ON LINE FAULT DETECTION FOR TRANSMISSION LINE USING
POWER SYSTEM STABILIZER SIGNALS
HAMZA ABUBEKER ALI FALIFLA
A project report submitted in partial fulfillment of the
requirements for the award of the degree
of Master Engineering
(Electrical-Mechatronics & Automation Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iii
To my late mother, my father, my brothers and sisters
For their support and care
iv
ACKNOWLEDGEMENT
First of all, Praise to Allah, the Most Gracious and Most Merciful, Who has
created the mankind with knowledge, wisdom and power.
I would like to take this opportunity to express my deepest gratitude to a
number of people who have provided me with invaluable help over the course of my
studies.
I would like to thank my supervisor Dr. Mohd Fauzi Othman, for his help and
encouragement during this project, and for reviewing this thesis. His wise
suggestions have always helped me and a great number of them have gone into the
thesis.
My sincere gratitude and thanks also goes to those who have contributed to
the completion of this research directly or indirectly.
v
ABSTRACT
It is a well known fact that power systems security is required to smooth
power operations and planning. This requires that power system operators at the
control centre appropriately handle information on faults and detect these faults
effectively. In this study, the “oscillation” for each of the four machines in “no fault
condition”, “fault with PSS” and “without PSS “are recorded at various fault
conditions for fault detection using a Multi Resolution Analysis (MRA) Wave
Transform. The MRA decomposes the signal where the components are analyzed for
their energy content and characteristic and then used as a feature for different classes
and condition of the fault. The same features are also fed to the Generalized
Regression Neural Network (GRNN) and Probabilistic Neural Network (PNN) as a
fault classifier and the results are compared for analyzing classification rate
performance. Once the fault is classified using the above classifier, its location is sent
to the lookup table using the online neuro- fuzzy control strategy the optimum value
of the gain and time constant for the PSS (Power System Stabilizer) are selected and
used to compensate the damping at various fault conditions. Then by using
PST(Power System Toolbox) to build state variable models in small signal analysis,
and for modeling of machines and control system for performing transient stability
simulation of a power system, These dynamic models are coded as MATLAB
functions. The expected results will show that the control action of PSS (Power
System Stabilizer) using this method is more robust in damping the oscillation
compared to the fixed conventional PSS. Hence, this study will show that not only
the PSS able to compensate the damping due to the disturbance but also by using the
developed algorithm it succeeds to detect and classify the fault conditions on the
parallel transmission lines.
vi
ABSTRAK
Sudah diketahui bahawa keselamatan bagi sistem kuasa adalah perlu bagi
melicinkan operasi sistem kuasa dan perancangan. Ini memerlukan operator sistem
kuasa di pusat kawalan mengendali maklumat mengenai kerosakan dan mengesannya
dengan efektif. Di dalam kajian ini pengayunan bagi empat mesin di dalam keadaan
tiada kerosakan, keadaan kerosakan dengan PSS dan tanpa PSS direkodkan di dalam
pelbagai jenis keadaan kerosakan bagi mengesan kerosakan dengan menggunakan
“Multi Resolution Analysis (MRA) Wave Transform”. MRA menguraikan isyarat
dimana komponen-komponennya dianalisis kandungan tenaganya dan ciri-cirinya.
Ia digunakan sebagai satu kaedah untuk pembezaan kelas-kelas dan keadaan
kerosakan. Ciri yang sama digunakan untuk “Generalized Regression Neural
Network (GRNN)” dan “Probabilistic Neural Network (PNN)” sebagai pengkelasan
kerosakan dan hasil keputusannya dibandingkan untuk menganalisis prestasi kadar
pengkelasan.
Apabila kerosakan dikelaskan menggunakan pengkelasan diatas,
lokasinya dihantar untuk jadual carian menggunakan keadah kawalan “online neorofuzzy”untuk nilai optimum dan pemalar masa bagi “PSS (Power Sistem Stabilizer)”
dipilih dan digunakan untuk redaman pelbagai keadaan kerosakan.
dengan
menggunakan
“PST
(Power
System
Toolbox)”
untuk
Kemudian
membina
pembolehubah keadaan model-model dalam analisis isyarat kecil, dan untuk
memodelkan mesin-mesin dan sistem kawalan untuk disimulasikan kestabilan
sementara sistem kuasa. Model kestabilan ini dikodkan mengunakan fungsi-fungsi
MATLAB.
.
vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xi
LIST OF FIGURES
xii
LIST OF ABBREVIATIONS
xiv
LIST OF APPENDICES
xv
INTRODUCTION
1
1.1
Problem statement
1
1.2
Objective of the Project
2
1.3
Scope of the Project
3
1.4
Thesis outline
7
LITERTURE RVIEW
8
2.1
Introduction
8
2.2
Transient Protection of Transmission Line
9
Using Wavelet Transform
2.3
Wavelet Transform In The Accurate
detection of high Impedance Arcing Faults In
High Voltage Transmission Lines
14
viii
3
BACKGROUND OF THE PROJECT
20
3.1
Introduction
20
3.2
Rotor angle stability
21
3.2.1
Small-signal
22
3.2.2
Transient stability
24
3.3
Voltage stability and voltage collapse
24
3.4
Mid-term and long-term stability
25
3.5
Block Model
25
3.5.1 Classical Model
26
3.5.2
Detail model with out control
26
3.5.3
Detail Model of the excitation control
26
3.6
Basic concepts in applying PSS
27
3.7
Types of input signals which used in
30
power system stabilizer (PSS)
3.8
Control signal
30
3.9
Structure and Tuning of PSS
32
3.9.1 Washout circuit
33
3.9.2
Dynamic compensator
33
3.9.3
Torsional filter
35
3.9.4
Limiter
35
3.10
Wavelet Transforms
36
3.10.1 Short-Time Fourier analysis
38
3.10.2 The ability of the Wavelet Transforms
39
3.10.3 Easy steps to continuous Wavelet
41
Transform
3.10.4 The suitable selecting for the
43
Wavelet algorithms
3.10.4.1
Haar Wavelet algorithms
3.10.4.1.1
3.10.4.2
Limitation of the Haar
The Daubechies Wavelet
45
45
46
Transforms
3.10.4.3
Comparison between Haar
47
and Daubechies
3.11
Neural Networks
48
ix
4
3.11.1 Generalized Regression Neural Networks
48
3.11.2 Probabilistic Neural Networks(PNN)
49
3.11.2.1
The Problem with PNN
50
3.11.2.2
The Solution Weighted PNN
51
METHODOLOGY OF THE PROJECT
53
4.1
Methodology of the Project
53
4.2
Test the system
55
4.2.1
Run the Base Program
55
4.2.2
Run the Wavelet Program
55
4.3
Fault Detection
56
4.4
Fault Classification
57
4.5
Using GRNN As Fault Classifier
58
and Detector
4.6
Using PNN As Fault Classifier
58
and Detector
5
RESULTS
60
5.1
Introduction
60
5.2
Test the system
61
5.2.1
Run the Base Program
61
5.2.2
Run the Wavelet Program
63
5.3
Result and Discussion
67
5.4.
Fault Classification
71
5.4.1
72
5.5
Using GRNN as a Fault Classifier
5.4.2 Using PNN as a Fault Classifier
72
Fault Location
73
5.5.1
73
Using GRNN as a Fault Detector
5.5.2 Using PNN as a Fault Detector
6
74
CONCLUSION AND FURTHER
76
6.1
77
Conclusion
x
6.2
Further Work
78
REFERENCES
79
APPENDIX A
82
xi
LIST OF TABLES
TABLE
TITLE
PAGE
2.1
Comparison of speed of different wavelet
12
3.1
Comparison between Haar and Daubechies
47
5.1
The best features for db5-L5 analysis when
67
three phase fault occurs in generator 1.
5.2
The best features for Haar analysis when
67
three phase fault occurs. in generator 1.
5.3
The best features for db5-L5 analysis when
68
three phase fault occurs in generator 2.
5.4
The best features for Haar analysis when
68
three phase fault occurs in generator 2.
5.5
The best features for db5-L5 analysis when line to
69
line to ground fault occurs in generator 1.
5.6
The best features for Haar analysis when line to
69
line to ground fault occurs in generator 1.
5.7
The best features for db5-L5 analysis when line to
70
line to ground fault occurs. in generator 2.
5.8
The best features for Haar analysis when line to
71
line to ground fault occurs in generator 2.
5.9
Classification the Faults by using GRNN
72
5.10
5.11.
Classification the Faults by using PNN.
Using GRNN as a detection fault
73
74
4.12.
Using PNN as a detection fault
74
xii
LIST OF FIGURES
FIGURE
TITLE
PAGE
1.1
Single Line Diagram of Two-Area system
4
1.2
Flow chart for the methodology of the project.
6
2.1
system configuration
10
2.2
The a-g fault current waveform
11
2.3
Detail coefficients for a-g fault signal
13
2.4
Flow chart for wavelet algorithm.
14
2.5
The 154 kV Korean transmission system studied
16
2.6
A typical fault current waveform at relaying point
16
2.7
The coefficient of d1 under ‘a’-earth HIF using DWT
17
2.8
A block diagram of the fault detection technique
18
3.1
Synthesis of accelerating power signal
31
3.2
Configuration of PSS Function
32
3.3
Block diagram of PSS
32
3.4
Washout circuit.
33
3.5
The mother wavelet
36
3.6
Wavelet analysis
37
3.7
Short-Time Fourier analysis
39
3.8
ability the Wavelet to perform local analysis
40
3.9
Difference between Fourier and Wavelet Coefficients
40
3.10
Steps of the Continuous Wavelet Transform
41
3.11
A comparison of the DWT and CWT
43
3.12
Difference between Haar and Daubechies
44
for the same Data
3.13
MATLAB GRNN Network Architecture
49
xiii
(FROM MATLAB)
3.14
PNN is not robust with respect to a define
50
Transformations of feature space. Originally
3.15
The principle axes of the affine transformation
51
4.1
Flow chart for the methodology of the project.
54
5.1
Single Line Diagram of Two-Area system.
60
5.2
The Generator Speed Deviation following a three phase
62
fault at 5% of Transmission line
5.3
The Generator Speed Deviation following a Line to Line
63
to Ground fault at 5% of transmission Line
5.4
The DWT of a Speed Deviation of Generator No1,
63
using db5 analysis
5.5
The DWT of a Speed Deviation of Generator No 1
64
as an out put of the ezd5_33 program.
5.6
The DWT of a Speed Deviation of Generator No1,
65
using Haar analysis
5.7
The DWT of a Speed Deviation of Generator No.1.
as an out put of the abd_33 program.
66
xiv
LIST OF ABBREVIATIONS
PSS
-
Power System Stabilizer
PST
-
Power System Toolbox
AVR
-
Automatic Voltage Regulator
WT
-
Wavelet Transforms
MRA
-
Multi Resolution Analysis
GRNN
-
Generalized Regression Neural Networks.
PNN
-
Probabilistic Neural Networks
T.L
-
Transmission Line
F.T
-
Fourier Transform
CWT
-
Continuous Wavelet transform
DWT
-
Discrete Wavelet transform
ϕ
-
Mother Wavelet
δ
-
Smoothing Factor
FIR
-
Filter Implementation
HIF
-
High Impedance Faults
EMTP
-
Electromagnetic Transients Program
MVA
-
Mega Volt Amperes
CT
-
Current Transform
d
-
Detail Coefficient
a
-
Approximate Coefficient
FC
-
Fault Current
HVDC
-
High Voltage Direct Current
EMTP
-
Electro Magnetic Transients Program
xv
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
MATLAB PROGRAM
82
1
CHAPTER 1
INTRODUCTION
1.1
Problem statement
Electric power systems are among the largest structural achievements of
man. The generators within interconnected power systems usually produce
alternating current, and are synchronized to operate at the same frequency. In a
synchronized system, the power is naturally shared between generators in the ratio of
the rating of the generators, but this can be modified by the operator. Systems which
operate at different frequencies can also be interconnected, either through a
frequency converter or through a direct current tie.
Alternating current generators remain in synchronism because of the selfregulating properties of their interconnection. If one machine deviates from its
synchronous speed. Power is transferred from the other generators in the system in
such a way as to reduce the speed deviation. The moments of inertia of the
generators also come into play, and result in the speed overcorrecting in an
analogous manner to a pendulum swinging about its equilibrium; the pendulum
inertia is equivalent to generator inertia, and the torque on the pendulum due to
gravity is equivalent to synchronizing torque between the generators in the power
system. However, generators are much more complicated dynamic devices than
pendulums, and one must not be tempted to put too much emphasis on this analogy.
2
However, it is true to say that power system oscillations are as natural as those of
pendulums.
An interconnected power system can not operate without control. This is
affected by a combination of manual operator controls and automatic control. The
operators control the power that the generator supplies under normal operating
conditions and the automatic controls come into play to make necessary fast
adjustments so as to maintain the system voltage and frequency within design limits
following sudden changes in the system. Thus, most generators have speed
governing systems which automatically adjust the prime mover driving the generator
so as to keep the generator speed constant, and voltage regulating systems which
adjust the generators” excitation to maintain the generator voltages constant. The
automatic controls in power systems must, as with other automatic feedback controls,
be designed so that oscillations decay rather than grow.
It is well established that fast acting exciters with high gain AVR can
contribute to oscillatory instability in power systems. This type of instability is
characterized by low frequency (0.2 to 2.0Hz) oscillations which can persist( or even
grow in magnitude) for no apparent reason. This type of instability can endanger
systems security and limit power transfer. The major factors that contribute to the
instability are:
•
Loading of the generator or tie line.
•
Power transfer capability of transmission lines.
•
Power factor of the generator (leading power factor operation is more
problematic than lagging power factor operation).
1.2
Objective of the Project:
The main idea for this study is to propose a new method as supervised neural
network fault classifier. to integrate the application of PSS (power Systems
Stabilizer) as a the fault detection,. Power systems stability may be defined as “that
property of a power system that enables it to remain in a state of operating
3
equilibrium under normal operating conditions and to regain an acceptable state of
equilibrium after being subjected to a disturbance”.
Power systems stabilizers have been used for many years to add damping to
electromechanical oscillations. Essentially, they act through the generators excitation
system in such a way that a component of the electrical torque proportional to speed
change is generated (an addition to the damping torque).
A power systems stabilizer is used to add a modulation signal to a generator’s
automatic voltage regulator reference input. The idea is to produce an electrical
torque at the generator proportional to speed. Power systems stabilizer uses a simple
lead network compensator to adjust the input signal to give it the correct phase. The
most simple and typical type is the ΔΡ input type. And, recently Δ w input type
and/or Δ f input type PSS are also adopted in order to improve a stability of interarea mode due to the recent increase in power system and power re-routing.
For this, an output signal of the speed deviations of each generator of the
multi area multi machines system are taken as the input for wavelet analysis. The
basic concept in wavelet analysis is to select a proper wavelet, called mother wavelet
(analyzing wavelet or admissible), and then perform an analysis using its translated
and dilated versions. In this study the only Daubechies two wavelet transform (haar ,
db) are used to analyze the speed deviation measured on each of the generator in the
test system.
1.3
Scope of the Project
As we know, that the power system consists of components such as
generators, lines, transformers, loads, switches and compensators. The general
configuration of a modern power system is that power sources and loads are widely
dispersed. Generators and loads may be hundreds of miles away. The number of bulk
power exchanges over long distances has increased as a consequence of the
4
deregulation of the electric power industry. Usually, distributed control agents are
employed to provide reactive control at several places on the power network through
power system stabilizers (PSSs), automatic voltage regulators (AVRs), etc. Although
local optimization is realized by these agents, the lack of coordination among the
local agents may cause serious problems, such as inter-area oscillations. The power
system oscillations are complex, and they are not straightforward to analyze.
Therefore, before going into any detail, I will use an example to show the basic types
of oscillations that can occur. The example two – area system is artificial; its model
was created for a research report commissioned from Ontario Hydro by the Canadian
Electrical Association to exhibit the different types of oscillations that in both large
and small interconnected power systems. A single line diagram of the system is
shown in Figure 1.1 below [2].
1
10
20
11
3
13
2
120
110
12
101
14
4
Figure 1.1:
Single Line Diagram of Two-Area system.
Because of that, it is necessary to know fact that power system security
required to smooth power operation and planning. This requires that power system
operators at the control centre appropriately handle information on faults and detect
these faults effectively. In this study, The “ oscillation “ for each of the 4 machines in
“ no fault condition”, “ fault with PSS” and “without PSS” are recorded at various
fault conditions for fault detections for fault detection using a Multi Resolution
Analysis ( MRA ) Wavelet Transform.
5
The automatic controls in power systems must, as with other automatic
feedback controls, be designed so that oscillations decay rather than grow. This then
brings us to the reason for this study:
•
The nature of power systems oscillations.
•
The mathematical analysis techniques necessary to predict system
performance
•
Control methods to ensure that oscillations decay with time
Oscillations are observed in power systems as soon as synchronous
generators are interconnected to provide more power capacity and more reliability.
Originally, the interconnected generators are fairly close to one another, and
oscillations are at the frequencies in the order of 1 to 2 Hz. Damper windings on the
generator rotor are used to prevent the oscillations amplitudes increasing. Damper
windings act like a squirrel cage winding of an induction motor and produce a torque
proportional to the speed deviation of the rotor from synchronous speed. They absorb
the energy associated with the systems oscillations and therefore cause their
amplitudes to reduce.
As power systems reliability becomes increasingly important, the requirement
for a system to be able to recover from a fault cleared by rely action was added to the
systems design specifications. Rapid automatic voltage control is used to prevent the
system’s generators losing synchronism following systems fault. Fast excitation
systems however; tend to reduce the damping of system oscillations. Originally, the
oscillations most affected are those between electrically closely coupled generators.
Special stabilizing controls (Power System Stabilizers) are designed to damp these
oscillations.
The main steps procedures methodology for the project is presented by the
flow chart in the Figure 1.2
6
Figure 1.2:
Flow chart for the methodology of the project.
7
1.4
Thesis out line
Chapter 1: Consists of problem statement, objective, and the research scope,
methodology background is also presented.
Chapter 2: Includes introduction of fault detection on transmission lines
using wavelet transforms and explained in detail two previous papers which is
proposed by this approach, both the papers applied transient protection of
transmission line using wavelet transform; and wavelet transform in accurate
detection of high impedance arcing faults in high voltage transmission lines.
Chapter 3: This chapter consists of the background of the project; showing
up in detail the Basic concepts in applying PSS, Wavelet Transforms ,Generalized
Regression Neural Networks (GRNN) and Probabilistic Neural Networks (PNN).
Chapter 4: consists of the methodology process by showing up the detailed
diagram of the project methodology and highlights briefly the steps that have been
taken to meet the objective of this project.
Chapter 5: Presents the results of online fault detection on transmission line
using power system stabilizer signals, which applied two different types of fault;
three phase to ground and line to line to ground fault, then got signals as output from
the generators and got best features to enter it as input data to the Haar and db5 by
using MATLAB tool box to training it by using Generalized Regression Neural
Networks (GRNN) and Probabilistic Neural Networks (PNN)
Chapter 6: consist of the conclusion and some suggestions for future work.
8
CHAPTER 2
LITERTURE REVIEW
2.1
Introduction
In Electric power system, when transmission line fault occurring, plenty of
transient components of different frequency will be generated. Lots of fault
information is included in the transient components. So it can be used to predict the
fault or abnormity of equipments or power system, deal with the fault and analyze
the reason of fault or abnormity, the reliability of the power system will be
considerably improved. Today, to accurately obtain large amounts of various fault
transient information in time has become the reality. But the key problem is how to
use those transient signals to detect fault or to classify fault. Therefore the new
information mergence methods and the effective technology used in detection and
classification of electric power system faults transient is need to studied [11].
Information mergence usually includes a lot of information mergence
techniques, such as the estimation, statistics, information theory, artificial
intelligence and so on. The information theory technique includes clustering analysis,
relativity analysis, entropy theory and template methods, etc. Before analyzed by the
methods above, feature picking-up is an important approach to information
mergence, and the key to detection and identification of transient signals.
9
Recently there are many techniques used in picking-up the feature of signals,
e.g. time domain analysis, frequency domain analysis, time-frequency domain
analysis and bi-spectrum analysis. A lot of paper has been published in these fields,
e.g. time-sequence analysis and wavelet analysis etc. These technologies have
applied broadly in many industrial fields such as transmission lines fault detection,
here there are some of paper has been published in the same area.
2.2
Transient Protection of Transmission Line Using Wavelet Transform
The main function of digital relays in power system is to reduce the
consequences of faults by fault detection, localization and clearance of faults. The
main power signal analysis tools, which are currently used in the digital relay, have
proven very useful and efficient in power system steady state analysis. Among these
are (1) Kalman filtering based algorithms, (2) Fourier analysis based algorithms, (3)
Least squares methods based algorithms and (4) FIR filtering based protection.
However in presence of non-stationary signals, the performance of these techniques
is limited [11].
A more recent solution to the problem is the wavelet transform, That the
wavelet analysis has the capability of providing accurate transient information in
both time and frequency domain. The current and voltage signals obtained from a
transmission line when a fault occurs have long duration low frequency components
and short duration high frequency components. Wavelet transforms appear to offer
the right characteristics to analyze the information contained in these signals for the
purpose of line protection. Wavelet transform has a special feature of variable timefrequency localization, which is very different from windowed .Fourier transform.
This paper presents a new approach to developing protection techniques for
EHV overhead transmission lines. When a fault occurs, the fault current and voltage
waveforms contain significant high frequency transient signals. The presented
method is based on the detection of fault generated transient signals using wavelet
10
analysis., The simulation study is carried out, using ATP-EMTP software and
wavelet analysis, which indicates that the proposed approach has the potential of
developing high speed protection relays with accurate fault detection and
classification.
This paper presents a technique to detect and classify faults on EHV overhead
transmission line. The wavelet transform is applied to decompose the current signals
into series of detailed wavelet components, each of which is a time-domain signal
that covers a specific frequency band. Thus the time and frequency domain features
of the transient signals are extracted. The spectral energies of the wavelet
components are calculated and then employed to detect and classify the faults.
In this paper an extensive series of simulation studies have been carried out to
obtain various fault signals for subsequent analysis. A typical 400 kV power
transmission line system is shown in Figure 2.1. The simulation of this system was
carried out using ATP-EMTP software and it included study of system for different
fault types, fault positions and different fault inception angles.
Figure2.1:
system configuration
The wavelet transform is a recently developed mathematical tool for signal
analysis. It has become a very important tool for research in the field of mathematics,
physics and engineering. It transforms a time domain signal to time-scale domain.
This process of transformation is called signal decomposition because a signal is
11
decomposed into several other signals with different level of resolution. From these
decomposed signals, it is possible to recover the original time domain signal
perfectly. This reverse process is known as signal reconstruction. Given a signal or a
function x(t), its Continuous Wavelet transform (CWT) is defined as follows:
CWTx (a, b) =
1
a
∫
+∞
−∞
x(t )ϕ (
t −b
)dt
a
(2.1)
Where ‘a’ and b are the scaling (dilation) and translation (time shift)
parameters respectively and ϕ is the mother wavelet function. The choice of the
mother wavelet function is flexible provided that it satisfies the admissibility
conditions. There are many mother wavelet functions available and they have
different properties. The selection of mother wavelet depends on the given
application.
When a ground fault occurs, the measured current waveforms contain
significant transient components. Figure 2.2 shows a simulated a-g fault current
waveform. The a-g fault occurs at time 0.025 seconds. For the discrete wavelet
analysis of the signal there are many wavelets available with different properties.
Figure2.2:
The a-g fault current waveform
12
For the protection relaying purposes the wavelet should have properties like
availability of discrete transform, compact support, FIR filter implementation, fast
algorithm and orthogonal or biorthogonal analysis. For real time relaying purposes
the speed of algorithm and easy computation are very important. Since the wavelets,
Mexican hat, Morlet and Meyer do not have discrete transform and fast algorithm
implementation, so they were not considered. The wavelets designed by Daubechies
have the properties required so they were investigated using MATLAB s wavelet
toolbox. For the transmission line relaying purposes the speed is a very important
practical factor. Using different Daubechies wavelets and applying the moving
window the operating speed of different wavelets were measured in terms of number
of floating point operations (flops) required to analyze the current signal.
Table 2.1 compares the speed that some of the fastest wavelets are Haar, Db1,
Bior 1, Db2, Sym2, Db3, Coif 1 and Sym3. Out of these wavelets, considering speed
with suitability Db2, and Sym2 were more suitable, in the end Sym2 was chosen
because Sym2 is near symmetrical while the Db2 is asymmetric and Symmetry is a
very desirable property in signal processing applications because it leads to linear
phase response.
Table 2.1: Comparison of speed of different wavelet
Wavelet
Flops
Wavelet
flops
Db1
7387
Bior1.1
8035
Db2
13561
Bior1.3
20833
Db3
20293
Bior2.2
21103
Db4
28339
Bior2.4
36547
Coif1
20293
Bior3.1
14479
Coif2
46123
Bior3.3
29059
Coif3
78415
Bior4.4
37240
Coif4
120247
Bior5.5
45448
Haar
7225
Sym5
36277
Sym2
15561
Sym6
46123
Sym3
20293
Sym7
55951
Sym4
28339
Sym8
67669
13
The current signal was analyzed using the three level discrete wavelet
transform, which was implemented in MATLAB with the Sym2 wavelet and the
results are shown in Figure 2.3.
(a) Detail 1
(b) Detail 2
(c) Detail 3
Figure 2.3:
Detail coefficients for a-g fault signal
From previous Figure2.3 it can be seen that the fault generates big spikes in
the coefficient values (Coeff value), which can be used to detect and classify the
faults. The coefficient values before the fault occurs are near zero but after the fault
occur the coefficient values jump to a value. The actual value is dependent on the
fault condition. The coefficient values represent the spectral energy of the signal.
From the extensive analysis of many different signals, it was found that the
coefficient values for the fault signals were much higher for the fault signals than the
other transient signals without fault. Also the coefficient values for the faulty phase
are much higher than the other phases without fault. From these results a threshold
14
can be set to detect the faults and the details and approximations can be compared to
identify the ground faults. The proposed algorithm for detecting and classifying the
ground fault is shown in Figure 2.4. The fault signal from current transformers is
sampled first, and then three level discrete wavelet is carried out, which generates
details and approximation of the signal at three different levels. Then coefficient
values are compared with threshold values. If they exceed the threshold, the trip
signal is sent. The fault classification can be carried out by comparing the details and
approximation coefficients of signals of all three phases.
Figure 2.4:
2.3
Flow chart for wavelet algorithm.
Wavelet Transform In The Accurate detection of high Impedance Arcing
Faults In High Voltage Transmission Lines
High impedance faults (HIFs) are in general difficult to detect by
conventional protection such as distance or over current relays; this is principally due
to either relay insensitivity to the very low level fault currents and/or limitations on
other relay settings imposed by a HIF In the case of an over current relay, the low
15
levels of currents associated with HIF are below the sensitivity settings of the relay.
In the case of a distance relay, the accuracy of the estimation of impedance to fault
(particularly in terms of relay overreach/ under reach) can be significantly affected
by the HIF [12].
Most conventional fault detection techniques for HIF mainly involve
processing information based on the feature extraction of post-HIF current and
voltage signals. Hitherto, the algorithms developed include the current ratio method,
the high frequency method, the off-harmonic current method, the neural network and
Kalman filtering method. Although each of these techniques improves fault detection
to a certain extent, each has its own drawbacks.
It is well known that conventional Fourier transform (FT)-based techniques
do not possess the inherently time information associated with fault initiation. The
wavelet transform (WT) on the other hand, is useful in analyzing the transient
phenomena associated with transmission line faults and/or switching operations.
Unlike Fourier analysis, it also provides time information; it has the attribute of very
effectively realizing non-stationary signals comprising of both low and high
frequency components(such as those commonly encountered in power systems
networks) through the use of a variable window length of a signal.
This paper describes a new fault detection technique which involves
capturing the current signals generated in a transmission line under HIFs Its main
thrust lies in the utilization of the absolute sum value of signal components based on
the discrete wavelet transform (DWT). Sophisticated decision logic is also designed
for the determination of a trip decision.
The results presented relate to a typical 154kV Korean transmission system,
the faulted signals for which are attained using the well known Electromagnetic
Transients Program (EMTP) software. The simulation also includes an . embodiment
of a realistic nonlinear HIF model.
Figure 2.5 shows a typical 154 kV Korean Transmission System used in the
simulation studies presented herein. It comprises of a 26 km line length terminated in
16
two sources of 240 MVA and 180 MVA at ends P and Q, respectively; the nominal
power frequency is 60 Hz.
Figure 2.5:
The 154 kV Korean transmission system studied
The simulation of the power system has been carried out using the well
known EMTP software. Within the simulation has also been embodied an emulation
of the non-linear high impedance arcing faults. Fig. 2.6 typifies the actual current
waveform (measured at end P but which has been scaled down through a CT)
Figure.2.6:
A typical fault current waveform at relaying point
Is for an 'a'-earth HIF at 10 km from line end P and is for a fault near Va 0 ,the
distortion observed in the faulted 'a' phase can be directly attributed to the highly
complex and non-linear characteristic of the HIF arc path.
To aid the development of the fault detection technique using the DWT, WT
realization has been employed which determines a coefficient of dl (detail one) using
17
different mother wavelets for an actual current waveform. The mother wavelets
considered are Daubechies(db4), Biorthogonal(bior3. l), Coiflets (coif4) and
Symlets(sym5). Figure2.7. depicts the coefficient of dl for two of the four mother
wavelets with the DWT realization. As expected, the coefficients of d 1 increase on
fault inception. The performance of the DWT realization was evaluated under
different' fault types, fault inception angle and fault location and some of the results
have been shown below, but for brevity, only the coefficient associated with 'a'-earth
HIF at 10 km is shown in Figure 2.7.
(a) db4 Mother wavelet
Figure2.7:
(b) Ciof4 Mother wavelet
The coefficient of d1 under ‘a’-earth HIF using DWT
As a second step in fault detection technique, selection of the mother wavelet
is essential to enhance the performance of HIF detection technique to extract the
useful information rapidly. For the technique considered herein, this process leads to
an accurate classification between the faulted and healthy phases in the first instance,
thereby significantly improving the performance and speed of the HIF detection
process.
For comparison of the performances attained using different mother wavelets,
two conditions are compared as follows: (1) the magnitude of dl coefficient for
detecting the fault; (2) the classification ability between the faulted and healthy
phases.
18
Figigure2.8 shows the fault detection procedure of the proposed technique; FI
is a counter that signifies the sample number for which useful information through
DWT realization under HIF persists. SUM-d1 is sum value of the detailed output (d1
component) for one cycle period and is represented as an absolute value; FC is a
preset pick up level which needs to be exceeded by the absolute value of dl before
the counter FI can be incremented, and D is an integer that signifies a trip level.
Figure 2.8:
A block diagram of the fault detection technique
This decision logic has been specifically designed to discriminate between
HIF and non-fault transient events such as capacitor and line switching, arc furnace
loads etc. As can be seen, when SUM-d1 is greater than or equal to FC, the value of
FI is incremented and as soon as it attains the level D, this indicates an internal fault
and a trip signal is initiated he absolute sum value SUM-dl is based on summating
the dl coefficients over a 1 cycle period and the sampling rate employed is 3840 Hz
i.e., 64 samples/cycle at 60Hz. The whole process is based on a moving window
approach whereby the 1-cycle window is moved continuously by 1 sample. It is
19
apparent from the foregoing decision logic that the criteria for the protection relay to
initiate a trip signal is such that SUM-dl must stay above the threshold level FC
continuously for D samples(after fault inception). In this respect, an extensive series
of studies have revealed that in order to maintain relay stability for external faults
and also restrain under no-fault conditions, the optimal settings for FC and D are 0.1
and 128, respectively.
Finally, this paper described a novel technique for transmission line fault
detection under high impedance earth faults using the DWT. The technique presented
herein has a number of distinct advantages over other traditional HIF detection
techniques. For example, it is robust to a variation in different system and fault
conditions. It is stable for external faults and has the ability to discriminate clearly
between internal faults and non fault transient events such as capacitor and line
switching, arc furnace loads, etc; it has the inherent attribute of distinguishing
between the faulted phase(s) and healthy phase(s) and this is a significant advantage
for transmission systems in which single-pole tripping is employed, and which
therefore requires phase selection. The technique developed is based on current
signals only and therefore requires the use of CTs only.
.
20
CHAPTER 3
BACKGROUND OF THE PROJECT
3.1
Introduction
Power systems stability may be broadly defined as that property of a power
system that enables it to remain in a state of operating equilibrium under normal
operations and to regain an acceptable state of equilibrium after being subjected to a
disturbance [2].
Instability in a power system may be manifested in many different ways
depending on the system configuration and operating mode. Traditionally, the
stability problem has been one of maintaining synchronous operation. Since power
systems rely on synchronous machines for the generation of electrical power, a
necessary condition for satisfactory system operation is that all synchronous
machines remain in synchronism or, colloquially, “instep.” This aspect of stability is
influenced by the dynamics of generator rotor angles and power-angle relationships.
Instability may also be encountered without loss of synchronism. For
example, a system consisting of a synchronous generator feeding an induction motor
load through a transmission line can become unstable because of the collapse of load
voltage. Maintenance of synchronism is not an issue in this instance; instead, the
concern is stability and control of voltage. This form of instability can also occur in
loads converting an extensive area supplied by a large system. In the evaluation of
21
stability the concern is the behavior of the power system when subjected to a
transient disturbance. The disturbance may be small or large.
Small disturbance in the form of load changes take place continually, and the
system adjusts itself to the changing conditions. The system must be able to operate
satisfactorily under these conditions and successfully supply the maximum amount
of load. It must also be capable of surviving numerous disturbances of a severe
nature, such as short-circuit on a transmission line, loss of a large generator or load,
or loss of a tie between two subsystems. The system response to a disturbance
involves much of the equipment. For example, a short-circuit on a critical element
followed by its isolation by protective relays will cause variations in power transfers,
machine rotor speeds, and bus; the voltage variations will actuate both generator and
transmission system voltage regulators; the speed variations will actuate prime
mover governors; the change in tie line loadings may actuate generation controls; the
changes in voltage and frequency will affect on the system in varying degree
depending on their individual characteristics. In addition, devices used to protect
individual equipment may respond to variations in system variables and thus affect
the system performance. In any given situation, however, the responses of only a
limited amount of equipment may be significant. Therefore, many assumptions are
usually made to simplify the problem and to focus on factors influencing the specific
type of stability problem. The understanding of stability problems is greatly
facilitated by the classification of stability into various categories.
3.2
Rotor angle stability
Rotor angle stability is the ability of interconnected synchronous machines of
a power system to remain in synchronism. The stability problem involves the study
of the electromechanical oscillations inherent in power systems. A fundamental
factor in this problem is the manner in which the power outputs of synchronous
machines vary as their rotors oscillate. A brief discussion of synchronous machine
characteristic is helpful in developing the related basic concepts [2].
22
When two or more synchronous machines are interconnected, the stator
voltages and currents of all the machines must have the same frequency and the rotor
machine speed of each is synchronized to this frequency. Therefore, the rotors of all
interconnected synchronous machines must be in synchronized.
In a synchronous motor, the role of electrical and mechanical torques is
reversed compared to those in a generator. The electromagnetic torque sustains
rotation while mechanical load opposes. The effect of increasing the mechanical load
is to retard the rotor position with respect to the revolving field of the stator. The
Systems stability depends on the existence of both components of torque for each of
the synchronous machines. lack of sufficient synchronizing torque results in
instability through a periodic drift in rotor angle. On the other hand, lack of sufficient
damping torque results in oscillatory instability. For convenience in analyzing and
for gaining useful insight into the nature of stability problems, it is usual to
characterize the rotor angle stability phenomena in terms of the following two
categories:
3.2.1
Small- Signal (Small-Disturbance)
stability is the ability of the power system to maintain synchronism under
small disturbances. Such disturbances occur continually in the system because of
small variations in loads and generation. Instability that may result can be in two
forms [2]:
(i)
Steady increase in the rotor angle due to lack of sufficient synchronizing
torque.
(ii)
Rotor oscillation of increasing amplitude due to lack of sufficient
damping torque.
In today’s practical power systems, small-signal stability is largely a problem
of insufficient damping of oscillations. The stability of the following types of
oscillations is of concern:
23
•
Local modes or machine- system modes are associated with the swinging of
units at a generating station with respect to the rest of the power system. The
term local is used because the oscillations are localized at one station or a
small part of the power system.
•
Interarea modes are associated with the swinging of many machines in one
part of the system against machines in other parts. They are caused by two or
more groups of closely coupled machines being interconnected by weak ties.
•
Control modes are associated with generating units and other controls. Poorly
tuned exciters, speed governors,HVDC converters and static compensators
are the usual causes of instability of these modes.
•
Torsional modes are associated with the turbine-generator shaft system
rational components. Instability of torsional modes may be caused by
interaction with excitation controls, speed governors, HVDC controls, and
series-capacitor-compensated lines [18].
The system dynamic equations are liberalized about a steady state operating
point to get a linear set of state equation:
x = Ax +B u
(3.1)
y=Cx+Du
(3.2)
In some programs, and for the small signal stability, the state matrices are
calculated analytically from the Jacobians of the non-linear state equations. In the
power system toolbox, on the other hand, the linearization is performed by
calculating the Jacobians numerically.
A single driver, svm.mgen, for small signal stability is provided. It is
organized similar to the transient stability simulation driver s_simu. New models can
be designed to work satisfactorily in either driver. Generally, if a model is
satisfactory in s_simu, it will be satisfactory in svm_mgen[2].
24
3.2.2
Transient Stability
The ability of power systems to maintain synchronism when subjected to a
severe transient disturbance. The resulting systems response involves large
excursions of generator rotor angles and is influenced by the nonlinear power- angle
relationship. Stability depends on both the initial operating state of the system and
the severity of the disturbance. Usually, the system is altered so that post-disturbance
steady-state operations differ from that prior to the disturbance [2].
3.3
Voltage Stability and Voltage Collapse
Voltage stability is the ability of a power system to maintain steady
acceptable voltage at all buses in the system under normal operating conditions and
after being subjected to a disturbance. A system enters a state of voltage instability
when a disturbance, increase in load demand, or change in system condition causes a
progressive and uncontrollable drop in voltage. The main factor causing instability is
the inability of the power system to meet the demand for reactive power. The heart of
the problem is usually the voltage drop that occurs when active power and reactive
power flow through inductive reactance associated with the transmission network
[1].
A criterion for voltage stability is that, at a given operating condition for
every bus in the system, the bus voltage magnitude increases as the reactive power
injection at the same bus is increased. A system is voltage unstable if, for at least one
bus in the system, the bus voltage magnitude (V) decreases as the reactive power
injection (Q) at the same bus is increased. In other words, a system is voltage stable
if V-Q sensitivity is positive for every bus and voltage unstable if V-Q sensitivity is
negative for at least one bus.
Voltage instability is essentially a local phenomenon; however, its
consequences may have a widespread impact. Voltage collapse is more complex than
25
simple voltage instability and is usually the result of a sequence of events
accompanying voltage instability leading to a low-voltage profile in a significant part
of the power system.
3.4
Mid-Term and Long-Term Stability
The terms long-term stability and mid-term stability are relatively new to the
literature on power systems stability. They were introduced as a result of the need to
deal with problems associated with the dynamic response of power systems to severe
upsets. Severe system upsets result in large excursions of voltage, frequency, and
power flows that thereby invoke the action of slow processes, controls, and
protections not modeled in conventional transient stability studies [2].
Long-term stability analysis assumes that inter-machine synchronizing power
oscillations have damped out, the result being uniform system frequency, the focus is
on the slower and longer-duration phenomena that accompany large-scale system
upsets and on the resulting large, sustained mismatches between generation and
consumption of active and reactive power.
3.5
Block Model
The models available in this version of power system stabilizer (PST)
include.
26
3.5.1
Classical model
The generators are modeled as “classical”. Each classical generator model
has two dynamic variables:
•
The angle of the generator’s internal voltage
•
The generator’s speed deviation from synchronous speed.
3.5.2
Detail model with out control
The generator models are detailed, but with no additional automatic controls.
Each detailed generator model has six dynamic variables [2]:
•
The rotor angle
•
The rotor speed
•
The field flux linkage
•
The direct axis rotor damper winding flux linkage
•
The two flux linkages associated with the quadrature axis
•
The damper windings.
3.5.3
Detail model of the excitation control
The generator model is detailed and models of the excitation control
And speed governor are included. The excitation control is a fast acting thyristorbased system. The turbine is a steam turbine with a HP and LP stage and a fast acting
governor. This model has five additional dynamic variables [2]:
•
The output of the voltage transducer
•
The automatic voltage regulator output
•
Three governor / turbine variables.
27
3.6
Basic Concepts in Applying PSS
A brief review of the basic concepts of stabilization is undertaken here. The
power system, in general, is described by a set of nonlinear different ional and
algebraic equations. These can be expressed as [3]:
PX = F(X,Z), P =
d
dt
(3.3)
Y = H(X,Z)
(3.4)
0 = G(Y,Z)
(3.5)
The oscillatory instability can be viewed as stability of the operating point,
subjected to small, random perturbations which are always present. The analysis can
be performed by liberalizing the system equations around the operating point (X=X0,
Y=Y0, Z=Z0). Here X is the state variables, Y represent active and reactive power
injections (at buses), Z represent voltage magnitudes and angles at various buses
expressing
X = X0+ Δ X, Y = Y0+ Δ Y , Z = Z0+ Δ Z
(3.6)
It is possible to obtain the following equation
pΔX = [ A ] ΔX
(3.7)
Where
⎡ δF δF ⎛ δG δH δG ⎞ −1 δG δH ⎤
−
+
[A]= ⎢
⎜
⎟
⎥
⎣⎢ δX δZ ⎝ δY δZ δZ ⎠ δY δX ⎦⎥
(3.8)
It is to be noted that the elements of A are functions of the operating point.
The satiability of the operating point can be judged by the location of matrix A. If all
the real parts of the eigen values are negative, the system is stable. If one or more has
positive real part, then the system is unstable. While this criterion of stability is valid
for very small perturbations (which may not be in practice), the criterion indicates
28
problem areas but can not provide estimates for amplitudes of the oscillation
observed.
To give more insight into the problem, we can take up a multi-machine
system where generators are modeled by the “classical” model, neglecting flux
decay, saliency, damper windings and governor effects. In this case, the linear zed
system equations can be written as
[M] p2 Δδ = - [ K ] Δδ
(3.9)
Where
[M] is diagonal matrix with Mji=
2H j
ωB
( Hj is the inertia constant of jth synchronous
machine ) .
Kij=
δp ei
, where pei is the power output of 1st machine, δ j is the rotor angle of 1st
δδ j
machine referred to a rotating reference frame (with the operating speed ω 0 ).
If the network can be reduced by retaining only the internal buses of the
generators and the losses in the reduced network can be neglected. The
approximation assume that the voltages are around 1.0 p.u. and the bus angle
difference ( in steady-state) are small. The matrix [K] is singular and has rank ∠ (m1) where m is the size of k(also equal to the number of generators). This enables the
reduction of the number of angle variables by one by treating relative angles (with
respect to a reference machine which can be chosen as the first machine ) as state
variables.
The structure of the a vector Vj depicts the participation of various machines
in the oscillation mode whose frequency is ω j . It is to be noted that for a “m”
machine system, there are (m-1) oscillatory modes whose frequency varies in the
range of (0.2 to 3 Hz). The frequencies are obtained as square roots of the non-zero
and real eigen values of the matrix [M]-1 [K].
29
In a practical system, the various modes ( of oscillation ) can be grouped into
three broad categories.
•
Intra-plant modes.
Intra-plant modes in which only the generators in a power plant
participation. The oscillation frequencies are generally high in the range of
1.5 to 3.0 Hz.
•
Local modes.
Local modes in which several generators in an area participate. The
frequencies of oscillations are in the range of 0.8 to 1.8 Hz.
•
Interarea modes.
Inter-area modes, in which generators over an extensive area
participate. The oscillation frequencies are low and in the range of 0.2 to 0.5
Hz. The power system stabilizer (PSS) is designed mainly to stabilize local
and inter-area modes. However, care must be taken to avoid unfavorable
interaction with intra-plant modes or introduce new modes which can become
unstable. Depending on the system configuration, the objective of PSS can
differ. In Western U.S.A, PSS are mainly used to damp inter-area modes
without jeopardizing stability of local modes. In other systems such as
Ontario Hydro, the local modes were the major concern. In general, however,
PSS must be designed to damp both types of modes. The procedures for
tuning of PSS depend on the type of applications. The objective of PSS is to
introduce additional damping torque without affecting the synchronizing
torque.
30
3.7
Types of input signals used in power system stabilizer (PSS):
As mentioned before, a PSS detects the changing of generator output power
and controls the excitation value. The type of PSS is identified by the detecting
signal. The most simple and typical type is Δ p input type [3].
Recently, the Δ w input type and/or Δ f input types PSS are also adopted in
order to improve a stability of inter-area mode due to the recent increase in power
systems and power re-routing.
3.8
Control Signal
The obvious control signal (to be used as input to the PSS) is the deviation in
the deviation in the rotor velocity. However, for practical implementation, other
signals such as bus frequency electrical power, acceleration power are also used. The
latter signal is actually synthesized by a combination of electrical and mechanical
power signals. The mechanical power signal can be obtained from the gate position
in a hydraulic turbine or steam pressures in steam turbine. Nevertheless, it is difficult
to measure mechanical power. It can be argued that if mechanical power variations
are slow, then a signal derived from the electrical power approximates acceleration
power. However, this can pose problem during rapid increases of generation for
which PSS action leads to depression in the voltage [3].
A recent development is to synthesize acceleration power signal from speed
and electrical power signals. This is shown in Figure3.1 a similar approach is used at
Ontario Hydro and the power system stabilizer (PSS) utilizing these signals are
termed as Delta-P-Omega stabilizer. It is claimed that the new control signal has
eliminated the problem of torsional interaction and improved reliability.
31
Δω
Ms
1 + sT
+
∑
∑
(To pss )
+
1
1 + sT
Figure3.1:
pe
Synthesis of accelerating power signal
The choice of control signal for PSS can be based on the following criteria
•
.The signal must be obtained from local measurements and easily
synthesized.
•
The noise content of the signal must be minimal. Otherwise complicated
filters are required which can introduce their own problems.
•
The PSS design based on a particular signal must be robust and reject noise.
This implies that lead compensation must be kept to a minimum to avoid
amplifying the noise.
All the control signals considered- rotor speed, frequency, electrical power
are locally available. The speed signal can be obtained from a transducer using a
tooth wheel mounted on the shaft.
Alternately it can be obtained from the angle of the internal voltage which
can be synthesized. The bus frequency signal can be obtained from a Hall Effect
transducer [18].
32
Configuration of PSS Function
Figure3.2:
3.9
Structure and Tuning Of PSS
The block diagram of the PSS used in industry is shown in Figure3.3. It
consists of a washout circuit, dynamic compensator, torsional filter and limiter. The
function of each of the complements of PSS with guidelines for the selection of
parameters (tuning) are given next. It is to be noted that the major objective of
providing PSS is to increase the power transfer in the network, which would
otherwise be limited by oscillatory instability [3].
Dynamic
compensator
Washout
u
sTw
1 + sTw
u
Torsional Filter
Limiter
Vs
T(s)
FILT (S)
Figure3.3:
Block diagram of PSS
33
3.9.1
Washout circuit
The washout circuit is provided to eliminate steady-state bias in the output of
PSS which will modify the generator terminal voltage. The PSS is expected to
respond only to transient variations in the input signal (say rotor speed) and not to the
dc offsets in the signal. This is achieved by subtracting from it the low frequency
components of the signal obtained by passing the signal through a low pass filter (see
Figure3.4)[3].
+
u
∑
u
-
1
1 + sTw
Figure3.4:
Washout circuit.
The washout circuit acts essentially as a high pass filter and it must pass all
frequencies that are of interest. If only the local modes are of interest, the time
constant Tw can be chosen in the range of 1 to 2. However, if inter-area modes are
also to be damped, then Tw must be chosen in the range of 10 to 20. A recent study
has shown that a value of Tw = 10 is necessary to improve damping of the inter-area
modes. There is also a noticeable improvement in the first swing stability when Tw is
increased from 1.5 to 10. The value of Tw also improved the overall terminal voltage
response during system islanding conditions [3].
3.9.2 Dynamic Compensator
The dynamic compensator used in industry is made up of two lead-lag stages
and has the following transfer function [3]:
34
T(s) =
K s (1 + sT1 )(1 + sT3 )
(1 + sT2 )(1 + sT4 )
(3.10)
Where Ks is the gain of PSS and the time constants, T1 to T4 are chosen to
improve a phase lead for the input signal in the range of frequencies that are of
interest (0.1 to 3.0 Hz). In general, the dynamic compensator can be chosen with the
following transfer function
T(s) =
k s N (s)
D( s )
(3.11)
Where :
N(s) = 1 + a1 s + a2 s2 +…………….+ap sp
(3.12)
N(s) = 1 + b1 s + b2 s2 +…………….+bp sp
(3.13)
The zeros of D(s) should lie in the left half plane. They can be complex or
real. Some of the zeros N(s) can lie in the right half plane making it a non-minimum
phase. For design purposes, The PSS transfer function is approximated to T(s), the
transfer function of the dynamic compensator. The effect of the washout circuit and
torsional filter may be neglected in the design but must be considered in evaluating
performance of PSS under various operating conditions, there are two design criteria:
1. The time constant, T1 to T4 in the above equation (3.8) are to be
chosen from the requirements of the phase compensation to archive
damping torque.
2. The gain of PSS is to be chosen to provide adequate damping of all
critical modes under various operating conditions. It is to be noted
that PSS is tuned at a particular operating condition (full load
conditions with strong or weak AC system) which is most critical.
35
3.9.3
Torsional Filter
The torsional filter in the PSS is essentially a band reject filters to attenuate
the first torsional mode frequency. The transfer function of the filter can be
expressed as [3]
2
wn
FILT(s) = 2
s + 2ςwn s + w 2 n
(3.14)
For stabilizers derived from accelerating power, torsional filter can have a
simple configuration of a low pass filter independent of the frequency of the torsional
mode to be filtered out. A torsional filter is necessitated by the adverse interaction of
PSS with the torsional oscillations. This can lead to shaft damage, particularly at
light generator loads when the inherent mechanical damping is small. Even if shaft
damage does not occur, stabilizer output can go into saturation making it ineffective.
•
The criteria for designing of the torsional filter are:
1. The maximum possible change in damping of any torsional mode is less than
some fraction of the inherent torsional damping.
2. The phase lag of the filter in the frequency range of 1 to 3 Hz is minimized.
3.9.4
Limiter
The output of the PSS must be limited to prevent the PSS acting to counter
the action of Automatic Voltage Regulator (AVR). For example, when load rejection
takes place, the Automatic Voltage Regulator (AVR) acts to reduce the terminal
voltage when PSS action calls for higher value of the terminal voltage ( due to the
increase in speed or frequency ). It may even be desirable to trip the power system
stabilizer (PSS) in case of load rejection .The negative limit of power system
stabilizer (PSS) output is of importance during the back swing of the rotor (after
36
initial acceleration is over). The Automatic Voltage Regulator (AVR) action is
required to maintain the voltage (and thus prevent loss of synchronism) after the
angular separation has increased. The Power System Stabilizer (PSS) action in the
negative direction must be curtailed more than in the positive direction. Ontario
Hydro uses 0.05 p.u. as the lower limit and 0.1 to 0.2 as the higher limit. Recent
studies have shown that higher negative limit can impair first swing stability [3].
3.10
Wavelet Transforms
In mathematics, wavelets, wavelet analysis, and the wavelet transform refers
to the representation of a signal in terms of a finite length or fast decaying oscillating
waveform (known as the mother wavelet). This waveform is scaled and translated to
match the input signal. The translation and dilation operations applied to the mother
wavelet are performed to calculate the wavelet coefficients, which represent the
correlation between the wavelet and a localized section of the signal. The wavelet
coefficients are calculated for each wavelet segment, giving a time-scale function
relating the wavelets correlation to the signal. This process of translation and dilation
of the mother wavelet is depicted in Figure 3.5[5].
.
Figure 3.5:
The mother wavelet
37
There are a large number of wavelet transforms each suitable for different
applications. For a full list see list of wavelet related transforms but the common
ones are listed below:
•
Continuous wavelet transforms (CWT).
•
Discrete wavelet transforms (DWT).
•
Fast wavelet transforms (FWT).
•
Wavelet packet decomposition (WPD).
•
Stationary wavelet transform(SWT)
As a technique, Wavelet transform has a special feature of variable timefrequency location which is different from the windowed Fourier transform. The
wavelet transform is often compared with the Fourier transform, in which signals are
represented as a sum of sinusoids. The main difference is that the wavelets are
localized in both time and frequency whereas the standard Fourier transform is only
localized in frequency. The Short time Fourier transform (STFT) is also time and
frequency localized but there are issues with the frequency time resolution and the
wavelets often give a better signal representation using Multi resolution analysis,as
we know , the Fourier analysis has a serious drawback. When a signal is transformed
into the frequency domain, time information is lost. However If you are mainly
concerned with stationary signals, signals that do not change much over time, this
drawback is not very important [4].
The Short-Time Fourier Transform (STFT) maps a signal into a 2-D function
of time and frequency. However, the time and frequency information can only be
obtained with limited precision. The precision is determined by the size of the
window used to analyze the signal.
Figure3.6:
Wavelet analysis
38
Wavelet analysis is a windowing technique, similar to the STFT, with
variable-sized windows. It allows the use of long time intervals, where more low
frequency information is sought, and shorter regions, where more high frequency
information is what you are after. Wavelet analysis is capable of revealing aspects of
data that other signal analysis techniques miss, including aspects such as trends,
breakdown points, discontinuities, and self- similarity. It is also often used to
compress or de noise a signal without any appreciable degradation.
Wavelet algorithms process data at different scales so that they may provide
multiple resolutions in frequency and time, these are mainly being used in this study
to detect and classify faults. This property of multi resolution is particularly useful
for analyzing fault transients, which contain localized high frequency components
and superposed power frequency signals for continuous wavelet transform. On the
other hand, the discrete wavelet transform can also be defined for discrete time
signals. The basic concept in wavelet analysis is to select a proper wavelet, called
mother wavelet (analyzing wavelet or admissible), and then perform an analysis
using its translated and dilated versions.
Before explaining the some of the application of wavelet analysis, a brief
exposure on some of the basic concepts that make wavelet analysis such as a useful
signal processing tool will be presented. It is not the purpose of this thesis to provide
a detailed mathematical foundation for wavelet analysis is similar to Fourier analysis
in the sense that it breaks a signal down into its constituent parts the analyzed. Where
the Fourier transform breaks the signal into a series of sine waves of different
frequencies, the wavelet transform breaks the signal into its ‘wavelet’, scaled and
shifted versions of the “mother wavelet”.
3.10.1 Short-Time Fourier analysis
In an effort to correct this deficiency, Dennis Gabor (1946) adapted the
Fourier transform to analyze only a small section of the signal at a time, a technique
called windowing the signal. Gabor's adaptation, called the Short-Time Fourier
39
Transform (STFT), maps a signal into a two-dimensional function of time and
frequency [4].
Figure3.7:
Short-Time Fourier analysis
The STFT represents a sort of compromise between the time- and frequencybased views of a signal. It provides some information about both when and at what
frequencies a signal event occurs. However, you can only obtain this information
with limited precision, and that precision is determined by the size of the window.
While the STFT compromise between time and frequency information can be useful,
the drawback is that once you choose a particular size for the time window, that
window is the same for all frequencies. Many signals require a more flexible
approach one, where we can vary the window size to determine either time or
frequency more accurately.
3.10.2 The ability of the Wavelet Analysis
One major advantage afforded by wavelets as shown in Figure 3.8 is the
ability to perform local analysis that is, to analyze a localized area of a larger signal.
Consider a sinusoidal signal with a small discontinuity one so tiny as to be barely
visible. Such a signal could easily be generated in the real world, perhaps by a power
fluctuation or a noisy switch.
40
Figure3.8:
ability the Wavelet to perform local analysis
A plot of the Fourier coefficients (as provided by the fft command) of this
signal shows nothing particularly interesting: a flat spectrum with two peaks
representing a single frequency. However, a plot of wavelet coefficients clearly
shows the exact location in time of the discontinuity. as shown in the Figure 3.9.
Figure 3.9:
Difference between Fourier and Wavelet Coefficients
As show above the Wavelet analysis is capable of revealing aspects of data
that other signal analysis techniques miss, aspects like trends, breakdown points,
discontinuities in higher derivatives, and self-similarity. Furthermore, because it
affords a different view of data than those presented by traditional techniques,
wavelet analysis can often compress or de-noise a signal without appreciable
degradation.
41
3.10.3 Easy Steps to a Continuous Wavelet Transform
The continuous wavelet transform is the sum over all time of the signal
multiplied by scaled, shifted versions of the wavelet. This process produces wavelet
coefficients that are a function of scale and position. It is really a very simple
process. In fact, here are the five steps of an easy recipe for creating a CWT:
a. Take a wavelet and compare it to a section at the start of the original signal.
b. Calculate a number, C, that represents how closely correlated the wavelet is
with this section of the signal. The higher C is, the more the similarity. More
precisely, if the signal energy and the wavelet energy are equal to one, C may
be interpreted as a correlation coefficient. Note that the results will depend on
the shape of the wavelet you choose.
c. Shift the wavelet to the right and repeat steps 1 and 2 until you have covered
the whole signal.
d. Scale (stretch) the wavelet and repeat steps 1 through 3.
Figure3.10:
Steps of the Continuous Wavelet Transform
42
e. Repeat steps 1 through 4 for all scales. When you are done, you will have the
coefficients produced at different scales by different sections of the signal.
The coefficients constitute the results of a regression of the original signal
performed on the wavelets. How to make sense of all these coefficients? You
could make a plot on which the x-axis represents position along the signal
(time), the y-axis represents scale, and the color at each x-y point represents
the magnitude of the wavelet coefficient C. These are the coefficient plots
generated by the graphical tools. These coefficient plots resemble a bumpy
surface viewed from above. If you could look at the same surface from the
side, you might see something like this: The continuous wavelet transform
coefficient plots are precisely the time-scale view of the signal we referred to
earlier. It is a different view of signal data from the time-frequency Fourier
view, but it is not unrelated.
It is these properties of being irregular in shape and compactly supported that
make wavelets an ideal tool for analyzing signals of a non-stationary nature. Their
irregular shape lends them to analyze signals with discontinuity or sharp changes,
while their compactly supported nature enables temporal of a signal’s features. When
analyzing signals of a non-stationary nature, it is often beneficial to be able to
acquire a correlation between the time and frequency domains of a signal. The
Fourier transform, provides information about the frequency domain, however time
localized information is essentially lost in the process. The problem with this is the
inability to associate features in the frequency domain with their location in time, as
an alteration in the frequency spectrum will result in changes throughout the time
domain. In contrast to the Fourier transform, the wavelet transform allows
exceptional localization in both the time domain via translations of the mother
wavelet, and in the scale (frequency) domain via dilations (Vida, 1991).
It should be noted that the process examined here is the Discrete Wavelet
Transform (DWT), where the signal is broken into dyadic blocks (shifting and
scaling is based on a power of 2). The continuous wavelet transform (CWT) still uses
discretely sampled data, however the shifting process is a smooth operation across
the length of the sampled data, and the scaling can be defined from the
minimum(original signal scale) to a maximum chosen by the user, thus giving a
43
much finer resolution. The trade-off for this improved resolution is an increased
computational time and memory required to calculate the wavelet coefficients. A
comparison of the DWT and CWT representations of a signal is shown below in
Figure 3.11[5].
Figure3.11:
A comparison of the DWT and CWT representations of a signal
3.10.4 The suitable selecting for the wavelet algorithms
There are a wide variety of popular wavelet algorithms, But our concern will
only be for only two strategies of wavelet transform (Haar,db), including Daubechies
wavelets, Mexican Hat wavelets and Morlet wavelets. These wavelet algorithms
have the advantage of better resolution for smoothly changing time series. However,
they have the disadvantage of being more expensive to calculate than the Haar
wavelets. The higher resolution provided by these wavelets is not worth the cost for
financial time series, which are characterized by jagged transitions. The wavelet
44
literature covers a wide variety of wavelet algorithms, which are drawn from an
infinite set of wavelet algorithms. When I first started studying wavelets, one of the
many questions I had was "How does one decide which wavelet algorithm to use?"
There is no absolute answer to this question. The choice of the wavelet algorithm
depends on the application. The Haar wavelet algorithm has the advantage of being
simple to compute and easier to understand. The Daubechies D4 algorithm has a
slightly higher computational overhead and is conceptually more complex. As the
matrix forms of the Daubechies D4 algorithm above show, there is an overlap
between iterations in the Daubechies D4 transform step. This overlap allows the
Daubechies D4 algorithm to pick up detail that is missed by the Haar wavelet
algorithm.
As shown in the Figure 3.12, the red line in the plot below shows a signal
with large changes between even and odd elements. The pink line plots the largest
band of Haar wavelet coefficients. The green line plots the largest band of
Daubechies wavelet coefficients. The coefficient bands contain information on the
change in the signal at a particular resolution. In this version of the Haar transform,
the coefficients show the average change between odd and even elements of the
signal. Since the large changes fall between even and odd elements in this sample,
these changes are missed in this wavelet coefficient spectrum. These changes would
be picked up by lower frequency (smaller) Haar wavelet coefficient bands.
Red: data, Green: Daubechies, Pink: Haar
Figure3.12:
Difference between Haar and Daubechies foe the same Data
45
3.10.4.1
Haar Wavelet algorithms
The Haar wavelet algorithms published here are applied to time series where
the number of samples is a power of two (e.g., 2, 4, 8, 16, 32, 64...) The Haar wavelet
uses a rectangular window to sample the time series. The first pass over the time
series uses a window width of two. The window width is doubled at each step until
the window encompasses the entire time series.
Each pass over the time series generates a new time series and a set of
coefficients. The new time series is the average of the previous time series over the
sampling window. The coefficients represent the average change in the sample
window. For example, if we have a time series consisting of the values v0, v1, ... vn, a
new time series, with half as many points is calculated by averaging the points in the
window. If it is the first pass over the time series, the window width will be two, so
two points will be averaged:
For (i = 0; i < n; i = i + 2)
si = (vi + vi+1)/2;
The 3-D surface below graphs nine wavelet spectrums generated from the
512 point AMAT close price time series. The x-axis shows the sample number, the
y-axis shows the average value at that point and the z-axis shows log2 of the window
width.
3.10.4.1.1
Limitations of the Haar Wavelet Transform
The Haar wavelet transform has a number of advantages:
•
It is conceptually simple.
•
It is fast.
•
It is memory efficient, since it can be calculated in place without a temporary
array.
46
•
It is exactly reversible without the edge effects that are a problem with other
wavelet transforms.
The Haar transform also has limitations, which can be a problem for some
applications. In generating each set of averages for the next level and each set of
coefficients, the Haar transform performs an average and difference on a pair of
values. Then the algorithm shifts over by two values and calculates another average
and difference on the next pair. The high frequency coefficient spectrum should
reflect all high frequency changes. The Haar window is only two elements wide. If a
big change takes place from an even value to an odd value, the change will not be
reflected in the high frequency coefficients.
3.10.4.2
The Daubechies Wavelet Transform
The Daubechies wavelet transform is named after its inventor (or would it be
discoverer?), the mathematician Ingrid Daubechies. The Daubechies D4 transform
has four wavelet and scaling function coefficients. The scaling function coefficients
are:
Each step of the wavelet transform applies the scaling function to the data
input. If the original data set has N values, the scaling function will be applied in the
wavelet transform step to calculate N/2 smoothed values. In the ordered wavelet
transform the smoothed values are stored in the lower half of the N element input
vector.
The wavelet function coefficient values are:
g0 = h3 g1 = -h2 g2 = h1 g3 = -h0
47
Each step of the wavelet transform applies the wavelet function to the input
data. If the original data set has N values, the wavelet function will be applied to
calculate N/2 differences (reflecting change in the data). In the ordered wavelet
transform the wavelet values are stored in the upper half of the N element input
vector.
3.10.4.3
Comparison between Haar and Daubechies
As we have clearly seen, there are some differences between the two types of
the Wavelet (Haar and Daubechies), Table3.1 shown below will explain some other
differences.
Table3.1:
Comparison between Haar and Daubechies
Haar Wavelets
Daubechies Wavelets
General characteristics
Compactly supported
Compactly supported
Family
Haar
Daubechies
Short name
Haar
Db
Examples
DWT
Haar is the same as db1
Possible
db1 or haar, db4, db15
Possible
CWT
Possible
Possible
Orthogonal
Yes
Yes
Biorthogonal
Yes
Yes
Compact support
Yes
Yes
Support width
1
2N-1
Filters length
2
2N
Regularity
Haar is not continuous
about 0.2 N for large N
Symmetry
Yes
far from
Number of vanishing
moments for psi
1
N
48
3.11
Neural Networks
The use of pattern recognition for power systems security analysis was first
investigated in 1968. Since neural network can be fully applied for pattern
recognition, it has been widely investigated for transient classification (including
faults). All automatic learning approaches to system security analysis or transient’s
analysis are confronted by the obstacle of transients – set generation arising from the
large number of combinations of variables and topologies and the computation
required for each simulation. Besides the combinational problem, there is the
problem of generating a sufficiently large population of “insecure” states for a highly
meshed network. How to deal with this problem is still an open problem candidates
of this ideal neural network is Generalized Regression Neural Networks (GRNN).
3.11.1 Generalized Regression Neural Networks (GRNN).
Generalized Regression Network Networks (GRNN) has been developed by
Donald Specht and works as a multi-layer feed-forward network. GRNN is based on
localized basis function NN which uses the probability density functions and is quite
similar in principle to the RBF NN. The term general regressions imply that the
regression surface is not restricted to be linear. In control engineering, neural
network models are often used as dynamic plants emulators for controller design and
also as configurations when they estimate the future values of variables. In many
previous applications of the GRNN, the sigma (sigma) which is referred to as the
smoothing factor in the GRNN algorithm is usually fixed and thus not applicable in a
dynamic environment. To date, there has not been much work on the application of
GRNN for online prediction and classification.
Figure3.13 illustrates a MATLAB GRNN Network Architecture used in this
project. The nprod box (code function normprod) produces S2 elements in vector n2.
Each element is the dot product of a row of LW2,1 and the input vector a1, which are
all normalized by the sum of the elements of a1.
49
Figure3.13:
MATLAB GRNN Network Architecture (FROM MATLAB)
The first layer is just like that for newrbe network where it has as many
neurons as there are input/ target vectors in P. Specifically, the first layer weights are
set to P'. The bias b1 is set to a column vector of 0.8326/SPREAD. The user chooses
SPREAD, the distance an input vector must be from a neuron's weight vector to be
0.5. The operation is just like the newrbe radial basis layer described previously.
Each neuron's weighted input is the distance between the input vector and its weight
vector, calculated with dist. Each neuron's net input is the product of its weighted
input with its bias, calculated with netprod. Each of these net ‘neurons' output is its
net input passed through radbas. If a neuron's weight vector is equal to the input
vector (transposed), its weighted input will be 0, its net input will be 0, and its output
will be 1. If a neuron's weight vector is a distance of spread from the input vector, its
weighted input will be spread, and its net input will be sqrt(-log(0.5)) (or 0.8326).
Therefore its output will be 0.5.
3.11.2 Probabilistic Neural Networks (PNN)
PNN (Specht 1990) is a pattern classification algorithm which falls into the
broad class of “nearest-neighbor-like” algorithms. It is called a “neural network”
50
because of its natural mapping onto a two-layer feed forward network. It works as
follows.
Let the exemplars from class i be the k-vectors x i j for j = 1,…., N i . Then, the
Figure 3.14: PNN is not robust with respect to affine
transformations of feature space.
.
Note that the class likelihood functions are sums of identical isotropic
Gaussians centered at the exemplars. The single free parameter of this algorithm
is σ , the variance of the Gaussians (the rest of the terms in the likelihood functions
are determined directly from the training data). Hence, training a PNN consists of
optimizing σ relative to some evaluation criterion, typically the number of
classification errors during cross-validation
3.11.2.1
The Problem with PNN
The main drawback of PNN and other “nearest-neighbor-like” algorithms is
that they are not robust with respect to affine transformations (i.e., transformations of
→
→
→
the form x → A x + b ) of feature space. (Note that in theory affine transformations
should not affect the performance of back propagation, but the results of Section 3
show that this is not true in practice.) Figures 1a and 2a depict examples of how
affine transformations of feature space affect classification performance. In Figures
1a and 2a, the point A2 is closer (using Euclidean distance) to point A1, which is
51
also from class A, than to point B1, which is from class B. Hence, with a training set
consisting of the exemplars A1 and B1, PNN would classify A2 correctly.
Figure 3.15: The principle axes of the affine transformation
do not necessarily correspond
Figures with the coordinate axes.1b and 2b depict the feature space after
affine transformations. In both cases, A2 is closer to B1
than to A1 and would
hence be classified incorrectly. For the example of Figure 2, the transformation
matrix A is not diagonal (i.e., the principle axes of the transformation are not the
coordinate axes), and the adverse effects of this transformation cannot be undone by
any affine transformation with diagonal A.
This problem has motivated us to generalize the PNN algorithm in such a
way that it is robust with respect to affine transformations of the feature space.
3.11.2.2
The Solution Weighted PNN (WPNN)
This flaw of nearest-neighbor-like algorithms has been recognized before,
and there have been a few proposed solutions. They all use what Dasarathy (1991)
calls “modified metrics”, which are non-Euclidean distance measures in feature
space.
52
All the approaches to modified metrics define criteria which the chosen
metric should optimize. Some criteria allow explicit derivation of the new metrics
(Short and Fukunuga 1981; Fukunuga and Flick 1984). However, the validity of
these derivations relies on there being a very large number of exemplars in the
training set. A more recent set of approaches (Atkeson 1991; Kelly and Davis 1991)
(i) use criteria which measure the performance on the training set using leaving-one
out cross-validation , (ii) restrict the number of parameters of the metric to increase
statistical significance, and (iii) optimize the parameters of the metric using nonlinear search techniques. For his technique of “locally weighted regression”, Atkeson
(1991) uses an evaluation criterion which is the sum of the squares of the error using
leaving-one-out. His metric has the form d 2 = w1 ( x1 − y1 ) 2 + ....... + wk ( x k − y k ) 2 ,
and hence has k free parameters w1 .....wk . He uses Levenberg-Marquardt to optimize
these parameters with respect to the evaluation criterion. For their Weighted KNearest Neighbors (WKNN) algorithm, Kelly and Davis (1991) use an evaluation
criterion which is the total number of incorrect classifications under leaving-one-out.
Their metric is the same as Atkeson's, and their optimization is done with a genetic
algorithm.
53
CHAPTER 4
METHODOLOGY OF THE PROJECT
4.1
Methodology of the Project
In this section Figure 4.1 shows the procedure of main steps for fault
detection on transmission line using power system stabilizer, also it shows some
tools like wavelet transform (WT), Generalized Regression Neural Network (GRNN)
and Probabilistic Neural Network (PNN) are used to detect and classify the faults.
54
Figure 4.1:
Flow chart for the methodology of the project.
55
4.2
Test The System
Below are the steps and procedures for fault detection and classification using
the Wavelet Transform (WT) based Artificial Neural Network (ANN) input data the
original data file is modified to include the factious transmission line which acts as
the bus bar in the nearest line where the fault occurs. Bus bar 376 is introduced
between bus 3 and bus 101 and the various calculations of the line distance are
obtained. The Load low program calculates the new reference bus voltage and its
magnitude automatically.
4.2.1
Run the Base Program
The transient simulation program is run with the three phases to ground and
line to line to ground fault occurring on the transmission line between bus 376 and
bus 101. All out put programs are saved in a Mat file. The main output data for the
feature extraction is the speed deviation. The program is run for different lengths of
the transmission line distance i.e. 1%, 5%, 25%, 50%, 75%, 95% and 100%. A single
drive, svm_mgen, for small signal stability is provided. It is organized similar to the
transient stability simulation driver s_simu . New model can be designed to work
satisfactory in either driver. Generally, if a model is satisfactory in s_simu, it will be
satisfactory in svm_mgen.
4.2.2
Run the wavelet Program
Transient signals have some characteristics such as high frequency and
instant break, so wavelet transform is strong tool for them in feature picking-up, and
it satisfies the analysis need of electric power transient signals. Usually wavelet
transform of transient signal is expressed by multi-revolution decomposition fast
algorithm which utilizes the orthogonal wavelet bases to decompose the signal to
components under different scales. It is equal to recursively filtering the signal with a
56
high-pass and low-pass filter pair. Filtering by high pass filter produces details and
filtering by low-pass produces approximations. The band width of these two filters is
equal.
The wavelets possess multidimensional characters and are able to adjust their
scale to the nature of the signal features. Singularities and irregular structures in
signal waveform often carry important information from an informatics-theoretic
point of view. The WT analysis provides a kind of mathematical “microscope” to
zoom in or zoom out on those interesting structures Furthermore, wavelets can be
orthonormal and are able to capture deterministic features. Therefore, WT can
decompose a signal into localized contributions labeled by so-called dilation and
translation parameters. These parameters represent the information of different
frequency component contained in the analyzed signals
Since the wavelet transform with its ability allowing the localization both in
time and frequency domain. In this project, the MATLAB l Simulink is used to
generate fault signals and verify the correctness of the algorithm, dyadic wavelet
transform with Haar and db5 wavelets to analyze three phase to ground and line to
line to ground fault signals of transmission line. By The output signal of the speed
deviation is taken as the input to the wavelet program. The coefficients are generated
and the features from each signal are extracted, namely, the number of speaks that
occur in the signal, the minimum number of speaks of the signal, the frequency of the
signal and the energy functions. Each of the coefficients and features are saved for
further processing.
4.3
Fault Detection
As a framework of security control, fault detection is one of the important
tasks. Specifically, it requires that power system operators at control centers
appropriately handle information on faults and detect faults effectively. In other
words, more sophisticated fault detection techniques are necessary to maintain secure
power systems.
57
In this project study, a method is proposed for fault detection and classification. A
novel technique, called optimal feature selection in the wavelet domain and
supervised neural network-fault classifier is developed. An output signal of the speed
deviations of each generator of the multi area multi machines system are taken as the
input for the wavelet analysis which are then fed to the Generalized Regression
Neural Network (GRNN) and Probabilistic Neural Network (PNN) to give the
location and classification of the fault.
4.4
Fault Classification
Several algorithms have been reported for fault detection in transmission
lines. They are based on either artificial neural networks (ANNs) or wavelets
transform (WT). Most of them have been developed for relaying purposes and may
only distinguish a fault from the normal steady-state power system operation.
Fault classification algorithms based on ANN have been proposed in this
project. The WT has also used to get best features for the coefficients. Furthermore,
combined techniques have already been used, such as ANN and WT as a fault
detection, both fault detection and classification algorithms found in the project have
been developed from simulated data obtained using an MATLAB Program.
This project proposes a novel method for fault detection and classification in
transmission lines by using transient stability program and record the speed deviation
of every generator. Both fault signals are then fed to the wavelet program to extract
the features for fault classification using neural network.
58
4.5 Using GRNN As Fault Classifier and Detector
Generalized Regression Network Networks (GRNN) has been developed by
Donald Specht and works as a multi-layer feed-forward network. GRNN is based on
localized basis function NN which uses the probability density functions and is quite
similar in principle to the RBF NN. The term general regressions imply that the
regression surface is not restricted to be linear. In control engineering, neural
network models are often used as dynamic plants emulators for controller design and
also as configurations when they estimate the future values of variables. In many
previous applications of the GRNN, the sigma ( δ ) which is referred to as the
smoothing factor in the GRNN algorithm is usually fixed and thus not applicable in a
dynamic environment. To date, there has not been much work on the application of
GRNN for online prediction and classification.
4.6
Using PNN as a Fault Classifier and Detector
Probabilistic Neural Networks (PNN) is feed forward networks which are
built with three layers. They estimate the probability density function for each class
based on the training samples using either Parzen window or similar probability
density function which are calculated for each test vector. as it attention before that,
Probabilistic Neural Networks (PNN) are feed forward networks which are built with
three layers, one pass, learning network that uses sums of Gaussian distribution to
estimate the class probability functions as learned from training vector sets
Consequently, the PNN is able to make a classification decision in accordance with
the Bayes strategy for decision rules and to provide probability and reliability
measures for each classification, Learning involves choosing a single suitable
smoothing factor which is the common standard deviation for all the Gaussian. The
PNN uses one of a class of probability density function estimators which
asymptotically approaches the underlying parent density provided that it is smooth
and continuous. The network is tolerant of erroneous training vectors and sparse data
samples can be adequate for optimal performance. It is both easy to use and fast for
moderately sized data bases. The major disadvantage of the PNN is that all training
59
vectors must be stored and used to classify new vectors, thus requiring large
memories for many practical problems. This is not a severe disadvantage if the PNN
is implemented in a parallel hardware structure where memory is relatively
inexpensive. Among the advantages offered by PNN are that they train faster (more
than five times faster than back propagation), they converge to a Bayes classifier if
enough training example examples are provided, they enable a fast incremental
training and are robust to noisy example.
60
CHAPTER 5
RESULTS
5.1
Introduction
The two-area four machine system with a double circuit transmission lines
between two areas is modified to include a fictitious bus for the study. This system is
well-known and its data and analysis can be found in (Rogers, 2000), (see Figure
5.1)[2].
1
10
20 3
11
13 120
2
12
101
4
Figure5.1:
14
Single Line Diagram of Two-Area system.
110
61
5.2
Test the System
Below are the steps and procedures for fault detection and classification using
the proposed new techniques.
Data file Program- The original data file is modified to include the factious
transmission line which acts as the bus bar in the nearest line where the fault occurs.
Bus bar 376 is introduced between bus 3 and bus 101 and the various calculations of
the line distance are obtained. The Load low program calculates the new reference
bus voltage and its magnitude automatically.
5.2.1
Run the Base Program
The transient simulation program is run with the three- phase fault occurring
on the transmission line between bus 376 and bus 101.
All out put programs are saved in a Mat file. The main output data for the
feature extraction is the speed deviation. The program is run for different lengths of
the transmission line distance i.e. 1%, 5%, 25%, 50%, 75%, 95% and 100%.
A single drive, svm_mgen, for small signal stability is provided. It is
organized similar to the transient stability simulation driver s_simu . New model can
be designed to work satisfactory in either driver. Generally, if a model is satisfactory
in s_simu, it will be satisfactory in svm_mgen.
Figure 5.2 shows the generator speed deviation for each generator G1, G2,
G3 and G4 by doing some change in the delinevary.m to apply a three phase to
ground fault at 5% of transmission line between bus 376 and bus 101.
62
Figure5.2:
The Generator Speed Deviation following a three phase fault at
5% of Transmission line between bus 376 and bus 101.
Figure 5.3 shows the generator speed deviation for each generator G1, G2,
G3 and G4 by doing some change in the delinevary.m to apply a line to line to
ground fault at 5% of transmission line between bus 376 and bus 101.
63
Figure5.3:
The Generator Speed Deviation following a Line to Line to Ground
fault at 5% of transmission Line between bus 376 and bus 101
5.2.2
Run the wavelet Program
The output signal of the speed deviation is taken as the input to the wavelet
program. as shown in the Figure 5.4
Figure5.4:
The DWT of a Speed Deviation of Generator No1, using db5 analysis.
64
The coefficients d2 and d3 are generated and the features from each signal are
extracted, namely, the maximum value of speaks which occurs in the signal, the
standard deviation, the power and the energy functions of the signal. Each of the
coefficients and features are saved for further processing. The Multi Resolution
Analysis (MRA) Wavelet Transforms technique was used to extract the relevant
feature from the input signal i.e. the speed deviation. The MRA decomposes the
signal into different resolutions allowing a detailed analysis of the behavior of the
signal in its different spectral hands.
Figure 5.5 shows the Wavelet coefficient d2, d3 of the speed generator
deviation for the generator G1 as an output of the a small program ezd5_33.m, which
is programmed to produce all the coefficients for db5, level 3.when three phase fault
and line to line to ground fault occurs at variety percentage of the transmission line
between bus 376 and bus 101.
(a). Wavelet coefficient d2
(b). Wavelet coefficient d3
Figure5.5:
The DWT of a Speed Deviation of Generator number1
as an out put of the ezd5_33 program.
65
Figure 5.6 shows another Wavelet coefficient d1, d2 and d3 of the same
speed generator deviation for generator G1 as an output of the wavelet transform by
using an other method called Haar analysis at the same level.
Figure5.6: The DWT of a Speed Deviation of Generator No1, using Haar analysis.
Figure 5.7 shows the Wavelet coefficient d2, d3 of the speed generator
deviation for the generator G1 as an output of the small program abd_33.m, which
programmed to produce all the coefficients for Haar analysis, level 3.when three
phase fault and line to line to ground fault occurs at variety percentage of the
transmission line between bus 376 and bus 101.
66
(a). Wavelet coefficient d2
(b). Wavelet coefficient d3
Figure5.7:
The DWT of a Speed Deviation of Generator No1as
an out put of the abd_33 program.
67
5.3
Results and Discussions
Both Tables 1 and 2 show the best features extracted from the coefficients d2
and d3 for generator G1 as an output of the db5 and Haar analysis, when the three
phases to ground fault occurs on the transmission line.
Table 5.1:
The best features for db5-L5 analysis when three phase fault occurs.
Maximum
Value * E-5
5.3991
6.7149
7.6423
7.7731
7.8175
7.8363
7.8398
G 1 db5-L3. (three phase fault)
Power
Standard
Energy
Deviation * E-6 * E-11
* E-8
8.5258
7.2472
2.4278
1.2985
1.6811
5.6318
9.985
9.9403
3.33
9.543
9.0806
3.042
9.458
8.9201
2.9882
9.412
8.8335
2.9592
9.404
8.8180
2.954E
Percentage
of length
0.24140
0.22243
0.21448
0.21467
0.21474
0.21477
0.21478
0.51070
0.89064
0.59609
0.45503
0.44010
0.43546
0.43471
1%
5%
25%
50%
75%
95%
100%
Table 5.2:
.0026003
0.079087
0.035427
0.020644
0.019311
0.018906
0.018841
0.87111
0.026494
.0011868
0.069156
0.64691
0.63334
0.63118
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
The best features for Haar analysis when three phase fault occurs.
Maximum
Value * E-4
2.5540
3.3652
2.5601
2.5602
2.5602
2.5602
2.5602
G 1 Haar-L3. (three phase
Power
Standard
Deviation E-5 * E-9
9.4687
8.9389
0.12438
0.15423
8.6974
7.5420
8.4824
7.1735
8.4381
7.0990
8.4243
7.0757
8.4220
7.0718
fault)
Energy
* E-6
2.9945
5.1668
2.5266
2.4031
2.3782
2.3703
2.3691
Percentage
of length
4.8844
6.4081
4.9098
0.19202
0.24397
0.17563
0.12316
0.19881
0.10302
1%
5%
25%
0.36763
0.59346
0.30753
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
68
4.9103
4.9105
4.9106
4.9106
0.16992
0.16939
0.16922
0.16919
0.28788
0.28606
0.28550
.28541
9.6438
9.5830
9.5643
9.5613
50%
75%
95%
100%
Both Tables 3 and 4 show the beast features extracted from the coefficients
d2 and d3 for generator G2 as an output of the db5 and Haar analysis, when the three
phases to ground fault occurs on the transmission line.
Table 5.3:
The best features for db5-L5 analysis when three phase fault occurs.
Maximum
Value
5.3991
1.9635
2.0611
2.0886
2.0980
2.1019
2.1026
G 2 db5-L3. (three phase fault)
Standard
Power
Energy
DeviationE-5 E-10
E-7
2.0734
4.2863
1.4359
2.4323
5.8985
1.9760
2.3592
5.5490
1.8589
2.3290
5.4079
1.8117
2.3201
5.3667
1.7978
2.3194
5.3637
1.7968
2.3194
5.3635
1.7968
Percentage
of length
2.4140
5.2797
5.3329
5.3419
5.3452
5.3467
5.3469
0.10973
0.15650
9.5728
9.5447
9.2423
9.1689
9.1574
1%
5%
25%
50%
75%
95%
100%
0.012005
0.024419
0.91364
0.90829
0.85165
0.83818
0.83607
0.40215
0.81802
0.30607
0.30428
0.28530
0.28079
0.28008
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
Table 5.4: The best features for Haar analysis when three phase fault occurs.
Percentage
G 2 Haar-L3. (three phase fault)
of length
Coefficient
Maximum
Standard
Power
Energy
Value E-4
Deviation E-4 E-8
E-6
8.0436
1.5073
2.2652
7.5884
1%
6.6166
1.8761
3.5091
0.11755
5%
6.0858
1.3088
1.7078
7.5210
25%
6.0133
1.2747
1.6201
5.4274
50%
5.9890
1.2666
1.5995
5.3582
75%
5.9788
1.2637
1.5921
5.3335
95%
5.9768
1.2632
1.5908
5.3291
100%
D2
69
0.0015
0.0011
9.8667
9.6843
9.6229
9.5970
9.5922
3.0027
3.5508
2.4685
2.4256
2.4083
2.4025
2.4014
8.9891
0.12571
6.0755
5.8658
5.7828
5.7547
5.7497
0.30113
0.42112
0.20353
0.19651
0.19372
0.19278
0.19262
1%
5%
25%
50%
75%
95%
100%
D3
Both Tables 5 and 6 show the best features extracted from the coefficients d2
and d3 for generator G1 as an output of the db5 and Haar analysis, when the line to
line to ground fault occurs on the transmission line.
Table5.5
The best features for db5-L5 analysis when line to line to ground
fault occurs.
G 1 db5-L3. (Line To Line To Ground Fault )
Maximum
Standard
Power
Energy
Value E-5
Deviation E-6 E-11
E-9
88.923
1.0945
0.11943
40.010
3.7986
9.6096
9.2071
0.30844
2.5591
5.1737
2.6687
8.9401
2.6448
4.1676
1.7317
5.8011
2.6743
3.9275
1.5380
5.1522
2.6869
3.8136
1.4500
4.8575
2.6893
3.7910
1.4329
4.8002
Percentage
of length
2.4886
0.23170
8.5817
7.2864
7.3697
7.4052
7.4119
1%
5%
25%
50%
75%
95%
100%
4.1475
0.78580
0.33085
0.23671
0.18934
0.16789
0.16402
1.7150
0.061564
0.010914
0.55867
0.35743
0.28086
0.26823
5.7453
0.0020624
0.036561
0.018715
0.011974
0.94088
0.89856
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
Table5.6: The best features for Haar analysis when line to line to ground fault
occurs.
G 1 Haar-L3. (Line To Line To Ground Fault )
Maximum
Standard
Power
Energy
Value E-4
Deviation E-5 E-9
E-7
39.135
87.943
771.08
25.831
2.9034
9.3716
8.7565
0.29334
1.1259
4.0561
1.6403
5.4949
Percentage
of length
1%
5%
25%
Coefficient
D2
70
1.1638
1.1769
1.1825
1.1835
3.6096
3.5195
3.4931
3.4889
1.2990
1.2350
1.2165
1.2136
4.3517
4.1373
4.0753
4.0656
50%
75%
95%
100%
71.795
5.3610
2.1179
2.1905
2.2156
2.2263
2.2283
1.7005
17.694
7.7383
6.9837
6.8277
6.7772
6.7697
28.831
0.31215
5.9703
4.8627
4.6479
4.5793
4.5692
96.584
1.0457
2.0001
1.6290
1.5570
1.5341
1.5307
1%
5%
25%
50%
75%
95%
100%
D3
Both Tables 7 and 8 show the best features extract from the coefficients d2
and d3 for the generator G2 as an output of the db5 and Haar analysis, when the line
to line to ground fault occurs on the transmission line.
Table5.7:
The best features for db5-L5 analysis when line to line to ground fault
occurs.
G 2 db5-L3. (Line To Line To Ground Fault )
Maximum
Standard
Power
Energy
Value E-5
Deviation E-6 E-11
E-8
1.1927
1.4469
20.873
699.24
5.3169
0.10171
0.010315
3.4554
3.5747
7.2086
0.51810
1.7356
3.7025
5.5309
3.0499
1.0217
3.7467
4.9799
2.4725
82.828
3.7656
4.8798
2.3742
79.535
3.7692
4.8661
2.3608
79.087
Percentage
of length
3.4120
0.37590
0.10059
0.10403
0.10521
0.10572
0.10582
1%
5%
25%
50%
75%
95%
100%
5.9111
0.011984
0.36360
0.34217
0.25354
0.22447
0.21962
3.4837
0.01431
0.013181
0.011673
0.064089
0.50237
0.48090
1.1670
479.67
44.158
0.39104
0.21470
0.16829
0.16110
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
71
Table 5.8:
The best features for Haar analysis when line to line to ground fault
occurs.
G 2 –Haar-L3. (Line To Line To Ground Fault )
Maximum
Standard
Power
Energy
Value E-4
Deviation E-5 E-9
E-6
48.720
90.239
811.87
0.027198
3.5591
0.14507
0.20982
7.0290
1.4853
4.8631
2.3580
78.992
1.5392
4.0221
1.6129
54.033
1.5579
3.8272
1.4604
48.922
1.5658
3.7617
1.4109
47.264
1.5673
3.7507
1.4026
46.987
Percentage
of length
80.819
6.5295
2.5978
2.6967
2.7309
2.7455
2.7483
1%
5%
25%
50%
75%
95%
100%
5.4
1.6679
0.27059
8.3616
7.7223
7.3110
7.1876
7.1672
27.737
0.73000
6.9708
5.9456
5.3291
5.1507
5.1215
929.18
0.24455
2.3352
1.9918
1.7852
1.7255
1.7157
1%
5%
25%
50%
75%
95%
100%
Coefficient
D2
D3
Fault Classification
For generating a small fault we use a small signal stability program to
generate the change of load by giving a small step impulse response in the time
domain. Then, using the state-space program we calculate the speed deviation of the
generator after the response to simulate. For the three-phase fault and line to line to
ground fault we simulate using transient stability program and record the speed
deviation of every generator. Both fault signals are then fed to the wavelet program
to extract the features for fault classification using neural network.
72
5.4.1 Using GRNN as a Fault Classifier
All the inputs and output data for both the faults are stored. We now run the
designed GRNN program with sigma=0.00001 and succeed in obtaining a 100%
correct classifier for the above training data. As for the testing data we train various
combinations of faults and of the GRNN algorithms and succeed in classifying all the
false correctly. If one of the values is out of range, GRNN will give NaN which will
stop the program. Hence, we use a range checking subroutine to modify the distance
between the minimum and maximum data points. The output results of the program
for the test patterns gave a 100% correct classification of the fault types.
1%
5%
Classification the Faults by using GRNN.
Classification of Line To Line To
Classification of Three phase Fault
Ground Fault
Db5
Result
Haar
Result
Db5
Result
Haar
Result
1%
1%
1%
1%
5%
5%
5%
5%
25%
25%
50%
50%
75%
75%
75%
75%
75%
95%
95%
95%
95%
95%
100%
100%
100%
100%
100%
Table 5.9:
%
5.4.2
25%
100 %
50%
25%
25%
100 %
50%
100 %
50%
100 %
Using PNN as a Fault Classifier
Using the same data, the simulation is now carried out using the PNN fault
classifier. But for the PNN, the output target index must first be converted into a
vector by the command T=ind2vec(Tc). By using the command net=newpn(P,T ), the
network is constructed and simulated. It is interesting to note that the value of sigma
( i.e 1) is the only default value and the simulation which gives a 100% correct
classification for the training pattern. When the testing is done using the unseen data,
it also gives a 100% correct classification.
73
25%
Classification the Faults by using PNN.
Classification of Line To Line To
Classification of Three phase Fault
Ground Fault
Db5
Result
Haar
Result
Db5
Result
Haar Result
1%
1%
1%
1%
5%
5%
5%
5%
25%
25%
25%
25%
100 %
50%
50%
75%
75%
75%
75%
75%
95%
95%
95%
95%
95%
100%
100%
100%
100%
100%
Table 5.10:
%
1%
5%
5.5
100 %
50%
100 %
50%
100 %
50%
Fault Location
From the data points with 4 different input variables, only the coefficients of
level 3 from generators 1 and 2 are taken. From the simulation observation,
generators 1 and 2 are taken from the same area. Relatively, when fault occurs close
to the generator, the bigger the amplitude, the smaller the energy functions. Thus, to
make the network simpler we use only the data generated from these two generators.
5.5.1
Using GRNN as a Fault Detector
For the simulation results shown in the Table 5-11,which get it at difference
length line of 4%, 23%, 27%, 80%, 83%, 85%, 89%, 90% and 99% only the data
from generators 1 and 2 of wavelet coefficient no 5 is used as the input training
dataset. For GRNN the data must be exact or near to the training data, otherwise,
when they are out of range, it can be difficult to get a hundred percent. The result
shows that for the training data the algorithm succeeds in obtaining 100% accuracy.
For testing an unseen data of a small variation in the original signal, a noise of 0.01%
is added to the signal and thus we run the simulation again for the whole location.
The result of correct classification is 100%.
74
Table 5.11:
Using GRNN as a detection fault.
Line To Line To Ground Fault
Three phase Fault
%
Db5
Error
Haar
Error
Db5
Error
Haar
Error
4
23
27
80
5
25
25
87.1957
1%
2%
2%
7.19%
5
25
25.005
83.9925
1%
2%
1.99
3.99%
1
50
25
75.8501
4%
27%
2%
4.15%
50
25
25
89.00
46%
2%
2%
9%
83
88.3711
5.37%
98.0833
15%
81.823
1.17%
89.69
6.69%
85
89.7845
4.78%
84.687
0.31%
89.6281
4.62%
90.08
5.08%
89
91.3699
2.36
98.1095
9.10%
95.8591
6.85%
90.76
1.76%
90
91.7025
1.70%
85.2386
4.76%
96.2145
6.21%
90.91
0.91%
99
96.9255
2.07%
85.990
13%
97.7
1.3%
92.97
6.03%
5.5.2
Using PNN as a Fault Detector
This result is for all data from generators 1 and 2 of the wavelet coefficient
level 3 for length line of 4%, 23%, 27%, 80%, 83%, 85%, 89%, 90% and 99% as an
input training and adding 0.1% for testing and checking data. Otherwise, if it is out
of range, it can be difficult to obtain a 100% correct classification.
From the training data, we succeed in obtaining a 100% correct classification.
The program will convert automatically the class position to find the exact location.
A 0.01% noise is applied to the original system data and the transient simulation is
run again for the whole set of classes. Then, we run the PNN algorithm and find out
that it gives a 100% correct classification and location.
Table 5.12:
Line To Line To Ground Fault
Three phase Fault
%
4
23
27
Using PNN as a detection fault.
Db5
Error
Haar
Error
Db5
Error
Haar
Error
5
25
25
1%
2%
2%
5
25
25
1%
2%
2%
1
50
25
3%
27%
2%
50
25
25
46%
2%
2%
75
80
83
75
75
5%
8%
75
100
5%
17%
75
75
5%
8%
75
75
5%
8%
85
95
10%
95
10%
95
10%
95
10%
89
95
6%
100
11%
95
6%
95
6%
90
95
5%
95
5%
95
5%
95
5%
99
100
1%
100
1%
100
1%
100
1%
The training time of PNN algorithm for a small scale problem is a few
seconds. With the optimal window size (presently adjusted by a trial and error
operation), the GRNN can classify samples in the testing set within the range
between 80% to 99%, during this range it is clear to observe that the error of
accuracy by using the features which got from Haar analysis was more than the error
in the db5 analysis for both of the two types of fault ( Three phase to ground and line
to line to ground fault), This means that the accuracy by using db5 is better than
Haar.
From the previous results in the Table 5.12 it can seen that PNN could not
give exactly value for its detection but it just gave approximately values, with other
meaning PNN give the nearest value for the target, because of that it has a very fast
training speed and at the same time achieves a low accuracy compared to the GRNN.
However, it has a drawback in the finding of the optimal value of sigma which
controls the width of the area of influence and would typically be set to smaller
values for larger sample sizes. However, it would not be reasonable to initially search
a range of 0.00001 to 10.0 for sigma.
Figure 5.11 has shown the results by using GRNN as a detection fault when
three phase and line to line to ground fault occurs on the transmission line, It is clear
to note that GRNN could give exactly values for its detection .Hence, it has accuracy
better than using PNN. In the same time could observe that the error of accuracy by
using the features which got from Haar analysis was more than the error in the db5
analysis for both of the two types of fault.
76
Also from the previous results in the Table 5.12 it can seen that, when the
percentage of transmission line was 89%, 90% and 99%, the PNN could give
95%,95% and 100% respectively because of all the previous percentage of
transmission line was near to the it is target for both two analysis transform Haar and
db5.But at the same percentage of transmission line in the Table 5.11 it can note that,
when GRNN is used to detect the three phases fault could give more accurate as
following 91.36%,91.70%and 96.92 respectively when db5 is used and
98.10%,85.23%and 85.99%, this mean that db5 could give better accurate than Haar.
77
CHAPTER 6
CONCLUSION AND FURTHER WORK
6.1 Conclusion
A method for classification and fault detection of a transmission line using
intelligent technique is proposed in this study. The use of Multi Resolution Analysis
(MRA) Wavelet transform is expected to be very efficient in extracting relevant
features from the signal, which produced by the generator The MRA decomposition
components were analyzed for their energy content classes and locations of the fault
to be classified.
A wavelet based approach is presented in this project, which can be used for
detecting and classifying faults on transmission line when three phase and line to line
to ground fault occurs. After detailed analysis of the properties of different mother
wavelets of Haar and db5 analysis to select best features of the coefficients, which
presented in the d2 and d3 to enter it as an input to the PNN and GRNN to training
for classification and detection the faults. After training the information it is found
that db5 is better than Haar in both detectors PNN and GRNN. It is also found that
db5 is more accurate than Haar.
A neural network is very popular as classifiers and has proved efficient;
however, they need a large number of data to be effective. But using GRNN and
PNN with small amounts of data give quite a good classification rate. The result
78
shows that for the training data the algorithm succeeds in obtaining 100% accuracy.
For testing an unseen data of a small variation in the original signal, a noise of 0.01%
is added to the signal, thus the simulation ran again for the whole location. The result
of correct classification is 100%.
From all the previous results , which shown in the Tables 5-9,5-10,5-11 and
5-12 during both types of fault three phases and line to line to ground fault can
conclude that ,the classification and detection fault for the transmission line by using
the best features of the signal at coefficient d2 was better than using coefficient d3.
Hence, this study found a novel way of detecting, classifying and locating the
transmission line fault using intelligent technique based on PSS input signals as
compared to the traditional method such as the traveling wave and the impedance
methods which are already established.
6.2 Further Work
Some of programs were programmed to calculate Power, Energy, Standard
Deviation and Maximum value as the best features of the signals. Which are fed to
both Probabilistic Neural Network (PNN) and Generalized Regression Network
(GRNN).
In future, some other programs need to be added to compute other features of
those signals like Mean Value, Variance, Frequency, Number of up peaks, Number
of down peaks, Trapezoid Integration to get more accuracy for classifying and
detecting all types of fault that will occur on the Transmission Line.[iu8]
A further work should consider finding the optimal value of sigma such that it
can vary throughout the epoch for an optimal selection.
79
REFERENCES
[1] Kundur. P, “Power System Stability and Control”, McGraw-Hill, 1994, ISBN 007-035958-X.
[2] Graham Rogers,” Power System Oscillations “, New York:Kluwer, 2000.
[3] Padiyar. K. R, “ Power System Dynamics : Stability and Control “,printed in the
Republic of Singapore,1996, ISBN: 0-471-19002-0
[4] M.F.Othman, M.Mahfouf, and D. A. Linkens, “Transmission Lines fault
detection, classification and location using power system stabilizer”, IEEE.
International conference on Electric utility Deregulation, Restructuring and power
technologies, April 2004, Hong Kong.
[5] Mohd Fauzi Othman, “A Hybrid System Approach to Control and Fault
Detection and Accommodation in Power Systems “Department of Automatic Control
and Systems Engineering University of Sheffield, March 2004, UK.
[6] Graham J.Rogers,“The Application of Power System Stabilizers to a
Multigenerator Plant”, IEEE Transaction On Power System-c,vol.15,no. 1,February
2000.
[7]. Lakmeeharan & M.L. Coker, “Optimal Placement and Tuning of Power
System Stabilisers”, IEEE Technology Goup, 1999, pp40175, South Africa.
[8] Xiaomeng Li, Student Member,” A Neural Network Based Wide Area Monitor
for a Power system, IEEE 2005.
80
[9] K. M. Silva, Student Member, IEEE, B. A. Souza, Senior Member, IEEE, and N.
S. D. Brito, “ Fault Detection and Classification in Transmission Lines Based on
Wavelet Transform and ANN ” IEEE Transactions On Power Delivery, vol. 21, no.
4, October 2006
[10] K. Harish Kashyap and. U.Jayachandra Shenoy, “Classification of Power
System Faults Using Wavelet Transforms And Probabilistic Neural Networks”,
IEEE, Department of Electrical Engineering.
[11] M Solanki,, Y H Song, S Potts and A Perks, “Transient Protection of
Transmission Line Using Wavelet Transform” IEEE, Developments in Power
System Protection, Conference Publication No.479 0, 2001 .
[12] C.H.Kim, H.Kim and R.K. Aggarwal, A.T. Johns, “Wavelet Transform In The
Accurate Detection of High Impedance Arcing Faults In High Voltage Transmission
Lines” IEEE, Developments in Power System Protection,
Conference Publication
No.479 0, 2001 .
[13] Marek Michalik, Waldemar Rebizant,, Miroslaw Lukowicz, , and Sang-Hee
Kang, “High-Impedance Fault Detection in Distribution Networks With Use of
Wavelet-Based Algorithm ” IEEE Transactions On Power Delivery, vol. 21, no. 4,
October 2006
[14] Ali-Reza Sedighi, Mahmood-Reza Haghifam, and Mohammad-Hassan
Ghassemian “High Impedance Fault Detection Based on Wavelet Transform and
Statistical Pattern Recognition ” IEEE Transactions On Power Delivery, vol. 20, no.
4, October 2005.
[15] Joe-Air Jiang, , Ping-Lin Fan, Ching-Shan Chen, Chi-Shan Yu, and Jin-Yi Sheu
“A Fault Detection and Faulted-Phase Selection Approach for Transmission Lines
with Haar Wavelet Transform ” IEEE 0-7803-8 1 10-6/03/$17.00 , 2003.
[16] M Solanki and Y.H. Song “Transient Protection of EHV Transmission Line
Using Discrete Wavelet Analysis ” IEEE, 0-7803-7989-6/03/$17.00 2003 .
81
[17] Shyh-Jier Huang, Cheng-Tao Hsieh and Ching-Lien Huang, “Application of
Morlet Wavelets to Supervise Power System Disturbances ” IEEE, Transactions on
Power Delivery, Vol. 14, No. 1, January 1999.
[18] MITUBISHI Corporation, Power System Stabilizer (PSS).http://www.power
system stabilizer.
82
APPENDIX A
A-1 Test Data 1
% Two Area Test Case
% With simple exciters and turbine governors on all generators
% PSS on generator 1
>> ls
.
d2aload2.m
signal44.mat
svc_indx.m
..
d_brpti.m
signal995.mat
svm_mgen.m
Pst_var.m
d_testdc.m
signal996.mat
swcap.m
calc.m
da3m9bst.m
signal997.mat
switch.m
cdps.m
data16m.m
signal998.mat
tg.m
chq_lim.m
data2a.m
signal999.mat
tg_indx.m
d19jlm.m
signall.mat
sim_fle.mat
untitled_acc.dll
d19jold.m
data50m.m
slprj
dmaneu.exe
lmod.m
loadflow.m
p_file.m
p_m_file.m
mac_em.m
p_pss.m
exc_dc12.m
mac_ib.m
p_tg.m
exc_indx.m
mac_igen.m
pss.m
exc_st3.m
mac_ind.m
pss_des.m
form_jac.m
mac_indx.m
pss_indx.m
i_simu.m
mac_park.m
pss_phse.m
ind_ldto.m
mac_sub.m
pst2ud.exe
insimit.m
mac_tra.m
pstdat.exe
untitled.fig
data3mIg.m
d19k.m
dc_load.m
data3m9b.m
d19jnew.m
sinal22.mat
dc_line.m
untitled_accel_rtw
datadaag.m
vsdemo.m
d19l.m
datadarl.m
smpexc.m
y_sparse.m
inv_lf.m
mdc_sig.m
pstdmeu.doc
83
d2a_det1.m
speed1.mat
datalaag.m
mexc_sig.m
pstlfeu.doc
lfdcs.m
ml_sig.m
rbus_ang.m
y_switch.m
d2a_em.m
speed2.mat
lfdc.m
datalam.m
ybus.m
d2a_pflf.m
datane.m
lfdemo.m
datanp48.m
lfmaneu.exe
mpm_sig.m
rec_lf.m
speed3.mat
d2a_pss.m
msvc_sig.m
red_ybus.m
mtg_sig.m
rlm_indx.m
speed4.mat
d2a_pssd.m
datasmib.m
lftap.m
dc_cont.m
line_cur.m
stab_d.m
d2a_sub.m
nc_load.m
rlmod.m
stabf.m
d2a_tra.m
dc_cur.m
line_dpq.m
ns_file.m
rml_sig.m
dc_indx.m
line_pq.m
p_cont.m
s_simu.m
statef.m
d2a_vs.m
step_res.m
d2aload.m
dc_lf.m
lm_indx.m
p_exc.m
signal16.mat
svc.m
% This program (dlinevary.m) to find the speed deviation for thr four Generators
% G1,G2,G3 and G4 by change the percentage of length for the transmission line,
%also by this program can apply any fault need to detection it and classify.
% Two Area Test Case
% with simple exciters and turbine governors on all generators
% pss on generator 1
% d2a_pss.m
% bus data format
% bus:
% col1 number
% col2 voltage magnitude(pu)
% col3 voltage angle(degree)
% col4 p_gen(pu)
% col5 q_gen(pu),
% col6 p_load(pu)
% col7 q_load(pu)
84
% col8 G shunt(pu)
% col9 B shunt(pu)
% col10 bus_type
%
bus_type - 1, swing bus
%
- 2, generator bus (PV bus)
%
- 3, load bus (PQ bus)
% col11 q_gen_max(pu)
% col12 q_gen_min(pu)
bus = [ 1
1.03 18.5 7.00 1.61 0.00 0.00 0.00 0.00 2 10.0 -10.0;
2
1.01 8.80 7.00 1.76 0.00 0.00 0.00 0.00 2 10.0 -10.0;
3
0.9781 -6.1 0.00 0.00 9.76 1.00 0.00 0.00 3 0.0 0.0;
10
1.0103 12.1 0.00 0.00 0.00 0.00 0.00 0.00 3 0.0 0.0;
11
1.03 -6.8 7.16 1.49 0.00 0.00 0.00 0.00 2 10.0 -10.0;
12
1.01 -16.9 7.00 1.39 0.00 0.00 0.00 0.00 2 10.0 -10.0;
13
0.9899 -31.8 0.00 0.00 17.67 1.00 0.00 0.00 3 0.0 0.0;
20
0.9876 2.1 0.00 0.00 0.00 0.00 0.00 0.00 3 0.0 0.0;
101
1.00 -19.3 0.00 1.09 0.00 0.00 0.00 0.00 1 0.0 0.0;
110
1.0125 -13.4 0.00 0.00 0.00 0.00 0.00 0.00 3 0.0 0.0;
120
0.9938 -23.6 0.00 0.00 0.00 0.00 0.00 0.00 3 0.0 0.0];
% line data format
% line: from bus, to bus, resistance(pu), reactance(pu),
%
line charging(pu), tap ratio
%please vary the percentage length to 25%, 50%, 75% and 100%
perc=5;
linelength=110*perc;
xr=0.0001; xl=0.001; %pu/km bc=0.00175 pu/km
rnew=xr*linelength; xnew=xl*linelength;
line =
[1 10 0.0
2 20 0.0
0.0167 0.00 1. 0.;
0.0167 0.00 1. 0.;
3 20 0.001 0.0100 0.0175 1. 0.;
3 101 0.011 0.0010 0.1925 1. 0.;
3 101 rnew xnew 0.1925 1. 0.;
10 20 0.0025 0.025 0.0437 1. 0. ;
85
11 110 0.0
0.0167 0.0
1. 0. ;
12 120 0.0
0.0167 0.0
1. 0. ;
13 101 0.011 0.11
0.1925 1. 0. ;
13 101 0.011 0.11
0.1925 1. 0. ;
13 120 0.001 0.01
0.0175 1. 0. ;
110 120 0.0025 0.025 0.0437 1. 0. ;];
>>dlinevary.m
>> non-linear simulation
enter the base system frequency in Hz - [60]
enter system base MVA - [100]
Do you want to solve loadflow > (y/n)[y]
inner load flow iterations
4
tap iterations
1
Performing simulation.
constructing reduced y matrices
initializing motor,induction generator, svc and dc control models
initializing other models
generators
generator controls
Warning: Requested axes limit range too small; rendering with minimum range
allowed by machine precision.
> In C:\MATLAB6p5\toolbox\matlab\graph2d\title.m at line 27
In F:\RESULT\s_simu.m at line 379
Warning: Requested axes limit range too small; rendering with minimum range
allowed by machine precision.
> In C:\MATLAB6p5\toolbox\matlab\graph2d\xlabel.m at line 27
In F:\RESULT\s_simu.m at line 380
Warning: Requested axes limit range too small; rendering with minimum range
allowed by machine precision.
> In C:\MATLAB6p5\toolbox\matlab\uitools\allchild.m at line 23
In C:\MATLAB6p5\toolbox\matlab\graphics\private\clo.p at line 49
86
In C:\MATLAB6p5\toolbox\matlab\graphics\newplot.p (ObserveAxesNextPlot) at
line 117
In C:\MATLAB6p5\toolbox\matlab\graphics\newplot.p at line 72
In F:\RESULT\s_simu.m at line 378
elapsed time = 19.016s
You can examine the system response
Type 1 to see all machine angles in 3D
2 to see all machine speed deviation in 3D
3 to see all machine turbine powers
4 to see all machine electrical powers
5 to see all field voltages
6 to see all bus voltage magnitude in 3D
7 to see the line power flows
0 to quit and plot your own curves
enter selection >> 0
% by this command plot(t,mac_spd(t,:1)it can find speed deviation for the four
%Generators
>> plot(t,mac_spd(t,:1);
>> plot(t,mac_spd(t,:2);
>> plot(t,mac_spd(t,:3);
>> plot(t,mac_spd(t,:4);
% To save the speed deviation for Generator G1 in to mat.file use the follwing
%comand
>>x = mac_spd(t,:1);
>>save s1, x.mat;
>>wavemenu;
%This programme (ezd5_3.m) use db5 transform to analysis the speed deviation
%for Generator G1
% also by this program can calculate all the best features of the d2 and d3 like
%standard deviation, maximum value, power and energy of signal.
% Load original 1D signal.
s1=mac_spd(1,:)-1;
s=s1(1:168);
ls = length(s);
87
[ca1,cd1] = dwt(s,'db5');
a1 = upcoef('a',ca1,'db5',1,ls);
d1 = upcoef('d',cd1,'db5',1,ls);
% Now plot a1 + d1.
% Invert directly decomposition of s using coefficients.
a0 = idwt(ca1,cd1,'db5',ls);
plot(a1)
grid
plot(d1)
grid
% Perform decomposition at level 3 of s using db1.
[c,l] = wavedec(s,3,'db5');
% Extract approximation coefficients at level 3,
% from wavelet decomposition structure [c,l].
ca3 = appcoef(c,l,'db5',3);
% Extract detail coefficients at levels 1, 2 and 3,
% from wavelet decomposition structure [c,l].
cd3 = detcoef(c,l,3);
cd2 = detcoef(c,l,2);
cd1 = detcoef(c,l,1);
% Reconstruct approximation at level 3,
% from wavelet decomposition structure [c,l].
a3 = wrcoef('a',c,l,'db5',3);
% Reconstruct detail coefficients at levels 1,2 and 3,
% from the wavelet decomposition structure [c,l].
d3 = wrcoef('d',c,l,'db5',3);
d2 = wrcoef('d',c,l,'db5',2);
d1 = wrcoef('d',c,l,'db5',1);
% Reconstruct s from the wavelet decomposition structure [c,l].
a0 = waverec(c,l,'db5');
% Standard deviation of the detail coefficients at levels 1,2 and 3
s1=std(d1);
s2=std(d2);
s3=std(d3);
88
% Covariance of the detail coefficients at levels 1,2 and 3
c1=cov(d1);
c2=cov(d2);
c3=cov(d3);
% Variance of the detail coefficients at levels 1,2 and 3
var1=var(d1);
var2=var(d2);
var3=var(d3);
% Minimu value of the detail coefficients at levels 1,2 and 3
mn1=min(d1);
mn2=min(d2);
mn3=min(d3);
% Maximum value of the detail coefficients at levels 1,2 and 3
mx1=max(d1);
mx2=max(d2);
mx3=max(d3);
% Average of mean value of the detail coefficients at levels 1,2 and 3
mmn1=mean(d1);
mmn2=mean(d2);
mmn3=mean(d3);
% Median value of the detail coefficients at levels 1,2 and 3
median1=median(d1,n);
median2=median(d2,n);
median3=median(d3,n);
% Trapezoidal numerical integration of the detail coefficients at levels 1,2 and 3
z1=trapz(d1);
z2=trapz(d2);
z3=trapz(d3);
% Next power of the detail coefficients at levels 1,2 and 3
p1=nextpow2(d1);
p2=nextpow2(d2);
p3=nextpow2(d3);
% Correlation coefficients of the detail coefficients at levels 1,2 and 3
R11=corrcoef(d1);
89
R22=corrcoef(d2);
R33=corrcoef(d3);
R12=corrcoef(d1,d2);
R13=corrcoef(d1,d3);
R23=corrcoef(d2,d3);
R32=corrcoef(d3,d2);
R31=corrcoef(d3,d1);
R21=corrcoef(d2,d1);
% Discrete Fourier transform of the detail coefficients at levels 1,2 and 3
F1=fft(d1);
F2=fft(d2);
F3=fft(d3);
T1=peaks(d1);
T2=peaks(d2);
T3=peaks(d3);
A1=max(T1);
A2=max(T2);
A3=max(T3);
B1=min(T1);
%This programme (abd_33.m) use Haar transform to analysis the speed deviation
%for Generator G1
% also by this program can calculate all the best features of the d2 and d3 like
%standard deviation, maximum value, power and energy of signal.
% This Program to compute all the coefficients for the out put signal of the
Generator 1 by using Haar analysis
% Load original 1D signal.
s1=mac_spd(1,:)-1;
s=s1(1:335);
ls = length(s);
[ca1,cd1] = dwt(s,'haar');
a1 = upcoef('a',ca1,'haar',1,ls);
d1 = upcoef('d',cd1,'haar',1,ls);
% Now plot a1 + d1.
% Invert directly decomposition of s using coefficients.
90
a0 = idwt(ca1,cd1,'haar',ls);
plot(a1)
grid
plot(d1)
grid
% Perform decomposition at level 3 of s using db1.
[c,l] = wavedec(s,3,'haar');
% Extract approximation coefficients at level 3,
% from wavelet decomposition structure [c,l].
ca3 = appcoef(c,l,'haar',3);
% Extract detail coefficients at levels 1, 2 and 3,
% from wavelet decomposition structure [c,l].
cd3 = detcoef(c,l,3);
cd2 = detcoef(c,l,2);
cd1 = detcoef(c,l,1);
% Reconstruct approximation at level 3,
% from wavelet decomposition structure [c,l].
a3 = wrcoef('a',c,l,'haar',3);
% Reconstruct detail coefficients at levels 1,2 and 3,
% from the wavelet decomposition structure [c,l].
d3 = wrcoef('d',c,l,'haar',3);
d2 = wrcoef('d',c,l,'haar',2);
d1 = wrcoef('d',c,l,'haar',1);
% Reconstruct s from the wavelet decomposition structure [c,l].
a0 = waverec(c,l,'haar');
% Standard deviation of the detail coefficients at levels 1,2 and 3
s1=std(d1);
s2=std(d2);
s3=std(d3);
% Covariance of the detail coefficients at levels 1,2 and 3
c1=cov(d1);
c2=cov(d2);
c3=cov(d3);
% Variance of the detail coefficients at levels 1,2 and 3
91
var1=var(d1);
var2=var(d2);
var3=var(d3);
% Minimu value of the detail coefficients at levels 1,2 and 3
mn1=min(d1);
mn2=min(d2);
mn3=min(d3);
% Maximum value of the detail coefficients at levels 1,2 and 3
mx1=max(d1);
mx2=max(d2);
mx3=max(d3);
% Average of mean value of the detail coefficients at levels 1,2 and 3
mmn1=mean(d1);
mmn2=mean(d2);
mmn3=mean(d3);
% Median value of the detail coefficients at levels 1,2 and 3
median1=median(d1,n);
median2=median(d2,n);
median3=median(d3,n);
% Trapezoidal numerical integration of the detail coefficients at levels 1,2 and 3
z1=trapz(d1);
z2=trapz(d2);
z3=trapz(d3);
% Next power of the detail coefficients at levels 1,2 and 3
p1=nextpow2(d1);
p2=nextpow2(d2);
p3=nextpow2(d3);
% Correlation coefficients of the detail coefficients at levels 1,2 and 3
R11=corrcoef(d1);
R22=corrcoef(d2);
R33=corrcoef(d3);
R12=corrcoef(d1,d2);
R13=corrcoef(d1,d3);
R23=corrcoef(d2,d3);
92
R32=corrcoef(d3,d2);
R31=corrcoef(d3,d1);
R21=corrcoef(d2,d1);
% Discrete Fourier transform of the detail coefficients at levels 1,2 and 3
F1=fft(d1);
F2=fft(d2);
F3=fft(d3);
T1=peaks(d1);
T2=peaks(d2);
T3=peaks(d3);
A1=max(T1);
A2=max(T2);
A3=max(T3);
B1=min(T1);
% This program (alisaka.m )Use Generalized Regressions Neural Network to classify
%(GRNN)and detection three phase fault for transmission line by analysis the speed
%deviation for the Generator G1
spread=0.1;
net = newgrnn(P,Tc,spread);
Y = sim(net,P)
%Yc = vec2ind(Y);
%testing network
%P2 = [4.8844 1.9202 3.6763 1.2316]'
%1
%P2 = [6.4081 2.4397 5.9346 1.9881]'
%5
%P2 = [4.9098 1.7563 3.0753 1.0302]'
%25
%P2 = [4.9103 1.6992 2.8788 9.6438]'
%50
%P2 = [4.9105 1.6939 2.8606 9.5830]'
%75
%P2 = [4.9106 1.6922 2.8550 9.5643]'
%95
%P2 = [4.9106 1.6919 2.8541 9.5613]'
P2 = [5.3991 8.5258 7.2472 2.4278]'
Y = sim(net,P2)
%Yd = vec2ind(Y)
%100
%1
93
%out=Yd
out=Y
>> alisaka.m
>>
P=
Columns 1 through 9
5.3991 2.4140 2.5540 4.8844 6.7149 2.2243 3.3652 6.4081 7.6423
8.5258 5.1070 9.4687 1.9202 1.2985 8.9064 1.2438 2.4397 9.9850
7.2472 2.6003 8.9389 3.6763 1.6811 7.9087 1.5423 5.9346 9.9403
2.4278 8.7111 2.9945 1.2316 5.6318 2.6494 5.1668 1.9881 3.3300
Columns 10 through 18
2.1448 2.5601 4.9098 7.7731 2.1467 2.5602 4.9103 7.8175 2.1474
5.9609 8.6974 1.7563 9.5435 4.5503 8.4824 1.6992 9.4587 4.4010
3.5427 7.5420 3.0753 9.0806 2.0644 7.1735 2.8788 8.9201 1.9311
1.1868 2.5266 1.0302 3.0420 6.9156 2.4031 9.6438 2.9882 6.4691
Columns 19 through 27
2.5602 4.9105 7.8363 2.1477 2.5602 4.9106 7.8398 2.1478 2.5602
8.4381 1.6939 9.4127 4.3546 8.4243 1.6922 9.4044 4.3471 8.4220
7.0990 2.8606 8.8335 1.8906 7.0757 2.8550 8.8180 1.8841 7.0718
2.3782 9.5830 2.9592 6.3334 2.3703 9.5643 2.9540 6.3118 2.3691
Column 28
4.9106
1.6919
2.8541
9.5613
Tc =
94
Columns 1 through 15
1
1
1
1
5
5
5
5 25 25 25 25 50 50 50
Columns 16 through 28
50 75 75 75 75 95 95 95 95 100 100 100 100
Y=
Columns 1 through 9
1.0000 1.0000 1.0000 1.0000 5.0000 5.0000 5.0000 5.0000 25.0000
Columns 10 through 18
25.0000
25.0000
25.0000
51.8211
50.0000
71.2217
75.7621
83.6675
95.2378
85.6819
83.2211
94.1011
96.1855
80.5539
Columns 19 through 27
83.1011
81.6933
93.0957
86.0556
Column 28
83.4523
P2 =
5.3991
8.5258
7.2472
2.4278
Y=
1
out =
1
% This program (email4.m )Use Probabilistic Neural Networks (PNN)to classify and
%detection three phase fault for transmission line by analysis the speed deviation for
%the Generator G1
%p=tafh;
>>T = ind2vec(Tc);
%training network
net = newpnn(P,T);
Y = sim(net,P)
Yc = vec2ind(Y);
%testing network
95
%P2 = [5.3991 8.5258 7.2472 2.4278]'
%1
%P2 = [6.7149 1.2985 1.6811 5.6318]'
%5
%P2 = [7.6423 9.9850 9.9403 3.3300]'
%25
%P2 = [7.7731 9.5435 9.0806 3.0420]'
%50
%P2 = [7.8175 9.4587 8.9201 2.9882]'
%75
%P2 = [7.8363 9.4127 8.8335 2.9592]'
%95
P2 = [5.3991 8.5258 7.2472 2.4278]'
%1
Y = sim(net,P2)
Yd = vec2ind(Y)
out=Yd
>> email4.m
P=
Columns 1 through 9
5.3991 2.4140 2.5540 4.8844 6.7149 2.2243 3.3652 6.4081 7.6423
8.5258 5.1070 9.4687 1.9202 1.2985 8.9064 1.2438 2.4397 9.9850
7.2472 2.6003 8.9389 3.6763 1.6811 7.9087 1.5423 5.9346 9.9403
2.4278 8.7111 2.9945 1.2316 5.6318 2.6494 5.1668 1.9881 3.3300
Columns 10 through 18
2.1448 2.5601 4.9098 7.7731 2.1467 2.5602 4.9103 7.8175 2.1474
5.9609 8.6974 1.7563 9.5435 4.5503 8.4824 1.6992 9.4587 4.4010
3.5427 7.5420 3.0753 9.0806 2.0644 7.1735 2.8788 8.9201 1.9311
1.1868 2.5266 1.0302 3.0420 6.9156 2.4031 9.6438 2.9882 6.4691
Columns 19 through 27
2.5602 4.9105 7.8363 2.1477 2.5602 4.9106 7.8398 2.1478 2.5602
8.4381 1.6939 9.4127 4.3546 8.4243 1.6922 9.4044 4.3471 8.4220
7.0990 2.8606 8.8335 1.8906 7.0757 2.8550 8.8180 1.8841 7.0718
2.3782 9.5830 2.9592 6.3334 2.3703 9.5643 2.9540 6.3118 2.3691
96
Column 28
4.9106
1.6919
2.8541
9.5613
Tc =
Columns 1 through 15
1
1
1
1
5
5
5
5 25 25 25 25 50 50 50
Columns 16 through 28
50 75 75 75 75 95 95 95 95 100 100 100 100
Y=
(1,1)
1
(1,2)
1
(1,3)
1
(1,4)
1
(5,5)
1
(5,6)
1
(5,7)
1
(5,8)
1
(25,9)
1
(25,10)
1
(25,11)
1
(25,12)
1
(50,13)
1
97
(50,14)
1
(50,15)
1
(50,16)
1
(75,17)
1
(75,18)
1
(75,19)
1
(75,20)
1
(95,21)
1
(95,22)
1
(95,23)
1
(95,24)
1
(100,25)
1
(100,26)
1
(100,27)
1
(100,28)
1
P2 =
5.3991
8.5258
7.2472
2.4278
Y=
(1,1)
Yd =
1
out =
1
>>
1