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Determination of reservoir daily operation policies by stochastic dynamic programming Tsou, C. Anthony 1970

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DETERMINATION OF RESERVOIR DAILY OPERATION POLICIES BY STOCHASTIC DYNAMIC PROGRAMMING  by C. ANTHONY TSOU B.A. Hon., O x f o r d U n i v e r s i t y , 1967.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n t h e Department of C i v i l Engineering  We a c c e p t t h i s t h e s i s as conforming t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1970  In p r e s e n t i n g  this  thesis in p a r t i a l  f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree 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  I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e for  s c h o l a r l y purposes may  by h i s r e p r e s e n t a t i v e s .  thesis for f i n a n c i a l  written  permission.  Department of  Civil  gain s h a l l  Engineering  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  Columbia  thesis  Department or  It i s understood t h a t c o p y i n g or  of t h i s  Study.  c o p y i n g of t h i s  be granted by the Head of my  that  publication  not be a l l o w e d w i t h o u t  my  - ii  -  ABSTRACT  Reservoir operation p o l i c i e s are o f t e n formulated on the b a s i s o f c r i t i c a l f l o w h y d r o l o g y .  deterministically  However, i f a dynamic r i v e r d a i l y  flow  f o r e c a s t system i s a v a i l a b l e f o r t h e whole season, t h e f o r e c a s t i n f o r m a t i o n s h o u l d be f u l l y u t i l i z e d i n r e s e r v o i r r e g u l a t i o n . two approaches t o d e t e r m i n i n g  G i v e n such a f o r e c a s t system,  optimal d a i l y operation p o l i c i e s f o r a s i n g l e  purpose f l o o d c o n t r o l r e s e r v o i r a r e suggested.  Both approaches use s t o c h a s t i c  dynamic programming: one i n v o l v e s t h e m i n i m i z a t i o n o f t h e e x p e c t e d v a l u e o f f l o o d damages, and t h e o t h e r i n v o l v e s m i n i m i z i n g t h e p r o b a b i l i t y o f o c c u r r e n c e of an u n d e s i r a b l e e v e n t , w h i c h i s a f l o o d damage e x c e e d i n g  a c e r t a i n amount.  The p r o b a b i l i s t i c approach n o t o n l y o f f e r s a s e t o f a l t e r n a t i v e o p t i m a l  daily  operation p o l i c i e s , but a l s o i n d i c a t e s the p r o b a b i l i t i e s of being able to a c h i e v e t h e o b j e c t i v e s , and thus i t forms a b a s i s f o r comparing and e v a l u a t i n g the a l t e r n a t i v e o b j e c t i v e s .  - iii -  TABLE OF CONTENTS Page No.  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  BASIC PROBABILITY AND STATISTICS  5  CHAPTER 3  FORECAST SYSTEM  11  CHAPTER 4  DYNAMIC PROGRAMMING  17  CHAPTER 5  APPLICATION OF DYNAMIC PROGRAMMING TO A SINGLE PURPOSE FLOOD CONTROL RESERVOIR  31  REFERENCES  52  APPENDICES  54  - iv-  LIST OF FIGURES FIGURE  TITLE  PAGE NO.  2.1  PROBABILITY DENSITY FUNCTION  7  8  3.1  HYPOTHETICAL HYDROGRAPH  11  3.2  MARGINAL PROBABILITY DENSITY FUNCTION  13  3.3  JOINT BIVARIATE NORMAL DISTRIBUTION DENSITY FUNCTION  14  3.4  CONDITIONAL PROBABILITY DENSITY FUNCTION  16  4.1  ONE-STAGE DECISION MODEL  17  4.2  MULTI-STAGE DETERMINISTIC DECISION MODEL  20  4.3  MULTI-STAGE STOCHASTIC DECISION MODEL  24  5.1  PROBABILITY FUNCTION  32  5.2  FLOOD DAMAGE FUNCTION  34  5.3  HYPOTHETICAL RESERVOIR RULE CURVE  36  5.4  MULTI-STAGE STOCHASTIC RESERVOIR OPERATION MODEL ...  39  5.5  POSSIBLE RESULTS FROM A PARTICULAR DECISION  42  5.6  OUTPUT OF THE PROBABILISTIC APPROACH TO DETERMINING OPTIMAL RESERVOIR DAILY OPERATION POLICIES  44  - v -  LIST OF TABLES TABLE  5.1  TITLE  PAGE NO.  OUTPUT OF THE PROBABILISTIC APPROACH TO DETERMINING OPTIMAL RESERVOIR DAILY OPERATION POLICIES  5.2  43  OUTPUT OF THE EXPECTED VALUE APPROACH TO DETERMINING OPTIMAL RESERVOIR DAILY OPERATION POLICIES  46  - vi -  ACKNOWLEDGEMENT  The a u t h o r wishes t o thank P r o g e s s o r L.G. M i t t e n , Department of Commerce and B u s i n e s s A d m i n i s t r a t i o n , U.B.C, f o r h i s h e l p and guidance i n dynamic programming, and P r o f e s s o r  S.W. Nash, Department o f M a t h e m a t i c s ,  U.B.C, f o r h i s k i n d a s s i s t a n c e i n t h e f i e l d o f p r o b a b i l i t y t h e o r i e s .  The a u t h o r a l s o w i s h e s t o thank Mr. S.O. R u s s e l l , Department of C i v i l E n g i n e e r i n g , this thesis.  U.B.C, f o r h i s a s s i s t a n c e i n t h e p r e p a r a t i o n o f  -1-  C H A P T E R  1  INTRODUCTION  Dynamic programming was developed by R. B e l l m a n i n the 1950's ( r e f e r e n c e s 1, 2) b u t t h e a p p l i c a t i o n o f dynamic programming to w a t e r r e s o u r c e s systems was f i r s t i n t r o d u c e d by W.A. H a l l and N. Buras i n 1961 ( r e f e r e n c e 3 ) .  S i n c e t h e n dynamic programming has  proved t o be a most p o w e r f u l and v e r s a t i l e o p t i m i z a t i o n the f i e l d  technique i n  o f w a t e r r e s o u r c e s b o t h f o r d e s i g n and development o f  o p e r a t i n g p o l i c i e s f o r a s i n g l e purpose r e s e r v o i r , a m u l t i p l e r e s e r v o i r , o r a system o f such r e s e r v o i r s  ( r e f e r e n c e s 4, 5, 6, 7, 8 ) .  However, t h e o p e r a t i n g p o l i c y o r r u l e c u r v e o f a r e s e r v o i r developed on t h e b a s i s ' c r i t i c a l period'  of a deterministic  purpose  input, usually  i s usually  that of the  w h i c h can be e i t h e r o b t a i n e d from t h e h i s t o r i c a l d a t a  or from g e n e r a t e d s y t h e t i c f l o w s .  While t h i s i s usually  satisfactory  f o r d e s i g n p u r p o s e s , i t i s l e s s s a t i s f a c t o r y f o r o p e r a t i n g purposes where f l o w s a r e n o t known i n advance.  I n t h e case o f a r e s e r v o i r  used  f o r f l o o d c o n t r o l an o p e r a t i o n p o l i c y based on a sound r u l e c u r v e can m i n i m i z e f l o o d damage f o r c e r t a i n r i v e r f l o w s , b u t i t i s n o t l i k e l y t o m i n i m i z e f l o o d damage f o r u n u s u a l p a t t e r n s o f f l o o d h y d r o g r a p h s .  In  c e r t a i n c a s e s , p a r t i c u l a r l y i n an a r e a such as B r i t i s h Columbia where a l a r g e p r o p o r t i o n o f the r u n o f f comes from snowmelt, i t i s p o s s i b l e t o forecast  f l o w s w i t h some degree o f a c c u r a c y .  possible  t o use f l o w f o r e c a s t s  H o p e f u l l y , i t s h o u l d be  t o improve o p e r a t i n g p o l i c i e s .  -2-  The  purpose o f t h i s s t u d y i s t o demonstrate t h e f e a s i b i l i t y  of u s i n g dynamic programming t o determine t h e o p t i m a l d a i l y strategy The  operating  f o r a r e s e r v o i r , given a long-range r i v e r flow f o r e c a s t  system.  example used i n t h e s t u d y i s a s i n g l e purpose f l o o d c o n t r o l r e s e r v o i r  with l i m i t e d storage, but unlimited  discharge c a p a c i t y , which i s operated  to m i n i m i z e t h e damage downstream o f t h e r e s e r v o i r f o r a whole f l o o d season.  Assuming t h a t when t h e r i v e r f l o w downstream o f t h e r e s e r v o i r exceeds a c e r t a i n v a l u e to the r e s e r v o i r  Q  QL,  f l o o d damage w i l l o c c u r , then once the i n f l o w  b e g i n s t o exceed  QL,  d e c i d e whether t o s t o r e the excess f l o w  t h e r e s e r v o i r o p e r a t o r must  (Q - QL)  t o p r e v e n t any immediate  damage, t o s t o r e p a r t o f t h e excess f l o w , o r even n o t t o s t o r e any water now i n o r d e r t o r e s e r v e s t o r a g e t o be a b l e t o p r e v e n t g r e a t e r  damage i n the  f u t u r e , when t h e n a t u r a l r i v e r f l o w reaches i t s maximum v a l u e . make an a p p r o p r i a t e  I f he i s t o  d e c i s i o n , and n o t o p e r a t e s i m p l y on a r u l e c u r v e b a s i s ,  the r e s e r v o i r o p e r a t o r must have some knowledge o f t h e p o s s i b l e f u t u r e i n f l o w s f o r t h e whole f l o o d p e r i o d .  P r o v i d e d w i t h such a l o n g - r a n g e  f o r e c a s t , and r e a l i z i n g t h e u n c e r t a i n t i e s i n the f o r e c a s t v a l u e , programming c a n a s s i s t t h e r e s e r v o i r o p e r a t o r t o seek a " b e s t " operating  daily  flow  dynamic  reservoir  p o l i c y such t h a t t h e p o s s i b i l i t y and amount o f f l o o d damage  during  the whole season i s m i n i m i z e d .  As l o n g as 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 n o t g r e a t e r  than  QL,  t h e r e i s no a c t u a l d e c i s i o n t o make b u t t o l e t the water pass t h e r e s e r v o i r . However when t h e i n f l o w t o the r e s e r v o i r reaches  QL  the f l o o d season, a d e c i s i o n on r e s e r v o i r o p e r a t i n g  a t the beginning of p o l i c y has t o be made on  the b a s i s o f e s t i m a t e s o f how much and f o r how l o n g t h e f u t u r e r i v e r  flows  -3-  may  exceed QL.  I t i s assumed t h a t the f o r e c a s t system g i v e s a s e t of  i n f l o w s from then on to the end of the p e r i o d . p e r f e c t , i . e . i f the v a l u e s  daily  I f the f o r e c a s t system i s  of d a i l y f l o w i t p r e d i c t s w i l l a c t u a l l y o c c u r ,  then i t i s a s i m p l e m a t t e r to d e c i d e the r e s e r v o i r o p e r a t i n g p o l i c y t h a t makes the b e s t use of the l i m i t e d r e s e r v o i r s t o r a g e the f l o o d damage caused d u r i n g the whole season.  c a p a c i t y to m i n i m i z e  Such a d e c i s i o n p r o c e s s  is deterministic.  However, i n r e a l i t y , no f o r e c a s t system can ever a c h i e v e  p e r f e c t accuracy.  Hence, u n c e r t a i n t i e s i n the p r e d i c t e d f u t u r e f l o w s must  be c o n s i d e r e d considered  i n developing  a reservoir operating p o l i c y .  Criteria  i n t h i s study were t h a t e i t h e r the e x p e c t e d t o t a l damage or  the  p r o b a b i l i t y of o c c u r r e n c e of a f l o o d g r e a t e r than a c e r t a i n magnitude be m i n i m i z e d f o r the c o n s i d e r e d  season.  f o r e c a s t f u t u r e d a i l y i n f l o w s and of those p r e d i c t e d f l o w s .  P o l i c i e s are d e r i v e d on the b a s i s  the c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s  Such a d e c i s i o n p r o c e s s i s s t o c h a s t i c .  s t u d y i s based on d a i l y d i s c h a r g e s  This  of the F r a s e r R i v e r a t Hope, w h i c h  b e l i e v e d t o be r e p r e s e n t a t i v e of snowmelt f l o o d d i s c h a r g e B r i t i s h Columbia, and  of  f o r which flow records  patterns  are  in  are r e a d i l y a v a i l a b l e .  The  r e s e r v o i r i s assumed to be j u s t upstream of Hope, an assumption w h i c h s i m p l i f i e d the c o m p u t a t i o n but does not i n v a l i d a t e the approach. no l o n g range d a i l y f l o w f o r e c a s t system was system based on l i n e a r r e g r e s s i o n was optimization analysis.  Two  a v a i l a b l e , a simple  improvised  Since forecast  to p r o v i d e d a t a f o r the  types of f l o o d damages are c o n s i d e r e d .  One  i s a r e c u r s i v e type of damage, i . e . the amount of damage done i n the p a s t may  be r e p e a t e d i n the f u t u r e i f the f l o o d reaches the same l e v e l .  other i s a non-recursive  type of damage, i . e . u n l e s s  The  the f u t u r e f l o o d  l e v e l exceeds t h a t of the p a s t , no e x t r a damage w i l l be caused i n t h a t p a r t i c u l a r f l o o d season.  For the case of r e c u r s i v e type of damage, an  -4-  o p t i m a l s t r a t e g y i s determined by dynamic programming w h i c h m i n i m i z e s the e x p e c t e d t o t a l damage f o r t h e whole season.  F o r t h e case o f n o n - r e c u r s i v e  type o f damage, a g a i n by dynamic programming, an o p t i m a l s e t o f d e c i s i o n s i s formed such t h a t t h e p r o b a b i l i t y o f t h e t o t a l damage f o r t h e e n t i r e season e x c e e d i n g a c e r t a i n amount i s m i n i m i z e d .  To s t a t e t h i s  criterion  i n r e v e r s e , t h e p r o b a b i l i t y t h a t t h e t o t a l s e a s o n a l damage w i l l n o t exceed a c e r t a i n amount i s maximized.  In are  t h e f o l l o w i n g c h a p t e r s , some a p p l i c a b l e p r o b a b i l i t y  theories  f i r s t o u t l i n e d , f o l l o w e d by a d e s c r i p t i o n o f t h e s i m p l i f i e d f l o w f o r e -  c a s t system w h i c h was developed f o r purposes o f t h e s t u d y .  Next, a b r i e f  summary o f dynamic programming and t h e b a s i c r e c u r s i v e f o r m u l a e a r e g i v e n , s i n c e t h e s t u d y r e l i e s so h e a v i l y on t h i s r e l a t i v e l y new o p t i m i z a t i o n technique.  F i n a l l y , the two above mentioned approaches a r e developed and  p r e s e n t e d t o g e t h e r w i t h r e s u l t s and comments.  The computer programmes  d e v e l o p e d i n t h e c o u r s e o f t h e s t u d y a r e g i v e n i n Appendices 1, 2, and 3.  -5-  C H A P T E R  2  BASIC PROBABILITY AND STATISTICS  Some o f t h e main concepts o f p r o b a b i l i t y and s t a t i s t i c a l used i n t h i s s t u d y a r e o u t l i n e d below.  analysis  D e t a i l s can be found i n r e f e r e n c e s 9,  10, and 11. 2.1 are  I n t h e u s u a l language o f p r o b a b i l i t y t h e o r i e s ,  the f o l l o w i n g  symbols  defined: Af\B : t h e i n t e r s e c t i o n o f A and B PCAftB) : t h e p r o b a b i l i t y o f event A and event B o c c u r r i n g PCA) : t h e p r o b a b i l i t y o f event A  occurring  PCB) :. t h e p r o b a b i l i t y o f event B  occurring  P(A/B) : t h e c o n d i t i o n a l p r o b a b i l i t y of event A o c c u r r i n g  together  given  t h a t event B has a l r e a d y o c c u r r e d PCB/A) : t h e c o n d i t i o n a l p r o b a b i l i t y of event B o c c u r r i n g t h a t event A has a l r e a d y o c c u r r e d then, PCAftB) = P(A) * PCB/A) = PCB) * P(A/B) I f A and B a r e independent e v e n t s , then PCB/A) = P(.B), PCA/B) =. PCA) therefore, P(Af\B) = PCA) * PCB)  given  -6-  2.2  P(A/)Bfl  fl MflNn  ...flYAZ)  = PCARBfl ... HM) * P(NH ...ftYilZ/AflB  OM)  i . e . t h e p r o b a b i l i t y o f events A,B,...,M....Y,Z o c c u r r i n g t o g e t h e r t o t h e p r o b a b i l i t y o f events A,B,...M o c c u r r i n g t o g e t h e r c o n d i t i o n a l p r o b a b i l i t y o f events N, e v e n t s A,B,  m u l t i p l i e d by the  >Y,Z o c c u r r i n g t o g e t h e r  ,M have a l r e a d y o c c u r r e d .  i s equal  given  that  S i m i l a r l y i f events A,B,...,M,  N,....,Y,Z a r e i n d e p e n d e n t , then P(.Al\B(N  (\MftNfl .... r\Y<\Z)  = PCA) * PCB) *  2.3  *P(M) *' PCN)*  * PCY) * B(Z)  I f X i s a d i s c r e t e random v a r i a b l e w i t h p r o b a b i l i t y f u n c t i o n P ( X ) ,  the e x p e c t e d v a l u e o f H ( X ) , w r i t t e n as E [ H C X ) J , i s d e f i n e d a s : E[H(X)] = E X  2.4  H(X) * PCX)  I f A,B,....,Y,Z a r e independent d i s c r e t e random v a r i a b l e s w i t h  p r o b a b i l i t y f u n c t i o n s P ( A ) , PCB),...., P O O , P ( Z ) , t h e e x p e c t e d v a l u e o f H(A,B  ,Y,Z) i s d e f i n e d a s :  E[H(A,B,  ,Y,Z)] = E E  EE  A B  Y Z  [HCA,B, . . . ,Y ,Z) * P(A)  * P ( B ) * . . .*P(Y) * PCZ  N.B. i f t h e random v a r i a b l e s a r e c o n t i n u o u s r a t h e r than d i s c r e t e , then t h e summation s i g n s a r e r e p l a c e d by i n t e g r a l s i g n s . In engineering  s t u d i e s i t i s o f t e n found more c o n v e n i e n t t o c o n v e r t  random v a r i a b l e t o a d i s c r e t e random v a r i a b l e .  a continuous  I t w i l l be found t h a t throughout  t h i s s t u d y , t h e summation s i g n s r a t h e r than t h e i n t e g r a l s i g n s a r e used, and t h a t a c o n t i n u o u s f u n c t i o n Cor v a r i a b l e ) i s t r e a t e d as a d i s c r e t e f u n c t i o n Cor variable).  -7-  The expected v a l u e o f a random v a r i a b l e X i s c a l l e d the mean v a l u e  2.5 of X,  i s written y  2.6  y , and i s d e f i n e d a s : x  = E(X)  X  o f a random v a r i a b l e X , w r i t t e n  The v a r i a n c e  a  2  = E[(X  X  - y J ] = E(X ) 2  2  -y  X  2  as a , i s d e f i n e d a s : x  2  X  I t s p o s i t i v e square r o o t  i s denoted by a , and i s c a l l e d the s t a n d a r d  deviation  X  of  X. I f t h e r e e x i s t two random v a r i a b l e s X and Y then the c o v a r i a n c e o f  2.7  X and Y i s d e f i n e d a s :  a  xy  = E[X - u ) * (Y - u ')] = E(XY) x y  E(X)  * E(Y)  a measure o f how X and Y tend t o v a r y  The c o v a r i a n c e g i v e s  = E(XY)  - y * y x y  together.  The c o r r e l a t i o n c o e f f i c i e n t of X and Y i s d e f i n e d a s :  2.8  . M  has and  -  xy  . a  xy_  =  a  * a  x  y  the p r o p e r t y t h a t  |P  I  <  1 f°  r  Y a r e independent random v a r i a b l e s , a  xy  = p  xy  A N  Y  random v a r i a b l e s X and Y.  then  = 0  I f a random v a r i a b l e X i s Normally D i s t r i b u t e d ,  2.9  function of X i s :  density  f (X)  =  1  C  e  /2Tra  where y i s the mean o f the random v a r i a b l e X a  2  If X  i s the v a r i a n c e  of random v a r i a b l e X.  the p r o b a b i l i t y  Fig.  2.1  The p r o b a b i l i t y o f h a v i n g a p a r t i c u l a r v a l u e o f X such t h a t X < X i s Q  X  P(X  X )  <  0  0  I  =  f ( x ) dx  J* 2.1.  i . e . t h e shaded a r e a shown i n F i g . 2.10  u , u and a , x y x  a  2  variables  2  y  respectively,  and t h e c o r r e l a t i o n c o e f f i c i e n t o f random  X and Y i s p , and i f X and Y a r e j o i n t l y n o r m a l l y d i s t r i b u t e d , t h e  j o i n t p r o b a b i l i t y density  function i s  i  f(X,Y) =  X and Y w i t h means and v a r i a n c e s  I f t h e r e e x i s t two random v a r i a b l e s  1  2^ a  x  a  x-y  i  -== 1i-p N  exp{ 2  y  - —±— 2(1-P  )  [ ( — ^ ) cr  x  x-y  2  Y-y  2  - 2P—^—Z+ a a  x  <U-*)]} a  y  y  Hence i f random v a r i a b l e X i s n o r m a l l y d i s t r i b u t e d and random v a r i a b l e s are j o i n t l y n o r m a l l y d i s t r i b u t e d , t h e n t h e c o n d i t i o n a l p r o b a b i l i t y function  of Y given X = X  f (Y|X = ' X ) = p -  0  42TT a KJI^P  2  y  density  is: exp {  0  ~=-T: 2a  2  y  (l-p ) 2  [-Y-  y v  ^  - ^  (Xo - y ) ] x  a  2  }  x  x  S and Y  ^  -9-  w h i c h can be r e a r r a n g e d a s : Y  f (Y'|'X = X ) 0  = -r  ±—T.  2TT o . J l - p y  exp -h  -  [y  y +  (x;  ^  P  {  ~  a 1-p y  2  H  -y )] x  •  }  2  2  i . e . the c o n d i t i o n a l 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 of Y given X = X  0  i s a l s o normally d i s t r i b u t e d with a mean v a l u e = y + P (X - y ) y a x x 0  and variance  2.11  =  a (1 - p ) y 2  2  Simple l i n e a r r e g r e s s i o n e q u a t i o n : Y = A + B*X  + E  where E i s a random e r r o r component w i t h mean = 0, and v a r i a n c e = a  2  A and B a r e c o n s t a n t s The e s t i m a t e s o f A and B can be found e i t h e r by t h e L e a s t Squares  procedure  o r by t h e Maximum L i k e l i h o o d f u n c t i o n approach: N a••= E Y./N i-l x  a regression coefficient b = p — xy ax , correlation coefficient =  p xy  C  ' oo x y  t  m  ;  . _ N = E (X.Y.) - NXY ... x l x=l a  Standard d e v i a t i o n o f Y, a 7  = E[(Y - y ) ] = 2  7  x  a  y  N E ( Y . ) - NY i=l 2  1  2  -10-  s t a n d a r d d e v i a t i o n o f X, a  X  = E[(X  -  )] 2  u  X  =  N  £ . 1=1  (x. ) 2  1  -  _  NX  2  The e s t i m a t e d v a l u e of Y i s g i v e n by:  Y = a + b*(X-X)  The above b r i e f o u t l i n e o f p r o b a b i l i t y t h e o r y and  statistical  a n a l y s i s i s g i v e n as background t o t h e development o f t h e l o n g range d a i l y r i v e r f l o w f o r e c a s t model w h i c h i s p r e s e n t e d i n the f o l l o w i n g  chapter.  -11-  C H A P T E R  3  FORECAST SYSTEM  3.1  P r o f e s s o r J.F. M u i r a t the C i v i l E n g i n e e r i n g Department, t h e  U n i v e r s i t y o f B r i t i s h Columbia has d e v e l o p e d a r i v e r f l o w f o r e c a s t system u s i n g m u l t i p l e r e g r e s s i o n a n a l y s i s , r e f e r e n c e 12. system o n l y g i v e s f i v e day f o r e c a s t s .  However, t h i s f o r e c a s t  But, i n o r d e r t o determine an  optimum o p e r a t i o n p o l i c y f o r a f l o o d c o n t r o l r e s e r v o i r w h i c h would m i n i m i z e the damage f o r the e n t i r e f l o o d season, a l o n g range f l o w f o r e c a s t c o v e r i n g the whole season must be used f o r the o p t i m i z a t i o n analysis.  Since Professor Muir's  f o r e c a s t system c o u l d not b e used, a  s i m p l e f o r e c a s t model was i m p r o v i s e d  f o r the purpose * o f " t h i s  I f t h e p e r i o d , o f i n t e r e s t i s from m t o n_as  m  Fig.  3.1  then knowing.the r i v e r f l o w s on and p r e y i o u s pack r e m a i n i n g  shown i n F i g . 3 A.,  Time t  n  HYPOTHETICAL  study.  . j  HYDROGRAPH to m, and knowing the snow  t o be. m e l t e d and o t h e r m e t e o r o l o g i c a l i n f o r m a t i o n , i t i s :  c o n c e i v a b l e t h a t from h i s t o r i c a l records,, m a t h e m a t i c a l e q u a t i o n s may b e e s t a b l i s h e d t o p r e d i c t f l o w s on f u t u r e days (m + i ) ,  where i = 1, 2,  , (n - m)  The s i g n i f i c a n c e o f these e q u a t i o n s tests.  c a n be d e t e r m i n e d b y s t a t i s t i c a l  F o r s i m p l i c i t y , i t i s assumed t h a t t h e r e e x i s t  linear  relationships:  QCk + i ) = A + B * Q(k) + E where  Q : daily river  (3.1)  flow  , Cn - 2) , ( n - 1 )  k = m, (m + 1) , (m + 2) , i = 1, 2,  , (n - k )  E : a random e r r o r component w h i c h i s n o r m a l l y d i s t r i b u t e d w i t h mean zero and v a r i a n c e a  2  T h i s n e c e s s i t a t e s knowing t h e a c t u a l r i v e r f l o w QCk)  on day k, and p r e d i c t s  t h e f l o w on day (k + i ) , w i t h QCk + i ) depending o n l y on QCk).  E q u a t i o n 3.1  i s _ o f t h e s t a n d a r d s i m p l e l i n e a r r e g r e s s i o n model o u t l i n e d i n s e c t i o n 2.12. I n t h i s s t u d y , t h e p e r i o d o f i n t e r e s t i s a r b i t r a r i l y chosen from May 1 5 t h to June 3 0 t h , s i n c e d a i l y f l o w r e c o r d s a t t h e gauging s t a t i o n l o c a t e d a t Hope a r e r e a d i l y a v a i l a b l e f o r t h i s p e r i o d f r o m 1953 t o 1967. They were used as h i s t o r i c a l d a t a t o d e t e r m i n e t h e s e t s o f c o n s t a n t s A's and B's. F o r t y — f i v e s e t s o f r e g r e s s i o n e q u a t i o n s were o b t a i n e d . c o n s i s t s of 4 6 equations  i n t h e form: + b  M  1  5  j  M  1  6  * CQ(M15) - Q(M15))  QPCJ30) = Q(J30) + b  M  1  5  j  J  3  0  * (Q(M15) - Q(M15))  QPCM16)  where  The f i r s t s e t  =  QCM16)  QP i s t h e p r e d i c t e d r i v e r  flow  Q  i s t h e mean v a l u e o f d a i l y  M  f o r May  J  f o r June  flow  -13-  The  second  set consists  of 45 equations r e l a t i n g the a c t u a l  flow on May  16th  to p r e d i c t e d flows on the f u t u r e days, e t c .  The n a t u r a l r i v e r flow on any day k i s a random v a r i a b l e  3.3  follows a c e r t a i n determined,  distribution.  The shape of t h i s d i s t r i b u t i o n may  i f d a t a are a v a i l a b l e ,  h i s t o g r a m and d e r i v i n g  by p l o t t i n g  However, f o r  o f t h i s s t u d y i t i s assumed t h a t the d a i l y f l o w Q(k) and v a r i a n c e a,  2  be  the d a i l y flows as a  an e q u a t i o n f o r t h i s c u r v e .  d i s t r i b u t i o n w i t h mean  which  purposes  f o l l o w s a normal  as shown i n F i g .  3.2.  A f (Q(K))  MK \ Now  Fig. 3 : 2  MARGINAL DENSITY F U N C T I O N OF RANDOM VARIABLE Q(K) ."  c o n s i d e r t h e r i v e r flows on two  t h e assumption  Q(K)  made i n s e c t i o n  3.1,  different  days k and  the p r e d i c t i o n  + i ) a r e b o t h random v a r i a b l e s ,  of QCk)  and QCk  + i).  revealed i f available an e q u a t i o n may  then t h e r e e x i s t s  The n a t u r e of t h i s b i v a r i a t e data are p l o t t e d  Ck + i ) .  Based  of the r i v e r flow on  Ck + i ) w i l l depend o n l y upon the r i v e r f l o w on day k, QCk  ;  and s i n c e QCk) a joint  on day  and  distribution  d i s t r i b u t i o n may  on a t h r e e d i m e n s i o n a l s c a l e  be and  be. d e r i v e d to d e s c r i b e , the curved s u r f a c e w h i c h r e p r e s e n t s  the j o i n t p r o b a b i l i t y  density function.  I n t h i s s t u d y , the j o i n t  i s assumed to be b i v a r i a t e l y n o r m a l l y d i s t r i b u t e d W i t h t h i s assumption  the c o n d i t i o n a l  probability  distribution  as shown i n F i g . 3.3. d e n s i t y f u n c t i o n of  the  Fig. 3 . 3  JOINT OF  BIVARIATE DENSITY  RANDOM  VARIABLES  FUNCTION  Q ( K )  AND  Q(K+i)  -15-  random v a r i a b l e QCk  + i ) g i v e n t h a t random v a r i a b l e Q(k) has  v a l u e i s e a s i l y o b t a i n e d as o u t l i n e d i n s e c t i o n 2.11. the c o n d i t i o n a l d i s t r i b u t i o n f u n c t i o n , and  The  r e l a t i o n s h i p between  the s i m p l e l i n e a r r e g r e s s i o n  the j o i n t b i v a r i a t e d i s t r i b u t i o n i s shown i n F i g . 3.4.  seen t h a t  the p r e d i c t e d  flow QPCk + i ) on day  e l s e b u t one of the p o s s i b l e f o r e c a s t model, has  3.4  flows on day  From F i g . 3.4  f o r e c a s t model, say f o r the f l o w on day  distribution.  i s nothing  occurrence.  I f those o t h e r f a c t o r s were a l s o random v a r i a b l e s ,  constitute a multivariate  i t is  t h a t QPCk + i ) , i n the  depend not o n l y upon the f l o w on day k, b u t a l s o upon a l l o t h e r factors.  equation,  (k + i ) by e q u a t i o n 3.1  (k + . i ) , except  the maximum p r o b a b i l i t y of  A sophisticated  taken a p a r t i c u l a r  Ck + i ) would  important  then they would  However, f o r p r e s e n t purposes  our  i n t e r e s t i s not i n the i n d i v i d u a l d i s t r i b u t i o n of Q(k + i ) , b u t i n the c o n d i t i o n a l d i s t r i b u t i o n of QCk  + i).  The  t h r e e assumptions made i n t h i s  f o r e c a s t system a r e : Cl)  the r i v e r f l o w on any day k f o l l o w s  C2)  the r i v e r flow on day  a normal d i s t r i b u t i o n  (k + i ) depends o n l y on r i v e r flow  on day k, and a l i n e a r r e l a t i o n s h i p e x i s t s between them (3)  t h e j o i n t , d i s t r i b u t i o n of the r i v e r flows on day  .......  Ck + i )  i s a b i v a r i a t e normal d i s t r i b u t i o n I t i s r e a l i z e d that  these assumptions o v e r s i m p l i f y  the r e a l s i t u a t i o n , but  s i n c e t h e emphasis of t h i s study i s on t h e o p t i m i z a t i o n forecast  analysis,  this  system i s c o n s i d e r e d adequate as a s o u r c e of d a t a to i l l u s t r a t e  optimization  process.  If a better  f o r e c a s t model were developed,  optimization  p r o c e s s would not change a t a l l .  the  the  -16-  1 f ( Q ( K ) , Q ( K + i))  JOINT DENSITY  BIVARIATE FUNCTION  OF R A N D O M Q(K)  VARIABLES  CONDITIONED  DENSITY  FUNCTION  OF  RANDOM  VARIABLE  Q ( K - H ) G I V E N Q(K)  A N D Q ( K + i) .  FIG.  3. 4  -17-  C H A P T E R  DYNAMIC  4  PROGRAMMING  Dynamic programming i s one o f the o p t i m i z a t i o n O p e r a t i o n s Research s t u d i e s . several feasible alternatives. Cl)  Optimization  methods employed i n  means f i n d i n g a b e s t s o l u t i o n among  Dynamic programming i s based on:  the p h i l o s o p h y o f b r e a k i n g a complex problem i n t o a s e r i e s of s m a l l e r  problems, and then combining the r e s u l t s of the  s o l u t i o n o f the s m a l l e r  problems to o b t a i n  the s o l u t i o n o f  the whole complex problem. C2)  the p r i n c i p l e o f o p t i m a l i t y ,  as s t a t e d by Bellman:  "An  o p t i m a l p o l i c y has the p r o p e r t y t h a t whatever the i n i t i a l s t a t e and d e c i s i o n a r e ,  the r e m a i n i n g d e c i s i o n s  must  c o n s t i t u t e an o p t i m a l p o l i c y w i t h r e g a r d to the s t a t e r e s u l t i n g from the f i r s t  decision."  Some o f the concepts of dynamic programming, as used i n t h i s study a r e o u t l i n e d below.  D e t a i l s can be found i n r e f e r e n c e  13.  The t e r m i n o l o g y f o l l o w s  usage  4.1  A one-stage d e c i s i o n system:  D (DECISION)  X(lNPUTS)  Y(OUTPUTS)  t  R(RETURN)  Fig.  4.1  normal  -18I n F i g . 4.1,  the box  represents  a stage and  the n o t a t i o n s  represent  the  following quantities: (1)  an i n p u t s t a t e v a r i a b l e X, t h a t g i v e s a l l the information  relevant  about i n p u t s to the s t a g e ; i t r e p r e s e n t s  the  c o n d i t i o n of the system at the b e g i n n i n g of t h i s stage (2)  an o u t p u t s t a t e v a r i a b l e Y, information  t h a t g i v e s a l l the  about output from the s t a g e ; i t r e p r e s e n t s  c o n d i t i o n of the system at the end (3)  relevant  a d e c i s i o n v a r i a b l e D,  the  of t h i s s t a g e  t h a t c o n t r o l s the o p e r a t i o n  of  the  stage (4)  a s t a g e r e t u r n R, a s i n g l e - v a l u e d and  outputs.  f u n c t i o n of i n p u t s ,  decision  I t i s a measure of the u t i l i t y o b t a i n e d  s t a g e when the i n p u t i s X,  from  the  and  the  the d e c i s i o n s e l e c t e d i s D,  r e s u l t i n g output i s Y. R = r(X,D,Y) (5)  (4.1)  a stage transformation  t , a single-valued  known as the s t a g e - c o u p l i n g the end and  transformation  function, expression  also  the s t a t e at  of the s t a g e as a f u n c t i o n of the d e c i s i o n v a r i a b l e  the s t a t e at the b e g i n n i n g of the  Y = t(X,D) S u b s t i t u t e Eq^  stage (4.2)  4.2  i n t o Eq.  4.1  R = r(X,D,Y) = r(X,D, t(X,D)) =  r(X,D)**  M a t h e m a t i c a l R i g o r would r e q u i r e R = r ( X , D, t ( X , D)) = g(X, D) t o d i s t i n g u i s h the r e t u r n f u n c t i o n w i t h two arguments from the f u n c t i o n w i t h t h e s e arguments, as the f u n c t i o n a l n a t u r e w i l l change. However, t o s i m p l i f y n o t a t i o n , we use r to r e p r e s e n t the stage r e t u r n f u n c t i o n , thus r ( X , D, Y ) , r ( X , D, t ( X , D ) ) , and r ( X , D) are a l t e r n a t i v e e x p r e s s i o n s f o r the stage return r.  -19The one-stage i n i t i a l s t a t e o p t i m i z a t i o n problem i s t o choose D such t h a t t h e s t a g e r e t u r n R i s maximized ( o r m i n i m i z e d ) .  D e n o t i n g f ( X ) as t h e o p t i m a l  r e t u r n and D* = D(X) as t h e o p t i m a l d e c i s i o n , we have r ( X , D*) = r ( X , D ( X ) ) = f ( X ) = max r ( X , D) D where  r ( X , D)  max r ( X , D) means the maximum v a l u e o f r ( X , D) among a l l t h e D p o s s i b l e v a l u e s the d e c i s i o n v a r i a b l e D can t a k e .  If  t h e i n i t i a l s t a t e X i s n o t f i x e d , and we a r e f r e e t o choose X, t h e n  there  may e x i s t a p a r t i c u l a r v a l u e o f X such t h a t f ( X * ) = max f ( X ) = max r ( X , D) X X,D By t h e t r a n s f o r m a t i o n Y = t ( X , D) the i n v e r s i o n o f t h i s t r a n s f o r m a t i o n  t will  give  X = t"(Y, D) then, R = r ( X , D, Y) = r(t"(Y, D) , D, Y) = r ( D , Y) therefore  the one-stage f i n a l s t a t e o p t i m i z a t i o n problem i s t o choose D  as a f u n c t i o n o f t h e f i n a l s t a t e Y, D* = D(Y) = f ( Y ) = max r ( Y , D) D 4.2  S e r i a l multistage  d e c i s i o n system:  A s e r i a l multistage  system c o n s i s t s o f a s e t o f s t a g e s j o i n e d t o g e t h e r  i n s e r i e s so t h a t output from one s t a g e becomes t h e i n p u t t o t h e n e x t stage r e t u r n and t r a n s f o r m a t i o n  as shown i r i F i g . 4 . 2 .  ..  - 2 0 -  ,D_  X  "X . N  X;  2  N-I  N  n H  x  R.  RN-I  R.  r~i ^  x  R.  D,  D,  i-l  x  I  2  r~i ' *  x  i i  x  ° •  *  R.  R.  Fig.4.2 For a g e n e r a l stage i , ( i = 1 , 2 ,  , N ) o f t h e N - s t a g e system, the stage  transformation i s X  i-1  f= c  i  ( X  i 'V  ( 4 - 3 )  and t h e s t a g e r e t u r n R  i  =  i  r  (  X  i ' °i  )  ( 4  '  4 )  S i m i l a r l y f o r stage ( i + l )  X = W I+1' D  (4.5)  X  ±  ± + 1 )  Combining Eq. 4 . 3 and Eq. 4 . 5  • h^i+v i + r  Vi  V  D  I f we t r a c e t h e s t a g e s f u r t h e r up, i . e . i n t h e d i r e c t i o n from s t a g e 1 t o s t a g e N i n F i g . 4 . 2 , e v e n t u a l l y we o b t a i n  Vi i.e.  =  i N'  fc (X  V  D  input state v a r i a b l e X  N  '  •> i + r D  V  -  (4  6)  ^ t o t h e ( i ^ l ) t h stage depends o n l y on t h e v e r y  i n i t i a l c o n d i t i o n o f t h e system and a l l t h e d e c i s i o n s made p r i o r t o t h e ( i - l ) t h stage.  From Eq. 4 . 4 i t i s seen t h e r e t u r n o f i t h stage i s a f u n c t i o n o f t h e  -21-  i n p u t to i t h stage and the d e c i s i o n made a t the i t h stage R. = r . (X. , D. ) x  x  x  x  now by Eq. 4.6 x  i  = W  i  = i  V  V  D  N-r  > i 2 > . i+1> D  D  +  Hence R  i.e.  r  (  X  N ' V  D  N-r  ' i 2 V i l ' V D  D  +  ( 4  +  t h e r e t u r n of the i t h s t a g e depends o n l y on the v e r y i n i t i a l  -  7 )  c o n d i t i o n of  the system and a l l the d e c i s i o n s made p r i o r t o and d u r i n g the i t h s t a g e .  In  o t h e r words, the d e c i s i o n made a t the i t h s t a g e i n f l u e n c e s o n l y those r e t u r n s from stage 1 t o s t a g e i .  The t o t a l r e t u r n R^ from s t a g e 1 to s t a g e N i s some  f u n c t i o n of the i n d i v i d u a l s t a g e r e t u r n s R,  =  C X  G [ r  N' N-1' ' X  N  C X  N' V '  The N-stage i n i t i a l  X  r  ' V  » °N' N-1'  l  D  > i  N - l ° W N-1>' P  r  s t a t e o p t i m i z a t i o n problem  C  X  l '  V  ( 4  8 )  i s to maximize (or minimize) the  N-stage t o t a l r e t u r n R^ over the d e c i s i o n v a r i a b l e s D^, D^, f i n d the o p t i m a l r e t u r n as a f u n c t i o n o f the i n i t i a l Denoting  '  , D^, i . e . to  s t a t e X^.  f ( X ^ as the maximum N-stage r e t u r n , D* = D ^ X ^ ) , the o p t i m a l i t h s t a g e d e c i s i o n , w i t h i X* =  t_^CX^) , i  =  1, 2,  ,N  the o p t i m a l i t h s t a g e Input s t a t e , w i t h =  1, 2,  (N-1)  ,  Hence we have  W  =  G l r  N ° V > N > N-l<*N-l> N - 1 ' D  }  r  D  }  = max G t ^ C ^ , D ) , r ^ Q ^ , N  D  N'"' l D  ' ^CX*, D*)J ,  r  ^  , D ^ J (4.9)  -22-  Subject t o i-1  X  =  t  ± ±'  V  iX  i = 1. 2,  ,N  I n o r d e r t o b r e a k t h e complex o p t i m a l N-stage r e t u r n f^CX^) as e x p r e s s e d i n Eq. 4.9 i n t o a s e r i e s o f s i m p l e r problems, we must a c h i e v e the c r u c i a l s t e p of moving t h e m a x i m i z a t i o n w i t h r e s p e c t t o^_-^> °N-2'  ' °1' '- '-^ '  D  Nth s t a g e r e t u r n .  T h i s i s known as 'decomposition'.  :  ns:  e t  ie  Sufficient conditions  f o r a c h i e v i n g t h i s i m p o r t a n t change i n t h e p o s i t i o n o f m a x i m i z a t i o n s have been g i v e n by L.G. M i t t e n (Reference 1 4 ) . (1)  These c o n d i t i o n s a r e :  Separability, i.e. G [ r  N  C X  N> V '  = W  V  N - l N - l > N-1>' C X  V '  G  ' i  D  2 N-l N-l> [ r  ( X  r  D  (  X  l' V  N-1>>  ' l r  (  ]  X  l ' V  ]  }  and G^ a r e r e a l - v a l u e d f u n c t i o n s , and  where (2)  r  Monotonicity, i . e . G^ i s a m o n o t o n i c a l l y n o n d e c r e a s i n g f u n c t i o n o f G^ f o r every r ^  When these two 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 v i t a l d e c o m p o s i t i o n can b e a c h i e v e d , namely, max D  N'Vl'"  =  G [ r ( X , D^), ^ ( X ^ , N  > D  N  ,r  ^  , D^j  1  max G ^ C X ^ , D ) ,  max  N  D  D^) ,  N  D  G^r^OC^, D^) ,  ,r  ^  ,  ]} (4.10)  N-1'--" 1 D  I t has a l s o been known t h a t i f t h e t o t a l N-stage r e t u r n i s e i t h e r t h e sum o f the N-stages i n d i v i d u a l r e t u r n s , i . e . R  T - N V r  C  V  +  r  N - l N - l > N-1> C X  D  +  +  r  l<*l»  V  -23-  o r t h e p r o d u c t o f t h e N-stages n o n - n e g a t i v e i n d i v i d u a l r e t u r n s , i . e . *T  =  r  N N' V *  r  C X  (X  N-1' V - l * *  * l r  then t h e d e c o m p o s i t i o n can always be a c h i e v e d .  ( X  l'  V  L e t t h e o p e r a t o r "o" denote  e i t h e r one o f t h e above m a t h e m a t i c a l o p e r a t i o n s ,  namely a d d i t i o n o r  m u l t i p l i c a t i o n , we have, W  -  n  "** N ' V l " "  {  r  n  D  D  N  (  V V  °  N-1 N-1' N - 1  r  (X  D  ° ••• ° i  }  r  ( X  l> °1  ) }  l  by d e c o m p o s i t i o n , we have,  W  -  { r  N  ( x  N' V  °  N  m  a  x  r n  U  N-1'"  , U  N-i  ( x  N-r  D  N-I>  °  ° i r  (  x  r  V  1  = max {rjjCXjj, D > o f ^ C X ^ ' ) }  (4.12)  N  4.3  D e t e r m i n i s t i c and s t o c h a s t i c d e c i s i o n making The  recursion equation derived  i n s e c t i o n 4.2 i s a p p l i c a b l e t o  d e t e r m i n i s t i c models where f o r each s t a g e once a d e c i s i o n i s made, t h e r e t u r n ( o p t i m a l o r not)  from t h a t s t a g e i s unambiguously s p e c i f i e d , and t h e  s t a t e r e s u l t i n g from t h a t d e c i s i o n i s a l s o unambiguously s p e c i f i e d . I n these c a s e s , d e c i s i o n making i s under c e r t a i n t y .  But t h e r e a r e s i t u a t i o n s  where f o r each s t a g e , once a d e c i s i o n i s made n e i t h e r t h e r e t u r n from t h a t s t a g e , n o r t h e s t a t e r e s u l t i n g from t h a t d e c i s i o n a r e unambiguously s p e c i f i e d , but r a t h e r t h e r e i s a s e t o f p o s s i b l e r e t u r n s occurrences.  and s t a t e s due t o chance  These cases a r e known as d e c i s i o n making under r i s k o r s t o c h a s t i c  d e c i s i o n making.  4.4  S e r i a l multistage  d e c i s i o n system w i t h  uncertainty  An N-stage s t o c h a s t i c system i s s i m i l a r t o an N-stage d e t e r m i n i s t i c s y s t e m e x c e p t t h a t a t each s t a g e t h e r e i s a random v a r i a b l e t h a t a f f e c t s t h e  -24-  stage r e t u r n and  K  D  N  K _, N  N  t r a n s f o r m a t i o n . a s shown i n F i g .  D .,  X  K  N  X  N-2  X  f  D  4.3.  K,_, D,.  ;  K  D  2  X  X.  i  K,  2  'N-I  :  I  R.  R:  D,  Fig.4.3 At any  stage i , i = 1,  2,  , N,  d e c i s i o n made a t t h a t stage D^,  t h e r e i s an i n p u t  a random v a r i a b l e k^,  to t h a t s t a t e X_^, and  a return  a  from  t h a t s t a g e as a f u n c t i o n of X., D., k. • x x i R. = r . ( X . , D. , X X X X The  t r a n s f o r m a t i o n a t each stage i s X. . = t . ( X . , D., x-1 X X X  The  C4.13)  k.) X  random v a r i a b l e k^ may  (4.14)  k.) X  take on a s e r i e s of v a l u e s which f o l l o w a c e r t a i n  p r o b a b i l i t y d i s t r i b u t i o n P ( k ^ ) ; hence, g i v e n a f i x e d i n i t i a l  input  state  to the i t h stage and exactly  a f i x e d d e c i s i o n D., R... and X. . cannot be s p e c i f i e d x x x-1 to the random v a r i a b l e k.. S i m i l a r to Eq. 4.14 we get  due  X  S i n c e k.,.. x+1 cannot be  i  =  ' i + i ^ i + r i+i» D  (4.15)  W  i s another random v a r i a b l e w i t h a p r o b a b i l i t y f u n c t i o n P(k specified exactly  either.  Hence, f o r a g i v e n i n i t i a l  x+1  ),  condition  X. x X^,  -25-  the s t a t e v a r i a b l e s X. x  2,  i - l , the stage r e t u r n s  i  ,  (N-1)  R_^,  2,  = 1,  N  are a l l random v a r i a b l e s .  i  R  =  (  i  r  •  (  t  i i  (  i r  x  +  k  D.,  +1  we  k. , + 1  t r a c e the s t a g e s f u r t h e r up,  s t a g e N i n F i g . 4.3,  i  R  Eq. 4.16  =  r  i  C X  reveals Cl)  4.15  i+ri+i>» V V  D  +  = r.CX., D. , If  and Eq.  V V  i V  r  Combining Eq. 4.13  i . e . i n the d i r e c t i o n from s t a g e 1 to  e v e n t u a l l y we  N>  D  k.)  obtain  'V V  N'  'V  *  (4  that  the d e c i s i o n made a t the i t h stage i n f l u e n c e s those r e t u r n s  only  from s t a g e 1 to s t a g e i Cas i n the  d e t e r m i n i s t i c model) C2)  the i t h s t a g e r e t u r n does n o t depend on random v a r i a b l e s k, , k  2  1 If  , k. ... x-1  the t o t a l r e t u r n from the N stages i s the sum  returns,  of the i n d i v i d u a l s t a g e  then R  =  =  T - N ' N-1' (  X  X  X  N E r . ( X . , D., . -, X X X x.=l r  N  ( x  N'  V  V + !  l ' °N' °N-1'  D  l ' N'  '"'  k  k  l ^  k.) X  r  N-i°W V r W  +  •••  +  r  i r ( x  V  k  i  }  16)  -26-  s u b j e c t t o X.  = t . ( X . , D., k . ) , i = 1, 2, ... N.  3.-1  1  1  1  1  Take t h e r e t u r n a t s t a g e (N-1) as an example.  If _^ i X  N  s  f i x e d , the r _ j N  i s a f u n c t i o n o f t h e random v a r i a b l e k^_^> hence b y s e c t i o n 2.3, t h e expected value of the ( N - l ) t h stage r e t u r n i s ^ N - A - l ' °N-1> where  E  =  ,  E  ^ N - l ^ - l ) * N-l r  C X  N - l ' °N-1'  W  ]  means t h e summation o f a l l t h e p o s s i b l e v a l u e s the random v a r i a b l e k^ ^ may t a k e .  •^N-l^N-l^  t  ^  ie  P ^ k a b i l i t y f u n c t i o n of  ^.  But s i n c e X ^ ^ = t C 3 ^ , D , k^) , t h e r e f o r e X ^ ^ i s i t s e l f a f u n c t i o n o f a n o t h e r random v a r i a b l e k^.  Hence t a k i n g c o n s i d e r a t i o n o f t h e u n c e r t a i n t y i n X^._^,  r e s u l t i n g from random e f f e c t s a t s t a g e N, t h e expected r e t u r n from s t a g e (N-1) i s  E[r _ CX _ , D _ N  =  =  1  N  1  N  15  k ^ ) ]  * N<*N-1> * J N-1°W ^-1 *N-1 P  * W  {  * <  E  P  P  * N - l N - l > °N-1' r  N-l N-l> * N - l C k  r  ( X  C X  N - l ' N-1> D  k  N-l  ) }  T h e r e f o r e , f o l l o w i n g a s i m i l a r argument i n t h e d e r i v a t i o n o f Eq. 4.17, the t o t a l expected r e t u r n from N s t a g e s i s  ( 4  -  1 7 )  -27-  T^Sl'  R  E  **' ^1' ^N' ""' ^1^  * [ E  W  N  r  k ^ ) ] +  ^N-l  *T&  +  V V  E  * I  N-l N-l> *N-1 E  TJ  •••  +  D _  * r ^ C ^ ,  ^ C ^ )  +  P  W  E  * t  V  s  [ E P. Ck ) * r ( X , D 1  1  1  W k.  C k  1  A  * i r  C X  i'V  V  ]  * ••• *  J  k ^ ] ... ]  lf  l  k  E  •••  +  W  +  * N N' V { r  V l  E  (  V  ( X  N - l  K  *  }  [  E  +  P  ,  E  N-2° N-2 K  [ E P.(k.) * r . ( X . , D., k . ) ] ... k.  * ^ . i V l ) TI-l  *  [  E  P  ^ - i V l N-1  N-2°W  )  5  * N - l N - l ' V l ' N-l> r  ( X  k  *  ]+...+  * ••• *  V2  [ E P ( k ) * r ( X , D , k ) ] ... ']} 1  k  1  1  1  1  1  l  S u b j e c t t o X. . = t . ( X . , D., k . ) , i = 1, 2, i-l 1 1 1 1  N  f ^ C X ^ ) , as a f u n c t i o n o f the i n i t i a l c o n d i t i o n X^, i s d e f i n e d t o be the t o t a l m i n i m i z e d e x p e c t e d r e t u r n from N s t a g e s , and by the p r i n c i p l e o f o p t i m a l i t y ,  -28-  we o b t a i n ,  f  CX^) =  max  I CX^,  V W " '  =  ™*  X  D  W  + Z P^Cy,) * W  * {r CX , D , y N  N  N  N'" l N  D  D  +  W  D _ k^) N  r  h-1  k  VlVP *  +E  D)  1  D  W  [ E  ^-1  W*  ^N-2  *  [ E P.Ck.) * r.CX., D., k.)] ... J + ... + \ 11 i i ' I l k. i  /  hi-i  W  W  *J [  P  N-2 W.*  ^-2  N  y  D  W  N -  P  2  V2  W  *  [  E v  '  •••  +  D^, k^)+  +  E  P  P  E  (  *  (  *  [  E  1  ( x  i' V  *  [ E  k  k  i  )  •••  ]  ] } }  *  N-l  W  W  W  * i r  D _ , k ^ ) + ... N  (  X  i '  2  V V  ]  ]  i  W  W  *  *  *N-3  [ E P ( k ) * r C X , D , k ) ] ... ]}}}} 1  r  N-1'-" 1  V 3  N-2 W  * i  D  P ^ C ^ )*  N-3 W  Tl-2  k  W i  »*' < * ^iV^  +  ^  <W  E  [E  k  = max { E P 0 ^ ) * {r (X^, D  +  *  (  1  1  l  S i n c e by d e f i n i t i o n  x  x  (4.18)  -29-  -  ^ D  ,  S  V +  P  N-2 V (  { n  J  N - 1 W *' N - l N - l ' °N-1' W  P  N-1'*" 1  C  * N-2 \ - 2 • V  )  r  2  { r  C X  +  V l  D  > W  (  2  +  2  E  V  P  N-2 W* C  1  ,  E  V3  2  + ••• / V2  P N  P  -2<W.*  [  N-3°W*  E  P  ••• *  J W  [  * i r  i  k  N-3 N-3> * ' ' ' * Ck  [  V 3  * W  (  X  i ' °i' V  * l r  1  (  X  l'  V  ]  1  k  l  )  ]  ] } >  C4.19). Combining Eq. 4.18  ?  N V C  and Eq. 4.19,  -  J N  S  we  W  *  To g e n e r a l i z e t h i s e x p r e s s i o n , we ?. (X.) = max{ E P . O O X X ~ , D. k. 1  x  i = 1, 2,  1  obtain,  [ r  N N' V  V  C X  +  W V l > '  ]  }  obtain, * [ r . ( X . , D., x  x  x  x  k.) + J.  x-1  _ (X.  .)]}  x-1  x  , N  (4.20)  T h i s i s the s t o c h a s t i c r e c u r s i o n f o r m u l a i n dynamic programming. by A. Kauffman (Reference  As p o i n t e d  out  15) i n the f a c e of an u n c e r t a i n f u t u r e , the o p t i m i z a t i o n  can o n l y be c a r r i e d out i n one d i r e c t i o n :  by p r o c e e d i n g  from the f u t u r e towards  the p a s t , namely t o proceed a g a i n s t the d i r e c t i o n of the a r r o w s , i . e . from s t a g e 1 t o s t a g e N.  T h i s i s known as backward computation or upstream f l o w .  T h i s r u l e i s the consequence of the s t o c h a s t i c m u l t i s t a g e method by which t h i s expected  value i s c a l c u l a t e d .  However, i n a d e t e r m i n i s t i c model, i f the  s t a g e — c o u p l i n g f u n c t i o n s t_^  X. x-1  = t . (X., D.) x x x  i = 1, 2, ... •'  '  N  -30-  have i n v e r s e f u n c t i o n s t . 1  X. = t T ( X . 1  1 5  D.)  i = 1, 2,  ... N  then the o p t i m i z a t i o n can be c a r r i e d out i n e i t h e r f o r w a r d o r backward sequence.  Engineering  r i v e r f l o w s as one  s t u d i e s of w a t e r r e s o u r c e s  of the i n p u t d a t e .  v a r i a b l e , problems of w a t e r r e s o u r c e s  systems o f t e n e n c o u n t e r  S i n c e r i v e r f l o w i s a random systems are f r e q u e n t l y ,  s t o c h a s t i c by  nature. 4.5  C o m p u t a t i o n a l Scheme I n the above d e r i v a t i o n of the r e c u r s i v e f o r m u l a e of d e t e r m i n i s t i c  and  s t o c h a s t i c dynamic programming, b l o c k diagram r e p r e s e n t a t i o n s  s e q u e n t i a l d e c i s i o n p r o c e s s were shown i n F i g s . 4.2 numbering of s t a g e s i s q u i t e s t a n d a r d such p r a c t i c e p r o b a b l y b e i n g multistage  d e c i s i o n problem.  and  4.3.  The  of  the  backward  i n dynamic programming, the r e a s o n f o r  the sequence of d e c i s i o n s made i n s o l v i n g a As i n F i g . 4.2,X  0  q u i t e o f t e n denotes the  f i n a l system c o n d i t i o n and X^ denotes the i n i t i a l system c o n d i t i o n ;  and  f r e q u e n t l y i n dynamic programming a p r o b l e m can o n l y be s o l v e d i f one proceeds from the f i n a l c o n d i t i o n X  l0  t o determine a s e t of d e c i s i o n s w h i c h  w i l l l e a d t o the i n i t i a l c o n d i t i o n X^ o p t i m a l l y .  Hence p r a c t i c a l c o m p u t a t i o n a l  scheme i s a l s o o f t e n proceeded b a c k w a r d l y from s t a g e 1 t o s t a g e N. backward approach w i l l be used i n the f o r m u l a t i o n i n Chapter 5.  The  -31CHAPTER 5 APPLICATION OF DYNAMIC PROGRAMMING TO A SINGLE FLOOD CONTROL RESERVOIR The f i c t i t i o u s r e s e r v o i r i s assumed t o have a s t o r a g e c a p a c i t y o f 500,000 s f d and f o r s i m p l i c i t y i t i s a l s o assumed t h a t t h e r e s e r v o i r i s a b l e to d i s c h a r g e as much w a t e r as d e s i r e d . The o p t i m i z a t i o n as a s e q u e n t i a l  o f t h e r e s e r v o i r o p e r a t i n g proceedure i s t r e a t e d  decision process.  The p r o c e s s o f o p e r a t i n g a dam i s c o n t i n u o u s  s i n c e i n f l o w v a r i e s c o n t i n u o u s l y w i t h t i m e , and t h e r e s e r v o i r s t o r a g e i s also continuous. for  function  However, t h e most c o n v e n i e n t way t o f o r m u l a t e t h e problem  computation on a d i g i t a l computer i s t o c o n s i d e r a s e r i e s o f time  i n c r e m e n t s , and t o t r e a t t h e i n f l o w s  -as . d i s c r e t e v a r i a b l e s .  The volume  of w a t e r i n s t o r a g e w i l l thus a l s o appear t o change i n s t e p w i s e manner. s i z e o f t h e increment i n f l o w and i n t h e volume o f w a t e r s t o r e d r e s e r v o i r depends on t h e a c c u r a c y one w i s h e s t o o b t a i n .  The  i n the  I n t h i s study, the  i n f l o w i s c o n s i d e r e d h a v i n g an increment o f 10,000 c f s , and t h e t o t a l  reservoir  s t o r a g e i s d i v i d e d i n t o 50 i n c r e m e n t s , each increment e q u a l s t o 10,000 s f d .  5.1  D i s c r e t i z a t i o n of the inflow to the r e s e r v o i r . As d e s c r i b e d i n s e c t i o n 3.3, knowing t h e a c t u a l r i v e r f l o w on day k,  the c o n d i t i o n a l d i s t r i b u t i o n s o f f l o w s on f u t u r e days a r e a l l assumed t o be normally d i s t r i b u t e d . For normal d i s t r i b u t i o n , t h e p r o b a b i l i t y o f X h a v i n g a v a l u e l e s s than or e q u a l t o X  Let t dt  Q  i s ( s e c t i o n 2.9)  xzu a  i a  dx X -y 0  -32-  The f o l l o w i n g  approximation formula f o r t h i s i n t e g r a t i o n i s proposed by  C. H a s t i n g s J r . ( r e f e r e n c e 13)  P(X ^ X )  = 1.0 - Z ( t ) * ( a  D  X  where  1  * r- Y + a  2  * Y  2  +  * Y ) 3  ...  (5.1)  y  0  Xory t = Y = 1.0 / ('1.0 + p * t )  -ht  2  1  Z(t)  J 2 ^ p  a N.B.  If X  0  < y,  =  x  0 . 3 3 2 6 7  = 0.43618  a  0  = -0'. 12016  X -y  then l e t t  P(X <_ X )  2  0  = Z(t) * ( a  1  3  = -,,93729  and  * Y + a  P[Q(K+i)]  a  2  * Y  2  + a  3  * Y  3  )  A P[Q(K+i)]  Q(K+i)  Q(K + i)  QP(K+i)  QP(K+i)  F i g . 5.1  U s i n g Eq. 5.1, the shaded a r e a under a normal d i s t r i b u t i o n curve i s r e a d i l y o b t a i n a b l e , hence c o n v e r t i n g the continuous c o n d i t i o n a l d i s t r i b u t i o n to a d i s c r e t e c o n d i t i o n a l d i s t r i b u t i o n t o c a l c u l a t e the c o n d i t i o n a l p r o b a b i l i i t i e s associated  w i t h the p o s s i b l e r i v e r flow on any f u t u r e  day.  F o r example,  knowing  -33the a c t u a l r i v e r f l o w on May 1 9 t h Q(M19) = 200,000 c f s by Eq. 3.1 t h e mean v a l u e of t h e f l o w on May 20th i s 210,000 c f s w i t h a standard d e v i a t i o n equal to  W  J - M19,M20 = 52'°°° * J 1 " ^ 9 7 2 5 ) 2 1  p  = The  52,000 * 0.234 = 12,000 c f s .  c o n d i t i o n a l c o n t i n u o u s normal d i s t r i b u t i o n o f f l o w on May 20th i s d i s c r e t i z e d  to  give:  Possible r i v e r flow on May 2 0 ( 1 0 c f s )  170  2  .. , P r o b a b i l i t y of occurrence  5.2  :  0  Q  1  180  190 Q 8 7  200 ^  210 ^  220  230 Q g 7  240  250  ^  F l o o d damage f u n c t i o n . F l o o d damage i s u s u a l l y measured i n terms o f money, and t h e a c t u a l  d e t e r m i n a t i o n o f t h e damage f u n c t i o n i s q u i t e a f o r m i d a b l e t a s k .  Since the  o b j e c t o f t h i s study i s t o demonstrate t h e a p p l i c a t i o n o f dynamic programming to  determine  an o p t i m a l r e s e r v o i r o p e r a t i o n p o l i c y , the s p e c i f i c n a t u r e o f t h e  damage f u n c t i o n i s not examined i n d e t a i l .  I t i s assumed t h a t when t h e r i v e r  d i s c h a r g e downstream from t h e r e s e r v o i r exceeds QL = 200,000 c f s , damage w i l l b e g i n t o o c c u r and damage w i l l reach a maximum when t h e f l o w exceeds QU = 500,000 cfs.  The v a l u e s .of QU and QL used i n t h i s study a r e a r b i t r a r i l y chosen, as i s  the damage f u n c t i o n i t s e l f .  *  S i n c e t h e a c t u a l form o f t h e damage f u n c t i o n i s  The p r o b a b i l i t y o f o c c u r r e n c e o f a f l o w o f say 180,000 c f s i s t a k e n as the p r o b a b i l i t y t h a t t h e f l o w w i l l f a l l between 175,000 and 185,000 c f s .  -34-  i m m a t e r i a l as f a r as t h e approach t o o p t i m i z a t i o n a n a l y s i s i s concerned, f o r s i m p l i c i t y i t i s assumed here t h a t t h e l a r g e r t h e r i v e r f l o w i s t h e g r e a t e r i s the f l o o d  ( i . e . t h e damage done by r i v e r f l o w Q l i s g r e a t e r than t h e damage  done by r i v e r f l o w s Q2, Q3,  , Qn, i f Q l >_  Q2 + Q3 + ... + Qn, and  Q i > QL, f o r i = 2, 3, .... , n ) . Hence t h e f l o o d damage f u n c t i o n i s o f t h e shape shown i n F i g . 5 . 2 .  H UJ  o <  < Q Q O O _l QL= 2 0 0 , 0 0 0  QU= 5 0 0 , 0 0 0 .  Q(cfs) DISCHARGE  Fig. 5 . 2 Two types o f damage a r e c o n s i d e r e d i n t h e s t u d y , namely: (1) n o n - r e c u r s i v e damage, w h i c h i s d e f i n e d as f o l l o w s :  unless the r i v e r  discharge  downstream from t h e r e s e r v o i r exceeds t h e maximum d i s c h a r g e e x p e r i e n c e d so far  i n t h a t y e a r , no e x t r a damage w i l l be done even i f t h e f l o w exceeds QL.  (2) r e c u r s i v e damage, w h i c h i s d e f i n e d as f o l l o w s :  as l o n e as t h e d i s c h a r g e  downstream o f t h e r e s e r v o i r exceeds QL, damage w i l l be caused i n accordance w i t h t h e g i v e n f l o o d damage f u n c t i o n . 5.3  General d e s c r i p t i o n of a f l o o d c o n t r o l r e s e r v o i r operation p o l i c y . The purpose o f a f l o o d c o n t r o l r e s e r v o i r i s t o r e t a i n a l l t h e excess  water which would cause damage ( i f t h e s t o r a g e c a p a c i t y i s l a r g e enough), o r to  s t o r e p a r t o f t h e excess w a t e r t o m i n i m i z e  s t o r a g e i s n o t g r e a t enough).  t h e i n e v i t a b l e damage ( i f t h e  -35I f the i n f l o w s d u r i n g the e n t i r e f l o o d season are known w i t h c e r t a i n t y , .then the r e s e r v o i r o p e r a t i n g p o l i c y i s s i m p l e t o b e g i n c l o s i n g the gates a t p o i n t c, and m a i n t a i n  t o form, namely  a constant  discharge  such  t h a t the r e s e r v o i r w i l l be f u l l a t p o i n t d, as shown i n F i g . 5 . 3 .  Although  damage i s done d u r i n g p e r i o d ab, the maximum downstream d i s c h a r g e  i s only  r a t h e r than Q  max  Q  0  : t h e r e f o r e the i n e v i t a b l e damage i s m i n i m i z e d ,  When the i n f l o w s f o r the e n t i r e f l o o d season are not known w i t h c e r t a i n t y , then i t i s a d i f f i c u l t t a s k t o d e c i d e when t o s t o r e water and  how  much excess w a t e r to s t o r e , s i n c e an immediate a v o i d a n c e of a s m a l l damage may  r e s u l t i n a much l a r g e r damage i n the f u t u r e .  T h e r e f o r e the r e s e r v o i r  o p e r a t i o n p o l i c y must have the e n t i r e f l o o d season i n mind, and s h o u l d be such t h a t the t o t a l s e a s o n a l  the p o l i c y  damage i s as s m a l l as p o s s i b l e .  S i n c e the f u t u r e r i v e r i n f l o w i s u n c e r t a i n , a l t h o u g h the range of p o s s i b l e v a l u e s  the f u t u r e i n f l o w may  take on any p a r t i c u l a r day  known from the f o r e c a s t , i t i s i m p o s s i b l e w i l l r e s u l t due  to a p a r t i c u l a r d e c i s i o n .  t o know beforehand how  is  much damage  An example i s g i v e n below t o  i l l u s t r a t e this point. I f a f o r e c a s t system i n d i c a t e s t h a t the i n f l o w on a p a r t i c u l a r day may  take any  Q(10  P(Q) and  3  cfs)  one  of the f o l l o w i n g v a l u e s w i t h the a s s o c i a t e d p r o b a b i l i t i e s  170  180  190  200  210  220  230  240  250  .001  .017  .087  .233  .323  .233  .087  .017  .001  suppose the r e s e r v o i r to be h a l f f u l l at the b e g i n n i n g  a d e c i s i o n made on t h a t day not t o s t o r e any  inflow.  of t h a t day,  Hence, due  d e c i s i o n , the r e s e r v o i r w i l l s t i l l be h a l f f u l l at the end  c o r r e s p o n d i n g damage) r e s u l t i n g from t h a t d e c i s i o n ?  Two  to t h i s  of t h a t  But what w i l l be the r i v e r f l o w downstream of the r e s e r v o i r (and  and  day.  the  p o s s i b l e ways of  -37assessing t h i s s i t u a t i o n are: (1)  expected v a l u e approach: S i n c e t h e r i v e r f l o w on t h a t day may take any v a l u e from 170,000 c f s t o 250,000 c f s w i t h a s s o c i a t e d p r o b a b i l i t i e s of o c c u r r e n c e  (thus t h e  o u t f l o w i s o f t h e same n a t u r e ) , t h e expected v a l u e of t h e r i v e r f l o w on t h a t day i s  250,000 I Q=170,000  > P(Q)*Q = 10  * [170*0.001 + 180*0.017 +  + 240*0.017  + 250*0.001] = 210,000 c f s . The expected v a l u e has a s t a t i s t i c a l s i g n i f i c a n c e , i n t h a t i t r e p r e s e n t s < anaestimate  of t h e average v a l u e o f r i v e r f l o w s l i k e l y t o occur on t h a t  day, even though the a c t u a l r i v e r f l o w on t h a t day may n o t c o i n c i d e w i t h the expected (2)  value  p r o b a b i l i t y approach: Since t h e p r o b a b i l i t y of f l o w b e i n g e q u a l t o 170,000 c f s i s 0;001 and the p r o b a b i l i t y of f l o w b e i n g e q u a l t o 180,000 c f s i s 0.017, one can say t h a t t h e p r o b a b i l i t y of f l o w b e i n g not g r e a t e r than 180,000 c f s i s 0.018.  When one d e a l s w i t h p r o b a b i l i t y , u n l e s s t h e v a l u e i s 1.0, one can never be c e r t a i n whether an event w i l l t a k e p l a c e o r n o t . From t h i s example, i t i s seen t h a t t h e d e f i n i t e r i v e r f l o w on t h a t day cannot be p i n - p o i n t e d .  S i n c e t h e f l o o d damage i s dependent on t h e o u t f l o w  from r e s e r v o i r , which i n t u r n depends on t h e i n f l o w t o t h e r e s e r v o i r , t h e damage i t s e l f i s a random v a r i a b l e .  T h e r e f o r e , t h e damage r e s u l t i n g from any  p a r t i c u l a r d e c i s i o n cannot be p i n - p o i n t e d e i t h e r .  The p r o b a b i l i s t i c n a t u r e  of t h e answer t o such problems may seem r a t h e r i n p r e c i s e t o a  deterministic  -38-  frame o f mind, b u t t h i s i s a t y p i c a l c h a r a c t e r i s t i c o f any s t o c h a s t i c p r o c e s s . As d i s c u s s e d optimization  i n t h e f o l l o w i n g s e c t i o n s , dynamic programming as an  t e c h n i q u e can be used t o f i n d o p t i m a l r e s e r v o i r o p e r a t i n g  w h i c h i s e i t h e r t o m i n i m i z e t h e t o t a l expected damage,  policy  o r t o maximize t h e  p r o b a b i l i t y o f a damage n o t e x c e e d i n g a c e r t a i n amount f o r t h e e n t i r e f l o o d season.  5.4  N o n - r e c u r s i v e type of damage. S i n c e t h e maximum p o s s i b l e damage done on a c e r t a i n day i s g i v e n  i n F i g . 5 . 2 , and s i n c e t h e damages a r e n o n - r e c u r s i v e , t h e maximum  possible  damage t h a t might r e s u l t f o r t h e e n t i r e f l o o d season i s a l s o known. A v a l i d c r i t e r i o n f o r the r e s e r v o i r operating  p o l i c y would be  to guarantee t h a t t h e damage f o r t h e whole f l o o d season does n o t exceed a c e r t a i n p e r c e n t a g e o f t h e maximum p o s s i b l e damage.  T h i s means t h a t our d e s i r e  i s t o o p e r a t e t h e r e s e r v o i r i n such a manner t h a t t h e i n e v i t a b l e damage i s c o n t a i n e d w i t h i n ct% o f t h e maximum p o s s i b l e damage. to f i n d a r e s e r v o i r operating  Hence t h e o b j e c t i v e i s  p o l i c y w h i c h w i l l maximize t h e p r o b a b i l i t y  of o c c u r r e n c e o f an event w h i c h we w i s h t o o c c u r , i . e . maximize t h e p r o b a b i l i t y of l i m i t i n g t h e damage t o some f i g u r e w h i c h has been d e c i d e d upon. In the f o l l o w i n g formulation, are d e f i n e d  a s t a g e r e f e r s t o a day. The v a r i a b l e s  as f o l l o w s :  s t a t e v a r i a b l e X_^:  l e v e l o f water i n r e s e r v o i r a t t h e b e g i n n i n g o f stage i  d e c i s i o n v a r i a b l e D_^:  d e c i s i o n made a t s t a g e i t o o p e r a t e t h e r e s e r v o i r  such t h a t t h e aim i s t o l e t t h e water l e v e l i n r e s e r v o i r at t h e end o f stage i be a t random v a r i a b l e Q_^: r i v e r i n f l o w t o t h e r e s e r v o i r d u r i n g state variable X  v  ., :  x-1  stage i  t h e a c t u a l water l e v e l i n r e s e r v o i r a t t h e end o f  stage i ,  although the d e c i s i o n  l e v e l Y. , x-1  i s made t o aim f o r a  -39X.  = t . (X., D.,  x-1 a P(R  X  Q.) X  X  X  : p e r c e n t a g e o f maximum p o s s i b l e damage : p r o b a b i l i t y o f c a u s i n g damage n o t g r e a t e r than ct% o f  < a/ X., D.) i — x x  the maximum p o s s i b l e damage d u r i n g s t a g e i o n l y , g i v e n t h a t a t t h e b e g i n n i n g o f stage i t h e water l e v e l i n r e s e r v o i r i s a t X., and a d e c i s i o n D. i s made t o aim x x f o r a water l e v e l Y.  • a t t h e end o f the stage i  °  Xr-1  f^(X^,a)  : t o t a l maximized p r o b a b i l i t y o f damage n o t e x c e e d i n g a% o f t h e maximum p o s s i b l e damage from s t a g e i t o stage 1, where stage 1 i s the end o f t h e f l o o d  R e t u r n R^  season.  : t h e f l o o d damage s u f f e r e d a t stage i o n l y  D X  Di  it  QN  N  N  X  N  N-I  X  ^  i  R,  Di_,  Qi  R  Qi-,  ft  i-l  X  i_  D, p  X  i  l  Q, l  i-l  If R  :  i-l  •  R  F i g . 5.4  By d e f i n i t i o n o f f / ( X ,a) : f.(X.,a) = 1  1  max D  P(R., R. -, D  1  1  -  1  R  n  < a  1 —  1  X. , D. , D. ^ , x x x-1  ., D ) 1  As p o i n t e d out i n s e c t i o n 4.4, t h e d e c i s i o n made a t s t a t e i i n f l u e n c o n l y those r e t u r n s from s t a g e i "to stage 1, o r t o s t a t e i t i n r e v e r s e , t h e d e c i s i o n s made d u r i n g stage 1 t o stage ( i - l ) have no i n f l u e n c e whatsoever on the r e t u r n from stage i .  -40Hence,  P(R., R. 1 x-1 = P(R. < x —  R, < 1—  a  a \ X., D., D. 1  1  x  X., D.) * P(R. x x x-1  I 1  = P(R. < a I X., D.) x — x x 1  D.)  x-1  X  ,R < 1 —  a  I 1  *X, J- P(X. ., R. ., x-1 x-1  R.< ct , X., D., . . ., D,) l — x x ' 1 , R. < a I R. < a , X., D., 1— x— x x  = P(R. < a I X., D.) *l [ P ( X . , I R. < a, X., D.) * P(R. x — x- l X. .. i-l x— x x x-1 x-1 1  , D,) 1  1  x-1  R^alX. 1— x-1  1  therefore, max { P ( R . < a  f.(X.,a) =  x x  x —  I X . , D.) * P ( R .  '  x  x  R . <a  x-1  1—  R . < a,X.,D..;.,D,)}  I  l  1  —  i  i  1  1  1  = max { P ( R . < a | x . , D . ) * ^ [ P ( X . . IR. <a , X . ,D. )*max P ( R . . ,. . .R. <a IX. i — x x x-1 l — i i i - l 1— i - l x x D _ ,...,D 1  1  1 5  1  i  1  D. .....Dj]} x-1 1  1  max { P ( R . < a | x . , D . ) * T [ P ( X . J R . <a,X. ,D . ) * f . . (X. a ) ] } x— X x „ x-1 x— x x x-1 x-1 D. X. . x x-1 1  1 5  1  .... (5.1) where  means summing through a l l t h e p o s s i b l e s t a t e s X._ , r e s u l t i n g X  i-1  1  from t h e random i n f l o w Q. and t h e d e c i s i o n D. t o aim f o r Y. . x x x-1 at t h e end o f s t a g e i  max D.  means t o seek an o p t i m a l d e c i s i o n D* among a l l t h e p o s s i b l e 1  decisions  a t stage i  The random i n f l o w Q., a c c o r d i n g t o t h e f o r e c a s t may have v a l u e Q. 1 x, j j = 1 , 2,  where  m (Q. ._ <Q. . ) , w i t h a s s o c i a t e d p r o b a b i l i t y P(Q. . ) . I f a t 1  the b e g i n n i n g o f s t a g e i , X^ i s known and a d e c i s i o n  i s made t o o p e r a t e  the r e s e r v o i r on t h a t day such t h a t t h e aim i s t o have t h e water l e v e l i n t h e r e s e r v o i r a t t h e end o f s t a g e i a t possible:  then three types o f d e c i s i o n  are  -41Type ( 1 ) : d e c i s i o n D. such t h a t Y. , < X., i . e . t h e d e c i s i o n i s t o lower 1 x-1 x the w a t e r l e v e l i n t h e r e s e r v o i r no m a t t e r what t h e random inflow  i s . Such a d e c i s i o n i s always a t t a i n a b l e , i . e .  X. . = Y. . i s always t r u e , and t h e r e f o r e x-1 x-1 Prob(X. = Y. , / R.<a, X., D.) = 1.0 x-1 x-1 x— X X Prob(X. = Y / R.<a, X., D.) = 0.0 x-1 x-1 x— X X Hence Eq.5.1 becomes f  (X ,a) = max {P(R <ct / X , D ) * f  (X  ,a)} ...  (5.2)  ^i Type ( 2 ) : d e c i s i o n D. such t h a t Y. , = X., i . e . t h e d e c i s i o n i s t o x x-1 x r  m a i n t a i n t h e water l e v e l i n r e s e r v o i r no m a t t e r what t h e random inflow Q  i s . S i n c e such a d e c i s i o n i s a l s o always a t t a i n a b l e ,  X^_^ = Y_^_^ i s always t r u e , and t h e r e f o r e Eq.5.1 becomes Eq.5.2. Type ( 3 ) : d e c i s i o n D. such t h a t Y. . > X., i . e . t h e d e c i s i o n i s t o s t o r e x x-1 x more w a t e r up t o t h e l e v e l Y. ,. There a r e t h r e e p o s s i b i l i t i e s x-1 a s s o c i a t e d w i t h such a d e c i s i o n , depending on t h e a c t u a l i n f l o w v a l u e o f t h e random v a r i a b l e Q. x (3a): Q. - > (Y. - X . ) i . e . t h e minimum p o s s i b l e v a l u e of i n f l o w Q. 1,1 — x-1 x x n  is sufficient  t o f i l l up t h e volume; t h e n such a d e c i s i o n i s  always a t t a i n a b l e ,  = Y_^_^ i s always t r u e and Eq.5.1 becomes  Eq. 5.2. (3b): Q. - < (Y. . - X.)< Q. , i . e . t h e d e c i s i o n a i m e d a t l e v e l Y. . x,l x-1 l — i,m i - l may o n l y be a t t a i n a b l e i f i n f l o w exceeds a c e r t a i n v a l u e Q. . , i,k l<k<m  -42An example o f such a case i s shown i n Fig.5.5.  k=4, m = l l , namely t h e random i n f l o w Q. may t a k e any v a l u e Q. ., j =  1,2,  ,11.  I f Q. = Q. ,, then i n s t e a d o f r e a c h i n g t h e aimed l e v e l Y. l i , l i - l  a t the  end o f s t a g e i , i t w i l l be a t s t a t e t . T h e r e f o r e Y. , = X. , = t . i-l i - l I n s h o r t , X^_^ may be any one o f t h e 4 p o s s i b l e s t a t e s , t , u, v , and Y. .• Hence t h e summation c o n s i s t s o f 4 terms: P ( X . , = t/R.<a, X., i-l i - l l — l 1  D ) = P(Q ±  ± ) 1  ), P(X  V :  P  (  p  I f when  =  i - l  X  (  x  = u/R.<a  ± - 1  i - i  =  v  /  R  X., D.) - P C Q ^ ) ,  5  il ' i ' V  = i - i Y  a  /  X  R  i ^ '  V  a  =  p  ^±,3^'  V  = %  a  n  p (  d  Q , i  ) j  g, t h e damage done w i l l be a% o f t h e maximum p o s s i b l e  damage, t h e n P(R.<a / X i> i — l  (3c):  V  =  I  j=l  P(Q  ) '  J  (Y. , - X.) > Q. , i . e . t h e maximum p o s s i b l e v a l u e o f i n f l o w Q.'.is i-l l i,m i x  s t i l l not s u f f i c i e n t  t o f i l l up t h e volume, then t h e d e c i s i o n aimed  -43at  l e v e l Y. , i s n o t a t t a i n a b l e . x-1  T h e r e f o r e P(X. .. = Y. ., / R. <ct, x-1 x-1 x —  X., D.) = 0.0 and no damage w i l l be p o s s i b l e .  Hence P(R.<aIX.,D.) = 1.0.  R e f e r e n c e 15 g i v e s good g r a p h i c a l i l l u s t r a t i o n s o f s i m i l a r  stochastic  processes. 5.5  A digital  computer programme has been w r i t t e n (appendix 1) f o r the  computation o f Eq.5.1 w i t h t h e i n i t i a l c o n d i t i o n t h a t t h e r e s e r v o i r i s empty and the f i n a l c o n d i t i o n t h a t t h e r e s e r v o i r i s f u l l . T y p i c a l f i n a l output i s shown i n F i g . 5 . 6 .  T y p i c a l f i n a l output i s shown i n  F i g . 5 . 6 , p l o t t e d a c c o r d i n g t o o p t i m a l s o l u t i o n s t a b u l a t e d i n T a b l e 5.1. T a b l e 5.1  Percentage of damage a Maxxmum Probabxlxty  Q  Q  Decision (% o f r e s e r v o i r to f i l l )  Q  1 Q  Q  Q  Q  2.0  2 Q  0  Q  0.0  Q  3 Q  Q  Q  0.0  Q  4 Q  Q  Q  2.0  ±  5 Q  Q  Q  5  8.0  6 Q  J  Q  g  34.0  ? 0  A  1  Q  g Q  Q  34.0  1  Q  9 Q  Q  34.0  1  Q  1 Q Q  Q  34.0  1  Q  Q  34.0  34.0  The r e s u l t shows t h a t a c c o r d i n g t o t h e f o r e c a s t f u t u r e i n f l o w s t h e p r o b a b i l i t y of h a v i n g damage n o t g r e a t e r than 30% o f t h e maximum p o s s i b l e damage i s p r a c t i c a l l y zero.  level  However, t h e r e i s 0.1 p r o b a b i l i t y o f h a v i n g damage not  e x c e e d i n g 40%, i f t h e r e s e r v o i r i s o p e r a t e d a c c o r d i n g t o a c e r t a i n p o l i c y , w h i c h i n v o l v e s a i m i n g f o r t h e r e s e r v o i r b e i n g 8% f u l l the v e r y f i r s t day. There i s 0.57 p r o b a b i l i t y of h a v i n g damage n o t e x c e e d i n g 50% o f t h e maximum p o s s i b l e damage, i f t h e r e s e r v o i r i s o p e r a t e d a c c o r d i n g t o a n o t h e r p o l i c y i.e.  a i m i n g f o r t h e r e s e r v o i r b e i n g 34% f u l l t h e v e r y f i r s t day.  This i n  f a c t shows t h a t t h e r e s u l t o f dynamic programming t o determine o p t i m a l r e s e r v o i r  Fig. 5.6  -45o p e r a t i o n p o l i c y by a p r o b a b i l i t y approach which i s o p t i m i z e d .  5.6  I t a l s o p r o v i d e s a b a s i s to e v a l u a t e these p o l i c i e s .  R e c u r s i v e type of damage: The  o f f e r s a s e t of p o l i c i e s , each of  expected v a l u e  approach.  s t o c h a s t i c r e c u r s i v e formula developed  i n s e c t i o n 4.4  is  most r e a d i l y a p p l i c a b l e i n t h i s case to f i n d an o p t i m a l r e s e r v o i r o p e r a t i n g s t r a t e g y such t h a t the t o t a l expected damage f o r the e n t i r e f l o o d season i s minimized. f . ( X . ) = min D  i  = min D  i  {£  P(Q.)*  [R.(Q. X.,D.) + f _ ( X . ^ ) ] } ±  ±  ^ i P(Q )* R ^ C ^ / X . ^ )  {£ Q  f.£  ±  i  X  P(\_ /\,D ) ±  ±  ^ i - i ^ t - l ^  i-1  where f . (X.) i s the minimum expected damage over a l l p e r i o d s s t a r t i n g i i from stage i to the end of the f l o o d season, namely stage 1 R(Q^/X^,D^) i s the damage done at stage i o n l y , i f the random inflow  takes on a p a r t i c u l a r v a l u e and g i v e n t h a t at  the b e g i n n i n g of stage i the water l e v e l i s at a d e c i s i o n D. l P(Q^)  and  i s made  i s the p r o b a b i l i t y of  t a k i n g a p a r t i c u l a r v a l u e , thus  c a u s i n g the c o r r e s p o n d i n g amount of damage P (X^_^/X^ ,D_^)  i s the p r o b a b i l i t y of r e a c h i n g s t a t e ^ _ ^ x  at the  end of stage i , g i v e n t h a t the water l e v e l at b e g i n n i n g of stage i i s at X. and a d e c i s i o n D. i s made. l l A computer programme has a l s o been w r i t t e n (appendix 2) f o r t h i s o p t i m i z a t i o n . Given the f o r e c a s t s , as o u t l i n e d i n Chapter 3, the p o s s i b l e d e c i s i o n s on the very f i r s t table  5-2.  day and the p r o b a b l e r e s u l t s from each d e c i s i o n are shown on  -46T a b l e 5-2 Possible decisions: r e s e r v o i r percentage full  M i n i m i z e d expected damage from 2nd day t o t h e end o f t h e f l o o d season.  Minimized expected damage done i n t h e 1st day.  Total minimized expected damage f o r t h e whole season.  2  135.2  1.2  137.4  4  135.5  0.5  136.0  6  135.9  0.1  136.0  8  136.2  0.0  136.2  10  136.5  0.0  136.5  12  136.9  0.0  136.9  14  137.3  0.0  137.3  16  137.7  0.0  137.7  18  137.9  0.0  137.9  20  138.0  0.0  138.0  22  138.2  0.0  138.2  24  138.4  0.0  138.4  26  138.6  0.0  138.6  28  138.9  0.0  138.9  30  139.1  0.0  139.1  32  139.1  0.0  139.1  34  139.1  0.0  139.1  36  139.1  0.0  139.1  38  139.2  0.0  139.2  40  139.2  0.0  139.2  42  139.3  0.0  139.3  44  137.4  0.0  137.4  46  137.7  0.0  137.7  48  137.9  0.0  137.9  50  138.2  0.0  138.2  52  138.6  138.6  -47F r o m t h e a b o v e t a b l e 5-2 i t c a n b e s e e n t h a t t h e o p t i m a l for  the e n t i r e f l o o d season r e q u i r e s  be t o l e t t h e r e s e r v o i r f i l l expected seasonal  and t h i s w i l l  or the p o s s i b i l i t y when t h e o p t i m a l  Recursive  strategy  day, t h e aim should  g i v e a minimum t o t a l  Although the expected  m e a n i n g f u l , t h e d i s a d v a n t a g e o f f i n d i n g an  s t r a t e g y b a s e d on t h e m i n i m i z a t i o n  no i n d i c a t i o n i s g i v e n  5.7  t o 4% f u l l  damage o f 136 u n i t s o f v a l u e .  approach i s s t a t i s t i c a l l y operating  that f o r the f i r s t  operating  regarding  the risk  of actually achieving  value'  optimal  o f t h e e x p e c t e d damage i s t h a t  involved i n this optimal  strategy,  t h i s minimum t o t a l e x p e c t e d damage,  strategy i s carried out.  t y p e o f damage:  suggested p r o b a b i l i t y approach.  An a l t e r n a t i v e m e t h o d t o d e t e r m i n e t h e o p t i m a l  operating  policy  w i t h a damage f u n c t i o n o f t h e r e c u r s i v e t y p e w o u l d b e t o f o l l o w a p r o b a b i l i t y approach, s i m i l a r  to that described  i n s e c t i o n 5.4 b u t m o d i f y t h e r e c u r s i o n  e q u a t i o n s as f o l l o w s : S t a t e v a r i a b l e X^:  system c o n d i t i o n at the beginning  D e c i s i o n v a r i a b l e D_^: State v a r i a b l e X R e t u r n R_^ a  ^:  of stage i  d e c i s i o n made a t s t a g e i system c o n d i t i o n a t t h e end o f s t a g e i  :  damage s u f f e r e d a t s t a g e i o n l y  :  t o t a l damage s u f f e r e d f r o m s t a g e i t o s t a g e 1, the  end o f t h e f l o o d season  P r o b ( R . + R. . + . . .+R..<ct/X. , D., D. x x-1 1— x x x-1  ,D.) 1  1 ?  a  =  7 P r o b ( R . < g / X . , D.) * P r o b ( R . ,,+R. + . . .+R. <ct-g/X. , D.,...,D_) _fv x— x x x - 1 x-2 1— x x 1 p-U 0  a  =  T  Prob(R.<6/X., D.) *[TY 1  P r o b ( X . . I X . , D . ) * P r o b ( R . -+R. R  „+...+•  <a-B|X _ ,D _ ,...D )] i  1  i  1  1  -48L e t f . ( X . , a ) be t h e optimum p r o b a b i l i t y t h a t t o t a l damages done from s t a g e i till  s t a g e 1 a r e n o t g r e a t e r than a, t h e n  f.(X.,a) = 11  max D D i ' i - l ' " U  U  max D. l  5.8  D  Prob(R.+R. _ +...+R_<a/X.,D.,D. l i - l 1— l l i - l 1  D..) 1  a { I p.(r.<B/x.,d.)*[ I Prob (X. _ /X. ,D. ) * f . _ (X. _ , (ct-g) ) ] } g^O. X. . i i i i i l i l • i - l 1  1  1  1  Summary I n b o t h t h e e x p e c t e d v a l u e and the p r o b a b i l i t y approaches t o f i n d i n g  o p t i m a l r e s e r v o i r o p e r a t i o n p o l i c i e s , t h e o p t i m i z a t i o n s a r e a l l based upon t h e p r o b a b i l i t y d i s t r i b u t i o n s of future r i v e r flows.  I n t h i s s t u d y t h e assumptions  of: (1)  S i n g l e purpose f l o o d c o n t r o l r e s e r v o i r , and  (2)  Unlimited reservoir outlet capacity  were made t o s i m p l i f y t h e a n a l y s i s o f t h e main p r o b l e m , w h i c h was t o develop a p o l i c y f o r d e c i s i o n making under u n c e r t a i n t y .  However, a r e a l r e s e r v o i r  w i t h a l i m i t e d o u t l e t c a p a c i t y would m e r e l y add a c o n s t r a i n t , i n t h a t t h e outflow  from t h e r e s e r v o i r c o u l d not exceed the d i s c h a r g e c a p a c i t y .  It i s  one o f the most a t t r a c t i v e f e a t u r e s of dynamic programming t h a t such c o n s t r a i n t s can be e i t h e r l i n e a r o r n o n - l i n e a r , and they w i l l not l i m i t t h e use o f dynamic programming.  I n f a c t , c o n s t r a i n t s u s u a l l y s h o r t e n t h e computing time  s i n c e they reduce the number o f a v a i l a b l e a l t e r n a t i v e s .  For a multipurpose  r e s e r v o i r , t h e r e t u r n from any s t a g e , R^, w i l l n o t c o n s i s t o f o n l y one term as i n t h i s s t u d y , i . e . f l o o d damage, b u t r a t h e r a composite o f a l l t h e r e t u r n s from i r r i g a t i o n , w a t e r s u p p l y , hydropower,  flood control, etc.  can be r e a d i l y h a n d l e d i n dynamic programming. and f o r m u l a t i o n s  But a g a i n t h i s  I n summary, the b a s i c c o n c e p t s  d e v e l o p e d i n t h i s s t u d y c o u l d be r e a d i l y adapted t o m u l t i p l e  -49purpose r e s e r v o i r w i t h l i m i t e d o u t l e t A r e s e r v o i r d a i l y operating period"  The  as the  input  constantly most up  advantage of u s i n g to the  revised  from May  the 16th  to the  the f l o w on May  end  16th  the  To  clarify  of the  16th,  p o l i c y can  a c t u a l flows which o c c u r and  forecast  l e t us  imagine that  system p r e d i c t s  the  the  d a i l y flows  with t h i s information  i s determined.  The  as a r e s u l t of t h i s f l o w and  an  reservoir  itself  by  When assuming  the d e c i s i o n executed  change from b e i n g empty to a new  be  on  a c c o r d i n g to t h i s o p t i m a l p o l i c y .  state  on  (either  information  a v a i l a b l e to  the  system namely the a c t u a l v a l u e of the  f l o w on May  16th  of  random flow on flows from May  that 17th  day,  the  to the  end  be more a c c u r a t e s i n c e more i n f o r m a t i o n With the new  s e t of p r e d i c t e d  a c t u a l water l e v e l i n the determine a new can be  system  From the new  originally of f u t u r e  operation  a random v a r i a b l e o r i g i n a l l y , r e v e a l s  reservoir w i l l  the  d e f i n i t e sequence of r i v e r  the  this point,  r e s e r v o i r operation  remain empty, or p a r t l y f i l l e d ) . forecast  i s that  f l o o d season, and  i s c a r r i e d out  a p a r t i c u l a r v a l u e and 16th,  analysis  "critical  model and  output from a d a i l y flow f o r e c a s t  r e s e r v o i r i s empty, the  on May  f o r that  to take i n t o account the  o p t i m a l p o l i c y f o r the operation  the  optimization  to date f o r e c a s t s .  15th  i t i s a deterministic  p o l i c y so formed i s o p t i m a l only  flows.  May  p o l i c y based on a n a l y s i s of the  of r e c o r d s i s inadequate, s i n c e  operation  May  capacity.  v a l u e s and  the v e r y end  r e s e r v o i r d a i l y operating a v a i l a b l e on each  day.  strategy  of the  system can  of the p e r i o d .  the new 16th,  strategy.  give  This  i s a v a i l a b l e to the  r e s e r v o i r on May  optimal d a i l y operation  repeated t i l l  forecast  instead  the  predictions  prediction  forecast  i n i t i a l condition,  should  system. i.e.  dynamic programming can  the again  L i k e w i s e the whole p r o c e s s  f l o o d season.  In t h i s way,  makes the b e s t p o s s i b l e use  of the  the informati  -50From t h e above a n a l y s i s , i t i s seen t h a t dynamic programming as an optimization  t e c h n i q u e i s e x t r e m e l y p o w e r f u l and r e a d i l y a p p l i c a b l e  Resources systems.  However, m e a n i n g f u l r e s u l t s from any m a t h e m a t i c a l method  b a s i c a l l y r e l y upon t h e v a l i d i t y o f i n p u t d a t e . model and r e a l i s t i c u t i l i t y f u n c t i o n s etc. are required  t o Water  Thus a r e a l i s t i c  forecast  such as f l o o d damage, i r r i g a t i o n b e n e f i t ,  b e f o r e a dynamic programming approach such as those developed  i n t h i s s t u d y can be v e r y u s e f u l i n t h e d e t e r m i n a t i o n o f r e s e r v o i r  operating  policies.  Conclusions Two d i f f e r e n t approaches t o d e t e r m i n i n g the o p t i m a l d a i l y o p e r a t i n g strategy  f o r a r e s e r v o i r have been developed i n t h i s s t u d y , and a p p l i e d  t o the  case o f o p e r a t i n g a s i n g l e purpose f l o o d c o n t r o l r e s e r v o i r when t h e i n f l o w s can be f o r e c a s t but  w i t h some a c c u r a c y .  one i n v o l v e s  Both approaches use dynamic programming,  the m i n i m i z a t i o n o f t h e expected v a l u e o f t h e " r e t u r n " i n  t h i s case f l o o d damage, and t h e o t h e r i n v o l v e s occurrence of a desirable  maximizing the p r o b a b i l i t y of  e v e n t , namely r e s t r i c t i n g t h e i n e v i t a b l e damage t o  a c e r t a i n amount. The  p r o b a b i l i t y approach o f f e r s a s e t of- a l t e r n a t i v e o p t i m a l p o l i c i e s ,  each one based on m a x i m i z i n g t h e p r o b a b i l i t y o f r e s t r i c t i n g t h e damage t o a certain selected  f i g u r e , and i t a l s o i n d i c a t e s  a c h i e v e each o f the s e l e c t e d evaluating  objectives  the a l t e r n a t i v e p o l i c i e s .  the p r o b a b i l i t y of being able t o  which give a b a s i s  f o r comparing and  Thus i t g i v e s more i n f o r m a t i o n t o t h e  d e c i s i o n maker than t h e e x p e c t e d v a l u e approach, which s i m p l y o f f e r s one p o l i c y to m i n i m i z e t h e expected v a l u e o f damage b u t g i v e s no i n d i c a t i o n o f t h e r i s k s involved.  Given a d a i l y forecast  system, and a r e a l i s t i c u t i l i t y f u n c t i o n f o r  f l o o d damage, an approach such as one o f those developed i n t h i s study has t h e  -51advantage o v e r the t r a d i t i o n a l r u l e curve approach (which i s u s u a l l y based on the c r i t i c a l p e r i o d of r e c o r d ) , t h a t the p o l i c y so formed i s o p t i m a l the i n f o r m a t i o n a v a i l a b l e t o t h a t t i m e , and new  i n f o r m a t i o n becomes a v a i l a b l e .  given  can be c o n t i n u o u s l y r e v i s e d as  -52References (1)  R. B e l l m a n , "Dynamic Programming", 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 , New J e r s e y , 1957.  (2)  R. Bellman and S.E. D r e y f u s , " A p p l i e d Dynamic Programming", 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 , New J e r s e y , 1962.  (3)  W.A. H a l l and N. Buras, "The Dynamic Programming Approach t o Water Resources Development", J o u r n a l o f G e o p h y s i c a l .  Y  y  Research Vol.66, No.2, Feb.1961, pp.517-520.  (4)  W.A. H a l l and D.T. H o w e l l , " O p t i m i z a t i o n o f S i n g l e - p u r p o s e R e s e r v o i r D e s i g n w i t h t h e A p p l i c a t i o n o f Dynamic Programming t o S y n t h e t i c Hydrology Sample", J o u r n a l o f H y d r o l o g y 1,  1063,  pp.355-363.  (5)  W.A. H a l l , " O p t i m a l D e s i g n o f M u l t i - p u r p o s e R e s e r v o i r " , J o u r n a l o f H y d r a u l i c D i v i s i o n , A.S.C.E., 90(HY4), 1964, pp.141-149.  (6)  W.A. H a l l and T.G. R o e f s , "Hydropower P r o j e c t Output O p t i m i z a t i o n " , J o u r n a l o f Power D i v i s i o n , A.S.C.E., J a n u a r y 1966, pp.67-79.  (7)  W.L. M e i e r , J r . and C.S. B e i g h t l e r , "An O p t i m i z a t i o n Method f o r B r a n c h i n g M u l t i s t a g e Water Resources.System", Water Resources Research V o l . 3 , No.3, t h i r d q u a r t e r 1967,  pp.645-652.  (8)  W.A. H a l l , W.S. B u t c h e r , and A. Esogbue, " O p t i m i z a t i o n o f t h e O p e r a t i o n of a M u l t i - p u r p o s e R e s e r v o i r by Dynamic Programming", Water Resources R e s e a r c h , V o l . 4 , No.3, June 1968,  pp.471-477.  (9)  N.R. Draper and H. Smith, " A p p l i e d R e g r e s s i o n New Y o r k , 1966.  A n a l y s i s " , John W i l e y ,  (10)  H.J. L a r s o n ,  " I n t r o d u c t i o n t o P r o b a b i l i t y Theory and S t a t i s t i c a l I n f e r e n c e " , John W i l e y , New Y o r k , 1969.  (11)  A.M. Mood and F.A. G r a y b i l l , " I n t r o d u c t i o n t o t h e Theory o f S t a t i s t i c s " , 2nd e d i t i o n , M c G r a w - H i l l , 1963.  (12)  J . F . M u i r , " F r a s e r R i v e r F l o o d Flow F o r e c a s t i n g " , Department o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Columbia, Water Resources D i v i s i o n P u b l i c a t i o n No.4, J u l y 1969.  (13)  G.L. Nemhauser, " I n t r o d u c t i o n t o Dynamic Programming", John W i l e y and Sons, 1966.  (14)  L.G. M i t t e n , " C o m p o s i t i o n P r i n c i p l e s f o r S y n t h e s i s o f O p t i m a l M u l t i s t a g e P r o c e s s " , O p e r a t i o n s R e s e a r c h , 12, 1964, pp.610-619.  -53R e f e r e n c e s - cont:  (15)  A. Kauffmann, "Graphs, Dynamic Programming, and F i n i t e Games", t r a n s l a t e d by H.C. Sneyd, Academic P r e s s , New Y o r k , 1967.  (16)  M. Abramowitz and I.A. Stegun ( e d i t e d ) , "Handbook o f M a t h e m a t i c a l F u n c t i o n s w i t h F o r m u l a s , Graphs, and M a t h e m a t i c a l T a b l e s " , Dover, 1965.  - 54 -  APPENDIX 1  COMPUTER PROGRAMME FOR THE PROBABILISTIC APPROACH TO NON-RECURSIVE TYPE OF FLOOD DAMAGE  A A, A AA AX A A X X X X X X X X X X XX X X X A A X A A A AAAA AAAA A A A A A A A  RFS  NO.  017744  UNIVERSITY  OF  B C COMPUTING  CENTRE  MTS (AN 120)  i  P L E A S E RETURN TG COMPUTING CENTRE $ S I G TSOU * * L A S T SIGNON WAS: 1 2 ; 5 Q : 3 3 08-14-70 USER "TSOU" SIGNED ON AT 1 2 : 5 0 : 4 1 ON 0 8 - 1 4 - 7 0  $LIST  STAFF * * * * * * * * * * * * * * * * * * * * ;  .  TAXIlt155)  _ _ 1 . ^ _ ^ ^ _ ^ _ „ D J ^ ^ 2 COMMON Q M I D ( 6 0 ) , P ( 6 0 ) , F ( 4 3 , 5 1 , l l ) 3 INTEGER D, Qt DO, QMAX,. QM ID 4 R E A D ( 5 , 1 0 1 ) QP 5 101 FORMAT(13F6.0) 6 READ(5,102) R .7 L02„F0.Ri1.AJJ_1.3.rL6„.3J . _ 7.1 111 = 0 8 DO 1 I = 1, 42 9 DO 1 J = I t 51 10 DO 1 K = I t 11 11 1 D ( I t J t K) = 0 JL2. DO 2 I = I t 43 13 DO 2 J = 1, 51 14 DO 2 K = I t 11 15 2 F 1 I , J , K) = -1.0 15.1 A = -1.0 16 00 3 K - 1, 11 .1.7 , 3._FJ_4 3.t. 5.1_tJl.L_= ...1..JQL 18 DO 18 1 = 1 , 5 1 19 18 I S T A T E ( I) = I 20 L = 41 21 DO 4 I = 1, 42 22 I I = 1. + I 23 SFGMA •= 52.0 * S QRT ( 1.0 - R 1 I I ) * * 2 ) 24 C A L L F L O W ( Q P ( I I ) ,SEGMA,IMAX) 25 DO 6 INT = 1,- 51 26 DO 7 I P '= I t 11 27 QMAX = 2 0 0 + 30 * ( I P - 1 ) 28 DO 8 INTPRE =1,51 _2_9 I F ( F ( I I + l . I N T P R E . I P ) . EQ. A > GO TO 8 30 I F ( INTPRE - I N T ) 9, 10, 11 31 9 DQ = ( INT - INTPRE ) * 10 32 Q = DQ + QM ID ( 1 ) 33 I F ( 0 .GT. QMAX ) GO TO 12 34 14 SUM = 0.0 35. __DU__L5 IQ = 1, I MAX 35 .1 Q = DQ + QM ID( IQ ) 35.2 I F ( 0 .GT. QMAX ..AND. IP .NE. 11 ) GO TO 16 35.3 15 SUM = SUM + P( IQ) 35.4 16 FTEMP = SUM * F( I 1 +1, I N T P R E , I P ) 36 GO TO 13 ...3.7 ..^ ^ ..12...EIE.MP. =.. .0...0 , 38 GO TO 13 39 10 I F ( Q M I D ( l ) .GT. QMAX ) GO TO 12 40 DQ = 0 ; 41 GO TO 14 45 11 DQ = ( INT - I N T P R E ) * 10 Jt5_.J 100 = ( I N T P R E - INT ) * 10 46 I F ( Q M I D U M A X ) . LT . IDQ ) GOTO 7 55 v CALL FLOODtDQ,QMAX,IMAX,IP,FTEMP,INT,INTPRE,II ) 56 13 I F ( FTEMP . L T . F ( I I , INT-t IP > ) GO TO 8  .  AAAAAA;  55  -  F{I I , I N T , I P ) = FTEMP D i l l , I N T , I P ) = INTPRE 8 CONTINUE 7 CONTINUE IF < I I . E Q . 1 ) GO TO 6 CONTINUE WRITE(-6 201) II 201 FORMAT{1 H I , ' S T A G E = • » 1 2 )  57 58 59 60 61 62 63 64  1__6.5 65.1 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 &2_ . 83. . 84 85 86 86.1 87 88 88.1 89 90 91 92 93 94 95 96 97 98 99 . 99.1 100 102 103 _IQ3_._1 104 105 106 107 108 108.1 108.2 108 . 3 108.31,  17  f  :  DjD__5_lJl_^^_JJ_r_2 ^_ _J = : IIP = IP - 1 WRITE(6 ,20 2) I IP, ( 1 S T A T E ( J ) , J = 1 » 1 7 ) , ( D ( I I , J , IP) ,J = 1,17) , It F t I I , J , I P ) , J = 1 , 1 7 ) ; 2 0 2 F O R M A T ( I X , 1 2 , *0% D A M . • , 3 X , • S T A T E • , 1 7 1 6 , / , 1 0 X , D E C I S I O N ' 1 7 ( 4 X , 1 2 ) , 1/,10X,'.0PT.PR0.',17F6.2,/) i • . WRITE(6,203) (I S T A T E ( J ) , J = 1 8 , 3 4 ) , ( D ( I I , J , I P ) , J = 1 8 , 3 4 ) , 1 ( F t I I, J , I P ) » J = 1 8 * 3 4 ) 2 03 F O R M A T ( 1 3 X , S T A T E ' , 1 7 I 6 , / , 1 0 X , ' D E C I S I O N ' , 1 7 ( 4 X , I 2 ) , / , 1 0 X , ' O P T . P R O . I',17F6.2,/) ; " 5 W R I T E ( 6 , 2 0 3 ) t I S T A T Et U ) , J = 3 5 , 5 1 ) , ( D ( I I , J , I P ) , J = 3 5 , 5 1 ) , 1(F(II,J,IP) ,J=35,51) ' ' 4 L = L - 2 : , __ 1 7 W R I T E ( 6 , 2 0 4 ) I I I, ( I ST AT E ( J ) , J-= 1 , 10 ) , ( D < 1 , 1 , K ) , K= 1 , 1 1 ) , 1(F{1,1,Kl,K=1,11) 2 0 4 F O R M A T t 1 H i , / / / , ' S T A GE=1 * , 5 X , ' I N I T I A L C O N D I T I O N : RESERVOIR E M P T Y ' , 5 I X , ' F I N A L CONDITION: RESERVOIR F U L L ' » / / / / » 1 2 X , ' P E R C E N T A G E DAMAGE',1 2 1 C 5 X , 1 2 , » 0 ' ) , / / / / , 2 1 X , ' D E C I S I G N ' , 1 1 < 6 X , 1 2 ) , / / / / , 1 0 X , • M A X I M U M PROBA 3BILITY' .11F8.2) . _____ ___ ___ . . STOP . END S U B R O U T I N E FLOOD ( D Q , Q M A X , I M A X , I P , F T E M P , I N T , I N T P R E , I I ) COMMON Q M I D ( 6 0 ) , P ( 6 0 ) , F ( 4 3 , 5 1 , l l ) INTEGER DQ, QMAX, QMID, Q SUM - 0 . 0 ; . SUMPRE = 0 . 0 A = 1.0 Q = DQ + QM I D ( 1 ) . ; I F ( Q - O ) 1 , 2 , 3 1 DO 4 IQ = 1 , I M A X Q = DQ + Q M I D ( I Q ) . . . j ; . . IF ( Q . G T . 0 ) GO TO 5 , J = INT + Q M I D U Q ) / 10 IF ( Ft I l +l t J , I P ) .EQ. A ) 6 0 TO 12 SUMPRE = SUMPRE + P ( I Q ) * F(II+1,J,IP) 1 2 SUM = SUM + PtIQ) GO TO 4 ; . . 5 I F I Q . G T . QMAX . A N D . IP . N E . 11 ) GO TO 6 SUMPRE = SUMPRE + P ( I Q ) * F ( 1 1 + 1 , I N T P R E , IP ) SUM = SUM + P t I Q ) 4 CONTINUE 6 F T E M P = SUM * S U M P R E IF ( II . E Q . 4 2 ) F T E M P = SUM GO TO 7 2 DO 8 IQ = 1 , I M A X 0 = DQ •+ QM I D( IQ ) _ ^ I F ( Q . . G T . QMAX .AND. IP . N E . 11 ) GO' TO 9 8 SUM = SUM + P t I Q ) 9 F T E M P = S U M * F ( I 1+ 1 , I N T P R E , I P ) ; , GO TO 7 3 . 0 = DO • + QM I D t 1 ) I F < Q . G T . QMAX .AND. I P . N E . II ) GO TO 1 3 ; 1  :  1  :  :  ;  :  w • "  r  - -  108 .32 108.4 108.5 108 .6 r 108.61 . 108.62 108.63 108.7 1OR.8 108.9 109  no -  111 112 1 13 114 115 116 116.1 117 11 8 -' 119 120 120.1 121 122 1 2 7 . 1  122.2 122.3 123 125 126 1 77 128 129 130 131 132  j 1  -j  -,'  i  '13 3  134 135 136 137 138 1  39  140 141 142 143 144 145 146 147 148 149 150 151  I  152 153 154  DO 10 IQ = 1, IMAX 57 Q - OQ + QtM I D ( I Q ) IF ( Q .GT. QMAX .ANO. IP .NE. 11 ) GO TO 11 10 SUM = SUM + P ( IQ) GO TO l l 13 FTEMP = 0.0 GO.TO 7 , 1 1 FTEMP . = SUM * F ( I I + l , I N T P R E , I P ) 7 RETURN END FUNCTION HAST(T) X = 1'iO-V'TT.O + 0.3326*7 * T ) HAST = 1.0 - 0.39894 * EXP{-0.5*T**21 * { 0.43618*X - 0. 12016* X**2 1+ 0. 9'3729*X**3.) RETURN END SUBROUTINE FLOW(QP,SEGMA,I MAX) COMMON QMID(60), P ( 6 0 ) , F ( 4 3 , 5 1 , l l ) i INTEGER QMID, QTEMP PTOTAL = 0.0 DO 1 I = l i 30 A. I =. I QMID(I) = QP + 10.0 * ( AI - 1.0 ) QDUMMY ,= QMID(I) TU = ( QDUMMY + 5.0 - QP ) / SEGMA TL = ( QDUMMY - 5.0 - QP ) / SEGMA IF ( I .NE. 1 ) GO TO 6 P ( I ) = { HA ST{TU) - 0.5 ) * 2.0 GO TO 7 6 P<I) = HA ST { TU) - HAST(TL) PTOTAL = PT QT AL + 2.0 *. P ( I ) t GO TO 8 7 PTOTAL = PTOTAL + P ( 1 ) 8 IF { PTOTAL .LT. 0.99 ) GO TO 1 . QMI D ( I + l ) = QMIDCI) + 10 P(I+1> = 0.5 * .( 1.0 - PTOTAL ) JJ = 2 * ( I * 1 ) - 1 J = JJ / 2 + 1 PTFMP = P ( I + l ) QTEMP = QMID(I + 1) GO TO 2 4 CONTINUE 2 K = 1 L =J 4 P { L ) = P ( K) QMID(L) = QMID(K) IF { L .EQ. J J - 1 ) GO TO 3 K •= K + 1 L = L + 1 GO TO 4 3 N = J - 1 L = L + 1 P(L )• '= PTEMP QMID(L) = QTEMP DO 5 M = 1 , N QMID(M) = 2 * Q M I D U ) - QMID(L) P(M) = P ( L ) 5 L = L - 1 I MAX = J J RETURN >>  155 END OF  >  ii  $SIG  END FILE  58  - 59 -  APPENDIX 2  COMPUTER PROGRAMME FOR THE EXPECTED VALUE APPROACH TO RECURSIVE TYPE OF FLOOD DAMAGE  RFS  NO.  017751  UNIVERSITY  OF  B C COMPUTING  CENTRE  MTS(AM120)  * * * * * * * * * * * * * * * * * * * * P t - C A S C R C T U R N TO C O M P U T I N G C E N T R E — S - T A F F * * * * * * * * * * * * * * * * * * * * $ S I G TSOU **1AST SIGNON WAS: 1 6 : 2 6 : 3 5 08-13-70 U S E R " T S O U " S I G N E D ON A T 1 2 : 4 9 : 3 9 ON 0 8 - 1 4 - 7 0 $LIST S T 0 i l , 2 3 0 ) 1 D T M F N S I O N QMI D t 6 0 ) . P ( 6 0 ) , QP ( 4 2 ) , R ( 4 2 ) , F ( 4 3 . 5 1 ) , D ( 4 2 1 5 1 ) 2 1 , N O ( 4 2 , 5 1 ) , I STA TE ( .5 1) 3 INTEGER D 4 QO = 2 0 0 . 0 5 QU = 4 0 0 . 0 6 DOOM = 3 9 . 0 _7___ DJD_3J> L_=_1J_£2: 8 DO 3 0 J = 1 , 51 9 D U , J) = 0 10 3 0 NO ( I t J ) = 0 10.02 6 0 2 F O R M A T ( I X , • 1ST AT E= * , I 2 ) 11 0 0 31 1 = 1 , 51 .12 3 1 I ST AT E H ) = I 13 R E A D ( 5 , 1 0 1 ) QP 14 101 FORMAT ( 1 3 F 6 . 0 ) 15 R E A D ( 5 , 10 2 ) R 16 1 0 2 FORMAT I 1 3 F 6 . 3 ) 17 DO 10 I = 1, 43 ..L8 D,0.._1.0 J__-=_l.t.._§_l, 19 10 F ( I , J) = 1 0 . 0 * * 1.0 20 F(43,51) = 0.0 21 L = 41 22 . . DO 6 I = 1 , 42 23 11= I _24 IF ( I I . N E . 1) GO TO 2 8 25 WRITE(6,201) II 26 2 0 1 FORMAT( I X , ' S T A G E = 1 2 , / , 1 G X , ' P O S . D E C . , 5 X , « S T A G E * 2 OPT. •, 5 X , ' E X P . R 1  27  28 29  1 E T . ' , 5 X ,* E X P . T A T A L " )  ;  28 SEGMA= 5 2 . 0 * S Q R T I 1 . 0 RUI)**2) CALL FLOWIQP(I I ) , S E G M A , P , Q M I D , I M A X ) _3JO QD_U J-NJ = L, 51 31 DO 12 INTPRE, = 1 , 51 32 IF ( F { I I + l , I N T P R E ) .EQ. 10.0**10 ) GO TO 12 33 SUM = 0 . 0 1_ 34 SP = 0 * 0 35 DUMMY = 0 . 0 3.6 TF { T N T P R F - I NT ) 13_,_Jj4 _L5 37 13 P T E M P = 1 . 0 38 DQ = ( I N T I N T P R E ) * 10 39 . I F ( DQ . G T . QO .AND. DQ . L T . QU ) GO TO 1 6 40 I F ( DQ . L E . QO ) GO TO 3 6 41 D E F R E T = DOOM ^2-_________. Ga_jm_3.5__^^ 43 3 6 DO 17 IQ = 1 , I MAX, 44 Q = DQ + Q M I D ( I Q ) 45 I F ( Q . L E . QO ) G O TO 1 7 46 I F ( Q . G T . QO .AND. Q . L T . QU ) GO TO 2 4 47 SUM = SUM + P ( I Q ) * DOOM A3 GO TO 1 7 ! 49 24 CALL FLOOD{Q,CAM) 50 SUM = SUM + D A M * P U Q ) 51 17 C O N T I N U E :  J  6  0  c s  /  !  1  \  •  ]  1  1 •  1 1 •  II  —1  —i  1  —1  •  -  52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98. 99 100 101 102 103 104 105 106 107 108 109 110 111  F T E M P = SUM + F ( I I + l , I N T P R E ) D T E M P = SUM IF ( II .NE. 1 ) GO TO 18 W R I T E ( 6 , 2 0 4 ) I N T P R E , F { 2 , I NT P R E ) , D T E M P , F T E M P 204 F O R M A T ( 1 0 X , 1 8 , 5 X , F 1 2 . 1 , 5 X , F 8 . 1 , 5 X f F 9 . 1 ) GO TO 1 8 16 C A L L F L O O D { D Q , DEFRET) 3 5 DO 1 9 . IQ = 1 , IMAX  IF  32 19  14.  33 20 .  15  22  34  23  21  (  OM10(10)  .LE.  CO )  GO TO  61  1  19  IF ( QMID(IQ) .LT. QU ) GO TO 3 2 SUM = SUM + P ( I Q ) * DOOM -. GO TO 1 9 • ' ' ' CALL F L O O D l ' Q M I D U Q ) ,DAM) . SUM = SUM + . D A M * P(IQ) CONTINUE D T E M P - SUM + D E F R E T F T E M P = DTEMP + F( I I + l , I N T P R E ) IF (II .NE. 1 ) GO TO 18 WRITE(6,204) INTPRE , F ( 2 , I N T P R E ) ,DTEMP,FTEMP GO TO 1 8 PTEMP = 1 . 0 DO 2 0 IQ = 1 , IMAX IF ( QMID(IQ) . L E . QO ) GO TO 2 0 IF ( QMID(IQ) .LT. QU ) GO T O ' 3 3 SUM = SUM + P ( I Q ) * DOOM GO TO 2 0 CALL FLOOD!QMID<IQ),DAM) S U M = S U M + DAM * P(IQ) CONTINUE ' F T E M P = SUM + F ( I I + l , I N T P R E ) D T E M P = SUM 'IF {' I I .NE. 1 ) GO TO 18 WRITE(6,204) INTPRE,F(2,INTPRE),DTEMP,FTEMP GO TO 1 8 DQ = ( I N T P R E INT ) * 10 IF ( Q M I D ( I M A X ) .LT. DQ ) GO TO 2 5 DO 2 1 IQ'= 1, IMAX Q = Q M I D ( I Q ) - DQ IF ( Q . G E . 0 . 0 ) GO TO 2 2 LUCK = INTPRE + ( Q / 1 0 . 0 ) I F ( F ( 11 + I V L U C K ) . E Q . 10.0**10 ) GO TO 2 1 SUM= S U M + F ( I I + 1 , L U C K ) * P (.1Q ) GO TO 2 1 SP = S P + P(IQ) IF ( Q . G T . QO .AND. Q . L T . QU ) GO TO 2 3 I F ( Q . G E . QU ) GO TO 3 4 SUM = SUM + P ( I Q ) * F( I I + l , I N T P R E ) GO TO 2 1 SUM = SUM + P ( I Q ) * (DOOM + F ( I I + 1 , I N T P R E ) ) DUMMY = DUMMY + DOOM * . P M Q ' ) GO TO 2 1 CALL FLOOD(Q,DAM) SUM= S U M + P ( I Q ) * ( DAM + F ( I I + 1 , I N T P R E ) ) DUMMY = DUMMY + DAM * P(IQ) CONTINUE P T E M P = SP F T E M P - SUM D T E M P = DUMMY IF J II .NE. 1 ) GO TO 1 8 WRITE<6, 204) INTPRE ,F( 2 , INTPRE) ,DTEMP,FTEMP  -  112 113 114 115 116 1 17 118 119 1 20 121 122 ''123 124 125 1 26 127 128 129 130 131 13? 133 134 135 136  18 26  27 12 25 11  IF ( FTEMP - F i l l , I N T F(I I,INT ) = FTEMP D( I I , INT ) = INTPRE PDEC - PTEMP D A M EX P = D T E M P N O ( 1.1 , I N T ) GO TO 12 N0( I I , INT)  )  =  1  =  NO ( I I , I N T )  CONTINUE IF ( I I .EQ.  1  CONTINUE " WRITE ( 6 ,202 )  )  GO  TO  )  26,  +  27,  12  62  1  29  I I , ( I S T A T E ( J ) , J = l ,2 5 ) , ( D ( I I , J ) , J = l , 2 5 ) , ( N 0 ( I I , J ) V  1J=1,25) F O R M A T ! I X , ' S T A G E = ' , 1 2 , / , 1 0 X , * S T A T E ' , 3 X , 2 5 I 4 , / , 1 0 X , ' D I C I SI 202 1/.10X.'CHOICE',2.X,25I4) W R I T E ( 6, 2 0 6 ) 206  ON*,2514,  ( IS T A T E ( J ) , J = 2 6 , 5 1 ) , { D ( 1 1 , J ) , J=2 6,51) ,(NO(11» J)»J = 26 ,  151). FORMAT( 10X,»STATE'  ,3X , 2 6 1 4 , / , 1 O X , • D l C I S I O N ' ,26  14,/,10X,•CHOICE',2X  1,2614) 6 29  L = L - 2 K = ( 0 ( 1 , 1 ) 1 ) * 2 W R I T E ( 6 , 2 0 5 ) ( QM I D ( I ) , 1= 1 , I M A X ) , ( P ( I ) , I = 1 , I M A X ) , K ,F ( 1 , 1 ) 2 0 5 F O R M A T ( / / , ' I N F L O W D I S T R I BUT I O N ' , / , 2 G X , 9 F 1 0 . 1 , / , 2 0 X , 9 F 1 0 . 3 , / / , 1 0 X , 1 ' THE B E S T S T R A T E G Y I S TO A L L O W RE S E R V O I R ' , 1 3 , ' ! . F U L L . T H I S GIVES 2THE  MINIMUM  EXPECTED  DAMAGE  0F',F10.1)  137 1 38  STOP END  139  FUNCTION HAST{T) X = 1. 0 / '( 1.0 + 0.33267 * T ) H A S T = 1.0 - 0.39894 * EXP(-0.5*T*#2)  140 141  1+  142  RETURN END SUBROUTINE  146 147  IQ GO  0.12016*X**2  FLOOD(Q,OAM) v  170) ,  IQ  1.0  51  151  52  152 153 154  DAM GO  53  DAM = GO T O  3.3 9  155 1 56  54  DAM = GO TO  4.6 9  157 158  55  159  56  160 161 1 62  57  DAM = GO TO  .  9  = 2.1 TO"9  = 6 . 0  GO TO 9 DAM = 7.5 GO T O 9 DAM = 9 . 1 GO T O 9 DAM =10.8 GO TO 9 DAM = 12.6  163 164 165 166  58  167 168 169  60 61  170 171  DAM = GO TO  16.5 9  62  DAM  18.6  59  -  = { Q - 200.0 ) / 10.0 • . TO (51,52,53,54,55,56,57,5 8,59,60,61,62,63,64,6 5,66,67,68,69,  149 1 50  DAM  (0. 43618*X  0.93729*X**3)  143 144 145  148  *  GO TO 9 DAM = 1 4 . 5 GO T O 9  =  .  .. •  '  •  ''  172 17 3 174 175 176 177 178 179 1 80 181 1 82 183 184 185 186 187 188 189 190 191 19? 193 194 195 196 197  63 64 65 66 67 -  68  . 69 70 9  198  199 200 201 202 203 2 04 205 206 2 07 208 209 210 211 212 213 214 215 216 217 218 219 2 20 221 2 72 223 224 225 226 227 728 229 230 END OF  7 8  •'  1 2 4  3  5  FILE  GO TO 9 DAM = 2 0 . 8 GO TO 9 DAM = 2 3 . 1 GO TO 9 DAM = 2 5 . 5 GO TO 9 DAM = 2 8 . 0 GO TO 9 DAM = 3 0 . 6 GO TO 9 DAM = 3 3 . 3 . GO TO 9 DAM = 3 6 . 1 GO TO 9 DAM = 3 9 . 0 RETURN END SUBROUTINE F L O W ( Q P , S E G M A , P , Q M I D , I M A X ) D I M E N S I O N Q M I D ( 6 0 ) , P<60> PTOTAL = 0 . 0 DO 1 I = 1 , 30 AI = I Q M I D ( I ) = QP + 1 0 . 0 * ( AI 1.0 ) TU = < Q M I D U ) + 5 . 0 - QP ) / SEGMA TL = ( Q M I D U ) - 5 . 0 - QP ) / S E G M A P U ) = HAST (TU) HAST(TL) IF ( I . E Q . 1 ) GO TO 7 PTOTAL = PTOTAL + 2 . 0 * P U ) GO TO 8 PTOTAL = PTOTAL + P ( 1 ) I F ( P T O T A L . L T ' . 0 . 9 9 )' GO TO 1 QMID( I + l ) = Q MID( I) + 10.0 PI 1 + 1 1 = 0 . 5 * ( 1 . 0 - P T O T A L ) J J = 2 * ( I + 1 ) - 1 J = JJ / 2 + 1 PTEMP = P U + 1 ) QTEMP = Q M I D ( I + l ) GO TO 2 CONTINUE K = 1 L = J P(L ) = P(K) QMID(L) = QMID(K) i IF ( L . E Q . J J - 1 ) GO TO 3 K = K + 1 L = L + 1 GO TO 4 N = J -r 1 L = L + 1 P ( L ) = PTEMP Q M I D I L ) = QTEMP DO-5 M = 1, N QMID(M) = 2 . 0 * QMID(J) QMID(L) P(M) = P ( L ) L = L - 1 IMAX = J J RETURN END  63  ........  •  

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