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Signal Processing Tools for

Power Quality Event Classification

RAFAEL FLORES

Department of Signals and SystemsDepartment of Electric Power EngineeringSchool of Electrical Engineeringchalmers university of technology

Goteborg, Sweden 2003


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Thesis for the degree of Licentiate of Engineering

Technical Report No. 477L

Signal Processing Tools forPower Quality Event Classification

by

RAFAEL FLORES

Department of Signals and SystemsDepartment of Electric Power Engineering

School of Electrical EngineeringChalmers University of Technology

S-412 96 Goteborg, Sweden

Goteborg 2003


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RAFAEL FLORES

Signal Processing Tools forPower Quality Event Classification.

This thesis has been prepared using LATEX.

Copyright c 2003, RAFAEL FLORES.All rights reserved.

Technical Report No. 477LDepartment of Signals and SystemsDepartment of Electric Power EngineeringChalmers University of TechnologySE-412 96 Goteborg, Sweden

ISSN 1651-4998

Printed in Sweden by Chalmers ReproserviceGoteborg, November 2003


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.....to Cecilia, my lovely wife

and Renzo, my dear son.


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Abstract

The aim of this work is the development of signal processing tools for au-tomating power system event classification when three-phase voltage andcurrent waveforms are available. Most of the power quality monitoring pro-grams result in a huge amount of measurement data which makes analysisdifficult. Therefore, the development of such tools is required.

The use of a complex Kalman filter for the estimation of positive- and

negative-sequences from three-phase voltages or currents is proposed. A com-plex voltage or current is obtained by applying the dq-transform using arotational operator. The algorithm for estimating positive- and negative-sequences from three-phase voltages or currents containing K harmonics isalso given. The proposed method keeps the same accuracy as the conven-tional method at reduced computational complexity since the number ofstate-variables is reduced by 2/3. The performance of the conventional andthe proposed estimators is evaluated under noisy environments. The Cramer-Rao lower bound of the estimation of positive- and negative-sequences isfound. The performance of these two methods is assessed as well.

An extension of the model used to estimate positive- and negative-sequencesfrom three-phase voltage samples is proposed to estimate the time-varyingfundamental frequency. An analysis on the stability, initial conditions andaccuracy is performed for this model which is implemented with an extendedcomplex Kalman filter.

An algorithm to directly compute the active and reactive power from thevoltage and current waveforms is proposed as a way of combining them. Un-der the conventional approach the voltage and current phasors are computed,followed by the calculation of the apparent, active and reactive power. Theadvantage of the proposed approach is that the power is computed withoutcomputing the voltage and current phasors as an intermediate step which

makes the method suitable for fast tracking of power changes. The tech-nique is based on the least squares estimator which performs well for bothnoise free and noise-corrupted three-phase signals.

Keywords power quality, power system monitoring, power system eventclassification, voltage dips, Kalman filter, least squares, symmetrical compo-nents, active and reactive power.

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Contents

Abstract i

Contents iii

Acknowledgements vii

Abbreviations ix

Notation xi

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . 4

2 An Overview on Power-System Event Classification 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Most Used Techniques . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Rms - Root Mean Square . . . . . . . . . . . . . . . . 7

2.2.2 FFT - Fast Fourier Transform . . . . . . . . . . . . . . 72.2.3 Filter Banks and Adaptive Filters . . . . . . . . . . . . 82.2.4 Kalman Filters . . . . . . . . . . . . . . . . . . . . . . 82.2.5 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Automatic Classification Systems . . . . . . . . . . . . . . . . 92.3.1 Rule-Based Expert Systems . . . . . . . . . . . . . . . 102.3.2 Pattern Recognition Systems . . . . . . . . . . . . . . . 10

2.4 What Is the Next? . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Topics for Continuing Research in this Field . . . . . . . . . . 13

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2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Power-System Database 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Many Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Building the Tool . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Designing a Simple Database . . . . . . . . . . . . . . 183.3.2 Graphical User Interface - GUI . . . . . . . . . . . . . 20

3.4 Accessing the Waveforms from the Internet . . . . . . . . . . . 223.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Three-phase Approach:Positive- and Negative-Sequence Estimator 23

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Positive- and Negative-Sequence Estimation: Direct Method . 25

4.2.1 The - and the dq-Transforms . . . . . . . . . . . . . 264.2.2 Estimation of Positive- and Negative-Sequences . . . . 284.2.3 Extension to Three-Phase Systems Containing

K Harmonics . . . . . . . . . . . . . . . . . . . . . . . 294.2.4 Currents in the dq-frame . . . . . . . . . . . . . . . . . 31

4.3 Performance of Indirect and Direct Methods . . . . . . . . . . 32

4.3.1 Cramer-Rao Lower Bound (CRLB) . . . . . . . . . . . 324.3.2 Computational Complexity of Indirect and Direct Meth-ods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Test Results for Simulated and Measured Data . . . . . . . . . 394.4.1 Estimation Under Noisy Conditions . . . . . . . . . . . 394.4.2 Simulated Unbalanced Voltage Dip: Square Error and

Detection Time . . . . . . . . . . . . . . . . . . . . . . 424.4.3 Measured Data . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Estimation of Time-Varying Frequency in Power-Systems 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Defining the Problem . . . . . . . . . . . . . . . . . . . . . . . 555.3 Extending the Previous Model . . . . . . . . . . . . . . . . . . 565.4 Extended Complex Kalman Filter (ECKF) Algorithm . . . . . 58

5.4.1 Tuning the Extended Complex Kalman Filter . . . . . 585.5 Extension to Three-Phase Systems Containing K Harmonics . 595.6 Test Results for Simulated Data . . . . . . . . . . . . . . . . . 61

5.6.1 Three-Phase Systems: Simulated Voltages . . . . . . . 615.6.2 Constant Frequency Variation . . . . . . . . . . . . . . 62

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5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Active and Reactive Power Estimation 65

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Single-Phase Instantaneous Power . . . . . . . . . . . . . . . . 666.3 Defining the Problem: Three-Phase Approach . . . . . . . . . 68

6.3.1 Least Squares Estimator for Active and Reactive Power 716.4 Three-Phase Measured Voltages and

Currents Corrupted by Noise . . . . . . . . . . . . . . . . . . . 736.5 Test Results for Simulated Data . . . . . . . . . . . . . . . . . 73

6.5.1 Three-Phase Systems: Simulated Voltages and Currents 73

6.5.2 Simulated Voltage Dip Conditions . . . . . . . . . . . . 766.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Conclusions and Future Research 81

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.2.1 Bayes Detection . . . . . . . . . . . . . . . . . . . . . . 847.2.2 A Three-Phase and Multilevel Approach . . . . . . . . 857.2.3 Noise Model for Power-System Signals . . . . . . . . . 867.2.4 Direction Finding . . . . . . . . . . . . . . . . . . . . . 86

A The Cramer-Rao Lower Bound of the dq-Transform 87A.1 Complex Noise of the dq-Transform . . . . . . . . . . . . . . . 87A.2 Cramer-Rao Lower Bound: Single Sinusoid . . . . . . . . . . . 89A.3 Mean and Variance of the dq-Transform . . . . . . . . . . . . 90A.4 Cramer-Rao Lower Bound of the dq-Transform . . . . . . . . . 91

List of Figures 95

List of Tables 99

References 101

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Acknowledgements

My grateful thanks to Professor Math Bollen, Docent Irene Gu, and ProfessorMats Viberg from Chalmers University of Technology for their expertise andsupport. Thanks to each member of the steering group: Magnus Erikssonfrom Trinergi, Helge Seljeseth from Sintef Energy Research, Daniel Karlssonfrom ABB Automation Products, Gosta Bengtsson from Goteborg EnergiNat and Ingemar Andersson from Goteborg Energi Research for their com-ments and contributions to this work.

This work has been supported by Goteborg Energy Research Foundation

and it has been developed as a collaboration of the Electric Power Engi-neering Department and the Signal and Systems Department at ChalmersUniversity of Technology.

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Abbreviations and Acronyms

ac alternate current.CFG configuration file: COMTRADE.CKF complex Kalman filter.COMTRADE common format for transient data exchange for

power-systems.CRLB Cramer-Rao lower bound.CWGN complex white Gaussian noise.DAT data file: COMTRADE.

DFT discrete Fourier transform.ECKF extended complex Kalman filter.EKF extended Kalman filter.ESPRIT estimation of signal parameters via rotational

invariance technique.FBLP forward-backward linear prediction.FFT fast Fourier transform.GUI graphical user interface.HDR general information file: COMTRADE.IPP independent power producer.ISO independent system operator.i.i.d. independent identically distributed.KF Kalman filter.LP linear prediction.LMS least mean squares.LS least squares.LSE least squares estimator.MLE maximum likelihood estimator.MSE mean square error.MUSIC multiple signal classification method.

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PHD Pisarenko harmonic decomposition.

PQDIF power quality data interchange format.rms root mean square.RE relative error.QMF quadrature mirror filters.SE square error.SNR signal to noise ratio.SQL standard query language.STFT short time Fourier transform.SVD singular value decomposition.WGN white Gaussian noise.

WLS weighted least squares.WSS wide sense stationary.

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Notation

In this thesis, the symbols for discrete signals: voltages, currents and othersare always mentioned; subscripts are used to distinguish between electricalphases: e.g. a, b and c. The symbols for continuous signals are explicitlymentioned.Vectors are written in boldface lowercase letters and matrices are written asboldface uppercase letters. Complex signals would have a tilde and vectorsand matrices with complex signals would have tilde as well.The meaning of the following symbols are, if nothing else is stated:

xT

transpose operator.x complex conjugate.xH hermitian transpose, i.e. complex conjugate transpose.Re{.} real part of a complex quantity.Im{.} imaginary part of a complex quantity.tr{X} trace operator.|X| determinant operator.|x|2 magnitude operator.[X]ij the (i, j) th element of the matrix X.V phasor - complex quantity.

|V

|absolute value of a phasor.

phase-angle of a phasor.diag[a1, a2, . . . , ak] a kxk diagonal matrix with diagonal

elements a1, . . . , ak.Im identity matrix of order mxm,

usually, m is omitted.

an estimate of the parameter .E[.] the expectation of a random variable.var[x] the variance of x.cov[x] the covariance of x.

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Cx the covariance of matrix x.

log(x) the natural logarithm of x.e unitary vector.w.r.t. with respect to.W GN(, 2) white Gaussian noise with mean and variance 2.CWGN(, 2) complex white Gaussian noise with mean and variance 2.

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Chapter1

Introduction

1.1 Background

The term Power Quality covers a broad list of electromagnetic phenom-ena concerning with the interaction of power-system networks and end-userequipment. End-user equipment is sensitive to certain types of voltage dis-turbances in the system, but the equipment on its turn may produce some

current disturbances, which pollute the system. As many sensitive processesin industrial systems do care about the disturbances in the supplied voltages,industries are more concerned about the operational and economic aspectsof these disturbances.

Running extensive power quality monitoring programs is important inorder to understand, identify and solve problems regarding power quality. Inmany cases, such monitoring programs end up in a huge amount of measureddata which makes analysis difficult. Therefore, the development of automatictools for assessment of the measured data is required to help utilities, regu-lators and customers to have a clear understanding of what is happening in

their networks.In a previous project, relevant work has been done on defining how to

identify events automatically by applying a rule-based expert system[1]; sev-eral disturbances, such as voltages dips: fault or non-fault induced, trans-former saturation, motor starting, interruptions and capacitor switching werestudied. Additionally, a first step to understand transients in power-systemsthrough ESPRIT has been done. All this analysis has been performed usingthree-phase voltages, through separately studying each voltage from a singlepoint in the entire network.

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CHAPTER 1. INTRODUCTION

1.2 Motivation

Although a rule-based expert system was developed to analyze the distur-bances from single-site voltage waveforms, a three-phase approach and amultilevel voltage scheme are more desirable. Joint analysis of both voltageand current waveforms will provide more understanding on the underlyingevent. With all this information we may try to establish mapping functionsfrom the time domain to the information domain that describe power-systemdisturbances in a three-phase system. Finding possible functions is one ofthe objectives in this PhD project. The emphasis in this work will be put onthe development of automatic processing using signal-processing and pattern

recognition techniques combined with power-system knowledge. A three-phase approach will be investigated. Three-phase current records added tothe three-phase voltage approach will be tested and defined giving the exist-ing system additional tools to enable identification of disturbances by usingmultilevel approach and exploring the current waveforms.

With the deregulation of the energy systems, energy has become a com-modity. Utilities sell energy and power, the term power quality includes byitself voltages and currents so we will try to distinguish between differenttype of disturbances by looking into the power: active and reactive power.

The final motivation for this work is the need to develop an open ac-

cess software tool for power quality research and education. This softwaremay enable sharing of power quality information such as voltage and cur-rent waveforms, feature extraction algorithms or final statistics in an openarchitecture.

1.3 Outline

This thesis is organized in seven chapters. Chapter 1, is introductory and itsummarizes the background, the motivation, the organization of this thesisand finally, it lists publications.

Chapter 2 is a review on applied signal processing techniques for auto-matic classification of power quality events, future trends are outlined aswell. The time when power quality monitoring equipment just collected rawwaveforms has gone. In the near future power quality monitoring systemswill be demanded to be able to identify and classify events automatically inorder to solve the problems in electrical networks in a smarter way.

Chapter 3 deals with how to organize data (event waveforms) which comefrom different utilities. Initially recordings were stored in many different fileformats. The need to standardize the information and to make data sharable

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1.3. OUTLINE

and easy accessible is justified. Moreover, an open architecture database is

designed and developed to manage and share all recordings. Furthermore, agraphic user interface in Matlab was designed to query this database. Mat-lab has shown to be a suitable platform to develop and test algorithms.Additionally, an open access module has been proposed to write and readinformation in the database from anywhere in the world though the Internet.This scheme will allow users to gather new event records in an easier, saferand faster way than today.

Chapter 4 proposes the use of a complex Kalman filter for the estimationof positive- and negative-sequences from three-phase voltages or currents. Acomplex voltage or current is obtained by applying the -transformfollowed

by the dq-transformusing a rotational operator. The algorithm for estimat-ing positive- and negative-sequences from three-phase voltages or currentscontaining K harmonics is also given.

Estimation of positive- and negative-sequences is performed through twosteps: the magnitude and phase-angle of each individual electrical phase ofthe voltage or current are first estimated and the symmetrical componenttransformation is then applied. This is referred as the indirect method. Theproposed method - called the direct method - offers a direct estimation of thepositive- and negative-sequences keeping the same accuracy. The proposedmethod has a reduced computational complexity since the number of state-

variables is reduced by 2/3 as compared to that in the indirect method. Theperformance of the indirect and direct methods is evaluated under noisy en-vironments. The Cramer-Rao lower bound of the estimation of the positive-and negative-sequences is found. The performance of both methods is as-sessed as well.

Chapter 5 proposes the extension of the previous model used to estimatepositive- and negative sequences from three-phase voltage samples, presentedin Chapter 4, to estimate the time-varying fundamental frequency. An analy-sis of the stability, initial conditions and accuracy is performed for this modelwhich is implemented with an extended complex Kalman filter. Simulationresults are presented.

Chapter 6 proposes an algorithm to directly compute the active and re-active power from the voltage and current waveforms as a way of combiningthem. Under the conventional approach, voltage and current phasors arecomputed, followed by the calculation of the apparent, active and reactivepower. The advantage of this approach is that the power is computed with-out computing the voltage and current phasors as an intermediate step. Thetechnique is based on the least squares estimator which performs well forboth noise-free and noise-corrupted three-phase signals.

Chapter 7 summarizes the main conclusions of this thesis and discusses

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CHAPTER 1. INTRODUCTION

on the proposed methods. Future research in this area is outlined. A combi-

nation of techniques such as Kalman filter and Bayesian classification is pro-posed. Power-system features, e.g. amplitude and phase-angle of positive-and negative-sequences extracted by Kalman filters shall be used. Bayesclassification schemes defined from the extracted features and the ability todetect, select and classify power-systems events shall be treated. Addition-ally, when an event occurs in a network, many of the power quality monitorsinstalled will register the event in different locations at about the same time.The joint analysis of these records should be done by taking into accountprobabilistic techniques and combining features obtained at different loca-tions and from several power quality monitors. The same observed event

from different points in the network would give more information about thelocation and the network conditions that caused this disturbance. A difficulttask is to determine whether an event recorded at the same time by sev-eral power quality monitors is the same event, because time stamps in theserecordings are not synchronized exactly. The need to study noise in power-system waveforms in more detail is treated. The ability to improve detection,classification and protection schemes may be related with the properties ofnoise. And finally, a method to find the direction of an event based in thechange of rate for voltage and current in three-phase system is mentioned.The direction can be found by modelling it with Kalman filters as one step

ahead estimators.

1.4 List of Publications

1. R. A. Flores, P. A. David and J. Szczupak, Real time precise harmonicestimation, 13th Power Systems Computation Conference, vol. 2, pp.1063-1069, July 1999.

2. R. A. Flores, State of the art in the classification of power qualityevents, an overview, Proceedings of 10th International Conference inHarmonics and Quality of Power, October 2002.

3. R. A. Flores, I. Y. H. Gu and M. H. J. Bollen, Positive and negativesequence estimation for unbalanced voltage dips, Proceedings of 2003IEEE Power Engineering Society General Meeting, July 2003.

4. R. A. Flores, I. Y. H. Gu and M. H. J. Bollen, Performance assess-ment of positive/negative-sequence estimation method to three-phaseunbalanced voltages and currents, submitted to IEEE Transactionson Power Delivery.

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Chapter2An Overview on Power-System Event

Classification

This chapter gives a brief review on the most frequently used techniques inautomatic classification of power quality events and as well as outlines theresearch trends. The time when power quality monitoring equipment justtook pictures of raw waveforms has gone. Power quality monitoring systemsare demanded to be able to identify and classify events automatically in orderto solve problems in electrical networks in a smarter and faster way.

2.1 Introduction

To gather the first glimpse of which is the current state of power qualityevent classification techniques, lets take a look at the evolution of powerquality monitoring in terms of technology and users. In Fig.2.1, a time-linehas been plotted1. In the 90s, the technology applied in classification tendedto merge power-system engineering knowledge mainly with signal processingtechniques. In the latest years, pattern recognition, data mining, decision-

making and networking were incorporated as new technologies for automaticclassification. This entire advance tends to process raw data and extractinformation to obtain knowledge in order to make decisions [3] and solveproblems with less human aid.

Moreover, users of power quality event classification schemes have spreadfrom a few field-service engineers in the 70s to hundreds of people in the2000s; in power utilities, consultant companies and governmental agencies;working to assess power networks and to include power quality indexes in

1The first 30 years of the diagram has been extracted from [2]

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CHAPTER 2. AN OVERVIEW ON POWER-SYSTEM EVENT CLASSIFICATION

2000s90s80s70s

Voltmeter

Tape

TECHNOLOGY

Oscilloscopes

Oscillographs

GraphicsPaper

Digital Signal Processing

Computers

Mass storage devices

Communitacions Networkinginternet

Decisionmaking

Data mining

Pattern recognition

USERS

Field service engineer

Power Quality groups

Industrial/plant/facilities

Uti li ty companies Regulatory agencies

ISO

IPP

Figure 2.1: Time-line of power quality monitoring equipment evolution

power-system economic performance studies. This chapter will summarizethe most-used signal processing techniques and will point out three tech-niques which are expected to be developed and improved in the coming years.The chapter is organized as follows: section 2.2 describes the most-used sig-nal processing techniques at current time, section 2.3 describes techniques inautomatic classification, section 2.4 analyzes and identifies the next steps inautomatic classification of power quality events, section 2.5 describes three

potential topics for research in this field and finally, section 2.6 presents con-clusions on this review.

2.2 Most Used Techniques

This section attempts to give a quick glance at signal processing techniquesused in power quality event classification.

A great quantity of work has been focused on the estimation of am-plitude, phase-angle and frequency of the fundamental voltage or currentsignals and its related harmonics. Although this is a well studied subject insignal processing, these techniques do not explain by themselves or give moreknowledge about the underlying causes in power-system events. Let a singlesinusoidal, representing a voltage(or current) of interest whose parametersare changing in time, be described by

s(n) = A(n)sin(n + (n)) (2.1)

where A(n) is the amplitude of the voltage (or current) changing in time, (n)is the angle-phase changing in time, and is the angular frequency which is

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2.2. MOST USED TECHNIQUES

function of the network frequency in the discrete-time domain. Finally, the

index n denotes the discrete-time.A brief summary on the basic techniques for estimation of model param-

eters in (2.1) is as follows.

2.2.1 Rms - Root Mean Square

A simple approximation of the amplitude of the sinusoidal in (2.1) is usingthe root mean square (rms), rms can be evaluated over a cycle or a half cyclewindow and it is described by

srms(n + N) = 1

N

Ni=1

s2(n + i) (2.2)

where, N is the window length.Rms is the most used tool and it gives a rough approximation of the

fundamental frequency amplitude profile of a waveform. A great advantageof this method is its simplicity, speed of calculation and less requirement ofmemory, because rms can be stored periodically instead of sample per sample[4]. However, its dependency of window length is considered as a disadvan-tage: one cycle window length will give better results in terms of profilesmoothness than a half cycle window at the cost of a lower time-resolution.

Moreover, rms does not distinguish between fundamental frequency, harmon-ics or noise components, therefore accuracy will depend of the harmonics andnoise content.

In a recent work, rms three-phase voltage profiles can be used for eventanalysis and automatic classification with a high degree of certainty [5]. Rmsclassification of energizing, non-fault interruption, fault interruption, stepchange and voltage dip due to faults is shown to be feasible. When using rmstechnique phase-angle information is lost. Therefore, transformer saturation,induction motor starting and capacitor switching can not be distinguishedaccurately.

2.2.2 FFT - Fast Fourier Transform

Another well established tool for estimation of fundamental amplitude andphase-angle of a signal is the discrete Fourier transform (DFT). DFT trans-forms the signal from time-domain to the frequency-domain. The basic equa-tion that describes the DFT is

S(k) =1

N

N1n=0

s(n)ejkn (2.3)

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where S(k) is the DFT evaluated at the frequency k, N is the window length,

k = { 2kN } is a set of fixed and equally spaced frequencies, 0 k N 1.The fast Fourier transform (FFT) is the DFTs computational efficient

implementation, its fast computation is considered as an advantage. Withthis tool it is possible to have an estimation of the fundamental amplitudeand its harmonics with a reasonable approximation. FFT performs well forestimation of periodic signals in stationary state; however it doesnt per-form well for detection of sudden or fast changes in waveform e.g. transientsor voltages dips. In some cases, results of the estimation can be improvedwith windowing, i.e. Hanning, Hamming, Kaiser windows or filtering, e.g.low-pass or high-pass filters. Window length dependency resolution is a dis-

advantage e.g. the longer the data window (N) the better the frequencyresolution.

2.2.3 Filter Banks and Adaptive Filters

Filters seem to be suitable to extract signals in a specified band-width, e.g.low-pass band, band-pass and high-pass band filters. Filter banks are a morecompact and powerful implementation of single filters. Filter banks havebeen used to study in more detail a specific band of the frequency spectrum.A combination of quadrature mirror filters (QMF) arranged in binary trees

combined with the DFT for estimation of each band components is shownto be efficient for harmonic estimation. This technique was used in differentapplications to detect rapid changes in the waveform or to estimate specificsub-band components[6], e.g. harmonic contents between 500 to 1000 Hz,may due to capacitor switching.

Another refined filtering technique is adaptive filtering, which estimatesparameters in a windows based - least squares (LS), recursive least squares(RLS) or least mean squares (LMS). These techniques have the advantage ofperforming well if the signal is noise free or noise corrupted. Other techniqueshave been derived from this approach e.g. weighted least squares (WLS).

2.2.4 Kalman Filters

A well-known technique is the so-called Kalman Filter. This technique isdefined as a state-space model, and can be used to track amplitude andphase-angle of fundamental frequency and its harmonics in real time undernoisy environment, which was proposed in [7]. Since then, many applicationshave come up, e.g. detection of harmonics sources and optimal localizationof power quality monitors [8].

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2.3. AUTOMATIC CLASSIFICATION SYSTEMS

2.2.5 Wavelets

Since 1994, the use of wavelets has been applied to non-stationary harmonicsdistortion in power-systems[9]. This technique is used to decompose the sig-nal in different frequency bands and study its characteristics separately. Asdescribed in [10, 11], wavelets perform better with non-periodic signals thatcontain short duration impulse components as is typical in power-systemtransients. Many different types of wavelets have been applied to iden-tify power-system events such as: Daubechies, Dyadic, Coiflets, Morlet andSymlets wavelets. These were found more suitable for power-system studies.Furthermore, Wavelet based techniques were proposed for detection and mea-

suring of power-system disturbances [12, 13]. Although, wavelet type shouldbe chosen accordingly to the specific event to study, making this techniquewavelet-dependent and less general.

Finally, the short time Fourier transform (STFT) is commonly known asa sliding window version of the FFT, which has shown better results in termsof frequency selectivity compared with wavelets which have center frequenciesand bandwidths fixed. However, STFT has a fixed frequency resolution for allfrequencies, and has shown be more suitable for harmonic analysis of voltagedisturbances than binary tree filters or wavelets when applied to voltage dips[14].

2.3 Automatic Classification Systems

A general scheme for automatic classification systems is depicted in Fig. 2.2.Its application for power-quality event classification is shown to be feasible[15, 16].

In the figure, Block I represents a pre-processing stage. In this blockestimation of the signal components is performed. Then, an algorithm forsignal segmentation in different stages is applied, e.g. pre-event, during-eventand post-event stages. Transition stages are defined as well.

Block II represents a feature extraction stage. Feature extraction canbe done through any of the techniques described before such as wavelets orKalman filter. Wavelets are mainly used to quantify features for differenttypes of power-system events. However, researchers do not fully agree withwavelets universal use, due to features obtained with wavelets are highly de-pendent on the type of mother wavelet chosen. Another drawback is thatmost of the work was done with simulated data. Therefore, event type fea-tures and its extraction procedure using wavelets are still subjective. Stan-dardization will be necessary in order to continue working in this area.

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Block III represents the classification stage based on defined rules, e.g.

knowledge based expert systems, pattern recognition or any logic to discrim-inate different types of events.

Finally, Block IV represents the decision making stage. In this stage thetype event is assigned to an actual type event. In many proposed algorithms,Blocks III and IV are merged in one process held by neural networks, fuzzylogic, Bayesian or any other pattern recognition technique.

Preprocessing

II III IV

Kalman filter

Wavelets

I

Parametersestimation

Rules or logic

Pattern

recognitionSegmentation

Feature extraction Classification Decision making

Knowledge based

Figure 2.2: Automatic classification scheme

A more detailed view of the techniques for automatic classification ofpower quality events will be presented as follows:

2.3.1 Rule-Based Expert SystemsExpert systems were proposed to identify, classify and diagnose power-systemevents successfully for a limited number of events [17], i.e. an expert systemfor classification and analysis of voltage dips using Kalman filter for estima-tion of the amplitude has been shown in [18]. Rule-based expert systemsare highly dependant on if ..then clauses. If many event types or featureswere analysed, the expert system would become more complicated and therisk of losing selectivity would increase (ambiguity would increase). Anotherdrawback is that these expert systems are not always portable due to settingsdepend mostly on the designer or operator of the system for a particular set

of events in a particular power-system. It would become hard to adjust theexpert system parameters for a different power-system and operator require-ments. Therefore, the application of this kind of expert systems is limited inreal practice. Self-learning ability is desired.

2.3.2 Pattern Recognition Systems

Under the assumption that energy contents of the non-fundamental compo-nent in a signal change depending on the type of event, wavelets are widely

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2.4. WHAT IS THE NEXT?

applied for detection, quantification and classification of a variety of power

quality disturbances, i.e. harmonics and transients. In [19, 20, 21], auto-matic classification systems were proposed based on wavelets feature extrac-tion. However, feature extraction based on wavelets transform has manydisadvantages, which were previously shown.

Recently, more sophisticated algorithms for automatic classification wereproposed. An on-line power quality disturbance detector was proposed in[15], despite, using wavelet feature extraction the novel idea in this applica-tion is the use of a Bayesian classifier. This algorithm analyzes the missingvoltage, which is decomposed using wavelets, features are extracted from thescaled signals. Different patterns are found for different event types. Then,

the system classifies these features using Bayesian approach. A drawback ofusing Bayes formula is that a-priori probability density function (pdf) ofeach event must be known in advance.

Although this kind of classification algorithms work well for Gaussianpdfs, its performance in power quality events is not fully accepted due to thenon-Gaussian nature of the event pdfs and the free choice of event features.

Time-domain analysis has shown better results for automatic classifica-tion of voltage dips and swells. For fast changes including phase-angle jumps,like caused by capacitor switching, it is better to apply frequency methods.Another novel algorithm is found in [16], even though, its feature extraction

stage is wavelet-based.Up to here, all algorithms were applied just to voltage waveforms in asingle-site or one node in a network. A recent algorithm combining three-phase voltage and current waveforms and applying non-supervised classifica-tion techniques is shown in [22], e.g. clustering using K-nearest neighbors.

2.4 What Is the Next?

Hereto, a glance of the most used techniques oriented to automatic classifi-cation of power-system events has been presented.

More work has to be done to obtain a better understanding of voltageand current behavior under different event conditions. Additionally, a betterunderstanding of relations between voltages and currents in different voltagelevels and its propagation in the network is still required.

Moreover, feature definition and extraction have to be enhanced in orderto improve classification algorithms. As a result of this work, a more generalset of accepted features for each type of event could be drawn. Although, thework done in the definition and proposition of classifiers based on non-power-system techniques is still immature, statistical approach for classification

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algorithms have shown interesting results.

Topics which shall become interesting in order to continue research in thisfield are listed as follows:

1. Segmentation is based on the three single-phase voltage waveformslooking into them separately. A segmentation scheme taking into ac-count three-phase signals in just one index should be more easy tohandle.

2. What kind of information can be obtained by the discrete or continuousrms and which information is lost? e.g. phase-angle information is lost.

What will happen if current rms is added to the classification scheme?

3. Using classification based on voltage rms can not distinguish differencesbetween motor starting and transformer saturation. Can unbalance bedetected based on rms?

4. For a given small number of power quality monitors how to arrangethem to obtain the optimal allocation to cover a broader area.

5. Based on the assumption that energy content of the signal would changedepending on the type of disturbance, faster and higher degree of dis-

crimination regarded conventional voltage-based disturbance detectionapproach can be designed. Pdfs use to study these particularities isfeasible.

6. Event waveforms at a single-site can be studied easily, two or more setof waveforms of the same event are to be discriminated or selected insome way to characterized a particular event e.g. the same voltage diprecorded at different locations in a network.

7. Faster voltage dip detection is required in order to make protectiondevices more accurate and to increase selectivity index, the numbers offalse alarm triggers should be minimized.

8. A more general mapping function for event characterization is missed,its definition is relevant in order to design and define a more intelligentautomatic classification scheme compared with the rules-based ones.

Finally, advances in automatic classification in power quality events shouldinfluence or change the way that hardware for power quality is currently de-fined.

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2.5. TOPICS FOR CONTINUING RESEARCH IN THIS FIELD

2.5 Topics for Continuing Research in thisField

In this section, three potential topics are analysed in more detail in order toexplore its advantages and disadvantages.

1. Three-phase approach: Most of the voltage analysis is done phase byphase separately, a three-phase approach is missed. A three-phase in-dex is not available and could be implemented for example by theintroduction of the symmetrical components.

2. Including current information: Most of the techniques presented in thisreview are based in voltages waveforms, voltage dips originated by overcurrent could be detected earlier if a current stage analysis is addedto the actual classification schemes. e.g. over-current will produce andecrease in voltage that trigs a disturbance detection scheme in volt-age. Additionally, event direction can be detected from the additionalinformation given by currents.

3. Pdfs discrimination: Energy content of the signal would change de-pending on the type of disturbance. Faster and higher degree of dis-crimination regarded conventional voltage based disturbance detection

approaches may use pdf of the residual signal. After extracting thefundamental frequency. Studying these particularities based on thisassumption could lead to design new and better classification schemes.

2.6 Conclusions

From the above overview, conclusions can be drawn as follows:

1. What is required is not a more accurate estimation of the voltage sig-nal components phase by phase, e.g. amplitude, phase-angle or net-

work frequency. Rather, what is required is to study the relationsbetween three-phase voltages or currents and the underlying power-system event. A three-phase approach is missing.

2. The introduction of the currents to perform power quality event clas-sification studies is strongly needed.

3. Studying the pdfs of the residual error after estimating the main pa-rameters may have a good potential in future research.

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Chapter3Power-System Database

This chapter deals with how to organize all data(event waveforms) whichcome from different utilities. Data come from different monitors which arestored in different file formats. The need to standardize the informationand to make it sharable and easy accessible is justified. Moreover, an openarchitecture database is proposed to manage and share all the information.Furthermore, a graphics user interface (GUI) was designed in Matlab to querythis database. Matlab has shown to be a suitable platform to develop andtest algorithms. Additionally, an open access module is being proposed to

write and read information stored in the database from anywhere in the worldthough the Internet. This scheme will allow to gather new event waveformsin an easy, safe and fast way.

3.1 Introduction

A single event has been studied from measurements in a single-site at thepower-system network. As the need to study more points of the network atthe same time, the requirement for more refined tools appears. Managingthe recorded information in plain text files is a simple and easy way to accessthe information. But limitations are with the ability to correlate, or find in-formation i.e. the same event observed from different power quality monitorsat the same time. This task could be more complicated if a single utilityhandles more than one power quality device, which means data of the sameutility stored in more than one format.

Two basic issues must be addressed:

Event waveforms in different files and Event waveforms in different formats.

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CHAPTER 3. POWER-SYSTEM DATABASE

If a wide or big scale study is to be carry out, the need to read and

study information coming from many different sources in many file formatscould be complicated to handle and distracting us from the project objective.Therefore, event waveforms are to be managed in such a way that allowcompactness and easy access. Queries shall be able to be done accordinglyto this new setup. A third and maybe the most important problem to beaddressed is

the need to handle information of more than one location at the sametime.

Single-phase and single-site waveforms are not useful anymore.If the propagation of an event through the power-system is to be studied,and its cause to be found, the stored information can not be accessed sequen-tially in plain text files anymore. More systematic and compact organizationof the data is required e.g. in a database. This approach will make possibleto get a list of all the events which have had happened in a specific date, timeand area or voltage level. For this new requirements storing the informationin a database is justified.

A simple relational database should be used to store the minimum infor-mation, e.g. geographical, electrical, and actual waveforms.

3.2 Many Formats

As power quality monitors are available in the market, many formats haveappeared; plain text(ASCII), binary and other formats are currently availableto store event information coming from a wide variety of capturing devices.Most recently, databases to store and manage waveforms are available. Amajor number of these databases are proprietary 1.

The most common feature of all commercial databases is high costs (ex-

pensive initial cost and maintenance) and these databases are attached tospecific power quality monitor manufacturer. It means once a utility buy apower quality monitor, the utility must use the proprietary software. Thisclosed architecture makes impossible to access or share the waveforms fromother utilities or even worst from any other application unless a specific re-quirement have been done to the manufacturer which could lead to expensivefees.

1for exclusive use of customers paying license fees - closed architecture: not allow toshare or access data from third parties

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3.3. BUILDING THE TOOL

In this project we have been collecting event waveforms from many utili-

ties worldwide e.g. Scotish Power (UK), Goteborg Energy (Sweden), Sintef(Norway), Enersur and Egemsa(Peru). A need to unified and interchangeof information is strong for gathering real waveforms to study and test newalgorithms, we should have access to the plain data. We can not afford topay expensive licenses for different software tools.

That is why a common format to interchange information has been cho-sen, COMTRADE which is defined as a standard [23] and its review [24].

This standard basically defines three files for interchange of information.

1. Header File (xxxxxxxx.HDR): which includes some general information

of the event.

2. Configuration File (xxxxxxxx.CFG): which includes information of theconfiguration of the monitor.

3. Data File (xxxxxxx.DAT): the plain data in text format.

For the purpose of this project, only CFG and DAT files are used. Butstill, managing two text files per event in a big number of events is difficult.Standardizing the use of COMTRADE files solves the first two problemspresented before. To solve the third problem, information about all events

must be stored in a database with an open structure.How to build this database and what it should handle are treated in the

next section.Due to the increasing awareness on the difficulty of comparing results ob-

tained by researchers and utilities when trying to characterize power qualitywith many types of instruments. A recommended practice has been devel-oped by IEEE[25], commonly known as PQDIF, which stands for PowerQuality Data Interchange Format. This practice recommends the unifor-mity of basic algorithms and data reduction methods that instrument manu-facturers should apply. Details of this document are out of the scope of this

chapter, however, we think that it is important to take into account its useas a general practice.

3.3 Building the Tool

An open-architecture database is required to store and manage a large num-ber of event waveforms. This database can store not only measured databut also simulated data coming from different sources in a common formatCOMTRADE.

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In Fig. 3.1 is shown the interactivity of this database. Waveforms could

come from many sources and the information stored could be used to testand develop algorithms through a graphical user interface (GUI). Storing theinformation in a standard database will allow to share the information in asafe and easy way. Moreover, reading and writing in the database should bepossible through the Internet.

Openarchitecture

database

Power system

real data

Power system

Simulated

data

Matlab

Test and

development platform

GUI

real data

Sweden

real data

UK

real data

Per

Internet

Figure 3.1: Open architecture application: dashed - in our computer system

3.3.1 Designing a Simple Database

The minimum information to be stored would include basic information:three voltage channels and four current channels are required. Additionalinformation regarding company, location of record, configuration of the powerquality monitor.

Database is being designed to handle the minimum information regard-

ing power-system measurements. Five basic tables are required which aredescribed as follows (Q prefixes mean quantity and C prefixes mean charac-ter):

1. config: contains information regarding the configuration of the powerquality monitor and defines the following fields: index, cod site, C description,Q Vfactor, Q Ifactor, Q freq sample and C typ monitor

2. diag: contains information about the event and defines the followingfields: index, cod for, C typ event, C desc event

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3. site: contains information about the location where the waveforms

were acquired and defines the following fields: index, cod site, C company,C location, C net nom voltage

4. records head: contains information about the header of the recordsand defines the following fields: index, cod site, C date, C start time,cod for

5. records det: contains the raw data and defines the following fields:index, cod for, C time, Q Va, Q Vb, Q Vc, Q Vn, Q Ia, Q Ib, Q Icand Q In

The relationships between all tables and their fields in this database isshown in Fig. 3.2 where * means key fields2.A prototype database has been implemented in Access-Microsoft because

is available in all Chalmers computers under Windows XP Pro operativesystem.

config

index

cod_site*

C_description

Q_Vfactor

Q_Ifactor

Q_freq_sampleC_typ_monitor

records_head

cod_site

C_date

C_start_time

cod_for*

diag

index

cod_for*

C_typ_event

C_desc_event

records_det

index

cod_for*

C_time

Q_Va

Q_VbQ_Vc

Q_Ia

Q_Ib

Q_Ic

Q_In

site

index

cod_site*

C_company

C_location

Q_net_nom_voltage

Figure 3.2: Power Quality database relationship scheme

2Key fields handle information that is unique - No more than one record can have thesame information

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3.3.2 Graphical User Interface - GUI

Once the database is available, event waveforms can be analysed and man-aged with an graphical user interface (GUI) developed in Matlab which isshown in Fig.3.3.

Figure 3.3: Graphical User Interface - Query Module

This interface allows to choose:

a database: each company can have its own database. a location: a specific point or voltage level in the network. queries: for specific date and time. raw or processed waveforms: plotting or printing.The plot option shown in Fig. 3.4; shows six windows which are describe

as follow:

1. time-domain three-phase voltages;

2. time-domain three-phase currents;

3. amplitude of three-phase voltages;

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0 50 100 150 200 250 300 3501.5

1

0.5

0

0.5

1

1.5

msec

p.

u.

0 50 100 150 200 250 300 3503

2

1

0

1

2

3

msec

p.

u.

0 50 100 150 200 250 300 3500.4

0.5

0.6

0.7

0.8

0.9

1

1.1

msec

p.

u.

abs Fundamental Voltage

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

msec

p.

u.

abs Fundamental Current

50 100 150 200 250 300

40

45

50

55

60

msec

o

degrees

Angle Fundamental Voltage

50 100 150 200 250 300 350

40

20

0

20

40

60

msec

o

degrees

Angle Fundamental Current

Figure 3.4: Graphical User Interface - Plot Module

4. amplitude of three-phase currents;

5. phase-angles of three-phase voltages;

6. phase-angles of three-phase currents.

Additionally, an auxiliary plot shown in Fig.3.5 shows six more windowswhich describe:

1. positive and negative-sequences: three-phase voltages;

2. positive and negative-sequences: three-phase currents;

3. instantaneous active three-phase power;

4. instantaneous reactive three-phase power;

5. instantaneous three-phase power factor;

6. Kalman filter three-phase voltage residuals for segmentation.

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0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1

1.2

1.4

msec

p.u.

Symmetrical Components Voltage

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

msec

p.u.

Symmetrical Components Current

0 50 100 150 200 250 300 3502

1.5

1

0.5

0

0.5

1

1.5

2

msec

p.u.

Active Power

0 50 100 150 200 250 300 3501.5

1

0.5

0

0.5

1

1.5

2

msec

p.u.

Reactive Power

0 50 100 150 200 250 300 3501

0.5

0

0.5

1

msec

Cos

Power Factor

0 50 100 150 200 250 300 3500.03

0.02

0.01

0

0.01

0.02

KF Residuals

msec

Error(p.u.)

Figure 3.5: Graphical User Interface - Auxiliary Plot Module

3.4 Accessing the Waveforms from the Inter-net

Once an open-architecture database is available and its development andtest platform is being set up. The need to get more information coming frompower-system in the world has shown up. An open access module can allowto read and write information in this database from anywhere. That is whythe remote access to the database has been developed.

3.5 ConclusionsThe waveform database developed in this thesis will allow researchers andstudents to manage information coming from different companies world wide.Data is organized in an open-architecture an easily accessible through Matlab,which allows development of algorithms. Our purpose is not competing withcommercial products, we just like to have an easy and cheap way to get morewaveforms from many companies and to share results.

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Chapter4

Three-phase Approach:Positive- and Negative-Sequence

Estimator

This chapter proposes the use of a complex Kalman filter for the estimationof positive- and negative-sequences from three-phase voltages or currents. Acomplex voltage or current is obtained by applying the -transformfollowedby the dq-transform using a rotational operator. The algorithm for three-phase voltages or currents containing K harmonics is also given.

Estimation of positive- and negative-sequences is performed through twosteps: the magnitude and phase-angle of each individual electrical phase ofthe voltage or current are first estimated and the symmetrical componenttransformation is then applied, this is called the indirect method. The pro-posed method - called the direct method - offers a direct estimation of the

positive- and negative-sequences with the same accuracy at a reduced com-plexity. The proposed method has a lesser complexity since the number ofstate-variables is reduced by 2/3 as compared to that in the indirect method.The complexity of the proposed algorithm is also analysed.

The Cramer-Rao Lower Bound is given. Further, how well the indirectand direct estimators approach to this bound is examined. The performanceof the positive- and negative-sequence estimator under noisy conditions isevaluated.

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CHAPTER 4. THREE-PHASE APPROACH:

POSITIVE- AND NEGATIVE-SEQUENCE ESTIMATOR

4.1 Introduction

Positive- and negative-sequence voltages were originally introduced to speedup calculations involving non-symmetrical faults, and as such are introducedin almost any text book on electric power-systems[26]. More recent ap-plication lies in the diagnostics of power-systems during non-symmetricaloperation including faults. A method for characterizing the unbalancedvoltage dips from the positive- and negative-sequences has been proposedin [27, 28, 29]. The two-component method is based on the idea that thepositive- and negative-sequence source impedances are equal for static cir-cuits. To achieve reliable and fast characterization and thereby a correct

identification of voltage dips, a good estimation and fast detection of thepositive- and negative-sequence voltages is required.

Several methods for estimating positive- and negative-sequences havebeen proposed: A well known method to calculate symmetrical componentsin a power-system is based on two stages as shown in Fig.4.1, [30] proposesthe use of FFT to estimate phasors, [31] proposes least squares (LS) filters,[32] proposes weighted least squares (WLS) filters, [33] proposes a non-linearleast squares filters and recently, [34] have proposed to use Kalman filters.Although, FFT based algorithms have shown to have finite robust to noise.The problem with FFT is limited resolution when there are multi-component

signals. Least squares(LS) based methods are block based, they lack on sta-tistical assumptions about the signals, i.e. only a model signal is assumed.Besides, LSs performance will undoubtedly depend upon the properties ofthe corrupting noise as well as the modelling errors. In the last work, themagnitudes and phase-angles were first estimated using three Kalman filters(one for each phase), and the positive- and negative-sequences were then ob-tained by using the symmetrical-component transformation(T3), as shown inFig.4.1. Note that the angle between positive- and negative-sequence voltageis an important parameter in determining the dip type [35]. Therefore, morereliable and more efficient methods for estimating positive- and negative-sequences are desirable. Symmetrical component voltages and currents are

also a suitable tool for describing load behavior during voltage dips and otherdisturbances [36, 37, 38].

In this chapter an improved approach for estimating the positive- andnegative-sequences using a complex Kalman filter is proposed and tested.Comparing it with the two-step method, this method offers a direct esti-mation of the positive- and negative-sequences, and reduces the computa-tion complexity. As shown in Fig.4.2, the proposed method simplifies thestructure used for the estimation: instead of three Kalman filters, only oneKalman filter and one transformation is required. Further, the total number

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4.2. POSITIVE- AND NEGATIVE-SEQUENCE ESTIMATION: DIRECT METHOD

of variables to be estimated in Fig.4.1 is reduced by 2/3 of that required in

Fig.4.2.

va(n)

vb(n)

vc(n) Ac, c

Filter

Filter

Filter

Ab, b T3

Adaptive

Adaptive

Adaptive

Aa, a

V0

V1

V2

Figure 4.1: Indirect method: three adaptive filters and one transformation

va(n)

vb(n)

vc(n)

ej0n

Xv(n) vdq(n) Adaptive

FilterT1V1

V2

Figure 4.2: Direct method: one adaptive filter and one transformation

In the proposed method, the sampled values of three-phase voltages va(n),vb(n) and vc(n) are -transformed to a complex voltage v(n). Then, thiscomplex voltage v(n) is dq-transformed, resulting in a complex voltagevdq(n), as it is shown in Fig.4.2. The result contains the embedded positive-and negative-sequences. Thereby, the positive- and negative-sequences aremodelled by a state-space equation, and the complex Kalman filter is used toestimate the state vector iteratively. In this chapter the method will be usedto estimate positive- and negative-sequence voltages. The same method canbe applied to estimate positive- and negative-sequence currents.

4.2 Positive- and Negative-Sequence Estima-tion: Direct Method

In this section, the -transform and the dq-transform (or, the Parks trans-form) will be described. The complex Kalman filter for estimating thepositive- and negative-sequences will also be described, where the three-phasevoltage is modelled as consisting of voltage of fundamental frequency (50 Hzin Europe or 60 Hz in USA) under harmonics and noise distortions.

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4.2.1 The - and the dq-Transforms

Consider a three-phase system with the following voltages:

va(n) =

2Va cos(0n + a) + Na(n)

vb(n) =

2Vb cos(0n + b) + Nb(n) (4.1)

vc(n) =

2Vc cos(0n + c) + Nc(n)

where va(n), vb(n) and vc(n) are the sampled voltages; Va, Vb and Vc are therms or effective value; a, b and c are the phase-angles; 0 is the discreteangular frequency (0 = 2f0/fs), being f0 the fundamental frequency ofvoltage, i.e. 50 Hz in Europe and 60 Hz in USA, and f

s

the sampling fre-quency; Na(n), Nb(n) and Nc(n) are i.i.d. white Gaussian noise with zeromean and 2 variance [WGN(0,2)]; and n is the discrete time index. Definethe -transform for the three-phase voltages as follows:

v(n) =2

3[va(n)ea + vb(n)eb + vc(n)ec] (4.2)

where ea = 1, eb = ej 23 = a and ec = e

j 23 = a2 = a, and a is a rotationaloperator. (4.2) can be interpreted as the projection of three voltage phasorsonto the -space, as sketched in Fig.4.3 for any time n. The dq-transform

rotating at

ec vc(n)

speed.

eb vb(n)

ea va(n)

v(n)

Figure 4.3: Representing three-phase voltages in -space

is then applied as follows

vdq(n) = v(n)ej0n, (4.3)

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it should be noted that both v(n) and vdq(n) are complex voltages as a

function of time. The last transformation could be interpreted as a syn-chronization of the -space at angular frequency o set to be equal to thefundamental voltage frequency in this study.

q

rotating atd

2 speed.

V1

V2

Figure 4.4: Positive- and negative-sequences in the dq-space

The symmetrical component voltagesare defined from the voltage phasors:Va = Vae

ja, Vb = Vbejb and Vc = Vce

jc and then, it follows

V0V1V2

= 13

111

1aa2

1a2

a

VaVbVc

(4.4)where V0 = V0e

j0 , V1 = V1ej1 and V2 = V2e

j2 are the symmetrical compo-nent voltages (zero-, positive- and negative-sequences), which are defined asphasors as well. Re-writing (4.2) in the complex form and considering (4.4),it follows,

v(n) =

2

V1ej0n + V2

ej0n

+ N(n)ej0n (4.5)

then, synchronizing (4.5) at ejon leads to re-write (4.3) as

vdq(n) =

2 V1 + V2ej20n + N(n). (4.6)

For sinusoidal phase voltages defined as in (4.1), the first component V1in (4.6) is a complex constant number, while the second component V2 is acomplex number counter rotating at an angular speed of 20, which are thepositive- and negative-sequence voltages, respectively. Any unbalance in thethree-phase system will appear as an non-zero rotating V2 in the dq spaceas shown in Fig. 4.4. The complex noise N(n), is a linear combination ofnoise in three-phases, has zero mean and 4

32 variance, see Appendix A.1 for

more details. The positive- and negative-sequences are estimated separatelyin the dq space from (4.6) as detailed in the next section.

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4.2.2 Estimation of Positive- and Negative-Sequences

Using state-space modelling, the positive- and negative-sequence voltages in(4.6) can be estimated by a complex Kalman filter. The state equation andobservation equation associated with a Kalman filter can be described asfollows

X(n + 1) = A(n)X(n) + U(n)

Y(n) = H(n)X(n) + V(n) (4.7)

where X(n) is a complex state vector sized 2x1, A(n) is a transition matrix

sized 2x2, U(n) is a vector containing zero-mean white noise with 2u variance,

and V(n) is the observation noise which is white zero-mean with 2v variance.

Strictly speaking a complex signal should hold a tilde x(n), however theestimated value of a complex signal would have both symbols x(n). And ifthe state number is added the notation gets more complicated. Therefore,we decided to relax the notation and not use tilde in all sections related toKalman filter.

State-space modelling of the three-phase system

For the three-phase system described in (4.1), the dq-transform can be foundby defining the following state vector

X(n) = [x1 x2]T

n =V1 V2

ej20nT

n, (4.8)

it follows that the state equation of the Kalman filter is

x1x2

n+1

=

10

0ej20

x1x2

n

+

u1u2

n

(4.9)

and the observation equation becomes

y(n) =

2[ 1 1]

x1x2

n

+ v(n) (4.10)

and the negative-sequence is found by applying

V2 = x2e

j20n. (4.11)

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4.2.3 Extension to Three-Phase Systems Containing

K Harmonics

The model used for the phase voltages in the previous subsection is that ofnon-distorted sinusoids, e.g. voltage without harmonic distortion. In realitythe power-system voltage is always distorted. This distortion is interpretedby the Kalman filter as a fast fluctuation in the amplitude and phase-angleof the complex phase voltages. The distortion of the voltages is generallydescribed as the superposition of a fundamental and a number of harmonicsat integer multiples of the fundamental frequency. This harmonic model canbe used as a basis for a Kalman filter estimation of the positive- and negative-

sequences. Such a model would not only estimate the fundamental complexvoltages but also the complex voltages for the harmonic components.Consider a three-phase system containing K harmonics, to be modelled

through the following expression:

va(n) =

2K

k=1

Vka cos(nk + ka) + Na(n)

vb(n) =

2K

k=1

Vkb cos(nk + kb ) + Nb(n)

vc(n) =

2K

k=1

Vkc

cos(nk + k

c

) + Nc(n) (4.12)

using the similar way as in the previous case, we may define the symmetricalcomponents for each harmonic k as [39]

Vk0Vk1Vk2

= 1

3

11

1

1aa2

1a2

a

VkaVkbVkc

. (4.13)

After some algebraic manipulations using (4.2), (4.3) and (4.13), and apply-ing the dq-transform to the phase voltages modelled by (4.12), it follows

vdq(n) = 2 Kk=1

Vk1 ej(k1)0n +K

k=1

Vk2 ej(k+1)0n

+ N(n) (4.14)

where the relation k = k0 is applied.

Model for the three-phase system containing K harmonics

Considering the system described in (4.14), the state vector of the complexKalman filter contains 2K elements and is defined as

X(n) =x1 x2 x(2K1) x2K

Tn

(4.15)

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= V11 (V12 )ej20n Vk1 e

j(K1)0n (Vk2 )

ej(K+1)0nTn

.

The state equation associated with the system becomes

X(n + 1) = A(n)X(n) +

u1...

u2K

n

(4.16)

considering

A(n) =

1

0...00

0

ej20

...00

. . .

0

0...ej(K1)0

0

0

0...0

ej(K+1)0

(4.17)

and the observation equation for the system becomes

y(n) =

2 [1 . . . 1]

x1...

x2K

n

+ v(n) (4.18)

The negative-sequence associated with the fundamental (k = 1) and theharmonics (k [2, K]) can be found by applying

V2k(n) = x2k(n)e

j(k+1)0n. (4.19)

Kalman filter algorithm

The algorithm for iteratively estimating the state vector is summarized step by stepas follows, more details can be found in [40]. The covariance matrix of modelnoise is Q = E[U(n)UT(n)] = 2uI, where Im is the identity matrix of order m

equal to the number of state variables. The covariance matrix of observationnoise is C = E[V(n)VT(n)] = 2v Il, where Il is the identity matrix of order lequal to the number of observation equations. The initial conditions are setas x(0|0) = [0 0]T and P(0|0) = RIn and R 1.Step 1 Prediction:

x(n|n 1) = A(n 1)x(n 1|n 1)Step 2 One step prediction error covariance matrix:

P(n|n 1) = A(n 1)P(n 1|n 1)A(n 1)H + Q

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Step 3 Kalman gain matrix:

K(n) = P(n|n 1)HT[HP(n|n 1)HT + C]1Step 4 Filtering:

x(n|n) = x(n|n 1) + K(n)[y(n) Hx(n|n 1)]Step 5 Filtering error covariance matrix:

P(n|n) = [Im K(n)H]P(n|n 1)An interesting property of the Kalman filter is that the Kalman gain K(n)and the error covariance matrix P(n|n) do not depend on the data x(n),they are dependent on the model and observation noises. In this work, the

variances of both noises were chosen to be fixed under the assumption that thenoises were stationary, no adaptive update was done, which is not probablytrue when dealing with real waveforms.

4.2.4 Currents in the dq-frame

From the measured three-phase currents a dq-transform can be easily found.A summary of the relevant equations regarding positive- and negative-sequenceestimation for currents is as follows

ia(n) =

2Ia cos(on + a) + Ma(n)

ib(n) = 2Ib cos(on + b) + Mb(n) (4.20)ic(n) =

2Ic cos(on + c) + Mc(n)

where Ia, Ib and Ic are the rms or effective value for each electrical phase; a,b and c are the phase-angles, o is the angular frequency; Ma(n),Mb(n)andMc(n) are i.i.d. WGN(0,

2) and n is the discrete time index. The dq-transform is

idq(n) =2

3[ia(n)ea + ib(n)eb + ic(n)ec] e

j0n (4.21)

where ea = 1, eb = ej2

3 = a and ec = ej2

3 = a2 = a, and a is a rotationaloperator representing a rotation over 120. Consider a three-phase currentsystem containing K harmonics, to be modelled through the following ex-pression:

ia(n) =

2K

k=1

Ika cos(nk + ka ) + Ma(n)

ib(n) =

2K

k=1

Ikb cos(nk + kb ) + Mb(n) (4.22)

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ic(n) = 2K

k=1 Ik

c cos(nk + k

c ) + Mc(n)

where k = k0. The resulting complex dq current can be re-written as

idq(n) =

2

K

k=1

Ik1 ej(k1)0n +

Kk=1

Ik2ej(k+1)0n

+ M(n) (4.23)

where M(n) has a zero mean and variance equal to 43

2, see Appendix A.1for more details. The direct method based on Kalman filter can be appliedto currents in the same way as it was applied to voltages in Sections 4.2.2

and 4.2.3., for the fundamental frequency and harmonics case, respectively.

4.3 Performance of Indirect and Direct Meth-ods

This section analyzes the Cramer-Rao Lower Bound (CRLB) for the dq-transform model. Additionally, a comparison between the indirect and directmethods is performed in terms of computational complexity.

4.3.1 Cramer-Rao Lower Bound (CRLB)The Cramer Rao lower bound(CRLB) will give the lowest bound for thevariance of the estimated parameter from a given model. One way to assesshow well a parameter can be estimated is by finding how close are the varianceof the estimations to this lower bound. In the next Section, CRLB for thepositive- and negative-sequences is found from the dq-transform.

CRLB for a single sinusoid

A single sinusoid is modelled as

x(n) = 2Acos(0n + ) + N(n) (4.24)

where, A is the rms value of the sinusoid, is phase-angle, 0 is the known an-gular frequency, and N(n) is WGN(0,2). Having N samples (n = 0, 1,...,N1) define a vector x = [x(0) . . . x(N 1)]T. If A and are parameters to beestimated define a parameter vector as = [A ]T. The Cramer-Rao LowerBound (CRLB) for a single sinusoid in vector form follows as (Theorem 3.2[40])

var()i [I()]1ii (4.25)

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4.3. PERFORMANCE OF INDIRECT AND DIRECT METHODS

where the Fisher information matrix I() for a signal corrupted by WGN is

simplified to (Section 3.9 [40])

[I()]ij =1

2

N1n=0

x[n; ]

i

x[n;]

j. (4.26)

The CRLB for each estimated parameter, as shown in Appendix A.2, isknown to be equal to

var(A) 2

N(4.27)

var() 2

NA2 . (4.28)

CRLB for dq-transform

Since the dq-transform is a linear combination of three-phase signals resultingin a complex signal embedded in complex white Gaussian noise(CWGN).Then, if |V1|, |V2|, 1 and 2 are the parameters to be estimated define aparameter vector = [|V1| |V2| 1 2]T. The Cramer-Rao Lower Bound(CRLB), for such a complex signal, in vector form is known to be equal to(Section 15.7 [40])

var()i

[I()]1ii

(4.29)

where I() is the complex Fisher information matrix defined as

[I()]ij = tr

C1x ()

Cx()

iC1x ()

Cx()

j

+ 2Re

H()

iC1x ()

()

j

. (4.30)

The CRLB for each estimated parameter, as shown in Appendix A.4, isknown to be equal to

var(|V1|) 2

3N(4.31)

var(|V2|) 2

3N(4.32)

var( 1 ) 2

3|V1|2N(4.33)

var( 2 ) 2

3|V2|2N. (4.34)

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Table 4.1: Simulated unbalanced three-phase voltages

Amplitude (p.u.) Phase-angle()

va(n): 1.00 5

vb(n): 0.10 125

vc(n): 1.00 245

Approaching the CRLB with indirect and direct methods

Unbalanced three-phase voltages were simulated applying data presented in

Table.4.1 to the model described in (4.1), assuming fundamental frequency50 Hz and sampling frequency 3800 Hz, for three-phase signals with a sam-pling window of ten cycles, i.e. a window of 76 samples per cycle. WGN wasadded to each voltage phase varying the signal to noise ratio(SNR) from 6to 60 dB (= 0.5 to 0.001 p.u.). The variances of the positive- and negative-sequence estimations were found for a total of 1000 independent realizations,i.e. the same three-phase voltage data with different corrupting noise. Thevariances are found over a window of one cycle. And then, the resultingvariances from the different realizations are averaged arithmetically. Fig.4.5to Fig.4.8 show the CRLB and the bound reached for both methods whenestimating parameters under different noise contents [41]. The results show

that the indirect method approaches the CRLB sightly better than the di-rect method although the differences are small. These results suggest thatthe two methods provide similar estimation accuracy however a slightly bet-ter performance is reached with the indirect method at more computationalcomplexity.

4.3.2 Computational Complexity of Indirect and Di-rect Methods

This subsection analyzes the performance of both methods in terms of com-putational complexity.

Computational complexity

Two stages are required to implement the indirect method. The first stagerequires six real state-variables when estimating a three-phase system. Afterestimating these variables a second stage is performed to compute three com-plex variables; positive-, negative- and zero-sequences. The direct methodonly takes one stage which has the dq-transform built in. This block re-

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101520253035404550556090

80

70

60

50

40

30

20

10

SNR

10log10[var(|V

1|)]

Figure 4.5: CRLB: true(solid), indirect(dot) and direct(dash) method for positive-sequence amplitude estimation

101520253035404550556080

70

60

50

40

30

20

10

0

SNR

10

log10[var(

1)]

Figure 4.6: CRLB: true(solid), indirect(dot) and direct(dash) method for positive-sequence phase-angle estimation

35


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101520253035404550556090

80

70

60

50

40

30

20

10

SNR

10log10[var(|V

2|)]

Figure 4.7: CRLB: true(solid), indirect(dot) and direct(dash) method for negative-sequence amplitude estimation

101520253035404550556080

70

60

50

40

30

20

10

0

SNR

10

log10[var(

2)]

Figure 4.8: CRLB: true(solid), indirect(dot) and direct(dash) method for negative-sequence phase-angle estimation

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Table 4.2: Computational complexity of the Kalman filter

Number of indirect method direct method

state-variables(real): m 6 4

observation-equation(real): l 3 2

quires four real state-variables (two complex state-variables) to compute twocomplex variables; the positive- and negative-sequences. In Table.4.2 theminimum number of variables to estimate with both methods is summa-rized. The main difference between both methods relies on the number of

state-variables to estimate.The computational complexity is traditionally measured in terms of flops -floating point operations; such as addition or multiplication of real (scalar)numbers. Each addition or multiplication is counted as one flop[42]. Thenumber offlops required to implement the Kalman filter algorithm step by stepis summarized in Table.4.3, where m is the number of real state-variablesto estimate and l is the number of real observation-equations required. e.g.Kalman filter order = 1 - means only fundamental frequency; indirect methodrequires: m = 6 and l = 3; and direct method requires m = 4 and l = 2; foronly voltages. In case that voltages and currents are to be estimated in thesame Kalman filter the given numbers (m and l) shall be doubled.

Table 4.3: Number of flops (Kalman filter)

Step Additions Multiplications(1): (m 1)m m2(2): m2(2m 1) 2m3(3)a: 2(m 1)ml + (m 1)l2 + l2 + l3 + (l 1)ml 2m2l + 2ml2 + l3(4): (m 1)l + l + (l 1)m + m 2ml(5): m2(l + m 1) m2l + m3a

matrix inverse implemented by LU Decomposition. [43].

Fig.4.9 shows the total number ofKflops (= 1000 flops) required to performKalman filter each time n when m state-variables are to be estimated. If bothmethods are implemented using Kalman filter, it is shown from this figurethat the direct method is lesser computational complex than the indirectmethod. 1

1These results are for the general Kalman filter implementation. Specific algorithmoptimization can be done for each method, which is out of the scope of this chapter.

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4 6 8 120

0.5

1

1.5

2

0

0.5

1

1.5

2

number of state variables m

Kflops

Figure 4.9: Total flops = number of Additions plus Multiplications

100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

N samples

sec

Figure 4.10: Computation time when estimating: indirect method: KF order 5(solid) -KF order 1(dot); direct method: KF order 5(dash) - KF order 1(dash-dot)

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4.4. TEST RESULTS FOR SIMULATED AND MEASURED DATA

Finally, Fig.4.10 shows the required time to estimate the positive- and

negative-sequence, for a Kalman Filter of orders 1 and 5, using both meth-ods. i.e. order 1 means only fundamental frequency and order 5 meansfundamental frequency plus second, third, fourth and fifth harmonics. Testswere performed for different simulated voltage signals of 100 to 1000 samplesin 100 sample steps. All tests were done with Matlab in a 1 GHz PC, there-fore these tests are just a relative indication of the running time. Results canvary depending on the type of computer used and the algorithm program-ming techniques. 2 From this figure, it is possible to conclude that the directmethod runs faster than the indirect method in computation terms.

4.4 Test Results for Simulated and Measured

Data

In this section, a comparison between the direct and the indirect methodsbased on the average relative error for the estimation of the positive- andnegative-sequences in a noisy environment is presented. Then a simulatedvoltage dip is tested under controlled situation to assess the accuracy of bothmethods in terms of square error and detection time. And finally, a real dataset is taken from a distribution utility power quality database in order to find

the positive- and negative-sequences. Square error and detection time are notpossible to evaluate for this record because the true values are unknown.

4.4.1 Estimation Under Noisy Conditions

The direct method was developed for estimating only positive- and negative-sequences under noisy conditions. The relative error is computed between thetrue positive-, negative-sequence magnitudes and the phase-angle differencebetween the positive- and negative-sequence [x(n)] and the estimated [x(n)]by the indirect and the direct methods. When the relative error (RE) is

defined asRelative Error(n) =

|x(n)| |x(n)||x(n)| 100(%) (4.35)

Two cases have been simulated with data presented in Table.4.1, under theassumption that the system is unbalanced and suffers a voltage dip due toa fault in phase-b which has a magnitude of 1.0 p.u. for the pre- and afterdisturbance segments. The magnitude of the voltage dip(retained voltage)is set from 0.10 to 0.90 p.u. (in steps of 0.01 p.u.). Estimated values for a

2Algorithm programs were not optimized for these tests.

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0 0.2 0.4 0.6 0.8 110

3

102

101

100

Retained voltage amplitude (p.u.)

AverageRelativeError%

SNR = 10 dB

SNR = 30 dB

Figure 4.11: Average relative error of magnitude in positive-sequence: indirect(solid)and direct(dash) methods under noise

total of 1000 independent realizations were averaged. When the voltage dipis shallow (retained voltage around 0.90 p.u.) a big error is expected from(4.35).

The noise level has set to SNR 10 dB for the first case and SNR 30 dB

for the second. Other possible scenarios would be between these two val-ues. Fig.4.11 shows the average relative error of estimating the magnitudeof the positive-sequence |V1| with both methods and for the two cases. Theaverage relative error of the estimated magnitude of the positive-sequence issmall because the positive-sequence energy is dominant in the signal. Theaccuracy of estimating the positive-sequence (magnitude and phase-angle) isacceptable with both methods. Even for SNR = 10 dB the resulting erroris less than 1%. The performance of both methods is very similar. Fig.4.12shows the average relative error of estimating the magnitude of the negative-sequence |V2|, with both methods. The average relative error of the estimatedmagnitude of the negative-sequence is bigger than that shown in Fig.4.11.The higher relative error for the negative-sequence voltage is due to its abso-lute value being low, especially for high retained voltage. This is related tothe general problem of estimating small difference between two noisy signals.The differences between the two methods are marginal(less than 0.5%). Itmeans that almost the same accuracy is reached with both methods whenestimating the negative-sequence magnitude. Then, Fig.4.13 shows the er-ror of the phase-angle difference between

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