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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from the published and unpublished work of others has been acknowledged in the text and a list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



3.5.2 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



3.6.2 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


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


Figure 3.5 (b): Power estimation (MUSIC & Root MUSIC) for clean signal


Figure 3.6 (a): Power estimation (MUSIC & Root MUSIC) for noisy signal


Figure 3.6 (b): Power estimation (MUSIC & Root MUSIC) for noisy signal


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


Figure 3.8 (b): Frequency estimation (MUSIC & Root MUSIC) for clean signal


Figure 3.9 (a): Frequency estimation (MUSIC & Root MUSIC) for noisy signal


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


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


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


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

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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: �� = Г�U

(13)

�� = Г�U

The rotational invariance between both subspaces leads to the equation:

�� = Ф�� (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

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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.




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


data of amplitude estimation for noisy signal in terms of MSE for increasing data length can be


Table 3.2: The amplitude estimation for noisy signal in terms of MSE for increasing data length


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




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:


Random noise is generated by using randn function in Matlab



Page|22


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


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




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


(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:


Conditions:



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


data of power estimation for clean signal in terms of MSE for increasing data length can be


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



method results are similar). Root MUSIC method performs better for power estimation in terms

of MSE.


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:





Page|27


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.


data of power estimation for noisy signal in terms of MSE for increasing data length can be


Table 3.5: The power estimation for noisy signal in terms of MSE for increasing data length


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



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:






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.


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


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


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


(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:


Conditions:




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


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.


data of Frequency estimation for clean signal in terms of MSE for increasing data length can be


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


Data length

Me

an

Sq

ua

re E

rro

r

MUSICRootMUSICESPRIT

Page|34


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.


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.




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


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


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


Data length

Mean S

quare

Err

or

MUSICRootMUSICESPRIT

Page|36


data of frequency estimation for noisy signal in terms of MSE for increasing data length can be


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


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:


Sampling frequency is taken as 300 Hz.


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.


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


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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