UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

General equations for short-range optimization of a combined hydro-thermal electric system Arismunandar, Raden Artono 1960

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1960_A7 A7 G3.pdf [ 5.71MB ]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0105072.json
JSON-LD: 1.0105072+ld.json
RDF/XML (Pretty): 1.0105072.xml
RDF/JSON: 1.0105072+rdf.json
Turtle: 1.0105072+rdf-turtle.txt
N-Triples: 1.0105072+rdf-ntriples.txt
Original Record: 1.0105072 +original-record.json
Full Text
1.0105072.txt
Citation
1.0105072.ris

Full Text

GENERAL EQUATIONS FOR SHORT-RANGE OPTIMIZATION OF A COMBINED HYDRO-THERMAL ELECTRIC SYSTEM by RADEN ARTON0 ARISMUNANDAR B . A . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f E l e c t r i c a l E n g i n e e r i n g We 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 t o t h e s t a n d a r d s r e q u i r e d f r o m c a n d i d a t e s f o r t h e d e g r e e o f M a s t e r o f A p p l i e d S c i e n c e . Members o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA March, 1960 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t permission f o r e xtensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of E l e c t r i c a l Engineering, The U n i v e r s i t y of B r i t i s h Columbia, Vancouver S, Canada. Date March 3, I960 ABSTRACT T h i s t h e s i s o f f e r s a r e v i e w and a n a n a l y s i s o f a l l e x c e p t t h e l e s s i m p o r t a n t a d v a n c e s o f t h e p r e v i o u s l y d e v e l o p e d methods and e q u a t i o n s f o r o p t i m i z i n g t h e o p e r a t i o n o f an e l e c t r i c s y s t e m o f m t h e r m a l and n h y d r o p l a n t s . I n t h i s a n a l y s i s b o t h s h o r t - r a n g e ( t w e n t y - f o u r h o u r s , s e v e n d a y s ) and l o n g - r a n g e (one y e a r ) p e r i o d s a r e i n v o l v e d . The p r i m a r y o b j e c t i v e o f t h i s t h e s i s i s t o d e r i v e , u s i n g t h e C a l c u l u s o f V a r i a t i o n s , g e n e r a l d i f f e r e n t i a l e q u a t i o n s f o r s h o r t - r a n g e o p t i m i z a t i o n o f combined h y d r o - t h e r m a l s y s t e m s . The b a s i c c r i t e r i o n f o r c h o o s i n g t o s o l v e t h e s h o r t - r a n g e i n  s t e a d o f t h e l o n g - t e r m p r o b l e m l i e s i n t h e t h e o r y o f f o r e c a s t  i n g i n g e n e r a l , t h e t h e o r y o f f o r e c a s t i n g o f s t r e a m f l o w s i n p a r t i c u l a r , and i s b a s e d on t h e a f o r e m e n t i o n e d a n a l y s i s . T e s t s f o r e s t a b l i s h i n g t h e f a c t t h a t t h e above g e n e r a l e q u a t i o n s a c t u a l l y p r o d u c e t h e d e s i r e d minimum c o s t o f o p e r a t i o n a r e g i v e n i n t h e f o r m o f t h r e e o t h e r n e c e s s a r y c o n d i t i o n s and t h r e e s u f f i c i e n t c o n d i t i o n s . T hese c o n d i t i o n s a r e known i n t h i s b r a n c h o f m a t h e m a t i c s as t h e a n a l o g u e o f L e g e n d r e ' s c o n  d i t i o n , t h e W e i e r s t r a s s 1 a n a l o g u e o f t h e J a c o b i ' s c o n d i t i o n and t h e W e i e r s t r a s s ' E - f u n c t i o n c o n d i t i o n f o r a minimum. A we l l - k n o w n example i s worked o u t u s i n g t h e s e c o n d i t i o n s . I n a d d i t i o n t o t h e above, t h i s t h e s i s a l s o p r o v e s t h a t a l l p r e v i o u s l y d e v e l o p e d methods and e q u a t i o n s f o r s h o r t - t e r m o p t i  m i z a t i o n a r e e s s e n t i a l l y e q u i v a l e n t , and t h a t t h e s e .formulas a r e m e r e l y s i m p l i f i e d forms o f t h e g e n e r a l e q u a t i o n s d e v e l o p e d i n t h i s t r e a t i s e . i i i TABLE OP CONTENTS Page L i s t o f I l l u s t r a t i o n s v i Ack.HOWX © d^J 6331©H"t/ 0 o o o o o « * e f t o « » o o o o o o o * o o o O « o * * * « e « * « » o > » * * vx X Ch£Lp*fc©r X • I n t r ocLuc "fcx on o « « « o » « * o « « « » « « « « « o » » » » » « « « « « « « X C h a p t e r I I . Review and A n a l y s i s o f P r e v i o u s Work ........ 4 2.1 O p t i m i z a t i o n o f a T h e r m a l S y s t e m 4 2.2 O p t i m i z a t i o n o f a Hydro System 7 2.3 O p t i m i z a t i o n o f a H y d r o - T h e r m a l System . 9 2.3.1 I n t r o d u c t i o n 9 2.3.2 P r e v i o u s Long-Term Methods ..... 10 2.3.3 P r e v i o u s S h o r t - T e r m Methods and E q u a t i o n s 15 C h a p t e r I I I . The S h o r t - T e r m H y d r o - T h e r m a l P r o b l e m and I t s G e n e r a l S o l u t i o n s 19 3.1 S t a t e m e n t o f t h e P r o b l e m 19 3.2 M a t h e m a t i c a l F o r m u l a t i o n o f t h e P r o b l e m 21 3.3 G e n e r a l S o l u t i o n s o f t h e P r o b l e m : The F i r s t N e c e s s a r y C o n d i t i o n . . . . . . . . . . . . . 25 303.1 I n t r o d u c t i o n . . . . . o o . . . . . . . . . . . . 25 3.3.2 T h e r m a l - P l a n t E q u a t i o n s ........ 28 3.3.3 H y d r o - P l a n t E q u a t i o n s 30 3.3.4 S u b s t i t u t i o n o f L o s s F a c t o r s and C o s t F u n c t i o n s i n t h e G e n e r a l 3.3.5 Two S i m p l i f i e d C a s e s 35 C h a p t e r IV. O t h e r N e c e s s a r y and S u f f i c i e n t C o n d i t i o n s f o r a Minimum o f t h e I s o p e r i m e t r i c P r o b l e m ...... 42 i v Page 4 « 1 XXl'bjT'OCLlXC'fcl Oil ooooooooooooooooooooooooooe 42 4.2 The Second N e c e s s a r y C o n d i t i o n ......... 42 4.3 The T h i r d N e c e s s a r y C o n d i t i o n .......... 43 4.4 The F o u r t h N e c e s s a r y C o n d i t i o n ......... 45 4.5 The T h r e e S u f f i c i e n t C o n d i t i o n s ........ 46 Ch a p t e r V. Example o f a L o s s l e s s T w o - P l a n t P r o b l e m ..... 47 5.1 S t a t e m e n t o f t h e P r o b l e m ....... .... . 47 5.2 S o l u t i o n s o f t h e P r o b l e m : The F i r s t N e c e s s a r y C o n d i t i o n o . . o U o . . o o o o o o o « . . o o 48 5.3 T e s t s f o r N e c e s s a r y and S u f f i c i e n t OOIlCLX'tj'XOIlS ooooooooooooooooooooooooooooo 51 5©4 Conelixsxons O O O O O G O O « O O O O O « O Q O © O O O O O « O O Q 53 5.4.1 D i s c u s s i o n on t h e S o l u t i o n s ...o. 53 5.4.2 D i g i t a l Computer A p p l i c a t i o n .... 54 Ch a p t e r V I . The G e n e r a l E q u a t i o n s Compared w i t h P r e v i o u s l y D e v e l o p e d F o r m u l a s a..<,.o...o.ooo»os...o...o. 56 6.1 I n t r oduc t i on . o o o o o . . 0 0 0 0 0 0 0 0 0 ° . . . . . . D O . . 56 6.2 Co m p a r i s o n w i t h K r o n ' s E q u a t i o n .» „«o» 0» 56 6.3 C o m p a r i s o n w i t h R i c a r d ' s E q u a t i o n s ...oo 58 6.4 C o m p a r i s o n w i t h CDGK's and GK's E q u a t i o n s 60 6.5 C o m p a r i s o n w i t h E q u a t i o n s o f t h e MIT GrrOUjp O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 61 6.5.1 I n t r o d u c t i on 0 «. .» . o „ o o « . . « . . . . .. 61 6.5.2 Power O u t p u t s as V a r i a b l e s ....... 62 6.5.3 P l a n t - S t o r a g e V a l u e s as V a r i a b l e s 63 6.6 C o m p a r i s o n w i t h Watchorn's E q u a t i o n s ... 65 Ch a p t e r V I I . C o n c l u d i n g Remarks — F u t u r e Work . . o . . . . . . 68 Ri OfOlTOnCOS oooooooooooooooooooooooooooooooooooooooooooao 70 RfOfflOXlC X £t"tUX @ eoooooooooooooooooooooooooooooooooeooooooooo A p p e n d i x A A p p e n d i x B oooooooooooooooooooooooosooooooeoooooooooooooo O O O O O O O O O O O O O O O O O O O O O O O V U O O O O O O O O O O O O O O O O O O O v i L I S T OP ILLUSTRATIONS F i g u r e Page 1. " F r e e " S p i l l C u r v e « „ . < , < , » . » . . . < > . . < . . o o . . . . « . . . . . . . <.. 87 2. " C o n t r o l l e d " S p i l l C u r v e 0 . . . 0 0 0 . 0 . . . 0 0 0 0 . 0 0 0 0 . . 0 . . 87 3© "Vfitri 8*t X On Of f Ct) « O O O C > O O 6 I » O O O O O I > O I > O O O O O O o o o o o o o o o © 90 v i i ACKNOWLEDGEMENT T h i s i n v e s t i g a t i o n i s s p o n s o r e d by t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada. The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o D r . F r a n k Noakes, t h e s i s s u p e r v i s o r , f o r g u i d a n c e , s t i m u l u s and a d v i c e d u r i n g t h e c o u r s e o f t h i s r e s e a r c h - w o r k , t o D r . A. D. Moore f o r h i s a s s i s t a n c e i n many a s p e c t s o f t h i s work, and a l s o t o D r . R. F. H o o l e y o f t h e Department o f C i v i l E n g i n e e r i n g who a d v i s e d t h e a u t h o r on s e v e r a l h y d r a u l i c v i e w p o i n t s o f t h e p r o b l e m . The a u t h o r i s g r a t e f u l t o Mr. J o h n H. D r i n n a n , P l a n n i n g E n g i n e e r o f t h e B r i t i s h C o l u m b i a E n g i n e e r i n g Company, Van c o u v e r , B. C , and Mr. J . S. W i n d s o r o f t h e P l a n n i n g D i v i s i o n o f t h e " B r i t i s h C o l u m b i a Power Commission, V i c t o r i a , B. C , w i t h whom t h e a u t h o r has had many i n f o r m a t i v e d i s c u s s i o n s on t h e p r a c t i c a l s i d e o f t h e p r o b l e m s o l v e d i n t h i s t h e s i s . The f o r  mer company, t h r o u g h Mr. L. A. T a y l o r , S u p e r v i s o r o f t h e N e t  work A n a l y z e r S e c t i o n , and t h e l a t t e r company, t h r o u g h Mr. A. W. L a s h , C o n s u l t i n g E n g i n e e r , have s u p p l i e d t h e a u t h o r w i t h t h e n e c e s s a r y d a t a f o r t h e a f o r e m e n t i o n e d p r o b l e m . The a u t h o r s h o u l d l i k e t o t h a n k D r . E. L e i m a n i s o f t h e Department o f M a t h e m a t i c s f o r h i s g e n e r o u s a s s i s t a n c e i n r e - r i e w i n g t h e m a t h e m a t i c a l a s p e c t s o f a p r o b l e m w h i c h , sometimes, does n o t seem t o l e n d i t s e l f t o an e x a c t s o l u t i o n . Thanks a r e due t o D r . J . F. S z a b l y a and t h e a u t h o r ' s v i i i c o l l e a g u e s i n t h e power gro u p o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g f o r much e n l i g h t e n i n g and s t i m u l a t i n g d i s c u s s i o n . The a u t h o r i s i n d e b t e d t o t h e Colombo P l a n A d m i n i s t r a t i o n i n Canada f o r a s s i s t a n c e r e c e i v e d i n t h e f o r m o f a S c h o l a r s h i p awarded d u r i n g h i s s t a y f r o m 1956 t o 1960. CHAPTER I INTRODUCTION F o r a number o f y e a r s power s y s t e m e n g i n e e r s have made s e v e r a l a t t e m p t s t o s o l v e t h e p r o b l e m o f how t o o p e r a t e an e l e c t r i c s y s t e m most e c o n o m i c a l l y , i . e . , t o o b t a i n a s e t o f e x t r e m a l s o f o p e r a t i o n . These e x t r e m a l s c a n be g i v e n i n e i t h e r o f t h e two f o r m s : (1) as a s e t o f minima, f o r example, m i n i m i z a t i o n o f c o s t o v e r a p r e d e t e r m i n e d f u t u r e i n t e r v a l , m i n i m i z a t i o n o f l o s s e s (due t o s p i l l a g e , due t o v i o l a t i o n s o f c e r t a i n l i m i t a  t i o n s , due t o u n r e l i a b i l i t y o f s e r v i c e w h i c h r e s u l t s i n l o s s o f c u s t o m e r s ) , e t c . , o r (2) as a s e t o f maxima, s u c h as m a x i m i z a t i o n o f h y d r o - e n e r g y o v e r a c e r t a i n p e r i o d . The s e t o f e x t r e m a l s d i s c u s s e d i n t h i s t h e s i s i s e i t h e r t h a t o f m i n i m i z a t i o n o f c o s t o r m a x i m i z a t i o n o f h y d r o - e n e r g y . The p r o b l e m o f o p t i m i z a t i o n d i f f e r s i n c o m p l e x i t y w i t h t h e t y p e o f s y s t e m and w i t h t h e l e n g t h o f t i m e i n w h i c h t h e o p t i m i z a t i o n i s c o n s i d e r e d . The t h e r m a l p r o b l e m d i f f e r s f r o m t h e e x c l u s i v e l y h y d r o p r o b l e m , w h i c h i s a g a i n d i f f e r e n t f r o m t h e p r o b l e m o f a combined h y d r o - t h e r m a l e l e c t r i c s y s t e m . The l a t t e r i s more d i f f i c u l t t o s o l v e t h a n t h e s e c o n d w h i c h i s i n t u r n h a r d e r t h a n t h e f i r s t . The p r o b l e m f u r t h e r i n c r e a s e s i n c o m p l e x i t y i f o p t i m i z a t i o n i s d e s i r e d o v e r a l o n g p e r i o d due t o * " O p t i m i z a t i o n o f o p e r a t i o n " w i l l o f t e n be a b b r e v i a t e d a s ^ simply, " o p t i m i z a t i on". 2 t h e v e r y many u n c e r t a i n t i e s o f t h e f u t u r e . P r o b a b i l i s t i c meth ods have been u s e d w i t h t h e a i d o f dynamic programming t e c h  n i q u e s t o cope w i t h t h e above u n c e r t a i n t i e s , b u t none o f t h e s e a p p r o a c h e s have come c l o s e t o t h e d e s i r e d minimum r e s u l t s . The s h o r t - t e r m p r o b l e m i s t h e r e f o r e r e l a t i v e l y s i m p l e r t h a n t h e l o n g - r a n g e one, m a i n l y b e c a u s e i t i n v o l v e s l e s s v a g u e n e s s t h a n t h e l a t t e r . I r r e s p e c t i v e o f t h e c o n s i d e r a t i o n o f l o n g - r a n g e o r s h o r t - r a n g e p e r i o d s t h e h y d r o - t h e r m a l p r o b l e m i s g e n e r a l l y d i f f i c u l t i f t h e number o f p l a n t s i n t h e s y s t e m , e s p e c i a l l y t h e h y d r o p l a n t s , i s l a r g e , and i f t h e s y s t e m i s s p r e a d o v e r a l a r g e g e o g r a p h i c a l a r e a . I f t h e s t o r a g e e l e v a t i o n s v a r y i n t h e i r o r d e r o f m a g n i t u d e j f r o m p l a n t t o p l a n t , t h e s y s t e m u s u a l l y c a n  n o t be s i m p l i f i e d and, t h e r e f o r e , p r e s e n t s e x t r a d i f f i c u l t i e s . I n a d d i t i o n , i n v a r i o u s systems many w e l l - e s t a b l i s h e d l o a d s c h e d u l e s must be d r a s t i c a l l y a l t e r e d t o accomodate h e a v y v e s  s e l s on t h e r i v e r . The above a r e some o f t h e p r o b l e m s i n a h y d r o - t h e r m a l c a s e . I t i s e v i d e n t t h a t t h e economic p r o b l e m as a whole seems t o be m a s s i v e and, h e n c e , a l m o s t u n s o l v a b l e . F o r t h i s r e a s o n o n l y p a r t o f t h e whole p r o b l e m w i l l be s o l v e d . I n t h e n e x t p a r a g r a p h , t h e p u r p o s e and s c o p e o f t h e t h e s i s a r e i n t r o d u c e d . T h i s t h e s i s d i s c u s s e s t h e v a r i o u s t y p e s o f s y s t e m p r o b  lems i n v o l v e d f o r b o t h t h e l o n g - r a n g e and s h o r t - t e r m p e r i o d s , and a n a l y z e s a l l p r e v i o u s methods d e v e l o p e d by d i f f e r e n t a u t h o r s . The p r i m a r y o b j e c t i v e o f t h i s t h e s i s i s t o d e r i v e , u s i n g t h e C a l c u l u s o f V a r i a t i o n s , g e n e r a l d i f f e r e n t i a l equa-3 t i o n s f o r t h e s h o r t - r a n g e p r o b l e m , r e a l i z i n g t h a t , b a s e d on t h e above a n a l y s i s , no e x a c t m a t h e m a t i c a l s o l u t i o n c an be ob t a i n e d f o r t h e l o n g - r a n g e p r o b l e m . T h i s t r e a t i s e a l s o p r o v e s t h a t t h e s e g e n e r a l e q u a t i o n s c an be r e d u c e d t o t h e s e v e r a l s h o r t - t e r m e q u a t i o n s d e v e l o p e d by d i f f e r e n t a u t h o r s p r e v i o u s  l y , when c e r t a i n s i m p l i f y i n g a s s u m p t i o n s a r e a p p l i e d . A t t h e same t i m e a s e t o f p r o o f s and r e a s o n i n g s , i n d i c a t i n g t h e e q u i  v a l e n c e o f one p r e v i o u s l y d e v e l o p e d e q u a t i o n and a n o t h e r , i s g i v e n . I n a d d i t i o n , f o r t h e f i r s t t i m e i n t h i s f i e l d t h e s e c o n d , t h i r d and f o u r t h n e c e s s a r y c o n d i t i o n s , and t h e f i r s t , s e c o n d and t h i r d s u f f i c i e n t c o n d i t i o n s f o r t h e m i n i m i z a t i o n p r o b l e m a r e worked o u t . F o r t h e sake o f c l a r i f i c a t i o n a s i m p l e s y s t e m o f one h y d r o and one t h e r m a l p l a n t i s c o n s i d e r e d i m p o s i n g t h e above c o n d i t i o n s , a l o n g w i t h a wel l - k n o w n example p r e p a r e d by a u t h o r i t i e s i n t h e f i e l d o f economic l o a d - d i s p a t c h i n g . 4 CHAPTER I I REVIEW AND ANALYSIS OP PREVIOUS WORK 2.1 O p t i m i z a t i o n o f a Thermal System The p r o b l e m o f o p t i m i z i n g t h e o p e r a t i o n o f an e x c l u s i v e l y t h e r m a l s y s t e m , i . e . , a power s y s t e m o f m t h e r m a l p l a n t s s u p p l y  i n g a g i v e n l o a d , i s r e l a t i v e l y s i m p l e r t h a n t h a t o f o p t i m i z i n g a p u r e h y d r o s y s t e m due t o v a r i o u s r e a s o n s . I t i s g e n e r a l l y 1* known t h a t i n t h e l a t t e r p r o b l e m many u n c e r t a i n t i e s , s u c h as w e a t h e r c o n d i t i o n s , t h e amount o f f u t u r e i n f l o w t o t h e r e s e r  v o i r s and c o n s e q u e n t l y t h e amount o f w a t e r a v a i l a b l e i n s t o r a g e , a r e i n v o l v e d . C o n v e r s e l y , i n t h e t h e r m a l p r o b l e m t h e amount and t y p e o f f u e l " a t hand" a n d / o r " i n o r d e r " i s t h e amount p u r  c h a s e d , and t h e r e f o r e , c a n be d e t e r m i n e d more p r e c i s e l y . F u r  t h e r m o r e , i t i s more c o n v e n i e n t t o a s s i g n d o l l a r v a l u e s t o t h e amount o f oil„ gas o r c o a l b u r n e d t o g e n e r a t e c e r t a i n megawatts o f t h e r m a l power. I t i s t h e r e f o r e p o s s i b l e t o p l o t a t h e r m a l c o s t c u r v e , l 0 e 0 o f u e l c o s t i n d o l l a r s p e r hour v e r s u s t h e r m a l power o u t p u t i n megawatts, f r o m w h i c h t h e i n c r e m e n t a l t h e r m a l c o s t c u r v e c a n be d e r i v e d . On t h e o t h e r hand, i t t a k e s a g r e a t d e a l o f guess-work t o compute t h e i n c r e m e n t a l w a t e r v a l u e o f any h y d r o p l a n t . The t h e r m a l p r o b l e m may become d i f f i c u l t , however, i f t h e f u e l and i t s p r i c e a r e unknown» I n some systems t h e r e i s a c h o i c e o f o i l and gas c a r r y i n g a peak e s c a l a t i o n c h a r g e w i t h * The s u p e r s c r i p t n u m e r a l s r e f e r t o t h e l i s t of R e f e r e n c e s on pp.70 t o 79 i n c l u s i v e . t h e added c o m p l i c a t i o n t h a t e i t h e r d o m e s t i c n a t u r a l gas or l i q u i d p e t r o l e u m c a n be u s e d 0 I n o p e r a t i n g t h e t h e r m a l p l a n t s t h e shape o f t h e f u e l c o s t c u r v e may d i c t a t e a u n i f o r m l o a d f o r t h e t h e r m a l s o u r c e s , b u t i t i s by no means c e r t a i n t h a t i t w i l l p r o d u c e t h e d e s i r e d o p t i m i z a t i o n , s i n c e t h e p o s s i b i l i t y o f p e a k i n g w i t h o i l t o a s s i s t t h e d o m e s t i c peak gas l o a d must be i n c l u d e d . F u r t h e r m o r e , w i t h d i f f e r e n t c o s t and h e a t c o n  t e n t between u n i t s w i t h i n a p l a n t , and between one p l a n t and a n o t h e r , t h e i n c r e m e n t a l r a t e o f one t h e r m a l p l a n t depends on b o t h t h e power o u t p u t o f t h a t p l a n t and t h e a n t i c i p a t e d l o a d i n g 2 o f o t h e r t h e r m a l p l a n t s w i t h i n t h e s y s t e m as w e l l . I t i s a l s o known, t h a t t h e r e i s t h a t d i f f i c u l t y i n d e t e r m i n i n g p r e  c i s e l y what t h e i n c r e m e n t a l r a t e i s o v e r a s m a l l band o f o u t - 3 p u t . T h e r e i s , t h e r e f o r e , a complex i n t e r c o n n e c t i o n o f s e  v e r a l p r o b l e m s a f f e c t i n g t h e optimum t h e r m a l s c h e d u l e . I f t h e t h e r m a l p r o b l e m i s n o t as complex as t h e one men t i o n e d i n t h e p r e v i o u s p a r a g r a p h , t h e n t h e q u e s t i o n o f t h e r m a l o p t i m i z a t i o n i s t h a t o f m i n i m i z i n g t h e t o t a l f u e l c o s t £]C- J o v e r a c e r t a i n p r e - d e t e r m i n e d and f i x e d f u t u r e t i m e i n t e r v a l T 5 J m Y. C . (P™ .) d t = minimum, . „. (1-1) j = l «J T 3 where P,^ i s t h e t h e r m a l p l a n t o u t p u t (Mw), when s u p p l y i n g a c e r t a i n l o a d demand P^ and l o s s e s P^s m ^ P T j = P L + P D ° ^ " 2 ^ * F o r N o m e n c l a t u r e see a l s o pp.80-84 The s o l u t i o n o f t h i s p r o b l e m i s g i v e n by t h e c o n d i t i o n t h a t a l l p l a n t s s h o u l d be o p e r a t e d a t e q u a l i n c r e m e n t a l r a t e s when t r a n s - 4-6 m i s s i o n l o s s e s a r e n e g l e c t e d , and a t e q u a l i n c r e m e n t a l c o s t o f d e l i v e r e d power when t r a n s m i s s i o n l o s s e s a r e t a k e n i n t o 7—18 a c c o u n t . ~ I n t h e f o r m e r c a s e t h e c o n d i t i o n c a n be p r o v e d when a s i m p l e t w o - p l a n t t h e r m a l s y s t e m i s c o n s i d e r e d . The t o t a l f u e l c o s t t o be m i n i m i z e d i s , t h e n , g i v e n b y C^ , = + Cg • ••• (1—3) w i t h t h e s u b s i d i a r y c o n d i t i o n t h a t PD = P T 1 + P T 2 ' or P T 2 = Pp - P T 1 . ... (1-4) The t o t a l d i f f e r e n t i a l i s g i v e n by d P T 2 = - d P T 1 P ... (1-5) s i n c e dPp = 0 ... (1-6) w i t h a c o n s t a n t l o a d demand a t any one t i m e . To m i n i m i z e C,p i t s d e r i v a t i v e w i t h r e s p e c t t o t h e two v a r i a b l e s must v a n i s h , d C T d C T • J " P — = j p 1 — 0. . . . (1—7) a r T 2 a r T l U s i n g e q u a t i o n (1-5) one o b t a i n s t h e e q u a l i n c r e m e n t a l r a t e c o n d i t i o n m e n t i o n e d e a r l i e r s dC, d C 2 -=ns— s= ,w = c o n s t a n t . ... (1-8) d P T 1 d P T 2 When t r a n s m i s s i o n l o s s e s a r e i n c l u d e d , t h e p r o b l e m becomes s l i g h t l y more complex. However, w i t h t h e de v e l o p m e n t o f t h e 19—30 31—34 v a r i o u s t y p e s o f co m p u t e r s , b o t h a n a l o g u e ~ and d i g i t a l , 7 35 and n e t w o r k a n a l y z e r s t h e t h e r m a l p r o b l e m i s s o l v a b l e . 2.2 O p t i m i z a t i o n o f a Hydro System The p r o b l e m o f d e t e r m i n i n g t h e most e c o n o m i c a l method o f o p e r a t i n g a h y d r o - e l e c t r i c s y s t e m o f n p l a n t s s u p p l y i n g a g i v e n l o a d has b e e n a t t e m p t e d f o r o v e r f o r t y y e a r s ? ^ " " ^ S e v e r a l i n t e r p r e t a t i o n s were g i v e n by many a u t h o r s t o t h e t e r m "op timum" u s e d i n t h i s t r e a t i s e , a l t h o u g h v e r y few p r o v e t h a t t h e i r r e s u l t s p r o d u c e t h e a c t u a l d e s i r e d minimum c o s t o f o p e r a  t i o n . Some o f t h e i r f i n d i n g s c o n t a i n i n g t h e more i m p o r t a n t a d v a n c e s o f o p t i m i z a t i o n a r e d i s c u s s e d b e l o w . 45 I n 1929, S t r o w g e r s t a t e d t h a t f o r b e s t economy t h e p l a n t s h o u l d be o p e r a t e d a t maximum e f f i c i e n c y and i n s u c h a manner t h a t maximum p r o d u c t i o n i s r e a l i z e d . S t r o w g e r f u r t h e r assumed t h a t t h i s o p e r a t i n g p r o c e d u r e i s t h e o n l y way t o make t h e utmost u s e o f t h e a v a i l a b l e r e s o u r c e . Schamberger^ i n 1935, o p e r a t e d 46 h i s h y d r o s t a t i o n s on t h e b a s i s o f m i n i m i z a t i o n o f l o s s c a u s e d b y i n e f f i c i e n t l o a d i n g s o f t h e u n i t s and i m p r o p e r l o a d i n g s o f t h e v a r i o u s c o n n e c t e d s t a t i o n s . F o u r f u n d a m e n t a l r u l e s o f o p e r a t i o n were e s t a b l i s h e d , t o be f o l l o w e d i n o r d e r t h a t l o s s e s a t t h e s t a t i o n s can be m i n i m i z e d . An a t t e m p t t o s o l v e one o f t h e more d i f f i c u l t h y d r o s y s t e m p r o b l e m s was made by B u r r i n 1941, i n a M a s t e r ' s t h e s i s a t t h e 47 M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y . The p u r p o s e o f h i s t h e s i s was t o d e v e l o p g e n e r a l p r i n c i p l e s i n d e t e r m i n i n g t h e l o a d i n g o f "common-flow"* h y d r o - e l e c t r i c s t a t i o n s , i . e . , s t a  t i o n s s i t u a t e d on t h e same s t r e a m ( r i v e r ) . R e a l i z i n g t h e com p l e x i t y o f s u c h a p r o b l e m , no e f f o r t was made t o s o l v e any 8 s p e c i f i c c a s e , a l t h o u g h B u r r d i d c o n s i d e r a s i m p l e i l l u s t r a  t i v e example w i t h two p l a n t s on a s t r e a m and a number o f s i m p l i  f y i n g a s s u m p t i o n s . T en y e a r s l a t e r , J o h n s o n o f t h e U n i v e r s i t y 48 o f W a s h i n g t o n , e x t e n d e d t h e p r o b l e m t o a t h r e e - p l a n t c a s e . B u r r ' s work was c o n t i n u e d f o r t h e g e n e r a l c a s e o f one- p l a n t - o n - o n e - s t r e a m by two o f h i s c o l l e a g u e s a t t h e same i n - 49 s t i t u t e , who i n 1950, w r o t e a j o i n t M a s t e r ' s t h e s i s on econo my l o a d i n g o f h y d r o s y s t e m s . I n t h i s t h e s i s C h a n d l e r and G a b r i e l l e e s t a b l i s h e d some m a t h e m a t i c a l c r i t e r i a f o r economy l o a d i n g and a p p l i e d them t o a s i m p l e h y d r o s y s t e m i n o r d e r t o o b t a i n g e n e r a l p r i n c i p l e s and c o n c l u s i o n s . The above c r i t e r i a were a l s o a p p l i e d t o a s y s t e m p r o b l e m w i t h a c t u a l n u m e r i c a l d a t a . One i n t e r e s t i n g f e a t u r e o f C h a n d l e r ' s and G a b r i e l l e ' s work i s t h e i n c l u s i o n o f a number o f major f a c t o r s s u c h as head, f l o w , s t o r a g e , p l a n t c h a r a c t e r i s t i c s ( e l e c t r i c a l and h y d r a u l i c ) , t i m e d e l a y o f f l o w between p l a n t s and t r a n s m i s s i o n l o s s e s . A more t h o r o u g h s t u d y on t h e e f f e c t o f t h e l a s t f a c t o r on optimum p l a n t l o a d i n g was c o n s i d e r e d i n a M a s t e r ' s T h e s i s 50 51 by Bobo o f t h e U n i v e r s i t y o f P i t t s b u r g h . J o h a n n e s s e n o f M.I.T. w r o t e a s i m i l a r t h e s i s t o s t u d y t h e r e l a t i v e changes i n c o s t s when t r a n s m i s s i o n l o s s e s a r e i n c l u d e d and changes i n t h e p r e d i c t e d s t r e a m f l o w . A n o t h e r t y p e o f o p t i m i z a t i o n i s d e a l t w i t h i n a p a p e r by 52 M c l n t y r e , B l a k e and C l u b b i n t h e f o r m o f "an e f f i c i e n t s c h e d  u l e f o r a d a i l y , w e e k l y , m o n t h l y o r s e a s o n a l b a s i s " t o meet an e s t i m a t e d l o a d . A g e n e r a l p u r p o s e d i g i t a l computer ( B e n d i x G-15A) i s u s e d f o r t h i s p r o b l e m w h i c h t a k e s as i n p u t d a t a a s e t o f s t r e a m f l o w s and maximum c a p a b i l i t y l i m i t s a t a l l p l a n t s , s p e c i f y i n g t h e i n i t i a l s t o r a g e v a l u e a t e a c h r e s e r v o i r . The o u t p u t f o r r u n - o f - r i v e r p l a n t s c o n s i s t s o n l y o f peak and a v e r a g e c a p a b i l i t y and p l a n t d i s c h a r g e , w h i l e f o r s t o r a g e p l a n t s v a l u e s f o r " c h a n g e - i n - s t o r a g e - c o n t e n t " , "end s t o r a g e c o n t e n t " and c o n  v e r s i o n f a c t o r s i n MW p e r t h o u s a n d s e c o n d f e e t a r e t y p e d out i n a d d i t i o n . The c r i t e r i o n f o r t h i s t y p e o f o p t i m i z a t i o n i s t h a t i n t h e e v e n t o f a n o v e r d r a f t , o v e r f i l l , o r t h e v i o l a t i o n o f a r e s e r v o i r o u t l e t r e s t r i c t i o n r e s u l t i n g f r o m an i n v a l i d o p e r a t i n g i n s t r u c t i o n , t h e computer w i l l t y p e out an i n d i c a t o r a l o n g w i t h a r e s e r v o i r i d e n t i f i c a t i o n code and h a l t . The en g i n e e r must t h e n s p e c i f y some o p e r a t i o n a l p r o c e d u r e w h i c h w i l l a l l o w t h e computer t o p r o c e e d w i t h a new i t e r a t i o n . 'i • „ , 2.3 O p t i m i z a t i o n o f a H y d r o - T h e r m a l System 2.3.1 I n t r o d u c t i o n The p r o b l e m o f o p t i m i z i n g t h e o p e r a t i o n o f an e l e c  t r i c power s y s t e m o f m t h e r m a l and n h y d r o p l a n t s s u p  p l y i n g a g i v e n l o a d , i s n o t e x a c t l y e q u i v a l e n t t o a com b i n a t i o n o f a p u r e t h e r m a l p r o b l e m and an e x c l u s i v e l y 53—55 h y d r o p r o b l e m . I n a h y d r o - t h e r m a l p r o b l e m t h e o b j e c t i s e q u i v a l e n t t o t h e p u r e t h e r m a l c a s e o f m i n i m i z i n g t h e t o t a l c o s t o f o p e r a t i o n o f t h e t h e r m a l p l a n t s o n l y . On t h e o t h e r hand, when t h e s e t h e r m a l p l a n t s a r e ex c l u d e d , t h e p r o b l e m becomes t h a t o f m a x i m i z i n g h y d r o e n e r g y a t a l l p l a n t s o v e r a f u t u r e t i m e i n t e r v a l . T h i s 45 i s a l m o s t e q u i v a l e n t t o S t r o w g e r ' s c r i t e r i a o f "making t h e u t m o s t u s e " o f t h e a v a i l a b l e w a t e r r e s o u r c e . A l l o f t h e above s t a t e m e n t s mean t h a t i n t h e c a s e o f a l i o -10 e a t i n g l o a d f o r t h e d i f f e r e n t p l a n t s , t h e o p t i m i z i n g e q u a t i o n s can be u s e d b o t h f o r t h e r e s t r i c t e d t h e r m a l and t h e combined h y d r o - t h e r m a l s y s t e m . The r e v e r s e i s n o t a l w a y s true., however, s i n c e t h e s c h e d u l i n g e q u a t i o n s o f a h y d r o s y s t e m a r e n o t n e c e s s a r i l y e q u a l t o t h e s c h e d  u l i n g e q u a t i o n s o f a h y d r o - t h e r m a l s y s t e m w i t h t h e t h e r m a l e q u a t i o n s o b l i t e r a t e d . I n g e n e r a l t h e h y d r o — t h e r m a l p r o b l e m c a n be d i v i d e d i n t o two g e n e r a l g r o u p s : ( i ) s h o r t - t e r m o r s h o r t - r a n g e , and ( i i ) l o n g - t e r m o r l o n g - r a n g e , b o t h h a v i n g e n t i r e l y d i f f e r e n t c h a r a c t e r i s t i c s and, h e n c e , r e q u i r e c o m p l e t e l y d i f f e r e n t s o l u t i o n s . The f i r s t p r o b l e m i s r e l a t i v e l y s i m p l e as i t d e a l s o n l y w i t h a s h o r t f u t u r e t i m e i n t e r  v a l (24 h o u r s , one week) and, hence, c a n be t r e a t e d w i t h c e r t a i n t y . C o n v e r s e l y , t h e p r o b l e m o f l o n g - r a n g e o p t i  m i z a t i o n i s much more complex s i n c e i t d e a l s w i t h many u n p r e d i c t a b l e v a r i a b l e s i n a much l o n g e r f u t u r e t i m e i n t e r v a l (one y e a r ) . F o r t h i s r e a s o n t h e l a t t e r c a s e i s more p r o b l e m a t i c a l t h a n t h e f i r s t one and,''cons e— i q u e n t l y , more d i f f i c u l t t o a s s i m i l a t e m a t h e m a t i c a l l y . 2.3.2 P r e v i o u s Long-Term Methods One o f t h e w e l l - k n o w n a n a l y t i c a l methods f o r t h e 56 57 l o n g - r a n g e p r o b l e m i s t h a t o f C y p s e r ' whose a p p r o a c h i s o u t l i n e d i n a d o c t o r a t e t h e s i s a t M. I . T. C y p s e r d e a l t w i t h l a r g e systems h a v i n g l a r g e s t o r a g e s where p l a n t e f f i c i e n c i e s depend on p a s t o p e r a t i o n s o f s t o r a g e , and where p r e s e n t o p e r a t i o n s a r e b a s e d on a whole s e t 11 o f p r e d i c t i o n s o f f l o w and l o a d demand f o r t h e whole f u t u r e l o n g - r a n g e p e r i o d t o be o p t i m i z e d . T h i s method i s o b j e c t i o n a b l e p r i m a r i l y b e c a u s e , w h i l e p r e d i c t i n g s t r e a m f l o w f o r a s h o r t p e r i o d i s s t i l l p o s s i b l e , l o n g - r a n g e p r e d i c t i o n s , p a r t i c u l a r l y i n t h e west c o a s t , a r e u s u a l l y n o t v e r y a c c u r a t e . The o p e r a t i n g scheme ob t a i n e d f o r t h a t y e a r c a n t h e r e f o r e be g r o s s l y m i s l e a d i n g . I n a d d i t i o n , t h e f a c t t h a t t h e r a t i o o f maximum t o extreme low r i v e r f l o w i n t h i s p a r t o f N o r t h A m e r i c a i s v e r y h i g h ( e . g . , B r i d g e R i v e r , B r i t i s h C o l u m b i a : 158 t o 1) compared t o t h e r a t i o i n t h e e a s t e r n p a r t o f t h i s c o n t i n e n t ( e . g . , S t . Lawrence R i v e r : 2 t o 1) a l m o s t a n n i h i l a t e any v a l u e 69 o f f o r e c a s t i n g o f s t r e a m f l o w i n t h e west c o a s t . Cyp- s e r o b t a i n e d h i s r e s u l t s by d e v e l o p i n g a p r o c e d u r e , u s i n g t h e "method o f s t e e p e s t d e s c e n t " , f o r s u c c e s s i v e l y im p r o v i n g a p r o p o s e d mode o f o p e r a t i o n s u c h t h a t t h e e f f e c  t i v e c o s t c a n be c o n t i n u o u s l y r e d u c e d . C o n t r a r y t o C y p s e r ' s a p p r o a c h , L i t t l e f ^ i n a t h e s i s s u b m i t t e d f o r t h e d e g r e e o f D o c t o r o f P h i l o s o p h y i n P h y s  i c s a t t h e M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , d i d n o t assume t h a t f u t u r e r i v e r f l o w i s known i n d e t a i l a y e a r i n a d v a n c e . F o r t h i s r e a s o n p r o b a b i l i s t i c methods a r e u s e d t o m i n i m i z e t h e e x p e c t e d c o s t , b u t n o t t h e c o s t i t  s e l f . A s i m p l e m a t h e m a t i c a l model c o m p r i s e s one h y d r o - i e l e c t r i c p l a n t and a r e s e r v o i r , one t h e r m a l p l a n t , a g i v e n l o a d demand, and a s e t o f s t r e a m f l o w s c h a r a c t e r  i z e d by p r o b a b i l i t y d e n s i t i e s . I n o p t i m i z i n g t h e o p e r a -12 t i o n t h e p l a n n i n g p e r i o d i s d i v i d e d i n t o N s m a l l e r i n t e r  v a l s ( L i t t l e u s e d N = 26; one i n t e r v a l i s 2 w e e k s ) . A t t h e b e g i n n i n g o f e a c h i n t e r v a l a d e c i s i o n i s made ab o u t t h e u s e o f s t o r a g e i n t h a t i n t e r v a l t a k i n g i n t o c o n s i d  e r a t i o n t o d a y ' s r e s e r v o i r l e v e l and t h e r i v e r f l o w p a t t e r n i n t h e i m m e d i a t e l y p r e c e d i n g i n t e r v a l . U s i n g a t h i r t y - n i n e - y e a r r e c o r d o f r i v e r f l o w s o f t h e C o l u m b i a R i v e r f o r t h e p r o b a b i l i t y d e n s i t y f u n c t i o n and a s s u m i n g a c o n s t a n t l o a d t h r o u g h o u t t h e y e a r , t w e n t y - s i x d e c i s i o n f u n c t i o n s were o b t a i n e d . E a c h i s r e p r e s e n t e d by a s e t o f g r a p h s w h i c h t e l l how much s t o r e d w a t e r s h o u l d be u s e d f o r t h e n e x t two weeks as a f u n c t i o n o f t h e volume o f t h e p r e  c e d i n g two weeks o f f l o w , w i t h t h e p r e s e n t volume o f w a t e r as p a r a m e t e r s . The a u t h o r o f t h i s t h e s i s f e e l s t h a t t h e method d e s c r i b e d above i s d e f i c i e n t i n c e r t a i n ways. W h i l e t h i s a p p r o a c h may g i v e a t r u e minimum o v e r - a l l c o s t f o r t h e whole y e a r , o p t i m i z a t i o n o f s e v e r a l i n t e r v a l s a r e u n c e r t a i n . I t i s t r u e t h a t even i f i t i s p o s s i b l e t o know what t h e a c t u a l volume o f s t o r a g e i s a t t h e b e g i n n i n g o f e ach i n t e r v a l , t h e volume a t any t i m e w i t h i n t h e i n t e r  v a l i s e n t i r e l y guess-work. T h i s i s due t o sudden c h a n g e s . o f w e a t h e r c o n d i t i o n s w h i c h c a u s e d r a s t i c changes i n r a i n f a l l and, h e n c e , r i v e r f l o w d u r i n g a p r e c e d i n g s h o r t p e r i o d w i t h i n t h e i n t e r v a l i t s e l f . I f t h i s phenomenon o c c u r r e d , t h e whole p a t t e r n o f p l a n n e d - s t o r a g e - u s e * i n t h a t i n t e r v a l s h o u l d be a l t e r e d c o n s i d e r a b l y , o r e l s e a L i t t l e ' s t e r m . 13 c e r t a i n amount o f w a t e r must be s p i l l e d o r f l o o d l i m i t a  t i o n s v i o l a t e d . C o n s e q u e n t l y , t h e whole p a t t e r n o f s t o r a g e u s e o f t h e n e x t i n t e r v a l s h o u l d be a l t e r e d , w h i c h i n t u r n w i l l change a l l s e t s o f p l a n n e d - s t o r a g e - u s e s o f a l l s u c c e e d i n g i n t e r v a l s . L i t t l e ' s c o m p a r i s o n o f h i s method w i t h t h e w e l l - e s t a b l i s h e d r u l e - c u r v e * o p e r a t i o n i s a l s o o b j e c t i o n a b l e , p r i m a r i l y b e c a u s e t h e r u l e - c u r v e u s e d i s b a s e d on t h e d r i e s t y e a r o f t h e above t h i r t y - n i n e — y e a r r e c o r d . I f t h e f l o w c o n d i t i o n s a r e n o t a s s e v e r e , t h e r u l e - c u r v e o p e r a t i o n may g i v e a h i g h e r p e r c e n t a g e o f s a v i n g s t h a n t h e one p e r c e n t c a l c u l a t e d b y L i t t l e . L i t t l e ' s c h o i c e o f a c o n s t a n t l o a d f o r a whole y e a r i s n o t p r a c t i c a l , and h i s c l a i m t h a t c h a n g i n g t h e ^ l o a d f r o m t i m e t o t i m e w i l l o n l y add a l i t t l e c o m p l i c a t i o n i s n o t j u s t i f i e d . When t h e s y s t e m c o n s i s t s o f n h y d r o p l a n t s (n l a r g e r t h a n 1) t h e p r o b l e m i s d i f f i c u l t t o s o l v e s i n c e i t w i l l c o n s i s t o f a t l e a s t n x N d e c i s i o n f u n c t i o n s o f two v a r i a b l e s e a c h . 59 L i t t l e ' s work was f o l l o w e d by Koopmans' o f t h e Cowles F o u n d a t i o n f o r R e s e a r c h i n E c o n omics a t T a l e U n i  v e r s i t y . Koopmans' p a p e r d e a l s w i t h a s i m p l e t w o - p l a n t s y s t e m s i m i l a r t o t h a t o f h i s p r e d e c e s s o r , and i t s p u r p o s e i s t o c o n s t r u c t a " f e a s i b l e w a t e r s t o r a g e p o l i c y " w h i c h m i n i m i z e s t h e t h e r m a l c o s t o v e r a p r e d e t e r m i n e d p l a n n i n g p e r i o d , w h i l e m e e t i n g a g i v e n l o a d demand. T h i s method D i s c u s s e d i n a l a t e r p a r a g r a p h . Koopmans 1 t e r m . 14 i s u n i q u e i n t h a t i t o f f e r s a n a d d i t i o n a l f e a t u r e o f a s s o c i a t i n g w i t h t h e above o p t i m a l s t o r a g e p o l i c y i m p u t e d " e f f i c i e n c y p r i c e s " o f t h e power g e n e r a t e d and o f t h e w a t e r u s e d and i n s t o r a g e , and i m p u t e d " e f f i c i e n c y r e n t s " f o r t h e u s e o f t h e p l a n t and t h e r e s e r v o i r . However, i t has a number o f l i m i t a t i o n s i n t h a t c e r t a i n i m p r a c t i  c a l s i m p l i f y i n g a s s u m p t i o n s a r e made: ( i ) f u t u r e l o a d and f l o w c o n d i t i o n s a r e known w i t h c e r t a i n t y ( i i ) v a r i a t i o n s i n h e a d c a n be n e g l e c t e d . One o f t h e most w i d e l y u s e d methods f o r l o n g - r a n g e o p t i m i z a t i o n and w h i c h w i l l p r o b a b l y p r o d u c e t h e c l o s e s t t o t h e d e s i r e d minimum c o s t i s c a l l e d t h e r u l e - c u r v e methodi T h i s method i s e x t e n s i v e l y c o v e r e d i n a t w e n t y - page t r a n s a c t i o n s p a p e r by B r u d e n e l l and G i l b r e a t h o f t h e 60 T e n n e s s e e V a l l e y A u t h o r i t y . T h i s p a p e r d e a l s w i t h t h e s u b j e c t o f "economic i n t e g r a t i o n " * o f h y d r o and t h e r m a l p l a n t s i n d e l i v e r i n g t h e r e q u i r e d l o a d t o t h e h i g h v o l t  age t r a n s m i s s i o n s y s t e m . I n s u p p l y i n g t h i s l o a d t h e b a s i c c r i t e r i o n i s t h a t o f m i n i m i z i n g t h e a v e r a g e a n n u a l p r o d u c t i o n c o s t u n d e r t h e most a d v e r s e c o n d i t i o n s o f w a t e r . To a r r i v e a t t h e d e s i r e d economic r e s u l t s e v e r a l g u i d e s i n t h e f o r m o f c u r v e s a r e u s e d : ( i ) t h e b a s i c r u l e c u r v e i s t h e d i a g r a m w h i c h shows t h e e x p e c t e d o r p l a n n e d r e s e r v o i r l e v e l s o r t h e p l o t o f r e m a i n i n g s t o r a g e a t any g i v e n t i m e , t a k i n g i n t o a c c o u n t t h e most c r i t i c a l c o n d i t i o n s o f s t r e a m f l o w ; ( i i ) t h e " n o - s p i l l r u l e c u r v e " *TVA's t e r m . i s t h e c u r v e o f t h e s u r p l u s between t h e f i r m l o a d and t h e e n e r g y a v a i l a b l e d u r i n g t h e maximum f l o w p e r i o d . I f t h e s t o r a g e c o n t e n t i s above t h i s c u r v e , t h e l o s s due t o s p i l l  i n g s h o u l d be b a l a n c e d a g a i n s t t h e g a i n i n e n e r g y f r o m t h e u s e o f a v a i l a b l e f l o w ; ( i i i ) t h e "economy g u i d e - l i n e " i s a c u r v e b e l o w w h i c h t h e v a l u e o f i n c r e m e n t a l s t o r a g e i s g r e a t e r t h a n t h e c o s t o f o p e r a t i n g t h e t h e r m a l power a t any one t i m e . U s i n g t h i s d i a g r a m , the^ e n g i n e e r would s i m p l y s h u t down o r o p e r a t e t h e t h e r m a l s o u r c e a c c o r d i n g t o t h e p o s i t i o n o f t h e s t o r a g e c o n t e n t , whether below o r above t h e l i n e ; ( i v ) t h e "economy g u i d e c u r v e " i s a s e t o f f a m i l y c u r v e s u s e d i n t h e same manner as t h e p r e v i o u s  l y d e s c r i b e d c u r v e b u t o v e r a p e r i o d o f t i m e . Thus f o r any day and a c t u a l s t o r a g e c o n t e n t t h e amount o f a u x i l i a r y power t o be u s e d i s i n d i c a t e d . .3.3 P r e v i o u s S h o r t - T e r m Methods and E q u a t i o n s I n t h e f i e l d o f s h o r t - t e r m o p t i m i z a t i o n s e v e r a l meth ods and e q u a t i o n s have been d e r i v e d t o s o l v e t h e p r o b l e m o f s c h e d u l i n g t h e v a r i o u s g e n e r a t i n g p l a n t s t o meet a g i v e n l o a d a t one p a r t i c u l a r t i m e . T h e s e e q u a t i o n s a r e o u t l i n e d b elow i n t h r e e s e c t i o n s a c c o r d i n g t o t h e i r s i m i  l a r i t i e s . ( i ) R i c a r d * 6 1 R i c a r d d e r i v e d i n 1940, a s e t o f o p e r a t i n g s c h e d u l e s f o r a h y d r o - t h e r m a l s y s t e m w i t h no l o s s e s . H i s work was R i c a r d i s t h e f i r s t p e r s o n t o d e r i v e t h e t y p e o f equa t i o n s d e v e l o p e d i n t h i s s e c t i o n , hence, t h e name. 16 c o n t i n u e d b y C h a n d l e r , Dandeno, G l i m n and K i r c h m a y e r ( h e n c e f o r t h a b b r e v i a t e d : CDGK) i n 1953, who i n c l u d e d t r a n s m i s s i o n l o s s e s b u t w i t h c o n s t a n t head. The l a t t e r g o method was i m p r o v e d by G l i m n and K i r c h m a y e r ( h e n c e f o r t h a b b r e v i a t e d : GK) who i n c l u d e d t r a n s m i s s i o n l o s s e s and v a r i a b l e - h e a d p l a n t s . K r o n o f t h e G e n e r a l E l e c t r i c Company d e v e l o p e d e q u i v a l e n t e q u a t i o n s . T h e i r e q u a t i o n s a r e as f o l l o w s : C "1 R i c a r d ' s E q u a t i o n s (September 1940) t h e r m a l : d C / d P T = X, ... (2-9) h y d r o : / Q e x p [ [ 2^ <g ] |§- . X, ... (2-10) where X = c o n s t a n t , L a g r a n g i a n m u l t i p l i e r , i n c r e m e n t a l c o s t o f d e l i v e r e d power ($/Mw-hr), YQ = c o n v e r s i o n c o n s t a n t ( $ / f t ), Q = h y d r o plant d i s c h a r g e ( c f s ) , h = n e t head ( f t ) , o A = s u r f a c e a r e a o f r e s e r v o i r ( f t , a c r e ) , PJJ = h y d r o p l a n t o u t p u t (Mw). CDGK's E q u a t i o n s ( O c t o b e r 1 9 5 3 ) 6 2 dC. £ P . t h e r m a l : - - i - + X *L- = x , j = n+l,...,n+m ... (2-11) a ± T j 0 i T j h y d r o : / i |&L- + \ ^ - L . - \, i = 1, n ... (2-12) ttiHi d * H i GK's E q u a t i o n s (December 1 9 5 8 ) 6 3 dC t h e r m a l : • — - + X = X, ... (2-13) 17 t h y d r o : % exp J jg « + X ^ = X, . . . ( 2 - 1 4 ) .0 J rl n C O K r o n ' s E q u a t i o n s (December 1958) b P T -|bP„ * t I OP T I 0 P t h y d r o : j oP^ J o h - + d t [x j1 - O P H J T Q 0. ... (2-15) ( i i ) M. I . T. , The p e r s o n n e l o f t h i s s e c t i o n have a l l b e e n members o f t h e Economy L o a d i n g R e s e a r c h Group a t t h e M a s s a c h u s e t t s I n  s t i t u t e o f T e c h n o l o g y , T h e i r work i n t h i s f i e l d was i n i - 56 t i a t e d by C y p s e r who d e v e l o p e d a s e t o f s c h e d u l i n g equa t i o n s u s i n g t h e a s s u m p t i o n t h a t v a r i a t i o n s i n e l e v a t i o n s and p l a n t e f f i c i e n c i e s c a n be n e g l e c t e d , C y p s e r ' s e q u a t i o n s a r e n o t l i n e a r and, t h e r e f o r e , n o t s o l v a b l e by n u m e r i c a l i t e r a  t i o n s o r by means o f an a n a l o g u e computer o f t h e ne t w o r k 64 a n a l y z e r t y p e . C a r e y ' s t h e s i s s u g g e s t e d an a p p r o a c h w h i c h w i l l l i n e a r i z e C y p s e r ' s e q u a t i o n s . The e q u a t i o n s d e v e l o p e d I n t h i s s e c t i o n a r e : 56 C y p s e r ' s S h o r t - R a n g e E q u a t i o n s ( F e b r u a r y 1953) b e . bP, t h e r m a l : ^ 4 — _ ^5— = - | i ( t ) , 3 = n+l,...,n+m (2-16) bC. b P T h y d r o : - X - u ( t ) W ^ - = - u ( t ) , i = l , . . . , n (2-17) where X^ and y,(t) a r e L a g r a n g i a n m u l t i p l i e r s . C a r e y ' s E q u a t i o n s (June 1 9 5 3 ) 6 4 ' 6 5 b e . b P L t h e r m a l : "Tr, + P * T p — = -n("k)> 3 = n+l,...,n+m (2-18) 0 ± T j ^ T j b P t h y d r o : 6 - r ^ - = X. - u-(t), i = l , . . . , n (2-19) ° r H i 1 18 where |3 = a v e r a g e i n c r e m e n t a l f u e l c o s t (S/Mw-hr). •x- 56 57 C y p s e r ' s Long-rltange E q u a t i o n s ( F e b r u a r y 1953) ' b(c.+c ) h y d r G J b ^ J - ^ ^(c.+cp i ^(dSi/at ) J = °» ...,,(2-20) where = s t o r a g e volume a t h y d r o p l a n t i ( f t ' ) , ( i i i ) W a t c h o r n ( A p r i l 1 9 5 5 ) 6 6 W a t c h o r n d e f i n e d * * t h a t maximum economy w i l l be o b t a i n e d i f t h e f o l l o w i n g e q u a t i o n s a r e s a t i s f i e d : dC dP^ L f d P D/dt dQ/dt d P H ] = N, w i t h where 53i d y T oQ- .... (2-21) .. (2-22) y = r e s e r v o i r o r pond e l e v a t i o n ( f t ) , y^= head l o s s ( f t ) , y^, s= t a i l w a t e r e l e v a t i o n ( f t ) , F = w a t e r i n f l o w t o r e s e r v o i r ( c f s ) , N = i n c r e m e n t a l w a t e r v a l u e , ( $ / h r / l 0 0 0 c f s ) . I n c l u d e d h e r e f o r c o m p a r i s o n w i t h GK's e q u a t i o n s . W a t c h o r n d i d n o t p r o v e h i s d e f i n i t i o n . 1 19 CHAPTER I I I THE SHORT-TERM HYDRO-THERMAL PROBLEM AND ITS GENERAL SOLUTION 3.1 S t a t e m e n t o f t h e P r o b l e m Of t h e v a r i o u s t y p e s o f e x t r e m a l s i n v o l v e d ( s e e C h a p t e r I ) i n s o l v i n g t h e s h o r t - r a n g e o p t i m i z a t i o n p r o b l e m , t h i s t h e s i s i s l i m i t e d t o t h e d e r i v a t i o n o f t h e g e n e r a l s o l u t i o n o f t h e m a t h e m a t i c a l p r o b l e m d e f i n e d i n t h e f o l l o w i n g p a r a g r a p h s . The o b j e c t i v e i s t o d e t e r m i n e a s e t o f g e n e r a t i n g s c h e d u l e s f o r an e l e c t r i c power s y s t e m o f m t h e r m a l and n h y d r o p l a n t s s u c h t h a t t h e t o t a l o p e r a t i n g c o s t o v e r a p r e d e t e r m i n e d s h o r t - t e r m f u t u r e i n t e r v a l c a n be m i n i m i z e d , when i t i s d e s i r e d t o s u p p l y a g i v e n l o a d demand. I n t h i s p r o b l e m t h e f o l l o w i n g l i m i t a  t i o n s a r e im p o s e d : (1) The o p e r a t i n g c o s t s i n v o l v e d a r e o n l y t h o s e c o s t s w h i c h v a r y d i r e c t l y w i t h t h e p l a n t s ' power o u t p u t due t o t h e f a c t t h a t i n t h e p r o c e s s o f c o m p u t a t i o n s u s e d , o n l y t h o s e t e r m s w i t h t h e d e r i v a t i v e o f t h e c o s t w i t h r e s p e c t t o - t h e power o u t p u t w i l l a p p e a r i n t h e e q u a t i o n s . C o n s e q u e n t l y , c a p i t a l c o s t s on r e s e r  v o i r and g e n e r a t i n g s t a t i o n s , o r l a b o u r c o s t s may be o m i t t e d . M a i n t e n a n c e c o s t s a r e p u r p o s e l y o m i t t e d s i n c e t h e y a r e s m a l l and r e l a t i v e l y i n d e f i n i t e . The d o m i n a t i n g c o s t w i l l t h e r e f o r e be t h e c o s t o f f u e l a t t h e t h e r m a l p l a n t s . (2) One s e t o f i n f o r m a t i o n i s assumed t o be known, and t h a t i s , t h e amount o f e n e r g y a v a i l a b l e a t e v e r y h y d r o p l a n t d u r i n g t h e p e r i o d o f o p t i m i z a t i o n . T h i s i n f o r m a t i o n c a n be 60 o b t a i n e d f r o m t h e r u l e - c u r v e , where t h e e n e r g y a v a i l a b l e i s 20 computed f r o m t h e knowledge o f ( i ) r e s e r v o i r e l e v a t i o n a t p r e s  e n t , c a l l e d t h e b e g i n n i n g o f t h e o p t i m i z i n g p e r i o d , ( i i ) r e s e r  v o i r e l e v a t i o n a t t h e end o f t h e p l a n n i n g p e r i o d , and ( i i i ) t h e t o t a l amount o f i n f l o w i n t o t h e r e s e r v o i r . The l a s t two f a c  t o r s e x p l a i n t h e whole c r i t e r i o n f o r c h o o s i n g t o s o l v e t h e s h o r t - t e r m r a t h e r t h a n t h e l o n g - r a n g e p r o b l e m . W h i l e i t i s p r a c t i c a l l y i m p o s s i b l e t o d e t e r m i n e t h e e x p e c t e d p a t t e r n o f f l o w f o r a whole y e a r , i t i s , however, h i g h l y p r o b a b l e " t o g u e s s w i t h e x a c t n e s s " what t h e t o t a l f l o w i s g o i n g t o be, f o r t h e whole day tomorrow o r t h e whole week n e x t week. The above s t a t e m e n t r u n s p a r a l l e l w i t h t h e g e n e r a l t h e o r y o f f o r e c a s t i n g : l o n g - r a n g e f o r e c a s t i n g ( e . g . , w e a ther, b u s i n e s s c o n d i t i o n s , e t c . ) i s more d i f f i c u l t t h a n f o r e c a s t i n g o v e r a s h o r t p e r i o d o f t i m e . (3) The p e r i o d c o n s i d e r e d c o u l d v a r y f r o m one day o r 24 h o u r s t o one week o r s e v e n d a y s , d e p e n d i n g on t h e r e l i a b i l i t y o f f o r e c a s t i n g o f w a t e r r e s o u r c e s o f t h e s y s t e m . I t s h o u l d be n o t e d , however, t h a t g e n e r a l l y a l o n g e r o p t i m i z i n g p e r i o d i s d e s i r a b l e f o r t h e o b v i o u s r e a s o n t h a t t h e l o n g e r t h e p e r i o d t h e more e c o n o m i c a l i t i s , s i n c e l e s s c o m p u t a t i o n s a r e t o be p e r  f o r m e d . On t h e o t h e r hand, t h e s h o r t e r t h e p e r i o d t h e more a c c u r a t e a r e t h e r e s u l t s . (4) T h i s t h e s i s i s l i m i t e d a l s o t o t h e c o n s i d e r a t i o n o f f i x e d p e r i o d s , i . e . , a p e r i o d w i t h f i x e d e n d - p o i n t s . T h i s means t h a t i f t h e p e r i o d t o be o p t i m i z e d e x t e n d s f r o m , s a y , F r i d a y , J a n u a r y 1, 1960, a t 01.00 a.m. t i l l F r i d a y , J a n u a r y 8, O p t i m i z i n g p e r i o d s u s u a l l y s t a r t on a F r i d a y f o r t h e s i m p l e r e a s o n t h a t t h e y w i l l i n c l u d e t h e week-end's l o a d p a t t e r n w h i c h i s d i f f e r e n t f r o m t h a t o f any week-day. 21 1960, a t 01.00 a.m., t h e n i t must s t a r t and end e x a c t l y a t t h e t i m e s i n d i c a t e d . I f t h a t i s n o t t h e c a s e , t h e p r o b l e m becomes much more c o m p l i c a t e d t h a n t h e p r e s e n t one, s i n c e more d i f f i c u l t " v a r i a b l e e n d - p o i n t " c a s e s w i l l e n t e r t h e p i c t u r e . (5) O n l y one h y d r o p l a n t w i l l be c o n s i d e r e d on any one s t r e a m . The common-flow p r o b l e m i s more complex due t o a d d i  t i o n a l r e s t r i c t i o n s w h i c h must be s e t f o r t h e o p e r a t i o n o f t h e h y d r o p l a n t and f o r s e v e r a l r e a s o n s o f i n t e r d e p e n d e n c y : ( i ) t h e amount o f w a t e r r e l e a s e d f r o m any u p s t r e a m p l a n t w i l l o f f - s e t t h e o p e r a t i o n o f any downstream p l a n t , and ( i i ) t h e o p e r a t i o n o f a m i d d l e p l a n t i s g o v e r n e d by b o t h t h e o p e r a t i o n o f i t s up s t r e a m and downstream p l a n t s . (6) V a r i o u s h y d r o l i m i t a t i o n s a r e assumed, g i v e n i n t h e f o r m s o f p r o j e c t and o p e r a t i n g r e s t r a i n t s i n a s u b s e q u e n t s e c t i o n . 3.2 M a t h e m a t i c a l F o r m u l a t i o n o f t h e P r o b l e m The p r o b l e m p r e v i o u s l y s t a t e d w i l l be f o r m u l a t e d i n t h e f o l l o w i n g p a r a g r a p h s . The o b j e c t i s t o m i n i m i z e t h e i n t e g r a l I o f t h e f u e l c o s t C. o v e r a f i x e d f u t u r e s h o r t - t i m e i n t e r - v a l T, i . e . , where t r t and t a r e t h e two f i x e d e n d - p o i n t s o f t h e i n t e r v a l . U e * ' The p r o b l e m a d m i t s two s e t s o f r e s t r i c t i o n s , f p r e n e r g y and • • • (3-1) 22 f o r l o a d r e q u i r e m e n t s , i . e . J l = f P H 1 (Qi'V*) d t = Bi» ••• ( 3 - 2 * 1 ) 4) T Jn = f P H n (Qn>Vt) d t = B n ' ••• < 3-2.n) J 0 o r , i n g e n e r a l , f o r h y d r o p l a n t i rT J i = ^ P H i ^ i , * 1 ! ' " ^ d t = B i » ( 3 - 2 . i ) and n n+m * S £ P H i ( Q i ' h i ^ )  + J5S+1PTj <*> - P L <*Hi' PTi'*> - * » < * > = °- ... (3-3) The above r e s t r i c t i o n s c a n be changed i n t o a more s u i t a b l e f o r m ( s e e A p p e n d i x A) i f t h e f o l l o w i n g t r a n s f o r m a t i o n s , u s i n g f l o w v a r i a b l e s F^ and s t o r a g e v a r i a b l e s S^, a r e p e r f o r m e d : P H i ( 2 i ' h i ' t ) = ^ i ^ i ' 8 ! ? 8 ! ' * * ' 1 = i * - - - 1 1 (3-4) where £L = d S ^ / d t = t h e r a t e o f change o f s t o r a g e ( c f s ) . * S i n c e t h e n a t u r a l i n f l o w s t o t h e r e s e r v o i r s a r e " a l i e n " v a r - 67 i a b l e s , t h e y a r e u n c o n t r o l l a b l e and i n d e t e r m i n a b l e . F o r t h i s r e a s o n s o l u t i o n s c o n t a i n i n g t h e s e v a r i a b l e s w o u l d be mean i n g l e s s . Assuming t h a t t h e y a r e known t h e y c a n be e l i m i n a t e d f r o m t h e d e t e r m i n i n g e q u a t i o n s . The h y d r o and l o a d r e s t r i c - * . d q u I n g e n e r a l , q = — , where q i s a t i m e v a r i a b l e . ° ' d t u 23 t i o n s can t h e r e f o r e be g i v e n i n t h e f o l l o w i n g f o r m s : T J1 = j P H 1 ( S l f S l f t ) d t = B i r ... (3-5.1) 0 T p W J S „ , S - t ) d t = i , ... (3-5.n) ''-I o r , i n g e n e r a l , f o r h y d r o p l a n t i T J± = J P H i ( S i , S j . , t ) d t = B i , i = l , . . . n ... ( 3 - 5 . i ) and n n+m 05 I P H . ( S . , S . , t ) + I P T , ( t ) i = l n i 1 1 j = n + l A 3 - P L ' [ P H i ( S i > S i ) , P T j , t ] - P D ( t ) = 0 . . . . ( 3 - 6 ) I n a d d i t i o n t h e f o l l o w i n g p r o j e c t and o p e r a t i n g l i m i t a t i o n s f o r t h e h y d r o p l a n t s must be o b s e r v e d : ( l ) P r o j e c t L i m i t a t i o n s , a f u n c t i o n o f t h e d e s i g n and l o  c a t i o n o f t h e p l a n t / r e s e r v o i r , c h a n n e l s , t u r b i n e s , e t c . , b u t i n d e p e n d e n t o f t h e o p e r a t i o n o f t h e s y s t e m i t s e l f ; a t h y d r o p l a n t i , where i = 1, ... , n:* a. Maximum t u r b i n e d i s c h a r g e a t maximum g a t e o p e n i n g as f u n c t i o n o f t h e n e t h e a d : ^ 6 ' ^ ~ - - Q i - 2iMT ( h i > - - - < 3 " 7 ) b . Minimum s t o r a g e e l e v a t i o n due t o l o c a t i o n o f i n t a k e g a t e : ' h = ^ i u G - ••• ( 3 - 8 ) * S u b s c r i p t s M and \x s t a n d f o r maximum and minimum r e s p e c t i v e l y . S t o r a g e a t l e v e l s b e l o w t h e t u r b i n e i n t a k e g a t e s i s c a l l e d " d e a d s t o r a g e " (Koopmans' t e r m 5 9 ) . 24 G. Minimum st o r a g e ' e l e v a t i o n due t o l i m i t s o f s t o r a g e b a s i n : 5 6 * 6 8 y± - y i | 4 B . ••• ( 3 ~ 9 ) 56 d . Maximum o f w a t e r f l o w t h r o u g h c o n d u i t s : Qi ^ Q i M C . ... (3-io) e. Maximum power o u t p u t a t e ach h y d r o p l a n t : P H i * P H i M ' **• ( 3 " 1 1 ) (2) O p e r a t i n g L i m i t a t i o n s , r e s t r i c t i n g f a c t o r s i n t h e o p e r a t i o n o f t h e s y s t e m ; a t h y d r o p l a n t i , where i = 1,..., n:* e g g o a. Maximum s t o r a g e e l e v a t i o n due t o f l o o d p r o s p e c t s : ' y i S y i M F * (3-12) b . Minimum p l a n t d i s c h a r g e and s p i l l a g e f o r t h e p r o t e c t i o n o f f i s h : 5 6 ' 6 8 a± + Q± = Q± { . ... (3-13) c. Minimum p l a n t d i s c h a r g e and s p i l l a g e f o r n a v i g a t i o n a l 3,56 p u r p o s e s : * 0 i + Q i = 2 i u N ' ••• ( 3 - 1 4 > d. Minimum p l a n t d i s c h a r g e and s p i l l a g e f o r i r r i g a t i o n a l 70 p u r p o s e s : ai + Q i = Q i u l * (3-15) e. Minimum p l a n t d i s c h a r g e t o a l l o w t h e p l a n t t o be o p e r a t e d 68 a t minimum l o a d f a c t o r d u r i n g p e a k i n g : 2 i * Q i | x P * (3-16) See c o r r e s p o n d i n g f o o t - n o t e on page 23. f . Maximum s t o r a g e d r a f t a t c e r t a i n l a k e s f o r r e c r e a t i o n a l p u r p o s e s : S i = S i M R ' ( 3 - 1 7 ) Most o f t h e o p e r a t i n g r e s t r i c t i o n s a p p l y o n l y d u r i n g c e r t a i n p e r i o d s o f t h e y e a r ; some a r e d i s t i n c t l y s e a s o n a l i n n a t u r e s u c h t h a t t h e r e s u l t s o b t a i n e d f r o m t h e o p t i m i z i n g equa t i o n s s h o u l d be c h e c k e d a g a i n s t t h e s e r e s t r a i n t s . So l o n g as t h e o p t i m i z i n g p e r i o d i s s h o r t e r t h a n t h e p e r i o d s m e n t i o n e d above, t h e r e s u l t s m e n t i o n e d e a r l i e r w i l l be u s a b l e . On t h e o t h e r hand, a l l o f t h e p r o j e c t l i m i t a t i o n s a p p l y a t a l l t i m e s o f t h e y e a r . 3.3 G e n e r a l S o l u t i o n s o f t h e P r o b l e m : The F i r s t N e c e s s a r y  C o n d i t i o n 3.3.1 I n t r o d u c t i o n The p r o b l e m f o r m u l a t e d i n t h e p r e c e d i n g s e c t i o n w i l l be s o l v e d u s i n g t h e method o f t h e C a l c u l u s o f V a r i a - 71—75 t i o n s as o u t l i n e d i n s e v e r a l t e x t b o o k s and p a p e r s . A summary o f t h i s t y p e o f c a l c u l u s , w i t h s p e c i a l r e f e r  ence t o t h i s p r o b l e m , i s g i v e n i n A p p e n d i x B. The C a l c u l u s o f V a r i a t i o n s d e a l s w i t h p r o b l e m s o f d e t e r  m i n i n g extreme v a l u e s . However, w h i l e i n t h e o r d i n a r y t h e o r y o f maxima and minima, t h e p r o b l e m i s t o d e t e r  mine t h o s e i n d e p e n d e n t v a r i a b l e s x, y , z, ... w h i c h w i l l m aximize o r m i n i m i z e a g i v e n f u n c t i o n f = f (x, y , z, . . . ) , i n t h e C a l c u l u s o f V a r i a t i o n s d e f i n i t e i n t e g r a l s i n v o l v i n g one o r more unknowns a r e c o n s i d e r e d . The p r o b l e m i n t h e l a t t e r c a s e i s t o d e t e r m i n e t h e s e 26 unknown f u n c t i o n s s u c h t h a t t h e d e f i n i t e i n t e g r a l w i l l t a k e maximum o r minimum v a l u e s . The p r o b l e m o f t h i s t h e s i s i s more c o m p l i c a t e d t h a n t h e one p r e s e n t e d above due t o t h e r e s t r i c t i o n s g i v e n by e q u a t i o n s (3-5.1) t o (3-5.n) and ( 3 - 6 ) . The f i r s t s e t o f r e s t r a i n t s i n d i c a t e s an i s o p e r i - m e t r i c c a s e , w h i l e t h e l a t t e r i s a s p e c i a l c a s e o f t h e p r o b l e m o f L a g r a n g e . These two s e t s o f r e s t r a i n t s a r e q u i t e d i f f e r e n t i n n a t u r e when s o l u t i o n s f o r t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s a r e r e q u i r e d , so t h a t t h e p r o b  lem c a n o n l y be s o l v e d i f c e r t a i n t y p e s o f t r a n s f o r m a t i o n s a r e c o n s i d e r e d . These t r a n s f o r m a t i o n s w i l l a l t e r t h e com b i n e d p r o b l e m i n t o e i t h e r an e x c l u s i v e l y i s o p e r i m e t r i c one o r a p u r e L a g r a n g e p r o b l e m . Due t o v a r i o u s p r a c t i c a l r e a s o n s g i v e n i n A p p e n d i x B the- t r a n s f o r m a t i o n i n t o t h e i s o p e r i m e t r i c p r o b l e m i s c h o s e n . T h i s means, t h a t i n  s t e a d o f u s i n g e q u a t i o n (3-6) i t s e l f , t h e i n t e g r a l o f t h i s e q u a t i o n i s c o n s i d e r e d d u r i n g t h e p l a n n i n g p e r i o d T. T h i s i n t u r n i m p l i e s t h a t , p h y s i c a l l y , t h e c o n d i t i o n f o r l o a d r e q u i r e m e n t s i s now r e p l a c e d by t h e r e s t r a i n t s f o r e n e r g y r e q u i r e m e n t s . The p r o j e c t and o p e r a t i n g l i m i t a t i o n s a r e g i v e n i n t h e f o r m o f i n e q u a l i t i e s and, h e n c e , c a n n o t be i n c l u d e d i n t h e v a r i a t i o n a l c a l c u l u s p r o b l e m . They w i l l be u s e d , however, i n t h e f o l l o w i n g s e n s e : I f t h e r e s u l t s p r o d u c e d by t h e o p t i m i z i n g e q u a t i o n s d e r i v e d below v i o l a t e any o f t h e i r r e s t r a i n t s , t h e extreme v a l u e s (maxima o r minima, w h i c h e v e r s u i t a b l e ) s h o u l d be i n s e r t e d i n s t e a d , and t h e p r o c e d u r e r e  p e a t e d . 27 The t h e s i s p r o b l e m , h e n c e f o r t h c a l l e d t h e " c o n d i  t i o n e d " p r o b l e m , * i s e x a c t l y e q u i v a l e n t t o t h e f o l l o w i n g v a r i a t i o n a l p r o b l e m . I n s t e a d o f c o n s i d e r i n g t h e v a r i a t i o n o f t h e d e f i n i t e i n t e g r a l g i v e n by e q u a t i o n (3-1) w i t h t h e a u x i l i a r y c o n d i t i o n s (3-5.i) and t h e i n t e g r a l o f (3-6), c o n s i d e r t h e m o d i f i e d i n t e g r a l T I = f Hdt, ... (3-18) 0 where n+m n H = E C - i + Z > i P H i + K+-\ 0' ••• (3-19) jin+1 J f=l 1 H l n + 1 w i t h no a u x i l i a r y c o n d i t i o n s . I n e q u a t i o n (3-19) X^  and X n +^, t h e L a g r a n g i a n m u l t i p l i e r s , a r e t o be c o n s i d e r e d c o n s t a n t s r e l a t i v e t o t h e p r o c e s s o f v a r i a t i o n . The g e n e r a l s o l u t i o n o f t h e p r o b l e m o f m i n i m i z i n g I i s g i v e n by t h e E u l e r * s e q u a t i o n s : © H d OH 5q^ ~ d t b q u = 0, u = 1,2 ... (3-20) w i t h q u as v a r i a b l e s and u t h e t y p e of v a r i a b l e . F o r u = 1 one has q-^  •= , and f o r u = 2, q^ = . Thence, t h e r e a r e m v a r i a b l e s o f t h e f i r s t and n v a r i a b l e s o f t h e s e c o n d t y p e : 41 = P T , n + l ; q i = PT,n+2 ; 41 = P T , n + k ; ~-'> ^ = PT,n +m; — < 3" 2 1> B o l z a ' s term.a The u n c o n d i t i o n e d p r o b l e m i s t h a t w i t h no r e s t r i c t i o n s . 28 <4 = S 1 J *2 = S 2 ; 4 = %' * * *' -^2 = S n ; A ... (3—22) • 1 o ° 2 A «X ~„ • n i q 2 = S l S q 2 = S 2 ; q 2 = S £ ; ...; ^  = S n. The above E u l e r ' s d i f f e r e n t i a l e q u a t i o n s a r e t h e f i r s t n e c e s s a r y c o n d i t i o n f o r ah extremum, and t h e r e f o r e a l s o f o r a minimum. 3.3.2 T h e r m a l - P l a n t E q u a t i o n s F o r q^ = P^ , n + 1 ^ one s o l v e s , u s i n g e q u a t i o n s (3-6) and (3-19) SH f 3 — + Tx. + ^ P T n + l jin+1 ^ T j n + l m 1 ^ P T , n + l i = l d P T , n + l J^H-1 ^ T ^ + l ^ T j n + l ' ^ T ^ + l / and ... (3-23.1) 1 i bH W * C j f ^ P H i + X n + 1 g * P H i , V m * P T . j * \ * P D i = l d P T , n + l J = n + 1 d P T , n + l * P T , n + l *PT,n+J ... (3-24.1) I n t h e above e q u a t i o n s , a l l FJJ^'S, P^ ,^ .' s and P^ a r e f u n c  t i o n s o f t i m e . P T i s a f u n c t i o n o f b o t h P,T. and P m ., L H i T j ' and hence a l s o a f u n c t i o n o f t i m e . S u b s t i t u t i n g e q u a t i o n s (3-23.1) and (3-24.1) i n t h e E u l e r e q u a t i o n s (3-20) and c o m b i n i n g t h e summation terms 29 w h e r e v e r p o s s i b l e , one o b t a i n s : n+m j=n+l - X be b P ^ + Xn+1 b P bP T,n+1 i = l 1 n + 1 ^ n + l * P H i n+1 5 P T \ 1 » n+1 T, n+1 n d d t n+m J=n+1 + X. b P T i j n+1 * * T i _ * P T , n + l , V ( ? [ ? , : ^ P H i ^ ^ ~ • 51^77 + S P — • b l ^ T,n+1 n + 1 ~ A T , n + l ' b P T - X n+1 5Pn T,n+1 + ^ T . n + l / &* T f n + 1_ ^ P D ^ P T , n + l = 0, T,n+1 ... (3-25.1) w h i c h c a n be s i m p l i f i e d i n t o f l<*> " a t f l ( t ) . ^ » ± i T,n+1. = o, where „1 ( t ) = £ j=n+l .. (3-26.1) ^ /dC. \ b P T i OPH H i - X n+1 b P T b P S^TT+I^ D T,n+1 n+1 ... (3-27.1) t h I n g e n e r a l , one a c q u i r e s f o r t h e k t h e r m a l p l a n t , where k = 1, 2,....,m: 0P„ f k ( t ) " T,n+k b p T,n+k_ = 0, ... (3-26.k). w h i c h c a n be w r i t t e n as f k ( t ) - d f k ( t ) d t / b P T , n + k OP \ u r T , n + k j + f k ( t ) d_ d t bP, OP T,n+k 1 T, n+k/_ = 0, (3-28.k) 30 where f k ( t ) T,n+k + ^ - Vl ' ^ P L , * PD T,n+k u T,n+k ; ... (3-27.k) 3.3.3 H y d r o - P l a n t E q u a t i o n s As b e f o r e , f o r q E u l e r ' s e q u a t i o n s f o r t h i s v a r i a b l e : 1 .1 • As b e f o r e , f o r q 2 = and q 2 = S^, one s o l v e s t h e - x n + l * P L ^ PD| d 5 § ~ + " d t Xv si1 + ^i^r1 + n ( + / X 7\T> W b s b p - + , x - S i x ^ + n+i si""" + j j f c . i ^ + i as - W + - \ * P D n + l = 0. ... (3-29.1) R e a l i z i n g t h e dependence o f t h e c o s t on t h e t h e r m a l power o u t p u t , and t h e h y d r o p l a n t o u t p u t on i t s s t o r a g e v a l u e , one o b t a i n s t h e f o l l o w i n g r e l a t i o n s : b e . ^ b P H 1 ^ i = ^ r j ' SPiT * ' 0 P H i bP H i b P H 1 ... (3-30.1) • e « ( 3 " - 3 X • X) 31 _ ^ T i . * P H 1 S s p = 3 P bPT 0PT HI ss7 bp. O P ^ * b"^ HI ... (3-33.1) bPT bP, b C j ^ C . i ^ P T , j * P H 1 b s i b ^ ... (3-34.1), .. (3-35.1) b P H i :bP H i S I T = b P HI b P ^ b P ^ * PH1 b P H 1 b s . b l ^ bS 3 b T (3-36.1) .. (3^37.1) &*L * P L bP„ bPrv b s , ~ ^  HI * P H 1 b p HI b~s~ b S x b ^ b s i .. (3-38.1) . (3-39.1) U s i n g e q u a t i o n s (3-30.1) t o (3-39.1) i n e q u a t i o n s (3-29.1) one o b t a i n s s i m i l a r t o t h e p r e v i o u s c a s e : 8 (-t) - d t ' , bS g ( t ) ' S s f = 0, where g 1 ( t ) 5^ 7 + + X-J •El j = n + l (^+^.1 ) ... (3-40.1) bP„, H i i ™ n + l ; b P - X /bPj b P n + 1 l ^ P H l + b ^ r H l bP. c3si HI HI ... (3-41.1) 32 I n g e n e r a l , one o b t a i n s f o r t h e X h y d r o p l a n t where 1^ = 1,2!,...n : = 0, ... (3-40.. o r r.^<+\ d g ^ t ) d g ^ t } - [-^dt • sg^  + « <*>• d t = 0, ... (3-42./) where g ^ ( t ) = n+m j=n + l be. " V l 3 ! £ 7 + 5P7 5 T " b P H i  + X n + 1 > ^ + ... (3-41. Prom e q u a t i o n (3-41.>C) one o b s e r v e s t h e s i m i l a r i t y between k I f ( t ) and g ( t ) w h i c h c o n t a i n s an a d d i t i o n a l f a c t o r bP H £/bs£. 3.3.4 S u b s t i t u t i o n o f L o s s F a c t o r s and C o s t F u n c t i o n s i n t h e ,4-9 G e n e r a l E q u a t i o n s I t c an g e n e r a l l y be assumed""" t h a t t h e f u e l c o s t a t any t h e r m a l p l a n t i s a q u a d r a t i c f u n c t i o n o f i t s power o u t p u t , i . e . , a t t h e r m a l p l a n t j 1 2 C . = a . P m . + b . P„, . + c ., ... ( 3 - 4 3 . j ) where a., b. and c. a r e c o n s t a n t s . Hence J J 0 bC, dC. ^L- = a. P T . + b . = a g J - . .... (3-44.J ) T h e r e a r e s e v e r a l ways o f e x p r e s s i n g t r a n s m i s s i o n 76-90 l o s s e s i n terms o f p l a n t g e n e r a t i o n s f o r b o t h s i n g l e 33 91—96 and i n t e r c o n n e c t e d s y s t e m s , but o n l y some of t h e more i m p o r t a n t methods and d e v e l o p m e n t s a r e o u t l i n e d below. 7fi ( i ) The B - C o n s t a n t Method, i n i t i a t e d by George i n 77 1 9 4 3 , f u r t h e r d e v e l o p e d by E a t o n , Ward and H a l e , and 78 K i r c h m a y e r and S t a g g , i s made p o s s i b l e by f o u r b a s i c 78 a s s u m p t i o n s . The l o s s e q u a t i o n d e v e l o p e d u s i n g t h e s e a s s u m p t i o n s w i l l be r e f e r r e d t o as t h e s i m p l i f i e d l o s s e q u a t i o n , c o n t r a r y t o t h e i m p r o v e d and more g e n e r a l f o r - 84 mula by E a r l y , Watson and S m i t h (1955) w h i c h i g n o r e d t h e b a s i c a s s u m p t i o n s u s e d p r e v i o u s l y . E a r l y and Wat- 8 5 son d e v e l o p e d i n t h e same y e a r a new method o f d e t e r m i n  i n g c o n s t a n t s f o r t h e G e n e r a l T r a n s m i s s i o n L o s s E q u a t i o n ( G T L E ) . ( i i ) The V o l t a g e P h a s e - A n g l e Method d e v e l o p e d by 81 83 B r o w n l e e i n 1955 was u s e d by Cahn t o d e t e r m i n e i n c r e  m e n t a l and t o t a l l o s s f o r m u l a s . The use of power t r a n s  f e r e q u a t i o n s t o d e r i v e c o o r d i n a t i o n e q u a t i o n s e x p r e s s e d as f u n c t i o n s o f v o l t a g e p h a s e - a n g l e s r e s u l t e d i n t h e 89 M i l l e r e q u a t i o n s . ( i i i ) A g e n e r a l method o f c a l c u l a t i n g i n c r e m e n t a l t r a n s m i s s i o n l o s s e s and t h e GTLE was d e v e l o p e d by Watson and S t a d l i n 8 8 i n 1959. ( i v ) A new and r e v o l u t i o n a r y a p p r o a c h t o l o s s m i n i  m i z a t i o n i n power systems was d e v e l o p e d by C a l v e r t and 86 Sze i n 1958, and a p p l i e d t o a s i m p l e s y s t e m by C a l v e r t , 87 Sze and G a r n e t t i n 1959. (v) A n o t h e r new and f u n d a m e n t a l l y d i f f e r e n t method i n d e t e r m i n i n g l o s s f o r m u l a s f r o m d i g i t a l l o a d f l o w 34 90 s t u d i e s was d e v e l o p e d by George i n a r e c e n t p a p e r . Of a l l t h e above methods, t h e most w i d e l y u s e d B-Con- s t a n t a p p r o a c h w i l l be u s e d i n t h i s t h e s i s . The s i m p l i - 77 78 f i e d l o s s e q u a t i o n i s g i v e n by: ' m+n m+n P L = Z_, Brs P r Ps ' u r=l s=l r s r s ... (3-45) where B „ o i s t h e d e r i v e d l o s s f o r m u l a c o e f f i c i e n t ( c o n -r s x s t a n t a t one s p e c i f i c l o a d l e v e l ) , and P and P a r e e i t h e r t h e r m a l o r h y d r o p l a n t g e n e r a t i o n s . D i f f e r e n  t i a t i n g P^ w i t h r e s p e c t t o t h e s e g e n e r a t i o n s , one o b t a i n s 0PT T,n+k m+n n+k,s s ' ...,(3-46.k) bP T m+n H = 2 B ( s P s ... (3-47.//) S u b s t i t u t i o n o f e q u a t i o n s ( 3 - 4 4 . j ) , (3-46.k) and (3-47./) i n e q u a t i o n s (3-27.k) and ( 3 - 4 1 . A y i e l d s f o r t h e k t h e r m a l p l a n t : n=rn ( t ) = E ( ^ T i + V ' w ) te'1* + j i n + 1 J T j 3 n+1 o P T n + k OP H i - X . m+n 2 ) B n+k P -+-n+k,s s ... (3-48.k) and f o r t h e JL h y d r o p l a n t : 35 A m+n bP m , bp + n i=J < V X n + l > b P H i / m+n - X n+1 D s s ^g|- . ... (3-49.£) 3.3.5 Two S i m p l i f i e d C a s e s Two t y p e s o f s i m p l i f i e d c a s e s w i l l be c o n s i d e r e d i n t h i s s e c t i o n : t h e c a s e where o n l y one h y d r o and one t h e r  mal p l a n t e x i s t s and t h e c a s e where s c h e d u l i n g o f genera t i o n s i s p e r f o r m e d a t one p a r t i c u l a r t i m e , ( i ) Type A: The T w o - P l a n t P r o b l e m The o p t i m i z i n g e q u a t i o n s d e r i v e d i n t h e p r e c e d i n g s e c t i o n s become much s i m p l e r when t h e l o a d demand i s s a t i s f i e d by one h y d r o p l a n t (n = 1) and one t h e r m a l p l a n t (m = 1 ) . Hence, u s i n g e q u a t i o n s (3-26.1) and (3-27.1) one s o l v e s f o r t h e t h e r m a l p l a n t : " f t f V ) . b P T2 = 0, ... (3-50) w i t h f * ( t ) t a k i n g t h e s p e c i a l f o r m o f f A ( t ) = fbc S P T2 - X 0 ? 1 b P H 1 + x 2 ) + ( x 1 + x 2 ) 3 ^ + T2 b P L bP D S P ^ + SP T2 j ... (3-51) and f o r t h e h y d r o p l a n t , f r o m e q u a t i o n s (3-40.1) and (3-41.1): , T , bS, 1 0, ... (3-52) g ( t ) - d t g ( t ) * b ^ 36 where g X ( t ) tec. T2 SP" T2 HI X2|^7+ S P " * PH1 HI 3s: .. (3-53) The two e q u a t i o n s (3-50) and (3-52) can be combined i n t o one d i f f e r e n t i a l e q u a t i o n as i n A p p e n d i x B: U S , **1 o r T 2 / dP, T2 d t d 2 s x d 2 P T 2 ds x d t 2 d t 2 d t where H, = b 1 = ( s x ) 2 ^ 2 S H bP, T2, (p ) 2 T2 - 1 M . bs, b bH ( S X ) ( P T 2 ) 1^7 ^ PT2j = o, ... (3-54) •«• (3"* 55 «£t) ... (3-55.b) ... (3-55.c) and where H c a n be d e r i v e d f r o m t h e g e n e r a l f o r m o f equa t i o n ( 3 - 1 9 ) : H = C 2 + X 1 P R 1 + X 2 ( P H 1 + P T 2 - P L - P D ) . ... (3-56) I t c a n be s e e n t h a t t h e d i f f e r e n t i a l e q u a t i o n (3-54) i s o f t h e s e c o n d o r d e r . I t s g e n e r a l s o l u t i o n c o n t a i n s , t h e r e f o r e , two a r b i t r a r y c o n s t a n t s o f i n t e g r a t i o n a and 6, two i s o p e r i m e t r i c c o n s t a n t s and Xg. Hence P T 2 = P T 2 ^* X l ' X 2 ' ^ (3-57) 37 S 1 = S1 ( a , B, X l t X 2 , t ) . ... (3-58) d S l From e q u a t i o n ( 3 - 5 8 ) , = can be f o u n d w h i c h t o g e t h e r w i t h t h e f l o w F ^ ( t ) w i l l d e t e r m i n e P H 1 = PH1 ( a ' P ' V V P i » * ) • ••• ( 3 - 5 9 ) From t h e known i n I P H 1 S l » 8 1 » t ^ d t = B l » ( 3 _ 6 ° ) '0 f r o m t h e a u x i l i a r y c o n d i t i o n 0 = P H 1 + P T 2 - P L - P D = 0, ... (3-61) and t h e i n i t i a l c o n d i t i o n s s1 ( t = ©)' = s 1 0 , P T 2 ("t = °) = P T 2 0 ! ... (3-62) a, 6, X^ and X 2 can be f o u n d . ( i i ) Type B: The E q u i v a l e n c e o f t h i s Method w i t h a l l  P r e v i o u s l y Known Methods I n p r a c t i c e , s i n c e t h e c u r v e o f l o a d demand does n o t f o l l o w a p a t t e r n w h i c h i s p r e s e n t a b l e i n t h e f o r m o f a s i m p l e , c o n t i n u o u s and d i f f e r e n t i a b l e f u n c t i o n o f t i m e : P D = P D ( t ) , ... (3-63) t h e p r o b l e m can be r e v i s e d as f o l l o w s : I n s t e a d o f d e t e r m i n i n g what S ^ ( t ) , S 2 ( t ) , . . . , S £ ( t ) , . . . , S n ( t ) and P T > n + 1 ( t ) , . . . , P T j n + k ( t ) , . . . , P T ) n + m ( t ) a r e , when F - ^ ( t ) , ... , F ^ ( t ) , . . . , F Q ( t ) a r e g i v e n t o meet P p ( t ) 38 and P ^ ( t ) , one solves the problem of what c o n t r i b u t i o n each p l a n t should make i n order to meet the l o a d r e q u i r e  ments at any one time t where x = 0,1,...,e ( t n = 0 ; t = T ) . In other words, at load l e v e l Pr.(t ) what e • D x the values of V n + l ^ ' P T , n + k ^ X ) ' P T f n + » ^ x ) » '' ( 3 " 6 4 ) P H 1 ( V ' ' P H / ( t x } ' PHn ( tx>' •••(3-65) should be, and hence S - ^ t J , , S £ ( t x ) , , S n ( t x ) (3-66) combined with a given set of flows x' ' n x x' such t hat n n+m i = l J=n+1 J The elements of sets (3-64) to (3-67) above are no longer f u n c t i o n s of time, such t h a t expressions as a P H £ ( t x ) M , a p T j n + k ^ x ^ d ^ a n d d P j j ( t x ) / d t do not e x i s t . However, S /<V - dir t=t ••• ( 3 ~ 6 9 ) X does e x i s t , s i n c e i t i s one of the three b a s i c v a r i a b l e s P T n+k' ^ a n d ^ ("the ^ x ' s a r e • n e n c e - ^ o r t h omitted f o r convenience), remembering t h a t each one of them are d i f  f e r e n t independent v a r i a b l e s . In a d d i t i o n , b a s i c v a r i  ables of any one p l a n t are c h a r a c t e r i s t i c of t h a t p l a n t 39 o n l y . The s c h e d u l i n g e q u a t i o n s c a n t h e r e f o r e be s i m p l i  f i e d u s i n g t h e f o l l o w i n g r e l a t i o n s . F o r t h e t h e r m a l p l a n t s : DC. ^5— J — =0 f o r j £ n+k ° r T , n + k be. be , ••• < 3 - 7 0 ) - S T " - bp » f o r J - n + k ° r T j ° r T , n + k b P T i ^ - ^ J — = 0 f o r j f n+k = 1, f o r j = n+k (3-71) b p m n i — = 0 f o r a l l k = l,...m ... (3-72) ^ T , n+k ( s i n c e P ^ i s a f u n c t i o n o f F^, and S\ o n l y ) , b P D = 0, f o r a l l k = 1,...m ... (3-73) ^ PT,n+k b P T , — £ J = 0, f o r a l l k = l,...m ... (3-74) b P T , n + k ^ P W i * — = 0, f o r a l l k = l,...m ... (3-75) d P T , n + k b P T =0 f o r a l l k = l,...m ... (3-76) b p T , n + k ( s i n c e i s a f u n c t i o n o f P ^ and P^ .^ only, b u t n o t of t h e i r d e r i v a t i v e s ) , ^ P T ) =0. f o r a l l k = 1,...m ... (3-77) b P T , n + k S u b s t i t u t i n g t h e above e q u a t i o n s i n e q u a t i o n s (3-26.k) 40 and (3-27.k) one o b t a i n s f o r t h e k t h e r m a l p l a n t , o r t h e r m a l p l a n t j : bC . / b P L \ S i m i l a r l y , f o r t h e h y d r o p l a n t s : = 0, bs. = 0 = 1, bp D = o, 2 i = o, OS. = 0 = 1. f o r a l l -L = 1,.. .n f o r i ^ / f o r i = K f o r a l l $. = 1,.. ,n f o r a l l / i s = 1, ...n f o r i ^ / f o r i = ... ( 3 - 7 8 . j ) . (3-79) . (3-8-0) . (3-81) . (3-82) (3-83) Thus f o r h y d r o p l a n t i : U i + X n + 1 } S S T " ~ Xn+1 ^ 5 S ~ ~ + d_ d t * P H i ^ P H i i n + l ; ^ " An+1 = 0, i = 1,.. .n ... ( 3 - 8 4 . i ) w h i c h by c o m b i n i n g t h e l o s t terms can be w r i t t e n i n a s i m p l e r f o r m : 41 537 H i at as. + X n+1 0PT * P H i 5 s 7 ~ + d__ dt "n+1 1 - bP T ~p H i H i = 0. i = 1,...n ( 3 - 8 5 . i ) i J CHAPTER IV 42 OTHER NECESSARY AND SUFFICIENT CONDITIONS FOR A MINIMUM OF THE ISOPERIMETRIC PROBLEM 4.1 I n t r o d u c t i o n I n a d d i t i o n t o t h e f i r s t n e c e s s a r y c o n d i t i o n o f m i n i m i z i n g t h e c o n d i t i o n e d i n t e g r a l I g i v e n by e q u a t i o n (3-18) o f t h e p r e  v i o u s c h a p t e r , t h e r e a r e t h r e e o t h e r n e c e s s a r y and t h r e e s u f f i  c i e n t c o n d i t i o n s t o be s a t i s f i e d . To a v o i d c o m p l e x i t y o f symbols and n o t a t i o n s i n v o l v e d i n t h e g e n e r a l p r o b l e m , and t o f a m i l i a r i z e t h e r e a d e r w i t h t h e c o n c e p t and use o f s u c h c o n  d i t i o n s , t h e s i m p l i f i e d p r o b l e m o f t y p e A, C h a p t e r I I I , w i l l be c o n s i d e r e d . The e x t e n s i o n t o t h e g e n e r a l c a s e o f a s y s t e m o f 71 72 m t h e r m a l and n h y d r o p l a n t s i s o b v i o u s . ' The n o t a t i o n s u s e d i n t h e f o l l o w i n g s e c t i o n s a r e i d e n t i c a l w i t h t h e ones u s e d i n A p p e n d i x B, where t h e two v a r i a b l e s x and y a r e now r e p l a c e d by P,p2 and S-^  r e s p e c t i v e l y . 4.2 The Second N e c e s s a r y C o n d i t i o n The s e c o n d n e c e s s a r y c o n d i t i o n f o r a minimum o f I i s g i v e n 71 72 by t h e a n a l o g u e o f t h e L e g e n d r e ' s c o n d i t i o n : ' H x =T 0 ... ( 4 - 1 ) a l o n g t h e e x t r e m a l CQ, e x p r e s s e d i n t h e f o r m o f e q u a t i o n s (3-57) and ( 3 - 5 8 ) . H^ i s g i v e n by e i t h e r e q u a t i o n (3-55.a) o r (3-55.b) o r (3-55.c) w h i c h , u s i n g t h e o r i g i n a l E u l e r e q u a t i o n s f o r t h e t h e r m a l p l a n t ( o r h y d r o p l a n t ) , c an be w r i t t e n as i a - b P T 2 ... (4-2.a) 43 i _b_ (f> ) Z bs, V J r T 2 ' 1 " bs <*T2>'<fll> ^ 1 bF T2 J ... (4-2.b) ... (4-2.c) where f ^ ( t ) and g ^ ( t ) a r e g i v e n by e q u a t i o n s (3-51) and (3-53) r e s p e c t i v e l y . 4.3 The T h i r d N e c e s s a r y C o n d i t i o n The t h i r d n e c e s s a r y c o n d i t i o n i s g i v e n by W e i e r s t r a s s 1 s a n a l o g u e o f J a c o b i ' s c o n d i t i o n D ( t , t J = W W 2 ( V o2(t) w 3 ( t 0 ) w 3 (t) J0 t r '0 '0 t '0 t V o ^ d t '0 '0 u4(tQ) o 4 ( t ) t ^ t i J Uo^dt J Uco 2dt J Uw 3dt J Uw 4dt J V y 2 d t J " Yojgdt J '0 t Va> 4dt J0 £ 0, ...(4-3) where a ) 2 ( t ) w 3 ( t ) w 4 ( t ) b P T 2 b P T 2 P * b a * ' b t * So" ' bs ± b P T 2 ^ P T 2 b s , St- • b(3 " • b t • b T ' bS-^  b P T 2 b P T 2 bSj^ b~t~ * b ^ i • b t ° 5 * 7 ' b S x b P T 2 b P T 2 b S 1 b ^ ' b x 2 • b t (4-4) 44 and b U = S P ^ 1 zs. (bp b p T2 HI bS- dP, T2 d t d 2 S ] d t d2p m o dS1 T2 1 d t ' w i t h G, = 1 b ( S l)2 b P T 2 1 b 1^12/ ( P T 2 ) 2 M i 1 b !2»"HI ( f T 2 ) ( S l ) c * T 2 1 * 1 o o o o d t ... ( 4 - 5 ) .. (4-6.a) .. (4-6.b) (4-6.c) S i m i l a r l y ! V = b 0 bs^ | b P T 2 b p T2 dP, T2 d 2S- d2p T2 w i t h 0. = d t • dt2~ b / * 0 1 b p T 2 b ( 1^ ( P T 2 ) 2 b 0 d S 3 d t " ... (4-7) , (4-8.a) , (4-8.b) , (4-8.c) - ( f T 2 ) ( S l ) 5 i T 2 • where 0 i s g i v e n by e q u a t i o n ( 3 - 6 1 ) . Computing t h e d e r i v a t i v e s fbp v = HI b § x ^ 2 bP + 1 bP T S P T2 bP S P b P D \ b P H 1 ^ ~ ^ H T W + 0 ! © p rT2 o and S^ b T2/ ^ PT2 "dP T 2 d 2 S l d t d t b p T 2 i + + HI X 2 dt2 dS J L d t where 0 n = 1 b i = Jg-jz bP T2 'bP HI S P + i bP T b p D T2 55 T2 ^ PT2 bP b£ T2 T2j (4-9) (4=10oa) 45 i a b ? T 2 b P L  1 + ~ p ^ P D \ ^ PH1 HI " ^RI " " ^ H l / ^ 1 b I + O P T 2 b P L ~ p HI bP HI 1 D  "^iTlbs b P D \ b P R 1 ... (4-10.b) . (4-10.c) 1 J The • t h i r d n e c e s s a r y c o n d i t i o n c an a l s o be g i v e n as *. = T = *n » ... (4-11) where t g i s t h e c o n j u g a t e o f t h e p o i n t t g , t h e b e g i n n i n g o f t h e o p t i m i z i n g p e r i o d , and t t h e end o f t h i s p e r i o d . The c o n - j u g a t e p o i n t t ^ i s t h e r o o t n e x t g r e a t e r t h a n t ^ o f t h e equa t i o n D ( t , t Q ) = 0 . ... (4-12) 4.4 The F o u r t h N e c e s s a r y C o n d i t i o n The f o u r t h n e c e s s a r y c o n d i t i o n f o r a minimum i s g i v e n by W e i e r s t r a s s as E ( Pip 2> S^, Pip 2 i i S^, P^»2? 1' ^ " l 5 ^2 ^ ^ ^' * * ° (4-13) w h i c h must be f u l f i l l e d a l o n g t h e e x t r e m a l CQ f o r e v e r y d i r e c - t i o n P T 2 and S^. £ The E - f u n c t i o n i s g i v e n by H ( PIJ>2' 8 1 ' P T 2 ' 8 1 ' X l ' X 2 ^ 'P bH . T bH_ T2 bP~T 1 6s u r T 2 ° s ] (4-14) w i t h H ^ T 2 , ^ ] ) , ^ T 2 , ^ 1 '^1 9^2^ = ^2 ^ T 2 ' ^ ^1 "^Hl ^ l ' ^ l ' " ^ "*~ + X 2 I P H 1 ( S T , S l f t ) + P T 2 ( t ) - P T ( t ) - P B ( t ) , (4-15) * 71 72 B o l z a ' c a l l s i t t h e W e i e r s t r a s s ' c o n d i t i o n . 46 and W - ' ^ ^ - W 1 ' ...(4-16) T2 ° T2 2H = g ^ t j . — i , ... (4-17) oSj^ bSj^ 1 1 w i t h f ( t ) and g ( t ) g i v e n b y e q u a t i o n s (3-51) and (3-53) r e s p e c t i v e l y . 4.5 The T h r e e S u f f i c i e n t C o n d i t i o n s The t h r e e s u f f i c i e n t c o n d i t i o n s f o r a,minimum o f i n t e g r a l I a r e g i v e n b y t h e s e c o n d , t h i r d and f o u r t h n e c e s s a r y c o n d i t i o n s w i t h t h e e q u a l i t y s i g n o m i t t e d , i . e . , ^=-0, ... (4-18) t e * t 0 \ . (4-19) E (Prp2»8l , P T 2 ' 1' T2' 1' ^ i » ^ 2 ^ = " ^ ' 0 0 0 (4-20) where and t h e E - f u n c t i o n a r e g i v e n by e q u a t i o n s ( 4 - 2 „ a 0 b 9 c ) and (4-14) r e s p e c t i v e l y , a n d t Q d e f i n e d as p r e v i o u s l y . 47 CHAPTER V EXAMPLE OP A LOSSLESS TWO-PLANT PROBLEM 5.1 S t a t e m e n t o f t h e P r o b l e m As an i l l u s t r a t i o n o f how t h e f o u r n e c e s s a r y and t h e t h r e e s u f f i c i e n t c o n d i t i o n s can be ap p l i e d , t h e t w o - p l a n t p r o b l e m , d e f i n e d i n c h a p t e r H I as t h e s i m p l i f i e d Type A c a s e , w i l l be c o n s i d e r e d . I n t h i s i l l u s t r a t i o n i t i s f u r t h e r assumed t h a t t h e t r a n s m i s s i o n l o s s e s c an be n e g l e c t e d . The p r o b l e m i s e s s e n t i a l l y t h e same as t h a t o f G l i m n and K i r c h m a y e r ( a b b r e - 63 v i a t e d : GK) who assumed t h e f o l l o w i n g : (1) The i n c r e m e n t a l c o s t o f t h e r m a l power i s c o n s t a n t , o r t h e c o s t f u n c t i o n l i n e a r : C 2 = \ P T 2 + C 2 ( 5 - l } 6 C 2 / d P T 2 = b 2 = d C 2 / d P T 2 . ... (5-2) (2) The r e s e r v o i r i s a v e r t i c a l - s i d e d t a n k , and t h e t a i l - w a t e r e l e v a t i o n i n d e p e n d e n t o f f l o w , s u c h t h a t ( i f head l o s s and s p i l l a g e a r e n e g l e c t e d ) t h e n e t head c a n be e x p r e s s e d a s : t h = h Q + J ~ & d t , . „ . (5-3) 0 where F, t h e i n f l o w t o t h e r e s e r v o i r i s c o n s t a n t . (3) The l o a d i s c o n s t a n t , e.g., n u m e r i c a l l y e q u a l t o t h e c o n s t a n t s a and \ 2 : P D = a + X 2 . ... (5-4) (4) T r a n s m i s s i o n l o s s e s c an be n e g l e c t e d : P L = 0. ... (5-5) 48 5.2 S o l u t i o n s o f t h e P r o b l em: The F i r s t N e c e s s a r y C o n d i t i o n 6 3 GK's s o l u t i o n s a r e g i v e n i n t h e f o r m o f a l i n e a r r e l a  t i o n s h i p between h y d r o p l a n t o u t p u t and t i m e ( s e e F i g u r e 8 — GK's p a p e r ) and a l i n e a r f u n c t i o n o f head, and t h e r e f o r e s t o r a g e , w i t h t i m e , i . e . , f o r t h e s t o r a g e S1 = \ - Bt, ... (5-6) w i t h c o n s t a n t s and B c o n s i s t e n t l y c h o s e n w i t h t h e g e n e r a l s o l u t i o n g i v e n by e q u a t i o n ( 3 - 5 8 ) . Assuming a r b i t r a r i l y , t h a t t h e f u n c t i o n a l r e l a t i o n s h i p between t h e h y d r o p l a n t o u t p u t and t h e s t o r a g e and t h e change o f s t o r a g e i s g i v e n by P H 1 = " S l " S l ' ( 5 " 7 ) one o b t a i n s by u s i n g e q u a t i o n (5-6) t h a t P H 1 = ~ ^ + P + Pt, ••• (5-8) w h i c h i s c o n s i s t e n t w i t h b o t h e q u a t i o n s (3-59) and F i g u r e 8 o f GK's p a p e r . 6 3 Due t o a s s u m p t i o n (4) one a c q u i r e s f o r t h e t h e r m a l p l a n t P T 2 = P D ~ P H 1 ' ... (5-9) w h i c h combined w i t h e q u a t i o n s (5-4) and (5-8) y i e l d s P T 2 = (a + \ 2 + \±) - B ( t +.1), ... (5-10) w h i c h s a t i s f i e s e q u a t i o n ( 3 - 5 7 ) . S i n c e t h e p l a n t ' s o u t p u t and t h e s t o r a g e v a l u e a r e a l l p o s i t i v e p h y s i c a l q u a n t i t i e s , t h e f o l l o w i n g i n e q u a l i t i e s h o l d : 4 9 B = \ , . . . ( 5 - 1 1 ) a+X2+X1-p = 0 . The p h y s i c a l meaning o f e q u a t i o n s ( 5 - 8 ) and ( 5 - 1 0 ) a r e t h a t due t o t h e h i g h c o s t o f t h e r m a l power, i t i s d e s i r a b l e t o h o l d t h e h y d r o g e n e r a t i o n low a t t h e b e g i n n i n g o f t h e p e r i o d and t h e t h e r m a l g e n e r a t i o n h i g h a t t h e b e g i n n i n g , b u t low a t the end o f t h e p e r i o d . The n e x t s t e p i s t o p r o v e t h a t e q u a t i o n s ( 5 - 6 ) and ( 5 - 1 0 ) s a t i s f y t h e E u l e r ' s e q u a t i o n s . To do t h i s t h e f o l l o w i n g d e r i v a  t i v e s a r e f i r s t computed: Prom e q u a t i o n s ( 5 - 4 ) and ( 5 - 5 ) one o b t a i n s b P L b P L • V p — - — = ° » 6 0 0 ( 5 - 1 2 ) °RT2 ° H I b P D b P D 3PT: = w ~ = °' ••• ( 5 ~ 1 3 ) T 2 U A H T f r o m e q u a t i o n ( 5 - 9 ) : b P H 1 b P T 2 Xp = - 1 . N P = - 1 , . . . ( 5 - 1 4 ) ° R T 2 0 ± H 1 f r o m e q u a t i o n ( 5 - 1 0 ) : P T 2 = ^ T = - P * ( 5 ~ 1 5 > b P T 9 rrr^- = t + 1 , . . . ( 5 - 1 6 ) o P T 2 and u s i n g e q u a t i o n s ( 5 - 6 ) and ( 5 - 7 ) dS 5 1 = d t 1 S T = -rr— = - 6 , . . . ( 5 - 1 7 ) I f t i s h i g h , P,j,2 i s h i g h , and hence Cg i s h i g h . 50 s; OP oT HI and OP HI = - ( t + 1 ) , t+1 ... (5-18) ... -(5-19) ... (5-20) s i n c e V si ? H 1 = - ( x r p t ) + - ± - X - s. t+1 ^ ' v l j _ T- j_ - «•. (5—21) One may now s o l v e f 1 ( t ) and g * ( t ) g i v e n by e q u a t i o n s (3-51) and ( 3 - 5 3 ) : f 1 ( t ) t r - ( b 2 + X 2 ) - (X±+X2) = b 2 - X X , ... (5-22) g X ( t ) ( b 2 + X 2 ) + ( X 1 + X 2 ) t+1 /, , \ t+1 (5-23) S u b s t i t u t i o n o f t h e s e i n e q u a t i o n s (3-50) and (3-52) y i e l d s (W - ft (VV-***1) = ( b 2 - X 1 ) - ( b g - X ^ - l = 0, ... (5-24) and d_ d t ( V X i ) . t+1 . t ( b ^ j . i + i - (b.-X,) = .(Vxi) k - (5-25) E q u a t i o n (5-24) a l w a y s s a t i s f i e s E u l e r ' s e q u a t i o n s w h i l e equa t i o n (5-25) w i l l be i d e n t i c a l l y z e r o i f and o n l y i f , f o r t ?4 0, b 2 = * • ] _ . . ' ... (5-26) 51 S u b s t i t u t i o n o f t h i s v a l u e i n e q u a t i o n (5-24) i s a t r i v i a l s o l u t i o n . As m e n t i o n e d e a r l i e r , t h e n u m e r i c a l v a l u e s a r e c h o s e n a r b i t r a r i l y s u c h t h a t c o n d i t i o n (5-26) can be s a t i s f i e d i f X^, t h e i n i t i a l s t o r a g e v a l u e , i s c h o s e n a p p r o p r i a t e l y . 5.3 T e s t s f o r N e c e s s a r y and S u f f i c i e n t C o n d i t i o n s The r e s u l t s o f t h e p r e v i o u s p a r t w i l l now be t e s t e d a g a i n s t t h e t h r e e n e c e s s a r y c o n d i t i o n s a l o n g w i t h t h e t h r e e s u f f i c i e n t c o n d i t i o n s . ( l ) The Second N e c e s s a r y C o n d i t i o n T h i s c o n d i t i o n i s g i v e n by e q u a t i o n (4-1) t o g e t h e r w i t h e i t h e r o f t h e t h r e e e q u a t i o n s ( 4 - 2 . a ) , (4-2.b) o r ( 4 - 2 c ) . U s i n g t h e s i m p l e s t e q u a t i o n (4-2.a) and making use o f t h e r e  l a t i o n s ( 5 - 1 5 ) , ( 5 - 1 6 ) , (5-17) and (5-22) one o b t a i n s u 1 d ( b 2 - \ x ) (t+D ... (5-27) E q u a t i o n (5-27) i s i d e n t i c a l l y z e r o due t o c o n d i t i o n ( 5 - 2 6 ) . Hence, t h e s e c o n d n e c e s s a r y c o n d i t i o n i s s a t i s f i e d . (2) The T h i r d N e c e s s a r y C o n d i t i o n U s i n g e q u a t i o n s ( 4 - 4 ) , (5-6) and (5-10) t h e f o l l o w i n g f u n c t i o n s o f t i m e can be c a l c u l a t e d : W l ( t ) = ( - B ) . l - (-B) • 0 = -8, ( o 2 ( t ) = ( _ p ) . - ( t + l ) - ( - 6 ) ( - t ) = B, ... (5—28) U g ( t ) = (-B) • 1 - ( - B ) . l = 0, w 4 ( t ) = (-B) -1- (-B) • 0 = -B. S i n c e w ^ ( t ) , c o 2 ( t ) , W g ( t ) and w 4 ( t ) a r e c o n s t a n t , t h e f i r s t 52 and s e c o n d rows o f t h e d e t e r m i n a n t D ( t , t Q ) a r e i d e n t i c a l and t h e r e f o r e D ( t , t Q ) = 0, ... (5-29) and., c o n s e q u e n t l y , t h e t h i r d n e c e s s a r y c o n d i t i o n i s n o t s a t i s  f i e d . (3) The F o u r t h N e c e s s a r y C o n d i t i o n S u b s t i t u t i n g e q u a t i o n s ( 5 - 1 ) , ( 5 - 2 ) , ( 5 - 7 ) , ( 5 - 1 6 ) , ( 5 - 1 8 ) , (5-22) and (5-23) i n t h e W e i e r s t r a s s • s E - f u n c t i o n (4-14) one o b t a i n s E = b 2 P T 2 + c 2 + X 1 ( - S 1 - § 1 ) + x J ( - S 1 - ' l 1 ) + P T 2 - P ] ) + Pmo ( b ^ M t + l ) +S 1 ( b ^ ^ J . ^ i • t (5-30) T2 2 , v l I n t h i s e q u a t i o n S^ = - 6 = c o n s t a n t = P^ 2> and t h e r e f o r e S l = S l = P T 2 = P T 2 ' f r o m w h i c h , by s u b s t i t u t i n g e q u a t i o n s ( 5 - 4 ) , (5-8) and (5-10) (5-31) (5-32) i n equation ( 5 - 3 0 ) : E = c 2 + ( b 2-\ 1)(\ 1+6) + b 2 ( a + X 2 ) + BtCbg-Xj^), which, due to condit ion (5-26) reduces to E = c 2 + b 2 (a + \ 2 ) , ... (5-33) ... (5-34) and hence, s i n c e t h e c o n s t a n t s a r e a l l p o s i t i v e , t h e f o u r t h n e c e s s a r y c o n d i t i o n i s s a t i s f i e d . (4) The T h r e e S u f f i c i e n t C o n d i t i o n s A c c o r d i n g t o e q u a t i o n s ( 5 - 2 7 ) , and D ( t , t Q ) r e s p e c  t i v e l y a l w a y s v a n i s h . T h e r e f o r e , t h e f i r s t and s e c o n d 53 s u f f i c i e n t c o n d i t i o n s , w h i c h a r e e q u i v a l e n t t o t h e s e c o n d and t h i r d n e c e s s a r y c o n d i t i o n s w i t h o u t t h e e q u a l i t y s i g n , w i l l n e v  e r be s a t i s f i e d . However, t h e t h i r d s u f f i c i e n t c o n d i t i o n i s a l w a y s s a t i s f i e d s i n c e , due t o e q u a t i o n (5-34) E =»0. ... (5-35) 5.4 C o n c l u s i o n s 5.4.1 D i s c u s s i o n on t h e S o l u t i o n s F o r t h e s p e c i f i c p r o b l e m s t a t e d i n p a r t 5.1 o n l y t h e f i r s t , s e c o n d , and f o u r t h n e c e s s a r y c o n d i t i o n s , and t h e t h i r d s u f f i c i e n t c o n d i t i o n , a r e s a t i s f i e d i f e q u a t i o n (5-26) h o l d s . The s o l u t i o n s o f t h i s p r o b l e m , however, do n o t s a t i s f y W e i e r s t r a s s • s t h i r d n e c e s s a r y c o n d i t i o n and t h e f i r s t and s e c o n d s u f f i c i e n t c o n d i t i o n s . T hese l a s t t h r e e c o n d i t i o n s c o u l d be e a s i l y met, i f q u a d r a t i c o r h i g h e r o r d e r s o l u t i o n s were u s e d , i n t h e p l a c e o f e q u a t i o n s (5-6) and ( 5 - 1 0 ) . H^ would, t h e n , n o t have t o v a n i s h and t h e d e t e r m i n a n t D ( t , t Q ) w o u l d n o t c o n t a i n i d e n t i c a l rows l e a d  i n g t o a z e r o s o l u t i o n . N e v e r t h e l e s s , t h i s example was c h o s e n t o e n a b l e t h e r e a d e r t o compare t h e method employed i n t h i s t h e s i s w i t h methods d e v e l o p e d by a u t h o r i t i e s i n t h i s f i e l d who o b t a i n e d l i n e a r f u n c t i o n s o f t i m e f o r t h e i r s t o r a g e and t h e r m a l p l a n t v a l u e s . The example was made s i m p l e enough s u c h t h a t t h e n e c e s s a r y and s u f f i c i e n t c o n  d i t i o n s c o u l d be e a s i l y c a l c u l a t e d . I t i s t o be n o t e d r e g r e t f u l l y , t h a t even i f t h e s e v e n c o n d i t i o n s a r e s a t i s f i e d , t h e c o n d i t i o n e d p r o b l e m 54 w i l l o n l y l e a d t o a " s e m i - s t r o n g minimum" due t o t h e f a c t t h a t , w i t h t h e f i r s t and s e c o n d s u f f i c i e n t c o n d i t i o n s s a t i s f i e d , no g u a r a n t e e c a n be g i v e n as t o t h e p o s s i b i l i t y o f c o n s t r u c t i n g W e i e r s t r a s s ' E - f u n c t i o n . The p r o b l e m t h e r e f o r e d i f f e r s f r o m t h e u n c o n d i t i o n e d c a s e where t h e q u e s t i o n o f W e i e r s t r a s s ' c o n s t r u c t i o n c o u l d be answered i n t h e a f f i r m a t i v e . I n g e n e r a l , t h e p r o b l e m i s n o t f i n i s h e d a t t h e end of p a r t 5.3. I f p r o j e c t and o p e r a t i n g l i m i t a t i o n s e x i s t t h e r e s u l t s w o u l d s t i l l have t o be compared w i t h t h e s e l i m i t a t i o n s . I f v i o l a t i o n s o c c u r , tb.e c o r r e s p o n d i n g ex t r e m a l s s h o u l d be c h o s e n and t h e whole p r o c e d u r e r e p e a t e d . F o r t h e same o r d e r o f magn i t u d e o f q u a n t i t i e s o b t a i n e d , i . e . , f o r s m a l l v i o l a t i o n s t h e t e s t s f o r n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s w o u l d p r o b a b l y s t i l l be v a l i d , and hence, n o t t o be p e r f o r m e d a g a i n . 5.4.2 D i g i t a l Computer A p p l i c a t i o n I n s o l v i n g t h e above t w o - p l a n t p r o b l e m , o r , i n g e n e r a l , f o r . t h e s o l u t i o n o f a combined h y d r o - t h e r m a l p r o b  lem w i t h m t h e r m a l and n h y d r o p l a n t s , h i g h - s p e e d d i g i t a l computers c a n be employed most a d v a n t a g e o u s l y . These cpm- p u t e r s c a n be u s e d i n e i t h e r o f t h e two f o r m s : ( i ) d i r e c t l y , by a s s u m i n g one s o l u t i o n f o r each p l a n t ( i n t o t a l m+n s o l u t i o n s ) and s u b s t i t u t i n g t h e s o l u  t i o n s i n t h e g e n e r a l e q u a t i o n s , o r , * B o l z a ' s t e r m . 55 ( i i ) i n d i r e c t l y , by f i r s t s o l v i n g e i t h e r o f t h e l e s s complex G-K's, K r o n ' s , or C y p s e r ' s e q u a t i o n s and t h e n s u b  s t i t u t i n g them i n t h e g e n e r a l e q u a t i o n s d e r i v e d i n t h i s t h e s i s . T h e r e a r e s e v e r a l c r i t e r i a w h i c h c o u l d be u s e d i n making t h e above c h o i c e , a l t h o u g h t h e most l o g i c a l c r i  t e r i o n must be t h a t o f t h e s p e e d , i n w h i c h t h e c o r r e c t economic s o l u t i o n c a n be o b t a i n e d u s i n g t h e same computer. The d i r e c t method w o u l d n o t c r e a t e d i f f i c u l t i e s i f t h e o p e r a t i n g e n g i n e e r , b a s e d on h i s e x p e r i e n c e , knows how t o make r e l i a b l e i n i t a l e s t i m a t e s o f a l l s o l u t i o n s . However, t h e p r o b l e m becomes i n s u r m o u n t a b l e i f many p l a n t s a r e p r e  s e n t i n t h e s y s t e m . I n t h i s c a s e , t h e i n d i r e c t method wo u l d p r o d u c e t h e r e s u l t s f a s t e r . A f t e r t h e c o r r e c t r e s u l t s a r e o b t a i n e d t h e y a r e t o be t e s t e d a g a i n s t t h e t h r e e o t h e r n e c e s s a r y and t h e t h r e e s u f f i c i e n t c o n d i t i o n s . The computer program s h o u l d a l s o c o n t a i n t e s t s f o r t h e v a r i o u s p r o j e c t and o p e r a t i n g l i m i  t a t i o n s . I n a l l o f t h e above c a s e s i t e r a t i v e l o o p s s h o u l d be u s e d f o r r e p e a t i n g t h e p r o c e d u r e . 56 CHAPTER V I THE GENERAL EQUATIONS COMPARED WITH PREVIOUSLY DEVELOPED FORMULAS 6 • I n t r o d u c t i o n I n t h i s c h a p t e r t h e g e n e r a l e q u a t i o n s d e v e l o p e d i n c h a p  t e r I I I w i l l be Compared w i t h a l l known f o r m u l a s summarized i n s e c t i o n 2.3.3 o f c h a p t e r I I . I t i s e v i d e n t t h a t t h e methods g i v e n i n t h e above s e c t i o n c a n o n l y be compared w i t h t h e s i m p l i  f i e d t y p e B c a s e o f t h e g e n e r a l e q u a t i o n s , s i n c e o n l y s c h e d u l i n g o f g e n e r a t i o n s or l o a d a l l o c a t i o n s among p l a n t s a r e c o n s i d e r e d i n t h e s e methods, i n s t e a d o f t h e g e n e r a l c a s e u s i n g t i m e - f u n c  t i o n s d e r i v e d i n t h i s t h e s i s . I n o r d e r t h a t e a c h one o f t w e l v e e q u a t i o n s i n t h e t h r e e s h o r t - t e r m g r o u p s c a n be j u d g e d , t h e com p a r i s o n w i l l be made u s i n g t h e s i m p l i f i e d t w o - p l a n t model. F u r t h e r m o r e , t h e g r o u p i n g s o f e q u a t i o n s w i l l be i g n o r e d when p l a c i n g s i d e by s i d e t h e s i m p l i f i e d t y p e B e q u a t i o n s ( 3 - 7 8 . j ) and (3-85.1) w i t h t h e p r e v i o u s l y d e v e l o p e d f o r m u l a s (2-9) t o ( 2 — 2 2 ) . The c o m p a r i s o n w i l l commence w i t h t h e e a s i e s t and f i n  i s h w i t h t h e most d i f f i c u l t e q u a t i o n t o compare. 6.2 C o m p a r i s o n w i t h K r o n ' s E q u a t i o n 6 3 Kr o n ' s p r o b l e m i s n o t r e s t r i c t e d by t h e f i r s t a u x i l i a r y c o n d i t i o n ( 3 - 5 . i ) u s e d i n t h i s t r e a t i s e . F o r t h i s r e a s o n t h e L a g r a n g e ' s m u l t i p l i e r \^ w i l l d i s a p p e a r and e q u a t i o n ( 3 - 8 5 . i ) w i l l r e d u c e t o b P T \ bP , I , * J i d_ 2 1 1 ~ £ P H "*'dt X 2 I1 ~ ) b / = 0, ... (6-1) r e m o v i n g a l l numbered s u b s c r i p t s o f t h e p l a n t v a r i a b l e s . The '57 p r o b l e m t h e r e f o r e becomes t h a t o f c o m p a r i n g K r o n ' s e q u a t i o n (2-15) w i t h e q u a t i o n ( 6 - 1 ) . K r o n ' s v a r i a b l e s a r e q and q, o r q and Q u s i n g n o t a t i o n s o f t h i s t e x t , where t q = q ( t ) = q ( 0 ) + j Qdt. ... (6-2) G I f l e a k a g e and e v a p o r a t i o n a r e i g n o r e d t h e i n f l o w t o a r e s e r v o i r e q u a l s t h e o u t f l o w and t h e t i m e r a t e o f change o f s t o r a g e , * i . e . , F ( t ) = Q ( t ) + a ( t ) + S ( t ) , w h i c h c a n be w r i t t e n as Q ( t ) = F ( t ) - 0 ( t ) - S ( t ) . ... (6-3) The s t o r a g e a t any t i m e t can be g i v e n , a s t S ( t ) = S ( 0 ) +J S ( t ) d t . ... (6-4) 0 S u b s t i t u t i n g e q u a t i o n (6-3) i n e q u a t i o n (6-2) and sub s t i t u t i n g e q u a t i o n (6-4) i n t h e r e s u l t i n g e q u a t i o n one o b t a i n s t t q ( t ) = q ( 0 ) + j F ( t ) d t - f a ( t ) d t - S ( t ) + S ( 0 ) . ... (6-5) 0 G L e t t r "0 J F ( t ) d t = r ( t ) + r ( o ) , . . . (6-6) t f o ( t ) d t = i t ( t ) + 7i(0), ... (6-7) "0 See a l s o A p p e n d i x A. 58 t h e n K r o n ' s q ( t ) w i l l be e q u i v a l e n t t o t h e m a g n i t u d e o f t h e v o l  ume o f s t o r a g e S ( t ) i f q(G) = - I\(t) - r ( 0 ) + u ( t ) + u ( 0 ) - S ( 0 ) . ... (6-8) S i n c e q ( 0 ) i s n o t a f u n c t i o n o f t i m e , t h e t i m e f u n c t i o n s s h o u l d v a n i s h , t h u s r ( t ) = T t ( t ) , ... (6-9) and hence q ( 0 ) = - r ( 0 ) + Tl(0) - S ( 0 ) . ... (6-10) T h i s l a s t e q u a t i o n s t i p u l a t e s t h a t t h e volume o f s t o r a g e a t t h e b e g i n n i n g o f e a c h p l a n n i n g p e r i o d depends upon t h e i n t e g r a t e d f l o w and s p i l l a g e d u r i n g a l l p r e v i o u s p l a n n i n g p e r i o d s . S u b s t i t u t i n g e q u a t i o n s (6-6) t o (6-10) i n e q u a t i o n (6-5) g i v e s q ( t ) = - S ( t ) , ... (6-11) i f r o m w h i c h t h e e q u i v a l e n c e o f e q u a t i o n s (6-1) and (2-15) c a n be s e e n w i t h * X = — ^2' * * * (6—12) and H = ___H 1 A H ' - £ c p • ... (6-13) 6.3. C o m p a r i s o n w i t h R i c a r d ' s E q u a t i o n The b e s t way t o i d e n t i f y t h e e q u i v a l e n c e between R i c a r d ' s e q u a t i o n and t h e s i m p l i f i e d g e n e r a l e q u a t i o n s ( 3 - 8 5 . i ) i s by- Q u a n t i t i e s or v a r i a b l e s o f t h i s t h e s i s w i l l h e n c e f o r t h be w r i t t e n on t h e r i g h t - h a n d s i d e o f t h e e q u a l i t y s i g n , whenever an e q u i v a l e n c e i s p r o v e n . v 59 c o m p a r i n g t h e f o r m e r w i t h K r o n ' s e q u a t i o n (2-15). The l i n k 6 3 between t h e two i s p r o v i d e d by t h e e q u a t i o n i l X 7 A, p ' where C£ as a f u n c t i o n o f t i m e c a n be d e t e r m i n e d f r o m ... (6-14) y d £ _ M d t ... (6-15) I f t r a n s m i s s i o n l o s s e s a r e n e g l e c t e d , K r o n ' s e q u a t i o n r e d u c e s t o X * P H . d xfz = o, A'SF" + d t w i t h t h e e q u i v a l e n c e i n t h i s t h e s i s : bP„ 1 ... (6-16) X 2 * P H d A "SE. d t H L2 S T = 0. ... (6-17) S u b s t i t u t i n g e q u a t i o n (6-14) i n e q u a t i o n (6-15) and by r e - a r r a n g i n g one o b t a i n s X bQ/bh d__ A W c S P j d t X SQTSP H J = 0. ... (6—18) R e a l i z i n g t h e dependence o f t h e p l a n t d i s c h a r g e Q on t h e n e t head h and t h e p l a n t o u t p u t P^, one may w r i t e y = Q - Q (fa, P H ) = 0, and, hence ^ = 1 ^ 0 , ... (6-19) 6 3 f r o m . w h i c h one d e r i v e s bQ/bh ^ P H and bQ/0PH = "Sh" 1 * P H SQ/bPH ~ W ' ... (6-20) ... (6-21) 6Q S u b s t i t u t i n g t h e l a s t two e q u a t i o n s i n e q u a t i o n (6-18) r e s u l t s i n K r o n ' s e q u a t i o n (6-16) and i t s e q u i v a l e n c e ( 6 - 1 7 ) . 6-4 C o m p a r i s o n w i t h CDGK's and GK's E q u a t i o n s 6 2 CDGK's t h e r m a l p l a n t e q u a t i o n s a r e e x a c t l y e q u i v a l e n t 6 3 w i t h t h o s e d e v e l o p e d by G l i m n and K i r c h m a y e r . T h e i r equa t i o n s a r e a l s o t h e same w i t h t h e s i m p l i f i e d g e n e r a l e q u a t i o n ( 3 - 7 8 . j ) w i t h t h e o b v i o u s i d e n t i t y X = - ^ n + i » (6-22) w h i c h r e d u c e s t o i d e n t i y (6-12) f o r a s i n g l e h y d r o p l a n t p r o b  lem (n = 1 ) . W i t h t h e e x c e p t i o n o f t h e l o s s t e r m s , CDGK's and GK's h y d r o p l a n t s e q u a t i o n s a r e i d e n t i c a l w i t h R i c a r d ' s e q u a t i o n s whose i d e n t i t y w i t h t h e g e n e r a l e q u a t i o n s has been p r o v e n . The l o s s t e r m s c a n be i n s e r t e d i n t h e f o r m o f p e n a l t y f a c t o r s and w h i c h r e d u c e t o u n i t y when t h e l o s s e s v a n i s h , i . e . , T 1 L T ~ 1 - 6 P L / 6 P T ' • o • (6 —2 3) L H " l - 6 P L / b P H ' I n s t e a d o f e q u a t i o n (6-14) one t h e r e f o r e has X ^T 7 = 0-Q/c% L~ ' w h i c h by s u b s t i t u t i n g t h e v a l u e o f \ f o r t h e l o s s l e s s t h e r m a l p l a n t s y s t e m becomes dC/dP T L T 7 = 3 5 7 5 5 ^ L~ • ( 6 " 2 4 ) X i n i h i s c a s e i s l o s s l e s s . 61 I n t e g r a t i n g e q u a t i o n (6-15) one o b t a i n s / * 7 = 70 exp .. (6-25) f r o m w h i c h one d e r i v e s dC b g _ d P T ~ L T b P H t_ Q 7o - p ( / % .. (6-26) T h i s l a s t e q u a t i o n i s e x a c t l y e q u i v a l e n t t o t h e two GK's s c h e d u l i n g e q u a t i o n s (2-13) and (2-14) combined. I f t h e v a r i a t i o n s o f t h e n e t head c a n be n e g l e c t e d , h i s no l o n g e r a v a r i a b l e , t h u s 58 = 0, ' ... (6-27) and, t h e r e f o r e , f r o m e q u a t i o n (6-25) 7= 7) = c o n s t a n t . ... (6-28) T h i s i s t h e p r o b l e m s o l v e d by CDGK g i v e n by e q u a t i o n ( 2 - 1 2 ) . 6.5 C o m p a r i s o n w i t h E q u a t i o n s of t h e M. I . T. Group 6.5.1 I n t r o d u c t i o n 56 I n t h e s h o r t - r a n g e c a s e , C y p s e r s p e c i f i e s a p r e  d e t e r m i n e d amount o f wa t e r a t e a c h h y d r o p l a n t o v e r a s h o r t f u t u r e t i m e i n t e r v a l . The same s p e c i f i c a t i o n e s s e n t i a l l y a p p l i e s t o t h e p r o b l e m d i s c u s s e d i n t h i s t h e s i s and t o t h e 61 63 62 model u s e d by R i c a r d , GK and CDGK. C y p s e r , however, d i s t i n g u i s h e s two c a s e s f o r h i s s p e c i f i c a t i o n : ( i ) w i t h r u n - o f - r i v e r and pondage p l a n t s ( s m a l l s t o r a g e p l a n t s i n  c l u d e d ) t h e amount o f w a t e r s p e c i f i e d f o r s h o r t t e r m use a r e t h e " a n t i c i p a t e d a v a i l a b i l i t i e s " b a s e d on s h o r t - r a n g e 62 p r e d i c t i o n s o f s t r e a m f l o w ; ( i i ) w i t h l a r g e s t o r a g e p l a n t s where a p p r e c i a b l e v a r i a t i o n s i n s t o r a g e e l e v a t i o n s and e f f i c i e n c i e s o c c u r , t h e s p e c i f i c a t i o n s must be t h e r e s u l t o f a s h o r t - t e r m o p t i m i z a t i o n s c h e d u l e . F o r t h e above r e a s o n s , t h e p r o b l e m can be s o l v e d i n two ways: w i t h power outputs, as v a r i a b l e s and w i t h v a l u e s o f s t o r a g e as v a r i  a b l e s . A b r i e f o u t l i n e o f t h e two c a s e s and t h e i r s i m i  l a r i t y w i t h t h e g e n e r a l e q u a t i o n s o f t h i s t h e s i s a r e g i v e n i n t h e f o l l o w i n g s e c t i o n s . I n t h i s c o n n e c t i o n C a r e y ' s l i n e a r i z a t i o n 6 4 ' 6 5 - w i l l be m e n t i o n e d . 6.5.2 Power O u t p u t s as V a r i a b l e s The s p e c i f i e d amount o f watex w i l l g i v e a s p e c i f i e d amount o f e n e r g y o v e r a s h o r t f u t u r e t i m e i n t e r v a l and, t h e r e f o r e , an i n t e g r a t e d a v e r a g e power f r o m t h e h y d r o p l a n t . Hence, f o r t h e t w o - p l a n t p r o b l e m : T 1 ( P R - P R A ) d t = 0. ... (6-29) '0 The e f f e c t i v e c o s t t o be m i n i m i z e d i s composed o f two p a r t s : t h e t h e r m a l f u e l c o s t and t h e c o s t o f v i o l a t i n g one o r more h y d r o l i m i t a t i o n s , i . e . , T T $ = f C 2 ( P T ) d t + f C 1 ( P R ) d t . ... (6-30) 0 0 C o m b i n i n g t h e two e q u a t i o n s above t o g e t h e r w i t h t h e Of t h e R i c a r d s e c t i o n ( s e e c h a p t e r I I ) , o n l y K r o n d i d n o t use t h i s s p e c i f i c a t i o n . 56 c o n d i t i o n s f o r l o a d r e q u i r e m e n t s , one s o l v e s ^ • ^ - ^ S P " " ^ ••• ( 6 ~ 3 1 ) bC 2 b P L X P ~ • H Tp~ = ~ H° ««» (6-32) o r T 2 0 H U s i n g an a s s u m p t i o n s i m i l a r t o t h e one a d o p t e d i n t h i s t h e s i s t h a t no v i o l a t i o n s a r e a l l o w e d , one o b t a i n s bC, s ~ = 0 . . . (6-33) w h i c h r e v e a l s t h e e x a c t e q u i v a l e n c e between C y p s e r ' s equa t i o n s (6-31) and (6-32) and CDGK's e q u a t i o n s (2-11) and (2-12), w i t h t h e f o l l o w i n g c o n d i t i o n s : -\=J^ , (6-34) and -U. = X = -X^ ° o o o (6-35) C a r e y ' s e q u a t i o n s a r e o b t a i n e d by u s i n g e q u a t i o n (6-33), and by r e p l a c i n g a l l - u . ( t ) ' s on t h e l e f t hand s i d e o f t h e e q u a l i t y s i g n by (3 w h i c h i s e q u a l t o t h e w e i g h t e d a v e r a g e v a l u e o f t h e i n c r e m e n t a l f u e l c o s t . ^ 4 ' ^ C a r e y ' s e q u a t i o n s a r e , t h e r e f o r e , n e c e s s a r i l y t h e same as t h o s e o f C y p s e r and, henc e , e q u i v a l e n t w i t h t h e g e n e r a l equa t i o n s „ 6.5o3 P l a n t - S t o r a g e V a l u e s as V a r i a b l e s C y p s e r 1 s g e n e r a l p r o b l e m i n v o l v e s t h e use o f p l a n t s t o r a g e as v a r i a b l e s and d e f i n i n g e r r o r f u n c t i o n s s u c h t h a t t h e s t o r a g e a t any ti m e t e q u a l s t h e optimum s t o r a g e 64 p l u s t h i s e r r o r f u n c t i o n , i . e . , S ( t ) = S Q ( t ) + Vc{±), ... (6-36) where e ( t ) i s a f i x e d u n d e t e r m i n e d c u r v e c o n s t r a i n e d by t h e b o u n d a r y c o n d i t i o n s e ( t =0) = 0, ... (6-37) e ( t = T) = 0 . ... (6-38) T h r o u g h t h e use o f t h e C a l c u l u s o f V a r i a t i o n s i t can be p r o v e n t h a t t h e optimum s t o r a g e c u r v e w i l l be g i v e n by E u l e r ' s e q u a t i o n s b c 2 d_ d t be. = 0, i f no h y d r o v i o l a t i o n s a r e a l l o w e d , o b t a i n s bC, be, "si '2 bc_ ~ S ~ = b P n be bP,n bP H be bP H b P T bP H "PU * bs be bs ' ... (6-39) By e x p a n s i o n one ... (6-40) ... (6-41) w h i c h , f o r a l o s s l e s s s y s t e m ( s e e e q u a t i o n s ( 3 - 7 8 . j ) and (5-14 ) ) , r e d u c e t o bPT be _ , ~ S ~ K2 bS bS 2 b s . .. (6-42) ... (6-43) S u b s t i t u t i n g e q u a t i o n s (6-42) and (6-43) i n e q u a t i o n O n l y one t y p e o f c o s t i s now i n v o l v e d , hence s u b s c r i p t 2 c a n be o m i t t e d . (6-39) y i e l d s bP H l 2 3S~ " d t bP H = 0, (6-44) w h i c h i s t h e s i m p l i f i e d g e n e r a l e q u a t i o n (6-1) w i t h P^ = 0, 66* 6.6 C o m p a r i s o n w i t h Watchorn's E q u a t i o n s W a t c h o r n d e f i n e s f o r "maximum economy": dP N = % • w ° 0"0 ( 6 ~ 4 5 ) F o r a l o s s l e s s t r a n s m i s s i o n s y s t e m t h e power o u t p u t s a r e r e l a t e d by f r o m w h i c h one o b t a i n s t h e d i f f e r e n t i a l s . (6-46) d P T = d P D - d P H oo. (6-47) T a k i n g t h e d e r i v a t i v e s o f e q u a t i o n (6-47) w i t h r e s p e c t t o t h e p l a n t d i s c h a r g e one s o l v e s d P T d P D / d t d P H dQ~ " dQ/dt dQ~ (6-48) C o n s i d e r i n g a v a r i a b l e - h e a d p l a n t w i t h two v a r i a b l e s P H = P H ^ ' H ^ S (6=49) one d e r i v e s d P H S P H SP H D H dQ~ = bQ 3h~ ° dQ ° l b ^ b U j B u t t h e n e t h e a d i s g i v e n by ( s e e A p p e n d i x A) h = y T (6-51) Watchorn's t e r m . 66 .. (6-52) .. (6-53) .. (6-54) A s s u m i n g no s p i l l a g e , one has y = y ( S ) , y T = y T (Q), y L - y L (Q), f r o m w h i c h dh ay dS d y T d y L dQ = dS ° dQ " dQ ~ dQ ° D e p e n d i n g on whether o r n o t t h e i n f l o w e x c e e d s t h e amount b e i n g d i s c h a r g e d , one o b t a i n s f r o m e q u a t i o n (6-3) dS ... (6-55) dQ /n vs d t Q-F dS - (Q-F) JQ - dQ7at " — dQ ' .. (6-56) when a p o s i t i v e v a l u e o f (Q-F) i s assumed. S u b s t i t u t i n g equa t i o n (6-56) i n e q u a t i o n (6-55) and, t h e n , i n e q u a t i o n (6-50) y i e l d s dP H 0 P R bP a . ~ ~ H I J K ay dQ " ~Q~ ~ h ™ ( W d T ' d T dQ U s i n g e q u a t i o n (6-57) i n e q u a t i o n (6-48) and, t h e n , i n e q u a t i o n (6-45) one o b t a i n s f o r t h e Watchorn's economic e q u a t i o n : Q~F d 3 dy T d y L W (6-57) N dC d P D / d t d P H bP d P T LdQ/dt *" ~Q~ " ~h" H d y T dy T (6-58) 66 dQ/dt dS ~ dQ " dQ' To p r o v e t h e s i m i l a r i t y between t h e Watchorn's e q u a t i o n s and t h o s e o f G l i m n and K i r c h m a y e r , and hence o f t h i s t h e s i s , t h e f o l l o w i n g m a n i p u l a t i o n s a r e employed. W r i t i n g e q u a t i o n (6-45) as N d0_ W d P T dC dP, T (6-59) 67 and u s i n g e q u a t i o n (6-48) and R i c a r d ' s l o s s l e s s e q u a t i o n (2-9) one o b t a i n s 1 N 1 d P D / d t d P R dQ/dt = W~l = X. .o o (6-60) I f t h e v a r i a t i o n s o f l o a d demand w i t h t i m e c a n be n e g l e c t e d o r s m a l l compared w i t h t h e v a r i a t i o n s o f t h e p l a n t d i s c h a r g e , t h e n d P D •g-|~— = 0 o o o o (6—61 ) T h i s c o n d i t i o n c a n be made p o s s i b l e i f b l o c k s o f power a r e c o n  s i d e r e d d u r i n g t h e p l a n n i n g p e r i o d , o r i f i n d i v i d u a l l o a d s a r e c o n s i d e r e d s u c h as t h e t y p e B c a s e o f c h a p t e r III„ C o m b i n i n g e q u a t i o n s (6-60) and (6-61) g i v e s - N = X. oo. (6-62) H W i t h no l o s s e s GK's e q u a t i o n c a n be w r i t t e n as t [f bQ d t Jo e x P J 3h I ~ o w h i c h r e d u c e s t o CDGK's bQ _ ^ p - = X, < . » o (6-63) 0 d"p" = ^' "°° (6-64) H i f h ead v a r i a t i o n s a r e i g n o r e d . 6 6 Assuming t h a t Watchorn's c l a i m t h a t N i s a c o n s t a n t , t h e n e q u a t i o n s (6-62) and (6-64) a r e n e c e s s a r i l y e q u i v a l e n t s i n c e ^ i s a l s o a c o n s t a n t and b o t h q u a n t i t i e s s h a r e t h e same u n i t . The c o n n e c t i n g e q u a t i o n between t h e two methods i s g i v e n b y JQ = - N, ... (6-65) f r o m w h i c h t h e s o u g h t — f o r e q u i v a l e n c e c a n be made o b v i o u s . 68 CHAPTER V I I CONCLUDING REMARKS — FUTURE WORK I n t h e p r e c e d i n g c h a p t e r s , a l l p r e v i o u s l y known methods and e q u a t i o n s f o r o p t i m i z i n g o p e r a t i o n o f combined h y d r o - t h e r m a l e l e c t r i c s y s t e m s , f o r b o t h s h o r t and l o n g p e r i o d s o f t i m e , have b e e n r e v i e w e d and a n a l y z e d . I n c o n c l u s i o n t h e f o l l o w i n g com ments can be made: (1) The l o n g - r a n g e p r o b l e m i s n o t amenable t o an e x a c t i m a t h e m a t i c a l s o l u t i o n b e c a u s e o f t h e d i f f i c u l t y i n s t a t i n g t h e a n t i c i p a t e d s t r e a m f l o w s . (2) F o r t h e s h o r t - r a n g e p r o b l e m g e n e r a l d i f f e r e n t i a l equa t i o n s have been d e r i v e d f o r o p t i m i z a t i o n o f h y d r o - t h e r m a l e l e c t r i c s y s t e ms e m p l o y i n g t h e C a l c u l u s o f V a r i a t i o n s . (3) W i t h r e g a r d t o t h e p r e v i o u s l y d e v e l o p e d s h o r t - r a n g e methods i t has been shown ( i ) t h a t t h e s e methods a r e e q u i v a l e n t p r o v i d i n g c e r t a i n c o n d i t i o n s a r e s a t i s f i e d , and ( i i ) t h a t t h e e q u a t i o n s d e r i v e d by t h e a u t h o r s o f t h e s e methods a r e m e r e l y s i m p l i f i e d f orms o f t h e a f o r e m e n t i o n e d g e n e r a l e q u a t i o n s . (4) I n a d d i t i o n t o t h e g e n e r a l e q u a t i o n s , s e v e r a l n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r s h o r t - r a n g e o p t i m i z a t i o n p r o b l e m s have been d e r i v e d . (5) The s o l u t i o n s o f a s t a n d a r d p r o b l e m u s e d by w e l l - k n o w n a u t h o r i t i e s i n t h i s f i e l d have been t e s t e d a g a i n s t t h e v a r i o u s c o n d i t i o n s m e n t i o n e d above. (6) The use o f h i g h - s p e e d d i g i t a l computers f o r s o l v i n g t h e g e n e r a l e q u a t i o n s and t e s t i n g them f o r t h e n e c e s s a r y and 69 s u f f i c i e n t c o n d i t i o n s has been d i s c u s s e d . I n v i e w o f t h e above, i t i s o b v i o u s t h a t much r e m a i n s t o be done. I n t h e f o l l o w i n g p a r a g r a p h s , s e v e r a l o f t h e more p e r t i n e n t f i e l d s o f work r e q u i r i n g f u r t h e r s t u d y a r e l i s t e d . ( i ) D i g i t a l computer a p p l i c a t i o n o f t h e method d e v e l o p e d i n t h i s t h e s i s , i . e . , s o l u t i o n s o f g e n e r a l e q u a t i o n s , t e s t s o f c o n s t r a i n t s and t e s t s f o r n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s . ( i i ) P o s s i b i l i t y o f d i g i t a l computer programming o f t h e l o n g - r a n g e TVA p r o c e d u r e , a f t e r b e i n g m a t h e m a t i c a l l y f o r m u l a t e d , T h i s method a p p e a r s t o o f f e r p r o b a b l e s o l u t i o n s . ( i i i ) F u r t h e r s t u d y o f s e v e r a l l o n g - r a n g e methods w i t h r e g a r d t o t h e i r d e f i c i e n c i e s , e.g., C y p s e r , L i t t l e and o t h e r s l i s t e d i n t h i s t h e s i s 0 ( i v ) F u r t h e r s t u d y o f t h e e f f e c t o f n e g l e c t i n g t r a n s - 51 m i s s i o n l o s s e s i n t h e l o n g - r a n g e p r o b l e m . (v) I n c l u s i o n o f i n c r e m e n t a l m a i n t e n a n c e c o s t i n t h e . . . . .. 97-101 o p t i m i z i n g e q u a t i o n s . ( v i ) Development o f a s e t o f d e f i n i t e and r e l i a b l e f o r e  c a s t i n g p r o c e d u r e s f o r p r e d i c t i n g s t r e a m f l o w s , ^ ^ ^ 70 REFERENCES The f o l l o w i n g r e f e r e n c e s a r e numbered a c c o r d i n g t o t h e i r o r d e r o f a p p e a r a n c e . They a r e g r o u p e d i n s e v e r a l s e c t i o n s i n w h i c h c h r o n o l o g i c a l l i s t i n g s , i f p o s s i b l e , a r e m a i n t a i n e d . A. G e n e r a l I n f o r m a t i o n and O p t i m i z a t i o n o f T h e r m a l Systems 1. J u s t i n , J o e l D. and M e r v i n e , W i l l i a m G., Power S u p p l y E c o n o m i c s , J o h n W i l e y and Sons, 1934. 2. K l a p p e r , D.B., D i s c u s s i o n on R e f e r e n c e 66. 3. V e n c i l l , G.J., D i s c u s s i o n on R e f e r e n c e 66. 4. S t e i n b e r g , M.J. and S m i t h , T.H., Economic L o a d i n g o f Steam Power P l a n t s and E l e c t r i c Systems, M c G r a w - H i l l Book Company, I n c . , New Y o r k , 1935. 5. Kaufmann, P a u l G. , "Load D i s t r i b u t i o n Between I n t e r c o n n e c t e d Power S t a t i o n s " , J o u r n a l o f t h e I E E , v o l . 90-91, 1943-1944, p t . I I (Power), pp. 119-140. 6. S t e i n b e r g , M.J. and S m i t h , T . H . , " I n c r e m e n t a l L o a d i n g o f G e n e r a t i n g S t a t i o n s " , E l e c t r i c a l E n g i n e e r i n g , v o l . 52, O c t o b e r 1933, pp. 674-678. 7. K i r c h m a y e r , L.K., Economic O p e r a t i o n o f Power Systems, J o h n W i l e y and Sons, I n c . , New Y o r k , 1958. 8. K i r c h m a y e r , L.K., Economic C o n t r o l o f I n t e r c o n n e c t e d Systems, J o h n W i l e y and Sons, New Y o r k , 1959. " 9. S t e i n b e r g , M.J., "How t o C a l c u l a t e I n c r e m e n t a l P r o d u c t i o n C o s t s " , Power, v o l . 90, J u l y 1946, pp. 76-78, p . 146. 10. K i r c h m a y e r , L.K. and M c D a n i e l , G.H., " T r a n s m i s s i o n L o s s e s and Economic L o a d i n g o f Power Systems", G e n e r a l E l e c  t r i c Review, S c h e n e c t a d y , New Y o r k , O c t o b e r 1951. 11. K i r c h m a y e r , L.K. and S t a g g , G.W., " E v a l u a t i o n o f Methods o f C o - o r d i n a t i n g I n c r e m e n t a l F u e l C o s t s and I n c r e m e n t a l T r a n s m i s s i o n L o s s e s " , A I E E T r a n s a c t i o n s , v o l . 71, p t . I l l , 1952, pp. 513-521. 12. K r o n , G a b r i e l , " T e n s o r i a l A n a l y s i s o f I n t e g r a t e d T r a n s m i s s i o n Systems - P a r t I I I . The ' P r i m i t i v e ' D i v i s i o n " , A I E E  T r a n s a c t i o n s , v o l . 71, pt„ I I I , O c t o b e r 1952, pp. 814- 822.- 13. Ward, J.B., "Economy L o a d i n g S i m p l i f i e d " , A I E E T r a n s a c t i o n s , v o l . 72, p t . I l l , December 1953, pp. 1306-1311. 71 14. B a r k e r , D . C , J a c o b s , W.E. , F e r g u s o n , R.W. and H a r d e r , E.L., "L o s s E v a l u a t i o n - P a r t I . L o s s e s A s s o c i a t e d w i t h S a l e Power", A I E E T r a n s a c t i o n s , v o l . 73, p t . I I I - A , pp. 709-716. 15. T r a v e r s , R.H., H a r k e r , D . C , Long, R.W. and H a r d e r , E.L., "L o s s E v a l u a t i o n - P a r t I I I . Economic D i s p a t c h S t u d i e s o f Steam E l e c t r i c G e n e r a t i n g Systems", A I E E T r a n s  a c t i o n s , v o l . 73, p t . I I I - B , O c t o b e r 1954, pp. 1091- 1104. 16o L e V e s c o n t e , L.B. and H i c k s , K.L., "The Common Sense o f I n  c r e m e n t a l L o a d i n g " , P r o c e e d i n g s o f t h e A m e r i c a n Power  C o n f e r e n c e , C h i c a g o , v o l . XIX., 1957, pp. 521-527. 17. Thomas, R.W., K i r c h m a y e r , L.K. and W i l s o n , J.R., "Economic C o n s i d e r a t i o n s i n G e n e r a t i o n S c h e d u l i n g f o r t h e S o u t h  w e s t e r n P u b l i c S e r v i c e Company System", A I E E T r a n s   a c t i o n s " , v o l . 76, p t . I l l , F e b r u a r y 1958, pp. 1545- 1552. 18. L u b i s i c h , P.G., " P e n a l t y F a c t o r s f r o m Power-System Equa t i o n s " , A I E E T r a n s a c t i o n s , v o l . 77, p t . I l l , Aug. 1958, pp. 494-501. B. Computers f o r L o a d D i s p a t c h i n g o f Thermal Systems 19. I m b u r g i a , C A . , K i r c h m a y e r , L.K. and S t a g g , G.W., "A T r a n s  m i s s i o n L o s s P e n a l t y F a c t o r Computer", A I E E T r a n s  a c t i o n s , v o l . 73, p t . I l l , pp. 567-571. 20. E a r l y , E.D., P h i l l i p s , W.E. and S h r e v e , W.T., "An I n c r e  m e n t a l C o s t o f Power D e l i v e r e d Computer", A I E E T r a n s   a c t i o n s , v o l . 74, p t . I l l , June 1955, pp. 529-535. 21. M o r r i l l , C D . and B l a k e , J.A., "A Computer f o r Economic S c h e d u l i n g and C o n t r o l o f Power Systems", A I E E T r a n s  a c t i o n s , v o l . 74, p t . I I I , 1955, pp. 1136-1142. 22. O s t e r l e , W.H. and H a r d e r , E .L., " L o s s E v a l u a t i o n - P a r t IV. Economic D i s p a t c h Computers P r i n c i p l e s and A p p l i c a  t i o n " ,. M M - I l i l ^ s a c J a o n s , v o l . 75, p t . I l l , 1956, pp. 387-394. 23. S q u i r e s , R.B., B y e r l y , R.T., C o l b o r n , H.W. and H a m i l t o n , W.R., " L o s s E v a l u a t i o n - P a r t V. Economic D i s p a t c h Computer: D e s i g n " , A I E E T r a n s a c t i o n s , v o l . 75, p t . I l l , 1956, pp. 719-727. 24. B u r n e t t , K„N., H a l f h i l l , D.W. and Sh e p a r d , B.R.", "A New A u t o m a t i c D i s p a t c h i n g System f o r E l e c t r i c Power Systems", A I E E T r a n s a c t i o n s , v o l . 75, p t . I l l , 1956, pp. 1049-1056. 72 25. G l i m n , A . F „ , K i r c h m a y e r , L.K., P e t e r s o n , V.R. and S t a g g , G.W., " A c c u r a c y C o n s i d e r a t i o n s i n Economic D i s  p a t c h i n g o f Power Systems - P a r t I " , A I E E T r a n s   a c t i o n s , v o l . 75, p t . I l l , 1956, pp. 1125-1137. 26. C o l b o r n , H.W., B y e r l y , R„T„ and S q u i r e s , R . B " A p p l i c a t i o n o f Economic D i s p a t c h Computers t o Power Systems", P r o c e e d i n g s o f t h e A m e r i c a n Power C o n f e r e n c e , v o l . X V I I I , 1956, pp. 515-522. 27. T r a v e r s , R.H., " A u t o m a t i c Economic D i s p a t c h i n g and L o a d C o n t r o l - Ohio E d i s o n System", AIEE T r a n s a c t i o n s , v o l . 76, June 1957, pp. 291-297. 28. F a r l e y , A „ P „ , D e s c r i p t i o n - W e s t i n g h o u s e E l e c t r o n i c D i s  p a t c h Computer, P e n n s y l v a n i a E l e c t r i c A s s o c i a t i o n , P i t t s b u r g h , Pa., O c t o b e r 24-25, 1957. 29. H a m i l t o n , I„R. and O s t e r l e , W.H.,"Operating E x p e r i e n c e w i t h West Penn Power Company's Economic D i s p a t c h Com p u t e r " , A I E E T r a n s a c t i o n s , v o l . 77, p t . I l l , O c t o b e r 1958, p p 0 702-706. 30. Brown, MoJ.,"An A u t o m a t i c D i s p a t c h i n g System", AIEE T r a n s  a c t i o n s " , v o l . 78, p t . I l l , O c t o b e r 1959, pp. 957- 963. 31. G l i m n , A „ F . r K i r c h m a y e r , L.K,, Habermann, R . J r . and Thomas, R.W., " A u t o m a t i c D i g i t a l Computer A p p l i e d t o G e n e r a  t i o n S c h e d u l i n g " , A I E E T r a n s a c t i o n s , v o l . 73, p t . I l l , O c t . 1954, pp. 1267-1275o 32. Dandeno, P.L., "Use o f D i g i t a l Computers i n C o o r d i n a t i o n o f S y stem G e n e r a t i o n " , A I E E C o n f e r e n c e P a p e r , CP 56 - 1000. 33. S h i p l e y , R.Bruce and H o c h d o r f , M a r t i n , " E x a c t Economic D i s p a t c h - D i g i t a l Computer S o l u t i o n " , A I E E T r a n s  a c t i o n s , v o l . 75, pt„ I I I , 1956, pp. 1147-1153. 34. Dandeno, P.L., " A p p l i c a t i o n s o f D i g i t a l Computers i n Power System A n a l y s i s " , P r o c e e d i n g s o f t h e A m e r i c a n Power  C o n f e r e n c e , C h i c a g o , v o l . X V I I I , 1956, pp. 530-536. 35. George, E.E., Page, H.W„ and Ward, J.B., " C o o r d i n a t i o n o f F u e l C o s t and T r a n s m i s s i o n L o s s by t h e Use o f t h e Network A n a l y z e r t o D e t e r m i n e P l a n t L o a d i n g S c h e d  u l e s " , A I E E T r a n s a c t i o n s , v o l . 68, p t . I I , 1949, pp. 1152-1163. C. O p t i m i z a t i o n o f Hydro Systems 36. C r e a g e r , W i l l i a m P. and J u s t i n , J o e l D., H y d r o - E l e c t r i c 73 Handbook, J o h n W i l e y and Sons, I n c . , New Y o r k , 1927, p p 0 1075-1126, 37. McCrea, J r . , f . S , , " S e g r e g a t i o n o f H y d r o - E l e c t r i c Power C o s t s " , A I E E T r a n s a c t i o n s , v o l , 52, March 1933, pp. 1-8. 38. A l l n e r , F „ A „ , "Economic A s p e c t s o f Water Power", A I E E T r a n s  a c t i o n s , v o l . 52, March 1933, pp» 156-166. 39. Frampton, A.H„ and F l o y d , G , D„,"Factors i n t h e Economic S u p p l y o f E n e r g y i n H y d r o e l e c t r i c Systems", A I E E  T r a n s a c t i o n s , v o l . 66, 1947, pp. 1117-1125. 40. Cooper, A.R., "Load D i s p a t c h i n g and t h e Reasons f o r I t , w i t h S p e c i a l R e f e r e n c e t o t h e B r i t i s h G r i d System", J o u r n a l o f t h e I E E , v o l o 95, p t , I I , 1948, pp. 713- 723, 41. G o r n s h t e i n , V 0 M o , Most A d v a n t a g e o u s D i s t r i b u t i o n o f Loads Among E l e c t r i c Power S t a t i o n s O p e r a t i n g i n P a r a l l e l , S t a t e E n e r g y P u b l i s h i n g House, Moscow, U.S.S.R,, 1949. 42. Hoard, B , V , , " A n a l y s i s o f S t o r a g e Use t o O b t a i n Maximum I n  c r e m e n t a l E n e r g y f r o m Two Hydro S t o r a g e P l a n t s " , A I E E T r a n s a c t i o n s , v o l . 73, p t . I I I - B , 1954, pp. 1041-1048. 43. Monroe, R o b e r t A., "Some F a c t o r s i n t h e Economic G e n e r a t i o n o f Power by t h e TVA System", P r o c e e d i n g s o f t h e A m e r i  c a n Power C o n f e r e n c e , Chicag©7 v o l . ' XIX, 1957, pp„ 361-368. 44. B e r n h o l t z , B „ , S h e l s o n , W. and K e s n e r , 0., "A Method o f S c h e d u l i n g Optimum O p e r a t i o n o f O n t a r i o H y d r o ' s S i r Adam B e c k - N i a g a r a G e n e r a t i n g S t a t i o n " , A I E E T r a n s  a c t i o n s , v o l , 77, pt„ I I I , December 1958, pp'. 981- 989, D. Methods o f O p t i m i z i n g Hydro Systems B e i n g R e v i e w e d and  A n a l y z e d 45. S t r o w g e r , E,B,, " O p e r a t i n g a H y d r o - E l e c t r i c S ystem f o r B e s t Economy and Means Employed f o r C h e c k i n g P e r  f o r m a n c e o f O p e r a t i o n " , NELA B u l l e t i n , v o l . 16, 1929, pp. 777-780, 46. Schamberger, S o 0 , , "Hydro S t a t i o n O p e r a t i o n w i t h Minimum L o s s " , E d i s o n E l e c t r i c I n s t i t u t e B u l l e t i n , v o l . 3, 1935, pp. 19-23. 47. B u r r , H.A.,"Load D i v i s i o n between Common-Flow H y d r o e l e c t r i c 74 S t a t i o n s " , S. M, T h e s i s , M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , 1941. 48. J o h n s o n , D a v i d L», " D i g i t a l Computer S o l u t i o n t o D e t e r m i n e E c o n o m i c a l Use o f Hydro S t o r a g e " , A IEE T r a n s a c t i o n s , v o l . 75, p t . I l l , December 1956, pp.; 1153-1156. 49. C h a n d l e r , J r . , Webster M. and G a b r i e l l e , A n thony F., "Econo my L o a d i n g o f H y d r o e l e c t r i c Systems", S, M. T h e s i s , M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , September 1950. 50. Bobo, P o w e l l E . , " E f f e c t o f T r a n s m i s s i o n L o s s e s on t h e Optimum P l a n t L o a d i n g S c h e d u l e o f the. West Penn E l e c  t r i c Company", M. S, T h e s i s , U n i v e r s i t y o f P i t t s  b u r g h , P i t t s b u r g h , Pa., 1951. 51. J o h a n n e s s e n , P a u l R,, " T r a n s m i s s i o n L o s s and S t r e a m F l o w s " , S. M. T h e s i s , M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , Cambridge, Mass., June 1953. 52. M c l n t y r e , H„M, , B l a k e , C.W. and C l u b b , J . S . , " D i g i t a l Com p u t e r A i d s i n Power P o o l O p e r a t i o n S t u d i e s " , A I EE  T r a n s a c t i o n s , v o l . 77, p t . I , Nov. 1958, pp. 652-657. E« O p t i m i z a t i o n o f H y d r o - T h e r m a l Systems 53. R u s k i n , V.W.^ "What Do L o s s e s C o s t i n Hydro, T h e r m a l , and Combined Systems", A I E E T r a n s a c t i o n s , v o l . 75, p t . I l l , 1956, pp. 332-339. 54. R u s k i n , V.W., "Thermal P l a n t s f o r ' F i r m i n g Up" Hydro", A I E E T r a n s a c t i o n s , v o l . 76, p t . I l l , A u g u s t 1957, pp„ 609- 612, 55. D r y e r , W a l t e r , " I n t e g r a t i n g Steam and Hydro Power i n N o r t h  e r n and C e n t r a l C a l i f o r n i a " , P r o c e e d i n g s o f t h e A m e r i   can Power C o n f e r e n c e , C h i c a g o , v o l , XIX, 1957, pp, 352-360, i F. Methods o f Long-Range O p t i m i z a t i o n o f H y d r o - T h e r m a l Systems  B e i n g Reviewed and A n a l y z e d 56. C y p s e r , R u d o l p h J o h n , "Optimum Use o f Water S t o r a g e i n H y d r o - T h e r m a l E l e c t r i c Systems", S c . D. T h e s i s , M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , Cambridge, Mass., F e b r u a r y 1953. 57. C y p s e r , R.J., "Computer S e a r c h f o r E c o n o m i c a l O p e r a t i o n o f a H y d r o t h e r m a l E l e c t r i c System", A I E E T r a n s a c t i o n s , v o l . 73, p t . I I I - B , 1954, pp. 1260-1267. 75 58. L i t t l e , J o h n D . C , "The Use o f S t o r a g e Water i n a H y d r o - E l e c t r i c System", J o u r n a l o f t h e O p e r a t i o n s R e s e a r c h  S o c i e t y o f A m e r i c a , B a l t i m o r e , Md., v o l . 3, No. 2, May 1955, pp. 187-197. 59. Koopmans, T j a l l i n g C , "Water S t o r a g e P o l i c y i n a S i m p l i f i e d H y d r o e l e c t r i c System", P a p e r No. 115, Cowles Founda t i o n f o r R e s e a r c h i n E c o n o m i c s , Y a l e U n i v e r s i t y , New Haven, Conn., 1958. 60. B r u d e n e l l , Ross N. and G i l b r e a t h , J a c k H., "Economic C o m p l i  m e n t a r y O p e r a t i o n o f Hydro S t o r a g e and Steam Power i n t h e I n t e g r a t e d TVA System", A I E E T r a n s a c t i o n s , v o l . 78, p t . I l l , , J une 1959, pp. 136-156. ~ " G. Methods o f S h o r t - R a a g e Optimiz.a,tion o f H y d r o - T h e r m a l Systems  B e i n g R e v i e w e d and A n a l y z e d 61. R i c a r d , J . , "The D e t e r m i n a t i o n o f Optimum O p e r a t i n g S c h e d u l e f o r I n t e r c o n n e c t e d Hydro and Thermal S t a t i o n s " , Revue  Ge'ngrale de 1 ' E l e c t r i c i t y , P a r i s , F r a n c e , S e p t . 1940, p. 167. 62. C h a n d l e r , W.G., Dandeno, P.L., G l i m n , A 6 F . and K i r c h m a y e r , L.K. , "Short-Range Economic O p e r a t i o n o f a Combined T h e r m a l and H y d r o - E l e c t r i c Power System", A I E E T r a n s  a c t i o n s , v o l . 72, p t . I l l , O c t o b e r 1953, pp. 1057- 1065. 63. G l i m n , A.F. and K i r c h m a y e r , L.K., "Economic O p e r a t i o n o f V a r i a b l e - H e a d H y d r o e l e c t r i c P l a n t s " , A I E E T r a n s  a c t i o n s , v o l . 77, p t . I l l , December 1958, pp. 1070- 1078. 64. C a r e y , J o h n J o s e p h , "Load A l l o c a t i o n on a H y d r o t h e r m a l E l e c t r i c System", S. M» T h e s i s , M a s s a c h u s e t t s I n  s t i t u t e o f T e c h n o l o g y , Cambridge, Mass., June 1953. 65. C a r e y , J o h n j . , " S h ort-Range L o a d A l l o c a t i o n a l H y d r o - T h e r m a l E l e c t r i c System", A I E E T r a n s a c t i o n s , v o l . 73, p t . I I I - B , 1954, pp. 1105-1111. 66. Watchorn, CW., " C o o r d i n a t i o n o f Hydro and Steam G e n e r a  t i o n " , A I E E T r a n s a c t i o n s , v o l . 74, p t . I l l , 1955, pp. 142-150. " : ' • H. M i s c e l l a n e o u s 67. Von Neumann, J o h n and M o r g e n s t e r n , O s k a r , T h e o r y o f Games  and Economic B e h a v i o u r , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1944. 76 68. W i n d s o r , J . S . , R e p o r t on A n n u a l O p e r a t i n g P l a n f o r t h e V a n c o u v e r I s l a n d System, P l a n n i n g D i v i s i o n , B r i t i s h C o l u m b i a Power Commission, V i c t o r i a , B.C., June 2, 1958. 69. Ingledow, T., " H y d r o - E l e c t r i c Power and H y d r o - E l e c t r i c Power Development i n t h e Lower M a i n l a n d C o a s t a l A r e a o f B r i t i s h C o l u m b i a " , ( p a p e r r e a d b e f o r e a J o i n t M e e t i n g o f t h e V a n c o u v e r S e c t i o n o f A I E E , EIC , B, C, E n g i n e e r i n g S o c i e t y , V a n c o u v e r B o a r d o f T r a d e , J a n u a r y 8, 1 9 59). 70. M i l l e r , R.H., D i s c u s s i o n on R e f e r e n c e 66. I . C a l c u l u s o f V a r i a t i o n s 71. B o l z a , O s k a r , L e c t u r e s on t h e C a l c u l u s o f V a r i a t i o n s , U n i  v e r s i t y o f C h i c a g o P r e s s , C h i c a g o , 1904. 72. B o l z a , O s k a r , V o r l e s u n g e n u e b e r V a r i a t i o n s r e c h n u n g , D r u c k und V e r l a g v o n B.G. T e u b n e r , L e i p z i g und B e r l i n , 1909. 73. L a n c z o s , C o r n e l i u s , The V a r i a t i o n a l P r i n c i p l e s o f M e c h a n i c s , U n i v e r s i t y o f T o r o n t o P r e s s , T o r o n t o , 1949. 74. S m i t h , O.M., " J a c o b i ' s C o n d i t i o n f o r t h e P r o b l e m o f L a g r a n g e i n t h e C a l c u l u s o f V a r i a t i o n s " , T r a n s a c t i o n s , A m e r i  c a n M a t h e m a t i c a l S o c i e t y , v o l . 17, 1916, pp. 459-475. 75. B l i s s , G.A., "The P r o b l e m o f L a g r a n g e i n t h e C a l c u l u s o f V a r i a t i o n s " , A m e r i c a n J o u r n a l o f M a t h e m a t i c s , v o l . 52, 1930, pp. 673-744. J» T r a n s m i s s i o n L o s s E q u a t i o n s and I n c r e m e n t a l T r a n s m i s s i o n  L o s s F o r m u l a s 76. George, E.E., " I n t r a s y s t e m T r a n s m i s s i o n L o s s e s " , A I E E T r a n s  a c t i o n s , v o l . 62, March 1943, pp. 153-158. 77. Ward, J.B., E a t o n , J.R. and H a l e , H.W., " T o t a l and I n c r e  m e n t a l L o s s e s i n Power T r a n s m i s s i o n N etworks", A I E E  T r a n s a c t i o n s , vol„ 69, p t . I , 1950, pp. 626-632. 78. K i r c h m a y e r , L.K. and S t a g g , G.W., " A n a l y s i s o f T o t a l and I n c r e m e n t a l L o s s e s i n T r a n s m i s s i o n Systems", A I E E  T r a n s a c t i o n s , v o l . 70, p t . I I , 1951, pp. 1197-1204. 79. K r o n , G a b r i e l , " T e n s o r i a l A n a l y s i s o f I n t e g r a t e d T r a n s  m i s s i o n Systems - P a r t I . The S i x B a s i c R e f e r e n c e Frames", A I E E T r a n s a c t i o n s , v o l . 70, p t . I I , 1951, pp. 1239-1248. 77 80. G l i m n , A.Po, Habermann, R., K i r c h m a y e r , L.K. and S t a g g , G. W., " L o s s F o r m u l a s Made E a s y " , A I E E T r a n s a c t i o n s , v o l . 72, p t . I l l , A u g u s t 1953, pp.730-737. 81. B r o w n i e e , W.R. , " C o o r d i n a t i o n o f I n c r e m e n t a l F u e l C o s t s and. I n c r e m e n t a l T r a n s m i s s i o n L o s s e s by F u n c t i o n s o f V o l t a g e Phase A n g l e s " , A I E E T r a n s a c t i o n s , v o l . 73, p t . I l l , June 1954, pp. 529-541. 82. H a r d e r , E.L., F e r g u s o n , R.W., J a c o b s , W.E. and H a r k e r , D.C, " L o s s E v a l u a t i o n - P a r t I I . C u r r e n t - and Power-Form L o s s F o r m u l a s " , A I E E T r a n s a c t i o n s , v o l . 73, p t . I l l , 1954, pp. 716-731*: ~ ~ 83. Cahn, C.R., "The D e t e r m i n a t i o n o f I n c r e m e n t a l and T o t a l L o s s F o r m u l a s f r o m F u n c t i o n s o f V o l t a g e Phase A n g l e s " , A I E E T r a n s a c t i o n s , v o l . 74, p t . I l l , A p r i l 1955, pp. 161-176. 84. E a r l y , E.D., Watson, R.E. and S m i t h , G.L., "A G e n e r a l T r a n s  m i s s i o n L o s s E q u a t i o n " , A I E E T r a n s a c t i o n s , v o l . 74, p t . I l l , 1955, pp. 510-520. 85. E a r l y , E.D. and Watson, R.E., "A New Method o f D e t e r m i n i n g C o n s t a n t s f o r t h e G e n e r a l T r a n s m i s s i o n L o s s E q u a t i o n " , A I E E T r a n s a c t i o n s , v o l . 74, p t . I l l , 1955, pp. 1417- 1423. 86. C a l v e r t , J . F . and S z e , T.W., "A New A p p r o a c h t o L o s s M i n i  m i z a t i o n i n E l e c t r i c Power Systems", A I E E T r a n s   a c t i o n s , v o l . 76, p t . I l l , 1957, pp. 1439-1446. 87. Sze, T.W., G a r n e t t , J.R. and C a l v e r t , J . F . , "Some A p p l i c a  t i o n s o f a New A p p r o a c h t o L o s s M i n i m i z a t i o n i n E l e c  t r i c a l U t i l i t y Systems", A I E E T r a n s a c t i o n s , v o l . 77, p t . I l l , F e b r u a r y 1959, pp .1577-1585. 88. Watson, R.E. and S t a d l i n , W.O., "The C a l c u l a t i o n o f I n c r e  m e n t a l T r a n s m i s s i o n L o s s e s and t h e G e n e r a l T r a n s  m i s s i o n L o s s E q u a t i o n " , A I E E T r a n s a c t i o n s , v o l . 78, p t . I l l , A p r i l 1959, pp. 12-18. 89. M i l l e r , A.R., Koen, J r . , H.R. and D e l i y a n n i d e s , J . S . , "The Use o f Power T r a n s f e r E q u a t i o n s t o D e r i v e Economic C o o r d i n a t i o n R e l a t i o n s h i p s E x p r e s s e d as F u n c t i o n s o f V o l t a g e Phase A n g l e s " , A I E E T r a n s a c t i o n s , v o l . 78, p t . I l l , O c t o b e r 1959, pp. 747-753. 90. George, E.E., "A New Method o f Making L o s s F o r m u l a s D i  r e c t l y f r o m D i g i t a l Power Flow S t u d i e s " , T r a n s a c t i o n s  P a p e r 59-1081, A I E E F a l l G e n e r a l M e e t i n g , O c t o b e r 11-16, 1959. 78 91. G l i m n , A.F., K i r c h m a y e r , L.K. and S t a g g , G.W., " A n a l y s i s o f L o s s e s i n I n t e r c o n n e c t e d Systems", A I E E T r a n s  a c t i o n s , v o l . 71, p t . I l l , O c t o b e r 1952, pp„ 796-808. 92. H a l e , H.W., "Power L o s s e s i n I n t e r c o n n e c t e d T r a n s m i s s i o n N e t w o r k s " , A I E E T r a n s a c t i o n s , v o l . 71, p t . I l l , December 1952, pp. 993-998. 93. K r o n , G a b r i e l , " T e n s o r i a l A n a l y s i s o f I n t e g r a t e d T r a n s m i s s i o n Systems - P a r t IV. The I n t e r c o n n e c t i o n o f T r a n s m i s s i o n Systems", A I E E T r a n s a c t i o n s , v o l . 72, p t . I l l , A u g u s t 1953, pp. 827-~839. 94. G l i m n , A.F., K i r c h m a y e r , L.K, and S t a g g , G,¥., " A n a l y s i s o f L o s s e s i n L o o p - I n t e r c o n n e c t e d Systems", A I E E T r a n s  a c t i o n s , v o l , 72, p t . I I I , O c t o b e r 1953, pp. 944-953, 95. G l i m n , A.P., K i r c h m a y e r , L.K. and S k i l e s , J . J . , "Improved Method o f I n t e r c o n n e c t i n g T r a n s m i s s i o n L o s s F o r m u l a s " , A I E E T r a n s a c t i o n s , v o l . 77, p t . I l l , O c t o b e r 1958, pp. 755-760. 96. K e r r , B.H. and K i r c h m a y e r , L.K., " T h e o r y o f Economic O p e r a  t i o n o f I n t e r c o n n e c t e d A r e a s " , A I EE T r a n s a c t i o n s , v o l . 78, p t . I l l , A u gust^1959, pp. 647-653, K. M a i n t e n a n c e C o s t and I n c r e m e n t a l M a i n t e n a n c e C o s t 97. R o b e r t s o n , A.S, and S t a n d i n g , R,0., "The M a i n t e n a n c e o f H y d r o e l e c t r i c G e n e r a t i n g U n i t s " , A I E E T r a n s a c t i o n s , v o l . 66, 1947, pp. 776-782. 98. S t e i n b e r g , M.J., " I n c r e m e n t a l M a i n t e n a n c e C o s t s o f Steam- E l e c t r i c G e n e r a t i n g S t a t i o n s " , A I E E T r a n s a c t i o n s , v o l . 76, F e b r u a r y 1958, pp. 1251-1254. 99. L i g h t , F . H . , " A p p l i c a t i o n o f D i g i t a l Computer T e c h n i q u e f o r Development o f t h e I n c r e m e n t a l M a i n t e n a n c e C o s t " , A I E E T r a n s a c t i o n s , v o l . 78, p t . I I I , F e b r u a r y 1959, pp. 1562-1568, 100. Z e l e n k a , D.B. and T r a v e r s , R.H.,"Fundamental C o n c e p t s o f I n c r e m e n t a l M a i n t e n a n c e C o s t s as Used by Ohio E d i s o n Company", A I E E T r a n s a c t i o n s , v o l , 78, p t c I I I , June 1959, pp. 163-165, 101. N i c h o l s , C. ( c h a i r m a n ) , A I E E Committee R e p o r t : " R e p o r t on P r e s e n t - D a y P r a c t i c e s o f H a n d l i n g I n c r e m e n t a l M a i n  t e n a n c e C o s t s as They A p p l y t o Economic D i s p a t c h o f Power", A I E E T r a n s a c t i o n s , v o l . 78, p t . I l l , J une 1959, pp. 279-282. 79 L• F o r e c a s t i n g P r o c e d u r e s U s i n g P r o b a b i l i t y T h e o r y 102. F o s t e r , H . A l d en, " T h e o r e t i c a l F r e q u e n c y C u r v e s and T h e i r A p p l i c a t i o n t o E n g i n e e r i n g P r o b l e m s " , T r a n s . Am. Soc. C. E. , v o l . LXXXVII, 1924, pp. 142-173": 103. Loane, E.S. and Watchorn, C.W., " P r o b a b i l i t y Methods A p p l i e d t o G e n e r a t i n g C a p a c i t y P r o b l e m s o f a Com b i n e d Hydro and Steam System", A I E E T r a n s a c t i o n s , v o l . 66, 1947, pp. 1645-1654. 104. M c l n t y r e , H e n d e r s o n N. and S a c h s , M i l t o n S., " F o r e c a s t i n g P r o c e d u r e s Advance E f f e c t i v e Water R o u t i n g s on t h e U.S. C o l u m b i a R i v e r H y d r o - e l e c t r i c System", A I E E  T r a n s a c t i o n s , v o l . 77, p t . I l l , F e b r u a r y 1959, pp. 1588-1593. 105. K a r t v e l i s h v i l i , N.A., " P r o b a b i l i t y T h e o r y and O p e r a t i n g Modes f o r Power Systems", I z v e s t i a Akad. Nauk O t d e l e - n i e T e k h. Nauk ( E n e r g e t i k a i A u t o m a t i k a ' ) N o . l | 3-10, J a n - F e b 1959 ("excerpts i n 'Power E x p r e s s ' , v o l . 1, No. 0, J u l y 1959, I n t e r n a t i o n a l P h y s i c a l Index, I n c . , New Y o r k ) . / 80 NOMENCLATURE The f o l l o w i n g n o t a t i o n s a r e g i v e n i n an a l p h a b e t i c a l o r d e r . N o t a t i o n s u s i n g s u b s c r i p t s i o r j i n d i c a t e g e n e r a l i t i e s ; t h e s e s u b s c r i p t s may be removed f o r c o n v e n i e n c e i f s i m p l i f i e d two- p l a n t p r o b l e m s a r e c o n s i d e r e d . U n i t s a c t u a l l y u s e d i n p r a c t i c e a r e t h e ones i n d i c a t e d between b r a c k e t s o r m u l t i p l e s o f t e n o f them. L a t i n A l p h a b e t s A^ = s u r f a c e area o f r e s e r v o i r a t h y d r o p l a n t i ( a c r e ) a . , b . , c . = c o n s t a n t s o f t h e c o s t f u n c t i o n a t t h e r m a l p l a n t j J J J B^ = c o n s t a n t o f J \ (Mw-hr) B. 1 = c o n s t a n t o f J \ (Mw-hr) o B = d e r i v e d l o s s f o r m u l a c o e f f i c i e n t (l/(Mw) ) r s C. = c o s t o f v i o l a t i n g h y d r o - l i m i t a t i o n s a t h y d r o p l a n t 1 i U / h r ) C. = f u e l c o s t a t t h e r m a l p l a n t j ( $ / h r ) J D ( t , t Q ) = d e t e r m i n a n t u s e d i n t h e t h i r d n e c e s s a r y c o n d i t i o n and t h e s e c o n d s u f f i c i e n t c o n d i t i o n E = W e i e r s t r a s s ' f u n c t i o n P. = w a t e r i n f l o w t o r e s e r v o i r o r pond a t h y d r o p l a n t 1 i ( c f s ) G^,U,V = t i m e f u n c t i o n s t o compute D ( t , t g ) H = i n t e g r a n d o f I ( $ / h r ) H-^  = s e c o n d p a r t i a l d e r i v a t i v e u s e d i n t h e s e c o n d n e c e s  s a r y c o n d i t i o n and f i r s t s u f f i c i e n t c o n d i t i o n h^ = n e t head a t hydro p l a n t i ( f t ) h ^ = g r o s s head a t h y d r o p l a n t i ( f t ) I = i n t e g r a l t o be m i n i m i z e d w i t h no a u x i l i a r y c o n d i  t i o n '($) I = i n t e g r a l t o be m i n i m i z e d w i t h a u x i l i a r y c o n d i t i o n s ($) J \ = i s o p e r i m e t r i c c o n d i t i o n w i t h h^ and as v a r i a b l e s 81 J \ = i s o p e r i m e t r i c c o n d i t i o n w i t h and as v a r i a b l e s = p e n a l t y f a c t o r a t h y d r o p l a n t i ( d i m e n s i o n l e s s ) Lipj = p e n a l t y f a c t o r a t t h e r m a l p l a n t j ( d i m e n s i o n l e s s ) m = number o f t h e r m a l p l a n t s N = i n c r e m e n t a l w a t e r v a l u e u s e d by W a t c h o r n ( $ / h r / l O O O c f s ) n = number o f h y d r o p l a n t s = l o a d , demanded or d e l i v e r e d power (Mw) PJJ^ = power o u t p u t o f h y d r o p l a n t i (Mw) P^ = power t r a n s m i s s i o n l o s s e s (Mw) Prj,j = power o u t p u t o f t h e r m a l p l a n t j (Mw) P,j,_j = d P ^ / d t = t i m e r a t e o f change o f P ^ (Mw/hr) = d i s c h a r g e a t h y d r o p l a n t i ( c f s ) q = K r o n ' s s t o r a g e v a r i a b l e ( f t ) k , t h . , , . t h , q u = k v a r i a b l e o f t h e u t y p e 4 ^ = d q k / d t = t i m e r a t e o f change o f q.k = s t o r a g e volume a t h y d r o p l a n t i ( a c r e - f t ) ss d S ^ / d t = t i m e r a t e o f change o f ( c f s ) T = l e n g t h o f t h e o p t i m i z i n g p e r i o d ( h r s , d a y s ) t = t i m e v a r i a b l e ( h r s ) i>Q = b e g i n n i n g o f o p t i m i z i n g p e r i o d i t ^ = c o n j u g a t e o f p o i n t i>Q t = end o f o p t i m i z i n g p e r i o d y ^ = r e s e r v o i r o r pond e l e v a t i o n a t h y d r o p l a n t i ( f t ) y , . = head l o s s due t o f r i c t i o n , e t c . a t h y d r o p l a n t i L l ( f t ) y,p^ = t a i l w a t e r e l e v a t i o n a t h y d r o p l a n t i ( f t ) 82 Gre e k A l p h a b e t s a, 8 = i n t e g r a t i o n c o n s t a n t s f o r t h e t w o - p l a n t p r o b l e m 8 = w e i g h t e d a v e r a g e i n c r e m e n t a l f u e l c o s t u s e d by t h e MIT Group ($/Mw-hr) = c o n v e r s i o n f a c t o r u s e d f o r t h e v a r i a b l e - h e a d c a s e x To 0)- = c o n v e r s i o n c o n s t a n t a t h y d r o p l a n t i f o r t h e c o n  s t a n t - h e a d c a s e ( $ / m i l l i o n ft<3) = c o n v e r s i o n c o n s t a n t f o r t h e v a r i a b l e - h e a d c a s e ( $ / m i l l i o n f t 3 ) r ( t ) , r ( 0 ) = t i m e f u n c t i o n s o f t h e i n t e g r a t e d f l o w ( f t 3 ) e ( t ) = e r r o r f u n c t i o n o f S. i X = L a g r a n g i a n m u l t i p l i e r , u s e d by t h e R i c a r d Group as i n c r e m e n t a l c o s t o f d e l i v e r e d power ($/Mw-hr) X^ = L a g r a n g i a n m u l t i p l i e r , u s e d i n t h i s t h e s i s as i n  c r e m e n t a l c o s t a t h y d r o p l a n t i ($/Mw-hr) -X^ = L a g r a n g i a n m u l t i p l i e r , u s e d by t h e MIT Group as p a r t o f t h e i n c r e m e n t a l c o s t a t h y d r o p l a n t i ($/Mw-hr) -X , = L a g r a n g i a n m u l t i p l i e r , u s e d i n t h i s t h e s i s as i n  c r e m e n t a l c o s t o f d e l i v e r e d power ($/Mw-hr) - u ( t ) = L a g r a n g i a n m u l t i p l i e r , u s e d by t h e MIT Group as i n c r e m e n t a l c o s t o f d e l i v e r e d power ($/Mw-hr) y = p a r a m e t e r o f e r r o r f u n c t i o n c ( t ) n ( t ) , T C(0) = t i m e f u n c t i o n s o f t h e i n t e g r a t e d s p i l l a g e ( f t ) = s p i l l a g e a t h y d r o p l a n t i ( c f s ) 0 = a u x i l i a r y c o n d i t i o n s f o r l o a d r e q u i r e m e n t s (Mw) = f u n c t i o n o f n e t head h^, d i s c h a r g e Q., and h y d r o p l a n t o u t p u t ( t ) , . . . = t i m e f u n c t i o n s t o compute D(t,.tQ) w 4 ( t ) ; 0 1 D e r i v a t i v e s dCj/dP,j,j = i n c r e m e n t a l f u e l c o s t a t t h e r m a l p l a n t j ($/Mw-hr) 83 bC/bPrr. = i n c r e m e n t a l c o s t a t h y d r o p l a n t i , u s e d by t h e 1 MIT Group ($/Mw-hr) dP„/dQ = i n c r e m e n t a l h y d r o e q u i v a l e n t , u s e d by Watchorn (Mw/lOOOcfs) OP^/bPjj^ s= i n c r e m e n t a l t r a n s m i s s i o n l o s s of h y d r o p l a n t i ( d i m e n s i o n l e s s ) bP^/OPrp. = i n c r e m e n t a l t r a n s m i s s i o n l o s s o f t h e r m a l p l a n t *• j ( d i m e n s i o n l e s s ) d Q ^ / d P ^ = i n c r e m e n t a l w a t e r r a t e a t h y d r o p l a n t i , u s e d by t h e R i c a r d Group (ft^/Mw-hr) S u b s c r i p t s and S u p e r s c r i p t s The f o l l o w i n g s u b s c r i p t s a n d / o r s u p e r s c r i p t s a r e u s e d i n connec t i o n w i t h : i , / C - h y d r o p l a n t s j , k , n - t h e r m a l p l a n t s r , s - g e n e r a t i o n s , e i t h e r t h e r m a l or h y d r o x - t i m e v a r i a b l e s u - c o n t r o l v a r i a b l e s The f o l l o w i n g s u b s c r i p t s s t a n d f o r : B - s t o r a g e b a s i n C - c o n d u i t s , c h a n n e l s D - demand, d e l i v e r e d e - end o f t i m e i n t e r v a l P - f l o o d ' f - f i s h G - i n t a k e g a t e s o r g r o s s head H - h y d r o p l a n t I - i r r i g a t i o n l o s s e s maximum minimum n a v i g a t i o n z e r o t i m e p e a k i n g r e c r e a t i o n t h e r m a l p l a n t , t o t a l or t u r b i n e 85 APPENDIX A THE PROOF OF EQUIVALENCE OF P ^ l Q ^ h ^ t ) AND P H i ( F i , S^ ,^ 8± , t ) I t i s g e n e r a l l y known t h a t f o r any h y d r o p l a n t i , c o n s i d  e r i n g one p l a n t on one s t r e a m o n l y , t h e r e i s a d e f i n i t e r e  l a t i o n s h i p between t h e h y d r o p l a n t o u t p u t P ^ , t h e n e t head h ^ s and t h e p l a n t d i s c h a r g e . T h i s r e l a t i o n s h i p c a n be g i v e n ... 56,68 e i t h e r as > 63 o r , as P H i = P H i ( Q i ' h i j t ) » ( A ~ 1 } Q. = Q . ( h . , P H i , t ) . ... ( A - 2 ) B o t h e q u a t i o n s a r e t h e same, and t h e r e f o r e o n l y t h e e q u i v a l e n c e o f e q u a t i o n ( A - l ) w i t h P H i = P R i ( F i , S . , S i , t ) ... (A-3) w i l l be p r o v e n . The r e a s o n f o r u s i n g e q u a t i o n (A-3) i s p u r e l y m a t h e m a t i c a l : ( l ) t o c o n f o r m w i t h t h e method u s e d t h e p r e s e n c e o f t h e f a c t o r = d S ^ / d t , t h e t i m e r a t e o f change of t h e s t o r  age v a l u e , i s e s s e n t i a l ; ( 2 ) t h e f l o w f a c t o r F^ i s an " a l i e n v a r i a b l e " w h i c h c a n n o t be c o n t r o l l e d and i n d e t e r m i n a b l e and, h e n c e , can be o m i t t e d f r o m t h e o p t i m i z i n g e q u a t i o n s , i . e . , i t i s w a s t e f u l t o d e t e r m i n e what t h e f l o w s h o u l d be i f i t i s known t h a t t h e r e s u l t o b t a i n e d i s g o i n g t o be u s e l e s s . T h i s l a s t a s s u m p t i o n w i l l g r e a t l y s i m p l i f y a r a t h e r complex p r o b l e m . The n e t head i s t h e d i f f e r e n c e between t b e r e s e r v o i r e l e  v a t i o n y ^ and t h e t a i l w a t e r e l e v a t i o n y ^ and t h e h e a d l o s s y L i ' 1 * 6 *' h . ( t ) = y, ( t ) - y „ , ( t ) - y T , ( t ) , ... (A-4) 86 which can be written as h.(t) = h G . ( t ) - y L i ( t ) , ... ( A i 5 ) where h G i ( t ) = y i ( t ) - y T i ( t ) ••• ( A " 6 ) = the gross head at hydro plant i . To investigate the dependence of the net head and the plant discharge on the flow and the storage factors, the f o l  lowing relations w i l l be revealed: (1) There i s a d e f i n i t e relationship between the reservoir 56 68 elevation-and i t s storage volume: ' y i = y i ^ i * * ) - ••• <A-7) (2) The tailwater elevation depends on the discharge of water through the plant and the amount being s p i l l e d through the spillways-? 6' 6 8* y T i = yTi(Qi,oi,t). ... (A-8) (3) The head loss i s composed of skin f r i c t i o n and eddy losses, the l a t t e r caused by sudden changes i n the di r e c t i o n of flow or by sudden changes i n v e l o c i t y . This head loss i s a 36 function of the discharge through the plant: y L i = y L i » ) • ««» U - 9 ) ( 4 ) Generally, the spillage depends only on one variable, 6 8 the elevation of the reservoir: ai = ai^y±'t^ ••• (A-10) *A very exceptional case occurs when water discharges, or i s being s p i l l e d , into a very large r i v e r or other very large drainage areas. Then, the tailwater elevation i s independent of any of these variables. 87 w h i c h combined, w i t h e q u a t i o n (A-7) g i v e s a± = c J i ( S i , t ) . ... ( A - l l ) I n t h e c a s e o f a " f r e e " s p i l l * a t p l a n t s w i t h no s p i l l - g a t e s ( e . g . , a t r u n - o f — r r i v e r p l a n t s ) , t h e s p i l l a g e i s a c o n t i n u o u s f u n c t i o n o f e l e v a t i o n , see P i g . 1. I n t h e c a s e o f a " c o n  t r o l l e d " P i g . 1. " F r e e " s p i l l c u r v e F i g . 2. " C o n t r o l l e d " s p i l l c u r v e s p i l l , * s p i l l a g e i s a d j u s t e d m a n u a l l y : when t h e r e s e r v o i r r e a c h e s a c e r t a i n h e i g h t , a c e r t a i n amount o f w a t e r must be s p i l l e d t h r o u g h t h e s p i l l w a y s ; t h e s p i l l c u r v e i s t h e r e f o r e a s t e p - f u n c t i o n o f e l e v a t i o n , see F i g . 2. ( 5 ) The r e l a t i o n between t h e d i s c h a r g e and a l l o t h e r v a r i  a b l e s a r e g i v e n by t h e c o n t i n u i t y e q u a t i o n a t any h y d r o p l a n t , i . e . , i g n o r i n g l e a k a g e and e v a p o r a t i o n , t h e i n f l o w must e q u a l t h e o u t f l o w p l u s t h e t i m e r a t e o f change o f s t o r a g e , * * o r , F . ( t ) = Q i ( t ) + 0 . ( t ) + S . ( t ) , ... (A-12) w h i c h can be w r i t t e n as Q . ( t ) = F . ( t ) - 0 i ( t ) - S . ( t ) , ... (A-13) * W i n d s o r ' s t e r m . 6 8 * * T h i s v a l u e can be e i t h e r p o s i t i v e o r n e g a t i v e . 88 whose f u n c t i o n a l r e l a t i o n s h i p i s g i v e n by Q. = Q . C F . ^ S . , t ) . ... (A-14) S u b s t i t u t i n g e q u a t i o n ( A - l l ) i n t h e l a s t e q u a t i o n y i e l d s Q. = Q . ( F . , S . , S 1 , t ) . ... (A-15) (6) S u b s t i t u t i n g e q u a t i o n s ( A - l l ) and (A-15) i n e q u a t i o n (A-8) g i v e s y T i = y T i ( F i , S i , S i , t ) , ... (A-16) w h i c h combined w i t h e q u a t i o n s ( A - 4 ) , ( A - 7 ) , (A-9) and (A-15) g i v e s t h e f u n c t i o n a l c h a r a c t e r o f t h e n e t head h i = h i ( F i , S i , S . , t ) . ... (A-17) To p r o v e t h e sameness o f e q u a t i o n ( A - l ) and e q u a t i o n (A-3) one s u b s t i t u t e s e q u a t i o n s (A-15) and (A-17) i n e q u a t i o n ( A - l ) . 89 APPENDIX B AN OUTLINE OF THE VARIATIONAL CALCULUS PROBLEM WITH AUXILIARY CONDITIONS 1, The U n c o n d i t i o n e d P r o b l e m and t h e F i r s t N e c e s s a r y C o n d i t i o n f o r an Extremum The v a r i a t i o n a l p r o b l e m i n w h i c h no a u x i l i a r y c o n d i t i o n s 71 a r e i n v o l v e d i s c a l l e d t h e u n c o n d i t i o n e d p r o b l e m . I t i s t h e c a s e o f f i n d i n g t h e extreme v a l u e (maximum o r minimum) o f a d e f i n i t e i n t e g r a l * t 1 = / F ( q . l f • . - j t i p , ^ * • • . f C l p » t ) d t f ... ( B - l ) t 0 w i t h t h e b o u n d a r y c o n d i t i o n s q ( t n ) and q ( t ) g i v e n ( u = l , . . . , p ) ; IX \J XL 0 t h u s , t h e i r v a r i a t i o n s a t t h e two e n d - p o i n t s must v a n i s h : 6 % ( t ) 0, 6 q u ( t ) Jt=t, = 0. ... (B-2) t=t . '0 J U - " e The v a r i a b l e s q.-^ » • • •»1p a r e unknown f u n c t i o n s o f t , t o be de t e r m i n e d s u c h t h a t t h e i n t e g r a l I has an extreme v a l u e , . 71-73 hence 61 = 0 ... (B-3) f o r i n d e p e n d e n t v a r i a t i o n s o f q u , s u b j e c t o n l y t o t h e bo u n d a r y c o n d i t i o n s ( B - 2 ) . C o n s i d e r a t t h i s moment one v a r i a b l e ( p = l ) , t h e n e q u a t i o n ( B - l ) r e d u c e s t o ^ I± =J F 1 ( q 1 , q 1 , t ) d t . ... (B-4) q u means d q u / d t . 90 Suppose t h a t t h e f u n c t i o n = f ( t ) i s an e x t r e m a l o f I , , t h e n 6 I X = 0, ... (B-5) I n o r d e r t o o b t a i n t h e f i r s t n e c e s s a r y c o n d i t i o n f o r an extremum, c o n s i d e r t h e m o d i f i e d f u n c t i o n f ( t ) = f ( t ) + eV(-t), ... (B-6) where M t ) i s some a r b i t r a r y , c o n t i n u o u s and d i f f e r e n t i a b l e new 71—73 f u n c t i o n . One must now p r o v e t h a t t h e change o f t h e i n  t e g r a l due t o t h e change i n t h e f u n c t i o n becomes z e r o . U s i n g t h e s m a l l v a r i a b l e p a r a m e t e r e one c a n m o d i f y t h e f u n c t i o n f ( t ) by a r b i t r a r i l y s m a l l amounts. Comparing t h e v a l u e s o f t h e o r i g i n a l f u n c t i o n f ( t ) w i t h t h e m o d i f i e d f u n c t i o n f ( t ) a t a c e r t a i n d e f i n i t e p o i n t t by f o r m i n g t h e d i f f e r e n c e between t h e two f u n c t i o n s , one o b t a i n s • 6 ^ = f ( t ) - f ( t ) = e l / ( t ) . ... (B-7) 73 T h i s d i f f e r e n c e i s c a l l e d t h e " v a r i a t i o n " of f ( t ) , see P i g . 3 b e l o w . *1 f ( t ) F i g . 3 V a r i a t i o n o f f ( t ) 91 U s i n g e q u a t i o n ( B - 7 ) , t h e v a r i a t i o n o f t h e i n t e g r a n d F^ o f t h e i n t e g r a l 1^, c a u s e d by t h e v a r i a t i o n o f q-^, can now be com- , , 71-73 p u t e d s 5F 1(q . 1 >4 1,t) = F1(q.1+oi441+e#,t) - Fi( (li » 4 1,t) = e ^ F l 1 b F « .. (B-8) when t h e h i g h e r o r d e r t e r m s o f t h e T a y l o r s e r i e s a r e n e g l e c t e d s i n c e e a p p r o a c h e s z e r o . S u b s t i t u t i n g e q u a t i o n (B-8) i n e q u a t i o n s (B-4) and (B-5) y i e l d s t '0 t 0 l < » i ' d t = 0, ( B - 9 ) w h i c h t h r o u g h i n t e g r a t i o n by p a r t s becomes 6 J F x d t = ej ^ d t + e '0 '0 d F t d t 0 '0 55 b q . I/dt = 0, ... (B-10) S i n c e ^ ( t ) v a n i s h e s a t t h e two e n d - p o i n t s due t o c o n d i t i o n ( B - 2 ) , t h e s e c o n d t e r m o f e q u a t i o n (B-10) d r o p s o u t . Thus, t h i s e q u a t i o n r e d u c e s t o 61 '0 '0 Mi d t b4].j V cit, ... ( B - l l ) w h i c h can be w r i t t e n as t . r e I E 1 ( t ) ^ ( t ) d t = 0, (B-12) J0 71 A c c o r d i n g to t h e F u n d a m e n t a l Lemma o f t h e C a l c u l u s o f V a r i a t i o n s , 9 2 i f E , ( t ) i s c o n t i n u o u s i n ( t Q , t @ ) , and i f ^ /(t) v a n i s h e s a t t ^ and t @ and a d m i t s a c o n t i n u o u s d e r i v a t i v e i n ( t Q , t g ) , t h e n E ^ t ) = 0 ... (B-13) i n ( t ~ , t ). T h i s l e a d s t o 0 e bP bP B i ( t ) i^5?7-°- ••• ( B ' 1 4 ) a d i f f e r e n t i a l e q u a t i o n d i s c o v e r e d by E u l e r (1744) and w i l l be r e f e r r e d t o as E u l e r " s ( d i f f e r e n t i a l ) e q u a t i o n . * The same r e s u l t may be o b t a i n e d f o r t h e g e n e r a l c a s e o f p ( p ^ l ) v a r i a b l e s by s e l e c t i n g one d e f i n i t e q u l e a v i n g t h e o t h e r v a r i a b l e s unchanged, and r e p e a t i n g t h e above p r o c e s s . I n t h i s c a s e , one o b t a i n s a s y s t e m o f s i m u l t a n e o u s d i f f e r e n t i a l equa t i o n s ™ / j . \ bP d b P _ .. . E u ( t ) = 5 q ^ _ d t & f " = °- u = l , . . . , p ... ( B - l 5 ) T h e s e e q u a t i o n s a r e t h e f i r s t n e c e s s a r y c o n d i t i o n f o r an ex tremum ( i . e . minimum o r maximum) f o r t h e u n c o n d i t i o n e d p r o b l e m . S i n c e t h e p r o b l e m o f t h e t h e s i s i s t h a t o f m i n i m i z a t i o n , o n l y t h e f i r s t n e c e s s a r y c o n d i t i o n f o r a minimum w i l l be c o n s i d e r e d i n t h e f o l l o w i n g s e c t i o n s . 2 . The C o n d i t i o n e d P r o b l e m and t h e F i r s t N e c e s s a r y C o n d i t i o n  f o r a Minimum V The p r o c e s s o f f i n d i n g , i n g e n e r a l , t h e extreme v a l u e , and i n t h i s t r e a t i s e , t h e minimum v a l u e o f t h e i n t e g r a l I g i v e n by e q u a t i o n ( B - l ) , w i l l now be combined w i t h two a u x i l i a r y * L a n c z o s c a l l s t h i s e q u a t i o n E u l e r - L a g r a n g e 5 s e q u a t i o n , K n e s e r and H i l b e r t c a l l i t L a g r a n g e ' s e q u a t i o n , b u t L a g r a n g e h i m s e l f a t t r i b u t e s i t t o E u l e r . 93 c o n d i t i o n s : ( i ) t h e c o n d i t i o n s o f t h e p r o b l e m o f L a g r a n g e and ( i i ) t h e i s o p e r i m e t r i c c o n d i t i o n s . The v a r i a t i o n a l p r o b l e m o f 71 t h i s t y p e i s c a l l e d t h e c o n d i t i o n e d p r o b l e m . 2.1 The P r o b l e m o f L a g r a n g e The p r o b l e m o f s e c t i o n 1 i s now m o d i f i e d s u c h t h a t t h e • * v a r i a b l e s , . .., and 4]_»»«»»9.p a r e n o l o n g e r i n d e p e n d e n t , b u t r e s t r i c t e d by t h e c o n d i t i o n s 0 F F ( , . . . , q. , 4 , » • • • »<lp 91) =0. a = l , . . . , a (a-=p) ... (B-16) I f t h e c u r v e s o f t h e f a m i l y q.u = q^ttje,, . . . ,e ) u = l , . . . , p ... (B-17) p a s s t h r o u g h t h e p o i n t s tQ and t , s a t i s f y e q u a t i o n ( B - 1 6 ) , and c o n t a i n , f o r t h e p a r a m e t e r v a l u e = 0, t h e m i n i m i z i n g c u r v e V <l u = < l u ( t , 0 , . o o , 0 ) , u = l , . . . , p ... (B-18) t h e n t h e f u n c t i o n t P [ t , q u ( e 1 , . . . , e ] p ) , 4 U ( e± , • • •, e p ) ] d t ... ( B - l 9) *0 74 must have a minimum f o r a l l e = 0. The a r c C~ must be an u O 72 e x t r e m a l , and a c c o r d i n g t o t h e E u l e r - L a g r a n g e M u l t i p l i e r R u l e , t h e r e e x i s t s a s e t o f f u n c t i o n A. ( t ) where a = l , ... , a(a-=p), s u c h t h a t i f a K = P + Y s • • • ( B - 2 ° ) o = l a a t h e n t h e d e r i v a t i v e I'(0) i s e x p r e s s i b l e i n t h e f o r m ^ t r x> 94 where t h e f u n c t i o n s T u ( t ) a r e t h e v a r i a t i o n s o f t h e f a m i l y ( B - 1 7 ) , d e f i n e d by t h e r e l a t i o n s b q . u ( t , e 1 , . . . , e_) "~e u = T u ( t ) . ... (B-22) e u ~ ° I f t h e a r c CQ m i n i m i z e t h e i n t e g r a l I , i t i s n e c e s s a r y t h a t t h e f i r s t v a r i a t i o n I ' ( 0 ) g i v e n by e q u a t i o n (B-21) v a n i s h e s , f r o m 72 74 75 w h i c h t h e E u l e r - L a g r a n g e R u l e i s o b t a i n e d : ' ' I f CQ i s a m i n i m i z i n g a r c t h e r e e x i s t s a s e t o f m u l t i p l i e r s \ a ( t ) s u c h t h a t a t e v e r y p o i n t o f CQ, t h e e q u a t i o n s d |K_ = Q u = l , . . . , p . . . (B-23) G 4 U a t 5q u a r e s a t i s f i e d , w i t h K g i v e n by e q u a t i o n (B-20). E q u a t i o n s (B-23) a r e t h e f i r s t n e c e s s a r y c o n d i t i o n s f o r a minimum o f t h e L a g r a n g e p r o b l e m , and a r e a n a l o g o u s t o e q u a t i o n s (B-15) f o r t h e u n c o n d i t i o n e d p r o b l e m o f s e c t i o n 1. 2.2 The I s o p e r i m e t r i c P r o b l e m A u x i l i a r y c o n d i t i o n s a p p e a r i n g i n t h e f o r m o f d e f i n i t e i n t e g r a l s J . w h i c h must have p r e s c r i b e d c o n s t a n t v a l u e s B. t J l ~ J - ° ° '^p'^a* °'' » 4 p>t) d t = B X , ' . . . (B-24) t r e J n = J G n ( i 1 » • • • j q - p * ^ * • • • »q. p»t) d t = BN, * 0 a r e c a l l e d " i s o p e r i m e t r i c " c o n d i t i o n s . * * T h i s t e r m i s d e r i v e d f r o m t h e f i r s t h i s t o r i c a l l y r e c o r d e d ex- tremum p r o b l e m o f f i n d i n g t h e maximum a r e a bounded by a p e r i  m e t e r o f a g i v e n l e n g t h . 7 3 95 R e p e a t i n g t h e p r o c e s s o f v a r y i n g t h e i n t e g r a l as i n t h e un c o n d i t i o n e d p r o b l e m , one o b t a i n s bG, d b G 1 J \ 1 6 J i -J b ^ " dt 0 4 I rtel'dGn - O G J « - - J k - - - : i X 5q^dt+...+J '0 :t. OGj^ d 6q d t = 0, ... (B-25) e / 0 G n d d & n A g a i n , m u l t i p l y i n g e q u a t i o n s (B-25) by some u n d e t e r m i n e d c o n - .s OG. s t a n t X^ and a d d i n g t h e r e s u l t t o 61, one o b t a i n 61 '0 ± _ bF_ / b G l d_ ^ ^ ~ dt bqJ+Al ~~{ ' dt 6 4 I ; n + . ».+X I ~ ~ — + n oq > u u n 6q^dt+... + ... f 6 OF d_ OF + J - q ~ ~ d t ~q" d n p/ + X ' ^ 1 d _ ^ l 11 b * p " d t ^ qp/ fbG. +...+X n bq.T + d t D4 6q d t = 0, u p (B-26) *p/J w h i c h can be w r i t t e n as t 6I' -tflblr (*+hGl+°°-+XnGJ~ ft Sfc- ( F + X 1 G 1 + . . . + X n G n ) 6 q 1 d t + '0 ~f" • • • t -/ e[^r(F+W---+W- ft - | - ( F + w - - + x n G n ) tg TI 6q dt=0. P ... (B-27) A c c o r d i n g t o t h e Fundamental Lemma o f s e c t i o n 1, t h e c o e f f i - c i e n t s o f 6 q u must v a n i s h , t h u s 3 ^ ( F + X 1 G 1 + . . . + X n G n ) - fa A . ( p + X L ( J 1 + „ . + V n ) r f , u = l , . . . , p ... (B-28) 96 The l a s t two e q u a t i o n s show t h a t t h e i s o p e r i m e t r i c p r o b l e m can be t r a n s f o r m e d i n t o a f r e e v a r i a t i o n a l p r o b l e m w i t h no a u x i l i a r y c o n d i t i o n s , by c h a n g i n g t h e o r i g i n a l F i n t o a new f u n c t i o n 7 2 ' 7 3 n L = F + V X. G., .. . (B-29) where X^ i s an u n d e t e r m i n e d c o n s t a n t . 2.3 A C o m b i n a t i o n o f t h e P r o b l e m o f L a g r a n g e and t h e I s o   p e r i m e t r i c P r o b l e m A l l , e x c e p t t h e f i r s t , n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e L a g r a n g e p r o b l e m a r e q u i t e d i f f e r e n t i n n a t u r e f r o m t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e i s o p e r i m e t r i c p r o b  lem. F o r t h i s r e a s o n , t h e combined p r o b l e m c a n o n l y be s o l v e d i f , e i t h e r t h e i s o p e r i m e t r i c p r o b l e m i s t r a n s f o r m e d i n t o a L a g r a n g e p r o b l e m , o r v i c e v e r s a . 75 The f i r s t t y p e o f t r a n s f o r m a t i o n i s p e r f o r m e d as f o l l o w s ? R e w r i t i n g , f o r c o n v e n i e n c e , t h e i s o p e r i m e t r i c c o n d i t i o n s (B-25) J i = / G . ( t , q , q ) d t = B., ... (B-30) t Q where t h e s e t ( t , q , , ... , q p , 4 ^ » • • • * 4 p ) I s n o w r e p r e s e n t e d by ( t , q . , 4 ) » o n e i n t r o d u c e s new v a r i a b l e s t f e . Z ± ( t ) = J G i ( t , q , 4 ) d t . ... (B-31) The p r o b l e m i s now t o f i n d two s e t s o f e x t r e m a l s q = q ( t ) and ... (B-32) Zj,= Z i ( t ) i = 1, ... ,n ... (B-33) 97 s a t i s f y i n g t h e c o n d i t i o n s dZ j [ G r i ( t J q , s q ) - •g-r- = 0, <l(-t 0) = q . ( t e ) = q.e, ... (B-34) Z . ( t 0 ) = 0, Z . ( t e ) = B., one w h i c h m i n i m i z e s I . The f u n c t i o n L° a n a l o g o u s t o t h a t g i v e n by e q u a t i o n (B-29) now has t h e f o r m L« = P + i = l dZ. r 1  G i " d t ~ ... (B-35) and t h e d i f f e r e n t i a l e q u a t i o n s d e t e r m i n i n g t h e e x t r e m a l s a r e bL• d b L 1 £>q ~. d t Sq The n e q u a t i o n s bL• d bL 0. ... (B-36) d X i b Z ~ " d t ^ ( d Z i / d t ) " d t ~ 0 • • • » n .oo (B-37) show t h a t i n t h i s c a s e t h e m u l t i p l i e r s ?u a r e a l l c o n s t a n t s . A s e c o n d t y p e o f t r a n s f o r m a t i o n i s a l s o p o s s i b l e p r o v i d e d t h a t each o f t h e f u n c t i o n s 0 f f i g i v e n by e q u a t i o n ( B - l 6 ) r e w r i t t e n as 0 a = 0 ( t , q , q ) = 0 a=l,,..,a(a<p) ... (B-38) c a n be i n t e g r a t e d . I n t e g r a t i n g e q u a t i o n (B-38) y i e l d s f j 0 a ( t ? q , q ) d t = 0, a = l , . . ., a(a«=p) ... (B-39) t 0 w h i c h i s a s i m p l i f i e d f o r m o f t h e i s o p e r i m e t r i c c o n d i t i o n . S e v e r a l c r i t e r i a must be o b s e r v e d i n making t h e c h o i c e o f 98 t h e f i r s t o r t h e s e c o n d t y p e o f t r a n s f o r m a t i o n m e n t i o n e d p r e  v i o u s l y ? ( i ) i n t e g r a b i l i t y o f t h e f u n c t i o n s 0 , ( i i ) t h e number o f i s o p e r i m e t r i c c o n d i t i o n s n compared t o t h e number o f L a g r a n g e c o n d i t i o n s a, and ( i i i ) ease o f a p p l i c a t i o n w i t h r e g a r d t o t h e v a r i o u s n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s . S i n c e i n t h e t h e s i s p r o b l e m n i s g e n e r a l l y l a r g e r t h a n a, where a = 1, t h e s e c o n d t r a n s f o r m a t i o n f r o m t h e L a g r a n g e p r o b l e m t o t h e i s o  p e r i m e t r i c p r o b l e m w i l l be u s e d . F u r t h e r m o r e , t h e f u n c t i o n s f o r t h e s p e c i a l p r o b l e m d i s c u s s e d h e r e a r e d e f i n i t e l y i n t e g r a b l e , as t h e y c o n t a i n s i m p l e summation terms and f u n c t i o n s w h i c h a r e i d e n t i c a l t o t h e i s o p e r i m e t r i c c o n d i t i o n s g i v e n by e q u a t i o n s (B-24) or ( B - 3 0 ) . A n o t h e r a d v a n t a g e of t h e i s o p e r i m e t r i c p r o b  l e m i s t h e f a c t t h a t i t c o n t a i n s c o n s t a n t L a g r a n g i a n m u l t i p l i e r s i n s t e a d o f t h e t i m e - v a r i a b l e m u l t i p l i e r s as i n t h e c a s e o f t h e p r o b l e m o f L a g r a n g e . 3. O t h e r N e c e s s a r y and S u f f i c i e n t C o n d i t i o n s f o r a Minimum o f  t h e I s o p e r i m e t r i c P r o b l e m 3.1 I n t r o d u c t i o n I n l i n e w i t h t h e t h e s i s p r o b l e m c o n d i t i o n s f o r m i n i m i z a  t i o n o f i n t e g r a l I w i t h i s o p e r i m e t r i c c o n d i t i o n s w i l l be c o n  s i d e r e d . F o r t h e sake o f s i m p l i c i t y , t h e d i s c u s s i o n w i l l be l i m i t e d t o t h e t w o - v a r i a b l e c a s e (p=2). The p r o b l e m can now be r e f o r m u l a t e d as f o l l o w s : M i n i m i z e t h e i n t e g r a l * ^ 0 d t *The p r i m e s i m m e d i a t e l y f o l l o w i n g t h e v a r i a b l e s x and y a r e d e r i v a t i v e s w i t h r e s p e c t t o t o f t h o s e v a r i a b l e s . 99 w i t h i s o p e r i m e t r i c c o n d i t i o n s t e J = | G ( x , y , x < , y ' , t ) d t = B ... (B-41) •I '0 and t J 6 0 ( x , y , x ' , y « , t ) d t = 0, ... (B-42) r e p l a c i n g q,, q 2 , q , a n < 1 4 2 ^ y x, y , x* and y ' r e s p e c t i v e l y < , The f i r s t n e c e s s a r y c o n d i t i o n s f o r a minimum o f I d e f i n e d b y e q u a t i o n (B-40) w i t h i s o p e r i m e t r i c c o n d i t i o n s (B-41) and (B-42) a r e g i v e n by e q u a t i o n s bH d_ bH n &x. ~ d t fcx' - u ' OH d_ bH_ _ n Sy " d t by' ~ ' (B-43) w h i c h a r e e q u i v a l e n t t o t h e one d i f f e r e n t i a l e q u a t i o n * SP Sx " SP" Sy + H i ( x ' y " ~ x " y , ) = °> ••• (B"44) where H = P + XjG + X20, ... (B-45) and TT _ 1 b (iH " l - (y« )2 Sx1* Sx7" = i V " 3 ^ SP* ( B ~ 4 6 ) _ 1 b bH (x' )2 bF7" SP" ' The g e n e r a l s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n (B-44) i s * A n a l o g o u s t o B o l z a ' s ' " p u r e " i s o p e r i m e t r i c p r o b l e m . 100 g i v e n by* x = x ( t , a , S } X 1 , X 2 ) , Cos y = y ( t , a , p , X 1 , X 2 ) , ... (B-47) where a and 8 a r e t h e two c o n s t a n t s o f i n t e g r a t i o n . * * Equa t i o n s (B-47) a r e t h e s e t o f e x t r e m a l s f o r t h e t h e s i s p r o b l e m . 3.2 N e c e s s a r y C o n d i t i o n s ( l ) The s e c o n d n e c e s s a r y c o n d i t i o n f o r a minimum i s t h a t * H l = o ... (B-48) a l o n g t h e e x t r e m a l CQ, d e f i n e d by e q u a t i o n (B-47), w h i c h s a t i s  f i e s t h e b o u n d a r y and a u x i l i a r y c o n d i t i o n s , and where i s g i v e n by e q u a t i o n (B-46). T h i s i s t h e a n a l o g u e o f L e g e n d r e ' s c o n d i t i o n f o r t h e u n c o n d i t i o n e d p r o b l e m . (2) The t h i r d n e c e s s a r y c o n d i t i o n f o r a minimum i s g i v e n by W e i e r s t r a s s 1 s a n a l o g u e o f J a c o b i ' s c o n d i t i o n * D ( t , t Q ) = w 1 ( t Q ) CO x ( t ) JuUldt J Jo t J Vttfjdt J '0 w 2(t 0) w 3(t 0) °4<V u>2(t) U)g(t) c o 4 ( t ) U o ^dt t J Uwgdt J Uto 4dt 0 + + + V w 2 d t ( V w 3 d t f Tu> 4dt 0 ^ 0 *\ 0 (B-49) f o r t Q «= t - t 71 72 * A n a l o g o u s t o B o l z a ' s ' " p u r e " i s o p e r i m e t r i c p r o b l e m . **The g e n e r a l s o l u t i o n s o f E u l e r e q u a t i o n s w i t h p v a r i a b l e s a r e de p e n d e n t on 2p-2 i n t e g r a t i o n c o n s t a n t s . 101 or t * 5 t ... (B-50) 0 e where t Q ' , c a l l e d t h e c o n j u g a t e p o i n t o f t Q , i s t h e r o o t n e x t g r e a t e r t h a n t g o f t h e e q u a t i o n D ( t , t Q ) = 0. ... (B-51) The t - f u n c t i o n s a r e g i v e n by t h e f o l l o w i n g r e l a t i o n s : , v / + \ _ by, ^5 . & & w l ( t ) = St 5o bt ba ' W 2 l t J " St bP bt bP w 3 ( t ) = st sx~ " s*- si^' (+ \ by. bx bx b y w 4 ( t ) = st SXT: - st sir:' u ( t ) i - ^ sf + G i ( x , y " - * " y , ) • - ( B ™ 5 3 ) where r 1 b bG r l = jp~)Z b x ' bx' _ ! b_ bG x'y' b x ' b y ' 1 b bG and where 0 _ 1 _b_ b 0 _ ^1 - ( y . ) 2 6 x ' bx' b bg. x » y ! bx' by" 1 b b 0 ( x ' ) 2 by' S F • (B-52) (B-54) " (x«)2 b y ' by' ' = S F s i - s h - s f + 0 i ( x ' y " - x " y , ) • - ( B ~ 5 5 ) (B-56) 102 (3) The f o u r t h n e c e s s a r y c o n d i t i o n s f o r a minimum i s g i v e n by W e i e r s t r a s s 1 i n e q u a l i t y * E (x, y , x', y \ x \ y 1 ; X±, \2) - 0 , ... (B-57) w h i c h must be f u l f i l l e d a l o n g t h e e x t r e m a l CQ, d e f i n e d by equa t i o n ( B - 4 7 ) , f o r e v e r y d i r e c t i o n x'5 and y 1 , where* E (x, y , x \ y ' , x» , y« ; X ^ \ g ) = H (x, y , x 5 , y 1 ; X-p X G ) x' £T H(x, y, x», y ' j \ ± , \ g ) + ... (B-58) + y 8 ^ y r H (x, y , x', y ' ; X±, Xg) 3.3 S u f f i c i e n t C o n d i t i o n s The e x t r e m a l C Q , d e f i n e d by e q u a t i o n ( B - 4 7 ) , f u r n i s h e s a " s e m i - s t r o n g " minimum f o r t h e i n t e g r a l I g i v e n by e q u a t i o n (B-39) w i t h a u x i l i a r y c o n d i t i o n s (B-40) and ( B - 4 1 ) , i f t h e c o n  d i t i o n s H x > 0 , ... (B-59) t < t „ ! , ... (B-60) e u ' E =• 0 ... (B-61) 71 72 a r e f u l f i l l e d . ' The above s u f f i c i e n t c o n d i t i o n s a r e , t h e r e  f o r e , e x a c t l y e q u i v a l e n t t o t h e l a s t t h r e e n e c e s s a r y c o n d i t i o n s e x c e p t f o r t h e e q u a l i t y s i g n . * A n a l o g o u s t o B o l z a ' s ' " p u r e " i s o p e r i m e t r i c p r o b l e m . 

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Japan 17 0
China 16 11
United States 3 0
France 1 0
Sweden 1 0
City Views Downloads
Tokyo 17 0
Beijing 16 0
Ashburn 2 0
Sunnyvale 1 0
Unknown 1 0
Stockholm 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

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.831.1-0105072/manifest

Comment

Related Items