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Optimal operation of a system of flood control reservoirs Flavell, David Richard 1974

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OPTIMAL OPERATION OF A SYSTEM OF FLOOD CONTROL RESERVOIRS By' DAVID RICHARD FLAVELL B.A.Sc, The U n i v e r s i t y o f B r i t i s h Columbia, 19 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t'h'e Department of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard.. THE UNIVERSITY OF BRITISH COLUMBIA November, 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s 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 C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f C i v i l E ngineering The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date September 25, 1974 A B S T R A C T This t h e s i s d e s c r i b e s a study of the a p p l i c a t i o n o f l i n e a r programming, a mathematical o p t i m i z a t i o n t e c h n i q u e , t o the problem of o p e r a t i n g a system of f l o o d c o n t r o l r e s e r v o i r s i n the most e f f i c i e n t way f o r maximum f l o o d peak re d u c t i o n . The r e s e r v o i r system s t u d i e d was one which i s proposed f o r f l o o d c o n t r o l i n the F r a s e r River Basin and f o r which p r e l i m i n a r y designs have been made. The proposed s i t e s are i n the headwater areas o f the r i v e r b a s i n , remote from the area to be p r o t e c t e d . A channel r o u t i n g method which was l i n e a r was r e q u i r e d f o r use with the l i n e a r programming technique. A method was developed which was based on monoclinal wave theory. Several d i f f e r e n t cases were s t u d i e d i n which the r e s e r v o i r c a p a c i t i e s and combinations of r e s e r v o i r s i n c l u d e d i n the system were v a r i e d . The r e s u l t s show the maximum r e g u l a t i o n which i s a v a i l a b l e from each combination of r e s e r v o i r s and i d e n t i f y the minimum storage c a p a c i t i e s r e q u i r e d at each s i t e . A p p l i c a t i o n of the technique i n p l a n n i n g s t u d i e s and i n r e a l - t i m e r e s e r v o i r operation i s d i s c u s s e d . i i TABLE OF CONTENTS Page LIST OF TABLES iv LIST OF FIGURES . . . . . . . . . . v Chapter I. INTRODUCTION 1 12. THE FLOODING PROBLEM . . 4 1x3, LINEAR PROGRAMMING . . . . . . . . 8 I4l. MODELLING THE SYSTEM 13 f. ROUTING EQUATIONS 18 Vt. RESULTS . . 27 VIZ. CONCLUSION . . . 37 BIBLIOGRAPHY 3 9 APPENDICES A. ROUTING EQUATIONS - CHANNEL STORAGE 403 B. REGULATION OF DESIGN FLOOD 42' i i i LIST OF TABLES Table Page 5.1 LINEAR ROUTING EQUATIONS 24 6.1 SUMMARY OF RESULTS: DESIGN FLOOD ..... 31 6.2 FLOOD PEAK. REDUCTIONS 34 i v LIST OF FIGURES Figure Page 2.1 SYSTEM E - LOCATION OF PROPOSED PROJECTS 6 3.1 GRAPHICAL SOLUTION OF LINEAR PROGRAMMING PROBLEM . 11 5.1 DERIVATION OF ROUTING EQUATIONS 21 6.1 FLOOD CONTROL REGULATION BY LINEAR PROGRAMMING 1967—CASE I . . 28 6.2 FLOOD CONTROL REGULATION BY LINEAR PROGRAMMING DESIGN FLOOD—GASES!.-!;. . 29 V ACKNOWLEDGEMENT The author wishes to express h i s a p p r e c i a t i o n to h i s s u p e r v i s o r , Mr. S.O. R u s s e l l , f o r h i s guidance and encouragement during the research and w r i t i n g of t h i s t h e s i s . He a l s o wishes to thank Dr. Michael C. Quick f o r h i s help and guidance, and Richard Brun, who drew the diagrams, both o f whom were most co o p e r a t i v e i n h e l p i n g to complete the t h e s i s i n time. v i Chapter 'l INTRODUCTION Th i s t h e s i s d e s c r i b e s a study of the a p p l i c a t i o n of a mathematical o p t i m i z a t i o n technique, l i n e a r programming, t o the s o l u t i o n of a problem i n water r e s o u r c e s , the o p e r a t i o n of a system of r e s e r v o i r s i n such a way as to o b t a i n maximum downstream f l o o d c o n t r o l . I t d e s c r i b e s how the p h y s i c a l system was modelled to f i t the requirements of the technique, and how the c a p a b i l i t y of the technique was expanded to i n c l u d e as much as p o s s i b l e the c o m p l e x i t i e s of the p h y s i c a l system. Results are presented and e v a l u a t e d to give some i n d i c a t i o n o f the u s e f u l n e s s o f the technique as a t o o l i n p l a n n i n g s t u d i e s and i n a c t u a l r e s e r v o i r o p e r a t i o n . The study was prompted by the present need f o r f l o o d c o n t r o l on the Fraser River i n B r i t i s h Columbia. A system o f f l o o d c o n t r o l r e s e r v o i r s i n the headwater area of the Frasgr basin has been proposed and i s p r e s e n t l y under study. To determine the maximum amount o f c o n t r o l o b t a i n a b l e , and the minimum number of r e s e r v o i r s r e q u i r e d , i t i s d e s i r e a b l e t o know how to operate the r e s e r v o i r s i n the most e f f e c t i v e manner p o s s i b l e . Determining optimal operat-i n g p o l i c i e s i s a d i f f i c u l t and complex task even with h i n d s i g h t data because the r e s e r v o i r s are remote from the downstream c o n t r o l p o i n t , because the r e g -u l a t i o n r e s u l t s f f r o m the combined e f f e c t of the s e v e r a l s i t e s i n the system, and because the storage i s l i m i t e d at some o f the s i t e s . P r e v i o u s l y there has not been an e f f e c t i v e a n a l y t i c a l t o o l f o r determining the best o p e r a t i o n of a system of f l o o d c o n t r o l r e s e r v o i r s or f o r comparing the e f f e c t i v e n e s s of d i f -f e r e n t r e s e r v o i r s i t e s f o r planning purposes. The l i n e a r programming technique 1 2 provides- s u c t t a t o o l . . L i n e a r programming i s a mathematical t o o l used i n o p e r a t i o n s r e s e a r c h . Operations: research- i s an approach, to decision-making which, i n v o l v e s m o d e l l i n g , u s u a l m a t h e m a t i c a l l y , the e s s e n t i a l elements of a p h y s i c a l or o r g a n i z a t i o n a l system and endeavouring t o f i n d , w i t h i n e x i s t i n g c o n s t r a i n t s , the l e v e l o f a c t i v i t y o f each, o f these elements which, w i l l maximize the o v e r a l l o b j e c t i v e o f the system. The problem i s , t y p i c a l l y , one o f a l l o c a t i n g s c a r c e resources among competing a c t i v i t i e s i n an optimal manner. L i n e a r programming deals with, such a problem by modelling the system as a s e t of l i n e a r a l g e b r a i c equations'and obtains a s o l u t i o n by a l g e b r a i c manipulation o f these equations. In the present a p p l i c a t i o n , the o b j e c t i v e i s t o minimize the f l o o d peak at a downstream p o i n t by determining an optimal s e t of d a i l y storage values at each o f s e v e r a l r e s e r v o i r s s u b j e c t to the c o n s t r a i n t s o f storage a v a i l a b l e at each s i t e , d a i l y r e s e r v o i r i n f l o w s , minimum r e s e r v o i r s p i l l requirements, r e s e r v o i r outflow c a p a c i t y , and r i v e r channel r o u t i n g e f f e c t s . Most elements and c o n s t r a i n t s o f the system are capable of being model-l e d i n the manner r e q u i r e d by the l i n e a r programming technique w i t h e s s e n t i a l l y no s i m p l i f i c a t i o n , and the standard technique i s capable of h a n d l i n g most aspects o f the system. There was however d i f f i c u l t y i n w r i t i n g the o b j e c t i v e f u n c t i o n i n such a way t h a t the maximum flow at the downstream c o n t r o l p o i n t i s minimized. In a d d i t i o n , r e g u l a t e d flows at the c o n t r o l p o i n t are r e l a t e d t o the d a i l y r e s e r v o i r storage values by reach r o u t i n g e f f e c t s which are not l i n e a r , and s i m p l i f i e d l i n e a r r e l a t i o n s h i p s had t o be developed t o meet the requirements of the technique. The modelling o f the system, t o g e t h e r with the methods developed t o meet the above d i f f i c u l t i e s , are d e s c r i b e d i n two main sections, o f the t h e s i s . Considerable work has been done on the development of the technique i n 3 a previous study by W. A. Johnson [1|. The contribution of the present writer has been primarily to improve the routing technique and to expand the applica-tion of the method from a hypothetical situation to a real system of several reservoirs. For the sake of completeness of the present work, however, some of the discussion includes material that was dealt with in the ear l ier thesis. The paper is divided into a number of chapters. Chapters22 and 3 are preliminary in nature. Chapter 2 contains a discussion of the flooding problem in the Fraser River Basin, the proposed flood control system, and the d i f f i c u l -ty in operating the system for most effective flood control , and Chapter 3 is an elaboration of the l inear programming technique. Chapter 4 shows how the linear programming technique was adapted to f i t the physical system, and describes the details of writing the objective function and the constraints. Chapter 5 describes the development of a l inear routing method. Chapter 6 discusses the results that were obtained, and Chapter 7 gives the conclusions. Chapter 2 THE FLOODING PROBLEM In order to provide a background f o r the a p p l i c a t i o n o f the o p t i m i -z a t i o n method d e s c r i b e d i n t h i s t h e s i s i t i s necessary to i n c l u d e a b r i e f d i s c u s s i o n of the f l o o d i n g problem, the r e s e r v o i r system proposed as a s o l u t i o n , and the d i f f i c u l t y of o p e r a t i n g the system i n an optimal manner. The study i s concerned with the f l o o d i n g problem which e x i s t s on the Fraser River i n B r i t i s h Columbia. The F r a s e r i s one of the major r i v e r s of Canada, and d r a i n s an area of some 90,000 square m i l e s . I t s r u n o f f p a t t e r n i s dominated by snowmelt fl o o d s i n the s p r i n g , with g e n e r a l l y low flows during the remainder o f the year. The f l o o d s are normally contained w i t h i n the r i v e r banks, but severe combinations of snowpack and temperature c o n d i t i o n s i n the s p r i n g sometimes cause the waters t o spread onto the f l o o d p l a i n s . The f l o o d hazard r e s u l t s from development of the f l o o d p l a i n s f o r a g r i c u l t u r a l , r e s i d e n t i a l , and i n d u s t r i a l use. In p a r t i c u l a r , the lower Frase r V a l l e y i s a h e a v i l y populated area. An example o f the damage which can be done by a major f l o o d was the f l o o d of 1948, which caused approximately $20,000,000 damage, and completely d i s r u p t e d t r a n s p o r t a t i o n and communication l i n k s i n the Lower F r a s e r V a l l e y . Because development has been e x t e n s i v e s i n c e t h a t time i t i s estimated t h a t f l o o d i n g of the same extent today would do over $200 m i l l i o n damage. Another f l o o d of the magnitude of 1948 i s not a h i g h l y u n l i k e l y occurrence. That f l o o d i s estimated to have a return p e r i o d of between 50 and 100 y e a r s . Several times s i n c e 1948 the snowpack has been as l a r g e or l a r g e r , so t h a t the r i g h t temperature c o n d i t i o n s would have p r e c i p a t e d a major f l o o d . 4 5 As a r e s u l t o f the 1948 f l o o d a study group, the F r a s e r River Board [ 2 ] , was s e t up to i n v e s t i g a t e f l o o d c o n t r o l and h y d r o - e l e c t r i c power i n the Fraser b a s i n . Although a l a r g e dam c l o s e to Hope on the F r a s e r would provide e x c e l l e n t f l o o d c o n t r o l and would produce l a r g e q u a n t i t i e s o f power, the Board d i d not recommend such a dam because there would be l i t t l e p o s s i b i l i t y of a v o i d i n g major damage to the F r a s e r salmon runs. Instead a system of s m a l l e r dams on the F r a s e r and c e r t a i n t r i b u t a r i e s , a l l l f a r upstream from Hope, was recommended, the s i t e s being chosen to avoid i n t e r f e r e n c e with the salmon. A map of the F r a s e r b a s i n showing the l o c a t i o n o f the p r o j e c t s i n the proposed system, c a l l e d "System E", i s shown i n Figure 2.1. I t i s d e s i r e a b l e to be able to operate f l o o d c o n t r o l r e s e r v o i r s i n the most e f f e c t i v e manner p o s s i b l e , i e . to be able to o b t a i n the maximum f l o o d peak r e d u c t i o n p h y s i c a l l y p o s s i b l e . This i s e s p e c i a l l y t r u e o f the proposed system on t h e - F r a s e r R i v e r , because, although the r e s e r v o i r s wouls! be used to produce needed power as w e l l as f o r f l o o d c o n t r o l , i t i s expected t h a t they would have c e r t a i n negative e f f e c t s on the n a t u r a l environment. I f the most e f f e c t i v e o p e r a t i o n can be determined, the number of r e s e r v o i r s r e q u i r e d can be minimized. I t would be r e l a t i v e l y easy t o operate a l a r g e r e s e r v o i r l o c a t e d c l o s e to Hope f o r maximum f l o o d c o n t r o l - simply reduce the s p i l l t o a l e v e l such t h a t the f l o o d l e v e l s i n the Lower Frase r V a l l e y are h e l d w i t h i n accept-able l i m i t s . The r e s e r v o i r would have s u f f i c i e n t c a p a c i t y to s t o r e the r e q u i r e d flow throughout the f l o o d i n g p e r i o d . However, the proposed system of remote r e s e r v o i r s would be c o n s i d e r a b l y more d i f f i c u l t t o operate i n an optimal manner. The major reason f o r the i n c r e a s e d d i f f i c u l t y i s t h a t some of the r e s e r v o i r s do not have s u f f i c i e n t c a p a c i t y t o s t o r e a l l of t h e i r i n f l o w throughout the f l o o d p e r i o d . I t t h e r e f o r e becomes c r u c i a l l y important t o time LEGEND POWER SITES POWER a STORAGE SITES SUB - BASIN DEVELOPED S U B - BASIN FRASER RIVER BASIN INDEX SITE HI GRAND CANYON 104 POWER MCGREGOR 89 CARIBOO FALLS 153 HOBSON LAKE I42A CLE AR WAT ER-AZURE 142 HEMP CREEK 141 CLEARWATER" I 94A GRANITE FALLS (PORTAGE MOUNTAIN ) S U B - B A S I N UPPER FRASER RIVER Mc GREGOR RIVER CARIBOO RIVER CLEARWATER RIVER CLEARWATER RIVER CLEAR WATER RIVER CLEARWATER RIVER CL EARWATER R I VER PEACE RIVER F i g . 2.1 S y s t e m " E " - L o c a t i o n of P r o p o s e d P r o j e c t s . 7; the storage of water c o r r e c t l y . S t o r i n g too soon may r e s u l t i n the a v a i l a b l e storage c a p a c i t y being f i l l e d before the f l o o d i s passed, while s t o r i n g too l a t e w i l l e q u a l l y r e s u l t i n a l o s s o f e f f e c t i v e n e s s . Determining the c o r r e c t s t o r i n g p a t t e r n f o r r e s e r v o i r s with a lack o f abundant storage i s made more complex by the r o u t i n g e f f e c t s o f the r i v e r chan-nel between the r e s e r v o i r s i t e s and the f l o o d hazard area. There i s a delay of s e v e r a l days Between the flow r e d u c t i o n at the s i t e and the r e s u l t i n g e f f e c t i n the Lower F r a s e r V a l l e y . In a d d i t i o n , the e f f e c t s o f flow r e d u c t i o n s at the s i t e s are reduced by water i n the r i v e r channel coming out of stor a g e . These e f f e c t s must be accounted f o r i n computing the r e s e r v o i r o p e r a t i o n . F i n a l l y , the d i f f i c u l t y o f the task i s i n c r e a s e d by the f a c t t h a t the f l o o d r e d u c t i o n r e a l i z e d downstream i s the r e s u l t o f the combined a c t i o n o f a number of r e s e r v o i r s . I t i s not easy t o determine the way i n which to combine the operations o f the r e s e r v o i r s i n order t o determine the most e f f e c t i v e use of the l i m i t e d storage i n each. In s p i t e o f the use of computer s i m u l a t i o n models, t o date the only method a v a i l a b l e t o determine a best o p e r a t i n g p o l i c y has been t r i a l and e r r o r . T h i s method i s e s p e c i a l l y onerous.when a number of r e s e r v o i r combinations must be compared i n order to minimize the number of p r o j e c t s b u i l t , as i n the present case. T h i s paper d e s c r i b e s the a p p l i c a t i o n o f the l i n e a r programming o p t i m i -z a t i o n technique t o the aKoye problem. Given the hydrographs at the sites, and at a p o i n t r e p r e s e n t i n g the f l o o d hazard a r e a , the most e f f e c t i v e o p e r a t i o n of a system-of r e s e r v o i r s f o r maximum f l o o d c o n t r o l i s e a s i l y computed. The method i s q u i t e f l e x i b l e , and s o l u t i o n s are r e a d i l y obtained f o r d i f f e r e n t r e s e r v o i r s i z e s , l o c a t i o n s , and combinations. Chapter 3 LINEAR PROGRAMMING Linear programming is a mathematical technique which deals with the problem of allocating scarce resources to a variety of competing act iv i t ies in such a way as to meet some objective in an optimal manner.- This chapter out-lines the formulation of the l inear program and the basic principles of the method of solution, and suggests the method's usefulness in the area of sensi -t i v i t y analysis. Linear programming deals with a problem by modeling the elements of the physical system as a set of l inear algebraic equations and obtains a s o l -ution by algebraic manipulation of these equations. The model is characterized by the objective which is to be maximized, the act iv i t ies undertaken to meet the objective, and the resources used or products produced by the ac t i v i t ies . The mathematical statement of a general form of the linear programming model is as follows: Find x n , x 2 , . . . x n which maximizes the objective function Z = c-|X-| + c 2 x 2 + . . . + c n x n subject to the restrictions a l l x l + a 1 2 x 2 + • ••• *ln Aftn x n 1 b l a 2 1 x l + a 2 2 x 2 + • • ' + A2n x n 1 b2 a n x n + a x + . . . + A x < b ml 1 m2 2 mn n - m and x-| > (D, X£ > 0 . . . x p > 0 8 9 where Z i s the value of the o b j e c t i v e eg. p r o f i t x- i s the l e v e l of a c t i v i t y j a^j i s the amount of resource/product i used/produced per u n i t of a c t i v i t y j b. . i s the amount of resource/product i a v a i l a b l e or r e q u i r e d c. i s the e f f e c t i v e n e s s of one u n i t of a c t i v i t y j 3 i n meeting the o b j e c t i v e Z The s o l u t i o n method can handle e q u a l l y w e l l , minimizing the o b j e c t i v e f u n c t i o n i n s t e a d of maximizing, and c o n s t r a i n t s t h a t resource usage be g r e a t e r than or equal t o , or e x a c t l y equal t o , some amount. L i n e a r i t y r e q u i r e s that the o b j e c t i v e f u n c t i o n and a l l of the c o n s t r a i n t s be l i n e a r f u n c t i o n s of the x.. In other words, the u n i t e f f e c t i v e n e s s and u n i t resource usage of each of the a c t i v i t i e s must not change with the l e v e l of the a c t i v i t y . A simple example w i l l serve t o i l l u s t r a t e the f o r m u l a t i o n of the problem, and a g r a p h i c a l s o l u t i o n t o the example w i l l make c l e a r the b a s i c p r i n c i p l e s of the method of s o l u t i o n . A f a c t o r y produces p i c t u r e frames and f i g u r i n e s . Each frame r e q u i r e s 1 u n i t of l a b o u r , 1 u n i t of wood, and 2 u n i t s of p a i n t . Each f i g u r i n e r e q u i r e s 2 u n i t s of l a b o u r , 1 u n i t of wood, and 1 u n i t of p a i n t . There i s a $3 p r o f i t f o r each frame, and $5 f o r each f i g u r i n e . The supply of labour i s l i m i t e d to 10 u n i t s , wood 6 u n i t s , and p a i n t 10. u n i t s . How many frames and how many f i g u r i n e s should be produced f o r maximum p r o f i t ? tn t h i s example the a c t i v i t i e s are the production of frames and f i g u r i n e s , and x-| and are the number of each produced. The o b j e c t i v e i s t o maximize p r o f i t , and the o b j e c t i v e f u n c t i o n i s w r i t t e n as Z = 3x-| + 5x£. The amount o f p a i n t used by the two a c t i v i t i e s must be l e s s than or equal to 10, so the 10 f i r s t c o n s t r a i n t i s x-, + 2 x 2 < 10 S i m i l a r l y , the other c o n s t r a i n t s are x-j + x 2 < 6 2x 1 + x 2 < 10 and x-| > 0, x 2 > 0 The usual method used t o s o l v e a l i n e a r programming problem i s the "simplex method", a process of a l g e b r a i c manipulation o f the o b j e c t i v e and c o n s t r a i n t f u n c t i o n s which p r o g r e s s i v e l y approach the optimal s o l u t i o n . The process i s normally c a r r i e d out on an e l e c t r o n i c computer. However, f o r the simple example d e s c r i b e d above i n which there are only two a c t i v i t i e s a gr a p h i c a l s o l u t i o n may be obtained which w i l l help t o make c l e a r what the simplex method does. In the g r a p h i c a l method x-j and x 2 axes are drawn, and the equations which form the outer bounds of the c o n s t r a i n t f u n c t i o n s are p l o t t e d (see Figure 3.1). The shaded area i n the diagram represents the region w i t h i n which a l l of the c o n s t r a i n t s are s a t i s f i e d . The equation Z = 3x^ + 5x 2 can be represented on the diagram as a s e r i e s o f p a r a l l e l l i n e s f o r d i f f e r e n t values of Z. In t h i s example the value of Z in c r e a s e s as the l i n e moves f u r t h e r from zthe o r i g i n . I t can r e a d i l y be seen t h a t the f u r t h e s t the l i n e can move from the o r i g i n while remaining w i t h i n the permissable region i s the i n t e r s e c t i o n p o i n t (2,4), and the r e s u l t i n g optimal value of Z i s 26. The simplex method reaches a s o l u t i o n by p r o g r e s s i v e l y moving from one i n t e r s e c t i o n p o i n t , or " b a s i c f e a s i b l e s o l u t i o n " , t o the next i n the d i r e c t i o n o f i n c r e a s i n g values o f Z. When no f u r t h e r step w i l l i n c r e a s e the value o f the o b j e c t i v e f u n c t i o n the optimal s o l u t i o n has been found. 11 L i n e a r programming i s useful i n the area o f s e n s i t i v i t y a n a l y s i s . I t often happens t h a t the parameters c^, a „ , or b.. are not known with c e r t a i n t y , or may change subsequent t o s o l u t i o n o f the o r i g i n a l problem. I t i s p o s s i b l e to r e l a t e changes t o the optimal value of the o b j e c t i v e f u n c t i o n t o these changes i n the parameters d i r e c t l y from the o r i g i n a l s o l u t i o n without rerunning . F igure 3.1 G r a p h i c a l S o l u t i o n of L i n e a r P rogramming P r o b l e m . the l i n e a r program. Again r e f e r r i n g t o the diagram i n Figure 3.1, i f , say, the p r o f i t from the p i c t u r e frames c^ was changed from $3 to any value i n the range $2.50 to $5, the s o l u t i o n would not change from ( x ^ x ^ ) = (2,4), and the p r o f i t would change by the optimal value of v a r i a b l e x^, i . e . $2 per u n i t change i n c.j I f the supply of wood, b 9 , was changed from 6, the p r o f i t would change 12 as a l i n e a r f u n c t i o n of w i t h i n the range where the s o l u t i o n occurs at t h e , i n t e r s e c t i o n of equations + 2x^ = 10 and x^ + x^ = b^, i . e . , i n the termin-ology of the simplex method, as long as the b a s i c s o l u t i o n i s not changed. I f t h i s example were s o l v e d by the simplex method, the change i n p r o f i t per u n i t change i n b^ could be e x t r a c t e d from the f i n a l form of the a l g e b r a i c equations r e s u l t i n g from the simplex s o l u t i o n . S i m i l a r l y , the changes i n p r o f i t per u n i t change i n any o f the parameters, and the ranges of the parameters w i t h i n which these rates of change apply, may be obtained from the simplex s o l u t i o n . Chapter 4 MODELLING THE SYSTEM Formulating the l i n e a r programming model r e q u i r e s a n a l y s i s of the p h y s i c a l system i n terms o f the o b j e c t i v e f u n c t i o n , a c t i v i t i e s , and c o n s t r a i n t s d e s c r i b e d i n the previous chapter. The f i r s t step i n the a n a l y s i s i s to i d e n -t i f y the b a s i c elements of the system and t h e i r i n t e r a c t i o n s . In the present study the system c o n s i s t s of a downstream c o n t r o l p o i n t , Hope, at which the r e g u l a t e d peak flow i s t o be minimized; f i v e r e s e r v o i r s , namely, Grand Canyon, Cariboo F a l l s , and three on the Clearwater R i v e r ; d a i l y i n f l o w s to each of the r e s e r v o i r s during the f r e s h e t ; and storage c a p a c i t i e s and maximum discharge rates at each r e s e r v o i r . Reductions to the flows at Hope are r e l a t e d t o the amounts of water s t o r e d at the r e s e r v o i r s by r i v e r channel r o u t i n g e f f e c t s . In t h i s system, the a c t i v i t i e s , or d e c i s i o n v a r i a b l e s , are the h o l d -outs, i e . the amounts o f water t o be s t o r e d , at each r e s e r v o i r each day. The c o n s t r a i n t s are the d a i l y r e s e r v o i r i n f l o w s which l i m i t the amount of water a v a i l a b l e f o r s t o r i n g ; the t o t a l storage c a p a c i t y o f each r e s e r v o i r ; and the maximum discharge c a p a c i t i e s . The o b j e c t i v e i s to minimize the peak o f the r e g u l a t e d hydrograph at Hope. A s p e c i a l problem i s c r e a t e d by the l o c a t i o n of the three r e s e r v o i r s on the Clearwater R i v e r i n s e r i e s on the same r i v e r . The l i n e a r program r e q u i r e s knowledge of i n f l o w s , but i n f l o w s t o downstream r e s e r v o i r s cannot be known u n t i l o p e r a t i o n o f upstream r e s e r v o i r s i s determined. T h i s problem was d e a l t with, by modelling the three r e s e r v o i r s as one l o c a t e d at the s i t e f u r t h e s t 13 14 downstream [Hemp Creek] and having a storage c a p a c i t y equal t o the combined c a p a c i t y o f the three a c t u a l r e s e r v o i r s . Subsequent to s o l u t i o n o f the l i n e a r program i t was t h e r e f o r e necessary t o check t h a t the volume s t o r e d c o u l d be d i v i d e d betv/een the three r e s e r v o i r s without exceeding the c a p a c i t y o f any one. The f i r s t s e t of c o n s t r a i n t s r e q u i r e s t h a t every holdout s h a l l be l e s s than or equal t o the i n f l o w t o the r e s e r v o i r on the c u r r e n t day: < • < hz • < X l n "ml < rml < lmZ rl mn < m^n where Hi., i s the holdout a t r e s e r v o i r i on day j I-.- i s the i n f l o w t o r e s e r v o i r i on day j m i s the number of r e s e r v o i r s i n the program n i s the number o f days i n c l u d e d i n the program The second c o n s t r a i n t s e t l i m i t s the volume s t o r e d i n each r e s e r v o i r , i.e. the sum of the ho l d o u t s , t o the r e s e r v o i r c a p a c i t y : (2] H.11 + H 1 2 + . . . + H L N < ST ^21 ^22 * ^2n - ^2 Hml + Hm2 + ' ' • + % - Sm where S.. i s the storage c a p a c i t y o f r e s e r v o i r i 14 15 The t h i r d s e t of c o n s t r a i n t s r e s t r i c t s the outflow t o the discharge c a p a c i t y o f the r e s e r v o i r . T h i s c o n s t r a i n t i s necessary t o provide f o r the p o s s i b i l i t y o f i n v o l u n t a r y s t o r a g e , i e . storage which was not scheduled by the l i n e a r program f o r optimal f l o o d c o n t r o l , but occurred as a r e s u l t o f i n f l o w g r e a t e r than outflow c a p a c i t y . Discharge c a p a c i t i e s are a f u n c t i o n o f r e s e r v o i r e l e v a t i o n , and were determined by r o u t i n g the i n f l o w s through the r e s e r v o i r with the o u t l e t gates wide open. Use of the values computed i n t h i s manner would be c o n s e r v a t i v e i f i n v o l u n t a r y storage occurred subsequent to planned s t o r a g e , but i t normally occurs e a r l y i n the f r e s h e t when r e s e r v o i r l e v e l s are low and inflo w s begin r i s i n g r a p i d l y . The c o n s t r a i n t s may be w r i t t e n as f o l l o w s : Hi F12 etc.. where 0. . i s the r e s e r v o i r outflow F.. i s the maximum discharge c a p a c i t y on day j Rewriting these i n e q u a l i t i e s i n terms of holdouts y i e l d s the f o l l o w i n g s e t of c o n s t r a i n t s : W H n ] >_ I „ - F l l H 12 :12 " F l 2 % ^ L l p - F l p Kml t- rml " F r a l Hm2 rm2 " Fm2 H > I F mp — mp mp 16 The parameter p, the number of days for which this set of constraints was written, was set at 12 because i t was found that involuntary storages rarely occurred beyond the f i r s t twelve days. The objective equation must be written in such a way that the peak of the regulated hydrograph is minimized. The regulated flow on a particular day is related to the holdouts at the reservoirs by an equation of the form Rj * Qj - D. where R. is the regulated flow on day j at Hope Q. is the unregulated flow on day j at Hope D. is the effect of the holdouts at a l l reservoirs routed J to Hope Development of the routing equations wi l l be discussed in the following chapter. The day of the peak of the regulated hydrograph is not known, and in fact the peak value may occur on many days. It is therefore obviously not adequate to write the objective function as minimizing the flow on any one day. No single equation can be written in terms of the holdouts whtch i f minimized would minimize the. peak of the regulated hydrograph. The solution to this d i f f i cu l t y was obtained by using a dummy variable, Y, which was constrained to be greater than or equal to the regulated flow on every day included in the program. The objective function Z was then set equal to Y. Minimizing Z therefore minimizes the peak subject to the constraints described above, and ensures that on no day is the flow higher than this optimum value. As a result of the manipulation described above a fourth set of constraints is added to the program. These require that on each day the dummy variable Y shall be greater than or equal to the unregulated flow, less the sum of the routed holdouts: 17 (4) Y > Q1 - D1 Y > Q 2 - D2 Y > Q - D The objective function is then written simply as: Minimize Z = Y It w i l l be noted that, with the exception of the fourth set of con-straints which are not yet written in terms of the holdouts H . . , a l l of the constraints and the objective function are l inear functions of the variables H..J and Y. Development of routing equations such that this last set of con-straints can be written as linear inequalities is discussed in the next chapter. It may also be seen that setting up the model for a given year requires know-ledge of the flow data for the entire freshet. The significance of this l imitation w i l l be discussed in the concluding chapter. Chapter 5 ROUTING EQUATIONS One of the major requirements of the linear programming solution method is that the objective function and the constraints be l inear functions of the decision variables. It was shown in the previous chapter that, with the poss-ible exception of the fourth constraint set which is not yet written in terms of the holdouts, this requirement is met. In the fourth set of constraints, the regulated flows at Hope are related to the holdouts by equations describing river channel routing effects. This chapter deals with the method developed to write the routing equations as l inear functions of the holdouts. Propagation of a flood wave down a ri'ver channel is essentially a non-linear phenomenon, since wave travel times and incremental channel storage vary with the level of flow. It was therefore necessary to develop a simplified l inear method of routing which would nevertheless give a good approximation of the true routing effects. In the present study the method developed was based on monoclinal wave theory. This theory is a simplif ication of the more general theory of unsteady flow in open channels [3]], in which the energy slope is assumed to overpower the other energy terms in the equation of motion. The equation of motion is then o) s 0 = _ i L C2R and the continuity equation is (2) ^ + B i X = o [ 4 ] , 9X 8t assuming lateral inflow is negligible within the river channel reach under consideration, 19 where S 0 i s the bed slope v i s the v e l o c i t y o f flow C i s Chezy's C, a f a c t o r of r e s i s t a n c e t o flow R i s the h y d r a u l i c radius Q i s .the discharge B i s the channel width y i s the channel depth x i s a measure of d i s t a n c e along the channel t i s time W r i t i n g the equation of motion i n terms of flow Q, and r e p l a c i n g h y d r a u l i c radius R with depth y (assuming a wide open ch a n n e l ) , equation (1) becomes (3) Q = v B y = B C y % S% D i f f e r e n t i a t i n g i ; ^equation (3) and s u b s t i t u t i n g i n equation (2) y i e l d s 0 or, removing B. (4) 1.5 B c So2 y2M. + B _sy 9X- 9t 1.5 C S o ^ /2 ay + 9^. = 0 9X I f the term 1.5 C S Q 2 y 2 i s equal t o dx (wave v e l o c i t y ) the l e f t hand s i d e dt of (4) i s the t o t a l d i f f e r e n t i a l which i s then equal to 0. In other words, f o r the case of a wide open channel, a given depth of water y i s propa-g a t i n g at the rate (5) Jx„ =• 1.5 C Sah yh = 1.5 v dt This r e s u l t allows computation o f the time o f t r a v e l of a given l e v e l of flow between two p o i n t s on a r i v e r given an average v e l o c i t y f o r the reach at t h a t flow l e v e l . A second c o n s i d e r a t i o n i n the development of the r o u t i n g equations i s 20 the channel storage which occurs with the passage of a flood wave. Given a change in flow A I at the upstream boundary of a reach, the corresponding change in outflow from the reach A O willbbe less than A I by the amount of flow which goes into or comes out of storage. This relation may be expressed as (6) A O = (1-C) A I where C is the fraction of the flow change which goes into or comes out of storage. Given the considerations of channel storage and wave travel time, the method by which the outflow from a reach may be computed from the inflows to the reach is developed as follows. Figure 5.1a shows the position of a wave on two successive days j - 1 and j , assuming, for the moment, no channel storage. It is assumed that the water surface .is a straight l ine between the stations at either end of the reach. The flows I are measured at the upstream station U, and the time of travel of flow level I • between stations U and D is known to be, T days. The outflow on day g is then computed as a l inear interpolation of the inflows: (7) 0. = I + (1-T) (I - I ) J J-1 0 J-1 Should the time of travel be greater than one day the interpolation becomes an extrapolation which could i f large enough introduce errors of a larger magnitude than the interpolation. Travel times for the reaches chosen are however less than or only s l ight ly greater than one day. If the flow at U increases from I. to I-, then the increase at D w i l l be reduced by the fraction C which goes into storage. This effect is accounted for by substituting (1 - C) (Ij - for (Ij - Ij_-|) in equation C7): C 8 1 03. = 1 ^ + (1-T) (1-C) [Ij - I j . , ) 21 Woter s u r f a c e on day j Water s u r f a c e on day j - I U 0 Wave Position Fig.5. la Water sur face on day j Water surface on day j - 1 U 0 Wave Position Fig.5.1 b Fig.5.1 Derivation of Routing Equations 22 Figure 5.1b i l lustrates the computation with the storage effect included. Equation (8) can be rewritten Oj = 0 " T)(l - C) Ij + (1 - (1 - T)(l - 0) Ij.-, or (9) 0 j - A Ij • B I . . , where A = (1 - T)(l - C) and B = 1 - A For the purpose of computing the routing equations the river was divided into a number of reaches, each bounded by Water Survey of Canada gauging stations. The travel times and storage constants were obtained from the stage - discharge relationships and channel cross-sectional areas at these gauging stations. Values for each reach were taken as the average of the values at the gauging stations at either end of the reach. The storage constant C was computed from the equation (10) C = dS = |_ da_ dQ 3Q where S is the volume of water in storage in the channel reach L is the length of the reach Q is the discharge da. is the change in cross-sectional area per unit change in flow The value of — at each gauging station was obtained by plotting the area as dQ a function of stage, then combining the stage - area and stage - discharge relationships to give a single area - discharge relationship. The travel times T were computed from the equation (11) T = _ L _ 1.5v where V i s the yelocity of flow 23 The flow velocity was computed as the discharge divided by the area.. Tn order that the routing equations tie l inear functions of the flows i t is necessary that the parameters C and T be constants not varying with the level of flow. It was found that C was very nearly constant at the gauging stations used in this study over a wide range of flow. Travel times do however vary to some degree. Travel times, and hence the routing constants A and B, were computed for discharges varying from 250,000 to 400,000 cfs . at Hope, and corresponding levels of flow at other gauging stations. (Only flood level flows are of interest in this study of flood control.) The constants varied only a few percent over this range, and i t was concluded that a reasonable approxima-tion could be made. Values of T were chosen at a discharge of 400,000 cfs . at Hope, and corresponding flows at other stations, as this flow is about mid-way between the minimum level of concern and maximum recorded levels. The constants A and B were computed using these values of T. A test was made comparing routing using the routing constants developed as described above, and the same method except with values of T computed as a function of flow. The routed hydrographs were almost ident ical . The f inal step in developing the routing equations was to combine the equations for individual reaches to give a single equation for each dam si te relating the flow at the si te to Hope. Because the equations are l inear , this could be done simply by successively substituting the equation for the adjacent upstream reach into the equation for the downstream reach. The resulting equations have the form 0 2 ] o, = • A I . + ' B i . , + e i . , + . . . J J J-1 J-2 The-equations for the individual reaches and the combined equations are l is ted in Table 5.1. Because these equations are l inear , holdouts, which are reductions 24 Table 5.1 Linear Routing Equations Individual Reaches Hemp Creek to Clearwater Station °i- .71 :J + .29 V Clearwater Station to McLure 0. = J .57 I. J + .43 I. J-McLure to Spences Bridge 0. = J .46 I. J + .54 I. J-Cariboo Falls to Marguerite 0. = J .06 I . J + .94 I . J-Grand Canyon to Shelley V .01 I. J +1 .01 I . J-Shelley to Marguerite 0. = J .12 I. J + .88 h-Marguerite to Texas Creek 0. = J .03 I. J + • 9f Texas Creek or Spences Bridge to Hope .30 I. J + .70 Reservoir Sites To Hope Grand Canyon Cariboo Falls Clearwater 0. = J 0. • J 0. = J .24 I .04 I. , + .3611. . + .60 I J-2 J-3 J-4 .03 I. . + .33 I. 9 + .64 I.-_o j-1 • . J ^  .05 t j + .26 I . . , + .40 ! . . 2 • 3-3 + -05 25 to the flow or negative flows, may be routed as readily as positive flows. This feature allows'the holdouts to be routed directly and subtracted from the unregulated hydrograph at the downstream control point, rather than routing the regulated reservoir outflows reach by reach and adding local inflows at appropriate points. In other words, the regulated flow at Hope can be related to the reservoir holdouts by a single linear equation, as required by the l inear programming method. (An additional advantage in routing the holdouts is that any errors in the routing method are applied, not to the fu l l flow in the r iver , but only to the holdouts which are of a much smaller magnitude.) The routing method described in this chapter was adapted from the method used in the UBC Flow Model, a routing model developed by M. C. Quick and A. Pipes of the University of British. Columbia Department of C iv i l Engin-eering 143. Storage constants and functions for computing travel times from discharges, were taken directly from the model and used for developing the routing equations used in the present study. The primary alteration made to the original method was the use of average values of the time of travel T instead of computing them at each level of flow.* Using the general form of the routing equations (12), the fourth set of constraints in the linear program (see Chapter 4) can be rewritten in terms of the holdouts: Y - % ~ A l i H l s " A12 H l , s - 1 " • • • ^ A l s "1,1 *The method of incorporating the effect of channel storage into the routing equations described in this chapter is approximate in that i t does not use the storage constants in precisely the same manner they are used in the UBC Flow Model. The method used to incorporate channel storage in the Flow Model is described in Quick and Pipes [4], and the derivation of l inear routing equations based on the more accurate method is given (in Appendix A. In view of the limited accuracy of the basic data, and the fact that average values of the routing parameters were used, rather than varying the parameters with the flow, the approximate method was considered to be suff ic ient ly accurate. \ 26 A 2l : H2;,ss " A22 H 2 , s - 1 - A n H 3 , s A32 H 3 , s - 1 " A 2 s H 2 , l " A3s H 3 , l etc. Rearranging so that a l l variables are on the le f t hand side, this set of equations becomes (4) A l l H l , s + A12 H l , s - 1 . + A l s V l + A 21 H 2,s + A22 H 2 , s - 1 + . . A2s H 2 ] l + A 3 1 H 3 , s + A32 H 3*j is -1 A3s H 3 , l + Y > Q-A l l H l ,s+1 + A 1 2 H l , s A l s H l , 2 + A 2 1 H2,s+1 + A22 H 2 , s . + A2S H 2 , 2 + A 3 1 H3,s+1 + A32 H 3 , s A3s H 3 , 2 + Y i Qs+i A l l lr,-n A12 H l , n - 1 A l s H l ,n -s+1 + A 2 1 H 2,n + A22 H 2,n -1 + . A2s H2qn -s+1 + A 3 1 H 3,n + A32 H 3,n -1 . . + A3s 3,n -s+1 + Y >. Qn where Q is the unregulated flow at Hope on day j H, i_k 1 S the holdout at reservoir i on day j - k A.j is the routing coefficient applied to holdout H s is the number of coefficients in the routing equation with the largest number of coefficients The objective function and a l l sets of constraints now satisfy the l inear i ty requirement of the linear programming technique. Chapter 6-RESULTS Linear programming solutions were obtained for two large floods on the Fraser River, one the freshet of 1967 and the other a design flood developed by the Fraser River Board [2]. The lat ter flood, with i t s high peak and very large volume, is a severe test of the reservoir systems' capacity to regulate a flood. Reservoir storage capacities were also taken from the Fraser River Board C.F.R.B.) designs [2]. These capacities are, respectively, 1 ,000,000 sfd. (second-foot-days, or cfs.-days) at Grand Canyon; 580,000 sfd. at Cariboo Falls;aand 1,630,000 sfd. at the Clearwater reservoir. The solutions were obtained using a University of Br i t ish Columbia Computing Centre l ibrary routine for solving large l inear programming problems entit led UBC LIP. This routine computes the values of the holdouts and the regulated peak. The regulated hydrograph at Hope was subsequently computed by routing the holdouts and subtracting from the unregulated hydrograph. Samples of the hydrograph at the sites and at Hope prior to and with regulation according to the linear programming solutions awe' shown in Figures 6.1 and 6.2. The F.R.B. design flood was computed by that board essentially by tak-ing the maximum discharge as corresponding to the peak of the 1894 flood (estimated at 600,000 cfs.) and scaling up the 1948 flood by the proportion of thekp.eaks. The design flood hydrograph at Hope used in the present study was the simplif ied hydrograph displayed in the F.R.B. Final Report [2]. Hydro-graphs at the reservoir sites were computed from percentage contributions 27 600 500 400 « 300 u O O O O iii 200 80 60 40 20 0 40 20 0 60 40 20 0 Discharge prior to regulation by linear programming DISCHARGE HYDROGRAPHS AT HOPE - Natura I discharge Regulated discharge RESERVOIR REGULATION AT GRAND CANYON Natura 1 d ischarge ^ — - R e g u l a t e d discharge RESERVOIR REGULATION AT CARIBOO FALLS NaturaI discharge Regulated discharge RESERVOIR REGULATION AT CLEARWATER • I • • • 1 • • 1 • • • • • • • • • • • • • i i i i i i i i i I • i i i i i i 28 June July Fig.6.1 Flood Control on the Fraser River 1967 - CASE I 600 500 4 0 0 • * 300 o O O O 0) o 200 80 60 40 20 0 40 20 0 60 40 20 0 Discharge prior to regulat ion by l inear p rog ramming Discharge with reservoi r regulat ion DISCHARGE HYDROGRAPHS AT HOPE — Natu ra l d i scharge R e g u l a t e d discharge . . RESERVOIR REGULATION AT GRAND CANYON -•Natural d i s c h a r g e A -— -Regulated discharge RESERVOIR REGULATION AT CARIBOO FALLS Natural d i s c h a r g e Regulated discharge RESERVOIR REGULATION AT CLEARWATER • i i i i i i i • i -i • i I i . • • . . • • • • t • i • • • • . . i i . • I . i M a y J u n e Fig .6 .2 Flood Control on the Fraser River FRB Design F l o o d - C A S E I 30 of the runoff at each site to the runoff at Hope. The hydrographs at Hope and at the sites are shown in Figure 6.2*. The 1967 flood was analyzed for the single case of a l l three reservoirs with capacities as designed. For this filio.od the maximum flow at Hope was reduced from 382,000 cfs. to 272,000 c f s . , a reduction of 110,000 cfs . The design f lood, however, was analyzed with a number of different reservoir capacities and with different reservoir combinations. The cases studied and the results obtained for this flood are summarized in Table 6 .1 . It may be seen that Cases I through V include a l l three reservoirs, but with varying storage capacities. In Case I a l l reservoirs have the design capacities. Cases VI and VII show one or more reservoirs removed from the system. In these last cases the reservoirs included a l l have the design capacities. Several useful results may be obtained from examination of the table. The f i r s t is the identif ication of which reservoirs, as designed, have adequate storage capacity. The flow reduction which can be effected by a given reser-voir i s limited by one of two possible factors. Because on any one day the reservoir 'caravstoremm6 more than that day's inflow, the inflows to the reservoir may be the l imit ing factor. However i f the reservoir lacks suff ic ient storage to accommodate a l l of the inflow over the desired storing period, then the regulation w i l l be limitedbbyystorage. In Case I, with the fu l l system as designed, the Cariboo Falls and Clearwater reservoirs used considerably less than their capacity, which would indicate that the linear program was able to make fu l l use of the inflows to these reservoirs. This is confirmed by exam-ination of Figure 6.2, which shows the hydrographs at the sites before and after regulation. At Cariboo Falls and Clearwater the flow is completely cut *See discussion in Appendix B.. 31 Table 6.1 Summary of Results : Design Flood Case Grand Canyon Cariboo Fa l l s Clearwater Maximum Regulated Flow at Hope Storage Storage Storage -AvaiIable Used Avai Table Used Avai lable Used I 1000 1000 580 406 1630 584 416 II 1000 1000 580 424 375 375 423 III 1000 1000 580 406 250 250 429 IV 1500 1430 580 187 1630 379 416 V 500 500 580 350 1630 975 424 VI 1000 829 580 64 0 471 VII 1000 601 0 - 0 492 Motes: The storage values are in k s f d . , flows are in kefs . The unregulated.peak flow at Hope i s 576,000 c f s . ( th is assumes diversion of Nechako discharges). off during the storing period. At these sites the l inear program identif ies the minimum period during which storing is required. Because considerably less than the available capacity was used at each of these s i t e s , i t is evident that, with the design capacities, they would be relat ively easy to operate for flood control without the guidance given by a linear programming solution. At Grand Canyon, on the other hand, a l l of the capacity is used. How-ever, examination of Case IV in Table 6.1 shows that no reduction to the peak was obtained by adding capacity at this s i te . It follows that a l l of the i n -flow was useful in reducing the peak at Hope. This is confirmed by examination of the hydrograph in Figure 6.2 which shows that a l l of the inflow was stored at the time of the peak. However optimal use of the storage at this site re-sulted in a regulated hydrograph which varied radically throughout the storing period. This reservoir is therefore identif ied as lacking abundant storage and would be more d i f f i c u l t to operate in an optimal manner in a real operating situation. Increased capacity at this s i t e , i f physically possible, would make for greater ease of operation for maximum effectiveness of the available ihf'low. A second result is the identif ication of,the minimum required reservoir capacities. It has already been shown that Grand Canyon would benefit from more than i t s 1,000,000 sfd. design capacity. However in no case does Cariboo Falls use more than 424,000 s f d . , or about two-thirds of i t s capacity, and Clearwater uses no more than 584,000 s f d . , or about one-third of i t s capacity, except when the capacity at Grand Canyon is reduced below design. Reservoir sizes at these sites could therefore be signif icantly reduced without increas-ing the regulated peak at Hope or increasing the d i f f i cu l t y of operation. Cases I I , I I I , and V show the ab i l i t y of the linear programming method to maximize the effectiveness of very limited amounts of storage. In Cases 33 II* and III the storage capacity at Clearwater is reduced considerably below the 584,000 sfd. used in Case I, yet the limited capacity i s used in such a way that the regulated peak is not s ignif icant ly increased [compare Case VI*). Case V shows a similar result when storage i s reduced to one-half the design capacity at Grand Canyon. Intthis case considerable storage is transferred to the Clearwater reservoir to replace the loss at Grand Canyon.. Cases VI and VII show the results that would be obtained i f only Grand Canyon and Cariboo Fa l l s , or-Grand Canyon by i t s e l f , were included in the system. Here the regulated peaks are increased considerably. It is note-worthy that in these cases the storage use i s considerably less than the ava i l -able design capacity, especially when Grand Canyon is operated by i t s e l f . In a real operating situation i t would therefore be easier.: to operate Grand Canyon By i t s e l f than in combination with other reservoirs. Table 6.2 shows, for Case I, the reductions to the peak contributed by each reservoir. It is evident that Grand Canyon gives much the largest reduc-t ion , followed by Clearwater and Cariboo Fal ls . A better idea of the relative value of the reservoirs would be given by analyzing a number of large floods. It should be noted also that, because- Grand Canyon does not have abundant storage relative to inflow, the linear programming solution may cause i t to appear somewhat better than i t would in a real operating situation. The f inal result to note from the data in Table 6.1 is the reduction to the peak obtained by the linear programming method compared to the reduction obtained in the Fraser River Board studies. The peak regulated flow of 393,000 cfs. obtained in that study i s not directly comparable to the 416,000 cfs . computed In Case I, because the F.R.E. study included regulation at Bridge and McGregor reservoirs. An estimate may however be obtained of the increased *Case II. shows the result that would be obtained i f Grand Canyon, Cariboo Fa l l s , and Hemp Creek, only of the Clearwater reservoirs were Included in the system. Table 6.2 Flood Peak Reductions Site Site Reduction Grand Canyon Cariboo Falls Clearwater 84 21 55 Note: The values in this table apply to the design f lood, Case I. The reductions are in units of kefs. 35 reduction made available using the l inear programming method. In the F.R.B. study, fixed amounts of flow were allowed to pass each reservoir during the freshet. These were 30,000 cfs. at Grand Canyon, 3,400 cfs. at Cariboo Fa l l s , 6,700 cfs. at Hemp Creek, and 8,000 cfs . at BrctdgeoRiver. It was shown above, however, that in Case I a l l of the inflow at a l l three sites was stored at the time of the peak, so that the regulated peak could not be further reduced with the available inflow. If Bridge and McGregor had been included in the linear program i t would theoretically have been possible to reduce the peak below the F.R.B. value by 48,000 c f s . , the sum of the flow passed at the sites in the F.R.B. study. In fact the increased reduction would probably have been somewhat less than that amount - the lower peak would require a longer storing period which could cause storage to become the l imit ing factor at Grand Canyon and reduce the effectiveness of that reservoir. Nevertheless a figure of 30,000 to 40,000 c fs . i s reasonable. This corresponds to approximately 0.75 f t . to 1.0 f t . on the guage of the Fraser at Mission, which is a signif icant amount. In Chapter 3 i t was shown that the effect on the optimal value of the objective function of limited changes to the constraint parameters could be computed directly from the original solution to the program. This is called sensit iv i ty analysis. The UBC LIP program prints out the changes to the objecr-tive function per unit change in each of the constaint parameters, and the range of values of the parameters within which these rates of change apply. For example, in the cases and at the s ites where less than the available capacity was used Csee Table 6.1} the printout shows that varying the available capacity within the range Cstorage used, + inf in i ty ) would have zero effect on the optimal value of the regulated peak. In cases where the f u l l storage was used the marginal effectiveness of changing the storage capacity within a 36 limited range may be read directly from the printout. Thus the effects of involuntary storage on the regulated peak may be easily computed. Chapter 7 CONCLUSION This paper has described the application of l inear programming to the problem of operating a system of remote reservoirs in an optimal manner for maximum reduction of downstream flood levels. It has been shown that a reser-voir system can readily be modelled in terms of the l inear objective function and constraints of the linear program, subject to the use of l inear channel routing equations. Given the hydrographs at the sites and at the downstream control point, the method computes the minimum regulated peak at the control point theoretically possible, and the operating policy at the reservoir or reservoirs necessary to obtain the optimal regulation. Solutions are easily computed for varying reservoir s izes, locations, and combinations. The linear programming solution identif ies which reservoirs have abun-dant storage capacity relative to inflow volumes during the freshet, and which have limited capacity. At sites with abundant storage, the solution shows the minimum storing period required for optimum regulations, and the smallest reser-voir capacity which would give the same flow reduction. At sites with limited capacity relative to inflow the l inear program determines the operating procedure, which, in combination with the other reservoirs in the system, most effectively reduces, downstream flows. It was noted in Chapter 4 that the l inear programming method requires knowledge of the flow data for the entire freshet. This technique cannot there-fore be used in a real-time operating s i tuat ion, because suff ic ient ly accurate forecasts: for periods beyond a few days are not available. Nor can i t be used in a planning study to determine the actual flood control benefits expected from 37.» 3<3 a given reservoir system. The expected Benefits must Be determined using reser-voir operating policies which would apply in a real-time operating situation. The linear programming analysis gives results which are to some degree Better than what would in fact be attainable for any system containing s i tes with limited storage capacity. However, Because the linear programming results are nevertheless fa i r l y accurate estimates of the regulation which is in fact attainaBle, the method does have value in the early stages of a planning study in providing information about a proposed reservoir system. It provides a ready means of comparing the effectiveness of various combinations of reservoirs. It identifies, the minimum reservoir capacities required at proposed sites and therefore which reservoirs would be d i f f i c u l t to operate due to the physical l imitations of storage capacity at the s i t e . F inal ly , It provides a way of setting a theoretical bound to the regulation available from a given system with which operating policies subject to the errors of forecasting may be compared. 39 BIBLIOGRAPHY Johnson, W. A. Optimal Operation of an Upstream Reservoir for Flood  Control. Department of C iv i l Engineering, The University of Br i t ish Columbia, October 1970. Final Report of the Fraser River Board on Flood Control and Hydro- Electr ic Power in the Fraser River Basin. V ic tor ia , B. C : The Queen's Printer, September 1963. Henderson, F. M. Open Channel Flow. MacMillan, 1966, Quick, M. C , and A. Pipes. Non-Linear Channel Routing by Computer. Department of C iv i l Engineering, The University of Br i t ish Columbia Can unpublished paper}. Linsley, R. K., M. A. Kohler, and J . L. Paulfrus. Hydrology for Enqin- eersocj- New York,TToronto, and London: McGraw H i l l Co. Inc. L t d . , 1958. U.B.C. Users Group Program. A Linear Programming Package - LIP. D. O'Reil ly. University of Br i t ish Columbia Computing Centre, September, 1970. H i l l i e r , F. S . , and G. J . Lieberman. Introduction to Operations  Research. San Francisco: Holden-Day, Inc. , 1969. 40 APPENDIX A. Routing Equations - Channel Storage Channel routing in the UBC Flow Model is accomplished in two separate steps (see Quick and Pipes [4]). The f i r s t step computes the outflow from a reach assuming pure translation of the flood wave. Writing the equation for outflow on day j in terms of inflow on days j and j - 1 , the outflow is computed as (!) OT. - + ( 1 - T ) (I d where OT is the outflow from the reach computed from pure translation of the flood wave without considering channel storage I is the inflow to the reach T is the travel time of the flood wave through the reach j is the day for which the flow is to be computed The second step incorporates the effects of channel storage. The ouflow is computed as (2) 0. = 0. + (OT. - 0. ,) _L J J-1 J J - T 1+h where -K 0 is the outflow with the effect of channel storage included K is the storage constant Using average values of the time of travel T and storage constant K, a single equation may be developed for each reach which is a l inear function of the inflows to the reach. Substituting (1) into (2) yields °j = ° j - l + ^ j - l - (1 " T) (1^ - I^-,) - Oj.-,) _ L (1 - T) I, + J _ (T) I. , + J i - 0 . , 1+K J 1+K h': J " 1 1+K J " 1 or (3) 0 j = C0 Ij • I • C2 0J., 4.1 where C = J _ 0 - T h C, = Qt , C =JC u 1 T I F c-HK7 1 T+K L T+K Extending equation (3) to N days gives (4) Oj = C 0 Ij + CC-, + c 2 c o ) + ( c 2 c n + c 2 c 0 ) I._z + . . . + C 2 n " ^ C l ^-n+l + C 2 n " 1 9 j - n + l Since the values of C Q , C-| , and C2 are less than one, cubic terms and higher may be dropped. This gives a result of the form C5) 0. = ^ Ij + A2 £ A 3 I . j _ 2 + A 4 I j _ 3 etc. Given the general form of the equation as developed above, the equations for the individual reaches may be computed using the appropriate values of travel time T and storage constant K, and combined to give the equations re lat -ing the holdouts at the damsites to Hope. 42-APPENDIX R. Regulation of Design Flood This appendix; contains a sample of the results of a l inear programming analysis using the actual design flood data developed in the Fraser River Board flood control study, rather than the simplified hydrograph shapes used in the present study. The hydrographs presented here give a more rea l i s t i c picture of the reservoir operations that would be required to regulate the design flood than do those shown in Figure 5 .1. The amounts of storage used and the flood peak reductions are, however, s imilar . The unregulated flow data used for the analysis were taken from the Fraser River Board Final Report and studies carried out for that report, and the regulated results were taken from studies carried out by the Inland Waters Directorate, Pacif ic Region, Department of the Environment, as preliminary work for other flood control studies. FLOOD CONTROL ON THE F R A S E R RIVER FRB DESIGN FLOOD-CASE I May June FLOOD CONTROL ON THE FRASER RIVER FRB DESIGN FLOOD-CASE H 44 6 0 0 5 0 0 4 0 0 « 3 0 0 u O O O • E 2 0 0 a> - 8 0 o <j tn O 6 0 4 0 2 0 0 4 0 2 0 0 6 0 4 0 2 0 0 DISCHARGE HYDROGRAPHS AT MISSION RESERVOIR REGULATION AT GRAND CANYON . 1 RESERVOIR REGULATION AT CARIBOO FALLS RESERVOIR REGULATION AT CLEARWATER * 1 • • • * • • 1 i i i . . i . I • . . May June FLOOD CONTROL ON THE FRASER RIVER FRB DESIGN FLOOD - C A S E 3 Z H 45 7 0 0 6 0 0 5 0 0 2 4 0 0 1 u O O O 5 3 0 0 k_ o u tn £ 2 0 0 — Discharge prior to regulation by linear programming DISCHARGE HYDROGRAPHS AT MISSION 1 0 0 8 0 6 0 4 0 2 0 N a t u r a l d i scharge Regu la ted d i s c h a r g e RESERVOIR REGULATION AT GRAND CANYON * * 1 • * * • '• I I May June 

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