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Signal Processing Tools forPower Quality Event Classification

RAFAEL FLORES

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

Goteborg, Sweden 2003

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

RAFAEL FLORESSignal 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, SwedenISSN 1651-4998

Printed in Sweden by Chalmers ReproserviceGoteborg, November 2003

.....to Cecilia, my lovely wife

and Renzo, my dear son.

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- andnegative-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 whichmakes 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.

i

Contents

Abstract i

Contents iii

Acknowledgements vii

Abbreviations ix

Notation xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . 4

2 An Overview on Power-System Event Classification 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Most Used Techniques . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Rms - Root Mean Square . . . . . . . . . . . . . . . . 72.2.2 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

iii

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

3 Power-System Database 153.1 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 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Positive- and Negative-Sequence Estimation: Direct Method . 25

4.2.1 The αβ- 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 . . . . . . . . . . 324.3.1 Cramer-Rao Lower Bound (CRLB) . . . . . . . . . . . 324.3.2 Computational Complexity of Indirect and Direct Meth-

ods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Test Results for Simulated and Measured Data . . . . . . . . . 39

4.4.1 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 535.1 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

iv

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

6 Active and Reactive Power Estimation 656.1 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 736.5.2 Simulated Voltage Dip Conditions . . . . . . . . . . . . 76

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Conclusions and Future Research 817.1 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

v

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

vii

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.

ix

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.

x

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.trX 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.φ 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.

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

xi

Cx the covariance of matrix x.log(x) the natural logarithm of x.~e unitary vector.w.r.t. with respect to.WGN(µ, σ2) white Gaussian noise with mean µ and variance σ2.CWGN(µ, σ2) complex white Gaussian noise with mean µ and variance σ2.

xii

Chapter 1Introduction

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 somecurrent 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 intheir networks.

In a previous project, relevant work has been done on defining how toidentify 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.

1

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 patternrecognition 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

2

1.3. OUTLINE

and easy accessible is justified. Moreover, an open architecture database isdesigned 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 αβ-transform followedby the dq-transform using 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

3

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 eventfrom 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 stepahead 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.

4

Chapter 2An 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, let’s 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 90’s, 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 70’s to hundreds of people in the2000’s; 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]

5

CHAPTER 2. AN OVERVIEW ON POWER-SYSTEM EVENT CLASSIFICATION

2000’s90’s80’s70’s

Voltmeter

Tape

TECHNOLOGY

Oscilloscopes

Oscillographs

GraphicsPaper

Digital Signal Processing

Computers

Mass storage devices

Communitacions Networking−internet

Decision−making

Data mining

Pattern recognition

USERS

Field service engineer

Power Quality groups

Industrial/plant/facilities

Utility 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 threepotential 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(ωn+ φ(n)) (2.1)

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

6

2.2. MOST USED TECHNIQUES

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

N∑

i=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

N−1∑

n=0

s(n)e−jωkn (2.3)

7


where S(k) is the DFT evaluated at the frequency ωk, N is the window length,ωk = 2kπ

N is a set of fixed and equally spaced frequencies, 0 ≤ k ≤ N − 1.

The fast Fourier transform (FFT) is the DFT’s computational efficientimplementation, 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 doesn’t 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 treescombined 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].

8

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.

9


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.

Pre−processingII III IV

Kalman filterWavelets

I

Parametersestimation

Rules or logicPatternrecognitionSegmentation

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 Systems

Expert 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 setof 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

10

2.4. WHAT IS THE NEXT?

applied for detection, quantification and classification of a variety of powerquality 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 Gaussianpdf ’s, its performance in power quality events is not fully accepted due to thenon-Gaussian nature of the event pdf ’s 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 extractionstage 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

11


algorithms have shown interesting results.Topics which shall become interesting in order to continue research in this

field 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. Pdf ’s 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.

12

2.5. TOPICS FOR CONTINUING RESEARCH IN THIS FIELD

2.5 Topics for Continuing Research in this

Field

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. Pdf ’s 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 detectionapproaches 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 pdf ’s of the residual error after estimating the main pa-rameters may have a good potential in future research.

13


14

Chapter 3Power-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 towrite 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.

15

CHAPTER 3. POWER-SYSTEM DATABASE

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

16

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 informationof 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 eventsmust be stored in a database with an open structure.

How to build this database and what it should handle are treated in thenext 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 thischapter, 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.

17


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

Open

architecture

database

Power system

real data

Power system

Simulated

data

Matlab

Test and

development platform

GUI

real data

Sweden

real data

UK

real data

Perú

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

18


3. site: contains information about the location where the waveformswere 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 becauseis available in all Chalmer’s computers under Windows XP Pro operativesystem.

config

indexcod_site*C_descriptionQ_VfactorQ_IfactorQ_freq_sampleC_typ_monitor

records_head

cod_siteC_dateC_start_timecod_for*

diag

indexcod_for*C_typ_eventC_desc_event

records_det

indexcod_for*C_timeQ_VaQ_VbQ_VcQ_IaQ_IbQ_IcQ_In

site

indexcod_site*C_companyC_locationQ_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

19


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 describeas follow:

1. time-domain three-phase voltages;

2. time-domain three-phase currents;

3. amplitude of three-phase voltages;

20


0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

msec

p.u.

0 50 100 150 200 250 300 350−3

−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 deg

rees

Angle Fundamental Voltage

50 100 150 200 250 300 350

−40

−20

0

20

40

60

msec

o deg

rees

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.

21


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 50 100 150 200 250 300 350−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

msec

p.u.

Active Power

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

2

msec

p.u.

Reactive Power

0 50 100 150 200 250 300 350−1

−0.5

0

0.5

1

msec

Cos

θ

Power Factor

0 50 100 150 200 250 300 350−0.03

−0.02

−0.01

0

0.01

0.02KF Residuals

msec

Err

or (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 Conclusions

The 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.

22

Chapter 4Three-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 αβ-transform followedby 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 thepositive- 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.

23

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 correctidentification 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-componentsignals. 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, LS’s 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 arealso 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

24

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 inFig.4.2.

va(n)

vb(n)

vc(n) Ac, φc

Filter

Filter

Filter

Ab, φb T3

Adaptive

Adaptive

Adaptive

Aa, φa

V0

V1

V2

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

va(n)

vb(n)

vc(n)

e−jω0n

Xvαβ(n) vdq(n) Adaptive

FilterT1

V1

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 αβ-transformed to a complex voltage vαβ(n). Then, thiscomplex voltage vαβ(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 αβ-transform and the dq-transform (or, the Park’s 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.

25



4.2.1 The αβ- and the dq-Transforms

Consider a three-phase system with the following voltages:

va(n) =√

2Va cos(ω0n+ φa) +Na(n)

vb(n) =√

2Vb cos(ω0n+ φb) +Nb(n) (4.1)

vc(n) =√

2Vc cos(ω0n+ φc) +Nc(n)

where va(n), vb(n) and vc(n) are the sampled voltages; Va, Vb and Vc are therms or effective value; φa, φb and φc are the phase-angles; ω0 is the discreteangular frequency (ω0 = 2πf0/fs), being f0 the fundamental frequency ofvoltage, i.e. 50 Hz in Europe and 60 Hz in USA, and fs the sampling fre-quency; Na(n), Nb(n) and Nc(n) are i.i.d. white Gaussian noise with zeromean and σ2 variance [WGN(0,σ2)]; and n is the discrete time index. Definethe αβ-transform for the three-phase voltages as follows:

vαβ(n) =2

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

where ~ea = 1, ~eb = ej 2π3 = a and ~ec = e−j 2π

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

β

rotating at

α

~ec vc(n)

ω speed.

~eb vb(n)

~ea va(n)

vαβ(n)

Figure 4.3: Representing three-phase voltages in αβ-space

is then applied as follows

vdq(n) = vαβ(n)e−jω0n, (4.3)

26


it should be noted that both vαβ(n) and vdq(n) are complex voltages as afunction of time. The last transformation could be interpreted as a syn-chronization of the αβ-space at angular frequency ωo set to be equal to thefundamental voltage frequency in this study.

q

rotating atd

2ω speed.

V1

V ∗

2

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

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

jφa , Vb = Vbejφb and Vc = Vce

jφc and then, it follows

V0

V1

V2

=

1

3

111

1aa2

1a2

a

Va

Vb

Vc

(4.4)

where V0 = V0ejφ0 , V1 = V1e

jφ1 and V2 = V2ejφ2 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,

vαβ(n) =√

2(

V1ejω0n + V2

∗ejω0n)

+ N(n)ejω0n (4.5)

then, synchronizing (4.5) at e−jωon leads to re-write (4.3) as

vdq(n) =√

2(

V1 + V2∗e−j2ω0n

)

+ N(n). (4.6)

For sinusoidal phase voltages defined as in (4.1), the first component V1

in (4.6) is a complex constant number, while the second component V2 is acomplex number counter rotating at an angular speed of 2ω0, 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 − space

as shown in Fig. 4.4. The complex noise N(n), is a linear combination ofnoise in three-phases, has zero mean and 4

3σ2 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.

27



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 matrixsized 2x2, U(n) is a vector containing zero-mean white noise with σ2

u variance,and V (n) is the observation noise which is white zero-mean with σ2

v variance.

Strictly speaking a complex signal should hold a tilde x(n), however theestimated value of a complex signal would have both symbols ˆx(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∗e−j2ω0n

]T

n, (4.8)

it follows that the state equation of the Kalman filter is

[

x1

x2

]

n+1

=

[

10

0e−j2ω0

] [

x1

x2

]

n

+

[

u1

u2

]

n

(4.9)

and the observation equation becomes

y(n) =√

2 [1 1]

[

x1

x2

]

n

+ v(n) (4.10)

and the negative-sequence is found by applying

V2 = x∗2e−j2ω0n. (4.11)

28


4.2.3 Extension to Three-Phase Systems ContainingK 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 modelledthrough the following expression:

va(n) =√

2K

∑

k=1

V ka cos(nωk + φk

a) +Na(n)

vb(n) =√

2K

∑

k=1

V kb cos(nωk + φk

b ) +Nb(n)

vc(n) =√

2K

∑

k=1

V kc cos(nωk + φk

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]

V k0

V k1

V k2

=

1

3

111

1aa2

1a2

a

V ka

V kb

V kc

. (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

[

K∑

k=1

V k1 e

j(k−1)ω0n +K

∑

k=1

V k2

∗e−j(k+1)ω0n

]

+ N(n) (4.14)

where the relation ωk = kω0 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(2K−1) x2K

]T

n(4.15)

29



=[

V 11 (V 1

2 )∗

e−j2ω0n · · ·V k1 e

j(K−1)ω0n (V k2 )

∗

e−j(K+1)ω0n]T

n.

The state equation associated with the system becomes

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

u1...

u2K

n

(4.16)

considering

A(n) =

10...00

0e−j2ω0

...00

· · ·· · ·. . .

· · ·· · ·

00...

ej(K−1)ω0

0

00...0

e−j(K+1)ω0

(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) = x∗2k(n)e−j(k+1)ω0n. (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)] = σ2

uI, where Im is the identity matrix of order mequal to the number of state variables. The covariance matrix of observationnoise is C = E[V (n)V T (n)] = σ2

vIl, 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

30


Step 3 Kalman gain matrix:K(n) = P (n|n− 1)HT [HP (n|n− 1)HT + C]−1

Step 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, thevariances 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(ωon+ ψa) +Ma(n)

ib(n) =√

2Ib cos(ωon+ ψb) +Mb(n) (4.20)

ic(n) =√

2Ic cos(ωon+ ψc) +Mc(n)

where Ia, Ib and Ic are the rms or effective value for each electrical phase; ψa,ψb and ψc are the phase-angles, ωo is the angular frequency; Ma(n),Mb(n)andMc(n) are i.i.d. WGN(0,σ2) 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

−jω0n (4.21)

where ~ea = 1, ~eb = ej 2π3 = a and ~ec = e−j 2π

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(nωk + ψk

a) +Ma(n)

ib(n) =√

2K

∑

k=1

Ikb cos(nωk + ψk

b ) +Mb(n) (4.22)

31



ic(n) =√

2K

∑

k=1

Ikc cos(nωk + ψk

c ) +Mc(n)

where ωk = kω0. The resulting complex dq current can be re-written as

idq(n) =√

2

[

K∑

k=1

Ik1 e

j(k−1)ω0n +K

∑

k=1

Ik2

∗e−j(k+1)ω0n

]

+ M(n) (4.23)

where M(n) has a zero mean and variance equal to 43σ2, see Appendix A.1

for 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 4.2.2and 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(ω0n+ φ) +N(n) (4.24)

where, A is the rms value of the sinusoid, φ is phase-angle, ω0 is the known an-gular frequency, andN(n) is WGN(0,σ2). HavingN samples (n = 0, 1, ..., N−1) define a vector x = [x(0) . . . x(N − 1)]T . If A and φ are parameters to beestimated define a parameter vector as θ = [A φ]T . The Cramer-Rao LowerBound (CRLB) for a single sinusoid in vector form follows as (Theorem 3.2[40])

var(θ)i ≥ [I(θ)]−1ii (4.25)

32

4.3. PERFORMANCE OF INDIRECT AND DIRECT METHODS

where the Fisher information matrix I(θ) for a signal corrupted by WGN issimplified to (Section 3.9 [40])

[I(θ)]ij =1

σ2

N−1∑

n=0

∂x[n; θ]

∂θi

∂x[n; θ]

∂θj

. (4.26)

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

var(A) ≥ σ2

N(4.27)

var(φ) ≥ σ2

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|, φ1 and φ2 are the parameters to be estimated define aparameter vector ξ = [|V1| |V2| φ1 φ2]

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(ξ)i ≥ [I(ξ)]−1ii (4.29)

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

[I(ξ)]ij = tr

[

C−1x (ξ)

∂Cx(ξ)

∂ξi

C−1x (ξ)

∂Cx(ξ)

∂ξj

]

+ 2Re

[

∂µH(ξ)

∂ξi

C−1x (ξ)

∂µ(ξ)

∂ξj

]

. (4.30)

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

var(|V1|) ≥ σ2

3N(4.31)

var(|V2|) ≥ σ2

3N(4.32)

var( φ1 ) ≥ σ2

3|V1|2N(4.33)

var( φ2 ) ≥ σ2

3|V2|2N. (4.34)

33



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 inTable.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 (σ= 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 showthat 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-

34


1015202530354045505560−90

−80

−70

−60

−50

−40

−30

−20

−10

SNR

10 lo

g10[

var (

|V1|)]

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

1015202530354045505560−80

−70

−60

−50

−40

−30

−20

−10

0

SNR

10 lo

g10[

var (

φ 1)]

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

35



1015202530354045505560−90

−80

−70

−60

−50

−40

−30

−20

−10

SNR

10 lo

g10[

var (

|V2|)]

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

1015202530354045505560−80

−70

−60

−50

−40

−30

−20

−10

0

SNR

10 lo

g10[

var (

φ 2)]

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

36


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 ofstate-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 of flops 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 +m3

a matrix inverse implemented by LU Decomposition. [43].

Fig.4.9 shows the total number of Kflops (= 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.

37



4 6 8 120

0.5

1

1.5

2

0

0.5

1

1.5

2

number of state variables − m

Kfl

ops

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)

38

4.4. TEST RESULTS FOR SIMULATED AND MEASURED DATA

Finally, Fig.4.10 shows the required time to estimate the positive- andnegative-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 findthe 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) isdefined as

Relative 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.

39



0 0.2 0.4 0.6 0.8 110−3

10−2

10−1

100

Retained voltage − amplitude (p.u.)

Ave

rage

Rel

ativ

e E

rror

%

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 dBfor 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 the positive- and negative-sequence(φ1 − φ2). The shallower the voltage dip the bigger the error. Both meth-ods were shown to be biased when estimating this parameter under noise

40


0 0.2 0.4 0.6 0.8 110−2

10−1

100

101

102


SNR = 10 dB

SNR = 30 dB Ave

rage

Rel

ativ

e E

rror

%

Figure 4.12: Average relative error of magnitude in negative-sequence: indirect(solid)and direct(dash) methods under noise

0 0.2 0.4 0.6 0.8 110−1

100

101

102

Ave

rage

Rel

ativ

e E

rror

% SNR = 10 dB

SNR = 30 dB


Figure 4.13: Average relative error of the phase-angle difference between the positive-and negative-sequences: indirect(solid) and direct(dash) methods under noise

41



Table 4.4: Unbalanced voltage dip: phase-a

I (p.u.) I () II (p.u.) II () III (p.u.) III ()

Fund: 1.0154 0 0.8762 0 1.0159 0

3rd: 0.0037 -30.92 0.0079 -149.61 0.0045 -50.20

5th: 0.0100 45.86 0.0022 47.38 0.0101 21.12

7th: 0.0044 103.54 0.0014 -80.25 0.0046 80.27

Table 4.5: Unbalanced voltage dip: phase-b


Fund: 1.0250 0 0.9444 0 1.0245 0

3rd: 0.0032 5.09 0.0021 133.74 0.0032 -6.73

5th: 0.0080 -63.05 0.0075 45.57 0.0090 -86.11

7th: 0.0034 123.87 0.0033 112.61 0.0033 81.95

conditions. Fortunately, almost always power-system waveforms (voltages orcurrents) will content harmonics (which can be modelled) and small levelsof noise (SNR>30dB), so from the figures shown, we can say that the esti-mated positive- and negative-sequence with the direct method reach almostthe same accuracy than the indirect method. These results confirm that thephase-angle difference (φ1 − φ2) is very sensitive to noise which have beenshown previously when the CRLB was studied in Section 4.3.1.

4.4.2 Simulated Unbalanced Voltage Dip: Square Er-ror and Detection Time

A three-phase voltage signal was simulated by applying (4.12) with param-eters shown in Tables.4.4 to 4.6, which describe amplitude(p.u.) and phase-angle() for the fundamental frequency and odd harmonics (3th, 5th and 7thharmonics) sampled at 4800 Hz and 50 Hz for the fundamental frequency.Simulated waveforms are set to have different stages or segments: pre-voltagedip (Stage I) from 0 to 104 msec, during-voltage dip (Stage II) from >104to <208 msec and after-voltage dip (Stage III) from >208 to 312 msec, withnoise (σ =0.01, SNR ≈ 40 dB) in each electrical phase. Transition betweenstages I and II is called ”Start of the voltage dip” (ST) and between stagesII and III is called ”End of the voltage dip” (ED). The resulting signal isdepicted in Fig.4.14, which shows the noiseless and noise corrupted signals.

42


Table 4.6: Unbalanced voltage dip: phase-c


Fund: 1.0079 0 1.0104 0 1.0096 0

3rd: 0.0052 -7.16 0.0061 -84.23 0.0059 -21.16

5th: 0.0078 -201.62 0.0065 3.80 0.0080 -219.67

7th: 0.0044 -227.46 0.0022 -289.70 0.0041 -264.65

Fig.4.15 shows the simulated voltage dip, the start of the voltage dip (ST)and the end of the voltage dip (ED) for the positive-sequence. And Fig.4.16shows the start of the voltage dip (ST) and the end of the voltage dip (ED)for the negative-sequence. These figures show different KF orders. The di-rect method is applied for KF orders 1, 3, 5 and 7. As shown, order 1 hasfaster response than order 7. When applying indirect method with the sameKF orders, results differ slightly from those obtained when applying directmethod.

Therefore, to assess these results, first, the square error (SE) is calculatedand defined as follows

SE =N−1∑

n=0

|x(n) − x(n)|2 (4.36)

where x(n) is the true value and x(n) is the estimated value. The exactvalue is reached asymptotically. Second, another important feature to con-sider is the detection time. The detection time can be defined as the timethat the estimator takes to come within a confidence interval around thetrue positive- and negative-sequence magnitudes and remain there duringa specified time. There is a tradeoff between confidence and the detectiontime, the more confident the longer the detection time. For this test only,the detection time is defined as the first instant that the estimator reachesthe true value after the start of the voltage dip (ST) or the end of the voltagedip (ED) have occurred as shown in Fig.4.17. This assumption may not befully true, however, it is used just to compare how fast these both methodscan track the true positive- and negative-sequence magnitudes. Table.4.7summarizes the SE for estimating positive- and negative-sequences with KForders 1 to 7 using indirect and direct methods. From this table the directmethod shows smaller SE than the indirect method, however differences arevery small. Tables.4.8 and 4.9 show the detection time when positive- andnegative-sequences are estimated with KF orders 1,3,5 and 7, for the startof the voltage dip (ST) and the end of the voltage dip (ED). From these two

43



100 120 140 160 180 200

−1

−0.5

0

0.5

1

msec

Am

plitu

de (p

.u.)

100 120 140 160 180 200

−1

−0.5

0

0.5

1

msec

Am

plitu

de (p

.u.)

Figure 4.14: Simulated unbalanced three-phase voltages; phase-a(solid), phase-b(dash)and phase-c(dash-dot): [Top] with harmonics noiseless [Bottom] with harmonics and noise40 dB

Table 4.7: Square error (SE): direct (d) and indirect (i) methods

Positive(d) Positive(i) Negative(d) Negative(i)

KF Order 1: 0.0491 0.0471 0.0401 0.0396

KF Order 3: 0.0671 0.0674 0.1274 0.1279

KF Order 5: 0.0778 0.0781 0.1409 0.1413

KF Order 7: 0.0840 0.0843 0.1480 0.1483

44


100 105 110 115 1200.64

0.66

0.68

0.7

0.72

0.74

msec

Am

plitu

de (p

.u.)

1

3

5

7

205 210 215 220 2250.64

0.66

0.68

0.7

0.72

0.74

msec

Am

plitu

de (p

.u.) 1

3 7

5

Figure 4.15: Positive-sequence estimated with direct method - order 1(solid), order3(dash), order 5(dash-dot), order 7(dot)and true value(solid-thin): [Top] start of the volt-age dip - ST [Bottom] end of the voltage dip - ED

45



100 105 110 115 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

msec

Am

plitu

de (p

.u.)

1

3

7

5

205 210 215 220 2250

0.02

0.04

0.06

0.08

0.1

msec

Am

plitu

de (p

.u.)

1

7

3

5

Figure 4.16: Negative-sequence estimated with direct method - order 1(solid), order3(dash), order 5(dash-dot), order 7(dot) and true value(solid-thin): [Top] start of thevoltage dip - ST [Bottom] end of the voltage dip - ED

46


0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

msec

x(n)

Detection time

Figure 4.17: Detection time: true value(solid) and estimated value(dash), confidenceinterval (dot)

Table 4.8: Detection time (start of the voltage dip-ST) [msec]: direct(d) and indirect(i)methods


KF Order 1: 0.1670 0.1894 0.1670 0.2566

KF Order 3: 12.4178 12.4626 14.1872 14.2096

KF Order 5: 13.5601 13.6049 14.6575 14.6799

KF Order 7: 13.7840 13.7616 15.3966 15.3294

tables, it is possible to conclude that both methods reach the true values atalmost the same detection time, resulting in small differences which are notconclusive. From Fig.4.18 and Fig.4.19, we can conclude that the lowerorder the lower the detection time and the higher the square error. It is atradeoff between detection time, and variance of the estimation. The lowervariance in the estimation the higher the order of the KF.

4.4.3 Measured Data

Three-phase voltage waveforms were chosen from a database in a distribu-tion network. In Fig.4.20 the voltage dip waveform is shown. The voltagedip starts at 105 msec and ends at 185 msec in phase-a. The true values ofthe positive- and negative-sequences are unknown in this case. Assessmentof both methods estimation in terms of SE or detection time is not feasi-ble. However, it is possible to estimate the positive- and negative-sequences

47



Table 4.9: Detection time (end of the voltage dip-ED) [msec]: direct(d) and indirect(i)methods


KF Order 1: 10.7652 10.8324 3.5088 3.5984

KF Order 3: 15.5805 15.6925 7.2714 7.3162

KF Order 5: 16.0956 16.1852 8.0553 8.0777

KF Order 7: 16.6107 16.6331 8.3240 8.3016

50 100 150 200 250 3000.64

0.66

0.68

0.7

0.72

0.74

msec

Eff

ectiv

e va

lue

(p.u

.)

Figure 4.18: Amplitude of the positive-sequence estimated with direct method - order1(dot), order 7(solid) and true (dash)

50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

msec

Eff

ectiv

e va

lue

(p.u

.)

Figure 4.19: Amplitude of the negative-sequence estimated with direct method - order1(dot), order 7(solid) and true (dash)

48

4.5. CONCLUSIONS

100 120 140 160 180 200

−1

−0.5

0

0.5

1

msec

Am

plitu

de −

p.u

.

Figure 4.20: Three-phase voltage waveforms: phase-a(solid), phase-b(dash) and phase-c(dash-dot)

by applying the direct and the indirect method. As shown in the previ-ous example, there are no major differences between estimation with bothmethods in terms of accuracy, therefore, only direct method estimations arepresented in this subsection. In this record the network frequency is foundto be 49.93 Hz, voltage waveforms were sampled at 4800 Hz. The Kalmanfilter model was set to order 7 (K=7, fundamental frequency, second, third,fourth, fifth, sixth and seventh harmonics). Fig.4.21 up to Fig.4.24 show thedirect method estimation results for the fundamental frequency only.

4.5 Conclusions

The direct method gives almost the same error variance as compared to theindirect method which uses complex phase voltages or currents as an inter-mediate step. The direct method is computationally more efficient becausethe number of state-variables is only 2/3 of that for the indirect method.The performance of the direct method and the indirect method are aboutthe same in terms of accuracy due noise variance, however, positive- andnegative-sequence estimation for the direct method requires less computa-tion.

Applying the direct method may lead to faster voltage dip detection andclassification schemes. Besides, an additional gain is to consider three-phasesjointly instead of individually. Estimation performance for both methods istested in terms of robustness to noise contents. When the voltage dip is shal-low (retained voltage around 0.90 p.u. in any phase) a big error can be ex-

49



50 100 150 200 250 300 3500.64

0.66

0.68

0.7

0.72

0.74

msec

effe

ctiv

e va

lue

(p.u

.)

Figure 4.21: Amplitude of the positive-sequence estimated with direct method

50 100 150 200 250 300 350

−140

−120

−100

−80

−60

−40

−20

0

20

msec

Phas

e an

gle

(deg

rees

)

Figure 4.22: Phase-angle of positive-sequence estimated with direct method

50

4.5. CONCLUSIONS

50 100 150 200 250 300 3500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

msec

effe

ctiv

e va

lue

(p.u

.)

Figure 4.23: Amplitude of negative-sequence estimated with direct method

50 100 150 200 250 300 35045

46

47

48

49

50

51

52

msec

Phas

e an

gle

(deg

rees

)

Figure 4.24: Phase-angle of negative-sequence estimated with direct method

51



pected from the estimation of the phase-angle of the negative-sequence whichis shown in the CRLB study and the simulations in noisy conditions. Theworst case is when the signal is corrupted by noise due operation conditions,faults, transients, harmonics and measurements errors. In this work, whiteGaussian noise was assumed to corrupt three-phase signals in the same mag-nitude, this assumption is not probably fully correct, but this is the first steptaken to understand three-phase signal noise content. However more workis needed to define and identify the type of noise present in power-systemwaveforms (voltages and currents), its importance, causes and effects.

An example is shown in which the direct and indirect methods have beenapplied to simulated voltage signals and differences on the estimation areassessed in terms of square error and detection time. Moreover, measuredvoltages obtained during a real event in a medium-voltage public distributionnetwork are studied using harmonic modelling.

The direct method has good potential to track specific voltage or currentcharacteristics, e.g. changes in the magnitude of the positive-sequence ofthe fifth harmonic. This kind of information may be useful for automaticidentification and classification schemes.

52

Chapter 5Estimation of Time-Varying Frequency

in Power-Systems

This chapter proposes the modification of the previous model used to esti-mate positive- and negative-sequences from three-phase voltage samples inChapter 4, to estimate the time-varying fundamental frequency by addingone state-variable. An analysis of the stability, initial conditions and accu-racy is performed for the method which is implemented with an extendedcomplex Kalman filter(ECKF). Results from simulated and measured datatests are presented.

5.1 Introduction

Most of the techniques used to estimate parameters for sinusoids (funda-mental frequency signal plus harmonics) assume that the fundamental fre-quency is constant. In reality, the power-system frequency is not constant,i.e. a power-system operates with a frequency close to so called ”nominalfrequency”. The nominal frequency is 50 Hz in Europe and 60 Hz in mostof North, Central and South America. During operation it is perfectly ac-ceptable to assume that the frequency is time-varying. Differences betweenthe real frequency and the assumed in the model may cause error when thepositive- and negative-sequence phase-angles are estimated. In the case ofestimating positive- and negative-sequences with the objective of detection,identification and classification of the event type, these errors could lead tomiss match or wrong identification. That is why we believe that time-varyingfrequency estimation may be of interest.

Signal frequency, amplitude and phase-angle estimation have been studied

53

CHAPTER 5. ESTIMATION OF TIME-VARYING FREQUENCY IN POWER-SYSTEMS

by signal processing and researches in other fields for many years, a hugenumber of books and scientific journal articles are available dealing withthis topic [44, 45, 46, 47, 48]. Zero-crossing and DFT techniques were usedto estimate frequency [49, 50, 51], unfortunately, their performance is poorunder noise. Therefore, longer sampling time is required. For these reasonsthe investigation of more refined techniques was needed. In this chapter, thegoal is to extent the model discussed in Chapter 4, in order to investigate itscapability for time-varying frequency estimation and/or become aware of itslimitations if it is the case.

When considering complex sinusoids in additive complex white noise de-scribed as follows

v(n) =√

2K

∑

k=1

V ke(ωkn+φk) + N(n) (5.1)

where V k is the rms of the fundamental frequency (k = 1) and harmonics(1 < k ≤ K), φk phase-angle, ωk = kω0 having ω0 = 2πf0/fs, with f0 as thefundamental frequency, e.g. 50 Hz in Europe or 60 Hz in Americas, fs as thesampling frequency and finally, N(n) assuming a complex white noise withzero mean and known variance σ2.

Non parametric techniques have shown to be more suitable for frequencyestimation, particularly Eigen methods. Pisarenko has demonstrated thatthe frequencies could be derived from the eigenvector corresponding to theminimum eigenvalue of the signal autocorrelation matrix, this method iscalled Pisarenko Harmonic Decomposition (PHD)[52]. The method is limitedin its usefulness due to its low sensitivity to noise, additionally it requiresto know a-priori the number of complex sinusoids K and the noise assumedmust be white. Therefore it is not used in practice. If the order is high thecomputation may result time consuming. Although, this method was thefirst step into the eigenvalue decomposition methods which are more robustto noise sensitivity.

MUltiple SIgnal Classification Method (MUSIC)[53] applies the conceptof decomposing the signal autocorrelation matrix into signal and noise sub-spaces. The autocorrelation function is inexact due to noise. The noisevariance is computed by averaging the smallest eigenvalues related to thenoise sub-space. In this method the effect of spurious peaks is reduced byaveraging. Finally, frequency is computed from the largest averaged peaks.

PHD and MUSIC estimation methods can be improved dramatically byusing Singular Value Decomposition (SVD) of the signal covariance matrixin the presence of noise [53]. By computing the eigenvalues and splittingthem on signal and noise sub-spaces [54]; then setting the noise eigenvalues

54

5.2. DEFINING THE PROBLEM

to zero and recomputing the spectral peaks which results in a higher degreeof accuracy for frequency estimation. Although, the computer complexity tocompute the SVD is extremely high.

On the other hand, considering parametric techniques for the model de-scribed in (5.1), the Forward-Backward Linear Prediction(FBLP) was pro-posed [55], this method tries to fit sinusoid signal models into the observeddata. Unfortunately, this method doesn’t perform well in the presence ofnoise; the location of the estimated spectral peaks can be greatly affected bya small amount of noise which produces false frequency mapping.

A Maximum Likelihood Estimator (MLE) was proposed for estimationof all signal parameters (amplitude, phase-angle and frequency) resultingin a non-linear estimator which can highly depend on the initial conditions.Therefore, the estimation could reach a local minimum making the frequencyestimation biased [56], by adding constraints to the method, the frequencyestimation was improved, efficiency of the method is reached asymptotically[57].

An extended complex Kalman filter (ECKF) used to estimate all param-eters (amplitude, phase-angle and frequency) of a single sinusoid is presentedin [41, 58]. Its properties are extensively analysed covering stability, accu-racy and initial conditions of this type of estimators, which give good resultsunder some constraints. Other Kalman filter estimators are reported in [59].Finally, modified techniques to track frequency have been proposed by ap-plying the αβ-transform [60, 61]. Their stability, accuracy and designingproperties were studied extensively by [62, 63]. These methods are highlynon-linear and depend strongly on the initial conditions. If initial conditionswere chosen incorrectly the performance of the estimator is poor, resultingin biased estimations [64].

This chapter is organized as follows: Section 5.2 defines the problem ofestimating time-varying fundamental frequency from the sampled voltages.Section 5.3 extends the previous model presented in the Chapter 4, for time-varying frequency estimation. Section 5.4 describes the extended complexKalman filter algorithm. Section 5.5 summarizes the model for the harmonicscase, Section 5.6 shows results based on simulated and measured data. Andfinally, conclusions are given in Section 5.7.

5.2 Defining the Problem

Having three-phase observed voltage such as

va(n) =√

2Va cos(ω0n+ φa) +Na(n)

55


vb(n) =√

2Vb cos(ω0n+ φb) +Nb(n) (5.2)

vc(n) =√

2Vc cos(ω0n+ φc) +Nc(n)

where Va, Vb and Vc are the rms; φa, φb and φc are the phase-angles; ω0

is the discrete angular frequency (ω0 = 2πf0/fs), being f0 the fundamentalfrequency of voltage, i.e. 50 Hz in Europe and 60Hz in USA and fs thesampling frequency; Na(n), Nb(n) and Nc(n) are i.i.d. WGN(0,σ2); and n isthe discrete time index. In this model, it is assumed the same frequency forthe three-phases. Combining these samples by using the αβ -transform toobtain an alternative expression as

vαβ(n) =√

2(

V1ejω0n + V2

∗e−jω0n)

(5.3)

where V1 and V2 are defined in (4.4) in Chapter 4 as the positive- andnegative-sequences, respectively. This expression can be synchronized bye−jωn to obtain the dq-transform as

vdq(n) = vαβ(n)e−jωn (5.4)

vdq(n) =√

2(

V1ejω0ne−jωn + V2

∗e−jω0ne−jωn)

. (5.5)

In the previous chapter, we assumed ω = ωo, so the expression was reducedto

vdq(n) =√

2(

V1 + V2∗e−j2ωon

)

. (5.6)

However, the unknown variable is the fundamental frequency so it is not pos-sible to compute the dq-transform because in (5.4) a synchronization or rota-tional operator is required. Therefore, the frequency must be computed fromthe αβ-transform in (5.3). The goal is to estimate frequency and positive-,negative-sequences from the sampled voltages which are combined into theαβ-transform. The next Section only consider the estimation of the funda-mental frequency, although the model is extended to estimate harmonics andits frequencies in Section 5.5.

5.3 Extending the Previous Model

Starting with (5.3) and using it to estimate our goal. The nonlinear systemconsidered would be as

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

Y (n) = H(n)X(n) + V (n) (5.7)

56

5.3. EXTENDING THE PREVIOUS MODEL

where aX(n) is a nonlinear function of the state-vector, U(n) the complexmodel noise with zero mean and variance σ2

u and V (n) has zero mean and a σ2v

variance which are complex and noises have real and imaginary componentsas [65]

σ2v = σ2

vr + iσ2vi

σ2vr = σ2

vi (5.8)

and

σ2u = σ2

ur + iσ2ui

σ2ur = σ2

ui. (5.9)

For the three-phase system described in (5.1), let’s define the state-vector as

X(n) = [x1 x2 x3]T

n (5.10)

x1(n) = ejωn

x2(n) = V1ejωn = V1 x1(n)

x3(n) = V2∗e−jωn = V2

∗ x−11 (n).

It follows that the state-equation becomes

x1

x2

x3

n+1

= ax(n) +

u1

u2

u3

n

(5.11)

with

ax(n) =[

x1 x2x1x3

x1

]T

n

(5.12)

and the observation-equation becomes

y(n) =√

2 [0 1 1]

x1

x2

x3

n

+ v(n). (5.13)

The time-varying frequency is computed by applying

f(n) =fs

2πnImlog[x1(n)] (5.14)

where fs is the sampling frequency. And the positive- and negative-sequencesare found as

V1 = x2(n)e−jωn = x2(n)x−11 (n) (5.15)

V2 = x∗3(n)e−jωn = x∗3(n)x−11 (n) (5.16)

57


5.4 Extended Complex Kalman Filter (ECKF)

Algorithm

As noticed the state-equation is nonlinear, the algorithm is summarized asfollows, more details can be found in [40](Chapter 13, Section 13.7 and Chap-ter 15). The algorithm would be modified as follows

• Step (1) Prediction:x(n|n− 1) = ax(n− 1|n− 1)

• Step (2) Error covariance matrix a-priori:P (n|n− 1) = A(n− 1)P (n− 1|n− 1)A(n− 1)H +Q

• Step (3) Kalman gain matrix:K(n) = P (n|n− 1)HT (n)[H(n)P (n|n− 1)HT (n) + C]−1

• Step (4) Filtering:x(n|n) = x(n|n− 1) +K(n)[y(n) −H(n)x(n|n− 1)]

• Step (5) Error covariance matrix a posteriori:P (n|n) = [Im −K(n)H]P (n|n− 1)

Taylor series expansion is applied to linearize the non-linear term a., thenthe first derivative is found to be as

A(n) =∂ax(n)∂x(n)

∣

∣

∣

∣

∣

x(n)=x(n|n)

=

1x2

−x3x−21

0x1

0

00x−1

1

(5.17)

Expanding ax(n) into Taylor series will show that a second derivative isnot zero and the third derivative doesn’t either, therefore, the non-linearitycould be considered strong and unavoidable. This means that to use the firstderivative to estimate the non-linear variable(frequency) would make thealgorithm weak to nonlinearities in the signals, e.g. jumps or discontinuitiessuch as voltage dips. To ensure stability and convergence properties withthis approximation, the chose of initial conditions is highly important whichis discussed next.

5.4.1 Tuning the Extended Complex Kalman Filter

The initial conditions could take any value x(0|0) close to the ideal val-ues i.e. x1 initial condition would be function of 50 Hz, just to start but

58

5.5. EXTENSION TO THREE-PHASE SYSTEMS CONTAINING K HARMONICS

never equal to zero. The covariance matrix P (0|0) = RI could be initi-ated at any value for R > 0. The covariance matrix of model noise isQ = E[U(n)UT (n)] = σ2

uI, where Im. The covariance matrix of observa-tion noise is C = E[V (n)V T (n)] = σ2

vIl. Then, Q and C could take anyvalue different from zero. However, the main weakness of ECKF frequencyestimators is the biased estimation. [62, 63] proposed the introduction of anew parameter ε, making [A(n)]11 = 1 − ε, for tuning the ECKF. [64] hasdetermined another important property to reduce bias, λ = σ2

u/σ2v and ε = 0.

5.5 Extension to Three-Phase Systems Con-

taining K Harmonics

As it was defined in Chapter 4, Section 4.2.3., consider a three-phase systemcontaining K harmonics, to be modelled through the following expression:

va(n) =√

2K

∑

k=1

V ka cos(nωk + φk

a) +Na(n)

vb(n) =√

2K

∑

k=1

V kb cos(nωk + φk

b ) +Nb(n) (5.18)

vc(n) =√

2K

∑

k=1

V kc cos(nωk + φk

c ) +Nc(n)

and defining the symmetrical components for each harmonic k as [39]

V k0

V k1

V k2

=

1

3

111

1aa2

1a2

a

V ka

V kb

V kc

(5.19)

then, after some algebraic manipulations to find the αβ-transform using (4.2)and (5.19), to the phase voltages modelled by (5.18), it follows

vαβ(n) =√

2

[

K∑

k=1

V k1 e

jkωn +K

∑

k=1

V k2

∗e−jkωn

]

+ N(n)ejωn (5.20)

where the relation ωk = kω is applied and ω is unknown.

Model for the three-phase system containing K harmonics

Considering the system described in (5.20), the state vector of the Kalman fil-ter contains (2K+1) elements, one additional state compared with the model

59


defined in (4.16), and it is defined as

aX(n) = [x1 x2 x3 · · · x2K x2K+1]T

n

=[

ejωn V 11 e

jωn (V 12 )

∗

e−jωn · · ·V K1 ejKωn (V K

2 )∗

e−jKωn]T

n

=[

x1 x2x1 x3x−11 · · · x2Kx

K1 x2K+1x

−K1

]T

n. (5.21)

The state equation associated with the system becomes

X(n+ 1) = aX(n) +

u1...

u2K+1

n

(5.22)

and the observation equation for the system becomes

y(n) =√

2 [0 1 . . . 1]

x1...

x2K+1

n

+ v(n) (5.23)

The time varying frequency is found as described in (5.14). The positive-and negative-sequences associated with the fundamental (k = 1) and theharmonics (k ∈ [2, K]) can be found by

V1k(n) = x2k(n)e−jkωn = x2k(n)x−k

1 (n) (5.24)

V2

k(n) = x∗2k+1(n)e−jkωn = x∗2k+1(n)x−k

1 (n). (5.25)

Kalman filter algorithm

The algorithm presented in Section 5.4 still holds, just it is required to com-pute the A(n) term, then Taylor series expansion is applied to linearize thenon-linear term a., the first derivative is found to be as

A(n) =∂ax(n)∂x(n)

∣

∣

∣

∣

∣

x(n)=x(n|n)

=

1x2

−x3x−21

...KxK−1

1 x2K

−Kx2K+1xK+11

0x1

0...00

00x−1

1...00

. . .

. . .

. . .

. . .

. . .

. . .

000...xK

1

0

000...0

x−K1

(5.26)

60

5.6. TEST RESULTS FOR SIMULATED DATA

Table 5.1: Unbalanced three-phase simulated data


va(n): 1.00 5

vb(n): 0.90 125

vc(n): 1.00 245

Table 5.2: Estimated frequency over a two cycle window: SNR 60 and 50 dB

λ mean(60dB) var(60dB) mean(50dB) var(50dB)

50 50.1028 0.0365 50.1308 0.3531

10 50.1053 0.0531 50.1348 0.4124

0 50.1064 0.0577 50.1116 0.4510

5.6 Test Results for Simulated Data

5.6.1 Three-Phase Systems: Simulated Voltages

An unbalanced three-phase voltage system was simulated from data shownin Table.5.1. The simulated fundamental frequency is set up to 50.10 Hzand the sampling frequency is 3800 Hz. Noise was added to each phasevarying SNR from 40, 45, 50 and 60 dB (σ=0.01, 0.0055, 0.003 and 0.001p.u.). The ECKF estimator was tuned to be σ2

v = 0.001, ε = 0 and λ = 0, 10and 50. Table.5.2 and Table.5.3 show the resulting mean and the variancefor the estimated frequency when SNR 60, 50, 45 and 40 dB and differentλ. Results were averaged over a 2 cycle window. It is clear from thesetables that the noise produces bias on the estimations, and this bias cannot be eliminated totally even in the asymptotically sense(when N tends toInfinity). Therefore, choosing λ = 50 may reduce the bias and the variance.However this bias can not be eliminated totally. Unfortunately, when theSNR is low (≤45dB) the bias and the variance remain even with a high λ.

The lower bias and variance resulted when SNR is 60 dB and λ = 50,then Table.5.4, shows the relation between the window length and the vari-ance of the estimation. These results confirm that the frequency variance isreduced when the window size increased, as proved in (Equation 3.41) in [40]which confirms that the variance is inversely proportional to the SNR andthe window size.

61


Table 5.3: Estimated frequency over a two cycle window: SNR 45 and 40 dB

λ mean(45dB) var(45dB) mean(40dB) var(40dB)

50 50.1945 1.1106 50.4138 5.6240

10 50.2053 1.4328 50.4078 4.8135

0 50.2313 1.8456 50.4781 5.5150

Table 5.4: Window length vs Standard Deviation of the estimated frequency

Window length Standard Deviation-σ(10−3)

N/4: 0.2264

N/2: 0.2114

N : 0.2045

2N : 0.1965

30N : 0.1450

5.6.2 Constant Frequency Variation

Three-phase voltages were simulated from the same data in Table.5.1. Noisewas added (SNR =40, 45, 50 and 60 dB), the fundamental frequency is setup to be 50 Hz and its constant variation (±∆F ) was simulated. The ECKFwas tuned to σ2

v = 0.001, ε = 0 and λ = 50. Fig.5.1 and Fig.5.2 showthe estimated fundamental frequency with a constant variation under noisyconditions. When the SNR is high, i.e. SNR ≥ 60 dB, the algorithm is evencapable of tracking frequency with certain level of accuracy. However, whenthe SNR is low, i.e. SNR ≤ 45 dB, the algorithm is strongly biased and theestimations can not be trusted.

5.7 Conclusions

Frequency estimation based in ECKF is a highly non-linear approach, and re-sults in biased estimations. With high SNR(≥ 50 dB), frequency estimationsare less biased. However, with low SNR(≤ 45 dB) estimations are stronglybiased. Non-linearity can be reduced by increasing more terms to the Taylorseries expansion. However, a second, a third or any higher term may prob-ably not be necessary because the algorithm will become more complicatedand impractical.

Tuning the ECKF for frequency tracking is not an easy task. How to

62

5.7. CONCLUSIONS

20 40 60 80 100 120 140 160 18047

48

49

50

51

52

53

msec

Hz

+0.0040

+0.0068

∆ = 0.015 Hz/msec

−0.0040

−0.0068

−0.015

Figure 5.1: Constant frequency deviation: for SNR 60 dB (solid) and SNR 50 dB (dash)

20 40 60 80 100 120 140 160 18047

48

49

50

51

52

53

msec

Hz

∆ = 0.015 Hz/msec

+0.0068

+0.0040

−0.0040

−0.0068

−0.015

Figure 5.2: Constant frequency deviation: for SNR 60 dB (solid) and SNR 40 dB (dash)

63


solve this problem is still unclear and it may result in a difficult task to do.Using this estimator to track frequency under voltage dip conditions maycomplicate performance, resulting in a very sensitive and highly nonlinearestimator dependent on initial conditions, which are unknown after a newstage is reached. Choosing any initial condition when a voltage dip is de-tected, may result in erroneous estimations because the ECKF may reach alocal optimum instead of the global one. Finally, we have not found practicalapplications for the ECKF when referring to frequency estimation, however,we have found enough evidence to hold that the problem remains still un-solved.

64

Chapter 6Active and Reactive Power Estimation

This chapter presents an algorithm to directly estimate the active and reac-tive power from the sampled voltage and current waveforms under the leastsquares (LS) criterion. Conventional approach computes the voltages andcurrent phasors, then the apparent, active and reactive power are calculated.A least squares estimator (LSE) will be used to estimate active and reactivepower. LS methods are model based and its performance will depend on thenoise level. However, the advantage of this approach is that LSE is a simple,fast and cheap method that can be applied to track fast changes in power.

6.1 Introduction

A well known technique to compute the active and reactive power is basedon the DFT by first computing the voltage and current phasors [66] and thenusing classical power-system knowledge assuming that the signals are deter-ministic. Although the DFT technique is widely used in the power-systemcommunity, its resolution is limited when dealing with multi-component sig-nals, i.e. harmonics. Its accuracy is degraded when the signal is not a perfectdeterministic signal, i.e. when it is noise corrupted. The randomness of thevoltage and current signals are caused by the fluctuation of the electricalsystem parameters (load and generation units), as customers switch on oroff various devices. This randomness is increased when noise is added to thesignal due to nonlinearities in the sampling process.

Many researchers have focused their work on power quality and power-system events by studying voltage behavior: phase by phase or three-phases;there are available extensive studies on this topic, many of them where al-ready mentioned in Chapter 2. However, there are a few studies consid-

65

CHAPTER 6. ACTIVE AND REACTIVE POWER ESTIMATION

ering the current behavior, and studying currents by themselves may notgive enough information on the underlying event. On the other hand, inthe power-system protection area, a wide known and used device to protecttransmission lines is the distance relay, which bases its fundamental on thecombination of voltages and currents as impedance by applying Ohm’s law,where resistance (R) and reactance (X) are analysed to detect and trip whena possible fault is located in a protection zone with a certain degree of con-fidence [67, 68]. The combination of voltage and current measurements, asthe active and reactive power was not extensively studied in the power qual-ity field. However, a power signature analyzer based on active and reactivepower behavior was proposed [69] and a locator of power quality sources wasdefined by using this approach [70], both recently. In these works, voltageand current phasors were computed first, by using block based least squarestechniques: recursive [71] or sequential forms [72], then the standard activeand reactive power were found.

By using the definition of instantaneous power, we are able to define anactive and reactive power estimator. We will not follow the classical approachof computing first the voltage and current phasors. The active and reactivepower can be estimated directly from the voltage and current samples. Theleast squares (LS) method will be used to estimate active and reactive power.LS methods are model based and its performance will depend on the noiselevel. We will consider the LS as a simple first approach to investigate thisfield. Then the active-, reactive power and the power factor can be studiedunder different power-system events to understand their behavior.

In Section 6.2 the single-phase instantaneous power in continuous-timedomain is discussed. Section 6.3 discusses the problem to estimate activeand reactive power from the sampled three-phase voltages and currents inthe discrete-time domain. In Section 6.4 the instantaneous power is discussedwhen the sampled signals are corrupted by noise. Section 6.5 presents testresults using simulated data. Finally, conclusions are presented in Section6.6.

6.2 Single-Phase Instantaneous Power

The instantaneous power p(t) is defined in the continuous-time domain asthe multiplication of instantaneous signals: voltage v(t) and current i(t) fora single-phase ac system [73]. Note that the instantaneous power definitionchosen for this chapter is not the definition used in power electronics control,where voltages, currents and instantaneous power are referenced to the αβspace [74, 75, 76].

66

6.2. SINGLE-PHASE INSTANTANEOUS POWER

Load

v(t)

i(t)

Figure 6.1: Single phase ac circuit

The angular frequency is Ω = 2πfo, where fo is the network frequency,i.e. 50 Hz in Europe and 60 Hz in Americas. Define

v(t) =√

2V cos (Ωt) (6.1)

i(t) =√

2I cos (Ωt+ Φ) (6.2)

where the voltage v(t) is considered to be the reference signal, i.e. at 0, Vand I are the rms or effective values of voltage and current, respectively andΦ the phase-angle difference between voltage and current, i.e. the currentlag or lead w.r.t. the voltage. In Fig.6.1, the voltage applied to the loadresults in a current i(t) for the single-phase ac system. The instantaneouspower p(t) is defined as follows,

p(t) = v(t)i(t) (6.3)

= 2V I[cos(Ωt) cos(Ωt+ Φ)]

= 2V I[cos(Ωt)cos(Ωt) cos(Φ) − sin(Ωt) sin(Φ)]= 2V I[cos(Ωt)2 cos(Φ) − cos(Ωt) sin(Ωt) sin(Φ)]

= 2V I

[

1

2+

cos(2Ωt)

2

cos(Φ) − sin(2Ωt)

2sin(Φ)

]

= V I[cos(Φ) + cos(Φ) cos(2Ωt) − sin(2Ωt) sin(Φ)]

p(t) = V I cos (Φ)[1 + cos (2Ωt)]

−V I sin (Φ)[sin (2Ωt)]

p(t) = (instantaneous active power)

+(instantaneous reactive power)

where the mean value of the instantaneous active power is known as theactive power and the mean value of instantaneous reactive power is zero, butwith a maximum value of V I sin (Φ). This can be interpreted in electrical

67


terms as: the voltage source is supplying constant flow of energy to the loadin one direction only known as active power. At the same time an interchangeof energy is taking place between the source and the load of average zero butwith peak value of V I sin (Φ) known as the reactive power. The interchangeof energy between the source and the inductive or capacitive elements takesplace at twice the supply frequency. Therefore, it is possible to think of anactive power P (t), defined as P (t) = V I cos (Φ); and a reactive power Q(t),defined as Q(t) = V I sin (Φ). 1 Where Φ is the phase-angle between thephasors V = V and I = IejΦ and can be computed as

Φ = arctan

[

Q(t)

P (t)

]

. (6.4)

Then, the instantaneous power is defined as

p(t) = P (t)[1 + cos (2Ωt)] −Q(t)[sin (2Ωt)] (6.5)

An equally important electrical quantity is the power factor defined as cos(Φ).The active and reactive power are defined for sinusoidal quantities, thus forconstant V , I and Φ. To obtain their values some kind of averaging overat least one cycle is required. (6.5) is also derived under the assumptionthat voltages and currents are sinusoidal. However, from (6.5), it is possi-ble to obtain estimated values for P (t) and Q(t) at a sub-cycle time scale(N=Window length, e.g. N ≤ one cycle), and therefore, the power factor.

6.3 Defining the Problem: Three-Phase Ap-

proach

The definition presented in Section 6.2. is valid for the continuous-timedomain and it can be extended to the discrete-time domain, if the Nyquisttheorem is satisfied [77]. Then, the discrete angular frequency becomes ω =2πf0/fs, where f0 is the network frequency, and fs is the sampling frequency.In a three-phase system, voltages are defined as

va(n) =√

2Va cos (ωn+ φa)

vb(n) =√

2Vb cos (ωn+ φb) (6.6)

vc(n) =√

2Vc cos (ωn+ φc)

1The term (t) is adopted to emphasize that the quantity is defined in the continuous-time domain, and to avoid confusion with their discrete versions.

68

6.3. DEFINING THE PROBLEM: THREE-PHASE APPROACH

and currents as

ia(n) =√

2Ia cos (ωn+ ψa)

ib(n) =√

2Ib cos (ωn+ ψb) (6.7)

ic(n) =√

2Ic cos (ωn+ ψc)

and having N samples(n = 0, 1, ..., N − 1). Therefore, the instantaneouspower p(n), in the discrete time domain, for each phase may be computed as

pa(n) = va(n)ia(n)

pb(n) = vb(n)ib(n) (6.8)

pc(n) = vc(n)ic(n)

(6.5) was derived assuming that the voltage signal is always referenced orat least this reference is known. Unfortunately, most of the cases, whenmeasured voltages and currents, this reference is unknown, therefore, it isimportant to derive a more general expression for the discrete time domain,such as

v(n) =√

2V cos (ωn+ φ) (6.9)

i(n) =√

2I cos (ωn+ ψ) (6.10)

where the difference (ψ − φ) would be applied to compute the active andreactive power, for any electrical phase, as shown in Fig.6.2, where three-phase voltage and current phasors are depicted.

Vb

rotating at

Vc

0

120

240

ωn speed.

Va

Ia

Ib

Ic

ψa

φa

φb

ψb

φc

ψc

Figure 6.2: Representing three-phase voltage and current phasors

69


The instantaneous power for this general case, is computed as

p(n) = v(n)i(n)

= 2V I cos (ωn+ φ) cos (ωn+ φ)= 2V I cos (ωn+ φ) cos (ωn+ φ+ [ψ − φ])= 2V I cos (ωn+ φ) [cos (ωn+ φ) cos (ψ − φ)

− sin(ωn+ φ) sin(ψ − φ)]= 2V I

cos2 (ωn+ φ) cos (ψ − φ)

− sin (ωn+ φ) cos (ωn+ φ) sin (ψ − φ)

= 2V I[

1

2+

1

2cos (2ωn+ 2φ)

]

cos (ψ − φ)

−[

1

2sin (2ωn+ 2φ)

]

sin (ψ − φ)

= V I cos (ψ − φ) + V I cos (ψ − φ) cos (2ωn+ 2φ)

−V I sin (ψ − φ) sin (2ωn+ 2φ)

= P (n) + P (n) cos (2ωn+ 2φ) −Q(n) sin (2ωn+ 2φ) (6.11)

the active power P (n) and the reactive power Q(n) are defined as

P (n) = V I cos (ψ − φ) (6.12)

Q(n) = V I sin (ψ − φ) (6.13)

The problem is to estimate active power P (n) and reactive power Q(n), foreach phase, from the sampled voltages and currents when the phase-angledifference (ψ − φ) is unknown and over a window (N).As it is possible to notice, if φ = 0, then (6.11) is equivalent or analogousto (6.5). If φ is different than zero and it is unknown, it is necessary tore-write (6.11) in such a way that a LSE can be found. After applying sometrigonometric identities, the instantaneous power can be re-written as

p(n) = v(n)i(n)

= 2V I cos (ωn+ φ) cos (ωn+ ψ)= 2V I cos (ωn) cos (φ) − sin(ωn) sin(φ) cos (ωn) cos (ψ)

− sin (ωn) sin (ψ)= 2V I

cos2 (ωn) cos (φ) cos (ψ) − sin (ωn) cos (ωn) sin (φ) cos (ψ)

− sin (ωn) cos (ωn) cos (φ) sin (ψ) + sin2 (ωn) sin (φ) sin (ψ)

= 2V I

cos2 (ωn) cos (φ) cos (ψ) − sin (ωn) cos (ωn) [sin (φ) cos (ψ)

+ cos (φ) sin (ψ)] + sin2 (ωn) sin (φ) sin (ψ)

70

6.3. DEFINING THE PROBLEM: THREE-PHASE APPROACH

= 2V I

1

2cos (φ) cos (ψ) +

1

2cos (2ωn) cos (φ) cos (ψ)

−1

2sin (2ωn) cos (ψ + φ)

+1

2sin (φ) sin (ψ) − 1

2cos (2ωn) sin (φ) sin (ψ)

= V I cos (ψ − φ) + V I cos (ψ + φ) cos (2ωn)

−V I sin (ψ + φ) sin (2ωn)

p(n) = P (n) + V I cos (ψ + φ) cos (2ωn) − V I sin (ψ + φ) sin (2ωn) (6.14)

and P (n) is the active power defined in (6.12). From (6.14) a LSE can bedefined for estimation of P (n) and Q(n).

6.3.1 Least Squares Estimator for Active and ReactivePower

A LSE for the three-phase case can be defined by re-writing (6.14) as 2

p(n) = [1 cos (2ωn) − sin (2ωn)]

x1

x2

x3

(6.15)

where x1 = P (n), x2 = V I cos (ψ + φ) and x3 = V I sin (ψ + φ). Define θ asa vector containing the parameters to be estimated,

θ =

x1

x2

x3

=

Px2

x3

. (6.16)

It follows,

V I =√

x22 + x2

3 (6.17)

(ψ − φ) = arccos[

x1

V I

]

(6.18)

and finally, the reactive power can be computed by,

Q(n) = V I sin (ψ − φ). (6.19)

Then, (6.15) can be re-written in vector form as,

p = Hθ (6.20)

2Note that from (6.11) it is not possible to define a LSE because φ is unknown, howeverfrom (6.14) a LSE is defined straight forward.

71


where p = [p(0) p(1) ... p(N − 1)]T contains N samples over a window.And the transition matrix is defined as

H =

11...1

1cos (2ω)

...cos (2ω(N − 1))

0− sin (2ω)

...− sin (2ω(N − 1))

. (6.21)

Applying, the standard least squares method [40], the estimated θ becomes

θ = [HTH]−1HTp (6.22)

and the inverse [HTH]−1 has a closed form[78] equal to

[HTH]−1 =

1N

00

02N

0

002N

. (6.23)

Finally, P (n) and Q(n) are the estimated active and reactive power, for eachelectrical phase, and they are averaged over a window N . The active powerfor each phase is known to be as

Pa(n) = VaIa cos (ψa − φa)

Pb(n) = VbIb cos (ψb − φb) (6.24)

Pc(n) = VcIc cos (ψc − φc)

and the reactive power is

Qa(n) = VaIa sin (ψa − φa)

Qb(n) = VbIb sin (ψb − φb) (6.25)

Qc(n) = VcIc sin (ψc − φc)

(ψa − φa), (ψb − φb) and (ψc − φc) are the phase-angle differences betweenthe voltage and current for each electrical phase. The phase-angle differencefor each electrical phase, can be determined by

(ψa − φa)(n) = arctan

[

Qa(n)

Pa(n)

]

(ψb − φb)(n) = arctan

[

Qb(n)

Pb(n)

]

(6.26)

(ψc − φc)(n) = arctan

[

Qc(n)

Pc(n)

]

72

6.4. THREE-PHASE MEASURED VOLTAGES AND

CURRENTS CORRUPTED BY NOISE

6.4 Three-Phase Measured Voltages and

Currents Corrupted by Noise

The instantaneous power p(n) is estimated from the sampled voltages andcurrents. However, it is perfectly acceptable to assume that both signals arenoise corrupted due to, operation, interference or sampling. Then, (6.9) and(6.10) can be re-written as

v(n) =√

2V cos (ωn+ φ)

vs(n) = v(n) +Nv(n) (6.27)

i(n) =√

2I cos (ωn+ ψ)

is(n) = i(n) +Ni(n) (6.28)

where Nv(n) and Ni(n) are i.i.d. WGN with mean zero and variance σ2v and

σ2i ,respectively . This assumption can not be fully true, however, the Gaus-

sian case has extensively been studied in signal processing, therefore, it couldresult in a first step to understand the problem of having noisy signals whenstudying power quality events. The voltage and current noises are assumedto be uncorrelated between them, i.e. E[v(n)Ni(n)] = E[i(n)Nv(n)] = 0.When the estimated instantaneous power p(n) is corrupted by noise, then(6.14) can be re-written as

p(n) = E[p(n)]

= E[v(n) +Nv(n)v(n) +Nv(n)]= E[v(n)i(n)] + E[v(n)Ni(n)] + E[i(n)Nv(n)] + E[Nv(n)Ni(n)]

= v(n)i(n)

= P (n) + V I cos (ψ + φ) cos (2ωn) − V I sin (ψ + φ) sin (2ωn).(6.29)

In this case the LSE still holds for estimation of active and reactive power.However, a LS is model based and statistical properties on the signals arenot assumed, therefore, estimations are noise dependent.

6.5 Test Results for Simulated Data

6.5.1 Three-Phase Systems: Simulated Voltages andCurrents

A three-phase system was simulated with data presented in Table.6.1. Sam-pled signals were corrupted by noise WGN(0,σ2), for SNR ≈ 30, 40, 45, 50

73


Table 6.1: Unbalanced three-phase system: simulated data


va(n): 1.05 5

vb(n): 0.95 125

vc(n): 1.00 245

ia(n): 1.00 15

ib(n): 1.20 135

ic(n): 1.00 255

Table 6.2: True active and reactive power and power factor

Active Power(p.u.) Reactive Power(p.u.) Power Factor

Phase-a: 0.5170 0.0912 0.9848

Phase-b: 0.5613 0.0990 0.9848

Phase-c: 0.4924 0.0868 0.9848

and 60 dB. The signal was simulated with sampling frequency 3800 Hz; fun-damental frequency 50 Hz; total samples N=1292; averaged over a window(one cycle) of 76 samples. The true active and reactive power per phase isgiven in Table.6.2, where the power factor is given as well. To assess accuracyof the estimator the mean square error (MSE) was calculated. The MSE isdefined as follows

MSE =1

N

N−1∑

n=0

|x(n) − x(n)|2 (6.30)

where x(n) is the true value and x(n) is the estimated value. Table.6.3 showsthe MSE for active and reactive power estimations under different SNR, theMSE is very small resulting in a good estimator that minimizes the squareerror but it is noise dependent. Table.6.4 and Table.6.5 show the estimatedactive and reactive power and their variances under different SNR (30, 40, 45,50 and 60 dB). From these tables, it is possible to conclude that the LSE foractive and reactive power is good but it depends on the noise of the signals.Therefore, estimations for real measurements may be good if the signal noisecontent is low, e.g. SNR > 45 dB. The LSE will perform well for the caseof swells and voltage dips, when sudden variations on the amplitudes andphase-angles for the three-phase signals may occur.

74


Table 6.3: Mean square error(MSE) vs SNR

SNR (dB) Active Power Reactive Power

30 0.028969 0.716523

40 0.002072 0.054998

45 0.000483 0.017059

50 0.000132 0.005405

60 0.000017 0.000493

Table 6.4: Active power: estimated value and variance(10−3) for different SNR(dB)

SNR Pa(p.u.) var Pb(p.u.) var Pc(p.u.) var

30 0.518181 0.005630 0.559769 0.051898 0.493238 0.017523

40 0.516873 0.001852 0.560969 0.001503 0.492661 0.001151

45 0.517228 0.000577 0.561591 0.000634 0.492489 0.000398

50 0.517037 0.000114 0.561188 0.000106 0.492366 0.000168

60 0.517054 0.000015 0.561321 0.000020 0.492394 0.000013

Table 6.5: Reactive power: estimated value and variance(10−3) for different SNR(dB)

SNR Qa(p.u.) var Qb(p.u.) var Qc(p.u.) var

30 0.089554 0.306176 0.092075 0.821146 0.081937 0.390034

40 0.092089 0.031800 0.100236 0.094864 0.088642 0.028861

45 0.090220 0.018934 0.098425 0.001503 0.087671 0.012055

50 0.091233 0.004990 0.098539 0.003262 0.087211 0.007045

60 0.091403 0.000485 0.099091 0.000307 0.086811 0.000432

75


Table 6.6: Unbalanced voltage dip: voltages(v) and currents(c)


va: 1.0154 50.88 0.8762 43.90 1.0159 51.14

vb: 1.0250 -69.78 0.9444 -68.63 1.0245 -69.40

vc: 1.0079 170.14 1.0104 164.48 1.0096 175.64

ca: 0.9687 2.19 2.1718 4.96 0.9560 3.07

cb: 0.8819 -119.55 1.8550 -146.27 0.8643 -120.33

cc: 0.8834 125.09 1.0350 126.06 0.8592 124.48

Table 6.7: True active and reactive power per phase

I (p.u.) II (p.u.) III (p.u.)

Pa: 0.3243 0.7385 0.3226

Pb: 0.2892 0.1853 0.2758

Pc: 0.3137 0.4097 0.2707

Qa: 0.3689 0.5823 0.3554

Qb: 0.3435 0.8536 0.3448

Qc: 0.3109 0.3220 0.3267

6.5.2 Simulated Voltage Dip Conditions

A three-phase voltage and current signals were simulated by applying (4.1)with parameters shown in Table.6.6 for voltages and currents, which de-scribes amplitude(p.u.) and phase-angle() with 50 Hz for the fundamentalfrequency sampled at 4800 Hz. Simulated waveforms are set to have dif-ferent stages or segments: pre-voltage dip (Stage I) from 0 to 104 msec,during-voltage dip (Stage II) from >104 to <208 msec and after-voltage dip(Stage III) from >208 to 312 msec. Different noise content is applied to eachphase (SNR ≈ 30, 40, 45, 50 and 60 dB). Table.6.7 shows the true activeand reactive power for each stage (I, II and III). Finally, Table 6.8 showsthe MSE when estimating active and reactive power under different SNR.Again, the LSE produces good estimations but these estimations are noisedependent. Fig.6.3 to 6.5 show the estimated active, reactive power andpower factor when SNR 30 dB, for the simulated voltage dip. Fig.6.3 showschanges in the active power during the simulated voltage dip. Active powerin phase-a has a considerable increased, more than twice. The active powerin phase-b, has decreased in about 30% and active power in phase-c has in-

76


Table 6.8: Mean square error(MSE) vs SNR

SNR (dB) Active Power(MSE) Reactive Power(MSE)

30 0.014101 0.059656

40 0.001340 0.005661

45 0.000450 0.001783

50 0.000138 0.000540

60 0.000022 0.000069

50 100 150 200 250 3000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

msec

p.u.

Figure 6.3: Estimated active power: phase-a(solid), phase-b(dash) and phase-c(dash-dot)

creased in about 30%. In Fig.6.4, the reactive power per phase is shown.Reactive power in phase-b during the simulated voltage dip has an increasedof almost 50%, reactive power in phase-a almost double. And reactive powerremains almost constant for phase-c. Until here, we can not conclude muchon the changes in the active and reactive power per phase. However, by ana-lyzing the power factor plotted in Fig.6.5, it is possible to conclude that thevoltage dip was mainly involving phase-b where the power factor decreasesconsiderably. The power factor of phases-a and -c remain almost constant,small changes may probably are due to compensate active power decreasedin phase-b.

77


50 100 150 200 250 3000.2

0.3

0.4

0.5

0.6

0.7

0.8

msec

p.u.

Figure 6.4: Estimated reactive power: phase-a(solid), phase-b(dash) and phase-c(dash-dot)

50 100 150 200 250 3000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

msec

Cos

(φ)

Figure 6.5: Estimated power factor: phase-a(solid), phase-b(dash) and phase-c(dash-dot)

78

6.6. CONCLUSIONS

6.6 Conclusions

LSE gives a good solution for the estimation of active power and reactivepower from the three-phase sampled voltages and currents. One of the mainadvantages of LS method is that it can handle corrupted data and gener-ate the optimal solution under the LS criterion. Since the assumption thatnoise and signal are uncorrelated mostly unsatisfied, the estimation resultswill depend on the noise properties. However, this is considered as a goodapproximation that may be used to identify and classify power signals asa simpler approach. LSE can be used to track fast changes in active andreactive power.

79


80

Chapter 7Conclusions and Future Research

7.1 Conclusions

The work presented in this thesis aims to investigate signal processing tech-niques applied to power-system event classification when field measurementsare available: i.e. three-phase voltage and current waveforms.

The problem of dealing with data coming from different sources in manyformats arises when field voltage and current waveforms are available becausemany utilities have several power quality monitors installed at different pointson the network. In this thesis, we deal with real measurements and in someminor cases simulations. The technique or method used to sample the voltageand current waveforms is outside of the scope of this work. Our main concernare the discrete signals which are captured with any power quality monitor.

Power quality equipment manufacturers use their proprietary software tomanage all recorded waveforms. How this software processes them and howthese algorithms work is not given or specified in technical manuals. Be-cause of this, we propose an open architecture database to manage the min-imum necessary information regarding voltage and current waveforms andopen algorithms to process and treat the information. This software tool en-ables open access for collecting and sharing recorded waveforms. Algorithmsused to process this information are detailed as well. Moreover, graphicaluser interfaces are standardized for handling and presenting the information.Furthermore, accessing this software through the Internet is enabled givingflexibility to its open architecture.

Algorithms and procedures can be added as modules during the courseof the research project. The developed software can be used for researchor educational purposes as well, e.g. collected waveforms can be sharedamong researchers or students to investigate specific events. Additionally, the

81

CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH

software may allow algorithm testing when dealing with feature extractionor rules for classification. Finally, statistics can be obtained from a specificset of waveforms regarding low voltage records in certain power-systems.

A method for combining three-phase information is developed and stud-ied. The method is based in the dq-transform which allows the estimationof the positive- and negative-sequences for three-phase voltages or currents.The estimator is proposed and analysed for the fundamental frequency andharmonic cases. Moreover, the accuracy, stability and computational com-plexity are studied. The proposed method gives a direct way to computethe positive- and negative-sequences for voltages and currents for classifica-tion purposes and it is implemented by Kalman filters. The method is analternative for the phase by phase analysis. The Cramer-Rao lower bound isfound for the dq-transform model. The performance of the estimator undernoisy conditions is tested. In this work, and in most other studies it is as-sumed that three-phase signals are corrupted by Gaussian white noise. Thisassumption is probably not fully true. However, this is the first step towardsunderstanding three-phase signals corrupted by noise. More work is requiredto define and identify the type of noise present in power-system waveforms,its importance, causes and effects.

The direct method has good potential for tracking specific voltage orcurrent characteristics, e.g. fast changes in the magnitude of the positive-sequence of the fifth harmonic. This kind of information may be useful forautomatic identification and classification schemes, although such a schemeis not defined yet.

In the case of characterizing voltage dips based on positive- and negative-sequences, a shallow voltage dip can be identified wrongly due to an error inthe phase-angle of the negative-sequence if the voltage signals are corruptedby noise. By finding the Cramer-Rao Lower Bound of the dq-transformmodel, this assumption was proved to be correct. A small unbalance in thethree-phase system could lead to phase-angle error of the negative-sequencein voltage and/or current.

Another important conclusion drawn from the proposed algorithm is re-lated to the power-system frequency. The frequency must be known a-priorito accurately compute the phase-angle of positive- and negative-sequences.In most cases, the assumption of a constant frequency close to the nominalfrequency: 50Hz in Europe and 60 Hz in America, would be enough. How-ever, the system can operate at a slightly different frequency which may resultin apparent phase-angle shifting. This may cause incorrect classification ofthe underlying power-system event. In power quality monitors this problemis solved by synchronizing the measurements with an internal clock, and by azero-crossing method to find the frequency. In our case the problem may be

82

7.1. CONCLUSIONS

solved by using frequency trackers or phase lock loop (PLL)algorithms ap-plied to the voltage waveforms. With these algorithms the estimation errorof phase-angle may be reduced.

As the need to estimate frequency from the sampled waveforms (mainlyvoltages because they are less corrupted by harmonics or noise) arose, afrequency tracker based on extended complex Kalman filters (ECKF) fromvoltage waveforms is investigated. This algorithm is derived from the modelused for estimation of positive- and negative-sequences presented in Chapter4. Estimation based on ECKF is a highly non-linear approach, and the resultsare biased. Tuning the ECKF for frequency tracking is difficult. With thisapproach we demonstrate that the problem of estimating frequency fromthe signal using a ECKF is still unsolved. We have found that estimatingpositive-, negative-sequences and frequency with the same model may beeasier from the implementation point of view. However, it is not feasible yet.This remains as an interesting topic for continuing research.

Another interesting way of combining and studying voltage and currentwaveforms is given through the instantaneous power. From the definition ofinstantaneous power a simpler method to estimate active and reactive poweris derived and proposed. The method is analysed and implemented by usingLeast Squares Estimators (LSE). One of the main advantages of the LSE isthat it can handle noise corrupted signals and generate the optimal solutionunder the LS criterion. The problem of estimating the active and reactivepower from sampled voltages and currents corrupted by noise when thereis no phase reference is presented. This estimator may be a useful methodfor field applications when dealing with real waveforms corrupted by noise.Again, this algorithm was developed under the assumption that the systemis operating at the nominal frequency. However, when dealing with realmeasured waveforms an angular reference is not available. For this reason,the estimation of the phase-angle difference between the signals may be ofimportance, e.g. phase-angle of voltage and current waveforms.

83


7.2 Future Research

This section outlines four topics on which further research would be appre-ciated.

7.2.1 Bayes Detection

A combination of power-system features extracted by Kalman Filter tech-niques and Bayes Classification scheme is proposed. Power-system featuressuch as, amplitude and phase-angle of positive- and negative-sequences and/oractive, reactive power and power factor computed by applying signal process-ing methods, as those proposed in this report, should be used. Bayesian clas-sification could be defined based on these features. It is expected that theseschemes would have the ability to detect, select and classify power-systemevents under different disturbances.

Industrial customers are mainly affected by dips. Voltage dips may causemal-trip of equipment in power-system networks. Sometimes mal-trip resultsin economic loss due to the shut down of production processes and sometimesdue to the damage caused to end-user or generation equipment. In order toavoid this problem, different types of detection devices have been proposed.Detection schemes are based on the rate of change or threshold-crossing algo-rithms with a detection time less than 10msec, e.g. 1-5msec when combiningvoltage information into the dq-transform (as treated in Chapter 4). How-ever, during the detection time of the voltage dip, industrial components(motor drives which are more sensitive) are still exposed to the dip, whichcould still cause a production stoppage.

A possible new method for faster recognition of voltage dips is the useof detection schemes based on Kalman filters and Bayes classification tech-niques, e.g. for the voltage dip case [79]. This approach will allow devicesto detect any dip almost instantaneously and the expert system is able tolearn by itself. Preliminary tests have shown that detection of a voltage dipis possible in less than 10 msec even with a high confidence rate [15]. Thisframework could be enhanced for any power-system disturbance.

For a given set of events εi, for i = 1, ...,M , where M is the total num-ber of recorded events for a particular network or utility within a certainperiod of time by many power quality monitors. Let us define a vectorϑ = [d1 d2 . . . dk] that contains k features and characterizes any event. Thedefinition of dk features is still under study. Some features are considered in[5, 16, 17, 18, 19, 20, 21, 22, 79].

The first step should be to define a large number of features to charac-

84

7.2. FUTURE RESEARCH

terize these events. As more features are defined, the events will be betterunderstood. A second step is to recognize ζf classes, e.g. voltage dip typeA, capacitor switching, ..., interruptions, etc. The third step would be toreduce the number of nuisance features until the class ζf is re-defined andoptimized. A final and fourth is needed when a new class ζf+1 is not clas-sified by the expert system because its features are unknown. In this casethe implementation of the ability to self learn from unknown events wouldbe required.

Once a given set for a particular power-system is analysed, the expertsystem would learn from this preliminary set, and incoming events are goingto be classified with the previous knowledge. Definition of pre-knowledge foreach class ζf is required.

7.2.2 A Three-Phase and Multilevel Approach

The detection of the same power-system event at different locations in thesystem is another interesting issue for further research. The power qual-ity monitors installed will register the same event in different locations atdifferent time stamps because synchronization is not always possible. If asingle-site study is carried out, this may result in counting the same power-system event more than once. We believe that the combined analysis ofthese records should be done by taking into account probabilistic techniquesto solve the multilevel approach. Some features may be defined by takinginto account different voltage levels, e.g. voltage and current waveforms inmore than one buss in the power-system.

The same record seen from different points in the network would give moreinformation about the location and the network conditions that caused thisevent to happen, even if the network configuration is changing. A difficulttask is to determine and select that an event registered at different time stampwith many power quality monitors is the same event. When the time stampsis different, it is important to fix rules or a methodology to determine or inferabout the type and location of the event. The references that would be usedto continue this research are: [15, 16, 80, 81, 82, 83, 84, 85, 79, 86, 87, 88].

Characterization of voltages and currents measured at different bussesin a network is necessary. If just three-phase voltages would be analysed,determining the event type accurately is limited and determining its locationis difficult. Analyzing voltage waveforms allows one to infer the type of eventbut we would not be confident of its direction or location. When concerningthe same event recorded by different power quality monitors the task is evenmore difficult.

How can we determine if the recorded events which are labelled with

85


different time stamps are the same event or different events? How confidentare we about the type of the event? How confident are we about the locationof the event? These questions can not be answered without a combination ofsignal processing, pattern recognition and power-system knowledge. The useof redundancy in the measurements at different location could be of interestwhen correlating the information.

To perform this analysis we deal with real recordings at three-phase volt-age and current at multilevel voltage as well as simulations. Current record-ings will give us information about the direction of the event. Additionally,direction information may be found from voltage measurements at two con-secutive locations. The first step of this analysis will consider voltage dipsand voltage swells in a multi-source network. Gradually more events will beadded to be analysed.

7.2.3 Noise Model for Power-System Signals

The investigation of the noise when collecting field waveforms is needed. Allthe work done in power-systems assumes that waveforms are deterministicor noise free. This assumption could be a good approach for planning ordesigning a power-system. When dealing with real waveform collected fromfield, some kind of noise is present in many different proportions and havingdifferent statistical distributions. Although the noise can be neglected inmost cases, knowing its properties and being able to infer about its behaviorcan lead to improved estimation and classification schemes. Noise can beinvestigated to determine its importance, causes, effects. This may probablyhave direct impact in protection schemes; mal-function or mal-operation maybe reduced. A better understanding of noise under operation conditions maybe of importance when studying real waveforms.

7.2.4 Direction Finding

A considerable amount of work has been done on the determination of theorigin of a power-system disturbance when dealing with real waveforms. Webelieve that a combination of power-system and signal processing techniques,may be able to detect faster and more accurately the direction of the event.[30] proposes the basis for distance relaying with symmetrical componentsbased on the rate of voltage and current changes and derives one singleequation to handle all possible faults in a power-system. If this scheme isimplemented with Kalman filters that are one step ahead estimators, thedirection of the event and the event classification may be accomplished in afaster way.

86

Appendix AThe Cramer-Rao Lower Bound of the

dq-Transform

A.1 Complex Noise of the dq-Transform

(4.6) can be re-written as

vdq(n) =√

2(


)

+2

3[Na(n) + aNb(n) + a2Nc(n)]e−jωon (A.1)

where a = ej2π

3 is the rotating factor for computing the symmetrical compo-nents and Na(n), Nb(n) and Nc(n) are i.i.d. WGN(0,σ2). Let define vectorsas

Na = [Na(0) Na(1) . . . Na(N − 1)]T

Nb = [Nb(0) Nb(1) . . . Nb(N − 1)]T

Nc = [Nc(0) Nc(1) . . . Nc(N − 1)]T

and a matrix E = diag[1 e−jω0 . . . e−jω0(N−1)]. Then, the second term of(A.1) can be re-written in vector form as

N =2

3ENa + aNb + a2Nc (A.2)

where the complex noise N is a linear combination of the noise in the three-phase and it is still WGN, let’s find the mean and the covariance. Theexpected value of the complex noise would be

E[N] = E[

2

3E

Na + aNb + a2Nc

]

87

APPENDIX A. THE CRAMER-RAO LOWER BOUND OF THE DQ-TRANSFORM

=2

3E

[

E Na + aE Nb + a2E Nc]

= 0 (A.3)

and the covariance is computed as

Cov(N) = E[NNH

] (A.4)

=4

9E

[

ENa + aNb + a2NcNa + aNb + a2NcHEH]

=4

9E

[

ENa + aNb + a2NcNHa + a∗NH

b + a2∗NHc EH

]

considering

EE∗ = EEH = I

a = ej 2π3

a2 = a∗ = e−j 2π3

a3 = 1

a4 = a2∗ = a = ej 2π3

where I is the identity matrix. It follows,

Cov(N) =4

9E

[

NaNHa + a2NaN

Hb + aNaN

Hc (A.5)

aNbNHa + NbN

Hb + a2NbN

Hc

a2NcNHa + aNcN

Hb + NcN

Hc

]

however Na, Nb and Nc are i.i.d., therefore they are uncorrelated

E[NaNHb ] = E[NaN

Hc ] = E[NbN

Ha ] = E[NbN

Hc ]

= E[NcNHa ] = E[NcN

Hb ] = 0

and the following covariance matrices

E[NaNHa ] = E[NbN

Hb ] = E[NcN

Hc ] = σ2I.

It follows that (A.5) is equal to

Cov(N) =4

3σ2I. (A.6)

Since N is CWGN(0, 43σ2), (A.1) is simplified as

vdq(n) =√

2(


)

+ N(n). (A.7)

88

A.2. CRAMER-RAO LOWER BOUND: SINGLE SINUSOID

A.2 Cramer-Rao Lower Bound: Single Sinu-

soid

A single sinusoid can be modelled as

x(n) =√

2Acos(ω0n+ φ) +N(n) (A.8)

where, A is the rms value of the sinusoid, φ is phase-angle, ω0 is the knownangular frequency, and N(n) is a Gaussian white noise with zero mean andvariance σ2. Having N samples (n = 0, 1, ..., N − 1) define a vector x =[x(0) . . . x(N − 1)]T . If A and φ are parameters to be estimated define aparameter vector as θ = [A φ]T . The Cramer-Rao Lower Bound (CRLB)for a single sinusoid in vector form follows as (Theorem 3.2 [40])

var(θ)i ≥ [I(θ)]−1ii (A.9)

where the Fisher information matrix I(θ) is given as

[I(θ)]ij = −E[

∂2 ln p(x; θ)

∂θi∂θj

]

(A.10)

as the signal is corrupted by Gaussian white noise, (A.10) is simplified to(Section 3.9 [40])

[I(θ)]ij =1

σ2

N−1∑

n=0

∂x[n; θ]

∂θi

∂x[n; θ]

∂θj

(A.11)

and for simplification let r = ω0n+ φ and using some approximations [56]

N−1∑

n=0

sin (2r) ≈ 0 (A.12)

N−1∑

n=0

cos (2r) ≈ 0 (A.13)

then

[I(θ)]11 =2

σ2

N−1∑

n=0

cos2 r =2

σ2

N−1∑

n=0

(

1

2+

1

2cos(2r)

)

≈ N

σ2

[I(θ)]12 = [I(θ)]21 = − 2

σ2A cos (r) sin (r) = − A

σ2sin (2r) ≈ 0

[I(θ)]22 =2

σ2

N−1∑

n=0

A2 sin2 (r) =2

σ2

N−1∑

n=0

A2(

1

2− 1

2sin (2r)

)

≈ NA2

σ2

89


the Fisher information matrix becomes

I(θ) =1

σ2

[

N0

0NA2

]

,

therefore

[I(θ)]−1 = σ2

[

1N

001

NA2

]

,

from (A.9), the CRLB for the estimated parameters in a single sinusoid isobtained as

var(A) ≥ σ2

N(A.14)

var(φ) ≥ σ2

NA2. (A.15)

A.3 Mean and Variance of the dq-Transform

To re-write (A.7) in vector form, let’s define vectors as

v = [vdq(0) vdq(1) . . . vdq(N − 1)]T

V1 = [V1(0) V1(1) . . . V1(N − 1)]T

V2

∗ = [V ∗2 (0) V ∗

2 (1) . . . V ∗2 (N − 1)]T

and

E2 = diag[1 e−j2ω0 . . . e−j2ω0(N−1)]

then (A.7) in vector form becomes

v =√

2(

V1 + E2V2

∗)

+ N (A.16)

the expected value µ follows as

µ = E[v]

= E[√

2(

V1 + E2V2∗)

+ N]

= E[√

2(

V1 + E2V2∗)]

+ E[

N]

=√

2(

V1 + E2V2∗)

(A.17)

the covariance matrix is found as

cov(v) = E[(v − µ)(v − µ)H ] (A.18)

= E[NNH

] = cov(N)

90

A.4. CRAMER-RAO LOWER BOUND OF THE DQ-TRANSFORM

from (A.6)

cov(v) = Cv =4

3σ2I (A.19)

and the inverse

C−1v

=3

4σ2I. (A.20)

A.4 Cramer-Rao Lower Bound of the dq-Transform

If |V1|, |V2|, φ1 and φ2 are the parameters to be estimated then (A.7) can bere-written as

vdq(n) =√

2|V1|ejφ1 +√

2|V2|e−jφ2e−j2ωon + N(n) (A.21)

and the expected value (A.17) becomes

µ(n) =√

2|V1|ejφ1 +√

2|V2|e−jφ2e−j2ωon. (A.22)

Define a parameter vector ξ = [|V1| |V2| φ1 φ2]T , the Cramer-Rao Lower

Bound (CRLB), for such a complex signal v, in vector form is known to beequal to (Section 15.7 [40])

var(ξ)i ≥ [I(ξ)]−1ii (A.23)

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

[I(ξ)]ij = tr

[

C−1v

(ξ)∂Cv(ξ)

∂ξi

C−1v

(ξ)∂Cv(ξ)

∂ξj

]

+ 2Re

[

∂µH(ξ)

∂ξi

C−1v

(ξ)∂µ(ξ)

∂ξj

]

(A.24)

from(A.17) then

µ[n; ξ] =√

2|V1|ejφ1 +√

2|V2|e−jφ2e−j2ωn (A.25)

however Cv is not a function of ξ, the first term in (A.24) equals to zero and(A.24) becomes

[I(ξ)]ij = 2[C−1v

]ijRe

[

N−1∑

n=0

∂µ∗[n; ξ]

∂ξi

∂µ[n; ξ]

∂ξj

]

(A.26)

91


Using approximations (A.12) and (A.13) in (A.26), it follows,

[I(ξ)]11 = 2[C−1v

]11Re

[

N−1∑

n=0

√2e−jφ1ejφ1

√2

]

= 4[C−1v

]11N =3N

σ2

[I(ξ)]12 = 2[C−1v

]12Re

[

N−1∑

n=0

√2e−jφ1

√2e−jφ2e−j2ωn

]

= 4[C−1v

]12Re

[

N−1∑

n=0

e−j(φ1+φ2+2ωn)

]

= 4[C−1v

]12N−1∑

n=0

[cos (φ1 + φ2 + 2ωn)] ≈ 0

[I(ξ)]13 = 2[C−1v

]13Re

[

N−1∑

n=0

√2e−jφ1j

√2|V1|ejφ1

]

= 4[C−1v

]13Re

[

N−1∑

n=0

j|V1|]

≈ 0

[I(ξ)]14 = 2[C−1v

]14Re

[

N−1∑

n=0

√2e−jφ1 − j

√2|V2|e−jφ2e−j2ωn

]

= 4[C−1v

]14Re

[

N−1∑

n=0

−j|V2|e−j(φ1+φ2+2ωn)

]

= 4[C−1v

]14N−1∑

n=0

[

|V2| − sin (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]21 = 2[C−1v

]21Re

[

N−1∑

n=0

√2ejφ2ej2ωn

√2ejφ1

]

= 4[C−1v

]21Re

[

N−1∑

n=0

ej(φ1+φ2+2ωn)

]

= 4[C−1v

]21N−1∑

n=0

[cos (φ1 + φ2 + 2ωn)] ≈ 0

[I(ξ)]22 = 2[C−1v

]22Re

[

N−1∑

n=0

√2ejφ2ej2ωn

√2e−jφ2e−j2ωn

]

= 4[C−1v

]22N =3N

σ2

[I(ξ)]23 = 2[C−1v

]23Re

[

N−1∑

n=0

√2ejφ2ej2ωnj

√2|V1|ejφ1

]

92

A.4. CRAMER-RAO LOWER BOUND OF THE DQ-TRANSFORM

= 4[C−1v

]23Re

[

N−1∑

n=0

j|V1|ej(φ1+φ1+2ωn)

]

= 4[C−1v

]23N−1∑

n=0

[

|V1| − sin (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]24 = 2[C−1v

]24Re

[

N−1∑

n=0

√2ejφ2ej2ωn − j

√2|V2|e−j2φ2e−j2ωn

]

= 4[C−1v

]24Re

[

N−1∑

n=0

−j|V2|]

≈ 0

[I(ξ)]31 = 2[C−1v

]31Re

[

N−1∑

n=0

−j√

2|V1|e−jφ1√

2ejφ1

]

= 4[C−1v

]31Re

[

N−1∑

n=0

−j|V1|]

≈ 0

[I(ξ)]32 = 2[C−1v

]32Re

[

N−1∑

n=0

−j√

2|V1|e−jφ1√

2e−jφ2e−j2ωn

]

= 4[C−1v

]32Re

[

N−1∑

n=0

−j|V1|e−j(φ1+φ1+2ωn)

]

= 4[C−1v

]32N−1∑

n=0

[

|V1| − sin (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]33 = 2[C−1v

]33Re

[

N−1∑

n=0

−j√

2|V1|e−jφ1j√

2|V1|ejφ1

]

= 4[C−1v

]33Re

[

N−1∑

n=0

|V1|2]

=3|V1|2Nσ2

[I(ξ)]34 = 2[C−1v

]34Re

[

N−1∑

n=0

−j√

2|V1|e−jφ1 − j√

2|V2|e−jφ2e−j2ωn

]

= 4[C−1v

]34Re

[

N−1∑

n=0

−e−j(φ1+φ2+2ωn)

]

= 4[C−1v

]34N−1∑

n=0

[

|V1||V2| − cos (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]41 = 2[C−1v

]41Re

[

N−1∑

n=0

j√

2|V2|ejφ2ej2ωn√

2ejφ1

]

93


= 4[C−1v

]41Re

[

N−1∑

n=0

j|V2|ej(φ1+φ2+2ωn)

]

= 4[C−1v

]41N−1∑

n=0

[

|V2| − sin (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]42 = 2[C−1v

]42Re

[

N−1∑

n=0

j√

2|V2|ej2φ2ej2ωn√

2e−jφ2e−j2ωn

]

= 4[C−1v

]42Re

[

N−1∑

n=0

j|V2|]

≈ 0

[I(ξ)]43 = 2[C−1v

]43Re

[

N−1∑

n=0

j√

2|V2|ejφ2ej2ωnj√

2|V1|ejφ1

]

= 4[C−1v

]43Re

[

N−1∑

n=0

−|V1||V2|ej(φ1+φ2+2ωn)

]

= 4[C−1v

]43N−1∑

n=0

[

|V1||V2| − cos (φ1 + φ2 + 2ωn)]

≈ 0

[I(ξ)]44 = 2[C−1v

]44Re

[

N−1∑

n=0

j√

2|V2|ej2φ2ej2ωn − j√

2|V2|e−j2φ2e−j2ωn

]

= 4[C−1v

]44Re

[

N−1∑

n=0

|V2|2]

=3|V2|2Nσ2

.

Finally, the CRLB is obtained as

var(|V1|) ≥ σ2

3N(A.27)

var(|V2|) ≥ σ2

3N(A.28)

var( φ1 ) ≥ σ2

3|V1|2N(A.29)

var( φ2 ) ≥ σ2

3|V2|2N. (A.30)

94

List of Figures

2.1 Time-line of power quality monitoring equipment evolution . . 6

2.2 Automatic classification scheme . . . . . . . . . . . . . . . . . 10

3.1 Open architecture application: dashed - in our computer system 18

3.2 Power Quality database relationship scheme . . . . . . . . . . 19

3.3 Graphical User Interface - Query Module . . . . . . . . . . . . 20

3.4 Graphical User Interface - Plot Module . . . . . . . . . . . . . 21

3.5 Graphical User Interface - Auxiliary Plot Module . . . . . . . 22

4.1 Indirect method: three adaptive filters and one transformation 25

4.2 Direct method: one adaptive filter and one transformation . . 25

4.3 Representing three-phase voltages in αβ-space . . . . . . . . . 26

4.4 Positive- and negative-sequences in the dq-space . . . . . . . . 27

4.5 CRLB: true(solid), indirect(dot) and direct(dash) method forpositive-sequence amplitude estimation . . . . . . . . . . . . . 35

4.6 CRLB: true(solid), indirect(dot) and direct(dash) method forpositive-sequence phase-angle estimation . . . . . . . . . . . . 35

4.7 CRLB: true(solid), indirect(dot) and direct(dash) method fornegative-sequence amplitude estimation . . . . . . . . . . . . . 36

4.8 CRLB: true(solid), indirect(dot) and direct(dash) method fornegative-sequence phase-angle estimation . . . . . . . . . . . . 36

4.9 Total flops = number of Additions plus Multiplications . . . . 38

4.10 Computation time when estimating: indirect method: KForder 5(solid) - KF order 1(dot); direct method: KF order5(dash) - KF order 1(dash-dot) . . . . . . . . . . . . . . . . . 38

4.11 Average relative error of magnitude in positive-sequence: in-direct(solid) and direct(dash) methods under noise . . . . . . . 40

95

LIST OF FIGURES

4.12 Average relative error of magnitude in negative-sequence: in-direct(solid) and direct(dash) methods under noise . . . . . . . 41

4.13 Average relative error of the phase-angle difference betweenthe positive- and negative-sequences: indirect(solid) and di-rect(dash) methods under noise . . . . . . . . . . . . . . . . . 41

4.14 Simulated unbalanced three-phase voltages; phase-a(solid), phase-b(dash) and phase-c(dash-dot): [Top] with harmonics noise-less [Bottom] with harmonics and noise 40 dB . . . . . . . . . 44

4.15 Positive-sequence estimated with direct method - order 1(solid),order 3(dash), order 5(dash-dot), order 7(dot)and true value(solid-thin): [Top] start of the voltage dip - ST [Bottom] end of thevoltage dip - ED . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.16 Negative-sequence estimated with direct method - order 1(solid),order 3(dash), order 5(dash-dot), order 7(dot) and true value(solid-thin): [Top] start of the voltage dip - ST [Bottom] end of thevoltage dip - ED . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.17 Detection time: true value(solid) and estimated value(dash),confidence interval (dot) . . . . . . . . . . . . . . . . . . . . . 47

4.18 Amplitude of the positive-sequence estimated with direct method- order 1(dot), order 7(solid) and true (dash) . . . . . . . . . . 48

4.19 Amplitude of the negative-sequence estimated with direct method- order 1(dot), order 7(solid) and true (dash) . . . . . . . . . 48

4.20 Three-phase voltage waveforms: phase-a(solid), phase-b(dash)and phase-c(dash-dot) . . . . . . . . . . . . . . . . . . . . . . 49

4.21 Amplitude of the positive-sequence estimated with direct method 50

4.22 Phase-angle of positive-sequence estimated with direct method 50

4.23 Amplitude of negative-sequence estimated with direct method 51

4.24 Phase-angle of negative-sequence estimated with direct method 51

5.1 Constant frequency deviation: for SNR 60 dB (solid) and SNR50 dB (dash) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Constant frequency deviation: for SNR 60 dB (solid) and SNR40 dB (dash) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1 Single phase ac circuit . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Representing three-phase voltage and current phasors . . . . . 69

6.3 Estimated active power: phase-a(solid), phase-b(dash) andphase-c(dash-dot) . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Estimated reactive power: phase-a(solid), phase-b(dash) andphase-c(dash-dot) . . . . . . . . . . . . . . . . . . . . . . . . . 78

96

LIST OF FIGURES

6.5 Estimated power factor: phase-a(solid), phase-b(dash) andphase-c(dash-dot) . . . . . . . . . . . . . . . . . . . . . . . . . 78

97

LIST OF FIGURES

98

List of Tables

4.1 Simulated unbalanced three-phase voltages . . . . . . . . . . . 34

4.2 Computational complexity of the Kalman filter . . . . . . . . 37

4.3 Number of flops (Kalman filter) . . . . . . . . . . . . . . . . . 37

4.4 Unbalanced voltage dip: phase-a . . . . . . . . . . . . . . . . . 42

4.5 Unbalanced voltage dip: phase-b . . . . . . . . . . . . . . . . 42

4.6 Unbalanced voltage dip: phase-c . . . . . . . . . . . . . . . . . 43

4.7 Square error (SE): direct (d) and indirect (i) methods . . . . . 44

4.8 Detection time (start of the voltage dip-ST) [msec]: direct(d)and indirect(i) methods . . . . . . . . . . . . . . . . . . . . . . 47

4.9 Detection time (end of the voltage dip-ED) [msec]: direct(d)and indirect(i) methods . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Unbalanced three-phase simulated data . . . . . . . . . . . . . 61

5.2 Estimated frequency over a two cycle window: SNR 60 and 50dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Estimated frequency over a two cycle window: SNR 45 and 40dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Window length vs Standard Deviation of the estimated fre-quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Unbalanced three-phase system: simulated data . . . . . . . . 74

6.2 True active and reactive power and power factor . . . . . . . . 74

6.3 Mean square error(MSE) vs SNR . . . . . . . . . . . . . . . . 75

6.4 Active power: estimated value and variance(10−3) for differentSNR(dB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.5 Reactive power: estimated value and variance(10−3) for dif-ferent SNR(dB) . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6 Unbalanced voltage dip: voltages(v) and currents(c) . . . . . . 76

99

LIST OF TABLES

6.7 True active and reactive power per phase . . . . . . . . . . . . 766.8 Mean square error(MSE) vs SNR . . . . . . . . . . . . . . . . 77

100

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