Canal design by dynamic programming, computer programme CANDY G. Radovic Energoprojekt-Hidronizenjering, Bui. Lenjina 12, Belgrade, Yugoslavia

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1 Canal design by dynamic programming, computer programme CANDY G. Radovic Energoprojekt-Hidronizenjering, Bui. Lenjina 12, Belgrade, Yugoslavia Abstract The problem considered in this technical note presents an approach to irrigation and drainage canal design procedure which uses dynamic programming algorithm where objective function and constraints are related to criteria used in canal design. This method is the first step in development of an enhanced design concept and is based on the optimization of construction costs. Application of the dynamic programming allows sophisticated improvements and corrections of the procedure. Introduction Dynamic programming (DP) is a powerful and versatile tool for solving a wide range of sequential problems in water resources. The technical literature abounds in a variety of applications of discrete dynamic programming to water resource systems planning, design and operations. The canal design procedure presented in this text rely on dynamic programming procedure which basically creates a network of possible paths. Components of the network are related to the stages and states of dynamic programming. The canal design is based on the principle of limiting velocities of water flow. In irrigation&drainage practice we usually operate with two limiting velocities: minimum permissible velocity as a limit for sedimentation and maximum permissible velocity as a limit for erosion. In irrigation and drainage practice some of the variables are defined beforehand. The discharge (Q), the roughness coefficient (n) and the side slope

2 58 Hydraulic Engineering Software (m) are normally defined before the design procedure. The two variables that will actually determine a canal's characteristics are the longitudinal slope (s) and bottom width (b). By calculating these two values the canal dimensions are determined. Furthermore, there is one additional constraint that has to be satisfied: mean velocity of the water has to be within a range of minimum and maximum velocity. Once this has been satisfied the canal dimensions can be accepted as feasible ones. The design procedure is carried out section per section where new section begins where anyone of the defined elements (Q,n,m,s,b) changes its value. Canal Design by Dynamic Programming Dynamic Programming is a mathematical procedure designed primarily to improve the computational efficiency of select mathematical programming problems by decomposing them into smaller, and hence computationally simpler, subproblems. Each subproblem is then considered separately with the objective of reducing the volume of computations. In one dimensional DP models for each of the subproblems there is only one decision variable. However, since the subproblems are interdependent, a procedure must be devised to link the computations in a manner that guarantees that a feasible solution for each stage is also valid for the entire problem[l]. Elements of a DP model applied to the canal design can be seen as follows: DP stage.since the canal design is usually carried out for a certain canal section, the section represents a DP stage. DP objective function. The goal is to design the cheapest canal which is still able to meet the design criteria. In other words, the objective is to minimize construction costs of the canal. DP variables. If a prismatic cross section is assumed then the size of the canal will influence the costs. The two main characteristics are the bed elevation (which determines the bed slope) and the width of the canal's bed. These two variables influence the amount of work and the construction costs. DP constraints. In canal design procedure, the main constraint is the water surface level. No matter if it is a question of irrigation or drainage canal, there is always a certain criterion related to water surface level (WSL). In case of an irrigation canal WSL should be above a certain level at a given canal section, while in case of a drainage canal WSL should be below a certain level at a given canal section. Another, very usual constraint, is the velocity of water. The water is allowed to flow neither too slow nor too fast. As it has been mentioned already DP creates a network of possible paths. These paths define feasible solutions among which the optimum one is to be found. In Figure 1. the network of possible paths of canal bed positions for a given longitudinal ground profile is presented. For a longitudinal profile as given in Fig.l the following items will be defined, as an input for the design procedure: Canal longitudinal section is defined

Hydraulic Engineering Software 59 0 cross cross cross sect i on sect I on sect i on Sect i on #2 Section #3 Section #4 L=500m L=500m Q=0.8m3/s L=500m 0=0, 8m3/ L=500m Q=0.8m3/s n=0,025 m=1.5 n=0.

3 Hydraulic Engineering Software 59 0 cross cross cross sect i on sect I on sect i on Sect i on #2 Section #3 Section #4 L=500m L=500m Q=0.8m3/s L=500m 0=0, 8m3/ L=500m Q=0.8m3/s n=0,025 m=1.5 n=0.025 m=1.5 n=0.025 m=1.5 n=0.025 m=1.5 b1=1m b2=1.5m b1 = 1m b2=1.5m b1=1m b2=1.5m b1=1m b2= 1,5m for at I canal sections: Vmin m/s, Vmax = 0.8 m/s objective function: min C, vhere C is excavation cost [1$/m Figure 1. Network of Possible Paths. Number of sections (reaches) and for each section the values for discharge (Q), roughness coefficient (n), side slope (m) and maximum and minimum allowable velocities (v^^v^in).. Assumption is made that canal reaches are long enough so at the end of each reach normal depth (hj occurs.. Upper and lower limit for possible canal bed positions. Assuming a drainage canal upper limit determines maximum allowable water level. The canal bed is below this limit at least for the value of calculated water depth. The lower limit can be defined either as the maximum canal depth or by minimum allowable longitudinal slope and bed elevation at the most downstream cross-section. The canal bed has to be above the lower limit. In other words a strip defined by canal bed and calculated WSL has to be within the lines that define the upper and lower limit.. The canal is to be designed section per section where the possible bed positions at the beginning/end of each section are defined before (within the upper and lower limit). At these cross sections a group of points is defined as possible bed positions. The distance between each pair of these points is an increment in height, 6h. The value for 6h can vary from 1 to several centimeters. Taking into consideration only canals with subcritical flow, positions of canal bed are to be defined at the starting cross section. These positions are the values for the state variable ZB, at the beginning of the first DP stage. These starting points are depicted on the first canal cross section on Figure 1. They are

60 Hydraulic Engineering Software presented as a set of points representing bed elevation within the upper and lower limit line at interspatial distance of <5h.

4 60 Hydraulic Engineering Software presented as a set of points representing bed elevation within the upper and lower limit line at interspatial distance of <5h. Values for the next state variables ZB; are calculated using the state equation: where ZBj+i ZBj Si Lj ZB,+, = ZB; + % * L; (1) - bed elevation at the end of the canal section (i) - bed elevation at the beginning of the canal section (i) - longitudinal bed slope at the canal section (i) - length of the canal section (i). State variable in the above equation is elevation of the canal bed ZB;+i, ZB; which obtains connection between the stages. As it has already been mentioned for each DP stage there are two decision variables: longitudinal bed slope (s) and bed width (b). Due to this, one could think of this DP problem as two dimensional. Although these two are decision variables they are interdependent and can be sublimed into one decision variable. The objective is to find the cheapest solution. Therefore at a certain canal section (i), of length (LJ and bed elevations ZB,,ZB^i at its ends, the vector of longitudinal slopes {sj is defined. If one of the longitudinal slopes (s^) from vector {sj can be realized with several possible values for the bed width from vector {bj, it is logical that the minimum construction costs will be for the smallest value of {bj. As a consequence of the above notation the following is introduced: s^ which is the j* member of the longitudinal bed slope vector {sj for the alternate states for the i* canal section (i) with corresponding bed width bt=min{bj}. In this way the dynamic programming problem is being transformed into a one-dimensional procedure because the information of bed width (b) is incorporated in slope vector {sj where for each of the slopes the corresponding value for (b) is defined. The value for (b) is equal to the minimum one, where the conditions for lower and upper limit are fulfilled. Allowable slopes define the slope vector {s^} within the maximum (s^^x) and the minimum (s^in) slopes which correspond to the maximum and minimum permissible velocities. Feasible slopes are those from the allowable slopes vector meeting the condition that with a normal depth (hj, the water surface level at the ends of the considered canal section satisfies the upper limit criterion. As a summary of the canal design procedure presented herein the following should be noted:. the objective of the design is to determine the cheapest solution. main elements that are to be defined are. canal bed width (b). canal longitudinal slope(s). canal vertical position (ZB). several possible values for bed width are considered where the smallest one among those that render feasible solutions is chosen

5 . the critera used for the design are related to. limiting velocities of water flow. requested water surface levels. Mathematical Formulation Hydraulic Engineering Software 61 Following the design principles and assumption presented so far, the problem of canal design can be defined in mathematical terms in the following way: Objective function Constraints. allowable slopes min C (2) s<\ = (Uj*6h)/L; (3). feasible slopes LL; < (ZBi 4- h<\i) < ULj # where h%; is from Manning's equation Q; = 1/n; ^ A^\j * (R(\jf " (s("\j)'" (7) and the slopes s^ in the equation are M ^ g(k) ^»(k) /o\ S i,max ^ % jj a S i^in. (Q) The calculations are carried out through recursive equations: The state equation: f;(zbj)= min {C^s^J + f^(zb^)} 0 ZB; = ZB,i + s^ * Li OQ The notations used in the above equations have the following explanations: C - total canal construction cost <5h - increment in height Lj - canal section length

6 62 Hydraulic Engineering Software s^\j - slope (j) for stage (i) with bed width (k) s^jmax - maximum allowable bed slope for stage(i) with bed width (k) s^i,min ~ minimum allowable bed slope for stage (i) with bed width (k) - lower limit elevation at stage (i) - upper limit elevation at stage (i) ZBj - bottom elevation at stage (i) h%; - normal depth for stage (i) with bed width (k) A^jj - wetted area for stage (i) with bed slope (j) and bed width (k) R*^J - hydraulic radius for stage (i) with bed slope (j) and bed width (k) nj - Manning's roughness coeff. for stage (i) Qi - discharge for stage (i) N - number of stages kjuj - positive integer numbers - optimum solution for stage (i) and given bed elevation (ZBJ Computer Programme CANDY Programme CANDY has been developped on the basis of the theoretical background described in the previous chapters. The programme is written in the computer language PASCAL Version 5.5 following the algorithm presented in Figure 2. INPUT DATA DETERMINATION OF ALLOWABLE SLOPES AND CORRESPONDING BED WIDTHS DETERMINATION OF FEASIBLE SLOPES CALCULATIONS THROUGHOUT THE STAGES Figure 2.CANDY Flow Chart OPTIMUM SOLUTION THROUGH BACKWARD RECURSIVE PROCEDURE CANDY is menu driven software where input/output data are created/stored as files which can easily be accessed by text editors. Using PASCAL facilities CANDY is organized as main programme and three units containing procedures (subroutines) related to the main parts of the programme (figure 3). Apart from the tabular presentations, the output results can also be presented in the form of

7 Hydraulic Engineering Software 63 graphics using Drawing Exchange Files (.DXF) which are supported by AutoCAD programme package. 1 NPUT 1 NPUT DATA 1 [\ FILE 1 I V 1 c ; A N D Y MAIN 1 PROGRAMME UNIT1 READ INPUT 1 DATA 1 \_^ UNIT2 CALCULATION 1 PROCEDURES 1 "^_^^ Figure 3. CANDY Organization Chart 1 I [) ^ OUTPUT OPTIMAL SOLUTION "^_^^ QUANTITIES SCREEN I "-^^ AutoCADf.0x0 I I 1 1 The present programme capabilities allows four values for the bed widths for each canal section to be optimized. Also, this programme gives five best solutions including the optimum one and the following four near-best solutions. The reason for this is the wish to include the engineering judgement in finding the best solution, e.g. solutions can be similar regarding the costs but different in shape and profile of the canal. Therefore, it is on the design engineer to chose the most acceptable solution. As an example, for input data as depicted in figure 1. using CANDY the optimal solutions as presented in figure 4 is obtained. Figure 4. Optimal Solution

8 64 Hydraulic Engineering Software Conclusion Dynamic Programming has been used as a mean for achieving two purposes: (a) to create a standard design procedure for canal design by defining the steps that can easily be translated into one of the computer languages (b) to introduce the techniques of optimization in order to put together the objectives and constraints of the design in such a way that an iterative trial-error procedure is avoided. Such approach to canal design represents the first step in algorithm definition based on the DP method application. Not all the conditions and principles that may occur in canal design have been presented here. Nevertheless, the described procedure can be easily modified and extended by changing or introducing new terms in the objective function and constraints. References 1.Chapter in a book 1. Hamdy Taha,Operations Research, Chapter 9, Dynamic (Multistage) Programming, pp , Macmillan Publishing Company, New York, USA, MSc Thesis 2. Goran DJ.Radovic,An Approach to Canal Design by Dynamic Programming,Catholic University Leuven, Leuven, Belgium, 1992

GRADUALLY VARIED FLOW

CVE 341 Water Resources Lecture Notes 5: (Chapter 14) GRADUALLY VARIED FLOW FLOW CLASSIFICATION Uniform (normal) flow: Depth is constant at every section along length of channel Non-uniform (varied) flow: