Table Of ContentTable of Contents
Cover
Title Page
Preface
1 Introduction
1.1 From Fourier transform to wavelet transform
1.2 Application of time–frequency analysis in transient signal processing
1.3 wavelet transform application in transient signal processing of power systems
References
2 The Fundamental Theory of Wavelet Transform
2.1 Continuous wavelet transform [1–4]
2.2 The discretization of continuous wavelet transform [1, 5]
2.3 Multiresolution analysis and the Mallat algorithm
2.4 The properties of basic wavelets [5]
2.5 Frequently used properties of the mother wavelet in engineering signal analysis [5]
References
3 Wavelet Analysis and Signal Singularity
3.1 Lipschitz exponents
3.2 Characterization of signal singularity based on wavelet transform
3.3 The wavelet transform property of random noise
Further reading
4 Sampling Techniques in Wavelet Analysis of Transient Signals
4.1 Wavelet sampling in direct wavelet transform
4.2 Pre-sampling in indirect wavelet transform
4.3 Selection algorithms for pre-sampling filters
4.4 Problems in the implementation of wavelet transform
Reference
Further reading
5 Wavelet Basis Selection for Transient Signal Analysis of Power Systems
5.1 The sources and features of power system transient signals
5.2 Detection of a high-order singular signal
5.3 Detection of weak transients of low-frequency carriers
5.4 Detection of transients in narrow-band interference
5.5 Data compression and de-noising of transients
5.6 Location of transients
5.7 Selection of wavelet decomposition scales
Further reading
6 Construction Method of Practical Wavelets in Power System Transient Signal Analysis
6.1 The construction and application of a class of M-band wavelets [1–2]
6.2 The construction and application of a class of recursive wavelets
6.3 The application and structure of an optimal wavelet basis
References
7 Wavelet Postanalysis Methods for Transient Signals in Power Systems
7.1 Modulus maxima and the singularity exponents method
7.2 Energy distribution analysis method
7.3 Wavelet neural networks
7.4 Statistical and clustering analysis methods
References
8 Application of Wavelet Analysis in High-voltage Transmission Line Fault Location
8.1 Basic principle of traveling wave–based fault location for transmission lines
8.2 Traveling wave fault location method based on wavelet analysis for transmission
lines
8.3 Fault location method based on traveling wave natural frequencies for transmission
lines
8.4 Fault location method based on time–frequency characteristics of traveling wave
References
Further reading
9 Application of Wavelet Transform in Fault Feeder Identification in a Neutral
Ineffectively Grounded Power System
9.1 The basic principle of faulty line selection in neutral ineffectively grounded
systems
9.2 The application of wavelet analysis in faulty line selection in small current
grounded systems
References
10 Application of Wavelet Transform to Non-unit Transient Protection
10.1 Principle of transient protection for EHV transmission lines
10.2 Transient protection based on wavelet transform for EHV transmission lines
References
11 The Application of Wavelet Analysis to Power Quality Disturbances
11.1 Problem description of power quality disturbances
11.2 Power quality disturbance detection based on wavelet transform
11.3 Power quality disturbance identification basedon wavelet transform
11.4 Power quality disturbance source location based on wavelet analysis
References
12 Wavelet Entropy Definition and Its Application in Detection and Identification of Power
Systems’ Transient Signals
12.1 Definitions of six wavelet entropies
12.2 Detection and identification of power systems’ transient signals based on wavelet
entropy
12.3 Fault diagnosis based on wavelet entropy fusing model
References
Appendix A: Simulation Models
A.1 500 kV transmission line simulation model
A.2 The simplified 500 kV two-terminal power system: example 1
A.3 The simplified 500 kV two-terminal power system: example 2
A.4 Neutral ineffectively grounded power system
A.5 Distribution network with capacitor banks
A.6 Power quality disturbances
Author Index
Subject Index
End User License Agreement
List of Tables
Chapter 02
Table 2.1 Properties of special wavelets
Table 2.2 The time–frequency localization parameters of several frequently used
wavelets
Chapter 04
Table 4.1 The filter using the orthogonal wavelet method
Table 4.2 The filter using the optimized orthogonal wavelet method
Chapter 05
Table 5.1 The common disturbance types
Chapter 06
Table 6.1 Wavelet localization parameters (wavelet1 is ψ , wavelet2 is ψ , wavelet3
1 2
is ψ )
3
Table 6.2 The optimized four-coefficient and six-coefficient dyadic sequence h(k)
Chapter 07
Table 7.1 Lipschitz exponents of typical signals
Table 7.2 Processed data under different fault resistances
Table 7.3 Processed data under different fault distances
Chapter 08
Table 8.1 The WMM of the second transient waterfront on each scale
Chapter 11
Table 11.1 The filter bank coefficients of a quadratic spline wavelet
Table 11.2 The location of sharp variations and the corresponding modulus maxima
extracted by a quadratic spline wavelet
Table 11.3 The location of sharp variations and the corresponding modulus maxima
extracted by a db4 wavelet
Table 11.4 Reconstruction factors for de-noising voltage transient disturbances
Table 11.5 Possible state combinations of the capacitors that are switched off
Table 11.6 The classification results of SVM classifiers
Chapter 12
Table 12.1 Summary of results obtained in fault classification (t is the fault inception
A
angle of phase A; R is the fault resistance; and d is the fault location)
Table 12.2 The results of fault phase identification based on the fusion of two arbitrary
wavelet entropies
Table 12.3 The results of fault phase identification based on the fusion of three
arbitrary wavelet entropies
Table 12.4 The results of fault phase identification based on the fusion of four arbitrary
wavelet entropies
Table 12.5 Results of fusion of wavelet entropies’ evidence for fault line identification
Appendix A
Table A.1 Parameters of the power source
Table A.2 The load data
Table A.3 Transformer data
Table A.4 Models of power quality disturbances
List of Illustrations
Chapter 01
Figure 1.1 Diagram of short-time Fourier transform: (a) window in time domain, (b)
time–frequency plane division
Figure 1.2 Diagram of STFT
Chapter 02
Figure 2.1 The stretched or compressed versions of the mother wavelet and the
variation of the wavelet widths corresponding to different a and b parameters: (a) the
mother wavelet, (b) when , (c) when , while b is unchanged, (d) the
wavelet widths with different combinations of a and b
Figure 2.2 Demonstrations of Ψ(aω) versus a. (a) ; (b) ; (c)
Figure 2.3 The time-frequency domains of wavelet transform with different a
Figure 2.4 The classification of wavelets
Figure 2.5 The network structure of Equations (2.32) and (2.33): (a) lowpass
decomposition, (b) highpass decomposition, (c) the combination of the above
Figure 2.6 The implementation of filter banks based on multiresolution
Figure 2.7 Inverse wavelet transform: (a) the jth step, (b) the process in terms of j
decreasing from j to 0
Figure 2.8 The computation of the scale function and the wavelet function
Figure 2.9 Littlewood–Paley wavelet
Figure 2.10 Meyer wavelet
Figure 2.11 Morlet wavelet
Figure 2.12 The first kind of Daubechies wavelets
Figure 2.13 The second kind of Daubechies wavelets, also called Symlet wavelets
Figure 2.14 The third kind of Daubechies wavelets, also called Coiflets wavelets
Figure 2.15 Bubble wavelets
Figure 2.16 Fourth Bspline wavelet
Figure 2.17 Fourth Ospline wavelet
Figure 2.18 Fourth Cspline wavelet
Figure 2.19 The time–frequency localization parameter: (a) the first kind of Daubechies
wavelets, (b) the second kind of Daubechies wavelets, (c) the third kind of Daubechies
wavelets), (d) Bspline wavelets, (e) Ospline wavelets, (f) Cspline wavelets
Chapter 03
Figure 3.1 The graphic of Lipschitz exponent
Figure 3.2 The Lipschitz exponents of common singularities: (a) ramp function, (b) step
function, (c) impulse function
Figure 3.3 The singular signal and its triple decomposition: (a) Singular signal, (b)
decomposition at scale 1, (c) decomposition at scale 2, (d) decomposition at scale 3
Figure 3.4 A noisy sinusoidal signal and its maxima curve
Figure 3.5 Gaussian white noise and its modulus maxima at different scales (1–4)
Chapter 04
Figure 4.1 The sampling process using direct wavelet transform
Figure 4.2 Comparison of the DWT errors using four different methods
Figure 4.3 Variation of DWT errors (dB) under different filter lengths
Chapter 05
Figure 5.1 Wavelet decomposition of signal x (t)
1
Figure 5.2 Wavelet decomposition of signal x (t)
2
Figure 5.3 Wavelet decomposition of signal x (t)
3
Figure 5.4 Detection of transients in a low-frequency carrier with four wavelets
Figure 5.5 Detection of transients in narrow-band interference
Figure 5.6 Detection of an oscillatory damped transient in narrow-band interference
Figure 5.7 The property comparison of a Gauss-H wavelet and B-spline wavelet
Figure 5.8 Location property comparison of a Gauss-H1 wavelet, B-spline wavelet,
and db4 wavelet
Figure 5.9 Comparison of three wavelets for fault location
Figure 5.10 Frequency band diagram of discrete wavelet decomposition (f is the
s
sampling frequency)
Figure 5.11 Timescale spectra of two transients
Chapter 06
Figure 6.1 The decomposition and reconstruction of an M-band wavelet
Figure 6.2 The frequency domain characteristic of a six-band wavelet filter bank
Figure 6.3 The time domain characteristic of a six-band wavelet
Figure 6.4 The frequency domain characteristic of a six-band wavelet
Figure 6.5 The six-band wavelet decomposition of a high-frequency signal mixed with
short-time transient signals: (a) c (n), (b) d (1) (n), (c) d (2) (n), (d) d (3) (n), (e) d (4)
1 1 1 1
(n), (f) d (5) (n)
1 1
Figure 6.6 The dyadic wavelet transform of a high-frequency signal mixed with short-
time transient signals: (a) c (n), (b) d (1) (n), (c) d (2) (n), (d) d (3) (n), (e) d (4) (n),
1 1 1 1 1
(f) d (5) (n)
1
Figure 6.7 The six-band wavelet decomposition of a zero-mode fault current in an EHV
transmission line: (a) c (n), (b) d (1) (n), (c) d (2) (n), (d) d (3) (n), (e) d (4) (n), (f) d
1 1 1 1 1
(5) (n)
1
Figure 6.8 The time domain and frequency domain waveforms of three kinds of
recursive wavelets
Figure 6.9 The frequency measurement curve performed by DFT and the recursive
wavelet method when the frequency hops
Figure 6.10 The phasor measurement curve of amplitude error performed by DFT and
the recursive wavelet method when the frequency hops
Figure 6.11 The phasor measurement curve of angle error performed by DFT and the
recursive wavelet method when the frequency hops
Figure 6.12 The phasor measurement curve of the total error performed by DFT and the
recursive wavelet method when the frequency hops
Figure 6.13 The fault current measurement of a signal with decaying DC component
Figure 6.14 The amplitude measurement curve of a signal with declining DC
component using the wavelet transform method and recursive wavelet method
Figure 6.15 The optimization of the wavelet basis in the analysis of a transient fault
phase voltage signal: (a) voltage, (b) four-coefficient, (c) six-coefficient
Figure 6.16 The optimization of a wavelet basis in the analysis of a transient fault zero-
mode current signal: (a) voltage, (b) four-coefficient, (c) six-coefficient
Figure 6.17 The optimal wavelets corresponding to the fault voltage signals: (a) four-
coefficient, (b) six-coefficient
Figure 6.18 The optimal wavelets corresponding to the fault current signals: (a) four-
coefficient, (b) six-coefficient
Figure 6.19 Comparison of three low-pass filters employing different wavelets
Figure 6.20 Voltage transient compression based on optimal wavelets: (a) discrete
wavelet approximation , (b) wavelet detail , (c) wavelet detail,
(d) wavelet detail , (e) reconstruction signal, as
Figure 6.21 Current transient compression based on optimal wavelets: (a) discrete
wavelet approximation , (b) wavelet detail , (c) wavelet detail,
(d) wavelet detail , (e) reconstruction signal, as
Figure 6.22 Voltage transient compression based on db3: (a) discrete wavelet
approximation , (b) wavelet detail , (c) wavelet detail, (d)
wavelet detail , (e) reconstruction signal, as
Figure 6.23 Current transient compression based on db3: (a) discrete wavelet
approximation , (b) wavelet detail , (c) wavelet detail, (d)
wavelet detail , (e) reconstruction signal, as
Figure 6.24 The relation of compression rate and reconstruction error between an
optimal wavelet (Opt) and db3 wavelet (Db3): (1) fault phase voltage, (2) zero-mode
current
Chapter 07
Figure 7.1 Lipschitz exponents of each current harmonic
Figure 7.2 Lipschitz exponents of each current harmonic with stochastic interference
Figure 7.3 Lipschitz exponents of a fundamental component mixed with harmonics
Figure 7.4 Lipschitz exponents for internal and external faults (F1 and F2 are 1 km
away from the bus M, and the fault resistance is 0.5 ohm): (a) internal fault occurred at
F1 when the voltage magnitude is high, (b) external fault occurred at F2 when the
voltage magnitude is high
Figure 7.5 Lipschitz exponents for internal and external faults (F1 and F2 are at the
middle of the line, and the fault resistance is 0.5 ohm): (a) internal fault occurred at F1
when the voltage magnitude is high, (b) external fault occurred at F2 when the voltage
magnitude is high
Figure 7.6 The variation of energy fluctuation coefficients computed with the current in
Figure 7.4. Ratio , S *S *S ; Ratio , S *S *S *S ; Ratio , S *S *S *S
1 1 2 3 2 1 2 3 4 3 2 3 4 5
Figure 7.7 The variation of energy fluctuation coefficients computed with the current in
Figure 7.5. Ratio , S *S *S ; Ratio , S *S *S *S ; Ratio , S *S *S *S
1 1 2 3 2 1 2 3 4 3 2 3 4 5
Figure 7.8 The time domain waveform of simulation signals
Figure 7.9 Wavelet energy statistical diagram of signal 1 and signal 2
Figure 7.10 Wavelet energy moment statistical diagram of signal 1 and signal 2
Figure 7.11 Wavelet network induced by orthogonal wavelet transform
Figure 7.12 Wavelet network induced by a wavelet frame
Figure 7.13 Wavelet network induced by a wavelet basis
Figure 7.14 Flow chart of the algorithm
Chapter 08
Figure 8.1 The superposition principle for traveling wave analysis: (a) a fault occurs at
point F, (b) the equivalent faulted network, (c) a pre-fault network, (d) a pure-fault
network
Figure 8.2 Refraction and reflection of a traveling wave
Figure 8.3 Traveling wave fault location principle: (a) single-ended fault location
(total reflection), (b) single-ended fault location (partial reflection), (c) double-ended
fault location
Figure 8.4 Wavelet transform of an aerial mode current when a single phase-to-ground
fault occurs 100 km from the measuring end
Figure 8.5 Spectrum of the α mode current for a three-phase fault
Figure 8.6 Equivalent circuit of a line represented as two ports connecting to power
systems represented as lamped circuits
Figure 8.7 Flow chart of a fault location using traveling wave natural frequencies
Figure 8.8 The magnitude of complex wavelet transform of an α mode current
Figure 8.9 Curve of norm entropy for scale 1 to 512
Figure 8.10 The waveform of the α mode current traveling wave
Figure 8.11 The selected best: (a) decomposition scale and (b) arrival time of the
second transient wave front
Chapter 09
Figure 9.1 Simplified diagram of the neutral ungrounded system
Figure 9.2 Phasor diagram of a phase A–to-ground fault
Figure 9.3 The distribution diagram of the current after a single phase-to-ground fault
occurred
Figure 9.4 The zero-sequence equivalent network of a single phase-to-ground fault and
its phasor diagram: (a) Equivalent network, (b) phasor diagram
Figure 9.5 The current distribution of a neutral grounded via Peterson coil system after
