Design of optimal irrigation networks

IRRIGATION AND DRAINAGE

Irrig. and Drain. 55: 21–32 (2006)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/ird.214

DESIGN OF OPTIMAL IRRIGATION NETWORKSy

MENELAOS E. THEOCHARIS1*, CHRISTOS D. TZIMOPOULOS2, STAVROS I. YANNOPOULOS2

AND MARIA A. SAKELLARIOU-MAKRANTONAKI3

1Department of Crop Production, Technological Educational Institute of Epirus, 47100 Arta, Greece2Department of Rural and Surveying Engineering, A.U.TH, 54006 Thessaloniki, Greece

3Department of Agriculture, Crop Production and Rural Environment, University of Thessaly, 38334 Volos, Greece

ABSTRACT

The problem of selecting the best pattern of pipe diameters of an irrigation network has long been given

considerable attention by engineers when designing hydraulic works. The classical optimization techniques, which

have been proposed until now, are the following: (1) the linear programming method; (2) the nonlinear

programming method; (3) the dynamic programming method; and (4) Labye’s method. Mathematical research

of the problem, using the previous methods, is very complex, and for this reason the numerical solution calls for a

lot of calculations, especially in the case of a network with many branches. In this study, the formulation of a new

simplified nonlinear programming method is presented, which can replace the existing ones with the best results. A

comparative evaluation between the proposed simplified method and the general nonlinear programming method

in a particular irrigation network is also developed. The results of the proposed simplified method are fully

identical with the results of the nonlinear programming method. Consequently, this method can be equally used for

the classical methods. Copyright # 2006 John Wiley & Sons, Ltd.

key words: irrigation; network; minimum cost; optimal design; linear method; nonlinear method; dynamic method; simplified method

RESUME

Le probleme du choix du systeme de diametres des canalisations d’un reseau d’irrigation a ete depuis longtemps

l’objet d’une attention particuliere de la part des ingenieurs de conception d’ouvrages hydrauliques. Les

techniques classiques d’optimisation proposees jusqu’a present sont les suivantes: (1) la methode de program-

mation lineaire; (2) la methode de programmation non lineaire; (3) la methode de programmation dynamique; et

(4) la methode du Labye. La solution mathematique du probleme, en utilisant les methodes precedentes, est tres

complexe et demande beaucoup de calculs, particulierement dans le cas d’un reseau avec beaucoup de Ious-

divisions. Dans cette etude, la formulation d’une nouvelle methode simplifiee de programmation non lineaire est

presentee, qui peut remplacer les methodes existantes avec de meilleurs resultats. Une evaluation comparative

entre la methode simplifiee proposee et la methode de programmation non lineaire generale dans un reseau

particulier d’irrigation est egalement developpee. Les resultats de la methode simplifiee proposee sont entierement

identiques aux resultats de la methode de programmation non lineaire. En consequence, cette methode peut

egalement etre employee pour les methodes classiques. Copyright # 2006 John Wiley & Sons, Ltd.

mots cles: irrigation; reseau; cout minimum; conception optimale; methode lineaire; methode non lineaire; methode dynamique; methodesimplifiee

Received 26 August 2005

Copyright # 2006 John Wiley & Sons, Ltd. Accepted 19 October 2005

* Correspondence to: Menelaos E. Theocharis, Department of Crop Production Technological Educational Institute of Epirus, 47100 Arta,Greece. E-mail: [email protected] de reseaux optimaux d’irrigation.

INTRODUCTION

The problem of optimal design of hydraulic networks, using the nonlinear programming method, concludes in a

system of nonlinear equations, where the unknown variables are the available frictional head losses in the network

branches. This system has a precise solution only in the case of a network having pipes in sequence (Swamee et al.,

1973; Swamee and Khanna, 1974; Tzimopoulos, 1982; Theocharis, 2004). For each branched network the system,

which consists of nþ k equations (n is the number of the supplying branches and k is the number of the ends of the

network), is nonlinear, so its complete mathematical solution is impossible (Noutsopoulos, 1969; Watanatada,

1973; Tzimopoulos, 1982; Chiplunkar and Khanna, 1983; Vamvakeridou-Lyroudia, 1990; Theocharis, 2004).

Since the numerical solution of the system calls for a lot of laborious operations, especially in the cases of networks

having many branches, the use of computers is indispensable (Chiplunkar and Khanna, 1983). A method which

does not require the use of computers is based on the observation (Theocharis, 2004) that each supplied branch of

the 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

there is a brief presentation of the proposed simplified optimization method, as well as a comparative evaluation

between this and the general nonlinear programming method. Application and comparative evaluation in a

particular irrigation network are also developed. The results of this comparison show that the total cost of the

network, resulting from the simplified method, differs from the cost of the general method only by 0.022%.

THE NONLINEAR PROGRAMMING OPTIMIZATION METHOD

For further study of the problem the import of certain symbols concerning the network is necessary. The network

endings are called ‘‘ends of the network’’. Every outlet—except from those coinciding with the ends—and every

point of the network where a pipe is branched in more than one pipe are called ‘‘nodes of the network’’. The first

ones are termed ‘‘simple nodes’’ and the second ‘‘junction nodes’’. A junction node may coincide with an outlet.

The part of a network within two successive nodes, or a node and an end, is called a ‘‘pipe (or conduit) of the

network’’. The total number of pipes is equal to the total sum of the nodes and the ends of the network. Each pipe is

denoted by the number of the downstream node or end. In branched networks, every outlet is supplied by only one

pipe and every node or end is joined to the previous node by only one pipe as well. The total number of pipes

within two successive junction nodes or a junction node and an end is defined as a branch of the network. The total

number of branches is equal to the total sum of the junction nodes and the ends. Each branch that leads to an end of

the network is called a ‘‘supplied branch’’ and all those remaining are ‘‘supplying branches’’. The total number of

pipes from the intake of the network up to every outlet is termed ‘‘route of the outlet’’. If an outlet coincides with an

end of the network, the corresponding route is called a ‘‘complete route’’. Each junction node is denoted by Ni,

where i ¼ 1; 2 . . . r . . . n and n is the total number of the nodes within the network. Nodes are numbered in

ascending order from upstream to downstream as N1;N2 . . .Nr . . .Nn. Each end is denoted by Nrj where

r ¼ 1; 2; . . . ; n is the junction node by which the branch is supplied and j ¼ 1; 2; . . . ; p is the number of the

supplied branches beginning from the junction node r. They are numbered from upstream to downstream as

N11 . . .N1j . . .Nr1 . . .Nrj . . .Nn1 . . .Nnj. Each supplying branch is denoted by Si where i is the symbol of

the downstream junction node. They are numbered in ascending order from upstream to downstream as

S1; S2 . . . Sr . . . Sn. Each supplied branch is denoted by Lrj, where r and j are as defined above. They are numbered

from upstream to downstream as: L11 . . . L1j . . . Lr1 . . . Lrj . . . Ln1 . . . Lnj. Each pipe within the supplying branch Si is

22 M. E. THEOCHARIS ET AL.

Copyright # 2006 John Wiley & Sons, Ltd. Irrig. and Drain. 55: 21–32 (2006)

https://www.researchgate.net/publication/245298641_Optimal_Design_of_Branched_Water_Supply_Networks?el=1_x_8&enrichId=rgreq-5d70290f2f125a32a22d68870cf93eeb-XXX&enrichSource=Y292ZXJQYWdlOzIzNzU1OTI4NDtBUzoxMDE3Mjc2MDkwMzI3MTNAMTQwMTI2NTE2MDU4OA==


https://www.researchgate.net/publication/279756758_Least-Cost_Design_of_Water_Distribution_System?el=1_x_8&enrichId=rgreq-5d70290f2f125a32a22d68870cf93eeb-XXX&enrichSource=Y292ZXJQYWdlOzIzNzU1OTI4NDtBUzoxMDE3Mjc2MDkwMzI3MTNAMTQwMTI2NTE2MDU4OA==


https://www.researchgate.net/publication/279760396_Optimization_of_dead_end_water_distribution_mains?el=1_x_8&enrichId=rgreq-5d70290f2f125a32a22d68870cf93eeb-XXX&enrichSource=Y292ZXJQYWdlOzIzNzU1OTI4NDtBUzoxMDE3Mjc2MDkwMzI3MTNAMTQwMTI2NTE2MDU4OA==


denoted by the symbol sit where t ¼ 1; 2; . . . ; v is the total number of pipes within the supplying branch. Each pipe

within the supplied branch Lrj is denoted by the symbol ‘rjq where q ¼ 1; 2; . . . ; t is the total number of pipes

within the supplied branch. In Figures 1 and 2 a network including the symbols that were described above is

presented.

Consider a single pipe within a branched network. The cost of pipe according to Mandry’s cost function

(Mandry, 1967) is expressed by

C ¼ AD�l ð1Þ

where C is the pipe cost (euros), A and � are constants depending on the material of pipe and determined according

to the least square method, D is the internal diameter of pipe (m) and l is the length of pipe (m). Using the Darcy-

Weisbach formula the frictional head loss is calculated as

�h ¼ �lV2

2gDð2Þ

Figure 1. An ideal branched network

Figure 2. A real branched network

DESIGN OF OPTIMAL IRRIGATION NETWORKS 23


where �h is the frictional head loss (m), � is the friction coefficient calculated from the Colebrook–White equation

and V is the velocity (m s�1). From Equation (2) is produced

D ¼ �0:2 l0:2Q0:4

1:6465�h0:2ð3Þ

in which Q is the discharge in the pipe (m3 s�1). From Equations (1) and (3) is produced

C ¼ f !

�h!�1ð4Þ

where ! ¼ 1 þ 0:2� . In the above relationship the function f is called ‘‘the characteristic function’’ of the pipe

and expressed by

f ¼ Al�Q2

12:10

� �!�1" #1=!

ð5Þ

Consider a branched irrigation network under its ideal form (suppose that the network consists only of branches

and all the single nodes are neglected) as shown in Figure 1. The cost of the supplying branch Si can be calculated

as

Cit ¼f !it

�h!�1it

¼ f !it�HSi

FSi

f it

� �!�1¼ f it

�HSi

FSi

� �!�1) CSi ¼

Pvt¼1 f t

�HSi

FSi

� �!�1¼ FSi

�HSi

FSi

� �!�1¼

F!Si

�H!�1Si

ð6Þ

where CSi is the cost of the Si supplying branch (euros), �HSi is the frictional head loss (m) and FSi ¼Pv

t¼1 ft is the

characteristic function of the Si branch. Therefore the minimal cost of a branched hydraulic network according to

the nonlinear programming optimization method is obtained from the minimal value of the objective function Z

which is expressed by

Z ¼Xni¼1

F!Si

�H!�1Si

" #þXnr¼1

Xkj¼1

F!Lrj

�H!�1Lrj

" #ð7Þ

meeting the specific functional and non-negativity constraints.

The functional constraints of the problem are expressed by

Xri¼1

�HSi þ�HLrj ¼ hN0� hNrj

for all r and j ð8Þ

where �HLrj is the frictional head loss in Lrj supplied branch (m), hN0is the piezometric head at the water intake

(m) and hNrjis the piezometric head at the rjth end (m). If the number of terminations in the network is k, then the

number of functional constraints is k as well.

The non-negativity constraints are expressed by

�HSi > 0 and �HLrj > 0 ð9Þ

The minimization of the objective function is obtained using the Lagrangian multiplier technique (Noutsopou-

los, 1969; Watanatada, 1973; Tzimopoulos, 1982; Chiplunkar and Khanna, 1983; Vamvakeridou-Lyroudia, 1990;

Theocharis, 2004). This technique appends the equality constraints to the objective function (cost function)

through Lagrangian multipliers to obtain the Lagrangian function. The minimization of the Lagrangian function

amounts to the minimization of the objective function as the constraints are satisfied. The optimal solution is

achieved by taking the first partial derivative of the Lagrangian function with respect to the decision variables (head

losses) and the Lagrangian multipliers and equating the expressions thus obtained to zero. The solution of these





equations results in a minimum cost design if the Lagrangian multipliers are positive. The obtained system of

nonlinear equations is

FSi

�HSi

� �!¼Xnr¼i

Xkj¼1

FLrj

�HLrj

� �!ð10Þ

and

�HLrj ¼ ðhN0� hLrjÞ �

Xri¼1

�HSi ð11Þ

The mathematical solution of the system is very complex and for this reason the numerical solution is achieved

through an iteration procedure, which calls for a lot of calculations, especially in the case of a network with many

branches. The system is usually solved using the secant method (Tzimopoulos, 1982; Theocharis, 2004). Initial values

for �HSið1Þ and�HLrjð1Þ are obtained from the complete route presenting the minimum average gradient and then the

values of�HSið2Þ and�HLrjð2Þ of the first iteration are obtained. The procedure is repeated and the values of the second

iteration are obtained and so on, until the convergence of the values which is successful after at least 12 iterations.

THE PROPOSED SIMPLIFIED NONLINEAR PROGRAMMING OPTIMIZATION METHOD

If an arbitrarily complete route, N0–Nrj, of the network is selected as shown in Figure 1, and the rest of the network

is ignored, the optimal head losses, �H0Si

and �H0Lrj

, in all the branches that belong in this route can be calculated

using the following relations:

FSi

�H0Si

" #!¼

FLrj

�H0Lrj

" #!ð12Þ

and

�H0Lrj

¼ ðhN0� hLrjÞ �

Xri¼1

�H0Si

ð13Þ

The solution of the system (Tzimopoulos, 1982) leads to

�H0Si¼ FSiPr

i¼1 FSi þ FLrj

� hN0� hNrj

� �and �H0

Lrj¼

FLrjPri¼1 FSi þ FLrj

� hN0� hNrj

� �ð14Þ

Next it will be proved that, if the complete route presenting the minimum average gradient is selected, the �H0Si

and �H0Lrj

which have been calculated from Equation (14) do not differ considerably from the �HSi and �HLrj

which have been calculated according to the general nonlinear method (Equations 10 and 11).

Study of the problem parameters

It is proved (Theocharis, 2004) that each supplied branch of a network:

(a) Tends to raise the economic piezometric line at the junction points above the economic piezometric line of the

network which does not include this branch. That means that if only a random complete route N0�Nrj of a

branched network (Figure 1) is selected and �H0Si

that belong to this are calculated, the resulting piezometric

heads at the junction points are in every case smaller than the corresponding piezometric heads which have been

calculated from the complete network according to the general nonlinear optimization method using Lagrange

multipliers. Therefore the following relation is validated:Xmi¼1

�H0Si>Xmi¼1

�HSi for m ¼ 1; 2; . . . ; r ð15Þ



(b) Decreases the optimal frictional losses in the upstream supplying branches, while on the contrary it increases

the optimal piezometric losses in both the downstream supplying branches and in all, without exception, the

supplied branches of the network. That is to say that if the random supplied branch Lmj belonging to the

complete route N0�Nrj is selected and the �HSi and �H0Si

corresponding to the real network and the network

remains as if the Lmj is ignored, are calculated, Figure 1, it is proved that:

�H0Si> �HSi for i ¼ 1; 2; . . . ;m ð16Þ

�H0Si< �HSi for i ¼ mþ 1;mþ 2; . . . ; r ð17Þ

and

�HLij > �H0Lrj

for i ¼ 1; 2; . . . ;m; . . . ; r ð18Þ

Calculating the average square error

For the �HSi that have been calculated according to the general method (Equations 10 and 11) and the �H0Si

that

have been calculated according to the simplified proposed method from the random complete route (Equation 14),

the square error is calculated from (Theocharis, 2004)

eSi ¼�HSi ��H

0

Si

�HSi

) e2Si¼ 1 þ

�H0

Si

�HSi

!2

�2�H

0

Si

�HSi

ð19Þ

and the average square error for all the supplying branches of the complete route N0�Lrj (Figure 1), is

em ¼Pr

i¼1 eið Þ2

r¼ 1 þ C2

r

Xri¼1

Xnr¼i

Xpj¼1

Fprj

�Hprj

!!" #2!

� 2C

r

Xri¼1

Xnr¼i

Xpj¼1

Fprj

�Hprj

!!" #1!

ð20Þ

where

C ¼hN0

� hNrjPri¼1 FSi þ FLrj

ð21Þ

Theoretical documentation of the method

All the above lead to the following conclusions (Theocharis, 2004).

The optimal head losses in each route N0� Ni within the complete route N0�Nrj, which are calculated

according to the proposed method, are bigger than the corresponding head losses calculated according to the

general nonlinear programming method.

Each supplied branch (i) tends to raise the piezometric line at the junction points above the piezometric line of

the network, which does not include this branch and (ii) tends to decrease the optimal head losses in all the

upstream supplying branches, while on the contrary it tends to increase the optimal head losses in all the

downstream supplying branches. In the first supplying branch (because only downstream supplied branches exist) a

significant percentage reduction of the optimal head losses is expected during the passage from the simplified to the

general optimization method. In the next supplying branches, this percentage difference is continuously decreased

(because both upstream and downstream supplied branches exist) and in the last supplying branch this can also be

reversed.



Consequently, the calculation error is important only in the first supplying branch, while in the following it is

progressively decreased, tending to zero or even to its inversion. Consequently the two optimization methods

present progressive convergence in each complete route.

The cost of each supplying branch as calculated from Equation (6), for the usual variation of !, is influenced

mainly by its FSi and secondarily (but by far less) by its �HSi . The error, consequently, that results from the

imprecise estimation of �HSi , is very small and for this reason great precision in its calculation is not necessary.

If the above-mentioned conclusions are applied in the complete route of the network presenting the minimum

average gradient, it is realized that the two piezometric lines, which result from the general method and the

simplified one (Figure 3), are polygonal lines which hang from the complete route edges, the piezometric line of

the simplified method is a convex function and, as has been proved, it surrounds the piezometric line of the general

method. Under the effect of the remaining supplied branches, the piezometric line of the simplified method tends to

be raised at the junction points in order to meet the real economic piezometric line (the piezometric line of the

general method). Because the quantities S�H0Si� S�HSi are not particularly important, finally the real economic

piezometric line does not differ significantly from the piezometric line of the simplified method.

Calculating the head losses of the pipes

(i) Pipes in the supplying branches. The head losses in each supplying branch Si, which belongs to

the complete route presenting the minimum average gradient N0�Nrj (Figures 1 and 2), are calculated from

Equation (14). The head losses of the pipes within these branches are calculated by

�h0sit ¼f sitPvt¼1 f sit

�h0Si ¼f sitPvt¼1 f sit

FSiPri¼1 FSi þ FLrj

hN0� hNrj

� �¼

f sitPri¼1 FSi þ FLrj

hN0� hNrj

� �ð22Þ

For the other supplying branches (Figures 1 and 2) the next procedure is followed: all the complete routes

beginning from the junction point Nr and having as an end the downstream edges of the network are determined

and the one which presents the minimum average gradient results, e.g. Nr � Nuj.

The head losses of each supplying branch, which belongs in this complete route, are calculated by

�H0Si¼ FSiPu

i¼rþ1 FSi þ FLuj

� hNr� hNuj

� �ð23Þ

and the head losses of the pipes within these supplying branches are calculated by

�h0sit ¼f sitPvt¼1 f sit

�h0Si ¼f sitPvt¼1 f sit

FSiPui¼rþ1 FSi þ FLuj

hNr� hNuj

� �¼

f sitPui¼rþ1 FSi þ FLuj

hNr� hNuj

� �ð24Þ

This procedure is repeated until all the supplying branches of the network are calculated.

Figure 3. The piezometric line of the complete route presenting the minimum average gradient according to the general and the proposedmethod



(ii) Pipes in the supplied branches. Using the piezometric heads at the junction points, which have been

calculated by the above-mentioned procedure, the head losses in the pipes within every supplied branch can be

calculated from

�h0‘rjq ¼f ‘rjq

FLrj

h0Nr� hNrj

� �ð25Þ

in which the h0Nris determined from

h0Nr¼ hN0

�Xri¼1

�h0Si ð26Þ

Calculating the pipe diameters

The diameters of all the pipes within the supplying branches of the network are calculated (Tzimopoulos, 1982;

Theocharis, 2004) as

D0sit¼ �sit

lsitQ2sit

12:10�h0fsit

" #0:2

ð27Þ

where

�h0fsit¼

�h0sit1:10

Similarly the diameters of all the pipes, within the supplied branches of the network, are calculated from

D0‘rjq

¼ �‘rjq

l‘rjqQ2‘rjq

12:10�h0f‘rjq

" #0:2

ð28Þ

Calculating the minimum cost of the network

The minimum total cost of the network is calculated from

P0N ¼

Xni¼1

Xvt¼1

f !sit

�h0!�1sit

" #þXnr¼1

Xpj¼1

Xtq¼1

f !‘rjq

�h0!�1‘rjq

" #ð29Þ

IMPROVEMENT OF THE PROPOSED SIMPLIFIED METHOD

Although the optimal head losses, �H0Si

, that have been calculated according to the proposed method, give the

possibility of determining with a satisfactory approach the economic diameters of the pipes, they have significant

divergences from the real values that result using the general nonlinear method. This weakness of the method is

confirmed (Theocharis, 2004) as it is described next.

If the discharge of a supplied branch is nullified and respectively the discharge of all the upstream to this

supplying branches is decreased in an amount that it is influenced by the supplied branch, the head losses in these

supplying branches are not expected to be altered, because: (a) the nihilism of a supplied branch discharge tends to

increase the head losses of all its upstream supplying branches and (b) the reduction of the supplying branches’

discharge tends to decrease the corresponding head losses almost at the same percentage.



Consequently, if the discharge of all the supplied branches, except the supplied branch of the complete route

presenting the minimum average gradient, is set to zero and the discharge of the supplying branches is replaced at

the same time by the value of the discharge that would result in the case that the supplied branches have been

neglected, the resulting �H0Si

are not expected to differ from the corresponding �HSi . After these, for the

calculation of �H0Si

a network has to be created, which would be similar to the real one. In this network FSi are

substituted by F0Si which are calculated if the discharge of all the supplied branches is nullified except the one that

belongs to the complete route presenting the minimal average gradient.

Thus, the result from the system of nþ k nonlinear equations is

F0Si

�H0Si

" #!¼

FLrj

�H0Lrjq

" #!ð30Þ

and

�H0Lrj

¼ ðhN0� hNrjÞ �

Xri¼1

�H0Si

ð31Þ

from the solution of which (Tzimopoulos, 1982; Theocharis, 2004) �H0Si

are resulted.

�H0Si¼

F0SiPr

i¼1 F0Sii

þ FLrj

ðhN0� hNrj

Þ ð32Þ

Figure 4. The real network under solution



From the �H0Si

and �H0Lrj

, the head losses �h0sit and �h0‘rjq of all the pipes which belong to the complete route,

hN0�hNrj are calculated. Using the �h0sit and �h0‘rjq and the real discharges of the pipes, the diameters of the pipes

within the complete route hN0�hNrj

, are calculated. The piezometric heads at the junction points, which have been

calculated by the above-mentioned procedure, are used as heads of the supplied branches of the network; the head

losses and then the diameters of the pipes for all the next supplied branches are calculated.

APPLICATION

Figure 4 shows a network under solution with hN0¼ 89:30 m. The material of the pipes is PVC 10 atm. Figures 4

and 5 represent the real and ideal networks and provide geometric and hydraulic details.

Solving the problem using the general nonlinear programming method

The complete route of the network presenting the minimum average gradient is N0�N51. For N0 ¼ 89:30 m the

minimum average gradient is SmN51¼ 0:5397% and the initial values of first approximation are: �HN1

¼ 0:27 m,

�HN2¼ 1:08 m, �HN3

¼ 0:65 m, �HN4¼ 1:00 m, �HN5

¼ 1:03 m, �HL11¼ 12:63 m, �HL12

¼ 14:93 m,

�HL21¼ 10:05 m, �HL22

¼ 13:35 m, �HL31¼ 8:70 m, �HL32

¼ 11:75 m, �HL41¼ 6:80 m, �HL42

¼ 9:50 m,

�HL51¼ 4:48 m, �HL52

¼ 8:08 m.

Applying the secant method for the ideal network, after the 15th iteration the values of �HSi are reached and

then the values of the piezometric heads HSi are calculated as shown in Table I.

Solving the problem using the simplified method and the improved simplified method

Applying both the simplified and the improved simplified method for the complete route presenting the

minimum average gradient, the values of �H0Si

are calculated and then the values of the piezometric heads, H0i , as

shown in Table I.

12 L12 l =1195 m L11 l =770 m 12

h12=74.10 m h12=76.

=

40 m

22 L22 l =1175 m L21 l = 670 m 21

h22=74.60 m h21=77.90 m

32 L32 l =1000 m L31 l = 970 m 31

h32=75.55 m h31=78.60 m

42 L32 l =810 m L41 l = 915 m 41

h42=76.80 m h41=79.50 m

52 L52 l =810 m L51 l = 830 m 51

h52=76.80 m h51=80.80 m

N

0

S 1

N1

S 2

N2

S3

N

3 S

4

N4

S 5 N N

N

N

N

N

5

l 1=5

0 m

l 2=2

00 m

l 3 =

120

m

l 4 =

185

m l

5=19

0 m

m30.89h0N

N

N

N

N

N

Figure 5. The ideal network under solution



Selecting the economic diameters of the pipes

Using the piezometric heads at the junction points, as they have been calculated before with the three methods,

as heads of the supplied branches of the network, the frictional head losses as well as the diameters of the pipes

within the other supplied branches are calculated. The results are shown in Table II. From the table and using

Mandry’s cost function, the total cost of the network is calculated by the three methods as following: (a) according

to the general method: PN¼s 295 108; (b) according to the simplified method: PN¼s 300 887; and (c) according

to the improved simplified method: PN¼s 295 173.

Comparing the above costs, it is concluded that (a) the total cost resulting from the simplified method differs

from the cost of the general method by 1.958%; (b) the total cost resulting from the improved simplified method

differs from the cost of the basic method by only 0.022%.

Table I. Values of �Hi and Hi according to the general and to the proposed method

Branch General method Simplified method Improved simplified method

�Hi (m) Hi (m) �H0i (m) H0

i (m) �H0i (m) H0

i (m)

S1 0.31 88.99 0.54 88.76 0.31 88.99S2 1.22 87.77 1.97 86.79 1.25 87.74S3 0.73 87.04 1.04 85.75 0.75 86.99S4 1.13 85.91 1.31 84.44 1.16 85.83S5 1.32 84.59 1.10 83.34 1.19 84.64

Table II. Final values of �hi and Di according to the general and the proposed method

Pipe General Simplified Improved Pipe General Simplified Improvedmethod method simplified method method simplified

method method

�hrjq Drjq �h0rjq D0rjq �h0rjq D0

rjq �hrjq Drjq �h0rjq D0rjq �h0rjq D0

rjq

(m) (mm) (m) (mm) (m) (mm) (m) (mm) (m) (mm) (m) (mm)

s11 0.31 425 0.54 380 0.31 424 l314 1.39 123 1.21 126 1.42 122l111 3.14 141 3.03 142 3.09 141 l321 1.89 183 1.66 188 1.86 184l112 5.06 126 4.95 127 5.04 126 l322 3.23 172 2.84 177 3.18 173l113 4.39 104 4.38 105 4.46 104 l323 2.77 159 2.46 163 2.76 160l121 2.06 193 2.00 194 2.03 194 l324 2.48 143 2.22 146 2.50 143l122 3.67 184 3.58 185 3.63 184 l325 1.11 118 1.02 120 1.14 118l123 3.33 173 3.26 174 3.32 173 s41 1.13 309 1.31 299 1.16 307l124 2.74 160 2.71 160 2.75 160 l411 1.75 186 1.33 197 1.71 187l125 2.02 143 2.02 143 2.05 143 l412 1.95 172 1.49 182 1.92 173l126 1.07 119 1.09 118 1.11 118 l413 1.54 154 1.19 163 1.53 155s21 1.22 395 1.97 359 1.25 393 l421 1.16 128 0.92 134 1.18 128l211 2.63 145 2.34 148 2.59 145 l422 1.93 169 1.59 176 1.89 170l212 4.56 130 4.10 133 4.54 130 l423 2.99 156 2.48 162 2.94 157l213 2.67 107 2.45 109 2.71 107 l424 2.52 140 2.12 145 2.51 140l221 1.85 197 1.68 201 1.81 198 l425 1.67 116 1.44 120 1.70 116l222 3.28 188 3.01 191 3.24 188 s51 1.32 258 1.10 267 1.19 263l223 2.92 177 2.69 180 2.90 177 l511 0.78 203 0.51 220 0.78 203l224 2.45 163 2.28 166 2.46 163 l512 1.29 187 0.85 204 1.30 187l225 1.81 146 1.70 148 1.83 146 l513 1.04 168 0.70 182 1.06 167l226 0.87 121 0.83 122 0.90 120 l514 0.68 139 0.46 150 0.71 138s31 0.73 360 1.05 336 0.75 358 l521 1.42 178 1.15 186 1.40 179l311 2.66 179 2.22 186 2.61 180 l522 2.43 165 2.00 172 2.43 165l312 2.38 166 2.01 171 2.36 166 l523 1.84 148 1.53 153 1.86 147l313 2.00 148 1.71 153 2.01 148 l524 1.70 123 1.45 127 1.76 122



CONCLUSIONS

The results of the proposed simplified method and especially those of the improved method are fully identical with

the results of the nonlinear programming method. The calculating procedure required for the determination of the

available piezometric losses is much shorter when using the simplified method than when using the general one—

in the first case k independent linear equations have to be resolved, while in the other there is a system of nþ k

nonlinear equations. 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 method.

REFERENCES

Chiplunkar AV, Khanna P. 1983. Optimal design of branched water supply networks. Journal of Environmental Engineering Division, ASCE

109(3): 604–618.

Mandry JE. 1967. Design of pipe distribution for sprinkler and drainage. Journal of the Irrigation and Drainage Division, ASCE 93.

Noutsopoulos G. 1969. The economic piezometric line in gravity pipe distribution systems. Journal of Technika Chronika., Athens 1969(10):

661–676 (in Greek).

Swamee P, Kumar V, Khanna P. 1973. Optimization of dead end water distribution mains. Journal of Environmental Engineering Division,

ASCE 99(2): 123–134.

Swamee P, Khanna P. 1974. Equivalent pipe methods for optimizing water networks—facts and fallacies. Journal of Environmental

Engineering Division, ASCE 100(1): 93–99.

Theocharis M. 2004. Irrigation networks optimization. Economic diameter selection, PhD. thesis, Dept of Rural and Surveying Engineering,

A.U.TH., Salonika (in Greek with extended summary in English).

Tzimopoulos C. 1982. Agricultural Hydraulics. Vol. II, 51-94, A.U.TH., Salonika (in Greek).

Vamvakeridou-Lyroudia LS. 1990. Pressure water supply-irrigation networks. Solution-Optimization. Hydraulics Engineering Computer

Applications, Athens (in Greek).

Watanatada T. 1973. Least cost design of water distribution systems. Journal of the Hydraulics Division, ASCE 99(9): 1497–1513.








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