The designating factors in the design of branched irrigation networks are the cost of pipes and the cost of pumping. They both depend directly on the hydraulic pump head. It is mandatory for this reason to calculate the optimal pump head as well as the corresponding economic pipe diameters, in order the minimal total cost of the irrigation network to be produced. The classical optimization techniques, which have been proposed so long, are the following: the linear programming optimization method, the nonlinear programming optimization method, the dynamic programming optimization method and Labye’s method. The mathematical research of the problem using the above classical optimization techniques is very complex and the numerical solution calls for a lot of calculations, especially in the case of a network with many branches. For this reason, many researchers have developed simplified calculation methods with satisfactory results and with less calculation time needed. A simplified nonlinear optimization method has been developed at the Aristotle University of Thessaloniki — Greece by M. Theocharis. The required calculating procedure is much shorter when using Theocharis’ simplified method than when using the classic optimization methods, because Theocharis’ method requires only a handheld calculator and just a few numerical calculations. In this paper a comparative calculation of the pump optimal head as well as the corresponded economic pipe diameters, using: (a) Labye’s optimization method, (b) the linear programming optimization method and (c) Theocharis’ simplified nonlinear programming method is presented. Application and comparative evaluation in a particular irrigation network is also developed. From the study it is concluded that Theocharis’ simplified method can be equally used with the classical methods.
The problem of selecting both, the best arrangement for the pipe diameters and the optimal pump head, so that the minimal total cost of a project to be produced, received considerable attention many years ago by the engineers who study hydraulic works. The classical optimization techniques, which have been proposed so far, are the following: (a) The linear programming optimization method [1], [2], [3], [4], [5], [6], [7] and [8], (b) the nonlinear programming optimization method [4], [5], [8] and [9], (c) the dynamic programming method [4] and [8], and (d) Labye’s optimization method [4], [7], [8], [10], [11], [12], [13] and [14]. The common characteristic of all the above techniques is an objective function, which includes the total cost of the network pipes, and which is optimized according to specific constraints. The decision variables that are generally used are: the pipe diameters, the head losses, and the pipe lengths. As constraints are used: the pipe lengths, and the available piezometric heads in order to cover the friction losses. The mathematical research of the problem using the above classical optimization techniques is very complex and the numerical solution calls for a lot of calculations, especially in the case of a network with many branches. A simplified nonlinear programming optimization method which does not require the use of computers is based on the observation[4], [6], [15] and [16] that each supplied branch of a branched network tends to raise the economic hydraulic gradient at the junction points, above the economic hydraulic gradient of the network, which does not include this branch. Thus, if the complete route of the network presenting the minimum average hydraulic gradient is selected, the economic piezometric line corresponding to this (assuming that all the other supplied branches are neglected) tends to be raised under the influence of the supplied branches, which at first had been neglected, so that the final hydraulic gradient does not differ remarkably from this economic hydraulic gradient. But only one equation is needed to calculate the economic hydraulic gradient corresponding to the complete route of the network presenting the minimum average gradient. Using the piezometric heads at the junction points, which have been calculated by the above-mentioned procedure from the complete route presenting the minimum average gradient, as heads of the supplied branches, the frictional head losses and the diameters of the pipes within the supplied branches can be easily calculated.
In the present work a brief presentation of: (a) Labye’s optimization method, (b) the linear programming optimization method and (c) Theocharis’ simplified nonlinear programming optimization method, is presented. Application and comparative evaluation in a particular irrigation network are also developed. The results of this comparison show that the total annual cost of the project is in fact the same in any case. So Theocharis’ simplified method can be applied with no distinction in the study of the branched hydraulic networks.
From Table 4, Table 7 and Table 10 it is concluded that the economic pipe diameters, which are selected, have approximately the same values when the three optimization methods are applied.
From Table 3, Table 6 and Table 9 the minimal total cost of the project is calculated by the three methods as following: (a) according to the linear programming optimization method: View the MathML sourcePan.=41443€; (b) according to Labye’s optimization method: View the MathML sourcePan.=41660€; and (c) according to Theocharis’ simplified method: View the MathML sourcePan.=41296€. Comparing the above costs, it is concluded that the total cost resulting from Theocharis’ simplified method differs from the cost of: (a) the linear programming optimization method by 0.35%; and (b) Labye’s optimization method by only 0.87%. Similar differences have been detected for a great number of applications made by authors, using the above-mentioned methods.
The three optimization methods in fact conclude to the same result and therefore can be applied with no distinction in the studying of the branched hydraulic networks.
The selection of economic pipe diameters using the linear programming method as well as Labye’s method, results in a vast number of complex calculations. For this reason the application as well as the supervision of these methods and the control of the calculations is time consuming and difficult, especially in the case of a network with many branches.
The results of the proposed Theocharis’ method are fully identical with the results of the linear programming method as well as with the results of Labye’s method. The calculating procedure required is much shorter when using the simplified method than when using the other two methods. Therefore the proposed simplified method is indeed very simple to handle and for practical uses it requires only a handheld calculator and just a few numerical calculations. Consequently, the simplified method can be equally used for the classical methods.
