1st International Conference “From Scientific Computing to Computational Engineering”
1st IC-SCCE
Athens, 8-10 September, 2004
© IC-SCCE
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
Menelaos E. Theocharis* , Christos D. Tzimopoulos ** , Stauros I. Yannopoulos *** , Maria A.
Sakellariou - Makrantonaki ****
*
Dep. οf Crop Production Tech. Educ. Inst. of Epirus, 47100 Arta, Greece, e-mail: theoxar@teiep.gr
** Dep. of Rural and Surveying Engs, A.U.TH, 54006 Thessaloniki, Greece, e-mail: tzimop@vergina.eng.auth.gr
*** Dep. of Rural and Surveying Engs, A.U.TH, 54006 Thessaloniki, Greece, e-mail: giann@eng.auth.gr
**** Dep. οf Agric., Crop Prod., and Rural Envir., Univ. of Thessaly 38334,Volos,Greece, e-mail: msak@agr.uth.gr
Keywords: Pub station, head, network, cost, linear, nonlinear
Abstract. 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 head of the pump station. An increase of the head of
the pump involves a reduction of the construction cost of the network, as well as, an increase of the pumping
cost. It is mandatory for this reason to calculate the optimal head of the pump station, in order to derive the
minimum total cost of the irrigation network. The certain calculating methods in identified the above total
cost of a network, that have been derived are: the linear programming optimization method, the nonlinear
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 head using the linear programming method and the
nonlinear programming method is presented. Application and comparative evaluation in a particular irrigation
network is also developed.
1. INTRODUCTION
The problem of selecting the best arrangement for the pipe diameters and the optimal pumping head so as the
minimal total cost to be produced, has received considerable attention many years ago by the engineers who
study hydraulic works. The knowledge of the calculating procedure in order that the least cost is obtained, is a
significant factor in the design of the irrigation networks and, in general, in the management of the water
resources of a region. The classical optimization techniques, which have been proposed so long, are the
following: a) The linear programming optimization method [1,3,7,8,10], b) the nonlinear programming optimization
method [2,5,6,9,10,11], c) the dynamic programming method [10,13,14], and d) the Labye’s optimization method [4,10,12].
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 pipes diameters, the head losses, and the pipes’ lengths. As constraints are used: the pipe
lengths, and the available piezometric heads in order to cover the friction losses. In this study, a systematic
calculation procedure of the optimum pump station head is presented, using the linear programming method and
the nonlinear programming method is presented. Application and comparative evaluation in a particular
irrigation network is also developed.
2. THE VARIANCE OF THE HEAD OF THE PUMP STATION
The total cost of an irrigation network consists of the cost of the pipe network and the pumping cost. The cost
of the pipe network depends on the pipe diameters, which are related to the discharge and the available
piezometric head. As the available piezometric head of the pipes is proportional to the piezometric head of the
water intake, the cost of the pipe network directly depends on the pump station head.
The pumping cost consists of: a) the cost of the mechanical infrastructure and the expense for maintenance of
the pump station and b) the operational cost of the pump station. This cost is generally directly proportional to
the hydraulic head of the pump station.
Consequently, the increase of the pump station head involves an increase of the total pumping cost and a
reduction of the cost of the pipes network.
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
The optimal head of the pump station is the one that leads to the minimum total annual cost of the project.
In the studies of irrigation networks, the determination of the optimal head of the pump station is essential, so
that the most economic project to be planned.
As it is impossible to produce a continuous function between the optimal pump station head and the total cost
of the project, the calculation of the optimal pump station head is achieved through the following process:
a) the variance of the pump station head are determined, b) using one of the known optimisation methods, the
optimal annual total cost of the project for every head of the pump station is calculated, c) The graph ΡET.- HA is
constructed and its minimum point is defined, which corresponds in the value of Hman, This value constitutes the
optimal pump station head.
The variance of the values of the head is resulted as following [10]:
If in a complete route of an irrigation network the smallest allowed pipe diameters are selected, the maximum
value of the head of the pump station is resulted, in order to cover the needs of the complete route. A further
increase of this value, involves the setting of a pressure reducing valve on this route. Additionally, if in a
complete route the biggest allowed pipe diameters are selected, the minimum value of the pump station is
resulted, in order to cover the needs of the complete route. A further reduction of this value involves setting a
pump in this route. These two extreme values constitute the highest and the lowest possible value of the pump
station head respectively.
3. METHODS
3.1. The linear programming method
According to this method, the search for optimal solutions of hydraulic networks is carried out considering
that the pipe diameters can only be chosen in a discrete set of values corresponding to the standard ones
considered. Due to that, each pipe is divided into as many sections as there are standard diameters, the length of
these sections thus being adapted as decision variables. The least cost of the pipe network, PΔ, is obtained from
the minimal value of the objective function , meeting the specific functional and non negativity constraints.
a. The objective function
The objective function is expressed by [ 3,5]:
f (X)  CX
(1)
where C is the vector giving the cost of the pipes sections in Euro per meter and X is the vector giving the
lengths of the pipes sections in meter.
The vectors C and X are determined as:
C  C1....C i ... C n 

for i  1 , 2 ,..., n
C i  δ i1......δij ...δ ik

for i  1 , 2 ,..., n and
X  X1 ....X i ... X n T

X i  x i1......xij ... x ik
(2)
j  1 , 2 ,..., k
for i  1 , 2 ,..., n
T
for i  1 , 2 ,..., n and
(3)
(4)
j  1 , 2 ,..., k
(5)
where:
x11 , x12 , ... , x nk , are the decision variables in meter, ij is the cost of jth section of ith pipe in Euro per meter,
n is the total number of the pipes in the network , and k the total number of each pipe accepted diameters (= the
number of the sections in which each pipe is divided).
b. The functional constraints
The functional constraints are length constraints and friction loss constraints.
The length constraints are expressed by:
k
L i   x ij
(6)
 Δh i  H Α  h i
(7)
j1
The friction losses constraints are expressed by:
i
i 1
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
for all the nodes, i , where HA is the piezometric head of the water intake, hi is the minimum required
i
piezometric head at each node i. The sum  h i is taken along the length of every route i, of the network.
i 1
c. The non negativity constraints
The non negativity constraints are expressed by:
x ij  0
(8)
d. The optimization of the objective function
The minimal value of the objective function, min f(X)  C X , is obtained using the simplex method. The
calculation is carried out for :
min H A  H A  Z A  H man  max H A
(9)
3.2. The nonlinear programming method
a. The objective function
The minimal value of the pipe network, PN, is obtained from the minimal value of the objective function [8,10,11]:
 
i
Pi (h i )    y.z
i  1  h
 i
n




w
(10)
by determining the n number h i , meeting the specific functional and non negativity constraints, where i is the
random pipe of the network.
In the above relationship the function φ i is called “ the characteristic function” of the ith pipe and expressed by:
 
A
φi   ν 
C 
 o
1/w
z
(11)
Li Qi
where :
w
yν  2  x
,
2x
z

y  2  x
,
C0 
1.6465
f
0.2
(12)
and x = y = 0,5 are exponents in the Darcy – Weisbach formula .
A and ν are fitting coefficients of the Mandry’s cost function [5]:
ν
δ i  A.D i
(13)
The friction coefficient, f, is calculated using the Colebrook–White equation.
In each branch of the network:  i    i , where i denote the conduits of each branch.
b. The functional constraints
The functional constraints of the problem are expressed by:
i
 h i  H A  H i
(14)
i 1
for all i, where HA is the piezometric head of the water intake, Hi is the piezometric head of each lattice point i,
i
and the sum  h i is taken along the length of every consumption route i.
i 1
For the complete routes of the network the functional constraints of the problem are expressed by:
j
 h i  H A  H j
i 1
(15)
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
j
in which the sum
 h i is taken along the length of each complete route of the network and j is found at the
i 1
terminations of the network. If the number of terminations in the network is k , then the number of functional
constraints is k as well.
The solution, (calculation of the n-number h i ) after the mathematical analysis of the problem must also be
checked for the constraint of the nonintersecting of the resulting piezometric line with ground.
c. The non negativity constraints
The non negativity constraints are expressed by:
h i > 0
(16)
d. The optimization of the objective function
The minimization of the objective function that is obtained using Lagrange multipliers, concludes to the
system:
w
w
 
 
i

    j 


 H 
 H  j 
 i
(17)
and
H  j  (H A  H  j )  H i
(18)
where i is the random supplying branch and the sum includes all the secondary branches downstream the junction
point i.
e. The solution of the system
The system is solved using the secant method [8, 10,11]:
1. Initial values for H i and H  j are obtained from the relations:
 i  min S m L i
i).
(19)
where Li is the total length of the pipes of the branch i and :
min S m  min
HA  H j
(20)
j
 Li
i 1
which is the minimum average hydraulic gradient of the complete routes of the network and
H j  H A  H j  H i
ii).
(21)
where the sum denotes the total supplying branches of the route of the specific supplied branch.
2. Based on the initial values the following quantities are calculated:
 
Fi   i 
 Hi 
w
w  i 
i


F 

i
 H
H  H 
i
i 
i 
΄
F
w
(22a)
and
  j 

Gi   
 H  j 
w
 
w  j 
G 

i
 H
H   H  
i
j 
j 
΄
G
i
w
(22b)
and finally:
΄

F

i
F  Fi  G i . ΄

i
G

i
  F΄
 : 1  i
΄
 
  G i




(22c)
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
From F , H i 
i
i
1/ w
F
is calculated and from the second group of equations H  j . Thus the values of H i and
i
H  j of the first repetition are obtained. Using these values we peat the procedure and find the values of the
second repetition and so on, until the convergence of the values.
f. Calculating the linear and total losses of the pipes
The total losses of the pipes are calculated using the relation:
h   
i
(23)
i
for the supplier branches and the relation :
h ο i 
ΔH  j
 j
i
(24)
for the supplied branches. Local losses are equal to 10 % of the linear losses and therefore the linear losses are
  i
h  i
h f 
for the supplier branches and h f i 
for the supplied branches respectively.
i
1.10
1.10
g. Calculating the economic diameter Di of the pipes
The economic diameter of the pipes is calculated from [10]:

1   L i
Di 
Q
C 0 i  i  h f
i
 




0 .5 
0.4



(25)
e. Calculating the cost of the pipes Pi and cost of the network, PN
From the cost function, (eq.13), the cost of each pipe is calculated and then the total cost of the network:
PN 
i n
 Pi
(26)
i 1
4. THE TOTAL ANNUAL COST OF THE NETWORK
The total annual cost of the project is calculated by [10]:
Pan.  PN an.  PMan.  POan.  Ean.  0.0647767 PN  2388 .84QH man Euro
(27)
where PN an.  0.0647767 PN is the annual cost of the network materials , P M an.  265 .292 QH man is the annual
cost of the mechanical infrastructure , POan.  49.445 QΗ man
is the annual cost of the building infrastructure,
and Ε an.  2074 .11QΗ man is the annual operational cost of the pump station
5. THE SELECTION OF THE OPTIMAL HEAD OF THE PUMP STATION
For the variance of the pump station head min H A  H A  Z A  H man  max H A , Pan. is calculated. Then, the
graph Pan.    is constructed from which the minimum value of Pan. is resulted. Finally the Ηman = ΗA - ΖΑ
corresponding to minP an. is calculated, which is the optimal head of the pump station.
6. APPLICATION
The optimal pump station head of the irrigation network, which is shown in Figure 1, is calculated. The
material of the pipes is PVC 10 atm. The required minimum piezometric head at each node, hi, is resulted by the
elevation head at the network nodes, Zi, bearing in mind that the minimum required pressure head is 25 m at the
water consuming nodes and 4 m at the others. Figures 1 and 2 represent the real and ideal networks and provide
geometric and hydraulic details.
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
Figure 1. The real under solution network
6.1. The variance of the head of the pump station
The complete route presenting the minimum average gradient is A-39. The minimum and maximum friction
losses of all the pipes, which belong to this complete route, are calculated and the results are shown in table 1.
min H A  h 39   1.10 J i min L i   80 .80  2.77  83 .57 m and
39
From
the
table
is
resulted
that
i  K1
max H A  h 39   1.10 J i max L i   80 .80  21 .79  102 .59 m
39
i  K1
L= 885 m
h=77.20 m
π9
h=76.80 m
π6 πL=
9 1000 m
h=75. 55 m
π9 π 4
L= 1175 m
h=74.60 m
π2
h=74.10 m
π9
π9
L= 1195 m
L=200m L=120m L=185m
π8 L= 810 m
K36
L=150m
π10
L=50 m
L= 839 m π9
h= 80.80 m
K28
L= 915 m
π7
h= 79.50 m
K19
π5
L= 970 m
h= 78.60 m
K10 L= 670 m π3
h= 77.90 m
K1
L= 770 m
π1
h= 76.40 m
A ZA=52 m
Figure 2. The ideal under solution network
π9
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
6.2. The optimal head of the pump station
a.
The optimal cost of the network according to the linear programming method
The minimization of the objective function is obtained using the simplex method. The calculation is made for
pump heads: HA = 84.00 m till 102.00 m, where ΖΑ=52m. The results are presented in Table 2.
b.
The optimal cost of the network according to the nonlinear programming method
The optimal cost of the network, which is obtained using the secant method, which is applied using as the
initial values of ΔHi and ΔHπj, those which are resulted from the complete route presenting the minimum average
gradient. The calculation is made for pump heads: H A = 84.00 m till 102.00 m, where ΖΑ=52m. The results are
presented in Table 2.
Li
Pipe
[m]
39
38
37
36
Κ36
225
240
240
125
190
Minimum losses Maximum losses
maxDi min1,1Ji minDi max1,1Ji
[mm]
[%]
[mm]
[%]
160
0.249
110
1.607
225
0.167
140
1.774
280
0.120
160
1.966
315
0.114
200
1.103
450
0.071
250
1.342
Li
pipe
[m]
K28
K19
K10
K1
A
185
120
200
50
0
Minimum losses Maximum losses
maxDi min1,1Ji minDi max1,1Ji
[mm]
[%]
[mm]
[%]
500
0.090
315
0.909
500
0.194
355
1.083
500
0.309
400
0.951
500
0.450
450
0.766
0
0
Table 1. The minimum and maximum head losses of all the pipes, which belong to the complete route presenting
the minimum average gradient
HA
[m]
84
85
86
87
88
PN [€]
PN [€]
PN [€]
HA
HA
Linear Nonlinear [m] Linear Nonlinear [m] Linear Nonlinear
method
method
method
method
method
method
377362
373488
89 299360
297908
94 266292
261247
352158
351439
90 290911
288916
95 262157
255792
335087
334219
91 283707
280904
96 258885
250759
320760
320117
92 277403
273708
97 256757
246090
309074
308208
93 271466
267193
98 254770
241743
PN [€]
Linear Nonlinear
method
method
99 253397
237682
100 252340
233872
101 251850
230289
102 251522
226911
HA
[m]
Table 2. The optimal cost of the network, PN , for pump heads HA = 84.00 m till 102.00 m.
c. The optimal head of the pump station
The total annual cost of the project is calculated using the linear programming method, as well as the nonlinear
programming method, for pump heads: H A = ZA + Hman = 84.00 m till 102.00 m, where ΖΑ=52m. The results are
presented in Table 3. After that, the corresponding graphs Pan. - ΗA (Figure 3) are constructed and the min Pan. are
calculated. From the min Pan. the corresponding Ηman = ΗA - ΖΑ are resulted.
HA
[m]
84
85
86
87
88
PET [€]
PET [€]
PET [€]
HA
HA
Linear Nonlinear [m] Linear Nonlinear [m] Linear Nonlinear
method
method
method
method
method
method
43595
43344
89
41535
41443
94
42386
42059
42562
42515
90
41588
41457
95
42715
42304
42054
41998
91
41717
41536
96
43102
42576
41726
41683
92
41908
41669
97
43563
42872
41567
41510
93
42122
41845
98
44032
43189
PET [€]
Linear Nonlinear
method
method
99
44543
43524
100 45071
43876
101 45640
44243
102 46216
44622
HA
[m]
Table 3. The total annual cost of the project, Pan. , for HA = 84.00 m till 102.00 m.
From Table 3 and Figure 3 it is concluded that: (a) according to the linear programming method, the minimum
cost of the network is minPΕΤ. = 41443.87 €, produced for HA = 88.70 m with corresponding H man = 88.70 –
52.00 = 36.70 m and (b) according to the nonlinear programming method, the minimum cost of the network is
minPΕΤ. = 41296 €, produced for HA = 89,05 m with corresponding Ηman = 89.05 – 52.00 = 37.05 m.
Note: The values HA=88.70 m and 89.05 m resulted after the application of the above, for smaller subdivisions of H A value
which are not presented in Table 3.
3
P an.. [ 10 Euro]
Total annual cost of the project
Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makrantonaki
46,50
46,00
45,50
45,00
44,50
44,00
43,50
43,00
42,50
42,00
41,50
41,00
40,50
Linear method
Nonlinear method
83
84
85
86
87
88
89
90
91
92
93 94
95
96
97
98
99 100 101 102 103
Pump station head ΗΑ [m]
Figure 3. The optimal total annual cost of the network, P an, per the pump station head, HA.
7. CONCLUSIONS
The search of the optimal head of the pump station should be realized between the acceptable values, which
are resulted from the complete route presenting the minimum average gradient.
The optimal head of the pump station that results, according to the nonlinear programming method is
Hman = 37.05 m, while according the linear programming method is Hman = 36.70 m. These two values differ
only 0.95 %, while the corresponding difference in the total annual cost of the project is only 0.35 %.
The two 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.
8. REFERENCES
[1] Alperovits E. and Shamir U., 1977. Design of optimal water distribution. Water Res. Res. l3 (6): 885-900.
[2] Chiplunkar A. V., Khanna Ρ.,(1983). “Optimal design of branched water supply networks”, Jour. Env. Eng.
Div., ASCE, 109(3), 604-618
[3] Ioannidis, D. A., 1992. "Ανάλυση και εφαρμογή του γραμμικού προγραμματισμού σε συλλογικά δίκτυα υπό
πίεση και σύγκριση με τη μη γραμμική μέθοδο και τη μέθοδο του Labye", M.Sc. Thesis, A.U.TH., Salonika, (In
Greek).
[4] Labye Y., 1966, 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.
[5]Mandry, J. E., (1967). “Design of pipe distribution for sprinkler and drainage”, Jour. of the irrigation and
Drainage Div., ASCE. 93.
[6] Noutsopoulos, G. (1969). “ The economic piezometric line in gravity pipe distribution systems ”, Jour.
Technika Chronika., Athens, 1969(10), 661 – 676, (In Greek).
[7] Shamir U., 1974. Optimal design and operation. Water Res. Res, 10(1): 27-36.
[8] Smith D. V., 1966. Minimum cost design of linearly restrained water distribution networks. M. Sc. Thesis ,
Dept. of Civil Eng., Mass. Inst. of Techncl., Cambridge.
[9] Swamee, P.Κ., Kumar, V., and Khanna ,P.(1973a). “Optimization of dead end water distribution mains”,
Jour. Env. Eng. Div., ASCE, 99(2), 123-134.
[10] Theocharis, M. (2004). “Βελτιστοποίηση των αρδευτικών δικτύων. Επιλογή των οικονομικών διαμέτρων”,
Ph.D. Thesis, Dep. of Rural and Surveying Engin. A.U.TH., Salonika, (In Greek).
[11] Tzimopoulos, C. (1982). “Γεωργική Υδραυλική ” Vol. II, 51-94, Salonika, (In Greek).
[12] Tzimopoulos C., 1991. "Η Μέθοδος του Labye", Salonika, (In Greek).
[13] Vamvakeridou - Liroudia L., 1990. "Δίκτυα υδρεύσεων - αρδεύσεων υπό πίεση. Επίλυση – βελτιστοποίηση",
Athens, (In Greek).
[14] Yakowitz, S., 1982. Dynamic programming applications in water resources, Journal Water Resources.
Research, Vol. 18, Νο 4, Aug. 1982, pp. 673- 696.