06final paperic epsmsoathens,july6 9,2005

7/31/2019 06final Paperic Epsmsoathens,July6 9,2005

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1st International Conference on Experiments/Process/System Modelling/Simulation/Optimization

1st IC-EpsMsOAthens, 6-9 July, 2005

IC- EpsMsO

LINEAR PROGRAMMING AND DYNAMIC PROGRAMMING APPLICATION IN

IRRIGATION NETWORKS DESIGNMenelaos E. Theocharis1 , Christos D. Tzimopoulos 2 , Stauros I. Yannopoulos 2 , Maria A.

Sakellariou - Makrantonaki 3

1 Dep. f Crop Production Tech. Educ. Inst. of Epirus, 47100 Arta, Greece, e-mail:[email protected]

2 Dep. of Rural and Surveying Engs, A.U.TH, 54006 Thessaloniki, Greece

3 Dep. f Agric., Crop Prod., and Rural Envir., Univ. of Thessaly 38334,Volos,Greece

Keywords: Irrigation, head, network, cost, linear, dynamic

Abstract.The designating factors in the design of branched irrigation networks are the cost of pipes and the costof 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 ismandatory 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, thathave been derived are: the linear programming optimization method, the non linear programming optimization

method, the dynamic programming optimization method and the Labyes method. All above methods have grownindependently and a comparative study between them has not yet been derived. In this paper a comparative

calculation of the optimal head of the pump station using the linear programming method and the dynamicprogramming 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 theminimal 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 asignificant 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 thefollowing: a) The linear programming optimization method [1,2,4,5,6,7,8], b) the nonlinear programming optimization

method [6,7,8], c) the dynamic programming method [3,6,8,9], and d) the Labyes optimization method [3,6,7,8]. Thecommon 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 generallyused 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 calculationprocedure 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.

3. METHODS

3.1 The linear programming optimization method

According to this method, the search for optimal solutions of hydraulic networks is carried out consideringthat 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 ofthese sections thus being adapted as decision variables. The least cost of the pipe network, P, is obtained from

the minimal value of the objective function, meeting the specific functional and non negativity constraints.

a. The variance of the head of the pump station

The optimal head of the pump station is the one that leads to the minimum total annual cost of the project. Inthe studies of irrigation networks, the determination of the optimal head of the pump station is essential, so that
mailto:[email protected]#[email protected]:[email protected]#[email protected]:[email protected]#[email protected]


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Menelaos E. Theocharis, Christos D. Tzimopoulos , Stauros I. Yannopoulos , Maria A. Sakellariou - Makra ntonaki

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 ofthe known optimisation methods, the optimal annual total cost of the project for every head of the pump station

is calculated, c) The graph ET.- HA is constructed and its minimum point is defined, which corresponds in thevalue of Hman, This value constitutes the optimal pump station head. The variance of the values of the pump

station head is resulted as following[6]

: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 furtherincrease 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 isresulted, 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 pumpstation head respectively.

b. The objective function

The objective function is expressed by [ 2,6]:

CX)X(f = (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.......CCC ni1= for n,...,2,1i = (2)

.........C ikiji1i = for n,...,2,1i = and k,...,2,1j = (3)

( ) X.......XXX Tni1= for n,...,2,1i = (4)

( ) x.........xxX Tikiji1i = for n,...,2,1i = and k,...,2,1j = (5)

where:

nk1211 x,...,x,x , 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 (= thenumber of the sections in which each pipe is divided).

c. The functional constraints

The functional constraints are length constraints and friction loss constraints.

The length constraints are expressed by:

=

=k

1jiji xL (6)

The friction losses constraints are expressed by:

i

i

1ii hHh

=

(7)

for all the nodes, i , where HA is the piezometric head of the water intake, h i is the minimum required

piezometric head at each node i. The sum =

i

1iih is taken along the length of every route i, of the network.

d. The non negativity constraints

The non negativity constraints are expressed by:


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0xij (8)

e. The least cost of the pipe network

The least cost of the pipe network, P, is obtained from the minimal value of the objective function. The

minimal value of the objective function, XCf(X)min = , is obtained using the simplex method. The calculation

is carried out for :

AmanAAAHmaxHZHHmin += (9)

where min HA and max HA are as they are defined above in paragraph 3.1.a

f. The selection of the pump stationoptimal head

For the variance of the pump station head AmanAAA HmaxHZHHmin += , an.P is calculated. Then, the

graph an.P is constructed from which the minimum value of an.P is resulted. Finally the man = A -

corresponded to an.minP is calculated, which is the optimal head of the pump station.

e. Selecting the optimal pipe diameters

Using the simplex method the optimal pipe diameters, D, corresponded to the optimal head of the pumpstation are obtained.

3.2. The dynamic programming optimization 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 onesconsidered. The least cost of the pipe network 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 [ 6,8,9]:

*1)(iij

*i FminF ++= (10)

where *iF is the optimal total cost of the network downstream the node i, and ij is the cost of pipe i with a given

diameter j.

he decision variables, Dij, are the possible values for the accepted diameters of each pipe.

b. The functional constraints

The functional constraints [ 6,8,9] are specific functional constraints and non negativity constraints.The specific functional constraints for every complete route of the network, are expressed by:

i

ik k HhH

j

j + = (11)

where jH is the required minimal piezometric head value at j end, Hi is the required minimal

piezometric head value at node i, and the sum =

j

ikkh is the total friction losses from node i to the end j of

every complete route of the network.

The non negativity constraints are expressed by:

ih > 0 (12)

where i = 1... n, and n is the total number of the pipes in the network.

c. Calculating the minimal total cost of the network


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At first the minimal acceptable piezometric head value hi for every node of the network is determined. If the

node has a water intake, then hi = Zi+25 m, otherwise hi = Zi = 4 m.

Thereinafter the technically acceptable head values at the nodes of the network are obtained from the minimalvalue of the objective function [ 6,8,9]. The calculations begin from the end pipes and continue sequentially till the

head of the network, which is the edge of the last pipe. Because the decision variables can receive a limitednumber of distinguishable discontinuous values, the algorithm that will be applied is the complete model of

dynamic programming with backward movement (Backward Dynamic Programming, Full Discrete DynamicProgram-ming BDP, FDDP).

i). A network with pipes in sequence

The required piezometric head value at the upstream node of each end pipe is calculated [6], for every possible

acceptable commercial diameter, if the total friction losses are added to the acceptable piezometric head valuesfor the corresponding downstream end.

From the above computed head values, all those that have smaller value than the required minimum head

value at the upstream node, are rejected (Criterion A). The remaining head values are classified in declining orderand the corresponding costs that are calculated should be in ascending order. The solutions that do not meet this

restriction are rejected (Criterion B). The application of the B criterion is easier, if the checking begins from

the bigger costs. Each cost should be smaller than the precedent one otherwise it should be rejected. Thedescribed procedure is being repeated for the next (upstream) pipes. Further every solution that leads to a

diameter smaller than the collateral possible diameter of the downstream pipe is also rejected (Criterion C).

ii). Branched networks[6]

From now on, every pipe whose downstream node is a junction node will be referred as a branched pipe.

Defining of the downstream node characteristics

The heads at the upstream nodes of the contributing pipes are calculated. The solutions, which lead to a head

value that is not acceptable for all the contributing pipes, are rejected (Criterion D). The remaining head values,which are the acceptable ones for the branched node are classified in declining order.

The relevant cost of the downstream part of the network - which is the sum of the contributing branches costs - is

calculated for every possible node head value.

Defining of the upstream node characteristics

The total friction losses, corresponding to every diameter value of the branched pipe, are added to every headvalue at the downstream node and so the acceptable heads at the upstream node are calculated.

The total cost of the network part including the branched pipe is calculated for all the possible heads at theupstream node.

The acceptable characteristics of the upstream node are resulted using the same procedure with the one of thepipes in sequence (criteria B and C).

The described procedure is continued for every next branched pipe till the head of the network where the totalnumber of possible head values is resulted.

d. Selecting the pump stationoptimal headFor every calculated value of the pump station head, HA, that is resulted from the backwards dynamic

programming procedure, a corresponding value of the network cost, Pan, is computed[ 6,8,9]. Then, the graph Pan.-

HA is constructed and from that the minimum value of Pan is resulted. Finally the man = A - that

corresponds to minPan is calculated and this is the optimal head of the pump station.

e. Selecting the optimal pipe diameters[ 6,8,9]

The friction losses, corresponding to every acceptable diameter of the first pipe of the network, are subtractedfrom the optimal value of HA, and thus the possible head values at the end of the first pipe are calculated. The

biggest of these head values, which lies within the limits determined by the backwards procedure, is selected. Thediameter value that corresponds to this head value is the optimal diameter of the first pipe. This procedure has to

be repeated for every network pipe till the end pipes.

3.3 The total annual cost of the network

The total annual cost of the project is calculated by [6]:

=+++= .anOMN.na EPPPP .anan.an. manN QH.P.0 8423880647767 + Euro (13)


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where Nan.N P0647767.0P = is the annual cost of the network materials , manan.M QH292.265P = is theannual

cost of the mechanical infrastructure , Q.P manOan. 44549= is the annual cost of the building infrastructure,

and manan. Q11.2074 = is the annual operational cost of the pump station

4. 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 theelevation head at the network nodes, Z i, bearing in mind that the minimum required pressure head is 25 m at the

water consuming nodes and 4 m at the others. Figure 1 represents the real networks and provide geometric and

hydraulic details.

Figure 1. The real under solution network

4.1. The optimal cost of the network according to the linear programming method

a. The variance of the head of the pump station

The complete route presenting the minimum average gradient is A-39. The minimum and maximum frictionlosses of all the pipes, which belong to this complete route, are calculated and the results are shown in table 1.

From the table is resulted that ( ) m57.8377.280.80LJ10.1hHmin39

1Kiimini39A

=+=+==

and ( ) m59.10279.2180.80LJ10.1hHmax39

1Kiimaxi39A

=+=+==

Table 1. The minimum and maximum head losses of all the pipes, which belong to the complete route presentingthe minimum average gradient

PipeLi

[m]

Minimum losses Maximum losses

pipeLi

[m]

Minimum losses Maximum losses

maxDi

[mm]

min1,1Ji

[%]

minDi

[mm]

max1,1J i

[%]

maxDi

[mm]

min1,1Ji

[%]

minDi

[mm]

max1,1J i

[%]39 225 160 0.249 110 1.607 K28 185 500 0.090 315 0.909


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38 240 225 0.167 140 1.774 K19 120 500 0.194 355 1.083

37 240 280 0.120 160 1.966 K10 200 500 0.309 400 0.951

36 125 315 0.114 200 1.103 K1 50 500 0.450 450 0.766

36 190 450 0.071 250 1.342 A 0 0 0

b. The optimal cost of the network

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 =52m. The results are presented in Table 2.

Table 2. The optimal cost of the network, PN , for pump heads HA = 84.00 m till 102.00 m.

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

84 377362 87 320760 90 290911 93 271466 96 258885 99 253397 102 251522

85 352158 88 309074 91 283707 94 266292 97 256757 100 252340

86 335087 89 299360 92 277403 95 262157 98 254770 101 251850

c. The optimal head of the pump station

The total annual cost of the project is calculated using the linear programming method and the eq. (13), forpump heads: HA = ZA + Hman = 84.00 m till 102.00 m, where =52,00 m. The results are presented in Table 3.After all, the corresponding graphs Pan. - A (Figure 2) are constructed and the min Pan. is calculated. From the

minPan. the corresponding man = A - is resulted.

Table 3. The total annual cost of the project, Pan. , for HA = 84.00 m till 102.00 m.

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

84 43595 87 41726 90 41588 93 42122 96 43102 99 44543 102 46216

85 42562 88 41567 91 41717 94 42386 97 43563 100 45071

86 42054 89 41535 92 41908 95 42715 98 44032 101 45640

From Table 3 and Figure 2 it is concluded that the minimum cost of the network is minP. = 41443 ,produced for HA = 88.70 m with corresponding Hman = 88.70 52.00 = 36.70 m

Note: The value HA=88.70 m have been resulted after the application of the above, for smaller subdivisions of HA values

which are not presented in Table 3.

e. Selecting the optimal pipe diameters

Using the simplex method the optimal pipe diameters, D, corresponded to the optimal head of the pumpstation are obtained. The results are presented in Table 4.

Table 4. The optimal pipe diameters, DN , according to the linear programming method

Pipe

Length

[m]

DN

[mm] Pipe

Length

[m]

DN

[mm] Pipe

Length

[m]

DN

[mm] Pipe

Length

[m]

DN

[mm] Pipe

Length

[m]

DN

[mm]1 50 500 10 200 450 19 120 400 28 185 355 36 125 225

1 140 160 10 135 160 19 230 200 28 131 225 37 9 225

2 280 140 11 40 160 20 240 200 28 54 200 37 231 200

3 167 125 11 250 140 21 250 160 29 240 200 38 240 200

3 183 110 12 245 125 22 94 140 30 235 160 39 225 160

4 120 225 13 120 225 22 156 125 31 255 140 40 120 200

5 235 200 14 235 200 23 120 200 32 125 200 41 240 200

6 240 200 15 235 200 24 230 200 33 64 200 42 225 160

7 122 200 16 225 200 25 123 200 33 161 160 43 80 140

7 108 160 16 5 160 25 107 160 34 235 160 43 220 125

8 210 160 17 210 160 26 255 160 35 225 125

9 160 125 18 145 125 27 165 125 36 190 280


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4.2. The optimal cost of the network according to the dynamic programming method

a. The optimal cost of the network

The complete model of dynamic programming with backward movement (Backward Dynamic Programming,Full Discrete Dynamic Program-ming BDP, FDDP) is applied. The results are presented in Table 5.

Table 5. The optimal cost of the network, PN , according to the dynamic programming method

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

83,57 405033 85,79 345931 87,75 319724 89,50 300660 91,48 286134 93,38 274908 97,06 261429

83,74 395595 86,01 341371 88,06 316387 89,71 298800 91,75 284619 93,77 273359 97,57 260170

84,02 384593 86,25 338809 88,25 312555 90,02 297106 92,01 283533 94,06 271466 97,89 258266

84,49 371598 86,48 334756 88,50 310703 90,25 294676 92,23 281717 94,49 269746 98,79 256869

84,73 364916 86,76 330295 88,56 309127 90,50 292930 92,49 279850 95,04 267313 99,78 256232

85,00 360399 87,00 327143 88,79 307246 90,77 291689 92,82 279005 95,48 264974 100,24 255319

85,25 353925 87,23 324942 89,03 306172 91,04 290119 93,02 277241 95,99 261993 100,86 25442185,49 349940 87,48 321236 89,24 302955 91,24 288481 93,24 275839 96,48 261913 102,57 253858

c. The optimal head of the pump station

Using the eq. (13), the total annual cost of the project is calculated for pump heads, HA, the values given intable 5. The results are presented in Table 6. After all, the corresponding graphs Pan. - A (Figure 2) are

constructed and the min Pan. is calculated. From the minPan. the corresponding man = A - is resulted.

Table 6. The total annual cost of the project, Pan. , according to the dynamic programming method

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

HA[m]

PN[]

83,57 45130 85,79 42627 87,75 42104 89,50 41919 91,48 42166 93,38 42571 97,06 43900

83,74 44619 86,01 42465 88,06 42075 89,71 41922 91,75 42227 93,77 42706 97,57 44126

84,02 44073 86,25 42445 88,25 41937 90,02 41999 92,01 42313 94,06 42759 97,89 44197

84,49 43519 86,48 42321 88,50 41968 90,25 41978 92,23 42327 94,49 42905 98,79 44640

84,73 43228 86,76 42195 88,56 41907 90,50 42016 92,49 42360 95,04 43073 99,78 45192

85,00 43096 87,00 42139 88,79 41920 90,77 42095 92,82 42500 95,48 43187 100,24 45412

85,25 42823 87,23 42134 89,03 41993 91,04 42160 93,02 42509 95,99 43299 100,86 45723

85,49 42709 87,48 42040 89,24 41911 91,24 42175 93,24 42550 96,48 43583 102,57 46712

From Table 6 and Figure 2 it is concluded that the minimum cost of the network is minP . = 41907 ,produced for HA = 88.56 m with corresponding Hman = 88.56 52.00 = 36.56 m

d. Selecting the optimal pipe diameters

The optimal pipe diameters are calculated according to the procedure that has been described in paragraph3.2.e. The results are presented in Table 7.

Table 7. The optimal pipe diameters, DN , according to the dynamic programming method

PipeLength

[m]

DN[mm]

PipeLength

[m]

DN[mm]

PipeLength

[m]

DN[mm]

PipeLength

[m]

DN[mm]

PipeLength

[m]

DN[mm]

1 50 500 10 200 450 19 120 400 28 185 355 36 125 225

1 140 160 10 135 160 19 230 200 28 185 225 37 240 225

2 280 140 11 290 160 20 240 200 29 240 200 38 240 200

3 350 125 12 245 110 21 250 160 30 235 160 39 225 140

4 120 225 13 120 225 22 250 140 31 255 140 40 120 225

5 235 200 14 235 200 23 120 200 32 125 200 41 240 2006 240 200 15 235 200 24 230 200 33 225 200 42 225 160


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40,50

41,00

41,5042,00

42,50

43,00

43,50

44,00

44,50

45,00

45,50

46,00

46,50

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103

Pump station head [m]

Tota

lannualcostofthep

Pa

n..[

103E

uro]


7 230 200 16 230 200 25 230 200 34 235 160 43 300 125

8 210 140 17 210 160 26 255 160 35 225 110

9 160 140 18 145 140 27 165 110 36 190 280

Figure 2. The optimal total annual cost of the network, Pan, per the pump station head, HA.

5. CONCLUSIONS

The search of the optimal head of the pump station using the linear programming method, should be realizedbetween the acceptable values, which are resulted from the complete route presenting the minimum average

gradient.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 using the dynamic programmingmethod in a network with many branches.

The optimal head of the pump station that results, according to the linear programming method is Hman =

36.70 m, while according the dynamic programming method is Hman = 36,56 m. These two values differ only0.38 %. while the corresponding difference in the total annual cost of the project is only 1.12 %.

The two optimization methods in fact conclude to the same result and therefore can be applied with nodistinction in the studying of the branched hydraulic networks.

6. REFERENCES

[1] Alperovits E. and Shamir U., 1977. Design of optimal water distribution. Water Res. Res. l3 (6): 885-900.

[2] Ioannidis, D. A., 1992. "

Labye", M.Sc. Thesis, A.U.TH., Salonika, (In

Greek).

[3] Labye Y., 1966, Etude des proceds de calcul ayant pour but de rendre minimal le cout dun reseau de

distribution deau sous pression, La Houille Blanche,5: 577-583.

[4] Shamir U., 1974. Optimal design and operation. Water Res. Res, 10(1): 27-36.

[5] 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.

[6] Theocharis, M. (2004). . ,Ph.D. Thesis, Dep. of Rural and Surveying Engin. A.U.TH., Salonika, (In Greek).

[7] Tzimopoulos, C. (1982). Vol. II, 51-94, Salonika, (In Greek).

Dynamic method

Linear method

Pump station head HA [m]

Totalannu

alcostoftheproject

Pan[x103

]


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[8] Vamvakeridou - Liroudia L., 1990. " - . ",

Athens, (In Greek).

[9] Yakowitz, S., 1982. Dynamic programming applications in water resources, Journal Water Resources.Research, Vol. 18, 4, Aug. 1982, pp. 673- 696.

06final paperic epsmsoathens,july6 9,2005

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