single ended traveling wave based fault location using discrete wavelet transform

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Theses and Dissertations--Electrical and Computer Engineering Electrical and Computer Engineering

2014

SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION

USING DISCRETE WAVELET TRANSFORM USING DISCRETE WAVELET TRANSFORM

Jin Chang University of Kentucky, [email protected]

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Recommended Citation Recommended Citation Chang, Jin, "SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION USING DISCRETE WAVELET TRANSFORM" (2014). Theses and Dissertations--Electrical and Computer Engineering. 58. https://uknowledge.uky.edu/ece_etds/58

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The document mentioned above has been reviewed and accepted by the student’s advisor, on

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above.

Jin Chang, Student

Dr. Yuan Liao, Major Professor

Dr. Caicheng Lu, Director of Graduate Studies

SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION

USING DISCRETE WAVELET TRANSFORM

THESIS

A thesis submitted in partial fulfillment of the

requirements for the degree of Master of Science in Electrical Engineering

in the College of Engineering

at the University of Kentucky

By

Jin Chang

Lexington, Kentucky

Director: Dr. Yuan Liao, Professor of Electrical and Computer Engineering

Lexington, Kentucky

2014

Copyright © Jin Chang 2014

ABSTRACT OF THESIS

SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION USING

DISCRETE WAVELET TRANSFORM

In power transmission systems, locating faults is an essential technology. When a fault

occurs on a transmission line, it will affect the whole power system. To find the fault

location accurately and promptly is required to ensure the power supply. In this paper,

the study of traveling wave theory, fault location method, Karrenbauer transform, and

Wavelet transform is presented. This thesis focuses on single ended fault location

method. The signal processing technique and evaluation study are presented. The

MATLAB SimPowerSystem is used to test and simulate fault scenarios for evaluation

studies.

KEYWORDS: Fault Location, Traveling Wave, Single End, Karrenbauer Transform,

Discrete Wavelet Transform.

Jin Chang

Nov 17, 2014

SINGLE ENDED TRAVELING WAVE BASED FAULT LOCATION

USING DISCRETE WAVELET TRANSFORM

By

Jin Chang

Dr. Yuan Liao

Director of Thesis

Dr. Caicheng Lu

Director of Graduate Studies

Nov 17, 2014

Date

iii

ACKNOWLEDGEMENTS

I would like to express my special appreciation and thanks to my advisor Dr.

Yuan Liao, for his tremendous guidance. This thesis would not be finished without his

extensive knowledge and innovative ideas.

I would also like to thank Dr. Yuming Zhang, and Dr. Paul Dolloff, for

serving as my committee members, and for the invaluable comments and feedbacks

they gave.

Last but not least, I would like to express my deepest gratitude to my parents,

my wife and my baby daughter, for their endless love and support.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ......................................................................................... iii

Table of Contents .......................................................................................................... iv

List of Table .................................................................................................................. vi

List of Figure.............................................................................................................. viii

Chapter 1 Introduction ............................................................................................... 1

1.1 Background of Fault Location ......................................................................... 1

1.2 Fault Location Methods ................................................................................... 2

1.3 Main Problems of Fault Location Method ....................................................... 3

Chapter 2 The Theory of Travelling Wave Fault Location Method .......................... 5

2.1 Process of Transient Traveling Wave on Transmission Line ........................... 5

2.2 Wave Reflection and Refraction ...................................................................... 8

2.3 Traveling Wave Method ................................................................................. 10

2.4 Karrenbauer Transformation .......................................................................... 13

2.4.1 The Module Velocity of Traveling Wave ........................................................... 15

2.4.2 The Classification of Fault Type ........................................................................ 16

2.5 Wavelet Transform ......................................................................................... 20

Chapter 3 single ended Traveling Wave Fault Location method ............................. 25

3.1 System Introduction ....................................................................................... 25

3.2 Signal Processing ........................................................................................... 26

3.3 Testing Wave Velocity and Sample Rate........................................................ 32

3.3.1 The Selection of Sample Rate ............................................................................ 32

v

3.3.2 The Selection of Wave Velocity ......................................................................... 34

Chapter 4 Evaluation study...................................................................................... 42

4.1 Fault Type ...................................................................................................... 42

4.1.1 One Line to Ground Fault .................................................................................. 42

4.1.2 Line to Line Fault............................................................................................... 50

4.1.3 Double Line to Ground Fault ............................................................................. 58

4.1.4 Three Line Fault ................................................................................................. 65

4.2 Fault Resistance ............................................................................................. 69

4.3 Transmission Line Length ............................................................................. 73

Chapter 5 Conclusion .............................................................................................. 75

References .................................................................................................................... 76

VITA ............................................................................................................................ 79

vi

LIST OF TABLE

Table 2.1 : Fault Type Occurrence and Severity .......................................................... 17

Table 2.2: Characteristic of Different Fault Types ....................................................... 20

Table 3.1: Different Sample Rate to Record Voltage Signals and Results of Evaluation

...................................................................................................................................... 33

Table 3.2: Different Sample Rate to Record Current Signals and Results of Evaluation

...................................................................................................................................... 34

Table 3.3: The Fault Location Result with Line-Modal Velocity V1 = 2.9188*105 km/s

...................................................................................................................................... 36

Table 3.4: The Calculation Result of Wave Velocity in Different Fault Locations ...... 39


...................................................................................................................................... 40

Table 4.1: Fault Location Result for One Line to Ground Fault .................................. 49

Table 4.2: Fault Location Result for the L-L Fault by Using the Voltage Signal ........ 56

Table 4.3: Fault Location Result for the L-L Fault by Using the Current Signal ........ 57

Table 4.4: Fault Location Result for Double Line to Ground Fault ............................. 64

Table 4.5: Fault Location Result for Three Line to Ground Fault ............................... 68

Table 4.6: The Result for the AG Fault with Different Resistances............................. 69

Table 4.7: The Result for the L-L Fault with Different Resistances ............................ 70

Table 4.8: The Result for the L-L-G Fault with Different Resistances ........................ 71

vii

Table 4.9: The Result for the Three Line to Ground Fault with Different Resistances72

Table 4.10: Fault Location Estimation for A 200 KM 500 KV Line ........................... 73

Table 4.11: Fault Location Estimation for A 300 KM 500 KV Line ........................... 74

viii

LIST OF FIGURE

Figure 2.1: Superposition Principle [7] .......................................................................... 6

Figure 2.2：Schematic Diagram for Traveling Wave of a Fault on Transmission Line 8

Figure 2.3: Schematic for Wave Reflection and Refraction [8] ..................................... 9

Figure 2.4: Lattice Diagram for One Line to Ground Fault ......................................... 12

Figure 2.5: Simplified Diagram of Different Fault Types............................................ 18

Figure 2.6: Discrete Wavelet Transform Decomposition ............................................. 22

Figure 2.7: The ‘db4’ Mother Wavelet ......................................................................... 23

Figure 3.1: Simulation System for Single Ended Traveling Wave Fault Location

Method ......................................................................................................................... 25

Figure 3.2: The Flow Chart of Signal Processing ........................................................ 27

Figure 3.3: Half Period Window of Voltage and Current Signal ................................. 28

Figure 3.4: Wavelet Transformation for Voltage Modal Signal ................................... 29

Figure 3.5: Wavelet Transformation for Current Modal Signal ................................... 30

Figure 3.6: The Wavelet Modulus Maxima for Voltage Signal ................................... 31

Figure 3.7: The Wavelet Modulus Maxima for Current Signal ................................... 32

Figure 3.8: Voltage Signal for a AG Fault at 35km from bus A .................................. 37

Figure 3.9: The Wavelet Modulus Maxima for Voltage Signal for Fault Location at 35

km ................................................................................................................................ 38

ix

Figure 4.1: Voltage and Current Signal for the AG Fault at 20 km ............................. 43

Figure 4.2: The Modulus Maxima of the Voltage Signal for the AG Fault at 20 km .. 44

Figure 4.3: The Modulus Maxima of the Current Signal for the AG Fault at 20 km .. 45

Figure 4.4: Voltage and Current Signal for the AG Fault at 70 km ............................. 46

Figure 4.5: The Modulus Maxima of the Voltage Signal for the AG Fault at 70 km .. 47

Figure 4.6: The Modulus Maxima of the Current Signal for the AG Fault at 70 km .. 48

Figure 4.7: Voltage and Current Signal for the L-L Fault at 40 km ............................. 50

Figure 4.8: The Modulus Maxima of the Voltage Signal for the L_L Fault at 40 km . 51

Figure 4.9: The Modulus Maxima of the Current Signal for the L_L Fault at 40 km . 52

Figure 4.10: Voltage and Current Signal for the L-L Fault at 60 km ........................... 53

Figure 4.11: The Modulus Maxima of the Voltage Signal for the L_L Fault at 60 km 54

Figure 4.12: The Modulus Maxima of the Current Signal for the L_L Fault at 60 km55

Figure 4.13: Voltage and Current Signal for the L-L-G Fault at 15 km ...................... 58

Figure 4.14: The Modulus Maxima of the Voltage Signal for the L-L-G Fault at 15 km

...................................................................................................................................... 59


Figure 4.16: Voltage and Current Signal for the L-L-G Fault at 90 km ...................... 61


...................................................................................................................................... 62


x

Figure 4.19: Voltage and Current Signal for the L-L-L-G Fault at 55 km ................... 65

Figure 4.20: The Modulus Maxima of the Voltage Signal for the LLL-G Fault at 55 km

...................................................................................................................................... 66

Figure 4.21: The Modulus Maxima of the Current Signal for the LLL-G Fault at 55 km

...................................................................................................................................... 67

CHAPTER 1 INTRODUCTION

1.1 Background of Fault Location

The power industry is an essential part of a country. Typically, there are three

parts of a power system: generation, transmission, and distribution. All of them are

indispensable for the entire power system. In recent years, the power system is

increasingly sophisticated. The transmission lines are longer and longer. The voltage

level on transmission line is also higher and higher. However, the faults of the

transmission lines are difficult to avoid. Those faults endanger the safety and the

reliability of the power system. The longer the power outages, the greater the damage

caused. Therefore, quickly pinpointing the fault location and restoring power supply

is required by all utilities. In order to fix the fault rapidly, finding the accurate location

of fault point is the key. Thus, the subject of fault location estimation for the

transmission line is especially interesting to electric engineers and utilities.

With the development of power system, the fault of the transmission line has been

more complex. In general, there are four different types of faults that occur on

transmission lines, such as single-phase to ground fault, line-to-line fault, double-line

to ground fault, and three-line fault. Generally, faults on transmission lines are caused

by lightning, storms, freezing rain, snow, insulation breakdown. In addition, external

objects such as tree branches and birds are factors that can cause short circuits [1]. In

the past, utilities had to send staff to seek the transmission line, which might need

2

several hours to find the location. Especially flashover transient faults are more

difficult to find. Thence, how to find the fault location promptly and accurately has

been a popular subject in the power industry for many years.

1.2 Fault Location Methods

Nowadays, fault location methods are developed in many different ways. The

keys of fault location methods are accuracy and reliability. Methods of locating faults

on a transmission line can be classified into two fundamental categories: techniques

based on power-frequency components, and the other utilizing the higher-frequency

components of the transient fault signals. The former one is using voltage and current

measured and necessary system parameters to calculate fault location. The fault

location methods with voltage and current measurements are proposed in [2] and [3].

Methods are based on impedance matrix that can establish the equations to govern the

relationship of the measurements and fault location are proposed in [4], [5] and [6].

The latter one is the traveling wave method. In this paper, the traveling wave method

is mainly discussed.

As a fast and accurate method, travelling wave method has been reported since

1931. According to the superposition principle, the forward and backward wave will

generate at the fault point after the fault occurs. The essential idea of traveling wave

method is based on the correlation between those forward and backward traveling

3

waves. The advantage of using the traveling wave method is that the power frequency

phenomena such as power swings and current transformer (CT) saturation, and

insensitivity to fault type, fault resistance, fault-inception angle, and source

parameters of the system can be avoided [1].

1.3 Main Problems of Fault Location Method

Using the impedance calculation method is much easier to implement than

traveling wave method. It is also much cheaper than the traveling wave method since

its hardware investment is low. Nevertheless, the accuracy of power-frequency

method can be affected by many factors such as transition resistance, nonlinear

voltage transformers, and asymmetrical transmission lines. The traveling wave

method uses transient signal, which is less affected by those factors. Thus, the error of

traveling wave method is much lower than impedance calculation method.

Since it is unaffected by fault resistance, transmission line types, and two-end

systems, traveling wave method is more reliable and accurate than impedance

calculation method. The main factors to the accuracy of travelling wave fault location

method are the propagation time and the velocity of the fault travelling wave. There

still are some problems for the traveling wave method. The first, to distinguish

between the traveling waves reflected from the fault point and from the remote end of

the line is a problem. The second, the uncertainty of the traveling wave can affect the

4

accuracy, such as the randomness of fault types and fault transition time. The third,

the velocity of traveling wave is affected by the different climates and environments.

The fourth, the sample rate of recording signals is also a significant factor which can

affect the accuracy of traveling wave method.

5

CHAPTER 2 THE THEORY OF TRAVELLING WAVE FAULT

LOCATION METHOD

The principle of the traveling wave method is based on the traveling high

frequency signals of voltage and current, which are recorded at end bus. In particular,

using arrived time difference between the initial and few subsequent wave peaks at

end bus and the velocity of the traveling wave calculates the distance of fault point.

2.1 Process of Transient Traveling Wave on Transmission Line

Assume there is a fault that occurs on a single phase transmission line of a two

end bus system. According to superimposed theory, there are two equal amplitude and

opposite direction voltage sources superimposed at fault point. The amplitude of those

two voltage sources is equal to the amplitude of the pre-fault voltage at fault point F.

The post fault network of a single phase transmission line power system can be regard

as a superimposed circuit of a pre-fault normal operation network and an additional

fault component network.

6

Figure 2.1: Superposition Principle [7]

(a): two end power system with a fault on single-phase transmission line (b): equivalent

circuit (c): pre-fault normal operation network (d): additional fault component network

Figure 2.1 presents the superposition principle. The figure 2.1 (a) is a one line

diagram of a two end bus power system with a fault on transmission line at the point F.

The figure 2.1 (b) is the equivalent circuit of figure 2.1 (a), the ef is equal to pre-fault

voltage at the fault point F. The figure 2.1 (c) is the pre-fault normal operation network.

The ef is equal to pre-fault voltage of point F. The figure 2.1(d) is the fault component

network. The –ef is equal in amplitude and opposite in direction of pre-fault voltage of

the fault point F. This figure only appears after fault.

When a fault occurs on the transmission line, the voltage and current transients will

travel towards end buses. According to the principle of reflection and refraction of the

traveling wave, these transient signals will continue to bounce back and forth between

the fault point and end buses [1]. The following equations are from Jan Izykowski,

A B

Ea E

b

F

A B

Ea

F

Eb

ef

-ef

A B

Ea

F

ef

Eb

A B

Za

F

-ef Z

b

(a)

(b)

(c)

(d)

7

Murari Mohan Saha and Rosolowski, Eugeniuszs’ book [1]. In single-phase lossless

transmission line, the voltage u and current i of the traveling wave at point x and time

t obey the partial differential equations:

t

iL

x

u (2-1)

t

uC

x

i

(2-2)

In equation (2-1) and (2-2): L and C are the inductance and capacitance of the line per

unit length.

The equation (2-1) and (2-2) can be deformed from partial differential by x and t:

t

uLC

x 2

2

2

2u

(2-3)

t

iCL

x

i2

2

2

2

(2-4)

Resistance is assumed to be negligible. The solutions of equation (2-3) and (2-4) are:

)(u)(t)u(x, vtxvtxu rf (2-5)

)(1

)(1

),( vtxuZ

vtxuZ

txi r

c

f

c

(2-6)

In equation (2-6): 𝑍𝑐 = √𝐿

𝐶 is the characteristic impedance of the transmission line,

v = √1

𝐿𝐶 is the velocity of propagation.

8

Figure 2.2：Schematic Diagram for Traveling Wave of a Fault on Transmission Line

Figure 2.2 represents the forward and the reverse waves, which leave the fault

point F and travel in opposite direction on transmission line toward to the end bus. In

equation (2-6), uf(x-vt) is the forward voltage traveling wave, and ur(x+vt) is the

reverse voltage traveling wave. The current traveling wave is similar to the voltage

traveling wave.

2.2 Wave Reflection and Refraction

When a fault occurs on the transmission line, there will be a forward and a backward

traveling wave towards two sides of transmission line. During this time, at the point of

wave impedance change, the traveling wave will reflect and transmit.

A B

F

uf ur

9

Figure 2.3: Schematic for Wave Reflection and Refraction [8]

In figure 2.3, η1 and η2 are wave impedance, e, r, t representing Incident, reflected,

and transmitted wave.

The following equations are from Yaozhong Ge’s book [9].

At point A:

ret

ret

iii

uuu

(2-7)

Meanwhile:

1

1

2

ee

rr

tt

iu

iu

iu

(2-8)

Solution of two formulas above:

2

Incident wave (e) Transmitted wave (t)

Reflected wave (r)

η1 A

η2

1

10

er

et

uu

uu

21

12

21

22

(2-9)

R is the reflection coefficient, T is the transmission coefficient, and 1 + R = T.

21

12R

(2-10)

21

22

T (2-11)

The relation between current reflection coefficient 𝑅𝑖 and voltage reflection

coefficient 𝑅𝑢 is:

Ri = -Ru (2-12)

2.3 Traveling Wave Method

Traveling wave method is divided into three modes [1]:

Single-ended mode: using traveling wave transients that produced by fault

point.

Double-ended mode: using traveling wave transients that produced by fault

point.

Single-ended mode: using pulse signal by devices after fault.

11

Single-ended mode uses traveling waves generated by fault, according to the first

two traveling waves which are recorded at end bus to determine the fault location.

Devices of this mode are only installed at one end of the line, and the communication

system is not required on the other buses. The single-ended mode demands simpler

devices than double-ended; however, this mode is subjected from transmitted wave

generated at the fault point and reflection waves from the other buses.

There are various methods to identify the reflected wave either from the fault

point or the remote end. One is based on wave polarity. The first reflected wave from

remote end has an opposite polarity of the first reflected wave from fault point. Thus,

the polarities of first and second reflected waves can be used to identify if the fault

point is located in first half or the second half of the transmission line. The other one

is based on impedance method. Single ended impedance method is used to estimate a

location in order to get a small range of fault location. Then, the range is used to

indicate if the reflected wave is from the fault point or from the remote end.

12

Figure 2.4: Lattice Diagram for One Line to Ground Fault

Figure 2.4 (a) shows a two bus transmission line system, which has a fault point

F1 on the first half of the line AB, and the second arrived wave is the reflected wave

from fault point. A traveling wave detector is placed at bus A. For the fault F1, the

time difference between the first two wave peaks will record as t2 – t1. Figure 2.4 (b)

A

B

F1

t1 t

2

(a)

A

B

F2

t1’ t

2’

(b)

13

shows the line with the fault F2, which occurs on the second half of the line AB, and

the second arrived wave is the reflected wave from remote end. A traveling wave

detector is still placed at bus A. For the fault F2, the time difference between the first

two wave peaks is t2’- t1’.

When the second arrived wave is the reflected wave from fault point, calculating

the distance ‘X’ between bus A and F1 uses the formula:

)(2

112 ttvX (2-13)

In equation (2-13): v is the wave velocity, which is close to velocity of light.

When the second arrived wave is the reflected wave from remote end, calculating

the distance ‘X’ between bus A and F2 uses the formula:

)''(2

112 ttvLX (2-14)

In equation (2-14): L is the length of transmission line.

2.4 Karrenbauer Transformation

Detailed description of the Karrenbauer transformation theory can be referred to

Yaozhong Ge’s book [8]. The equations in this section are from Ge’s book. In the three

phase lossless transmission line, the voltage and current equations:

14

t

it

it

i

LLL

LLL

LLL

x

ux

ux

u

c

b

a

smm

msm

mms

c

b

a

(2-15)

t

ut

ut

u

KKK

KKK

KKK

x

ix

ix

i

c

b

a

smm

msm

mms

c

b

a

(2-16)

In equation (2-15) and (2-16), Ls is the self-inductance, Lm is the mutual inductance,

Cs is the self-capacitance, Cm is the mutual capacitance, Ks=Cs+2Cm, Km=-Cm.

Because of electromagnetic coupling, the matrixes of L and C are not diagonal

matrixes. In addition, the LC and CL are also not diagonal matrixes. Thus, it is not

easy to calculate the traveling wave equations. In order to decouple the three phase

system, the phase domain signals need to be transformed into modal components.

mSUpU (2-17)

mp QII (2-18)

In equation (2-17) and (2-18), S and Q are modules transformation matrixes, Up and Ip

are phase values, Um and Im are modal values.

A well-known Karrenbauer phase-to-modal transformation matrix S is given:

211

121

111

S Q ,

101

011

111

3

1S 11- Q (2-19)

15

Using Karrenbauer transformation changes phase value to modals value:

c

b

a

u

u

u

u

u

u

101

011

111

3

1

(2-20)

c

b

a

i

i

i

i

i

i

101

011

111

3

1

(2-21)

Therefore,

u

u

u

U m

0

,

i

i

i

I m

0

(2-22)

In equation (2-22), the uα and the iα are α modal; the uβ and the iβ are β modal; the

uo and the i0 are 0 modal.

The α modal and the β modal are called “line-modal”, since the α modal

represents the “line-modal” between phase A and phase B, and the β modal represents

the “line-modal” between phase A and phase C. The 0 modal is called “zero-modal”.

2.4.1 The Module Velocity of Traveling Wave

The wave velocity is a significant component for traveling wave fault location

method. The accuracy of fault location is affected by the wave velocity. In commons,

the wave velocity is approximate the light speed, and it is smaller than the light speed.

16

Since the voltage and current signals are transformed from phase value to modal value,

the modal velocity should be used for calculating the fault locations.

The modal voltage and modal current equations [8] are:

t

ULCSS

x

U mm

2

21

2

2

(2-23)

t

ILCQQ

x

I mm

2

21

2

2

(2-24)

11

11

00

11

00

00

00

CL

CL

CL

LCQQLCSS (2-25)

Thus, the modal α and β have the same velocity V1, and zero modal velocity V0:

11

1

1

CLV (2-26)

00

0

1

CLV (2-27)

2.4.2 The Classification of Fault Type

In general, there are four types of fault commonly occurs on the transmission line.

The occurrence and the severity for each fault type are different. The table 2.1 is from

the paper [10].

17

Table 2.1 : Fault Type Occurrence and Severity

Type of Fault Symbol Occurrence (%) Severity

Line to Ground L – G 75-80 much less severe

Line to Line L – L 10 - 15 less severe

Double Line to

Ground 2L - G 5 - 10 severe

Three Line

(Ground) 3 - ϕ 2 - 5 very severe

Table2.1 shows the one line to ground fault is the most common fault type occurs

on the transmission line. The four fifths faults are one line to ground fault. The

occurrence of the three phase fault is small, but it is the most severe fault type.

Using the Karrenbauer transform can identify the fault types. The following

materials of this section are from Xinzhou, Dong’s paper [11]. The matrix of

Karrenbauer transform is a full-order matrix. In the matrix, the α-modal is the

line-modal between phase A and phase B; the β-modal is the line-modal between

phase A and phase C. For representing the line-modal between phase B and phase C,

the γ-modal is constructed in the matrix. Thus, the new Karrenbauer Transformation

matrix is:

18

c

b

a

110

101

011

111

3

1

0

(2-28)

Figure 2.5: Simplified Diagram of Different Fault Types

(a): one line to ground fault; (b): ling to line fault;

(c): double line to ground fault; (d): three line to ground fault.

1) One line to Ground Fault: The simplified diagram is shown in figure (a).

Assuming the grounding resistance is zero, the instantaneous boundary conditions

are:

0

0

A

CB

u

ii

(2-29)

Then the instantaneous modal components are:

A

B

C

(a)

A

B

C

(b)

A

B

C

(c)

A

B

C

(d)

19

0

3

1

0

A

A

A

i

i

i

i

i

i

i

(2-30)

2) Line to Line Fault: The simplified diagram (phase B to phase C) is shown in

figure (b). Assuming the grounding resistance is zero, the instantaneous boundary

conditions are:

CB

CB

A

uu

ii

i

0

(2-31)


B

B

B

i

i

i

i

i

i

i

2

0

3

1

0

(2-32)

3) Double Line Grounding Fault: The simplified diagram (phase B to phase C

grounding) is shown in figure (c). Assuming the grounding resistance is zero, the

instantaneous boundary conditions are:

0

0

CB

A

uu

i (2-33)


CB

C

B

CB

ii

i

i

ii

i

i

i

i

3

1

0

(2-34)

4) Three ling Grounding Fault: The simplified diagram is shown in figure (d).

Assuming the grounding resistance is zero, the instantaneous boundary conditions

are:

0

0

CBA

CBA

uuu

iii (2-35)


20

CB

CA

BA

ii

ii

ii

i

i

i

i 0

3

1

0

(2-36)

Table 2.2: Characteristic of Different Fault Types

0-modal α-modal β-modal γ-modal

AG iA iA iA 0

BG iB - iB 0 iB

CG iC 0 -iC -iC

AB 0 2 iA iA - iA

BC 0 - iB iB 2 iB

CA 0 -iC -2 iC -iC

ABG iA + iB iA - iB iA iB

BCG iB + iC - iB -iC iB - iC

CAG iA + iC iA iA – iC -iC

ABC(G) 0 iA - iB iA – iC iB – iC

*All modals shown above are multiplied by 3.

Table 3.2 represents the characteristics of various fault types in different situations.

2.5 Wavelet Transform

Wavelet transform is an effective mathematical method for transient traveling

wave. It has been widely used in faulted phase identification, traveling wave fault

location, and transformer protection [12]. By using Fourier transform, a sinusoid wave

21

is broken up into sine waves of various frequencies. Analogously, Wavelet transform

can break up a signal into shifted and scaled versions of the mother wavelet. However,

different from the Fourier transform, the wavelet transform allows time localization of

different frequency components of a signal. In other words, wavelets can adjust their

time-widths to their frequency. For example, the time-widths of the higher frequency

wavelets will be narrow, and the time-widths of the lower frequency wavelets will be

broader [13]. Since it applies to irregular, asymmetric, and oscillatory waveforms, the

wavelet transform is used in this paper as a mathematical tool for analyzing the

transient voltage and current traveling wave.

In the recent years, discrete wavelet transform (DWT) has been an attractive

signal processing tool. When a discrete signal is translated by discrete dyadic wavelet

transform, it will be decomposed into the approximation coefficients (A) and the

detail coefficients (D). In addition, the original signal also can be reconstructed by its

discrete wavelet approximations and discrete wavelet details as well [14]. By using

the discrete wavelet transform, the approximation coefficients are obtained by

convolving the original signal with the low-pass filter; the detail coefficients are

obtained by convolving the original signal with the high-pass filter.

The wavelet transform of sampled waveforms can be obtained by implementing

the discrete time wavelet transform. The equation of the discrete wavelet transform is

from [13].

22

m

m

m a

nakg

akm

0

0

n0

nx 1

),(DWT (2-37)

In equation (2-37), g [] is the mother wavelet; the ma0

represents scaling factors and

the ma0n represents time shifting factors.

Figure 2.6: Discrete Wavelet Transform Decomposition

In figure 2.6, S is the original signal; F is the output signal of the low-pass filter; G is

the output signal of the high-pass filter; the block after those two filters is down

sampling. This operation is keeping the even indexed elements.

There are many different types of mother wavelets, such as Harr, Daubichies (db),

Coiflet (coif) and Symmlet (sym) wavelets. To choose which type of mother wavelet

is significant for detecting and localizing different types of fault transients. The choice

also depends on a particular application. For short and fast transient disturbances, db4

High-pass filter

Low-pass filter 2

2

Approximation

Coefficients

(A1)

Detail

Coefficients

(D1)

S

F

G

23

and db6 are better [15]. Since the traveling wave is short and fast fault transient, the

db4 mother wavelet is used in this paper.

Figure 2.7: The ‘db4’ Mother Wavelet

Figure 2.7 shows the db4 mother wavelet. The figure is made by the display wavelets

of wavelet tool box of the MATLAB.

Based on the wavelet coefficients, the modulus maxima of a wavelet transform

contain the singularity information of a signal. The position where the modulus

maxima appear is the exact position where the peak of traveling wave occurs.

According to this principle, finding the traveling wave peaks converts into finding the

wavelet modulus maxima. In this paper, the DWT ‘db4’ is chosen as wavelet

24

transform. After the transformation, the wavelet modulus maxima of the detail

coefficient are found. Those modulus maxima are used as the peaks of the traveling

wave.

25

CHAPTER 3 SINGLE ENDED TRAVELING WAVE FAULT L

OCATION METHOD

3.1 System Introduction

In this paper, MATLAB® is chosen as the simulation programs. The following

system was constructed in SimPowerSystem™.

Figure 3.1: Simulation System for Single Ended Traveling Wave Fault Location

Method

26

System Parameters:

System Base Voltage: 500 KV

System Base Power: 100 MVA

System Frequency: 60 Hz

Source P Voltage: 500 KV

Source Q Voltage: 500 KV

Source P Positive Sequence Impedance: 17.177+j45.5285 Ω

Source P Zero Sequence Impedance: 2.5904+14.7328 Ω

Source Q Positive Sequence Impedance: 15.31+j45.9245 Ω

Source Q Zero Sequence Impedance: 0.7229+15.1288 Ω

Positive Sequence Line Resistance: 0.15573Ω/km

Zero Sequence Line Resistance: 0.37650625Ω/km

Positive Sequence Line Inductance: 9.7066e-4 H/km

Zero Sequence Line Inductance: 0.003 H/km

Positive Sequence Line Capacitance: 1.2093e-8F/km

Zero Sequence Line Capacitance: 7.4949e-9 F/km

Line Length: 100 km

3.2 Signal Processing

The voltage and current signal data are recorded by the simulation system. Those

data are phase values. In order to change the phase values to modal values, the

Karrenbauer Transformation is used. From the Karrenbauer Transformation, the 0, α,

and β modal values are calculated. Then, using the DWT, which is with the db4 mother

wave, translates those modal values into the approximation coefficients and detail

coefficients. Then, the points where the modulus maxima of detail coefficients appear

27

are the exact wave peak positions, so that the sample number of wave peaks are the

point positions of modulus maxima.

Figure 3.2: The Flow Chart of Signal Processing

Figure 3.2 shows all the steps of the signal processing of the single ended traveling

wave fault location method. Figure3.2 also presents the order and the function of each

step.

Assume there is one line to the ground fault that occurs at 30 km from bus A; the

fault transition time is from 0.041 s to 1 s; the fault and ground resistances are

negligible. From the simulation system, the three phase voltage and current signal are

recorded.

Recorded Voltage &Current Signals

Half Period Sample Window from Fault Point

DWT with db4

Karrenbauer Transformation

Sample Number for First and Second Peak

Calculation

Distance of Fault Location

28

Figure 3.3: Half Period Window of Voltage and Current Signal

Figure 3.3 represents the half period three phase voltage and current traveling

wave signals. The initial point is the point when the fault occurs. This figure shows

the results of the first and the second steps of the signal processing.

The third step of the signal processing is using Karrenbauer transformation to

translate the phase value to modal value. The fourth step is using the ‘db4’ DWT to

decompose the modal value into approximation coefficients and detail coefficients.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-6

-4

-2

0

2

Points

Corr

ent

(pu)

29

Figure 3.4: Wavelet Transformation for Voltage Modal Signal

Figure 3.4 represents the signal processing of the voltage signal. The first plot

shows the α-modal signal for the voltage signal; the second plot shows the

approximation coefficients at level 1 from the wavelet transformation; the last plot

shows the detail coefficients at level 1 from the wavelet transformation. This figure

shows the results of the third and the fourth step of the signal processing.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.5

0

0.5

Points

V-m

ode

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.5

0

0.5

Points

A1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.05

0

0.05

Points

D1

30

Figure 3.5: Wavelet Transformation for Current Modal Signal

Figure 3.5 represents the signal processing of the current signal. The first plot

shows the α-modal signal for the current signal; the second plot shows the

approximation coefficients at level 1 from the wavelet transformation; the last plot

shows the detail coefficients at level 1 from the wavelet transformation. This figure

shows the results of the third and the fourth step of the signal processing.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

0

2

Points

I-m

ode

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

0

2

Points

A1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-5

0

5x 10

-5

Points

D1

31

The fifth step is finding the wavelet modulus maxima of the detail coefficients.

Therefore, the sample number of the first and second wave peaks are found.

Figure 3.6: The Wavelet Modulus Maxima for Voltage Signal

Figure 3.6 represents the wavelet modulus maxima for the voltage signals. This

figure shows the sample number for the first and second traveling wave peaks: the

initial wave peak number is 103, and the second wave peak number is 305. This

figure shows the result of the fifth step of signal processing.

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 103

Y: 0.9516

Points

Voltage (

pu)

X: 305

Y: 0.4233

32

Figure 3.7: The Wavelet Modulus Maxima for Current Signal

Figure 3.7 represents the wavelet modulus maxima for the current signals. This

figure shows the sample number for the first and second traveling wave peaks: the

initial wave peak number is 105 and the second wave peak number is 307. This figure

shows the result of the fifth step of signal processing.

3.3 Testing Wave Velocity and Sample Rate

3.3.1 The Selection of Sample Rate

The accuracy of traveling wave method is affected by the sampling rate. In general,

the frequency of the transient traveling wave is over hundreds of KHz. Therefore, to

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 307

Y: 0.3953

Points

Curr

ent

(pu)

X: 105

Y: 1

33

ensure the accuracy of fault locations, the sample rate is at least hundreds of KHz. In

order to find a high accuracy sample rate, the various sample rates are tested by the

simulation system in this paper.

Assume there is one line to the ground fault at 30 km from bus A, the fault

resistance is negligible and the velocity of the traveling wave is 2.9188*105 km/s. The

fault transition time is 0.041 s to 1 s. The following table shows the result of using

different sample rates to record voltage signals.

Table 3.1: Different Sample Rate to Record Voltage Signals and Results of Evaluation

Sample Rate

(KHz)

Second Peak

Point (p2)

First Peak

Point (p1)

Calculate Fault location

(km)

Error

(%)

245.760 78 29 29.10 0.90

491.520 157 53 30.88 0.88

983.040 307 105 29.99 0.01

1966.080 609 205 29.99 0.01

Table 3.1 represents the calculation results of fault locations by using voltage

signal. This table shows that the smallest error, which is 0.01%, is obtained when

sample rate are 983.040 KHZ and 1966.080 Hz.

34

Table 3.2: Different Sample Rate to Record Current Signals and Results of Evaluation

Sample Rate

(KHz)

Second Peak

Point (p2)

First Peak

Point (p1)

Calculate Fault location

(km)

Error

(%)

245.760 78 29 29.10 0.90

491.520 154 55 29.39 0.61

983.040 305 103 29.99 0.01

1966.080 611 204 30.21 0.21

Table 3.2 represents the calculation results of fault locations by using the current

signals. This table shows the smallest error, which is 0.01%, is obtained when sample

rate is 983.040 KHz.

Table 3.1 and 3.2 shows the smallest error is 0.01% when the sample rate is

983.040 KHz for both voltage and current signals. Thus, the 983.040 KHz is selected

as the sample rate for the simulation system in this paper.

3.3.2 The Selection of Wave Velocity

The modal α and β have the same velocity V1, which is called line-modal velocity,

and zero module velocity is V0, which is called zero-modal velocity. In the section 3.1,

the equations (2-26) and (2-27) are given:

11

1

1

CLV

00

0

1

CLV

35

The traveling wave velocity can be calculated by system parameters. From section

5.1: the positive sequence line inductance L1 is 9.7066e-4 h/km; zero sequence line

inductance L0 is 0.003 h/km; the positive sequence line capacitance C1 is

1.2093e-8f/km; and the zero sequence line capacitance C0 is 7.4949e-9 f/km. Thus,

the wave velocity of α and 0 modules:

skm /101089.2 50

skm /109188.2 5

According to V. Kale, S. Bhide and P. Bedekars’ paper [16], the wave impedance

of the 0-modal component is large, so the velocity of the 0-modal is smaller than the

line-modal velocity. In addition, the 0-modal component is significant only when the

fault is a grounding fault. Therefore, the 0-modal signal cannot be used for all types of

faults. The velocity of line-modal is close to light speed because its wave impedance

is small and it is not affected by frequency and other conditions. Thus, the line-modal

velocity V1=2.9188*105 km/s is selected as the traveling wave velocity in this paper.

Assume there is one line to ground fault on transmission line, the fault transition

time is 0.041 s to 1 s, and the fault resistance is negligible. By using the voltage

transient signal and the V1 = 2.9188*105 km/s, the following table shows the result of

single ended traveling wave fault location method.

36


Table 3.3 represents the result of traveling wave fault location method by using

line-modal velocity.

In this paper, a method, which is utilize historical data, is provided to calculate

traveling velocity.

Actual Fault

Location (km)

Second Peak

Point (p2)

First Peak

Point (p1)

V1=2.9188*105km/s

Calculation

Fault

Location (km)

Error (%)

5 54 19 5.20 0.20

10 105 37 10.10 0.10

15 155 53 15.14 0.14

20 204 69 20.04 0.04

25 255 87 24.94 0.06

30 305 103 29.99 0.01

35 357 121 35.04 0.04

40 409 137 40.38 0.38

45 459 155 45.13 0.13

50 506 171 49.73 0.27

55 490 187 55.02 0.02

60 475 205 59.92 0.08

65 458 221 64.82 0.18

70 441 239 70.01 0.01

75 425 255 74.76 0.24

80 406 273 80.26 0.26

85 389 289 85.15 0.15

90 374 305 89.76 0.24

95 357 323 94.95 0.05

Average Error 0.14%

37

Assume there is one line to ground fault occurs 35 km from Bus A, the fault

transition time is 0.041 s to 1 s, and the fault resistance is negligible. The wave velocity

can be found by the following steps:

Step 1: using the system to simulate the fault location, thus, the voltage traveling wave

signals are recorded.

Figure 3.8: Voltage Signal for a AG Fault at 35km from bus A

Figure 3.8 shows the voltage waveforms represent half of a cycle following the

inception of the fault which is 35 km from Bus A.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-1.5

-1

-0.5

0

0.5

1

1.5

Points

Voltage (

pu)

38

Step 2: using the Karrenbauer transformation and the wavelet transform to get the time

difference between the arriving time of the initial wave and the arriving time of the

reflect wave from the fault point.

Figure 3.9: The Wavelet Modulus Maxima for Voltage Signal for Fault Location at 35

km

Figure 3.9 shows the wavelet modulus maxima focused view on the first 200 points.

This figure shows the sample number of the initial wave peak (p1) is 121, and the

sample number of the second wave peak (p2) is 357.

Step 3: Calculation

The sample time is 983040

1 s and the x = 35 km.

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 121

Y: 1

Points

Voltage (

pu)

X: 357

Y: 0.3866

39

t

xv

2 (3-1)

12 ppt (3-2)

The solution of (3-1) and (3-2) is:

skmv 5109158.2

121357

983040352

Table 3.4: The Calculation Result of Wave Velocity in Different Fault Locations

Fault Location

(km)

Wave Velocity

(105km/s)

Fault Location

(km)

Wave Velocity

(105km/s)

5 2.8087 55 2.9199

10 2.8913 60 2.9127

15 2.8913 65 2.9035

20 2.9127 70 2.9199

25 2.9257 75 2.8913

30 2.9199 80 2.9565

35 2.9158 85 2.9491

40 2.8913 90 2.8494

45 2.9103 95 2.8913

50 2.9344 Average Velocity 2.9050

Table 3.4 shows the calculation results for traveling wave velocity with various

fault locations. Assume the fault locations are known, using equation (3-1) and (3-2) to

calculate the wave velocity. This table shows the average of all traveling wave

velocities is 2.9050*105 km/s.

40

As a result, the traveling wave velocity is v= 2.9050*105 km/s. Assume there is one

line to ground fault on transmission line, the fault transition time is 0.041 s to 1 s, and

the fault resistance is negligible.


Actual Fault

Location (km)

Second Peak

Point (p2)

First Peak

Point (p1)

V=2.9050*105 km/s

Calculation Fault

Location (km)

Error (%)

5 54 19 5.17 0.17

10 105 37 10.05 0.05

15 155 53 15.07 0.07

20 204 69 19.95 0.05

25 255 87 24.82 0.18

30 305 103 29.85 0.15

35 357 121 34.87 0.13

40 409 137 40.19 0.19

45 459 155 44.92 0.08

50 506 171 49.50 0.50

55 490 187 55.23 0.23

60 475 205 60.11 0.11

65 458 221 64.98 0.02

70 441 239 70.15 0.15

75 425 255 74.88 0.12

80 406 273 80.35 0.35

85 389 289 85.22 0.22

90 374 305 89.80 0.20

95 357 323 94.98 0.02

Average Error 0.16%

41

Table 3.5 represents the result of traveling wave fault location method by using the

traveling wave velocity v= 2.9050*105 km/s. This table shows the average error is

0.16%, which is close to the average error by using the line-modal velocity v=

2.9188*105 km/s.

42

CHAPTER 4 EVALUATION STUDY

In this chapter, all kinds of faults are estimated to test the performance of the single

ended traveling wave fault location method. Those faults can be divided into three

aspects. The first aspect is fault type. In this section, four different fault types are

simulated. The second aspect is fault resistance. In this section, various fault resistances

are simulated with different fault types. The third aspect is the length of transmission

line. In this section, the 200 and 300 KM transmission lines are respectively simulated.

4.1 Fault Type

From section 3.2, the fault type can be classified. The single ended traveling wave

fault location method works for all fault types. The length of the transmission line in

this simulation system is 100 km. when the fault point is at 50 km from bus A, the first

peak point is 171 and the second peak point is 506. The initial point is captured when a

fault occurs. In this simulation system, when the first peak point is over than 171 and

second peak point is less than 506, the second wave is the reflected wave from the

remote end bus. Otherwise, the second peak is the reflected wave from fault point.

4.1.1 One Line to Ground Fault

Assume there is one line to ground fault that occurs 20 km from Bus A, the fault

transition time is from 0.041 s to 1 s, and the fault and ground resistance is negligible.

43

Figure 4.1: Voltage and Current Signal for the AG Fault at 20 km

The figure 4.1 shows the three-phase voltage and current signal which are recorded

by the simulation system. In this figure, the voltage and current waveforms represent

half of a cycle following the inception of fault, which is at 20 km from Bus A.

According to the signal processing section, those voltage and current phase values

are transformed to modal values by Karrenbauer transformation. Then, the α-modal

values of voltage and current are translated by DWT ‘db4’. In the end, the wavelet

modulus maxima are found from the detail coefficients.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-10

-5

0

5

Points

Corr

ent

(pu)

44

By using the voltage signal data:

Figure 4.2: The Modulus Maxima of the Voltage Signal for the AG Fault at 20 km

Figure 4.2 shows the wavelet modulus maxima for voltage signal. This figure

shows the initial wave peak of voltage signal is p1 = 69 and the second wave peak of

voltage signal is p2 = 204, so the second peak is the reflected wave from fault point.

Therefore, the following equation is used to estimate the fault location.

kmvx 04.20692042

1

%04.0%100100

2004.20

error

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 69

Y: 1

Points

Voltage (

pu) X: 204

Y: 0.5089

45

By using the current signal data:

Figure 4.3: The Modulus Maxima of the Current Signal for the AG Fault at 20 km

Figure 4.3 shows the wavelet modulus maxima for current signal. This figure

shows the initial wave peak of current signal is the initial wave peak of current signal is

p1’ = 71 and the second wave peak of voltage signal is p2’ = 204.

kmvx 74.19712042

1

%26.0%100100

2074.19

error

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 71

Y: 1

Points

Curr

ent

(pu)

X: 204

Y: 0.2105

46

Assume there is one line to ground fault that occurs 70 km from Bus A，the fault

transition time is from 0.041 s to 1 s, and the fault and ground resistances are negligible.

Figure 4.4: Voltage and Current Signal for the AG Fault at 70 km

Figure 4.4 shows the three phases voltage and current signal which are recorded by

the simulation system. In this figure, the voltage and current waveforms represent half

of a cycle following the inception of fault, which is at 70 km from Bus A.

From the signal processing, the modulus maxima are obtained from the voltage and

the current signals.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-6

-4

-2

0

2

Points

Corr

ent

(pu)

47

By using the voltage signal:

Figure 4.5: The Modulus Maxima of the Voltage Signal for the AG Fault at 70 km

Figure 4.5 shows the wavelet modulus maxima for voltage signal. In this figure, the

initial wave peak of voltage signal is p1 = 239 which is over 171, and the second wave

peak of voltage signal is p2 = 441 which is less than 506, so the second peak is the

reflected wave from remote end. Therefore, the following equation is used to estimate

the fault location.

kmvLx 01.702394412

1

%01.0%100100

7001.70

error

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 239

Y: 1

Points

Voltage (

pu)

X: 441

Y: 0.3572

48

By using the current signal:

Figure 4.6: The Modulus Maxima of the Current Signal for the AG Fault at 70 km

Figure 4.6 shows the wavelet modulus maxima for current signal. In this figure, the

initial wave peak of current signal is p1’ = 239 and the second wave peak of voltage

signal is p2’ = 441.

kmvLx 01.702394412

1

%01.0%100100

7001.70

error

0 50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X: 239

Y: 0.3569

Points

Curr

ent

(pu)

X: 441

Y: 0.3366

49

Table 4.1: Fault Location Result for One Line to Ground Fault

Fault Location

(km)

Voltage Signal Current Signal

Estimation

(km)

Error (%) Estimation

(km)

Error (%)

5 5.20 0.20 4.60 0.40

10 10.10 0.10 10.10 0.10

15 15.14 0.14 15.29 0.29

20 20.04 0.04 19.74 0.26

25 24.94 0.06 25.24 0.24

30 29.99 0.01 29.99 0.01

35 35.04 0.04 34.89 0.11

40 40.38 0.38 40.08 0.08

45 45.13 0.13 44.69 0.31

50 49.73 0.27 50.18 0.18

55 55.02 0.02 55.31 0.31

60 59.92 0.08 59.62 0.38

65 64.82 0.18 64.96 0.04

70 70.01 0.01 70.01 0.01

75 74.76 0.24 74.61 0.39

80 80.26 0.26 80.26 0.26

85 85.15 0.15 84.86 0.14

90 89.76 0.24 89.90 0.10

95 94.95 0.05 94.80 0.20

Table 4.1 represents the estimations and errors of traveling wave fault location

method by using both voltage and current signal. In this table, all the errors of the

estimated fault locations are below 1%. This table shows that the error is large

50

compared to the others when the fault point is close to the end buses and the middle

point of the transmission line.

4.1.2 Line to Line Fault

Assume there is line-to-line fault that occurs 40 km from Bus A, the fault transition

time is from 0.041 s to 1 s, and the fault resistance is negligible.

Figure 4.7: Voltage and Current Signal for the L-L Fault at 40 km

Figure 4.7 represents the three-phase voltage and current signal for the L-L fault

which is at 40 km from Bus A. In this figure, the voltage and current waveforms

represent half of a cycle following the inception of fault.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-5

0

5

Points

Corr

ent

(pu)

51

After the signal processing, the wavelet modulus maxima for this fault will be

found.


Figure 4.8: The Modulus Maxima of the Voltage Signal for the L_L Fault at 40 km

Figure 4.8 shows the wavelet modulus maxima of voltage signal for the L-L fault

which is at 40 km from bus A. In this figure, the initial wave peak of voltage signal is p1

= 137 which is less than 171 and the second wave peak of voltage signal is p2 = 409, so

the second peak is the reflected wave from fault point. Therefore, the following

equation is used to estimate the fault location.

kmvx 38.401374092

1

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 137

Y: 1

Points

Voltage (

pu)

X: 409

Y: 0.6784

52

%38.0%100100

4038.40

error


Figure 4.9: The Modulus Maxima of the Current Signal for the L_L Fault at 40 km

Figure 4.9 shows the wavelet modulus maxima of current signal for the L-L

fault which is at 40 km from bus A. In this figure, the initial wave peak of current signal

is p1’ = 136 and the second wave peak of voltage signal is p2’ = 406.

kmvx 06.401364062

1

%06.0%100100

4006.40

error

0 50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

X: 136

Y: 0.4993

Points

Curr

ent

(pu)

X: 406

Y: 0.4544

53

Assume there is a line-to-line fault that occurs at 60 km from Bus A, the fault

transition time is from 0.041 s to 1 s, and the fault resistance is negligible.

Figure 4.10: Voltage and Current Signal for the L-L Fault at 60 km

The figure 4.10 represents the three-phase voltage and current signal for the L-L

fault which is at 60 km from Bus A. In this figure, the voltage and current waveforms



found. Different from the AG fault, the second wave of the L-L fault, which is recorded

at the end bus, is the reflected wave from the fault point when the fault point is in the

second half of the transmission line.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-1

-0.5

0

0.5

1

1.5

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-4

-2

0

2

4

Points

Corr

ent

(pu)

54


Figure 4.11: The Modulus Maxima of the Voltage Signal for the L_L Fault at 60 km

Figure 4.11 shows the wavelet modulus maxima of voltage signal for the L-L fault

which is at 40 km from bus A. In this figure, the initial wave peak of voltage signal is p1

= 205 which is over 171, and the second wave peak of voltage signal is p2 = 609 which

is over 506. In this situation, the second wave is not a wave reflection from remote end,

but a wave reflection from fault point.

kmvx 98.592056092

1

%02.0%100100

60-98.59error

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 205Y: 1

Points

Voltage (

pu)

X: 609Y: 0.7251

55



Figure 4.12 shows the wavelet modulus maxima of current signal for the L-L fault

which is at 60 km from bus A. In this figure, the initial wave peak of current signal is p1’

= 204 and the second wave peak of voltage signal is p2’ = 608.

kmvx 98.592046082

1

%02.0%100100

60-98.59error

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X: 204Y: 0.4142

Points

Curr

ent

(pu)

X: 608Y: 0.5678

56

Table 4.2: Fault Location Result for the L-L Fault by Using the Voltage Signal

Actual Fault

Location (km)

Second Peak

Point (p2)

First Peak

Point (p1)

Estimation

(km)

Error

(%)

5 54 19 5.20 0.20

10 105 37 10.10 0.10

15 155 53 15.14 0.14

20 204 69 20.04 0.04

25 255 87 24.94 0.06

30 305 103 29.99 0.01

35 357 121 35.04 0.04

40 409 137 40.38 0.38

45 459 155 45.13 0.13

50 506 171 49.73 0.27

55 558 187 55.08 0.08

60 609 205 59.98 0.02

65 660 221 65.17 0.17

70 711 239 70.07 0.07

75 761 255 75.12 0.12

80 810 273 79.72 0.28

85 861 289 84.92 0.08

90 911 305 89.97 0.03

95 963 323 95.01 0.01

Table 4.2 shows the result of line to line fault by using the voltage signal. In this

table, the error is large when the fault point is close to end buses.

57

Table 4.3: Fault Location Result for the L-L Fault by Using the Current Signal

Actual Fault

Location (km)

Second Peak

Point (p2)

First Peak

Point (p1)

Estimation

(km)

Error

(%)

5 52 21 4.60 0.40

10 105 37 10.10 0.10

15 155 52 15.29 0.29

20 204 71 19.74 0.26

25 257 87 25.24 0.24

30 307 105 29.99 0.01

35 357 122 34.89 0.11

40 406 136 40.08 0.08

45 456 155 44.69 0.31

50 509 171 50.18 0.18

55 558 189 54.78 0.22

60 608 204 59.98 0.02

65 661 223 65.02 0.02

70 711 239 70.07 0.07

75 761 254 75.27 0.27

80 810 273 79.72 0.28

85 863 289 85.21 0.21

90 913 307 89.97 0.03

95 964 322 95.31 0.31

Table 4.3 shows the result of line to line fault by using the current signal. In this

table, the error is large when the fault point is close to end buses.

58

4.1.3 Double Line to Ground Fault

Assume there is a double line to ground fault that occurs at 15 km from Bus A, the

fault transition time is from 0.041 s to 1 s, and the fault resistance is negligible.

Figure 4.13: Voltage and Current Signal for the L-L-G Fault at 15 km

Figure 4.13 represents the three-phase voltage and current signal for the L-L-G



After the signal processing, the wavelet modulus maxima for this L-L-G fault will

be plotted.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-10

-5

0

5

10

Points

Corr

ent

(pu)

59



Figure 4.14 shows the wavelet modulus maxima of voltage signal for the L-L-G

fault which is at 15 km from bus A. In this figure, the initial wave peak of voltage signal

is p1 = 53 and the second wave peak of voltage signal is p2 =155, so the second peak is

the reflected wave from fault point. Therefore, the following equation is used to

estimate the fault location.

kmvx 14.15531552

1

%14.0%100100

1514.15

error

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 53Y: 1

Points

Voltage (

pu)

X: 155Y: 0.8189

60



Figure 4.15 shows the wavelet modulus maxima of current signal for the L-L-G



kmvx 29.15521552

1

%29.0%100100

1529.15

error

Assume there is a line-to-line to ground fault that occurs 90 km from Bus A, the


0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X: 52Y: 0.8076

Points

Curr

ent

(pu)

X: 155Y: 0.6424

61

Figure 4.16: Voltage and Current Signal for the L-L-G Fault at 90 km

Figure 4.16 represents the three-phase voltage and current signal for the L-L-G



After the signal processing, the wavelet modulus maxima for this L-L-G fault will

be plotted.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-2

-1

0

1

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-4

-2

0

2

4

Points

Corr

ent

(pu)

62



Figure 4.17 shows the wavelet modulus maxima of voltage signal for the L-L-G

fault which is at 15 km from bus A. In this figure, the initial wave peak of voltage signal

is p1 = 305 which is over 171, and the second wave peak of voltage signal is p2 = 913

which is over 506, so the second peak is the reflected wave from fault point. Therefore,

the following equation is used to estimate the fault location.

kmvx 26.903059132

1

%26.0%100100

9026.90

error

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X: 305Y: 0.6928

Points

Voltage (

pu)

X: 913Y: 0.7308

63



Figure 4.18 shows the wavelet modulus maxima of current signal for the L-L-G



kmvx 97.893079132

1

%03.0%100100

9097.89

error

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X: 307Y: 0.7371

Points

Curr

ent

(pu)

X: 913Y: 0.7219

64

Table 4.4: Fault Location Result for Double Line to Ground Fault

Fault Location

(km)


Estimation

(km)


(km)

Error (%)

5 5.20 0.20 4.60 0.40

10 10.10 0.10 10.10 0.10

15 15.14 0.14 15.29 0.29

20 20.04 0.04 19.74 0.26

25 24.94 0.06 25.24 0.24

30 29.99 0.01 29.99 0.01

35 35.04 0.04 34.89 0.11

40 40.38 0.38 40.08 0.08

45 45.13 0.13 44.69 0.31

50 49.73 0.27 50.18 0.18

55 55.08 0.08 54.78 0.22

60 59.98 0.02 59.98 0.02

65 65.17 0.17 65.02 0.02

70 70.07 0.07 70.07 0.07

75 75.12 0.12 75.27 0.27

80 79.72 0.28 79.72 0.28

85 84.92 0.08 85.21 0.21

90 90.26 0.26 89.97 0.03

95 95.01 0.01 95.31 0.31

Table 4.4 shows the result of double line to ground fault by using voltage and current

signal. This table is same as the result table of the line to line fault.

65

4.1.4 Three Line Fault

Assume there is a three line to ground fault that occurs at 55 km from Bus A, the


Figure 4.19: Voltage and Current Signal for the L-L-L-G Fault at 55 km

Figure 4.19 represents the three-phase voltage and current signal for the three line

to ground fault which is at 55 km from Bus A. In this figure, the voltage and current

waveforms represent half of a cycle following the inception of fault.


plotted.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-4

-2

0

2

Points

Voltage (

pu)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-5

0

5

Points

Corr

ent

(pu)

66


Figure 4.20: The Modulus Maxima of the Voltage Signal for the LLL-G Fault at 55 km

Figure 4.20 shows the wavelet modulus maxima of voltage signal for the three line

to ground fault which is at 55 km from bus A. In this figure, the initial wave peak of

voltage signal is p1 = 187 which is over 171 and the second wave peak of voltage signal

is p2 = 558 which is over 506, so the second peak is the reflected wave from fault point.

Therefore, the following equation is used to estimate the fault location.

kmvx 08.551875582

1

%08.0%100100

5508.55

error

0 100 200 300 400 500 600 700 800 9000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 187

Y: 0.7141

Points

Voltage (

pu)

X: 558

Y: 1

67


Figure 4.21: The Modulus Maxima of the Current Signal for the LLL-G Fault at 55 km

Figure 4.21 shows the wavelet modulus maxima of current signal for the three line

to ground fault which is at 55 km from bus A. In this figure, the initial wave peak of

current signal is p1’ = 189 and the second wave peak of voltage signal is p2’ = 558.

kmvx 78.541895582

1

%22.0%100100

5578.54

error

0 100 200 300 400 500 600 700 800 9000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 189

Y: 1

Points

Curr

ent

(pu)

X: 558

Y: 0.3566

68

Table 4.5: Fault Location Result for Three Line to Ground Fault

Fault Location

(km)


Estimation

(km)


(km)

Error (%)

5 5.20 0.20 4.60 0.40

10 10.10 0.10 10.10 0.10

15 15.14 0.14 15.29 0.29

20 20.04 0.04 19.74 0.26

25 24.94 0.06 25.24 0.24

30 29.99 0.01 29.99 0.01

35 35.04 0.04 34.89 0.11

40 40.38 0.38 40.08 0.08

45 45.13 0.13 44.69 0.31

50 49.73 0.27 50.18 0.18

55 55.08 0.08 54.78 0.22

60 59.98 0.02 59.98 0.02

65 65.17 0.17 65.02 0.02

70 70.07 0.07 70.07 0.07

75 75.12 0.12 75.27 0.27

80 79.72 0.28 79.72 0.28

85 84.92 0.08 85.21 0.21

90 90.26 0.26 89.97 0.03

95 95.01 0.01 95.31 0.31

Table 4.5 shows the result of three line to ground fault by using the voltage and

current signal.

69

4.2 Fault Resistance

In this part, the fault resistances and ground resistances are added to test the

accuracy of single ended traveling wave fault location method.

Table 4.6: The Result for the AG Fault with Different Resistances

Fault

Resistance

(Ω)

Actual Fault

Location

(km)


Estimated

Fault

Location

(km)

Error (%)

Estimated

Fault

Location

(km)

Error (%)

1

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.01 0.01 70.01 0.01

95 94.95 0.05 94.80 0.20

10

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.01 0.01 70.01 0.01

95 94.95 0.05 94.80 0.20

50

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.01 0.01 70.01 0.01

95 94.95 0.05 94.80 0.20

100

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70

70 70.01 0.01 70.01 0.01

95 94.95 0.05 94.80 0.20

200

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.01 0.01 70.01 0.01

95 94.95 0.05 94.80 0.20

Table 4.6 shows the result for the AG fault by using the single ended traveling wave

fault location method. The accuracy is same as the system without fault resistances.

Table 4.7: The Result for the L-L Fault with Different Resistances

Fault

Resistance

(Ω)

Actual Fault

Location

(km)


Estimated

Fault

Location

(km)

Error (%)

Estimated

Fault

Location

(km)

Error (%)

1

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

10

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

71

Table 4.7 shows the result for the line to line fault by using the single ended traveling

wave fault location method. The accuracy is same as the system without fault

resistances.

Table 4.8: The Result for the L-L-G Fault with Different Resistances

Fault

Resistance

(Ω)

Actual Fault

Location

(km)


Estimated

Fault

Location

(km)

Error (%)

Estimated

Fault

Location

(km)

Error (%)

Ron:1

Rg:1

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

Ron:1

Rg:200

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

Ron:10

Rg:1

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

Ron:10

Rg:200

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

72

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

Where Ron is the fault resistance; Rg is the ground resistance

Table 4.8 shows the result for the double line to ground fault by using the single ended

traveling wave fault location method. The accuracy is same as the system without fault

resistances.

Table 4.9: The Result for the Three Line to Ground Fault with Different Resistances

Fault

Resistance

(Ω)

Actual Fault

Location

(km)


Estimated

Fault

Location

(km)

Error (%)

Estimated

Fault

Location

(km)

Error (%)

1

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

10

5 5.20 0.20 4.60 0.40

30 29.99 0.01 29.99 0.01

50 49.73 0.27 50.18 0.18

70 70.07 0.07 70.07 0.07

95 95.01 0.01 95.31 0.31

73

Table 4.9 shows the result for the three line to ground fault by using the single ended

traveling wave fault location method. The accuracy is same as the system without fault

resistances.

4.3 Transmission Line Length

In this section, the single ended traveling wave method is tested with different

transmission line lengths. Several AG faults are respectively simulated on the 200 km

and the 300 km transmission lines.

Table 4.10: Fault Location Estimation for A 200 KM 500 KV Line

Fault Location

(km)


Estimation

(km)


(km)

Error (%)

5 5.20 0.10 4.60 0.20

10 10.10 0.05 10.10 0.05

15 15.14 0.07 15.29 0.15

95 95.01 0.01 95.76 0.38

100 100.21 0.10 99.62 0.19

105 104.99 0.01 104.99 0.01

185 184.86 0.07 185.01 0.01

190 189.90 0.05 190.50 0.25

195 194.80 0.10 195.40 0.20

Table 4.10 shows the fault location estimations for a 200 km transmission line.

Table 4.10 represents the single ended traveling wave fault location method is feasible

for the 200 km line.

74

Table 4.11: Fault Location Estimation for A 300 KM 500 KV Line

Fault Location

(km)


Estimation

(km)


(km)

Error (%)

5 5.20 0.07 4.60 0.13

10 10.10 0.03 10.10 0.03

15 15.14 0.05 15.29 0.10

145 144.89 0.04 145.19 0.06

150 150.24 0.08 150.09 0.03

155 155.11 0.04 154.66 0.11

285 285.01 0.00 285.45 0.15

290 289.61 0.13 289.46 0.18

295 294.95 0.02 294.95 0.02

Table 4.11 shows the fault location estimations for a 300 km transmission line.

Table 4.11 represents the single ended traveling wave fault location method is feasible

for the 300 km line.

75

CHAPTER 5 CONCLUSION

The objective of this thesis is to present the single ended traveling wave fault

location method and to evaluate the performance of this method. Since the method is

based on the traveling wave, the fundamental theory and properties of the traveling

wave are introduced in this thesis. Using the traveling wave theory of transmission

line, the voltage and the current transient signals are translated into their modal

signals by Karrenbauer transformation. In addition, the traveling wave velocity and

fault type can be found by the Karrenbauer transformation. The modal signals are

transformed into the approximate coefficient and the detail coefficient by the wavelet

transform. The maxima modules of the detail coefficient are used to determine the

sample number of the peaks. Using the first two peaks sample number estimates the

fault location. In addition, the evaluation study presents that the single ended traveling

wave fault location method is applicable to all fault types, various fault resistances,

and different transmission line lengths.

76

REFERENCES

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Springer, 2010.

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77

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[12] C. Kim and R. Aggarwal, "Wavelet transforms in power systems," IET Power

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[13] F. H. Magnago and A. Abur, "Fault Location Using Wavelets," Power Delivery,

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[14] S. Mallat and W. Hwang, "Singularity Detection and Processing with Wavelets,"

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IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 617-643, Mar 1992.

[15] M. T. N. Dinh, M. Bahadornejad, A. Shahri and N.-K. Nair, "Protection schemes

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[16] V. Kale, S. Bhide and P. Bedekar, "Fault Location Estimation Based on Wavelet

Analysis of Traveling Waves," Power and Energy Engineering Conference

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Location," IEEE, pp. 1113-1117, 2000.

79

VITA

Author

Name: Jin Chang

Birthplace: Zhengzhou, Henan, China

Education

08/2010—05/2013

Bachelor of Engineering

Bachelor of Science in Electrical Engineering

University of Kentucky

Awards

Graduate Certificate of Power and Energy by Power and Energy Institute of

Kentucky. Spring 2014

single ended traveling wave based fault location using discrete wavelet transform

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