master of science in eee project by greg clawson and mira lopez spring 2012.pdf

7/29/2019 Master of Science in EEE Project by Greg Clawson and Mira Lopez Spring 2012.pdf

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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 SCIENCEin

Electrical and Electronic Engineering

and

MASTER OF SCIENCE

inElectrical and Electronic Engineering

by

Greg Clawson

Mira Lopez

SPRING2012


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2012

Greg Clawson

Mira Lopez

ALL RIGHTS RESERVED


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A Project

by

Greg Clawson

Mira Lopez

Approved by:

_____________________________________, Committee Chair

Turan Gnen, Ph.D.

_____________________________________, Second Reader

Salah Yousif, Ph.D.

____________________

Date


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Student: Greg Clawson

Mira Lopez

I certify that these students have met the requirements for the format contained in 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


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Abstract

of



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 developed 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 necessary 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

program.

_____________________________________, Committee ChairTuran Gnen, Ph.D.

_____________________Date


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DEDICATION

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


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TABLE OF CONTENTS

Page

Dedication...vi

List of Tables .xii

List of Figures...xiii

Chapter

1.INTRODUCTION.......12.LITERATURE SURVEY ....3

2.1.Introduction ....32.2.Support Structure ....32.3.Line Spacing and Transposition.5

2.3.1.Symmetrical Spacing.....62.3.2.Asymmetrical Spacing.......72.3.3.Transposed Line. ..10

2.4.Line Constants..112.5.Conductor Type and Size..122.6.Extra-High Voltage Limiting Factors...16

2.6.1.Corona ..162.6.2.Line Design Based on Corona.....192.6.3.Advantages of Corona.192.6.4.Disadvantages of Corona .....20


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2.6.5.Prevention of Corona...202.6.6.Radio Noise.. 202.6.7.Audible Noise ..21

2.7.Line Modeling...212.8.Line Loadability242.9.Fault Events..252.10.Fault Analysis.262.11.Single Line-to-Ground (SLG) Fault...272.12.Line-to-Line (L-L) Fault.....272.13.Double Line-to-Ground (DLG) Fault.282.14.Three-Phase Fault...292.15.The Per-Unit System...30

3.MATHEMATICAL MODEL....313.1.Introduction...313.2.Geometric Mean Distance (GMD)...313.3.Geometric Mean Radius (GMR)..333.4.Inductance and Inductive Reactance.....343.5.Capacitance and Capacitive Reactance.353.6.Long Transmission Line Model....353.7.Sending-End Voltage and Current....403.8.Power Loss....42


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3.9.Transmission Line Efficiency...443.10.Percent Voltage Regulation....443.11.Surge Impedance Loading (SIL)453.12.Sag and Tension .46

3.12.1.Catenary Method.... 463.12.2.Parabolic Method...50

3.13.Corona Power Loss.513.13.1.Critical Corona Disruptive Voltage...513.13.2.Visual Corona Disruptive Voltage.533.13.3.Corona Power Loss at AC Voltage54

3.14.Method of Symmetrical Components.....553.14.1.Sequence Impedance of Transposed Lines59

3.15.Fault Analysis.....613.16.Per Unit...623.17.Single Line-to-Ground (SLG) Fault...633.18.Line-to-Line (L-L) Fault....663.19.Double Line-to-Ground (DLG) Fault.....693.20.Three-Phase Fault...72

4.APPLICATION OF MATHEMATICAL MODEL....764.1.Introduction..764.2.Design Criteria......77


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4.3.Geometric Mean Distance (GMD).......774.4.Geometric Mean Radius (GMR)......784.5.Inductance and Inductive Reactance....784.6.Capacitance and Capacitive Reactance........794.7.Long Line Characteristics.........804.8.ABCD Constants......814.9.Sending-End Voltage and Current....834.10.Power Loss......844.11.Percent Voltage Regulation.. ..............864.12.Transmission Line Efficiency.....864.13.Surge Impedance Loading (SIL)864.14.Sag and Tension ......87

4.14.1.Catenary Method874.14.2.Parabolic Method...89

4.15.Corona Power Loss ....894.15.1.Critical Corona Disruptive Voltage.......894.15.2.Visual Corona Disruptive Voltage.....914.15.3.Corona Power Loss at AC Voltage....924.15.4.Corona Power Loss forFoul Weather Conditions.94

4.16.Per Unit.......974.17.Fault Analysis Outline....98


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4.18.Procedure Using Symmetrical Components.......994.19.Fault Analysis at the End of Transmission Line.......100

4.19.1.Single Line-to-Ground (SLG) Fault........1014.19.2.Line-to-Line (L-L) Fault..1044.19.3.Double Line-to-Ground (DLG) Fault..1094.19.4.Three Line-to-Ground (3LG) Fault......113

5.CONCLUSIONS.....117Appendix A. Conductor and TowerCharacteristics.......119

Appendix B. Aspen Simulation Model and Analysis.....120

Appendix C. Aspen Fault Analysis Summary....123

Appendix D. MATLABAspen Fault Analysis Results.........131

Appendix E. MATLAB Code.........137

Bibliography182


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LIST OF TABLES

Tables Page

1. Table 2.1 Typical conductor separation.52. Table 2.2 Aluminum vs. copper conductor type..133. Table 3.1 Corona Factor..554. Table 3.2 Power and functions of operatora...565. Table 4.1 Design parameters776. Table 4.2 System data for power system model......997. Table 4.3 Fault analysis of SLG fault at receiving end of line..1048. Table 4.4 Fault analysis of L-L fault at receiving end of line...1089. Table 4.5 Fault analysis of DLG fault at receiving end of line11210.Table 4.6 Fault analysis of 3LG fault at receiving end of line..116


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LIST OF FIGURES

Figures Page

1. Figure 2.1 Three-phase line with symmetrical spacing.....62. Figure 2.2 Cross section of three-phase line with horizontal tower configuration....63. Figure 2.3Three-phase line with asymmetrical spacing84. Figure 2.4A transposed three-phase line.105. Figure 2.5 Equivalent circuit of short transmission line..226. Figure 2.6 Nominal-T circuit of medium transmission line....227. Figure 2.7 Nominal- circuit of medium transmission line.238. Figure 2.8 Segment of 1-phase and neutral connection for long transmission line.239. Figure 2.9 Practical Loadability for Line Length2510.Figure 2.10 General representation for single line-to-ground fault.2711.Figure 2.11 General representation for line-to-line fault.2812.Figure 2.12 General representation of double line-to-ground fault.2913.Figure 2.13 General representation for three-phase fault....3014.Figure 3.1 Bundled conductors configurations....3215.Figure 3.2 Cross section of three-phase horizontal bundled-conductor..3216.Figure 3.3 Segment of 1-phase and neutral connection for long transmission line.3617.Figure 3.4 Parameters of catenary...4818.Figure 3.5 Parameters of parabola...5019.Figure 3.6 Sequence components....56


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20.Figure 3.7 Single line-to-ground faultsequence network connection.6421.Figure 3.8 Line-to-line fault sequence network connection....6622.Figure 3.9 Double line-to-ground fault sequence network connection7023.Figure 3.10 Three-phase fault sequence network connection..7224.Figure 4.1 3H1 wood H-frame type structure......7625.Figure 4.2 One line diagram of power system model......9826.Figure 4.3 Power system model with fault at end of line......10027.Figure 4.4 Equivalent sequence networks.10128.Figure 4.5 Sequence network connection for SLG fault...10129.Figure 4.6 Sequence network connection for L-L fault.10530.Figure 4.7 Sequence network connection for DLG fault...10931.Figure 4.8 Sequence network connection for 3LG fault ...113


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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 end 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 determines 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

regulation, 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 structure 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 compared for selecting the optimal

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 conditions. The system model will interconnect the transmission line to a typical


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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 transmission 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 flow 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) Section 1


Equation Section (Next)


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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 analysis 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 support 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 factors 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,

rainfall, snow, ice, relative 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


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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 structure fabrication are wood, 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 structures 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 less 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 structures without defining specific load support requirements.


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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 analysis: 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 cross-section will swing out further than a conductor of large cross-section. There

are several formulas in use to determine right spacing [8].This is NESC, USA formula

3.6812

LD 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 in cm

Voltage between

conductors

Minimum horizontal

spacing

Minimum vertical

spacing

Up to 8700V 12in 16in

8701 to 50,000V12in, plus 0.4in for each1000 V above 8700V*

40in

Above 50,000V 12in, plus 0.4in for each1000 V above 8700V

40in, plus 0.4in for each1000 V above 50,000V

*This is approximate.

Table 2.1 Typical conductor separation [11].


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Electrical Aspect: When increasing spacing (GMD = geometric mean distance between

the phase conductors in ft) Z1 (positive-sequence impedance) increases and Z0 (zero-

sequence impedance) decreases. If the neutral is placed closer to the phase conductors it

will reduce Z0 but may increase the resistive component of Z0. A small neutral with high

resistance increases the resistance part of Z0 [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:

0a b c

I I I (2.2)

(a) (b)

Figure 2.1 Three-phase line with symmetrical spacing: a) geometry; b) phase inductance [8].

b ca

D12 D23

D31

Figure 2.2 Cross section of three-phase line with horizontal tower configuration.

La

neutral

D

D

D

Ia

IbIcr


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The total flux linkage of phase conductor is:

7

'

1 1 1

2 10 ( )a a b cI ln I ln I lnDr D

(2.3)

b c aI I I (2.4)

7

'

1 12 10 a a aI ln l

rI n

D

(2.5)

72 10'

a aln

r

DI (2.6)

Because of symmetry:

a b c (2.7)

and the three inductances are identical.

The inductance per phase per kilometer length:

0.2s

D mHL ln

D km (2.8)

r= the geometric mean radius, GMR, and is shown by Ds

For a solid round conductor:

1

4s

D r e

(2.9)

Inductance per phase for a three-phase circuit with equilateral spacing is the same as for

one conductor 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


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the phases, the voltage drop due to line inductance will be unbalanced even when the line

currents are balanced. The distances between the phases are denoted byD12,D32 andD13.

The following flux linkages for the three phases are obtained:

D13

D23

D12

a

b

c

Figure 2.3 Three-phase line with asymmetrical spacing [8].

7

12 13

1 1 12 10 ( )

'a a b cI ln I ln I ln

r D D (2.10)

7

12 23

1 1 12 10 ( )

'b a b cI ln I ln I ln

D r D (2.11)

7

13 23

1 1 12 10 (

' )c a b cI ln I ln I ln

D D r (2.12)

In matrix form:

L I (2.13)


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The symmetrical inductance matrix:

12 13

7

12 23

13 23

1 1 1

'

'

'

1 1 12 10

1 1 1

ln ln lnr D D

L ln ln lnD

r

r D

ln ln lnD D

(2.14)

With Ia as a reference for balanced three-phase currents:

2240b a aI I a I (2.15)

120c a aI I a I (2.16)

The operatora:

120aa I (2.17)

2240aa I

(2.18)

The phase inductances are not equal and they contain an imaginary term due to the

mutual inductance:

7

12 13

1 1 1 2 10 ( )

'

aa a b c

a

L I ln I ln I lnI r D D

(2.19)

7

12 23

1 1 1 2 10 (

')

bb a b c

b

L I ln I ln I lnI D r D

(2.20)

7

13 23

1 1 1 2 1'

0

cc a b c

c

L I ln I ln I lnI D D r

(2.21)


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2.3.3 TRANSPOSED LINE

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 line and is shown in Figure 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

b

b

b

c

c

c

1

2

3

D23

D12

S/3

Length of line, S

S/3 S/3

1

2

3

1

2

3

Section IISection I Section III

Figure 2.4 A Transposed 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 (La,Lb andLc) previously

calculated in (2.19) to (2.21). Consequently,

3a b cL L LL

(2.22)


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Since, 2 1 120 1 240 1o oa a (2.23)

The average of a b cL L L

come to be

7

'

12 23 13

2 10 1 1 1 1 3

3L ln ln ln ln

r D D D

(2.24)

1

37 12 23 31( )2 10

'

D D DL ln

r

(2.25)

The inductance per phase per kilometer length:

0.2s

GMD mHL ln

D km (2.26)

Ds is the geometric mean radius, (GMR). For stranded conductorDs is obtained from the

manufactures data. However, for solid conductor:

1

4'sD r r e

(2.27)

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

312 23 31GMD 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 important cause of power loss in a transmission line. The ac

resistance or effective resistance of a conductor is


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2 Lac

PR

I (2.29)

where the real power loss (PL) 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 impedance, 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 AND SIZE

A conductor consists of one or more wires appropriate for carrying electric current. Most

conductors are made of either aluminum or copper.


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Aluminum (Al) Copper (Cu) ObservationMelting Point 660C 1083C

Annealing

starts

Most rapidly

above 100C 100C200C and 325C

Both soften and lose

tensile strength.

Resistance to

corrosionGood Very Good

Al corrodes quicklythrough electricalcontact with Cu or

steel. This galvaniccorrosionaccelerates in the

presence of salt.

OxidationWhen exposed to the

atmosphere

Al thin invisibleoxidation film

protects against

most chemicals,weather and evenacids.

Resistivity Very low

Cu conductor hasequivalent

ampacity of analuminumconductor that istwo AWG sizes

larger. A larger Alcross-sectional areais required to obtainthe same loss as in aCu conductor

Usage

Al is lighter, lessexpensive and so it has

been used for almost all

new overheadinstallations

Cu is widely used as apower conductor, butrarely as an overheadconductor. Cu isheavier and more

expensive than Al

The supply of Al isabundant, 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 all new overhead installations. Aluminum for power


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conductors is alloy 1350, which is 99.5% pure and has a minimum conductivity of 61.0%

IACS [10].

Different types of aluminum conductors are available:

AACall-aluminum conductor

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

overhead conductors [10].

ACSRaluminum conductor, steel reinforced

Because of its highmechanical strength-to-weight ratio, ACSR has equivalent or higher

ampacity for the same size conductor. The steel 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 conductor 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 conductors are bare (no insulating cover) [8].

AAACall-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 of excessive corrosion [10].


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ACARaluminum conductor, alloy reinforced

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 they 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, so 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 106 in2. In the metric system, these figures are a

diameter of 0.0254 mm and an area of 506.71 106 mm2

[11]. Wire sizes are given in

gauge numbers, 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, conductors are generally given in circular mils (cmil) or in

thousands of circular mils (cmil 103); stranded conductors for distribution purposes

usually range from a minimum of no. 6 to a maximum of 1,000,000 cmil (or 1000 cmil

103) and may consist of two classes of strandings. Gauge numbers may be determined

from the formula:

0.3249

1.123nDiameter in (2.30)


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105,500Cross sectionalarea

1,261ncmil (2.31)

where n is the gauge number (no. 0 = 0; no. 00 =1; no. 000 =2; no.0000 =3) [11].

2.6 EXTRA HIGH VOLTAGE LIMITING FACTORS

Limiting factors for extra high voltage are:

a) Coronab)

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 air dielectrics when the

electrical field reaches the critical surface gradient is known as corona. Corona effect

causes significant power loss and a high frequency current. Corona comes in different

forms: visual corona as violet or blue glows, audible corona 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 TVI) 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) Electrical2) Atmospheric


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3) Conductor1) 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 disruptivevoltage. Corona and line voltage are directly proportional.

c) Conductor electrical field: Conductor electrical field depends on the voltage andconductor configuration i.e., vertical, horizontal, delta 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, temperature and pressure have aneffect 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 other weather conditions. The 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:


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a) Radius or size of the conductor: The larger the size of the conductor (radius) thelarger 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 cP V V (2.32)

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

Vc = disruptive (inception) critical voltage kV (rms)

b) Spacing between conductors: The larger the spacing between conductors thesmaller the power loss. This can be observed from power loss approximation:

lossr

PD

(2.33)

r = conductor radius

D = distance (spacing) between conductors

c) Number of conductors / Phases: In case of a single conductor per phase for highervoltages there is a significant corona loss. In order to reduce corona loss two 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 orprofiles. 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.


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e) Surface conditions of the conductors:The disruptive voltage is higher for smoothcylindrical 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 conductorfrom the ground. Corona loss is greater for smaller clearances.

g) Heating of the conductor by load current:Load currentcauses heating of theconductor 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 line in

fair weather condition corona loss of 1kW per three-phase mile and foul weather loss of

20 kW per three-phase mile is acceptable [7].

2.6.3 ADVANTAGES OF CORONA

Corona reduces the magnitude of high voltage waves due to lightning by partially

dissipating as a corona loss. In this case it has a purpose of a safety valve.


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2.6.4 DISADVANTAGES OF CORONA

a) Loss of powerb) The effective capacitance of the conductor is increased which increases the

flow of charging current.

c) Due to electromagnetic and electrostatic induction field corona interferes withthe 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 conductorb) increasing spacing of the conductorsc) selecting proper type of the conductord) 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).

Radio interference(RI): It affects amplitude modulated (AM) radio waves within the

standard broadcast band (0.5 to 1.6 MHz). Frequency modulated (FM) waves are less

affected.


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Television interference (TVI): 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, requires a sufficiently accurate model of a line. How a transmission line is

modeled depends on the line length. There are three classes of line lengths. For line

lengths that are classified as short, up to 50 miles, the model is simplified because shunt

capacitance 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 can be a

simple, lumped impedance value, as shown in Figure 2.5 [1].


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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 shunt 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

network, as shown in Figures 2.6 and 2.7.

Figure 2.6 Nominal-T circuit of medium transmission line [1].

I S Z = R + jX L I R

+

V S-

+

V R-

a a

N N

S e n d i n g

e n d

( s o u r c e )

R e c e i v i n g

e n 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 Ya a

N N

V Y


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Figure 2.7 Nominal- 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 defined 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 connection for long transmission line [5].

I S R + jX L I R

+

V S-

+

V R-

C / 2 G / 2

IC 1

a a

N N

C / 2 G / 2

I C 2

I I


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This model accounts for the changes 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 this project will only cover the mathematical model used for

designing long line lengths. For details of the long transmission line mathematical 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 purely resistive load equal to its 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 = 90, for the terminal voltages. However, for reasons

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 practical 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.


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Figure 2.9 Practical 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 stability limit.

2.9 FAULT EVENTS

A fault event in an 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, the network can experience either an open circuit or a short circuit. Short-circuit


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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 typically faults are simulated at different points throughout the transmission

system. It is crucial to have precise analysis of the designed system to prevent faults

interruption. In general the three phase faults can be classified as:

1. Shunt faults (short circuits)1.1.Unsymmetrical faults (Unbalanced)

1.1.1. Single line-to-ground (SLG) fault1.1.2. Line-to-line (L-L) fault1.1.3. 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


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2.11 SINGLE LINE-TO-GROUND (SLG) FAULT

Seventy percent of all transmission line faults are attributable 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 phase conductor, generally defined as phase a, is

connected to ground by an impedance value Zf. Figure 2.10 shows the general

representation of an SLG fault.

Zf

a

b

c

n

F

+

Vaf-

Ibf= 0af Icf= 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., ionization of air,


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flashover, or bad insulation. Figure 2.11 shows the general representation of an LL fault

[1].

a

b

c

F

Ibfaf=0 Icf= -Ibf

Zf

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

Zfand a common ground impedance value Zg.


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Figure 2.12 shows the general representation of a DLG fault.

Zf

a

b

c

n

F

Ibfaf= 0 Icf

Zf

Zg

N

Ibf+Icf

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

2.14 THREE-PHASE FAULT

A three-phase (3) 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 3

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].


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Figure 2.13 shows the general representation of a 3 fault.

Zf

a

b

c

n

F

Ibfaf Icf

Zf

ZgN

Iaf+Ibf+ Icf= 3Ia0

Zf

Figure 2.13 General representation for three-phase (3) 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.


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Chapter 3

MATHEMATICAL MODEL


3.1 INTRODUCTION

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 structures 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 DISTANCE (GMD)

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


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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 conductors configurations: (a) two-conductor bundle; (b) three-conductorbundle; (c) four-conductor bundle [1].

b'b

d

c'c

d

a'a

d

D12 D23

D31

Figure 3.2 Cross section of bundled-conductor three-phase line with horizontal towerconfiguration [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.Deq, the cube root of the product of the three-phase

spacings, is the geometric mean distance (GMD) between phases:

312 23 31 eq mD D D D D ft (3.1)


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3.3 GEOMETRIC MEAN RADIUS (GMR)

Geometric Mean Radius (GMR) of bundled conductors for

Two-conductor bundle:

bS SD D d ft (3.2)

Three-conductor bundle:

23 bS SD D d ft (3.3)

Four-conductor bundle:

34 bS SD D d ft (3.4)

where:

= GMR of subconductors

distance between two subconductors

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

forDeqis obtained by using the distances between bundle centers. If the conductors are

stranded and the bundle spacing dis large compared to the conductor outside radius, 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:

bSCD r d ft (3.5)

Three-conductor bundle:

3 2 bSC

D r d ft (3.6)


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Four-conductor bundle:

34

1.09

b

SCD r d ft

(3.7)

where:

= outside radius of subconductors

= distance between two subconductors.

3.4 INDUCTANCE AND INDUCTIVE REACTANCE

For three-phase transmission lines that are completely transposed, Equation (3.1) can be

used to find the equivalent equilateral spacing for the line. Thus, the average inductance

per phase is

72 10 ln eq

a

s

D HL

D m

(3.8)

or

100.7411 log eq

a

s

D mHL

D mi

(3.9)

and the inductive reactance is found by

2L a

X f L per phase (3.10)

or

0.1213 ln eq

L

s

DX

D mi

per phase (3.11)


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3.5 CAPACITANCE AND CAPACITIVE REACTANCE

The average line-to-neutral capacitance per phase is

10

0.0388 F toneutral

log

N

eq

CD mi

r

(3.12)

where

1

3 eq m ab bc caD D D D D ft (3.13)

radius of cylindrical conductor in feet.

The capacitive reactance is calculated by

1

2C

N

Xf C

(3.14)

or

100.06836 log .eq

C

DX 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 point along the line [1]. Figure 3.3 depicts a

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


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Figure 3.3 Segment of one phase and neutral connection for long transmission line. [5]

The following derivation is given by Saadat [5]. The series impedance per unit length is

z, and the shunt admittance per phase is y, wherez = r + jL andy = g + jC. Consider

one small segment of line x 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 Kirchhoffs voltage law

( )V x x V x z xI x (3.16)

or

( )( )

V x x V xzI x

x

(3.17)


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Taking the limit as , we have

( )( )

dV x

zI xdx (3.18)

Also, from Kirchhoffs current law

I x x I x y xV x x (3.19)

or

( )( )

I x x I xyV x x

x

(3.20)

Taking the limit as , we have

( )( )

dI xyV x

dx (3.21)

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

2

2

( ) ( )( )

d x dI xz z x

dx

VyV

dx (3.22)

Let

2zy (3.23)

The following second-order differential equation will result.

2

2

2

( )0

d V xV x

dx (3.24)

The solution of the above equation is

1 2x x

V x Ae A e (3.25)

where , known as thepropagation constant, is a complex expression given by (3.23) or

( ) ( )j zy r j L g j C (3.26)


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The real part is known as the attenuation constant, and the imaginary component is

known as thephase constant. is measured in radian per unit length. From (3.18), the

current is

1 2 1 21 ( ) x x x xdV x yI x Ae A e Ae A ez dx z z

(3.27)

or

1 21 x x

C

I x A e A eZ

(3.28)

whereZc is known as the characteristic impedance, given by

C

zZ

y (3.29)

To find the constants and, we note that when , () , and () .

From (3.25) and (3.28) these constants are found to be

12

R C RV Z IA (3.30)

22

R C RV Z IA

(3.31)

Upon substitution in (3.25) and (3.28), the general expressions for voltage and current

along a long transmission line become

2 2x xR C R R C R

V Z I V Z I V x e e

(3.32)

2 2

R RR R

x xC C

V VI I

Z ZI x e e

(3.33)


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

2 2

x x x x

R C R

e e e eV x V Z I

(3.34)

1

2 2

x x x x

R R

C

e e e eI x V I

Z

(3.35)

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

follows:

cosh sinhR C RV x x V Z x I

(3.36)

1

sinh coshR RC

I x x V x IZ

(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 sinhS R C RV l V Z l I (3.38)

1

sinh coshS R RC

I l V l IZ

(3.39)

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

S R

S R

V VA B

I IC D

(3.40)

where

cosh cosh coshA l YZ (3.41)

sinh sinh sinhC CZ

B Z l YZ ZY

(3.42)


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sinh sinh sinhC CY

C Y l YZ Y Z

(3.43)

cosh cosh coshD A l YZ (3.44)

where

LZ r jx l (3.45)

is total line series impedance per phase

SY g jb l (3.46)

is total line shunt admittance per phase.

Note that A D (3.47)

and 1AD 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 design may still be adjusted to match preferred input

parameters. After the ABCDconstants are determined, as shown in the previous section,

the following steps can be used to find the sending-end voltage and current.


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Using the receiving-end design requirements for load power, voltage, and power factor,

the receiving-end line-to-neutral voltage and current magnitude are determined by the

following equations:

( )

3

R L L

R L N

VV (3.50)

and

( )3

R

R L L

S

IV

(3.51)

where:

() receiving-end line-to-neutral voltage (kV),

magnitude of receiving-end line current (A).

The receiving-end current phasor can be found by

(cos sin )R R R Rj I I (3.52)

where:

angle difference between () and

and can be found by taking the inverse cosine of the power factor.

Using the calculated values for ABCD constants and receiving-end voltage and current,

we can use Equation (3.40) to determine the corresponding sending-end voltage and

current.

The sending-end voltage and current can be equated by

( )( ) ( )S L N RR L N V A V B I (3.53)


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and

( ) ( )

S RR L N

I C V D I (3.54)

where:

sending-end line current (A),

() sending-end line-to-neutral voltage (kV).

The sending-end line-to-line voltage is

( ) 3 1 30S L L S L N V V (3.55)

where:

() sending-end line-to-line voltage (kV).

Note that an additional is added to the angle since the line-to-line voltage is

ahead of its line-to-neutral voltage.

3.8 POWER LOSS

Typically, referring to power loss in a transmission line means the difference in real

power between the sending- and receiving- ends. To calculate the power loss, the first

step is to determine the power factor at each end. For the receiving-end, the power factor

is normally specified per design criteria. For the sending-end, the power factor is found

by determining the angle S between the sending-end current and voltage phasors. The

expression for sending-end power factor is

cos( os) c

SSS IV L N s

pf

(3.56)


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where:

sending-end power factor,

() = angle of sending-end line-to-neutral voltage phasor,

angle of sending-end current phasor,

angle difference between () and .

Then, using the value of, the equation for calculating real power at the sending-end is

( )3 3 cosS L L S S SP V I (3.57)

where:

() sending-end real power in the line (MW).

A similar equation for calculating the receiving-end real power is

( )3 3 cosR L L R RRP V I (3.58)

where:

() receiving-end real power in the line (MW).

Using the calculated values from the above equations, real power loss in the line is found

by

(3 ) (3 ) (3 )L S RP P P (3.59)

where:

() total real power loss in the line (MW).

The majority of power loss in a transmission line is a result of real power loss due to the

resistance of the line. A good design will minimize the total real power loss in the line.


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3.9 TRANSMISSION LINE EFFICIENCY

Performance of transmission lines is determined by efficiency and regulation of lines.

Transmission line efficiency is:

100R

S

P

P (3.60)

where:

= transmission line efficiency

= receiving-end power

= sending end power

% 100

Powerdeliverd at receiving endtransmissionlineefficiency

Powersent fromthe sendingend (3.61)

% 100R

S

Ptransmissionlineefficiency

P (3.62)

The end of the line where source of supply is connected is called the sending end and

where load is connected is called the receiving end [1].

3.10 PERCENT VOLTAGE REGULATION

Voltage regulation of the line is a measure of the decrease in receiving-end voltage as

line current increases. In mathematical terms, percent voltage regulation is defined as the

percent change in receiving-end voltage from the no-load to the full-load condition at a

specified power factor with sending-end voltage VS held constant, that is,

, ,

,

- | |

| |

R NL R FL

R FL

V VPercentVR 100

V (3.63)


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where:

|VR,NL|=magnitude of receiving-end voltage at no-load,

|VR,FL|=magnitude of receiving-end voltage at full-load with constant |Vs|,

|VS|=magnitude of sending-end phase (line-to-neutral) voltage at no load.

3.11 SURGE IMPEDANCE LOADING (SIL)

In power system analysis of high frequencies or surges caused by lightning, losses are

typically ignored and surge impedance becomes important. A line is lossless when its

series resistance and shunt conductance are zero [6]. The surge impedance of a lossless

line, also known as characteristic impedance, is a function of line inductance and

capacitance, and can be expressed as

CL

ZC

(3.64)

or

C C LZ X X (3.65)

where:

shunt capacitive reactance (),

series inductive reactance (),

characteristic impedance ().

Surge impedance loading (SIL), a measure of the amount of power the line delivers to a

purely resistive load equal to its surge impedance [6], is found for a three-phase line by

the following equation:


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2

( )| |Lr L

c

kVSIL MW

Z

(3.66)

where: SIL = surge impedance loading (MW).

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. SIL, or natural loading, is a

function of the line-to-line voltage, line inductance and line capacitance. Since the

characteristic impedance is based on the ratio of inductance and capacitance, SIL is

independent of line length. The relationship between SIL and voltage explains why an

extra-high voltage line has more power transfer capability than lower voltage lines.

3.12 SAG AND TENSION

3.12.1 CATENARY METHOD

Sag-tension calculations predict the behavior of conductors based on recommended

tension limits under varying loading conditions. These tension limits specify certain

percentages of the conductors rated breaking strength that are not to be exceeded upon

installation or during the life of the line. These conditions, along with the elastic and

permanent elongation properties of the conductor, provide the basis for defining the

amount of resulting sag during installation and long-term operation of the line.

Accurately determined initial sag limits are essential in the line design process. Final sags

and tensions depend on initial installed sags and tensions and on proper handling during

installation. The final sag shape of conductors is used to select support point heights and

span lengths so that the minimum clearances will be maintained over the life of the line.


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If the conductor is damaged or the initial sags are incorrect, the line clearances may be

violated or the conductor may break during heavy ice or wind loadings [1].

( )max

T w c d (3.67)

( )max

T w c d (3.68)

minT w c (3.69)

H w c (3.70)

Hcw

(3.71)

minT H (3.72)

T= the tension of the conductor at any point P in the direction of the curve

w = the weight of the conductor per unit length

H= the tension at origin 0

c = catenary constant

s = the length of the curve between points 0 and P

v = that the weight of the portions is ws

L = horizontal distance.


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d

y

AB

L

0H

2

ls

2

ls

Hc

w

V ws

0' (Directrix) x

y

T=wyTy=ws

Tx=wc

Figure 3.4 Parameters of catenary [1].

An increase in the catenary constant, having the units of length, causes the catenary curve

to become shallower and the sag to decrease. Although it varies with conductor


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temperature, ice and wind loading, and time, the catenary constant typically has a value

in the range of several thousand feet for most transmission-line catenaries.

For equilibrium

x

T H (3.73)

yT w s (3.74)

Tx = the horizontal component

Ty= the vertical component.

The total tension in the conductor at any pointx:

w xT H cos

H

(3.75)

The total tension in the conductor at the support:

2

w LT H cos

H

(3.76)

The sag or deflection of the conductor for a span of length L between supports on the

same level:

cosh 12

H w Ld

w H

(3.77)


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3.12.2 PARABOLIC METHOD

The conductor curve can be observed as a parabola for short spans with small sags.

x

y

A B

L

0 Hwx

d

Ty

H

T Ty

Tx

P

Figure 3.5 Parameters of parabola [1].

The following assumptions can be taken into consideration when using parabolic method:

1. The tension is considered uniform throughout the span.2. The change in length of the conductor due to stretch or temperature is the same as

the change of the length due to the horizontal distance between the towers [1].

Approximate value of tension by using parabolic method can be calculated as

2

8

w LT

d

(3.78)


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or

2

8

w Ld T

(3.79)

when

1

2x L (3.80)

y d . (3.81)

3.13 CORONA POWER LOSS

3.13.1 CRITICAL CORONA DISRUPTIVE VOLTAGE

The maximum stress on the surface of the conductor is given by:

ln

LNmax

V kVE

D cmm r

r

(3.82)

VLN = the phase or line-to-neutral voltage in kV

D = is equivalent spacing in cm

r = radius of the conductor in cm

mc = surface irregularity factor ( )

mc = 1 for smooth, solid, polished round conductor

mc = 0.930.98 for roughened or weathered conductor

mc = 0.800.87 for up to seven strands conductor

mc = approx. 0.90 for large conductor with more than seven strands [12]

Mean voltage gradient can be calculated from:


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ln 3

LNmean

V kVE

D cmm r

r

(3.83)

The air density correction factor is defined as:

3.9211

273

p

t

(3.84)

where:

p = the barometric pressure in cm Hg

t = temperature inC

The critical disruptive voltage (corona inception voltage) Vc is voltage at which complete

disruption of dielectric occurs. The dielectric stress is 30 kV/cm peak or 21.1 rms at

NTP i.e., 25C and 76 mmHg. Vc is minimum conductor voltage with respect to earth at

which the corona is expected to start. At Vc corona is not visible [7].

30 c cD

V m r ln kV peak r

(3.85)

21.1 c cD

V m r ln kV rmsr

(3.86)

Vc = the critical disruptive voltage in kV

The critical disruptive voltage Vc line-to-line is

( ) ( )3

c L L c rmsV V kV

(3.87)


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3.13.2 VISUAL CORONA DISRUPTIVE VOLTAGE

In order to observe corona visually the inception voltage has to exceed the critical

disruptive voltage Vc. The visual critical voltage Vv is given by:

0.301

30 1 vD

V m r ln kV peak rr

(3.88)

0.301

21.1 1 vD

V m r ln kV rmsrr

(3.89)

Vv = the visual critical voltage in kV

D = equivalent spacing of conductors in cm

r = radius of the conductor in cm [7]

mv = surface irregularity factor ( )

m = 1 for smooth, solid, polished round conductor

For local and general visual corona:

m = 0.930.98 for roughened or weathered conductor

For local visual corona:

m = 0.700.75 for weathered stranded conductor

For general visual corona:

m = 0.800.85 for weathered stranded conductor


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3.13.3 CORONA POWER LOSS AT AC VOLTAGE

For AC transmission lines empirical equations are used to determine corona loss.

According to Peek corona power loss can be determined from:

2 5241 25 10 / LN c

r kWP f V V phase peak

D km

(3.90)

P = corona loss in kW/km/phase

= density correction factor

VLN = the phase or line-to-neutral voltage in kV

Vc = the critical disruptive voltage in kV

f = frequency

r = radius of the conductor in cm

D = equivalent spacing of conductors in cm.

It is desirable to design transmission line with corona loss between 0.10 and 0.21

kW/km/phase for fair weather conditions. For lower loss range i.e., when

V 1.8LN

cV (3.91)

Peeks formula is not accurate [7].

According to Petersons corona power loss formula:

2

5

10

2.1 10 /cV kW

P f F phaseD km

logr

(3.92)

F = corona loss function


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VLN / Vc 1.0 1.2 1.4 1.5 1.6 1.8 2.0

F 0.037 0.082 0.3 0.9 2.2 4.95 7.0

Table 3.1 Corona factor [7].

Above stated formulas are used for fair weather conditions. For wet weather conditions

critical disruptive voltage is approximately 0.80 of the fair weather calculated value. The

calculated disruptive critical voltage for three-phase horizontal conductor configuration

can be determined as:

3 0.96c c fair V V (3.93)

for the middle conductor and

3 1.06

c c fair V V

(3.94)

for the two outer conductors [1].

3.14 METHOD OF SYMMETRICAL COMPONENTS

According to Charles Fortescue, a set of three-phase voltages are resolved into the

following three sets of sequence components:

1. Zero-sequence components: consisting of three phasors with equal magnitudes and

with zero phase displacement

2. Positive-sequence components, consisting of three phasors with equal magnitudes,

120 phase displacement

3. Negative-sequence components, consisting of three phasors with equal magnitudes,

120 phase displacement[1].


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Vb0 Vc0 = V0Va0

Vc2

Vb2 Va2 = V2

Va1 = V1

Vb1

Vc1

(a) (b) (c)

Figure 3.6 Sequence components: ( a) zero ( b) positive ( c) negative [8 ]

0 1 2

a a a a V V V V (3.95)

0 1 2

b b b b V V V V (3.96)

0 1 2

c c c c V V V V (3.97)

Operatora is a complex number with unit magnitude and a 120 phase angle. When any

phasor is multiplied by a, that phasor rotates by 120 (counterclockwise).

A list of some common powers, functions and identities involving a:

Power or Function In Polar Form In Rectangular Form

a 1120 -0.5+j0.866

a2 1240=1-120 -0.5-j0.866

a3

1360=10 1.0+j0.0

a4 1120 -0.5+j0.866

1+a= -a2 160 0.5+j0.866

1- a -30 1.5-j0.8661+ a

2= -a 1-60 0.5-j0.866

1- a2 30 1.5+j0.866

a -1 150 -1.5+j0.866a+ a2 1180 -1.0+j0.0

a - a2 90 0.0+j1.732a

2- a -90 0.0-j1.732

a2- 1 -150 -1.5-j0.866

1 + a+ a2 0 0.0+j0.0ja 1210 -0.884+j0.468

Table 3.2 Power and functions of operatora[1].


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1 120 a (3.98)

1 31 1202 2

j a (3.99)

Similarly, when any phasor is multiplied by

2 1 120 1 240 a (3.100)

the phasor rotates by 240.

The phase voltages in terms of the sequence voltages i.e.synthesis equations:

0 1 2 0a a a a V V V V (3.101)

2

0 1 2 0b a a a V a V aV V (3.102)

2

0 1 2 0c a a a V aV a V V (3.103)

The sequence voltages in terms of phase voltages i.e. analysis equations:

01

3a a b c V V V V (3.104)

211

3a a b c

V V aV aV (3.105)

221

3a a b c V V aV aV (3.106)

In matrix form the phase voltages can be expressed as

0

2

1

2

2

1 1 1

1

1

a a

b a

c a

V V

V a a V

V a a V

(3.107)


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and the sequence voltages can be expressed as

02

1

2

2

1 1 11 13

1

a a

a b

a c

V V

V a a V

V a a V

(3.108)

or

012abc V A V (3.109)

1

012 abc

V A V (3.110)

where

22

1 1 1

1

1

A a a

a a

(3.111)

1 2

2

1 1 11

13

1

A a a

a a

(3.112)

Similarly, the phase currents in matrix form can be expressed as

0

2

1

2

2

1 1 1

1

1

a a

b a

c a

I I

I a a I

I a a I

(3.113)

and the sequence currents can be expressed as

0

2

1

2

2

1 1 11

13

1

a a

a b

a c

I I

I a a I

I a a I

(3.114)


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or

012abcI A I

(3.115)

1

012 abc

I A I (3.116)

3.14.1 SEQUENCE IMPEDANCES OF TRANSPOSED LINES

In order to attain equal mutual impedances the line should be transposed or conductors

should have equilateral spacings.

Hence, for the equal mutual impedances

ab bc ca m Z Z Z Z (3.117)

In case when the self-impedances of conductors are equal to each other

aa bb cc s Z Z Z Z . (3.118)

Therefore,

s m m

abc m s m

m m s

Z Z Z

Z Z Z Z

Z Z Z

(3.119)

where,

0.1213ln es a es

Dr r j l

D

Z (3.120)

and

0.1213ln em eeq

Dr j l

D

Z . (3.121)

ra = resistance of a single conductora


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re= resistance of Carsons equivalent earth return conductor which is a function of

frequency

31.588 10er f

. (3.122)

At 60 Hz,

0.09528e

r

(3.123)

At 60 Hz frequency and for 100

average earth resistivity

2788.55eD ft. (3.124)

The equilateral spacings of the conductors can be calculated as

3eq m ab bc caD D D D D (3.125)

TheDs is geometric mean radius (GMR) of the phase conductor.

The sequence impedance matrix of a transposed transmission line can be expressed as

012

2 0 0

0 0

0 0

s m

s m

s m

Z Z

Z Z Z

Z Z

(3.126)

where, by definition,

Z0 is zero-sequence impedance at 60Hz

0 00 2s mZ Z =Z Z (3.127)

3

0 23 0.1213ln ea e

s eq

Dr r j l

D D

Z , (3.128)


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Z1 is positive-sequence impedance at 60Hz

1 11 s m-Z Z =Z Z (3.129)

1 0.1213lneq

a

s

Dr j l

D

Z , (3.130)

Z2 is negative-sequence impedance at 60Hz

2 22 s m-Z Z =Z Z (3.131)

2 0.1213ln

eq

a

s

D

r j lD

Z . (3.132)

Therefore, the sequence impedance matrix of a transposed transmission line can be

expressed as

0

012 1

2

0 0

0

0 0

Z

Z Z

Z

(3.133)

3.15 FAULT ANALYSIS

Three-phase faults can be balanced (i.e., symmetrical) or unbalanced (i.e.,

unsymmetrical). The unbalanced faults are more common. In order to resolve an

unbalanced system the method of symmetrical components can be applied by converting

the system into positive, negative and the zero-sequence fictitious networks. After

defining positive, negative and zero-sequence currents for specific fault phase currents,

sequence and phase voltages can calculated.


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3.16 PER UNIT

Power, current, voltage, and impedance are often expressed in per-unit or percent of

specified base values.

The per-unit values are calculated as:

-

actualquantityper unit quantity

basevalueof quantity (3.134)

where actual quantity is the value of the quantity in the actual units and the base value

has the same units as the actual quantity, forcing the per-unit quantityto be

dimensionless. The actual value may be complex but the base value is always a real

number. Consequently, theangle of the per-unit value is the same as the angle of the

actual value [8] .

In a given power system two independent base values can be arbitrarily selected at one

point. Typically the base complex power Sbase1 and the base voltage VbaseLN are chosen

for either a single-phase circuit or for one phase of a three-phase circuit. In order to

preserve electrical laws in the per-unit system, the following equations must be used for

other base values:

1 1 1base base baseP Q S (3.135)

11

basebase

baseLN

SI

V

(3.136)

2

1

baseLN baseLNbase base basebase base

V VR X

I SZ

(3.137)

1 base base base

base

Y G BZ

(3.138)


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2

basebasebase

kVZ

MVA (3.139)

( )

3

basebase

base LL

MVAI

kV (3.140)

The subscriptsLNand 1represent line-to-neutraland per-phase respectively, for

three-phase circuits. Equations (2.35) and (2.36) are also effective for single-phase

circuits by omitting the subscripts.

By agreement, the following two rules for base quantities are assumed:

1) The value of Sbase1 is the same for the entire power system2) The ratio of the voltage bases on either side of a transformer is selected to be the

same as the ratio of the transformer voltage ratings.

As a result per-unit impedance remains unchanged when referred from one side of a

transformer to the other [8].

3.17 SINGLE LINE-TO-GROUND (SLG) FAULT

An SLG fault generally occurs when one phase conductor either falls to the ground or

makes contact with the neutral wire. Figure 3.7 depicts the typical representation of an

SLG fault at a fault point F with a fault impedance Zf. It is customary to show the fault

occurring on phase a. If the fault actually takes place on another phase, the phases of the

system can simply be relabeled in the appropriate sequence.


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(a)

(b)

Figure 3.7Single line-to-ground fault: (a) general representation; (b) sequence networkconnection [1]

From inspection of Figure 3.7a, the currents for phases b and c are

0bf cf

I I (3.141)

Zf

a

b

c

n

F

+

Vaf-

Ibf= 0af Icf= 0

F0 Z0 N0

+ VA0 -

F1 Z1 N1

+ VA1 -

F2 Z2 N2

+ VA2 -

1.0+ -

0o

3ZfIa1

Ia1a0 Ia2

Zero-sequence

network

Positive-sequence

network

Negative-sequence

network


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Substituting values into Equation (3.114), the symmetrical components for

the currents are given as

0

2

1

2

2

1 1 11

1 03

1 0

a af

a

a

a a

a a

I I

I

I

(3.142)

Using the above equation, the sequence currents for phase a are

0 1 2

1

3a a a af I I I I (3.143)

and can be rewritten as

0 1 23 3 3 af a a a I I I I (3.144)

By inspection of Figure 3.7b, the zero-, positive-, and negative-sequence currents are

equal and can be determined by

0 1 2

0 1 2

1.0 0

3a a a

f

I I I

Z Z Z Z(3.145)

Substituting (3.145) into Equation (3.144) gives

0 1 2

1.0 03

3af

f

IZ Z Z Z

(3.146)

With the sequence current values, Equation (3.155) can be used to find the sequence

voltages, and then Equation (3.107) can be used to find the phase voltages.


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3.18 LINE-TO-LINE (L-L) FAULT

A line-to-line (L-L) fault occurs when two conductors are short-circuited. Figure 3.8a

represents the characteristic representation of a L-L fault at a fault point F with a fault

impedance Zf. Figure 3.8b indicates the interconnection of resulting sequence networks.

It is presumed that L-L fault is between phases b and c.

(a)

(b)

Figure 3.8Line-to-line fault: (a) general representation; (b) sequence network connection [1].

a

b

c

F

Ibfaf=0 Icf= -Ibf

Zf

F0

Z0

N0

+

VA0 = 0-

F1

Z1

N1

+

VA1-

F2

Z2

N2

+

VA2-

+

1.0-0o

Ia1a0=0 Ia2

Zf

master of science in eee project by greg clawson and mira lopez spring 2012.pdf

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