DESIGN AND FAULT ANALYSIS OF

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DESIGN AND FAULT ANALYSIS OF

A 345KV 220 MILE OVERHEAD TRANSMISSION LINE


A Project


Presented to the faculty of the Department of Electrical and Electronic Engineering

California State

University, Sacramento


Submitted in partial satisfaction of

the requirements for the degree of




MASTER OF SCIENCE

in

Electrical and Electronic Engineering


and

MASTER OF SCIENCE

in

Electrical and Electronic Engineering


by


Greg Clawson

Mira Lopez



SPRING

2012

ii





































© 2012


Greg Clawson


Mira Lopez


ALL RIGHTS RESERVED

iii


DESIGN AND FAULT ANALYSIS OF


A 345KV 220 MILE OVERHEAD TRANSMISSION LINE




A Project



by



Greg Clawson


Mira Lopez















Approved by:


_____________________________________, Committee Chair

Turan

Gönen
,

Ph.D.


_____________
________________________,
Second Reader

Salah Yousif
,

Ph.D
.



___
_____________
____

Date



iv











Student:

G
reg Clawson


Mira Lopez



I certify that these students have met the requirements for
the
format contained i
n the
University format manual
and that this project is suitable for shelving in the Library and
that
credit is to be awarded for the project.















_____________________________, Graduate Coordinator _________________

B. Preetham. Kumar
,

Ph.D.


Date



Department of Electrical and Electronic Engineering




v


Abstract


of


DESIGN AND FAULT

ANALYSIS OF


A 345KV 220 MILE OVERHEAD TRANSMISSION LINE


by


Greg Clawson


Mira Lopez



Efficient and reliable transmission of bulk power economically benefits both the power
company and consumer. This report gives clarification to concept and procedure

in
design of an overhead 345 kV long transmission line. The project will find an optimum
design alternative which meets certain criteria including transmission efficiency, voltage
regulation, power loss,
line sag and tension
. A MATLAB script will be dev
eloped to
assess which alternative solutions can fulfill the criteria.

Integration of protective devices is a fundamental part of achieving power system
reliability. To determine the sizing and setting of protective devices, analysis of potential
fault
conditions provide the necess
ary current and voltage data.
A fault analysis for the
final
transmission line
design will

be simulated two ways: 1)

by using a MATLAB script
that
was developed for this project
and 2) by using an available Aspen One Liner
pro
gram.



_____________________________________, Committee Chair

Turan Gönen, Ph.D.



_____________________

Date

vi


DEDICATION


I dedicate
my work
to my sister, Mandica Konjevod for
inspiring

me.















vii


TABLE OF CONTENTS

Page

Dedicati
on…………………………………………………………………………

...…vi

List of Tabl
es……………………………………………………………………………
.
x
ii

List of Figures…………………………………………………………………………
...x
iii

Chapter

1.

INTRODUCTION
……………………………………………………………
..

..
….
.
.
1

2.

LITERATURE SURVE
Y…………………………………………………………
..

.
.
3

2.1.

Introduction
………………………………………………………………
.

….
..
3

2.2.

Support Structure
………………………………………………………
.
………...
3

2.3.

Line Spacing and Transposition………………………………………
.
…………
5

2.3.1.

Symmetrical Spacing
……………………………………………….……
.
.
.
.6

2.3.2.

Asymmetrical Spacing
………………………………………………..…
.
.
...7

2.3.3.

Transposed Line
…………………………………………
…………….
.

.
10

2.4.

Line Constants…………………………………………………………………..
11

2.5.

Conductor Typ
e and Size…………………………………………………….

.
12

2.6.

Extra
-
High Voltage Limiting Factors……………………………..
…………

.
16

2.6.1.

Corona
…………………………………………………………
………
..
…16

2.6.2.

Line Design Based on Corona
……………………………………

..
…...
19

2.6.3.

Advantages of Corona
……………………………………
……………
.

19

2.6.4.

Disadvantages of Corona
……………………………………………
..

...
20

viii


2.6.5.

Prevention of Corona
…………………………………………………

..
.
20

2.6.6.

Radio Noise
…………………………………………………
…………
…..
20

2.6.7.

Audible Noise
…………………………………………………………
..

21

2.7.

Line Modeling…………………………….
………………
……………

.
…….
21

2.8.

Line Loadability
……………………………………………………………
……
24

2.9.

Fault Events……………………………………………..
……………………

25

2.10.

Fault Analysis…………………………………………
………

…….
………
26

2.11.

Single
Line
-
to
-
Ground (SLG) Fault………...
………………
……
…………

27

2.12.

Line
-
to
-
Line

(L
-
L) Fault…………………………….
……
……..
…………

..
27

2.13.

Double
Line
-
to
-
Ground (DLG) Fault
……………………
…….
………………28

2.14.

Three
-
Phase Fault…………………………………….

…….
…………
…….
29

2.15.

The Per
-
Unit System…………………………………..

……
……
……
.
……
30

3.

MATHEMATICAL MODEL
……………………………..………
.
…………………
.
31

3.1.

Introduction
……………………………………………………………
……..
.

31

3.2.

Geometric Mean Distance
(GMD)…………………………………………...…
31

3.3.

Geometric Mean Radius (GMR)
………………………………
………………..
33

3.4.

Inductance and Inductive Reactance……………………………………
.
..
……
..
34

3.5.

Capacitance and Capacitive Reactance…………………………………

.


35

3.6.

Long Transmission Line Model………………………………………...

.
……
35

3.7.

Sending
-
E
nd

Voltage and Current
……………………………………………
.
...
40

3.8.

Power Loss…………………………………………………………..
………
.
….
42

ix


3.9.

Transmission Line Efficiency
……………………………
………..…………….
44

3.10.

Percent Voltage Regulation………………………………
…….
…...
…………44

3.11.

Surge Impedance Loading (SIL)
………………………………
……
………

45

3.12.

Sag and
Tension
…………………………………………………
……
………
.
46

3.12.1.

Catenary Method………………
.
………...…………………………

46

3.12.2.

Parabolic Method…………………
..
…………
……………………

….
50

3.13.

Corona Power Loss……………………
…….
………
………………………

51

3.13.1.

Critical Corona Disruptive Voltage
…………………………………
…...
51

3.13.2.

Visual Corona Disruptive Voltage
………
…………………………
…….
53

3.13.3.

Corona Power Loss at AC Voltage
………………………………
………
54

3.14.

Method of Symmetrical Components…………………………………
……...
..
55

3.14.1.

Sequence Impedance of Transposed Lines………………………………59

3.15.

Fault Analysis……………………………………………
…………...…
…….
.
61

3.16.

Per Unit………………………………………………………..……
…….
……62

3.17.

Single
Line
-
to
-
Ground (SLG) Fault
………………………..
………
…….
……
6
3

3.18.

Line
-
to
-
Line (L
-
L) Fault…………………………
.
……...………

……
……66

3.19.

Double
Line
-
to
-
Ground (DLG) Fault
…………………...…………
…….
…….
69

3.20.

Three
-
Phase Fault………………………………………………

…….
……
..
72

4.

APPLICATION OF MATHEMATICAL MODEL
……………
.……………
.
……..76

4.1.

Introduction
……………………………………………………..……

………
76

4.2.

Design Criteria………………………………………
…………………..
……
.
.
..
77

x


4.3.

Geometric Mean Distance (GMD)……
………..
……………...………………
..77

4.4.

Geometric Mean Radius (GMR)………………………………………
….…..
.
..78

4.5.

Inductance and Inductive Reactance
.
………………..
.…………………………78

4.6.

Capacitance and Capacitive Reactance
…………………...……...…..
………
…79

4.7.

Long Line Characteristic
s……………………
...

…………………………..
..
..80

4.8.

ABCD Constants……………………
…….………………………..
………….
..8
1

4.9.

Sending
-
E
nd Voltage and Current
…………………………………..
…………
..83

4.10.

Power
Loss…………………………………….
……………..
………
…….

..84

4.11.

Percent Voltage Regulation
…………………………………………..
.
..
........
.
..86

4.12.

Transmission Line Efficiency……………
…………..
…………………
…….
..
8
6

4.13.

Surge Impedance Loading (SIL)
……………………………………
……

…86

4.14.

Sag and Tension
……………………………………………………
……
..
.
….
..87

4.14.1.

Catena
ry Method……………………………
……………………………87

4.14.2.

Parabolic Method…………………………………
……………...
……
…89

4.15.

Corona Po
wer Loss …………………………………………………
…….

.
..89

4.15.1.

Critical Corona Disruptive Voltage
…………………………………..
...
..89

4.15.2.

Visual Corona Disruptive Voltage…………………
………………...

..91

4.15.3.

Corona Power Loss at
AC Voltage
………………………………..
……
..92

4.15.4.

Corona Power Loss for
Foul Weather Conditions.………………
……

94

4.16.

Per Unit……………………
…………………………………
……..
.
….…….
..97

4.17.

Fault Analysis Outline
………………………………………
…….
…………
.
..98

xi


4.18.

Procedure Using Symmetrical Components……
………………..
…..…
…….
..99

4.19.

Fault
Analysis at the End of Transmission Line
……………..
..……
……..

.100

4.19.1.

Single Line
-
to
-
Ground (SLG) Fault
………………………...
……...
.

.101

4.19.2.

Line
-
to
-
Line (L
-
L) Fault………………
.
…………………………

….
104

4.19.3.

Double Line
-
to
-
Ground (DLG) Fault
………
.
……………………
…….
109

4.19.4.

Three Line
-
to
-
Ground (3LG)

Fault……………………………
.
.
..
.…….
113

5.

CONCLUSION
S………………………..
……………………………………
..
…….
117

Append
ix A.
Conductor and Tower

Characteristics………………
.
………………..
.
..
.
119

Appendix B.
Aspen Simulation Model and Analysis………
.
………
.
……
……..
.……
1
20

Appendix C.
Aspen Fault Analysis Summary
……………

…..

.
….………………
123

Appendix
D.
MATLAB

Aspen Fault Analysis Results..
………………
.
………
.
…...
..
1
31

Appendix E.
MA
TLAB Code……………………
...
……………………………
...
…..
.
1
37

Bibliog
raphy………………………………………………
…………………
…………
18
2









xii


LIST OF TABLES

Tables

Page

1.

Table 2.1 Typical conducto
r separation…………………………………….…………5

2.

Table 2.2 Aluminum vs. copper conductor type…………………………….……
.
…13

3.

Table 3.1

Corona Factor………………………………………………………..……55

4.

Table 3.2

Power and functions of operator
a
………………………………………
.
..56

5.

Table

4.1 Design pa
rameters…………………………………………………………77

6.

Table 4.2

System data for power
s
ystem model.…………………………...………..99

7.

Table 4.3

Fault analysis of SLG fault
at receiving end of line.……………….
……10
4

8.

Table 4.
4

Fault analysis of L
-
L fault at rec
eiving end of line.………….…….……108

9.


Table 4.5

Fault analysis of DLG fault at
receiv
ing end of line……………………112

10.

Table 4.6

Fault analysis of 3LG fault

at receiving end of line……………………
.
.116










xiii


LIST OF FIGURES

Figures




Page

1.

Figure 2.1 Three
-
phase line with symmetrical
spacing…………………………..…...6

2.

Figure 2.2 Cross section of three
-
phase line with horizont
al tower configuration…....6

3.

Figure 2.3

Three
-
phase line with asy
mmetrical spacing………………………………8

4.

Figure 2.4

A transposed thre
e
-
phase line…………………………………………….10

5.

Figure 2.5
Equivalent circuit of short

transmission line…………………….……
.
…22

6.

Figure 2.6 Nominal
-
T circuit of medium
transmission line……………………...….22

7.

Figure 2.7
Nominal
-
π circuit of medi
um transmission line………………………
.
…23

8.

Figure 2.8
Segment of 1
-
phase and neutral connecti
on
for long transmission line
.
…23

9.

Figure 2.9 Practical Loadability

for Line Length……………………………………25

10.

Figure 2.10 General representation for singl
e line
-
to
-
ground fault………………….27

11.

Figure 2.11 General representation for
line
-
to
-
line fault…………………………….28

12.

Figure 2.12

General representation of doubl
e line
-
to
-
ground fault………………….29

13.

Figure 2.13 General representation for three
-
phase fault………………………...
….30

14.

Figure 3.1 Bundled conductors co
nfigurations………………...…………………….32

15.

Figure 3.2 Cross section of three
-
phase horiz
ontal bu
ndled
-
conductor……………..32

16.

Figure 3.3
Segment of 1
-
phase and neutral connecti
on for long transmission line…
.
36

17.

Figure 3.4 Parameters of cat
enary…………………………………...………………48

18.

Figure 3.5 Parameters of par
abola…………………………………………………...50

19.

Figure 3.6
Sequence c
omponents…
…………………………
..
……………………..56

xiv


20.

Figure 3.7 Single line
-
to
-
ground fault

seq
uence network connection……………….6
4

21.

Figure 3.8 L
ine
-
to
-
line
fault sequence network connection………...……
………….66

22.

Figure 3.9 Double line
-
to
-
ground fault seq
uence network connection………………70

23.

Figure
3.10 Three
-
phase fault sequence

network connection………………………..72

24.

Figure 4.1 3H1 wood H
-
frame type

structure.
…………………………………...…..76

25.

Figure 4.2 One line diagram of power system model.…………………………...
…..98

26.

Figure 4.3 Power system model with fault
at end of line.
……………………….....100

27.

Figure 4.4 Equivalent seque
nce networks.
…………………………………………101

28.

Figure 4.5 Sequence network connecti
on for SLG fault…
………………………...10
1

29.

Figure 4.6 Sequence network connection for L
-
L

fault.
……………………………10
5

30.

Figure 4.7 Sequence network connecti
on fo
r DLG fault.
……………………..……109

31.

Figure 4.8 Sequence network connectio
n for 3LG fault ……………………..…….113



1




Chapter 1

INTRODUCTION


The purpose of this project is to design an overhead long transmission line that operates
at an extra
-
high voltage (EHV), and effectively supplies power to a specified load. The
line will have a length of 220 miles, and operate at 345 kV. The receiving e
nd of the line
will be connected to a load of 100 MVA with a lagging power factor of 0.9.

Design of an overhead transmission line is an intricate process that essentially involves a
complete study of conductors, structure, and equipment

[1]
. The study d
etermines the
potential effectiveness of a proposed system of components in satisfying design criteria.
The design criteria for this project are primarily focused on electrical performance
requirements. The criteria include transmission line efficiency,
power loss, voltage
regulati
on, line sag and tension.
To simplify the design process for this project, the same
support structure will be used for all design options, and for the final solution. The
options in conductor size with the predetermined struct
ure will provide alternative
solutions. A MATLAB program will be used to determine the performance of all
alternative solutions with respect to each design criteria. Amongst the options, the ones
that meet all design criteria will be considered and compar
ed for selecting the opt
imal

final solution.

A fault analysis will be completed for the final solution in order to demonstrate the
electrical behavior and performance of the transmission line system, when subjected to
fault condi
tions. The system model
will in
terconnect the transmission line to a typical
2




generator source, via a step
-
up transformer, in order to supply power to the line. On the
receiving end of the line, the load will be connected. This fault study will be completed
twice, one time via a

MATLAB program, and another time via the ASPEN One
-
Liner
software. Current and voltage conditions will be found during the different fault events.
The study will cover fault events occurring at the following three locations: 1) beginning
of the transmis
sion line, 2) midpoint
on the transmission line

and 3) end of the
transmission line. At each location, the four classical fault types will be considered.

The last analysis for this project will use the ASPEN One
-
Liner software to simulate the
load f
low for the final line design using the same system model as described for the fault
study. The results will indicate performance of the final line design under normal
operating conditions.



Equation Chapter (Next) Sec
tion 1

Equation Chapter (Next) Section 1

Equation Section (Next)





3




Chapter 2

LITERATURE SURVEY


2.1 INTRODUCTION

This chapter succinctly introduces and explains
important fundamental concepts and
terminology involved with transmission line design and fault analysis. Some basic theory
is provided as circumstantial information that leads to general questions and issues that
must be addressed during the design and a
nalysis processes.


2.2 SUPPORT STRUCTURE

A line design usually has structure support requirements that are very similar to
requirements of some existing lines

[1]
. Thus, an existing structure design can likely be
found and leveraged to accommodate the su
pport requirements. For this reason, most of
the work associated with the structure involves defining the configuration and mechanical
load requirements that the structure must support in order to select the appropriate
existing structure design.

Many f
actors must be considered when defining the configuration and mechanical load of
an overhead transmission line. First, data about the environmental conditions and climate
must be gathered and reviewed. Parameters such as air temperature, wind velocity,
r
ainfall, snow, ice, relati
ve humidity

and solar radiation must be studied

[9]
.
Subsequently, other factors are assessed, including conductor weight, ground shielding
needs, clearance to ground, right of way, equipment mounting needs, material
4




availability
, terrain to be crossed, cost of procurement, and lifetime upgrading and
maintenance

[1]
.

Conductor load is found by calculating sag/tension on the conductor. The amount of
tension depends on the conductor's weight, sag, and span. In addition, wind and
ice
loading increases the tension and must be included in the load specifications

[9]
. For safe
operation of conductors, the structure must have a margin of strength under all expected
load/tension conditions. For all conditions, the structure must also

provide adequate
clearance between conductors.

The three main types of structures are pole, lattice, and H
-
frame. The lattice and H
-
frame
types are stronger than the pole type, and provide more clearance between conductors.
Common materials used for str
ucture fabrication are wo
od, steel, aluminum

and concrete

[1].

For an extra
-
high or ultra
-
high voltage line, conductors are larger and heavier, so the
structure must be stronger than ones that are used for lower voltage lines. Steel lattice
type structur
es are the most reliable, having advantages in strength

of structure type and
material

and in additional clearance between conductors. In comparison, wood and
concrete pole type structures are suited more for lower load stresses. Wood has
advantages of l
ess procurement cost and natural insulating qualities

[1]
.

Since the scope of this project is primarily focused on the electrical design criteria of
transmission lines, a structure for this design will be selected from a group of existing
345kV support str
uctures without defining specific load support requirements.



5




2.3 LINE SPACING AND

TRANSPOSITION

When designing a transmission line the spacing between conductors should be taken into
consideration. There are two aspects of spacing analysis: mechanical and electrical.

Mechanical Aspect:

Wing conductors usually swing synchronously. However, in cases

of small size conductors and long spans there is the likelihood that
conductors

might
swing non
-
synchronously
.

In
order to determine correct conductor spacing
the
following
factors should be included into analy
sis: the material, the diameter

and the size
of the
conductor, in addition to maximum sag at the center of the span. A conductor with
smaller cr
os
s
-
section will swing out furthe
r than a conductor of large cross
-
section. There
are several formulas in
use to determine right spacing [8].
This is NESC, US
A formula


3.681
2
L
D A S
  

(
2
.
1
)

D

=

horizontal spacing in cm

A

=

0.762 cm per kV line voltage

S

=

sag in cm

L

=

length of insulator string i
n cm

Voltage between
conductors

Minimum horizontal
spacing

Minimum vertical
spacing

Up to 8700V

12in

16in

8701 to 50,000V

12in, plus 0.4in for each
1000 V above 8700V
*

40in

Above 50,000V

12in, plus 0.4in for each
1000 V above 8700V

40in, plus 0.4in for each
1000 V above 50,000V


*
This is approximate.

Table
2.
1 Typical conductor separation

[11]
.

6




Electrical Aspect
:
When increasing spacing (GMD
Φ

= geometric mean distance between
the phase conductors in ft) Z
1
(positive
-
sequence impedance) increases and Z
0
(zero
-
sequence impedance)
decreases. If the neutral is placed closer to the phase conductors it
will reduce Z
0

but may increase the resistive c
omponent of Z
0
. A small neutral with high
resistance increases the resistance part of Z
0

[8].


2.3.1
SYMMETRICAL SPACING

Three
-
phase line with symmetrical spacing forms an equilateral triangle with a distance D
between conductors. Assuming that the currents are balanced:


0
a b c
I I I
  

(
2
.
2
)

(
a)




(
b)






Figure 2.1

Three
-
phase
line

with symmetrical spacing
:

a)
g
eometry;

b)
phase i
nductance

[
8
]
.



b
c
a
D
12
D
23
D
31

Figure 2.2

Cross section of three
-
phase line with horizontal tower configuration
.

L
a
neutral
D
D
D
I
a
I
b
I
c
r
7




The total flux linkage of phase conductor is:


7
'
1 1 1
2 10 ( )
a a b c
I ln I ln I ln
D
r
D


  


  

(
2
.
3
)


b c a
I I I
  

(
2
.
4
)


7
'
1 1
2 10
a a a
I ln l
r
I n
D


 
    






(
2
.
5
)


7
2 10
'
a a
ln
r
D
I



  

(
2
.
6
)

Because of symmetry:


a b c
  
 

(
2
.
7
)

and the three inductances are identical.

The inductance per phase per kilometer length:



0.2
s
D mH
L ln
D km
 

(
2
.
8
)

r

=
the geometric mean

radius, GMR, and is shown by D
s

For a solid round conductor:




1
4
s
D r e

 

(
2
.
9
)

Inductance per phase for a three
-
phase circuit with equilateral spa
cing is the same as for
one condu
ctor of a single
-
phase circuit.

2.3.2
ASYMMETRICAL SPACING

While constructing a transmission line it is necessary to take into account

the

practical
problem
of how
to maintain symmetrical spacing. With asymmetrical spacing
between
the phases, the voltage drop due to line inductance will be unbalanced even when the line
8




currents are balanced. The distances between the phases are denoted by
D
12
,
D
32

and
D
13
.
The following flux linkages for the three phases are obtained:

D
13
D
23
D
12
a
b
c

Figure 2.3

Three
-
phase line with asymmetrical spacing

[
8
].


7
12 13
1 1 1
2 10 ( )
'
a a b c
I ln I ln I ln
r D D


   

  

(
2
.
10
)


7
12 23
1 1 1
2 10 ( )
'
b a b c
I ln I ln I ln
D r D


      


(
2
.
11
)



7
13 23
1 1 1
2 10 (
'
)
c a b c
I ln I ln I ln
D D r


      


(
2
.
12
)

In matrix form:



L I

 

(
2
.
13
)






The symmetrical inductance matrix:

9





12 13
7
12 23
13 23
1 1 1
'
'
'

1 1 1
2 10
1 1 1

ln ln ln
r D D
L ln ln ln
D
r
r D
ln ln ln
D D

 
 
 
 
 
 
 
 
 
 


(
2
.
14
)

With

I
a
as a reference for balanced three
-
phase currents:


2
240
b a a
I I a I

  


(
2
.
15
)


120
c a a
I I a I

  



(
2
.
16
)

The operator
a
:


120
a
a I




(
2
.
17
)


2
240
a
a I




(
2
.
18
)

The phase inductances are not equal and they contain an imaginary term due to the
mutu
al inductance:


7
12 13
1 1 1
2 10 ( )
'

a
a a b c
a
L I ln I ln I ln
I r D D



       

(
2
.
19
)


7
12 23
1 1 1
2 10 (
'
)

b
b a b c
b
L I ln I ln I ln
I D r D



       

(
2
.
20
)


7
13 23
1 1 1
2 1
'
0

c
c a b c
c
L I ln I ln I ln
I D D r


 
       






(
2
.
21
)



2.3.3
TRANSPOSED LINE

10




In most power system analysis a per
-
phase model of the transmission line is required.
The previously
above stated inductances are unwanted because they result in an
unbalanced circuit configuration. The balanced nature of the circuit can be restored by
exchanging the positions of the conductors at consistent intervals. This is known as
transposition of li
ne and is shown in Fig
ure 2.4
. In this

example

each segment of the line
is divided into three equal sub
-
segments. Transposition involves interchanging of the
phase configuration every one
-
third the length so that each conductor is moved to occupy
the next
physical position in a regular sequence.


a
a
a
a
a
a
b
b
b
b
b
b
c
c
c
c
c
c
1
2
3
D
23
D
23
D
12
D
12
S
/
3
Length of line
,
S
S
/
3
S
/
3
1
2
3
1
2
3
Section II
Section I
Section III

Figure
2.4

A
T
ransposed three
-
phase
line

[
7
].

In a transposed line, each phase takes all the three positions. The inductance per phase
can be found as the average value of the three
inductances (
L
a
,
L
b

and
L
c
) previously
calculated in (2.19) to (2.21
). Consequently,


3
a b c
L L L
L
 


(
2
.
22
)

Since,

2
1 120 1 240 1
o o
a a
      

(
2
.
23
)

11




The average of
a b c
L L L
 

come to be


7
'
12 23 13
2 10 1 1 1 1
3
3
L ln ln ln ln
r D D D

 

     
 
 

(
2
.
24
)



1
3
7
12 23 31
( )
2 10
'
D D D
L ln
r

 
 

(
2
.
25
)

The inductance per phase per kilometer
length:


0.2
s
GMD mH
L ln
D km
 

(
2
.
26
)

D
s
is the geometric mean radiu
s, (GMR). For stranded conductor
D
s
is obtained from the
manufacture’s data. However, for solid conductor:


1
4
'
s
D r r e

  

(
2
.
27
)

GMD (geometric mean distance) is the equivalent conductor spacing:


3
12 23 31
GMD D D D
  

(
2
.
28
)

For the modeling purposes it is convenient to treat the circuit as transposed.


2.4 LINE CONSTANTS

Transmission lines have four
basic constants: series resistance, series inductance, shunt
capacitance, and shunt conductance

[8]
.

Series resistance is the most impo
rtant cause of
power loss in a transmission line. The
ac

resistance or effective resistance of a conductor is



2

Ω
L
ac
P
R
I


(
2
.
29
)






12




where the real power loss (
P
L
) in the conductor is in watts, and
the conductor's rms
current (I) is in amperes

[8]
. The amount of resistance in the line depends mostly upon
conductor material resistivity, conductor length, and conductor cross
-
sectional area.

The inductance of a transmission line is calculated as flux
linkages per ampere. An
accurate measure of inductance in the line must include both flux internal to each
conductor and the external flux that is produced by the current in each conductor

[5]
.
Both series resistance and series inductance, i.e. series im
pedance, bring about series
voltage drops along the line.

Shunt capacitance produces line
-
charging currents. Shunt capacitance in a transmission
line is due to the potential difference between conductors

[1]
.

Shunt conductance causes, to a much lesser

degree, real power losses as a result of
leakage currents between conductors or between conductors and ground. The current
leaks at insulators or to corona

[8]
. Shunt conductance of overhead lines is usually
ignored.


2.5 CONDUCTOR TYPE A
ND SIZE

A cond
uctor consists of one or more wires appropriate for carrying electric current. Most
conductors are made of either aluminum or copper.





Aluminum (Al)

Copper (Cu)

Observation

13




Melting Point

660

C

1083

C


Annealing
starts

Most rapidly

above 100

C


100

C

200

C and 325

C


Both soften and lose
tensile strength.

Resistance to
corrosion

Good

Very Good

Al corrodes quickly
through electrical
contact with Cu or
steel. This galvanic
corrosion
accelerates in the
presence of salt.

Oxidation

When exposed to the
atmosphere


Al thin invisible
oxidation film
protects against
most chemicals,
weather and even
acids.

Resistivity


Very low

Cu conductor has
equivalent

ampacity of an
aluminum
conductor that is
two AWG sizes

larger. A la
rger Al
cross
-
sectional area
is
required to obtain
the same loss as in a
Cu conductor

Usage

Al is lighter, less
expensive and so it has
been used for almost all
new overhead
installations

Cu is widely used as a
power conductor, but
rarely as an overhead
conductor. Cu is

heavier and more

expensive than Al

The supply of Al is
abundant, whereas
that of Cu is limited.


Table 2.2 Aluminum vs. copper conductor type
.

Since aluminum is lighter and less expensive for a given current
-
carrying capability it has
been used by utilities for almost al
l new overhead installations. Aluminum for power
conductors is alloy 1350, which is 99.5% pure and has a minimum conductivity of 61.0%
IACS

[10]
.

14




Different types of aluminum conductors are a
vailable:

AAC


all
-
aluminum conductor

Aluminum grade 1350
-
H19
AAC has the highest conductivity
-
to
-
weight ratio of all
overhead conductors

[10]
.

ACSR


aluminum conductor, steel reinforced

Because of its high

mechanical strength
-
to
-
weight ratio, ACSR has equivalent or higher
ampacity for the same size conductor. The s
teel adds extra weight, normally 11 to 18% of
the weight of the conductor. Several different strandings are available to provide different
strength levels. Common distribution sizes of ACSR have twice the breaking strength of
AAC. High strength means the c
onductor can withstand higher ice and wind loads.

Also, trees are less likely to break this
conductor

[10
]
.

Stranded conductors

are easier to
manufacture, since larger conductor sizes can be obtained by simply adding successive
layers of strands. Stranded
conductors are also easier to handle and more flexible than
solid conductors, especially in larger sizes. The use of steel strands gives ACSR
conductors a high strength
-
to
-
weight ratio. For purposes of heat dissipation, overhead
transmission
-
line conductor
s are ba
re (no insulating cover) [8
]
.

AAAC


all
-
aluminum alloy conductor

This alloy of aluminum, the

6201
-
T81 alloy, has high strength and equivalent ampacities
of AAC or ACSR. AAAC finds good use in coastal areas where use of ACSR is
prohibited
because o
f excessive corrosion

[10]
.

ACAR


aluminum conductor, alloy reinforced

15




Strands of aluminum 6201
-
T81 alloy are used along with standard 1350 aluminum. The
alloy strands increase the strength of the conductor. The strands of both are the same
diameter, so t
hey can be arranged in a variety of configurations. For most urban and
suburban applications, AAC has sufficient strength and has good thermal characteristics
for a given weight. In rural areas, utilities can use smaller conductors and longer pole
spans, s
o ACSR or another of the higher
-
strength conductors is more appropriate

[10]
.

Conductor Sizes

The American Wire Gauge (AWG) is the standard generally employed in this country
and where American practices prevail. The
circular mil
(cmil) is usually used as
the unit
of measurement for conductors. It is the area of a circle having a diameter of 0.001 in,
which works out to be 0.7854 × 10

6 in
2
. In the metric system, these figures are a
diameter of 0.0254 mm and an area of 506.71 × 10

6 mm
2

[11
]
.

Wire sizes are given in
gauge n
umbers, which, for distribution
system purposes, range from a minimum of no. 12
to a maximum of no.0000 (or 4/0) for solid type conductors. Solid wire is not usually
made in sizes larger than 4/0, and stranded wire for sizes

larger than no. 2 is generally
used. Above the 4/0 size, co
nductors are generally given in
circular mils (cmil) or in
thousands of circular mils (cmil × 10
3
); stranded conductors for distribution purposes
usually range from a minimum of no. 6 to a maximum

of 1,000,000 cmil (or 1000 cmil ×
10
3
) and may consist

of two classes of strandings.
Gauge numbers may
be determined
from the formula:


0.3249

1.123
n
Diameter in


(
2
.
30
)

16





105,500
Cross sectional area
1,261
n
cmil
 

(
2
.
31
)

where
n
is the gauge number (no. 0 = 0; no. 00 =


1;

no. 000 =


2; no.0000 =


3)

[11]
.


2.6
EXTR
A HIGH VOLTAGE LIMITING FACTORS

Limiting factors for extra high voltage are:

a)

Corona

b)

Radio noise (RN)

c)

Audible Noise (AN)


2.6.1
CORONA

Air surrounding
conductors act
as an insulator between them.

Under certain conditions
air gets ionized and
its partial breakdown occurs. Disruption of a
ir dielectrics when
the
electrical field
reaches the critical surface gradient is known as corona.
Corona effect
causes significant power lo
ss and a high frequency current.

Corona comes in different
forms:
visual corona

as violet or blue glows,
audible cor
ona

as high pitched sound

and
gaseous corona

as ozone gas which can be identified by its specific odor.

In addition
high conductor surface gradient causes the emission of radio
and television
interference
(RI

and T
V
I
)

to the surrounding antennas

known as

radio corona
.

In order to
design
corona free lines
it is necessary to take into consideration
following

factors:

1)

Electrical

2)

Atmospheric

17




3)

Conductor

1)

Electrical
Factors
:

a)

Frequency and waveform of the supply:

Corona loss is a function of frequency
.
For that

reason

the

higher the frequency of
the
supply

voltage

the higher is corona
loss
.

This means that corona loss at
60 Hz
is greater

than at
50

Hz.
As a result
direct current (DC) corona loss is less than the alternate current (AC).

b)

Line Voltage
:
Line voltage

factor is significant for voltages higher than disruptive
voltage. Corona and line voltage are directly proportional.

c)

Conductor electrical field:

Conductor electrical field depends on the voltage and
conductor configuration i.e., vertical, horizontal, del
ta etc. In horizontal
configuration the middle conductor has a larger
electrical
field than the outsides
ones. This means that the critical disruptive voltage is lower
for the middle
conductor and therefore corona loss is larger.


2)

Atmospheric Factors:

Air

density, humidity, wind, t
emperature
and
pressure have an
effect on the corona loss.
In addition

rain, snow, hail and dust can reduce the critical
disruptive voltage and hence increase the corona loss.
Rain has more effect on the
corona loss than any othe
r weather conditions.
T
he most
influential are temperature
and pressure.

Atmospheric condition

such as air density is directly proportional to the
air strength breakdown.


3)

Conductor Factors:

Several different conductor factors
affect

the corona loss:

18




a)

Radius

or size of the conductor
:
The larger the size of the conductor (radius) the
larger the power
lower
loss.
For a certain voltages the larger the conductor size,
the larger the critical disruptive voltage and therefore the smaller the power loss.


2
( )
loss ln c
P V V
 

(
2
.
32
)

V
ln

= line
-
to
-
neutral (phase) operating voltage in kV

V
c

= disruptive (incep
tion) critical voltage kV (rms)

b)

Spacing between conductors
: T
he larger the spacing between conductors the
smaller the power loss. This can be observed from

power loss approximation
:



loss
r
P
D


(
2
.
33
)

r = conductor radius

D = distance (spacing) between conductors

c)

Number of conductors / Phases
:
In case of a single conductor per phase for higher
voltages there is a significant corona loss. In order to reduce corona loss t
wo or
more conductors are bundled together
.

By bundling conductors
the
self
-
geometric mean distance (
GMD
)

and
the
critical disruptive
voltage are
greater
than in case of a single conductor per phase which leads to reducing corona loss.

d)

Profile
or shape
of the conductor
:
Conductors can have different

shapes or
profiles. The profile of the conductor (cylindrical, oval, flat, etc.,) affects the
corona loss. Cylindrical shape has better field uniformity than any other shape
and
hence less corona loss.


19




e)

Surface conditions of the conductors
:

The disruptiv
e voltage is higher for smooth
cylindrical conductors. Conductors with uneven surface have more deposit (dust,
dirt, grease, etc.,) which lowers the disruptive voltage and increases corona.

f)

Clearance from ground
: Electrical field is affected by the height of the conductor
from the ground. Corona loss is greater for smaller
clearances
.


g)

Heating of the conductor by
load current
:

Load current

causes heating of the
conductor which accelerates the drying of the
conductor surface after rain. This
helps to minimize the time of the wet conductor and indirectly reduces the corona
loss

[12]
.


2.6.2 LINE DESIGN BASED ON CORONA

When designing a long transmission line (TL) it is desirable to have corona
-
free lines for
fair weather conditions and to minimize corona loss under wet weather conditions. The
average corona value is calculated by finding out corona loss per kilometer at various
points at long transmission line and averaging them out. For typical transmission l
ine in
fair weather condition corona loss of 1kW per three
-
phase mile and foul weather loss of
20 kW per three
-
phase mile is accepta
ble

[7]
.


2.6.3
ADVANTAGES OF CORONA

Corona reduces the magnitude of high voltage waves due to lightning by partially
dissip
ating as a corona loss. In this case it has a purpose of a safety valve.


20




2.6.4
DISADVANTAGES OF CORONA

a)

Loss of power

b)

The effective capacitance of the conductor is increased which increases the
flow of charging current.

c)

Due to electromagnetic and electrost
atic induction field corona interferes with
the communication lines which usually run along the same route as the power
lines

[7]
.



2.6.5 PREVENTION OF CORONA

Corona loss can be prevented by:

a)

increasing the radius of conductor

b)


increasing spacing of the
conductors

c)

selecting proper type of the conductor

d)

using bundled conductors

[7].


2.6.6 RADIO NOISE

Radio noise (RN)
happens

due to corona and gap discharges

(sparking)
. It is
unwanted
interference within radio frequency band. RN includes radio
interference (RI) and
television interference (TVI).

R
adio interference

(RI)
:
It affects amplitude modulated (AM) radio waves within the
standard broadcast band (0.5 to 1.6 MHz)
.

F
requency modulated (FM) waves

are less
affected.


21




Television interference (T
VI)
: In general TVI is caused by sparking within VHF (30
-
300MHz) and UHF (300
-
3000MHz) bands.
Two types
of TVI are recognized due to
weather conditions:
fair and foul

[1]
.



2.6.7

AUDIBLE NOISE

Audible noise (AN) takes place predominantly during foul weather conditions due to
corona. AN sounds like a hiss or sizzle.
In addition
corona produces low
-
frequency
humming

tones (
120
-

240Hz
)

[1]
.



2.7 LINE MODELING

To understand the electrical
performance of a transmission line, electrical parameters at
both ends of a line must be evaluated. When voltage and current is given at one end of a
line, an accurate calculation of voltage and current at the other end, or at some point
along the line, r
equires a sufficiently accurate model of a line. How a transmission line is
model
ed depends on the line length.
There are t
hree classes of line lengths.
For line
lengths that are classified as short, up to 50 miles, the model is simplified because shunt
c
apacitance and shunt admittance can be omitted because they have little effect on the
accuracy of the model. Because the line impedance is constant throughout the line, the
current will be the same from the sending end to the receiving end, so the model c
an be a
simple, lumped impedan
ce value, as shown in Figure 2.5

[1]
.

22





Figure 2.5
Equivalent circuit
of short transmission line [1].

For line lengths that are classified as medium, between 50 and 150 miles, there is enough
current leaking through the shun
t capacitance that shunt admittance must be included in
order for the model to be an acceptable representation. However, a medium line is still
short enough that lumping the shunt admittance at some points along the line is a
sufficiently accurate model

[
1]
. Typically, a medium line is modeled either as a T or π
network, as shown in Figures 2.6 and 2.7
.


Figure 2.
6

Nominal
-
T circuit o
f medium transmission line [1].

I
S
Z
=
R
+
jX
L
I
R
+
V
S
-
+
V
R
-
a
a’
N ’
N
S en d i n g
en d
(
s o u r c e
)
R ec ei v i n g
en d
(
s o u r c e
)
l
I
S
R
/
2
+
j
(
X
L
/
2
)
R
/
2
+
j
(
X
L
/
2
)
I
R
+
V
S
-
+
V
R
-
C
G
I
Y
a
a’
N ’
N
V
Y
23





Figure 2.7

Nominal
-
π circuit of medium transmission line [1].

For line lengths that
are classified as long, above 150 miles, the needed accuracy from the
model requires that the series impedance and shunt admittance be represented by a
uniform distribution of the line parameters

[1]
. Each differential length is infinitely small
and defin
ed as a unit length. The series impedance and shunt admittance is represented
for each unit length

of line, as shown in Figure 2.8
.


Figure 2.8

Segment of one phase and neutral connecti
on for long transmission line [5
].

I
S
R
+
j X
L
I
R
+
V
S
-
+
V
R
-
C
/
2
G
/
2
I
C
1
a
a’
N ’
N
C
/
2
G
/
2
I
C
2
I
I
24




This model accounts for the chan
ges in voltage and current throughout the line exactly as
the series impedance and shunt admittance affect them. In this way, the difference
between voltage and current at the sending end and receiving end can be analyzed
accurately

[5]
. The scope of thi
s project will only cover the mathematical model used fo
r
designing long line lengths.
For details of the long transmission line mat
hematical model,
see Chapter 3.


2.8 LINE LOADABILITY

The characteristic impedance of a line, also known as surge impedance,

is a function of
line inductance and capacitance. Surge impedance loading (SIL) is a measure of the
amount of power the line delivers to a pu
rely resistive load equal to it
s surge impedance.
SIL provides a comparison of the capabilities of lines to
carry load, and permissible
loading of a line can be expressed as a fraction of SIL.

The theoretical maximum power that can be transmitted over a line is when the angular
displacement across the line is δ = 90

, for the terminal voltages. However, for rea
sons
of system stability, the angular displacement across the line is typically between 30


and
45


[8]
. Figure 2.9

illustrates the differences in curve plots for the theoretical steady
-
state stability limit and a practical line loadability. The practica
l line loadability is
derived from a typical voltage
-
drop limit of











and a maximum angular
displacement of 30


to 35


across the line

[8]
. The loadability curve is generally
applicable to overhead 60
-
Hz lines with no compensation.

25





Figure
2.9

Practica
l Loadability for Line Length [8
].


As indicated by the chart, for lines classified as short, the power transfer capability is
determined by the thermal loading limit. For medium and long lines, maximum power
transfer is determined by the stab
ility limit.


2.9 FAULT EVENTS

A fault event in a
n

electric power transmission system is any abnormal change in the
physical state of a transmission system that impairs normal current flow. Typically, a
fault in a transmission line occurs when an external object or force causes a short circuit.
Examples

of external objects that intrude upon an overhead transmission line are
lightning

strikes, tree limbs, animals, high winds, earthquakes, and local structures.
Other faults occur when components or devices in a transmission system fail. During a
fault, t
he network can experience either an open circuit or a short circuit. Short
-
circuit
26




faults impose the most risk of damaging elements in a power system. Open circuit faults
are typically not a threat for causing damage to other network elements.


2.10
FAULT
ANALYSIS

Important part of TL designing includes fault analysis.
In order to have well protected
network

t
ypically faults are simulated
at different points throughout the
transmission
system
. It is crucial to have precise analysis of the
designed
s
ystem to prevent fault’s
interruption.

In general
the
three phase
faults
can be classified as:

1.

Shunt faults (short circuit
s
)

1.1.

Unsymmetrical faults

(Unbalanced)

1.1.1.

Single l
ine
-
to
-
ground

(SLG)

fault

1.1.2.

Line
-
to
-
line

(L
-
L)

fault

1.1.3.

Double line
-
to
-
ground

(DLG)

fault

1.2.

Symmetrical
fault

(Balanced)

1.2.1.

Three
-
phase
-
fault

2.

Series Faults (open conductor)

2.1.

Unbalanced faults

2.1.1.

One line open (OLO)

2.1.2.

Two lines open (TLO)

3.

Simultaneous faults



27




2.11 SINGLE LINE
-
TO
-
GROUND

(SLG)

FAULT

Seventy percent of all transmission line faults are attri
butable to when a single conductor
is physically damaged and either lands a connection to the ground or makes contact with
the neutral wire

[1]
. This fault type makes the system unbalanced and is called a single
line
-
to
-
ground (SLG) fault. The failed pha
se conductor, generally defined as phase a, is
connected to ground by an impedance value Z
f
. Figure 2.10

shows the general
representation of an SLG fault.

Z
f
a
b
c
n
F
+
V
af
-
I
bf
=
0
I
af
I
cf
=
0

Figure 2.10

General representation for single line
-
to
-
ground fault [1].


2.12 LINE
-
TO
-
LINE
(L
-
L)
FAULT

A line
-
to
-
line fault

is unsymmetrical (unbalanced) fault and it
takes place when two
conductors are short
-
circuited. This can happen for various reasons i.e.,
ioniz
ation of air
,

28




flashover,

or bad insulation.

Figure 2.11

shows

the general representation of an LL
fault

[
1]
.

a
b
c
F
I
bf
I
af
=
0
I
cf
=
-
I
bf
Z
f

Figure 2.11

General representation for

line
-
to
-
line fault [1].


2.13 DOUBLE LINE
-
TO
-
GROUND
(DLG)
FAULT

Ten percent
of all transmission line faults are attributable to when two conductors are
physically damaged and both of them land a connection through the ground or both
contact the neutral wire

[1]
. This fault type makes the system unbalanced and is called a
double
line
-
to
-
ground (DLG) fault. The failed phase conductors, generally defined as
phases b and c, are each connected to ground by their own separate fault impedance value
Z
f

and a common ground impedance value Z
g
.





29





Figure 2.12

shows the general represent
ation of a DLG fault.

Z
f
a
b
c
n
F
I
bf
I
af

=
0
I
cf
Z
f
Z
g
N
I
bf
+
I
cf

Figure 2.12 General representation of d
ouble line
-
to
-
ground fault [1].


2.14 THREE
-
PHASE FAULT

A three
-
phase (3Φ) fault
occurs when all three phases of a TL are short
-
circuited

to each
other or earthed.

It
is
a symmetrical (balanced) fault and the most severe one.

Since 3Φ
fault is balanced it is sufficient to identify the positive sequence network. As all three
phases carry 120


displaced equal currents the single line diagram
can be

used for the
analysis.

Three
-
phase faults make 5% of the initial faults in a power system

[1]
.





30




Figure 2.13

shows
the general representation of a

3Φ fault.

Z
f
a
b
c
n
F
I
bf
I
af
I
cf
Z
f
Z
g
N
I
af
+
I
bf
+
I
cf
=
3
I
a
0
Z
f

Figure 2.13 General representation
for three
-
phase (3Φ) fault
[1].


2.15 THE PER
-
UNIT SYSTEM

In power system analysis it is beneficial to normalize or scale quantities because

of

different
ratings of the equipment used.
Usually the impedances of machines and
transformers are specified in per
-
unit or percent of nameplate rating.
Using per
-
unit
system has more than a few advantages
such
as simplifying hand calculations,
elimination of ideal transformers as circuit component
, bringing voltage from beginning
to end of the system close to unity,

and
simplifies analysis of the system overall.
Particular

disadvantages are that sometimes phase shifts are eliminated and equivalent
circuits look more abstract. In spite of this per
-
unit system is widely used in

industry
.

31




Chapter 3

MATHEMATICAL MODEL

Equation Chapter (Next) Section 1

3.1 INTRODUCT
ION

This chapter steps through the mathematical approach which is used for design and
analysis of an overhead extra
-
high voltage long transmission line. Some information
about the physical solution is included to relate the mathematical model and physical

solution.

After design requirements are established, the first step in preliminary design is to choose
a standardized support structure that can be adapted to provide the best solution for the
given job. The selection should be taken from a group of st
ructures that have been
categorized as standard designs for the transmission voltage level that matches the design
requirement. A selected structure will define the spacing between conductor phases and
the limits on conductor size that can be supported.

The next step in preliminary design is to choose a conductor type and size that has
adequate capacity to handle the load current. With a preliminary selection of support
structure and conductor type and size, a detailed design analysis can be undertaken
, as
shown in the mathematical approach from the following sections.


3.
2 GEOMETRIC MEAN DIS
TANCE (GMD
)



Bundling of conductors is used for extra
-
high voltage (EHV) lines instead of one large
32




conductor

per phase. The bundles used at the EHV range usually have two, three, or four
subconductors

[1]
.

(a)

(b)


(c)

d
d
d
d
d
d
d
d

Figure 3
.1
Bundled

c
o
nductors configurations
: (a) two
-
conductor bundle; (b) three
-
conductor
bundle; (c) four
-
conductor bundle

[1
].



b
'
b
d
c
'
c
d
a
'
a
d
D
12
D
23
D
31

Figure 3.2 Cross section of
bundled
-
conductor
three
-
phase
line with
horizontal

tower
configuration

[1
].


The three
-
conductor bundle has its conductors on the vertices of an equilateral triangle,
and the four
-
conductor bundle has its conductors on the corners of a square.

For balanced three
-
phase operation of a completely transposed three
-
phase line only one
phase needs to be considered.
D
eq
, the cube root of the product of the three
-
phase
spacings, is the geometric mean distance (GMD) between phases:


3
12 23 31

eq m
D D D D D ft
  

(
3
.
1
)

33




3.3
GEOMETRIC MEAN RADIU
S (GMR)

Geometric Mean Radius (GMR) of bundled conductors for

Two
-
conductor bundle:



b
S S
D D d ft
 

(
3
.
2
)

Three
-
conductor bundle:


2
3

b
S S
D D d ft
 

(
3
.
3
)

Four
-
conductor bundle:


3
4

b
S S
D D d ft
 

(
3
.
4
)

w
here:



= GMR of subconductors




distance between two subconductors

If the phase spacings are large compared to the bundle spacing, then sufficient accuracy
for
D
eq

is obtained by using the distances between bundle centers. If the conductors are
stranded and the bundle spacing

d

is large compared to the conductor outside r
adius, each
stranded conductor is replaced by an equivalent solid cylindrical conductor with
GMR=


.


The modified GMR of bundled conductors used in capacitance calculations for

Two
-
conductor bundle:


b
SC
D r d ft
 

(
3
.
5
)

Three
-
conductor bundle:

3 2

b
SC
D r d ft
 

(
3
.
6
)

34




Four
-
conductor bundle:


3
4
1.09
b
SC
D r d ft
 

(
3
.
7
)

w
here:



= outside radius of subconductors



= dist
ance between two subconductors.


3.4 INDUCTANCE AND INDUCTIVE REACTANCE

For three
-
phase transmission lines t
hat
are completely transposed
, E
quation
(3.1)

can

be
used to find the equivalent equilateral spacing for the line. Thus, the average inductance
per phase is


7
2 10 ln
eq
a
s
D
H
L
D m

 
  
 
 

(
3
.
8
)



or


10
0.7411 log
eq
a
s
D
mH
L
D mi
 
 
 
 

(
3
.
9
)


and the inductive reactance is found by



2
L a
X f L

  

per phase

(
3
.
10
)

o
r



0.1213 ln

eq
L
s
D
X
D mi
 

 
 
 
per phase

(
3
.
11
)




35




3.5 CAPACITANCE AND CAPACITIVE REACTANCE


The average line
-
to
-
neutral capacitance per phase is


10
0.0388
μF
to neutral
log
N
eq
C
D
mi
r

 
 
 

(
3
.
12
)

where




1
3

eq m ab bc ca
D D D D D ft
  

(
3
.
13
)





radius of cylindrical conductor in feet.

The capacitive reactance is calculated by


1
2
C
N
X
f C


 

(
3
.
14
)

o
r


10
0.06836 log .
eq
C
D
X M mi
r
 
  
 
 

(
3
.
15
)


3
.6 LONG TRANSMISSION LINE MODEL

For lines 150 miles and longer, i.e. long lines, modeling with lumped parameters is not
sufficiently accurate for
representing the effects of the parameters’ uniform distribution
throughout the length of the line. An acceptable model provides mathematical
expressions for voltage and current at any p
oint along the line

[1]
. Figure 3.3

depicts a
segment of one phase o
f a three
-
phase transmission line of length
l
.

36





Figure 3.
3
Segment of one phase and neutral connectio
n for long transmission line. [5
]


The following d
erivation is given by Saada
t [5
]. The series impedance per unit length is
z, and the shunt admittance per phase is y, where
z = r + jωL
and
y = g + jωC
. Consider
one small segment of line Δx at a distance x from the receiving end of the line. The
phasor voltages and currents on both
sides of this segment are shown as a function of
distance
. From Kirchhoff’s voltage law






( )
V x x V x z xI x
   

(
3
.
16
)

o
r




( )
( )
V x x V x
zI x
x
 



(
3
.
17
)


37




Taking the limit as




, we have


( )
( )
dV x
zI x
dx


(
3
.
18
)

Also, from Kirchhoff’s current law








I x x I x y xV x x
    

(
3
.
19
)

o
r




( )
( )
I x x I x
yV x x
x
 
 


(
3
.
20
)

Taking the limit as




, we have


( )
( )
dI x
yV x
dx


(
3
.
21
)

Differentiating (3.18) and substituting from (3.21
), we get


2
2
( ) ( )
( )
d x dI x
z z x
dx
V
yV
dx
 

(
3
.
22
)

Let


2
zy



(
3
.
23
)

The following second
-
order differential equation will result.




2
2
2
( )
0
d V x
V x
dx

 

(
3
.
24
)

The solution of the above equation is




1 2
x x
V x Ae Ae
 

 

(
3
.
25
)

where γ, known as the
propagation constant
, is a
complex expression given by (3.23
) or


( ) ( )
j zy r j L g j C
    
      

(
3
.
26
)

38




The real part

α

is known as the
attenuation constant
, and the imaginary component β is
known as the
phase constant
. β is measured in r
adian per unit length. From (3.18
), the
current is








1 2 1 2
1 ( )
x x x x
dV x y
I x Ae A e Ae A e
z dx z z
   

 
    

(
3
.
27
)

o
r






1 2
1
x x
C
I x Ae A e
Z
 

 

(
3
.
28
)

where
Z
c

is known as the
characteristic impedance
, given by


C
z
Z
y


(
3
.
29
)

To find the constants



and


, we note that when



,

(

)



, and

(

)



.
From (3.25) and (3.28
) these constants are fo
und to be


1
2
R C R
V Z I
A



(
3
.
30
)


2
2
R C R
V Z I
A



(
3
.
31
)

Upon substitution in (3.25) and (3.28
), the general expressions for voltage and current
along a long transmission
line become




2 2
x x
R C R R C R
V Z I V Z I
V x e e
 

 
 

(
3
.
32
)




2 2
R R
R R
x x
C C
V V
I I
Z Z
I x e e
 

 



(
3
.
33
)


39




The equations for voltage and currents can be rearranged as follows:




2 2
x x x x
R C R
e e e e
V x V Z I
   
 
 
 

(
3
.
34
)




1
2 2
x x x x
R R
C
e e e e
I x V I
Z
   
 





(
3
.
35
)

Recognizing the hyperbolic functions sinh, and cosh, the above equations are written as
follows:








cosh sinh
R C R
V x x V Z x I
 
 

(
3
.
36
)








1
sinh cosh
R R
C
I x x V x I
Z
 
 

(
3
.
37
)

We are particularly interested in the relation between the sending
-
end and the receiving
-
end on the line. Setting



,

(

)




and

(

)



, the result is






cosh sinh
S R C R
V l V Z l I
 
 

(
3
.
38
)






1
sinh cosh
S R R
C
I l V l I
Z
 
 

(
3
.
39
)

Rewriting the above equations in terms of ABCD constants, we have


S
R
S
R
V
V
A B
I
I
C D
 
 
 

 
 
 
 
 
 

(
3
.
40
)

w
here




cosh cosh cosh
A l YZ
 
  

(
3
.
41
)




sinh sinh sinh
C C
Z
B Z l YZ Z
Y
 
  

(
3
.
42
)

40







sinh sinh sinh
C C
Y
C Y l YZ Y
Z
 
  

(
3
.
43
)




cosh cosh cosh
D A l YZ
 
   

(
3
.
44
)

w
here







L
Z r jx l
  

(
3
.
45
)

is
total line series impedance per phase




S
Y g jb l
 

(
3
.
46
)

is

total line shunt admittance per phase
.

Note that

A D


(
3
.
47
)

and

1
AD BC
 
.

(
3
.
48
)

For a long transmission line, conductance is very small compared to susceptance, and can
be omitted for simplicity. Thus,


can be reduced to the following equation:




1
C
Y jb l j l
X
  

(
3
.
49
)


3.7
SENDING
-
END VOLTAGE AND CURRENT

One step in line design is analyzing what power input,
i.e. voltage and current, is needed
at the sending
-
end in order to deliver the load power requirements. If the resulting power
input needs are within parameters that are acceptable to the overall power system, the
design is viable. However, the line desi