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UBC Theses and Dissertations

A multiconductor transmission line with distributed compensation Orton, Harry Ernest 1968

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A MULTICONDUCTOR TRANSMISSION LINE WITH DISTRIBUTED COMPENSATION by HARRY ERNEST ORTON B.E., U n i v e r s i t y o f New South Wales, 1966. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OP MASTER OF APPLIED SCIENCE i n the Department o f E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d Research S u p e r v i s o r Members of the Committee the Department Members of the Department o f E l e c t r i c a l E n g i n e e r i n g UNIVERSITY OF BRITISH COLUMBIA December, 1968. Head of THE In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e ^LSL^t^Zsij^ "^5 , (^ k©/ ABSTRACT . The t h e s i s presents an a n a l y s i s of the steady-s t a t e o p e r a t i o n of a multiconductor power t r a n s m i s s i o n l i n e with d i s t r i b u t e d compensation. The a n a l y s i s o u t l i n e s two methods f o r d e r i v i n g the v o l t a g e and c u r r e n t equations along the l i n e s t a r t i n g from Maxwell's c o e f f i c i e n t s and the T e l e -graph equations. C o r r e c t i o n f a c t o r s due to an imperfect earth and conductor i n t e r n a l impedance are d i s c u s s e d . A l s o , by making the assumption t h a t the system of conductors i s completely transposed i t i s shown that c o n s i d e r a b l e s i m p l i f i c a t i o n of the equations i s p o s s i b l e . Numerical r e s u l t s are given f o r a two and a f o u r -conductor, s i n g l e phase t r a n s m i s s i o n l i n e with earth r e t u r n and f o r a three-phase system having four conductors per phase. The r e s u l t s show that s u c c e s s f u l o p e r a t i o n of the system i s prevented by e l e c t r i c f i e l d l i m i t a t i o n s at the surface of the conductors. i i TABLE OF CONTENTS Page ABSTRACT . i i TABLE 0? CONTENTS i i i LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENT v i i i 1. INTRODUCTION '.. 1 2. SYSTEM STABILITY AND POWER TRANSFER IMPROVEMENT ... 3 2.1 System Expansion 3 2.2 S t a b i l i t y and Power T r a n s f e r Improvement .... 3 2.3 I n t r o d u c t i o n to D i s t r i b u t e d Compensation .... 4 3. BASIC ASSUMPTIONS 7 3.1 General D e s c r i p t i o n 7 3.2 The Completely Transposed System 8 3.3 Admittance Array C o r r e c t i o n F a c t o r s 9 3.4 Impedance Array C o r r e c t i o n Factors 10 3.4.1 Conductor I n t e r n a l Impedance 10 3.4.2 Earth C o r r e c t i o n F a c t o r s 12 3.5 E r r o r A n a l y s i s 14 4. MATHEMATICAL BACKGROUND . 19 4.1 Maxwell's C o e f f i c i e n t s 19 4.2 General D i f f e r e n t i a l Equations 24 4.2.1 General D i f f e r e n t i a l Equations Method I ..... 25 4.2.2 General D i f f e r e n t i a l Equations Method II ... . 30 i i i Page 4.3 The Completely Transposed and Lossless Line . 32 4.4 D i f f e r e n t i a l Equation Solution 37 4.4.1 Solution and Boundary Conditions 37 4.4.2 Comparison of Solution with the Generalized C i r c u i t Constants 43 4.5 Addition of Lumped Parameters 45 4.6 Computer Programming 51 4.7 Maximum Allowable Voltage Difference 54 5. RESULTS OP ANALYSIS OP A 500 kV TRANSMISSION LINE WITH DISTRIBUTED COMPENSATION 59 5.1 Description of the Transmission Line Systems. 59 5.2 Two-Conductor System 61 5.3 Four-Conductor System 67 5.3.1 I n i t i a l Investigations 67 5.3.2 Optimizing the Conductor Diameter and Spacings 69 5.3.3 Use of Lumped Parameters 74 5.4 Three-Phase System 80 5.5 Physical Limitations 83 5.6 Comparison of Series and Distributed Compensation 85 6. CONCLUSION 86 APPENDIX. A. Numerical Evaluation of Conductor Coefficients 88 A . l Two-Conductor Case 88 A.2 Four-Conductor Case 89 i v Page APPENDIX B. N u m e r i c a l E v a l u a t i o n o f Maximum A l l o w a b l e V o l t a g e D i f f e r e n c e be tween C o n d u c t o r s 92 B. 1 T w o - C o n d u c t o r Case 92 B.2 F o u r - C o n d u c t o r Case 95 REFERENCES 99 v LIST OP ILLUSTRATIONS F i g u r e Page 2.1 D i s t r i b u t i o n of Lumped C a p a c i t o r s 5 2.2 The Open-Ended L i n e 6 3.1 E a r t h C o r r e c t i o n Parameters 13 4.1 A System of n P a r a l l e l Conductors Above P e r f e c t Ground . . . „ 19 4.2 n-Conductor System w i t h Mutual C o u p l i n g 25 4.3 S i n g l e - C o n d u c t o r T r a n s m i s s i o n L i n e w i t h D i s t r i b u t e d Parameters 30 4.4 Two-Port Networks 44 4.5 Lumped P a r a l l e l C a p a c i t o r s 46 4.6 Lumped S e r i e s R e a c t o r s 47 4.7 P a r a l l e l R e a c t o r s 48 4.8 Computer Flow Chart 53 4.9 C r o s s - S e c t i o n of a System of n P a r a l l e l Conductors ' 55 5.1 V o l t a g e and C u r r e n t P r o f i l e s Along the L i n e (Two-Conductor Case) 62 5.2 Conductor Diameter and Maximum V o l t a g e D i f f e r e n c e (Two-Conductor Case) 66 5.3 I n s t a l l a t i o n of P a r a l l e l C a p a c i t o r s A l o n g the L i n e a t 50 M i l e I n t e r v a l s (Two-Conductor Case) 66 5.4 V o l t a g e and C u r r e n t P r o f i l e s Along the L i n e (Four-Conductor Case) 68 5.5 O p t i m i z a t i o n of Conductor Spacings (Four-Conductor Case) 70 v i F i g u r e Page 5.6 O p t i m i z a t i o n o f C o n d u c t o r D i a m e t e r ( F o u r - C o n d u c t o r Case) •••• 70 5.7 V o l t a g e and C u r r e n t P r o f i l e s A l o n g t h e L i n e ( F o u r - C o n d u c t o r O p t i m i z e d Case) 73 5.8(a) L o c u s P l o t o f V o l t a g e s and C u r r e n t s A l o n g t h e L i n e 75 5.8(b) Lumped P a r a m e t e r S e c t i o n o f a Two-Conductor L i n e 75 5.9 I n s t a l l a t i o n o f P a r a l l e l R e a c t o r s A l o n g t h e L i n e 79 5.10 T h r e e - P h a s e System Above P e r f e c t G r o u n d......... 82 5.11 E f f e c t o f I n c r e a s e d L i n e V o l t a g e on t h e Maximum V o l t a g e D i f f e r e n c e . 84 A . l T wo-Conductor L i n e Above P e r f e c t G round 88 A. 2 F o u r - C o n d u c t o r L i n e Above P e r f e c t Ground 89 B. l C r o s s - S e c t i o n o f a Two-Conductor L i n e 92 B.2 C o n d u c t o r S u r f a c e V o l t a g e G r a d i e n t s ( T wo-Conductor Case) 94 B.3 C r o s s - S e c t i o n o f a F o u r - C o n d u c t o r L i n e 95 B.4 C o n d u c t o r S u r f a c e V o l t a g e G r a d i e n t s ( F o u r - C o n d u c t o r ) 97 v i i ACKNOWLEDGEMENT T h i s t h e s i s could not have been completed without the i n s p i r a t i o n and encouragement of my s u p e r v i s i n g p r o f e s s o r s Dr. F. Noak.es and Mr. R. Jamieson, whose i n f l u e n c e and experience helped shape the ideas expressed. I am a l s o indebted to Dr. H. R. Chinn f o r h i s i n v a l u a b l e a s s i s t a n c e with the computer programming, to Mr. A. MacKenzie f o r drawing the graphs and diagrams and to my w i f e Carmen f o r t y p i n g the t h e s i s and proof reading the d r a f t . x G r a t e f u l acknowledgement i s extended to t h e U n i v e r s i t y of B r i t i s h Columbia and the N a t i o n a l Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support of the p r o j e c t . v i i i 1 1. INTRODUCTION The aim of t h i s t h e s i s i s to determine the s t e a d y -s t a t e performance of a m u l t i c o n d u c t o r power t r a n s m i s s i o n l i n e w i t h d i s t r i b u t e d compensation. The concept of a d i s t r i b u t e d - c o m p e n s a t i o n l i n e a r o s e from c o n s i d e r a t i o n o f w e l l - e s t a b l i s h e d techniques' o f s e r i e s compensation. A lumped s e r i e s c a p a c i t o r on a l o n g t r a n s m i s s i o n l i n e i s o f t e n r e p l a c e d by s e v e r a l c a p a c i t o r s spaced a l o n g t h e l i n e , i n o r d e r to reduce o v e r v o l t a g e s near t h e c a p a c i t o r s . I t i s c l e a r t h a t i f a l a r g e number of s e r i e s c a p a c i t o r s c o u l d be e v e n l y d i s t r i b u t e d a l o n g the l i n e then t h e o v e r v o l t a g e s would be e l i m i n a t e d . For economic reasons t a i s would be i m p o s s i b l e . However, i f an open-ended l i n e were used then the d i s t r i b u t e d c a p a c i t a n c e between two or more con d u c t o r s c o u l d be used t o a c h i e v e compensation, thus a v o i d i n g the problems a s s o c i a t e d w i t h s e r i e s elements such as have been d e s c r i b e d by Marbury (2) (3) and Johnson* ' and Harder. ' P r e v i o u s work by P i p e s ^ ^ ^ and B e w l e y ^ ^ on l o s s l e s s , m u l t i c o n d u c t o r t r a n s m i s s i o n l i n e s w i t h d i s t r i b u t e d parameters forms a c o n v e n i e n t s t a r t i n g - p o i n t f o r d e v e l o p i n g t h e system e q u a t i o n s . In the t h e s i s a s i n g l e - p h a s e , two-c o n d u c t o r , open-ended l i n e w i t h e a r t h r e t u r n i s f i r s t a n a l y s e d u s i n g the m a t r i x methods suggested by P i p e s . The m a t r i x approach a l l o w s an easy e x t e n s i o n of t h e a n a l y s i s f o r more complex systems. 2 As i s t h e c a s e f o r a l l EHV t r a n s m i s s i o n l i n e s , t h e v o l t a g e g r a d i e n t a t t h e s u r f a c e o f t h e b u n d l e d c o n d u c t o r s p r o v e s t o be an i m p o r t a n t f a c t o r i n t h e n u m e r i c a l a n a l y s i s of t h e t r a n s m i s s i o n s y s t e m . A l s o , c o r r e c t i o n f a c t o r s due t o an i m p e r f e c t e a r t h and c o n d u c t o r i n t e r n a l impedance a r e d i s c u s s e d . 3 2. SYSTEM STABILITY AND POWER TRANSFER IMPROVEMENT 2.1 System Expansion As t h e modern power system expanded as a r e s u l t of the i n c r e a s i n g s i z e of l o a d b l o c k s , n a t u r a l sources were more f u l l y e x p l o i t e d . In c o u n t r i e s such as Canada, t h e U n i t e d S t a t e s of America, R u s s i a and Sweden the s e n a t u r a l sources a re g e n e r a l l y a v a i l a b l e i n the form o f h y d r o e l e c t r i c g e n e r a t i o n p o t e n t i a l a t d i s t a n c e s o f t e n i n excess of 300 m i l e s from the l o a d c e n t r e s . To t r a n s p o r t t h e heavy l o a d s e c o n o m i c a l l y under s t a b l e c o n d i t i o n s , e x t e n s i v e use has been made of the e x t r a h i g h - v o l t a g e t r a n s m i s s i o n l i n e and t h e s e r i e s c a p a c i t o r . 2.2 S t a b i l i t y and Power T r a n s f e r Improvement The r e l a t i o n s h i p between economy and i n c r e a s e d t r a n s m i s s i o n v o l t a g e was w e l l known as e a r l y as 1919 when S i l v e r ^ ^ recommended the i n t r o d u c t i o n of 220 kV as a s t a n d a r d v o l t a g e s u i t a b l e f o r t h e t r a n s f e r of l a r g e amounts of power. For s h o r t l i n e s ( l e s s than 300 m i l e s i n l e n g t h ) t h e amount o f power which can be t r a n s f e r r e d w i t h o u t l o s s o f s t a b i l i t y v a r i e s as t h e square of the l i n e v o l t a g e ( l i n e r e s i s t a n c e n e g l e c t e d ) . However, as t h e l e n g t h of the l i n e i n c r e a s e s t h i s i s no l o n g e r the case and t h e r e l a t i o n s h i p between economy, amount o f power, v o l t a g e and the l i n e l e n g t h must be c o n s i d e r -ed. O v e r a l l system s t a b i l i t y has been improved by f a s t s w i t c h i n g , good r e l a y i n g , i n t e r m e d i a t e s w i t c h i n g s t a t i o n s , almost l i g h t n i n g - p r o o f l i n e s and good system d e s i g n . The most i m p o r t a n t f a c t o r l i m i t i n g power t r a n s f e r and l i n e s t a b i l i t y on l o n g l i n e s , however, i s t h a t of t h e h i g h s e r i e s i n d u c t i v e r e a c t a n c e . Many methods have been suggested t o reduce the s e r i e s r e a c t a n c e , some of which are g i v e n below: 1. R e d u c t i o n of the f r e q u e n c y , 2. Use of synchronous condensers, 3. P a r a l l e l o p e r a t i o n of t r a n s m i s s i o n l i n e s , 4. Use of bundled conductors f o r each phase, and 5. S e r i e s c a p a c i t o r compensation. The a d d i t i o n of s e r i e s c a p a c i t o r s improves not o n l y the power t r a n s f e r and s t a b i l i t y , but a l s o the v o l t a g e r e g u l a t i o n of the l i n e , s i n c e the i n d u c t i v e r e a c t a n c e of the l i n e i s reduced. T h i s improvement i n v o l t a g e r e g u l a t i o n i s d i s c u s s e d i n References 14 and 15. 2.3 I n t r o d u c t i o n to D i s t r i b u t e d Compensation S e r i e s c a p a c i t o r compensation i n t r o d u c e s l i n e o v e r v o l t a g e s at t h e c a p a c i t o r s . To reduce th e s e o v e r v o l t a g e s i t would be of c o n s i d e r a b l e advantage to d i s t r i b u t e t h e c a p a c i t a n c e u n i f o r m l y a l o n g the l e n g t h of t h e l i n e i n a l a r g e number of s m a l l e r u n i t s as shown i n F i g u r e 2.1 (a) and ( b ) . At p r e s e n t , economical c o n s i d e r a t i o n s l i m i t t h e number of s m a l l e r u n i t s to t h r e e or f o u r f o r a l i n e o f 600 m i l e s i n l e n g t h . 5 O-- o —o H h H H H H h TRANSMISSION LINE CONFIGURATION (a) (b) VOLTAGE - TO-6R0UND PROFILE FIGURE 2.1 D i s t r i b u t i o n o f Lumped C a p a c i t o r s I f t h e r e were no economic l i m i t the c a p a c i t a n c e c o u l d be d i s t r i b u t e d a l o n g t he l i n e i n an i n f i n i t e number of u n i t s . T h i s would e n t i r e l y e l i m i n a t e normal o p e r a t i n g o v e r -v o l t a g e s as shown i n F i g u r e 2.1. , These c o n s i d e r a t i o n s have i n f a c t suggested the concept of the open-ended l i n e . Here, i n s t e a d of u s i n g an i n f i n i t e number of lumped c a p a c i t o r s , the d i s t r i b u t e d c a p a c i t a n c e between two or more conductors i s used t o a c h i e v e compensation, as shown i n F i g u r e 2.2. 6 LINE 1 LINE 2 -DISTRIBUTED CAPACITANCE 0 FIGURE 2.2 The Open-Ended Line 7 3.. BASIC ASSUMPTIONS 3.1 G e n e r a l D e s c r i p t i o n C o n s i d e r a t i o n of every c o n t i n g e n c y i n v o l v e d i n t r a n s m i s s i o n l i n e problems would make the mathem-a t i c a l t r e a t m e n t of an n-conductor system v e r y complex. To reduce t h e c o m p l e x i t y of the problem a number of s i m p l i f y i n g assumptions are i n t r o d u c e d . The t r a n s m i s s i o n l i n e c o n s i d e r e d i n t h e a n a l y s i s c o n s i s t s of n p a r a l l e l c onductors surrounded by a homogeneous d i e l e c t r i c and suspended above a p e r f e c t ground p l a n e , A " p e r f e c t ground" i s one h a v i n g a u n i f o r m (17) r e l a t i v e p e r m e a b i l i t y of u n i t y and i n f i n i t e c o n d u c t i v i t y . I n a c t u a l i t y the ground p l a n e i s not p e r f e c t as i s (27) (31) d i s c u s s e d by Carson and Wise. ' ' However, the assumption o f a p e r f e c t ground i s j u s t i f i e d f o r a m u l t i -c o n d u c t o r system a t power f r e q u e n c i e s as the e r r o r i n v o l v e d i s s m a l l (see S e c t i o n 3.5). The e f f e c t o f c o n d u c t o r sag and ground i r r e g u l a r i t i e s has been n e g l e c t e d so t h a t each conductor i s assumed t o be s t r a i g h t and p a r a l l e l t o a h o r i z o n t a l ground p l a n e w i t h an average h e i g h t H. E f f e c t s of s t e e l towers and i n s u l a t o r s on the lea k a g e c u r r e n t s may be (25) assumed n e g l i g i b l e . 8 3.2 The C o m p l e t e l y Transposed System The terra " c o m p l e t e l y t r a n s p o s e d " i n f e r s t h a t t h e n c o n d u c t o r s of t h e t r a n s m i s s i o n l i n e occupy each o f the n p o s i t i o n s of t h e c o n d u c t o r s f o r an e q u a l d i s t a n c e and i n c y c l i c o r d e r such t h a t t h e impedance and a d m i t t a n c e a r r a y s ( d e f i n e d i n Chapter 4) are "doubly symmetric" (symmetric about b o t h d i a g o n a l s ) . A doubly symmetric m a t r i x i s of the form i l l u s t r a t e d i n E q u a t i o n 3.1 f o r a f o u r conductor system. b a d c c d a b d c b a (3.1) To determine t h e " c o m p l e t e l y t r a n s p o s e d " impedance and admittance a r r a y s i t w i l l be n e c e s s a r y t o determine the a r r a y s f o r the n p o s s i b l e c o n f i g u r a t i o n s and e v a l u a t e t h e average o f t h e n a r r a y s . T h i s may be e x p r e s s e d m a t h e m a t i c a l l y by n Z r s ( t r a n s p o s e d ) = 1 r S n where r and s have v a l u e s from 1 t o n. 9 A numerical e v a l u a t i o n f o r the four-conductor system i s given i n Appendix A. 3.3 Admittance Array C o r r e c t i o n F a c t o r s The p o t e n t i a l c o e f f i c i e n t array d e s c r i b e d i n S e c t i o n 4.1 f o r an n-conductor system may be determined from the e q u a t i o n s ^ ^ r r In 2TtC 2H R + (M + jN ) r r " r r (3.2a) rs In 2ne. a rs rs + 1 HG (M + jN ) x rs 0 rs (3.2b) where M and N are the r e a l and imaginary c o r r e c t i o n rs rs to J f a c t o r s f o r the r s element of the ar r a y . The admittance a r r a y i s determined by a p p l y i n g (18) [Y] = 0 0) [p] -1 (3.3) The c o r r e c t i o n f a c t o r s M and N are dependent upon the p e r m i t t i v i t y and p e r m e a b i l i t y of f r e e space, the r e l a t i v e p e r m i t t i v i t y , r e l a t i v e p e r m e a b i l i t y and c o n d u c t i v i t y of the ground, the radius and geometric p o s i t i o n of the conductor and f i n a l l y the frequency of the system. 10 Doench, i n i i i s work on g r o u n d c o n d u c t i v i t y and (32 ) p e r m i t t i v i t y , has p l o t t e d c u r v e s s h o w i n g t h e e f f e c t o f f r e q u e n c y v a r i a t i o n on t h e c o r r e c t i o n f a c t o r s . T hese c u r v e s ( R e f e r e n c e 32, page 48) show t h a t b e l o w 1 KHz t h e c o r r e c t i o n f a c t o r s a r e e f f e c t i v e l y z e r o . S i n c e we a r e o n l y c o n s i d e r i n g s i n u s o i d a l v o l t a g e s a t power f r e q u e n c i e s t h e c o r r e c t i o n f a c t o r s i n E q u a t i o n s 3.2 may be n e g l e c t e d g i v i n g t h e s i m p l i f i e d E q u a t i o n s 4.2. 3.4 Impedance A r r a y C o r r e c t i o n F a c t o r s 3.4.1 C o n d u c t o r i n t e r n a l Impedance I n a m u l t i c o n d u c t o r s y s t e m where some o r a l l o f t h e c o n d u c t o r s a r e r e l a t i v e l y c l o s e t o e a c h o t h e r , t h e " p r o x i m i t y e f f e c t " may i n t r o d u c e c o r r e c t i o n f a c t o r s i n t h e i m p e d a n c e a r r a y . T h e s e c o r r e c t i o n f a c t o r s depend on t h e f r e q u e n c y , t h e d i s t a n c e b e t ween c o n d u c t o r s , c o n d u c t o r s i z e and p e r m e a b i l i t y . P r o x i m i t y e f f e c t i s p r e s e n t f o r t h r e e -p h a s e c i r c u i t s as w e l l as s i n g l e - p h a s e c i r c u i t s . C a r s o n ^ ^ has shown t h a t t h e i n c r e a s e i n i n t e r n a l i m p e d a n c e o f one c o n d u c t o r due t o t h e p r o x i m i t y o f a r e t u r n c o n d u c t o r i s o n l y one p e r c e n t a t h i g h f r e q u e n c i e s and w i t h a r a t i o o f c o n d u c t o r s p a c i n g t o c o n d u c t o r r a d i u s o f 15 t o 1. 11 For bundled c o n d u c t o r s at power f r e q u e n c i e s the p r o x i m i t y e f f e c t w i l l be s m a l l e r and so may be n e g l e c t e d . The conductor i n t e r n a l impedance of a c y l i n d r i c a l c onductor a t power f r e q u e n c i e s i s ^ ^ ^ ^ ' ^ ^ ^ ' ^ ^ ^ Z i = R i + ^ w L i ^ 3* 4^ where R. = d.c. r e s i s t a n c e of t h e c o n d u c t o r i n ohms/mile l - ' and L. = _u_ = i n d u c t a n c e of t h e conductor due t o 1 8tt i n t e r n a l f l u x —7 = 0.5. x 10 henry/meter or = 0.806 x 1 0 ~ 4 h e n r y / m i l e The i n t e r n a l i n d u c t a n c e (L^) has been d e r i v e d by S t e v e n s o n ^ " ^ assuming t h a t the c u r r e n t d e n s i t y of t h e c y l i n d r i c a l conductor i s u n i f o r m ( t h a t i s s k i n e f f e c t n e g l i g i b l e ) and the conductor i s non-magnetic w i t h a (26) r e l a t i v e p e r m e a b i l i t y of u n i t y . B u t t e r w o r t h ' ( i n h i s work on e l e c t r i c a l c h a r a c t e r i s t i c s of overhead l i n e s ) has taken t h e e f f e c t o f s t r a n d i n g i n t o account by u s i n g (15) t h e method of geometric mean r a d i u s . However, f o r t h e p r e s e n t s t u d y , s u f f i c i e n t a c c u r a c y i s o b t a i n e d by (32) c o n s i d e r i n g the conductor t o be c y l i n d r i c a l . ' 12 3.4.2 E a r t h C o r r e c t i o n Factors To c o r r e c t the impedance matrix f o r an earth of f i n i t e c o n d u c t i v i t y , r e s i s t a n c e and reactance c o r r e c t i o n f a c t o r s R and X a s s o c i a t e d with the earth e e r e t u r n path are determined by using an i n f i n i t e s e r i e s (27) developed by Carson. Real and imaginary c o r r e c t i o n component matrices P and Q, r e s p e c t i v e l y , are c a l c u l a t e d i n terms of r and 0, two parameters such t h a t r = /cou.' . a (3.5) rs rs and G r g = the angle subtended at conductor r by the images of conductors r and s (see Figure 3.1). E v a l u a t i o n of r depends on rs ' ifi = 2nf = angular frequency, .tive (31) (j, = r e l a t i v p e r m e a b i l i t y of the ground = 1.0, p. — absolute p e r m e a b i l i t y = u. u , (28) p = ground r e s i s t i v i t y = 100,0 ohm.meters and a r g = d i s t a n c e from conductor r to the image of conductor s. 13 GROUND PLANE OF FINITE CONDUCTIVITY IMAGES FIGURE 3.1 E a r t h C o r r e c t i o n P a r a m e t e r s c a n The C a r s o n c o r r e c t i o n f a c t o r s due t o t h e e a r t h be w r i t t e n ^ ' ^ 2 7 ^ R rs P wu r s * 7t ( 3 . 6 a ) 71 ( 3 .6b) 1 4 The i n f i n i t e s e r i e s used t o e v a l u a t e the c o r r e c t i o n component m a t r i c e s P and Q depends upon the v a l u e o f r . Por our case a t a f r e q u e n c y of 60 Hz and w i t h a ground r e s i s t i v i t y of 100 ohm.meters r 1 2 = 0.066 .(27). R e f e r r i n g t o Carson's paper P and Q may be e v a l u a t e d by P = 1 1 r S 8 1 r cos 0 + r s cos 20 (0.6728 + I n 2 ) 372 r s r S T 6 r s — . . , - r s + ( r r s ) 2 Q s i n 20 (3.7a) 16 r S r S and Q = -0.0386 + 1 l n r s 2 r s + 1 3^/2 r cos 0 (3.7b) r s r s 3.5 E r r o r A n a l y s i s C o n s i d e r a b l e s i m p l i f i c a t i o n can be o b t a i n e d i n the m a t h e m a t i c a l s o l u t i o n of the t r a n s m i s s i o n l i n e problem i f the c o r r e c t i o n components due t o the e a r t h r e t u r n p ath and the i n t e r n a l impedance of each conductor can be n e g l e c t e d . The e r r o r i n v o l v e d i n n e g l e c t i n g t h e s e components i s determined from the f o l l o w i n g a n a l y s i s . I t i s shown i n S e c t i o n s 4 . 2 and 4 . 4 t h a t D 2 [v] = [ z ] • [ Y ] • [ V ] (Eqn. 4 . 1 6 ) 15 D 2 [i] = [ i j • [z] • [ i ] ( E q n . 4.17) [z] • [Y] = [ l ] • [z] = [ j ] (Eqn 4.20) and X = - Jj~ ( E q n . 4.40) r r w here t h e s y s t e m i s assumed t o be l o s s l e s s and " c o m p l e t e l y t r a n s p o s e d " . However, t h e above e q u a t i o n s , w i t h t h e e x c e p t -i o n o f E q u a t i o n 4.40, a p p l y t o any m u l t i c o n d u c t o r , " c o m p l e t e l y t r a n s p o s e d " s y s t e m / W h e t h e r t h e s y s t e m i s assumed l o s s l e s s o r n o t . r (28) F o r a l o s s y s y s t e m t h e e l e m e n t s i n t h e \ (25) i m p e d a n c e m a t r i x w i l l c o n t a i n f i v e components o f t h e f o r m Z' = R. + R + j ( X + X. + X ) (3.8) l e g i e • where t h e s u f f i x e s h ave t h e f o l l o w i n g s i g n i f i c a n c e : -g = t h e c o n t r i b u t i o n due t o t h e p h y s i c a l g e o m e t r y o f t h e s y s t e m ( i n t h e l o s s l e s s c a s e [zj = [^g]' S e c t i o n 4.3) i = t h e c o n t r i b u t i o n due t o f a c t o r s i n t e r n a l t o t h e c c o n d u c t o r ( S e c t i o n 3.4.1) and e = t h e c o n t r i b u t i o n due t o an i m p e r f e c t e a r t h r e t u r n p a t h ( S e c t i o n 3.4.2). 16 The p r o p a g a t i o n c o n s t a n t s f o r a l o s s y s y s t e m ( 2 8 ) may be d e t e r m i n e d f r o m M = { H • W ) l / 2 (3.9) where t h e p r o p a g a t i o n c o n s t a n t m a t r i x [ j i j i s d i a g o n a l w i t h t h e d i a g o n a l e l e m e n t s d e t e r m i n e d by and \ a r e t h e e i g e n v a l u e s o f [z ] * [ Y " ] , Now a t any p o i n t a l o n g t h e l i n e t h e v o l t a g e s and c u r r e n t s a r e r e l a t e d by t h e c h a r a c t e r i s t i c i m p e d a n c e o f t h e l i n e ^ 2 8 ) where f o r a c o m p l e t e l y t r a n s p o s e d s y s t e m [ z j - [ i ] "I J [*•] . [T] I 1 / 2 ( 3 . 1 2) 17 The complete s o l u t i o n f o r the v o l t a g e and (28) c u r r e n t can be w r i t t e n as [ v ] = c " W x [ v o ] ( 3 . 1 3 ) [I]' = „-M* [ z j " ' [!o] (a.14) where the m a t r i c e s V and I are the t e r m i n a l c o n d i t i o n s o o of the l i n e a t x = 0 ( t h a t i s a t the r e c e i v i n g end). I t i s c l e a r from E q u a t i o n s 3.13 and 3.14 t h a t an e r r o r a n a l y s i s of the s o l u t i o n may be o b t a i n e d by c o n s i d e r i n g the m a t r i c e s Z q and K. The v a l u e s of b o t h Z and Y were determined c w i t h the impedance c o r r e c t i o n f a c t o r s i n c l u d e d ( E q u a t i o n 3.8) and compared w i t h the u n c o r r e c t e d v a l u e s determined by p u t t i n g [>'] = [ 2 ] = [ X g ] (3.15). T a b l e 3.1 summarises t h e e r r o r s o b t a i n e d i n b o t h Z and V by n e g l e c t i n g the impedance c o r r e c t i o n c f a c t o r s . 18 FACTOR °/o ERROR fo ERROR NEGLECTED Z c EARTH 1 10.0 10.0 CORRECTION 2 5.5 5.5 3 3.0 3.0 INTERNAL 1 2.8 2.8 IMPEDANCE 2 3.4 3.4 3 3.6 3.6 TABLE 3.1 E r r o r A n a l y s i s where 1. r e f e r s to the two-conductor case ( S e c t i o n 5.2) 2. r e f e r s to the four-conductor case (S e c t i o n 5.3) and 3. r e f e r s to the three-phase case (S e c t i o n 5.4) The aim of t i i i s t h e s i s i s to consider a three-phase multiconductor system; the two-and four-conductor systems are used only f o r preparatory c a l c u l a t i o n s . Since the e r r o r s i n v o l v e d i n the three-phase case are small i t can be concluded that the c o r r e c t i o n f a c t o r s may be neglected i n a l l s t u d i e s i n v o l v e d i n t h i s t h e s i s . I t w i l l be shown i n Chapter 4 that t h i s assumption c o n s i d e r a b l y s i m p l i f i e s the mathematical s o l u t i o n of a multiconductor t r a n s m i s s i o n system. 4. MATHEMATICAL BACKGROUND 4 01 Maxwell's C o e f f i c i e n t s (a) P o t e n t i a l C o e f f i c i e n t s C o n s i d e r a system of n p a r a l l e l , c y l i n d r i c a l c o n d u c t o r s of f i x e d p o s i t i o n as shown i n F i g u r e 4.1. FIGURE 4.1 A System of n P a r a l l e l Conductors Above P e r f e c t Ground 20 . I f a u n i t charge i s p l a c e d on conductor r then t h e p o t e n t i a l r e l a t i v e t o ground a c q u i r e d by each conductor may be w r i t t e n P l r> P£r' ••••» ? r r ' ••••» P n r # F u r t h e r , i f cond u c t o r r a c q u i r e s a charge then the p o t e n t i a l s would be Q r P l r » Q rP2r' » 2 r P n r « I t f o l l o w s u s i n g the p r i n c i p l e of s u p e r p o s i t i o n t h a t t he p o t e n t i a l s caused by sim u l t a n e o u s charges Q^, Q^, Q n on each of the n con d u c t o r s would be V l = P l l S l + P l 2 2 2 + •••• + P l n Q n V 2 = P 2 l 2 l + P 2 2 2 2 + •••• + ^ n ^ n « v n = vnlQ± + P n 2 Q 2 + .... + P n n Q n ( 4 a ) The l i n e a r c o e f f i c i e n t s or p o t e n t i a l c o e f f i c i e n t s (PJJ_» p 1 2 ' ••••> P n n ^ depend s o l e l y on the geometry of the system. As shown i n F i g u r e 4.1 t h e c y l i n d r i c a l c o n d u c t o r s a r e c o n s i d e r e d t o be w i d e l y s e p a r a t e d and to be suspended over an e q u i p o t e n t i a l ground p l a n e so t h a t c o n d u c t o r s 1, 2, n have images 1', 2', n' r e s p e c t i v e l y . By a p p l i c a t i o n of the method of images t o F i g u r e 4.1, the eq u a t i o n s f o r the p o t e n t i a l c o e f f i c i e n t s may be o b t a i n e d ; -211 p = 1 In r d a r a f / meter (4.2 a) 2nc R ' o r 21 and p r s 1 i n r s d a r a f / m e t e r (4.2 b) 2nc r s w h e re II = t h e h e i g h t o f c o n d u c t o r r above t h e g r o u n d p l a n e , R r t h e r a d i u s o f c o n d u c t o r r , a — t h e d i s t a n c e b e t w e e n t h e c o n d u c t o r r and t h e image o f c o n d u c t o r s, = t h e d i s t a n c e b e t w e e n c o n d u c t o r s r and s, -12 r s r s and c = 8.85 x 10 f a r a d s p e r m e t e r . E q u a t i o n s 4.1 may be w r i t t e n i n m a t r i x form:-[V] = [ P ] . [ Q ] (4.3) where n '11 '12 ' i n '21 P 2 2 '2n n l n2 nn 22 and Q I t i s p o s s i b l e , however, t o express the v a l u e o f t h e charges i n terras of c a p a c i t y c o e f f i c i e n t s C , where such I* s c o e f f i c i e n t s a re f u n c t i o n s o f t h e geometry of t h e conductor arrangement. Q n C^V.. + + C, V 11 1 I n n c v n l v l + + C V nn n The above e q u a t i o n s may be w r i t t e n i n m a t r i x form:-[Q] - [c] • W (4.4) where [ c j = '11 .. C In C i • • # • c n l nn t h a t I t i s c l e a r by comparing E q u a t i o n s 4.3 and 4.4 [C] = [ p ] - 1 (4.5) (b) E l e c t r o m a g n e t i c C o e f f i c i e n t s I f a c u r r e n t i f l o w s i n co n d u c t o r r , then t h e r ' f l u x ( 0 r ) l i n k i n g the c i r c u i t c o n t a i n i n g c o n d u c t o r r may be w r i t t e n L r r.i r» where L i s c a l l e d t h e " e l e c t r o m a g n e t i c (17) c o e f f i c i e n t " of c o n d u c t o r r . • Then a p p l i c a t i o n of the s u p e r p o s i t i o n p r i n c i p l e t o a m u l t i c o n d u c t o r system g i v e s t h a t 0 1 = hi 1! + L 1 2 A 2 + •••• + h^ n K = K\H + ^ 2 * 2 + •••• + Wn (4'6> where the c o e f f i c i e n t s L are c a l l e d the " c o e f f i c i e n t s of r r s e l f i n d u c t a n c e " and the c o e f f i c i e n t s L the " c o e f f i c i e n t s r s of mutual i n d u c t a n c e " . In m a t r i x n o t a t i o n [ 0 ] - h] • [i] (4.7) 24 For the system of n p a r a l l e l c o n d u c t o r s s i t u a t e d over a p e r f e c t l y c o n d u c t i n g ground p l a n e , as shown i n F i g u r e 4.1, these c o e f f i c i e n t s have the v a l u e s : r r o 4u 1 2 + 2 i n R r / henry/meter (4.8a) and r s U -i a r o l n r s 2n b r s henry/meter (4.8b) where = 4n x 10 henry/meter ( p e r m e a b i l i t y of f r e e space) The above e q u a t i o n s have been d e r i v e d by (21) Kuznetsov and S t r a t o n v i c h by comparison of Maxwell's e q u a t i o n s and the T e l e g r a p h e q u a t i o n s ( E q u a t i o n s 4.9 and 4.12), In t h e i r o r i g i n a l d e r i v a t i o n of E q u a t i o n 4.8 K u t z n e t s o v and S t r a t o n v i c h i n c l u d e d a term due t o the p r o x i m i t y e f f e c t . T h i s has been o m i t t e d here f o r reasons o u t l i n e d i n S e c t i o n 3.4. E q u a t i o n 4.8(a) i n c l u d e s a c o r r e c t i o n f a c t o r ( l / 2 ) t o account f o r the i n t e r n a l i n d u c t a n c e of t h e c o n d u c t o r . ' (17 ) {18) * ' T h i s c o r r e c t i v e f a c t o r i s n e g l e c t e d i n S e c t i o n 4.3 f o r reasons g i v e n i n S e c t i o n 3.4. 4.2 G e n e r a l D i f f e r e n t i a l E q u a t i o n s Two methods f o r d e r i v i n g the t r a n s m i s s i o n l i n e d i f f e r e n t i a l e q u a t i o n s were s t u d i e d . The f i r s t ( d e s c r i b e d 25 i n S e c t i o n 4.2.1) was g i v e n by Bewley i n h i s work on m u l t i c o n d u c t o r systems. The e q u a t i o n s ure d e r i v e d by assuming lumped parameters f o r an i n f i n i t e s i m a l l e n g t h of l i n e as shown i n F i g u r e 4.2. The second method ( d e s c r i b e d i n S e c t i o n 4.2.2) i s an o r i g i n a l method o b t a i n e d by t h e e x t e n s i o n of a s o l u t i o n (15) suggested by Stevenson. T h i s method has been found t o y i e l d the r e q u i r e d e q u a t i o n s much more d i r e c t l y t han the method used by Bewley. 4.2.1 G e n e r a l D i f f e r e n t i a l E q u a t i o n s Method 1 eg d x ts» FIGURE 4.2 n-Conductor System w i t h Mutual C o u p l i n g 26 F i g u r e 4.2 shows an n-conductor t r a n s m i s s i o n l i n e whose co n d u c t o r s are assumed t o be p a r a l l e l t o each o t h e r and t o the ground p l a n e . The c o n d u c t o r s are i n t e r c o n n e c t e d e l e c t r o m a g n e t i c a l l y and e l e c t r o s t a t i c a l l y as shown i n the f i g u r e . A s s o c i a t e d w i t h each u n i t l e n g t h of l i n e , c o n d u c t o r r has the f o l l o w i n g p arameters:— L = s e l f i n d u c t a n c e of conductor r , r r ' L = mutual i n d u c t a n c e between c o n d u c t o r s r and s, r s ' C r r = s e l f c a p a c i t a n c e c o e f f i c i e n t of c o n d u c t o r r , C r g = mutual c a p a c i t a n c e c o e f f i c i e n t between co n d u c t o r s r and s, R = s e r i e s r e s i s t a n c e of conductor r , r r ' g = leakage conductance of conductor r t o ground, g = leakage conductance between c o n d u c t o r s r and s, I* s Q r r = sum o f leakage conductances on c o n d u c t o r r , G r s = n e g a t i v e o f leakage conductance gj.s» Z r r = s e l f impedance of c o n d u c t o r r , Z r g = mutual impedance between c o n d u c t o r s r and s, T = admittance of c o n d u c t o r r and r r Y = admittance between c o n d u c t o r s r and s. r s The l a s t s i x parameters are d e f i n e d by t h e e q u a t i o n s G r r = «rl + g r 2 +•••• + 8 r n G = G " = -g = -g r s s r r s 6 s r 27 Z = R + pL r r r * r r Z = pL r s r s Y = G + pC and . r r r r r r r Y = Y = pC = pC r s s r r s r s r where p = JOJ f o r s i n u s o i d a l waves. I n t h e f o l l o w i n g a n a l y s i s , b o t h v o l t a g e s and c u r r e n t s a r e assumed t o be s t e a d y s i n u s o i d a l waves. (17) The d i f f e r e n t i a l e q u a t i o n v ' ( T e l e g r a p h e q u a t i o n ) f o r t h e p o t e n t i a l on c o n d u c t o r 1 i s = + R 1 i 1 ( 4 . 9 ) W W D i f f e r e n t i a t i n g E q u a t i o n 4.6 g i v e s b 0 x = p L n i 1 + p L 1 2 i 2 + .... + p L l n i n ( 4 > 1 0 ) W By c o m b i n a t i o n o f E q u a t i o n s 4.9 and 4.10 and m a k i n g t h e n e c e s s a r y s u b s t i t u t i o n f r o m t h e above d e f i n i t i o n s - ^ 1 = hi1! + Z12^2 + •"' + Zlnln ( 4 ' U ) ox 28 ' (17) S i m i l a r l y , the c u r r e n t T e l e g r a p h e q u a t i o n ' i s 4 i , = >>Q 1 + xl (4.12) 6 t where i ^ i s the sum o f the leakage c u r r e n t s f l o w i n g from c o n d u c t o r 1 t o the ground and t o t h e o t h e r c o n d u c t o r s . From E q u a t i o n 4.12 we can o b t a i n " S - Y 1 1 V 1 + Y 1 2 V 2 + •• + Y l n V n (4.13) S i m i l a r v o l t a g e and c u r r e n t e q u a t i o n s e x i s t f o r every c o n d u c t o r . By g r o u p i n g them and a p p l y i n g the m a t r i x t e c h n i q u e , E q u a t i o n s 4.11 and 4.13 may be w r i t t e n i n s i m p l i f i e d form:-- D [v] (4,14a) -D [ l ] M " [v] (4.14b) where Sx" '11 J l n J n l nn and l l n l I n nn ( M a t r i c e s I and V have been d e f i n e d i n S e c t i o n 4.1) D i f f e r e n t i a t i n g E q u a t i o n 4.14 (a) w i t h r e s p e c t t o x we o b t a i n -°2 w (4.15) I t may be noted t h a t t h e m a t r i x Z i s independent of t h e v a r i a b l e x ( d i s t a n c e from the r e c e i v i n g end a l o n g t h e l i n e ) . S u b s t i t u t i o n o f E q u a t i o n 4.14 (b) i n t o 4.15 g i v e s D 2 [ v ] . [z] . [,] . [V] (4.16) By r e p e a t i n g the above sequence of o p e r a t i o n s u s i n g E q u a t i o n 4.14 (b) one o b t a i n s D 2 [ j ] [ I ] - - [z] . [ I ] Now f o r a l l t r a n s m i s s i o n l i n e systems Z = Z r s s r (4.17) and Y ~ Y r s s r T h e r e f o r e t h e m a t r i c e s Z and Y w i l l be s y m m e t r i c . Hence [z] « [ Y ] = [ l ] • [z] ( 4 . 1 8 ) E q u a t i o n s 4.16 and 4.17 may now be w r i t t e n i n s i m p l i f i e d f o r m : -D 2 [ v ] ( 4 . 1 9 a ) and D 2 [ I ] (4.19b) w h e r e (4.20) 4.2.2 G e n e r a l D i f f e r e n t i a l E q u a t i o n s Method I I /+ Ai I I •Q-Q e s e+21e • ! e 1 i -X- - C B » L 0 A D FIGURE 4.3 S i n g l e - C o n d u c t o r T r a n s m i s s i o n L i n e w i t h D i s t r i b u t e d P a r a m e t e r s 31 F i g u r e 4.3 shows a s i n g l e - c o n d u c t o r t r a n s m i s s i o n l i n e w i t h a ground r e t u r n . The l i n e parameters are not shown as lumped c i r c u i t elements as they are c o n s i d e r e d t o be d i s t r i b u t e d a l o n g the l i n e . E a r t h r e t u r n parameters have been n e g l e c t e d ( S e c t i o n 3.4). The v o l t a g e and c u r r e n t increments (Ae a n d A i ) may be w r i t t e n Z i and Ye r e s p e c t i v e l y , where Z i s the s e r i e s impedance per u n i t l e n g t h and Y i s the admittance to ground p e r u n i t l e n g t h . I f t h e s m a l l increment A x approaches z e r o , the v o l t a g e g r a d i e n t per u n i t l e n g t h f o r i n c r e a s i n g x w i l l become ^e = Z i (4.21) *x and s i m i l a r l y the c u r r e n t g r a d i e n t may be w r i t t e n : -j&i = Ye (4.22) cot D i f f e r e n t i a t i n g E q u a t i o n 4.21 w i t h r e s p e c t t o x and s u b s t i t u t i n g E q u a t i o n 4.22 we o b t a i n D 2e = ZYe (4.23) s i m i l a r l y D 2 i = Y Z i (4.24) where D = j ) _ 3x I n t h e d e r i v a t i o n o f t h e above e q u a t i o n s , o n l y one c o n d u c t o r has been c o n s i d e r e d . E x p a n s i o n t o a m u l t i -c o n d u c t o r s y s t e m may be a c h i e v e d by r e p l a c i n g t h e v o l t a g e , c u r r e n t , i m p e d a n c e and a d m i t t a n c e by t h e a p p r o p r i a t e m a t r i c e s A d e v e l o p m e n t a l o n g t h e l i n e s o f t h a t g i v e n above y i e l d s where ( E q n ' s . 4 . 1 9 ) as b e f o r e . ( N o t e : The s i g n o f t h e v o l t a g e and c u r r e n t g r a d i e n t s i n E q u a t i o n s 4 . 2 1 and 4 . 2 2 i s a c o n s e q u e n c e o f t h e r e f e r e n c e ( 1 5 ) c o n d i t i o n s c h o s e n f o r A e and A i i n F i g u r e 4 . 3 . ) 4 . 3 The C o m p l e t e l y T r a n s p o s e d and L o s s l e s s L i n e The s o l u t i o n o f E q u a t i o n s 4 . 1 9 may be s i m p l i f i e d b y a s s u m i n g t h a t t h e n c o n d u c t o r s a r e " c o m p l e t e l y t r a n s p o s e d " The t e r m " c o m p l e t e l y t r a n s p o s e d " a p p l i e d t o an n - c o n d u c t o r s y s t e m i m p l i e s t h a t b o t h t h e i m p e d a n c e and a d m i t t a n c e a r r a y s a r e s y m m e t r i c a b o u t t h e i r d i a g o n a l s ( s e e S e c t i o n 3 . 2 ) . F o r t h e t r a n s p o s e d s y s t e m we may w r i t e f o r any r and s Z r r Z s s Z r s Z s r Y r r Y s s and Y r s Y s r 33 Further s i m p l i f i c a t i o n i s possible i f the system i s considered to be lossless (see Section 3.5), that i s R = 0 r r and B = 0 6 r s By neglecting the terra due to the internal flux i n Equations 4.8 (see Section 3.4) one obtains L = JJo ln 2**r henry/meter (4.25a) r r 2it R and L = ]j> In a r s henry/meter (4.25b) R S 2 1 1 b rs where L i s the self-inductance c o e f f i c i e n t and L i s the r r rs mutual inductance c o e f f i c i e n t for the symmetric system. Now by comparison of Equations 4.2 and 4.25 i t i s clear that 1 2 P r r - r r - V D r r (4.26a) U c Ko o and p = 1 L = v 2 L (4.26b) r r s rs o rs U c po o 34 where v 1 t h e v e l o c i t y o f l i g h t i n f r e e s p a c e , fVo" u o = p e r m e a b i l i t y o f f r e e space and C q = p e r m i t t i v i t y o f f r e e s p a c e . I t i s u s e f u l a t t h i s s t a g e t o c o n s i d e r t h e s i m p l e c a s e o f two p a r a l l e d c o n d u c t o r s s i t u a t e d above a g r o u n d p l a n e . F o r t h i s case t h e r e l a t i o n s h i p between t h e p o t e n t i a l c o e f f i c i e n t m a t r i x and i n d u c t a n c e m a t r i x i s [ P ] -*11 *12 where L and L , '12 = v '11 = L , = L 11 = L 12 22 = L 21 (4.27) Now t h e r e f o r e [ l ] = jco [c] = jco [p] •2 ' -1 n - l P e r f o r m i n g t h e i n d i c a t e d m a t r i x i n v e r s i o n y i e l d s (4.28) "2" 2 2 L - L / - L , 35 where Y = I n = Y 2 2 and Y1 = Y 1 2 = T 2 1 Prom E q u a t i o n 4.28 one may o b t a i n Y = j a) v 2 L 2 - L 2 o i (4.29a) and Y i = - 0 _JjL_ L l (4.29b) Now, by. expansion of E q u a t i o n 4.20 f o r the s i m p l e two cond u c t o r case OJ J l l J 1 2 J 1 2 J l l z x Z and by m a t r i x m u l t i p l i c a t i o n we o b t a i n ZY + Z XY X ZYX + Z1Y ZYX + ZXY ZY + ZXY (4.30) where Z = jwL (4.31 a) and Z 1 = j w L 1 (4.31 b) 3 6 On s u b s t i t u t i n g E q u a t i o n s 4.31 and 4.29 i n t o E q u a t i o n 4.30 t h e m a t r i x J becomes -co 0 —to ( 4 . 3 2 ) S i m i l a r l y f o r a s y s t e m o f n p a r a l l e l c o n d u c t o r s E q u a t i o n 4.27 may be e x t e n d e d t o g i v e p n . . . p l n 'In • • • *nn = v L . . . . . L I n L l n . . . L and as b e f o r e Y ... Y I n Y Y x l n * ' * 1 J co v~2~ L . . . L I n L l n " -1 A g a i n e x p a n d i n g E q u a t i o n 4.20 and s u b s t i t u t i n g t h e v a l u e s f o r Z and Y one o b t a i n s f o r a s y s t e m o f n c o n d u c t o r s to 0 0 0 - w 0 o (4 .33 ) (n x n) I t i s i m p o r t a n t t o n o t e t h a t t h e m a t r i x J i s a d i a g o n a l m a t r i x w i t h e q u a l d i a g o n a l e l e m e n t s . The v a l u e o f e a c h d i a g o n a l e l e m e n t i s t h e same and i n d e p e n d e n t o f t h e c o n d u c t o r p h y s i c a l g e o m e t r y . The above o b s e r v a t i o n s a p p l y t o any number o f c o n d u c t o r s p r o v i d e d t h e s y s t e m i s assumed l o s s l e s s and s y m m e t r i c . 4.4 D i f f e r e n t i a l E q u a t i o n S o l u t i o n 4.4.1 S o l u t i o n and B o u n d a r y C o n d i t i o n s E q u a t i o n s 4.19 r e p r e s e n t a s y s t e m o f n o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f t h e s e c o n d o r d e r . I t w i l l be n e c e s s a r y t h e r e f o r e t o s o l v e f o r 2n a r b i t r a r y c o n s t a n t s . 3 8 A p o s s i b l e s o l u t i o n i s [v] = [ M ] • [cosh U x ] + [ N ] • [sinh*x ] ( 4 . 3 4 a) W = [ S] * [cosh^x] + [ T ] • [sinh^x] ( 4 . 3 4 b) where M,N,S and T are matrices-^ w i t h c o n s t a n t terms. D i f f e r e n t i a t i n g E q u a t i o n 4 . 3 4 ( a ) w i t h r e s p e c t t o x, t h e f o l l o w i n g second o r d e r d i f f e r e n t i a l e q u a t i o n may be o b t a i n e d ; -D 2 [v] = [ M ] • [*2] « [coshXx] + [ N ] • [*2]° [sinh*x] ( 4 . 3 5 ) where a i + J ^ i ^*ie P r o P a g a ' t ' i o n c o n s t a n t the a t t e n u a t i o n c o n s t a n t of cond u c t o r 1 the phase c o n s t a n t of c o n d u c t o r 1 o *2 0 n ( 4 . 3 6 ) arid where 39 P o r a c o m p l e t e l y t r a n s p o s e d and l o s s l e s s s y s t e m t h e p r o p a g a t i o n c o n s t a n t w i l l be t h e same f o r e a c h c o n d u c t o r . T h e r e f o r e t h e m a t r i x [ > 2 ] ( E q u a t i o n 4.36) w i l l r e d u c e t o a s c a l a r (^ ) and E q u a t i o n 4.35 c a n t h e n be w r i t t e n : -D 2 [v] = * 2 [hi] c o s h t f x + X2 [N] s inhYx (4.37) S u b s t i t u t i n g E q u a t i o n 4.34 ( a ) i n t o E q u a t i o n 4.37 we o b t a i n D 2 [v] = Y 2 [v] (4.38) By c o m p a r i s o n o f E q u a t i o n s 4.38 and 4.19 ( a ) i t i s p o s s i b l e t o w r i t e r e a r r a n g i n g , i>a<> - A • [v] = [o] where U i s t h e u n i t m a t r i x . From t h e c h a r a c t e r i s t i c e q u a t i o n [o i t i s p o s s i b l e t o o b t a i n t h e e i g e n v a l u e s o r c h a r a c t e r i s t i c r o o t s o f t h e m a t r i x J . The p r o p a g a t i o n c o n s t a n t f o r e a c h c o n d u c t o r i s t h e n t h e s q u a r e r o o t o f t h e r e s p e c t i v e e i g e n v a l u e 40 I n S e c t i o n 4.4 i t has been diown t h a t t h e J m a t r i x i s a d i a g o n a l m a t r i x w i t h e q u a l d i a g o n a l e l e m e n t s J r r « Hence t h e c h a r a c t e r i s t i c e q u a t i o n y i e l d s , f o r t h e p r o p a g a t i o n c o n s t a n t o', Y r = + | J r r where r = 1 ,n o r sim. .. .. r r i m p l y X = + j J r r (4.40) The c o n s t a n t m a t r i c e s S and T ( E q u a t i o n 4.34(b)) a r e d e p e n d e n t upon t h e m a t r i c e s M and N; t o f i n d t h e r e l a t i o n -s h i p b e t ween t h e s e m a t r i c e s , i t i s n e c e s s a r y t o d i f f e r e n t i a t e E q u a t i o n 4.34(a) w i t h r e s p e c t t o x and o b t a i n D [v] =. V [M] s i n h T x + X [N] c o s h ^ x (4.41) E q u a t i o n 4.21 may be exp a n d e d t o t h e m a t r i x f o r m D [V] = [Z] • [i] (4.42) By c o m p a r i s o n o f E q u a t i o n s 4.41 and 4.42 we o b t a i n [z] * [ i ] = t [M] s i n h Y x + [N] coshXx 41 0] -1 On p r e m u l t i p l y i n g b o t h s i d e s o f t h i s e q u a t i o n by t h e s o l u t i o n f o r Iij may be w r i t t e n [i] = u [z] _ 1 [M] s inhVx + Y [z] ~ 1 [N] cosh^x ( 4 . s p e c i f y t h e r e c e i v i n g end c o n d i t i o n s . The r e c e i v i n g end v o l t a g e s and c u r r e n t s a r e h e r e u s e d as t h e b o u n d a r y c o n d i t i o n s t o o b t a i n a c o m p l e t e s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s as f o l l o w s : -when x = 0 ( a t t h e r e c e i v i n g end) I t i s u s u a l i n t r a n s m i s s i o n l i n e p r o b l e m s t o (4 . 4 4 a ) and ( 4 . 4 4 b ) where V. n and a t t h e r e c e i v i n g e n d . I n 42 By s u b s t i t u t i o n of the boundary c o n d i t i o n s i n t o E q u a t i o n 4.34 (a) and 4.44 we o b t a i n and M M (4.45) (4.46) The f i n a l s o l u t i o n may be o b t a i n e d by s u b s t i t u t -i n g E q u a t i o n s 4.45 and 4.46 i n t o E q u a t i o n s 4.34 (a) and 4.43. [v] = JVJ cosh &x + [z] . [ i j s i n h & x (4.47a) [ i j = * [z] _ 1 [VJ s i n h X x + |^IJ c o s h S x (4.47b) T h i s may be w r i t t e n i n t h e f a m i l i a r form of A , B , . C,D parameters (by u s i n g p a r t i t i o n e d m a t r i c e s ) :-"v" "A : — i~ B" — —1 \ I c ' . 1 D • (4.48) where [ A ] M [c] [uJ cos Px j _[z] s i n px - 1 s i n Px (4.49a) (4.49b) (4.49c) 43 and [u] cos Bx (4.49d) Equations 4.48 and 4.49 are used i n the computer program d e s c r i b e d i n S e c t i o n (4.6). 4.4.2 Comparison of S o l u t i o n with the G e n e r a l i z e d C i r c u i t Constant .(15) Standard c i r c u i t theory d e s c r i b e s how any two-port network may be represented by the f o l l o w i n g equations V s A B~ I s_ C D — _ where V and R R = sending end v o l t a g e of the two-port network, sending end c u r r e n t of the two-port network, r e c e i v i n g end v o l t a g e of the two-port network, r e c e i v i n g end c u r r e n t of the two-port network. In general f o r a two-port network the f o l l o w i n g r e l a t i o n between A,B,C,D parameters a p p l i e s AD - BC = 1 but f o r the more s p e c i f i c case of a symmetric two-port network A = D 44 The above equations are r e s t r i c t e d by r e q u i r i n g t h a t the c u r r e n t at one terminal of a p a i r must always be equal and opposite to the c u r r e n t at the other t e r m i n a l of that p a i r (See Figure 4.4a). Two-port network theory may be a p p l i e d to a multicbnductor t r a n s m i s s i o n system by i n c r e a s i n g the number of l i n e s from one to n as shown i n Figure 4.4(b). The r e s t r i c t i o n s t i l l holds that the sum of the c u r r e n t s e n t e r i n g at the upper t e r m i n a l s of a p a i r must always be equal to the sum of the c u r r e n t s at the lower t e r m i n a l s . T h i s i s the case f o r our t r a n s m i s s i o n system where the upper t e r m i n a l s represent the conductors and the lower t e r m i n a l s the ground r e t u r n paths. (a) n o o n FIGURE 4.4 Two-l'ort Networks 45 F o r t h e s y m m e t r i c , m u l t i c o n d u c t o r s y s t e m , t h e a b o v e e q u a t i o n s may be w r i t t e n i n m a t r i x f o r m as [A] = [D] ( 4 . 5 0 a ) and [A] • [D] - [B] • [C] = [u] ( 4 . 5 0 b ) By. c o m p a r i s o n o f E q u a t i o n s 4 . 4 9 ( a ) and (d) i t i s c l e a r t h a t E q u a t i o n s 4.50 a r e s a t i s f i e d . S u b s t i t u t i o n o f E q u a t i o n s 4.49 i n t o 4 . 5 0 ( b) p r o d u c e s [u] c o s 2 6 + [u l s i n 2 B = [u] so t h a t o u r s o l u t i o n ( E q u a t i o n 4.49) s a t i s f i e s t h e e x p r e s s -i o n s f o r a s y m m e t r i c t w o - p o r t n e t w o r k . 4.5 A d d i t i o n o f Lumped P a r a m e t e r s To o b t a i n s a t i s f a c t o r y o p e r a t i o n o f t h e m u l t i -c o n d u c t o r t r a n s m i s s i o n l i n e w i t h d i s t r i b u t e d c o m p e n s a t i o n , i t was f o u n d ( S e c t i o n 5.3) t h a t a d d i t i o n a l s e r i e s o r p a r a l l e l , l umped components were r e q u i r e d t o be i n s t a l l e d a t c e r t a i n p o i n t s a l o n g t h e l i n e . To a l l o w t h e s i m p l e s t c o m p u t e r s o l u t i o n f o r a ( 1 9 ) s y s t e m c o n t a i n i n g lumped c o m p o n e n t s , p a r t i t i o n e d m a t r i c e s were d e r i v e d f o r e a c h c o n f i g u r a t i o n . F o r t h e t w o - c o n d u c t o r 46 case the method i s as f o l l o w s : -(a) P a r a l l e l C a p a c i t o r s V1 V1 l1 O o L ^- O 2 72 h FIGURE 4.5 Lumped P a r a l l e l C a p a c i t o r s The equations f o r the system i l l u s t r a t e d i n Figu r e 4.5 may be w r i t t e n V 1 - V Y l ~ 1 V2 = V !• = h + I c = I , + Y c ( V l - V 2 ) *2 = h ~ h = h ~ Yc ( Yl ~ V (4'51) and i n p a r t i t i o n e d m a t r i x f o r m 4 7 V 2 1%2 0 0 ' 0 0 0 0 Y -Y c c -Y Y c c 1 0 0 1 ( 4 . 5 2 ) ( b ) S e r i e s R e a c t o r s FIGURE 4 . 6 Lumped S e r i e s R e a c t o r s The v o l t a g e and c u r r e n t e q u a t i o n s f o r t h e s y s t e m i n F i g u r e 4 . 6 a r e : -V i = V l + hhl V • - V J- T 7 ¥ 2 ~ 2 + 2 S2 H = h and i n p a r t i t i o n e d m a t r i x form ri 0 1 1 7 0 ~ v l V 2 0 0 ZS2 I I - 0 0 1 0 h } 2 0 0 0 1 (c) P a r a l l e l R e a c t o r s FIGURE 4 . 7 P a r a l l e l R e a c t o r s The v o l t a g e and current equations f o r the system i n Figure 4.7 are:-V' = V, where and J2 L l L2 Q Q L l L2 (4.55) (Q* and Q* are the values of r e a c t i v e power a s s o c i a t e d with r e a c t o r s 1 and 2 r e s p e c t i v e l y i n Figure 4.6) In p a r t i t i o n e d matrix form v i 1 0 ' 1 0 0 0 1 1 1 1 0 0 \ l — r 0 i i 1 0 0 i Y 1 L2 I 0 1 The three c o n f i g u r a t i o n s given above are a l l symmetric two-port networks, and f o r each case the p a r t i t i o n e d matrix s a t i s f i e s Equation 4.50. F o r i n s t a n c e by c o n s i d e r a t i o n o f E q u a t i o n 4.56 ( E x a m p l e ( c ) ) i t i s p o s s i b l e t o w r i t e [ A ] . [ „ ] . 1 0 0 1 and [ A ] - [ D J - [ B J . [c] - [u] wh e r e [ B ] . 0 0 0 0 and Y L 1 ° 0 L2 t h u s p r o v i d i n g a c h e c k o f t h e v a l i d i t y o f E q u a t i o n 4.56. Now f o r a t r a n s m i s s i o n l i n e o f f o u r o r more c o n d u c t o r s , E q u a t i o n s 4.52, 4.54 and 4.56 may be expan d e d f o r e x ample i f t h e number o f c o n d u c t o r s i n F i g u r e 4.6 was i n c r e a s e d t o f o u r , E q u a t i o n 4.56 w o u l d become 1 0 0 0 I 0 0 ' 0 0 ~ v l ~ V 2 0 1 0 0 | 0 0 0 0 V 2 V 3 0 0 1 0 0 0 0 0 V 3 V 4 0 0 0 1 0 0 0 0 _V_4 H Y L 1 0 0 0 1 1 1 0 0 0 J l H 0 Y L 2 0 0 1 1 0 1 0 0 J 3 0 0 Y L 3 1 0 | 1 0 0 1 0 h 0 0 0 0 0 0 1 h 51 W i t h b o t h t h e t r a n s m i s s i o n l i n e e q u a t i o n ( E q u a t i o n 4.48) and t h e lumped component e q u a t i o n s ( E q u a t i o n s 4.52, 4.54 and 4.56) i n p a r t i t i o n e d m a t r i x f o r m , t h e c o m p u t e r p r o g r a m may be d e v e l o p e d w i t h e a s e . 4.6 Computer P r o g r a m m i n g An u n d e r s t a n d i n g o f t h e c o m p u t e r p r o g r a m may be g a i n e d by r e f e r r i n g t o b o t h t he f l o w - c h a r t g i v e n i n F i g u r e 4.8 and t h e f o l l o w i n g s t e p - b y - s t e p d e s c r i p t i o n . The c o m p u t e r p r o g r a m i s i n i t i a t e d by r e a d i n g i n t h e s y s t e m d a t a i n c l u d i n g t h e g e o m e t r i c a l p o s i t i o n and d i a m e t e r o f e a c h c o n d u c t o r , t h e t r a n s m i s s i o n l i n e l e n g t h (LGTH) and t h e i n c r e m e n t a l l e n g t h (ILGTH) o v e r w h i c h one v o l t a g e and c u r r e n t c a l c u l a t i o n i s p e r f o r m e d . From t h e g e o m e t r i c a l p o s i t i o n and t h e d i a m e t e r of e a c h c o n d u c t o r , t h e p o t e n t i a l and e l e c t r o m a g n e t i c c o e f f i c i e n t s a r e d e t e r m i n e d u s i n g E q u a t i o n s 4.2 and 4.8. A f t e r c o m p u t i n g t h e Z and Y m a t r i c e s , t h e J m a t r i x i s c a l c u l a t e d u s i n g b o t h methods g i v e n i n E q u a t i o n 4.18. I f t h e J m a t r i x i s n o t t h e same i n b o t h c a s e s t h e n t h e p r o g r a m w i l l be s t o p p e d us t h e s y s t e m i s no l o n g e r s y m m e t r i c ( t h i s i s a n e c e s s a r y c o n d i t i o n as d e s c r i b e d i n S e c t i o n 4 . 2 ) . F u r t h e r t e s t s a r e c o n d u c t e d on t h e J m a t r i x t o d e t e r m i n e w h e t h e r i t i s a d i a g o n a l m a t r i x w i t h e q u a l d i a g o n a l e l e m e n t s ( E q u a t i o n 4 . 3 3 ) . O n l y a f t e r a l l t h e s e r e s t r i c t i o n s a r e s a t i s f i e d w i l l t h e c o m p u t a t i o n p r o c e e d . 52 By using Equation 4.40 to c a l c u l a t e the phase constant (fi), i t i s p o s s i b l e to determine and s t o r e the A,B,C,D:arrays (Equations 4.49) f o r the t r a n s m i s s i o n l i n e . The t e r m i n a l v o l t a g e and current matrices (V^ and 1^) and the d i s t a n c e from the r e c e i v i n g end (NGLTH) where the f i r s t lumped component i s i n s e r t e d , are read i n t o the computer and s t o r e d . Prom t h i s data i t i s p o s s i b l e , u s i n g Equation 4.48, to determine the v o l t a g e s and currents along the multiconductor t r a n s m i s s i o n l i n e . The c a l c u l a t i o n s w i l l proceed towards the sending end u n t i l M times ILGTH equals NLGT1I. (M i s a counter w i t h i n the program to record the number of incremental c a l c u l a t i o n s performed around loop 2). When the program a r r i v e s at a d i s t a n c e NLGTH from the r e c e i v i n g end the computer w i l l bypass loop 2 and decide whether the sending end has been reached. Computations w i l l cease when N times ILGTH equals the t o t a l l ength of the l i n e , t a a t i s at the sending end. (N i s a c o u n t e r w i t h i n the program to r e c o r d the t o t a l number of c a l c u l a t i o n s along the l i n e ) . However, i f the sending end has not been reached (N.ILGTH ^ LGTH) the program w i l l add the necessary lumped components by proceeding around loop 1. A f t e r reading i n the i n f o r m a t i o n regarding the a d d i t i o n a l lumped component and NLGTH (the d i s t a n c e to the next component), the computer decides which type of lumped component has been added, determines the a p p r o p r i a t e A,B,C and'D arrays and c a l c u l a t e s the v o l t a g e s and c u r r e n t s on the sending end s i d e of the component by using one of the START \ READ LGTH, ^XlLGTH AND \SYSTEM DATA COMPUTE Z AND Y MATRICES COMPUTE AB CD MATRICES M = 0 i COMPUTE VOLTAGES AND CURRENTS COMPUTE J MATRIX COMPUTE V AND I ALONG LINE N = N + 1 M=M + I READ ADDITIONAL COMPONENT .AND NLGTH NO YES iWRITE AND PLOT SOLUTION STOP FIGURE 4 . 8 Computer Flow Chart 54 E q u a t i o n s 4.52, 4,54 o r 4,56. ' The v o l t a g e s and c u r r e n t s d e t e r m i n e d i n l o o p 1 ( F i g u r e 4.8) now become t h e t e r m i n a l c o n d i t i o n s V and I Li LI ( E q u a t i o n 4.48) when t h e c o m p u t e r r e t u r n s t o t h e main b o d y o f t h e p r o g r a m . The p r o c e d u r e i s r e p e a t e d u n t i l N t i m e s I L G T H e q u a l s L G T H , t h a t i s t h e p r o g r a m has r e a c h e d t h e s e n d i n g end o f t h e l i n e , a t w h i c h p o i n t t h e s o l u t i o n i s p r i n t e d , s c a l e d and p l o t t e d and t h e c o m p u t e r p r o g r a m i s t e r m i n a t e d . 4.7 Maximum A l l o w a b l e V o l t a g e D i f f e r e n c e When a t r a n s m i s s i o n l i n e has one c o n d u c t o r p e r p h a s e , i t c a n be assumed t h a t t h e p o t e n t i a l g r a d i e n t i s u n i f o r m a r o u n d t h e s u r f a c e o f t h e c o n d u c t o r p r o v i d e d t h e r a t i o o f t h e c o n d u c t o r d i a m e t e r t o t h e s p a c i n g b e t w e e n p h a s e s i s s m a l l . However, when s e v e r a l c o n d u c t o r s o f a m u l t i -c o n d u c t o r l i n e a r e i n c l o s e p r o x i m i t y t o e a c h o t h e r , a p p r e c i a b l e f i e l d d i s t o r t i o n may r e s u l t , a l l o w i n g much h i g h e r l i n e v o l t a g e s w i t h o u t e x c e e d i n g t h e d i s r u p t i v e c r i t i c a l i 4 ( 2 4 ) v o l t a g e . The f o l l o w i n g a n a l y s i s d e t e r m i n e s t h e maximum a l l o w a b l e v o l t a g e d i f f e r e n c e between any two c o n d u c t o r s o f a p a r t i c u l a r p h a s e w h i l e m a k i n g u s e o f t h e a s s u m p t i o n s (5) i n t r o d u c e d by S. C r a r y i n h i s work on b u n d l e d c o n d u c t o r s . 55 T h e s e a s s u m p t i o n s a r e : -(a ) C o n t r i b u t i o n t o t h e e l e c t r i c f i e l d i n t e n s i t y due t o t h e o t h e r p h a s e s a n d . t h e g r o u n d p l a n e i s n e g l e c t e d , ( b ) The e l e c t r i c f i e l d a s s o c i a t e d w i t h c h a r g e s on t h e o t h e r c o n d u c t o r s i n t h e same ph a s e i s u n i f o r m n e a r t h e c o n d u c t o r u n d e r c o n s i d e r a t i o n a n d ( c ) The movement o f c h a r g e s on each c o n d u c t o r t o e n s u r e t h a t t h e c o n d u c t o r s u r f a c e r e m a i n s an e q u i p o t e n t i a l i s n e g l e c t e d . T h i s phenomenon i s e x p l a i n e d a d e q u a t e l y i n R e f e r e n c e 22. I t i s assumed t h a t a l l t h e c h a r g e i s c o n c e n t r a t e d a t t h e c e n t r e o f a c y l i n d r i c a l c o n d u c t o r . hy F I G U R E 4.9 C r o s s - S e c t i o n o f a S y s t e m o f n P a r a l l e l C o n d u c t o r s 56 The e l e c t r i c f i e l d i n t e n s i t y at the p o s i t i o n of conductor 1 due to charges on conductors 2 , 3 , . . . . , n may be w r i t t e n E , (2E ) 2 + ( S E V ) 2 (4.57) where E ^ and E are the e l e c t r i c f i e l d i n t e n s i t y components i n the X and Y d i r e c t i o n s r e s p e c t i v e l y (See Figure 4.9). The e l e c t r i c f i e l d i n t e n s i t y at conductor 1 due to conductor n i s wh ere and E l n Qn 2 l x e o d l n (4.58) n = charge on conductor n, d ^ n = d i s t a n c e between conductors 1 and n. C q = p e r m i t t i v i t y of f r e e space. The x and y components of E ^ N are and E = E , cos 0, x l n In E = E , s i n 9, y l n l n (4.59a) (4.59b) where 0^ n i s the angle between the x a x i s and the e l e c t r i c f i e l d i n t e n s i t y v e c t o r at conductor 1. 5 7 Expansion of Equation 4 . 5 7 gives E x = 1 \ I ( Q 2 cos 0 1 2 + . . . + Q N cos 9 l n 2 +. 2itc o ! I U i n d l n Q2 sin 6 1 2 + ... + sin ) 2 ( 4 < 6 Q ) d m S i m i l a r l y En « l/Al COS enl + + Vl C0S °n,n-l V + N % VVdnl d n , n - l 0, sin 0 , + ... + Q , sin 0 , l 2 *1 n l *n-l n,n-l d , d , nl n,n-l (4.61) Now the maximum e l e c t r i c f i e l d intensity at the surface of an uncharged conductor 1 (considered to be a cylinder) placed in a previously uniform f i e l d E^ can be w r i t t e n ^ E m l = 2 E1 (4.62) If this conductor is given a charge , the maximum f i e l d intensity w i l l become E , = 2 E, + 81 (4.63) m l 1 2tt c R, o 1 58 S i m i l a r l y f o r t h e n t h c o n d u c t o r Eran = 2 E n + Qn ( 4 . 6 4 ) where 2 ire 11 o n R = t h e r a d i u s o f t h e n t h c o n d u c t o r n To e v a l u a t e t h e maximum p o t e n t i a l g r a d i e n t n e a r e a c h c o n d u c t o r f r o m E q u a t i o n 4 . 6 3 , i t w i l l be n e c e s s a r y t o d e t e r m i n e t h e c h a r g e on e a c h c o n d u c t o r f r o m E q u a t i o n 4 . 3 : -[v] = [p] [ p j ( E q u a t i o n 4 . 3 ) I f we p r e m u l t i p l y b o t h s i d e s by JjpJ we o b t a i n [ c ] = H " 1 M < 4 - 6 5 > To c o m p l e t e t h e s o l u t i o n i t i s n e c e s s a r y t o assume v a l u e s o f , ....,V n so t h a t t h e l i n e c h a r g e s c a n be c a l c u l a t e d , E v a l u a t i o n o f t h e maximum c o n d u c t o r p o t e n t i a l g r a d i e n t s f o r t h e c o n d u c t o r s t h e n f o l l o w s i m m e d i a t e l y f r o m E q u a t i o n 4 . 6 3 . D e t e r m i n a t i o n o f t h e maximum a l l o w a b l e v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s n e c e s s i t a t e s a g r a p h i c a l s o l u t i o n as d e m o n s t r a t e d i n A p p e n d i x B. S o l u t i o n s a r e g i v e n f o r b o t h a t w o - a n d a f o u r - c o n d u c t o r c a s e . 59 5. RESULTS OF ANALYSIS OF A 500 kV TRANSMISSION LINE WITH DISTRIBUTED COMPENSATION 5.1 D e s c r i p t i o n of the Transmission Line Systems Modern E1IV, long d i s t a n c e t r a n s m i s s i o n l i n e s of l a r g e c a p a c i t y have a v o l t a g e r a t i n g of around 500kV ( l i n e to l i n e ) . Although much grea t e r v o l t a g e r a t i n g s have been suggested ( S e c t i o n 5.5), i t w as decided to study a 500kV v o l t a g e l i n e so that a d i r e c t comparison could be made between e x i s t i n g s e r i e s compensated l i n e s and the proposed distributed-cornpensation system. Before the three-phase system was considered i n i t s e n t i r e t y , p r e l i m i n a r y c a l c u l a t i o n s were performed on two- and four-conductor, single-phase t r a n s m i s s i o n l i n e s with ground r e t u r n . For the two-conductor case the l i n e parameters were based on those suggested by the (14) (25) Westinghouse Engineers and Galloway. Conductor diameter 0.950 inches Conductor s i z e 556.6 MCM ACSR Conductor r e s i s t a n c e . . . . 0.168 ohms/mile Conductor spacing 1.5 f t . Average conductor height 50 f t . Transmission l i n e l ength 600 miles Both conductors are the same height above the ground (see Figure A.1) and h e l d i n a f i x e d p o s i t i o n r e l a t i v e to each other by i n s u l a t e d s p a c e r s . The above v a l u e s o f c o n d u c t o r d i a m e t e r and s p a c i n g have been s e l e c t e d f o r t h e i n i t i a l c a l c u l a t i o n o n l y . The o p t i m i z i n g p r o c e d u r e f o r d e t e r m i n i n g t h e l i n e p a r a m e t e r s i s g i v e n i n S e c t i o n 5.2 The t h r e e - p h a s e t r a n s m i s s i o n s y s t e m u s e d i n t h e c a l c u l a t i o n s , d e s c r i b e d i n S e c t i o n 5.4, c o n s i s t s o f : -P o u r c o n d u c t o r s p e r p h a s e C o n d u c t o r d i a m e t e r .....0.806 i n c h e s C o n d u c t o r s i z e 250 MCM ACSR C o n d u c t o r r e s i s t a n c e .....0.235 o h m s / m i l V e r t i c a l s p a c i n g b e t w e e n c o n d u c t o r s ...1.0 f o o t H o r i z o n t a l s p a c i n g 1.5 f e e t S p a c i n g b e t w e e n p h a s e s 40 f e e t A v e r a g e c o n d u c t o r h e i g h t 50 f e e t T r a n s m i s s i o n l i n e l e n g t h 600 m i l e s The p h a s e s were a t e q u a l h e i g h t s above t h e g r o u n d i n a f l a t a r r a y ( s e e F i g u r e 5 . 1 0 ) . W i t h i n e a c h p h a s e t h e two u p p e r as w e l l as t h e two l o w e r c o n d u c t o r s were c o n n e c t e d t o g e t h e r a t r e g u l a r i n t e r v a l s a l o n g t h e l i n e . The two c o n d u c t i v e g r o u p s were i n s u l a t e d f r o m e a c h o t h e r so t h a t t h e d i s t r i b u t e d c a p a c i t a n c e b e t w e e n them c o u l d be u s e d t o p r o d u c e a c o m p e n s a t i o n e f f e c t . The c o n d u c t o r d i a m e t e r and s i z e g i v e n above were s e l e c t e d f r o m (14 ) a m a n u f a c t u r e r ' s t a b l e by u s i n g t h e c o n d u c t o r c u r r e n t a t t h e r e c e i v i n g end as t h e s e l e c t i o n c r i t e r i o n . C o n d u c t o r s p a c i n g s have been s e t a r b i t a r i l y a t t h e above v a l u e s f o r t h e i n i t i a l c a l c u l a t i o n s . Optimum v a l u e s o f b o t h t h e d i a m e t e r and t h e c o n d u c t o r s p a c i n g s w i l l be d e t e r m i n e d i n S e c t i o n 5.3. 5.2 T w o - C o n d u c t o r S y s t e m I n i t i a l c a l c u l a t i o n s were p e r f o r m e d f o r a t w o -c o n d u c t o r s i n g l e - p h a s e t r a n s m i s s i o n l i n e w i t h e a r t h r e t u r n ( s e e F i g u r e 2.5) h a v i n g t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s a t t h e r e c e i v i n g end. L o a d v o l t a g e V, 288.7 kV (500 kV) V o l t a g e o f c o n d u c t o r 2, V 2 288.7 kV L o a d 250 MVA L o a d power f a c t o r 0.94 l a g g i n g The v o l t a g e d i f f e r e n c e b e t ween c o n d u c t o r s 1 and 2 a t t h e r e c e i v i n g end was a r b i t a r i l y s e t t o z e r o . C o n d u c t o r s p a c i n g f o r t h i s c a l c u l a t i o n was 1.5 f e e t . By s u b s t i t u t i o n o f t h e above b o u n d a r y c o n d i t i o n s and s y s t e m p a r a m e t e r s ( g i v e n i n S e c t i o n 5.1) i n t o t h e c o m p u t e r p r o g r a m d e s c r i b e d i n S e c t i o n 4.6, t h e v o l t a g e s and c u r r e n t s a l o n g t h e l i n e were d e t e r m i n e d . T h e s e r e s u l t s a r e p l o t t e d i n F i g u r e 5.1 w h i l e t h e s a l i e n t p o i n t s may be s u m m a r i s e d as f o l l o w s Power a n g l e ( 6 ) .48.4° S e n d i n g end p h a s e a n g l e 41.6° l e a d i n g Maximum v o l t a g e b e t w e e n c o n d u c t o r s a t any p o i n t a l o n g t h e l i n e 177 kV V 1 s e n d i n g end 257 kV Vg s e n d i n g end 426 kV 62 DISTANCE FROM SENDING END (MILES) FIGURE 5.1(b) C u r r e n t P r o f i l e s A l o n g t h e L i n e ( T w o - C o n d u c t o r C a s e ) 63 374 amps 393 amps The r e s u l t o f p r i m a r y i n t e r e s t i s t h e maximum v o l t a g e d i f f e r e n c e o f 177 kV b e t w e e n t h e two c o n d u c t o r s . By e x a m i n a t i o n o f F i g u r e B.2 i t i s c l e a r t h a t i f t h e v o l t a g e V e x c e e d s a p p r o x i m a t e l y 290 kV, t h e n t h e c o r o n a l o s s e s w i l l be e x c e s s i v e w h e n e v e r t h e v o l t a g e b e tween c o n d u c t o r s i s g r e a t e r t h a n z e r o . T h e r e f o r e , t h e above s y s t e m i s n o t s a t i s f a c t o r y as b o t h t h e v o l t a g e (426 kV) and t h e maximum v o l t a g e d i f f e r e n c e (177 kV) e x c e e d t h e l i m i t a t i o n s . To c o r r e c t t h e s i t u a t i o n s e v e r a l methods c a n be u s e d : -1. The v o l t a g e d i f f e r e n c e b e t ween c o n d u c t o r s may be r e d u c e d by i n s t a l l i n g l umped p a r a l l e l c a p a c i t o r s a t r e g u l a r i n t e r v a l s a l o n g t h e l i n e , 2. The v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s may a l s o be r e d u c e d by d e c r e a s i n g t h e c o n d u c t o r s p a c i n g and i n c r e a s i n g t h e c o n d u c t o r d i a m e t e r and 3. The maximum a l l o w a b l e v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s may be i n c r e a s e d by v a r y i n g t h e c o n d u c t o r s p a c i n g and i n c r e a s i n g t h e c o n d u c t o r d i a m e t e r . Now a s t u d y o f t h e above methods must i n c l u d e a l l o w a n c e f o r o b t a i n i n g maximum c o m p e n s a t i o n . I n o t h e r w o rds any method u s e d t o overcome t h e v o l t a g e l i m i t a t i o n s must a l s o t e n d t o r e d u c e t h e power a n g l e o f t h e t r a n s m i s s i o n s y s t e m . I s e n d i n g end I s e n d i n g end 6 4 Method 1 given above may be used to reduce the vo l t a g e g r a d i e n t at the conductor surface by reducing the v o l t a g e d i f f e r e n c e between conductors (Figure 5.3). How-ever, t h i s method w i l l a l s o i n c r e a s e the power angle 6, as shown i n the f i g u r e , because the c u p a c i t i v e c u r r e n t flowing between conductors has been reduced. Although the c a p a c i t -ance between conductors has been in c r e a s e d by adding lumped p a r a l l e l c a p a c i t o r s i t .does not r e s u l t i n an o v e r a l l i n c r e a s e i n c a p a c i t i v e c u r r e n t . A p p l i c a t i o n of Method 2 to the t r a n s m i s s i o n system produces the d e s i r e d r e d u c t i o n of the vo l t a g e d i f f e r e n c e while the power angle remains constant. I t w i l l be shown i n Se c t i o n 5.3.3 that a l a r g e v o l t a g e d i f f e r e n c e between conductors w i l l g ive the best compensation. To maximize the allowable v o l t a g e d i f f e r e n c e between conductors and hence maximize the system compensation, Method 3 may be used. By combining both Methods 2 and 3 (Figure 5.2) the optimum system parameters may be found such that the system w i l l be operating at maximum allowable v o l t a g e d i f f e r e n c e . This maximum allowable v o l t a g e d i f f e r e n c e w i l l g i v e the maximum p o s s i b l e compensation (minimum power angle) while s a t i s f y i n g the vo l t a g e g r a d i e n t c o n d i t i o n at t h e conductor s u r f a c e f o r a l l p o i n t s along the l i n e . The optimum values obtained from Figure 5.2 are given i n Table 5.1 Conductor Conductor Max. Voltage Spacing (Ft.) Diam. (Ins.) (kV) 1.5 2. 55 130 1.0 2. 50 110 0.5 2.40 78 TABLE 5.1 Two-Conductor Line Parameters Reference to the manufacturers t a b l e f o r expanded conductors (Reference 14, pp.. 50-51) shows that the l a r g e s t expanded conductor made to date has a diameter of approximately 2.1 inches. I t would t h e r e f o r e be necessary to reduce the v o l t a g e g r a d i e n t at the conductor s u r f a c e to a value corresponding to a p r a c t i c a l conductor diameter. A common method of e f f e c t i v e l y reducing the vo l t a g e g r a d i e n t i s to use " s p l i t " or "bundled" conductors. T h i s i s d i s c u s s e d : i n S e c t i o n 5.3. 6 6 CONDUCTOR DIAMETER (INCHES) FIGURE 5.2 Conductor Diameter and Maximum Voltage D i f f e r e n c e (Two-Conductor Case) o o Q\ i i : 1 1 '30 0-0 50 10-0 150 20-0 25-0 PARALLEL CAPACITANCE (JJF) FIGURE 5.3 I n s t a l l a t i o n of P a r a l l e l C a p a c i t o r s Along the Line at 50 Mile I n t e r v a l s (Two-Conductor Case) 67 5.3 P o u r - C o n d u c t o r S y s t e m 5.3.1 I n i t i a l I n v e s t i g a t i o n s The b o u n d a r y c o n d i t i o n s assumed i n S e c t i o n 5.2 were a g a i n u s e d as t h e l o a d c o n d i t i o n s a t t h e r e c e i v i n g end, e x c e p t t h a t t h e l o a d was now d i v i d e d b e t w e e n two c o n d u c t o r s i n p a r a l l e l s u c h t h a t t h e l o a d p e r c o n d u c t o r was 125 MVA a t a l a g g i n g power f a c t o r o f 0.94 (minus 20 d e g r e e s ) . F o r t h e i n i t i a l c a l c u l a t i o n t h e h o r i z o n t a l and v e r t i c a l s p a c i n g s •were assumed t o be 1.5 f e e t and 1.0 f o o t r e s p e c t i v e l y , as shown i n F i g u r e B.3. The r e s u l t s o f t h e i n i t i a l c a l c u l a t i o n a r e p l o t t e d i n F i g u r e 5.4 and may be s u m m a r i s e d as f o l l o w s : -Power a n g l e (6) 49° S e n d i n g end p h a s e a n g l e 41° ( l e a d i n g ) Maximum v o l t a g e d i f f e r e n c e ( V g - V ^ ) . . . 8 7 kV V 1 and V 2 s e n d i n g end 262 kV and s e n d i n g end 344 kV 1^  and I 2 s e n d i n g end 215 Amps Ig and 1^ s e n d i n g end 219 Amps A d v a n t a g e s o v e r t h e t w o - c o n d u c t o r s y s t e m i n c l u d e : -1. The maximum v o l t a g e d i f f e r e n c e b e t w e e n any two c o n d u c t o r s has been r e d u c e d by 100 kV w i t h o u t s i g n i f i c -a n t i n c r e a s e o f power a n g l e 6. The c o n d u c t o r s u r f a c e v o l t a g e g r a d i e n t has b e e n r e d u c e d b o t h by t h i s r e d u c t i o n i n v o l t a g e and by t h e i n c r e a s e i n t h e number o f c o n d u c t o r s ( s e e A p p e n d i x B ) . 68 FIGURE 5.4( a ) V o l t a g e P r o f i l e s A l o n g t h e L i n e ( F o u r - C o n d u c t o r C a s e ) 0 100 200 300 400 500 DISTANCE FROM SENDING END (MILES) FIGURE 5.4(b) C u r r e n t P r o f i l e s A l o n g t h e L i n e ( F o u r - C o n d u c t o r C a s e ) 6 9 2. An i n c r e a s e i n t h e number o f c o n d u c t o r s h a s an e f f e c t s i m i l a r t o i n c r e a s i n g t h e c o n d u c t o r d i a m e t e r s . However, by u t i l i z i n g a l a r g e r number o f c o n d u c t o r s i t i s p o s s i b l e to keep t h e c o n d u c t o r d i a m e t e r w i t h i n p r a c t i c a l l i m i t s . The o n l y d i s a d v a n t a g e a p p e a r s t o be a s l i g h t i n c r e a s e i n t h e power a n g l e c a u s e d b y t h e d e c r e a s e o f t h e v o l t a g e d i f f e r e n c e . However, t h i s d i s a d v a n t a g e may be overcome by m a x i m i z i n g t h e v o l t a g e d i f f e r e n c e , b e t w e e n c o n d u c t o r s as d i s c u s s e d i n S e c t i o n 5.3.2. 5.3.2 O p t i m i z i n g t h e C o n d u c t o r D i a m e t e r and S p a c i n g s I n o r d e r t o o b t a i n maximum c o m p e n s a t i o n ( i . e . minimum power a n g l e ) t h e maximum a l l o w a b l e v o l t a g e must e x i s t b e t w e e n t h e two u p p e r and t h e two l o w e r c o n d u c t -o r s . T h i s maximum a l l o w a b l e v o l t a g e w i l l be l i m i t e d by t h e c o n d u c t o r s u r f a c e v o l t a g e g r a d i e n t c o n d i t i o n as d i s c u s s e d i n S e c t i o n 4.7 and e v a l u a t e d i n A p p e n d i x B. The r e l a t i o n s h i p between t h e s y s t e m c o m p e n s a t i o n (power a n g l e ) , t h e c a p a c i t i v e c u r r e n t and t h e v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s w i l l become a p p a r e n t i n S e c t i o n 5.3.3. By a p p l i c a t i o n o f t h e e q u a t i o n s g i v e n i n S e c t i o n 4.7 i t i s p o s s i b l e t o o p t i m i z e t h e c o n d u c t o r s p a c i n g s t o o b t a i n t h e maximum v o l t a g e d i f f e r e n c e . The maximum a l l o w a b l e v o l t a g e d i f f e r e n c e and t h e r e q u i r e d c o n d u c t o r s p a c i n g s may be d e t e r m i n e d f r o m F i g u r e 5.5. 70 ki k 40 ki CD S o 1 30-20 10 0-0 DIAM = 0-606 INS = 320 KV 0-5 1-0 1-5 2-0 HORIZONTAL SPACING (ft) 2-5 FIGURE 5.5 O p t i m i z a t i o n o f C o n d u c t o r S p a c i n g s ki o 5: ki cr. ki k k Q ki to o 140 1 120 106 100 60 0-25 HORIZONTAL SPACING 7-0(ft) VERTICAL SPACING 2-0 (ft) 0-50 0-75 1-00 1-25 Y35 CONDUCTOR DIAMETER (INCHES) 1-50 FIGURE 5.6 O p t i m i z a t i o n o f C o n d u c t o r D i a m e t e r 71 Figure 5.5 has been d e r i v e d by determining the maximum all o w a b l e v o l t a g e d i f f e r e n c e at a p a r t i c u l a r h o r i z o n t a l and v e r t i c a l conductor spacing using the vo l t a g e g r a d i e n t at the conductor s u r f a c e as the l i m i t i n g c r i t e r i o n (Appendix B). Both the h o r i z o n t a l and v e r t i c a l spacings were v a r i e d from 0.5 to 2.5 f e e t as shown i n t h e f i g u r e . Figure 5.5 shows t h a t the spacings r e q u i r e d f o r the maximum p e r m i s s i b l e v o l t a g e d i f f e r e n c e are:-H o r i z o n t a l spacing 1.0 foot V e r t i c a l spacing 2.0 f e e t With these spacings, the maximum p e r m i s s i b l e v o l t a g e d i f f e r e n c e between conductors i s 36 kV. Our aim at t h i s p o i n t i s to use every p o s s i b l e means to i n c r e a s e the maximum allowable v o l t a g e d i f f e r e n c e between conductors to maximise the compensation e f f e c t (that i s to reduce the power angle 6). R e f e r r i n g again to S e c t i o n 4.7, i t i s obvious that the conductor diameter £>lays an important p a r t i n the e v a l u a t i o n of the conductor surface v o l t a g e g r a d i e n t . In f a c t , a s the diameter i n c r e a s e s the vo l t a g e g r a d i e n t w i l l decrease (Equation 4.63), so th a t the maximum all o w a b l e v o l t a g e d i f f e r e n c e between conductors may be in c r e a s e d . T h i s i s shown g r a p h i c a l l y i n Figure 5.6 f o r values of V^ and V 4 v a r y i n g from 280 kV to 350 kV, where and V 4 are the r e s p e c t i v e v o l t a g e s of conductors 3 and 4 (see Figure B. 3) . As the diameter of the conductors i s i n c r e a s e d the maximum vo l t a g e d i f f e r e n c e between conductors at any po i n t along the l i n e w i l l decrease. T h i s curve i s a l s o p l o t t e d i n Figure 5.6. The minimum conductor diameter s a t i s f y i n g the maximum v o l t a g e g r a d i e n t c o n d i t i o n at a l l p o i n t s along the l i n e i s determined by the i n t e r s e c t i o n of the two curves i n the f i g u r e . Under the load t e s t , v o l t a g e s and do not exceed 350 kV (Figure 5.4(a)) so the minimum diameter w i l l be 1.35 inches. For t h i s diameter i t i s p o s s i b l e to have a max-imum of 106 kV between any two conductors. I f the v o l t a g e at any p o i n t along the l i n e exceeds t h i s value corona discharge w i l l commence. (This c r i t i c a l v o l t a g e w i l l be lower f o r moist a i r ) . The question now a r i s e s whether an i n c r e a s e i n conductor diameter w i l l e f f e c t the curves of Figure 5.5 and hence a l t e r the conductor spacings. On i n v e s t i g a t i n g .this q uestion i t has been found t h a t , as the conductor diameter i s increased, the curves of Fig u r e 5.5 are d i s p l a c e d v e r t i c a l l y but s t i l l r e t a i n the same r e l a t i v e p o s i t i o n to each other and approximately the same shape. Therefore, the maximum vo l t a g e g r a d i e n t w i l l s t i l l occur at a v e r t i c a l spacing of 2.0 f e e t and a h o r i z o n t a l spacing of 1.0 f o o t . A second i n v e s t i g a t i o n was performed f o r the four-conductor l i n e with a conductor diameter of 1.35 inches and with v e r t i c a l and h o r i z o n t a l spacings of 2.0 f e e t and 73 600r 0 100 200 300 400 500 DISTANCE FROM SENDING END (MILES) 600 FIGURE 5.7(a) V o l t a g e P r o f i l e s Along the L i n e (Four-Conductor O p t i m i z e d Case) 100 DISTANCE 200 FROM 300 SENDING 400 END 500 (MILES) FIGURE 5.7(b) C u r r e n t P r o f i l e s A long the L i n e (Four-Conductor O p t i m i z e d Case) 1.0 f o o t r e s p e c t i v e l y . The v o l t a g e d i f f e r e n c e a t t h e r e c e i v -i n g end was s e t t o 106 kV i n p h a s e w i t h t h e l o a d v o l t a g e and Y .. 4 The r e s u l t s o f t h i s i n v e s t i g a t i o n a r e shown g r a p h i c a l l y i n F i g u r e 5.7 and may be s u m m a r i s e d as f o l l o w s : -Power a n g l e (6) 41.1° S e n d i n g end p h a s e a n g l e 48.9° ( l e a d i n g ) Maximum v o l t a g e d i f f e r e n c e 106 kV V1 and V 2 s e n d i n g end 268 kV Vg a nd s e n d i n g end . . .' 341 kV 1^ and I 2 s e n d i n g end 456. Amps and I s e n d i n g end 146 Amps 5.3.3 Use o f Lumped P a r a m e t e r s As i n d i c a t e d by t h e above r e s u l t s , o n l y a s l i g h t i m p r o v e m e n t has been o b t a i n e d by o p t i m i z i n g t h e c o n d u c t o r s p a c i n g s and d i a m e t e r . The p r e s e n t s y s t e m i s s t i l l u n s a t i s -f a c t o r y b e c a u s e o f t h e f o l l o w i n g r e a s o n s : -1. Power a n g l e i s s t i l l q u i t e l a r g e t 2. S e n d i n g end p h a s e a n g l e i s a l a r g e l e a d i n g a n g l e a n d 3. The c u r r e n t s i n c o n d u c t o r s 3 and 4 ( 1 ^ and 1^, s e e F i g u r e B.3) have been r e d u c e d , b u t n o t s i g n i f i c a n t l y . I n an a t t e m p t t o overcome t h e s e o b s t a c l e s , F i g u r e 5 . 8 ( a ) w i l l be u s e d as an a i d t o g a i n i n s i g h t i n t o t h e p r o b -l e m and h o p e f u l l y l e a d t o p o s s i b l e s o l u t i o n s . F i g u r e 5.8 ( a ) i l l u s t r a t e s t h e l o c u s p l o t s o f t h e v o l t a g e and c u r r e n t v e c t o r s a l o n g t h e l i n e . The change i n a v e c t o r b e t ween numbered p o i n t s on t h e l o c u s i s t h e change e x p e r i e n c e d CURRENT SCALE: 1" =200 AMPS >10 VOLTAGE SCALE: 7" = 700 KV FIGURE 5.8(a) L o c u s P l o t o f V o l t a g e s and C u r r e n t s A l o n g t h e L i n e FIGURE 5.8(b) Lumped P a r a m e t e r S e c t i o n o f T w o - C o n d u c t o r L i n e 76 by t h a t p a r t i c u l a r v e c t o r o v e r a d i s t a n c e o f 50 m i l e s . V e c t o r s have been drawn f o r t h e change e x p e r i e n c e d by e a c h v e c t o r f r o m t h e 100 m i l e t o t h e 150 m i l e p o i n t . A l t h o u g h t h e f i g u r e has been drawn f o r t h e t w o - c o n d u c t o r c a s e d e s c r i b e d i n S e c t i o n 5.2, t h e s i m p l i f -i c a t i o n i s j u s t i f i e d as t h e u n d e r l y i n g p r i n c i p l e s i n b o t h t h e two and f o u r - c o n d u c t o r c a s e s a r e t h e same. ( N o t e : The c u r r e n t and t h e v o l t a g e V^ i n t h e f i g u r e a r e e q u i v a l e n t t o t h e c u r r e n t s and v o l t a g e s o f c o n d u c t o r s 3 and 4 f o r t h e f o u r - c o n d u c t o r c a s e . ) The s y m b o l s u s e d i n t h e l o c u s p l o t a r e d e f i n e d i n F i g u r e 5.8(b) w h i c h i l l u s t r a t e s a 50 m i l e s e c t i o n o f t h e l i n e . I n t h i s f i g u r e i t has b e e n assumed t h a t t h e d i s t r i b u t e d p a r a m e t e r s may be lumped i n t o t h e c i r c u i t p a r a m e t e r s shown. P o s s i b l e methods t o overcome t h e above o b s t a c l e s may now b^ Pound f r o m F i g u r e 5 . 8 ( a ) . R e d u c t i o n o f t h e l a r g e power a n g l e (5) may be a t t a i n e d by r e d u c i n g t h e v o l t a g e component 1^ Z^^ ( F i g u r e 5 . 8 ( a ) ) . The power a n g l e i s t h e a n g l e b e t w e e n t h e v o l t a g e V 9 a t t h e r e c e i v i n g end ( l o a d v o l t a g e ) and t h e v o l t a g e V ^ 1 a t t h e s e n d i n g end ( g e n e r a t o r v o l t a g e ) . I t i s c l e a r f r o m t h e f i g u r e t h a t t h e power a n g l e w i l l i n c r e a s e as t h e component I ^ Z ^ i n c r e a s e s . Hence 1^ must be r e d u c e d i f p o s s i b l e . Two~ i m m e d i a t e methods s u g g e s t t h e m s e l v e s : r e d u c t i o n o f Z^ Q, and r e d u c t i o n o f 1^. By r e f e r e n c e t o E q u a t i o n 4 . 8 ( b ) , r e d u c t i o n o f Z-^ 1 S p o s s i b l e i f t h e s p a c i n g b e t w e e n c o n d u c t o r s 1 and 2 ( b , 0 ) i s d e c r e a s e d o r t h e h e i g h t a bove g r o u n d ( a p p r o x i m a t e l y ^ r s / 2 ) o f t h e t r a n s m i s s i o n l i n e i s i n c r e a s e d . Such c h a n g e s w o u l d be u n s a t i s f a c t o r y b e c a u s e t h e c o n d u c t o r s p a c i n g s have been f i x e d by t h e v o l t a g e g r a d i e n t c o n d i t i o n a t t h e c o n d u c t o r s u r f a c e , and i n c r e a s i n g c o n d u c t o r h e i g h t s w i l l i n c r e a s e l i n e c o s t s . G r a d u a l r e d u c t i o n o f I g may be made p o s s i b l e b y i n c r e a s i n g t h e m a g n i t u d e o f I c o r by c h a n g i n g i t s p h a s e a n g l e t o a p p r o x i m a t e l y z e r o d e g r e e s ( ~ I C w i l l t h e n have a p h a s e a n g l e o f 180°). The m a g n i t u d e o f I c i s d e p e n d e n t upon t h e v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s 1 and 2 (Y^g b e i n g f i x e d by t h e c o n d u c t o r s s p a c i n g ) , T h i s v o l t a g e d i f f e r e n c e has a l r e a d y been m a x i m i z e d , so t h a t t h e a l t e r n a t -i v e o f c h a n g i n g t h e p h a s e a n g l e o f 1^ must be c o n s i d e r e d . To v a r y t h e p h a s e a n g l e o f - I c i t i s n e c e s s a r y t o v a r y t h e p h a s e a n g l e o f t h e v o l t a g e d i f f e r e n c e b e t ween c o n d u c t o r s . The v o l t a g e d i f f e r e n c e i s i n t u r n d e p e n d e n t upon t h e p h a s e a n g l e b e t w e e e n V^ and . V a r y i n g t h e p h a s e a n g l e o f V j ) i p p U P ^° ^5° l a g g i n g , a t t h e s e n d i n g end, was i n f a c t f o u n d t o p r o d u c e a s m a l l e r power a n g l e . B u t , s i n c e t h e p h a s e r e l a t i o n s h i p c h a n g e d between V^ and V^,so t o o d i d t h e m a g n i t u d e o f t h e v o l t a g e d i f f e r e n c e . U n f o r t u n a t e l y the l a t t e r v o l t a g e e x c e e d e d i t s p e r m i s s i b l e v a l u e and so t h e i n v e s t i g a t i o n was d i s c o n t i n u e d . 78 The use of p a r a l l e l lumped c a p a c i t o r s i n s t a l l e d between conductors at 50 mile i n t e r v a l s along the l i n e has been d i s c u s s e d i n S e c t i o n 5.2. T h e i r use has been proved i n e f f e c t i v e . Lumped s e r i e s r e a c t o r s i n s t a l l e d i n l i n e 1 (see Fi g u r e 4.6) have proved u n s u c c e s s f u l as they i n c r e a s e the power angle and the maximum vo l t a g e d i f f e r e n c e of the system by i n c r e a s i n g the s e l f impedance of that conductor. V/hen the t r a n s m i s s i o n l i n e l e n g t h exceeds 200 miles, leakage c u r r e n t to ground becomes s u b s t a n t i a l . To compensate f o r t h i s l a r g e leakage c u r r e n t , p a r a l l e l r e a c t o r s must be i n s t a l l e d along the l i n e . U n f o r t u n a t e l y when p a r a l l e l r e a c t o r s are added to the system the magnitude of the v o l t a g e at the sending end (generation v o l t a g e ) must be i n c r e a s e d to such an extent t h a t i t w i l l impose a major r e g u l a t i o n problem. Figure 5.9 has been p l o t t e d f o r the a d d i t i o n of p a r a l l e l r e a c t o r s to l i n e 1. Two advantages are evident from t h i s f i g u r e : the power angle i s reduced s u b s t a n t i a l l y and so too i s the sending end power f a c t o r angle. Two of the three o b s t a c l e s have been d i s c u s s e d so f a r . The remaining o b s t a c l e i s that of reducing 1^ and 1^ ( i g i n the two-conductor l i n e ) . I t i s c l e a r t h a t i f t h i s c urrent i s not reduced to zero, an a d d i t i o n a l generator w i l l need to be connected to the system at the sending end so that the r e c e i v i n g end c o n d i t i o n s (boundary c o n d i t i o n s ) of the system can be main-t a i n e d . 79 FIGURE 5.9 I n s t a l l a t i o n o f A l o n g t h e L i n e P a r a l l e l R e a c t o r s 80 O n l y one method h a s been f o u n d s u c c e s s f u l i n r e d u c i n g t h e c u r r e n t i n l i n e s 3 and 4; t h a t i s i n s t a l l i n g s e r i e s r e a c t o r s i n l i n e s 3 and 4 t o i n c r e a s e t h e s e l f i m p e d -a n c e o f t h e s e c o n d u c t o r s a t t h e s e n d i n g end. U n f o r t u n a t e l y , i n s t a l l a t i o n o f s e r i e s r e a c t o r s i n c r e a s e d t h e maximum v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s t o s u c h an e x t e n t t h a t t h e max-imum p e r m i s s i b l e v o l t a g e d i f f e r e n c e b e t w e e n c o n d u c t o r s was e x c e e d e d . T h e r e f o r e , t h i s method was a l s o u n s a t i s f a c t o r y . I n c o n c l u s i o n , t h e t h r e e d i s a d v a n t a g e s l i s t e d on Page 74 c a n n o t be c o r r e c t e d w i t h o u t i n t r o d u c i n g o v e r v o l t a g e s w h i c h e x c e e d t h e l i m i t a t i o n s o f t h e s y s t e m . 5.4 T h r e e - P h a s e S y s t e m The t h r e e - p h a s e s y s t e m s t u d i e d u s e s t h e p h y s i c a l d a t a g i v e n i n S e c t i o n 5.1 and t h e p a r a m e t e r s d e t e r m i n e d i n S e c t i o n 5.3 ( s e e F i g u r e 5.10). The v a l u e s a r e s u m m a r i s e d as f o l l o w s : -P o u r c o n d u c t o r s p e r p h a s e C o n d u c t o r d i a m e t e r 1.35 i n c h e s V e r t i c a l s p a c i n g 2.0 f e e t H o r i z o n t a l s p a c i n g 1.0 f o o t S p a c i n g b e t ween p h a s e s 40.0 f e e t A v e r a g e c o n d u c t o r h e i g h t 50.0 f e e t T r a n s m i s s i o n l i n e l e n g t h 600 m i l e s The c o m p u t e r p r o g r a m d e s c r i b e d i n S e c t i o n 4.6 was w r i t t e n so t h a t i t c o u l d be u s e d f o r a t h r e e - p h a s e s y s t e m as w e l l as t h e s i n g l e - p h a s e m u l t i c o n d u c t o r s y s t e m s d i s c u s s e d i n S e c t i o n s 5.2 and 5.3. S u b s t i t u t i o n o f t h e above d a t a i n t h e p r o g r a m gave t h e f o l l o w i n g r e s u l t s ; -Power a n g l e ( 6 ) 36.6° S e n d i n g end p h a s e a n g l e 53.4° ( l e a d i n g ) Maximum v o l t a g e b e t w e e n 106 kV c o n d u c t o r s V x and V g s e n d i n g end 23 5 /36.6° kV V 3 and V 4 s e n d i n g end 30 5 /51.8°kV 1^ and s e n d i n g end ........ 514.6/93.4°Amps I„ and I . s e n d i n g end 166.6 /31.3°Amps The above r e s u l t s a r e g i v e n f o r one p h a s e o f t h e t h r e e - p h a s e s y s t e m . V o l t a g e s and c u r r e n t s o f t h e two r e m a i n i n g p h a s e s a r e o b t a i n e d by d i s p l a c i n g t h e v e c t o r s b y 120 and 240 d e g r e e s . The m a g n i t u d e s w i l l r e m a i n c o n s t a n t as t h e s y s t e m i s assumed t o be c o m p l e t e l y t r a n s -p o s e d ( s e e S e c t i o n 3 . 2 ) . A l t h o u g h some o f t h e above r e s u l t s may be i m p r o v e d by t h e a d d i t i o n o f p a r a l l e l r e a c t o r s on l i n e s 1 and 2, f o r t h e f o u r - c o n d u c t o r c a s e ( s e e F i g u r e 5.9) t h e c u r r e n t s I„ and I . w i l l r e m a i n c o n s t a n t . 3 4 I t a p p e a r s t h a t any f u r t h e r i n v e s t i g a t i o n s h o u l d be p o s t p o n e d u n t i l a method has been f o u n d t o ease t h e v o l t a g e g r a d i e n t l i m i t a t i o n on t h e s y s t e m . FIGURE 5.10 Three-Phase System Above P e r f e c t Ground 83 5.5 P h y s i c a l L i m i t a t i o n s The l i m i t a t i o n p r e v e n t i n g a s u c c e s s f u l c o n c l u s -i o n t o t h e p r o b l e m i s t h e c o n d u c t o r s u r f a c e v o l t a g e g r a d i e n t l i m i t a t i o n d e s c r i b e d i n S e c t i o n 4.7 and e v a l u a t e d i n A p p e n d i x B. I f t h e p o t e n t i a l g r a d i e n t a t t h e s u r f a c e o f e a c h c o n d u c t o r c o u l d be i n c r e a s e d above 3000 kV/meter, i t may be p o s s i b l e t o r e a c h a s a t i s f a c t o r y s o l u t i o n . The v o l t a g e g r a d i e n t a t t h e c o n d u c t o r s u r f a c e may be i n c r e a s e d s u b s t a n t i a l l y by i n c r e a s i n g t h e r e l a t i v e p e r m i t t i v i t y o f t h e i n s u l a t i n g m e d i a . However, t h i s p r o v e d u n s u c c e s s f u l b e c a u s e o f t h e l a r g e i n c r e a s e i n t h e s y s t e m power a n g l e . F o r example, i f t h e r e l a t i v e p e r m i t t i v i t y i s i n c r e a s e d t o 2.5 ( a t y p i c a l v a l u e f o r r u b b e r ) t h e power a n g l e w i l l i n c r e a s e t o 118° w h i c h i s i n e x c e s s o f t h e s t a b l e r e g i o n f o r t h e s y s t e m . P r e s e n t day t r e n d s i n EIIV power t r a n s m i s s i o n a r e f o r e v e r - i n c r e a s i n g l i n e v o l t a g e s ; t h e h i g h e s t v o l t a g e s u g g e s t e d . t o d a t e i s 1500 kV, l i n e t o l i n e , The q u e s t i o n may now a r i s e w h e t h e r d i s t r i b u t e d c o m p e n s a t i o n w i l l be a d v a n t a g e o u s f o r s u c h l i n e s . The m a j o r d i s a d v a n t a g e w i l l o b v i o u s l y be e x c e s s i v e v o l t a g e g r a d i e n t s a t t h e c o n d u c t o r s u r f a c e s . I n d e e d as t h e l i n e v o l t a g e i n c r e a s e s t h i s p r o b l e m w i l l become more a c u t e , a s i s shown i n F i g u r e 5.11 f o r o n e - p h a s e c o n s i s t i n g o f f o u r c o n d u c t o r s o f 2.0 i n c h e s i n d i a m e t e r . The c o n d u c t o r s p a c i n g s f o r t h e f i g u r e w ere t a k e n t o be 1.0 f o o t h o r i z o n t a l and 2.0 f e e t v e r t i c a l . 84 LINE VOLTAGE (KV) CONDUCTOR DIAM 2-0 INCHES CONDUCTOR SPACING 1-0* HORIZONTAL 2-0 VERTICAL FIGURE 5.11 E f f e c t s o f I n c r e a s e d L i n e V o l t a g e on t h e Maximum V o l t a g e D i f f e r e n c e 85 F o r t h e above c o n d u c t o r s p e c i f i c a t i o n t h e maximum l i n e v o l t a g e p o s s i b l e i s 84 5 kV t o g r o u n d o r a p p r o x i m a t e l y 1450 kV l i n e t o l i n e . A t t h i s l i n e v o l t a g e t h e v o l t a g e d i f f e r e n c e b e t ween c o n d u c t o r s must be z e r o , so t h a t d i s t r i b u t -ed c o m p e n s a t i o n w o u l d n o t be f e a s i b l e a t t h i s h i g h v o l t a g e . A l l o w i n g a v o l t a g e d i f f e r e n c e o f 106 kV b e t w e e n c o n d u c t o r s w o u l d g i v e a l i n e t o g r o u n d v o l t a g e o f 575 kV o r a l i n e t o l i n e v o l t a g e o f 1000 kV. L i n e v o l t a g e s above 1000 kV c o u l d be made p o s s i b l e by i n c r e a s i n g t h e number o f c o n d u c t o r s p e r b u n d l e and e x p a n d i n g t h e c o n d u c t o r d i a m e t e r t o r e d u c e t h e c o n d u c t o r s u r f a c e v o l t a g e g r a d i e n t . 5.6 C o m p a r i s o n o f S e r i e s and " D i s t r i b u t e d C o m p e n s a t i o n The r e s u l t s o b t a i n e d i n t h i s c h a p t e r may be s u m m a r i s e d by c o m p a r i s o n o f t h e two s y s t e m s o f c o m p e n s a t i o n . F o r 500 kV s y s t e m s and s i m i l a r l o a d c o n d i t i o n s t h e f o l l o w i n g power a n g l e s were o b t f . i n e d : -T y p i c a l s e r i e s c o m p e n s a t i o n s y s t e m 28° D i s t r i b u t e d c o m p e n s a t i o n s y s t e m ...37° V a r i a t i o n o f t h e power a n g l e due t o a l o a d c hange on t h e s e r i e s c o m p e n s a t i o n s y s t e m may be c o r r e c t e d b y i n c r e a s i n g o r d e c r e a s i n g t h e v a l u e o f t h e s e r i e s c a p a c i t o r as r e q u i r e d ( s e e S e c t i o n 2 . 2 ) . However, f o r t h e d i s t r i b u t e d s y s t e m t h e p r o b l e m i s more d i f f i c u l t . I t has been f o u n d f o r t h i s s y s t e m t h a t t h e power a n g l e i n c r e a s e s w i t h t h e s y s t e m l o a d , b u t i n t h i s c a s e t h e r e a r e no p a r a m e t e r s t h a t can be r e a d i l y v a r i e d t o c o r r e c t t h e . s i t u a t i o n . 86 6. CONCLUSION .A m u l t i c o n d u c t o r d i s t r i b u t e d - c o m p e n s a t i o n t r a n s -m i s s i o n l i n e has been a n a l y s e d u s i n g a m a t r i x method b a s e d on M a x w e l l ' s p o t e n t i a l and i n d u c t a n c e c o e f f i c i e n t s . I t has b e en shown t h a t , f o r t h e t h r e e - p h a s e c a s e a t power f r e q u e n c i e s , t h e s y s t e m c a n be assumed t o be l o s s l e s s . T h i s h a s a l l o w e d a c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e a n a l y s i s . The s y s t e m v o l t a g e and c u r r e n t e q u a t i o n s have b e e n d e r i v e d f o r a m u l t i c o n d u c t o r t r a n s m i s s i o n l i n e and w r i t t e n so t h a t any number o f c o n d u c t o r s may be c o n s i d e r e d . S i m p l i f i c a t i o n o f t h e a n a l y s i s . w a s made p o s s i b l e by t h e u s e o f A,B,C,D p a r a m e t e r s and p a r t i t i o n e d m a t r i c e s . The p o s s i b l e a d d i t i o n o f lumped e l e m e n t s a t any p o i n t a l o n g t h e l i n e was a l l o w e d f o r i n t h e s o l u t i o n . ( 5 ) C r a r y i n h i s work on v o l t a g e g r a d i e n t s a t t h e s u r f a c e o f b u n d l e d c o n d u c t o r s was a b l e t o s i m p l i f y h i s e q u a t i o n s b e c a u s e t h e c h a r g e on e ach c o n d u c t o r was t h e same. However, f o r an o p e n - e n d e d l i n e t h e c h a r g e s on t h e c o n d u c t o r s a r e no l o n g e r e q u a l so t h a t a g r a p h i c a l method i s n e c e s s a r y t o o b t a i n a s o l u t i o n o f t h e e q u a t i o n s . Two n u m e r i c a l e x a m p l e s have been s t u d i e d i n an a t t e m p t t o d e v e l o p a s u i t a b l e t h r e e - p h a s e s y s t e m . U n f o r t u n -a t e l y b o t h t h e two and f o u r - c o n d u c t o r e x a m p l e s p r o v e d t o be i m p r a c t i c a l b e c a u s e o f t h e l i m i t a t i o n i m p o s e d on e a c h s y s t e m by t h e b r e a k d o w n v o l t a g e o f a i r n e a r the c o n d u c t o r s u r f a c e s . 87 A l t h o u g h t h e u s e o f lumped s e r i e s i n d u c t o r s and p a r a l l e l r e a c t o r s was f o u n d t o i m p r o v e s y s t e m o p e r a t i o n , t h e i r i n t r o d u c t i o n i n c r e a s e d t h e o v e r v o l t a g e s a l o n g t h e l i n e s t i l l f u r t h e r . An a t t e m p t t o r e d u c e t h e v o l t a g e g r a d i e n t . l i m i t a t i o n a t t h e c o n d u c t o r s u r f a c e by i n c r e a s i n g t h e p e r m i t t i v i t y o f t h e i n s u l a t i n g m e d i a p r o v e d i n e f f e c t i v e as t h e power a n g l e o f t h e s y s t e m was f o u n d t o i n c r e a s e t o an u n s t a b l e v a l u e . I t may be c o n c l u d e d t h a t t h e u s e o f d i s t r i b u t e d c o m p e n s a t i o n i n l a r g e power s y s t e m s i s i m p r a c t i c a l and u n e c o n o m i c a l . APPENDIX A N u m e r i c a l E v a l u a t i o n o f C o n d u c t o r C o e f f i c i e n t s A . l T w o - C o n d u c t o r Case GROUND PLANE IMAGES FIGURE A . l T w o - C o n d u c t o r L i n e Above P e r f e c t G r o u n d I n F i g u r e A . l t h e mean c o n d u c t o r h e i g h t ( l i ) i s 50 f e e t w h i l e t h e c o n d u c t o r d i a m e t e r and s p a c i n g i s 0.950 i n c h e s and 1.5 f e e t r e s p e c t i v e l y . From E q u a t i o n s 4.2 and 4.5 i t f o l l o w s t h a t 6.03 -3.23 = jw [c] = j x 10" -3.23 6.03 mhos/mil< ( A . l ) 89 By n e g l e c t i n g t h e i n t e r n a l f l u x t e r m i n E q u a t i o n s 4.8, t h e Z m a t r i x becomes "0.9 52 0. 510" [z] = j u [L ] = J 0.510 0.952 o h m s / m i l e (A.2) E v a l u a t i o n o f t h e J m a t r i x f r o m E q u a t i o n 4.20 g i v e s M - W • [ i ] 4.093 0.000 0.000 4.093 10 - 6 (A.3) By a p p l i c a t i o n o f E q u a t i o n 4.40 we o b t a i n t h e p r o p a g a t i o n c o n s t a n t — 3 t = j 2.02 x 10~ r a d i a n s / m i l e A.2 F o u r - C o n d u c t o r Case H a 3 4 ^TROUND PLANE IMAGES FIGURE A.2 F o u r - C o n d u c t o r L i n e Above P e r f e c t G r o u n d I n F i g u r e A.2 II = 5 0 . 0 f e e t 13" 1 . 0 f o o t v e r t i c a l c o n d u c t o r s p a c i n g b^2= 1 * 5 f e e t h o r i z o n t a l c o n d u c t o r s p a c i n g The c o n d u c t o r d i a m e t e r i s 0 . 8 0 6 i n c h e s . The a d m i t t a n c e a r r a y may be d e t e r m i n e d f r o m E q u a t i o n s 4 . 2 and 4 . 5 as b e f o r e : -" 7 . 3 0 2 7 - 1 . 7 9 0 2 - 2 . 6 5 6 3 - 1 . 2 4 6 0 " - 1 . 7 9 0 2 7 . 3 0 2 7 - 1 . 2 4 6 0 - 2 . 6 5 6 3 - 2 . 6 5 6 3 - 1 . 2 4 6 0 7 . 3 1 1 8 - 1 . 7 8 1 1 - 1 . 2 4 6 0 - 2 . 6 5 6 3 - 1 . 7 8 1 1 [ Y ] = j x 1 0 - 6 mho mi 7.3118_ (~A.4) A l s o t h e Z m a t r i x may be w r i t t e n M -0 . 9 7 2 2 0 . 5 1 1 0 0 . 5 5 9 0 0 . 4 8 7 5 0 . 5 1 1 0 0 . 9 7 2 2 0 . 4 8 7 5 0 . 5 5 9 0 0 . 5 5 9 0 0 . 4 8 7 5 0 . 9 6 9 7 0 . 5 0 8 6 0 . 4 8 7 5 0 . 5 5 9 0 0 . 5 0 8 6 0 . 9 6 9 7 ohms/mi 1< ( A . 5 ) E v a l u a t i o n o f J f r o m E q u a t i o n 4 . 2 0 g i v e s 4 . 0 9 2 6 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 4 . 0 9 2 6 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 4 O 0 9 2 6 0 . 0 0 0 0 1 0 and as i n S e c t i o n A . l *K = j B = j 2 . 0 2 x 1 0 - 3 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 4 . 0 9 2 6 r a d i a n s / m i l e - 6 (A.6 ) To o b t a i n t h e Z and Y m a t r i c e s f o r a " c o m p l e t e l y t r a n s p o s e d " s y s t e m f r o m E q u a t i o n s A.4 and A.5 i t i s n e c e s s a r y t o d e t e r m i n e t h e a v e r a g e e l e m e n t s f o r e a c h a r r a y as i n d i c a t e d i n S e c t i o n 3.2. The Z and Y a r r a y s f o r a " c o m p l e t e l y t r a n s p o s e d " f o u r - c o n d u c t o r s y s t e m a r e |_Y]= j x 10 -6 7.3073 -1.7857 -2.6563 -1.2460 -1.7857 7.3073 -1.2460 •2.6563 -2.6563 -1.2460 7.3073 -1.7857 -1.2460 -2.6563 -1.7857 7.3073 m h o s / r a i l e H = 0.9710 0.5098 0.5590 0.4873 0.5098 0.9710 0.4873 0.5590 0.5590 0.4873 0.9710 0.5098 0.4873 0.5590 0.5098 0.9710 ohms/mi1e APPENDIX B N u m e r i c a l E v a l u a t i o n o f Maximum A l l o w a b l e V o l t a g e D i f f e r e n c e b e t ween C o n d u c t o r s B . l T w o - C o n d u c t o r Case FIGURE B . l C r o s s - S e c t i o n o f a T w o - C o n d u c t o r L i n e The t w o - c o n d u c t o r c a s e i l l u s t r a t e d i n F i g u r e B . l has t h e f o l l o w i n g p a r a m e t e r s C o n d u c t o r d i a m e t e r = 0.95 i n c h e s t h e r e f o r e C o n d u c t o r r a d i u s = 0.0396 f e e t E v a l u a t i o n o f E q u a t i o n s 4.60 t o 4.64 f o r t h e t w o - c o n d u c t o r c a s e g i v e s 12 d 2 1 = 1 . 5 f e e t E ml Jm2 •ne 12.6316 0.6667 0.6667 12.6316 (B.I: 93 By use o f t h e n u m e r i c a l v a l u e s c a l c u l a t e d i n S e c t i o n A . l E q u a t i o n 4.65 may be w r i t t e n ; -Q, = 10 -12 3.0241 -1.6190 -1.6190 3.0241 (B.2) V 4.3801 -2.1753 _Em2 -2.1753 4.3801 S u b s t i t u t i o n o f E q u a t i o n B.2 i n t o E q u a t i o n B . l and p e r f o r m i n g t h e m a t r i x m u l t i p l i c a t i o n y i e l d s ( B . 3 ) The maximum p e r m i s s i b l e v o l t a g e d i f f e r e n c e may be e v a l u a t e d f r o m E q u a t i o n B.3 by a g r a p h i c a l method d e s c r i b e d b e l o w and i l l u s t r a t e d i n F i g u r e B.2. W h i l e v a r y i n g v o l t a g e and k e e p i n g c o n s t a n t , v a l u e s o f E -, and E „ a r e d e t e r m i n e d . The r e s u l t s a r e ml m2 p l o t t e d a g a i n s t t h e v o l t a g e d i f f e r e n c e (V - ) as shown i n t h e f i g u r e . A d d i t i o n a l c u r v e s were o b t a i n e d f o r s e v e r a l v a l u e s o f v o l t a g e . The maximum p e r m i s s i b l e v o l t a g e d i f f e r e n c e may be d e t e r m i n e d f o r each v a l u e o f V„ when e i t h e r E , o r E „ d ml m^ (23) e x c e e d s t h e c o r o n a v o l t a g e g r a d i e n t f o r d r y a i r , w h i c h i s E ^ d r y a i r = 646 k V / f o o t (R.M.S.) The c o r o n a v o l t a g e g r a d i e n t f o r a i r i s p l o t t e d a s a h o r i z o n t a l s t r a i g h t l i n e i n t h e f i g u r e . F o r v a l u e s o f g r e a t e r t h a n V i t i s s e e n t h a t when V 0 = 260 kV, YnT1,„ = 16 kV and 2 D I F F when V 2 = 280 kV, D I F F 6 kV Ema = 646-6 KV/ft (RMS) m2 20 0 20 VOLTAGE DIFFERENCE (KV) DECREASING V1 INCREASING V1 FIGURE B.2 Conductor Surface Voltage Gradients (Two-Conductor Case) CO 95 However, i f e x c e e d s a p p r o x i m a t e l y 292 kV t l i e n t h e a i r a r o u n d t h e c o n d u c t o r s w i l l i o n i z e . S i n c e t h e c u r v e s E , and E ,, have d i f f e r e n t nil in 2 s l o p e s , i t f o l l o w s t h a t i t i s p o s s i b l e t o have a l a r g e r v o l t -age d i f f e r e n c e b e t ween t h e c o n d u c t o r s i f i s l e s s t h a n V^. From F i g u r e B.2 when V_ = 260 kV V^ ,. " = 33 kV 2 D I F F and when V g = 280 kV V D I F F = 1 3 k V a g a i n when e x c e e d s 292 kV t h e a i r w i l l i o n i z e . B.2 F o u r - C o n d u c t o r Case FIGURE B.3 C r o s s - S e c t i o n o f a F o u r - C o n d u c t o r L i n e I n t h e f o u r - c o n d u c t o r c a s e , c o n d u c t o r s 1 and 2 w i l l be c o n d u c t i v e l y c o n n e c t e d as w i l l be c o n d u c t o r s 3 and 4, hence 96 and V l V 2 Q 3 = Q 4 = v. (B.4) 12 d 1 3 .= 1.0 f o o t d, „ . = 1.802 f e e t 14 3 • ~ '4 The p a r a m e t e r s o f F i g u r e B.3 f o r c o n d u c t o r 1 a r e 1.5 f e e t I t i s a s t r a i g h t f o r w a r d m a t t e r t o d e t e r m i n e t h e c o r r e s p o n d i n g p a r a m e t e r s f o r t h e r e m a i n i n g c o n d u c t o r s . By s u b s t i t u t i o n o f t h e above n u m e r i c a l v a l u e s i n E q u a t i o n s 4.60 and 4.61 i t i s p o s s i b l e t o w r i t e t h e f o l l o w -i n g i n d e p e n d e n t e q u a t i o n s . 1 E l = and E 3 = 2%c | ^0.6667 Q x + 0.4622 Q 3J 2 + ^ 1.3071 Q 3 j 2 ( B . 5 a ) | ^0.4622 Q 1 + 0.6667 Q 3J 2 +^1.3071 Q^j 2 (B.5b) S u b s t i t u t i n g t h e n u m e r i c a l v a l u e s d e t e r m i n e d i n S e c t i o n A.2 i n t o E q u a t i o n 4.65 y i e l d s t h e f o l l o w i n g m a t r i x e q u a t i o n = 10 12 0.7216 0.5091 0 . 5 0 9 l ! 0.7192 (B.6) By p r e m u l t i p l i c a t i o n o f E q u a t i o n B.6 by t h e i n v e r s e o f t h e n u m e r i c a l m a t r i x we o b t a i n Qi = 10 -12 2.7683 •1.9596 -1.9596 2.7775 1 (B.7) B e c a u s e a d i r e c t r e l a t i o n s h i p does n o t e x i s t b e t w e e n E q u a t i o n s B.5 and B.7 t h e s o l u t i o n p r o c e d u r e a d o p t e d i n S e c t i o n B . l n e e d s t o be m o d i f i e d . 1000 800 Ema. = 64 6-6 KV/ft to 600Y 400 U J Co 5 o ir>3 200Y 20 0 20 40 VOLTAGE DIFFERENCE 4 CONDUCTORS DECREASING V 7 INCREASING V 1 FIGURE B.4 Conductor S u r f a c e V o l t a g e G r a d i e n t s (Four-Conductor Case) cc -4 A n u m e r i c a l s o l u t i o n can be f o u n d by c a l c u l a t i n , t h e l i n e c h a r g e s f r o m E q u a t i o n B.7 ( f o r a p a r t i c u l a r v o l t a g c o n d i t i o n ) and s u b s t i t u t i n g t h e s e v a l u e s i n t o E q u a t i o n s B.5 The maximum f i e l d i n t e n s i t 3 r may be e v a l u a t e d f r o m E q u a t i o n 4.64. D e t e r m i n a t i o n o f t h e maximum p e r m i s s i b l e v o l t a g d i f f e r e n c e may be o b t a i n e d g r a p h i c a l l y f r o m F i g u r e B.4. When t h e v o l t a g e s and a r e g r e a t e r t h a n v o l t a g e s V„ and V., t h e f o l l o w i n g a r e t h e maximum a l l o w a b l e ° 3 4 v o l t a g e s b e t w e e n c o n d u c t o r s when Y 0 = 280 kV V n T T I T l = 42 kV 3 D I F F V 3 = 300 kV V D I p p = 35 kV and V 3 = 320 kV V D I p p = 27 kV However, when v o l t a g e s and a r e l o w e r t h a n V 3 and t h e f o l l o w i n g c o n d i t i o n s may be o b t a i n e d f r o m F i g u r e B.4:-when V 0 = 280 kV V „ T T 1 „ = 60 kV 3 D I F F V 3 = 300 kV V D I p p = 50 kV and V Q = ' 320 kV V 1. T„ T 1 = 40 kV 3 D I F F The above v o l t a g e c o n d i t i o n s a p p l y f o r h o r i z o n t a l a n d v e r t i c a l c o n d u c t o r s p a c i n g s o f 1.5 f e e t and 1.0 f o o t r e s p e c t i v e l y . ( R e f e r t o S e c t i o n 5.3 f o r t h e s p e c i f i c a t i o n o f c o n d u c t o r s p a c i n g s ) . 99 REFERENCES 1. C l a r k e , E. and C r a r y , S.B., S t a b i l i t y and L i m i t a t i o n s o f L ong D i s t a n c e A.C. Power - T r a n s m i s s i o n S y s t e m s , A I E E T r a n s a c t i o n s , V o l . 60, 1941, p. 1051. 2. M a r b u r y , R.E., and J o h n s o n , F.D., A 24,000 K i i o v a r S e r i e s C a p a c i t o r i n 230 kV T r a n s m i s s i o n L i n e , A I E E T r a n s a c t i o n s , V o l . 70, 1 9 5 1 , p. 1621 3. H a r d e r , E.L., e t a l , S e r i e s C a p a c i t o r s d u r i n g F a u l t s a n d R e c l o s i n g , A I E E T r a n s a c t i o n s , V o l . 70, 1951, p. 1627. 4. P i e r c e , R.E., and G e o r g e , E.E., E c o n o m i c s o f Long D i s t a n c e E n e r g y T r a n s m i s s i o n , A I E E T r a n s a c t i o n s , V o l . 67, 1948, pp. 1089-1094. 5. 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