Development of a wavelet-based algorithm to detect and determine ...

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6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 35 gorithms etc. Therefore, the situation is that the wavelet transform applications in power systems are very meaningful but just at the beginning, and there is a great deal of concerning research to do. The wavelet transform with its special characteristics, such as the time-frequency lo- cation, the auto-adaptivity to frequency, the flexible selection and the fast algorithms etc., opens a door towards a lot of possibilities and large developing field in power systems. However, it is not very easy to start understanding and using the wavelet transform because the wavelet transform is built up on the theories of modern mathematics and modern signal processing [7] and the most the of published books about wavelet transform are the contributions of mathematicians. In this chapter, the first section gives the basic ideas of the wavelet transform. In the second section, the continuous wavelet transform (CWT) is introduced. The discrete wavelet transform (DWT) as the key point will be introduced in the third section. The forth section displays how the wavelet transform works through an example. The fifth section then describes and discusses the general problems and corresponding solutions in practical applications. 6.2 Basic ideas of the wavelet transform 6.2.1 Representation of signals through transforms The idea in mathematics and in engineering is to represent and to analyze a signal or a system in different domains. The changes from on domain to another are called transforms. An useful transform is to decompose a signal into elementary functions for a space. These elementary functions should be chosen with some care to be sure that the transform is invertible. Generally, they should form an orthogonal base or a base for the respective space.
6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 36 Suppose that a sequence {ϕn(t)} is a base for a space, and then any functions f(t) in the space can be represented with this base. Namely, there is Equation 6.1. f(t) = � Inϕn(t) n ∈ Z (6.1) Where In is a scalar sequence generated through the inner product of f(t) and ϕn(t), as shown in equation 6.2. {In} = 〈f(t), ϕn(t)〉 = Where ϕ ∗ n(t) is the complex conjugation of the function ϕn(t). � +∞ −∞ f(t)ϕ ∗ n(t)dt (6.2) Equations 6.1 and equation 6.2 form a pair of transforms. Equation 6.2 is called the (for- ward) transform or the decomposition because it breaks a function into pieces through the base for the space. Equation 6.1 is called the inverse transform or the reconstruction because it puts the pieces back together to retrieve the original signal. It is clear that the property of a base is determinate for the transform. A transform theory always involves how to break a function apart (transform) and then to put it back together (inverse transform). The reason for doing this is that sig- nificant insights, clear patterns, and efficiencies can be obtained through the operation on the pieces rather than the original function. Electrical engineers are familiar to many transforms: Fourier transform, Laplace transform, z transform, symmetric components transform and so on. These transforms all have own well working areas. The wavelet transform is nothing else but another transform. It represents signals through the space base that have the expected characteristics. Representation of signals through the wavelet transform Wavelets and the admissible condition: A wavelet belongs to one kind of functions which have the following characteristics: • The amplitude quickly decays to zero in both directions.
Page 1 and 2: Development of a wavelet-based algo
Page 3 and 4: Eidesstattliche Versicherung Ich ve
Page 5 and 6: 2.5.2 Software modules . . . . . .
Page 8 and 9: Chapter 1 Introduction Electrical p
Page 10 and 11: Chapter 2 Power Quality Recorder Si
Page 12 and 13: 2.2. FUNCTIONS 5 2.2.2 Power data r
Page 14 and 15: 2.3. INTERFACES 7 The binary inputs
Page 16 and 17: 2.4. OPERATION MODES 9 cessed. Figu
Page 18 and 19: 2.5. SOFTWARE (OSCOP P) 11 2.5.2 So
Page 20 and 21: 2.5. SOFTWARE (OSCOP P) 13 Figure 2
Page 22 and 23: 3.1. SIMEAS R CONFIGURATION 15 •
Page 24 and 25: 3.2. LABORATORY USE 17 Then the SIM
Page 26 and 27: 4.2. SOURCES OF TRANSIENT DATA 19 c
Page 28 and 29: 4.3. FILES FOR DATA EXCHANGE 21 4.3
Page 30 and 31: 4.3. FILES FOR DATA EXCHANGE 23 p=
Page 32 and 33: 4.3. FILES FOR DATA EXCHANGE 25 ft
Page 34 and 35: 4.4. DIFFERENCES BETWEEN THE 1991
Page 36 and 37: Figure 5.1: Structure chart of the
Page 38 and 39: 5.1. DATA STRUCTURE OF DATA STORAGE
Page 40 and 41: Chapter 6 Wavelets 6.1 Introduction
Page 44 and 45: 6.2. BASIC IDEAS OF THE WAVELET TRA
Page 60 and 61: Chapter 7 Algorithm to determine ce
Page 62 and 63: 7.2. ANALYZATION OF THE DISTURBANCE
Page 68 and 69: 7.3. MATLAB IMPLEMENTATION 61 7.3.2
Page 70 and 71: Chapter 8 Conclusion and Look-Out T
Page 72 and 73: LIST OF FIGURES 65 6.8 Scale level
Page 74 and 75: Bibliography [1] Ronald W. Schafer

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