Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The development of a reactive power management technique for a planning environment Garrett, Bretton Wayne 1978

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1978_A7 G37.pdf [ 4.06MB ]
Metadata
JSON: 831-1.0065613.json
JSON-LD: 831-1.0065613-ld.json
RDF/XML (Pretty): 831-1.0065613-rdf.xml
RDF/JSON: 831-1.0065613-rdf.json
Turtle: 831-1.0065613-turtle.txt
N-Triples: 831-1.0065613-rdf-ntriples.txt
Original Record: 831-1.0065613-source.json
Full Text
831-1.0065613-fulltext.txt
Citation
831-1.0065613.ris

Full Text

THE DEVELOPMENT OF A REACTIVE POWER MANAGEMENT TECHNIQUE FOR A PLANNING ENVIRONMENT by BRETTON WAYNE GARRETT B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia 1974  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department o f E l e c t r i c a l  Engineering  We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA Jufcy, 1978  (cp B r e t t o n Wayne G a r r e t t , 1978  In p r e s e n t i n g t h i s  thesis  an advanced degree at the I  Library shall  f u r t h e r agree  for  f u l f i l m e n t o f the requirements f o r  the U n i v e r s i t y of B r i t i s h  make i t  freely available  that permission  for  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  that  study. thesis  s c h o l a r l y purposes may be granted by the Head of my Department or  by h i s of  in p a r t i a l  representatives.  this  thesis  It  is understood that copying o r , p u b l i c a t i o n  f o r f i n a n c i a l gain shall  not be allowed without my  written permission.  Department of The  Electrical  University of B r i t i s h  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  March 12, 1973  Engineering  Columbia  Abstract  A computer-aided a l g o r i t h m i s developed f o r t h e management of r e a c t i v e power f l o w i n an e l e c t r i c power system. technique  i s designed  The  to a s s i s t transmission planning  engineers  i n e s t a b l i s h i n g s a t i s f a c t o r y base-case power f l o w s o l u t i o n s . The o b j e c t i v e i n t h e a l g o r i t h m i s t o reduce r e a l power l o s s i n t h e system t h r o u g h c o n t r o l o f r e a c t i v e power f l o w , and so i s d i f f e r e n t than conventional  "VAr a l l o c a t i o n "  algorithms.  The m i n i m i s a t i o n i s performed by a s p e c i a l l y adapted g r a d i e n t search w i t h a sub-optimal  s t e p - s i z e , which can be s i m p l y  i n c o r p o r a t e d i n t o a s t a n d a r d Newton-Raphson power f l o w program. A s p e c i a l feature of t h i s t h e s i s i s the presentation of a s e t of contour p l o t s of the o b j e c t i v e f u n c t i o n versus v a r i o u s p a i r s o f c o n t r o l v a r i a b l e s . An a n a l y s i s o f these p l o t s i s presented,  and i s used t o demonstrate t h e v a l i d i t y o f t h e  steepest descent m i n i m i s a t i o n technique  f o r t h i s problem.  Comments a r e g i v e n on t e s t s conducted w i t h t h i s  technique  on a t y p i c a l B r i t i s h Columbia Hydro and Power A u t h o r i t y power f l o w s i m u l a t i o n c o n s i s t i n g o f 245 b u s s e s and 327 b r a n c h e s , w i t h 47 c o n t r o l l a b l e g e n e r a t o r s transformers.  and 44 c o n t r o l l a b l e v a r i a b l e - t a p  The a l g o r i t h m i s c l a i m e d t o be e f f e c t i v e and  e f f i c i e n t f o r studies of t h i s  size.  TABLE OF CONTENTS  INTRODUCTION  1  Chapter I . PRINCIPLES AND TECHNIQUES OF REACTIVE POWER MANAGEMENT  5"'  Manual Techniques O p t i m a l (Automatic) Techniques I I . THE MANAGEMENT OF REACTIVE POWER USING LOSS REDUCTION  22  The Development o f an O b j e c t i v e F u n c t i o n I I I . THE INVESTIGATION OF THE CONSTRAINED OBJECTIVE FUNCTION  33  Contours o f C o n s t a n t Loss The E f f e c t o f t h e P e n a l t y Terms R e a c t i v e Power L i m i t s on G e n e r a t o r s IV. THE CHOICE OF A SUITABLE MINIMISATION TECHNIQUE  45  The Treatment o f C o n s t r a i n t s The Method o f M i n i m i s a t i o n Summary V. THE PERFORMANCE OF THE TECHNIQUE  5  7  -iv-  V I . CONCLUSIONS V I I . DIRECTIONS FOR FURTHER WORK  67 68  REFERENCES  70  APPENDIX  74  -V-  LIST OF TABLES  I . P r o g r e s s o f Convergence f o r 245 Bus Problem ... 60  -vi-  LIST OF ILLUSTRATIONS  1. Three-bus example system  90  2. Contours o f c o n s t a n t l o s s f o r system o f figure 1  91  3. Contours o f c o n s t a n t l o s s w i t h reduced generation 4. Three-bus example system w i t h added b r a n c h ....  91 92  5. Contours o f c o n s t a n t l o s s f o r system o f figure 4  93  6. Contours of c o n s t a n t l o s s w i t h reduced l o a d ... 93 7. Contours of c o n s t a n t l o s s w i t h added transformer  94  8. Contours of c o n s t a n t l o s s w i t h t r a n s f o r m e r and b r a n c h o f f i g u r e 4  94  9. Contours o f c o n s t a n t l o s s w i t h reduced power factor  95  10. Contours o f c o n s t a n t l o s s w i t h reduced power f a c t o r and l o a d 11. Contours o f c o n s t a n t l o s s w i t h bus 3 shunt ....  95 9  ^  12. Contours o f c o n s t a n t l o s s w i t h t r a n s f o r m e r and bus 3 shunt  ^6  13. C o n t o u r s o f c o n s t a n t l o s s w i t h bus 3 shunt and b r a n c h o f f i g u r e 4  97  -vii-  14. Contours o f f i g u r e 2 w i t h added v o l t a g e penalty  97  15. Contours o f f i g u r e 3 w i t h added v o l t a g e penalty  98  16. C o n t o u r s o f f i g u r e 11 w i t h added v o l t a g e penalty  98  17. Contours o f f i g u r e 16 w i t h added r e a c t i v e power p e n a l t y  99  18. Contours o f f i g u r e 14 w i t h added p e n a l t y f o r g e n e r a t o r r e a c t i v e power  99  19. Contours o f f i g u r e 14 w i t h o p t i m i s a t i o n path 20. Contours o f f i g u r e 7 w i t h v o l t a g e  .100 penalty  and o p t i m i s a t i o n p a t h  100  21. Contours o f f i g u r e 17 w i t h o p t i m i s a t i o n path  101  -viii-  NOTATION  Notes: 1)  S c a l a r q u a n t i t i e s a r e w r i t t e n as s i m p l e o r s u b s c r i p t e d variables  (e.g. V ) . V e c t o r q u a n t i t i e s a r e w r i t t e n as  s i m p l e v a r i a b l e s w i t h an overhead b a r (e.g. V ) .  Matrix  q u a n t i t i e s a r e w r i t t e n as s i m p l e v a r i a b l e s w i t h i n square b r a c k e t s (e.g. [YJ ) . 2)  The n u l l m a t r i x i s w r i t t e n matrix i s written  3)  [l] .  [o] , w h i l e t h e i d e n t i t y -  The z e r o v e c t o r  A "A" p r e c e d i n g some v a r i a b l e v a r i a t i o n i n that variable.  i s w r i t t e n 0*.  (e.g. AV) i n d i c a t e s a The r e a l and r e a c t i v e power  mismatches a r e t h e e x c e p t i o n s t o t h i s r u l e . w r i t t e n AP and AQ. always be p o i n t e d 4)  A subscript  They a r e  That t h e s e a r e mismatches w i l l out i n the t e x t .  ( i , j , o r k) a p p l i e d t o a s c a l a r v a r i a b l e  (e.g. Vi ) i n d i c a t e s t h a t t h e v a r i a b l e p e r t a i n s t o t h e bus i n d i c a t e d b y t h e s u b s c r i p t .  Similarly, a variable  w i t h two s u b s c r i p t s i n d i c a t e s t h a t t h e v a r i a b l e to the p a i r o f busses i n d i c a t e d . of s u b s c r i p t e d  pertains  A partial derivative  v a r i a b l e s indicates that the d e r i v a t i v e  i s an element o f t h e d e r i v a t i v e m a t r i x (e.g. — — i s t h e aVj ( i , j ) element o f t h e m a t r i x  jjjy^J) -  A  s, i s used t o i n d i c a t e t h e s l a c k b u s .  special subscript,  -ix-  The  u n q u a l i f i e d term "power" r e f e r s t o t h e complex power  S. A b s o l u t e v a l u e i s i n d i c a t e d w i t h two v e r t i c a l b a r s ( e . g . j.-lj  = 1) , w h i l e  the Euclidean  norm i s i n d i c a t e d  two p a i r s o f v e r t i c a l b a r s (e.g.  with  |Jl + j ljj = Yl) .  Summations a r e g i v e n over elements o f a s e t .  F o r example,  the sum o f t h e squares o f t h e r e a c t i v e power mismatches 2 f o r t h e power system i s ^t(AQi)• The  superscript  "des" i n d i c a t e s t h e d e s i r e d v a l u e o f t h e  variable.ntSimilarly,  "sched" i n d i c a t e s t h e s c h e d u l e d  v a l u e o f t h e v a r i•a b l e . value of the voltage The  superscript  magnitude a t bus i .  "max" i n d i c a t e s t h e maximum d e s i r e d  of the v a r i a b l e . desired  els s i n d i c a t e s t h e d e s i r e d E.g. Vj_  Similarly,  "min" i n d i c a t e s t h e minimum  value.  Where a v e c t o r o r v e c t o r - v a l u e d greater  An a s t e r i s k s u p e r s c r i p t S  e x p r e s s i o n i s shown t o be  t h a n 0, t h i s i s i n t e n d e d t o mean t h a t each  element o f t h e v e c t o r  e.g.  value  (or e x p r e s s i o n ) i s g r e a t e r  t h a n 0.  i n d i c a t e s t h e complex c o n j u g a t e ,  i s t h e complex c o n j u g a t e o f t h e power.  A "T" s u p e r s c r i p t  i n d i c a t e s the transpose of the vector  or m a t r i x .  E.g. V  i s the transpose of the voltage  vector.  A subscript  "o" i n d i c a t e s t h e i n i t i a l v a l u e o f t h e v a r i a b l e .  -X-  e.g. 14)  x  A "V"  Q  i s the i n i t i a l v a l u e of v e c t o r  p r e c e d i n g a v a r i a b l e i n d i c a t e s the g r a d i e n t  v a r i a b l e , e.g. V F r e s p e c t t o u. the 15)  x.  U  i n d i c a t e s the g r a d i e n t  A "rj2."  s  i ii m  a  r  iy  of F  of  that  with  i n d i c a t e s the H e s s i a n of  variable.  A s u p e r s c r i p t such as  "k" or "k+1"  i s used t o i n d i c a t e  v a l u e o f the v a r i a b l e a t s t e p k or k+1 process.  i n the  iterative  the  -xi-  Symbols: S  -  complex power. matrix  P  -  |jj^-j,  Sometimes used f o r t h e s e n s i t i v i t y  b u t always i n d i c a t e d as such.  r e a l ( a c t i v e ) power = Re{sj.  P o s i t i v e from bus i n t o  system (or ground). Q  -  i m a g i n a r y ( r e a c t i v e ) power = Irn^S^.  P o s i t i v e from  bus i n t o system (or ground). V  ,-  voltage  magnitude.  8  -  voltage  a n g l e r e l a t i v e t o an a r b i t r a r y r e f e r e n c e  voltage  ( u s u a l l y t h e s l a c k bus v o l t a g e ) .  Z  -  impedance (complex).  R  -  resistance = Re{zj.  X  -  reactance =  Y  -  a d m i t t a n c e (complex) = Z~^~.  G  -  conductance = Re £YJ.  B  -  s u s c e p t a n c e = Im^Y^.  -  transmission  f  -  o b j e c t i v e f u n c t i o n t o be m i n i m i s e d ,  g  -  the vector of e q u a l i t y c o n s t r a i n t s  h  -  the vector of i n e q u a l i t y c o n s t r a i n t s  Im[z].  r e a l power l o s s ,  (equal t o 0 ) . (greater  than  0) . u  -  vector of c o n t r o l v a r i a b l e s : generator  voltages,  t r a n s f o r m e r t a p s , and a l l o c a t e d r e a c t i v e power.  -Xll-  v e c t o r o f independent v a r i a b l e s : l o a d bus v o l t a g e s , voltage  angles.  weighting  f a c t o r s f o r v o l t a g e p e n a l t y term,  w e i g h t i n g f a c t o r s f o r r e a c t i v e a l l o c a t i o n term, tap s e t t i n g f o r v a r i a b l e tap transformers  of the  s e t NT. s e t o f a l l busses i n t h e system. s e t o f a l l l o a d and o t h e r busses i n t h e system f o r w h i c h t h e bus v o l t a g e i s n o t f i x e d . set of a l l generators  i n t h e system e l i g i b l e f o r  v o l t a g e adjustment d u r i n g o p t i m i s a t i o n . s e t o f a l l busses i n t h e system e l i g i b l e as l o c a t i o n s f o r shunt r e a c t i v e banks d u r i n g o p t i m i s a t i o n . s e t o f a l l busses a t w h i c h t h e v o l t a g e i s t o be h e l d t o w i t h i n some t o l e r a n c e o f n o m i n a l . set of a l l v a r i a b l e transformer  taps e l i g i b l e f o r  adjustment d u r i n g o p t i m i s a t i o n . s e t o f a l l busses w i t h a c o n n e c t i n g b r a n c h t o bus i . s u b s e t o f NV f o r w h i c h v o l t a g e s a r e h i g h e r t h a n t h e d e s i r e d maximum. s u b s e t o f NV f o r w h i c h v o l t a g e s a r e lower t h a n t h e d e s i r e d minimum.  ACKNOWLEDGEMENT  The a u t h o r would l i k e t o thank t h e System P l a n n i n g Department  o f t h e B r i t i s h Columbia Hydro and Power A u t h o r i t y  f o r p e r m i s s i o n t o undertake t h i s p r o j e c t w h i l e employed b y t h e Department,  and f o r t h e system d a t a used f o r t h e two t e s t c a s e s .  The a u t h o r would p a r t i c u l a r l y l i k e t o thank Dr. B.A. D i x o n of B.C. Hydro f o r h i s u n c e a s i n g encouragement and h e l p , w i t h o u t w h i c h t h i s p r o j e c t would never have begun. The a u t h o r would a l s o l i k e t o thank D r s . H.W. Dommel and G.F. the  S c h r a c k o f t h e Department  of E l e c t r i c a l Engineering of  U n i v e r s i t y o f B r i t i s h Columbia, f o r t h e i r v a l u e d e f f o r t s as  supervisors of this project.  -1-  INTRODUCTION  W i t h power systems becoming l a r g e r and more t i g h t l y meshed, t h e development o f good power f l o w s i m u l a t i o n s f o r each y e a r and l o a d i n g c o n d i t i o n o f a c a p r i c i o u s t e n - y e a r system p l a n i s becoming one o f t h e most t e d i o u s involved i n planning transmission  procedures  networks.  The p r o c e d u r e i n v o l v e d i n p r o d u c i n g a good power  flow  s o l u t i o n i s s u f f i c i e n t l y w e l l defined t h a t the m a j o r i t y of the adjustments c a n be done b y an automated t e c h n i q u e .  Someone  f a m i l i a r w i t h t h e power system ( e . g . a good t e c h n o l o g i s t ) c a n s e t up t h e c o n s t r a i n t s f o r t h e p r o c e s s and e v a l u a t e  the r e s u l t s .  I f t h e r e s u l t s a r e n o t e x a c t l y r i g h t t h e f i r s t t i m e , as i s likely  (depending on t h e a b i l i t y o f t h e u s e r ) , t h e t e c h n i q u e  can be used i t e r a t i v e l y .  Provided  the person "at the c o n t r o l s "  understands what c o n s t i t u t e s a good s o l u t i o n , i t i s p o s s i b l e t o a c h i e v e a good s i m u l a t i o n i n much l e s s t i m e , and w i t h much l e s s f r u s t r a t i o n , than i s p o s s i b l e w i t h standard simulation  manual power  flow  techniques.  There a r e b a s i c a l l y o n l y t h r e e s t e p s flow simulation.  i n p r o d u c i n g a power  F i r s t , t h e t r a n s m i s s i o n and g e n e r a t i o n  be m o d e l l e d , and t h e system l o a d s must be a d j u s t e d  must  t o the load  f o r e c a s t f o r t h e y e a r and l o a d i n g c o n d i t i o n (e.g. 1984 heavy  -2-  w i n t e r peak) under c o n s i d e r a t i o n .  Second, a r e a s o n a b l e genera-  t i o n schedule, i n c l u d i n g import/export schedules, e s t a b l i s h e d t o s u p p o r t the t o t a l l o a d .  must be  T h i s s c h e d u l e must t a k e  i n t o account the e f f e c t of upstream hydro p l a n t s on downstream plants  (e.g. B.C.  Hydro and Power A u t h o r i t y ' s S i t e One  dependent on the upstream G.M.  Shrum p l a n t ) , the  of water f o r h y d r o p l a n t s , the d e s i r a b i l i t y of  plant i s  availability operating  t h e r m a l p l a n t s , the m e r i t o r d e r of a v a i l a b l e hydro and  thermal  plants, etc.  transform-  T h i r d , the g e n e r a t o r v o l t a g e s , v a r i a b l e  er taps, switchable must be a d j u s t e d w i t h i n safe  c a p a c i t o r / r e a c t o r banks and the l i k e a l l  so t h a t the v o l t a g e s around the system are  (and s t a b l e ) o p e r a t i n g  then f u r t h e r adjusted power s o u r c e s ,  limits.  voltages  T h i s f i n a l t r i m m i n g of v o l t a g e s  voltage  profile.  i s c a r r i e d out i n i n c r e a s i n g  t o the nearness of the s t u d y date t o  the  a c t u a l d a t e , w i t h the most a t t e n t i o n b e i n g g i v e n t o the y e a r of the  first  plan.,  T h i s t h i r d and  *  are  u s i n g a v a i l a b l e and p l a n n e d r e a c t i v e  so as t o o b t a i n a r e a s o n a b l e  d e t a i l according  The  l a s t s t e p , the adjustment of  generators,  U s u a l l y t h i s i s a s u b j e c t i v e e v a l u a t i o n , which i n v o l v e s many i n t e r - r e l a t i n g f a c t o r s , such as whether the s i m u l a t i o n i s f o r a normal or emergency (outage) c o n d i t i o n , and the l o c a t i o n , s i z e , and n a t u r e of the a f f e c t e d l o a d .  -3-  v a r i a b l e t r a n s f o r m e r t a p s , s w i t c h e d r e a c t i v e banks, e t c . , and p a r t i c u l a r l y the d e t a i l e d i n v e s t i g a t i o n i n v o l v e d i n e a r l y budget y e a r s , g e n e r a l l y r e q u i r e s t h e most work i n any power f l o w s i m u l a t i o n , and i t i s thus f o r t h e r e s o l u t i o n o f t h i s s t e p t h a t many computer-aided  t e c h n i q u e s have been proposed.  T h i s t h e s i s summarizes s e v e r a l o f t h e v a r i o u s  techniques  proposed f o r r e a c t i v e power management , and t h e n p r e s e n t s t h e development and a n a l y s i s o f an o b j e c t i v e f u n c t i o n d i f f e r e n t from t h a t adopted b y most o t h e r a u t h o r s .  Instead of minimising  the d e v i a t i o n o f system v o l t a g e s from " s t a n d a r d " v a l u e s as i s u s u a l l y done, t h e t e c h n i q u e p r e s e n t e d here reduces t h e t r a n s m i s s i o n l o s s e s i n t h e system, which,  i n the o p i n i o n of the author,  p r o v i d e s t h e s o l u t i o n w h i c h good " v o l t a g e d e v i a t i o n " t e c h n i q u e s o n l y approximate.  T h i s amounts t o s o l v i n g t h e r e a c t i v e h a l f o f  the g e n e r a l o p t i m a l power f l o w problem addressed by Dommel [ l j  *  R e a c t i v e power management i s d e f i n e d here t o be t h e c o n t r o l of generator v o l t a g e s , v a r i a b l e transformer tap s e t t i n g s , and e x i s t i n g s w i t c h a b l e shunt c a p a c i t o r and r e a c t o r banks, . . and t h e a l l o c a t i o n o f new c a p a c i t o r and r e a c t o r banks, i n a manner w h i c h b e s t a c h i e v e s t h e d e s i r e d g o a l o f v o l t a g e c o n t r o l , l o s s r e d u c t i o n , o r b o t h . T h i s i s more g e n e r a l than r e a c t i v e power a l l o c a t i o n ("VAr a l l o c a t i o n " ) w h i c h d e a l s o n l y w i t h t h e a l l o c a t i o n o f new and e x i s t i n g b l o c k s o f shunt compensation.  -4-  and Sasson  [2] .  F i n a l l y , the a n a l y s i s  o f the o b j e c t i v e  function  i s used t o  s e l e c t and t e s t a c o m p u t a t i o n a l l y e f f i c i e n t s o l u t i o n t e c h n i q u e which w i l l r e q u i r e  a minimum o f u s e r i n t e r a c t i o n t o be e f f e c t i v e .  -5-  CHAPTER I  P r i n c i p l e s and Techniques o f R e a c t i v e Power Management  A l t h o u g h r e a c t i v e power does n o t u s e f u l l y c o n t r i b u t e t o t h e f l o w o f energy i n t h e power system, i t has a s i g n i f i c a n t e f f e c t on power system performance and e f f i c i e n c y .  By r e d u c i n g unne-  c e s s a r y r e a c t i v e power f l o w s , i t i s p o s s i b l e t o i n c r e a s e therma l l y r e s t r i c t e d a c t i v e power c a p a c i t i e s o f l i n e s ,  transformers,  and g e n e r a t o r s . System s t a b i l i t y i s improved by r e d u c i n g the wide v a r i a t i o n i n bus v o l t a g e s w h i c h c h a r a c t e r i s t i c a l l y accompany h i g h r e a c t i v e power f l o w s .  The l o w e r c u r r e n t s improve  v o l t a g e r e g u l a t i o n on d i s t r i b u t i o n c i r c u i t s , and reduce energy l o s s e s t h r o u g h o u t t h e system. There a r e s e v e r a l ways i n which u n n e c e s s a r y r e a c t i v e power f l o w s c a n be reduced.  The most common way i s t o s u p p l y t h e  n e c e s s a r y r e a c t i v e power g e n e r a t i o n a t t h e l o a d i t s e l f .  This  i s u s u a l l y done u s i n g shunt c a p a c i t o r s o r r e a c t o r s . A l t h o u g h , i d e a l l y , shunt r e a c t i v e d e v i c e s s h o u l d be p r o v i d e d a t a l l busses where t h e power f a c t o r i s l e s s t h a n u n i t y , the c o s t of these d e v i c e s p r o h i b i t s t h i s p r a c t i c e . I n s t e a d , i t i s u s u a l t o i n s t a l l shunt d e v i c e s i n such a way t h a t t h e b e n e f i t s a r e s h a r e d over s e v e r a l a d j a c e n t b u s s e s .  A  -6-  t r a d e o f f i s thus made between t h e e f f e c t i v e n e s s o f t h e c o r r e c t i o n , and t h e c o s t o f t h e equipment.  The a l l o c a t i o n o f new and  e x i s t i n g shunt d e v i c e s i n a manner w h i c h o f f e r s t h e g r e a t e s t b e n e f i t f o r t h e l o w e s t c o s t i s t h e problem a d d r e s s e d by most "VAr a l l o c a t i o n "  techniques.  Some t e c h n i q u e s  p e r f o r m t r u e r e a c t i v e power  management,  o f f e r i n g a d d i t i o n a l ways o f c o n t r o l l i n g r e a c t i v e power f l o w . W i t h i n s t a t o r c u r r e n t , f i e l d c u r r e n t , and s t a b i l i t y  restrictions,  g e n e r a t o r v o l t a g e s may be c o n t r o l l e d , a l t e r i n g b o t h t h e r e a c t i v e power p r o d u c t i o n  (or a b s o r p t i o n )  a t the generator  i t s e l f , and  the r e a c t i v e power f l o w t h r o u g h a d j a c e n t , s t r o n g l y connected busses. A n o t h e r method o f c o n t r o l l i n g t h e r e a c t i v e power f l o w i n a system i s t r a n s f o r m e r  t a p adjustment.  networks, t h e l i n e i n d u c t a n c e l i n e resistance.  I n overhead t r a n s m i s s i o n  i s t y p i c a l l y much g r e a t e r t h a n  Under these c o n d i t i o n s , t h e a c t i v e power f l o w  a l o n g a t r a n s m i s s i o n l i n e i s n e a r l y independent o f t h e d i f f e r e n c e between t h e v o l t a g e magnitudes a t t h e ends o f t h e l i n e ,  while  the r e a c t i v e power f l o w i s n e a r l y p r o p o r t i o n a l t o t h i s  voltage  difference.  By u s i n g t r a n s f o r m e r  t a p s t o a l t e r one o r more bus  v o l t a g e s , i t i s p o s s i b l e t o c o n t r o l t h e f l o w o f r e a c t i v e power t h r o u g h these b u s s e s i n d e p e n d e n t l y  o f t h e f l o w o f r e a l power.  (Note t h a t o n l y t h e r e a c t i v e power f l o w i n g t h r o u g h a bus may be  -7-  controlled.  Any r e a c t i v e power absorbed b y a l o a d must always  be s u p p l i e d , i r r e s p e c t i v e o f any adjustments i n t r a n s f o r m e r t a p s . )  Manual Techniques  Perhaps t h e most o b v i o u s method f o r a l l o c a t i n g  reactive  power s o u r c e s i s by i n s p e c t i o n o f the power f l o w s on v a r i o u s lines.  Adjustments t o g e n e r a t o r v o l t a g e s , e x i s t i n g  (switchable)  shunt r e a c t i v e banks, and v a r i a b l e t r a n s f o r m e r t a p s e t t i n g s can be d e t e r m i n e d from c a r e f u l s t u d y o f bus v o l t a g e s and c i r c u i t r e a c t i v e power f l o w s .  A f t e r t h e s e adjustments have been determ-  i n e d and checked w i t h a new power f l o w s i m u l a t i o n , r e m a i n i n g r e g i o n s o f h i g h o r low v o l t a g e , and t r a n s m i s s i o n l i n e s w i t h h i g h r e a c t i v e power f l o w s a r e i d e n t i f i e d from t h e newly c a l c u l a t e d results.  New shunt r e a c t i v e d e v i c e s can t h e n be l o c a t e d a t  busses c e n t r a l t o t h e problem a r e a s .  This procedure i s repeated  f o r each l o a d i n g c o n d i t i o n t o determine t h e t o t a l  system  requirements f o r the year b e i n g s t u d i e d . T h i s t e c h n i q u e , which i s o f t e n r e f e r r e d t o as the t r i a l and e r r o r method, can a l s o be used t o p l a n v o l t a g e s u p p o r t f o r outage c a s e s .  L i n e s can be removed from t h e s t u d y , as n e c e s s a r y ,  t o i n v e s t i g a t e each system c o n t i n g e n c y .  The power f l o w s a r e  c a l c u l a t e d as f o r t h e normal c o n d i t i o n base c a s e , and r e a c t i v e  -8-  compensation l o c a t e d so as t o c o r r e c t f o r a d v e r s e v o l t a g e s , and t o m i n i m i s e the power f l o w s a l o n g h e a v i l y l o a d e d t r a n s m i s s i o n lines.  The shunt compensation r e q u i r e m e n t s f o r a l l cases would  t h e n be a s s i m i l a t e d i n t o a s i n g l e system p l a n . W h i l e t h i s method o f a l l o c a t i n g r e a c t i v e power s o u r c e s g i v e s e x a c t power f l o w s o l u t i o n s , i t does have s e v e r a l drawbacks. F i r s t l y , a t l e a s t two power flows=r-one . i n i t i a l t o f i n d the u n c o r r e c t e d v o l t a g e s and power f l o w s , and a t l e a s t one o t h e r t o t e s t the a l l o c a t i o n s c h e m e — a r e  r e q u i r e d f o r each l o a d i n g and  contingency c o n d i t i o n . Secondly, i t i s d i f f i c u l t for  t o determine the c o r r e c t s i z e  each r e a c t i v e i n s t a l l a t i o n , p a r t i c u l a r l y when i t i s n e c e s s a r y  t o keep v o l t a g e s a t s e v e r a l a d j a c e n t b u s s e s w i t h i n some t o l e r a n c e of  nominal.  T h i s problem i s f u r t h e r c o m p l i c a t e d by the need t o  combine r e a c t i v e r e q u i r e m e n t s f o r a l l c o n t i n g e n c i e s i n t o a s i n g l e p l a n f o r the system, i n t h a t compensation added f o r one c o n t i n g e n c y w i l l a f f e c t the amount o f compensation needed f o r other contingencies. In  s p i t e o f the amount o f work i n v o l v e d i n t h i s type o f  r e a c t i v e power a l l o c a t i o n s t u d y , the t r i a l and e r r o r t e c h n i q u e i s the one used by most Canadian u t i l i t i e s the  [3] f o r s t u d i e s o f  f i r s t one o r two y e a r s o f the system p l a n . A n o t h e r manual t e c h n i q u e which i s sometimes used (and i s  -9-  p r e s e n t l y i n use a t B.C. Hydro) i s t h e s e n s i t i v i t y £4].  technique  I n t h i s method, t h e p a r t i a l d e r i v a t i v e s o f v o l t a g e change  w i t h r e s p e c t t o r e a c t i v e power network d a t a  (-^7^)  a r e c a l c u l a t e d from the  (these d e r i v a t i v e s can e a s i l y be o b t a i n e d from t h e  J a c o b i a n m a t r i x o f a Newton-Raphson power f l o w program).  I ti s  then p o s s i b l e t o f i n d the bus v o l t a g e s w h i c h accompany a change i n r e a c t i v e power:  AV =  (1)  [Sj] AQ  where AV = e x p e c t e d  v o l t a g e changes  AQ = changes i n r e a c t i v e power out o f busses dV 4Qj  ( s e n s i t i v i t y m a t r i x d e r i v e d from J a c o b i a n matrix).  From t h i s e q u a t i o n , i t i s p o s s i b l e t o c a l c u l a t e t h e changes i n bus r e a c t i v e shunts w h i c h w i l l a d j u s t the v o l t a g e a t bus i by AV^.  The most e f f e c t i v e l o c a t i o n f o r a shunt r e a c t i v e bank i s  t h e n a t t h e bus w i t h t h e h i g h e s t s e n s i t i v i t y  coefficient  (element o f [ s j ] ) f o r t h e v o l t a g e a t bus i ( i . e . bus k, where l ikl s  =  m a x  ^l irj| ' S  f o r  a  1  1  3 ^ QJ) • N  B o t h t h e l o c a t i o n and t h e  s i z i n g o f the bank a r e thus o b t a i n e d  directly.  I t i s i n t e r e s t i n g t o note t h a t f o r d i f f e r e n c e s (9^ - 0 j ) ~  0,  -10-  v o l t a g e magnitudes V ^ a 1 p.u., and l o w r e s i s t a n c e branches ( ij R  < < :  x  ij) '  t  h  e  n  which i s j u s t t h e s h o r t - c i r c u i t r a t i o o f t h e b r a n c h between busses i and j . T h i s , a l o n g w i t h t h e a p p r o x i m a t i o n  has been e x p l o i t e d by M a l i s z e w s k i , Garver, and Wood £5]. Dommel ^6] has shown t h a t t h e s e n s i t i v i t y t e c h n i q u e c a n m o d i f i e d t o determine t h e amount o f r e a c t i v e compensation Qj_ r e q u i r e d a t a s i n g l e bus i t o m i n i m i s e t h e o b j e c t i v e f u n c t i o n  f =  S(Vi- v  d e s  )  2  where V i = v o l t a g e a t bus i des V"i = d e s i r e d v o l t a g e a t bus 1.  I n s e r t i n g t h e f i r s t - o r d e r s e n s i t i v i t y r e l a t i o n o f Vj_ about V  (  gives  -11-  f =  ^(v  o i  + s  ± i  AQ,  - vf  e s  )  2  where s  i j ^Qj  i s  e x p r e s s i o n f o r &V  t h e  L  = ^AQA  from  (1)  A t the minimum  d f  dAQj  from  - 0. o r  which AQ,  '  =  1  3  J  Is*  The s e n s i t i v i t y t e c h n i q u e has the advantage o f b e i n g a t r u e r e a c t i v e power management t e c h n i q u e , and can be used t o c a l c u l a t e r e q u i r e d adjustments v o l t a g e s i n the same way  as was  i n t r a n s f o r m e r t a p s and done above f o r shunt  generator compensation.  The major d i s a d v a n t a g e o f the s e n s i t i v i t y t e c h n i q u e i s t h a t i t assumes t h a t the r e l a t i o n s h i p between AV and AQ i s a l i n e a r one, w h i l e i n p r a c t i c e i t i s v e r y complex.  The  calculated  v a l u e s , t h e r e f o r e , are v a l i d o n l y f o r s m a l l changes i n bus  -12-  voltage.*  A f t e r an a l l o c a t i o n scheme i s s e l e c t e d , i t must be  t e s t e d w i t h a power f l o w t o f i n d t h e e x a c t bus v o l t a g e s .  If  the v o l t a g e s a r e s t i l l u n s a t i s f a c t o r y , t h e p r o c e d u r e can be repeated. Note t h e s i m i l a r i t y between t h i s t e c h n i q u e  and the t r i a l  and e r r o r approach: t h e y b o t h b e g i n w i t h t h e s o l u t i o n o f an i n i t i a l power f l o w , and end w i t h t h e s o l u t i o n o f a c o n f i r m i n g power f l o w .  The d i f f e r e n c e i s s o l e l y i n t h e way t h e r e a c t i v e  i n s t a l l a t i o n s a r e l o c a t e d and s i z e d . technique,  the planning engineer  I n t h e t r i a l and e r r o r  s e l e c t s t h e l o c a t i o n on t h e  b a s i s o f t h e r e a c t i v e power f l o w s and v o l t a g e l e v e l s . t h e n use judgement t o s i z e t h e i n s t a l l a t i o n — a t a s k  He must  complicated  by t h e d i f f i c u l t y o f p r e d i c t i n g t h e e f f e c t o f t h e new  installa-  t i o n on v o l t a g e s a t a d j a c e n t b u s s e s .  technique  provides t h i s information d i r e c t l y  The s e n s i t i v i t y  (albeit approximately),  and  much more q u i c k l y . The s e n s i t i v i t y t e c h n i q u e  i s f a s t , f a i r l y simple, handles  a l l c o n t i n g e n c i e s , a l l l o a d i n g c o n d i t i o n s , and a l l types o f *  The a c c u r a c y o f t h e p r e d i c t i o n s always improves as t h e magnitude of voltage c o r r e c t i o n decreases. F o r example, u s i n g t h i s t e c h n i q u e on a t y p i c a l B.C. Hydro system, t o change a bus v o l t a g e from 0.998 p.u. t o 1.025 p.u., t h e p r e d i c t e d change i n Q a t t h a t bus b r i n g s t h e v o l t a g e t o 1 . 0 2 6 — a n e r r o r o f 0.12%. F o r a s t a r t i n g v o l t a g e ( a t t h e same bus) o f 1.007 p.u., t h e new p r e d i c t e d change i n Q b r i n g s t h e v o l t a g e t o 1.0253 p . u . — a n e r r o r o f o n l y 0.03%.  -13-  r e a c t i v e power management (shunt compensation, and g e n e r a t o r v o l t a g e s ) .  transformer taps,  Unfortunately, i t i s only  approximate,  and i t s h a r e s w i t h a l l manual t e c h n i q u e s , and many o f the " o p t i m a l " a u t o m a t i c t e c h n i q u e s , the d i f f i c u l t y of  combining  a l l o c a t i o n s f o r a l l l o a d i n g c o n d i t i o n s and c o n t i n g e n c i e s i n t o a s i n g l e , l e a s t - c o s t system p l a n .  Optimal  (Automatic)  Techniques  The major d i f f e r e n c e between manual and o p t i m a l r e a c t i v e power a l l o c a t i o n t e c h n i q u e s i s t h a t o p t i m a l t e c h n i q u e s b o t h l o c a t e and s i z e shunt r e a c t i v e banks a u t o m a t i c a l l y . The  term  " o p t i m a l " i m p l i e s — i n some cases c o r r e c t l y — t h a t the r e s u l t a n t a l l o c a t i o n scheme i s the b e s t p o s s i b l e , s u b j e c t t o c o n s t r a i n t s such as bus v o l t a g e l i m i t s , maximum and minimum a c c e p t a b l e installation sizes, etc. O p t i m a l r e a c t i v e power a l l o c a t i o n t e c h n i q u e s can be g e n e r a l l y d i v i d e d i n t o two t y p e s .  The s i m p l e s t , and the one on which  the  m a j o r i t y o f the l i t e r a t u r e has been w r i t t e n , i s the group o f l i n e a r i z e d t e c h n i q u e s : l i n e a r programming, i n t e g e r programming, and 0-1 programming.  These t e c h n i q u e s u s u a l l y d e a l o n l y w i t h  t h e a l l o c a t i o n o f shunt r e a c t i v e banks, and always assume a linear "objective function"  (which i n t h i s case means t h a t AV  -14-  and AQ  are assumed p r o p o r t i o n a l ) , s u b j e c t t o l i n e a r c o n s t r a i n t s .  (There i s a v a r i a t i o n of l i n e a r programming known as q u a d r a t i c programming, which a l l o w s the use o f a q u a d r a t i c o b j e c t i v e function subject to l i n e a r constraints.) The  second type of o p t i m a l t e c h n i q u e ,  i s the group of n o n l i n e a r t e c h n i q u e s programming.  These t e c h n i q u e s  and the most complex,  o f t e n termed n o n l i n e a r  d e a l w i t h a l l types of r e a c t i v e  power management, can h a n d l e a wide range of o b j e c t i v e f u n c t i o n s , and can have b o t h l i n e a r and n o n l i n e a r c o n s t r a i n t s .  This  great  f l e x i b i l i t y i s not w i t h o u t p e n a l t y , however, as the c o m p u t a t i o n time r e q u i r e d f o r s o l u t i o n i n c r e a s e s r a p i d l y w i t h problem c o m p l e x i t y , and the s o l u t i o n t e c h n i q u e  used must o f t e n be  tailored  t o the p r o b l e m i n o r d e r t o f i n d any s o l u t i o n a t a l l . Nevertheless,  i f the e x t r a f l e x i b i l i t y i s r e q u i r e d , i t i s n e a r l y  always p o s s i b l e t o d e v e l o p a w o r k a b l e a l g o r i t h m .  Since  o b j e c t i v e f u n c t i o n i s u n r e s t r i c t e d , these t e c h n i q u e s  the  produce as  e x a c t an answer as the p r e c i s i o n o f the computer, the d a t a , the convergence b e h a v i o u r  of the a l g o r i t h m w i l l  and  permit.  L i n e a r O p t i m a l Techniques  The b a s i c l i n e a r o p t i m a l r e a c t i v e power a l l o c a t i o n i s l i n e a r programming.  technique  I n the l i n e a r programming approach, the  -15-  voltage  changes i n t h e system a r e assumed t o be p r o p o r t i o n a l t o  the changes i n r e a c t i v e power;  AV = [s] AQ  (2)  where AV = e x p e c t e d changes i n bus v o l t a g e s AQ = changes i n r e a c t i v e power o u t o f busses [s^] = c o n s t a n t m a t r i x , w h i c h may o r may not be t h e [ S j ] o f e q u a t i o n (1) .  The c o n s t r a i n t s on AV,  AV  > V  m i n  ±  m  ±  ^ v  - V  ±  f o r a l l i € NV Av  are combined w i t h  a  x  - V  ±  (2) t o o b t a i n t h e s e t o f i n e q u a l i t i e s  .SgijAQj * v f  J  6  N  Q m  a  x  ^>S±A AQ-i < V i  J6NQ  J  J  The c o n s t r a i n t s on AQ a r e :  n  -  V  i  j  \ f o r a l l i £ NV - Vi) 7  (3) (4)  -16-  AQj ^ Q  - Qj  m a x  (5)  A Q i 2t 0, o r -AQj —  f o r a l l j £ NQ  0  (6a) (6b)  where equation  (6a) i s used f o r i n d u c t i v e a l l o c a t i o n s , and  equation  (6b) i s used f o r c a p a c i t i v e a l l o c a t i o n s .  The o b j e c t i v e f u n c t i o n t o be m i n i m i s e d  i s t h e sum o f t h e a b s o l u t e  values of a l l the r e a c t i v e a d d i t i o n s :  f(Q)  The  =  (7)  £|AQj| •  s e r i e s of equations  (3) - (7) form a s t a n d a r d  linear  programming problem, and c a n be s o l v e d by any o f t h e a v a i l a b l e l i n e a r programming a l g o r i t h m s .  The r e s u l t o f t h e o p t i m i s a t i o n  i s a s e t o f r e a c t i v e a l l o c a t i o n s AQ f o r t h e busses chosen b y t h e planning engineer.  I n some methods, AQ i s p e r m i t t e d t o assume  any v a l u e , w h i l e i n o t h e r s i t i s c o n s t r a i n e d t o be one o f a s e t of standard s i z e s .  I n t h e case where any v a l u e i s p e r m i s s i b l e ,  the s i z e s must l a t e r be rounded t o t h e n e a r e s t s t a n d a r d s i z e b y the p l a n n i n g e n g i n e e r .  A power f l o w c a n then be r u n w i t h t h e  s t a n d a r d i z e d a l l o c a t i o n s t o o b t a i n t h e e x a c t bus v o l t a g e s . A l t h o u g h d i f f e r i n g i n some p o i n t s — e s p e c i a l l y i n t h e  -17-  f o r m u l a t i o n o f e q u a t i o n ( 2 ) — a l l l i n e a r programming a l l o c a t i o n t e c h n i q u e s have t h i s g e n e r a l form [ 5 , 7 J .  They a r e u s u a l l y  s o l v e d w i t h the Simplex l i n e a r programming a l g o r i t h m , f o r which s t a n d a r d code i s a v a i l a b l e . These t e c h n i q u e s can be made t o handle m u l t i p l e c o n t i n g e n c i e s a u t o m a t i c a l l y . M a l i s z e w s k i e t a l . [5]  accomplish  this  by s o l v i n g a l l c o n t i n g e n c i e s t o g e t h e r as one m a s s i v e l i n e a r programming problem. for  I f n e c e s s a r y , some o f t h e c a p a c i t y needed  one c o n t i n g e n c y can be removed d u r i n g o t h e r c o n t i n g e n c i e s  t o p r e v e n t the v o l t a g e from e x c e e d i n g t h a t e x p e r i e n c e d under normal c o n d i t i o n s . The i n s t a l l e d c a p a c i t y i s p e r m i t t e d t o take any v a l u e , l a t e r b e i n g rounded t o the n e a r e s t s t a n d a r d v a l u e by the u s e r . The l i n e a r programming t e c h n i q u e s based on (2) t h r o u g h  (7)  use l i n e a r i z e d s e n s i t i v i t y i n f o r m a t i o n . A d i f f e r e n t t e c h n i q u e , b y K o h l i and K o h l i ^ 8 ] , uses an i n t e g e r programming t e c h n i q u e to a l l o c a t e c a p a c i t o r s i n u n i t s i z e s .  This technique b u i l d s a  "tree" of a l l p o s s i b l e c a p a c i t o r c o n f i g u r a t i o n s , s u b j e c t t o the r e s t r i c t i o n s determined by the p l a n n i n g e n g i n e e r .  These  c o n f i g u r a t i o n s a r e then s y s t e m a t i c a l l y t e s t e d t o see i f t h e y s a t i s f y bus v o l t a g e c o n s t r a i n t s .  The s e a r c h i s o r d e r e d so t h a t  the f i r s t f e a s i b l e s o l u t i o n w i l l be o p t i m a l ( l e a s t number o f capacitors).  The a u t h o r s suggest t h a t the e x t e n t o f t h e  -18-  ( o t h e r w i s e s u b s t a n t i a l ) s e a r c h can be reduced b y assuming t h a t the v o l t a g e on a bus w i l l be a f f e c t e d t h e most b y c a p a c i t o r s a t t h a t b u s . U n f o r t u n a t e l y t h i s i s n o t always t r u e .  (The v o l t a g e  i n c r e a s e w h i c h accompanies c a p a c i t o r a d d i t i o n s i s due t o a reduc t i o n o f c u r r e n t i n t h e t r a n s m i s s i o n f e e d i n g t h e bus i n q u e s t i o n . The c u r r e n t r e d u c t i o n — a n d  hence v o l t a g e i n c r e a s e — w i l l be t h e  g r e a t e s t when t h e c a p a c i t y i s added d i r e c t l y a t t h e p o i n t o f t h e r e a c t i v e l o a d , due t o t h e s a v i n g i n r e a c t i v e power l o s s e s betwee the bus i n q u e s t i o n and t h e l o a d p o i n t .  The e x c e p t i o n t o t h i s i  when t h e r e i s a c o n s t a n t v o l t a g e bus i n t h e v i c i n i t y , i n w h i c h case p r e d i c t i o n i s complex, and i s b e s t done d i r e c t l y from t h e sensitivities.) As a r e s u l t , t h e s e a r c h remains s u b s t a n t i a l , and so i s o f little,  i f any, p r a c t i c a l v a l u e f o r use w i t h o t h e r than s m a l l  subsystems o f u s u a l power networks. Convergence o f a l l o f these t e c h n i q u e s may be hampered b y automatic  t r a n s f o r m e r t a p s w i t c h i n g o r g e n e r a t o r Q l i m i t s , as  these u p s e t t h e assumed network l i n e a r i t y .  Nonlinear Optimal  Techniques  N o n l i n e a r o p t i m a l r e a c t i v e power management t e c h n i q u e s  may  t r e a t t h e r e a c t i v e power p r o b l e m separately,,, o r as a p a r t o f t h e  -In-  complete o p t i m a l power f l o w problem, which would a l s o o p t i m i s e r e a l power f l o w s . t r e a t t h e r e a c t i v e power problem separately.  They m i n i m i s e  the t o t a l c a p a c i t y a l l o c a t e d , subject  t o v o l t a g e and shunt c a p a c i t y c o n s t r a i n t s . This technique  solves the n o n l i n e a r problem w i t h a s e r i e s  of l i n e a r approximations  designed  t o make each g r a d i e n t s t e p  t e r m i n a t e on a c o n s t r a i n t boundary.  A f t e r each g r a d i e n t s t e p ,  a power f l o w i s s o l v e d t o f i n d t h e e x a c t bus v o l t a g e s .  By  assuming t h a t t h e o p t i m a l s o l u t i o n l i e s a t t h e i n t e r s e c t i o n o f c o n s t r a i n t boundaries, programming [lo]  t h i s technique  applies linear  t o f i n d the s o l u t i o n .  approximation  This technique  i s very  s i m i l a r t o t h e l i n e a r programming approach o f [5] . Sachdeva and B i l l i n t o n  | l l ) s o l v e t h e r e a c t i v e power man-  agement p r o b l e m as a p o r t i o n o f a complete o p t i m a l power f l o w . T h i s t e c h n i q u e p e r f o r m s t r u e r e a c t i v e power management i n c l u d i n g t r a n s f o r m e r t a p s and g e n e r a t o r v o l t a g e s .  The shunt compensation  is allocated i n unit sizes. Although  t h e t e c h n i q u e does handle m u l t i p l e c o n t i n g e n c i e s ,  t h i s i n c r e a s e s the data storage requirements  considerably.  m o d i f i e d method which uses l e s s s t o r a g e i s p r e s e n t e d  A  i n \l2~].  T h i s m o d i f i e d t e c h n i q u e s o l v e s t h e o p t i m a l r e a l power f l o w f i r s t , and then t h e o p t i m a l r e a c t i v e power f l o w ( w i t h r e a c t i v e shunt  -20-  allocation).  This c y c l e i s repeated  i f the o p t i m a l r e a c t i v e  s t e p a l t e r s the v o l t a g e a n g l e s by an amount s u f f i c i e n t t o change the r e a l power f l o w s i g n i f i c a n t l y from the optimum. Both of these methods, the f u l l and the d e c o u p l e d , use Fletcher-Reeves  or F l e t c h e r - P o w e l l technique  a  f o r the m i n i m i s a t i o n ,  w i t h a l l c o n s t r a i n t s t r e a t e d as p e n a l t y f u n c t i o n s ( s i m i l a r l y t o [2])Two  o t h e r o p t i m a l power f l o w t e c h n i q u e s  t o r e a c t i v e power management. s i m i l a r t o t h a t of functions.  are a l s o a p p l i c a b l e  Sasson e t a l . [13]  use a  technique  [ l l ] , w i t h a l l c o n s t r a i n t s h a n d l e d as p e n a l t y  The c o e f f i c i e n t s o f t h e p e n a l t y terms are a l t e r e d  d u r i n g the s o l u t i o n t o speed convergence t o a f e a s i b l e The major d i f f e r e n c e between the t e c h n i q u e s  of  [l3]  and  i n the m i n i m i s a t i o n , f o r w h i c h Sasson e t a l . use the w h i c h i s computed from the Newton-Raphson J a c o b i a n Dommel and T i n n e y [ l ] use a g r a d i e n t s e a r c h  solution. [ll] i s  Hessian,  matrix.  (and a l s o a  H e s s i a n a p p r o x i m a t i o n ) f o r m i n i m i s a t i o n , t r e a t i n g the power flow equations The  ( e q u a l i t y c o n s t r a i n t s ) w i t h Lagrange m u l t i p l i e r s .  i n e q u a l i t y c o n s t r a i n t s are t r e a t e d as a b s o l u t e l i m i t s f o r  independent v a r i a b l e s (e.g. g e n e r a t o r  v o l t a g e s ) , and as p e n a l t y  f u n c t i o n s f o r dependent v a r i a b l e s (e.g. l o a d bus v o l t a g e s ) . The  g r a d i e n t t e c h n i q u e , w h i l e sometimes s l o w e r t o converge t h a n  F l e t c h e r - P o w e l l or H e s s i a n  techniques,  requires considerably  -21-  l e s s s t o r a g e and c o m p u t a t i o n A l l of the techniques  time p e r s t e p . [ l , 2,11-13] s o l v e t h e o p t i m a l power  f l o w (and t h e r e a c t i v e power management e x t e n s i o n ) w i t h o u t assuming o b j e c t i v e f u n c t i o n l i n e a r i t y .  -22-  CHAPTER I I  The Management of R e a c t i v e Power U s i n g Loss  Reduction  Most of the many papers w r i t t e n on r e a c t i v e power a l l o c a t i o n propose the use o f l i n e a r programming and r e l a t e d t e c h n i q u e s determine  to  the l e a s t amount of shunt c a p a c i t a n c e which must be  added a t v a r i o u s busses t o ensure t h a t system v o l t a g e s a r e above u s e r - s p e c i f i e d minimums.  Most of these l i n e a r programming  t e c h n i q u e s are a d a p t a b l e t o shunt i n d u c t o r s w i t c h i n g a l s o , a l l o w i n g the s c h e d u l i n g o f a l l shunt r e a c t i v e d e v i c e s i n a r o u g h l y - l e a s t c o s t way.  The computer program thus does i n an  o r d e r l y f a s h i o n e x a c t l y what a p l a n n i n g e n g i n e e r would do i n an "educated"  manual way:  add c a p a c i t o r s or i n d u c t o r s as  to adjust a l l voltages to w i t h i n set l i m i t s .  necessary  If this isa l l  t h a t i s d e s i r e d , these t e c h n i q u e s work v e r y w e l l : t h e y are  fast,  easy t o use, and i n e x p e n s i v e t o r u n . The a l l o c a t i o n o f new  shunt i n s t a l l a t i o n s cannot r e a s o n a b l y  b e g i n , however, u n t i l a l l g e n e r a t o r v o l t a g e s , v a r i a b l e t r a n s f o r m e r t a p s , s t a t i c and synchronous compensators, and  existing  s w i t c h e d shunt r e a c t i v e banks have been a d j u s t e d t o o b t a i n the *  These t e c h n i q u e s m i n i m i s e the t o t a l a d d i t i o n a l c a p a c i t y , n e g l e c t i n g the f a c t t h a t the c o s t o f a new i n s t a l l a t i o n i s l i k e l y much more than the c o s t o f e n l a r g i n g an e x i s t i n g one.  -23-  b e s t p o s s i b l e base c a s e .  Only then i s i t r e a s o n a b l e t o attempt  i d e n t i f y i n g s i z e s and l o c a t i o n s f o r any new i n s t a l l a t i o n s . The t a s k o f o b t a i n i n g the i n i t i a l  "optimum" base case i s  c o n s i d e r a b l y more d i f f i c u l t — a t l e a s t f o r major t r a n s m i s s i o n networks--than  l o c a t i n g new r e a c t i v e banks.  A transmission-  system i s p l a n n e d f o r c o n t i n u o u s e x p a n s i o n . . W i t h a l o a d growth of 7% or l e s s p e r y e a r , i t s h o u l d never be n e c e s s a r y t o add e x t r a r e a c t i v e banks a t more than a few busses a t a i n l a r g e systems.  time—even  T h i s i s e s p e c i a l l y t r u e i n v i e w o f the f a c t  t h a t i t i s cheaper t o add one o r two l a r g e banks (or expand e x i s t i n g banks) t h a n t o add s e v e r a l s m a l l e r ones, due m a i n l y t o the c o s t o f the s w i t c h g e a r , buswork, c o n c r e t e pads, e t c .  that  are r e q u i r e d t o t r a n s f o r m a symbol on a diagram i n t o a r e a l i t y on the system.  I t i s even r e a s o n a b l e t o do t h i s  u s i n g V-Q s e n s i t i v i t i e s when n e c e s s a r y .  manually,  The major d i f f i c u l t y ,  not o n l y f o r r e a c t i v e power a l l o c a t i o n , b u t whenever power f l o w s t u d i e s a r e made, i s t o o b t a i n the b e s t p o s s i b l e base c a s e . To many p e o p l e , p a r t i c u l a r l y p r a c t i s i n g power e n g i n e e r s , a "good" base case i s one w i t h a u n i f o r m , o r n e a r l y s o , v o l t a g e profile.  T h i s e x p l a i n s t h e f r e q u e n t appearance i n the l i t e r a t u r e  of "VAr a l l o c a t i o n " t e c h n i q u e s d e s i g n e d t o c o r r e c t bus v o l t a g e s t o w i t h i n s p e c i f i e d bounds.  The need f o r s p e c i f y i n g these bounds,  however, and i n p a r t i c u l a r the d i f f i c u l t y and importance o f  -24-  d e t e r m i n i n g t h e " c o r r e c t " bounds t o use, l i m i t s t h e e f f e c t i v e ness o f t h i s approach.  A poor c h o i c e o f v o l t a g e l i m i t s f o r any  o f t h e l i n e a r programming t e c h n i q u e s d i s c u s s e d e a r l i e r c a n p r e v e n t convergence b y p l a c i n g i n c o m p a t i b l e c o n s t r a i n t s on t h e solution.  Some b u s s e s , p a r t i c u l a r l y those near c o n s t a n t v o l t a g e  busses, w i l l be i n s e n s i t i v e t o changes i n shunt r e a c t i v e power a t busses t o which t h e y a r e connected branch  impedances.  i t s neighbour  through r e l a t i v e l y l a r g e  I f t h e v o l t a g e c o n s t r a i n t s on t h e bus and  do n o t a l l o w f o r a r e a l i s t i c r e l a t i o n s h i p between  the two v o l t a g e s , the problem w i l l be e i t h e r u n s o l v a b l e , o r t h e s o l u t i o n w i l l r e q u i r e an u n r e a s o n a b l y  l a r g e amount o f shunt  compensation. The avoidance  o f t h i s type o f d i f f i c u l t y i s g e n e r a l l y l e f t  as t h e r e s p o n s i b i l i t y o f t h e u s e r . unreasonable  Although i t i s not,  perhaps,  t o assume t h a t an e x p e r i e n c e d e n g i n e e r can s e t  s a t i s f a c t o r y v o l t a g e c o n s t r a i n t s , one o f t h e most common a r guments g i v e n i n f a v o u r o f r e a c t i v e power a l l o c a t i o n  techniques  i n g e n e r a l i s t h a t t h e y r e q u i r e l e s s e x p e r i e n c e w i t h t h e power system t o use e f f e c t i v e l y than do manual t e c h n i q u e s .  The d i f f -  i c u l t y o f s e t t i n g p r o p e r v o l t a g e l i m i t s weakens t h i s argument considerably for v o l t a g e - c o r r e c t i o n techniques. Although there i s l i t t l e question that v o l t a g e - c o r r e c t i o n t e c h n i q u e s c a n a s s i s t w i t h t h e a l l o c a t i o n p r o c e s s , and may even  -25-  o c c a s i o n a l l y p r o v i d e a l l o c a t i o n schemes r e q u i r i n g l i t t l e o r no f u r t h e r m o d i f i c a t i o n , the r e q u i r e m e n t f o r u s e r - s e l e c t e d v o l t a g e l i m i t s can l e a d t o an u n n e c e s s a r y amount o f p r e l i m i n a r y work, and the r e s u l t i n g a l l o c a t i o n scheme i s always s e n s i t i v e t o the limits selected. The more e x p e r i e n c e  one g a i n s w i t h power f l o w s t u d i e s , the  more apparent i t becomes t h a t u n i f o r m tages a r e not o f p r i m a r y  (or o t h e r s e l e c t e d )  vol-  importance a t t r a n s m i s s i o n , and t o a  lesser extent subtransmission  system l e v e l s .  Of more importance  are t h e magnitudes, and t h e r e l a t i v e magnitudes, o f t h e r e a l and r e a c t i v e power f l o w s . Whereas r e a l power f l o w s a r e u s u a l l y d e t e r m i n e d by a v a i l a b i l i t y o f (extremely e x p e n s i v e ) g e n e r a t i o n s o u r c e s ,  energy  r e s e r v e s , and t h e c u r r e n t system l o a d , r e a c t i v e power f l o w s a r e more f l e x i b l e .  As. there: i s no energy r e q u i r e d f o r t h e g e n e r a t i o n  o f r e a c t i v e power, i t i s much e a s i e r and l e s s e x p e n s i v e  t o gen-  e r a t e t h a n i s r e a l power, and so c a n be produced n e a r e r the l o a d , r e d u c i n g t r a n s m i s s i o n l o s s e s , equipment l o a d i n g s , and v o l t a g e drops.  By r e d u c i n g u n n e c e s s a r y r e a c t i v e power f l o w s , i t i s  p o s s i b l e t o approximate a. u n i f o r m v o l t a g e c o n d i t i o n .  Indeed,  i t i s by c o n t r o l l i n g r e a c t i v e power f l o w s t h a t v o l t a g e - c o r r e c t i o n type  "VAr a l l o c a t i o n " t e c h n i q u e s  constraints.  attempt t o s a t i s f y v o l t a g e  -26-  R e a c t i v e power f l o w i s , i n t h i s way, more fundamental t o a "good" base case than i s a s e l e c t e d v o l t a g e p r o f i l e .  It is  always p o s s i b l e t o m i n i m i s e r e a c t i v e power f l o w s ; i t i s n o t always p o s s i b l e t o a c h i e v e a d e s i r e d v o l t a g e p r o f i l e  (given  normal o p e r a t i n g c o n s t r a i n t s ) . I t i s f o r t h e s e reasons t h a t minimum r e a c t i v e power f l o w , and not v o l t a g e p r o f i l e , w i l l  be  used i n t h i s t h e s i s as the p r i m a r y c r i t e r i o n f o r the s e l e c t i o n of a "good" base c a s e .  The Development o f an O b j e c t i v e  Function  From the f o r e g o i n g d i s c u s s i o n , i t s h o u l d be c l e a r t h a t a s u i t a b l e objective function f o r a minimisation process w i l l  be  r e l a t e d d i r e c t l y t o the r e a c t i v e power f l o w i n the t r a n s m i s s i o n system.  There are s e v e r a l such f u n c t i o n s which would be  suit-,  able: (a)  the sum o f the squares o f the c u r r e n t s i n each b r a n c h  (b)  the sum of the squares o f the r e a c t i v e power f l o w s i n each b r a n c h  (c)  the r e a c t i v e power l o s s i n the system (sum of | | l | | | x j 2  for  a l l b r a n c h e s , which i s e f f e c t i v e l y a w e i g h t e d  of squares o f c u r r e n t s )  sum  -27-  (d)  t h e r e a l power l o s s i n t h e system (sum o f  Rfor  a l l branches, s i m i l a r l y t o (c)) (e)  t h e s l a c k bus r e a l power ( c o n s t a n t term + r e a l power l o s s i f r e a l power i n j e c t i o n s  are not altered)  or any s i m i l a r f u n c t i o n . Function  (a.) i s g e n e r a l , s i n c e i t i s t h e c u r r e n t w h i c h  produces b o t h t h e r e a l and t h e r e a c t i v e power l o s s e s .  Unfort-  u n a t e l y , i t would tend t o e q u a l i s e b r a n c h c u r r e n t s i r r e s p e c t i v e o f t h e equipment r e p r e s e n t e d b y t h e b r a n c h .  S i n g l e - c i r c u i t , low  c a p a c i t y branches would thus be l o a d e d a t c u r r e n t l e v e l s comparable t o m u l t i - c i r c u i t , high capacity branches—obviously untenable  an  prospect:  Function  (b) p a r t i a l l y a v o i d s t h e problem, s i n c e i t p l a c e s  no r e s t r i c t i o n s on t h e r e a l power f l o w .  I t does t e n d t o b a l a n c e  r e a c t i v e power f l o w s between branches i n t h e same way as (a) b a l a n c e s c u r r e n t s , however, w h i c h , a l t h o u g h n o t r e s u l t i n g i n q u i t e such an u n r e a s o n a b l e s i t u a t i o n as ( a ) , i s s t i l l an u n d e s i r able c h a r a c t e r i s t i c  f o r reasons analogous t o those g i v e n above  for (a). F u n c t i o n (c) i s a v a r i a t i o n  o f (a) w h i c h tends t o a v o i d  the problem o f e q u a l i s i n g c u r r e n t f l o w s by w e i g h t i n g t h e squares o f t h e c u r r e n t s i n each b r a n c h w i t h a f a c t o r e q u a l t o t h e b r a n c h reactance.  Effectively,  t h i s f u n c t i o n tends t o e q u a l i s e t h e  -28-  p r o d u c t (|l|{|x|*.  A b r a n c h of t w i c e the r e a c t a n c e o f a n o t h e r  b r a n c h w i l l , t h e r e f o r e , be s c h e d u l e d t o c a r r y about h a l f of  the  c u r r e n t i n the low r e a c t a n c e b r a n c h (assuming t h a t the branches are e f f e c t i v e l y i n p a r a l l e l , and stant).  t h a t the t o t a l c u r r e n t i s con-?  T h i s i s a near i d e a l s h a r i n g of c u r r e n t , and has  advantage of m i n i m i s i n g which must be p r o v i d e d Function  the t o t a l r e a c t i v e power  generation  t o meet a .(fixed) r e a c t i v e power l o a d .  (d) i s an i n t e r e s t i n g a n a l o g t o ( c ) .  a r e s u l t s i m i l a r t o t h a t of the r a t i o X/R  i s constant  I t produces  (c) i n most c a s e s , and  f o r every branch.  The  identical i f  advantage of  t h i s f u n c t i o n over (c) l i e s i n the c o s t d i f f e r e n c e , environmentally,  the  economically,  and s o c i a l l y , between r e a l and r e a c t i v e power.  R e a l power comes from l a r g e dams and r e s e r v o i r s , thermal s t a t i o n s burning  conventional  i r r e p l a c e a b l e and e x p e n s i v e s u p p l i e s  of  c o a l or p e t r o l e u m , or n u c l e a r t h e r m a l s t a t i o n s w h i c h produce unmanageable, or n e a r l y so, f i s s i o n b y - p r o d u c t s .  Reactive  i n c o n t r a s t , i s g e n e r a t e d n a t u r a l l y by t r a n s m i s s i o n  lines  power, and  c a b l e s , and can be g e n e r a t e d d e l i b e r a t e l y i n shunt c a p a c i t o r banks.  Function  *  See  (d) s h a r e s w i t h f u n c t i o n s  appendix s e c t i o n A6  (a) t o (c) a d i f f i c u l t y  f o r a p r o o f of  this.  -29-  of c a l c u l a t i o n , i n t h a t c u r r e n t s must be c a l c u l a t e d f o r each b r a n c h , and t h e n summed a c c o r d i n g Function  (e) i s more e l e g a n t ,  t o t h e f u n c t i o n employed.  as t h e s l a c k bus power may be  c a l c u l a t e d w i t h t r i v i a l e f f o r t from t h e s o l u t i o n v o l t a g e s .  For  c o n s t a n t r e a l power i n j e c t i o n s a t busses o t h e r t h a n t h e s l a c k bus,  as i s t h e case h e r e , t h i s f u n c t i o n produces r e s u l t s i d e n -  t i c a l t o those o f ( d ) . I t i s t h i s o b j e c t i v e f u n c t i o n w h i c h w i l l be used i n t h i s t h e s i s . In order t o prevent the s o l u t i o n a l g o r i t h m the bus v o l t a g e s  from d r i v i n g  e x c e s s i v e l y high, i t i s necessary e i t h e r t o  augment t h e o b j e c t i v e f u n c t i o n w i t h a term d e s i g n e d t o i n c r e a s e or " p e n a l i s e " t h e o b j e c t i v e f u n c t i o n as t h e bus v o l t a g e s from n o m i n a l , o r t o f o r m a l l y c o n s t r a i n t h e o b j e c t i v e w i t h a voltage c o n s t r a i n t d u r i n g the m i n i m i s a t i o n . technique permits a voltage  to deviate  deviate  function The former  f a r from nominal i f t o  do so s i g n i f i c a n t l y reduces e i t h e r o t h e r v o l t a g e d e v i a t i o n s , o r the p r i m a r y  objective function  (system r e a l power l o s s ) .  l a t t e r t e c h n i q u e w i l l not p e r m i t t h e v o l t a g e s  The  to violate their  r e s p e c t i v e c o n s t r a i n t s even i f t h e c o n s t r a i n t s a r e p r e v e n t i n g s o l u t i o n o f t h e problem.  The augmented o b j e c t i v e  function  ("penalty f a c t o r " ) t e c h n i q u e i s t h e r e f o r e p r e f e r a b l e use,  as i t a t l e a s t ensures t h e e x i s t e n c e I f the algorithm  for this  ,  of a s o l u t i o n .  i s t o be u s e f u l f o r t h e a l l o c a t i o n o f new  -30-  shunt r e a c t i v e power s o u r c e s , the o b j e c t i v e f u n c t i o n w i l l  require  an a d d i t i o n a l term a c c o u n t i n g f o r the c o s t o f the a d d i t i o n a l shunt c a p a c i t y which i s t o be s u p p l i e d .  T h i s may e a s i l y be  a c c o m p l i s h e d by a d d i n g a term s i m i l a r t o the v o l t a g e  (penalty)  term, which w i l l add t o the o b j e c t i v e f u n c t i o n an amount e q u a l to  the w e i g h t e d sum of squares of the compensation added. The f i n a l  form of the o b j e c t i v e f u n c t i o n i s , t h e r e f o r e ,  f ( u , x ) = P_(u,x) + 2lw<cr  2_w (v  i n  )  max. 2 ,  -V-;  /TT  i  (v, - V ?  i  2  +  2  ) + 2zk k B  (8)  where term w i ( V i - V i  i n  )  term Wj (Vj - V j * ) 1 3  2  2  i s low v o l t a g e p e n a l t y f o r bus i i s h i g h v o l t a g e p e n a l t y f o r bus j  2 . term z^B^ i s shunt c a p a c i t y p e n a l t y f o r bus k P  s  = s l a c k bus r e a l power  Wj_ = v o l t a g e w e i g h t i n g f a c t o r f o r bus i = shunt c a p a c i t y w e i g h t i n g f a c t o r f o r bus k B^. = r e a c t i v e shunt added a t bus k The c o n s t r a i n t s on f ( u , x ) a r e :  g =  = 0  f o r a l l busses  (9)  -31-  w h i c h a r e t h e power f l o w e q u a t i o n s r e q u i r i n g t h e power mismatches t o be z e r o a t a l l busses (AP and AQ a r e t h e v e c t o r s o f r e a l and r e a c t i v e power mismatches, r e s p e c t i v e l y ) ,  V - V™ * > 0 1  1  "max ~" V - V > 0  (10a) f o r a l l busses € NG (10b)  w h i c h a r e t h e minimum and maximum v o l t a g e l i m i t s f o r a l l controlled voltage  (generator) b u s s e s ,  — . mxn — T - T > 0 "max — — T - T > 0  (Ha) f o r a l l b u s s e s € NT (lib)  w h i c h a r e t h e minimum and maximum t a p l i m i t s f o r a l l c o n t r o l l e d transformers,  — mxn — B - B > 0 B  max  — — - B > 0  f o r a l l busses €. NQ  (12a) (12b)  w h i c h a r e t h e minimum and maximum l i m i t s o f r e a c t i v e compensation t o be added t o e l i g i b l e b u s s e s ( o f t h e s e t NQ), and f i n a l l y  -32-  min > Q - Q Q  max  0  - Q > 0  (13a)  for a l l generator busses  (13b)  w h i c h a r e the minimum and maximum r e a c t i v e power l i m i t s f o r the generators.  The  i n e q u a l i t i e s (10) - (13) c o l l e c t i v e l y  the i n e q u a l i t y c o n s t r a i n t s e t h >  0.  Note t h a t the l i m i t s on the c o n t r o l v a r i a b l e s , (10) through simpler  constitute  equations  (12) are l i n e a r , which w i l l p e r m i t the use of a  constrained  optimisation  technique.  -33-  CHAPTER I I I  The I n v e s t i g a t i o n o f t h e C o n s t r a i n e d  The c h o i c e  Objective  of a n u m e r i c a l m i n i m i s a t i o n  knowledge o f t h e n a t u r e o f t h e c o n s t r a i n e d  Function  technique objective  One o f t h e s i m p l e s t ways o f g e t t i n g t h i s i n f o r m a t i o n  requires function. i s through  the use o f c o n t o u r p l o t s , i n w h i c h c o n t o u r s o f c o n s t a n t  objec-  t i v e f u n c t i o n value are p l o t t e d versus the various c o n t r o l v a r iables.  Such c o n t o u r p l o t s can be produced by u s i n g a power  f l o w program t o e v a l u a t e  the o b j e c t i v e f u n c t i o n f o r various  v a l u e s o f two c o n t r o l v a r i a b l e s .  The c o n t o u r p l o t s i n t h i s  t h e s i s were produced by a. power f l o w program w h i c h  automatically  v a r i e d t h e c o n t r o l v a r i a b l e s on t h e two axes t h r o u g h each o f eight values, evaluations).  g i v i n g a t o t a l o f 64 power f l o w s  (or f u n c t i o n  The v a l u e s o f t h e o b j e c t i v e f u n c t i o n were t h e n  i n t e r p o l a t e d between these p o i n t s t o o b t a i n t h e c o n t o u r s f o r plotting. The o b j e c t i v e f u n c t i o n chosen ( e q u a t i o n •k  8) i s composed o f  t h r e e p a r t s : t h e s l a c k bus r e a l power , t h e v o l t a g e  *  penalties,  The r e a l power l o s s i s e q u a l t o t h e s l a c k bus r e a l power p l u s a constant. The c o n t o u r s o f r e a l power l o s s thus have i d e n t i c a l form t o t h e c o n t o u r s o f s l a c k bus r e a l power. The c o n t o u r s o f r e a l power l o s s w i l l be used h e r e i n .  -34-  and the a l l o c a t e d r e a c t i v e power p e n a l t i e s .  A l t h o u g h i t i s not  d i f f i c u l t t o p l o t the c o n t o u r s f o r t h i s o b j e c t i v e f u n c t i o n , i t w i l l be d i f f i c u l t t o a n a l y s e such a complex s e t o f c o n t o u r s directly.  I t i s s i m p l e r t o analyse the l o s s contours f i r s t ,  and t h e n a n a l y s e the e f f e c t t h a t the p e n a l t i e s w i l l have on these c o n t o u r s .  Contours o f C o n s t a n t Loss  The system f o r which c o n t o u r s have been o b t a i n e d i s shown i n f i g u r e 1.  T h i s t h r e e - b u s system i s a s i m p l i f i e d r e p r e s e n t a -  t i o n o f the p o r t i o n of B.C.  Hydro's 230 kV system from B r i d g e  R i v e r (bus 2) t h r o u g h Cheekeye (bus 3) t o Vancouver  (bus 1 ) .  The c o n t o u r s o f c o n s t a n t l o s s v e r s u s the v o l t a g e s a t busses 1 and 2 a r e shown i n f i g u r e 2.  These c o n t o u r s appear t o be  s t r o n g l y p a r a b o l i c , w i t h the a x i s p a r a l l e l t o and  slightly  d i s p l a c e d from the l i n e V-j_ = V 2 . The r e d u c t i o n o f l o s s w i t h i n c r e a s i n g v o l t a g e a t busses 1 and 2 i s due t o the r e s u l t a n t i n c r e a s e i n v o l t a g e a t bus  3,  w h i c h reduces the c u r r e n t n e c e s s a r y t o p r o v i d e t h e l o a d P and  Q.  T h i s reduced c u r r e n t f l o w t h e n r e s u l t s i n reduced l o s s i n the two b r a n c h e s . On e i t h e r s i d e o f the a x i s the l o s s i n c r e a s e s .  T h i s i s not  -35-  due t o a v a r i a t i o n i n t h e v o l t a g e  a t bus 3, w h i c h i s e s s e n t i a l l y -  constant along a locus a t r i g h t angles t o the a x i s the d i r e c t r i x ) .  (parallel to  F o r example, a l o n g t h e l i n e AB d e f i n e d by A =  (1.20,0.85) and B = (0.85,1.20), t h e p e r u n i t bus 3 v o l t a g e s a r e : 1.00, 1.00, 1.00, 1.00, 1.00, 1.01, 1.01, 1.01 f o r i n c r e m e n t s o f 0.05 pu i n  and V .  (The f a c t t h a t t h e v o l t a g e s  2  s l i g h t l y as V 2 i n c r e a s e s l i n e AB d e f i n e d  increase  and V-^ d e c r e a s e s i n d i c a t e s t h a t the  above a c t u a l l y i n t e r s e c t s the a x i s a t an a n g l e o  s l i g h t l y l e s s t h a n 90 , w h i c h means t h a t t h e a x i s i s not q u i t e p a r a l l e l t o t h e l i n e V-^ = V 2 . ) The r e a s o n f o r t h e i n c r e a s e  of l o s s t o e i t h e r side of the  a x i s becomes a p p a r e n t upon c l o s e e x a m i n a t i o n o f the power f l o w s c o r r e s p o n d i n g t o each p o i n t on t h e p l o t : p o i n t s  o f f the a x i s  c o r r e s p o n d t o a t r a n s f e r o f r e a c t i v e power (and hence from one g e n e r a t o r t o t h e o t h e r . the l o c u s o f s o l u t i o n v o l t a g e s r e a c t i v e power i s z e r o .  current)  The a x i s o f t h e c o n t o u r s i s  f o r which t h i s interchange of  The f o l l o w i n g t a b l e shows t h i s e f f e c t  f o r t h r e e p o i n t s on t h e p l o t o f f i g u r e 2:l • 2 (pu) (pu) v  0.95 1.00 1.05  V  1.10 1.05 1.00  °- from 2 t o 3 (pu)  Q from 1 t o 3 (pu)  system l o s s (pu)  2.364 0.746 -0.718  1.014 0.485 2.128  0.239 0.216 0.250  -36-  The a x i s i n t h i s case runs a p p r o x i m a t e l y a l o n g the l i n e d e f i n e d by C = (0.85,0.93) and D = (1.15,1.20).  The  CD  reactive  power i s seen t o f l o w from g e n e r a t o r 2 t o g e n e r a t o r 1 f o r v o l t a g e (0.95,1.10), and from 1 t o 2 f o r v o l t a g e (1.05,1.00).  The  loca-  t i o n o f the a x i s i s a f f e c t e d by such parameters as the r e l a t i v e impedances o f the b r a n c h e s , the r e a l power f l o w s a l o n g each o f the b r a n c h e s , and the R/X r a t i o s of the b r a n c h e s .  By way  of  example, f o r the case o f f i g u r e 2, the r e a l power f l o w s from bus 2 t o bus 1.  F o r the case o f f i g u r e 3, however, b o t h gen-  e r a t o r s s u p p l y power t o bus 3 e q u a l l y , r e s u l t i n g i n a s m a l l e r d i s p l a c e m e n t o f the a x i s from the l i n e  = V  than i s the case  f o r f i g u r e 2. F i g u r e 3, which i s f o r a case i d e n t i c a l t o t h a t of f i g u r e 2 e x c e p t f o r the g e n e r a t i o n , a l s o shows more e l o n g a t e d c o n t o u r s due t o the reduced power, and so c u r r e n t , f l o w i n g on the b r a n c h from bus 2 t o bus 3. The l o s s  (£) thus d e c r e a s e s l e s s r a p i d l y  w i t h i n c r e a s i n g v o l t a g e t h a n f o r the more h e a v i l y l e a d e d case (the d e r i v a t i v e -j-j"  =  2IR and so i s p r o p o r t i o n a l . t o the c u r r e n t  f l o w ) , w h i l e the l o s s due t o the i n t e r c h a n g e o f r e a c t i v e power between g e n e r a t o r s i s a f f e c t e d r e l a t i v e l y l e s s by the r e d u c t i o n i n l o a d c u r r e n t . Hence the g r e a t e r e l o n g a t i o n o f the c o n t o u r s . The r e l a t i o n s h i p between the shape o f the c o n t o u r s and the l i n e l o a d i n g i s f u r t h e r a p p a r e n t from i n s p e c t i o n o f f i g u r e  5.  T h i s i s t h e c o n t o u r p l o t f o r t h e s i t u a t i o n o f f i g u r e 4, w i t h a  ( h y p o t h e t i c a l ) b r a n c h from bus 1 t o bus 2 p a r a l l e l i n g t h e  p r e v i o u s p a t h . Any r e a c t i v e power f l o w i n g from g e n e r a t o r generator bus  1 t h e r e f o r e has an a l t e r n a t e p a t h around t h e l o a d  3. T h i s reduces s i g n i f i c a n t l y t h e c u r r e n t on t h e o t h e r  l i n e s . The c o n t o u r s have been e l o n g a t e d they are v i r t u a l l y p a r a l l e l •Ki  2 to  t o t h e p o i n t where  lines.  T h i s extreme e l o n g a t i o n o f t h e c o n t o u r s  i s due, as w i t h  the case o f f i g u r e 3, t o t h e f a c t t h a t t h e c u r r e n t s a s s o c i a t e d w i t h t h e t r a n s f e r o f r e a l power a r e now v e r y s m a l l (due t o e q u a l l o a d - s h a r i n g b y t h e two g e n e r a t o r s ) , and t h e v a r i a t i o n i n l o s s w i t h c u r r e n t i s s i g n i f i c a n t l y l e s s t h a n would be t h e case f o r a h i g h e r c u r r e n t f l o w . The v a r i a t i o n i n l o s s due t o the exchange o f r e a c t i v e power between t h e two g e n e r a t o r s , however, i s much g r e a t e r t h a n f o r t h e case o f f i g u r e 3. T h i s i s due t o t h e f a c t t h a t w i t h a reduced impedance between t h e two g e n e r a t o r b u s s e s , a given d i f f e r e n c e i n v o l t a g e magnitude produces h i g h e r c u r r e n t t h a n p r e v i o u s l y . S i n c e t h e c u r r e n t i n c r e a s e s more r a p i d l y t h a n b e f o r e , t h e square o f t h e c u r r e n t i n c r e a s e s much more r a p i d l y , w i t h a c o r r e s p o n d i n g l y r a p i d increase i n loss. F i g u r e 6 shows t h e c o n t o u r s  f o r t h e same case:  as f i g u r e 2,  but w i t h a l o a d a t bus 3 o f o n l y 4 0 % o f t h e p r e v i o u s  value.  -38-  Here a g a i n the d r a m a t i c e l o n g a t i o n o f the c o n t o u r s i s verye v i d e n t . T h i s i s a g a i n due  t o the s u b s t a n t i a l decrease  in  l o a d i n g o f the t r a n s m i s s i o n l i n e between bus 2 and the l o a d at bus  3.  The c o n t o u r s o f f i g u r e 7 were o b t a i n e d by p l a c i n g a t r a n s former o f z e r o impedance i n the t r a n s m i s s i o n l i n e between busses 2 and 3 ( w i t h the t a p a t bus  3 ) . I n t h i s way  the t a p o f the  t r a n s f o r m e r c o u l d c o n t r o l the v o l t a g e a t l o a d bus 3 w i t h o u t d i r e c t l y a f f e c t i n g the c u r r e n t f l o w a l o n g the l i n e between busses 2 and 3. The c o n t o u r s of f i g u r e 7 t h e r e f o r e , are  essen-  t i a l l y the c o n s t a n t l o s s c o n t o u r s f o r the t r a n s m i s s i o n l i n e between bus 3 and bus 1. These c o n t o u r s a r e v e r y s i m i l a r t o the l i g h t l o a d c o n t o u r s f o r the whole system. F i g u r e 8 shows c o n t o u r s f o r a case i d e n t i c a l t o f i g u r e b u t w i t h a zero-impedance t r a n s f o r m e r as f o r f i g u r e 7.  4,  The  c o n t o u r s are a g a i n e l o n g a t e d , b u t t h i s time t h e y appear t o be e l l i p t i c a l , w i t h a minimum w i t h i n the p l o t range.  The  voltage  at bus 2 f o r t h i s c a s e , as f o r the case o f f i g u r e 7, i s one p e r - u n i t . T h i s e x p l a i n s the minimum l o c a t e d about a t a p o f and a v o l t a g e on bus  one  1 of one p e r - u n i t . The c l o s e d c o n t o u r s i n  t h i s example r e s u l t from the f a c t t h a t the t a p now c i r c u l a t i n g r e a c t i v e power f l o w throughout  controls a  the system.  The  minimum l o s s c o n d i t i o n o c c u r s f o r a c i r c u l a t i n g Q f l o w o f z e r o ,  -39-  w h i c h o c c u r s f o r a t a p s e t t i n g o f 1 and f o r no t r a n s f e r o f r e a c t i v e power between g e n e r a t o r s — i . e . when t h e v o l t a g e a t bus 1 a p p r o x i m a t e l y e q u a l s t h e v o l t a g e o f bus 2 a t one p e r - u n i t . One o f t h e o b j e c t i v e s o f t h e a l g o r i t h m b e i n g d e v e l o p e d i s t o a l l o c a t e r e a c t i v e power i n such a way as t o m i n i m i s e the t r a n s m i s s i o n l o s s e s i n t h e system, and so i t i s r e a s o n a b l e t o i n v e s t i g a t e t h e e f f e c t o f t h e power f a c t o r o f t h e l o a d a t bus 3 on t h e t r a n s m i s s i o n l o s s e s . F o r t h e cases o f f i g u r e s 9 and 10 the power f a c t o r o f t h e l o a d was reduced from t h e 9 5 % o f t h e p r e v i o u s c a s e s t o 80%. As can be seen from comparison w i t h t h e p r e v i o u s c a s e s ( f i g u r e s 2 and 6, r e s p e c t i v e l y ) , a l t e r i n g t h e power f a c t o r has a n e g l i g i b l e e f f e c t on t h e shape o f t h e c o n tours . T h i s i s f u r t h e r demonstrated by the c o n t o u r s o f f i g u r e s 11 t o 13. I n each c a s e , and p a r t i c u l a r l y i n t h e case o f f i g u r e 13 (branch between bus 1 and bus 2 ) , t h e c o n t o u r s a r e v e r y e l o n g a t e d f o r t h e v a r i a t i o n i n shunt a t bus 3. T h i s i s t r u e because, i n t h i s p a r t i c u l a r c a s e , t h e v o l t a g e magnitude r e l a t i v e l y i n s e n s i t i v e t o changes  a t bus 3 i s  i n t h e shunt a t bus 3. S i n c e  the r e a c t i v e power f l o w , and so t h e system l o s s , i s d e t e r m i n e d by t h e v o l t a g e o f bus 3 r e l a t i v e t o t h e v o l t a g e s o f busses 1 and 2, t h e system l o s s i s a l s o r e l a t i v e l y i n s e n s i t i v e t o changes  i n t h e shunt a t bus 3. (For V"2 = 1.0 pu and  = 1.0 pu,  -40-  V  3  0.0  v a r i e s from 0.9791 pu t o 0.9893 pu over the range o f shunts t o 0.7 pu f o r the case o f f i g u r e  11.)  There a r e f o u r c o n c l u s i o n s , t h e n , t h a t can be drawn from t h e s e r e s u l t s . F i r s t l y , i n t h e absence o f extreme c o n d i t i o n s , the l o s s w i l l depend more on g e n e r a t o r v o l t a g e s and on t r a n s former t a p s than-on l o a d power f a c t o r s . S e c o n d l y , the l o s s decreases w i t h a simultaneous increase i n generator v o l t a g e s . T h i r d l y , the l o s s d e c r e a s e s as the g e n e r a t o r v o l t a g e magnitudes approach a u n i f o r m v a l u e . * And l a s t l y , the degree o f e l o n g a t i o n o f the c o n t o u r s i n c r e a s e s v e r y r a p i d l y as the l o a d i n g d e c r e a s e s . ( C o n v e r s e l y , the c o n t o u r s become more c i r c u l a r as t h e l o a d i n g i n c r e a s e s . T h i s l a s t p o i n t suggests t h a t the o p t i m i s a t i o n p r o c e s s w i l l be more d i f f i c u l t under l i g h t - l o a d c o n d i t i o n s because o f t h e e l o n g a t i o n o f t h e c o n t o u r s , i n d i c a t i n g poor s c a l i n g o f the v a r i a b l e s . As the magnitude of the l o s s i s l e s s under l i g h t - l o a d , however, the o p t i m i s a t i o n can be t e r m i n a t e d before  .- c o m p l e t i o n w i t h l i t t l e p e n a l t y ; t h i s d i f f i c u l t y i s  not a s e r i o u s one.)  * T h i s p o i n t i s i m p o r t a n t i n t h a t i t spans the gap between the "equal v o l t a g e " c r i t e r i o n used by most u t i l i t y e n g i n e e r s and the "minimum l o s s " c r i t e r i o n p r e s e n t e d h e r e . G e n e r a l l y the two c r i t e r i a produce v e r y s i m i l a r r e s u l t s , as t h e y must i f the"minimum l o s s " c r i t e r i o n i s t o c a r r y c r e d i b i l i t y .  -41-  The E f f e c t o f the P e n a l t y Terms The e f f e c t of the p e n a l t y terms depends t o a l a r g e e x t e n t on the p e n a l t y f a c t o r s used. By a d j u s t i n g the p e n a l t y f a c t o r s , the b a l a n c e  i n the o b j e c t i v e f u n c t i o n between the l o s s and  two p e n a l t y terms can be s h i f t e d t o o b t a i n d i f f e r e n t  the  perform-  ance c h a r a c t e r i s t i c s . Because the dependence o f the o h j e c t i v e f u n c t i o n on l o s s i s l i n e a r , w h i l e the dependence on the p e n a l t i e s i s q u a d r a t i c , i t i s n o t . p o s s i b l e t o equate, f o r example, a l o s s o f 1 MW w i t h a v o l t a g e d e v i a t i o n o f 0.01  pu or a  r e a c t i v e shunt a l l o c a t i o n o f 1 MVAr. N e v e r t h e l e s s , i t i s p o s s i b l e t o determine the g e n e r a l e f f e c t of the p e n a l t y terms on the c o n t o u r s by u s i n g v a l u e s of 7.5  and 1.0  f o r the v o l t a g e  and r e a c t i v e power p e n a l t i e s , r e s p e c t i v e l y . (These v a l u e s were found t o g i v e a c c e p t a b l e performance i n the f i n a l program.) F i g u r e 14 i l l u s t r a t e s the e f f e c t o f the v o l t a g e p e n a l t y on the c o n t o u r s o f f i g u r e 2. The c o n t o u r s have been c l o s e d a t the h i g h v o l t a g e end of the p l o t range, and have become g e n e r a l l y rounded. The most extreme e f f e c t o f the v o l t a g e p e n a l t y i s observed  i n f i g u r e 15, w h i c h corresponds  t o the l i g h t - l o a d case  o f f i g u r e 6. Here the e l o n g a t e d c o n t o u r s have been rounded t o n e a r - c i r c u l a r . I n b o t h c a s e s , the minimum i s c l e a r l y bounded, as opposed t o the o r i g i n a l  cases.  The r o u n d i n g o f the c o n t o u r s o c c u r s i n a l l cases  except  -42-  those  i n w h i c h one parameter i s the shunt on bus 3.  shows the c o n t o u r s a l t y term.  F i g u r e 16  o f f i g u r e 11 augmented w i t h the v o l t a g e pen-  This l a c k of rounding  i s due m a i n l y t o the r e l a t i v e  i n s e n s i t i v i t y o f t h e bus 3 v o l t a g e t o t h e bus 3 s h u n t — a c h a r a c t e r i s t i c w h i c h was. p o i n t e d out e a r l i e r .  This voltage  i n s e n s i t i v i t y r e s u l t s i n o n l y a v e r y s m a l l change i n v o l t a g e p e n a l t y w i t h shunt, so t h a t the m o d e r a t i n g e f f e c t o f the v o l t a g e perialtyt;on::the r.coh.tour . shape i s m i n i m a l . The e f f e c t o f t h e r e a c t i v e power p e n a l t y c a n be seen i n f i g u r e 17, which c o r r e s p o n d s t o the same case as f i g u r e 16. The  contours  have now become n e a r l y c i r c u l a r .  This i s again  due  t o the r e l a t i v e i n s e n s i t i v i t y o f t h e v o l t a g e o f bus 3 t o  the shunt, which r e s u l t s i n the r e a c t i v e power p e n a l t y term dominating  t h e o b j e c t i v e f u n c t i o n i n t h e absence o f r e a c t i v e  power t r a n s f e r between g e n e r a t o r s .  Both t h e r e a c t i v e power  p e n a l t y term and t h e l o s s due t o an i n t e r c h a n g e  of r e a c t i v e  power a r e q u a d r a t i c terms, and so w i l l produce c i r c u l a r with a suitable  choice of r e a c t i v e penalty  contours  factor.  I t may appear from t h e f o r e g o i n g d i s c u s s i o n t h a t t h e r e i s l i t t l e p o i n t i n a l t e r i n g bus s h u n t s . this i s quite true.  F o r the example used h e r e ,  I n cases where the s e n s i t i v i t y o f a l o a d  bus v o l t a g e t o a bus shunt i s g r e a t , however, t h e r e w i l l be a s t r o n g v a r i a t i o n o f l o s s w i t h shunt, and t h e a s s o c i a t e d  contours  -43-  w i l l be d i s t i n c t l y more rounded t h a n f o r t h i s example system. As i t i s o n l y f o r cases e x h i b i t i n g t h i s s t r o n g s e n s i t i v i t y t h a t shunt c o n t r o l would be used, t h e r e i s no cause t o doubt t h e e f f e c t i v e n e s s o r u s e f u l n e s s o f shunt c o n t r o l from t h e p r e c e d i n g results.  R e a c t i v e Power L i m i t s on G e n e r a t o r s  The s e t o f e q u a t i o n s (8) t o (12) does n o t q u i t e d e s c r i b e the r e a c t i v e power management problem c o m p l e t e l y . A l l g e n e r a t o r s have l i m i t s on the r e a c t i v e power t h e y may absorb o r produce, and so a r e c o n s t r a i n e d by e q u a t i o n s ( 1 3 ) , analogous t o t h e equations  (12) f o r a l l o c a t i o n b u s s e s .  Because o f t h e analogous  s i t u a t i o n w i t h the r e a c t i v e a l l o c a t i o n b u s s e s , i t may seem desirable  t o t r e a t g e n e r a t o r r e a c t i v e power l i m i t s i n t h e same  m a n n e r — i . e . as p e n a l t y terms. T h i s would, i n f a c t , be a bad c h o i c e , as t h e c o n t o u r s o f f i g u r e 18 i n d i c a t e . penalised  G e n e r a t o r b u s s e s 1 and 2 i n t h i s case were  ( q u i t e l i g h t l y , i n f a c t ) t o 0.95 power f a c t o r .  The  e l o n g a t i o n o f t h e c o n t o u r s — e v e n w i t h t h e v o l t a g e term i n c l u d e d — i s c l e a r l y the worst y e t encountered.  The e x p l a n a t i o n i s t h a t  the r e a l and r e a c t i v e l o s s e s i n a power system a r e v e r y c l o s e l y r e l a t e d — i n f a c t , t h e y a r e p r o p o r t i o n a l f o r each b r a n c h i n t h e  -44-  system.  A l t h o u g h t h e sum o f t h e r e a c t i v e l o s s e s i n t h e system  w i l l n o t be q u i t e p r o p o r t i o n a l t o the sum o f t h e r e a l l o s s e s ( u n l e s s t h e X/R r a t i o i s c o n s t a n t f o r e v e r y b r a n c h ) , i t i s n e v e r t h e l e s s t r u e t h a t t h e minimum r e a c t i v e power l o s s  situation  w i l l c o r r e s p o n d g e n e r a l l y t o t h e minimum r e a l power l o s s s i t u a t i o n . S i n c e t h e minimum r e a c t i v e power l o s s s i t u a t i o n i s the minimum r e a c t i v e power g e n e r a t i o n c o n d i t i o n , t h e g e n e r a t o r s w i l l be c a l l e d upon t o generate t h e most r e a c t i v e power when t h e r e a l power l o s s i s h i g h , and v i c e v e r s a .  P e n a l i s i n g the generator Q  v i o l a t i o n s as a square term i s thus v e r y much l i k e u s i n g an o b j e c t i v e f u n c t i o n of  f =oC + k<£  a  where k = constant  which e x h i b i t s v e r y e l o n g a t e d c o n t o u r s . I t i s b e t t e r , t h e n , t o handle t h e g e n e r a t o r Q l i m i t s i n another way, t h e r e b y a v o i d i n g t h e m i n i m i s a t i o n problems which a t t e n d v e r y poor s c a l i n g .  An e f f e c t i v e method o f h a n d l i n g t h e  g e n e r a t o r l i m i t s w i l l be p r e s e n t e d i n the f o l l o w i n g c h a p t e r .  CHAPTER I V  The  The  Choice of a S u i t a b l e M i n i m i s a t i o n  problem described  the m i n i m i s a t i o n  b y e q u a t i o n s (8) t h r o u g h (13) r e q u i r e s  of a nonlinear  l i n e a r and n o n l i n e a r  Technique  objective function subject t o  constraints.  There a r e a g r e a t many p o s s -  i b l e approaches t o t h e s o l u t i o n o f t h i s type o f problem.  These  many approaches d i f f e r m a i n l y i n t h e degree t o which t h e y u t i l i s e information second, and h i g h e r  about t h e o b j e c t i v e f u n c t i o n  (such as f i r s t ,  o r d e r d e r i v a t i v e s ) , and t h e manner i n w h i c h  they t r e a t the c o n s t r a i n t s .  Constrained  o p t i m i s a t i o n i s gen-  e r a l l y much more complex than u n c o n s t r a i n e d o p t i m i s a t i o n , and i t i s therefore  i m p o r t a n t t h a t t h e c o n s t r a i n t s be t r e a t e d i n  the way l e a s t l i k e l y t o u p s e t t h e m i n i m i s a t i o n  The  procedure.  Treatment o f C o n s t r a i n t s  Both e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s c a n be h a n d l e d i n e i t h e r o f two ways.  F i r s t l y , t h e y c a n be h a n d l e d d i r e c t l y ,  wherein the e q u a l i t y c o n s t r a i n t s are solved along with the other conditions  f o r a minimum as a s e t o f s i m u l t a n e o u s e q u a t i o n s , and  the a c t i v e i n e q u a l i t y c o n s t r a i n t s a r e observed a t each s t e p as  -46-  a d d i t i o n a l equations  on t h e c o n t r o l v a r i a b l e s .  I n each case,  the c o n s t r a i n t s a r e met t o w i t h i n t h e p r e c i s i o n o f t h e c a l c u l a t i o n s a t e v e r y s t e p , so t h a t a l l i n t e r m e d i a t e s o l u t i o n s a r e f e a s i b l e . T h i s approach i s e s s e n t i a l l y i d e n t i c a l whether i t i s implemented as Lagrange m u l t i p l i e r s , g r a d i e n t . p r o j e c t i o n , o r g r a d i e n t  reduc-  tion. The second approach i s t o use t h e p e n a l t y f u n c t i o n t e c h n i q u e , i n w h i c h t h e s t e p s a r e p e r m i t t e d t o e n t e r and l e a v e t h e f e a s i b l e region at w i l l .  For steps terminating outside the f e a s i b l e  r e g i o n , a " p e n a l t y term" i s added t o t h e o b j e c t i v e f u n c t i o n . T h i s p e n a l t y term u s u a l l y i n c r e a s e s q u a d r a t i c a l l y as t h e s t e p leaves the f e a s i b l e region.  The s t e p s a r e t h u s "encouraged",  and n o t , as w i t h t h e p r e v i o u s approach, " f o r c e d " t o remain w i t h i n the boundaries of the f e a s i b l e r e g i o n . t i o n progresses,  As t h e m i n i m i s a -  t h e p e n a l t y term i s o f t e n m u l t i p l i e d by an  e v e r - i n c r e a s i n g f a c t o r , w h i c h t e n d s t o keep t h e i n t e r m e d i a t e s o l u t i o n s s u c c e s s i v e l y l e s s i n f e a s i b l e u n t i l , a t the s o l u t i o n p o i n t , t h e s o l u t i o n i s f e a s i b l e t o w i t h i n some t o l e r a n c e . Note t h a t b o t h t e c h n i q u e s course  are e s s e n t i a l l y the same—as of  t h e y must b e — i n t h a t t h e second approach uses t h e  minimisation process  t o s o l v e t h e same e q u a t i o n s  a l g e b r a i c a l l y by the f i r s t technique. solve the c o n s t r a i n t equations  as a r e s o l v e d  Because t h e two t e c h n i q u e s  d i f f e r e n t l y , i t i s reasonable t o  -47-  e x p e c t one  t e c h n i q u e t o be s u p e r i o r t o the o t h e r f o r c e r t a i n  forms o f c o n s t r a i n t s .  I f the e q u a t i o n s of c o n s t r a i n t can  s o l v e d a n a l y t i c a l l y w i t h o u t an e x t r a o r d i n a r y  be  amount of computa-  t i o n , the f i r s t method i s c l e a r l y s u p e r i o r , e s p e c i a l l y as intermediate  the  s o l u t i o n s f o r t h i s method are a l l f e a s i b l e , and  so  u s a b l e - - a l t h o u g h ' s u b - o p t i m a l . . An i m p o r t a n t d i s a d v a n t a g e of  the  1  penalty  f u n c t i o n approach i s t h a t the p e n a l t y  terms d i s t o r t  c o n t o u r s , o f t e n making the r e s u l t i n g augmented o b j e c t i v e  function  much more d i f f i c u l t t o m i n i m i s e (the g e n e r a t o r Q l i m i t s of l a s t c h a p t e r are an example). w h i c h may  the  the  For cases of c o n s t r a i n t e q u a t i o n s  not be e a s i l y s o l v e d a n a l y t i c a l l y , however, the  penalty  f u n c t i o n approach i s p r e f e r a b l e . The  c o n s t r a i n t s on the c o n t r o l v a r i a b l e s — t r a n s f o r m e r  generator voltages,  and  taps,  shunt r e a c t i v e s o u r c e a l l o c a t i o n s — c a n  be e a s i l y t r e a t e d as a b s o l u t e  l i m i t s on the a l l o w a b l e v a r i a t i o n  i n the c o n t r o l v a r i a b l e s , and  so h a n d l e d d i r e c t l y .  c o n s t r a i n t s , however, must be t r e a t e d by one discussed  above.  The  of the two  W h i l e the e q u a l i t y c o n s t r a i n t s c o u l d  be t r e a t e d w i t h e i t h e r of the two  other methods likely  t e c h n i q u e s , the i n e q u a l i t y  c o n s t r a i n t s on the g e n e r a t o r r e a c t i v e power l i m i t s s h o u l d be h a n d l e d as p e n a l t y  terms.  As was  demonstrated i n the  c h a p t e r , t h e s e p e n a l t i e s have a s e v e r e i n f l u e n c e on the f u n c t i o n c o n t o u r s , even w i t h a s m a l l m u l t i p l y i n g f a c t o r .  not last  objective Both  -48-  e q u a l i t y and  g e n e r a t o r r e a c t i v e power i n e q u a l i t y c o n s t r a i n t s  will  thus be t r e a t e d here u s i n g the a n a l y t i c method of c o n s t r a i n t handling. I t i s not p o s s i b l e t o s o l v e e x p l i c i t l y the n o n l i n e a r tion (9)  equa-  f o r the dependent v a r i a b l e s as a f u n c t i o n of the  dent v a r i a b l e s . the c u r r e n t  I f the f u n c t i o n g ( e q u a t i o n  intermediate  9)  i s expanded about  solution point i n a first-order  e x p a n s i o n , however, i t i s p o s s i b l e t o s o l v e the new t i o n e x p l i c i t l y f o r the independent v a r i a b l e s . between the dependent and  f o r the g r a d i e n t  f u n c t i o n t o o b t a i n a reduced e x p r e s s i o n  Taylor  linear  rela-  This r e l a t i o n s h i p  independent v a r i a b l e s may  s t i t u t e d i n t o the e x p r e s s i o n  indepen-  of the  then be  sub-  objective  i n w h i c h the g r a d i e n t  a f u n c t i o n of the independent v a r i a b l e s o n l y .  The  equations  Because o f the l i n e a r i z a t i o n of the c o n s t r a i n t  equation,  is are  c a l c u l a t e d i n appendix s e c t i o n A 2 .  each i n t e r m e d i a t e i b l e r e g i o n , and  s t e p w i l l not: n e c e s s a r i l y end w i t h i n the  feas-  i t i s thus n e c e s s a r y t o a d j u s t the s o l u t i o n  v e c t o r a t each s t e p t o c o r r e c t t h i s . do t h i s f o r the e q u a l i t y c o n s t r a i n t s  The (9)  most e f f i c i e n t way  to  i s t o s o l v e the s e t of  I t i s w o r t h n o t i n g here t h a t i f the o b j e c t i v e f u n c t i o n were l i n e a r i z e d a l o n g w i t h the c o n s t r a i n t s , and i f the minimum was assumed t o l i e a l o n g a c o n s t r a i n t boundary, the p r o b l e m c o u l d be s o l v e d u s i n g a s t a n d a r d l i n e a r programming program. This i s the method used i n [ 9 ] .  -49-  simultaneous  n o n l i n e a r e q u a t i o n s u s i n g a. c o n v e n t i o n a l Newton-  Raphson power f l o w program (the r e a d e r w i l l r e c a l l t h a t c o n s t r a i n t s (9) a r e j u s t t h e s e t o f power f l o w e q u a t i o n s ) . There a r e a. number o f advantages t o t h i s approach.  First,  the power f l o w program p r o v i d e s t h e s l a c k bus power and l o a d bus v o l t a g e s which a r e n e c e s s a r y f u n c t i o n a t each s t e p .  f o r the e v a l u a t i o n o f the o b j e c t i v e  Second, as was p o i n t e d o u t b y Dommel and  Tinney i n \l\ , t h e g r a d i e n t may be e a s i l y f o r m u l a t e d a t each s t e p from terms o f t h e J a c o b i a n m a t r i x  ( d e r i v e d i n appendix s e c t i o n  A l ) produced i n t h e power f l o w program.  As an a d d i t i o n a l advan-  t a g e , t h i s approach p e r m i t s t h e b u s - t y p e s w i t c h i n g p o r t i o n o f t h e power f l o w program t o ensure t h e s a t i s f a c t i o n o f t h e g e n e r a t o r r e a c t i v e power c o n s t r a i n t s o f e q u a t i o n ( 1 3 ) . Bus-type s w i t c h i n g i s t h e most common way o f e n s u r i n g t h e o p e r a t i o n o f g e n e r a t o r b u s s e s w i t h i n t h e i r r e a c t i v e power generation restrictions.  I t acts by s w i t c h i n g generator r  (constant  P, c o n s t a n t V) busses t o l o a d ( c o n s t a n t P, c o n s t a n t Q) busses whenever t h e y a r e no l o n g e r a b l e t o h o l d t h e s c h e d u l e d without exceeding  voltage  a Q l i m i t ( i . e . when t h e c o n s t r a i n t s become  active).  When t h e s c h e d u l e d v o l t a g e may a g a i n be h e l d w i t h o u t  exceeding  a Q l i m i t , t h e s w i t c h e d bus i s p e r m i t t e d t o r e v e r t  back t o c o n s t a n t v o l t a g e . S w i t c h i n g busses from type P,V t o P,Q amounts t o c h a n g i n g  -50-  the a c t i v e i n e q u a l i t y c o n s t r a i n t s i n t o e q u a l i t y c o n s t r a i n t s , reducing  t h e d i m e n s i o n o f t h e s o l u t i o n space by one f o r each bus  switched.  T h i s moves one bus v o l t a g e  from t h e s e t o f independent  v a r i a b l e s t o t h e s e t o f dependent v a r i a b l e s , t h e r e b y r e d u c i n g t h e dimensionality  o f t h e g r a d i e n t by one (which i s t h e same as  p r o j e c t i n g the f u l l gradient boundary).  onto t h e a p p r o p r i a t e  constraint  The bus-type s w i t c h i n g t e c h n i q u e used i n power  programs i s t h e r e f o r e  i d e n t i c a l i n i t s e f f e c t t o gradient  t i o n or gradient p r o j e c t i o n i n a constrained  flow reduc-  optimisation.  Summarizing, i t i s n o t p o s s i b l e t o use p e n a l t y  function  methods on t h e g e n e r a t o r i n e q u a l i t y c o n s t r a i n t s due t o t h e adverse e f f e c t s these p e n a l t i e s have on t h e o b j e c t i v e  function  contours.  reduction  I t i s p o s s i b l e , however, t o use a g r a d i e n t  t e c h n i q u e , by u s i n g t h e l i n e a r i z e d e q u a t i o n s f o r t h e e q u a l i t y and  a c t i v e i n e q u a l i t y c o n s t r a i n t s t o reduce t h e d i m e n s i o n a l i t y  of t h e g r a d i e n t ,  and then c o r r e c t i n g f o r t h e e f f e c t s o f t h e  l i n e a r i z a t i o n a t t h e end o f each s t e p . done u s i n g a c o n v e n t i o n a l  I f this correction i s  Newton-Raphson power f l o w program, t h e  o b j e c t i v e f u n c t i o n and i t s g r a d i e n t may be e v a l u a t e d  with  little  e x t r a e f f o r t , and t h e g e n e r a t o r r e a c t i v e power ( i n e q u a l i t y ) c o n s t r a i n t s a r e a u t o m a t i c a l l y s a t i s f i e d by b e i n g c o n v e r t e d t o e q u a l i t y c o n s t r a i n t s , when t h e y become a c t i v e , by t h e bus-type switching  algorithm.  -51-  The Method o f M i n i m i s a t i o n  I t has been presumed above t h a t the g r a d i e n t would be an e s s e n t i a l p a r t o f any s e l e c t e d m i n i m i s a t i o n t e c h n i q u e .  While  the m i n i m i s a t i o n c a n , o f c o u r s e , be p e r f o r m e d w i t h o u t knowledge of the g r a d i e n t , b e t t e r performance can u s u a l l y be r e a l i z e d by t a k i n g advantage o f t h i s and any o t h e r i n f o r m a t i o n about the objective function.  W i t h t h e method o f c o n s t r a i n t h a n d l i n g  d e s c r i b e d above, the c a l c u l a t i o n o f t h e g r a d i e n t r e q u i r e s a r e l a t i v e l y minor amount o f c o m p u t a t i o n (most o f w h i c h c o n s i s t s of one r e p e a t s o l u t i o n w i t h t h e f a c t o r i z e d J a c o b i a n m a t r i x  from  the power f l o w s t e p ) . I t i s a l s o p o s s i b l e , as has been p o i n t e d out by Sasson (13J t o c a l c u l a t e the m a t r i x o f second p a r t i a l  derivatives—called  the H e s s i a n m a t r i x — u s i n g the terms o f the J a c o b i a n m a t r i x as f o r the g r a d i e n t . ,  Using the Hessian matrix the m i n i m i s a t i o n  problem may be s o l v e d u s i n g a g e n e r a l i z e d v e r s i o n o f t h e NewtonRaphson method o f s o l v i n g c n o n l i n e a r e q u a t i o n s . . F o r o b j e c t i v e f u n c t i o n s w i t h e l o n g a t e d c o n t o u r s , the g e n e r a l i z e d NewtonRaphson method e x h i b i t s more r a p i d and r e l i a b l e convergence t h a n s t e e p e s t d e s c e n t o r m o d i f i e d s t e e p e s t d e s c e n t methods,  *  See appendix s e c t i o n A3.  -52-  w h i c h make use o f the g r a d i e n t o n l y . The d i s a d v a n t a g e  of Hessian-based techniques  a p p l i c a t i o n i s the l a r g e amount o f s t o r a g e and r e q u i r e d t o produce the H e s s i a n m a t r i x . [l4J  in this  computation  F u r t h e r , Himmelblau  p o i n t s out t h a t , w h i l e H e s s i a n - b a s e d t e c h n i q u e s  exhibit  q u a d r a t i c convergence i n the v i c i n i t y of the minimum, s t e e p e s t descent methods may  be s u p e r i o r f a r away from the minimum.  t h i s a p p l i c a t i o n , i t i s not n e c e s s a r y  For  t o know the optimum  e x a c t l y , b u t o n l y t o w i t h i n , p e r h a p s , a few p e r c e n t , so t h a t the major p o r t i o n o f the o p t i m i s a t i o n e f f o r t w i l l o c c u r away from the minimum.  B e a r i n g i n mind the o b s e r v a t i o n s o f the  last  c h a p t e r , where the o b j e c t i v e f u n c t i o n c o n t o u r s were found t o be only moderately e l l i p t i c a l  ( f o r a reasonable  s e l e c t i o n of  p e n a l t y f a c t o r s ) , w i t h no i r r e g u l a r i t i e s i n shape t o cause convergence f a i l u r e , the s t e e p e s t d e s c e n t method appears t o be a s l i g h t l y b e t t e r choice f o r t h i s a p p l i c a t i o n than based methods.  I t was  with a Lagrangian  Hessian-  the s t e e p e s t descent method, c o u p l e d  treatment  of e q u a l i t y c o n s t r a i n t s , which  was  chosen by Dommel and Tinney i n [ l j . In  o r d e r t o g a i n the maximum improvement a t each s t e p ,  s t e e p e s t descent  searches  g e n e r a l l y use s t e p l e n g t h s c a l c u l a t e d  t o t e r m i n a t e each s t e p a t the f u n c t i o n minimum i n each s u c c e s s i v e search d i r e c t i o n .  These s e a r c h e s  a r e termed " o p t i m a l s t e p - s i z e "  -53-  searches. The o p t i m a l s t e p l e n g t h can be approximated  from t h e v a l u e  o f t h e o b j e c t i v e f u n c t i o n , and perhaps i t s d e r i v a t i v e s , a t one o r more p o i n t s i n t h e c u r r e n t d i r e c t i o n o f s e a r c h .  The number  o f v a l u e s needed i s dependent upon t h e d e s i r e d a c c u r a c y o f t h e a p p r o x i m a t i o n , w h i c h determines for  interpolation  direction.  (or e x t r a p o l a t i o n ) i n t h e c u r r e n t s e a r c h  I f o n l y t h e f u n c t i o n v a l u e i s known a t each p o i n t ,  v a r i o u s types o f d i r e c t s e a r c h e s \l5^)  t h e o r d e r o f t h e p o l y n o m i a l used  (see, f o r example, Himmelblau  may be used. The  f i r s t d e r i v a t i v e of the o b j e c t i v e f u n c t i o n i n the  d i r e c t i o n of steepest descent i s the negative of the g r a d i e n t of the o b j e c t i v e f u n c t i o n ( t h i s i s the d i r e c t i o n a l d e r i v a t i v e of the o b j e c t i v e f u n c t i o n i n the d i r e c t i o n of search). g r a d i e n t i s e v a l u a t e d a t each p o i n t t o determine  As t h e  t h e next  direc-  t i o n o f s e a r c h , b o t h t h e v a l u e o f t h e o b j e c t i v e f u n c t i o n and i t s f i r s t d i r e c t i o n a l d e r i v a t i v e are a v a i l a b l e immediately, f u r t h e r work. (the  To i n t e r p o l a t e w i t h a second-order  without  polynomial  lowest order polynomial which can reasonably d e s c r i b e the  o b j e c t i v e f u n c t i o n i n t h e d i r e c t i o n o f s e a r c h ) , one o t h e r p i e c e of information i s r e q u i r e d . W h i l e t h i s m i s s i n g p i e c e o f i n f o r m a t i o n c o u l d be o b t a i n e d from a f u r t h e r f u n c t i o n e v a l u a t i o n i n t h e d i r e c t i o n o f s e a r c h ,  -54-  i t i s c o m p u t a t i o n a l l y more e f f i c i e n t t o approximate the second d i r e c t i o n a l d e r i v a t i v e o f the o b j e c t i v e f u n c t i o n .  The  reason  the second d i r e c t i o n a l d e r i v a t i v e must be approximated i t s e x a c t c a l c u l a t i o n would r e q u i r e the H e s s i a n gradient) matrix.  i s that  (second-order  I f t h i s m a t r i x were a v a i l a b l e , w h i c h would  r e q u i r e c o n s i d e r a b l e e f f o r t , i t c o u l d be used d i r e c t l y f o r a Hessian-based Two  minimisation.  o b s e r v a t i o n s p e r m i t the H e s s i a n t o be e a s i l y  F i r s t , Smirnov  [l6] has p o i n t e d out t h a t , as can be  g r a p h i c a l l y , f o r each t w o - d i m e n s i o n a l  approximated.  illustrated  p r o j e c t i o n o f the  (ellip-  t i c a l ) c o n t o u r space, a. s t e e p e s t descent s e a r c h converges t o the optimum a l o n g the major a x i s o f the e l l i p s e .  I n the m u l t i -  d i m e n s i o n a l c a s e , the s e a r c h w i l l converge a l o n g the major a x i s o f the h y p e r - e l l i p s o i d , w h i c h i s i n the d i r e c t i o n of the e i g e n v e c t o r c o r r e s p o n d i n g t o the minimum e i g e n v a l u e o f the matrix. may,  Hessian  The second d e r i v a t i v e i n subsequent d i r e c t i o n s o f s e a r c h  t h e r e f o r e , be approximated  as a c o n s t a n t e q u a l t o the  minimum e i g e n v a l u e . Second, as has a l r e a d y been p o i n t e d o u t , the  contours  d i s c u s s e d i n the p r e v i o u s c h a p t e r are n e a r l y c i r c u l a r .  This  i n d i c a t e s t h a t the H e s s i a n m a t r i x has d i a g o n a l terms w h i c h are a l l o f the same o r d e r o f magnitude, and o f f - d i a g o n a l terms which are r e l a t i v e l y s m a l l .  T h i s f u r t h e r improves the u s e f u l n e s s of  -55-  of Smirnov's o b s e r v a t i o n f o r t h i s type o f problem. The second d i r e c t i o n a l d e r i v a t i v e may, t h e r e f o r e , be c a l c u l a t e d on t h e b a s i s o f the p r e v i o u s s t e p , and t h e n used i n a second-order T a y l o r e x p a n s i o n t o p r e d i c t t h e optimum s t e p l e n g t h f o r the c u r r e n t step  (see appendix s e c t i o n A 4 ) .  Although  this  method f o r c a l c u l a t i n g t h e s t e p - s i z e -is approximate, t h e approxi m a t i o n improves d u r i n g t h e m i n i m i s a t i o n , and s u b s t a n t i a l l y l e s s computer time can be r e q u i r e d t h a n f o r t h e c a l c u l a t i o n o f the H e s s i a n  matrix.  Summary  The b e s t scheme f o r t h e s o l u t i o n o f e q u a t i o n s  (8) t h r o u g h  (13) i s , t h e r e f o r e , 1)  Newton-Raphson power f l o w s o l u t i o n , s a t i s f y i n g e q u a l i t y c o n s t r a i n t s , and e v a l u a t i n g t h e o b j e c t i v e f u n c t i o n and i t s g r a d i e n t .  (2)  steepest-descent  search, using a sub-optimal  s i z e c a l c u l a t e d from t h e p r e c e d i n g  s t e p assuming  c i r c u l a r o b j e c t i v e f u n c t i o n contours t h a t the Hessian k = constant).  i s o f t h e form  step-  ( i . e . assuming  -56-  (3)  equality constraints  (9) a r e h a n d l e d u s i n g  gradient  reduction. (4)  inequality constraints absolute  (10) - (12) a r e h a n d l e d as  l i m i t s on c o n t r o l v a r i a b l e v a r i a t i o n s  (analogous t o g r a d i e n t r e d u c t i o n  for active  constraints). (5)  inequality constraints  (13) a r e a u t o m a t i c a l l y  using gradient r e d u c t i o n by a bus-type  handled  switching  f e a t u r e i n t h e power f l o w program used a t s t a g e ( 1 ) . T h i s approach i s v i r t u a l l y i d e n t i c a l t o t h e g e n e r a l  approach  of Dommel and Tinney, and i s i n c o n t r a s t t o t h e more computat i o n a l l y complex scheme o f Sasson e t a l .  I t i s worth  noting  t h a t , a l t h o u g h Dommel and T i n n e y t r e a t e d t h e e q u a l i t y c o n s t r a i n t s as Lagrange terms, t h e e q u a t i o n s f o r t h e o p t i m i s a t i o n a r e m a t h e m a t i c a l l y i d e n t i c a l t o those d e v e l o p e d f o r t h e g r a d i e n t r e d u c t i o n approach used h e r e . t o be c o n s i d e r e d  I f the e q u a l i t y c o n s t r a i n t s are  u s i n g Lagrange m u l t i p l i e r t h e o r y ,  then the bus-  type s w i t c h i n g scheme i n t h e power f l o w r o u t i n e causes t h e a c t i v e i n e q u a l i t y c o n s t r a i n t s o f (13) t o be t r e a t e d as Kuhn-Tucker terms. here.  The approach o f [Y] i s thus i d e n t i c a l t o t h a t o u t l i n e d  -57-  CHAPTER V  The Performance o f t h e Technique  Because t h e t e c h n i q u e o u t l i n e d i n t h e l a s t c h a p t e r i s e s s e n t i a l l y an enhancement ( t o account f o r g e n e r a t o r r e a c t i v e power c o n s t r a i n t s and shunt c a p a c i t o r and r e a c t o r a l l o c a t i o n s ) of t h e r e a c t i v e power o p t i m i s a t i o n t e c h n i q u e d e s c r i b e d and  by Dommel  Tinney, i t was implemented b y m o d i f y i n g an a v a i l a b l e program  based on t h e t e c h n i q u e d e s c r i b e d  i n £l} .  As t h e r e was an i n t e r p o l a t i o n scheme i n ! t h e o r i g i n a l p r o gram, i t was r e t a i n e d on t h e assumption t h a t i t would improve the e s t i m a t e o f o p t i m a l s t e p - s i z e near t h e minimum,  thereby  s p e e d i n g convergence. As i t i s i n t h e v i c i n i t y o f t h e minimum that steepest  descent e x h i b i t s the worst performance, the  i n t e r p o l a t i o n p r o c e s s was a c t i v a t e d o n l y near t h e end o f t h e optimisation process. T h i s i n t e r p o l a t i o n r o u t i n e c a l c u l a t e s t h e approximate second d e r i v a t i v e assuming t h a t t h e f i r s t d i r e c t i o n a l d e r i v a t i v e depends o n l y on t h e c o n t r o l v a r i a b l e s equations derived  ( u j j , using the  i n appendix s e c t i o n A5.  T h i s was t h e o n l y m o d i f i c a t i o n t o t h e scheme d e v e l o p e d i n the l a s t c h a p t e r .  -58-  The p r o g r e s s o f t h e o p t i m i s a t i o n  t e c h n i q u e when p e r f o r m i n g  an u n c o n s t r a i n e d m i n i m i s a t i o n on t h e t h r e e - b u s system o f f i g u r e 1 i s p l o t t e d i n f i g u r e 19 on the o b j e c t i v e  function contours of  f i g u r e 14 ( f o r t h e c o n t r o l o f two g e n e r a t o r v o l t a g e s ) .  In  f i g u r e 20, t h e p r o g r e s s i s p l o t t e d on t h e c o n t o u r s f o r t h e c o n t r o l o f one t r a n s f o r m e r t a p and one g e n e r a t o r v o l t a g e (as for  f i g u r e 7 ) , and i n f i g u r e 21, i t i s p l o t t e d on t h e c o n t o u r s  of f i g u r e 17 f o r the c o n t r o l o f one bus v o l t a g e and one r e a c t i v e shunt ( a t bus 3 ) . As can be seen from these p l o t s , t h e optimisation  p r o g r e s s e s w e l l i n t h e f i r s t few s t e p s ,  r e a c h e s the minimum  and  (to w i t h i n p r a c t i c a l tolerances) a f t e r 3  steps. To t e s t t h e p r o c e d u r e on a r e a l i s t i c ,  fully  problem, a 1976 w i n t e r heavy l o a d r e p r e s e n t a t i o n  constrained o f t h e B.C.  Hydro system was used, c o n s i s t i n g o f 245 busses and 327 b r a n c h e s , w i t h 47 c o n t r o l l a b l e g e n e r a t o r s , and 44 c o n t r o l l a b l e (on-load tap-changing) t r a n s f o r m e r s . A f t e r 16 i t e r a t i o n s o f t h e m i n i m i s a t i o n ,  which r e q u i r e d  a  t o t a l o f 72 power f l o w i t e r a t i o n s , t h e s o l u t i o n was a c c e p t a b l y c l o s e t o t h e optimum.  C a r e f u l o b s e r v a t i o n o f the p r o g r e s s o f  convergence r e v e a l e d t h a t o c c a s i o n a l l y an i t e r a t i o n would a p p a r e n t l y d i v e r g e , r e s u l t i n g i n an i n c r e a s e the o b j e c t i v e  i n the value of  f u n c t i o n and/or t h e d e r i v a t i v e , b o t h o f which  -59must reduce f o r t h e p r o c e s s t o be c o n v e r g e n t .  This  apparent  d i v e r g e n c e would o c c u r for one o r two s t e p s , w i t h t h e n e x t s e v e r a l steps converging normally.  T h i s p r o c e s s may  occur  s e v e r a l times d u r i n g a m i n i m i s a t i o n (see Table I ) . The most p r o b a b l e cause f o r t h i s p e c u l i a r b e h a v i o u r  i s that  the s t e p - s i z e chosen a t each i t e r a t i o n i s o n l y an a p p r o x i m a t i o n to  the o p t i m a l s t e p .  This approximation  i s based on t h e  assumptions t h a t : a)  the o b j e c t i v e f u n c t i o n i s of order 2 or l e s s i n the d i r e c t i o n of search.  b)  t h e second d e r i v a t i v e o f t h e o b j e c t i v e f u n c t i o n i s constant f o r a l l d i r e c t i o n s of search  ( i . e . the  H e s s i a n m a t r i x i s d i a g o n a l , w i t h a l l d i a g o n a l terms equal). c)  the e q u a l i t y and a c t i v e g e n e r a t o r r e a c t i v e power i n e q u a l i t y c o n s t r a i n t s are approximately l i n e a r  over  the r e g i o n o f t h e s t e p . d)  no i n a c t i v e i n e q u a l i t y c o n s t r a i n t s w i l l become  active  d u r i n g t h e s t e p , and no a c t i v e i n e q u a l i t y c o n s t r a i n t s w i l l become  inactive.  I f any o f these assumptions a r e i n v a l i d f o r a g i v e n s t e p — a s , i n g e n e r a l , a t l e a s t one w i l l be-- the c a l c u l a t e d s t e p - s i z e w i l l be s u b - o p t i m a l .  Depending on t h e degree t o w h i c h t h e assump-  t i o n s a r e i n v a l i d , t h e c a l c u l a t e d s t e p - s i z e may become s u b -  TABLE I  P r o g r e s s o f Convergence f o r 245 Bus Problem  Step No.-  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  First Directional Derivative  7.37 2.68 2.23 3.27 2.61 1.03 0.853 0.674 0.797 9.37 9.05 1.14 0.911 1.56 1.73 0.382 0.271 0.238 0.364 0.175 0.122 0.098  S l a c k Bus Power plus penalties  15.88 15.58 15.52 15.50 15.48 15.46 15.45 15.44 15.42 15.59 15.64 15.44 15.43 15.42 15.43 15.41 15.41 15.41 15.41 15.41 15.41 15.41  -61-  o p t i m a l t o the p o i n t of b e i n g  divergent.  While--for . a steepest descent s e a r c h — a  s t e p - s i z e of l e s s  t h a n the o p t i m a l amount w i l l g e n e r a l l y o n l y h o l d the c o n v e r gence r a t e down somewhat, t o o l a r g e a s t e p - s i z e can cause the new  v a l u e f o r the o b j e c t i v e f u n c t i o n and/or g r a d i e n t t o be  g r e a t e r t h a n the p r e v i o u s v a l u e , g i v i n g a d i v e r g e n t s t e p . a s t e p - s i z e l a r g e r t h a n the optimum, whether or not the i t s e l f w i l l be d i v e r g e n t depends on by how i s too l a r g e , and on how  step  much the s t e p l e n g t h  r a p i d l y the o b j e c t i v e f u n c t i o n and  g r a d i e n t change i n the new  d i r e c t i o n of  search.  Of the above f o u r a s s u m p t i o n s , t h e r e are two which are most l i k e l y t o d i s t u r b the a p p r o x i m a t i o n size.  For  the  t o the optimum s t e p -  For problems w i t h a l a r g e number of i n e q u a l i t y c o n -  s t r a i n t s , the f o u r t h assumption w i l l l i k e l y be v i o l a t e d a t most s t e p s .  The  s w i t c h i n g of c o n s t r a i n t s from the i n a c t i v e  s e t t o the a c t i v e s e t produces a d i s c o n t i n u i t y i n the o p t i m i s a t i o n process which c o u l d give r i s e t o sporadic T h i s c o n s t r a i n t s w i t c h i n g was  divergence.  o b s e r v e d t o be o c c u r r i n g a t most  s t e p s i n the m i n i m i s a t i o n . The  second assumption i s a l s o known t o be sometimes  u n r e l i a b l e , as i t i m p l i e s t h a t the o b j e c t i v e f u n c t i o n c o n t o u r s must be c i r c u l a r , whereas t h e y are known t o be always e l l i p t i cal  t o some degree.  From t h i s o b s e r v a t i o n , we can p r e d i c t  -62-  t h a t t h e c a l c u l a t e d s t e p - s i z e w i l l be t o o s m a l l i n some d i r e c t , t i o n s , and t o o l a r g e i n o t h e r s . S i n c e the e s t i m a t e f o r t h e o p t i m a l s t e p - s i z e i s based upon the r a t e o f change o f the g r a d i e n t i n t h e d i r e c t i o n o f s e a r c h f o r t h e p r e v i o u s s t e p (and e v a l u a t e d over t h e span o f t h a t s t e p ) , a g r e a t e r - t h a n - o p t i m a l s t e p - s i z e w i l l most l i k e l y r e s u l t when the new d i r e c t i o n o f s e a r c h i s more p e r p e n d i c u l a r t o t h e major a x i s o f t h e c o n t o u r s t h a n was t h e l a s t .  The e f f e c t i s more  pronounced, and more r e a d i l y l e a d s t o d i v e r g e n c e , when t h e contours are s t r o n g l y e l l i p t i c a l .  I t i s m i t i g a t e d somewhat by  c o n s t r a i n t a c t i v a t i o n , i n t h a t the successive search d i r e c t i o n s are t h e r e b y a l t e r e d from t h e u s u a l n e a r - o r t h o g o n a l s e a r c h d i r e c t i o n s o f an u n c o n s t r a i n e d  (nearly-optimal step-size) search.  Of the two assumptions l i s t e d above as p o t e n t i a l causes o f the observed  s p o r a d i c d i v e r g e n c e , i t i s t h e second w h i c h i s the  p r o b a b l e major c o n t r i b u t o r . T h i s would account  f o r the f a c t  t h a t t h e d i v e r g e n t s t e p s o c c u r r a r e l y ; the s t e p s would u s u a l l y be convergent  u n t i l t h e c o n t o u r s became t o o e l l i p t i c a l .  One way t h i s c o u l d o c c u r , f o r example, i s when a g e n e r a t o r on a l o n g , l i g h t l y l o a d e d f e e d e r "came o f f " a minimum-Q  limit  ( t h a t i s , the g e n e r a t o r had been, b u t i s no l o n g e r , l i m i t e d t o i t s minimum a v a i l a b l e r e a c t i v e power o u t p u t ) .  I n the p l a n e o f  t h i s and some o t h e r g e n e r a t o r v o l t a g e , i t i s r e a s o n a b l e t h a t t h e  -63-  second g e n e r a t o r v o l t a g e may a f f e c t t h e o b j e c t i v e f u n c t i o n c o n s i d e r a b l y more than t h e f i r s t — l e a d i n g t o e l l i p t i c a l  contours  i n t h e p l a n e c o r r e s p o n d i n g t o these two g e n e r a t o r v o l t a g e s . When t h e f i r s t g e n e r a t o r reaches a mdnimum-Q l i m i t more l i k e l y ,  (which i s  under t h e c i r c u m s t a n c e s , than a high l i m i t ) ,  this  p l a n e v a n i s h e s because t h e f i r s t g e n e r a t o r v o l t a g e i s no l o n g e r a control variable.  T h i s would then  ( e f f e c t i v e l y ) reduce t h e  e l l i p t i c i t y of the m u l t i - d i m e n s i o n a l contours, s t a b i l i z i n g the a d a p t i v e s t e p - s i z e and thus t h e convergence. I t may appear from t h e above d i s c u s s i o n t h a t a b e t t e r a p p r o x i m a t i o n t o t h e o p t i m a l s t e p - s i z e , o r perhaps even a more p o w e r f u l u n c o n s t r a i n e d m i n i m i s a t i o n t e c h n i q u e i s needed.  These  are n o t n e c e s s a r i l y s o l u t i o n s , however, i n t h a t t h e improvement g a i n e d w i l l be much l e s s t h a n would be e x p e c t e d due t o t h e e f f e c t of  t h e i n e q u a l i t y c o n s t r a i n t s , and may n o t be s u f f i c i e n t t o  warrant the e x t r a c a l c u l a t i o n  necessary.  The e f f e c t o f t h e i n e q u a l i t y c o n s t r a i n t s i s r a t h e r h a r d t o p r e d i c t , other than t h a t they are l i k e l y  t o d i s t u r b the steady  convergence o f t h e same u n c o n s t r a i n e d problem.  This disturbance  i s so p o w e r f u l t h a t improvements i n t h e u n c o n s t r a i n e d o p t i m i s a t i o n t e c h n i q u e used a t each s t e p do n o t n e c e s s a r i l y speed up t h e overall solution.  I n p a r t i c u l a r , knowing t h e o p t i m a l s t e p - s i z e  i s o f l i t t l e advantage i f t h e s e a r c h d i r e c t i o n i s d e f l e c t e d from  -64-  the n e g a t i v e g r a d i e n t d i r e c t i o n by i n e q u a l i t y c o n s t r a i n t s .  The  same i s t r u e f o r H e s s i a n and r e l a t e d s e a r c h e s , i n t h a t t h e s e a r c h d i r e c t i o n c a l c u l a t e d may b e a r l i t t l e r e l a t i o n t o t h e f i n a l d e f l e c t e d search  direction.  A n o t h e r problem w h i c h e x h i b i t e d i t s e l f was t h e e f f e c t o f the v o l t a g e p e n a l t y f a c t o r s on t h e convergence r a t e . for  Values  t h e v o l t a g e p e n a l t y f a c t o r s which a r e t o o l a r g e c a n l e a d t o  e r r a t i c convergence, p r o b a b l y due t o t h e r e s u l t i n g of the o b j e c t i v e f u n c t i o n t o the v o l t a g e p e n a l t i e s .  sensitivity This i s  not u n d u l y s u r p r i s i n g , s i n c e i f the " c i r c u l a r i z i n g " e f f e c t o f the v o l t a g e p e n a l t i e s demonstrated i n c h a p t e r I I I i s c a r r i e d t o extremes, t h e c o n t o u r s w i l l become e l l i p t i c a l a g a i n , t h i s w i t h t h e minor and major axes i n t e r c h a n g e d .  time  Machine p r e c i s i o n  may a l s o be a p r o b l e m w i t h l a r g e v o l t a g e p e n a l t y m u l t i p l i e r s due to  t h e l a r g e r second d e r i v a t i v e terms and<-consequent s h o r t e r  step lengths. The use o f moderate v o l t a g e p e n a l t y f a c t o r s p e r m i t t e d t h e 500 kV busses at. remote g e n e r a t o r s i t e s t o r i s e w e l l over t h e d e s i r e d v a l u e o f 1.05 p e r u n i t t o v a l u e s as h i g h as 1.10 pu. L a r g e r v a l u e s o f v o l t a g e p e n a l t y f a c t o r s on t h e s e b u s s e s worsened convergence w i t h o u t d e c r e a s i n g t h e f i n a l v o l t a g e s  signif-  i c a n t l y , i n d i c a t i n g t h a t t h e h i g h remote s i t e v o l t a g e s were important t o the m i n i m i s a t i o n of the balance  of the t o t a l  -65-  o b j e c t i v e f u n c t i o n — i ,e. t h e maintenance o f r e a s o n a b l e  voltages  on o t h e r system busses and t h e r e d u c t i o n o f system l o s s . The b e s t way t o reduce these v o l t a g e s , where n e c e s s a r y , i s t o c o n t r o l t h e maximum v a l u e o f g e n e r a t o r v o l t a g e on t h e a s s o c i a t e d generator busses. as an a b s o l u t e l i m i t ,  Since the generator voltage i s t r e a t e d  t h i s w i l l p r e v e n t t h e h i g h - v o l t a g e bus  from e x c e e d i n g s a f e v o l t a g e l e v e l s .  T h i s p r o b l e m i s most l i k e l y  t o a r i s e when t r a n s f o r m e r t a p s , r e a c t o r banks, e t c . which were not made c o n t r o l l a b l e a r e p o o r l y s e t .  T h i s was, i n f a c t , t h e  problem w i t h t h e t e s t c a s e , as t o o many r e a c t o r s had been used a t s t a t i o n s between t h e remote g e n e r a t i o n and t h e l o a d c e n t e r . I t i s i m p o r t a n t t o r e a l i z e t h a t , w h i l e problems such as t h e h i g h v o l t a g e s noted above appear s e r i o u s , t h e y c a n i n f a c t be i m p o r t a n t c l u e s t o d e f i c i e n c i e s i n t h e power system on w h i c h the m i n i m i s a t i o n was o p e r a t i n g .  The o p t i m i s a t i o n p r o c e s s  acts  on t h e c o n t r o l v a r i a b l e s i n any way n e c e s s a r y t o a c h i e v e i t s objective.  P r o v i d e d always t h a t t h e o b j e c t i v e i s a r e a s o n a b l e  one, u n c o n v e n t i o n a l s o l u t i o n s may i n d i c a t e poor adjustment o f o t h e r parameters n o t c o n t r o l l a b l e b y t h e program, o r — a s i s perhaps t o o o f t e n t h e c a s e — m e r e l y  i n v e t e r a t e t h i n k i n g on t h e  p a r t of the person e v a l u a t i n g the s o l u t i o n . Other t h a n these i s o l a t e d h i g h v o l t a g e s , t h e r e s u l t i n g system s t a t e was much b e t t e r t h a n t h e a u t h o r had been a b l e t o  -66-  a c h i e v e when u s i n g t h i s case p r e v i o u s l y f o r system s t u d i e s w i t h a c o n v e n t i o n a l manual power f l o w , one major improvement b e i n g the system v o l t a g e  profile.  -67-  CHAPTER VI  Conclusions  The o p t i m a l r e a c t i v e power f l o w p r o b l e m can be s o l v e d u s i n g a s t e e p e s t d e s c e n t s e a r c h w i t h g r a d i e n t r e d u c t i o n (or e q u i v a l e n t l y , Lagrange) c o n s t r a i n t terms, u s i n g an o b j e c t i v e f u n c t i o n composed o f t h e s l a c k bus r e a l power, and v o l t a g e and r e a c t i v e power p e n a l t y terms f o r l o a d bus v o l t a g e s and a l l o c a t i o n respectively.  (load)  busses,  This o b j e c t i v e f u n c t i o n i s g e n e r a l l y w e l l - s c a l e d ,  and a s u b - o p t i m a l  s t e p - s i z e s e a r c h p e r f o r m s e f f e c t i v e l y on t h e  f u l l y c o n s t r a i n e d problem, p r o v i d e d t h a t t h e v o l t a g e p e n a l t y f a c t o r s are not excessive. The r e s u l t i n g s e t o f g e n e r a t o r and t r a n s f o r m e r  settings,  and s i z i n g s f o r shunt r e a c t i v e compensation banks produce a g e n e r a l l y good power f l o w case w i t h l e s s e n g i n e e r i n g t h a n would be r e q u i r e d u s i n g c o n v e n t i o n a l manual  effort  adjustments.  -68-  CHAPTER V I I  D i r e c t i o n s f o r F u r t h e r Work  The most i m p o r t a n t technique  r e m a i n i n g work i s t h e e v a l u a t i o n o f t h e  i n a p r o d u c t i o n environment.  e n c o u n t e r e d — t h e sporadic divergence  The two problems  and t h e d i f f i c u l t y i n  h o l d i n g down remote bus v o l t a g e s w i t h p e n a l t y terms not thought t o be s e r i o u s .  alone—are  The o n l y way o f c o n f i r m i n g  b e l i e f however, i s by o b t a i n i n g p r o d u c t i o n  this  experience.  In a d d i t i o n t o the aforementioned p r o d u c t i o n t e s t i n g , the t e c h n i q u e may need t o be extended t o c o v e r t h e many  automatic  f e a t u r e s a v a i l a b l e i n modern power f l o w programs (e.g. b l o c k s of shunt r e a c t i v e c a p a c i t y a c t i v a t e d by v o l t a g e magnitude, g e n e r a t o r s w i t h r e a c t i v e power a d j u s t e d t o h o l d remote bus voltages w i t h i n l i m i t s , e t c . ) .  I n many c a s e s , these  features  of power f l o w programs w i l l be made u n n e c e s s a r y by an o p t i m a l r e a c t i v e power management f e a t u r e .  Nevertheless,  allowance  must  be made f o r p o s s i b l e c o n f l i c t s between t h e o p t i m i s a t i o n p r o c e s s and t h e ( g e n e r a l l y r a t h e r crude) automatic  c o n t r o l p r o v i d e d by  such f e a t u r e s , and t h e f e a t u r e s s h o u l d be removed, i n h i b i t e d d u r i n g o p t i m i s a t i o n , or f o r m a l l y i n c o r p o r a t e d i n t o the o p t i m i s a tion  process.  -69-  One f u r t h e r problem w h i c h s h o u l d be i n v e s t i g a t e d i s m u l t i p l e contingency o p t i m i s a t i o n .  T h i s i s somewhat d i f f e r e n t t h a n  o p t i m i s a t i o n f o r a normal o p e r a t i n g c o n d i t i o n , i n t h a t the outage c o n d i t i o n s must meet c e r t a i n minimum o p e r a t i n g l i m i t s on l o a d bus v o l t a g e )  with only on-load transformer  (generally  t a p s , gen-  e r a t o r v o l t a g e s , and s w i t c h a b l e banks o f shunt compensation being adjusted  from normal o p e r a t i n g s e t t i n g s .  o f f - l o a d transformer  T h i s means t h a t  taps must be s e t so t h a t i t i s p o s s i b l e t o  a c h i e v e t h e minimum o p e r a t i n g l i m i t s u s i n g o n l y t h e a d j u s t a b l e parameters.  A l t h o u g h the o p t i m i s a t i o n may be p e r f o r m e d sep-  a r a t e l y on each c o n t i n g e n c y c o n d i t i o n , a method i s r e q u i r e d f o r e f f i c i e n t l y c o m b i n i n g t h e s e p a r a t e c o n t i n g e n c y optima i n t o a single  result.  -70-  REFERENCES  H.W.  Dommel and W.F. T i n n e y . "Optimal Power F l o w  S o l u t i o n s . " IEEE Trans., v o l . PAS-87, O c t . 1968, pp. 1866-76.  A l b e r t M. Sasson. "Combined Use o f t h e P o w e l l and F l e t c h e r - P o w e l l N o n l i n e a r Programming Methods f o r O p t i m a l Load F l o w s . " IEEE T r a n s . , vol. ^PAS-88, ;  No. 10, O c t . 1969, pp. 1530-1537.  A S u r v e y o f Canadian U t i l i t y P r a c t i c e s i n P l a n n i n g & A p p l i c a t i o n o f S t a t i o n Shunt C a p a c i t o r Banks. By K. N i s h i k a w a r a , Chairman. T o r o n t o , O n t a r i o : Canadian E l e c t r i c a l A s s o c i a t i o n Power System P l a n n i n g & O p e r a t i o n S e c t i o n , 1976.  W.F. T i n n e y and H.W.  Dommel. "Steady S t a t e S e n s i t i v i t y  A n a l y s i s . " Report No. 3.1/10, 4 t h Power Systems Computation C o n f e r e n c e , Grenoble 1972.  ( F r a n c e ) , Sept. 11-16,  -71-  Raymond M. M a l i s z e w s k i , Len L. G a r v e r , and A l l e n J . Wood. " L i n e a r Programming as an A i d i n P l a n n i n g K i l o v a r Requirements."  IEEE Trans., v o l . PAS-87,  No. 12, Dec. 1968, pp. 1963-67.  H.W. Dommel. "Input-Output S e n s i t i v i t i e s . " U n p u b l i s h e d notes.  A. K i s h o r e and E.F. H i l l .  " S t a t i c O p t i m i z a t i o n of  R e a c t i v e Power Sources b y Use o f S e n s i t i v i t y Parameters." IEEE T r a n s . , v o l . PAS-90, No. 3, May/June 1971, pp. 1166-73.  N.P. K o h l i and J.C. K o h l i . Allocation  "Optimal C a p a c i t o r  by 0-1 Programming." Paper A 75 476-2,  IEEE Summer Power M e e t i n g , San F r a n c i s c o , C a l i f . , J u l y 20-25, 1975.  A. K u p p u r a j u l u and K. Raman Nayar. " M i n i m i s a t i o n o f Reactive-Power I n s t a l l a t i o n i n a Power System." P r o c . I E E , v o l . 119, No. 5, May 1972, pp. 557-563.  -72-  10  D a v i d M. Himmelblau. A p p l i e d N o n l i n e a r Programming. M c G r a w - H i l l , New York, 1972, c h a p t e r 6.  11  S.S. Sachdeva and R. B i l l i n t o n .  "Optimum Network VAr  P l a n n i n g by N o n l i n e a r Programming." IEEE Trans., v o l . PAS-92, J u l y / A u g . 1973, pp. 1217-25.  12  . "Optimum Network VAr P l a n n i n g U s i n g R e a l and R e a c t i v e Power D e c o m p o s i t i o n N o n - l i n e a r A n a l y s i s . " P r o c . 8 t h Power I n d u s t r y Computer A p p l i c a t i o n s C o n f e r e n c e , M i n n e a p o l i s , Minn., 1973, pp. 339-347.  13  A.M.  Sasson, F. V i l o r i a , and F. A b o y t e s . "Optimal  Load F l o w S o l u t i o n U s i n g the H e s s i a n M a t r i x . " IEEE T r a n s . , v o l . PAS-92, Jan./Feb. 1973, pp.31-41.  14  Himmelblau. A p p l i e d N o n l i n e a r Programming, pp. 87-88.  15  I b i d . C h a p t e r 2.  16  K.A.Smirnov. " O p t i m i z a t i o n o f the Performance o f a Power System by the D e c r e a s i n g G r a d i e n t Method." I z v e s t i y a A k a d e m i i Nauk SSSR, s e r . E n e r g e t i k a i  -73-  Transport  (News o f t h e Academy o f S c i e n c e s USSR, Power  E n g i n e e r i n g and T r a n s p o r t S e r i e s ) , Moscow, No. 2, 1966, pp. 19-28, i n R u s s i a n . T r a n s l a t e d f o r 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 by t h e J o i n t P u b l i c a t i o n s Research S e r v i c e s , May 1967.  -74-  APPENDIX  D e r i v a t i o n of Relevant  Al  Equations  The Terms o f t h e J a c o b i a n M a t r i x  The power e n t e r i n g the system a t bus i i s g i v e n by  S* = (P. - j Q.) = V  ±  (G.. + j B..) +  V  i  (cos ©  2  i  - j sin6  i  ) *  2  (G.. + j B..) V. (cos 9. + j s i n 9.) ;ewcr J  J  J  J  J  B r e a k i n g t h i s e q u a t i o n i n t o r e a l and i m a g i n a r y components  V^j [cos  Pi = V  sin e  ±  Q±  ( G ^ cos  (G j s i n 9j +  - B-j-j s i n 9^) + cos 9 j ) ] + V ? G  ±  i;L  Q i = - V i ^ V ^ r c o s 0. {G-- s i n 0. + B , cos 9.) H  sin 9  ±  (G  ± j  cos 9 j -  s i n 9j)] -  B i j  The terms o f the J a c o b i a n m a t r i x a r e :  H l l  -ae,:  N  "  "  V  i  ^  J  l  1  " ^e;  L  l  l  "  V  l  ^  -75-  where AP and AQ are the r e a l and r e a c t i v e power mismatches (positive into bus), respectively  Define  «ij  =  G  ij  /^ij  =  G  i j ^-  c o s  s  n  9  j " ij  e  j  B  +  B  ii  s i n  c o s  j  e  i  9  so ^4-  Now,  = -3- •  = o(.  since ^ i ~ gen^ ~ l o a d ^ p  p  ^Qi = Q vl'.r  ~  p  - Qioad  g e n i  i  p  ±  - Qi  where p  ge  n i  '  P  load ' i  Qgen.^'  Qloadi  a n d  a r e  constant,  then  H  i i  =  " V ^ V j Q. +  V  i  v V .  j  cos 9.  s i n 9.)  -76-  Nij = - v i i  N  * • = - V i V j (oCij cos 9 i + ^ i j f t  = - v.  a P i  -= ~V $ V .  s i n 6i)  (*.. cos 9-  H  sin6  -2 V G.. = - P. - V G.. l l l l ll l 2  ij  J  = - ^  = V  V j (<X  ±  J . . =-. ll  ..  =  cos 9  ij  +  i  sin 0 ) ±  V. 2.V. (of. . cos 0. +/3\ . s i n G.) 1J£A/CIJ ij x f xj i — 1P. + — —j l  L  2  - V.4& = j avT  L. . = - V l®i' = 11  v  V G.. ll l 2  i J v  pencil -2 V ? B  c o s  ± i  i  e  /"ID = -Q  s i n  i + V?B  ±  e  i)  i ±  The Terms o f the G r a d i e n t  The o b j e c t i v e f u n c t i o n i s  f(x,u) = P (x,u) + .^.w (V s  i  ^z.  .  i  - ?  c  v  h  e  d  )  2  +  sched.2  ( B . - B.  )  where -T x _T u  lev]  V  = voltage  B  c  [t vc  B]  on busses o f t h e s e t NG  = r e a c t i v e shunt on busses o f the s e t NQ  -77-  L e t F(u) = f ( x , u ) .  The g r a d i e n t 7 F  U  may be found by  o b s e r v i n g t h a t the f i r s t - o r d e r v a r i a t i o n o f F i s g i v e n by  A F = -|j-Au + ~ A x  Now,  e x p a n d i n g the e q u a l i t y  = V F ^ Au  constraint  (A2.1)  (9) i n a f i r s t - o r d e r  Taylor expansion:  g(x,u) = g ( x , u ) + j j J ^ A x +[-|J]AU 0  Q  S i n c e g(x,u) = g ( x , u ) = 0, Q  Substituting  0  (A2.2) i n t o  (A2.1)  A  The g r a d i e n t V F  U  "  = VFJ  AU  i s thus g i v e n by  = i^_r42.r^_ri£ i  (  „.  3)  -78-  T h i s i s t h e same e x p r e s s i o n f o r t h e g r a d i e n t as Dommel and T i n n e y o b t a i n f o r t h e g r a d i e n t i n £lj u s i n g the Lagrange m u l t i p l i e r approach. The terms o f t h i s e x p r e s s i o n a r e : ZAP  ae  2AQ (A2.4)  iAf  av  i£_ ax  T  I  L  dV  av/  Mf at ^>AP  au  + 2w(V - V  S C h e d  i _  aaa at  iAQ  (A2.6)  avc  -0.-  (A2.5)  aaa ae  E q u a t i o n (A2.3) c a n be c a l c u l a t e d  e a s i l y by t a k i n g advantage o f  the f a c t t h a t t h e e x p r e s s i o n  (which c o r r e s p o n d s t o t h e v e c t o r o f Lagrange m u l t i p l i e r s  i n the  -79-  Dommel and T i n n e y approach) can be o b t a i n e d from one  repeat  s o l u t i o n u s i n g the t r a n s p o s e d f a c t o r i z e d J a c o b i a n m a t r i x from the Newton-Raphson power f l o w . E x p r e s s e d i n terms o f t h e J a c o b i a n m a t r i x , t h i s e x p r e s s i o n becomes -tf  - J  The terms f o r (A2.4) and (A2.5) can be o b t a i n e d d i r e c t l y from the J a c o b i a n m a t r i x .  MPi it  The terms f o r (A2.6) and (A2.7) a r e  W i j cos 0 i H - ^ i j s i n e ) = - ML ±  i f i i s non-tap s i d e bus, o r 2  \jf Gij  i f i i s t a p s i d e bus.  Similarly, Li  i f i i s non-tap s i d e bus, or  'J  +  V  i f i i s t a p s i d e bus  -80-  - N. N. . ll  = LID  = 0 - Vi  A3  The Terms o f t h e H e s s i a n M a t r i x  F o l l o w i n g t h e same p r o c e d u r e as f o r t h e c a l c u l a t i o n  of the  terms o f t h e g r a d i e n t , l e t F(u) = f ( x , u ) , so t h a t  A F = PF^Au + h A u ] > F ] Au T  u u  U s i n g t h e same e x p r e s s i o n  f o r A x i n terms o f 4u as i n s e c t i o n A2,  -81-  The e v a l u a t i o n o f t h i s m a t r i x r e q u i r e s a c o n s i d e r a b l y  greater  amount o f c o m p u t a t i o n t h a n does t h e e v a l u a t i o n o f t h e g r a d i e n t The a d d i t i o n a l m a t r i c e s needed a r e :  <7 f "uu 2  £-P  0  s  2Z  0  aP aea*/ 2  s  <J8a© V f 2  xx  + 2(  avai/  7 f  aeat  ae av  ^avat  a^Ps av^i/  c  2  xu  0 c  -82-  f o r w h i c h the terms a r e :  <r  P  S  \2 P 5_£s_  p. Ms i J 2%  - _  ? j'  gs/  Gs  s  s - non-tap s i d e bus, o r = t a p s i d e bus.  .0 _  -  A/5/' N  *i  s = non-tap; s i d e bus, o r + -WsGsj  A/sJ ^Ps  A/sJ  _  3P _ 2 G 3 V/ ^ ss 2  S  ^  _  0 " sj N  f%  - - p„ + s ss s v  G  Nsj <?V,  _ 0  S-H  -  V B s ss  fPs  _  *-y  s  _  s  i^  e  >j  U S >  -83-  A4  The A p p r o x i m a t i o n  t o the Optimal  Step-size  L e t F(u) = f ( x , u ) as i n s e c t i o n A2.  Now expand F (u) t o  second o r d e r :  F(u) = F  +  V F„ An + h A u  O For s t e e p e s t  ^  T  U  T  T^F I-  ~1 Au  UU  J  descent  =  _ c£AJ>Tu  where [A] r e p r e s e n t s t h e r o t a t i o n and s c a l i n g o f flu as a r e s u l t of c o n s t r a i n t r e f l e c t i o n  (diagonal  matrix).  Substituting,  F(u) = F  so t h a t  - ^cCAiVF^  j_  C ' ^ ^ M ' C ^ U W ^  -84-  do  jlCAl^^f  1  Assuming t h a t the o b j e c t i v e f u n c t i o n i s o f o r d e r 2 o r l e s s i n the d i r e c t i o n o f s e a r c h ,  d£  k+/  _  i£  so t h a t 7 - ^ = G ( c o n s t a n t ) , t h e n  k  The d e r i v a t i v e a t s t e p k.+ 1 i n t h e d i r e c t i o n o f s t e p k i s  so  that  G =  -  The o p t i m a l s t e p - s i z e w i l l ensure t h a t by  k+1  V fu T  =  LA]  — ac  = 0, and i s g i v e n  -85-  and  r-k+l _ ~  -c L A l t7Fu KlAl**' VF +'[\ k  S u b s t i t u t i n g [AJVF*  A t s t e p k+1, DT  =  '^^^1  d  an  !ldu |/ k  i s s t i l l unknown, and so we  1  2  = (c ) k  2  assume  - [i]  giving  — k+1 4u  For the f i r s t s t e p , t h e r e i s no p r e v i o u s t h e r e i s no i n f o r m a t i o n f o r — — — i n a cr  information,  so  -86-  F  = F  2  -H^FJU C  1  In order t o solve t h i s t h a t an o p t i m a l  + hie ) 1  1  for c  0 -  2  w i t h o u t knowing  s t e p w i l l reduce t h e o b j e c t i v e f u n c t i o n by an  a r b i t r a r y amount.  E x p e r i e n c e i n d i c a t e s t h a t 2% i s r e a s o n a b l e ,  so  F  2  - F  1  = -0.02 F  1  and  -0.02 F  1  = -grille  1  + ^ ( c  1  )  ^ - a c** 2  Since the step i s optimal,  which i m p l i e s  that  and -0.02 F  1  , we assume  = -fj^ul/c  1  + h c ^ T F u l l = -h c ^ T P J K  -87-  Thus  and  A5 '.' I n t e r p o l a t e d  Step-size  I f the s i g n o f any p a r t i a l d e r i v a t i v e changes, the c o r r e s p o n d i n g v a r i a b l e can be i n t e r p o l a t e d , as the minimum i n t h a t d i r e c t i o n has been p a s s e d . Assuming  = Rj_ ( c o n s t a n t )  3u*  then  1E1  k+1  6ui  k+2  For an o p t i m a l  step,  IE1 a«*i  = 0 and  -88k+1  •if_  k+1  A6  P r o o f t h a t the O b j e c t i v e F u n c t i o n  ^)<l(( |x( for a l l 2  System Branches E q u a l i z e s t h e P r o d u c t /jljf )x|  C o n s i d e r t h e f o l l o w i n g s e c t i o n o f a power system: 3  The t o t a l c u r r e n t I  T  = 1^ +  i s assumed c o n s t a n t , w h i l e  the two component c u r r e n t s Ij_ and I  2  may be a l t e r e d by a d j u s t -  ment o f the t r a n s f o r m e r t a p t . The r e a c t i v e power l o s s i n these two branches i s g i v e n by  = ! l + *2 2 = 1 1 + < I I  x  X  The v a l u e o f I j f o r which ^  X  X  T  "  1  2 x  2-  - 2 ( I - I• l)'X2 = 0 T T  by  w i t h r e s p e c t t o 1^ e q u a l t o  zero.  1  l)  i s a minimum can be determined  g  s e t t i n g the f i r s t d e r i v a t i v e of  = 2I X  I  n  A  9  -89-  which i m p l i e s that  l l x  -  I  2 2' X  Q.E.D.  = 5.14 pu 5 =-2.076 pu Q =-0.535 pu  Y„ = 4.989-j 29.72 pu Y, =-4.989 + j 29.84 pu Y = 6.063-j 28.4 pu Y„=-6.063 + j 28.67 pu ^3=11.05-j 58.12 pu 3  3  on 100 MVA  M  a  230 kV  F i g u r e 1. Three-bus example system.  B U S I VOLTAGE  F i g u r e 2. Contours o f c o n s t a n t l o s s f o r system o f f i g u r e 1. Voltages are per unit based on 230 kV, and c o n t o u r v a l u e s a r e MW.  BUS I VOLTAGE  F i g u r e 3. Contours o f c o n s t a n t l o s s f o r system o f f i g u r e 1, except t h a t P2 = 1 . 0 pu, so t h a t b o t h g e n e r a t o r s p r o v i d e a p p r o x i m a t e l y h a l f o f t h e bus 3 r e a l power each.  o  o  t  £ = 1.0 pu £ = - 2 . 0 7 6 pu Q =-0.535 pu  Y = 6.985-j 41.37 pu Y =-l.996 + j 11.94 pu Y = -4.989 + j 29.84 pu Y = 8.059 - j 40.05 pu Y „ = - 6 . 0 6 3 + j 28.67 pu fl  I2  3  3  2l  Y = 1105-j 58.12 pu 33  on 100 MVA  a  230 kV  F i g u r e 4.^ Three-bus example system o f f i g u r e 1, w i t h a d d i t i o n a l h y p o t h e t i c a l b r a n c h between busses 1 and. 2, and P.2""•= 1.0 pu.  ?  0 0.85  BUS  I VOLTAGE  F i g u r e 5. Contours o f c o n s t a n t system o f f i g u r e 4.  0.95  BUS  loss f o r  1.05  100  VOLTAGE  F i g u r e 6. Contours o f c o n s t a n t l o s s f o r system o f f i g u r e 1, b u t w i t h o n l y 4 0 % o f the l o a d a t bus 3 ( P = 2.056 pu, S3 = -0.83 - j 0.214 p u ) . 2  BUS I VOLTAGE F i g u r e 7. Contours o f c o n s t a n t l o s s v e r s u s v o l t a g e a t bus 1, and t a p s e t t i n g o f h y p o t h e t i c a l , z e r o impedance t r a n s f o r m e r i n s e r t e d a t t h e bus 3 end o f t h e b r a n c h between busses 1 and 3. The t a p i s on t h e bus 3 s i d e . Otherwise t h e system i s i d e n t i c a l t o t h a t o f f i g u r e 1.  BUS I VOLTAGE F i g u r e 8. Contours o f c o n s t a n t l o s s v e r s u s v o l t a g e a t bus 1, and t a p s e t t i n g o f h y p o t h e t i c a l transformer i d e n t i c a l t o that f o r f i g ure 7. Otherwise t h e system i s i d e n t i c a l t o t h a t o f f i g u r e 4.  42.0  38.0  C M  30.0  270  24.0  21.0  18.0  0.95  B U S I VOLTAGE  BUS  F i g u r e 9. Contours o f c o n s t a n t l o s s v e r s u s ' v o l t a g e . System i s t h a t o f f i g u r e 1, e x c e p t -that t h e power f a c t o r o f t h e l o a d a t bus 3 i s o n l y 8 0 % (so t h a t S = -1.715 - j 1.286 pu) . 3  15.0  12.0  1.0  9.0  1.05  I VOLTAGE  F i g u r e 10. Contours o f c o n s t a n t l o s s v e r s u s v o l t a g e . System i s t h a t o f f i g u r e 1, e x c e p t t h a t t h e l o a d a t bus 3 has been reduced t o o n l y 4 0 % (as f o r f i g u r e 6 ) , and t h e power f a c t o r reduced t o 80%. Thus S = -0.686 j 0.515 pu. 3  -28.0 -27.0 - 26.0  - 25.0  ro O I-  UJ  §  CVJ  o >  " o cr. 25.5  CO 03  24.0  I  lj_  27.0  0_  • 28.5  I—  25.0  I  26.0  30.0  BUS  3 SHUNT  F i g u r e 11. Contours o f c o n s t a n t l o s s v e r s u s v o l t a g e a t bus 2, and t h e v a l u e o f r e a c t i v e shunt a t bus 3. Other t h a n t h e bus 3 s h u n t , the system i s t h a t o f f i g u r e 1.  (J\  27.0  BUS  3  SHUNT  F i g u r e 12. Contours o f c o n s t a n t l o s s v e r s u s the t a p on t h e h y p o t h e t i c a l t r a n s f o r m e r (as f o r f i g u r e 7 ) , and bus 3 r e a c t i v e shunt (as f o r f i g u r e 1 1 ) . Otherwise t h e system i s t h a t o f f i g u r e 1.  F i g u r e 13. Contours of c o n s t a n t l o s s v e r s u s v o l t a g e a t bus 2, and the v a l u e of r e a c t i v e shunt a t bus 3. Other t h a n the bus 3 shunt, the system i s t h a t o f f i g u r e 4.  F i g u r e 14. Contours of f i g u r e 2 augmented w i t h a v o l t a g e p e n a l t y term f o r bus 3 v o l tage. The p e n a l t y f a c t o r i s 7.5, and the maximum and minimum u n p e n a l i z e d v o l t a g e s are 1.05 and 1.00 pu, r e s p e c t i v e l y .  F i g u r e 15. Contours o f f i g u r e 3 augmented w i t h a v o l t a g e p e n a l t y term as f o r f i g u r e 14.  F i g u r e 16. Contours o f f i g u r e 11 augmented w i t h a v o l t a g e p e n a l t y term as f o r f i g u r e 14.  BUS  3  SHUNT  F i g u r e 17. C o n t o u r s o f f i g u r e 16 augmented w i t h a p e n a l t y term f o r the shunt r e a c t i v e power i n j e c t e d a t bus 3. Any amount o f r e a c t i v e i n j e c t i o n i s p e n a l i z e d , w i t h the p e n a l t y f a c t o r b e i n g 1.0.  F i g u r e 18. Contours of f i g u r e 14 augmented w i t h a p e n a l t y term f o r t h e r e a c t i v e power produced o r absorbed by g e n e r a t o r s 1 and 2. The g e n e r a t o r s a r e a l l o w e d t o produce (or absorb) r e a c t i v e power t o a power f a c t o r o f 0.95, e x c e s s r e a c t i v e power b e i n g p e n a l i z e d w i t h a p e n a l t y f a c t o r of 1.0.  F i g u r e 19. Contours o f f i g u r e 14, on which has been p l o t t e d t h e p r o g r e s s o f t h e p r o g rammed o p t i m i s a t i o n method.  Figure with a 14, on of the  20. Contours o f f i g u r e 7 augmented v o l t a g e p e n a l t y term as f o r f i g u r e which has been p l o t t e d the p r o g r e s s programmed o p t i m i s a t i o n method.  F i g u r e 21. Contours o f f i g u r e 17, on which has been p l o t t e d the p r o g r e s s o f the p r o g rammed o p t i m i s a t i o n method.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065613/manifest

Comment

Related Items