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Optimal operation of an upstream reservoir for flood control Johnson, Wayne Adrian 1970

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OPTIMAL OPERATION OF AN UPSTREAM RESERVOIR FOR FLOOD CONTROL by WAYNE ADRIAN JOHNSON B.A.Sc, The University of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 In p r e s e n t i n g t h i s t h e s i s in 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 a d v a n c e d degree at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a g r e e 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 p u r p o s e s may be g r a n t e d by the Head o f my Depar tment 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 . Depar tment o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date if /97e> A B S T R A C T This thesis describes the development of a method for deter-mining the most e f f i c i e n t way to operate an upstream flood control r e s e r v o i r for maximum flood peak reduction at a downstream point. Linear programming i s used as the optimization technique. A s i m p l i -f i e d case i s studied, namely that of a s i n g l e storage r e s e r v o i r approximately 500 miles upstream from the area to be protected. A channel routing method which was l i n e a r was required f o r use with l i n e a r programming. For t h i s reason a Muskingum type channel routing method was chosen. The r e s u l t s for the three years that were studied are presented i n graphical form. They i n d i c a t e the extent that the. downstream peak could be reduced and the operation of the upstream re s e r v o i r which would be required to bring about this reduction i n peak flow. Procedures for extending the technique to more complex systems and possible applications of the method are discussed. i i TABLE OF CONTENTS Page LIST OF TABLES i v LIST OF FIGURES v Chapter I . INTRODUCTION 1 I I . THE FLOOD CONTROL PROBLEM 5 I I I . CHANNEL ROUTING THEORY 9 IV. DETERMINATION OF ROUTING CONSTANTS 15 V. EMPLOYING LINEAR PROGRAMMING 27 VI. RESULTS 33 V I I . SUMMARY . . 41 BIBLIOGRAPHY 42 APPENDIX 44 i i i LIST OF TABLES Table Page IV.1 ROUTING PARAMETERS FOR THE DIFFERENT CASES 23 i v LIST OF FIGURES Figure Page 2.1 TYPICAL DISCHARGE HYDROGRAPH AT HOPE DURING FLOODING SEASON 5 2.2 TYPICAL DISCHARGE HYDROGRAPH AT SHELLEY DURING FLOODING SEASON 6 3.1 VARIOUS TYPES OF STORAGE DURING THE PASSAGE OF A FLOOD WAVE 10 4.1 MUSKINGUM STORAGE LOOPS 19 4.2 INFLOW AND OUTFLOW HYDROGRAPHS FOR A TYPICAL FLOOD . . . 19 6.1 HOLDOUTS NECESSARY TO ACCOUNT FOR INVOLUNTARY STORAGE . . 35 v A C K N O W L E D G E M E N T The author wishes to express h i s gratitude to his supervisor, Mr. S. 0. R u s s e l l , f o r h i s valuable guidance,and encouragement during the research, development and preparation of th i s t h e s i s . v i CHAPTER I INTRODUCTION The Fraser River originates i n the i n t e r i o r of B r i t i s h Columbia and empties into the P a c i f i c Ocean near Vancouver, B.C. Between Hope, B.C. and the r i v e r ' s mouth, i t flows through r i c h a g r i c u l t u r a l land and some highly i n d u s t r i a l i z e d areas, both of which are quite heavily populated. Due to the extensive development along the banks of the Fraser River, any excessive flooding of the r i v e r c a r r i e s with i t the threat of considerable property damage. The most recent flood of the Fraser River to cause s i g n i f i c a n t damage occurred on May 31, 1948. The r i v e r had a peak flow of 536,000 cfs at Hope which i s the highest recorded flow. The only flow to exceed t h i s value occurred i n 1894 when the flow was roughly estimated to be about 600,000 c f s . (The mean annual peak flow at Hope i s 313,000 c f s ) . This floo d of 1948 caused an estimated $20 m i l l i o n damage to the Lower Mainland area (the area between Hope and the r i v e r mouth). The extensive development on the Fraser River flood plains since 1948 has made the area much more v a l -uable and i t has been estimated that a flood of the 1948 magnitude would now cause approximately $200 m i l l i o n damage. The p o s s i b i l i t y of getting a flood of the 1948 magnitude i n the near future i s very r e a l . Several years since 1948 have had equal or greater accumulated snowpacks and had there been a sustained hot s p e l l at the c r i t i c a l time severe flooding would have occurred. 1 2 Due to the severe flood threat, a j o i n t F e d e r a l - P r o v i n c i a l Fraser River Flood Control Board was set up i n 1955 to study and report on the problems of flooding. The most s u i t a b l e permanent s o l u t i o n to the Fraser River flood problem would be the construction of one or more r e s e r v o i r s . A suggested so l u t i o n which had r e l a t i v e l y few supporters was the construction of a si n g l e very large dam at Moran, B.C. This would solve the Lower Mainland flooding problems but the idea was rejected as i t was in c o n f l i c t with the salmon f i s h i n g industry. The Fraser River supports a m u l t i - m i l l i o n d o l l a r salmon run every year. The construction of a large dam such as that pro-posed at Moran on the main stem of the r i v e r would eliminate a large part of the B.C. salmon f i s h i n g industry. As a r e s u l t , the Fraser River Flood Control Board suggested an a l t e r n a t i v e s o l u t i o n [1] of b u i l d i n g several smaller dams i n the headwaters of the r i v e r which would have a minimal i n -fluence on the f i s h while s t i l l being e f f e c t i v e for flood c o n t r o l , i f pro-perly regulated. The Fraser River f l o o d problem prompted the study described i n t h i s t h e s i s . A s i n g l e r e s e r v o i r near the flood plains i s r e l a t i v e l y sim-ple to regulate but several r e s e r v o i r s i n the headwaters remote from the area to be protected are much more d i f f i c u l t to operate for e f f i c i e n t flood c o n t r o l . To the present time there have been no adequate a n a l y t i c a l tools to determine the best method of c o n t r o l l i n g upstream reservoirs or for comparing the r e l a t i v e effectiveness of reservoirs at d i f f e r e n t l o c a -tions . In t h i s study, which represents a f i r s t step i n developing such an a n a l y t i c a l procedure, a method i s developed for f i n d i n g the most e f f e c -3 t i v e way of operating a flood control r e s e r v o i r given the natural flows at the reservoir s i t e and the l o c a t i o n to be protected. The procedure which i s used i s not l i m i t e d by the number of reservoirs nor by t h e i r l o c a t i o n . That i s , complex systems can be handled as r e a d i l y as simple systems. Such a procedure w i l l eliminate many of the approximations now required for both planning and operating remote res e r v o i r s for flood con-t r o l . Problem Formulation The method makes use of l i n e a r programming which i s one of the most powerful optimization techniques a v a i l a b l e and one for which very good computer routines e x i s t . However, the f a c t that l i n e a r programming i s being used imposes c e r t a i n demands on the types of problem that can be handled. In p a r t i c u l a r , a l l the terms i n the objective function and the constraints must be l i n e a r . Consequently, i t i s necessary to formulate the problem to meet these requirements. I n i t i a l l y the problem has been broken down to a simpler form which s t i l l incorporates the e s s e n t i a l features of a flood regulation problem. A s i n g l e t h e o r e t i c a l r e s e r v o i r has been assumed at Shelley, B.C., approximately 500 miles upstream from Hope on the main branch of the Fra-ser River. The reservoir has a given storage volume and an unlimited d i s -charge capacity. The object i s to use the r e s e r v o i r volume as e f f i c i e n t l y as possible to minimize the maximum flow at Hope. The study i t s e l f can be divided into several d i s t i n c t parts. The f i r s t part deals with channel routing and the determination of constants 4 for use i n routing equations for the chosen reaches of the r i v e r . The second part describes the computation procedure which made use of a l i n e a r programming formulation and the f i n a l part discusses the r e s u l t s and p o s s i -ble applications and future extensions of the method. CHAPTER II THE FLOOD CONTROL PROBLEM The problem which requires optimizing can best be shown with the aid of several discharge hydrographs. There are two hydrographs of importance: one at Hope, B.C. (the downstream hydrograph), and the one at the l o c a t i o n of the hypothetical r e s e r v o i r which i n this case i s Shelley, B.C. (the upstream hydrograph). A t y p i c a l hydrograph at Hope during the flooding season has the shape shown below. i I i April May June July I •Fig. 2.1 A Typical Discharge Hydrograph at Hope During Flooding Season. 5 A corresponding t y p i c a l upstream hydrograph would have the shape shown i n F i g . 2.2. By storing the correct amount of water i n the re s e r v o i r at the correct time (as depicted by the dotted l i n e i n F i g . 2.2) the peak flow at Hope can be reduced a maximum amount as shown by the corresponding dotted l i n e i n F i g . 2.1. It should be emphasized that only i f the reser v o i r i s operated i n the exact optimal manner w i l l the peak flow at Hope be reduced by the maximum amount. By using l i n e a r programming t h i s condition i s e a s i l y s a t i s f i e d . The l i n e a r programming s o l u t i o n gives the new maximum peak flow at Hope and i t also gives the d a i l y values to be stored at the 7 r e s e r v o i r to ensure that t h i s maximum peak i s not exceeded. In order to determine the e f f e c t that a flow at Shelley w i l l have on the flow at Hope, i t i s necessary to route the Shelley flows down the r i v e r channel to Hope. (Due to channel storage, the flood flow i s de-layed and reduced and i s not simply tra n s l a t e d ) . In the present study i t i s not p o s i t i v e flows but the holdouts at the r e s e r v o i r , or negative flows, that must be routed down the r i v e r . However, since a l i n e a r rout-ing method i s used, i t doesn't matter whether the flows are p o s i t i v e or negative. The negative flows are simply routed down the r i v e r channel us-ing the constants derived from the analyses of p o s i t i v e flows and these routed values are then subtracted from the natural hydrograph at Hope to obtain the new hydrograph. The ordinate Q' shown i n F i g . 2.1 i s a discharge value of the new hydrograph at Hope (flood control included). This ordinate i s made up of two components: the ordinate of the natural hydrograph at Hope (Q ) n minus the routed holdouts at Shelley (D ). That i s , n Q' = Q - D (2-1) n n n D^ i s the routed holdouts for day 'n' and i s composed of: D = c.H + c„H . + c.H „ + . . . . + c, H , n I n 2 n-1 3 n-2 k n-k+1 (2-2) where the H's represent d a i l y holdouts. Combining these equations, Q' = Q " [c.H + c 0H + c H 0 + . . n ' . n I n 2 n-1 3 n-2 new natural hydrograph hydrograph + c]jH (2-3) ordinates ordinates routed holdouts 8 This i s the b a s i c equation f o r determining the new hydrograph at Hope. This f i r s t p a r t of the study was concerned w i t h determining the con-s t a n t s c , c , . . . ,c and the second part was concerned w i t h f i n d i n g the values of the holdhout H , H , , . . . H , ,,. n n-1 n-k+1 When the r o u t i n g constants were being derived using r e a l flows of record, i t was necessary to be c a r e f u l w i t h t r i b u t a r y flows so that the volume of i n f l o w could be made equal to the volume of outflow over the per-i o d i n question. When r o u t i n g negative flows t h i s i s not necessary and no concern .need be paid to t r i b u t a r y f l o w s . A l l t r i b u t a r y flows are accounted f o r by the n a t u r a l hydrograph at Hope. I t ' i s only a negative flow from one par t of the r i v e r which i s being routed. A l l other flows i n the r i v e r , apart from the holdouts, c o n t r i b u t e to the n a t u r a l hydro-graph at Hope e x a c t l y the same as before. This f o l l o w s from the assump-t i o n of l i n e a r i t y . Thus, as long as the constants were o r i g i n a l l y derived t a k i n g account of t r i b u t a r y f l o w s , t r i b u t a r y flows need no longer be con-si d e r e d i n r o u t i n g the holdouts. CHAPTER III CHANNEL ROUTING THEORY* Flood routing, whether i t be channel routing or r e s e r v o i r routing, can be defined as the procedure whereby the hydrograph of a flood wave at a point on a : stream i s determined from known or assumed data at one or more points upstream. As the discharge i n a channel increases, stage also increases and with i t the volume of water i n temporary storage i n the channel. During the following portion of a flood an equal volume of water must be released from storage. As a r e s u l t of t h i s , a flood wave moving down a channel appears to have i t s time base lengthened and i t s crest lowered. Channel routing i s simply an a n a l y t i c a l technique used i n Hydrology to compute the e f f e c t of channel storage on the shape and movement of a flood wave. Routing i s the s o l u t i o n of the storage equation which i s an expression of continuity, I - 0 = | | (3-1) where I = inflow, 0 = outflow, d S — = change i n storage/unit of time. This can be converted to a more usable form as shown below. -S--2- • t — ' t = S 2 - S± (3-2) 9 See [2], [3] and [4] i n the bibliography. 10 where the subscripts 1 and 2 represent the beginning and end, respective-l y , of the time i n t e r v a l ' t ' . For r e s e r v o i r routing there i s a second equation r e l a t i n g storage and discharge which i s of the form S = f(0) (3-3) where S = storage, 0 = outflow. That i s , storage i s a function of outflow only. Channel routing i s made more complicated by the fa c t that storage i s not a function of outflow only as i n r e s e r v o i r s . This can be seen from a s i m p l i f i e d p r o f i l e of a r i v e r channel as shown i n Figure 3.1." Negative F i g . 3.1 Var ious Types of Storage During The Passage of a Flood Wave. The storage beneath a l i n e p a r a l l e l to the r i v e r bed i s c a l l e d 'prism' storage while the storage between the p a r a l l e l l i n e and the actual p r o f i l e i s c a l l e d 'wedge' storage. During r i s i n g stage, a large volume of wedge storage may e x i s t before any appreciable increase i n outflow occurs. During f a l l i n g stage, the inflow decreases much more quickly than the out-11 flows and hence the wedge storage volume becomes negative. Routing i n channels requires a storage r e l a t i o n s h i p which adequately represents t h i s wedge storage. This i s usually done by including inflow as a para-meter i n the storage equation. Thus, storage becomes a function of i n -flow and outflow. Although there are several d i f f e r e n t channel routing procedures they are a l l based on some knowledge of the r i v e r reach under consideration. In some instances t h i s knowledge may be l i m i t e d to that available from top-ographic maps, or i t may include a f a i r l y complete h i s t o r y of past floods, as i s the case with the Fraser River. As has previously been shown, routing procedures are generally based on the r e l a t i o n s h i p between discharge and storage. The most common method used to obtain t h i s r e l a t i o n s h i p i s by the analysis of floods of record, assuming that the r e l a t i o n s h i p s established w i l l be v a l i d f o r future flows. The data necessary for t h i s analysis are the flow records at the upstream, downstream, and t r i b u t a r y gauges, and t h i s was used i n the present analysis. Although there are various d i f f e r e n t channel routing procedures a v a i l a b l e , they are b a s i c a l l y the same, the differences a r i s i n g to a con-siderable extent from minor v a r i a t i o n s in algebraic manipulation or from refinements i n the basic assumptions. The choice of a p a r t i c u l a r rout-ing method depends on several f a c t o r s , such as the nature of the a v a i l -able data, personal preference, accuracy requirements, and intended a p p l i -cations of the r e s u l t s . The l a t t e r proved to be the most important f a c -tor i n the present study. The routing method that was chosen had to meet two important 12 requirements. Since l i n e a r programming requires that the objective func-t i o n as well as a l l of the constraints be l i n e a r , and the derived routing equations were going to be used as constraints, a completely l i n e a r rout-ing method was required. Also, i t was necessary that one be able to com-bine the d a i l y routing equations so that outflows from a reach could be expressed as a function of inflows only. The routing method which best met these requirements was a Muskingum type of channel routing. As do a l l channel routing methods, the Muskingum method solves the continuity and storage equations simultaneously. The continuity equa-t i o n used i s given by equation (3-2) and the storage equation used i s given below. S = K [x l + (1-x) 0] (3-4) or S 2 - S 1 = K [x ( I 2 - I 1 ) + (l-xKC^-C^)] (3-5) where S = storage, I = inflow, 0 = outflow. x i s a constant which expresses the r e l a t i v e importance of inflows and outflows i n determining storage. K i s known as the 'storage constant' and i s the r a t i o of storage to discharge, with the dimension of time. (K i s approximately equal to the t r a v e l time through the reach). If equation (3-5) i s substituted for ^ - S ^ i n equation (3-2), the r e s u l t i n g equation becomes: 13 = C I , + C.I. + c_o. o 2 1 1 2 1 (3-6) where C Kx - 0.5t o K - Kx + 0.5t Kx + 0.5t (3-7) K - Kx + 0.5t K - Kx - 0.51 K - Kx + 0.51 By a d d i n g the e q u a t i o n s f o r C , and C^, i t w i l l be found t h a t + + = 1. E q u a t i o n (3-6) shows the Muskingum r o u t i n g e q u a t i o n i n the form t h a t i t i s g e n e r a l l y a p p l i e d . I t can be seen t h a t i n o r d e r t o use t h e e q u a t i o n t h e c o n s t a n t s C , C. , and C„ must f i r s t be d e t e r m i n e d . T h i s i s o 1 2 done by c h o o s i n g a time i n t e r v a l , t , and h a v i n g p r e v i o u s l y s o l v e d f o r K and x ( t o be d i s c u s s e d l a t e r ) s u b s t i t u t i n g i n t o e q u a t i o n s (3-7) t o o b t a i n the r e q u i r e d c o n s t a n t s . Thus, d e t e r m i n i n g the r o u t i n g c o n s t a n t s f o r a r e a c h e s s e n t i a l l y r e d u c e s down t o s o l v i n g f o r K and x f o r t h a t r e a c h . s i o n c o n t a i n i n g , s a y , 0^, 0 and i n f l o w v a l u e s c o u l d be o b t a i n e d as shown below. By com b i n i n g the r o u t i n g e q u a t i o n s f o r s e v e r a l days an e x p r e s -0 (3-8) 14 Since , and are a l l less than unity, w i l l be a very small num-ber and thus the 0^ term should have a n e g l i g i b l e contribution to 0^. (If the 0^ term i s not n e g l i g i b l e , the process can be extended to include more days u n t i l i t becomes so). Eliminating 0 and combinining c o e f f i -cients , 0^ can be written as: °4 " V 4 + V 3 + V2 + V i ( 3'" 9 ) Thus, the outflow from the reach i s now expressed as a function of i n -flows to the reach only. The a b i l i t y of the chosen routing method to lend i t s e l f to t h i s technique was very important i n the a p p l i c a t i o n to the l i n -ear programming part. CHAPTER IV DETERMINATION OF ROUTING CONSTANTS General From Shelley (the l o c a t i o n of the hypothetical reservoir) to Hope, the Fraser River travels approximately 500 miles, which i s much too far for a s i n g l e reach using the Muskingum routing equation. For t h i s reason the r i v e r was divided into three reaches, approximately equal i n length. (They were not exactly the same length as each reach started and ended at a well-established discharge gauge and these gauges were not evenly spaced). The d i v i s i o n into three d i f f e r e n t reaches meant that three sets of Muskin-gum constants had to be evaluated, one set for each reach. In order to use the Muskingum routing method (or to evaluate the constants) the t o t a l inflow volume to a reach must equal the t o t a l outflow volume from the same reach over the duration of a flood ( i . e . , C + C + o 1 C 2 = 1). This means that e i t h e r there must be no tr i b u t a r y flows through-out the reach or else the t r i b u t a r y flows must be separated out and the main stem inflows adjusted so that the inflow volume does equal the out-flow volume. It was the presence of t r i b u t a r y flows which made the routing con-stant d i f f i c u l t to determine on the Fraser River. Throughout i t s entire length, the Fraser River has numerous t r i b u t a r y streams, some of which are quite substantial i n size compared to the flow i n the main stem. It i s the t r i b u t a r y flows which can cause the routing constants to show 15 16 v a r i a b i l i t y from year to year. If a l l the t r i b u t a r y flows and the main stem flow had the same r e l a t i v e values each year, then the routing con-stants would be consistent. However, due to the v a r i a b i l i t y i n c l i m a t i c conditions over t r i b u t a r y basins, there w i l l not be a d i r e c t r e l a t i o n s h i p between flows and the routing constants w i l l appear to vary from flood to f l o o d . This proved to be the biggest s i n g l e problem i n obtaining the Muskingum routing c o e f f i c i e n t s . Due to the inaccuracies i n the constants which l a t e r a l inflows could cause, a simple method of pro-rating the inflows so that the i n -flow volume would equal the outflow volume was needed. This was done by choosing the l a r g e s t t r i b u t a r y flow i n the reach and adding the t r i b u -tary flow times a constant to the main stem inflows. This constant was obtained by using the following formula: ^ ^Outflows - i n f l o w s (4-1) ^ T r i b u t a r y Flows where the summations extended over a 115 day period. The above method gave a set of inflows and outflows (with inflow volume equal to outflow volume) which could be used i n an analysis to determine the routing constants. A l l that remained was to choose a method for t h i s a n a l ysis. There are several alternate ways of carrying out the analysis. Three methods were t r i e d as discussed below. Regression Analysis On f i r s t looking at the Muskingum routing equation i t would appear as though i t were Ideall y suited to multiple regression techniques [5] to 17 determine the routing c o e f f i c i e n t s . A multiple regression equation has the form: Y i " b o + V i i + b 2 X 2 i + b 3 X 3 i ( 4 " 2 ) where b , b., b_ = regression coefficients, o 1 z X^, , = independent variables, Y = dependent variable. M u l t i p l e Regression packages are available which w i l l carry out a regres-sion analysis without the constant b Q c o e f f i c i e n t . The form of the regres-sion equation then becomes Y. = b.X 1. + b„X 0. + b . X - . (4-3) x l l x 2 2 i 3 3x This i s analagous with the routing equation 0. = C I. + C.I. . + C.O. . (4-4) x o x 1 i - l 2 i - l Thus, by l e t t i n g 0. be the dependent v a r i a b l e and I., I. .. 0. . the i n -J ° x l i - l , i - l , dependent v a r i a b l e s , a multiple regression analysis can be c a r r i e d out and the c o e f f i c i e n t s C , C_ and C„ w i l l be determined f o r the p a r t i c u l a r o 1 2 set of data. Although t h i s would appear to be a quick and e f f i c i e n t method, the r e s u l t s proved to be incompatible with the r e a l system. When the best routing c o e f f i c i e n t s were obtained, at l e a s t one of the c o e f f i c i e n t s would con s i s t e n t l y come out negative. It was usually that had the negative value. Any of the c o e f f i c i e n t s coming out negative i s not consistent with what i s a c t u a l l y happening. It i s not f e a s i b l e f o r any of the va r i a b l e s 18 I., I. , , or 0. , to have a negative e f f e c t on the outflow 0.. They must l i - l i - l l have a p o s i t i v e e f f e c t or no e f f e c t at a l l . The method was abandoned without any further i n v e s t i g a t i o n . Storage Versus Weighted Discharge Method With t h i s method the values of K and x are f i r s t determined sim-ultaneously and then the routing c o e f f i c i e n t s , C , C. and C„ are calcu-J o 1 2 lated by s u b s t i t u t i n g into equations (3-7). By combining equations (3-2) and (3-5) we get 0.5t [(I± + I 2 ) - (0± + 0 2 ) ] = K [ x ( I 2 - I 1 ) + (1 - x) (0 2 - 0 1 ) ] (4-5) 0.5t [.(!, + I 2 ) - (0 1 + 0 2 ) ] AS (4-6) x ( I 2 - I 1 ) + (1 - x) (0 2 - 0 1) AQ The numerator represents the storage increment (AS) while the denominator represents the weighted flow increment (AQ). Successive values of the storage increment and the flow increment are computed using known i n -flows and outflows (which have been adjusted for volume) from a given flood of record. This procedure i s c a r r i e d out for various d i f f e r e n t values of the parameter x. The computed values of the accumulated storage increments and the accumulated flow increments are then p l o t t e d , i d e a l l y producing curves i n the form of loops as shown i n F i g . 4.1. 0 0 Accumulated Storage F i g . 4.1 Muskingum Storage Loops . The value of x that r e s u l t s i n a loop clo s e s t to a s i n g l e l i n e i s accepted as the correct value. The value of K i s given by the r e c i p r o c a l of the slope of the loop most c l o s e l y forming a sin g l e l i n e . Although t h i s would appear to be a good method i t did not work well on the Fraser River. In order to make the loops close i t was necessary to f i n d a flood that started and ended at the same flow value as shown i n Figure 4 . 2 . Inflow Hydrograph Outflow Hydrograph TIME Fig . 4 .2 Inflow and Outflow Hydrographs for a Typical Flood 20 If the outflow and inflow curves crossed at a d i f f e r e n t flow l e v e l at the beginning and end of the flood (as was usually the case) the loops would not close regardless of the x value chosen. Floods which met t h i s requirement were extremely d i f f i c u l t to f i n d on the Fraser River (espe-c i a l l y floods of high magnitude). Another problem was f i n d i n g floods composed of a si n g l e peak. If the flood peaked, receded and then peaked again at a d i f f e r e n t value, the loops would have secondary loops formed i n them. These two problems made t h i s method of determining routing constants impractical for the Fraser River. Sampling Technique The t h i r d and f i n a l method that was t r i e d i n determining rout-ing constants was a very simple sampling technique. Although the method may appear to be the l e a s t r e f i n e d of the three, i t proved to give the best r e s u l t s i n the shortest amount of time. The method consisted of choosing a possible range of x values (0.0 to 0.4 i n steps of .05) and a range of K values (0.6 to 2.5 i n steps of .1). A value for both x and K was then chosen, and using t h i s x and K value the routing constants , and for the reach were calculated using equations (3-7). The inflows were then routed down the reach, using the Muskingum routing equation to obtain a series of predicted outflows. The predicted outflows were subtracted from the corresponding known outflows to obtain a re s i d u a l value. The re s i d u a l values were squared and summed to give a value which r e f l e c t e d the accuracy with which the routing was car r i e d out. This procedure was employed for a l l 21 possible combinations of x and K. The values of x and K which gave the lowest sum of squares of the residuals were taken as the best values for that p a r t i c u l a r reach. Without the aid of a high speed computer t h i s method would have been hopeless due to the large number of cal c u l a t i o n s required. However, computers have made techniques such as t h i s p r a c t i c a l and e f f i c i e n t . Due to the l a t e r a l inflows and the nature of the Fraser River i t was i n i t i a l l y suspected that the r i v e r would not exhibit l i n e a r routing parameters. In p a r t i c u l a r , i t was expected that the storage constant K would vary depending upon the flow. Due to t h i s expected v a r i a b i l i t y , the flow was divided into l e v e l s which were 50,000 cfs wide. That i s , a best value for K and x were determined for the flow values l y i n g i n the range of 0 cfs - 50,000 c f s ; another best K and x were determined for the flow values f a l l i n g i n the range of 50,000 cfs - 100,000 c f s , and so on. By examining the best parameters i n each l e v e l i t was hoped to deter-mine the r e l a t i o n s h i p s which existed i n the r i v e r channel. It was found that the value of K varied quite e r r a t i c a l l y and the three years of data that were analysed proved i n s u f f i c i e n t to e s t a b l i s h any r e l a t i o n s h i p . However, the r e l a t i o n s h i p f or x proved to be much more consistent, with the best value always turning out to be zero. This was rather su r p r i s i n g as a value around .2 or .3 would be expected f o r a r i v e r of t h i s type. A value of zero f o r x means that storage i n the r i v e r i s a function of out-flow only, which implies that we have reservoir routing instead of channel routing. Since the use of d i f f e r e n t constants for d i f f e r e n t flow l e v e l s did 22 give better answers, i t would have been desirable to use th i s d i v i s i o n of flow values. However, t h i s would have complicated the l i n e a r program-ming considerably and i t was f e l t that the extensive changes required of the l i b r a r y routine were not warranted at t h i s stage of the study. The l i n e a r programming i s only applied during periods of flooding so that i t should be necessary to analyse only the flows which make up the peak of the flood to determine routing constants. Consequently, only records of high flows were used to evaluate the routing constants and i t was hoped that this method would minimize the non-linear e f f e c t without increasing the complexity of the l i n e a r programming. The above technique was car r i e d out i n the three d i f f e r e n t reaches f o r the three d i f f e r e n t years which were analyzed. Since i t was found that the parameter K varied considerably from year to year, minimum, maximum and average values that were used i n the routing are shown i n Table IV.1. For a p a r t i c u l a r reach, there was a maximum and a minimum value of K (which represented a v a r i a t i o n i n the re s i d u a l sum of squares of approximately three per cent). Since three years' data were analyzed, there were three maximum and three minimum values of K. The maximum and minimum values i n Table IV.1 represent the highest maximum and the lowest minimum for each reach. The average K values were weighted taking into account such factors as which K value turned up most often and what the best K value was as represented by the smallest r e s i d u a l sum of squares. The d i f f e r e n t values that K could take were divided into cases 1, 2 and 3 as shown i n the following table, f or l a t e r use i n a s e n s i t i v i t y a n a l y s i s . 23 TABLE IV.1 ROUTING PARAMETERS FOR THE DIFFERENT CASES REACH x MAX K [case 1] AVERAGE K [case 2] MIN K [case 3] 1 0.0 2.5 1.6 1.0 2 0.0 2.3 1.3 •8 3 0.0 2.5 1.6 .9 Conversion of Constants Once the Muskingum iconstants were obtained i t was necessary to convert these constants into ones which could be applied to the routing equations where the outflow i s expressed as a function of inflows only. This could be done by a s u b s t i t u t i o n procedure but a simpler method was used. To do th i s a set of inflows as shown below was devised. n-4 n-3 n-z n-1 n n+l n+2 n+3 . . . . 1000 1000 1000 1000 0 0 0 0 . . . . The value of 1000 i s completely a r b i t r a r y and was chosen only because i t was a convenient number. The inflow values of zero are not a r b i t r a r y and they must be made equal to zero. The above inflows are routed down the f i r s t reach using the Muskingum routing equation with the previously determined Muskingum routing constants applicable to the f i r s t reach. The outflows from the f i r s t reach are then used as the inflows to the 24 second reach and the routing procedure i s repeated, t h i s time on the second reach using the corresponding Muskingum constants. This procedure i s continued u n t i l the outflows from the t h i r d reach are obtained. It w i l l be r e c a l l e d that the Muskingum equation has the form 0 = C.I + C.I . + C O . (4-7) n I n 2 n-1 3 n-1 In order to s t a r t the routing procedure a value f o r 0 , must be known, n-1 for each of the three reaches. This value can be a r b i t r a r i l y chosen. However, i t i s better to choose a value of 1000 for each of the three unknown outflows as the routing procedure w i l l then s t a b i l i z e i n one i t e r a t i o n . ( The process i s considered to have s t a b i l i z e d when the out-flows from the t h i r d reach have a constant value of 1000 - the same as the inflows to the f i r s t reach). If a value other than 1000 i s chosen, more inflow values of 1000 w i l l be required u n t i l the process does s t a b i l i z e , and the further the chosen value i s from 1000 the larger w i l l be the number of i t e r a t i o n s u n t i l s t a b i l i z a t i o n occurs. Once s t a b i l i z a t i o n has occurred the outflow from the t h i r d reach w i l l have a constant value of 1000. Then, as soon as an inflow value of zero i s encountered, the outflows begin decreasing. For i l l u s t r a t i v e purposes a hypothetical set of outflows i s shown below. n - i n-2 n-1 n n+1 n+2 n+3 1000 1000 1000 980 940 860 800 Each decrease i s caused by another value of the inflow becoming zero and i t i s these decreases which make i t possible to calculate the 25 new c o e f f i c i e n t s . This can be shown best by the use of several equations, 0 = 1000 = c. T + c I + c I + . . . + c ! (4_ 8) n-1 1 n-1 2 n-2 3 n-3 te n-k 0 = 980 = c I + c I + . . . + c i ( 4 _ 9 ) n / I n 2 n-1 3 n-2 R n-k+1 The only factor which could cause the decrease of 20 units i n the out-flow would be l n equation (4-8) since the equations are i d e n t i c a l i n a l l other respects. (The fact that the constants are multiplying d i f f e r e n t inflow values makes no differ e n c e since a l l inflows are either zero or 1000. I f I and I . are both 1000, c- I _ has the same value n-1 n-2 2 n-1 as c o l „ ) . It can then be said that L n—/ 0 , - 0 =0.1 n n-1 n 1 n-1 . c 0 - 0 _ n-1 n n-1 1000 - 980 1000 = .02 S i m i l a r i l y , f o r day 0 n + ^ : n+l ^£ n+l ^Zn 3 n-1 k n-k+2 (4-10) The only difference between equation (4-9) and (4-10) i s that i n equation (4-9) C2"'-n_i ^ a s a value and i n equation (4-10) i t i s zero. Therefore, 26 i t must be c I '2 n-1 that i s accounting f o r the difference between 0 and n 0 n+1' Hence: 0 - 0 n n+1 c I " 2 n-1 c 0 - 0 n+1 2 n 1 i n-1 980 - 940 1000 .04 In a s i m i l a r manner as many constants as i s required to make them add up to one (or s u f f i c i e n t l y close to one) can be determined. Choice of Routing Method It i s r e a l i z e d that i f one were tr y i n g to choose the "best" method for routing flows i n the Fraser River the Muskingum method might not be chosen. For the present study, however, the prime require-ment i s for a " l i n e a r " routing method to provide data i n a foi~m s u i t a b l e for l i n e a r programming. The method described above met that need. CHAPTER V EMPLOYING LINEAR PROGRAMMING General Linear Programming [7] i s a mathematical technique used to o p t i -mize an objective i f t h i s objective can be expressed as a l i n e a r function and the :constraints can be expressed as l i n e a r e q u a l i t i e s or i n e q u a l i t i e s . A l l problems i n l i n e a r programming, when expressed mathematically, are si m i l a r to the general form shown below. Maximize Z = c.x, + c„x„ + . . . + c x (5-1) 1 1 2 2 n n subject to: a ^ x + a., „x + . . . + a, x < b 1 11 1 12 2 In n — 1 a„,x + a„„x + . . . + a x < b. 21 1 22 2 2n n — 2 • • a ..x + a „x. + . . . 4- a x < b (5-2) ml 1 m2 2 mn n — m x, > 0, x > 0, . . . x > 0 1 — z — n — Z represents the optimal value and equation (5-1) i s any l i n e a r expres-sion containing any number of v a r i a b l e s . Equations (5-2) are the l i n e a r c o n s traints, a l l of which must be s a t i s f i e d simultaneously while an opt i -m a ± s o l u t i o n i s determined. An optimal, s o l u t i o n may be considered a so l u t i o n which maximizes the objective function or i t may be a so l u t i o n which minimizes the objec-27 28 t i v e function. Both are optimal, solutions and i t depends on the p a r t i c u l a r problem that i s being solved j u s t which type of optimal s o l u t i o n i s de-s i r e d . The above example i s one i n which the objective function i s to be maximized. However, t h i s problem could very e a s i l y be changed to a minimization problem by changing the sign of the objective function. If l i n e a r programming problems are small they can quite often be solved using graphical techniques or by manually carrying out the sim-plex method of s o l u t i o n . However, large l i n e a r programming problems are v i r t u a l l y impossible to solve by hand and require computer sol u t i o n s , which usually solve using the simplex method or some v a r i a t i o n of i t . The computer program which was used i n the present study was a new rou-t i n e [6] which has ju s t recently been made ava i l a b l e to the U.B.C. l i b r a r y routines. This program o f f e r s very fast solutions to quite large l i n e a r programming problems at reasonable cost and proved to be an extremely valuable asset. In the present problem of optimizing flood c o n t r o l , the objective function and a l l of the constraints could quite e a s i l y be expressed as l i n e a r equations so that l i n e a r programming proved to be an i d e a l o p t i -mization technique. Formulation of Constraints In order to discuss the development of the constraints, i t i s f i r s t necessary to outline b r i e f l y the form of the objective function. The objective function can be expressed simply as 29. Min Z = Y where Y i s a dummy v a r i a b l e . Since i t i s the maximum flow at Hope that i s to be minimized by storing water i n the re s e r v o i r , i t can be seen that Y must represent the maximum value of the natural flow at Hope minus the routed holdout, i . e . , (Q - D ) . For each day of the flood there i s n n max going to be a value of Q - D and i t i s the f i r s t set of constraints n n which ensures that the maximization a c t u a l l y does take place with (Q - D ) n n These constraints are: max Y > Q - [c.1 + c„I + c,I _ + . . . + c. I . ...] — in 1 m 2 m-1 3 m-2 k m-k+1 Y > (1 - [c.I + c.I + c.I . + . . . + c. I . ..] — Tii+1 1 m+1 2 m 3 m-1 k m-k+2 : : : ( 5 - 3 ) Y > Q - [ C l I + c-I . + C _ I 0 + . . . + c. I . , J — n I n 2 n-1 3 n-2 k n-k+1 where Q = natural discharge at Hope, I = d a i l y holdouts, °1' °2' ' " * " Ck = rout-'-n8 constants. These constraints ensure that the maximum value of Q - D i s chosen f or n n the objective function. The second set of constraints concerns the li m i t e d amount of water entering the r e s e r v o i r . Sine the d a i l y holdouts cannot possibly be greater than the d a i l y flow into the r e s e r v o i r , the l i n e a r programming must be constrained so as to choose a holdout less than or equal to the 30 inflow f o r that day. This i s accomplished by the following set of equa-tions . H l ^ l ( 5 - 4 ) H < I n — n where H = value of d a i l y holdouts, I = natural d a i l y inflows to the r e s e r v o i r . A t h i r d constraint (only becomes'a set of constraints when there i s more than one reservoir) r e s t r i c t s the t o t a l amount of water stored i n the r e s e r v o i r . Obviously the volume of water stored cannot exceed the volume of the r e s e r v o i r . The following constraint makes sure t h i s con-d i t i o n i s s a t i s f i e d . E1 + H 2 + H 3 + + H < Volume of Reservoir n — ( 5 - 5 ) Using the objective function and the constraints described above, the l i n e a r programming w i l l carry out a s o l u t i o n which w i l l y i e l d the minimum peak flow at Hope and the values of the d a i l y holdouts to achieve t h i s minimum peak. 31 Routing Holdouts Although the l i n e a r programming s o l u t i o n gives considerable i n -formation, i t does not d i r e c t l y give the shape of the downstream hydro-graph.: In order to get t h i s the holdouts must be routed down the r i v e r using the same routing constants that were used i n the l i n e a r programming. These routed values are then subtracted from the natural hydrograph to get the new hydrograph with flood control included. Factors L i m i t i n g Peak Flow Reduction . The extent of the reduction i n peak flow that w i l l occur at Hope w i l l be governed by two d i f f e r e n t and unrelated r e s t r i c t i o n s ; of which one or the other w i l l be the c o n t r o l l i n g r e s t r i c t i o n depending upon the circumstances. One of these r e s t r i c t i o n s w i l l be due to the f i n i t e storage capacity of the hypothetical r e s e r v o i r at Shelley. The reser-v o i r can only store a l i m i t e d amount of water and once i t has f i l l e d there can be no more reductions i n the flow at Hope. However, t h i s does not mean that everything w i l l be stored u n t i l the r e s e r v o i r i s f u l l and noth-ing stored from then on. The l i n e a r programming sol u t i o n determines the most advantageous way of making use of the l i m i t e d storage so that the peak flow i s reduced a maximum amount while the r e s e r v o i r does not f i l l u n t i l the end of the f l o o d , i f at a l l . The second r e s t r i c t i o n i s due to the l i m i t e d amount of water flowing into the r e s e r v o i r on any p a r t i c u -l a r day. The flow at Hope cannot possibly be reduced by more than the value which i s stored at the r e s e r v o i r , which i s i t s e l f l i m i t e d by the 32 amount of flow entering the r e s e r v o i r . Thus, i f a l l the water entering the r e s e r v o i r i s stored, t h i s obviously w i l l set a l i m i t on the reduction i n flow at Hope, regardless of the amount of water i n the r e s e r v o i r . These two r e s t r i c t i o n s are going to govern the problem at d i f f e r e n t times, depending upon the type of f l o o d . If the flood i s f a i r l y high and l a s t s f o r a considerable length of time i t w i l l be the f i n i t e r e s e r v o i r volume which w i l l l i m i t the peak flow reduction. However, i f the flood only l a s t s f o r a short length of time, then the maximum amount that can be stored on any p a r t i c u l a r day ( l i m i t i n g inflow hydrograph) w i l l probably be the governing l i m i t a t i o n . CHAPTER VI RESULTS The technique of optimizing flood control that has been described was carried out on three d i f f e r e n t floods of record on the Fraser River. The three years that were used were 1955, 1964 and 1967, these years consisting of the three highest recorded flows since the 1948 f l o o d . The r e s u l t i n g hydrographs at Hope as well as the hydrographs showing the operation of the r e s e r v o i r can be seen i n the Appendix. Since the routing constants for any p a r t i c u l a r reach did vary from year to year, three d i f f e r e n t sets of routing constants were used to test the s e n s i t i v i t y of the method to d i f f e r e n t values. The three d i f f e r e n t sets of routing con-stants resulted i n three d i f f e r e n t downstream hydrographs and three d i f f e r -ent methods of regulating the upstream reservoir f or each year analyzed. These three d i f f e r e n t s i t u a t i o n s are depicted by cases 1, 2 and 3. The routing parameters which were used i n determining the three d i f f e r e n t sets of routing constants are shown i n Table IV.1. The p l o t s of the downstream hydrographs show that the minimum peak varies s u r p r i s i n g l y l i t t l e even with widely d i f f e r i n g routing con-stants. In other words, the maximum peak reduction i s r e l a t i v e l y insen-s i t i v e to the routing constants. However, the operation of the upstream r e s e r v o i r v aries considerably for the d i f f e r e n t cases and tends to be quite s e n s i t i v e to the routing constants. It can be seen that the l i n e a r programming determines what the minimum peak flow w i l l be and then stores as l i t t l e water as possible 3 4 to maintain t h i s peak throughout the flooding period. In a l l of the years analyzed, the t o t a l r e s e r v o i r volume was used. However, by increasing the volume of the hypothetical r e s e r v o i r , the r e s t r i c t i o n caused by li m i t e d inflow to the reservoir could be made to govern. Correcting for A Limited Discharge Capacity The r e s u l t s which have been presented have an inherent assumption included i n them which should be explained. This assumption i s that the res e r v o i r has an i n f i n i t e discharge capacity, which means that involun-tary storage (shown by the shaded area of Fi g . 6.1) could be eliminated. It can be seen that involuntary storage r e s u l t s when the inflow to the res e r v o i r exceeds the maximum discharge capacity of the re s e r v o i r . When th i s s i t u a t i o n does occur, the involuntary storage must be accounted f o r . This i s not a problem for a t h e o r e t i c a l r e s e r v o i r , but a r e a l r e s e r v o i r does not have an i n f i n i t e discharge capacity. Therefore, before this method i s applied to a r e a l s i t u a t i o n , changes w i l l have to be made to take account of the l i m i t e d discharge r e s t r i c t i o n s of r e s e r v o i r s . ( A l -though the a l t e r a t i o n s required are quite simple, i t was f e l t they were not necessary i n developing the basic methodology of t h i s f l o o d control method). The technique used to overcome a f i n i t e discharge capacity would be, f i r s t , to route the inflows through the r e s e r v o i r with the res e r v o i r at i t s maximum discharge capacity (reservoir routing). The outflows would then form the maximum discharge curve f o r that p a r t i c u l a r r e s e r v o i r . (Fig. 6.1). 3 5 \> Natural Inflow Hydrograph Involutary-^ Storaqe ^ / 1 X — Maximum Discharge Curve Hn y<^~ I 1 S / / / I . // F TIME Fig. 6.1 Holdouts Necessary to Account for Involuntory Storage . Due to storage i n the re s e r v o i r , the maximum discharge curve w i l l be less than or equal to the input hydrograph up to the point where the two curves cross. During the period when the inflow hydrograph exceeds the maximum discharge curve there w i l l have to be a set of constraints forcing the l i n e a r programming to have minimum holdouts. That i s , dur-ing t h i s c r i t i c a l period water i s being stored due to involuntary s t o r -age and th i s volume stored must be accounted f o r by constraining the pro-gram to have holdouts which are at le a s t as large as the differe n c e be-tween the two curves. Referring to F i g . 6.1, I represents the inflow n value for day 'n' and F^ represents the maximum discharge possible on that day. Thus, 1^ minus F^ represents the involuntary storage occurring on day 'n'. It can be seen that the holdout on day 'n' must be made greater than or equal to the value of I - F^ i n order to account for a l l the water stored. This s i t u a t i o n occurs each day the input hydro-36 graph exceeds the maximum discharge curve and so the following constraints would be needed. H l ^ h ~ F l H 2 > I 2 - F 2 (6-1) H > I - F n — n n where H = d a i l y holdout, I = d a i l y inflow, F = d a i l y maximum discharge. Limitations of the Method At t h i s point i t should be emphasized that employing t h i s method of obtaining optimal flood control to i t s f u l l e s t advantage requires the knowledge of a l l discharge hydrographs i n advance. In e f f e c t t h i s means that a perfect forecasting method i s required to achieve the ab-solute minimum peak flow at Hope. A forecast of three or four days would be of l i t t l e value to th i s method. It would be possible to use a forecast of several days and determine a best reservoir operation p o l i c y but t h i s operation p o l i c y would vary considerably depending on what happens l a t e r on i n the flooding period. An analysis using a short period forecast w i l l suggest storing begin immediately and w i l l y i e l d a minimum peak during the forecast period which most c e r t a i n l y w i l l not be the minimum peak for the e n t i r e f l o o d . An analysis using a t o t a l flood 37 forecast would reveal the most opportune time to begin storing water i n order to get a minimum downstream peak for the e n t i r e flood. Although a forecast f o r the e n t i r e flood i s required to give a minimum peak flow for the whole flooding season, there are possible ways of minimizing t h i s deficiency which w i l l be discussed l a t e r . Given a perfect forecast (or data of record), the accuracy of t h i s flood control method becomes completely dependent on the accuracy of the routing constants, since the l i n e a r programming introduces essen-t i a l l y no e r r o r s . Since the routing constants did vary considerably from year to year, average routing values may produce considerable errors i n the holdouts being routed. However, t h i s problem can be minimized also. If the method were being used for flood control on a forthcoming f l o o d , an average value of routing constants from analysis of past floods must be used. In t h i s s i t u a t i o n a forecast hydrograph must also be used and the errors i n the forecast w i l l e a s i l y overshadow any errors i n the rout-ing constants. If the method were being used for planning purposes, analysis of past records for the p a r t i c u l a r year or year would reveal the constants which f i t each year's data best. By using the best values ( i f they are appreciably better than the average values) the routing procedure should be of about the same accuracy as the available data. So, f o r either of the above a p p l i c a t i o n s , the errors i n routing constants w i l l not s i g -n i f i c a n t l y a f f e c t the f i n a l r e s u l t s . Possible Future Extensions Now that the basic l i n e a r programming method has been developed 38 using a s i m p l i f i e d model there are several simple extensions which could be applied i n order to make the method more complete and f l e x i b l e . One of the possible extensions would be the i n c l u s i o n of several more reservoirs to take i n a l l the proposed dam s i t e s on the headwaters of the Fraser River. This would enable one to carry out a complete flood control study of the Fraser River system. Adding more reservoirs would not change the channel routing except that considerably more analysis would be required to determine the routing constants f o r the other branches of the r i v e r . The l i n e a r programming would remain b a s i c a l l y the same except that there would be more sets of constraints. For each reser v o i r there would be sets of constraints s i m i l a r to equations (5-4) and (5-5). The-other set of constraints (equations 5-3) would simply be-come larger as shown below. Y > Q - [c.H + c.H + . . .• + c.H ] - [1 J + 1 J , + — n I n 2 n-1 k n-k+1 I n 2 n-1 . . . + 1 'J _ , , ] - . . . . m n-m+1 J 1 v f i r s t r e s e r v o i r second res e r v o i r where H = holdouts from f i r s t r e s e r v o i r , J = holdouts from second r e s e r v o i r . Thus, i t can be seen that although more res e r v o i r s w i l l increase the siz e of the problem considerably, the basic methodology remains the same. Another addition which could possibly prove valuable would be the use of stochastic data. When attempting to apply t h i s flood control 3 9 method to a forthcoming flood there are severe forecast l i m i t a t i o n s . I t may be f e a s i b l e to s y n t h e t i c a l l y generate many years of flood data. These synthetic floods might then be s t a t i s t i c a l l y analysed so that when a flo o d i s forecast i t could be adjusted to account for the amount of r i s k one i s w i l l i n g to accept. Although stochastic data w i l l not eliminate the prob-lem of forecasting a flood i t might enable one to employ a r i s k function which would be quite u s e f u l . Possible Applications of the Method Although most of the possible uses of t h i s flood control method have been mentioned throughout the text, a b r i e f summary of the a p p l i c a -tions may prove valuable. One of the two primary ways of employing t h i s f l o o d control method would be i n determining the best way to regulate reservoirs to give the best flood control during a forthcoming flood. As has been discussed, the biggest b a r r i e r to t h i s type of a p p l i c a t i o n i s the require-ment of a forecast for the e n t i r e f l o o d , which w i l l probably l i m i t pre-sent applications i n t h i s area. Due to the considerable f l e x i b i l i t y of t h i s a n a l y t i c a l procedure, i t can be used to decide on the best r e s e r v o i r operation to s u i t many varied flood problems. For example, there may be more snow i n one sub-basin than there i s i n the others and t h i s method could be used to take t h i s into account when planning flood control operation. The second primary a p p l i c a t i o n i s i n the f i e l d of planning. Due to the fact that the state of forecasting i s considerably behind the de-40 velopment of a n a l y t i c a l tools requiring forecasts, any present a p p l i c a -tions would most probably be i n the planning areas. In planning prob-lems, floods of record or possibly s y n t h e t i c a l l y generated floods are used which eliminates the need f or a forecast. Thus, by analyzing many sets of data the effectiveness of the r e s e r v o i r system under d i f f e r e n t flooding conditions would be determined. Although t h i s would be the primary objective, there are other parameters which can be examined. By analyzing recorded and synthetic floods i t would also be possible to determine the maximum reservoir s i z e required at the selected l o c a t i o n s . For each flood analyzed, the l i n e a r programming gives the minimum re s e r v o i r volume required. Thus, by analyzing many d i f f e r e n t floods, the largest minimum r e s e r v o i r volume required could be determined and the reservoir could then be designed f o r t h i s value, i f p h y s i c a l l y p o s s i b l e . Another parameter related to the planning aspects would be the determination of the most e f f e c t i v e l o c a t i o n f o r proposed r e s e r v o i r s . Once again, by using recorded and stochastic flood data, and by varying the l o c a t i o n of the proposed r e s e r v o i r s , i t would be possible to deter-mine the r e l a t i v e effectiveness of the reservoirs i n reducing the peak flow at Hope due to the varied l o c a t i o n s . CHAPTER VII S U M M A R Y The problem of determining the optimal way to operate a remote reser v o i r to maximize the reduction i n downstream peak flow for flood control purposes has been studied. The problem has been s i m p l i f i e d by assuming a s i n g l e r e s e r v o i r with a given storage, an unlimited discharge capacity, and approximately 500 miles from the area to be protected. As long as a l i n e a r channel routing method i s used, l i n e a r programming has been shown to be an excellent optimization technique which solves the problem quite e a s i l y . By using a Muskingum type of channel routing, the requirement of a l i n e a r routing method i s s a t i s f i e d . Widely d i f f e r e n t channel routing parameters have been used to test the s e n s i t i v i t y of the developed flood control method to d i f f e r e n t routing constants. I t has been shown that the reduction i n peak flow i s r e l a t i v e l y i n s e n s i t i v e to routing constants but the operation of the r e s e r v o i r i s quite dependent on the routing constants. Although the method i s e a s i l y extended to more complex systems, forecasts hamper i t s use i n an operational sense. To obtain the best flood control of a forthcoming flood, a l l upstream and downstream hydro-graphs for the e n t i r e flooding period must be forecast. The method there-fore i s more useful for planning purposes as data of record are then used, thus eliminating the need of a forecast. 41 B I B L I O G R A P H Y Final Re-port of the Fraser River Board on Flood Control and Hydro-Eleotrio Power in the Fraser River Basin. V i c t o r i a , B.C.: The Queen's P r i n t e r , September 1963. L i n s l e y , R. K., Kohler, M. A., and Paulhus, J . L. Hydrology for Engineers. New York, Toronto and London: McGraw-Hill Co. Inc. Ltd., 1958. Chow, V. T. et al. Handbook of Applied Hydrology. New York: McGraw-Hill Co. Inc. Ltd., 1964. "Routing of Floods Through River Channels," Engineering and Design Manual3 EM 1110-2-1408, U.S. Corps of Engineers, March 1, 1960. Draper, N. R. and Smith, H. Applied Regression Analysis. New York John Wiley and Sons, 1966. U.B.C. Users' Group Program. A Linear Programming Package - LIP. D. O'Reilly. U n i v e r s i t y of B r i t i s h Columbia Computing Centre, September 1970. H i l l i e r , F. S. and Lieberman, G. J . Introduction to Operations Research. San Francisco, C a l i f . : Holden-Day, Inc., 1969. 42 A P P E N D I X 44 FLOOD CONTROL ON THE FRASER RIVER ( 1955 ). 400 375 350 325 300 275 250 225 U -o o o o 200 150 Discharge Hydrographs at Hope 100 UJ o tr < i o 50 0 150 100 50 0 150 100 50 Natural Discharge Reservoir Regulation at Shelley. Case Natural Discharge Reservoir Regulation at Shelley. Case 2 Natural Discharge Reservoir Regulation at Shelley. Case 3. • • • i i i i i i i i i i I i i *i i i i 'i i i i i i i i i i i i i i i i i i i i i • i i i i i i i i -i i i i i i i i ' i i • i i i i MAY JUNE JULY 4 2 5 4 0 0 3 7 5 3 5 0 3 2 5 3 0 0 275 2 5 0 225 2 0 0 Lu O I 5 0 O O O FLOOD CONTROL ON THE FRASER RIVER ( 1 9 6 4 ) ^ m 45 100 50 LU or < I o o - 160 100 50 0 150 100 50 Discharge Hydrographs at Hope Natural Discharge Reservoir Regulation at Shelley. Case Natural Discharge Reservoir Regulation at Shelley. Cose 2 :-— Natural Discharge i i i l i i i l i i M A Y Reservoir Regulation at Shelley. Case 3 i i i i i i i i i i , , , i i i i J U N E J U L Y 400 FLOOD CONTROL ON THE FRASER RIVER ( 1 9 6 7 ) . 46 Discharge Hydrographs at Hope Nature I Discharge Reservoir Regulation at Shelley. Case Natural Dischorge Reservoir Regulat ion at Shel ley . Case 2. Natural Discharge Reservoir Regulation at Shel ley . Case 3 . . ' ' ' ' ' I * ' i i i i i • • • '''''''''''''''•I''''' i i i i i i MAY JUNE JULY 

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