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) j1 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 , 2x 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.