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UBC Theses and Dissertations

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 . S c , The U n i v e r s i t y 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 Civil  We accept t h i s the r e q u i r e d  THE  Engineering  t h e s i s as conforming t o  standard  UNIVERSITY OF BRITISH COLUMBIA October,  1970  In  presenting  this  thesis  an a d v a n c e d d e g r e e the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  this  written  at  the U n i v e r s i t y  make i t  for  gain  of  if  of  Columbia,  British  by  Columbia  /97e>  for  shall  the  requirements  reference copying  of  I agree and this  that  not  copying  or  for that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  for extensive  p u r p o s e s may be g r a n t e d It  fulfilment  available  permission.  Department  Date  freely  that permission  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t  my  A B S T R A C T  This thesis describes mining the most e f f i c i e n t way  the development of a method f o r  to operate an upstream f l o o d c o n t r o l  r e s e r v o i r f o r maximum f l o o d peak r e d u c t i o n  at a downstream p o i n t .  L i n e a r programming i s used as the o p t i m i z a t i o n fied  technique.  case i s s t u d i e d , namely t h a t of a s i n g l e s t o r a g e  a p p r o x i m a t e l y 500  miles  upstream from the area  channel r o u t i n g method which was l i n e a r programming. method was  chosen.  For The  l i n e a r was  A simpli-  reservoir  to be p r o t e c t e d .  required  f o r use  A  with  t h i s reason a Muskingum type channel r o u t i n g r e s u l t s f o r the three y e a r s t h a t were s t u d i e d  are p r e s e n t e d i n g r a p h i c a l form.  They i n d i c a t e the extent  downstream peak c o u l d be reduced and r e s e r v o i r which would be  deter-  required  the o p e r a t i o n  of the  that  the.  upstream  to b r i n g about t h i s r e d u c t i o n  in  peak flow.  Procedures f o r e x t e n d i n g the technique to more complex  systems and  p o s s i b l e a p p l i c a t i o n s of the method are  ii  discussed.  TABLE OF CONTENTS Page  LIST OF TABLES  iv  LIST OF FIGURES  v  Chapter I. II. III. IV. V. VI. VII.  INTRODUCTION  1  THE FLOOD CONTROL PROBLEM  5  CHANNEL ROUTING THEORY  9  DETERMINATION OF ROUTING CONSTANTS  15  EMPLOYING LINEAR PROGRAMMING  27  RESULTS  33  SUMMARY . .  41  BIBLIOGRAPHY  42  APPENDIX  44  iii  LIST OF TABLES  Table  IV.1  Page  ROUTING PARAMETERS FOR THE DIFFERENT CASES  iv  23  LIST OF FIGURES  Figure  2.1  Page  TYPICAL DISCHARGE HYDROGRAPH AT HOPE DURING FLOODING SEASON  5  TYPICAL DISCHARGE HYDROGRAPH AT SHELLEY DURING FLOODING SEASON  6  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  6.1  HOLDOUTS NECESSARY TO ACCOUNT FOR INVOLUNTARY STORAGE . .  2.2  3.1  v  . . .  19  35  A C K N O W L E D G E M E N T  The author wishes t o express h i s g r a t i t u d e to h i s s u p e r v i s o r , Mr. S. 0. R u s s e l l , f o r h i s v a l u a b l e guidance,and encouragement the r e s e a r c h , development and p r e p a r a t i o n of t h i s  vi  thesis.  during  CHAPTER I  INTRODUCTION  The F r a s e r R i v e r o r i g i n a t e s i n the i n t e r i o r of B r i t i s h Columbia and  empties i n t o  the P a c i f i c Ocean near Vancouver, B.C.  and  the r i v e r ' s mouth, i t flows through  highly industrialized Due  Between Hope,  rich agricultural  B.C.  l a n d and some  a r e a s , both of which are q u i t e h e a v i l y p o p u l a t e d .  to the e x t e n s i v e development along the banks of the F r a s e r R i v e r ,  any  e x c e s s i v e f l o o d i n g of the r i v e r c a r r i e s w i t h i t the t h r e a t of c o n s i d e r a b l e p r o p e r t y damage. The most r e c e n t f l o o d damage o c c u r r e d on May  of the F r a s e r R i v e r to cause s i g n i f i c a n t  31, 1948.  The  at Hope which i s the h i g h e s t recorded v a l u e o c c u r r e d i n 1894  (The mean annual  flood  caused  area  on the F r a s e r R i v e r f l o o d  cause approximately  o n l y flow to exceed to be  $20 m i l l i o n damage to the Lower  the r i v e r mouth).  that a flood  The  this  about This Mainland  e x t e n s i v e development  has made the area much more v a l of the 1948  magnitude would  of the 1948  magnitude i n the  $200 m i l l i o n damage.  The p o s s i b i l i t y of g e t t i n g a f l o o d near f u t u r e i s v e r y r e a l .  S e v e r a l y e a r s s i n c e 1948  g r e a t e r accumulated snowpacks and had the c r i t i c a l  The  r o u g h l y estimated  p l a i n s s i n c e 1948  i t has been e s t i m a t e d  a peak flow of 536,000 c f s  peak flow at Hope i s 313,000 c f s ) .  an e s t i m a t e d  (the area between Hope and  u a b l e and  flow.  when the flow was  600,000 c f s . of 1948  r i v e r had  have had  t h e r e been a s u s t a i n e d hot s p e l l at  time s e v e r e f l o o d i n g would have o c c u r r e d . 1  equal or  now  2  Due River Flood  to the s e v e r e f l o o d t h r e a t , a j o i n t F e d e r a l - P r o v i n c i a l F r a s e r  C o n t r o l Board was  set up  i n 1955  to study and  r e p o r t on  the  problems of f l o o d i n g . The most s u i t a b l e permanent s o l u t i o n to the F r a s e r R i v e r f l o o d problem would be  the c o n s t r u c t i o n of one  s o l u t i o n which had s i n g l e very  relatively  l a r g e dam  supporters  at Moran, B.C.  f l o o d i n g problems but salmon f i s h i n g  few  the i d e a was  industry.  salmon run every y e a r .  The The  or more r e s e r v o i r s . was  A suggested  the c o n s t r u c t i o n of a  T h i s would s o l v e the Lower Mainland  r e j e c t e d as i t was  i n c o n f l i c t with  the  F r a s e r R i v e r supports a m u l t i - m i l l i o n d o l l a r  c o n s t r u c t i o n of a l a r g e dam  such as that  pro-  posed at Moran on the main stem of the r i v e r would e l i m i n a t e a l a r g e p a r t of the B.C.  salmon f i s h i n g  industry.  As  a r e s u l t , the F r a s e r R i v e r  Flood  C o n t r o l 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 s e v e r a l smaller  dams i n the headwaters of the r i v e r which would have a minimal i n -  f l u e n c e on perly  the f i s h w h i l e  thesis.  F r a s e r R i v e r f l o o d problem prompted the study d e s c r i b e d A s i n g l e r e s e r v o i r near the f l o o d p l a i n s i s r e l a t i v e l y  p l e to r e g u l a t e but area  e f f e c t i v e for flood c o n t r o l , i f pro-  regulated. The  this  s t i l l being  to be p r o t e c t e d  flood  control.  To  s e v e r a l r e s e r v o i r s i n the  headwaters remote from  are much more d i f f i c u l t  to operate f o r e f f i c i e n t  the p r e s e n t  time t h e r e have been no adequate  in simthe  analytical  t o o l s to determine the b e s t method of c o n t r o l l i n g upstream r e s e r v o i r s or f o r comparing the r e l a t i v e e f f e c t i v e n e s s of r e s e r v o i r s at d i f f e r e n t l o c a tions . In t h i s study, which r e p r e s e n t s  a first  step i n d e v e l o p i n g  such  an a n a l y t i c a l p r o c e d u r e , a method i s developed f o r f i n d i n g the most e f f e c -  3  t i v e way  of o p e r a t i n g  a f l o o d c o n t r o l r e s e r v o i r given  at the r e s e r v o i r s i t e and which i s used i s not location.  the l o c a t i o n to be p r o t e c t e d .  l i m i t e d by  The  flows  procedure  the number of r e s e r v o i r s nor by  their  That i s , complex systems can be handled as r e a d i l y as  systems. required  the n a t u r a l  simple  Such a procedure w i l l e l i m i n a t e many of the approximations f o r both p l a n n i n g  and  operating  now  remote r e s e r v o i r s f o r f l o o d con-  trol.  Problem F o r m u l a t i o n  The  method makes use  most p o w e r f u l o p t i m i z a t i o n good computer r o u t i n e s i s being  of l i n e a r programming which i s one  techniques a v a i l a b l e and  exist.  In p a r t i c u l a r , a l l the  c o n s t r a i n t s must be  linear.  f o r which  the  very  However, the f a c t t h a t l i n e a r programming  used imposes c e r t a i n demands on the  handled.  one  of  types of problem that can  terms i n the o b j e c t i v e f u n c t i o n and  be  the  Consequently, i t i s n e c e s s a r y to formulate  the problem to meet these r e q u i r e m e n t s . Initially which s t i l l problem.  incorporates  been broken down to a s i m p l e r  The  charge c a p a c i t y .  form  the e s s e n t i a l f e a t u r e s of a f l o o d r e g u l a t i o n  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  a p p r o x i m a t e l y 500 ser River.  the problem has  been assumed at S h e l l e y ,  m i l e s upstream from Hope on the main branch of the  r e s e r v o i r has The  object  a given i s to use  storage  volume and  B.C., Fra-  an u n l i m i t e d  the r e s e r v o i r volume as  dis-  efficiently  as p o s s i b l e to minimize the maximum f l o w at Hope. The  study i t s e l f  can be d i v i d e d i n t o s e v e r a l d i s t i n c t p a r t s .  f i r s t p a r t d e a l s w i t h channel r o u t i n g and  the d e t e r m i n a t i o n  of  constants  The  4  f o r use  i n r o u t i n g equations f o r the chosen reaches of the r i v e r .  second p a r t d e s c r i b e s  the  programming f o r m u l a t i o n b l e a p p l i c a t i o n s and  computation procedure which made use  and  the  f i n a l part discusses  future extensions  of the method.  The  of a l i n e a r  the r e s u l t s and  possi-  CHAPTER I I  THE FLOOD CONTROL PROBLEM  The problem which r e q u i r e s o p t i m i z i n g the a i d of s e v e r a l d i s c h a r g e importance:  hydrographs.  one a t Hope, B.C.  during  B.C.  There are two hydrographs of  (the downstream h y d r o g r a p h ) , and the one  at the l o c a t i o n of the h y p o t h e t i c a l Shelley,  can b e s t be shown w i t h  r e s e r v o i r which i n t h i s case i s  (the upstream h y d r o g r a p h ) .  A t y p i c a l hydrograph a t Hope  the f l o o d i n g 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  5  Season.  A corresponding  t y p i c a l upstream hydrograph would have the shape shown  i n F i g . 2.2.  By s t o r i n g the c o r r e c t  amount of water i n the r e s e r v o i r at the  c o r r e c t time (as d e p i c t e d by the d o t t e d l i n e i n F i g . 2.2) the peak f l o w at Hope can be reduced a maximum amount as shown by the c o r r e s p o n d i n g d o t t e d l i n e i n F i g . 2.1. operated  I t should be emphasized t h a t o n l y i f the r e s e r v o i r i s  i n t h e exact o p t i m a l manner w i l l t h e peak f l o w at Hope be reduced  by t h e maximum amount. easily satisfied.  By u s i n g l i n e a r programming t h i s  condition i s  The l i n e a r programming s o l u t i o n g i v e s the new maximum  peak f l o w at Hope and i t a l s o g i v e s the d a i l y v a l u e s t o be s t o r e d a t the  7  r e s e r v o i r to ensure  t h a t t h i s maximum peak i s not  In o r d e r to determine  the e f f e c t  have on the flow at Hope, i t i s n e c e s s a r y  t h a t a flow at S h e l l e y w i l l to r o u t e the S h e l l e y flows down  the r i v e r channel  to Hope.  l a y e d and  and i s not simply t r a n s l a t e d ) .  reduced  (Due  exceeded.  to channel s t o r a g e , the f l o o d f l o w i s deIn the p r e s e n t study i t  i s not p o s i t i v e flows but the h o l d o u t s a t the r e s e r v o i r , or n e g a t i v e f l o w s , t h a t must be r o u t e d down the r i v e r . i n g method i s used, negative.  However, s i n c e a l i n e a r r o u t -  i t doesn't matter whether the flows are p o s i t i v e or  The n e g a t i v e flows are simply r o u t e d down the r i v e r channel  i n g the c o n s t a n t s d e r i v e d from  the a n a l y s e s of p o s i t i v e flows and  r o u t e d v a l u e s a r e then s u b t r a c t e d from the n a t u r a l hydrograph o b t a i n the new The new  hydrograph  up of two  these  at Hope to  hydrograph. o r d i n a t e Q'  shown i n F i g . 2.1  i s a d i s c h a r g e v a l u e of  at Hope ( f l o o d c o n t r o l i n c l u d e d ) .  components:  Q'  n  = Q  n  - D  D  = c.H I n  n  at Hope (Q ) n  That i s ,  (2-1)  n  i s the r o u t e d h o l d o u t s f o r day  the  T h i s o r d i n a t e i s made  the o r d i n a t e of the n a t u r a l hydrograph  minus the r o u t e d h o l d o u t s at S h e l l e y (D ). n  D^  us-  'n' and  i s composed o f :  + c„H . + c.H „ + . . . . + 2 n-1 3 n-2  c, H , k n-k+1 (2-2)  where the H's  represent d a i l y holdouts.  Q'  =  Q ' . n new natural hydrograph hydrograph ordinates ordinates n  Combining these  "  [c.H I n  equations,  + c H + c H + 2 n-1 3 n-2 0  0  + ]jH  . .  (2-3)  c  routed  holdouts  8  T h i s i s t h e b a s i c e q u a t i o n f o r d e t e r m i n i n g the new hydrograph a t Hope.  T h i s f i r s t p a r t o f the study was concerned  s t a n t s c , c , . . . ,c  w i t h d e t e r m i n i n g the con-  and t h e second p a r t was concerned  w i t h f i n d i n g the  v a l u e s o f t h e h o l d h o u t H , H , , . . . H , ,,. n n-1 n-k+1 When t h e r o u t i n g c o n s t a n t s were b e i n g d e r i v e d u s i n g r e a l f l o w s of r e c o r d , i t was n e c e s s a r y  t o be c a r e f u l w i t h t r i b u t a r y f l o w s so t h a t t h e  volume o f i n f l o w c o u l d be made e q u a l t o t h e volume of o u t f l o w over t h e p e r iod i n question.  When r o u t i n g n e g a t i v e f l o w s t h i s i s n o t n e c e s s a r y and  no concern .need be p a i d t o t r i b u t a r y f l o w s . accounted  A l l t r i b u t a r y flows are  f o r by t h e n a t u r a l hydrograph a t Hope.  I t ' i s only a negative  f l o w from one p a r t of t h e r i v e r which i s b e i n g r o u t e d .  A l l other flows  i n t h e r i v e r , a p a r t from the h o l d o u t s , c o n t r i b u t e t o t h e n a t u r a l hydrograph a t Hope e x a c t l y the same as b e f o r e . t i o n of l i n e a r i t y . t a k i n g account  T h i s f o l l o w s from t h e assump-  Thus, as l o n g as the c o n s t a n t s were o r i g i n a l l y d e r i v e d  o f t r i b u t a r y f l o w s , t r i b u t a r y f l o w s need no l o n g e r be con-  s i d e r e d i n r o u t i n g the holdouts.  CHAPTER I I I CHANNEL ROUTING THEORY*  F l o o d r o u t i n g , whether i t be channel r o u t i n g o r r e s e r v o i r can be d e f i n e d as the procedure whereby the hydrograph a p o i n t on a stream  i s determined  :  more p o i n t s upstream.  o f a f l o o d wave at  from known or assumed d a t a a t one or  As the d i s c h a r g e i n a channel i n c r e a s e s , stage  a l s o i n c r e a s e s and w i t h i t the volume of water i n temporary the channel.  During  wave moving down a channel appears  used i n Hydrology  storage i n  the f o l l o w i n g p o r t i o n of a f l o o d an equal volume  of water must be r e l e a s e d from s t o r a g e .  i t s c r e s t lowered.  routing,  Channel  As a r e s u l t of t h i s , a f l o o d  to have i t s time base lengthened and  r o u t i n g i s simply an a n a l y t i c a l  t o compute the e f f e c t  technique  of channel s t o r a g e on the shape  and movement o f a f l o o d wave. Routing i s the s o l u t i o n of the s t o r a g e e q u a t i o n which i s an e x p r e s s i o n of c o n t i n u i t y , I - 0 = ||  (3-1)  where I = inflow, 0 = outflow, dS —  = change i n s t o r a g e / u n i t o f time.  T h i s can be converted to a more u s a b l e form as shown below.  — ' t = S - S (3-2)  -S-- - • t 2  2  9 See  [ 2 ] , [3] and [4] i n the b i b l i o g r a p h y .  ±  10  where the s u b s c r i p t s 1 and ly,  2 r e p r e s e n t the b e g i n n i n g and end, r e s p e c t i v e -  of the time i n t e r v a l ' t ' . F o r r e s e r v o i r r o u t i n g t h e r e i s a second  and  d i s c h a r g e which i s of the  equation r e l a t i n g  storage  form  S = f(0)  (3-3)  where S = storage, 0 = outflow.  That i s , s t o r a g e i s a f u n c t i o n of o u t f l o w o n l y . more c o m p l i c a t e d by  the f a c t  o n l y as i n r e s e r v o i r s . river  Channel r o u t i n g i s made  t h a t s t o r a g e i s not a f u n c t i o n of  T h i s can be seen from a s i m p l i f i e d  channel as shown i n F i g u r e  outflow  p r o f i l e of a  3.1."  Negative  F i g . 3.1  V a r i o u s Types of Storage During The Passage of a Flood The  Wave.  s t o r a g e beneath a l i n e p a r a l l e l to the r i v e r bed  'prism' s t o r a g e w h i l e the s t o r a g e between the p a r a l l e l l i n e and profile i s called  'wedge' s t o r a g e .  wedge s t o r a g e may  exist  During f a l l i n g  i s called the  actual  During r i s i n g s t a g e , a l a r g e volume of  b e f o r e any a p p r e c i a b l e i n c r e a s e i n o u t f l o w o c c u r s .  s t a g e , the i n f l o w decreases much more q u i c k l y than the  out-  11  flows and hence the wedge s t o r a g e volume becomes n e g a t i v e . channels  r e q u i r e s a s t o r a g e r e l a t i o n s h i p which adequately  t h i s wedge s t o r a g e .  in  represents  T h i s i s u s u a l l y done by i n c l u d i n g i n f l o w as a p a r a -  meter i n the s t o r a g e e q u a t i o n . flow and  Routing  Thus, s t o r a g e becomes a f u n c t i o n of i n -  outflow. Although  t h e r e are s e v e r a l 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 r e a c h under c o n s i d e r a t i o n . In some i n s t a n c e s t h i s knowledge may o g r a p h i c maps, o r i t may  be l i m i t e d  include a f a i r l y  as i s the case w i t h the F r a s e r R i v e r . procedures storage.  to t h a t a v a i l a b l e from t o p -  complete h i s t o r y of p a s t f l o o d s ,  As has p r e v i o u s l y been shown, r o u t i n g  are g e n e r a l l y based on the r e l a t i o n s h i p between d i s c h a r g e  and  The most common method used to o b t a i n t h i s r e l a t i o n s h i p i s by  the  a n a l y s i s of f l o o d s of r e c o r d , assuming t h a t the r e l a t i o n s h i p s e s t a b l i s h e d w i l l be v a l i d f o r f u t u r e f l o w s .  The  data necessary  the flow r e c o r d s at the upstream, downstream, and t h i s was  used i n the p r e s e n t Although  t r i b u t a r y gauges,  channel  routing  procedures  the same, the d i f f e r e n c e s a r i s i n g to a con-  s i d e r a b l e e x t e n t from minor v a r i a t i o n s i n a l g e b r a i c m a n i p u l a t i o n refinements ing  i n the b a s i c assumptions.  The  method depends on s e v e r a l f a c t o r s , such  able data, personal preference, accuracy c a t i o n s of the r e s u l t s . tor  i n the p r e s e n t The  The  and  analysis.  t h e r e are v a r i o u s d i f f e r e n t  a v a i l a b l e , they are b a s i c a l l y  f o r t h i s a n a l y s i s are  c h o i c e of a p a r t i c u l a r  rout-  as the nature of the  avail-  requirements,  l a t t e r proved  or from  and  intended  to be the most important  applifac-  study.  r o u t i n g method t h a t was  chosen had  to meet two  important  12  requirements.  S i n c e l i n e a r programming r e q u i r e s t h a t the o b j e c t i v e f u n c -  t i o n as w e l l as a l l o f the c o n s t r a i n t s be l i n e a r , and the d e r i v e d r o u t i n g e q u a t i o n s were going to be used as c o n s t r a i n t s , a completely l i n e a r r o u t i n g method was r e q u i r e d .  A l s o , i t was necessary  b i n e the d a i l y r o u t i n g equations expressed met  so t h a t outflows  as a f u n c t i o n of i n f l o w s o n l y .  t h a t one be a b l e to comfrom a reach c o u l d be  The r o u t i n g method which b e s t  these requirements was a Muskingum type of channel r o u t i n g . As do a l l channel r o u t i n g methods, the Muskingum method s o l v e s  the c o n t i n u i t y and s t o r a g e equations s i m u l t a n e o u s l y . t i o n used i s g i v e n by e q u a t i o n  The c o n t i n u i t y equa-  (3-2) and the s t o r a g e e q u a t i o n used i s g i v e n  below.  S = K [ x l + (1-x) 0]  (3-4)  or  S  2  - S  1  = K [x ( I - I ) + ( l - x K C ^ - C ^ ) ] 2  (3-5)  1  where S = storage, I = inflow, 0 = outflow. x i s a constant which expresses outflows i n determining is  storage.  the r e l a t i v e importance of i n f l o w s and K i s known as the 'storage c o n s t a n t ' and  the r a t i o of s t o r a g e t o d i s c h a r g e , w i t h the dimension  approximately (3-5)  equal t o the t r a v e l time through  i s s u b s t i t u t e d f o r ^ - S ^ i n equation  becomes:  of time.  the r e a c h ) .  (K i s  I f equation  (3-2), the r e s u l t i n g  equation  13  = C I , + C.I. o 2 11  + c_o. 2 1  (3-6)  where C  Kx - 0 . 5 t K - Kx + 0 . 5 t  o  Kx + 0 . 5 t K - Kx + 0 . 5 t K - Kx K - Kx + By a d d i n g = 1.  the equations Equation  for C  ( 3 - 6 ) shows  that i t i s generally applied. equation the constants C o done b y c h o o s i n g and  x  i twill  that  +  to use the  be d e t e r m i n e d .  This i s  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  substituting  down t o s o l v i n g  into equations  (3-7) t o o b t a i n  the r o u t i n g constants f o r a f o r K and x f o r t h a t  the r o u t i n g equations  s i o n c o n t a i n i n g , s a y , 0^, 0  be f o u n d  I t c a n be s e e n t h a t i n o r d e r  Thus, d e t e r m i n i n g  reduces  By c o m b i n i n g  and C^,  the Muskingum r o u t i n g e q u a t i o n i n the form  a time i n t e r v a l ,  the r e q u i r e d c o n s t a n t s .  0.51 0.51  , C. , a n d C„ m u s t f i r s t 1 2  ( t o be d i s c u s s e d l a t e r )  reach e s s e n t i a l l y  ,  (3-7)  reach.  f o r s e v e r a l d a y s an  expres-  and i n f l o w v a l u e s c o u l d b e o b t a i n e d as shown  below.  0 (3-8)  +  14  Since  ,  ber and  thus the 0^  the 0^  and  are a l l l e s s than u n i t y ,  term s h o u l d have a n e g l i g i b l e c o n t r i b u t i o n to 0^.  term i s not n e g l i g i b l e , the p r o c e s s  more days u n t i l c i e n t s , 0^  i t becomes s o ) .  can be extended  Eliminating 0  (If  to i n c l u d e  and combinining  coeffi-  can be w r i t t e n as:  °4 " V 4  +  V  + 3  V2  Thus, the o u t f l o w from the reach i s now flows to the reach o n l y . itself  w i l l be a v e r y s m a l l num-  to t h i s  The  technique was  ear programming p a r t .  +  V i  expressed  (3  '"  9)  as a f u n c t i o n of i n -  a b i l i t y of the chosen r o u t i n g method to l e n d v e r y important  i n the a p p l i c a t i o n to the  lin-  CHAPTER IV  DETERMINATION OF ROUTING  CONSTANTS  General  From S h e l l e y (the l o c a t i o n o f the h y p o t h e t i c a l r e s e r v o i r ) t o Hope, the F r a s e r R i v e r t r a v e l s approximately  500 m i l e s , which i s much too f a r  f o r a s i n g l e reach u s i n g the Muskingum r o u t i n g e q u a t i o n . the r i v e r was d i v i d e d i n t o t h r e e reaches,  approximately  For t h i s  reason  equal i n l e n g t h .  (They were not e x a c t l y the same l e n g t h as each reach s t a r t e d  and ended at  a w e l l - e s t a b l i s h e d d i s c h a r g e gauge and these gauges were not evenly The  d i v i s i o n i n t o three d i f f e r e n t reaches meant t h a t t h r e e s e t s of Muskin-  gum c o n s t a n t s had t o be e v a l u a t e d , one s e t f o r each  reach.  In order to use the Muskingum r o u t i n g method constants)  (or to e v a l u a t e the  the t o t a l i n f l o w volume t o a reach must equal the t o t a l  volume from the same r e a c h over the d u r a t i o n of a f l o o d  (i.e., C o  C  2  spaced).  = 1).  T h i s means t h a t e i t h e r t h e r e must be no t r i b u t a r y flows  outflow + C + 1 through-  out the reach o r e l s e the t r i b u t a r y flows must be separated out and the main stem i n f l o w s a d j u s t e d so t h a t the i n f l o w volume does e q u a l the o u t f l o w volume. I t was the presence stant d i f f i c u l t  o f t r i b u t a r y flows which made the r o u t i n g con-  to determine on the F r a s e r R i v e r .  Throughout i t s e n t i r e  l e n g t h , the F r a s e r R i v e r has numerous t r i b u t a r y streams, some of which are q u i t e s u b s t a n t i a l i n s i z e compared to the flow i n the main stem. is  the t r i b u t a r y flows which can cause the r o u t i n g c o n s t a n t s 15  to show  It  16  variability  from year to y e a r .  stem flow had  the same r e l a t i v e v a l u e s  s t a n t s would be conditions  consistent.  Muskingum r o u t i n g Due could  the b i g g e s t  will  a direct relationship  appear to vary  to the i n a c c u r a c i e s  i n the  constants  which l a t e r a l  times a constant  needed.  by u s i n g  the ^  to the main stem i n f l o w s .  following  T h i s was  done  adding the  tribu-  T h i s constant  was  formula:  ^Outflows - i n f l o w s ^Tributary Flows  where the summations extended over a 115 The  inflows  so t h a t the i n -  choosing the l a r g e s t t r i b u t a r y flow i n the reach and  obtained  the  coefficients.  cause, a simple method of p r o - r a t i n g the i n f l o w s  t a r y flow  from f l o o d to  s i n g l e problem i n o b t a i n i n g  f l o w volume would equal the o u t f l o w volume was by  con-  to the v a r i a b i l i t y i n c l i m a t i c  t h e r e w i l l not be  the r o u t i n g constants  T h i s proved to be  the main  each y e a r , then the r o u t i n g  However, due  over t r i b u t a r y b a s i n s ,  between flows and flood.  I f a l l the t r i b u t a r y flows and  day  (4-1)  period.  above method gave a s e t of i n f l o w s  and  outflows  (with  inflow  volume equal to o u t f l o w volume) which c o u l d be used i n an a n a l y s i s to determine the r o u t i n g c o n s t a n t s . method f o r t h i s a n a l y s i s . out  the a n a l y s i s .  Regression On  A l l that remained was  to choose a  There are s e v e r a l a l t e r n a t e ways of  Three methods were t r i e d as d i s c u s s e d  carrying  below.  Analysis first  l o o k i n g at the Muskingum r o u t i n g e q u a t i o n i t would appear  as though i t were I d e a l l y s u i t e d to m u l t i p l e r e g r e s s i o n  techniques  [5]  to  17  determine  the r o u t i n g c o e f f i c i e n t s .  A m u l t i p l e r e g r e s s i o n e q u a t i o n has the  form:  Y  i  " o b  Vii  +  +  b  2 2i X  +  b  3 3i X  ( 4  "  2 )  where  bo, b., b_ 1 z = regression coefficients, X^, , = independent variables, Y = dependent variable. M u l t i p l e R e g r e s s i o n packages are a v a i l a b l e which w i l l c a r r y out a r e g r e s s i o n a n a l y s i s without  the c o n s t a n t b  Q  coefficient.  The form of the regres-  s i o n e q u a t i o n then becomes  Y. = b . X . + b „ X . + b . X - . x l lx 2 2i 3 3x 1  T h i s i s analagous  (4-3)  0  w i t h the r o u t i n g e q u a t i o n  0.  x  = C I . + C.I. . + C.O. . ox 1 i-l 2 i-l  (4-4)  Thus, by l e t t i n g 0. be the dependent v a r i a b l e and I . , I . .. ° x l i-l, J  0. . the i n i-l,  dependent v a r i a b l e s , a m u l t i p l e r e g r e s s i o n a n a l y s i s 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 o 1 2  f o r the p a r t i c u l a r  s e t of d a t a . Although r e s u l t s proved  t h i s would appear to be a q u i c k and e f f i c i e n t method, the  t o be i n c o m p a t i b l e w i t h the r e a l system.  When the b e s t  r o u t i n g c o e f f i c i e n t s were o b t a i n e d , a t l e a s t one of the c o e f f i c i e n t s would c o n s i s t e n t l y come out n e g a t i v e . I t was u s u a l l y value.  t h a t had the n e g a t i v e  Any of the c o e f f i c i e n t s coming out n e g a t i v e i s not c o n s i s t e n t w i t h  what i s a c t u a l l y happening.  I t i s not f e a s i b l e f o r any of the v a r i a b l e s  18  I . , I. , , or 0. , to have a n e g a t i v e e f f e c t on the o u t f l o w 0.. 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 without  any  further  Storage Versus  at a l l .  The method was  Weighted D i s c h a r g e Method  u l t a n e o u s l y and  o  l a t e d by s u b s t i t u t i n g i n t o equations  0.5t  [(I  ±  determined  then the r o u t i n g c o e f f i c i e n t s , C , C. and C„ are  J  (3-5) we  abandoned  investigation.  With t h i s method the v a l u e s of K and x are f i r s t  and  They must  (3-7).  1  sim-  calcu-  2  By combining e q u a t i o n s  (3-2)  get + I ) 2  - (0  ±  + 0 )]  = K  2  [x(I  0.5t  [.(!, +  x(I  - I ) +  2  I )  1  2  2  - I ) +  - (0  (1 - x)  The numerator r e p r e s e n t s the s t o r a g e increment r e p r e s e n t s the weighted flow increment s t o r a g e increment flows and o u t f l o w s f l o o d of r e c o r d .  and  the f l o w increment  i d e a l l y producing  (AQ).  1  +  0 )]  (0  2  (0  - 0 )]  2  (4-5)  1  AS  2  - 0 )  (4-6)  AQ  1  (AS) w h i l e the denominator S u c c e s s i v e v a l u e s of  the  are computed u s i n g known i n -  (which have been a d j u s t e d f o r volume) from a g i v e n T h i s procedure  v a l u e s of the parameter x. s t o r a g e increments  (1 - x)  1  and  The  i s c a r r i e d out f o r v a r i o u s computed v a l u e s of the  the accumulated flow increments  different  accumulated are then  curves i n the form of loops as shown i n F i g .  plotted, 4.1.  0  Accumulated  Storage  0  F i g . 4.1 Muskingum Storage L o o p s . The v a l u e o f x t h a t r e s u l t s i n a loop c l o s e s t to a s i n g l e l i n e i s accepted as the c o r r e c t v a l u e .  The v a l u e of K i s g i v e n by the r e c i p r o c a l of the  s l o p e of the loop most c l o s e l y forming  a single  line.  A l t h o u g h t h i s would appear to be a good method i t d i d not work w e l l on the F r a s e r R i v e r . necessary  to f i n d a f l o o d  as shown i n F i g u r e  In order to make the loops c l o s e i t was t h a t s t a r t e d and ended at the same flow v a l u e  4.2.  Inflow Hydrograph Outflow Hydrograph  TIME  Fig. 4 . 2  Inflow and Outflow Hydrographs for a Typical Flood  20  If  the o u t f l o w and i n f l o w curves c r o s s e d a t a d i f f e r e n t f l o w l e v e l a t  the b e g i n n i n g and end o f the f l o o d  (as was u s u a l l y the case) the loops  would not c l o s e r e g a r d l e s s of the x v a l u e chosen. requirement cially  were extremely  difficult  f l o o d s of h i g h magnitude).  composed of a s i n g l e peak.  Floods which met t h i s  to f i n d on the F r a s e r R i v e r  Another problem was f i n d i n g f l o o d s  I f the f l o o d peaked, receded  and then peaked  again at a d i f f e r e n t v a l u e , the l o o p s would have secondary in  them.  loops formed  These two problems made t h i s method of determining r o u t i n g  constants i m p r a c t i c a l  Sampling  f o r the F r a s e r R i v e r .  Technique  The ing  (espe-  t h i r d and f i n a l method t h a t was t r i e d i n d e t e r m i n i n g r o u t -  c o n s t a n t s was a v e r y simple sampling  technique.  Although  may appear t o be the l e a s t r e f i n e d of the t h r e e , i t proved  the method  to g i v e the  b e s t r e s u l t s i n the s h o r t e s t amount o f time. The method c o n s i s t e d of choosing a p o s s i b l e range of x v a l u e s (0.0 to 0.4 i n s t e p s o f .05) and a range o f K v a l u e s  (0.6 to 2.5 i n  s t e p s o f . 1 ) . A v a l u e f o r both x and K was then chosen, x and K v a l u e the r o u t i n g constants c a l c u l a t e d u s i n g equations  (3-7).  ,  and  and u s i n g t h i s  f o r the reach were  The i n f l o w s were then r o u t e d down the  reach, u s i n g the Muskingum r o u t i n g e q u a t i o n to o b t a i n a s e r i e s o f p r e d i c t e d outflows.  The p r e d i c t e d outflows were s u b t r a c t e d from  known o u t f l o w s t o o b t a i n a r e s i d u a l v a l u e . squared  The r e s i d u a l v a l u e s were  and summed t o g i v e a v a l u e which r e f l e c t e d  which the r o u t i n g was c a r r i e d o u t .  the c o r r e s p o n d i n g  T h i s procedure  the accuracy w i t h was employed f o r a l l  21  p o s s i b l e combinations  of x and K.  lowest  of the r e s i d u a l s were taken as the b e s t v a l u e s f o r  sum  of squares  that p a r t i c u l a r  The v a l u e s of x and K which gave the  reach.  Without the a i d of a h i g h speed computer t h i s method would have been h o p e l e s s due  to the l a r g e number of c a l c u l a t i o n s r e q u i r e d .  computers have made techniques Due was  to the l a t e r a l i n f l o w s and  initially  parameters.  such as t h i s p r a c t i c a l and  suspected  In p a r t i c u l a r , i t was  expected Due  of the F r a s e r R i v e r i t  t h a t the s t o r a g e constant  to t h i s expected  range of 0 c f s - 50,000 c f s ; another the flow v a l u e s f a l l i n g  K  variability,  d i v i d e d i n t o l e v e l s which were 50,000 c f s wide.  b e s t v a l u e f o r K and x were determined  on.  efficient.  t h a t the r i v e r would not e x h i b i t l i n e a r r o u t i n g  would v a r y depending upon the f l o w . the flow was  the n a t u r e  However,  That  is, a  f o r the flow v a l u e s l y i n g i n the  b e s t K and x were determined  for  i n the range of 50,000 c f s - 100,000 c f s , and  By examining the b e s t parameters i n each l e v e l i t was  mine the r e l a t i o n s h i p s which e x i s t e d i n the r i v e r t h a t the v a l u e of K v a r i e d q u i t e e r r a t i c a l l y and  channel.  so  hoped to d e t e r I t was  found  the t h r e e years of  data  that were a n a l y s e d proved  insufficient  to e s t a b l i s h any  However, the r e l a t i o n s h i p  f o r x proved  to be much more c o n s i s t e n t , w i t h  the b e s t v a l u e always t u r n i n g out to be z e r o . as a v a l u e around  .2 o r .3 would be expected  T h i s was  relationship.  rather surprising  f o r a r i v e r of t h i s type.  v a l u e of zero f o r x means t h a t s t o r a g e i n the r i v e r i s a f u n c t i o n of f l o w o n l y , which i m p l i e s t h a t we  have r e s e r v o i r r o u t i n g i n s t e a d of  A  out-  channel  routing. S i n c e the use of d i f f e r e n t c o n s t a n t s f o r d i f f e r e n t  flow l e v e l s d i d  22  g i v e b e t t e r answers, i t would have been d e s i r a b l e to use t h i s of f l o w v a l u e s .  However, t h i s would have complicated  ming c o n s i d e r a b l y and i t was f e l t  the l i n e a r program-  t h a t the e x t e n s i v e changes r e q u i r e d  of the l i b r a r y r o u t i n e were not warranted The  division  a t t h i s stage of the study.  l i n e a r programming i s o n l y a p p l i e d d u r i n g p e r i o d s of f l o o d i n g  so t h a t i t should be n e c e s s a r y  to a n a l y s e o n l y the flows which make up  the peak of the f l o o d t o determine r o u t i n g c o n s t a n t s .  Consequently,  only  r e c o r d s of h i g h flows were used to e v a l u a t e the r o u t i n g c o n s t a n t s and i t was  hoped t h a t t h i s method would minimize the n o n - l i n e a r e f f e c t  i n c r e a s i n g the complexity The  of the l i n e a r  without  programming.  above technique was c a r r i e d out i n the t h r e e d i f f e r e n t  f o r t h e t h r e e d i f f e r e n t y e a r s which were a n a l y z e d .  reaches  Since i t was found  that  the parameter K v a r i e d c o n s i d e r a b l y from year t o y e a r , minimum, maximum and average v a l u e s  t h a t were used i n the r o u t i n g are shown i n Table IV.1.  a p a r t i c u l a r r e a c h , t h e r e was a maximum and a minimum v a l u e of K represented  a v a r i a t i o n i n the r e s i d u a l sum of squares  three per c e n t ) .  (which  o f approximately  Since three y e a r s ' data were a n a l y z e d ,  maximum and t h r e e minimum v a l u e s o f K.  For  there were three  The maximum and minimum v a l u e s  i n T a b l e IV.1 r e p r e s e n t the h i g h e s t maximum and the lowest minimum f o r each r e a c h . such  The average K v a l u e s were weighted t a k i n g i n t o  account  f a c t o r s as which K v a l u e turned up most o f t e n and what the b e s t K  v a l u e was as r e p r e s e n t e d by the s m a l l e s t r e s i d u a l sum of squares. The d i f f e r e n t v a l u e s t h a t K c o u l d take were d i v i d e d i n t o  cases  1, 2 and 3 as shown i n the f o l l o w i n g t a b l e , f o r l a t e r use i n a s e n s i t i v i t y analysis.  23  TABLE IV.1 ROUTING PARAMETERS FOR THE DIFFERENT CASES  REACH  x  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  MAX  K [case 1]  AVERAGE K [case 2]  MIN K [case 3]  of Constants  Once the Muskingum iconstants were o b t a i n e d i t was n e c e s s a r y to c o n v e r t these c o n s t a n t s i n t o ones which c o u l d be a p p l i e d to the r o u t i n g e q u a t i o n s where the o u t f l o w i s expressed  as a f u n c t i o n o f i n f l o w s o n l y .  T h i s c o u l d be done by a s u b s t i t u t i o n procedure but a s i m p l e r method was used.  .  To do t h i s a s e t of i n f l o w s as shown below was d e v i s e d .  n-4  n-3  n-z  n-1  n  n+l  . . . 1000  1000  1000  1000  0  0  The v a l u e of 1000 i s completely was a convenient  number.  0  n+3 0  . . . .  a r b i t r a r y and was chosen o n l y because i t  The i n f l o w v a l u e s of zero are not a r b i t r a r y and  they must be made equal to z e r o . first  n+2  The above i n f l o w s are routed down the  reach u s i n g the Muskingum r o u t i n g e q u a t i o n w i t h the p r e v i o u s l y  determined Muskingum r o u t i n g c o n s t a n t s a p p l i c a b l e to the f i r s t The  outflows from the f i r s t  reach.  reach are then used as the i n f l o w s to the  24  second  r e a c h and  second  r e a c h u s i n g the c o r r e s p o n d i n g Muskingum c o n s t a n t s .  is  the r o u t i n g procedure  continued u n t i l the o u t f l o w s from  w i l l be r e c a l l e d  the t h i r d  = C.I I n  n  time on This  the procedure  reach are o b t a i n e d .  t h a t the Muskingum e q u a t i o n has  0  In order to s t a r t  i s repeated, t h i s  the  form  + C.I . + CO . 2 n-1 3 n-1  the r o u t i n g procedure  f o r each of the t h r e e r e a c h e s .  (4-7)  a value f o r 0  , must be known, n-1  T h i s v a l u e can be a r b i t r a r i l y  However, i t i s b e t t e r to choose a v a l u e of 1000 unknown o u t f l o w s as the r o u t i n g procedure  chosen.  f o r each of the t h r e e  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 p r o c e s s i s c o n s i d e r e d to have s t a b i l i z e d when the flows from the t h i r d r e a c h have a c o n s t a n t v a l u e of 1000 i n f l o w s to the f i r s t  r e a c h ) . I f a v a l u e other than 1000  i n f l o w v a l u e s of 1000 and  out-  - the same as  until stabilization  i s chosen, more  the l a r g e r w i l l be  the  occurs.  Once s t a b i l i z a t i o n has o c c u r r e d the o u t f l o w from the t h i r d w i l l have a constant v a l u e of 1000. zero i s encountered, purposes  reach  Then, as soon as an i n f l o w v a l u e of  the o u t f l o w s b e g i n d e c r e a s i n g .  For i l l u s t r a t i v e  a h y p o t h e t i c a l s e t of o u t f l o w s 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 zero and  the  w i l l be r e q u i r e d u n t i l the p r o c e s s does s t a b i l i z e ,  the f u r t h e r the chosen v a l u e i s from 1000  number of i t e r a t i o n s  It  by another v a l u e of the i n f l o w becoming  i t i s these decreases which make i t p o s s i b l e to c a l c u l a t e  the  25  new c o e f f i c i e n t s .  T h i s can be shown b e s t by the use of s e v e r a l equations,  c. T  0 = 1000 = n-1  0  =  n  1 n-1  980 =  +  c I + 2 n-2  c I + 2 n-1  /In  c I + . . .+ c ! 3 n-3 te n-k  c I + . . . + 3 n-2  The o n l y f a c t o r which c o u l d cause t h e decrease flow would be  l  n  equation  i n a l l other respects. different  c i R n-k+1  8  (4_  9 )  of 20 u n i t s i n t h e o u t -  (4-8) s i n c e the e q u a t i o n s  (The f a c t  (4_ )  are i d e n t i c a l  t h a t the c o n s t a n t s are m u l t i p l y i n g  i n f l o w v a l u e s makes no d i f f e r e n c e s i n c e a l l i n f l o w s are e i t h e r  zero o r 1000.  If I and I . a r e both 1000, c- I _ has the same v a l u e n-1 n-2 2 n-1 as c o l „ ) . I t can then be s a i d t h a t L n—/ 0  , - 0 =0.1 n-1 n 1 n-1 n  .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 o n l y d i f f e r e n c e between e q u a t i o n (4-9)  C  2"'- _i ^ n  a  s  a  (4-9) and (4-10) i s t h a t i n e q u a t i o n  v a l u e and i n e q u a t i o n  (4-10) i t i s z e r o .  Therefore,  26  that i s a c c o u n t i n g f o r the d i f f e r e n c e between 0 and i t must be c I '2 n-1 n 0 n+1'  Hence:  0 - 0 n+1 n  c 2  c I  " 2 n-1 0 - 0 n+1 n 1  i n-1  980 - 940 1000 .04  In a s i m i l a r manner as many c o n s t a n t s as i s r e q u i r e d t o make them add up to one (or s u f f i c i e n t l y c l o s e t o one) can be determined.  Choice o f Routing Method It i s realized  t h a t i f one were t r y i n g t o choose the " b e s t "  method f o r r o u t i n g flows i n the F r a s e r R i v e r the Muskingum method might n o t be chosen.  F o r the p r e s e n t study, however, the prime r e q u i r e -  ment i s f o r a " l i n e a r " r o u t i n g method to p r o v i d e data i n a foi~m s u i t a b l e f o r l i n e a r programming.  The method d e s c r i b e d above met t h a t need.  CHAPTER V  EMPLOYING LINEAR PROGRAMMING  General  L i n e a r Programming  [7] i s a mathematical  technique used  mize an o b j e c t i v e i f t h i s o b j e c t i v e can be expressed and the : c o n s t r a i n t s can be expressed  to o p t i -  as a l i n e a r  function  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 m a t h e m a t i c a l l y , are s i m i l a r to t h e g e n e r a l form shown below.  Maximize Z = c.x, + c„x„ + . . . + c x 1 1 2 2 n n  (5-1)  subject t o : a^x + a., „x + . . . + a, x < b 11 1 12 2 In n — 1  1  a„,x + a„„x + . . . + a x < b . 21 1 22 2 2n n — 2 •  •  a ..x + a „x. + . . . 4- a x < b ml 1 m2 2 mn n — m  (5-2)  x, > 0, x > 0, . . . x > 0 1 — z — n — Z r e p r e s e n t s the o p t i m a l v a l u e and e q u a t i o n s i o n c o n t a i n i n g any number of v a r i a b l e s .  (5-1) i s any l i n e a r  Equations  expres-  (5-2) a r e the l i n e a r  c o n s t r a i n t s , a l l o f which must be s a t i s f i e d s i m u l t a n e o u s l y w h i l e an o p t 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 c o n s i d e r e d a s o l u t i o n which maximizes  the o b j e c t i v e f u n c t i o n or i t may be a s o l u t i o n which minimizes 27  the objec-  28  tive function. problem t h a t sired.  Both are o p t i m a l , s o l u t i o n s and i t depends on the p a r t i c u l a r  i s being solved  j u s t which type of o p t i m a l  s o l u t i o n i s de-  The above example i s one i n which the o b j e c t i v e  function i s to  be maximized. minimization  However, t h i s problem could v e r y e a s i l y be changed t o a problem by changing the s i g n of the o b j e c t i v e  I f l i n e a r programming problems a r e s m a l l be  solved  using  graphical  impossible  often  However, l a r g e l i n e a r programming problems a r e  t o s o l v e by hand and r e q u i r e  which u s u a l l y s o l v e u s i n g The  they can q u i t e  techniques or by manually c a r r y i n g out the sim-  p l e x method of s o l u t i o n . virtually  function.  computer s o l u t i o n s ,  the simplex method o r some v a r i a t i o n o f i t .  computer program which was used i n the p r e s e n t study was a new  tine  which has j u s t r e c e n t l y been made a v a i l a b l e to the U . B . C .  [6]  routines.  T h i s program o f f e r s v e r y f a s t s o l u t i o n s  roulibrary  to quite large l i n e a r  programming problems at r e a s o n a b l e c o s t and proved to be an extremely valuable  asset. In the p r e s e n t problem of o p t i m i z i n g  f u n c t i o n and a l l o f the c o n s t r a i n t s l i n e a r e q u a t i o n s so t h a t mization  could  f l o o d c o n t r o l , the o b j e c t i v e  q u i t e e a s i l y be expressed as  l i n e a r programming proved to be an i d e a l o p t i -  technique.  F o r m u l a t i o n of C o n s t r a i n t s  In o r d e r to d i s c u s s first  the development of the c o n s t r a i n t s , i t i s  n e c e s s a r y to o u t l i n e b r i e f l y  objective function  the form of the o b j e c t i v e  can be expressed simply as  function.  The  29.  Min  Z =  Y  where Y i s a dummy v a r i a b l e . is  to be minimized  S i n c e i t i s t h e maximum f l o w at Hope t h a t  by s t o r i n g water i n the r e s e r v o i r , i t can be seen t h a t  Y must r e p r e s e n t the maximum v a l u e of the n a t u r a l f l o w a t Hope minus the r o u t e d h o l d o u t , i . e . , (Q - D ) . F o r each day of the f l o o d t h e r e i s n n max going t o be a v a l u e of Q - D and i t i s t h e f i r s t s e t of c o n s t r a i n t s n n which ensures  that the maximization  a c t u a l l y does take p l a c e w i t h  (Q - D ) n n  These c o n s t r a i n t s a r e : max  Y > —  Q - [c.1 + c„I + c,I _ + . . . + in 1 m 2 m-1 3 m-2  Y > —  (1  - [c.I + c.I + c.I . + . . . + 1 m+1 2 m 3 m-1  Tii+1  :  :  Y > —  c. I . ...] k m-k+1 c. I . ..] k m-k+2 ( 5 - 3 )  :  Q - [ I + c-I . + _ I + . . . + c. I . , J n I n 2 n-1 3 n-2 k n-k+1 C l  C  0  where Q = n a t u r a l d i s c h a r g e a t Hope, I = daily °1'  °2' ' " * " k C  =  These c o n s t r a i n t s ensure  rout  holdouts,  -'- 8 c o n s t a n t s . n  t h a t the maximum v a l u e of Q - D i s chosen f o r n n  the o b j e c t i v e f u n c t i o n . The  second  s e t of c o n s t r a i n t s concerns  water e n t e r i n g t h e r e s e r v o i r .  the l i m i t e d  Sine the d a i l y h o l d o u t s  be g r e a t e r than t h e d a i l y f l o w i n t o  amount o f  cannot  possibly  t h e r e s e r v o i r , t h e l i n e a r programming  must be c o n s t r a i n e d so as to choose a h o l d o u t  less  than or e q u a l to the  30  i n f l o w f o r t h a t day.  T h i s i s accomplished by  the f o l l o w i n g s e t of equa-  tions .  l ^ l  H  (5-4)  H  < I n — n  where H = v a l u e of d a i l y  holdouts,  I = natural d a i l y inflows  A third i s more than one the r e s e r v o i r . volume of the  constraint  to the  reservoir.  (only becomes'a s e t of c o n s t r a i n t s when  r e s e r v o i r ) r e s t r i c t s the  t o t a l amount of water s t o r e d  O b v i o u s l y the volume of water s t o r e d reservoir.  The  following  there  cannot exceed  in  the  c o n s t r a i n t makes sure t h i s  con-  dition is satisfied. E  1  + H  2  + H  3  + H  +  n  < Volume of —  Reservoir (5-5)  U s i n g the o b j e c t i v e f u n c t i o n and the l i n e a r programming w i l l c a r r y out minimum peak f l o w at Hope and a c h i e v e t h i s minimum peak.  the  constraints described  a s o l u t i o n which w i l l y i e l d  the v a l u e s of the d a i l y h o l d o u t s to  above, the  31  Routing  Holdouts  Although the l i n e a r programming s o l u t i o n g i v e s c o n s i d e r a b l e i n formation, graph.:  i t does not d i r e c t l y g i v e the shape of the downstream h y d r o -  In order  to get t h i s the h o l d o u t s  u s i n g the same r o u t i n g c o n s t a n t s These routed v a l u e s get the new  must be routed  down the  t h a t were used i n the l i n e a r  river  programming.  are then s u b t r a c t e d from the n a t u r a l hydrograph to  hydrograph w i t h f l o o d c o n t r o l i n c l u d e d .  F a c t o r s L i m i t i n g Peak Flow Reduction .  The  extent  of the r e d u c t i o n i n peak f l o w t h a t w i l l occur  w i l l be governed by two one  different  or the o t h e r w i l l be  circumstances. storage  One  u n r e l a t e d r e s t r i c t i o n s ; of which  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  of these  r e s t r i c t i o n s w i l l be due  can be no more r e d u c t i o n s  s t o r e d from then on.  most advantageous way  amount of water and  i n the flow at Hope.  mean t h a t e v e r y t h i n g w i l l be The  to lar  the end  the l i m i t e d day.  The  finite The  once i t has  reser-  filled  there  However, t h i s does not  s t o r e d u n t i l the r e s e r v o i r i s f u l l  and  noth-  l i n e a r programming s o l u t i o n determines  of making use of the l i m i t e d s t o r a g e  peak flow i s reduced a maximum amount w h i l e until  to the  c a p a c i t y of the h y p o t h e t i c a l r e s e r v o i r at S h e l l e y .  v o i r can o n l y s t o r e a l i m i t e d  ing  and  at Hope  of the f l o o d , i f at a l l .  The  so t h a t  the r e s e r v o i r does not second r e s t r i c t i o n i s  the  the fill due  amount of water f l o w i n g i n t o the r e s e r v o i r on any p a r t i c u f l o w at Hope cannot p o s s i b l y be reduced by more than  v a l u e which i s s t o r e d 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 the  32  amount of f l o w e n t e r i n g the r e s e r v o i r .  Thus, i f a l l the water e n t e r i n g  reservoir i s stored, t h i s obviously w i l l  set a l i m i t  on the r e d u c t i o n i n  f l o w at Hope, r e g a r d l e s s of the amount of water i n the r e s e r v o i r . two  r e s t r i c t i o n s are going  depending upon the type for  to govern the problem at d i f f e r e n t  of f l o o d .  limit  times,  I f the f l o o d i s f a i r l y h i g h and  a c o n s i d e r a b l e l e n g t h of time i t w i l l be the f i n i t e  which w i l l  These  the peak flow r e d u c t i o n .  r e s e r v o i r volume  However, i f the f l o o d  only  l a s t s f o r a s h o r t l e n g t h of time, then the maximum amount t h a t can s t o r e d on any p a r t i c u l a r day be  the governing  limitation.  lasts  ( l i m i t i n g i n f l o w hydrograph) w i l l  be  probably  the  CHAPTER VI RESULTS  The  technique  of o p t i m i z i n g f l o o d  was  c a r r i e d out on three d i f f e r e n t  The  t h r e e y e a r s t h a t were used were 1955,  f l o o d s of r e c o r d on the F r a s e r R i v e r .  c o n s i s t i n g of the t h r e e h i g h e s t recorded The  c o n t r o l t h a t has been d e s c r i b e d  1964  and  1967,  these  flows s i n c e the 1948  years flood.  r e s u l t i n g hydrographs at Hope as w e l l as the hydrographs showing  the o p e r a t i o n of the r e s e r v o i r can be seen i n the Appendix.  Since  the  r o u t i n g c o n s t a n t s f o r any p a r t i c u l a r r e a c h d i d v a r y from year to y e a r , t h r e e d i f f e r e n t s e t s of r o u t i n g c o n s t a n t s were used to t e s t the of  the method to d i f f e r e n t v a l u e s .  The  three d i f f e r e n t  s e t s of r o u t i n g con-  s t a n t s r e s u l t e d i n three d i f f e r e n t downstream hydrographs and  three  ent methods o f r e g u l a t i n g the upstream r e s e r v o i r f o r each year These t h r e e d i f f e r e n t  s e t s of r o u t i n g c o n s t a n t s are shown i n Table  the t h r e e  differ-  analyzed.  s i t u a t i o n s are d e p i c t e d by cases 1, 2 and  r o u t i n g parameters which were used i n d e t e r m i n i n g  sensitivity  3.  The  different  IV.1.  The p l o t s of the downstream hydrographs show t h a t the minimum peak v a r i e s s u r p r i s i n g l y l i t t l e stants.  even w i t h w i d e l y d i f f e r i n g  r o u t i n g con-  In other words, the maximum peak r e d u c t i o n i s r e l a t i v e l y  s i t i v e to the r o u t i n g c o n s t a n t s .  insen-  However, the o p e r a t i o n of the upstream  r e s e r v o i r v a r i e s c o n s i d e r a b l y f o r the d i f f e r e n t  cases and  tends  to be  q u i t e s e n s i t i v e to the r o u t i n g c o n s t a n t s . It  can be seen t h a t the l i n e a r programming determines  minimum peak flow w i l l be and  then s t o r e s as l i t t l e  what the  water as p o s s i b l e  34  to m a i n t a i n t h i s peak throughout the f l o o d i n g p e r i o d . a n a l y z e d , t h e t o t a l r e s e r v o i r volume was used.  In a l l of the y e a r s  However, by i n c r e a s i n g  the volume of the h y p o t h e t i c a l r e s e r v o i r , the r e s t r i c t i o n caused by l i m i t e d i n f l o w to the r e s e r v o i r could be made to govern.  Correcting  for A Limited  The included  Discharge  r e s u l t s which have been p r e s e n t e d have an i n h e r e n t  i n them which should  be e x p l a i n e d .  r e s e r v o i r has an i n f i n i t e d i s c h a r g e tary storage  Capacity  assumption  T h i s assumption i s that the  c a p a c i t y , which means t h a t  involun-  (shown by t h e shaded area o f F i g . 6.1) c o u l d be e l i m i n a t e d .  I t can be seen that i n v o l u n t a r y s t o r a g e  r e s u l t s when the i n f l o w to the  r e s e r v o i r exceeds the maximum d i s c h a r g e  c a p a c i t y of the r e s e r v o i r .  t h i s s i t u a t i o n does o c c u r , the i n v o l u n t a r y  storage  When  must be accounted f o r .  T h i s i s n o t a problem f o r a t h e o r e t i c a l r e s e r v o i r , b u t 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 d i s c h a r g e method i s a p p l i e d  capacity.  though the a l t e r a t i o n s r e q u i r e d not n e c e s s a r y i n d e v e l o p i n g  this  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 -  are q u i t e simple,  i t was f e l t  t o route  the i n f l o w s  r e s e r v o i r a t i t s maximum d i s c h a r g e  capacity  through the r e s e r v o i r w i t h the  capacity  (reservoir routing).  o u t f l o w s would then form the maximum d i s c h a r g e ( F i g . 6.1).  they were  the b a s i c methodology of t h i s f l o o d c o n t r o l  The t e c h n i q u e used t o overcome a f i n i t e d i s c h a r g e  would be, f i r s t ,  reservoir.  before  to a r e a l s i t u a t i o n , changes w i l l have to be made t o  take account o f the l i m i t e d d i s c h a r g e  method).  Therefore,  The  curve f o r t h a t p a r t i c u l a r  35  \> Natural Inflow Hydrograph Involutary-^ Storaqe ^ /  1 X — Maximum Discharge Curve Hn y<^~  I  1 / //  /  /  S I. F  TIME  Fig. 6.1  H o l d o u t s N e c e s s a r y to Account for Involuntory  Due  t o storage  Storage.  i n the r e s e r v o i r , the maximum d i s c h a r g e  curve w i l l  be l e s s than o r e q u a l to the i n p u t hydrograph up to the p o i n t where the two  curves c r o s s .  During the p e r i o d when the i n f l o w hydrograph exceeds  the maximum d i s c h a r g e  curve t h e r e w i l l have to be a s e t of c o n s t r a i n t s  f o r c i n g the l i n e a r programming to have minimum h o l d o u t s . ing  t h i s c r i t i c a l p e r i o d water i s being  age  and t h i s volume s t o r e d must be accounted f o r by  gram to have  holdouts  tween the two curves.  value  greater all  stor-  c o n s t r a i n i n g the p r o -  R e f e r r i n g t o F i g . 6.1, I r e p r e s e n t s n the maximum d i s c h a r g e  Thus, 1^ minus F^ r e p r e s e n t s  on day 'n'.  s t o r e d due to i n v o l u n t a r y  which a r e at l e a s t as l a r g e as the d i f f e r e n c e be-  f o r day 'n' and F^ r e p r e s e n t s  t h a t day.  That i s , d u r -  the i n f l o w  p o s s i b l e on  the i n v o l u n t a r y s t o r a g e  occurring  I t can be seen t h a t the h o l d o u t on day 'n' must be made  than or e q u a l t o the v a l u e  the water s t o r e d .  of I  - F^ i n o r d e r  t o account f o r  T h i s s i t u a t i o n o c c u r s each day the i n p u t  hydro-  36  graph exceeds the maximum d i s c h a r g e curve and so t h e f o l l o w i n g  constraints  would be needed.  H  l  H  H  ^  2  h~  > I  2  l  F  - F  (6-1)  2  > I - F n — n n  where H = d a i l y holdout, I = daily  inflow,  F = d a i l y maximum  Limitations  At  discharge.  of the Method  t h i s point  of o b t a i n i n g  i t should be emphasized t h a t employing t h i s method  o p t i m a l f l o o d c o n t r o l to i t s f u l l e s t  advantage  the knowledge of a l l d i s c h a r g e hydrographs i n advance. means t h a t  a perfect  f o r e c a s t i n g method i s r e q u i r e d  s o l u t e minimum peak f l o w a t Hope.  reservoir  p o l i c y would v a r y c o n s i d e r a b l y  what happens l a t e r on i n the f l o o d i n g p e r i o d . period  or four  days  I t would be p o s s i b l e to use  a f o r e c a s t o f s e v e r a l days and determine a b e s t p o l i c y but t h i s o p e r a t i o n  In e f f e c t t h i s  to a c h i e v e the ab-  A f o r e c a s t of three  would be o f l i t t l e v a l u e to t h i s method.  requires  operation depending on  An a n a l y s i s u s i n g  a short  f o r e c a s t w i l l suggest s t o r i n g b e g i n immediately and w i l l y i e l d a  minimum peak d u r i n g  t h e f o r e c a s t p e r i o d which most  the minimum peak f o r the e n t i r e f l o o d .  c e r t a i n l y w i l l not be  An a n a l y s i s u s i n g  a total flood  37  f o r e c a s t would r e v e a l the most opportune time to b e g i n s t o r i n g water i n order to get a minimum downstream peak f o r the e n t i r e f l o o d .  Although  a  f o r e c a s t f o r the e n t i r e f l o o d i s r e q u i r e d to g i v e a minimum peak f l o w for  the whole f l o o d i n g season,  there are p o s s i b l e ways of m i n i m i z i n g  d e f i c i e n c y which w i l l be d i s c u s s e d Given this flood of  later.  a p e r f e c t f o r e c a s t (or data of r e c o r d ) , the accuracy  c o n t r o l method becomes completely  dependent on the  the h o l d o u t s b e i n g r o u t e d .  produce c o n s i d e r a b l e e r r o r s i n  However, t h i s problem can be minimized  I f the method were b e i n g used f o r f l o o d c o n t r o l on a forthcoming an average v a l u e of r o u t i n g c o n s t a n t s  also. flood,  from a n a l y s i s of p a s t f l o o d s must  In t h i s s i t u a t i o n a f o r e c a s t hydrograph must a l s o be used  the e r r o r s i n the f o r e c a s t w i l l ing  essen-  S i n c e the r o u t i n g c o n s t a n t s d i d v a r y c o n s i d e r a b l y from  year to y e a r , average r o u t i n g v a l u e s may  be used.  of  accuracy  the r o u t i n g c o n s t a n t s , s i n c e the l i n e a r programming i n t r o d u c e s  t i a l l y no e r r o r s .  constants.  this  e a s i l y overshadow any  and  e r r o r s i n the r o u t -  I f the method were being used f o r p l a n n i n g  purposes,  a n a l y s i s of past r e c o r d s f o r the p a r t i c u l a r year or year would r e v e a l the c o n s t a n t s which f i t each y e a r ' s data b e s t .  By u s i n g the b e s t v a l u e s ( i f  they are a p p r e c i a b l y b e t t e r than the average v a l u e s ) the r o u t i n g should be of about the same accuracy of  as the a v a i l a b l e d a t a .  procedure  So, f o r e i t h e r  the above a p p l i c a t i o n s , the e r r o r s i n r o u t i n g c o n s t a n t s w i l l not  nificantly affect  P o s s i b l e Future Now  the f i n a l  sig-  results.  Extensions  t h a t the b a s i c l i n e a r programming method has been  developed  38  using be  a s i m p l i f i e d model there  a p p l i e d i n order  a r e s e v e r a l simple e x t e n s i o n s  which  could  to make the method more complete and f l e x i b l e .  One of the p o s s i b l e e x t e n s i o n s  would be t h e i n c l u s i o n of s e v e r a l  more r e s e r v o i r s t o take i n a l l t h e proposed dam s i t e s on the headwaters of the F r a s e r R i v e r .  T h i s would enable one to c a r r y out a complete f l o o d  c o n t r o l study o f the F r a s e r R i v e r not  system.  Adding more r e s e r v o i r s would  change the channel r o u t i n g except t h a t c o n s i d e r a b l y  would be r e q u i r e d  t o determine the r o u t i n g constants  branches of the r i v e r .  more a n a l y s i s  f o r the other  The l i n e a r programming would remain b a s i c a l l y  the same except t h a t there would be more s e t s o f c o n s t r a i n t s .  F o r each  r e s e r v o i r t h e r e would be s e t s o f c o n s t r a i n t s s i m i l a r to e q u a t i o n s (5-4) and  (5-5). The-other s e t o f c o n s t r a i n t s  (equations  5-3) would simply be-  come l a r g e r 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  m  1 first  reservoir  'J  _ , , ] - . . . .  n-m+1  J  v  second r e s e r v o i r  where H = h o l d o u t s from f i r s t  reservoir,  J = h o l d o u t s from second r e s e r v o i r . Thus, i t can be seen that although more r e s e r v o i r s w i l l of the problem c o n s i d e r a b l y ,  increase  the s i z e  the b a s i c methodology remains the same.  Another a d d i t i o n which could p o s s i b l y prove v a l u a b l e would be the use  of s t o c h a s t i c data.  When attempting to apply  this flood control  39  method to a forthcoming f l o o d t h e r e are s e v e r e f o r e c a s t may  be  f l o o d s might then be  forecast  willing lem  It  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 f l o o d d a t a .  synthetic is  limitations.  i t could  to a c c e p t .  of f o r e c a s t i n g  which would be  be  s t a t i s t i c a l l y a n a l y s e d so t h a t when a f l o o d  a d j u s t e d to account f o r the  amount of r i s k one  A l t h o u g h s t o c h a s t i c d a t a w i l l not a f l o o d i t might enable one  quite  These  eliminate  t o employ a r i s k  the  is  prob-  function  useful.  Possible Applications  of the  Method  Although most of the p o s s i b l e uses of t h i s f l o o d c o n t r o l method have been mentioned throughout the t i o n s may  prove  One  give  primary ways of employing t h i s f l o o d  i n d e t e r m i n i n g the b e s t way  the b i g g e s t  ment of a f o r e c a s t sent a p p l i c a t i o n s Due it  two  the b e s t f l o o d c o n t r o l d u r i n g  discussed,  can be  varied basin  to r e g u l a t e  As has  f o r the  to  been require-  e n t i r e f l o o d , which w i l l p r o b a b l y l i m i t  pre-  i n t h i s area.  to the  considerable  f l o o d problems.  For  than t h e r e i s i n the  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 p r o c e d u r e ,  the b e s t r e s e r v o i r o p e r a t i o n example, t h e r e may o t h e r s and  the  to s u i t many  be more snow i n one  sub-  t h i s method could be used to take operation.  second p r i m a r y a p p l i c a t i o n i s i n the  f a c t that  reservoirs  a forthcoming f l o o d .  t h i s i n t o account when p l a n n i n g f l o o d c o n t r o l  to the  control  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  used to d e c i d e on  The  applica-  valuable.  of the  method would be  t e x t , a b r i e f summary of the  s t a t e of f o r e c a s t i n g  f i e l d of p l a n n i n g .  i s considerably  behind the  Due de-  40  velopment of a n a l y t i c a l t o o l s r e q u i r i n g f o r e c a s t s , any p r e s e n t t i o n s would most p r o b a b l y be i n the p l a n n i n g a r e a s . lems, f l o o d s of r e c o r d used which e l i m i n a t e s  or p o s s i b l y  flooding conditions primary o b j e c t i v e , By possible  Thus, by a n a l y z i n g  Although t h i s would be the  there are o t h e r parameters which can be examined.  analyzing  r e c o r d e d and s y n t h e t i c  floods  i t would a l s o be  t o determine the maximum r e s e r v o i r s i z e r e q u i r e d  at the  For each f l o o d a n a l y z e d , the l i n e a r programming  the minimum r e s e r v o i r volume r e q u i r e d .  Thus, by a n a l y z i n g  d i f f e r e n t f l o o d s , the l a r g e s t minimum r e s e r v o i r volume r e q u i r e d be  determined and the r e s e r v o i r could  if physically  many  o f the r e s e r v o i r system under d i f f e r e n t  would be determined.  selected locations. gives  In planning prob-  s y n t h e t i c a l l y generated f l o o d s a r e  the need f o r a f o r e c a s t .  s e t s o f d a t a the e f f e c t i v e n e s s  applica-  many could  then be designed f o r t h i s v a l u e ,  possible.  Another parameter r e l a t e d t o the p l a n n i n g a s p e c t s would be the d e t e r m i n a t i o n o f the most e f f e c t i v e l o c a t i o n f o r proposed Once a g a i n , by u s i n g the  r e c o r d e d and s t o c h a s t i c  reservoirs.  f l o o d d a t a , and by v a r y i n g  l o c a t i o n o f the proposed r e s e r v o i r s , i t would be p o s s i b l e  mine the r e l a t i v e e f f e c t i v e n e s s f l o w a t Hope due to the v a r i e d  to deter-  of the r e s e r v o i r s i n r e d u c i n g the peak locations.  CHAPTER V I I  S U M M A R Y  The  problem of d e t e r m i n i n g the o p t i m a l way  to operate a remote  r e s e r v o i r to maximize the r e d u c t i o n  i n downstream peak flow  c o n t r o l purposes has  The  been s t u d i e d .  assuming a s i n g l e r e s e r v o i r w i t h a g i v e n c a p a c i t y , and  a p p r o x i m a t e l y 500  miles  problem has storage,  for flood  been s i m p l i f i e d by  an u n l i m i t e d  from the area  discharge  to be p r o t e c t e d .  long as a l i n e a r channel r o u t i n g method i s used, l i n e a r programming been shown to be  an e x c e l l e n t o p t i m i z a t i o n  problem q u i t e e a s i l y .  By u s i n g  technique which s o l v e s  As has  the  a Muskingum type of channel r o u t i n g ,  the requirement of a l i n e a r r o u t i n g method i s s a t i s f i e d . Widely d i f f e r e n t channel r o u t i n g parameters have been used to t e s t the s e n s i t i v i t y of the developed f l o o d c o n t r o l method to d i f f e r e n t routing constants. is relatively  I t has  been shown t h a t the r e d u c t i o n  i n s e n s i t i v e to r o u t i n g constants  r e s e r v o i r i s q u i t e dependent on the r o u t i n g  but  i n peak  the o p e r a t i o n  flow of  the  constants.  Although the method i s e a s i l y extended to more complex systems, f o r e c a s t s hamper i t s use  i n an o p e r a t i o n a l sense.  To o b t a i n the  f l o o d c o n t r o l of a f o r t h c o m i n g f l o o d , a l l upstream and graphs f o r the e n t i r e f l o o d i n g p e r i o d must be f o r e i s more u s e f u l f o r p l a n n i n g  forecast.  downstream hydroThe  purposes as data of r e c o r d  thus e l i m i n a t i n g the need of a f o r e c a s t .  41  best  method are  there-  then used,  B I B L I O G R A P H Y  Final Re-port of the Fraser River Board on Flood Control and HydroEleotrio 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., K o h l e r , M. A., and Paulhus, J . L . Hydrology for Engineers. New York, Toronto and London: McGraw-Hill Co. I n c . L t d . , 1958.  Chow, V. T. et al. Handbook of Applied McGraw-Hill Co. I n c . L t d . , 1964.  Hydrology.  New York:  "Routing of F l o o d s Through R i v e r Channels," Engineering and Design Manual EM 1110-2-1408, U.S. Corps of E n g i n e e r s , March 1, 1960. 3  Draper, N. R. and Smith, H. Applied John W i l e y and Sons, 1966.  Regression  Analysis.  New York  U.B.C. U s e r s ' Group Program. A Linear Programming Package - LIP. D. O ' R e i l l y . U n i v e r s i t y o f B r i t i s h Columbia Computing C e n t r e , September 1970.  H i l l i e r , F. S. and Lieberman, G. J . Introduction to Operations Research. San F r a n c i s c o , C a l i f . : Holden-Day, I n c . , 1969.  42  A P P E N D I X  44  F L O O D C O N T R O L ON T H E  F R A S E R RIVER ( 1955 ).  400  375 350  325 300  275 250 225  U-  200  o o 150 o o  Discharge Hydrographs at Hope Natural Discharge  100 UJ  o tr < i o  50  0  Reservoir Regulation at Shelley. Case  150 Natural Discharge  100 50  0 150  Reservoir  Regulation at Shelley. Case 2 Natural  Discharge  100 50  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 iJULY i i i iii 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• JUNE MAY  I  i i iii i  F L O O D C O N T R O L ON THE F R A S E R RIVER ( 1 9 6 4 ) ^  425  400  375  350  325  300  275  250  225  200  Discharge Hydrographs at Hope  Lu  O  I 50  O O O  Natural Discharge 100  LU  50  or < I  o -  o  Reservoir  160  Regulation  at Shelley. Case Natural Discharge  100  50  0  Reservoir  150  Regulation at Shelley. Cose 2 :-— Natural Discharge  100  50  Reservoir  i i i l i i i l ii MAY  Regulation at Shelley. Case 3 i  i  i  JUNE  i  i  i  i  i  i  i  ,  ,  ,  i  i  JULY  i  i  m 45  46  FLOOD CONTROL ON THE FRASER RIVER ( 1 9 6 7 ) . 400  Discharge Hydrographs at Hope Nature I Discharge  Reservoir Regulation at Shelley. Case Natural Dischorge  Reservoir R e g u l a t i o n at S h e l l e y . Case 2. Natural Discharge  Reservoir Regulation at S h e l l e y . Case 3 . . ' ' ' ' '  MAY  I*'  i  i  i  i  i  '''''''''''''''•I'''''  JUNE  •  •  •  JULY  ii iii i  

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