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Voltage and current profiles and low-order approximation of frequency-dependent transmission line parameters Marti, Luis 1982

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VOLTAGE AND CURRENT PROFILES AND LOW-ORDER APPROXIMATION OF FREQUENCY-DEPENDENT TRANSMISSION LINE PARAMETERS  by  LUIS MARTI  E l e c . Engr., C e n t r a l U n i v e r s i t y of Venezuela, 1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l E n g i n e e r i n g  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982  ©  L u i s M a r t i , 1982  In p r e s e n t i n g  this  requirements  f o r an  of  British  it  freely  thesis  Columbia, available  understood for  by  that  financial  Library  shall  for reference  and  study.  I  for extensive copying of be  her  copying or shall  g r a n t e d by  not  be  E l e c t r i c a l Engineering  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date  DE-6  (.3/81)  A p r i l 27th  1982  of  Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives. publication  the  University  the  h i s or  gain  the  I agree that  s c h o l a r l y p u r p o s e s may  department or  f u l f i l m e n t of  advanced degree at  agree that permission for  in partial  written  ABSTRACT  In  this  electromagnetic  thesis project,  transients  The f i r s t  model has been designed to f i n d v o l t a g e s and c u r r e n t s e q u a l l y spaced p o i n t s a l o n g the  from the s o l u t i o n a t  frequency dependence of the  line  the end p o i n t s parameters).  r e p r e s e n t i n g a segment of  the  line.  line  ( t a k i n g i n t o account the  This " p r o f i l e  d e r i v e d from the cascade c o n n e c t i o n of n e q u i v a l e n t  internal  to the s i m u l a t i o n of  i n power systems, have been d e v e l o p e d .  a t a number of i n t e r m e d i a t e , ("profile"),  two models r e l a t e d  model"  is  c i r c u i t s , each one  The s o l u t i o n i s c a r r i e d out with an  time s t e p which i s an exact s u b m u l t i p l e  of the  t r a v e l time of  the  p r o p a g a t i o n mode i n c o n s i d e r a t i o n . A s e r i e s of  tests  model are more a c c u r a t e segmenting the  line  is  shows t h a t the r e s u l t s o b t a i n e d with  than those o b t a i n e d i f  the standard p r a c t i c e  of  followed.  With the a i d of the p r o f i l e the p r o p a g a t i o n of t r a n s i e n t s The second p a r t of low-order  this  approximation of  routines,  along a l i n e , this  a movie which  illustrates  has been produced as  well.  t h e s i s d e s c r i b e s the development of  the frequency-dependence of  from a reduced amount of i n f o r m a t i o n .  line  Using the e l e c t r i c a l  parameters, parameters  power frequency and dc conductor r e s i s t a n c e s , the tower c o n f i g u r a t i o n the  line  is reconstructed.  e v a l u a t i o n of the f u n c t i o n s are parameters  line  This equivalent  line  over a l i m i t e d  of the  Rational  the frequency-dependence of  these  frequency range with a reduced order model.  A n a l y t i c a l t e s t s and t r a n s i e n t  simulations, indicate  that  model i s r e a s o n a b l y a c c u r a t e over the frequency range of i n t e r e s t practical  at  c o n f i g u r a t i o n permits  parameters over a wide frequency r a n g e .  then used to approximate  a  applications. ii  the  i n most  Both models have been i n c o r p o r a t e d Electromagnetic  T r a n s i e n t s Program  (EMTP).  iii  i n t o the UBC v e r s i o n of  the  TABLE OF CONTENTS Page  ABSTRACT  i i  ACKNOWLEDGEMENT PART I :  v i i  VOLTAGE AND CURRENT PROFILES ALONG A TRANSMISSION LINE  INTRODUCTION  2  CHAPTER 1: THEORETICAL CONSIDERATIONS  4  1.1  D e s c r i p t i o n o f the P r o f i l e Model i n the Frequency Domain  4  1.2  S o l u t i o n i n the Time Domain  10  1.3  I n i t i a l Conditions  13  CHAPTER 2:  IMPLEMENTATION OF THE SOLUTION METHOD  15  2.1  General C o s i d e r a t i o n s  15  2.2  I n t e r n a l Time Step A t '  17  2.3  Approximation o f A-^(OJ)  19  2.4  Animated  Motion P i c t u r e o f T r a v e l l i n g Waves  along the Line CHAPTER 3:  22  NUMERICAL RESULTS  28  3.1  Introduction  28  3.2  Truncation  30  and Segmentation E f f e c t s  3.3  E n e r g i z a t i o n o f a L i n e Terminated w i t h a Lightning Arrester CONCLUDING REMARKS  i V  42 51  Page PART I I :  LOW-ORDER APPROXIMATION OF THE FREQUENCY DEPENDENCE OF TRANSMISSION LINE PARAMETERS 53  INTRODUCTION CHAPTER 1: 1.1  1.2  55  THEORETICAL CONSIDERATIONS  E v a l u a t i o n o f the T r a n s m i s s i o n  L i n e Parameters  1.1.1  S e r i e s Impedance M a t r i x  55  1.1.2  Carson's C o r r e c t i o n Terms  52  1.1.3  Shunt Admittance M a t r i x  54  1.1.4  Comparison between Exact and Approximate Formulas  55  E v a l u a t i o n o f the E q u i v a l e n t L i n e C o n f i g u r a t i o n from the Parameters a t Power Frequency  68  1.3  Skin E f f e c t C o r r e c t i o n F a c t o r  71  1.4  C o r r e c t i o n o f the E q u i v a l e n t L i n e C o n f i g u r a t i o n when R j i s Known  74  1.5  E v a l u a t i o n o f the C h a r a c t e r i s t i c and Propagation F u n c t i o n  77  (  CHAPTER 2:  c  Impedance  RATIONAL FUNCTIONS APPROXIMATION OF THE CHARACTERISTIC IMPEDANCE AND PROPAGATION FUNCTION  2.1  Rational Functions.  2.2  Asymptotic Approximation when t h e Number o f Poles  General  79 79  Considerations  i s Fixed  82  2.3  Approximation o f the C h a r a c t e r i s t i c Impedance Z (a>)  86  2.4  Approximation o f the P r o p a g a t i o n  88  2.5  Evaluation of C  93  2.6  Implementation o f t h e Method  94  c  v  F u n c t i o n A^to)  Page CHAPTER 3: NUMERICAL RESULTS  g  5  3.1  Recapitulation  g  5  3.2  E v a l u a t i o n o f the L i n e Parameters from the E q u i v a l e n t L i n e C o n f i g u r a t i o n  g  6  1  0  3  1  1  2  3.3  E v a l u a t i o n o f A^ (w)  3.4  Low-Order R a t i o n a l - F u n c t i o n s  3.5  Frequency Domain Response  3.6  and Z (to) c  3.5.1  Introduction  3.5.2  Open C i r c u i t Response  3.5.3  Short C i r c u i t Response  Transient Simulation  Approximations  1 2  i n the EMTP  1  3  8 1  ^39 1  CONCLUSIONS  5  4  161  APPENDICES APPENDIX I-A:  APPENDIX I-B:  APPENDIX I-C:  APPENDIX II-A:  GENERAL SOLUTION OF THE LINE EQUATIONS IN THE FREQUENCY DOMAIN JOSE MARTI'S MODEL  1  6  2  FREQUENCY-DEPENDENCE 167  USER'S GUIDE FOR THE VOLTAGE AND CURRENT PROFILE OPTION OF THE EMTP, AND FOR THE OUTPUT DISPLAY PROGRAM  179  SKIN EFFECT CORRECTION FOR ROUND CYLINDRICAL CONDUCTORS  187  APPENDIX I I - B : CARSON'S CORRECTION TERMS FOR EARTH RETURN EFFECT  1  APPENDIX I I - C : SIMULATION OF THE JOHN DAYLOWER MONUMENTAL TRANSMISSION LINE USING THE LOW-ORDER APPROXIMATION PROGRAM  196  APPENDIX II-D:  USER'S GUIDE FOR THE LOW-ORDER APPROXIMATION PROGRAM  BIBLIOGRAPHY  9  1  205 212  vi  ACKNOWLEDGEMENT  I t i s i m p o s s i b l e to thank a l l the i n d i v i d u a l s and d i r e c t l y or i n d i r e c t l y project.  c o l l a b o r a t e d i n the completion  However, r i s k i n g  I must g i v e s p e c i a l To the  institutions of t h i s  that thesis  i n g r a t i t u d e towards those not mentioned below,  thanks,  "girls",  i n the main o f f i c e of the Department of E l e c t r i c a l E n g i n e e r i n g , f o r o f t e n going out of t h e i r way  to give me  To Dr. Dommel, f o r h i s i n s i g h t , h e l p  a hand.  and  unshakeable p a t i e n c e . To my  b r o t h e r , Jose, whose genius shared  To Debby, f o r her To my  love.  parents.  vii  dream.  l i t the way  of  our  PART I  VOLTAGE AND  CURRENT P R O F I L E S  ALONG A TRANSMISSION  1  LINE  INTRODUCTION  When a power system i s d i s t u r b e d from i t s normal s t e a d y - s t a t e o p e r a t i o n , a t r a n s i t i o n between the o r i g i n a l s t a t e and  the new  steady-  s t a t e c o n d i t i o n s (with p o s s i b l e changes i n the network) must o c c u r . t r a n s i t i o n p e r i o d begins with very f a s t e l e c t r o m a g n e t i c  This  t r a n s i e n t s ( i n the  o r d e r of m i l l i s e c o n d s ) , which u s u a l l y i n v o l v e t r a n s i e n t o v e r v o l t a g e s high t r a n s i e n t c u r r e n t s . transients  T h i s i s f o l l o w e d by slower  ( i n the order of seconds),  or  electromechanical  which are not d i s c u s s e d i n t h i s  thesis. P r o t e c t i o n d e v i c e s are used to l i m i t the magnitude of o v e r v o l t a g e s a c r o s s expensive equipment and The  equipment i n order to p r e v e n t damage i n t h i s  subsequent s u b s t a t i o n f a i l u r e s . s e l e c t i o n and  d e t a i l e d system s t u d i e s . used f o r such  transient  the c o - o r d i n a t i o n of these d e v i c e s r e q u i r e  The  e f f i c i e n c y and  accuracy  s t u d i e s , become more important  of computer programs  as systems grow i n s i z e  and  complexity. Programs such as the E l e c t r o m a g n e t i c are capable  T r a n s i e n t s Program  (EMTP)  of s i m u l a t i n g a l a r g e v a r i e t y of t r a n s i e n t c o n d i t i o n s , f o r  systems of v a r y i n g s i z e and ed by Dommel [ 1 ] , [ 2 ] , has  complexity. evolved and  T h i s program, o r i g i n a l l y  develop-  grown as s i m u l a t i o n models have  been improved by other r e s e a r c h e r s . In the l a s t decade or so, much e f f o r t has a c c u r a t e m o d e l l i n g of t r a n s m i s s i o n l i n e s , quency dependence of t h e i r parameters. developed  and  implemented by J.R.  tional flexibility  t a k i n g i n t o account  One  such model has  Marti [ 3 ] .  of t h i s model have p r o v i d e d  The  the  to the fre-  r e c e n t l y been  s i m p l i c i t y and  computa-  the b a s i c t o o l s f o r the  development of other r e l a t e d models ( i n p a r t i c u l a r , 2  been devoted  those d i s c u s s e d i n  3  this  thesis). Most l i n e models, e i t h e r with  parameters, g i v e v o l t a g e and the  line.  tions.  frequency-dependent or  constant  c u r r e n t i n f o r m a t i o n o n l y a t the end  This information u s u a l l y s a t i s f i e s  points  the needs f o r most  simula-  There are s i t u a t i o n s , however, when the i n f o r m a t i o n a t the  p o i n t s of the l i n e may  The  end  not be enough to give a c l e a r p i c t u r e of the  a l l behaviour of the l i n e  (e.g., when l i g h t n i n g a r r e s t e r s are  u s u a l p r a c t i c e , when v o l t a g e s  and  currents at  over-  considered).  intermediate  p l a c e s are needed, i s to segment the l i n e i n t o s h o r t e r s e c t i o n s , and examine the response at the end however, can be i n a c c u r a t e and Very few (without The  o n l y model found  segmenting the l i n e ) can be [4], which takes  T h i s procedure,  and  current  found i n the  profile  literature.  frequency-dependence i n t o account,  Semlyen's approach to the frequency-dependence problem  In t h i s model, the v o l t a g e s ate p o i n t along  the l i n e .  to  i s c e r t a i n l y time consuming.  attempts to determine the v o l t a g e  explicitly  i s based on  p o i n t s of each segment.  of  and  c u r r e n t s are  A p r o f i l e can  c a l c u l a t i o n s with d i f f e r e n t segmentation  found a t o n l y one  thus be o b t a i n e d  by  [5].  intermedirepeated  ratios.  I t i s the purpose of the f i r s t p a r t of t h i s p r o j e c t , to d e v e l o p a p r o f i l e model f o r any  number of e q u a l l y spaced p o i n t s along  Such a model should 1 . I t should  be more a c c u r a t e  segmentation of the 2.  meet the f o l l o w i n g  the  requirements:  than the c o n v e n t i o n a l  "external"  line.  The  computer r o u t i n e s must be designed  to p e r m i t  and  e f f i c i e n t i n t e r c o n n e c t i o n with  frequency-dependence  the  a  simple  v e r s i o n of the EMTP. 3.  line.  The  r o u t i n e s should be easy to use,  minimum amount of e f f o r t on  and  should  the p a r t of the  require a  user.  CHAPTER 1  THEORETICAL CONSIDERATIONS  1.1  D e s c r i p t i o n of the P r o f i l e  Model i n the Frequency Domain  Consider an m-phase t r a n s m i s s i o n l i n e with ground r e t u r n , as r e p r e s e n t e d i n F i g u r e 1 .1  o-  -o  L<2 °-  -O  IDo  -o  m  -o  g  ki  k  m°-  g  Fig.  1.1:  c-  m  R e p r e s e n t a t i o n of an m-phase t r a n s m i s s i o n l i n e w i t h ground r e t u r n A f t e r the a p p r o p r i a t e modal t r a n s f o r m a t i o n s have been made, as  d e s c r i b e d i n appendix I-A,  the behaviour  of the l i n e can be analyzed i n  the modal domain, where i t can be s t u d i e d as a s e t of m, lines. its  Each of these s i n g l e - p h a s e l i n e s  modal surge  impedance Z ( u ) c  equations o b t a i n e d from l i n e mode, and  this  can  single-phase  then be d e s c r i b e d i n terms of  and p r o p a g a t i o n  function  A-|(u)).  The  s i n g l e - p h a s e model, are a p p l i c a b l e to  therefore s u f f i c i e n t  to d e s c r i b e the l i n e  any  i n the phase  domain. Consider  then, the r e p r e s e n t a t i o n of one  t i o n modes, shown i n F i g u r e  1.2  4  of the l i n e ' s  propaga-  5  i (t)  •m  U  (t) o m  vu(t)  Fig.  1.2:  v (t) m  Modal r e p r e s e n t a t i o n .  In the frequency domain, t h i s (frequency-dependent) model of F i g u r e  Fig.  1.3:  where,  l i n e can be d e s c r i b e d by the  1.3, as d e s c r i b e d  L i n e model i n the frequency domain.  i n appendix  I-B.  Bk = V  k  -  I Z K  I  Fk = V  K  +  (backward  e q  travelling  (1.1)  functions)  m eq z  I Z K  (forward t r a v e l l i n g  e q  functions)  The r e l a t i o n s h i p between forward and backward  (1  .2)  travelling  f u n c t i o n s i s g i v e n by Bk  =  A 1 ,c m  (1.3)  =  e  (1 .4)  F  where,  Y(CO)  = \/z' y  Z'  i s the modal s e r i e s impedance  ( R ' ( ) + jwL'(w) ) w  per u n i t l e n g t h (fi/km) Y'  i s the modal shunt admittance per u n i t l e n g t h  &  c  i s the l i n e  (G  1  + jaC ) 1  (mhos/km)  l e n g t h i n km  If we connect i n cascade  (n + 1) of the c i r c u i t s  i n F i g u r e 1.3,  each r e p r e s e n t i n g  a l i n e segment of l e n g t h £, we o b t a i n the e q u i v a l e n t  c i r c u i t of Figure  1.4  -eq  Fig.  1.4:  Cascade c o n n e c t i o n of (n + 1) e q u i v a l e n t l i n e  Note t h a t t o connect t o d i v i d e a l i n e of l e n g t h £  c  m o «  circuits.  (n + 1) segments of l e n g t h & i s e q u i v a l e n t =  £(n + 1) i n t o  (n + 1) segments of l e n g t h  I.  T h i s r e s u l t s i n n nodes w i t h n i n t e r m e d i a t e  v o l t a g e s and c u r r e n t s .  The f o l l o w i n g d e f i n i t i o n s can e a s i l y be extended from (1.1)  to (1.4) f o r the forward  and backward t r a v e l l i n g  equations  f u n c t i o n s of each  segment j . k = V  B  - I  k  Z  k  e q  (1 .5) &m - Ym  J  k,j  ~  v  j  -  m,j  ~  v  j  +  B  B  _  Fk = V m  F  v  z  m  +  I  =  m,j  = Vj  v  j  +  eq  !j eq z  (1 .6)  j eq  I  z  + I  k  k,j  F  F  _  m  Z eq  k  (1 .7)  m eq z  j eq  x  z  (1 .8)  *j eq» z  where Vj and I j ( j = 1,2,..., n) are the v o l t a g e s and c u r r e n t s a t each of the n i n t e r m e d i a t e  nodes.  The r e l a t i o n s h i p s between f u n c t i o n s can be summarized Bm,1 B  and backward  travelling  as f o l l o w s , (1.9)  = Ai 1=\  k,n = 1  F  A  B ,j  = 1  B  = A-j F  A  m  k >  forward  j  F  (1.10)  m  k,j-1 m f  j  + 1  (j = 2,  (1.11)  n  (1.12)  (j = 1, ..., n-1 ),  where, -Y(u)U  Ai  =  e  I  =  ft /(n  (1.13)  and, c  + 1)  Note t h a t I i s the l e n g t h of a segment, and l  c  of the complete l i n e .  i s the l e n g t h  A l s o note t h a t the r e l a t i o n s h i p s between  the end  8 p o i n t s of the l i n e  (equations (1.1) through  r e g a r d l e s s o f the number of segments  (1.4)) s t i l l  hold  true,  connected.  From the e q u i v a l e n t c i r c u i t o f F i g u r e 1.4, Vj = I j Z  e q  + B  k f  j  Vj = - I j Z q +  B j.  e  Adding  m>  up these two e q u a t i o n s , 2 Vj = B  A l s o , from  j + B  k >  (1.14]  ni/  the e q u i v a l e n t c i r c u i t , 2 Ij Z  = B^j - B  e q  k f j  .  (1.15)  A d d i t i o n o f (1.14) and (1.15) g i v e s , Vj + I j Z S u b t r a c t i o n of (1.14) from  = Bjj^ j .  (1.15) g i v e s ,  Vj - Uj ZZ Comparing the l a s t  e q  - B  e q  k f  j  two equations with Fk,j = B  (1.8) we f i n a l l y o b t a i n , (1.16)  m / j  Fm,j = B , j .  (1.17)  k  Equations backward t r a v e l l i n g  (1.11) and (1.12) can now be expressed  i n terms of  functions alone, Bm,j = 1  Bm,j-1  A  B ,j k  = 1 k,j+1 A  B  U (D  = > 2  =  1  ""  »  n  n  +  (1.18)  )  1  ) •  (1.19)  I f the i n t e r m e d i a t e backward t r a v e l l i n g f u n c t i o n s B j ^ j and B j ^ j a r e known, the i n t e r m e d i a t e v o l t a g e s and c u r r e n t s V j and I j a r e u n i q u e l y determined  by equations  determined  i f the forward t r a v e l l i n g f u n c t i o n s  a r e known.  (1.14) and (1.15).  For example, with F  k  Also, B  k >  j  and B ^ j can be  l\ and F  known from e q u a t i o n  m  (end p o i n t s )  (1.9), B j ^ i can be  o b t a i n e d ; w i t h BJJ,^ known, B ^ j (j=2,...,n) i s determined  by equation  (1.18).  starting  F  m«  An analogous  procedure  can be f o l l o w e d f o r B  k f  j,  from  9 At t h i s p o i n t i t can be seen t h a t the o n l y i n f o r m a t i o n needed t o evaluate permits  the i n t e r n a l v o l t a g e s  and c u r r e n t s i s g i v e n by  the e v a l u a t i o n of the i n t e r m e d i a t e  This presents The  and  used f o r the complete  c e r t a i n advantages t h a t w i l l be d i s c u s s e d  propagation  This  m  v o l t a g e s and c u r r e n t s i n a way  which i s independent of the s o l u t i o n method being line.  and F .  f u n c t i o n A-|(td) i s i d e n t i c a l  later.  f o r each segment,  i t i s approximated by r a t i o n a l f u n c t i o n s i n the same way as f o r the  propagation  f u n c t i o n A - | ( t u ) o f the complete l i n e  The  fC  partial  (see appendix  f r a c t i o n s expansion of the r a t i o n a l  II-B).  functions  yields,  A 1  Since line,  U)  _Jil_  =  _JlV  +  s + B-|  +  s + 3^  ...  +  _ J E  V  e "  J  U  C  (  K  2  0  )  s+Sm  the surge impedance i s independent of the l e n g t h of the  the same approximation  used f o r the s o l u t i o n of the complete l i n e i s  used f o r the e v a l u a t i o n of the p r o f i l e , r, i \ Z (to) c  =  v , o +  K  ki 1  s+ai  + A  that i s , kT , * — + . . .  s+a2  , +  (1.21)  k * — m  s  +  a  m  '  10 1 .2  S o l u t i o n i n the Time Domain The i n t e r m e d i a t e  backward t r a v e l l i n g from the forward  v o l t a g e s and c u r r e n t s are determined by the  f u n c t i o n s , and these can be s e q u e n t i a l l y determined  travelling  Consider  f u n c t i o n s a t the end p o i n t s of the l i n e .  equation  Vi  (1.9)  = 1 A  The time domain c o u n t e r p a r t  F  k«  i s obtained  u s i n g the Inverse  Fourier  Transform, b The c o n v o l u t i o n explained  m > 1  (t)  =  a i  * f (t).  (1 .23)  k  ( i n d i c a t e d by "*") can be s o l v e d by r e c u r s i v e methods, as  i n appendix I I - B , g i v i n g ,  bm,l(t>  =  m .Z b i  m / 1 > i  (t)  (1.24)  with, b  B,1,i(t) = 9i V  1 f i  (t-At) +  C i  f (t-r) k  + di f (t-r-At) K  where, g^, c^ and d i are the i n t e g r a t i o n c o n s t a n t s integral  (1.23),  (1.25)  from the c o n v o l u t i o n  and are g i v e n by, - S i At  9i hi =  1 - 9i Si At kj  di = k i and $j_ a r e o b t a i n e d  (1 - h j )  _ k j ( g j - hj)  h  from the r a t i o n a l - f u n c t i o n s approximation  f o r the l e n g t h of one segment (see e q u a t i o n Similarly, b  k / n  ( t ) i s obtained  (1.20)).  from e q u a t i o n  (1.10),  of A-] (u)  11 k,n(t) = 1 ( t ) * f ( t ) m k,n(t) = Z k,n,i(t) i=1  b  (1.26)  a  m  b  (1.27)  b  with, b  k,n,i  {t)  = *i  bk,n,i( "At) + c  f (t-r)  t  ±  m  + di f ( - r - A t ) . m  The i n t e r m e d i a t e f u n c t i o n s b j ^ j f t ) and b ^ j ( t ) a r e o b t a i n e d  i na similar  m  f a s h i o n from e q u a t i o n s  h  m,j,i^^  (1.28)  t  (1.18) and (1.19).  b  m,j  ( t )  b  m,j  ( t )  = l<t>*Vj-1 m = .1 m , j , i 1-1 a  b  = 9i b ,j,i(t-At)  +  m  C  ( -  ( t )  1  )  (1-30)  ( t )  b^j^tt-c)  i  2 9  +  d  ±  b  m #  j _ ( t - -A t ) , }  r  (1.31) and, b  k / j  (t)  =  a i  ( t ) * bj^j +^ t )  (1.32)  m  k , j (t) =  b  .|  bk.j.itt) = 9i b ,j,i(t-At) k  +  b 1  C  k,j,i(t)  b  i  k / j + 1  (1.33)  ( t - r ) + di  b (t-r-At) kfj+1  (1.34) As mentioned i n the p r e v i o u s s e c t i o n , with b j ^ j and b ^ j known, the i n t e r m e d i a t e v o l t a g e s a r e o b t a i n e d u s i n g e q u a t i o n  (1.14) i n the time  domain, Vj(t) =  -|(b  k > j  (t) + b  m #  j(t)  ).  The i n t e r m e d i a t e c u r r e n t s can be o b t a i n e d the time  (1.35) from equation  (1.15) i n  domain 2ej(t) = b  m >  j(t)  -b  k f j  (t),  (1.36)  where, ej(t)  = ij(t)  * z  e q  (t).  (1.37)  12 From e q u a t i o n z  e q  (t)  Introducing  (1.21),  = [k  Q  6 ( t ) + k, e " "  (1.38) i n t o  1 1  + ... + k  m  e"  a m , t  ] u(t).  (1.38)  (1.37) m | ej 1  ej(t)  = ej  ? 0  (t) +  f i  (t),  (1.39)  where, e e. . ( t ) = m,  i j ( t)  (1 .40)  e. • ( t - A t ) + p. i . ( t ) + g. i . ( t - A t ) .  (1.41)  j f 0  (t)  = k  D  The c o e f f i c i e n t s f o r the r e c u r s i v e c o n v o l u t i o n (1.37) (see appendix I-B)  mi, P i and g i are given -a,-  = h  e  ^  =  algebraic i.(t) 3  At  k j (1 - h j )  gi equations  by,  1 - mj aj_At  1  P  Introducing  of e q u a t i o n  al (1.40) and (1.41) i n t o (1.36) we o b t a i n ,  a f t e r some  manipulations, =  iri(b p L2  m  i  (t)  -b  J  k  (t))  j  - q ijlt-At)  -  m' E mi i =1  e  j  i(t-At) (1 .42)  where,  p = k  q =  j  0  +  m" S q  m* Z p 1=1  ±  i  13 1 .3  Initial  Conditions  With equations  (1.36) and (1.42),  the c u r r e n t and v o l t a g e  p r o f i l e s on a g i v e n l i n e can be o b t a i n e d when the p a s t h i s t o r y terms o f the forward  travelling  f u n c t i o n s f o r the end p o i n t s  ( f j and f ) are m  known. Note t h a t a r e c o r d of p a s t h i s t o r y terms f o r the i n t e r m e d i a t e parameters must be kept f o r the s o l u t i o n . v e c t o r s must be e v a l u a t e d If  When t = 0^ these  history  from the i n i t i a l c o n d i t i o n s of the s i m u l a t i o n .  the c u r r e n t s and v o l t a g e s a t the ends of the l i n e are zero, a l l the  h i s t o r y terms are s e t i n i t i a l l y conditions exist prior  to zero.  I f l i n e a r , ac s t e a d y - s t a t e  to t = 0, the p a s t h i s t o r y terms can be e v a l u a t e d  from the phasor q u a n t i t i e s a s s o c i a t e d with e j , <=J  ej (t) b  m,j  b  k  #  j  a n c  ^ ^k»  ^ Vj(co)  t ) - — ^ V j ( w )  ij(t) From e q u a t i o n  bk,j  1> Ej (co)  ( t ) < a  (  t>m,j'  oTjU)  «a  (1.20) A  1  m  =  (co)  1  E  A.  i=1  1  ,  .  (co),  ,  e  1  where, A  S i n c e F^ and F system, B  K >  m  »  1  1  .(co) =  ,  k  j  8^  a r e known from the s t e a d y - s t a t e s o l u t i o n of the  j ^-j_ ( a n d Bj^-j -j_ (<»>)  (1.9), (1.10),  l  co +  can be obtained  f  (1.18), and V i B  m  ,i  equations  (1.19), (  a  )  )  =  A  1 , i  (  w  )  . (co) = AT Am) H  from  *k B  (1.43)  ( U ) )  .  .(co)  (j=2,...,n)  (1.44)  14  Vn,i "> (  k,j,i  5  =  ( & ) )  =  A  1,i>> 1 , i  A  U  V > U  < '45> 1  ^ t , j+l , i  )  (j = 1i...»n-1)(1.46)  ( a ) )  The i n t e r m e d i a t e s t e a d y - s t a t e c u r r e n t s can be found equation  using  (1 .15) = V i ' " ) - Vj("> 2 Z (to)  I, (co)  ,  eq w i t h Z g((o) g i v e n by e q u a t i o n e  A l s o , from equation  (1.21). (1.21) 111 + ... +  Z (to) = o + k  eq  Z  Introducing  Z  e  q  /  i  (ai)  (1.47)  k  m'  =  (to) i n t o equation  (1.37) i n the frequency  domain  E-(to) = Z (to) 7 . (to) J 3 J  (1.48)  et  results i n , E.  .(to) = Z „ (to) e  H  (1.49)  7,(0)).  The time domain e q u i v a l e n t s of these phasor q u a n t i t i e s are then g i v e n by, b  b,D  Vj  ( t )  i  t)  e  j (  ( t )  cos (tot + arg(B • (to) ) ) = w . (to)| cos (tot + a r g ( B j (to)) ) = (to)| cos(tot + a r g d j (to)) ) • (to)|  "  1J  3,i  (  t  )  =  fi  1  k j  (to)| c o s (tot +  Note t h a t i n the case of dc i n i t i a l  a r g ( E ^j / - -(to)) ) H  1  conditions (i.e.,  c h a r g e ) , to i s simply s e t to zero i n the equations  above.  trapped  CHAPTER 2  IMPLEMENTATION OF  2.1  General  THE  SOLUTION METHOD  Considerations  The  model and  equations  d e s c r i b e d i n Chapter  1 have been  mented i n t o UBC's frequency-dependence v e r s i o n of the EMTP.  Since  implethe  p r o f i l e model r e q u i r e s a r e l a t i v e l y s m a l l amount of i n f o r m a t i o n from  the  s o l u t i o n of the l i n e a t the end p o i n t s , the i n t e r c o n n e c t i o n with the EMTP is  very simple  and  straightforward.  to be added to the main program, and  Less  than 90  a l l profile  FORTRAN statements  had  c a l c u l a t i o n s are made i n  subroutines. The  profile  t i o n f o r the end order  c a l c u l a t i o n s proceed  points  (actually,  to a v o i d numerical  there i s a d e l a y of one  i n s t a b i l i t y when the  the t r a v e l time of the complete  line).  to i n c r e a s e computational The  t i o n s with o n l y one  time  solu-  step i n  p r e f e r r e d over  memory  the  requirements  efficiency.  of the l i n e .  The  u s e r ' s p o i n t of view, i t i s e a s i e r and  several line  to decrease  p r o f i l e model p r e s e n t s some important  e x t e r n a l segmentation  the  time s t e p i s very c l o s e to  This method was  a l t e r n a t i v e of p o s t - p r o c e s s i n g , i n order and  s i m u l t a n e o u s l y with  most obvious faster  advantages over one  i s t h a t , from  to r e q u e s t p r o f i l e  command, (see appendix I-C)  the the  calcula-  than i t i s to s e t up  segments.  The  number of n u m e r i c a l  convolutions  (one of the most  time  consuming o p e r a t i o n s i n the frequency-dependent s o l u t i o n of a l i n e ) i s reduced  by a f a c t o r of  (3n + 4)/(4n + 4), where n i s the number of  mediate nodes.  15  inter-  16 The d i f f e r e n c e , although intermediate These savings intermediate  p o i n t s , can be c o n s i d e r a b l e  l a r g e values  of n.  node, t o 24.75% f o r 100  modes. advantage, however, i s the i n c r e a s e i n accu-  Some of the computational  are d i s c u s s e d  for relatively  range from 12.5% f o r 1 i n t e r m e d i a t e  The most important racy.  not s p e c t a c u l a r f o r a s m a l l number o f  next.  aspects  l e a d i n g to t h i s  improved  accuracy  17 2.2  At  I n t e r n a l Time Step  1  In the s o l u t i o n of t r a n s m i s s i o n the main exact case  source  of error  fact  that  f ( t - A t - r ) are  line  interpolation  be  reduced by u s i n g a s m a l l A t ,  The the c l o s e s t  submultiple profile  (and  of  f(t-r+At).  f ( t - C ) are e v a l u a t e d  t i o n method, i n c r e a s e s with  of  multiple  of  at  the end  line  of an  the number of time steps  internal  segments. the e x t e r n a l segmenta-  l i n e has  more than  The  The  one  phase,  which i s a  sub-  time s t e p slows down the p r o f i l e needed i s g r e a t e r  to s o l v e the  c u r r e n t s and  voltages  a t s l i g h t l y d i f f e r e n t p o i n t s i n time,  interpolation  i n v o l v e d when  modes.  necessary  Note t h a t the i n t e r m e d i a t e  considered.  the number of s e c t i o n s c o n s i d e r e d .  l f o r a l l propagation use  mode  that i s  p o i n t s , no a d d i t i o n a l  i n comparison with  t i o n s because the number of time s t e p s  evaluated  time step A t '  longer p o s s i b l e to choose an e x t e r n a l A t  The  At  e l i m i n a t e d by choosing  I f o r the  improvement becomes more e v i d e n t when the i t i s no  must  interpolation error  a c e r t a i n amount of i n t e r p o l a t i o n  gained,  terms  an i n t e g e r number,  The  internal  i s needed f o r the i n t e r m e d i a t e accuracy  an  T h i s e r r o r , i n the  For example, f(t-c)  or i t can be  l a r g e s t ) submultiple  f(t-At'-r)  The  A t i s not  when the p a s t h i s t o r y  i s needed.  r o u t i n e s s e l e c t an  is s t i l l  interpolation  intervals,  C.  Although there and  step  [6].  I f r / A t i s not  evaluated.  i n t e r p o l a t e d between f ( t - C - A t ) and  to be an exact  and  the time  of frequency-dependent s o l u t i o n s , occurs  l i n e a r or h i g h e r - o r d e r  can  the  f r a c t i o n of the t r a v e l time r of the  f ( t - r ) and  be  is  l i n e s a t d i s c r e t e time  line  than,  a t the end  calcula-  or equal  to  points.  f o r each mode w i l l  be  t h e r e f o r e , an a d d i t i o n a l  i s needed to o b t a i n the phase domain q u a n t i t i e s . interpolation  f o r phase v o l t a g e s and  currents introduces  minimal amount of e r r o r because i t i s not a cumulative  process:  a  the phase  18  voltages  f o r one s e c t i o n a r e n o t needed to e v a l u a t e  the next s e c t i o n ; and once a l l the v o l t a g e s step,  they a r e no longer When the l i n e  choose a time step propagation  are obtained  i s segmented e x t e r n a l l y , care must be taken to than the t r a v e l time of the f a s t e s t  mode of the s h o r t e s t s e c t i o n . as the number of i n t e r m e d i a t e  must be decreased a c c o r d i n g l y . er system, t h i s  reduction  prohibitive) increase  sections increases, A t  When the segmented l i n e  i s u s u a l l y not j u s t i f i a b l e  components of the network.  i n computing  the decrease i n A t i s c o n f i n e d  the p r o f i l e  to the p r o f i l e  mode) A t i s l a r g e r than the t r a v e l  racy i s d e s i r e d ,  I-C) .  1  (and o f t e n  r o u t i n e s , because  s o l u t i o n o n l y , while the When ( f o r a given  time of the s m a l l e s t s e c t i o n , the  i s automatically  the p r o f i l e  f o r the r e s t of the  costs.  r e s t of the network can be s o l v e d with a l a r g e r A t .  i n t e r n a l time s t e p A t  i s p a r t of a l a r g -  This r e s u l t s i n an unnecessary  This problem i s a l l e v i a t e d with  m  f o r one time  needed f o r any f u t u r e c a l c u l a t i o n s .  that i s smaller  Therefore,  to At'/ »  the phase v o l t a g e s o f  s e t equal  routines provide  t o i.  the o p t i o n  If larger accuto decrease A t '  where m i s an i n t e g e r number s u p p l i e d by the user  (see  appendix  19  2.3  Approximation  The  of A-| (co)  r a t i o n a l - f u n c t i o n s approximation of Ai(co) can be o b t a i n e d  w i t h v e r y h i g h a c c u r a c y over the frequency range studies.  of i n t e r e s t i n t r a n s i e n t  However, w i t h the b e s t approximations a t t a i n a b l e w i t h the (*) , peak e r r o r s of 0 . 5 %  a p p r o x i m a t i n g r o u t i n e s p r e s e n t l y a v a i l a b l e a t UBC  are not uncommon f o r values of | A-|(co)| between 1 . 0 and 0 . 1 in  the r e g i o n where | A-j(to)| < 0 . 1 are not s i g n i f i c a n t i n most a p p l i c a t i o n s ) . These e r r o r s  tend to accumulate  Suppose, f o r example, t h a t a 5 0 0 km L e t | A^co^)) co  =  and | A ^ c o ^ ]  CO f o r a 5 0 km q  L e t us now =  i s segmented.  l i n e i s segmented i n t o 1 0 s e c t i o n s .  line,  respectively.  I W l  50  =  0.980  I 1  500  =  0.81707  i s o f f by 0 . 5 1 % a t  suppose t h a t the approximation of | A.| (to )|  C0 ; Q  IV'VUo I f we  when the l i n e  be the a c t u a l magnitudes of A-| (co) a t  and 5 0 0 km  A  CO  (higher e r r o r s  calculate |A  1 1  =  0.985  • (OJ )| ^  (co )| ^ Q Q from | Q  I VK>>I  500  0  l 1<"o>l  [ A  =  Comparing | A i ' ( t o ) | 500 w i t h | A-|(co )| 0  0  i n the approximation of the 5 0 km  5  0  Q we  Q  we 5 0  ]  1  obtain, °  =  ° '  8  5  9  7  3  can see t h a t an e r r o r of 0 . 5 1 %  segment, y i e l d s an e r r o r of 5 . 2 2 % i n the  e s t i m a t e d value of j A-| (w )| 500. Q  (*) Jose R. M a r t i has been working (at the time t h i s t h e s i s i s being w r i t t e n ) on more r e f i n e d v e r s i o n s of the approximating r o u t i n e s a v a i l a b l e a t UBC. P r e l i m i n a r y r e s u l t s have been encouraging, and the new r o u t i n e s can be expected to y i e l d even more a c c u r a t e approximations i n the near future.  The p r e c e d i n g example i l l u s t r a t e s how a p p r o x i m a t i o n of A-|(io) accumulate cascade.  the e r r o r s  when s e v e r a l lirtes are connected i n  Note t h a t the e r r o r i n c r e a s e s w i t h the number of s e c t i o n s ,  the e r r o r of one  s e c t i o n i s t r a n s m i t t e d to the  line  sectioning.  as  next.  This i s an unavoidable source of e r r o r i n any involving  i n the  The model developed  calculation  i n Chapter  1 i s also  a f f e c t e d by the q u a l i t y of the approximation of A-|(u>), but to a l e s s e r e x t e n t than the e x t e r n a l segmentation the c o n v o l u t i o n s a-| (t) * b of the l i n e ,  j (t) and a-| ( t ) * b  (see s e c t i o n 1.2)  o n l y , but i t i s averaged visualized  k f  procedure. m #  Since the e v a l u a t i o n of  j (t) s t a r t a t o p p o s i t e ends  the e r r o r does not accumulate  along the l i n e .  m,j  "  A  l m,j-l  k,j  =  A  l k,j l  B  B  (  B  (  j  j  =  (1.18) and  =  2  n  )  1  n  -  X  +  k  B  B  B  "  A  l  m, 2  "  A  l  m,l  !  =  A  l  m, 2  =  A  l  ,  m, 3  B  m,10 S t a r t i n g from node m, B  k  m, 1  F  m,9  B  F  A l  F  2  A  A  =  l  A l  3  1 0  A  F  k  F  k  F  F  k  k,10  A  l  k,9  A  l  B  k,10  = A* 1  F  k,8  A  l  \,9  = A/ 1  F  k,l  A  l  B  k,2  = A. 1  m  10  m m  F  m  (1.19)  )  Suppose t h a t the l i n e has 10 i n t e r m e d i a t e nodes. node ,  end  T h i s can be more e a s i l y  i n the frequency domain from e q u a t i o n s B  a t one  Starting  from  I t can then be seen t h a t , when the e r r o r i s maximum i n i s minimum i n B j m/  and v i c e v e r s a .  The r e s u l t i s an a p p r o x i m a t e l y  constant error l e v e l f o r a l l s e c t i o n s .  22 2.4  Animated Motion P i c t u r e of T r a v e l l i n g Waves Along the Suppose t h a t the i n t e r m e d i a t e  time t are known. it  I f the number of i n t e r m e d i a t e  i s p o s s i b l e to p l o t these v o l t a g e s  o b t a i n a smooth curve,  Fig.  2.1:  (or c u r r e n t s )  at a  given  p o i n t s i s l a r g e enough,  the  l e n g t h of the  line,  and  2.1  V o l t a g e p r o f i l e 3.6 ms a f t e r the e n e r g i z a t i o n of an open ended, s i n g l e - p h a s e l i n e .  wave a t time t .  Jl can be v i s u a l i z e d as a s t i l l  I f s e v e r a l of these s t i l l  s e q u e n t i a l l y , f o r i n c r e a s i n g values could  against  as shown i n F i g u r e  This p l o t of v ( t ) vs voltage  voltages  Line  then be  p i c t u r e of a  frames were d i s p l a y e d  of t, the e f f e c t of  "a moving wave"  created.  To demonstrate the p o s s i b i l i t i e s  of t h i s procedure as a  t o o l , an animated movie of t r a v e l l i n g waves along produced as p a r t of t h i s p r o j e c t .  a transmission  teaching line  was  For the g e n e r a t i o n of t h i s movie, the  line  from John Day  Lower Monumental of the B o n n e v i l l e Power A d m i n i s t r a t i o n an example.  To demonstrate the behaviour  modes, the l i n e was  on the l i n e , points  and  v o l t a g e s and  (51 p o i n t s per  (as shown i n F i g u r e 2.1)  The  a t every  time step on the IBM  a 16 mm  c o - o r d i n a t e d with  r e s u l t i s a movie  shows a few  s i t u a t i o n simulated  p u l s e of 0.5  (13 min.  ms  d u r a t i o n i n t o the 500 km  movie camera.  which i s very d i f f i c u l t  The  number of  the time s t e p of the t r a n s i e n t  l o n g ) , i n which t r a n s i e n t on a l i n e  (*).  the i n j e c t i o n l i n e , with  of a u n i t  voltage  the r e c e i v i n g end case.  a v i s u a l i z a t i o n of the t r a n s i e n t phenomena i n to o b t a i n from the u s u a l p l o t s ( v o l t a g e or  f i x e d p o i n t on the l i n e ) .  t e a c h i n g a i d are c o n s i d e r a b l e , and lightly  length  3279 c o l o u r  P o s i t i v e sequence parameters were used i n t h i s p a r t i c u l a r  c u r r e n t vs time a t any  (*)  intermediate  s e l e c t e d frames e x t r a c t e d from the movie.  i n t h i s case was  T h i s movie permits  be  simulated  second.  F i g u r e 2.2  a way  or  c u r r e n t s were p l o t t e d a g a i n s t the l i n e  phenomena are seen as waves p r o p a g a t i n g  open.  a t 49  as  to c r e a t e a r e l a t i v e l y smooth motion a t a p r o j e c t i o n speed of  18 frames per  The  e i t h e r with zero  c u r r e n t s were o b t a i n e d  used  propagation  S e v e r a l t r a n s i e n t s i t u a t i o n s were  photographed with  exposures per p l o t was simulation  was  time step a f t e r i n c l u d i n g the end p o i n t s ) .  These v o l t a g e s and  t e r m i n a l , and  of the d i f f e r e n t  assumed to be s i n g l e - p h a s e ,  p o s i t i v e sequence parameters.  (BPA)  to  The  p o s s i b i l i t i e s as a  the b e n e f i t s as an a n a l y s i s t o o l  cannot  disregarded.  Copies of t h i s movie can be borrowed by c o n t a c t i n g Dr. the Department of E l e c t r i c a l E n g i n e e r i n g a t UBC.  H.W.  Dommel i n  24  0.67  2.0 1  ms  1.17  l.S l.S-  1.9  i.o -  o.s-J  o.s-  0  o•  -0.3  -0.5-  -1.0  -1.0-  -1.5 -2.0  ms  -l.5  o  so ico  150  z;o  OISr.lMCS  1.64  2.3 1  250 s : o 3 5 0 (RILOKEIERSl  <ao <so  500  ms  -  :  ,  3  '  0  5 0  1.85  l.S-J  1.9  1.0  _  C  2.0  1.5  _  , 3 ! 5 D 203 2Zi 333 ?:0 <'.! "S3 5:3 OlS'S-iCt tK;:_5r.iTE*Si  ms  0.5  -o.s -1.0-j  -1.0  -1.5  -1.5  -2.0  0  50  ICC  150 200 250 200 ;f,0 OlSrSKCE  IK:L3J-.£T£B5]  4150  450 500  -2.0  -0.5-  -1.0 -  -1.3 •  -1.5-  -1.5-  Fig.  r—.• i , , , 100 ISO 200 250 300 353 430 450 500 DISTANCE (KILOMETERS)  2.2:  50  100  150 200 250 300 3:0 400 433 530 IX:LEK£TESS!  OISTRKCE  -O.S-  -2.0 4——i 0 SO  4  -2.9  1 0  ,  SO  100  1  ,  ,  ,  J50 200 253 333 353 4C3 450 505 OISIRSCE tPCLO-.ETERS)  S e l e c t e d frames from the T r a v e l l i n g Waves Movie.  25  2.45 m s  2.0-  >S  '1  0.5  2.92 m s  2.0 -j  15-  1.0 0.5  4  -0.5-  -0.5  -1.0 •  -1.0  -1.5-  -i.s-3 S0  100  ISO 200 230 300 T.O 400 450 OISTSKCc IK.'LWETERSl  500  3.22 m s  -2.0  0  50  100 ISO 200 250 3:3 350 433 <50 OISTRNCt (KILCIETESS)  50:  3.63 m s  2.0  :  1-51.0 0.5  -0.5 -1.0 -1.5-3 S  50  100  150 203 250 300 350 433 -50 500 0IS7RNCE IKILCMETERSI  3.73 m s  2.0:  0  SO  i.s-i  l.o •  1.0-  0.S-;  0.5-  500  3.89 m s  2.0 -j  I.S-j  100 ISO 200 25" 300 350 "SO *S0 DISTANCE (KILOMETERS:  0-0.5-  -0-5 -  -1.0-1.5-2.0-  SC  100  ISO 200 2SC 300 350 400 4S0  oisrivcE UILGKETESSI  F i g . 2.2:  (continuation)  500  -2.0  0  SO  100 ISO 200 250 303 350 400 450 OISTRNCE  (KILOMETERS)  S00  4.70  2.0  ms  5.17  2.0  '1  ms  i.H 1.0  o.H  0.5  -0.5 -1.0  -i-H 0  50  I  0  50  100  100  F i g . 2.2:  150 200 2S0 300 350 0ISTRNCE iKilOMETEnS)  «0  <S0  150 200 250 300 350 DISTANCE (KIL3-.ETESS)  CO  OO  (continuation)  1  500  1  53J  -2.0  SO  -i.u-i  0  "i  50  !00  150 200 250 300 350 DISTANCE (KILOMETERS)  100  ISO 200 250 300 350 DIS73MCE (KILC.1ETERS1  i  1  i  1  27 For example, when a v o l t a g e l i m i t i n g d e v i c e a r r e s t e r i s connected to the end assumed t h a t the v o l t a g e s along  of the l i n e ,  such as a l i g h t n i n g  i t i s sometimes  the l i n e w i l l not exceed the  determined by the c h a r a c t e r i s t i c of the l i g h t n i n g a r r e s t e r .  erroneously voltage To prove  this  assumption wrong, i t i s s u f f i c i e n t  to segment the l i n e and  of the i n t e r m e d i a t e  overvoltages.  However, when the v o l t a g e wave i s seen  r e f l e c t i n g back and  f o r t h along  the  line,  to observe some  the i n t e r p r e t a t i o n of  the  phenomena becomes much s i m p l e r . An animated movie can o b v i o u s l y not be produced e v e r y  time a  t r a n s i e n t s i m u l a t i o n i s performed, but such movies can be made f o r s e l e c t e d cases  f o r t e a c h i n g purposes.  The  best s o l u t i o n  for routine  v i s u a l i z a t i o n of power t r a n s i e n t s would be to d i s p l a y the wave motion on a suitable graphics  terminal.  A t the time t h i s t h e s i s i s being w r i t t e n , some work i s being done i n t h i s d i r e c t i o n  i n the Department of E l e c t r i c a l E n g i n e e r i n g  u s i n g the Megatek 4000, f a s t - r e f r e s h , g r a p h i c be expected  station.  t h a t i n the near f u t u r e , r o u t i n e examination  w i l l become p o s s i b l e a f t e r  I t can  at  UBC,  therefore  of wave motion  the e x e c u t i o n of a t r a n s i e n t s i m u l a t i o n .  CHAPTER 3  NUMERICAL RESULTS  3.1  Introduction The  results  from a s e r i e s of t e s t s and comparisons  i n t h i s chapter i n o r d e r to assess the performance In these t e s t s  the parameters  of the p r o f i l e  of a t y p i c a l 500 KV l i n e w i l l  BPA's John Day t o Lower Monumental t r a n s m i s s i o n l i n e ) . line  i s assumed t o be 500 km.  3.1,  and the p h y s i c a l c h a r a c t e r i s t i c s  a)  w i l l be shown  The tower c o n f i g u r a t i o n  model.  be used  (i.e.,  The l e n g t h of the i s shown i n F i g u r e  of the conductors are l i s t e d  below:  Phase conductors dc r e s i s t a n c e = 0.032405 H/km tube  t h i c k n e s s / o u t s i d e diameter = 0.3636  (stranded conductor  i s approximated  w i t h the e f f e c t s of the s t e e l core  as a tube, ignored)  diameter = 4.0691 cm  b)  Ground wires  (assumed to be semented or "T-connected"  ^*^)  dc r e s i s t a n c e = 1.6218 fi/km tube t h i c k n e s s / o u t s i d e diameter  =0.5  diameter = 0.98044 cm  (*) Grounded a t one tower, and i n s u l a t e d , as w e l l as s e r i e s i n t e r r u p t e d a t the a d j a c e n t towers. T h i s arrangement p r o v i d e s e l e c t r o s t a t i c s h i e l d i n g (ground wires c o n s i d e r e d i n c a p a c i t a n c e c a l c u l a t i o n s ) , but e l i m i n a t e s c i r c u l a t i n g c u r r e n t s (ground wires i g n o r e d i n impedance calculations).  28  29 c)  Ground r e s i s t i v i t y = 100 Q-m  3.93 m  1  3.93 m  c»|^  c  30.02 m  23.62 m  6.55m  |  c» Ua  6.55m  0.457 m  15.24 m  Fig.  3.1:  Tower c o n f i g u r a t i o n of BPA's John Day t o Lower Monumental 500 KV t r a n s m i s s i o n l i n e . Height shown i s average h e i g h t above ground.  30 3.2  Truncation The  ate v o l t a g e s  and  two and  p o l a t i o n i n the  Segmentation E f f e c t s  major sources of e r r o r i n the c a l c u l a t i o n of currents  are  the  forward t r a v e l l i n g  i n t e g e r number), and  the  These two  open (see F i g u r e  to l i n e a r  inter-  (caused when r / A t i s not  s e c t i o n to the  e f f e c t s w i l l be  purpose, a simple e n e r g i z a t i o n being  functions  segmentation e r r o r s caused by  e r r o r s from the s o l u t i o n of one  end  t r u n c a t i o n e r r o r s due  intermedi-  illustrated  an  the accumulation  of  next. in this section.  t e s t w i l l be performed, with the  For  this  receiving  3.2)  t=0 m -o  COS  F i g . 3.2:  (ujt)  0  Energization  The  l i n e w i l l be assumed to be  sequence parameters. 9 intermediate  first  the  l e n g t h of the  to i s o l a t e  s e t of t e s t s the  1/20th of the is  The  nodes w i l l be  In order the  test.  line  i s 500  with p o s i t i v e  km,  and  10 s e c t i o n s  considered. t r u n c a t i o n from segmentation e f f e c t s , f o r  time s t e p A t has  t r a v e l time,- t h i s  t r a v e l time f o r a 50 km  w i l l be p r e s e n t  single-phase  i n the r e s u l t s .  implies  been chosen to be  t h a t A t =1^/2  s e c t i o n ) , and  t h a t no  exactly  (where truncation  r^  0  errors  or  31 I t was mentioned i n Chapter t i o n e r r o r s i s the approximation segment ( 5 0 km i n t h i s c a s e ) .  of the p r o p a g a t i o n  of j A-j ( OJ)|  significant  1.0  f o r t h e purposes  and  0.1.  of this  Graph 1.2 shows the e r r o r the r a t i o n a l - f u n c t i o n s approximation the  500  tween  and | A-] ( to)|  the l i n e c o n s t a n t s program being used  values o f | A]_ (to) | between  km l i n e  IO.C^Q  and  f u n c t i o n f o r each  (with the  as the r e f e r e n c e ) ,  |A(to)|  Lower v a l u e s o f  1  for  are not  project. f u n c t i o n f o r [| A-|(to)| of | A-j (to)| 500^*  i s 1 . 6 7 9 8 ms, while r^Q i s 1 . 6 7 9 ms. i s very s m a l l i n t h i s case  rE^QQ  line  Graph 1.1 shows the e r r o r f u n c t i o n of the  r a t i o n a l - f u n c t i o n s approximation output from  2 t h a t the main source of segmenta-  T  n  e  t  r  a  ]  1  v  e  0  l  compared t o time f o r  The d i f f e r e n c e be-  (approximately  -0.074%).  Graph 1 .3 shows the r e c e i v i n g end v o l t a g e when the l i n e i s e x t e r n a l l y segmented The  (compared to the unsegmented  line).  assessment of the accuracy of the p r o f i l e model p r e s e n t s a  practical difficulty  because i t i s not c l e a r what r e f e r e n c e or a c c u r a t e  r e s u l t s should be used s i g n a l s were i n j e c t e d  f o r comparison purposes. i n t o the l i n e ,  If only single-frequency  an exact t h e o r e t i c a l  be o b t a i n e d u s i n g the s o l u t i o n of the l i n e equations domain and  (see appendix I - A ) .  time consuming p r o c e s s .  response  could  i n the frequency  However, t h i s would be a r a t h e r i m p r a c t i c a l A simpler a l t e r n a t i v e  r a t e ) would be to use o n l y two l i n e  (although not as a c c u -  segments and a d j u s t the r e s p e c t i v e  l e n g t h s to the i n t e r m e d i a t e p o i n t of i n t e r e s t .  This should g i v e a r e a s o n -  a b l y a c c u r a t e r e f e r e n c e model, assuming t h a t the p a r t i t i o n i n g of the l i n e i n t o two segments does n o t i n t r o d u c e s i g n i f i c a n t e r r o r s . s i m u l a t i o n s such comparison  two-segment models w i l l be used  purposes.  In the f o l l o w i n g  as a r e f e r e n c e f o r  5.0 n  FREQUENCY  (HZ)  (a)  Graph 1.1:  E r r o r f u n c t i o n s f o r the magnitudes o f (a) 50 km. (b) 500 km.  (co)|.  FREQUENCY  Graph 1.3:  (HZ)  R e c e i v i n g end v o l t a g e . E x t e r n a l l y segmented vs unsegmented l i n e .  34 Graphs 1.4  through 1.9  from the r e c e i v i n g end,  show the  u s i n g both the p r o f i l e and  Note t h a t the e r r o r i n the distance  from the sending end,  constant  error  effect  (on  multiple 1.68).  The  larger  10-segment models.  a  the  relatively  Graph 1.10  a time step  t r a v e l time ( i n t h i s case At = 0.1  shows  t h a t i s not  ms,  gives  i s segmented i n t o 10 s e c t i o n s  the  a  sub-  that i s , f ^ / A t =  l a r g e f o r the unsegmented the e r r o r i s  profile  model i s not  very  line,  but  considerably  s e n s i t i v e to the e x t e r n a l  matter of f a c t i t i s only a f f e c t e d to the same e x t e n t is affected  (see Graphs 1.12  and  c a t i o n e r r o r s were i n t r o d u c e d by  km  (see Graph 1.11). The  not  50  10-segment model.  truncation e f f e c t s .  d i f f e r e n c e s are not very  line  and  10-segment model decreases with  the unsegmented l i n e ) of using  of the  when the  consider  the  250,  Also note that the p r o f i l e model  r e s u l t s than the  Let us now  a t 450,  while the p r o f i l e p r e s e n t s  (see s e c t i o n 2.3).  consistently better  voltages  the p r o f i l e  model).  by  1.10; the  the unsegmented  these graphs suggest that the s o l u t i o n of the  Graph 1.14,  the e x t e r n a l At i s not a s u b m u l t i p l e  At; as  line trun-  unsegmented l i n e  further i l l u s t r a t e s  and  that even when  of £ (as i t occurs i n three-phase  cases) the p r o f i l e model performs a d e q u a t e l y . The  running c o s t s f o r the p r e v i o u s  (based on UBC's r a t e s of $1200 per Number of v a r i a b l e s requested i n output  hour of CPU  10 segments (cc  $)  t e s t are  shown i n Table  3.1  time)  Main Program (cc $)  P r o f i l e Model Output Processing (cc $)  Total (cc $)  1  0.67  0.68  0.08  0.74  9  0.76  0.68  0.10  0.78  18  0.86  0.68  0.18  0.86  Table 3.1:  Computing c o s t s f o r the s i n g l e - p h a s e  a  energization  test.  35 Note t h a t the p r o f i l e appendix I-C)  and  a l l the v o l t a g e s and  f r e e format) i n an i n t e r m e d i a t e v a r i a b l e s are read  from t h i s  the a i d of a p o s t - p r o c e s s i n g case,  for  tively  3.2  The  intermediate  t h a t i n the  segmented  time the requested  the i d e n t i f i c a t i o n  (and  v a r i a b l e s becomes d i f f i c u l t  manipulation (at best)  When the number  c a l c u l a t i o n s are  of the output  compara-  almost becomes a  c o s t s f o r a two-phase, 1 0 - s e c t i o n case are shown  Main Program (cc $)  P r o f i l e Model Output Processing (cc $)  Total (cc $)  10 segments (cc  $)  18  0.90  0.70  0.10  0.80  36  1 .00  0.70  0.15  0.85  3.2:  output  below.  Number of v a r i a b l e s requested i n output  Table  A l s o note,  to be made every  the p o s t - p r o c e s s i n g  running  (written i n  d e s i r e d amount of  10 v a r i a b l e s are p r i n t e d i n the same r u n .  f a s t e r and  Table  The  (or branches) i n c r e a s e s , the p r o f i l e  necessity. in  program.  l a t e r p l o t t i n g ) of the output  lines  c u r r e n t s were s t o r e d  (see  and w r i t t e n i n a manageable form with  A l s o i n the segmented case,  when more than of  file.  file  a complete s i m u l a t i o n has  changes.  c a l c u l a t i o n s were o n l y performed once  Running c o s t s f o r the e n e r g i z a t i o n of a two-phase l i n e with 9 i n t e r m e d i a t e nodes.  36 4.00  -1  0.0050  0.0100 TIME  Graph 1.4:  0.0150  0.0200  (SECONDSI  V o l t a g e a t 450 km from the sending end. Two segments vs p r o f i l e model; At = r/20.  o CM  cr — t n  UJ  o L3 CO  CO > CO  I  CNJ  LU CJ d  O > 0.0050  0.0100  0.0150  TIME (SECONDS)  Graph 1.5:  V o l t a g e a t 450 km from the sending end. Two segments vs 10 segments; A t = r/20.  0.0200  37  Graph 1.6:  V o l t a g e a t 250 km from the sending end. Two segments vs p r o f i l e model; At = t/20.  4 . 0 0 -i  ^  -3.00  - 3 . 0 0 -\  -4.00 0.0050 TIME  Graph 1.7:  0.0100 (SECONDS!  0.0150  V o l t a g e a t 250 km from the sending end. Two segments vs 10 segments; A t = r / 2 0 .  0.0200  38  4.00 -i CM \  S  — i  3.00-  II  t—  S  2 • 00  LU CJ  £  -3.00  :  o > - 4 . 00 -j  •0  1  1  1  1  1  1  0.0050  1  1  1  1  1  1  0.0300 T J ME  1  j  1  0.0150  (SECONDS)  Graph 1.8:  V o l t a g e a t 50 km from the sending end. Two segments vs p r o f i l e model; At = r/20.  Graph 1.9:  V o l t a g e a t 50 km from the sending end. Two segments vs 10 segments; A t = r / 2 0 .  1  ,  ,  ,  1  0.0200  „  4.00-] 3.00  2.00  1 .00  A  0  4  -l .oo H o >  - 2 . 0 0 -\  -3.00  -4.00  -r  1  1  0.0050  0.0100 TIME  0.0150  (SECONDS)  Graph 1.10:  R e c e i v i n g end v o l t a g e , unsegmented At = t/20 vs At = r/16.8.  Graph 1.11:  R e c e i v i n g end v o l t a g e , 1 0 segments vs -unsegmented l i n e (C=At/16.8).  line,  '  0.0200  40  o >  -4.00 .0050  0 . 0100 TIME  Graph 1.12:  x  0.0150  0.0200  (SECOND5I  V o l t a g e a t 450 km from the sending end, p r o f i l e model. At = r/20 vs At = f/16.8  4 . 0 0 -,  CO CO  2 M  3.00  H  2.00  H  CO  to  n o  CO  o in  LU CJ)  cr  o >  -4.00 .0050  0.0100 TIME  Graph 1.13:  0.0150  [SECONDSI  V o l t a g e a t 450 km from the sending end. Two segments vs p r o f i l e model. At = c/16.8  0.0200  Graph 1.14:  V o l t a g e a t 450 km from the sending end. Two segments (At = r/20) vs p r o f i l e model  (At =  r/16.8).  42  3.3  E n e r g i z a t i o n o f a L i n e Terminated With a L i g h t n i n g A r r e s t e r Occasionally,  the i n f o r m a t i o n a t the end p o i n t s of the l i n e  g i v e s i n s u f f i c i e n t i n s i g h t i n t o the o v e r a l l performance of the l i n e . For example, i t i s sometimes assumed t h a t the o v e r v o l t a g e s a t the r e c e i v i n g end are l a r g e r than a t any i n t e r m e d i a t e line. linear.  p o i n t along the  T h i s i s true as long as a l l the components i n the system are When a n o n - l i n e a r , v o l t a g e - l i m i t i n g d e v i c e such as a l i g h t n i n g  a r r e s t e r i s connected a t the end of the l i n e ,  the v o l t a g e s a t the  i n t e r m e d i a t e p o i n t s can be s u b s t a n t i a l l y h i g h e r , and i n some l i n e s little  i n s u l a t i o n margin, f l a s h o v e r a t some i n t e r m e d i a t e  with  towers c o u l d  occur. In t h i s s e c t i o n , a s i t u a t i o n where i n t e r m e d i a t e higher  than the r e c e i v i n g end v o l t a g e s w i l l be s i m u l a t e d . Consider  F i g . 3.3:  the c i r c u i t  shown i n F i g u r e 3.3  S i m u l a t i o n o f a three-phase l i n e with l i g h t n i n g a r r e s t e r s .  terminated  v o l t a g e s are  43 The  simplified  model used f o r the  l i g h t n i n g a r r e s t e r s i s shown i n F i g u r e  3.4  F i g u r e 3.4:  The absolute  L i g h t n i n g a r r e s t e r model. (b) v - i c h a r a c t e r i s t i c .  (a) E q u i v a l e n t  v o l t a g e - c o n t r o l l e d switch of f i g u r e 3.4  value of the r e c e i v i n g end  circuit,  (a) c l o s e s when the  v o l t a g e exceeds v„  ., and  opens  again  S a t  as soon as the c u r r e n t goes through z e r o . For t h i s  test,  the parameters of the l i g h t n i n g a r r e s t e r were  chosen so t h a t under normal s w i t c h i n g o p e r a t i o n s , without trapped the o v e r v o l t a g e s  were below v_ . ( i . e . , v_ . = 2.6 s a L. sa u  sources i n F i g u r e 3.3  were s e t to 1.0  arrester's characteristic is  BPA's John Day  ( f o r v>v  sa  ^-)  p.u.  p.u.).  (peak) and  shows the p r o f i l e  v a l u e s ) when there i s no trapped  voltage  the slope of  d v / d i = 1.0 fi; the  to Lower Monumental f o r a l e n g t h of 500  Graph 1.15  The  charge,  line  the  simulated  km.  of maximum o v e r v o l t a g e s  (absolute  charge p r i o r to l i n e e n e r g i z a t i o n .  44 Graphs 1.16 t o 1.18 show the r e c e i v i n g end v o l t a g e s .  Note t h a t i n t h i s  case the maximum o v e r v o l t a g e a t the r e c e i v i n g end i s l a r g e r  than a t any  p o i n t a l o n g the l i n e . When the worst  c o n d i t i o n f o r trapped charge  l i g h t n i n g a r r e s t e r s are t r i g g e r e d  i s s i m u l a t e d , the  (see graphs 1.20 t o 1.22).  In t h i s  case  the maximum o v e r v o l t a g e s a t s e v e r a l i n t e r m e d i a t e p o i n t s along the l i n e are h i g h e r than a t the r e c e i v i n g end (see Graph 1.19). times  a t which they occur are l i s t e d  D i s t a n c e from sending end (km) 50 1 00 150 200 250 300 350 400 450 500 Table  Phase A Time Vmax (p.u.) (ms) -2.38 -2.79 -2.87 -2.92 -2.92 -2.90 -2.84 2.80 2.74 2.73  3.3:  7.39 7.36 7.53 7.51 7.56 7 .40 7.21 • 16.73 16.51 2.31  These v o l t a g e s and the  i n Table 3.3.  Phase B Time Vmax (p.u.) (ms) 1 .89 2.05 2.20 2.37 2.51 2.61 2.65 2.72 2.79 2.87  Maximum o v e r v o l t a g e s with trapped p r i o r to l i n e e n e r g i z a t i o n .  7.12 6.94 6.78 6.60 6.53 6.27 6.10 5.93 5.75 5.59  Phase C Vmax Time (ms) (p.u.) -2.21 -2.36 -2.49 -2.57 -2.60 -2.61 -2.64 -2.51 -2.65 -2.71  4.03 4.19 4.36 4.53 4.62 4.84 4.99 5.15 5.30 5.38  charge  Graphs 1.23 and 1.24 show the v o l t a g e s a t 50 and 300 km from the sending end when trapped charge  i s considered.  Graph 1.16:  R e c e i v i n g end v o l t a g e .  Phase A, no trapped  charge.  46  Graph 1.17:  R e c e i v i n g end v o l t a g e .  Phase B, no trapped  charge.  Graph 1.18:  R e c e i v i n g end v o l t a g e .  Phase C, no trapped  charge.  47  Graph 1.19:  P r o f i l e of maximum o v e r v o l t a g e s Trapped charge.  Graph 1.20:  Receiving  end v o l t a g e .  (absolute  Phase A, trapped  values).  charge.  Graph 1.21:  R e c e i v i n g end v o l t a g e .  Phase B, trapped  charge.  Graph 1.22:  R e c e i v i n g end v o l t a g e .  Phase C, trapped  charge.  4.00 - i  S  3.00  A  S zooH i=  \00  A  OH -I .00  -2.00  A  -3.00 H -4.00 0.0100 (SECONOSI  0.0)50  0.0200  0.0100 TIME (SECONDSI  0.0150  0.0200  Tirt  4.00  T  UJ  ct  3.00-  o  S  2-30  CL o_  £  i.oo H  CD £  0-  cc  r  CL  ^ -1.00je:  R  UJ  -2.00 •  5 o cc >  -3.00  -4.00 •  0.0050  4.00 • 3.00  S  2.00 A  i .oo A 0 1 -1 .00 -2.00 -3.00 H -4.00  Graph 1 .23:  0.0050  0.0100 TIME (SECONDS)  0.0150  0.0200  V o l t a g e a t 50 km from sending end. (a) Phase A. (b) Phase B. (c) Phase C.  4.  DO  0200  0 . 0 ) 0 0  TIME (SECONDSI 1.00 UJ  ID £  3 . 0 0  § tier or  2 . 0 0 -  X  1 . 0 0 <TJ  UJ  £  o-  X  a  r;  - 1 . 0 0  -  - 2 . 0 0  -  o "  <E  UJ  LD g  - 3 . 0 0 -  ••  -4.00 0 . 0 0 5 0  O.OOSO  Graph 1 . 2 4 :  1  r  0.01O0  0 . 0 1 5 0  0 . 0 2 0 0  0 . 0 ) 0 0  0 . 0 1 5 0  0 . 0 2 0 0  TIME (SECONDSI  TIME (SECONDSI  V o l t a g e a t 300 km from sending end. (a) Phase A. (b) Phase B. (c) Phase  C.  CONCLUDING REMARKS  In the f i r s t p a r t of t h i s t h e s i s p r o j e c t a model f o r the e v a l u a t i o n of v o l t a g e been p r e s e n t e d .  The model and s u p p o r t i n g  overcome the accuracy standard  and c u r r e n t p r o f i l e s along  and data  encountered by  segmentation methods f o r p r o f i l e c a l c u l a t i o n s .  The model and r o u t i n e s are g e n e r a l , capabilities  ii)  l i n e has  r o u t i n e s have been designed t o  management d i f f i c u l t i e s  The main advantages achieved  i)  a transmission  Truncation  can be summarized as f o l l o w s :  and l i m i t e d  only by the  of the main (host) program.  e r r o r s are e s s e n t i a l l y e l i m i n a t e d ,  and segmentation  e r r o r s are minimized.  iii)  The r o u t i n e s are as f a s t or f a s t e r than standard procedures, and data  management  capabilities  The p o s s i b i l i t y of r o u t i n e p r o f i l e  segmentation  are c l e a r l y  c a l c u l a t i o n s , i s very  u s e f u l i n some p r a c t i c a l a p p l i c a t i o n s (e.g., c o - o r d i n a t i o n and  superior.  of i n s u l a t i o n  protection devices). Profile  r e l a t e d procedures such as the dynamic v i s u a l d i s p l a y  of t r a v e l l i n g waves c o u l d prove to be an e x c e l l e n t t e a c h i n g a i d and permit a b e t t e r understanding of t r a n s i e n t phenomena.  51  PART I I  LOW-ORDER  APPROXIMATION  OF THE FREQUENCY  DEPENDENCE  OF TRANSMISSION L I N E  52  PARAMETERS  INTRODUCTION  Accurate models which take i n t o account of t r a n s m i s s i o n l i n e  parameters,  have become very important  modern a n a l y s i s of t r a n s i e n t phenomena. require detailed  the frequency  t o o l s i n the  Such a c c u r a t e models u s u a l l y  i n f o r m a t i o n about tower c o n f i g u r a t i o n , e a r t h  e t c . ; i f many l i n e s have to be modelled,  dependence  the approximation  resistivity,  of the l i n e  parameters becomes a t e d i o u s and time consuming p r o c e s s . When the e f f e c t of one or more l i n e s outcome of a p a r t i c u l a r neither j u s t i f i a b l e  study,  i s not c r i t i c a l  the a c c u r a t e m o d e l l i n g of these  nor d e s i r a b l e .  to the lines i s  In such cases, a f a s t e r and simpler  l i n e model would be p r e f e r r e d , i f i t s accuracy were not s e r i o u s l y compromised. A simplified a)  I t should be c o m p u t a t i o n a l l y f a s t e r , and e a s i e r existing  b)  l i n e model should meet the f o l l o w i n g requirements:  a c c u r a t e models.  I t should be more a c c u r a t e than models t h a t do not take i n t o the frequency dependence of l i n e  c)  to use than the  account  parameters.  I t should r e q u i r e l e s s i n f o r m a t i o n (or i n p u t data)  than  the more  a c c u r a t e models. d)  I t should be compatible  with the h o s t program of the more a c c u r a t e  models ( i n t h i s case, the frequency-dependence v e r s i o n of UBC's EMTP code * )• (  }  (*) T h i s and a l l f u t u r e r e f e r e n c e s to the frequency-dependence model and/ or r o u t i n e s o f the EMTP r e f e r o n l y to those developed by Jose R. M a r t i (as p a r t i a l requirement f o r h i s Ph.D. degree a t the U n i v e r s i t y of B r i t i s h Columbia [ 7 ] ) . I t i s not intended t o c r e a t e the i m p r e s s i o n t h a t t h i s i s the o n l y frequency-dependence model a v a i l a b l e to EMTP u s e r s .  53  54  Such a s i m p l i f i e d  line  model i s developed  i n the second p a r t of  this  t h e s i s p r o j e c t , where the f o l l o w i n g assumptions have been made:  (1)  The l i n e  i s balanced  or p e r f e c t l y transposed,  distinct  propagation  modes are c o n s i d e r e d ;  t h a t i s , o n l y two  zero sequence or ground  mode, and p o s i t i v e sequence o r sky mode. (2)  The c a p a c i t a n c e s and shunt the e n t i r e frequency  (3)  conductance of the l i n e  are c o n s t a n t  over  range.  The ground r e s i s t i v i t y i s c o n s t a n t , and i t s v a l u e , as w e l l as the values of the e l e c t r i c a l are known.  parameters R, L and C a t power  frequency,  CHAPTER 1  THEORETICAL CONSIDERATIONS  1.1  E v a l u a t i o n o f the T r a n s m i s s i o n L i n e Parameters  1.1.1  S e r i e s Impedance  Matrix  In the frequency domain, a t r a n s m i s s i o n terms of the f o l l o w i n g d  V  p h  =  d I  P  and Y  respectively system  z  =  h  dx  p n  can be d e s c r i b e d i n  equations,  ~3x  where Z  line  (1.1)  ph ph I  Y  V  p h  p h  (1 .2)  ,  a r e the s e r i e s impedance and shunt admittance m a t r i c e s ,  p n  ( s u b s c r i p t ^ stands f o r phase q u a n t i t i e s ) .  For an n-phase  p  (without ground w i r e s ) , Z  1,1  ( u , )  z  z (co)  1,2  ( w )  Z  1,n  ( a ) )  2/1  Z ,(to) ph  z  n,1  *1,1 '2,1  z  ( a , )  yi,2  ( U j )  (to)  y ,1  U  (to)  *1,n  ( u )  (1.3)  n, n  ( w )  (1 .4)  y -'njn (^)  )  n  The elements of Z h(w) can be c a l c u l a t e d p  from the p h y s i c a l  c h a r a c t e r i s t i c s of the conductors and tower c o n f i g u r a t i o n : z  k,k  ( U )  = k R  (  w  )  +  A  R  k,k  (  u  )  +  ^  ( a ) L  int,k<  w )  +  u  L  ext,k  +  Ax  k,k< > w)  i n fi/km (1 .5)  55  56 Z  k,j  ( u )  =  Z  j,k  (U}) =  A  V j  U  )  +  j ( U  m,k,j  L  +  A X  k,j  ( u ) )  i n fi/kra (1.6) where, R^djj)  i s the ac r e s i s t a n c e of conductor k. frequency i s due v' " ) 1  3  i S  th  to s k i n e f f e c t  I t s dependency on the  (see appendix  i n t e r n a l inductance of conductor k.  e  depends on the frequency because  II-A)  I t s value a l s o  of s k i n e f f e c t .  If s k i n  e f f e c t i s ignored, v  L  . .  = 2*10  -4  £n J 2 l _  i s the e x t e r n a l i n d u c t a n c e . of the tower,  i n H/km  ( 1  '  7 )  I t o n l y depends on the geometry  and i t i s given by,  L ext,k. = 2-10" In l iri i i  i n H/km  4  <'> 1  8  k  L  ,  i s the mutual  inductance between conductors k and j .  It  o n l y depends on the geometry of the tower and i t i s given by,  Vk,j  = 2-10-4 . n ^ i  i  n  H  /  k  d.9)  m  ki3 h^  i s the average h e i g h t of conductor k i n m.  GMR D,  k  i s the geometric mean r a d i u s of conductor k i n m.  .  i s the d i s t a n c e between conductor k and conductor j  d • r. k w = 2 T f " v  ARk,k  (U))  '  A R  i n m.  the image of  (see Fig.1.1)  i s the d i s t a n c e between conductors k and j i s the r a d i u s of conductor k i n m.  i n m.  where f i s the frequency i n Hz.  k,j  ( a ) )  '  are  Carson's  c o r r e c t i o n terms  f o r e a r t h r e t u r n e f f e c t s , i n ^/Km appendix  II-B).  (see  57  images  Fig.  1.1:  Tower geometry. In the case of bundled  and  conductors  i f equal c u r r e n t d i s t r i b u t i o n  assumed), the bundle  k  i s symmetrical,  i n the conductors of the bundle i s  can be t r e a t e d as a s i n g l e e q u i v a l e n t  r e p l a c i n g the GMRk and r an e q u i v a l e n t  ( i f the bundle  conductor by  i n (1.7) and (1.8) by an e q u i v a l e n t GMR and  radius:  N G M R  eq,k  =  v  _  / N  * GMR .A  (1.10)  N1  k  N /  r  e q  ,k  =yN-*k.A  N _ 1  (1.11) where  N  =  number of conductors i n the  GMR  k  =  geometric mean r a d i u s o f the i n d i v i d u a l conductor  A  =  r a d i u s o f the bundle  Note t h a t equations  bundle. i n m.  i n m.  (1.5) an (1.6) are g e n e r a l and account f o r the  frequency dependence o f the parameters. conductors are i d e n t i c a l .  L e t us now assume that a l l  I t then f o l l o w s  that  58 *eq,k G M R P  o  = eq,j= r  k =  R (co)  G  m  ^'  r  -i = GMR  < * n  (1.14).  s  If the l i n e i s p e r f e c t l y transposed, impedance matrix different tion  i f the l i n e  illustrated,  we can assume that the  Zp^ i s the average of the impedance m a t r i c e s  transposition sections. i s going  This i s a p a r t i c u l a r l y  transposed  o-  B  o-  C  o-  II  Fig.  1.2:  III  Transposition for a single c i r c u i t .  The impedance matrix z P  h  z  z  z  z  l l 12 21 22 31 32  J  f o r the e n t i r e l i n e i s ,  T<( ph i l  =  z  z  z  z  z  z  s  2m  m  z  s  13 23 33  *m m  z  good approxima-  i n two p l a c e s ,  1 .2.  A  of the  to be s t u d i e d a t the end p o i n t s o n l y .  f o r a three-phase l i n e  +  t >  z  z  z  XI  + [z  1  22 23 21 32 33 31 12 13 l l  p h  I I X  ]>  z  z  z  z  z  z  z  z  )  (1.13)  =Rj(co) = R (co)  k  1 2  z  33 31 32 13 l l 12 23 21 22 z  z  z  z  z  z  This i s  i n Figure  59 with,  22 z  In  m  =  2 ' 12 z  an n-phase system, z  elements and z  +  13  z  s  +  *33> z  23>  would be the average of a l l d i a g o n a l  the average of a l l o f f - d i a g o n a l elements.  m  1  n  -  Z  z  k=l  n  2  m  n(n-l)  T h e r e f o r e , f o r a balanced  k  (1.15) k  n-1  n  Z  Z  k  =  i j=2  (or p e r f e c t l y z z  m  z kk  transposed  (co) (co)  (1.16)  f o r k<j  z (co) . z (co) s  line), z (co) m  (1.17)  ph  where, z  z  z (co) s  z (co) m  s  m  (co) = R (co) + A R (co) + j(co L. s  s  (co) = A R (co) + j(coL m  (co) + co L  int  ^ + A X (co) )  ext  s  + AX (co) ) , m  (1.18) (1.19)  m  with,  L  L  L  2 icf  4  _  =  2 io"  4  int = 2 io"  4  ext m  2h r D In d r Zn GMR in  (1. 20) (1. 21) (skin e f f e c t  neglected)  (1. 22)  1/n  kSi n-1  k n  . n.  .n  h  k=l  j=2  n-1  n  k=l  j=2  (geometric mean h e i g h t )  o K  1 2/(n-l)n k  k<j  (geometric mean d i s t a n c e between k and and the image o f j )  (1.24)  k<j  (geometric mean d i s t a n c e among the n conductors)  (1.25)  'J 2/(n-l)n k,:  A R ( c o ) , AR (co) , Ax (co) and AX (co) , are the averages o f Carson's s  m  c o r r e c t i o n te rms,  (1.23)  s  m  60 AR (w)  E  = I  s  n k=l  Ax (co) g  AR (co).  n *=1 .-, ,  ,  k  n-1  . E  7  n ( n - l ) k=l  modal  E  A Rk  .E  Ax,  j=2 n  9  m  distinct  n  £  t  n ( n - l ) k=l n  Ax (a))  The balanced  '  t  K  3  m a t r i x of e q u a t i o n  =2  .(co)  j>k  . (co)  j>k  /D  k  (1.17) leads to only  p r o p a g a t i o n modes (zero and p o s i t i v e  series  t  *D  sequence).  two  T h e r e f o r e the  impedance matrix has the form, z ( co) Q  zA co) Z (co) = m  (1.26)  . o  z  (us)  where z(co)  =  z(co) +  z. (co) 1  =  z (co) - z ( co) s m  o  s  °  (n-1)z(co)  (1.28)  To o b t a i n the sequence impedances we into z  o  (1 .27) and  (co) = R  +  s  j(co  zA co)  R +  s  g  L.  mt  (1.19)  int  +  s  (co)  + AR  j ( c o L.  (n-1 ) A R ( c o )  +  s  + co L  (co)  (to)  -  ^  ext  +  (n-1)co L  m  +  AX  + to L  ext  (co)  — co L  m  +  AX  =  Ro(to)  (to)  =  RT (to) + j X i ( c o ) ,  Z-|  (co)  +  (n-1)  AX  m  (co)),  (1 .29)  m  z (co) Q  s  AR (co)  but,  then,  i n t r o d u c e (1.18) and  (1 .28)  AR  (co)  (1 .27)  m  +jX (co) 0  s  (to)  -  AX  m  (co)),  (1 .30)  61  R (co) o  = R (co) + AR s s  (co) = co  X 0  (co) +  (n-l)AR  (L. (co) + L + mt ext  m  (co)  (n-1) L ) + AX (co) + m s  (1.31) (n-1) AX (co) m (1.32)  R, (co) 1  = R (co) + AR (co) - AR (co) s s m  (1.33)  X  = co (L. (co) + L -L ) + AX (co) - AX (co). mt ext m s m  (1.34)  1  (co)  62 1.1.2  Carson's  C o r r e c t i o n Terms  Carson's  the e a r t h r e t u r n e f f e c t earth r e s i s t i v i t y the conductors  s  (see appendix  Ax ,  m  and Ax  s  Il'-B) .  p , on the frequency  m  account f o r  These terms depend on the  f , and on the r e l a t i v e p o s i t i o n of  and e a r t h p l a n e .  At power frequency is  terms A R , AR ,  correction  (50 o r 60 Hz) the term  s m a l l , so t h a t o n l y the f i r s t  term  i n equation  a i n equation  (II-B.1]  (II-B.2) needs to be  retained. If described  the l i n e  is perfectly  i n the p r e v i o u s s e c t i o n  transposed, the a v e r a g i n g  procedure  yields  -4  AR U) S  =  AR (oj) =  £_L_L_12—  AXs(co)  =  co'2'10  [0.61 59315- In  (2 h-k  AXm(oj)  =  (jj'2'10  [0.6159315-  (D  where  k  m  =  Introducing  -4  -4  4 TT fs  i n H/km  £n  ./—)  ]  J~^) ]  k-  (1.35) in  in  fy/km  ft/km  (1.36)  (1.37)  • 10~ . 4  (1.35) i n t o  (1.31) and  (1.33) -4  R (OJ)  =  R (u>)  R-l ( i o )  =  Rs(co)  0  If  s  +  n  V  '  1  0  (1  „„ L. . = 2*10 -4 Hn mt  (1.32) we  .39)  we assume t h a t the s k i n e f f e c t i n f l u e n c e on the i n t e r n a l  i n d u c t a n c e a t power f r e q u e n c i e s i s n e g l i g i b l e ,  Introducing  (1.38)  —  equation  (1.22)  r GMR  t h i s v a l u e , and e q u a t i o n s  obtain, after  then from  some a l g e b r a i c  (1.36) and  (1.37) i n t o  manipulations  equation  63  X (co) = 2 nu •' 1 0 ~  4  0  in  b  ,^  with  ^  p  /  (1.40)  f  1^1"  VGMR d  , 0.6159315 b = e  Similarly,  i n t r o d u c i n g equations  (1.36) and  (1.37) i n t o  equation  (1.34)  X-i(co) = 2'co'10~  4  Hn _JL_ GMR  (1.41)  64 1.1.3  Shunt Admittance  Matrix  To e v a l u a t e Y p  (to) i t i s convenient  n  Maxwell's p o t e n t i a l c o e f f i c i e n t m a t r i x l,l  P  Pph  where 1  k,j  If we Pph  ITT  =  p n  run  l,2  -1  r  •n,l  Pk,k  first  (1 .42)  = u  p  P  P  to e v a l u a t e  =Pj,k  2  h  i  in  "rf  £ n  2^z~  l  n  equals "the c a p a c i t a n c e  ,n  (1.43]  km/F  l  assume t h a t the shunt  n  n  (1.44)  M  conductance m a t r i x G i s z e r o , matrix  the i n v e r s e of  K h p  -1 ph  ph  ph If d i a g o n a l and  the  Ph  line  is perfectly  o f f - d i a g o n a l elements of P  whose elements are g i v e n o  =  "s  P  Introducing  transposed  (1.43) and  P  average out  t o o b t a i n a balanced  by  the  matrix,  (1 .45)  —  n(n-l)  s  can  n  m  p  p n  we  =  n-1  n  Lk=l  j=2  '  k<j  (1.46)  (1.44), 1 2TT£  -z  = ~ 2TTE  „ An  2 h r  An' ~ d  w i t h h, r , D and d d e f i n e d as i n the p r e v i o u s s e c t i o n s .  (1 .47)  (1.48)  The modal p o t e n t i a l P P  Introducing  = p  Q  l  = s P  equations  +  s  c o e f f i c i e n t s are (n-1) p  "  P  m  m  (1.47) and  (1.48),  the modal, Maxwell's  coefficients  become, 1 „ 2 h D Jin 2TTE ,n-l o rd  .„> (1.49)  n - 1  p  =-  o  P  =  1  -, 1 2 ue o  The i n v e r s e of P modal c a p a c i t a n c e s  m o  2 h d  n  r  de  are simply  i  s  (1.50)  p  K  mode»  a  n  d  since P  the r e c i p r o c a l s  d e i - d i a g o n a l , the s  m o  of the modal  potential  coefficients, 2Tre c  c  o  =  Jin  „ ,  2TTC  0  n-1  i = Jin- r f 7 r D  (1 .51 )  -  (1  52)  66 1.1.4  Comparison between Exact and Approximate Formulas In order t o i l l u s t r a t e  for  the sequence q u a n t i t i e s ,  the accuracy of the approximate formulas  a typical  500KV l i n e  d e s i g n w i l l be  considered,  12.19m  12.19 m  o  o  o  o  o  o  o  o  o  o  o  o  15.24m  Fig.  1.3:  O  O  O  O  CU57m  Tower c o n f i g u r a t i o n of r e f e r e n c e  line,  where, conductor  diameter  =  22.86 mm  conductor  GMR  =  conductor  dc r e s i s t a n c e  9.32688 mm =  0.104763  Assuming t h a t the l i n e bundle  fy/km.  i s perfectly  transposed, and t h a t the  i s t r e a t e d as a s i n g l e e q u i v a l e n t conductor,  r  =  198.253 mm  GMR  =  188.427 mm  we have  d  =  15.361 m  D  =  34.78 m  Rdc  =  0.02619 fi/km Table 1.1 shows the r e s u l t s of i n t r o d u c i n g  these q u a n t i t i e s i n  the approximate f o r m u l a s , compared t o the "exact" parameters o b t a i n e d from the l i n e ' s  tower geometry  ( a t 60 Hz)  u s i n g UBC's L i n e Constants  Program.  R-l (fiAm) L-i (mH/km) C-| (yF/km)  R L C  D Q  Q  (!)A"i) (mH/km) (yF/km)  Table 1 . 1 :  Exact Formulas  Approximate Formulas  0.02643 0.8801 0.0133 0.1974 3.307 0.008361  0.02643 0.8802 0.0132 0.2041 3.289 0.008341  Error (%)  0.009 0.84 -3.38 0.55 0.24  Comparison between exact and approximate parameters.  These r e s u l t s  illustrate  t h a t the approximate  a c c u r a t e enough a t power f r e q u e n c y . i n c r e a s e d , these formulas would  formulas a r e  However, i f the f r e q u e n c y were  no longer be adequate.  68 1.2  Evaluation of an Equivalent Line Configuration Fran the Parameters at Power Frequency In o r d e r to e v a l u a t e the l i n e parameters  over the f r e q u e n c y  range of i n t e r e s t ,  the p h y s i c a l c o n f i g u r a t i o n of the l i n e must be known.  I f L-] , C-) , LQ and C  Q  instance), be used GMR,  are known a t a s u f f i c i e n t l y  the approximate  to f i n d  of the l i n e ,  p h y s i c a l parameters line,  are the same as those of the  original  of the e q u i v a l e n t l i n e are c l o s e  C o n s i d e r the approximate L.  L  r  =  2-10  =  o  -4  An  equations f o r L-|, L  k =  1  2  V  '  Z  Q /  C-| and  D  2  h  =  4  ' IT  6  1  5  9  3  1  Q  n / ,/ GMR  <' > 1  • d  n-1  °  r  C  (1.53) 54  (1.55)  2lT~d  £n  should  line.  _^JIIl  4  r  k  I f the  -5GMR  2n.10- £n  £n  0  line.  to those of the  —  (1.56)  2TTEO  e  t h a t i s r , d,  the frequency v a r i a t i o n of i t s e l e c t r i c a l parameters  a l s o be c l o s e to that of the o r i g i n a l  =  can  These q u a n t i t i e s d e f i n e an e q u i v a l e n t l i n e whose power  frequency parameters  where b  (60 Hz, f o r  i n the p r e v i o u s s e c t i o n s  some of the p h y s i c a l parameters  h, and D .  original  formulas developed  low frequency  TjH-1  d  n-1  5  f~5 •  10~  4  These are f o u r e q u a t i o n s and  f i v e unknowns.  The a d d i t i o n a l e q u a t i o n  needed i s g i v e n by D  2  =  (2h)  2  + d .  The d i f f e r e n c e between D c a l c u l a t e d o b t a i n e d by f i n d i n g  2  from equation (1.57) and  (1 .57) the value  the geometric mean d i s t a n c e between the conductors  t h e i r images, i s l e s s than  2%.  and  69 Let us now d e f i n e a  _ "  i  (Li/2-10  .  ,_ , /^x / ( P/f) n  a  =  2  b  k  a  3  a  4  1 .58)  6  =  n  n  e  2  1 .59)  (Lo/2-10-*)  ( 2TT EQ/C i)  S  1 .60)  (2TT£ /C ) 0  Introducing  these  =  0  1 .61 )  6  definitions  i n t o the equations  above, we o b t a i n  d GMR  1 .62) n-1 d'  GMR  1 .63)  2 h d r D  1 .64)  n-1 2 h D'  1 .65)  r Dn T T 4 h After  2  + d  2  some a l g e b r a i c m a n i p u l a t i o n s GMR  =  d  =  D  =  a  1 .57) we f i n a l l y  obtain,  — l (a -a ) /n  1 .68)  1  1  a a  1 .69)  2  1/n  4  1.70)  3 ^ D  2  ,2 - d  1 .71 )  2 h d a D  1 .72)  3  Equations equivalent  (1.68) to (1.72) g i v e  the p h y s i c a l c o n f i g u r a t i o n of an  l i n e which has the same power frequency  parameters as the l i n e  we wish t o s i m u l a t e . A numerical  example  c o n f i g u r a t i o n o f the r e f e r e n c e c o n f i g u r a t i o n obtained parameters.  i s given i n t a b l e 1.2, where the tower line  i s compared  from e q u a t i o n s  t o the e q u i v a l e n t  (1.68) t o (1.72) u s i n g 60 Hz  line  70  Reference L i n e ( i n m)  Equivalent Line ( i n m)  15.361 0.188427 34.78 15.24 0.198253  14.334 0.175276 32.66 14.675 0.196490  d GMR D h r  Table  1.2:  the approximate equations f o r the e l e c t r i c a l  parameters were very a c c u r a t e obtained  ( l e s s than 1% e r r o r ) the p h y s i c a l  from the same equations shows e r r o r s up t o 7%.  to the l o g a r i t h m i c nature of the equations; produce r e l a t i v e l y  l a r g e changes i n a-| , a2,  (1.58) to (1.61)),  which i n turn r e f l e c t  tion .  6.69 6.98 6.09 3.71 0.89  Comparison between the a c t u a l l i n e c o n f i g u r a t i o n and the e q u i v a l e n t l i n e configuration.  Note t h a t while  tion  Error (%)  configura-  This  i s due  s m a l l v a r i a t i o n s i n L and C, and a^ (see equations  on the e q u i v a l e n t  line  configura-  71 1 .3  Skin E f f e c t C o r r e c t i o n  Factor  With the e q u i v a l e n t  l i n e c o n f i g u r a t i o n obtained  s e c t i o n , i t i s now  p o s s i b l e to use  the more e x a c t  (1.34) to e v a l u a t e  the sequence r e a c t a n c e s .  accounted f o r the v a r i a t i o n of Rg(a)) and If we  L^  _i n t e r n a l r a d i u s external radius  F i g . 1.4:  Let us now R^to)  Since A R ( c o ) and s  =  recall R (co) s  AR (to) m  c o n f i g u r a t i o n , and  s  have not  yet  to s k i n e f f e c t .  equation  + AR (LO) s  can  now  =  as,  q -± r  =  Tubular  -  conductor  (1.33) AR (to) m  be e v a l u a t e d  from the e q u i v a l e n t  line  R-|(o)) i s assumed to be known, the s e l f - r e s i s t a n c e of  the e q u i v a l e n t conductor can be R (co)  (1.32) and  the conductor's dc r e s i s t a n c e and on the c o r r e c t i o n  f a c t o r s (see appendix I I - A ) , which i s d e f i n e d  =  (aj) due  previous  assume t h a t the e q u i v a l e n t conductor i s t u b u l a r , s k i n  e f f e c t depends o n l y on  s  formulas  However, we n t  i n the  R-|(co)  determined,  -AR (to) s  +AR (to) m  (1.7  72  I f the dc r e s i s t a n c e o f t h e conductor i s a l s o known, i t i s p o s s i b l e to f i n d  ( n u m e r i c a l l y , u s i n g the s k i n e f f e c t s u b r o u t i n e ) t h e v a l u e  of s t h a t makes the s e l f - r e s i s t a n c e determined  vary from R^,-. t o R  s  ( co).  With s  i n t h i s manner, a l l the i n f o r m a t i o n needed t o e v a l u a t e the  e l e c t r i c a l parameters  (over the e n t i r e frequency  range) of the e q u i v a l e n t  l i n e i s known. If the dc r e s i s t a n c e i s not known, the value of s can be determined  from  the i n t e r n a l inductance I"i-n-t(to).  From e q u a t i o n  (1.34), L i ( i o ) i s g i v e n by n t  L. J i o ) = — [X (co) - AX (co) + AX (co)]-L + L int co 1 s m ext m n  (1.74)  where a l l the q u a n t i t i e s i n the r i g h t hand s i d e can be e v a l u a t e d with -the equivalent of  l i n e c o n f i g u r a t i o n . With Rg(co) and Li ^-(co) known, the values n  and s can be determined  n u m e r i c a l l y with the s k i n e f f e c t  sub-  routine . This a l t e r n a t i v e ,  although more d e s i r a b l e because l e s s  informa-  t i o n needs to be known, i s more i n a c c u r a t e than the d e t e r m i n a t i o n of s with R^  c  given.  The reason  the t o t a l i n d u c t a n c e ,  i s t h a t s i n c e L i t ( c o ) i s a s m a l l f r a c t i o n of n  the d i f f e r e n c e between the a c t u a l and e q u i v a l e n t  p h y s i c a l parameters produce r e l a t i v e l y Li (co).  T h i s can be i l l u s t r a t e d  reference  line.  n t  large errors  i n the e v a l u a t i o n of  with a numerical example u s i n g the  Let us assume t h a t a t 60 Hz Li (u>) can be approximated' using nt  equation  (1.22) L.i n t,  =  2.10" in GMR 4  r  We now i n t r o d u c e the a c t u a l and e q u i v a l e n t l i n e parameters i n t o e q u a t i o n . The r e s u l t s  are shown below.  this  73  Reference L i n e ( i n mH/km)  Equivalent Line ( i n mH/km)  Error (%)  0.0221495  -117.86  0.01016667  These r e s u l t s of 6.75%  illustrate  t h a t the r e l a t i v e l y  s m a l l e r r o r i n GMR  propagates e x p o n e n t i a l l y because of the s m a l l magnitude of L i It must be noted, however, t h a t even though the r e l a t i v e  is  n t  error  l a r g e , the d i f f e r e n c e i n magnitudes i s not, and the a c t u a l e r r o r s i n  the d e t e r m i n a t i o n of s and Rjj reference  line,  c  are c o n s i d e r a b l y s m a l l e r .  the values of s and  not given are o n l y  23.2%  and 16.9%  For the  c a l c u l a t e d when R £  smaller, r e s p e c t i v e l y .  c  is  .  74 1 .4  C o r r e c t i o n o f the E q u i v a l e n t L i n e C o n f i g u r a t i o n when R^,-, i s Known As mentioned i n the p r e v i o u s s e c t i o n ,  known, the value of s can be determined of  i s a l s o known.  f i n e r approximation  The knowledge  With t h i s a d d i t i o n a l p i e c e of i n f o r m a t i o n , a  of the e q u i v a l e n t l i n e can be o b t a i n e d  e s t i m a t e g i v e n by equations Consider L (co)  =  Q  L  1  inductances  L  int  int  these  (  W  )  +  L  ext  " ^  (  +  A x  (AXg(u)  (n-1) A x ^ c o n / t o  +  s ( ) - X (w))/o>. w  A  m  L (co) and L-](co) are the power frequency Q  and they are known d a t a .  The i n t e r n a l  inductance L ^  s u b r o u t i n e a t power f r e q u e n c y .  Carson's  terms Ax (co) and Ax (co) are e v a l u a t e d u s i n g the f i r s t m  the e q u i v a l e n t l i n e  mined from  equations  the f i r s t  (1.72).  + (n-1) 1^+  e x t  two e q u a t i o n s  s  from  (1.32) and (1.34),  (u>) + L  g i v e n by the s k i n e f f e c t tion  (1.68) through  equations  L  =  ( U ) )  In  of  with good a c c u r a c y .  an a c c u r a t e value of s i m p l i e s t h a t an a c c u r a t e value of the i n t e r n a l  inductance  of  i f the dc r e s i s t a n c e i s  parameters.  Therefore L ^  n t  and L  m  n t  (co) i s  correc-  approximation can be d e t e r -  (1.32) and (1.34) with a minimum ( d i r e c t ) knowledge  the tower c o n f i g u r a t i o n . Let + (AX (co) - Ax<,(co))/co °  a  =  L i (to)  - L. Au) int  b  =  L (co) - L. (co) - (AX (co) + (n-1) AX (co))/co ° int "  i  m  n  s  m 111  (1.75) (1.76)  then T Un  L  From e q u a t i o n  ext  - (--> b  a  _  a  +  ha •  (1.20) L  (1.77)  n  ext  =  A „ 2 h 2.10-4 An _  (1.78)  75 from which  where L  e x  ^ i s calculated  o b t a i n e d from e q u a t i o n From e q u a t i o n  JW/2-10  L  =  from e q u a t i o n  )  (1 .79)  (1.78) and r i s the value  (1.72).  (1.22) l n T  =  GMR  =  L  int -  2.10~  4  in  GMR  from which (Lint/2-10- ) 4  r  e  (  And f i n a l l y , d and D are o b t a i n e d from equations d  =  D  =  u  4  0  )  (1.62) and (1.65)  a i GMR [ra ] 1 2 h  8  (1.81) l/(n-l)  • d  (1.82)  The c o r r e c t e d tower c o n f i g u r a t i o n ( f o r the r e f e r e n c e l i n e ) i s shown i n t a b l e  1.3. Reference Line ( i n m)  E q u i v a l e n t Line ( i n m)  15.361 0.188427 34.78 15.24 0.198253  15.058 0.184440 34.01 14.99 0.196490  d GMR D h r  Table 1.3:  Second approximation c o n f i g u r a t i o n when  Error (%) 1 .97 2.12 2.22 1 .64 0.89  of the tower i s known.  This new e q u i v a l e n t l i n e i s a b e t t e r approximation original  line.  The improvement o b t a i n e d u s i n g t h i s procedure  the a c c u r a c y of the i n i t i a l initial of  of the depends on  estimate of the e q u i v a l e n t r a d i u s r .  This  e s t i m a t e of r i s u s u a l l y very good because of the p a r t i c u l a r  equations  (1.68) through  (1.72).  I t a l s o depends on the value of  form  76 L^ (to) obtained using nt  the s k i n e f f e c t s u b r o u t i n e , which i s a l s o  a c c u r a t e because l i t t l e e r r o r i s i n v o l v e d R  dc  ^  s  i n the e v a l u a t i o n  very  of s when  known.  This  f i n e adjustment procedure can be i n t e r p r e t e d  i t e r a t i v e s o l u t i o n of e q u a t i o n s  (1.53) through  as the  (1.57) exchanging, a f t e r  the f i r s t i t e r a t i o n , e q u a t i o n (1.57) w i t h the e q u a t i o n t h a t c a l c u l a t e s s k i n effect. not  If the dc r e s i s t a n c e  available  performed.  i s not known, the s k i n e f f e c t equation i s  ( a c t u a l l y not r e l i a b l e ) and only  the f i r s t  i t e r a t i o n can be  77 1.5  E v a l u a t i o n of the C h a r a c t e r i s t i c  Impedance and  Propagation  With the e q u i v a l e n t l i n e c o n f i g u r a t i o n o b t a i n e d section,  the modal parameters o f the l i n e over  Function  i n the  the frequency  previous  range of  i n t e r e s t can be e v a l u a t e d , and with them, the c h a r a c t e r i s t i c  impedance  and  These are  propagation  f u n c t i o n A-| f o r zero and p o s i t i v e  sequence.  Z  c  d e f i n e d as f o l l o w s R + jcoL l / G + jcoC  (1.83)  =  (co)  =  e-^  (1.84)  Y (co)  =  Z  Al  where R,  c  (  w  )  \] (R + jcoL) (G + jcoC) .  The  l e n g t h , and  range, u s i n g equations  c o n s t a n t over  configuration.  the frequency  given as i n p u t data The  zero.  line  modal r e s i s t a n c e s and  equivalent line  those  (1.85)  L, C are the sequence parameters d e f i n e d i n the  s e c t i o n s , & i s the  frequency  ( u , ) £  shunt  inductances  (1.31) through The  conductance.  are e v a l u a t e d over (1.34),  with  the  the  c a p a c i t a n c e s , s i n c e they are v i r t u a l l y  range of i n t e r e s t i n t r a n s i e n t a n a l y s i s ,  been assumed to be  the c h a r a c t e r i s t i c impedance goes to i n f i n i t y  goes to zero  (see equation  (1.83)).  This s i t u a t i o n ,  The  of Z (co) by r a t i o n a l f u n c t i o n s would r e q u i r e a s u p e r f l u o u s c  amount of p o l e s and  zeros to s i m u l a t e  Therefore, a f i n i t e  value of G i s assumed i n the e v a l u a t i o n of Z (co.)  the low frequency  KV l i n e i s 0.3 • 1 0 I t i s to be noted  region. c  (the same v a l u e of G i s used i n A-| (co ) f o r c o n s i s t e n c y ) . f o r a 500  as  apart  from being p h y s i c a l l y i m p o s s i b l e , i s c o m p u t a t i o n a l l y u n d e s i r a b l e . approximation  are  [10].  conductance, up to t h i s p o i n t , had  However, i f G=0,  the frequency  G i s the shunt  previous  - 7  A typical  S/km.  t h a t the magnitude of G does not  affect  value  78  significantly  the r e s u l t s of t r a n s i e n t s i m u l a t i o n s , however, p a s t ejgpeiri-  ence i n the s i m u l a t i o n of HVDC l i n e s contributes relatively  to the n u m e r i c a l s t a b i l i t y long s i m u l a t i o n s ( s e v e r a l  Up  to t h i s p o i n t the l i n e  no ground w i r e s . ors;  niques  (n + m)  x  (1.6)  under study has  been assumed to fawe  longer  a  n  d  as normal confflaactY ^ ( 0 1 ) wo.uIkiS  Using a p p r o p r i a t e r e d u c t i o n tecks-  p h  i s then  f o r Zpj^to),  Although  implicit (1.43) and  to (n x n ) .  The  i n these m a t r i c e s , and (1.44) f o r the  e f fecit <of  equations  fe-xwell's  {(11.5)  coefficient  true.  the o v e r a l l  e f f e c t of ground wires on  e t e r s i s s m a l l , the e q u i v a l e n t l i n e The  seconds).  Y ( w ) can be reduced  pn  m a t r i x are no  matrices.  v a l u e of Q  of the s o l u t i o n p r o c e s s during;  with m ground w i r e s , Z ^ ( O J ) and  (n + m)  [11], Z ( o ) ) and  the m ground wires and  that a f i n i t e  In g e n e r a l , ground wires are t r e a t e d  i n an n-phase system  then be  suggests  a c c u r a c y of the approximation  configuration  i s n o t i c e a b l y affeclbed.  i s t h e r e f o r e reduced,  the p o i n t of making i t u s e l e s s (see appendix  the l i n e paE.iaim-  II-C).  a l t h o u g h not foo  CHAPTER 2  RATIONAL FUNCTIONS APPROXIMATION OF THE CHARACTERISTIC IMPEDANCE AND PROPAGATION FUNCTION  2.1  Rational Functions.  General  Considerations  In the frequency-dependence v e r s i o n of the EMTP, Z (to ) and A^ (to) c  must be approximated by r a t i o n a l f u n c t i o n s i n order to o b t a i n c l o s e d mathematical forms which are compatible solution algorithm. B((o)  =  with  the frequency-dependent  These r a t i o n a l f u n c t i o n s have the g e n e r a l k  0  form  (s + z i ) ( + Z2) ... (s + z ) (s + p]^) (s + P 2 ) ... (s + p ) s  (2.1)  m  n  where,  n  s  =  jto  z^  =  zero of B(to)  Pi  =  pole of B(to)  k  =  positive,  0  = m  n > m  All  real  t o approximate  constant Z (to) c  t o approximate A-| (to )  the zeros and p o l e s i n e q u a t i o n  the l e f t - h a n d s i d e o f the complex p l a n e . belongs  Under these c o n d i t i o n s , B(to)  to the c l a s s of minimum p h a s e - s h i f t f u n c t i o n s , t h e r e f o r e ,  magnitude f u n c t i o n | B(to)| u n i q u e l y determines arg  (2.1) are r e a l and l i e i n  (B(to)).  T h i s p r o p e r t y reduces  its  i t s phase f u n c t i o n  the approximating  p r o c e s s t o the  l o c a l i z a t i o n o f the p o l e s and zeros o f B(to) so t h a t t h e magnitudes o f B(to) and t h e f u n c t i o n t o be approximated are matched.  79  80 A v e r y simple, and y e t p o w e r f u l , way t o f i n d of B(co) i s the use of Bode's asymptotes.  Consider  the p o l e s and zeros  the magnitude f u n c t i o n  of B(co), and take i t s l o g a r i t h m l o g | B(io)|  =  log k  0  + l o g | s + z-J  log | s + p j  + l o g | s + z j + ... + l o g | s + z j  - l o g | s + p | - ... - l o g | s + p j . 2  For s = jco each one of these terms has an asymptotic  behaviour  (2.2)  with  r e s p e c t to co. C o n s i d e r i n g , f o r i n s t a n c e , the term y it  =  -log | j  w + p\ ±  ,  follows that f o r co <<  P  f o r co >>  P ; L  l  y =  log p  (2.3)  y =  - l o g co ,  (2.4)  which, as a f u n c t i o n of l o g co, d e f i n e two s t r a i g h t t h a t i n t e r s e c t a t co = p . n  These are shown i n F i g u r e 2.1.  I  F i g . 2.1:  Equation  Asymptotic  l i n e s o r asymptotes  loglpj)  behaviour  of y =  log(oj)  - l o g | s + p^j .  (2.3) d e f i n e s a h o r i z o n t a l l i n e , and e q u a t i o n  (2.4) d e f i n e s a  l i n e o f s l o p e - 1 u n i t s / d e c , t h a t i s , whenoj-i/o^ = 10, y decreases by 1 unit.  81 Equation  (2.2) can then be v i s u a l i z e d as a s e t of b u i l d i n g  b l o c k s or s t r a i g h t traced.  l i n e s w i t h which the f u n c t i o n to be approximated can be  I t s h o u l d be noted t h a t these l i n e segments do not r e p r e s e n t the  a c t u a l form of j B(ui)| , but an asymptotic guide or s k e t c h t h a t d e f i n e s the boundaries where t h i s f u n c t i o n a c t u a l l y The  r o u t i n e s of the frequency-dependence  this principle approximating  to a l l o c a t e  because  Therefore, a s i m p l i f i e d in this  of t h i s  Furthermore,  (e.g., two or three p o l e s ) ,  approach  thesis project.  c o r n e r s are p o s s i b l e . next.  thesis project,  these r o u t i n e s are not  f o r approximations  of very low order  these r o u t i n e s can g i v e e r r a t i c  of p o l e s i s f i x e d beforehand,  discussed  zeros ( c o r n e r s ) of the  the user has no d i r e c t c o n t r o l over the order of the  approximation used.  developed  the p o l e s and  o p t i o n of the EMTP use  function [7].  For the purposes adequate  lies.  to the assignment With  of asymptotes  this simplified  and adequate  The b a s i c p r i n c i p l e s  results. has been  method,the number  approximations  with very  f o r t h i s approach  are  few  82  2.2  Asymptotic  Approximated on when the Number o f P o l e s i s F i x e d  C o n s i d e r H(aj), whose magnitude f u n c t i o n (shown i n F i g u r e 2 . 2 ) bounded and m o n o t o n i c a l l y d e c r e a s i n g over the frequency ( 0 t o 1 0 HZ). 6  (Note  range of i n t e r e s t  t h a t | H(co)| i s p l o t t e d on a l o g - l o g  scale)  |H(w)l  F i g . 2 . 2 : Magnitude f u n c t i o n of H ( L U )  to approximate H(to) w i t h a r a t i o n a l  As mentioned e a r l i e r ,  f u n c t i o n B(co), i t i s s u f f i c i e n t t o match t h e i r magnitude f u n c t i o n s . L e t B(co) be of order two, B(co)  =  k  n  (s +  (s +  Z j )  z  (s + P i ) (s +  2  (2.5)  ) .  P 2 )  Taking the l o g a r i t h m of | B(to)| , l o g | B(co)|  + log| s + z-J  =  log k  +  log | s +  Q  z  2  |  -  l  o  9 " |  s  - log | s + Pi| +  p  2l  (2.6)  is  83 Since | H(to)| i s bounded, i t can be s u b d i v i d e d i n t o f o u r e q u a l l y spaced h-) , 2 n  sections. a  n  d  through h  T  2  n  The h o r i z o n t a l asymptotes  and h^.  The  pass  with s l o p e - 1 u n i t s / d e c w i l l  asymptotes  e  of | B(co)| w i l l  through  pass  l o c a t i o n of the c o r n e r s of the asymptotic  approximation . w i l l be a t the i n t e r s e c t i o n of the asymptotes  (see F i g u r e  2.3).  |H(LJ)1  Fig.  2.3:  ,  4  Asymptotic a p p r o x i m a t i o n of | H ( O J ) | .  For  example, to f i n d p^ we  intersect  log  | H(to )|  =  log  | H( co)|  = - l o g co + l o g h  the s t r a i g h t  l o g h-, 2  + l o g to^.  S u b t r a c t i n g these e q u a t i o n s log  h-|  =  - log  log  (ui)  =  - l o g h-| + l o g h  0)  =  h2  OJ  + log h  + log  2  2  OJ2  + l o g tu  2  lines  84 T h i s i s the value of to a t which the i n t e r s e c t i o n o c c u r s , t h e r e f o r e Pi  Proceeding  =  "2  a n a l o g o u s l y with the other c o r n e r s , we f i n a l l y  h  P  =  2  h  z  2  =  h  3  4 h  obtain  t04  3  4 0)4 5  h  The  value o f k  can be found by matching | H(to)| and | B(to)| a t  Q  to = 0 H(0)|  =  |B(0)l  =  k  z n  l 2 z  r  from which k  =  Q  |  H(0)|  l 2.  P  P  Z  T h i s procedure and  l  Z  2  can be extended  zeros n , by s u b d i v i d i n g | H(to )|  to an a r b i t r a r y number of poles  i n t o 2n segments and then f i n d i n g the  i n t e r s e c t i o n of the h o r i z o n t a l asymptotes  (which w i l l  pass  through ^ , f o r  i  odd) w i t h the n o n - h o r i z o n t a l asymptotes  (which w i l l  pass  through h ^ , f o r  i  even).  These i n t e r s e c t i o n s  p.  = - ^ - " 2i-l  2  n  then d e f i n e the p o s i t i o n s of p^ and z^.  (2.7)  i  h  2i .. ^<*>2i W '2i+l  <'>  h  z,1  where The  = ,  2  8  i = 1,... ,n  constant k  k  Q  will  then by g i v e n by n  0  =  H ( O )  p  n  i=l  —  z  i  (2.9)  85 This method can p o s i t i v e and  negative  be  generalized  slopes, provided  to approximate f u n c t i o n s  they are  broken down i n t o segments of m o n o t o n i c a l l y functions  F i g . 2.4:  (see F i g u r e  s i n g l e valued  i n c r e a s i n g and  and  negative  can  decreasing  2.4).  A r b i t r a r y f u n c t i o n with p o s i t i v e and  with  slopes.  be  86 2.3  Approximation of the C h a r a c t e r i s t i c Impedance Z (co) c  If G i s assumed to be monotonically infinity.  decreasing  the c h a r a c t e r i s t i c impedance i s a  f u n c t i o n t h a t becomes c o n s t a n t  I f G i s non-zero, | Z (to)| c  w i t h p o s i t i v e s l o p e f o r low for  zero,  can  frequencies  show two and  the r e s t of the f u n c t i o n (see F i g u r e  as  to tends to  regions:  a segment  a segment with  negative  slope  2.5).  650 i  FREQUENCY  Fig.  2.5:  | Z (to)| , zero sequence. From BPA's John DayLower Monumental 500 KV t r a n s m i s s i o n l i n e . c  The G.  (HZ)  presence of the peak i n Figure 2.5  As mentioned e a r l i e r ,  significantly  the a c t u a l value  depends on  of G does not  the r e s u l t s of a t r a n s i e n t s i m u l a t i o n , as  the value  of  affect long as i t i s  non-zero. In the approximating r o u t i n e s developed i n t h i s p r o j e c t , the peak i n j Z (to)| i s e l i m i n a t e d , as shown by the dashed l i n e i n F i g u r e c  ( t h i s i s approximately  e q u i v a l e n t to d e c r e a s i n g  G).  Since  s l o p e r e g i o n i s e l i m i n a t e d , fewer p o l e s are needed i n the  2.5  the p o s i t i v e approximation,  87 and  the r e s u l t i n g m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n can be  with the method d e s c r i b e d i n S e c t i o n The shifting  o v e r a l l accuracy  between | Z (w)| c  and | B( LO)|  is  2.2.  of the approximation  p o l e - z e r o p a i r s around t h e i r  can be s h i f t e d u n t i l  distort  i s very s m a l l  (e.g., two  0  by the u s e r . response  area  Q  i s p r e f e r r e d to a f  Q  and | Z (co )| i s c  This procedure,  when the s p e c i f i e d  0  however,  may  number of poles  or three p o l e s ) .  the use of e i t h e r method  is optional.  the  by  the p o l e - z e r o p a i r t h a t i s c l o s e s t to  In the computer program developed project,  f  the d i f f e r e n c e between | B(ui )|  the o v e r a l l frequency  improved  minimized.  response,  below an e r r o r l e v e l s p e c i f i e d  can be  i n i t i a l positions, until  When c l o s e matching a t power frequency good o v e r a l l frequency  approximated  as p a r t of t h i s  thesis  ( s h i f t i n g or power frequency  matching)  88 2.4  Approximation of the P r o p a g a t i o n  The  g e n e r a l shape of | A-|(u))| i s approximately  different line be  F u n c t i o n A-] ( OJ)  lengths and p r o p a g a t i o n  modes.  the same f o r  Three d i f f e r e n t r e g i o n s can  observed: I.  A h o r i z o n t a l r e g i o n f o r low f r e q u e n c i e s  II.  An elbow or t r a n s i t i o n zone f o r m i d - f r e q u e n c i e s .  III.  A high slope region f o r high frequencies  FREQUENCY  F i g . 2.6:  A-|(to)  The constant  zero  (HZ  high a t t e n u a t i o n region  (see F i g u r e 2.6) p r e s e n t s  T h i s suggests  t h a t | A-j(co)|  approximated by a s i n g l e - p o l e r a t i o n a l f u n c t i o n 1  (  a  )  )  =  (high a t t e n u a t i o n ) .  sequence.  s l o p e of 1 u n i t / d e c .  B  (low a t t e n u a t i o n ) .  B-|(OJ),  ki  (s + P ) x  Such a s i n g l e - p o l e approximation  i s shown i n F i g u r e 2.7.  an almost  c o u l d be  FREQUENCY  Fig.  2.7:  S i n g l e - p o l e approximation  (HZ)  of | Ai(co)| .  In t h i s s i n g l e - p o l e approximation, be observed;  one  two  a t very h i g h f r e q u e n c i e s , and  l a r g e - e r r o r regions  can  the o t h e r , i n the  mid-frequency range. The  e r r o r i n the high a t t e n u a t i o n r e g i o n i s not c r i t i c a l .  frequencies i n a t y p i c a l and will  t r a n s i e n t case have r e l a t i v e l y  are r a p i d l y a t t e n u a t e d  by the l i n e .  be used to improve the approximation  High  low magnitudes,  T h e r e f o r e , no a d d i t i o n a l p o l e s f o r values of j A-j (co )j  below  0.4. The  other h i g h - e r r o r r e g i o n i s i n the t r a n s i t i o n  a c c u r a t e approximation a relatively frequency  of t h i s zone i s very important  low a t t e n u a t i o n and  One  way  The  because i t p r e s e n t s  i t u s u a l l y l i e s c l o s e to the power  range, t h e r e f o r e , a d d i t i o n a l p o l e s and  approximate t h i s  zone.  zeros are needed to  region. to determine the a l l o c a t i o n of these a d d i t i o n a l  corners  90 is  to examine  the d i f f e r e n c e  f u n c t i o n between | A-^to))  and the  a p p r o x i m a t i o n | B-j (cc)| .  single-pole  C o n s i d e r the a p p r o x i m a t i n g f u n c t i o n  B(to) w i t h n p o l e s  and n-1  zeros, =  B(co)  k  (s + Z j ) ... (s + z ) (s + p i ) ... (s + p _ i ) (s + p )  Q  (2.10)  n  n  Let  B-|(u)) be the s i n g l e - p o l e B  1  ( a ) )  =  function  t h a t approximates | A-|(co)|  kl (s + p ) '  (2.11)  n  w i t h k! = | A-, {0)|  P  .  n  f u n c t i o n D(co) w i l l  The d i f f e r e n c e  | A-|(OJ)|  log  =  | D(OJ)|  -  be d e f i n e d  | Bi(o))|  log  =  log  as, | D(co)|  | Ai (o))| .  (2.12)  j B {m)\ 1  Introducing  (2.11)  I( l  =  J _  I1  | D( )|  =  P  | Ai(M)|  D  w)  u  Let  us now  define  I  A  n  s  + Pn|  |s + p | .  D ^ c u ) as the r a t i o n a l  ( s u b s t i t u t i n g A ^ c o ) by B(ai) i n e q u a t i o n |B  i D l  equations  | D-, (to  function  t h a t approximates D ( t o )  (2.12))  (co) I  =  (co)  (2.14) |B  Introducing  (2.13)  n  )|  =  1  (CO)  J  (2.10) and  I  (2.11), we f i n a l l y  obtain  kp (s + z ) ... ( + z n _i)l k-| | (s + P-)) ... (s + p _ i )| x  (2.15)  S  n  The magnitude f u n c t i o n  of D(to) i s shown i n F i g u r e  2.9(a)  F i g . 2.9:  Difference (a) |D(to)|  f u n c t i o n |D(U>)| . Zero sequence. Reference (b) |D(to)j , |B (to)| and | A (OJ)| . 1  X  line  92 The assignment of the a d d i t i o n a l c o r n e r s of | B( to)| reduced  to the approximation  interested  of | D(to)| by | D-| (to)| .  Since we are  i n the a c c u r a t e m o d e l l i n g of the t r a n s i t i o n  p o r t i o n between f = 0 and  f  i s then  = f j_ w i l l be approximated.  zone, o n l y the T h i s segment  r e p r e s e n t s a m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n , and i t i s approximated with  n-1 p o l e s and z e r o s , u s i n g the procedure The accuracy  approximation  of A-| (to).  d e s c r i b e d i n s e c t i o n 2.3.  of a t r a n s i e n t s i m u l a t i o n depends s t r o n g l y on the i f an a c c u r a t e power frequency  response i s  d e s i r e d , an e x a c t matching can be o b t a i n e d by r e p o s i t i o n i n g pole-zero p a i r u n t i l  the c l o s e s t  the d i f f e r e n c e l i e s w i t h i n a p r e - s p e c i f i e d  range.  However, s h i f t i n g a s i n g l e p o l e - z e r o p a i r may,  i n some i n s t a n c e s ,  introduce d i s t o r t i o n  response.  i n the o v e r a l l frequency  When o v e r a l l accuracy all  p o l e - z e r o p a i r s are s h i f t e d  a r e a between | A-| (to)|  i s p r e f e r r e d to power frequency from t h e i r  matching,  i n i t i a l p o s i t i o n s u n t i l the  and | B-j C cu)| i s minimized.  2.5  Evaluation of t In the p r e v i o u s s e c t i o n  been approximated  the magnitude f u n c t i o n of A-| (co) has  by a r a t i o n a l f u n c t i o n B-j (co).  I t i s shown i n appendix  I-B t h a t AT (co) = P ( co) e "  3  ^  (2.16)  Since B( co) i s a f u n c t i o n o f the c l a s s o f minimum p h a s e - s h i f t , by approximating  the magnitude f u n c t i o n of A-| (co) w i t h | B(co)| , the phase  f u n c t i o n of B( co) w i l l c o i n c i d e with the phase f u n c t i o n of P( to) .  To  approximate both magnitude and phase o f A]_(co), c~ must be e v a l u a t e d so t h a t AT (co) = B(co) e " Taking arg  (2.17)  j W C  the argument f u n c t i o n of both s i d e s of e q u a t i o n  (A-|(to))  =  (2.17),  a r g (B(co)) - cor  from which, C  =  - i (arg (B(co)) - a r g (A-,(co))).  (2.18)  Since B(co) i s not e x a c t l y equal t o P(co), the value of c i n equation  (2.18) w i l l  vary with  accuracy of the approximation  the frequency  to a degree d i c t a t e d by the  of the magnitude  function.  The frequency dependence v e r s i o n of the EMTP e v a l u a t e s c by averaging  the values o b t a i n e d from  frequency  range ( i . e . ,  equation  (2.18) over a c e r t a i n  f o r values of | A-| (co )| between 0.9 and 0.7).  a h i g h - o r d e r approximation  i s used,  t h i s procedure  i s j u s t i f i a b l e because  the d i f f e r e n c e s between | A-| (to )| and | B(co)| are s m a l l . however, A-) (co) i s not known, b u t r a t h e r the p r o p a g a t i o n equivalent l i n e . original  line,  When  In t h i s  project,  f u n c t i o n of an  Even i f t h i s e q u i v a l e n t l i n e i s a good estimate of the  the low number of p o l e s and zeros i n B(co) o n l y guarantees a  good f i t over a r e l a t i v e l y  l i m i t e d frequency  range.  Because of these  c o n s i d e r a t i o n s , C i s e v a l u a t e d a t one p o i n t o n l y , t h a t i s , a t the power frequency.  2.6  Implementation o f the Method The e v a l u a t i o n of the e q u i v a l e n t  line  representation  and the  a p p r o x i m a t i o n of A-|(to) and Z (co), have been implemented i n a FORTRAN c  computer program.  T h i s program  (with a p p r o x i m a t e l y 1300 FORTRAN  statements) has been designed so t h a t a f u t u r e i n t e r c o n n e c t i o n with the e x i s t i n g EMTP code can be made w i t h minimum A guide f o r i t s use i s attached  effort.  i n appendix I I - D .  CHAPTER 3  NUMERICAL  3.1  RESULTS  Recapitulation From the power frequency parameters (and dc r e s i s t a n c e f o r higher  accuracy) the p h y s i c a l c o n f i g u r a t i o n of an e l e c t r i c a l l y has  been o b t a i n e d .  T h i s permits the e v a l u a t i o n  equivalent  line  of R( to) and L(to), from  which Z (w) and A-] (w) can be c a l c u l a t e d over the e n t i r e frequency c  range. The and  frequency-dependence v e r s i o n of the EMTP r e q u i r e s  A-|(co) be i n a c l o s e d mathematical form.  A-| ( to) are approximated by r a t i o n a l f u n c t i o n s techniques that permit the use of a very The using be  transmission  line described  the techniques d e s c r i b e d  compared with those obtained  the EMTP.  95  c  For t h i s purpose, Z (to)and c  using  approximating  low number of p o l e s  and z e r o s .  i n Chapter 1 w i l l be  i n this thesis project. using  that Z (to)  simulated  The r e s u l t s  will  the frequency-dependence v e r s i o n of  9.6 3.2 Evaluation of the Line Parameters Frcm the Equivalent Line Ccrifiguration The  i n p u t data necessary  t o o t a i n the e q u i v a l e n t  c o n f i g u r a t i o n f o r the r e f e r e n c e  The  line  (see F i g u r e 1.3) i s shown below:  Power frequency  =  60 Hz  Line  length  =  500 km  Shunt conductance  =  0.3-10  Earth r e s i s t i v i t y  =  100fi-m  dc  =  0.02619 0,/km  r e s i s t a n c e (R  dc  line  -7s  /  e l e c t r i c a l parameters a t 60 Hz (from  k  m  the output  of the Line  Constants Program) a r e : Positive  sequence: n/km  RT  =  0.02643  L-,  =  0.8808  mH/km  C  =  0.0133  UF/km  }  Zero sequence:  The  R  Q  =  0.1974  ft/km  L  Q  =  3.3308  mH/km  C  Q  =  0.008361 UF/km  r e s u l t i n g e q u i v a l e n t l i n e c o n f i g u r a t i o n i s summarized below.  Reference Line  E q u i v a l e n t Line R given R not given Parameters . Error Parameters Error ( i n m) (%) ( i n m) (%) d c  Parameters ( i n m) d GMR D h r  15.361 0.18843 34.780 15.240 0.19825  15.058 0.18444 34.070 14.990 0.196490  d c  1 .97 2.12 2.22 1 .64 0.89  14.334 0.17528 32.662 14.675 0.19649  6.69 6.98 6.09 3.71 0.89  The 10  Q  v a r i a t i o n of R and L over  Hz, i s c a l c u l a t e d  2.1 through  the frequency  range from 10~^  1 : 0  from the e q u i v a l e n t l i n e c o n f i g u r a t i o n s (see graphs  2.4). The output  from the Line Constants  Program i s used as  reference. From these graphs,  the f o l l o w i n g o b s e r v a t i o n s can be made:  a)  The e s t i m a t e d  values o f R and L a r e more a c c u r a t e when  i s known.  b)  P o s i t i v e sequence parameters p r e s e n t l a r g e r e r r o r l e v e l s  than  zero  sequence parameters. c)  The r e s i s t a n c e s p r e s e n t l a r g e r e r r o r s than the i n d u c t a n c e s . When R ^  lated  C  i s known, the s k i n e f f e c t c o r r e c t i o n f a c t o r s i s c a l c u -  from the i n c r e a s e i n the s e l f - r e s i s t a n c e  from i t s dc to i t s 60 Hz  value R-l(oj) = Since the term  R S ( O J )  + (AR (co) - AR (co)).  R^too)) i s n e g l i g i b l e a t 60 Hz, the i n c r e a s e i n  ( A R S ( O J ) - A  R (co) can be assumed to be caused s  value of s estimated  m  s  when R^  c  by s k i n e f f e c t o n l y .  i s known i s very a c c u r a t e  T h e r e f o r e , the (see s e c t i o n  1.3). When R^ it  i s calculated  i s not known, s cannot be estimated  from the i n c r e a s e i n the i n t e r n a l inductance  to i t s power frequency e v a l u a t i o n of L ^ inaccurate.  a c c u r a t e l y because  n t  from i t s dc  value; and as i t was shown i n s e c t i o n 1.3, the  u s i n g the e q u i v a l e n t l i n e c o n f i g u r a t i o n i s very  A l s o , when R^  i s not known, the f i n e r adjustment of the l i n e  c o n f i g u r a t i o n e x p l a i n e d i n s e c t i o n 1.4 cannot be made, and the e v a l u a t i o n of Carson's c o r r e c t i o n terms becomes l e s s  accurate.  The p o s i t i v e sequence r e s i s t a n c e i s more s e n s i t i v e t o d i f f e r e n c e s i n AR is  g  and A  R m  ,  than the zero sequence r e s i s t a n c e .  t h a t R-| depends on (ARg-AR^, while R  Q  depends on (AR  S  The reason  + (n-1) A R ) ; m  98 since AR  and AP^ a r e r e l a t i v e l y  S  l a r g e , and of comparable magnitudes  ( f o r mid to h i g h f r e q u e n c i e s ) , s m a l l e r r o r s i n A R relatively affected  l a r g e e r r o r s i n ( A R - A R ^ ) , while S  and A R  G  M  produce  (AR + (n-1)&R ) i s m  S  t o a l e s s e r degree. The  e v a l u a t i o n of the inductances  i s more a c c u r a t e  than the  e v a l u a t i o n of the r e s i s t a n c e s because s k i n e f f e c t o n l y a f f e c t s L. ^, which int is  only a s m a l l f r a c t i o n of L-| and L . Q  Furthermore, the magnitudes of  Carson's c o r r e c t i o n terms are c o n s i d e r a b l y s m a l l e r T h i s can be e a s i l y v i s u a l i z e d over 1 0  4  f o r the r e a c t a n c e s .  i f we note t h a t , f o r example, RQ i n c r e a s e s  times from i t s dc v a l u e , while L  of i t s dc v a l u e , when the frequency  0  d e c r e a s e s o n l y to 1 / 4  i s i n the MHz  range.  t n  10*  10"  10*  10  10-'  10 10 1 10 10 10 10 10 10 1010  io-'  i n u n i i in II  II .1 I ill ll ,1 I III II .1 m Ml iII ,1 I ill n . i i HI M i i in II ,i i in II ,i i in FREQUENCY  (HZ)  10"  'i  iff  3  Iff  5=  io-' 10-'  1 10111111 0 1 10 10 10 10 10 10 1010 j  i mill  IIIIIII  i i in II ,i l in II ,i i in II FREQUENCY  i l in n  i i in II M, Ui M ,i IIIIII .  (HZ)  tier o  e  Mini  io"  io  i 111111 I I I I I I I  i  io  i i in II , i I n m , i i in II .1 i in II ,1 m i II .1 i in II J i in n ,  io  io FREQUENCY  Graph 2 . 1 :  io  io  io  i or  io  (HZ)  P o s i t i v e sequence r e s i s t a n c e . (a) R d g i v e n (b) Rdc n o t g i v e n (c) E r r o r f u n c t i o n c  Graph 2 . 2 :  Zero sequence r e s i s t a n c e . (a) R ^ g i v e n (b) Rdcnot g i v e n (c) E r r o r f u n c t i o n .  ] .00  0.90  0.80  0.104  "  0.60  0.50  IIIIIII  10  i I ill l l , 1 I ill l l . 1 I in ll J Mil  10  10  10  10  ll  I I ill ll  10  FREQUENCY (HZ)  1  10  1,1111111  10  10  H  0.90  H  0.80  0.  5  IIIIIII  1  .00  1  -  IIIIIII  10  )0  A  0.60  0.50  1 IIIIIII  io  IIIIIII  io  IIIIIII  i  i nun  io  ,i nun  io  ,i Mini  io  i nnn  io  FREQUENCY (HZ)  ,i M U M  io  .I I I I I I I . I I I I I I I ,  io  io  icr  5.0 4.0 3.0 2.0  i.H  o  -i.oA -2.0  -3.0  A  -4.0 -5.0  IIIIII  10  Graph 2.3:  10"  J IIIIII  i nun  1  10  i  mil . 1 IIIIII ,  10  10  n• '  10  IIIIII , 1 IIIIII . 1 n u n  10  10  ,1 n u n  10  10  FREQUENCY (HZ)  P o s i t i v e sequence i n d u c t a n c e . (a) RcJc g i v e n (b) R(j not g i v e n (c) E r r o r f u n c t i o n . c  7.00 6.50 6.00 -f 5.50 5.00 4.50 4.00 3.50 3.00 2.50 2.00 1 50 1 .00  llllll  10""  Jllllll  10  imu 10"  M U M  1  IIIIIII  10  1  10  .1111111,1  10  l i m i t i m i n .i 10  10  1 .1 I III II ,1 I i l l II .1 I 111 i l ,  10 10 10 FREQUENCY I HZ)  IIIIII  ,i  i  .11  10  in n ,i i in II  10 10 FREQUENCY (HZ)  10  .1  10  10  I .  10  i in n ,i i n m . 10  10  5 0 4.0 3.0 2.0 1.0 0 -1.0 -Z.0H -3.0 -4.0 -5.0 10"  Graph 2.4:  I III II  IIIIIII  10"  1  IIIIIII  10  1111111,1111111,1111111,1111111,1111111.1111111,1111111 .  10  10 10 FREQUENCY (HZ)  10  10  10  10  Zero sequence i n d u c t a n c e . (a) R^ g i v e n (b) Rdc n o t g i v e n (c) E r r o r f u n c t i o n . c  103 3.3  Evaluation  of A-| (co) and Z (co) c  The magnitude and phase f u n c t i o n s  of A-| (co) and Z (co) are c  shown i n graphs 2.5 through 2.12. Note t h a t the e r r o r  l e v e l s i n the e v a l u a t i o n  of the p o s i t i v e  sequence r e s i s t a n c e , are a l s o r e f l e c t e d i n Z (co) and A-|(co). c  that  the h i g h  sequence  e r r o r region  (for frequencies  A l s o note  i n the phase f u n c t i o n of Z (to), p o s i t i v e c  above 100 Hz), i s not very s i g n i f i c a n t because  the phase angle i s very c l o s e  to zero,  and s m a l l d i f f e r e n c e s  between the  phase angles of the r e a l f u n c t i o n and the a p p r o x i m a t i o n , appear as relative errors  (see graph 2.9).  S i m i l a r l y , when the magnitude of A-|(co) i s below 0.05 dc value, error  large  function  relative errors f o r | A-|(co)|  Note t h a t  lose t h e i r s i g n i f i c a n c e .  the phase f u n c t i o n s  arg (A-j (co)) >TT ) .  times i t s  In graph 2.5 t h  <: 0.05 has been o m i t t e d .  degrees, while the phase f u n c t i o n s (allowing  large  of Z (co) have been p l o t t e d i n c  of A-| (co) have been p l o t t e d  i n radians  (c)  -5.0  1  10  Graph 2.5:  IIIIIII  10  IIIIIII  1  IIIIIII  10  i I I I I I I ,1 I I I I I I , i I I I I I I .1 I I I I I I ,i n u n .1 I I I I I I , i I I I I I I .  ID  10 10 FREQUENCY (HZ)  10  10  10  10  A-) magnitude f u n c t i o n , p o s i t i v e sequence. (a) qiven (b) not given (c) E r r o r dc ^ dc  function,  5.0  -,  4.3  i  FREQUENCY  (HZ)  FREQUENCY  (MZ)  3.0 H 2.0j  -2.0 -J -3.0 -4.0 -5.0  -1 I I I I I I I  I  10  Graph 2.6:  2  4  1 I I II  1  2  4  11  10  1  2  I III  I I ,  I  I I II  11 ,  I  I I II  4 10 2 4 10 FREQUENCY (HZ)  11 ,  I  I I II  2  4  10  2  4  11  10  A-| phase f u n c t i o n , p o s i t i v e sequence. (a) R^, g i v e n (b) R, not g i v e n (c) E r r o r  function  106 1.0 ->  FREQUENCY  Graph 2.7:  A-) magnitude f u n c t i o n , zero sequence. (a) R, given (b) R, not g i v e n (c) E r r o r  function.  107  35-|  (b)  (c)  - i — I I I I I I — i — I I I I I I — i — i  10"' 2 Graph 2.8:  1 2  4  4  10 2  M I  4  II,—i—i  10 2  FREQUENCY I HZ)  M I  4  I I . — i — I I I I I I  10 2  Ai phase f u n c t i o n , zero sequence. (a) R given (b) R not given d c  d c  4  10  .  (c) E r r o r  function.  108  (a)  300 H 10j  i in II i ,i n u n10,i n u n10,i n u n10 10 IIIIII] i n u n10 i m i n10,i n u n10,i i m u10.1 M i n10 #  FREQUENCY (HZ)  1000 900 H  (b)  200 -\ j i in II i i III II—i i in n—i i in II i m i II i m i II ,i i in II ,i i in II ,i I in n .1 i in n , 10  10'  I  10  10  10 10 10 FREQUENCY (HZ)  10  10  10  5.0 4.0 3.0 2.0 1.0  (c)  0 -1.0 -2.0 -3.0 -4.0 -5.0  Graph 2.9:  10  IIIIIII IIIIIII  10  1  IIIIIII  10  IIIIIII,  10  I ,1 IIIIII . I llllll ,1 IIIIII . I IIIIII , I IIIIII ,  10 10 10 FREQUENCY (HZ)  10  10  10  Z magnitude f u n c t i o n , p o s i t i v e sequence. (a) R given (b) R not given (c) E r r o r c  d c  d c  function,  109  (a)  -15  I IIIIIII  10  10"  IIIIIII  IIIIIII  1  i IIIIII .i  HUM  ,i  IIIIII .i  IIIIII ,i  nun  ,i  IIIIII ,i IIIIII  10 10 10 10 10 10 10 FREQUENCY (HZ I  10  15-1  (b)  -45  I IIIIIII  10  IIIIIII IIIIIII  10 1  i HUM  .i  min  ,i  IIIIII .i  m i l l ,I IIIIII i m m  ,i  IIIIII ,  10 10 10 10 10 10 10 FREQUENCY (HZ)  10  110 1000  900  BOO  700  (a)  600  500  400  300  200  I I I I I I I I I I I I I I I I I I I I I I 11 inn ,i i inn ,i i in n .1 i in II ,i n u n .1 m i I I - i nu II . 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (MZ)  (b)  i in II—i ,i n u n 10' ,1 m m 10 ,i i n m 10 ,i m m 10 j i in II 10_) i in II—i 1 10 n u n10i m m 10i n u n 10 t  }  FREQUENCY  (HZ)  10,  5.0 4.0 -| 3.0 2.0 1.0  (O  0 -1 .0 -2.0 -3.0 -4.0 -5.0 10  Graph 2.11:  j i ill II j i ill II 10 1  i mi ii i nu II ,i nu II ,i i in II j nu II ,i mi II .i nu II ,i mi II . 10 10 10 10 10 10 10 10 FREQUENCY (HZ)  Z magnitude f u n c t i o n , (a) R given (b) R c  d c  d c  zero sequence. not given (c) E r r o r  function.  Ill  I5n  (b)  o  -'  0  1 J I I I I I I I I I I I I I I I I I I I I I I I I I I I . I I I I I I I .1 I I I I I I .I I I I I I I , I I I I I I I I I I I I I I ,1 I I I I I I , 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ)  Graph 2 . 1 2 :  Z phase f u n c t i o n , zero sequence. (a) R, g i v e n (b) R, n o t g i v e n c  (c) E r r o r  function.  112 3.4  Low-Order R a t i o n a l - F u n c t i o n s In order  to assess  Approximations  the performance of the r a t i o n a l - f u n c t i o n s  a p p r o x i m a t i n g r o u t i n e s , A.] (to) and  Z (to) c a l c u l a t e d from the  equivalent  c  l i n e c o n f i g u r a t i o n (of the r e f e r e n c e  l i n e ) w i l l be approximated with  ous  these  numbers of p o l e s and  zeros.  For  approximations R ^  c  vari-  i s assumed  to be known. Graphs 2.13 t i o n s with and Chapter 2. allowed  t o 2.16  without  the p o l e - z e r o  When no s h i f t i n g  to exceed The  show the e r r o r f u n c t i o n s of these  number of p o l e s  i s used, the e r r o r a t 60  Hz has  needed to approximate a g i v e n  shape and  of the zero sequence p r o p a g a t i o n  the accuracy  above 300  Hz,  desired.  not  in been  function  there  i s i n c r e a s e d beyond f i v e .  is  little  Note that  the e r r o r remains unchanged r e g a r d l e s s  number of p o l e s used because no a d d i t i o n a l poles are added to model region. region  For f r e q u e n c i e s (| A-] (to )| <0.1)  the a c c u r a t e  de-  In the magnitude  function, for instance,  improvement when the number of poles frequencies  s h i f t i n g procedure e x p l a i n e d  0.5%.  pends on i t s p a r t i c u l a r  for  approxima-  above 1 KHz,  t h a t i s , f o r the h i g h  the e r r o r exceeds 5%,  approximation of t h i s  the this  attenuation  but as mentioned i n Chapter 2  r e g i o n i s not  important.  A l s o note t h a t the s h i f t i n g procedure reduces the o v e r a l l e r r o r , but slows c o n s i d e r a b l y the simple  the  power frequency The  f i t t i n g process matching  (see t a b l e 3.1)  as compared  to  process.  a p p r o x i m a t i o n of A-) (to) f o r p o s i t i v e sequence, i s p a r t i c u -  larly difficult  in this  i n the  Hz  1 to 100  u s u a l l y present  l i n e because of the s m a l l d e p r e s s i o n  region  (see graphs 2.13  f o r l e n g t h s of 500 km  and  2.17).  This  i n | Ai(to)| irregularity,  or more, i s r e s p o n s i b l e f o r the  i n c r e a s e i n the e r r o r of the 5-pole approximation over the 4-pole However, as the number of p o l e s  i n c r e a s e s beyond 5,  one.  the o v e r a l l e r r o r  113 decreases as expected.  I t s h o u l d be noted t h a t the number of s h i f t i n g  loops f o r these c a l c u l a t i o n s was l i m i t e d  to f o u r  (to decrease  costs).  If the s h i f t i n g p r o c e s s had been allowed t o c o n t i n u e a u t o m a t i cally  ( u n t i l the improvement between i t e r a t i o n s were l e s s  than 1%), a  b e t t e r approximation would have been o b t a i n e d . The  approximation of Z ( ) f o r zero sequence i s c o m p a r a t i v e l y c  more d i f f i c u l t because the endpoints  a l a r g e number of p o l e s i s n e c e s s a r y to approximate  (see graphs  2.16 and 2.20).  p o l e s t o a c c u r a t e l y approximate errors  The EMTP r o u t i n e s needed 17  the whole frequency range.  However,  i n excess of 1% o n l y occur f o r f r e q u e n c i e s above 20 KHz, t h e r e f o r e ,  a number of p o l e s l a r g e r this project. allowed  w  than e i g h t i s not j u s t i f i a b l e  i n the c o n t e x t of  Note that i n t h i s case, i f the s h i f t i n g  t o proceed beyond f o u r i t e r a t i o n s ,  f r e q u e n c i e s would have i n c r e a s e d s l i g h t l y  p r o c e s s had been  the e r r o r i n the low to mid to decrease  the e r r o r a t high  frequencies. The approximation of Z ( ) f o r p o s i t i v e sequence, c  w  i n t e r e s t i n g case where the p o l e - z e r o s h i f t i n g procedure for  the a c c u r a t e matching  of the f u n c t i o n .  i s an  i s very important  When no s h i f t i n g  i n c r e a s e i n the order of the approximation j u s t accumulates p o l e s i n the low frequency r e g i o n , d i s t o r t i n g  i s used, an the a d d i t i o n a l  the approximation  rather  than improving i t . In graphs shown: mation  2.17 through 2.20, three s e t s o f approximations a r e  a very low-order  approximation  (2 p o l e s ) , a more a c c u r a t e a p p r o x i -  (10 p o l e s ) , and a compromise between a c c u r a c y and number of p o l e s  (4 p o l e s f o r Z,-.^) and A-|(u>) p o s i t i v e sequence, sequence,  5 p o l e s f o r A i ( ) zero  and 6 p o l e s f o r Z (co) z e r o  sequence).  One  of the low-order  c  of the main advantages  w  approximation  r o u t i n e s , i s t h a t they are very i n e x p e n s i v e c o m p u t a t i o n a l l y .  A high-  114  Graph 2.13:  Error functions. Low-order approximation of A-| , pos i t i ve sequence.  115  Graph 2.14:  Error functions. z e r o sequence.  Low-order approximation of A-j ,  116  Graph 2.15:  Error functions. Low-order approximation of Z, p o s i t i v e sequence.  Graph  2.16:  Error functions. z e r o sequence.  Low-order approximation of  Z , c  118 a c c u r a c y approximation  produced  $10.00, w h i l e low-order  with the EMTP r o u t i n e s can c o s t over  approximation  c o s t s are u s u a l l y under $3.00 ( u s i n g  10 p o l e s i n Z (co) and A-|(co), and 4 s h i f t i n g  loops).  c  are based  These running c o s t s  on UBC's r a t e s o f $1,200.00 per hour of CPU time under  Terminal  use. Table 3.1 shows the c o s t of a high a c c u r a c y s i m u l a t i o n of the r e f e r e n c e l i n e u s i n g the EMPT r o u t i n e s .  The c o r r e s p o n d i n g  are shown i n graphs 2.21 t o 2.24, where the output Program i s used  from  approximations  the L i n e Constants  as r e f e r e n c e .  UBC's l i n e  c o n s t a n t s program  $  4.50  $  0.24  $  1 .23  $  1.11  $  1.18  $  1 .75  TRANFLIN.0 ( C a l c u l a t i o n of Z . (co ) and A-j (co)) c  TRANA1.0 (approximation of A-, (co ) )  Positive  sequence (18 p o l e s , 12 z e r o s )  Zero sequence (21 p o l e s , 15 zeros)  TRANZC.O (approximation of Z (co)> c  Positive  sequence (9 p o l e s , 9 z e r o s )  Zero sequence (17 p o l e s ,  17 zeros)  Total  Table 3.1:  $ 10.01  Costs i n $cc of the frequency--dependence r o u t i n e s of the EMTP.  119 Table 3.2 shows the running c o s t s of the low-approximation  program f o r  v a r i o u s numbers of p o l e s and z e r o s .  Cost $cc  Number o f Poles  Z (co)  A-, (co)  c  Positive Sequence  Zero Sequence  Positive Sequence  Zero Sequence  Shifting  2 3 4 5 6 7 8 9 10 4  2 3 4 5 6 7 8 9 10 6  2 3 4 5 6 7 8 9 10 4  2 3 4 5 6 7 8 9 10 5  0.25 0.34 0.51 0.93 1 .27 1 .63 2.04 2.53 3.05 0.79  Table  3.2:  Costs  i n $cc of the low-approximation  It i s i n t e r e s t i n g  to note  of A-|(co) have comparable e r r o r Therefore  the higher computational  can be a t t r i b u t e d Ai (co).  0.16 0.16 0.16 0.15 0.17 0.20 0.12 0.17 0.17 —  program.  that high and low-order  l e v e l s when | A^oo)]  No s h i f t i n g  approximations  i s g r e a t e r than  e f f o r t i n the h i g h - o r d e r  0.4.  approximation  to the m o d e l l i n g of the high a t t e n u a t i o n r e g i o n of  120 1.0 ~y  FREOUENCr  Graph 2.17:  (HZ)  Approximation of A-j, p o s i t i v e sequence. (a) 2 p o l e s (b) 4 p o l e s (c) 10 p o l e s  Graph 2.18:  Approximation o f Aj_, z e r o sequence. (a) 2 p o l e s  (b) 5 p o l e s  (c) 10 p o l e s  1000  -i  FREQUENCY  Graph 2.19:  (HZ)  Approximation of Z , p o s i t i v e sequence. (a) 2 p o l e s (b) 4 p o l e s (c) 10 p o l e s c  Graph 2.20:  Approximation o f Z , c  (a)  2 poles  (b)  zero  sequence.  6 poles  (c)  10  poles  Graph 2.21 :  A-] p o s i t i v e sequence, h i g h - o r d e r a p p r o x i m a t i o n . (a) Magnitude f u n c t i o n (b) E r r o r f u n c t i o n  l.O-i  FREOUENCT IH7J (b)  Graph 2.22:  Ai z e r o sequence, h i g h - o r d e r a p p r o x i m a t i o n . (a) Magnitude f u n c t i o n (b) E r r o r f u n c t i o n  Graph 2.23:  Z p o s i t i v e sequence, h i g h - o r d e r a p p r o x i m a t i o n . (a) Magnitude f u n c t i o n (b) E r r o r f u n c t i o n c  J 000  -i  FREQUENCY  (HZ)  (a)  5.0 1.0-1 3.0 2.0 1.0 04 -1.0 -2.0 -3.0 -j -4.0 -5.0  10  i I ill  II  10  )  i in  II—i  1  i in II—i I ill II 10 10  ,1 I III II ,1 I III II  10  FREQUENCY  10  J  I III II .1 I III II .1 I III II ,1 I III II ,  10  10  10  (HZ)  (b)  Graph 2 . 2 4 :  Z z e r o sequence, h i g h - o r d e r a p p r o x i m a t i o n . (a) Magnitude f u n c t i o n (b) E r r o r f u n c t i o n c  10  128 3.5  Frequency Domain Response  3.5.1  Introduction The  b e s t t e s t f o r the v a l i d i t y  of the approximating  techniques  d e s c r i b e d i n t h i s p r o j e c t , i s to simulate t r a n s i e n t phenomena i n the EMTP, and  to compare the r e s u l t s with those o b t a i n e d u s i n g the a v a i l a b l e  a c c u r a c y models. very l a r g e and s u l t s here.  U n f o r t u n a t e l y , the number of p o s s i b l e s i m u l a t i o n s i s  i t would not be p r a c t i c a l  s t e a d y - s t a t e , open and one  to p r e s e n t a l a r g e number of r e -  However, a s m a l l but w i s e l y chosen number of t e s t s can give a  good i d e a of the r e l a t i v e a c c u r a c y  at  high-  frequency;  of the models.  short c i r c u i t  with  the frequency  responses  Two  such  t e s t s are  to s i n u s o i d a l  v a r i e d over  the  excitation  the frequency  range of  interest. From the s o l u t i o n of the l i n e (see appendix I-A), the open c i r c u i t by  2 V V  m  (co)  =  Q  response  i n the frequency  (see F i g u r e 3.1),  (3.1) -y  hf  (co) and V  m  i s the  voltage.  Similarly, when the l i n e  i s given  A (co)  i s the peak magnitude of the v o l t a g e source,  r e c e i v i n g end  domain  — 1 +  where V  equations  the s h o r t c i r c u i t response,  i s s h o r t e d , i s given  or r e c e i v i n g end  current  by 2 V  I (co) m  These two (a)  Open and a line  A (co) ° Z (co) (1 - A (co)) c 1 2  t e r m i n a t i o n s are extreme and  model; i f the response  i s good f o r these  to assume t h a t the response  good.  (3.2)  1  t e s t s p r e s e n t s e v e r a l advantages:  short c i r c u i t  reasonable a l s o be  =  f o r any  two  t r y i n g cases f o r cases, i t i s  other t e r m i n a t i o n w i l l  129 (b)  The behaviour of the models over the whole f r e q u e n c y range readily  (c)  can be  observed.  The r e s u l t s of the d i f f e r e n t r a t i o n a l - f u n c t i o n s approximations can be compared to t h e o r e t i c a l r e s u l t s s i n c e A-| (to) and Z c ( t o )  i n equations  (3.1) and (3.2) can be e v a l u a t e d a t any g i v e n frequency from  the o u t -  put of UBC's L i n e Constants Program.  o m  cos  (wt)  V  m  COS(uot)  (a)  Fig.  3.1:  (b)  (a) Open c i r c u i t and  In  these t e s t s ,  (b) Short c i r c u i t  the l i n e w i l l be assumed to be s i n g l e - p h a s e ,  w i t h p o s i t i v e or zero sequence parameters, of  the approximations The  tests.  involved  i n order t o i s o l a t e the e f f e c t s  i n each p r o p a g a t i o n mode.  f o l l o w i n g models have been t e s t e d :  Model 1. - High-order approximation u s i n g the parameters  calculated  from  calculated  from  UBC's L i n e Constants Program. Model 2. - High-order approximation u s i n g the parameters the e q u i v a l e n t l i n e  r e p r e s e n t a t i o n with  given.  Model 3. - High-order a p p r o x i m a t i o n u s i n g the parameters the e q u i v a l e n t l i n e  r e p r e s e n t a t i o n w i t h R,  calculated  not g i v e n .  from  130 Model 4. - Low-order a p p r o x i m a t i o n  (2 p o l e s , R^  Model 5. - Low-order approximation  (10 p o l e s , R^  Model 6. - Low-order approximation  (4,5 and 6 p o l e s , R^  Model 7. - Constant parameters  given).  c  c  given). c  given).  o p t i o n of the EMTP.  Table 3.3 shows the number of p o l e s and zeros used different  approximations.  AT (co)  Z ( co) c  Positive sequence  Zero sequence Model  i n the  Poles  1 2 3 4 5 6 Table 3.3:  17 17 18 2 10 6  Zeros  Poles  17 17 18 2 10 6  9 9 9 2 10 4  Positive sequence  Zero sequence  Zeros  Poles  9 9 9 2 10 4  Zeros  21 21 25 2 10 5  15 15 19 1 9 4  Poles  Zeros  18 18 17 2 10 4  12 12 11 1 9 3  Number of p o l e s and zeros used i n the d i f f e r e n t models t e s t e d i n t h i s s e c t i o n . o  Models 2 and 3 are used f u n c t i o n s approximation from parameters  the e f f e c t of the r a t i o n a l -  the e f f e c t of the e s t i m a t i o n of the l i n e  (from the e q u i v a l e n t l i n e c o n f i g u r a t i o n ) .  show the e f f e c t s line.  to i s o l a t e  Models 4, 5 and 6  of the order of the approximation i n the response  of the  131 3.5.2  Open C i r c u i t Response The open c i r c u i t  response f o r the d i f f e r e n t  ed above, i s shown i n graphs  2.25 through 2.31.  on A-|(w) o n l y (see e q u a t i o n ( 3 . 1 ) ) , mation  l i n e models d e s c r i b -  Note t h a t V ( w ) depends m  t h e r e f o r e the e f f e c t s of the a p p r o x i -  of Z ( w ) are not p r e s e n t i n t h i s c  test.  The response of model 2 f o r zero sequence t o the t h e o r e t i c a l response; the p o s i t i v e sequence  is virtually  identical  response of the same  model shows s i g n i f i c a n t e r r o r s o n l y f o r f r e q u e n c i e s above 10 KHz. When R^  c  f o r z e r o sequence, reflecting  i s not known  (model 3 ) , the response i s s t i l l  good  but p r e s e n t s c o n s i d e r a b l e e r r o r s f o r p o s i t i v e  the e r r o r s  sequence,  i n the s i m u l a t i o n of A-|(co) (see graph 2.5).  It i s i n t e r e s t i n g  to note that the d i f f e r e n c e s between the  responses of models 5 and 6 a r e r e l a t i v e l y  s m a l l , a l t h o u g h the order of  the approximation i n model 6 i s almost twice that of model 5. The  response of the 2-pole approximation  (model 4) i s r e a s o n a b l y  good i n the low to mid f r e q u e n c i e s range, and o f f e r s a d e f i n i t e improvement over the c o n s t a n t parameters  model (model 7 ) .  overall  ~i—iMin—i  10" 2  4  1 2  i  III  4  I I — i — i III 10 2 4  II,—i—IIIIII, i 10 2 4 10 2  FREQUENCY  M U M .  4  (HZ)  (a)  2.25:  O/C response, model 1. (a) P o s i t i v e sequence  (b) Zero  sequence  10  i—rm  2  4  10'  ~\  2  I  III  I  II  I I 11  II  4 10 2 4 10" FREQUENCY (HZ) (a)  2.26:  O/C response, model 2. (a) P o s i t i v e sequence  (b) Zero  sequence  i—i i u r 2 4 ID'  134  F i g . 2.27:  0/C response, model 3. (a) P o s i t i v e sequence  (b) Zero  sequence  135  F i g . 2.28:  0/C response, model 4. (a) P o s i t i v e sequence  (b) Zero  sequence  136 ]0.o q CL  o  9.0 -  II  8.0 LU D O  o  7.0 6.0  LU to Ln O  5.0 -j 4.0 -  O  CL LO  LU a: O  CD  3.0 : 2.0 1 .0 -i-  I  10"' 2  M U M  4  1 M U M  1  2  4  1  10  2  M U M .  I  I I I I I I  4 10 2 4 10' FREQUENCY (HZ) (a)  IO.O q  9.0 II  F i g . 2.29:  0/C response, model 5. (a) P o s i t i v e sequence  (b) Zero  sequence  lO.O-i  9.0 H  F i g . 2.30:  O/C response, model 6. (a) P o s i t i v e sequence  (b) Zero  sequence  138  FREQUENCY  (HZ)  (b)  F i g . 2.31:  0/C response, model 7. (a) P o s i t i v e sequence  (b) Zero  sequence  139 3.5.3  S h o r t C i r c u i t Response The  i n graphs  2.32  short c i r c u i t through  In t h i s  responses of the d i f f e r e n t models are shown  2.45.  test,  the r e s u l t s are i n f l u e n c e d by the approximation  o f Z (co) (see e q u a t i o n ( 3 . 2 ) ) , and c  Ai(co) now  combine  the e r r o r s i n the approximation of  ( u s u a l l y d i s f a v o u r a b l y ) w i t h the e r r o r s  The r e l a t i v e l y  i n Z (co). c  l a r g e e r r o r s i n the low frequency range are due 2  to the n u m e r i c a l s e n s i t i v i t y of the f a c t o r A-|/(1  - A-]) i n e q u a t i o n  a t low f r e q u e n c i e s , where the magnitude of A-|(u)) i s very c l o s e to For example, a d i f f e r e n c e of 0.01% produce  e r r o r s up to 100%  (3.2) 1.0.  i n the approximation of A i ( ) can  i n the s h o r t c i r c u i t  w  response  [9].  These  e r r o r s , however, are not very important as long as the dc response i s matched i d e n t i c a l l y ,  which  i s the case when  i s known.  0.050 -f  FREQUENCY  Graph 2.32:  S/C response, model 1. (a) P o s i t i v e sequence  (HZ)  Mid to h i g h f r e q u e n c i e s , (b) Zero sequence  Graph  2.33:  S/C response, model 1. (a) P o s i t i v e sequence  Low f r e q u e n c i e s , (b) Zero sequence  Graph 2.34:  S/C response, model 2. (a) P o s i t i v e sequence  Mid to h i g h f r e q u e n c i e s , (b) Zero sequence  Graph 2.35:  S/C response, model 2. (a) P o s i t i v e sequence  Low f r e q u e n c i e s . (b) Zero sequence  O.OJO -j  Graph 2.36:  S/C response, model 3. (a) P o s i t i v e sequence  Mid to h i g h f r e q u e n c i e s , (b) Zero sequence  Graph 2.37:  S/C response, model 3. (a) P o s i t i v e sequence  Low f r e q u e n c i e s . (b) Zero sequence  Graph 2.38:  S/C response, model 4. (a) P o s i t i v e sequence  Mid to high f r e q u e n c i e s , (b) Zero sequence  0.100  T  .090 .080 .070 .060 .050 .040 .030 .020 .010 -0.000  10  H  I I Mill  3 4 6  10  ,  i i MIII  — I II Mill , — r—  4 6 10" 2 3 4 6 FREQUENCY (HZ)  1  (a)  Graph 2.39:  S/C r e s p o n s e , model 4. (a) P o s i t i v e sequence  Low f r e q u e n c i e s . (b) Zero sequence  -i—i  IIIIII  3 4 6  10  Graph  2.40:  S/C r e s p o n s e , model 5. (a) P o s i t i v e sequence  Mid to high f r e q u e n c i e s , (b) Zero sequence  0.100 -i 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010  H  . 000 - f  10  1 — i — i  IIIIII  2 3 4 6  ,  10  1—I—i  i I 1111 ,  2 3 4 6  I—I—I I I I I I I  10" 2 3 4 6  1  I—I—I  FREQUENCY (HZ)  (a)  Graph 2.41:  IIIIII  2 3 4 6  S/C response, model 5. Low f r e q u e n c i e s . (a) P o s i t i v e sequence (b) Zero sequence  10  0.050 H .  ii  _  0.040  _J UJ  o o  0.030  LU CO CO  o Q-  0.020 H  LU CO  o coco «  0 . 0 1 0 H  o  -s CO  FREQUENCY (HZ) (a)  0.010 CL CJ X CO  Ixl  O o  UJ CO  o cr. LU (VI  LU CO  o 0_  CO LU  0.002  £  0.001 -0.000  T  1  1—I  M i l l  2 3 4 6 10" FREQUENCY (HZ) (b)  Graph 2.42:  S/C response, model 6. (a) P o s i t i v e sequence  Mid to high f r e q u e n c i e s , (b) Zero sequence  Graph 2.43:  S/C response, model 6. (a) P o s i t i v e sequence  Low f r e q u e n c i e s . (b) Zero sequence  0.050 -*  *  0.040 -^  Graph 2.44:  S/C response, model 7. (a) P o s i t i v e sequence  Mid to high f r e q u e n c i e s , (b) Zero sequence  O.lOO-i 0.090 0.060 -1 ^  0.070 -I  LU  o  0.060 -1 0.050 0.040 0.030 0.020 0.010 0.000 10'  n—i  3 4  IIIIII  6  10  ,  n—I I I 11 ll -i—i i i 1111 3 4 6 1 3 4 6 10" FREQUENCY (HZ)  -|—I—I i I 11 II  2  (a)  Graph 2.45:  S/C response, model 7. (a) P o s i t i v e sequence  Low f r e q u e n c i e s , (b) Zero sequence  3 4  6  10  154 3.6  Transient Simulation  i n the EMTP  To complement the t e s t s i n s e c t i o n 3.5, a t r a n s i e n t s i m u l a t i o n u s i n g the EMTP, w i l l be shown i n t h i s s e c t i o n .  Consider  the c i r c u i t i n  F i g u r e 3.2.  t=0 m -o  C O S (IAJI)  Fig.  3.2:  0  Line e n e r g i z a t i o n  The  line  test.  will  be assumed to be s i n g l e - p h a s e  zero sequence parameters.  The 60 Hz s i n u s o i d a l v o l t a g e  peak) i s connected The  to the c i r c u i t  r e c e i v i n g end v o l t a g e s  with p o s i t i v e or source  (1.0 p.u.  a t t = 0, and the r e c e i v i n g end i s open.  f o r the d i f f e r e n t models used i n s e c t i o n 3.5  are shown i n graphs 2.46 t o 2.51.  Note t h a t the r e s u l t s  order  the frequency-dependence r o u t i n e s of  approximation,  obtained  with  the EMTP (model 1 ) , are used as a r e f e r e n c e .  from the high  Graph 2.46:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model  2.  156  a  -4.00  0.0050  0.0300  or.  0.0150 0.0200 TIME (SECONDS)  0.0250  0.0300  0.0250  0.0300  (a)  4.00 -i  0.0050  0.0100  0.0150 0.0200 TIME (SECONDSI (b)  Graph 2.47:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model 3.  157  Graph 2.48:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model  4.  158  Graph 2.49:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model  5.  159  Graph 2.50:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model  6.  160  Graph 2.51:  E n e r g i z a t i o n t e s t , r e c e i v i n g end v o l t a g e . (a) P o s i t i v e sequence (b) Zero sequence  Model  7.  CONCLUSIONS  A low-order  approximation  of a transposed, overhead second  p a r t of t h i s The  relatively  thesis  f o r the frequency-dependent  parameters  t r a n s m i s s i o n l i n e , has been o b t a i n e d i n the project.  amount of i n f o r m a t i o n r e q u i r e d f o r such an approximation i s  small.  Compared to the c o n s t a n t parameters o p t i o n of the EMTP,  the o n l y a d d i t i o n a l parameters needed to approximate the frequencydependent behaviour  of the l i n e , are the e a r t h r e s i s t i v i t y and the dc  r e s i s t a n c e of the c o n d u c t o r s .  The  main advantages of t h i s approximation  can be summarized as  follows: i)  I t i s c o m p u t a t i o n a l l y f a s t e r than the e x i s t i n g r o u t i n e s of the EMTP, both  frequency-dependence  i n the l i n e parameters  approximating  p r o c e s s and d u r i n g t r a n s i e n t s i m u l a t i o n s .  ii)  I t i s more a c c u r a t e than the c o n s t a n t parameters model.  iii)  I t g i v e s very a c c u r a t e r e s u l t s over the low to mid f r e q u e n c i e s range.  • Two main disadvantages i)  can be p o i n t e d o u t :  The e q u i v a l e n t l i n e c o n f i g u r a t i o n i s o n l y a c c u r a t e when the t r a n s m i s s i o n l i n e can be assumed to be p e r f e c t l y  ii)  transposed.  I t l o s e s a c c u r a c y when very f a s t t r a n s i e n t s are c o n s i d e r e d .  161  APPENDIX I-A  GENERAL SOLUTION OF THE LINE EQUATIONS IN THE FREQUENCY DOMAIN  162  163 The e q u a t i o n s r e l a t i n g v o l t a g e s and c u r r e n t s i n the f r e q u e n c y domain a r e d V h e— = Z , I , dx Ph ph D  -  d Iph - — — — = Ypj^  Where Z p admittance m a t r i c e s , quantities.  and Yp  n  d  V  d  x  2  I  h  9  d x^  (shunt c u r r e n t drop e q u a t i o n ) .  (I-A.2)  impedance and  shunt  r e s p e c t i v e l y , and the s u b s c r i p t p^ stands f o r phase (I-A.1) and  £ h = ( Z ^ Y_J 2 Ph Ph p  (I-A.1)  a r e the s e r i e s  n  Differentiating d  ( s e r i e s v o l t a g e drop e q u a t i o n )  (I-A.2) we  obtain  V_ ph  (I-A.3)  u  = (Y , Z . ) I . ph ph ph  (I-A.4)  To proceed with the s o l u t i o n of e q u a t i o n s (I-A.3) and it  i s convenient to d i a g o n a l i z e  (Z^  Y D n  ^  a  n  d  ^ p h ph^ i Y  Z  n  o  r  d  e  r  t  o  (I-A.4),  obtain  a decoupled s e t of d i f f e r e n t i a l e q u a t i o n s . L e t P and Q be the m a t r i c e s that d i a g o n a l i z e (Yp Z ), h  D h  respectively. P" n  1  -1  y  Let us now  Z  Y h  ph^  a  n  d  Then Z  ph  Y  ph  =  P  Y Z Q ph ph  D  =  zy  (diagonal)  D„„ y  (I-A.5)  (diagonal).  (I-A.6)  z  define P" Q  1  -1  V I  p h  =  V  (I-A.7)  =  I .  m  m  I n t r o d u c i n g these d e f i n i t i o n s and e q u a t i o n s (I-A.5) and (I-A.3) and  ( p  (I-A.4). we  obtain  (I-A.8) (I-A.6) i n t o  164 d  V,m = D_. V 2 zv x  2  =  d d  2  U  V m  Im = D._ I 2 yz x  d Introducing  (I-A.7)  m  (I-A.8)  m  the above d e f i n i t i o n s  i n t o equations  (I-A.1) and  (I-A.2) d V, m = D_ ii  d  i  =  x  d I.— d x  I  (I-A.9)  m  z  = y V,  (I-A.10)  D  m  where, 'z  D  y  = p  -1  Z_ Q "ph  = Z  K ;  P  = Q~'  (diagonal)  m  Y , P = Y ph  m  (diagonal),  (I-A.11)  (I-A.12)  and, Z  m  =  modal impedance matrix  Y  m  =  modal admittance matrix  D  Equations  zy  =  D  y'  yz  =  2  (I-A.7) to (I-A.10) can now  d  2  v  d d  m  Y  x  2 Z  V  m  be r e w r i t t e n ,  (I-A. 13)  2  2  d  x  m  x  Y  2  I  m  (I-A.14)  2  dv d x  m  =  Zm Im  (I-A.15)  165 —  d x  = Y  V  m  (I-A.16)  m  These equations d e f i n e the behaviour domain.  Since they are uncoupled,  of the l i n e i n the modal  the s o l u t i o n i s r e l a t i v e l y simple and  g i v e n by  V (x,u» m  .  »  I (x,co) m  =  A e"  =  —  A  Y ( a ) )  X +  -Y(OJ)  e  x  c  B e  Y  (  B  w  - - e c  )  (I-A.17)  X  Y(co)  x  .  '  (I-A.18)  where, \l m m z  Y  Zm Y  Note t h a t Z equations  c  m  (propagation  constant)  (surge impedance)  and y are d i a g o n a l m a t r i c e s ; t h i s  (I-A.17) and (I-A.18) have the same form when Z  e i t h e r matrices or s c a l a r s consideration).  and y are  ( i . e . , when o n l y one p r o p a g a t i o n mode i s under  A and B are a l s o d i a g o n a l matrices and they are d e f i n e d  by boundary c o n d i t i o n s . C o n s i d e r , f o r example,  the l i n e  below  'm  k c-  c  implies that  -o m  'm  166  where o n l y one p r o p a g a t i o n mode w i l l be c o n s i d e r e d , and s u b s c r i p t s m  K  and  now denote sending and r e c e i v i n g ends, r e s p e c t i v e l y . Taking, f o r i n s t a n c e , x = £ v  m =  _  I  A e  m -  —  A  z  -y-i Y£ + B e  e  c  -yl  - —  B Z  yl  e c  give v  A  =  B  =  v  m  c m 2 z  m  yl  I  e  + c m z  -yI e  I  ~  2  T h e r e f o r e the v o l t a g e and c u r r e n t a t the sending end become  Vk  =  I  = k  V  - Z  m  v  e  —  Z  Y£  -yl  - e ' 2  c  c  I  ^  m  yl  _ I  e'  m  -yl  + e ' 2  which i n terms o f h y p e r b o l i c f u n c t i o n s , g i v e  V  k  =  V  V = — c  m  cosh(yJc) - Z  I  c  T a  sinh(y£)  (I-A.19)  m  I  k  sinh(y^) - I  m  cosh(y^)  (I-A.20)  z  These r e p r e s e n t the " c l a s s i c a l " l i n e i n the frequency domain.  s o l u t i o n f o r a s i n g l e phase  For d e t a i l e d proof of the matrix  r e l a t i o n s h i p s s t a t e d here, the reader should r e f e r to [ 8 ] .  APPENDIX I-B  JOSE  MARTI'S  FREQUENCY-DEPENDENCE  167  MODEL  168 What f o l l o w s i s o n l y a b r i e f overview of the b a s i c concepts upon which based.  the frequency-dependence  model used i n t h i s  thesis project i s  Obvious space l i m i t a t i o n s do not permit a d e t a i l e d e x p l a n a t i o n of  the model, but r a t h e r a r e f e r e n c e guide f o r the reader who i s somewhat f a m i l i a r with the model. the reader should r e f e r  For a more complete e x p l a n a t i o n of the model to [3] and [ 7 ] .  169 C o n s i d e r the l i n e r e p r e s e n t e d  ( f o r a g i v e n mode) i n F i g u r e  I-B.1 below  im(t)  v (t) m  Fig.  I-B.1:  Modal r e p r e s e n t a t i o n i n the time  The a s s o c i a t e d model f o r t h i s Figure  Fig.  domain.  l i n e i n the frequency domain i s shown i n  I-B.2.  I-B.2:  L i n e model i n the frequency  Z q(co) i s approximated e  same response  domain.  as an R-C network which has, e s s e n t i a l l y , the  as the surge  impedance Z (co). c  The backward p r o p a g a t i o n  170 f u n c t i o n s B ( to) and k  B (to)  =  V (to) - Z ( t o ) I*(to)  (I-B.1)  Bm(to)  =  V (to) - Z ( t o ) I ( t o ) .  (I-B.2)  k  Let  us now  k  e q  m  e q  d e f i n e the forward p r o p a g a t i o n f u n c t i o n s F^to) and  F (to), m  =  V (to) + Z ( t o ) I (to)  (I-B.3)  m(u)  =  V (to) + Z ( t o ) I ( t o ) .  (I-B.4)  F  k  appendix  e q  m  The I-A)  k  e q  m  g e n e r a l s o l u t i o n of the l i n e  i n the frequency domain  k  =  cosh(Y&) V  I  k  =  — Z  m  - Z  sinh(Y)t) V  sinh  C  m  (yl) I  - cosh  (I-B.5)  m  (YJi) I .  (I-B.6)  m  c  r e l a t i o n s h i p between B (to) and F (to) i s found by k  (I-B.1)  B (to)  =  e  B^to)  =  e~  us now  k  to ( I - B . 6 ) .  t a k i n g i n t o account -Y J k  (see  i s g i v e n by  V  equations  Let  m  F (to) k  The  B^i to) are d e f i n e d as f o l l o w s  combining  A f t e r some a l g e b r a i c m a n i p u l a t i o n s ,  t h a t Z q(to) = e  Z (OJ)/  w  e  c  obtain,  F (oj)  (I-B.7)  F (to).  (I-B.8)  m  yl  and  k  d e f i n e the p r o p a g a t i o n f u n c t i o n A-| (^)  as  -Y I? A^to) where I  =  i s the l i n e  Equations  (I-B.1)  e  ,  (I-B.9)  l e n g t h and Y =  to (I-B.4)  \/ Z' Y'  can now  (see appendix  I-A).  be r e w r i t t e n as  B (to)  =  V (to) - Ek(to) = A-|F (to)  (I-B.10)  B (to)  =  V (to) - Emfto) = A-|F (to)  (I-B.11)  F (to)  =  V (to) + E (to)  (I-B.12)  F (to)  =  V (to) + Em(to) ,  (I-B.13)  k  m  k  m  k  m  m  k  k  k  m  where, E (to)  =  I (to) Z ( t o )  (I-B.14)  EmJto)  =  I (to) Z ( o ) ) .  (I-B.15)  k  k  m  e q  e q  171 Since Z g(co) i s the response of a l i n e a r R-C e  network, the time  domain r e p r e s e n t a t i o n of e q u a t i o n s (I-B.10) through by means of the I n v e r s e F o u r i e r Transform, b (t)  =  k  b  vfc(t) - e ( t ) = k  (t)*f (t)  (I-B.16)  m  =  k( )  =  z q( )* k( )  (I-B.18)  =  z  (I-B.19)  v  " em(t)  that i s ,  m( ) fc  m(t)  a i  (I-B.15) can be found  = a,(t)*f (t),  (I-B.17)  k  where e  f c  e (t) m  t  i  t  e  e q  (t)*i (t). m  (lower case l e t t e r s are used to i n d i c a t e upper  time domain q u a n t i t i e s , w h i l e  case i s used i n the frequency domain) These  equations d e f i n e the e q u i v a l e n t c i r c u i t of F i g u r e II-B.3  •m(t)  k o  o  m  v (t) m  Fig.  II-B.3:  Frequency dependence model i n the time domain.  Note t h a t i n the time domain, e ( t ) i s the v o l t a g e a c r o s s the e q u i v a l e n t k  R-C  network t h a t s i m u l a t e s Z ( u ) . c  The p r o p a g a t i o n f u n c t i o n a-|(t) i n the time domain can be expressed as a-,(t)  =  p(t-C),  (I-B.20)  172 where p ( t ) has the same shape as a-|(t) but d i s p l a c e d - t the  time u n i t s from  origin. domain A-|(to) can then be expressed  In the frequency AT ( t o )  =  P(to) e 3^ -  as (I-B.21 )  c  The f u n c t i o n P(to) can be approximated by r a t i o n a l f u n c t i o n s of the form P(s)  =  H  (s + z i ) (s + Z 2 ) (s + p-i ) (s + p )  ... (s + z ) , ... (s + p ) n  2  m  where z± and p^ are the zeros and p o l e s of P ( s ) i n the complex  plane;  these  s i n g u l a r i t i e s are r e a l and l i e i n the l e f t - h a n d s i d e of the complex  plane  (m>n). P(s)  =  Therefore,  P(s) can be expanded i n p a r t i a l 1 + s + p-|  2 + ... + s + p  k  k  2  i n the time domain  p(t)  =  [k, e~ l p  From equation  + k  t  (I-B.20) we  k  (I-B.22)  m  p ( t ) becomes e  2  m s + p  fractions  + ... + k  _ P 2 t  now  e"-^ ] u ( t ) . 1  m  o b t a i n a-|(t) (I-B.23)  a  l (  t)  = [k! e - P l  ( t  "  + k  C )  e-P2 " > (t  2  c  + ... + k  With a-|(t) i n the form of equation i n equations methods.  (I-B.16) and  Consider,  m  e^m^O  (I-B.23),  j  u ( t  _ , c  the c o n v o l u t i o n s  (I-B.17) can be s o l v e d by r e c u r s i v e i n t e g r a t i o n  f o r i n s t a n c e , the c o n v o l u t i o n i n t e g r a l of  equation  (I-B.16) OO b  k(t>  since  =  -co m (  f  f  ai(t)= 0 b (t) k  =  k  =  a  l  f o r t<r  /" f ( t - u ) m  I n t r o d u c i n g equation b (t)  t _ u )  (  u  )  this a i  d u  ' i n t e g r a l becomes  ( u ) du.  (I-B.23) i n t o  m E b. . ( t ) i=l k , i  (I-B.24) (I-B.24), (I-B.25)  where  b  k,l(t)  = f  •  °°  f ( t - u ) k. e  r  can  future confusion  (I-B.26)  u  m  Note t h a t i n e q u a t i o n to a v o i d  , * ( -r)  D  1  du  (I-B.26) p i has been s u b s t i t u t e d by 6 i  i n notation.  The i n t e g r a l i n e q u a t i o n  (I-B.26)  be broken i n t o two p a r t s ,  **,i<t>  = /,  fm(t-u) k  r + A t  e"  ±  e  i  (  u  -  C  )  du  +  /^  + A t  f (t-u) k m  ±  e  -  e i ( u  "  C )  du,  (I-B.27) but  the second i n t e g r a l i n e q u a t i o n  first  i n t e g r a l can be e v a l u a t e d  A f t e r some a l g e b r a i c b, . (t) k, 1  =  (I-B.27) i s e ^ i _  numerically  using  A t  b i(t-At), k  and the  the t r a p e z o i d a l r u l e .  manipulations,,  g-; bv -j(t-At) + c . f (t-C) + d. f ( t - C - A t ) , J. JS. ,x i m i m  (I-B.28)  where, 9i h  =  e 1 -  1  i  ±  BiAt k T7  d  g  =  i  -  *i  —  (1 " (  g i  - h ; ±  and b (t) k  m I h .(t) i=l k , i  =  k  From e q u a t i o n s  (I-B.11) and f (t) m  therefore, b  k >  values  /  of b  k  (I-B.13) i t can be seen t h a t =  i(t)  i n equation  i, b  and  m  2 v (t) - b (t), m  m  (I-B.28) depends o n l y on p a s t  history  v . m  The e v a l u a t i o n o f e ( t ) = i ( t ) * z q ( t ) proceeds i n a k  k  e  174 similar  f a s h i o n as the e v a l u a t i o n of b ( t ) .  The r a t i o n a l  k  fractions  that  approximate Z q(io) are g i v e n by, e  , . ( + 1 ) ( + 2 ) ... ( + n ) , (S) = n ;—j : ; r' eq (S + P l ) (s + p ) .•. (s + p ) s  2  s  2  s  z  H  Z  2  m  where, as i n the case of A-|(o>) the p o l e s are r e a l and l i e i n the left-hand  s i d e of the complex p l a n e , but i n t h i s case Expanding i n p a r t i a l  e q  (s)  =  k +  s + p. 1  therefore,  z eq  ( t  )  =  [k  k  1  Q  + ... +  6 ( t ) + k!  0  m.  fractions,  k Z  n =  e  (I-B.29),  m  s + p  - a i Lt  m  .  1  + ... + k  . - r0 ^ t .  e" ^]  m  u(t),  where p^ has been s u b s t i t u t e d by a j_ t o a v o i d ambiguity From equation  (I-B.30)  i n the n o t a t i o n .  (I-B.18) CO  e  k  =  '  where the lower l i m i t  i (t-u)  0  e q  ( u ) du,  i n the i n t e g r a l  Introducing equation e  z  k  k<t>  =  e  k,o^)  (I-B.31)  i s zero because z q ( t ) = 0 f o r t<0. e  (I-B.30) i n t o  (I-B.31), (I-B.32)  J ^ k , ! ^ '  +  where, e  k,o  e  Noting  k,i  ( t )  (  t  )  =  ±  =  f  )<.  Q  {t)  k  o  ik  (  t  =  -  u  )  R  k  t h a t the i n t e g r a l i n e q u a t i o n  i n equation e  k j i  (I-B.24) w i t h  (t)  =  where, -a.: At m^  =  h  -  e 1  '  ith e  k > i  o  i k  i  e  (  t  d u  (I-B.33)  )  -  (I-B.34  (I-B.34) i s analogous to the i n t e g r a l  1=0, (t-At) + p  ±  i (t) + q k  ±  i (t-At) k  (I-B.35)  175 k. -  Pi  M- i) h  ±  <li  (mi - h i )  1  Introducing e  k  equations  ( t )  =  R  k  L  k  (I-B.33) and  { t )  +  e  k,e  ( t _ A t )  (I-B.35) i n t o +  e  k,c  ( t _ A t )  (I-B.32) (I-B.36)  '  where, m R  k  =  R  o  +  J  x  ( e q u i v a l e n t constant  Pi m  e , (t-At)  =  e  =  k  k ( 6  c  (t-At)  i (t-At) w  m Z  mi e i ( t - A t ) k >  With equations of Figure  i  k  I-B.3  can  be  e  k o k  vwr  kc  +  e  (I-B.36),  the e q u i v a l e n t  e  k ( i  )  circuit  I-B.4  ke  —(^^)— b ( k  I-B.4:  transformed  „6  Equivalent c i r c u i t  This equivalent c i r c u i t  EMTP;  ( h i s t o r y of p a r t i a l v o l t a g e s  s i m p l i f i e d as shown i n Figure  ( t )  Fig.  currents)  ( t )  Rk  v  (I-B.28) and  (history of  resistance)  V m ( t )  i n the time domain  ( u s i n g elementary c i r c u i t  transformations)  i n t o the c i r c u i t of F i g u r e I-B.5, which i s compatible  can with  be the  176  'm(t)  cm  v (t) m  Fig.  I-B.5:  wherei  Simplifed c i r c u i t  , i s a past h i s t o r y eq,k  constant  i n the time domain.  c u r r e n t source,  resistance, ieq.k  f k,c  =  e  ( t  "  A t )  The e q u i v a l e n t c i r c u i t the p a st h i s t o r y values process are  and Rv i s an e q u i v a l e n t  these  +  e  k,e  ( t  "  i n Figure  of equation  A t  )  bk^^/R],  I-B.5 can be s o l v e d  (I-B.37).  are zero i f the i n i t i a l  +  (I-B.37) recording  At the beginning  of the  c o n d i t i o n s a t the ends of the l i n e  zero. If these  initial  c o n d i t i o n s are not zero,  g i v e n by p r e - t r a n s i e n t s t e a d y - s t a t e , tions  ( i . e . , trapped  charge),  t h a t i s , i f they are  or u s e r - s u p p l i e d dc i n i t i a l  the p a s t h i s t o r y c u r r e n t sources  condi-  can be  determined as f o l l o w s . The e q u i v a l e n t c i r c u i t Figure  I-B.6  f o r steady-state  c o n d i t i o n s i s shown i n  177  Fig.  I-B.6:  Equivalent c i r c u i t  Consider  f o r example  f o r steady-state  node k.  V  k  and I  voltages  and c u r r e n t s known from the i n i t i a l  system.  We  must now  find  the s t e a d y - s t a t e  e  k,i  <  • Ek,i  b  k,i  <  »  fm From equation  from equation z  eq  =  k  a r e the phasor  steady-state  s o l u t i o n of the  e q u i v a l e n t s of  \ , i  >• F m  <  (I-B.14) E  but  conditions.  Z  !k e q  =  z  (I-B.29)  + Z  Q  k  2  + ... + Z ,  (I-B.38)  m  where, Z z  =  D  .  k  k =  _  E  K  =  therefore,  (I-B.39) and  _  Z I Q  m  k  + (  i  •  (I-B.40)  jco + P i  1  Introducing  (I-B.39)  Q  (I-B.40) i n t o  _  I Zi) I 1=1  k  (I-B.14)  178 E  k,i  =  i *k  z  Partial history From e q u a t i o n  k >  i  are o b t a i n e d as f o l l o w s :  (I-B.10) %  but  sources B  =  from equations  l Fm-'  (I-B.21) and  l  A  A  =  A  (I-B.22)  l , l + l , 2 + •••  +  A  A  1 > N  where, A-, 1  Introducing  1  1  •  k. 1 — j co + p^  =  (I-B.42) i n t o _  (I-B.10),  _  *k  =  F  -]C0C  e  m  m  A  l,i>  '  and,  Finally, F  m  F  m  =  i s obtained v  m  +  z  eq  from e q u a t i o n  •'-m  The phasor q u a n t i t i e s i n the time =  | E , i | cos (cut +  arg(E  k f i  ))  b^ft)  =  |B  arg(B  k / i  ))  f (t)  =  | F  m  k  k/i  m  | cos (cot +  | cos (cot + a r g ( F ) ) .  When the i n i t i a l trapped  domain are given by  e ,i(t) k  charge,  the i n i t i a l  o b t a i n e d as b e f o r e by  (I-B.4)  m  c o n d i t i o n s are s u p p l i e d by the user, i . e . , v a l u e s o f e -j_, b k/  s e t t i n g co =  A s i m i l a r procedure  k  ^ and  0.  i s f o l l o w e d f o r node  m.  f  m  can  be  APPENDIX I-C  USER'S GUIDE FOR THE VOLTAGE AND CURRENT PROFILE OPTION OF THE EMTP, AND FOR THE OUTPUT DISPLAY PROGRAM  179  180 The  routines  that evaluate  the e l e c t r i c a l p r o f i l e o f a l i n e have  been implemented as p a r t of the frequency-dependence v e r s i o n o f UBC's EMTP The  o b j e c t code of t h i s  LUI:MDTRAN3.0.  v e r s i o n i s a v a i l a b l e t o UBC users  The i n t e r c o n n e c t i o n with  the EMTP r e q u i r e d  90 FORTRAN statements, and the s u p p o r t i n g less  subroutines  under the name approximately  required  slightly  than 660 FORTRAN statements. A typical  call  to t h i s  $ RUN LUI:MDTRAN3.0 0=PAR.INT The  v e r s i o n of the EMTP i s shown below  1=PAR  2=-2 3=-3 4=-4 5=TRAN.DA  only d i f f e r e n c e s between t h i s  6=-X  command and the standard  frequency-dependence command are l o g i c a l u n i t s 0 and 3. Logical unit 0 contains  the constants  k^ and 3^ from the r a t i o n a l -  f u n c t i o n s approximation of A-|(co) f o r the l e n g t h of one of the segments i n t o which the l i n e has been s u b d i v i d e d .  F i l e PAR.INT does not i n c l u d e  the approximation f o r z ( w ) s i n c e i t i s the same used f o r the complete c  line  (logical unit 1 s t i l l  file  i s requested The  contains  the same parameters whether the p r o -  or n o t ) .  currents  and v o l t a g e s  are w r i t t e n i n f r e e format  computer c o s t s and to ease the manipulation involved)  into logical To  of the l a r g e amounts of data  u n i t 3.  invoke the " p r o f i l e "  been i n t r o d u c e d  ( t o save  i n t o the l i n e c a r d  option,  the f o l l o w i n g m o d i f i c a t i o n s have  (frequency-dependence o p t i o n )  in file  A6 12 12 3 i s t 7  A6 e 9  Freq.  (Hz)  E62  | IDELT j  NODE m  1 NINT j  NODE NAMES NODE k  ] MODEL j  I TYPE |  TRAN.DA ( l o g i c a l u n i t 5)  | 11  12 12 12  10 11 U 13 14 15 IS 17 ia 19 20 21 22 2] 21 25 26 27 28 29 X 31 32 33 31 35 36 37 36 39 10 41 12 13 44 15 46 47 4B 49 so 5', 52 53 51  56 57 5* 5S 60 si 62 63 516556  1570 pi n 73 71  77 It 73  15  -  i  181 The a d d i t i o n a l parameters NINT  > 0  number of i n t e r m e d i a t e  are e x p l a i n e d  below:  v o l t a g e s and c u r r e n t s .  The minimum i s 1  and the maximum i s 99. = 0  disables "profile"  IDELT > 0  option,  number of times the i n t e r n a l  At i s to be segmented.  example, IDELT=2 reduces the i n t e r n a l time step IDELT = 0  the i n t e r n a l  At' t o At'/2.  At i s determined a u t o m a t i c a l l y by the program.  (Note that even i f IDELT > 0, the v o l t a g e s and c u r r e n t s w i l l p r i n t e d with  For  s t i l l be  the same number of time steps d e f i n e d i n the main  program).  tmax  E10.6  i  ? 3  i  I MAX  *s)  13  E10.6 6  7  B  9 10  ii I ? 13 U lb IC 17 18  19 70?i p ?3  ISKIP  At(s)  I PUNCH  A time saving o p t i o n has been i n c l u d e d i n the time c a r d ,  ?6 ?7  Af  O  (Hz)  *mox  | cr o iA  tHz)  / ) CL  E80  13  13  §5  £  73 1 0  31 3 ?  33  [1  11  E80  3415 X J7 36" 33 it ii  E  17 43 444b 45 47  ^0  80 y si  EE  S7  0  SO ' 0  11 11  Ll il  1  X  7*  7(1  77  1  -  f  T !!  ISORT = 0 o r 1  A l l v o l t a g e s and c u r r e n t s w i l l be w r i t t e n on l o g i c a l  unit  3. ISORT = 2  Only the i n t e r m e d i a t e  voltages w i l l  be w r i t t e n on  logical  currents, w i l l  be w r i t t e n on  logical  u n i t 3. ISORT = 3  Only the i n t e r m e d i a t e u n i t 3.  A sample of the d i f f e r e n t i n p u t f i l e s John Day - Lower Monumental l i n e . requested,  the l i n e  will  i s shown below f o r the  Only the i n t e r m e d i a t e  voltages are  be segmented i n t o 9 s e c t i o n s , and the i n t e r n a l  182 time s t e p w i l l be made two times s m a l l e r than the i n t e r n a l l y  calculated  time s t e p .  1 INTERMEDIATE PARAMETERS 2 .93322E-05.020 3 -1SWIA TWOA 4 -2SWIB TWOB 5 -3SWIC TWOC 6 7 ONEA SWIA .0005 8 ONEB SWIB .0003 9 ONEC SWIC .0006 10 92TW0A 1.0 11 92TW0B 1.0 12 92TW0C 1.0 13 14 140NEA 1 1.0 60. 15 140NEB 1 1.0 60. 16 140NEC 1 1.0 60. 17 18 TWOA TWOB TWOC 19 20 21 End of F i l e  Sample i n p u t parameters  To ease  file  ARRESTER  1.0 1.0 1.0  TEST. O  3PHASE MON LINE  2.6 2.6 2.6  60.00 60.00 60.00  -19 -1 9 -1 9  2.0 2.0 2.0  -10.8 -130.8 109.2  of the p r o f i l e  2  O.O 0.0 0.0  o p t i o n of the EMTP  the m a n i p u l a t i o n of the l a r g e amounts of output  generated d u r i n g p r o f i l e  calculations,  a separate program reads the output  data from l o g i c a l u n i t 3 ( o r i g i n a l l y w r i t t e n i n f r e e format and thus meaningless  under MTS LIST OR COPY commands) and w r i t e s o n l y the p a r t of  the output requested by the u s e r .  The o b j e c t code o f t h i s  auxiliary  program i s a v a i l a b l e under the name LUI:OUTINTL.O. A t y p i c a l run command i s shown below $ RUN LUI:OUTINTL.O The  3=-3  5=PAR.0UT  6=-A  7=-B  requested v o l t a g e s are w r i t t e n on l o g i c a l u n i t 6 and the  c u r r e n t s on l o g i c a l u n i t 7. P i l e PAR.OUT c o n t a i n s the output c o n t r o l parameters.  An example  183 is  shown below  (a sample  control f i l e  i s a l s o a v a i l a b l e on "READ ONLY"  b a s i s under t h e name LUI:PAR.0UT).  End  1 2 3 4 5 6 7 8 of  +0 -1 1 2 3 2  IALLV IALLI (BRANCH.SECTION) (VOLTAGES) (BRANCH,SECT ION) (VOLTAGES) (BRANCH,SECTION) (VOLTAGES) (BRANCH,SECTION) (VOLTAGES) (BLANK CARD) (VOLTAGES) (BLANK CARD) (CURRENTS)  1 1 1 9  File  Sample c o n t r o l parameters  file  The parameter IALLV c o n t r o l s  the output mode:  IALLV < 0  No v o l t a g e s w i l l be p r i n t e d .  IALLV = 1  A l l i n t e r m e d i a t e v o l t a g e s w i l l be p r i n t e d .  IALLV = 0  Only the v o l t a g e s s p e c i f i e d by the user w i l l be p r i n t e d . When IALLV = 0, the output i s s e l e c t e d by i n d i c a t i n g the  branch(es) and i n t e r m e d i a t e s e c t i o n ( s ) f o r which the v o l t a g e s are d e s i r e d . The branch number  i s determined by the order i n which the f r e q u e n c y  dependent l i n e s were s p e c i f i e d t h i s example). B of l i n e the  I f , f o r example,  1 w i l l be branch number  branch number would be 6.  specified  i n the i n p u t f i l e  t h i s way i s 300.  two three-phase l i n e s are s t u d i e d , 2.  I f phase C of l i n e  phase  two i s d e s i r e d ,  The maximum number of v o l t a g e s t h a t can be  A blank card s i g n a l s  v o l t a g e s . I f IALLV i s l e s s than zero o r 1, inserted,  f o r the EMTP (TRAN.DA i n  the end of the s p e c i f i e d  a b l a n k c a r d must a l s o be  t o s i g n a l the end o f u s e r s p e c i f i e d  voltages.  184 IALLI i s analogous rents.  Note t h a t i f ,  be i n s e r t e d a f t e r  t o IALLV, but used f o r the output of c u r -  f o r example IALLV = IALLI = 1 , two blank cards must  IALLI.  A sample output from file  shown above, i s l i s t e d  l o g i c a l u n i t s 6 and 7 , u s i n g the c o n t r o l  below.  names of the l i n e nodes (as used branch i s p r i n t e d  Note t h a t a r e f e r e n c e t a b l e with the  i n TRAN.DA), and the code number of each  f o r easy r e f e r e n c e ( a l s o u s e f u l to v e r i f y  put requested i s the output o b t a i n e d ) .  t h a t the out-  VOLTAGES  (BRANCH,SECTION)  REFERENCE TABLE LINE 1  FROM BUS SWIA  TO BUS TWOA  LINE 2  FROM BUS SWIB  TO BUS TWOB  LINE 3 FROM BUS SWIC TO BUS TWOC TIME 2 1 1 1 0.0 0. 0 -0.0 0.90589439E-05-0. 67144878E 76-0.6714487BE0.181 17888E-04-0. 20728855E 76- 0.20728855E0. 27 176832E-04-0. 71533231E 77- 0.71533231E0.36235776E-04-0 20253559E 77- 0.20253559E0.45294720E-04-0. 83561079E 78- 0.83561079E0.54353664E-04 0. 0 -0.0 0.63412607E-04 0. 0 -0.0 0.72471551E-04 O. O -0.0 0.81530495E-04 0 0 -0.0 0.90589439E-04 0 0 -0.0 0.99648383E-04 O. 0 -0.0 0. 10870733E-03 0. 0 -0.0  0.45294720E 0.46200614E 0.47106508E 0.48012403E 0.48918297E 0.49824192E 0.5073008GE O . 5 1635980E  -03 -03 -03 -03 -03 -03 -03 -03  0 . 17574351E -02 o. 17664941E -02 0. 17755530E -02 0. 17846120E -02 0. 17936709E -02 0. 18027298E -02 0. 18 1 17888E-02 0. 18208477E -02 o. 18299067E -02 0. 18389656E -02 0. 18557322E -02 o. 18650575E -02 0. 18743827E -02 0. 18837080E -02  3 1 -0.0 76-0.67144878E 76- 0.20728855E 77- 0.71533231E 77- 0.20253559E 78- 0.83561079E -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0  0.0 0.0 0.0 0.0 0.64531823E-01- 0.12906365E+00 0.13092492E+CO- 0.26216827E+00 O.13217002E+OO- 0.27711089E+00 0.97277730E-01- 0.30802471E+00 0.44892915E-01- 0.35641999E+00 0. 1 1844B03E-02-0.39645930E+OO 0 0 0 0 0 0 0 0 0 0 0 0 0 0  91578177E+00-0. 91442024E+00-0. 91304797E+00-0. 91166498E+O0-0. 91027129E+00-0. 90886692E+00-0. 90745188E+00-0. 90602620E+00-0. 90458989E+0O-0. 90314298E+O0-0. 90043776E+00-0. 89891700E+00-0. 89738506E+00-0. 89584197E+00-0.  11345900E+0011006518E+0010667015E+0010327395E+CO99876626E-0196478212E-0193078746E-0189678265E-0186276807E-0182874410E-0176574874E-0173069876E-0169564040E-0166057407E-01•  Sample output from l o g i c a l u n i t 6.  2 9 -0.0 •76-0.67144878E-76 •76-0.20728855E-76 •77-0.71533231E-77 •77-0.20253559E-77 •78-0.83561079E-78 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0  •0.0 -0.0 -0.0 -0.0 0.64531823E-01-0.0 0.13092492E+0O-0.O 0.13217O02E+0O-0.0 0.97277730E-01-0.0 0.44892915E-01-0.0 O.11844803E-02-0.0 0.80320761E+00 -0.0 0.80522853E+00 •0.0 0.80724016E+00 •0.0 O.80924245E+00 0.0 0.81123540E+O0 •0.0 0.8132 1896E*00 -0.0 0.81519312E+0O -0.28648792E-01 0.81715785E+00 •0.12252582E+00 0.8191 1312E*0O•0.19735267E+00 0.82105892E+00 -0.23084061E+00 0.82463582E+00 -0.24633897E+00 0.82661063E+00 -0.24802598E+00 O. 82857528E*00 •0.24856349E+00 0.83052975E+00 -0.24852947E+00  Output d i s p l a y  program.  186  CURRENTS  (BRANCH,SECTION)  REFERENCE TABLE  Sample output  from  LINE 1  FROM BUS  SWIA  TO BUS  TWOA  LINE 2  FROM BUS  SWIB  TO BUS  TWOB  FROM BUS SWIC TO BUS LINE 3 NO CURRENT OUTPUT REQUESTED  TWOC  logical  unit  7.  Output d i s p l a y  program.  APPENDIX II-A  SKIN EFFECT CORRECTION FOR ROUND CYLINDRICAL CONDUCTORS  187  188 As the frequency o f the c u r r e n t c i r c u l a t i n g increases,  the c u r r e n t d e n s i t y i n c r e a s e s near  conductor.  i n a conductor  the s u r f a c e of the  T h i s phenomenon i s known as s k i n e f f e c t and i t a f f e c t s the  r e s i s t a n c e and i n t e r n a l i n d u c t a n c e i n l a r g e power c o n d u c t o r s . r e s i s t a n c e i n c r e a s e s w i t h the frequency while the i n t e r n a l  The  inductance  decreases. The  formulas shown below permit the c a l c u l a t i o n of s k i n  for tubular conductors.  Stranded conductors can be approximated  conductors of the same c r o s s - s e c t i o n a l a r e a . can be approximated core i s n e g l i g i b l e  Steel reinforced  effect as s o l i d  conductors  as t u b u l a r conductors when the i n f l u e n c e of the s t e e l (note t h a t s o l i d  conductors are o n l y a p a r t i c u l a r  case  of t u b u l a r c o n d u c t o r s ) . The R  s  +  int  J a ) L  R  formula used _  ,1_ ~ 2 :  d c  f o r the c a l c u l a t i o n of s k i n e f f e c t i s 2\  m  ~  S  (ber mr + j b e i mr) + 0 (ker mr + j k e i mr) (ber' mr + j b e i ' mr) + 0 (ker' mr + j k e i ' mr)  with _  b e r ' mq + j b e i ' mq k e r ' mq + j k e i ' mq '  and, R R  =  s  dc  Li t  ac r e s i s t a n c e  =  ^  =  i n t e r n a l i n d u c t a n c e ( s k i n e f f e c t i n c l u d e d ) i n H/km  n  c  resistance  ( s k i n e f f e c t i n c l u d e d ) i n °./km i n ^/km  r  =  o u t s i d e r a d i u s o f conductor  q  =  i n s i d e r a d i u s of conductor  s  =  ^ • r  and, (mr)  2  =  k  — 1 - s  2  189  Fig.  II-A.1:  Tubular  conductor.  The m o d i f i e d B e s s e l f u n c t i o n s can be c a l c u l a t e d approximations.  A s u b r o u t i n e t h a t uses the above formulas  L i n e Constants Program. effect calculations the  The same s u b r o u t i n e i s used  in this  thesis.  A typical call  form CALL SKIN (S, RDC,  F, RS, XI)  where S  =  q/r  RDC  =  dc r e s i s t a n c e i n fi/km  F  =  frequency  i n Hz  with  polynomial  i s used  i n the  f o r a l l the s k i n to t h i s s u b r o u t i n e has  190 RS  =  ac r e s i s t a n c e  in  XI  =  ac r e a c t a n c e  i n ft/km  A sample c a l c u l a t i o n of R/R  dc  and  L  i  Q/km  / i L  n  t  n  t  /  dc  f  o  r  s  =  0  ,  ft/km i s shown i n F i g u r e II-A.2  10' =  FREQUENCY  Fig.  II-A.2:  (HZ)  V a r i a t i o n of the r e s i s t a n c e and i n d u c t a n c e due to s k i n e f f e c t .  internal  5  a  n  d  R  dc  =  0 , 0 1  APPENDIX I I - B  CARSON'S CORRECTION TERMS FOR EARTH RETURN EFFECT  191  192 Carson's c o r r e c t i o n  terms account  f o r the f a c t  does not behave as an i n f i n i t e and p e r f e c t l y c o n d u c t i n g  t h a t the e a r t h plane  return currents.  Carson's formula  systems s t u d i e s .  I t i s based on the f o l l o w i n g assumptions:  (a)  The  conductors  (this  permits  plane p e r p e n d i c u l a r to the (b)  The  e a r t h has  i s normally a c c u r a t e enough f o r power  are long enough so t h a t three d i m e n s i o n a l  can be n e g l e c t e d  uniform  The  conductors).  c o n d u c t i v i t y and  spacing between the conductors  so t h a t p r o x i m i t y e f f e c t s can be  can be are  represented  $ = 0.  i s much l a r g e r  an  than  their radius,  ignored.  f o r mutual impedance), and  1,  as  parallel.  Carson's c o r r e c t i o n terms depend on the angle impedance and  end-effects  the s o l u t i o n of the f i e l d problem on a  i n f i n i t e p l a n e , to which the conductors (c)  f o r ground  on  $($ = 0 f o r s e l f  the parameter a,  K  where (II-B.1 )  a  wi th  D  2  D  D  i,k  ( i n m) ^  n  m  )  for self f°  r  impedance  mutual impedance  f  frequency  P  e a r t h r e s i s t i v i t y i n fi«m.  i n Hz  193  h;  images  Fig.  II-B.1:  Tower  The c o r r e c t i o n  geometry.  term f o r the r e s i s t a n c e AR becomes i n f i n i t e  w h i l e AX/to = AL goes t o z e r o . is  f i n i t e , AR and Ax a r e very s m a l l .  conductor  (p  = 0 ) , AR and Ax become  Carson's i n f i n i t e AR  I f the e a r t h r e s i s t i v i t y  =  In the l i m i t ,  when f -»• °°,  i s very low and f  i f earth i s a perfect  zero.  s e r i e s when a < 5 can be w r i t t e n as,  4to-10~ {^-  AX  4  -b]_a« cosi^  4to- 10~ {-2-(0.6159315-lna) 4  +b]_a- coscj)  +b2[ (c2~lna) acos2<f>+4>asin2<t>] 2  2  -d2a cos2cf> 2  +b3a3coscf>  (C4~lna) a cos4<f)+<j>a sin4cj>]  -b4[ +b5a-'cos5cJ)  4  4  •f-bgC (c6-lna)a cos6<f>+cJ>a sin6cf>]  -dga^cosScj)  +b7a 'cos7c|>  + b7a^cos7<f)  -dga^cosScJ)  -bg[ (cg-lna) a^cos8<f>+<f>a^sin8<}>]  6  - .. .}  6  (II-B.3)  + .. .}  (II-B.4)  194 Each 4 s u c c e s s i v e terms form a r e p e t i t i v e p a t t e r n .  The c o e f f i c i e n t s b^,  C i and d i a r e c o n s t a n t s , which are obtained from the r e c u r s i v e  sign . b. = b. „ with the s t a r t i n g value l i-2 i(i+2)  . <  b, = - — 1 6  f o r odd s u b s c r i p t s .  1  . b = — <- 2 16 cl  = c  d • l  =  T  4  - + -T- + r ^ r with the s t a r t i n g i-2 l i+2  value c  f o r even  . b. , l  1,2,3,4; s i g n = -1 f o r i = 5,6,7,8, a > 5, the f o l l o w i n g cos<j>  AR'  a A X '  =  subscripts,  = 1 .3659315,  w i t h s i g n = ±1 changing a f t e r each 4 s u c c e s s i v e terms  For  formulas:  cos 6  finite  2 cos2§  cos3<j>  3cos 5<j>  etc.,).  s e r i e s i s used: 3cos5<;  45cos7(j)  a-"  a  a' cos3d  ( s i g n = +1 f o r i =  4LO-  10~  4  7  (II-B.4)  4co-10"'  45cos7<f>  (II-B.5)  The t r i g o n o m e t r i c f u n c t i o n s are c a l c u l a t e d d i r e c t l y h  cosd  ik  i  + k D  . , sm<j>  n  and  i k  and f o r h i g h e r terms i n the s e r i e s ,  x  ik  =  from the geometry, ik D  i k  from the r e c u r s i v e formulas  a^cos (i<j>) = [a^^-cos ( i-1) ())• cos<f) - a -'-sin (i-1) (j)* sinij)]. a 1-  a^cos (i<f>) = [ a  For  1  '''cos (i-1) (}>• sin<\> + a  power c i r c u i t s  1  "'"sin (i-1) <j>* cos<j)]-a  a t power frequency, a reasonable approxima-  t i o n can be o b t a i n e d i f o n l y two terms i n e q u a t i o n higher frequencies,  low r e s i s t i v i t y ,  (II-B.2) are used.  For  and wider s p a c i n g s , more terms need  to be taken i n t o account as the parameter a becomes l a r g e r .  The Line Con-  s t a n t s Program uses these formulas to account f o r ground r e t u r n  effects.  195 For the purposes o f t h i s Program  thesis project,  the s e c t i o n o f the Line Constants  t h a t c a l c u l a t e s Carson's terms has been r e w r i t t e n as a s u b r o u t i n e .  A typical call  to t h i s  s u b r o u t i n e has the form  CALL CARSON (HAV, DM, F, RHO, DRS, DRM, DXS, DXM) where, HAV  =  average h e i g h t i n m.  DM  =  geometric mean d i s t a n c e between c o n d u c t o r s  RHO  =  earth r e s i s t i v i t y  F  =  frequency i n Hz.  i n °.«m.  DRS, DRM, DXS, DXM are the r e s u l t i n g ft/km.  s e l f and mutual c o r r e c t i o n  terms i n  APPENDIX I I - C  SIMULATION OF THE JOHN DAY-LOWER MONUMENTAL TRANSMISSION LINE USING THE LOW-ORDER APPROXIMATION PROGRAM  196  197 BPA's John Day Figure 3 . 1 ,  to Lower Monumental 500 KV t r a n s m i s s i o n l i n e  p a r t I) has been modeled w i t h the low-order  r o u t i n e s developed i n p a r t I I of t h i s In o r d e r to i l l u s t r a t e  thesis  approximation  project.  the e f f e c t of ground  wires i n the  e v a l u a t i o n of the e q u i v a l e n t l i n e c o n f i g u r a t i o n , | A-|(to)|  and | Z (oo)| c  w i t h segmented or T connected ground wires (see f o o t n o t e , page 28) shown i n graphs  II-C.1 and  and | Z (co)| when the ground c  It i s i n t e r e s t i n g  II-C.2.  are  Graph II-C.3 and II-C.4 show | A-] (co) |  wires have been removed. to note i n graphs  II-C.3 and  II-C.4, that the  s l i g h t l y b e t t e r approximation o b t a i n e d f o r the r e f e r e n c e l i n e 1 .3, p a r t  (see  II) i s p r o b a b l y due  to the f a c t that the tower  (see F i g u r e  c o n f i g u r a t i o n of  the r e f e r e n c e l i n e i s h o r i z o n t a l , while the c o n f i g u r a t i o n of BPA's l i n e i s triangular. correction  T h i s would a f f e c t terms  the accuracy of the e s t i m a t i o n of  a t h i g h f r e q u e n c i e s , where the d i f f e r e n c e between  c o n s i d e r i n g an average h e i g h t or the h e i g h t of the i n d i v i d u a l  conductors  i s more marked. Graphs II-C.5 t o II-C.7 show the open and s h o r t responses of t h i s  Carson's  l i n e when there are no ground w i r e s .  circuit  •198  Graph II-C.1 :  A-| magnitude f u n c t i o n . Segmented ground w i r e s , R^ given. (a) P o s i t i v e sequence (b) Zero sequence.  c  199 9 5 0 TJ 900  y  -1  FREQUENCY  (HZ)  9 5 0 -q y LU  900  -i  8 5 0 -3  FREQUENCY  (HZ)  (b) Graph II-C.2:  Z magnitude f u n c t i o n . Segmented ground w i r e s , R$ given. (a) P o s i t i v e sequence (b) Zero sequence. c  c  Graph II-C.3:  A-] magnitude f u n c t i o n . (a) P o s i t i v e sequence  No ground w i r e s , R d (b) Zero sequence.  c  given,  Graph II-C.4:  Z magnitude f u n c t i o n . (a) P o s i t i v e sequence c  No ground w i r e s , R^,-. g i v e n , (b) Zero sequence.  0.060  t  Graph II-C.5:  O/C r e s p o n s e . No ground w i r e s , R^ g i v e n . (a) P o s i t i v e sequence (b) Zero sequence. c  O.OJO H .  y  0.009 -  11  LU  0.008 4  Graph II-C.6:  S/C response. Mid to h i g h f r e q u e n c i e s . No ground w i r e s , R$ g i v e n , (a) P o s i t i v e sequence (b) Zero sequence. c  204  0.150  -i  II  0. J20 H  CE  Q.  m  0.030  CO O Q.  0.060  LU CO z  o CL CO LU CC  0.030 ^  (_) \ CO  -0.000 JO"  ~T  2  I—I  IIIIII  ,  3 4 6 JO  1  1—I I I I I I I  ,  I  1—I I II I 11  2 3 4 6 10" 2 3 4 6 J FREQUENCY (HZ)  ~l  1—I I I I I IT  2  3 4 6  JO  (a)  0.150 CL CJ  II  0.3 20 H  cr  CD o  0.090  LU CO  o  §  0.060  o  CL CO  0.030  U tn  -0.000 1 10"  r I I IIIIII , 2 3 4 6 JO  I  1 I I I I I I I ,  1  1 IIIIIII  2 3 4 6 JO 2 3 4 6 FREQUENCY (HZ)  1  J  1 I I I I I IT  2 3 4 6  JO  (b) Graph II-C.7:  S/C r e s p o n s e . Low f r e q u e n c i e s . No ground w i r e s , R^c g i v e n . (a) P o s i t i v e sequence (b) Zero sequence.  APPENDIX II-D  USER'S GUIDE TO THE LOW-ORDER APPROXIMATION PROGRAM  205  206 The  sequential f i l e  LO.O c o n t a i n s  low-order a p p r o x i m a t i o n r o u t i n e s . to  UBC's MTS  the o b j e c t code of the  LO.O i s a v a i l a b l e with  READ ONLY s t a t u s  users.  A t y p i c a l run command i s shown below $ RUN LUI: LO.O  5 = DA.LO  6 = -6  7 = -7  11 = -11 DA.LO c o n t a i n s  8 = -8  9 = -9  12 = -12  the i n p u t data and the output  c o n t r o l parameters.  L o g i c a l u n i t s 6 through 12 c o n t a i n output  information.  file  has been p e r m i t t e d  i s shown below (an example data  ONLY b a s i s .  1 2 3 4 5 6 7 8 9  10 11 12 13 14 15 16 17 18 of  End  This f i l e  information  or a semi-colon Line is  20.  file  A typical  data  on a READ  under the name LUI: DA.LOT)  3 60.00 500 . 0 .OOD00003 100.0 .88080000-03 3.330800D-03 .01330000000-06 .00836 10000-06 .026190000 .02643000 .19740000 .01 10 10 6 4 5 4 0 5 4 3 .5.5 1 1 1 File  Sample i n p u t parameters  The  can be read  10 = -10  NUMBER OF PHASES FREQUENCY OF PARAMETERS LINE LENGTH CONDUCTANCE EARTH R E S I S T I V I T Y L K 6 0 H Z ) . L0I60HZ1 C 1 . CO RDC , RF ..RO FM I N NO OF DEC. NO OF POINTS/DEC NUMBER OF POLES IN ZCO AND ZC1 NUMBER OF POLES IN A10 AND A11 S H I F T I N G OF A10 AND A11 (0=N0) S H I F T I N G OF ZCO AND ZC1 (0=N0) ERROR (%) AT LINE FREQUENCY FOR ZC AND WRITE R(W), L(W) (1=YES 0=N0) WRITE ZC(W), *ZC(W)* (1=YES 0=N0) WRITE A1. *A1* ( 1 ^ Y E S 0=N0)  A1  file.  from t h i s f i l e  i s read  i n f r e e format, t h e r e f o r e a blank  can be used as d e l i m i t a t o r between q u a n t i t i e s . 1 i s the number of phases; the minimum i s 2 and the maximum  The v i o l a t i o n of these  limits  i s not d e t e c t e d  program, but the system w i l l d i s c o n t i n u e e x e c u t i o n  i n t e r n a l l y by the  due t o a r r a y  overflow.  207 L i n e s 2 through 8 are s e l f - e x p l a n a t o r y . r e s i s t a n c e i s not known r e q u i r e d , a f l a g value  ( l i n e 8) and the 'RDC not g i v e n ' o p t i o n i s of 0.0 must be i n t r o d u c e d  Line 9 contains  the s t a r t i n g  frequency  dependent response of the parameters i s to be L i n e 10 c o n t a i n s the d e s i r e d frequency Line  the number of poles  c  these  limits,  a t which the  R^ . c  frequency-  evaluated.  band.  11 c o n t a i n s  10 and the minimum  i n the p l a c e of  the number of decades and p o i n t s per decade o f  of Z (to) f o r zero and p o s i t i v e sequences. is  Note t h a t i f the dc  i s 1.  r e q u i r e d i n the s i m u l a t i o n  The maximum number  No warning messages d e t e c t  but an a r r a y o v e r f l o w  w i l l result  permitted  the v i o l a t i o n  and the e x e c u t i o n  of  will  stop. Line propagation 2.  1 2 i s the number of poles  f o r the a p p r o x i m a t i o n of the  f u n c t i o n A-| (to); the maximum number i s 10 and the minimum i s  As i n the case of Z (to) no warning messages are p r o v i d e d , c  limits  are v i o l a t e d , e x e c u t i o n Line  A-|(to).  loops.  Values g r e a t e r  o p t i o n f o r the approximation of  o p t i o n and matching a t l i n e  frequency  w i l l be  than zero s p e c i f y the maximum number of s h i f t i n g  I f an improvement of more than 1% i n the area between | A-](to)|  and | B(to)| shifting  stop.  13 c o n t r o l s the s h i f t i n g  Zero d i s a b l e s t h i s  assumed.  will  but i f the  i s not o b t a i n e d ,  process  stops  or i f the maximum number of loops i s met, the  and c o n t r o l i s d i r e c t e d t o the next p a r t of the  program. L i n e 14 c o n t a i n s  analogous i n f o r m a t i o n as l i n e  13 f o r the  a p p r o x i m a t i o n of Z ( t o ) . c  Line and  15 i s the e r r o r i n (%) between the r e a l  (obtained  from R,L  C g i v e n by the user) and approximated f u n c t i o n s Z (co) and A-] (to) a t  the l i n e frequency  c  (line  2).  Note t h a t the same e r r o r l e v e l w i l l be  assumed f o r both zero and p o s i t i v e sequence.  208 If the s h i f t i n g disabled. set  I f no s h i f t i n g  option  i s used,  the matching  procedure i s  and no matching  are d e s i r e d ,  this  e r r o r can be  override  the matching  to a l a r g e value ( f o r example 5 0 % ) . This w i l l  procedure. Line 7.  16 c o n t r o l s  the i n f o r m a t i o n  to be w r i t t e n i n l o g i c a l  unit  An example i s shown below.  FREQUENCY 0. 1000000E -01 0. 1258925E -01 0. 1584893E -01 0. 1995262E -01 0. 25 1 1886E -01 0. 3162278E -01 0. 3981072E -01 0 5 0 1 1 8 7 2 E -01 0. 6 3 0 9 5 7 3 E -01  1 2 3 4 5 6 7 8 9 10  Sample output from  Line Z (io). c  17 c o n t r o l s  RO (OHM S J 0. 262 1959E -01 0. 2622725E -01 0 .2623690E -01 0. 2624904E -01 0. 2626431E -01 0..2628355E -01 0 .2630776E -01 0. 2633823E -01 0..2637658E -01  LO (MH) 0. 5922672E+01 0. 5853623E+01 o. 5784579E+01 0 .5715537E+01 0 .5646500E+01 0. 5577470E+01 0 .5508444E+01 0., 5439424E + 01 0 . 537041 1E + 01  u n i t 7.  the output of the approximation process of i n l o g i c a l u n i t 9 and zero sequence  u n i t 8.  P O S I T I V E SEQUENCE SURGE IMPEDANCE *ZC 1 * FREQUENCY ZC 1 o. 9341646E+03 0 9339930E+03 0 .1000000E-01 0. 9338927E+03 0 .1258925E-01 0. 9340593E+03 0. 9338926E+03 0. 9337339E+03 0 .1584893E-01 0. 9334824E+03 0. 9336287E+03 0 .1995262E-01 0..9330845E+03 o 9332112E+03 O .2511886E-01 0 .9324553E+03 0. 9325514E+03 0 .3162278E-01 0. 9314622E+03 0. 9315105E+03 0 .3981072E-01 0 .9298981E+03 0. 9298727E+03 0 .5011872E-01  Sample output from  1 2 3 4 5 6 7 8 9 10  logical  L1 (MH) 0. 8808493E+00 0. 8808493E+00 0. 8808493E+00 0. 8808486E+00 0. 8808486E+00 0. 8808493E+00 0. 8808493E+00 0. 8808493E+00 0. 8808486E+00  P o s i t i v e sequence i s w r i t t e n  in logical  1 2 3 4 5 6 7 8 9 10  RI (OHMS) 0. 2619000E -01 0. 26 19000E -01 0. 2619000E -01 0. 26 19000E -01 0. 2619000E -01 0. 26190COE -01 0. 2619000E -01 0. 2619000E -01 0. 2619000E -01  logical  ERROR (%) -0. 1836608E- 01 -0. 1783459E- 01 -0. 1699431E- 01 -0. 1566774E- 01 -0. 1357827E- 01 -0. 1029905E- 01 -0. 5182284E- 02 0. 2728810E- 02  u n i t 9.  ZERO SEQUENCE SURGE IMPEDANCE ERROR (%) * ZCO* FREQUENCY ZCO 0.9363422E+03 -0.3129076E-02 0.1000000E-01 0. 9363715E+03 0.9363251E+03 -0.4958766E-02 0.1258925E-01 0. 9363715E+03 0.9362980E+03 -0.7857896E-02 0.1584893E-01 0. 9363715E+03 0.9362550E+03 -0.1245087E-01 0.1995262E-01 0. 9363715E+03 0.9361868E+03 -0.1972561E-01 0.2511886E-01 0. 9363715E+03 0.9360790E+03 -0.3124371E-01 0.3162278E-01 0. 9363715E+03 0.9359083E+03 -0.4946962E-01 0.3981072E-01 0. 9363715E+03 0.9356385E+03 -0.7828299E-01 0.5011872E-01 0. 9363715E+03  Sample output from  logical  u n i t 8.  209 In of  the output f o r | Z (co)| , zero sequence, ZCO i s the magnitude c  the c h a r a c t e r i s t i c impedance generated from the e q u i v a l e n t l i n e  u r a t i o n , *ZC0* i s the r a t i o n a l - f u n c t i o n s approximation of | Z (co)| c  configand  ERROR (%) i s the percentage e r r o r between them. Line 18 c o n t r o l s line  17.  magnitude  respectively.  ZERO SEQUENCE FREQUENCY O.IOOOOOOE-01 0.1258925E-01 0.1584893E-01 0.1995262E-01 O.2511886E-01 O.3162278E-01 0.3981072E-01 0.5011872E-01  Sample output from l o g i c a l  1 2 3 4 5 6 7 8 9 10  i n a s i m i l a r way as i n  U n i t s 10 and 11 c o n t a i n the zero and p o s i t i v e  functions,  1 2 3 4 5 6 7 8 9 10  the output of | A-) ( w)|  PROPAGATION FUNCTION A10 *A10* ERROR {%) 0.9860748E+00 O.9860747E+00 -O.2222977E-05 0.9860727E+00 0.9860747E+00 0.2027605E-03 0.98G0702E+00 0.9860747E+00 O.4G10163E-03 0.9860669E+00 0.9860747E+00 0.7865459E-03 0.9860628E+00 0.9860746E+00 0.1197158E-02 O.9860576E+00 0.9860745E+00 0.1715597E-02 0.9860510E+00 0.9860744E+00 0.2371082E-02 0.9860426E+00 0.9860742E+00 0.3201412E-02  unit  10.  P O S I T I V E SEQUENCE PROPAGATION FREQUENCY A11 0.1000000E-01 0.9860814E+00 0.1258925E-01 0.9860808E+00 O.1584893E-01 O.9860797E+00 0.1995262E-01 0.9860780E+00 0.2511886E-01 0.9860754E+00 0.3162278E-01 0.9860712E+00 0.3981072E-01 0.9860645E+00 0.5011872E-01 0.9860540E+00  Sample output from l o g i c a l  FUNCTION »A11 * ERROR (%) 0.986081OE+00 -O.4863591E-04 0.9860807E+00 -O.9226487E-05 O.9860802E+00 O.5319254E-04 0.9860795E+00 0.1520180E-03 0.9860784E+00 0.3083897E-03 0.9860766E+00 0.5555812E-03 0.9860738E+00 0.9457518E-03 0.9860694E+00 0.1560140E-02  u n i t 1 1.  L o g i c a l u n i t 6 c o n t a i n s the r e c o r d of the i n p u t parameters, equivalent line  r e p r e s e n t a t i o n , and approximation  process.  210 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 .36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 6 1 62 63 64 65 66 67 68 69 End o f F i l e  RECORD OF INPUT  PARAMETERS  NUMBER OF PHASES* 3 LINE LENGTH* 500.0000000 L I N E PARAMETERS PER UNIT LENGTH AT  6 0 . 0 0 0 HZ ARE:  ZERO SEQUENCE RESISTANCE INDUCTANCE CAPACITANCE EARTH R E S I S T I V I T Y  0.1974000E+00 OHMS 0 . 3 3 3 0 8 0 0 E - 0 2 HENRYS 0 . 8 3 6 1 0 0 0 E - 0 8 FARADS O.1000000E+03 OHMS  P O S I T I V E SEQUENCE RESISTANCE INDUCTANCE CAPACITANCE  0.2643000E-01 OHMS 0 . 8 8 0 8 0 0 0 E - 0 3 HENRYS O.133OOO0E-07 FARADS  FROM THE GIVEN LINE PARAMETERS. THE ESTIMATED LINE CONFIGURATION I S : EQUIVALENT GEOMETRIC MEAN RADIUS O.1844397E+00 RADIUS OF EQUIVALENT BUNDLED CONDUCTOR O.1964899E+00 AVERAGE DISTANCE BETWEEN CONDUCTORS •0.1508298E+02 AVERAGE HEIGHT OF CONDUCTORS O.1498967E+02 AVERAGE DISTANCE BETWEEN CONDUCTORS O.3400702E+02 AND THEIR IMAGES  CORRECTION FACTOR FOR SKIN E F F E C T S=(INTERNAL RADIUS)/!EXTERNAL RADIUS)* DC RESISTANCE 0 . 2 6 1 9 0 0 0 E - 0 1 OHMS  **'• APROXIMATION PROCESS FOR ZCO **** NUMBER OF POLES* 6 NUMBER OF ZEROS= 6 S H I F T I N G OPTION S E L E C T E D ERROR(%) AT 6 0 . 0 0 0 0 0 HZ IS REACHED AFTER 4 S H I F T I N G LOOPS  **** APROXIMATION PROCESS FOR A10 NUMBER OF POLES* 5 NUMBER OF ZEROS* 4 ERROR (%) AT 6 0 . 0 0 0 0 0 HZ IS REACHED AFTER O ITERATIONS  **** APROXIMATION PROCESS FOR ZC1 NUMBER OF POLES* 4 NUMBER OF ZEROS* 4 S H I F T I N G OPTION S E L E C T E D ERROR(% ) AT 60.OOOOO HZ IS REACHED AFTER 3 S H I F T I N G LOOPS  **** APROXIMATION PROCESS FOR A11 **** NUMBER OF POLES= 4 NUMBER OF ZEROS* 3 S H I F T I N G OPTION IS S E L E C T E D ERROR AT 6 0 . 0 0 HZ I S 0.016O% REACHED AFTER 2 S H I F T I N G LOOPS  S_ample o u t p u t from l o g i c a l u n i t 6.  METERS METERS METERS METERS METERS  O. 8095438E + 00  -O.7488213379  0. 2659203301  -O. 1 188697158  211 L o g i c a l u n i t 12 c o n t a i n s the c o e f f i c i e n t s of the p a r t i a l f r a c t i o n s expansion of the r a t i o n a l - f u n c t i o n s approximation of the l i n e . The output format i s compatible with the frequency-dependence the  v e r s i o n of  EMTP, and i t i s the i n f o r m a t i o n t o be i n t r o d u c e d i n t o l o g i c a l u n i t 1. I t i s important to take i n t o account t h a t the formatted w r i t i n g  of  i n f o r m a t i o n i s time consuming and c o m p u t a t i o n a l l y e x p e n s i v e .  fore, in  f o r most a p p l i c a t i o n s  lines  i t i s recommended  16 through 18 be s e t to z e r o .  that the c o n t r o l  Thereparameters  BIBLIOGRAPHY  [I]  H.W. Dommel, " D i g i t a l Computer S o l u t i o n of E l e c t r o m a g n e t i c T r a n s i e n t s i n S i n g l e - and Multiphase Networks". IEEE Trans., PAS-88, pp. 388-399, A p r i l 1969.  [2]  H.W. Dommel and W.S. Meyer, "Computation of E l e c t r o m a g n e t i c Transients". IEEE P r o c , V o l . 62 ( 7 ) , pp. 983-993, J u l y 1974.  [3]  J.R. M a r t i , "Accurate M o d e l l i n g o f Frequency-Dependent Transmission Lines i n Electromagnetic Transients Simulations". IEEE Power Industry Computer A p p l i c a t i o n s (PICA) Conference, P h i l a d e l p h i a , PA, 9 pages, May 1981.  [4]  H.L. Leon, P r o f i l e s of T r a n s i e n t V o l t a g e s along Overhead L i n e s (M.A.Sc. t h e s i s ) . Department of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of Toronto, December, 1979.  [5]  A. Semylen and A. Dabuleanu, " F a s t and Accurate S w i t c h i n g T r a n s i e n t C a l c u l a t i o n s on T r a n s m i s s i o n Lines with Ground Return u s i n g Recursive Convolutions". IEEE Trans., PAS-94, pp. 561-571, March/ A p r i l 1975.  [6]  Line Constants of Overhead L i n e s User's Manual. Methods A n a l y s i s Group, Branch o f System E n g i n e e r i n g , B o n n e v i l l e Power A d m i n i s t r a t i o n , P o r t l a n d , Oregon, Appendix 2 by H.W. Dommel, August 1977.  [7]  J.R. M a r t i , The Problem of Frequency Dependence i n T r a n s m i s s i o n M o d e l l i n g (PhD t h e s i s ) . Department of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1981.  [8]  I b i d . , pp. 189-193.  [9]  I b i d . , p. 106  [10]  D.E. Hedman, "Propagation  on Overhead T r a n s m i s s i o n  Line  Lines:  I. Theory of Modal A n a l y s i s ; I I . Earth Conduction E f f e c t s and Practical Results". IEEE Trans., PAS-84, pp. 200-211, March 1965. [II]  H.W. Dommel, Notes on Advanced Power Systems A n a l y s i s . of B r i t i s h Columbia, 1975.  [12]  UBC - Overhead L i n e Parameters Program. O r i g i n a l l y w r i t t e n by H.W. Dommel a t the B o n n e v i l l e Power A d m i n i s t r a t i o n , P o r t l a n d , Oregon. M o d i f i e d a t the U n i v e r s i t y of B r i t i s h Columbia (UBC), Canada, by I . I . Dommel, K.C. Lee, and T. Hung. User's Manual, UBC, August 1980.  212  University  BIBLIOGRAPHY  [13]  UBC - E l e c t r o m a g n e t i c T r a n s i e n t s Program (EMPT). Originally written by H.W. Dommel a t the B o n n e v i l l e Power A d m i n i s t r a t i o n , P o r t l a n d , Oregon. M o d i f i e d a t the U n i v e r s i t y of B r i t i s h Columbia (UBC), Canada, by H.W. Dommel. Frequency-dependence v e r s i o n o r i g i n a l l y w r i t t e n by J.R. M a r t i a t UBC. M o d i f i e d by L. M a r t i . User's Manual, UBC, August 1978 ( f i r s t p u b l i s h e d i n 1976).  [14]  J.P. B i c k f o r d , N. M u l l i n e u x , and J.R Reed, Computation of Power Systems T r a n s i e n t s (book). Peregrinus ( f o r the I E E ) , Herts (England), 1976.  [15]  L.V. Bewley, T r a v e l l i n g Waves on T r a n s m i s s i o n Systems Dover, New York, 1963 ( f i r s t p u b l i s h e d i n 1933).  [16]  EHV T r a n s m i s s i o n L i n e Reference Book. New York, N.Y., 1968.  213  Edison  Electric  (book).  Institute,  

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