

8

1.6

7

1.4

Magnitude

Amplitude

(a) 10

6 5 4

1.2 1 0.8

3

0.6

2

0.4

1

0.2

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

0

10 20 30 40 50 60 70 80 90 100

Time (s)

frequency (Hz)

Figure 4.8 (a) Exponential signal and (b) its magnitude spectrum.

decaying DC component. As the energy of the spectrum spreads along the frequency, the estimation of the phasor (fundamental sinusoid component) is affected by the DC component and a particular method of estimation is needed. Some of the methods used for the estimation of phasors in the presence of a decaying DC are discussed in Section 7.6

4.3.2 Fourier Transform Properties There are several important properties or theorems of the Fourier transform which are useful for signal processing applications. These properties can be used to determine the Fourier transform of a signal without the need to apply the analysis or synthesis equations. The tips in Table 4.3 show that these properties can be used as a shortcut to obtain the Fourier pair. This section will present the main theorems and demonstrate their application in examples. Table 4.4 presents the main theorems of a Fourier transform. In this table a, b, t0 and V0 are arbitrary real constants. Table 4.4 Fourier Transform properties. f ðtÞ $ FðjVÞ gðtÞ $ GðjVÞ

Row No.

Properties

1

Linearity

af ðtÞ þ bgðtÞ $ aFðjVÞ þ bGðjVÞ

2

Duality

FðjtÞ $ 2pf ðVÞ

3

Time shifting

4

Time scaling

f ðt  t0 Þ $ FðjVÞejVt0   f ðatÞ $ j1aj F jV a

5

Frequency shifting

ejV0 t f ðtÞ $ FðjV  jV0 Þ

6

Time convolution

f ðtÞ  gðtÞ $ FðjVÞGðjVÞ Ð1 where f ðtÞ  gðtÞ ¼ 1 f ðtÞgðt  tÞdt

7

Frequency convolution

f ðtÞgðtÞ $ FðjVÞ  GðjVÞ

8

Parseval’s formula

Ð 1 1 where FðjVÞ  gðjVÞ ¼ 2p 1 FðjyÞgðjV  jyÞdy Ð1 Ð 2 2 1 1 1 jf ðtÞj dt ¼ 2p 1 jFðjVÞj dV

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(b)

(a) πδ(Ω+Ω0)

π δ(Ω –Ω0) j

πδ(Ω –Ω0)

Ω

0

–Ω0

0

–

Ω0

Ω

π δ(Ω +Ω0) j

Figure 4.9 Spectrum of the (a) cosine function and (b) sine function.

Example 4.4 Find the Fourier transform of the following function: f ðtÞ ¼ A cos V0 t Solution: Using the Euler formula, f ðtÞ ¼ cos V0 t ¼

e jV0 t þ ejV0 t 2

and the linear property of Table 4.4, 1  1  Fð jVÞ ¼ = e jV0 t þ = ejV0 t : 2 2 Using the Fourier transform pair in Table 4.4 (pair 4) we obtain: Fð jVÞ ¼ pdðV  V0 Þ þ pdðV þ V0 Þ: Figure 4.9 depicts the plot of the Fourier transform of the cosine and sine function respectively. Note that the sinusoid functions are represented as impulse functions. This fact leads us to an important conclusion: any periodic function can be written as a sum of its sinusoids (Fourier series), and the Fourier transform of any periodic function can be represented as a set of its impulse functions.

4.4 The Sampling Theorem Power system signals are continuous time signals. However, discrete-time signal processing algorithms are used more often for processing power system signals. For example, the

Discrete Transforms

105

substitution of analog protection relays by numerical relays which contain a digital signal processor (DSP) running a digital signal processing algorithm. The modern harmonic analyzers are all digital equipment, running DSP algorithms such as the fast Fourier transform (FFT). We therefore need to convert the continuous-time signals to a discrete-time signal, and there needs to be certainty that the DSP algorithm will produce the same results as its analog processing. The link between the analog and digital world is given by the sampling theorem, derived in the following. The sampling theorem states that a band-limited signal xðtÞ for which Xð jVÞ ¼ 0 for

jVj 

Vs 2

(4.28)

where Vs ¼ 2pF s ¼ 2p=T s is the sampling frequency in rad/s, F s is the sampling frequency in Hz and T s is the sampling period, can be uniquely determined from its samples xðnT s Þ. To demonstrate the sampling theorem, we refer to the analog signal as xa ðtÞ and the samples of the sampling signal x½n. We then have: x½n ¼ xa ðnT s Þ:

(4.29)

The Fourier transform of xa ðtÞ as defined by Equation (4.25) is X a ð jVÞ ¼

ð1 1

xa ðtÞejVt dt:

(4.30)

Mathematically, the sampling signal can be represented by the product of the analog signal and a periodic impulse train pðtÞ, where 1 X

pðtÞ ¼

dðt  nT s Þ

(4.31)

n¼1

xp ðtÞ ¼ xa ðtÞpðtÞ ¼

1 X

xa ðnT s Þdðt  nT s Þ:

(4.32)

n¼1

Figure 4.10 depicts the mathematical representation of the uniform sampling processing. In Figure 4.10a an ideal impulse modulator is represented. The analog signal (Figure 4.10b) is multiplied by the train of impulse (Figure 4.10c), resulting in the sampled signal (Figure 4.10d). Applying the Fourier transform definition in Equation (4.32), taking into account that xa ðnT s Þ is a constant in t, yields X p ð jVÞ ¼

1 X n¼1

xa ðnT s Þ=fdðt  nT s Þg:

(4.33)

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(a) xa(t)

xa(t)

(b)

xp(t)

X p(t)

t

p(t)

(c)

–2Ts –Ts 0

xp(t)

(d)

Ts 2Ts

–2Ts –Ts 0

t

Ts 2Ts

t

Figure 4.10 Generation of a sampled signal: (a) ideal sampling model; (b) analog signal; (c) impulse train; (d) sampled signal.

According to Table 4.3, line 4, 1 X

X p ð jVÞ ¼

xa ðnT s ÞejVnT s :

(4.34)

n¼1

Using Equation (4.29), X p ð jVÞ ¼

1 X

x½nejVnT s :

(4.35)

n¼1

Equation (4.35) is the Fourier transform of the sampled signal, which is very close to the discrete-time Fourier transform (DTFT). However, before defining the DTFT we need to understand what happens with the spectrum of the sampled signal X p ð jVÞ compared to the spectrum of the original signal X a ð jVÞ. To arrive at the relationship between the two spectrum representations, the fact that the impulse train is a periodic function of fundamental period T s needs to be remembered; consequently, it can be rewritten as an exponential Fourier series: pðtÞ ¼

1 1 X e jVs kt : T s k¼1

(4.36)

Equation (4.32) can then be rewritten: xp ðtÞ ¼ xa ðtÞpðtÞ ¼

1 1 X e jVs kt xa ðtÞ: T s k¼1

(4.37)

Using the frequency shifting property in Table 4.4, we arrive at the following Fourier transform relationship: X p ð jVÞ ¼

1 1 X X a ð jV  kVs Þ: T s k¼1

(4.38)

Discrete Transforms

107

X p ð jVÞ is therefore a periodic function of Vs , consisting of a sum of shifted and scaled replicas of X a ð jVÞ. It is referred to as the baseband spectrum, shifted by integers that are multiples of Vs and scaled by 1/Ts. The frequency range Vs =2  V  Vs =2 is referred to as the baseband or Nyquist band. Equation (4.38) describes the relation between the spectrum of the original signal and the sampled signal, and is the basis for defining the sampling theorem. Figure 4.11 illustrates the effect of the time-domain sampling in its frequency domain. Figure 4.11a depicts the spectrum of the original signal, which is assumed to be a band-limited signal with no energy above frequency Vm. Figure 4.11b depicts the spectrum of the sampling signal, assuming Vs > 2Vm . The correspondent spectrum is periodic in frequency and repeats at each multiple of Vs ; only two replicas of the baseband spectrum are represented. Note that there is no mix of individual replicas and it is possible to obtain the original signal if the sampled signal is passed through a low-pass filter with rejection frequencies equal to Vs  2Vm . Figure 4.11c depicts the case

(a)

Xa( jΩ)

– Ωm

Ωm

0

(b)

Ω

Xp( jΩ) 1 Ts Xa(jΩ–jΩs)

1 X (jΩ) Ts a

1 X (jΩ+jΩs) Ts a

–Ωs –Ωm –Ωs –Ωs+Ωm–Ωm

Ωm Ωs –Ωm Ωs

0

(c)

Ωs+Ωm

Ω

Xp( jΩ)

1 X (jΩ+jΩs) Ts a

1 X (jΩ) Ts a

–Ωs –Ωm –Ωs –Ωm

0

–Ωs+Ωm

Ωm Ωs Ωs– Ωm

1 X (jΩ–jΩs) Ts a

Ωs+Ωm

Ω

Figure 4.11 Illustration of the effect of time-sampling in the frequency domain: (a) spectrum of original signal; (b) spectrum of the sampled signal when Vs > 2Vm ; and (c) spectrum of the sampled signal when Vs < 2Vm :

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when Vs < 2Vm . As observed in this case, there is an overlap between the individual replicas and part of the original information is lost; consequently, the original signal cannot be recovered by filtering. This effect of spectrum overlap is known in literature as aliasing. The above result is more commonly known as the sampling theorem or Nyquist sampling theorem, which can be summarized as follows. Given a band-limited signal xðtÞ, Xð jVÞ ¼ 0 for

jVj  Vm

can be uniquely determined from its samples values xðnT s Þ if Vs > 2Vm . This relationship is also known as the Nyquist theory. The highest frequency in the signal is known as the Nyquist frequency, since it determines the minimum sampling frequency that must be used to sample the signal and recover it from the illustration.

4.5 The Discrete-Time Fourier Transform Equation (4.34) represents the Fourier transform of a sampled signal. Performing the following change of variable, v ¼ VT s

(4.39)

where v is in radians, Equation (4.35) can be rewritten: X p ð jv=T s Þ ¼

1 X

x½nejvn :

n¼1

The information in parentheses on the left-hand side of the above equation indicates that the new independent variable is v. Frequently in literature regarding discrete transforms, the information regarding the time step is omitted and a more informative notation is used to represent the Fourier transform of the sampled signal: 1 X

Xðe jv Þ ¼

x½nejvn :

n¼1

The previous equation is referred to as the discrete-time Fourier transform (DTFT) which represents both the Fourier transform of the samples of the analog signal as well as the Fourier transform of the sequence fx½ng; this is independent of whether this sequence was obtained from current sampling. Power engineers must be comfortable working with this notation; although it omits the time step information, it carries other useful information about the discrete transforms which is discussed in the following section. Equation (4.39) must be kept in mind because it is the main link between digital frequency (in radians) and analog frequency (in radians per second). As mentioned at the beginning of Section 4.3, all transforms must be governed by analysis and synthesis equations. For DTFT the analysis and synthesis equations are: Xðe jw Þ ¼

1 X n¼1

x½nejvn ;

analysis equation or DTFT

(4.40)

Discrete Transforms

109

X (e jω ) X (e j (ω –2π ))

X (e j (ω +2π ) )

–2π –ωm –2π –2π +ωm –ωm

0

ωm 2π –ωm 2π

ω

2π +ωm

Figure 4.12 DTFT spectrum.

x½n ¼

1 2p

ðp p

Xðe jv Þe jvn dv;

synthesis equation or IDTFT:

(4.41)

The synthesis equation is also known as the inverse discrete-time Fourier transform or IDTFT. In a DTFT equation, the independent variable is v and the notation e jv carries useful information such as the periodicity of the DTFT, which is 2pk where k is an integer. We have Xðe jðvþ2pkÞ Þ ¼

1 X

x½nejðvþ2pkÞn ¼

n¼1

1 X

x½nejvn e2pkn ¼

n¼1

1 X

x½nejvn

n¼1

where we have used the fact that e2pkn ¼ cosð2pknÞ  j sinð2pknÞ ¼ 1 for integers k and n. If both axes of Figure 4.11b are multiplied by T s, then Vs =f s ¼ 2p and the factor 1=T s is omitted. Figure 4.12 represents the Fourier spectrum of Xðe jv Þ; note the periodicity of the spectrum. Other information carried by the notation e jv is that the DTFT is a complex function of v and can be represented in a rectangular (real and imaginary) or polar form. The polar form is commonly used:    

X e jv ¼ X e jv e juðvÞ

(4.42)

where jX ðe jv Þj is called the magnitude function or magnitude spectrum and uðvÞ ¼ argfX ðe jv Þg is the phase function or phase spectrum. Furthermore, the spectrum produced by sequence fx½ng is a continuous function in v.

4.5.1 DTFT Pairs The existence of the DTFT depends on the convergence of Equation (4.40), and the summation must be finite. If the DTFT pair exists, then the IDTFT must satisfy Equation (4.41). As mentioned in Section 4.3, some pairs do not appear to obey these equations,

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that is, their existence cannot be proven using the analysis and/or the synthesis equations directly. Again, the limitation lies in the mathematical background needed to circumvent some non-convergence summation or integral. Instead of immersing ourselves in such a challenging mathematical issue, we instead chose to list in Table 4.5 the most common DTFT pairs and highlight those which cannot be obtained directly through the definition.

4.5.2 Properties of DTFT As with the continuous-time Fourier transform, a variety of properties of the DTFT provide further insight into the transform and are often useful during the evaluation of transforms and their inverse. Table 4.6 lists the main properties of the DTFT.

4.6 The Discrete Fourier Transform (DFT) In practice, the frequency analysis of discrete-time signals is most conveniently performed on a digital signal processor rather than through analog processing. By definition, the DTFT is performed over an infinite sequence length, and frequency is a continuous variable. These constraints make the processing of such sequences not feasible in a digital processor. To overcome these limitations, the discrete Fourier transform (DFT) is used. The analysis and synthesis equations are defined: X½k ¼

N1 X

x½nej2pkn=N

analysis equation or DFT

(4.43)

synthesis equation or IDFT:

(4.44)

n¼0

x½n ¼

N1 1X X½ke j2pkn=N N k¼0

where the synthesis equation is also known as the inverse discrete Fourier transform (IDFT). Equation (4.43) is the direct DFT and the summation is performed over N samples of x½n. X½k is the discrete Fourier transform of x½n. The DFT has the same length as the IDFT; that is, in Equation (4.43) n ¼ 0, 1, . . . , N  1 and in Equation (4.44) k ¼ 0, 1, . . . , N  1. The first connection between the DFT and DTFT can be made considering a sequence of length N: Xðe jv Þ ¼

N 1 X n¼0 jv

x½nejvn

(4.45)

X½k ¼ Xðe Þjv¼2pk=N : The DFT is obtained by the uniform sampling of the DTFT at N equally spaced frequencies vk ¼ 2pk=N where 0  k  N  1 on the v axis from 0  v < 2p. The v axis can be represented as shown in Figure 4.13 for a sequence of length N ¼ 32 samples. In this figure, only the k values on the upper semicircle are marked but its variation is 0  k  31. The value of 2p=N is known as the frequency resolution or frequency bin, and represents the smaller value for vk . This representation is very useful, because it highlights the facts that: (1) the periodicity of

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Table 4.6 Discrete-time Fourier transform properties. Row No.

Properties

x½n $ Xðejv Þ y½n $ Yðejv Þ

1 2 3 4

Linearity Time shifting Frequency shifting Convolution

ax½n þ by½n $ aXðejv Þ þ bYðejv Þ x½n  n0  $ ejvn0 Xðejv Þ ejv0 n x½n $ Xðejðvv0 Þ Þ x½n  y½n $ Xðejv ÞYðejv Þ 1 P x½ky½n  k where x½n  y½n ¼

5 6

Time reversal Differentiation in frequency

7

Symmetry

8

Multiplication

9

Parseval’s formula

x½n $ Xðejv Þ dXðejv Þ nx½n $ j dv 8  jv jv > < Xðe Þ ¼ X ðe Þ x½nreal ! jXðejv Þj ¼ jXðejv Þj > : ^ Xðejv Þ ¼ ^ Xðejv Þ x½n real and even ! Xðejv Þ real and even x½n real andðodd ! Xðejv Þ purely imaginary and odd 1 x½ny½n $ Xðeju ÞYðejðvuÞ Þdu 2p 2p ð 1 P 1

jv

2 2 Xðe Þ dv jx½nj ¼ 2p n¼1

k¼1

2p

X½k is N; and (2) the value of F s =2 corresponds to p. The second piece of information is important when the frequency in Hertz corresponding to each value of X½k is required: p ! F s =2 vk ! f : From the previous relationship, pk Fs: (4.46) f ¼ N k =6

k =3 k =2 ω2

k =1

k =16 ω k = 2πk / N

Figure 4.13 Correspondence between the frequency DTFT and DFT.

Discrete Transforms

113

A common representation of the DFT pair uses the simplification of the exponential term: W N ¼ ej2p=N :

(4.47)

The DFT and IDFT can then be represented: X½k ¼

N 1 X

x½nW kn N

analysis equation or DFT

(4.48)

synthesis equation or IDFT:

(4.49)

n¼0

x½n ¼

N 1 1X X½kW kn N N k¼0

The DFT is a mathematical tool to obtain the Fourier spectrum by using computational processing, different to the DTFT where the computation is performed manually. However, as in the following examples, some DFTs are solved manually for a better understanding of some important points. The first step is to find the DFT of N samples of a sinusoid sequence with the digital frequency of vr ¼ 2pr=N.

Example 4.5 Find the N-points DFT of the length sequence: x½n ¼ cosð2prn=N Þ 0  n  N  1 and 0  r  N  1, r is an integer number. Using the Euler identity, we can write  1  j2prn=N e  ej2prn=N : x½n ¼ 2 In finding the DFT of the single N-points of the exponential x1 ½n ¼ e j2prn=N , X 1 ½k ¼

N 1 X

e j2pðrkÞn=N ¼

n¼0

1  e j2pðrkÞ 1  e j2pðrkÞ=N

(4.50)

when using the formula of the sum of the first N terms of a geometric series. The above expression is 0 for r  k 6¼ lN, l an integer, because e j2pðrkÞ ¼ 1 and the denominator is nonzero. However, when r  k ¼ lN, we have an indetermination of the type 0=0. Applying L’H^ opital’s rule to eliminate the indetermination yields:  X 1 ½k ¼

N;

for r  k ¼ lN;

0;

otherwise:

l an integer

(4.51)

By using this result in the initial equation, taking into account that the DFT is a linear transform, we have 8 > < N=2; for k ¼ r (4.52) X½k ¼ N=2; for k ¼ N  r > : 0; otherwise:

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Magnitude spectrum 16 14

Magnitude

12 10 8 6 4 2 0

0

5

10

15

20

25

30

35

k

Figure 4.14 Magnitude spectrum of 32 samples of cosine sequence (N ¼ 32; r ¼ 6).

Figure 4.14 depicts the DFT of the previous signal for N ¼ 32 and r ¼ 6. Note that there are two pulses at indices r ¼ 6 and N – r ¼ 26. This is consistent with Table 4.5 line 5 where the DTFT of a sinusoid signal of infinite length is a train of impulses, with the first two centered at frequencies v0 and v0 . If v0 corresponds to a bin, that is, v0 ¼ 2pr=N, then the DFT will correctly represent the DTFT of the sequence x½n ¼ cosvr n; otherwise, the DFT will not be able to accurately represent the DTFT. The magnitude plot presented in Figure 4.14 is symmetric regarding the central sample, which is true if the sequence is real. Additionally, the phase is anti-symmetric regarding the central sample. The property that states this condition is:   (4.53) X½k ¼ X  hkiN where the symbol h iN means ‘modulo N’. The modulo N operator functions in the sequence index in order for the results to be kept within the interval 0 to N – 1. If r ¼ hniN , then r ¼ n þ lN;

(4.54)

where l is an integer such 0  r  N  1. If the index is negative as in Equation (4.53,) l ¼ 1 yields: r¼kþN

for k < 0:

We then have the following symmetry in the previous example: X½1 ¼ X  ½31 X½2 ¼ X  ½30 .. . X½15 ¼ X  ½17 X½16 ¼ X  ½16

Discrete Transforms

115

The central term k ¼ 16 must be real in order for Equation (4.53) to be satisfied. Figure 4.13 was generated using the MATLAB1 code: N=32; r=6; tet=2*pi/N; n=0:N-1; x=cos(tet*r*n); figure(1) X=fft(x); stem(n,abs(X),’k’,’LineWidth’,3) title({’Magnitude spectrum’},’FontSize’,16,’FontName’,’Verdana’) xlabel({’k’},’FontSize’,14,’FontName’,’Verdana’) ylabel({’Magnitude’},’FontSize’,14,’FontName’,’Verdana’)

using the FFT algorithm to compute the DFT. The FFT is an elegant algorithm for the computation of the DFT that drastically reduces the related computational effort. While the DFT requires the N2 complex, multiplications and N (N  1) complex additions, the FFT requires N log2 N complex multiplications and ðN=2Þlog2 N complex additions.

Example 4.6 Compute the N-point DFT of the sequence:

x½n ¼

X½k ¼

8 < 1;

0nN1

:

otherwise

0;

N 1 X 1  ej2pk ej2pnk=N ¼ ; 1  ej2pk=N n¼0

0  k  N  1:

In the expression above there is an indetermination for k ¼ 0. By application of the L’H^ opital rule,  X½k ¼

N; 0;

k¼0 otherwise:

Example 4.7 Computing the inverse DFT. In several applications we need to find the sequence from the DFT. For this purpose we can use the IDFT Equation (4.44) or, more easily, apply the inverse

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FFT algorithm (IFFT). The command in MATLAB1 is ifft(X). A common error can however occur at the time of building the vector X[k] because it must obey the symmetry property described by Equation (4.53).

4.6.1 Sampling the Fourier Transform Equation (4.45) shows the DFT as a sampling frequency of the DTFT. The question that needs to be answered is whether the sequence obtained using the inverse DFT is the same as that obtained by using the inverse of DTFT. If the sequences are the same, then we can obtain any point of the DTFT using only samples of the DFT. As such, consider a sequence x½n and its DTFT Xðe jv Þ. The sampling of the DTFT at N equally spaced points vk ¼ 2pk=N, k ¼ 0; 1;    N  1 and the IDFT of these samples will give the inverse sequence y½n, n ¼ 0; 1;    N  1. This sequence can be equal or not to x½n. Note that there was no imposition of any restriction to the length of x½n. 1 X

Y½k ¼ Xðe j2pk=N Þ ¼

x½lej2pkl=N

(4.55)

l¼1

The N-points inverse DFT of Y½k from Equation (4.44) is y½n ¼

N1 1X Y½ke j2pnk=N ; N k¼0

n ¼ 0; 1;    N  1:

(4.56)

Substituting Equation (4.55) into Equation (4.56), N1 X 1 1X x½lej2pkl=N e j2pnk=N N k¼0 l¼1 " # 1 N 1 X 1X j2pkðnlÞ=N ¼ x½l  e : N k¼0 l¼1

y½n ¼

(4.57)

The last summation is the identity presented in Equation (4.51). Making use of this identity yields y½n ¼

1 X

x½n þ rN;

n ¼ 0; 1;    N  1:

(4.58)

r¼1

Equation (4.58) is the desired relation between y½n and x½n. It indicates that the sequence y½n is obtained from x½n by adding an infinite number of shifted replicas of x½n to get the results only from the interval n ¼ 0; 1; . . . ; N  1. The replicas are shifted by an integer multiple of N.

4.6.2 Discrete Fourier Transform Theorems There are a number of important theorems for the DFT that are useful in digital signal processing; the most important are listed in Table 4.7.

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117

Table 4.7 Discrete Fourier transform properties. Row No.

x½n $ X½k y½n $ Y½k

Properties

1 2

Linearity Circular time shifting

3

Circular frequency shifting

4

N-point circular convolution

5

Modulation

6

Parseval’s formula

ax½n þ by½n $ aX½k þ bY½k   0 x hn  n0 iN $ W kn X½k  N  k0 W N x½n $ X hk  k0 iN x½n  y½n $ X½kY½k N1   P x½ky hn  kiN where x½n  y½n ¼ x½ny½n $ N1 P

2

1 N

jx½nj ¼

n¼0

N1 P

k¼0

  X½mY hk  miN

n¼0 N1 P 1 jX½kj2 N k¼0

4.7 Recursive DFT In real-time applications where the DFT components need to be computed at each new sample, the recursive DFT represents an efficient way to compute these components. To derive the basic expression to the kth component, consider Figure 4.15 which depicts two consecutive data windows. Data window n – 1 is defined as xn1 ¼ fx½n  N; x½n  N þ 1 . . . x½n  2; x½n  1g and data window n as xn ¼ fx½n  N þ 1; x½n  N þ 2 . . . x½n  1; x½ng. The DFT for the kth harmonic for the window n is given by: X k ½ n ¼

N1 X

kðNm1Þ

x½n  mW N

;

k ¼ 0; 1; . . . ; N  1:

(4.59)

m¼0

Changing the variables yields: X k ½n ¼

n X

kðNþln1Þ

x½lW N

:

l¼nNþ1

Figure 4.15 Two consecutive windows.

(4.60)

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The DFT for window n – 1 is easily obtained as: n1 X

X k ½n  1 ¼

kðNþlnÞ

x½lW N

:

(4.61)

l¼nN

Multiplying both sides by W k N results in: W k N X k ½n  1 ¼

n1 X

kðNþln1Þ

x½lW N

:

(4.62)

l¼nN

Re-arranging the second term of Equation (4.62) yields: W k N X k ½n  1 ¼

n X

kðNþln1Þ

x½lW N

þ W k N ðx½n  N  x½nÞ

(4.63)

l¼nNþ1

by making use of the identity of W kN N ¼ 1. As such, the summation in Equation (4.63) can be easily identified as X k ½n and the final expression for a recursive DFT is: X k ½n ¼ W k N ðX k ½n  1 þ x½n  x½n  NÞ:

(4.64)

The block diagram for Equation (4.64) is illustrated in the shaded part of Figure 4.16. Note that the input signal is a real signal and the output of the filter is a complex signal, because of the complex multiplier W kN . Another important point to mention is the phase of X k ½n. As observed from Figure 4.15, the phase is shifted forwards by u radians every new sampling, which means that the phase refers to the window n instead of the initial reference. To move the reference back to the initial point, we need to subtract u radians from the phase of each new DFT sample. This can be accomplished using the final section of the block diagram of Figure 4.16. In this part of the figure, the initial value for the memory is assumed equal to 1. ^ k ½n for a signal x½n ¼ cos½un  1:5. Figure 4.17 shows the phase response of X k ½n and X Using the DFT method, the frequencies of the components calculated by Equation (4.64) are functions the window size N. The frequency of the component of order k ¼ 1, X 1 ½n, is

Figure 4.16 Block diagram for recursive DFT.

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119

Figure 4.17 Phase estimation using DFT: solid line is the phase of X k ðnÞ and dashed line is the phase of X k ðnÞ.

given by: f1 ¼

1 Fs ¼ NT s N

(4.65)

and the other components have frequencies: f k ¼ kf 1 ;

where

k ¼ 2; 3; . . . ; N:

(4.66)

As an example, let x[n] be a real periodic signal with fundamental frequency f0 sampled with L samples per cycle, that is, using a sampling frequency of F s ¼ Lf 0 , where L is an integer. This process is called ‘synchronous sampling’ because the sampling frequency is an integer multiple of the fundamental frequency of the signal. The sampling period can then be written as T s ¼ T 0 =L, where T 0 is the fundamental period of x[n]. From Equation (4.65), we have: f1 ¼

1 N ðT 0 =LÞ

(4.67)

if the window size is equal to the number of samples per cycle (i.e. N ¼ L); then f1 ¼ f0.

Example 4.8 Consider that x[n] was sampled at a rate of 16 samples per cycle. Determine the window length L in order that all the terms can be estimated using DFT. x½n ¼ A1 cosðv0 nÞ þ A1:5 cosð1:5v0 nÞ þ A3 cosð3v0 nÞ v0 ¼ 2pf 0 T s :

(4.68)

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Figure 4.18 Correspondence between the index k of DFT and the real frequency in Hertz.

The signal in this example is composed of a fundamental component, the third harmonic and an interharmonic. If f0 is 60 Hz, then the frequency of the interharmonic is 90 Hz. Choosing N ¼ 2L leads to Equation (4.67), and hence f 1 ¼ 30 Hz. The value of k for the components in the previous signal corresponds to k ¼ 2, 3 and 6, respectively. Figure 4.18 depicts an illustrative representation between the DFT index k, the real frequency in Hz and the angle in radians. The angle of p radians corresponds to half of the sampling frequency (Fs/2 Hz), and corresponds to k ¼ N/2. In this situation the DFT gives the correct estimation for each component of the signal. However, instead of taking N ¼ 2L let N ¼ L; f 1 ¼ 60 Hz and the interharmonic term cannot then be estimated and, moreover, the fundamental and the third harmonic estimations will not be accurate because the interhamonic will interfere with the accuracy of the final results. This effect is known as picket-fences [7], and is caused by interharmonics in the signal (see Chapter 8). The frequency resolution of a DFT is related to its window’s length. If synchronous sampling of a signal is assumed, the small frequency component can be obtained from DFT: f1 ¼

f0 Nc

(4.69)

where Nc (¼ N/L) is the number of integer cycles of the fundamental frequency. In the IEC 61100-3-60 standard the number of cycles used for a 60 Hz power system is 12 (10 for 50 Hz system). The frequency resolution is 5 Hz, which means that the DFT is computed every 5 Hz up to the maximum frequency given by Fs/2. The price to be paid for this small frequency resolution and the reduction of the picket-fences effect is the increase of the convergence time for the estimation of each component. In the expression given by Equation (4.64) the convergence takes Nc cycles. This time is prohibitive for typical control and protection applications. Chapter 7 will focus on accurate and fast estimators.

4.8 Filtering Interpretation of DFT The DFT can be interpreted as a process of modulation and filtering, allowing an easy interpretation of some common phenomena that occur in DFT as well as introducing a new

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121

method for the computation of the DFT that can overcome some of its limitations. The starting point for this interpretation is the DFT definition (Equation (4.59)), which can be simplified as: X k ½n ¼ W k N

N1 X

x½n  me j2pkm=N :

(4.70)

m¼0

The constant W k N is responsible for the modification of the phase of the term generated by the summation in Equation (4.70). For the simplification of the analysis this constant is left out, that is: ^

X k ½n ¼

N 1 X

x½n  me j2pkm=N :

(4.71)

m¼0

Equation (4.71) can be interpreted as the convolution of the sequence x½n and the function h½n ¼ y½ne j2pkn=N

(4.72)

where y[n] is a rectangular window of length N. Equation (4.72) represents the impulse response of a complex filter. Figure 4.19 shows the DFT filtering interpretation of Equation (4.71). Note that despite x[n] being a real sequence, the output of the filter is a ^ complex signal. The modulus of X k ½n is the modulus of the kth DFT term taking the xn window, and its phase is the phase of the DFT unless a phase correction needs to be applied according to Equation (4.70). Figure 4.20 depicts the magnitude response for hk ½n; k ¼ 0; 1; 2. These plots were generated considering L ¼ N ¼ 16, that is, the sampling rate of F s ¼ 16 60 Hz. In this case k ¼ 0, 1, 2 corresponds to DC, the fundamental and the second harmonic component, respectively. The first filter is readily identified as the moving average filter (MAF) (Section 5.3) the other filters are complex filters since their magnitude frequency response is not a even function. It can be observed that the fundamental filter rejects all harmonic components, including the DC component. The same is observed for the other filters, for example the hk ½n rejects all components except for the component of frequency f k. A fundamental doubt may occur at this moment. If for example x½n ¼ A cosðw1 nÞ, how can ^ the absolute value of X 1 ½n be constant and equal to A? Isn’t the idea of filter processing a specific range of frequencies that passes throughout the filter and blocks others? If so,

Figure 4.19 DFT filtering interpretation.

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Figure 4.20 Magnitude response for DFT complex filter for DC, fundamental and second harmonic filter.

shouldn’t the output of h1½n filter be a sinusoidal signal? The answer is no, because the filter is complex. Figure 4.21 illustrates what happens when x½n passes through the filter. Using the Euler identity, A A (4.73) x½n ¼ e jv1 n þ ejv1 n 2 2 or, in the frequency domain,   A A X e jw ¼ dðv  v1 Þ þ dðv þ v1 Þ: 2 2

Figure 4.21 The filtering operation of a complex filter.

(4.74)

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Figure 4.22 Frequency response of the filter: (a) magnitude response and (b) phase response.

The second impulse of Equation (4.74) is the left impulse in Figure 4.21. Note that it falls where the filter has zero gain and then it is filtered. The output of the filter is only the first term of Equation (4.74) or, in the time domain, y ½ n ¼

A jv1 n e : 2

(4.75)

The filter output is a rotating phasor, but its module is the desired amplitude of the signal divided by 2 and its phase is the phase of the input signal. The spread effect due the presence of an interharmonic can be understood by observing Figures 4.21 and 4.22. Consider for example a signal composed of the fundamental and second harmonics plus an interharmonic with frequency of 117 Hz. By observing the frequency response of the filters in Figure 4.22, it can be noted that the gains of this filter are not 0 in the frequency of 117 Hz, so the estimation of each component is affected by the interharmonics (mainly the second harmonic) because of the lower attenuation for this component.

4.8.1 Frequency Response of DFT Filter Note that filters hk ½n are obtained by multiplying the basic window y½n by a complex exponential function. This corresponds to the modulation theorem, so the frequency response of each kth filter is obtained as H k ðe jv Þ ¼ H 0 ðe jðvvk Þ Þ

(4.76)

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where H 0 ðe jv Þ is the frequency response of the Nth MAF filter, defined: H 0 ðe jv Þ ¼

1 sinðNv=2Þ jðN1Þv=2 e : N sinðv=2Þ

(4.77)

The magnitude of H 1 ðe jv Þ is depicted in Figure 4.22a. Remember that the frequency in Hertz is related to the frequency v by the fundamental relationship v ¼ 2pf T s .

4.8.2 Asynchronous Sampling In addition to the error caused by the presence of interharmonics, the DFT method can also fail when the number of samples per cycle is not an integer number, that is, when the sampling is asynchronous. This is a common situation in power systems because the fundamental frequency experiences a frequency variation due to changes in load and generation imbalances, the inertia of the generator or the controller of the generator. When in a strong system the frequency variation is very small (less than 0.5 Hz); in a weak system or in exceptional situations when a large amount of load is disconnected from the system, the frequency variation can reach higher values (as large as 10 Hz). From the point of view of signal processing, when the sampling rate is not synchronous the phenomenon referred to as leakage is responsible for the introduction of error in the estimation components. An easy interpretation of errors caused by asynchronous sampling can be obtained from Figure 4.21. Assume the input signal is a single cosine given by xðtÞ ¼ Acosð2pf tÞ where f ¼ f 1 þ Df is the off-nominal frequency, then the discrete signal using an integer multiple of f 1 as sampling rate is given by: x½n ¼

A jðv1 þDvÞn A jðv1 þDvÞn e þ e : 2 2

(4.78)

After passing this signal by H 1 ðe jv Þ, the steady-state output is given by A A ^ X ½n ¼ H 1 ðe jðv1 þDvÞ Þ e jðv1 þDvÞn þ H 1 ðe jðv1 DvÞ Þ ejðv1 þDvÞn ¼ X f þ ½n þ X f  ½n: (4.79) 2 2 Equation (4.79) is the addition of two phasors, and is obtained by the multiplication of the real phasor by the frequency response of the filter in f þ ¼ f 1 þ Df and f  ¼ ðf 1 þ Df Þ frequencies. While X f þ ½n rotates counterclockwise, X f  ½n rotates clockwise. This sum is ^ illustrated in Figure 4.22, where it is obvious that the amplitude of resultant phasor

X½n will oscillate with frequency 2f. In addition, the mean value of amplitude is X f þ while the amplitude of oscillation is X f  . Figure 4.23a shows the magnitude estimation using DFT for off-nominal frequencies. The frequency deviation ranges from 5 Hz to 5 Hz. The sampling rate is F s ¼ 32 60 Hz. This figure shows that the error in magnitude is very small and can be neglected for practical applications. The problem however resides in the phase estimation. In this case, if the reference frequency is 60 Hz but the local frequency presents a deviation, the phase difference between the reference signal (a cosine signal of 60 Hz and zero phase, cosðv1 nÞ) and a real

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Figure 4.23 Error due to off-nominal frequency using DFT: (a) magnitude estimation for frequency deviation from 5 Hz to 5 Hz; (b) phase difference between a reference signal (at 60 Hz) and an offnominal signal (61 Hz).

signal increases with time. Figure 4.23b presents the phase difference for a real signal given by cosð2pvn þ 45 Þ and a frequency deviation of 0.5 Hz. The x axis is shown from sample 32 in order to discard the transient response produced by the filtering processing. Note that the phase difference increases continually. This example uses a reference signal that assumes the exact system frequency, when the signal is acquired at the same site however, the signals will undergo the same frequency variation and the phase difference will be as depicted in Figure 4.24, where a oscillation behavior is observed. In this example the phase difference is measured between the signals (for a frequency deviation of 0.5 Hz): xr ½n ¼ cosðvnÞ xm ½n ¼ cosðvn þ 120 Þ:

(4.80)

The error depicted in Figure 4.24 is insignificant. The error related to off-nominal frequency is significant when the frequency deviation is bigger, which only occurs in weak power systems, or in phasor measurement unit (PMU) applications when distant sites can undergo different frequency deviations. In both situations the frequency estimation is an essential part

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Figure 4.24 Phase difference between a reference signal and a measuring signal, both off-nominal frequency signals (60.5 Hz).

of correcting the phase error. Frequency estimation and robust phase and amplitude estimators are considered in Chapter 7.

4.9 The z-Transform The Laplace transform is generally used to simplify the analysis of continuous differential equations in the time domain. The application of the Laplace transform to a ordinary differential equation describing the input–output relation changes the integral-differential relation to an algebraic relation in the variable s. Correspondingly, discrete-time systems deal with difference equations relating input–output and the use of the z-transform converts the changes in the difference relationship to an algebraic relationship in the z-domain. The s-transform and the z-transform are parallel techniques with several similarities. Figure 4.25 presents the plane s and the plane z; while the s-plane is arranged in a rectangular

s-plane

z-plane jΩ = Im(s)

Im(z)

re jω

e jω s = σ + jΩ

r ω

Re(z)

σ = Re(s)

(a)

(b)

Figure 4.25 The s- and z-plane.

1

Discrete Transforms

127

coordinate system, the z-plane uses a polar format. While the jV axis (the imaginary axis) carries information about the Fourier transform, the unity circle carries information about the DTFT. We can obtain the Fourier transform from the Laplace transform by the substitution: Xð jVÞ ¼ XðsÞjs¼jV :

(4.81)

Similarly, we can derive the DTFT from the z-transform through the substitution: Xðe jv Þ ¼ XðzÞjz¼e jv :

(4.82)

Both derivations are only possible if the region of convergence (ROC) of the s-transform contains the jV axis and the ROC of the z-transform contains the unity circle e jv . This will be true for most cases, but there are some exceptions that must be considered. The analysis and synthesis equations of the Laplace transform are well known, but are repeated here for convenience: 1 ð

XðsÞ ¼

xðtÞest dt;

analysis equation ðLaplace transformÞ

(4.83)

1

1 xðtÞ ¼ 2pj

sþ1 ð

XðsÞest dt

synthesis equation ðinverse transformÞ:

(4.84)

sj1

The s-transform pair is denoted xðtÞ $ XðsÞ. This synthesis equation states that x(t) can be recovered from its Laplace transform evaluated for a set of values of s ¼ s þ jv in the ROC, with s fixed and v varying from 1 to 1. The z-transform analysis and synthesis equations are defined: XðzÞ ¼ x½n ¼

1 2pj

I

1 X

x½nzn

analysis equation ðz-transformÞ

(4.85)

synthesis equation ðinverse z-transformÞ:

(4.86)

n¼1

GðzÞzn1 dz C

The z-transform pair is denoted x½n $ XðzÞ. The synthesis equation states the sequence x½n can be recovered from its z-transfom evaluated along a contour z ¼ re jv in the ROC with the radius fixed and v varying over a 2p interval. The evaluation of the inverse transform (Laplace or z-transform) requires the use of contour integration in the complex plane. Thankfully, the synthesis equation is rarely needed in the evaluation of the inverse transform; the same technique as the partial fractions used to compute the inverse of an s-transform can be used to compute the inverse z-transform. To use the partial fractions expansion, the requirement in the z-transform is a rational function of z.

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z-plane

Im(z)

1/2

1

X

Re(z)

Figure 4.26 Example 4.9: ROC.

Example 4.9 Find the z-transform of x½n ¼ an u½n: The z-transform is obtained using the analysis equation: XðzÞ ¼

1  X

n az1 :

n¼0

The above equation represents

the sum of the terms of a geometric series with infinite terms, which only converges if az1 < 1. Then, XðzÞ ¼

1 ; 1  az1

for

1

az < 1:

(4.87)

The condition for the convergence defines the ROC: j zj > j a j Figure 4.26 illustrates the ROC for a ¼ 1=2; the dark region is the ROC. Note that the value a ¼ 1=2 is the root of the denominator of Equation (4.87); the roots of the denominator of a rational function of z are known as function poles and are marked in the figure with the symbol . The zeroes of the numerator are the zeroes of the function and are marked in the figure by 0s. Note that the zero of Equation (4.87) is located at the origin of the z-plane.

4.9.1 Rational z-Transforms A single-input, single-output (SISO) discrete-time system is described in the time domain by a difference equation. The general format of the SISO discrete-time system is given by the recursive equation: y½n ¼

M X i¼0

bi x½n  i 

N X i¼1

ai y½n  i

(4.88)

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129

where y½n is the output, x½n is the input and bi ði ¼ 0; 1; . . . ; MÞ and ai ði ¼ 1; 2; . . . ; NÞ are the system coefficients. This equation represents the algorithm that will be programmable in the digital signal processor or synthesized in hardware (for example in a field programmable gate array or FPGA), and it describes the relationship between the input and output. Just as continuous systems are controlled by differential equations, discrete time systems are controlled or described by the difference equations. However, several important aspects of the system cannot be directly observed from looking at the difference equation, such as the stability of the system and the frequency response characteristics. These aspects are better understood and observable using the z-domain. To change Equation (4.88) for the z-domain, we must first derive the important property of the z-transform of a time shifting sequence. Let X½z be the z-transform of x½n; we want to find the z-transform of x½n  n0 . X 1 ðzÞ ¼

1 X

x½n  n0 zn :

n¼1

Letting n  n0 ¼ m we obtain: X 1 ðzÞ ¼

1 X

1 X

x½mzðmþn0 Þ ¼ zn0

m¼1

x½mzm ¼ zn0 XðzÞ:

m¼1

This property can be represented: x½n  n0  $ zn0 XðzÞ:

(4.89)

Appling the z-transform in Equation (4.88) and making use of the property above yields: Y½z ¼

M X

bi zi X½z 

i¼0

N X

ai zi Y½z:

i¼1

The transfer function HðzÞ ¼ YðzÞ=XðzÞ can be rewritten: M P

H ½ z ¼

i¼0 N P

bi zi ;

(4.90)

ai zi

i¼0

where a0 ¼ 1. In power system applications, the transfer function is ‘proper’ or ‘strictly proper’, which means N  M. Furthermore, the poles are all single, that is, there are no multiple poles in Equation (4.90). Equation (4.90) can be factored in different forms, the expansion in partial fractions being the most practical form. The expansion in partial fractions, considering simple poles and proper functions, can be written [8]: H ½ z ¼

N X l¼1

rl ; 1  ll zl

(4.91)

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where z ¼ ll are the poles of HðzÞ, 1  l  N, ll are distinct and the constants rl are called residues, defined   rl ¼ 1  ll z1 HðzÞjz¼ll :

(4.92)

Example 4.10 Find the impulse response of the system with transfer function:

HðzÞ ¼

1  0:8z1  0:15z2 : 1  1:3z1 þ 0:4z2

The impulse response is obtained by taking the inverse z-transform of YðzÞ ¼ HðzÞXðzÞ Table 4.8 shows the z-transform of the unit impulse sequence as d½n $ 1. To find the impulse response, the transfer function must be written in the form of Equation (4.91). The MATLAB1 command ‘residuez’ returns the residues and poles of the function; an example of the required code is as follows: b = [1 0.8 0.15]; %numerator coefficients a = [1 1.3 0.4]; %denominator coefficients [r,p] = residuez(b,a); % r is the vector contained the residue % p is the vector contained the poles

which would generate the result: r ¼ [0.625 2.0] and p ¼ [0.8 0.5]. Table 4.8 Some commonly used z-transform pairs. Row No.

z-transform

Sequence

ROC

1

d½n

1

All values of z

2

u½n

1 1  z1

jzj > 1

3

an u½n

1 1  az1

jzj > jaj

4

rn cosðv0 nÞu½n

1  ðr cosðv0 ÞÞz1 1  ð2r cosðv0 ÞÞz1 þ r2 z2

jzj > jrj

5

rn cosðv0 nÞu½n

ðr sinðv0 ÞÞz1 1  ð2r cosðv0 ÞÞz1 þ r2 z2

jzj > jrj

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131

s-plane

z-plane jΩ = Im(s)

Im(z )

x

X

X

x

1/2

σ = Re(s) x

X X

(a)

1

Re(z )

X

(b)

Figure 4.27 Stability region in the (a) s-plane and (b) z-plane.

The partial-fraction expansion is: YðzÞ ¼

0:625 2:0 þ : 1  0:8z1 1  0:5z1

To compute the inverse transform, the results of Example 4.10 and the fact that the z-transform is linear are used, to obtain: 1 ; the ROC is jzj > jaj; 1  az1 y½n ¼ ½0:625ð0:8Þn þ 2ð0:5Þn u½n:

a n u½ n $

4.9.2 Stability of Rational Transfer Function The previous example shows that the poles are responsible for the exponential terms in the time domain. If the modulus of the pole is ðr½n  s½nÞ  r ½n h:  n¼0 n¼0

N1 X

2

N1 X

2

(11.8)

H0

Applying logarithms to both sides of Equation (11.8), H1 is chosen if N 1 X n¼o

r½ns½n > s 2 ln h þ

N 1 1X s2 ½n: 2 n¼0

(11.9)

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As s[n] is known and deterministic, the energy term can be incorporated in a new threshold and Equation (11.9) can be written N 1 X

r½ns½n > h0 :

(11.10)

n¼o

It can be observed that the correlation between the known signal s[n] and the received signal r[n] is calculated and compared with the threshold (h0 ). This detector is therefore known as correlator. Equation (11.10) can also be written as the convolution between the received signal r[n] and the flipped version of the known signal (h[n] ¼ s[N  1  n]) at time N  1, called a matched filter. The output of the matched filter at time n ¼ N  1 is given by y½N  1 ¼

N1 X

r½ns½n:

(11.11)

n¼0

Consider a deterministic sinusoidal signal s[n] that is corrupted by an additive Gaussian white noise (AWGN) after transmission, which should be detected at the receiver end of a communication device. Figure 11.4 shows s[n] and Figure 11.5 shows four realizations of the received signal r[n], where the upper two are related to the signal while the lower two are related to noise. Visually it is very difficult to distinguish between signal and noise. Ten thousand realizations for both hypotheses can be generated by using the following coding in MATLAB1

Figure 11.4 Example of a deterministic sinusoidal signal s[n] that should be detected.

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Figure 11.5 Samples of the received signal. The two realizations at the top are related to signal while the two at the bottom are related to noise.

%Detection Example n=1:11; %deterministic sinusoidal signal s=sin(2 pi 1/10 (n-1)); t=0:10; stem(t,s) pause %Gaussian White Noise w=randn(10000,11); w1=randn(10000,11); sn=repmat(s,10000,1); %received signals r1=sn+w; r0=w1; %examples of signals and noise at receiver figure subplot(4,1,1),stem(t,r1(1,:)) subplot(4,1,2),stem(t,r1(2,:)) subplot(4,1,3),stem(t,r0(3,:)) subplot(4,1,4),stem(t,r0(4,:))

The optimum detector in this case is the correlator. Its estimated ROC curve can be seen in Figure 11.6. If both hypotheses are equiprobable, the optimum threshold (minimum probability error) can be calculated according to

362

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Figure 11.6 ROC for the optimum detection system.

  N 1 PðH 1 Þ 1X þ h0 ¼ s 2 ln s2 ½n: PðH 0 Þ 2 n¼0

(11.12)

This results in a probability of detection of around 87% for a false alarm around 13%. The detection system can be implemented in MATLAB1 using %Correlator Detector Det=s; %Correlator outputs rH1=r1 Det’; rH0=r0 Det’; %Histogram for rH0 (noise) and rH1 (signal+noise) figure hist(rH0,30) hold hist(rH1,30) title('Histogram') hold %ideal threshold for equal a priori probabilities TH=log(1)+1/2 sum(s.^2); %probability of detection and false alarm for the optimum threshold Pd=length(rH1(rH1>=TH))/10000 Pf=length(rH0(rH0>TH))/10000 %building the ROC curve THRoc=[-5:17/200:12]; For i=1:201 PdRoc(i)=length(rH1(rH1>=THRoc(i)))/1000;

Detection

363

PfRoc(i)=length(rH0(rH0>THRoc(i)))/1000; end figure plot(PfRoc,PdRoc) title('ROC curve')

11.3.5 Deterministic Signals with Unknown Parameters The power system fundamental component signal f [n] can be modeled as a deterministic signal with unknown parameters as f ½n ¼ A0 cosð2pf 0 =f s n þ u0 Þ. We may not be sure of the values of amplitude A0, fundamental frequency f0 and the phase u0, but the sampling frequency fs is usually known. There are two main approaches to solve this issue. The first considers the signal as deterministic and uses the generalized likelihood ratio test (GLRT). The second is a Bayesian approach, where the signal is modeled as a realization of a random process. In the classical case of unknown deterministic signal parameters, an optimal detector usually does not exist. A suboptimal detector is therefore required and the GLRT will usually demonstrate good detector performance. If the Bayesian approach is adopted, the resulting detector can be said to be optimal. The difficulty of using the Bayesian approach is in specifying the prior probability density functions and carrying out the integration in practical scenarios. In this section we discuss the case of unknown signal amplitude in order to illustrate the design of the detector test using both approaches. First, we discuss the importance of signal knowledge for the design of a detector system.

11.3.5.1 Unknown Signal Consider a detection problem H 0 : r½n ¼ w½n n ¼ 0; 1; . . . ; N  1 H 1 : r½n ¼ s½n þ w½n n ¼ 0; 1; . . . ; N  1

(11.13)

where s[n] is deterministic but completely unknown and w[n] is the WGN with a variance of s 2. A GLRT would select H1 if f r=^s;H 1 ðr=^s½0; . . . ^s½N  1; H 1 Þ >h f r=H 0 ðr=H 0 Þ

(11.14)

where ^s½n is the maximum likelihood estimation (MLE) under H1. To determine the MLE, the likelihood function should be maximized over the signal samples "

# N1 1 X 2 exp  2 ðr½n  s½nÞ ; f r=^s;H 1 ðr=^s½0; . . . ^s½N  1; H 1 Þ ¼ 2s n¼0 ð2ps 2 ÞN=2 1

(11.15)

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resulting in ^s½n ¼ r½n. Thus, from Equation (11.14): 1 ð2ps 2 ÞN=2 " # > h: N 1 1 1 X 2 exp  2 r ½n 2s n¼0 ð2ps 2 ÞN=2

(11.16)

After taking logarithms of both sides, this yields N 1 X

r2 ½n > h0

select H 1 :

(11.17)

n¼0

It can be seen from Equation (11.17) that this is just an energy detector. The performance of the energy detector is worse than the correlator (optimal), which is due to the design of the energy detector where the signal was considered unknown. A comparison between both detectors can be performed using the example of Section 11.3.4, where a sinusoidal deterministic signal should be detected. Figure 11.7 shows the ROC curves for both detectors (correlator and energy detector), where it is possible to verify that the correlator presents a much better performance than the energy detector. These curves can be reproduced in MATLAB1 using %Detection Example n=1:11; %deterministic sinusoidal signal s=sin(2 pi 1/10 (n-1)); t=0:10; %Gaussian White Noise

Figure 11.7 Comparison between the correlator and the energy detector for determinist signal detection under WGN.

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w=randn(10000,11); w1=randn(10000,11); sn=repmat(s,10000,1); %received signals r1=sn+w; r0=w1; %Correlator Detector Det=s; %correlator outputs rH1=r1 Det’; rH0=r0 Det'; %building the ROC curve THRoc=[-5:17/200:12]; For i=1:201 PdRoc(i)=length(rH1(rH1>=THRoc(i)))/10000; PfRoc(i)=length(rH0(rH0>THRoc(i)))/10000; end figure plot(PfRoc,PdRoc) title('ROC curves') %Energy Detector r2H1=sum(r1.^2,2); r2H0=sum(r0.^2,2); THRoc=[0:55/200:55]; For i=1:201 Pd2Roc(i)=length(r2H1(r2H1>=THRoc(i)))/10000; Pf2Roc(i)=length(r2H0(r2H0>THRoc(i)))/10000; end hold plot(Pf2Roc,Pd2Roc,'r')

11.3.5.2 Unknown Amplitude Consider the problem of detecting a known deterministic signal with unknown amplitude in WGN. The problem can be formulated: H 0 : r½n ¼ w½n

n ¼ 0; 1; . . . ; N  1

H 1 : r½n ¼ As½n þ w½n n ¼ 0; 1; . . . ; N  1

(11.18)

where s[n] is known and the amplitude A is unknown. The w[n] is WGN with a variance of s 2. In order to build the detector, the LRT " # N 1 1 1 X 2 exp  2 ðr½n  As½nÞ 2s n¼0 ð2ps 2 ÞN=2 " # (11.19) > h select H 1 N 1 1 1 X 2 exp  2 r ½n 2s n¼0 ð2ps 2 ÞN=2

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can be used. Taking logarithms of both sides results in A

N 1 X

r½ns½n > s 2 ln h þ

n¼0

N 1 A2 X s2 ½n ¼ h0 : 2 n¼0

(11.20)

If the sign of A is known and A > 0, the test for whether H1 should be selected is reduced to: N 1 X

r½ns½n >

h0 ¼ h00 : A

(11.21)

r½ns½n <

h0 ¼ h00 : A

(11.22)

n¼0

If A < 0, we select H1 if N 1 X n¼0

The tests are optimal and the detector reduces to the correlator. If the amplitude sign is unknown however, it is not possible to construct a unique test and the detector should be designed using the GLRT or the Bayesian approach. 11.3.5.3 GLRT Design Using the generalized likelihood ratio test, H1 is selected if ^ f r=A;H ^ 1 ðr=A; H 1 Þ f r=H 0 ðr=H 0 Þ

> h;

(11.23)

^ is the maximum likelihood estimator of A under the assumption of H1, given by where A N1 P

^¼ A

r½ns½n

n¼0 N1 P

:

(11.24)

s2 ½n

n¼0

From Equation (11.20), H1 is selected if ^ A

N 1 ^2 X A r½ns½n > s ln h þ s2 ½n: 2 n¼0 n¼0

N 1 X

2

(11.25)

From Equation (11.24), we have ^A ^ A

N 1 X n¼0

s2 ½n 

N 1 ^A ^X A s2 ½n > s 2 ln h: 2 n¼0

(11.26)

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The GLRT design therefore leads to the selection of H1 if vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2s 2 ln h ^ >u : jAj uN1 tP 2 s ½n

(11.27)

n¼0

Equivalently, Equation (11.27) can be written ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  v   u X N 1 N 1 u X   r½ns½n > t2s 2 ln h s2 ½n    n¼0 n¼0

(11.28)

which is a correlator that accounts for the unknown amplitude sign. It is important to stress that the detector performance has deteriorated in comparison to the correlator (Equation (11.22)), but is much better than the energy detector (Equation (11.17)). The performance of the GLRT detector in comparison to the correlator and the energy detector can be evaluated using the previous example (detection of a sinusoidal signal). Figure 11.8 shows the ROC curves for the three detectors. It is possible to see that the performance of the GLRT detector lies between that of the correlator and the energy detector. This behavior is expected since the amplitude sign must be known for the correlator detector, but not for the GRLT. 11.3.5.4 Bayesian Design The Bayesian approach considers the signal s[n] as a random process and requires knowledge of the prior probability density function of A. Assuming that A is a random variable following a Gaussian distribution with mean mA and variance s 2A , independent of w[n], the decision to

Figure 11.8 Comparison between the correlator, the GLRT and the energy detector.

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select H1 is determined by (see details in reference [2]):

s 2 þs

mA N1 P 2 A

n¼0

N 1 X

s2 ½n n¼0

r½ns½n þ

s 2A

  N1 P 2 2 2 2 2s s þ s A s ½n

N 1 X

!2 r½ns½n

> h0 :

(11.29)

n¼0

n¼0

This detector is a combination of the correlator and the second-order correlator (GRLT). Its performance is optimal in the Newman-Person sense [2].

11.4 Detection of Disturbances in Power Systems The increasing levels of harmonic distortion in power line signals is driving the development of signal processing tools capable of performing more detailed and precise detection of disturbances and events. One of the major requirements for power quality monitoring equipment is the detection of PQ disturbances fixed in time, with the ability to detect both the initiation and end of the fault. The necessity for improved detection performance in continuous electric signals monitoring devices has motivated the development of several techniques that have a good tradeoff between computational complexity and performance [3–5]. Much research has been performed on wavelet-transform-based techniques for the purposes of detection. There are also several techniques that make use of second-order information about the error signal, which results from the subtraction of the fundamental component from the electric signal. Analysis of the error signal is an attractive and interesting solution to characterize the presence of disturbances [4]. The techniques proposed in references [3,6,7] are very similar in the sense that all of them make use of second-order statistics of the error signal for the detection of the occurrence of disturbances.

11.4.1 The Power System Signal The discrete version of the monitored power line signals can be divided into non-overlapped frames of N samples and the discrete sequence in a frame. These can be expressed as an additive contribution of several types of phenomena: x½n ¼ xðtÞjt¼nT s :¼ f ½n þ h½n þ i½n þ t½n þ v½n;

(11.30)

where n ¼ 0, . . . , N  1, Ts ¼ 1/fs is the sampling period and the sequences f ½n, h½n, i½n, t½n and v½n denote the power system fundamental component signal, harmonics, interharmonics, transient and background noise, respectively. These signals are defined:   f ½n f ½n ¼ A0 ½ncos 2p 0 n þ u0 ½n ; fs h½n ¼

M X m¼1

hm ½n;

(11.31)

(11.32)

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i½n ¼

J X

ij ½n;

(11.33)

j¼1

t½n ¼ tspi ½n þ tnot ½n þ tcas ½n þ tdam ½n;

(11.34)

and v½n is an independently and identically distributed (i.i.d.) noise as normal Nð0; s 2v Þ and independent of f ½n, h½n, i½n and t½n. In Equation (11.31), A0 ½n, f 0 ½n and u0 ½n refer to the magnitude, fundamental frequency and phase of the power system fundamental signal, respectively. In Equation (11.32) and (11.33), hm ½n and ij ½n are the mth harmonic and the jth interharmonic, respectively, which are defined   f 0 ½n n þ u m ½ n hm ½n ¼ Am ½ncos 2pm fs   f I;j ½n n þ uI;j ½n ij ½n ¼ AI;j ½ncos 2p fs

(11.35)

(11.36)

where Am ½n is the magnitude and um ½n is the phase of the mth harmonic and AI;j ½n, f I;j ½n and uI;j ½n are the magnitude, frequency and phase of the jth interharmonic, respectively. In Equation (11.34), tspi ½n, tnot ½n, tcas ½n and tdam ½n are transients named spikes, notches, decaying oscillations, and damped exponentials.

11.4.2 Optimal Detection The detection of an event or disturbance can be performed using the N samples frame that can be represented by vector x. When the power system signal is in its normal operation, x ¼ f n þ v;

(11.37)

where f n is the fundamental component vector of the power system signal in its normal range of operation (amplitude and frequency) and v is the background noise vector. When a disturbance occurs, vector x can be expressed as x ¼ f n þ Df n þ h þ i þ t þ v;

(11.38)

where Df n represents the abnormal conditions related to the fundamental component (sag, swell, interruptions or frequency variation), h is the vector related to the harmonics, i interharmonics and t transients. A disturbance occurs if any of these signals are present in x. The detection of disturbances can be formulated as a binary hypothesis test: H0 : x ¼ fn þ v H 1 : x ¼ f n þ Df n þ h þ i þ t þ v

(11.39)

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where H 0 refers to the normal operation of the power system signal (absence of disturbance) and H 1 refers to the occurrence of a disturbance on the power system signal. The first step in the design of a detection system is to generate and select representative features p from the signal frame x that can map the input data into a feature space where the separation between both hypotheses is increased. In order to design the optimum detector without assigning costs for the decisions, Equation (11.1) should be used. It requires knowledge of f x=H 0 ðx=H 0 Þ, f x=H 1 ðx=H 1 Þ, PðH 0 Þ and PðH 1 Þ. These quantities are not normally known, but they can be estimated using the training dataset [7]. Probability density function estimation is not an easy task in a high-dimensional space. This is a common situation in several detection problems. Additionally, this approach can lead to a detection system of high computational complexity. Sub-optimum detection algorithms are therefore commonly used, for example the techniques presented in Chapter 10 including LS methods, distance-based classifiers, neural networks and SVM.

11.4.3 Feature Extraction Several disturbance detection techniques make use of the wavelet transform and second-order statistics as a feature extraction technique due to the nature of the power system signal (see Equation (11.30)). The wavelet transform divides the input signal into different frequency bands. For each band the energy is measured (second-order statistics) and used as input to the detection algorithm. Another common approach for feature extraction is the removal of the fundamental component f n from the input signal vector x. As can be seen in Equation (11.39), the fundamental component is common to both hypotheses; its removal from x therefore increases the separation between the hypotheses. As such, instead of using x for the design of the detection system, the error signal e ¼ x  ^f n is used where ^f n is the estimation of the fundamental component of the power system signal. The error signal can be generated using a notch filter (Chapter 5) or a fundamental component estimation technique. Some detection techniques directly use the error signal as an input for the detection algorithm while others measure the energy of the error signal, reducing the dimension of the feature space. Another relevant approach for feature extraction is the use of higher-order statistics (HOS) cumulants, since these are blind to any kind of Gaussian process (background noise) whereas second-order information is not. Cumulant-based signal processing techniques can handle colored Gaussian noise automatically, whereas second-order techniques may not. Cumulantbased techniques therefore boost signal-to-noise ratio when electric signals are corrupted by Gaussian noise [8].

11.4.4 Commonly Used Detection Algorithms Concerning detection algorithms, several techniques have been employed for power system disturbance detection. Techniques are usually suboptimal as the complexity of the problem impedes the development of the likelihood ratio detector designed using the Bayesian approach. Several authors have made use of specialized knowledge to perform the feature extraction, and the detection threshold is chosen using the training dataset. Such an example is to detect transients; for this the cycle-by-cycle difference [9] could be used followed by an energy detector.

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When several features are extracted from the power system signal in order to perform the disturbance detection, the training dataset should be used to design a detector based on distance (e.g. Euclidian distance), LS, ANN or SVM or a fuzzy-based detector, among other techniques.

11.5 Examples 11.5.1 Transmission Lines Protection The first task in the protection of transmission lines is fault detection. This is very important for the design of the effective isolation of the electric power system that may be experiencing such a fault. Basically, a fault detection algorithm must differentiate the normal state from the faulted state. This need to be carried out within a few milliseconds and its performance (speed and accuracy) directly affects the performance of the protection system. In order to reduce the risk of a false alarm, a counter is used to confirm and differentiate between a fault and non-fault. A fault is detected only if a consecutive number of samples that overpass a threshold are higher than a predefined value. In reference [10], three consecutive samples are considered. Fault detection algorithms can use the characteristic fault components to perform their task while maintaining their ability to identify an occurrence even in the presence of harmonic frequencies, white noise, notches, spikes and frequency variations. One of the simplest algorithms used for fault detection is described in reference [10], called the short Fourier filter. In this case, a FIR filter with zeros located at each harmonic is used and its output y[n] is expressed as the difference between two samples spaced by one cycle (that corresponds to N samples) of the current signal, that is s½n ¼ jy½n  y½n  N þ 1j:

(11.40)

An even simpler form is the detection based on the comparison of two consecutive samples of the signal i[n] [11]. The equation in the discrete-time domain is: s½n ¼ ji½n  i½n  1j:

(11.41)

However, there is a more robust method of performing a detection task: a long Fourier filter [11]. The main difference between a long and a short Fourier filter is that the former has double zeros on the unit circle spaced every 2f0 (twice the fundamental frequency). This process makes it somewhat more immune to low-frequency variations, especially the frequencies around 60 Hz (or 50 Hz). It is implemented by the equation:        N 3N  þ 1 : s½n ¼ i½n þ i n  þ 1  i½n  N þ 1  i n  2 2

(11.42)

A very interesting class of parameters is the HOS or cumulants which has been successfully applied as a pre-processing tool in detection issues. In power quality, it provides an appropriate tool for detecting and classifying disturbances [12–14]. The remarkable results of [15–17] for detection purposes, parameter estimation and classification applications must be mentioned.

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Figure 11.9 The block diagram structure for fault detection.

Figure 11.9 shows a detector stage as described in reference [17]. The detection process starts with the filtering process performed by an IIR notch filter (see Chapter 5), which has a transfer function HðzÞ ¼

1þa 1  2bz1 þ z2 : 2 1  bð1 þ aÞz1 þ az2

(10.43)

This filter removes the fundamental frequency component f0. Its use is based on the fact that various techniques of detection are performed using an error signal. The formulation based on hypotheses test: H 0 : ev ½n ¼ rv ½n H 1 : ev ½n ¼ rv ½n þ tv ½n

(11.44)

can be used for fault detection. The error signal is composed of a Gaussian noise rv[n] if the H0 hypothesis is true. This situation is equivalent to normal operation of the electrical power system (EPS). On the other hand, the error signal is composed of a Gaussian noise component rv[n] added to a transient component tv[n], considering that the H1 hypothesis is true. This situation is equivalent to the operation of the EPS under fault conditions. This is a reasonable formulation because, after the attenuation of the fundamental component, only a noise component is expected at the output of the filter (and possibly very small harmonics). After the filtering stage, the HOS block is used to calculate the cumulants of the error signals. The cumulants are obtained at this stage and applied to the N-length sliding windows of signals eA ½n, eB ½n and eC ½n. Since the third-order cumulants of a symmetrically distributed random process are zero, only the second- and fourth-order cumulants are used. It is worthwhile mentioning that the detection is made in this approach using the voltage signal. A combination of cumulants for phases A, B and C is the input of the third and last stage. This stage is composed of an artificial neural network. Simulations were performed using a sampling frequency fs ¼ N  f0 with N ¼ 256. The fault distances considered were d ¼ {5, 10, 15, . . . , 145} with fault inception angles of {0 , 30 , 60 , . . . , 330 } and fault resistances between phase and ground Rg ¼ {0.1, 1.0, 10, 100, 400}. The resistance between phases Rp ¼ {0.1, 1, 10, 50} in the case of faults involving phases. The system used to train the detector and all other functions of the relay system is depicted in

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Figure 11.10 The simulated electrical power system.

Figure 11.10. For the detection, the data windows have a length of 1/8 (one-eighth) of the fundamental period. Figure 11.11 shows the comparative results of the detection using the methodology discussed above and the traditional methodologies, which involve sample by sample (ss) and cycle by cycle (sc). For the example presented in Figure 11.11, the system frequency was varied over the range 59–61 Hz. The main point to be highlighted in the detection of faults based on neural networks and cumulants is that after training the NN the threshold does not need to be adjusted as in the case of traditional methods; the threshold is established by the neural network optimization.

11.5.2 Detection Algorithms Based on Estimation The information obtained from the estimation algorithms can be used for detection, despite the fact that this procedure is not commonly used in protection systems. However, if the estimation algorithm is able to indicate quickly that there is a significant deviation from the expected estimation value, it can be used to detect a fault. The estimation algorithm presented

Amplitude

(a) 1 Phase B

0 –1

Phase C

Phase A

0

64

128

192

256

Amplitude

(b) 0.92

C4,vb(128)

0.9 0.88

C4,va(128)

C4,vc(64) 0

64

128

192

256

192

256

192

256

Amplitude

(c) 1 0.5 0

0

64

128

Amplitude

(d) 1

sc

0.5 0

sa 0

64

128 Sample Index

Figure 11.11 Performance of the detection method proposed in reference [17]: (a) fault signal; (b) cumulants used in detection process; (c) output of the proposed detector; (d) and output of the detection sa and sc.

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Figure 11.12 Detection method based on parameter estimation.

in Chapter 5 shows this property. Figure 11.12 shows the basic block diagram of the estimation and detection method. The algorithm presented in Figure 11.12 estimates the amplitude of the current signal. The term I w0 ðejv0 Þ can be used to indicate the fault because its values change abruptly during the fault. In fact, at the first sample after the fault its value is high enough to overpass the threshold. The detection algorithm consists of obtaining the absolute value of I w0 ðejv0 Þ to compare this with a threshold and increment a counter for consecutive overpasses; when the counter reaches a specified number of overpasses, the detector issues an alarm. To analyze and compare the performance of the proposed method and other methods in literature, simulations were performed by inserting non-fault components in the system such as frequency variations, harmonic components, noise, notches and spikes. The changes that occur in response to the estimation algorithm in such circumstances do not indicate a fault occurrence. In this case, the adjacent minimum values Lmin should be chosen to be higher than the values of the responses given by the algorithms. Simulations of fault occurrence, characterized as changes in the amplitude and the existence of a DC exponential with decaying or oscillating sub-synchronous values, indicate that the minimum of the first three samples after the fault occurrence should be adopted as the maximum value of borderline Lmax. Clearly, Lmin must be lower than Lmax. If the opposite happens, this is an indication that the algorithm in question is not immune to that type of power quality disturbance and the detector may indicate a false alarm. Several simulations were performed while varying the signal frequency over the range 48–72 Hz, the harmonic amplitude by 10–40% of the fundamental value, the SNR over the range 20–50 dB, the fundamental amplitude variation in fault condition 20–100%, the time

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Table 11.1 The worst case for defining the threshold. Lmin

Lmax

Frequency Harmonics Noise Notches Spikes Fundamental Exponential amplitude decaying amplitude Proposed 2.3040 method

1.1081

0.5998 0.6004 1.6125 32.2500  103

t

Subsynchronous oscillation

8.0768 13.5621 0.9017  106

constant of an exponential from 0.5 up to 10 cycles, the amplitude of the exponential by 0–10 times the value of the nominal current and, finally, the sub-synchronous frequency from 10 to 50 Hz and its time constant from 0.01 to 0.1 s. Table 11.1 presents the worst-case scenario used to define the threshold of the detection algorithm. It can be observed that any value higher than 2.304 and smaller than 8.0763 can be used as threshold. The development of new algorithms for use in digital protection devices, monitoring and control requires validation in real time. In order to validate the system under conditions as close as possible to that of the power system where it is going to be installed, it is necessary that both the algorithm under test and the grid where it is going to be installed can operate in real time. The real-time digital power system simulator (RTDS) is one of the most-used real-time electromagnetic digital simulators worldwide. It has traditionally been used to test protective relays and control equipment, but its main feature is the ability to interact in real time with external components and equipment that, through the I/O cards interface (analog to digital or A/D card and digital to analog or D/A card) become part of the simulation loop. This feature is referred to as hardware-in-the-loop (HIL) [18] and provides, among other features, the ability to validate prototypes of new equipment and components as well as the study of real-time transients in an integrated system. On the other hand, when developing new algorithms for protection, control and monitoring, the implementation of these on dedicated hardware may require a long development time of the software or, in the case of a field-programmable gate array (FPGA), a long time to synthesize the digital circuit. An interesting approach that reduces the development time is the use of virtual hardware, such as dSPACE. dSPACE allows the algorithms previously tested in MATLAB1/Simulink to be compiled according to a real-time operating system, running on high-performance processors. The virtual instrument is then connected to the RTDS through the A/D and D/A cards of both instruments. Figure 11.13 shows the HIL implemented to test the detection algorithm described above. The estimation and detector algorithms were compiled from MATLAB1 and transferred to the dSPACE board. The real time simulation was performed in the RTDS and the network is presented in Figure 11.14. Figure 11.15 shows an example of a detection algorithm working in real time. The threshold used for this example was 3; the fault is applied at cycle 79 and then removed at cycle 84. Fault elimination is obtained directly by removing the fault resistance. The figure shows the current signal, the amplitude estimation, the output I w0 ¼ I w0 ðejv0 Þ and the trigger signal I w0 > 3.

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Figure 11.13 HIL to test the detection and estimation algorithm.

Compensator

XL1

R1

XL2

XC1

1 pu

F1

R2

XC2

F2

1-10º pu

Figure 11.14 Network implemented in RTDS to test the detection and estimation algorithm.

200 Input Signal i[n] Detection Signal Iw 0

150

Detection Signal (Iw 0>3) Estimation

magnitude

100 50 0 –50 –100 70

72

74

76

78

80 cycles

82

84

86

Figure 11.15 Example of the detection algorithm in real time.

88

90

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11.5.3 Saturation Detection in Current Transformers The importance of correcting the current distortion to prevent operation errors of protective devices connected to the current transformer (CT) secondary operating in saturation has been discussed extensively in the literature. The first step in making this correction is to know precisely where the saturation region starts and finishes. The correction methods known from literature are highly dependent on accurate knowledge of the saturated and unsaturated regions, due to the fact that in non-saturated areas the secondary current is a true copy (scale) of the primary fault current. On the other hand, the secondary current behaves differently in saturated regions depending on the non-linearity of the magnetizing inductance of the CT. Several techniques exist for detecting the saturation of transformers. The main theory behind the detection algorithm is that the saturation causes a breakdown of the current waveform. Any method that then detects the breakdown in a signal would appear to be able to detect the saturation region. The most commonly used methods are based on wavelets [19] and derivatives [20]. Both work very well if there is no added noise in the signal. However, if a small noise equivalent to 50 dB of SNR is added to the system, the detection performance deteriorates completely. Figure 11.16 shows the distorted secondary current and the respective detail coefficient d1 when the wavelet db4 is used. Note that the detail coefficient is a good indicator of the saturation region; in this case, the signal does not contain noise. When noise is added, it is noted from Figure 11.17 that the first level of detail of the discrete wavelet transform (DWT) is seriously affected. There is no safe threshold level for the coefficient d1 that could be used to detect the beginning and the end of the saturated regions. Further study of the second, third and fourth levels of detail (d2, d3 and d4, respectively) shown in Figure 11.17 however indicate a possible way to detect this error. Since the coefficient values oscillate over a large number of samples around the start and end points of saturation, the simple choice of a threshold is not

Figure 11.16 Distorted part of the secondary current revealed in the detail coefficient at level d1. No noise added.

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Figure 11.17 Distorted portion of the secondary current and detail coefficients with noise.

enough to identify saturated regions with accuracy and further processing is required to try to better identify such points. A sudden change in current behavior at the points where the saturation region begins and ends has been observed. This feature makes the first derivative of the current suffer a discontinuity at these points. In attempting to solve this problem, the third derivative method was proposed in reference [20] in which the behavior of the third derivative of the current signal is observed, demonstrating that it reaches sufficiently large values at the points of interest to be used in the detection process. The third derivative is represented by the third difference (del3[n]) and is given by del3 ½n ¼ is ½n  3is ½n  1 þ 3is ½n  2  is ½n  3

(11.45)

where is is the secondary current. Figure 11.18 shows the behavior of the third difference calculated from a secondary current generated by the CT simulation algorithm in MATLAB1, without the presence of noise. The start and stop points of saturation are shown clearly by a large increase in the absolute value of the third difference function in the four subsequent samples. In contrast, at points outside the transition area (between the saturated and unsaturated regions) the value of the third difference function takes much lower values. The theoretical threshold value can be established by considering the output of Equation (11.45) in the unsaturated region. The secondary current in the unsaturated region

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40

Current (p.u.)

30 20 10 0 –10 –20 –30 0

1

2

3

1

2

3

4 Cycle

5

6

7

8

5

6

7

8

20

Third difference

15 10 5 0 –5 –10 –15 –20 0

4 Cycle

Figure 11.18 Distorted part of secondary current and the third derivative function when no noise is added to the signal.

can be written i s ½ n ¼

    I pf max 2p cos n  expðnT s =tÞ ; Kn N

(11.46)

where I pf max is the maximum current at the primary side, K n is the transformer relation and t is the time constant of the DC component. The maximum value of the third difference can be obtained by considering the hypotheses of zero DC component. The final value is defined: maxðdel3 ½nÞ ¼

 p i3 I pf max h 2 sin ; Kn N

(11.47)

which can be used to set the threshold for detecting the start and the stop of the saturation function of the third difference. In order to prevent filter inaccuracies of the inherent sensitivity of the algorithm, a margin factor k is adopted so that the threshold TshTD (third difference) can be expressed: TshTD ¼ k

 p i3 I pf max h 2sin : Kn N

(11.48)

Despite good results obtained by the previous method, for the detection of faults in noisy scenarios the above will be inaccurate. In reference [20], the authors used a first-order RC lowpass filter to smooth the effect of noise caused by the converter process (digital to analog and analog to digital) connected to the output of the ATP and input of the detector. The studies performed showed that the correct choice of cutoff frequency associated with the adjustment of the margin factor retains the efficiency of the method of the third difference. However, no

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380 40 Current (p.u.)

30 20 10 0 –10 –20 –30 0

1

2

3

4 Cycle

5

6

7

8

1

2

3

4 Cycle

5

6

7

8

Third difference

15 10 5 0 –5 –10 –15 –20 0

Figure 11.19 Distorted region of the secondary current and the third derivative function when noise is added to the signal and the signal is pre-filtered.

studies have been conducted regarding the background noise from the power system and electromagnetic interference in the CT secondary circuit. The derivative has the property of increasing high-frequency noise. For example, Figure 11.19 shows the same example of Figure 11.18 with noise of 50 dB added to the signal and filtered by a second-order low-pass Butterworth filter, with cutoff frequency of 500 Hz. Further studies indicate that the use of more complex filters, such as the Chebchev [21] of 36 samples (for a sampling rate of 200 samples per cycle) can restore the effectiveness of the method of the third difference. However, these types of filters increase the computational effort and the delay in 18 samples.

11.6 Smart-Grid Context and Conclusions Signal detection is used in power systems to monitor disturbances and protection activities. Detection theory is related to the random nature of physical signals and its statistical properties are used in order to design optimum detection systems. In power systems protection, detection can be performed with the adequate signal processing of waveforms that extract the information and detect parameters and thresholds related to the statistical nature of the event. The higher complexity of smart grids will produce more intricate and compound signals that will make the detection process more difficult. This chapter introduced the basic aspects of detection theory using the Bayesian framework and discussed the deterministic signal detection in white Gaussian noise. We have also described example applications of power system signals in transmission line protection and saturation detection in current transforms. The more complex the smart grid of the future, the more detailed and advanced the detection methods will have to be in order to monitor the network and mitigate problems for the proper and continuous operation of the grid.

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References 1. Van Trees, H.L. (2001) Detection, Estimation and Modulation Theory, Part I: Detection, Estimation and Linear Modulation Theory, John Wiley and Sons. 2. Kay, S.M. (1998) Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, Signal Processing Series. 3. Duque, C.A., Ribeiro, M.V., Ramos, F.R. and Szczupak, J. (2005) Power quality event detection based on the principle divided to conquer and innovation concept. IEEE Transactions on Power Delivery, 20 (4), 2361–2369. 4. Gu, I.Y.H., Ernberg, N., Styvaktakis, E. and Bollen, M.J.H. (2004) A statistical-based sequential method for fast online detection of fault-induced voltage dips. IEEE Transactions on Power Delivery, 19 (2), 497–504. 5. Ece, D.G. and Gerek, O.N. (2004) Power quality event detection using joint 2-D-wavelet subspaces. IEEE Transactions on Instrumentation and Measurement, 53 (4), 1040–1046. 6. Abdel-Galil, T.K., El-Saadany, E.F. and Salama, M.M.A. (2003) Power quality event detection using Adaline. Electric Power Systems Research, 64 (2), 137–144. 7. Theodoridis, S. and Koutroumbas, K. (2008) Pattern Recognition, 4th edn, Academic Press. 8. Mendel, J.M. (1991) Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. Proceedings of the IEEE, 79 (3), 278–305. 9. Bollen, M.H.J. and Gu, I.Y.H. (2006) Signal Processing of Power Quality Disturbances, IEEE Press, Power Engineering Series. 10. Mohanty, S.R., Pradhan, A.K. and Routray, A. (2008) A cumulative sum-based fault detector for power system relaying application. IEEE Transactions on Power Delivery, 23 (1), 79–86. 11. Wiot, D. (2004) A new adaptive transient monitoring scheme for detection of power system events. IEEE Transactions on Power Delivery, 19 (1), 42–48. 12. Gerek, O.N. and Ece, D.G. (2006) Power-quality event analysis using higher order cumulants and quadratic classifiers. IEEE Transactions on Power Delivery, 21 (2), 883–889. 13. Ribeiro, M.V., Marques, C.A.G., Duque, C.A., Cerqueira, A.S. and Pereira, J.L.R. (2007) Detection of disturbances in voltage signals for power quality analysis using HOS. EURASIP Journal on Advances in Signal Processing, 1, 177. 14. Ferreira, D.D., Cerqueira, A.S., Duque, C.A. and Ribeiro, M.V. (2009) HOS-based method for classification of power quality disturbances. Electronics Letters, 45 (3), 183–185. 15. Giannakis, G.B. and Tsatsanis, M.K. (1990) Signal detection and classification using matched filtering and higher order statistics. IEEE Transactions on Acoustics, Speech and Signal Processing, 38 (7), 1284–1296. 16. Colonnese, S. and Scarano, G. (1999) Transient signal detection using higher order moments. IEEE Transactions on Signal Processing, 47 (2), 515–520. 17. Carvalho, J.R., Coury, D.V., Duque, C.A. and Jorge, D.C. (2011) Development of detection and classification stages for a new distance protection approach based on cumulants and neural networks. IEEE Power Energy Soc. Gen. Meet., 1–7. 18. Liu, Y., Steurer, M. and Ribeiro, P.F. (2009) Real-time simulation of time-varying harmonics. Chapter 18 in TimeVarying Waveform Distortions in Power Systems (Ribeiro, P.F., ed.), Wiley-IEEE Press. 19. Li, F., Li, Y. and Aggarwal, R.K. (2002) Combined wavelet transform and regression technique for secondary current compensation of current transformers. IEE Proceedings: Generation, Transmission and Distribution, 149 (4), 497–503. 20. Kang, Y.-C., Ok, S.-H. and Kang, S.-H. (2004) A CT saturation detection algorithm. IEEE Transactions on Power Delivery, 19 (1), 78–85. 21. Mitra, S.K. (2003) Digital Signal Processing: A Computer-Based Approach, McGraw Hill Publishing.

12 Wavelets Applied to Power Fluctuations 12.1 Introduction As discussed in detail in Section 9.6 of Chapter 9 the wavelet transform is a mathematical tool that analyzes time-varying signals. Unlike conventional frequency analysis methods, wavelets give information about the range of frequency components of a signal as a function of time. Wavelets applications are of value when there are large time-varying discrepancies of the frequencies to be analyzed. This is especially true when the signal to be analyzed is time-varying and non-periodic. In power systems, wavelets are usually applied as a diagnostic tool for power quality identification; as for Fourier series, the flexibility of the tool allows application to many scientific and technological areas. The advent of smart grids intensified the need for tools to support the balancing of loads both in generation and distribution. Under these circumstances, wavelets can be an enabling tool. In this chapter wavelets are applied to analyze power fluctuations in load and generation profiles. As the generation and load in AC power systems must always be balanced, wavelets can be applied to identify the required balancing capacity of the system by zooming into unforseen frequency variations. This is achieved by observing the time-varying nature of the dominant frequencies that are contained in the signals. To guarantee a stable supply of an AC power system, a continuous and precise balance between the electric power generation and load demands is needed. To achieve this equilibrium, the frequency must continuously remain within very strict limits. In general, there are three reasons for imbalance: (a) deviation of the load from its predicted values; (b) the loss of generation or network failures; and (c) the deviation of the generation from its expected values. With the advent of smart grids, defined as electricity networks that can intelligently integrate the behavior and actions of all users connected to it in order to efficiently deliver sustainable, economic and secure electricity supplies, the dynamics and complexity of an electricity grid has been substantially increased. Due to the introduction of renewable energy sources (RES) for a more distributed and renewable generation, unbalances can occur more Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo M
arcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

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frequently due to the increase in deviations between generation and load. Methods used to detect these variances are usually based on time-series analysis, a Fourier transform or fuzzy sets [1–3]. All deal with the uncertainty of generation and loads. Traditionally, the need for the balancing capacity of a network [4–10] is defined with the help of probabilistic methods. These techniques specify a certain value for expected losses or the non-delivery of the specific load (energy), which are all based on a certain availability of balancing capacities of that load and its system [11–12]. The increase of distributed and renewable generators requires the development of new techniques. These should be able to define the fluctuation and variability of generation instead of its unavailability. Furthermore, these techniques should be applied to determine and size the specific requirements for generation and electricity storage characteristics. In references [13–15], a method that uses a fast Fourier transform (FFT) to evaluate the needs of different classes of reserve capacity is established. One of the properties of FFT is the assumption of periodicity of a load; changes in the frequency domain over time will therefore not be reflected in the results. On the other hand, wavelet theory (WT) can be applied to identify fluctuations of nonperiodic generators. WT uses different window lengths to identify the different frequencies in a signal. Short windows are applied for high frequencies and long windows are used for low frequencies. The suggestion of using WT in power systems was first made during a meeting of the IEEE Working Group on Harmonic Modeling Simulation in 1993 and subsequently published in 1994 [16]. Wavelet transforms are used in different fields of expertise, including mathematics, quantum physics, seismic geology and electrical engineering [17]. In the field of electrical power engineering, wavelets are mainly applied to identify transients in power systems and for power system protection analysis [18]. Other research focuses on the application of wavelets in wind power forecasting [19,20]. In this chapter, we propose the application of wavelet theory to identify fluctuations in generation by RES. With the use of RES, the results can be used to define the needs for extra balancing capacity or for electricity storage concerns and provide the information needed to balance the power output of specific generators.

12.2 Basic Theory As detailed in Section 9.6, wavelet theory is based on small (oscillatory) transitorily finite waves are used for the construction or reconstruction of time-varying signals. A wavelet is a function that must satisfy two basic conditions: it must be oscillatory and it must decay rapidly to zero [21]. The assumption is that any signal P(t) can be represented by the superposition of scaled variations of a basic wavelet (the so-called mother wavelet). For this to happen, the mother wavelet is expanded in time (or space) and translated. A mother wavelet is defined as cðtÞ, where the original signal is a superposition such as [22]: PðtÞ ¼

XX k

PDWT ðn; kÞck;n ðtÞ

(12.1)

n

where P(t) is the original signal converted to wavelets, PDWT ðn; kÞ are the wavelet coefficients to be found using the transformed wavelets Wm,n and the wavelets ck;n ðtÞ are transformations

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of the mother wavelet cðtÞ using [22–24]: t  t  ct;s ðtÞ ¼ jsj1=2 c ; s

385

t; s 2 R; s 6¼ 0:

Alternatively, assuming s ¼ 2k and t ¼ 2k n results in [23]   t  2k n k=2 ; t; s 2 R; s 6¼ 0: cn;k ðtÞ ¼ ð2Þ c 2k

(12.2)

(12.3)

The wavelet coefficients PDWT ðn; kÞ can be found by taking the inner product of P(t) and cn;k ðtÞ: D E (12.4) PDWT ðn; kÞ ¼ PðtÞ; cðtÞk;n : The mother wavelet is dilated and translated. The dilation depends on k and is equal to 2k. The translation depends on both k and n and is equal to 2kn [25]. By dilating and translating the mother wavelet cðtÞ, the wavelet analysis allows for the breakdown of a signal according to scale (long window: low frequency; short window: high frequency). By changing the scaling factor k the mother wavelet cðtÞ is measured in time by a factor of 2k, where different frequency ranges can be selected. To some extent the scaling factor can be regarded as a filter in the frequency domain [25]. The relation between the frequency range Fk and the scale factor k can be found from Equation (12.5) as presented in Section 9.6.5, repeated below for convenience: 8  Fs Fs > > > ; < kþ1 k ; k ¼ 1; 2; . . . ; L  1 2 2  (12.5) Fk ¼  > Fs > > 0; ; k ¼ L; : 2k where L is the maximum scaling factor in the analysis of a signal and Fs is the sampling frequency of that signal P(t). Usually the sampling frequency Fs is predetermined by the dataset to be analyzed, where the lowest frequency to be analyzed follows from the purpose of analysis.

12.3 Application of Wavelets for Time-Varying Generation and Load Profiles This section explores the applications of WT in generation and load profiles. Through the analysis of relevant signals, the variations in the signals of different frequency ranges can be characterized. This section also discusses three practical case scenarios: a system load, the power output of a wind farm and the power fluctuations in a small 10 kV distribution network (acting as a microgrid).

12.3.1 Fluctuation Analyses with FFT The use of a FFT to determine and quantify variations in generation and load profiles was suggested in references [13,14]. By applying FFT, an impression of the variations in the

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Figure 12.1 System load during one week.

frequency domain can be acquired. This can be seen in Figures 12.1 and 12.2 for a load pattern of one week, where the FFT provides the components in the frequency domain. In Figure 12.2, different peaks in the frequency domain are indicated by gray marks corresponding to frequencies 1/(1 week) Hz and 1/(1 day) Hz. According to reference [15], large power variations occur over one-quarter of this time frame. By integrating the frequency components over different frequency ranges, the required amount of balancing capacity to balance the corresponding power variations can be determined. As FFT transforms time series into the frequency domains, any time information in the original signal is lost as periodicity is assumed in FFT. Short and rare events therefore average out if a long time series is analyzed. Shortening the time series by applying the short-time Fourier transform (STFT) with suitable time windows partially solves this issue. However, it leads to a reduction of its resolution in the frequency domain because of fixed windows in time.

12.3.2 Methodology For a signal P(t) the factors PDWT ðn; kÞ indicate the presence of transformed wavelet components at different scaling factors in the original signal at different time shifts. For all signals mentioned here, the signal is normalized using Equation (12.6) to be within [0,1] and the Meyer wavelet is applied for the wavelet transform: Pnorm ðtÞ ¼

PðtÞ  min½PðtÞ : max½PðtÞ  min½PðtÞ

(12.6)

The maximum scaling factor L for the wavelet decomposition follows from the lowest frequency to be analyzed and its sampling frequency Fs, which can be found from Equation (12.5). Any prior knowledge about fluctuation patterns that may perhaps be present in the signal should be considered in order to correctly define the required maximum scaling

Figure 12.2 Frequency components of a system load.

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factor L and the desired sampling period. As a next step, the signal is decomposed and the components PDWT ðn; kÞ are those of interest. From all wavelet components PDWT ðn; kÞ that represent a certain frequency range, a number of components are selected that contribute most to the original signal P(t) according to Equation (12.1). The selection of these components is based on the RMS value of each individual factor. The components considered to contribute most are those with the highest RMS values, as derived from Equation (12.7). As such, the most relevant scaling factors PDWT ðn; kÞ and the original signal can be approximated by a synthetic signal. This is based on the superposition of the most relevant scaling factors as, indicated in Equation (12.1). These relevant scaling factors can also be investigated, as each component holds information about the original signal within a certain frequency bandwidth. When the individual relevant scaling factors are determined, they reveal additional information about fluctuations in the original signal at specific time periods. From this point onwards the wavelet coefficients PDWT ðn; kÞ are referred to as An;k for simplicity. We therefore have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u L u 1 X (12.7) A2 : Ak;RMS ¼ t L þ 1 n¼0 k;n

12.3.3 Load Fluctuations To illustrate the application of wavelets in power systems, a system load was analyzed. The measurement of power data from a particular week in the year 2008 was used. A sampling period of 4 s was used, acquired from the Dutch utility. The data represent the aggregated production data from all large (i.e. > 60 MW) power plants in the Netherlands and is assumed to be equal to the aggregation of loads and losses countrywide. These data were first normalized to be within the domain [0,1] using Equation (12.6). After normalizing the data, a maximum scaling factor M of 15 was chosen using Equation (12.5) to find the components with the highest fundamental period of 1/FM of 3 days. The three components An,k with the highest RMS values were identified from Equation (12.7). Both the original and the synthetic load profiles are displayed in Figure 12.3. A selection of the most relevant scaling factors as well as the moving average are displayed in Figure 12.4. A number of conclusions can be drawn from the wavelet analysis of this system load. The periods that are most relevant in the original signal are the daily and the half-daily fluctuations. It can be seen that specific daily patterns are less present during the weekend

Figure 12.3 Original and synthesized load profiles based on relevant scaling factors of Am,n.

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Figure 12.4 Most relevant scaling factors for a load profile during a week.

(days 6 and 7) than during weekdays (days 1 to 5). The weekly pattern can also be recognized by the moving average. It is interesting to note that these conclusions could not be drawn using conventional FFT analysis. In addition to analysis of the system load described above, the active power load of one of the phases of a distribution transformer was monitored for a number of days. There are approximately 50 households connected to this phase. The sampling period of the measurements was of 10 min. The measured signal contained more high-frequency components than the system load due to the significantly lower aggregation level. To analyze these faster fluctuations, the data were first normalized to be within the domain [0,1] using Equation (12.6) and were subsequently analyzed through wavelets (FFT) with a maximum scaling factor of 4 to find components with a maximum fundamental period (1/FM) of 3 hours. The value of 3 hours was selected and their RMS values were chosen, based on experience. The three most relevant components of Am,n and AM,n were selected to reconstruct the signal. The original and reconstructed synthetic signals are given in Figure 12.5. Both AM,n and the most relevant components were used to recreate the synthetic signal in Figure 12.5 and are displayed in Figure 12.6. From A1 and A2, the half-hourly and hourly fluctuating components in the load are clearly seen. It can be observed that these components are less evident during night-time (when the load is low). Again, this is a conclusion that cannot be drawn using conventional FFT analysis.

Figure 12.5 Original and synthesized load profiles during three days based on relevant scaling factors.

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Figure 12.6 Most relevant scaling factors for a load profile during three subsequent days.

12.3.4 Wind Farm Generation Fluctuations For this example a 25.5 MW onshore wind farm close to the city of Rotterdam in the Netherlands was selected. The wind farm consists of 17 identical wind turbines of 1.5 MW each. The turbines were connected to the 23 kV network via two connections with 8 and 9 turbines each. For both connections the aggregate current was measured during the month of May 2009. With the voltage assumed as constant, the aggregate power was calculated. The data, sampled at a period of 1 min, were available from the responsible grid operator and were normalized using Equation (12.6). A sampling period of 1 min is assumed to be short enough to recognize fluctuations in wind power for load balancing [3]. As such, wavelet analysis was performed with a maximum scaling factor M of 14 with a sampling period of 1 min; this leads to the lowest traceable period (1/FM) of 3 days. Three components Am,n with the highest RMS value Ak,RMS were identified using Equation (12.7) and considered as the most relevant. By summing only these most relevant scaling factors and AM,n, the original signal was synthesized. Both the original and the synthesized signals are displayed in Figure 12.7. AM,n and the three most relevant components (A11,n, A12,n and A13,n) are displayed in Figure 12.8. From Figure 12.8 it can be concluded that, based on the RMS values, the three components that contribute most to the original signal have fundamental periods of 1.4–2.8, 2.8–5.7 and 5.7–11.4 days. It can therefore also be concluded that the main fluctuations of this wind farm occur on a daily scale. This also means that no electricity storage devices could be properly sized to handle these fluctuations in order to balance the power output of this wind farm.

Figure 12.7 Original and synthesized wind farm generation profiles.

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Figure 12.8 Most relevant wavelet components for a wind farm time-series.

12.3.5 Smart Microgrid In this section, wavelet methodology is applied to to a smart microgrid to analyze power fluctuations. The microgrid in this study consists of a number of loads connected to a small 10 kV radial distribution network. A 2 MW wind turbine (W) and a conventional generator (G) are also connected to the network. The microgrid is to be operated in island mode, so the conventional generator needs to be able to deal with the aggregated fluctuations of load and wind turbine. The network topology is illustrated in Figure 12.9. The power which is a conventional generator G needs to produce PG(t) is given by: PG ðtÞ ¼

X

Pload ðtÞ þ

X

Plosses ðtÞ  Pwind ðtÞ

(12.8)

where Pload(t) is the power of each load, Plosses(t) are the network losses and Pwind(t) is the power generated by the wind turbine.The load profile in the network correlates to a month of national production data from the Dutch Tranmission Systems Operator (TSO) and is scaled to have a maximum value of 5 MW. The aggregated load was divided equally over the 8 loads in the network with a power factor of 1. The generation from the wind turbine was obtained by

Figure 12.9 Structure of the smart microgrid.

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Figure 12.10 Original and synthesized power profiles of generator G during one month.

taking one month of aggregated wind data as described in Section 3.4. The wind power is scaled to have a maximum value of 2 MW and is also assumed to have a unit power factor. A month was simulated for each minute using a load flow simulation of the network in order to find the network losses Plosses(t) and to determine the power PG(t) to be generated by the conventional generator G in order to balance the power in the network. After completing the load flow simulations and normalizing the power profile using Equation (12.6), the power to be generated by generator G was investigated using the wavelet methodology in order to determine the characteristic fluctuations to be managed by generator G. The load flow simulations were performed for each minute during a month; the sampling period 1/Fs of PG(t) is 1 min. Using Equation (12.4,) the maximum scaling factor was decided to be 14 so that a maximum traceable period 1/FM of 3 days occurs. As in the previous sections, the three components of Am,n with the highest RMS values were chosen to be most relevant. Using these components and only AM,n the power profile for generator G can be summarized. The original and reconstructed power profiles are given in Figure 12.10. It can be concluded from this figure that the components in the signal are equal to or smaller than 1 day and contribute most to the original signal. Based on their RMS values calculated from Equation (12.7), the three most relevant wavelet components are A10,n, A11,n and A12,n. These, as well as the moving average AM,n are given in Figure 12.11.

Figure 12.11 Most relevant wavelet components for the microgrid simulation during one month.

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From Figure 12.11 it can be concluded that the component with the daily profile has the largest share in the original signal. Generator G must therefore be able to ramp up and down within this period. If Generator G is able to follow the fluctuations within the 1 day period it will be able to produce the synthetic profile as shown in Figure 12.10. To provide the difference in power between the synthesized and the original profiles, an electricity storage device can be added to the microgrid under study. As shown in this example, wavelet analysis can be used to characterize both the generator G and the required electricity storage device.

12.4 Conclusions The increasing complexity of the electricity grid requires new signal processing techniques which can be used to properly and effectively analyze and diagnose the system conditions. Wavelet analysis is used more and more in power systems applications, and it is proposed that wavelet analysis is applied to determine fluctuation patterns in generation and load profiles. This is achieved by the filtering of its wavelet components based on their RMS values, and it is possible to identify the most relevant scaling factors from the analysis. Three different case studies – a load profile, a wind farm and the operation of a microgrid – were analyzed in the context of a smart grid, demonstrating that the wavelet method can be applied to identify the most effective time range for assessment of both generation and load fluctuations. While conventional FFT algorithms only give the information as a function of frequency, STFT gives information in terms of both time and frequency. Unfortunately, it has disadvantages concerning the frequency resolution. However, since wavelet analysis yields the present frequency components as a function of time by using variable windows in the time domain, the issue of frequency resolution can be resolved. The application of wavelet analysis as described in this chapter may prove useful both for the characterization of possible electricity storage devices and the determination of the required balancing capacities. It can also improve the bids of energy companies in energy markets by having specific information on the characteristic fluctuations of its renewable generation, providing them with the ability to counteract these by using conventional generation and electricity storage. Experience in selecting the number of scaling factors and main frequency ranges to be identified is important for accurate results. Furthermore, any prior knowledge concerning characteristic fluctuations present in the signal should be considered to draw valid conclusions based on the results from wavelet analysis. In a future of smart grids, the application of wavelets to analyze generation and load signals may prove very useful for agents responsible for the operation and control of the network. These agents could use wavelet analysis to improve their performance and to investigate price signals [26].

References 1. Holttinen, H. (2005) Impact of hourly wind power variations on the system operation in the Nordic countries. Wind Energy, 8, 197–218. 2. Papaefthymiou, G., Schavemaker, P.H., van derSluis, L., Kling, L., Kurowicka, D. and Cooke, R.M. (2006) Integration of stochastic generation in power systems. Electric Power Energy Systems, 28, 655–667. 3. Bansal, R.C. (2003) Bibliography on the fuzzy set theory applications in power systems (1994–2001). IEEE Transactions on Power Systems, 18, 1291–1299. 4. Doherty, R. and O’Malley, M. (2005) A new approach to quantify reserve demand in systems with significant installed wind capacity. IEEE Transactions on Power Systems, 20, 587–595.

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5. Allen, E.H. and Ilic, M.D. (2000) Reserve markets for power systems reliability. IEEE Transactions on Power Systems, 15, 228–233. 6. Havel, P., Hor
acek, P., Cern
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ık, J. (2008) Optimal planning of ancillary services for reliable power balance control. IEEE Transactions on Power Systems, 23, 1375–1382. 7. Amjady, N. and Keynia, F. (2010) A new spinning reserve requirement forecast method for deregulated electricity markets. Applied Energy, 87, 1870–1879. 8. Booth, R.R. (1972) Power system simulation model based on probability analysis. IEEE Transactions on Power Systems, 91, 62–69. 9. Fotuhi-Firuzabad, M., Bilinton, R. and Aboreshaid, S. (1996) Spinning reserve allocation using response health analysis. Generation, Transmission and Distribution, IEE Proceedings, 143, 337–343. 10. Arce, J.R., Ilic, M.D. and Garc
es, F.F. (2001) Managing short-term reliability related risks. In Proceedings of Power Engineering Society Summer Meeting, Vancouver, British Columbia, Canada, July 15–19, pp. 516–522. 11. Billinton, R. and Allan, R.N. (1984) Operating reserve. In Reliability Evaluation of Power Systems, Pitman Publishing Limited, pp. 139–171. 12. Mazumdar, M. and Bloom, J.A. (1996) Derivation of the Balerieux formula of expected production costs based on chronological load considerations. Electric Power Energy Systems, 18, 33–36. 13. Alvarado, F.L. (2002) Spectral analysis of energy-constrained reserves. In Proceedings of 35th Hawaii International Conference on Systems Science, pp. 749–756. 14. Frunt, J., Kling, W.L. and Myrzik, J.M.A. (2009) Classification of reserve capacity in future power systems. In Proceedings of 6th International Conference on European Energy Market, Leuven, Belgium, May 27–29. 15. Frunt, J., Kling, W.L. and van den Bosch, P.P.J. (2010) Classification and quantification of reserve requirements for balancing. Electric Power Systems Research, 80, 1528–1534. 16. Ribeiro, P.F. (1994) Wavelet transform: an advanced tool for analyzing non-stationary distortions in power systems. ICHPS VI/94 Italy. 17. Graps, A. (1995) An introduction to wavelets. IEEE Computing in Science and Engineering, 2, 1–18. 18. Galli, A.W., Heydt, G.T. and Ribeiro, P.F. (1996) Exploring the power of wavelet analysis. IEEE Computer Applications in Power, 9, 37–41. 19. Dong, L., Wang, L., Liao, X., Gao, Y., Li, Y. and Wang, Z. (2009) Prediction of wind power generation based on time series wavelet transform for large wind farm. In Proceedings of 3rd International Conference on Power Electronric Systems and Applications, Hong Kong, May 20–22, pp. 1–4. 20. Lei, C. and Ran, L. (2008) Short-term wind speed forecasting model for wind farm based on wavelet decomposition. In Proceedings of 3rd International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjuing, China, April 6–9, pp. 2525–2529. 21. Masson, P.J., Silveira, P.M., Duque, C. and Ribeiro, P.F. (2008) Fourier series: Visualizing Joseph Fourier’s imaginative discovery via fea and time-frequency decomposition. 13th International Conference on Harmonics and Quality of Power, ICHQP. 28 Sept–1 Oct 2008, 1–5. 22. Lee, C.H., Wang, Y.J. and Huang, W.L. (2000) A literature survey of wavelets in power engineering applications. Proceedings of the National Science Council, Republic of China(A), 24, 249–258. 23. Lebedeva, E.A. and Protasov, V.Y. (2008) Meyer wavelets with least uncertainty constant. Mathematical Notes, 84, 680–687. 24. Wilkinson, W.A. and Cox, M.D. (1996) Discrete wavelet analysis of power system transients. IEEE Transactions on Power Systems, 11, 2038–2044. 25. Xu, J., Senroy, N., Suryanarayanan, S., and Ribeiro, P.F. (2006) Some techniques for the analysis and visualization of time-varying waveform distortions. In Proceedings of 38th North American Power Systems Symposium, pp. 257–261. 26. Frunt, J., Kling, W.L. and Ribeiro, P.F. (2011) Wavelet decomposition for power balancing analysis. IEEE Transactions on Power Delivery, 26 (3), 1608–1614.

13 Time-Varying Harmonic and Asymmetry Unbalances 13.1 Introduction Traditional harmonic distortion analysis assumes balanced and steady-state conditions. However, in the real world this is far from the physical reality and experience of engineers that deal with unbalanced and time-varying components. In theory, certain harmonic components are associated with specific sequences such as positive, negative or zero. However, in real life harmonic distortions can cause a gamut of sequences for every possible time-varying frequency. The readings of these conditions complicate any analysis and need to be properly modeled. The time-varying nature of these unbalances at higher frequencies has not yet been studied in detail. This chapter introduces these unbalances with the use of the sliding-window recursive Fourier transform (SWRFT), with which the frequencies of the time-varying harmonic components are computed, calculated and then plotted for their positive, negative and zero sequences and parameters. The recorded signals used are from a real system. These time-varying parameters are used to investigate the nature of its transient phenomena and to provide information for protection and control applications. In the past, harmonic analyses was performed taking into account only a few harmonicproducing devices such as power electronic (PE) inverters and converters, HVDC, SVC, and so on. As their supply systems are well balanced and symmetrical, most studies are carried out in the steady state and based on positive sequence network representations. However, as more and more harmonic-producing loads are connected to the grid, this ‘steady-state’ condition has changed significantly [1–3]. Due to the randomness of the loads and the dynamics of three-phase systems, two kinds of phenomena have become common: (1) unbalanced harmonics and (2) continuous variability of harmonics [1]. New concepts have been published, including suggestions that combine probabilistic and spectral methods (also referred as evolutionary spectrum) [4]. However, most of the techniques applied rely on Fourier transform methods that implicitly assume stationary conditions and a linearity of components. Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo M
arcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

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Utilities and industries have therefore focused their attention on methods of analysis with the ability to provide correct assessments of time-varying harmonic distortions. This issue has become crucial for control, protection, supervision and the proper diagnosis of problems. Harmonic distortion studies in electric systems when significant variations are observed due to load or system variations have been performed using a probabilistic approach and assuming that the harmonic components vary slowly enough to affect the accuracy of the analytical and monitoring process. Another relevant issue is related to the unbalanced harmonics in three-phase systems. Finding the sequence harmonic components in balanced systems is a well-studied subject. However, when the supply voltage and loads are unbalanced, there will be large deviations from the traditional pattern. Under these circumstances, symmetrical component theory cannot be applied for accurate identification of the sequence components used for power quality assessment. The same occurs for other important applications such as protection and control of power systems [4]. Publications on time-varying harmonic unbalances is limited [5–8]. In order to analyze distorted waveforms that vary continuously in the time domain the concept of time-varying waveform distortions is discussed [9]. A sliding-window discrete Fourier transform (SWDFT) can be a useful tool as it provides the capacity to analyze and visualize voltage and current waveforms and graphically illustrate the time-varying harmonic components. This chapter describes and illustrates how this approach can be used to determine harmonic sequence components with some practical examples. Some useful parameters in the time domain for each of these situations are discussed, knowledge of which can lead to a better understanding of unbalances and the asymmetries related to each of the harmonic frequencies.

13.2 Sequence Component Computation Computation of the symmetrical components is depicted in Figure 13.1, based on the slidingwindow recursive DFT SWRDFT presented in Chapter 9. For each phase the signal is split into its harmonics of order h using the SWDFT architecture. The magnitude and angle of each harmonic is subsequently used for the computation of its symmetric components. The result

Figure 13.1 Symmetrical components computation using SWDFT.

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Figure 13.2 Architecture to reconstruct the components in the time domain.  0 are vectors of positive (Sþ hk ), negative (Shk ) and zero (Shk ) sequences representing the phasor of harmonic h at instant k. The notation S can be used for voltage or current. The symmetrical components are computed according to:

2 3 1 1 S0hk 6 þ 7 16 4 Shk 5 ¼ 4 1 a 3 1 a2 S hk 2

32 3 SAhk 1 76 7 a2 54 SBhk 5 a

(13.1)

SChk

 0 where Sþ hk , Shk and Shk are the sequence components, h is the harmonic order and a can be represented by the phasor:

a ¼ 1ff120

(13.2)

a2 ¼ 1ff 240 :

(13.3)

The symmetrical components can then be reconstructed in the time domain by using the architecture shown in Figure 13.2. The quadrature term is obtained from the symmetrical component vector. Then, using Equation (13.2) of the Fourier theory, this term is used to obtain the component in its own time domain.

13.3 Time-Varying Unbalance and Harmonic Frequencies As defined by the European standards, the degree of unbalance or the voltage unbalance factor (VUF) is the ratio of the negative sequence voltage to the positive sequence voltage, calculated: %VUF ¼

V  100 Vþ

(13.4)

where Vþ and V are the positive and negative sequence voltages, respectively. This parameter is used at several harmonic frequencies for a better understanding of unbalances in nonsinusoidal and time-varying situations:

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%VUFh ðkÞ ¼

8  V hðk Þ > >  100; for positive sequence; > > < Vþ hðk Þ > Vþ > hðk Þ > > : V   100; for negative sequence:

(13.5)

hðk Þ

The current unbalance factor (IUF) for non-sinusoidal and time-varying conditions is defined:

%IUFh ðkÞ ¼

8  I hðkÞ > > > þ  100; > < I hðkÞ

if positive sequence;

> Iþ > hðkÞ > > : I   100;

if negative sequence:

(13.6)

hðkÞ

13.4 Computation of Time-Varying Unbalances and Asymmetries at Harmonic Frequencies A simulated signal was used to evaluate the proposed methodology. The main concern is to evaluate the accuracy of the signal decomposition of its harmonics and then to compute the symmetrical components and unbalances at these different harmonic frequencies. Furthermore, the three-phase signal simulated is time-varying. The scenario below illustrates the main use of this methodology. A signal was generated according to following equations: pffiffiffi 2M A1 sinðwt þ uA1 Þ pffiffiffi V B1 ¼ 2M B1 sinðwt þ 1B1 Þ pffiffiffi V C1 ¼ 2M C1 sinðwt þ aC1 Þ pffiffiffi V A5 ¼ 2M A5 sinð5 wt þ uA5 Þ pffiffiffi V B5 ¼ 2M B5 B sinð5 wt þ 1B5 Þ pffiffiffi V C5 ¼ 2M C5 sinð5 wt þ aC5 Þ pffiffiffi V A7 ¼ 2M A7 sinð7 wt þ uA7 Þ pffiffiffi V B7 ¼ 2M B7 sinð7 wt þ 1B7 Þ pffiffiffi V C7 ¼ 2M C7 sinð7 wt þ mC7 Þ V A1 ¼

(13.7) (13.8) (13.9) (13.10) (13.11) (13.12) (13.13) (13.14) (13.15)

where M A1 ¼ M B1 ¼ M C1 ¼ 1; M A5 ¼ M C5 ¼ 0:3; M A7 ¼ M B7 ¼ M C7 ¼ 0:2; uA1 ¼ uA5 ¼ uA7 ¼ 0; 1B1 ¼ 1B5 ¼ aC5 ¼ 120 ; aA7 ¼ aC7 ¼ 120 and A, B and C are different phases.

Time-Varying Harmonic and Asymmetry Unbalances

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Figure 13.3 Unbalance computation at 5th harmonic for the simulated signal.

The unbalance was produced by changing the magnitude of phase B at the 5th harmonic and the phase angle of the 7th harmonic according to M B5 ¼ 0:3 et

(13.16)

aB7 ¼ 120 et :

(13.17)

The signal is the sum of the three components: V A ¼ V A1 þ V A5 þ V A7

(13.18)

V B ¼ V B1 þ V B5 þ V B7

(13.19)

V C ¼ V C1 þ V C5 þ V C7 :

(13.20)

Figure 13.3 shows the 5th harmonic decomposed using the SWDFT and its unbalance is analyzed using the methodology above. Equation (13.19) describes the magnitude of phase B at the 5th harmonic and it decreases exponentially. Figure 13.3 shows two points in the time domain to better illustrate the unbalance calculations; two points of the phase angle at 7th harmonic changes exponentially are shown in Figure 13.4.

Figure 13.4 Unbalance computation at 7th harmonic for the simulated signal.

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Figure 13.5 Error measurement by using the proposed methodology to calculate the unbalance: (a) 5th harmonic and (b) 7th harmonic.

The signal was decomposed according to the methodology above, and the unbalance was calculated using the symmetrical components from the signal VA, VB and VC. The error was computed as the difference between the calculated theoretical value using the equations and the obtained results, and is depicted in Figure 13.5. As can be seen, the error is less significant if the unbalance is higher. Figure 13.6 shows the sum of all zero-sequence components. The result is compared to the theoretical value, and the error shown in Figure 13.7.

Figure 13.6 Zero-sequence component in time domain.

Figure 13.7 Error computing the zero-sequence component in time domain.

Time-Varying Harmonic and Asymmetry Unbalances

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Figure 13.8 Time varying phasor of: (a) fundamental, (b) 5th harmonic and (c) 7th harmonic.

Figure 13.8 shows the phasors as a function of time. The fundamental is balanced and can easily be seen. The 5th harmonic presents magnitude variations in one of the phases and the 7th harmonic has angle variations. This graphs is useful to observe the time-varying phasors and is used in the examples described in the following section.

13.5 Examples Examples of the methodology applied to real signals under time-varying conditions are described in the following. Sample conditions include a three-phase inrush current, a threephase voltage during a sag and the unbalance of an electronic converter system. The computations include both the time-varying unbalance and the asymmetry during these transients. These parameters can also be applied to support control, protection, supervision and accurate diagnosis of possible network problems.

13.5.1 Inrush Current For illustrative purposes a real three-phase inrush current was used and is shown in Figure 13.9. From the figure it can be seen that the transient behavior remains after the

Figure 13.9 Three-phase inrush current during a transformer energization.

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Figure 13.10 Zoom view of a three-phase inrush current during a transformer energization.

signal sample is used. The graph represents 4 s in time, so that the shape of the signal can be clearly seen and the time-varying behavior highlighted. Details can be seen in Figure 13.10. The signal was decomposed using SWDFT. The harmonics were reconstructed and the sum of all components was computed in order to compare it to the original signal. The result of phase A is shown in Figure 13.11. A small deviation can be seen during the fast transient, but this can be disregarded. The components are subsequently used to calculate the symmetric components for each harmonic and the related unbalances and asymmetries as a function of time. Figure 13.12 shows the sum of all zero-sequence components, representing the actual neutral to ground current which is subsequently used for the proper diagnoses of unsafe conditions. Figure 13.12b shows the asymmetry ratio (I0/I1) for the fundamental frequency. These parameters clarify the nature of the phenomena and can be used to set improved protection systems. The unbalance is verified by calculating the negative-sequence component over the positive sequence, according to Equation (13.13). The results for the fundamental, second and fifth harmonics are shown in Figures 13.13–13.15 respectively. A high degree of unbalance can be

Figure 13.11 Reconstructed signal compared to the original signal.

Time-Varying Harmonic and Asymmetry Unbalances

Figure 13.12 (a) Zero-sequence component in time domain and (b) asymmetry ratio.

Figure 13.13 Time-varying current unbalance for fundamental component.

Figure 13.14 Time-varying current unbalance for 2nd harmonic.

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Power Systems Signal Processing for Smart Grids

404

Figure 13.15 Time-varying current unbalance for 5th harmonic.

observed for the fundamental and its harmonic frequencies, with higher values during the initial transient. Such parameters can be used for better performance of control and protection systems during transient time. The time-varying phasors of the 5th harmonic are shown in Figure 13.16, which correspond to the case of a high unbalance ratio of around 50% (a) and an unbalance ratio of around 10% (b).

13.5.2 Voltage Sag A voltage sag was used as another example to understand the nature of unbalances in timevarying signals. A real signal in an aluminum sheet facility of 88 kV was used. Figure 13.17 depicts the three-phase voltage that has harmonics and experiences a sag. This signal was decomposed and its unbalances were calculated for all frequencies using the methodology described. Three frequency components are shown as a function of time: Figure 13.18 shows the unbalances at the fundamental frequency; Figure 13.19 at the 5th

Figure 13.16 Time-varying phasors of 2nd harmonic around (a) 50% unbalanced and (b) 5% unbalanced.

Time-Varying Harmonic and Asymmetry Unbalances

405

Figure 13.17 Real signal of voltage during a sag.

harmonic; and Figure 13.20 at the 7th harmonic. As expected, analysis of the sag decomposition reveals that the fundamental frequency initially has a greater unbalance, after which it slowly recovers. However, the 5th and 7th harmonics demonstrate completely different behavior during the sag with a variation range from 10% to 100%. For instance, the 5th harmonic is almost balanced at 0.6 s, although at approximately 1.1 s the unbalance is >60%. It is important to note that the sequence changes during the sag. Initially the 5th harmonic has a negative sequence, but during the sag it becomes positive and then again negative after the event. The same analysis can be performed for the 7th harmonic: the phase sequence is

Figure 13.18 Fundamental component decomposed by SWRDFT and unbalance computation during a sag transient.

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Figure 13.19 Three-phase 5th harmonic decomposed by SWRDFT and the unbalance computation during a sag transient.

switched twice during the sag and is changed again when the unbalance of the three-phase signal achieves 100%. The phasor behavior of the 5th and 7th harmonics is depicted in Figure 13.21. As can be seen, the magnitude of 5th harmonic phases A and B increase and phase C decreases. The phase angles of phases A and B demonstrate large changes during this period. For the 7th

Figure 13.20 Three-phase 7th harmonic decomposed by SWRDFT and unbalance computation during a sag transient.

Time-Varying Harmonic and Asymmetry Unbalances

407

Figure 13.21 Three-phase voltage symmetrical component for (a) 5th harmonic and (b) 7th harmonic.

harmonic, notice that at 0.6 s the signal is almost balanced but, due to variations of the phase angles and magnitudes, the unbalance increases.

13.5.3 Unbalance in Converters The method proposed in Sections 13.2–13.4 is also used to observe the unbalances during a voltage sag when converters are connected to the grid. The schematic of the three-phase system is shown in Figure 13.22. The first trial was conducted with a three-phase short circuit of 1.5 s. The waveforms of the three-phase voltage and current are shown in Figures 13.23 and 13.24, respectively.

Figure 13.22 Schematic diagram of the experiment with converters.

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Figure 13.23 Three-phase voltage during three-phase short circuit and zoom view of the beginning of the event.

Figure 13.24 Three-phase current during three-phase short circuit and zoom view of the beginning of the event.

The behaviors of unbalances at harmonic frequencies are estimated using the method (Sections 13.2–13.4) and depicted in Figure 13.25. The fundamental current magnitude during the sag is higher but balanced. However, the harmonics of each phase have different responses; for instance, the 5th and 7th (Figure 13.26) harmonics have the unbalance ratio increased by about 20%. The high values of imbalance are achieved at the beginning and end of the short circuit. The second test was conducted with a short circuit between two phases; Figures 13.27 and 13.28 show the voltage and current for this situation. As the short circuit has a direct impact on two phases, the fundamental presents high current unbalance deviation during the

Figure 13.25 Current unbalance during voltage sag.

Time-Varying Harmonic and Asymmetry Unbalances

Figure 13.26 Unbalances at 7th harmonic during a three-phase short circuit.

Figure 13.27 Voltage during sag for phase-to-phase short circuit.

Figure 13.28 Current during voltage sag for phase-to-phase short circuit.

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410

Figure 13.29 Unbalance for phase-to-phase short circuit.

Figure 13.30 Unbalances at 5th harmonic during phase-to-phase short circuit.

event. However, the current harmonics unbalances experience particular variations during the short circuit similar to that of the first test, as can be seen in Figure 13.29. Although the phaseto-phase short circuit produces 50% of the unbalance for the fundamental component, the 5th harmonic has the same unbalance rate before and during the event as shown in Figure 13.30. The 7th harmonic has high unbalance variations due to the low magnitude of this component that experiences greater interference from harmonics and limitations of the filter.

13.6 Conclusions This chapter has described a method to assist in the observation and evaluation of unbalances and asymmetries in power systems based on a time-varying decomposition of harmonic frequencies. For this, the signal processing decomposition method SWDFT is used. The timevarying harmonics and their positive-, negative- and zero-sequence components are calculated for each frequency. A simulated signal is used to verify the accuracy of this methodology. Real currents and voltage signals are used in order to understand the unbalances during transients. These derived parameters can be applied to support control, supervision and proper diagnosis of possible power quality and network events.

Time-Varying Harmonic and Asymmetry Unbalances

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To the knowledge of the authors, this is the first time that such a time-varying decomposition has applied in this context and explicitly illustrated. This method may become a very useful tool in aiding the engineer to better understand the harmonic distortions of certain phenomena under time-varying and unbalanced system conditions.

References 1. Bonner, A., Grebe, T., Gunther, E., Hopkins, L., Marz, M.B., Mahseredjian, J., Miller, N.W. et al. (1996) IEEE task force on harmonics modeling and simulation: modeling and simulation of the propagation of harmonics in electric power networks. Part i: concepts, models and simulation techniques. IEEE Transactions on Power Delivery, 11 (1), 452–465. 2. Baghzouz, Y., Burch, R.F., Capasso, A., Cavallini, A., Emanuel, A.E., Halpin, M., Langella, R. et al. (Probabilistic Aspects Task Force of Harmonics Working Group) (2002) Time-varying harmonics. Part ii: harmonic summation and propagation. IEEE Transactions on Power Delivery, (1), 279–285. 3. Morrison, R.E. (1984) Probabilistic representation of harmonic currents in AC traction systems. Proceedings of IEEE, Part B, 131 (5), 181–189. 4. Ribeiro, P.F. (2003) A novel way for dealing with time-varying harmonic distortions: the concept of evolutionary spectra. Power Engineering Society General Meeting, 13–17 July, 2, pp. 1153. 5. Marei, M.I., Saadany, E.F.E. and Salama, M.M.A. (2004) A processing unit for symmetrical components and harmonics estimation based on a new adaptive linear combiner structure. IEEE Transactions on Power Delivery, 19 (3), 1245–1252. 6. Henao, H., Assaf, T. and Capolino, G.A. (2003) The discrete Fourier transform for computation of symmetrical components harmonics. IEEE Bologna PowerTech Conference, June 23–26, Bologna, Italy. 7. Silveira, P.M., Duque, C.A., Baldwin, T. and Ribeiro, P.F. (2008) Time-varying power harmonic decomposition using sliding-window DFT. IEEE International Conference on Harmonics and Quality of Power, Wollongong. 8. Hartley, R. and Welles, K. (1990) Recursive computation of the Fourier transform. IEEE International Symposium on Circuits and Systems, 3, 1792–1795. 9. Jouanne, A. and Banerjee, B. (2001) Assessment of voltage unbalance. IEEE Transactions on Power Delivery, 16 (4), October.

Index accuracy classes 62, 67–69 AC decaying of short circuit 22 acquisition, analysis, detection, extraction AC saturation 48 ADC 6 resolution 77, 80 aggregate power 389 aggregated wind data 391 Aliasing error 82 analog conditioning 6 analog to digital conversion 75 analyzers 5 anti-aliasing filter 82 artificial intelligence 7 asymmetries 4, 6 asymmetry unbalances 395 back-propagation algorithm 337 back-to-back switching 18, 20 balanced and steady-state 395 Bayes Deciscion Theory 323 Butterworth filter 83, 84 capacitor banks 1 classification of the waveforms classifier LMS algorithm 335 sum error square 334 collective RMS 6 communication 3 confusion area 328

5

5

consumers 1, 4, 8 control information technologies 2 control/supervision applications 78 cubic B-Spline Interpolation 180 cummulants 325 current during voltage sag for phase-to-phase short circuit 409 current unbalance during voltage sag 408 current unbalance factor (IUF) 398 curve fitting techniques 6 cyber-physical smart grid of the future 6, 9 DC saturation 49 decimator 174 decomposition 4, 5 denoising 6 deregulation 2 detection on the Bayesian Framework 356 detection correlator 360 third difference 378 DFT 6, 7 diagnostic tool 383 diagnostics 1 differential equation 6 digital fault record (DFR) 59 recorders 1 digital relays 1 dilated 385 dimensional complexity 3 distorted waveforms 396

Power Systems Signal Processing for Smart Grids, First Edition. Paulo Fernando Ribeiro, Carlos Augusto Duque, Paulo Marcio da Silveira and Augusto Santiago Cerqueira. Ó 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd. Companion Website: http://www.wiley.com/go/signal_processing/

Index

414

distribution 1, 2, 6, 9 Down- sampler 168 frequency domain analysis dynamic range 77

histogram 6 home area networks (HANs) HVDC, SVC 395

170

effective value (RMS) 324 electricity storage 384, 389, 392 energy 2, 6 equipment diagnostics 1 performance 4 euclidian distance 329 evolutionary spectrum 395 factor of increase of voltage 27 false alarm 357 Faraday effect 73, 74 fast transients 6 fault location 6 ferroresonance suppression circuit 66 FFT 6, 7, 384–386, 388, 392 filter 410 banks 6, 7 fitting techniques 6 flicker 27 fluctuation analyses 385 fluctuations of nonperiodic generators 384 Fourier series 5, 6, 383 transform 6, 395, 396 fractional sampling rate alteration 175, 176 Ferranti effect 12 frequency domain 384–386 spectrum 6 fundamental component 403, 405, 410 generalized likelihood ratio test (GLRT) generation 1, 8, 9 Gibbs phenomenon 30, 31 grid of the future 1, 2, 6, 9 hardware in the loop (HIL) 375 harmonic sequence 33, 34, 35 harmonic spectrum 32 harmonic distortion analysis 395 frequencies 396 orders 6 producing loads 395

363

6

IEEE Working Group on Harmonic Modeling Simulation 384 imbalance 383 imbalances and asymmetries 4 impedance 5, 6 induction motors 1 information and communications technology (ICT) 6 inrush current 43–45, 401, 402 instrument transformer (s) 159 intelligent electronic devices (IEDs) 1, 59 interconnected system 1 interharmonics 40 interpolator 170 inverters and converters 395 Kalman filter 6 knee-point of excitation curve 70 load flow simulations 391 fluctuations 387, 392 profiles 385, 387, 388, 392 local area networks (LANs) 6 loss of generation 383 low-pass filters 6 LSM, derivatives 6 Mahalanobis distance 329 management system 9 market 6 matched filter 360 maximum likelihood estimator 366 measurement and analysis 4 model system 6 modeling 4 monitoring and control of power 4, 8 mother wavelet 384, 387, moving average 387, 388, 390, 391 multilayer perceptron 336 multirate systems 166 Nearest neighbor classifier 329 negative sequence 397, 398, 402, 405 Netherlands 387, 389

Index

415

power variations 386, 392 PQ identification and classification 6 principal Components Analysis 326 probabilistic analysis 6 probabilistic and spectral methods 395 probabilistic parameters 6 probability of detection 357 of Missing Signal 357 producers 3 Prony method: modified Least-square aproach 259 Prony spectral estimation 254 proper transfer function 129 protection applications 79 protection of Transmission Lines 371 pseudospectrum 266

network failures 383 topology 390 Newman-Person Criterion 357 Nobles identities 172 noise model 80 non-conventional transducers 59 non-linear time-varying loads 8 non-periodic 383 non-sinusoidal 398 normalization 327 mapping the minimum and maximum Values 328 notch 6 operation 3, 6 other time-frequency techniques 6 other transformations - Walsh, Hilbert outlier removal 328

6

parallegram of accuracy 63, 69 parametric estimation and identification 4 parasitic capacitances 72 pattern recognition 6 patterns in generation 392 perceptron 330 phase 4, 6 phasor 6 phasor measurement unit (PMU) 6 pockel effect 72 positive negative and zero sequences 395 negative or zero 395 sequence 385, 397, 398, 402 power electronic (PE) 1, 395 power factor 5, 6 power fluctuations 383, 385, 390 power line carrier 67 power quality 3, 6, 13 applications 79 assessment 396 disturbance classification 350 identification 383 power system analysis 1 control 1 load forecasting 351 protection 1, 384 security assessment 353 power transformers 1

quantization SNR 80 quase stationary 90 rate of rise of TRV 17 real time digital simulator (RTDS) 223 real-time sampling rate alteration 176 receiving operating characteristics (ROC) reconstructed signal 402 recursive DFT 119 reliability 1 renewable energy 2 renewable energy sources (RES) 383 resampling 175 residue 130 residuez function 130 RMS 6, 384, 386–389, 391, 392 estimation 144 harmonic group 249 harmonic subgroup 249 interharmonic subgroup 250 sag transient sampling alteration 166 sampling frequency 385, 386 sampling period 387–389, 391 sampling rate 81 sampling segmentation 7 sampling theorem 81, 104, 108 saturation detection of current transforms saturation factor 50 scalegram 6 secure electricity supplies 383

358

377

Index

416

short-term frequency estimator based on zero crossing 195 side lobe 239 signal and noise subspace techniques 262–269 signal decomposition using a notch filter 161 signal processing framework 4 signal processing techniques 1 signal subspace 262 sine and cosine fir filters 163 sliding-window recursive 395 sliding-window recursive Fourier transform (SWRFT) 395 smart energy grid 2 smart grids 383, 392 smart microgrid 390 pecial filters 6s spectral estimation: parametric methods 254 spectral leakage 229–231 spectrum estimation 227 spline interpolation 177 stakeholder complexity 3 standard burdens 63, 68 stationary signal 90 steady-state components 6 STFT 6, 7 subgroup THD (THDS) 249 sum, LSM, derivatives 6 superposition 384, 387, support vector machines (SVM) 342 surge arrester 14 sustainable and low-carbon 2 sustained overvoltage 12 SWDFT 6 SWRDFT 396, 405, 406 symmetrical components 4, 6, 396–398, 400 vector 397 sympathetic inrush 46 synchronous machines 1 synthetic load profiles 387 system evaluation holdout method 350 leave-one-out method 350 technological complexity 3 telecommunication 6, 9 THD 6, 159 time-frequency decompositions 5 techniques 6

time-varying 398, 401–404, 410 components 6, 395 harmonic 4, 397 asymmetry unbalances 395 components 395, 396 distortions 396 time-varying load 247 time-varying phasors 6 5th harmonic 404 time-varying signals 383, 384 time-varying unbalance and asymmetries 398 and harmonic frequencies 397 time-varying waveform distortions 396 toeplitz matrix 257 total demand distortion (TDD) 97 total harmonic distortion (THD) 40, 96 time domain 159 tranmission systems operator (TSO) 390 transducers 6 transfer function in Z domain 129 transients in power systems 384 transient recovery voltage (TRV) 17 translated 384, 385 transmission 1, 6, 9 transport of electrical energy 2 trapezoidal integration 146 travelling waves 6 unbalance for phase-to-phase short circuit 410 unbalance in Converters 407 unbalanced 395, 396, 404, 411 unbalances 6 asymmetries and 396 at higher frequencies 395 unbiased estimator 186 up-sampler 168 frequency domain analysis 169 vandermonde matrix 256 variable speed drives (VSD) 246 voltage control 1 voltage during sag for phase-to-phase short circuit 409 voltage sag 25, 404, 407–409 voltage swells 27 voltage source inverter (VSI) 246

Index

wavelet (s) 7, 383–392 wavelet theory (WT) 384 wavelet transform (s) 6, 383, 384, 386 wide area networks (WANs) 6 wind farm generation fluctuations 389 wind power forecasting 384 wind turbines 247 Window (s) 236 magnitude responses 239

417 in MATLAB1 239 rectangular; Barlett; Hanning and Hamming 237–239 z-transform 126, 127 m-grid 9 pairs 130 properties 132