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Optimization of pipe sizes in open-ended sprinkler irrigation systems Burkholder, David R. 1980

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O P T I M I Z A T I O N OP P I P E S I Z E S I K O P E N - E N D E D S P R I N K L E R I R R I G A T I O N S Y S T E M S b y D A V I D R . BURKHOLDER B . A . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1976 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( T h e D e p a r t m e n t o f 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 a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 1980 © D a v i d R . B u r k h o l d e r , 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 DE-6 B P 75-51 1 E ABSTRACT A sprinkler irrigation system is typically designed by f i r s t choosing the set of operating conditions and the location for the pipe network, and then selecting the optimal pipe sizes for the network so that the system cost is minimized. An efficient optimizing technique would greatly improve this design process. This thesis examines a number of procedures for optimizing pipe sizes in open-ended (tree-like) sprinkler irrigation systems. The Linear Program-ming Technique was found to be by far the most suitable procedure for solving complex pipe sizing problems. A computer model u t i l i z i n g a general linear programming routine is presented for optimizing pipe sizes in gravity systems, and pumping systems with both constant-speed and variable-speed power units. A comprehensive discussion is included, describing a l l input parameters, and the function and operation of each major segment of the model. An example is used to reinforce ideas formulated in the discussion and to i l l u s t r a t e how the model can be implemented. A complete l i s t i n g of the model, and a copy of both the input data and the output, for the example, are given in the indices. i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES ix ACKNOWLEDGMENT x i i CHAPTER 1. INTRODUCTION 1 2. PIPE SIZING METHODS INVESTIGATED 4 2.1 Cost Index Methods 4 2.1.1 Gravity Systems, Perold's Method and Russell's Method . 4 2.1.2 Pumping Systems, Keller's Method and Russell's Method . 6 2.1.3 Evaluation of Cost Index Methods 8 2.1.4 Economic Pipe Theory 8 2.2 Minimum Length Method and a Cost Index Example 11 2.2.1 Minimum Length Method Derivation 11 2.2.2 Negate Minimum Length Method Using a Cost Index Example 13 2.2.3 Evaluation of Minimum Length Method 17 2.3 Dynamic Programming Method 18 2.3.1 Dynamic Programming Method Derivation 18 2.3.2 Negate Dynamic Programming Method 22 2.3.3 Evaluation of Dynamic Programming Method 25 2.3.4 Decreasing, Concave Up Cost Function 25 2.4 Linear Programming Method 29 2.4.1 Linear Programming Formulation and Background Theory .. 29 2.4.2 Constant Flow Example 30 2.4.3 Time-variable Flow Example 32 2.4.4 Pumping Example 33 2.4.5 Evaluation of Linear Programming Method 34 i i i TABLE OF CONTENTS (continued) Page CHAPTER 3. PIPE SIZING OPTIMIZATION MODEL 35 3.1 Define Model Input Parameters 35 3.1.1 General Input Data 35 3.1.2 Pump Input Data 39 3.1.3 Feeder (N) and Conveyance Feeder (N) Input Data .... 40 3.1.4 Conveyance Section Input Data 44 3.1.5 Required Formats for Input Data 45 3.2 Define Input Parameters for the General Linear Programming Routine (LIPSUB) 46 3.2.1 Background Definitions 46 3.2.2 Define Input Parameters for LIPSUB 46 3.2.3 SLACK Parameters (Sensitivity Measurements) 48 3.3 Background Theory for Model Discussion 49 3.3.1 Minimum Allowable Hydraulic Head at Control Points . 49 3.3.2 Maximum Number of Pipe Sizes in a Section 50 3.3*3 Discontinuities in Pipe Systems 52 3.3.4 Pump Operating and Fixed Costs 57 3.3.4.1 Operating Pump Cost 57 3.3.4.1.1 Constant-speed Power Units 58 3.3.4.1.2 Variable-speed Power Units 59 3.3.4.2 Fixed Pump Cost 60 3.4 Model Discussion 62 3*4.1 Introduction 62 3.4.2 Eliminate Uneconomic Pipe Sizes 63 3.4.3 Determine the Total Number of Time Intervals 63 3.4.4 Conveyance Section Discharge Computation 64 3.4.5 Eliminate Redundant Head Loss Equations 64 3.4.5.1 Zero Flow Redundancy 65 iv TABLE OF CONTENTS (continued) Page CHAPTER 3. 3.4.5.2 Head Loss Redundancy 65 3.4.5.3 Control Point Elevation Redundancy 67 3.4.5.4 Feeder Head Loss Equation Index Redundancy and Single Stage Redundancy 67 3.4.5.4.1 Feeder Head Loss Equation Index Redundancy ...................... 67 3.4.5.4.2 Single Stage Redundancy ......... 68 3.4.6 Determine the Minimum Allowable Inlet Head for Given Pipe Sizes 68 3.4.7 Generate I n i t i a l Tableau 68 3.4.8 Determine Inlet Head Values 69 3.4.9 Print Pump Output and Calculate Fixed Pump Cost 69 3.4.10 Subroutine LIP 70 3.4.11 Generate New Tableau 70 3.4.12 Subroutine LIPSUB and Optimal Pipe Location, Size and Length 71 3.4.12.1 Subroutine LIPSUB 71 3.4.12.2 Define Output Parameters from LIPSUB 71 3.4.12.3 Discontinuity 72 3.4.13 Variable Reduction Algorithms 72 3.4.13.1 Introduction 72 3.4.13.2 Feeder Algorithm 73 3.4.13.3 Conveyance Section Algorithm 77 3.4.13.4 Operating Details 78 3.4.14 Sensitivity Measurements 79 3.4.15 Calculate Pump Operating Cost 82 4. MODEL EXAMPLE 83 4.1 Introduction 83 4.2 Explanation of Feeder Input Data Derivation 83 v TABLE OP CONTENTS (continued) Page CHAPTER 4. 4.3 Explanation of Conveyance System Input Data Derivation 86 4.4 Explanation of Computer Printout 86 4.5 Pump and Pipe Size Input Data Derivation 86 4.5.1 Raw Data 86 4.5.2 Mainline Pipe Cost and Head Loss Coefficient 87 4.5.3 Fixed Pump Cost 89 4.5.4 Operating Pump Cost 90 4.6 Sprinkler Irrigation System Layout, and Summary of General Input Data and Pump Input Data 91 4.7 Feeder Input Data Derivation 93 4.8 Conveyance System Input Data Derivation 97 4.9 Optimal Pipe Size Design 100 4.10 Discussion of Optimal Pipe Size Design 101 5. CONCLUSIONS AND FUTURE DEVELOPMENTS 104 5.1 Conclusions 104 5.2 Future Developments 106 BIBLIOGRAPHY 108 INDEX A 109 - Listing of the Pipe Size Optimization Model 109 INDEX B 134 - Input Data for Model Example outlined in Chapter 4 135 - Output for Model Example outlined in Chapter 4 136 v i LIST OP TABLES TABLE Page 2.1 Economic Pipe Sizes 9 2.2 Uneconomic Pipe Sizes 9 2.3 Pipe Costs and Head Loss Coefficients, Minimum Length Example (2) 13 2.4 Cost Index Values, Minimum Length Example (2) 15 2.5 Cost Index Comparison for Minimum Length Solution, Minimum Length Example (2) 15 2.6 Cost Index Comparison for New Solution, Minimum Length Example (2) 16 2.7 Pipe Costs and Head Losses, Linear PrograimTiing Constant Plow Example 31 2.8 Formulation Tableau, Linear Programming Constant Flow Example 31 2.9 Formulation Tableau, Linear Programming Time-variable Flow Example 32 2.10 Formulation Tableau, Linear Programming Pumping Example 33 3.1 Conveyance Section Matrix 38 3.2 Discharging Outlets TH(J,I ,2), J=1, ...NC(N), 1=1, ...IC(N) . 43 3.3 Minimum Hydraulic Head 43 3.4 Required Formats for Input Data 45 4.1 Mainline Pipe Cost 87 4.2 Irrigation Pump and Motor Prices 87 4.3 Annual Cost of Pipe 88 4.4 Head Loss Coefficients 89 4.5 Annual Fixed Pump Cost 90 4.6 Conveyance Section Matrix 92 4.7 Pipe Costs and Head Loss Coefficients 92 4.8 Fixed Pump Cost Function 92 4.9 Discharging Outlets, Feeder Number (jj) 93 v i i LIST OF TABLES (continued) Page TABLE 4.10 Discharging Outlets, Feeder Number(T) 94 4.11 Discharging Outlets, Feeder Number(?) 95 4.12 Discharging Outlets, Feeder Number(8) 96 4.13 Discharging Outlets, Feeder Number(7) 96 4.14 Conveyance Section Discharge, Conveyance Line 1 97 4.15 Discharge from Conveyance Feeders, Conveyance Line 1 97 4.16 Conveyance Section Discharge, Conveyance Line 2 98 4.17 Discharge from Conveyance Feeders, Conveyance Line 2 98 4.18 Total Number of Significant Head Loss Equations 99 4.19 Minimum Allowable Hydraulic Elevation, at Conveyance Nodes .. 99 4.20 Conveyance Section Length 99 4.21 Slope of the Optimal Pipe Cost Function 103 v i i i LIST OF FIGURES FIGURE Page 2.1 Gravity System 5 2.2 Pumping System 7 2.3(a) Cost vs. Head Loss Graph for Economic Pipe Sizes 9 2.3(b) Cost vs. Head Loss Graph for Uneconomic Pipe Sizes ........ 9 2.4 Decreasing, Concave Up Cost Function 10 2.5 Define System, Minimum Length Example (l) 11 2.6(a) Design Procedure Minimum Length Example (1), Step 1 12 2.6(b) , Step 2 12 2.6(c) , Step 3 12 2.7 Optimal Solution, Minimum Length Example (1) 13 2.8 Define System, Minimum Length Example (2) 13 2.9 Minimum Length Solution, Minimum Length Example (2) . . . . . . 14 2.10 Optimal Solution, Minimum Length Example (2) 14 2.11 Define System, Dynamic Programming Example (1) 18 2.12 Cost vs Head Loss Graph, Dynamic Programming Example (1) . 19 2.13 Design Procedure, Dynamic Programming Example (1) 21 2.14 Define System, Dynamic Programming Example (2) 22 2.15 Optimal Solution, Dynamic Programming Example (2) 22 2.16(a) Independence Constraint, Dynamic Prograinming Example (2), Route 1 24 2.16(b) Route 2 24 2.17(a) Two Specific Decreasing, Concave Up Cost Functions,C^ vs h^ 26 2.17(b) ,C2 vs h 2 26 2.18 Cumulative Cost Function, C^ vs h 2, when h^=h1+h2=4 26 2.19 Formulation and Optimal Solution, Linear Programming Constant Flow Example 31 2.20 Formulation and Optimal Solution, Linear Programming Time-variable Flow Example 32 ix LIST OF FIGURES (continued) FIGURE Page 3.1 T y p i c a l System Layout 38 3.2 T y p i c a l Feeder Layout 43 3.3 I n s e r t i n g a Dummy Node at Point B 50 3.4 Maximum Number of Pipe Sizes i n a Section 51 3.5( a) D i s c o n t i n u i t y i n a Series System, Define System 53 3.5(b) , Continuously Decreasing Pipe Sizes 53 3.5(c) , Random Pipe Expansions and Contractions 53 3.6(a) Dis c o n t i n u i t y i n a Branched System, Define System 55 3.6(b) , Optimal Solution 55 3.6(c) , Solution with D i s c o n t i n u i t y 55 3.7(a) D i s c o n t i n u i t y i n a Series System with Unequal Flow, Define System 56 3.7(b) , Optimal Solution 56 3.7(c) , Solution with D i s c o n t i n u i t y 5& 3.8 Ideal Pump Operating Cost Formulation, Constant-speed Power Unit 58 3.9 S i m p l i f i e d Pump Operating Cost Formulation, Constant-speed Power Unit 59 3.10 Ideal Pump Operating Cost Formulation, Variable-speed Power Unit 60 3.11 T y p i c a l T o t a l Pump Cost Formulation , 61 3.12 Define a l l separate Flow Condition Comparisons, f o r Two Flow Patterns 66 3.13 V a r i a b l e Reduction i n Feeders f o r Maximum I n l e t Head ....... 73 3.14 V a r i a b l e Reduction i n Feeders f o r Minimum I n l e t Head ....... 74 x LIST OF FIGURES (continued) FIGURE Page 3.15(a) Va r i a b l e Reduction, Deviate Condition, Define System 75 3.15(b) , Optimal Solution when In l e t Head = 2.4m . . . 75 3.15(c) , Optimal Solution when I n l e t Head = 3»0m . . . 75 3.16 V a r i a b l e Reduction i n Conveyance Sections f o r Maximum Inl e t Head 77 3.17 V a r i a b l e Reduction i n Conveyance Sections f o r Minimum I n l e t Head 77 3.18 Define a l l separate Head Loss Conditions given the Optimal Pipe Size Solution and a p a r t i c u l a r Flow .Pattern 80 3.19(a) Procedure f o r Defining Individual Head Loss Conditions i n Feeders, Condition 1 81 3.19(b) , Condition 2 81 4.1 Define Values and Redundancy Symbols i n the Discharge Outlet Table 85 4.2 S p r i n k l e r I r r i g a t i o n System Layout 91 4 .3( a ) L a t e r a l Movements Diagram, Feeder Number (jj) 93 4.3(b) Hydraulic Grade Line Diagram, Feeder Number 93 4 .4(a) L a t e r a l Movements Diagram, Feeder Number(?) 94 4.4(b) Hydraulic Grade Line Diagram, Feeder Number(7) 94 4 .5( a ) L a t e r a l Movements Diagram, Feeder Number(2) 95 4.5(b) Hydraulic Grade Line Diagram, Feeder Number (g) 95 4.6 Hydraulic Grade Line Diagram, Conveyance Line 1 97 4.7 Hydraulic Grade Line Diagram, Conveyance Line 2 98 4.8 Optimal Pipe Size Design 100 4.9 Cumulative Cost Function f o r Model Example 101 4.10(a) Define the Minimum Cumulative Cost of the System,Condition 1 102 4.10(b) ,Condition 2 102 x i ACKNOWLEDGEMENT T h e a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o h i s a d v i s o r , D r . S . 0. R u s s e l l , f o r h i s g u i d a n c e a n d e n c o u r a g e m e n t . S p e c i a l t h a n k s a l s o t o D r . T . P o d m o r e o f t h e B i o - R e s o u r c e ; : E n g i n e e r i n g D e p a r t m e n t a n d t o M r . V i n c e H e l t o n o f P a c i f i c I r r i g a t i o n L t d . f o r t h e i r h e l p i n p r o v i d i n g d a t a f o r t h e m o d e l e x a m p l e . F i n a l l y I w o u l d l i k e t o t h a n k t h e C i v i l E n g i n e e r i n g D e p a r t m e n t f o r f i n a n c i a l s u p p o r t a n d t h e u s e o f t h e c o m p u t e r f a c i l i t i e s . x i i Chapter 1 1 INTRODUCTION' The design of irrigation schemes has traditionally been carried out in a haphazard way; Design problems are usually solved by selecting the most economic design from a number of t r i a l designs. This i s often a tedious procedure and i t does not ensure an overall optimum. Considerable s k i l l is required, on the part of the designer, to separate and assimilate the important data that governs the design, from the mass of available inform-ation. Irrigation design can be divided into two main parts. In the f i r s t part the designer selects a set of operating conditions and locations for a l l mainlines and outlets. This selection may well be based on considerations other than purely economic ones. In the second part the designer attempts to optimize pipe sizes (ie. minimize pipe cost) for the system defined in the f i r s t part. In contrast to the f i r s t part, this second part is purely dependent upon economic considerations, and lends i t s e l f well to mathematical analysis. Ideally the designer should reexamine his layout in the light of his analysis; but since the analysis is time consuming this i s usually not done. An efficient method for optimizing pipe sizes allows more layouts to be examined and optimized. This would f a c i l i t a t e comparisons between alternatives and would greatly improve the overall design process. Optimizing pipe sizes for an open-ended (tree-like), as opposed to a network, sprinkler irrigation system is the topic of this thesis. There are two types of irrigation systems, gravity systems and pumping systems. The approaches for optimizing pipe sizes-differ slightly for these two systems. In simple gravity systems, mainlines are designed to minimize pipe cost while absorbing the available head in f r i c t i o n losses. These losses are 2 d e p e n d e n t u p o n f l o w r a t e . I n p u m p i n g s y s t e m s , t h e sum o f p i p e c o s t a n d p u m p i n g c o s t m u s t b e m i n i m i z e d . I n t h i s c a s e , t h e h e a d a t t h e i n l e t t o t h e i r r i g a t i o n s c h e m e b e c o m e s a v a r i a b l e . A s t h i s i n l e t h e a d i s i n c r e a s e d , t h e p u m p i n g c o s t w i l l a l s o i n c r e a s e , b u t p i p e c o s t w i l l d e c r e a s e . A t some o p t i m a l v a l u e o f h e a d t h e o v e r a l l s y s t e m c o s t w i l l b e m i n i m i z e d . B e s i d e s b e i n g d e p e n d e n t u p o n t h i s i n l e t h e a d , p u m p i n g c o s t i s a l s o d e p e n d e n t u p o n f l o w r a t e a n d a n n u a l o p e r a t i n g t i m e . I n a p u m p i n g s y s t e m , i n o r d e r t o c o m p a r e p i p e c o s t w i t h p u m p i n g c o s t i t i s n e c e s s a r y t o r e d u c e a l l v a l u e s t o t h e i r a n n u a l e q u i v a l e n t s , w h i c h d e p e n d u p o n l i f e e x p e c t a n c y a n d i n t e r e s t r a t e . P u m p i n g c o s t i s u s u a l l y -s e p a r a t e d i n t o f i x e d ( e q u i p m e n t ) c o s t a n d o p e r a t i n g ( f u e l a n d m a i n t e n a n c e ) c o s t . A n u m b e r o f d i f f e r e n t m e t h o d s f o r o b t a i n i n g o p t i m a l p i p e s i z e s w e r e i n v e s t i g a t e d , w i t h v a r y i n g d e g r e e s o f s u c c e s s . I n m o s t c a s e s t h e m e t h o d s e x a m i n e d e i t h e r b e c a m e t o o c o m p l e x o r t h e i r l o g i c b r o k e down u n d e r s c r u t i n y , w h e n c o m p l i c a t e d s p r i n k l e r s y s t e m e x a m p l e s w e r e a t t e m p t e d . T h e C o s t I n d e x M e t h o d s f o r m u l a t e d b y P e r o l d , K e l l e r a n d R u s s e l l , a r e n o t s u i t a b l e a s g e n e r a l p i p e s i z i n g s o l u t i o n s , b u t t h e i n d e x i t s e l f i s a u s e f u l c o n c e p t . T h e C o s t I n d e x c a n b e u s e d t o i d e n t i f y u n e c o n o m i c p i p e s i z e s a n d i s h e l p f u l i n u n d e r s t a n d i n g o p t i m a l p i p e s i z i n g b e h a v i o r . L i n e a r P r o g r a m m i n g i s t h e o n l y m e t h o d i n v e s t i g a t e d t h a t c a n h a n d l e c o m p l i c a t e d e x a m p l e s w i t h n o s e r i o u s d i f f i c u l t i e s . L i n e a r P r o g r a m m i n g i s a v e r y p o w e r f u l o p t i m i z a t i o n t e c h n i q u e ; I t c a n b e u s e d t o s o l v e p i p e s i z i n g p r o b l e m s f o r a n y o p e n - e n d e d p i p e n e t w o r k a n d o p e r a t i n g c o n d i t i o n . I t i s p o s s i b l e t o USB l i n e a r p r o g r a m m i n g f o r s o l v i n g p i p e s i z i n g p r o b l e m s b e c a u s e b o t h h e a d l o s s a n d p i p e c o s t a r e l i n e a r l y p r o p o r t i o n a l t o p i p e l e n g t h ; I f t h e r e w a s a s i m p l e l i n e a r r e l a t i o n s h i p b e t w e e n p u m p i n g c o s t a n d i n l e t h e a d , t h e n p u m p i n g c o s t c o u l d a l s o b e i n c l u d e d i n t h e l i n e a r p r o g r a m m i n g f o r m u l a t i o n . U n f o r t u n a t e l y t h i s i s n o t n o r m a l l y t h e c a s e . I t i s s t i l l p o s s i b l e t o u s e L i n e a r P r o g r a m m i n g f o r s o l v i n g p u m p i n g s y s t e m s b u t t h e p r o c e d u r e i s m o r e 3 i n v o l v e d . The a n a l y s i s needs to be repeated a number of times, to o b t a i n optimal pipe c o s t , f o r a range of i n l e t head v a l u e s . For each i n l e t head, pumping cost i s c a l c u l a t e d s e p a r a t e l y , and then added to the optimal pipe cost to get the t o t a l cost of the system. T o t a l system cost a t v a r i o u s head values are compared, and the head t h a t produces the minimum cost i s the optimum;" Each of the methods i n v e s t i g a t e d , i n c l u d i n g the L i n e a r Programming Technique are described i n Chapter 2. A model has been developed to s o l v e the pipe s i z i n g problem f o r g r a v i t y systems and pumping systems w i t h both constant-speed and v a r i a b l e - s p e e d power u n i t s , u t i l i z i n g a general l i n e a r programming r o u t i n e . In Chapter 3 a H input parameters f o r the model and the l i n e a r programming r o u t i n e are defi n e d . The model i s made up of a number of d i f f e r e n t p a r t s . The background theory, f u n c t i o n , and workings of each p a r t are a l s o discussed i n Chapter 3« An example, u s i n g a t y p i c a l i r r i g a t i o n system i s o u t l i n e d i n Chapter 4» to show how the model operates. T h i s example i l l u s t r a t e s how input data i s formulated, and discusses the optimal p i p e ^ i z i n g r e s u l t s . Conclusions and suggestions f o r f u t u r e developments are presented i n Chapter % The s e c t i o n on f u t u r e developments describes how the model c o u l d be improved to make i t more e f f i c i e n t and useful-as a design a i d . Index A contains a l i s t i n g of the model, and Index B contains a copy of the input data and the output f o r the example o u t l i n e d i n Chapter 4« Chapter 2 PIPE SIZING METHODS INVESTIGATED 21.1 Cost Index Methods Upon close examination a method proposed by Russell i s found to be identical to those published by Keller and Perold. The following sections prove that the approaches are identical and discuss why none of them are particularly acceptable, 2.1.1 Gravity Systems. Perold fs Method and Russell's Method For a given gravity system 5Perold attempts to formulate a procedure for determining the most economic pipe sizes for single and branched systems. The procedure evaluates the financial gain and loss resulting from changing pipe sizes in such a way as to maintain a constant head loss. To compare the various possible combinations of pipe size changes Perold uses a Cost Difference Index which is identical to an index proposed by Russell, Perold's Method: Cost Difference Index for pipe sizes 1 and 2 = C ACm^ x A L m 2 , l = A C m l , 2 / A H m 2 1 = ( C m l " C m2^ / ^ 2 " Eml^ = C l - ° 2 = A C1.2 = A C1.2 H2 " H l (K 2Q n - K^n) A K 2 t l Q n this derivation assumes a head loss formula of the form sH = KQ, • = KmQ11! ACm - cost difference per meter length, in dollars.^ Aim - length of pipe changed per meter head loss, in meters*, AHm - change in head loss per meter length, in meters Cm - cost per meter length, in dollars Hm - head loss per meter length, in meters C - total cost, in dollars H - total head loss, in meters AC - total cost difference, in dollars K - head loss coefficient, in meters/(l/sec) n Q, - flow rate, in l/sec n - exponential AK - change in head loss coefficient, in meters / ( l / s e c ) n Km - head loss coefficient per meter length, in meters/(l/sec) n L - length of pipe, in meters Russell's Method: Gravity System |Head=Ht , A • ^ • Head=0 FIG. 2.1 Objective: minimize C^  + C^  - pipe cost in section A and B _1 Given constraint + S^B* " H t = 0 1 From 1 & 2; minimize G = C^ + Cg + ^(\^ + - H : t) 1 Take derivatives and solve = b2k + = °; x= - 3 ° A i 3K A ^K A a c _ = ^ B + A o J 1 = 0; A= - d C B S 3 K B d K B dW From 4 & 1; - 3 C A _ - 2> C B a KA^A n 3 KB^B n the value for this expression must be constant for every section in the optimal pipe size solution Iii real terms, the change in cost and head loss coefficient depend upon the particular pipe sizes that are present in the section. Assuming optimal pipe sizes 1 and 2 in a given section, the expression can be written as follows: -AC _ Ac This result i s identical to the one found from Perold*s analysis; Russell has made a similar derivation for a "branch system which is also consistent with the procedure followed by Perold. 2.1.2 Pumping Systems. Keller's Method and Russell's Method For a pump system Keller proposes a Flow Index to compare the economic f e a s i b i l i t y of the various possible combinations of pipe, sizes. This Flow Index defines the flow rate at which the power-cost saving due to the .lower f r i c t i o n loss in the larger pipe is equal to the fixed-cost addition due to the higher i n i t i a l cost of the larger pipe; This index can also be described as the flow rate at which the sums of the fixed and power costs are equal (for a given pump discharge) for the two pipe sizes in question. The Flow Index i s formulated as follows for a given pipe section. The analysis assumes that only pipe cost and pump operating cost are significant and ignores capital pump cost. Keller's Method: Pump Operating Cost Difference for pipe sizes 1 and 2 = ACp 2 1 = (Ks x Qp) x AH2 1 = Kst X AH2 1 - — 6 Acp - pump operating cost difference, i n dollars Ks - cost per l/sec per meter head loss, in dollars Qp - pump flow rate, in l/sec AH - change in head loss, in meters Kst - cost per meter head loss, in dollars At a particular flow rate Q, the fixed cost difference equals the pump operating cost difference. From 6 ; AG^ 2 = Kst x AH2 1 = Ks t x ( A K ? 1QN) Q = AC \ 1 n Kst XAK 2,1 t h i s i s the Flow Index Value f o r pipe s i z e s 1, and 2 Ru s s e l l ' s Method : Pumping System Head=H •Head=Ht FIG. 2.2 Ob j e c t i v e : minimize Cp + C^ - pump ope r a t i n g cost and f i x e d cost i n s e c t i o n A Given c o n s t r a i n t n KAQA + H T ~ H = 0 From 8 & _2; minimize P = Cp + C A + ^ ( K A Q A n + Ht - H) Take d e r i v a t i v e s and so l v e ap =i^A+AQAN = 0 ; A = - ! ° A d X D KA 3K AQ A n From JI & 12; i°J2 = - A <3P = dCp. + X(-1) = 0 ; A= acp_ 2>H 2>H 2H ac„ 7iE n 8 10 12 In r e a l terms, assuming pipe s i z e s 1 and 2 i n the s e c t i o n , the expression r j becomes: A C p 2 1 - A c 2 A ^ g *~ ' A K o V = A K o ,V AH 2,1 ^2,r AC. *2f1' From 6 and 13_ K s t = 1.2 ; Q = 2,1 ^  AC 1i.2 Kst AK, 2,1/ This r e s u l t i s i d e n t i c a l to the one found from K e l l e r ' s a n a l y s i s 8 2.1.3 Evaluation of Cost Index Methods A discussion by T. A l Austin on P e r o l d 1 s method may well sum up the main c r i t i c i s m of a l l three Cost Index Methods; authored by R u s s e l l , Perold, and K e l l e r . While i t i s true that the procedures converge r a p i d l y f o r r e l a t i v e l y simple systems, i t i s questionable whether the procedures would converge as r a p i d l y f o r large multibranch systems or time-variable flow systems. For any but the simplest of problems the methods become inc r e a s i n g l y cumbersome and would require s i g n i f i c a n t s k i l l on the part of the user to wade through a l l the possible pipe change combinations i n a l o g i c a l manner. Another c r i t i c i s m brought forward i n Austin's discussion i s that i n Perold's procedure there i s no guarantee that a l o c a l optimum i s not obtained instead of the desired global optimum. 2.1.4 Economic Pipe Theory Although the Cost Index Methods are not s u i t a b l e as general pipe s i z i n g s o l u t i o n s , the index i t s e l f i s u s e f u l . The Cost Index i n the form (Cu^ - Cm2)/(Rm2 - Hn^) can be used to eliminate uneconomic pipe s i z e s , and i s u s eful i n understanding optimal pipe s i z i n g behavior. A pipe s i z e i s only "economic" i f f o r some value of head l o s s over a section, the pipe cost i s l e s s expensive by i n c l u d i n g the pipe s i z e than i t i s by excluding i t . In economic pipe s i z e s the Cost Index w i l l decrease with decreasing pipe s i z e (Fig.2.3 (a) and ( b ) ) . Economic Pipe S i z e s Table 2.1 Pipe Head Loss Cost Cost Index (m/lOOm) ($/l00m) 1 0.02 80. 1000. 2 0.04 60. 500. 3 0.08 40. 250. 4 0.16 20. Cost Index decreases w i t h decreasing pipe s i z e Cost vs. Head Loss Graph f o r Economic Pipe S i z e s o o Oi o o 80. 60. 40. N o t i c e t h a t the Cost Index i s simply the (-slope) of the graph. 20. 0.02 0J04 Uneconomic Pipe S i z e s Table 2.2 Pipe Head Loss Cost Cost Index (m/l00m) ($/l00m) 1 0.02 80. 1200. 2 0.04 56. 100. 3 0.08 52. 400V 4 0.16 20. I f pipe 3 i s e l i m i n a t e d (given by a broken l i n e ) the cost of pipe i n the head/loss range '(0.04-0.16 m/l00m) w i l l i n c r e a s e . Pipe 3 i s an economic pipe s i z e . F I G . 2 . 3 ( a ) Cost Index does not decrease w i t h decreasing pipe s i z e Cost vs-Head Loss Graph f o r Uneconomic Pine S i z e s I f pipe 3 i s e l i m i n a t e d (given by a broken l i n e ) the cost of pipe i n the head/Loss range (0.04—0.16 m/l00m) w i l l decrease. Pipe 3 i s an uneconomic pipe s i z e . 2.3 (b) 0.02 0.04 From Figures 2.3(a) and (b) i t is easy to see why pipe sizes are economical i f their Cost Index Values decrease with decreasing pipe size. From the preceding discussion and Figures 2.3(a) and (b), i t becomes evident that a series of economic pipe sizes will form a decreasing, concave up cost function, for any given pipe section. Decreasing. Concave Up Cost Function Head Loss(m) 11 2.2 Minimum Length Method and a Cost Index Example It i s intuitive that a small diameter pipe must be less expensive than a large diameter pipe* From this, i t was i n i t i a l l y thought that absolute  cost formed an index which controlled optimal design, and not the Cost Index proposed by Russell and Perold. This oversight led to the development of the Minimum Length Method; 2.2.1 Minimum Length Method Derivation Assuming that an Absolute Cost Index governs design, i t is possible to minimize cost by minimizing the length of each largest remaining pipe size, starting with the largest (ie. most expensive) pipe size and continuing with each progressively smaller pipe size. The following simple example demonstrates this method: Define System. Minimum Length Example (l) A B C Head = h QA FIG. 2.5 Head = 0 Flow Rate - Q Design procedure: 1. For a given reservoir head, draw in the Hydraulic Grade Line, for the largest pipe size using the reservoir elevation as the upstream starting point (Fig. 2.6 (a)). 2. Using the next largest pipe size, draw a Hydraulic Grade Line from each control point upstream towards the reservoir using the control point's minimum allowable head as the downstream starting point. These Hydraulic Grade Lines should intersect the Hydraulic Grade Line for the previous larger pipe size. The control point whose Hydraulic Grade Line has the furthest downstream intersection, governs design. The point of inter-section forms the downstream boundary for the previous larger pipe size (Fig. 2.6 (b)). 3; Repeat design procedure 2. until the entire system is defined (Fig. 2.6 (c)), 12 Design Procedure, Minimum Length Example (1) Step 1 FIG. 2.6 (a) Step 2 downstream i n t e r s e c t i o n coniirol FIG. 2.6 (b) Step 5 FIG. 2.6 (c) f o r l a r g e s t pipe; A l l o w a b l e Head t i | o l P o i n t Grade L i n e from p o i n t , f o r est pipe; 2. downstream i n t e r s e c t i o n Grade L i n e from c o n t r o l p o i n t , f o r pipe; 3. l a r g e s t 13 Optimal Solution, Minimum Length Example (1) Optimal Hydraulic Grade Line Diagram FIG. 2.7 Notice how this procedure produces the minimum length for each progressively largest remaining pipe size. 2.2.2 Negate Minimum Length Method Using a Cost Index Example The problem with this method can be illustrated using another example. The example also serves to show the value of the Cost Index in understanding optimal pipe sizing behavior. Define System. Minimum Length Example (2) Pipe Costs and Head Loss  Coefficient-? Minimum Length Example (2) Pipe Head Loss Coef. (m/100 m) 30 l/sec 20 l/sec 10 l/sec Res.Ht. 1 2 3 4 0.00082 0.0029 0.0074 0.023 Table 2.3 Cost (ft/100 m) 32.4 18.4 14.3 8.9 A 3 m 10 1/ 100 m B sec io U 100 m sec 10 \h sec X (Um 100 m FIGV 2.8 14 15 Using the Cost Index Values i t i s easy to see why the Minimum Length Solution i s not optimal and how to go about obtaining the optimal solution. Cost Index Values. Minimum Length Example (2) Pipe Head Loss Coef. (m/lOOm) Cost ($ / l00m) 1 0.00082 32.4 2 0.0029 18.4 3 0.0074 14.3 4 0.023 8.9 ACm Alta Cost Index A (0=30) B (Q=20) C (0=10) 6731 7.48 16.83 67.31 911 1.01 2.28 9.11 346 0.384 0.865 3.46 Cost Index -ACm AHm" = ACm AKmQ2 Table 2.4 The Cost Index must be constant for every section in the optimal pipe size solution (page 5 ). Compare Cost Index Values for the Minimum Length Solution to see i f they are constant. Cost Index Comparison for Minimum Length Solution. Minimum Length Example (2) Table 2.5 Pipe 1 Pipe 2 Pipe 3 Pipe 4 A (Q = 30) B(Q = 20) C(Q = 10) (.4) /67V3X 1.01 0.384 0.865 3.46 Q = flow rate (l/sec) 1. Section A is made up of pipe 1 and pipe 2; Cost Index = 7*48 2. Section B i s made up of pipe 2; 16.83 > Cost Index > 2.28 3. Section C is made up of pipe 2j 67.31 > Cost Index > 9.11 Notice that statement 1 and 3 are not compatable. The Minimum Length Solution does not provide the optimal-pipe sizing solution. If the solution is not optimal, consider a l l the various pipe size combinations and use Cost Index Values to pick the one with the highest cost saving. T ne only v a l i d combination available i s to replace pipe 2 with pipe 1 i n s e c t i o n A, and r e p l a c e pipe 2 w i t h pipe 3 i n s e c t i o n C. This replacement proceeds u n t i l one of the pipe s i z e s i s completely r e p l a c e d , then the Cost Index Values are compared again to see i f any f u r t h e r changes-are economic. 1_ Replace a l l of s e c t i o n C w i t h pipe 3» f i n d the change i n head and the change i n c o s t . 2 ^AKrn^ 2 x Q c 2 x L c = ( 0 . 0 0 7 4 - 0 . 0 0 2 9 ) / l 0 0 x 1 0 2 x 100 = 0 . 4 5 A C 5 ^ 2 = A C m 5 j 2 x. L c = (14.3 - 18 .4)/100 x 100 = (-4 .10) 2 F i n d the l e n g t h of replacement i n s e c t i o n A which has the equ i v a l e n t change i n head ( i e . 0 . 4 5 m ) and a l s o f i n d the change i n c o s t . 2 = 0.45m =AKm 2 1 x Q^2 x = (0.0029 - 0.00082)/ l00 x 3 0 2 x L A LA =f 0 .45 j = 24.1 m AC. m 1 . 2 10.01872/ ( 3 2 . 4 - 18 .4)/100 x 24.1 = 3.37 3, F i n d Net Cost Reduction, the New Cost, and the New Pipe S i z e S o l u t i o n . Net Cost Reduction = Ac^ 2 +AC 1 g = (-4.10) + 3.37 = $(-0.72) New Cost = Old Cost + Net Cost Reduction = 63.12 + (-0.72) = $62.4o New Pipe S i z e S o l u t i o n i s i d e n t i c a l to the one shown i n F i g u r e 2.10, page 14 . 4 Compare New S o l u t i o n w i t h the Cost Index. Cost Index Comparison f o r New S o l u t i o n . jMnimum Length Example (2) Table 2.6 Pipe 1 Pipe 2 Pipe 3 Pipe 4 A (Q = 30) B (Q, = 20) C (Q = 10) ( . 4 ) /TC85A 67.31 1.01 WW 0.384 0.865 WW Q = f l o w r a t e ( l / s e c ) 1. S e c t i o n A i s made up of pipe 1 and pipe 2; Cost Index = 7 .48 2. S e c t i o n B i s made up of pipe 2; 16.83 > Cost Index > 2.28 3 . S e c t i o n C i s made up of pipe 3> 9.11 > Cost Index > 3.46 A l l three r e s t r i c t i o n s are compatible. The new s o l u t i o n i s o p t i m a l . 17 2.2.3 E v a l u a t i o n of Minimum Length Method The Minimum Length Method does provide a c l o s e approximation of the optimal s o l u t i o n i n a number of cases. I t was f i r s t thought that t h i s method might be u s e f u l i n p r o v i d i n g a s t a r t i n g p o i n t f o r a Cost Index a n a l y s i s , s i m i l a r to the one o u t l i n e d i n S e c t i o n 2.2.2. U n f o r t u n a t e l y t h i s a n a l y s i s s u f f e r s from the same dilemma as Perold's Method; f o r l a r g e , complex systems i t simply becomes too cumbersome to be of any r e a l use. 2.3 Dynamic Programming Method Dynamic Programming i s a simple but powerful o p t i m i z i n g technique. I t - , i s b a s i c a l l y j u s t a systematic way of s e l e c t i n g an optimal "path", given a s e r i e s of a l t e r n a t i v e s . T his technique can be used to determine optimal pipe s i z e s i n simple i r r i g a t i o n systems, but i n more complex systems, problems a r i s e . 2.3.1 Dynamic Programming Method D e r i v a t i o n The example used to describe the Minimum Length Method i s employed again here to i l l u s t r a t e the Dynamic Programming Method. Define System.Dynamic Programming Example ( l ) 30 l / s e c 20 l / s e c , 10 l / s e c B 10 l / s e c 10 l / s e c 10 l / s e c FIG. 2.11: For Pipe Costs and Head Loss C o e f f i c i e n t s , see Table 2.3, page 13. With the use of t h i s data draw a Head Loss C o e f f i c i e n t v s . Cost Graph 2 LOSS and then u s i n g the equation H = KQ, convert the graph to a Head'vs. Cost Graph ( F i g . 2.12). 0.005 0.010 Head Loss C o e f f i c i e n t (m) 0.020 Cost vs Head l o s s Graph. Dynamic Programming Example (1) 1.5 2.0 S e c t i o n A, 30 l / s e c 1.'5 2.'0 S e c t i o n C, 10 l / s e c FIG. 2.12 MD Design procedure: 1. At a given pipe s e c t i o n (stage v a r i a b l e ) and upstream head ( s t a t e v a r i a b l e ) there are a number of p o s s i b l e a l t e r n a t i v e routes from the previous s e c t i o n ( h y d r a u l i c grade l i n e s ) . P r i n t the pipe cost along a l l h y d r a u l i c grade l i n e s . Given the head l o s s over a s e c t i o n , F i g u r e 2.12 i s used to determine pipe c o s t . 2. Work upstream from s e c t i o n C to s e c t i o n A. For each s e c t i o n , and upstream head v a l u e , determine the minimum cumulative pipe c o s t . There i s a cumulative pipe cost v a l u e a s s o c i a t e d w i t h each r o u t e . This v a l u e can be separated i n t o two p a r t s : 1. the cost of pipe over the present s e c t i o n which depends upon head l o s s 2. the minimum cumulative pipe cost from the previous s e c t i o n , upstream head v a l u e , to the downstream end p o i n t . 3. A f t e r completion, t r a c e back through the system to f i n d the optimal s o l u t i o n ( F i g . 2.13). 30.4+34.1=64.5 Design Procedure. Dynamic Programming Example (1) 26.7+37.6=64.3 F I G . 2.13 r o 22 2.3.2 Negate Dynamic Progranmiing Method A simple t i m e - v a r i a b l e f l o w example demonstrates why dynamic programming does not work f o r a l l systems. Time-variable f l o w simply means that f l o w i n the i r r i g a t i o n system changes w i t h time; there i s more than one time i n t e r v a l and f l o w i s not constant f o r a l l time i n t e r v a l s . Add another time i n t e r v a l to the previous example and examine the new system u s i n g dynamic programming. Define System. Dynamic Programming Example (2) Time I n t e r v a l Head = 3-=;-m FIG. 2.14 /^Flow Rate ( l / s e c ) I 20 I 20 I 20 , I I 10 11.20 I I 10 Head = A B C I o i 1 0 I 20 I I 10 I I 10 I I 10 100m 100m 100m For Pipe Costs and Head Loss C o e f f i c i e n t s , see Table 2 .3 , page 13. Optimal S o l u t i o n . Dynamic Programming Example (2) Head = 3 m FIG. 2.15 . 0 1 2 57.7 m 242.3 m The optimal s o l u t i o n i s given i n F i g u r e 2.15 but u n f o r t u n a t e l y t h i s example and others l i k e i t are not e a s i l y handled by the Dynamic Programming Technique. Problems a r i s e because now the r e s u l t a n t pipe system must be compatible w i t h both time i n t e r v a l s . I t i s not p o s s i b l e to work upstream from s e c t i o n C to s e c t i o n A because the downstream end p o i n t head c o n d i t i o n f o r both time i n t e r v a l s i s not completely d e f i n e d . 1 . The downstream head f o r the c r i t i c a l time i n t e r v a l = G.O, but the c r i t i c a l time i n t e r v a l i s unknown. 2. The downstream head f o r the n o n - c r i t i c a l time i n t e r v a l >0.0'j-but the exact v a l u e i s unknown. This problem i s overcome by s t a r t i n g a t the upstream r e s e r v o i r and work-i n g downstream. The upstream end p o i n t ( i e . r e s e r v o i r ht.) isknown f o r both time i n t e r v a l s and does not cause any d i f f i c u l t i e s . For t h i s example, pipe s e c t i o n i s used as the stage v a r i a b l e , and the downstream head v a l u e f o r time i n t e r v a l I I i s used as the s t a t e v a r i a b l e . U n f o r t u n a t e l y t h i s arrangement ignores the e f f e c t of time i n t e r v a l I and because of t h i s omission the example turns out to be not s t r i c t l y independent. Independence i s the fundamental c h a r a c t e r i s t i c which makes dynamic programming so powerful. Simply s t a t e d independence means that d e c i s i o n s f o r the present stage only depend upon: 1. c o s t over the present stage 2. cumulative cost from the previous stage and does not depend upon how these cumulative cost values were der i v e d . In the stage, s t a t e arrangement described above, the optimal r e s u l t f o r a given stage, s t a t e may not n e c e s s a r i l y be the one of minimum c o s t , depending on the downstream boundary c o n d i t i o n s i n subsequent stages. Consider the f o l l o w i n g two routes- - look s p e c i f i c a l l y a t one stage, s t a t e d e c i s i o n : s e c t i o n B, downstream head = 0,5 m. Independence Constraint, Dynamic JProgrananing Example (2) 24 Route 1 0 . 0 FIG 2.16 Route 2 0 . 0 FIG 2.16 B 3-r-a.nl i r. Gr-arl 3 Line—HGL 3 0 . 4 17 .6+30.4=48.0 2 0 ^ 2 ^ ? ' 48.0+20.2=68.2 A B C 3 . 0 2 . 0 a +> -& "LO - H 0 . 0 sta, ?e. section B 25,2 \ \ sta &e-^downstre am head=0.5m \ I 25.2 ^ 2 4 ^ ^ / \ 24 .5+25.2=49.7 49.7^+17.8=67.5 -3.0 2.0 +> 4.0 ** £.0 Comparing these two alternative routes, i f the dynamic programming technique i s followed,the minimum cost alternative (ie. Route 1 ) , for section B, downstream head = 0,5 m, w i l l be retained, while Route 2 i s labelled as an uneconomic choice and discarded. In the end result i t turns out that Route 2 has the overall minimum cost and is the optimal choice of the two alternatives. The dynamic programming technique does not work because the effect of time interval I has essentially been ignored in the stage, state derivation. The only way to include time interval I, i s to add another state variable to the program. This approach involves multi-dimensional dynamic programming. In a new derivation using multi-dimensional dynamic programming, the pipe section stays as the stage variable, but there are two state variables, one to define head loss in both time intervals and another to define the 25 downstream head of one time i n t e r v a l i n r e l a t i o n to the other. 2.3.3 E v a l u a t i o n of Dynamic Programming Method This type of s o l u t i o n i s not s u i t a b l e because each a d d i t i o n a l time i n t e r v a l i ntroduced i n t o the example r e q u i r e s another dimension i n the dynamic program f o r m u l a t i o n . The Dynamic Programming Method works w e l l f o r constant f l o w problems but breaks down when t i m e - v a r i a b l e f l o w problems are attempted. COST 2.3.4 Decreasing. Concave Upv Function For the optimal pipe s i z i n g problem i t i s p o s s i b l e to e x p l o i t the w e l l d efined r e l a t i o n s h i p between pipe cost and head, to g r e a t l y reduce the computational l o a d i n the dynamic program. In a dynamic program, to o b t a i n cumulative cost f o r the c u r r e n t stage, s t a t e , two separate costs must be added; 1. the cost over the present stage. 2. the minimum cumulative c o s t , from the previous stage, s t a t e , to the end p o i n t . For the optimal pipe s i z i n g problem i t turns out t h a t both these costs are a f u n c t i o n of head, and t h a t both r e s u l t i n g cost f u n c t i o n s are decreasing and concave up. 1. a f u n c t i o n ( f ) i s decreasing on an i n t e r v a l I , i f f* < 0 on I 2. a f u n c t i o n i s concave up on I , i f f " > 0 on I For Decreasing, Concave Up : Cost F u n c t i o n Diagram see F i g . 2.4, page 10. From the d i s c u s s i o n of economic pipe and from F i g . 2.12 i t i s obvious t h a t the cost over a stage always forms a decreasing, concave up cost f u n c t i o n . I t i s not so obvious t h a t the minimum cumulative cost f o r the previous stage a l s o forms t h i s type of cost f u n c t i o n . Assuming f o r the moment that both costs do form cost f u n c t i o n s t h a t are decreasing, concave up, i t i s very simple to determine the minimum cumulative cost f o r the present stage f o r any s t a t e v a l u e . Consider the f o l l o w i n g decreasing, concave up cost f u n c t i o n s : 26 C1 = h 1 20 ^ + 100 C 2 = 5 h 2 " 3 6 h 2 + 1 0 8 ^ - — 1 - ^ < 0; when < 10) C 1 = 2h. 20 C ' = 6h - 36 < 0; when (h„ < 6) cy = 2 > 0 C 2" = 6 > 0 Two S p e c i f i c Decreasing. Concave Up Cost Functions 100 80 C 1 60 40 20 Cj v s . h 1 h1 C1 0 100 1 81 2 64 3 49 4 36 -J 1—r FIG. 2.17 (a) 100 80 c 2 60 -f 40 + 20 C 2 v s . h 2 h 2 °2) 0 108 1 75 2 48 3 27 4 12 1 t—r FIG. 2.17 (b) F i n d the minimum summation of C^  and C 2 ( i e . minimize C^ = C 1 + C 2) f o r a given h^, such that h^ = h^ + h 2» P l o t values of C, v s . fa0 given h, = 4, and choose the minimum v a l u e . 3 2 3 Cumulative Cost F u n c t i o n . C^ v s . hg, when h^ = h^ + h 2 = 4 160 140 C^ 120 100 80 FIG. 2.18 C, . = 108 3mm 0 1 The minimum cost ( i e . C j m i n = 1 0 8 ) o c c u r s a t n 2 = 3 ( i e ' ^ = 1) when = 0. The slopes of the two cost f u n c t i o n s a t t h i s p o i n t are equal. ah c ^  = 2 (1) - 20 = C 2» =6 (3) - 36 = (-18) 3 C ^ C In general the minimum cost of CL occurs when \ = \ = 0 or when 1 = 2 , given h^ = h ? + h. . ah 1 ah 2 5 ^ ' The above i s only p o s s i b l e when the two input costs form cost f u n c t i o n s that are decreasing, concave up. The above a n a l y s i s i s f o r a s i n g l e head value h... What i s r e a l l y 3 d e s i r e d i s a general cost f u n c t i o n which gives the minimum cost (CU . ) f o r any gi v e n value of head ( h ^ ) . In general = C, + C 2 = (h., 2 - 20 h 1 + 100) + ( 3h 2 2 - 36h2 + 108) 1_ Subs h 3 = h1 + h 2 h1 = h 3 " n 2 1 t i t u t e 2 i n t o 1.; C ? = ( h ? - h 2 ) 2 - 20( h ? - h g) +100 + 3 h 2 2 - 36h2 +108—^ To minimize C ; 3_ = 2(h - h ) ( - l ) - 20 (-1) + 6h„ - 36 = 0 ? 3h 5 d 1 = -2h, + 2h_ + 20 + 6h 0 - 36 = 0 5 <L 2 — h = ^ + 2 4 4 S u b s t i t u t e _4_ i n t o _3_; C 3min = ( h 3 " ( ^ +2>^ " 2 0K ~ & +2)) + 1 0 0 + 3 & +2)2 - 3 6 & +2) + 108 4 0 4 4 4 = 1 h 2 - 24 h + 192 4 0 5 Check : f i n d C T . when h v = 4 ; 3mm 3 C 3min(h 5= 4) = f ( 4 ) 2 " 2 4 ( 4 ) + 1 9 2 = 1 0 8 28 The above cost f u n c t i o n can be used to f i n d the minimum cost f o r any-s t a t e v a l u e ( h ^ ) , i n a given stage. In a conventional dynamic program a l a r g e number of routes must be t e s t e d to o b t a i n t h i s minimum cos t . Even more important, i s the f a c t that t h i s cost f u n c t i o n , which i s a l s o the input minimum cumulative cost f u n c t i o n f o r the next stage, i s always decreasing, concave up. C , min = % h, - 2 4 < 0 when (h < 1 6 ) C", min = ^ > 0 J ' • 2 ^ ^ 2 This means t h a t f o r the next stage and a l l subsequent stages, an i d e n t i c a l procedure can be f o l l o w e d . This concept has powerful connotations that are not j u s t r e s t r i c t e d to dynamic programming. On page 10 the concept was des c r i b e d f o r a s i n g l e pipe s e c t i o n . Here i t has been expanded to i n c l u d e a number of pipe s e c t i o n s i n s e r i e s . Although i t i s d i f f i c u l t to prove, i t seems th a t t h i s concept can be expanded s t i l l f u r t h e r . In g e n e r a l , given any pipe network and any  set of op e r a t i n g c o n d i t i o n s , the pipe cost and i n l e t head w i l l always form a  decreasing, concave up cost f u n c t i o n . 2 9 2 . 4 L i n e a r Programming Method L i n e a r Programming i s one of the most powerful o p t i m i z i n g techniques. I t u s u a l l y requires, the use of a computer because the method of s o l u t i o n i s f a i r l y complex and the computational l o a d i s otherwise p r o h i b i t i v e . I t s only other disadvantage i s that i t can only be used i f the r e l a t i o n s h i p between v a r i a b l e s i s l i n e a r . 2 . 4 . 1 L i n e a r Programming Formulation and Background Theory A general f o r m u l a t i o n of a l i n e a r programming problem, w i t h (n) v a r i a b l e s , i s o u t l i n e d below: Minimize or Maximize C.X.. + C 0X„ . . . . + C X Z (optimum value) 1 1 2 2 n n subj e c t to the f o l l o w i n g (m) CONSTRAINTS: A 1 1 X 1 t A 1 2 X 2 • * • • ' + A 1 n X S ~ b 1 A 2 1 X 1 + A 2 2 X 2 - - - • + A 2 A A m 1 X 1 t A m 2 X 2 ' where ~ represents <, >, or =. + A X — b mn n m and the NON-NEGATIVITY CONDITION X j ^ O f o r a l l values of ( j ) where: Xj are DECISION VARIABLES Cj are OBJECTIVE FUNCTION COEFFICIENTS A i j are ACTIVITY COEFFICIENTS b i are RIGHT-HAND SIDE COEFFICENTS L i n e a r programming algorithms, such as the Simplex Algorithm, s o l v e problems i n much the same manner as presented by P e r o l d and R u s s e l l , but more e f f i c i e n t l y . A t each i t e r a t i o n i n the s o l u t i o n , the a l g o r i t h m evaluates the e f f e c t of a u n i t i n c r e a s e i n each d e c i s i o n v a r i a b l e on the o b j e c t i v e f u n c t i o n . The v a r i a b l e that produces the s m a l l e s t increase i n the o b j e c t i v e f u n c t i o n i s added to the s o l u t i o n . The v a r i a b l e i s i n t r o d u c e d i n t o the s o l u t i o n by the maximum amount, without v i o l a t i n g any of the c o n s t r a i n t s . The m a t r i x i s then rearranged to accommodate the i n c l u s i o n of the new v a r i a b l e . The procedure i s repeated u n t i l no f u r t h e r improvement i n the o b j e c t i v e f u n c t i o n s o l u t i o n i s p o s s i b l e . The L i n e a r Programming Technique can be used to s o l v e optimal pipe s i z i n g problems i n an open-ended ( t r e e - l i k e ) g r a v i t y system, i f the f l o w r a t e f o r each s e c t i o n i s known. Given a number of p r e - s e l e c t e d pipe s i z e s f o r each s e c t i o n , the lengths of these pipe s i z e s become the d e c i s i o n v a r i a b l e s . Both head l o s s and pipe cost are l i n e a r l y p r o p o r t i o n a l to pipe l e n g t h , and are thus compatible w i t h the l i n e a r i t y r e s t r i c t i o n . 2.4.2 Constant Flow Example The f o l l o w i n g example i s i d e n t i c a l to the one used to d e s c r i b e the Minimum Length and Dynamic Programming Methods. The p e r m i s s i b l e pipe s i z e s i n s e c t i o n A are 1 and 2, i n s e c t i o n B, 2 and 3 and i n s e c t i o n C, 2 and 3« Thus there are a t o t a l of s i x d e c i s i o n v a r i a b l e s - , X^, X^, X^, X,., and which represent the l e n g t h of pipe f o r the d i f f e r e n t pipe s i z e s . 31 Formulation and Optimal S o l u t i o n . L i n e a r PrograiMiing Constant Flow Example 30 l / s e c 20 l / s e c 10 l / s e c Head=3m A 10 l/t 100 i B ec 10 1/ 100 m C sec 10 1, 100 m Head=0.0m Pipe S i z e 1 2 2 3 2 3 T o t a l Unknown Length (m) X1 X2 X 3 X4 X5 X6 Optimal Length (m) 80.7 19.3 100 0 0 100 Head Loss (m) 0.596 0.504 1.160 0 0 0.74 3.00 Optimal Sc j l u t i o n 0 1 3 I 80 .7 m 119.3 m 100 m FIG. 2.19. Pipe Costs and Head Losses. L i n e a r Programming Constant Flow Example Pipe Head Loss Coef. (m/l00m) Km Cost (1/100m) Head Loss (m/l00m) Hm 30 l / s e c 20 l / s e c 10 l / s e c 1 0.00082 32.4 0.738 8:328 0.082 2 0.0029 18 .4 2.61 1.16 0.29 3 0.0074 14.3 6.66 2.96 0.74 Hm=KmQ Table 2.7 Formulation Tableau. L i n e a r Programming Constant Flow Example V a r i a b l e s X1 X2 X3 X4 X5 X6 Objective F u n c t i o n 32.4 18 .4 18 .4 14 .3 18 .4 14 .3 — Min Head Loss C o n s t r a i n t 0.738 2.61 1.16 2.96 0.29 0.74 < 3.0 Length C o n s t r a i n t s 1.0 1.0 = 1.0 1.0 1.0 _ 1.0 1.0 1.0 1.0 Table 2.8 32 2 . 4 . 3 Time-variable Flow Example The t i m e - v a r i a b l e f l o w example which caused so much d i f f i c u l t y i n the Dynamic Programming Method causes very l i t t l e d i f f i c u l t y i n the L i n e a r Programming Case. Using L i n e a r Programming and the same s i x d e c i s i o n v a r i a b l e s , examine the f o l l o w i n g t i m e - v a r i a b l e f l o w example. Formulation and Optimal S o l u t i o n , L i n e a r Programming Time-variable Flow Example I 20 I 20 I 20--Time I n t e r v a l -Flow Rate ( l / s e c ) Head~3m I I 30 I I 20 I I 10 „ . _ A 100m a I 0 I I 10 100m u I 0 LII 10 y 100m I 20 I I 10 T o t a l Pipe S i z e 1 2 2 3 2 3 Unknown Length (m) X1 X2 X3 X 4 X 5 X6 Optimal Length (m) 5 7 . 7 42.3 100. 0 . 100. 0 . Head Loss I 0.189 0.491 1.16 0 . 1.16 0 . 3.00 Head Loss I I 0.426 1.104 1.16 0 . 0 .29 0 . 2.98 Optimal S o l u t i o n 1 2 FIG. 2 .2 157.Ym dip, 3m Formulation Tableau, L i n e a r Progrannning Time-variable Flow Example "Variables X1 X2 X3 X4 X5 X6 O b j e c t i v e Function 32.4 18 .4 18 .4 14.3 18 .4 14.3 = Min. Head Loss C o n s t r a i n t s 0.738 2.61 1.16 2.96 0.29 0.74 < 3.0 0.328 1.16 1.16 2.96 1.16 2.-96 < 3.0 Length C o n s t r a i n t s 1.0 1.0 = 1.0 1.0 1.0 = 1.0 1.0 1.0 = 1.0 Table 2.9 Notice t h a t the only d i f f e r e n c e between the constant f l o w and the time-v a r i a b l e f l o w f o r m u l a t i o n i s the a d d i t i o n of a s i n g l e e x t r a head l o s s c o n s t r a i n t which defines head l o s s e s i n the new time i n t e r v a l . 2 , 4 . 4 Pumping Example U n t i l now the d i s c u s s i o n has only i n c l u d e d g r a v i t y systems. In g e n e r a l , pumping systems can not be handled e x c l u s i v e l y i n a l i n e a r program because pumping costs aren't u s u a l l y l i n e a r l y p r o p o r t i o n a l to i n l e t head. I f pumping costs were l i n e a r l y p r o p o r t i o n a l to i n l e t head the i n l e t head co u l d be i n c l u d e d as an e x t r a d e c i s i o n v a r i a b l e and the l i n e a r program co u l d be used to f i n d the optimum i n l e t head, and the minimum t o t a l system cost i n c l u d i n g both the pipe and pumping c o s t . Take the t i m e - v a r i a b l e f l o w example and i n s e r t a pump i n s t e a d of the constant r e s e r v o i r h e i g h t . Assume that the annual pumping costs can be descr i b e d simply as $ l , 0 / ( l / s e c ) / ( m ) . With a known pumping flow r a t e of 30,0 l / s e c , and an i n t a k e water l e v e l of 0,0 m, t h i s pumping cost becomes simply $ 3 0 » 0 / ( m of i n l e t head). The i n l e t head i s i n c l u d e d i n the l i n e a r programming f o r m u l a t i o n by adding i t as a seventh d e c i s i o n v a r i a b l e , X^. Formulation Tableau, L i n e a r Programming Pumping Example V a r i a b l e s X1 X2 X3 X4 X5 X6 X7 Objective Function 32 .4 18 . 4 18 . 4 1 4 . 3 18 . 4 1 4 . 3 30.0 Min. Head Loss C o n s t r a i n t s 0.738 2 .61 1 .16 2 . 9 6 0 . 2 9 0 . 7 4 - 1 . 0 0.0 0.328 1.16 1 .16 2 . 9 6 1.16 2 . 9 6 - 1 . 0 < 0.0 Length C o n s t r a i n t s 1 .0 1 .0 = 1 .0 1 .0 1 .0 = 1 .0 1 .0 1 .0 = 1 .0 Table 2.10 I f pumping costs are not l i n e a r l y p r o p o r t i o n a l to i n l e t head, which i s g e n e r a l l y the case, then the above approach cannot be used. The pumping system-, can s t i l l be so l v e d , but pumping costs cannot simply be i n c l u d e d i n the l i n e a r program. The system i s t r e a t e d as a g r a v i t y system, but the a n a l y s i s i s repeated f o r a number of i n l e t head v a l u e s . This g i v e s the optimal pipe cost f o r d i f f e r e n t values of i n l e t head. For each given i n l e t head v a l u e , the pump op e r a t i n g and f i x e d costs are c a l c u l a t e d and added to the optimal pipe cost to f i n d the t o t a l cost of the system. The t o t a l cost f o r the v a r i o u s head values are compared to a r r i v e a t the minimum t o t a l cost and optimum pump discharge head. 2.4.5 E v a l u a t i o n of L i n e a r Programming Method The L i n e a r Programming Technique provides an e f f i c i e n t method f o r s o l v i n g the optimal pipe s i z i n g problem f o r both g r a v i t y and pump systems. Complex problems c o n t a i n more d e c i s i o n v a r i a b l e s and more head l o s s and l e n g t h c o n s t r a i n t s , but the b a s i c f o r m u l a t i o n i s i d e n t i c a l to the one o u t l i n e d i n the examples. Branched systems, t i m e - v a r i a b l e f l o w and v a r i a b l e - s p e e d power u n i t s create no major d i f f i c u l t i e s . The only remaining problem i s to e f f i c i e n t l y organize the system's input data, i n order to minimize the computational l o a d i n the l i n e a r prograjnming a l g o r i t h m . 35 Chapter 3 PIPE SIZING OPTIMIZATION MODEL 3.1 Define Model Input Parameters A computer model has been developed f o r s o l v i n g optimal pipe s i z i n g problems u t i l i z i n g the L i n e a r Programming Technique. This s e c t i o n defines a l l i nput parameters i n the model, and a l s o e x p l a i n s t h e i r l i m i t a t i o n s and use. 3.1.1 General Input Data NSECT - number of conveyance s e c t i o n s . The primary f u n c t i o n of conveyance s e c t i o n s i s simply to convey water, although there may be o u t l e t s (conveyance feeders) a t some of the nodes. A conveyance s e c t i o n and the node immediately downstream, are both d e f i n e d by the same value ( I ) , ( 1 = 1 , ... NSECT). IPIPE - number of pipe s i z e s . The program checks f o r uneconomic pipe s i z e s . NF - number of fe e d e r s , e x c l u d i n g conveyance feeders. The e x c l u s i v e f u n c t i o n of a feeder i s to d i s t r i b u t e water to l a t e r a l s through one or more o u t l e t s . Feeders are l o c a t e d a t the downstream end of every branch i n the system. A feeder w i t h more than one o u t l e t w i l l always form a s e r i e s ( i e . a s i n g l e l i n e of o u t l e t s , w i t h no branches). NFT - number of feeders, i n c l u d i n g conveyance feeders. A conveyance feeder i s an o u t l e t l o c a t e d a t a conveyance node. ITLT - t o t a l number of time i n t e r v a l s . I f ITLT = 0, i t s v a l u e i s c a l c u l a t e d w i t h i n the model. There i s a d i f f e r e n t time i n t e r v a l f o r every separate f l o w c o n d i t i o n i n the system. T y p i c a l l y , the farmer changes an i r r i g a t i o n system once or twice a day and there i s a d i s t i n c t time i n t e r v a l f o r each change. The program sets an a r b i t r a r y maximum l i m i t of 50 time i n t e r v a l s . This may not completely define every p o s s i b l e case but should be s u f f i c i e n t to i n c l u d e a l l cases t h a t may govern design. EXP - head l o s s e x p o n e n t i a l ; HL = HK * Q ** EXP. HL - head l o s s (m/100 m). HK - head l o s s c o e f f i c i e n t (the u n i t s f o r t h i s parameter are a c t u a l l y (rn/lOO m)/ ( l / s e c ) but are shortened to (m/100 m) f o r convenience). QfC- - flow r a t e ( l / s e c ) . The next three parameters r e f e r s p e c i f i c a l l y to a pumping system where the a n a l y s i s i s repeated a number of times, f o r v a r i o u s i n l e t head v a l u e s . They s p e c i f y the range and i t e r a t i o n i n t e r v a l f o r the i n l e t head v a l u e s . EMIN - minimum d e s i r e d i n l e t head (m). I f EMIN = 0.0, the minimum a l l o w a b l e i n l e t head i s c a l c u l a t e d . EMAX - maximum d e s i r e d i n l e t head (m). TINT - change i n head ( m / i t e r a t i o n ) . The a n a l y s i s i s repeated a t o t a l of ( - ^ ^ ^ ^ ^ + l ) times. Head values vary between EMIN and EMAX, and head i s i n c r e a s e d by TINT before each subsequent a n a l y s i s . I f a g r a v i t y system i s to be analyzed, simply set EMIN and EMAX both equal to the r e s e r v o i r h e i g h t . I t i s obvious from the above equation that TINT must be gr e a t e r than 0.0. The next three parameters h e l p to organize the s t r u c t u r e of the conveyance system. They d e f i n e the routes from a l l feeders and conveyance f e e d e r s , to the upstream source (pump or r e s e r v o i r ) . ISTAGE - number of stages = (NSECT - NP + l ) . Stages d i v i d e the conveyance system i n t o a number of d i f f e r e n t p a r t s . Each conveyance node defines a stage. A conveyance node i s a node w i t h one or more conveyance s e c t i o n s immediately downstream. 37 Stages must be numbered from the downstream end p o i n t s to the upstream source. I f a stage i s downstream from another, i t must be numbered f i r s t . Conveyance feeders t h a t form a s e r i e s must have stages t h a t f o l l o w one another. MXTJJ - maximum number of upstream and downstream conveyance s e c t i o n s at a stage (conveyance node). This i s e s s e n t i a l l y equal to the maximum number of downstream conveyance s e c t i o n s plus one, s i n c e there i s always only one upstream conveyance s e c t i o n . I N ( l , j ) - the conveyance s e c t i o n matrix. At each stage ( j ) , I N ( l , j ) defines the upstream s e c t i o n and a l l the downstream s e c t i o n s . IN(1,J) defines the upstream s e c t i o n , and I N ( l , J ) , I = 2,...MXIN, defines a l l the downstream s e c t i o n s . I f the number of downstream s e c t i o n s a t a given stage i s l e s s than MXLN'j'1 the remaining spaces i n the a r r a y are f i l l e d w i t h zeros. In the f i n a l stage there i s no upstream s e c t i o n . The source i s immediately upstream of the conveyance node, and IK(l,ISTAGE) i s set equal to zero. HK(l) - head l o s s c o e f f i c i e n t (m/100 m), f o r pipe s i z e ( i ) . COST ( i ) - cost ($/l00 m), f o r pipe s i z e ( i ) . These costs are annual cost values and depend upon l i f e expectancy and i n t e r e s t r a t e as w e l l as i n i t i a l c o s t . 3 8 T y p i c a l System Layout CONVEYANCE FEEDER OUTLETS FEEDER OUTLETS Feeder Number (5) ( f o u r t h feeder) FIG. 3.1 CONVEYANCE SECTIONS (numbers shown a t the downstream node) NSECT = 8 MXIN = 3 (Maximum number of downstream conveyance s e c t i o n s equals 2 ) NF = 4 NFT = 4 + 2 conveyance feeders ='6 ISTAGE = NSECT - NF+1 = 8 - 4 + 1 = 5 Conveyance S e c t i o n M a t r i x  Table 3.1 STAGE 1 2 3 4 5 u/S D/S U/S NODE 6 4 7 upstream downstream D/S NODE 1 5 6 7 8 D/S NODE 2 3 0 4 0 3 9 3.1.2 Pump Input Data APC - annual pump operating cost ($/(l/sec)/(m)), I f APC = 0.0, the input data defines a g r a v i t y system, or a pumping system where computation of pumping costs are ignored by the program. IVHP - i n d i c a t e s the type of pumping system to be used. I f IVHP = 0 , there i s a constant-speed power u n i t . I f IVHP £ 0, there i s a va r i a b l e - s p e e d power u n i t . P0R(T_T) - f r a c t i o n of annual pump use, during time i n t e r v a l ( I T ) . Each time i n t e r v a l need not n e c e s s a r i l y be the same l e n g t h of time. This parameter i s i n t r o d u c e d to g i v e the user the op t i o n of v a r i a b l e time i n t e r v a l d u r a t i o n . I f ITLT ( t o t a l number of time i n t e r v a l s ) = 0, i n d i c a t i n g i t s v a l u e i s to be c a l c u l a t e d i n the model, POR(lT) values need not be defined, they are a u t o m a t i c a l l y set equal to (1/lTLT ( c a l c u l a t e d ) ) . I f ITLT £ 0, POR(lT) values must be s p e c i f i e d . DAT - i n t a k e water l e v e l (m). The model assumes th a t there are no head l o s s e s between the i n t a k e and the pump. The head imparted by the pump i s then-simply the d i f f e r e n c e between the known i n l e t head and the i n t a k e water l e v e l . INCPC - number of p o i n t s to de f i n e f i x e d pump cost f u n c t i o n . The shape of the f i x e d pump cost f u n c t i o n i s l e f t e n t i r e l y up to the user. The complexity of the curve and the accuracy d e s i r e d w i l l d e f i n e the number of po i n t s used to describe the f u n c t i o n . Between p o i n t s , the f u n c t i o n i s assumed to be l i n e a r . I f INCPC = 0 , the f i x e d pump cost i s set equal to zero, and values f o r CPC(l) and HE(l) need not be defin e d . CPC(l) - f i x e d pump cost {%) at head( H E ( l ) ) , I = 1,INCPC. The cost i s an annual cost v a l u e , and depends upon l i f e expectancy and i n t e r e s t r a t e as w e l l as i n i t i a l c o s t . HE(l) - represents the pump discharge head (m), whose f i x e d pump cost i s C P C ( l ) . 40 3,1.3 Feeder (N) and Conveyance Feeder (N) Input Data NODE(N) - node number a s s o c i a t e d w i t h a feeder or conveyance feeder. In a feeder, HODE(N) r e f e r s to the f u r t h e s t upstream feeder o u t l e t . In a conveyance feeder, NODE(N) r e f e r s to the conveyance feeder o u t l e t . As mentioned befo r e , a conveyance s e c t i o n and the node immediately downstream, both have the same v a l u e . Thus i n a d d i t i o n to d e f i n i n g the appr o p r i a t e o u t l e t , NODE(N) a l s o defines the conveyance s e c t i o n immediately upstream. The next f o u r parameters d e f i n e a l l p o s s i b l e f l o w patterns o r i g i n a t i n g from the feeder. A l l d i s c h a r g i n g o u t l e t s and t h e i r discharge values are defi n e d f o r each feeder time i n t e r v a l . KC(N.) - number of i r r i g a t i o n c y c l e s . IC(N) - number of s e t t i n g s per i r r i g a t i o n c y c l e ( i e . number of feeder time i n t e r v a l s ) . In a conventional i r r i g a t i o n system there are a number of l a t e r a l s f o r each feeder. Each l a t e r a l has a s p e c i f i e d r o u t e through the f i e l d , d i s c h a r g i n g from a p a r t i c u l a r o u t l e t during each feeder time i n t e r v a l . This route defines the i r r i g a t i o n c y c l e f o r the l a t e r a l . When a l l feeder time i n t e r v a l s are completed the i r r i g a t i o n c y c l e i s repeated. The number of i r r i g a t i o n c y c l e s i s equal to the number of l a t e r a l l i n e s o p e r a t i n g a t any one time ( i e . the number of o u t l e t s d i s c h a r g i n g s i m u l t a n e o u s l y ) . The number of s e t t i n g s per i r r i g a t i o n c y c l e i s equal to the number of separate d i s c h a r g i n g o u t l e t c o n d i t i o n s along a l a t e r a l ' s e n t i r e r o u t e ? ( i e . the number of time i n t e r v a l s i n the f e e d e r ) . JH(j,I,N) - d i s c h a r g i n g o u t l e t during c y c l e ( J ) , a t s e t t i n g ( i ) . O u t l e t s are numbered from the downstream end of the feeder to the upstream end. S i m i l a r to conveyance s e c t i o n s , the o u t l e t number a l s o defines the feeder s e c t i o n immediately upstream U of the o u t l e t . Q(j,I,N) - discharge ( l / s e c ) from o u t l e t IE(J,I,N). 41 I f a l a t e r a l i s not discharging- during a given feeder time i n t e r v a l , the values of TJR(j,I,N) and Q(j,I,N) must he zero. The columns of the IR(j,I,N) m a t r i x , J = 1,...NC(N), d e f i n e a l l the d i s c h a r g i n g o u t l e t s d u r ing a s p e c i f i c feeder time i n t e r v a l ( i ) . The rows of the IR(j,I,N) m a t r i x , I = 1,...IC(N), d e f i n e a l l the d i s c h a r g i n g o u t l e t s f o r a s p e c i f i c i r r i g a t i o n c y c l e ( j ) , IR(J,I,N) i s subsequently rearranged so;;that a l l d i s c h a r g i n g o u t l e t s f o r a given feeder time i n t e r v a l are i n i n c r e a s i n g order, Q(j,I,N) i s then changed from i n d i v i d u a l o u t l e t discharge .to t o t a l discharge immediately upstream of the o u t l e t , SM(N) - feeder s e c t i o n l e n g t h (100 m). SD(l,N) - l e n g t h (100 m) of each feeder s e c t i o n ( i ) . I f SM(N) = 0.0, feeder s e c t i o n l e n g t h i s not a constant value throughout the feeder, and i n d i v i d u a l SD(l,N) v a l u e s , f o r each feeder s e c t i o n ( i ) , must he de f i n e d i n the input data. I f SM(N) ^ 0.0, the feeder s e c t i o n l e n g t h i s a constant v a l u e and in p u t data f o r SD(l,N) i s not r e q u i r e d . For every feeder s e c t i o n ( i ) , the program sets SD(l fN) equal to SM(N). SM(N) i s subsequently changed to equal t o t a l feeder l e n g t h , QM(N) - o u t l e t discharge ( l / s e c ) Q,(J,I,N) - discharge ( l / s e c ) from o u t l e t UR(j,I,N). I f QM(N) = 0.0, o u t l e t discharge from every o u t l e t i s not a constant value throughout the feeder, and i n d i v i d u a l Q(j,I,N) v a l u e s , f o r each i r r i g a t i o n c y c l e ( j ) and s e t t i n g ( i ) , must be de f i n e d i n the in p u t data. I f QM(N) ^ 0.0, the o u t l e t discharge i s a constant v a l u e and input data f o r Q(J,I,N) i s not r e q u i r e d . For every i r r i g a t i o n c y c l e ( j ) and s e t t i n g ( i ) , the program sets Q(J,I,N) equal to QM(N). ICP(N) - number of c o n t r o l p o i n t s ( i e . o u t l e t s i n the f e e d e r ) . I f ICP(N) = 1, there are no feeder s e c t i o n s . NC(N) w i l l n e c e s s a r i l y equal one. SM(N) becomes a dummy parameter and can be s e t to any v a l u e . Both SD(l,w) and XJEt(j,I,N) can be e l i m i n a t e d from the input data. The program set s IR(1,I,N) equal to one or zero depending upon whether or not the o u t l e t i s d i s c h a r g i n g f o r a p a r t i c u l a r feeder time i n t e r v a l ( i ) . CP(l,N) - minimum a l l o w a b l e h y d r a u l i c e l e v a t i o n (m) f o r c o n t r o l p o i n t ( i ) . A l l c o n t r o l p o i n t s ( o u t l e t s ) have minimum a l l o w a b l e head values equal to t h e i r ground e l e v a t i o n plus a s p e c i f i c r e q u i r e d pressure head (page 4 9 ) . The program assumes t h a t a l l CP(l,N) values must be g r e a t e r than or equal to zero. The input data f o r conveyance feeders d i f f e r s s l i g h t l y from t h a t f o r feeders, A conveyance feeder i s a s i n g l e o u t l e t . The number of i r r i g a t i o n c y c l e s (NC(N)) i s n e c e s s a r i l y equal to one. The minimum h y d r a u l i c e l e v a t i o n f o r the conveyance feeder i s defined w i t h those of other conveyance c o n t r o l p o i n t s i n C P T ( l ) . As a r e s u l t , values f o r NC(K), ICP(N), CP(N), SM(N), J> SD(W) and T_R(j,I,N) can be omitted from the input data. 4 3 T y p i c a l Feeder Layout Feeder Number (T) (second feeder) ( i e . ii = 2) © L a t e r a l 1 (10 l / s e c ) / C o n t r o l P o i n t ( i e . o u t l e t ) a l s o defines feeder s e c t i o n immediately upstream 6 5 4 5 2 1 100m I 100m 100m I 100m ' 100m 1 100m FIG. 3.2 L a t e r a l 2 (10 ]J/sec) D i s c h a r g i n g Outlets IR(j.I,2). J=1....NC(N). I=1,...IC(N) ^^-^Time I n t e r v a l C y c l e ^ ^ ( l ) ( l a t e r a l ) ( j > - \ 1 2 3 4 5 6 7 1 7 6 5 4 3 2 1 2 1 2 3 4 5 6 7 Table 5.2 Minimum H y d r a u l i c Head (Required pressure head equals 25.0 m) Cont r o l P o i n t Ground E l e v a t i o n (m) (CP(I,N)) H y d r a u l i c Head (m) 1 2.9 27.9 2 3.2 28.2 3 3.7 28.7 4 4.3 29.3 5 3.8 28.8 6 4.2 29.2 7 4.6 29.6 N = 2 N0DE(2) = © IC(2) =7 (the l a t e r a l s discharge from every o u t l e t , down the f u l l l e n g t h of the f i e l d ) NC(2) = 2 (number of l a t e r a l s ) ICP(2) = 7 SM(2) = 1.0(l00m) QM(2) = 10.0(l/sec) S D ( l , 2 ) , 1=1,...6; i s not i n c l u d e d , s i n c e SM(2) £ 0.0 Q(J,I , 2 ) , J=1,..2; 1=1, . . .7 ; i s not in c l u d e d , s i n c e QM(2) ^  0.0 Table 5.5 44 3.1.4 Conveyance S e c t i o n Input Data CPT ( i ) - minimum h y d r a u l i c e l e v a t i o n (m) f o r conveyance node ( i ) . A l l s t r i c t l y conveyance nodes ( i e . not a conveyance feeder) w i l l have minimum a l l o w a b l e head values equal to t h e i r ground e l e v a t i o n . A l l conveyance feeders have minimum a l l o w a b l e head values equal to t h e i r ground e l e v a t i o n plus a s p e c i f i c r e q u i r e d pressure head (page 4 9 ) . The program assumes t h a t a l l CPT(l) values must be g r e a t e r than or equal to zero. DIST(l) - l e n g t h (m) of conveyance s e c t i o n ( i ) . N o t i c e t h a t DIST(l) i s i n (meters) w h i l e the feeder s e c t i o n l e n g t h i s i n (100 meters). 45 3.1.5 Required Formats f o r Input Data DATA TYPE CONTENTS FORMAT GENERAL DATA NSECT,IPIPE,NF,NFT,MXIN,ISTAGE,ITLT EXP,EMIN,EMAX,TINT,APC ((IN(I,J),J=1,MXIN),1=1,ISTAGE) (7I6,5F6.2) (2014) PUMP DATA INCPC,IVHP,DAT (POR(lT),IT=1,ITLT) (HE(I),CPC(I),1=1,INCPC) (2I4,F6.1) (12F6.2) (12F6.2) GENERAL DATA (HK(I),COST(I),1=1,IPIPE) (12F6.2) FEEDER DATA N=1,NF NODE(N),IC(N),NC(N),ICP(N),SM(N),QM(N) (CP(l,N),I=1,ICPN); ICPN=ICP(N) (SD(I,N),1=1,ICPM1); ICPM1=ICP(N)-1 ((IR(J,I,N),1=1,ICN),J=1,NCN); ICN=IC(N) . NCN=NC(N) ((Q(j,I,N),I=1,ICN),J=1,NCN) (4I4,2F4.2) (12F6.2) (12F6.2) (2014) (12F6.2) CONVEYANCE FEEDER M T A NV=NFP1,NFT; NFP1=NF+1 NODE(NV),IC(NV),QM(NV) (Q(1,I,NV),I=1,ICNV); ICNV=IC(NV) (2I4,F4.2) (12F6.2) CONVEYANCE SECTION DATA (CPT(l),I=1,ISM1); ISM1=ISTAGE-1 (DIST(l),1=1,NSECT) (12F6.2) (12F6.2) Table 3 . 4 46 3.2 Define Input Parameters f o r the General L i n e a r Programming Routine (LIPSUB) The l i b r a r y r o u t i n e (LIPSUB) c a r r i e s out the l i n e a r programming. I t contains a s e t of FORTRAN IV subroutines. 3.2.1 Background D e f i n i t i o n s The f o l l o w i n g are d e f i n i t i o n s f o r terms used i n the LIPSUB d e s c r i p t i o n . Only terms which r e f e r to the pipe s i z i n g model s p e c i f i c a l l y , are g i v e n . Terms p r e v i o u s l y d e f i n e d i n the L i n e a r Programming Formulation (page 29 ) are excluded. primal s o l u t i o n - defines the optimal value f o r each d e c i s i o n v a r i a b l e . unbounded f u n c t i o n - i n d i c a t e s that the o b j e c t i v e can be improved i n d e f i n i t e l y without v i o l a t i n g any c o n s t r a i n t s . REAL *8 - defines r e a l , double p r e c i s i o n v a r i a b l e s . The model i t s e l f does not need such p r e c i s e accuracy but these v a r i a b l e s must be s p e c i f i e d double p r e c i s i o n to s a t i s f y the requirement by LIPSUB. 3.2.2 Define Input Parameters f o r LIPSUB The d e s c r i p t i o n s below overlook some d e t a i l s about parameters and options i n LIPSUB that don't p e r t a i n to the model. For a more general d i s c u s s i o n see (UBC L I P , February, 1977). Parameters t h a t are de f i n e d n u m e r i c a l l y i n the model, have t h e i r values w r i t t e n below the c a l l statement. CALL LIPSUD(TE,NDIMTB,M,NG,NE,m,NOBJ,NIfflS,TOL,NCBX,NCHK1,A1,iA24A3,A4, & nn) I I I I I I 301 0 0 0 (50.D-6) 228 TE - i s a REAL *8, two-dimensional a r r a y , dimensioned a t l e a s t to M + 1, by NG + 1. On entry, i t contains the tableau of the problem. The f i r s t row contains 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 . The NG + 1th column contains the r i g h t - h a n d s i d e s of the c o n s t r a i n t s . On output, v a r i o u s elements of TE 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 , and the value of the p r i m a l s o l u t i o n v a r i a b l e s . See the d e s c r i p t i o n of NCHK 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 TE(1,NG+1). i s the f i r s t dimension of the ar r a y TE. i s the number of c o n s t r a i n t s . M+1 >NDIMTB. on e n t r y , i t contains the number of v a r i a b l e s . On e x i t , i t contains 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 . i s the number of e q u a l i t y c o n s t r a i n t s , =•',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. = 0 i f o b j e c t i v e f u n c t i o n r a n g i n g i s not attempted. = 0 i f r i g h t - h a n d s i d e ranging i s not attempted. i s a REAL *8 v a r i a b l e . Numbers sm a l l e r than TOL i n absolute v a l u e are considered zero. i s an INTEGER one-dimensional a r r a y , dimensioned a t l e a s t M+1. On e x i t , NCHK(l), 1=2,...M+1 contains the index of a v a r i a b l e which i s i n c l u d e d i n the pr i m a l s o l u t i o n . The v a r i a b l e ' s v a l u e i s l o c a t e d i n TE(I,NG+1) where the value NG has been re t u r n e d by the subroutine, i s an INTEGER one-dimensional a r r a y , dimensioned a t l e a s t NG. I t contains the index of the dual s o l u t i o n v a r i a b l e . The dual s o l u t i o n i s only c a l c u l a t e d when NE = 0. In the model NE>0, so NCHK1 i s not used. are REAL *8, one-dimensional a r r a y s . They co 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 . When NOBJ = 0 , they are not used and may be dimensioned to 1. are REAL *8, one-dimensional a r r a y s . They co n t a i n the lower and upper bounds on the ri g h t - h a n d s i d e s . When NRHS = 0 , they are not used and may be dimensioned to 1. 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 u n f e a s i b l e . 4 8 3 . 2 . 3 SLACK Parameters ( S e n s i t i v i t y Measurements) The optimal s o l u t i o n defines a s e r i e s of h y d r a u l i c grade l i n e s f o r the system. I t i s o f t e n d e s i r a b l e to know how f a r these l i n e s are s i t u a t e d above the minimum al l o w a b l e head values f o r c e r t a i n c o n t r o l p o i n t s . SLACK parameters from LIPSUB conta i n t h i s i n f o r m a t i o n , f o r each head l o s s c o n s t r a i n t , but unfortunately,problems a r i s e which make these SLACK parameters unusable. Besides d e f i n i n g the optimal values f o r each d e c i s i o n v a r i a b l e , the prim a l s o l u t i o n a l s o contains SLACK parameters. For a given c o n s t r a i n t the SLACK value defines the d i f f e r e n c e between the inp u t r i g h t - h a n d s i d e c o e f f i c i e n t , and the a c t u a l r i g h t - h a n d s i d e c o e f f i c i e n t obtained by s o l v i n g the c o n s t r a i n t u s i n g the r e s u l t s from the primal s o l u t i o n . For the model's head l o s s c o n s t r a i n t s these SLACK values define the d i f f e r e n c e between maximum a l l o w a b l e head l o s s and the a c t u a l head l o s s . U n f o r t u n a t e l y , r e s u l t s from the l i n e a r prpgram co n t a i n d i s c o n t i n u i t i e s (page 5 2 ) i D i s c o n t i n u i t i e s tend to lower h y d r a u l i c grade l i n e s , and reduce the SLACK parameters below t h e i r maximum v a l u e s . Thus, SLACK values from the model are of no use and the d e s i r e d i n f o r m a t i o n must be c a l c u l a t e d independently a f t e r the optimal s o l u t i o n has been rearranged to exclude d i s c o n t i n u i t i e s . 4 9 3.)3 Background Theory f o r Model D i s c u s s i o n 3.3.1 Minimum All o w a b l e H y d r a u l i c Head a t Co n t r o l P o i n t s There are two types of c o n t r o l p o i n t s , each w i t h i t s own minimum a l l o w a b l e head v a l u e . I f the c o n t r o l p o i n t i s s t r i c t l y a conveyance node ( i e . not an o u t l e t ) , i t s minimum a l l o w a b l e head i s equal to i t s ground e l e v a t i o n (pipe p r o f i l e ) . In general i t i s u n d e s i r a b l e f o r h y d r a u l i c grade l i n e s to f a l l below the pipe p r o f i l e a t any p o i n t . I f the c o n t r o l p o i n t i s an o u t l e t ( i e . feeder or conveyance feeder o u t l e t ) the minimum a l l o w a b l e head valu e depends upon whether or not the o u t l e t i s d i s c h a r g i n g f o r a given time i n t e r v a l . I f the o u t l e t i s not d i s c h a r g i n g , i t s minimum a l l o w a b l e head i s equal to i t s ground e l e v a t i o n . I f the o u t l e t i s d i s c h a r g i n g , i t s minimum a l l o w a b l e head i s equal to i t s ground e l e v a t i o n plus a s p e c i f i c r e q u i r e d pressure head. T y p i c a l l y i n s p r i n k l e r systems t h i s r e q u i r e d pressure head i s between 25m and 50m; a co n s i d e r a b l y l a r g e head. Under these circumstances i t i s reasonable to suppose t h a t the head l o s s equations o r i g i n a t i n g from non-discharging o u t l e t s w i l l not govern pipe s i z e design. In order to decrease the number of head l o s s c o n s t r a i n t s i n the l i n e a r program f o r m u l a t i o n , the model assumes t h a t only d i s c h a r g i n g o u t l e t s w i l l produce s i g n i f i c a n t head l o s s c o n s t r a i n t s and a l l non-discharging o u t l e t s are ignored. In the r e s u l t i n g optimal pipe s i z e s o l u t i o n i t i s u n l i k e l y that head l o s s c o n s t r a i n t s f o r non-discharging o u t l e t s w i l l be v i o l a t e d i f a l l head l o s s c o n s t r a i n t s f o r d i s c h a r g i n g o u t l e t s have been s a t i s f i e d . However, i t i s a d v i s a b l e t h a t the user check the s u p p o s i t i o n by p l o t t i n g the h y d r a u l i c grade l i n e s produced by the optimal pipe s i z e s o l u t i o n . In c e r t a i n i n s t a n c e s , a s i g n i f i c a n t c o n t r o l p o i n t can occur i n a l o c a t i o n other than a t a conveyance node or an o u t l e t . A dummy node can be i n s e r t e d i n order to i n c l u d e the i n f l u e n c e of t h i s e x t r a c o n t r o l p o i n t . This dummy node 50 i s i n i t i a l i z e d i n e x a c t l y the same way as a d i s c h a r g i n g o u t l e t except i t w i l l have zero f l o w f o r a l l time i n t e r v a l s . I n s e r t i n g a Dummy Mode a t P o i n t B. Minimum A l l o w a b l e Head a t .Control P o i n t B. Hy d r a u l i c Grade L i n e - HGL Minimum All o w a b l e Head a t C o n t r o l P o i n t A. Required Pressure Head a t C o n t r o l P o i n t A. Con d i t i o n 1 C o n d i t i o n 2 FIG. 5.3 C o n d i t i o n 1 i s the optimal s o l u t i o n i f the c o n t r o l p o i n t a t B i s ignored and i t s minimum a l l o w a b l e head c o n s t r a i n t i s v i o l a t e d . C o n d i t i o n 2 i s the optimal s o l u t i o n i f the c o n t r o l p o i n t a t B i s i n c l u d e d i n the a n a l y s i s . The cost of pipe w i l l be gr e a t e r f o r Co n d i t i o n 2 than f o r C o n d i t i o n 1. The dummy node a l s o provides a u s e f u l way of d e a l i n g w i t h the r a r e case o u t l i n e d above where a non-discharging o u t l e t may define a s i g n i f i c a n t head l o s s c o n s t r a i n t . An e x t r a dummy node co u l d be i n c l u d e d immediately upstream or downstream of the o u t l e t , w i t h zero distance s e p a r a t i n g the two, and have the ground e l e v a t i o n as i t s minimum a l l o w a b l e head v a l u e . 3.3.2 Maximum Number of Pipe S i z e s i n a S e c t i o n A pipe s e c t i o n i s de f i n e d as a l e n g t h of pipe w i t h no intermediate c o n t r o l p o i n t s ( o u t l e t s ) . From Figures 2.3(a) and (b) i t can be noted t h a t f o r a given head l o s s value over a s e c t i o n , there i s an optimal ( i e . minimum cost) pipe s i z e s o l u t i o n t h a t contains no more than two pipe s i z e s . Assuming economic pipe s i z e s , i f three pipe s i z e s are present, i t w i l l always be economical to r e p l a c e the l a r g e s t and s m a l l e s t pipe s i z e s w i t h the medium pipe s i z e , u n t i l the l e n g t h of e i t h e r the l a r g e s t or s m a l l e s t pipe s i z e becomes zero. Maximum Number of Pipe S i z e s i n a S e c t i o n H y d r a u l i c Grade L i n e - HGL A h = A H a „ „ = A H b : For u n i t s to parameters see page 4 . Cost of r e p l a c i n g pipe 1 w i t h pipe 2 over l e n g t h La = (Cm 2 -Cm^) x La Cost of r e p l a c i n g pipe 3 w i t h pipe 2 over l e n g t h Lb = (Cm 2 -Cm^) x Lb Head l o s s from pipe 1 over La = Ha^ = Hm^  x La pipe 2 over La = Ha 2 = Hm2 x La pipe 2 over Lb = Hb 2 = Em^ x Lb pipe 3 over Lb = Hb, Hm., x Lb 3 — 2 A h = A Ha 2 1 = (Ha 2 - H a ^ = (Hm2 - Hm^La or La = A n/(Hm 2 - Hm.,) — A h =AHb^ 2 = (Hb^ - Hb 2) = (Hm^ - Hm 2)Lb or Lb =^h/(Em^ - Hm2) — Change i n cost between s e c t i o n 1 and s e c t i o n 2, from J_ and 2 ACost = (Cm 2 - Cn^) x La + (Cm 2 - Cm^) x Lb S u b s t i t u t e _3_ and 4 i n t o _5_ ACost = (Cm 2 - Cn^) x A n + (Cm 2 - Cm^) x A h = (Hm2 - Hm1) ^ I i (Hm3 - Hm2) (ACnu J £i2 AHm /AC: 3.2 \ l i 2 A H m 2 , l J \ x A h By d e f i n i t i o n of economic pipe AH""1" 2,1 It".follows thaiACost i s nega t i v e , or t h a t s e c t i o n 2 costs l e s s than s e c t i o n 1. 3.3.3 D i s c o n t i n u i t i e s i n Pipe Systems The problem of d i s c o n t i n u i t y a r i s e s because the l i n e a r program does not d i s t i n g u i s h between a p i p e l i n e network w i t h continuously decreasing pipe s i z e and one w i t h random pipe c o n t r a c t i o n s and expansions along i t s l e n g t h , as l o n g as a l l the c o n s t r a i n t s have been s a t i s f i e d * ( F i g . 3.5 ( a ) , (b) and ( D i s c o n t i n u i t y i n a S e r i e s System 53 Define System Head = 3«0m FIG. 5.5 (a) I 20 I I 25 I 20 I I 20 I 20 I I 5 I 0 I I 5 I 0 Head = 0.0m I 20 I I 15 t I I 5 For Pipe Costs and Head Loss C o e f f i c i e n t s , see Table 2.3, page13 • H y d r a u l i c Grade L i n e - HGL 5.0m Continuously Decreasing Pipe S i z e s 57.7m TODm Optimal (post .3 100m 242.3m 100m Head Loss I I 2.3m 0.7m -0.0m 3.0m Random Pipe Expansioi LS Optimal ( lost 463.3 and Contractions FIG. 3.5 (c) '<•/ \W \>\ / 's> 1 2 1 2 Head Loss I I 2.8m D.2m 17.7m 182.3m 40.0m 60.0m As f a r as the l i n e a r program i s concerned, the s o l u t i o n s i n F i g . 3.5 (b) and (c) are i d e n t i c a l . They have the same cost and n e i t h e r case v i o l a t e s any c o n s t r a i n t . From a p r a c t i c a l p o i n t of view there are a number of reasons why d i s c o n t i n u i t y i s u n d e s i r a b l e : 1. Added head l o s s due to pipe c o n t r a c t i o n s and expansions. 2. Added cost of f i t t i n g s and complications i n c o n s t r u c t i o n procedure. 3. Lowered h y d r a u l i c grade l i n e i n the c r i t i c a l time i n t e r v a l (the time i n t e r v a l which governs pipe s i z e design). 4. Increased head l o s s e s i n n o n - c r i t i c a l time i n t e r v a l s . This 4"th reason becomes important , i f the r e s e r v o i r i s r e p l a c e d by a v a r i a b l e - s p e e d power u n i t . For v a r i a b l e - s p e e d power u n i t s , i t i s d e s i r a b l e to minimize the head l o s s e s i n each time p e r i o d , i n order to reduce the r e q u i r e d pump discharge head and input power requirement. To minimize head l o s s e s i n n o n - c r i t i c a l time i n t e r v a l s i t i s necessary to e l i m i n a t e d i s c o n t i n u i t i e s . I t i s e a s i e s t to look a t t h i s phenomenon by c o n f i n i n g the a n a l y s i s to two s e c t i o n s . D i s c o n t i n u i t y can only occur i f : 1 . there are no branches between the two s e c t i o n s . 2. the two s e c t i o n s have the same f l o w , during the c r i t i c a l time i n t e r v a l . Each branch has at l e a s t one head l o s s equation to d e f i n e i t s f l o w p a t t e r n ( s ) and thus i t w i l l i n f l u e n c e the optimal s o l u t i o n i n upstream s e c t i o n s . This i n f l u e n c e w i l l n e c e s s a r i l y o b s t r u c t the formation of a d i s c o n t i n u i t y . The f o l l o w i n g example shows t h a t , given the optimal s o l u t i o n , any attempt to form a discontinuity., w i l l always make the system l e s s economic ( F i g . 3.6 ( a ) , (b) and ( c ) ) . 55 D i s c o n t i n u i t y i n a Branched System For Pipe Cost and Head Loss C o e f f i c i e n t s see, Table 2.3, page13• I 20 Head=2m I I 15 Define System  FIG. 3.6 (a) 100m 1-20 I I 0 Head=0m I 0 I I 15 Head=0m 100m 100m 38.5m Optimal S o l u t i o n  FIG. 5.6 (b) 61.5m 100.0m 13.6m B B6.4m Head Loss I = 2.0m Head Loss I I = 2.0m Cost = $57.04 1.8.5m, 81.5m 20.0m 80.0m S o l u t i o n w i t h D i s c o n t i n u i t y FIG. 3.6 (c) 22.8m 77.2m B Head Loss I = 2.0m Head Loss I I = 2.0m Cost = $57.42 56 Forming a d i s c o n t i n u i t y i n s e c t i o n s A and B w i l l not change the head l o s s f o r time i n t e r v a l I , but i t w i l l always in c r e a s e the head l o s s i n time i n t e r v a l I I , To compensate f o r the inc r e a s e d head l o s s i n s e c t i o n A, the head l o s s i n s e c t i o n C must decrease, which n e c e s s i t a t e s a net cost i n c r e a s e . Given two s e c t i o n s i n s e r i e s , i f the flows i n both s e c t i o n s are not the same during the c r i t i c a l time i n t e r v a l , a d i s c o n t i n u i t y cannot occur. D i s c o n t i n u i t y i n a S e r i e s System w i t h Unequal Flow Define System  FIG. 5.7 (a) = 1.0m 20 l / s e c 10 l / s e c Head = 0.0m 10 l / s e c ,10 l / s e c 100m 100m For P i pe Cost and Head Loss C o e f f i c i e n t s see, Table 2.3, page 13. Optimal Solutio: 1 1 2 FIG. 3.7 (b) 54.1m 145.9m Head Loss = 1.0m S o l u t i o n w i t h 1 2 1 ? D i s c o n t i n u i t y 34.1m ' 65.9m 20.0m 80.0m FIG. 3.7 (c) Head Loss =1.12 > maximum a l l o w a b l e head l o s s The formation of a d i s c o n t i n u i t y always increases head l o s s beyond i t s maximum a l l o w a b l e v a l u e . 57 3,3.4 Pump Operating and F i x e d Costs I t should be pointed out from the outset that every i r r i g a t i o n system i s unique. The pump discharge head and flow r a t e requirements may vary tremendously from one scheme to another. General methods f o r c a l c u l a t i n g pump operating and f i x e d costs are described below, but they won't n e c e s s a r i l y f i t a l l cases. These general methods were chosen because the input data i s e a s i l y o b t a i n a b l e and easy to work w i t h . In order to formulate these methods a number of s i m p l i f y i n g assumptions have been made which may not h o l d f o r every s i t u a t i o n . The user should look a t h i s p a r t i c u l a r system to see i f the s c a l e of the problem, the set of operating c o n d i t i o n s , and the r e q u i r e d p r e c i s i o n are compatible w i t h these assumptions. I f the designer f e e l s h i s system i s not compatible w i t h these assumptions, the model can s t i l l be used to determine the optimum pipe cost f o r d e s i r e d values of i n l e t head. This i s done by simply t r e a t i n g the system as a g r a v i t y system w i t h v a r i a b l e i n l e t head c o n d i t i o n s . Given these optimum pipe costs at v a r i o u s head v a l u e s , the user can add i n h i s own pump costs to complete the a n a l y s i s . 3.3.4.1 Operating Pump Cost The purpose of t h i s a n a l y s i s i s to see how the pump ope r a t i n g cost v a r i e s w i t h respect to changing i n l e t head. I t i s assumed that power cost i s the only s i g n i f i c a n t o p e r a t i n g cost t h a t v a r i e s w i t h head. Operating Cost = f (Power, Time of Operation) Power = f (Flow Rate, Pump Discharge Head, E f f i c i e n c y ) .". Operating Cost = f (Flow Rate, Pump Discharge Head, E f f i c i e n c y , Time of Operation) Operating pump cost depends upon the set of flo w c o n d i t i o n s and the type of pumping system. For a given i n l e t head, each i n d i v i d u a l time i n t e r v a l has i t s own flow c o n d i t i o n . The r e q u i r e d flow r a t e and i n l e t head vary from one time i n t e r v a l to another because d i f f e r e n t o u t l e t s are d i s c h a r g i n g . There are two main types of pumping systems, one w i t h a constant-speed power u n i t and the other w i t h a v a r i a b l e - s p e e d power u n i t . The model has been developed to handle both types. 3 . 3 « 4 « 1 » 1 Constant-speed Power U n i t s Constant-speed power u n i t s have a f i x e d head-discharge curve. For a given f l o w r a t e the pump w i l l provide a s p e c i f i c head valu e which i s independ-ent upon the needs of the s p r i n k l e r i r r i g a t i o n system downstream of the pump. The pump must be chosen to f u n c t i o n f o r the most demanding op e r a t i n g c o n d i t i o n . At other operating c o n d i t i o n s the excess head i s d i s s i p a t e d by t h r o t t l i n g v a l v e s . Each valu e of i n l e t head w i l l d e f i n e a d i f f e r e n t pump and/or i m p e l l e r s i z e w i t h i t s own i n d i v i d u a l head-discharge and e f f i c i e n c y - d i s c h a r g e curves. As flow c o n d i t i o n s change from one time i n t e r v a l to another, the pump w i l l a d j u s t i t s e l f to the new flow r a t e by moving along the head-discharge curve to a new pump op e r a t i n g c o n d i t i o n . This new ope r a t i n g c o n d i t i o n defines the r e q u i r e d head, f l o w , and e f f i c i e n c y f o r the pump. Knowing these values and the time d u r a t i o n , f o r each time i n t e r v a l , and the power c o s t , i t i s p o s s i b l e to c a l c u l a t e a value f o r o p e r a t i n g c o s t . Pump Operating C o n d i t i o n s , during time i n t e r v a l ( n ) : Qn - flow r a t e ( l / s e c ) hn - minimum r e q u i r e d pump discharge head (m) Hn - a c t u a l pump discharge head (m) en - e f f i c i e n c y Tn - time d u r a t i o n (hr/season) Operating Pump Cost = K x Id e a l Pump Operating Cost Formulation.  Constant-speed Power U n i t T h r o t t l e d h e a d 4 K n=1 ,4 o s t / ( l / s e c ) / ( m ) / ( h r ) ) 'Hn x On x Tn^ en FlowU) FIG. 5.8' For the power range examined (20-60kw), the head-discharge curves and the e f f i c i e n c y - d i s c h a r g e curves are f a i r l y f l a t , over a l a r g e range of flo w r a t e s . I t i s assumed th a t head and e f f i c i e n c y are constant f o r a l l values of f l o w . This g r e a t l y s i m p l i f i e s the ope r a t i n g cost computation by e l i m i n a t i n g the need to define the head-discharge curve and the e f f i c i e n c y - d i s c h a r g e curve f o r d i f f e r e n t i n l e t head va l u e s . S i m p l i f i e d Pump Operating Cost Formulation.  Constant-speed Power U n i t Operating Pump Cost h = [K X * 2 L ( Q n x T n ) ] x h \ % n=1,4 - Constant x h where h i s the i n l e t head value minus the i n t a k e water l e v e l . Flow FIG. 5.9 As the i n l e t head i s i n c r e a s e d , the pump discharge -head (h) w i l l i ncrease but a l l other values s t a y constant. Thus, f o r a constant-speed power u n i t the o p e r a t i n g pump cost can be expressed as a simple l i n e a r r e l a t i o n s h i p w i t h respect to i n l e t head. 3.3.4.1.2 Variable-speed Power U n i t s Variable-speed power u n i t s don't have a f i x e d head-discharge curve. Given a s p e c i f i c flow r a t e , the pump discharge head can be v a r i e d by changing the speed of the pump. This type of pump can c a t e r to the needs of the s p r i n k l e r i r r i g a t i o n system by supplying only the amount of head a c t u a l l y r e q u i r e d by the system f o r any p a r t i c u l a r time i n t e r v a l . Reducing the pump discharge head w i l l reduce input power requirements and ope r a t i n g pump co s t . Each valu e of i n l e t head may define a d i f f e r e n t pump and/or i m p e l l e r s i z e w i t h i t s own set of head-discharge curves, a t v a r i o u s pump speeds, and e f f i c i e n c y curves. The set of head-discharge curves are not e x p l i c i t l y 60 r e q u i r e d f o r the op e r a t i n g cost c a l c u l a t i o n , but the e f f i c i e n c y curves are r e q u i r e d . As flow c o n d i t i o n s change from one time i n t e r v a l to another, the pump can be adjusted to the new pump op e r a t i n g c o n d i t i o n , by changing the speed of the pump. The o p e r a t i n g c o n d i t i o n , i n every case, w i l l be d e f i n e d by the pump flow r a t e and the minimum r e q u i r e d pump discharge head, f o r the new time i n t e r v a l . This o p e r a t i n g c o n d i t i o n defines the e f f i c i e n c y of the pump. V/ith these values and those f o r time i n t e r v a l d u r a t i o n and power c o s t , i t i s p o s s i b l e to c a l c u l a t e the pump op e r a t i n g c o s t . I d e a l Pump Operating Cost Formulation,  Variable-speed Power U n i t Pump Speed - PS Operating Pump Cost = K x n=1,4 hn x Qn x Tn en Flow FIG. 5.10 I t i s assumed that e f f i c i e n c y i s constant f o r the range of head and flow values being considered. This s i m p l i f i e s the op e r a t i n g cost computation by e l i m i n a t i n g the need t o de f i n e the e f f i c i e n c y curves f o r d i f f e r e n t i n l e t head va l u e s . S i m p l i f i e d Pump Operating Cost Formulation ( v a r i a b l e - s p e e d power u n i t ) Operating Pump Cost = K (hn x Qn x Tn) (where e i s the value f o r e n=1,4 vc * constant e f f i c i e n c y ) 3.3.4.2 F i x e d Pump Cost The user has the freedom to define how the f i x e d pump cost w i l l vary w i t h pump discharge head. The model reads i n a number of cost and head values which are used to de f i n e the f i x e d cost f u n c t i o n . F i x e d pump costs must be reduced to t h e i r annual cost e q u i v a l e n t s . 61 I t i s assumed th a t the cost of the pump, motor combination, i s the only s i g n i f i c a n t f i x e d pump cost that v a r i e s w i t h changing i n l e t head. Both constant-speed and v a r i a b l e - s p e e d power u n i t s are produced i n d e f i n i t e s i z e s according to a s p e c i f i c power r a t i n g . They are designed to to supply power up to t h i s r a t i n g but not normally above i t . I f the maximum r e q u i r e d power f a l l s anywhere between the r a t e d values f o r two power u n i t s , the l a r g e r u n i t must be used. Thus, the f i x e d pump cost g e n e r a l l y forms a stepped f u n c t i o n w i t h r e s p e c t to maximum-:required power. The maximum r e q u i r e d power i s a f u n c t i o n of flo w r a t e , pump discharge head and e f f i c i e n c y . In the pipe s i z i n g a n a l y s i s , flow r a t e s and e f f i c i e n c y are assumed constant f o r a l l values of i n l e t head. This means that maximum r e q u i r e d power i s simply a f u n c t i o n of pump discharge head. In t h i s i n s t a n c e , i t i s p o s s i b l e to de f i n e a stepped, f i x e d pump cost f u n c t i o n w i t h r e s p e c t to pump discharge head. T y p i c a l T o t a l Bump Cost Formulation -p w o o Stepped!, f i x e d pump nnst, f i mot, inn Pump Discharge Head FIG. 5.11 3 . 4 Model Discussion 3.4»1 Introduction The model reads in parameters which define the pipeline system. The input data must be rearranged for use in the general linear programming routine (LIPSUB). Routine LIPSUB requires an input tableau which is similar to the linear programming formulation outlined on page 29 . It is desirable to reduce the size of the tableau by eliminating constraints which are not capable of governing pipe size design. In a typical irrigation system there are a great number of potential head loss constraints, but only a small portion are capable of governing design. Redundant head loss constraints are eliminated from the analysis. The remaining significant head loss constraints, together with the pipe length constraints and the objective function form the required linear programming tableau. After the linear programming analysis, for the desired inlet head, the results must be rearranged into a more presentable form. Discontinuities which may exist in the optimal pipe size solution must be removed. Minimum required inlet head values are calculated and printed for each individual time interval. These sensitivity values help to determine what flow patterns and control points govern design and help to inter pre-^optimal pipe sizing behavior in the system. Optimal pipe sizes for the entire system and the minimum pipe cost are also printed in the output. In a gravity system the execution is then terminated. In a pumping system the linear programming analysis must be repeated a number of times to find the optimal pipe cost for a range of inlet head values. In this instance, i t is possible to reduce the number of variables in the linear programming tableau by analyzing the results from previous inlet head iterations. Decreasing the number of variables significantly reduces the computational load in the linear program. In addition to optimum pipe cost, the model also calculates pump operating cost and pump fixed cost for each inlet head. The model concludes by printing 63 a s u m m a r y o f p i p e , p u m p , a n d t o t a l s y s t e m c o s t f o r e a c h i n l e t h e a d v a l u e . T o b e t t e r a p p r e c i a t e t h e c o n t e n t a n d c o n t e x t o f t h e f o l l o w i n g d i s c u s s i o n , i t s h o u l d b e r e a d a n d i n t e r p r e t e d i n c o n j u n c t i o n w i t h t h e m o d e l l i s t i n g i n I n d e x A . 3.4.2 E l i m i n a t e U n e c o n o m i c P i p e S i z e s On p a g e 8 i t w a s s h o w n h o w t h e C o s t I n d e x c o u l d b e u s e d t o e l i m i n a t e u n e c o n o m i c p i p e . T h e C o s t I n d e x i s d e f i n e d a s ( C m ^ C m g ^ C H m ^ - H n L j ) f o r p i p e s i z e s 1 a n d 2. Cm - c o s t p e r 100 m e t e r l e n g t h , i n d o l l a r s Hm - h e a d l o s s p e r 100 m e t e r l e n g t h , i n m e t e r s F o r e c o n o m i c p i p e t h e C o s t I n d e x m u s t d e c r e a s e w i t h d e c r e a s i n g p i p e s i z e . I n t h e m o d e l t h e C o s t I n d e x i s d e f i n e d a s (Cm^-CrngVCKm^-Km^) . T h i s i s p o s s i b l e b e c a u s e : Hm = Q n x Km w h e r e Q n i s a c o n s t a n t Km - h e a d l o s s - . . c o e f f i c i e n t p e r 100 m l e n g t h , i n m e t e r s Q - f l o w r a t e , i n l / s e c n - e x p o n e n t i a l I f a s m a l l d i a m e t e r p i p e c o s t s m o r e t h a n a l a r g e d i a m e t e r p i p e , i t i s i n t u i t i v e t h a t t h e s m a l l d i a m e t e r p i p e m u s t b e u n e c o n o m i c . T h i s c a s e , w h e r e t h e C o s t I n d e x i s l e s s t h a n z e r o , i s n o t e x p l i c i t l y d e a l t w i t h i n t h e a n a l y s i s . I t i s a s s u m e d t h a t t h e u s e r w i l l e l i m i n a t e t h e s e o c c u r r e n c e s . 3.4.3 D e t e r m i n e t h e T o t a l N u m b e r o f T i m e I n t e r v a l s T h e t o t a l n u m b e r o f t i m e i n t e r v a l s f o r t h e e n t i r e s y s t e m i s d e f i n e d a s t h e l e a s t common m u l t i p l e , f o r a l l t h e v a l u e s o f IC(N), w h e r e I C ( N ) i s t h e n u m b e r o f t i m e i n t e r v a l s f o r e a c h i n d i v i d u a l f e e d e r (N) o r c o n v e y a n c e f e e d e r (N). T h e l e a s t common m u l t i p l e i s f o u n d b y t a k i n g e a c h IC(N) v a l u e , d e t e r m i n i n g i t s p r i m e f a c t o r s , a n d m u l t i p l y i n g t h e p r e s e n t l e a s t common 6 4 m u l t i p l e by any prime f a c t o r s not already i n c l u d e d during previous d e r i v a t i o n . I n c r e a s i n g the number of time i n t e r v a l s g r e a t l y increases the computational l o a d . A maximum value of 50 i s allowed i n the program. This value has no r e a l s i g n i f i c a n c e and i s simply introduced to make the user aware of the p o t e n t i a l l y p r o h i b i t i v e computational burden. I d e a l l y the user should draw up a complete i r r i g a t i o n schedule f o r the system, showing the p o s i t i o n of a l l d i s c h a r g i n g o u t l e t s f o r every time i n t e r v a l , thus a s s u r i n g an e f f i c i e n t o v e r a l l p l a n . 3 . 4 . 4 Conveyance S e c t i o n Discharge Computation In each feeder (N), d i s c h a r g i n g o u t l e t s IR(j,I,N), f o r every feeder time i n t e r v a l ( i ) , are rearranged i n i n c r e a s i n g order ( i n an upstream d i r e c t i o n ) . The flow r a t e Q(j,I,N) i s then changed from the o u t l e t discharge, to the t o t a l flow upstream of the o u t l e t . At a conveyance node there i s one i n l e t flow and one or more o u t l e t f l o w s . To determine the flow through the conveyance s e c t i o n immediately upstream of the conveyance node ( i n l e t flow) simply add up a l l the r e q u i r e d flows to s e c t i o n s immediately downstream of the conveyance node, and the f l o w d i s c h a r g i n g from the conveyance node i t s e l f ( o u t l e t f l o w s ) . 3 . 4 . 5 E l i m i n a t e Redundant Head Loss Equations Each time i n t e r v a l defines a s p e c i f i c flow p a t t e r n . This flow p a t t e r n passes through a number of c o n t r o l p o i n t s . T h e o r e t i c a l l y , each c o n t r o l p o i n t defines a head l o s s c o n s t r a i n t that should be i n c l u d e d i n the f o r m u l a t i o n of the l i n e a r program. In p r a c t i c e , only a small p o r t i o n of the t o t a l number of p o t e n t i a l head l o s s c o n s t r a i n t s are a c t u a l l y s i g n i f i c a n t ( i e . capable of governing pipe s i z e design). The f o l l o w i n g d i s c u s s i o n o u t l i n e s the v a r i o u s methods used to separate and remove head l o s s c o n s t r a i n t s which are o b v i o u s l y n o n - s i g n i f i c a n t . An 65 e x a m p l e i s o u t l i n e d o n p a g e 83. 3.4.5.1 Z e r o F l o w R e d u n d a n c y T h r e e r u l e s a r e p r e s e n t e d b e l o w . T h e y r e f e r t o c o n v e y a n c e c o n t r o l p o i n t s d u r i n g a n y g i v e n t i m e i n t e r v a l . 1. I f t h e r e i s z e r o d i s c h a r g e f r o m a c o n v e y a n c e f e e d e r c o n t r o l p o i n t , t h e n n o s i g n i f i c a n t h e a d l o s s c o n s t r a i n t e x i s t s f o r t h a t c o n t r o l p o i n t * ( p a g e t y q ) . 2. A dummy c o n t r o l p o i n t , o n e t h a t h a s z e r o o u t l e t f l o w f o r a l l t i m e p e r i o d s , a n d a m i n i m u m a l l o w a b l e h e a d e q u a l t o i t s g r o u n d e l e v a t i o n , i s n o t a f f e c t e d b y t h e r u l e 1. 3. I f a s e c t i o n u p s t r e a m o f a c o n t r o l p o i n t h a s z e r o f l o w , t h e n n o s i g n i f i c a n t h e a d l o s s c o n s t r a i n t e x i s t s f o r t h a t c o n t r o l p o i n t . A n o t h e r t y p e o f Z e r o F l o w R e d u n d a n c y o c c u r s i n a f e e d e r , w h e n LR(j, I,N)= 0; T h i s c o n d i t i o n i n d i c a t e s t h a t a l a t e r a l i s n o t d i s c h a r g i n g d u r i n g a p a r t i c u l a r t i m e i n t e r v a l . 3.4.5.2 H e a d L o s s R e d u n d a n c y A h e a d l o s s c o n s t r a i n t o r i g i n a t i n g f r o m a c e r t a i n c o n t r o l p o i n t i s r e d u n d a n t i f a n o t h e r h e a d l o s s c o n s t r a i n t f r o m t h e s a m e c o n t r o l p o i n t c o n t a i n s e x c l u s i v e l y g r e a t e r o r e q u a l f l o w i n e v e r y u p s t r e a m s e c t i o n . T h e r e may b e a l a r g e n u m b e r o f d i f f e r e n t f l o w p a t t e r n s o r i g i n a t i n g f r o m t h e s ame c o n t r o l p o i n t . T h e p r o g r a m c o m p a r e s e a c h i n d i v i d u a l f l o w p a t t e r n w i t h e v e r y o n e o f t h e o t h e r p o s s i b l e f l o w p a t t e r n s . T h e h e a d l o s s r e d u n d a n c y a l g o r i t h m f o r f e e d e r c o n t r o l p o i n t s i s c o n s i d e r -a b l y m o r e c o m p l e x t h a n t h e o n e f o r c o n v e y a n c e c o n t r o l p o i n t s , b u t t h e b a s i c c o n c e p t s a r e t h e s a m e . F o r a g i v e n f e e d e r (N), t h e m a t r i x IR(J,I,N), w h i c h c o n t a i n s d i s c h a r g i n g o u t l e t s , f o r c y c l e ( j ) , a n d t i m e i n t e r v a l ( i ) , i s s e a r c h e d t o p i c k o u t a l l f l o w p a t t e r n s w i t h t h e s ame c o n t r o l p o i n t . T h e p a r a m e t e r MAX i s u s e d t o f a c i l i t a t e t h i s s e a r c h i n g p r o c e s s b y t e r m i n a t i n g t h e s e a r c h i f n o f u r t h e r f l o w p a t t e r n s a r e p o s s i b l e . F o r e a c h p a i r o f f l o w p a t t e r n s , f l o w s a r e c o m p a r e d , f i r s t w i t h i n t h e f e e d e r , a n d t h e n i n a l l upstream conveyance s e c t i o n s . Flow w i t h i n the feeder i s not e x p l i c i t l y d e f i n e d f o r each s e c t i o n of the feeder, but r a t h e r as fl o w between d i s c h a r g i n g o u t l e t s . The upstream d i s c h a r g i n g o u t l e t s may be d i f f e r e n t f o r the two flo w p a t t e r n s . Instead of comparing flows i n each s e c t i o n , the flows are compared f o r each change i n flow c o n d i t i o n s . Define a l l separate Flow Condition Comparisons T f o r Two Flow Patterns •Flow Rate ( l / s e c ) Time I n t e r v a l I 10 Time I n t e r v a l I I 10 HGL FIG. 5.12 i n Flow Cond i t i o n Comparison 2 Flow I I > Flow I i n Flow C o n d i t i o n Comparison 3 Flow I > Flow I I .'. n e i t h e r f l o w p a t t e r n i s redundant Flow C o n d i t i o n Comparisons 67 3.4.5.3 C o n t r o l P o i n t E l e v a t i o n Redundancy For a given feeder and f l o w p a t t e r n , i f the minimum a l l o w a b l e head va l u e a t one d i s c h a r g i n g o u t l e t ( i e . c o n t r o l p o i n t ) i s g r e a t e r than or equal to tha t of a second d i s c h a r g i n g o u t l e t f u r t h e r upstream from the f i r s t , then the head l o s s c o n s t r a i n t o r i g i n a t i n g from the upstream d i s c h a r g i n g o u t l e t i s redundant and can be e l i m i n a t e d from the a n a l y s i s . For any p o s s i b l e pipe s i z e arrangement, the r e s u l t i n g h y d r a u l i c grade l i n e w i l l always be i n t e r c e p t e d by the minimum a l l o w a b l e head f o r the downstream c o n t r o l p o i n t before i t can reach the minimum a l l o w a b l e head f o r the upstream c o n t r o l p o i n t . S i m i l a r reasoning can be used f o r conveyance s e c t i o n s w i t h no conveyance feeders. U n l i k e f e e d e r s , zero f l o w i n conveyance s e c t i o n s cause problems and r e q u i r e s p e c i a l p r o v i s i o n s . Even i f the above redundancy c o n d i t i o n i s met, the upstream head l o s s c o n s t r a i n t should not be e l i m i n a t e d i f there i s zero f l o w i n the conveyance s e c t i o n immediately upstream of the downstream c o n t r o l p o i n t . In a conveyance system which i n c l u d e s conveyance fe e d e r s , some conveyance c o n t r o l p o i n t s have minimum a l l o w a b l e head values equal to t h e i r ground e l e v a t i o n , and others have minimum a l l o w a b l e head values equal to t h e i r ground e l e v a t i o n plus some r e q u i r e d pressure head. The redundancy procedure f o r t h i s case becomes unduly complexand has been excluded from the a n a l y s i s . 3.4.5.4 Feeder Head Loss Equation Index Redundancy and S i n g l e Stage Redundancy 3.4.5.4.1 Feeder Head Loss Equation Index Redundancy The t o t a l number of time i n t e r v a l s (ITLT) i s some m u l t i p l e (KN) of the number of time i n t e r v a l s i n an i n d i v i d u a l feeder ( l C ( N ) ) . ( i e . The time c y c l e and s e t of flow p a t t e r n s f o r the feeder, repeats i t s e l f KH times f o r each f u l l time c y c l e of the e n t i r e system;) During each of these KN times (K = 1,2,3 KN)» the s e t of fl o w patterns w i t h i n the feeder w i l l be the same but those outside the feeder, i n conveyance s e c t i o n s , w i l l g e n e r a l l y be d i f f e r e n t . I f flows are the same, then head l o s s e s w i l l a l s o be the same. For p a r t i c u l a r time i n t e r v a l s where flows ( i e . head losses) within the feeder are i d e n t i c a l , the Head Loss Redundancy Algorithm f o r feeders can ignore the comparison of head losses within the feeder and i s o l a t e the head l o s s comparison to the upstream conveyance sections. 3.4.5*4.2 Single Stage Redundancy In some cases a feeder i s separated from the i n l e t '(reservoir or pump) by a s i n g l e conveyance se c t i o n . The flow through any conveyance section immediately u'pstream of a feeder i s defined e x c l u s i v e l y by that feeder. Since no other upstream conveyance s e c t i o n e x i s t s , there i s no need to employ the head l o s s comparison of the upstream conveyance sections. 3.4.6 Determine the Minimum Allowable I n l e t Head f o r Given Pipe Sizes Using the l a r g e s t pipe s i z e , the model generates hydraulic grade l i n e s f o r a l l s i g n i f i c a n t head loss equations, as defined i n the above section, and determines the minimum allowable head at the i n l e t that doesn't v i o l a t e any constraints. Head loss equations o r i g i n a t i n g from both conveyance and •a. and feeder control points are tested. The head loss equations contain head Q l o s s components f o r every section (both feeder and conveyance) between the control point and the i n l e t . 3.4.7 Generate I n i t i a l Tableau Generate a tableau s i m i l a r to the formulation on page29 > f o r use i n the general l i b r a r y routine (LIPSUB). The f i r s t row contains the c o e f f i c i e n t s f o r the objective function. The next NNE rows contain c o e f f i c i e n t s f o r the non-equality (head loss) constraints. The l a s t NE rows contain the c o e f f i c i e n t s f o r the equality (section length) constraints. For t h i s i n i t i a l tableau the right-hand side column remains empty because the desired i n l e t head i s s t i l l unknown. The number of columns ( i e . v a r i a b l e s ) i n the tableau i s equal to the number of sections i n the system, both conveyance and feeder, m u l t i p l i e d by the number of d i f f e r e n t pipe s i z e s . The main purpose of this section is to organize the significant head loss equations into a form that can be used in LIPSUB. For a given significant head loss equation, the values of head loss per 100m length are calculated for every section, pipe size combination, and entered into the tableau as activity coefficients. Care must be taken to assure that each activity coefficient is assigned to the proper column (ie. variable) corresponding to the appropriate section and pipe size. In feeders, flow rates are defined as the flow between two discharging outlets. It is necessary to determine the flows for every feeder section. 3.4.8 Determine Inlet Head Values Given values for minimum desired inlet head (EMIN), maximum desired inlet head (EMAX), and change in head per iteration (TINT), the model selects specific head values to be analyzed and arranges them in a particular order. If EMIN ^ 0 . 0 , its value is taken as the minimum desired inlet head. If EMIN = 0.0 the model calculates the minimum allowable inlet head, and EMIN is redefined as the minimum permissable head given that (EMAX-EMIN) must divide evenly into TINT. EMAX is the fir s t head to be analyzed, followed by the minimum allowable head and/or EMIN, depending upon the original value of EMIN. After EMIN, the model simply iterates from EMIN to EMAX increasing the previous head by TINT before each subsequent analysis. 3.4.9 Print Pump Output and Calculate Fixed Pump Cost Besides simply printing output, this section also determines the fixed pump cost for the given inlet head HT(LP). The program calculates the maximum required head imparted by the pump (HT(LP)-DAT), and then uses the fixed pump cost function, defined by costs CPC(l), and heads HE(l), I = 1,..INCPC, to find the corresponding fixed pump cost. If the number of points used to define the fixed pump cost function (INCPC) is equal to zero, the f i x e d pump cost i s set equal to zero f o r a l l values of head. I f the r e q u i r e d pump discharge head i s l e s s than the minimum head d e f i n i n g the cost f u n c t i o n (HE(l))^ the f i x e d pump cost i s set equal to CPC(1). I f the r e q u i r e d pump discharge head i s g r e a t e r than the maximum head d e f i n i n g the cost f u n c t i o n (HE(lNCPC)), the f i x e d pump cost i s set equal to a l a r g e number ( i e . 1,000,000). This i s more to i d e n t i f y t h i s c o n d i t i o n than to represent the r e a l cost f u n c t i o n . I f the r e q u i r e d pump discharge head i s i n between two heads d e f i n i n g the cost f u n c t i o n , t h e f i x e d pump cost i s obtained by assuming that the cost f u n c t i o n i s l i n e a r between the two head v a l u e s . 3.4.10 Subroutine L IP For every d e s i r e d i n l e t head v a l u e , subroutine L IP completes the ta b l e a u r e q u i r e d by the l i n e a r programming r o u t i n e (LIPSUB), c a l l s LIPSUB, rearranges output from LIPSUB i n t o a more presentable form, checks and r e d e f i n e s boundary c o n d i t i o n s f o r the two v a r i a b l e r e d u c t i o n a l g o r i t h m s , determines and records the minimum al l o w a b l e i n l e t head and a c r i t i c a l c o n t r o l p o i n t , f o r each time i n t e r v a l ( i e . s e n s i t i v i t y measurements), and p r i n t s optimal pipe s i z e r e s u l t s . 3.4.11 Generate Hew Tableau The purpose of t h i s s e c t i o n i s to generate a tab l e a u f o r use i n the general l i b r a r y r o u t i n e (LIPSUB). The only d i f f e r e n c e between t h i s t a b l e a u and the i n i t i a l t a b l e a u i s the e l i m i n a t i o n of i n s i g n i f i c a n t v a r i a b l e s and the i n c l u s i o n of the rig h t - h a n d - s i d e c o e f f i c i e n t s . For a pumping system the l i n e a r programming a n a l y s i s i s repeated f o r a number of i n l e t head v a l u e s . From the r e s u l t s of previous head values i t i s p o s s i b l e to e l i m i n a t e v a r i a b l e s which are ob v i o u s l y not s i g n i f i c a n t to the present a n a l y s i s . The approach f o r determining a v a r i a b l e ' s s i g n i f i c a n c e i s d i s c u ssed i n a f o l l o w i n g s e c t i o n . I t was explained p r e v i o u s l y t h a t care must be taken to assure that a c t i v i t y c o e f f i c i e n t s are assigned to the proper s e c t i o n and pipe s i z e . With some v a r i a b l e s being e l i m i n a t e d and the others being rearranged, even more care i s r e q u i r e d . Two parameters,JP(j) and JC(j) are used to keep t r a c k of a v a r i a b l e ' s pipe s i z e , and i t s conveyance s e c t i o n or feeder number. Upon e x i t from LIPSUB a v a r i a b l e ' s i n d i v i d u a l feeder s e c t i o n i s not important, only i t s feeder number i s of any s i g n i f i c a n c e . The r i g h t - h a n d s i d e c o e f f i c i e n t f o r a given head l o s s c o n s t r a i n t contains the maximum al l o w a b l e head l o s s f o r t h a t c o n s t r a i n t . The maximum a l l o w a b l e head l o s s i s equal to the i n l e t head minus the minimum a l l o w a b l e head at the c o n t r o l p o i n t from which the head l o s s c o n s t r a i n t o r i g i n a t e d . The r i g h t -hand s i d e c o e f f i c i e n t f o r a given s e c t i o n l e n g t h c o n s t r a i n t simply contains the l e n g t h of the s e c t i o n . 3.4.12 Subroutine LIPSUB, and Optimal Pipe L o c a t i o n . S i z e and Length 3.4.12.1 Subroutine LIPSUB This i s the l i b r a r y r o u t i n e used to perform the a c t u a l l i n e a r programming a n a l y s i s . The main purpose of the model thus f a r has been to organize the input data f o r use i n LIPSUB. The output from LIPSUB must be rearranged i n t o a more s u i t a b l e form. 3.4.12.2 Define Output Parameters from LIPSUB The output contains three d i f f e r e n t pieces of i n f o r m a t i o n : 1. the minimum cost of the pipe system, f o r the given i n l e t head valu e . 2. the v a r i a b l e numbers (NCHK(I)) i n c l u d e d i n the optimal s o l u t i o n . Using the parameters JP(NCHK(l)) and JC(NCHK(l)), these v a r i a b l e numbers d e f i n e pipe s i z e and l o c a t i o n ( i e . conveyance s e c t i o n or feeder number). 3. the v a r i a b l e number's v a l u e . These values c o n t a i n the optimal l e n g t h f o r the pipe s i z e , and conveyance s e c t i o n or feeder number de f i n e d by the v a r i a b l e number. For a given feeder, r e s u l t s from a l l i n d i v i d u a l feeder s e c t i o n s are 72 combined to form an optimal pipe s i z e , l e n g t h , conglomerate. The reason f o r t h i s i s explained below. 3.4»12.3 D i s c o n t i n u i t y D i s c o n t i n u i t i e s are u n d e s i r a b l e and must be e l i m i n a t e d . As o u t l i n e d on page 52 , d i s c o n t i n u i t i e s can only occur when the s e c t i o n s are i n s e r i e s and f l o w i s constant ( i e . intermediate o u t l e t discharge i s zero) during the c r i t i c a l time i n t e r v a l . These c o n d i t i o n s occur f r e q u e n t l y i n feeders. For a given feeder, d i s c o n t i n u i t i e s are e l i m i n a t e d by simply combining the r e s u l t s from a l l feeder s e c t i o n s and arranging them according to pipe s i z e . This i s done w h i l e r e a r r a n g i n g the output from LIPSUB. There are two other sources of d i s c o n t i n u i t y . I t may occur w i t h i n a feeder and the conveyance s e c t i o n immediately upstream, or w i t h i n s e c t i o n s bounded by a s e r i e s of conveyance feeders. For t h i s second case i t i s necessary to search the system f o r conveyance feeders and determine upstream and downstream boundaries f o r s e c t i o n s p o s s i b l y a f f e c t e d by d i s c o n t i n u i t y . D i s c o n t i n u i t y can only e x i s t i f conveyance feeders occur i n a s e r i e s ( i e . w i t h no branches at any of the conveyance nodes). The upstream and down-r stream boundaries are defined by the occurence or non-occurrence of branches at the conveyance nodes. The f i r s t stage i s ignored because by d e f i n i t i o n i t must c o n t a i n one or more branches. To e l i m i n a t e d i s c o n t i n u i t y , the program simply combines pipe s i z e r e s u l t s from a l l p o s s i b l y a f f e c t e d s e c t i o n s and rearranges them according to decreasing pipe s i z e ( i e . the l a r g e s t pipe s i z e at the upstream boundary, and p r o g r e s s i v e l y s m a l l e r pipe s i z e s downstream). 3.4.13 V a r i a b l e Reduction Algorithms 3»4»13«1 I n t r o d u c t i o n These algorithms are only of v a l u e f o r pumping systems, where the a n a l y s i s i s repeated f o r a number of i n l e t head values. Both the a l g o r i t h m s , one f o r feeders and one f o r conveyance s e c t i o n s , are used to reduce the number of v a r i a b l e s i n the general l i n e a r programming r o u t i n e LIPSUB, by u s i n g r e s u l t s from previous i t e r a t i o n s to i d e n t i f y v a r i a b l e s that are o b v i o u s l y not s i g n i f i c a n t . The algorithms are based on the premise that as i n l e t head i s i n c r e a s e d , pipe i n t e r s e c t i o n s i n the r e s u l t i n g optimal pipe s i z e s o l u t i o n s , move upstream. U n f o r t u n a t e l y , t h i s i s not always the case, and the premise' must be v e r i f i e d before a c c e p t i n g any r e s u l t s . I f a pipe i n t e r s e c t i o n moves downstream as i n l e t head i s i n c r e a s e d , the boundary c o n d i t i o n s which define v a r i a b l e r e d u c t i o n are r e e v a l u a t e d , and i f the previous a n a l y s i s d i d not i n c l u d e a c r i t i c a l v a r i a b l e , the a n a l y s i s i s repeated. The two algorithms d i f f e r s l i g h t l y i n t h e i r approach, b a s i c a l l y because feeder s e c t i o n s form a s e r i e s w h i l e conveyance s e c t i o n s p r i m a r i l y form a branched system. The feeder a l g o r i t h m analyzes the e n t i r e feeder as a complete u n i t w h i l e the conveyance s e c t i o n a l g o r i t h m analyzes each s e c t i o n i n i s o l a t i o n . 3.4.13.2 Feeder A l g o r i t h m The maximum i n l e t head value i s f i r s t to be analyzed. In the r e s u l t i n g optimal pipe s i z e s o l u t i o n a l l pipe i n t e r s e c t i o n s should be i n t h e i r f u r t h e s t upstream p o s i t i o n . Given a p a r t i c u l a r pipe s i z e w i t h i t s upstream pipe i n t e r s e c t i o n a t p o i n t A, v a r i a b l e s which r e f e r to t h i s pipe s i z e and any s e c t i o n upstream of A, can be e l i m i n a t e d . V a r i a b l e Reduction i n Feeders f o r Maximum I n l e t Head P o i n t A i s the upstream pipe i n t e r s e c t i o n f o r the given pipe s i z e . FIG. 5.13 P o i n t A (pipe 3) P o i n t A (pipe 4) ' 2 1 1 31 1—^ s e c t i o n 3 2 1 pipe s i z e 1 2 3 4 1 2 3 4 1 2 3 4 * e l i m i n a t e d pipe s i z e s • * - shown i n b l a c k The minimum i n l e t head value i s analyzed next. In the r e s u l t i n g optimal pipe s i z e s o l u t i o n a l l pipe i n t e r s e c t i o n s should be i n t h e i r f u r t h e s t down-stream p o s i t i o n s . Given a p a r t i c u l a r pipe s i z e w i t h i t s downstream pipe i n t e r s e c t i o n a t B, v a r i a b l e s which r e f e r to t h i s pipe s i z e and any s e c t i o n s downstream of B, can be e l i m i n a t e d . V a r i a b l e Reduction i n Feeders f o r Minimum I n l e t Head P o i n t B i s the downstream pipe i n t e r s e c t i o n f o r the given pipe s i z e . P o i n t B (pipe 1) P o i n t B (pipe 2) FIG. 3 . U 1 r - 2 —n ! 3 • s e c t i o n 3 2 1 pipe s i z e 1 2 3 4 1 2 3 4 1 [2 3 4 e l i m i n a t e d pipe s i z e s • For a l l remaining i t e r a t i o n s the program increases the previous i n l e t head by some head change increment and repeats the a n a l y s i s . The v a r i a b l e r e d u c t i o n method f o r these subsequent i t e r a t i o n s i s i d e n t i c a l to the one used f o r the minimum i n l e t head valu e . U n f o r t u n a t e l y not a l l pipe i n t e r s e c t i o n s move upstream when i n l e t head i s i n c reased ( F i g . 3 . 1 5 ( & ) , ( b ) , and ( c ) ) . 75 V a r i a b l e Reduction. Deviate C o n d i t i o n Define System  FIG. 5.15 (a) I 30 I 10 I 1 p > - ^ ^ I I 25 I I 25 / l i 0 I 20 I 0 I I 0 ,11 25 ' 100m 100m 100m Head=0.0m ,Time I n t e r v a l ^ v f ' ^ F l o w Rate ( l / s e c ) I 10 II 0 Head=2.4m= For Pipe Costs and Head Loss C o e f f i c i e n t s see Table 2 .3 , page 1; Optimal S o l u t i o n when I n l e t Head = 2.4 m Hyd r a u l i c Grade L i n e - HGL FIG. 5.15 (b) (Head I at C o n t r o l P o i n t 2 = 1.26 m Head=0.0m 94.2m 105.8m ' 66.4m '33.6m 76 As inlet head is increased from 2.4 to 3.0 m a l l the pipe intersections should move upstream hut pipe intersection (3»4) actually moves downstream. By examining the hydraulic grade lines for the system the reason for this deviation becomes clear. When the inlet head increases, the head at control point 2 for Hydraulic Grade Line I actually decreases, thus allowing less head loss in section 1 . The above example is a f a i r l y isolated incident however, and for the vast majority of cases no deviations of this kind occur. Before accepting any results from the linear program, they are checked to see that a l l pipe intersections move upstream with increasing inlet head. This checking procedure can be divided into two parts, one that compares the present i n l e t head results with those of the preceding lower inlet head, and the other that compares the present inlet head results with those of the maximum inlet head. The underlying theory and solution procedure for both parts are essentially the same. The following explanation only deals with the f i r s t part, the analogous explanation for the second part is omitted. If a pipe intersection (n,n+l) moves downstream with increasing i n l e t head, but doesn't attempt to move into a downstream section ( j ) , where the variable associated with pipe size (n) has previously been eliminated, then there is nothing wrong with the results from the linear program. Although the results are correct, a deviate condition has occured and steps are taken to r e i n i t i a t e variables associated with pipe size (n), in a number of downstream sections, in case the condition continues for the next higher iterative head. If a pipe intersection (n,n+l) moves downstream with increasing head, and does attempt to move into a downstream section ( j ) , then the results from the linear program are wrong and the analysis must be repeated. If the pipe intersection (n,n+l) is located immediately upstream of section ( j ) , i t i s assumed that the pipe intersection would move further downstream into the section i f i t could. Before repeating the analysis, the variables associated with pipe size (n) are reinitiated in a number of downstream sections. 77 3 . 4 . 1 3 . 3 Conveyance Section Algorithm The conveyance section algorithm again uses the premise that as inlet head is increased, a l l pipe intersections move upstream, but in addition i t uses', the fact that only two pipe sizes can appear in a section at any one time (page 50 ) . For the maximum inlet head value a l l pipe intersections in the optimal pipe size solution should be in their furthest upstream position. The analysis finds the largest pipe size (k-1) present in the section, sets the smallest possible pipe size for the section equal to pipe (k), and eliminates variables that refer to pipe sizes smaller than pipe (k). Variable Reduction in Conveyance Sections for Maximum Inlet Head FIG. 3.16 » 2 3 pipe size 1 2 3 4 "77 eliminated pipe sizes • * - shown in black For the minimum inlet head value a l l pipe intersections in the optimal pipe size solution should be in their furthest downstream position. The analysis finds the smallest pipe size (m+1) present in the/section, sets the largest possible pipe size for the.section equal to pipe (m), and eliminates variables that refer to pipe sizes j. larger than pipe (m). Variable Reduction in Conveyance Sections for Minimum Inlet Head FIG. 5.17 2 pipe size 1 2 3 4 eliminated pipe sizes H The variable reduction method in subsequent higher head iterations is identical to the one used for the minimum inlet head value. Before accepting results, they are checked for deviations. This checking procedure has two parts, one comparing the present inlet head r e s u l t s to the preceding lower i n l e t head r e s u l t s , and the other comparing the present i n l e t head r e s u l t s to the maximum i n l e t head r e s u l t s . An explanation i s given f o r the f i r s t p a r t , but the analogous explanation f o r the second p a r t i s omitted. I f a pipe i n t e r s e c t i o n (m,m+l) moves downstream, w i t h i n s e c t i o n ( i ) as i n l e t head i s i n c r e a s e d , a d e v i a t i o n has occured but the a l g o r i t h m does not recognize i t u n t i l the pipe i n t e r s e c t i o n (m,m+l) moves outside of s e c t i o n ( i ) . Once pipe (m+l)leaves the s e c t i o n i t i s then p o s s i b l e t h a t pipe (m-1) may attempt to enter the s e c t i o n , but cannot because the v a r i a b l e a s s o c i a t e d w i t h pipe (m-1) has p r e v i o u s l y been e l i m i n a t e d . I f pipe i n t e r s e c t i o n (m,m+l) leaves s e c t i o n ( i ) but pipe (m-1) does not attempt to enter the s e c t i o n , then the r e s u l t s from the l i n e a r program are completely c o r r e c t , but the program r e i n i t i a t e s the v a r i a b l e a s s o c i a t e d w i t h pipe (m-1). I f pipe (m-1) does attempt to enter s e c t i o n ( i ) , then the r e s u l t s from the l i n e a r program are wrong and the a n a l y s i s must be repeated. I f the le n g t h of pipe (m) i n the s e c t i o n immediately upstream from s e c t i o n ( i ) i s zero ( i e . the pipe i n t e r s e c t i o n (m-1,m) i s l o c a t e d immediately upstream of s e c t i o n ( i ) ) , i t i s assumed that the pipe (m-1) would move f u r t h e r downstream i n t o s e c t i o n ( i ) i f i t could. Before r e p e a t i n g the a n a l y s i s , the program r e i n i t i a t e s the v a r i a b l e a s s o c i a t e d w i t h pipe (m-1). 3.4.13.4 Operating D e t a i l s The parameter 12 i s used to i n d i c a t e i f , i n the previous a n a l y s i s , a c r i t i c a l v a r i a b l e has been excluded, ( i e . a pipe s i z e wishes to enter a s e c t i o n , but the v a r i a b l e a s s o c i a t e d w i t h t h a t pipe s i z e and s e c t i o n , has p r e v i o u s l y been eliminated) and i f the a n a l y s i s must be repeated. I f f o r some reason an unstable c o n d i t i o n a r i s e s , and f o r a given i n l e t head v a l u e , the a n a l y s i s has already been repeated twice and i s to be repeated a t h i r d time, parameter 11, i s used to .terminate execution. A f t e r the c o r r e c t l i n e a r programming s o l u t i o n has been found and any deviate c o n d i t i o n s have been d e a l t w i t h , the program r e d e f i n e s the v a r i a b l e r e d u c t i o n boundary c o n d i t i o n s f o r use i n the next h i g h e r i n l e t head v a l u e . In some i n s t a n c e s , a pipe i n t e r s e c t i o n may move downstream f o r a number of i n c r e a s i n g i n l e t head v a l u e s . Parameter DP2(lW,N), i s used to assure s a f e boundary c o n d i t i o n s i n these i n s t a n c e s . U n f o r t u n a t e l y , i n both v a r i a b l e r e d u c t i o n algorithms the a n a l y s i s i s complicated by roundoff e r r o r which tends to make the computation s l i g h t l y more messy and clumsy than would otherwise be necessary. 3 . 4 . 1 4 S e n s i t i v i t y Measurements Using the optimal pipe s i z e r e s u l t s , the program generates h y d r a u l i c grade l i n e s f o r a l l v a l i d head l o s s equations, and determines and records the minimum al l o w a b l e head a t the i n l e t , f o r each time i n t e r v a l . The c o n t r o l p o i n t a s s o c i a t e d w i t h the minimum al l o w a b l e head v a l u e , i s a l s o given as an output parameter. The output only defines one such c r i t i c a l c o n t r o l p o i n t f o r every time i n t e r v a l , but there may w e l l be more than one. Head l o s s equations from both conveyance and feeder c o n t r o l p o i n t s are t e s t e d . The t e s t procedure f o r conveyance c o n t r o l p o i n t s i s q u i t e s t r a i g h t forward. Perhaps the only s u r p r i s i n g f e a t u r e i s the i n c l u s i o n of the zero flow redundancy method. I t :has been i n c l u d e d here to e l i m i n a t e i n v a l i d head l o s s c o n s t r a i n t s (page 6 5 ) . The t e s t procedure f o r feeders i s c o n s i d e r -a b l y more complex, and r e q u i r e s f u r t h e r e x p l a n a t i o n . When t e s t i n g feeder c o n t r o l p o i n t s , there are head l o s s c o n t r i b u t i o n s by both conveyance s e c t i o n s and feeder s e c t i o n s . The program c a l c u l a t e s each of these c o n t r i b u t i o n s s e p a r a t e l y . As e x p l a i n e d on page 6 7 , the t o t a l number of time i n t e r v a l s (ITLT) i s some m u l t i p l e (KN), of the number of time i n t e r v a l s i n an i n d i v i d u a l feeder (IC(N)). The program takes advantage of the f a c t t h a t head l o s s e s w i t h i n the feeder are constant f o r each of the KN times (K=1,2,3...KN). For each of the feeder's i n d i v i d u a l time i n t e r v a l s 80 (l= 1 , 2 , 3 , . . .IC(N)), the program c a l c u l a t e s the minimum a l l o w a b l e head val u e a t the upstream end of the feeder, and then uses t h i s constant v a l u e , KN times. For every time i n t e r v a l (IT), where IT = (K-1) X IC(N) + I , K=1,2,3,...KN, the constant i s added to the c o n t r i b u t i o n by conveyance s e c t i o n s to get the head value a t the i n l e t . In order to determine the c o n t r i b u t i o n by feeder s e c t i o n s , the program takes the optimal pipe s i z e s o l u t i o n and the feeder flow p a t t e r n f o r the given time i n t e r v a l , and d i v i d e s the feeder up i n t o i n d i v i d u a l head l o s s c o n d i t i o n s , each w i t h i t s own i n d i v i d u a l pipe s i z e and fl o w r a t e . Define a l l separate Head Loss Conditions given the Optimal Pipe S i z e S o l u t i o n  and a p a r t i c u l a r Flow P a t t e r n Flow Conditions 3 2 1 \& Optimal Pipe S i z e 1 2 — ^ — S o l u t i o n Head Loss Conditions 1 2 3 4 5 FIG. 5.18 81 There are a number of important parameters used to separate each i n d i v i d u a l head l o s s c o n d i t i o n . The a n a l y s i s works downstream from the upstream end of the feeder. XI - Length of remaining downstream fl o w c o n d i t i o n s , i n c l u d i n g the present one, to be analyzed. XS - Length of the present f l o w c o n d i t i o n . DT - Length of remaining downstream pipe s i z e s , e x c l u d i n g the present one, to be analyzed. DB - Length of the present f l o w c o n d i t i o n already analyzed. Look a t these parameters f o r the f i r s t two head l o s s c o n d i t i o n s . Procedure f o r D e f i n i n g I n d i v i d u a l Head Loss Conditions i n Feeders C o n d i t i o n 1 FIG. 5 . 1 9 (a) YT YP: DT DB=0 1 2 r - ^ r -present flow c o n d i t i o n 3 present pipe s i z e 1 (XI-DT) < XS - In t h i s case, the program c a l c u l a t e s head l o s s f o r the present head l o s s c o n d i t i o n , then r e d e f i n e s the present pipe s i z e , and updates values f o r DB and DT. Con d i t i o n 2 FIG. 3 . 1 9 (b) 1 X l «_ XS , DB . T DT .  1 2 — 3 — present f l o w c o n d i t i o n 3 present pipe s i z e 2 (XI-DT) > XS - In t h i s case, the program c a l c u l a t e s head l o s s f o r the present head l o s s c o n d i t i o n , then r e d e f i n e s the present f l o w c o n d i t i o n , and updates values f o r X I , DB, and XS. 82 3. 4.15 Calculate Pump Operating Cost This section i s divided into two parts, one f o r a variable-speed power u n i t and the other f o r a constant-speed power u n i t . In the variable-speed power u n i t case, the pumpflow r a t e and minimum required pump discharge head f o r each i n d i v i d u a l time i n t e r v a l , are used i n the c a l c u l a t i o n of the pump operating cost. In the constant-speed power un i t case, the pump discharge head i s assumed constant (HT(LB)-DAT), f o r a l l values of pump flow r a t e . Since the pump discharge head i s f i x e d , any excess head must be consumed by va l v i n g , and no power saving i s afforded. Since, i n both the variable-speed and constant-speed case, the flow rate f o r an i n d i v i d u a l time i n t e r v a l (IT) i s s i g n i f i c a n t , the time i n t e r v a l ' s duration (PQR(lT)), i s also s i g n i f i c a n t . Chapter 4 MODEL EXAMPLE 4.1 Introduction A specific sprinkler system example is used to demonstrate the operation of the model. Besides recording the required input data, the example also shows how this input data relates to what is physically happening in the system. Results from the various methods used to eliminate redundant head loss • equations are also recorded. A layout plan of the sprinkler system defines a l l the structural components of the system. It includes a l l feeders, conveyance feeders, and conveyance sections and nodes. 4.2 Explanation of Feeder Input Data Derivation For each of the three main feeders (ie. Feeder Number 1, 2, and 3) a sketch showing lateral movements and a hydraulic grade line diagram are included to help i l l u s t r a t e what is actually taking place within the feeder. The sketch shows the position of each lateral during every time interval. Discharging outlets are defined by these lateral positions. The hydraulic grade line diagram is simply an aid to visualize and compare head losses within the feeder for different time intervals. The slopes of the hydraulic grade lines are proportional to flow but don't pertain to any particular pipe size. In the diagram, the hydraulic grade lines converge at the conveyance node immediately upstream of the feeder. In practise this i s not necessarily the case. They are plotted in such a manner simply to help point out the differences in various lines. For each feeder a table i s included to define discharging outlets and discharge upstream of the outlets, for every time interval. This table also defines redundant head loss equations and explains the reason for their 84 redundancy. Explanations f o r parameters and symbols used i n the t a b l e are given below. IT - time i n t e r v a l number f o r the e n t i r e system, IT=1,....ITLT I - time i n t e r v a l number f o r the i n d i v i d u a l feeder, 1=1,....IC(N). ISS(lT,N) - value of the Feeder Head Loss Equation Index (page 67 ), f o r time i n t e r v a l (IT) and feeder (N). Time i n t e r v a l ( i ) f o r an i n d i v i d u a l feeder i s repeated KN times, KN=ITLT/IC(N), f o r every complete time c y c l e , IT=1, ITLT. Employing a l l KN time i n t e r v a l s , where the feeder time i n t e r v a l ( i ) i s a constant, the program c a r r i e s out the Head Loss Redundancy Method f o r conveyance s e c t i o n s upstream of the feeder. I f the flows i n one time i n t e r v a l (11) i s e x c l u s i v e l y l e s s than or equal to the flows i n a second time i n t e r v a l , f o r a l l upstream conveyance s e c t i o n s , then ISS(l1,N) i s s e t equal to zero. This means that every head l o s s equation o r i g i n a t i n g from the feeder during time i n t e r v a l ( i l ) i s redundant. Flows i n conveyance s e c t i o n s , f o r every time i n t e r v a l are shown on page 97 . The values of ISS(lT,N) may become c l e a r e r upon examination of these conveyance s e c t i o n f l o w s . J - o u t l e t rank number. O u t l e t s are ranked according to i n c r e a s i n g o u t l e t number ( i e . moving upstream). For each o u t l e t rank number ( j ) and time i n t e r v a l ( I T ) , the t a b l e provides three pieces of i n f o r m a t i o n : _1_ the d i s c h a r g i n g o u t l e t number 2 the discharge immediately upstream of the o u t l e t ( l / s e c ) 2, a redundancy symbol. This symbol i n d i c a t e s whether or not the head l o s s equation o r i g i n a t i n g from the d i s c h a r g i n g o u t l e t i s redundant, and the cause of redundancy. Define Values and Redundancy Symbols i n the Discharge O u t l e t Table Disc h a r g i n g Redundancy Out l e t Number Symbol Discharge Upstream of O u t l e t ( l / s e c ) Redundancy Symbols Zero Flow Redundancy Head Loss Redundancy C o n t r o l P o i n t E l e v a t i o n Redundancy Feeder Head Loss Equation Index Redundancy S i g n i f i c a n t Head Loss Equation FIG. 4.1 86 4.3 Explanation of Conveyance System Input Data D e r i v a t i o n The conveyance system i s d i v i d e d i n t o two l i n e s : l i n e 1 - POMP, © , a n d © node numbers l i n e 2 - PUMP, © , (Q), © , and(D^^"^ A h y d r a u l i c grade l i n e diagram and a t a b l e d e f i n i n g flows i n conveyance s e c t i o n s are presented f o r each of the two conveyance l i n e s . The h y d r a u l i c grade l i n e diagram i s again used to help v i s u a l i z e and compare head l o s s e s during jeach time p e r i o d . The t a b l e d e f i n i n g conveyance s e c t i o n f l o w , i s a l s o used to define redundant head l o s s equations and the cause of redundancy.. The redundancy symbols are i d e n t i c a l to those used f o r feeders. 4.4 Explanation of Computer P r i n t o u t Values p r i n t e d i n the computer output a l l have headings which i n most cases are s e l f explanatory. The only values that may r e q u i r e f u r t h e r explan-a t i o n are the ;'pnes a s s o c i a t e d w i t h the headings "NO. OF ITERATIONS=" and "NO. OF VARIABLES=". These two values are i n c l u d e d to show whether or not the V a r i a b l e Reduction Algorithms are working i n a d e s i r a b l e manner. The NO. OF ITERATIONS i n d i c a t e s the number of times the l i n e a r programming a n a l y s i s i s repeated f o r a given i n l e t head valu e . I f the HO; OF ITERATI0NS=1, x no v a r i a b l e r e d u c t i o n problem has occured.„ The NO. OF VARIABLES i s simply the number of v a r i a b l e s s t i l l remaining i n the l i n e a r programming a n a l y s i s . 4.5 Pump and Pipe S i z e Input Data D e r i v a t i o n 4.5.1 Raw Data Pipe and pump costs were obtained from Vince Helton of P a c i f i c I r r i g a t i o n L t d . on February 25, 1980. 87 M a i n l i n e Pipe Cost Cost per 100 f e e t (based on 40' lengths) P l a i n Aluminum w i t h couplings Pipe Outside S i z e Diameter Cost ($/ioo f t ) 1 8 " 452. 2 7" 364. 3 6" 267. 4 5" 206. 5 4" 154. 6 3" 116. Table 4.1 Vari a b l e - s p e ed power u n i t s were not even I r r i g a t i o n Pump and Motor P r i c e s Constant-speed power u n i t s E l e c t r i c 3 Phase Horsepower Cost($) 25 1500. 30 1750. 40 2100. 50 2600. 60 3200. Table 4.2 system s i z e , t h e i r cost i s i n h i b i t i v e ( i e . —$12000.). An estimate of op e r a t i n g pump cost was obtained from B.C. Hydro, I n d u s t r i a l Use Department on February 26, 1980. During the i r r i g a t i o n season the cost of power f o r i r r i g a t i o n i s 1.6 0 / k i l o w a t t - h r . A f a i r amount of work i s r e q u i r e d to change t h i s raw data i n t o a form that can be used by the model. 4.5.2 M a i n l i n e Pipe Cost and Head Loss C o e f f i c i e n t A l l f i x e d costs are equated to t h e i r annual cost e q u i v a l e n t s . This annual cost depends upon the l i f e expectancy and compound i n t e r e s t r a t e . The C a p i t a l Recovery F a c t o r , combines l i f e expectancy and i n t e r e s t r a t e i n t o one number. To determine the equipment costs per year, simply m u l t i p l y the i n i t i a l cost of equipment by the C a p i t a l Recovery F a c t o r . The pipe cost i s f i r s t converted to a cost per 100 m l e n g t h f i g u r e and then to i t s annual cost f i g u r e . Annual Cost of Pipe L i f e expectancy of pipe = 15 years I n t e r e s t r a t e = 10 % C a p i t a l Recovery Rate = 0.1315 (From Rain B i r d S p r i n k l e r I r r i g a t i o n Handbook, page 14.) Pipe S i z e Outside Diameter Cost ($/l00ft) Cost (VOOm) Annual Cost($) 1 8" 452. 1483. 195.0 2 7" 364. 1194. 157.0 5 6" 267. 876. 115.2 4 5" 206. 676. 88.9 5 4" 154. 505. 66.4 6 3" 116. 381. 50.1 Table 4 .5 On page 39 of the Rain B i r d S p r i n k l e r I r r i g a t i o n Handbook there i s a t a b l e that shows Head Loss i n f e e t per 100 f e e t of Aluminum Pipe w i t h Coupler f o r v a r i o u s f l o w r a t e s and pipe s i z e s . The head l o s s equation used i n the model i s of the form: Hm = Km x Q,n where Hm - head l o s s per 100meter l e n g t h , i n meters Km - head l o s s c o e f f i c i e n t per 100meter l e n g t h , i n meters Q - flow r a t e , i n l / s e c n - exponential From the t a b l e i t i s p o s s i b l e to estimate values f o r the two unknowns, Km and n. Km i s dependant on pipe s i z e but n i s constant f o r a l l pipe s i z e s . A value f o r n i s estimated/ i n the f o l l o w i n g manner. n v Hm — — Km = a n Hm = Km Q d ct Hi = Km n Km = H m b n i t f o l l o w s that Hm, H: ; mb Hm a = n H: n 89 /Hm \ . /Q \ log(Hm /Hm. ) or l o g I al = n x l o g f_a _ n = o v a' V Using the t a b l e and the above equation, i t i s found that n — 1 . 8 9 . With t h i s v a l u e f o r n, i t i s a simple matter to determine the Km value f o r each pipe s i z e . Km = Hm - flow values need to be converted from U.S. gal/minute to l / s e c before computation of Km. - Hm values are dimensionless, so no conversion from ( f t / l O O f t ) to (m/lOOm) i s r e q u i r e d . Head l o s s C o e f f i c i e n t s Pipe Outside Head Loss S i z e Diameter C o e f f i c i e n t 1 8 " 0 . 0 0 0 7 6 2 7 " 0 . 0 0 1 4 4 3 6 " 0 . 0 0 3 0 8 4 5 " 0 . 0 0 7 5 3 5 4 " 0 . 0 2 3 2 6 3 " 0 . 0 9 8 0 Table 4 . 4 4 . 5 . 3 F i x e d Pump Cost The pump and motor costs are given as a f u n c t i o n of horsepower. This must be converted to values of t o t a l head imparted by the pump. Using the S i m p l i f i e d Pump Operating Cost Formulation i l l u s t r a t e d i n Fi g u r e 3 * 9 (page 5 9 ) » the maximum r e q u i r e d power w i l l occur a t the maximum flow r a t e . maximum brake horsepower, i n horsepower maximum pump flow r a t e , i n l / s e c ( 4 8 l / s e c ) v e r t i c a l l i f t , i n meters e f f i c i e n c y = 0 . 7 0 HPmax = Qmax x h HPmax -7 6 e n „ c Qmax -h = 7 6 e x HPmax J c h Qmax = 1.11 x HPmax ° The equipment cost must be reduced to i t s equivalent annual co s t . The l i f e expectancy i s 20 y e a r s , the i n t e r e s t r a t e i s 10 %, and the C a p i t a l Recovery Rate i s 0.1175. 90 Annual F i x e d Pump Cost Horsepower Head F i x e d Annual (m) Cost($) Cost($) 25. 28. 1500. 176.3 30. 33. 1750. 205.6 40. 44. 2100. 246.8 50. 56. 2600. 305.5 60. 67. 3200. 376.0 Table 4.5 4.5.4 Operating Pump Cost An annual pump ope r a t i n g cost (0PC) can be der i v e d from the known power co s t , and u s i n g the S i m p l i f i e d Pump Operating Cost Formulation i l l u s t r a t e d i n F i g u r e 3.9 (page 59) . HP - power, i n horsepower 0PC = $ 0 . 0 l 6/(kilowatt-hr) x T x KP KP - power, i n k i l o w a t t s T - ope r a t i n g time per year, i n hrs (2000 hr) KP = 0.7457 k i l o w a t t x HP = 0.7457 x /Qavg x ti horsepower |76 e c Qavg- average flow r a t e , f o r t o t a l number of time i n t e r v a l s , i n l / s e c . = 0.449 x Qavg x h 0PC = $0,016 x 2000 x 0.7457 Qavg x h 76 x 0.7 APC = OPC = $0.449/(l/sec)/m input parameter Qavg x h 91 4 . 6 Sprinkler;. I r r i g a t i o n System Layout, and Summary of General Input Data  and Pump Input Data,, S p r i n k l e r I r r i g a t i o n System Layout Puffix FIG. 4 . 2 Feeder O u t l e t s CP e ^ ® 0 ° ® o^ © © Feeder O u t l e t s SCALE Om 200m 400m 92 General Input Data NSECT = 13 IPIPE = 6 NF = 5 NFT = 11 ITLT = 0 EXP = 1.89 EMIN = 0.0 m EMAX = 60.0 m TINT = 5.0 m ISTAGE = 9 MXIN = 3 Pipe Costs and Head Loss C o e f f i c i e n t s Pipe Head Loss Coef. Cost (m/l00m) ($/l00m) 1 0.00076 195.0 2 0.00144 157.0 3 0.00308 115.2 4 0.00753 88.9 5 0.0232 66.4 6 0.0980 50.1 Table 4.7 Pump Input Data Conveyance S e c t i o n M a t r i x upstream - U/S downstream - D/S tage U/S Node D/S Node D/S Node 1 3 2 1 2 4 3 0 3 5 4 0 4 6 5 0 5 9 8 7 6 10 9 0 7 11 10 0 8 12 11 6 9 0 13 12 Table 4.6 F i x e d Pump Cost Function No. Head(m) Cost(|) 1 28 .0 176.3 2 28.0 205.6 3 33.0 205.6 4 33.0 246.8 5 44.0 246.8 6 44.0 305.5 7 56.0 305.5 8 56.0 376.0 9 67.O 376.0 Table 4.8 APC = $0.449/(l/sec)/m INCPC = 9 IVHP = 0 DAT = 0.0m 93 4.7 Feeder Input Data D e r i v a t i o n  Feeder Number (ij) L a t e r a l Movements Diagram  L a t e r a l 1 (4 l / s e c ) l e n g t h of a l l feeder sections=90m L a t e r a l , 2 (4 l / s e c ) during FIG. 4.3 (b) co 7 CM CM CO CM CM CM CO CM CO CM 00 CM 00 CM CO CM CO CM c— CM General Information N = 1 N0DE(1) = © IC(1) = 8 NC(1) = 3 ICP(1) = 12 SM(1) = 0.9(l00m) QM(1) = 4 . 0(l/sec) Disc h a r g i n g O u t l e t s . Feeder Number (ij) Discharging O u t l e t Redundancy Symbol Discharge upstream of O u t l e t ( l / s e c ) Table 4.9 8 s i g n i f i c a n t equations Explanations f o r values and symbols are given on page85« Feeder Number (1 L a t e r a l Movements Diagram FIG. 4.4 (a) Lateral»1 (4 l / s e c ) FIG. 4.4 (b) CM General Information N = 2 N0LE(2) - © IC(2) = 4 NC(2) = 3 ICP(2) = 8 SM(2) = 0.0(100m) QM(2) = 0 . 0(l/sec) L a t e r a l £ (4 l / s e c ) h l / s e c ) vo CM CM CM CM MD CM LT\ CM CM D i s c h a r g i n g O u t l e t s , Feeder Number ( l ) IT 1 2 3 4 5 6 7 8 I 1 2 3 4 1 2 3 4 \ISS 1 1 1 1 0 0 0 0 1 • • 5.0 • oO 4.0I 0.0 1 0 2+-5.0 2 — 4.0 0 + 0.0 2 9*0 3 ^ 9.0 3 ™ 9.0 4 ^ 5.0 4 < ?. 0 5 9.0 3 + 9.0 4 5.0 3 13.0 6 ^ 13.0 5 ^ 8 9.0I 1 5. 0 7 13.0 6 13.0 5 9.0 Table 4.10 6 s i g n i f i c a n t equations Explanations f o r f i g u r e s given on t h i s page are shown on page 93. Feeder Number (2 FIG. 4.5 (a) FIG 4.5 (b) L a t e r a l Movements Diagram L a t e r a l 1 (4 l / s e c ) l e n g t h of a l l feeder ''sections = 90 m C\J MD OJ K CM H y d r a u l i c Grade l i n e DLagran M D CM 3 L a t e r a l 2 ($ l/ s e c ) co CM >vO K> "^t" CO MD t — General Information? ^ ^ vo* ^ ^ CM CM CM CM CM CM MD MD MD MD CM CM CM N NODE ( 3 ) = © IC(3) = 8 NC(3) = 2 ICP(3) = 9 SM(3) = 0.9(l00m) QM(3) = 0 . 0(l/sec) Di s c h a r g i n g Outlets, Feeder Number (g) Table 4.11 IT 1 2 3 4 5 6 7 8 I 1 2 3 4 5 6 7 8 \rss * \ 1 1 1 1 4 I 5.0 1 1 5.0 1 1 1 i 1 1 • 5.0 • 5.0 • 5.0 4 * 0 ^ 4 ^ 0.0 0 S > 0.0 1 7 2 9 \ 9.0 8 \ 9.0 7 ^ 9.0 6 / 9.0 5 \ 9.0 9 ! ^ 5 4.0 2 Z 4.0 7 s i g n i f i c a n t equations Explanations f o r f i g u r e s given on t h i s page are shown on page 93 . 96 Feeder Number (g) General Information N = 4 N O D E U ) - ® I C (4 ) = 2 NC (4) = 1 ICP(4) = 1 SM(4) = O.O(lOOm) «M(4) = 0.0(l/sec) CP(1 ,4 ) = 28.0(m) Discha r g i n g O u t l e t s , Feeder Number (&) IT 1 2 3 4 5 6 7 8 I 1 2 1 2 1 2 1 2 \ISS 1 1 0 0 . 0 0 0 0 1 • 4.0l 0.0 1 4,0 0 0.0 1 4.0 0 + 0.0 1 4.0-0 0.0 Table 4.12 1 s i g n i f i c a n t equation Feeder Number Cf) General Information N = 5 NODE ( 5 ) -® IC(5) = 2 NC(5) = 1 ICP (5 ) = 1 SM(5) = O.O(lOOm) QM(5) = 0 . 0(l/sec) CP(1,5) = 28.3(m) Discha r g i n g O u t l e t s . Feeder Number Cf) IT 1 2 3 4 5 6 7 8 I 1 2 1 2 1 2 1 2 \ i s s 1 1 0 0 0 0 0 0 1 0 ® 0.0 1 ^ 0 + 4-01 0.0 1 4.0 0 0.0 1 + 4.0 0 0.0 1 + 4.0 Table 4.15 1 s i g n i f i c a n t equation Explanations f o r f i g u r e s given on t h i s page are shown on page 93. 97 4.8 Conveyance System Input Data D e r i v a t i o n  Conveyance L i n e 1 © © © © © H y d r a u l i c Grade 1 ine Dis gram Explanations f o r f i g u r e s given on t h i s page are shown on page 93. FIG. 4.6 Conveyance S e c t i o n Discharge. Conveyance L i n e 1 Se c t i o n - S 1 2 3 • 0 4 5 6 7 1 J 8 7 © • 22.0 22. / 0 22. 18 .0 22. / .0 22. 0 / 17.0 • 1 — 13. / 0 © 24.0 22 0 .0 22. 0 0 22. / 0 24 / .0 0 22.0 Is 17.0 V-17.0 © 0 24.0 24 / .0 • 26.0 22, 0 .0 0 24.0 24. / 0 21. / 0 0 17.0 © 0 9A.CI 28 • 0 28 / 0 22 0 24 0 .0 28, / 0 23, / 0 17 0 © a 36 .o| 36.0 0 36.0 32 / .0 iz 36.0 36 0 .0 31 0 .0 27 / .0 sc-145 '2&6 Table 4.14 6 s i g n i f i c a n t equations General Information f o r Conveyance Feeders :i..N = 6 N =7 NODE(6) - © N0DE(7) = © IC(6) = 4 IC(7) - 4 QM(6) = 0.0(l/sec) QM(7) = 0.0(l/sec) N = 8 N =9 N0DE(8) - © N0DE(9) = @ IC(8) = 4 IC(9) = 4 QM(8) = 0.0(l/sec) QM(9) = 0.0(l/sec) Discharge from Conveyance  Feeders. Conveyance L i n e 1 ( l / s e c ) Time I n t e r v a l Conveyance Feeder © © © © 1 2.0 0.0 0.0 4.0 2 0.0 2.0 4.0 0.0 3 0.0 4.0 2.0 0.0 4 4.0 0.0 0.0 2.0 Table 4.15 98 Conveyance L i n e 2 Pud FIG. 4.7 Explanations f o r f i g u r e s given on t h i s page are shown on page 93. © © © © H y d r a u l i c Grade L i n e Diagram 4 -1,3,&5 -2&6 Conveyance S e c t i o n Discharge. Conveyance L i n e 2 S e c t i o n - S \ I T s \ 1 2 3 4 5 6 7 8 ® / 4.0 4 . f l 4.< / 3 4.< / 3 4.( / 3 4.< / 4.( / 3 4. / 3 © 0 4.0 8 .0 4.C O ) 8.C / ) 4.C <s> ) 8.C / ) 4.C 0 ) 8.C / ) • 8 .0 S> 8 .0 8.C / ) 8.C 0 ) 8.C / ) 8.C 0 ) 8.C / ) 8.C 0 ) © 36.0 36.0 36.0 32.0 36.0 36.0 31.0 27.0 Table 4.16 3 s i g n i f i c a n t equations General Information f o r Conveyance Feeders N = 10 N0DE(10)J = ® IC(10) = 2 QM(10) = 0.0(l/sec) N = 11 N0DE(11) = © IC(11) = 2 QM(11) = 0.0(l/sec) Discharge from Conveyance Feeders.  Conveyance L i n e 2 ( l / s e c ) 0 Time I n t e r v a l Conveyance Feeder © © 1 0,0 4.0 2 4.0 0.0 Table 4.17 99 T o t a l Number of S i g n i f i c a n t  Head Loss Equations Minimum A l l o w a b l e H y d r a u l i c  E l e v a t i o n , a t Conveyance Nodes Feeder 1 2 3 4 5 Conveyance L i n e 1 2 T o t a l Equations 6 7 1 1 Equations 6 3 32 Stage Ups trearn Node E l e v a t i o n (m) 1 3 1.6 2 4 26.5 3 5 27.4 4 6 28.1 5 9 3.1 6 10 28 .4 7 11 28.2 8 12 28.8 Table 4.19 Table 4.18 Conveyance S e c t i o n Length S e c t i o n Length(m) 1 120.0 2 90.0 3 120.0 4 120.0 5 120.0 6 120.0 7 120.0 8 120.0 9 120.0 10 120.0 11 120.0 12 240.0 13 360.0 Table 4.20 100 4.9 Optimal Pipe S i z e Design I n l e t Head = 55m S o Pump 401.7m 490.4m 120m. © © <2)4-a o M D KN B o OJ SCALE 0m 200m 588.3m PIPE SIZES 398.0m 5 © (5) © © © o, J-f2 a ON . ON C— 400m 3 4 5 6 511.6m a o co a o 00 FIG. 4.8 101 4.10 D i s c u s s i o n of Optimal Pipe S i z e Design 5000" 4000-3000--p 8 2000-1000-0. 35 Cumula Cumulative ( i e . T o t a l System) C i v e Cost Function f o r Model Example pst Function 'timal £££fCost Fun °tion Operating Pump Cost F u n c t i o i i F i x e d Rimp Cost Function 60 40 45 50 55 FIG. 4.9 I n l e t Head(m) - F i x e d Pump Cost increases i n a stepped manner. W i t h i n the range of head values between each s t e p , one s p e c i f i c pump motor s i z e governs design, and F i x e d Pump Cost i s constant. - Operating Pump Cost ( i e . Power Cost) i s l i n e a r l y p r o p o r t i o n a l to head. - Optimal Pipe Cost forms a decreasing, concave up cost f u n c t i o n w i t h respect to head,, (page 28 ). 65 102 For t h i s type of pump data the minimum cumulative cost can only occur a t one of two head v a l u e s . J_ a t the head where the absolute slope of the optimal pipe cost f u n c t i o n equals t h a t of the op e r a t i n g pump cost f u n c t i o n . Define the Minimum Cumulative Cost of the System Con d i t i o n 1 -p m o o Cost Function Head where absblut cost slope equal pump cost slope e pipe s o p e r a t i n g Minimum Cost FIG. 4.10 (a) I n l e t Head 2 a t the head where a step occurs i n the f i x e d pump cost f u n c t i o n , and the absolute slope of the optimal pipe cost f u n c t i o n equals t h a t of the ope r a t i n g pump cost f u n c t i o n a t some head valu e before the next step. Head where s i n the f i x e d s lope equals slope Minimum Cost Head where absolute pipe cost o p e r a t i n g pump cost I n l e t Head 103 The t a b l e below shows that the optimal pipe cost does indeed form a decreasing, concave up cost f u n c t i o n . I t a l s o shows that the absolute slope of the optimal pipe cost f u n c t i o n equals that of the op e r a t i n g pump cost f u n c t i o n a t an i n l e t head of approximately 55 m. Slope of the Optimal Pipe Cost Function Head(m) Pipe Cost($) 65. 2732. 60. 2803. 55. 2896. 50. 3027. 45. 323L 40. 3509. 35. 3985. A P i p e ($) Cost Pipe Cost Slope ($/m) - 71 . -14.2 -93. -18.6 -131. -26.2 -204. -40.8 -278. -55.6 -476. -95.2 Table 4.21 Head APC = 0.449 / ( l / s e c ) / m Qavg = 45.75 l / s e c Slope of Operating Pump Cost Function = Operating Pump Cost = 0.449 x 45.75 H e a d =20.54 a/m F u r t h e r a n a l y s i s i s r e q u i r e d i n the 50 to 60 m head range, u s i n g a sma l l e r i t e r a t i v e head, to get a more accurate r e s u l t . C o n s i d e r i n g the general nature of the input cost v a l u e s , the closeness of t o t a l cost values i n the 50 to 60 m head range, and r e g u l a r shape of the optimal pipe cost f u n c t i o n , i t i s doubtful t h a t any f u r t h e r a n a l y s i s would be of any r e a l s i g n i f i c a n c e . 1 0 4 Chapter 5 CONCLUSIONS AND FUTURE DEVELOPMENTS 5.1 Conclusions I r r i g a t i o n designers are o f t e n f a c e d w i t h a d i f f i c u l t task. In an i r r i g a t i o n system, economic c o n s i d e r a t i o n s are r a r e l y the only design c r i t e r i a . Boundary and topographic c o n d i t i o n s , l a n d use c o n s i d e r a t i o n s , and labour e f f i c i e n c y may a l l i n f l u e n c e design. Every i r r i g a t i o n design i s unique. Considerations t h a t may be s i g n i f i c a n t f o r one, may not n e c e s s a r i l y be s i g n i f i c a n t f o r another. Because of the v a r i e d and i n t a n g i b l e nature of the problem i t i s not p r a c t i c a l to analyze a l l design c r i t e r i a together i n one general model. For the purpose of t h i s t h e s i s , i r r i g a t i o n design has been d i v i d e d i n t o two main p a r t s . In the f i r s t p a r t , a s e t of op e r a t i n g c o n d i t i o n s and the l o c a t i o n of a l l , ;mainlines and o u t l e t s are defined. The s e t of op e r a t i n g c o n d i t i o n s i n c l u d e s i n l e t head v a l u e ( s ) , discharge data t h a t defines f l o w patterns f o r the e n t i r e system f o r every time p e r i o d , and pumping i n f o r m a t i o n , i f a p p l i c a b l e . The data f o r m a i n l i n e and o u t l e t l o c a t i o n defines the s t r u c t u r e and s c a l e of the p i p e l i n e network and the r e q u i r e d h y d r a u l i c head values f o r every c o n t r o l p o i n t i n the network. In the second p a r t , pipe s i z e s i n the defi n e d m a i n l i n e system are optimized. The f i r s t p a r t may w e l l depend upon a v a r i e t y of i n t a n g i b l e design c r i t e r i a but the second p a r t , the o p t i m i z a t i o n of pipe s i z e s , i s only dependent upon economic c o n s i d e r a t i o n s . This t h e s i s has r e s t r i c t e d i t s e l f to t h i s second p a r t of the design process. A computer model has been developed which i s capable of o p t i m i z i n g pipe s i z e s f o r any open-ended s p r i n k l e r i r r i g a t i o n system. There are no i n h i b i t i n g r e s t r i c t i o n s p l a c e d on e i t h e r the p i p e l i n e network or the op e r a t i n g c o n d i t i o n s . The model has been programmed to handle any c o n d i t i o n of time v a r i a b l e f l o w . Time v a r i a b l e f l o w defines a system where there i s more than one time i n t e r v a l , and flo w patterns f o r d i f f e r e n t time i n t e r v a l s are not n e c e s s a r i l y the same. In some cases s i g n i f i c a n t c o n t r o l p o i n t s may be s i t u a t e d i n a l o c a t i o n other than an o u t l e t or branch i n t e r s e c t i o n . The model can i n c l u d e any number of these e x t r a c o n t r o l p o i n t s . The model can be used f o r g r a v i t y systems or f o r pumping systems w i t h e i t h e r constant-speed or v a r i a b l e - s p e e d power u n i t s . I f the pump f l o w r a t e and o p e r a t i n g head are f a i r l y constant f o r a l l time i n t e r v a l s , then a constant-speed power u n i t i s u s u a l l y the most economic, but i f the pump flow r a t e and op e r a t i n g head, change r a d i c a l l y from one time p e r i o d isto another, a v a r i a b l e - s p e e d power u n i t may become more economic. A constant-speed power u n i t has a lower f i x e d ( i e . c a p i t a l ) c o s t , but a v a r i a b l e - s p e e d power u n i t , because i t can vary i t s o p e r a t i n g head f o r a given flow r a t e , has a lower o p e r a t i n g c o s t . The i n n o v a t i o n o f t h i s model w i l l g r e a t l y improve the o v e r a l l design process. The designer can ob t a i n the optimal system cost f o r a number of p o t e n t i a l m a i n l i n e systems. This optimal system cost defines the economic design c r i t e r i o n f o r the given m a i n l i n e system. Using these economic c r i t e r i a t o gether"with the l e s s t a n g i b l e design c r i t e r i a , the designer can r a t i o n a l l y compare the v a r i o u s a l t e r n a t i v e s , and determine an optimal s o l u t i o n . The designer s t i l l r e q u i r e s a f a i r amount of sound judgement i n order to weigh the i n t a n g i b l e s a g a i n s t the economic c r i t e r i a , but at l e a s t now, system cost f i g u r e s are e a s i l y o b t ainable and completely r e l i a b l e . In the past, pipe o p t i m i z a t i o n problems f o r a gi v e n m a i n l i n e system were f r e q u e n t l y s o l v e d by examining a s e r i e s of t r i a l designs and choosing the one w i t h minimum cos t . The model f r e e s the designer from these tedious hand c a l c u l a t i o n s and provides him w i t h the absolute minimum cost s o l u t i o n ; a g l o b a l optimum. The time s a v i n g a f f o r d e d by the model allows the designer to concentrate on endeavors more s u i t e d to h i s e x p e r t i s e , such as e v a l u a t i n g more p o t e n t i a l m a i n l i n e systems or examining the i n f l u e n c e of i n t a n g i b l e design c r i t e r i a more c l o s e l y . 106 The model provides the designer w i t h a powerful t o o l ; a s o l i d base from which the e n t i r e design process can develop. 5.2 Future Developments The f o l l o w i n g s e c t i o n describes a number of improvements that can be i n t r o d u c e d to make the model more e f f i c i e n t and u s e f u l to the designer. For a given pumping system, i t may be p o s s i b l e that the t o t a l pumping cost, i n c l u d i n g both f i x e d and o p e r a t i n g c o s t , i s approximately l i n e a r l y p r o p o r t i o n a l to the i n l e t head. In t h i s p a r t i c u l a r i nstance the i n l e t head can be i n c l u d e d as a d e c i s i o n v a r i a b l e i n a l i n e a r programming f o r m u l a t i o n . This allows the pumping cost to be i n c l u d e d i n the o b j e c t i v e f u n c t i o n . This procedure i s much more e f f i c i e n t than the one p r e s e n t l y used i n the model, which assumes a n o n - l i n e a r r e l a t i o n s h i p between i n l e t head and t o t a l pumping c o s t . I t would be advantageous to modify the present model to accommodate t h i s s p e c i a l case. P r e s e n t l y the C o n t r o l P o i n t E l e v a t i o n Redundancy Method i s only a p p l i e d to conveyance networks w i t h no conveyance feeders and to feeders i n i s o l a t i o n from the r e s t of the system. I t would be b e n e f i c i a l to extend t h i s redundancy method to i n c l u d e conveyance networks w i t h conveyance f e e d e r s , and feeders i n combination w i t h t h e i r upstream conveyance networks. The f o l l o w i n g recommendation i n v o l v e s the i n i t i a l boundary c o n d i t i o n s used i n the V a r i a b l e Reduction Algorithms. The present model assumes th a t n o t h i n g i s known about the optimal pipe s i z e s i n the network f o r the given range of i n l e t head v a l u e s . I f the designer does have some p r i o r knowledge about the expected optimal pipe s i z e s , the model should be a l t e r e d to r e c e i v e such i n f o r m a t i o n and use i t to redu&e the number of v a r i a b l e s e n t e r i n g the l i n e a r programming a n a l y s i s . The model a u t o m a t i c a l l y i n c l u d e s a l l d i s c h a r g i n g o u t l e t s and branch i n t e r s e c t i o n s as p o t e n t i a l c o n t r o l p o i n t s , but unless e x p l i c i t l y d e f i n e d i n the input data, other p o t e n t i a l c o n t r o l p o i n t s along the l o n g i t u d i n a l s e c t i o n 107 w i l l not be i n c l u d e d . Given the optimal pipe s i z e s o l u t i o n i t i s d e s i r a b l e to check that the h y d r a u l i c grade l i n e s have not dropped below the e l e v a t i o n p r o f i l e a t any p o i n t along the e n t i r e l e n g t h of the network. A p l o t t i n g r o u t i n e could be introduded to draw the h y d r a u l i c grade l i n e s and the e l e v a t i o n p r o f i l e . Such a p l o t would a l s o be h e l p f u l i n understanding optimal pipe s i z i n g behavior. 108 BIBLIOGRAPHY A u s t i n , T. A l , "Di s c u s s i o n of a r t i c l e by Ronald P. P e r o l d , Economic Pipe S i z i n g f o r G r a v i t y S p r i n k l e r Systems," J o u r n a l of the I r r i g a t i o n and  Drainage D i v i s i o n , Proceeding of the ASCE, V o l . 101, No. LR2, June 1975, P. 115. B i r d , C., A L i n e a r Programming Package. UBC LIP, Computing Centre, U n i v e r s i t y of B r i t i s h Columbia, February 1977. B r i s b i n , P a t r i c k E., et a l . S p r i n k l e r I r r i g a t i o n Workshop Manual, V i c t o r i a , B. C., Department of A g r i c u l t u r e , 1975* F r y , A. W., et a l . S p r i n k l e r I r r i g a t i o n Handbook. Glendora, C a l i f o r n i a , Rain B i r d S p r i n k l e r Mfg. Corp., 1971. K e l l e r , Jack, " S e l e c t i o n iof Economical Pipe S i z e s f o r S p r i n k l e r I r r i g a t i o n Systems.""Transactions of the ASAE. V o l . 8, No. 2, 1965, p. 186. P e r o l d , Ronald P., "Economic Pipe S i z i n g f o r G r a v i t y S p r i n k l e r Systems," J o u r n a l of the I r r i g a t i o n and Drainage D i v i s i o n , Proceeding of the ASCE, V o l . 100, No. IR2, June 1974, p. 107. P e r o l d , Ronald P., "Economic Pipe S i z i n g i n Pumped I r r i g a t i o n Systems," J o u r n a l of the I r r i g a t i o n and Drainage D i v i s i o n . Proceeding of the  ASCE, V o l . 100, No. IR4, December 1974, p. 425. R u s s e l l , S. 0., ;Notesifrom course on Water Resource System A n a l y s i s , U n i v e r s i t y of B. C., January 1979. Stephenson, David, " P i p e l i n e System A n a l y s i s and Design,"in P i p e l i n e Design  f o r Water Engineers. Amsterdam, the Netherlands, E l s e v i e r S c i e n t i f i c Publishing ;}Co., 1976, p. 27. INDEX A L i s t i n g of the Pipe S i z e O p t i m i z a t i o n Model C GIVEN A CONVENTIONAL OPEN ENDED (TREE-LIKE) IRRIGATION SYSTEM, C THIS PROGRAM DETERMINES THE OPTIMAL PIPE SIZES FOR ALL THE C MAINLINES. THE PROGRAM CAN HANDLE THE FOLLOWING TYPES OF SYSTEMS: C 1/- SINGLE FEEDERIMAINLINE WITH MULTIPLE OUTLETS). C 2/- BRANCHED SYSTEM, WITH MORE THAN ONE FEEDER, BUT WITH NO C OUTLETS ALONG CONVEYANCE SECTIONS. C 3/- BRANCHED SYSTEM, WITH MORE THAN ONE FEEDER, AND OUTLETS C ALONG CONVEYANCE SECTIONS. C 4/- GRAVITY SYSTEM. C 5/- PUMP SYSTEM, WITH A CONSTANT-SPEED POWER UNIT. C 6/- PUMP SYSTEM, WITH A VARIABLE-SPEED POWER UNIT. C C MAIN - THE PURPOSE OF THE MAIN PROGRAM IS TO READ IN C PARAMETERS WHICH DEFINE THE PIPELINE SYSTEM, ARRANGE C THE DATA FOR USE IN A GENERAL LINEAR PROGRAMMING C ROUTINE!LIPSUB), AND PRINT OUT INFORMATION DESCRIBING C THE OPTIMAL SOLUTION, FOR DESIRED HEAD VALUES. C -SEE (UBC LIP!FEB,1977) AND PROGRAM WRITEUP FOR MORE C INFORMATION)• C C GENERAL INPUT DATA. C NSECT - NO. OF CONVEYANCE SECTIONS=NQ. OF NODES. C IPIPE - NO. OF PIPE SIZES. C NF - NO. OF FEEDERS. C NFT - NO. OF FEEDERS fINCLUDING CONVEYANCE)• C ITLT - TOTAL NO. OF TIME INTERVALS. C IF ( ITLT=0), ITS VALUE IS CALCULATED. C EXP - HEAD LOSS EXPONENTIAL IHL=HK*Q**£XP). C EMIN - MIN. DESIRED U/S HEAD!Ml. C IF !EMIN=0.0), MIN. ALLOWABLE U/S HEAD IS CALCULATED. C EMAX - MAX. DESIRED U/S HEADIM). C TINT - CHANGE IN HEAD!M)/ITERATION. C ISTAGE - NO. OF STAGES=NO. OF CONVEYANCE NODES. C (NODES WITH CONVEYANCE SECTIONS D/S). C MXIN - MAX. NO. OF U/S AND D/S CONVEYANCE SECTIONS AT A STAGE. C INI I,J) - THE CONVEYANCE SECTION MATRIX. C {DEFINES U/S AND D/S CONVEYANCE SECTIONS AT ST AG E( I) ) . C H K U i - HEAD LOSS COEF.CM/IOOM) FOR PIPE SIZE! I ) . C COST I I I - - COST($/100M) FOR PIPE S I Z E ! I ) . C C PUMP INPUT DATA C APC - ANNUAL PUMPING COST « $/IL/S EC)/CM)). C !APC=Q.O) INDICATES A GRAVITY SYSTEM. C IVHP - IIVHP.EQ.O) INDICATES A CONSTANT-SPEED POWER UNIT. C UVHP.NE.O) INDICATES A VARIABLE-SPEED POWER UNIT. C DAT - INTAKE WATER LEVEL(M). C POR(IT) - FRACTION OF ANNUAL PUMP USE, DURING TIME INTERVAL! I T ) . C INCPC - NO. OF POINTS TO DEFINE PUMP CAPITAL COST FUNCTION. C CPCU) - PUMP CAPITAL COST!$> AT HEADIM), HE(I) . C C FEEDER!N) AND CONVEYANCE FEEDER!N) INPUT DATA. C NODEIN) - NODE NO. ASSOCIATED WITH FEEDER OR CONVEYANCE FEEDER. C IC(N) - NO. OF SETTINGS/I RRI. CYCLE i NO. OF TIME INTERVALS). C NC(N) - NO. OF IRRIGATION CYCLES. C ICP(N) - NO. OF CONTROL POINTS {OUTLETS). C SMfN) - SECTION LENGTH; CHANGES TO TOTAL FEEDER L ENGTH! LOOM). C IF (SM!N) = 0.0), SECTION LENGTH IS NOT CONSTANT. C QM( N) - OUTLET DISCHARGECL/SEC). C IF (QM(N)=0.0), OUTLET DISCHARGE IS NOT CONSTANT. C CPII,N) - MIN. HYDRAULIC ELEVATION!M) FOR CONTROL PT.CII. C SDUrNi - LENGTHdOOM) OF EACH SECTION!!). -IU C I R U f l t N ) - DISCHARGING OUTLET DURING CYCLEJJI, AT SETTING! I ) . C Q!J,I,N)- DISCHARGE !L/SEC) FROM OUTLET!IRIJ »I»N) ) » CHANGES TO C TOTAL DISCHARGE UP STREAM OF THE OUTLET. C C CONVEYANCE SECTION INPUT DATA. C CPTM) - MIN. HYDRAULIC ELEVATIONIM) FOR CONVEYANCE NODE(I). C DISTII) - LENGTH! M) OF CONVEYANCE SECTION!!). C DOUBLE PRECISION TE, A l , A2, A3, A4, DABS COMMON TE! 301,301) ,AU1) , A2 ! 1 ) , A3U ) , A4C1), CP! 20,10 ) , COST! 6) , *HKI6),D!6, 10 ) tDS!6,301,NCHKH 300) , NCHKI 301) ,IN (20 ,4) , DI ST!301, *IC!20),NC(10),ICP!10),SMI10),Q{10,10,20),N0DE{20),IRC 10,10,10}, •HE d 0) , C PC I 10) * I S ! 1 0 , 50 ,10) , T B ( 3 01,3 01 ) , H T12 5) , C S T (2 5 ) , *DP1(6,10 ),DP2!6,10) ,DMC6,10),JG(3003 ,JPI 300},LBX,EXP,M,M1,NE, *I PI PE » LB, NSECT, NF, I STAGE, MX I N»CPT 120) , ITL»CPU 1 25}» APC, QTQT !5 0) , *ISS!50, 10), I S I ! 50,20) ,QT( 50,30) ,ELT! 50), ICR (50) , I RC! 50) » INR! 50 ), *DAT,IVHP,II!30)»IO!30)»NFT,SD!20»1Q)»QM!20),PORI50) C C GENERAL INPUT DATA. C READ<5,2)NSECT,IPIPE,NF,NFT,MXIN,ISTAGE,ITLT,EXP,EMIN,EMAX,TINT, *APC 2 F0RMAT!7I6,5F6.2) WRITE{6,4)NSECT,IPIPE,NF,NFT,EXP,EMIN,EMAX 4 FORM ATI * 1 • » * NO . OF CONVEYANCE SECTI0NS = ' ,I2,3X, *'N0. OF ORIGINAL PIPES=*,12,5X,'NO. OF FEEDERS=*,12/ *'TOTAL NO. OF FEEDERS {INCLUDING CONVEYANCE)=• ,12,15X , *'HEAD LOSS EXP0NENTIAL=',F4.2/ *»MIN. HEAD=» ,F5.1,MM)« ,14X,'MAX. HEAD= •, F5 .1, • !M ) • } READ!5,6H! I N d . J I , J=i,MXI N) , 1 = 1,1 STAGE) 6 FORMAT 12014) WRITE! 6, 8) 8 FORMAT! • «,'CONVEYANCE SECTION MATRIX'/ ** STAGE U/S NODE'»3!5X,* D/S NODE')) DO 10 I=1,ISTAGE 10 WRITE!6,12)I,IIN!I,J),J=1,MXIN) 12 F0RMAT!2X,I2,4!8X,I2,3X) ) IFIAPC.EQ.O.OGO TO 28 C C PUMP INPUT DATA. C READ(5,14}INCPC,IVHP,DAT 14 FQRMAT!2I4,F6.l) IFdVHP.NE.O)WRIT£(6,16) IFdVHP.EQ.0)WRITE<6,18) 16 FORMAT!'0','VARIABLE-SPEED PUMPING PLANT.') 18 FORMAT!»0','CONSTANT-SPEED PUMPING PLANT.*) WRITE(6,20)APC,DAT 20 FORMAT!' + » »33X,'ANNUAL PUMPING COST=$»,F6.4,•/(L/SEC)/IM)•, *4X,'INTAKE WATER LEVEL=»,F6.1,»IM).•) IF!ITLT.EQ.0)G0 TO 24 READ15,30)!POR!IT),IT=1,ITL T) WRITE!6,22)11T,PGR! IT I, IT =1,ITLI) 22 FORMAT!* ',•FRACTION OF ANNUAL PUMP USE, FOR EACH TIME INTERVAL.•/ *'TIME INTERVAL',3X,* FRACTION*/50!6X,12,10X,F6.4/)) 24 IF!INCPC.EQ.0)G0 TO 28 READ!5,30)! HE! I ),CPC! I ),I=1,I NCPC) WRITE!6,26)!HE!I),CPC!11,1=1,INCPC) 26 FORMAT!» »,'PUMP CAPITAL COST.*/'HEAD!M)',10X, •COST I $ ) » / 28 30 C C C *10(F6.2,12X,F6.2/)) READ(5*301 (HK(I),C0STU)»I=1, IPIPE) FORMAT(12F6.2) 112 ELIMINATE UNECONOMIC PIPE SIZES FROM THE ANALYSIS. NP=0 C1=<C0ST!1)-C0ST<2))/!HK!2)-HK(l)) IPMl=IPIPE-l DO 36 I=2,IPM1 IP=I-NP C2=!C0STIIP)-C0ST! IP*-1))/(HK( IP+l)-HK! IP J J IFK1.LE.C2) GO TO 32 C1=C2 GO TO 36 32 IPIPE=IPIPE-1 DO 34 IT=IP,IPIPE C0STUT)=C0STUT+1 ) 34 HK(IT)=HKlIT+1) Cl=!COST{ IP-D-COST! IP))/(HK( I P)-HK(IP-l)) NP=NP+1 36 CONTINUE WRITE(6,38)(I,HK{I),COST(I 3,1=1,I PIPE) 38 FORMAT!* 0* » 1 PIPE*,7X,'HEAD LOSS COEF.(M/100M) 1,7X, **COST($/IOOM)«/6UX,12,17X,F6.5,18X,F6.2/J) C C FEEDER INPUT DATA. C DO 86 N=It NF READC5,40)NODE(N),IC(N),NCCM!, TCPCN), SM! N),QM{N) 40 FORMAT1414,2F4.2) WRITE!6,42)N0DE!N)»ICCNI,NCIN)*SMIN) ,QM( N) 42 FORMAT!///* FEEDER NO.* , 12 / • NO. SETT INGS/I RR I. CYCLE-• ,12,4X, *»N0. IRRI. CYCLES=« ,12,4X, "SECTION LENGTH='»F4.2,'!1 QOM)*»4X» ••OUTLET FLOW RATE=•,F4.1,»CL/SEC)«) ICPN=ICPCN) ICN=ICCN ) NCN=NC!N) READ (5,30) (CP! I , N), I=1,ICPN) WRITEI6,44) 44 FORMAT{• • , » CONT RQL PT.•,7X,•ELEVAT ION IM)•) ICPM1=ICPN-1 DO 46 I=l,ICPN 46 WRITE!6,48)I,CP!I,N) 48 F0RMAT!4X,I2,16X,F6.2) IF!ICP(N).EQ.1)G0 TO 70 IF(SM!N).NE.O.O)GO TO 54 READ (5, 30)! SOU ,N) ,I=1,ICPM1) WRITE!6,50) 50 FORMAT!* SECTION*,11X,•LENGTH IiOOM)* ) DO 52 1=1, ICPM1 52 WRITEI6,48)I,SD!I,N) GO TO 58 54 DO 56 1=1,ICPM1 56 SD(I,N)=SM!N) 58 CONTINUE SMIN)=0.0 DO 60 I=1,ICPM1 60 SM(N) = SM(N)+SD! I,N) READ!5,6)(!IR!J,I,N),1=1,ICN),J=1,NCN) WRITE!6,62)!I,I=l,ICN) 1 1 3 62 FORMAT(' ' »18X,»DISCHARGING OUTLETS•/»TIME INTERVAL....', *1013X,I2,» /• ) ) DO 66 J=1,NCN WRITE!6,64)J,IIRIJ,I ,N ) , 1= 1, I CN) 64 FORMAT!' »,11X,12,4X,10!3X,12,IX)) 66 IF!J.EQ.1)WRITE!6,68) 68 F O R M A T ! ' + » , ' C Y C L E . ) * 70 IF!QM!N).NE.0.0)G0 TO 78 READ(5,30)<!Q!J,I,N),I=1,1CN),J=1,NCN) WRITE 16,72)(I,1 = 1,ICN) 72 FORMAT!• *,18X,'FLOW FROM DISCHARGING OUTLETS!L/SEC)•/ *'TIME INTERVAL.... 1,10!3X, 12, •/•)) DO 76 J=1,NCN WRITE!6,74)J,!Q!J,I,N),1=1,ICN) 74 FORMAT!' »,11X, I2,4X,10!1X,F5.1)) 76 IF! J.EQ.1)WRITE f6,68) GO TO 82 78 DO 80 1=1,ICN DO 80 J=1,NCN 80 Q!J,I,N)=QM(N) 82 CONTINUE IF!ICP!N).NE.1)G0 TO 86 DO 84 1=1,ICN IR!l,I,N) = l 84 IFIQ!1,I,N).EQ.O.O)IRU,I,N) = 0 86 CONTINUE IF!NF.EQ.NFT)GO TO 100 C C CONVEYANCE FEEDER INPUT DATA. C NFP i=NF+l DO 98 NV=NFP1,NFT READ!5,88)N0DE!NV),ICINV),QM!NV): 88 FORMAT!214,F4.2) WRITE!6,90)N0DEINV),IC!NV),QM<NV) 90 FORMAT!//'/'CONVEYANCE FEEDER N0.',I2/'N0. SETTINGS=' , 12, *3X,»OUTLET FLOW RAT E= », F4. I, • ( L/SEC ) • ) ICNV=IC!NV) IF!QM!NV).NE.0.0)G0 TO 94 READ!5,30)IQ!l,I,NV), 1=1,ICNV) WRITE!6,72)!I,1=1,ICNV) WRITE! 6, 92 ) !Q! 1,1, NV), 1=1, ICNV) 92 FORMAT ! * • ,17X,10! 1 X, F5 .1 ) ) GO TO 98 94 DO 96 1=1,ICNV 96 Q(1,I,NV)=QM!NV) 98 CONTINUE 100 CONTINUE C C DETERMINE THE TOTAL NO. OF TIME INTERVALSU TL) . C IF! ITLT.EQ.O)GO TO 102 ITL=ITLT GO TO 114 102 ITL=IC(1) IF1NFT.EQ.1)G0 TO 114 DO 112 N=2,NFT IF!ITL/ICCN)*IC!N).EQ.ITLIGO TO 112 J=IC!N) 104 IF( J/INT*INT.NE. J)G0 TO 110 ITL=INT*ITL IF( ITL.LE.50IG0 TO 108 WRITE16.106) 106 FORMAT('0* , •NO. OF TIME INTERVALS IS GREATER THAN 50. •) GO TO 404 108 IF!ITL/IC!N)*IC!N).EQ.ITL)GO TO 112 J=J/INT IF{ J.EQ.l )G0 TO 112 GO TO 104 110 INT=INT+1 GO TO 104 112 CONTINUE 114 WRITE! 6, 116.) I TL 116 FORMAT!*-','TOTAL NO. OF TIME INTERVALS=»,I 2) C C CONVEYANCE SECTION DISCHARGE COMPUTATIONS AND INPUT DATA. C QT11T»I)- DISCHARGE IN CONVEYANCE SECTION! I ) , DURING TIME C INTERVAL!IT). DO 124 N=1,NF ICN=IC!N) NCN=NCCN) C CHANGE QIJ,I,NI TO THE TOTAL FLOW UPSTREAM OF OUTLET!IR!J,I,N)). IF!NCN.EQ.l)G0 TO 122 NCM1=NCN-1 DO 118 1=1,ICN DO 118 J=l,NCMl J1=J+1 DO 118 K=J1,NCN IF!IR!K,I,N).GE.IRiJ,I,N))GO TO 118 IRS=IR1J,I,N) IR!J,I,N)=IR(K,I,N) IR!K,I,N)=IRS QS=Q!J,I,N) Q1J,I,N)=Q!K,I,N) Q!K,I,N)=QS 118 CONTINUE DO 120 1=1,ICN DO 120 J=2,NCN 120 Q!J,I,N)=Q!J,I,N)+QCJ-1,I,N) 122 CONTINUE 124 CONTINUE C COMPUTE CONVEYANCE SECTION DISCHARGE. DO 126 N=1,NF DO 126 IT=1,ITL KB=(IT-1)/IC!N)+l IB=IT-IC!N)*(KB~1) 126 QT!IT,NODE(N))=Q!NC(N),IB,N) ISMl=ISTAGE-i IF! ISMl.EQ.O)GO TO 146 READ! 5 , 30 ) ICPT! I ), I = 1, ISM1) DO 128 IT=l,ITL DO 128 I=l,ISMl 128 OT!IT,INII,1))=0.0 IF!NF.EQ.NFT)GO TO 132 DO 130 NV=NFP1,NFT DO 130 I T - l t I T L KB =! I T-1) /1C1NV) -»• 1 IB=IT-ICINV)*1KB-1) 130 QT(IT,N0DE<NV))=Q(1,IB,NV) 1 1 5 132 CONTINUE DO 134 IT=1,ITL 00 134 1=1,ISM1 DO 134 L=2,MXIN IFIINU,L).EQ.O)GO TO 134 QT1IT*IN!I»1) )=QT1 IT,IN( 1*1) )+QT11T»INlI *L) ) 134 CONTINUE WRITE(6,138)11,1=1,I SMI) WRITEC6,1401(INI 1,1),1 = 1,1 SMI) WRITEI6,142) DO 136 IT=1,ITL 136 WRITE!6*144) IT* CQT1IT*IN1I*1))»I=1»ISM1) 138 FORMAT!*DISCHARGE IN CONVEYANCE SECT IONS(L/SEC).'/ ••STAGE.... ',20112,4X)) 140 FORMAT!•NODE ',20!12,4X1) 142 FORMAT!* TIME INTERVAL') 144 FORMATI5X,I2*6X»201F4.0*2XI) 146 IF!APC.EQ.O.O)GO TO 150 C TOTAL PUMP FLOW!QTOT!IT)), DURING TIME INTERVAL(IT). . DO 148 IT=1,ITL IFUTLT.EQ.0)PORCIT) = l.0/ITL QTOTCIT)=0.0 00 148 L=2,MXIN IF!IN!I STAGE,L).EQ.0)G0 TO 148 QTOT!IT3=QT0T!IT)+QTIIT*INlI STAGE»L)) 148 CONTINUE 150 READ!5,30)(DIST!I),1=1,NSECT) WRITE(6,152) 152 FORMAT !• 0* ,» CONVEYANCE SECTION LENGTHIM), AND ELEV.(M)* •/•SECTI0N»,8X,«LENGTH»,8X,»ELEV.») DO 162 1=1,NSECT IF(ISM1.EQ.0)G0 TO 158 DO 154 IT=1, ISMi 154 IF!I.EQ.IN(IT,l))WRITE!6,156)I,DISTCI),CPT1 IT) 156 F0RMATC3X,I2,12X,F5.0,8X,F6.2) 158 DO 160 N=1,NF 160 IF!I.EQ.NODEfN))WRITEC6,156)I,DIST(I),CP!ICP!N),N) 162 CONTINUE C C ELIMINATE REDUNDANT HEAD LOSS EQUATIONS1HLE). C ISII I T , I ) - CONVEYANCE CONTROL PT.CNODE), 1HLE) INDEX. C ISS! IT,NH FEEDER (HLE) INDEX; USED TO HELP DE TERMINE 11SIJ ,1 T, N) ) C ISIJ,IT,N)-FEEDER CONTROL PT.10UTLET), C HLE) INDEX• C C C C c c c c c c c c c c ISI! IT, I ) , ISSI IT,N), IS1J,IT,N)=1 - SIGNIFICANT EQN. I S K I T t l l t ISSIIT,N), ISCJ,IT,N) = 0 - REDUNDANT EQN. IT TIME INTERVAL. I CONVEYANCE CONTROL PT. INDICATOR. N FEEDER NO. J IRRIGATION CYCLE. 1ST - NO. OF REDUNDANT EQNS. IH NO. OF SIGNIFICANT EQNS. NNE NO. OF NON-EQUALITY CONSTRAINTS TOTAL NO. OF SIG.CHLE). NE NO. OF EQUALITY CONSTRAINTS=TOTAL NO. OF SECTIONS. M TOTAL NO. OF CONSTRAINTS. NB TOTAL NO. OF VARIABLES. NNE=0 NE=NSECT WRITEC6,164) H 6 164 FORMATC//'NO. OF SIGNIFICANT HEAD LOSS EQNS.CHLE);•/ *'NODE»,IX,'TOTALCHLE)*,IX,•NON-SIGC HLE)•,IX,'SIGC HLE)•I IFCISM1.EQ.OJGO TO 210 C CONVEYANCE CONTROL PT. C HLE) INDEXCISI(IT,I)). DO 166 I=1,1 SMI DO 166 IT=1,ITL 166 I S I f I T , I ) = i C ZERO FLOW REDUNDANCY. DO 206 1 = 1,I SMI IFCITL.EQ.1)G0 TO 202 IFfNF.EQ.NFT)G0 TO 176 DO 174 NV=NFPl,NFT IFCNODECNV).NE.IN{l,i))GO TO 174 ICNV=ICC NV) DO 168 IB=1,ICNV IFCQC1,IB,NV).NE.O.O)GO TO 170 168 CONTINUE GO TO 176 170 DO 172 IT=1,ITL K8=(IT-i)/ICCNV)*l IB=IT-ICCNV)*CKB-l) IFCQU,IB,NV).NE.O.O)GO TO 172 ISI(IT,I)=0 172 CONTINUE GO TO 176 174 CONTINUE 176 CONTINUE DO 178 IT=1,ITL IF{QTlIT,IN(I,l)).NE.0.0)GO TO 178 ISICIT,I)=0 178 CONTINUE C HEAD LOSS REDUNDANCY. ITLM1=ITL-1 DO 190 IT=i,ITLMl IFCISICIT,I).EQ.O)GO TO 190 IT1=IT+1 DO 188 IV=IT1,ITL IFCISICIV,I).EQ.O)GO TO 188 IP=I J1 = 0 J2=0 180 IFCQT{IT,INCIP,1)).LT.QT(IV,INCIP,1)))J1=1 IF1QT(IT,INC IP,1)).GT.QT{IV,IN(IP,1))) J2=l IFCJl.EQ.LAND.J2.EQ.DG0 TO 188 IP1=IP+1 DO 182 IW=IP 1,1 STAGE DO 182 L=2,MXIN IFCINMP,1).EQ. INCI W,L) ) GO TO 184 182 CONTINUE 184 IP=IW IFCI P.NE.I STAGE)GO TO 180 IFCJl.EQ.0)G0 TO 186 ISI CIT,I ) = 0 GO TO 190 186 ISItIV,I)=0 188 CONTINUE 190 CONTINUE IFCNF.NE.NFT)GO TO 200 C CONTROL PT. ELEVATION REDUNDANCY. IZ=I IP1=IZ+1 DO 194 IW= IP i t ISTAGE DO 194 L=2,MXIN IF!IN(IZtll.EQ.IN(IWtL))GO TO 196 CONTINUE IFCIW.EQ.ISTAGE)GO TO 200 IPl=IW+l IZ=IW IFCCPTCINCIW»1)) •GT.CPTCINI I»1 )) )G0 TO 192 DO 198 IT=1, ITL IFIQT!ITfIN(I,1) ).EQ.0.0IGO TO 198 I SI ! IT,IW)=0 CONTINUE GO TO 192 CONTINUE IST=0 DO 204 IT=1,ITL IFC I S I U T , I).EQ.0)IST=IST+1 IH=ITL-IST NNE=NNE+IH WRITE(6,280)INC 1,1),ITL,I ST,IH CONTINUE WRITEC6,208) FORMAT C• 1 ) DO 282 N=1,NF ICN=IC!N) NCN=NC<N) FEEDER CHLE) INDEX!ISS!IT,N)). DO 212 1=1,ITL ISS! I,N)=1 SINGLE STAGE REDUNDANCY. DO 214 IW=1,ISTAGE DO 214 L=2,MXIN IFCNODECNI.EQ.INCIWtL) )G0 TO 216 CONTINUE CONTINUE KN=ITL/ICN IFCKN.EQ.l)G0 TO 234 IF!IW.NE.ISTAGE)GO TO 220 DO 218 1=1,ICN DO 218 K=2,KN IT=CK-1)*ICN+I ISS!IT,N)=0 GO TO 234 KNM1=KN-1 HEAD LOSS REDUNDANCY. DO 232 1=1,ICN DO 232 K=1,KNM1 IT=CK-1)*ICN+I IFCISSCIT,N)iEQ.O)GO TO 232 K1=K+1 DO 230 KP=K1,KN IV=IKP-1)*ICN+I IF!ISSCIV,N).EQ.O)GO TO 230 IP=IW J1 = 0 J2=0 IFCQTCITtlNCIPti )).LT.QT!IVtINCIP,l)) ) J i = l IF! QT!IT, INCIP,1)).GT.QTCI V, INCIP,1)))J2 = l IFC JLEQ.LAND.J2.EQ.DG0 TO 230 IP1=IP+1 DO 224 IZ=IPitISTAGE DO 224 L=2,MX IN IFUNCIP,i).EQ.INCIZ,L))GO TO 226 224 CONTINUE 226 IP=IZ IF!IP.NE.ISTAGEIGO TO 222 IF!J1.EQ.O)GO TO 228 ISSCIT,N)=0 GO TO 232 228 ISSUV,N)=0 230 CONTINUE 232 CONTINUE 234 CONTINUE C FEEDER CONTROL PT. CHLE) INDEXUSC J , IT,N)). DO 236 IT=1,ITL DO 236 J=i,NCN 236 l S I J t I T t N ) = l C HEAD LOSS REDUNDANCY. DO 270 ND=1,NCN NNI=NCN-ND+1 DO 268 1=1,ICN DO 268 K=1,KN IT=IK-l)*ICN+I C FEEDER (HLE) INDEX REDUNDANCY, AND ZERO FLOW REDUNDANCY. IF!ISSCIT,N).NE.0.AND.IR CNNI,I,N).NE.0) GO TO 238 IS(NNItITtN)=0 238 IFCISCNNI,IT,N).EQ.O)GO TO 268 DO 266 NS=ltNNI MAX=0 IFd.EQ.ICN. AND.NS.EQ.l )G0 TO 264 NN=NNI-NS-H IB=I+1 IFCNS.EQ.DGO TO 240 IF(NS.NE.NNI)MAX=1 IB=l 240 DO 262 IG=IB,ICN IF(MAX.NE.O.AND.IR(NN,IGtN).SE.IR{NNI,I,N1)MAX=0 IFCIG.EQ.I)G0 TO 262 IFfIRCNNtIG,N).NE.IRCNNI,I,N)) GO TO 262 DO 260 KB=l,KN IV=CKB-1)*ICN+IG IFCISSCIVt N).E Q.O.OR.IS CNN11V t N).EQ.O)G0 TO 250 C COMPARE HEAD LOSSES IN CONVEYANCE SECTIONS. IY=0 IPY=0 IFCIW.EQ.ISTAGE)GO TO 248 IP=IW 242 IFCQTCITtINC IP,1)).GT.QTCIV,INCIP,1)))IY=l IFCQTCITtINCIP,i)).LT.QTCIV,INC IP,1)J)IPY=1 IFCIY.EQ. LAND. IPY.EQ.DGO TO 260 IP1=IP+1 DO 244 IZ=IPI,ISTAGE DO 244 L=2,MX1N IFCIN(IP,1).EQ.INCIZ,L))G0 TO 246 244 CONTINUE 246 IP=IZ IFCIP.NE.ISTAGE)GO TO 242 C COMPARE HEAD LOSSES IN FEEDER SECTIONS. 248 KI=NNI 1 1 9 KG=NN 250 IF (Q(KI,I,N).GT.Q{KG,IG,N))IY=L IF < QCKI,I,N).LT.Q(KG,IG,N ))IPY=1 IF IIY.EQ.I.AND.IPY.EQ.l) GO TO 260 IRI=ICP(N)+1 IRIP=ICP(N)+1 IF(KI.NE.NCN)IRI=IRtKI+l»I ,N) IFCKG.NE.NCNHRIP = IR< KG+1, IG,N) IFIIRI.GT.IRIP)GO TO 254 IF(IRI.EQ.IRIP)GO TO 252 Kl=KI+l GO TO 250 252 IFURI.EQ.ICP{N)+l.AND.IRIP.EQ.ICP<N)+l)GD TO 256 KI=KI+1 254 KG=KG+1 GO TO 250 256 IFIIPY.EQ.OJGO TO 258 ISINNI,IT,N)=0 GO TO 268 258 ISINN,IV,N) = 0 260 CONTINUE 262 CONTINUE 264 IFIMAX.EQ.l) GO TO 268 266 CONTINUE 268 CONTINUE 270 CONTINUE IFCNCN.EQ.1J GO TO 276 C CONTROL PT. ELEVATION REDUNDANCY* DO 274 IP=1,ICN NCM1=NCN-1 DO 274 NP=1,NCM1 NP1=NP+1 DO 274 NPT=NP1,NCN IFCIR(NPT,IP,N).EQ.0.OR.IR(NP,IP,N).EQ.0)G0 TO 274 IF{CPCIR(NPT,IP,N),N).GT.CP{IR{NP,IP,N),N))GO TO 274 DO 272 K=1,KN ITMK-l)*ICN+IP 272 ISCNPT,IT,N)=0 274 CONTINUE 276 CONTINUE ITH= ITL*NCN IST=0 DO 278 IT=l,ITL DO 278 J=1,NCN 278 IF(ISfJ,IT,N).EQ.O) IST=IST+l IH= ITH-IST N£=NE*-ICPIN)-l NNE=NNE+IH WRITEI6,280)NODE{N3,!TH,IST V IH 280 FORMAT!* »,I2,6X,12,10X,12,9X,12) 282 CONTINUE M=NE*NNE NB=IPIPE*NE NBl=NB+l M1=M+1 WRITE(6,284)NE,NNE,M,NB 284 FORMAT (• 0' , •NO. OF SECTIONS^ 1,12,20X,•TOTAL NO. OF SIG. HEAD •, *«LOSS EQNS.=«,I3/ **TOTAL NO. OF CONSTRAINTS=•,I 3,10X,•TOTAL NO. OF VARIABLES =• ,13) IFIM.GT.300.GR.NB.GT.3003 GO TO 404 IFIEMIN.NE.O.0)G0 TO 320 C C DETERMINE THE MINIMUM ALLOWABLE HEAD FOR GIVEN PIPE SIZES. C EXX - MIN. ALLOWABLE HEADIM). C EXX=0.0 IFIISM1.EQ.03GG TO 294 C TEST CONVEYANCE CONTROL PTS. 00 292 IP=1,ISM1 DO 292 IT=1,ITL IFI ISIt IT,IP).EQ.0)GO TO 292 I=IP EX=0.0 286 EX=EX+HK(l )*QT{IT,INII ,1) )**EXP*DISTIIN(I,1) )/100 . IP 1=1 + 1 DO 288 IZ=IP1,ISTAGE DO 288 L=2,MXIN IFIINII,1).EQ.INIIZ,L)}G0 TO 290 288 CONTINUE 290 1=17. IFI I.NE.ISTAGE)GO TO 286 IFIEX+CPT1IP).LE.EXX)GQ TO 292 EXX=EX+CPTUP) 292 CONTINUE C TEST FEEDER CONTROL PTS. 294 DO 318 N=1,NF ICN=IC(N) NCN=NC(N) KN=ITL/ICN C CONTRIBUTION BY CONVEYANCE SECTIONS. DO 296 IP=1,ISTAGE DO 296 L=2,MXIN IFINODEIN)•EQ.INI IP•L))G0 TO 298 296 CONTINUE 298 DO 316 1=1,ICN EL=0. DO 308 K=1,KN IT=IK-1)*ICN+I IFI ISSI IT, N) .EQ.O.OR.QT UT ,NODE{ N) ) .EQ.0.0)GO TO 303 EX=0. EX=EX+HK(i)*QTIIT.NODEIN))**EXP*DIST(NODE IN))/100. IFIIP.EQ.ISTAGE)G0 TO 306 IW=IP 300 EX=EX+HK(11*QTCIT,INIIW,1))**EXP*DISTI INIIW,111/100. IP1=IW-H DO 302 IZ=IP1,ISTAGE DO 302 L=2,MXIN IFIINIIW,11.EQ.INIIZ,L)1 GO TO 304 302 CONTINUE 304 IW=IZ IFIIW.NE.ISTAGE) GO TO 300 306 IFIEX.LE.EL)G0 TO 308 EL=EX 308 CONTINUE C CONTRIBUTION BY FEEDER SECTIONS. DO 314 J=1,NCN NN=NCN-J+l IF!IR(NN,I,N).EQ.O)GO TO 316 IPM=ICPIN)-l IFINN. NE.NCN):IPM=IR<NN+1 ,1 ,N>-1 IM=IR(NN»I,N3 XS=0.0 IFfIM.GT.IPMJGO TO 312 DO 310 IX=IM,IPM 310 XS=XS«-SD(IX,N) 312 EL=EL+HK(1)*Q(NN,I,N)**EXP*XS IFIEL+CPnR<NN,I,N),NJ.LE.EXX)GO TO 314 EXX=EL+CP(TR(NN,I,N1,N1 314 CONTINUE 316 CONTINUE 318 CONTINUE 320 CONTINUE C C GENERATE INITIAL TABLEAU (TB(K,J ) ) . C TBIK,J) - THESE ARRAYS BOTH CONTAIN THE LINEAR PROGRAMMING C STECK.J) TABLEAU USED IN THE GENERAL LIBRARY ROUT INE(LIP SUB). C THE FIRST ROW CONTAINS THE COEFS. FOR THE OBJECTIVE C FUNCTION. THE NEXT (NNEJ ROWS CONTAIN THE COEFS. FOR C THE NON-EQUALITY CONSTRAINTS. THE LAST (NE) ROWS C CONTAIN THE COEFS. FOR THE EQUALITY CONSTRAINTS. IN C TB(K,J) THE CNB+1TH) COLUMN REMAINS EMPTY BUT IN C TE(K,J), WHEN THE DESIRED HEAD IS KNOWN* IT CONTAINS C THE RIGHT—HAND-SIDES OF THE CONSTRAINTS. C UPON EXIT FROM SUBROUTINE(LIPSUB), ELEMENTS OF C TE(K,J) CONTAIN VARIOUS OUTPUT PARAMETERS. TO SAVE C THE ORIGINAL VALUES OF THE TABLEAU FOR USE IN THE NEXT C DESIRED HEAD VALUE, BOTH ARRAYS ARE REQUIRED. C C K r i . REFERS TO THE TABLEAU ROWS! CONSTRAINTS ) . C J - REFERS TO THE TABLEAU COLUMNS(VARIABLESJ. C DO 322 K=i,Ml DO 322 J=1,NB JT=(J-1)/IPIPE+1 JS=J-IPIPE*CJT-1) TB(K,J)=0.0 C OBJECTIVE FUNCTION COEFS-IF (K.EQ. IITBI K» J)=COST( JS) C EQUALITY CONSTRAINT!SECTION LENGTH) COEFS. 322 IF{K.GT.NNE+1.AN0.JT.EQ.K-NNE-1)T8(K,J)=1.0 C NON-EQUALITY CONSTRAINT (HEAD LOSS(HD) COEFS. K= 1 IF(ISM1.EQ.0)G0 TO 334 C CONVEYANCE CONTROL PT. (HL) COEFS. DO 332 IW=1,ISMl DO 332 IT=1,ITL IF( ISHIT,IW).EQ.O)GO TO 332 K=K+1 IX=IW 324 DO 326 JS=i,IPIPE J=(INUX,1)-1)*IPIPE+JS 326 TB(K,J)=HKI JS)*QT{IT , INUX ,11 )**EXP IP1=IX*1 DO 328 IZ=IP1,ISTAGE DO 328 L=2,MXIN I F d N U X , D.EQ.INC IZ,L))GO TO 330 328 CONTINUE 330 IX=IZ IFCIX.NE.ISTAGE1G0 TO 324 332 CONTINUE C FEEDER CONTROL PT. (HL)COEFS. 334 N=0 ISECT=NSECT GO TO 338 336 IT=IT+1 IFilT.NE.ITL+IJGO TO 344 IT=1 JB=JB+1 I F ( J B . N E . N C ( N ) + l ) G O T O 344 IF(N.EQ.NFJGO TO 368 I SECT*! SECT* I CP IN)-1 338 IT=0 JB=1 N=N+1 DO 340 1 = If I STAGE DO 340 L=2 , M X I N IFINODEINI ;EQ.INU,L))GO TO 342 340 CONTINUE 342 CONTINUE GO TO 336 344 IF(ISCJB,IT,N).EQ.0IGO TO 336 C CONTRIBUTION BY FEEDER SECTIONS. K=K+1 IFCICP(N).EQ.lJGO TO 356 JT=1 KB=(IT-1)/IC(N)-U IB=IT-IC(N1*(KB-1} 346 DO 352 JS=1,IPIPE J=USECT+JT-1)*IPIPE+JS NCN=NC(N) DO 350 NN=JB,NCN IF(NN.EQ.NCN)GO T O 348 I F ( JT.GEi.IR<NN, I B, N ) . AND. JT.LT .IR<NN+1, IB t N )) TBI K, J) = HK( JS)* *QINN,IB,N)**EXP GO TO 350 348 IF ( JT.GE.IRCNCN,IB,N))TB(K, J )=HK(JS)*Q(NN, IB,N)**EXP 350 CONTINUE IFITBCK,J).EQ.0.0)G0 TO 354 352 CONTINUE 354 IF( JT.EQ.ICPt N)-rll GO: TO 356 JT=JT+l GO TO 346 356 CONTINUE C CONTRIBUTION BY CONVEYANCE SECTIONS. IX=I LX=L 358 DO 360 JS=1, I PIPE J=CIN(IX,LX)-1)*IPIPE+JS 360 TB(K,J)=HK<JS)*QTUT,IN(IX,LX))**EXP IFILX.NE.DGO TO 366 IP1=IX+1 DO 362 12=IPItISTAGE DO 362 LZ=2,MXIN IFUNC IX,l)„EQ.INflZ,LZ))GO TO 364 362 CONTINUE 364 IX=IZ 366 LX=1 IFMX.EQ.ISTAGEJGO TO 336 GO TO 358 368 CONTINUE 123 C C INITIAL-BOUNDARY CONDITIONS FOR VARIABLE REDUCTION ALGORITHMS. C DPllI,N).DP2!I,N), AND DMII,N) ARE USED TO REDUCE THE NO. OF C VARIABLES IN FEEDER(N). C DP1(I,NJ- MIN. EXPECTED U/S LOCATION FOR D/S BOUNDARY OF P I P E ( I ) . C DP2U ,N)- LOCATION OF D/S BOUNDARY OF PIPE l i l t AT PREVIOUS C LOWER HEAD VALUE. C DM!I•N) — MAX. EXPECTED U/S LOCATION FOR U/S BOUNDARY OF PIPE!!). C C I H J ) AND IO!JJ ARE USED TO REDUCE THE NO. OF VARIABLES IN C CONVEYANCE SECTIGN(J). C I K J ) - LARGEST EXPECTED PIPE SIZE IN SECTIQNU). C IOIJ) - SMALLEST EXPECTED PIPE SIZE IN SECTION! J) .- . C DO 370 1=1 11 PIPE DO 370 N=1,NF DP1II,N)=0. DP2!I,N) = 0. 370 DM!I,N)=SM!N) DO 372 J=l,NSECT I I ! J ) = 1 IFIDIST! J).EQ.O.O)IIIJ)=IPIPE 372 IO!JI=IPIPE C C CALCULATE MIN. COST OF SYSTEM, OPTIMAL PIPE SIZES, AND SOME C SENSITIVITY MEASUREMENTS, FOR DESIRED HEADIHTILB)). C LB - HEAD ITERATION NO. C LBX - MAX. HEAD ITERATION NO. C IFIEMIN.NE.O.OIGO TO 374 INT=EXX/TINT4-1 EM=TINT*INT BL= ! EM AX-EM ) HINT+2 GO TO 376 374 BL=!EMAX-EMIN)/TINT*1 376 LBX=BL DIFF=BL-LBX IF!DIFF*GT.0.001)LBX=LBX+l IFCLBX.GT.OJGO TO 380 WRITE!6,378) •378 FORMAT! •PROBLEMS WITH EMIN AND EMAX. *) GO TO 404 380 CONTINUE !• LB=LBX HTILBJ=EMAX CALL LIP LB=l IF!LB.EQ.LBX)G0 TO 388 382 IF!EMIN.NE.0.0)G0 TO 384 EMIN=EM HTILB)=EXX+0.0001 CALL LIP LB=LB+1 IF!LB.EQ.LBX)GO TO 388 384 CONTINUE HTILB)=EMIN 386 CALL LIP LB=LB-H IFfLB.EQ.LBX)G0 TO 388 HT(LB)=HT(LB—1)+TINT 1 2 4 GO TO 386 388 IF{APC.EQ.O.O)GO TO 404 C C PRINT PUMP OUTPUT. C WRITE(6,390) 390 FORMAT(•-* , 10X,'TOTAL OPERATIONAL PIPE PUMP'/ •• HEADIM) COST($) CQST($) COSTC$) COST($)•/) TMCT=1000000. DO 400 LT=1,LBX CPCL=0.0 LP=LB-LT+1 IF(INCPC.EQ.O)GO TO 396 IFIHT(LP3-DAT.GT.HE(1))GO TO 392 CPCL=CPCU) GO TO 396 392 DO 394 1=2,1NCPC IF{HTILP)—DAT.GT.HElI))G0 TO 394 CPCL= CHT {LP)-DAT-HE ( I-i1 )/(HE(I)-HE(1-1)3*ICPC 11 3-rCPCI 1-1).) + *CPC(1-1) GO TO 396 394 CONTINUE CPCL=1000000. 396 CONTINUE CTO=CPU(LP)+CST(LP)+CPCL WRITEC6,398)HT(LP),CT0,CPU(LP),CST(LP),CPCL 398 F0RMAT(1X,F6.2,1X,F9.1,3X,F7;1,IX,F9.1,IX,F9.1) IFICTO.GE.TMCTJGO TO 400 TMCT=CTO 400 CONTINUE WRITEI6,402)TMCT 402 FORMAT(«-»,« MIN. COST OF IRRIGATION SCHEME=$',F8.2//) 404 CONTINUE STOP END C C LIP - GENERATE A NEW TABLEAU TEIK,J), USING TWO VARIABLE C REDUCTION ALGORITHMS TO SIMPLIFY THE COMPUTATION IN C SUBROUTINEILIPSUB). REARRANGE OUTPUT.FROM LIPSUB IN A C MORE PRESENTABLE FORM. C SUBROUTINE LIP DOUBLE PRECISION TE,A1,A2,A3,A4,DABS COMMON TEC301,301),A1C1),A2(I),A3(1),A411),CP( 20,10),C0ST(6) , *HK(6)•DI6,103,DS(6,30),NCHK1C300),NCHK1301),IN(20,4),DIST(33), *IC(20),NCC10),ICP(10),SM(10),Q!10,10,20),N0DE(20),IR(10,10,10), *HE(10),CPC(10),IS(10,50,10),TB(301,301),HT(25),CST(25), *DPlf 6,10), DP2{ 6,10) ,DM (6,10) ,JC( 300) ,JP( 300),LBX, EXP ,M,Ml, NE, *IPIPE,LB,NSECT,NF,ISTAGE,MXIN,CPT(20),ITL,CPU(25),APC,QT0T(50), *I SSI 50,10) , I SI (50,2 0) ,QT(50,30),ELT(50 ), ICR (50) ,T RCI 50), INRI 50), •DAT,IVHP,11(30),10(3 0),NFT,SD{ 20,103 ,QM(20),POR(5 0) 11=0 C C GENERATE NEW TABLEAU (T E ( K , J 3 ) . C INFORMATION ON TE(K,J) IS GIVEN WITH TB(K,J 3, IN THE MAIN PROGRAM. C JNN - VARIABLE NO., ASSOC. WITH CONVEYANCE SECTIONS. C JN - VARIABLE NO., ASSOC. WITH FEEDER SECTIONS. C JP(J) - PIPE SIZE ASSOC. WITH VARIABLE NO.iJ). C JCCJNN) - CONVEYANCE SECTION ASSOC. WITH VARIABLE NO.(JNN). C JCtJNJ - FEEDER NO. ASSOC. WITH VARIABLE NO.UN). 125 C NG TOTAL NO. OF VARIABLES. C 2 11=11+1 JNN=0 C CONVEYANCE SECTION VARIABLES!JNN). DO 6 JT=l,NSECT IIP=II!JT) IOP=IQIJT) DO 4 JS=IIP,IOP JNN=JNN+1 JPtJNN)=JS JCIJNN)=JT J=IJT-1)*IPIPE+JS DO 4 K=1,M1 4 TE(Kt JNNJ = TBIK f J) 6 CONTINUE C FEEDER SECTION VARIABLESCJN). JN=JNN N=l ISECT=NSECT 8 IF(TCP!N).EQ.l)GO TO 26 C DETERMINE IF FEEDER VARIABLES ARE SIGNIFICANT. OO 24 JS=1,IPIPE JT=1 IDM=1 C USING DM(JSfN) FIND U/S BOUNDARY CONDITION!IDM) FOR PIPE(JS). SDM=SD!IDM,N) 10 IF!SDM.GT.DM!JS,NI+0.0001)G0 TO 12 IDM=IDM+1 IF!IDM.EQ.ICPIN)JGO TO 12 SDM=SDM+SD!IDM,N) GO TO 10 12 IDP=0 C . USING DPllJSfN) FIND D/S BOUNDARY CONDITION HDP) FOR PIPE!JS). SDM=SD!IDP+lfN) 14 IF!SDM.GE.DPl!JSfN)-0.0001)G0 TO 16 IDP=IDP+1 IF!IDP+1.EQ.ICPIN))GO TO 16 SDM=SDM+SD!I0P+1 ,N) GO TO 14 16 CONTINUE 18 IF! JT.LE.IDP.OR.JT.GT. IDM)GO TO 22 JN=JN+l JP(JN)=JS JC! JN)=N J=(ISECT+JT-1}*IPIPE+JS DO 20 1=1tMl 20 TE!I,JN)=TB!I,J) 22 IF!JT.EQ.ICP(N)-1)G0 TO 24 JT=JT+1 GO TO 18 24 CONTINUE 26 IFCN.EQ.NFJGQ TO 28 ISECT=ISECT+ICP!N!-1 N=N+1 GO TO 8 28 CONTINUE C RIGHT-HAND-SI DEf RHS ) COEFS. TE!1,JN+1)=0.0D0 C NON-EQUALITY CONSTRAINT<HEAD LOSS(HLl) (RHS) COEFS. 1 9 < K=l 1 2 6 ISM1=ISTAGE-1 IFIISMI.EQ.OJGO TO 32 C CONVEYANCE CONTROL PT. (HL) IRHS) COEFS. DO 30 1=1 * I SMI DO 30 IT=1,ITL IFI IS I MT, I).EQ.O)GO TO 30 K=K+1 TE(K,JN+1)=HT(L8)-CPT(I) 30 CONTINUE C FEEDER CONTROL PT. (HL) (RHS) COEFS. 32 N=0 GO TO 36 34 IT=IT+1 IF{IT.NE.ITL + 11 GO TO 38 IT=1 JB = JB+1 IF(JB.NE.NC(N)+1>G0 TO 38 IFIN.EQ.NF)G0 TO 40 36 IT=0 JB=1 N=N+1 GO TO 34 38 I F U S I J B , IT,N).EQ.O)GO TO 34 K=K+l K B = (IT-l)/IC(N)+l IB=IT-IC(N)*(KB-1) TE(K,JN+i>=HT{LB)-CP(TR(JB,IB,N),N) GO TO 34 40 CONTINUE C EQUALITY CONSTRAINTISECTION LENGTH) (RHS) COEFS. 1=1 42 K=K+i TE( K»JN+1)=DIST(11/100. IFII.EQ.NSECT)G0 TO 44 1 = 1 + 1 GO TO 42 44 N=l 1 = 1 46 IF(ICPCN);EQ.11G0 TO 48 K=K+1 TE(K,JN+1)=SD(I,N) I F I I . E Q . I C P I N l - l l GO TO 48 1=1 + 1 GO TO 46 48 IFIN.EQ.NFIGO TO 50 N=N+1 1=1 GO TO 46 50 CONTINUE NG=JN JN1 = JN+1 C C LIPSUB IS A LIBRARY SUBROUTINE WHICH CARRIES OUT THE LINEAR C PROGRAMMING TECHNIQUE.- (SEE UBC LIP(FE81977) FOR DOCUMENTATION.) C CST1LB) - MIN. COST OF PIPE SYSTEM FOR DESIRED HEAD!HT(LB)). C NCHKII) - VARIABLE NO. IN PRIMAL SOLUTIONIPIPE LOCATION £ S I Z E ) . C TEU»NG+1 T-CONTAINS THE VARIABLE'S VALUER ENGTH OF PIPE SIZE). C CALL LIPSUB(TE,301,M,NG,NE,0,0,0,50.D-6,NCHK,NCHK1,A1,A2,A3,A4,127 *S228) CSTCLB)=DABSCTE( i,NG+l)) DO 52 1=1, IPIPE DO 52 JT=1,NSECT 52 DSCI,JT) = 0.0 DO 54 1=1,IPIPE DO 54 N=1,NF 54 DCI,N)=0.0 DO 58 1=2,Ml IFINCHKCI) .GT.JN)GO TO 58 JS=JPCNCHKC D ) IFCNCHKCI).GT.JNN)GO TO 56 JT=JCCNCHKU)) DS(J S,JT J=TE(I,NG+i)*1OO. GO TO 58 56 N=JCCNCHKCI) ) DCJS,N)=D<JS,N)«-TE(I,NG+1)*100. 58 CONTINUE C DISCONTINUITY BETWEEN FEEDERCN) AND CONVEYANCE SECTION! NODE! N) ). DO 66 N=1,NF DST=0.0 DO 60 1=1, IPIPE DSS=DSfIfNODEIN))+D<I,N) DST=DST+DSS IFCDST.GE.DISTCNODECN)))GO TD 62 DSCI,NODE(N)>=DSS DCI,N)=0.0 60 CONTINUE 62 DSC ItNODEC N))=DISTC NQDEC N))+DSS-DST DCI,N)=DST-DISTCNODECN)) IFCI.EQ.IPIPEIGO TO 66 IP1=I+1 DO 64 I=IPi, IPIPE DC I,N)=DS (I , NODECN) JtOH,N) 64 DSC I»NQDECN))=0.0 66 CONTINUE C FOR SYSTEMS WITH OUTLETS ALONG CONVEYANCE SECTIONS, NEED TO AVOID C DISCONTINUITY IN THE CONVEYANCE SECTIONS AFFECTED BY THESE C CONVEYANCE FEEDERS. C KS - D/S BOUNDARY FOR SECTIONS AFFECTED BY DISCONTINUITY. C KF - U/S BOUNDARY FOR SECTIONS AFFECTED BY DISCONTINUITY. IFC NF.EQ.NFT)GG TO 94 IW=l 68 IW=IW+1 .IFIIW.GE.ISTA6E160 TO 94 LT=0 DO 70 L=2,MXIN 70 IFCIN(IW,L);NE.0)LT=LT+1 IFCLT.GT.l)G0 TO 68 KS=IW-1 IWPl=IW+i DO 76 IWP=IWP1,ISTAGE LT=0 DO 72 L=2,MXIN C C C C C OPTIMAL PIPE LOCATION, SIZE, AND LENGTH. DS(I,JT)- LENGTH OF PIPE SIZECI), IN CONVEYANCE SECTIONCJT). OCI,N) - LENGTH OF PIPE SIZECI), IN FEEDERCN). I F ( I N C I W P , L ) . N E . 0 ) L T = L T + 1 1 2 8 I F C L T . G T . l ) G 0 T O 7 8 DO 7 4 L = 2 , M X I N I F C I N C I W P - 1 , 1 ) . E Q . I N C I W P , L ) ) G O T O 7 6 C O N T I N U E G O T O 7 8 C O N T I N U E K F = I W P - 1 D O 9 2 1 = 1 , I P I P E D S T = 0 . 0 DO 8 0 K = K S » K F D S T = D S T * D S C I , I N C K , 1 ) J K A = K F D R = D I S T C I N ( K A , l l ) I F 11.EQ.I)GO T O 8 6 I M 1 = I - 1 D O 8 4 I B = l , I M l D R = D R - D S C I B , I N C K A , 1 ) > C O N T I N U E I F C D S T . G E . D R ) G O TO 9 0 D S C I , I N C K A » 1 ) ) = D S T I F C K A . E Q . K S J G Q T O 9 2 K A M I = K A - 1 DO 8 8 K R = K S » K A M I DSC I » I N C K R , 1 1 ) = 0 . 0 GO T O 9 2 D S C I ,INf K A , 1 ) ) = D R D S T = D S T - D R K A = K A - 1 I F C K A . L T . K S J G O T O 9 2 G O T O 8 2 C O N T I N U E I W = I W P G O T O 6 8 C O N T I N U E O E F I N E B O U N D A R Y C O N D I T I O N S F O R T H E V A R I A B L E R E D U C T I O N A L G O R I T H M S . B O T H T H E A L G O R I T H M S , O N E F O R C O N V E Y A N C E S E C T I O N S A N D O N E F O R F E E D E R S , A R E B A S E D O N T H E P R E M I S E T H A T A S H E A D I S I N C R E A S E D A L L T I N T E R S E C T I O N S B E T W E E N TWO P I P E S I Z E S W I L L M O V E U P S T R E A M . B E C A U S E OF C O M P L I C A T I N G F A C T O R S T H I S I S N O T A L W A Y S T H E C A S E , S O T H E R U L E I S C H E C K E D B E F O R E A C C E P T I N G A N Y R E S U L T S . I F I N C R E A S I N G T H E H E A D M O V E S A P I P E I N T E R S E C T I O N D O W N S T R E A M , T H E B O U N D A R Y C O N D I T I O N S A R E R E E V A L U A T E D A N D I F T H E P R E V I O U S A N A L Y S I S D I D N O T I N C L U D E A C R I T I C A L V A R I A B L E , T H E A N A L Y S I S I S R E P E A T E D C 1 2 = 1 ) . D P L C I , N ) , D P 2 C I F N ) , D M C L , N ) - A R E E X P L A I N E D I N T H E M A I N P R O G R A M , I U J ) , A N D I O C J K 1 2 = 0 V A R I A B L E R E D U C T I O N A L G O R I T H M F O R C O N V E Y A N C E S E C T I O N S . U S E S F A C T T H A T A N O P T I M A L S E C T I O N C A N N O T C O N T A I N M O R E T H A N TWO P I P E S I Z E S . D O 1 2 4 I S E = 1 , I S T A G E DO 1 2 4 L S E = 2 , M X I N L S = I N ( I S E , L S E I I F C L S . E Q . O . O R . D I S T C L S J . E Q . O . O I G O T O 1 2 4 C O M P A R E R E S U L T S W I T H T H O S E F R O M M A X . U / S H E A D . DO 1 1 2 1 = 1 , I P I P E I F C D S C I , L S ) . L E . 0 . 0 0 0 5 I G O T O 1 1 2 C DEFINE MAX. U/S HEAD RESULTS. IFCLB.EQ.LBX.AND.H-i.LT.IOCLS))IOCLS)=I+1 IFU-n.LE.IO(LS}.OR.IO(LS).EQ.IPIP£)GO TO 114 C DEVIATION OCCURRED, CHECK AND REEVALUATE RESULTS. DO 104 N=1,NF 104 IFCNODECN).EQ.LS.AND.DCIOCLS),N).LE.0.0005)12=1 IFCISMl.EQ.0)GO TO 110 DO 108 I MM,I SMI IFCINCIM,13.NE.LS)G0 TO 108 KW=0 JW=0 DO 106 L=2,MXIN IFCINCIM,L).EQ.0)GO TO 106 KW=KW+1 IFCDSCIOCLS),INIIM,L)).LE.0.0005)JW=JW+1 106 CONTINUE IFCKW.EQ.JW)I2=1 108 CONTINUE 110 I0CLS)=I0CLS) + 1 GO TO 114 112 CONTINUE 114 IFC LB.EQ.LBX)GO TO 124 C COMPARE RESULTS WITH THOSE FROM PRECEDING LOWER U/S HEAD. DO 120 IW=1,IPIPE I=IPIPE-IW+l IFIDSCI,LS).LE.0.0005)GO TO 120 IF(I^l.GEiIICLS).OR.HCLS).EQ.l)GO TO 124 C DEVIATION OCCURRED, CHECK AND REEVALUATE RESULTS. IFCISE.EQ.ISTAGEJGO TO 116 IFCDSCIICLS},INCISE,1)).GT.0.0005)GO TO 118 116 12=1 118 IICLS ) = I I ( L S ) - l GO TO 124 120 CONTINUE 124 CONTINUE C VARIABLE REDUCTION ALGORITHM FOR FEEDER SECTIONS. IPM1-IPIPE-1 00 136 N=l,NF ICPMl=ICPfN)-l IFCICPCN).EQ.l)G0 TO 136 IFCLB.EQ.LBX)GO TO 130 C COMPARE RESULTS WITH THOSE FROM PRECEDING LOWER U/S HEAD. DT=0.0 DO 128 I=1,IPM1 IW=IPM1-1+1 DT=DT+D(IW+1,N)/100. IFCDP1CIW,N)-DT.LT.0.00005)20 TO 128 C: DEVIATION OCCURRED, CHECK AND REEVALUATE RESULTS. SDM=0. 0 DO 126 IL=1,ICPM1 SDM=SDM+SDCIL,N) 126 IFCABSCDT-SDM).LT.0.00005)12=1 DP1C IW,N)=DP1CIW+1,N) 128 CONTINUE C COMPARE RESULTS WITH THOSE FROM MAX. U/S HEAD. 130 DT=SMCN) DO 134 1=2,IPIPE DT=DT-DCI-1,N)/100. C DEFINE MAX. U/S HEAD RESULTS. IF(LB.EQ.LBX)DM(I,N)=DT IFIDT-DMU,N).LT.0.00005)GO TO 134 ..,0 C DEVIATION OCCURRED, CHECK AND REEVALUATE RESULTS. 5 SDM=0.0 DO 132 IL=1,ICPM1 SDM=SDM+SD!IL,N) 132 IF! ABS t DT-SDM).LT.0.00005)12=1 DM!I,N)=DM1I-1,N) 134 CONTINUE 136 CONTINUE IFILB.EQ.LBX)G0 TO 142 C IF CRITICAL VARIABLE HAS BEEN EXCLUDED, ANALYSIS IS REPEATED. IF{I2.EQ.I.AND.I1.EQ.2JG0 TO 232 IF(I2.EQ.1)G0 TO 2 C REDEFINE IIILS) FOR USE IN NEXT HIGHER HEAD VALUE. DO 138 ISE=1,ISTAGE DO 138 LSE=2,MXIN LS=IN1ISE,LSE) IFILS.EQ.O.OR.DISTILS).EQ.O.O)GO TO 138 DO 137 IW=1,IPIPE 1=1P IPE-IW+1 IFIDS!I,LS).LE.0.0005)GO TO 137 IF!I-l.GT.II1LS))II{LS)=I-1 GO TO 138 137 CONTINUE 138 CONTINUE C REDEFINE DPL(IMtN) S DP2! IW, N) FOR USE IN NEXT HIGHER HEAD VALUE. DO 140 N=l, NF IF(ICPINI.EQ.1)G0 TO 140 DT=0.0 DO 139 I=1,IPM1 IW=IPMl-I+l DT=DT+DIIW+1,N)/100. IFIDP1CIW,N)-DT.LT.0.00005.AND.DP2(IW,N)-DT.LT.0.30005) *DP1UW,N)=DT DP2!IW,N)=DT 139 CONTINUE 140 CONTINUE 142 CONTINUE C C CARRY OUT SENSITIVITY MEASUREMENTS; COMPUTE MIN. ALLOWABLE HEAD C AND CRITICAL CONTROL PT.• FOR EACH TIME INTERVALI I T ) . C a T l I T l - MIN. ALLOWABLE HEADIM). C INRITT) - FEEDER NODE NO. FOR CRITICAL CONTROL PT.( ICR! IT)) . C IRC(IT) - CONVEYANCE CRITICAL CONTROL PT. C WRITE(6,144)HTCLB),CSTILB),I1,JN 144 FORMAT!///* HEAD = ,,F 8.4,* IM)•,7X,'MIN. COST=$»,F10.4, *7X,'N0. OF ITERATIONS= «,I1,7X,«N0. OF VARIABL ES= *, *I3) WRITE!6,146) 146 FORMAT I• ',»SENS ITIVITY MEASUREMENTS.•/10X,•TIME',10X,'ALLOWABLE', *14X,'CRITICAL CONTROL PT.*/8X,•INTERVAL',7X,'MIN. H£ADIM)',8X, •'FEEDER',IX,'OUTLET',4X,'CONVEYANCE•) DO 148 IT=1,ITL 148 ELTIIT)=0. IFIISM1.EQ.0)G0 TO 168 C TEST CONVEYANCE CONTROL PTS. C ZERO FLOW REDUNDANCY. DO 166 1=1,I SMI DO 166 IT=1,ITL IF(QTUT, IN! I, I )).EQ.O.O)GG TO 166 IF!NF.EQ.NFT)GQ TO 156 NFP1=NF+ 1 DO 154 NV=NFP1»NFT IFiNGDEtNV) ;NE.IN(I ,l))GO TO 154 ICNV=IC!NV) DO 150 IB=1,ICNV IF!Q!1,IB,NV).NE.0.0)G0 TO 152 150 CONTINUE GO TO 156 152 KBM IT-D/IC !NV)+i I8=IT-IC!NV)*IKB-1) IF! QU,IB,NV).EQ.0.0)GO TO 166 GO TO 156 154 CONTINUE 156 EX=0. IW=I 158 DO 160 JS=1,IPIPE 160 EX=EX+HK! JS) *QT! IT• INIIW,1) )**EXP*DS ! JS, IN! IW, 1))/100. IP1=IW + 1 DO 162 IZ=IPI,ISTAGE DO 162 L=2,HXIN iF!IN!IW,l).EQ.INIIZ,L))GO TO 164 162 CONTINUE 164 IW=IZ IF!IW.NE.ISTAGE)GO TO 158 IFIEX+CPT!I).LE.ELTUT))GO TO 166 ELT! I T) = EX+CPT!I) IRCUT)=IN!I,1) INRUT) = 0 ICR!IT)=0 166 CONTINUE C TEST FEEDER CONTROL PTS. 168 DO 198 N=1,NF ICN=IC(N) NCN=NCIN) C CONTRIBUTION BY CONVEYANCE SECTIONS. DO 170 IPP=1,ISTAGE DO 170 L=2,MXIN IF!NODE!N).EQ.IN!IPP»L))G0 TO 172 170 • CONTINUE 172 KN=ITL/ICN DO 198 1=1,ICN DO 198 K=1,KN IT=(K-1)*ICN+I IF!QTMT,NODE!N) ).EQ.0.0)GO TO i98 EX=0. IX=IPP LX=L 174 DO 176 JS=1, IPIPE 176 EX=EX+HK!JS)*QT!IT,IN!IX,LX)}**EXP*DS1JS,IN!IX,LX))/100. IFILX.NE.DGO TO 182 IP 1=IX + 1 DO 178 IZ=IPl,ISTAGE DO 178 LZ=2,MXIN IF!IN!IX,1).EQ.IN!IZ,LZ))G0 TO 180 178 CONTINUE 180 IX=IZ 182 LX=i I F U X.NE.ISTAGE)GO TO 174 IF(K.NE.1)GQ TO 196 1 3 2 C CONTRIBUTION BY FEEDER SECTIONS. ED=-0.0001 EL=0. IW=1 XI=SM(N) DT=XI-DIIW,N1/100. DO 194 J=1,NCN NN=NCN-J+1 IFIIR(NN»I»N).EQ.O)GO TO 198 DB=0. IPM=ICP(NI-1 IF!NN.NE.NCN)IPM=IR(NN+1,ItN)-l IM=IR(NN, I,N) XS=0.0 IFdM.GT.IPMTGO TO 186 DO 184 IX=IM,IPM 184 XS=XS+SD(IX,N) 186 IFIXI-DT.GE.XS1GO TO 188 EL=EL+HK(IWJ*Q(NN,I,N)**EXP*(XI-OT-DB) IW=IW+1 IFCIW.EQ.IPIPE+1)GO TO 190 OB=XI-DT DT=DT-D(IW,N)/100. GO TO 186 188 £L=EL-»-HKCIW)*QJNN, I , N) **EXP*( XS-DB1 XI=XI-XS 190 IFIEL+CPCIR(NN,I,N),N).LE.ED)GO TO 192 ED=EL+CP(TR(NN,I,N),N) IED=IR(NN,I, N) 192 IF(IW.EQ.IPIPE+i)GO TO 196 194 CONTINUE 196 CONTINUE IFIEX+ED.LE.ELTCIT))G0 TO 198 ELTC IT )=EX+ED IRCCIT) = 0 INRUT)=NODE(N) ICRIIT)=IED 198 CONTINUE DO 200 IT=1,ITL 200 WRITE(6,202I IT,ELT(IT)»INRIIT),ICRIIT),IRC(IT) 202 F0RMATC11X,I2,12X,F7.3,13X,I 2,5X,I 2,1IX,I 2) IFCAPC.EQ.0.01G0 TO 210 C C CALULATE PUMP OPERATING COSTI CPU(LB))i FOR HEADIHTCLB)). C IFIIVHP.EQ.01G0 TO 206 CPUILB)=0. DO 204 IT=1,ITL IFIELTIIT).LE.DAT1GO TO 204 CPU(LB)=CPU(LBI +(ELTIIT1-DAT1*QTOT(IT)*POR(IT)*ARC 204 CONTINUE GO TO 210 206 QSUM=0. DO 208 IT=1,ITL 208 QSUM=QSUM+QTOT(IT)*POR(ITI CPU(LB)=(HT(LB)-DAT)*QSUM*APC 210 CONTINUE C C PRINT OUTPUT. WRITE!6,2121(I*1=1»IPIPE) 212 FORMAT 11 «,*PIPE LENGTH!Ml FOR CONVEYANCE SECTIONS. 1 */»SECTI0N*,6!4X,*PIPE*,I2,5X)) IF!ISMl.EQ.O)GG TO 220 DO 214 IP=1,ISMi 1=1 STAGE-IP 214 WRITEI6,216)INII,1),!DS!JS,IN!I,1))»JS=1*IPIPE) 216 FORMAT! * « ,2X, 12 ,2X ,614X ,F6.1 , 5X.) ) WRITE!6, 218) 218 FORMAT I• •) 220 DO 222 N=1,NF 222 WRITE!6,216)NODEIN)»(DS(JS»NODE!N))»JS=1,IPIPE)-WRITE(6,224)!I,1=1,IPIPEI 224 FORMAT!* *,»PIPE LENGTH FOR FEEDER PIPES1M).'/ *'• N0DE*,2X,614X,»PIPE*,I2,5X)I DO 226 N=1,NF IFCICP!N).EQ»1)G0 TO 226 «RITEI6,216)N0DE1N) , ID! JS, N) , JS=1, IPIPE) 226 CONTINUE GO TO 236 228 WRITE16,230)HTILB) 230 FORMAT!•-*,*HEAD= »,F7.4,6X, **FUNCTION IS UNBOUNDED OR UNFEASIBLE *) CST!LB)=1000000. GO TO 236 232 WRITE! 6, 234) 234 FORMAT! • » , * PROBLEM IN REDUCING NO. OF VARIABLES.*) . STOP 236 CONTINUE RETURN END 134 INDEX B Input Data and Output f o r Model Example o u t l i n e d i n Chapter 4» - I n p u t D a t a f o r M o d e l E x a m p l e o u t l i n e d i n C h a p t e r 4, 13 6 5 11 3 9 0 1.89 0 . 0 6 5 . 3 2 1 4 3 0 5 4 0 6 5 0 9 8 7 0 12 11 6 0 13 12 5 C 0 .0 246 .8 2 6 . 1 7 6 . 3 2 8 . 2 0 5 . 6 3 3 . 205.6 3 3 . 2 4 6 . 8 4 4 . 5 6 . 30 5.5 5 6 . 3 7 6 . 0 6 7 . 3 7 6 . 0 00076 . 1 5 5 . 0 . 0 0 1 4 4 1 5 7 . 0 . 00308 1 1 5 . 2 . 0 0 7 5 3 8 8 . 9 .0232 6 6 . 4 13 8 3 12 .90 4 . 0 2 7 . 9 28 .4 2 8 . 2 2 8 . 5 2 8 . 4 28.1 2 8 . 7 2 9 . 3 29 .1 28.1 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 9 10 11 12 1 4 3 8 C O C O 2 5 . 6 2 5.8 2 6 . 2 2 5 . 5 25 .6 26 .0 2 6 . 2 2 6 . 4 0 .90 0 . 5 0 0 . 9 C 0 .90 1.20 1.20 1.20 8 7 6 5 4 3 2 0 1 2 3 4 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 4 .0 4 . 0 0 . 0 5 .0 5.0 2 8 2 9 .90 0 . 0 2 6 . 6 2 6 . 4 2 6 . 7 2 6 . 7 2 6 . 6 26 .8 26 .4 2 6 . 3 2 6 . 6 9 8 7 6 5 4 3 2 1 2 3 4 5 6 0 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 4 . 0 5 .0 5.0 5.0 5.0 0 .0 0 . 0 8 2 1 1 C O 0 .0 2 8 . 0 4 .C C O 7 2 1 1 C O 0 .0 2 8 . 3 O.C 4 . 0 4 4 C O 2.0 C C 0 . 0 4 . 0 5 4 C O 0 .0 2.C 4 . 0 0 .0 6 4 C C 0 . 0 4 . 0 2.0 0 . 0 12 4 C O 4 .C C C O.C 2.0 10 2 C O 0 .0 4 . 0 11 2 C C 4 . 0 C O 1.6 2 6 . 5 2 7 . 4 28 .1 3.1 2 8 . 4 2 8 . 2 2 8 . 8 1 2 0 . s o . 1 2 0 . 1 2 0 . 1 2 0 . 120. 1 2 0 . 1 2 0 . 1 2 0 . 120 . 3 6 0 . 5. 0 .449 10 9 0 11 10 4 4 . 3 0 5 . 5 .0980 50.1 2 8 . 9 29 .5 4 5 6 5.0 5.0 0 5.0 5.0 1 2 0 . 240. NO. OF CCNVEVANCE SECTIONS-13 NO. OF ORIGINAL PIPES' 6 NODE TOTAL NO. OF FEEDERS (INCLUDING CONVEYANCEI-11 MIN. HEAD- 0.O(MI MAX. HEAD- 65.01M) CONVEYANCE SECTION MATRIX STAGE U/S NODE D/S NODE D/S NODE D/S 1 3 2 1 2 4 3 0 3 5 4 0 4 6 5 0 5 9 8 7 6 10 9 0 7 11 10 0 8 12 11 6 9 0 13 12 NO. OF FEEDERS- 5 HEAD LOSS EXPONENTIAL-1.89 CONSTANT-SPEED PUMPING PLANT. PUMP CAPITAL COST. ANNUAL PUMPING COST»$0.4490/(L/SECI/(MI INTAKE WATER LEVEL- O.OtMl, HEAO(M) COSTUI 28.00 174.30 28.00 20S.60 33.00 203.60 33.00 246.80 44.00 246.80 44.00 305.50 96.00 305.50 36.00 376.00 67.00 376.00 PIPE HEAD LOSS COEF.«P/100MI C0STII/100MI 1 .00076 195.00 2 .00144 157.00 3 .00308 115.20 4 .00753 88.90 5 .02320 66.40 6 .C9800 50.10 P f i c + d f i c + Hj O H o P> <D 1 I—1 CD O fi FEEDER NO. 13 NO. SETTINGS/I RRI. CONTROL PT. 1 2 3 4 5 6 7 8 9 10 11 12 CYCLE- 8 ELEVATION(Ml 27.90 28.40 28.20 28.50 28.40 28.10 28.70 29.30 29.10 28.10 28.90 29.50 NO. IRRI. CYCLES- 3 SECTION LENGTH-0.90U00MI OUTLET FLOW RATE- 4.0IL/SECI DISCHARGING OUTLETS TIME INTERVAL.... 1/ 2/ 3/ 4/ 5/ 6/ 7/ 8/ CYCLE.... 1 12 11 10 9 8 7 6 5 2 4 3 2 1 1 2 3 4 3 5 6 7 8 9 10 11 12 CD" O I <D 4 ON FEEDER NC. 1 MO. SETTINGS/IRRI CONTROL PT. CYCLE " 4 ELEVAT IOMMI NO. IRRI . C Y C L E S ' 3 SECTION LENGTH=0.0 (lOOMI OUTLET FLOM RATE* 0.OIL/SEC) 1 25.60 2 25 .80 3 26.20 4 25 .50 5 25.60 6 26.00 7 26.20 8 26 .40 ION LENGTHI1C0NI 1 0 .90 2 6.90 3 0.90 4 0 .90 5 1 .20 . . 6 1.20 7 1.20 DISCHARGING TIME INTERVAL. . . C Y C L E . . . . 1 2 3 TIME INTERVAL. . . C Y C L E . . . . 1 2 3 1/ 2/ 7 3 2 3/ 6 2 3 4 / 5 0 4 FLOM FROM DISCHARGING O U T L E T S « L / S E C I If 4 . 0 4 . 0 5.0 2/ 4 .0 4 . 0 5.0 3 / 4 . 0 4 .0 5.0 4 / 4 . 0 0 . 0 5.0 FEEDER NO. 2 NO. SETTINGS/IRRI CONTROL PT. 1 2 3 4 5 6 7 8 9 TIME I N T E R V A L . . . . C Y C L E . . . . 1 2 TIME I N T E R V A L - . . . C Y C L E . . . . 1 2 CYCLE* 8 NO. IRRI . CYCLES* 2 ELEVATION(Ml 26 .60 26 .40 26 .70 26 .70 26 .60 26 .80 26 .40 26 .30 26.60 DISCHARGING OUTLETS 1/ 2/ 3/ 4 / 5/ 6/ 9 8 7 6 5 4 I, 2 3 4 5 6 FLOW FROM DISCHARGING OUTLETS IL/SECI 1/ 2/ 3/ 4/ 5/ 6/ 4 . 0 4 .0 4 .0 4 . 0 4 . 0 4 . 0 5.0 5.0 5.0 5.0 5.0 5.0 SECTION LENGTH*0.90(100MI OUTLET FLOW RATE* O.OtL/SECI 7/ 3 0 7/ 4.0 0.0 8/ 2 0 8/ 4.0 0.0 " " ^ " s E T T I N G S / I R R I . CYCLE* 2 NO. IRRI. CYCLES* I SECTION LENGTH*0.0 I100MI OUTLET FLOM RATE* 0 .0 (L /SEC I NO CONTROL PT. ELEVATIOMMI 1 28.00 FLOW FROM DISCHARGING OUTLETSIL/SECI TIME INTERVAL.... 1/ 2/ CYCLE.... 1 4.0 0.0 FEEDER NO. 7 HO. SETTINGS/I RRI. CYCLE" 2 NO. IRRI. CYCLES- 1 SECTION LENGTH=0.0 (100MI OUTLET FLOW RATE- O.OU/SEC) CCNTROL PT. ELEVATIONIMI 1 28.30 FLCW FROM DISCHARGING OUTLETSIL/SECI TIME INTERVAL.... 1/ 2/ CYCLE.... 1 0.0 4.0 CONVEYANCE FEEDER NC. 4 NO. SETTINGS" 4 OUTLET FLCW RATE" O.OIL/SECI FLCW FROM'DISCHARGING OUTLETS(L/SECI TIME INTERVAL.... 1/ 2/ 3/ 4/ 2.0 0.0 0.0 4.0 CONVEYANCE FEEDER NO. 5 NO. SETTINGS" 4 OUTLET FLOW RATE" O.OIL/SECI FLCW FROM DISCHARGING OUTLETSIL/SECI TIME INTERVAL.... 1/ 2/ 3/ 4/ . 0.0 2.0 4.0 0.0 CONVEYANCE FEEDER NO. 6 NO. SETTINGS" 4 OUTLET FLCW RATE" O.OIL/SECI FLOW FROM DISCHARGING OUTLETS«L/SECI TIME INTERVAL.... 1/ 2/ 3/ 4/ 0.0 4.0 2.0 0.0 CONVEYANCE FEEDER NO.12 NO. SETTINGS- 4 OUTLET FLOW RATE- 0.0(L/SECI FLCW FROM DISCHARGING OUTLETSIL/SECI TIME INTERVAL.... 1/ 2/ 3/ 4/ 4.0 0.0 0.0 2.0 CONVEYANCE FEEDER NO.10 NO. SETTINGS- 2 OUTLET FLOW RATE- O.OIL/SECI FLOW FROM DISCHARGING OUTLETSIL/SECI TIME INTERVAL.... 1/ 2/ 0.0 4.0 CONVEYANCE FEEDER NO.11 NO. SETTINGS- 2 OUTLET FLOW RATE- O.OIL/SECI CO FLCW FROM DISCHARGING OUTLETS(L/SECI TIME INTERVAL.... 1/ 2/ 4.0 0.0 TOTAL NO. OF TIME INTERVALS- 8 DISCHARGE IN CONVEYANCE SECTIONSIL/SECI. STAGE. ... 1 2 3 4 5 6 7 8 NODE 3 4 5 6 9 10 11 12 TIME INTERVAL 1 22. 24. 24. 24. 4. 4. 8. 36. 2 22. 22. 24. 28. 4. 8. 8. 36. 3 22. 22. 26. 28. 4. 4. 8. 36. 4 18. 22. 22. 22. 4. 8. 8. 32. 5 22. 24. 24. 24. 4. 4. 8. 36. 6 22. 22. 24. 28. 4. 8. 8. 36. 7 17. 17. 21. 23. 4. 4. 8. 31. 8 13. 17. 17. 17. 4. 8. 8. 27. CONVEYANCE SECTION LENGTH!*!, AND ELEV.(Ml SECTION LENGTH ELEV. 1 120. 26.40 2 90. 26.60 3 120. 1.60 4 120. 26.50 5 120. 27.40 6 120. 28.10 7 120. 28.30 8 120. 28.00 9 12C. 3.10 10 120. 28.40 11 120. 28.20 12 240. 28.80 13 360. 29.50 NO. OF SIGNIFICANT HEAO LOSS EQNS.(HLE1. NODE TOTALIHLEI NON -SIGIHLEI SIGIHLEI 3 8 6 2 4 8 7 1 5 8 7 1 6 8 7 1 9 8 7 1 10 8 7 1 11 8 7 1 12 8 7 1 13 24 16 8 1 24 18 6 2 16 9 7 8 8 7 1 7 8 7 1 NO. OF SECTIONS =39 TOTAL TOTAL NO. OF CONSTRAINTS- 71 TOTAL OF SIG. HEAD LOSS EONS.- 32 HEAD- 65.0000IMI MIN. COST=$ 2732.3735 SENSITIVITY MEASUREMENTS. NO. OF ITERATIONS- 1 NO. OF VARIABLES-234 I i TIME ALLOWABLE CRITICAL CONTROL PT. INTERVAL MIN. f-EAOCMI FEEDER OUTLET CONVEYANCE 1 63.785 13 4 0 2 63.259 1 2 0 3 65.00C 1 2 0 4 65.000 13 1 0 5 65.000 13 1 0 6 63.259 1 2 0 7 62.642 13 3 0 8 63.785 13 4 0 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 12 0.0 0.0 240.0 0.0 0.0 0.0 11 0.0 0.0 0.0 0.0 0.0 120.0 10 0.0 0.0 0.0 0.0 0.0 120.0 9 0.0 0.0 0.0 0.0 0.0 120.0 6 0.0 0.0 0.0 120.0 0.0 0.0 5 0.0 0.0 0.0 120.0 0.0 0.0 4 0.0 0.0 0.0 120.0 0.0 0.0 3 0.0 .0.0 , 0.0 120.0 0.0 0.0 13 0.0 0.0 0.0 0.0 360.0 0.0 1 0.0 0.0 0.0 0.0 120.0 0.0 2 0.0 0.0 0.0 0.0 90.0 0.0 8 0.0 0.0 0.0 0.0 0.0 120.0 7 0.0 0.0 0.0 0.0 0.0 120.0 PIPE LENGTH FOR FEEDER PIPES1MI. NOOE PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 13 0.0 0.0 0.0 0.0 171.0 819.0 1 0.0 0.0 0.0 0.0 517.9 202.1 2 0.0 0.0 0.0 0.0 125.0 595.0 HEAO* 30.3939(M> MIN. COST** 5732.2031 NO. OF ITERATIONS' 1 NO. OF VARtABLES=222 SENSITIVITY MEASUREMENTS. TIME ALLOWABLE CRITICAL CONTROL PT. INTERVAL MIN. KEAOIK) FEEDER OUTLET CONVEYANCE 1 30.394 13 4 0 2 30.394 7 1 0 3 30.394 13 2 0 4 30.394 13 8 0 5 30.394 13 8 0 6 30.394 13 2 0 7 30.083 13 3 0 8 30.394 13 4 0 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 12 240.0 0.0 0.0 0.0 0.0 0.0 11 0.0 0.0 120.0 0.0 0.0 0.0 10 0.0 0.0 120.0 0.0 0.0 0.0 9 0.0 ' • 0.0 120.0 0.0 0.0 0.0 6 120. 0 0.0 0.0 0.0 0.0 0.0 5 120.0 0.0 0.0 0.0 0.0 0.0 4 120.0 0.0 0.0 0.0 0.0 0.0 3 120.0 0.0 0.0 0.0 0.0 0.0 13 0.0 360.0 0.0 0.0 0.0 0.0 1 0.0 120.0 0.0 0.0 0.0 0.0 2 0.0 90.0 0.0 0.0 0.0 0.0 B 0.0 0.0 0.0 0.0 107.5 12.5 7 0.0 0.0 83.7 36.3 0.0 0.0 6 30.7 PIPE LENGTH FOR FEEDER PIPESIMI. NODE PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE ., 0 Q '19.3 84.9 372.3 182.7 30. l \ 0 0 434 7 105.3 0.0 167.6 12.4 2 0.0 116.9 423.1 149.4 30.6 0.0 HEAD" 35.0000IMI MIN. COST"* 3985.3701 SENSITIVITY MEASUREMENTS. TIME INTERVAL 1 2 3 4 5 6 7 ALLOWABLE MIN. HEAC!Ml 35.000 35.00C 35.000 35.000 35.000 35.000 , 34.608 35.000 NO. OF ITERATIONS" 1 CRITICAL CONTROL PT. NO. OF VARIABLES=163 SECTION PIPE 1 PIPE 2 12 240.0 0.0 11 0.0 0.0 10 0.0 0.0 9 0.0 0.0 6 0.0 120.0 5 0.0 120.0 4 0.0 120.0 3 0.0 120.0 13 0.0 0.0 1 o.c 0.0 2 0 .0 0.0 8 0.0 0.0 7 0.0 0.0 PIPE LENGTH FOR FEEDER PIPESIMI. NODE PIPE 1 P IPS 2 13 0.0 0.0 1 0.0 0.0 2 o.c 0.0 FEEDER OUTLET CONVEYANCE 13 4 0 7 1 0 1 3 0 13 1 0 13 1 0 13 2 0 13 3 0 13 4 0 PIPE 3 PIPE 4 PIPE 5 0.0 0.0 0.0 0.0 0.0 120.0 0.0 0.0 120.0 0.0 0.0 93.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 45.0 315.0 0.0 111.4 8.6 0.0 0.0 90.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 PIPE 3 PIPE 4 PIPE 5 0.0 498.7 484.7 0.0 540.0 89.7 0.0 164.3 510.0 PIPE 6 0.0 0.0 0.0 26.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 120.0 120.0 PIPE 6 6.6 90.3 45.7 HEAD" 40.00001 Ml MIN. COST"! 3508.5769 SENSITIVITY MEASUREMENTS. NO. OF ITERATIONS" 1 NO. OF VARIABLFS"116 TIME ALLOWABLE INTERVAL MIN. HE A C(MI 1 40.000 2 ' 40tOOG 3 40.000 4 40.000 5 40.000 6 40.000 7 38.634 8 38.733 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 CRITICAL CONTROL PT. FEEDER OUTLET CONVEYANCE 2 7 1 13 13 7 13 7 PIPE 3 PIPE 4 PIPE 5 PIPE 6 12 0.0 240.0 0.0 11 0.0 0.0 0.0 10 0.0 0.0 0.0 9 0.0 0.0 0.0 6 0.0 46.3 73.7 5 0.0 0.0 120.0 4 0.0 0.0 120.0 3 o.o 0.0 120.0 13 0.0 0.0 0.0 1 0.0 0.0 0.0 2 0.0 0.0 0.0 8 0.0 0.0 0.0 7 0.0 0.0 0.0 PIPE LENGTH FOR FEEDER PIPESIMI. NODE PIPE 1 PIPE 2 PIPE 3 13 0.0 0.0 0.0 1 0.0 0.0 0.0 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 120.0 0.0 0.0 51.4 68.6 0.0 0.0 120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 360.0 0.0 0.0 120.0 0.0 0.0 90.0 0.0 0.0 0.0 0.0 120.0 0.0 0.0 120.0 PIPE 4 PIPE 5 PIPE 6 270.0 347.1 372.9 540.0 79.0 101.0 83.4 556.4 80.2 HEAD" 45.00001Kl MIN. COST"* 3231, SENSITIVITY MEASUREMENTS. 3132 NO. OF ITERATIONS" 1 NO. OF VARIABLES«104 TIME INTERVAL 1 2 3 ALLOWABLE MIN. HEAOIMI 45.000 45.000 45.000 CRITICAL CONTROL PT. FEEDER OUTLET CONVEYANCE 13 7 1 4 45.000 13 5 45.000 13 6 45.000 7 7 43.356 13 8 45.00C 13 PIPE LENGTH(Ml FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 PIPE 3 12 0.0 185.2 54.8 11 0.0 0.0 0.0 10 0.0 0.0 0.0 9 0.0 CO 0.0 6 0.0 0.0 120.0 5 0.0 0.0 120.0 4 0.0 0.0 120.0 3 0.0 0.0 120.0 13 0.0 0.0 0.0 1 0.0 0.0 0.0 2 0.0 0.0 0.0 8 0.0 0.0 0.0 7 0.0 0.0 0.0 PIPE LENGTH FOR FEEDER PIPESIMI. NODE PIPE 1 PIPE 2 PIPE 3 13 0.0 0.0 0.0 1 0.0 0.0 0.0 2 0.0 0.0 0.0 HEAD* 50.0000CI MIN. COST'S 3027. 1296 PIPE 4 PIPE 5 PIPE 6 0.0 0.0 0.0 0.0 60.7 59.3 0.0 0.0 120.0 0.0 0.0 120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 360.0 0.0 0.0 120.0 0.0 0.0 0.0 90.0 0.0 0.0 0.0 120.0 0.0 0.0 120.0 PIPE 4 PIPE 5 PIPE 6 54.5 436.0 499.5 240.0 300.0 180.0 0.0 465.2 254.8. NO. OF ITERATIONS" 1 NO. OF VARIABLES" 92 SENSITIVITY MEASUREMENTS. IV) 1 TIME ALLOWABLE CRITICAL CONTROL PT. INTERVAL MIN. HEAOCMI FEEDER OUTLET CONVEYANCE 1 5C.0OO 13 4 0 2 49.965 7 1 0 3 50.000 1 2 0 4 50.000 13 1 0 5 50.000 13 1 0 6 49.965 7 1 0 7 48.857 13 3 0 8 50.000 13 4 0 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE I PIPE 2 PIPE 3 PIPE 4 PIPE 5 12 0.0 0.0 240.0 0.0 0.0 11 0.0 0.0 0.0 0.0 0.0 10 0.0 0.0 0.0 0.0 0.0 9 0.0 0.0 0.0 0.0 0.0 6 0.0 0.0 120.0 0.0 0.0 5 0.0 0.0 120.0 0.0 0.0 4 0.0 0.0 120.0 0.0 0.0 3 0.0 120.0 0.0 0.0 13 0.0 0.0 0.0 134.1 225.9 1 0.0 0.0 0.0 120.0 0.0 2 0.0 0.0 0.0 0.0 90.0 8 0.0 0.0 0.0 0.0 0.0 7 0.0 0.0 0.0 0.0 0.0 PIPE LENGTH FOR FEEDER PIPESIMt. NOOE PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 13 0.0 0.0 0.0 0.0 472.5 1 0.0 0.0 0.0 122.5 417.5 2 0.0 0.0 0.0 0.0 315.4 HEAD" 55.00001M) MIN. COST»» 2896.0633 NO. OF ITERATIONS" 1 SENSITIVITY MEASUREMENTS. TIME ALLOWABLE CRITICAL CONTROL PT. INTERVAL MIN. hEADIMI FEEDER OUTLET CONVEYANCE 1 55.000 13 4 0 2 53.857 13 3 0 3 55.000 I 2 0 4 53.030 13 1 0 5 55.000 2 ' 5 0 6 53.613 1 2 0 7 53.857 13 3 0 8 55.000 13 4 0 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 12 0.0 0.0 240.0 0.0 0.0 11 0.0 0.0 0.0 0.0 0.0 10 0.0 0.0 0.0 0.0 0.0 9 0.0 0.0 0.0 0.0 0.0 6 0.0 0.0 120.0 0.0 0.0 5 0.0 0.0 120.0 0.0 0.0 4 0.0 0.0 10.4 109.6 0.0 3 0.0 0.0 0.0 120.0 0.0 13 0.0 0.0 0.0 0.0 360.0 1 0.0 0.0 0.0 120.0 0.0 2 o.o 0.0 0.0 0.0 90.0 PIPE 6 0.0 120.0 120.0 120.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 120.0 120.0 PIPE 6 517.5 180.0 404.6 OF VARIABLES" 88 PIPE 6 0.0 120.0 120.0 j 120.0 ( 0.0 0.0" j 0.0 i 0.0 ] 0.0 i i 0.0 0.0 MM 8 0.0 0.0 0.0 0.0 0.0 120.0 7 0.0 0.0 0.0 0.0' 0.0 120.0 PIPE LENGTH FOP. FEEDER PIPESIM). NODE PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 13 0.0 0.0 0.0 0.0 401.7 588.3 1 0.0 0.0 0.0 48.4 491.6 180.0 2 0.0 0.0 0.0 0.0 240.1 479.9 HEAD* 60.00001Kl MIN. COST'S 2803.4417 NO. OF ITERATIONS- 1 NO. OF VARIABLES SENSITIVITY MEASUREMENTS. TIME ALLOWABLE CRITICAL CONTROL PT. INTERVAL MIN. HEACIMI FEEOER OUTLET CONVEYANCE 1 60.000 13 4 0 2 58.657 13 3 0 3 60.000 1 2 0 4 56.871 13 1 0 5 60.000 • 2 5 0 6 53.406 1 2 0 7 58.857 13 3 0 8 60.000 ' 13 4 0 PIPE LENGTHIMI FOR CONVEYANCE SECTIONS. SECTION PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 12 0.0 0.0 240.0 0.0 0.0 0.0 11 0.0 0.0 0.0 0.0 0.0 120.0 10 0.0 0.0 0.0 0.0 0.0 120.0 9 0.0 0.0 0.0 0.0 0.0 120.0 6 0.0 0.0 120.0 0.0 0.0 0.0 5 0.0 0.0 49.7 70.3 0.0 0.0 4 0.0 0.0 0.0 120.0 0.0 0.0 3 0.0 0.0 0.0 120.0 0.0 0.0 13 0.0 0.0 0.0 0.0 360.0 0.0 1 0.0 0.0 0.0 0.0 120.0 0.0 2 0.0 0.0 0.0 0.0 90.0 0.0 8 0.0 0.0 0.0 0.0 0.0 120.0 7 0.0 0.0 0.0 0.0 0.0 120.0 PIPE LENGTH FOR FEEOER PIPES(MI. NODE PIPE 1 PIPE 2 PIPE 3 PIPE 4 PIPE 5 PIPE 6 13 0.0 0.0 0.0 0.0 270.4 719.6 1 0.0 0.0 0.0 0.0 540.0 180.0 2 0.0 0.0 0.0 0.0 165.7 554.3 TOTAL OPERATIONAL PIPE PUMP HEADJMI CGSTCSI COST«$l COST!SI COST!SI 65.00 4443.6 1335.2 2732.4 376.0 60.00 4411.9 1232.5 2803.4 376.0 55.00 . 4331.4 1129.8 2896.1 305.5 50.00 4359.7 1027il 3027.1 305.5 45.00 4461.2 924.4 3231.3 305.5 40.00 4577.0 821.7 3508.6 246.8 35.00 4951.1 719.0 3985.4 246.8 30.39 6562.1 624.3 5732.2 205.6 MIN. COST CF IRRIGATION SChEME-S 4331.38 

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