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I
Performance Comparison of Power Quality
Evaluation Using Advanced High Resolution
Spectrum Estimation Methods
Md. Ziaul Hoque
Department of Electrical and Electronic Engineering
Dhaka University of Engineering & Technology, Gazipur
August 2017
II
Performance Comparison of Power Quality
Evaluation Using Advanced High Resolution
Spectrum Estimation Methods
A dissertation submitted in partial fulfillment of the requirements for the
degree of
Master of Engineering in Electrical and Electronic Engineering
By
Md. Ziaul Hoque
Student No. 132204F
Under Supervision of
Dr. Md. Raju Ahmed
Professor, Dept. of EEE, DUET, Gazipur
Department of Electrical and Electronic Engineering
Dhaka University of Engineering & Technology, Gazipur
August 2017
III
Declaration
I declare that this project is my own work and has not been submitted in any form for another
degree or diploma at any university or other institute of tertiary education. Information derived
from the published and unpublished work of others has been acknowledged in the text and a list
of references is given.
Md. Ziaul Hoque Date: 24/08/2017
IV
Acknowledgements
All praise is for Almighty Allah for having guided us at every stage of our life.
I would like to convey my sincere feeling and profound gratitude to my supervisor Dr. Md.
Raju Ahmed for his guidance, encouragement, constructive suggestions, and support throughout
the span of this project. Among many things I learned from Dr. Md. Raju Ahmed, that
persistent effort is undoubtedly the most important one, which enables me to sort out several
important issues in the area of signal processing. I also want to thank Dr. Md. Raju Ahmed for
spending so many hours with me in exploring new areas of research and new ideas and
improving the writing of this dissertation. I am also thankful to all the teachers and staffs of the
Department of EEE of Dhaka University of Engineering and Technology for their support and
encouragement.
Most importantly, I wish to thank my parents, in particular Advocate Ekramul Hoque for being
my driving force and standing with me through thick and thin. Without them I would never have
come so far in pursuing my dream.
After all, I would like to express my gratitude to the Electrical & Electronic Engineering
Department of Dhaka University of Engineering & Technology, Gazipur for providing an
excellent environment for research.
V
Abstract
Most of the conventional methods of power quality assessment in power systems are almost
exclusively based on Fourier Transform that suffer from various inherent limitations. First
limitation of an FFT based method is that of frequency resolution, whereas the second limitation
is due to coherent signal sampling of data which proves itself as a leakage in spectrum domain.
These two performance limitations of FFT or other similar methods become particularly
troublesome while analyzing short data records. To overcome this problem high regulation
spectrum estimation methods can be used where resolution problem is not found. In this project,
high resolution methods, such as MUSIC, root MUSIC and ESPRIT are discussed that use a
different approach to spectral estimation; instead of trying to estimate the power spectral density
(PSD) directly from the data, they model the data as the output of a linear system driven by white
noise, and then attempt to estimate the parameters of that linear system. Detail Matlab
simulations are carried out in order to investigate the performance of Multiple Signal
Classification (MUSIC), Root MUSIC and Estimation of Signal Parameter via Rotational
Invariance Technique (ESPRIT) methods in estimating amplitude, power (squared amplitude)
and frequency estimation of synthetic power signal both in clean and noisy conditions. Using
mean square error (MSE) as the evaluation criterion, the variation of amplitude, power (squared
amplitude) and frequency estimation are shown with respect to data sequence length and signal
to noise ratio (SNR) and their influences on MSE are compared for the different methods
mentioned above.
VI
List of Abbreviations
MUSIC Multiple Signal Classification
ESPRIT Estimation of Signal Parameter via Rotational Invariance Technique
SNR Signal to Noise Ratio
MSE Means Square Error
DFT Discrete Fourier transforms
FFT Fast Fourier transforms
STFT Short time Fourier transforms
PSD Power spectral density
VII
List of Publications
[1] Md. Ziaul Hoque, Md. Raju Ahmed; “Performance Comparison of Power Quality
Evaluation using advanced high Resolution Spectrum Estimation Methods” IOSR Journal of
Electrical and Electronics Engineering (IOSR-JEEE); e-ISSN: 2278-1676,p-ISSN: 2320-
3331, Volume 12, Issue 1 Ver. I (Jan. – Feb. 2017), PP 57-67, DOI: 10.9790/1676-09522839.
[2] Md. Raju Ahmed, Md. Ziaul Hoque; “Spectral Analysis of Power Quality Evaluation Using
Different Estimation Methods” Journal of Electrical Engineering (JEE): Volume 16/2016
Edition 1, Article 16.2.20, 2016.
[3] Md. Ziaul Hoque, Md. Raju Ahmed; “Performance Evaluation of an OFDM Power Line
Communication System with Impulsive Noise and Multiple Receiving Antenna” 2nd
International Conference on Electrical & Electronic Engineering (ICEEE 2017), RUET.
VIII
List of Contents
Acknowledgments V
Abstract VI
List of Abbreviations VII
List of Publications VIII
List of Figures XIII
List of Tables XIV
1. Introduction 1
1.1 Signal 1
1.1.1Amplitude 2-3
1.1.2 Frequency 3-4
1.2 Importance of Amplitude and Frequency Estimation in Power Signal 4-5
1.3 Conventional Methods 5
1.4 High Resolution Methods 5-6
1.4.1 Nonparametric Methods 6
1.4.2 Parametric Methods 6
1.5 Motivation of using High Resolution Methods 6-7
1.6 Objective of the Project 7
1.7 Organization of the Project 7
1.8 Conclusion 7
2. High Resolution Spectrum Estimation Methods 8
2.1 Introduction 8
2.2 MUSIC Method 8-9
IX
2.3 Root MUSIC Method 10
2.4 ESPRIT Method 11-12
2.5 Conclusion 12
3. Simulation Results and Performance Comparison 13
3.1 Introduction 13
3.2 Working Procedure and Simulation Conditions 13-15
3.3 Performance Evaluation Criteria 15
3.3.1 Data Length 15
3.3.2 SNR 16
3.4 Amplitude Estimation 16
3.4.1 Effect of Data Length on MSE for Amplitude Estimation 16
3.4.1.1 Clean Signal 17-19
3.4.1.2 Noisy Signal 19-21
3.4.2 Effect of SNR on MSE for Amplitude Estimation 21-23
3.5 Power Estimation 23
3.5.1 Effect of Data Length on MSE for Power Estimation 24
3.5.1.1 Clean Signal 24-26
3.5.1.2 Noisy Signal 26-29
3.5.2 Effect of SNR on MSE for Power Estimation 29-31
3.6 Frequency Estimation 31
3.6.1 Effect of Data Length on MSE for Frequency Estimation 31-32
3.6.1.1 Clean Signal 32-34
3.6.1.2 Noisy Signal 34-36
3.6.2 Effect of SNR on MSE for Frequency Estimation 36-39
3.7 Conclusion 39
X
4. Conclusion 40
4.1 Concluding Remarks 40-41
4.2 Scopes for Future Work 41
References 42-45
XI
List of Figures
Figure 1.1 (a): Amplitude. 2
Figure 1.1 (b): Amplitude measurement of sine waves. 2
Figure 1.2 (a): High and low fequency ratio waves. 3
Figure 1.2 (b): Frequency measurement of sine waves. 3
Figure 3.1 (a): Source signal for clean condition in linear scale. 14
Figure 3.1 (a): Source signal for noisy condition in linear scale. 14
Figure 3.2 (a): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to data length for clean signal in log scale. 17
Figure 3.2 (b): Amplitude estimation (MUSIC & Root MUSIC) for clean signal
in terms of MSE with respect to data length in linear scale. 18
Figure 3.3 (a): Amplitude estimation (MUSIC & Root MUSIC) for noisy signal
in terms of MSE with respect to data length in log scale. 20
Figure 3.3 (b): Amplitude estimation (MUSIC & Root MUSIC) for noisy signal
in terms of MSE with respect to data length in linear scale. 20
Figure 3.4 (a): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to SNR in log scale. 22
Figure 3.4 (b): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to SNR in linear scale. 22
Figure 3.5 (a): Power estimation (MUSIC & Root MUSIC) for clean signal
in terms of MSE with respect to data length in log scale. 25
Figure 3.5 (b): Power estimation (MUSIC & Root MUSIC) for clean signal
in terms of MSE with respect to data length in linear scale. 25
Figure 3.6 (a): Power estimation (MUSIC & Root MUSIC) for noisy signal
in terms of MSE with respect to data length in log scale. 27
Figure 3.6 (b): Power estimation (MUSIC & Root MUSIC) for noisy signal
in terms of MSE with respect to data length in linear scale. 28
Figure 3.7 (a): Power estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to SNR in log scale. 30
XII
Figure 3.7 (b): Power estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to SNR in linear scale. 30
Figure 3.8 (a): Frequency estimation (MUSIC & Root MUSIC) for clean signal
in terms of MSE with respect to data length in log scale. 32
Figure 3.8 (b): Frequency estimation (MUSIC & Root MUSIC) for clean signal
in terms of MSE with respect to data length in linear scale. 33
Figure 3.9 (a): Frequency estimation (MUSIC & Root MUSIC) for noisy signal
in terms of MSE with respect to data length in log scale. 35
Figure 3.9 (b): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) for noisy
Signal in terms of MSE with respect to data length in linear scale. 35
Figure 3.10 (a): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) in terms
of MSE with respect to SNR in log scale. 37
Figure 3.10 (b): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) in terms
of MSE with respect to SNR in linear scale. 38
XIII
List of Tables
Table -3.1: The amplitude estimation for clean signal in terms of MSE
for increasing data length. 18
Table -3.2: The amplitude estimation for noisy signal in terms of MSE
for increasing data length. 21
Table -3.3: Amplitude estimation (MUSIC & Root MUSIC) in terms
of MSE for changing SNR. 23
Table -3.4: The power estimation for clean signal in terms of MSE
for increasing data length. 26
Table -3.5: Power estimation (MUSIC & Root MUSIC) in terms of MSE
for increasing data length 28
Table -3.6: Power estimation (MUSIC & Root MUSIC) in terms of MSE
for changing SNR. 31
Table -3.7: The Frequency estimation for clean signal in terms of MSE
for increasing data length. 33
Table -3.8: The amplitude estimation (MUSIC, Root MUSIC & ESPRIT) for noisy
signal in terms of MSE for increasing data length. 36
Table -3.9: Frequency estimation (MUSIC, Root MUSIC & ESPRIT) in terms of
MSE for changing SNR. 38
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Chapter 1
Introduction
The quality of power or voltage waveforms has become a matter of great importance for power
utilities, electrical energy customers and additionally for manufacturers of electrical and
electronic equipment. The energy markets strengthen the competition and are expected to drive
down the prices of energy which is the main part for the requirements concerning the power
quality. Normally, the power and voltage waveforms are expected to be a pure sinusoidal with a
given amplitude and frequency which is very difficult. Because the harmonic, inter harmonic
components, faults or other switching transients distort waveforms of power and voltage. Most
of the fashionable frequency power converters generate a large spectrum of harmonic elements
that indicate the standard of delivered energy, however, increase the energy. The proliferation of
nonlinear loads connected to power systems has triggered a growing concern with power quality
issues. The inherent operation characteristics of these loads deteriorate the quality of delivered
energy, and increase the energy losses as well as decrease the reliability of a power system [1],
[2], [6]. In some cases, large convertor systems generate not solely characteristic harmonics
typical for the perfect convertor operation, however additionally a substantial quantity of non
characteristic harmonics and inter harmonics, which can powerfully verify the standard of the
power-supply voltage [7], [8], so that higher management control and protection depend on the
estimation of the power signal elements.
1.1 Signal
A signal is a physical quantity which varies with respect to time, space and contains information
from source to destination. A signal as referred to in communication systems, signal processing,
and electrical engineering is a function that conveys information about the behavior or attributes
of some phenomenon"[9]. In the physical world, any quantity exhibiting variation in time or in
space (such as an image) is potentially a signal that might provide information on the status of a
physical system or convey a message between observers, among other possibilities [10].
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1.1.1 Amplitude
In particular, for sound analysis, amplitude is the objective measurement of the degree of change
(positive or negative) in atmospheric pressure (the compression and rarefaction of air molecules)
caused by sound waves. Sounds with greater amplitude will produce greater changes in
atmospheric pressure from high pressure to low pressure. Amplitude is almost always a
comparative measurement, since at the lowest-amplitude end (silence), some air molecules are
always in motion and at the highest end, the amount of compression and rarefaction though finite
is extreme. In electronic circuits, amplitude may be increased by expanding the degree of change
in an oscillating electrical current. The figure below represents amplitude measurement of sine
waves.
Figure 1.1 (a) Amplitude.
Amplitude is directly related to the acoustic energy or intensity of a sound. Both amplitude and
intensity are related to sound's power shown in Figure1.1 (a).
Figure 1.1 (b) Amplitude measurement of sine waves.
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If we tried to calculate the average amplitude of a sine wave, it would unfortunately equal zero,
since it rises and falls symmetrically above and below the zero reference as seen in Figure1.1 (b).
This would not tell us very much about its amplitude, since low-amplitude and high-amplitude
sine waves would appear equivalent.
1.1.2 Frequency
The number of cycles per unit of time is called the frequency. For convenience, frequency is
most often measured in cycles per second (cps) or the interchangeable Hertz (Hz) (60 cps = 60
Hz).Frequency also describes the number of waves that pass a fixed place in a given amount of
time. So if the time it takes for a wave to pass is 1/2 second, the frequency is 2 per second. If it
takes 1/100 of an hour, the frequency is 100 per hour. In Figure 1.2 (a) high and low fequency
ratio waves are shown and in Figure 1.2 (b) frequency measurement of sine waves is shown.
Figure 1.2 (a) High and low fequency ratio waves.
Figure 1.2 (b) Frequency measurement of sine waves.
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Usually frequency is measured in hertz unit. The hertz measurement, abbreviated Hz, is the
number of waves that pass by per second. For example, an "A" note on a violin string vibrates at
about 440 Hz (440 vibrations per second).
1.2 Importance of Amplitude and Frequency Estimation in Power Signal
Power could be defined as the actual physical power, or more often, for convenience with
abstract signals, could be defined as the squared value of the signals. (Statisticians study the
variance of a set of data, but because of the analogy with electrical signals, still refer to it as the
power spectrum).The total power of the signals would be a time average since power has units of
energy/time.
P = lim �→�
1
2T� x(t)��
��dt
Also, the power of a signal may be finite even if the energy is infinite. Power also can be defined
as the squired amplitude of a particular signal.
P =��∑ A������
Where, A� is the amplitude of a particular signal.
In power system, frequency, amplitude and phasor are the most important parameters for
monitoring, control and protection. All of these can reflect the whole power system situation. In
electrical power system, it is of utmost important to keep the frequency as close to its original
value as possible. In order to control the power system frequency, it needs to be measured
quickly and accurately. Normally, power system voltage and current waveforms are distorted by
harmonic and inter harmonic components, particularly during system disturbance. Faults or other
switching transients may change the magnitude and phase angles of the waveforms. However,
voltage and current can also be distorted by non-linear loads, power electronic components and
inherent non-linear nature of the system elements [8]. Not only that, the assessment of power
quality can be done either by calculating, measuring or estimating power quality indices
(frequency, spectrum, harmonic distortion etc.). Thus, the estimation of power quality indices is
Page|5
still an important and yet challenging part in power system. The estimation techniques alone can
improve the accuracy of measurement of spectral parameters of distorted waveforms that occur
in power systems, in particular, the estimation of the power quality indices [9]. However, more
reliable methods are required for power quality monitoring and estimation.
1.3 Conventional Methods
Most of the conventional methods of power quality assessment in power systems are almost
exclusively based on Fourier Transform. These methods suffer from various inherent limitations.
These are following:
• Estimation error depends on the data length (phase dependence of the estimation error).
• Resolution problem particularly troublesome when analyzing short data records.
• The high degree dependency of the resolution on signal-to-noise ratio.
• The high degree dependency of the resolution on the initial phase of the harmonic
components.
• Due to coherent signal sampling of the data there is a leakage in spectrum domain.
• The improvement of the accuracy of measurement of spectral parameters of distorted
waveforms encountered in power systems, in particular the estimation of the power quality
indices is difficult [9].
In order to overcome these problems, high resolution spectrum estimation method can be used
where resolution problem is not found.
1.4 High Resolution Methods
High-resolution methods are generally defined to be high-performance methods for estimating
and detecting the desired and undesired signal components present in a given set of data. The
term “high-resolution” conjointly implies a decent ability to resolve terribly “Similar” signal
elements. One of the most common issues in signal processing is known as frequency estimation.
In frequency estimation, “high-resolution” typically refers to a decent ability to resolve two or
additional closely set frequencies within the given data. On the other hand, in amplitude
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estimation, “high-resolution” typically refers to a decent ability to resolve two or additional
closely set amplitudes within the given data. There are two types of high-resolution methods,
parametric methods and non-parametric methods.
1.4.1 Nonparametric Methods
The non-parametric high-resolution methods maximize the output of some desired information
with little knowledge of the data structure. The choice between parametric methods and non-
parametric methods largely depends on one’s confidence in the assumed data model [10]
1.4.2 Parametric Methods
The parametric high-resolution methods result from ingenious exploitations of known data
structures. Example: MUSIC method, ESPRIT method etc. Parametric methods can yield higher
resolutions than nonparametric methods in cases when the signal length is short. These methods
use a different approach to spectral estimation; instead of trying to estimate the PSD directly
from the data, they model the data as the output of a linear system driven by white noise, and
then attempt to estimate the parameters of that linear system [10]. The ESPRIT and the root-
Music spectrum estimation methods are based on the linear algebraic concepts of sub-spaces
which are called “subspace methods” [11]; the model of the signal in this case is a sum of
sinusoids in the background of noise of a known covariance function.
1.5 Motivation of using High Resolution Methods
Accuracy of all spectrum estimation methods is not the same. Fourier transform is one of these
methods that suffer much in forms of resolution. In the normal method like fast Fourier
Transform (FFT) that has some limitations. First limitation of this method is that of frequency
resolution, i.e. the ability to distinguish the spectrum responses of two or more signals. Second
limitation of such method is due to coherent signal sampling of the data which proves itself as a
leakage in spectrum domain. We can reduce this leakage by windowing but it introduces
additional distortions. These two performance limitations of FFT or similar methods are
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particularly troublesome for analyzing short data records. To overcome these problems high
regulation spectrum estimation method can be used where no resolution problem is found.
MUSIC, ESPRIT, and root MUSIC are termed as high resolution spectrum estimation methods.
These high resolution methods can develop the accurate measurement of spectral parameters of
distorted waveforms of the power quality indices [9].
1.6 Objective of the Project
The objectives of this project are:
1. To implement MUSIC, ESPRIT and Root MUSIC algorithms by simulations using
MATLAB as a tool.
2. To investigate and compare the amplitude estimation performance of MUSIC and Root
MUSIC in terms of MSE with respect to data length and SNR.
3. To investigate and compare the power (squared amplitude) estimation performance of
MUSIC and Root MUSIC in terms of MSE with respect to data length and SNR.
4. To investigate and compare the Frequency estimation performance of MUSIC, Root
MUSIC and ESPRIT in terms of MSE with respect to data length and SNR.
1.7 Organization of the Project
This project is organized as follows; chapter 1 presents the introduction of the overall project.
Chapter 2 provides a comprehensive overview of high-resolution spectrum estimation methods
for synthetic power signals. Simulation results and quantitative performance of the proposed
methods are shown in detail in chapter 3. Finally, concluding remarks, contribution and
suggestions for future works of the project are described in chapter 4.
1.8 Conclusion
In this chapter, the introduction and organization of the overall project are described in details.
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Chapter 2
High Resolution Spectrum Estimation Methods
2.1 Introduction
There are various kinds of methods that are used in power, amplitude and frequency estimation
to improve the accurate measurement of spectral parameters from distorted waveforms
encountered in power systems, in particular the estimation of the power quality indices of which
the conventional high resolution methods are more effective. In this chapter, the description of
the methods of high resolution especially MUSIC, ESPRIT and ROOT MUSIC are provided.
2.2 MUSIC Method
The idea of MUSIC (Multiple Signal Classification) was developed in [15] [19] where the
averaging was proposed for improvement of the performance of Pisarenko estimator [16].
Instead of using only one noise eigenvector, the MUSIC method uses many noises Eigen filters
and presents signal sub space. The number of computed eigen values M > K +1 where, M is the
largest noise subspace correspond to the signal subspace.
The MUSIC method assumes the model of the signal as:
x=∑ ��� � + ŋ; �=|�|�� � (1)
Where �� = [� ���
�….��(���)�� ]�- projection of the signal vector, ��-complex amplitudes of the
signal components, N-number of signal samples, p-number of components, ŋ-noise, �� -
components frequency;
If the noise is white, the correlation matrix is:
� �∑ �{����∗}����
��� + ���� (2)
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N-p smallest eigen values of the correlation matrix (matrix dimension N>p+1) correspond to the
noise subspace and p largest (all greater than ��- noise variance) correspond to the signal sub
space.
The matrix of noise eigenvectors of the above matrix (2) is defined by:
������=[���� ���� … . . ��] (3)
To compute the projection matrix for the noise subspace:
������ = ������������∗� (4)
The squared magnitude of the projection of an auxiliary vector W= [� �� � … . . ��(���) �]�
allows computation of projection of vector w onto the noise subspace is given by:
�∗������� � = �∗�������������∗� � =
=∑ ��(�� )������ ��∗(�� )!→∑ �������∗(�/�∗)������ (5)
Since each of the elements of the signal vector is orthogonal to the noise subspace, the quantity
goes to zero for the frequencies where w=��. So the MUSIC pseudo spectrum is defined as:
P= [�∗�������������∗� �]-1
The last polynomial in (5) has p double roots lying on the unit circle, which angular positions
correspond to the frequencies of the signal components. This method of finding the frequencies
is therefore called root-MUSIC.
After the calculation of the frequencies, the powers of each component can be estimated from the
eigen values and eigenvectors of the correlation matrix, using the relations
��∗���� = �� and � = ∑ ����∗����� + ���� (6)
And solving for p� − components power.
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2.3 Root MUSIC Method
Root-MUSIC method facilitates the same ideas with MUSIC and differs only in the second step
of the MUSIC algorithm. The main advantage of Root-MUSIC over MUSIC is its lower
computational complexity [19] [20].
The MUSIC spectrum is an all pole function of the form
�"#��� =�
$%�[&'()� *�*��&(()] (7)
Let C=����+ using equation (7) written as
�"#�� = ∑ ∑ �����'"��)�,-���'(). � �"� /���/"�� (8)
Where A=���0�� ������������
1, and �"� is the entry in the !23 row and "23column of C.
Combination of two sums into one gives equation (9):
�"#�� = ∑ ��/��� �������,-4���'(). � (9)
Where �� = ∑ �"�"���4 is the sum of the entries of C. Along the #23 diagonal polynomial
representation D (z) will
D(z)=∑ �����/��4��/�� (10)
If the eigen decomposition corresponds to the true spectral matrix, then MUSIC spectrum
becomes equivalent to the polynomial D(z) on the unit circle and peaks in the MUSIC spectrum
exists as ROOTs of the D(z) lie close to the unit circle [17]. A pole of D (z) at z=$� = │$�│exp
(jarg ($�)) will result in a peak in the MUSIC spectrum at � = %&��({λ/2rd} arg [��]).
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2.4 ESPRIT Method
The original ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) was
developed by another one as in example [13]. It is based on naturally existing shift invariance
between the discrete time series which leads to rotational invariance between the corresponding
signal subspaces. The shift invariance is illustrated below. After the Eigen-decomposition of the
autocorrelation matrix as:
� = �∗� (11)
It is possible to partition a matrix by using special selector matrices which select the first and the
last (M-1) columns of a (M ×M) matrix, respectively:
Г� = [�/��|'(/��)×�]'/��)×/ (12)
Г� = ['(/��)×� |�/��]'/��)×/
By using selector matrices Г two subspaces are defined, spanned by two subsets of eigenvectors
as follows: �� = �U
(13)
�� = �U
The rotational invariance between both subspaces leads to the equation:
�� = �� (14)
Where, (5;)*+, representing different frequency components and matrix ф defined as:
Ф = ,-67� 00 -67� ⋯ 0
0 0⋮ ⋮0 0
⋮ ⋮⋯ -675. (15)
The following relation can be proven [14]:
[Г��]ф=Г� (16)
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The matrix ф contains all information about p components’ frequencies. In order to extract this
information, it is necessary to solve (15) for ф. By using a unitary matrix (denoted as T), the
following equations can be derived:
Г�(�/)ф=Г�(�/)
Г���/ф/∗�� = Г� (17)
In the further considerations the only interesting subspace is the signal subspace, spanned by
signal eigenvectors U8. Usually it is assumed that these eigenvectors correspond to the largest
Eigen values of the correlation matrix and U8 = [u�, u�, … … … . , u9 . ESPRIT algorithm
determines the frequencies -675as the eigenvalues of the matrix ф. In theory, the equation (15) is
satisfied exactly. In practice, matrices 0� and 0 � are derived from an estimated correlation
matrix, so this equation does not hold exactly, it means that (15) represents an over-determined
set of linear equations.
2.5 Conclusion
In this chapter, the theoretical developments of conventional high resolution methods, especially
Multiple Signal Classification (MUSIC), Root MUSIC and Estimation of Signal Parameter via
Rotational Invariance Technique (ESPRIT) methods are provided in details.
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Chapter 3
Simulation Results and Performance Comparison
3.1 Introduction
In power system an accurate knowledge of the spectral components of non-stationary current and
voltage waveforms is required. Basically power quality refers to the calculation of waveform
distortion indices [2], [3]. Several indices are in common use for the characterization of
waveform distortions. In this chapter, the performance of MUSIC, Root MUSIC and ESPRIT are
investigated and compared by using Matlab simulations.
3.2 Working Procedure and Simulation Conditions
To Investigate and represent the high resolution methods, Firstly it is designed a signal both for
clean and noisy conditions by using Matlab as an experimental tool. Secondly, model the data as
the output of a linear system driven by white noise and then estimate the parameters of that non
linear system. Using mean square error (MSE) the variation of amplitude, power (squared
amplitude) and frequency estimation are shown and their influences on MSE are compared for
the different methods. After that detail Matlab simulations are carried out in order to investigate
the performance of these methods. Using MSE the variation of amplitude, power (squared
amplitude) and frequency estimation are shown. Also their influences on mean square error
(MSE) are compared for the different methods. Finally, the performance of high resolution
methods (MUSIC, Root MUSIC & ESPRIT) are Investigated and compared in terms of MSE for
amplitude, power and frequency estimation with respect to data length and SNR.
To evaluate the performance of proposed methods, the simulation conditions are following:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ f1� + .97 ∗ sin �2 ∗ π ∗ t ∗ (2 ∗ f1)� is used for clean
condition and signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ f1� + .97 ∗ sin �2 ∗ π ∗ t ∗ (2 ∗ f1)) +
randn(size(t)� is used for noisy condition.
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In Figure 3.1(a) the signal is plotted for clean condition in linear scale and in Figure 3.1(b) the
signal is plotted for noisy condition in linear scale.
Figure 3.1(a): Source signal for clean condition in linear scale.
Figure 3.1(b): Source signal for noisy condition in linear scale.
0 10 20 30 40 50 60 70 80 90 100-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Signal for clean condition
Signal length
Time
0 10 20 30 40 50 60 70 80 90 100-5
-4
-3
-2
-1
0
1
2
3
4
Signal for noisy condition
Signal length
Time
Page|15
Random noise is generated by using randn function (to generate random noise) in Matlab.
Sampling frequency 50< Fs <1000 Hz.
SNR varies from 100 to 0 dB.
Signal length from 50 to 400 samples is considered.
3.3 Performance Evaluation Criterion
The criterion used for evaluating the MUSIC, ESPRIT and Root MUSIC is mean square error
(MSE) with respect to data length and SNR. The equation of MSE is given by,
For amplitude estimation, A:;< = √[(��)∑ (A�=8> − A?@A)�]���� .
For Frequency estimation, F:;< = √[(��)∑ (F�=8> − F?@A)�]���� .
And for power estimation, P:;< = √[(��)∑ (A�=8>� − A?@A�)]����
Where,
A�=8> = Estimated amplitude
A?@A = Original amplitude
F�=8> = Estimated frequency
F?@A = Original frequency
3.3.1 Data Length
Data length or data sequence length is the length of a power signal. Parametric high resolution
methods (MUSIC, ESPRIT and Root MUSIC) methods use a different approach to spectral
estimation; instead of trying to estimate the frequency, amplitude and power directly from the
data, they model the data as the output of a linear system driven by white noise, and then attempt
to estimate the parameters of that linear system [20]. In addition, the parametric high resolution
methods (MUSIC, ESPRIT and Root MUSIC) lead to a system of linear equations which is
relatively simple to solve. MUSIC, ESPRIT and Root MUSIC methods yield higher resolutions
than nonparametric methods in cases when the signal length is short.
Page|16
3.3.2 SNR
Signal-to-noise ratio (often abbreviated SNR or S/N) is a measure used in science and
engineering that compares the level of a desired signal to the level of background noise. It is
defined as the ratio of signal power to the noise power. It is the power ratio between a signal and
noise. A ratio higher than 1:1 indicates more signal than noise [16]. While SNR is commonly
quoted for electrical signals, it can be applied to any form of signal (such as isotope levels in an
ice core or biochemical signaling between cells). It can be derived from the formula
0+1 = 2B�CDEF 2GH�IJ = µ ⁄⁄
Where,
µ = The signal mean or expected value
= The standard deviation of the noise
3.4 Amplitude Estimation
The parameter that can reflect the whole power system situation is called amplitude. It is very
necessary for the power system monitoring, control and protection. In electrical power system it
is of utmost importance to keep the amplitude as close to its nominal value as possible. In order
to control the power system amplitude it needs to be measured quickly and accurately. For this
the estimation of amplitude is still an important and yet challenging part. The effects of data
length and SNR on MSE for amplitude estimation are following:
3.4.1 Effect of Data Length on MSE for Amplitude Estimation
The data sequence length influences the mean square error and therefore, the accuracy of high
resolution methods depends on data length. The performance of the high resolution methods
(Root MUSIC, MUSIC and ESPRIT) could be identified by comparing the mean square error of
amplitude estimation both for shorter and higher data length.
Both for clean and noisy signal, the performance of the mean square error of amplitude
estimation of the high resolution methods (Root MUSIC, MUSIC, and ESPRIT) are shown
below:
Page|17
3.4.1.1 Clean Signal
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150� is used.
Sampling frequency 300 Hz.
Signal length from 50 to 400 samples are considered.
In Figure 3.2(a) the results are shown where the amplitude estimation (MUSIC & Root MUSIC)
in terms of MSE with respect to data length for clean signal is plotted in log scale and in Figure
3.2(b) the results are shown where the amplitude estimation (MUSIC & Root MUSIC) in terms
of MSE with respect to data length for clean signal is plotted in linear scale.
Figure 3.2(a): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect to
data length for clean signal in log scale.
50 100 150 200 250 300 350 40010
0
101
102
X: 50
Y: 10.17X: 100
Y: 8.989X: 150
Y: 7.142
X: 200
Y: 6.557
X: 250
Y: 6.183
X: 300
Y: 6.742
X: 350
Y: 5.973
MSE of amplitude estimation (MUSIC)depending on data length
Data length
log10(M
SE)
X: 400
Y: 4.307
50 100 150 200 250 300 350 400-10
0
-10-1
-10-2
X: 50
Y: -0.04504X: 100
Y: -0.0845 X: 150
Y: -0.1277X: 200
Y: -0.1753X: 250
Y: -0.2282X: 300
Y: -0.2876X: 350
Y: -0.3549
MSE of amplitude estimation (RootMUSIC) depending on data length
Data length
log10(M
SE)
X: 400
Y: -0.4315
Page|18
Figure 3.2(b): Amplitude estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE
with respect to data length in linear scale.
When roughly summarizing different results from the Figure 3.2(a) and Figure 3.2(b), a list of
data of amplitude estimation for clean signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.1.
Table 3.1: The amplitude estimation for clean signal in terms of MSE for increasing data length
Data Length MSE of MUSIC
Method
MSE of Root MUSIC
Method
50 10.17 -0.04498
100 8.989 -0.08449
150 7.142 -0.12770
200 6.557 -0.17530
250 6.183 -0.22820
300 6.742 -0.28760
350 5.973 -0.35490
400 4.307 -0.43100
50 100 150 200 250 300 350 400
-6
-4
-2
0
2
4
6
8
10
12
14
MSE of amplitude estimation depending on data length
Data length
log10(M
SE)
MUSIC
Root MUSIC
Page|19
Comments: In Figure 3.2(a) and Figure 3.2(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC and Root MUSIC
method results are similar). Root MUSIC method performs better for amplitude estimation in
terms of MSE.
3.4.1.2 Noisy Signal
The performance of the mean square error estimation of amplitude of the high resolution
methods (MUSIC and Root MUSIC) for noisy signal are shown below:
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Random noise is generated by using randn function in Matlab.
Sampling frequency 300 Hz.
Signal length from 50 to 300 samples are considered.
In Figure 3.3(a) the results are shown where the amplitude estimation (MUSIC & Root MUSIC)
in terms of MSE with respect to data length for noisy signal is plotted in log scale and in Figure
3.3(b) the results are shown where the amplitude estimation (MUSIC & Root MUSIC) in terms
of MSE with respect to data length for noisy signal is plotted in linear scale.
Page|20
Figure 3.3(a): Amplitude estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE
with respect to data length in log scale.
Figure 3.3(b): Amplitude estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE
with respect to data length in linear scale.
50 100 150 200 250 300
100.8
100.9
X: 50
Y: 9.685X: 100
Y: 8.981
X: 150
Y: 7.142
X: 200
Y: 6.557X: 250
Y: 6.183
MSE of amplitude estimation (MUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300-10
0
-10-1
-10-2
X: 300
Y: -0.2876
X: 250
Y: -0.2282
X: 200
Y: -0.1753
X: 150
Y: -0.1277
X: 100
Y: -0.08455
MSE of amplitude estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
X: 50
Y: -0.02793
50 100 150 200 250 300-6
-4
-2
0
2
4
6
8
10
12
MSE of Amplitude estimation depending on data length
Data length
log10(MSE)
MUSIC
Root MUSIC
Page|21
When roughly summarizing different results from the Figure 3.3(a) and Figure 3.3(b), a list of
data of amplitude estimation for noisy signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.2.
Table 3.2: The amplitude estimation for noisy signal in terms of MSE for increasing data length
Data Length MSE of MUSIC
Method
MSE of Root MUSIC
Method
50 9.685 -0.04504
100 8.981 -0.08450
150 7.142 -0.12770
200 6.557 -0.17530
250 6.183 -0.22820
300 6.742 -0.28760
Comments: In Figure 3.3(a) and Figure 3.3(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC and Root MUSIC
method results are similar). Root MUSIC method performs better for amplitude estimation in
terms of MSE.
3.4.2 Effect of SNR on MSE for Amplitude Estimation
There is a strong dependency of the accuracy of the frequency estimation on SNR. The
performance of the high resolution methods (MUSIC & Root MUSIC) could be identified by
comparing the mean square error of amplitude estimation both for very low and very high noise
levels.
Both for low and very high noise level the performance of the mean square error of amplitude
estimation of the high resolution methods (MUSIC & Root MUSIC) are shown below:
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Random noise is generated by using randn function in Matlab
Sampling frequency 300 Hz.
SNR varies from 100 to 0 dB.
Page|22
In Figure 3.4(a) the results are shown where the amplitude estimation (MUSIC & Root MUSIC)
in terms of MSE with respect to SNR is plotted in log scale and in Figure 3.4(b) the results are
shown where the amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect
to SNR is plotted in linear scale.
Figure 3.4(a): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect to
SNR in log scale.
Figure 3.4(b): Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect to
SNR in linear scale.
0 10 20 30 40 50 60 70 80 90 100
100.56
100.57
X: 30
Y: 3.632
X: 10
Y: 3.635
X: 20
Y: 3.631
X: 40
Y: 3.632
X: 50
Y: 3.632
X: 60
Y: 3.632
X: 70
Y: 3.632
X: 80
Y: 3.632
X: 90
Y: 3.632
X: 0
Y: 3.798
MSE of Amplitude estimation (MUSIC)depending on SNR
SNR [db]
log10(MSE)
0 10 20 30 40 50 60 70 80 90 100-0.61
-0.6
-0.59
-0.58
X: 0
Y: -0.5897
X: 10
Y: -0.5998
X: 20
Y: -0.6003
X: 30
Y: -0.6004
X: 40
Y: -0.6004
X: 50
Y: -0.6004
X: 60
Y: -0.6004
X: 70
Y: -0.6004
X: 80
Y: -0.6004
MSE of Amplitude estimation (Root MUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 90
Y: -0.6004
0 10 20 30 40 50 60 70 80 90 100-1
-0.5
0
0.5
1
1.5
2
2.5
MSE of amplitude estimation depending on SNR
SNR [db]
log10(MSE)
MUSIC
Root MUSIC
Page|23
When roughly summarizing different results from the Figure 3.4(a) and Figure 3.4(b), a list of
data of amplitude estimation in terms of MSE for changing SNR can be represented, as shown in
Table 3.3.
Table 3.3: Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
Data Length MSE of
MUSIC Method
MSE of Root
MUSIC Method
0 3.798 -0.5897
10 3.635 -0.5998
20 3.631 -0.6003
30 3.632 -0.6004
40 3.632 -0.6004
50 3.632 -0.6004
60 3.632 -0.6004
70 3.632 -0.6004
80 3.632 -0.6004
90 3.632 -0.6004
100 3.632 -0.6004
Comments: In Figure 3.4(a) and Figure 3.4(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC and Root MUSIC
method results are similar). Root MUSIC method performs better for amplitude estimation in
terms of MSE in terms of MSE.
3.5 Power Estimation
In general, power system voltage and current waveforms are distorted by harmonic and inter
harmonic components, particularly during system disturbance. Faults or other switching
transients may change the magnitude and phase angles of the waveforms. However, voltage and
current can also be distorted by non-linear loads, power electronic components and inherent non-
linear nature of the system elements [3]. Not only that the assessment of power quality can be
done either by calculating, measuring or estimating power quality indices (frequency, spectrum,
harmonic distortion etc.).Only the estimation techniques can improve the accuracy of
measurement of spectral parameters of distorted waveforms encountered in power systems, in
particular the estimation of the power quality indices [4]. For these reason the power estimation
is very important. The effects of data length and SNR on power estimation are following:
Page|24
3.5.1 Effect of Data Length on MSE for Power Estimation
The sequence of data length influences the mean square error and therefore, the accuracy of high
resolution methods depends on data length. The performance of the high resolution methods
(MUSIC & Root MUSIC) could be identified by comparing the mean square error of power
estimation both for shorter and higher data length.
Both for clean and noisy signal, the performance of the mean square error of power estimation of
the high resolution methods (MUSIC & Root MUSIC) are shown below:
3.5.1.1 Clean Signal
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150� is used.
Sampling frequency 300 Hz.
Signal length 50 to 400 samples are considered.
In Figure 3.5(a) the results are shown where the power estimation (MUSIC & Root MUSIC) in
terms of MSE with respect to data length for clean signal is plotted in log scale and in Figure
3.5(b) the results are shown where the power estimation (MUSIC & Root MUSIC) in terms of
MSE with respect to data length for clean signal is plotted in linear scale.
Page|25
Figure 3.5(a): Power estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE with
respect to data length in log scale.
Figure 3.5(b): Power estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE with
respect to data length in linear scale.
50 100 150 200 250 300 350 40010
0
101
102
X: 50
Y: 20.37 X: 200
Y: 13.23
X: 100
Y: 18.13X: 150
Y: 14.41X: 250
Y: 12.43
X: 300
Y: 13.63
X: 350
Y: 12.07
X: 400
Y: 8.673
MSE of Power estimation(MUSIC)depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400-10
0
-10-1
-10-2 X: 100
Y: -0.02924X: 150
Y: -0.04442X: 200
Y: -0.06647X: 250
Y: -0.09625X: 300
Y: -0.135X: 350
Y: -0.1844
X: 400
Y: -0.2464
X: 50
Y: -0.02034
MSE of Power estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
-10
-5
0
5
10
15
20
25
30
MSE of power estimation depending on data length
Data length
log10(MSE)
MUSIC
Root MUSIC
Page|26
When roughly summarizing different results from the Figure 3.5(a) and Figure 3.5(b), a list of
data of power estimation for clean signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.4.
Table 3.4: The power estimation for clean signal in terms of MSE for increasing data length
Data
Length
MSE of MUSIC
Method
MSE of Root MUSIC
Method
50 20.39 -0.02034
100 18.13 -0.02924
150 14.41 -0.04442
200 13.23 -0.06647
250 12.43 -0.09625
300 13.63 -0.13500
350 12.07 -0.18440
400 8.673 -0.24640
Comments: In Figure 3.5(a) and Figure 3.5(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC and Root MUSIC
method results are similar). Root MUSIC method performs better for power estimation in terms
of MSE.
3.5.1.2 Noisy Signal
The performance of the mean square error estimation of power of the high resolution methods
(MUSIC and Root MUSIC) for noisy signal is shown below:
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Random noise is generated by using randn function in Matlab.
Sampling frequency 300 Hz.
Signal length 50 to 300 samples are considered.
Page|27
In Figure 3.6(a) the results are shown where the power estimation (MUSIC & Root MUSIC) in
terms of MSE with respect to data length is plotted in log scale and in Figure 3.5(b) the results
are shown where the power estimation (MUSIC & Root MUSIC) in terms of MSE with respect
to data length is plotted in linear scale.
Figure 3.6(a): Power estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with
respect to data length in log scale.
50 100 150 200 250 300
101.1
101.2
X: 50
Y: 17.09
X: 300
Y: 13.63
X: 100
Y: 18.12
X: 150
Y: 14.41
X: 200
Y: 13.23
MSE of Power estimation(MUSIC)depending on data length
Data length
log10(M
SE)
X: 250
Y: 12.43
50 100 150 200 250 300-10
0
-10-1
-10-2
X: 50
Y: -0.0191
X: 250
Y: -0.09625
X: 300
Y: -0.135
X: 200
Y: -0.06647
X: 150
Y: -0.04441
MSE of Power estimation (RootMUSIC) depending on data length
Data length
log10(M
SE)
X: 100
Y: -0.02921
Page|28
Figure 3.6(b): Power estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with
respect to data length in linear scale.
When roughly summarizing different results from the Figure 3.6(a) and Figure 3.6(b), a list of
data of power estimation for noisy signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.5.
Table 3.5: The power estimation for noisy signal in terms of MSE for increasing data length
Data Length MSE of MUSIC
Method
MSE of Root MUSIC
Method
50 17.09 -0.01910
100 18.12 -0.02921
150 14.41 -0.04441
200 13.23 -0.06647
250 12.43 -0.09625
300 13.63 -0.13500
350 5.973 -0.35490
400 4.307 -0.39690
50 100 150 200 250 300
-5
0
5
10
15
20
25
MSE of Power estimation depending on data length
Data length
log10(M
SE)
MUSIC
Root MUSIC
Page|29
Comments: In Figure 3.6(a) and Figure 3.6(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC and Root MUSIC
method results are similar). Root MUSIC method performs better for power estimation in terms
of MSE.
3.5.2 Effect of SNR on MSE for Power Estimation
There is a strong dependency on the accuracy of the power estimation on SNR. The performance
of the high resolution methods (MUSIC and Root MUSIC) could be identified by comparing the
mean square error of power estimation both for very low and very high noise levels.
Both for low and very high noise levels the performance of the mean square error of power
estimation of the high resolution methods (MUSIC & Root MUSIC) are shown below:
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Random noise is generated by using randn function in Matlab.
Sampling frequency 300 Hz.
SNR varies from 100 to 0 dB.
In Figure 3.7(a) the results are shown where the power estimation (MUSIC & Root MUSIC) in
terms of MSE with respect to SNR is plotted in log scale and in Figure 3.7(b) the results are
shown where the power estimation (MUSIC & Root MUSIC) in terms of MSE with respect to
SNR is plotted in linear scale.
Page|30
Figure 3.7(a): Power estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR
in log scale.
Figure 3.7(b): Power estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR
in linear scale.
When roughly summarizing different results from the Figure 3.7(a) and Figure 3.7(b), a list of
data of power estimation in terms of MSE for changing SNR can be represented, as shown in
Table 3.6.
0 10 20 30 40 50 60 70 80 90 100
100.85
100.88
X: 10
Y: 7.27
X: 0
Y: 7.69
X: 30
Y: 7.273
X: 50
Y: 7.274
X: 70
Y: 7.274
X: 90
Y: 7.274
X: 20
Y: 7.269
X: 40
Y: 7.274
X: 60
Y: 7.274
MSE of Power estimation(MUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 80
Y: 7.274
0 10 20 30 40 50 60 70 80 90 100
-10-0.411
-10-0.406
X: 10
Y: -0.3913
X: 70
Y: -0.3923
X: 20
Y: -0.3921X: 30
Y: -0.3924
X: 40
Y: -0.3923
X: 50
Y: -0.3923
X: 60
Y: -0.3923
X: 80
Y: -0.3923
X: 0
Y: -0.3876
MSE of Power estimation(RootMUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 90
Y: -0.3923
0 10 20 30 40 50 60 70 80 90 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
MSE of Power estimation depending on SNR
SNR [db]
log10(MSE)
MUSIC
Root MUSIC
Page|31
Table 3.6: Power estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
Data Length MSE of MUSIC
Method
MSE of Root MUSIC
Method
0 7.69 -0.3876
10 7.27 -0.3967
20 7.273 -0.3943
30 7.274 -0.3933
40 7.274 -0.3923
50 7.274 -0.3923
60 7.274 -0.3923
70 7.274 -0.3923
80 7.274 -0.3923
90 7.274 -0.3923
100 7.274 -0.3923
Comments: In Figure 3.7(a) and Figure 3.7(b), it is seen that there is a sharp decrease of the
estimation error for changing SNR (pattern for MUSIC and Root MUSIC method results are
similar). Root MUSIC method performs better for power estimation in terms of MSE.
3.6 Frequency Estimation
Frequency is the most important parameter in power system monitoring, control, and protection.
It can reflect the whole power system situation. In electrical power system it is of utmost
importance to keep the frequency as close to its nominal value as possible. In order to control the
power system frequency it needs to be measured quickly and accurately. But in general, power
system voltage and current waveforms are distorted by harmonic and inter harmonic
components, particularly during system disturbance. Faults or other switching transients may
change the magnitude and phase angles of the waveforms. So the estimation of frequency is still
an important and yet challenging part in power system. The effect of data length and SNR on
MSE for frequency estimation is shown below:
3.6.1 Effect of Data Length on MSE for Frequency Estimation
The data sequence length influences the mean square error and therefore, the accuracy of high
resolution methods depends on data length. The performance of the high resolution methods
(MUSIC, Root MUSIC and ESPRIT) could be identified by comparing the mean square error of
frequency estimation both for shorter and higher data length.
Page|32
Both for clean and noisy signal, the performance of the mean square error of frequency
estimation of the high resolution methods (MUSIC, Root MUSIC and ESPRIT) are shown
below:
3.6.1.1 Clean Signal
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .98 ∗ sin �2 ∗ π ∗ t ∗ 100� is used.
Sampling frequency 300 Hz.
Signal length 50 to 400 samples are considered.
In Figure 3.8(a) the results are shown where the frequency estimation (MUSIC & Root MUSIC)
in terms of MSE with respect to data length for clean signal is plotted in log scale and in Figure
3.8(b) the results are shown where the frequency estimation (MUSIC & Root MUSIC) in terms
of MSE with respect to data length for clean signal is plotted in linear scale.
Figure 3.8(a): Frequency estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE
with respect to data length in log scale.
50 100 150 200 250 300 350 400
100.32326
100.32327
100.32328
100.32329
100.3233
100.32331
100.32332
100.32333
MSE of frequency estimation depending on data length
Data length
log10(MSE)
MUSIC
RootMUSIC
ESPRIT
Page|33
Figure 3.8(b): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) for clean signal in terms
of MSE with respect to data length in linear scale.
When roughly summarizing different results from the Figure 3.8(a) and Figure 3.8(b), a list of
data of Frequency estimation for clean signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.7.
Table 3.7: The Frequency estimation for clean signal in terms of MSE for increasing data length
Data Length MSE of MUSIC MSE of Root
MUSIC
MSE of ESPRIT
50 1.897 1.898 1.898
100 1.896 1.898 1.898
150 1.895 1.898 1.898
200 1.891 1.898 1.898
250 1.891 1.898 1.898
300 1.886 1.898 1.898
350 1.885 1.898 1.898
400 1.883 1.898 1.898
50 100 150 200 250 300 350 4002.105
2.105
2.1051
2.1052
2.1052
2.1052
2.1053
2.1054
2.1054
2.1054
MSE of frequency estimation depending on data length
Data length
Me
an
Sq
ua
re E
rro
r
MUSICRootMUSICESPRIT
Page|34
Comments: In Figure 3.8(a) and Figure 3.8(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC, Root MUSIC
and ESPRIT method results are similar). Though it is seen that MUSIC method performs better
for frequency estimation but for many simplifications, different assumptions and the complexity
of the problem ESPRIT method is better than MUSIC and Root MUSIC.
3.6.1.2 Noisy Signal
The performance of the mean square error estimation of amplitude of the high resolution
methods (MUSIC, Root MUSIC, and ESPRIT) for noisy signal is shown below:
Conditions:
Signa y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Random noise is generated by using randn function in Matlab.
Sampling frequency 300 Hz.
Signal length from 50 to 300 samples are considered.
In Figure 3.9(a) the results are shown where the frequency estimation (MUSIC & Root MUSIC)
in terms of MSE with respect to data length for noisy signal is plotted in log scale and in Figure
3.9(b) the results are shown where the frequency estimation (MUSIC & Root MUSIC) in terms
of MSE with respect to data length for noisy signal is plotted in linear scale.
Page|35
Figure 3.9(a): Frequency estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE
with respect to data length in log scale.
Figure 3.9(b): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) for noisy signal in terms
of MSE with respect to data length in linear scale.
50 100 150 200 250 300 350 400
100.32326
100.32327
100.32328
100.32329
100.3233
100.32331
100.32332
100.32333
MSE of frequency estimation depending on data length
Data length
log10(MSE)
MUSIC
RootMUSIC
ESPRIT
10
20
30
40
50
60
50 100 150 200 250 300 350 4002.105
2.105
2.1051
2.1052
2.1052
2.1052
2.1053
2.1054
2.1054
2.1054
MSE of frequency estimation depending on data length
Data length
Mean S
quare
Err
or
MUSICRootMUSICESPRIT
Page|36
When roughly summarizing different results from the Figure 3.9(a) and Figure 3.9(b), a list of
data of frequency estimation for noisy signal in terms of MSE for increasing data length can be
represented, as shown in Table 3.8.
Table 3.8: The amplitude estimation for noisy signal in terms of MSE for increasing data length
Data Length MSE of MUSIC MSE of ESPRIT MSE of Root MUSIC
50 1.897 1.898 1.898
100 1.896 1.898 1.898
150 1.895 1.898 1.898
200 1.891 1.898 1.898
250 1.891 1.898 1.898
300 1.886 1.898 1.898
350 1.885 1.898 1.898
400 1.883 1.898 1.898
Comments: In Figure 3.9(a) and Figure 3.9(b), it is seen that there is a sharp decrease of the
estimation error for increasing length of the data sequence (pattern for MUSIC, Root MUSIC
and ESPRIT method results are similar). Though it is seen that MUSIC method performs better
for frequency estimation but for many simplifications, different assumptions and the complexity
of the problem ESPRIT method is better than MUSIC and Root MUSIC.
3.6.2 Effect of SNR on MSE for Frequency Estimation
There is a strong dependency of the accuracy of the frequency estimation on SNR. The
performance of the high resolution methods (MUSIC, Root MUSIC, and ESPRIT) could be
identified by comparing the mean square error of frequency estimation both for very low and
very high noise levels.
Both for low and very high noise levels the performance of the mean square error of frequency
estimation of the high resolution methods (MUSIC, Root MUSIC, and ESPRIT) are shown
below:
Page|37
Conditions:
Signal y = .99 ∗ sin �2 ∗ π ∗ t ∗ 100� + .97 ∗ sin �2 ∗ π ∗ t ∗ 150) + randn(size(t)� is used as a source signal.
Sampling frequency is taken as 300 Hz.
SNR varies from 100 to 0 dB.
In Figure 3.10(a) the results are shown where the frequency estimation (MUSIC & Root
MUSIC) in terms of MSE with respect to SNR is plotted in log scale and in Figure 3.10(b) the
results are shown where the frequency estimation (MUSIC & Root MUSIC) in terms of MSE
with respect to SNR is plotted in linear scale.
Figure 3.10(a): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) in terms of MSE with
respect to SNR in log scale.
0 10 20 30 40 50 60 70 80 90 100
100.32326
100.32328
100.3233
100.32332
100.32334
MSE of frequency estimation depending on SNR
SNR [dB]
log(M
SE) MUSIC
RootMUSIC
ESPRIT
10
20
30
40
50
60
Page|38
Figure 3.10(b): Frequency estimation (MUSIC, Root MUSIC & ESPRIT) in terms of MSE with
respect to SNR in linear scale.
When roughly summarizing different results from the Figure 3.10(a) and Figure 3.10(b), a list of
data of frequency estimation in terms of MSE for changing SNR can be represented, as shown in
Table 3.9.
Table 3.9: Frequency estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
SNR MSE of MUSIC
Method
MSE of Root MUSIC
Method
MSE of ESPRIT
Method
0 2.1051 2.10543 2.10537
10 2.1045 2.10541 2.10537
20 2.1045 2.10541 2.10537
30 2.1045 2.10541 2.10537
40 2.1045 2.10541 2.10537
50 2.1045 2.10541 2.10537
60 2.1045 2.10541 2.10537
70 2.1045 2.10541 2.10537
80 2.1045 2.10541 2.10537
90 2.1045 2.10541 2.10537
100 2.1045 2.10541 2.10537
0 10 20 30 40 50 60 70 80 90 100
2.105
2.1051
2.1051
2.1052
2.1052
2.1053
2.1053
2.1054
2.1054
2.1055
MSE of frequency estimation depending on SNR
SNR [dB]
Mean
Sq
uare
Err
or
MUSIC
RootMUSIC
ESPRIT
Page|39
Comments: In Figure 3.10(a) and Figure 3.10(b), it is seen that there is a sharp decrease of the
estimation error for changing SNR (pattern for MUSIC, Root MUSIC and ESPRIT method
results are similar). MUSIC method performs better for frequency estimation in terms of MSE.
3.7 Conclusion
In this chapter, the performance evaluation criterion and simulation results of the proposed
method are described in details. It is found that both methods (Root MUSIC & ESPRIT) are
similar. Root MUSIC uses both of noise and signal subspace to estimate the signal components
and for this reason Root MUSIC performs better for amplitude and power estimation in terms of
MSE with respect to data length and SNR. Because of simplifications and the complexity of the
problem the performance of ESPRIT method is better for frequency estimation in terms of MSE
with respect to data length. Finally the performance of MUSIC method is better for frequency
estimation with respect to signal to noise ratio (SNR).
Page|40
Chapter 4
Conclusion
4.1 Concluding Remarks
This project represents the high resolution methods MUSIC, RootMUSIC and ESPRIT that
model the data as the output of a linear system driven by white noise and then attempt to estimate
the parameters of that non linear system. Matlab simulations are carried out in details due to
investigate the performance of MUSIC, Root MUSIC and ESPRIT methods in estimating
amplitude, power (squared amplitude) and frequency estimation of synthetic power signal both
in clean and noisy conditions. Using mean square error (MSE) as the evaluation criterion, the
variation of amplitude, power (squared amplitude) and frequency estimation are shown with
respect to data sequence length and SNR and their influences on MSE are compared for the
different methods.
In previous work [19] [20] [30], the performance of MUSIC and ESPRIT methods were
compared for frequency estimation only, they did not analyze MUSIC, Root MUSIC and
ESPRIT methods at a time. Most importantly, the major contribution of this project is to
investigate and compare the performance of MUSIC, Root MUSIC and ESPRIT methods for
amplitude, power and frequency estimation.
It is concluded that both methods (Root MUSIC & ESPRIT) are similar in the sense that they are
both eigen decomposition based methods which rely on decomposition of the estimated
correlation matrix into two subspaces: noise and signal subspace. On the other hand, MUSIC
uses the noise subspace to estimate the signal components while ESPRIT uses the signal
subspace and Root MUSIC uses both of noise and signal subspace. In addition, Root MUSIC
uses both of noise and signal subspace to estimate the signal components and for this reason
Root MUSIC performs better for amplitude and power estimation in terms of MSE with respect
to data length and SNR.
Page|41
In previous work [19] [20] [23] [30], only the response and performance of MUSIC and ESPRIT
were calculated and compared in log scale and they only focused on ESPRIT method. In this
project, the response and performance of MUSIC, RootMUSIC and ESPRIT are calculated and
compared both in log and linear scale with varying from very low value to very high value to
make a comprehensive discussion and focused among all of the mentioned methods.
Finally it is concluded that due to many simplifications, different assumptions and the
complexity of the problem, simulation results represent the performance of ESPRIT method is
better for frequency estimation in terms of MSE with respect to data length. MUSIC uses the
noise subspace that is why the performance of MUSIC method is better for frequency estimation
with respect to signal to noise ratio (SNR).
Though this study is mainly based on comparison of three advanced high resolution estimation
methods, it will likely to be applicable for developing a new idea that will be helpful for future
research in power system analysis.
4.2 Scopes for Future Work
There are still some scopes for future research as mentioned below:
• In particular cases, it can be investigated and compared the performance of MUSIC, Root
MUSIC and ESPRIT methods in terms of MSE with respect to the size of the correlation
matrix.
• In particular cases, it can be investigated and compared the performance of amplitude &
Frequency estimation of Parametric and Non Parametric methods in terms of MSE with
respect to data length and SNR.
• The performance of amplitude, power and frequency estimation in terms of MSE of all
the Methods mentioned above can be compared with respect to other high resolution
methods.
Page|42
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