Design of Branched Water Supply Networks Using Labye’s
Method and Nonlinear Programming Optimization Method
Menelaos Theocharis 1) , Christos Tzimopoulos 2), Maria SakellariouMakrantonaki 3) , Stavros Yannopoulos 2), and Ioannis Meletiou 1)
1)
Technical Educational Institution of Epirus, Dep. of Crop Production, 47100 Arta, Greece, Tel. +302681050128,
e-mail: theoxar@teiep.gr
2)
Aristotle University of Thessaloniki, Dep. of Rural and Surveying Engineers, 54006 Thessaloniki, Greece.
3) University
of Thessaly, Dep. of Agricultural Crop Production and Rural Environment, 38334 Volos, Greece.
Introduction
The needed factors in the design of branched irrigation networks are the pipes and the
pumping costs. They both depend directly on the hydraulic head of the pump. It is mandatory
to calculate the optimal pump head as well as the corresponding economic pipe diameters, in
order to derive the minimal total cost of the irrigation network. The certain calculating
methods in identified the above total cost, which have been derived are: the linear
programming optimization method, the non linear programming optimization method, the
dynamic programming optimization method and the Labye’s method. All above methods
have grown independently and a comparative study between them has not yet been derived.
In this paper a comparative calculation of the optimal pump head as well as the
corresponded economic pipe diameters, using the Labye’s optimization method and the
nonlinear programming optimization method, is presented. Application and comparative
evaluation in a particular irrigation network is also developed.
Methods
The Labye’s optimization method
According to this method [1,2,3,4], the optimal solution of hydraulic networks is obtained
considering that the pipe diameters can only be chosen in a discrete set of values
corresponding to the standard ones considered. It consists of the tracing of a zigzag line in a
coordinates diagram, from which the minimal cost of the network can be obtained as a
function of the total piezometric losses of the network.
The nonlinear programming optimization method
According to this method [4,5], the minimal value of the pipe network is obtained from the
minimal value of the objective function by determining the optimal friction losses meeting the
specific functional and non negativity constraints. The minimization of the objective function
is obtained using the Lagrangian Multiplier Technique. This technique appends the equality
constraints to the objective function through Lagrangian multipliers to obtain the Lagrangian
function. The optimal solution is achieved by taking the first partial derivative of Lagrangian
function with respect to the decision variables (friction losses) and the Lagrangian multipliers
and equating the expressions thus obtained to zero.
Conclusions – Discussion
It is concluded that the total annual cost of an irrigation project which is obtained using the
above mentioned optimization methods is in fact the same in any case. The selected
economic pipe diameters have the same value when the two optimization methods are
applied. From the studied application it is concluded that the optimal annual cost of the
project according to the two optimization methods differs only 0.5 %. Similar differences have
been detected for a great number of applications made by authors, using the abovementioned methods. Therefore the two-optimization methods can be applied with no
distinction in the studying of branched hydraulic networks. The selection of optimal pipe
diameters using both the optimization methods, results 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. For this reason, various 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 in 2004 by M. Theocharis [6]. The proposed Theocharis’ method
requires only a handheld calculator and just a few numerical calculations and gives results of
similar accuracy to the ones of the classic optimization methods in much shorter calculation
time Consequently, the Theocharis’ simplified method can also be applied to irrigation
network design.
References
[1] Labye, Υ., Methodes permettant de determiner les caracteristiques optimales d 'un reseau de distribution
d'eau - Methode discontinue. Βull. Techn. du Genie Rural, Νο 50, Apr. 1961
[2] Labye, Y., Etude des procedés de calcul ayant pour but de rendre minimal le cout d’un reseau de distribution
d’eau sous pression. La Houille Blanche, 5: 577-583, 1966.
[3] Livaditis, E., Méthode discontinue de U. Labye pour la recherche d’une solution de coordination économique
des divers diamètres dans un d’irrigation sous pression. Jour. Tech. Chronika., Athens, 1969(10), 661-676.
[4] Theocharis, M., Irrigation networks optimization. Economic diameters selection. Ph.D. Thesis, Dep. of Rural
and Surveying Engin. A.U.TH., Salonika, 2004, (In Greek with extended summary in English).
[5] Theocharis, M., et al, Comparative Calculation of the Optimal Head of the Pump station of the Irrigation
Networks using a) The Linear Programming Method and b) The Nonlinear Programming method. Proc. 1st
IC-SCCE , Sept. 2004, Athens, Greece.
[6] Theocharis, M., et al, Dynamic method and a simplified nonlinear method in irrigation networks optimization,
Jour. WSEAS Transactions on Advances in Eng. Education, Issue 3, Vol. 2 , July 2005 , pp. 156–165.