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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 IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n 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 c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA October 1985 © Timothy P. B a t t l e , 1985 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the The U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permi s s i o n . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 1985 A b s t r a c t A computer-aided design scheme f o r both man-made and n a t u r a l runoff systems i s presented. The model uses l i n e a r programming to s o l v e Muskingum r o u t i n g equations through a drainage system, and p r o v i d e s design i n f o r m a t i o n through p o s t - o p t i m a l i t y ( s e n s i t i v i t y ) a n a l y s i s . With the o b j e c t i v e of minimizing the peak outflow from the system and using hydrograph o r d i n a t e s as the d e c i s i o n v a r i a b l e s , the output of the l i n e a r programming a n a l y s i s shows the extent that each flow o r d i n a t e at every node in the network i n f l u e n c e s the peak flow at some downstream l o c a t i o n . T h i s i n f o r m a t i o n can a i d the user i n speeding up the design process to a r r i v e at an e f f i c i e n t design - i . e . , one which e i t h e r minimizes c o n s t r u c t i o n c o s t s or reduces the p o t e n t i a l r i s k of f l o o d damage. Table of Contents A b s t r a c t i i L i s t of Tab l e s v L i s t of F i g u r e s v i No t a t i o n v i i Acknowledgement ix 1. INTRODUCTION 1 1.1 Design Refinement 3 1.2 E x i s t i n g Models ..4 1.3 Computer-Aided Design 6 2. PROGRAM FORMULATION 9 2.1 Muskingum Channel Routing 9 2.2 L i n e a r Programming and S e n s i t i v i t y A n a l y s i s 13 2.3 S o l v i n g the Muskingum Equation u s i n g L i n e a r Programming 14 2.3.1 O b j e c t i v e F u n c t i o n Formulation 15 2.3.2 Minimax C o n s t r a i n t s 16 2.3.3 S i n g l e Reach 17 2.3.4 S e r i e s of Reaches 20 2.3.5 Branching 23 3. PROGRAM DESCRIPTION 28 3.1 Input 28 3.2 Formulating the Tableau 31 3.3 L i n e a r Programming Subroutine 32 3.4 Output 34 4. ALTERNATE FORMULATION 35 4.1 S i n g l e Reach 35 4.2 Two Reaches in S e r i e s 36 4.3 S e r i e s of Reaches 40 4.4 Program D e s c r i p t i o n 44 4 . 5 Comment 46 5. NUMERICAL EXAMPLES 47 5.1 Case 1: S i n g l e Reach 48 5.2 Case 2: Three Reaches i n S e r i e s 62 5.3 Case 3: Branched Network 76 5.4 Case 4: Complex Network 84 6. DISCUSSION OF RESULTS 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 91 6.2 Dual S o l u t i o n Vector 92 6.3 Right Hand Side Ranging 94 6.4 Cost A n a l y s i s 95 6.5 Graphics I n t e r f a c e ......96 REFERENCES 97 APPENDIX A: PROGRAM LISTING 99 APPENDIX B: LIPSUB DESCRIPTION 105 i v L i s t of Tables I ROUTING CONSTRAINTS FOR A SINGLE REACH 21 II LP TABLEAU FOR A SINGLE REACH 22 III LP TABLEAU FOR 3 CHANNELS IN SERIES 24 IV LP TABLEAU FOR A NETWORK WITH ONE BRANCH 27 V SAMPLE INPUT FILE 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 v L i s t of F i g u r e s 1 . EXAMPLE 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 80 15. STATION 3: COMPARISON OF EXAMPLES 3.1 & 3.2 82 16. STATION 7: COMPARISON OF EXAMPLES 3.1 & 3.3 83 17. STATION 10: COMPARISON OF EXAMPLES 3.1 & 3.4 85 1 8 . EXAMPLE 4.1 86 v i N o t a t i o n c k i n e m a t i c wave s p e e d . cln a r e c u r s i v e s e r i e s e x p r e s s i n g a c o e f f i c i e n t f o r t h e n 1"* 1 o r d i n a t e o f t h e / r e a c h . / t h e i d e n t i f i c a t i o n number o f a r e a c h , o r o f t h e n o d e i m m e d i a t e l y u p s t r e a m . I t h e number o f n o d e s i n t h e d r a i n a g e s y s t e m . s t o r a g e c o n s t a n t ; t h e r a t i o o f s t o r a g e t o d i s c h a r g e f o r ' a c h a n n e l r e a c h o r r e s e r v o i r . L . t h e l e n g t h o f a c h a n n e l r e a c h . n t h e i d e n t i f i c a t i o n number o f a h y d r o g r a p h o r d i n a t e . N t h e t o t a l number o f o r d i n a t e s i n a h y d r o g r a p h . p(a,b) a c o e f f i c i e n t o f t h e sum o f t h e p r o d u c t s o f /3 a n d 7, w h e r e 0 o c c u r s a t i m e s a n d 7 o c c u r s b t i m e s . q t h e d i s c h a r g e p e r u n i t c h a n n e l w i d t h . Qln t h e d i s c h a r g e a t t h e o r d i n a t e o f t h e h y d r o g r a p h a t node / . v i i the peak outflow at the downstream p o i n t of a drainage system. the storage in a channel reach or r e s e r v o i r , the slope of the channel bed. the r o u t i n g p e r i o d (the time i n t e r v a l between su c c e s s i v e o r d i n a t e s ) . a dummy v a r i a b l e used i n LP minimax formulat i o n . t+ 2k{ t-2kx f o r reach /. t~2k(1-x) t -2kx f o r reach /. t +2kx t -2kx f o r reach /. v i i i Acknowledgement I would l i k e to express my a p p r e c i a t i o n to the N a t u r a l Sciences and E n g i n e e r i n g Research C o u n c i l f o r t h e i r funding support f o r t h i s r e a s e a r c h . I would e s p e c i a l l y l i k e to express my a p p r e c i a t i o n to Dr. W. F. C a s e l t o n f o r h i s encouragement and guidance d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . ix Chapter 1 INTRODUCTION In recent years, the need f o r more thorough design of drainage and f l o o d c o n t r o l systems has been recognized as i n c r e a s i n g l y important. Higher c o n s t r u c t i o n c o s t s , i n c r e a s e d p o t e n t i a l f o r f l o o d damage, and l a r g e r land areas a f f e c t e d by developments have demanded that drainage aspects be given g r e a t e r c o n s i d e r a t i o n . To a l a r g e extent, the i n c r e a s e d a v a i l a b i l i t y of computers has allowed f o r d e t a i l e d d e s i gns to be produced more r e a d i l y , and has l e d to more complex watershed-wide p l a n n i n g approach (Yen & Suvak, 1975). The usual approach i n d e s i g n i n g a drainage system i s to : 1. s e l e c t a l a y o u t ; 2. determine design flows e n t e r i n g each element of the system; 3. estimate the channel parameters; 4. c a l c u l a t e the flows throughout the system; and 5. modify the channel parameters and repeat steps 3 and 4 u n t i l a s a t i s f a c t o r y design i s achieved. Step 1 depends e n t i r e l y on s i t e c o n d i t i o n s , and of course does not apply to n a t u r a l systems. For man-made systems, the lay o u t depends l a r g e l y on t o p o g r a p h i c a l f e a t u r e s and the arrangement of the development which the system serves. Step 2 i s u s u a l l y performed based on 1 2 p r e c i p i t a t i o n records or flow records upstream of the system. The design c r i t e r i a , such as the frequency of the event f o r which the system i s to be designed, must be chosen. The t h i r d step i s based l a r g e l y on judgement. C e r t a i n design v a r i a b l e s such as g r a d i e n t have a dependence on s i t e c o n d i t i o n s , but most parameters such as channel width and type are l e f t to the d e s i g n e r ' s d i s c r e t i o n , p r o v i d e d they adequately convey the flows from the upstream s e c t i o n . Many r o u t i n g procedures are a v a i l a b l e for p r e d i c t i n g , the passage of a f l o o d through a r e s e r v o i r or channel reach as r e q u i r e d f o r step 4. These methods can be c l a s s i f i e d i n two major c a t e g o r i e s , h y d r o l o g i c r o u t i n g and h y d r a u l i c r o u t i n g (Viessman et a l . , ' 1977). H y d r o l o g i c r o u t i n g uses both the c o n t i n u i t y equation and a s t o r a g e - d i s c h a r g e r e l a t i o n s h i p f o r the reach. H y d r o l o g i c r o u t i n g methods i n c l u d e Muskingum and G r a p h i c a l I n t e g r a t i o n methods for channels, and Storage I n d i c a t i o n (or M o d i f i e d Pulse) and l i n e a r r e s e r v o i r methods f o r r e s e r v o i r s . H y d r a u l i c methods use the c o n t i n u i t y equation combined with an equation of motion, o f t e n the momentum equation. These types of r o u t i n g methods use p a r t i a l d i f f e r e n t i a l equations to d e s c r i b e the usteady flow i n open channels. 3 .1 . 1 DESIGN REFINEMENT The f i n a l step i n the d e s i g n process, where the designer attempts to produce a more e f f i c i e n t d e s i g n , can be repeated as many times as deemed necessary. The o b j e c t i v e of t h i s step i s to a r r i v e a t a design capable of conveying the design flows at a minimum c o s t , or one r e s u l t i n g i n the l e a s t f l o o d damage f o r a given design budget. Since c o n s t r u c t i o n c o s t s are g e n e r a l l y r e l a t e d to the s i z e of the f a c i l i t i e s , reducing the maximum flows which the elements of the- system must convey w i l l r e s u l t i n a decrease i n c o s t s . However, there are u s u a l l y a l a r g e number of design v a r i a b l e s , even i n a modest runoff system, and very few g u i d e l i n e s to suggest which v a r i a b l e s to modify. The consequences of a change in the design of a pipe or channel i n one branch on the peak outfow from the system (e.g., whether to i n c r e a s e or decrease and by how much) are d i f f i c u l t to p r e d i c t s h o r t of performing a r e - a n a l y s i s of the e n t i r e system. For example, d e c r e a s i n g the s i z e s of a conduit or channel i n one branch may r e s u l t in the requirement of a much l a r g e r c r o s s - s e c t i o n a l area i n a downstream p o r t i o n of the system. A l t e r n a t i v e l y , a s l i g h t i n c r e a s e i n c r o s s - s e c t i o n of one of the branches may r e s u l t i n a great decrease i n the maximum flow f a r t h e r downstream. I n t r o d u c t i o n of storage r e s e r v o i r s complicates the problem of f i n d i n g the o p t i m a l design even more. Storage i s t y p i c a l l y used because of the a t t e n u a t i o n ( r e d u c t i o n ) e f f e c t on the peak flow. However, as flows are routed through a 4 reservoir, translation of the peak to a later point in time also occurs. In other words, the result of temporary storage is that the hydrograph of a flood passing through the reservoir has a lower peak flow which occurs at a later time than i f there had been no storage. As a result, the translated peak in one branch may coincide with the peak flow from another branch, resulting in a much higher peak flow downstream of the junction. Thus the time lag imposed by the introduction of storage can counteract the benefits of the attenuation of peak flow, and i t is not always obvious, especially when dealing with a complex network, when storage w i l l reduce overall peak flows. The fact that detention storage is not always helpful in reducing peak flows is becoming increasingly recognized (Curtis & McCuen, 1977; Duru, 1981; Dendrou & Delleur, 1982). The model presented in this thesis will point out the locations and times at which flow modifications will reduce the peak flow at some downstream reference point. It wi l l also indicate the effectiveness of any such modifications in reducing the peak outflow, thereby allowing the designer to choose a design which will result in the greater benefit to the overall system. 1.2 EXISTING MODELS Many types of computer models for designing drainage systems are in use today, especially in the f i e l d of urban stormwater runoff management. There are four broad 5 c a t e g o r i e s of such models, each with an i n c r e a s i n g l e v e l of s o p h i s t i c a t i o n and complexity: screening models, which estimate the complexity of the problem; p l a n n i n g models, which assess the o v e r a l l runoff problem; design models, which perform a d e t a i l e d s i m u l a t i o n of a s i n g l e storm event, and; o p e r a t i o n a l models, which produce a c t u a l c o n t r o l d e c i s i o n s d u r i n g a storm event (Huber et a l . , 1981). An example of a model s u i t a b l e f o r use at the planning l e v e l the Stormwater Runoff Model (STORM), developed by the United S t a t e s Army Corps of Engineers. T h i s model can be used f o r p l a n n i n g r e q u i r e d storage and treatment c a p a c i t y for storm runoff from s i n g l e catchments (Perks, 1977). For f i n a l design a p p l i c a t i o n s of s i n g l e - e v e n t storms and f o r m u l t i p l e catchments, the Storm Water Management Model (SWMM) i s among the most popular in North America. T h i s model, developed by the U n i t e d S t a t e s Environmental P r o t e c t i o n Agency, i s c l a s s i f i e d as a p r e d i c t i o n and management s i m u l a t i o n model (Yen & Suvak, 1975). It uses a s l i g h t l y s i m p l i f i e d kinematic-wave approximation to route flows flows through a runoff system, s t a r t i n g with small pipe diameters and i t e r a t i n g with l a r g e r s i z e s u n t i l f r e e s u r f a c e flow c o n d i t i o n s are a t t a i n e d . Another model in common use i n North America i s the I l l i n o i s Storm Sewer S i m u l a t i o n Model (ISS). T h i s model i s used f o r both flow p r e d i c t i o n and design of sewer s i z e s f o r a given system layout and s p e c i f i e d s l o p e . The method employs unsteady flow equations and accounts f o r the mutual 6 backwater e f f e c t s between the sewers i n the network. Many other models are c u r r e n t l y a v a i l a b l e and there i s a great d e a l of l i t e r a t u r e expanding on the r e l a t i v e m e r its of each. Good comparisons of v a r i o u s models and t h e i r a p p l i c a b i l i t y to c e r t a i n design s i t u a t i o n s are found i n Perks (1977), Fok et a l . (1979) and Stephenson (1981). While some of the models a r r i v e at design parameters for i n d i v i d u a l segments ( f o r example, SWMM i t e r a t e s u n t i l i t reaches the s m a l l e s t p o s s i b l e pipe s i z e w i t h i n each branch), none w i l l perform o p t i m i z a t i o n of the e n t i r e system. For the designer to determine i f the investment i n a storage f a c i l i t y or s l i g h t l y l a r g e r c o nduit s i z e at an upstream l o c a t i o n w i l l produce g r e a t e r b e n e f i t s i n cost savings downstream, the e n t i r e program would need to be re-run. What i s needed i s a method for determining how to a l t e r a complex drainage system design i n order to achieve g r e a t e r b e n e f i t s . T h i s i s the o b j e c t i v e of the computer-aided design (CAD) scheme presented here. 1.3 COMPUTER-AIDED DESIGN Computer aided design has so f a r had l i t t l e impact on design procedures i n h y d r o t e c h n i c a l design and i t remains an open q u e s t i o n as to how f a r design processes f o r w a t e r - r e l a t e d p r o j e c t s might be handled by computers in the f u t u r e . C e r t a i n l y the p o t e n t i a l b e n e f i t s i n the use of CAD techniques i n the design or m o d i f i c a t i o n of runoff systems, both n a t u r a l and man-made, have yet to be e s t a b l i s h e d or 7 even demonstrated. Because of the geometric and dynamic complexity of the o v e r a l l runoff system, a completely n o n - i n t e r a c t i v e computer approach appears improbable. I t i s a l s o perhaps u n d e s i r a b l e in view of the l a r g e l y i n t a n g i b l e s o c i a l , economic and p o l i t i c a l 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 f i n a l d e s i g n . C o n s i d e r a b l e human designer i n t e r v e n t i o n and i n t e r a c t i o n i s t h e r e f o r e a n t i c i p a t e d i n any r u n o f f system CAD approach. The design of a CAD scheme i s i t s e l f a type of problem whose, s o l u t i o n p r e s e n t l y i n v o l v e s more a r t than s c i e n c e . The computerized design process u s u a l l y f a l l s i n t o one of three c a t e g o r i e s : i t e r a t i v e method; d i r e c t method; or design s e l e c t i o n (Besant, 1983). A n a l y t i c a l support beyond s t r a i g h t f o r w a r d equation s o l v i n g can be expected i n c i v i l e n g i n e e r i n g CAD schemes and may i n c l u d e f i n i t e element methods, s i m u l a t i o n , or o p t i m i z a t i o n depending on the type of d e s ign problem i n v o l v e d (Encarnacao, 1983). An i t e r a t i v e d e s i gn approach i s c o n s i d e r e d i n t h i s t h e s i s , with the d e s i g n e r being guided toward a "good" design through an o p t i m i z a t i o n scheme. The purpose of the r e s e a r c h presented here i s to a s c e r t a i n the nature and value of the i n f o r m a t i o n which such a scheme might provide the d e s i g n e r . S e n s i t i v i t y i n f o r m a t i o n 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 techniques i s the primary source of i n f o r m a t i o n communicated to the d e s i g n e r . Although o p t i m i z a t i o n i s i n v o l v e d there i s no suggestion that a design which can be c o n s i d e r e d to be o p t i m a l i n any g l o b a l sense i s being sought. A CAD scheme 8 which leads to designs which can be s a i d to be both r a t i o n a l and s u p e r i o r with a reasonable degree of conf i d e n c e would seem to be a r e a l i s t i c and p r a c t i c a l g o a l . Both l i n e a r and n o n - l i n e a r o p t i m i z a t i o n methods have been i n c o r p o r a t e d i n t o CAD schemes. In the work d e s c r i b e d here a s t r i c t l y l i n e a r model f o r the runo f f system i s adopted. T h i s r e p r e s e n t s a r e l a t i v e l y s i m p l i s t i c approach to m o d e l l i n g r u n o f f , and more a c c u r a t e n o n - l i n e a r models do e x i s t , p a r t i c u l a r l y i n the context of urban runoff systems. However, the l i n e a r model compensates by allowing, the a p p l i c a t i o n of l i n e a r programming, the most r e l i a b l e and powerful of o p t i m i z a t i o n methods with the most r e f i n e d p o s t - o p t i m a l i t y techniques. The l i n e a r model does have a b a s i s i n p r a c t i c e i n the form of Muskingum channel r o u t i n g , and i n r e p r e s e n t i n g storage by l i n e a r r e s e r v o i r s . The combination of l i n e a r models and l i n e a r programming was t h e r e f o r e f e l t to o f f e r the most f e r t i l e technique f o r i n v e s t i g a t i n g sources of a n a l y t i c a l i n f o r m a t i o n which might be e x p l o i t e d i n the context of CAD. Chapter 2 PROGRAM FORMULATION As mentioned i n the p r e v i o u s chapter, the model d e s c r i b e d i n t h i s t h e s i s uses l i n e a r programming to solve Muskingum-type r o u t i n g equations. T h i s chapter w i l l provide a d e s c r i p t i o n of each of these components and of the way they were combined to p r o v i d e the d e s i r e d s e n s i t i v i t y i n f o r m a t i o n . 2.1 MUSKINGUM CHANNEL ROUTING The Muskingum method of channel r o u t i n g has been widely used s i n c e i t s i n t r o d u c t i o n by G. T. McCarthy i n 1938 (Nash, 1959). As a h y d r o l o g i c type of r o u t i n g technique, i t uses a st o r a g e - d i s c h a r g e r e l a t i o n s h i p and a form of the c o n t i n u i t y equation. The storage equation f o r a s i n g l e reach of a stream i s : S = k[xQi + (l-x)Q2] (2.1) where S i s the storage i n the reach; k i s the storage constant, the r a t i o of storage to dis c h a r g e ; Q 1 i s the in f l o w to the reach; Q 2 i s the outflow from the reach; and x i s a constant f o r the reach e x p r e s s i n g the importance of in f l o w i n determining storage. The other equation forming the b a s i c f o r m u l a t i o n of the Muskingum method i s a form of the c o n t i n u i t y equation: 9 10 (Ql+Qj) < Q ? + Q i ) , x f - J = 8 2 - 8 , (2.2) where « i s the r o u t i n g p e r i o d (the time increment); Q] , Qj, and S, are the inflow, outflow, and storage at the s t a r t of the r o u t i n g p e r i o d ; and Q j , Q j , and S 2 are the i n f l o w , outflow, and storage at the end of the r o u t i n g p e r i o d . T h i s f i n i t e - d i f f e r e n c e form of the c o n t i n u i t y equation assumes a l i n e a r r e l a t i o n s h i p between storage and d i s c h a r g e d u r i n g the routing, p e r i o d . The usual approach used i n the Muskingum method i n v o l v e s combining equations (2.1) and (2.2) to o b t a i n the f o l l o w i n g e x p r e s s i o n : Q 2 = t+2kx Q i + t-2kx Q , _ t-2k{\-x) Q 2 > ( 2 > 3 ) t+2k(\-x) t+2k{\-x) t+2k(\-x) Then, with the values of the complete inflow hydrograph and the f i r s t outflow o r d i n a t e known, the remaining o r d i n a t e s of the outflow hydrograph can be determined s e q u e n t i a l l y . The r o u t i n g p e r i o d t and the storage c o n s t a n t k are both expressed i n the same u n i t s of time. The range of a p p l i c a b l e values of t has been suggested by Viessman et a l . (1977) as: k/3 < t < k, (2.4) and by Chow (1964) as: 2kx < t < k. (2.5) 11 I t has been suggested by Nash (1959) that the i n t e r v a l t should be kept small r e l a t i v e to k to i n s u r e the accuracy of the f i n i t e d i f f e r e n c e c a l c u l a t i o n s . If t were g r e a t e r than k, a f l o o d wave c o u l d pass through the channel without the peak being r e g i s t e r e d by the r o u t i n g e q u a t i o n . The value of k f o r a p a r t i c u l a r channel i s u s u a l l y determined e m p i r i c a l l y , and a commonly used c a l i b r a t i o n equation i s noted by Weinmann (1979), Koussis (1978), and many others as: where L i s the l e n g t h of the reach, and c i s the kinematic wave speed. T h e r e f o r e , k has sometimes been i n t e r p r e t e d as the p r o p a g a t i o n time i n the reach (Viessman et a l . , 1977; K o u s s i s , 1978). The value of the weighting f a c t o r x i s o f t e n determined g r a p h i c a l l y , using t r i a l - a n d - e r r o r u n t i l the best f i t i s achieved. The bounds f o r x are given by Viessman and Koussis as: If x = 0.5, there would be no a t t e n u a t i o n , and i f x > 0.5, the outflow peak would be g r e a t e r than the i n f l o w . If x = 0, Equation (2.3) reduces to the s p e c i a l case of a l i n e a r r e s e r v o i r , and the outflow i s determined by: (2.6) 0 < x < 0 . 5. (2.7) 12 Qi = l + 2k t-2k t +2k Qi • (2.8) For n a t u r a l channels, x t y p i c a l l y v a r i e s from 0 to 0.3, and has an average value of approximately 0.2 (Viessman et a l . , 1977). I t seems apparent that t h i s value of x was assumed when s e t t i n g the l i m i t s f o r t i n Equation (2.4), s i n c e s u b s t i t u t i n g t h i s value i n t o Equation (2.5) makes the two e x p r e s s i o n s n e a r l y e q u i v a l e n t . Attempts to c a l i b r a t e x i n terms of p h y s i c a l and h y d r a u l i c parameters were i n i t i a t e d by J . A. Cunge in 1969, and have been repeated by Koussis (1978) and Ponce (1979) i n the form: where q i s the d i s c h a r g e per u n i t width and S0 i s the channel bed s l o p e . I t was mentioned above that i f x was a s s i g n e d a value of 0.5, the r e s u l t i n g outflow hydrograph would e x h i b i t pure t r a n s l a t i o n - that i s , i t would have the same shape as the i n f l o w hydrograph, but s h i f t e d on the time a x i s . The s i t u a t i o n where there i s no a t t e n u a t i o n or t r a n s l a t i o n -where the i n f l o w and outflow hydrographs are i d e n t i c a l both s p a t i a l l y and temporally - can be a t t a i n e d by s e t t i n g k = 0 and Ql = Q 2. The u s e f u l n e s s of t h i s technique w i l l be demonstrated i n Chapter 5. x (2.9) 2 2S 0cL 1 3 2.2 LINEAR PROGRAMMING AND SENSITIVITY ANALYSIS Li n e a r programming (LP) w i l l f i n d the optimum (maximum or minimum) of a l i n e a r o b j e c t i v e f u n c t i o n ( c o n t a i n i n g a s e r i e s of d e c i s i o n v a r i a b l e s ) subject to any number of l i n e a r c o n s t r a i n t equations. In so doing, i t a l s o determines the values of the d e c i s i o n v a r i a b l e s r e q u i r e d to achieve the optimal value of the o b j e c t i v e f u n c t i o n . S e n s i t i v i t y a n a l y s i s can r e v e a l a great d e a l of inf o r m a t i o n on the i n f l u e n c e of the v a r i o u s c o e f f i c i e n t values i n the LP form u l a t i o n at the optimal s o l u t i o n . The dual s o l u t i o n vector has a v a r i a b l e a s s o c i a t e d with each p a r t i c u l a r c o n s t r a i n t , the value of which represents the r e l a t i v e change i n the optimum value of the o b j e c t i v e with respect to a change i n the r i g h t hand sid e (RHS) c o e f f i c i e n t of that c o n s t r a i n t . In other words, the dual s o l u t i o n v e c t o r c o n t a i n s , for each c o n s t r a i n t , the value by which the o b j e c t i v e f u n c t i o n w i l l change i f a u n i t change were to occur i n each of the c o n s t r a i n t c o e f f i c i e n t s . RHS c o e f f i c i e n t ranging determines the amount by which the value of each RHS c o e f f i c i e n t can be changed before the optimal b a s i s i s no longer f e a s i b l e . Outside of t h i s range, a d i f f e r e n t set of d e c i s i o n v a r i a b l e s w i l l comprise the b a s i s . The dual s o l u t i o n value i s only v a l i d w i t h i n the boundaries e s t a b l i s h e d by RHS ranging. O b j e c t i v e f u n c t i o n c o e f f i c i e n t ranging (OFCR) i s performed to determine how f a r each o r i g i n a l o b j e c t i v e f u n c t i o n (cost) c o e f f i c i e n t can be i n d i v i d u a l l y v a r i e d 1 4 before the b a s i s i s no longer o p t i m a l . Within the bounds e s t a b l i s h e d by OFCR, any v a r i a t i o n of a s i n g l e o b j e c t i v e f u n c t i o n c o e f f i c i e n t w i l l o n l y change the value of the o b j e c t i v e f u n c t i o n optimum, but the values of the optimal b a s i s v a r i a b l e s w i l l remain unchanged. 2.3 SOLVING THE MUSKINGUM EQUATION USING LINEAR PROGRAMMING As an o p t i m i z i n g t o o l , l i n e a r programming i s probably most o f t e n used with the primary o b j e c t i v e of minimizing c o s t s , time,., q u a n t i t i e s , e t c . In t h i s f o r m u l a t i o n , however, the d i r e c t r e s u l t of the LP w i l l be to reproduce the outflow hydrograph. The a n a l y t i c a l value of the procedure w i l l come through i n s p e c t i o n of the s e n s i t i v i t y a n a l y s i s i n f o r m a t i o n . The purpose of using LP to so l v e a r o u t i n g equation i s to determine the p o i n t s i n a r u n o f f system i n which the flow has the g r e a t e s t i n f l u e n c e on the peak flow at some ref e r e n c e p o i n t downstream. I f the o b j e c t i v e f u n c t i o n i s formulated to minimize the downstream peak flow ( i . e . , the maximum o u t f l o w ) , and the v a r i a b l e s are d e f i n e d as the upstream f l o w s , then the s e n s i t i v i t y i n f o r m a t i o n w i l l show the l o c a t i o n s where reducing the flow w i l l r e s u l t i n the gr e a t e s t decrease of peak outflow, Q^. I t w i l l show the l o c a t i o n and time of the flow o r d i n a t e s to which Q i s the P most s e n s i t i v e to change. The c o n s t r a i n t s , formulated from the r o u t i n g equations, w i l l ensure that the c o r r e c t r e l a t i o n s h i p between inflow and outflow hydrographs are maintained. 15 In f o r m u l a t i n g the program, the sim p l e s t case of a s i n g l e l i n e a r r e s e r v o i r was f i r s t c o n s i d e r e d , and was run under a v a r i e t y of i n i t i a l c o n d i t i o n s and design parameters to determine which o r d i n a t e s of the in f l o w hydrograph c o n t r i b u t e most to the downstream peak. The program was then expanded to a s e r i e s of l i n e a r r e s e r v o i r s . The same was then done f o r a s e r i e s of Muskingum channels, and f i n a l l y a network of channels was developed. Since l i n e a r r e s e r v o i r s are simply a s p e c i a l case of Muskingum channels, only the; l a t t e r f o rmulation w i l l be' d e s c r i b e d here. 2.3.1 OBJECTIVE FUNCTION FORMULATION In a l l cases, the for m u l a t i o n i s of the "minimax" type - that i s , the o b j e c t i v e i s to minimize the maximum o r d i n a t e of the outflow hydrograph (the peak o u t f l o w ) . To enable t h i s f o r m u l a t i o n , a dummy v a r i a b l e , Y, was i n t r o d u c e d . T h i s v a r i a b l e was set equal to the peak outflow by a s e r i e s of c o n s t r a i n t s , wherein Y i s set gr e a t e r than or equal to each o f . t h e outflow o r d i n a t e s in t u r n . In a minimax f o r m u l a t i o n , the o b j e c t i v e i s u s u a l l y s o l e l y to minimize Y, the dummy v a r i a b l e . In t h i s f o r m u l a t i o n , however, i t was found that when there was more than one reach, some of the outflow o r d i n a t e v a r i a b l e s were a s s i g n e d values by the LP package that were higher than would be expected from normal r o u t i n g procedures. The v a r i a b l e s a f f e c t e d always occ u r r e d a f t e r the peak, but were never assi g n e d values higher than the peak. In other words, 16 the receeding part of the outflow hydrograph was o ver-estimated, although the value of the peak was determined a c c u r a t e l y i n every case. The problem arose because the Muskingum r o u t i n g equations were entered i n the LP f o r m u l a t i o n as i n e q u a l i t i e s i n order to f a c i l i t a t e the dual s o l u t i o n . To minimize the number of c o n s t r a i n t s , only the > i n e q u a l i t i e s were i n c l u d e d , which r e s u l t e d i n u n d e r c o n s t r a i n i n g the r o u t i n g process. T h i s problem was overcome, without the n e c e s s i t y of doubling the number of c o n s t r a i n t s , by i n c l u d i n g a penalty f u n c t i o n to b r i n g the outflow o r d i n a t e s i n t o the o b j e c t i v e f u n c t i o n . These v a r i a b l e s were m u l t i p l i e d by a f a c t o r ( i n t h i s case, 10" 1 0 ) to prevent them from becoming the dominant v a r i a b l e s i n the o b j e c t i v e f u n c t i o n . Thus, the" o b j e c t i v e f u n c t i o n i s : N I Minimize Y + Z (10" 1°Q ) (2.10) n=2 n where n i s the o r d i n a t e number; N i s the t o t a l number of o r d i n a t e s i n c l u d e d i n the r o u t i n g equation; I i s the number of the downstream re f e r e n c e node; and Q 1 i s the o r d i n a t e n of the hydrograph at p o i n t I. 2.3.2 MINIMAX CONSTRAINTS As mentioned above, the dummy v a r i a b l e Y i s made to equal by s e t t i n g Y > each of the outflow o r d i n a t e s i n t u r n . In other words, the d i f f e r e n c e between Y and each outflow o r d i n a t e must be p o s i t i v e , as: 17 Y - Qn > 0 (2.11) f o r n = 2,...,N. 2.3.3 SINGLE REACH To e x p l o i t the s e n s i t i v i t y a n a l y s i s c a p a b i l i t i e s of l i n e a r programming, the Muskingum r o u t i n g equation had to be re-formatted such that the i n f l o w s (the known values) appeared on the r i g h t hand sid e of the c o n s t r a i n t equations, while the-- unknown outflow- v a r i a b l e s (other than the f i r s t o r d i n a t e ) were t r e a t e d as d e c i s i o n v a r i a b l e s . The dual s o l u t i o n then g i v e s the r e l a t i v e e f f e c t of a change i n an inflow o r d i n a t e on the outflow peak, i n d i c a t i n g the p o i n t on the inflow hydrograph that most a f f e c t s the peak. Therefore, by r e v i s i n g Equation (2.3) and i n t r o d u c i n g new n o t a t i o n f o r the sake of c l a r i t y , the r o u t i n g equation fo r the f i r s t r o u t i n g p e r i o d becomes: 0Q| = Qi + <t>Q] ~ 7 Q 1 , (2.12) where j3 = t+2£(1-x). t -2kx (2.13) „ = t-2k(\-x). (2.14) t -2kx and <p = t +2kx t -2 kx (2.15) Note that the second and t h i r d terms on the r i g h t hand s i d e 18 of Equation (2.12) are c o n s t a n t s , s i n c e the f i r s t o r d i n a t e of each hydrograph must be known before performing any r o u t i n g . The Qj term, being part of the inflow hydrograph, must a l s o be known before r o u t i n g i s performed, and t h e r e f o r e Qj i s the only v a r i a b l e . Equation (2.12) forms the f i r s t c o n s t r a i n t i n the l i n e a r programming problem. For the second r o u t i n g p e r i o d , the equation was d e r i v e d i n the same manner and i s : 7QJ + 0Q1 =• QJ + 4>Ql (2.16) where Ql and Q| are the t h i r d o r d i n a t e s of the in f l o w and outflow hydrographs, r e s p e c t i v e l y . In t h i s step, both outflow o r d i n a t e s are on the l e f t hand s i d e , s i n c e they are both v a r i a b l e s . Equation (2.16) was f u r t h e r reduced by s u b s t i t u t i n g the expr e s s i o n f o r Qj , which i s obtained from Equation (2.12). The r e s u l t i n g e x p r e s s i o n i s : iKJl + 0Q§ = Q . 3 ~ * -UQ, " TQi) (2.17) where \p i s d e f i n e d as: \li = 7 " P>- (2.18) Again, the r i g h t hand s i d e c o n t a i n s only the inflow parameter, Ql, and a f u n c t i o n of the two constant terms Q] and Q 2. 19 E x p r e s s i o n s f o r the t h i r d and f o u r t h r o u t i n g i n t e r v a l s were determined i n a s i m i l a r manner, and t h e i r f i n a l form i s presented here. For the t h i r d r o u t i n g p e r i o d , the equation becomes: - ^ Q i + * Q i + PQl = Ql + 4>2(4>Q}~yQV ( 2 . 1 9 ) and for the f o u r t h r o u t i n g p e r i o d , the equation i s : 02<^QI - 0^Qi- + *Ql + PQl = Ql - * . 3 U Q ] - 7 Q - ? ) . ( 2 . 2 0 ) The same procedure was repeated f o r each r o u t i n g p e r i o d , with an a d d i t i o n a l outflow v a r i a b l e being introduced at each r o u t i n g p e r i o d , u n t i l a l l inf l o w and outflow v a r i a b l e s were i n c l u d e d i n the f o r m u l a t i o n . The d e c i s i o n v a r i a b l e s c o n s i s t of the outflows, Q2 , where n ranges from 2 to N , the number of hydrograph o r d i n a t e s . The r i g h t hand s i d e c o n s i s t s of the f i r s t i n f l o w and outflow o r d i n a t e s , Q ] and Q ?, and one of the in f l o w o r d i n a t e s Q 1 . The c o n s t r a i n t equation f o r any o r d i n a t e n can be expressed g e n e r a l l y as: V[(-0)""-'"V QJ] + PQ2n * Q'n + (-<*>)*~2UQ] - yQV ( 2 . 2 1 ) j-2 where n v a r i e s from 2 to N ; i s the n t^ 1 o r d i n a t e of hydrograph 1 (the upstream hydrograph); i s the n o r d i n a t e of hydrograph 2 (the downstream hydrograph); and N i s the number of o r d i n a t e s i n the storm hydrograph. 20 The r o u t i n g c o n s t r a i n t s f o r a case where there are 6 hydrograph o r d i n a t e s are given i n Table I. Note that in order to o b t a i n the s e n s i t i v i t y a n a l y s i s , the c o n s t r a i n t s must be i n > form, ra t h e r than e q u a l i t y c o n s t r a i n t s . As d i s c u s s e d e a r l i e r , e quivalence to an e q u a l i t y i s only achieved i f a s i n g l e e q u a l i t y c o n s t r a i n t i s represented by both < and > i n e q u a l i t i e s . By adding pe n a l t y terms to the o b j e c t i v e f u n c t i o n , i t was not necessary to in c l u d e the < c o n s t r a i n t s . With, the i n c l u s i o n of the minimax c o n s t r a i n t s and the o b j e c t i v e f u n c t i o n , the complete for m u l a t i o n f o r s o l v i n g the Muskingum r o u t i n g equation through a s i n g l e channel reach i s shown i n Table I I . 2.3.4 SERIES OF REACHES  To simulate the s i t u a t i o n where there may be a change i n channel c h a r a c t e r i s t i c s at some p o i n t i n the flow ( f o r example, a change i n pipe diameter or s l o p e ) , the fo r m u l a t i o n must allow f o r reaches with d i f f e r e n t values of k and x. (The r o u t i n g p e r i o d t must of course be the same fo r a l l reaches). To add more reaches to the system, a d d i t i o n a l c o n s t r a i n t equations are r e q u i r e d , and these c o n s t r a i n t s are s i m i l a r to the ones d i s c u s s e d p r e v i o u s l y i n equations (2.12) and (2.17) through (2.21). The only d i f f e r e n c e now i s that the i n f l o w s to the downstream reach (except the f i r s t o r d i n a t e ) have a l r e a d y been d e f i n e d as v a r i a b l e s (the 21 TABLE I ROUTING CONSTRAINTS FOR A SINGLE REACH CONST Ql Q! Ql Qi Qi RHS 1 P > Q2 + (0Ql - 7 Q 2 ) 2 4> > Q] -0(0Q; - 7 Q 2 ) 3 - H > Q i + 0 2 ( 0 Q l - 7 Q 2 ) 4 <t>2^ > - 7 Q 2 ) 5 4>2i> /5 > Q ^ + 0 4 ( 0 Q 1 - 7 Q 2 ) outflows from the upstream reaches), and so are placed on the l e f t hand s i d e of the c o n s t r a i n t e q u a t i o n s . T h e r e f o r e , the equation f o r the f i r s t r o u t i n g p e r i o d i s : -Q ' 2 + 0.Q' 2 + 1 = 0FQ'i - 7 , - Q i * 1 , (2.22) where / i s the reach number. For the second r o u t i n g p e r i o d , the equation becomes: - Q i + ^-QI*1 + P ^ Q ^ 1 = -0 (- (0 £-Qi - 7,-Q'i*1). (2.23) In general terms, the c o n s t r a i n t equation f o r the or d i n a t e of a hydrograph at node / i s : j=2 * ( - ^ ) " " 2 ( ^ Q i - 7 , 0 1 * 1 ) . (2.24) 2 2 TABLE II LP TABLEAU FOR A SINGLE REACH CONST Y Ql Qi Ql Qi Qi RHS Obj . 1 10" 1 0 10" 1 0 10- 1 0 10" 1 0 10" 1 0 (Minimize) 1 > Ql + UQ}~yQV 2 > Ql -«(0Ql-7Q?) 3 > Q W 2 U Q ] - T Q ? ) 4 > Q W 3 U Q ] - T Q I ) 5 <p2xP Q ^ + c / ) 4 (0Ql-7Qi) 6 1 -1 > 0 7 1 -1 > 0 8 1 -1 > 0 9 1 -1 > 0 10 -1 > 0 Note that the RHS c o n t a i n s only the f i r s t o r d i n a t e s of the hydrographs, which are c o n s t a n t s . From i n s p e c t i o n of the RHS of Equation ( 2 . 2 4 ) , i t may appear that no u s e f u l s e n s i t i v i t y i n f o r m a t i o n can be obtained. However, as w i l l be d i s c u s s e d i n Chapter 5 , i t was found that the dual s o l u t i o n a c t u a l l y provided s e n s i t i v i t y i n f o r m a t i o n f o r Q^. That i s , the dual value o b t a i n e d by the l i n e a r programming r o u t i n e r e p r e s e n t s the change i n the optimum per u n i t change i n Q^. T h i s r e s u l t can be e x p l a i n e d by the nature of Equation ( 2 . 2 4 ) , where by adding Q^ to each s i d e , the equation takes on the same form as the s i n g l e channel case 23 shown i n Equation (2.21). The complete l i n e a r programming for m u l a t i o n f o r a case with three reaches and s i x o r d i n a t e s i s given i n Table I I I . In a d d i t i o n to the r o u t i n g c o n s t r a i n t s (Numbers 1-12), the l a s t four are the minimax c o n s t r a i n t s . These c o n s t r a i n t s ensure that Y i s equal to Q^. 2.3.5 BRANCHING The f o r m u l a t i o n d e s c r i b e d thus f a r w i l l route flows through a s e r i e s arrangement of Muskingum channels, l i n e a r r e s e r v o i r s , or a combination of both. I t w i l l a l s o i n d i c a t e the r a t e of change of the peak outflow per u n i t change in flow o r d i n a t e s at a l l upstream p o i n t s . T h i s i n tu r n w i l l i n d i c a t e the l o c a t i o n s and flow o r d i n a t e s where a m o d i f i c a t i o n i n flow w i l l have the g r e a t e s t e f f e c t in changing the peak flow at the o u t l e t . To model a more complex runo f f s i t u a t i o n , however, i t i s necessary to i n c o r p o r a t e m u l t i p l e branches which enable incoming runoff from s e v e r a l sources to flow i n t o one channel or pipe reach, while a l s o m a i n t a i n i n g the s e n s i t i v i t y i n f o r m a t i o n f o r each brajnch. T h i s s e c t i o n d e s c r i b e s how these o b j e c t i v e s were met. When d e a l i n g with a complex network, i t i s important to have a c o n s i s t e n t numbering system to i d e n t i f y nodes. With a simple s e r i e s of channels, the numbering was s t r a i g h t f o r w a r d - the uppermost p o i n t was l a b e l l e d '1' and the numbers i n c r e a s e d i n a downstream d i r e c t i o n u n t i l the TABLE III LP TABLEAU FOR 3 CHANNELS IN SERIES T 02 Qi Ql Ql Q l Qi Qi Qi Qi QS Qi Ql RHS Objective 1 , 0 - i e lO"10 lO"10 10-' 0 (Minimize) Constraint 1 Constraint 2 Constraint 3 Constraint 4 fi, i>, fi, *, fi, 2 * , * t * t fi, * Qj * <*.Q!-7iQ?> * Q i - # i U i Q ! - 7 . Q ? ) * Qj*#?(#.Q!-7iQ?) * Q}-#?(# 1Q!-7.Q?) Constraint 5 Constraint 6 Constraint 7 Constraint 8 -1 -1 -1 -1 fi* *i fit -4*^2 *a fit 2 *» * i <Pt fit * ( • i Q i - 7 i Q ? ) i -• 2(*JQ?-7IQI) i • ? ( # l Q i - 7 2 Q ? ) * -•?(0»Q?-7JQ?) Constraint 9 Constraint 10 Constraint It Constraint 12 -1 -1 -1 -1 fi, $1 fii fit 2 * i * i -it*, *J fit i- (•JQ?-7JQI) * -•JU»Q|-7JQI) * •?(#i Q ? - 7 i Q " ) i -*?(•,Q?-7iQ') Constraint 13 Constraint 14 Constraint 15 Constraint 16 1 1 1 -1 -1 -1 -1 IV IV IV IV o o o o to 25 f i n a l node (the o u t l e t ) was reached. To i d e n t i f y branches and j u n c t i o n s , the f o l l o w i n g procedure was used. Numbering began at the uppermost p o i n t as before, and proceeded downstream to the f i r s t j u n c t i o n . The p o i n t immediately above the j u n c t i o n was a s s i g n e d a number, and the next c o n s e c u t i v e number was a s s i g n e d to the uppermost p o i n t of the adjacent branch. The numbering then c o n t i n u e d downstream u n t i l another j u n c t i o n was reached, where the numbering co n t i n u e d at the top of another branch. T h i s process continued u n t i l the< o u t l e t of. the network was reached. With t h i s numbering system, the j u n c t i o n s are a u t o m a t i c a l l y i d e n t i f i e d f o r simple branches. Each reach i s numbered a c c o r d i n g to i t s upstream node. A reach below a j u n c t i o n takes the h i g h e s t number of the two nodes immediately above the j u n c t i o n . When second or higher order branches were added, i t was necessary f o r the j u n c t i o n s to be e x p l i c i t e l y i d e n t i f i e d to a v o i d ambiguity. ( T h i s a c t u a l l y allows f o r g r e a t e r f l e x i b i l i t y i n numbering the branches). Except f o r i d e n t i f y i n g the j u n c t i o n s , the numbering system i s i d e n t i c a l to the p r e v i o u s case. The l i n e a r programming f o r m u l a t i o n was g e n e r a l l y the same as that f o r a s e r i e s of channels, the o n l y e x c e p t i o n s o c c u r i n g at the j u n c t i o n s . Here, the flows from both branches had to be superimposed and routed downstream. At the same time, the s e n s i t i v i t y i n f o r m a t i o n f o r each branch had to be preserved. 26 At a j u n c t i o n where reaches m and / j o i n and flow i n t o reach z'+1, the r o u t i n g equation becomes: -(QT+Q!) + P^Q!*1 = * , ( 0 7 + Q',) - 7,-Q^1. (2.25) Equation (2.25) i s simply Equation (2.22) with the a d d i t i o n of two e x t r a inflow terms from the branch. Here, Qm and Ql n n are the in f l o w s immediately above the j u n c t i o n , and Q^+1 i s the flow at the f i r s t node below the j u n c t i o n . For the second r o u t i n g p e r i o d at a j u n c t i o n the ex p r e s s i o n , d e r i v e d from Equation (2.23), i s : -<Q?+Q'3) + ^ . Q l + 1 + /3.Q1 + 1 = -tf f [* | .(Q7+Q / 1)-7 fQi + 1 ] . (2.26) In general terms, the exp r e s s i o n f o r a c o n s t r a i n t r e p r e s e n t i n g flow at a j u n c t i o n , d e r i v e d from Equation (2.24), i s : - ( C f + Q ' ) + V [ ( - 0 . ) " - ; - V Q / + 1 ] + 0 . Q / + 1 J =2 J * ( - 0 x - ) n " 2 [ * f (Q7 + Q\) - 7 fQi + 1 ] . (2.27) An example of the t a b l e a u f o r a branched network i s given i n Table IV. TABLE IV LP TABLEAU FOR A NETWORK WITH ONE BRANCH T Qi Q i Qi Q i Qi Q; Q: QS Q i Q i Qi Qi RHS Objective 1 1 0 - i o 1 0 " 1 0 1 0 " 1 0 1 0 " 1 0 (Minimize) Constraint 1 Constraint 2 Constraint 3 Constraint 4 * i 0, -•1*1 * i 0i 2 * i *i \f>i 0, * Q i • U.Ql-riQ?) * Qj-*.UiQl-7iQ5> * QJ**?U.Q!-7.Q?> i Q}-#?(#lQ!-7iQ?> Constraint 5 Constraint 6 Constraint 7 Constraint 8 0, 0, "•J^J * i 0i # i 2 ^ j . Pt * Q i + U j Q i - 7 i Q i > * Qi-«>,(#iQ?-7.Qi) * Qi+#?U>Q?-7iQD * Qi-#?(#jQ'-7>Q') Constraint 9 Constraint 10 Constraint 11 Constraint 12 -1 -1 -1 -1 -1 -1 -1 -1 0, <I>, 0, -*«^« 0t 2 «• <l>* * i 0% * [•.(Q?+Qi)-7.Q?1 * -•.[#.<Q?*Qi)-7.Qi) i •?[#.(Q?+Qi)-7.Qi) i - • ? ( • . ( Q i + Q l ) - 7 « Q 1 ) Constraint 13 Constraint 14 Constraint 15 Constraint 16 1 1 1 1 -1 -1 -1 2 0 2 0 2 0 2 0 ro Chapter 3 PROGRAM DESCRIPTION The p r e v i o u s chapter d e a l t with the f o r m u l a t i o n of an LP t a b l e a u to s o l v e the Muskingum r o u t i n g equation through a branched network. T h i s chapter w i l l present a b r i e f d e s c r i p t i o n of the F o r t r a n program that accepts the data, prepares i t f o r l i n e a r programming, and outputs the r e s u l t s . A l i s t i n g of the program i s i n c l u d e d i n Appendix A. 3.1 INPUT The program accepts input p e r t a i n i n g to the p h y s i c a l network and flow c h a r a c t e r i s t i c s from a standard data f i l e . An i n t e r a c t i v e v e r s i o n of the program was a l s o developed, but i t was found that the compiler was unable to o b t a i n s u f f i c i e n t a r r a y space to handle any but the s i m p l e s t networks. However, i t was found that the e a s i e s t way to handle data was to use data f i l e s , which c o u l d then be manipulated using the v i s u a l e d i t o r . An example of such a data f i l e i s i n c l u d e d as Table V . The f i r s t two l i n e s of input are the t i t l e of the data f i l e and a set of v a r i a b l e s which s p e c i f y output o p t i o n s . The next i n f o r m a t i o n read i s the number of s t a t i o n s (nodes) and j u n c t i o n s i n the network, and the number of o r d i n a t e s being used to represent the hydrographs at each s t a t i o n . The number of channel reaches i s then computed by the f o l l o w i n g e quation: 28 29 T A B L E V SAMPLE INPUT F I L E 1 EXAMPLE 3 . 1 : ONE JUNCTION 2 1 1 0 3 1 1 1 1 2 4 1 4 8 5 5 1 1 1 1 6 15.000 7 18.000 0 .200 K,X (REACH 1) 8 0.000 0 .000 K,X (REACH 2) (RESERVOIR) 9 21.000 0 .175 K,X (REACH 3) 1 0 19.500 0 . 1 60 K,X (REACH 5) 1 1 0.000 0 .000 K,X (REACH 6) (RESERVOIR) 12 20.250 0 .185 K,X (REACH 7) 1 3 16.400 0 .230 K,X (REACH 8) 1 4 0.000 0 .000 K,X (REACH 9). (RESERVOIR) 1 5 17.200 0 .215 K,X (REACH 10) 16 5.365 Q(2,1) 1 7 5.365 Q(3,1) 18 5.145 Q(4,1) 1 9 2.550 Q(6,1) 20 2.550 Q(7,1) 21 2. 100 0/(8,1) 22 7.885 Q(9,1) 23 7.885 Q(10,1) 24 7.350 Q(11,1) 25 5.975 QO,l) 26 7.650 Q(1,2) 27 12.250 Q<1,3) 28 9.450 Q(1,4) 29 8 .250 Q(1,5) 30 6.900 Q(1,6) 31 5.705 Q(1,7) 32 5.450 Q(1,8) 33 5.320 Q(1,9) 34 5.225 Q(1,10) 35 5. 175 Q(1,11) 36 5. 1 30 Q(1,12) 37 2.615 Q(5,1) 38 2.850 Q(5,2) 39 3.125 Q(5,3) 40 3.450 Q(5,4) 41 3.950 .. Q(5,5) 42 4.850 Q(5,6) 43 4.615 Q(5,7) 44 3.720 Q(5,8) 45 3. 1 50 Q(5,9) 46 3.015 Q(5,10) 47 2.945 48 2.920 30 NCHAN = NSTA-NJUNCT-1 (3.1) where NCHAN i s the number of channels; NSTA i s the number of s t a t i o n s , or nodes; and NJUNCT i s the number of j u n c t i o n s i n the network. Next, i n f o r m a t i o n r e g a r d i n g the branches and j u n c t i o n s are input to complete the d e s c r i p t i o n of the network l a y o u t . T h i s i n f o r m a t i o n i s s t o r e d i n v a r i a b l e a r r a y s f o r use i n s e t t i n g up the LP t a b l e a u . The value of the r o u t i n g p e r i o d i s then ente r e d . The v a l u e s of k and x f o r each reach are then read by the program and a s s i g n e d to t h e i r r e s p e c t i v e a r r a y s , a l l o w i n g f o r the v a r i a t i o n i n reach numbers that occur at j u n c t i o n s . At the same time, the values of 0, 7, 0, and \p are c a l c u l a t e d and s t o r e d i n a r r a y s . The execution stops and a message i s p r i n t e d i f the denominator of the c o e f f i c i e n t s i s negative or c l o s e to zero. With t h i s i n f o r m a t i o n , the p h y s i c a l d e s c r i p t i o n of the model network i s complete. At t h i s p o i n t , the only i n f o r m a t i o n remaining to be input are the d i s c h a r g e o r d i n a t e s - the i n f l o w hydrographs at the upper end of each branch, and the f i r s t o r d i n a t e of each i n t e r m e d i a t e and downstream node. The branching and j u n c t i o n i n f o r m a t i o n p r e v i o u s l y input i s used to ensure that the v a l u e s are a s s i g n e d to t h e i r proper l o c a t i o n i n the a r r a y . 31 It should be noted that the data f i l e s can be e a s i l y e d i t e d to i n c l u d e only a p o r t i o n of the network, i f d e s i r e d . A l l that i s r e q u i r e d i s to change the l i n e i n d i c a t i n g the number of nodes and d e l e t e the i n i t i a l i n f l o w s of the redundant reaches. T h i s would be u s e f u l i f , i n s t e a d of examining the network o u t l e t , the designer wished to examine the e f f e c t s of m o d i f i c a t i o n s on some intermediate p o i n t . 3.2 FORMULATING THE TABLEAU The tableau- s i z e i s c a l c u l a t e d by the program as a f u n c t i o n of the number of channel reaches and the number of o r d i n a t e s : NVARS = NCHAN*(NORDS-1)+1 (3.2) NCONST = (NCHAN+1)*(NORDS-1) (3.3) where NVARS i s the number of v a r i a b l e s and NCONST i s the number of c o n s t r a i n t s i n the LP t a b l e a u , and NORDS i s the number of o r d i n a t e s in each hydrograph. Each c o e f f i c i e n t i n the ta b l e a u i s i n i t i a l l y set to zero. The o b j e c t i v e f u n c t i o n i s then formulated to minimize the sum of the dummy v a r i a b l e Y plus the pen a l t y f u n c t i o n of the l a s t reach, as shown by the f i r s t row of Tables III and IV. The outflow hydrograph from each reach i s determined, using the standard Muskingum r o u t i n g procedure. T h i s step 32 i s not a c t u a l l y r e q u i r e d s i n c e r o u t i n g i s performed l a t e r in the LP segment, but i s n e v e r t h e l e s s i n c l u d e d to v e r i f y the accuracy of the LP r e s u l t s . Under c e r t a i n c o n d i t i o n s , a cumulative e r r o r may occur i n the r o u t i n g performed by the LP r o u t i n e , and t h e r e f o r e these r e s u l t s are compared with the s t r a i g h t f o r w a r d r o u t i n g r e s u l t s . If a descrepancy of 0.1% or more e x i s t s between the two values, a message g i v i n g the extent of the e r r o r i s p r i n t e d a l o n g s i d e the s o l u t i o n . The t a b l e a u i s then generated, one c o n s t r a i n t at a time-. F i r s t , the routing, c o n s t r a i n t s , are formulated., using the procedure d i s c u s s e d i n Chapter 2. C o r r e l a t i o n i s made between the c o n s t r a i n t number and the o r d i n a t e i t represents through a s e r i e s of t e s t s . F i n a l l y , the minimax c o n s t r a i n t s are added to complete the t a b l e a u . 3.3 LINEAR PROGRAMMING SUBROUTINE The l i n e a r programming problem i s solved u s i n g the UBC l i b r a r y r o u t i n e 'LIPSUB', a set of F o r t r a n IV s u b r o u t i n e s . A d e s c r i p t i o n of the r o u t i n e i s i n c l u d e d as Appendix B. On e n t r y , LIPSUB accepts the t a b l e a u and i n f o r m a t i o n regarding the o p e r a t i o n s to be performed. On output, the t a b l e a u c o n t a i n s the optimal v a l u e of the o b j e c t i v e and the values of the values of the p r i m a l and dual s o l u t i o n v a r i a b l e s . These values are addressed i n d i r e c t l y through an index. The RHS ranging i n f o r m a t i o n i s i n c l u d e d i n separate a r r a y s . On output, the upper and lower bounds give the values of the RHS c o e f f i c i e n t s which must be input to cause 33 a change i n the elements of the optimal s o l u t i o n v e c t o r . Because of the e x t r a terms c o n t a i n e d i n the RHS of the input t a b l e a u , some f u r t h e r m a n i p u l a t i o n i s necessary to o b t a i n the r e q u i r e d RHS ranging i n f o r m a t i o n . The a d d i t i o n a l terms on the RHS (other than a s i n g l e o r d i n a t e with u n i t c o e f f i c i e n t ) must be s u b t r a c t e d from the ranging r e s u l t s . For the c o n s t r a i n t s corresponding to an upstream s t a t i o n , these values w i l l then represent the true value f o r that reach. For example, i n the f i r s t c o n s t r a i n t i n Table IV, the value of (4>iQ]-7iQ 2) i s subtracted, from both the lower and upper bounds. The r e s u l t i n g values of the bounds t h e r e f o r e r e l a t e d i r e c t l y to Q j . For i n t e r m e d i a t e p o i n t s , where the o r d i n a t e does not appear on the RHS but i n s t e a d occurs as a negative value on the LHS, the value of the o r d i n a t e must be added to the RHS ranging r e s u l t s . For example, the value of Ql i n c o n s t r a i n t number 9 of Table IV must be added to the bounds to give the r i g h t hand s i d e the same form as c o n s t r a i n t 1. In e f f e c t , t h i s i s the same as r e d e f i n i n g the Ql from a d e c i s i o n v a r i a b l e to a RHS c o e f f i c i e n t . The o r i g i n a l RHS c o e f f i c i e n t , [ </>3 ( Q 2 + Q i )~7 3Qi ] , must a l s o be s u b t r a c t e d from the bounds. Again, the r e s u l t i n g upper and lower bounds r e l a t e d i r e c t l y to Q\. A p p l y i n g these types of manipulations f o r each c o n s t r a i n t where i t i s r e q u i r e d r e s u l t s i n RHS ranging i n f o r m a t i o n being a p p l i c a b l e t o a l l flow o r d i n a t e s throughout the network, with the exception of the i n i t i a l 34 o r d i n a t e s and the system outflow. As e x p l a i n e d i n Chapter 2, the ranging i s s t i l l v a l i d even though the o r d i n a t e does not appear on the RHS. 3.4 OUTPUT The output c o u l d be produced i n e i t h e r a t a b u l a r or g r a p h i c a l form. The l i s t i n g i n Appendix A c o n t a i n s output commands f o r a t a b u l a r d i s p l a y , but c o u l d be adapted to g r a p h i c a l d i s p l a y depending on the c a p a b i l i t i e s of the computer system used. The examples c i t e d i n Chapter 5 in c l u d e i l l u s t r a t i o n s of how g r a p h i c a l output might appear on a t e r m i n a l screen. The output i n c l u d e s the p r i m a l s o l u t i o n c o n t a i n i n g the outflow hydrograph o r d i n a t e s and the message i f any value i s not c l o s e to the value determined by the s t r a i g h t f o r w a r d r o u t i n g . The s e n s i t i v i t y i n f o r m a t i o n i s a l s o output, which i n c l u d e s the dual s o l u t i o n v e c t o r , the amount by which w i l l change f o r a u n i t change i n any o r d i n a t e , and the RHS ranging, which giv e s the bounds f o r each o r d i n a t e f o r which the dual value i s v a l i d . Chapter 4 ALTERNATE FORMULATION A d i f f e r e n t LP f o r m u l a t i o n i n v o l v i n g s i g n i f i c a n t l y fewer v a r i a b l e s but s t i l l m a i n t a i n i n g the same minimax fo r m u l a t i o n was a l s o developed, and assessed f o r purposes of sup p o r t i n g runoff system d e s i g n . Both f o r m u l a t i o n s give i d e n t i c a l LP r e s u l t s f o r a simple s e r i e s of reaches, although the simplex tableaus generated i n each case are very d i f f e r e n t . In the a l t e r n a t e f o r m u l a t i o n d e s c r i b e d here, the tableau s i z e depends s o l e l y on the number of hydrograph o r d i n a t e s , and does not i n c r e a s e as the number of reaches i n c r e a s e s . However, the complexity of the c o e f f i c i e n t s generated i n c r e a s e s r a p i d l y as the number of reaches i n c r e a s e s . The more complicated equations were v e r i f i e d using the UBC l i b r a r y program REDUCE. T h i s i n t e r a c t i v e LISP program performs expansion and o r d e r i n g of a l g e b r a i c e x p r e s s i o n s , a l l o w i n g the r e g u l a r but complex runoff equations to be developed a u t o m a t i c a l l y . 4.1 SINGLE REACH The s i n g l e reach case i s the same i n both f o r m u l a t i o n s . The e x p r e s s i o n which forms the b a s i s of the for m u l a t i o n s i s the one given i n Equation (2.3), expressed g e n e r a l l y as: 35 36 The t a b l e a u i s then formed, with the d e c i s i o n v a r i a b l e s being Qi,...,Q* and the minimax v a r i a b l e Y. The RHS c o e f f i c i e n t s are the in f l o w s Q] r...,Q^ plus a f u n c t i o n of Q] and Qj , which are both c o n s i d e r e d c o n s t a n t s . The LP ta b l e a u f o r a s i n g l e channel reach i s the same as that i n Table I I . 4.2 TWO REACHES IN SERIES R e l a t i n g the s e n s i t i v i t y a n a l y s i s from i n f l o w s to the outflows from a s i n g l e reach was- s t r a i g h t f o r w a r d : d e f i n i n g the outflow o r d i n a t e s as d e c i s i o n v a r i a b l e s i n such a way as to equate upstream inflow o r d i n a t e s with l i n e a r f u n c t i o n s of the outflow o r d i n a t e s y i e l d s the d e s i r e d s e n s i t i v i t y output. When f o r m u l a t i n g the case of two reaches i n s e r i e s , the problem arose of how to t r e a t the o r d i n a t e s of the hydrograph at the int e r m e d i a t e p o i n t . When r o u t i n g through the f i r s t reach, the int e r m e d i a t e hydrograph o r d i n a t e s would need to be t r e a t e d as d e c i s i o n v a r i a b l e s , but when r o u t i n g through the second reach, they would need to be p l a c e d i n the RHS. The fo r m u l a t i o n d e s c r i b e d i n Chapter 2 overcame t h i s problem by adding a set of c o n s t r a i n t s and v a r i a b l e s for each a d d i t i o n a l reach. The approach taken here, however, was to remove the inte r m e d i a t e hydrograph from the f o r m u l a t i o n . T h i s was achieved by t a k i n g advantage of the l i n e a r i t y of the r o u t i n g method to combine r o u t i n g equations f o r two (or more) reaches and form a s i n g l e e q u a t i o n . I t i s p o s s i b l e to route 37 a storm hydrograph through more than one reach to o b t a i n the outflow hydrograph, while g a i n i n g no i n f o r m a t i o n on the intermediate hydrographs. A l l that i s needed i s the f i r s t o r d i n a t e of each downstream hydrograph, i n a d d i t i o n to the complete inflow hydrograph. In other words, i t i s p o s s i b l e to d e f i n e the outflow from the second reach i n terms of the in f l o w to the f i r s t reach. R e c a l l that f o r r o u t i n g , the r e q u i r e d input i s the e n t i r e inflow hydrograph and the f i r s t o r d i n a t e of subsequent (downstream) hydrographs. T h e r e f o r e , the d e s i r e d f o r m u l a t i o n w i l l express Q^ as a f u n c t i o n of the o r d i n a t e s shown: From Equation (2.12), the r o u t i n g equation f o r the f i r s t reach i s : Q 1 = / ( Q 3 , Q 3 n- 1 , . • . , Qi , Q i ?, Ql>. (4.2) Qi = ^ Q n + "HQS n- 1 " « i Q n- 1 (4.3) and f o r the second reach i s : Q* = /32Q* + 7 2Q^ n- 1 n- 1 * (4.4) By combining equations (4.3) and (4.4), the Q^ term can be e l i m i n a t e d : 38 Q'n = PiP2Q3n + Pn2Q3n_] ~ (0,02-7! )Q 2 2. 1 - 0 i Q ; _ r (4.5) At the end of the f i r s t r o u t i n g p e r i o d («=2), Equation (4.3) becomes: Ql = 0,Ql + 7 i Q 2 " <t>^Q} , (4.6) Equation (4.4) becomes: Ql = /32Q| + 7 2Q? " 0 2 Q i , (4.7) and Equation (4.5) becomes: Ql = 0 i 0 2 Q l + 0,720? " ( 0 i 0 2 " 7 i ) Q 2 " 0 i Q ] . (4.8) At the end of the second r o u t i n g p e r i o d (n=3), Equation (4.5) becomes: Ql = 0 i 0 2 Q i + 0 i 7 2 Q l " ( 0 i 0 2 - 7 i ) Q ! " 0 i Q 2 - (4.9) By s u b s t i t u t i n g Equation (4.7) f o r Ql and Equation (4.8) f o r Ql i n Equation (4.9), the r e s u l t i n g e x p r e s s i o n f o r Q3 1 i s : Ql = 01020! " t ( 0 i + 0 2 ) 0 i 0 2 - ( 0 i 7 2 + 7 i 0 2 ) ]Qi "[(01+0 2)01-71l7 2Qi + (0i+0 2)(0102-71)Qi + 01 2Q]. (4.10) 39 For subsequent r o u t i n g p e r i o d s , more s u b s t i t u t i o n s are made, always reducing the terms to o b t a i n an e x p r e s s i o n i n the form of Equation (4.2). The r e s u l t i n g equation f o r the end of the t h i r d r o u t i n g p e r i o d (the f o u r t h o r d i n a t e ) i s : Qi = 010 2Q« - [(0i+02)^ 1/3 2-(/3 17 2 +7i/3 2)]Ql + [ ( 0 I 2 + 0 I 0 2 + 0 2 2 ) P M 3 2 - U I + 0 2 ) ( 0 , 7 2 + 71 /32 )+7 1 72 ^ + [ ( 0 l 2 + 0 l 0 2 + 0 2 2 ) j 3 , - ( 0 1 + 0 2 ) 7 i ] 7 2Q? " U l 2 + 01</>2 + </>22) ( 0 1 0 2 - 7 1 ) Q l " 0 1 3Q 1 - (4.11) For the f i f t h o r d i n a t e , at the end of the f o u r t h r o u t i n g p e r i o d , the equation becomes: Ql = 0 i 0 2 Q i - [ ( 0 i + 0 2 ) 0 i 0 2 - ( 0 1 7 2 + 7 i 0 2 ) ] Q 2 + [ ( c 6 1 2 + 0,0 2+<A 2 2) 0, 0 2 - (0 ,+0 2 ) (01 72+7102 ) + 7 l 72 ]Q| - [ (</.13 + 0 l 2 0 2 + 0 1 0 2 2 + 0 2 3 ) 0 1 0 2 " ( 0 1 2 + 0 1 0 2 + 0 2 2 ) ( 0 1 7 2 + 7 1 0 2 ) + ( 0 1 + 0 2 ) 7 l 7 2 ] Q l " t ( 0 1 3 + 0 1 2 0 2 + 0 l 0 2 2 + 0 2 3 ) 0 , - ( 0 , 2 + 0 , 0 2 + 0 2 2 ) 7 l ] 7 2 Q ? + ( 0 , 3 + 0 1 2 0 2 + 0 1 0 2 2 + 0 2 3 ) ( 0 , 0 2 - 7 1 ) Q ? + 0 i 4 Q l - (4.12) It i s obvious that f o r each s u c c e s s i v e r o u t i n g p e r i o d the c o e f f i c i e n t s i n c r e a s e i n complexity as more s u b s t i t u t i o n s are r e q u i r e d . However, by n o t i c i n g the r e c u r r i n g p a t t e r n s formed f o r each s u c c e s s i v e o r d i n a t e , a method of a u t o m a t i c a l l y generating the c o e f f i c i e n t s can be de v i s e d . The method used f o r automatic g e n e r a t i o n of c o e f f i c i e n t s f o r two reaches i n s e r i e s i s d e s c r i b e d here. 40 F i r s t of a l l , the <j> terms can be expressed as a r e c u r s i v e s e r i e s , c 2 , where: n • eg = 1 (4.13) c 2 = = 0 2cg+0, (4.14) c 2 = 4>i2 + (t>i<t>2 + <l>22 = </>2c2 + 0, 2 (4.15) C\ = <t>,3 + <t>l2<t>2 + <P,<l>22 + <p23 = <t>2C22+4>3 (4.16) or i n g e n e r a l , f o r two reaches, c\ = 0 2 ( c 2 _ 1 ) + (<*>,)". (4.17) T h e r e f o r e , f o r any o r d i n a t e n, the reduced r o u t i n g equation f o r two channels i s : tZ {(-))"-J[(cl ) 0 , / 3 2 - ( c 2 ._ > ( 0 l 7 2 + 7 , 0 2 ) + ( c 2 . _ 2 ) 7 l T 2 ] Q p + ( - D n [ ( c 2 _ 2 ) 0 1 - ( C 2 _ 3 ) 7 , ] 7 2 Q ? " ( - i ) ' 7 [ ( c 2 _ 2 ) ( 0 1 0 2 - 7 , ) Q 2 + U , ) " ~ 1 Q l J . (4.18) 4.3 SERIES OF REACHES As d i s c u s s e d i n Chapter 2, the complexity of the c o e f f i c i e n t s i n c r e a s e with each o r d i n a t e . In a d d i t i o n , the degree of complexity grows with each a d d i t i o n a l reach 41 through which the flow i s routed. In d e s c r i b i n g the fo r m u l a t i o n of the gene r a l case of automatic c o e f f i c i e n t g e n e r a t i o n f o r any number of reaches, i t i s perhaps best t o f i r s t present the case of three reaches. Then, by comparison with the one and two reach cases, the general f o r m u l a t i o n can be shown. The b a s i c r o u t i n g equation f o r the t h i r d reach i s : Q> = /33QJ + T a O ^ - * , Q j _ r (4.19) For the f i r s t r o u t i n g p e r i o d , where n = 2, Equation (4.19) becomes: QI = PsQl + 7 3 Q 1 " 03Qi*. ' (4.20) By s u b s t i t u t i n g Equation (4.20) f o r Q| i n Equation (4.8), the f o l l o w i n g e x p r e s s i o n i s d e r i v e d : Ql = /3,02 03Q§' + 010273Qf - 0, (/32<2>3-72)Q^ " (0i02-7i)Q 2 - 0 i Q i . (4.21) For the next r o u t i n g p e r i o d , a f t e r a l l the r e q u i r e d s u b s t i t u t i o n s are made, the r e s u l t i n g equation i s : 42 Ql = 0 i 0 2 / 3 3 Q § " [ 0 , 0 2 0 3 ( 0 1 + 0 2 + 0 3 ) - ( 0 1 0 2 7 3 + 0 l 7 2 0 3 + 7 l 0 2 0 3 ) ] Q 2 " [ 0 , 0 2 ( 0 1 + 0 2 + 0 3 ) " ( 0 1 7 2 + 7 1 0 2 ) ] 7 3 Q l + [ 0 1 ( 0 1 + 0 2 + 0 3 ) - 7 l ] ( 0 2 0 3 - 7 2 ) O j + ( 0 , 0 2 - 7 , ) ( 0 , + 0 2 ) Q i + 0 , 2 Q l . (4.22) For the next r o u t i n g p e r i o d , the r e s u l t i n g e x p r e s s i o n i s : Qi = 0 , 0 2 0 3 Q a - [ 0 , 0 2 0 3 ( 0 1 + 0 2 + 0 3 ) " ( 0 1 0 2 7 3 + 0 , 7 2 0 3 + 7 , 0 2 0 3 ) ] Q 3 + [ 0 1 0 2 0 3 ( 0 1 2 + 0 1 0 2 + 0 1 0 3 + 0 2 2 + 0 2 0 3 + 0 3 2 ) " ( 0 , 027 3 + 01.72 03+71 0 203 ) (01+02 + 03 ) + (017273+710273+71720 3 )1Q§ + [ 0 , 0 2 ( 0 1 2 + 0 1 0 2 + 0 1 0 3 + 0 2 2 + 0 2 0 3 + 0 3 2 ) - ( 0 i 7 2 + 7 i 0 2 ) ( 0 1 + 0 2 + 0 3 ) + 7 , 7 2 ] 7 a Q i 2 2 2 " [ 0 1 ( 0 2 0 3 - 7 2 ) ( 0 , +0102+0103+02 +0203+03 ) ~ 7 1 ( 0 2 0 3 - 7 2 ) ( 0 1 + 0 2 + 0 3 )]Ql " ( 0 1 0 2 - 7 1 ) ( 0 , 2 + 0 , 0 2 + 0 2 2 ) Q i " 0 i 3 Q i - (4.23) By comparing the above equations f o r Ql, Q ], and Qi with the e q u i v a l e n t equations f o r one and two reaches, i t can be seen that there are two d i s t i n c t r e p e a t i n g s e r i e s . The f i r s t i s that f o r the 0 terms, which i s s i m i l a r to t h a t f o r the two-reach case, but now c o n t a i n s a t h i r d element 0 3 . T h i s s e r i e s can be summarized as f o l l o w s : 43 c% = 1 (4.24) c] = + = <t>3cl + c2i (4.25) c\ = 0,2+0102+0103+022+0203+032 = 0 3c?+cl (4.26) or expressed g e n e r a l l y , f o r three reaches, By comparison with Equation (4.17), i t can be shown that the general e x p r e s s i o n of the s e r i e s f o r / reaches can be writen as: cl = (p.c1 , + c'"1 (4.28) n i n-1 n i f c] = </>i, c\ = <f>2, e t c . , and c° = 0 f o r a l l values of n. The other s e r i e s i n the f o r m u l a t i o n c o n t a i n s the 0 and 7 terms. T h i s s e r i e s can be d e s c r i b e d as the sum of the product of a l l permutations of p\ and 7 , as i l l u s t r a t e d , by the f o l l o w i n g examples: />0 , 1 ) = ( 0 i 7 2 + 7 i / 3 2 ) ( 4 . 2 9 ) p(2,\) = (/3 1 / 3 2 7 3 + / 3 i 7 2 0 3 + 7 i 0 2 / 3 3 ) (4 . 30 ) />(1,2) = ( / 3 i 7 2 7 3 + 7 i / 3 2 7 3 + 7 i 7 2 0 3 ) ( 4 . 3 1 ) p{2,2) = ( / 3 , 0 2 7 3 7 « + i 3 i 7 2 / 3 3 7 4 + 7 i / 3 2 0 3 7 i . + ( 3 i 7 2 7 3 0 a + 7 i 0 2 7 3 0 « + 7 i 7 2 03 0 « ) . ( 4 . 3 2 ) 44 Note that the f i r s t parameter i n d i c a t e s the number of j3 c o e f f i c i e n t s i n each term, and the second parameter i n d i c a t e s the number of 7 c o e f f i c i e n t s . The number of terms in a s e r i e s p{a,b) i s given by Walpole (1982) as: { a + b ) l (4.33) albl Using the above n o t a t i o n , the general equation f o r the t h n o r d i n a t e of a system with I nodes (1-1 reaches) i s : n n-m+1 . . T _ . T m=2 j=1 J + ( - 1 ) h [ n 2 1 c J : J _ l ^ ( I - y - l , y - l ) ] 7 l - l Q 5 ; = 1 " J 1 1 ' _ 1-1 7-1 + (-D n _ 1{ Z [ L c J n _ m _ ] P ( j - m - l ,m-))](p <j> - 7 . .JQ7,} j = 2 m=] J J J + ( - l ) " " 1 ^ . ^ ? . (4.34) 4.4 PROGRAM DESCRIPTION The input i s s i m i l a r to the program d e s c r i b e d i n Chapter 3. The drainage network i s d e f i n e d by f i r s t r e a d i n g the number of reaches, and then the values of k and x f o r each reach. The time frame f o r r o u t i n g the flows i s then e s t a b l i s h e d by reading the r o u t i n g p e r i o d and the number of o r d i n a t e s . The i n i t i a l flow c o n d i t i o n s are a l s o read i n . 45 The t a b l e a u s i z e i s c a l c u l a t e d by: NVARS = NORDS (4.35) NCONST = 2*NORDS-2 (4.35) where NVARS i s the number of LP v a r i a b l e s , NCONST i s the number of c o n s t r a i n t s , and NORDS i s the number of o r d i n a t e s . Note t h a t , u n l i k e the fo r m u l a t i o n d e s c r i b e d i n chapter 2 and 3, the t a b l e a u s i z e i n t h i s f o r m u l a t i o n i s independent of the number of channel reaches. The UBC l i b r a r y subroutine REPERM i s used to generate the complex c o e f f i c i e n t s . T h i s r o u t i n e outputs a l l the p o s s i b l e permutations of any number of marks (up to a maximum of 50) with r e p e t i t i o n s . The ta b l e a u i s then generated, f i r s t on the RHS and then the d e c i s i o n v a r i a b l e s . The LIPSUB r o u t i n e i s c a l l e d to perform the l i n e a r programming. The output i s s i m i l a r to that d e s c r i b e d i n Chapter 3, i n c l u d i n g the pri m a l s o l u t i o n (the outflows) and the s e n s i t i v i t y a n a l y s i s (the dual s o l u t i o n vector and the RHS r a n g i n g ) . Because of the d i f f e r e n t method used i n t h i s f o r m u l a t i o n , there was no problem with cumulative e r r o r that o c c u r r e d i n the other f o r m u l a t i o n , so there was no reason to compare with routed flows. 46 4.5 COMMENT The f o r m u l a t i o n presented in Chapter 2 provided by f a r the most comprehensive s e n s i t i v i t y r e s u l t s f o r branched networks, but the dramatic r e d u c t i o n in the number of v a r i a b l e s i n t h i s a l t e r n a t i v e f o r m u l a t i o n could be i n v a l u a b l e when performing o p t i m i z a t i o n - b a s e d a n a l y s i s on h i g h l y complex drainage systems. As p r e v i o u s l y demonstrated f o r a s e r i a l system, the number of v a r i a b l e s and c o n s t r a i n t s are determined s o l e l y by the number of time i n t e r v a l s (ordinates). used in the routing, process, and. are- independent of the number of reaches. The a l g e b r a i c manipulations r e q u i r e d to determine the c o e f f i c i e n t s are complex, but as has been demonstrated above, are amenable to automatic g e n e r a t i o n and can t h e r e f o r e be e s t a b l i s h e d e - f f i c i e n t l y even f o r l a r g e systems. Extensions of the f o r m u l a t i o n presented i n t h i s chapter might t h e r e f o r e serve as a u s e f u l paradigm i n the development of other f o r m u l a t i o n s f o r the a n a l y s i s of runoff systems. T h i s would be of p a r t i c u l a r p r a c t i c a l s i g n i f i c a n c e in a runoff system a n a l y s i s scheme based on a n o n - l i n e a r formulat i o n . Another p o s s i b i l i t y , not e x p l o r e d here, would be the e s t i m a t i o n of the v a l u e s of the c o e f f i c i e n t s i n formulations of the type d e s c r i b e d i n t h i s chapter by some approximation method. The method f o r g e n e r a t i n g the c o e f f i c i e n t s a c c u r a t e l y , as d e s c r i b e d above, would s t i l l be necessary f o r v e r i f i c a t i o n of an e s t i m a t i o n method. Chapter 5 NUMERICAL EXAMPLES The model d e s c r i b e d in chapters 2 and 3 was t e s t e d on a v a r i e t y of s i t u a t i o n s which may occur i n a design s e t t i n g . Some of the r e s u l t s are presented here as examples of the behaviour of the program. It i s u s e f u l to r e c a l l t h a t the r e s u l t s of the s e n s i t i v i t y a n a l y s i s (dual s o l u t i o n and RHS ranging) show which of the inflow o r d i n a t e s , i f mo d i f i e d , w i l l have the g r e a t e s t e f f e c t i n reducing the peak outflow. The dual s o l u t i o n value f o r a p a r t i c u l a r c o n s t r a i n t g i v e s the decrease i n peak outflow per u n i t decrease i n the corresponding i n f l o w o r d i n a t e , w i t h i n the determined l i m i t s . The RHS ranging gives the bounds beyond which the peak w i l l occur at a d i f f e r e n t o r d i n a t e on the outflow hydrograph. The purpose of the model should a l s o be kept i n mind while reviewing the f o l l o w i n g examples. The i n t e n t of the model i s not to produce a f i n a l d e t a i l e d d e s i g n , but ra t h e r to a i d in the i n i t i a l p l a n n i n g p r o c e s s . The model's aim i s to q u i c k l y p o i n t out to the designer the areas of the drainage network which c o u l d be m o d i f i e d i n order to e f f e c t the g r e a t e s t r e d u c t i o n i n peak flow at a downstream l o c a t i o n , and consequently reduce the r e q u i r e d channel dimensions or f l o o d consequences. I t i s intended to provide such i n f o r m a t i o n q u i c k l y and without the cumbersome c a l c u l a t i o n s t h a t design models r e q u i r e . 47 48 5.1 CASE 1: SINGLE REACH The s i m p l e s t case i n v o l v e s r o u t i n g a storm hydrograph through a s i n g l e reach, with no other i n f l o w s . T h i s example corresponds approximately to a t r a p e z o i d a l e a r t h channel with a bottom width of 2.0 m, s i d e slopes of 2h:1v, a gra d i e n t of 0.1%, and a Manning's n value of 0.022. The channel l e n g t h i s approximately 500 m, with no i n f l o w p o i n t s other than at the upstream end. The value of k, taken as e q u i v a l e n t to the propagation time, i s 8.0 minutes. The r o u t i n g p e r i o d t was s e l e c t e d as 5.0 minutes, s l i g h t y l e s s than the propagation time so that the f l o o d peak c o u l d not pass through the reach undetected. The value of x was a r b i t r a r i l y set at 0.20. I n i t i a l c o n d i t i o n s were uniform throughout the channel, with the f i r s t time p e r i o d inflow equal to the outflow. A f l o o d event was then routed through the reach. The f o l l o w i n g examples demonstrate some of the b a s i c f e a t u r e s of the model. Example 1.1: N a t u r a l C o n d i t i o n s The f i r s t run of the program was to perform the r o u t i n g under n a t u r a l c o n d i t i o n s and observe the s e n s i t i v i t y i n f o r m a t i o n and i t s r e l a t i o n to the peak outflow. The r e s u l t s of the f i r s t run are given i n Table VI and shown g r a p h i c a l l y i n F i g u r e 1. The i n f l o w hydrograph reaches a peak flow of 5.05 m3/s at the f o u r t h o r d i n a t e , and due to t r a n s l a t i o n and a t t e n u a t i o n , the outflow reaches a peak of 49 TABLE VI EXAMPLE 1.1 RESULTS EXAMPLE 1.1: SINGLE REACH, TRAPEZOIDAL CHANNEL. THERE ARE 2 STATIONS AND 1 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA GAMMA PHI PS I 1 8.000 0.200 9.8889 -4.3333 4.5556 -49.383 THERE ARE 10 HYDROGRAPH ORDINATES, 18 CONSTRAINTS, AND 10 VARIABLES. S T A . 1 2 3 4 5 6 7 8 9 10 1: 0.500 1.450 3.675 5.050 4.175 3.620 3.160 2.420 2.020 1.850 2: 0.500 0.596 1.301 2.774 3.964 4.026 3.752 3.344 2.785 2.338 PEAK OUTFLOW = 4.026426 PRIMAL SOLUTION: VARIABLE VALUE Q( 2, 2) 0.596 Q( 2, 3) 1 .301 Q( 2, 4) 2.774 Q( 2, 5) 3.964 Q( 2, 6) 4.026 Q( 2, 7) 3.752 Q( 2, 8) 3.344 Q( 2, 9) 2.785 Q( 2,10) 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 RHS RANGING BOUNDS VALUE FLOW LOWER UPPER 0.0425 1 .4500 -1 .1259 2.5955 0.0970 3.6750 -1.3702 4. 1770 0.2213 5.0500 2.8392 5.2700 0.5050 4.1750 4.0205 0. 1 0047E+19 0.1011 3.6200 3.0029 4.3005 0.0000 3. 1600 -3.4627 4.5106 0.0000 2.4200 -3.0943 4.8791 0.0000 2.0200 -2.6096 5.3638 0.0000 1.8500 -21.269 18.548 50 F I G U R E 1 E X A M P L E 1.1 E X A M P L E 1.1 S I N G L E R E A C H 10 - i 1 1-e 7 T I M E I N T E R V A L 10 51 4.03 m 3/sec f i v e minutes l a t e r ( F i g u r e 2). The dual s o l u t i o n vector i n d i c a t e s t h a t o r d i n a t e s 2 to 6 of the i n f l o w hydrograph i n f l u e n c e the peak outflow, and that the f i f t h o r d i n a t e (Ql) has the g r e a t e s t i n f l u e n c e . If one of these i n f l o w o r d i n a t e s i s a l t e r e d ( e i t h e r i n c r e a s e d or decreased), then the outflow peak w i l l be changed by an amount equal to the change of i n f l o w m u l t i p l i e d by the dual value (provided that the r e s u l t i n g d i s c h a r g e i s w i t h i n the l i m i t s given by the RHS r a n g i n g ) . The f o l l o w i n g three examples demonstrate t h i s p r i n c i p l e . Example 1.2: Ql Modified T h i s example i s i d e n t i c a l to the p r e v i o u s example, except t h a t Ql i s now reduced by 1 m3/s., from 5.05 to 4.05 m 3/s, as shown i n Table VII and F i g u r e 3. T h i s c o n d i t i o n r e p r e s e n t s a temporary d i v e r s i o n or storage of the flows above a c e r t a i n l e v e l . The r e s u l t i n g peak outflow i s 3.085 m 3/s, a decrease of 0.2213 from Example 1.1. T h i s d i f f e r e n c e i n e x a c t l y equals the dual value a s s o c i a t e d with Ql, as expected, s i n c e there was a u n i t decrease i n Ql. Since the new value of Ql i n Example 1.2 i s s t i l l w i t h i n the bounds given i n the RHS ranging i n f o r m a t i o n i n Example 1.1, the optimal b a s i s i s unchanged; i . e . , the peak outflow occurs at the same time p e r i o d i n both examples and the dual s o l u t i o n v e c t o r s are a l s o unchanged. I t i s i n t e r e s t i n g to note that although the upper and lower bounds fo r some of the o r d i n a t e s have changed from those i n Example 52 FIGURE 2 CASE 1 OUTFLOW HYDROGRAPHS C A S E 1 O U T F L O W H Y D R O G R A P H S 5 53 TABLE VII EXAMPLE 1.2 RESULTS EXAMPLE 1.2: SINGLE REACH, Q(l,4) reduced by 1. THERE ARE 2 STATIONS AND 1 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA GAMMA 1 8.000 0.200 9.8889 -4.3333 PHI 4.5556 PSI -49.383 THERE ARE 10 HYDROGRAPH ORDINATES, 18 CONSTRAINTS, AND 10 VARIABLES. STA. 1 2 3 4 5 6 7 8 9 1 0 1: 0.500 1.450 3.675 4.050 4.175 3.620 3.160 2.420 2.020 1.850 2: 0.500 0.596 1.301 2.673 3.459 3.805 3.655 3.302 2.766 2.330 PEAK OUTFLOW = 3.805139 PRIMAL SOLUTION: VARIABLE VALUE Q( 2, 2) 0.596 Q( 2, 3) 1 .301 Q( 2, 4) 2.673 Q( 2, 5) 3.459 Q( 2, 6) 3.805 Q( 2, 7) 3.655 Q( 2, 8) 3.302 Q( 2, 9) 2.766 Q( 2,10) 2.330 SENSITIVITY ANALYSIS: DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1, 2) 0.0425 1.4500 -1.1259 6.8648 2 Q( 1, 3) 0.0970 3.6750 0.91183 6.4508 3 Q( 1, 4) 0.2213 4.0500 2.8392 5.2700 4 Q( 1, 5) 0.5050 4.1750 3.6444 0.10000E+76 5 Q( 1, 6) 0.1011 3.6200 0.19742 3.9927 6 Q( 1, 7) 0.0000 3.1600 -3.3786 4.1565 7 Q( 1, 8) 0.0000 2.4200 -3.0574 4.4777 8 Q( 1, 9) 0.0000 ' 2.0200 -2.5934 4.9417 9 Q( 1,10) 0.0000 1.8500 -21.188 16.440 54 FIGURE 3 EXAMPLE 1.2 E X A M P L E 1 . 2 4 t h O R D I N A T E R E D U C E D 55 1.1, the bounds f o r the m o d i f i e d o r d i n a t e have remained c o n s t a n t . Example 1.3: Ql M o d i f i e d T h i s example i l l u s t r a t e s the e f f e c t s of changing an i n f l o w o r d i n a t e beyond the l i m i t s of the RHS ranging. The r e s u l t s are l i s t e d i n Table VIII and d i s p l a y e d i n Fig u r e 4. I n i t i a l flow c o n d i t i o n s are i d e n t i c a l to Example 1.1, with the exception of a u n i t decrease in Ql. In t h i s case, the new value of i s l e s s than the lower bound for that o r d i n a t e given i n Example 1.1. The peak outflow, which o c c u r r e d at the s i x t h r o u t i n g p e r i o d i n Example 1.1, now occurs at the f i f t h p e r i o d i n t h i s example, and there i s a new optimal b a s i s and dual v e c t o r . The r e s u l t i n g peak outflow i n t h i s example i s l e s s than i n the o r i g i n a l example by 0.1635 m 3/s. Comparison of the dual v e c t o r s f o r Examples 1.1 and 1.3 show that the values f o r each c o n s t r a i n t have simply s h i f t e d by one time i n t e r v a l . For example, the dual v a l u e a s s i g n e d to Ql i n Example 1.1 i s assign e d to Ql i n Example 1.3, and Qs takes on the value p r e v i o u s l y assigned to Ql. S i m i l a r l y , a l l other outflow v a r i a b l e s take on dual v a l u e s p r e v i o u s l y a s s i g n e d to an adjacent v a r i a b l e . Since the b a s i s changes from one example to the next, the dual s o l u t i o n does not g e n e r a l l y provide i n f o r m a t i o n to p r e d i c t the e f f e c t of a change on the optimal s o l u t i o n . In t h i s case, however, i t i s p o s s i b l e to p r e d i c t the change i n 56 TABLE V I I I EXAMPLE 1.3 RESULTS EXAMPLE 1.3: SINGLE REACH, Q(l,5) reduced by 1. THERE ARE 2 STATIONS AND 1 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA 1 8.000 0.200 9.8889 GAMMA •4.3333 PHI 4.5556 PSI •49.383 THERE ARE 10 HYDROGRAPH ORDINATES, 18 CONSTRAINTS, AND 10 VARIABLES. STA. 1 : 2: 1 0.500 0.500 2 1 .450 0.596 3 3.675 1 .301 PEAK OUTFLOW = 3.862899 PRIMAL SOLUTION: VARIABLE VALUE Q( 2, 2) 0.596 Q( 2, 3) 1 .301 Q( 2, 4) 2.774 Q( 2, 5) 3.863 Q( 2, 6) 3.521 Q( 2, 7) 3.530 Q( 2, 8) 3.247 Q( 2, 9) 2.742 Q( 2,10) 2.319 SENSITIVITY ANALYSIS: 4 5 6 7 8 9 10 5.050 3.175 3.620 3.160 2.420 2.020 1.850 2.774 3.863 3.521 3.530 3.247 2.742 2.319 DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1, 2) 0.0970 1 .4500 -1.1259 7.7294 2 Q( 1, 3) 0.2213 3.6750 1.8147 7.5144 3 Q( 1, 4) 0.5050 5.0500 4.2348 0.31205E+18 4 Q( 1, 5) 0.1011 3.1750 -3.7983 4.0205 5 Q( 1, 6) 0.0000 3.6200 -3.3709 4.2786 6 Q( 1, 7) 0.0000 3.1600 -3.2707 4.3788 7 Q( 1, 8) 0.0000 2.4200 -3.0101 4.6394 8 Q( 1, 9) 0.0000 2.0200 -2.5727 5.0768 9 Q( 1,10) 0.0000 1.8500 -21.085 17.115 FIGURE 4 EXAMPLE 1.3 EXAMPLE 1.3 5th ORDINATE REDUCED 57 10 CD CC < o CO Q 4 6 6 7 T I M E I N T E R V A L 10 58 Qp i n advance by p r o - r a t i n g the dual value f o r Qj before and a f t e r the s h i f t i n b a s i s . While the flow o r d i n a t e i s above the lower bound (4.0205), the dual value of 0.5050 a p p l i e s and the decrease i n Q when Qi decreases from 4.1750 to P 4.0205 i s (0.5050)•(0.1545), or 0.0780. When Q^ i s below the lower bound, the change i n peak outflow i s given by (4.0205-3.1750)•(0.1011), or 0.0855. The t o t a l d i f f e r e n c e in peak outflow from one example to the next i s the sum of the two v a l u e s , or 0.1635 as shown above. Example 1.4: Zero Ordinates A run of the model was made to c o n f i r m that Q was not P changed when the v a l u e s of the o r d i n a t e s with dual v a l u e s of zero were m o d i f i e d . The i n f l o w c o n d i t i o n s are i d e n t i c a l to Example 1.1, except that the seventh through tenth o r d i n a t e s of the i n f l o w hydrograph (each with a dual value of zero) are set to z e r o . I n s p e c t i o n of Table IX and F i g u r e 2 r e v e a l s that the peak outflow (and a l l preceeding o r d i n a t e s ) are the same i n both cases, as are the dual s o l u t i o n v a l u e s . Example 1.5: New Inflow Hydrograph In t h i s example, the same channel c h a r a c t e r i s t i c s were used, but with d i f f e r e n t i n i t i a l flow c o n d i t i o n s and i n f l o w hydrograph. Table X and F i g u r e 5 show the r e s u l t s . Comparison with p r e v i o u s examples shows that although t h i s produces s i g n i f i c a n t l y d i f f e r e n t flows, the dual v e c t o r i s i d e n t i c a l to that i n Example 1.1. 59 TABLE IX EXAMPLE 1.4 RESULTS EXAMPLE 1.4: SINGLE REACH, Q(l,7) to Q(1,10) = 0. THERE ARE 2 STATIONS AND 1 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA GAMMA PHI PSI 1 8.000 0.200 9.8889 -4.3333 4.5556 -49.383 THERE ARE 10 HYDROGRAPH ORDINATES, 18 CONSTRAINTS, AND 10 VARIABLES. STA. 1 2 3 4 5 6 7 8 9 1 0 1: 0.500 1.450 3.675 5.050 4.175 3.620 0.0 0.0 0.0 0.0 2: 0.500 0.596 1.301 2.774 3.964 4.026 3.432 1.504 0.659 0.289 PEAK OUTFLOW = 4.026426 PRIMAL SOLUTION: VARIABLE VALUE Q( 2, 2) 0.596 Q( 2, 3) 1 .301 Q( 2, 4) 2.774 Q( 2, 5) 3.964 Q( 2, 6) 4.026 Q( 2, 7) 3.432 Q( 2, 8) 1 .504 Q( 2, 9) 0.659 Q( 2,10) 0.289 SENSITIVITY ANALYSIS: DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1, 2) 0.0425 1 .4500 -1.1259 2.5955 2 Q( 1, 3) 0.0970 3.6750 -1.8176 4.1770 3 Q( 1, 4) 0.2213 5.0500 0.26875 5.2700 4 Q( 1, 5) 0.5050 4.1750 4.0205 0.10047E+19 5 Q( 1, 6) 0.1011 3.6200 3.0029 5.0918 6 Q( 1, 7) 0.0000 0.0 -2.9781 4.9952 7 Q( 1, 8) 0.0000 0.0 -1.3050 6.6683 8 Q( 1, 9) 0.0000 0.0 -0.57187 7.4015 9 Q( 1,10) 0.0000 0.0 -2.8558 36.961 60 TABLE X EXAMPLE 1.5 RESULTS EXAMPLE 1.5: SINGLE REACH, D i f f e r e n t Inflow. THERE ARE 2 STATIONS AND 1 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA 1 8.000 0.200 9.8889 GAMMA -4.3333 PHI 4.5556 PS I -49.383 THERE ARE 10 HYDROGRAPH ORDINATES, 18 CONSTRAINTS, AND 10 VARIABLES. STA. 1 1: 1.250 2: 1.250 PEAK OUTFLOW 2 .250 ,250 3 4.450 1 .574 6.252089 PRIMAL SOLUTION: VARIABLE VALUE Q( 2, 2) 1 .250 Q( 2, 3) 1 .574 Q( 2, 4) 3.435 Q( 2, 5) 5.405 Q( 2, 6) 6.252 Q( 2, 7) 5.632 Q( 2, 8) 4.705 Q( 2, 9) 3.911 Q( 2,10) 3.392 SENSITIVITY ANALYSIS: 4 5 6 7 8 9 10 6.875 7.250 5.375 4.120 3.350 3.025 2.820 3.435 5.405 6.252 5.632 4.705 3.911 3.392 DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1, 2) 0.0425 1.2500 -1.8661 1 1 .366 2 Q( 1, 3) 0.0970 4.4500 -2.3517 1 1 .261 3 Q( 1, 4) 0.2213 6.8750 1 .8906 9.8594 4 Q( 1, 5) 0.5050 7.2500 5.1535 0.10000E+76 5 Q( 1, 6) 0.1011 5.3750 -2.9977 6.9093 6 Q( 1, 7) 0.0000 4.1200 -5.1969 7.1838 7 Q( 1, 8) 0.0000 3.3500 -4.3945 7.9862 8 Q( 1, 9) 0.0000 3.0250 -3.6929 8.6878 9 Q( 1,10) 0.0000 2.8200 -30.728 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: THREE REACHES IN SERIES The f o l l o w i n g examples represent a s i t u a t i o n where there are three d i s t i n c t reaches i n s e r i e s , p l u s two p o t e n t i a l s i t e s f o r c o n s t r u c t i o n of r e s e r v o i r s below the f i r s t and second reaches ( F i g u r e 6). The f i r s t channel i s i d e n t i c a l to the channel i n Case 1. The second channel i s s i m i l a r , but has a higher Manning's n value of 0.027 to represent i n c r e a s e d roughness, and a steeper g r a d i e n t of 0.2%. The k value f o r t h i s channel i s 11 minutes, and x was chosen as 0.175. The l a s t reach i n the' system represents, a. concrete c u l v e r t with a 2 m diameter, a roughness c o e f f i c i e n t of 0.013, and a g r a d i e n t of 0.15%. The value of k i s 6.4 minutes and x i s 0.23. P r o v i s i o n was made f o r the i n c l u s i o n of l i n e a r r e s e r v o i r s a f t e r the f i r s t and before the l a s t reaches. T h e r e f o r e , the channel reaches are numbered 1, 3, and 5. I n i t i a l l y , the two p o t e n t i a l r e s e r v o i r s i t e s (reaches 2 and 4) are set up to behave as ' n u l l reaches', which allow f o r d i r e c t t r a n s f e r of the i n f l o w s with no t r a n s l a t i o n or attenuat i o n . Given that the designer has the choice of modifying the flow (through storage, d i v e r s i o n , or some other means) e i t h e r before i t e n t e r s the system or by d e t e n t i o n storage at reach 2 or 4, the dual s o l u t i o n values should a i d i n determining the most e f f i c i e n t method of reducing the peak outflow from the system. The f o l l o w i n g examples demonstrate the r e s u l t s from s e v e r a l types of system m o d i f i c a t i o n , and 63 STATION 1: INFLOW FIGURE 6 EXAMPLE 2.1 STATION 2 TIMC INTCWVM. STATION 3 TIMl INTCKV*L STATION S T1KIC INTCPVAU A2 / \ 3 16 TIME INTERVAL STATION A TTMC INTERVAL STATION 6 : O U T F L O W TtMt INTERVAL 64 t h e r e s u l t i n g h y d r o g r a p h s a t t h e n e t w o r k o u t l e t a r e i l l u s t r a t e d i n F i g u r e 7 . Example 2.1: N a t u r a l C o n d i t i o n s The f i r s t e x a m p l e i s s i m p l e r o u t i n g t h r o u g h t h r e e r e a c h e s . I n s p e c t i o n o f F i g u r e 6 a n d t h e o u t p u t i n T a b l e X I r e v e a l s t h a t , t h e u p p e r r e a c h e s g e n e r a l l y e x h i b i t a n e v e n l y d i s t r i b u t e d s e t o f d u a l v a l u e s f o r s e v e r a l o r d i n a t e s , w h i l e i n t h e l o w e r r e a c h e s one o r d i n a t e t e n d s t o h a v e an e x c e p t i o n a l l y h i g h d u a l v a l u e . T h i s i s t o be e x p e c t e d . , s i n c e f l o w s i n t h e l o w e r r e a c h e s h a v e a much more d i r e c t i n f l u e n c e on t h e o u t f l o w p e a k t h a n do f l o w s i n t h e more r e m o t e u p p e r r e a c h e s . To m e e t t h e o b j e c t i v e o f r e d u c i n g t h e o u t f l o w . p e a k , t h e a p p r o a c h t a k e n i s t o d e c r e a s e t h e f l o w a t t h e t i m e s w h i c h i n d i c a t e h i g h s e n s i t i v i t y , w h i l e p e r m i t t i n g t h e f l o w t o i n c r e a s e a t t h o s e t i m e s w h i c h w o u l d h a v e l i t t l e o r no e f f e c t on Q ^ . I n a s t r i c t s e n s e , p r e d i c t i n g a c h a n g e i n a d o w n s t r e a m r e a c h a s a r e s u l t o f c h a n g e s i n s e v e r a l f l o w o r d i n a t e s on t h e b a s i s o f t h e s e n s i t i v i t y i n f o r m a t i o n i s o n l y c o r r e c t f o r c h a n g e s i n f l o w o r d i n a t e s i n c o m b i n a t i o n w h i c h do n o t r e s u l t i n a c h a n g e i n o p t i m a l b a s i s ( S o l o w , 1984). Example 2.2: Ql M o d i f i e d The same i n f l o w h y d r o g r a p h was r o u t e d t h r o u g h t h e same s y s t e m , w i t h one i n f l o w o r d i n a t e c h a n g e d ( F i g u r e 8). Ql was 65 FIGURE 7 CASE 2 OUTFLOW HYDROGRAPHS C A S E 2 O U T F L O W H Y D R O G R A P H S 3.5-C3 or. < x o CO Q 2.5-2 -1.6 -1 -0.6 LEGEND O R i a i N A L 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 0-L 1 - i 1 1 1 r-4 6 6 7 8 T I M E I N T E R V A L 10 TABLE XI EXAMPLE 2.1 RESULTS 66 EXAMPLE 2.1: 3 REACHES, 2 NULL RESERVOIRS THERE ARE 6 STATIONS AND 5 CHANNEL REACHES. THE ROUTING PERIOD IS 5.000 REACH K X BETA GAMMA PHI PS I 8. 000 0 .200 9. 8889 -4.3333 4.5556 -49. 383 2 0. 0 0 .0 1 . 0000 1.0000 1 .0000 -0.0 3 1 1 . 000 0 .175 20 .130 - 11.435 7.6957 -166 .35 4 0. 0 0 .0 1 . 0000 1.0000 1 .0000 -0.0 5 6. 400 0 .230 7. 2257 -2.3619 3.8638 -30. 281 THERE ARE 10 HYDROGRAPH ORDINATES * 54 CONSTRAINTS, AND 46 VARIABLES • STA. 1 2 3 4 5 6 7 8 9 10 1 2. 375 5 .050 2 .775 2. 120 1 .545 1 .220 1 .045 0 .865 0.755 0 .680 2 0. 500 1 .824 3 .406 2.985 2 .441 1 .905 1 .502 1 .227 1 .013 0 .860 3 0. 500 1 .824 3 .406 2.985 2 .441 1 .905 1 .502 1 .227 1.013 0 .860 4 0. 500 0 .566 1 .188 2. 125 2 .470 2 .431 2.184 1 .876 1 .585 1 .330 5 0. 500 0 .566 1 .188 2. 125 2 .470 2 .431 2.184 1 .876 1 .585 1 .330 6 0. 500 0 .509 0 .633 1 . 136 1 .850 2 .262 2.341 2 . 1 92 1 .939 1 .665 SENSITIVITY ANALYSIS: DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1 , 2) 0. 1724 5.0500 1.8115 7.5006 2 Q( 1 , 3) 0.2049 2.7750 -0.95419 0.26091E+19 3 Q( 1 r 4) 0.1846 2.1200 1.4065 9.4592 4 Q( 1 5) 0.0731 1.5450 0.23844 2.8786 5 Q( 1 6) 0.0121 1.2200 -1.7551 3.5524 6 Q( 1 7) 0.0007 1.0450 -1.3852 4.7200 7 Q( 1 8) 0.0000 0.86500 -1.1402 10.117 8 Q( 1 9) 0.0000 0.75500 -0.94861 56.473 9 Q( 1 10) 0.0000 0.68000 -7.8274 972.86 10 Q( 2 2) 0.0978 1.8239 0.0 3.1477 1 1 Q( 2 3) 0.1579 3.4062 0.93212 4.4455 12 Q( 2 4) 0.2345 2.9854 1.0430 4.7425 13 Q( 2 5) 0.2798 2.4411 2.0311 0.61159E+11 14 Q( 2 6) 0.0856 1 .9048 0.89401 2.6710 15 Q( 2 7) 0.0069 1.5024 0.0 2.9765 16 Q( 2, 8) 0.0000 1.2272 0.0 3.6430 1 7 Q( 2, 9) 0.0000 1.0126 0.0 8.9063 18 Q( 2, 10) 0.0000 0.86030 0.0 99. 171 19 Q( 3, 2) 0.0978 1.8239 -1.0697 3.1477 20 Q( 3, 3) 0.1579 3.4062 0.93212 4.4455 21 Q( 3, 4) 0.2345 2.9854 1.0430 4.7425 22 Q( 3, 5) 0.2798 2.441 1 2.0311 0.61159E+11 23 Q( 3, 6) 0.0856 1.9048 0.89401 2.6710 24 Q( 3, 7) 0.0069 1.5024 -3.0667 2.9765 25 Q( 3, 8) 0.0000 1.2272 -2.6336 3.6430 26 Q( 3, 9) 0.0000 1.0126 -2.2276 8.9063 27 Q( 3, 10) 0.0000 0.86030 -25.915 99.171 28 Q( 4, 2) 0.0066 0.56576 0.0 3.5446 29 Q( 4, 3) 0.0203 1.1878 0.0 3.0961 30 Q( 4, 4) 0.0620 2.1252 0.0 2.7490 31 Q( 4, 5) 0.1896 2.4697 1.3040 2.6736 32 Q( 4, 6) 0.5800 2.4307 2.2505 0.39078E+16 33 Q( 4, 7) 0.1384 2.1835 1.6084 2.5204 34 Q( 4, 8) 0.0000 1.8756 0.0 2.5692 35 Q( 4, 9) 0.0000 1.5849 0.0 2.7502 36 Q( 4, 10) 0.0000 1.3301 0.0 6.2138 37 Q( 5, 2) 0.0066 0.56576 -0.52625 3.5446 38 Q( 5, 3) 0.0203 1.1878 -0.77142 3.0961 39 Q( 5, 4) 0.0620 2.1252 -1.0640 2.7490 40 Q( 5, 5) 0.1896 2.4697 1.3040 2.6736 41 Q( 5 6) 0.5800 2.4307 2.2505 0.39078E+16 42 Q( 5 7) 0.1384 2.1835 1.6084 2.5204 43 Q( 5 8) 0.0000 1 .8756 -1.4676 2.5692 44 Q( 5, 9) 0.0000 1.5849 -1.2866 2.7502 45 Q( 5 10) 0.0000 1.3301 -10.703 6.2138 67 FIGURE 8 EXAMPLE 2.2 STATION 1: INFLOW STATION 2 T5 STATION 5 STATION 6: OLTTR-OW 16 TlMt INTERVAL 68 reduced by one flow u n i t to 4.05, a value s t i l l above the lower bound determined in Example 2.1. The r e s u l t i n g r e d u c t i o n of Qp by 0.1724 c o u l d be p r e d i c t e d by n o t i c i n g that t h i s amount i s a l s o the value of the dual v a r i a b l e a s s o c i a t e d with Ql (Table X I I ) . T h i s p r e d i c t i o n i s p o s s i b l e even though intermediate hydrograph o r d i n a t e s were changed as a r e s u l t of the i n f l o w p e r t u r b a t i o n . The optimal b a s i s and dual v e c t o r from Example 2.1 remained unchanged i n Example 2.2. Example 2.3: R e s e r v o i r at Reach 4 A l i n e a r r e s e r v o i r with a k value of ten minutes was added at reach 4 to approximate the behaviour of a d e t e n t i o n storage b a s i n . T h i s r e s u l t e d i n a t r a n s l a t i o n of the peak flow e n t e r i n g reach 5 by two r o u t i n g p e r i o d s , and a n o t i c e a b l e a t t e n u a t i o n (Figure 9). I n c l u s i o n of the r e s e r v o i r a l s o had an e f f e c t on the s e n s i t i v i t i e s of a l l hydrograph o r d i n a t e s , even at p o i n t s above the r e s e r v o i r where the flow wasn't a l t e r e d . Example 2.4: R e s e r v o i r at Reach 2 A l i n e a r r e s e r v o i r was i n s e r t e d at reach 2 i n s t e a d of reach 4, and compared with the p r e v i o u s example. The storage c h a r a c t e r i s t i c s of the r e s e r v o i r s in the two examples were i d e n t i c a l ( i e , they both had the same value of k). Comparing F i g u r e 10 with F i g u r e 9 i t can be seen that the system outflow peak and the e n t i r e outflow hydrograph 69 TABLE XII PEAK OUTFLOWS FOR CASE 2 EXAMPLES Example D e s c r i p t i o n Reduction 2.1 N a t u r a l c o n d i t i o n s 2.34 -2.2 Q« reduced 2.17 7.3 % 2.3 Re s e r v o i r at Reach 4 2.01 14.1 % 2.4 Re s e r v o i r at Reach 5 2.01 14.1 % 2.5 C o n t r o l l e d withdrawal at Reach 2 2.08 11.1 % are i d e n t i c a l i n both cases, even though reach 2 e x h i b i t e d d i f f e r e n t dual v a l u e s than reach 4 i n Example 2.1. T h i s r e s u l t i s to be expected, s i n c e the system i s a l i n e a r one and t h e r e f o r e the outcome i s not dependent on the order of the components. Example 2.5: D i r e c t M o d i f i c a t i o n of Flow The previous two examples demonstrated the e f f e c t s of channel m o d i f i c a t i o n s and how the model attempts to p r e d i c t the e f f e c t s on the magnitude of the peak outflow. I t may a l s o be d e s i r a b l e to p r e d i c t the consequences of c o n t r o l l e d withdrawals, and t h i s example demonstrates one method by which the model can p r e d i c t the e f f e c t s of such a d i r e c t flow m o d i f i c a t i o n . In t h i s example, a l l flows above 2.5 m3/s at the t h i r d node are e x t r a c t e d from the system. Since the model i n i t s present form only a l l o w s f o r d i r e c t flow m o d i f i c a t i o n at FIGURE 9 EXAMPLE 2.3 STATION 1: INFLOW e.* STATION S j j i j i sr-i TIME INTCftVUkL 70 STATION 2 TIME INTERVAL STATION 4. i TIMC INTERVAL STATION 6 : O U T F L O W TIME I N T C B V . L 71 FIGURE 10 EXAMPLE 2.4 S T A T I O N * I N F L O W S T A T I O N 2 ... ..4 TIME INTERVAL S T A T I O N A. i i i i t i ! ! i i t i 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 was simulated by t r a n s f e r r i n g the hydrograph at S t a t i o n 3 i n Example 2.1 to S t a t i o n 1 and a l t e r i n g the flow at the a p p r o p r i a t e o r d i n a t e s . To a v o i d unwanted t r a n s l a t i o n and a t t e n u a t i o n e f f e c t s , the f i r s t two reaches were converted to n u l l reaches by changing the k values back to zero. By t h i s method, the model only d e a l s with the part of the drainage system downstream from S t a t i o n 3, without changing the i d e n t i t y numbers of each reach. The r e s u l t s given i n Table XII and Fig u r e 11 show that the flow m o d i f i c a t i o n s at S t a t i o n 3 r e s u l t e d i n a peak outflow of 2.084 m 3/s, a r e d u c t i o n of 0.257 from the o r i g i n a l case presented i n Example 2.1. The model was able to a c c u r a t e l y p r e d i c t t h i s r e s u l t , as can be seen by examining the output i n Table XI. Ql and Ql were the only o r d i n a t e s modified, and m u l t i p l y i n g the change i n each o r d i n a t e by the a p p r o p r i a t e dual value and summing the products g i v e s the exact value of the peak flow r e d u c t i o n which a c t u a l l y o c c u r r e d . Example 2.6: New Inflow Hydrograph The same system as Example 2.1 was used here, but with a completely d i f f e r e n t i nflow hydrograph. Comparing Table XIII with Table XI and F i g u r e 12 with F i g u r e 6 r e v e a l s that the dual s o l u t i o n i s the same as that f o r Example 2.1. T h i s reconfirms that the dual s o l u t i o n v e c t o r depends s o l e l y on the values of k and x and the time of the peak o r d i n a t e at the outflow p o i n t , and has no d i r e c t dependence on the 73 STATION 1: INFLOW FIGURE 11 EXAMPLE 2.5 STATION 2 TIMC INTERVAL STATION 3 TIMC INTERVAL STATION 5 TIMC M T C f W A L A 3 4 A TIME INTERVAL STATION 4 TIMC INTLRN4*L STATION 6 : O U T F L O W TIME INTERVAL TABLE XIII EXAMPLE 2.6 RESULTS 74 E X A M P L E 2 . 6 : 3 R E A C H E S , 2 N U L L R E S E R V O I R S T H E R E A R E 6 S T A T I O N S A N D 5 C H A N N E L R E A C H E S . T H E R O U T I N G P E R I O D I S 5 . 0 0 0 R E A C H K X B E T A G A M M A P H I P S I 1 8 . 0 0 0 0 . 2 0 0 9 . 8 8 8 9 - 4 . 3 3 3 3 4 . 5 5 5 6 - 4 9 . 3 8 3 2 0 . 0 0 . 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 - 0 . 0 3 1 1 . 0 0 0 0 . 1 7 5 2 0 . 1 3 0 - 1 1 . 4 3 5 7 . 6 9 5 7 - 1 6 6 . 3 5 4 0 . 0 0 . 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 - 0 . 0 5 6 . 4 0 0 0 . 2 3 0 7 . 2 2 5 7 - 2 . 3 6 1 9 3 . 8 6 3 8 - 3 0 . 2 8 1 S T A . 1 2 3 4 5 . 6 7 8 9 1 0 1 : 1 . 7 8 5 3 . 2 5 0 6 . 2 5 5 3 . 7 2 0 2 . 3 5 5 1 . 7 6 5 1 . 3 4 5 1 . 1 6 5 0 . 9 5 5 0 . 8 9 5 2 : 0 . 7 5 0 1 . 4 8 0 2 . 7 7 8 4 . 4 7 5 3 . 9 1 3 2 . 9 7 8 2 . 2 5 4 1 . 7 2 5 1 . 3 8 9 1 . 1 3 9 3 : 0 . 7 5 0 1 . 4 8 0 2 . 7 7 8 4 . 4 7 5 3 . 9 1 3 2 . 9 7 8 2 . 2 5 4 1 . 7 2 5 1 . 3 8 9 1 . 1 3 9 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 . 8 2 6 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 D U A L C O N S T . O R D I N A T E V A L U E 1 Q ( 1 , 2 ) 0 . 1 7 2 4 2 Q( 1 , 3 ) 0 . 2 0 4 9 3 Q ( 1 , 4 ) 0 . 1 8 4 6 4 Q ( 1 , 5 ) 0 . 0 7 3 1 5 Q( 1 , 6 ) 0 . 0 1 2 1 6 Q ( 1 , 7 ) 0 . 0 0 0 7 7 Q ( 1 , 8 ) 0 . 0 0 0 0 8 Q ( 1 , 9 ) 0 . 0 0 0 0 9 Q ( 1 , 1 0 ) 0 . 0 0 0 0 1 0 Q ( 2 , 2 ) 0 . 0 9 7 8 1 1 Q ( 2 , 3 ) 0 . 1 5 7 9 1 2 Q ( 2 , 4 ) 0 . 2 3 4 5 1 3 Q ( 2 , 5 ) 0 . 2 7 9 8 1 4 Q ( 2 , 6 ) 0 . 0 8 5 6 1 5 Q ( 2 , 7 ) 0 . 0 0 6 9 1 6 Q ( 2 , 8 ) 0 . 0 0 0 0 1 7 Q ( 2 , 9 ) 0 . 0 0 0 0 1 8 Q ( 2 , 1 0 ) 0 . 0 0 0 0 1 9 Q ( 3 , 2 ) 0 . 0 9 7 8 2 0 Q ( 3 , 3 ) 0 . 1 5 7 9 2 1 Q ( 3 , 4 ) 0 . 2 3 4 5 2 2 Q ( 3 , 5 ) 0 . 2 7 9 8 2 3 Q ( 3 , 6 ) 0 . 0 8 5 6 2 4 Q ( 3 , 7 ) 0 . 0 0 6 9 2 5 Q ( 3 , 8 ) 0 . 0 0 0 0 2 6 Q ( 3 , 9 ) 0 . 0 0 0 0 2 7 Q ( 3 , 1 0 ) 0 . 0 0 0 0 2 8 Q ( 4 , 2 ) 0 . 0 0 6 6 2 9 Q ( 4 , 3 ) 0 . 0 2 0 3 3 0 Q ( 4 , 4 ) 0 . 0 6 2 0 3 1 Q ( 4 , 5 ) 0 . 1 8 9 6 3 2 Q ( 4 , 6 ) 0 . 5 8 0 0 3 3 Q ( 4 , 7 ) 0 . 1 3 8 4 ^ 3 4 Q ( 4 , 8 ) 0 . 0 0 0 0 3 5 Q ( 4 , 9 ) 0 . 0 0 0 0 3 6 Q ( 4 , 1 0 ) 0 . 0 0 0 0 3 7 Q ( 5 , 2 ) 0 . 0 0 6 6 3 8 Q ( 5 , 3 ) 0 . 0 2 0 3 3 9 Q ( 5 , 4 ) 0 . 0 6 2 0 4 0 Q ( 5 , 5 ) 0 . 1 8 9 6 4 1 Q ( 5 , 6 ) 0 . 5 8 0 0 4 2 Q ( 5 , 7 ) 0 . 1 3 8 4 4 3 Q ( 5 , 8 ) 0 . 0 0 0 0 4 4 Q ( 5 , 9 ) 0 . 0 0 0 0 4 5 Q ( 5 , 1 0 ) 0 . 0 0 0 0 R H S R A N G I N G B O U N D S F L O W L O W E R U P P E R 3 . 2 5 0 0 2 . 7 7 7 7 1 6 . 5 4 8 6 . 2 5 5 0 5 . 5 8 7 1 0 . 8 6 1 4 5 E + 1 8 3 . 7 2 0 0 - 0 . 1 5 1 7 3 4 . 7 9 0 3 2 . 3 5 5 0 - 3 . 5 4 2 2 2 . 5 4 9 5 1 . 7 6 5 0 - 2 . 6 9 8 6 2 . 1 2 1 1 1 . 3 4 5 0 - 2 . 0 7 1 2 3 . 2 4 2 2 1 . 1 6 5 0 - 1 . 5 8 6 0 1 0 . 8 4 9 0 . 9 5 5 0 0 - 1 . 3 0 0 9 5 9 . 2 7 2 0 . 8 9 5 0 0 - 1 0 . 3 7 1 1 0 1 8 . 4 1 . 4 7 9 6 0 . 9 3 1 6 0 8 . 6 6 2 7 2 . 7 7 8 1 2 . 4 1 7 3 8 . 4 1 7 4 4 . 4 7 5 1 4 . 1 9 1 8 1 4 . 0 0 9 3 . 9 1 2 8 3 . 4 3 3 9 0 . 1 7 5 3 5 E + 1 3 2 . 9 7 8 0 0 . 0 3 . 0 8 9 7 2 . 2 5 4 1 0 . 0 2 . 5 2 9 6 1 . 7 2 5 2 0 . 0 4 . 2 5 3 6 1 . 3 8 9 2 0 . 0 9 . 6 5 1 0 1 . 1 3 9 2 0 . 0 1 0 4 . 0 3 1 . 4 7 9 6 0 . 9 3 1 6 0 8 . 6 6 2 7 2 . 7 7 8 1 2 . 4 1 7 3 8 . 4 1 7 4 4 . 4 7 5 1 4 . 1 9 1 8 1 4 . 0 0 9 3 . 9 1 2 8 3 . 4 3 3 9 0 . 1 7 5 3 5 E + 1 3 2 . 9 7 8 0 - 2 . 5 0 6 7 3 . 0 8 9 7 2 . 2 5 4 1 - 4 . 4 1 8 2 2 . 5 2 9 6 1 . 7 2 5 2 - 3 . 8 3 9 6 4 . 2 5 3 6 1 . 3 8 9 2 - 3 . 2 0 3 3 9 . 6 5 1 0 1 . 1 3 9 2 - 3 6 . 8 1 2 1 0 4 . 0 3 0 . 7 8 6 2 4 0 . 0 4 . 7 5 3 3 1 . 1 5 0 3 0 . 0 4 . 6 2 9 8 1 . 9 3 7 7 1 . 4 1 7 6 4 . 3 9 2 8 3 . 0 0 5 8 2 . 8 3 5 8 4 . 1 1 2 2 3 . 3 5 1 2 3 . 2 9 5 6 0 . 3 5 3 3 2 E + 1 4 3 . 1 5 4 0 0 . 3 3 2 0 5 E - 0 1 3 . 2 0 3 1 2 . 7 3 9 0 0 . 0 2 . 8 9 5 8 2 . 2 8 4 4 0 . 0 3 . 5 0 4 1 1 . 8 8 5 3 0 . 0 6 . 9 9 6 7 0 . 7 8 6 2 4 - 0 . 6 3 8 6 8 4 . 7 5 3 3 1 . 1 5 0 3 - 0 . 4 4 0 9 3 4 . 6 2 9 8 1 . 9 3 7 7 1 . 4 1 7 6 4 . 3 9 2 8 3 . 0 0 5 8 2 . 8 3 5 8 4 . 1 1 2 2 3 . 3 5 1 2 3 . 2 9 5 6 0 . 3 5 3 3 2 E + 1 4 3 . 1 5 4 0 0 . 3 3 2 0 5 E - 0 1 3 . 2 0 3 1 2 . 7 3 9 0 - 2 . 0 6 6 9 2 . 8 9 5 8 2 . 2 8 4 4 - 1 . 8 4 2 6 3 . 5 0 4 1 1 . 8 8 5 3 - 1 5 . 4 1 0 6 . 9 9 6 7 75 STATION 1: INFLOW / A \ a.« ^  TIME INTERVAL STATION 3 TIMC INTERVAL STATION 5 TIME INTERVAL FIGURE 12 EXAMPLE 2.6 STATION 2 TIME IMTCRVAL STATION 6 : O U T F L O W 16 TIME t N T E R W . 76 i n f l o w s . 5.3 CASE 3; BRANCHED NETWORK Th i s set of examples i n v o l v e s a network with a s i n g l e j u n c t i o n , as i l l u s t r a t e d i n F i g u r e 13. Once again, p r o v i s i o n i s made f o r the i n c l u s i o n of d e t e n t i o n storage s t r u c t u r e s . The reaches have each been a s s i g n e d d i f f e r e n t values of k and x to represent v a r i o u s combinations of channel s i z e , s l o p e , and d e t e r i o r a t i o n . The inflow hydrograph-entering the f i r s t branch i s g e n e r a l l y l a r g e r and peaks sooner than that e n t e r i n g the second branch. Example 3.1 The f i r s t example i l l u s t r a t e s a s i t u a t i o n where there i s no d e t e n t i o n s t o r a g e . As with the Case 2 examples, the high e s t dual value g e n e r a l l y occurs at an o r d i n a t e i n the downstream reach (Table XIV, F i g u r e 13). The f o l l o w i n g examples demonstrate the e f f e c t s of adding storage r e s e r v o i r s at v a r i o u s l o c a t i o n s i n the network, and how the model i s able to p r e d i c t such e f f e c t s . The r e s u l t i n g outflow hydrographs f o r each example are i l l u s t r a t e d i n F i g u r e 14. Example 3.2 Th i s example i l l u s t r a t e s the s i t u a t i o n t h a t occurs when a storage b a s i n i s i n s t a l l e d at Reach 2. T h i s s i t u a t i o n r e s u l t s i n a net in c r e a s e i n the peak outflow, d e s p i t e the 77 FIGURE 13 EXAMPLE 3.1 S T A T I O N 5 Tut Ttar wazir**. T»* IMTCW>*U. 78 TABLE XIV EXAMPLE 3.1 RESULTS EXAMPLE 3.1: ONE JUNCTION BRANCH NO. 1: STATION 1 TO STATION 4 JUNCTION: 4. 8 BRANCH NO. 2: STATION 5 TO STATION 11 JUNCTION: 11,11 THERE ARE 11 STATIONS AND 9 CHANNEL REACHES. THE ROUTING PERIOD IS 15.000 REACH K X BETA GAMMA PHI PSI 1 18 000 0 200 5.6154 - 1.7692 2.8462 -17.751 2 0 0 0 0 1.0000 1.0000 1.0000 -o.o 3 21 OOO 0 175 6.4902 -2.5686 2.9216 -21.530 5 19 500 0 160 5.4521 -2.0274 2.4247 -15.247 6 0 0 0 0 1.0000 1.0000 1.OOOO -0.0 7 20 250 0 185 6.3946 -2.3986 2.9960 -21.557 8 16 400 0 230 5.3991 - 1 .3755 3.0236 -17.700 9 0 0 0 0 1.0000 1.0000 1.0000 -0.0 10 17 200 0 215 5.5239 - 1 .5786 2.9453 -17.848 THERE ARE 12 HYDROGRAPH ORDINATES, 110 CONSTRAINTS, AND 100 VARIABLES. STA. 1 2 3 4 5 6 7 8 9 10 1 1 12 1 5 975 7 650 12 250 9 450 8 250 6 900 5 705 5 450 5 320 5 .225 5 175 5 . 130 2 5 365 6 081 7 975 10 404 9 537 8 415 7 165 6 1 19 5 638 5 .403 5 272 5 . 198 3 5 365 6 081 7 975 10 404 9 537 8 415 7 165 6 1 19 5 638 5 .403 5 .272 5 . 198 4 5 145 5 388 6 099 7 607 9 163 9 2 16 8 540 7 548 6 610 5 .987 5 .614 5 . 396 5 2 615 2 850 3 125 3 450 3 950 4 850 4 615 3 720 3 150 3 .015 2 .945 2 .920 6 2 550 2 634 2 820 3 071 3 401 3 911 4 458 4 392 3 865 3 .391 3 . 142 3 .014 7 2 550 2 634 2 820 3 071 3 401 3 911 4 458 4 392 3 865 3 .391 3 . 142 3 .014 8 2 100 2 394 2 573 2 767 3 009 3 333 3 780 4 193 4 235 3 .930 3 .554 3 .277 9 7 885 7 508 7 877 8 785 10 302 1 1 766 12 307 12 209 1 1 694 10 .890 10 026 9 .295 10 7 885 7 508 7 877 8 785 10 302 1 1 766 12 307 12 209 1 1 694 10 .890 10 .026 9 . 295 1 1 7 350 7 664 7 619 7 968 8 826 10 145 1 1 400 12 030 12 065 1 1 .655 10 .952 10 . 158 PEAK OUTFLOW = 12.06485 PRIMAL SOLUTION: VARIABLE VALUE 0( 1 1 2) 7 664 0( 1 1 3) 7 619 0( 1 1 4) 7 968 0( 1 1 5) 8 826 0( 1 1 6) 10 145 Q( 11 7) 1 1 400 0( 11 8) 12 030 0( 11 9) 12 065 0( 11 10) 11 655 0(11 11) 10 952 0( 1 1 12) 10 158 SENSITIVITY ANALYSIS: DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 Q( 1, 2) O 0841 7 . 6500 -2.7608 8.2311 2 0( 1. 3) 0 1434 12.250 5.3364 12.766 3 0( 1, 4) O 2102 9 . 4500 3.3093 10.681 4 0( 1, 5) 0 2382 8 2500 7.7880 0. 11333E+ 15 5 0( 1. 6) 0 1636 6 9000 6.5608 12.397 6 0( 1. 7) 0 0619 5 7050 5.0152 9.7410 7 0( 1. 8) 0 01 19 5 4500 2.3236 12.789 8 0( 1, 9) O 0009 5 3200 -4.2779 17.041 9 Q( 1. 10) 0 0000 5 2250 -4.1403 36.009 10 0( 1,11) 0 0000 5 1750 -4.0577 164.73 1 1 0( 1, 12) 0 0000 5 1300 -24.057 2077.5 12 0( 2. 2) 0 0310 6 081 1 0.0 7.0707 13 Q( 2, 3) 0 0658 7 9749 0.0 8.5196 14 0( 2, 4) 0 1291 10.404 3.9238 10.779 15 0( 2, 5) 0 2213 9 5370 5.0855 10.035 16 Q( 2. 6) 0 2906 8 4151 8.0970 0.44747E+13 17 0( 2, 7) 0 1822 7 1645 6.9021 10.949 18 0( 2, 8) 0 0508 6 . 1194 5.3633 9.2419 19 0( 2, 9) 0 0052 5 .6378 0.0 11.926 20 0( 2. 10) 0 0000 5 4032 0.0 15.870 21 Q( 2, 11) 0 OOOO 5 .2722 0.0 42.827 22 0( 2, 12) 0 0000 5 . 1976 0.0 374.24 TABLE XIV (continued) DUAL CONST. ORDINATE VALUE 23 0( 3. 2) 0 03 10 24 0( 3, 3) 0 0658 25 0( 3. 4) 0 1291 26 0( 3, 5) 0 2213 27 0( 3, 6) 0 2906 28 0( 3, 7) 0 1822 29 0( 3. 8) 0 0508 30 Q( 3. 9) 0 0052 31 0( 3. 10) 0 0000 32 0( 3,11) 0 0000 33 0( 3.12) 0 0000 34 0( 5, 2) 0 0870 35 0( 5, 3) 0 1450 36 0( 5. 4) 0 2080 37 0( 5. 5) 0 2321 38 0( 5, 6) 0 1603 39 0( 5, 7) 0 0618 40 0( 5. 8) 0 0122 41 0( 5, 9) 0 0010 42 0( 5. 10) 0 0000 43 0( 5.11) 0 0000 44 0( 5,12) 0 0000 45 0( 6, 2) 0 0286 46 0( 6, 3) 0 062G 47 0( 6. 4) 0 1262 48 0( 6. 5) 0 2215 49 0( 6. 6) 0 2965 50 0( 6. 7) 0 1865 51 0( 6, 8) 0 0518 52 0( 6, 9) 0 0052 53 0( 6, 10) 0 0000 54 0( 6.11) 0 0000 55 0( 6. 12) 0 0000 56 0( 7, 2) 0 0286 57 0( 7, 3) 0 0626 58 0( 7. 4) 0 1262 59 0( 7. 5) 0 2215 60 0( 7. 6) 0 2965 61 0( 7. 7) 0 1865 62 0( 7, 8) 0 0518 63 0( 7. 9) 0 0052 64 0( 7, 10) 0 0000 65 0( 7,11) 0 OOOO 66 0( 7,12) 0 0000 67 0( 4. 2) + 0( 8, 2) 0 0032 68 0( 4, 3) + 0( 8, 3) 0 0099 69 0( 4, 4) + 0( 8. 4) 0 0293 70 0( 4, 5) + 0( 8, 5) 0 0823 71 0( 4, 6) + 0( 8. 6) 0 2080 72 0( 4, 7) + 0( 8. 7) 0 4141 73 0( 4, 8) + 0( 8, 8) 0 2183 74 0( 4, 9) + 0( 8, 9) . 0 0335 75 0( 4,10) + 0( 8, 10) 0 0000 76 0( 4,11) + 0( 8,11) 0 0000 77 0( 4.12) + 0( 8,12) 0 0000 78 0( 9, 2) 0 0003 79 0( 9. 3) 0 001 1 80 0( 9. 4) 0 0039 81 0( 9, 5) 0 0137 82 0( 9. 6) 0 0478 83 0( 9. 7) 0 1672 84 0( 9. 8) 0 5849 85 0( 9. 9) 0 1810 86 0( 9, 10) 0 0000 87 0( 9.11) 0 OOOO 88 0( 9. 12) 0 0000 89 0( 10. 2) 0 0003 90 0( 10, 3) 0 001 1 91 0( 10, 4) 0 0039 92 0( 10, 5) 0 0137 93 0( 10, 6) 0 0478 94 0( 10. 7) 0 1672 95 0( 10. 8) 0 5849 96 0( 10, 9) 0 1810 97 0( 10.10) 0 OOOO 98 0( 10.11) 0 OOOO 99 0( 10.12) 0 OOOO RHS RANGING BOUNDS FLOW LOWER UPPER S.081 1 -5.8507 7.0707 7.9749 -3.7994 8.5196 10.404 3.9238 10.779 9.5370 5.0855 10.035 8.4151 8.0970 0.44747E+13 7.1645 6.902 1 10.949 6.1194 5.3633 9.2419 5.6378 - 1.0366 11.926 5.4032 -5.5801 15.870 5.2722 -5.2847 42.827 5.1976 -29.823 374.24 2.8500 -2.6480 3.4450 3.1250 -2.8626 3.6722 3.4500 -3.0604 4.8797 3.9500 3.4698 0.45668E+13 4.8500 4.4999 10.563 4.6150 3.9195 8.7799 3.7200 0.64959 11.235 3.1500 -3.4616 15.117 3.0150 -3.1108 33.B79 2.9450 -2.9305 159.32 2.9200 -13.511 1985.3 2.6339 0.0 3.6476 2.8201 0.0 3.3623 3.0712 0.0 3.4331 3 . 4009 0.0 3.8608 3.9109 3.5975 0.68122E+13 4.4577 4.2016 8.1859 4.3923 3.6518 7.4391 3.8655 0.0 10.007 3.3913 0.0 13.617 3.1421 0.0 39.943 3.0137 0.0 366.62 2.6339 -2.2471 3.6476 2.8201 -2.4281 3.3623 3.0712 -2.6357 3.4331 3.4009 -0.90404 3.8608 3.9109 3.5975 0.68122E+13 4.4577 4.2015 8.1859 4.3923 3.6518 7.4391 3.8655 -2.7 106 10.007 3.3913 -3.35 10 13.617 3.1421 -3.0734 39.943 3.0137 -17.939 366.62 7.7826 -5.1904 12.955 8.6719 -5.7954 10.443 10.373 -6.5925 11.025 12.172 4.4227 12.446 12.550 9.2862 12.717 12.319 12.143 0.66458E+13 11.741 11.554 13.835 10.846 9.8173 13.066 9.9166 -6.595 1 14.520 9.1683 -6.1396 17.904 8.6726 -41.512 65.535 7.5076 0.0 15.112 7.8772 0.0 14.895 8.7846 0.0 12.321 10.302 0.0 11.312 11.766 0.0 12.054 12 . 307 8 .8711 12.390 12.209 12.124 0.94121E+15 11.694 11.504 12.710 10.890 0.0 12.792 10.026 0.0 13.285 9.2950 0.0 19.827 7.5076 -5.5183 15.112 7.8772 -5.7446 14.895 8.7846 -6.3043 12.321 10. 302 -7.0423 11.312 11.766 -0.25819 12.054 12.307 8.871 1 12.390 12.209 12.124 0.94121E+15 11.694 11.504 12.710 10.890 -7.8340 12.792 10.026 -7.3409 13.285 9.2950 -46.819 19.827 FIGURE 14 CASE 3 OUTFLOW HYDROGRAPHS O U T F L O W H Y D R O G R A P H S C A S E 3: B R A N C H E D N E T W O R K L E G E N D O R I G I N A L  E X A M P L E 3 . 2 E X A M P L E 3 . 3 < I I 1 1 I 1 1 1 1 1 f 1 2 3 4 6 8 7 8 9 10 11 12 T I M E I N T E R V A L 81 a t t e n u a t i o n e f f e c t of the r e s e r v o i r . T h i s r e s u l t c o u l d be p r e d i c t e d from Example 3.1 by comparing the hydrographs and dual v e c t o r s at S t a t i o n 3 (the o u t l e t of the r e s e r v o i r ) f o r examples 3.1 and 3.2 (F i g u r e 15). When there i s no r e s e r v o i r present, the flow at s t a t i o n 3 i s g r e a t e s t at the f o u r t h time p e r i o d (Q«), while the g r e a t e s t d u a l values are a s s o c i a t e d with the f i f t h and s i x t h o r d i n a t e s (Qi and Q|). In other words, the model p r e d i c t s that a decrease i n flow f o l l o w i n g the maximum flow at node 3 w i l l have a gr e a t e r e f f e c t i n reducing Q^ than w i l l a decrease i n the maximum flow i t s e l f . By examining the r e s u l t s f o r Example 3.2, i t can be seen that the e f f e c t of i n s e r t i n g the r e s e r v o i r was to decrease Ql (an o r d i n a t e w i t h r e l a t i v e l y low s e n s i t i v i t y ) and in c r e a s e both Ql and Q| (o r d i n a t e s with high dual v a l u e s ) . T h e r e f o r e , the o v e r a l l e f f e c t i s a net i n c r e a s e i n those o r d i n a t e s with high d u a l v a l u e s , and consequently an i n c r e a s e i n Q^. Example 3.3 Thi s example r e p r e s e n t s a s i t u a t i o n where a r e s e r v o i r i s added to Reach 6 i n the second branch. Here the outflow peak Q^ i s reduced by about 8% from the case with no r e s e r v o i r s . Examination of the dual values i n Example 3.1 show how the model was able to p r e d i c t the d i f f e r e n t e f f e c t s from examples 3.2 and 3.3 (Figure 16). At S t a t i o n 7 i n Example 3.1 (no r e s e r v o i r ) , the hig h e s t dual value occurs e a r l i e r than the peak flow at the reach. T h e r e f o r e , the 82 FIGURE 15 STATION 3: COMPARISON OF EXAMPLES 3.1 AND 3.2 FIGURE 16 STATION 7: COMPARISON OF EXAMPLES 3.1 AND 3.3 S T A T I O N ~7 - I M O R E S E R V O I R « <3 " T I N / I E I N T E R V A L S T A T I O N V - W I T H R E S E R V O I R 84 e f f e c t of adding a r e s e r v o i r ( d e c r e a s i n g the flow at the peak and i n c r e a s i n g flow at some p o i n t s a f t e r the peak) r e s u l t s i n lowering those o r d i n a t e s corresponding to high dual v a l u e s . T h i s leads to a net decrease i n the peak outflow Qp ( F i g u r e 14). Example 3.4 T h i s example c o n t a i n s a r e s e r v o i r l o c a t e d at Reach 9 below the j u n c t i o n . As shown i n F i g u r e 14, i t a l s o r e s u l t e d in a decrease i n Q^, even though the peak occurs before the most s e n s i t i v e o r d i n a t e . T h i s can be e x p l a i n e d by the f l a t t e n e d shape of the hydrograph at t h i s p o i n t ( F i g u r e 17) whereby the e f f e c t s of r o u t i n g through another r e s e r v o i r have minimal e f f e c t on any p a r t i c u l a r o r d i n a t e . 5.4 CASE 4: COMPLEX NETWORK A mul t i - b r a n c h network i s c o n s i d e r e d i n the f o l l o w i n g examples, as i l l u s t r a t e d i n F i g u r e 18. Once again, the reaches each have d i f f e r e n t v a l u e s of k and x to represent a v a r i e t y of channel parameters. Example 4.1 The r e s u l t s f o r t h i s example are given i n Table XV and shown in F i g u r e 18. (To p r e s e r v e c l a r i t y , only some of the hydrographs are i n c l u d e d i n F i g u r e 18.) A number of c o n c l u s i o n s can be a r r i v e d a t from examining the r e s u l t s , p a r t i c u l a r l y r e g a r d i n g the upstream inflow p o i n t s . 85 FIGURE 17 STATION 10: COMPARISON OF EXAMPLES 3.1 AND 3.3 86 FIGURE 18 EXAMPLE 4.1 5A / \ / \ / \ ^ — _—A S T A T I O N 3 TMC IMTCKVKL 8 i'2 1 4 1 5 -nut iwrcirvjw. TIMC HTt*\*L 87 TABLE XV EXAMPLE 4.1 RESULTS EXAMPLE 4.1 (COMPLEX NETWORK, NO RESERVOIRS) BRANCH NO. BRANCH NO. BRANCH NO. BRANCH NO. STATION STATION STATION 1 TO STATION 3 TO STATION 8 TO STATION STATION 10 TO STATION JUNCTION JUNCTION JUNCTION JUNCTION 2. 4 7, 12 9.11 15. 15 THERE ARE 15 STATIONS AND 11 CHANNEL REACHES. THE ROUTING PERIOD IS 15.000 REACH K X BETA GAMMA PHI PSI 1 18 500 0 250 7.4348 -2 .2174 4 .2174 -33.573 3 21 000 0 175 6.4902 -2 .5686 2 .9216 -21.530 4 19 250 0 235 7.4679 -2 .4280 4 .0399 -32.597 5 0 0 0 0 1.OOOO 1 OOOO 1 .0000 - 0 . 0 6 21 550 0 190 7.3280 -2 .9234 3 . 4046 -27.873 8 20 750 0 215 7.8285 -2 .8922 3 .9362 -33.707 10 19 500 0 220 7.0748 -2 .4019 3 .6729 -28.387 1 1 18 700 0 265 8.3492 -2 .4541 4 .8951 -43.324 12 17 875 0 3 10 10.126 -2 . 4678 6 . 6579 -69.884 13 0 0 0 0 1 . OOOO 1 • OOOO 1 .0000 -0 .0 14 20 450 0 220 7.8144 -2 .8161 3 .9983 -34.061 THERE ARE 12 HYDROGRAPH ORDINATES. 132 CONSTRAINTS, AND 122 VARIABLES. STA. 1 2 3 4 5 6 7 8 9 10 1 1 12 1 7 975 12 650 10 850 9 450 8 250 6 900 5 70S 5 450 5 320 5 . 225 5 . 175 5 130 2 4 525 7 575 10 894 10 675 9 654 8 487 7 213 6 120 5 632 5 .400 5 .271 5 197 3 1 615 1 850 2 125 2 450 3 250 4 145 4 615 3 720 3 150 3 .015 2 .945 2 920 4 1 455 1 588 1 789 2 042 2 412 3 056 3 786 4 149 3 802 3 .387 3 . 152 3 023 5 5 885 6 375 8 728 1 1 402 12 202 12 040 1 1 632 1 1 107 10 430 9 .672 9 .026 8 591 6 5 885 6 375 8 728 1 1 402 12 202 12 040 1 1 632 1 1 107 10 430 9 .672 9 .026 8 59 1 7 5 480 5 790 6 463 8 189 10 229 1 1 393 1 1 726 1 1 598 1 1 21 1 10 .638 9 .969 9 343 8 2 015 3 270 3 835 4 010 3 650 3 205 3 700 3 B75 3 545 3 .350 3 .315 3 280 9 1 70S 2 061 2 895 3 510 3 779 3 641 3 429 3 622 3 740 3 .592 3 .435 3 355 10 1 450 1 690 1 825 1 900 1 860 1 785 1 650 1 525 1 415 1 .350 1 .200 1 . 155 1 1 1 440 1 481 1 638 1 772 1 851 1 846 1 787 1 679 1 562 1 .456 1 . 365 1 250 12 3 065 3 169 3 551 4 334 5 045 5 441 5 44 1 5 292 5 299 5 .270 5 .083 4 860 13 8 265 8 518 8 956 10 004 12 18 1 14 675 16 34 1 16 939 16 864 16 . 536 15 .977 15 . 194 14 8 265 8 518 8 956 10 004 12 181 14 675 16 34 1 16 939 16 864 16 .536 15 . 977 15 . 194 15 8 075 a 229 8 470 8 915 9 890 1 1 674 13 807 15 504 16 412 16 .659 16 .509 16 .068 PEAK OUTFLOW - 16.65948 SENSITIVITY ANALYSIS: DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 1 0( 1, 2) 0 . 1 184 12.650 9.2935 17.459 2 0( 1. 3) 0 . 1698 10.850 7.9251 17.761 3 0( 1. 4) 0 .2056 9.4500 5.2470 0.47599E+12 4 0( 1. S) 0 1900 8.2500 5.1569 17.895 5 0( 1. 6) 0 1 10O 6.9000 3.4935 8.7812 6 0( 1. 7) 0 0374 5.7050 -2.4920 7.7768 7 0( 1. 8) 0 0073 5.4500 -3.8235 10.435 8 0( 1. 9) 0 0007 5.3200 -3.5715 21.432 9 0( 1. 10) 0 OOOO 5.2250 -3.4528 86.936 10 0( 1. 1 1 ) 0 OOOO 5.1750 -3.3824 BOO. 07 1 1 0( 1. 12) 0 OOOO 5.1300 -33.512 19039. 12 0( 3. 2) 0 1233 1.8500 -1.6494 7.3134 13 0( 3. 3) 0 1686 2.1250 - 1. 1977 11.200 14 0( 3. 4 ) 0 1958 2.4500 -2.2685 0.76756E+12 15 0( 3. 5) 0 1763 3.2500 -0.15208 10.950 16 0( 3. 6) 0 1036 4.1450 O.44849 6.2140 17 0( 3. 7) 0 0367 4.6150 -3.5027 6.8631 18 0( 3. 8) 0 0075 3.7200 -3.7185 8.8716 19 0( 3. 9) 0 0008 3.1500 -3.4770 19.627 20 0( 3. 10) 0 OOOO 3.0150 -3.1508 82.278 21 0( 3, 1 1 ) 0 OOOO 2.94S0 -2.9691 733.40 22 0( 3. 12) 0 OOOO 2.9200 -16.699 16618. 23 0( 2. 2) + 0( 4, 2) 0 0599 9.1627 3.8238 14.625 24 Q( 2. 3) + 0( 4, 3) 0 1052 12.683 9.3612 16.730 25 0( 2. 4) + 0( 4, 4) 0 1663 12.717 10. 255 17.071 26 0( 2. 5) + 0( 4, 5) 0 2231 12.066 9.4176 24 1.08 27 0( 2. 6) + 0( 4, 6) 0 2242 11.543 9. 1431 0.69208E+12 28 0( 2. 7) • 0( 4, 7) 0 121 1 10.999 8.1788 12.459 29 0( 2. 8) + 0( 4, 8) 0 0335 10.270 1 .7317 11.985 30 0( 2. 9) + 0( 4 , 9) 0 0045 9.4345 -7 .1 t20 14.503 31 0( 2.10) + 0( 4. 10) 0 0002 8.7877 -6.6546 26.584 32 0( 2,11) + 0( 4, 11) 0 OOOO 8.4221 -6.2765 140.19 33 0( 2.12) + 0( 4, 12) 0 OOOO 8.2203 -55.940 2568.3 34 0( 5. 2) 0 0204 6.3753 0 .0 17.542 35 0( 5. 3) 0 0425 8.7279 1 .9367 14.687 36 0( 5, 4) 0 0840 11.402 7.7771 15.010 37 0( 5. 5) 0 1525 12.202 10.008 15.032 38 0( 5. 6) 0 2399 12.040 10.319 17.888 39 0( 5. 7) 0 2822 11.632 9 .9635 0.60989E+12 40 0( 5, 8) 0 1340 11.107 8.8171 12.122 41 0( 5. 9) 0 0260 10.430 0.23393 11.823 42 0( 5, 10) 0 0017 9 .6715 0 .0 14. 142 43 0( 5. 1 1 ) 0 OOOO 9.0261 0 .0 31 .7S3 44 0( 5. 12) 0 OOOO 8.5915. 0 .0 351.40 TABLE XV (continued) DUAL RHS RANGING BOUNDS CONST. ORDINATE VALUE FLOW LOWER UPPER 45 0 ( 6. 2 0 .0204 6 . 3753 -6.0763 17.542 46 0 ( 6, 3 0 0425 8. 7279 1 .9367 14.687 47 0 ( 6, 4 0 0840 1 1 .402 7.7771 15.010 48 0 ( 6, 5 0 1525 12 .202 10.008 15.032 49 0( 6. 6 0 2399 12 .040 10.319 17.888 50 0 ( 6. 7 0 2822 1 1 .632 9.9635 0.60989E+12 51 0( 6, 8 0 1340 1 1 . 107 8.8171 12.122 52 0 ( 6, 9 0 0260 10 .430 0.23393 11.823 53 0 ( 6. 10) 0 0017 9. 6715 -9.5348 14.142 54 0 ( 6, 11) 0 OOOO 9. 0261 -8.974 1 31.783 55 0 ( 6, 12) 0 OOOO 8. 5915 -59.873 351.40 56 0( 8, 2) 0 0548 3. 2700 -1.9944 8.6731 57 0( 8. 3 0 1005 3. 8350 0.54902 7.6288 58 0 ( 8, 4 0 1657 4 . 0100 1.7027 7.8237 59 0 ( 8, 5 0 2306 3. 6500 1.3306 42.394 60 Q( 8, 6. 0 2369 3. 2050 1.027O O.19979E+13 61 0 ( 8, 7 0 1234 3. 7000 0.99293 5.0246 62 0 ( 8. 8 0 0320 3. 8750 -2.9240 5.5214 63 0( 8, 9) 0 0040 3. 5450 -2.9858 8.4973 64 0 ( 8, 10) 0 0002 3. 3500 -2.8953 21.914 65 0( 8.11 0 OOOO 3. 3150 -2.7846 150.64 66 0 ( 8,12) 0 OOOO 3. 2800 -22.983 3060.9 67 0 ( 10, 2) 0 0509 1 . 6900 -1.1981 7.1697 68 0( 10, 3) 0 0961 1 . 8250 -1.2996 5.5456 69 0 ( 10, A. 0 1625 1. 9000 -0.36274 5.4672 70 0 ( 10, 5 0 2319 1 . 8600 -0.30946 22.212 71 0 ( 10, 6 0 2440 1 . 7850 -0.38152 0.19614E+13 72 0 ( 10, 7 : 0 1299 1 . 6500 -0.94003 2.9676 73 0 ( 10, 8 0 0344 1 . 5250 -1.2285 3. 1002 74 0 ( 10. 9 0 0044 1 . 4 150 -1 . 1516 6. 1267 75 0 ( 10. 10) 0 0002 1 . 3500 -1.0562 18.658 76 0( 10. 1 1 ) 0 OOOO 1 . 2000 -1.0O32 136.25 77 0 ( 10, 12) 0 OOOO 1 . 1550 -7.6852 2764.4 78 0( 9. 2 1 + 0( 1 1. 2) 0 0135 3. 5413 -2.1718 17.158 79 0( 9, 3 I + 0( 1 1 . 3) 0 0316 4. 5334 -2.4405 10.954 80 0( 9. 4 1 + 0 ( 1 1 . 4) 0 0701 5. 2823 1.3778 8.7071 81 0( 9. 5 1 + 0( 1 1 . 5) 0 1423 5. 6303 3.5475 7.9649 82 0( 9. 6 1 + 0 ( 1 1 . 6) 0 2483 5. 4873 4.0675 9 .0605 83 0 ( 9. 7 i + 0 ( 1 1 . 7) 0 3175 5. 2160 3.S131 0.18449E+13 84 0 ( 9. 8 + 0( 1 1 . 8) 0 1412 5. 301 1 3.1749 6.1543 85 0 ( 9. 9 + 0( 1 1 , 9) 0 0249 5. 3012 -3.1784 6.5942 86 0( 9. 10 i + 0( 1 1 . 10) o 0015 5. 0475 -3.1315 9.2796 87 0( 9. 11 i + 0( 11.11) 0 OOOO 4. 7996 -3.0196 28.544 88 0 ( 9. 12 I + 0 ( 11. 12) 0 OOOO 4. 6044 -35.969 395.18 89 0( 7, 2 i + 0 ( 12. 2) 0 0025 8. 9593 -4.1801 27.451 90 0( 7. 3 i + 0( 12. 3) 0 0066 10 .014 -4.6632 26.338 91 0( 7. 4 i + 0 ( 12. 4) 0 0174 12 .523 -1.3760 21.586 92 0 ( 7 . 5 i + 0 ( 12. 5) 0 0447 15 .275 9.7629 19.110 93 0( 7. 6 + 0 ( 12. 6) 0 1092 16 .834 14.502 18.696 94 0( 7. 7 + Q( 12. 7) 0 2420 17 . 167 16.035 18.547 95 0 ( 7, 8 i + 0 ( 12, 8) 0 4213 16 .890 16.051 0.20144E+13 96 0( 7, 9 + 0( 12, 9) 0 1423 16 .509 14.602 17.048 97 0 ( 7 , 10 t + 0 ( 12.10) 0 0126 15 .908 -3.6605 17.068 98 0 ( 7,11 + 0 ( 12.11) o OOOO 15 .052 -7.2389 19.207 99 0 ( 7, 12 + 0 ( 12,12) o OOOO 14 .202 -139.64 60.983 100 0 ( 13, 2) 0 0004 8. 5177 0.0 23.212 lOI 0 ( 13, 3! 0 0012 8 . 9558 0.0 22.871 102 0 ( 3, 4! o 0034 10 .004 0.0 22.215 103 0( 13, 5) 0 0094 12 . 181 0.0 21.272 104 0 ( 13. 6: 0 0261 14 .675 5.6667 20.012 105 Q( 13, 7] 0 0724 16 .341 13.095 18.264 106 0 ( 13, B : 0 2010 16 .939 15.769 17.632 107 0 ( 3, 9! 0 5578 16 .864 16.443 0.24698E+15 108 0 ( 3. 10) 0 1280 16 .536 14.604 16.886 109 0( 3, 11) 0 OOOO 15 .977 0.0 17.037 1 10 0 ( 3, 12) 0 OOOO 15 . 194 0.0 19.814 1 1 1 0 ( 4, 2) 0. 0004 8. 5177 -6.6670 23.212 112 0 ( 4. 3) 0 0012 8. 9558 -7.0267 22.871 113 0 ( 4, 4) 0. 0034 10 .004 -7.7272 22.215 1 14 0 ( 4, 5) 0.0094 12 . 181 -8.7493 21.272 1 15 0 ( 4, 6) 0 0261 14 .675 5.6667 20.012 1 16 0 ( 1 4, 7) 0. 0724 16 .341 13.095 18.264 117 0 ( 4. 8) 0. 2010 16 .939 15.769 17.632 118 0 ( 4, 9) 0. 5578 16 .864 16.443 0.22409E+15 1 19 0( 4, 10) 0. 1280 16 .536 14.604 16.886 120 0 ( 4, 11) 0. OOOO 15 .977 -12.831 17.037 121 0 ( 1 4, 12) 0.OOOO 15 . 194 -110.37 19.814 89 The i n f o r m a t i o n given f o r s t a t i o n s 3 and 10 are w e l l s u i t e d f o r a n a l y s i s , s i n c e (except f o r one o r d i n a t e at S t a t i o n 3) the flows can be reduced to zero without a f f e c t i n g the optimal b a s i s . I f i t were p o s s i b l e to c o n t r o l the flow at e i t h e r of these two inflow l o c a t i o n s , then F i g u r e 18 g i v e s some v a l u a b l e i n f o r m a t i o n f o r o p e r a t i o n p o l i c y . C o n t r o l l e d withdrawals or temporary d i v e r s i o n of flow at 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 d u r i n g the times a s s o c i a t e d with high dual v a l u e s ( i . e . , o r d i n a t e s 2 to 6). A f t e r about the seventh o r d i n a t e , flow c o u l d be allowed to pass through without i n c r e a s i n g the downstream peak; i n f a c t , some of the s t o r e d water c o u l d a l s o be s a f e l y r e l e a s e d at that time. Another way the model would be h e l p f u l i s i n d e c i d i n g on the best l o c a t i o n f o r a flow c o n t r o l s t r u c t u r e . I f a designer had a choice of i n s t a l l i n g a d i v e r s i o n or storage device at e i t h e r S t a t i o n 3 or S t a t i o n 10, the r e s u l t s i n Table XV can a i d i n determining the l o c a t i o n which most e f f i c i e n t l y reduces the downstream flows. For example, i f the flow at S t a t i o n 3 was t e m p o r a r i l y s t o r e d between the f i r s t and seventh o r d i n a t e s , t h i s would represent a storage requirement of about 16,600 m3. The model t e l l s us (by m u l t i p l y i n g the dual v a l u e s by the o r d i n a t e s r e t a i n e d by storage) that t h i s w i l l r e s u l t in a downstream peak re d u c t i o n of 2.34 m 3/s. On the other hand, s t o r i n g the f i r s t seven o r d i n a t e s of the flow at S t a t i o n 10 would r e q u i r e a h o l d i n g c a p a c i t y of 9640 m3 and would reduce Q by 90 1.65 m 3/s. Although the l a t t e r r e d u c t i o n i s l e s s than the former, i n terms of peak flow r e d u c t i o n per volume of water s t o r e d the S t a t i o n 10 l o c a t i o n turns out to be over 20% more e f f i c i e n t than S t a t i o n 3 (0.17 m3/s per thousand m3 s t o r e d as opposed to 0.14). Of course other f a c t o r s need to be con s i d e r e d as w e l l , such as the r e l a t i v e c o s t s of c o n s t r u c t i n g f a c i l i t i e s at e i t h e r l o c a t i o n , and whether or not the r e d u c t i o n i n peak outflow i s s u f f i c i e n t to allow smaller channel or pipe s e c t i o n s . These f a c t o r s c o u l d a l l be i n c o r p o r a t e d i n t o the model, depending on the s p e c i f i c a p p l i c a t i o n . The r e s u l t s of the s i n g l e pass through the model a l s o y i e l d v a l u a b l e i n f o r m a t i o n about the other s e c t i o n s of the drainage network. For example, i f water was withheld from passin g through S t a t i o n 6 at a constant r a t e of 1.5 m 3/s between the f o u r t h and ei g h t h time i n t e r v a l s , then f o r a t o t a l storage requirement of 6750 m3, a r e d u c t i o n of 1.34 m3/s would be gained. The e f f e c t i v e n e s s of storage at t h i s l o c a t i o n i n reducing the peak outflow i s then almost 0.2 m3/s per thousand m3, almost another 20% greater than storage at S t a t i o n 10. S i m i l a r l y , a l l other p o t e n t i a l storage l o c a t i o n s c o u l d be examined s y s t e m a t i c a l l y to a i d i n a r r i v i n g at a r a t i o n a l design and o p e r a t i o n p o l i c y . Chapter 6 DISCUSSION OF RESULTS As mentioned in previous chapters, the dual v e c t o r and r i g h t hand s i d e c o e f f i c i e n t ranging are the two mechanisms by which the model provides the most i n f o r m a t i o n i n a i d i n g d e s i g n . These f e a t u r e s and t h e i r i m p l i c a t i o n s w i l l be d i s c u s s e d in t h i s chapter, along with a d i s c u s s i o n of how c o s t s can be analyzed. 6.1 MODEL LIMITATIONS AND CAPABILITIES The choice of flow o r d i n a t e s as the p r i n c i p a l d e c i s i o n v a r i a b l e s in the f o r m u l a t i o n , as opposed to channel c h a r a c t e r i s t i c s , was e n t i r e l y a r b i t r a r y . I t i s p o s s i b l e that other c h o i c e s of d e c i s i o n v a r i a b l e s would l e a d q u i t e d i f f e r e n t schemes with other d i f f e r e n t advantages f o r CAD purposes, and these should of course be e x p l o r e d . Flow o r d i n a t e s are, however, a fundamental and e a s i l y understood i n d i c a t o r of runoff system behaviour. The form of s e n s i t i v i t y i n f o r m a t i o n which a r i s e s from the f o r m u l a t i o n r e l a t e s upstream flow o r d i n a t e changes to changes i n the peak outflow from the runoff . system - a n a t u r a l way to express s e n s i t i v i t y to a design engineer. S e c t i o n 6.4 d e s c r i b e s how t h i s s e n s i t i v i t y i n f o r m a t i o n can be e a s i l y converted to r e f l e c t the c o s t e f f e c t i v e n e s s of d e s i g n changes. 91 92 Perhaps the most s i g n i f i c a n t b e n e f i t r e v e a l e d i n the f o r m u l a t i o n presented i n t h i s t h e s i s i s that f u l l s e n s i t i v i t y i n f o r m a t i o n i s produced at every reach and every r e s e r v o i r e n t r y and d i s c h a r g e p o i n t i n the r u n o f f system from a single l i n e a r programming pass through the system. T h i s complete a n a l y t i c a l c a p a b i l i t y was not a n t i c i p a t e d d u r i n g the p r e l i m i n a r y f o r m u l a t i o n phase of the r e s e a r c h but, once confirmed, s u b s t a n t i a l l y enhances the CAD p o t e n t i a l of the approach presented here. The only a l t e r n a t i v e to producing comparable, i n f o r m a t i o n would be some kind of p e r t u r b a t i o n approach. However, t h i s would appear to i n v o l v e so many r e p e t i t i o n s of the f u l l system runo f f a n a l y s i s t h a t , even f o r a network of modest complexity, i t would prove to be i m p r a c t i c a l . 6.2 DUAL SOLUTION VECTOR The s i g n i f i c a n c e of the dual values has been s t r e s s e d i n p r e v i o u s c h a p t e r s . The dual value a s s o c i a t e d with a p a r t i c u l a r flow o r d i n a t e r e f l e c t s the amount by which a p e r t u r b a t i o n of the o r d i n a t e w i l l a f f e c t the magnitude of the peak outflow at the downstream end of the drainage network. T h i s p r i n c i p l e has been demonstrated in the examples c i t e d i n Chapter 5. There are two ways of p e r t u r b i n g flows when a n a l y z i n g a drainage network. One method i n v o l v e s d i r e c t m o d i f i c a t i o n of an upstream hydrograph o r d i n a t e (or set of o r d i n a t e s ) through withdrawal or by some form of a c t i v e l y c o n t r o l l e d 93 d i v e r s i o n , d e t e n t i o n , or r e t e n t i o n f a c i l i t y . The other method, simpler to achieve but more i n d i r e c t , i s to a l t e r the p h y s i c a l dimensions or c h a r a c t e r i s t i c s of a channel or r e s e r v o i r , which in turn causes an a l t e r a t i o n of the general hydrograph shape at i t s downstream end. For example, i n c r e a s i n g pipe diameter or roughness, d e c r e a s i n g the slope, or r e p l a c i n g a channel s e c t i o n with a r e s e r v o i r w i l l r e s u l t in a t t e n u a t i o n of the peak o r d i n a t e s and i n c r e a s e o r d i n a t e s f o l l o w i n g the peak. That i s , s e v e r a l ( i f not a l l ) of the hydrograph o r d i n a t e s w i l l be changed si m u l t a n e o u s l y . The l a t t e r method, however, does not produce simple p a t t e r n s of flow o r d i n a t e m o d i f i c a t i o n which can be p r e d i c t e d without performing f u l l r o u t i n g a n a l y s e s . Since the dual v a l u e s r e l a t e Q d i r e c t l y to the flows and not to channel c h a r a c t e r i s t i c s , the s e n s i t i v i t y i n f o r m a t i o n i s best t r a n s l a t e d by the designer i n the context of the f i r s t method of d i r e c t flow m o d i f i c a t i o n . With experience, however, the s e n s i t i v i t y values w i l l a l s o convey u s e f u l i n f o r m a t i o n to the designer on the consequences of channel m o d i f i c a t i o n and other p h y s i c a l perturbances. In a s t r i c t sense, the dual s o l u t i o n values are v a l i d p r e d i c t i o n s only i f a s i n g l e o r d i n a t e i s m o d i f i e d , while i n p r a c t i c e i t i s d e s i r a b l e to a l t e r more than one o r d i n a t e to simulate the behaviour of r e s e r v o i r s , e t c . which a f f e c t the whole hydrograph. Changing more than one o r d i n a t e at once, however, i n v a l i d a t e s the RHS boundary l i m i t s , and may r e s u l t in a change i n the optimal b a s i s . T h i s corresponds to the 94 downstream peak outflow o r d i n a t e s h i f t i n g to an e a r l i e r or l a t e r time i n t e r v a l . F o r t u n a t e l y , i t was recognized that the s e n s i t i v i t y i n f o r m a t i o n was s t i l l a v a i l a b l e when i t was observed t h a t , i f only flow o r d i n a t e s are being changed ( i . e . , not channel c h a r a c t e r i s t i c s ) , then the dual values a l s o s h i f t along the time a x i s . The method to determine the d i r e c t i o n which the dual v e c t o r w i l l s h i f t i s e x p l a i n e d i n the f o l l o w i n g s e c t i o n . When channel parameters are a l t e r e d , the l i n e a r programming, problem becomes; an e n t i r e l y new f ormulati-on, and i s no longer simply a m o d i f i c a t i o n of an e x i s t i n g problem as i s the case when only flow o r d i n a t e s are pert u r b e d . A whole new set of dual s o l u t i o n values and RHS bounds are generated fo r each reach. However, as demonstrated by the examples, the d i f f e r e n c e s are s t i l l small enough to allow g e n e r a l i z e d p r e d i c t i o n s to be made. 6.3 RIGHT HAND SIDE RANGING As d i s c u s s e d i n pre v i o u s c h a p t e r s , RHS' ranging provides the l i m i t s beyond which the dual s o l u t i o n v a l u e s are no longer v a l i d . If the o r d i n a t e a s s o c i a t e d with a p a r t i c u l a r c o n s t r a i n t i s perturbed enough to b r i n g the r i g h t hand sid e of that c o n s t r a i n t o u t s i d e of the bounds given by RHS ranging, then the b a s i s w i l l change and the system peak outflow w i l l occur at a d i f f e r e n t time. The upper bound p o r t i o n of the RHS ranging i n f o r m a t i o n p r o v i d e s an important c l u e i n p r e d i c t i n g the s h i f t i n 95 optimal b a s i s . I n s p e c t i o n of the examples i n the pr e v i o u s chapter shows that one upper bound i n each reach tends to i n f i n i t y . T h i s anomalously high upper bound may occur before, a f t e r , or concurrent with the peak flow f o r the reach, but i t always occurs at the time p e r i o d which e x h i b i t s the h i g h e s t dual v a l u e . At that p a r t i c u l a r time p e r i o d , the flow c o u l d be i n c r e a s e d i n d e f i n i t e l y ( s u b j e c t to flow c a p a c i t y c o n s t r a i n t s ) and the time at which the system peak outflow occurs would not be a l t e r e d . The i n c r e a s e i n can 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 the i n c r e a s e i n the upstream o r d i n a t e by i t s dual v a l u e . If an o r d i n a t e p r i o r to the o r d i n a t e with the i n f i n i t e upper bound i s i n c r e a s e d , then the system peak outflow w i l l occur e a r l i e r . I f an upstream o r d i n a t e a f t e r the ' i n f i n i t e bound' o r d i n a t e i s i n c r e a s e d , then Q w i l l occur at a l a t e r P time. T h i s type of i n f o r m a t i o n i s h e l p f u l when us i n g the channel m o d i f i c a t i o n approach which, with i n c r e a s e d storage, w i l l reduce e a r l y o r d i n a t e s and i n c r e a s e l a t e r ones. By examining the r e l a t i v e l o c a t i o n s of the peak flow f o r the reach and the ' i n f i n i t e bound' o r d i n a t e , i t i s p o s s i b l e to make p r e d i c t i o n s about the net e f f e c t on the system outflow peak. 6.4 COST ANALYSIS Although not e x p l i c i t e l y d e s c r i b e d i n the examples, i t i s f a i r l y easy t o t r a n s l a t e the s e n s i t i v i t y i n f o r m a t i o n to r e f l e c t c o s t s . For example, i f a u n i t c o s t f o r withdrawal 96 or storage of flow i n each reach were known or c o u l d be estimated, then these values c o u l d be a p p l i e d d i r e c t l y to the dual v a l u e s d e r i v e d here to o b t a i n the c o s t per u n i t r e d u c t i o n of the system outflow peak. A l t e r n a t i v e l y , i f the p o t e n t i a l damages due to f l o o d i n g c o u l d be estimated i n terms of Q^, then that c o u l d be compared to the c o s t per u n i t peak r e d u c t i o n to determine i f design m o d i f i c a t i o n at any p a r t i c u l a r p o i n t i n the network i s cost e f f e c t i v e . 6.5 GRAPHICS INTERFACE As has been demonstrated, the amount of i n f o r m a t i o n 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 i s s u b s t a n t i a l , and would overwhelm the designer i f i t was presented i n t a b u l a r form. The l e n g t h of time r e q u i r e d to handle and a s s i m i l a t e such p r i n t e d output would counteract much of the b e n e f i t s of using t h i s model and slow down the i t e r a t i v e design c y c l e . As shown by the f i g u r e s i n Chapter 5, the i n f o r m a t i o n generated i s of a type which r e a d i l y lends i t s e l f to p r e s e n t a t i o n i n the form of a set of simple two-dimensional p l o t s superimposed on a schematic of the drainage system. Such a p r e s e n t a t i o n c o u l d e a s i l y be generated by a h i g h r e s o l u t i o n computer gra p h i c s d i s p l a y . Colour and zoom c a p a b i l i t i e s would s i g n i f i c a n t l y enhance the r e a d a b i l i t y of the d i s p l a y envisaged. REFERENCES Besant, C. B. Computer-Aided Design and Manufacture. E l l i s Harwood S e r i e s i n E n g i n e e r i n g Science, 1983. Chow, Ven Te, e d i t o r . Handbook of A p p l i e d Hydrology. McGraw-Hill Book Company, New York, 1964. C u r t i s , D. C. and R. H. McCuen. Design E f f i c i e n c y of Stormwater Detention B a s i n s . J o u r n a l of the Water Resources Planning and Management D i v i s i o n , American S o c i e t y of C i v i l Engineers, Volume 103, Number WR1, May 1977. Dendrou, S. A. and J . W. D e l l e u r . Watershed-Wide Planning of Detention Basins. Proceedings of the Conference on Stormwater Detention F a c i l i t i e s . New England C o l l e g e , New Hampshire, August 1982. Duru, J . 0. On-Site D e t e n t i o n : A Stormwater Management or Mismanagement Technique? Second I n t e r n a t i o n a l Conference on Urban Storm Drainage. Urbana, I l l i n o i s , June 1981. Encarnacao, J . and E. G. S c h l e c h t e n d a h l . Computer Aided Design: Fundamentals and System A r c h i t e c t u r e s . S p r i n g e - V e r l a g , New York, 1983. Fok, A. T. K., Perks, A. R. and L. A. Pataky. A p p l i c a t i o n of Computer Models on an Urban Drainage R e l i e f Study. Proceedings of the I n t e r n a t i o n a l 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., Nix, S. J . , D i c k i n s o n , R. E. and D. J . Polmann. Storm Water Management Model User's Manual V e r s i o n III ( F i n a l D r a f t ) . U n i t e d S t a t e s Environmental P r o t e c t i o n Agency, November 1981. K o u s s i s , A. D. T h e o r e t i c a l E s t i m a t i o n s of F l o o d Routing Parameters. J o u r n a l of the H y d r a u l i c s D i v i s i o n , American S o c i e t y of C i v i l Engineers, Volume 104, Number HY1, January 1978. Nash, J . E. A Note on the Muskingum Flood-Routing Method. J o u r n a l of Geophysical Research, Volume 64, Number 9, 1959. Perks, Alan R. A Review of Urban Runoff Models. Modern Concepts in Urban Drainage; Conference Proceedings Number 5, Toronto, March 1977. Ponce, V. M. S i m p l i f i e d Muskingum Routing Equation. J o u r n a l of the H y d r a u l i c s D i v i s i o n , American S o c i e t y of 97 98 C i v i l Engineers, Volume 105, Number HY1, January 1979. Solow, D. L i n e a r Programming: An I n t r o d u c t i o n to F i n i t e Improvement Al g o r i t h m s . E l s e v i e r Science P u b l i s h i n g Company, Inc., New York, 1984. Stephenson, D. Stormwater Hydrology and Drainage. E l s e v i e r Press, New York, 1981. Viessman, W., Knapp, J . W., Lewis, G. L. and T. E. Harbaugh. I n t r o d u c t i o n to Hydrology (3rd e d i t i o n ) . Harper and Row, New York, 1977. Walpole, R. E. I n t r o d u c t i o n to S t a t i s t i c s (3rd e d i t i o n ) . MacMillan P u b l i s h i n g Company Inc., New York, 1982. Weinmann, P. E. and E. M. Laurenson. Approximate Flood Routing Methods: A Review. J o u r n a l of the H y d r a u l i c s D i v i s i o n , American Socie.t.y, of C i v i l Engineers, Volume. 105, Number HY12, December 1979. Yen, B. C. and A. S. Sevuk. Design of Storm Sewer Networks. J o u r n a l of the Environmental En g i n e e r i n g D i v i s i o n , American S o c i e t y of C i v i l E ngineers, Volume 101, Number EE4, August 1975. APPENDIX A: PROGRAM LISTING 1 C 2 C *** THIS PROGRAM WILL EVALUATE MUSKINGUM CHANNEL ROUTING *** 3 C *** FOR A BRANCHED NETWORK, USING LINEAR PROGRAMMING *** 4 C 5 IMPLICIT REAL*8(A - H,0 - Z) 6 REAL*8 K(50),X(50),DT,BETA(50),GAMMA(50),PHI(50),PSI(50), 7 1 Q(50,50),TABLO(640,640),BBOBJ(640),UBOBJ(640), 8 2 RHS,RHS1(640),RHS2,BBRHS(640),UBRHS(640) 9 INTEGER NVIN(640),NVOUT(640),IFIRST(50),ILAST(50),ILAST2(50) -10 DIMENSION TITLE(8) 1 1 C 12 C *** READ AND PRINT THE TITLE *** 13 C 14 1.0 READ (5,20,END=810) TITLE 15 20 FORMAT (8A8) 16 WRITE (6,30) TITLE 17 30 FORMAT ("1 ',///,8A8/) 18 C 19 C *** READ THE PARAMETERS *** 20 C 21 READ (5,40) IF1, IF2, IF3 22 READ (5,40) NSTA, NJUNCT, NORDS 23 40 FORMAT (414) 24 NCHAN = NSTA-NJUNCT-1 25 DO 50 1=1, 50 26 IFIRST(I) =0 27 ILAST(I) =0 28 ILAST2(I) = 0 29 50 CONTINUE 30 IF (NJUNCT .EQ. 0) GOTO 80 31 NJP1 = NJUNCT + 1 32 DO 60 1=1,NJP1 33 READ (5,40) ITEMP1,ITEMP2, ITEMP3 34 IFIRST(ITEMP1) = 1 35 ILAST(ITEMP3) = 1 36 ILAST2(ITEMP3) = ITEMP2 37 IF (IF1 .EQ. 1) WRITE (6,70) I,ITEMP1,ITEMP2,ITEMP2,ITEMP3 38 60 CONTINUE 39 70 FORMAT (* BRANCH NO.',12,': STATION ',12,' TO STATION ',12, 40 1 ' JUNCTION: ',12,',',12) 41 80 IFIRST(1) = 1 42 READ (5,90) DT 43 90 FORMAT (3F9.3) 44 DO 110 J=2, NSTA 45 IF (IFIRST(J) .EQ. l) GO TO 110 46 I = J-1 47 READ (5,90) K ( l ) , X ( l ) 48 DENOM = DT-2.*K(I)*X(I) 49 IF (DENOM .GT. 0.10) GOTO 100 50 WRITE (6,120) I, DENOM, K ( l ) , X(I), DT 51 STOP 52 100 BETA(I) = (DT+2.*K(I)*(1-X(I)))/DENOM 53 GAMMA(I) = (DT-2.*K(I)*(1-X(I)))/DENOM 54 PHI(I) = (DT+2.*K(I)*X(I))/DENOM 55 PSI(I) = GAMMA(I)-BETA(I)*PHI(I) 56 110 CONTINUE 57 120 FORMAT ('0THE DENOMINATOR IN REACH',12,' = ',F9.3,'.*/, 58 1 ' DENOMINATOR MUST BE SIGNIFICANTLY GREATER THAN ZERO.'/ 99 100 Appendix A (continued) 59 . 2 ' CHOOSE NEW VALUES OF K, X, OR DT ' /, 60 3 * SUCH THAT 2KX « DT. ',/ 61 4 ' AT PRESENT, K=*,F6.3,' X=',F6.3,* DT=',F6.3,'. 62 NORM1 = NORDS-1 63 NVARS = NCHAN*NORM1+1 64 NCONST = (NCHAN+1)*NORM1 65 NVM1 = NVARS - 1 66 C 67 G *** ECHO THE PARAMETERS *** 68 C 69 I F (IF1 .NE. 1) GOTO 180 70 WRITE (6,130) NSTA, NCHAN, DT 71 130 FORMAT ('0THERE ARE ',12,' STATIONS AND ',12, 72 1 • ' CHANNEL REACHES. THE ROUTING PERIOD I S ' , F 7 . 3 ) 73 WRITE (6,140) 74 140 FORMAT('0REACH',T11,'K',T18,'X',T26,'BETA',T38,'GAMMA*, 75 1 T50,'PHI',T63,'PSI') 76 DO 150 J=2, NSTA 77 IF ( I F I R S T ( J ) .EQ. 1) GOTO 150 78 I = J-1 79 WRITE (6,160) I , K ( I ) , X ( l ) , B E T A ( I ) , G A M M A ( I ) , P H I ( I ) , P S I 80 150 CONTINUE 81 160 FORMAT (14,1X,2F7.3,3X,4G12.5) 82 WRITE (6,170) NORDS, NCONST, NVARS 83 170 FORMAT ('0THERE ARE',13,' HYDROGRAPH ORDINATES,', 84 1 14,' CONSTRAINTS, AND',14,' VARIABLES.') 85 C 86 C *** ZERO THE TABLEAU *** 87 C 88 180 NVP1 = NVARS + 1 89 NCP1 = NCONST + 1 90 DO 200 J=1, NVP1 91 DO 190 1=1, NCP1 92 T A B L O ( l , J ) = 0.0D0 93 190 CONTINUE 94 200 CONTINUE 95 C 96 C *** FORMULATE THE MINIMAX OBJECTIVE FUNCTION (MINIMIZE Y) 97 C 98 TABLO(1,1) = 1. 99 DO 210 J=1, NORM1 100 TABLO(1,NVARS-NORM1+J) = 1.E-10 101 210 CONTINUE 102 C 103 C *** READ THE FIRST ORDINATE OF EACH DOWNSTREAM HYDROGRAPH 104 C 1 05 DO 220 1=2, NSTA 106 I F ( I F I R S T ( I ) .EQ. 1) GOTO 220 107 READ (5,90) Q ( I , 1 ) 108 220 CONTINUE 109 C 1 10 C *** READ THE INFLOW HYDROGRAPHS *** 1 1 1 C 1 12 DO 240 1=1, NSTA 1 13 IF ( I F I R S T ( I ) .NE. 1) GOTO 240 1 14 DO 230 N=1, NORDS 115 READ (5,90) Q(I,N) 116 230 CONTINUE 101 Appendix A (continued) 1 17 240 CONTINUE 1 18 C 119 C * * * ROUTE THE FLOWS AND PRINT THE HYDROGRAPHS * * * 1 20 C 121 I F ( I F 2 .NE. 1) GOTO 270 122 WRITE (6,250) (N, N= 1 , NORDS) FORMAT ( ' 0 S T A . * , 5 0 ( I 5 , 2 X ) ) 1 23 250 124 WRITE (6,260) ( Q ( 1 , N ) , N=1,NORDS) 125 260 FORMAT (' 1: ',50F7.3) 126 270 DO 320 1=2,NSTA I F ( I F I R S T ( I ) .EQ. 1) GOTO 310 1 27 128 IM1 = 1-1 1 29 IF ( I L A S T ( I M I ) .EQ. 1) GOTO 290 130 DENOM = D T + 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) 131 DO 280 N=2,NORDS Q ( I , N ) = ( ( D T + 2 . * K ( I M 1 ) * X ( I M 1 ) ) * Q ( I M 1 , N - 1 ) 1 32 1 33 1 +(DT-2.*K(IM1)*X(IM1))*Q(IM1,N) 134 2 - ( D T - 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) ) * Q ( I , N - 1 ) ) 135 3 /DENOM 136 280 CONTINUE 1 37 GOTO 310 138 290 DENOM = D T + 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) 1 39 DO 300 N=2, NORDS ITEMP = I L A S T 2 ( I M 1 ) 1 40 141 Q(I,N)=((DT+2.*K(IM1)*X(IM1))*(Q(IM1,N-1)+Q(ITEMP,N-1 1 + ( D T - 2 . * K ( I M 1 ) * X ( I M 1 ) ) * ( Q ( I M 1 , N ) + Q ( I T E M P , N ) ) 1 42 143 2 - ( D T - 2 . * K ( I M 1 ) * ( 1 . - X ( I M 1 ) ) ) * Q ( I , N - 1 ) ) 1 44 3 /DENOM 1 45 300 CONTINUE 1 46 310 IF ( I F 2 .EQ. 1) WRITE (6,330) I , (Q(l,N),N=1,NORDS) 147 320 CONTINUE 1 48 330 FORMAT (' ',12,': ',50F7.3) 149 C 150 C * * * FORMULATE THE ROUTING CONSTRAINTS *** 151 C 152 NROW = 2 1 53 DO 410 I=2,NSTA IF ( I F I R S T ( I ) .EQ. 1) GOTO 410 1 54 1 55 IM1 = 1-1 156 IF ( I L A S T ( I M I ) .EQ. 1) GOTO 340 157 RHS = PHI(IM1)*Q(IM1,1)-GAMMA(IM1)*Q(I,1) 1 58 GOTO 350 159 340 ITEMP = I L A S T 2 ( I M 1 ) 160 RHS = PHI(IM1)*(Q(ITEMP,1)+Q(IM1,1))-GAMMA(IM1)*Q(I,1) 161 350 DO 400 N=2, NORDS 162 RHS2 = R H S * ( - P H I ( I M 1 ) ) * * ( N - 2 ) 163 RHS1(NROW) = RHS2 1 64 I F ( I F I R S T ( I M I ) .EQ. 1) RHS2 = RHS2+Q(IM1,N) 165 TABLO(NROW,NVP1) = -RHS2 166 TABLO(NROW,NROW) = -BETA(IM1) 167 I F (N.EQ.2) GOTO 370 168 NM2 = N-2 169 DO 360 L=1,NM2 • TABLO(NROW,NROW-L) = -PSI(IM1 ) * ( - P H I ( I M 1 ) ) * * ( L - 1 ) 1 70 • 171 360 CONTINUE 172 370 I F ( I F I R S T ( I M I ) .EQ. 1) GOTO 390 173 NCOL = NROW-NORM1 174 TABLO(NROW,NCOL) = 1. 102 Appendix A (continued) 175 I F ( I L A S T ( I M I ) .NE. 1) GOTO 390 176 ITEMP = ILAST2(IM1) 177 NCOL2 = (ITEMP-1)*NORM1+N 178 DO 380 J=1,ITEMP 179 I F ( I F I R S T ( J ) .EQ. 1) NCOL2 = NCOL2-NORM1 180 380 CONTINUE 181 TABLO(NROW,NCOL2) = 1. 182 390 NROW = NROW+1 183 400 CONTINUE 184 410 CONTINUE 185 C 186 C *** ADD THE MINIMAX CONSTRAINTS *** 187 C 188 DO 420 N=2 r NORDS 189 TABLO(NCHAN*NORM1+N,1) = - 1 . 190 TABLO(NCHAN*NORM1+N,(NCHAN-1)*NORM1+N) = 1. 191 420 CONTINUE 192 C 193 C *** PRINT THE INPUT TABLO *** 194 C 195 I F ( I F 3 .NE. 1) GOTO 4 60 196 WRITE (6,430) 197 430 FORMAT ('OTHE INPUT TABLO:'/) 198 DO 440 1=1,NCP1 199 WRITE (6,450) (TABLO(I,J),J=1,NVP1) 200 440 CONTINUE 201 450 FORMAT (' ',100(G8.1)) 202 C 203 C *** CALL THE SUBROUTINE VERSION OF L I P *** 204 C 205 460 CALL LIPSUB(TABLO, 640, NCONST, NVARS, 0, 0, 0, 1, 1.D-6, 206 1 NVIN, NVOUT, BBOBJ, UBOBJ, BBRHS, UBRHS, &790) 207 C 208 C *** PRINT THE OUTPUT TABLO *** 209 C 210 I F ( I F 3 .NE. 1) GOTO 500 211 WRITE (6,470) 212 470 FORMAT ('OTHE OUTPUT TABLO:'/) 213 DO 480 1=1, NCP1 214 WRITE (6,490) (TABLO(I,J),J=1,NVP1) 215 480 CONTINUE 216 490 FORMAT (* ',100(G8.1)) 217 C 218 C *** PRINT THE OPTIMUM VALUE OF THE OBJECTIVE FUNCTION *** 219 C 220 500 OPTIM = -TABLO(1,NVP1) 221 WRITE (6,510) OPTIM 222 510 FORMAT ('0PEAK OUTFLOW =', 1PG15.7) 223 C 224 C *** PRINT THE PRIMAL SOLUTION *** 225 C 226 WRITE (6,520) 227 520 FORMAT CO PRIMAL SOLUTION:'//, ' VARIABLE' , 7X, ' VALUE') 228 NCPV = NCONST + NVARS 229 KOUNT1 = 2 230 KOUNT2 = 2 231 DO 560 1=2, NVARS 232 DO 530 J=2, NCP1 103 Appendix A (continued) 233 I F (NVIN(J) .EQ. I) GOTO 540 234 530 CONTINUE 235 540 DIFF = TABLO(J,NVP1)-Q(KOUNT1,KOUNT2) 236 ALLOW = 0.001*Q(KOUNT1,KOUNT2) 237 I F (DIFF .GT. ALLOW .OR. DIFF .LT. -ALLOW) WRITE (6,570) 238 1 KOUNT1,KOUNT2,TABLO(J,NVP1),DIFF 239 I F (KOUNT1 .LT. NSTA) GOTO 550 240 I F (D I F F . LE. ALLOW .AND. DIFF .GE. -ALLOW) WRITE (6,580) 241 1 KOUNT1,KOUNT2,TABLO(J,NVP1) 242 550 KOUNT2 = KOUNT2 + 1 243 I F (KOUNT2 .LE. NORDS) GOTO 560 244 KOUNT1 = KOUNT1+1 245 I F (IFIRST(KOUNT1) .EQ. 1) KOUNT1 = KOUNT1+1 246 KOUNT2 = 2 247 560 CONTINUE 248 570 FORMAT(1X,' Q(',12,',',12,')',1X,F12.3,2X, 2 49 1 '** DIFFERS FROM ROUTED VALUE BY ',F12.3,' **') 250 580 FORMAT(IX,' Q ( ' , I 2,',',I 2 , ' ) ' ,1X,F12.3) 251 C 252 C *** PRINT THE REDUCED COSTS, I F ANY *** 253 C *** ( O n l y i f an e r r o r o c c u r s ) *** 254 C 255 DO 590 1=1, NVARS 256 I F (NVOUT(I) .LE. NVARS) GOTO 600 257 590 CONTINUE 258 GOTO 640 259 600 WRITE (6,610) 260 610 FORMAT ('0REDUCED COSTS:'/' VARIABLE', 8X, 'VALUE') 261 DO 620 1 = 1 , NVARS 262 I F (NVOUT(I) .LE. NVARS) WRITE (6,630) N V O U T ( I ) , T A B L O ( 1 , I ) 263 620 CONTINUE 264 630 FORMAT (1X,I 5,7X,1PG12.5) 265 C 266 C *** PRINT THE DUAL SOLUTION AND RHS RANGING *** 267 C 268 640 WRITE (6,650) 269 650 FORMAT ('0 SEN S I T I V I T Y ANALYSIS:'// 270 1 T32,'DUAL',T51,'RHS RANGING BOUNDS'/' CONST.', 271 2 T13,'ORDINATE',T31,'VALUE',T41,'FLOW',T52,'LOWER', 272 3 T64,'UPPER'/) 27 3 NCMN1 = NCONST-NORM1 274 KOUNT1 = 1 275 KOUNT2 = 2 276 DO 740 J=1,NCMN1 277 DO 660 1=1, NVARS 278 INDEX = NVOUT(I)-NVARS 279 I F (INDEX .EQ. J ) GOTO 680 280 660 CONTINUE 281 WRITE (6,670) 282 670 FORMAT (' INDEX ERROR') 283 STOP 284 680 BBR = -UBRHS(J)-RHS1(J+1) 285 UBR = -BBRHS(J)-RHS1(J+1) 286 I F ( I F I R S T ( K 0 U N T 1 ) .EQ. 1) GOTO 690 287 I F ( I L A S T ( K 0 U N T 1 ) .EQ. l ) GOTO 700 288 BBR = BBR+Q(KOUNT1,KOUNT2) 289 UBR = UBR+Q(KOUNT1,KOUNT2) 290 I F (BBR . L E . 1.E-5 .AND. BBR .GE. -1.E-5) BBR = 0.00 1 04 Appendix A (continued) 291 I F (UBR .LE. 1.E-5 .AND. UBR .GE. -1.E-5) UBR = 0.00 292 690 WRITE(6,750) INDEX,KOUNT1,KOUNT2,TABLO(1,1), 293 1 Q(KOUNT1,KOUNT2),BBR,UBR 294 GOTO 710 295 700 ITEMP = ILAST2(KOUNT1) 296 QS = Q(ITEMP,KOUNT2)+Q(KOUNT1,KOUNT2) 297 BBR = BBR+QS 298 UBR = UBR+QS 299 I F (BBR .LE." 1 .E-5 .AND. BBR .GE. -1.E-5) BBR = 0.00 300 I F (UBR .LE. 1.E-5 .AND. UBR .GE. -1.E-5) UBR = 0.00 301 WRITE(6,760) INDEX,ITEMP,KOUNT2,KOUNT1,KOUNT2, 302 1 TABLO(1,I),QS,BBR,UBR 303 710 I F (KOUNT2 .EQ. NORDS) GOTO 720 304 KOUNT2 = KOUNT2+1 305 GOTO 740 306 720 KOUNT2 = 2 307 KOUNT1 = KOUNT1+1 308 WRITE(6,730) 309 730 FORMATC ') 310 I F (IFIRST(KOUNT1+1) .EQ. 1) KOUNT1 = KOUNT1+1 311 740 CONTINUE 312 750 FORMAT (IX,14,5X,' Q(' ,I 2,' , 1 ,12,')* ,T28,F9.4,T38,3G12.5) 313 760 FORMAT (IX,14,' Q ( ' , 12 , ' , ' , I 2 , ' ) + Q(',12,',',I 2 , ' ) ' ,T28, 314 1 F9.4,T38,3G12.5) 315 STOP 316 770 WRITE (6,780) IROW, JCOL, COEFF 317 780 FORMAT (' DATA INCORRECT*, 214, G12.5) 318 STOP 319 790 WRITE (6,800) 320 800 FORMAT (' NO OPTIMUM') 321 810 STOP 322 END 105 APPENDIX B - THE SUBROUTINE VERSION OF LIP PURPOSE To s o l v e l i n e a r programming problems using the same method as the s e l f - c o n t a i n e d program i n *LIP. TYPE OF ROUTINE A set of FORTRAN IV su b r o u t i n e s . 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, two-dimensional a r r a y , dimensioned at l e a s t M+1 by N+1. On e n t r y , i t c o n t a i n s the t a b l e a u of the problem. The f i r s t row c o n t a i n s the c o e f f i c i e n t s of the o b j e c t i v e f u n c t i o n . The l a s t NE rows c o n t a i n the c o e f f i c i e n t s of the e q u a l i t y c o n s t r a i n t s ( i f there are any). A l l intermediate rows c o n t a i n < type c o n s t r a i n t s only (> type c o n s t r a i n t s must be changed to < c o n s t r a i n t s by m u l t i p l y i n g through by - 1 ) . The N+1th column c o n t a i n s the right-hand s i d e s of the c o n s t r a i n t s ( s t a r t i n g at the second element of the column). 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,... N N+1 1 OBJECTIVE FUNCTION 2 < TYPE CONSTRAINTS R H S M NE EQUALITY CONSTRAINTS M+1 On output, v a r i o u s elements of TB are used to s t o r e the optimum value of the o b j e c t i v e f u n c t i o n , the value of the primal s o l u t i o n v a r i a b l e s , the value of the reduced c o s t s , and value of the dual s o l u t i o n v a r i a b l e s ( i f NE=0). See the d e s c r i p t i o n of NCHK and NCHK1 on the f o l l o w i n g page f o r d e t a i l s . The value of the optimum i s l o c a t e d i n TB(1, N+1). 106 NDIMTB i s the f i r s t dimension of the a r r a y TB. M i s the number of c o n s t r a i n t s . M+1<NDIMTB. N i s an INTEGER v a r i a b l e . On en t r y , i t c o n t a i n s the number of v a r i a b l e s . On e x i t , i t i s set to the number of v a r i a b l e s minus the number of e q u a l i t y c o n s t r a i n t s . NE i s the number of e q u a l i t y c o n s t r a i n t s . MAX =1 i f the o b j e c t i v e f u n c t i o n i s to be maximized. =0 i f the o b j e c t i v e f u n c t i o n i s to be minimized. NOBJ =1 i f o b j e c t i v e f u n c t i o n ranging i s to be attempted. The o b j e c t i v e f u n c t i o n can be ranged only i f NE=0. =0 otherwise. NRHS =1 i f right-hand s i d e ranging i s to be attempted. Right-hand s i d e ranging i s done only f o r i n e q u a l i t y c o n s t r a i n t s . =0 otherwise. TOL i s a REAL*8 v a r i a b l e . I t should be set to the to l e r a n c e f o r the problem. Numbers smal l e r than TOL in absolute value w i l l be c o n s i d e r e d zero . If TOL i s set to 0.D0 by the user, i t w i l l be r e s e t by the subroutine to 1.D-6. NCHK i s an INTEGER one-dimensional a r r a y , dimensioned at l e a s t M+1. On e x i t , NCHK(I), 1=2, ...,M+1 cont a i n s 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 pri m a l s o l u t i o n . The v a r i a b l e ' s value i s l o c a t e d i n TB(I,N+1) where the value N has been ret u r n e d by the subrou t i n e . NCHK1 i s an INTEGER one-dimensional a r r a y , dimensioned at l e a s t N (number of v a r i a b l e s ) . On e x i t , i f NCHK1(I)<number of v a r i a b l e s , then TB(1,I) giv e s the reduced cost of the v a r i a b l e with index NCHKI(I). If 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 s o l u t i o n v a r i a b l e with index, NCHK1(I) - number of v a r i a b l e s . I v a r i e s from 1 to N. The dual s o l u t i o n i s c a l c u l a t e d only i f NE=0. BBOBJ UBOBJ are REAL*8, one-dimensional a r r a y s , each dimensioned at l e a s t N. On e x i t , i f NOBJ=1 and NE=0, they c o n t a i n the lower and upper bounds on the c o e f f i c i e n t s of the o b j e c t i v e f u n c t i o n . I f NOBJ=0 they are not used and so may be dimensioned to 1. 1 07 BBRHS UBRHS are REAL*8, one-dimensional a r r a y s , each dimensioned at l e a s t M. On e x i t , i f NRHS=1 they c o n t a i n the lower and upper bounds on the right-hand s i d e s . Bounds are giv e n only f o r i n e q u a l i t y c o n s t r a i n t s . Bounds s t a r t i n the second l o c a t i o n of the a r r a y . I f NRHS=0 they a r e not used and so may be dimensioned to 1 . nn i s the statement number i n the user's c a l l i n g program to which c o n t r o l i s t r a n s f e r r e d i f the f u n c t i o n i s unbounded or i n f e a s i b l e . RESTRICTIONS 1. If NET*0, the dual s o l u t i o n i s not c a l c u l a t e d . 2. If NET*0, o b j e c t i v e r a n g i n g i s not done. 3. Right-hand s i d e ranging i s done only f o r i n e q u a l i t y c o n s t r a i n t s . 

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