Power systems signal processing for smart grids

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; đ ã 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 Tai Lieu Chat Luong 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 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) Electric power systems Signal processing–Digital techniques 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 2014 Contents About the Authors xiii Preface xv Accompanying Websites xxi Acknowledgments xxiii Introduction 1.1 Introduction 1.2 The Future Grid 1.3 Motivation and Objectives 1.4 Signal Processing Framework 1.5 Conclusions References 1 10 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 Contents vi 2.13 Other Kinds of Phenomena and their Signals 2.14 Conclusions References 56 57 58 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 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 Contents 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 vii 123 124 126 128 131 131 133 133 133 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 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 Contents viii 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 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 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 Power Systems Signal Processing for Smart Grids 386 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 tị ẳ Ptị  minẵPtị : maxẵPtị  minẵPtị (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 Wavelets Applied to Power Fluctuations 387 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 X (12.7) A2 : Ak;RMS ẳ t L ỵ nẳ0 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 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 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 388 Power Systems Signal Processing for Smart Grids Figure 12.4 Most relevant scaling factors for a load profile during a week (days and 7) than during weekdays (days 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 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 to find components with a maximum fundamental period (1/FM) of hours The value of 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 Wavelets Applied to Power Fluctuations 389 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 and 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 min, were available from the responsible grid operator and were normalized using Equation (12.6) A sampling period of 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 min; this leads to the lowest traceable period (1/FM) of 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 Power Systems Signal Processing for Smart Grids 390 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 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 tị ẳ X Pload tị ỵ X Plosses tị  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 MW The aggregated load was divided equally over the loads in the network with a power factor of The generation from the wind turbine was obtained by Figure 12.9 Structure of the smart microgrid Wavelets Applied to Power Fluctuations 391 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 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 Using Equation (12.4,) the maximum scaling factor was decided to be 14 so that a maximum traceable period 1/FM of 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 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 392 Power Systems Signal Processing for Smart Grids 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 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 Holttinen, H (2005) Impact of hourly wind power variations on the system operation in the Nordic countries Wind Energy, 8, 197–218 Papaefthymiou, G., Schavemaker, P.H., van derSluis, L., 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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/ 396 Power Systems Signal Processing for Smart Grids 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 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 Time-Varying Harmonic and Asymmetry Unbalances 397 Figure 13.2 Architecture to reconstruct the components in the time domain  are vectors of positive (Sỵ 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: 1 S0hk ỵ 16 Shk ¼ a a2 S hk 32 SAhk 76 a2 54 SBhk a (13.1) SChk  where Sỵ 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 Vỵ (13.4) where Vỵ 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: Power Systems Signal Processing for Smart Grids 398 %VUFh kị ẳ  V hk ị > >  100; for positive sequence; > > < Vỵ hk ị > Vỵ > hk ị > > : V   100; for negative sequence: (13.5) hðk Þ The current unbalance factor (IUF) for non-sinusoidal and time-varying conditions is defined: %IUFh kị ẳ  I hkị > > > ỵ  100; > < I hkị if positive sequence; > Iỵ > hkị > > : 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: p 2M A1 sinwt ỵ uA1 ị p V B1 ẳ 2M B1 sinwt ỵ 1B1 ị p V C1 ẳ 2M C1 sinwt ỵ aC1 ị p V A5 ẳ 2M A5 sin5 wt ỵ uA5 Þ pffiffiffi V B5 ¼ 2M B5 B sinð5 wt ỵ 1B5 ị p V C5 ẳ 2M C5 sin5 wt ỵ aC5 ị p V A7 ẳ 2M A7 sin7 wt ỵ uA7 ị p V B7 ẳ 2M B7 sin7 wt ỵ 1B7 ị p V C7 ẳ 2M C7 sin7 wt ỵ mC7 ị V A1 ẳ (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 399 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 ẳ V A1 ỵ V A5 ỵ V A7 (13.18) V B ẳ V B1 ỵ V B5 ỵ V B7 (13.19) V C ẳ V C1 ỵ V C5 ỵ 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 400 Power Systems Signal Processing for Smart Grids 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