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

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A MULTICONDUCTOR TRANSMISSION L I N E WITH DISTRIBUTED COMPENSATION  by HARRY ERNEST ORTON B.E., U n i v e r s i t y  o f New S o u t h W a l e s , 1 9 6 6 .  A THESIS SUBMITTED IN PARTIAL  FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OP MASTER OF APPLIED SCIENCE i n t h e Department o f Electrical  We a c c e p t  this  t h e s i s as c o n f o r m i n g  required Research  Engineering  to the  standard  Supervisor  Members o f t h e C o m m i t t e e  Head o f t h e  Department  Members o f t h e D e p a r t m e n t of E l e c t r i c a l  Engineering  THE UNIVERSITY OF B R I T I S H COLUMBIA December, 1968.  In p r e s e n t i n g  this thesis  in p a r t i a l f u l f i l m e n t o f the requirements 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 , 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 I further agree that permission  f o r extensive  I agree  that  and Study.  copying of this  thesis  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 .  It i s understood that copying or p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l written  gain  s h a l l n o t b e a l l o w e d w i t h o u t my  permission.  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  ^LSL^t^Zsij^  ^"5  Columbia  ,  (^k©/  ABSTRACT The state  thesis presents  compensation.  methods f o r d e r i v i n g line  graph  starting  from  steady-  The  analysis  outlines  line two  current equations  Maxwell's c o e f f i c i e n t s  and  along  the T e l e -  equations.  conductor  internal  assumption  transposed the  of the  power t r a n s m i s s i o n  t h e v o l t a g e and  Correction  the  analysis  o p e r a t i o n of a multiconductor  with d i s t r i b u t e d  the  an  .  due  t o an  imperfect  impedance a r e d i s c u s s e d .  t h a t the  system of conductors  earth  A l s o , by is  completely  single  results  are given  f o r a two  phase t r a n s m i s s i o n l i n e  and  f o r a three-phase  The  results by  of  i s possible.  Numerical conductor,  and  making  i t i s shown t h a t 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  equations  prevented  factors  system having  with  field  four conductors  limitations  conductors.  ii  at the  a four-  earth return  show t h a t s u c c e s s f u l o p e r a t i o n o f t h e electric  and  per  phase.  system i s  s u r f a c e of  the  TABLE OF CONTENTS Page ABSTRACT  .  i i  TABLE 0? CONTENTS  i i i  L I S T OF ILLUSTRATIONS  v i  viii  ACKNOWLEDGEMENT 1.  INTRODUCTION  2.  SYSTEM STABILITY AND POWER TRANSFER IMPROVEMENT  3.  2.1  System E x p a n s i o n  2.2  Stability  2.3  Introduction  1  ...  3 3  and Power T r a n s f e r  Improvement  ....  3  t o D i s t r i b u t e d Compensation  ....  4  BASIC ASSUMPTIONS  7  3.1  General  7  3.2  The C o m p l e t e l y T r a n s p o s e d System  8  3.3  Admittance Array  9  3.4  Impedance A r r a y  3.4.1  Conductor  3.4.2 E a r t h 3.5 4.  '..  Description  Correction Correction  Internal  Correction  Factors Factors  10  Impedance  10  Factors  12  Error Analysis  14  MATHEMATICAL BACKGROUND  .  19  4.1  Maxwell's C o e f f i c i e n t s  19  4.2  General  Differential  Equations  24  4.2.1  General  Differential  E q u a t i o n s Method  I .....  25  4.2.2 G e n e r a l  Differential  E q u a t i o n s Method  I I ... .  30  iii  Page 4.3  The Completely Transposed and L o s s l e s s Line  4.4  Differential  .  Equation S o l u t i o n  4.4.1 S o l u t i o n and Boundary Conditions 4.4.2 Comparison Generalized  5.  32 37 37  of S o l u t i o n with the C i r c u i t Constants  43  4.5  A d d i t i o n of Lumped Parameters  45  4.6  Computer Programming  51  4.7  Maximum Allowable V o l t a g e D i f f e r e n c e  54  RESULTS OP ANALYSIS OP A 500 kV TRANSMISSION LINE WITH DISTRIBUTED COMPENSATION  59  5.1  D e s c r i p t i o n of the Transmission Line Systems.  59  5.2  Two-Conductor  61  5.3  Four-Conductor System  5.3.1 I n i t i a l  System  67  Investigations  5.3.2 Optimizing  67  the Conductor Diameter and  Spacings  69  5.3.3 Use of Lumped Parameters  74  5.4  80  Three-Phase System  5.5 5.6  Physical Limitations Comparison of S e r i e s and D i s t r i b u t e d Compensation 6. CONCLUSION APPENDIX. A. Numerical E v a l u a t i o n of Conductor Coefficients A.l  Two-Conductor  A.2  Four-Conductor Case  83 85 86 88  Case  88 89  iv  Page A P P E N D I X B.  Numerical  Voltage  Evaluation  Difference  B. 1  Two-Conductor  B.2  Four-Conductor  of  between  Case  Maximum  Allowable  Conductors  92 92  Case  95  REFERENCES  99  v  L I S T OP  ILLUSTRATIONS  Figure  Page  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  5  2.2  The  6  3.1  Earth 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 C o n d u c t o r s Above P e r f e c t G r o u n d . . . „  19  4.2  n-Conductor System w i t h Mutual C o u p l i n g  25  4.3  Single-Conductor Transmission D i s t r i b u t e d Parameters  30  Open-Ended L i n e  Line  with  4.4  Two-Port Networks  44  4.5  Lumped P a r a l l e l  46  4.6  Lumped S e r i e s R e a c t o r s  47  4.7  Parallel  48  4.8  Computer Flow C h a r t  4.9  Cross-Section Conductors  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 L i n e (Two-Conductor Case)  5.2 5.3  5.4 5.5  Capacitors  Reactors  53  of a System of n  Parallel '  Along  55  the 62  C o n d u c t o r D i a m e t e r and Maximum V o l t a g e D i f f e r e n c e (Two-Conductor Case)  66  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 t h e L i n e a t 50 M i l e I n t e r v a l s ( T w o - C o n d u c t o r Case)  66  V o l t a g e and C u r r e n t P r o f i l e s L i n e ( F o u r - C o n d u c t o r Case)  68  O p t i m i z a t i o n of Conductor (Four-Conductor Case)  Along  Spacings  vi  the  70  Figure 5.6  5.7  5.8(a)  5.8(b)  5.9  Page Optimization of (Four-Conductor  Conductor Case)  Diameter ••••  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 ( F o u r - C o n d u c t o r O p t i m i z e d Case) Locus P l o t the Line Lumped Line  of V o l t a g e s  Line 73  Currents  Along 75  Parameter  Section  of  a  Two-Conductor 75  Installation the  and  the  70  of  Parallel  Reactors  Along  Line  79  5.10  Three-Phase  5.11  Effect  of  System  Above  Increased  Maximum V o l t a g e A.l  Two-Conductor  A. 2  Four-Conductor  B. l  Cross-Section  B.2  Conductor  Voltage  on  Line  Above  Line of  a  Above  .  P e r f e c t Ground P e r f e c t Ground  Two-Conductor Voltage  Line  Cross-Section  of  B.4  Conductor Surface (Four-Conductor)  a  88 89 92  94  Four-Conductor Voltage  84  Gradients  Case)  B.3  82  the  Difference  Surface  (Two-Conductor  Line  P e r f e c t Ground.........  Line  95  Gradients 97  vii  ACKNOWLEDGEMENT This without  the  professors and  inspiration Dr.  experience I  invaluable A.  also  indebted  Carmen f o r t y p i n g  the  have been  completed  R.  J a m i e s o n , whose  ideas  t o Dr.  the  supervising  expressed. H.  R.  Chinn  for his  computer programming,  g r a p h s and  t h e s i s and  influence  diagrams  proof  and  to  Mr.  to  my  reading  the  acknowledgement i s e x t e n d e d t o  the  x  Grateful University Council  Mr.  shape t h e  f o r drawing the  draft.  not  encouragement o f my  F. Noak.es and helped  am  and  assistance with  MacKenzie  wife  thesis could  of B r i t i s h  Columbia  o f Canada f o r t h e i r  and  the N a t i o n a l  financial  viii  support  Research  of the  project.  1 1.  The  aim  INTRODUCTION  of t h i s  t h e s i s i s to determine the  s t a t e performance of a m u l t i c o n d u c t o r with distributed The arose  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  spaced along  the  capacitors.  overvoltages For  A lumped s e r i e s c a p a c i t o r on a  capacitance achieve  evenly  w o u l d be  d i s t r i b u t e d along  the  line  e c o n o m i c r e a s o n s t a i s w o u l d be  b e t w e e n two  (2) J o h n s o n * ' and  o r more c o n d u c t o r s  Harder.  Previous  impossible. distributed  c o u l d be u s e d t o associated  matrix  and  Bewley^^  transmission lines with  p a r a m e t e r s forms a convenient system equations.  Marbury  (3) '  work by P i p e s ^ ^ ^  l o s s l e s s , multiconductor  starting-point for  developing  more c o m p l e x  systems.  of the  two-  first  u s i n g t h e m a t r i x methods s u g g e s t e d by P i p e s . an e a s y e x t e n s i o n  on  distributed  In the t h e s i s a s i n g l e - p h a s e ,  open-ended l i n e w i t h e a r t h r e t u r n i s  approach allows  then  eliminated.  w i t h s e r i e s e l e m e n t s s u c h as h a v e been d e s c r i b e d by  analysed  near  I t i s c l e a r t h a t i f a l a r g e number o f s e r i e s  compensation, thus a v o i d i n g the problems  conductor,  long  l i n e , i n order to reduce o v e r v o l t a g e s  H o w e v e r , i f an open-ended l i n e were u s e d t h e n t h e  the  line  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  c a p a c i t o r s c o u l d be  and  line  from c o n s i d e r a t i o n of w e l l - e s t a b l i s h e d techniques' of  transmission line  the  power t r a n s m i s s i o n  compensation.  s e r i e s compensation.  the  steady-  The  analysis for  2  As the  voltage  gradient  proves  to  of  the  transmission  an  imperfect  discussed.  be  i s the  an  case at  the  important  earth  for  surface  factor  system. and  a l l EHV  i n the  Also,  conductor  of  transmission the  bundled  numerical  lines, conductors  analysis  c o r r e c t i o n f a c t o r s due  internal  impedance  are  to  3 2.  2.1  SYSTEM S T A B I L I T Y AND POWER TRANSFER IMPROVEMENT  System Expansion As t h e modern power s y s t e m e x p a n d e d as a r e s u l t  of t h e i n c r e a s i n g s i z e o f l o a d b l o c k s , n a t u r a l s o u r c e s more f u l l y United  States  sources  I n c o u n t r i e s s u c h as Canada, t h e  o f A m e r i c a , R u s s i a and Sweden t h e s e  natural  a r e 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  generation miles  exploited.  were  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 e x c e s s o f 300  from the l o a d  centres.  To t r a n s p o r t t h e h e a v y l o a d s  economically  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 u s e has been made o f t h e e x t r a high-voltage  2.2  transmission line  and t h e s e r i e s c a p a c i t o r .  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  increased  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 Silver^^ standard of  recommended voltage  s u i t a b l e f o r t h e t r a n s f e r o f l a r g e amounts  power. For  the  t h e i n t r o d u c t i o n o f 220 kV as a  short l i n e s  ( l e s s t h a n 300 m i l e s  i n length)  amount o f power w h i c h c a n be t r a n s f e r r e d w i t h o u t  s t a b i l i t y v a r i e s as t h e s q u a r e o f t h e l i n e v o l t a g e resistance  increases  no l o n g e r t h e c a s e and t h e r e l a t i o n s h i p b e t w e e n  amount o f power, v o l t a g e ed.  (line  neglected). However, as t h e l e n g t h o f t h e l i n e  is  loss of  and t h e l i n e  this  economy,  l e n g t h must be  consider-  Overall  s y s t e m s t a b i l i t y has  been i m p r o v e d  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 stations,  almost l i g h t n i n g - p r o o f l i n e s  design.  The  and  s t a b i l i t y on  line  high  most i m p o r t a n t  series inductive  and  long l i n e s ,  switching good  factor limiting  by  system  power t r a n s f e r  however, i s t h a t of  the  reactance.  Many methods h a v e been s u g g e s t e d t o r e d u c e series reactance,  and  some o f w h i c h a r e g i v e n of the  Reduction  frequency,  2.  Use  3.  Parallel  4.  Use  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  of synchronous  s t a b i l i t y , but  line  This  2.3  of t r a n s m i s s i o n  of bundled conductors  l i n e , s i n c e the  i s reduced.  condensers,  operation  r e g u l a t i o n of the  discussed  below:  1.  o n l y t h e power t r a n s f e r and  and  lines,  f o r each phase,  a l s o the  of  the  regulation is  15.  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  overvoltages it  not  voltage  i n d u c t i v e reactance  improvement i n v o l t a g e  i n R e f e r e n c e s 14  the  w o u l d be  capacitance  at the c a p a c i t o r s .  reduce these  advantage to d i s t r i b u t e  uniformly  the  along  l e n g t h of the  shown i n F i g u r e 2.1  the  line i n a large (a) and  (b).  e c o n o m i c a l c o n s i d e r a t i o n s l i m i t t h e number o f  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 length.  overvoltages  of c o n s i d e r a b l e  number o f s m a l l e r u n i t s as At p r e s e n t ,  To  line  o f 600  miles  in  5  HhHHHHh  -o  O-  —o  TRANSMISSION  LINE  CONFIGURATION  (a)  (b) VOLTAGE - TO-6R0UND  FIGURE 2.1  Distribution  PROFILE  o f Lumped  I f t h e r e were no economic l i m i t c o u l d be d i s t r i b u t e d units. voltages  along  This would e n t i r e l y  theline  Capacitors  the capacitance  i n an i n f i n i t e  e l i m i n a t e normal o p e r a t i n g  as shown i n F i g u r e 2 . 1 .  c o n c e p t o f t h e open-ended l i n e .  over-  ,  T h e s e c o n s i d e r a t i o n s have i n f a c t  infinite  number o f  suggested t h e  H e r e , i n s t e a d o f u s i n g an  number o f lumped c a p a c i t o r s , t h e d i s t r i b u t e d  capacitance  between two o r more c o n d u c t o r s  c o m p e n s a t i o n , as shown i n F i g u r e 2.2.  i s used t o achieve  6  LINE 1  -DISTRIBUTED  LINE 2  0 FIGURE 2.2  The Open-Ended  Line  CAPACITANCE  7  3.. BASIC ASSUMPTIONS  3.1  General Description Consideration of every contingency involved  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 w o u l d make t h e mathematical  t r e a t m e n t o f an n - c o n d u c t o r s y s t e m v e r y  complex.  To r e d u c e t h e c o m p l e x i t y o f t h e p r o b l e m a number o f s i m p l i f y i n g assumptions are introduced. The t r a n s m i s s i o n l i n e a n a l y s i s consists of n p a r a l l e l by a homogeneous ground p l a n e ,  dielectric  considered i n the conductors surrounded  and s u s p e n d e d  above a p e r f e c t  A " p e r f e c t g r o u n d " i s one h a v i n g a u n i f o r m  r e l a t i v e p e r m e a b i l i t y o f u n i t y and i n f i n i t e  (17) conductivity.  I n a c t u a l i t y t h e g r o u n d p l a n e i s n o t p e r f e c t as i s (27) (31) d i s c u s s e d by C a r s o n and W i s e .  '  '  However, t h e  assumption o f a p e r f e c t ground i s j u s t i f i e d  for a multi-  c o n d u c t o r s y s t e m a t power f r e q u e n c i e s as t h e e r r o r is  small  involved  (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 s a g and g r o u n d  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 is  assumed t o be s t r a i g h t and p a r a l l e l  g r o u n d p l a n e w i t h an a v e r a g e h e i g h t H.  each c o n d u c t o r  to a horizontal E f f e c t s of steel  t o w e r s and i n s u l a t o r s on t h e l e a k a g e c u r r e n t s may (25) assumed n e g l i g i b l e .  be  8  3.2  The  Completely Transposed The  "completely transposed" infers  that  t h e n c o n d u c t o r s of the t r a n s m i s s i o n l i n e occupy  each  of  terra  System  t h e n p o s i t i o n s o f t h e c o n d u c t o r s f o r an  d i s t a n c e and  in cyclic  o r d e r such t h a t the  equal 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 C h a p t e r 4) "doubly symmetric"  (symmetric  both  diagonals).  A doubly symmetric  m a t r i x i s of the form  illustrated  i n E q u a t i o n 3.1  about  are  f o r a four conductor  system.  b  c  d  a  d  c  d  a  b  c  b  a  (3.1)  To d e t e r m i n e t h e " c o m p l e t e l y t r a n s p o s e d " i m p e d a n c e and to and  admittance arrays i t w i l l  be  determine the arrays f o r the n p o s s i b l e e v a l u a t e t h e average  of the n arrays.  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  where r and  r s  (transposed)  =  1  r S  n  s h a v e v a l u e s f r o m 1 t o n.  necessary configurations T h i s may  be  9  A numerical system i s g i v e n  3.3  evaluation  i n Appendix  Admittance Array  f o r the  A.  Correction  Factors  The p o t e n t i a l c o e f f i c i e n t Section  4.1  from t h e  f o r an n - c o n d u c t o r  2H  In  system may  rs  2ne. rs  and N  (M  +  R  2TtC  factors  array  described i n be  determined  equations^^  rr  where M  four-conductor  In a r s  1  +  rs rs  (M x  HG  are the r e a l  rr  + jN ) " rr  (3.2a)  rs  + jN ) rs  (3.2b)  0  and i m a g i n a r y c o r r e c t i o n  to  J  f o r t h e r s element o f t h e a r r a y . The a d m i t t a n c e  [Y]  array  = 0  0)  i s d e t e r m i n e d by  [p]  applying  -1  (3.3)  The c o r r e c t i o n f a c t o r s M and N a r e d e p e n d e n t upon t h e 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 the  relative  conductivity position the  permittivity, relative of the ground,  space,  permeability  the r a d i u s  o f t h e c o n d u c t o r and f i n a l l y  system.  of free  and  and g e o m e t r i c the frequency of  (18)  10  Doench,  i n iiis  work  on  ground  conductivity  and  (32 ) permittivity, of  frequency  curves  has  variation  (Reference  correction  may  3.4  be  effectively  the  sinusoidal  correction factors in  Correction  Factors  the  conductors  impedance  distance  may  as  conductor  effect  w e l l as  Carson^^ impedance  frequencies  introduce  Equations  4.2.  each  other,  correction factors  c o r r e c t i o n f a c t o r s depend  phase  a return  Equations  some o r a l l  close to  between conductors,  circuits  voltages  array.  Proximity  internal  system where  are r e l a t i v e l y  effect"  the  Impedance  a multiconductor  "proximity the  internal  These  b e l o w 1 KHz  Impedance A r r a y Conductor  effect  zero.  only considering  permeability.  of  show t h a t  simplified  These the  p a g e 48)  are  the  correction factors.  g i v i n g the  In  in  the  showing  neglected  3.4.1  the  we  power f r e q u e n c i e s  3.2  of  32,  on  f a c t o r s are  Since at  p l o t t e d curves  has of  conductor and  with  radius  of  i s present  single-phase  a ratio to  the  conductor  i s only  15  conductor  shown t h a t  one  1.  one of  on  due  the  frequency,  size  and  for  three-  circuits. increase to  percent  conductor  the at  in proximity  high  spacing  to  11  For bundled  c o n d u c t o r s a t 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  neglected. The cylindrical  c o n d u c t o r i n t e r n a l impedance o f a c o n d u c t o r a t power f r e q u e n c i e s  Z  where R. l  =  d.c.  and  =  _u_  L.  i  =  R  i  ^  +  w  L  is^^^^'^^^'^^^  i  ^ * ^ 3  4  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 ' =  i n d u c t a n c e o f t h e c o n d u c t o r due  to  8tt  1  internal  flux  —7  or  =  0.5.  =  0.806  The by  x  10 x  henry/meter 10~  4  henry/mile  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  Stevenson^"^  assuming  that the current d e n s i t y  the c y l i n d r i c a l conductor i s uniform (that i s skin negligible)  and  of effect  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 .  Butterworth  w o r k 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  '(in his lines)  has t a k e n t h e e f f e c t o f s t r a n d i n g i n t o a c c o u n t by u s i n g (15) t h e method o f g e o m e t r i c 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 t h e c o n d u c t o r t o be c y l i n d r i c a l . '  12  3.4.2  Earth Correction To  of  finite  correct  t h e impedance m a t r i x f o r an e a r t h  conductivity,  correction  factors  Factors  R  resistance  and X  e  and  reactance  a s s o c i a t e d with the  earth  e  r e t u r n path are determined  by  u s i n g an  infinite  series  (27) developed  by C a r s o n .  Real  and  imaginary  component m a t r i c e s P and  Q,  in  parameters  terms o f r and r  and  G  =  r g  0,  two =  rs  the angle  images o f c o n d u c t o r s  a  /cou.'  . a  subtended r and  such  are  (3.5)  rs  at conductor  s (see F i g u r e  ifi  =  2nf  = angular  (j,  =  r e l a .ttiivvee p e r m e a b i l i t y o f t h e 1.0,  depends '  the  on  frequency, ground  (31)  p.  —  a b s o l u t e p e r m e a b i l i t y = u. u ,  p  =  ground  =  d i s t a n c e from  r g  r by  3.1).  of r  rs  calculated  that  Evaluation  =  and  respectively,  correction  resistivity  conductor  s.  =  conductor  100,0  ohm.meters  r to the  (28)  image o f  13  GROUND PLANE OF FINITE CONDUCTIVITY  IMAGES  F I G U R E 3.1  The can  be  Earth  Carson  w r i t t e n ^ ' ^  Correction  correction 2 7  Parameters  factors  due t o t h e e a r t h  ^  R rs  P  wu rs * 7t  (3.6a)  (3.6b) 71  1 4  The  infinite  s e r i e s used to e v a l u a t e  c o r r e c t i o n component m a t r i c e s P and value of r .  Por our case  w i t h a ground r e s i s t i v i t y r  =  1 2  Q depends upon  at a frequency o f 100  the  o f 60 Hz  evaluated  P  0.066  8  r  372  cos  r  0  s  +  rs T6  r S  . +  and  Q r s  3.5  2  20  cos  r  may  (0.6728 + I n  ,  r  16  r  +  S  rs  rs  (3.7a) r S  r  1  cos 0 rs  3^/2  —  s i n 20  Q  2  2  s  .  ( rs)  = -0.0386 + 1 l n  (3.7b) rs  Error Analysis Considerable  i n the mathematical  simplification  solution  r e t u r n path  and  the i n t e r n a l  c a n be n e g l e c t e d .  The  can be  to the  impedance of each  [v]  earth conductor  from the f o l l o w i n g  I t i s shown i n S e c t i o n s 4 . 2 and  2  line  error involved in neglecting  components i s d e t e r m i n e d  D  obtained  of the t r a n s m i s s i o n  p r o b l e m i f t h e c o r r e c t i o n components due  these  Q  by  1  =11 r S  and  ohm.meters  .(27). R e f e r r i n g t o Carson's paper P and be  the  =  [z]  •  [ Y ]  •  [ V ]  analysis.  4 . 4 that  (Eqn.  4 . 1 6 )  )  15  D  [z]  [i]  =  [ i j • [z] • [ i ]  (Eqn.  • [Y]  =  [ l ] • [z]  (Eqn  =  -  2  and  X  where t h e system  i s a s s u m e d t o be  =  [j]  Jj~r r  4.20)  (Eqn.  l o s s l e s s and  4.17)  4.40)  "completely  transposed". However, t h e above e q u a t i o n s , ion  of Equation  "completely assumed  4.40,  apply  transposed"  t o any  system  l o s s l e s s o r n o t . (28) For  a lossy  the elements  \  will  contain five  Z'  + R  + j(X  =  R. l  e  g  + X.  =  the contribution the  i  =  system  + X  the contribution  )  significance:-  due t o t h e p h y s i c a l c a s e [zj =  due t o f a c t o r s  geometry of [^g]' S e c t i o n  i n t e r n a l to the  c  conductor e  =  (Section  the contribution path  (Section  3.4.1) a n d  due t o an i m p e r f e c t  3.4.2).  (25)  (3.8) •  e  i  ( i n the lossless  i n the  components o f t h e form  where t h e s u f f i x e s have t h e f o l l o w i n g g  the system i s  /Whether  impedance m a t r i x  except-  multiconductor,  r  system  with the  earth  return  4.3)  16  The p r o p a g a t i o n may  be d e t e r m i n e d  M  from  =  and  the diagonal  \  and of  elements  the  line^  determined  of  2  8  [z]  system  /  2  (3.9)  diagonal  by  * [Y"] ,  the line  the voltages  by t h e c h a r a c t e r i s t i c  impedance  )  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  [zj  l  matrix [jijis  a t any p o i n t a l o n g  currents are related  W)  •  constant  are the eigenvalues  Now  f o r a lossy  (28)  { H  where t h e p r o p a g a t i o n with  constants  -  [ i ] "I J [*•]  system  . [] I T  1 / 2  (3  . ) 12  17  The c o m p l e t e s o l u t i o n f o r t h e v o l t a g e current  c a n be w r i t t e n  as  [v]  =  c"W  [I]'  =  „-M*  where t h e m a t r i c e s V  (28)  [v ]  x  It  [zj"'  and I  [! ]  (a.14)  o  are the terminal  conditions  o 0 (that i s at the r e c e i v i n g end).  =  i s c l e a r f r o m E q u a t i o n s 3.13 and 3.14  an e r r o r a n a l y s i s considering  (3.13)  o  o of the l i n e a t x  and  that  o f t h e s o l u t i o n may be o b t a i n e d by  the matrices Z  and K.  q  The v a l u e s o f b o t h Z  and Y were  determined  c with  t h e impedance c o r r e c t i o n f a c t o r s  ( E q u a t i o n 3.8) and compared w i t h d e t e r m i n e d by  T a b l e 3.1 c  factors.  the uncorrected  values  putting [>']  both Z  included  =  [2]  =  [X ] g  summarises t h e e r r o r s  and V by n e g l e c t i n g  t h e impedance  (3.15).  obtained i n correction  18  °/o ERROR Z c  FACTOR NEGLECTED 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  where 1.  and  fo ERROR  Error  Analysis  r e f e r s t o t h e two-conductor case  2.  refers to the four-conductor  3.  r e f e r s to the three-phase The  aim o f t i i i s  phase m u l t i c o n d u c t o r systems a r e u s e d o n l y Since case are small factors  ( S e c t i o n 5.2)  case  case  (Section  (Section  t h e s i s i s to consider  s y s t e m ; t h e two-and f o rpreparatory  the errors  involved  5.4)  a three-  four-conductor  calculations.  i n the three-phase  i t can be c o n c l u d e d t h a t t h e c o r r e c t i o n  may be n e g l e c t e d  i n a l l studies  involved  i n this  thesis. It w i l l assumption solution  5.3)  be shown i n C h a p t e r 4 t h a t  considerably  this  s i m p l i f i e s the mathematical  of a multiconductor  transmission  system.  4.  4 1 0  Maxwell's (a)  MATHEMATICAL BACKGROUND  Coefficients  Potential  Coefficients  C o n s i d e r a system o f n p a r a l l e l ,  cylindrical  c o n d u c t o r s o f f i x e d p o s i t i o n as shown i n F i g u r e  FIGURE 4.1  A System  of n P a r a l l e l  Above P e r f e c t G r o u n d  4.1.  Conductors  20  . I f a u n i t charge the p o t e n t i a l  i s p l a c e d on c o n d u c t o r r t h e n  r e l a t i v e t o ground  a c q u i r e d by e a c h  may be w r i t t e n P > P£r' ••••» ? r r ' ••••» P conductor r a c q u i r e s a charge Q P r  l r  »  Q P2r'  » 2 P  r  r  n r  «  Further, i f  #  l r  conductor  n r  t h e n t h e p o t e n t i a l s w o u l d be  I  t  follows using thep 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 h e p o t e n t i a l s c a u s e d by s i m u l t a n e o u s c h a r g e s Q^, Q^,  V  V  Q  on each  n  Pl 2  2  P  2  o f t h e n c o n d u c t o r s w o u l d be  l  =  PllSl  +  2  =  P l2l  +  =  v Q  +  linear  coefficients or potential  2  2  2 2  2  +  ••••  +  +  ••••  +  P  ln n Q  ^n^n  «  v  n  The  nl  (PJJ_» 1 2 ' ••••> P p  n n  ±  P  n 2  Q  +  2  ....  +  P  n n  Q  n  (  4  a  )  coefficients  ^ depend s o l e l y on t h e g e o m e t r y o f t h e  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  conductors  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 a n d t o be s u s p e n d e d o v e r an e q u i p o t e n t i a l g r o u n d  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 o f t h e method o f images  to  F i g u r e 4.1, t h e e q u a t i o n s f o r t h e 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 ; -  p  =  211 1 2nc  In o  r r  R  d a r a f / meter '  (4.2 a)  21 and  p  1  rs  a  r rs rs  and  c  rs  2nc  w h e r e II R  in  =  (4.2  daraf/meter  b)  rs  the height of conductor ground plane,  r above  the  r,  radius of conductor  the  —  t h e d i s t a n c e between t h e conductor and t h e i m a g e o f c o n d u c t o r s,  =  the d i s t a n c e between  =  8.85  x 10  Equations  -12  4.1  [V]  conductors  farads per  r  r and  meter.  may  be w r i t t e n  =  [P]  .  i n matrix  n  '11  nl  P  form:-  (4.3)  [Q]  where  '21  s,  '12  'in  22  '2n  n2  nn  22  and  Q  It the  i s p o s s i b l e , however, t o express t h e v a l u e o f  c h a r g e s i n terras o f c a p a c i t y  coefficients C  , where I*  c o e f f i c i e n t s are functions  such  s  o f t h e geometry of t h e conductor  arrangement. C^V.. 11  Q  n  +  +  C, V In n  +  +  C  1  c  v nl l v  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 f o r m : -  [Q]  -  [c] • W '11  where  [cj  ..  (4.4)  C  In  =  C  i  nl  ••# •  c nn  It i s clear  by c o m p a r i n g E q u a t i o n s  4.3 and 4.4  that  [C]  (b)  =  Electromagnetic  flows i n conductor  r  ( 0 ) l i n k i n g the c i r c u i t rr  r  where L  r , then t h e '  c o n t a i n i n g conductor  r  w r i t t e n L .i »  (4.5)  1  Coefficients  If a current i flux  [p] -  r may be  i s c a l l e d the "electromagnetic (17)  coefficient"  of conductor  r.  • Then a p p l i c a t i o n  superposition p r i n c i p l e to a multiconductor  0  1  = hi ! 1 +  L  +  A  K = K\H ^2*2 +  system g i v e s t h a t  •••• h^n  +  12 2  of the  +  •••• Wn +  '>  (4 6  where t h e 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 rr s e l f i n d u c t a n c e " and t h e 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 rs of m u t u a l 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 t h e system over a p e r f e c t l y  of n p a r a l l e l  c o n d u c t i n g ground  conductors  situated  p l a n e , as shown i n F i g u r e  4.1, t h e s e c o e f f i c i e n t s have t h e v a l u e s :  and  rr  o 4u  1 2  rs  U o 2n  ln  -i  r  where  =  2 in  +  a b  R  rs  henry/meter  (4.8a)  r /  (4.8b)  henry/meter  rs  4n x 10  henry/meter  ( p e r m e a b i l i t y of free  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 c o m p a r i s o n  e q u a t i o n s and t h e T e l e g r a p h In t h e i r Kutznetsov  equations  of  Maxwell's  ( E q u a t i o n s 4.9  o r i g i n a l d e r i v a t i o n of Equation  and 4.12), 4.8  and S t r a t o n v i c h i n c l u d e d a t e r m due t o t h e  proximity effect.  T h i s has been o m i t t e d h e r e  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 ) to account  f o r the i n t e r n a l  inductance of the  conductor.'  (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 r e a s o n s g i v e n i n S e c t i o n 3.4. 4.2  General D i f f e r e n t i a l Two  differential  Equations  methods f o r d e r i v i n g 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 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 multiconductor  g i v e n by B e w l e y  systems.  The  equations  a s s u m i n g lumped p a r a m e t e r s f o r an l i n e as shown i n F i g u r e The  i n h i s work ure d e r i v e d  infinitesimal  on  by  length  of  4.2.  s e c o n d 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  extension  of a  solution  (15) s u g g e s t e d by S t e v e n s o n .  T h i s method has  y i e l d the  much more d i r e c t l y t h a n  required equations  method u s e d by  Bewley.  4.2.1  Differential  General  d x  eg  FIGURE 4.2  Equations  been f o u n d t o the  Method 1  ts»  n-Conductor System w i t h Mutual  Coupling  26  F i g u r e 4.2 shows an n - c o n d u c t o r  transmission line  whose c o n d u c t o r s a r e assumed t o be p a r a l l e l  t o each o t h e r and  to  t h e ground  plane.  The c o n d u c t o r s a r e 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 t h e  figure. 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 o f l i n e , c o n d u c t o r r has t h e f o l l o w i n g L L  rr rs  = self  parameters:—  inductance of conductor r , '  =  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, '  C  r r  =  s e l f capacitance coefficient  C  r g  =  mutual  of conductor r ,  capacitance coefficient  between  c o n d u c t o r s r and s, R  rr  g g  I* s  =  series  r e s i s t a n c e of conductor r ,  =  leakage conductance  of conductor r t o ground,  =  leakage conductance  b e t w e e n c o n d u c t o r s r and s,  '  Q  r r  =  sum o f l e a k a g e c o n d u c t a n c e s  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 o f c o n d u c t o r r ,  Z  r g  =  mutual  =  a d m i t t a n c e o f c o n d u c t o r r and  =  a d m i t 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.  T Y  rr  rs  The l a s t G  rr  G  rs  i m p e d a n c e b e t w e e n c o n d u c t o r s r and s,  s i x parameters =  «rl  =  G  sr  +  "=  g  r2 -g  a r e d e f i n e d by t h e e q u a t i o n s +•••• rs  =  -g s r  6  +  8 rn  27  Z  =  R  =  pL rs  Y . r r  =  G  Y  =  Y  Z  where p  rs  rs  +  r  pL * r r  +  r r  r  =  sr  pC r r  and  pC  =  rs  JOJ f o r s i n u s o i d a l  =  analysis,  rr  r  pC sr  waves.  In the  following  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  sinusoidal  t o be  steady  waves. (17)  The d i f f e r e n t i a l for  the potential  equation  on c o n d u c t o r  +  W  p L  n  i  Equation  +  1  R  1  i  (4.9)  1  W  Differentiating  =  equation)  1 i s  =  b0x  ' (Telegraph  v  p L  1  2  i  2  4.6  +  gives  ....  +  p L  l  n  i  n  (  4  >  1  W By c o m b i n a t i o n the necessary  - ^ 1 ox  =  of Equations  substitution  hi ! 1  +  12^2  Z  4.9  a n d 4.10  and making  from t h e above d e f i n i t i o n s  +  •"'  +  ln n  Z  l  ( 4  '  U )  0  )  28  Similarly,  the current Telegraph  4i,  =  >>Q 1  ' (17) equation 'is  (4.12)  l  +  x  6 t  where i ^  i s t h e sum o f t h e l e a k a g e c u r r e n t s f l o w i n g f r o m  conductor  1 t o t h e g r o u n d 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 c a n o b t a i n  " S  -  Y  11 1 V  +  Y  12 2 V  +  ••  +  Y  ln n V  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 every conductor.  (4.13)  exist for  By g r o u p i n g them and a p p l y i n g t h e 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  in simplified  form:-  where  -D  [v]  -D  [l]  (4,14a)  M " [v]  (4.14b)  Sx"  '11  J  nl  J  ln  nn  and ll  In  nl  nn  ( M a t r i c e s I and V have been d e f i n e d D i f f e r e n t i a t i n g Equation t o x we  i n Section 4.14  4.1)  (a) w i t h  respect  obtain  -° w 2  I t may  (4.15)  be n o t e d 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  the  line). S u b s t i t u t i o n of Equation  D [v] 2  .  [z]  .  [,]  By r e p e a t i n g t h e above using Equation  4.14  D [j] 2  Now  (b) one [I]-  and  Y  rs  rs  .  (b) i n t o 4.15  [V]  sequence of  gives  (4.16) operations  obtains  -  [z]  f o r a l l transmission  Z  4.14  =  Z sr  ~  Y sr  .  line  [I]  systems  (4.17)  Therefore  [z]  Hence  «  the matrices  [ Y ]  Equations simplified  =  Z and Y w i l l  [ l ] •  be  symmetric.  [z]  (4.18)  4.16 a n d 4.17 may now b e w r i t t e n i n  form:-  D  D  and  2  2  [v]  (4.19a)  [I]  (4.19b)  where  (4.20)  4.2.2  General  Differential  Equations  Method I I  /+ Ai •QI Q  e  s  e  +21e •  I  1  L  !  0  A D  e  i -X-  F I G U R E 4.3  Single-Conductor with  Distributed  -CB»  Transmission Parameters  Line  31  F i g u r e 4.3 line  shows a s i n g l e - c o n d u c t o r  w i t h a ground r e t u r n .  as lumped  circuit  d i s t r i b u t e d along  The  line  p a r a m e t e r s a r e n o t shown  e l e m e n t s as t h e y are c o n s i d e r e d the l i n e .  transmission  t o be  E a r t h r e t u r n p a r a m e t e r s have  b e e n 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 i n c r e m e n t s (Ae may  andAi)  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 t h e s e r i e s  impedance  p e r u n i t l e n g t h and Y i s t h e a d m i t t a n c e t o g r o u n d  per u n i t length. I f the small increment A x  approaches  zero,  voltage gradient per u n i t length f o r i n c r e a s i n g x w i l l  ^e *x and s i m i l a r l y  =  Zi  =  4.22  be w r i t t e n : -  Ye  (4.22)  D i f f e r e n t i a t i n g Equation and s u b s t i t u t i n g E q u a t i o n  we  become  (4.21)  t h e c u r r e n t g r a d i e n t may  j&i cot  the  4.21  with respect to x  obtain  De 2  =  ZYe  (4.23)  D i  =  YZi  (4.24)  similarly 2  where  D  = j)_ 3x  In the one  conductor  conductor current,  derivation  has  been  s y s t e m may impedance  the  above  considered.  be and  of  achieved  A development along  Expansion  by  admittance  equations, a  r e p l a c i n g the  by  the  to  the  lines  only multi-  voltage,  appropriate of  that  matrices  given  above  (Eqn's.  4 . 1 9 )  yields  as  where  (Note:  The  Equations  s i g n of the  4 . 2 1  and  voltage  4 . 2 2  is a  and  current  consequence  before.  gradients  of  the  in  reference  ( 1 5 ) conditions  4 . 3  The  chosen  Completely The  by The  assuming term  system are  that  the  implies that  symmetric  Transposed  solution  "completely  n  the  in Figure  and  conductors  transposed" both  the  Z ss Z rs  sr Y ss  r r and  Y  Y rs  (see  s y s t e m we  rr  Y  Line may  a p p l i e d to  s  sr  )  be  "completely  i m p e d a n c e and  transposed  Z  4 . 1 9  are  diagonals  Z  4 . 3 .  Lossless  of E q u a t i o n s  about t h e i r  For and  f o r A e and A i  an  simplified transposed"  n-conductor  admittance  arrays  Section 3.2). may  write  f o r any  r  33  Further i s considered  s i m p l i f i c a t i o n i s p o s s i b l e i f the system  to be l o s s l e s s  R  and  6B  i n Equations 4.8  and  r  R  S  = 0  rs  the terra due to the i n t e r n a l  (see S e c t i o n 3.4) one  L r  = 0  rr  By n e g l e c t i n g  (see Section 3.5), that i s  L  flux  obtains  =  JJo l n **r 2it R  henry/meter  (4.25a)  =  ]j> In r s 211 b rs  henry/meter  (4.25b)  2  a  where L  i s the s e l f - i n d u c t a n c e c o e f f i c i e n t and L i s the rr rs mutual inductance c o e f f i c i e n t f o r the symmetric system. Now is clear  by comparison of Equations 4.2  that  P  r r  -  1 U K  and  and 4.25 i t  p rs  c  r r  -  D  r  r  (4.26a)  o o 1  =  r  U p  2  V  o co  L  = rs  v L o rs 2  (4.26b)  34  where  the v e l o c i t y  1  v  of  light  i n free  space,  fVo" u  o  =  permeability  C  q  =  permittivity of  It  is  useful  of  free free  case the  stage to consider situated  matrix  simple  above a g r o u n d  plane.  coefficient  is  '12  *11  -  = *12  (4.27)  v  '11  where  L  and  L, =  Now  =  =  L , 11  =  L  L  =  L  [c]  =  12  jco  22 21  -1  jco [ p ]  n therefore  the  r e l a t i o n s h i p between the p o t e n t i a l  m a t r i x and i n d u c t a n c e  [P]  space.  at t h i s  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 For t h i s  s p a c e and  [l]  Performing the  - l  •2 '  indicated matrix inversion  yields (4.28)  "2"  2  2  L - L /  -L,  35  where  Y  and  Y  1  =  I  n  =  Y  =  Y  1 2  =  T  Prom E q u a t i o n 4.28 one may  =  Y  j  Y  Now,  by.  OJ  l l  J  12  and  (4.29a) L - L  2  2  o  _JjL_  2  i  (4.29b)  Ll  o f E q u a t i o n 4.20 f o r t h e  12  z  l  Z  x  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  ZY  l  obtain  case  J  J  -0  expansion  s i m p l e two c o n d u c t o r  J  i =  2 1  a)  v and  2 2  X  +  Z Y  +  ZY  X  ZY  X  ZY  X  where  Z  and  Z  1  =  jwL  =  jwL  X  +  ZY  +  ZY  1  X  (4.30)  (4.31  1  (4.31  a)  b)  36  On Equation  4.30  substituting the matrix J  Equations  4.31  and  4.29  into  becomes  -co (4.32) 0  Similarly Equation  4.27  may  be  p ... n  —to  f o r a system extended  p  as  .....  LI n  v L  'In • • • *nn  conductors  give  L  l n  =  and  to  of n p a r a l l e l  ...  l n  L  before  Y  ...  Y  L  In  ...  J co v~2~ Y x  l n *'*  Y  1  L  ln  "  L  In  -1  Again the  values  expanding  Equation  f o r Z a n d Y one o b t a i n s  4.20 a n d  substituting  f o r a system of n  conductors  to  0  0  0  - w  0  (4.33)  o (n  It diagonal  matrix with equal  each d i a g o n a l conductor to  i s important  t o note diagonal  physical  4.4 4.4.1  and  geometry.  The v a l u e  of  of the  The above o b s e r v a t i o n s provided  Equation  apply  t h e s y s t e m i s assumed  Solution  S o l u t i o n and Boundary Equations  necessary  elements.  symmetric.  Differential  differential  that the matrix J i s a  e l e m e n t i s t h e same a n d i n d e p e n d e n t  any number o f c o n d u c t o r s  lossless  x n)  equations  4.19  Conditions  represent  of t h e second  a system o f n order.  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  ordinary  It will  be  constants.  38  A possible solution i s  [v]  =  [M]  W  =  [ ] * [cosh^x]  • [cosh U x ]  S  •  [sinh*x]  +  [N]  +  [ T ] • [sinh^x]  w h e r e M,N,S and T a r e m a t r i c e s - ^ w i t h c o n s t a n t  (4.34  a)  ( 4 . 3 4 b)  terms.  D i f f e r e n t i a t i n g Equation 4.34(a) with respect to  x,  t h e f o l l o w i n g second  order d i f f e r e n t i a l  equation  may  be o b t a i n e d ; -  D  2  [v]  =  [ M ] • [* ] 2  « [coshXx]  +  [ N ] • [* ]° [sinh*x] 2  (4.35)  where  o *2  (4.36)  0  arid where  a  i  +  J^i  ^*  n  ie  P  r o  P g 't'i a  a  o n  constant  the a t t e n u a t i o n constant of conductor the phase c o n s t a n t of c o n d u c t o r  1  1  39  Por the  propagation  a completely constant  will  Therefore the matrix [> ]  D  [v]  2  4.35  (^ ) a n d E q u a t i o n  =  *  2  4.37  +  X  2  Equation  conductor.  reduce t oa  be w r i t t e n : -  sinhYx  [N]  4.34  will  system  (a) i n t o  (4.37)  Equation  we o b t a i n  D  2  [v]  = Y  2  By c o m p a r i s o n is  4.36)  can t h e n  [hi] c o s h t f x  Substituting  and l o s s l e s s  b e t h e same f o r e a c h  (Equation  2  scalar  transposed  (4.38)  [v]  o fEquations  4.38  a n d 4.19  (a) i t  possible towrite  rearranging,  i><> - A • [v] a  where U i s t h e u n i t  matrix.  From t h e c h a r a c t e r i s t i c it  = [o]  i s possible t o obtain the eigenvalues  roots  o fthe matrix J .  conductor  The p r o p a g a t i o n  i s then the square  [o  equation or characteristic constant  f o r each  root o fthe respective  eigenvalue  40  In matrix  J «  propagation  Y  or  r  si i mmp.l . y.  the characteristic  ship  =  D  +  equal  the  J  diagonal  elements  equation y i e l d s ,  f o r the  where  J  r  r  =  1  ,n  (4.40)  rr r  constant  between t h e s e  [v]  | r r  X = + j..J  dependent upon  Equation  diown t h a t  c o n s t a n t o',  The are  i t has been  i s a diagonal matrix with Hence  r r  4.4  Section  matrices  the matrices  S a n d T ( E q u a t i o n 4.34(b))  M a n d N; t o f i n d  matrices, i t i s necessary  to  the relationdifferentiate  4.34(a) w i t h r e s p e c t t o x a n d o b t a i n  =. V [M] s i n h T x  Equation  D  By  4.21  +  X [N] c o s h ^ x  may  [V] =  comparison  (4.41)  be e x p a n d e d t o t h e m a t r i x  [Z] • [i]  of Equations  (4.42)  4.41  a n d 4.42 we  obtain  [z] * [ i ]  =  t  [M] s i n h Y  x  +  form  [N] coshXx  41  On p r e m u l t i p l y i n g  0]  -1  [i]  the  =  solution  u [z] It  _  1  i s usual  the receiving  voltages  and c u r r e n t s  as  obtain  f o r Iij  a complete  sides  may b e  [M] sinhVx  specify  to  both  + Y [z] ~  end c o n d i t i o n s . are here used  e q u a t i o n by  written  i n transmission  solution  of this  1  line  [N] cosh^x  (4.  problems t o  The r e c e i v i n g e n d  as t h e boundary  of the differential  conditions  equations  follows:-  when x = 0 ( a t t h e r e c e i v i n g  end)  (4.44a)  and  (4.44b)  where V.  n  at  and I  n  the receiving end.  42 By s u b s t i t u t i o n E q u a t i o n 4.34  ( a ) and 4.44  and  [v]  =  JVJ  [ij  =  * [z]  M  (4.46)  solution  and 4.46  c o s h &x  _  obtain (4.45)  1  T h i s may C,D  we  may  i n t o E q u a t i o n s 4.34 [z]  +  [VJ  be o b t a i n e d by  .  sinhXx  substitut-  ( a ) and  sinh&x  |^IJ  4.43.  (4.47a)  coshSx  (4.47b)  be w r i t t e n i n t h e f a m i l i a r f o r m o f A , B , .  —  :  "A —  I  [ij  +  p a r a m e t e r s (by u s i n g p a r t i t i o n e d  "v"  into  M  The f i n a l i n g E q u a t i o n s 4.45  o f t h e boundary c o n d i t i o n s  c  .  B"  i~  '  D  \  matrices  ) :-  —1  (4.48)  •  1  where [A]  M [c]  [uJ j  c o s Px  _[z]  (4.49a) (4.49b)  s i n px -1  sin  Px  (4.49c)  43  [u]  and  Equations  4.48  and  cos  4.49  program d e s c r i b e d  in Section  4.4.2  of S o l u t i o n w i t h  Comparison Circuit  (4.49d)  Bx  a r e used  i n the  computer  (4.6).  the  Generalized  Constant  Standard t w o - p o r t n e t w o r k may  c i r c u i t theory be  .(15)  represented  by  describes the  how  any  following  equations  V  A  B~  C  D  s I  —  s_  where V  R and  R  =  sending  end  voltage  of the  two-port  network,  sending  end  current  of the  two-port  network,  receiving  end  voltage  of the  two-port  network,  receiving  end  current  of t h e  two-port  network.  In g e n e r a l relation  f o r a t w o - p o r t network t h e  between A,B,C,D p a r a m e t e r s AD  but  _  f o r the  -  more s p e c i f i c  BC  =  applies 1  c a s e o f a symmetric  network A  =  D  following  two-port  44  The that  the  equal that  and pair  above e q u a t i o n s  current  a t one  opposite (See  terminal  to the  Figure  are of  current  restricted  a p a i r must a l w a y s at the  of  lines  restriction entering equal This  the  one  still  at the  to the i s the  terminals  transmission  from  t o n as holds  of the  c a s e f o r our  represent  ground r e t u r n  the  may  s y s t e m by  be  terminal  that the  sum  at the  transmission conductors  4.4(b).  of the  of a p a i r  currents  a p p l i e d to  i n c r e a s i n g the  shown i n F i g u r e  upper t e r m i n a l s  sum  other  be of  4.4a).  Two-port n e t w o r k t h e o r y multicbnductor  by r e q u i r i n g  and  a number  The  currents  must a l w a y s  lower  terminals.  s y s t e m where t h e the  be  lower  upper  terminals  paths.  no o n  (a)  FIGURE 4.4  Two-l'ort  Networks  45  For above  the symmetric,  e q u a t i o n s may  and  [A]  •  [D]  be w r i t t e n  -  [B]  •  By. c o m p a r i s o n clear  that  multiconductor  system,  i n matrix  as  form  [A]  =  [D]  (4.50a)  [C]  =  [u]  (4.50b)  o f E q u a t i o n s 4 . 4 9 ( a ) and  E q u a t i o n s 4.50  the  are  (d) i t i s  satisfied.  Substitution  o f E q u a t i o n s 4.49  [u]  +  into  4.50(b)  produces  so t h a t  our  cos 6 2  f o r a symmetric  4.5  Addition  was  two-port  o f Lumped  To  it  obtain  2  4.49)  (Section  line 5.3)  with that  points  along  system  were d e r i v e d  the  express-  of the m u l t i -  d i s t r i b u t e d compensation, a d d i t i o n a l s e r i e s or t o be  installed  parallel,  at c e r t a i n  line.  allow  containing  satisfies  Parameters  components were r e q u i r e d  To  [u]  network.  lumped  the  =  satisfactory operation  transmission  found  sin B  s o l u t i o n (Equation  ions  conductor  [ul  the  lumped  f o r each  simplest  computer  components,  configuration.  solution for a  partitioned For the  matrices  (19)  two-conductor  46  c a s e t h e method i s as f o l l o w s : (a)  Parallel  Capacitors  1  1  V  l  V  o  O  1  O 2  ^7  h  2  FIGURE 4.5  Lumped  The e q u a t i o n s Figure  L  4.5 may  V Y  1  l  Capacitors  f o r t h e system i l l u s t r a t e d i n  be w r i t t e n  -  V  ~  2  =  !•  =  V  Parallel  1  V h  +  I  c  =  I,  +  Y ( c  V l  - V ) 2  *2 = h ~ h = h ~ c l ~ V Y  (Y  '  (4  51)  47 and  i n partitioned  matrix  form  0 V  (b)  Series  0  0  0  1  0  0  1  0  2  Y  1%  ' 0  -Y  2  -Y  c  Y  c  c  c  (4.52)  Reactors  FIGURE 4 . 6  Lumped  Series  Reactors  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 system  i n F i g u r e 4.6 a r e : -  i  =  V¥ •  -  V  2  ~  l  V  V  2  H = h  hhl  +  +  J-  7 2 S2  T  f o r the  and  in partitioned  m a t r i x form  0  i  r  V  2  I  I-  } 2  (c)  Parallel  0 0 0  0  0  1 1  7  0  0  Z  1  0  0  1  ~ l v  S2  h  Reactors  FIGURE 4 . 7  Parallel  Reactors  The  voltage  s y s t e m i n F i g u r e 4.7  V'  =  and  current  equations  for  the  are:V,  Ll J  2  where  Q  L2  and  Q*  reactors  are the  1 and  values  o f r e a c t i v e power a s s o c i a t e d  2 r e s p e c t i v e l y in Figure  In p a r t i t i o n e d  v  (4.55)  Q  Ll  and  (Q*  L2  i  1  0  0  1  matrix  ' 1 1 1  with  4.6)  form  0  0  0  0  1  0  0  1  1  \ l 0 The symmetric  three  —  r  0  i i i  Y  1  L2  configurations given  two-port networks,  partitioned  matrix  I  satisfies  and  above a r e a l l  f o r each c a s e  Equation  4.50.  the  For (Example  instance  by c o n s i d e r a t i o n  of Equation  4.56  (c))i t i s possible to write  [A]  .  [„]  .  1  0  0  1  and [A]-[DJ  wh e r e  .  [B]  thus providing  [ B J . [c]  -  0  0  0  0  -  [u]  and  L1 0  a check o f the v a l i d i t y  Now  f o r a transmission  Equations  for  i f t h e number o f c o n d u c t o r s  increased  V  2  V  3  V  4  H H J  3  to four,  4.52,  4.54  Equation  of four  a n d 4.56  4.56  would  L2  of Equation  line  conductors, example  °  Y  4.56.  o r more  may b e  i n Figure  expanded 4.6  was  become  1  0  0  0  I 0  0'  0  0  0  1  0  0  |0  0  0  0  V  2  0  0  1  0  0  0  0  0  V  3  0  0  0  1  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  1  0  0  0  1  Y  L1  0 0 0  Y  L2  0 0  1  1 10  ~ l~ v  __4 V  J  l  1  Y  L3  0  0  1 0 | 1  0  h h  51  With 4.48)  and  and  4.56)  may  be  4.6  the  Computer  the  by  and  the  diameter  with  (LGTH) and  understanding  including  each the  of the  (Equations  the  4.52,  computer  flow-chart given  4.54  program  program the  conductor,  incremental  is initiated  geometrical  the  is  From t h e  geometrical  position  potential  using  After  i s calculated  4.18.  I f the  J  Equations  computing  matrix  using  matrix  be  are  us  tests  whether  i t i s a diagonal 4.33).  satisfied  will  the  both  the  conducted  Only  on  matrix  after  and  the  and  which  and  the  one  diameter  same i n b o t h i s no  described J matrix  with  of  coefficients  4.8.  Y matrices,  system as  in  length  electromagnetic  equal  a l l these  computation  line  methods g i v e n  the  reading  performed.  Z and  condition  Further  (Equation  4.2  the  i s not  stopped  i s a necessary  and  by  (ILGTH) over  calculation  the  Figure  position  transmission  length  in  be  description.  current  program w i l l  (Equation  c o m p u t e r p r o g r a m may  step-by-step  data  determined  (this  form,  t o b o t h t he  computer  each conductor, are  matrix  The  and  equations  equation  ease.  following  of  line  Programming  referring  system  voltage  transmission  in partitioned  developed  gained  the  lumped component  An  4.8  both  proceed.  the  in  J  Equation  cases  longer  then  symmetric  in Section to  the  4.2).  determine  diagonal  restrictions  elements are  52  By constant  using Equation  (Equations  The 1^)  and  the  using Equation  will  the  NLGT1I. of  lumped component stored.  4.48,  (M i s a c o u n t e r  r e c e i v i n g end  whether the  sending  c e a s e when N t i m e s taat to  i s at t h e  r e c o r d the  by  end  has  until  end.  the  bypass  the program w i l l  add  the  necessary  lumped  and'D arrays and end  ILGTH  equals  r e c o r d the  number  l o o p 2 and  decide  l e n g t h of the  been r e a c h e d  i n the  lumped  information  the  will line,  program line).  (N.ILGTH ^ components  regarding  the  NLGTH ( t h e d i s t a n c e t o  the  computer d e c i d e s  which t y p e  been added, d e t e r m i n e s t h e c a l c u l a t e s the v o l t a g e s  s i d e of t h e  2).  1.  component and  component), t h e  component has  M times  (N i s a c o u n t e r w i t h i n t h e  not  reading  currents  Computations  total  has  additional  and  number o f c a l c u l a t i o n s a l o n g  loop  into  a t a d i s t a n c e NLGTH f r o m  end  around  read  The c a l c u l a t i o n s  program t o  been r e a c h e d .  ILGTH e q u a l s  total  After  sending  end  (V^  i t is possible,  line.  sending  proceeding  next  data  computer w i l l  sending  However, i f t h e LGTH)  i s i n s e r t e d , are  program a r r i v e s  the  line.  (NGLTH)  c a l c u l a t i o n s performed around loop  When t h e the  transmission  r e c e i v i n g end  w i t h i n the  the  current matrices  transmission sending  phase  store  to determine the v o l t a g e s  towards t h e  incremental  and  Prom t h i s  multiconductor  proceed  f o r the  d i s t a n c e from the  computer and  along  4.49)  terminal voltage  where t h e f i r s t the  to c a l c u l a t e the  ( f i ) , i t i s p o s s i b l e t o d e t e r m i n e and  A,B,C,D:arrays  and  4.40  component by  of  lumped  appropriate  and  using  c u r r e n t s on one  of  the  A,B,C the  START  \ READ LGTH, ^XlLGTH AND \SYSTEM DATA  COMPUTE  Z AND Y MATRICES  COMPUTE J MATRIX  0  M=  COMPUTE V AND I ALONG LINE  i  COMPUTE VOLTAGES AND CURRENTS  N =N+1 M=M + I  COMPUTE AB CD MATRICES  READ ADDITIONAL COMPONENT .AND NLGTH  NO  YES  iWRITE AND PLOT SOLUTION  STOP  FIGURE 4 . 8 Computer F l o w C h a r t  54  Equations  4 . 5 2 , 4,54 o r 4 , 5 6 . The  (Figure  voltages  4 . 8 ) now  '  and c u r r e n t s  determined  become t h e t e r m i n a l  i n loop and I  conditions V  LI  Li  (Equation  4 . 4 8 ) when t h e c o m p u t e r  returns  1  t o t h e main  body o f t h e program. The equals end  procedure  that  LGTH,  of the l i n e ,  scaled 4.7  i s repeated  at which point  Voltage  phase,  around  the surface  of the conductor  line  has one c o n d u c t o r p e r  of the conductor provided the  diameter  to the spacing  between  i s small.  conductor  line  appreciable  conductors  are i n close proximity  field  voltages  voltage.  distortion  without  may  exceeding  of a multi-  t o each  result,  other,  a l l o w i n g much  the d i s r u p t i v e  higher  critical  (24) The  allowable a  i s terminated.  Difference  H o w e v e r , when s e v e r a l  i4  sending  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  uniform  line  the  ILGTH  the solution i s printed,  and p l o t t e d and t h e computer program  When a t r a n s m i s s i o n  phases  N times  i s t h e program has reached  Maximum A l l o w a b l e  ratio  until  voltage  particular  introduced  f o l l o w i n g a n a l y s i s determines d i f f e r e n c e between  phase w h i l e  b y S. C r a r y  t h e maximum  any two c o n d u c t o r s  of  making use o f t h e assumptions  i n h i s w o r k on b u n d l e d  conductors.  (5)  55  These (a) due  on the  Contribution to the electric  to the other (b) the other  assumptions a r e : -  phases  The e l e c t r i c  conductors  conductor  under (c)  and.the ground  ensure t h a t t h e conductor  is  neglected. 22.  at t h e c e n t r e  i s uniform  near  of charges surface  on e a c h  remains  conductor  an e q u i p o t e n t i a l  phenomenon i s e x p l a i n e d a d e q u a t e l y  I t i s assumed  that a l l the charge  of a cylindrical  conductor.  hy  F I G U R E 4.9  charges  c o n s i d e r a t i o n and  The movement  This  intensity  i s neglected,  associated with  i n t h e same p h a s e  to  Reference  field  plane  field  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  in  i s concentrated  56  The of may  conductor be  electric  1 due t o c h a r g e s  i n t e n s i t y at the on c o n d u c t o r s  position  2,3,....,n  written  E ,  )  (2E  where E ^ and E in  field  (SE  +  2  are the e l e c t r i c  t h e X and Y d i r e c t i o n s The  electric  V  )  field  respectively field  (4.57)  2  intensity  components  (See F i g u r e 4 . 9 ) .  i n t e n s i t y at conductor 1  due t o c o n d u c t o r n i s  E  Qn  ln 2 l x e  (4.58)  o ln d  wh e r e n d^ C  and  n  q  =  charge  on c o n d u c t o r  =  distance  =  permittivity  between  and  E  where 0 ^ field  n  x y  =  E ,  =  E ,  conductors  of free  The x and y components  E  n,  of E ^  1 and n.  space. N  are  ln  cos 0, In  (4.59a)  ln  s i n 9, ln  (4.59b)  i s t h e a n g l e between  intensity vector  the x axis  at conductor  1.  and t h e  electric  57  Expansion of Equation 4 . 5 7 g i v e s  E  x  =  1  2itc  \ I ( Q  !IU  o  Q  2  cos  0  + . . . + Q  1 2  d  i n  sin 6  2  E  n«  l/Al nl COSe  +  +  VV nl  N %  d  sin  0, *1 d , nl  sin 0 , nl  2 l n  +.  )  2 (  4  <  6  Q  )  m  Vl d  9  l n  + ... +  1 2  d  Similarly  cos  N  C0S  n,n-l  °n,n-l V  + ... + Q , sin 0 , *n-l n,n-l d , n,n-l  +  l  2  (4.61)  Now the maximum e l e c t r i c f i e l d surface of an uncharged  i n t e n s i t y at the  conductor 1 (considered to be a  c y l i n d e r ) placed i n a p r e v i o u s l y uniform f i e l d E^ can be written^ E  =  m l  2 E  (4.62)  1  If t h i s conductor i s given a charge maximum f i e l d  intensity will  E , m l  =  , the  become  2 E, 1  +  1 2tt c R, o 1 8  (4.63)  58  Similarly  E  ran  f o r the nth conductor  =  2  E  n  (4.64)  Qn 2 ire 11 o n  +  where R  n  To each  conductor  determine  = the radius of t h e n t h conductor  e v a l u a t e t h e maximum p o t e n t i a l Equation 4 . 6 3 , i tw i l l  from  t h e charge  on e a c h  [v] =  If  [p]  To c o m p l e t e of  Evaluation the  , ....,V  n  as  then  between  demonstrated  two-and  both  so t h a t  follows  JjpJ  by  we o b t a i n  <- > 4 65  i ti s necessary  the line  charges  potential  immediately  from  t o assume  c a n be  calculated,  gradientsf o r Equation 4.63.  o f t h e maximum a l l o w a b l e v o l t a g e  conductors  i n Appendix  a four-conductor  sides  the solution  Determination difference  (Equation 4.3)  o f t h e maximum c o n d u c t o r  conductors  Equation 4 . 3 : -  from  M  1  near  be n e c e s s a r y t o  [pj  we p r e m u l t i p l y  [c] = H" values  conductor  gradient  B.  case.  necessitates Solutions  a graphical are given  solution  f o rboth  a  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  Modern E1IV,  long  to  line).  Although  have been s u g g e s t e d a  500kV v o l t a g e  Systems  distance transmission  o f l a r g e c a p a c i t y have a v o l t a g e (line  Line  rating  of around  much g r e a t e r v o l t a g e  ( S e c t i o n 5.5), i t w s  line  so  made between e x i s t i n g  that a d i r e c t  series  comparison  compensated l i n e s  500kV  ratings  decided  a  lines  to  study  could  and  be  the  proposed d i s t r i b u t e d - c o r n p e n s a t i o n system. Before in  i t s entirety,  on  two-  lines the  and  with  line  the  three-phase  four-conductor, single-phase ground  return.  parameters  For the  were b a s e d on (14)  and  two-conductor those  case  suggested  Galloway.  by  the  (25)  Conductor  556.6 MCM  size  Conductor r e s i s t a n c e . . . .  0.168  Conductor  1.5  spacing  conductor  Both conductors F i g u r e A.1)  to each  transmission  0.950  Transmission  relative  performed  Conductor diameter  Average  (see  considered  p r e l i m i n a r y c a l c u l a t i o n s were  Westinghouse Engineers  ground  s y s t e m was  other  are  and by  line  the  held  ft.  50 f t .  length  600  same h e i g h t  above  insulated spacers.  miles  the  position The  ACSR  ohms/mile  height  in a fixed  inches  above  values  of  conductor  for  the  initial  for  determining  calculation the  The calculations, Pour  diameter  line  spacing  only.  The  parameters  three-phase  described  conductors  and  optimizing  i s given  5.4,  Conductor  resistance  250  between  Spacing  between  Average  conductor  The  phase the  two  array  upper  as  that the  could  be  used  conductor  feet  length  600  miles  above  the  at  heights  Figure  5.10).  well  as  two  at  to produce and  equal  (see  the  Within  lower  regular intervals groups were  distributed  diameter  feet  50  conductive  so  foot  height  line. other  ohms/mil  feet  together  two  ACSR  40  were connected The  ...1.0 1.5  phases were  in a flat  conductors  phases  line  the  inches  MCM  .....0.235  spacing  Transmission  ground  .....0.806  size  spacing  in  5.2  phase  Conductor  Horizontal  in Section  consists of:-  Conductor diameter  Vertical  procedure  t r a n s m i s s i o n system used  in Section  per  have been s e l e c t e d  a compensation  size  given  conductors along  insulated  capacitance  the  from  between  effect.  above were  each  each  them The  selected  from  (14) a manufacturer's  table  at  end  the  receiving  spacings the  have  initial  diameter  and  as  been s e t  by the  conductor  the  selection  arbitarily  calculations. the  using  at  conductor  criterion. the  Optimum v a l u e s spacings  will  current Conductor  above v a l u e s of be  both  for  the  determined  in  Section  5.2  5.3.  Two-Conductor  System  Initial conductor (see at  c a l c u l a t i o n s were p e r f o r m e d  single-phase  Figure  transmission  2.5) h a v i n g  line  Voltage  t h e f o l l o w i n g boundary  V,  return  conditions  288.7 kV ( 5 0 0 k V )  of conductor  2, V  288.7 kV  2  Load Load  250 power The  factor voltage  spacing By  plotted  (given  the line  i n Figure  summarised  calculation  set to  was 1.5  i n Section  i n Section  Sending  4.6, t h e v o l t a g e s  the salient  (6)  These points  angle  between  along  41.6° conductors  at  any p o i n t  the line  177 kV  V  sending  end  257 kV  sending  end  426 kV  and  results are may b e  .48.4°  end phase  Maximum v o l t a g e  Vg  conditions  as f o l l o w s  Power angle  1  zero.  5.1) i n t o t h e  were determined.  5.1 w h i l e  1  feet.  s u b s t i t u t i o n o f t h e above boundary  computer program d e s c r i b e d along  lagging  d i f f e r e n c e between conductors  for this  system parameters  currents  MVA  0.94  2 a t t h e r e c e i v i n g e n d was a r b i t a r i l y  Conductor  and  earth  t h e r e c e i v i n g end. Load v o l t a g e  and  with  f o r a two-  leading  62  DISTANCE  FROM  FIGURE 5.1(b)  SENDING  END  (MILES)  Current P r o f i l e s Along (Two-Conductor Case)  the Line  63  I  sending  end  374  amps  I  s e n d i n g end  393  amps  The voltage By  difference  examination  voltage V losses  will  of primary  approximately  not s a t i s f a c t o r y  To  correct  that i f the t h e corona  the voltage Therefore,  the voltage  t h e above  system  (426 kV) and t h e  (177 kV) exceed  the situation  between  the limitations.  several  methods c a n  used:1.  may b e r e d u c e d regular  The v o l t a g e d i f f e r e n c e by i n s t a l l i n g  intervals 2.  may a l s o and  zero.  as b o t h  conductors.  290 kV, t h e n  be e x c e s s i v e w h e n e v e r i s greater than  i s t h e maximum  t h e two  o f F i g u r e B.2 i t i s c l e a r  maximum v o l t a g e d i f f e r e n c e  be  interest  o f 177 kV b e t w e e n  exceeds  conductors is  result  The v o l t a g e d i f f e r e n c e  conductors  spacing  and i n c r e a s i n g Now  system.  diameter  spacing  and  the conductor  difference  diameter.  o f t h e above methods  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 .  tend  conductors  may be i n c r e a s e d b y v a r y i n g t h e c o n d u c t o r  a study  any method  must a l s o  between  T h e maximum a l l o w a b l e v o l t a g e  between  words  capacitors at  by d e c r e a s i n g t h e c o n d u c t o r  the conductor  3.  allowance  lumped p a r a l l e l  conductors  along the l i n e ,  be r e d u c e d  increasing  between  used  t o overcome  t o reduce  t h e power  the voltage angle  must i n c l u d e In other limitations  of the transmission  64  Method 1 g i v e n  above may be used  voltage  gradient  voltage  d i f f e r e n c e between c o n d u c t o r s  ever,  this  at t h e conductor  method w i l l  shown i n t h e f i g u r e ,  surface  by r e d u c i n g t h e  (Figure  because t h e c u p a c i t i v e c u r r e n t  6, as flowing  Although the c a p a c i t -  ance between c o n d u c t o r s has been i n c r e a s e d  in  5 . 3 ) . How-  a l s o i n c r e a s e t h e power a n g l e  between c o n d u c t o r s has been r e d u c e d .  parallel  to reduce the  c a p a c i t o r s i t .does n o t r e s u l t  by a d d i n g  i n an o v e r a l l  lumped increase  capacitive current. Application  o f Method 2 t o t h e t r a n s m i s s i o n  system produces t h e d e s i r e d r e d u c t i o n d i f f e r e n c e while It voltage  t h e power a n g l e  will  of the voltage  remains  be shown i n S e c t i o n  constant. 5.3.3 t h a t  d i f f e r e n c e between c o n d u c t o r s w i l l  compensation.  To maximize t h e a l l o w a b l e  give  voltage  between c o n d u c t o r s and hence maximize t h e s y s t e m  a large  the best difference compensation,  Method 3 may be u s e d . By the  optimum  system w i l l difference. give while  c o m b i n i n g b o t h Methods 2 and 3 ( F i g u r e 5.2)  system p a r a m e t e r s may be f o u n d be o p e r a t i n g  a t maximum a l l o w a b l e  T h i s maximum a l l o w a b l e  voltage  t h e maximum p o s s i b l e c o m p e n s a t i o n satisfying  conductor  surface  such t h a t t h e  the voltage  gradient  f o r a l l points  along  voltage  difference  (minimum power  will angle)  c o n d i t i o n at t h e the l i n e .  The  optimum v a l u e s  o b t a i n e d from  F i g u r e 5.2  a r e g i v e n i n T a b l e 5.1  Conductor Spacing ( F t . )  2. 55  130  1.0  2. 50  110  0.5  2.40  expanded c o n d u c t o r s  of  (Reference  2.1 i n c h e s .  n e c e s s a r y t o reduce surface  to the manufacturers  expanded c o n d u c t o r  approximately  78  Two-Conductor L i n e Parameters  Reference  largest  Max. V o l t a g e (kV)  1.5  TABLE 5.1  the  Conductor Diam. ( I n s . )  table f o r  14, pp.. 50-51) shows  that  made t o d a t e has a d i a m e t e r I t would t h e r e f o r e be  the v o l t a g e gradient at the conductor  to a value corresponding  to a p r a c t i c a l  conductor  diameter. A common method o f e f f e c t i v e l y v o l t a g e g r a d i e n t i s t o use " s p l i t " This  i s d i s c u s s e d : i n S e c t i o n 5.3.  reducing the  or "bundled"  conductors.  66  CONDUCTOR FIGURE 5.2  DIAMETER  (INCHES)  C o n d u c t o r D i a m e t e r and Maximum V o l t a g e D i f f e r e n c e (Two-Conductor C a s e )  o  o Q\  i  0-0  i  50  10-0  PARALLEL FIGURE  5.3  :  CAPACITANCE  1  1  150  20-0  '30 25-0  (JJF)  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 t h e L i n e a t 50 M i l e I n t e r v a l s (TwoC o n d u c t o r Case)  67 5.3  Pour-Conductor  5.3.1  Initial  Investigations  The were again except in a  parallel lagging  initial  boundary  used  that  System  conditions  as t h e l o a d  the load  conditions  i n Section  5.2  a t t h e r e c e i v i n g end,  was now d i v i d e d b e t w e e n t w o c o n d u c t o r s  such t h a t  power  assumed  the load  p e r c o n d u c t o r was 1 2 5 MVA a t  f a c t o r o f 0.94 ( m i n u s  20 d e g r e e s ) .  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  •were a s s u m e d t o b e 1.5 f e e t  a n d 1.0 f o o t  F o rt h e  spacings  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 plotted  results of the i n i t i a l  i n Figure  Power  angle  Sending  5.4 a n d may b e s u m m a r i s e d (6)  1  and V  difference  (Vg-V^)...87 kV 262 kV  sending  end  344 kV  sending  end  2 1 5 Amps  a n d 1^ s e n d i n g  end  2 1 9 Amps  2  Advantages 1. two  conductors  ant  increase  and  41° ( l e a d i n g )  end  1^ a n d I  gradient  angle  sending  2  and  Ig  as f o l l o w s : 49°  end phase  Maximum v o l t a g e V  c a l c u l a t i o nare  over the two-conductor  T h e maximum v o l t a g e has been  o f power  reduced  angle  has been reduced  by t h e i n c r e a s e  6.  system  include:-  d i f f e r e n c e between any  b y 100 kV w i t h o u t The c o n d u c t o r  both by t h i s  signific-  surface  reduction  i n t h e number o f c o n d u c t o r s  voltage  i n voltage (seeAppendix B ) .  68  FIGURE  0  5.4(a)  100 DISTANCE  Voltage P r o f i l e s Along (Four-Conductor Case)  200 FROM  FIGURE 5.4(b)  300 SENDING  400 END  the Line  500 (MILES)  Current P r o f i l e s Along (Four-Conductor Case)  the Line  69 2. has  an e f f e c t  An i n c r e a s e similar  i n t h e number o f  conductors  to increasing the conductor  However,  by u t i l i z i n g  possible  to keep t h e c o n d u c t o r  diameters.  a l a r g e r number o f c o n d u c t o r s diameter w i t h i n  i t i s  practical  limits. The increase voltage  difference.  In  must ors. the  minimum exist  The  i n Section  the Conductor  order  to obtain  power a n g l e )  surface  i n Section  voltage 4.7  the capacitive current  between  conductors By  Section  4.7  spacings  maximum  compensation  voltage  will  gradient  spacings  i n Appendix  B.  (power  difference  i n Section  5.3.3.  given i n  the conductor difference.  The  d i f f e r e n c e and t h e r e q u i r e d  may b e d e t e r m i n e d  by  c o n d i t i o n as  and t h e v o l t a g e  t h e maximum v o l t a g e  conduct-  be l i m i t e d  compensation  become a p p a r e n t  voltage  voltage  and t h e two l o w e r  i t i s possible to optimize  maximum a l l o w a b l e conductor  and Spacings  a p p l i c a t i o n of the equations  to obtain  may b e  5.3.2.  and e v a l u a t e d  angle),  of the  difference,between  r e l a t i o n s h i p between t h e system  will  slight  t h e maximum a l l o w a b l e  maximum a l l o w a b l e  conductor  t o be a  disadvantage  Diameter  between t h e two u p p e r  This  discussed  this  the voltage  as d i s c u s s e d  Optimizing  appears  caused by t h e decrease  However,  by m a x i m i z i n g  conductors  (i.e.  disadvantage  i n t h e power a n g l e  overcome  5.3.2  only  from  Figure  5.5.  70  ki 40  DIAM  = 0-606 INS = 320 KV  k ki  30-  CD  S o  20  1 10 0-0  0-5  1-0  1-5  HORIZONTAL F I G U R E 5.5  SPACING  Optimization  2-0  2-5  (ft)  o f Conductor  Spacings  ki o 5: ki 140 cr. ki k k  Q  ki 120 to  o  106 100  HORIZONTAL VERTICAL  SPACING SPACING  7-0(ft)  2-0 (ft)  1 60 0-25  0-50  0-75  CONDUCTOR F I G U R E 5.6  1-00 DIAMETER  Optimization  1-25 Y35 (INCHES)  of Conductor  Diameter  1-50  71  Figure the  5.5 has been d e r i v e d  maximum a l l o w a b l e  horizontal gradient  voltage  and v e r t i c a l  (Appendix B ) . were v a r i e d  surface  spacing  using  the voltage  as t h e l i m i t i n g  B o t h t h e h o r i z o n t a l and v e r t i c a l  spacings  5.5 shows t h a t t h e s p a c i n g s  f o r t h e maximum p e r m i s s i b l e Horizontal Vertical  voltage  spacing  2.0 f e e t  spacings,  aim a t t h i s  means t o i n c r e a s e  are:-  1.0 f o o t  t h e maximum  d i f f e r e n c e between c o n d u c t o r s Our  required  difference  spacing  With these  point  permissible  i s 36 kV.  i s t o use e v e r y  t h e maximum a l l o w a b l e  voltage  i s to reduce  again  to Section  4.7, i t i s o b v i o u s  t h a t t h e c o n d u c t o r d i a m e t e r £>lays an i m p o r t a n t  In  evaluation  of the conductor  f a c t , a s the diameter  decrease voltage This  (Equation  surface  increases  4.63),  so t h a t  t h e maximum  d i f f e r e n c e between c o n d u c t o r s may be  and  V  are  the respective voltages  Figure  varying  B. 3) .  part i n  voltage  the voltage  i s shown g r a p h i c a l l y i n F i g u r e 4  effect  t h e power a n g l e 6 ) .  Referring  the  possible  difference  between c o n d u c t o r s t o maximise t h e c o m p e n s a t i o n (that  criterion  from 0.5 t o 2.5 f e e t as shown i n t h e f i g u r e . Figure  voltage  determining  d i f f e r e n c e at a p a r t i c u l a r  conductor  at the conductor  by  gradient.  gradient allowable increased.  5.6 f o r v a l u e s  f r o m 280 kV t o 350 kV, where of conductors  will  o f V^  and V  4  3 and 4 ( s e e  As  the diameter  of the conductors  t h e maximum v o l t a g e d i f f e r e n c e point  along the l i n e  plotted  will  between c o n d u c t o r s  decrease.  T h i s curve i s a l s o  minimum c o n d u c t o r  diameter  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  the  i s determined  s a t i s f y i n g the  at a l l p o i n t s along the  by t h e i n t e r s e c t i o n  o f t h e two c u r v e s i n  figure. Under t h e l o a d  exceed  350 kV ( F i g u r e  1.35 i n c h e s .  test,  5.4(a))  For t h i s  voltages  and  point  diameter  will  along the l i n e  commence.  (This  i t i s possible  exceeds t h i s  critical  do n o t  so t h e minimum d i a m e t e r  imum o f 106 kV between any two c o n d u c t o r s . any  a t any  i n F i g u r e 5.6. The  line  i s increased  will  be  t o have a max-  I f the voltage at  v a l u e corona d i s c h a r g e  voltage w i l l  be l o w e r  f o r moist  air) . The conductor  diameter  hence a l t e r question  q u e s t i o n now a r i s e s will  effect  the conductor  whether an i n c r e a s e i n  the curves  spacings.  i t has been f o u n d  that,  o f F i g u r e 5.5 and  On i n v e s t i g a t i n g . t h i s  as t h e c o n d u c t o r  i n c r e a s e d , t h e c u r v e s o f F i g u r e 5.5 a r e d i s p l a c e d but  still  and  approximately  voltage 2.0 f e e t  retain  t h e same r e l a t i v e t h e same shape.  gradient w i l l  and a h o r i z o n t a l A second  four-conductor and  still  line  with v e r t i c a l  position  vertically  t o each  other  T h e r e f o r e , t h e maximum  occur at a v e r t i c a l  s p a c i n g o f 1.0  investigation  spacing of  foot.  was p e r f o r m e d  with a conductor  and h o r i z o n t a l  diameter i s  diameter  f o r the  o f 1.35 i n c h e s  s p a c i n g s o f 2.0 f e e t and  73  600r  0  100 DISTANCE  200 300 FROM SENDING  FIGURE 5 . 7 ( a )  100 DISTANCE  200 FROM  FIGURE 5 . 7 ( b )  400 500 END (MILES)  600  Voltage 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)  300 SENDING  400 END  500 (MILES)  Current P r o f i l e s Along the Line ( F o u r - C o n d u c t o r O p t i m i z e d Case)  1.0 f o o t ing  respectively.  The v o l t a g e  difference at the receiv-  e n d was s e t t o 1 0 6 k V i n p h a s e w i t h  the load  voltage  a n d Y .. 4 The  results  of this  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 a n d may b e s u m m a r i s e d a s f o l l o w s : Power a n g l e (6) 41.1° Sending  end phase  Maximum v o l t a g e V  and V  Vg  and  1^  and I  1  and  5.3.3  2  2  I  angle  48.9°  difference  106 k V  sending  end  268 kV  sending  e n d . . .'  341 kV  sending  end  456. Amps  sending  end  1 4 6 Amps  U s e o f Lumped  Parameters  As i n d i c a t e d by t h e a b o v e improvement spacings factory  and  has been  obtained  and diameter.  1.  Power angle  2.  Sending  3.  The c u r r e n t s Figure  5.8(a) w i l l  and h o p e f u l l y  (a)  illustrates  the line.  numbered p o i n t s  quite  angle  large  been  i s a large  unsatis-  leading  angle  3 a n d 4 ( 1 ^ a n d 1^, s e e  reduced, b u t n o t s i g n i f i c a n t l y .  t o overcome  these  obstacles,  insight  into  plots of the voltage The change  on t h e l o c u s  slight  t  to possible solutions.  the locus  a  reasons:-  as an a i d t o g a i n  lead  only  system i s s t i l l  i n conductors  an a t t e m p t  be u s e d  along  i s still  B.3) have  lem  vectors  The p r e s e n t  end phase  results,  by o p t i m i z i n g t h e c o n d u c t o r  because of t h e f o l l o w i n g  In  (leading)  i n a vector  i s t h e change  Figure  the prob-  Figure  5.8  and c u r r e n t between  experienced  CURRENT >10  VOLTAGE  FIGURE  5.8(a)  SCALE: SCALE:  Locus P l o t o f Voltages Along t h e Line  FIGURE 5.8(b)  1" =200 AMPS 7" = 700 KV  and Currents  Lumped P a r a m e t e r S e c t i o n o f Two-Conductor L i n e  76  by  that  Vectors  particular  vector  have been drawn  each v e c t o r  two-conductor  case  described  is justified  the  two  and f o u r - c o n d u c t o r  The  current  in the  In this  parameters  b^  5.8(a)). V  9  ^  at the  if  the  simplif-  principles  i n both (Note:  i n the figure are of conductors  3  case.) i n the locus  p l o t are  a 50 m i l e  be  lumped i n t o  defined  section of  the  that the circuit  shown. methods  Pound f r o m  by r e d u c i n g The  power  Figure  the sending figure that I^ Z^  possible.  end  t o overcome t h e above  the voltage angle  (load voltage)  t h e power  voltage).  angle  increases.  angle  component  i s the angle  (generator  obstacles  5.8(a).  o f t h e l a r g e power  a t t h e r e c e i v i n g end  component  V^  by  f o r the  a r e t h e same.  illustrates  p a r a m e t e r s may  Reduction attained  miles.  point.  f i g u r e i t has been assumed  Possible now  5.2,  and v o l t a g e s  symbols used  5.8(b) w h i c h  distributed  may  cases  to the currents  line.  i n Section  as t h e u n d e r l y i n g  4 f o r the four-conductor  Figure  t o t h e 150 m i l e  and t h e v o l t a g e  The  o f 50  t h e f i g u r e has been drawn  ication  and  a distance  f o r t h e change e x p e r i e n c e d  f r o m t h e 100 m i l e Although  equivalent  over  will  Hence  1^  (5) may  1^ Z^^  between the  (Figure voltage  and t h e v o l t a g e I t i s clear  increase  be  V  1  from  as t h e  m u s t be  reduced  Two~ i m m e d i a t e m e t h o d s reduction Equation between above is  o f Z^ ,  and  Q  4.8(b),  reduction  conductors  ground  1 and  Such  the conductor  voltage  gradient  1  phase  angle  The  o f 180°).  o f I g may  of I  degrees  by t h e c o n d u c t o r s  of changing  vary  the phase  conductors.  The  upon t h e phase angle in  U  found  the  phase  the  magnitude  latter  ^°  investigation  surface, line  will  This  o f 1^ m u s t b e of - I  c  V^  and  lagging,  changed  voltage  Varying  power a n g l e . V^  difference.  discontinued.  alternat-  between  dependent  at the sending  between  2  necessary  difference  .  a  considered.  i t is  the  phase  end,  was  But,  since  and V^,so  too d i d  Unfortunately  exceeded i t s p e r m i s s i b l e value was  i t s phase  1 and  difference i s i n turn  of the voltage  costs.  then have  conductors  angle  and  i s dependent  c  spacing),  to produce a smaller  relationship  voltage  angle  betweeen ^5°  C  of I  of the voltage  voltage  line  by t h e  m a x i m i z e d , so t h a t t h e  the phase  angle  angle  of Vj)ipp P  fact  been  t h e phase  To v a r y to  fixed  changing  (~I  magnitude  (Y^g  already  height  b e made p o s s i b l e  o r by  c  d i f f e r e n c e between  ive  been  increase  upon t h e v o l t a g e fixed  or the  of the transmission  have  will  reduction  zero  spacing  w o u l d be u n s a t i s f a c t o r y  spacings  to approximately  to  p o s s i b l e i f the  S  ^rs/2)  conductor heights  has  reference  0  angle  difference  By  (b, ) i s decreased  i n c r e a s i n g the magnitude  being  themselves:  c o n d i t i o n at the conductor  Gradual by  2  changes  because  o f 1^.  o f Z-^  (approximately  increased.  increasing  reduction  suggest  and  so t h e  the  78 The  use of p a r a l l e l  between c o n d u c t o r s been d i s c u s s e d  a t 50 m i l e  i n Section  lumped  capacitors  intervals  5.2.  along  installed  t h e l i n e has  T h e i r u s e has been  proved  ineffective. Lumped s e r i e s r e a c t o r s Figure  4 . 6 ) have p r o v e d u n s u c c e s s f u l  power a n g l e  and t h e maximum v o l t a g e  s y s t e m by i n c r e a s i n g t h e s e l f  leakage current  compensate  forthis  must be i n s t a l l e d  t h e system  end that has  i twill  line  l a r g e leakage  along  length  current,  1.  when p a r a l l e l  sending  i s reduced  must be i n c r e a s e d  are evident  at the sending t o s u c h an  so  far.  1^  ( i g i n the two-conductor i s clear that  zero,  to  t h e system a t the sending  tained.  obstacles  obstacle  to  Figure  5.9  reactors to figure:  the  have been  i s that  discussed  of reducing  1^ and  line). i f this  an a d d i t i o n a l g e n e r a t o r  conditions  from t h i s  extent  angle.  Two o f t h e t h r e e  It  reactors  s u b s t a n t i a l l y and so t o o i s t h e  end power f a c t o r  The r e m a i n i n g  To  r e a c t o r s a r e added  a major r e g u l a t i o n problem.  Two a d v a n t a g e s  power a n g l e  exceeds 200  parallel  been p l o t t e d f o r t h e a d d i t i o n o f p a r a l l e l  line  conductor.  the l i n e .  voltage)  impose  increase the  d i f f e r e n c e of the  the m a g n i t u d e o f t h e v o l t a g e  (generation  1 (see  t o g r o u n d becomes s u b s t a n t i a l .  Unfortunately to  as t h e y  in line  impedance o f t h a t  V/hen t h e t r a n s m i s s i o n miles,  installed  will  current  i s not reduced  need t o be c o n n e c t e d  end so t h a t t h e r e c e i v i n g end  (boundary c o n d i t i o n s )  o f t h e s y s t e m can be main-  79  FIGURE 5.9  I n s t a l l a t i o n of P a r a l l e l Along the Line  Reactors  80 Only reducing series ance  one method h a s been  the current  reactors  of these  i n lines  a t the sending  conductors  imum p e r m i s s i b l e v o l t a g e exceeded.  3 a n d 4; t h a t  of series reactors  between  Therefore, In  t o such  this  5.4  The data  given  Section  Unfortunately,  an e x t e n t  that  voltage t h e max-  conductors  was  the three  disadvantages introducing  listed  on  overvoltages  o f t h e system.  System  three-phase  i n Section  5.3  imped-  m e t h o d was a l s o u n s a t i s f a c t o r y .  the limitations  Three-Phase  the self  t h e maximum  d i f f e r e n c e between  conclusion,  exceed  i s installing  end.  increased  P a g e 74 c a n n o t b e c o r r e c t e d w i t h o u t which  successful i n  3 and 4 t o i n c r e a s e  conductors  installation difference  i n lines  found  system  s t u d i e d uses  5.1 a n d t h e p a r a m e t e r s  (see Figure  5.10).  The v a l u e s  the physical determined i n  a r e summarised as  follows:Pour  conductors  Conductor Vertical  diameter  1.35  spacing  Horizontal  spacing  2.0  feet  1.0  foot  between phases  40.0  feet  Average  conductor  height  50.0  feet  length  600  The  line  computer program d e s c r i b e d  w r i t t e n so t h a t  i tcould  as  w e l l as t h e s i n g l e - p h a s e  in  Sections  the  inches  Spacing  Transmission  was  p e r phase  5.2  a n d 5.3.  be u s e d  miles  i n Section  4.6  f o r a three-phase  multiconductor  systems  system  discussed  S u b s t i t u t i o n o f t h e above d a t a i n  program gave t h e f o l l o w i n g  results;-  Power angle Sending  (6)  36.6°  end phase  Maximum v o l t a g e  angle  53.4°  between  (leading)  106 kV  conductors V V  x  3  and V and V  g  sending  end  2 3 5 /36.6° k V  4  sending  end  30 5 /51.8°kV  sending  e n d ........  514.6/93.4°Amps  end  1 6 6 . 6 /31.3°Amps  1^ a n d  I„ a n d I . s e n d i n g The three-phase remaining by  above r e s u l t s  system. phases  posed  are obtained  as t h e system  improved  currents  by d i s p l a c i n g  the vectors remain  i s a s s u m e d t o be c o m p l e t e l y  some o f t h e a b o v e r e s u l t s  by t h e a d d i t i o n of p a r a l l e l  I„ a n d I . w i l l 3 4 It  the  of the  and c u r r e n t s o f t h e two  The m a g n i t u d e s w i l l  2, f o r t h e f o u r - c o n d u c t o r  should  f o r one p h a s e  trans-  (see Section 3.2). Although  and  Voltages  120 and 240 d e g r e e s .  constant  are given  appears  remain  gradient  (see Figure  1  5.9) t h e  constant. investigation  a method has been  limitation  be  r e a c t o r s on l i n e s  t h a t any f u r t h e r  be p o s t p o n e d u n t i l  voltage  case  may  found  on t h e s y s t e m .  t o ease  FIGURE 5.10  T h r e e - P h a s e S y s t e m Above P e r f e c t  Ground  83 5.5  Physical Limitations The  ion  to  the  problem  limitation Appendix  limitation  described  may  the  conductor be  be  p o s s i b l e to  because  will  stable  the  For  to  2.5  now  (a t y p i c a l to  118°  f o r the  date  arise  advantageous will  of the  day  voltage  evaluated  f o r such  conductor  be  gradient  in  this  problem w i l l  5.11  f o r one-phase  were taken  to  be  surface relative  However, t h i s  increase relative  i n the  proved  system  permittivity power  which  i s i n excess  the  i n EIIV p o w e r  voltages; line  to  of  is  transmission  the  highest  line,  The  distributed  compensation  lines.  major  The  voltage as  the  c o n s i s t i n g of  line  The foot  four  conductor  voltage question  will  be  disadvantage  gradients  become more a c u t e , a s  1.0  conductor  f o r rubber) the  kV,  Indeed  i n diameter.  solution.  i n c r e a s i n g the  media.  of  3000 kV/meter, i t  at the  by  surface  value  line  excessive  surfaces.  at the  above  large  trends  i s 1500  whether  obviously  inches  conclus-  system.  for ever-increasing  suggested.to may  gradient  example, i f t h e  Present are  and  satisfactory  insulating  increase  region  a  substantially  unsuccessful  increased  increased  voltage  increased  angle.  be  reach  of  angle  4.7  surface  potential gradient  permittivity  power  in Section  could  The may  conductor  a successful  B. If  each  i s the  preventing  at  voltage  the increases  i s shown i n conductors  spacings  h o r i z o n t a l and  2.0  Figure  of  f o r the feet  2.0 figure  vertical.  84  LINE  VOLTAGE  CONDUCTOR CONDUCTOR  FIGURE 5.11  (KV)  DIAM 2-0 INCHES SPACING 1-0* HORIZONTAL 2-0 VERTICAL  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  For  the above  conductor s p e c i f i c a t i o n  line  voltage  p o s s i b l e i s 84 5 k V  1450  kV  to l i n e .  line  difference ed  compensation  would  conductors would line  1000  to l i n e  kV  could  conductors to  be  by  results  500  f o l l o w i n g power  kV  and  systems  required  but  (see S e c t i o n  readily  in this  between  o f 575  voltages  kV  or  above of  and  Compensation  c h a p t e r may  systems similar  of  be  compensation.  load  conditions  system  28°  system  ...37°  o f t h e power a n g l e due  as  load,  kV  gradient.  in this  s e r i e s compensation  that  voltage.  i n c r e a s i n g t h e number  o f t h e two  i n c r e a s i n g or decreasing  system  Line  voltage  obtained  by  this  high  voltage  "Distributed  compensation  system the problem  at t h i s  distribut-  a n g l e s were o b t f . i n e d : -  Variation the  so t h a t  expanding the conductor diameter  s e r i e s compensation  Distributed  c h a n g e on  kV.  made p o s s i b l e b y  of Series  voltage  d i f f e r e n c e o f 106  o f 1000  and  the  zero,  to ground  maximum  or approximately  voltage  feasible  a line  comparison  For  Typical  be  conductor surface  The  the  give  voltage  Comparison  summarised  not  a voltage  per bundle  reduce the  5.6  line  b e t w e e n c o n d u c t o r s m u s t be  Allowing  a  At t h i s  to ground  the  s y s t e m may  the value  2.2).  of the  However,  i s more d i f f i c u l t .  a r e no  be  series  I t has  capacitor  distributed  been found f o r  with  parameters  v a r i e d to c o r r e c t the . s i t u a t i o n .  load  corrected  f o r the  t h e power a n g l e i n c r e a s e s case there  to a  the that  system can  be  86 6.  .A mission on  multiconductor  line  has  Maxwell's  a  system  potential  can  be  considerable  distributed-compensation  been a n a l y s e d and  b e e n shown t h a t , f o r t h e the  CONCLUSION  three-phase  a s s u m e d t o be of  system v o l t a g e  so  t h a t any  Simplification  of  number o f  the  and  addition allowed  of  lumped  and  f o r i n the  This  It  has  allowed  equations  have  may  be  and  considered.  made p o s s i b l e b y  any  has  frequencies,  transmission line  conductors  at  based  analysis.  partitioned  elements  a t power  current  analysis.was  o f A,B,C,D p a r a m e t e r s  case  the  method  coefficients.  lossless.  been 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 written  a matrix  inductance  simplification  The  using  trans-  matrices.  point  along  the  The the  use  possible  line  was  solution.  (5) Crary surface  of  equations However, are to  no  bundled  i n h i s w o r k on conductors  because the f o r an  longer  open-ended  equal  so  obtain a solution Two  attempt ately  the  impractical s y s t e m by s u r f a c es.  two  because  the  line  of the  each the  simplify  conductor charges  was  on  examples have been three-phase  four-conductor  of the  is  his the  the  the  same.  conductors necessary  equations.  a suitable and  on  able to  t h a t a g r a p h i c a l method  numerical  to develop  both  charge  was  voltage gradients at  limitation  breakdown v o l t a g e  of  studied in  system.  Unfortun-  examples proved imposed  on  a i r n e a r the  an  to  each conductor  be  87 A l t h o u g h t h e use and  parallel  reactors  was  found  their  introduction increased  still  further. An  limitation  at the  permittivity the  power  unstable  attempt  angle  to  the  improve  of the system  by  found  operation,  along  the  line  gradient .  increasing  media proved was  inductors  system  the voltage  surface  insulating  series  overvoltages  to reduce  conductor  of the  of lumped  the  ineffective  to increase  to  as an  value. It  compensation uneconomical.  may  be  i n large  concluded power  t h a t t h e use  systems  of  distributed  i s impractical  and  APPENDIX A  Numerical  A.l  Two-Conductor  Evaluation  of Conductor  Coefficients  Case  GROUND  PLANE  IMAGES  FIGURE A . l  In 50 f e e t inches  while  Two-Conductor  Figure  Above P e r f e c t  A . l t h e mean c o n d u c t o r  the conductor  a n d 1.5 f e e t  Line  diameter  and spacing  4.2 a n d 4.5 6.03  jw [c]  height  (li)  i s  i s 0.950  respectively.  From E q u a t i o n s  =  Ground  =  j x  i tfollows  that  -3.23  10"  mhos/mil< -3.23  6.03 (A.l)  89 By 4.8,  neglecting the internal  the Z matrix  =  j  [L]  u  term  in  Equations  becomes  "0.9 52  [z]  flux  =  0. 510" ohms/mile  J  0.510  0.952 (A.2)  Evaluation  of the J matrix  from  Equation  4.20  gives  0.000  4.093  M  - W  10  • [i] 0.000  -6  4.093 (A.3)  By propagation  application  of Equation  4.40  we  obtain the  constant  —3 t A.2  =  j  Four-Conductor  2.02  x 10~  radians/mile  Case  H a  3 4  ^TROUND  PLANE  IMAGES F I G U R E A.2  Four-Conductor  Line  Above P e r f e c t Ground  In  F i g u r e A.2  II  =  50.0 feet 1.0  13"  conductor  1*5 f e e t h o r i z o n t a l conductor  b^2=  Equations  foot vertical  The  conductor  diameter  The  admittance  array  i s0.806  spacing spacing  inches.  may b e d e t e r m i n e d  from  4.2 and4 . 5 as before:-  [Y] = j x 10  -6  " 7.3027  -1.7902  -2.6563  -1.2460"  -1.7902  7.3027  -1.2460  -2.6563  -2.6563  -1.2460  7.3118  -1.7811  -1.2460  -2.6563  -1.7811  7.3118_  mho mi  (~A.4) Also  M -  the Z matrix  0.9722  0.5110  0.5590  0.4875  0.5110  0.9722  0.4875  0.5590  0.5590  0.4875  0.9697  0.5086  0.4875  0.5590  0.5086  0.9697  Evaluation  and  may b e w r i t t e n  o fJ from Equation  o h m s / m i 1<  (A.5)  4.20 gives  4.0926  0.0000  0.0000  0.0000  0.0000  4.0926  0.0000  0.0000  0.0000  0.0000  4O0926  0.0000  0.0000  0.0000  0.0000  4.0926  10  as i n Section A . l  *K =  j B  =  j  2.02  x  1 0- 3  radians/mile  -6  (A.6)  To transposed"  o b t a i n t h e Z and Y m a t r i c e s  system  to  determine  in  Section  from  Equations  the average  elements  four-conductor  H =  a n d A.5  "completely  i t i s necessary  f o r each a r r a y as  indicated  3.2. The Z and Y a r r a y s  |_Y]= j x 10  A.4  for a  -6  system  f o r a "completely  transposed"  are  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  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  mhos/raile  ohms/mi1e  APPENDIX B  Numerical  E v a l u a t i o n o f Maximum A l l o w a b l e  Difference  B.l  Two-Conductor  FIGURE B . l  between  Cross-Section  has t h e f o l l o w i n g  therefore  o f a Two-Conductor  case  illustrated  d i a m e t e r = 0.95  Conductor  radius  d  Evaluation two-conductor  case  2  1  i n Figure  =1.5  inches  = 0.0396  feet  feet  o f E q u a t i o n s 4.60 t o 4.64  f o rthe  gives  12.6316  ml  Line  parameters  Conductor  12  E  Conductors  Case  The t w o - c o n d u c t o r B.l  Voltage  0.6667 (B.I:  •ne J  m2  0.6667  12.6316  93 By u s e o f t h e n u m e r i c a l Section  A . l Equation  4.65 may  calculated i n  be w r i t t e n ; -  3.0241  -1.6190  -12 10  =  (B.2) 3.0241  -1.6190  Q,  Substitution and  values  performing  of Equation  the matrix  V  B.2  into  multiplication  Equation B . l  yields  -2.1753  4.3801  (B.3) 4.3801  -2.1753  _ m2 E  T h e maximum be  evaluated  described  permissible voltage  from Equation  B.3 b y a g r a p h i c a l m e t h o d  below and i l l u s t r a t e d  i n Figure  While varying voltage values  the  against  figure.  values  the voltage  -  obtained  ) as shown i n f o r several  .  T h e maximum be d e t e r m i n e d  were  constant,  The r e s u l t s a r e  d i f f e r e n c e (V  A d d i t i o n a l curves  of voltage  B.2.  and keeping  o f E -, a n d E „ a r e d e t e r m i n e d . ml m2  plotted  d i f f e r e n c e may  permissible voltage  f o r each v a l u e  o f V „ when  d i f f e r e n c e may  either  d  E , or E „ ml m^ (23)  exceeds  the corona voltage E^ d r y a i r = The  as  when V  0  =  For values 260 kV,  2  =  280 kV,  2  f o r d r y a i r , which i s  646 k V / f o o t  corona voltage  a horizontal straight  when V  gradient  line  of  Y D I,„ FF nT1  DIFF  gradient  (R.M.S.) f o r a i ri s plotted  i n the figure. greater than V i t i s seen = 16 k V and 6 kV  that  Ema = 646-6 KV/ft  (RMS)  m2  20 DECREASING  V  1  VOLTAGE  0 DIFFERENCE  20 (KV)  INCREASING  V  1  CO  FIGURE B.2  Conductor S u r f a c e V o l t a g e G r a d i e n t s (Two-Conductor Case)  95  However, the  a i raround  age  difference  and  V_ 2  when  V  again  B.2  =  g  when  i t i s possible  different  t o have i f  a larger  i s less  than  voltV^.  B.2 V^,. "= DIFF V  DIFF  =  3 3 kV 1  292 kV t h e a i r w i l l  3  k  V  ionize.  Case  C r o s s - S e c t i o n of a Four-Conductor  the four-conductor case,  be c o n d u c t i v e l y c o n n e c t e d  hence  tlien  ionize.  E , a n d E ,, h a v e nil in 2  280 kV  exceeds  In  will  a p p r o x i m a t e l y 292 kV  between t h e conductors  Four-Conductor  will  that  From F i g u r e = 260 kV  F I G U R E B.3  4,  the curves  i tfollows  when  exceeds  the conductors  Since slopes,  i f  as w i l l  conductors  Line  1 and 2  be c o n d u c t o r s  3 and  96  Q and  l  V  V  1.5 It  By  ing E  l  4.60  1  2%c  o f F i g u r e B.3 .=  1 3  parameters  =  v. '4  1.0  a n d 4.61  4  (B.4)  f o r conductor  foot  d, „ . = 14  1 are  1.802  to determine  f o r the remaining  substitution  independent =  Q  i s a s t r a i g h t f o r w a r d matter  corresponding  Equations  d  feet  =  3• ~  2  The p a r a m e t e r s 12  3  feet the  conductors.  o f t h e above n u m e r i c a l  values i n  i t i s possible to write the  follow-  equations. x  +  0.4622  QJ  2  | ^0.6667 Q  1  +  0.6667  QJ  2  | ^0.4622 Q  3  + ^1.3071  Qj  +^1.3071  Q^j  2  (B.5a)  3  and E  3  =  Substituting Section  A.2  into  3  the numerical  Equation  4.65 y i e l d s  values  2  (B.5b)  determined  the following  in  matrix  equation 0.509l!  0.7216 =  10  12  (B.6) 0.5091 By  inverse  0.7192  premultiplication  of the numerical  m a t r i x we  =  B.6  by the  obtain  -1.9596  2.7683  Qi  of Equation  -12 10  1 (B.7)  •1.9596 Because  2.7775  a direct  B.5  a n d B.7  relationship  between  Equations  the solution  adopted  i n S e c t i o n B . l n e e d s t o be  does n o t  exist  procedure  modified.  1000  800  Ema.  = 64 6-6  KV/ft  600Y  to  400  UJ  Co  5  ir>3  o  200Y  20 VOLTAGE DECREASING  V  7  FIGURE B.4  0 DIFFERENCE  20 4  40 CONDUCTORS INCREASING  Conductor  Surface Voltage Gradients  (Four-Conductor  Case)  V 1  cc -4  A numerical the  line  charges  condition) The  solution  from Equation  and s u b s t i t u t i n g  maximum f i e l d  intensit3  r  c a n be f o u n d  B.7  these  by  calculatin,  (for a particular values  into  may b e e v a l u a t e d  voltag  Equations from  B.5  Equation  4.64. Determination difference  o f t h e maximum p e r m i s s i b l e v o l t a g  may b e o b t a i n e d  graphically  When t h e v o l t a g e s  and  v o l t a g e s V „ a n d V., t h e f o l l o w i n g ° 3 4 voltages when  and  Y  between  3  conductors  =  280 kV  V  V  3  =  300 kV  V  D  I  p  V  3  =  320 kV  V  D  I  p  and  than  a r e t h e maximum a l l o w a b l e  0  3  F i g u r e B.4.  are greater  the following  =  42 kV  p  =  35 k V  p  =  27 k V  DIFF  n T T I T l  H o w e v e r , when v o l t a g e s V  from  and  are lower  c o n d i t i o n s may b e o b t a i n e d  than  from  F i g u r e B.4:when  V V  and  V  0  =  280 kV  V„ „ DIFF  =  60 kV  3  =  300 kV  V  =  50 k V  =  40 k V  3  3 Q  = ' 320 kV The  and v e r t i c a l respectively. of  T T 1  conductor  I  p  p  V . „ DIFF 1  above v o l t a g e conductor  D  T  T 1  conditions apply  spacings  o f 1.5 f e e t  f o r horizontal a n d 1.0  foot  ( 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 spacings).  99 REFERENCES  1.  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