MULTICONDUCTOR METHOD IN THE STUDY OF UNBALANCES IN EHV L1NES

L. Y -M. Yu, B. Sc. (Utopia), M. Sc. (Manitoba)

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THE MULTICONDUCTOR METHOD IN THE STUDY OF UNBALANCES IN EHV TRANSMISSION LlNES

by

luke Y.M. Yu, B.Sc. (Utopia),M.Sc. (Manitoba)

Department of Electricol Engineering

McGili University,

Montrea l, Quebec •

THE MULTICONDUCTOR METHOD IN THE STUDY OF UNBALANCES

IN EHV TRANSMISSION UNES

by

Luke Y.M. Yu, B.Sc. (Utopia), M.Sc. (Manitoba)

A thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of

Doctor of Ph i 1 osophy •

Department of Electrical Engineering,

McGi11 University,

Montreal, Canada.

November, 1969~

(ê) Luke Y.:;. Yu 1970

• ABSTRACT

Asymmetry is a problem of growing importance in modern transmission

lines. The trend towards higher voltage levels, greater number of subconductors

and the use of long untransposed 1 ines are contributing factors.

A general method for studying the voltages and currents in the subconductors

of a transmission line is developed and is called the Multiconductor Method. This

method is used to study unbalances due to asymmetry and to fault conditions, including

faults between subconductors of the same bundle as weil as faults between phases. In

( ::. ."

addition, the method is applied to an investigation of the electrical effect of spacers

on the current and voltage distribution. The advantage of the Mu\ ticonductor Method ,

lies in its ability to take into account the distributed parameters of the line and any

additional circuit components. Its appl ication is possible irrespective of the number

of subconductors per phase, the-nurTiber of phases or the number and type of faults.

While the Multiconductor Method is more general and allows the study of

cases which cannot be easily treated by the traditional method of symmetrical com-

ponents, a comparison is made of the two methods for the field where the latter is

normally applicable. The theory is illustrated by numerical examples of various bundle

configuration.

ACKNDWLEDGEMENTS

The author wishes to express his deep appreciation to Professor R. P. Comsa

for his guidance throughout this research.

To Dr. G. W. Farnell, Chairman of the Department, the author is sincerely

grateful for his kindness in reading the final manuscript and for many invaluable in­

structions and he Ipful suggestions.

The ciuthor's thanks are extended to his colleagues for many helpful

discussions, especially Mr. B.T. Doi ; to Mr. Alain Lefèvre for his assistance in

proof-reading the manuscript and to Mrs. P. Hyland for her excellent typing.

To my wife, Pacita, and our beloved son, Henry, Jr., 1 owe mu ch appre-

ciotion for their understanding and encouragement.

The author 0150 wishes to thank the National Research Council of Canada

for the financial assistance.

ii

.-iii

TABLE OF CONTENTS

Page

ABSTRACT

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS Hi

CHAPTER INTRODUCTION

CHAPTER " ANALYSIS OF BUNDLED CONDUCTOR TRANSMISSION UNES 5

2.1 Asymmetry Between Subconductors and Sub-conductor Parameters 5

2.2 Analysis 7 2.3 Practical Example and Computed Results 12 2.4 Discussions 21

CHAPTER III FAULT CONDITIONS WITHIN THE BUNDLE OF TRANSMISSION UNES 32

3.1 Analysis of the Faults 32 3.2 Practical Example and Computed Results 40 3.3 Discussions 44

CHAPTER IV THE STUDY OF SPACERS 50

4.1 Nature of the Problem 50 4.2 Chain Matrix 53 4.3 Practical E>5~mple and Computed Results 61 4.4 Discussions 67

CHAPTER V ANAL YSIS OF THREE PHASE TRANSMISSION UNES 70

5.1 Asymmetry Between Phases 70 5.2 GMR of Bundled Conductors 72 5.3 Computed Results and Discussions 74

CHAPTER VI FAULTS IN THREE PHASE TRANSMISSION LlNES 83

6.1 Symmetrical Compone nt Method 83 6.2 Multiconductor Analysis 86 6.3 Comparison of the Computed Results Obtained by the

Multiconductor Method and by the Symmetrical Com-ponent Method 92

6.4 Discussions 100

·. iv

Page

CHAPTER VII ANALYSIS OF FAULTS ON TRANSMISSION L1NES WITH FAULTED CONDUCTORS UNINTERRUPTED 103

7.1 Introduction 103 7.2 Analysis and Chain Matrix 104 7.3 Computed Results and Discussions 109

. CHAPTER VIII FAULT CURRENT CALCULATION WITH GROUND

WIRES INCLUDED 112

8.1 Analysis 112 8.2 Numerical Example 114

CHAPTER IX COMPUTER METHOD AND GENERAL PROCEDURE OF MANIPULA TING THE MULTICONDUCTOR METHOD BY COMPUTER PROGRAMS 119

9.1 General Procedure of Manipulation 119 9.2 Sorne Experiences in Using a Digital Computer 121

CHAPTER X CONCLUSIONS 125

Claim of Originality 126

APPENDIX A GENERALIZED TELEGRAPH EQUATIONS 128

APPENDIX B DETERMINATION OF L1NE CONSTRAINTS FOR A TRANSMISSION L1NE 135

APPENDIX C FORTRAN PROGRAM LISTINGS 140

APPENDIX D L1NE CONSTANTS OF THREE PHASE BUNDLED CONDUCTQR lRANSMISSION L1NES 168

APPENDIX E FAULT CALCULATIONS 174

BIBLIOGRAPHY 181

LIST OF PUBLICATIONS 186

•

o

CHAPTER 1

1 NTRO DUCT ION

The use of bundled conductors for overhead transmission lines operating

at voltages greater than 220 KV has greatly increased in recent years 1 due to numerous

2 advantages over the single conductor per phase configuration.

Asymmetry of the subconductors (due to their physical positions relative

to each other and relative to the ground) causes unbalance in the subconductor current

distribution and in the voltage between subconductors of the same phase as we Il as be-

tween phases. This unbalance has become a problem of growing importance in view

of the trend .toward.s higher operating voltage levels and greater number of subconductors

per phase for long, untransposed transmission line.2

Until now there has been no analytical

method available to study the effect of asymmetry of bundled conductor transmission 1 ines

and most calculations are based on the Geometrical Mean Radius eqyivalent circuit. 1,43

Wave propagation on a parai le 1 conductor transmission 1 ine has been the

object of continuous study in the field of electrical communications during the pa st few

d d 4,5,6,7,8,9 d hl' • h d' ob' f eca es. . Base upon t e genera equatlons governJng t e Istn utlon 0

voltages and currents in the uniform and parallel multiconductor transmission line, the

"Generalized Telegraph Equations" were formulated and their steady-state solution

relates the voltages and currents between any two points along the line. 4

ln the present

study, the "Generalized Telegraph Equations Il are applied with appropriate modifications

to power transmission 1 ines, and the resulting general method to determine the steady-

state voltage and current distribution of a transmission line is termed the Multiconductor

Method.

• 2

The relationship of the currents and voltages between any two points of a

transmission line can be expressed as

(1-1 )

where El ' Il ' E2

and 12

are voltage and current vectors at the two points. (Note

that, Il contains as many current components as there are conductors).

The matrix [J.112 ] is called the chain matrix between the points land

2. If the 1 ine between the points land ,2 is subdivided into m sections then

(1-2)

where [J.1. ] is the chain matrix of any subsection. A subsection could be just a 1

point on the line, e.g. a fault point. For such a case the chain matrix would be the

relationship between the voltages and currents just before the point to those just after.

A transmission line under study could be divided into any number of convenient sections.

ln the present study, fault-point chain matrices for various types of faults are

presented and calculations are also shown for chain matrices of normal healthy sections

for a few representative configurations.

The Multiconductor Method can take into account the distributed parameters

of the line and additional components in the line, for example, the resistance and

capacitance of spacers in a bundled conductor transmission 1 ine. It considers each sub-

conductor as a separate entity. The effect of changes of any one subconductor of a

•

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3

bundled conductor, under both normal or fault condition, on the behaviour of the

rest of the subconductors of a transmission line is taken into account by the Multi-

conductor Method. The method enables the determination of the voltage and current

distribution in individual subconductors along the transmission line, even very slight

unbalances in voltage and current can be calculated. The Multiconductor Method

also enables us to take into account the effect of ground wires without resulting to any

approximations. It can take the ground wires into account as separate conductors or

include them with the ground plane in the conventional methods as shown in Appendix B.

Here the method is applied to study the effect of the electrical characteris-

tics of spacers on current and voltage distribution in bundled conductor transmission

lines and the problem of faults in the three phase transmission line usually solved by

19 symmetrical components. The main purpose of the present study has been to illustrate

the versatility of the Multiconductor Method in handling the steady state problems of

transmission lines. The numerical examples have been chosen mainly to demonstrate

the procedure in dealing with various cases and do not represent any specific real system.

The numerical example considered is of a representative 735 KV three-phase

bundled conductor transmission line with four subconductors per phase. In addition, for

greater generality, two subconductors per phase and six conductors per phase are also

studied for cèrtain cases of current and voltage distribution. The transmission 1 ine is

assumed to be uniform. The effect of electromagnetic radiation, the proximity effect,

the skin effect, etc. 20 are neglected and the three-phase voltage supply at the sending

end is assumed to be obtained from an infinite bus of constant voltage and having no in-

• 21 22 ternal Impedance.' ln order to simplify calculations, the series resistance and the

leakage conductances are also neglected.

4

• ln the examples considered, the ground wires are treated by the con-

ventional method as shown in Appendix B.

o

•

C'\· . ,

5

CHAPTER Il

ANAL YSIS OF BUNDLED CONDUCTOR TRANSMISSION UNES

201 . Asymmetry between Subconductors and Subconductor Parameters

Si nce the sp li t conductor was proposed in 1909 by P. H. Thomas, 3 the

bundled conductor has been recognized as the best form of conductor in the design of

EHV transmission lines026

It has been the object of excellent studies,27,28 especially

29 10 30,31 fO Id d 1 dO 32,33,34,35,54 1 on corona, me constants, le an vo tage gra lents, e ec-

. 2 trostatic unbalances, etc 0

Although the bundled conductor transmission line is five to ten percent

more costly than the equivalent single conductor transmission line (as it requires a wider

and taller tower, more complicated and expensitle hardware, and higher stringing

26 charges), the use of bundled conductors in EHV transmission lines results in many

advantages over a single conductor line. Among them are:

1. Reduced corona loss.

2. Reduced radio interference level.

3. Reduced inductance.

The modern trend in transmission lines is to use long, untransposed lines at

higher operating voltages and a greater number of subconductors per phase,

thus forming a complex configuration. At the present, the maximum operating

voltage available in practice for three-phase bundled conductor transmission lines with

6

• four subconductors per phase is of the order of 765 Kv in the United States and 735 Kv

in Quebec, Canada. It is not unrealistic to expect that in the near future the three-

phase operating voltage wi Il be apprpximately 1.5 MV and the number of subconduc-.. ----.. ~":.

tors per phase will be of the order of twelve.

Due to the physical arrangement of subconductors in a transmission line,

a geometrical asymmetry exists between subconductors of the same phase as weil as

between phases. The asymmetry of subconductors results in an unbalance of line para-

meters which in turn causes a certain degree of unbalance in voltage and current between

subconductors of the same phase, although the line is operated under balanced applied

voltage and balanced load. (The degree of asymmetry between subconductors depends

upon their relative positions to each other and to the ground).

o ln the past, the unbalance between parameters due to asymmetry, being

small, was ignored in the general method of handling transmission line problems. In a

modern long transmission line, however, the unbalance in line parameters and, as a

consequence, in current and voltage distribution of subconductors is more pronounced

and its effect should not be neglected.

Since asymmetry is a problem of growing importance in the long EHV

transmission lines, it has become necessary to find sorne appropriate method to solve the

unbalances between subconductors as weil as between phases, permitting the determination

of the voltage and current distribution at any point along a multiconductor line. The

usual method in power engineering does not take into account the asymmetry between

subconductors of the same phase and does not solve the problems of unbalance between

•

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7

subconductors. The Multiconductor Methodpresented here can be used to analyze

the asymmetry between subconductors of a multi conductor transmission line. It is

valid for studying the unbalances, no matter how slight they are.

2.2 Analysis

For any untransposed transmission line with n conductors, the relation-

ship of voltages and currents between sending and receiv,ing end, is shown in Equation

(A -5) of Appendix A as

where E s

E r

1 s

1 r

[ ... J

E s

E r

1 1 s r

is the (n xl) 1

sending end voltage matrix,

is the (n xl) receiving end voltage matrix,

is the (n xl) sending end current matrix,

is the (n xl) receiving end current matrix,

is the so-called (2n x 2n) chain matrix of the line with

length 1, which can be determined by Equation (A-6) of

Appendix A without taking the account of spacers.

(2-1 )

o

8

A three-phase bundled conductor tral'lsmission line with b subconductors

per phase may be considered as a multiconductor system with a three-phase voltage

applied to the sending end. The treatment of an untransposed transmission line with

n conductors, can then be applied to the three-phase bundled conductor transmission

line with n = 3b conductors.

The effect of the ground wires is, in the present analysis, taken into account

by the method shown in Appendix B, that is, by considering only the line conductors

and modifying their parameters to allow for the ground-wire effect. However, the

Multiconductor Method is capable of handl ing the ground wires as additional conduc-

tors as shown in the set of calculations in Chapter VII', which indicate that the approxi-

mate method of accounting for the ground-wire effect is quite accurate.

For a precise calculation it should be taken into account that the transmission

line is not at the same height from the ground and, therefore, the line constants vary

along the line. Thus, the chain matrix [1..1 ] , more precisely, should be the product

of the chain matrices of several sections of transmission line, each with different height

to ground.

o

ce

9

(2-2)

where [1-111 ] to [1-1, t ] are the (2n x 2n) chain matrices of t sections of line,

which may be determined by its own constants respectively as shown in Appendix A.

This method can be used to calculate the chain matrix of a line section with sage

Each span is divided into a number of subsections in each of which the height above

ground is assumed constant. The accuracy of the chain matrix of the 1 ine section

with sag depends on the number of subsections. Thus, an accurate chain matrix of

a line section with sag can be obtained to whatever accuracy is desired for anyappli-

cation under consideration. In the illustrative examples here the line height in each

section is assumed constant.

The three-phase sending end voltage matrix is :

<è E =

s

and

EA

=

=

EA

EB

EC

v a

v a

v a

v c

v c

v c

10

(2-3)

{2-4}

where E A 1 EB

and EC are the sending end phase voltage (b xl) matrices of

phases A, Band C, v ,vb

and v are voltages of equal magnitude and at 1200

a c

phase displacement.

The three phase (n x n) load matrix is :

:.

11

ZLp 0 0

ZL = 0 ZLp 0 (2-5)

0 0 ZLp

where ZLp is the (b x b) load matrix per phase. It is given by :

ZLp = (2-6)

where zi is the load per phase.

For the determination of sending end currents, receiving end currents and

voltages, or currents and voltages at any point K along the transmission line the pro-

cedure indicated in the Appendix A [Equations (A-6 to A-9) ] can be used.

Then E r

= 1 r

and Ek

= Ik

[~k ] -1

E s

1 s

E s

1 s

(2-7)

(2-8)

where [~k ] is the (2n x 2n) chain matrix of the section of length Ik

from the

o

12

sending end to the point k which can be determined by Equation (A-6) of

Appendix A. For a given configuration of the bundled conductor transmission line,

the line constants can be obtained from its geometry as shown in Appendix B.

For the case of unbalanced applied voltages, the solution is easily

obtained by substituting the respective voltages into Equation (2-4), the rest of the

equations being the same as in solving the balanced system.

ln practice, the receiving end loads could be unbalanced. In this case 1

the individual load matrix per phase is set up as shown in Equation (2-6). Then

the three-phase load matrix ZL shown in Equation (2-5) will consist of three dif­

ferent matrices, while the rest of the equations remains unchanged, as used for the

balanced system. For the case when the load is a short ch~uit or an open circuit, the

same method is used. Thus, the Multiconductor Method is convenient to solve both

balanced and unbalanced systems.

2.3 Practical Example and Computed Results

A 735 KV three-phase four bundled conductor transmission line of

length 204.8 miles and configurations shown in Figure 2-1 was taken as the basic

13

(ft.

...... n-====z:

('J' l', \

\'\' (."....). \. ,. ,!-..,

\ . -. ,,--,..?J ~.

C~.:':'.:"l f\1"""= ....... ,.·.:~r:;j '-:)" "1 ,'"" ~r~

~ 0-·~~:·\~~ .. :':, ~ L' '"'~J~~~0~

'\ ') 'C' ."\ C'- ,', ,-;:- "= ·\.)0~ u-i'-:~0~

,,/ V\ fJ7' vU

FIGURE 2-1. CONFIGURATION OF A THREE PHASE 735 KV BUNDLED CONDUCTOR .

TRANSMISSION LlNE WITH FOUR SUBCONDUCTORS PER PHASE.

c~

14

example for the calculations. The spacers are not taken into account in this section

but their electrical characteristics will be studied separately in Chapter IV. For

simplicity, the transmission line was assumed to be at a constant height above the

ground as shown in Figure 2-1 and both the applied voltage at the sending end and the

load at the receiving end were assumed to be balanced. Loads of 300 (.8 + i .6)

ohms/phase and 600 (.9 + i ..; .19) ohms/phase were used. For other calculations,

various lengths, i.e. 51.2 miles, 102.4 miles, 409.6 miles, etc., and various loads,

i.e. 150 (.8 + j.6) ohmS/phase, 600 (.8 + j.6) ohmS/phase, etc., were also considered.'

ln order to investigate the different configurations of bundled conductor

transmission lines, calculations were also made on two subconductors per phase and six

subconductors per phase with the same relative positions to ground and between phases,

the same spacing between subconductors, the sa me diameter of the subconductors and the

sa me dimensions of the skywires. (See sketch in Appendix D).

For different configurations, the voltage across the load impedance at

the receiving end is

Assuming a" the subconductor currents are equal, then

o

15

where is the load impedance per phase and

is the number of subconductors per phase.

ln order to make a comparison of the current distribution for different numbers

of subconductors per phase, the load impedances are adjusted so as to have the same

value of (b. zl) •

From the geometrical dimensions of a given transmission line, the line constants

(Inductive reactance and capacitive susceptance) of the subconductors can be obtained

by the use of an appropriate digital computer program(see Appendix C). The values of the

line parameters of different bundled conductor configurations are listed in Appendix D.

The general principles for the determination of the line parameters using the geometri-

cal dimensions of a transmission line with n subconductors are shown in Appendix B •

The computed results in Tables 2-1, 2-2 and 2-3 give the terminal sub-

conductor currents for two subconductors per phase, four subconductors per phase and

six subconductors per phase respectively for an identical length of 204.8 miles of the

line. Two loads were used, namely an inductive load of power factor 0.8 and a

nearly resistive load of powei factor 0.9. In the Tables, l, IR and Il represent

the magnitude, the real part and the imaginary part of the current respectively.

16

TABLE 2-1.

SENDING END AND RECEIVING END SUBCONDUCTOR CURRENTS

.oF TWO SUBCONDUCTORS PER PHASE TRANSMISSION UNE

(AM PERES " 'R '

,,)

SENDING END RECEIVING END

Subcondu'ctor Phase Curr.e:nt Subconducto'r Phase Current 1 2 (Vector Sum) 1 2 (Vector Sum)

LOAD PER 'PHASE, : 600 '(.8, + ; .6) OHMS

Phase A 1 237.3 239~6 476.9 330.3 333.6 633.9 '1 R

-·150.6 -153.7 90.4 88.7

" 183.4 183.8 317.7 321.6

(~~ Phase B '1 260.7 261. 521.7 336.5 336.7 673.1

'R 256.7 256. . 231 .5 236.8

I, 45.4 50.8 -244.2 -239.3

Phase C 275.5 272.9 548.4 339.6 336.4 676.

'R -111 .5 -108.7 -330.9 -327.2

" -252. -250.3 - 76.4 - 78.5

LOAD PER PHASE: 1200 (.9 + i ~) OHMS

Phase A 246.4 248.9 495.4 181.2 183.1 364.3

'R -244.1 -246.9 5. 3.6

" 33.9 32.1 181. 1 183.

Phase B 269.6 269.8 539.4 183.5 183.7 367.2

'R 173.6 169.5 155.6 157.9

" 206.3 210. - 97.3 - 94.

"_ Phase C 272.9 270.2 543.1 185.2 183.6 368.8 ..... 'R 71.6 72.8 -163.6 -161.4

" -263.4 -260.2 - 86.9 - 87.5

17

• TABLE 2-2.

SEN DING END AND RECEIVING END SUBCONDUCTOR CURRENTS

OF FOUR SUBCONDUCTORS PER PHASE TRANSMISSION UNE

(AMPERES, l, IR' Il)

LOAD.PER PHASE: .300 (.8 +,j.6) ORMS

Subconductor Phase Current 2 3 4 (Vector Sum)

SENDING END

Phase A 206.5 207.2 210. 210.6 834.2

IR -34.4 - 34.1 - 38.4 - 38.1

Il 203.6 204.3 206.4 207.2

Phase B 225.2 225.6 225.9 226.3 903.0

(~~,. IR 213.3 213.6 216.7 217.1

Il -72.4 -72.6 - 63.7 - 63.9

Phase C 241. 241 .8 236.9 237.7 957.0

IR . -190. -190.3 -184.8 -185.2

Il -148.2 -149.1 -148.1 -149.

RECEIVING END

Phase A 296.1 297.1 301.4 302.4 '.11.97.

IR 102.3 103.2 99.9 100.8

Il 277.9 278.6 284.4 285.1

Phase B 1 307.8 308.3 308.2 308.8 1233.

IR 191.4 191 .8 201 .6 201.9

Il -241. -241.4 ·-233.1 -233.6

Phase C 1 311 .2 312.3 305.9 307. 1236.

~e .... IR -308.6 -309.6 -302.7 -303.7

Il - 39.9 - 40.6 - 43.6 -44.4

18 (. TABLE 2-2 (continued)

LOAD PER PHASE : 600 (.9+ i r:T9) OHMS

Subcônductor Phase Current 1 2 3 4 (Vector Surn)

SEND'NG END

Phase A 156.8 157.3 159.7 160 .1 633.9

'R -130.4 -130.6 -134.2 -134.5

" 87.1 87.7 86.5 87.

Phase B 178.8 179.1 179.4 179.6 716.7

'R 160.7 161. 158. 158.3

" 78.3 78.4 84.9 85.

Phase C 1 187.2 187.8 183.7 184.4 743.

'R - 31.6 - 31 .4 - 28.7 - 28.5 ( .

\...~. '

" -184.5 -185.1 -181 .5 -182.2

RECEIVING END

Phase A , 172. 172.6 175.1 175.7 . 695.4

IR 14.6 15. 12.3 12.7

l, 171.4 172. 174.7 175.2

Phase B 176.7 177. 177. 177.3 707.9

IR 143.4 143.7 148. 148.2

'1 -103.2 -103.4 - 97.2 - 97.4

Phase C 179.4 180. 176.4 177. 712.7

IR -163.9 -164.3 -160. -160.4

" -73. -73.6 - 74.2 -74.8

19 ", (e TABLE 2-3.

SENDING END AND RECEIVING END SUBCONDUCTOR CURRENTS

OF SIX SUBCONDUCTORS PER PHASE TRANSMISSION LlNE

(AMPERES l, IR' Il)

LOAD PER PHASE: 200 (.8 + i .6) OHMS

Subcondu'ctor Phase Current 1 2 3 4 5 6 (Veèfor Sun:'!)

SENDING END

Phase A 196.3 197.5 199. 20'l.Z 203. 204.5 1201.9

IR 10. 7.8 8.7 4. 5. 2.5

l, 196.1 197.4 198.8 201.7 203. 204.5

Phase B 215.2 '214.8' 215.6 215.6 216.4 216.9 1293.7

( IR 181. 183.6 184.2 189.7 190.2 193.2

Il -116.4 -111.5 -112. -102.6 -103.1 - 98.4

Phase C 231.8 228.5 230. 223.7 225.4 222.3 1361.7

IR -209.7 -206.1 -207.1 -200.2 -201.3 -198 .

Il - 98.7 - 98.7 -100.2 -100. -101.4 -101 • 1

RECEIVING END

Phase A 270. 271 .8 273.8 277.8 279.7 282. 1655.1

IR 108.5 106.5 108.4 104.2 106.1 103.8

Il 247.3 250.1 251.5 257.5 258.8 262.2

Phase B 287.3 286.4 287.4 287. 288. 288.3 1723.2

IR 159.3 164.7 165.3 176.6 177 .1 183.

Il -239.1 -234.3 -235.1 -226.2 -227. -222.8

Phase C 293.7 289.4 291.5 283.3 285.5 281.5 1724.7

-- IR -293.3 -289. -290.9 -282.5 -284.5 -280.5

Il - 15.2 - 17.1 - 18.3 - 21.7 - 23.1 - 24.5

20

(j TABLE· 2-3 (continued)

LOAD PER PHASE: 400 (.9 + i /":T9) OHMS

Subcondudor Phase Current 2 3 4 5 6 (Vector Sllm)

SENDING END

Phase A 1 132.9 133.8 134.8 136.8 137.6 138 .7 814.6

'R - 84.1 - 85.9 - 86. - 89.9 - 90. - 92.1

" 103. 102.7 103.8 103.1 104.1 103.8

Phase B 152.9 152.7 153.2 153.5 154. 154.4 920.

'R 150.8 150. 150.5 149.2 149.7 149.2

" 24.8 28.5 28.5 36. 35.9 39.8

Phase C 165.5 163. 164.2 159.5 160.7 158.5 971 .3

'R - 69.9 - 68. - 67.8 - 64.2 - 64.1 - 62.5

" -150. -148.2 -149.5 -146. -147.4 -145.6

( \ ,

RECEIVING END

Phase A 163.7 164.8 166. 168.4 169.5 170.9 1003.2

'R 22.4 20.8 21.7 18.3 19.2 17.3

" 162.2 163.5 164.6 167.4 168.5 170.

Phase B 171 .6 171 . 1 171 .7 171.4 172. 172.2 1029.4

'R 131 .6 134. 134.5 139.6 140. 142.8

" -110 . -106.3 -106.7 - 99.4 - 99.9 - 96.4

Phase C 176.7 174.2 175.4 170.6 171.9 169.5 1038.2

'R -165.4 -162.6 -163.5 -158.1 -159.1 -156.5

" - 62. - 62.4 - 63.5 - 64. - 65.1 - 65.2

21

2.4 Discussion

By the use of the Multiconductor Method, many interesting characteristics

of bundled conductor transmission 1 ines can be obtained based on the computed results.

From Tables 2-1, 2-2 and 2-3, it appears that the subconductor currents for various

configurations are unequal for the same value of (b. z,). The reason why the values

are not the same, as might be expected, is due to the fact that the equivalent self-

inductance per phase for a bundled conductor configuration is not varied inversely

proportionally to the number of subconductors per phase. Therefore, as the number of

subconductors per phase is increased from four to six, the equivalent self-inductance

per phase is reduced by less than 1/3 of the original value. It is seen that the current

per subconductor in the two subconductors per phase configuration is greater than in the

four or six subconductors per phase configuration as the (b. zl) is the same.

Generally speaking, the subconductor current distribution of the same phase

is quite equal, i.e. around one percent unbalance in the case of two subconductors per phase

22

configuration, two percent in the case of four subconductors per phase and four and a

half percent in the case of six subconductors per phase. (See Tables 2-1, 2-2 and

2-3). The load as weil as the length of the line appears to have negligible effect on the

. unbalance in the current distribution between the subconductors of the same phase.

The maximum voltage difference between subconductors of the same phase

for different numbers of subconductors per phase is illustrated in Figures 2-2, 2-3 and

2-4. Obviously, the value of the maximum voltage difference is negligible compared

to the operating voltage of the transmission line and is, in general, located at the middle

of the line.

Among the three phases, Phase B shows the least unbalance in the sub-

( conduc~or current distribution, and also the least voltage difference between subconductors.

\.

The unbalance in subconductor current distribution and the maximum voltage

difference between subconductors of the same phase are ,increased as the number of

subconductors per phase is increased and are affected by the geometrical asymmetry of 1

the transmission line. This unbalance can be explained by the fact that the maximum

distance between subconductors in the case of six subconductors per phase, is greater

than that in the case of two or four subconductors per phase, a Il confi gurations havi ng the

same spacing of 1.5 ft. The longer the transmission 1 ine, the higher is the value of the

maximum voltage difference as is shown in Figure 2-5.

Figure 2-6 shows the phase current verying almost Iinearly along the length

of the 1 ine with four subconductors per phase. It shows that the sending end current is

( '-.

3.0

2.0

.fl.t!.O Il il j j

23.

102.4 "

1"\) fi (:>":.

tt. ",/I .• r. ,; I~ .(~ . . ~ '-~. G, "'"

. . . .

FIGURE 2-2. MAXIMUM VOLTAGE DIFFERENCE BETWEEN SUBCONDUCTORS OF

SAME PHASE OF TWO SUBCONDUCTORS PER PHASE CONFIGURA­

TION VS. DISTANCE MEASURED FROM SENDING END.

(0) LOAD PER PHASE : ·600 (.8 + i .6) OHMS.

(b) LOAD PER PHASE: 1200 (.9 + i 1 .i 9) OHMS.

c

(

2V'

llV

o FIGURE 2-3.

24

.=

3 '" ~ SJIr1 ~ &21 .lkt!1 .t;ZC .. ~_t?\

wc ===-r-.~

.102.4 ~/· (1".: / • "l Uri., ~.

~ -.-.v ÎVI" .::: ~-.::: i i __ :..-"-'

MAXIMUM VOLTAGE DIFFERENCE BETWEEN SUBCONDUCTORS OF SAME PHASE OF FOUR SUBCONDUCTORS PER PHASE CONFIGURAïlON VS. DISTANCE MEASURED FROM SENDING END.

(0) LOAD PER PHASE: 300 (.8 + i .6) OHMS.

(

VOLTS 25

C:::::::::::===~" Dr:-~ u'\ ~rl:_ ~ M u ,M~.:.... .ri

.~

20l~.8 MILES FIGURE 2-4. MAXIMUM VOLTAGE DIFFERENCE BETWEEN SUBCONDUCTORS

OF SIX SUBCONDUCTORS PER PHASE CONFIGURATION VS; DISTANCE MEASURED FROM SENDING END.

(0) LOAD PER PHASE: 200 (.8 + i .6) OHMS.

(b) LOAD PER PHASE: 400 (.9 + i r.T9) OHMS.

(

10. -

r=~ j;.;.,·U,I' ... ~. . .~o

a

-

26

j, Ot .... ~ ",' 1 r ......

" ,~,I u" ..... ...,;/.

(e FIGURE 2-5. MAXIMUM VOLTAGE DIFFERENCE BETWEEN SUBCONDUCTORS OF SAME PHASE OF FOUR SUBCON DUCTORS PER PHASE TRANSMISSION lINE VS. LENGTH OF THE TRANSMISSION L/NE WITH LOAD 300 (.8 + i .6) OHMS PER PHASE.

AMPERES

• 1250-

1200·

1100

700

o

27

102 L~ 20L~8 MILES FIGURE: 2-6. THE VARIATION OF PHASE CURRENTS ALONG THE LENGTH OF FOUR

SUBCONDUCTORS PER· PIiASE TRANSMTSSJON· LlNr ·W/TH LOAD 300 (.8 + i .6) OHMS PER PHASE.

c-'

28

smaller than the receiving end current which is due to the fact that the former consists

of both the load current, which is usually inductive, and the capacitive current of the

line which compensates the inductive part of the load curren~.

As illustrated in Figure 2-7, the maximum current unbalance at the

sending end between phases is more pronounced in the case of the 1 ight load for a four

subconductors per phase transmission line. Further, Figure 2-8 shows that this maxi­

mum unbalance is increased as the length of the line is increased.

From the computed results, it is important to observe that the bundled

conductor transmission lines have almost a uniform current distribution in the subconductors

of the same phase and very slight voltage difference between subconductors of the same

phases.

As mentioned, both the resistance and the leakage conductance of the

subconductors have been disregarded in the numerical examples as they have negligible

effect on the computed results. However, it is possible, without any difficulty, to

include both the resistance and leakage conductance of the subconductors in the chain

matrix of the 1 ine •

The chain matrix of the transmission line shown in the Multiconductor

Method can be modifjed according to the physical position of the transmission 1 ine.

For instance, since the transmission line does not have the same height From the ground

at ail points, its line parameters will not be exactly the same along the line. For pre­

cise computations, the line has to be divided into several sections and the chain matrix

of the line will be the product of several chain matrices corresponding to the individual

sections.

20

15

5

FIGURE 2-7." MÀXIMUM CURRENT UNBALANCE BÈTWEEN PHASES IN % OF

CURRENT IN PHASE A VS. THE RATIO OF LOAD TO REFERENCE

LOAD, 300 (.8 + i .6) OHMS PER PHASE, FOR A FOUR SUB­

CONDUCTORS PER PHASE TRANSMISSION UNE.

29

1

% 20

10

o 51.2

30

15306 20~Q8

MILES

. FIGURE 2-8. MAXIMUM CURRENT UNBALANCE BETWEEN PHASES

IN % CURRENT IN PHASE A VS. LENGTH OF A

FOUR "SUBCONDUCTORS PER PHASE TRANSMISSION LlNE.

31

ln addition to taking into account the variation of height, anyadditional

compone nt in the line con be represented bya chain matrix and in turn be included in

the chain matrix of the line. Jn practice, the existence of insulators, pins and cross arms

which are essential to the support of the lines, may introduce additional capacitance,

which may change the circuit of the transmission line. The transmission 1 ine con then

be represented bya ladder network.17

The total chain matrix of the line will be the

productof several chain matrices. This idea will be applied also ir. the solution of fouit

. problems and the determination of the effect of spacers in the following chapters.

32

CHAPTER III

FAULT CONDITIONS WITHIN THE BUNDLE OF TRANSMISSION LlNES

3.1 Analysis of the Faults

Unbalance caused by fault conditions may occur in bundled conductor trans­

mission lines. This fault may take place in an individual subconductor, as in the case

of an open circuit or a short circuit to ground or it can take the form of a short circuit

between two or more subconductors of the same phase or of different phases.· Due to

the larger number of subconductors in the bundled conductor transmission line, the

possibility of fouit occurrence is increased.

ln the case of an abnormal condition caused by a fault along the line, the

same analysis by the Multiconductor Method can be applied if the chain matrix of the

line and the load matrix are modified accordingly. In general, the chain matrix of the

line consists of three parts: the chain matrix from the sending end to the fault point,

the fault point chain matrix and the chain matrix from the fault point to the receiving

end. In this first example of fault analysis, the faulted conductors are assumed to he in­

terrupted at the fault point and are not included in the chain matrix from the fault point

to the receiving end, but these sectionA are inc\uded later. The chain matrix becomes

[ ~ ] = [~11 ] [~f ] [~12 ] (3-1 )

where [~11 ] is the chain matrix from the sending end

to the fault point of length '1 ' which

can be determined by Equation (A-6) of

Appendix A.

[J.lf

] is the fouit point chain matrix, and

[J.l12J is the chain matrix from the fault point to the receiving

end of length 12

•

33

Of the numerous fouit possibil ities which may occur in the individual subcon-

ductor, only the most common faults are considered, su ch as open circuit or short circuit

to ground of one single subconductor or short circuit between two subconductors of the

same phase. The same form of analysis is applicable to other faults as weil if the ,

appropriate chain matrix is obtai~ed accor~ing to the respective fouit condition.

(a) Open Circuit or Short Circuit to Ground of Subconductor Al

rn fhis case, shown in Figure 3-1 (a), the (2nx2n) faultpointchain

matrix [J.lfJ ls the uni't matrix, and

0 ,,0 0

0 cosh (12

..JliYi) 0 sinh (1 JZ'Y') Z' 2 0

[ J.l12

] = 0 0 1 0

0 sinh (1 JY'Z')Z,~l, 0 2 0

cosh (1 ,jy'Z') 2

where Z' is the [(n-1) x (n-1) ] impedance matrix of the unfaulted subconductors, per mile,

y' is the [(n-1) x (n-1) ] admittance matrix of the unfaulted subconductors, per mile r

Z' is the characteristic impedance matrix of the unfaulted sub­o conductors and ZI = [Jz'y' ] -1 Z' = [y'f1 [,jY'ZI J. o

(3-2)

u " u

Ü 7··· ...

il :

$W'ze=-= -=m-z:rzr

,~ ~ C<J p=-'" -- .- •.••.. -•. -

D

DUU • r-r=

D

0 0 G c ~ C -

t i

~

~ fi ~

'1 ~ .. ~

" ~ -._==

ra ç

FIGURE 3.1. FAULT CONDITIONS.

(a) OPEN CIRCUIT OR SHORT CIRCUIT TO GROUND OF A SINGLE SUBCONDUCTOR (Al) .

(b) SHORT CIRCUIT BETWEEN TWO SUBCONDUCTORS OF PHASE A. (Al AN D A2) •

34-

The (n x n) i"hree phase load impedance matrix is

Z = L

where ZLf =

ZLf 0 0

0 ZLp 0

0 0 ZLp

Zf 1 0 --:-~----------

1 zi

1 1 •

1 o 1

1 1 •

~ zi

is the (b x b) load matrix of the phase A containing the fault,

and ZLp =

is the (b x b) load matrix of any of the two unfaulted phases Band C

as derived.

ZI is the normal load impedance per phase and

zf is the fouit impedance to ground which is infinity or zero according

to whether the fault is an open circuit or a short circuit to ground.

35

(3-3)

(3-4)

(b) Short Circuit of Two Subconductors of Sorne Phase (Al and A2

)

and

ln th is case 1 shown in Figure 3-1 (b)

0' 0 [ fJ

f ] = U +

fJfs 0

where U is the (2n x 2n) unit matrix and [J.lfsJ is a (n x n) matrix

fJfp 0 0

[fJ ] = fs

0 0 0

0 0 0

where [ J.lfp

] is (b x b) fault point chain matrix for phase A and

[fJ ] = fp

__ 1 ,

Z,f zf 1 l 1 f 0

-- -1 zf zf

-----f--o 10

Z,f is the fault impedance between two subconductors.

0 0 0 0 0

0 0 0 0 0

Finally [fJ12J = 0 0 cosh (12 J Z'Y') 0 0 sin h (1 J Zlyl) Z' 2 0

0 0 0 0 0

0 0 0 0 0

0 o sinh (12J?Zi)Z~-1 0 0 cosh (12

Jyl ZI)

36

(3-5)

(3-6)

(3-7)

(3-8)

1

Both Z', Y' and Z' are defined as in (a) except that they are matrices o •

of the order of [(n-2) x (n-2) ] •

The load matrix is given by

ZL =

ZLf 0

0 ZLp

0 0

z s

o 1 1

o z 1

0

0

ZLp

o

where 'ZLf = s ---i-;' ------ ---. 1 1 zl

o

1 •

1 • 1 1 • 1 • 1 1 zl

is the (b x b) matrix of phase A •

Here z s

10 is assumed to be a very Jarge impedance (10 ohms) to

represent open circuit.

ln order to show the generaljzed analysis of a short circuit between two

37

(3-9)

subconductors, either in the sa me phase or in different phases, the subconductor land

subconductor m of a bundled conductor transmission line with 'n subconductors are

assumed to be the faulted ones. Then in Equation (3-6), ail the elements of [J.1fs

] , a

(n x n) matrix, are zero except the e lements corresponding to the subconductor land

subconsuctor m as follows :

,'" e

[~ J= fs

m

n

,zf 0

0

o

m n

l 0--'

zf 0 . 0

0

o o . o

o o

For short circuit between subconductors 1 and 2 (Al and A2), the same

form of [~fs J is obtained as shown in Equaf'ion (3-7).

The submatrices of [~12 J are

cosh (12 jz' Y') =

sinh (1 ..JliYi)Z' = 2 0

A7ï=7i -1 sinh(l II/T 'Z')Z' = 2 0

. 38

(3-10)

<...

-

cosh (1 Jy'Z') = 2

where Z' Y"Z' are defined as in(a)except they are [(n-2) x (n-2) ] , 0 ,

39

matrices and not including the elements related to subconductor number land number in.

Al 1 A2, A3 and A4' are submatrices ofthe order of [(m-2~ x (m-2) ] •

B1

1 B2, B3 and B 4 ' are submatrices ofthe order of [(m-2) x (n-m) ].

Cl 1 C2

, C3

and C4, are submatrices ofthe order of [(n-m) x (m-2) ].

Dl 1 D2' D3 and D 4' are submatrices ofthe order of [(n-m) x (n-m) ] •

Therefore, the (2n x 2n) cha i n matrix i s

0 0 0 0 0 0 0

0 Al 0 B1

0 A2

0 B2

m 0 0 0 0 0 0 0

[fJ12 J = 0 Cl 0 Dl 0 C

2 0 D

2 (3-11 )

n+1 0 0 0 0 0 0 0

0 A3 0 B3 0 A4 0 84

n+m 0 0 0 0 0 0 0

0 C3

0 D3 0 C4

0 D4

Fault conditions other than those shown in this analysis can also be studied by

the Multiconductor Method, the corresponding ehain matrix being derived similarly taking

into account the specifie conditions.

1

40

3.2 Practical Example and Computed Results

The typical 204.8 mile long 735 KV bundled conductor transmission

line with four subconductors per phase, as shown in Figure 2-1 of Chapter JI, was con,-

sidered. The load impedance per phase was 300 (.8 + j.6) ohms.

The calculations were made for the case of an open circuit and a short cir­

cuit to ground of single subconductor Al in phase A and for the case of a short circuit

between two subconductors Al and A2

of phase A. In order to observe the effect of

the location of the fault on the fault current distributions, several fault points were chosen.

For practical computational purposes, the short circuit is mode lied bya resistance of 0.5

ohms and the open circuit by' 1010 ohms.

Table 3-1 shows the currents at the sending and receiving ends in both

phases Band C in the case of an open circuit and a short circuit to ground of subcon­

ductor Al of phase A .. Values are based upon the computed results of various fault

points. In the case of a short circuit between Al and A2

of phase A, the correspond­

ing values of the above currents are also listed in Table 3-1. Other calculated

results show that the effect on the unfaulted phase currents lB and IC of moving

the fault location is insignificant.

ln Figure 3-2 the variation of the short circuit current in subconductor Al

determined at the sending end as a function of the distance of the fault point measured

from the sending end is illustrated. In Figure 3-3, the same currents are represented"for

an open circuit of subconductor Al •

41

TABLE 3 - 1.

SEN DING AND RECEIVING END PHASE eURRENTS

IN PHASES B AND e, (AMPERES).

Sending End Re ceiv.in"g. End:

Balanced System

Al Short circuited

to ground

lB le

903.0 957.0

882.0 - 876.0 985.0

Al Open circuited 892.0 - 902.0 957.5

Short circuit be-

lB le

1233.0 .1236.0

1259.0 - 1251.0 1 256.0

1231.5 1234.0 - 1236.5

twèen Al and A2

879. - 900. 962. - 957.5 1230.0 1 234 • - 1 237 •

1

Kiloamperes

22

20

18

16

14

12

10

8

6

4

2

o 20.48 102.4

FIGURE 3-2. SENDING END SHORT CIRCUIT TO GROUND CURRENT

IN SUBCONDUCTOR Al VS. LOCATION OF FAULT

POINT MEASURED FROM SENDING END.

42

I~

204.8

Miles

Amperes

(e 200

150

100

50

o 20.48 102.4

FIGURE 3-3. SENDING END OPEN CIRCUIT CURRENT IN

SUBCONDUCTOR Al VS. LOCATION OF

FAULT POINT MEASURED FROM SEN DING END.

43

204.8

Miles

44

Figure 3-4 gives the sending end and receiving end currents in phase A,

not including the current in subconductor Al' with short circuit on Al as a function of

the location of the fault point. In Figure 3-5, the sa me currents are represented for

an open circuit of subconductor Al •

Figure 3-6 shows the variation of subconductor current in Al (or A2

) with

short circuit between Al and A2

determined at the sending end as a function of the

location of the fault, while Figure 3-7 gives sending end and receiving end currents

in Phase A, not inçluding currents in Al and A2 ' with short circuit between Al and

A2

• It is interesting to note that in this case, the sending end current per subconductor

may reach values as high as twice the normal current( see Figure 3-7). Moreover,

Table 3-1 shows that the unfaulted subconductcrs of the faulted phase may be over-

loaded for single subconductor or two subconductor faults. In design work, proper

attention to protect the subconductors against overload during faults is nessary.

3.3 Discussions

The Multiconductor Method has proved to be adequate for the purpose of

studying the fault conditions in a bundled conductor transmission line.

ln this anal ysis, only the most frequent fault conditions have been studied.

Due to the numerous possibilities of fault combination which would occur in a bundled

conductor transmission line with n subconductors, Equation (3-1) can be assumed to

...

..

c-" i.~

v':'.," ~~~=-­c:=:~==--=-----

45

l' ~ ~t __ = ____ ~ __ ~~,-~i ____ ~J=-~.~"'b.-====~Tbf==-=~'====~-===~=====~=-~

FIGURE 3-4.

1(J2·4 .... --,0'"

• f'" ,-.~ , , .. .,\ .. . .., ... ","'v-.JV SENDING AND RECEIVING END CURRENTS IN PHASE

A (SUBCONDUCIOR Al SHORl CIRCUITED TO GROUND) VS. LOCATION OF FAULT POINT MEA5URED FROM 5ENDING END.

Amperes

r n === " -=;=

46

f' -~ ~ n r, ('> L..I.., .;~·0v

FIGURE 3-5. SfNDING AND RECEIVING END CURRENTS IN PHASE A. (SUBCONDUCTOR Al OPEN CIRCUIT) VS. LOCATION OF FAULT POINT MEASURED FROM SENDING END.

. Amperes

• ·200

150

100

50

0-- - _.n

o 2:0-49

FIGURE 3-6.

47

!! .

. f\ .n~~.., ,-...-.. L", .. ~ ... v'-.J

SEN DING END CURRENT IN SUBCONDUCTOR Al OR

A2

(SHORT CIRCUIT BETWEEN Al AND A2) VS. LocA­

TION OF FAULT POINT MEASURED FROM SENDING END.

Amperes.

48

p ? rtΠ_ ===,r.==, . ~ == emr "== = == P..,,_

ft , -r-

FIGURE 3-7. r··I'l~~'J0 .;~ .. "' ... \.J v SEN DING AND RECEIVING END CURRENTS IN PHASE A

(SHORT CIRCUIT BETWEEN Al AND A2 ) VS. LOCATION OF FAULT POINT MEASURED FROM SENDING END.

49

be the general ized form for determining the chain matrix of the transmission line. Any

other fouit condition con be analyzed by the sorne Multiconductor Mèthod provided the

corresponding chain matrix is set up.

It should be pointed out that there is a possibility of having more than one

fault occurring in the line, either in different locations or of different types. This com-

plicated fault condition con 0150 be studied by the Multiconductor Method. In such a

case, the chain matrix of the line will be the product of several fault chain matrices and

severa 1 chain matrices of the line sections separated by the fault points.

50

CHAPTER IV

THE STUDY OF SPACERS

4.1 Nature of the Problem

The spacer is an important integral part of bundled conductor transmission

lines. The principal purpose of having spacers along the bundled transmission line is to

maintain, within acceptable limits and under ail conditions of normal service, the designed

spacing between subconductors of each phase. 36 Generally, a spacer is installed every

250 to 300 feet along the line, so that the number of spacers required for a three-

phase transmission line of two hundred miles will be of the order of ten thousand units,.

,,--. \

Whi le the design and mechanical properties of spacers have been the object of ex-

. 37 38 39 cellent studles, " little attention has been given to their electrical charac-

'_.~

teristics.

Although a spacer could be of many different types, it usually consists

of both metallic and insulation materials which separate electrically two subconductors of

the same phase. The rigid-bar spacer of the two subconductor bundle, has a neoprene

bushing which lines the clamping surface and is used to protect the conductor from

b • d 'd Of 1 0 0 h . l' 40 a raslon an to provl e unI orm c ampmg pressure ln t e camping area. Another

new design is the "Spacer-damper" with a silicone rubber bushingo 40 The four subconductor

bundle spacer used in Canada has silicone rubber mounted pivots between a cast frame and

1 ..1

51

cast clamps.41 The type used in England has siliconerubber mounted pivots and silicone

. rubber lined c1amps.41

According to available information, the resistance of the four subconductor

bundle spacer, measured between the clamp and the centre of a spacer, should range from

0.1 to 10 M-ohms. This range represents the acceptable limit for the silicone rubber

pivot. The capacitance of the insulation part of the spacer is in the range of several pf

to severa 1 hundred pf depending on the material used.

From the electrical point of view 1 a spacer of any type can be represented

bya circuit mode 1 consisting of a capacitance in parai lei with a resistance, connected

across two subconductors as shown in Figure 4-1. Therefore, the presence of spacers

alon9 a bundled conductor transmission line makes the whole transmission line complicated,

as it possesses electrical characteristics which change the configuration of the circuit of

the 1 ine. It should be pointed out that in a study of the role of spacers in creating un-

balanced condition in a transmission line, the fact has to be accepted that their charac-

teristics may vary over a wide range. Such variations occur not only between spacers of

different types but also between those of the same design. The degree of deterioration is,

of course, an important factor.

According to field reports, spacer parts have been known to get worn out

due to many causes, such as the effect of oscillation of the subconductors, abrasion of the

metallic parts and other factors.40

A spacer with a completely damaged insulation would

change the normal circuit configuration of the transmission line, as it may cause a short

circuit between 2, 3, •••• or b subconductors of the same phase within the same spacer

of a "b" subconductor bundle.

52

, e2 ,2--~--~------r-------------~~D . l '

l' 2

" 'e3 ,?---,~------------~~--------------~~--~~ J .' 1.1 , 13 1 13

1 1

'1

~--~~~,-~-------~----~----

b~e>b __________________________ ~ __ ~~~ __ • • 1

"lb ,lb

FIGURE 4-1. THE CIRCUIT MODEL OF A SPACER IN A BUNDLE OF

b SUBCONDUCTORS RELATIVE TO SUBCONDUCTOR 1.

54

Equation (A-6) of Appendix A.

[tJ2 J is the (2n x 2n) chain matrix of one spacer.

The electrical model circuit of a spacer can be represented by 0 resistonce

in parallel with a capacitance. Figure 4-1 shows the voltages and the currents asso-

ciated with a system of b subconductors.

For subconductor 1,

.................

)

1 = il + el (b - 1) (r + i 10) c)

where r is the parollel resistance of the spacer,

c is the parollel capacitance of the spacer.

Similar equations can be written for the other s~bconductors.

For .four subconductors per phase, the relationship is shown in Equation

(4-2) •

., ' ........ ./ ~'

el 1 0 0 0 0 0 0 0 el

e2 1 O· 1 . 0 0 0 0 0 0 e2

e3 1 0 0 1 0 0 0 0 0 e3

e4 1 0 0 0 1 0 0 0 0 e4 = 1 (4-2)

o 1 3 °a 1 0 1 0 1 0 1 0 0 0 0' 1 - + 1 wc - - - IWC - - - IWC - - - IWC 1 1 r r r r 1

o 1 1 1 0 3 03 1 0 1 0 0 1 0 0 0' 1 - - - IWC - + 1 WC - - - IWC - - - IWC 1 2 r' r .r r 2

o 1 1 1 0 1 0 3 03 1 0 0 0 1 0 0' 1 - - - IWC - - - IWC - + 1 wc - - - IWC 1 3 r rr r 3

o 1 1 1 0 1 0 1. 0 3 03 0 0 0 1 0' 1 - - - IWC - - - IWC - - - IWC - + 1 WC 1 4 r r r r 4

VI VI

56 .

• For b subconductors per phase, the chain matrix of a spacer per ........

phase is

o o o o + il.) (4-3)

o o

where

m 1 - -- --r r. .r

m r r r.

l m 1 r r r -,

[~ J = r r r (4-4) r

. r r

- -r r

1 m --r r . r r

1 m -r r r

and

(,. 57

me -c -c

-c me -c

-c -c

-c -c

[ /Je ] = (4-5) -c -c

-c -c

-c -c

-c -c me

are (b x b) matrices.

'" -" . U is the unit matrix of the 2b order and m = b - 1. ........

For a three phase system with n = 3 b subconductors

0 1 0 0 1 0

-------1-- -----.l--1

1

/Jr 0 0 /Je 0 0 1

1 1 1

[JJ2

] = u + 0 JJr 0 1 0 +jw 0 JJc

0 1 0 1 1

0 0 JJr 1 0 0 l JJ c 1

(4-6)

where U is the unit matrix of the 2 n order.

o

58

ln the damaged spacer case, only the damaged spacer is taken into con-

sideration, ignoring the normal spacers. Its resistance, r , is the resistance between . s

two subconductors when the spacer insulation is completely damaged. The value of r s

.. 3 -6 may lie between 10 and 10 ohms.

·Inthis case, the chain matrix ["'] consists of the ,three parts:

The chain matrix from the sending end to the location of the

fault (damaged spacer),

thechain·matrix of the point of the damaged spacer, and the

chain matrix from the location of the damaged spacer to the

receiving end.

where ["'11] is the chain matrix from the sending end to the

location of the damaged spacer of length Il '

is 'the chain matrix of the point of the damaged

spacer,

and [tJ12 ] is the chain matrix From the location of the damaged

spacer to the rece iving end.

80th ["'11 ] and ["'12] can be determined by Equation (A-6) of

Appendix A •

(4-7)

Because r has a very small value which means that the damaged spacer has s

a very large conductance, the capacitance of the damaged spacer can be neglected.

(-"'" ;

'-j

e,.

59

Thus, the resistance chain matrix for the damaged spacer can be obtained by re-

placing r by r in Equation (4-4) , wherever two subconductors form a metallic contact s

due to the damaged spacer.

Accordingly, the resistance matrix [Ilr

] of Equation (4-6) for the

damaged spacer can be denoted by [Il J. rf .

Then the three phase chain matrix of the point of the damaged spacer is

1 0 1 0

-----f--1

[ Il ] =' Ilrf 0 0 1 (4-8) f

1 0 Ilr 0 1 0

0 0 1 Ilr 1

ln the Equation (4-8), the damaged spacer is assumed in phase A. If

it is located in other,phases, the,pr.oper [llrfJ should be used to substitute [llrJ of

that phase with the damaged spacer.

For a bundle of four subconductors per phase, the chain matrix [llrfJ

will be :

1. For a spacer deterioration between two subconductors, (Al and A2

)

60

~ 2 l 1 1 -+ r r r r r

s s

1 2 l -+-

r r r r r [ ... ] =

s s (4-9) rf 1 3 1

r r r r

1 3 --r r r ,r

2. For a spacer deterioration between three subconductors, (Al' A2 and A3)

1.+ 2 1 1 -- --r r r r r

s s s

~, 1 2 1 -+-

__ .1 r r r r r

[ ... ]= s s s (4-10)

rf 1 2 +-

r r r r r s s s

3 r r r r

'3. For a spacer deterioration between four subconductors, (Al' A2' A3 and

61

3 r r r r s s s s

3 1 1 --r r r r s s s S

1 [J.I J= {4-11} rf 1 1 3 1

r r r r s s s s

1 . 1 3 r r r r s s s s

The three phase load matrix is given by :

Since the load remains connected in this case irrespective of the damaged

spacer, the load matrix does not change its form.

4.3 Practical Example and Computed Results

A typical 735 KV three-phase four bundled conductor transmission line of

204.8 miles, as shown in Figure 2-1 of Chapter Il was taken as the basis of the numerical

example. The load is assumed to be 300 (.8 + i .6) ohms per phase.

62

The spacers were assumed to be located at spans of 1 /20 miles along

the transmission li ne.

The effects of resistanceand capacitance of spacers were studied separate Iy.

First, the·capacitance was varied from 1 pf to 100,000 pf while the resistance was kept "

. constant at 1000 M-ohms. The computed results show that the capacitance has practically

no effecton thecurrent distribution in the subconductors. Next, the resistai1ce of the

spacers was varied from 100 M-ohms to a few K-ohms while the capacitance was kept

constant at 100 pf. From the computed results, it was found that there was no unbalance

in the current distribution in subconductors when the resistance per spacer exceeds 4 1< -ohms.

As the·resistance is reduced below 4 K-ohms, the current distribution at the receiving end

becomes unbalanced. The unbalance becomes more pronounced as the resistance decreases.

It should be pointed out that the vector sum of the subconductor currents is, in spite of their

unbalance, the same as the total phase current of the balanced system. Table 4-1

illustrotes the phosor of the subconductor currents at the receiving end for spocer

resistonces of 3.75 K-ohms.

No unbalonce of the subconductor currents at the sending end cou Id

be noticed for spacer resistonces obove J.5 K-ohms. Table 4-2 shows the phosor

of the subconductor currents ot the sending end for spocer resistonces of 1.3 K-ohms.

It should be, however, pointed out that the unbolonces in the current distribution

in subconductors ot the sending end ore insignificont compared to those at the recei­

ving end.

-.

Phase A

Phase B

Phase C

Phase A

Phase B

Phase C

TABLE 4...;1. RECEIVING END SUBCONDUCTOR CURRENTS

299.2'66°

258.8 [301 __ 50

293.9 i 180e.

2

SPACER RESISTANCE 3.75 K - OHMS

{AMPERES, MAGNITUDE}

Subconductor

3

0

4

268.2 (65.2 0

337.1 J63.~ 296.3l65.3 0 0 0 317.3 1 311. 2 320.2/307.5 341.41316

0 0

0

301.51 185 303.7~ 343. 1 /193.5

TABLE 4-2. SENDING END SUBCONDUCTOR CURRENTS

1

207.1 199.5 0

224.3/340 0

0 239.9~

SPACER RESISTANCE 1.3 K- OHMS

(AMPERES, MAGNITUDE)

Subconductor

2 3 4 0 0 0 206.1199.4 211.2 L!Ql 209. 9 Ll.Q.l.

0 0 225.41 341.5 226.7/343.5 226.8 1343.5 0 0 0 241.9~ 236.81 219 238.6 t1JL

0

Phase Currerit (Vector Su.m)

1197.0

1233.0 0

1236.0

Phase Current (Vector Sum)

834.2

903.0

957 .1 ,

~.

.,

0-CA)

• 64

ln order to compute the effect of one damaged spacer on the suhconductor

current distribution, many points along the line were chosen as the location of the hypothe­

tical damaged spacer. The resistance ·r is assumed to be 1010 ohms which has

-3 -6 practicallyno effect at 011, and the resistance r was assumed to be 10 and 10 s

ohms. A third value of .5 ohms for r was taken at random in order to observe the s

effect of a considerably higher value of r on the subconductor current distribution. s

These calculations were carried out on the assumption that the damaged spacer is always

in phase A.

From the computed results, Table 4-3 was set up to illustrate the influence

of the·resistance of the damaged spacer on the subconductor current distribution of phase A.

It shows that r of 0.5 ohms has no effect on the subconductor current distribution for s . this specific transmission line. -3 For r of 10 ohms there is no effect on the current s

distribution of the subconductors at a point close to the receiving end but in general, it

may cause unbalance in the subconductor current distribution.

ohms appears to be the most pronounced.

The effect of r of 10-6 5

Table 4-3 also shows that only currents in the subconductors forming a

metallic oontact due to the damaged spacer become unbalanced ; the currents in the rest

. of the subconductors of the some phase and in the subconductors of the other two phases

remain the same.

...,. \ } • TABLE 4-3.

RECEIVING END CURRENT DISTRIBUTION IN SUBCONDUCTORS

OF PHASE A WITH THE DAMAGED SPACER

Resistance Location of Dam- D{lmoged Subconductor of Damaged age , Miles from Connection (Ampere, Real and Imaginary Parts) .Spacerlohms Sending End Between 1 2 3 4 Normal System 102.3/277.9 103.2/278.6 99.9/284.4. 100.8/285 •1

0.5 Same as Normal System for any fouit location Al and ~

204.8 Al ' A2 and A3 Sorne a3 Normal System for any fault loccltion Al ' A2' A3 and A4

Al and A2

171.4/121 •3 34.1/4.35 •3 99.9/284.4 100.8/285 •1 10-3 184.32 Al' A2 and A3 187.5/144 •2 147;8/36•6 -29. 9 /660~ 1 1 00.8 /285 ~ 1

Al ' A2, A3 and A4 High degree of unbalance

102.4 Ali fault cases High degree of unbaJance

20.48 Ali fault cases High degree of unba!ance

~ eirst J Al and A2 102,2/277 .9 103.3/278 •6 99.9/284-.4 100.8/285 •1 20.

Al ' ~ and A3 104jL75. 103.1/279. 98.4/287. 100.8/285 •1

Spacer

Al 1 ~, A3 and A4 98ï286o 102.6/2800 103.6/277 .6 102.2/282.4 0--111

·'. Re;istonce Location of Dam­of Damaged age , Mi les from Spo\::erp ohm'3 Sending End

204.8

Damaged Connection BeiVleer.

o TABLE 4-3 (CONTiD).

Subconductor (Ampere, Real and Imaginary Ports) ·t 2 3

.A, and

A, ' A2 and A3

A 3l85.6/T 14-9.9 2 -2980.1/_593 .4 99.9/284,4

-64865.5/-41843.8 54630.4/35595.3 1054-0.6;7089.5

• 4

100.8 fi85.1

100.8/285.1

~Jd' ~!' A3 -13424.4;_4421.9218185'/120365. -126368'/_.57490.6 -77986.4/_57326.9 10-6

184.32 Ali fouit cases

Al and A2

102.4 Al ' A2 and A3

Al ' A2 ' A3 and A4

Al and A2

20.48 Al ' A2

..h 20.

and A3

Al ' A2 ' A3 and A4

~First J Al and A2

Al ' A2 Spacer . and A3

Al ' A2, A3 and A4

High

High

High

100.3/274 •7 94i171 •2

60.3/268.7

degree

degree

degree

105.2/281.8 110.4/285 .. 7

109.6/284•3

of

of

of

99.9/284•4 101/284 .•

110.8/283 •4

unbalance

unbalance

unbalance

100.8/285•1 100.8/285.1

125.5/289.6

0.. 0..

67

4.4 Discussions

The spacer of the bundled conductor transmission line can be represented

bya model consisting of a resistance in parallel with a capacitance, therefore, its

electrical effect on the wh61e circuit of a transmission line may be studied by the Multi-

·conductor Method with the proper chain matrix to include the spacers. This analysis

illustrates that the capacitance of the spacer does not have an important effect on the

circuit due to its high capacitive reactance (100 pf is equivalent to approximately

30 M-ohms at 60 cycles per second power frequency), whereas the resistance of the

spacer is found in certain cases to influence the network considerably.

Under normal conditions, spacers with very high resistance, have no effect

on the subconductor current distribution. However, the analysis reveals a high degree of ........ /

unbalance for low spacer resistance. In the numerical example, the Iimit below which

large unbalance between subconductor currents was obtaîned at the receiving end, was

4 K-ohms. It should be poînted out that the limit value of the resîstance of spacers

dîffers for each specifie line, depending on the length of line, load, configuration,

etc.

From the computed numerical results of the example considered, the

features of the phenomena of unbalance in the subconductor current distribution due

to low spacer resistances can be summarized. When the resistance of spacers is

higher than the 1 imit value of a given transmission line, the spacers do not affect the

subconductor current distribution. Under normal operation, the phasors of the suo-

68

conductor currents of the same phase are very close to each other (see Tables 2- 1

to 2-3). However, unbalance in subconductor currents, changes in both magnitude

and phase angle, can be noticed when the spacer resistances are below the limit

value. This unbalance becomes more pronounced for lower spacer resistances. As

the spacer resistance reaches a certain low value, a circulating current may be pro-

duced between one pair of subconductors and the other pair of the bundle, but the'

phase currents at the receiving end ( the sum of the subconductor currents ) mey remain

unchanged. The subconductor currents at the sending end will also become unba-

lanced but only for a much lower value of spacer resistance.

This phenonon can be explained as follows. The sending end current

distribution is not so fiable to become unba/anced as is the receiving end current when

the resistance of spacer drops. From the equations shown in Appendix A, it is seen

that the sending end currents are determined by the chain matrix of the 1 ine and the

load matrix for a given sending end voltage, while the receiving end currents are deter-

mined by the sending end currents and the cha in matrix of the 1 ine. The spacers of the

line may be treated as a single e/ectrical component which is included in the chain

matrix of the line and if the resistance of the spacers drops, the chain matrix changes.

1 f the load matrix is the dominant part then the sending end currents experience only

negligible changes due to the variation of the resistance of spacers. On the other hand,

changes of the receiving end currents may be noticed due to the different chain matrix

of the line for the same sending end currents. Furthermore, the current in an individual

subconductor at the receiving end is determined by the value of its corresponding elements

of the chain matrix of the line. Unequal subconductor currents in a bundle can be

expected. Under sorne extreme cases, one subconductor current may be very small or

even opposite to the current of the balanced system.

69

With regard to the det~rioration of spacers, it is important to consider the

possibility of having a damaged spacer as the analysis shows that unbalance in subconductor

currents may result. The unbalance of current distribution of the subconductors having

metalliccontacts across a damaged spacer is quite similar to the phenomena of unbalance

insubconductor currentdistribution due to low resistance of spacers.

It is important to point out that the sending end subconductor currents may

remain the same or change slightly while the·receiving end subconductor currents experience

some drastic changes which might weil be beyond the possible currentcarrying capacity of

the individual subconductor. Under such circumstances, the sending end current will not

·trip the circuit breaker to interrupt the supply. Such unbalances should be taken into con-

sideration in"the design work to provide proper protection for the subconductors of the

transmission li ne •

The analysis has shown not only that it is possible to study the electrica/

characteristics of spacers by the Mu/ticonductor Method, but a/50 that in fact such a study is

. of great importance in order to determine the proper current distribution in the subconductors.

Since a low spacer resistance or a damaged spacer may cause serious un­

balances in the subconductor current distribution, it is suggested that consideration shou/d

be given to the use of spacers having very high resistance. And since the spacer should

have rather high resistance even under conditions of damaged insu/ation, the use of non­

metallic spacers appears to be a solution which should be given further attention.

70

CHAPTER V

ANAL YSIS OF THREE PHASE TRANSMISSION UNES

5. 1 Asymmetry between Phases

Except for those bund/ed conductor transmission /ines operating at voltages

greater than 220 KV, most of the transmission line configurations are made of a single

·conductor per phase 1 referred to as the "three-phase three-wire system ". For the pur-

pose of distinguishing these from the bundled conductor configuration, only the latter are

specified by the term "Bundled"; otherwise the term "three-phase transmission line Il

implies that the line is of the single ~onductor per phase configuration. As the three-

phase transmission lines are not spaced equilateral/y and may not be transposed, there must ,,-.

he sorne degree of asymmetry between phases. If the resulting asymmetry between phases

is sI ight, the three phases are considered to be balanced provided the appl ied sending end

voltage and the receiving end load are balanced. Therefore the single phase equivalent

•• he d h h .• l' 22,42 .Clrcult can use to represent a t ree-p ase transml.SSlon me.

The single phase equivalent circuit of a three-phase transmission line may

he of the form that includes only the series resistance and inductance of the. transmission

line. They are shown as lumped parameters and are not uniformly distributed along the

line, moreover , the distributed capacitances of the line are neglected. This kind

of single phase equivalent circuit is acceptable for a short transmis'sion line as the

capacitances are very smal/ and have negligible effect on the accuracy of results.

•

--.. , . . ./'

71

The nominal - 1r circuit is often used to r~present a transmission line of a medium

length, and the equivalent - 1r circuit is often used to represent a long transmission

line. Bath of these models use the lumped parameters, instead of the distributed

parameters, of the transmission line, and are accurate only as far as the voltages

and currents at each end of the transmission line are concerned. 22 However the

exact solution of such a single phase transmission line, irrespective of the length of the

transmission 1 ine, can be obtained and the voltage and the current at any point along

the transmission line are obtained by using the equation which takes the distributed

parameters into account.

With regard to a three-phase system, it is necessary to point out that by

using the single phase equivalent circuit the asymmetry between phases is not taken into

account. Therefore, the results from the equations of a single phase equivalent circuit

of a three-phase transmission line with distributed parameters are unsatisfactory for many

appl ications. No equivalent sigle phase method has ever been developed to investi-

gate and calculate the voltages and currents of a three- phase transmission line taking

into account the effects of both distributed parameters and phase asymmetry.

If the asymmetry between phases is assumed to be more important than the distri-

buted capacitances of the line,then a three-phase circuit can be formed without taking the dis-

tributed capacitances into account. This three-phase circuit consists of only the lumped resistan-

ces and inductances, and the problem is then to solve an unbalanced three-phase circuit.

-f •. . • 0.

72

Since the Multiconductor Method is valid for the study of the bundled

conductor transmission line, it is. possible to use the Multiconductor Method to study three-

phase three-wire transmission lines with n=3 including both distributed parametersand asymmetry.

It is the pUfpose of this investigation to find out the differences between

the results calcul.ated by the Multiconductor Method with the phase asymmetry included,

and the results calculated by the single phase equivalent circuit without taking the phase

. asymmetry into account.

5.2 GMR of Bundled Conductors

The "Geometrical Mean Radius" for bundled conductors is defined as the

radius of a single conductor which would have the same inductance and capacitance as the

bundle, when located at the bundle axis. The introduction of GMR, in fact, reduces ail the

calculation regarding inductance and capacitance for a bundled conductor to that of a

single conductor of radius GMR.1,43

The relation between GMR and the radius of the subconductor of a bundle, R

• 43 IS

73

".- where S is the spacing between subconductors of a bundle,

b is the number of subconductors per bundle of one phase,

K is a constant and the values of K, for capacitive equi-

valence, are listed as follows :

b 2 ?" .. ' 4 5 ·6

K 1. 1. 1.09 1.212 1.348

(A correction factor is needed for the inductive equivalence).

This formula is applied to a three-phase bundled conductor transmission

line to obtain a three .. phase GMR equivalent transmission line with a single conductor

per phase. Although th is method is frequently used in practice, it should be pointed out

that here the asymmetry between subconductors of the same phase is completely ignored.

74

5.3 Computed Results and Discussions

The computations were' based upon a three phase GMR equivalent of a

four subconductors per phase transmission 1 ine as shown in Figure 2-1. The three-phase

GMR configuration is shown in Figure 5-1 and can be assumed to be a normal three-phase

three-wire transmission line. For the purpose of comparison, different lengths of trans­

mission lines, e.g. 204.8 miles, 102.4 miles and 51.2 miles,were employed. The

three-phase balanced load wasassumed to be300 (.8 + i .6) ohms, 600 (.8 + i .6) ohms and

600 (.9 + i . r.T9) ohms per phase respectively.

The line constants of the three phase GMR equivalent line is shown in

Table 5-1 together with the line constants obtained from the bundled configuration. It

shows that there is some discrepancy between them with a maximum difference 3.9% .for

the self-inductive reactance of phase B.

Table 5-2 shows the sending end currents obtained from the phasor sum of

the subconductor currenl's, the phase currents .. of the three-phase GMR equivalent line

and the current of the single phase equivalent circuit. The unbalance between phase

currents in a three-phase bundled conductor transmission line has already been shown in

Fi gure 2-8 of Chapter Il .

Figure 5-2 shows the maxmum error of the sending end phase current obtained

from the three-phase GMR equivalent line in percentage of phase current of a .bundled

conductor transmission 1 ine plotted as a function of the length of the transmission 1 ine.

ln Figure 5-3 the average sending end phase current of a three-phase transmission line

and the current obtained from its single phase equivalent circuit is shown as a funçtion

CQ

~. • ·1

@.JI .~~ .

A

FIGURE. 5-1.

75

B c

"r.-.::. 0':' 0 bJ c .... n r=") ~ C:::7/.:'· CC-I D '., \. '; - . 0 'I--'.~~O ~~-) _'-.\

" .. .J,J \. \d) Û GI;,,;;I . , .

THREE PHASE GMR EQUIVALENT OF A 735 KV

BUNDLED CONDUCTOR TRANSMISSION UNE.

'.

.. ./

B = c

B = c

76

TABLE 5-1.

UNE CONSTANTS (PER PHASE)

BY BUNDLED CONFIGURATION

0.59339 .083824 .030842

0.083824 0.59058 .083824 ohms / mile

0.030842 .083824 0.59339

7.13771 -.987865 -0.231219

-.987865 7.30210 -0.987865 micro-mhos / mile

-.231219 -.987865 7.13771

BY THREE PHASE GMR EQUIVALENT CIRCUIT

0.616268 0.083824 0.030842

0.083824 0.613461 0.083824- ohms- / mile

0.030842 0.083824 0.616268

7.13574 -.987431 -.230908

-.987431 7.29987 -.987431 micro-mhos / mile

-.230909 -.987431 7.13574

77

ctt TABLE 5-2.

SENDING END PHASE CURRENTS (AMPERES)

Length of Load / Phase By Bundled ' 'By 3 Phase By Single Phase

Line, Miles Ohms Configuration GMR Equivalent Equivalent Circuit

'A 'B 'C . 'A 'B 'C

300(.8 + j .6) 834. 903. 957 825. 893. 947. 869.

204.8 600(.8 + j .6) 512. 590. 618. 510. 587. 616. 533.

600(.9+ j./T9) 634. 717. 74~. 632. 715. 74l. 655.

300(.8 + j .6) 1071 . _ 11 04 . 1140. 1066. 1098. 1134. 1096.

102.4 600 (.8 + j. 6) 510. 539. 570. 509. 538. 569. 540.

600(.9+ i./T9) 579. 614. 642. 578. 612. 64l. 606.

c:; 300(.8 + i .6) 1230. 1247. 1267. 1227. 1243. 1264. 1244.

51.2 600(.8 + i .6) 589. 600. 618. 589. 599. 618. 607.

600 (. 9 + i /.T9) 623 . 637. 654. 622. 636. 654. 639.

c

% 78

a. ,LOAD=300L8-:-jo6) O~-iMS b. LOAD=600C8+j.6)OI-~MS

c. LOAD= 600(~9-:- jJ.19')O:-lMS

2.0

'1.0 1

o

«-~.

10- 1 L.. --Î

FIGURE 5-2. MAXIMUM ERROR OF SENDING END PHASE CURRENT OF GMR EQUIVALENT IN % OF PHASE CURRENT OF BUNDLED CONDUCTOR CONFIGURATION VS. LENGTH OF THE TRANSMISSION UNE.

~ . ('

'''- #

1200

.. 1100

1000

9 00

800

700

500 ~

r 1

o. 51.2

a· LOAD~300(.8"'J·6)OHM5 b. LOAD= GOO( .. 9+jf.i9) OHMS 79

. C. LOAD = 600 (. 8+j.6) OHMS

1 1 1 cP EQU-CIRCUIT . _____ 3cP SYSTEM

/ ~/ .,-

---- , -' ....... ..,......-----.- . ---" ----- ......

102.4 20!~.B MllES

FIGURE 5-3. AVERAGE VALUE OF SENDING END PHASE CURRENT OF A THREE PHASE TRANSMISSION UNE AND THE CURRENT . ,

FROM ITS SINGLE PHASE EQUIVALENT CIRCUIT VS. LENGTH 'OF THE TRANSMISSION L/NE.

1

15 Cl. LOAD=600 (.8-:-J:.6) Oi-}j\t~s

80 LOAD=600 (.9~-jli9) Or-~MS

, . iD.

·C. LOAD=300 (.8-}-j.6) Ol-iMS.

10

.

5

o 51.2 20L: .. 8 MI l r=-_1-~

FIGURE 5-4. MAXIMUM ERROR OF SENDING END CURRENT OF SINGLE PHASE EQUIVALENT CIRCUIT IN % OF PHASE CURRENT IN A THREE PHASE TRANSMISSION lINE VS. ltNGTH OF THE TRANSMISSION LlNE.

81

of the length of the transmission line. Figure 5-4 demonstrates the maximum error

of sending end current of the single phase equivalent circuit in percentage of phase current

in a three-phase three-wire transmission line again as a function of the length of the trans-

mission line. From the computed results, it seems that the three phase GMR equivalent . circuit offers satisfactory results in calculating the phase currents of a bundled conductor .

transmission line, if only the phase currents cire considered (ignoring the current dist-ribu-

tion in subconductors of the same phase). The current obtained from'the single phase

equivalent circuit of a three-phase three-wire transmission line is always lower than the

average value of the phase currents of a three phase system. The difference between the

average value of the three phase currents and the current calculated from its single phase

equivalent circuit is small especially for short or medium transmission line, but the error

incurred in using the current obtained from the single phase equivalent circuit for indivi-

dual phase currents of a three-phase system is quite significant, having a value of more

than ten percent for a 200 mi le long transmission 1 ine as shown in Figure 5-4. From

Figure 5-3, it is seen that the current of 204.8 mile line with light load of 600 ohms/phase

is greater than the current of 102.4 mi le 1 ine due to the fact that the larger capacitive

current of the longer line is much greater than the inductive part of the load current with

a light lood of 600 ohms /phase. Therefore, the sum of load current and capacitive

current of the line results in a higher value of current.

It is important to point out that in using the GMR te, calculate the phase

value of line constants of a bundled conductor configuration, sorne error may occur.

82

ln order to find accurate values of line constants, it is suggested that the bundled conductor

configuration be used, as shown in Appendix ~.

The Multiconductor Method may be employed for the study of the three-

phase three-wire transmission fine with distributed parameters and phase asymm~try per-

mitting the determination of the voltage and current at any point of . interest along the line.

Moreover, it is very convenient to study the system with unbalanced applied volatges or

unbalanced loads simply by substituting the unbalanced voltage phasors and unbalanced

load impedances into the sending end voltage matrix and load matrix respectively.

ln practice, the three-phase transmission line may not be of a ,constant

height from the ground throughout, or may have some external components in the line;.

then the chain matrix can be modified in order to include them as discussed in Chapter Il.

;For the transposed line, the chain matrix of the linè will be the product of severaJ matrices,

of which each represents a section of the transmission 1 ine.

83

CHAPTER VI

FAULTS IN THREE PHASE TRANSMISSION L1NES

6.1 Symmetrical Component Method

The symmetri ca 1 component method was presented by Dr. C. L. Fortescue

in 1918. 19 Since then, it has been the subject of many important articles and experi-

1 . .. 44,45,46,47 d . . d f th t fui menta investigatIons an IS recognlze as one 0 e mos power

tools for dealing with unbalances in the three-phase system.

ln fact, the symmetrical component method is similor in concept to the

method of Tensor Analysis, introduced by Dr. °Gabrie.l Kron oto the electriéaJ

. . f· Id 48 englneenng le • The rules of Tensor Analysis are applicable to the symmetrical com-

ponent method. By changing the "Reference Frame ", any three vectors con be analyzed

into three sets of balanced vectors. Appl ied to three-phose currents, any three currenl:

phasors con be analyzed into three sets of balanced currentsi any three voltage phasors

carl be analyzed into three sets of bala~ced voltages.

Thus, the voltages or currents at a point of fouit in a three-phase system

are analyzed into three sets of symmetrical components known a:; the positive phase

sequence, the negative pha~e se9uence and the zero phase sequence. Positive phase se-

quence voltages and currents are then determined throughout the system; this is a problem

of balanced voltages and currents and hence is easy. Negative phase sequence voltages

and currents and zero phase sequenc.e voltages and currents are determined independently

84

as weil. Then, finally, the actual unbalanced three-phase voltages or currents at any

fault point of interest in the system are found by adding the positive, negative and zero

phase sequence voltages or currents respectively. Thus the solution of a difficult problem

involving u'nbalanced currents and voltages is reduced to the solution of three easy pro-

blems involving only balanced currents and voltages. 13 One of the unique advan-

tages of using symmetrical components is that a solution generally can be determined

by desk calculations.

As mentioned in Chapter V, the three-phase transmission lines. are

assumed to be batanced in dealing with many types of analyses.22

,42 The same assumption

applies to the unbalanc:ed three phase system without taking into account the distributed

capacitances when the symmetrical components method is adopted. Thrs approximation

leads to a simplification of the problem and general/y offers an acceptable result without

appreciabfe error. If it is necessary to toke into account the distributed capacitances' of

a long transmission line, the problem becomes more complicated. Clarke has written one

chapter in her book showing how to include the distributed capacitances of the line in'

the sequence .constémts by using the correction factors from sorne chorts, developed on 'the basis

of a single phase equivalent circuit of an .unsymmetrical three phase system. 46 ln fact,

for the accurate analysis of the unbalance problem of a system, both the asymmetry be-

tween phases and the distributed capacitances of the line should be considered,leading

to a more tedious and complicated procedure, 49 and the results may still not be com-

pletelyaccurate. In practice, the single phase equivalent circuit is the most frequently

adopted in connection with the appl ication of symmetrical components.

, -'

85

The Multiconductor Method is applicable in bundled conductor system,

either under normal operation or under fault, and in balanced three-phase systems as

weil. The purpose of th is investigation is to apply the Multiconductor Method in

dealing with the unbalanced three-phase transmission line instead of using the popular

symmetrical components.

It is important to point out that the chain matrix of the Mu:ltlconductor

Method involves ail the electrical components in the circuit of the system. Thus the

distributed capacitances of the line and the asymmetry between phases of the line can

ail be included conveniently and accurately in this method. The Multiconductor

Method can determine the current and voltage in the fault point and the voltages and

currents in the faultless phases as weil. Moreover, if there is more than one fault

occurring simultaneously in a system, the method is still applicable.

The analysis is carried out on a three-phase transmission line, and yet

the Multiconductor Method can be extended to the study of more' complicated power

networks.

ln this chapter, the numerical example is based on the same three-phase

transmission line configuration as shown in Figure 5-1 of Chapter V. The calculations

are carried out using both the Multiconductor Method and the symmetrical component

86

method under the seme fault condition. The computed results are used to show the

validity of applying the Multiconductor Methodin solving the unbalanced three-

phase transmission line.

6.2 Multiconductor Analysis

Appendix A gives the calculation of the sending end currents as weil os

the receiving end currents and voltages for a transmission line with n conductors of

known line constants and sending end voltages which are assumed to be constant. In this

case of a three-phase thr'ee-wire transmission line, n= 3 and b= 1.

ln the case of a fault condition, the chain and load matrices have to be

modified.

The transmission line is divided into three parts:

1. The unfaulted line of length Il'

2. the section at the fouit point and

3. the faulted line 12

containing the unfaulted phases of length 12 •

For each of these parts a chain matrix is set up and the product of the

individual matrices gives the total chain matrix of the line:

(6-1 )

• where [fJ11 J is the chain matrix from the sending end to the

fault point of length Il ' which is determined

for ail fault cases by E9uation (A-6) of Appendix A.

[fJf J is the chain matrix ot the fault-point, and

[fJ12 J is the chain matrix of the unfaulted phases from

the fault point to the receiving end of length 12

•

The fault conditions, which will be considered, are:

87

(a) One phase open circuit or short circuit to ground, Figure 6-1 (a).

(b) Short circuit between two phases, Figure 6-1 (b).

(c) Short circuit to ground of two phases, Figure 6-1 (c).

(d) Short circuit to ground of three phases, Figure 6-1 (d).

ln the load matrix the impedances belonging to the faulted phases are re-

ploced by fouit impedances to ground which, theoretically, are zero or infinity according

to whether they have to represent a short circuit or an open circuit. For practical compu­

tational purposes the short circuit is modelled bya 0.5 ohm resistance and the open circuit

by a resistance of 1010

ohms.

(a) One phase open circuit or short circuit to ground.

Forthiscase, [fJfJ isthe (6x6) unitmatrixand

••

--.. ~

~'

! ~ -=~Zr fi . a ~ 1

S fi ~. . C-, '%Zr --,.----,

8 f 1 1

1 1 f ! --- --, R

. f , 1 , . __ J_

.... ·_····-Fe9 ,

1

1 ~ D ,1

2 1

~GJ

f>~ é~

~ .~ .,,1'" "") . t y.~ J 'v-- ...,~ .. ~ 9 t· n . ~ ..... c=nr= ... __ = ___ ' ....J!'

! i i ) ft !

!

FIGURE 6-1. FAULT CONDITIONS.

(0) SiNGLE PHASE FAULT.

CE i .-' ~ ft' 1 J 2

. (D»

Il ,./ ') lL;J

SHORT CIRCUIT BETWEEN TWO PHASES.

88

, J 1 ]R 1 , , "

(c) SHORl CIRCUIT TO GROUND OF TWO PHASES.

(d) SHORT CIRCUIT TO GROUND OF THREE PHASES.

C_

[~12 ]

where

where

89

O 0 0

(12

1 Zl 0 cosh (1 ;zryl) 0 sinh yi )Zl 2 0

= 0 0 0

0 sinh (1 ./ yi ZI )ZI -1 2 0

0 cosh (12

./ yi t l)

ZI = is the (2 x 2) impedance matrix of the unfaulted

line, per mile,

yi = is the (2 x 2) admittance matrix of the unfaulted

line, per mile,

ZI = is the characte ristic impedance of the unfaulted 0

line, and

ZI = [./ ZI y'J-1 ZI = [y'fl [/y' ZIJ 0

The lood impedance matrix is

o o

o o

o o

Zf is the fouit point impedance to ground and is assumed

to be very large or very sma Il according to whether

the fouit is an open circuit or short circuit to ground.

ZI is the load impedance per phase for the unfaulted phases.

(6-2)

(6-3)

90 (-(b) Short circuit between two phases.

The fault matrix [~fJ is given by

1

U 1 0'

- - -I-

-' - 0 I

zf zf 1

[~ ] = 0 U (6-4) f zf zf

0 0 0

where zf is the short circuiting impedance between

phases A and B at the fault point 1 and

U is the (3 x 3) unit matrix.

The chain'matrix of the unfaulted phase C of length 12

is

0 '0 0 0 0

0 0 0 0 0

0 0 cosh O2 Izy) 0 0 sinh O2 IZy )zo [~12J = (6-5)

0 0 0 0 0

0 0 0 0 0

0 0 -1 sinh {I2 IYz)Zo 0 0 cosh (12 1Yz)

where Z and y are the impedance and admittance per mile - of phase C,

Z is the characteristic impedance and is equal to 1 z 1 y • 0

ce ln this case the load matrix

Z S

o

91

o

Z = L

o Z S

o (6-6)

where

o o

Z is a very large impedance (1010

ohms) and s

replaces the load impedance per phase for the

faulted phases A and B. It is used for

digital computationa' purposes to simulate an

open circuit,

Z, is the load impedance for phase C.

(c) Short circuit to ground of two phases.

For this case, [l-IfJ is the (6 x 6) unit matrix and [1-1,2 J is the

sorne as in Equation (6-5).

The load matrix is given by

z = L

where

o

o

o o

Zf is the fault point impedance to ground,

z, is the load impedance for phase C •

(6-7)

(d) Short circuit to ground of three phases.

For this case, both [fJf] and [fJ,2J are (6 x 6) unit matrices.

The load matrix is given by

where Zf is the fouIt point impedance to ground.

6-3. Comparison of Computed Results obtained by the Multiconductor Method and by the Symmetrical Component Method

92

(6-8)

The three-phase transmission line of 204.8 miles with balanced 735 KV

sending end voltage and balanced load of 300 (.8 + i .6) ohms / phase was taken as the

basis of the computations. Its configuration is shown in Figure 5-1 of Chapter V and

its line constants are listed in Table 5-1 of Chapter V as weil.

The calculations covered both single phase faults (open circuit

and short circuit) and faults of two phases. Both the Multiconductor Method and the

symmetrical compone nt method were employed for the fouIt computations. By use of the

93

Multiconductor Method the sending end and the receiving end voltages and currents of

ail three phases can be obtained.

Figures 6-2, 6-3 and 6-4, show the dependence of fault current on the

distance of the fault point measured from the sending end, for short circuit of one line to

ground, between two lines and two lines to ground. respectively.

The results are compared to those obtained by symmetrical components in

Table 6-1. For this method which assumes a symmetrical configuration of the three phases,

average values of 0.6153327 ohms/ mile and 0.0661633 ohms/ mile for self and mutual

inductive reactances are used. A maximum discrepancy of 6% betwéen the results

of the two methods is noticed for the case of short circuit between two phases in Table

'. 6-1 (b). , .. ./

One of the advantages of the Multiconductor Methods is that the currents

in the faultless phases are obtained simultaneously with the fault currents. These currents

are shown, both for the sending and receiving ends in Table 6-2. The current values

were obtained for ail possible locatïons of the fault point between sending end and re-

ceiving end.

An interesting result of the present calculations is the current in phase A

when this phase is interrupted and connected to ground through a very large impedance

f(',\ V

,-'>

1 ,

• .;>

·0

1 i

r- f'?,~ .- .-.... .. \. .. ti ... "';...J

.' ,., 1.' H?/':', ''-''', \....,; \,J. ~...,w~

~o r." .... ,-:-,,# · ... , .. -:-·r:-1 V'V<tJW"'-..,,,,,..,,,,"..j

FIGURE 6 - 2. FAULT CURRENT (ONE PHASE GRQUNDED) VS. LOCATION OF FAULT POINT.

94

~.

. c:.".;::;:.. ~u

'':'I!~\ GV

..

[0

o O·

l j . . p;.. !

5"

.., 0 !.) "'" ....... '4'-10

FIGURE 6 - 3. FAULT CURRENT (SHORT CIRCUIT BETWEEN TWO PHASES) VS. LOCATION OF FAULT POINT.

95

,-= ,. '1 i; 1.

'.

r '" i , , 1

~

()

..

o

ij

I~ j

l J ., J Î

)

1 J 1 . , • .J

'1 '.1

1 'J

f'I ., ..

',,~,., r, ,'" 1: ,! /., ,.

G0 L .:·U

FIGURE 6 - 4. FAULT CURRENTS (rWO PHASES GROUNDED) VS.

LOCATION OF FAULT POINT.

96

.., ) " .... ', . -.. '.,----.. --.... ' ----··-'--,--~e

TABLE 6-1. FAULT CURRENTS AT THE FAULT POINT, IN KILO-AMPERES -' FouIt Point Measured From Sending, End, Miles

Type of FouI t 20.48 40.96 61.44 81.92 102.4 ,122.88 143.36 '163.84 1.84.32

Multiconductor Method

(0) Phase A to Ground (lA) 33.6 16.83 11.24 8.46' 6.79 5'.69 4.91 4.32 3.87

(b) Short Circuit be-tween Phases A and 33.75 16.88 11.25 8.45 6.77 5.66 4.87 4.27 3.82 ' B(I =-1)

A B

* (c) Short Circuit to

Ground of Phases 37.78 18.7 12.44 9.34 7.49 6.26 " ,5.39 4.79 4.24 A and B" lA

lB 35.94 18.2 12.19 9.17 '7.36 6.16 5.3 4.66 4.17

Current to Ground (Phosor Sum of' lA ~nd lB), 29.68 14.94 10.02 7.57 6.11 5.14 4.45 3.94 3.55

~

• 1 '-/ • TABLE 6-1. FAULT' CURRENTS AT THE FAULT POINT, IN· KILO-AMPERES (CONTlNUED)

Fault Point Meosured From Sending End, Miles

Type of Fault 20.:tf.8 40.96 61.44 81.92 102.4 122 .88 . 143.36 163.84 184.32

Symmetrica" Component Method'

(a) Phase A to Ground (I,J .3~ .65 . 16.83 11.22 8.42 6.73 5.6 .4.8 4.2 .. 3.74

(b) Short Circuit 00-tween Phases A 32.67 16.34 10.89 8.17 6.53 5.45 4.67 . 4.08 3.63 . and B (lA = -lB)

* (c) Short Circuit to Ground of Phases . A and B, . lA

36.99 18.26 12.12 . 9.07 7.25 6.03 5.17. 4.52 .4.0

'B . 35:05 17.77 1-1.9 8.95 7.17 5.98'· 5.13 4.49 . .·3~99

Current to Ground (Ppasor Sum of lA 9fld 'B) 30.33 15.19 10.13 7.6 6.08 5.07 . 4.24 ·3.8 3.38 .

* lA 'ags V by 84° . A

'B lags Vs by 95° .

•

--o. ex>

TABLE 6-2.

Currents in Faultless Phases, in Amperes.

(The fouit point was moved from scnding to receiving Em~

Type of Fôult

O. (a) Phase A Short circuited

Current in Phase B Current iOn Phase ë­Send.End Rec.End Send.End Rec.End

to ground 856.-859. 1258.-1261. 991. .1258.

o (b) Short cir­cuit between phases·À and B

O(c) . Phase A and B short circuited toground

Normal condi­tions 893.

850.-869. 1185.

94·( .. 950. 123).-1285.

947. 1227.

99

100

(1010 ohms), as shown in Figure 6-1 (a). The fault point was moved from the sending

end to the receiving end and the current through the 1010 ohms impedance was calcu-

lated by the multiconductor method. The results are.shown in the graph of Figure 6-5.

The current, represented by curve m, shows an increase as the fault point moves away

From the sending end, due to the distributed capacitance of the line, which is not ne-

glected. The sa me current determined by symmetrical components, neglecting stray

capacitances, which is given by curve S, shows invariance.

6.4 Discussions

The use of the Multiconductor Method for the analysis of the faults of a

three phase system, supplemented by the digital computer, has been proved successfully.

This method determines the fault current directly without the need of

symmetrical components. The results are accurate since phase asymmetries and ail distri-

buted parameters of the line can be·-taken into account. Further, the analysis supplies

both faulted voltages and currents, and voltages and currents in the faultless phases

simultaneously. It is important to point out that the currents in the faultless phases are

changed slightly as shown in Table 6-2 during the fault of a three phase transmission

line. The assumption that the currents in the faultless phases remain the same is not

correct.

This method also allows the determination of the voltage and current of

any point of interest along the unbalanced three phase transmission line. This is made

101

/1':1 . 0~ ~»··~1

. \1 ~

~T_ Cs) - - -n,. At ==-

ê-.• n\ =-v V

---,

FIGURE 6 - 5. OPEN CIRCUIT CURRENT VS. LOCATION OF FAULT.

(m) BY THE MULTICONDUCTOR METHOD.

(s) BY SYMMETRICAL COMPONENT METHOD.

102 feasible by Equation (A-9) shown in Appendix A.

If more than one fault occur simultaneously (either in different locations

or of different types) in an unbalanced system, It is very convenient to use the Multi­

conductor Method to solve the problem by obtaining an adequate chain' matrix according

to the actual fouit conditions of the system. An example of the calculation for two

simu/taneous faults ( phase to phase and phase to ground) by the Multiconductor Method

is shown in Appendix E.

For a de/ta-connected resistive load with a single phase fault occurring

in the middle of the line, a numerical example is also shown in Appendix E.

It is realized that the circuit of the transmission line shown in this chapter

is the simplest of ail three- phase circuits. Moreover, the sending end of the line is

assumed to be connected to an infinite bus of constant voltage and zero impedance as

mentioned. ln practice, the power system network is usuallyvery complicated with

interconnection between several sources instead of the single transmission line assumed

here. Further, the impedance of each separate generator is no longer zero and its

output voltage is varied under fault conditions. The analysis of such ~ complicated

problem is beyond the scope of the present investigation. The purpose of the analysis

of the simple circuit of the unbalanced three-phase transmission line is to verify the

validity of the application of the Multiconductor Method in dea/ing with the unba/anced

three-phase system. The analysis shown in this chapter may be considered as a good

reference for further application of the Multiconductor Method to the complicated power ~

networks.

103

CHAPTER VII

ANALYSIS OF FAULTS ON TRANSMISSION UNES

WITH THE FAULTED CONDUCTORS UNINTERRUPTED

7. 1 Introduction

The fault analysis on bundled conductor transmission lines in Chapter Ifl

and the fault analysis on three-phase three-wire transmission 1 ines in Chapter VI are based

upon the assumption that the faulted conductor or conductors are interrupted at the fault

point. This is true for the open circuit case, but may not be the case for the rest. As the

short circuit impedance is always the dominant portion compared to the load impedance,

the said assumption is applicable to the fault case with the faulted conductors uninterrupted

() without any appreciable error.

It is possible to analyze the faults of a transmission 1 ine, (either a bundled

conductor configuration or single conductor per phas~ configuration) including the portion

of the fau Ited conductors From the fault point to the receiving end. This analysis is put

into a generalized form i therefore, it may be applicable to either a bundled conductor

transmission line or to a three-phase three-wire transmission line.

The anàlysis will show sorne complexity in dealing with the fault of short

circuit to ground of one or severa! conductors. ln th i s case, the short

circuit to ground impedance matrix in parai lei with the chain matrix of the portion of the

line from the fault point to receiving end and the load matrix becomes the equivalent load

matrix with respect to the fault point.

[J

104

7.2 Analysis and Chain Matrix

Figure 7-1 is the circuit representation of three typical fault conditions

in a transmission line, either a bundled conductor configuration or a single conductor per

phase configuration, with n conductors. For a three-phase three-wire transmission

line, n = 3 and b = l, thus the number of conductors shown in Figure 7-1 will be equal

to the number of phases.

For the faulted transmission line, either a bundled conductor configuration

or a single conductor per phase configuration, with the faulted conductor or conductors un-

interrupted ot the fault point, the chain and load matrices have to be modified.

(0) Fault is not short circuited to ground :

The chain matrix of the line is

where [fJllJ is the chain matrix from the sending ènd "to the

fault point of length Il '

[fJfJ is the fault point chain matrix,

[fJ12 J is the chain matrix from the fault point to

the receiving end of length 12

•

(7-1 )

Both [fJll J and [fJ 12 J can be determined by substituting thei r respecti ve

lengths into Equation (A-6) of Appendix A •

[/-1 J can be obtained exactly as shown in Chapter Il r for bundled con­f

ductor transmission lines and Chapter VI for three-phase three-wire transmission lines. In

(tt '1 1 1 5 ,2

. , 1 ln 1 f

1 , l1

1 1 1 1 . r2

5 1 .: 1 1 . 1 n !

1 . b

~ <':;. <~ Sf <1 '. Y 1 , . 1 1

. 1 . 1

Î> ~ ~:; <; (JI U cr i 1 J ft

-&000

1 D

i 1

a i 1 1 1 ·1 1

1 (al) . l2 1

. 1 ~ 1 ~ j 1

1 1 1 1 1

(10) b ~ 1

i!: 1 sa· 1 ";) <" J" .~~I ____ ~~~~_i _________ ~ ___________ ~

R

R

12 l l s 1 : 1:-' ----r-: ---1 R

ln 1 1 .1 1 1 1 1

J l1.1 (C) 12 l FIGURE 7-1. FAULT CONDITIONS.

.105

(o) SHORT CIRCUIT TO GROUND OF SINGLE CONDUCTOR. ~) SHORT CIRCUIT BETWEEN TWO CONDUCTORS. (c) SHORT CIRCUIT TO GROUND OF TWO CONDUCTORS.

S = SENDING END. R = RECEIVING END.

~

this cose, the three phose lood motrix will be unchonged •

(b) Fouit isshort circu ited to ground :

As shown in Figure 7-1 (c), the current ot the fouit point of the line will

be expressed as

=

where

Ef

"

where

1 9

+

'f is the current matrix ot the fouit point of the line,

1 is the motrix of the fo ul ted current to ground, 9

" is the motrix of the lood current flowing to the

receiving end from the fouit point of the line.

ZL

= [~12 ] , r

U

Pl , = r

P2

Ef is the fouit point voltage motrix,

[~12 ] is the chain motrix of the section of the 1 ine from

the fouit point to the receiving end,

ZL is the lood matrix,

, is the receiving end current motrix, r

(7-2)

(7-3)

e

~ ,-

Pl and P2

are the submatri ces of the product

[PI2] [~l] Therefore, [ P

2 -1

" = Pl ] Ef

[Pl -1

or Ef = P2 J"

The relation between Ef and 'g may be expressed as

where Zf is the faulted impedance to ground matrix,

and

in which z , a high impedance, is used s

to substitute the e'ements corresponding to

the open circuit conductors in matrix Zf"

" in ternis of 'g is:

"

[p2

-1 ] Zf

, = P 1 g

'f = ,

+ '1 9

-1 = , + [P2 Pl zfJ

, g g

[U -1 = + P2 P 1 Zf ] 'g

Ef Zf

= , g

'f U + -1

P2 Pl Zf

107-

(1-4)

(1-5)

(1-6)

(1-7)

108

Since the voltage and current reJationsh ips between sending end and fault point con be

expressed os

where

E s

1 5

=

=

[1-'11 ]

91

9· 2

Zf

1 9

U+ -1

P2 Pl Zf

1 9

1 = 5

Es is the sending end voltage matrix,

J is the sending end current matrix, 5

91

and 92

ore the submatri ces of the product

For a given E ,1 con be determined. 5 5

(7-8)

(7-9)

~

Subsequently 1 the Ef 1 If 1 ,~ 1 'I and other pertinent values can

. be determined from the known values.

Ali the orders of the matrices are identical to those as described in

Appendix A •

7.3 Computed Results and Discussions

109

The numerical calculations are based upon the faults in:thr~& phase trans-

mission lines. The same specifie transmission line as described in Chapter VI. Was employed.

The faults of short circuit to ground of phase A and short circuit . t~ ground of phases

 and B were considered.

ln the case of the fault with short circuit to ground of phase A,

o

o

o

Z S

o

o

o

Z S

(7 .. 10}

ln the case of the fault with short circuit to ground of phasés A and

B,

Zf Z 0 f

Zf = zf zf 0 (7-11)

0 0 Z S

110

where zf is the short circuit to ground impedance, 0.5 ohms,

Z S

is the high impedance, 1010 ohms, to simulate

an open circuit for the unfaulted phases at the

faulted point of the 1 ine.

The computed results are listed in Tables 7-1 and 7-2. A comparison

of the results from Tables 6-1 and 6-2 and Tables 7-1 and 7-2 indicates that the

difference of the fouIt current, based upon the same fault condition, is·negligible. This

proves that the effect of the portion of the faulted conductor or conductors from the fault

point to the receiving end is 'nsi.gnif.icémt . .on the cQmputed :resuHs in genercl.

111

TABLE 7-1;

FAULT CURRENT AT THE FAULT POINT, (IN KILO-AMPERES).

Fault Point Measured From Sending End, Miles.

Type of Fault 20 .48 40. 96 61.44: 81. 92 102 .4 128 .88 143.36 163 .84 184 . 32

(a) Phase A to Ground 'A 33.6 16.83 11 .24 8 .46 6.79 5.69. 4.91 4.32 3.87

(b) Short Circuit to Ground of Phases A and B

'A 37.79 18.70 12.44 9.34 7.49 6.26 5.39 4.74 4.24

'B 35.94 18.20 12.19 9.17 7.36 6.16 5.30 4.66 4.17

Current to Ground 29.7 14.98 10.06 7.6 6.14 5.16 4.46 3.95 3.55 ·(Phasor Sum of 'A and lB)

TABLE 7-2.

SENDING END CURRENT IN FAULTLESS PHASES, (IN AMPERES).

Type of Fouit Phase Current

(0) Phase A to Ground 'B 859.

'C 99l.

(b) Short ci rcu it to

Ground of Phases

A and B IC 947. - 950.

112

CHAPTER VIII

FAULT CURRENT CALCULATION WITH THE GROUND WIRES INCLUDED

8. l Analysis

For a given transmission line with n conductors and ( w - n ) ground

wires, the analysis can be made by the Multiconductor Method in which the ground

wires are treated as individual conductors but of zero applied voltages. Thus the

transm ission 1 ine is assumed to have w conductors instead of n conductors.

two towers.

The chain matrix of the line with T towers is

( 8 - 1 )

where [!JI] is the ( 2w x 2w ) chain matrix of the line section between

It can be determined by Equation (A-6 ) of Appendix A.

[!Jw

] is the (2w x 2w ) chain matrix at the point of a tower (at

which the ground wires are grounded ).

= w+n

and [fJwgl is the

U 1 0 --1--1-o 0 _~ __ IU (8-2) o 1 fJwgl

1 1

(w-n)x(w-n) ground-wire chain matrix in which the grounded

condition is represented.

For a single ground wire, shown in Figure 8 - 1 (a),

[fJwg] = r1+rg ( 8 - 3 )

For two ground wires, shown in Figure 8 - 1 (b),

[

r 1 +r 9 , r 2]-1

[fJWg] = r 1 r2+r 9

(8 - 4 )

where r 1 and r 2 are the resistances of the connections from the

•

-,

G.W.

l '1 + r

9 --(a)

G.W.1

G.W.2

(b)

FIGURE 8-1. GROUND WIRE CONNECTIONS.

(a) SINGLE GROUND WIRE.

(b) TWO GROUND WIRES.

113

114

the ground wires to the tower foot, and r is the effective ground resistance 9

including the tower footing resistance and the resistance of the counterpoises.

8.2 Numerical Example

The numerical example is based upon the three-phase transmission

line shown in Figure 6 - 1 of Chapter VI, in which ground wire on the left hand

side is assumed to be the number 1 and the other one number 2. The transmission

line with a load of 300( .8 + j.6) ohms/ phase is assumed to be 25.6 miles long.

Its 1 ine constants are as follows:

B c

=

=

.637819 .105226 .0469018 .147401 .0723756 . 105226 .637819 . 105226 . 125063 . 125063

.0469018 .105226 .637819 .0723756. 147401

.147401 .125063 .0723756 1.15463 .129612

.0723756 . 125063 . 147401 . 129612 1. 15463

7. 13575 -. 987433 - • 987433 7. 29988 -.230908 -.987433 -.784453 -.555009 -.228812 -.555009

A B

-.230908 -.784453 -.228812 -.987433 -.555009 -.555009 7. 13575 - . 228812 -. 784453 -.228812 3.85007 -.301614 -.785543 -.301614 3.85007

C G.W.1 G.W.2

ohms/ mile

micro-mhos/ mile

The towers are assumed to be located .32 mi les apart from each

other. The values of both r 1 and r 2 are assumed to be 5. ohms. Ca Icul ations

are carried out for three values of r : 30. ohms, 10. ohms 5 and 5. ohms. 9

The fault conditions, short circuit between Phase A and ground

and short circuit between Phase A and Phase B, ore considered. The locations

of the fouit point are indicated with the results. The results are compared to

those obto ined when the effect of the ground wires was combined into the conductor

constants of the three-phase transmission line as shown in Appendix B.

H5

• Considering the results in Tables 8 - 1 to 8 - 3, it is seen that those values

are very nearly the same as those obtained by incorporated the ground wire effect into

the fine constants as shown in Appendix B. This suggests that the method of approxima-

ting the effect of ground wires is satis factory for most appl ication. However the

Mu/ticonductor Method may be used to include the ground wires explicitly where greater

accuracy may be dictated by the specific problem.

116

• TABLE 8-1 FAULT CURRENTS AT THE SENDING END, IN KILO-AMPERES Fouit Point Measured from Sending End, Miles

Type of Fouit 10.24 15.36 20.48

a. Three-Wire Transmission Une. (a) lA 67.03 44.76 33.59 (b) lA 67.96 45.49 34.24

lB 66.93 44.45 33. 19

b. Transmission Une with Two Ground Wires.

r = 30. ohms 9

(a) lA 66.70 44.62 33.52 (b)

'A 67.95 45.48 34.23

lB 66.92 44.44 33. 19

r = 10. ohms 9

(a) lA 66.83 44.67 33.54

(b) lA 67.95 45.48 34.23 ( -. -.,-" ,~ lB 66.92 44.44 33.19

r = 5. ohms 9

(a) lA 66.87 44.69 33.54 (b) lA 67.95 45.48 34.23

lB 66.93 44.44 33. 19

Note: (a) Phase A to Ground

(b) Short Circuit between Phases A and B

117

" TABLE 8-2 SENDING END CURRENTS IN FAULTLESS PHASES, IN AMPERES

Fau/t point Measured from Sending End, Miles Type of Fault 10.24 15.36 20.48

.lB 'C 'B 'C 'B 'C a. Three-Wire Transmission Une.

(a) 1421. 1372. 1418. 1373. 1416. 1374. (b) 1336. 1335. 1334.

b. Transmission Une with Two Ground Wires.

r = 30. ohms 9

(a) 1421. 1370. 1416. 1373. 1413. 1376. (b) 1336. 1334. 1333.

r = 10. ohms 9

(a) 1421. 1371. 1418. 1373. 1415. 1374. (b) 1336. 1335. 1334.

r = 5. ohms ".-.... 9 '--' (a) 1421. 1372. 1418. 1373. 1415. 1374.

(b) 1336. 1335. 1334. Note: (a) Phase A to Ground

(b) Short Circuit between Phases A and B

118 TABLE 8-3 RECEIVING END CURRENTS, IN AMPERES (_ Fouit Point Meosured From Sending End, Mi les

Type of Fouit 10.24 15.36 20.48 lA lB IC lA lB IC lA lB IC

a. Three-Wire Transmission Une. (a) 109. 1500. 1430. 70. 1480. 1410. 54. 1476. 1415. (b) 742. 634. 1386. 724. 652. 1385. 715. 660. 1384.

b. Transmission Une with Two Ground Wires. r = 30. ohms 9

(a) 110. 1480. 1410. 73. 1480. 1410. 54. 1473. 1417. (b) 742. 634. 1386. 724. 652. 1384. 716. 66l. 1383.

r = 10. ohms 9

(a) 108. 1480. 1410. 73. 1480. 1410. 55. 1475. 1415. (b) 742. 634: 1386. 724. 65l. 1385. 715. 660. 1384.

r = 5. ohms 9

(a) 107. 1480. 1410. 72. 1480. 1410. 55. 1476. 1415. (b) 742. 634. 1386. 724. 65l. 1385. 715. 660. 1384.

Note: (a) Phase A to Ground

(b) Short Circuit between Phases A and B

1

119

CHAPTER IX

COMPUTER METHOD AND GENERAL PROCEDURE OF MANIPULATING

THE MULTICONDUCTOR METHOD SY COMPUTER PROGRAMS

9. l General Procedure of Manipulation

OThe use of the Multiconductor Method to solve transmission line

problems, involving complicated matrices, is made feasible by the use of the

digital computer. The general procedure of manipulating the Multiconductor

Method by computer programs may be summarized as follows:

(1) From the given geometrical dimensions of a transmission

line configuration, insert the data into the main program

for solving the 1 ine constants, inductive reactance and

capacitive susceptance of the conductors.

(2) Insert the foJ/owing data into the required main program:

(a)

(b)

(c)

(d)

(e)

line constants,

length of the 1 ine,

sending end applied voltages,

load impedance per phase,

other necessary data, if required, e.g. r, r , s

)

(3) Find the chain matrix [1-1 ] of the 1 ine based upon the

case under analysis.

(4) Find Is for a given Es and known [1-1 ] and ZL •

(5) Find E and 1 from known E and 1 . r r s s

(6) Any values of Ek and 'k

can be obtained from the known Es,ls and

[l-1k ] by Equation (A-9) of Appendix A •

The above procedure is not applicable to solve grounded fault with the

faulted conductors uninterrupted at the fault point. In this case, [1-112] and ZL

should be combined first as shown in Chapter VII.

The three typical main programs and the subroutines, in FORTRAN IV

120

Language, Level G, suitable for use without modification on IBM 360 series computers,

are listed in Appendix C for the purposes of illustration and refe·rence.. The writing of the

main programs is based upon the matrix equations in the various analyses. The listed pro-

grams are applicable to the different bundled conductor configurations of transmission line

by proper changes on the dimensions of the arrays and vectors (matrices) shown in the be-

ginning of the programs.

(1) Main pro gram SBPC is the program having ,the data of the

potential coefficients of a three phase bundled conductor trans­

mission line with six subconductors per phase as the input and

its line constants, inductive reactance and capacitive susceptance ,

of subconductors as the output. The program wi Il be va 1 id by

proper modification for other bundled conductor configurations,

such as two or four subconductors per phase.

" ... ,

(2) Due to the complicated manipulation of matrix equations in the

Multiconductor Method, the computing time usually is quite

h igh. According to he record, the average CPU time for one

single calculation of main rogra SBVC, shown in Appendix C,

exceeds th i rty seconds on the 1 BM 360 / 75 computer.

For the sa me program with six subconductors per phase instead

of four subconductors per phase configuration, the CPU time

is more than seventy seconds. The CPU time for a single cal­

culation of finding the Hmit resistance of spacers in a four

subconductors per phase configuration is about one hundred seconds

on the IBM 360;15 computer. Therefore, efficient programs with

least computing times are required in practice.

(3) Inversion of a complex matrix is one of the important steps in the

application of the Multiconductor Method • Accuracy is a

big problem when dealing with sorne matrices having peculiar

.combination of elements. There are several means that can be

used to achieve better results, su ch as re-arranging the elements of

the matrix, raising the values of the elements, etc. In this appli­

cation, the value of the elements of the matrix in finding limit

resistance of spacers has been raised by 1012

; good results have

been obtained.

'122

ln order to mise the accuracy of the inverted matrix, a subroutine COR

or 'CRR was used based upon the following principles :

Let B be the inversed matrix of A and B = A -1 + E ,

where E be the matrix of error introduced

Then A B = U + A E

-1 B A E = [A + E] A E = E + E A E

(e 123

ln order to raise the accuracy of B,

-1 -1 Let BI = B - B A E = A + E - E + E A E = A - E A E ,

B' should be more accurate than B.

ln general,' this subroutine will improve the accuracy except in sorne cases

where B has a very peculiar combination of numbers in its elements.

(4) The chain matrix [iJ ] of the line consists of four submatrices,

hyperbolic functions of 1 Z y or IY Z as shown in Equation (A-6) of Appendix A. The [iJJ may be determined provided that thOe value of the square root of a matrix, IZ y or IY Z ,

can be obtained.

The square root of a matrix can be obtained by the eigenvalüe method"

Let A be the matrix,

D be the n th order diagonal matrix of eigenvalues,

M be the so-ca"ed Modal Matrix,

A = M D M -1 and AP = M DP M-1

Inthiscase, Pis 0.5.

ln fact, it is difficult to obtain the eigenvalues of a matrix with very high

accuracy. Thus, the results of the square root of a matrix will not be satisfactory.

ln this application, the value of the hyperb9'ic functions of the square root

of a matrix can be obtained satisfactorily from expanding the hyperbolic functions by

Maclaurinls expansion method as shown in Appendix A.

• (5) There is no subroutine available to toke out the imaginary part

of a complex double precision number. In this application, a

complex double precision number is supposed to be a function

of which the real and imaginory parts are taken out separately

by subroutihes DRL and DIM successfully. In fact, the

real part of a complex dOl,lble precision number may easily be

taken out by assigning any real number to be equal . to that

complex double precisfon number instead of using DRL •

124

(5) There is no subroutine available to take out the imaginary part

. of a complex double precision number. In this application, a

complex double precision number is supposed to be a function

of which the real and imaginary parts are taken out separately

by subroutihes DRL and DIM successfully. In fact, the

real part of a complex dOl,lble precision humber may easily be

taken out by assigning any real number to be equal . to that

complex double precisï'on number instead of using DRL •

124

CHAPTER X

CONCLUSIONS

125

ln the present work a new approach for the analysis of transmission lines,

called the Multiconductor Method, has been developed. This method permits the study

of voltage and current distribution in individual subconductors for a bundled conductor

transmission line. Unbalances between phases can also be analyzed by this method.

The versati lit y of the Mu Iticonductor Method has been demonstrated by its

application t~ the study of electrical characteristics of spacers, fault conditions of the

bundled conductor and fouit conditions of the three phase three wire system.

It is important to point out that the Multiconductor Method can take into

account not only ail the distributed parameters of a transmission line, but also any addi­

tional circuit component su ch as the impedance of a spacer, the short circuit impedance,

etc., which con be included in the chain matrix of the system. Therefore, the method

offers results of high accuracy.

It is realized that the Multiconductor Method cannot be used for desk

calculations due to the complicated matrices involved, but its application is made possible

by the use of the di gital computer. Moreover, the method is applicable irrespective of

the number of subconductors per phase or the number of phases.

ln solving for unbalcnced fcults in three-phase systems, the M~lticonductor

Method appears to be superior to the classical method of symmetrical components, since it

is valid when there are more than one fault occurring simultaneously in a transmission line.

126

ln practice, the power system network is more complicated than the single

transmission line considered in this thesis. Further development of the Multiconductor

Method should allow the application of such to the complicated network systems.

Some characteristics of various bundled conductor configurations, as

affected by number of subconductors, spacing, load and length of line, have been studied.

ln addition to the known characteristics of bundled conductor transmission lines, this inves-

tigation has led to the conclusion that there is a balanced current distribution in subcon-

ductors and only a very slight voltage difference between subconductors of the same phase,

under normal operating conditions.

The effect of the electrical characteristics of spacers has been studied

by the Multiconductor Method. It is important to point out that the high resistance of

spacers and the prevention of the deterioration of their insulation are of prime importance

in the normal operation of power transmission lines.

Finally, due to its general character and versatility, it is anticipated that

the Multiconductor Method will be extensively used for the solution of many problems

occurring in power systems.

Claim of Originality

To the best of the author's knowledge, the following contributions are

original:

127

(1) Adaptation of the "Generalized Telegraph Equations" to the

study of asymmetry in modern power transmission lines and thus

establ ishment of a new method termed the "Mu 1 ti conductor Method Il • * 1

(2) Investigation of the voltage and current distribution in subconductors

of bundled conductor transmission lines, operating under normal

. conditions. *

(3) Study of the fault conditions in bundled conductor transmission lines

caused by individual subconductors. **

(4) Study the effect of the eiectrical characteristics of spacer3.*,**

(5) Application of the Multiconductor Method to study the fault condi-

tions in three phase three wire transmission 1 i nes as a substitute to

the method of symmetrical components. ***

(6) Investigation of the accuracy in using the GMR to calculote the

line constants and the phase currents of bundled conductor trans-

mission lines.

* Paper to be presented at the IEEE, Winter Power Meeting, 1970.

** Paper to be presented at the IEEE, Winter Power Meeting, 1970.

*** Paper to be presented at the IEEE, Winter Power Meeting, 1970.

(-

/' ..... !

128

APPENDIX A

GENERALIZED TELEGRAPH EQUATIONS

Figure A-1 shows the transmission fine with n conductors. The general

equations governing the distribution of voltages and currents in the on conductor trans-

• • bd·· f 8,17 mission system can e expresse ln matnx orms as :

and

where

and

where

- 0: 1 = Y (P) E

o --E = Z (P) 1 OX

E is the voltage matrix of on ° conductors,

is the current matrix of n conductors,

y (P) is the admittance matrix of the fine,

Z (P) is the impedance matrix of the fine,

y (P) = G +

Z (P) = R + L

d Cdi"

d dt

G is the leakage conductance matrix of the fine per unit length,

C is the capacitance matrix of the line per unit length,

R is the resistance matrix of the line per unit length,

L is the inductance matrix of the line per unit length.

(A-1)

(A-2)

o 129

FIGURE A-l. VOLTAGE AND CURRENT DISTRIBUTIONS OF A

on CONDUCTOR TRANSMISSION UNE.

130

Equations (A-1) and (A-2) are therefore called the IIGeneralized Telegraph

Equations'! •

The steady state equations for the case when alternating voltages are im-

pressed on the lines at the sending end are

d Y (jw) E --1 = dt

(A-3)

d Z (jw) 1 - d't E = (A-4)

where

y = y (jw) = G + j w C = G + j B c

Z = Z (i w) = ~ + j w L = R + i XI .

B is the capacitance susceptance matrix of the line per unit length, c

and XI is the inductive reactance matrix of the line per unit length.

These two equations determine the values of matrices E and 1 subject to

the terminal boundary conditions.

The equation relating the sending end voltages and currents to the receiving

end voltages and currents is

E s

1 s

= [fl ]

E r

(A-5)

1 r

where E is the sendi ng end (n xl) vol toge motrix, s

. E is the receiving end (n x 1) voltage matrix, r

J is the sending end (n xl) current motrix, s

J is the receiving end (n x 1) current matrix, r

cosh (1 1 Z y ) sinh (1 l'ZY) Z o

and [~-] ~

-1 sinh (IIYZ) Z cosh (1 Iy Z) o

is the (2n x 2n) chain matrix of the line,

and where 1 is the total length of the transmission line,

Z is the (n x n) characteristic impedance matrix and o

131

The hyperbolic functions as shown in Equation (A-6) can be determined by

Maclaurin's power series expansion method :

where 12

(ZY) 14

(ZY}2 cosh {I 1 Z Y} = U + + + • • • . 2 ! 4 !

similarly for cosh 0 IYZ}

sinh (I 1 Z y) Z = (U + 12

(Z Y) 14 (Z y)2 + •.••• )(IZ) + 0 3 ! 5 !

and /Y2 -1 (U+

12 r;r Z) 14 \'( Z}2 + •.•• ) 0 Y)

sinh (1 Y Z) Z = 3! + 5! 0

132

The termination of the infinite .series of hyperbolic function depends on its

parameters-ond the accuracy required. ln general, if the infinite series converges

rapadily, the sum of the infinite series can be replaced by the first few terms.

ln the numerical examples with four subconductors per phase of this thesis,

the maximum absolute value of the elements of the matrix [12

(ZY)] is 0.0113552, the -3 -7 rest being of order 10 to 10 • Calculations inc/uding the fifth term of the infinite

series of the hyperbolic function have been made. The results show that the fifth term

has negligible effect on the calculated results. Thus, only four terms have been taken

into account in the calculation as shown in program SBVC of Appendix C. ln other

applications, a proper test to find out the termination of the infinite series of the hyper-

bolic function will be necessary in order to ensure the desired accuracy.

The line constant matrices result from the geometrical configuration of .. . 17,46 the transmission 1 me.

133

For a given (n x n) load matrix ZL

E = ZL 1 r r

Therefore E ZL Pl s

= [fJJ J = J r r

1 U , P2 s

where u is the (n x n) unit matrix,

Pl and ,P2 are the (n x n) submatrices of the product [fJ] [UZL

]

It con also be shown that

Thus,' 'the sending end current matrix can be obtained for a given E • s

(A-7)

From the known sending end voltage and current matrices, the currents and

voltages at the receiving end con be determined by

E r

E s

1 s

(A-8)

The equation re lating currents and voltages at the sending end and at any

point K along the transmission line, at a distance Ik

from the sending end is :

(-

c

134

E El( s

= [Ill< ]

1 II< s

or

~1< E s

-1 = [~ 1 (A-9) r< IK 1

s

where [lA", ] is the (2n x 2n) chain matrix between the sending end and the given point K, and can be determined by substituting 'K for , in the Equation (A-6) •

135

APPENDIX' B

DETERMINATION OF LlNE CONSTANTS OF A TRANSMISSION LlNE

For a given transmission line with "n" conductors and (w - n) skywires

(or ground wires), line constants, self and mutual inductances and self and mutual

capacitances, can be determined by its geometrical dimensions as shown in Figure B - 1 •

(a) By Maxwell IS equation, the linear relations between potentials and charges may be expressed in matrix form as

yi = P' Q'

where y' is the potential matrix of w conductors,

where

Q' is the charge matrix of w conductors,

pl is the matrix of coefficient of potentials, and is

determined by the geometrical dimensions of the

conductors in a transmission line.

Its elements may be expressed as

18 x 1011 2h

P ln r.

darafs / cm, = -R-' -rr r

18 x 1011 a

p: ln rs

darafs / cm , = ~ rs

rs

R = the radi us of the conductor r , ft, r

h = the height of the conductor r , ft, r

a = distance between conductor r and image of conductor s, ft 1 rs

b = distance between conductor rand conductor s, ft. rs

nr . ·1· .. w,v/.?W"..@'T..@;v,@;rM7"~$W'"J.@'..@.

-Hr ars

l r'

s',

FIGURE B-1. GEOMETRICAL DIMENSIONS OF TWO CONDUCTORS

OF A ' . n CON DU\.. '!"OR TRANSMISSION LlNE.

136

137

Since the ground wires ,are of zero potential, therefore q n+1 to q w can

be expressed in terms of ql

of n i nstead of w. 2, 30,46

to q • n

Thus V = P Q is formed and they are in order

Thus Q = P -1 V = C V

and C = P -1

where C is the matrix of coefficients of capacity, in farads 1 cm.

(b) By Flux Linkage:' relationship in a transmission line with "n"

conductors and (w - n) ground wires,

4)1 = LI Il

where 4)1 is the flux linkage matrix of w conductors,

Il is the current matrix of w conductors,

LI is the matrix of coefficients of induction.

Its e lements may be expressed as

1 2h

-9 1 = S 2. + 21.1:\

__ r} x 10 henries / cm.,

rr R r

a 10-9

1 = 21n~ x henries / cm • rs rs

Since the result of flux linkage in the ground wires is assumed to be zero,

then in

+ 1 to i,w may be expressed in terms of il to in' Thus q, = L 1 is formed

and the y are in order of n instead of w. 53

ft

138

ln practice, Land C may be expressed in henriesl mile and micro-

farads / mile ; and XI ' inductive reactance matrix and B , capacitive susceptance , c matrix, can be obtained in ohms / mile and micro-mhos / mile respectively.

(c) The rlne constants of a bundred t.onductor transmission' line,may be

where

é.xpres~ed fn pe r poose va 1 ue • 22

Assume n = 3b,

where n is the total number of subconductors in a transmission line,

Thus,

b is the number of subconductors per phase.

CAA

C = CBA

CCA

b b

CAA

= -l' L i=l j=1

2b b

CBA

= L l i=:=b+1 ;=1

n b

CCA = L l i=2b+1 j=1

CAB

CBB

CCB

c .. ,

" c .. ,

IJ

c .. , Il

b 2b

CAB = l L 'c .. " i=1 j=b+1 "

2b 2b

C = L L C .. , BB i=b+1 ;=b+1 "

n 2b

b : n

CAC = l . l C .• • i=1 j=2b+l "

2b n

CSC = L l C ... i=b+l j=2b+1 "

n n

CCB = l L c .. , CCC = l l C .. , i=2b+1 j=b+1 " i=2b+1 j=2b+1 "

(- 139

And 'AA 'AB 'AC

l = 'BA ,

'BC BB

'CA 'CB 'CC

where b b b 2b b n

LI'·· If

1

b2 i=1 ;=1

L 'I i=1 ;=b+1

'., , 'AC = --!r LI',· . If 'bl: Il i=1 ;=2b+1

2b b 2b 2b 2b n

'SC = ~ L L 'ij' i=b+ 1 ;=1

1 BB = b ~ L L 1 i i ' i=b+1 ;=b+1

1 \' -, 'BC =.:z-' L L' i; .

i=b+1 ;=2b+1

n :b n 2b

'CA = ~ IL' .. , b If i=2b+1 ;=1

'CB = ~ \' l , .. ,

bl: L If i=2b+1 ;=b+1

Note that,

Subconductors 1 to b are in Phase A,

Subconductors (b + 1) to 2 b are in Phase B,

and Subconductors (2b + 1) to n are in Phase C •

APPENDIX C

FORTRAN PRO GRAM LISTINGS

Three typical programs together with the subroutines are listed for the

purpose of illustration and reference.

(1) Main Programs

SBPC and Subroutine PCVY

SBVC

TFVC

(2) Subroutines

CINVRT

GMPRD

DRL and' DIM

COR ana CRR --

140

'.' SBPe

DOUBLE PRECISION P(20,20',Q(20~20',Z{lS,I~),Y(18,is) 'DOUBLE PRECISION R,Rt,0,00,H,Hl,H2,H3,H4,H12,HI3,HI4,H23,H24,

'2Al,A2,A3,A4,A5,A6,A7,A8,A9,AlO,All,AI2,A13,A14,A15,AI6,AI7, ' 9 A1S,Bl,BZ,S3,B4,85,B6,S7,BB,,89,BlO,Bll,S12,BI3,B14,Bl5,BI6, 3 B17,818,012,013,014,015,D18,019,010,0111,0114,0115,0116,0117,

141

4 0213,OZI7,027,037,047,049,057,0513,0313,D413,'E18,EI0,EI14,El16, 5 E213,E217,E313,E413,E27,E37,E47,E49,E57,E513 . N=20' ' M=N-2 $=1.500 R=I.3800/.2402 RG=7.00/(.16002*.24002) DO Il I=l,N DO Il J=l, N P ( 1, J ~ =0. Q(I,J)=O.

'11 CONT 1 N UE :0=.75DO*OSQRT(3.DO) OD=2.00*0 H=42.D0+1.DO/3.00 Hl=IC8.DO H2=Hl+2.00*H H3=Hl+00 H4=H 1-00 H13={Hl+H3)/Z.OO HI4={Hl+H4)/2.0û H23=(H2+H3}/2.00 H24=(H2+H4)/2.00 HIZ=(Hl+HZ)/Z.DO 0IZ=OSQRT(H13**2+.75DO**2) 013=DSQRT(HI4**Z+.75D0**2) CI4=DSQRT(HI3**2+Z.Z5DO**2) 015=DSQRT(H14**Z+Z.2500**1) 018=DSQRT(HI3**2+50.75**2) 019=DSQRT(H14**Z+50.750G**Z) 0111=OSQRT{H14**Z+52.Z5D9**~) 0114=DSQRT{H13**2+100.7500**Z) ~114=OSQRT(0**Z+lOG.75DO**Z). 0115=DSQRT(H14**Z+lCO.75DO**Z) 0116=DSQkT(H13**Z+102.2500**2) E116=O$QRT~D**2+102~2500**2) 0117=OSQRT(H14**Z+102.Z5DO**Z) E213=DSQRT(O**2+99.25DC**2) D217=OSQRT(Hl**2+101.5DC**2) E217=DSQRT(DO**2+101.5DC**2) D313=OSQKT(H14**Z+99.Z500**2) E313=DSQRT(D**2+99.25DO**2) 0413=QSQRT(H13**2+97.75DO**Z) E413=DSQRT(O**2+97.75D0**2) C27=DSCRT{H13**2+49.2500**2) E27=DSQRT(O**2+49.25DO**2) 037=DSQ~T(H14**2+49.25D~**2) E37=OSQRT{D**Z+49.2500**2) 047=DSCRT(H13**2+47.75DC**2)

'e E47=DSQRT(D**2+47.75DO**2) D49=DSQRT(Hl**Z+4805DO**2) E49=DSQRT{DO**2+48.5DO**2) D57=DSQRT(H14**Z+47075DP**2) E57=DSQRT(D**2+47.75DO**2) D513=OSQRT(H14**2+97075DC**2) E513=DSQRT(D**2+97.75DO**Z} E18=DSQRT(50.75DO**Z+D**2) EIQ=OSQRT(52.Z5DO**2+D**2) OlO=DSQRT(52.2500**2+H13**2) OZ13=DSQRT(H13**2+99.25DO**2) Al=OSQRT(H12**2+16.25DC**2) Bl=OSQRT(H**2+16.25DO**2) A2=OSQRT(15.50C**2+H23**2) B2=OSQ~T(15.500**2+(H-Ol**2) A3=DSQRT(lSoS00**2+H24**2) B3=OSQRT(15.5DO**2+(H+D)**2l A4=OSQRTC14.DO**2+H23**Z) B4=OSQRT(14.00**Z+(H-D)**2) A5=OSQRT{14.00**2+H24**2) B5=OSQRT(14.00**2+(H+Dl**Z) A6=OSQRTC13.25DC**2+H12**2) B6=OSQRT{13.25DO**2+H**2) A7=DSQRT(33~7500**Z+HIZ**2) B7=OSQRT(33.7500**Z+H**2) A8=OSQRT(34.5DO**2+H23**Z) B8=OSQRT(34.5DO**Z+(H-D)**Z) A9=DSQRT(34.5DO**2+HZ4**2) B9=DSQRT(34.5DO**2+(H+D}**Z) A16=DSQRT(36.00**2+HZ3**Z) BIO=DSQRT(36.DO**2+(H-D)**2) All=DSQRT(36.00**Z+H24**Z) Bll=DSQRT(36.DO**2+(H+D)**ZI A12=OSQRT(36.75DO**2+HIZ**Z) B12=DSQRT(36.75DC**2+H**2) A13=DSQRT(83.7500**Z+H12**Z) B13=OSQRT(83.75DO**Z+H**2) A14=DSQRT(84.5DO**2+H23**2l B14=OSQRT(84.5DC**2+(H-Dl**2) A15=OSQRT{84.5DO**Z+HZ4**2) B15=DSQRT(ô4.500**Z+(H+D)**2) Al6=DSQRT(86.DO**2+HZ3**2) A16=DSQRT(36.DO**2+(H-D)**2) B17=DSCRT(86.DO**2+(H+O)**2) A17=OSCRT(86.DO**Z+H24**2) A18=DSQRT(86.75DC**2+H12**2)

'B18=DSQRT(S6.75DO**2+H**2) P(2,3)=DLOGIO(HI/DO) P(1,2)=DLOGIO(012/1.5DO) P(1,3)=OLOG10(D13110500) P(1,4)=DLOG10(D14/DD) P(l,5)=OLOGIO<D15/DD) P(l,6)=DLOGIO(DSORT(Hl**2+3.DO**2)/3.DO) P(l,7)=DLUGlr(OSC~T(Hl**2+50.DC**2)/5a.DO)

.142 .

~ P (1,8 , =D L OG 1 0 ( Dl 81 E 18 , P(1,9)=DLOGlO(D19/E18) P(l,lO'=DLOGIOCOIC/EIO)

. P{l,ll}=DLOGIO(ülll/ElO) P{lt12'=DLOG10{DSQRT(Hl**2+S3.DO**2'/53.DO) P(1,13)=DlOGIO{DSQRT(Hl**2+100.00**2)/lOO.DO) P{l,14)=DLOGIO(Dl14/El14' P(1,'lS'=DLOGIO(Dl15/El14) P(1,16}=DLOGIO(Dl16/Ellb) 'PCl,17'=DLOGlOCDl17/El16} .'

.. P(2,4)=DLOGIO(OSQRT(H3**2+1.5DO**2'/1.5DO' P{2,5)=DlOGIOCDSQRT(Hl**2+l.5DO**2,/3.DOI P(2,7'=DLOGIOCD27/E27J P(2,lO'=DLOGIO(DSQRT{H3**2+51.5DO**2)/51.5DO) P{2,11)=DlOGIOCDSQRT(Hl**2+51.5DO**2)/DSQRT(DD**2+S1.SDO**2» P(2,13'=OLOGI0(0213/E213) PC2,14)=DLOGIC(DSQRT(H3**2+100.DO**2'/lOO.DO' P(2~lS)=DLOGIO(DSQRTCHl**2+1CO.DO**2'/DSQRT(lOO.DO**2+DO**2" P(2,16'=DlOGIO(OSQRT(H3**2+101.5DO**2)/lOl.SDO) P(2,17}=DLOGIO(D217/E217) P(3,S'=DlOGIOCOSCRTCH4**2+1.5DO**2)/1.SDO) PC3,7'=DLOG10C037/E37) PC3,9'=DLOGIOCDSQRTCH4**2+50.DO**2)/SO.DO) P(3,13)=DLOSIO(D313/E313) P(3,lS)=DLOGIO(DSQRT(H4**2+100.DO**2)/100.DO) P ( 3, 1 7 ) = 0 L DG 10 ( 0 S (.) RTe H 4* ~c 2 + 1 01. S D0';c,;c 2 ) / 1.0 1. S DO ) P(4,7'=DLOGIO(D47/f.47) P(4,9)=DLOGIO(D49/E49)

143

P(S,7)=DLOG10(D57/E57' P(6,71=DLOG10CDSQRT(Hl**2+47.DO**2)/47.DO) P(4,13'=DLOG10(0413/E413) P(4,14)=DLOGIO(OSQRT(H3**2+98.SDO**21/98.SDO) P(4,lS)=DLOGIO(DSQRTCHl**2+98.SDO**2)/DSQRTCOD**2+96.SDO**2') P{S,9)=DLOGIOCDSQRT(H4**2+48.5DC**2)/48.5DO) P(~,13)=DLOGIO(D513/E513) P(6,13)=DLOGIO(OSQkT(Hl**2+97.DO**2)/97.DO) P(6,7)=DLOGIO(DSQRT(Hl**2+47.DO**2)/47.DQ) P(3,11)=DLOGIO(DSQRT(H4**2+51.SDO**2)/Sl.SDC) P(2,8)=OLOG10(DSCRT(H3**2+50.DO**2'/SO.DO) P(2,91=DLOGIO(CSORT(Hl**2+50.D0**2)/OS(.)RT(DD**2+50.DO**2»

.P(4,S)=DLOGIO(DSQRT(H3**2+4805DO**2)/4B.5DO) P(S,15)=DLUGIO(DSQRT(H4**2+98.5DO**2'/98.SDOl P(1,18)=OLOGIO(DSQRT(Hl**2+103.DO**2'/103.DC) P(1,19)=DLOGIO(Al/~1) P(2,19)=OLOGIO(A2/62) P(3,19)=DLOG~O(A3/B3) P(4,19)=DLOGIO(A4/B4) P(5,19)=OLOGIO(A5/B5) P(6,lg)=DLOGIO(A6/B6) P(7,19)=DLOGIO(A7/B7' P(B,19'=DLOGln(A~/Bô) P(9,19)=DLOGIO(A9/B9) P(lO,19).=DLOGIO(AIC/BIO' P(11,19)=DLOGIO(All/Oll)

•• ..

,P(12,19)=DLOGIO(A12/B12' ·P(13,19)=OLOGIOCA13/B13)

P(14,19)=OLOGIOCA14/B14) P(lS,19)=OLOGIO(A15/B15) P(16,19)=OLOGIO(A16/Bl·6) P(1!,19)=OLOGIO(A17/B17) P(18,19)=OLOGIOCA18/BlS) P(19,20)=DLOGIC{OSQRT(H2**2+70.SDO**2)/70.SDO)

.P(14,20)=P(4,19) P( 16,20)=P( 2,19) P( lO,20)=P(8,19) P(18,20)=P(1,191 P ( 1 ~,2 0) =P ( 6 ,19·) P ( 12,20) =P ( 7,19 ) P(17,20)=P(3,19) P(15,20)=P(5,19) P(11,20)=P{9,19) PC8,20)=P(lO,19) P(7,20 )=P( 12,191 P(9,20)=P<l1,19) P(4,20 )=p{ 14,19) P(6,20)=P(13,19) P(.5,20)=P(15,19) P{2,20)=P(16,19) P(!,20)=PC18,19) P(3,20)=PC17,19) P(8,9)=P(2,3)

. P(4,5)=PC2,3) P(lO,lU=PC2,3) P( l4,1S)=P(2,3) P(16,17)=P(2,3) P(4,6)=PCl,2) P(7,8)=PC1,2) P(10,12)=P(1,2) P( 13,14)=P(1,2)

. P ( 16,18) =P ( l ,2 ). P(S,6)=PCl,3) P(7,9)=P(1,3) P(11,12)=P(1,3) P(13,lS)=P(ldl p.( 1 7 , 1 S ) = P ( l , 3 ) P{2,6)=PCl,4) P(7,lO)=P(l,4) P(S,12)=P(1,4) P(13,16)=PCl,4) P(14,18)=P(l,4) P(3,6)=P(1,5) P(7,11)=P(1,5) P(9,12)=P(1,5) P(13,17)=P(1,5) P( 15,lS)=P( 1,5) P(7,12)=P(l,6) P(13,lS)=P(1,6) P(6,12)=P(1,7)

144

P(7,13)=P(1,7) P ( 12, i S) =P (1,7) P(7,14)=P(l,8) P(4,12)=P(1,S) P ( 10 , 18) =P ( 1 , S ) P(1,lS)=P(1,9) P(S,12)=P(1,9) P{11,lS)=P(1,9' P(7,16)=P(l,lOJ P(S,lS)=P(l,lO' P(2,12)=P(1,.lO) P(7,lS)-=P(l,12) P(7,17)=P(1,ll} P(9,lS)=P(l,lU P(3,12)=Pll,111 P{6,lA)=P(1,13) P(4,lS)=P(1,14' P(S,18)=P(1,lS) P(2,lS)=P(1,16) P(3,lS)=P<1,17) P(S,lO)=P(2",4) P{14,16)=P(2,4) P(3,4)=P{2,5) P(S,lU=P(2,S) P(9,lO)=P(Z,5) P( 14tl7,=P(2,S) P(lS,16)=P(2,S) P(6,lO)=P(Z,7' P(S,13'=P(Z,7) P(12,16)=P(Z,7) P(4,lO'=P(Z,S) P{8,14)=P(2,8J P ( 10, 16' = P ( 2 ,8) P(S,11)=P(3,9) P(9,lS)=P{3,9) P( 11; 17)=P( 3,9) P{3,8)=P(2,9) P(4,11)=P(2,9) P(S,lO)=P(Z,9) P(9,14'=P{Z,9) P(S,15)=P(Z,9) P ( 10 ,1 7) -= P ( 2 ,9' P(11,16'=P(Z,9)

" P(8,16'=P(Z,lO) P(9,17)=P(3,lU P(9,11 )=P(3,5) P{ lS,17)=P(3,S) P(9,16)=P(2,11) p(3,lr)=p(Z,11l P(S,17)=P(Z,11l P(6,16)=P(?',13l P(4,l6)=P(Ztl4) P(S,16)=P(Z,lS' P(4,17)=P(Z,15)

145

e P ( 3 , 14 ) == P( 2. , 1'5 ) P(3,16)=P(2.,17) P(6,ll)=P(3,7) P{9,13)=:=P{3,7). P(l2,17)=P(3,7) P ( 6 , 1,7 ) = P ( 3, 13 ) P<5,17)=P{3,15) P(6,S)=P(4,7) P(lZ,14)=P(4,7) P[lO,13)=P(4,7) P ( 10 , 14) =P ( 4 ,8 ) P(5,S)::=P{4,9) P(11,14)=P(4,9) P(lO,15)=P(4,9) P(6,9'=P(5,7) P(1l,13)=P(5,7) P(12.,15)=P(S,7) P(6,14)=P(4,13) P(S,14)=P(4,15)

,'P(11,15)=P(5,9) P(6,15)=P(5,13) P(lZ,13'=P(6,7) DO 22 1= 1,N DO 22 J= 1 ,N Q(I-,J)=P(J,I)

22 CONTINUE 00 33 l=l,N

, '00 33 J=1,N P(I,J)=P(I,J)+QII,J)

33 CONTINUE P(l,l)=DLOGlO(Hl/R) P(2~2)=DLOGIO(H3/R) P(3,3)=DLOGIO(H4/RI P(l9,19}=DLOGIO(H2/RG) P{6,6)=P(1,1) P(7,7)=P(l,l) P(12,12,=P(1,U P(13t13)=P(1,1l P(18tlS)=P(ld) P(4,4'=P(2,2) P(8,8)=P(2,2) P(lO,l(')=P(Z,Z) P{14t14)=P(2,2' P{16,16)=P(2.,2) P(5,51=P(3,3) P(9,9)=P(3,3) P<l1,11l=P{3,3) P(15,lS)=P(3,3) P(17,17)=P{3,3) p(20,2n,=p(19,19) CALL PCVY(P,Z,Y)

2 FORMAT{2X,4D16.6) DO 6 l 3 1 = l , ,'1 WRITE(6,614)(Z{I,J),J=1,M)

146

.613 CONT l NÙE 614 FORMAT(~X,4DI5.6/10X,4015.61

W~ITE( 6,6(13) 603 FORMAT(10X,'VALUE OF BC@)

DO 615 I=l,M WRITE(6,614) (YCI,J) ,J=I,M)

615 CONTINUE WRITE(7,2)Zr,Y STOP END

TOTAL MEMORY REQUIREMENTS 00543E BYTES

-, .. '

147

,

. SUBROUTINE PCVY(P,V,S) 'COMPLEX*16 CO(18,18)

'148 ,

.' .

DOUBLE PRECISION ,V{18,18),S(18,18),P(20,20),PO(18,18),QO(18,18), 2 Z{20,20).

OPUBLE PRECISION O,DO,H,H3,H4 DOU~LE PRECISION AN,Hl,H2,R,RG,ZD,PCD,'ZS(3),YS(3) DOUBLE, PRECISION ZSS(2),YSS(2) D=.75DO*OSQRT(3.0C)

. 00=2. DOi.'D H=42.DO+l.DO/3.DO Hl=108.D0 H2=Hl+2.DO*H H3=/il+DD H4=HI-DD R=1.38DO/.24D2*.7788DO RG=7.DO/(.16D02*.24D021*.778800 N=20 M=N-l NN=N-2 N3=NN/3 NN3=N3*N3 ' AN=NN3

2 FORMAT(2X,4016.6) DO 4fl I=l,N DO 40 J=l, N

40 Z(I,J)=.74110-3*P(I,J)*.37703 Z(7,71=.74110-3*DLOGIO(H2/RG)*.377D3 Z(20,20)=Z(7,7) Z(19,19)=Z(7,7) Z{1,1)=o7411D-3*OLO~10(HI/R)*.~77D3 Z(2,2)=.74110-3*OLOGI0(H3/R)*.377D3 Z(3,3)=.7411D-3*OLOGIO(H4/R)*.377D3 Z(6,6)=Z(1,1) Z(7,7}=Z(1,1) l ( 12 f 12) =Z ( l ,1) Z(13,131=Z(1,1) Z{18,18)=Z(1,l) Z(4,4)=Z{2,2) Z{8,8'=Z(2,2) Z( 10 tl 0) =Z ( 2 ,2) Z( 14,14)=Z{2,2) Z( 16,16)=Z(2,2) Z(5,S)=Z(3,3) Z(9,9)=Z(3,3) l( 1l,lU=Z(3,3) Z(15,15)=Z(3,3) Z(17,17)=Z(3,3) ZD=Z(M,M)*Z(N,N)-Z(M,N)**2 PCD=P(M,MIOP{N,N)-P(M,N)**2 DO 60 I=l,NN DO 60 J=l,NN PD(I,J)=P(I,J,-(P(N,N)*P(M,J)-P(M,N,*P(N,J»*P(I,Ml/PC0

2 -(P(M,M,*P(N,J)-P(M,N)*P(M,J»*P(I,N)/PCD QO<I,J)=O.

60 CONT INUE 00 . 70 1 = 1 ~ N N DO 70 J=l,NN CO(I,J'=DCMPlX{POCI,JJ ,QO(I;JJ'

·70 CONTINUE . CAlL CINVRT(CD,NN,NN,~N) DO 50 l=l,NN DO 50 J=l,NN ..

149

V(I,J}=Z(I,J)-(Z(N,N'*Z(M;J)-Z(M,Nl*Z(N,J)'*Z(I,M)/ZO 2-(Z(M,M>*Z(N,J)-Z(M,N'*Z(M,J»*Z(J,N>/ZD S(I,J'=CO(I,J)*.03881DO*.377D-3 .

50 CONTINUE YS(l)=O. YS (2) =0. YS(3)=O. lS (1) =0. ZS(2)=O. lS(3'=(). YSS(l)=O. YSS(2)=O. ZSS(l)=O. ZSS( 2) =00 00,511 I=1,N3 DO 511 J = l , N 3 II=I+N3 JJ=J+N3 III=II+N3 JJJ=JJ +N3 YS(l)=YS(l}+$(I;J) YS(2)=YS(2)+S(Il,JJ) YS(3)=YS(3)+$(III,JJJl Z${l)=ZS(l)+V(I,J)/AN lS(2)=ZS(2'+V(II,JJ'/AN l$(3)=Z$(3'+V(III,JJJ,/AN YSS(l,=YSS(l)+S(I,JJ) YSS(2)=YS${2)+$tI,JJJ) ZSS(l)=lSS(l)+V(I,JJ'/AN lSS(2'=lSS(2)+V(I,JJJ)/AN

511 CONTINUE WRITE(6,523'

523 FORMAT(6X,'V~LUE OF lSS AND YS$@) WR Ir E ( 6,2 )l $ , Y S WRITE(6,2'ZSS WRITE(6,2)YSS RETURN END

TOTAL MEMORY REQUIREMENTS C04380 BYTES

SBve

CO~PLEX*16 AXy,ZZl(24,24),P(12,12) DOUBLE PRECISION ZZ(12,12),YY(lZ,lZ)

150

DOUBLE PRECISION ACU(12),ARIC12,1),AR2(12,1),AR{12,1),AIl(12,1', . 2 AIZ(12,1),AI(12,1),E(12,1),F(12,1),EIR(24,1"EII(24,1',EI3(Z4,1',

3 EI4(Z4,1),EI5{24,l',EI6C24,1),EIT{Z4,1),EISCZ4,1),EIW(24,1J, 4 Z (1Z , 12 , , y ( lZ, 12 , ,A <12 , 12 ) • B ( 12 , 12 ) " S l( i 2, 1 2) ,S 2 ( 12 ,1Z ) , 5 S3(12,IZl,S4(12,12),S7(12,lZ),S8(12,12J,S9(12,lZ),SlO(12,12), ~ Pl(12,12},P2(12,lZ"Ql(12,12),Q2(12,12),YR(i2,12),Y!(12,12), 7 V(24, 12 J,WC Z4,1Z) ,V1(24,lZ) ,V2{Z4,lZ) ,Wl{Z4,lZl ,W2( Z4, lZ), 8 A·l(Z4,24} ,Bl< 24,Z4) ,T(Z4,24) ,X(24,24) ,T3(Z4,2'd ,T4(24,24), q T5(2'-t,24) ,T6(Z4,Z4),Y1R(Z4,24),YII(24,24)

DOUBLE PRECISION DIM,ORL,XL,VKl,VK2,AT~(3)~ATI(3),ATT(3) EXTERNAL'OIM REAO(5,299'ZZ,YV

2q9 FORMAT(4D1506' N=24 M=N/2 Nl=M/3 N2=Nl*2 NNl=Nl+l NNZ=N2+1 DO 440 I=1,Nl E(I,1'=-.3675D06/DSORT(53DOl' F CI, 1 , = 0.3675 006

440 CONTINUE· DO 44~ I=NN1,NZ E(I,1'=.735D06/0SQRT(.3001) F( 1,1)=0000

44.1 CONTINlE DO 442 I=NN2,M E(I,1)=-o3675D06/0SQRTto3ÇOl) F(I~1)=-o3675006

442 CONTIl\lF. Z FORMAT(ZX,4D1506)

XL=51.200 WRITE(6,2'XL DO 998 I=l,M DO 998 J=I,fJ l.CI,JI=Zl(I,J) Y( !,J)=YY( l ,J)

998 CONTINLE CALL G~PRO(l.,y,A,~,M,M'

DO Z73 I=l,N DO 273 J=1,N Al CI, J '=0. BUI,J'=O.

273 CONTI ~LE '00 Il 1=1, M DO Il J=I, M

re ,ZCI,J':Z<I,J'*XL V( 1,Jt=XL*V{! ,J) . ACI,J)=-A(I,J)*XL**2

Il CONTINUE . DO 101 l=l,M DO 101 J=l,M Sl(I,J'=CoODO S 2 CI, J ) = Co OD 0 S3(I,J}~000DO

S4(I,J}=CoO[)0 S7fI,J)=OoODO S8 (I ,J )=00 ODO

101 B(J,I)=A(I,J) DO t'O 1 = 1 , M . SI ( 1 ,1 ) =01 DOl S2 ( l ,1 )=01001 S30,I )=olDOl S40,1)=01001

10 CONT 1 NUE CALL G~PRD(A,A,S7,M,~,M)

CALL G~PRD(B,B,S8,M,M,M) CALL G~PRD(S7,A,S9,~,M,~)

CALL G~PRO(S8,B,S10,~,M,~)

00 15 1=1, M DO 15 J=1,M . . . S2 (1 ,J )=S2( l ,J) +A( l, J) 10 6001+S7( l, J) 1.,12003+S9( l ,J )/0 504004 S3(I,J)=S3(I,J'+BCI,J)!o60Cl+S8(I,J)!o12003+S10(I,J'!o504D04 S l ( 1 t J ) = A ( l , J , 1 0 20 1 + SI ( l ,J ) + S 7 ( 1 , J ) ! 0 24 D 0 2 + S 9 ( l , J ) ! 0 72 D 03 S4(I,J)=B(I,J)/o2Dl+S4(I,J)+S8{I,J)!o24D02+S10(I,J)!072003

15 CONTINUE CALL G~PRD(S2,Z,S7,M,M,M) CALL G~PRD(S3,Y,S8,M,M,M' DO 105 I=l,M DO 105 J=l,~ II=I+M JJ=J+tJ. Al(I,J)=Sl< I,J) Al<II,JJ)=S4(I,J) BIC l ,JJ)=S7( I,J t

105 Bl(II,J)=S8( I,J) 472 DO 474 I=l,N

DO 474 J=l,N T ( l , J 1 =A l ( J , J, XCI,J'=Bl(I,J'

. 474 CONTI NUE DO 773 1 JH=l,2 CALL G~PRD(T.T,T3,N,~tN) CALL G~PRD(X,X,T4,N,N,N)

CALL G~PRD(X;T,T5tN,N,N)

CALL G~PRD(T,X,T6tN,N,N)

DO 797 I=l,N DO 797 J=l,N T( l, J} =T 3 ( l , J ) - T 4 ( l , J) XCI, J) =T 5 ( l , J , + T 6 ( l , J)

151

797 CONTI NUE 773 CONTI~LE

VKl=o 24003 VK2=o 18003 WRrTE(6,2'VKl,VK2 DO 466 I=1,N DO 466 J=l,M V(I,J'=OoDO WCI,J)=OoDO

466 CONTI NLE MM-=M+l 00,467 l=MM,N J=I-ti. VCI,J)=olOOI

,467 CONTI NUE DO 14 1=1,Nl DO 14 J-=1,N1 V( I,J'=VKI WCI,JJ=VK2

14 CONTI NUE DO 414 II=NNl,N2 DO 414 JJ=NNl,N2 V ( II , J J J' =V K 1 h'CII,JJJ=VK2

414 CONTINUE DO 514 III=NN2,M DO 514 JJJ=NN2,M V(III,JJJ)=VKI W ( 1 1 r , J J J ) =V K 2

514 CONTINUE CALL G~PRD(T,V,Vl,N,N,~J CALL G~PRD(X,W,V2,N,N,M) CALL G~PRDCT,W,Wl,N,~,M) CALL G~PRD(X,V,W2,N,N,M) DO 90 I=1,N DO 90 J=1, M VU l ,J )=Vl{ l ,J)-V2( l ,J)

90 WIn ,J )=r.l (1 ,J)+W2U ,J) DO <)9 1=1,"1 DO 99 J=1, M P2( l ,J '=Vl (1 ,J) 02 (I ,J' =Wl ( l ,J 1

99 CONT 1 NUE C INVERSE THE PARTITION MATRICS CF li

, DO 17 I=l,M DO 17 J=1,M V1(I,J'=Vl(I,J)*~lD07

17 Wl{I,J)=Wl([,J'*o1007 DO 16 1=1, M DO 16 J=l,M P ( r , J) =0 C f) PL X ( VI ( l ,J ) ,W 1 CI, J) )

16 CONTINLE C .I~PECAC\CF,I1FQ&.JQ

CALL CINVRT(P,~,M,~)

152

DO 93 I=l,M II:I+M 00·93 J:l, M PUI ,J)=Vl( II,J) Ql(I,J)=WUII,J) AXY=P CI, JI YR(I,J)=DRLCAXY)*olD07 YIn ,J}=DIM(AXY}*olD07

93 CONTINLE \\RITE(é,546)

.546 FORMAT(2X,' THE INTER-SUBCCNDUCTORS CURRENT'@) 333 CALL G~PRD(YR,E,ARl,~,M,l) CALL G~PRO{YI,F,AR2,M,M,1) CALL G~PRO(YI,t,AIl,~,M,l) CAlL G~PRD(YR,F,A12,M,~,1) DO 222 I=l,M DO 222 J=l,l AR (I , J 1 =A R 1 ( l , J 1 - AR 2 ( l , J )

222 AI(I,J,=AIl{I,J)+AI2(I,J) 332 CALL G~PRD(Pl,AR,ARl,M,M,l)

CAlL G~PRD(Ql,AI,AR2,M,M~1) CALL G~PRo<Pl,AI,Arl,M,M,l) CAlL G~PRD(Ql,AR,AI2,M,M,1) DO 200 I=l,M DO 200 J=l,l AR (l ,J ) = A RI ( l , J ) -·A R 2 ( l ,J )

200 AIn ,J)=AIU r,J)+AI2<I,J) WR IT E ( é, 2) AR ,A 1 00 444 I=l,M ACU( I)=DSQRTCAR(I,1)**2+AI(I,I)**21

444 CONTINLE WRITECé,2)ACU DO 551 1 = 1,3 A·TRC l' -=0. AT I( 1 ) =0.,

551 CONTINeE DO 552 I=l,Nl II=I+Nl III=II+Nl ATR(l)=ATRCl)+AR(I,l) ATR(21=ATR(21+ARCII,1) ATR(3)=ATRC3)+ARCIII,I) AT 1 ( 1 ) =A TIC 1) +A 1 ( l ,1) AT 1 C 2 1 =A Tl ( 2 1 + AI ( 1 1 , 1 , ATI(3'=ATI(3)+AI(III,11

552 CONTINUE DO 553 1 =1,3 ATTCI'=DSQRTCATR(II**2+ATI(II**21

553 COf\TINLE WRITE(é,2IATR ~/R 1 TE C 6,2) AT r WRITECé,2'ATT DO 806 I=l,M II =1 + M

153

, .

EIR( l, l)-=E( l ,1) EII(I,l)=F<I,l) EIRCII,ll=AR(I,I)

806 EI1( II ,1 )=AI (1 ,U . WR 1 T E ( 6 , 8 3 3 )

833 FORMAT(5X,'VAlUE OF 1 AT GENo SIDE@' 00 984 I=1,N 00 984 J=1,N TC l, J ) =T ( 1 , J ) >:c <:1 1010

984·· X ( l , J ) = X ( l ,J ) *" 1 0 1 0 00 2 1 9 1 = 1 ,N . 00 219 J=l,N

219 lllCI,J)=DCMPlX{T(I,J),X(I,J» DO 776 I=l,N DO 776 J=l,N TCI,J)-=Oo XC I,J)=Oo

776 CONTINt:E DO 774 l=l,N TCI,1)=1,,00

774 CONTINuE CO 771 IGF=I,4 IF{IGF-l)999,888,999

999 CONTINUE WRITE{6,668)

668 FORMAT(5X,'VALUE CF EACH SECTION@)· 888 CONTINLE

CAlL CINVRT(lll,N,N,N) DO 808 I=l,N DO R08 J=l,N AXY=ZZ 1{ l, J)

YIR{I,J)=DRL(AXY)*"lDI0 YII(I,J)=DIM{AXY)*olDIO

808 CONTINUE 335 CALL G~PRC(YII,EIR,EI6,N,N,I)

CAlL G~PRD(YIR,EIR,EI3,N,N,1) CAlL G~PRD(YII,EII,EI4,N,N,1) CALl G~PRO(YIR,EII,EI5~N,N,1) DO 810 1 =1, N DO 810 J=l,l EIS(I,J)=ËI3(I,J)-EI4{I,J). EIT<I,J)=EI5(!,J)+EI6(I,J)

810 CONTINUE WRITE(6,862)

867. FORMAT(SX,'VAlUE OF Vo A~o C IN REAL AND IMAGn@) DO 820 1 = l, N DO 820 J=l,1

820 Elh( I,J'=OSQRT(EIS(I ,J)>(c*2+EIT{I,J)"~*2) hRITE(6,2)EIS,EIT \oiRITEC"6,8671

867 FORMAT(SX,'VALUEOF ~AGo OF V AND I@) hR Il E ( f. ,2) El W DO A 3 1 1 = 1 , r-1 DO 831 J=l,l

154

,..

II=I+M ARCI,J)=EIS(II,J)

831 AI{I,J)=EIT(II,J) DO 661 1=1,3 ATR(I)=Oo ATI(I)=Oo

661 CONTINUE DO 662 1 =1 t Nl II=I +N 1 ItI=II+Nl ATR(l)=ATR(l)+ARfI,l) ATRC21=ATR(Zt+AR(II,11 ATR(3)=ATR(3)+AR(III,1) AT 1( 11 =A TI ( 1 ) + AI ( l ,1) ATI(2'=ATI(2)+AI(II,1) AT 1 ( 3 ) =A TI ( 3 ) + AI ( 1 1 l , J.) ,

662 CONTINUE DO 663 1=1,3 ATT(II=oSQRT(ATR(I)**2+ATICI'**2)

663 CONTINUE WRITE(6,2)ATR WRITE{é,Z'ATI WRITE(6,2)ATT CALL G~PRD{TtAl,T3,NtN,N) CALl G~PRO(X,81tT4,NtN,NI CALL G~PRO(X,Al,T5,N,N,NI CAll G~PRO<T,81,T6,N,N,N) DO 664 I=l,N 00 664 J=l,N ' T( I,J )=( T3( l ,J )-T4( l ,J) 1*01010 X (l, J) =( T5 ( 1 t J) +T6 (I ,J 1 ) *0 1010

664 CONT 1 N tJE DO 665 I=l,N 00 665 J=l,N Z Z l( l , J ) = DC ~ P LX ( T ( l , JI, X ( l ,J ) )

665 CONTINUE 00 988 I=l,N DO 988 J=l,N T ( l , J ) :: T ( 1 , J ) * 0 l D- 8 X ( l , J ) :: X U , J ) * 0 10-8

988 CONTINLE 111'CONTINLE

STOP END

TOTAL MEMORY REQUIREMENTS 019F62 BYTES

155

1

1. 1 1

1 1 1 1

1 1 : , 1 1 1

1(~: i ! 1

! 1 1 1

1 "1 !

1 1 j , ; : i )

1 ! 1

TFVC

COfJPLEX*16 AXY COMPLEX*16 CO(3,3),P{3,3),PP(3,3) ,ZZ(6,6) DOUBLE PRECISION XLL,OIR,DII,DIT COUBLE PRECISION ABF1,A8F2,ABF3 000BLE PRECISION C(3,31,C(3,3),S(3,3) DOUBLE PRECISION DI~,ORL,YRR(6,6),YRI(6,6) COUBLE PRECISION VKl,VK2. DOUBLE PRECISICN lPO,U4,U5,Z4,Z5 DOUBLE PRECISION UF(6,6),RSS DOUBLF. PRECISION Q3(3,3),Q4(3,3),P3(3,3),P4(3,31,

156

2 B(3,3"SI(3,3),S2(3,31,S4(3,31,S7(3,3),$3(3,3),$8(3,3),S9(3,3), 3 S10(3,3), R,RG,Hl,H2,H3,SC,012,DI3,014,024,015,045 '.' DO U BLE P 1= E C 1 S ION T ( 6 , 6 ) ,X ( 6 , 6 1 , AC U ( 3 ) ,A R l( 3) ~ AR 2 ( 3 , ,A Il ( 3 ) , . 2 AI2(3),AR(3',AI(3),YR(3,3),YIC3,3),E(3),F(3JiVl6,31,W(6,31,· .3 V l( 6, 3 ) , V 2 ( 6 ,3) ,W 1 ( 6 , 3) , W 2 ( 6 ,3) ,P 1 ( 3 , 3) , P 2 ( 3 , 3 ) , CH ( 3 , 3 J ,Q 2 ( 3 , 3 , , 4 PQ(3,3) ,Q(3,3) ,RSY DOUBLE PRECISICN PC(g,5' ,PD(3,3',QD(3,31,l(3,3),Y(3,3),A(3,3', 2 U(S,S),XL,VV . DOUBLE PRECISION EIT(~,lJ,EIX{6,l"EIC(6;11,EI3(6,1),~I4~6,1), 2 EI5(6,l',EI6(6,1) . DOUBLE PRECISION EIR(6,1),EII{6,1) DOUBLE PPECISION XTL,RS,TT(6,6),XX{6,6),T3C6,6"T4(6,6', 2 T5{6,61,T6(6,6) . DOUBLE PRECISION V\H6,3) ,WW(6,3) DOURLE PRECISION ZF(3,3) EXTE RN AL 011·1 N=6 M=3 RSS=o1C1/o5DO DO 981 1=1, N DO 981 J-=1,N UF ( l ,J )=00

981 CONTINUE DO 982 I=l,N UFCI,I)=01D1

982 CONTINUE U F C 4·, 1 ) = R S S UF(4,2)=-HSS UF C 5,1 )=-RSS UFCS,2)=RSS CC1t11=.,616268DO CC 3,3) =C ( 1 ,1 1 CC2,2)=061346100 C( 2, 1)=0 83U24[)-1 C ( 1 , 2 1 =C ( 2 , 1 ) C ( 2 ,3 ) =C ( l , 2 ) CC3,2)=CI2,3) C(1,3)=0308420-1 C ( 3, 1 ) =C ( 1, 3 )

"

( \

0(1,1)=.7135740-5 0(3,3)=0(1,1) 0(2~2)=o7299870-5 0(1,2)=-09874310-6 0(2.1)=0(1,2) 0(2,3)=0(1,2' 0(3,2)=0(1,2' 0(3,1)=~02309080-6 o ( 1,3) =0 ( 3, 1 )

,2 FORMAT(2X,3016.6) WRITE(6,325'

325 FORMAT(lOX,'VALUE OF Z@) DO 622 1 =1,,3

622 hRITECé,2){CCI,J),J=1,3' \-iR 1 T E C é , 324 ,

324 FORMAT(19X,'VALUE OF Y@) DO 621 1=1,3

621 WRITE(6,2)(OCI,J),J=1,3) VV=0735006/0SQRT(03001) E( 1) =-o5*VV E(2)=-o5*VV E( 3)=VV F(1)=-o5*VV~DSQRTCo~001) F ( 2) =-" F ( 1)

F(3'=Oo 'WRITE(6,2)E,F XTL~o2048003-o2048D02 DO 888 IJK=2,10 WRITE(6,2'XTL XL=XTL DO 756 IUO=1,2 DO 378 I=l,M

'DO 378 J=1,M Y(I,J)=O(I,J'*XL Z ( l , J ) =C ( l , J '* X L

378 CONTINUE CO 10 l=l,M, DO la J=1,M Sl(I,J'=O~ 52CI,J)=0. S3 ( l, J )=00 S4(I,J)=Oo

10 CONTINCE DO 20' 1=1,3 Sl(l~I)=olOl 52(1,1)=0101 S3(1,1)=0101 S4(1,1)=0101

20 CONTINUE JF(tUO~2)901,902,901

902 corn t NUE DO çe3 1=1,2 DO 983 J=1,3 52(1,1)=00

,157

ce ' S3(I,I'=0. Z ( 1, J) =00 YCI,J)=Oo

'Y(J,I)=O. ZCJ,I'=Oo

983 CONTINUE 901 CONTINUE

CALL'GMPRo(Z,y,A,3,3,3' DO 11 I=l,M 00 11 J=l, M

, A ( l , J' =- A ( 1 , J ) , Il CONTINUE

844

00 844 'I=l,M 00 844 J=l,M 8(J,I)=A(I,J' CONTINUE CALL G~PROCA,AtS7,3,3,3) CALL G~PRO(B,B,S8,3,3,3' CALL GMPRo(S7,A,S9,3,3,3' CALL G~PRoCS8,B,S10,3,3,3) 00 70 I=l,M 00 70 J=l,M ,

"

S l{ 1 ~ J , = S l( 1 , J , + A { l , J, /0 201 + S 7 ( l , J ) /0 24002 + S 9 ( l , J , /0 72003 S2{I,J'=S2(I,J)+S7(I,J)/o1203+A(I,J,/.,6D1+S9CI,J''o504004 S3 CI, J , = S 3 ( l ,J , + S 8 ( l , J ) / ., 12 D 3 + B ( l , J ) /06 D l + S 10 ( l , J ) /0 5041)04 S4CI,J)=S4(I,J)+B(I,J)/o2Dl+S8(I,J)/o24D02+S10CI,~)/o72003 70 CONTINUE' ' CALL G~PRDCS2tZ,S7,3,3,3) CALL G~PRO(S3,y,S8,3,3,3) 00 18 I=l,N 00 18 J=l,N T( I,J'=Oo X(I,J)=Oo

18 CONTINuE

,17

758

117

00 17 I=l,M 00 17 J=l,t-l II=I+M JJ=J+M TC I,J'=SlCI,J) T( II ,JJ)=S4( I,J) X(I,JJ)=S7(I,J' XCIl,J)=S8(I,J) IFCIUO-l)757,758,757 CONTINLE DO 117 1 = 1, N 00 117 J=l,N TT(l,J,=TCI,J' XX(I,J)=X(I,J) CONTINUE XL=o204~003-XTL

757 CONTINlE 756 cnNTINLF.

CALL G~PRo(TT,T,T3,6,6,6) CALL G~PRD(XX,X,T4,6,6,6)

158

CALL G~PRD(XX,T,T5,6,6t6) CALL G~PRo(TT,X,T6,6,6,6)'

,00 118 1 = l , N DO 118 J=l,N T(I,J)=T3(I,J)-T4(I,J) X ( l , J ) =T 5 ( l ,J ) +,T 6 ( l , J "

118 CONT.! NUE DO !9 I=l,N, DO 19 J=1,M V( I,J"=Oo WCl,J)=Oo

19 CONTI NeE DO 22 1=4,6 J=I-3

22 V( I,J'=!o VK!=24Co VK2=18Co RS=" 500 V(1,1)=RS V(2,2)=V(1,1' V(1,2)=V(1,1)

,VC2,1)=V(1,1' Vt3,3)=VK1 W(3,3)=VK2 WRITE(6,664)

664 FORMAT(2X,'VALUE CF THE lOAD@' WRITE(6,2)VK1,VK2 CALL G~PRo(T,V,Vl,6,6,3) CALL G~PRo(X,W,V2,6,6,3)

,CALL G~PRo(T,W,W1,6,6t3) CALL G~PRD(X,V,W2,6,6,3)

DO 90 I=1,N DO 90 J=l, M

, Vl(I,J)=VUI,J)-V2<I,J), 90 WUI,J)=Wl(I,J)+W2<I,JL _

C INVERSE THE PARTITION MATRICS CF Zl 00 116 I=1,t-l 00 116 J=1.M Il=I +M P ( 1 , J , =0 C tJ. PL X (V 1 ( 1 l , J ) , W l( lIt J ) )

116 CONTINUE CALL CINVRT(P,3,3,3)

,C IMPEOANCEflPQ&JQ 00 93 1=1,M 00 93 J=1,M Pl<I,J)=Vl(I,J' QI ( l , J ) =w 1 ( l ,J , AXY=P( I,J) fi2(T ,J)=CRUAXY)

93 Q2(I,J)=CI~(AXY) CALL G~PRD(Pl,P2,P3,3,3,3' CALL G~PPO{Ql,Q2,P4,3,3t3)

CALL G~PRO(Pl,C2,03,3,3,3) CALL G~PRD(Ql,P2,Q4,3,3,3)'

159

DO 200.I=I,M DO 200 J=l,M 9( I,J)=Q3(I,J)+04(I,J)

200 PQCI,J)=F3( l ,J)-P4CI ,J) DO 300 I=l,M DO 300 J=l,M

300 PP(I,J)=DCMPLX(~Q{I,J) ,Q(I,J)) CALl CINVRT(.PP,3,3,3) DO 400 I=!,M DO 400 J=I,r~ AX Y=P PC 1 , J ) YR(I,J)=CRL(AXY)

400 YICI,JJ=DIM{AXY) CALL GMPRD(YR,E,ARl.,3,3,1) CALL GMPRD{YI,F,AR~,373,1) CALL G~PRD(YI,E,All,3,3,1) CALL G~PRD(YR,F,AI2,3,3,lJ DO 222 I=l,M AR CIl =AR U 1) -A R 2 (! )

222 AI(l)=AI1(IHAI2(I) WRITE(6,546)

546 FORMAT(2X,' THE INTER-SUeCCNéuCTORS CURRENT#@) WR 1 T E ( é , 7 86 )

786 FORMAT(lOX,'SENDING END@) WRITE(6,2)AR,AI DO 4't4 1 = l, M

444 ACUCI)=DSQRT(AR(I)**2+AI(I 1**2) \otR 1 TE ( 6 , 2 ) AC U DO 503 I=!,N DO 503 J=l,N

503 ZZU,J)=DCMPLX(T<I,J),X(I,J» CALL CINVRT(ZZ,6,6,6) DO 854 I=l,N DO 854 J=l,N AXY=ZZ (1 ,J) YRRCI,J)=DRLCAXY) YRI(I,J)=DIMeAXY)

854 CONTINUE 01) 119 1=1,3 11=1+3 EIRCI,ll=ECII EIHl,U=F(J) EIR(II,l)=AR(I) El 1 ( 1 l ,1 ) =A 1 ( 1 )

119 CONTINLE CALL G~PRD(YRR,EJR~EI3,6,é,1) CALl G~PRD(YRI,EII,EI4,6,6,1) CALL G~PRD(YRI,tIR,EI5,6,6,1) CALL G~PRD(YRR,EII,EI6,6,6,1) DO 456 I=l,N DO 456 J=l,l

EIr< l ,JI=EI3( I,J)-EI4(I,J) EIX(I,J)=EI5(I,J)+EI6(I,J)

456 CONTINUE

160

C---, ..

DO 411 I=l,N EICtI,1)=DSQRT(fITCI,1)**2+EIX(!,1'**2)

411 CONTINUE WRITE(6,787)

787, FORMAT(lOX,'LOAO ENO@) WRITE(6;2)EIT,EIX WRITE«(:,2)EIC XT~=XTL-02048002

, 888 C a NT 1 NUE 666 CONTINUE

STOP , END

TOTAL MEMCRY REQUIREMENTS 003C56' BYTES

161

"

Ice

j

! ,. i

1. 1 i 1

1

1· 1 1

1

.i e 1 1 1

SUBROUTINE CINVRT CA,IK,IJ,N) IMPLICIT . REAL*S(A-H,O-Z) CO~PLEX*16 13(25) ,.C(25) ,PIVOT,Z,A o 1 ME N S 101'. 1 P ( 2.5) , 1 Q ( 2 5 ) , A ( 1 K , 1 J ) EPS=loC~.l2

1 DO 221 K=1,N PIVoT=(OoOO,o.oO) DO 61 I=K,N 00.61 J= K,N Cl =C 0 A ES ( A ( l , J ) ) C2=CDA8StPIVOT) IF(Çl-Ç2'61,6,é

6 PIVOT= A(!,JI IP{K)= 1

IQ(K)=J 61 CONTINUE

7 Cl=CDABS(PIVOT) IF(C1-EPS)39,B,S

S 1 F (1 P ( K ). E Q. K) GO T 0 Il 9'IPK= IP(K'

DO 10 J=1,N z= A(IPK,J) A{IPK,J)= A(K,J)

10 A(K,J)= Z Il IF ·(lQ(K).EQoK) GO Ta 14 12 1 Q K= lQ ( K)

DO 13 1 = 1, N . Z= A ( l ,1 QK ,

AtI,IQI<)= A(I,K) 13 A(l,K'= Z 14 DO 191 J= l,N

IF (J.t\E.K) GO TG 18 17 B(J)=(.lD01~OoDO)/PIVOT

C(J)=(.lD01,OoDO) GO TO 19

18 B(J'= -A(K,J)/ PIVOT C(J)= A(J,KI

19 A(J,K)=(O.DO,O.DO) 191 A(K,JI=(O.OO,O.DO)·

DO 221 J=l,N DO 22 1= 1,N

22 A( I,JI= A( l,JI +C(I'*B(J' 221 CONTINUE

DO 30 K= 1,N KK= N-I<+ 1 IF (IP(KK).EQoKK) GO TO 27

25 IPK= 1 P(KK) 00 26 1= 1,N z= A ( l ,1 PK) A(I,IPI<I= A(I,KK'

26 A( I,KK)= Z 27 IF (IO(KKI.EQ.KK) GO TC 30 2A IQK= IC(KKI

DO 29 J= 1,N

162

°Î " ' ..... J

z= A(IQK,J' A(IQK,J)= A(KK,J)

29 A( KK"J)= Z 30 CONTINUE

RE rURN 39 WRITE(é,40)Cl 40 FORMAT(lHO,15HPIVOT lESS THA~,DI002)

CALl EXIT END

TOTAL MEMORV REQUIREMENTS 000D5A BYTES

163

•

,e 1 1

SUBROUTINE GMPRD(A,B,C,N,M,K) DO UB LEP P E C t S ION A ( N ,M) ,B ( M, K , , C C N, K , DOUBLE PRECISION X,XP,XM

'DO l J=l,K DO 1 I=l,N

1, ct I,J)=OoOO DO 2 1 = l, N DO 2 J=l,K ' XP=OoDC XM=OeOO

8 DO 3 J J=l , M, X=A( I,JJ):>:cBCJJ,J) IF(XoGToOoDO)XP=XP+X IF(XoLTeOoDO)XM=XM+X

3 CONTINUE 2 CCI,J)=XF+XM

RETURN END

TOTAL ME~ORY REQUIREMENTS 0003A4 BYTES

164

165 DOUBLE PRECISION FUNCTION DRU AXY) le REI\L*8 AXY(l) OkL=AXYCl) RETURN END

TOTAL MEMORY REQUrRE~1ENTS G00122 BYTES

DOUBLE PRECISION FUNCTION DIM(AXY) REAL * 8 A X Y ( l ) . ' DIM=AXY(2. RETURN END

TOTAL MEr-10RY REQUIREMENTS n00122 BYTES

166 SUBROUTINE COR(P2,Q2,R,S,N,KK) DOUBLE PRECISION P2(N,N),Q2(N,N),R(N,N),S(N,N) ,RR(24,24), 2 RI(24,24), Tl{24,24),T2(24,24),T3(24,24I,T4(24,24) DOUBLE PRECISION C(24,24),0(24,24) DO 2 I:l,N 00 2 J=l,N Tl<"I,J)=OoDO T2fI,J)=OoDO T3(1,J'=OoDO T4( l ,J )=0000 , O( I,J)=OoDO 'c fI, J ) =00 DO

2 CONTI NU: DO 3 I =1, N

3 DCld)=olDOl 00 77 KLH=l,KK CALL G~PRD{P2,R,Tl,N,N,N) GALL G~PRD(Q2,S,T2,N,N,N)

, GALL,G~PROCP2,S,T3,N,N,N) CALL G~PRD(Q2,R,T4,N,N,N) DO 4 I=l,N ' DO 4 J=l~N

, RR ( l , J ) = 0 ( l , J ) - Tl ( l , J ) + T 2 ( l , J ) 4 'RI (1 ,J )=C( l ,J)-T3( l ,J)-T4( I,J)

CALL G~PRO(R,RR,Tl,N,N,N) GALL G~PRO(S,RI,T2,N,N,N) CALL G~PRD(RfRI,T3,N,N,NI CALL G~PRO(S,RR,T4,N,N,N) DO Il 1 = l, N DO' 11 J= 1 ,N R ( l , J ) =R ( l , J ) + Tl ( l , J ) - T 2 ( l , J )

Il SCI,J)=SCI,JI+T3(I,J)+T4<I,J) 77 GONTINLE

RETURN END

TOTAL MEMORY REQUIREMENTS 0096F6 BYTES

167 . . . SUBROUTINE CRR(P2,Q2,R,S,N,KK) DOUBLE PRECISION P2(N,N),Q2(N,N),~(N,N),S.(N,N),RR(12,12),

2 RI C 12 ,12) , Tl( 12,12) , T 2 ( 12 ,12) ,T 3 C 12,12 ) , T 4 ( 12, 12) ,C C 12 ,12 ) 30t12,12) .

DO 2 1 =1 ,N DO 2 J=l,N CCI,J)=OoOO O(I,J)=OoDO

2 CONTINLE DO 3 I=l,N

3 OC f, 1 )=0 1001 DO 77 KLH=l,KK CAll GMPRDCP2,R,T1,N,N,N} CALl G~PRD(Q2,S,T2,N,N,N)

CAlL G~PRD(P2,S,T3,N,N,N)

CAlL G~PRO(Q2,R,T4,N,N,N)

DO 4 1=1, N DO 4 J=1, N RR(I,J)=O(I,J)-Tl(I,J)+T2(I,J'

4 RI CI ,J ) =C ( l , J ) - T 3 ( l , J ) - T 4 ( l ,J ) CAll G~PRO(R,RR,Tl,N,N,N) CAll G~PRO(S,RI,T2,N,N,N) CAlL G~PRDCR,RI,T3,N,N,N)

CAlL G~PRD(S,R~,T4,N,N,N) DO Il 1=1, N DO Il J=l,N RCI,J)=R(I,J)+TlCI,J)-T2(I,J)

Il S(I,J)=S{1,J)+T3(l,J)+T4(l,J) 77 CONTINUE

RETURN END

TOT~l MEMORY REQUIREMENTS 002A5A BYTES

168

APPENDIX D

UNE CONSTANTS OF THREE PHASE BUNDLED CONDUCTOR

TRANSMISSION UNES

The computed inductive reactance matrix, XI and capacitive susceptance

matrix, B , for configurations having different numbers of subconductors per phase as c

described in Chapter ", are 1 isted in this Appendix.

follows :

XI =

where

The elements of matrix XI related to the subconductors of the line are as

-" ,

X12 - - - - Xl r - - - -.- - - - - Xl n

, '\.

X -rl

, Xs1 1

1

Xn1 - - - - - -

,

,

- - -

, '\.

" '\. '\.

'\. ,

- - - - ... ,

X is the self-inductive: reactance of subconductor r, rr

'\. -x nn

ohms 1. '1 ,mIe

X is the mutual - inductive reactance between subconductors rand s. rs

Similor/y, for B , in mhos/mile. c

The numbering of the subconductors of the line is From Phase A to

Phase C where the numbering sequence for subconductors, in a given phase for the

various configurations used, is shown in Figure D - 1 •

1 2 0 0

'2 Subcoriductor

Buriëfle

1 3

0 0

·0 0 2 4

:4 Subcondu'ctor

Bundle

2 4

0 0 1 6 0 0

0 0 3 5

'6 $ubcondüctor

Bundle

FIGURE '0'-'1: NUMBERING SEQUENCE OF SUBCONDUCTORS PERPHASE.

169

170 TW() SUACOr\'[)I.ICTnp.s ps;p. [)'-lI\SF CO\:F tGURI\Tlr:1t\,J

• VALUE OF INDUCT IVE REACTlINCE rvlATR IX

0 .. Q2521D 00 0.49QOlO 01) O.q()~fnO-01 O.P:8310r}-01 0.41R1/tD-01 O.40982D-Ol "" ,... ,rt ,.

0.908fl"3D-01 O.93786:1-('1.0.QZ:J79f) 00 00502740 OC O.91092D-Ol ·O.i38310[)-01 . __ .. ··-··O.e8310D-01· o. 91092D-01·-0.:'0274;)· 00·1). 92S79n 06 O.937B60-01 0.908830-01 . . 0.41R140-01 0.426720-01 0.Q10028-01 O~q~78~O~01 0.Q2493D· 00 0.499010 00 r.--· ·-·0.40QR2f)-01-0.4181.4ù-O'·-0.8~310D-OI-O.90R831)-01- 0.49901000 0.925210- 00 1 VALUE OF GAQ~CrT!V:. Sf!SGr:P;'\~IGE "'\T~IX

-O.6~577D-05-0.366R6D-05-0.184~30-06-0.16541D-06-0.66616D-07-0.630560-07 -O. 366R 61)-0 5 0.666'371)-05- 0.206958-06- 0 .1,R 4750- 06-0. 705 9 /+D-07-0. 6'>61' 61)-07 --0.lB4~30-06-0.206950~0~Oo6~727D-05-0.36579D-05-0.1~475f)-06-0.16541~-06 -0.16541J-D6-0.1~4750-06-0.~65790~05 0.66727D-OS-0.20695Q-06-0.1Q433D-06. I----"~~ ...... ~--l--I>d 07 G. 703 ~ 1.') () 7 0 d 1'3475 '! 06- (). 20695 D-Q'J G,-{~6 5 7 o......g..ç>-",:, 0 • 3(~l--G-5-­-O.f3C56D-07-0.66616D-07-0.16541D-06-0.18433D-06-0.366R6D-05 0.66577D-05 ... ______ ~~ •. ~"." •• ",. .. ~~_ •• _ .•. _ ....... __ ..... ~ ••. _ ..... _ •. _.~ •• .-_ r"" _"u"'. _; ____ .• w _~ ... __ .~ ...... _~_.~ ..... , ............. __ ••••••••• __ .......... _ •••• _ ......... ~.... ..~. __ • _",_ ~ ............ : ..... - .... _ ......... -'< ........ -:

•• ":'(00, "*. ______ ... _ ...... _- •• _ ... _. __ ._~ _____ .... _ ....... _ •••• _ .. ~_. __ .. _~. ___ "._ •• _._._~ ___ ._",_,,_ ..... ____ ....... ___ •• '" ... _.~ ... ___ ......... ,_. ___ •• _~.~ .... __ ~ ___ ~ ••.•••••••• ,. _'0;- ._ ....... _ •..•• ~ ' ...... _ .... __ .... • __ ._ ••.•

" ....... -._. -.- ... -.. -....... _~_._- ... __ .. _,._ ....... _ .. _. ___ ._., ..... _ .. " .... ____ ..... _ ...... -:--...... _: .... ____ ._ ... _. __ ..... ... · .. _._h·'.~ _. __ .. _~ ... _"~." ._.,. '"_' .. _,_ .... ~

•• -_ •• " --.~ •• -- .. -_ .... ,,.. - ___ • ___ ._ ........ _ ._ • __ ....... _._ •• _~ _~~._ ..... ~.-; __ •• ____ ._. ___ .... _--.. ______ •• _ ..... _ ••• , __ • _ ....... _ .......... _"" _._._ ..... _._._ •• _ •• 4 .................. "_' ........ _ ........... .

.. _-- -......... .. .... "_.-_ .. _--~ .. _ ... _~--_.' .. -.. ", ... - ... , ..... _~ ... --." ................. _ .. ,.- .,-" ..... -- ... , ..... -, .. -.. ---.~. ..... . "'-.

VALU~ OF INDUCTIVE REACTANCE MATRIX

0.924640 00 O.49759n 00 0.498.36D DO 0.455360 00 0.B4605D-01 O.B~BIOO-01 -_.- ---- ·C.·81 79'+D-Ol· o. BIO 10 D- 01 0 .. 31284[1-010 .. 3083 3D'-0 1 .0.30.5170-01. 0.300711)- 0 1.

0.49759D 00 0.922650 00 0.4553?D 00 0.496390 00 O.83133~D-01 0.P'3120ihOl .. _----- -0.81 039D-O 1··0. tiO '32 60- 0 1 ·0. :~083W- 01 o. 3039'tO-01 .0.300710:-01 0.29637 D-Ô 1

0.49836D 00 0.45535D 00 0.~2420D 00 004971AO 00 0.87435U-Ol 0.~66290-01 0.134503.0 01 O.H37G7") JI O.";.-2077f)~1 0.31-')l Q D 01 O.31-.-:~4D":'Ql !). ~Q.B.3.L,)-O . .I-<lL.--_

0.45536D 00 0.49639D 00 0.49718D 00 0.92?25D 0:) 0.86b6?1)-01 0.859391)-01 ----...:.---0. 83741D-0 1-·00830230-01 .. ·0 .. 31619D-0 1 0 .. 311 760-0 10. 3r) H3 31)-01 O. 30394D-Ol

0.84605D-01 O.8383R!)-Ol 0·.fl7435D-Ol 0.86662D-Ol 0.92160r; 00 0.49't570 00 ----:-····--.. ··0.4:95'53D 00-0. 452 ~ 5iJOO-·O. a45()30-Q le. 83 741 0- 0 1 o. BI 7':flf D-O 1 .·0. ::no 3q:)- 0 1 '.

0.83810D-01 0.831200-01 0.86629~-01 O.A5939D-Ol 0.49457D 00 0.Q196AD 00 0.452558 oé O./1':'3~'?I) 'JO 0.81707'')-01 0.830-2 3 (1-('\1 0.8 1 0lCH'-fl1 O.3032b·) 01

0.817940-01 0.8103QD-Ol 0.845030-01 0.837410-01 O,49553D 00 O.452~5D 00 - ...... --0 •. 92l6QD 00· 0.'+94571)· 00 0 • .'37'.351)-01 0.86662D':'01.0.84605D-OI0.B383BD-01

0.R10l0D-01 0.803260-01 0.f33707D-01 0.83023D-01 O.45255D (JO 0.'+91{,0c> 00 .~_._~- .. -o. 494570 · .. 000.919660 ·00·0 .. ?66 291)..., 0 1 ... 0 e 8S 93 9D-0 1.0. d3 81 o C,:,: 01. 0.8 31Z0 0-0 1 .

'. 0.31284D-Ol 0.308311)-01 0.3207,7D-Ol 0.31619D-01 0.845030-01 O.H'~70ThOl· O.i37'IY;O al Oof?'J,291 qt 'L??!.::')".' 0')-0.I,Q718 0 1'0 a ItO\~3'·r ad fI.'·SC;--jrj) 00

0.30833D-01 0.30394[')-010.316198-01 Q.31176D-01 0.837'tlD-:)1 O.P,~02V)-()1 . 0.866620-01 0.8593( 1)-01 0.49718D 00 0.1.J222~!) 00 0.4553(-,f)· 00 CJ.4963'·Ji1 0') 0~30517D-Ol 0.300710-01 0.31284D-Ol 0.30833D-01 0.B17S~D-Ol 0.91010D-Ol 0.846050-01 O.~3BI00-01 0.498360 00 0.4553~C> 00 O.92464b no O.~975YU 00

0.30071D-Ol 0.29637D-Ol O.309310~01 0.303940-01 O.~1039D-Ol O.80326U-Ol O. ? 3 g 3 Fm - 9 1 0.:'1'3 l l' Q ') rH -), 4. 5 5 'l 'j Q 1) 1) 0 , !.. Cl 6 ~ Cf!) Q Q Q. l; () 7 " C, n f\fl .,. Cf 2 2 (., 5 -) Q Q

VALUE OF CAPACITIVE SUSCEPTA~CE MATRIX . .

0.79987D-05-0.2~459D-05-0.2~4670-05-0.133190-05-:).~4!R4D-07-0.610180-07 -0.533a6D-07-0.~0334D-07-0;156821)-07-0.14333D-67-0.l~177D-07-0.12R~30-07 0.2'1'.593 ')) O.:lG9':'?') ':':;').~ )'3:'~1~-1)5 ~. :'d9 S Ci-:Ot;. O.6Pf,(-?O °7-0,5'=;lc:;cH~,) 07 -0.50572 D-O 7-0. 'd 9590-0 7-0. 14302;:1- 0 {-O. 131951)-07,...0.12 P. 5 3!)-Ü7:'O .. 1 l 763 0)-0 7

_ .. ----·-0.2'+4679-05-0. 133391)-05 ·0.80 041 0-05-0. 2440RD-05-0 .. 768001)-07- 0.73 4H -3D -0 7 -0.632990-07- 0.601 (.1 2.)-07-0. 1 7 4~28~ C 7':' C .. 16074:)-07-0.1':)692 (J-07 -O. 1'+3ù2;)-0 7

. -_. -- 0 • t 33380 - 0 5- o. 24"39 R D - ü 5 - 0 • ? 4 4 0 8 [) - 0 5 O. 80 1 Ob ü - 0 5 -:). 7 3 !J P. 5 D - 0 7 - O. 72 1 16:) - 0 7 - O. 60 390 [) - 0 7 - 0 • 5 !37 77 0 - 0 7 - O. 160 7 1+ f) - 0 ., - 0:. 149 20 D -0 7 - 0 • 1 4" 3 3 D - 0 7 - Cl. 1 31 9 ') ') - 0 7 O. (_ 4 l S 4 i) 07 C. 6 11 .s ':' p - (J7- 0 • 7 (1 ')lJ Q 9- 0 7 :). 73 é, as n .." .. O..L.ll-W-1-2-:-l.-l~) 5 - 'l • ~ u.:~.o.r).:-l)j~~ __ -0.243620-05-0. H 2370-05-0 .. 61299;)-f)7-0. 6CJ390D-07-0 .• 5 n-.-1t~D-07-J. 50'5 72!J-Q7

. - .. - 0 • 0 10 18 D-O 7 - O. 5 95 H Q i) - 0 7 - 0 • 7 34 Ü 3 f) - 0 7 - O. 7 2 1 16 D - 0 7 - () • 24 l 30 fJ - Ù 5 C • 8 J UVH) - 0 5 -0.132370-05-0.24300D-05-0.60102~-07-0.58777i)-07-0.5~334D-07-0.4895q!)-07

-0.531860-07-0.50572 ù-IJ 7- 0 .63 ~')9n- 07-0. hQ3<10D-f)7 -o. 2'dé 70:"'05-0. l ')?3 7:) -0 '; . 0.'801210-05-0.21+ 330D-05- 00 76HüO f)- 0 7-0.7368 5D-O 7-0. fA 1 HI, 1)-f)7';"~" (J 1161 i)-~, 7 o • 50 33 /11') 0 7~.!.Q..5 ç J - 0 7 1). (..L~ J) (')7-:).::: '3XJ.21~lD - o. ] ., ? ~ 7..D.::.Q ,) - Îl .... 2!d'Oill_=..O..:i __ - o. 2 lt 3 30 D - 0 5 (1. HO 113 '+':) - 0 5 - o. 7 ~ t,. 8 3 [) - 0 7 - 0 • 7 2 11 b 0 - 'J 7 - 0 • 6 1 ° 1.'3 :J - (1 7 - n. s ~ .') >l').) - 0 7

- o. 1 5 6 82 [)- 0 7 - O. 1'+ 3 :) 2 1) - 0 7 - 0 • l 7 4 p, 2 i) - 0 7 - G. 1 6 () 7 '.1) - 0 7 - 0 • 63 ? 9 C:1 [) - ~)7 - 0 • ~,J l 0 2 ~j - 0 7

-O. 76 g 0 0 iJ - 07 - 0 • 73 4 B 3.) - t)7 o.:-'{ 0 n 41 :) - uS - 0 • 24 4 (H.; 1) -1] 5 - (\ • 2 Il ,. Ô 7')-05 - 'J • l 33?> :, J - () :> - 0, 143'3 3lJ- 07- O. 131 9 S f)- 0 7 - () • 1 ~ 07 '+!) - 0 7 - o. l'. q Z oc: - 07- r. • '" 0 ~ '?~) [')-'.)7 - o. 5 d-n 7·) - 'J 7

_ -O. 73 6B SC - 0 7 - G • 72 Il i,:) - Q 7 - 0 e ;U~ 1 .. 0 ") n-I)·; O. ~j 1) 1 0:-< :) - 0 5 - [) 0 l ':\ 3 ~ El) - fJ ') -!) • ~ /, :3 C:3 ,)- () '; :-.o-.-l+H-1~')-{}-7-'}r-l-&~"-;,~.t+-..!'-7--~..s..~~"?'..l.j=Jl-L..,-o....1-f.3-33ll.~:.JJ .. .s13-d6D..-.JI~_'J .. 50 3.3.ft.J_~.u.7 ____ _

-O. 64 1 :3 'tO - () 7- r. • (; 1 0 l F, ;J- 0 7 - 0 • < ,:,. L. (~ 7 ') - ') ') - 0 • l .~ 3~' E Cl - t.1 S O. rF' ;~. 7 1:- 'J;; - 'J •. ) I+!~ ~.~ ) - ~~"

- 0 • -1 2 (j 5~ [)- 0 7 - O. l 1 7(,3;)- L> 7 -0 " 1/0 :.'. f)? r)- 0 7- ~). l ~ l .} 'ï U - 07 - ') • ') û '.J r 2 iJ - J 7 - o. l, :; :15 <) ) - 0 7 -O. 611 ':> qi)- 07 - (; • :, '-) S? 9J-:-l)7 -:) • 1 -n 3 <)1)- (l:; - C • 24 ? 9 S 1) -05 -0 .2 /,4,) ( 1)- () '; :).:1:.l 0(· )) - f)')

J72 SIX SUBCONOUCTORS PFP PHASF CONFIGURATION

VALUE OF INDUCTIVE REACTANCE MATRIX

00839280-01 0 0 831190-01 00A18140-01 00d036SD-01 0079079U-01 0078437D-01 0030846U-01 0030~31n-Ol Oa~0067D-01 0030079D-01 O~29322D-Ol 002934S~-Ol Oo49B55D CO 0092532D 60 0 0430940 06 O~4~904C 00 0~41330D 00 ·0043149D 00 00BS9070-01 0085125D-01 00H36~aD-Ol 00eZ30SU-01 008U904D-01 008ozeqD-01 00316130-01 00316010-01 0 0 209120-01 00308310-01 Oo300~OO-Ol 00 300790-01 I----f}-"Q -4,..c:;.o-::3z-~{1-9-1~~-t-A-S-o 0 0 lh~-3-2-'7-B-{)t}-O-~/...-9-5ry-3 ;r-Q0-0-o-4 .:?-9-7-90-00·-00846380-01 0083748D-tJl 0~8?552D-0J. 00oOS55D-0J..O o 797650-01 0~7c)0510-01 ·00308350-01 0030809D-Ol O~30D60D-Ol 0030050D-01 00ZQ3080-01 0~2932~D-Ol 0043170D CO 00499040 00 0041329D 00 00924680 00 0~~3053D 60 0049790~ 00 0~B8795D-Ol 0087958U-01 D,865090-01 00850200-01 00835940-01 0~8~93~D-Dl 00324170-01 00323980-01 00315970-01 0~31601D-Ol 0~30r090-01 0030R310-01 l----O1;l 4 30 O{)I)--BO 0,) 413 3 GB-BB-e~~-5-3-B-GG-{h...ç-2-1-50 f)-OO-{)~·4-':;'6-2 2-i)-Oo--00875210-01 00~6566D-0J. .O~85366D-Ol 0023G53Q-01 00824570-01 00816ÇOD-01 00316310-01 Oa31597D-01 003083RO-Ol Oo3U812D-01 C~30060D-Ol Oa300b70-01' 00413330 00 0043149D 00 0042979D 00 004979QO DO 00496220 00 00·92305D 00 00897070-01 0~e877eO-Ol 0.,57447D-01 0065786D-01 00844é,2D-Ol 00e3728D-Ol 00324490-01 00324170-01 0031631D-01 00316130-01 Go30a3~D-Ol 00308460-01 i----A-Q..-o-H8-3-9-2-BD 01 0) 8 5'3070 Oi-'~"fl, 3 3D G-l-e-rS-H-~--01 O~ 87-C)-?-1-e-~_-o1J&c;..7-0·7·!1-0-1--00920640 00 004954C9 00 Co 493720 00 OoL28780 00 O~42710D 00 004105~D 00 0083728D-01 O~829340-01 0081590D-01 .00S028~D-Ol 007qO~19-01 00784370-01 0083l190-01 00851250-01 00837430-01 00079580-01 0026566C-01 0?88778~-01 ·00~95400 00 O~92227D 00 C~427930 00 00496210 00.00410490 00 0 0428780 00 0085786J-01 0085020D-01 0 0836530-01 009230SD-Ol 0 0309550-01 00~0369D-01 I----F-l=o ++. -t ....... H.i-J":fi) 0 1 0) 8~ COl 00 2 2 5 :: i: DOl. O·) [: 6 5 0 9 E' G ::. (~-5-37.t-::.'~-{H-:--Eh-E-7-'r<'.-7-f)-{)-l--0049372D 00 Oa427930 00 CoS1892D 00 0~4104cD 00 Oo40?86D 00 Ook2710D 00 .. 0034468D-01 0,835949-01 O~r24570-01 O~80S04D-Ol 00797650-01 O~79079D-Ol Oo80369D-01 0~3230S0-01 0090SS5D-OI 00E5U20D-01 00f3653D-01 00857860-01 Oo4~87RO 00 Oo49621~ 00 Co~1049D 00 00922270 00 00427930 00 0049540D 00 00887780-01 0087958~-01 0~86566D-Ul O~e5125C-OI ü~83748D-Ol 0083J19D-O] 1------f-h<-~H-+.yB__B lOi) :)-e-9~11 0 0 7~~H-G-l---V~C;4-P-<H-{)""O-&2../t-5-7-&-{}l----Q,,+44';.) 8D-0-l---00427100 00 0041049D 00 00492860 00 Oo427S3D 00 Oo91892~ 00 Oa k9372Q 00 --_ ....... 0087447D-01 Oo865090-OJ OoR53tS6D-OI 00836SéJD-Ol 00825S2D-OI O,,8U!l4D-Ol 00784370-01 0080289D-01 Co 7S051D-01 Ooe29340-61 00F1690D-01 00P3728D-01 .'.- ._. __ .. 00410=20 00 0,)42878D 00 0,,427100 00 00495400 00 004~372f) 00 00S2064D 00 . 00897070-01 00~·87C;?D-01 Oo~n521[l-01 00859070-01 O,84(dBD-Ol 0\,g39~iW-01 ----iO~o~3 G44-&B-{H O:s 3 1-6-1-3~3GB-59-{}-1--D.r3-?4-l-7-G-O-l-{7cr3--1-6-3·1-D-0 -J-{}o-3·?4 4 Ç!. )-01---­Ooa3728~-01 00P57860-01 C~84468D-Ol OoB877fU-01 Oce7447D-Ol 00[97070-01 009230S0 00 00497900 00 0049622D 00 Oo4314sn 00 ·O~429790 00 ü a 4 1313D 0U ·Oo30831n-Ol 0031601D-01 On30809C-01 Oo323Ç&C-01 Oo315~7D-Ul Oo32417D-Ul Oo82Q340-01 C085070D-01 Oog3s c 4D-01 U067S~ED-01 ODe~5C~D-Ol o~en795n-Ol 00497S0D 00 00924SAD 00 00430~30 00 004SS04C 00 00413290 00 On43170n 00 ---~G-.s-3 OV6-7D-';H--{h;-3-{"~-1-?-B--{1-}:--0-,,-3-H O~J (-)-IJ-;l.-{!-o-3-1-5-S-=7-f:--01-0,,-30 +3 H~-0-1-·-·0 0 -3 lt· 3 J :) - (1 J._. -- -0~e1690D-01 0083653D-Ol Oo8?457D-01 O?8h56(O-01 O~~~3060-01 OaC7~2Jn-Ol Oo49622D 00 0~43053Cl 00 O~9?150~ 00 O~'+1330D 00 0~4o~;~:3D 00 O;>(~3f)UO() no 00 3 0 0 7 9~) - 0 1 0 ? 3 Ü 8 :3 1 D - 01 0 0 :3 00 'i 0 U - 03. 0 n j 1 b 0 ll: - 0 l 0 0 3 0 ·'H 2 0 - 0 : 00 :. 1 b J ~! i) - (H O? e. 0 2~ <;.1 D - 0 1 00 8 2 3 0 <: D - 0 1 ()" c~ 0 9 li 1. 0 - U 1 0 0 ~ ~ l 2 5 D - 0 ::. O,) ':l 3 ô t;. ~, f) - 0 1 0" G ') r. (l 7 D - U 1 0~4~14<;Ù CO 0,,4Q n O'-tD 00 0?41330n no Oo'12~"3?n 00 O~43(lCj4[) 00 oo/~ql'r;~I' 00 =---O-.,2~3-2-2 n-o·!-o-o-3·0 G5Dl)-{) 1-0~)-;:-0-3 0':; 0-U~--{1·n-3-C 8 Ôc: F- û l-{} 0-:'.· o·)(~ I)n -01- - ùo}I))3 ?:) [~- 0 l . O,7'7051D-()l 00P'f)(~5?U-Ol C,,7(n(J~:)-Ol ü:>(:"){'ifl"'-Ol 0->8?5~?[)-ül V.,i"'4(,::.;'[)-O'. 0,.,42979:) 00 Oolt132 C) 00 Cl a 'tS rj63 1) VU O.,430 C '4{) 00 00Q21;J;;n 00 c.,/1C?(·)l?[, or 00 2 I~ ? 4 ~ n - 0 1 O? 3 00 Fl D - ,) l U r', ? 9 :1 2 2 C - (j 1 U.., j C (i 3 1 C - 0 l ü.)? l> 0 (, on· -0 !. 00:; \) ;.Jt (., n - Ü l O.,7~"~~7~-()1 O,,8031.J'::ID-O~ C~7(";~)7C::D-OJ 0., 2~11Sfl-Ul u.,~lliil'11J-0J. ü.,ü:'1?··lD-Ol Oc'.,.1333!1 cc O~'+~17V' Of-' O"l-3]()OU U() (;.~/~C:t.l(·)~)I) co l',,!~9(-~:;?O 00 f)."-.??)f.:~: ()()

~ ________________________________________ ~ __________________________ 173 ____ _

et VALUE OF CAP.ACITiVE SUSCEPTANC~ MATRIX

0~82S040-05-0.,~3030D-05-0~22992D-05-0086317C-06-00859460-06-00 63644D-06 -0., 332270- 07- 0., 3143 éCl-O 7- Co~9.081 D - 0 7-0., 259030 -07- 0 0 235991)-07-0., 221 C) 4D-0 7 -'L_ ~ ~Jl[,'J O~ 0.,(:3(759 G~; 0_6&324C O~} O.Jf)&<~-2[) OS O,,5-':+.{~9D O~ -0023030D-05 Oo82784D-05-UnB6217D-06-0023021D-05-00636390-06-0~861390-06 ···-0~38194D-07-0.,36724n-07-0032R13D-07-ü030277D-07-0o26575D-07-00254R7D-07 -0.,R4356D-OA-0 0 873990-08-0 o 72879D-OP-Oo79207C-Oe-0.,650420-08-0~6e324D-08 -Oo2299~D-05-0.,86217D-06 Oo82B58D-05-0?63653C-C6-0~22950D-05-0.,A58040-06 -00359140-07-00 32984D-07-00 321070-07-0026830D-07-0 0 258450-07-0.,23476D-07 1------f-ho,-(-J.f-fT~.y--f-l-e-e-07_2-6-2-BB-{~-H-B-0-&-&:s-&SiJ4i-r::'r-O &-{h .. 5-5-2-7 -9-D-{}6-(h-J.)'6&-2-2{"-D 8-­'. -Ooa6317D-06-0023021n-05-00636~3n-06 OoB2S20t-05-0~B589~D-06-002297cD-05 .... ····-0~46688D-07-004436QD-07-Go402070-07-0~3617ÇD-07-0032246D-07-00305040-07 -0094860D-08-0097302D-Oe-OnC214eD-06-00a7399D-Oe-0~726200-0e-0 0 7541?D-OO -0085946D-06-0063630D-06-0022S50D-05-0 0 A5896C-06 008?~8eD-05-0022~460-05 -00 4 43 88D- 07- 0040 if 2 311-07- Co 3 0,660D-O 7- O? 32 54 7D -07-0" 316Li 70-0 7-00 2A/~6 3D -0 7 --'-----jGo--&-1.:.a-7-6-f)~.,+'~-e-D C-f:-B"':?-1·}4-~:fl--{}&--Oo-7-2-8-7..cr{,!-{j·8-G-?-6-? .. -1·1-70-08-GQ·6-34·7-5D-D8--0063644D-06-0~B6139D-06-00R5804D-06-0.,229790-05-002294fO-05 0.,A?36SD-05 ". -Oo50239D-07-00465558-07-Co~~063D-07-0~375~3D-07-0035095D-Q7-0~3Z2150-07 -Oo93B65D-08-00Ç4060U-0~-Oo81376D~08-0d84356C-Oe-Oo71t43D-OR-Oo7~120D-OS -Oo33227D-07-0038194D-07-0.,35°14D-07-0 0 4668RD-07-0 o 4438GD-07~01)502390-07 Oo82902D-05-0022938Q-05-0022S07D-C5-0 o 856UCD-06-0o B5227D-06-C061045D-06 0 0 3 22 1 5 D c-q-e t) J G"51Yrf.\-{:)4--B-c Z -B.{~-f.;--3-8-{)1-B-.,n./·-e-=7-D-{H-O,)~-3/-r"7-61)-{)-7-û-~-2-2-1-';4 G-fJ-7-­-Oo31436D-07-0?36724D-07-0.,~2qe4D-07-0~44369G-07-00404~30-07-0046555~-07 -0022938D-05 Oo82868D-05-0~r5401D-06-0a?29~~U-ü5~Oo63036D-06-008560OD-O~ -00375430-07-0036~79D-07-0o32547D-07-0.,302770-07-0026e30U-07-0025003~-07 ''-Oo2Ç081~-07-0032813D-07-0n32107D-07-0~40207G-07-0o39660D-07-0.,440~39-07 -00229070-05-0085431D-06 C~B2931D-05-0of3C36D-06-0022d930-05-0oH~287D~06 o ~ 3 50 950 Of 0., 32-2-/rf:-f) 0 7 ~~~-f?-{H--G-:,-2-{~-5+;-O-7~-fl.45B-O·7-{)-j)-2-?rJ:-.-99r;-o:r-­-0025903D-07-0030277D-07-0026~30D-07-0~36179C-07-0032547D-07-0.,37543D-07 --0.., S 5 6000 -06- 0 0 2295 6D-() 5- Co 6303 (:(j-06 00 ~2 e 6 8.0-0';)-0,) 8!j4311)-06-{)o 22 Q3 G()-J J) • -Oo46555ü-C7-00443AcD-07-0~40423D-07-0036724D-07-0.,32S84D-07-0 0 314368-07 -Oo2~599D-07-00265750-07-0o25f~5D-07-0o32246c~07-0o31647D-07-0 0 350950-07 -0085287D-06-Co63036D-06-C.,2~a?3D-ù5-0.,85431D-06 Oo82931D-05-0.,?2Ç07D-05 ----0-;s-4-'.-O !;)I) 07 Oo-'r0-2-D 78 07-003?', 6 GH~·:r-O-1'>-+?-&l-3-H-{H-O-~·2-!:O-7·1:)-0-7-{)··,-?-S0-:H-n-07----00221q4D-07-0025487D-07-0o23~76()-07-0c30504C-07-0o28~63D-07-0 0 32215D-07 -0063045D-06-0085600D-06-0ub~2~7~-O~-0~22936U-05-00229070-05 Oof290zn-05 -Oo50239D-07-00466G8D-07-0.,~438HD-07~0.,3PIÇ4D-07-üo359140-07-0.,33227D-07 .. '-Oo73120D-08-0 o R4356G-Ce-0"71A 4 3G-OH-O o Q4A60C-06-0o81~7QD-08-0o~3~65D-oa -00 322150-07- 0 0 37 543D -0 7- C" 3:; 0 C') 50-0 7- 0,,46555 [) -07~Oo 1+1+063 1)-07 -0 c 5 Û 2? C; [J-O 7 ---~Ofh-.,· H-2-ô-6-5-3-t)5~2.z-97-9.!7-cr.-r-00-2-.2-4-1·~B-{)-5- (}-,-8-(;-l .. 3-S f}--{) (~{) ,-+15[ .. 0';' 0-0 {,_·O c 1.> 3 (. 4L.'. ~ - 0 G-- ... -Oo7541eD-08-0.,R739çu-aS-0072~20Q-Ur-0.,g730üD-oe-00R21hal)-00-0oS4~60n-08 ~Oo30504D-07-0n3617qD-07-0,3?246D-07-0a4430SD-07-0~4U2C7D-07-0~46&8~D-07 -Oo22S7?D-05 Oo82820D-05-0~85gS6D-O~-O~2~0?lD-05-0,63&5;0-Ob-ODBb317D-06 -O~ 6 34 7 50- C8-0" 7 2 2 7f.? (;-02-() 1) 6271 70-0 Ü-O ~ .'~ ?14G 0-0 8-0071·:1410-02 -0 0 f) ~l7f n-(In -00 ?c346 30-07- 0.,3254 71)-()7- 0-:> 3 l 64 rD - U 7- 0" [,0'12 3D-0 7-0" 3q~t.· OIJ-O 7-0'34 /13,0 8f) -l) 7 ----j(};;-2-2-G4-6f:)-D5-D~.;.;..t.;-.y-)--'J-f:,-{lT'l-~2--3~ t< D-O:'·-{}-;::--6·3{;-?'· Si}-O{,-{):>-Z;: 11 ~vD-O 5-0 ~ -! Cj C;:4{·[,-{)(' - _.-- 0 0 6 r 12 4 [1- 08 - 0 ., 7 c:: 2 07 (J - iJ ~~ - 0,., f. ~) 0 4 ;: D - ù .v - 0 u <.; 7 3'~ <7 l; - 0 2 - 0" 7 2 :J 79 D - Ù 0 - ()" t' 4 3 5 t.~ D - 0 F; -00254870-07-0n30277~-07-C.,2~~75D-07-0~36726D-07-0n32B13U-u7-0n~~lG40-07 - 0 ~ 86 13 Q ~ - 06- 0.., 2 30 ;: lu- 0 '5 - 0" t- J ~ ~ C; " - 0 6 0 0 El ~ -f t 4 D - 0 ? - 0 u .;), f., 2 l 7 ~J - 0 0 - 0 ., 2 :: 0 :. 0 l' - 0 :: fit -0., 5 A 1) ? 2 1) - 0 P, - 0 ~ 6 'j 0 4 2 D - 'J i) - () ,., :; l) 2 7'-' D - 0 r: - 0 '.) 7 2 6 2 û C - 0 0 - O·.) (, 2 7 ~ 7 n - 0 s - () ') r l 6 '';1 f) - Ci G :-0 Q 2 3 4 7 1; [1 - 0 7 - Ù., 2,., i3 -:< 0 D - 0 7 - C., 2 ~, 8!+ 5 U - 0 7 - 0 ~ :3? C) r;. 1. f.') - 0 7 - 0., j; l 0 ï r) - 0 7 - Ct., -~ ~ ') u. 1 ) -:) 7 ----{);~-'?5th}Lt-0-0f,-{).,+~f- ~-3H-0":;"-O .,-?-2~·r)O{)-05-U ,,-~1{~2 J-7 D-Gl) - O.,:~? ~~) :;i)-O:;- O.~ J? Cl -:: ~!'-G ')-- 0 0 5 c 1 69 D - 0 :1 - 0 ., f_, '3 :3 2 l.. ~) - 1) .r' - 0 ., :- {e h 2 7 D - 0 1:. - 0 0 7 'j 'i J f [; - J .~ - 0 " ~ 3ft 7 ~: Ù - J e - (] n 7 ~ l ? 0: l - ~"\ 1: - 0 0 ? n C) li [) - 0 7 - I)? 2~) <) u ~ 1 \ - 0 7 - 0" ? 3 ? -:1 CI!) - u 7 - D? 3 li· "1(: D - 0 7 - 0 .) ~ q ()fI l ~1- (1 l - () ~ :- ;, 2 2' 7 L' - û 7 -0., ()36'i4iJ-Oh-O., -::r.:,3 J. 7f)-Of--O.., ~:-;(Î/+('Llï():'-C" 2::"0::8:]-0')-0·, _~2\(j 2~)-()r:; 0., ;,;'f·;J/.n_ ü ,:>

ce

-e

APPENDIX E

FAULT CALCULATIONS

(a) Delta Connected Resistive Load with Single-Phase to Ground Fault

(Figure E - 1 (a»

The chain matrix of the line '

where [Ill 1 ] is the chain matrix from the sending

end to the faultpoint of length Il

'

[Ilf

] is the fouit point chain matrix,

.. -'0,

[1l12J is the chain matrix from the fault point

to the rece,iving end of length 12

•

174

(E-l)

80th [1l11] and [Ill 2 ] can be determined by introducing the respective

lengths into Equation (A-6) of Appendix A •

U 1

0

- - - - -1

0 0

Aiso [Ilf

] = zf 1 (E-2)

0 0 0 U

0 0 0

A 1

B z,

C

Il '2 S R

(a) _.

q A

p

B

~'~-~--------~------'2------C--1 S

(b) R

FIGURE E-1. FAULT CONDITIONS

(a) DELTA CONNECTED RESISTIVE LOAD WITH SINGLE PHASE FAULT.

(b) TWO SIMUL TANEOUS FAULTS. (PHASE TO PHASE AND PHASE TO GROUND)

175

177

The sending end current matrix can be obtained from. Equation (A-7)

of Appendix A.

A numerical example is calculated bere for the three-phase transmission

line specified on page 92 by considering this time a delta connected resistive load

(Z, = 1,000 ohms). The Phase A to ground fault (zf = 0.5 ohm) is assumed to be

located at the middle of the line.

The values of the receiving end fault current and the current in the

same Phase A at normal condition together with the receiving end voltages are

1 isted as follows :

PHASE A VOLTAGE AND CURRENT AT RECEIVING END, (IN KILOVOLTS, AMPERES)

Real Part / Imaginary Part.

Normal Condition Fault ·Condition

VOLTAGE 400. / -142._ - 1 3 . s~ / - 30.3

CURRENT 1209. / -439. 505. / - 130.

From the phasor values of voltage and current, during the fault, it is

seen that the power meosured at the receiving end in Phase A is negative, i.e.

power flows from the other two phases, through the load, to the fault point.

178

The fault current in Phase A at various locations are shown in the

following diagram.

86.3/-647l. 112.5/-6659. 493./-92.5 505./-130. S :> ;. R

.. --,',

(b) Two Faults (Figure (E-1 (b))

There are numerous possibilities of simultaneous faults in a system. For

the purpose of illustrating the application of the Multiconductor Method, a two fault

case with short circuit between Phase A and Phase B at point "q" and Phase A

to ground fault at point "p" is considered. (see Figure E-l (b)). It is intended to

- -calculate the line currents at the fault points. Generally the analysis of this case is

quite similar to that shown on page 106 of Chapter VIII.

Consideringthe fault point "pli first, then Equations (7-2) to (7-7)

are applicable.

As shown in Equation (7-7), the final expression is

t

~

179

= l 9 (E-7)

-1 U + P2 Pl Zf

which includes the chain matrix of the section of the 1 ine with length 12

, [fJ12 J ,

the faulted impedance to ground matrix, Zf' and the load matrix, ZL •

where [fJ11

J is the ehain matrix of the section of the 1 ine'

from the sending end to the fault point IIp Il.

[ fJ ll a J and [fJll b J are the chain matri ces of the specifie

section of the line and can be determined by substi-

tuting their respective lengths into Equation (A-6)

of Appendix A .

[fJfq

J is shown in Equation (6-4) and

U o

[fJ J= 0 Zfq Zfq fq

1 0 U

Zfq Zfq

0 0 0

(E-8)

(E-9)

(e Then E

s

1 s

= [!J11 ]

= e 1

180

Zf

1 9

U + P p-1 2 1 Zf

1 9 (E-10)

-' . For a given E ,the 1 can be determined, which will permit the

s s

determination of voltage and current at any point k along the transmission 1 ine.

A numerical example is worked out by the same transmission line in-

dicated on page 92 . The location of the fault between Phase A and Phase 6

is assumed to be at 20!1-8 miles from the sending end and the location of the Phase

A to ground is assumed to be at the middle of the line. The values of the phase

currents measured at the sending end and at the fault point to ground are listed as

follows :

FAULT CURRENTS (IN KILO-AMPERES)

At Fault Point P Sending End (Phase A to Ground)

Phase A 33.9 3.86

Phase B 33.6 0.39

Phase C 0.9 1.06

"

1 /

SUBROUTINE CINVRT (A,IK,IJ,N' IMPLICIT REAL*8CA-H,O-l' COtlPLEX*16 13(25) ,.C(2S) ,PIVOT,l,A 01 ME N S la t\ 1 P ( 2.5) ,1 Q ( 2 5 ) , A ( 1 K , . 1 J ) EPS=loC~12

1 00 221 K= l, N PIVoT=COGoO,OoOO) 00 61 I=K,N 00.61 J= K,N Cl =C 0 A es C A ( l , J ) ) C2=COA es {P 1 VOT) IFCÇl-Ç2)61,6,é

6 PIVOT= ACI,J) IP{K)= 1

10(K)=J 61 CONTlt\UE

7 C1=CDABS(PIVOT) IF(C1-EPS)39,8,8

8 IF (IPCK )oEQoK) GO TO 11 9'IPK= IP(K)

DO 10 J=l,N l= A(IPK,J) ACIPK, J) = AC K, J)

10 A( K,J)= Z Il IF ·(IQ(K)oEQoK) GO TG 14 12 lQK= lQ(K'

DO 13 1 = 1, N . Z= A ( l ,1 QK )

A(I,IQIO= A(I,K) 13 A(I,K'= Z 14 DO 191 J= 1,N

IF (Jot\E.K) GO TG 18 17 B(J'=(olD01~OoDO'/PIVOT

C(J)=(olDOl,OoDO) GO TO 19

lB BCJ)= -A(K,J)/ PIVOT C(J)= A(J,K)

19 A(J,K'=(OoDO,O.DO' 191 ACK,J'={OoDO,OoDO'·

DO 221 J=l,N 00 22 1= 1,N

22 A{I,J'= IdI,J) +C(I)*B(J) 221 CONTINUE

00 30 K= 1,N KK= N- 1<+ 1 IF CIP(KK)oEQoKK) GO TO 27

25 IPK= IP{KK) DO 26 1= 1, N z= A( l ,IPK) A( I,IPI<I= A(!,KK)

26 AC I,KK)= Z 27 IF (IOCKK)oEQ.KK) GO TC 30 2A 10K= IC(KKI

DO 29 J= 1,N

162

./

l= ACIQK,J' ACIQK,J)= A(KK,J)

29 A ( KK" J ) = Z 30 CONTINUE

RE rJJRN 39 WRITECé,40)Cl 40 FORMAT(lHO,15HPIVOT lESS THAN,DIOo2)

CAll EXIT END

TOTAL MEMORY REQUIREMENTS OOOD5A BYTES

163

• SUBROUTINE GMPRD(A,B,C,N,M,K) 00 UB LEP FE C 1 S ION A ( N ,M ) ,8 ( fi, , K) , C ( N , K ) DOUBLE PRECISION X,XP,XM

'DO 1 J=l,K DO 1 1 = 1, N

1, C ( l, J ) = 0 0 00 00 2 1 =1, N DO 2 J=l,K ' XP=Oo DO XM=OoOO

8 DO 3 J J=l ,M, X=A( I,JJ)~cB(JJ,J) IF(X.GT.OoDO)XP=XP+X IF(XoLToOoDO)XM=XM+X

3 CONTINUE 2 C(I,JI=XF+XM

RETURN END

TOTAL MEMORY REQUIREMENTS 0003A4 BYTES

164

DOUBLE PRECISION FUNCTION DRL(AX~) REAL*8 AXY(l) DkL=AXY(11 RETURN END

TOTAL MEMORY REQUTREMENTS 000122 BYTES

DOUBLE PRECISION REAL * 8 A X Y ( 1 ) DIM=AXY(2) RETURN

FUNCTION DIM(AXY) ", "

END

TOTAL MEMORY REQUIREMENTS 000122 BYTES

165

" '

166 SU8ROUTINE CORtP2,Q2,R,S,N,KK) DOUBLE PRECISION P2(N,N),02(N,N),RCN,N),SCN,N),RRC24,241, 2 RI(24,24), Tlt24,24"T2C2 4 ,24"T3(24,241,T4{24,24) DOUBLE PRECISION C(24,241,CC24,24) 00 2 I-=l,N DO 2 J=l,N T1fI,J)=OeDO T2CI,J)=00DO T3(1,J)=00DO T4(I,J)=OoDO, oc l, J )=O~OO 'c ( l, J) =00 DO

2 CONTINU: DO 3 1 =1, N

3 0(1,1)=01001 DO 77 KLH=l,KK CALL G~PRO{P2,R,Tl,N,N,N) CAll G~PRO(Q2,S,T2,N,N,N) CALL ,G~PRO(P2,S,T3,N,N,N) CALL G~PRD(Q2,R,T4,N,N,N} DO 4 I=l,N ' DO 4 J:l~N

, RR ( l , J 1 = 0 ( l , J ) - T 1 ( l , J ) + T 2 CI, J ) 4 'R 1 (I , J ) = C ( l , J ) - T 3 ( l , J ) - T 4 ( l , J )

CALL G~PRD(R,RR,Tl,N,N,N) CALL G~PROCS,Rr,T2,N,N,N) CALL G~PRO(R,RI,T3,N,N,N) CALL G~PRD(S,RR,T4,N,N,N) DO 11 1=1, N DO' Il J=1, N RCI,J)=R(I,J)+TICI,J)-T2(I,J)

11 SCI,J}=SCI ,J)+T3<I,J)+T4<I,J) 77 CONTI NLE

RETURN END

TOTAL MEMORY REQUIREMENTS 0096F6 BYTES

e. 167

. . SUBROUTINE CRRCP2,Q2,R,$,N,KK) DOUBLE PRECISION P2(N,N),Q2(N,N),~(N,N),S.(N,N),RR(12,12),

2 RI C 12,12) , Tl( 12,12) , T 2 ( 12 ,12) , T 3 ( 12,12 ) , T 4 ( 12,12) ,C ( 12 , 12) , 3 0(12,12) .

DO 2 1=1 ,N DO 2 J=I,N C(I,J)=OoDO o{I,J)=OooO

2 CONTINLE DO 3 l=l,N

3 OC f, 1 }=o 1001 DO 77 KLH=1,KK CALL GMPROCP2,R,T1,N,N,N) CALL GPPRDCQ2,S,T2,N,N,N) CALL G~PRD(P2,S,T3,N,N,N)

CALl GPPRO(Q2,R,T4,N,N,N) DO 4 I=l,N DO 4 J=l,N RR(I,J)=0(I,J)-TICI,J)+T2(I,J)

4 RI (I ,J ) =C CI, J ) - T 3 ( l , J ) - T 4 Cl, J ) CALL G~PRO(R,RR,T1,N,N,N)

CAlL G~PRO(S,RI,T2,N,N,N)

CALL G~PRO(R,RI,T3,N,N,N)

CAlL G~PRO(S,R~,T4,N,N,N) DO 11 I=l,N DO Il J=l,N R(I,J)=RCI,J)+Tl(I,J'-T2(I,J)

Il S(I,J'=S(I,J'+T3<I,J)+T4(I,J) 77 CONTINUE

RETURN END

TOT~L MEMORY REQUIREMENTS 002A5A BYTES

168

APPENDIX D

UNE CONSTANTS OF THREE PHASE BUNDLED CONDUCTOR

TRANSMISSION UNES

The computed inductive reactance matrix, X, and capacitive susceptance

matrix, B , for confi gurations having different numbers of subconductors per phase as c

described in Chapter Il, are /isted in this Appendix.

follows :

XI =

where

The elements of matrix X, related to the subconductors of the line are as

, , , X -r1

1

Xs1 1

- - - - - -

X __ __ X rr rs , , , , , , , , ,

- - - -,

-',

"". , - -X

X is the self-inductive: reactance of subconductor r, rr

nn

ohms J. 01 {ml e

X is the mutual - inductive reactance between subconductors rand s 0 rs

Similar/y, for B , in mhos/mileo c

The numbering of the subconductors of the line is from Phase A to

Phase C where the numbering sequence for subconductors, in a given phase for the

various configurations used, is shown in Figure D - 1 •

1 2 0 0

'2 Subcoriductor

Buridle

1 3

0 0

,0 0 2 4

A. Subcondu'ctor

Bundle

2 4

0 0 1 6 0 0

0 0 3 5

0';" ,

'6 Subcondùctor

Bundle

FIGURE D' -' 1: NUMBERING SEQUENCE OF SUBCONDUCTORS PER 'PHASE.

169

170 TWr) SURCnN!)I,ICTnp,5 r>~R 111~"S F cn'.:F IGURlI TI 1Jt\,)

• VALUE OF INDUCTIVE REAÇ.Tl\NCE MATRIX

1·

0.92521D 00 0.499010 00 O.gC:HHnD-Ol 0.883100-01 O.I·IRl'tD-()l 0.40982D-Ol -' ., . . "

0.90SP30-01 0.93786:1-(\1,0 .C)2)J790 no 1).502740 OC o. 91092D-Ol 'O.Qf3310D-Ol '''''O.88310D-Ol' 0.91092D-01"·0.:;0274;)' 00,'0. 92S79fi 06 0.937860-01 0.90883i)-01 0.41R140-01 0.42672~-01 O.C)10S2D-Ol O~C)~7g~D~01 0.Q24930' 00 0.499010 00 -'··0.~0c)R2!)-Ol·0.41814J-O'··O.8:)310D-Ol·0.90R83/)-Ol· 0.49901D 00 0.92521D 00

- 0.6~5770-05-0.366R~D-05-0.1A4~3G-06-0.16541D-06-0.666160-07-0.630560-07 -O. 366R 61)- 0 5 0.666'3 71)-OS-0. 2069 5D- 0 6- O. ,l.R 4 750-06-0. 70~ 9 / ... 0-1)7-0.666 f 60-07 -0.lB4~3D-06-0.206950~0~ Oo6~727D-05-0.3~579D-05-0.134750-06-0.16541ry-06 -0.16541J-()6-0.184750-06-0.165790~05 O.66727D-05-0.20695D-06-0.1~433D-06. I------'H-~ r.;....o.{4J:,;) 0 7 (1. 7 0:3 ~ " ') () 7 0 al \~ " 7 5 '! - (' 6 - g. ? 0 :3 9 5 0 q S (h-66 é 5 70 -.g.ç>--:- 0 • 3 (~J--..G-.5-­-0.630560-07-1). 66616j)-07-0 .16541D-06-0. U343;3D-06-0. 366R6D-05 0.665 77D-05 ... ___ • ___ ~ •. _ .•..•. , ~"''' __ ~_''''''~'_'_ .......... _ ........... ~_ .... _.'~.~ ........ ~ .•• ,._# .• ,.~~ •• :_-.- .• __ ••• _ ........ _ •. _ ...... ~_ ••• .,. •..•••• __ ........... _._ •••.•• __ ................ _ .••• _._ .••.•••..••• __ " _ •.••. !' ...... _-. ....... _.h ......... _ ..... .

• • ":to" , .... _._ . .,. ._. __ .... , "._ ..... ...... ___ . ..l._ ... __ .......... __ ... _ .... _. __ .. _ .. "._ ... _ .. _ ......... _ .... _ .............. _____ ..... ___ ._ ....... _ ..... ___ .. _ .. ,_._._ .. _ .. _ .... __ .. _ ............ , ... ~ .. : ........... _ ..... ~ ....... : .. __ ... _ ........ ' ~._ ... _,

" ....... _._ ....... -......... -._. __ ._---,._ ... -_ ........... _-_............. . .... __ .... -...... _.--:'._ ... -: ..... __ ....... __ ._- ............... __ ...... _. __ .. _; .......... . '~' .. _.--._ .. ~ .- .:. ........ ~ ........ _.~_ .. ~ ..... " .. ".

••• _ ••• _ ..... ~ ... - •• _. ~ •• - ._ .... _____ ........... _._ .......... _._ •• _ •• _. ____ ._ ... ....: __ • ____ ........ _. __ ._ .... _______ .. _ ... __ ••• '_ ........ u •• _._ ...... _ ••• , •• _._ ...... --.._. '.0 ...................... _ ..... _ .•• _ ............. .

. •• _.- • ....,.:.. .. •..• .'_.-- .• _. __ .• _ .... _ ... _ ..... _ .. _ ...... _~ .•.• ' ..••••.•• _ ..... ~ •. , ............ ~ .. _ .. __ .r'_" __ ..................... ..--........ • ... _

,/

0.92464D 00 O.49759n 00 O.498.36D 00 0.455360 00 0.~4605D-Ol 0.838100-01 '-"- ·····0.·81 79'+!)-01· 0.810100- 01 0.31284[1-010.3083 3D-0 1 0.30.5170-01 0.300710-01.

0.497590 00 0.922650 00 0.4553'5D 00 0.496390 00 O. 83133~D-01 O.P'3120i)-..Ol -------0.81 039D-O 1··0. ~~032 60- 0 1 ·0. :W83W-O 1· O." 03 9'1 1)-0 1 .0.300 71 D~O 1 0.29637 D-a 1

0.498360 00 0.45535D 00 0.924200 00 O.4971RD OO.0.H7435U-Ol 0.~6h290-01 . . . 0.84503,8 91 O.~~3?C7') )1 O.'">r2077D--G-l O.:nJ)lDO 01-fJ.....3..l~:..;.L4Q 01 !). ~Q...B3-l,.-,=>.=-I),,j..lL--_

0.455360 00 0.496390 00 0.49718D 00 0.92?25D 0:) 0.8666?D-Ol 0.859391)-01 --"':---0.837410-01-·0·.830230-01···0 .. 316190-01 0 .. 311 760-0 10. 3~) H3 30-01 0.303 94D-Ol

O. 8 4605 D - 0 1 O. 83 8 3 R !) - 0 1 0 '. n 74 3 5 D - 0 1 0 • .3 6 6 6 ? D - 0 1 0 • 92 1 60 r; 00 O. 4 <;1 115 70 00 - ~ ... ---_ .. O. 4:9553 D 00-0.45255 iJ 00 -·0. i3 4:lb 3D - t) 10 • 83741 1)- 0 1 0.81 7 9 /t D -0 1 .. · (). 3103 ') ~),... 01· ..

C.83810D-Ol 0.831200-01 0.8~6290-01 0.R5939D-Ol 0.49457D 00 0.919~~D 00 0.452558 06 O.',':'~~'?I) ')0 9.81707'')_f)l O.83().23D (1] O.8'OlCH'-n] O.3032b·) 0]

0.817940-01 0.8103QO-Ol 0.84S03~-OI 0.837410-01 0049553D 00 0.452~5D 00 -_··_···0. ·9216QD 00· 0 ./~945 7D 00 . 0.;-37 /.351)-01 o. 8!.>662 D~O 1 . 0.8460 5D-0 10. d383 g D-O 1

0.BI0100-01 0.803260-01 0./B707D-01 0.83023D-01 0.45255D 00 ').1+=)3(,01) 00 ._,._- .. {). 494~ 70'.000.9196(1) ·00 ··0. ?66 291)..,.0 1.·.0 .. as 93<:'D-0 10.83 B 1 00,::,01. 0.831200-0 L .

. , 0.312840-01 0.308311)-01 0.3207,7D-01 0.31619D-01 0 .. 845030-01 O.WHOTh01· 0.i37',;'S9 al 0".8'.".29') ,>'1 ').'P";:'Y.' 00'°.49718 0 ''') 0.,*Oq3l,C 00 f'l.'t?"ï"rj) 00

o • 3 0 S 3 3 D - 0 1 O. 30 "394 Q - 0 l O. 3 16 l '::)!) - 0 1 o. 3 11 76 D - 0 l O. 83 7 'd f.) -:)1 o. FU 02 V) - 0 1 , 0.866620-01 0.8593(1)-01 0.49718D 00 0.11222?!J 00 0.4553N)· 00 0.4963'·)i) 0') 0~305170-01 0.30071D-Ol 0.312840-01 0.30833D-Ol 0.817S~O-01 0.31010D-01 0.84605D-Ol 0.83810U-Ol 0.49036D 00 0.4553~,) 00 O.92464b 00 O.~975YU 00

0.30071D-01 O.29~37i)-Ol O.308310~01 0.30394C-Ol 0.~1039D-Ol O.B0326U-Ol O.?3S3fH)-Ol O.:P1('Q .... fH),4.55"lSO DO O,!"Q6~qr) QO O.l.:07"r.;n np Q.922(~5·'1 00

VALUE OF CAPACITIVE SUSCEPTANCE M~TRIX

. . 0.79987D-05-0.2~459D-05-0.~44670-05-0.1331gD-0~-0.~41R4D-07-0,610180-07 -0 .. 533a6D-07-0.~0334J-07-0.156R20-07-0.14333D-07-0.1~177D-07-0.12R530-07 0.24'.593 SJ O.i~G9~'?·1 ~:;').~ n:'~'~ ... !)5 ').t."d9SCi a') Q.1)1~.9CO-07-0.S''?~c.:'~,) 0 7

-o. 505720-07-0 .'t:39590-07-0.14302~-O (-o. D 19?1j-07~O.12?53!)-07~a.l1763J-07 _·-'··-0.2'+46 7!)-05-0. 13'3391)-0'5 ,0.800411)-05-0. 2440FlD-05-0. 76P, OQf)-O 7-0.73 4H ·;0-07

-0.632990- 07- O. 601 (.1 2,)-1) 7- û. 17 /1"f} 2f):-Ç 7~C. 16074:)-07-0.1') ô9? [}-07-0. 1/+ ,ù2;)- 07 --'-0.t33380-05-0.2439RD-05-0.24408D-05 O.~OlObD-05-0.736R5D-07-0.72116J-07

- O. 60 390 [) - 0 7 - Q • 5 !37 77 0 - 0 7 - o. 161} 7 1+ !) - 0 '( - 0:. 149 20 0 -0 7 - 0 • 1 4 ':)3 3 n - 0 7 - Cl • 1 31 9', ') - 0 7 O. (.4 l 84:) 07 O. 611 (, ~ P - 0 7- 0 • 7 '') .) ') Q rd 0 7 :). 73 ha s n .. ,. .. O.L_1l-8.()..1-2-;l . ..l~)5-=.!l. ... ~-~WI)~-Dj~5"---­-o. 24 3620-05-0. 1·~737D-05-Ü. 61299i)-f)7-0. 60390D-07-0 .• 5 33',-It;D-!)7-J. 50'5 72!J-r)7

.- '·-0.010 l '3 0-0 7-0.5 95H G u-0 7-0.7 34H 3D-O 7-0. 721 16D-0 7-0.243 30fJ-ù 5 C. 8.11 R1ti)-O 5 -o. 132370-05-0.24 300Q- 05- o. (,01 0 2~- 0 7- 0.53777 D-O 7-0. 5~ 33 !tf)-O 7-'). ! .. ':'9 5q:)- 07

. -0.5318 6D-07-0. 505720-1)7-0.6"3 ~')9()- 07-0. f)(B <10D-()7 -O. 2'i·~ 62 D;"'05-0. 1 3 ?17:) -05 . 0.·80 1 21 0 - 0:; - 0 • 21+ 33 () D - 0 :; - 0 0 76 g Ü 0 f) - 0 7 - 0 • 736 a 5 D - 07 - 0 • fA 1 f-l/. 1) - f) 7 .:.. 0 Il (J lU,? i ) - ~)7

O.5033 /d) 9 7-4,,-4.P~').5 9 J 07- ') .r.>i.~ 1)- Cl 7-;). ~ s:z.:J.J~ -o. ] 12 ~ 71).:::.0 ';- O.....2:J..3Dill.=:ll.5, __ _ - O. 2 lt 3 30 D - 0 5 O. g 0 l B 1 .. ') - 0 5 - O. 7 1 t.. P. 3 [) - 0 7 - 0 • 7 2 Il (, 0 - ,) 7 - 0 • (, 1 0 l ,0. J - (1 7 - :.1. 5 U .') '-! '.) , ) - 0 7

- o. 1 56 H 2 0- 0 7 - o. 1'+ 3 :) 2 i) - 0 7 - 0 • l 7 4 ;:~ 2 j) - 0 7 - G. 1 6f)7 l, 1) - 0 7 - 0 • 63 ? 9 c:) D - J 7 - 0 • ~) ~H 02 C) - 0 7 -O. 76 Po 0 0 i)- 07 - 0 • 7 34 B 3,) - U 7 0 .~ 0 () 4 1 :1- i) 5 - 0 • 244 rH: !) -1) :, - t' • 2!t 1. Ô 7 f)- 05 - 'J • l 33::' =:> .) ... 0 :3

- O. l 4 3 3 3 1) - 07- o. 1 3 1 9 S f) - 0 7 - () • 1 t, 0 7 I+!) - 0 7 - O. 1 It 9 2 0 Ci - 0 7 - C • 1"-, 0 ~ '; :) {! -'J 7 - () • 5 (n 7 7 . ) - 'J 7 _ -ù. 736858- 07 - o. 721 ] ',:)- Q 7 - 0 0 ;U~ 1 .. 0 '.) rJ-l); (). }10 1 0:-' ~)-O 5-1) 0 1 :',3 ~ p, I)-(r'> -!) • ~ l, :3 q~ ,)- [)';

:---{h ... 14H-=7~')-{).1-9....-l-2-=-1-~~.!+-.1'-7 ... -~..s.."-'~-?-J.kllL..,..0-.1-' •. 3...":) 3.[L-_ÙL~ .... 533_d {; L:..-~ I~_~ .. 5 Ù 33.'t .J.7:.ü .7_ - --- ... -o. 6'"' 1 iJ 't Cl - () 7 - [' • (. l 0 l F, ~) - 0 7 - 0 • <.~ 4 () 7 ') - ') " - 0 • l .~ 3 ~~ ~5 ~) ... t.15 o. 7 '"F' ;J 7 IJ - 0 S - IJ. -' I+!+ :;.~ ) - ~; s

- 0 .'1 2 (~ 5~ [) - 0 7 - o. l 1 7 6 3;) - Ll 7 - n • l', 3 Il ? r 1- 0 7 - ;). L ~ l .} li U - 07 - ') • ') Û 'J 7 2 iJ - J 7 - \) • f, 3:1:; 9 ) - 0 7 -0.6116 ql)- 07 - (; • 5 '''; S? 9J~ l)7 -:) • 1 ·n 3 ')1)- u:; - c • 24 ? 9 S Cl -(1 S -0 .2 /.4) <:)I)-{)'; ;).:1 'lO(· n - ()')

172 SIX SUBCONOUCTOR$ PFP PHASF CONFIGURATION

VALUE OF INDUCTIVE REACT~NCE MATRIX

0"B39280-01 0,,831190-01 OQAIBI4Q-Ol 0"d036SD-01 0079079U-01 00784370-01 0,,308460-01 0.,30A31n-Ol 0030067D-01 0,,30079D-01 ü~29322D-01 0.,2934S~-OJ 00498550 CO Oo92532D 00 0043094D OÔ O . .,{,S·904D 00 0t)L~1330D 00 '00431 /t90 00' OoaS9070-01 00851250-01 0,,836~SO-01 O"eZ30SD-01 008U9040-01 O.,B02eQO-01 00316131)-01 0~31601D-OJ. Co~Og12D-Ol 00308310-01 Oo3005t)t)-O~. 0,,300790-01 1----f:-"",,-4~-::3-2-e-e~-B4I~_2+A_&B 0 0 lh~2-'TB-o-B-û-~/~5"r3 ;r-oO--{}::,-4-2-9-7-9l)-{)O'--0.,e4638D-Ol 008374êD-tH 0~8?552D-0J. Oo80S55D-Ol.0079765D-Ol 0~7c)0510-01 . 00308351)-01 0~30809D-Ol o~30D60D-Ol 0,,300500-01 OoZQ30BO-OI O~2932~D-01 0-:>431700 CO 0.,499040 00 0-:>41329D 00 0,,9248BO 00 00-43053D 00 Oo4S'790 rJ 00 0~88795D-Ol 0087958U-01 D)B65090-01 0.,850200-01 0~835940-01 0~8?93~D-Ol 0,,324170-01 0,,323980-01 00315970-01 0~31601D-01 OQ30r090-01 0~30A310-01 !------90.,..4-3-9-0{}l) 00 O.) 4 ::. 3 3 o-P--G~~~-~-8-{3e--o-o--S-2-1--50f)-OO-Db-4-':;'6-.~2-D-0v-----. 00875210-01 0,,865660-01 .0085366D-01 0023653Q-01 00824570-01 00816900-01 0 0 316310-01 0~315970-01 O~3083AO-Ol Oa30812U-01 0030060D-01 0~300b7D-Ol· 00413330 00 0043149D 00 0042979D 00 Oo497SQO DO 004962Zn 00 00'92305D 00 00897070-01 OQeS77BO-01 0~67447D-Ol 00657860-01 0.,84462D-Ol 00e3728D-01 0032449D-01 0032417D-01 0,,31631u~01 00316130-01 Oo3083~O-01 00308460-01 ----flOl7;o-HO~ 0 0 l Os S 5 S 0 7 D-{H--IO-oB-'".., (. 3 3 0 o--l-e~-~--0~-7-5-?-1-8-~:---01J8-S-7-{)-7H-o-I---. 00920640 00 004954CO 00 C0493720 00 00~28780 00 0~42710D 00 Oo4105?D 00 00637280-01 Oa829~40-01 6~81590D-oi .0?S028~D-Ol 0,,790519-01 00784370-01 00831190-01 00851250-01 0083743]-01 Oo87S58D-01 00265660-01 0?88778~-01 '00~9540D 00 O~92227D 00 C~42793~ 00 0049621D oO.Oo4i049D 00 00428780 00 0.,85786J-Ol 0~B5020D-Ol 0?836530-01 009230SD-01 00809550-01 00~03690-01

0049372D 00 00427930 00 CoS13~20 00 0041049D CO Oo49?860 00 Oo~2710D 00 ... 0034468D-01 0)835940-01 0~r2457D-01 0~80S04D-01 Oa797650-01 00790790-01 0080369D-01 0?323090-01 0080S55D-Ol 0?E5U20r-Ol 00f36S'3D-01 00857860-01 004~B7RO 00 Oo49621~ 00 C.,~1049D 00 00922270 00 0.,42793D 00 0049540D 00 0 0 887780-01 00B7958~-01 0~865660-Ul 0~B5125C-Ol O~8374HD-Ol 0083)19D-OJ 0679079B-€1 01) q-G9~'=H.-e0=7~)-H-9-l ~)" P.35-Ç4f't-0-±--{)1)-&?-4--5-7{}-o-~I)-&4-4"".) 00427100 00 0 0 410490 00 0049286D 00 0~427Ç3D 00 00918928 00 Oo4S372~ 00 0087447f)-01 0~S6509D-0J. OoH531::60-01 00836C;aG-OI 0082S:i2D-(J! O..,ôIF!14D-OJ 0~784370-01 00802890-01 Co7S0510-01 00e29340-61 00P169CD-Ol 00P3728D-01 00410~2D 00 0,,428780 00 (l~42710D 00 00495400 00 0~47372D 00 (JoS2064D 00 . 0 0 897070-01 00~87S:,;)-Ol Oo~~752H'-01 O..,85907Q-Ol O,8463BD-Ol 0,,3392iW-01 ---HO~-&B--O-l 0 " 3 l-frl-3~e-&3-5G-{}-l-o~4-l-7-t-O-l-v0-3-1-{'-3·1-!)--o -J.-{}o-3·?44 S·)-0 008372B~-Ol 00P57860-01 C~8446~D-01 OoBC77fU-01 Oa~74470-01 00E97070-0J. O,,9230S0 00 00497900 00 0~49622D 00 004314Sn 00 'O~429790 00 ü~~13l38 OU ·0,,30331n-Ol 00316010-01 On30809C-01 00323S6C-01 Oo315Ç7Q-Ul 0032417J-Ul Oo82Q340-01 C0 85020D-01 0~~35Q4D-Ol Oo67S~ED-Ol Ooe~5GGD-Ol Ooe8 7 Q5n-01 0?497S0D 00 0092488) 00 0t)430~3D 00 004SS04D 00 Oo41~29D 00 Oo43170n 00 ----,03-3 6%-7D-',:H---G0--3-G-8-l-2-8--G-l:---0--,,-3-{} O"~-H t)-lJ-i-e-o : -l-5-S-1-[-0 l-{h-3 0 +3 f: D-0-1---·0 Cl -3 H.- 3 J :)-() J.- ---0~e16900-01 Oo8365~D-Ol Oc8?4578-01 0.,8656(D-Ol O~fS3060-01 OaC7~2JO-ùl Oo49622D 00 0:l43053~ 00 O,,9?15CJC'l 00 (t~/+1330D 00 O,,40~~:3D 00 O;>'<,f)(){)D 00 0:) 3 0 0 79 'Cl - 0 1 0 0 3 Ü 8 3 1 D - 01 0 0 :3 00 SOU - Ü 3. 0 0 j 1 b Û 1 [: - 0 1 0" 3 0 ·:n 2 D - o!. 0" :.1!) J :~ i) - U_ 0., e. 0 2 i3 <:) D - 0 1 00 B 2 30 <: Ci - 0 1 (l" t~ 0 9 li {. D - ü 1 00 t: ~ l 2 :; [) - 0 l O,~ 8 3 Ô t1 !- f) - Ü IOn G '; r. (l 7 G - U 1 0.,4l14S0 CO Oo4Q1l0/ .. D 00 0041330n 00 0;:,'12,3?rl 00 O~430l;4f) 00 0,,1 .. <:;1'1)':/' 00 O-.,2~3-2-2 n-{}-!:-00--3-0e-5'Jl)--{; î.--{::-.,-;:-0-.30'3[\-01---ih-3-C e OC: p- Û l-{) o·-:?:O,)(~ I)r'l -01· - 0 0 :) Iy:.?~; l:- 0 l . o , 7'.;. 0 5 10 - 0 1 0 ~ PI) c 5 ~ U - 0 1 C., 7 <;7 (J ~;; - 0 1 Ü" (:"3 7 'do: r' - u 1 0., 8? ~ ~ ? [) - û l C.l r- 4 (, ::' .' D - () .'_ O,,42:.j79:) 00 0,,'t132 C Q 00 (j~/tsr)631) (}Ü O.,4?OC'{+D 00 Û,,'~21J;H1 00 C.)/1 C?(,}l?( or 0., ?':l ~4 ~ rl- 0 1 0", 3 007~"l D- i.) l Ü r-. 7. 9322[;- Ü 1 Un j Ct; 31 [-(H IJ " ? l)O(, -/(~' -0). 00 3\)i..tt hn -(i 1 1)" 7 r, I~ ~ 7 r.; - () 1 0 .,~ 0 3 G ~I [) - U ~ C:> 7 C; ~J7 <:·il - 0 } Ù., ê'. 1 l C; fi - 0 lu" >q il l 1. Il - o}. U.., c.i : CI ? :~ n - U 1 Oc / .. 13:,3f1 GO O.)/<~17l ___ ~ 0(.' O,,'·31UfJU Ut) i;:>/+C::.l!·,~)I) (JO l',,!~l)(::20 OU f),'c.?:<,'::.': ()(l _. - -~m: ?

__________________________________________________________________________ 173 ____ __

e VALUE OF CAPACITIVE SUSCEPT/lNC~ MATRIX

O~82B04D-05-0Q~3030D-05-0~229920-05-0096317C-06-0oB59460-06-0~63644D-06 -0.,33227D-07-0.,3143é[)-07-C.,~9.081D-07-0.,259030-07-0023599/)-07-0.,221940-07 0,,73120[' 0° ~'.I~':J e~ ü J3'. 759 C:; O~btj~2(,G OC O,,:&f-.;2-28 OS; Ou5-B-<~9D O~ -0023030D-05 Oo8278~D-05-UoB6217D-06-0023021C-05-0063639D-06-0~661398-06 ···-0~38194D-07-0?36724n-07-0032R13D-07-0o30277D-07-0o265750-07-00254R7D-07 -0.,R4356D-OR-0 0 87399D-Or-O o 72879D-OP-0079207C-Oe-O?650420-08-0?68324D-Oa -O~2299~D-05-0?86217D-06 Oo828580-05-0063653C-C6-0 J 229500-05-0 0 R58ü4 0-06 '-Oo359140-07-00 32984D-07-00321070-07-0e268300-07-00258450-07-0 023476D-07 ------f'Oho:-7.f-J..-!-1 .fT.:' 4-3-9--B-P.--en-2-6-2-e-B--G"B-e~-1-7-fi-{,~n)4i-l?'-{) &-{'~-5.2-7·9-8-t}B-(h-+6 6-2-z.P-O &-­'. -00863170-06-00230210-05-0 0636530-06 0.,82H20t-05-0 0 85896D-06-0 0 2297C O-05 "'-0~466880-07-004436°D-07-Co40207D-07-0?3617SD-07-0o322460-07-0 0 30504D-07 -0.,94860D-Oa-0.,97302D-Oe-Onn214CO-06-0.,û73990-0e-O o 72620D-Oe-Oo7541~C-OO -00859460-06-0 06363 0 D-06-00229500-05-00R5896C-06 008?~8eD-05-UQ22~46D-05 -00 4l~3 880- 07- 00 40l, 2 3n-o 7- C., 3 S660D-0 7- 0., 3254 7D -07-0" 316470-07-00 2n/té "30 -07 ..:...----1Go&F8-7-6-8--G-8-(h-&?+.',-2-13-B F o., 7-3:-A-4-?-f\-{JF.-O-o7-7.-8-7..crf..~-8-G-,)-6.'?-7-17D-O R-{j,y 6·3 4-7-5 ~-o 6-­-00636440-C6-0?86139D-06-0oA5804D-06-0?229790-05-0~2294fO-05 OoA?365D-05 .' -Oo50239D-07-0046555G-07-Co~~0630-07-0?375~3D-07-0o35095D-Q7-0 0 32215D-07 -00938650-0B-OoÇ4D600-0~-Oa818760-08-0d84356C-Oe-0071643D-OR-O o 73120D-OS -0033227D-07-0.,38194D-07-0.,35°14D-07-0 0 4668RD-07-0o 4438&0-07~0?S02390-07 0082902D-05-0022938Q-05-0.,22S070-05-0,,ü560CD-06-0oB5227D-06-C,,630450-06 -----+0·1-;0;-3+27-_ cr'"' +-l"'S·B-C-1--e" :' G-5V'T8-B-=7-B-c :-W~8-fF7-B""n.J'·-e-:7-n-o-7-{h)-~-3/-!"7-bH--O-7-0-()-2-2'1-'?4 G-(}-1---0031436D-07-0036724D-07-0?32q840-07-0~44369G-07-0o404~3D-07-0046555~-07 ~Oo22938D-05 OoB2868D-05-0~r5401D-06-0a?29~~U-05~Oo63036D-06-00856000-06 -Oo37543D-07-0036~79D-07-0.,32547D-07-0030277D-07-0326e30D-07-00Z5~03~-07 '-Oo2Ç081~-07-00328130-07-0n32107D-07-0a40207Q-07-0o396AOD-07-0o~40~3G-07 -Oo22907D-05-0085431D-06C.,B2931D-05-0of3C36D-ü6-00Z2cl930-05-0QH~287D~06 0<) 3 50 95 D o=t 0 0 32-è'ré+) 0 ï' rr.,~~-P.-&-7--(}y2~-?-1-5 .. E-{}-7--Eh-2-5->~5B-G·7-{)-')-2-3J:..99r,-{)-1-­-0025903D-07-0030277D-07-002é~30D-07-0a36179C-07-00325470-07-0 0 37543D-07 '-Ot;! S 5 600D -06- 0:> 2295 6[)-{J 5- Co 6303 (:(j- 06 0" 82 e 6 80-0 :'-0,) 8:> ~3 J.I)-O~-() 022 Q3 '30-;) 1) • -Oo66555D-07-0 0 443AcQ-07-0 0 40423D-07-0036724D-07-0 o 32SB4D-07-00314~68-07 -Oo2~599D-07-00265750-07-0n25f~50-07-tio32246c~07-0o31647D-07-00350Y50-07 -00852B7D-06-Co63036D-06-Co2~a93D-ü5-0085431D-06 0,,829310-05-0,,72~07D-05 ----(};;-4-' .. O ~,31) 0 -; 00 'r{)-t{) 7 D 0 7--0-0~f:~--{t0-3--?-B+3t!-O 1-0~·2-!:{}-7f)-O·7-{)-.)-? -SÛ"!.H-f)-07 .. -- --Oo2?lq4D-07-0025487D-07-0023~760-07-0030504C-07-0o28~63D-07-0 0 32215D-07 -Oo63045D-06-008560CU-06-0u&~2~7~-0~-Oo22936U-05-0o22907n-05 0~62çU2n-05 -Oo502390-07-00466G8D-07-0oL438IiD-07~0,,3PIÇ4D-07-0o~59140-07-00332270-07 '-Oo73120D-Oe-O"P4356n-Ce-On71A 4 30-UR-O:>Q4A60C-06-0o81~7oD-06-0;S3~65D-08 -0., 32 215D-0 7- 0 0 37 543D-07- C" :; 509 50-0 7- 0 0 46 55 5()-07~Oo It!+06 3 1)-07-0., 5û 23 ':i D-O 7 ----fiG? tj-Z3.s S-?--{}5-{h--2..z-97-9n,-or{h;-2~~·~ft-H-5-~(-1-3-Sn-{;(,..-o ,415[·040-0{'-·Oc 113(·6L-;:~-O ()----007541~D-08-0?R739ÇD-OS-0072~20G-Ur-0?97302D-oe-Oo~21hal)-OP-OaS4~60n-08 ~Oo30504D-07-0n3617qD-ü7-0~~?2460-ü7-0a643~SD-07-0~4U2C7D-07-0~46&8~D-07 -Oo22S7?D-05 Oo82820)-05-0~853Ç~D-06-0~2~0?lD-05-0~63&5;D-Ü6-0a~63170-06 -0)634 75D-C8-01) 72c79[~-02-0" 62717D-Ofj-()~ n?14GII-orJ-Oo 71d41D-02 -0 0 f)i17f n-(Id -Oo?a663D-07-0Q32547D-U7-0~31~470-U7-0040423D-U7-0~3q6~OO-07-004~3?8D-J7 ---00-2-?-9L-f:ri:)-{}5-'"J~.;..;.y.r·)-":,J&-OI'l-':'2-3~ t< i)-Ù~-.v-;Y-{~·3{)-3 ç::-{)<,-Ü:)-Z2'-:1 ~vD-O :;-0" ;J SÇ4{,l'-{:~, _ .. --006 f'."32 40- 08- 0., 7'/207 0- û~~- 0" f 504~ D -ù':>-Dv (; 7 3\~<7 L-oe -0" 72 J 79 D-ù n -0", i:'4"3 St:. ~)- OP. -00254870-07-0030277~-07-Uo2~~75D-U7-üa36726D-07-0n32n130-u7-0n~glG~~-07 - u? 8 6 1 3 Q !J - 06 - 0.., 2 3 0 ? l U - 0 5 - 0" t· J f, ~ 91' - 0 6 0 0 F3 t:' 7 f 4 I~ - 0 ~ - 0.., .:..1, A 2 1 7 CJ - 0 0 - 0 ., 2 :: 0 "3 C)f' - 0 ? e -00 5 f., ~? 2 [)- a P. - O? (- 'J 04 2 D - IJ!? - ()" 51) 2 7'-' 0- U F. - 0" 7 2 6 ;~ Û C- - 0 0 - () . .) (, 2 7 : 7 [) - 0 S -0" 716 /.;l :"1- Ci S :- 0 1) 2 3 4 7 f; Cl - 0 7 - ù . ., 2,., i3 .. 0 D - 0 7 - C., 2 ;. a!+ :> u - () 7 - 0 .) 3 2 C) oS /.1.) - U 7 - 0" J? 1 0 7 !) - () 7 - U., "1 :ï C) l "i l , - 'J 7 ---O,,''?5 fh.)ltJj-Of,--{) ,,+-3f: L;. 3H--l'..;-,-t) .,-{l-2·:l·f)()!)-{) 5-\) ,,-~!{. 2 J-7 [1-(:/,. 0., :~ ? ~ ~l:; CI -0:;- 0" ")? c ~ ~ !'- 0 C). -0 <) 5 C'. 6 9 ~)- 0 n- 0., f:. 8:3 2 t. r)_ fF - 0., ~ (,..., 2 ;;> [J- 0 1":. - 00 7 'j ii } r [; - o·~ -0 0 f, Yt 7 ~~ ù- 0 ('- (] n 7~) 1 ? 0: 1-~"\ 0 - 0 0 ?.?l C) lt [) - 0 7- 'JI) 2:) '-! u JI \ - Ù 7 - 0 ~ t.' 3 ? -j q!.1 - U 7 - \)" 3 1'. "" (, r"J - 0 7 -0 ,) 2 () Of' l ~~ - (1 7 - () ~ :"'1 :? ê 7 L' - iJ 7 - 0., f> 3f) It 4 iJ - 0 h - 0,) f"! '-' 3 J. 70- 0 1- - 0 ,., ~ 'i () 1. (, 11 ï U ~. - 0" 2 ::. 0 :; ~ :] - Ù C; - O,) ,~2 C:' Cj :2:-1 - ü ') 1)., ,1 :'1.; JI. n - Ü ~

-e

APPENDIX E

FAULT CALCULATIONS

(a) Delta Connected Resistive Load with Single-Phase to Ground Fouit

(Figure E - 1 (a»

The chain matrix of the 1 ine '

where [fJllJ is the chain matrix from the sending

end to the fault'point of length Il '

[fJf] is the fault point chain matrix,

0'-'·'

[fJ12 ] is the chain matrix from the fouit point

to the rece,iving end of length 12

•

174

(E-l )

Both [fJl1] and [fJ 12] can be determined by introducing the respective

lengths into Equation (A-6) of Appendix A •

U 1

0

- - - - -1

0 0

Also [fJf ] = zf 1 (E-2)

0 0 0 u

0 0 0

A 1

B

C

5 (0)

R

q A

p

B

~'-I~-b-------4-------1-2------C--1 5

(b) R

FIGURE E-l. FAULT CONDITIONS

(0) DELTA CONNECTED RESISTIVE LOAD WITH SINGLE PHASE FAULT.

(b) TWO SIMUL TANEOUS FAULTS. (PHASE TO PHASE AND PHASE TO GROUND)

175

' ..• 176

For a balanced three-phase delta connected resistive load, the relation

between E and , may be expressed as r r

where Y L is the (3 x 3) admittance matrix of the load.

Therefore

YL

E r

, r

E s

=

2

-1 z, -1

u

=

u

=

-1 -1 .. -":'."""

2 -1

-1 2

E r

E r

E r

(E-3)

(E-4)

(E-5)

(E-6)

(A ,:.

-"

177

The sending end current matrix can be obtained from Equation (A-7)

of Appendix A .

A numerical example is calculated bere for the three-phase transmission

line specified on page 92 by considering this time a de/ta connected resistive load

(ZI = 1,000 ohms). The Phase A to ground fault (zf = 0.5 ohm) is assumed to be

located at the middle of the 1 ine.

The values of the receiving end fault current and the current in the

--. same Phase A at normal condition together with the receiving end voltages are

listed as follows :

PHASE A VOLTAGE AND CURRENT AT RECEIVING END, (IN KILOVOLTS, AM PERES)

Real Part / Imaginary Part.

Normal Condition Fouit Condition

VOLTAGE 400. / -142._ - 13.5~ / - 30.3

CURRENT 1209. / -439. 505. / - 130.

From the phasor values of voltage and current, during the fault, it is

seen that the power meosured at the receiving end in Phase A is negative, i.e.

power flows from the other two phases, through the load, to the fouit point.

The fault current in Phase A at various locations are shown in the

following diagram.

86.3/-6471 • S ~

112.5/-6659. 493./-92.5

.. 1-;SO.S,L6566.S

(b) Two Faults (Figure (E-1 (b»

505./-130. ;. R

..... :,

178

There are numerous possibilities of simultaneous faults in a system. For

the purpose of illustrating the application of the Multiconductor Method, a two fouit

case with short circuit between Phase A and Phase B at point "q" and Phase A

to ground fault at point IIpll is considered. (see Figure E-l (b». It is intended to

- -calculate the line currents at the fault points. Generally the analysis of this case is

quite similar to that shown on page 106 of Chapter V"I.

Consideringthe fault point "pli first, then Equations (7-2) to (7-7)

are applicable.

As shown in Equation (7-7), the final expression is

179

= 1 9 (E-7)

-1 U + P2 Pl Zf

which includes the chain matrix of the section of the line with length 12

, [fJ12 ] ,

the faulted impedance to ground matrix, Zf' and the load matrix, ZL .

where [fJ11

] is the chain matrix of the section of the line'

from the sending end to the fault point IIp Il.

-.

[ fJ 11 a ] and [fJ11 b ] are the chain matri ces of the specifie

section of the line and can be determined by substi-

tuting their respective lengths into Equation (A-6)

of Appendix A .

[fJfqJ is shown in Equation (6-4) and

u o

[fJ J= 0 fq Zfq Zfq

l 1 0 U --

Zfq Zfq

0 0 0

(E-a)

(E-9)

180

Then E Zf s

= [ fJ 11 ] 1 9

1 U + P p-1 Zf s 2 1

-= 9

1 1 9 (E-10)

92

For a given E ,the 1 can be determined, which will permit the s s

determination of voltage and current at any point k along the transmission line.

A numerical example is worked out by the same transmission line in-

dicated on page 92 . The location of the fault between Phase A and Phase B

is assumed to be at 20.48 miles from the sending end and the location of the Phase

A to ground is assumed to be at the middle of the li ne • The values of the phase

currents measured at the sending end and at the fault point to ground are 1 isted as

follows :

FAULT CURRENTS (IN KILO-AMPERES)

At Fault Point P Sending End (Phase A to Ground)

Phase A 33.9 3.86

Phase B 33.6 0.39

Phase C 0.9 1.06

181

BIBLIOGRAPHY

"1. Sreenivasan, LV., "H.V. Bundle Conductor System", Electrical Times,

August 1961, pp. 291 - 292.

2. Gross, Eric T. B. and McNutt, W.J., "Electrostatic Unbalance to Ground of

Twin Conductor Lines", AIEE Transactions, Pt. III, Vol. 72, "1~53,; ~

pp. 1288 - 1297.

3. Thomas, P~ H., "Output and Regulation in Long Distance Line", AIEE Tran­

sactions, Pt. l, Vol. 28, 1909, pp. 615 - 640.

4. Carson, J.R • and Pioyt.., R. S., "Propagation of Periodic Currents over a System

of Parallel Wires", Bell Systems Telephone Journal, Vol. 6, 1927, pp. 495 - 545.

5. Bewley, L.V., Travelling Waves on Transmission Lines, New York,. John Wiley

and Sons, 1951.

6. Pipes, L .A., "Direct Computation of Transmission Matrices of Electrical Trans­

mission Lines, Pt. 1", Journal of Franklin Institute, Vol. 281, No."4, 1966,

pp. 275 - 292.

7. Pipes, L. A., "Matrix Theory of Multiconductor Transmission Lines", Phil. Mag.,

Series 7, Vol. 24,1937, pp. 97-113.

8. Rice, S. O., "Steady State Solutions of Tran~mission Line Equations", Bell r

Systems Telephone Journal, Vol. 20, 1941, pp. 131 - 178.

9. Amemiya, H., IITime-domain Analysis of Multiple Parallel Transmission Lines",

RCAReview, Vol. 28,1967, pp. 241 -276.

10. Pipes, L. A., Applied Mathematics for Engineers and Physicists, New York,

McGraw - Hill Book Co. Inc., 1958.

11. Hildebrand, F.B., Methods of Applied Mathematics, Englewood Cliffs, N.J.,

Prentice Hall Inc., 1965.

12. Bellman, R., Introduction to Matrix Analysis, New York, McGraw-Hill

Book Co., Inc., 1960.

182

13. Skill ing, H. H., Electrical Engineering Circuits, New York, John Wiley &

Sons, Inc., 1966.

14. Guillemin, E.A., Synthe sis of Passive Networks, New York, John Wiley &

Sons, Inc., 1957.

15. St!gant, S. A., Matrix and Tensor Analysis in Electrical Network Theory,

London, MacDonald & Co. Ltd., 1964.

16. LePage, W.R. and Seely, S., General Network Analysis, New York, McGraw­

Hill Book Co. Inc., 1952.

17. Pipes, L. A., Matrix Methods for Engineering, Englewood Cliffs, N.J.,

Prentice Hall, Inc., 1963.

18. McCracken, 0.0., A Guide to Fortran IV programming, New York, John Wiley

& Sons, Inc., 1965.

19. Fortescue, C.L., IIMethod of Symmetrical Coordinates Applied to the Solution

of Polyphase Networks", AIEE Transactions, ·Pt. Il, Vo/"., 37, 1918,

pp. 1 027 - 1140.

20. Carson, J. R., IIThe Rigorous and Approximate Theories of Electrical Transmission

Along Wires", Bell Systems Telephone Journal, Vol. 7, 1928, pp. 11 - 25.

21. Zaborszky, J. and Rittenhouse, J.W., Electric Power Transmission, New York,

The Ronald Press Co., 1954.

22. Stevenson, W. O., Jr., Elements of Power System Analysis, New York, McGraw -

Hill Book Co. Inc., 1962.

23. Comsa, R.P. and Rene, J.G., "An Air Model for the Study of Electrostatic Induc­

tion by Transmission Unes", IEEE Transactions on Power Apparatus and Systems,

Vo 1. 87, 1 968, pp. 1 002 - 1 010 .

.. '., 24.

183

Rene, J.R. and Comsa, R.P., IIComputer Analysis of Electrostatically Induced

Currents on Finite Objects by E. H. V. Transmission Unes Il, IEEE Transactions on

Power Apparatus and Systems, Vol. 87, 1968, pp. 997 - 1002.

25. Comsa, R. P. and Yu, L. Y.M., IITransient Electrostatic Induction by E. H. V.

Transmission Unes ll, presented to IEEE Winter Power Meeting, New York,

January 1 969.

26. Abett, P.A., Lindh, C.B. and Simmons, Jr., H.O., IIEconomics of Single

and Bundle : Conductors for Extra-High Voltage Transmission ll, AIEE Transactions,

Pt. Il l, Vol. 79, 1960, pp. 138 - 153.

27. Sandell, D.H., Shealy, A.N. and White, H.B., IIBibliographyon Bundled

Conducto,rs Il, IEEE Transactions on Power Apparatus and Systems, Vol. 82, 1963,

pp. 11 15 - 1128.

28. Alexandrov, G.N., IITheoryof Bundle: Conductors ll, presented to IEEE

Winter Meeting, New York, January 1969.

29. Miller, C.J., Jr., IIMathematical Prediction of Radio and Corona Characteristics

of Smooth, Bundled Conductors ll, AIEE Transactions, Pt. III, Vol. 75,1956,

pp. 1029 - 1037.

30. Rao·, V.S. Subba, IIBalancing Bundle ·-Conductor Transmission-Une Constants

without Transpositions ll, Proc. IEE, Vol. 112, No. 5,1965, pp. 931 - 940.

31. Clarke, E., IIThree Phase Multiple-Conductor Circuits ll, AIEE Transactions,

Pt. III, Vol. 51,1932, pp. 809 -820.

32. Temoshok, M., IIRelative Surface Voltage Gradients of Grouped Conductors ll,

AIEE Transactions, Pt. III, Vol. 67,1948, pp. 1583 -1591.

33. Gross, E. T • B .. and Stensland, L.R., IICharacteristics of Twin Conductor Arrange-1

ments ll, AIEE Transactions, Pt. III, Vol. 77,1958, pp. 721 -725.

1

'184

34. Timascheff, A. S., "Equigradient Unes in the Vicinity of Bundle ' Conductors", IEEE Transactions on Power Apparatus and Systems, Vol. 82, 1963, pp. 104 - 110.

35. Timascheff, A. S., "Field Pattern of Bundle : Conductors and their Electrostatic Properties", AIEE Transactions, Pt. III, Vol. 80, 1961, pp. 590 - 597.

36. Edwards, A. T. and Boyd, J .M., "Bundle' Conductor - Spacer Design Require­ments and Development of Spacer - Vibration Damper", Ontario Hydro Research Quarterly, Vol. 16, Second Quarter, No. 2,1964, pp. 1 -12.

37. Ruhlman, J. R., Eucker, R .A. and Swart, R. L., "Electrodynamic Studies of Bundled Conductor Spacers", IEEE Transactions on Power Apparatus and Systems, Vol. 82, 1963, pp. 750 -760.

38. Retallack, R. L., Fry, T. R. and Popeck, C. A., "Fault and Load Current Testing of a Bundle : Conductor Spacer", IEEE Transactions on Power Apparatus and Systems, Vol. 82, 1963, pp. 646 - 652.

39. Zaffanella, L. E., "Investigation on Bundle: Conductor Oscillations", presented to IEEE Winter Power Meeting, New York, January 1969.

40. Krahn, R. A. and Bessler, M. B., "Evaluation of Spacers for Bundle : Conductor Transmission Unes", presented to IEEE Winter Power Meeting, New York, January 1969.

41. Samuelson, A. J., Retallack, R. L. and Kravitz, R. A., "American Electric Power 765 KV Une Design ", IEEE Winter Power Meeting Papers, January 1969, pp. 67 - 73.

42. Johnson, W. C., Transmission Unes and Networks, New York, McGraw - Hill Book Co., Inc., 1950.

43. Varshney, M. P., "Bundle Conductor Calculations", Electrical Times, September 1961 .

44. Wagen, C.F. and Evans, R. O., Symmetrical Compon~tlts, New York, McGraw - Hill Book Co., Inc., 1933.

185

45. Calabrese, G. O., Symmetrical Components, New York, The Ronald Press Co., 1959.

46. Clarke, E. , Circuit Analysis of A. C. Power Systems, Vol. l, New York, John Wiley and Sons, Inc., 1943.

47. Westinghouse Electric Corp., Electrical Transmission and Distribution Reference Book, 1950.

48. Kron, G., Tensor for Circuits, New York, Dover Publications, Inc., 2nd • . Edition, 1958.

49. Messerle, H.K., Dynamic Circuit Theory, Oxford, Pergamon Press, 1965.

50. Stagg, G. W. and EI-Abiad, A. H., Computer Methods in Power System Analysis, New York, McGraw - Hill Book Co., 1968.

51 • Thomas, A. O., "Calculation of Transmission - Une Impedance by Digital Computer", AIEE Transactions, Pt. III, Vol. 78, 1959, pp. 1270 - 1275.

52. Coleman, D., Watts, F. and Shipley, R. B., "Digital Calculation of Over­head - Transmission - Line -Constants", AIEE Transactions, Pt. III, Vol. 78, 1959, pp. 1266 - 1269 •

53. Lawrence, R. F. and Povejsil, O. J., "Determination of Inductive and Capaci­tive Unbalance for Untransposed Transmission Lines", AIEE Transactions, Pt. III, Vol. 71, 1952, pp. 547 - 556.

54. Sarma, M. P. and Janischewskyj, W., "ElectrostaticFieldofaSystemof ParaI/el Cylindrical Conductors", presented to IEEE Winter Power Meeting, New York, January 1969.

186

LIST OF PUBLICATIONS

1. Three Phase Induction Motor, book in Chinese, Hong Kong, Chi - Cheng ----------------------, Book Co., 1962.

2. "Constant Starting Torque Control of Wound Rotor Inductor Motors", IEEE

Transactions Paper, presented to IEEE Summer Power Meeting, 1969.

3. "Transient Electrostatic Induction by EHV Transmission Unes", presenteJ to

IEEE Winter Power Meeting, 1969. ..,' ,

4. "Asymmetries in Bundled Conductor Transmission Unes", to be presented to IEEE

Winter Power Meeting, 1970,.

5. "Fault Calculations in EHV Transmission Unes by Multiconductor Analysi~", to

be presented to IEEE Winter Power Meeting, 1970.

6. "Abnormal Conditions between Subconductors in EHV Transmission Unes", to be

presented to IEEE Summer Power Meeting, 1970.