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A computer-aided design scheme for drainage and runoff systems Battle, Timothy P. 1985

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COMPUTER-AIDED DESIGN SCHEME FOR DRAINAGE AND RUNOFF SYSTEMS  by TIMOTHY P. BATTLE A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE  in THE FACULTY OF GRADUATE. STUDIES Department of C i v i l E n g i n e e r i n g  We a c c e p t t h i s  t h e s i s as conforming  to the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA O c t o b e r 1985 ©  T i m o t h y P. B a t t l e , 1985  In  presenting  this  requirements  Columbia,  freely  available  permission  scholarly  partial  I agree  the  for reference  purposes or  understood  that gain  that  and s t u d y .  by  may his  copying  shall  not  be or  granted her  be  of C i v i l  of  the  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date: October  1985  Columbia  make  I further this  by  the  Head  of t h i s  without  of it  agree  thesis  representatives.  allowed  Engineering  shall  of  or p u b l i c a t i o n  permi s s i o n .  Department  fulfilment  Library  for extensive copying  Department  financial  in  f o r an a d v a n c e d d e g r e e a t t h e The U n i v e r s i t y  British  that  thesis  of It  for my is  thesis for my  written  Abstract  A computer-aided  design  runoff  is  systems  programming drainage  to  of m i n i m i z i n g hydrograph of  the  each  flow  can at  efficient  construction damage.  as  i n speeding  costs  or  up  shows  location. the d e s i g n  - i . e . , one reduces  through  which  a  through  W i t h the o b j e c t i v e system  variables,  node i n t h e  some downstream  design  the  the d e c i s i o n  linear  information  (sensitivity) analysis. from  natural  uses  routing equations  programming a n a l y s i s  at  model  design  t h e peak o u t f l o w  a i d the user an  provides  o r d i n a t e at every  flow  The  s o l v e Muskingum  ordinates  linear  t h e peak  presented.  s y s t e m , and  post-optimality  scheme f o r b o t h man-made and  the  and  using  the  output  extent  that  network i n f l u e n c e s This information process either  the p o t e n t i a l  risk  to  arrive  minimizes of  flood  Table  of Contents  Abstract  i i  List  of Tables  List  of F i g u r e s  v vi  Notation  v i i  Acknowledgement 1.  ix  INTRODUCTION  1  1.1 D e s i g n  3  Refinement  1.2 E x i s t i n g  2.  Models  ..4  1.3 C o m p u t e r - A i d e d D e s i g n  6  PROGRAM FORMULATION  9  2.1 Muskingum  9  2.2 L i n e a r  Channel Routing  Programming  and S e n s i t i v i t y  2.3 S o l v i n g t h e Muskingum Programming 2.3.1  3.  Objective  Function  using  Formulation  Linear  13 14 15  2.3.2 Minimax C o n s t r a i n t s  16  2.3.3  17  S i n g l e Reach  2.3.4 S e r i e s o f R e a c h e s  20  2.3.5  Branching  23  PROGRAM DESCRIPTION  28  3.1  28  Input  3.2 F o r m u l a t i n g 3.3 L i n e a r  4.  Equation  Analysis  the Tableau  Programming  Subroutine  31 32  3.4 O u t p u t  34  ALTERNATE FORMULATION  35  4.1 S i n g l e  35  Reach  4.2 Two R e a c h e s i n S e r i e s  36  4.3  Series  of Reaches  40  4.4  Program D e s c r i p t i o n  44  4 . 5 Comment 5.  6.  46  NUMERICAL EXAMPLES  47  5.1  Case  1: S i n g l e  5.2  Case  2: T h r e e Reaches  5.3  Case  3: B r a n c h e d Network  76  5.4  Case  4: Complex  84  DISCUSSION OF  Reach  48 in Series  Network  RESULTS  62  91  6.1  Model L i m i t a t i o n s  and C a p a b i l i t i e s  6.2  Dual  6.3  R i g h t Hand S i d e R a n g i n g  94  6.4  Cost  95  6.5  Graphics  Solution Vector  91 92  Analysis Interface  ......96  REFERENCES  97  APPENDIX A: PROGRAM LISTING  99  APPENDIX B: LIPSUB  DESCRIPTION  iv  105  List  of Tables  I  ROUTING CONSTRAINTS FOR A SINGLE REACH  21  II  LP TABLEAU FOR A SINGLE REACH  22  III  LP TABLEAU FOR 3 CHANNELS  24  IV  LP TABLEAU FOR A NETWORK WITH ONE BRANCH  27  V  SAMPLE  29  VI  EXAMPLE  1.1 RESULTS  49  VII  EXAMPLE  1.2 RESULTS  53  VIII  EXAMPLE  1.3 RESULTS  56  IX  EXAMPLE  1.4 RESULTS  59  X  EXAMPLE  1.5 RESULTS  60  XI  EXAMPLE  2.1 RESULTS  66  XII  CASE 2 PEAK OUTFLOWS  69  XIII  EXAMPLE  2.6 RESULTS  74  XIV  EXAMPLE  3.1 RESULTS  78  XV  EXAMPLE  4.1 RESULTS  87  IN SERIES  INPUT F I L E  v  List  1 .  EXAMPLE  of F i g u r e s  1.1  50  2.  CASE 1 OUTFLOW HYDROGRAPHS  52  3.  EXAMPLE  1.2  54  4.  EXAMPLE  1.3  57  5.  EXAMPLE  1.5  61  6.  EXAMPLE  2.1  63  7.  CASE 2 OUTFLOW HYDROGRAPHS.  65  8.  EXAMPLE  2.2  67  9.  EXAMPLE  2.3  70  10.  EXAMPLE  2.4  71  1 1 .  EXAMPLE  2.5  73  12.  EXAMPLE  2.6  75  13.  EXAMPLE  3.1  77  14.  CASE 3 OUTFLOW HYDROGRAPHS  15.  STATION 3: COMPARISON  OF EXAMPLES  3.1 & 3.2  82  16.  STATION 7: COMPARISON  OF EXAMPLES  3.1  83  17.  STATION 10: COMPARISON  1 8.  EXAMPLE  OF EXAMPLES  4.1  80  3.1  & 3.3 & 3.4  85 86  vi  Notat ion  kinematic  c  c  l n  /  series expressing  for  ordinate  the  n "* 1  the  node  the  number  the  of  of  nodes  constant;  for'a  length  of  the  N  ordinate. the total hydrograph.  p(a,b)  a c o e f f i c i e n t of  a  coefficient  / r e a c h .  of  a r e a c h , or  of  upstream.  in  the  the  channel  identification  drainage  ratio  of  reach or  a channel  n  of  t h e sum o f  0 occurs  system.  storage  to  reservoir.  reach.  number  number  a n d 7, w h e r e  the  number  immediately  discharge  .  1  identification  storage  L  speed.  a recursive  the  I  wave  of  a  hydrograph  ordinates  in  a  t h e p r o d u c t s o f /3  a times  and 7 occurs  b  times.  q  Q  the  l n  the  d i s c h a r g e per  discharge  hydrograph  at  at node  vi i  unit  the / .  channel  width.  ordinate  of  the  t h e peak o u t f l o w drainage  a t t h e downstream p o i n t o f a  system.  the  storage  in a channel  the  s l o p e of t h e c h a n n e l  the  routing period  reach  or  reservoir,  bed.  ( t h e time  interval  between  successive ordinates).  a  dummy  variable  used  formulat ion. t+ 2k{ t-2kx  for  reach  /.  t~2k(1-x) t -2kx  for  reach  /.  t +2kx f o r t -2kx  reach  /.  vi i i  in  LP  minimax  Acknowledgement  I  would  Sciences  and  support  for  express  my  encouragement  like  t o e x p r e s s my  Engineering this  and  Research C o u n c i l  reasearch.  appreciation  a p p r e c i a t i o n to the  I  to  guidance  Dr.  ix  funding  would e s p e c i a l l y W.  during  thesis.  for their  F. the  Caselton  Natural  like  for  preparation  of  to his  this  Chapter 1 INTRODUCTION  In  recent  drainage  years,  t h e need  flood  control  and  increasingly increased affected be  important.  potential  increased  greater  to  To  complex  watershed-wide  costs, land  areas  drainage aspects  a large  extent, the  has a l l o w e d f o r d e t a i l e d  more r e a d i l y , planning  of  r e c o g n i z e d as  damage, and l a r g e r  have demanded t h a t  produced  design  construction  of computers  designs  has been  Higher  consideration.  availability be  systems  for flood  by d e v e l o p m e n t s  given  f o r more t h o r o u g h  and h a s l e d t o more  approach  (Yen  &  Suvak,  1975). The  usual approach  in designing  a  drainage  system  is  to: 1.  select  2.  determine the  a  layout; design  flows  entering  system;  3.  estimate the channel parameters;  4.  calculate  5.  modify and  Step course  t h e f l o w s t h r o u g h o u t t h e s y s t e m ; and  t h e c h a n n e l p a r a m e t e r s and r e p e a t  4 until  not  apply  systems,  the  layout  features  and  the  system  a satisfactory  1 depends e n t i r e l y  does  each element of  serves.  to  on  Step  2  is  site  conditions, systems.  largely  arrangement  on  usually  and  of  F o r man-made topographical  of the development  1  3  design i s achieved.  natural  depends  steps  performed  which the based  on  2 precipitation system. event  The for  records  or  design c r i t e r i a , which  the  The  Certain  design variables  site  width  third  step  conditions, and  provided  type  they  but  are  is  as  is  the  to  based  upstream  of  the  frequency  of  the  be  designed,  largely  on  left  parameters to  convey  the the  such  from  be  dependence as  designer's flows  must  judgement.  s u c h a s g r a d i e n t have a most  adequately  records  such  system  chosen.  on  flow  channel  discretion,  the  upstream  section. Many r o u t i n g the as  p a s s a g e of a f l o o d  major  routing both  the  include  the  Muskingum  channels, linear  for  and  and  Storage  reservoir  flow  a  Hydrologic  (or  e q u a t i o n combined  partial i n open  differential channels.  and  uses  storage-discharge routing  methods  methods f o r Pulse)  and  H y d r a u l i c methods an  equation  These types  equations  in  hydraulic  routing  Modified  with  reach  classified  Integration  o f t e n t h e momentum e q u a t i o n .  methods use usteady  and  Graphical  Indication  be  Hydrologic  equation  predicting,  or c h a n n e l  routing  methods f o r r e s e r v o i r s .  the c o n t i n u i t y  motion,  hydrologic  reach.  for  a reservoir  e t a l . , ' 1977).  continuity  relationship  available  T h e s e methods can  categories,  (Viessman  are  through  r e q u i r e d f o r s t e p 4.  two  use  procedures  to  of  of  routing  describe  the  3 .1 . 1 DESIGN REFINEMENT The  final  attempts  step  i n the d e s i g n  t o p r o d u c e a more e f f i c i e n t  a s many t i m e s  as  step  arrive  is  to  design  flows  least  flood  facilities,  deemed n e c e s s a r y . at  a  damage  for  c o s t s are reducing  the  there  variables,  even  guidelines  to  will are  whether  to  suggest  on  difficult  to  the  system.  conduit  or  For  channel of a  downstream  portion  in  decrease  Introduction p r o b l e m of typically on  the  finding  i n the  design  decrease  from  and  the  by  the Since  of  the  elements  of  in costs. design  very  few  modify.  of a p i p e  or  channel  system  how  The  (e.g.,  much)  are  a r e - a n a l y s i s of  example, d e c r e a s i n g  the  one  branch  the  may  system. one  of  of  storage  the  optimal the  the  area  design  a  b r a n c h e s may farther  the in  a  slight result  complicates  routed  the  Storage  (reduction)  flows are  a  downstream.  even more.  attenuation  of in  Alternatively,  reservoirs  However, as  sizes  result  cross-sectional  i n t h e maximum f l o w  flow.  in  size  and  to  the  s h o r t of p e r f o r m i n g  used because of peak  system,  variables  this  budget.  to the  which  i n c r o s s - s e c t i o n of  a great  of  a l a r g e number of  runoff  larger  of  repeated  resulting  in a decrease  usually  in  much  be  of c o n v e y i n g  design  peak outfow  or  predict  requirement  increase  the  increase  entire  result  i n a modest  branch  one  given  can  designer  objective  capable  or  the  maximum f l o w s w h i c h t h e  c o n s e q u e n c e s of a c h a n g e one  design,  generally related  the- s y s t e m must c o n v e y However,  a  where  The  design  a t a minimum c o s t ,  construction  in  process,  is  effect  through  a  4  reservoir,  translation of the peak to a later point in time  also occurs. storage  In  i s that  other  words,  the result  of  temporary  the hydrograph of a flood passing through  the reservoir has a lower peak flow which occurs at a time  than  i f there had been no storage.  translated peak in one branch may flow  from  another  the  As a result, the  coincide  with  the peak  branch, resulting in a much higher peak  flow downstream of the junction. by  later  Thus the time lag imposed  introduction of storage can counteract the benefits  of the attenuation of obvious,  especially  peak when  flow,  and  dealing  i t i s not always  with a complex network,  when storage w i l l reduce overall peak flows. detention  storage  The fact  that  i s not always helpful in reducing peak  flows i s becoming increasingly recognized (Curtis & McCuen, 1977; Duru, 1981; Dendrou & Delleur, 1982). The model presented in this thesis w i l l point locations  out the  and times at which flow modifications w i l l reduce  the peak flow at some downstream reference point.  It  will  also indicate the effectiveness of any such modifications in reducing the peak outflow, thereby allowing the designer choose  to  a design which w i l l result in the greater benefit to  the overall system.  1.2 EXISTING MODELS Many types of computer models systems  are  stormwater  for designing  drainage  in use today, especially in the f i e l d of urban  runoff  management.  There  are  four  broad  5 categories  of  such  sophistication estimate which  the  which p e r f o r m  the  An level  a detailed  the  the  Stormwater  States  storm For for  which  event  multiple  catchments,  model,  developed  by  Protection  Agency,  management  s i m u l a t i o n model  flows  simplified  flows  pipe diameters surface  and  used a  the  planning  (Yen  by  the  T h i s model can  be  capacity  1977).  single-event  storms  Storm Water Management in  the U n i t e d  North  States  as  a  America.  Environmental  prediction  & Suvak,  1975).  starting  larger  sizes  and  I t uses a  approximation  system,  with  at the  (Perks,  popular  runoff  iterating  event, control  treatment  of  classified  a  storm  to  route  with  small  until  free  flow c o n d i t i o n s are a t t a i n e d . i n common use  in  North  Storm Sewer S i m u l a t i o n M o d e l  for both  given  and  kinematic-wave  through  A n o t h e r model Illinois  is  models,  (STORM), d e v e l o p e d  applications  This  models,  actual  f o r use  catchments  (SWMM) i s among t h e most  which  planning  produce  required storage  design  models,  of  (Huber e t a l . , 1981).  Model  single  level  problem; d e s i g n  suitable  Model  slightly  problem;  C o r p s of E n g i n e e r s .  r u n o f f from final  increasing  s i m u l a t i o n of a s i n g l e  Runoff  Army  an  screening  runoff  models,  d u r i n g a storm  for planning  and  of  example of a model  United used  complexity:  overall  operational  decisions  for  and  complexity  assess  and;  models, each w i t h  flow p r e d i c t i o n  system  employs u n s t e a d y  layout flow  and  and  equations  America  (ISS).  d e s i g n of  specified and  the  T h i s model i s  sewer s i z e s  slope.  accounts  is  The  f o r the  for  method mutual  6 backwater e f f e c t s Many  each.  literature  Good  to c e r t a i n  Perks  Fok  (1977), While  none  the  will  or  location  will  downstream,  i s needed  design  1.3  i s a method  This  (CAD)  the  entire  branch),  system.  in  For  a t an  upstream  in  cost  savings  need t o be  in order  how  to  re-run. alter  to achieve the  a  greater  computer-aided  here.  COMPUTER-AIDED DESIGN Computer  aided  d e s i g n has  so  f a r had  little  design procedures  i n h y d r o t e c h n i c a l d e s i g n and  open  as  question  water-related future. techniques both  until i t  size  for determining  scheme p r e s e n t e d  parameters  storage  would  of  in  a  benefits  objective  their  found  w i t h i n each  the  merits  (1981).  design  conduit  system d e s i g n is  are  SWMM i t e r a t e s  size  program  there i s  m o d e l s and  investment  greater  entire  drainage  benefits.  i f the  larger  produce the  at  and  relative  Stephenson  o p t i m i z a t i o n of  slightly  the  situations  ( f o r example,  to determine  available on  the m o d e l s a r r i v e  perform  facility  complex  design  segments  network.  various  smallest possible pipe  the d e s i g n e r  What  of  e t a l . (1979) and  some of  individual  reaches  i n the  expanding  comparisons  applicability  for  sewers  o t h e r models a r e c u r r e n t l y  a g r e a t d e a l of of  between t h e  to  p r o j e c t s might  Certainly  and  be  far  design  handled  the p o t e n t i a l  i n the d e s i g n  natural  how  by  on  i t remains  an  processes  computers  in  b e n e f i t s i n t h e use  or m o d i f i c a t i o n of  man-made,  impact  runoff  have y e t t o be  of  for the CAD  systems,  e s t a b l i s h e d or  7  even  demonstrated. B e c a u s e of  overall  runoff  the  view  of  political  design  of a CAD  selection  design  (Besant,  straightforward engineering  design  equation  CAD  problem  approach  designer  optimization  guided  scheme.  such  information the  The  optimal  source  in  of  toward  a  p u r p o s e of nature  and  the  any  which  g l o b a l sense  can  i s being  with  Sensitivity techniques  considered  sought.  an  presented  to  involved there be  the  through  communicated  is  type  information  the d e s i g n e r .  information  element  iterative  research of  civil  the  thesis,  value  design  in  An  of  beyond  finite  1983).  the  one  support  "good" d e s i g n  optimization  that a design  science.  method; or  include  this  problem  into  d e r i v e d from p o s t - o p t i m a l i t y a n a l y s i s  Although  suggestion  in  a scheme m i g h t p r o v i d e  primary  designer.  than  o p t i m i z a t i o n d e p e n d i n g on  considered  i s to a s c e r t a i n the  which  may  of  expected  i n v o l v e d (Encarnacao,  is  being  be  is  approach.  usually f a l l s  can  and  design.  interaction  art  Analytical  and  economic  a type  method; d i r e c t  solving  schemes  more  process  the  undesirable  final  s y s t e m CAD  is itself  1983).  s i m u l a t i o n , or  design  runoff  scheme  categories: iterative  methods,  social,  i n t e r v e n t i o n and  presently involves  computerized  here  intangible  i n any  of  n o n - i n t e r a c t i v e computer  It i s a l s o perhaps  human d e s i g n e r  whose, s o l u t i o n  is  largely  anticipated  The  three  dynamic c o m p l e x i t y  o b j e c t i v e s which a l s o i n f l u e n c e the  therefore  of  improbable.  the  Considerable  The  and  system, a c o m p l e t e l y  approach appears in  geometric  A CAD  the i s no  to  be  scheme  8 which and  leads  to designs  superior  seem  with  w h i c h c a n be s a i d  a reasonable  t o be a r e a l i s t i c Both  been  linear  and p r a c t i c a l  and n o n - l i n e a r  incorporated  into  CAD  here a s t r i c t l y  linear  adopted.  represents  This  to m o d e l l i n g exist,  runoff,  particularly  However,  the  powerful  optimization  and  in practice in  investigating be e x p l o i t e d  the  The  to  storage models offer  the  of  system  is  s i m p l i s t i c approach  non-linear  models  by  t h e most  with linear  the  do  systems.  allowing, reliable most  the and  refined  model d o e s have a channel  routing,  by  linear  reservoirs.  The  and  linear  programming  was  most  s o u r c e s of a n a l y t i c a l i n the context  have  described  of u r b a n r u n o f f  programming, methods  would  methods  runoff  i n t h e form of Muskingum  of l i n e a r felt  for  compensates  techniques.  representing  combination therefore  linear  confidence  In t h e work  a relatively  model  of  basis  schemes.  i n the c o n t e x t  rational  goal.  and more a c c u r a t e  linear  post-optimality  of  optimization  model  application of  degree  t o be b o t h  CAD.  fertile  information  technique for which  might  Chapter PROGRAM  As  mentioned  described  in  this  Muskingum-type a  were  FORMULATION  the  thesis  previous  of  each  combined  to  chapter,  uses l i n e a r  routing equations.  description  they  in  2  the  programming  This chapter  model  to solve  will  provide  of these  components and of t h e way  provide  the  desired  sensitivity  information.  2.1 MUSKINGUM CHANNEL The used  ROUTING  Muskingum method o f c h a n n e l  r o u t i n g has been  s i n c e i t s i n t r o d u c t i o n by G. T. M c C a r t h y  1959).  As a h y d r o l o g i c t y p e  storage-discharge equation. stream  The  of r o u t i n g technique,  equation  where S i s t h e s t o r a g e constant,  the  to the reach;  is  constant  for  i n determining  The  other  in  ratio  inflow  inflow  for  a single  i t uses a continuity reach  of a  is: S = k[xQi  a  i n 1938 (Nash,  r e l a t i o n s h i p a n d a form of t h e storage  widely  of Q  2  (2.1)  2  the  reach;  storage  to  i s the outflow  the reach  k  is  the  discharge; Q  storage 1  from t h e r e a c h ;  expressing  i s the and x  the importance of  storage.  equation  Muskingum method  + (l-x)Q ]  forming  the b a s i c f o r m u l a t i o n of the  i s a form o f t h e c o n t i n u i t y e q u a t i o n :  9  10  (Ql+Qj)  where « i s t h e and  S,  the  are  f  routing period  the  and  form  of  and  Qj,  s t o r a g e a t t h e end  finite-difference linear  Qj,  increment);  storage at  and of  S  the  storage  are  2  the the  x  (2.2)  Q] ,  Qj,  start  and  equation discharge  of  inflow,  routing period.  the c o n t i n u i t y  r e l a t i o n s h i p between  ,  = 8 2 - 8 ,  J  (the time  inflow, outflow,  r o u t i n g p e r i o d ; and  outflow,  <Q?+Qi)  -  This  assumes during  a the  routing, p e r i o d . The  usual  approach  i n v o l v e s combining following  Q  2  Then,  with  the  first  the  outflow The  both  Q  i  t-2kx t+2k{\-x)  +  t h e v a l u e s of  outflow  and  Q  ,  the  Muskingum  (2.2)  hydrograph  expressed  _ t-2k{\-x) t+2k(\-x) Q  the complete  o r d i n a t e known, t h e  in  can  the  v a l u e s of  to  method  obtain  the  be  same  t has  remaining  been  (  2  >  3  )  of  and  o r d i n a t e s of  sequentially.  storage  units  >  inflow hydrograph  determined the  2  constant  time.  suggested  by  The  k  are  range  Viessman  of et  (1977) a s :  k/3  and  (2.1)  r o u t i n g p e r i o d t and  applicable al.  equations  in  expression:  t+2kx t+2k(\-x)  =  used  by  Chow  < t < k,  (2.4)  (1964) a s : 2kx  < t <  k.  (2.5)  11  It  has  been  s h o u l d be  suggested  kept  the  finite  k,  a  peak b e i n g k  for  registered a  by  by  the  to k to  i n s u r e the  channel  the channel  is  used  (1979),  interval  t  accuracy  of  I f t were g r e a t e r  routing equation.  a commonly  Weinmann  that  through  the  particular and  (1959)  calculations.  wave c o u l d p a s s  empirically, noted  Nash  small r e l a t i v e  difference  flood  by  Koussis  without The  usually  calibration (1978),  than the  value  of  determined equation  and  many  is  others  as:  (2.6)  where L wave  i s the  speed.  l e n g t h of Therefore,  the p r o p a g a t i o n Koussis,  reach,  k has  i n the  and  c is  the  sometimes been  reach  (Viessman  kinematic  interpreted et  al.,  value  graphically, achieved.  1977;  of t h e w e i g h t i n g using  The  bounds  factor  trial-and-error for  x  are  x i s often until  given  the by  determined best  f i t is  Viessman  x  = 0.5,  the outflow 0,  and  as:  0 < x < 0 . 5.  If  as  1978).  The  Koussis  time  the  peak would be  Equation  reservoir,  t h e r e would be  and  (2.3)  no  a t t e n u a t i o n , and  g r e a t e r than  reduces  the o u t f l o w  (2.7)  to the  the  inflow.  special  i s determined  if x >  by:  case  If  of a  0.5, x  =  linear  12  Qi =  For has  natural  l + 2k  channels, x t y p i c a l l y  (2.8)  Qi •  varies  from 0 t o 0.3,  an a v e r a g e v a l u e o f a p p r o x i m a t e l y 0.2  1977).  It  seems a p p a r e n t  when s e t t i n g  the l i m i t s  substituting  this  expressions nearly terms  of p h y s i c a l  J.  Cunge  A.  (1978) and  q  where channel  is bed  It 0.5,  was  and  assumed  (2.4),  since  to c a l i b r a t e  parameters been  were  repeated  two  x  in  initiated by  by  Koussis  form:  (2.9)  0  mentioned  -  that  where  per  and  and Ql = Q . 2  demonstrated  above t h a t  unit  w i d t h and  i s , i t would shifted  there  is  S  0  i s the  assigned a value  on  the  time  no a t t e n u a t i o n  - can be a t t a i n e d  usefulness  i n Chapter  5.  of  exhibit  have t h e same shape  outflow hydrographs are  temporally The  i f x was  o u t f l o w h y d r o g r a p h would  but  i n f l o w and  spatially  Equation  Attempts  have  discharge  hydrograph,  where t h e  v a l u e o f x was  2S cL  the r e s u l t i n g  situation  et a l . ,  slope.  translation inflow  (Viessman  and  i n t o E q u a t i o n (2.5) makes t h e  (1979) i n t h e  the  in  hydraulic  2  this  t  equivalent.  i n 1969, Ponce  that  for  value  and  x  of  t-2k t +2k  this  pure as  axis.  the The  or t r a n s l a t i o n identical by  setting  technique  -  both k = 0  will  be  13 2 . 2 LINEAR PROGRAMMING Linear or  programming  minimum)  of  a  s e r i e s of d e c i s i o n linear  determines  on  particular relative respect of  can  influence  constraint,  In  of  a variable  the value  of  In  other  f o r each c o n s t r a i n t ,  occur  each  the  constraint  of  ranging determines  no  longer  s e t of d e c i s i o n  The  solution value  dual  established  great  deal  solution.  to  of  The  with  represents  each the with  (RHS) c o e f f i c i e n t the dual  i f a unit  solution  change were t o  coefficients.  RHS  t h e amount by w h i c h t h e v a l u e  feasible.  different  also  t h e v a l u e by w h i c h t h e  e a c h RHS c o e f f i c i e n t c a n be changed is  of  function.  which  words,  change  basis  required  associated  i n t h e r i g h t hand s i d e  will  of  i t  a  the various c o e f f i c i e n t  function  coefficient  a  number  doing,  variables  reveal  objective in  so  (containing  any  at the optimal  has  constraint.  contains,  to  (maximum  i n t h e optimum v a l u e o f t h e o b j e c t i v e  t o a change  that  vector  analysis  vector  change  subject  of the o b j e c t i v e  i n t h e LP f o r m u l a t i o n solution  t h e optimum function  of t h e d e c i s i o n  the  ANALYSIS  find  equations.  the optimal value  information  dual  variables)  the values  Sensitivity  values  (LP) w i l l  linear objective  constraint  achieve  AND SENSITIVITY  Outside  variables i s only  before  will  valid  the  optimal  of t h i s range, a  comprise  within  the b a s i s .  the boundaries  by RHS r a n g i n g .  Objective  function  performed  to  determine  function  (cost)  coefficient how  coefficient  f a r each can  be  ranging original  (OFCR)  is  objective  individually  varied  14 before  the  established  basis  i s no l o n g e r  by OFCR, any v a r i a t i o n  function  coefficient  objective  function  basis  2.3  variables  an  often  costs, the  only  optimum,  will  change  single  objective  the  but the values  used  with  tool,  the  time,., q u a n t i t i e s ,  value  of  the  primary etc.  The  purpose  to determine the  of using  the points  point  formulated  to  minimize  analysis  on  system the  I f the  and  upstream  then the s e n s i t i v i t y  greatest location  where  decrease  variables  reducing of  t h e flow  peak  peak  are  ordinates  come  information. equation i s  flow  at  function  flow  will  some is  ( i . e . , the  defined  information  o u t f l o w , Q^.  and time of t h e flow  will  i n which t h e flow  objective  maximum o u t f l o w ) ,  locations  however,  the outflow  a routing  t h e downstream peak  the  minimizing  of the procedure  i n a runoff  influence  of  be t o r e p r o d u c e  LP t o s o l v e  downstream.  flows,  optimal  i s probably  In t h i s f o r m u l a t i o n ,  of the s e n s i t i v i t y  greatest  reference  objective  The a n a l y t i c a l v a l u e  inspection  of the  PROGRAMMING  l i n e a r programming  d i r e c t r e s u l t o f t h e LP w i l l  through  the  a  t h e bounds  remain u n c h a n g e d .  optimizing  hydrograph.  has  will  of  Within  SOLVING THE MUSKINGUM EQUATION USING LINEAR As  most  optimal.  as the will  show  result  i n the  It will  show t h e  t o which Q  i s the P  most the  s e n s i t i v e t o change. routing  relationship maintained.  equations, between  The c o n s t r a i n t s , will  inflow  ensure and  that  outflow  formulated the  from  correct  hydrographs are  15 In  formulating  single  linear  t h e program,  reservoir  was f i r s t  under a v a r i e t y of i n i t i a l to  determine  which  the  was  expanded then  done  finally  a  reservoirs only  c o n d i t i o n s and d e s i g n  parameters  of  the  series  inflow  peak.  a s e r i e s of l i n e a r of  The  a special  formulation  hydrograph program  reservoirs.  Muskingum  Since  c a s e o f Muskingum  will  be' d e s c r i b e d  was  The same  channels,  o f c h a n n e l s was d e v e l o p e d .  a r e simply  a  and linear  channels,  here.  OBJECTIVE FUNCTION FORMULATION In  that  of  network  of  and was r u n  ordinates  for a  the; l a t t e r  2.3.1  -  to  case  considered,  c o n t r i b u t e most t o t h e downstream then  simplest  a l l cases,  outflow  was  constraints, of.the  to  formulation,  t o t h e peak o u t f l o w  however,  Y,  the  To e n a b l e  i t was f o u n d  by t h e LP  values  This  by a s e r i e s o f  from  package normal  higher  than  objective  dummy  a f f e c t e d always o c c u r r e d  assigned  ordinate  than o r e q u a l  the  to  is  variable.  t h a t when t h e r e  some o f t h e o u t f l o w  expected  type  each  in turn.  formulation,  minimize  values  be  variables never  ordinates  one r e a c h ,  assigned  ( t h e peak o u t f l o w ) .  wherein Y i s s e t g r e a t e r  a minimax  solely  t h e maximum  a dummy v a r i a b l e , Y, was i n t r o d u c e d .  s e t equal  outflow  In  would  hydrograph  formulation,  variable  than  i s o f t h e "minimax"  i s , the o b j e c t i v e i s t o minimize  the  this  the formulation  ordinate that  after  In  this  was  more  v a r i a b l e s were  were  routing  usually  higher  than  procedures.  The  t h e peak, b u t  t h e peak.  In other  were  words,  16 the  receeding  part  over-estimated,  of  although  the the  determined  accurately  because the  Muskingum r o u t i n g  LP  formulation  dual  as  solution.  the  >  To  in  overcome,  constraints,  by  every  minimize  the  without  were  number of  were m u l t i p l i e d by  into  the  n i s the  ordinates of  the  included  downstream  was  problem  arose the  only in  problem the to  (in this  was  number bring  function.  dominant  the  resulted  doubling  function  a factor  in  of the  These  case,  10"  variables  in  1 0  )  the  function i s :  I  Z  (10" °Q 1  n=2  ordinate  peak  constraints,  objective  N  where  of  was  facilitate  This  T h u s , the" o b j e c t i v e  Minimize Y +  to  process.  them from b e c o m i n g the  function.  The  which  including a penalty  variables  objective  the  case.  necessity  ordinates  prevent  of  included,  routing  the  hydrograph  e q u a t i o n s were e n t e r e d  the  outflow  to  value  i n e q u a l i t i e s i n order  inequalities  underconstraining  outflow  number; N  i n the  reference  n  )  is  routing  (2.10)  the  total  equation;  node; and  Q  I i s the  i s the  1  number  of  number  ordinate  n of  the  2.3.2  hydrograph at p o i n t  I.  MINIMAX CONSTRAINTS As  equal  mentioned by  above,  the  s e t t i n g Y > e a c h of  turn.  In  other  outflow  ordinate  words, must be  the  dummy v a r i a b l e Y the  outflow  i s made  ordinates  d i f f e r e n c e between Y and  positive,  as:  to in  each  17 Y - Q  > 0  n  for  n = 2,...,N.  2.3.3  SINGLE REACH To  linear  exploit  appeared while  the  programming,  re-formatted  sensitivity  such  that  solution  then  gives  ordinate  inflow  as  notation  for  the f i r s t  0Q|  the r e l a t i v e  (other  =  Qi  +  „  =  <p =  period  <t>Q]  ~  known  values)  equations,  than  The  dual  e f f e c t of a change  i n an  variables.  i n d i c a t i n g the p o i n t  on  a f f e c t s t h e peak. (2.3)  f o r t h e sake of c l a r i t y , routing  had t o be  first  by r e v i s i n g E q u a t i o n  where j3 =  and  (the  of  the  decision  most  equation  of t h e c o n s t r a i n t  on t h e o u t f l o w peak,  Therefore,  that  side  hydrograph that  new  inflows  outflow- v a r i a b l e s  were t r e a t e d  Note  the  on t h e r i g h t hand  the-- unknown  inflow  analysis capabilities  t h e Muskingum r o u t i n g  ordinate)  the  (2.11)  and  introducing  the r o u t i n g  equation  becomes:  7Q1,  t+2£(1-x). t -2kx t-2k(\-x). t -2kx t +2kx t -2 kx  t h e s e c o n d and t h i r d  (2.12)  (2.13)  (2.14)  (2.15)  t e r m s on t h e r i g h t hand  side  18  of of  Equation  each hydrograph  routing. must  the  must  be  also  be Qj  first  known is  the second  t h e same manner  7QJ  before  + 0Q1 =• QJ  variables.  iKJl  +  programming  and  (2.12)  forms  problem. was d e r i v e d  (2.16)  o r d i n a t e s of  respectively.  o r d i n a t e s a r e on t h e l e f t Equation  the expression  (2.12).  Equation  + 4>Ql  hydrographs,  Equation  performed,  and i s :  outflow  substituting  is  any  hydrograph,  r o u t i n g p e r i o d , the equation  and Q| a r e t h e t h i r d  both  routing  ordinate  performing  of the i n f l o w  i n the l i n e a r  where Ql  outflow  part  before  the o n l y v a r i a b l e .  constraint  For  s i n c e the f i r s t  known  The Qj term, b e i n g  therefore  in  (2.12) a r e c o n s t a n t s ,  *-UQ,  hand  (2.16)  this  side,  was  f o r Qj , w h i c h  The r e s u l t i n g  0Q§ = Q . 3 ~  In  the  inflow step,  and both  s i n c e they  further is  reduced  obtained  are by from  expression i s :  " TQi)  (2.17)  where \p i s d e f i n e d a s :  \li = 7 " P>-  Again,  the  p a r a m e t e r , Ql, and  Q . 2  right  hand  (2.18)  side  and a f u n c t i o n  contains  only  o f t h e two c o n s t a n t  the  inflow  terms  Q]  19  Expressions  f o r the t h i r d  were d e t e r m i n e d presented  in a similar  here.  and f o u r t h  routing  manner, a n d t h e i r  For the t h i r d  routing  intervals  final  period,  form i s  the equation  becomes:  -^Qi  and  +  2  The period,  0^Qi-  with  each  + *Ql  2  period,  PQl  procedure  were  the  period, included  consist  until  of hydrograph  of the f i r s t  V[(-0)""-'"V  Q J ] + PQ  where n v a r i e s  from  j-2  ordinate  1  inflow  n  *  Q'  f o r each  v a r i a b l e being a l l inflow  N;  ( t h e upstream  and  outflow  The d e c i s i o n  2  from 2  ordinates.  The r i g h t  and o u t f l o w  ordinates,  ordinates  Q  .  1  + (-<*>)*~ UQ]  is  the  hydrograph);  i n t h e storm  hand Q]  The c o n s t r a i n t generally as:  -  n ^ t  (2.21)  yQV  1  ordinate is  2 ( t h e downstream h y d r o g r a p h ) ;  of ordinates  routing  introduced  Q , where n r a n g e s  2  n  (2.20)  3  n c a n be e x p r e s s e d  2 to  of hydrograph  t h e number  2  *. UQ]-7Q-?).  -  repeated  of the o u t f l o w s ,  number  the equation i s :  i n the formulation.  f o r any o r d i n a t e  hydrograph  = Ql  was  Q ? , a n d one o f t h e i n f l o w  equation  is  +  (2.19)  4> (4>Q}~yQV  +  an a d d i t i o n a l o u t f l o w  side consists and  = Ql  routing  routing  variables N,  -  same  variables  to  PQl  +  for the fourth  0 <^QI  at  *Qi  hydrograph.  of  the n and  N  20 The  routing  constraints  hydrograph ordinates Note the  that  are given  i n order  c o n s t r a i n t s must  constraints. equality  As  i s only  represented  >  the s e n s i t i v i t y  form,  discussed  achieved  there  are 6  i n Table I.  to obtain  be i n  rather  earlier,  analysis,  than  equality  equivalence  to  an  i f a single equality constraint i s  by b o t h < and > i n e q u a l i t i e s .  terms t o the o b j e c t i v e include  f o r a c a s e where  function,  i t  was  By a d d i n g not  penalty  necessary  to  the < c o n s t r a i n t s .  With, t h e i n c l u s i o n o f t h e minimax objective  function,  Muskingum  routing  constraints  the complete f o r m u l a t i o n  equation  and  the  f o r s o l v i n g the  through a s i n g l e channel  reach i s  shown i n T a b l e I I .  2.3.4 SERIES OF REACHES To  simulate  t h e s i t u a t i o n where  there  i n c h a n n e l c h a r a c t e r i s t i c s a t some p o i n t example,  a  change  formulation  must  k  (The r o u t i n g  for  and  x.  add  constraint similar  the  pipe  diameter  the or  flow  (for  slope),  f o r reaches with d i f f e r e n t values period  t must  the of  of c o u r s e be t h e same  a l l reaches). To  and  allow  in  in  may be a change  more  reaches  equations are required,  t o t h e ones d i s c u s s e d  (2.17) t h r o u g h inflows  ordinate)  to  to  have  (2.21). the  The o n l y  been  system,  additional  and t h e s e c o n s t r a i n t s a r e  previously  downstream  already  the  i n equations  difference  reach  defined  now  (except as  (2.12)  is  that  the f i r s t  variables  (the  21  TABLE I ROUTING CONSTRAINTS FOR A SINGLE REACH CONST 1  Ql  2  Q!  >  4>  >  Q] -0(0Q; - 7 Q  >  Qi+0 (0Ql -7Q )  >  -7Q )  3  - H  4  <t> ^ 2  4> i>  from t h e u p s t r e a m  left  equation  f o r the f i r s t  +  2  0.Q'  +  1  2  where / i s t h e r e a c h equation  -Qi  In ordinate  reaches),  >  a n d so  (0Ql - 7 Q )  +  Q2  2  2  +  =  2  )  2  2  Q^ 0 (0Q +  4  are  1  -7Q ) 2  placed  hand s i d e o f t h e c o n s t r a i n t e q u a t i o n s .  -Q'  the  /5  2  outflows  the  RHS  Qi  P  5  the  Qi  Ql  on  Therefore,  routing period i s :  0 Q'i -  number.  (2.22)  7,-Qi* , 1  F  For the second  routing period,  becomes:  ^-QI* + P ^ Q ^ 1  general  terms,  1  =  -0 - (0 -Qi (  £  -  (2.23)  7,-Q'i* ). 1  the c o n s t r a i n t equation  o f a h y d r o g r a p h a t node /  f o r the  i s :  j=2  *  (-^)""  2  (^Qi-7,01* ). 1  (2.24)  22  TABLE II LP  CONST  Y  Obj .  1  Ql 10"  1  TABLEAU FOR A SINGLE REACH  Qi 0  10"  Ql 1  10-  0  1  Qi 10"  0  RHS  Qi 1  0  10"  1  (Minimize)  0  1  >  Ql  2  >  Ql  3  >  4  >  <p xP 2  5 6  1  7  1  8  1  9  1  -1 -1 -1 -1  t h a t t h e RHS c o n t a i n s  hydrographs,  the  which  RHS o f E q u a t i o n  information  be  i n Chapter  discussed  solution That  actually  i s , the dual  routine  represents  in  This  Q^.  Equation equation  result  ( 2 . 2 4 ) ,  takes  i t may  2  U Q ] - T Q ? )  Q W  3  U Q ] - T Q I  5,  i t  0  >  0  >  0  >  0  >  0  )  no  useful  However, a s w i l l that  the  information linear  be  explained  where  by  adding  Q^  by to  on t h e same f o r m a s t h e s i n g l e  dual  f o r Q^.  programming  i n t h e optimum p e r u n i t  can  of  i n s p e c t i o n of  that  found  by t h e  ordinates  From  appear  sensitivity  obtained  t h e change  was  >  the f i r s t  c a n be o b t a i n e d .  provided value  only  are constants.  ( 2 . 2 4 ) ,  sensitivity  Q W  4  10  the  -«(0Ql-7Q?)  Q ^ + c/) (0Ql-7Qi)  -1  Note  UQ}~yQV  +  the  change  nature  of  each s i d e , the channel  case  23 shown  i n Equation The  with In last  complete  three  addition four  linear  programming  r e a c h e s and s i x o r d i n a t e s t o the r o u t i n g  a r e t h e minimax  ensure that  2.3.5  (2.21).  Y i s equal  formulation i s given  constraints  (Numbers  constraints.  These  formulation  described  thus  constraints  farwill  o f Muskingum  reservoirs,  of both.  rate  or a combination  ordinates the  modification  at  in  flow  changing  t h e peak  flow  complex  runoff  incorporate several  will at  sources  This  flow have  the  the  To i t  linear indicate  i n turn  a  more  incoming  to  runoff  i n t o one c h a n n e l o r p i p e  describes  a  effect in  necessary  information  in will  where  model  is  flows  change  greatest  outlet.  the s e n s i t i v i t y  section  unit  ordinates  however,  t o flow  also  This  branches which enable  while also maintaining brajnch.  and  situation,  multiple  per  a l l upstream p o i n t s .  locations  route  channels,  It will  o f change o f t h e peak o u t f l o w  indicate  from  1-12), t h e  t o Q^.  through a s e r i e s arrangement  flow  i n Table I I I .  BRANCHING The  the  f o r a case  reach,  for  each  how t h e s e o b j e c t i v e s  were  met. When d e a l i n g have a c o n s i s t e n t a  simple  numbers  numbering  series  straightforward the  w i t h a complex n e t w o r k ,  of  system  to identify  channels,  - t h e uppermost p o i n t  increased  i t i s important t o  the  nodes.  numbering  was l a b e l l e d  With was  '1' a n d  i n a downstream d i r e c t i o n u n t i l t h e  TABLE I I I LP TABLEAU FOR Qi  Ql  Ql  Ql  Qi  Qi  Qi  Qi  1  , -ie 0  QS  Qi  lO" lO" 10  RHS  Ql 10  10-'  fi,  * Q j * <*.Q!-7iQ?> * Qi-#iUiQ!-7.Q?)  fi,  * Qj*#?(#.Q!-7iQ?)  fi,  *,  *t  *  t  * Q}-#?(# Q!-7.Q?)  fi,  -1  1  fi*  -1  *i -1  -4*^2 -1  2  fit  *a  fit <Pt  *» * i  fit  fi,  -1 -1  $1  fii fit  -1  1 1  2 *i  *i -1  (•iQi-7iQ?)  i  -• (*JQ?-7IQI)  -it*,  *J  fit  -1 -1 -1  2  i  •?(#lQi-7 Q?)  *  -•?(0»Q?-7JQ?)  i-  -1  1  *  2  (•JQ?-7JQI)  *  -•JU»Q|-7JQI)  *  •?(#iQ?-7iQ")  i  -*?(•,Q?-7iQ')  o o o o  *,  2  IV IV  i>,  (Minimize)  0  IV  Objective Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5 Constraint 6 Constraint 7 Constraint 8 Constraint 9 Constraint 10 Constraint It Constraint 12 Constraint 13 Constraint 14 Constraint 15 Constraint 16  02  IV  T  3 CHANNELS IN SERIES  to  25 final  node To  ( t h e o u t l e t ) was  identify  procedure  was  as  before,  The  point  number,  branches  used. and  and  Numbering  proceeded  immediately and  reached.  the  continued  until  This  reached.  continued  With t h i s  automatically  A  reach  nodes  below a j u n c t i o n  avoid  f o r the  in  identifying  the  the  was only from  exceptions both  same as  branches  had  of. t h e the  network  junctions  was are  branches. i t s upstream  explicitely  actually  node.  the to  At  the  same t i m e ,  each branch  had  t o be  allows  system  series  junctions. be  two  preserved.  is  to  for  identical  formulation  Here, the  sensitivity  was  greater  of c h a n n e l s ,  superimposed  the  for  Except  programming  that for a  it  identified  branches).  linear  occuring at  then  branch.  b r a n c h e s were a d d e d ,  the  The  the  t o p of a n o t h e r  j u n c t i o n s , the numbering  g e n e r a l l y the  downstream. for  (This  previous case.  numbering  a  junction. order  numbering  to  t h e h i g h e s t number of t h e  j u n c t i o n s t o be  ambiguity.  flexibility  to  or h i g h e r  assigned  reached,  system,  takes  junction.  was  the< o u t l e t  f o r simple  above t h e  When s e c o n d necessary  until  point  assigned  The  i s numbered a c c o r d i n g t o  immediately  was  junction  at the  numbering  identified  Each reach  branch.  another  numbering c o n t i n u e d  process  junction  first  c o n s e c u t i v e number was adjacent  where t h e  following  began a t t h e uppermost  u p p e r m o s t p o i n t of t h e downstream  the  downstream t o t h e  above  the next  junctions,  and  the  flows routed  information  26 At reach  a j u n c t i o n where r e a c h e s  z'+1, t h e r o u t i n g e q u a t i o n  -(QT+Q!) +  Equation of  P^Q!*  1  (2.25) i s s i m p l y  Q',)  Equation  two e x t r a i n f l o w terms  and flow  into  becomes:  *,(07 +  =  m and / j o i n  from  7,-Q^ . 1  -  (2.25)  (2.22) w i t h  the addition  Here, Q  the branch.  and Q  m  n  are  the i n f l o w s immediately  the  flow a t the f i r s t For  the  ^.Ql  +  3  second  In  + 1  routing  /3.Q1  +  general  representing  +  at  period  Equation  at  a  a n d Q^  is  +1  /  f  the  a  junction  the  (2.23), i s :  = -tf [* .(Q7+Q )-7 Qi  1  terms,  flow  n  node below t h e j u n c t i o n .  e x p r e s s i o n , d e r i v e d from  -<Q?+Q' )  above t h e j u n c t i o n ,  l  |  1  expression  junction,  +  1  f  ].  (2.26)  for a constraint  derived  from  Equation  (2.24), i s :  -(Cf  +  Q')  + V[(-0.)"-;-VQ/+1] + J =2  0.Q  / + 1  J  *  An given  (-0 -) " [* n  x  2  f  (Q7  +  Q\)  example o f t h e t a b l e a u f o r a  i n T a b l e IV.  -  7 Qi ] . +  1  f  branched  (2.27)  network  is  TABLE IV LP TABLEAU FOR A NETWORK WITH ONE BRANCH T  Objective Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5 Constraint 6 Constraint 7 Constraint 8 Constraint 9 Constraint 10 Constraint 11 Constraint 12 Constraint 13 Constraint 14 Constraint 15 Constraint 16  Qi  Qi  Qi  Qi  Qi  Q;  Q: QS  1  Qi 1  0  - i o  Qi 10"  1  Qi 0  10"  1  Qi 0  10"  1  RHS  (Minimize)  0  * *i  -•1*1 * i *i  0, * i  0i  2  \f>i  0,  0, 0, "•J^J  * i  0i  #i ^j  . Pt  2  -1  -1 -1 -1  1 1 1  <I>,  0,  -*«^«  -1 -1  1  0, -1  -1  0t  2  *i  «• <l>* -1 -1  -1  0%  Qi •  U.Ql-riQ?)  *  Qj-*.UiQl-7iQ5>  *  QJ**?U.Q!-7.Q?>  i  Q}-#?(#lQ!-7iQ?>  *  Qi +  UjQi-7iQi>  *  Qi-«>,(#iQ?-7.Qi)  *  Qi+#?U>Q?-7iQD  *  Qi-#?(#jQ'-7>Q')  *  [•.(Q?+Qi)-7.Q?1  *  -•.[#.<Q?*Qi)-7.Qi)  i  •?[#.(Q?+Qi)-7.Qi)  i  -•?(•.(Qi Ql)-7«Q1) +  2  0  2  0  2  0  2  0  ro  Chapter  3  PROGRAM DESCRIPTION  The LP  previous  network.  description prepares  3.1  of  with  of  the  routing equation  chapter  t h e F o r t r a n program  will that  outputs  a  a  brief  the  data,  the  i s i n c l u d e d i n Appendix  an  through  present accepts  programming, and  t h e program  f o r m u l a t i o n of  results.  A.  INPUT The  network  program a c c e p t s and  it  was  sufficient  v e r s i o n of found  array  However,  handle  was  manipulated data  using  to  handle  was  data  the v i s u a l  i s i n c l u d e d as T a b l e  The  first  file  and  The  next  lines  of  a s e t of v a r i a b l e s i n f o r m a t i o n read  used  number of  following  to  represent channel  was  physical data  also  but  An  which  the  specify  i s t h e number of  the  28  simplest  could  way  to  then  example of  the  title  be  such  of t h e  output  a  number  of  is  then  computed  data  options.  stations  the hydrographs a t each  equation:  to obtain  V.  input are  reaches  file.  developed,  unable  which  editor.  j u n c t i o n s i n t h e n e t w o r k , and  being  the  t h a t the e a s i e s t  files,  file  two  was  any  found  to  a standard  the c o m p i l e r  it  t o use  from  the program  that  space  networks. data  input p e r t a i n i n g  flow c h a r a c t e r i s t i c s  interactive  but  The  This  i t for linear  A listing  and  dealt  t a b l e a u t o s o l v e t h e Muskingum  branched  An  chapter  (nodes)  ordinates station. by  the  29 TABLE V SAMPLE INPUT F I L E  1 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 1 4 1 5 16 1 7 18 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48  E X A M P L E 3 . 1 : ONE J U N C T I O N 1 1 0 11 1 1 2 1 4 8 5 1 1 1 1 15.000 18.000 0 .200 K,X ( R E A C H 0.000 0 .000 K,X ( R E A C H 21.000 0 .175 K,X ( R E A C H 19.500 0 . 1 60 K,X ( R E A C H 0.000 0 .000 K,X ( R E A C H 20.250 0 .185 K,X ( R E A C H 16.400 0 .230 K,X ( R E A C H 0.000 0 .000 K,X ( R E A C H 17.200 0 .215 K,X ( R E A C H 5.365 Q(2,1) 5.365 Q(3,1) 5.145 Q(4,1) 2.550 Q(6,1) 2.550 Q(7,1) 2. 100 0/(8,1) 7.885 Q(9,1) 7.885 Q(10,1) 7.350 Q(11,1) 5.975 QO,l) 7.650 Q(1,2) 12.250 Q<1,3) 9.450 Q(1,4) 8 .250 Q(1,5) 6.900 Q(1,6) 5.705 Q(1,7) 5.450 Q(1,8) 5.320 Q(1,9) 5.225 Q(1,10) 5. 175 Q(1,11) 5. 1 30 Q(1,12) 2.615 Q(5,1) 2.850 Q(5,2) 3.125 Q(5,3) 3.450 Q(5,4) 3.950 .. Q ( 5 , 5 ) 4.850 Q(5,6) 4.615 Q(5,7) 3.720 Q(5,8) 3. 1 50 Q(5,9) 3.015 Q(5,10) 2.945 2.920  1) 2) (RESERVOIR) 3) 5) 6) (RESERVOIR) 7) 8) 9). ( R E S E R V O I R ) 10)  30  NCHAN = NSTA-NJUNCT-1  where NCHAN i s the stations, the  or  n o d e s ; and  are  input  This  information to complete  information  setting then  up  the  the  i s the  NJUNCT i s t h e  number of  regarding  b r a n c h e s and  values  stored  a  At  number  of  junctions in  the  The  value  of  junctions  network  in variable arrays  information,  assigned  the  is  x f o r each  variation  message  coefficients  k and  and  c a l c u l a t e d and  and  the  d e s c r i p t i o n of  tableau.  of  f o r the  junctions. are  the  is  LP  program  allowing  the  layout.  f o r use  routing  in  period  the  the  or  if  are  values  read  that 0,  of  The  the  close  then  respective  numbers  in arrays.  printed  negative  their  in reach  same t i m e ,  stored is  to  reach  arrays, occur  7,  zero.  p h y s i c a l d e s c r i p t i o n of  0,  the  stops of  With model  at  and \p  execution  denominator to  by  the this  network  complete. At  input at each  this  are  the  the upper  point,  end  array.  are  only  information  ordinates  - the  of  e a c h b r a n c h , and  and  downstream node.  information  values  the  discharge  intermediate  junction the  NSTA  entered.  The  is  channels;  network. Next,  is  number o f  (3.1)  previously  assigned  to  input  their  remaining  inflow the  first  The  to  hydrographs ordinate  branching  i s used to ensure  proper  be  location  in  of and  that the  31 It edited All  should  be  to include  that  is  and  that  the data  a portion  files  This  the  initial  would  be  of m o d i f i c a t i o n s  on some  easily  i f desired.  i n d i c a t i n g the  inflows  useful  t h e network o u t l e t , t h e d e s i g n e r  effects  c a n be  of the network,  i s t o change t h e l i n e  delete  reaches.  examining the  only  required  number o f n o d e s redundant  noted  of  the  i f , i n s t e a d of  w i s h e d t o examine  intermediate  point.  3.2 FORMULATING THE TABLEAU The  tableau-  function  size  is  calculated  o f t h e number of c h a n n e l  by t h e p r o g r a m a s a  r e a c h e s and t h e number  of  ordinates:  NVARS = NCHAN*(NORDS-1)+1  (3.2)  NCONST = (NCHAN+1)*(NORDS-1)  (3.3)  where NVARS  i s t h e number  number  constraints  of  number o f o r d i n a t e s Each zero.  in  NCONST  is  the  and NORDS i s t h e  hydrograph. the  function  tableau  i s then  the  sum o f t h e dummy v a r i a b l e Y p l u s  the  last  reach,  and  i n t h e LP t a b l e a u ,  i n each  coefficient  The o b j e c t i v e  of v a r i a b l e s  is initially  set to  formulated  to minimize  the penalty  f u n c t i o n of  a s shown by t h e f i r s t  row o f T a b l e s  III  and  IV. The using  outflow  the  hydrograph  standard  from each r e a c h  Muskingum r o u t i n g  is  procedure.  determined, This  step  32 is  not  the  LP  a c t u a l l y required segment, but  accuracy  of  cumulative  the  e r r o r may  routine,  the  straightforward  0.1% the  results. occur  therefore routing  extent  time-.  of  the  error  tableau  First,  the  procedure  between t h e  is  3.3  LINEAR PROGRAMMING  A d e s c r i p t i o n of  the  of  operations  the  the  values  These v a l u e s The arrays. values  the  the  of  RHS  are  the  On  output,  the  RHS  the  by  the  compared  with of  a message g i v i n g the  solution.  c o n s t r a i n t at  formulated.,  a  using  C o r r e l a t i o n i s made  ordinate the  it  represents  minimax c o n s t r a i n t s  tableau.  SUBROUTINE  the  problem set  primal  using IV  the  of  and  information  On  output,  the  and  indirectly  upper  coefficients  is  the  and  solution  On  tableau  the  values  variables.  t h r o u g h an included  B.  regarding  o b j e c t i v e and  dual  UBC  subroutines.  i s i n c l u d e d as A p p e n d i x  information the  i s solved  of F o r t r a n  tableau  value  addressed  ranging  the  performed.  optimal  verify  in  descrepancy  one  2.  Finally,  routine  t o be  of  a  values,  i n Chapter  'LIPSUB', a  LIPSUB a c c e p t s  contains  two  If  generated,  programming  routine  entry,  the  c o n s t r a i n t number and  added t o c o m p l e t e  library  r e s u l t s are  results.  tests.  later  r o u t i n g performed  these  then  are  to  i s p r i n t e d alongside  discussed  linear  included  routing, c o n s t r a i n t s , are  t h r o u g h a s e r i e s of  The  i s performed  Under c e r t a i n c o n d i t i o n s , a  i n the  o r more e x i s t s between  The  the  and  routing  is nevertheless LP  LP  since  index. in  separate  lower bounds g i v e  w h i c h must be  input  to  the  cause  33  a change  i n t h e elements of  the  optimal  Because of t h e e x t r a terms c o n t a i n e d tableau, the on  some f u r t h e r m a n i p u l a t i o n  r e q u i r e d RHS r a n g i n g the  RHS  (other  coefficient) For  values  reach. the  will  value  appear  relate  intermediate  LHS, t h e v a l u e results.  an  9 of Table  right  hand  values  p o i n t s , where t h e  to  coefficient,  [ </> ( Q  Applying  throughout  Table  2 +  the  bounds  does  be added  t o the  RHS  of Q l i n c o n s t r a i n t  the  Ql  coefficient.  In e f f e c t ,  from The  a  decision  original  the  resulting  upper and lower  RHS from  3  bounds  Q\.  these  where i t  not  v a l u e on  Q i ) ~ 7 Q i ] , must a l s o be s u b t r a c t e d  to  being  t h e lower  ordinate  the value  redefining  Again,  IV,  be added t o t h e bounds t o g i v e t h e  RHS  3  directly  information  IV must  a  bounds.  constraint  of  f o r that  as a n e g a t i v e  o f t h e o r d i n a t e must  as  results.  the true value in  unit  station,  s i d e t h e same form a s c o n s t r a i n t 1.  i s t h e same  variable  with  upstream  constraint  terms  to Qj.  F o r example,  number  relate  to  resulting  directly  obtain  The a d d i t i o n a l  i s s u b t r a c t e d , from b o t h  2  The  to  ordinate  on t h e RHS b u t i n s t e a d o c c u r s  ranging  the  i n the f i r s t  bounds.  necessary  from t h e r a n g i n g  represent  o f (4>iQ]-7iQ )  upper  For  this  subtracted  then  F o r example,  therefore  the  be  is  single  the c o n s t r a i n t s corresponding  these  and  must  a  vector.  i n t h e RHS o f t h e i n p u t  information.  than  solution  types is  of  required  applicable  the network, w i t h  manipulations results to  in  RHS  a l l flow  the exception  of  for  each  ranging ordinates  the  initial  34 ordinates  and t h e s y s t e m o u t f l o w .  2,  the ranging  not  a p p e a r on t h e RHS.  3.4  OUTPUT The  is still  output  graphical commands  could  form.  be p r o d u c e d  The l i s t i n g  display  depending  computer  system  include  illustrations  a terminal The  outflow not  used.  The  includes  the  value  The s e n s i t i v i t y  includes  the  the  dual  dual  which g i v e s value  a  but on  could  the  be  in  or  output  adapted  capabilities cited  does  tabular  i n Appendix A c o n t a i n s  the primal  to  of t h e  Chapter  might  5  appear  s o l u t i o n c o n t a i n i n g the  and t h e message  i f any v a l u e i s  d e t e r m i n e d by t h e s t r a i g h t f o r w a r d information  solution vector,  change f o r a u n i t change  ranging,  either  o f how g r a p h i c a l o u t p u t  routing.  will  in  examples  hydrograph ordinates to  Chapter  screen.  output  close  in  even t h o u g h t h e o r d i n a t e  for a tabular display,  graphical  on  valid  As e x p l a i n e d  i s a l s o output,  t h e amount by w h i c h  i n any o r d i n a t e , and  t h e bounds f o r e a c h o r d i n a t e  is valid.  which  the  RHS  f o r which  Chapter ALTERNATE  A  different  fewer v a r i a b l e s formulation  FORMULATION  formulation  but  still  runoff  identical  LP  system  very  different.  here,  the tableau  for  a  tableaus  In  the  size  Both  simple  depends  same  minimax  f o r purposes of  formulations series  generated  alternate  significantly  the  and a s s e s s e d  design.  results  the simplex  involving  maintaining  was a l s o d e v e l o p e d ,  supporting  although  LP  4  in  of  reaches,  each  case  formulation  solely  on  the  give  are  described number  of  h y d r o g r a p h o r d i n a t e s , and does n o t i n c r e a s e a s t h e number o f reaches  increases.  coefficients reaches The UBC  allowing developed  4.1  generated  the  increases  complexity  rapidly  of  the  a s t h e number of  increases. more c o m p l i c a t e d  library  performs  However,  equations  program REDUCE.  expansion  and  the r e g u l a r but  This  were v e r i f i e d  interactive  o r d e r i n g of a l g e b r a i c complex  runoff  using the  LISP  program  expressions,  equations  to  be  automatically.  SINGLE REACH The  single  The  expression  the  one g i v e n  reach  case  i s t h e same  i n both  formulations.  w h i c h forms t h e b a s i s o f t h e f o r m u l a t i o n s i n Equation  (2.3),  35  expressed  generally as:  is  36 The being  tableau  Qi,...,Q*  coefficients and  and  that  formed, w i t h  the  minimax  the d e c i s i o n v a r i a b l e s variable  Y.  The  RHS  a r e t h e i n f l o w s Q ] . . . , Q ^ p l u s a f u n c t i o n o f Q] r  Qj , w h i c h a r e b o t h The  as  i s then  LP  tableau  considered  constants.  fora single  channel  reach  i s t h e same  II.  i n Table  4.2 TWO REACHES IN SERIES Relating outflows the  the s e n s i t i v i t y  from  outflow  a single  outflow  When  the  first  need  to  through the  the  of  case  how  to  reach,  with  linear  f u n c t i o n s of  the desired s e n s i t i v i t y of  two r e a c h e s  treat  at the intermediate  the  point.  the intermediate  the second  RHS.  The  reach,  they  formulation  by a d d i n g  ordinates  of  When r o u t i n g  described  be  would  routing  placed  i n Chapter  constraints  the  through  hydrograph o r d i n a t e s  would need t o  a set of  output.  i n s e r i e s , the  be t r e a t e d a s d e c i s i o n v a r i a b l e s , b u t when  t h i s problem for  ordinates yields  arose  hydrograph  t o the  was- s t r a i g h t f o r w a r d : d e f i n i n g  inflow ordinates  formulating  problem  reach  inflows  o r d i n a t e s a s d e c i s i o n v a r i a b l e s i n s u c h a way a s  to equate upstream the  a n a l y s i s from  in  2 overcame  and  variables  each a d d i t i o n a l reach. The  approach  intermediate achieved  taken  here,  hydrograph  from  however, the  was  to  combine  routing  reaches  a n d form a s i n g l e  equations  equation.  remove  formulation.  by t a k i n g a d v a n t a g e o f t h e l i n e a r i t y  method  to  the  T h i s was  of the r o u t i n g  f o r two  ( o r more)  I t i s p o s s i b l e t o route  37 a storm hydrograph outflow  t h r o u g h more t h a n one  hydrograph,  while  intermediate hydrographs. ordinate  of  complete to  inflow  hydrograph.  Recall  that  inflow  subsequent  the  first  in addition  to the  i t is  possible  r e a c h i n terms of the  reach. for  routing,  hydrograph  will  is  on  the  the  In o t h e r words,  the  and  e x p r e s s Q^  required  the  first  (downstream) h y d r o g r a p h s .  formulation  information  i s needed  t h e o u t f l o w from t h e s e c o n d  to the f i r s t  entire  A l l that  no  e a c h downstream h y d r o g r a p h ,  inflow  define  gaining  reach to obtain  input  ordinate  Therefore,  as a f u n c t i o n  i s the  of  the  the  of  desired  ordinates  shown:  Q  1  = /(Q ,  From first  3  n-  1  , . • . ,Qi ,  Equation  Q ?, i  (2.12),  (4.2)  Ql>.  the  routing  equation f o r the  reach i s :  Qi and  Q  3  =  ^  Q  n  f o r the second  Q*  By c o m b i n i n g eliminated:  +  "  «iQ  n-  1  n-  1 *  (4.3)  reach i s :  = /3 Q* + 2  "HQSn- 1  7 Q^ 2  n-  1  e q u a t i o n s (4.3) and  (4.4),  (4.4)  t h e Q^  term  can  be  38  Q'  At  n  = PiP Q  3  2  Pn2Q _  +  ~ (0,02-7! ) Q .  3  n  n  2  ]  t h e end o f t h e f i r s t  2  routing period  1  - 0iQ;_  (4.5)  r  ( « = 2 ) , E q u a t i o n (4.3)  becomes:  Ql  Equation  = 0,Ql + 7 i Q  At  = /3 Q| + 7 Q? " 0 Q i , 2  Equation  Ql  2  (4.7)  2  (4.5) becomes:  = 0i0 Ql  + 0,720?  2  the  (4.5)  (4.6)  (4.4) becomes:  Ql  and  " <t>^Q} ,  2  end  of  the  "  (0i0 "7i)Q 2  second  2  " 0 i Q ] .  (4.8)  r o u t i n g p e r i o d (n=3),  Equation  becomes:  Ql  =  0i0 Qi 2  +  0i7 Ql 2  Equation  "  (0i0 -7i)Q! 2  " 0iQ -  (4.7) f o r Ql a n d E q u a t i o n  By  substituting  Ql  i n Equation  Ql  = 01020! " t ( 0 i 0 ) 0 i 0 2 - ( 0 i 7 2 7 i 0 ) ] Q i  (4.9), the r e s u l t i n g  +  expression  (4.8)  f o r Q3  1  for  i s :  +  2  "[(01+0 )01-71l7 Qi 2  (4.9)  2  2  2  + (0i+0 )(0102-71)Qi 2  +  01 Q]. 2  (4.10)  39 For  subsequent  made, a l w a y s the end  Qi  reducing  of the t h i r d  = 010 Q«  -  2  [(0I  + [( 0 "  U l  the  2+  (4.2).  routing period  for  the  (the fourth ordinate) i s :  2  1  2  2  0 I 0 2 0 2 ) P M 3 - U I 0 ) ( 0 , 7 2 + 7 1 / 3 2 ) + 7 1 72 +  2  2  2 2  2  )j3,-(0 +02)7i  ]7 Q?  1  ordinate,  period,  the equation  Ql  2  2  " 01 Q 1 -  (4.11)  3  2  fifth  ^  +  01</>2 + </>2 ) ( 0 1 0 2 - 7 1 ) Q l  = 0i0 Qi  equation  in  +  1  l  2 +  expression  [(0i+02)^ /3 -(/3 7 7i/3 )]Ql  +0 0 +0  2  an  The r e s u l t i n g  2  l  more s u b s t i t u t i o n s a r e  t h e terms t o o b t a i n  form o f E q u a t i o n  +  For  routing periods,  a t t h e end o f t h e f o u r t h  routing  becomes:  [(0i 0 )0i0 -(0 72 7i0 )]Q2  -  +  +  2  +  [ (c6  -  [ (</. + 0  1  2  2  + 0,0 +<A ) 0, 0 - ( 0 , + 0 ) (01 7 2 + 7 1 0 2 ) 7 l 7 2 ]Q|  2  2  1  2  3  1  2 l  +  2  2  2  02+0102 + 02 )0102 2  3  "(01 +0102+02 )(0172+7102)+(01+02)7l7 ]Ql 2  2  2  "  t(01 +01 0 +0 0 3  2  +  (0, +0  It  is  the  2  3  2 1  0 +0 0 2  1  substitutions  method devised.  +0  3 2  coefficients  3 2  )0,-(0, +0,0 +0 2  2  )(0,02-71)Q?  f o r each  formed  method  f o r two r e a c h e s  (4.12)  4  by  successive  generating  2  + 0i Ql-  However,  for  )7l]7 Q?  complexity  f o r each  used  2 2  successive  in  required.  automatically The  +0  increase  are  patterns  of  2 2  2 2  obvious that  coefficients  recurring  l  routing as  more  noticing  the  ordinate,  the c o e f f i c i e n t s  automatic  period  c a n be  generation  in series i s described  a  here.  of  40  First recursive  = 1  c  2  =  c  2  \  <j> t e r m s  a l l , the  series,  eg  C  or  of  c ,  (4.14)  2  = 4>i + (t>i<t> + <l> 2  2  = </>c + 0,  2  2  2  = <t>, <t>l <t>2 <P,<l>2 <p2 3+  2  +  2+  f o r two  Therefore, f o r two  2  (<*>,)".  any  (4.17)  ._  2  2  the reduced  >(0 2+7,02 ) ( c +  (-D [(c _ )0 -(  "  (-i)' [(c _ )(0 02-7,)Q  n  2  2  1  2  7  C  routing  ._ ) 2  7  l  T  2  ]Qp  _ )7,]72Q? 3  2  2  2  l 7  +  1  2+  U,)"~ QlJ. 1  (4.18)  REACHES  As d i s c u s s e d coefficients  n,  ordinate  )0, /3 -(c  SERIES OF  (4.16)  3  channels i s :  {(-))"-J[(cl  of  = <t>2C 2+4>  3  reaches,  1  for  (4.15)  2  2  2  degree  as a  = 0 cg+0,  2  4.3  expressed  where:  2  n•  c\ = 0 ( c _ ) +  tZ  be  (4.13)  in general,  equation  can  in  Chapter  increase with complexity  2,  the  complexity  each o r d i n a t e .  grows  with  each  of  the  In a d d i t i o n , additional  the  reach  41 through  which  formulation generation first  the  of  flow  the  is  routed.  general  case  comparison  the  with  formulation  case  coefficient  i t i s perhaps best  of  three  reaches.  t h e one a n d two  reach  cases,  Then, the  to by  general  c a n be shown.  The b a s i c r o u t i n g e q u a t i o n  Q> For  d e s c r i b i n g the  of automatic  f o r any number o f r e a c h e s ,  present  In  =  the f i r s t  /3QJ 3  f o r the t h i r d  + T a O ^ - *,Qj_  r o u t i n g p e r i o d , where  reach i s :  (4.19)  r  n = 2,  Equation  (4.19)  becomes:  QI = PsQl  +  73Q1  By  substituting  the  following expression  Ql  "  +  ' (4.20)  3  (4.20) f o r Q| i n E q u a t i o n  Equation  = /3,02 03Q§'  0 Qi*.  i s derived:  01027 Qf 3  - 0, (/3<2>-7)Q^ " (0i02-7i)Q 2  For  the  substitutions  next  (4.8),  3  2  routing  period, after  a r e made, t h e r e s u l t i n g  2  - 0iQi.  (4.21)  a l l the required  equation i s :  42  Ql  =  0i0 /3 Q§ 2  3  "  [0,0203(01+02+03)-(010 7 +0l7203+7l0203)]Q  "  [0,02(01+02+03)"(0172+7102)]7 Ql  +  [01(01+02+0 )-7l](0 03-7 )Oj  +  (0,02-7,)(0,+0 )Qi  2  3  2  3  3  2  2  For  the  next  2  + 0, Ql.  (4.22)  2  routing  period,  the r e s u l t i n g  expression  is:  Qi  = 0,020 Qa 3  +  [0,0203(01+02+03)"(010273+0,7203+7,0203)]Q [010203(01 +0102+010 +02 +0203+0 2  2  3  2 3  3  )  " ( 0 , 0 2 7 3 + 01.72 03+71 0 203 ) (01+02 + 03 ) +(017273+710273+71720 )1Q§ 3  +  [0,02(01 +0102+0103+02 +0203+03 ) 2  2  2  -(0i72+7i0 )(0 +02+0 )+7,7 ]7aQi 2  1  3  2 "  [01(0203 72)(0, -  2  2 +0102+0103+02  2 +0203+03  )  ~71(0203-72)(01+02+03)]Ql "  (0102-71)(0, +0,02+02 )Qi  By  2  comparing  the  above  with  the e q u i v a l e n t  equations  can  be  there  The  first  for  the two-reach case,  This  seen  that  i s that  series  c a n be  "  2  (4.23)  3  f o r Ql,  equations f o r one and  a r e two  but now as  two  distinct  f o r the 0 terms,  summarized  0i Qi-  which  contains follows:  Q ] , and Q i reaches,  repeating  is similar a third  i t  series. to  element  that 0 . 3  43  c% = 1 c]  (4.24)  =  +  = <t> cl + c i  c\ = 0, +0 0 +0 03+0 +0 0 +03 2  2  1  2  1  2  2  g e n e r a l l y , f o r three  By  with  be  comparison  the general  = (p.c  l  of the s e r i e s  (4.26)  i t c a n be shown  f o r / reaches  n  , + c'" n  1  can  (4.28)  1  n-1  i  2  other  terms.  product the  />0  (4.17),  c] = </>i, c\ = <f> , e t c . , and c° = 0 f o r a l l v a l u e s The  7  3  reaches,  Equation  expression  = 0 c?+cl  writen as:  c  if  2  3  or e x p r e s s e d  that  (4.25)  2  3  p(2,\)  (0i72 7i/3 )  c a n be d e s c r i b e d a s t h e sum o f t h e o f p\ and 7 ,  =  (/3  2  /3 7 2  + 3  /3i7203 7i02/33 +  (/3i7273 7i/3 7 +  2  =  as  illustrated,  by  (4.29)  +  1  and  examples:  =  />(1,2) = p{2,2)  series  of a l l permutations  following  , 1)  i n the formulation contains the 0  series  This  o f n.  + 3  )  7i7203)  (4 . 30 ) (4.31)  (/3,0 7 7« i3i72/3374 7i/32037i. +  2  +  +  3  ( 3 i 7 2 7 3 0 a 7 i 0 2 7 0 « + 7 i 7 2 03 0 « ) . +  3  (4.32)  44 Note t h a t  the f i r s t  coefficients  each  term,  i n d i c a t e s t h e number o f j3  and  the  t h e number o f 7 c o e f f i c i e n t s .  indicates in  in  parameter  p{a,b)  a series  {  i s given  a  +  b  )  by W a l p o l e  second  parameter  The number o f t e r m s (1982) a s :  (4.33)  l  albl Using  t h e above n o t a t i o n , t h e g e n e r a l  equation  f o r the  th n  n  n-m+1  m=2 +  (-1)  .  T  h  [  2 cJ:J_ ; =1 "  _ n _ 1  n  1  J  1-1  1  l  reaches) i s :  _.  T  J  ^(I-y-l,y-l)]  7 1  l  - Q5 ' l  7-1  { Z [ L c _ _ ( j - m - l ,m-))](p j = 2 m=] J  n  m  ] P  <j> - 7 . .JQ ,} J J 7  J  (-l)"" ^.^?.  +  (4.34)  1  4.4 PROGRAM The Chapter  each  .  (1-1  nodes  j=1  + (-D  the  I  o r d i n a t e of a s y s t e m w i t h  DESCRIPTION  input 3.  is  similar  The d r a i n a g e  number o f r e a c h e s , reach.  The  by r e a d i n g  ordinates.  The i n i t i a l  network  and t h e n  time  established  to  frame  the  program  described  i s d e f i n e d by f i r s t the values  of k  and  f o r r o u t i n g the flows  reading x for i s then  t h e r o u t i n g p e r i o d a n d t h e number flow  in  c o n d i t i o n s a r e a l s o read i n .  of  45 The  tableau  where NVARS  size  i s c a l c u l a t e d by:  NVARS = NORDS  (4.35)  NCONST = 2*NORDS-2  (4.35)  i s t h e number o f LP  variables,  NCONST  i s the  number o f c o n s t r a i n t s , and NORDS i s t h e number o f o r d i n a t e s . Note t h a t ,  u n l i k e the formulation  3,  the  tableau  the  number o f c h a n n e l The  the  UBC  complex  possible  first  with  The  This any  formulation,  is  output  analysis  number  there  routed  i s independent of  called is  of  a l l  marks  to  similar  (up t o a  tableau  is  perform  the  then  linear  to that described i n  s o l u t i o n (the outflows)  (the dual  formulation,  the  the d e c i s i o n v a r i a b l e s .  was no p r o b l e m w i t h  flows.  outputs  The  s o l u t i o n vector  Because of t h e d i f f e r e n t  i n the other  compare w i t h  routine  repetitions.  3, i n c l u d i n g t h e p r i m a l  RHS r a n g i n g ) .  2 and  REPERM i s u s e d t o g e n e r a t e  on t h e RHS and t h e n  sensitivity  occurred  of  LIPSUB r o u t i n e  programming.  the  subroutine  permutations  generated,  formulation  i n chapter  reaches.  coefficients.  50)  Chapter  in this  library  maximum o f  The  size  described  method used cumulative  so t h e r e  and  and t h e  in  this  error  that  was no r e a s o n t o  46  4.5  COMMENT The  the  formulation  most  comprehensive  n e t w o r k s , but variables  for  a  are  the  in  invaluable highly  presented  when p e r f o r m i n g  serial  determined  ( o r d i n a t e s ) . used  i n the  of  of  required has  been  for  large  and  can  development  in  a  the  number  The  of  could  be  analysis  on  are  constraints  time  intervals  and. are- i n d e p e n d e n t  algebraic  coefficients above,  of  p r e v i o u s l y demonstrated  routing, process,  t h e r e f o r e be  of  the  therefore  systems.  the  branched  number  formulation  As  far  manipulations  a r e complex, amenable  to  but  as  automatic  established e-fficiently  even  systems.  Extensions might  by  demonstrated  generation  the  optimization-based  reaches.  to determine  in  by  for  number of v a r i a b l e s and  solely  number  reduction  systems.  system, the  2 provided  results  alternative  complex d r a i n a g e  the  sensitivity  dramatic  this  in Chapter  of  formulation  serve  other  a  useful  formulations  f o r the  T h i s would be  runoff  as  presented  of p a r t i c u l a r  s y s t e m a n a l y s i s scheme  in this  paradigm a n a l y s i s of  practical  based  on  chapter in  the  runoff  significance a  non-linear  formulat ion. Another p o s s i b i l i t y , estimation of  the  of  type  method.  the v a l u e s  described The  accurately, verification  as  not of  the  in this  method  explored  for  estimation  would  coefficients  chapter  by  method.  in  some  generating  d e s c r i b e d a b o v e , would  of an  here,  still  the be  be  the  formulations approximation coefficients necessary  for  Chapter NUMERICAL  The variety  model d e s c r i b e d of  situations  Some o f t h e r e s u l t s behaviour It  5  EXAMPLES  in chapters  2 and 3 was t e s t e d on a  w h i c h may o c c u r  are presented  here  in a design as  examples  of  the  results  of  the  of t h e program. is  sensitivity  useful  to  analysis  recall (dual  that  the  solution  and RHS  which of t h e i n f l o w o r d i n a t e s , i f m o d i f i e d , greatest  effect  solution  value  decrease  in  corresponding The  while model  purpose  constraint  per  unit  of  t h e model s h o u l d a l s o  the f o l l o w i n g examples.  point  planning process.  out  to  the  greatest  the  in  reduction  and  dimensions  or f l o o d  in  consequently  information  peak  limits.  reduce  consequences. quickly  and  at  a  to  i n mind of  the  the  is  of t h e effect  downstream  required  I t i s intended without  will  but r a t h e r  the areas  i n order  the  t h a t d e s i g n models r e q u i r e .  47  be k e p t  The m o d e l ' s aim  flow  the  hydrograph.  design,  designer  network w h i c h c o u l d be m o d i f i e d  calculations  gives  The i n t e n t  detailed  the  The d u a l  decrease  o r d i n a t e on t h e o u t f l o w  i s not t o produce a f i n a l  quickly  have  peak o u t f l o w .  particular outflow  will  show  g i v e s t h e bounds beyond w h i c h t h e peak  location,  such  the  ranging)  inflow o r d i n a t e , w i t h i n the determined  reviewing  drainage the  a  peak  a i d i n the i n i t i a l  to  reducing  for  at a different The  to  in  RHS r a n g i n g  occur  setting.  channel  to provide cumbersome  48 5.1  CASE  1: SINGLE REACH  The through  simplest a single  corresponds  case  reach,  channel other  of  0.1%,  than  routing  at  a  m,  the  period  arbitrarily  side  flood  time  event  slopes  m,  time, as  so t h a t  with  The  5.0 the  8.0  This  example  earth  channel  of  2h:1v, 0.022.  The  of  k,  taken  minutes.  flood  a  inflow points  minutes,  undetected.  hydrograph  of  no  value  is  The  slighty  less  could  not  peak  The  as  value  of x  was  0.20.  c o n d i t i o n s were u n i f o r m  first  following  500  selected  reach  set at  Initial the  the  trapezoidal  upstream end.  t was  inflows.  a Manning's n v a l u e  t o the p r o p a g a t i o n  through  with  to  2.0  and  the p r o p a g a t i o n  pass  the  of  other  length i s approximately  equivalent  than  w i t h no  approximately  w i t h a bottom width gradient  i n v o l v e s r o u t i n g a storm  throughout  time  p e r i o d inflow equal  was  then  routed  examples demonstrate  to the  through  some of  the  the  channel,  outflow.  the  reach.  A The  b a s i c f e a t u r e s of  model.  Example 1.1: The under  Natural  first  run of  natural  information results  of  the  program was  conditions  and the  Conditions  its  first  and  relation  run  observe to  are given  5.05  m /s  and  a t t e n u a t i o n , the o u t f l o w  translation  the  the r o u t i n g  the  sensitivity  peak o u t f l o w .  Table  peak  at  The  in  in  3  1.  the  graphically f l o w of  Figure  to perform  VI  and  i n f l o w hydrograph  fourth ordinate,  and  reaches  The shown  reaches  a  due  to  a peak  of  49  TABLE VI EXAMPLE 1.1 RESULTS  EXAMPLE  1.1: SINGLE REACH, TRAPEZOIDAL CHANNEL.  THERE ARE REACH 1  2 STATIONS AND  K 8.000  X 0.200  1 CHANNEL REACHES.  BETA 9.8889  GAMMA -4.3333  THERE ARE 10 HYDROGRAPH ORDINATES, S T A . 1 1: 0.500 2: 0.500  2 1.450 0.596  PEAK OUTFLOW = PRIMAL  3 3.675 1.301  4 5.050 2.774  THE ROUTING PERIOD IS PHI 4.5556  PS I -49.383  18 CONSTRAINTS, AND 5 4.175 3.964  6 3.620 4.026  7 3.160 3.752  5.000  10 VARIABLES. 8 2.420 3.344  9 2.020 2.785  10 1.850 2.338  4.026426  SOLUTION:  VARIABLE Q( 2, 2) Q( 2, 3) Q( 2, 4) Q( 2, 5) Q( 2, 6) Q( 2, 7) Q( 2, 8) Q( 2, 9) Q( 2,10)  VALUE 0.596 1 .301 2.774 3.964 4.026 3.752 3.344 2.785 2.338  SENSITIVITY ANALYSIS: CONST. 1 2 3 4 5 6 7 8 9  ORDINATE Q( Q( Q( Q( Q( Q( Q( Q( Q(  2) 3) 4) 5) 6) 7) 8) 9) 10)  DUAL VALUE 0.0425 0.0970 0.2213 0.5050 0.1011 0.0000 0.0000 0.0000 0.0000  FLOW 1 .4500 3.6750 5.0500 4.1750 3.6200 3. 1600 2.4200 2.0200 1.8500  RHS RANGING BOUNDS LOWER UPPER -1 .1259 -1.3702 2.8392 4.0205 3.0029 -3.4627 -3.0943 -2.6096 -21.269  2.5955 4. 1770 5.2700 0. 1 0047E+19 4.3005 4.5106 4.8791 5.3638 18.548  50 FIGURE EXAMPLE  E X A M P L E S I N G L E  1 1.1  1.1  R E A C H  10-i  1  e TIME  INTERVAL  1-  7  10  51 4.03  m /sec The  6  five  3  of  dual  the  minutes l a t e r  solution  vector  inflow hydrograph (Ql)  (Figure  2).  indicates that ordinates i n f l u e n c e the  that  the  fifth  ordinate  one  of  these  inflow ordinates  is altered  then  peak w i l l  or d e c r e a s e d ) , amount value  equal  t o the  (provided  limits  the  given  that by  Ql  Example 1.2: This except m /s,  as  3  Ql  shown  represents  a  a  3  Ql,  Since  is  level.  new  bounds  Example  1.1,  the  occurs  dual  be  The  VII  Figure  and  by  is  If  increased  changed  by  an  the  dual  within  the  following  solution  or  the  three  was  Example  RHS  ranging  to note that although  for  the  also the  o r d i n a t e s have c h a n g e d  4.05  the  flows  is  3.085  1.1.  value  a unit  to  condition  of  Example  i s unchanged;  are  5.05  peak o u t f l o w  same t i m e p e r i o d  vectors  example,  This  storage  dual  in  i n the  basis  3.  from  Ql  of  given  the  3  resulting  0.2213  previous  1 m /s., from  diversion  since there  value  the  by  The  optimal at  to  interesting some of  influence.  (either  discharge  reduced  of  expected,  the  the  identical  i s now  within  outflow  greatest  ranging).  e x a c t l y equals  the  and  principle.  temporary  in  as  RHS  peak o u t f l o w ,  inflow m u l t i p l i e d  resulting  this  decrease  difference with  the  in Table  above a c e r t a i n m /s,  change of  the  to  Modified  example  that  outflow  the  examples d e m o n s t r a t e  has  2  This  associated  decrease 1.2  is  Ql.  in  still  information i . e . , the  i n both  u p p e r and  peak  examples  unchanged. lower  from those  in  It  and is  bounds  i n Example  52  FIGURE 2 CASE 1 OUTFLOW HYDROGRAPHS  C A S E  5  1 O U T F L O W  H Y D R O G R A P H S  53 TABLE V I I EXAMPLE 1.2 RESULTS  EXAMPLE  1.2: SINGLE REACH,  THERE ARE  2 STATIONS AND  REACH K 1 8.000  X 0.200  Q ( l , 4 ) reduced by 1. 1 CHANNEL REACHES.  BETA 9.8889  GAMMA -4.3333  THERE ARE 10 HYDROGRAPH ORDINATES, STA. 1: 2:  1 0.500 0.500  2 3 1.450 3.675 0.596 1.301  PEAK OUTFLOW = PRIMAL  4 4.050 2.673  THE ROUTING PERIOD IS PHI 4.5556  PSI -49.383  18 CONSTRAINTS, AND 5 4.175 3.459  6 3.620 3.805  7 3.160 3.655  5.000  10 VARIABLES.  8 2.420 3.302  9 1 0 2.020 1.850 2.766 2.330  3.805139  SOLUTION:  VARIABLE Q( 2, 2) Q( 2, 3) Q( 2, 4) Q( 2, 5) Q( 2, 6) Q( 2, 7) Q( 2, 8) Q( 2, 9) Q( 2,10)  VALUE 0.596 1 .301 2.673 3.459 3.805 3.655 3.302 2.766 2.330  SENSITIVITY ANALYSIS: CONST. 1 2 3 4 5 6 7 8 9  ORDINATE Q( Q( Q( Q( Q( Q( Q( Q( Q(  1, 2) 1, 3) 1, 4) 1, 5) 1, 6) 1, 7) 1, 8) 1, 9) 1,10)  DUAL VALUE 0.0425 0.0970 0.2213 0.5050 0.1011 0.0000 0.0000 0.0000 ' 0.0000  FLOW 1.4500 3.6750 4.0500 4.1750 3.6200 3.1600 2.4200 2.0200 1.8500  RHS RANGING BOUNDS LOWER UPPER -1.1259 0.91183 2.8392 3.6444 0.19742 -3.3786 -3.0574 -2.5934 -21.188  6.8648 6.4508 5.2700 0.10000E+76 3.9927 4.1565 4.4777 4.9417 16.440  54  FIGURE 3 EXAMPLE 1.2  E X A M P L E 4th  O R D I N A T E  1.2 R E D U C E D  55 1.1,  the  bounds  for  the  modified  o r d i n a t e have  remained  constant.  1.3: Ql  Example  Modified  T h i s example  illustrates  inflow  ordinate  results  are l i s t e d  Initial with  the exception  the  new  value given  occurred  at  occurs new  i n Table VIII  of a u n i t  of  the  sixth  in this  example  by 0.1635  Comparison  and  example  one t i m e  to  Ql  Qs t a k e s all  assigned  the  dual  predict this  for that which  1.1, now  example, a n d t h e r e  is  The r e s u l t i n g  than  vectors  i s assigned  in  the  f o r Examples  a  peak  original  v a r i a b l e s take  simply  t o Ql  on d u a l  t o Ql.  1.3, and  Similarly,  values previously  variable.  does n o t g e n e r a l l y p r o v i d e  however,  shifted assigned  i n Example  b a s i s c h a n g e s f r o m one example  the e f f e c t  1.1 and 1.3  the d u a l v a l u e  previously assigned  t o an a d j a c e n t  case,  1.1, case,  outflow,  i n Example  vector.  F o r example,  1.1  outflow  solution  bound  f o r e a c h c o n s t r a i n t have  interval.  the  peak  routing period  of t h e d u a l  on t h e v a l u e  Since  the lower  The  i s less  In t h i s  4.  3  i n Example  other  The  t o Example  i n Ql.  an  m /s.  show t h a t t h e v a l u e s by  than  1.1.  dual  ranging.  and d i s p l a y e d i n F i g u r e  period in this  basis  changing  of t h e RHS  decrease  i s less  i n Example  outflow  of  conditions are identical  at the f i f t h  optimal  effects  beyond t h e l i m i t s  flow  ordinate  the  information  o f a change on t h e o p t i m a l i t i s possible to predict  t o the next, to  solution.  In  t h e change  in  56  TABLE V I I I EXAMPLE 1.3 RESULTS  EXAMPLE 1.3: SINGLE REACH, Q ( l , 5 ) reduced by 1. THERE ARE REACH 1  2 STATIONS AND K  X  8.000  1 CHANNEL REACHES.  BETA  0.200  GAMMA •4.3333  9.8889  THERE ARE 10 HYDROGRAPH ORDINATES, STA. 1 : 2:  1  2 1 .450 0.596  0.500 0.500  PEAK OUTFLOW =  3 3.675 1 .301  4 5.050 2.774  THE ROUTING PERIOD IS PHI 4.5556  PSI •49.383  18 CONSTRAINTS, AND 5 3.175 3.863  6 3.620 3.521  7 3.160 3.530  5.000  10 VARIABLES. 8 2.420 3.247  9 2.020 2.742  10 1.850 2.319  3.862899  PRIMAL SOLUTION: VARIABLE Q( 2, 2) Q( 2, 3) Q( 2, 4) Q( 2, 5) Q( 2, 6) Q( 2, 7) Q( 2, 8) Q( 2, 9) Q( 2,10)  VALUE 0.596 1 .301 2.774 3.863 3.521 3.530 3.247 2.742 2.319  SENSITIVITY ANALYSIS: CONST. 1 2 3 4 5 6 7 8 9  ORDINATE Q( Q( Q( Q( Q( Q( Q( Q( Q(  1, 2) 1, 3) 1, 4) 1, 5) 1, 6) 1, 7) 1, 8) 1, 9) 1,10)  DUAL VALUE 0.0970 0.2213 0.5050 0.1011 0.0000 0.0000 0.0000 0.0000 0.0000  FLOW 1 .4500 3.6750 5.0500 3.1750 3.6200 3.1600 2.4200 2.0200 1.8500  RHS RANGING BOUNDS LOWER UPPER -1.1259 1.8147 4.2348 -3.7983 -3.3709 -3.2707 -3.0101 -2.5727 -21.085  7.7294 7.5144 0.31205E+18 4.0205 4.2786 4.3788 4.6394 5.0768 17.115  57 FIGURE 4 EXAMPLE 1.3  E X A M P L E 1.3 5th ORDINATE REDUCED 10  CD CC <  o  CO Q  4  6  6  TIME INTERVAL  7  10  58 Q  p  i n a d v a n c e by  after  the  the  lower  and  the  p r o - r a t i n g the  shift bound  in basis. (4.0205),  decrease  in  dual value  While  the  the d u a l  Q  f o r Qj  flow  value  before  and  o r d i n a t e i s above  of  0.5050  when Q i d e c r e a s e s  from  applies 4.1750 t o  P  4.0205 i s ( 0 . 5 0 5 0 ) • ( 0 . 1 5 4 5 ) , o r the  lower  bound,  the  change  ( 4 . 0 2 0 5 - 3 . 1 7 5 0 ) • ( 0 . 1 0 1 1 ) , or in  peak  the  two  outflow  from  v a l u e s , or  Example  1.4:  A run  Zero  0.0780.  one  Q^  i n peak o u t f l o w  0.0855.  example  0.1635 as  When  shown  The  is  i s given  total  t o the  next  below by  difference  i s the  sum  of  above.  Ordinates  o f t h e model was  made t o c o n f i r m t h a t Q  was  not  P  c h a n g e d when t h e v a l u e s z e r o were m o d i f i e d . Example of  the  are  1.1,  except  of t h e  The  set  to  inflow c o n d i t i o n s are  t h a t the  i n f l o w hydrograph zero.  Inspection  t h a t t h e peak o u t f l o w  are  same i n b o t h  Example  1.5:  In used,  New  Inflow  Comparison  Table  the  X  (and  significantly  identical  to that  value  of  IX and  a l l preceeding the d u a l  to  tenth ordinates  Table  same c h a n n e l  initial  zero)  Figure 2  ordinates)  solution  values.  and  Figure  different  i n Example  1.1.  characteristics  f l o w c o n d i t i o n s and  w i t h p r e v i o u s examples  produces  through  identical  of  Hydrograph  with d i f f e r e n t  hydrograph.  of  c a s e s , as are  t h i s example,  but  seventh  (each w i t h a dual  reveals the  ordinates with dual values  5  show  shows t h a t  the  were  inflow results.  although  this  flows, the d u a l v e c t o r i s  59 TABLE IX EXAMPLE 1.4 RESULTS  EXAMPLE  1.4: SINGLE REACH, Q ( l , 7 ) t o Q(1,10) = 0.  THERE ARE  2 STATIONS AND  REACH K 1 8.000  X 0.200  1 CHANNEL REACHES.  BETA 9.8889  GAMMA -4.3333  THERE ARE 10 HYDROGRAPH ORDINATES, STA. 1: 2:  1 0.500 0.500  2 1.450 0.596  PEAK OUTFLOW =  3 3.675 1.301  4 5.050 2.774  THE ROUTING PERIOD IS PHI 4.5556  PSI -49.383  18 CONSTRAINTS, AND 5 4.175 3.964  6 3.620 4.026  7 0.0 3.432  5.000  10 VARIABLES. 8 0.0 1.504  9 1 0 0.0 0.0 0.659 0.289  4.026426  PRIMAL SOLUTION: VARIABLE Q( 2, 2) Q( 2, 3) Q( 2, 4) Q( 2, 5) Q( 2, 6) Q( 2, 7) Q( 2, 8) Q( 2, 9) Q( 2,10)  VALUE 0.596 1 .301 2.774 3.964 4.026 3.432 1 .504 0.659 0.289  SENSITIVITY ANALYSIS: CONST. 1 2 3 4 5 6 7 8 9  ORDINATE Q( Q( Q( Q( Q( Q( Q( Q( Q(  1, 2) 1, 3) 1, 4) 1, 5) 1, 6) 1, 7) 1, 8) 1, 9) 1,10)  DUAL VALUE 0.0425 0.0970 0.2213 0.5050 0.1011 0.0000 0.0000 0.0000 0.0000  FLOW 1 .4500 3.6750 5.0500 4.1750 3.6200 0.0 0.0 0.0 0.0  RHS RANGING BOUNDS LOWER UPPER -1.1259 -1.8176 0.26875 4.0205 3.0029 -2.9781 -1.3050 -0.57187 -2.8558  2.5955 4.1770 5.2700 0.10047E+19 5.0918 4.9952 6.6683 7.4015 36.961  60 TABLE X EXAMPLE 1.5 RESULTS  EXAMPLE  1.5: SINGLE REACH, D i f f e r e n t  THERE ARE  2 STATIONS AND  REACH K 1 8.000  X 0.200  1 CHANNEL REACHES.  BETA 9.8889  GAMMA -4.3333  THERE ARE 10 HYDROGRAPH ORDINATES, STA. 1: 2:  1 1.250 1.250  2 .250 ,250  PEAK OUTFLOW  3 4.450 1 .574  Inflow.  4 6.875 3.435  THE ROUTING PERIOD IS PHI 4.5556  PS I -49.383  18 CONSTRAINTS, AND 5 7.250 5.405  6 5.375 6.252  7 4.120 5.632  5.000  10 VARIABLES. 8 3.350 4.705  9 3.025 3.911  10 2.820 3.392  6.252089  PRIMAL SOLUTION: VARIABLE Q( 2, 2) Q( 2, 3) Q( 2, 4) Q( 2, 5) Q( 2, 6) Q( 2, 7) Q( 2, 8) Q( 2, 9) Q( 2,10)  VALUE 1 .250 1 .574 3.435 5.405 6.252 5.632 4.705 3.911 3.392  SENSITIVITY ANALYSIS: CONST. 1 2 3 4 5 6 7 8 9  ORDINATE Q( Q( Q( Q( Q( Q( Q( Q( Q(  1, 2) 1, 3) 1, 4) 1, 5) 1, 6) 1, 7) 1, 8) 1, 9) 1,10)  DUAL VALUE 0.0425 0.0970 0.2213 0.5050 0.1011 0.0000 0.0000 0.0000 0.0000  FLOW 1.2500 4.4500 6.8750 7.2500 5.3750 4.1200 3.3500 3.0250 2.8200  RHS RANGING BOUNDS LOWER UPPER -1.8661 -2.3517 1 .8906 5.1535 -2.9977 -5.1969 -4.3945 -3.6929 -30.728  1 1 .366 1 1 .261 9.8594 0.10000E+76 6.9093 7.1838 7.9862 8.6878 31.099  61  FIGURE 5 EXAMPLE 1.5  E X A M P L E 1.5 N E W INFLOW H Y D R O G R A P H  62 5.2  CASE 2: The  there  THREE REACHES IN  following  are  potential first  examples  three sites  and  SERIES represent  distinct  reaches  for construction  second  reaches  t o the c h a n n e l  similar,  but  represent  i n c r e a s e d roughness,  0.2%.  The  c h o s e n as  k value 0.175.  concrete  i s 6.4  of  was  the  two  reach  a  second  plus  two  below  the  channel  for  and  of  x  was  a  a.  roughness  The  inclusion  of  linear  last  reaches.  and  before  reaches  are  numbered  reservoir  the 1,  sites  reaches',  inflows  0.027 t o  0.15%.  first  'null  is  represents,  diameter,  the  is  gradient  i s 11 m i n u t e s ,  m  where  channel  i n the' s y s t e m  2  made  the  first  steeper  a g r a d i e n t of  potential  of  The  The  channel  a  t o behave as  transfer  series,  reservoirs  1.  and  0.23.  the  of  situation  M a n n i n g ' s n v a l u e of  x is  the c h a n n e l  s e t up  direct  with  after  Initially, 4) a r e  last  0.013, and  Provision  Therefore,  higher  The  m i n u t e s and  reservoirs  i n Case  for this  culvert  coefficient k  a  in  (Figure 6).  identical  has  a  with  3,  and  (reaches  which no  value  of  5.  2  and  allow  for  translation  or  attenuat ion. Given flow  (through  either at  t h a t t h e d e s i g n e r has  reach  storage,  before 2 or  determining  the  outflow  the  the  from  results  diversion,  i t e n t e r s the 4,  the d u a l  solution  from  The  c h o i c e of m o d i f y i n g or  some  s y s t e m or by  most e f f i c i e n t system.  the  values  method of  other  means)  detention  storage  should  of  aid  r e d u c i n g the  f o l l o w i n g examples  s e v e r a l types  the  in peak  demonstrate  s y s t e m m o d i f i c a t i o n , and  63 FIGURE 6 EXAMPLE 2.1 STATION 1: I N F L O W STATION 2  TIMC INTCWVM.  TIME I N T E R V A L  A  2  / STATION 3  \  3  STATION A  TIMl INTCKV*L  TTMC I N T E R V A L  STATION S STATION 6 : O U T F L O W  16  T1KIC INTCPVAU  TtMt INTERVAL  64 the  resulting  illustrated  Example  Figure  Natural  first  reaches.  that,  distributed the  set  lower  of  since  in  remote  upper  approach  the  indicate  high  increase  at  on Q ^ .  In  ordinates  routing  through  three  6 and the  for  one  output  lower  in  several ordinates,  while  tends is  to  to  be  have  than  do  flows  an  expected.,  r e a c h e s have a much more  peak  XI  evenly  This  exhibit  Table an  ordinate  value.  objective is  to  those a  times  in  correct  a  the  changes in in  the  direct  the  more  flow  outflow.peak, at  the  permitting  predicting  result  b a s i s of  result  while  sense,  as  for  reducing  of  a  flow  a change  in  or  flow no  in  to  effect in  several  information  ordinates optimal  the which  change  changes in  sensitivity  times  the  which would have l i t t l e  strict  on t h e  of  decrease the  sensitivity,  reach  w h i c h do n o t  simple  values  outflow  taken  downstream  are  reaches.  To meet  only  the  on t h e  outlet  reaches generally  reaches  high dual  influence  network  Figure  dual  exceptionally flows  is  upper  of  the  7.  example  the  at  Conditions  Inspection  reveals  in  in  2.1:  The  hydrographs  a  flow is  combination  basis  (Solow,  1984).  Example The system,  2.2:  Ql  Modified  same i n f l o w with  one  hydrograph  inflow  ordinate  was r o u t e d  through  changed (Figure  8).  the Ql  same was  65  FIGURE 7 2 OUTFLOW HYDROGRAPHS  CASE  C A S E  3.5-  2  O U T F L O W  H Y D R O G R A P H S  LEGEND ORiaiNAL E X A M P L E 2.2 E X A M P L E S 2.8  J!.l  E X A M P L E 2.6  2.5-  or. < x o C3  2-  CO Q  1.6 -  1-  0.6  0-L 1  -i 4  TIME  1  6  1  6  INTERVAL  1  7  r-  8  10  TABLE XI EXAMPLE 2.1 RESULTS  66  EXAMPLE 2.1: 3 REACHES, 2 NULL RESERVOIRS THERE ARE REACH 2 3 4 5  6 STATIONS AND  K 8. 000 0. 0 1 1 000 . 0. 0 6. 400  X 0 .200 0 .0 0 .175 0 .0 0 .230  5 CHANNEL REACHES.  BETA 9. 8889 1 . 0000 20 .130 0000 1 . 7. 2257  GAMMA 4.3333 1.0000 11.435 1.0000 - 2.3619  -  THERE ARE 10 HYDROGRAPH ORDINATES * STA. 1 2 3 4 5 6  1 2. 375 0. 500 0. 500 0. 500 0. 500 0. 500  2 5 .050 1 .824 1.824 0 .566 0 .566 0 .509  3 2 .775 3 .406 3 .406 1 .188 1 .188 0 .633  4 2. 120 2.985 2.985 2. 125 2. 125 1 . 136  THE ROUTING PERIOD IS PHI 4.5556 1 .0000 7.6957 1 .0000 3.8638  54 CONSTRAINTS, 5 1 .545 2 .441 2 .441 2 .470 2 .470 1 .850  6 1 .220 1 .905 1 .905 2 .431 2 .431 2 .262  PS I -49. 383 -0.0 -166 .35 -0.0 -30. 281  AND  7 1 .045 1 .502 1 .502 2.184 2.184 2.341  5.000  46 VARIABLES • 8 9 0 .865 0.755 1 .227 1 .013 1 .227 1.013 1 .876 1 .585 1 .876 1 .585 2 . 1 92 1 .939  10 0 .680 0 .860 0 .860 1 .330 1 .330 1 .665  SENSITIVITY ANALYSIS: CONST.  ORDINATE  1 2 3 4 5 6 7 8 9  Q( Q( Q( Q( Q( Q( Q( Q( Q(  10 1 1 12 13 14 15  1, 1, 1r 1 1 1 1 1  DUAL VALUE  FLOW  RHS RANGING BOUNDS LOWER UPPER  2) 3) 4) 5) 6) 7) 8) 9) 10)  0. 1724 0.2049 0.1846 0.0731 0.0121 0.0007 0.0000 0.0000 0.0000  5.0500 2.7750 2.1200 1.5450 1.2200 1.0450 0.86500 0.75500 0.68000  1.8115 -0.95419 1.4065 0.23844 -1.7551 -1.3852 -1.1402 -0.94861 -7.8274  7.5006 0.26091E+19 9.4592 2.8786 3.5524 4.7200 10.117 56.473 972.86  16 17 18  Q( Q( Q( Q( Q( Q( Q( Q( Q(  2 2) 2 3) 2 4) 2 5) 2 6) 2 7) 2, 8) 2, 9) 2, 10)  0.0978 0.1579 0.2345 0.2798 0.0856 0.0069 0.0000 0.0000 0.0000  1.8239 3.4062 2.9854 2.4411 1 .9048 1.5024 1.2272 1.0126 0.86030  0.0 0.93212 1.0430 2.0311 0.89401 0.0 0.0 0.0 0.0  3.1477 4.4455 4.7425 0.61159E+11 2.6710 2.9765 3.6430 8.9063 99. 171  19 20 21 22 23 24 25 26 27  Q( Q( Q( Q( Q( Q( Q( Q( Q(  3, 2) 3, 3) 3, 4) 3, 5) 3, 6) 3, 7) 3, 8) 3, 9) 3, 10)  0.0978 0.1579 0.2345 0.2798 0.0856 0.0069 0.0000 0.0000 0.0000  1.8239 3.4062 2.9854 2.441 1 1.9048 1.5024 1.2272 1.0126 0.86030  -1.0697 0.93212 1.0430 2.0311 0.89401 -3.0667 -2.6336 -2.2276 -25.915  3.1477 4.4455 4.7425 0.61159E+11 2.6710 2.9765 3.6430 8.9063 99.171  28 29 30 31 32 33 34 35 36  Q( Q( Q( Q( Q( Q( Q( Q( Q(  4, 2) 4, 3) 4, 4) 4, 5) 4, 6) 4, 7) 4, 8) 4, 9) 4, 10)  0.0066 0.0203 0.0620 0.1896 0.5800 0.1384 0.0000 0.0000 0.0000  0.56576 1.1878 2.1252 2.4697 2.4307 2.1835 1.8756 1.5849 1.3301  0.0 0.0 0.0 1.3040 2.2505 1.6084 0.0 0.0 0.0  3.5446 3.0961 2.7490 2.6736 0.39078E+16 2.5204 2.5692 2.7502 6.2138  37 38 39 40 41 42 43 44 45  Q( Q( Q( Q( Q( Q( Q( Q( Q(  5, 2) 5, 3) 5, 4) 5, 5) 5 6) 5 7) 5 8) 5, 9) 5 10)  0.0066 0.0203 0.0620 0.1896 0.5800 0.1384 0.0000 0.0000 0.0000  0.56576 1.1878 2.1252 2.4697 2.4307 2.1835 1 .8756 1.5849 1.3301  -0.52625 -0.77142 -1.0640 1.3040 2.2505 1.6084 -1.4676 -1.2866 -10.703  3.5446 3.0961 2.7490 2.6736 0.39078E+16 2.5204 2.5692 2.7502 6.2138  1  67 FIGURE 8 EXAMPLE 2 . 2 STATION 1: I N F L O W STATION 2  T5  STATION 5 STATION 6 : O L T T R - O W  16  TlMt I N T E R V A L  68  reduced lower  by bound  reduction that  this  as  of  the  reach  storage  basin.  entering  noticeable reservoir  Example A reach  an  flow  2.4:  and  dual  ordinates  5  variable  were  vector  from  changed  k value  of  Example  with  by  a  two  ten minutes  b e h a v i o u r of a  routing  (Figure  effect  2.1  4  resulted in a translation  on  even  9).  the  at  was  detention  of  the  periods,  peak  and  a  Inclusion  of  the  sensitivities  of  all  p o i n t s above the r e s e r v o i r  wasn't a l t e r e d .  Reservoir  linear  the  2.2.  a t Reach  ordinates,  4,  storage  dual  i n Example  reach  resulting  This prediction is possible  4 to approximate the  a l s o had  The  the  perturbation.  b a s i s and  This  above  p r e d i c t e d by n o t i c i n g  of  hydrograph  inflow  still  2.1.  be  value  XII).  attenuation  hydrograph where t h e  (Table  reservoir  added a t  flow  0.1724 c o u l d  Reservoir  linear  a value  Example  i s a l s o the  unchanged  Example 2.3: A  in  intermediate  optimal  remained  u n i t t o 4.05,  by  p  Ql  with  a result The  Q  amount  though  flow  determined  of  associated even  one  a t Reach  2  r e s e r v o i r was  i n s e r t e d at  compared  the  characteristics  e x a m p l e s were i d e n t i c a l  with of  ( i e , they  k).  Comparing F i g u r e  10 w i t h  the  system  peak and  outflow  the  previous reservoirs  b o t h had  Figure the  reach  the  9 i t can  2  instead  of  example.  The  in  two  the  same v a l u e be  e n t i r e outflow  seen  of  that  hydrograph  69  TABLE X I I PEAK OUTFLOWS FOR Example  Description  2.1  Natural  2.2  Q«  2.3  Reservoir  2.4  Reservoir  2.5  Controlled  are  different  dual  therefore  the  components.  2.17  7.3  %  a t Reach 4  2.01  14.1  %  a t Reach 5  2.01  14.1  %  2.08  11.1  %  w i t h d r a w a l a t Reach 2  values  channel  reach  since  the  4 i n Example  model  to predict  one  effects  t h e model a t t e m p t s  demonstrates  predict  the e f f e c t s  one  of  to p r e d i c t  outflow.  t h e c o n s e q u e n c e s of  example  can  This  of Flow  on t h e m a g n i t u d e of t h e peak  and t h i s  2.1. linear  two examples d e m o n s t r a t e d t h e  be d e s i r a b l e  exhibited  i s n o t d e p e n d e n t on t h e o r d e r of  m o d i f i c a t i o n s and how  withdrawals,  2  the system i s a  2.5: D i r e c t M o d i f i c a t i o n  the e f f e c t s  which  even t h o u g h r e a c h  than  t h e outcome  The p r e v i o u s  flow  -  i n both cases,  and  also  2.34  conditions  i s t o be e x p e c t e d ,  Example  Reduction  reduced  identical  result  CASE 2 EXAMPLES  It  may  controlled method  of s u c h a  by  direct  modification. In  this  example, a l l f l o w s above 2.5 m /s 3  node a r e e x t r a c t e d present  form  only  from the system. allows  at the  S i n c e t h e model  for direct  third  in i t s  flow m o d i f i c a t i o n a t  70 FIGURE 9 EXAMPLE 2.3 STATION 1: I N F L O W STATION 2 e.*  TIME I N T E R V A L  STATION 4.  i  TIMC I N T E R V A L  STATION S STATION 6 : O U T F L O W j j  i j i  sri  TIME INTCftVUkL TIME I N T C B V . L  71 FIGURE 10 EXAMPLE 2 . 4 STATION * INFLOW  STATION  2  ... ..4  TIME INTERVAL  S T A T I O N A.  i i i it i i i ti t ! !  1—5—j—:—!—v\ 4 • • TIME MTCRVC4L  STATION 6 : O U T F L O W  TH4C H4TCRVKL  72 upstream p o i n t s , the s i t u a t i o n the  was s i m u l a t e d  h y d r o g r a p h a t S t a t i o n 3 i n Example  altering  the  flow  unwanted  translation  reaches  were  values with  back t o z e r o .  3, w i t h o u t The the  results  of  given  original  case  to  2.084  a  3  examining  the output  ordinates  modified,  this  i n Table and  appropriate  products  the  which a c t u a l l y  Example  exact  dual value  2.6: New  completely  XIII the  with dual  the  values  the  outflow  a  peak  from t h e  The model was can  Q l and Ql the  value  of t h e peak  be  were  able  seen by the  change  and  only  i n each  summing flow  the  reduction  2.1 was u s e d h e r e ,  inflow hydrograph.  XI and F i g u r e  solution  reconfirms  in  Inflow Hydrograph  different  Table  11 show t h a t  occurred.  The same s y s t e m a s Example a  deals  reach.  0.257  as  multiplying  o r d i n a t e by t h e gives  2.1.  result, XI.  of  the k  from S t a t i o n  resulted  reduction  i n Example  predict  3  two  only  of each  X I I and F i g u r e  Station  and  To a v o i d  by c h a n g i n g  t h e model  numbers  i n Table  m /s,  presented  accurately  reaches  1  the f i r s t  s y s t e m downstream  the i d e n t i t y  flow m o d i f i c a t i o n s a t  outflow  null  By t h i s method,  of t h e d r a i n a g e  changing  ordinates.  and a t t e n u a t i o n e f f e c t s , to  transferring  2.1 t o S t a t i o n  at the appropriate  converted  the part  by  12 w i t h  i s t h e same a s t h a t  that the dual  solution  Figure  but  with  Comparing  Table  6 reveals  that  f o r Example  vector  depends  2.1. solely  This on  o f k and x a n d t h e t i m e o f t h e peak o r d i n a t e a t p o i n t , and  has  no  direct  dependence  on  the  73 FIGURE 11 EXAMPLE 2.5 STATION 1: I N F L O W STATION 2  TIMC I N T E R V A L  TIME I N T E R V A L  STATION 3  A 3  TIMC I N T E R V A L  A  STATION 4  4 TIMC INTLRN4*L  STATION 5 STATION 6 : O U T F L O W  TIMC M T C f W A L  TIME INTERVAL  TABLE X I I I EXAMPLE 2.6 RESULTS E X A M P L E  T H E R E  2 . 6 :  A R E  R E A C H 1  0 .  11  3  6  R E A C H E S ,  S T A T I O N S  K 8 .  2  3  X 2 0 0  0  0 .  0  1  0 .  1 7 5  2 0 1  0 .  0  0 .  0  5  6 .  4 0 0  0 .  2 3 0  1  1  :  1  9 .  3 .  C H A N N E L  1  - 1 1  . 0 0 0 0  3  P S  5 .  6 .  2 5 5  3 .  7 2 0  2 . 3 5 5  1  . 4 8 0  2 .  7 7 8  4 .  4 7 5  3 . 9 1 3  2 . 9 7 8  2 .  7 7 8  4 .  4 7 5  3 . 9 1 3  . 7 6 5  5 . 0 0 0  I  . 3 8 3  - 1 6 6 . 3 5  1 . 0 0 0 0  - 0 . 0  3 . 8 6 3 8  - 3 0  7  6  2 5 0  I S  - 0 . 0  7 . 6 9 5 7  - 2 . 3 6 1 9  4  P E R I O D  - 4 9  1 . 0 0 0 0  . 4 3 5  1 . 0 0 0 0  2 2 5 7  R O U T I N G  4 . 5 5 5 6  1 . 0 0 0 0  3 0  T H E  P H I  - 4 . 3 3 3 3  . 0 0 0 0 .  R E A C H E S .  G A M M A  8 8 8 9  7 .  2  . 7 8 5  5  R E S E R V O I R S  B E T A  0 .  . 0 0 0  N U L L  A N D  0 0 0  4  S T A .  2  74  . 2 8 1  8  9  1 0  1  . 1 6 5  0 . 9 5 5  0  2 . 2 5 4  1  . 7 2 5  1  . 3 8 9  1  .  1  3 9  2 . 9 7 8  2 . 2 5 4  1  . 7 2 5  1  . 3 8 9  1  .  1  3 9  1  . 3 4 5  . 8 9 5  2 :  0 .  7 5 0  1  3 :  0 .  7 5 0  1 .4  4 :  0 .  7 5 0  0 .  7 8 6  1  . 1  5 0  1  . 9 3 8  3 . 0 0 6  3 . 3 5 1  3 .  1 5 4  2  . 7 3 9  2 . 2 8 4  1  . 8 8 5  5 :  0 .  7 5 0  0 .  7 8 6  1  . 1  5 0  1  . 9 3 8  3 . 0 0 6  3 . 3 5 1  3 .  1 5 4  2  . 7 3 9  2 . 2 8 4  1  . 8 8 5  6 :  0 .  7 5 0  0 .  7 5 5  0 .  1  . 1 5 3  1 . 8 2 9  2 . 6 6 9  3 . 1 0 1  3  . 0 7 9  2 . 7 8 7  2  . 3 9 4  8 0  8 2 6  D U A L C O N S T .  O R D I N A T E  R H S  V A L U E  F L O W  R A N G I N G  L O W E R  1  Q (  1,  2 )  0 . 1 7 2 4  3 . 2 5 0 0  2 . 7 7 7 7  2  Q(  1,  3 )  0 . 2 0 4 9  6 . 2 5 5 0  5 . 5 8 7 1  B O U N D S U P P E R  1 6 . 5 4 8 0 . 8 6 1 4 5 E + 1 8  3  Q (  1,  4 )  0 . 1 8 4 6  3 . 7 2 0 0  - 0 . 1 5 1 7 3  4 . 7 9 0 3  4  Q (  1,  5 )  0 . 0 7 3 1  2 . 3 5 5 0  - 3 . 5 4 2 2  5  Q(  2 . 5 4 9 5  1,  6 )  0 . 0 1 2 1  1 . 7 6 5 0  - 2 . 6 9 8 6  2 . 1 2 1 1  6  Q (  1,  7 )  0 . 0 0 0 7  1 . 3 4 5 0  - 2 . 0 7 1 2  3 . 2 4 2 2  7  Q (  1,  8 )  0 . 0 0 0 0  1 . 1 6 5 0  - 1  8  Q (  1,  9 )  0 . 0 0 0 0  0 . 9 5 5 0 0  - 1 . 3 0 0 9  5 9 . 2 7 2  9  Q (  1 , 1 0 )  0 . 0 0 0 0  0 . 8 9 5 0 0  - 1 0 . 3 7 1  1 0 1 8 . 4  1 0  Q (  2 ,  2 )  0 . 0 9 7 8  1 . 4 7 9 6  0 . 9 3 1 6 0  8 . 6 6 2 7  1  Q (  2 ,  3 )  0 . 1 5 7 9  2 . 7 7 8 1  2 . 4 1 7 3  8 . 4 1 7 4  1 2  Q (  2 ,  4 )  0 . 2 3 4 5  4 . 4 7 5 1  4 . 1 9 1 8  1 3  Q (  2 ,  5 )  0 . 2 7 9 8  3 . 9 1 2 8  3 . 4 3 3 9  1 4  Q (  2 ,  6 )  0 . 0 8 5 6  2 . 9 7 8 0  0 . 0  1 5  Q (  2 ,  7 )  0 . 0 0 6 9  2 . 2 5 4 1  0 . 0  2 . 5 2 9 6  1 6  Q (  2 ,  8 )  0 . 0 0 0 0  1 . 7 2 5 2  0 . 0  4 . 2 5 3 6  9 )  1  1  7  . 5 8 6 0  1 0 . 8 4 9  1 4 . 0 0 9 0 . 1 7 5 3 5 E + 1 3 3 . 0 8 9 7  Q (  2 ,  0 . 0 0 0 0  1 . 3 8 9 2  0 . 0  9 . 6 5 1 0  1 8  Q (  2 , 1 0 )  0 . 0 0 0 0  1  0 . 0  1 0 4 . 0 3  . 1 3 9 2  1 9  Q (  3 ,  2 )  0 . 0 9 7 8  1 . 4 7 9 6  0 . 9 3 1 6 0  8 . 6 6 2 7  2 0  Q (  3 ,  3 )  0 . 1 5 7 9  2 . 7 7 8 1  2 . 4 1 7 3  8 . 4 1 7 4  2 1  Q (  3 ,  4 )  0 . 2 3 4 5  4 . 4 7 5 1  4 . 1 9 1 8  2 2  Q (  3 ,  5 )  0 . 2 7 9 8  3 . 9 1 2 8  3 . 4 3 3 9  1 4 . 0 0 9 0 . 1 7 5 3 5 E + 1 3  2 3  Q (  3 ,  6 )  0 . 0 8 5 6  2 . 9 7 8 0  - 2 . 5 0 6 7  3 . 0 8 9 7  2 4  Q (  3 ,  7 )  0 . 0 0 6 9  2 . 2 5 4 1  - 4 . 4 1 8 2  2 . 5 2 9 6  2 5  Q (  3 ,  8 )  0 . 0 0 0 0  1 . 7 2 5 2  - 3 . 8 3 9 6  4 . 2 5 3 6  2 6  Q (  3 ,  9 )  0 . 0 0 0 0  1 . 3 8 9 2  - 3 . 2 0 3 3  9 . 6 5 1 0  2 7  Q (  3 , 1 0 )  0 . 0 0 0 0  1  - 3 6 . 8 1 2  2 8  Q (  4 ,  2 )  0 . 0 0 6 6  0 . 7 8 6 2 4  0 . 0  2 9  Q (  4 ,  3 )  0 . 0 2 0 3  1 . 1 5 0 3  0 . 0  3 0  Q (  4 ,  4 )  0 . 0 6 2 0  1 . 9 3 7 7  1  3 1  Q (  4 ,  5 )  0 . 1 8 9 6  3 . 0 0 5 8  2 . 8 3 5 8  3 2  Q (  4 ,  6 )  0 . 5 8 0 0  3 . 3 5 1 2  3 . 2 9 5 6  .  1 3 9 2  1 0 4 . 0 3  4 . 7 5 3 3 4 . 6 2 9 8 . 4 1 7 6  4 . 3 9 2 8 4 . 1 1 2 2 0 . 3 5 3 3 2 E + 1 4  3 3  Q (  4 ,  7 )  0 .  3 . 1 5 4 0  0 . 3 3 2 0 5 E -  3 4  Q (  4 ,  8 )  0 . 0 0 0 0  2 . 7 3 9 0  0 . 0  2 . 8 9 5 8  3 5  Q (  4 ,  9 )  0 . 0 0 0 0  2 . 2 8 4 4  0 . 0  3 . 5 0 4 1  3 6  Q (  4 ,  1 0 )  0 . 0 0 0 0  1 . 8 8 5 3  0 . 0  6 . 9 9 6 7  1 3 8 4 ^  0 1  3 . 2 0 3 1  3 7  Q (  5 ,  2 )  0 . 0 0 6 6  0 . 7 8 6 2 4  - 0 . 6 3 8 6 8  4 . 7 5 3 3  3 8  Q (  5 ,  3 )  0 . 0 2 0 3  1 . 1 5 0 3  - 0 . 4 4 0 9 3  4 . 6 2 9 8  3 9  Q (  5 ,  4 )  0 . 0 6 2 0  1 . 9 3 7 7  1  4 0  Q (  5 ,  5 )  0 . 1 8 9 6  3 . 0 0 5 8  2 . 8 3 5 8  4 1  Q (  5 ,  6 )  0 . 5 8 0 0  3 . 3 5 1 2  3 . 2 9 5 6  4 2  Q (  5 ,  7 )  0 . 1 3 8 4  3 . 1 5 4 0  0 . 3 3 2 0 5 E -  4 3  Q (  5 ,  8 )  0 . 0 0 0 0  2 . 7 3 9 0  - 2 . 0 6 6 9  2 . 8 9 5 8  4 4  Q (  5 ,  9 )  0 . 0 0 0 0  2 . 2 8 4 4  - 1 . 8 4 2 6  3 . 5 0 4 1  4 5  Q (  5 , 1 0 )  0 . 0 0 0 0  1 . 8 8 5 3  - 1 5 . 4 1 0  6 . 9 9 6 7  . 4 1 7 6  4 . 3 9 2 8 4 .  1 1 2 2  0 . 3 5 3 3 2 E + 1 4 0 1  3 . 2 0 3 1  75 FIGURE EXAMPLE  12 2.6  STATION 1: I N F L O W STATION 2  /A\  a.« ^  TIME I N T E R V A L  STATION 3  TIMC I N T E R V A L  TIME IMTCRVAL  STATION 5 STATION 6 : O U T F L O W  16  TIME INTERVAL  TIME t N T E R W .  76 inflows.  5.3  CASE 3; BRANCHED NETWORK This  set  o f examples  junction,  as  illustrated  provision  is  made  structures. values  of  channel  in  Figure  13.  the i n c l u s i o n  and  size,  x  to  slope,  hydrograph-entering than  represent  and  that  Once  assigned  various  branch  entering  again,  different  The  i s generally  t h e second  storage  combinations of  deterioration.  the f i r s t  a single  of d e t e n t i o n  The r e a c h e s have e a c h been k  peaks sooner  for  i n v o l v e s a network w i t h  inflow  l a r g e r and  branch.  Example 3.1 The is  no  highest  first  example  detention dual  The adding  storage.  value  downstream r e a c h  illustrates  following  resulting  illustrated  Example  examples  reservoirs  storage  results  outflow  in Figure  a t an  there  2 examples, the  ordinate  i n the  demonstrate at  various  i s able  the  effects  locations  t o p r e d i c t such  hydrographs  f o r each  of  i n the effects.  example a r e  14.  3.2  This a  t h e Case  where  XIV, F i g u r e 1 3 ) .  n e t w o r k , and how t h e model The  As w i t h  generally occurs  (Table  storage  a situation  example basin  illustrates is  the s i t u a t i o n  installed  i n a net increase  that  a t Reach 2.  i n t h e peak o u t f l o w ,  occurs  This  when  situation  despite  the  77 FIGURE 13 EXAMPLE 3.1 STATION 5  Tut  Ttar  wazir**.  T»* M I TCW>*U.  78 TABLE XIV EXAMPLE 3.1 RESULTS EXAMPLE 3.1: ONE JUNCTION BRANCH NO. 1: STATION 1 TO STATION 4 BRANCH NO. 2: STATION 5 TO STATION 11  JUNCTION: 4. 8 JUNCTION: 11,11  THERE ARE 11 STATIONS AND 9 CHANNEL REACHES.  THE ROUTING PERIOD IS 15.000  REACH 1 18 2 0 3 21 5 19 6 0 7 20 8 16 9 0 10 17  PHI 2.8462 1.0000 2.9216 2.4247 1.OOOO 2.9960 3.0236 1.0000 2.9453  K 000 0 OOO 500 0 250 400 0 200  0 0 0 0 0 0 0 0 0  X 200 0 175 160 0 185 230 0 215  BETA 5.6154 1.0000 6.4902 5.4521 1.0000 6.3946 5.3991 1.0000 5.5239  GAMMA - 1.7692 1.0000 -2.5686 -2.0274 1.0000 -2.3986 - 1 .3755 1.0000 - 1 .5786  PSI -17.751  -o.o  -21.530 -15.247 -0.0 -21.557 -17.700 -0.0 -17.848  THERE ARE 12 HYDROGRAPH ORDINATES, 110 CONSTRAINTS, AND 100 VARIABLES. STA. 1 2 3 4 5 6 7 8 9 10 11  5 5 5 5 2 2 2 2 7 7 7  1 975 365 365 145 615 550 550 100 885 885 350  7 6 6 5 2 2 2 2 7 7 7  PEAK OUTFLOW =  2 3 4 5 6 650 12 250 9 450 8 250 6 900 081 7 975 10 404 9 537 8 415 081 7 975 10 404 9 537 8 415 388 6 099 7 607 9 163 9 2 16 850 3 125 3 450 3 950 4 850 634 2 820 3 071 3 401 3 911 634 2 820 3 071 3 401 3 911 394 2 573 2 767 3 009 3 333 508 7 877 8 785 10 302 1 1 766 508 7 877 8 785 10 302 1 1 766 664 7 619 7 968 8 826 10 145  7 8 9 10 11 12 5 705 5 450 5 320 5 .225 5 175 5 . 130 7 165 6 1 19 5 638 5 .403 5 272 5 . 198 7 165 6 1 19 5 638 5 .403 5 .272 5 . 198 8 540 7 548 6 610 5 .987 5 .614 5 . 396 4 615 3 720 3 150 3 .015 2 .945 2 .920 4 458 4 392 3 865 3 .391 3 . 142 3 .014 4 458 4 392 3 865 3 .391 3 . 142 3 .014 3 780 4 193 4 235 3 .930 3 .554 3 .277 12 307 12 209 1 1 694 10 .890 10 026 9 .295 12 307 12 209 1 1694 10 .890 10 .026 9 . 295 1 1 400 12 030 12 065 1 1.655 10 .952 10 . 158  12.06485  PRIMAL SOLUTION: VARIABLE 0( 1 1 2) 0( 1 1 3) 0( 1 1 4) 0( 1 1 5) 0( 1 1 6) Q( 11 7) 0( 11 8) 0( 11 9) 0( 11 10) 0(11 11) 0( 1 1 12)  VALUE 7 664 7 619 7 968 8 826 10 145 1 1 400 12 030 12 065 11 655 10 952 10 158  SENSITIVITY ANALYSIS: CONST.  ORDINATE  DUAL VALUE  FLOW  RHS RANGING BOUNDS LOWER UPPER  1 2 3 4 5 6 7 8 9 10 11  Q( 0( 0( 0( 0( 0( 0( 0( Q( 0( 0(  1, 2) 1. 3) 1, 4) 1, 5) 1. 6) 1. 7) 1. 8) 1, 9) 1. 10) 1,11) 1, 12)  O 0 O 0 0 0 0 O 0 0 0  0841 1434 2102 2382 1636 0619 01 19 0009 0000 0000 0000  7. 6500 12.250 9. 4500 8 2500 6 9000 5 7050 5 4500 5 3200 5 2250 5 1750 5 1300  -2.7608 5.3364 3.3093 7.7880 6.5608 5.0152 2.3236 -4.2779 -4.1403 -4.0577 -24.057  8.2311 12.766 10.681 0. 11333E+ 15 12.397 9.7410 12.789 17.041 36.009 164.73 2077.5  12 13 14 15 16 17 18 19 20 21 22  0( Q( 0( 0( Q( 0( 0( 0( 0( Q( 0(  2. 2) 2, 3) 2, 4) 2, 5) 2. 6) 2, 7) 2, 8) 2, 9) 2. 10) 2, 11) 2, 12)  0 0 0 0 0 0 0 0 0 0 0  0310 0658 1291 2213 2906 1822 0508 0052 0000 OOOO 0000  6 081 1 7 9749 10.404 9 5370 8 4151 7 1645 6. 1194 5 .6378 5 4032 5 .2722 5 . 1976  0.0 0.0 3.9238 5.0855 8.0970 6.9021 5.3633 0.0 0.0 0.0 0.0  7.0707 8.5196 10.779 10.035 0.44747E+13 10.949 9.2419 11.926 15.870 42.827 374.24  TABLE XIV ( c o n t i n u e d ) RHS RANGING BOUNDS LOWER UPPER  ORDINATE  DUAL VALUE  23 24 25 26 27 28 29 30 31 32 33  0( 0( 0( 0( 0( 0( 0( Q( 0( 0( 0(  3. 2) 3, 3) 3. 4) 3, 5) 3, 6) 3, 7) 3. 8) 3. 9) 3. 10) 3,11) 3.12)  0 0 0 0 0 0 0 0 0 0 0  03 10 0658 1291 2213 2906 1822 0508 0052 0000 0000 0000  S.081 1 7.9749 10.404 9.5370 8.4151 7.1645 6.1194 5.6378 5.4032 5.2722 5.1976  -5.8507 -3.7994 3.9238 5.0855 8.0970 6.902 1 5.3633 - 1.0366 -5.5801 -5.2847 -29.823  7.0707 8.5196 10.779 10.035 0.44747E+13 10.949 9.2419 11.926 15.870 42.827 374.24  34 35 36 37 38 39 40 41 42 43 44  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  5, 2) 5, 3) 5. 4) 5. 5) 5, 6) 5, 7) 5. 8) 5, 9) 5. 10) 5.11) 5,12)  0 0 0 0 0 0 0 0 0 0 0  0870 1450 2080 2321 1603 0618 0122 0010 0000 0000 0000  2.8500 3.1250 3.4500 3.9500 4.8500 4.6150 3.7200 3.1500 3.0150 2.9450 2.9200  -2.6480 -2.8626 -3.0604 3.4698 4.4999 3.9195 0.64959 -3.4616 -3.1108 -2.9305 -13.511  3.4450 3.6722 4.8797 0.45668E+13 10.563 8.7799 11.235 15.117 33.B79 159.32 1985.3  45 46 47 48 49 50 51 52 53 54 55  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  6, 2) 6, 3) 6. 4) 6. 5) 6. 6) 6. 7) 6, 8) 6, 9) 6, 10) 6.11) 6. 12)  0 0 0 0 0 0 0 0 0 0 0  0286 062G 1262 2215 2965 1865 0518 0052 0000 0000 0000  2.6339 2.8201 3.0712 3 . 4009 3.9109 4.4577 4.3923 3.8655 3.3913 3.1421 3.0137  0.0 0.0 0.0 0.0 3.5975 4.2016 3.6518 0.0 0.0 0.0 0.0  3.6476 3.3623 3.4331 3.8608 0.68122E+13 8.1859 7.4391 10.007 13.617 39.943 366.62  56 57 58 59 60 61 62 63 64 65 66  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  7, 2) 7, 3) 7. 4) 7. 5) 7. 6) 7. 7) 7, 8) 7. 9) 7, 10) 7,11) 7,12)  0 0 0 0 0 0 0 0 0 0 0  0286 0626 1262 2215 2965 1865 0518 0052 0000 OOOO 0000  2.6339 2.8201 3.0712 3.4009 3.9109 4.4577 4.3923 3.8655 3.3913 3.1421 3.0137  -2.2471 -2.4281 -2.6357 -0.90404 3.5975 4.2015 3.6518 -2.7 106 -3.35 10 -3.0734 -17.939  3.6476 3.3623 3.4331 3.8608 0.68122E+13 8.1859 7.4391 10.007 13.617 39.943 366.62  0 0032 0 0099 0 0293 0 0823 0 2080 0 4141 0 2183 . 00335 0 0000 0 0000 0 0000  7.7826 8.6719 10.373 12.172 12.550 12.319 11.741 10.846 9.9166 9.1683 8.6726  -5.1904 -5.7954 -6.5925 4.4227 9.2862 12.143 11.554 9.8173 -6.595 1 -6.1396 -41.512  12.955 10.443 11.025 12.446 12.717 0.66458E+13 13.835 13.066 14.520 17.904 65.535  CONST.  67 68 69 70 71 72 73 74 75 76 77  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  4. 2) + 4, 3) + 4, 4) + 4, 5) + 4, 6) + 4, 7) + 4, 8) + 4, 9) + 4,10) + 4,11) + 4.12) +  0(8, 2) 0(8, 3) 0(8. 4) 0(8, 5) 0(8. 6) 0(8. 7) 0(8, 8) 0(8, 9) 0(8, 10) 0(8,11) 0(8,12)  FLOW  78 79 80 81 82 83 84 85 86 87 88  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  9, 2) 9. 3) 9. 4) 9, 5) 9. 6) 9. 7) 9. 8) 9. 9) 9, 10) 9.11) 9. 12)  0 0 0 0 0 0 0 0 0 0 0  0003 001 1 0039 0137 0478 1672 5849 1810 0000 OOOO 0000  7.5076 7.8772 8.7846 10.302 11.766 12 . 307 12.209 11.694 10.890 10.026 9.2950  0.0 0.0 0.0 0.0 0.0 8 .8711 12.124 11.504 0.0 0.0 0.0  15.112 14.895 12.321 11.312 12.054 12.390 0.94121E+15 12.710 12.792 13.285 19.827  89 90 91 92 93 94 95 96 97 98 99  0( 10. 2) 0( 10, 3) 0( 10, 4) 0( 10, 5) 0( 10, 6) 0( 10. 7) 0( 10. 8) 0( 10, 9) 0( 10.10) 0( 10.11) 0( 10.12)  0 0 0 0 0 0 0 0 0 0 0  0003 001 1 0039 0137 0478 1672 5849 1810 OOOO OOOO OOOO  7.5076 7.8772 8.7846 10. 302 11.766 12.307 12.209 11.694 10.890 10.026 9.2950  -5.5183 -5.7446 -6.3043 -7.0423 -0.25819 8.871 1 12.124 11.504 -7.8340 -7.3409 -46.819  15.112 14.895 12.321 11.312 12.054 12.390 0.94121E+15 12.710 12.792 13.285 19.827  FIGURE 14 CASE 3 OUTFLOW HYDROGRAPHS  O U T F L O W C A S E  3:  H Y D R O G R A P H S  B R A N C H E D  N E T W O R K  LEGEND ORIGINAL E X A M P L E 3.2 E X A M P L E 3.3  <  1  I  I  2  3  1  4  1  6  I  8  1  7  1  8  TIME INTERVAL  1  9  1  10  1  11  f 12  81 attenuation predicted dual  effect  of the r e s e r v o i r .  from Example  This  3.1 by c o m p a r i n g  vectors at Station 3 (theoutlet  result  c o u l d be  the hydrographs  and  of the r e s e r v o i r )  for  e x a m p l e s 3.1 a n d 3.2 ( F i g u r e 1 5 ) . When t h e r e  i s no r e s e r v o i r  3 i s g r e a t e s t at the f o u r t h greatest  dual  ordinates that  a decrease  will  have a g r e a t e r  results  in  the  effect maximum  with  high dual  values,  and c o n s e q u e n t l y  Example  3.3  is  example  the f i f t h  words, t h e  in  reducing  flow  itself.  and  i n those  Q^  is  reservoirs.  reduced  by  Examination  about  earlier  than  (no  than  Therefore,  t h e peak  flow  from  to predict  at  effect  Ql  of with  and  Q|  high  dual  reservoir  Here the  outflow  the case  w i t h no  in  Example  the d i f f e r e n t  16).  At  reach.  value  3.1  effects  Station  the highest dual the  a  the o v e r a l l  where a  of the dual values  reservoir),  will  i n Q^.  8%  from e x a m p l e s 3.2 and 3.3 ( F i g u r e 3.1  a t node 3  both  represents a situation  show how t h e model was a b l e  Example  flow  (an o r d i n a t e  added t o R e a c h 6 i n t h e s e c o n d b r a n c h .  peak  predicts  ordinates with  an i n c r e a s e  and s i x t h  By e x a m i n i n g t h e  increase  values).  the  model  Q^  Ql  was t o d e c r e a s e  i s a net increase  This  (Q«), while  3.2, i t c a n be s e e n t h a t t h e  low s e n s i t i v i t y )  (ordinates  period  f o l l o w i n g t h e maximum  the reservoir  relatively  effect  In o t h e r  i n flow  f o r Example  inserting  time  the flow a t s t a t i o n  values are a s s o c i a t e d with  (Qi and Q | ) .  decrease  present,  7  in  occurs  Therefore,  the  82  FIGURE 15 STATION 3: COMPARISON OF EXAMPLES 3.1 AND  3.2  STATION  FIGURE 16 7: COMPARISON OF EXAMPLES  S T A T I O N  ~7  -  I M O  3.1 AND 3.3  R E S E R V O I R  « <3  "TIN/IE  S T A T I O N  V  -  I N T E R V A L  W I T H  R E S E R V O I R  84 effect peak  of a d d i n g and  results dual  a reservoir (decreasing  increasing  in lowering  values.  outflow  Q  Example  those  This  (Figure  p  flow  leads  the  whereby  CASE 4:  shown  effects  decrease  the  peak)  to  high  i n the  on  the  This  14,  at  peak  peak o c c u r s  can  be  this  Reach  9  i t also resulted before  explained point  by  the the  (Figure  of  r o u t i n g through another  any  particular  17)  reservoir  ordinate.  COMPLEX NETWORK  as  network  illustrated  e a c h have d i f f e r e n t  variety  of c h a n n e l  shown  in Figure  hydrograph at  reaches  The  net  reservoir located  even t h o u g h  the  A multi-branch examples,  a  ordinate.  shape of the  As  i n Q^,  have m i n i m a l e f f e c t  Example  a  the  14).  junction.  sensitive  flattened  5.4  to  corresponding  at  3.4  in a decrease most  flow  some p o i n t s a f t e r  ordinates  T h i s example c o n t a i n s below  at  the  i s considered in  Figure  values  of  in  the  18.  following  Once a g a i n ,  k and  the  x to represent  a  parameters.  4.1 results  in Figure  hydrographs conclusions particularly  18.  are can  for this  (To p r e s e r v e  included be  example are  arrived  regarding  the  in at  given  clarity,  Figure from  upstream  in Table  only  18.)  examining  XV  some of A the  inflow points.  number  and the of  results,  85 FIGURE 17 STATION 10: COMPARISON OF EXAMPLES 3.1 AND  3.3  86  FIGURE 18 EXAMPLE 4.1  8  STATION  3  5A /  /  / \  \  \  TMC M I TCKVKL  ^ — _—A  i'2  14  15  -nut iwrcirvjw.  TM I C HTt*\*L  87 TABLE XV EXAMPLE 4.1 RESULTS EXAMPLE 4.1 BRANCH BRANCH BRANCH BRANCH  (COMPLEX NETWORK, NO RESERVOIRS)  NO. NO. NO. NO.  STATION 1 STATION 3 STATION 8 STATION 10  TO TO TO TO  JUNCTION JUNCTION JUNCTION JUNCTION  STATION STATION STATION STATION  THERE ARE 15 STATIONS AND 11 CHANNEL REACHES. REACH 1 18 3 21 4 19 5 0 6 21 8 20 19 10 1 1 18 17 12 13 0 14 20  K 500 000 250 0 550 750 500 700 875 0 450  X 250 175 235 0 190 215 220 265 3 10 0 220  0 0 0 0 0 0 0 0 0 0 0  GAMMA -2 .2174 -2 .5686 -2 .4280 1 OOOO -2 .9234 -2 .8922 -2 .4019 -2 .4541 -2 . 4678 1 • OOOO -2 .8161  BETA 7.4348 6.4902 7.4679 1.OOOO 7.3280 7.8285 7.0748 8.3492 10.126 1 . OOOO 7.8144  THERE ARE 12 HYDROGRAPH ORDINATES. STA. 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15  7 4 1 1 5 5 5 2 1 1 1 3 8 8 8  1 2 3 975 12 650 10 850 7 575 10 894 525 1 850 615 2 125 455 1 588 1 789 885 6 375 8 728 885 6 375 8 728 480 5 790 6 463 015 3 270 3 835 70S 2 061 2 895 1 690 450 1 825 1 481 1 638 440 065 3 169 3 551 265 8 518 8 956 265 8 518 8 956 075 a 229 8 470  PEAK OUTFLOW -  4 9 450 10 675 2 450 2 042 1 1402 1 1402 8 189 4 010 3 510 1 900 1 772 4 334 10 004 10 004 8 915  2. 4 7, 12 9.11 15. 15  THE ROUTING PERIOD IS PHI 4 .2174 2 .9216 4 .0399 1.0000 3 . 4046 3 .9362 3 .6729 4 .8951 6 . 6579 1 .0000 3 .9983  15.000  PSI -33.573 -21.530 -32.597 -0.0 -27.873 -33.707 -28.387 -43.324 -69.884 -0.0 -34.061  132 CONSTRAINTS, AND 122 VARIABLES.  8 9 3 2 12 12 10 3 3 1 1 5 12 12 9  5 250 654 250 412 202 202 229 650 779 860 851 045 18 1 181 890  6 6 900 8 487 4 145 3 056 12 040 12 040 1 1393 3 205 3 641 1 785 1 846 5 441 14 675 14 675 1 1674  7 5 70S 7 213 4 615 3 786 1 1632 1 1632 1 1726 3 700 3 429 1 650 1 787 5 44 1 16 34 1 16 34 1 13 807  8 5 450 6 120 3 720 4 149 1 1107 1 1107 1 1598 3 B75 3 622 1 525 1 679 5 292 16 939 16 939 15 504  9 5 320 5 632 3 150 3 802 10 430 10 430 1 1 21 1 3 545 3 740 1 415 1 562 5 299 16 864 16 864 16 412  10 1 1 12 5 . 225 5 . 175 5 130 5 .400 5 197 5 .271 3 .015 2 920 2 .945 3 .387 3 . 152 3 023 9 .672 9 .026 8 591 9 .672 9 .026 8 59 1 10 .638 9 .969 9 343 3 .350 3 280 3 .315 3 .592 3 .435 3 355 1 .350 1 .200 1 . 155 1 .456 1. 365 1 250 5 .270 5 .083 4 860 16 . 536 15 .977 15 . 194 16 .536 15 . 977 15 . 194 16 .659 16 .509 16 .068  16.65948  SENSITIVITY ANALYSIS: CONST.  DUAL VALUE  ORDINATE  1 2 3 4 5 6 7 8 9 10 1 1  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  1, 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.  2) 3) 4) S) 6) 7) 8) 9) 10) 1 1) 12)  0 . 1 184 0 . 1698 0 .2056 0 1900 0 1 10O 0 0374 0 0073 0 0007 0 OOOO 0 OOOO 0 OOOO  12 13 14 15 16 17 18 19 20 21 22  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  3. 3. 3. 3. 3. 3. 3. 3. 3. 3, 3.  2) 3) 4) 5) 6) 7) 8) 9) 10) 1 1) 12)  0 0 0 0 0 0 0 0 0 0 0  23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  0( 2. 2) Q( 2. 3) 0( 2. 4) 0( 2. 5) 0( 2. 6) 0( 2. 7) 0( 2. 8) 0 ( 2. 9 ) 0( 2.10) 0( 2 , 1 1 ) 0( 2 . 1 2 ) 0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  + + + + + • + + + + +  0( 0( 0( 0( 0( 0( 0( 0( 0( 0( 0(  4, 4, 4, 4, 4, 4, 4, 4, 4. 4, 4,  5. 2) 5. 3) 5, 4) 5. 5 ) 5. 6) 5. 7) 5, 8) 5. 9 ) 5, 10) 5. 1 1 ) 5. 12)  2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)  FLOW  RHS RANGING BOUNDS LOWER UPPER  12.650 10.850 9.4500 8.2500 6.9000 5.7050 5.4500 5.3200 5.2250 5.1750 5.1300  9.2935 7.9251 5.2470 5.1569 3.4935 -2.4920 -3.8235 -3.5715 -3.4528 -3.3824 -33.512  17.459 17.761 0.47599E+12 17.895 8.7812 7.7768 10.435 21.432 86.936 BOO. 07 19039.  1233 1686 1958 1763 1036 0367 0075 0008 OOOO OOOO OOOO  1.8500 2.1250 2.4500 3.2500 4.1450 4.6150 3.7200 3.1500 3.0150 2.94S0 2.9200  -1.6494 - 1. 1977 -2.2685 -0.15208 O.44849 -3.5027 -3.7185 -3.4770 -3.1508 -2.9691 -16.699  7.3134 11.200 0.76756E+12 10.950 6.2140 6.8631 8.8716 19.627 82.278 733.40 16618.  0 0 0 0 0 0 0 0 0 0 0  0599 1052 1663 2231 2242 121 1 0335 0045 0002 OOOO OOOO  9.1627 12.683 12.717 12.066 11.543 10.999 10.270 9.4345 8.7877 8.4221 8.2203  3.8238 9.3612 10. 255 9.4176 9. 1431 8.1788 1 .7317 -7.1t20 -6.6546 -6.2765 -55.940  14.625 16.730 17.071 24 1.08 0.69208E+12 12.459 11.985 14.503 26.584 140.19 2568.3  0 0 0 0 0 0 0 0 0 0 0  0204 0425 0840 1525 2399 2822 1340 0260 0017 OOOO OOOO  6.3753 8.7279 11.402 12.202 12.040 11.632 11.107 10.430 9.6715 9.0261 8.5915.  0.0 1 .9367 7.7771 10.008 10.319 9.9635 8.8171 0.23393 0.0 0.0 0.0  17.542 14.687 15.010 15.032 17.888 0.60989E+12 12.122 11.823 14. 142 31 .7S3 351.40  TABLE XV  (continued)  ORDINATE 6. 2 0( 0( 6, 3 0( 6, 4 0( 6, 5 0( 6. 6 0( 6. 7 0( 6, 8 6, 9 0( 0( 6.10) 0( 6,11) 0( 6,12)  DUAL VALUE 0.0204 0 0425 0 0840 0 1525 0 2399 0 2822 0 1340 0 0260 0 0017 0 OOOO 0 OOOO  56 57 58 59 60 61 62 63 64 65 66  0( 8, 2) 0( 8. 3 0( 8, 4 0( 8, 5 Q( 8, 6. 0( 8, 7 8. 8 0( 0( 8, 9) 8, 10) 0( 0( 8.11 0( 8,12)  0 0 0 0 0 0 0 0 0 0 0  67 68 69 70 71 72 73 74 75 76 77  10, 2) 0( 10, 3) 0 ( 10, A. 0 ( 10, 5 0 ( 10, 6 0 ( 10, 7 : 0 ( 10, 8 0 ( 10. 9 0 ( 10. 10) 0( 10. 1 1 ) 0 ( 10, 12)  0 0 0 0 0 0 0 0 0 0 0  CONST. 45 46 47 48 49 50 51 52 53 54 55  0(  FLOW 6 .3753 8.7279 1 1.402 12 .202 12 .040 1 1.632 1 1. 107 10 .430 9.6715 9.0261 8.5915  RHS RANGING LOWER -6.0763 1 .9367 7.7771 10.008 10.319 9.9635 8.8171 0.23393 -9.5348 -8.974 1 -59.873  BOUNDS UPPER 17.542 14.687 15.010 15.032 17.888 0.60989E+12 12.122 11.823 14.142 31.783 351.40  0548 1005 1657 2306 2369 1234 0320 0040 0002 OOOO OOOO  3. 2700 3. 8350 4 .0100 3.6500 3. 2050 3. 7000 3. 8750 3. 5450 3.3500 3. 3150 3. 2800  -1.9944 0.54902 1.7027 1.3306 1.027O 0.99293 -2.9240 -2.9858 -2.8953 -2.7846 -22.983  8.6731 7.6288 7.8237 42.394 O.19979E+13 5.0246 5.5214 8.4973 21.914 150.64 3060.9  0509 0961 1625 2319 2440 1299 0344 0044 0002 OOOO OOOO  1. 6900 1. 8250 1.9000 1. 8600 1. 7850 1. 6500 1. 5250 1. 4 150 1. 3500 1. 2000 1 .1550  -1.1981 -1.2996 -0.36274 -0.30946 -0.38152 -0.94003 -1.2285 -1 . 1516 -1.0562 -1.0O32 -7.6852  7.1697 5.5456 5.4672 22.212 0.19614E+13 2.9676 3. 1002 6. 1267 18.658 136.25 2764.4  78 79 80 81 82 83 84 85 86 87 88  9. 2 1 9, 3 I 9. 4 1 9. 5 1 9. 6 1 0( 9. 7 i 0( 9. 8 0( 9. 9 0( 9. 10i 0( 9. 11i 0( 9. 12I  0( 1 1. 2) 0( 1 1 .3) 0( 1 1 . 4) 0( 1 1 .5) 0( 1 1 . 6) 0( 1 1 . 7) 0( 1 1 .8) 0( 1 1 ,9) 0( 1 1 10) . 0( 11.11) 0 ( 11. 12)  0 0 0 0 0 0 0 0  0135 0316 0701 1423 2483 3175 1412 0249 o 0015 0 OOOO 0 OOOO  3. 5413 4.5334 5. 2823 5. 6303 5. 4873 5.2160 5. 301 1 5. 3012 5.0475 4.7996 4.6044  -2.1718 -2.4405 1.3778 3.5475 4.0675 3.S131 3.1749 -3.1784 -3.1315 -3.0196 -35.969  17.158 10.954 8.7071 7.9649 9 .0605 0.18449E+13 6.1543 6.5942 9.2796 28.544 395.18  89 90 91 92 93 94 95 96 97 98 99  0( 7, 2 i + 0 ( 12. 2) 0( 7. 3 i + 0( 12. 3) 0( 7. 4 i + 0 ( 12. 4) + 0 ( 12. 5) 0( 7. 5 i 0( 7. 6 + 0 ( 12. 6) 0( 7. 7 + Q( 12. 7) 0( 7, 8 i + 0 ( 12, 8) 0( 7, 9 + 0( 12, 9) 0( 7 , 10 t + 0 ( 12.10) 0( 7,11 + 0 ( 12.11) 0( 7, 12 + 0 ( 12,12)  0 0 0 0 0 0 0 0 0  0025 0066 0174 0447 1092 2420 4213 1423 0126 OOOO o o OOOO  8.9593 10 .014 12 .523 15 .275 16 .834 17 . 167 16 .890 16 .509 15 .908 15 .052 14 .202  -4.1801 -4.6632 -1.3760 9.7629 14.502 16.035 16.051 14.602 -3.6605 -7.2389 -139.64  27.451 26.338 21.586 19.110 18.696 18.547 0.20144E+13 17.048 17.068 19.207 60.983  100 lOI 102 103 104 105 106 107 108 109 1 10  13, 2) 13, 3! 0( 3, 4! 0( 13, 5) 0 ( 13. 6: Q( 13, 7] 0 ( 13, B : 0( 3, 9! 0( 3. 10) 0( 3, 11) 0( 3, 12)  0 0004 0 0012 o 0034 0 0094 0 0261 0 0724 0 2010 0 5578 0 1280 0 OOOO 0 OOOO  8.5177 8 .9558 10 .004 12 . 181 14 .675 16 .341 16 .939 16 .864 16 .536 15 .977 15 . 194  0.0 0.0 0.0 0.0 5.6667 13.095 15.769 16.443 14.604 0.0 0.0  23.212 22.871 22.215 21.272 20.012 18.264 17.632 0.24698E+15 16.886 17.037 19.814  111 112 113 1 14 1 15 1 16 117 118 1 19 120 121  4, 2) 4. 3) 0( 4, 4) 0( 4, 5) 0( 4, 6) 0 ( 1 4, 7) 0( 4. 8) 0( 4, 9) 0( 4,10) 0( 4,11) 0 ( 1 4, 12)  0.0004 0 0012 0.0034 0.0094 0 0261 0.0724 0.2010 0.5578 0. 1280 0.OOOO 0.OOOO  8.5177 8.9558 10 .004 12 . 181 14 .675 16 .341 16 .939 16 .864 16 .536 15 .977 15 . 194  -6.6670 -7.0267 -7.7272 -8.7493 5.6667 13.095 15.769 16.443 14.604 -12.831 -110.37  23.212 22.871 22.215 21.272 20.012 18.264 17.632 0.22409E+15 16.886 17.037 19.814  0( 0( 0( 0( 0(  + + + + +  + + + + + +  0(  0(  0( 0(  89  The  information  suited  for  Station  the  flow  Figure  the  the optimal  18  gives  to  of  reduced  two  A f t e r about  or  dual  values  designer  diversion  flow  during the  ordinates  could  of  be  2 to  allowed  peak;  in  a l s o be s a f e l y r e l e a s e d  way  t h e model w o u l d  l o c a t i o n f o r a flow  had  a choice  be h e l p f u l i s i n d e c i d i n g  control  of i n s t a l l i n g  a t e i t h e r S t a t i o n 3 or S t a t i o n  Table  XV  can  efficiently flow  aid  in  determining  r e d u c e s t h e downstream at  Station  of  multiplying storage) reduction  about  the dual that  o f 2.34 m /s.  seven  3  ordinates  require a holding  this  16,600  values  this  structure.  will  by  10,  the  flows.  storage  results  For  example,  in  m.  a  storage  The model t e l l s  3  the  ordinates  result  in  the  other  of  the  flow 3  if  s t o r e d between t h e  would r e p r e s e n t  o f 9640 m  a  t h e l o c a t i o n w h i c h most  On  capacity  If  a d i v e r s i o n or  3 was t e m p o r a r i l y  and s e v e n t h o r d i n a t e s ,  requirement  first  f o r operation  i n c r e a s i n g t h e downstream  device  first  (i.e.,  then  time.  the best  the  without  locations,  temporary  some o f t h e s t o r e d w a t e r c o u l d  that  zero  information  the seventh o r d i n a t e ,  pass through without  well  one o r d i n a t e a t  to  inflow  valuable  high  are  I f i t were p o s s i b l e t o c o n t r o l  these  some  for  10  e i t h e r l o c a t i o n would be most e f f e c t i v e  Another on  be  C o n t r o l l e d withdrawals  at  fact, at  can  (except  basis.  times a s s o c i a t e d with 6).  f o r s t a t i o n s 3 and  since  flows  at either  policy. flow  analysis,  3)  affecting  given  a  retained  downstream  hand, at  us (by  storing  Station  and would  by peak the  10 would  reduce Q  by  90 1.65 m /s.  Although  3  former, stored  i n t e r m s o f peak the S t a t i o n  efficient as  the l a t t e r  than  3  Of c o u r s e  as  constructing  facilities  smaller be  well,  reduction  channel  such  or p i p e  incorporated  factors  the  peak o u t f l o w sections.  into  20% more stored  3  need  to  be  costs  of  whether  or  relative and  is sufficient  These  the  p e r volume of w a t e r  at either l o c a t i o n ,  in  than  per thousand m  other as  i s less  o u t t o be o v e r  S t a t i o n 3 (0.17 m /s  considered  the  reduction  10 l o c a t i o n t u r n s  o p p o s e d t o 0.14).  not  flow  reduction  factors  to allow  could a l l  t h e model, d e p e n d i n g on t h e s p e c i f i c  applicat ion. The yield  results  valuable  drainage passing  of the s i n g l e pass t h r o u g h  information  network. through  between  the  F o r example,  fourth  and  storage  m /s  would be g a i n e d .  location m /s 3  requirement  i n reducing  per  thousand Station  eighth  of  i f w a t e r was w i t h h e l d  time  o f 6750 m , 3  t h e peak o u t f l o w m, 3  at  10.  storage  locations could at a rational  sections  rate  of  intervals, a  almost  another  Similarly,  then 20%  the from m /s  then  for a  of  1.34  reduction  is  also  1.5  The e f f e c t i v e n e s s o f s t o r a g e  storage  arriving  the other  Station 6 at a constant  total 3  about  t h e model  3  at this  almost  0.2  greater  than  a l l other  potential  be examined s y s t e m a t i c a l l y t o a i d i n design  and o p e r a t i o n  policy.  Chapter DISCUSSION OF  As right by  mentioned  hand s i d e c o e f f i c i e n t  which  design.  the  6.1  can  features  in  this  be  of  in  other  different  flow  was  choices  schemes w i t h  indicator  of  sensitivity  information  these  two  mechanisms  information  with  and  in aiding  implications  will  be  a d i s c u s s i o n of  how  to  the as  principal opposed  arbitrary.  different  should  runoff  flow  from the  sensitivity how  as  this  of  system  is  be  to  a  design  the  changes.  91  -  a  the  effectiveness  Flow  form  of  formulation  natural  can  CAD  understood  The  engineer.  information  cost  for  changes to changes  r u n o f f . system  quite  explored.  behaviour.  ordinate  lead  easily  from  channel possible  advantages  course  which a r i s e s  sensitivity  reflect  decision  to  It  however, a f u n d a m e n t a l and  upstream  peak o u t f l o w  ordinates  other  are,  converted  vector  of d e c i s i o n v a r i a b l e s would  ordinates  describes  the  dual  CAPABILITIES  entirely  and  express  along  formulation,  purposes,  relates  most  their  chapter,  the  characteristics, that  the  are  the  analyzed.  choice  variables  chapters,  and  MODEL LIMITATIONS AND The  RESULTS  ranging  model p r o v i d e s  These  discussed costs  in previous  6  in way  Section  the to 6.4  be  easily  of  design  92 Perhaps  the  most  formulation  presented  sensitivity  information  reservoir from  entry single  a  during  the  in  discharge  analytical  confirmed, of  alternative  to producing  some  of  appear  to  runoff  capability  was  formulation  approach  comparable,  perturbation  that,  even  the  system.  research  here.  information  for  system  anticipated  enhances  approach.  every  and  the  not  p h a s e of  presented  full  runoff  through  the  CAD  The  only  would  However, t h i s of  a  the  the  be  would  full  system  of  modest  network  i t would p r o v e t o be i m p r a c t i c a l .  s i g n i f i c a n c e of  previous  of  This  examples c i t e d There are drainage an  the at  network. upstream  dual  ordinate the  i n Chapter  values  reflects  has  the  will  end  been  method  by  stressed  a s s o c i a t e d with by  the of  a  which  a  magnitude  of  the  drainage  demonstrated  in  the  5.  hydrograph  t h r o u g h w i t h d r a w a l or  been  amount  affect  ways of p e r t u r b i n g One  has  value  downstream  principle  two  dual  The  ordinate  peak o u t f l o w  network.  the  chapters.  flow  perturbation  of  the  analysis  particular  the  pass  that  reach  the  in  DUAL SOLUTION VECTOR The  in  in  i n v o l v e so many r e p e t i t i o n s  complexity,  6.2  point  is  every  substantially  potential  kind  thesis  programming  preliminary  once  this  benefit revealed  i s produced at  linear  T h i s complete  but,  and  significant  involves ordinate  some form  of  flows  when a n a l y z i n g  direct  a  modification  (or s e t of  ordinates)  actively  controlled  93  diversion, method, the  detention,  simpler  which  hydrograph  achieve  at  its  pipe diameter  r e p l a c i n g a channel a t t e n u a t i o n of  following  The  the  more i n d i r e c t ,  predicted  without  dual  channel  relate Q  translated  by  method  direct  however,  of  the  information  the  and  In a s t r i c t predictions practice simulate  only  routing  in  on  the  i n the  optimal  slope, result  all)  of  the  produce  simple  which  can  Since  f l o w s and  context  of  With  be  not  to  i s best  the  first  experience,  a l s o convey u s e f u l of  channel  perturbances.  dual  solution  ordinate  values  valid  while  in  ordinate  to  e t c . which a f f e c t  the  one  one  o r d i n a t e at  boundary l i m i t s ,  basis.  are  i s modified,  more t h a n  reservoirs,  t h e RHS  the  information  will  C h a n g i n g more t h a n  invalidates  a change  of  example,  will  the consequences  physical  i f a single  behaviour  the  values  general  analyses.  to the  sensitivity  i t i s d e s i r a b l e to a l t e r the  not  modification.  designer  sense,  the  or  simultaneously.  does  full  the  flow  alter  increase ordinates  modification  designer  other  whole h y d r o g r a p h . however,  changed  directly  sensitivity  to the  modification  be  ordinate  characteristics,  i s to  For  i s , s e v e r a l ( i f not  performing  values  other  channel  of  end.  peak o r d i n a t e s and  That  flow  alteration  The  roughness, d e c r e a s i n g  method, however,  of  of a  s e c t i o n with a r e s e r v o i r  the  peak.  latter  patterns  in  facility.  downstream  or  hydrograph o r d i n a t e s w i l l  the  but  i n t u r n c a u s e s an  shape  increasing  in  to  retention  p h y s i c a l d i m e n s i o n s or c h a r a c t e r i s t i c s  reservoir,  or  or  and  may  This corresponds  once, result to  the  94  downstream peak o u t f l o w later the  time  (i.e., also  if  not channel  shift  along  only  flow  the time  available  axis.  then  channel  no l o n g e r  is  the case  vector w i l l  parameters  simply  s e t of d u a l each  solution  reach.  was  changed  dual  values  shift  is  explained  in  are  altered,  that  ranging, outflow The provides  new f o r m u l a t i - o n , and problem as  ordinates are perturbed.  values  A whole  and RHS bounds a r e g e n e r a t e d  s m a l l enough t o a l l o w  generalized  RANGING  i n previous  beyond w h i c h t h e  valid.  constraint  linear  t o be made.  discussed  limits  the  However, a s d e m o n s t r a t e d by t h e examples,  RIGHT HAND SIDE  longer  of  flow  d i f f e r e n c e s are s t i l l  As the  when i t  a m o d i f i c a t i o n o f an e x i s t i n g  when o n l y  predictions  6.3  that  following section.  is  the  the  or  The method t o d e t e r m i n e t h e  programming, p r o b l e m becomes; an e n t i r e l y  for  earlier  ordinates are being  characteristics),  which the dual  When  new  t o an  F o r t u n a t e l y , i t was r e c o g n i z e d  i n f o r m a t i o n was s t i l l  that,  direction the  interval.  sensitivity  observed  ordinate shifting  constraint  will  dual  RHS' r a n g i n g  solution  the b a s i s w i l l occur  of  the  change  at a d i f f e r e n t  bounds  and  the  important  clue  in  are  no  particular hand  given system  side by RHS peak  time.  upper bound p o r t i o n o f t h e RHS r a n g i n g an  a  enough t o b r i n g t h e r i g h t  outside  provides  values  I f the o r d i n a t e a s s o c i a t e d with  i s perturbed  then  chapters,  predicting  information the  shift in  95  optimal  basis.  chapter  shows t h a t  infinity.  This  before,  after,  reach,  but  exhibits period, flow  of t h e examples  one u p p e r bound anomalously  or concurrent  i t always  the highest the flow  occurs  dual  could  i n each  reach  tends  upper  bound  may  high with  the at  value.  peak  the  At  be i n c r e a s e d  i n the previous  flow  time  that  outflow  can  occurs  occur  period  which  particular  time  indefinitely  (subject to  w o u l d n o t be a l t e r e d .  system  The i n c r e a s e i n  b e - d i r e c t l y p r e d i c t e d by m u l t i p l y i n g t h e i n c r e a s e  upstream If  ordinate  an o r d i n a t e  upper  bound  occur  earlier.  bound'  by i t s d u a l  prior  then  then Q  i s increased,  with  the  infinite  t h e s y s t e m peak o u t f l o w  I f an u p s t r e a m o r d i n a t e  ordinate  in  value.  t o the o r d i n a t e  i s increased,  to  f o r the  c a p a c i t y c o n s t r a i n t s ) and t h e t i m e a t w h i c h t h e  peak  the  Inspection  after will  the  will  'infinite  occur  at a  when  using  later  P  time.  This  type  of information  i s helpful  channel m o d i f i c a t i o n approach which, with will  reduce e a r l y o r d i n a t e s  examining  the  relative  and  increased  increase  later  l o c a t i o n s o f t h e peak  reach  and t h e ' i n f i n i t e  bound' o r d i n a t e ,  make  p r e d i c t i o n s about  the net e f f e c t  i t is  storage,  ones. flow  By  f o r the  possible  on t h e s y s t e m  the  to  outflow  peak.  6.4 COST ANALYSIS Although not e x p l i c i t e l y is  fairly  reflect  easy  costs.  described  i n the examples,  t o t r a n s l a t e the s e n s i t i v i t y F o r example,  i f a unit cost  i t  information to for  withdrawal  96 or  storage  of  estimated, the  dual  flow  then  values  i n each reach these  derived  reduction  of  potential  damages due  terms unit  of  the  Q^,  peak  reduction  6.5  GRAPHICS has  been  form.  The  could  i n the  the  length  by  generated  is  presentation  of  of  time  Such  a  per  to  unit the  estimated to the  in  cost  per  modification  at  effective.  a  type  in  computer  which  t o h a n d l e and  a schematic could graphics  much of  iterative  of of  easily  5,  be  display.  the  in  the  assimilate benefits  of  cycle.  information itself  to  two-dimensional  drainage  system.  g e n e r a t e d by Colour  enhance t h e  and  tabular  design  lends  simple the  information  substantial,  presented  readily  would s i g n i f i c a n t l y envisaged.  is  Chapter  form of a s e t  on  amount of  i f i t was  required  figures  presentation  display  be  i s cost  the  slow down the  i n the  superimposed  capabilities  cost  i f design  would c o u n t e r a c t  the  plots  resolution  be  Alternatively, if  analysis  designer  t h i s model and  shown  the  compared  network  could  applied directly  could  be  demonstrated,  such p r i n t e d output  the  that  from p o s t - o p t i m a l i t y overwhelm  As  flooding  or  INTERFACE  would  using  be  peak.  to determine  p a r t i c u l a r point  derived  to  then  could  known  here to o b t a i n  system outflow  any  As  values  were  and  a  high zoom  r e a d a b i l i t y of  REFERENCES B e s a n t , C. B. C o m p u t e r - A i d e d D e s i g n and M a n u f a c t u r e . Harwood S e r i e s i n E n g i n e e r i n g S c i e n c e , 1983. Chow, Ven Te, editor. Handbook M c G r a w - H i l l Book Company, New  of Applied York, 1964.  Ellis  Hydrology.  Curtis, D. C. and R. H. McCuen. Design Efficiency of Stormwater Detention Basins. Journal of t h e Water R e s o u r c e s P l a n n i n g and Management Division, American S o c i e t y o f C i v i l E n g i n e e r s , Volume 103, Number WR1, May 1977. Dendrou, S. A. and J . W. D e l l e u r . Watershed-Wide P l a n n i n g of D e t e n t i o n B a s i n s . P r o c e e d i n g s of t h e C o n f e r e n c e on Stormwater Detention F a c i l i t i e s . New E n g l a n d C o l l e g e , New H a m p s h i r e , August 1982. Duru,  J . 0. O n - S i t e D e t e n t i o n : A Stormwater Management o r Mismanagement Technique? Second International Conference on U r b a n Storm D r a i n a g e . Urbana, I l l i n o i s , J u n e 1981.  Encarnacao, J . and E . G. S c h l e c h t e n d a h l . Design: Fundamentals and System S p r i n g e - V e r l a g , New Y o r k , 1983. Fok,  Computer Aided Architectures.  A. T. K., P e r k s , A. R. and L . A. P a t a k y . Application of Computer M o d e l s on an U r b a n D r a i n a g e R e l i e f S t u d y . P r o c e e d i n g s of t h e International Symposium on Urban Storm Runoff. U n i v e r s i t y of Kentucky, J u l y 1979.  Huber, W. C , Heaney, J . P., N i x , S. J . , D i c k i n s o n , R. E . and D. J . Polmann. S t o r m Water Management M o d e l U s e r ' s Manual Version III (Final Draft). United States E n v i r o n m e n t a l P r o t e c t i o n A g e n c y , November 1981. K o u s s i s , A. D. Theoretical E s t i m a t i o n s of Flood Routing Parameters. Journal of the Hydraulics Division, A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , Volume 104, Number HY1, J a n u a r y 1978. Nash, J . E . A Note on t h e Muskingum Flood-Routing Method. Journal of G e o p h y s i c a l R e s e a r c h , Volume 64, Number 9, 1959. Perks, Alan R. A Review o f U r b a n R u n o f f M o d e l s . Modern Concepts i n Urban Drainage; Conference Proceedings Number 5, T o r o n t o , March 1977. Ponce, V. M. Journal  Simplified Muskingum of the H y d r a u l i c s D i v i s i o n , 97  Routing American  Equation. S o c i e t y of  98 Civil  E n g i n e e r s , Volume  105, Number HY1,  January  1979.  Solow, D. Linear Programming: An I n t r o d u c t i o n t o F i n i t e Improvement A l g o r i t h m s . Elsevier Science Publishing Company, I n c . , New Y o r k , 1984. S t e p h e n s o n , D. P r e s s , New  Stormwater H y d r o l o g y and D r a i n a g e . Y o r k , 1981.  Elsevier  V i e s s m a n , W., Knapp, J . W., L e w i s , G. L. and T. E . Harbaugh. Introduction to Hydrology (3rd e d i t i o n ) . H a r p e r and Row, New Y o r k , 1977. Walpole, R. E . Introduction to S t a t i s t i c s (3rd e d i t i o n ) . M a c M i l l a n P u b l i s h i n g Company I n c . , New Y o r k , 1982. Weinmann, P. E . and E. M. L a u r e n s o n . Approximate Flood R o u t i n g Methods: A Review. J o u r n a l of the Hydraulics Division, American Socie.t.y, of C i v i l E n g i n e e r s , Volume. 105, Number HY12, December 1979. Yen,  B. C. and A. S. Sevuk. D e s i g n o f Storm Sewer N e t w o r k s . Journal of the Environmental Engineering Division, A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , Volume 101, Number EE4, A u g u s t 1975.  APPENDIX A: PROGRAM 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  LISTING  C C C C  *** THIS PROGRAM WILL EVALUATE MUSKINGUM CHANNEL ROUTING *** *** FOR A BRANCHED NETWORK, USING LINEAR PROGRAMMING ***  C C C  *** READ AND PRINT THE TITLE ***  C C C  *** READ THE PARAMETERS ***  IMPLICIT REAL*8(A - H,0 - Z) REAL*8 K(50),X(50),DT,BETA(50),GAMMA(50),PHI(50),PSI(50), 1 Q(50,50),TABLO(640,640),BBOBJ(640),UBOBJ(640), 2 RHS,RHS1(640),RHS2,BBRHS(640),UBRHS(640) INTEGER NVIN(640),NVOUT(640),IFIRST(50),ILAST(50),ILAST2(50) DIMENSION TITLE(8)  1.0 READ (5,20,END=810) TITLE 20 FORMAT (8A8) WRITE (6,30) TITLE 30 FORMAT ("1 ',///,8A8/)  READ (5,40) IF1, IF2, IF3 READ (5,40) NSTA, NJUNCT, NORDS 40 FORMAT (414) NCHAN = NSTA-NJUNCT-1 DO 50 1=1, 50 IFIRST(I) = 0 ILAST(I) = 0 ILAST2(I) = 0 50 CONTINUE IF (NJUNCT .EQ. 0) GOTO 80 NJP1 = NJUNCT + 1 DO 60 1=1,NJP1 READ (5,40) ITEMP1,ITEMP2, ITEMP3 IFIRST(ITEMP1) = 1 ILAST(ITEMP3) = 1 ILAST2(ITEMP3) = ITEMP2 IF (IF1 .EQ. 1) WRITE (6,70) I,ITEMP1,ITEMP2,ITEMP2,ITEMP3 60 CONTINUE 70 FORMAT (* BRANCH NO.',12,': STATION ',12,' TO STATION ',12, 1 ' JUNCTION: ',12,',',12) 80 IFIRST(1) = 1 READ (5,90) DT 90 FORMAT (3F9.3) DO 110 J=2, NSTA IF (IFIRST(J) .EQ. l ) GO TO 110 I = J-1 READ (5,90) K ( l ) , X ( l ) DENOM = DT-2.*K(I)*X(I) IF (DENOM .GT. 0.10) GOTO 100 WRITE (6,120) I , DENOM, K ( l ) , X ( I ) , DT STOP 100 BETA(I) = (DT+2.*K(I)*(1-X(I)))/DENOM GAMMA(I) = (DT-2.*K(I)*(1-X(I)))/DENOM PHI(I) = (DT+2.*K(I)*X(I))/DENOM P S I ( I ) = GAMMA(I)-BETA(I)*PHI(I) 110 CONTINUE 120 FORMAT ('0THE DENOMINATOR IN REACH',12,' = ',F9.3,'.*/, 1 ' DENOMINATOR MUST BE SIGNIFICANTLY GREATER THAN ZERO.'/ 99  100 Appendix A ( c o n t i n u e d )  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 1 05 106 107 108 109 1 10 11 1 1 12 1 13 1 14 115 116  . 2 ' CHOOSE NEW V A L U E S OF K, X, OR D T ' / , 3 * SUCH T H A T 2KX « DT. ' , / 4 ' AT PRESENT, K=*,F6.3,' X=',F6.3,* DT=',F6.3,'. NORM1 = NORDS-1 NVARS = NCHAN*NORM1+1 NCONST = (NCHAN+1)*NORM1 NVM1 = NVARS - 1 C G C  ***  130  140  150 160 170 C C C  ***  190 200 ***  210 C C C  THE  PARAMETERS  ***  I F ( I F 1 .NE. 1) GOTO 180 WRITE ( 6 , 1 3 0 ) NSTA, NCHAN, D T FORMAT ('0THERE A R E ' , 1 2 , ' S T A T I O N S AND ' , 1 2 , 1 • ' CHANNEL R E A C H E S . THE ROUTING PERIOD I S ' , F 7 . 3 ) WRITE (6,140) FORMAT('0REACH',T11,'K',T18,'X',T26,'BETA',T38,'GAMMA*, 1 T50,'PHI',T63,'PSI') DO 150 J = 2 , NSTA I F ( I F I R S T ( J ) .EQ. 1) GOTO 150 I = J-1 WRITE ( 6 , 1 6 0 ) I , K ( I ) , X ( l ) , B E T A ( I ) , G A M M A ( I ) , P H I ( I ) , P S I CONTINUE FORMAT ( 1 4 , 1 X , 2 F 7 . 3 , 3 X , 4 G 1 2 . 5 ) WRITE ( 6 , 1 7 0 ) NORDS, NCONST, NVARS FORMAT ('0THERE A R E ' , 1 3 , ' HYDROGRAPH O R D I N A T E S , ' , 1 1 4 , ' CONSTRAINTS, AND',14,' V A R I A B L E S . ' )  180  C C C  ECHO  ***  ZERO  THE TABLEAU * * *  NVP1 = NVARS + 1 NCP1 = NCONST + 1 DO 200 J = 1 , NVP1 DO 190 1=1, NCP1 T A B L O ( l , J ) = 0.0D0 CONTINUE CONTINUE FORMULATE  T H E MINIMAX  OBJECTIVE  T A B L O ( 1 , 1 ) = 1. DO 210 J = 1 , NORM1 TABLO(1,NVARS-NORM1+J) CONTINUE READ  THE F I R S T ORDINATE  220 ***  READ DO  230  THE  INFLOW  (MINIMIZE  Y)  1.E-10  OF  E A C H DOWNSTREAM  DO  C C C  220 1=2, NSTA I F ( I F I R S T ( I ) .EQ. READ ( 5 , 9 0 ) Q ( I , 1 ) CONTINUE  =  FUNCTION  1) GOTO  HYDROGRAPHS  2 4 0 1=1, NSTA I F ( I F I R S T ( I ) .NE. 1) GOTO DO 2 3 0 N=1, NORDS READ ( 5 , 9 0 ) Q ( I , N ) CONTINUE  220  ***  240  HYDROGRAPH  101 Appendix A (continued)  1 17 1 18 119 1 20 121 122 1 23 124 125 126 1 27 128 1 29 130 131 1 32 1 33 134 135 136 1 37 138 1 39 1 40 141 1 42 143 1 44 1 45 1 46 147 1 48 149 150 151 152 1 53 1 54 1 55 156 157 1 58 159 160 161 162 163 1 64 165 166 167 168 169 1 70 • 171 172 173 174  240 C C C  * * * R O U T E T H E FLOWS  250 260 270  280 290  300 310 320 330 C C C  CONTINUE AND P R I N T T H E HYDROGRAPHS * * *  I F ( I F 2 . N E . 1) GOTO 2 7 0 W R I T E ( 6 , 2 5 0 ) ( N , N= 1 , NORDS) FORMAT ( ' 0 S T A . * , 5 0 ( I 5 , 2 X ) ) W R I T E ( 6 , 2 6 0 ) ( Q ( 1 , N ) , N=1,NORDS) FORMAT (' 1: ' , 5 0 F 7 . 3 ) DO 3 2 0 1 = 2 , N S T A I F ( I F I R S T ( I ) .EQ. 1) GOTO 3 1 0 IM1 = 1-1 IF ( I L A S T ( I M I ) .EQ. 1) GOTO 2 9 0 DENOM = D T + 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) DO 2 8 0 N=2,NORDS Q(I,N)=((DT+2.*K(IM1)*X(IM1))*Q(IM1,N-1) 1 +(DT-2.*K(IM1)*X(IM1))*Q(IM1,N) 2 -(DT-2.*K(IM1)*(1.-X(IM1)))*Q(I,N-1)) 3 /DENOM CONTINUE GOTO 3 1 0 DENOM = D T + 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) DO 3 0 0 N=2, NORDS ITEMP = I L A S T 2 ( I M 1 ) Q(I,N)=((DT+2.*K(IM1)*X(IM1))*(Q(IM1,N-1)+Q(ITEMP,N-1 1 +(DT-2.*K(IM1)*X(IM1))*(Q(IM1,N)+Q(ITEMP,N)) 2 -(DT-2.*K(IM1)*(1.-X(IM1)))*Q(I,N-1)) 3 /DENOM CONTINUE I F ( I F 2 .EQ. 1) W R I T E ( 6 , 3 3 0 ) I , ( Q ( l , N ) , N = 1 , N O R D S ) CONTINUE FORMAT (' ' , 1 2 , ' : ',50F7.3)  * * * FORMULATE T H E ROUTING  340 350  360 370  CONSTRAINTS * * *  NROW = 2 DO 4 1 0 I = 2 , N S T A I F ( I F I R S T ( I ) .EQ. 1) GOTO 4 1 0 IM1 = 1-1 IF ( I L A S T ( I M I ) .EQ. 1) GOTO 3 4 0 RHS = PHI(IM1)*Q(IM1,1)-GAMMA(IM1)*Q(I,1) GOTO 3 5 0 ITEMP = I L A S T 2 ( I M 1 ) RHS = PHI(IM1)*(Q(ITEMP,1)+Q(IM1,1))-GAMMA(IM1)*Q(I,1) DO 4 0 0 N=2, NORDS RHS2 = R H S * ( - P H I ( I M 1 ) ) * * ( N - 2 ) R H S 1 ( N R O W ) = RHS2 IF ( I F I R S T ( I M I ) .EQ. 1) R H S 2 = R H S 2 + Q ( I M 1 , N ) T A B L O ( N R O W , N V P 1 ) = -RHS2 TABLO(NROW,NROW) = - B E T A ( I M 1 ) I F ( N . E Q . 2 ) GOTO 3 7 0 NM2 = N-2 DO 3 6 0 L=1,NM2 • TABLO(NROW,NROW-L) = - P S I ( I M 1 ) * ( - P H I ( I M 1 ) ) * * ( L - 1 ) CONTINUE IF ( I F I R S T ( I M I ) .EQ. 1) GOTO 3 9 0 N C O L = NROW-NORM1 T A B L O ( N R O W , N C O L ) = 1.  102 Appendix A  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  380 390 400 410 C C C  (continued)  I F ( I L A S T ( I M I ) . N E . 1) GOTO 3 9 0 ITEMP = I L A S T 2 ( I M 1 ) NCOL2 = (ITEMP-1)*NORM1+N DO 3 8 0 J = 1 , I T E M P I F ( I F I R S T ( J ) .EQ. 1) NCOL2 = CONTINUE T A B L O ( N R O W , N C O L 2 ) = 1. NROW = NROW+1 CONTINUE CONTINUE  * * * ADD T H E MINIMAX  NCOL2-NORM1  CONSTRAINTS * * *  DO 4 2 0 N = 2 NORDS TABLO(NCHAN*NORM1+N,1) = - 1 . TABLO(NCHAN*NORM1+N,(NCHAN-1)*NORM1+N) 420 CONTINUE r  C C C  * * * PRINT  THE INPUT TABLO  = 1.  ***  I F ( I F 3 .NE. 1) GOTO 4 60 WRITE ( 6 , 4 3 0 ) 4 3 0 FORMAT ('OTHE I N P U T T A B L O : ' / ) DO 4 4 0 1=1,NCP1 WRITE ( 6 , 4 5 0 ) (TABLO(I,J),J=1,NVP1) 440 CONTINUE 4 5 0 FORMAT (' ' , 1 0 0 ( G 8 . 1 ) ) C C C  *** CALL 460 CALL 1  C C C  T H E SUBROUTINE VERSION  OF L I P * * *  LIPSUB(TABLO, 6 4 0 , NCONST, NVARS, N V I N , NVOUT, B B O B J , U B O B J , BBRHS,  * * * PRINT  0, 0, 0, 1, 1.D-6, UBRHS, & 7 9 0 )  THE OUTPUT TABLO * * *  I F ( I F 3 .NE. 1) GOTO 5 0 0 WRITE ( 6 , 4 7 0 ) 4 7 0 FORMAT ('OTHE O U T P U T T A B L O : ' / ) DO 4 8 0 1=1, NCP1 WRITE ( 6 , 4 9 0 ) (TABLO(I,J),J=1,NVP1) 480 CONTINUE 4 9 0 FORMAT (* ' , 1 0 0 ( G 8 . 1 ) ) C C C  * * * PRINT  T H E OPTIMUM  VALUE  OF T H E O B J E C T I V E  500 OPTIM = -TABLO(1,NVP1) WRITE ( 6 , 5 1 0 ) OPTIM 5 1 0 FORMAT ('0PEAK OUTFLOW =', C C C  * * * PRINT  FUNCTION * * *  1PG15.7)  THE PRIMAL SOLUTION * * *  WRITE ( 6 , 5 2 0 ) 5 2 0 FORMAT CO PRIMAL SOLUTION:'//, ' N C P V = NCONST + NVARS KOUNT1 =2 KOUNT2 = 2 DO 5 6 0 1=2, NVARS DO 5 3 0 J = 2 , NCP1  V A R I A B L E ' , 7X, '  VALUE')  103  Appendix  233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 2 49 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 27 3 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  A  (continued)  I F ( N V I N ( J ) .EQ. I ) GOTO 540 CONTINUE DIFF = TABLO(J,NVP1)-Q(KOUNT1,KOUNT2) ALLOW = 0 . 0 0 1 * Q ( K O U N T 1 , K O U N T 2 ) I F ( D I F F .GT. ALLOW .OR. D I F F . L T . -ALLOW) WRITE ( 6 , 5 7 0 ) 1 KOUNT1,KOUNT2,TABLO(J,NVP1),DIFF I F (KOUNT1 . L T . N S T A ) GOTO 550 I F ( D I F F . L E . ALLOW .AND. D I F F .GE. -ALLOW) W R I T E ( 6 , 5 8 0 ) 1 KOUNT1,KOUNT2,TABLO(J,NVP1) 550 KOUNT2 = KOUNT2 + 1 I F (KOUNT2 . L E . NORDS) GOTO 560 KOUNT1 = KOUNT1+1 I F ( I F I R S T ( K O U N T 1 ) .EQ. 1) KOUNT1 = KOUNT1+1 KOUNT2 = 2 560 CONTINUE 570 FORMAT(1X,' Q(',12,',',12,')',1X,F12.3,2X, 1 ' * * D I F F E R S FROM R O U T E D V A L U E BY ' , F 1 2 . 3 , ' **') 580 F O R M A T ( I X , ' Q(',I 2,',',I 2 , ' ) ' ,1X,F12.3)  530 540  C C C C  * * * P R I N T T H E R E D U C E D C O S T S , I F ANY * * * *** (Only i f an e r r o r o c c u r s ) *** DO  590 600 610  620 630 C C C  5 9 0 1=1, NVARS I F ( N V O U T ( I ) . L E . NVARS) GOTO 6 0 0 CONTINUE GOTO 6 4 0 WRITE ( 6 , 6 1 0 ) FORMAT ('0REDUCED C O S T S : ' / ' V A R I A B L E ' , 8 X , ' V A L U E ' ) DO 6 2 0 1 = 1 , NVARS I F ( N V O U T ( I ) . L E . NVARS) W R I T E ( 6 , 6 3 0 ) N V O U T ( I ) , T A B L O ( 1 , I ) CONTINUE FORMAT ( 1 X , I 5 , 7 X , 1 P G 1 2 . 5 )  * * * PRINT T H E DUAL SOLUTION  AND  RHS  RANGING  ***  640 WRITE ( 6 , 6 5 0 ) 6 5 0 FORMAT ('0 SENSITIVITY ANALYSIS:'// 1 T 3 2 , ' D U A L ' , T 5 1 , ' R H S RANGING BOUNDS'/' CONST.', 2 T13,'ORDINATE',T31,'VALUE',T41,'FLOW',T52,'LOWER', 3 T64,'UPPER'/) NCMN1 = NCONST-NORM1 KOUNT1 = 1 KOUNT2 = 2 DO 7 4 0 J=1,NCMN1 DO 6 6 0 1=1, NVARS INDEX = NVOUT(I)-NVARS I F ( I N D E X .EQ. J ) GOTO 6 8 0 660 CONTINUE WRITE ( 6 , 6 7 0 ) 670 FORMAT (' I N D E X ERROR') STOP 680 BBR = - U B R H S ( J ) - R H S 1 ( J + 1 ) UBR = - B B R H S ( J ) - R H S 1 ( J + 1 ) I F ( I F I R S T ( K 0 U N T 1 ) .EQ. 1) GOTO 6 9 0 I F ( I L A S T ( K 0 U N T 1 ) .EQ. l ) GOTO 7 0 0 BBR = B B R + Q ( K O U N T 1 , K O U N T 2 ) UBR = U B R + Q ( K O U N T 1 , K O U N T 2 ) I F ( B B R . L E . 1.E-5 .AND. BBR .GE. - 1 . E - 5 ) BBR = 0.00  1 04 Appendix A  291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322  (continued)  I F (UBR . L E . 1.E-5 .AND. UBR .GE. - 1 . E - 5 ) UBR = 0.00 WRITE(6,750) INDEX,KOUNT1,KOUNT2,TABLO(1,1), 1 Q(KOUNT1,KOUNT2),BBR,UBR GOTO 710 700 ITEMP = ILAST2(KOUNT1) QS = Q ( I T E M P , K O U N T 2 ) + Q ( K O U N T 1 , K O U N T 2 ) BBR = BBR+QS UBR = UBR+QS I F (BBR .LE." 1 .E-5 .AND. BBR .GE. - 1 . E - 5 ) BBR = 0.00 I F (UBR . L E . 1.E-5 .AND. UBR .GE. - 1 . E - 5 ) UBR = 0.00 W R I T E ( 6 , 7 6 0 ) INDEX,ITEMP,KOUNT2,KOUNT1,KOUNT2, 1 TABLO(1,I),QS,BBR,UBR 710 I F (KOUNT2 .EQ. NORDS) GOTO 720 KOUNT2 = KOUNT2+1 GOTO 740 720 KOUNT2 = 2 KOUNT1 = KOUNT1+1 WRITE(6,730) 730 FORMATC ') I F ( I F I R S T ( K O U N T 1 + 1 ) .EQ. 1) KOUNT1 = KOUNT1+1 740 C O N T I N U E 750 FORMAT ( I X , 1 4 , 5 X , ' Q ( ' ,I 2,' , , 1 2 , ' ) * , T 2 8 , F 9 . 4 , T 3 8 , 3 G 1 2 . 5 ) 760 FORMAT ( I X , 1 4 , ' Q ( ' , 12 , ' , ' , I 2 , ' ) + Q ( ' , 1 2 , ' , ' , I 2 , ' ) ' , T 2 8 , 1 F9.4,T38,3G12.5) STOP 770 W R I T E ( 6 , 7 8 0 ) IROW, J C O L , C O E F F 780 FORMAT (' DATA I N C O R R E C T * , 2 1 4 , G 1 2 . 5 ) STOP 790 W R I T E ( 6 , 8 0 0 ) 800 FORMAT (' NO OPTIMUM') 810 S T O P END 690  1  105 APPENDIX B - THE SUBROUTINE VERSION OF L I P PURPOSE To s o l v e l i n e a r programming p r o b l e m s u s i n g the s e l f - c o n t a i n e d program i n *LIP.  t h e same method a s  TYPE OF ROUTINE A s e t o f FORTRAN IV  subroutines.  AVAILABILITY •LIBRARY HOW  TO  USE  CALL LIPSUB(TB,NDIMTB, M ,N, NE,MAX,NOBJ,NRHS,TOL,NCHK,NCHK1, BBOBJ,UBOBJ,BBRHS,UBRHS , &nn) where: TB  i s a REAL*8, t w o - d i m e n s i o n a l a r r a y , dimensioned at least M+1 by N+1. On e n t r y , i t c o n t a i n s t h e t a b l e a u of the problem. The first row contains the coefficients o f t h e o b j e c t i v e f u n c t i o n . The l a s t NE rows contain the c o e f f i c i e n t s of the equality constraints ( i f there are any). A l l intermediate rows contain < type constraints only (> type constraints must be c h a n g e d t o < c o n s t r a i n t s by multiplying through by - 1 ) . The N+1th column contains the right-hand sides of t h e c o n s t r a i n t s ( s t a r t i n g a t t h e s e c o n d e l e m e n t of t h e c o l u m n ) . For a p a r t i c u l a r problem the t a b l e a u w i l l look l i k e : 1,2,... 1  N  N+1  OBJECTIVE FUNCTION  2 < TYPE  CONSTRAINTS R H S  M M+1  NE  EQUALITY  CONSTRAINTS  On o u t p u t , v a r i o u s e l e m e n t s o f TB a r e u s e d t o s t o r e t h e optimum v a l u e of the o b j e c t i v e function, the v a l u e of the p r i m a l s o l u t i o n v a r i a b l e s , the value of the reduced costs, and v a l u e of t h e d u a l s o l u t i o n v a r i a b l e s ( i f NE=0). See t h e d e s c r i p t i o n of NCHK and NCHK1 on t h e f o l l o w i n g page f o r d e t a i l s . The value of t h e optimum i s l o c a t e d i n T B ( 1 , N+1).  106  NDIMTB  i s the f i r s t  M  i s t h e number o f c o n s t r a i n t s .  N  i s an INTEGER v a r i a b l e . On e n t r y , i t c o n t a i n s t h e number of v a r i a b l e s . On exit, i t i s s e t to the number of v a r i a b l e s minus t h e number o f e q u a l i t y constraints.  NE  i s t h e number o f e q u a l i t y  MAX  =1 i f t h e o b j e c t i v e  function  i s t o be m a x i m i z e d .  =0 i f t h e o b j e c t i v e  function  i s t o be m i n i m i z e d .  NOBJ  TB.  M+1<NDIMTB.  constraints.  =1 i f o b j e c t i v e f u n c t i o n r a n g i n g i s t o be a t t e m p t e d . The o b j e c t i v e f u n c t i o n c a n be r a n g e d o n l y i f NE=0. =0  NRHS  dimension of the array  otherwise.  =1 i f r i g h t - h a n d Right-hand side constraints. =0  side ranging i s to r a n g i n g i s done o n l y  be attempted. for inequality  otherwise.  TOL  i s a REAL*8 variable. I t should be s e t t o t h e tolerance f o r t h e p r o b l e m . Numbers s m a l l e r t h a n TOL i n a b s o l u t e v a l u e w i l l be c o n s i d e r e d z e r o . I f TOL i s s e t t o 0.D0 by t h e u s e r , i t w i l l be r e s e t by t h e s u b r o u t i n e t o 1.D-6.  NCHK  i s an INTEGER o n e - d i m e n s i o n a l a r r a y , d i m e n s i o n e d a t l e a s t M+1. On e x i t , NCHK(I), 1=2, ...,M+1 contains the index of a v a r i a b l e which i s i n c l u d e d i n the p r i m a l s o l u t i o n . The v a r i a b l e ' s v a l u e i s l o c a t e d i n TB(I,N+1) where t h e v a l u e N h a s been r e t u r n e d by t h e subroutine.  NCHK1  i s an INTEGER o n e - d i m e n s i o n a l a r r a y , d i m e n s i o n e d a t least N (number of v a r i a b l e s ) . On exit, if NCHK1(I)<number o f v a r i a b l e s , t h e n T B ( 1 , I ) g i v e s t h e reduced cost of the v a r i a b l e w i t h index NCHKI(I). I f NCHK1(I) > number of v a r i a b l e s , then TB(1,I) c o n t a i n s the value of the dual solution variable w i t h i n d e x , NCHK1(I) - number of v a r i a b l e s . I v a r i e s f r o m 1 t o N. The d u a l s o l u t i o n i s c a l c u l a t e d o n l y i f NE=0.  BBOBJ UBOBJ  a r e REAL*8, o n e - d i m e n s i o n a l a r r a y s , e a c h d i m e n s i o n e d at least N. On exit, i f NOBJ=1 and NE=0, t h e y contain the lower and upper bounds on the coefficients o f t h e o b j e c t i v e f u n c t i o n . I f NOBJ=0 t h e y a r e n o t used and so may be d i m e n s i o n e d t o 1.  1 07 BBRHS UBRHS  nn  a r e REAL*8, o n e - d i m e n s i o n a l a r r a y s , e a c h d i m e n s i o n e d a t l e a s t M. On e x i t , i f NRHS=1 they contain the lower and upper bounds on t h e r i g h t - h a n d s i d e s . Bounds a r e g i v e n o n l y for inequality constraints. Bounds s t a r t i n t h e s e c o n d l o c a t i o n o f t h e a r r a y . I f NRHS=0 they a r e n o t u s e d and s o may be d i m e n s i o n e d to 1 . i s the program function  statement number i n the user's t o which control i s transferred i s unbounded o r i n f e a s i b l e .  calling i f the  RESTRICTIONS 1. I f NET*0,  the dual  2. I f NET*0,  objective  3. R i g h t - h a n d side constraints.  solution ranging ranging  i s not c a l c u l a t e d . i s n o t done. is  done  only  for  inequality  

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