- Email: [email protected]
Optimization of irrigation water distribution networks, layout included
agricultural water management 88 (2007) 110–118
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/agwat
Optimization of irrigation water distribution networks, layout included ´ lvarez, J.M. Tarjuelo Martı´n-Benito * P. Planells Alandı´, J.F. Ortega A Centro Regional de Estudios del Agua (Regional Centre of Water Research), Castilla-La Mancha University, Campus Universitario, s/n, E02071 Albacete, Spain
article info
abstract
Article history:
The purpose of this paper is to develop a procedure which takes into account both the
Accepted 12 October 2006
network layout and the pipe size of an on-demand water branched network in order to
Published on line 15 November 2006
obtain the lowest total cost (investment and energy cost). The process begins with a main ring network which is obtained by giving consideration to the possible alternatives of
Keywords:
branched irrigation networks, taking into account the limitations imposed by plot bound-
Irrigation networks
aries, gravel roads, etc. The optimization process progressively eliminates pipes to obtain
Layout
the most inexpensive branched irrigation network, verifying the criteria of the economic
Economic optimization
series methodology for sizing the pipes and the design flow obtained in the pipelines according to Cle´ment’s methodology. The dimensioning through the Economic Series Method determines the optimum diameter as a function of flow. Three stages are considered in the optimization process. In the first stage, the network layout and the pipe size of the lowest cost are obtained for different K values, as well as the necessary upstream head. In the second stage, the pumping station is dimensioned for each K value and annual costs for investment and energy consumption are obtained, according to the operation hours of the pumping station to satisfy crop water needs. In the third stage, the K value which leads to the lowest total cost (investment and energy cost) is determined. In the procedure, the network can be fed by direct injection (i.e. a fixed and a variable speed pump discharge directly into the network) or with a water tank (i.e. placed at an adequate head). The different network layouts and associated cost can be studied depending on the water source or sources in the network. This method can also be used for the layout of the main ring networks, establishing some minimum diameters as a reference to eliminate the pipelines. An interesting case could be the analysis of the network supply from several points. The sizing of the pipes that is performed, using the solutions that lead to the optimum network layout, is only a first approximation. Therefore, once the network with the optimum layout is determined, a new procedure for pipe size optimization should be used. # 2006 Elsevier B.V. All rights reserved.
1.
Introduction
Once the water source, the pressure necessities, the hydrant discharge and its approximate location in the plot are defined,
the design of a collective pressurized on-demand irrigation network can be summarized in five stages (Labye et al., 1988): (a) optimum network layout in order to minimize the total cost of the network (Awumah et al., 1989; Bhave and Lam, 1983;
* Corresponding author. Tel.: +34 967 599304; fax: +34 967 599238. E-mail address: [email protected] (J.M. Tarjuelo Martı´n-Benito). 0378-3774/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2006.10.004
agricultural water management 88 (2007) 110–118
Nomenclature ai
A, a Aar CRF Ct CT D DOT f g Dh H Hc Hg hi Hr Ir ki L N Ns Ne Nee
Nes
OQ p P Pi/g
Pti/g
Q qa qd qri Qi Qc Qp R Rr T td tr
coefficient that can be 0 or 1 depending on whether the hydrant is closed or open at that given time constants in the costs equation of the pipes average application rate of the system (mm h1) capital recuperation factor cost of the pipes (s) total investment annuity cost of network (s) internal diameter (m) the daily operation time (h) friction factor acceleration of gravity (m s2) available energy for head losses in the pipe series to be sized (m) pumping pressure (m) required head at upstream (m) hydrant elevation (m) number of downstream hydrants in the pipeline i friction head losses (m m1) irrigation interval (day) the number of downstream pipelines from the yet to be sized pipeline within the series studied pipe length (m) number of existing hydrants in the network number of irrigation subunits per plot number of equivalent hydrant discharges number of incoming equivalent hydrant discharges in the node (including those corresponding to the pipes and punctual inflow) number of outgoing equivalent hydrant discharges in the node (including those corresponding to the pipes and punctual outflow) network operating quality average hydrant operation probability electric power (kW) pressure head excess in the nodes i with respect to the setting pressure in the reducing valve (Pti/g) (m) reducing valve setting pressure, according to the minimum pressure necessary in the hydrant (m) flow (m3 s1) average hydrant discharge in the network (L s1) hydrant discharge (L s1) real discharge of hydrant i (L s1) design flow of pipeline i (m3 s1) flow in the upstream pipeline for the peak period (m3 s1) pre-fixed upstream discharge (m3 s1) drag coefficient (s2 m5) irrigation requirements (mm day1) number of pipes in the main ring network is the average daily operation time of the hydrant in the network (h) irrigation time in the peak period (h)
U v Vd
111
percentile of the normal distribution function associated with a definite operating quality (OQ) average water speed in the pipe (m s1) irrigation water volume to be applied per day (m3 day1)
Greek letter h
pump efficiency (decimal)
Granados, 1990); (b) calculation of the hydrant discharge according to plot sizes (Planells et al., 2001); (c) determination of design flow per pipeline (associated with a determined supply guarantee) (Cle´ment and Galand, 1979; Pulido-Calvo et al., 2003); (d) calculation of the optimum pipe size diameters, minimizing the investment and energy cost (Labye et al., 1988; Lansey and Mays, 1989; Pe´rez et al., 1996; Diopram, 2003); (e) analysis of the network performance under different operating conditions (Rossman, 1997; Aliod et al., 1997) to determine the possible supply failure situations of the network or of the pumping plant (Lamaddalena and Sagardoy, 2000). The network layout performance usually begins with a non-restricted layout (Girette method) using the approaches by proximity and minimum length. Afterwards, some restrictions such as layout for roads or plot boundaries are considered (Granados, 1990). The connection of the nodes in the network can be carried out according to geometric criterion (proximity), hydraulic criteria (pressure, discharge, etc.) or a combination of both (Martı´nez et al., 1993). These authors also propose the criteria of the radial union of the hydrants, which is a more advantageous variant of the proximity criteria, even though the result is conditioned by the location of the water source in the network. The algorithm consists of: ‘‘(a) placing the hydrants in increasing order of distance to the initial point, and (b) join the first one to the second, and then the third one to the closest of those already joined’’. Likewise, the remaining hydrants should be joined according to the order established at the beginning. Some additional conditions to bear in mind for the network layout are: Geological–geotechnical: rocks, unstable land, groundwater table, etc. Orographic: cavitation risk originated by the topography, placement of airholes, drainage, etc. Geographical: plot distribution, communication routes, buildings, etc. Social: right-of-way creation, temporal occupation, expropriation, etc. To design the network layout by means of the abovementioned methods, the pipeline cost is supposed to be only function of the pipe lengths, without taking into account the pipe diameters. The purpose of this paper is to develop a procedure which takes into account both the network layout and the most economic pipe size, in order to obtain the least amount of the total cost of the design (investment and energy cost). In the procedure, the network can be fed by direct
112
agricultural water management 88 (2007) 110–118
injection (i.e. fixed and variable speed pump discharge directly into the network) or with a water tank (i.e. placed at an adequate head). The different network layouts and associated cost can be studied depending on the water source or sources in the network.
2.
Methodology
It is possible to optimize an on-demand network layout by selecting the layout and the most economic pipe sizing at the same time. In the network design optimization process, the following costs should be considered: cost of the network ð1Þ þ cost of the pumping plant ð2Þ þ energy cost ð3Þ These three costs are not unrelated, so both the network (1) (its layout and pipe size) and the pumping plant (2) (number of fixed and variable speed pumps, automation equipment for regulation and control, etc.) are designed for the peak period of irrigation requirements. However, the annual energetic cost (3) is a function of the upstream consumed power, which varies during the irrigation season according to the discharge and head. This also depends on the efficiency of the pumping plant, which is a function of its design and its capacity of adaptation to the demand conditions of the network, according to the adopted regulation system (Planells et al., 2005). The optimization process can be structured in three stages. First, the cost of the pipes is determined using simultaneously both the network layout and pipe size for the most unfavorable situation (conditions of highest flow demanded in the network) simultaneously. In the second stage, the energy and the annual pumping investment costs are evaluated. In the third stage, the lowest total cost is determined.
2.1.
First stage
Optimization of the network layout and the economic pipe size at the same time. Once the optimum branched irrigation network and its pipe size are defined, the necessary upstream head is determined to develop the second stage. The process begins with a main ring network which contains all the nodes included in the network. The network is obtained by giving consideration to the possible alternatives of branched irrigation networks, while taking into account the limitations imposed by plot boundaries, gravel roads, etc. At this stage, non-outflow nodes can be created. From the network, the exact lengths of the pipelines and the topography level of the nodes will be known. The discharge of the hydrants could also be calculated for crop rotation in the peak consumption period. The optimization process use the economic series methodology for sizing the pipes (Pe´rez et al., 1996) and the design flow obtained in the pipelines according to Cle´ment’s methodology (Cle´ment, 1966). In order to simplify the process, a number of equivalent hydrant discharge (Nei) assigned to each selected pipe ‘‘i’’ can be deduced. The Nei is found by dividing the sum of the hydrant downstream discharge (qdk)
within pipeline i, by the average hydrant discharge of the network (qa). Both are calculated as PN qa ¼
j¼1
qd j
N
Phi and Nei ¼
k¼1
qdik
(1)
qa
where qd is the hydrant discharge (L s1); N the number of hydrants existing in the network; hi is the number of hydrants downstream from the pipeline i. The design flow of each pipeline (Qi) can be determined using Cle´ment’s methodology (Cle´ment and Galand, 1979), with the average probability ( p) of the set of hydrant (ki) downstream from the pipeline yet to be sized. The result is as follows (Planells et al., 2001): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u hi hi X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuX Q i ¼ p qdik þ U pð1 pÞt q2di k¼1
k¼1
k
(2)
with p¼
td DOT
where p is the average hydrant operation probability; td the average daily operation time of the hydrant in the network (h); DOT the daily operation time (h); qdj the hydrant j discharge (L s1); U is the percentile of the normal distribution function associated with a definite operating quality (OQ). Substitution of Eqs. (3) into (2) results in Eq. (4): hi X qdik ¼ Nei qa
(3)
k¼1
Q i ¼ pNei qa þ U
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi pð1 pÞ Nei q2a
(4)
Ph
where the quadratic mean k¼1 q2di =Nei has been replaced, in k first approximation, by the arithmetic mean qa to simplify the calculations. To verify Clement’s formula and find all the nodes linked within the branched irrigation network, the continuity of the continuous hypothetical flows in the nodes is imposed as a restriction (Planells et al., 2001): X
pNee qa ¼
X
pNes qa
(5)
where Nee is the number of incoming equivalent hydrants discharge in the node (including those corresponding to the pipes and punctual inflow); Nes is the number of outgoing equivalent hydrants discharge in the node (including those corresponding to the pipes and punctual outflow). Since p and qa are constants in the network, the following formula could be deduced: X
Nee ¼
X
Nes
(6)
Having established the hydrant discharge, Nes values are evident in the nodes that directly supply the hydrants. Ne values in the rest of the pipelines are raised as unknown quantities to be determined during the optimization process.
agricultural water management 88 (2007) 110–118
Since the design flows in the pipelines of the network are obtained according to Cle´ment’s methodology, the continuity in the nodes is verified for the number of equivalent hydrant discharges (Ne) (Eq. (6)) but not by the flow rates nor by the equilibrium of friction head loss in the rings. The optimization process is set out using SOLVER (a nonlinear integer programming methodology included in EXCEL). The objective function is to search for the lowest cost of the pipes, within the restriction of Ne continuity in the nodes. Nee and Nes are changeable variables, which are obtained in the optimization process. The design flow in the network pipelines, from the assigned Ne to each pipeline, is obtained according to Cle´ment’s methodology. Having established the design flow, the most economic pipe sizing can be calculated using the economic series method. In some pipelines, Ne is very close to 0 and, when they are deleted, the branched irrigation network with lower cost is obtained. In the process, the cost of the pipes, which depends on their diameter, will be established (Eq. (7)): Ct ¼ ADa
(7)
The investment annuity necessary in the pipes will be:
CT ¼
T X CRFLi ADai
(8)
where Ct is the cost of the pipes (s); CT the total cost of network (s); CRF the capital recovering factor; Li and Di the pipe length (m) and diameter (m) of each pipeline in the network; A and a the constants in the costs equation; T is the number of pipes in the main ring network. The optimal diameter in each pipeline (Di) could be obtained using the economic series method for a branched irrigation network with the equation (Pe´rez et al., 1996): 2 31=5 ki 0:0826 f 1=5 4X 2a=5þa 5 2=5þa 2=aþ5 L j Ql j Qli ¼ KQi Di ¼ Dh j¼1
function of the K value. However, the final branched network layout changes if the supplying point of the network is modified. Once the final branched irrigation network is established, the most unfavorable series and the necessary upstream head, as well as the power necessities can be determined. In this way, the K values and the necessary power are linked in the supplying point of the network. The initial K value (Eq. (10)) can be obtained by substituting in the Eq. (9) the diameter Dc deduced from the continuity equation applied to the upstream pipeline Dc ¼ ð4Q c =vpÞ1=2 , being Qc the flow in the upstream pipeline for the peak period (m3 s1) and v the average water speed in the pipe (m s1): K¼
4 vp
1=2
aþ1=2ðaþ5Þ
Qc
(9)
where Dh is the available energy for head losses in the series to be sized (m); f the friction factor; ki the number of pipelines downstream of the pipeline to be sized in the series studied; Qi is the flow of the pipeline to be sized in the series studied (m3 s1). Applying the economic series method to the main ring network (Pe´rez et al., 1996), Eq. (9) can be replaced in Eq. (8). Since K value is initially unknown, the calculation process can be simplified using different K values. Finally the K value that leads to the optimal network layout and pipe size will be obtained. Firstly, the same K value is supposed for all the main ring network pipes, maintaining the same value during the optimization process until the resulting branched network is obtained. Subsequently, the K value for the secondary networks can be adjusted. Using different K values in the main ring network leads to the same final branched network layout, although the network diameters (and consequently its cost) are a
(10)
The Eq. (10) shows the relationship between the K and v values. The diameter of each pipeline can now be determined by substituting Eqs. (4) in (9):
Di ¼ Kð pNei qa þ U
i¼1
113
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=aþ5 pð1 pÞNei q2a Þ
(11)
and by substituting Di in Eq. (8) the objective function to be minimized is obtained (Eq. (12)):
CT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=aþ5 a T X CRFLi A Kð pNei qa þ U pð1 pÞNei q2a Þ
(12)
i¼1
For each of the supposed K values, the optimization process determines the number of equivalent hydrants (Nei) in each pipeline of the network and the final branched network of least cost. Once the network layout and the branched irrigation network pre-size are known, different methods can be used for calculating the economic optimum pipe size of the network, minimizing the investment and energy cost (Labye et al., 1988; Lansey and Mays, 1989; Pe´rez et al., 1996; Diopram, 2003). So, the lowest network cost with normalized diameters is determined to each K value, as well as the necessary upstream head.
2.2.
Second stage
Evaluation of the annual energetic and the pumping station costs: 1. The first step is to determine the demand curve ‘‘headdischarge at upstream’’ (Hc Qc) (Planells et al., 2001), for the different K values (different upstream water speed into the pipe). The final objective is to obtain the K value that leads to the total minimum cost (investment and energy cost). To determine the maximum and minimum heads that correspond to a certain upstream flow, the discharge of each hydrant in the network (qdi) is multiplied by a coefficient ai. This coefficient can be 0 or 1 depending
114
agricultural water management 88 (2007) 110–118
(d) The upstream discharge Qc:
on whether the hydrant is closed or open at that given time. In this way the real discharge of each hydrant (qri) for each of the operating conditions supposed is obtained as
Qc ¼
n X ai qdi Q p
(19)
i¼1
(13)
where Qp is the pre-fixed upstream discharge (m3 s1); n is the number of hydrants in the network.
Since the same identification number is assigned to each pipeline and also to the node located immediately downstream, the flow in pipeline i (Qi) is the sum of discharges for the hydrants which pipeline i supplies: X qri (14) Qi ¼
In this way, the values ai that maximize the upstream head are identified, indicating the open or closed hydrants.
qri ¼ ai qdi
In a similar way, the minimum of Hc is obtained, with the same restrictions mentioned above, except (d), since Qc Qp is necessary for the converging process.
i 2 hi
where hi is the number of hydrants fed by pipeline i. To calculate friction head losses in the network pipes, the Darcy–Weisbach equation is used: Hri ¼ f i
8Li Qi2 ¼ Ri Qi2 p2 gD5i
(15)
where Hr is the friction head losses in the pipe (m m1); f the friction factor; L the pipe length (m); g the acceleration of gravity (9.81 m s2); D the internal diameter (m); Q the flow (m3 s1); R the drag coefficient (s2 m5); i is each pipe in the network. The coefficients ai that maximize or minimize the upstream head needed to guarantee minimum pressure in each node for different assumed discharges, can be identified using non-linear integer programming (Rios, 1988). This can be done by analyzing all the possible placements of open hydrants, which, when taken together, add up to the upstream discharge. To find the maximum of the upstream head (Hc), the method indicated above is used with the following restrictions: (a) Energetic (for all pipelines in the network): 0 12 X a jq A Hi ¼ Han Ri @ dj
(16)
j 2 ki
where Hi is the head downstream pipeline (m); Han is the head upstream pipeline i (m). (b) Pressure head excess in the nodes (Pi/g) with respect to the setting pressure in the reducing valve (Pti/g), is defined for each hydrant as Pi P ¼ Hi Hgi ti g g and the minimum function is applied: P Min i ) 0 g
(17)
(18)
where Hg is the hydrant elevation (m); (Pti/g) is the reducing valve setting pressure (m), according to the minimum pressure necessary in the hydrant. Pti/g can be different for each hydrant depending on the size and shape of the plot that it supplies and the type of emitter used (sprinkler or dripper, for example if, in the same network, both irrigation methods are used in different plots). (c) The coefficients ai.
2. The second step is the pumping plant design, determining the type and number of fixed and variable speed pumps necessary and estimating their costs. The power and efficiency of the pumping plant are calculated according to the discharge (Qc) and the head (Hc) required at upstream for the different K values. 3. After that, the absorbed power curves as a function of K are determined, by following the demand curves (Qc Hc) and taking into account the resulting combination of fixed or variable speed pumps. Depending on the volume of water distributed in the network in each period (daily, weekly, monthly, etc.) and the hypothesis of the demanded discharge evolution within each period, the operating time of the installation and the energy consumed (absorbed power multiplied by operation time) can be established, thus deducing the annual energy cost.
2.3.
Third stage
Following the preceding steps, for each K value, the investment annuity and the annual energy cost are obtained. Then, the total annual cost curve is determined for the different K values, obtaining the K value that leads to the minimum total cost. This method can also be used for the network layout of main ring networks, establishing some minimum diameters so that no pipeline disappears. An interesting case could be the analysis of the network supply from several points. Not only this matter but also the analysis of ring networks in general, can be tackled with this method.
3.
Results
In order to apply the methodology, the same network as in Planells et al. (2001, 2005) on design and regulation of the pumping plants was adopted. It should be mentioned that an on-demand irrigation network that irrigates 127.7 ha of citrus fruits in Valencia (Spain) has been used, with a plantation spacing of 5 m 4 m. In the agronomic design, six drips per plant with a nominal discharge of 4 L h1 and a pressure of 10 m are necessary. These conditions originate a minimum pressure of 25 m in the hydrants. In Fig. 1, the location of the hydrants and the superposition of the possible layouts that configure the initial main ring network are shown. The basic characteristics of the irrigation network are included in Table 1.
agricultural water management 88 (2007) 110–118
115
Fig. 1 – Nodes and pipes at the initial main ring network.
The cost equation adopted for the pipes was (Eq. (20)): Ct ¼ 163:69D1:53
(20)
where Ct is the cost of pipelines (s m1); D is the diameter (m). For the peak period (July) the following data were assumed: a daily operation time (DOT) = 18 h, an irrigation time tr = 3.5 h, a number of irrigation subunits per plot Ns = 1 (Planells et al., 2001, 2005) (then, the irrigation time of the plot tp = tr), an irrigation interval Ir = 1 day, irrigation requirements Rr = 4.2 mm day1, and an average application rate of the
system Aar = 1.2 mm h1. With this data it is possible to obtain a first approximation to the problem. The hydrant operation probability will be: p¼
td tr Ns 3:5 1 ¼ 0:194 ¼ ¼ DOT Ir DOT 1 18
(21)
where td is the daily operation time of the hydrant (h). The average discharge of the hydrant will be: PN qa ¼
j¼1
N
qd j
¼
425:7 ¼ 17:03 L s1 25
Fig. 2 – Nodes and pipes at the most inexpensive branched irrigation network.
(22)
116
7.02 23.40 5.76 19.20 5.70 19.00 0.00 0.00 0.33 1.10 1.89 6.30 0.00 0.00 4.47 14.90
33 32 31 30 29 28 27
5.40 18.00
In addition, and in order to simplify, an operating quality of OQ = 99% was used for Cle´ment’s method in all cases, so that U = 2.33. The results obtained lead to the optimum branched irrigation network of Fig. 2. Once the optimum network layout is established, a more accurate procedure could be applied to carry out the optimum size diameters of pipes using, Diopram for instance (Pe´rez et al., 1996), model which is based on the lineal programming technique. Considering different K values (different water speed (v) in the upstream pipe), the respective diameters and the cost of the branched irrigation network are obtained. The solution obtained with different K values is always the same branched irrigation network, although with different pipe diameters and costs. For each sized of the network according to the different K values, a maximum demand curve (Qc Hmax) is determined. This curve corresponds to the discharge concentration in the pipelines which requires the highest upstream head (Hmax), in order to guarantee that all the hydrants have a pressure higher than 25 m, which is the established minimum. The demand curves H = f(Q) resulting for each K value, adjusted to a second degree equation, with Q (m3 s1) and H (m) are shown in Fig. 3 The pumping station is dimensioned for each K value for the upstream discharge Qc (m3 s1) and Hc (m) in the peak period, establishing the type and number of fixed and variable speed pumps to be adjusted to the demand curve. In order to calculate the energy cost, the pumping plant is assumed to be composed of four parallel pumps, their characteristic curves being: H ¼ 54:2 7588Q 2
(23)
h ¼ 47:1Q 690:8Q 2
(24)
where H is the pumping pressure (m); Q the flow (m3 s1); h is the pumping efficiency (decimal). In a first approximation, a constant upstream discharge was supposed, equal to the continuous discharge referred to the DOT for the peak demand month (July) (Planells et al.,
Area (ha) Discharge (qdj) (L s1)
5.49 18.30
8.07 26.90
5.22 17.40
0.00 0.00
7.02 23.40
5.82 19.40
4.98 16.60
26 25 19 18
20
21
22
23
24
Node
5.22 17.40
34
2.58 8.60 4.92 16.40 0.00 0.00 0.00 0.00 2.55 8.50 3.93 13.10 3.84 12.80 5.16 17.20 8.37 27.90 0.00 0.00 5.04 16.80 Area (ha) Discharge (qdj) (L s1)
0.00 0.00
0.00 0.00
8.64 28.80
5.91 19.70
0.00 0.00
4.38 14.60
9 8 7 4 3 2 1
Table 1 – Basic characteristics of the irrigation network
5
6
Node
10
11
12
13
14
15
16
17
agricultural water management 88 (2007) 110–118
Fig. 3 – Maximum demand curves for different K values.
K = 0.6 ðv ¼ 1:81 m s1 Þ
320.6 168.3 369.1 589.5 649.2 1050.5 2202.1 831.1 719.2 404.9 328.1 332.8
7965.3
75691.4
K = 0.7 ðv ¼ 1:33 m s1 Þ
312.5 164.0 359.9 574.7 632.8 1024.0 2146.5 810.1 701.1 394.7 319.8 324.5
7764.6
95824.1
117
117544.6 156376.9 Annual investment cost (s)
140755.7
7532.7 7109.5 7038.9 Annual energy cost (s)
2001). This was calculated by dividing the irrigation water volume to be applied per day (Vd) (Table 2) by DOT. The same discharge was maintained in the remaining months, changing the DOT for each month according to the irrigation volume to be applied (Vd). The average probability of open hydrant ( p = td/DOT) must be constant so that the upstream design flows obtained according to Cle´ment’s methodology remain constant. Since DOT changes each month, td also needs to be changed each month to obtain the same p. To obtain the monthly consumed energy, the power is multiplied by the operation time of the pumping station in that month, and divided by the global efficiency. Assuming that there is one variable speed pump and the remaining three are fixed speed pumps, the electric power (P, in kW) absorbed for each K value is shown in Fig. 4. The monthly and annual energy cost, as well as the annual investment cost resulting, is shown in Table 2. Total costs according to the K value have been represented in Fig. 5, considering the capital recuperation factor CRF = 0.1. From this, it can be deduced that the minimum annual cost is 156,92 s, when K = 0.66.
664.1 383.3 919.5 1519.8 2464.8 4125.0 5360.0 4853.0 2822.4 1008.9 702.4 689.6
2.2 1.3 3.1 5.1 8.3 13.9 18.0 16.3 9.5 3.4 2.4 2.3
0.4 0.3 0.6 1.0 1.6 2.7 3.5 3.2 1.8 0.7 0.5 0.5
283.5 148.9 326.3 520.9 573.6 928.1 1945.4 734.3 635.5 357.8 290.1 294.3
286.3 150.4 329.5 526.2 579.4 937.4 1965.0 741.7 641.9 361.4 293.0 297.2
303.3 159.2 349.1 557.5 613.9 993.3 2082.2 785.9 680.1 382.9 310.3 314.8
Fig. 4 – Electric power absorbed by the network for different K values.
January February March April May June July August September October November December
K = 0.8 ðv ¼ 1:02 m s1 Þ K = 0.9 ðv ¼ 0:80 m s1 Þ K = 1.0 ðv ¼ 0:65 m s1 Þ
td (h) DOT (h) Vd (m3 day1) Month
Table 2 – Monthly and annual energy costs, and annual investment cost in the network
Monthly energy cost (s)
agricultural water management 88 (2007) 110–118
Fig. 5 – Investment, annual energy cost and total cost curves, according to K values.
118 4.
agricultural water management 88 (2007) 110–118
Conclusions
With this methodology, the network layout and the most economic pipe sizing can be optimized at the same time. The procedure allows us to take into account all the possible network layouts, performing the optimization in a direct way (minimizing the total investment cost), obtaining the optimum layout of the branched irrigation network, and the necessary upstream head. The sizing of pipes using the solutions that lead to the optimum network layout is only a first approximation. Therefore, once the network with the optimum layout is determined, a new procedure for the pipe size optimization should be used. The same procedure can be used by simply changing the location of the network supply, thus obtaining the supply location that leads to a lower total cost (investment and energy). An interesting case could be the analysis of the network supply from several points using this methodology. This method could also be used in the network layout optimization of main ring networks, establishing some minimum diameters so that no pipeline disappears.
Acknowledgement The authors wish to thank the Spanish National Research Plan which funded this research through Project AGL2001-1180C02.
references
Aliod, R., Eizaguerri, A., Estrada, C., Perna, E., 1997. Dimensionado y ana´lisis hidra´ulico de redes de distribucio´n a presio´n en riego a la demanda: Aplicacio´n del programa GESTAR. Riegos y Drenajes XXI 92, 22–38. Awumah, K., Bhatt, S.K., Goulter, I.C., 1989. An integer programming model for layout design of water distribution networks. Eng. Optimiz. 15, 57–70.
Bhave, P.R., Lam, C.F., 1983. Optimal layout for branching distribution networks. J. Transp. Eng. 109, 534–547. Cle´ment, R., 1966. Calcul des de´bits dans les re´seaux d’irrigation fonctionnant a` la demande. La Houille Blanche 20 (5), 553–575. Cle´ment, R., Galand, A., 1979. Irrigation par aspersion et re´seaux collectifs de distribution sous pression. Eyrolles, Paris. ´ ptimo de Redes Ramificadas 3.0. Diopram, 2003. Disen˜o O Manual de Usuario. Grupo Multidisciplinar de Modelacio´n de Fluidos. Universidad Polite´cnica de Valencia, Valencia. Granados, A., 1990. Infraestructura de regadı´os. Redes colectivas de riego a presio´n. (2 edicio´n). Escuela Te´cnica Superior de Ingenieros de Caminos Canales y Puertos. Universidad Polite´cnica, Madrid. Labye, Y., Olson, M.A., Galand, A., Tsourtis, N., 1988. Design and Optimisation of Irrigation Distribution Network. Irrigation and Drainage Paper 44. FAO, Rome. Lamaddalena, N., Sagardoy, J.A., 2000. Performance Analysis of On-demand Pressurized Irrigation Systems. Irrigation and Drainage Paper 59. FAO, Rome. Lansey, K.E., Mays, L.W., 1989. Optimization model for water distribution system design. J. Hydraul. Eng. 115, 1401–1418. Martı´nez, J., Gil, A., Rolda´n, J., 1993. Estudio del trazado de redes ramificadas de riego. XI Jornadas Te´cnicas sobre riegos, 2–4th June, Valladolid. Pe´rez, R., Andreu, M., Izquierdo, J., 1996. Disen˜o de redes de distribucio´n. In: Cabrera, E., Espert, V., Garcı´a-Serra, J., Martinez, F., Andris, M., Garcı´a, M. (Eds.), Curso de ingenierı´a hidra´ulica aplicada a los sistemas de distribucio´n de agua. Universidad Polite´cnica de Valencia, Valencia, Spain, pp. 653–727. Planells, P., Tarjuelo, J.M., Ortega, J.F., Casanova, M.I., 2001. Design of water distribution networks for on-demand irrigation. Irrig. Sci. 20, 189–201. Planells, P., Carrio´n, P., Ortega, J.F., Moreno, M.A., Tarjuelo, J.M., 2005. Pumping selection and regulation for water distribution networks. J. Irrig. Drain. Eng. 131 (3), 273–281. Pulido-Calvo, I., Roldan, J., Lo´pez-Luque, R., Gutie´rrez-Estrada, J.C., 2003. Water delivery system planning considering irrigation simultaneity. J. Irrig. Drain. Eng. 129, 247–255. Rios, S., 1988. Investigacio´n operativa. Optimizacio´n. Centro de Estudios Ramo´n Areces S.A., Madrid. Rossman, L.A., 1997. EPANET: User Manual. Risk Reduction Engineering Laboratory Office of Research and Development. United States Environmental Protection Agency, Cincinnati, OH.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/agwat
Optimization of irrigation water distribution networks, layout included ´ lvarez, J.M. Tarjuelo Martı´n-Benito * P. Planells Alandı´, J.F. Ortega A Centro Regional de Estudios del Agua (Regional Centre of Water Research), Castilla-La Mancha University, Campus Universitario, s/n, E02071 Albacete, Spain
article info
abstract
Article history:
The purpose of this paper is to develop a procedure which takes into account both the
Accepted 12 October 2006
network layout and the pipe size of an on-demand water branched network in order to
Published on line 15 November 2006
obtain the lowest total cost (investment and energy cost). The process begins with a main ring network which is obtained by giving consideration to the possible alternatives of
Keywords:
branched irrigation networks, taking into account the limitations imposed by plot bound-
Irrigation networks
aries, gravel roads, etc. The optimization process progressively eliminates pipes to obtain
Layout
the most inexpensive branched irrigation network, verifying the criteria of the economic
Economic optimization
series methodology for sizing the pipes and the design flow obtained in the pipelines according to Cle´ment’s methodology. The dimensioning through the Economic Series Method determines the optimum diameter as a function of flow. Three stages are considered in the optimization process. In the first stage, the network layout and the pipe size of the lowest cost are obtained for different K values, as well as the necessary upstream head. In the second stage, the pumping station is dimensioned for each K value and annual costs for investment and energy consumption are obtained, according to the operation hours of the pumping station to satisfy crop water needs. In the third stage, the K value which leads to the lowest total cost (investment and energy cost) is determined. In the procedure, the network can be fed by direct injection (i.e. a fixed and a variable speed pump discharge directly into the network) or with a water tank (i.e. placed at an adequate head). The different network layouts and associated cost can be studied depending on the water source or sources in the network. This method can also be used for the layout of the main ring networks, establishing some minimum diameters as a reference to eliminate the pipelines. An interesting case could be the analysis of the network supply from several points. The sizing of the pipes that is performed, using the solutions that lead to the optimum network layout, is only a first approximation. Therefore, once the network with the optimum layout is determined, a new procedure for pipe size optimization should be used. # 2006 Elsevier B.V. All rights reserved.
1.
Introduction
Once the water source, the pressure necessities, the hydrant discharge and its approximate location in the plot are defined,
the design of a collective pressurized on-demand irrigation network can be summarized in five stages (Labye et al., 1988): (a) optimum network layout in order to minimize the total cost of the network (Awumah et al., 1989; Bhave and Lam, 1983;
* Corresponding author. Tel.: +34 967 599304; fax: +34 967 599238. E-mail address: [email protected] (J.M. Tarjuelo Martı´n-Benito). 0378-3774/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.agwat.2006.10.004
agricultural water management 88 (2007) 110–118
Nomenclature ai
A, a Aar CRF Ct CT D DOT f g Dh H Hc Hg hi Hr Ir ki L N Ns Ne Nee
Nes
OQ p P Pi/g
Pti/g
Q qa qd qri Qi Qc Qp R Rr T td tr
coefficient that can be 0 or 1 depending on whether the hydrant is closed or open at that given time constants in the costs equation of the pipes average application rate of the system (mm h1) capital recuperation factor cost of the pipes (s) total investment annuity cost of network (s) internal diameter (m) the daily operation time (h) friction factor acceleration of gravity (m s2) available energy for head losses in the pipe series to be sized (m) pumping pressure (m) required head at upstream (m) hydrant elevation (m) number of downstream hydrants in the pipeline i friction head losses (m m1) irrigation interval (day) the number of downstream pipelines from the yet to be sized pipeline within the series studied pipe length (m) number of existing hydrants in the network number of irrigation subunits per plot number of equivalent hydrant discharges number of incoming equivalent hydrant discharges in the node (including those corresponding to the pipes and punctual inflow) number of outgoing equivalent hydrant discharges in the node (including those corresponding to the pipes and punctual outflow) network operating quality average hydrant operation probability electric power (kW) pressure head excess in the nodes i with respect to the setting pressure in the reducing valve (Pti/g) (m) reducing valve setting pressure, according to the minimum pressure necessary in the hydrant (m) flow (m3 s1) average hydrant discharge in the network (L s1) hydrant discharge (L s1) real discharge of hydrant i (L s1) design flow of pipeline i (m3 s1) flow in the upstream pipeline for the peak period (m3 s1) pre-fixed upstream discharge (m3 s1) drag coefficient (s2 m5) irrigation requirements (mm day1) number of pipes in the main ring network is the average daily operation time of the hydrant in the network (h) irrigation time in the peak period (h)
U v Vd
111
percentile of the normal distribution function associated with a definite operating quality (OQ) average water speed in the pipe (m s1) irrigation water volume to be applied per day (m3 day1)
Greek letter h
pump efficiency (decimal)
Granados, 1990); (b) calculation of the hydrant discharge according to plot sizes (Planells et al., 2001); (c) determination of design flow per pipeline (associated with a determined supply guarantee) (Cle´ment and Galand, 1979; Pulido-Calvo et al., 2003); (d) calculation of the optimum pipe size diameters, minimizing the investment and energy cost (Labye et al., 1988; Lansey and Mays, 1989; Pe´rez et al., 1996; Diopram, 2003); (e) analysis of the network performance under different operating conditions (Rossman, 1997; Aliod et al., 1997) to determine the possible supply failure situations of the network or of the pumping plant (Lamaddalena and Sagardoy, 2000). The network layout performance usually begins with a non-restricted layout (Girette method) using the approaches by proximity and minimum length. Afterwards, some restrictions such as layout for roads or plot boundaries are considered (Granados, 1990). The connection of the nodes in the network can be carried out according to geometric criterion (proximity), hydraulic criteria (pressure, discharge, etc.) or a combination of both (Martı´nez et al., 1993). These authors also propose the criteria of the radial union of the hydrants, which is a more advantageous variant of the proximity criteria, even though the result is conditioned by the location of the water source in the network. The algorithm consists of: ‘‘(a) placing the hydrants in increasing order of distance to the initial point, and (b) join the first one to the second, and then the third one to the closest of those already joined’’. Likewise, the remaining hydrants should be joined according to the order established at the beginning. Some additional conditions to bear in mind for the network layout are: Geological–geotechnical: rocks, unstable land, groundwater table, etc. Orographic: cavitation risk originated by the topography, placement of airholes, drainage, etc. Geographical: plot distribution, communication routes, buildings, etc. Social: right-of-way creation, temporal occupation, expropriation, etc. To design the network layout by means of the abovementioned methods, the pipeline cost is supposed to be only function of the pipe lengths, without taking into account the pipe diameters. The purpose of this paper is to develop a procedure which takes into account both the network layout and the most economic pipe size, in order to obtain the least amount of the total cost of the design (investment and energy cost). In the procedure, the network can be fed by direct
112
agricultural water management 88 (2007) 110–118
injection (i.e. fixed and variable speed pump discharge directly into the network) or with a water tank (i.e. placed at an adequate head). The different network layouts and associated cost can be studied depending on the water source or sources in the network.
2.
Methodology
It is possible to optimize an on-demand network layout by selecting the layout and the most economic pipe sizing at the same time. In the network design optimization process, the following costs should be considered: cost of the network ð1Þ þ cost of the pumping plant ð2Þ þ energy cost ð3Þ These three costs are not unrelated, so both the network (1) (its layout and pipe size) and the pumping plant (2) (number of fixed and variable speed pumps, automation equipment for regulation and control, etc.) are designed for the peak period of irrigation requirements. However, the annual energetic cost (3) is a function of the upstream consumed power, which varies during the irrigation season according to the discharge and head. This also depends on the efficiency of the pumping plant, which is a function of its design and its capacity of adaptation to the demand conditions of the network, according to the adopted regulation system (Planells et al., 2005). The optimization process can be structured in three stages. First, the cost of the pipes is determined using simultaneously both the network layout and pipe size for the most unfavorable situation (conditions of highest flow demanded in the network) simultaneously. In the second stage, the energy and the annual pumping investment costs are evaluated. In the third stage, the lowest total cost is determined.
2.1.
First stage
Optimization of the network layout and the economic pipe size at the same time. Once the optimum branched irrigation network and its pipe size are defined, the necessary upstream head is determined to develop the second stage. The process begins with a main ring network which contains all the nodes included in the network. The network is obtained by giving consideration to the possible alternatives of branched irrigation networks, while taking into account the limitations imposed by plot boundaries, gravel roads, etc. At this stage, non-outflow nodes can be created. From the network, the exact lengths of the pipelines and the topography level of the nodes will be known. The discharge of the hydrants could also be calculated for crop rotation in the peak consumption period. The optimization process use the economic series methodology for sizing the pipes (Pe´rez et al., 1996) and the design flow obtained in the pipelines according to Cle´ment’s methodology (Cle´ment, 1966). In order to simplify the process, a number of equivalent hydrant discharge (Nei) assigned to each selected pipe ‘‘i’’ can be deduced. The Nei is found by dividing the sum of the hydrant downstream discharge (qdk)
within pipeline i, by the average hydrant discharge of the network (qa). Both are calculated as PN qa ¼
j¼1
qd j
N
Phi and Nei ¼
k¼1
qdik
(1)
qa
where qd is the hydrant discharge (L s1); N the number of hydrants existing in the network; hi is the number of hydrants downstream from the pipeline i. The design flow of each pipeline (Qi) can be determined using Cle´ment’s methodology (Cle´ment and Galand, 1979), with the average probability ( p) of the set of hydrant (ki) downstream from the pipeline yet to be sized. The result is as follows (Planells et al., 2001): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u hi hi X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuX Q i ¼ p qdik þ U pð1 pÞt q2di k¼1
k¼1
k
(2)
with p¼
td DOT
where p is the average hydrant operation probability; td the average daily operation time of the hydrant in the network (h); DOT the daily operation time (h); qdj the hydrant j discharge (L s1); U is the percentile of the normal distribution function associated with a definite operating quality (OQ). Substitution of Eqs. (3) into (2) results in Eq. (4): hi X qdik ¼ Nei qa
(3)
k¼1
Q i ¼ pNei qa þ U
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi pð1 pÞ Nei q2a
(4)
Ph
where the quadratic mean k¼1 q2di =Nei has been replaced, in k first approximation, by the arithmetic mean qa to simplify the calculations. To verify Clement’s formula and find all the nodes linked within the branched irrigation network, the continuity of the continuous hypothetical flows in the nodes is imposed as a restriction (Planells et al., 2001): X
pNee qa ¼
X
pNes qa
(5)
where Nee is the number of incoming equivalent hydrants discharge in the node (including those corresponding to the pipes and punctual inflow); Nes is the number of outgoing equivalent hydrants discharge in the node (including those corresponding to the pipes and punctual outflow). Since p and qa are constants in the network, the following formula could be deduced: X
Nee ¼
X
Nes
(6)
Having established the hydrant discharge, Nes values are evident in the nodes that directly supply the hydrants. Ne values in the rest of the pipelines are raised as unknown quantities to be determined during the optimization process.
agricultural water management 88 (2007) 110–118
Since the design flows in the pipelines of the network are obtained according to Cle´ment’s methodology, the continuity in the nodes is verified for the number of equivalent hydrant discharges (Ne) (Eq. (6)) but not by the flow rates nor by the equilibrium of friction head loss in the rings. The optimization process is set out using SOLVER (a nonlinear integer programming methodology included in EXCEL). The objective function is to search for the lowest cost of the pipes, within the restriction of Ne continuity in the nodes. Nee and Nes are changeable variables, which are obtained in the optimization process. The design flow in the network pipelines, from the assigned Ne to each pipeline, is obtained according to Cle´ment’s methodology. Having established the design flow, the most economic pipe sizing can be calculated using the economic series method. In some pipelines, Ne is very close to 0 and, when they are deleted, the branched irrigation network with lower cost is obtained. In the process, the cost of the pipes, which depends on their diameter, will be established (Eq. (7)): Ct ¼ ADa
(7)
The investment annuity necessary in the pipes will be:
CT ¼
T X CRFLi ADai
(8)
where Ct is the cost of the pipes (s); CT the total cost of network (s); CRF the capital recovering factor; Li and Di the pipe length (m) and diameter (m) of each pipeline in the network; A and a the constants in the costs equation; T is the number of pipes in the main ring network. The optimal diameter in each pipeline (Di) could be obtained using the economic series method for a branched irrigation network with the equation (Pe´rez et al., 1996): 2 31=5 ki 0:0826 f 1=5 4X 2a=5þa 5 2=5þa 2=aþ5 L j Ql j Qli ¼ KQi Di ¼ Dh j¼1
function of the K value. However, the final branched network layout changes if the supplying point of the network is modified. Once the final branched irrigation network is established, the most unfavorable series and the necessary upstream head, as well as the power necessities can be determined. In this way, the K values and the necessary power are linked in the supplying point of the network. The initial K value (Eq. (10)) can be obtained by substituting in the Eq. (9) the diameter Dc deduced from the continuity equation applied to the upstream pipeline Dc ¼ ð4Q c =vpÞ1=2 , being Qc the flow in the upstream pipeline for the peak period (m3 s1) and v the average water speed in the pipe (m s1): K¼
4 vp
1=2
aþ1=2ðaþ5Þ
Qc
(9)
where Dh is the available energy for head losses in the series to be sized (m); f the friction factor; ki the number of pipelines downstream of the pipeline to be sized in the series studied; Qi is the flow of the pipeline to be sized in the series studied (m3 s1). Applying the economic series method to the main ring network (Pe´rez et al., 1996), Eq. (9) can be replaced in Eq. (8). Since K value is initially unknown, the calculation process can be simplified using different K values. Finally the K value that leads to the optimal network layout and pipe size will be obtained. Firstly, the same K value is supposed for all the main ring network pipes, maintaining the same value during the optimization process until the resulting branched network is obtained. Subsequently, the K value for the secondary networks can be adjusted. Using different K values in the main ring network leads to the same final branched network layout, although the network diameters (and consequently its cost) are a
(10)
The Eq. (10) shows the relationship between the K and v values. The diameter of each pipeline can now be determined by substituting Eqs. (4) in (9):
Di ¼ Kð pNei qa þ U
i¼1
113
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=aþ5 pð1 pÞNei q2a Þ
(11)
and by substituting Di in Eq. (8) the objective function to be minimized is obtained (Eq. (12)):
CT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=aþ5 a T X CRFLi A Kð pNei qa þ U pð1 pÞNei q2a Þ
(12)
i¼1
For each of the supposed K values, the optimization process determines the number of equivalent hydrants (Nei) in each pipeline of the network and the final branched network of least cost. Once the network layout and the branched irrigation network pre-size are known, different methods can be used for calculating the economic optimum pipe size of the network, minimizing the investment and energy cost (Labye et al., 1988; Lansey and Mays, 1989; Pe´rez et al., 1996; Diopram, 2003). So, the lowest network cost with normalized diameters is determined to each K value, as well as the necessary upstream head.
2.2.
Second stage
Evaluation of the annual energetic and the pumping station costs: 1. The first step is to determine the demand curve ‘‘headdischarge at upstream’’ (Hc Qc) (Planells et al., 2001), for the different K values (different upstream water speed into the pipe). The final objective is to obtain the K value that leads to the total minimum cost (investment and energy cost). To determine the maximum and minimum heads that correspond to a certain upstream flow, the discharge of each hydrant in the network (qdi) is multiplied by a coefficient ai. This coefficient can be 0 or 1 depending
114
agricultural water management 88 (2007) 110–118
(d) The upstream discharge Qc:
on whether the hydrant is closed or open at that given time. In this way the real discharge of each hydrant (qri) for each of the operating conditions supposed is obtained as
Qc ¼
n X ai qdi Q p
(19)
i¼1
(13)
where Qp is the pre-fixed upstream discharge (m3 s1); n is the number of hydrants in the network.
Since the same identification number is assigned to each pipeline and also to the node located immediately downstream, the flow in pipeline i (Qi) is the sum of discharges for the hydrants which pipeline i supplies: X qri (14) Qi ¼
In this way, the values ai that maximize the upstream head are identified, indicating the open or closed hydrants.
qri ¼ ai qdi
In a similar way, the minimum of Hc is obtained, with the same restrictions mentioned above, except (d), since Qc Qp is necessary for the converging process.
i 2 hi
where hi is the number of hydrants fed by pipeline i. To calculate friction head losses in the network pipes, the Darcy–Weisbach equation is used: Hri ¼ f i
8Li Qi2 ¼ Ri Qi2 p2 gD5i
(15)
where Hr is the friction head losses in the pipe (m m1); f the friction factor; L the pipe length (m); g the acceleration of gravity (9.81 m s2); D the internal diameter (m); Q the flow (m3 s1); R the drag coefficient (s2 m5); i is each pipe in the network. The coefficients ai that maximize or minimize the upstream head needed to guarantee minimum pressure in each node for different assumed discharges, can be identified using non-linear integer programming (Rios, 1988). This can be done by analyzing all the possible placements of open hydrants, which, when taken together, add up to the upstream discharge. To find the maximum of the upstream head (Hc), the method indicated above is used with the following restrictions: (a) Energetic (for all pipelines in the network): 0 12 X a jq A Hi ¼ Han Ri @ dj
(16)
j 2 ki
where Hi is the head downstream pipeline (m); Han is the head upstream pipeline i (m). (b) Pressure head excess in the nodes (Pi/g) with respect to the setting pressure in the reducing valve (Pti/g), is defined for each hydrant as Pi P ¼ Hi Hgi ti g g and the minimum function is applied: P Min i ) 0 g
(17)
(18)
where Hg is the hydrant elevation (m); (Pti/g) is the reducing valve setting pressure (m), according to the minimum pressure necessary in the hydrant. Pti/g can be different for each hydrant depending on the size and shape of the plot that it supplies and the type of emitter used (sprinkler or dripper, for example if, in the same network, both irrigation methods are used in different plots). (c) The coefficients ai.
2. The second step is the pumping plant design, determining the type and number of fixed and variable speed pumps necessary and estimating their costs. The power and efficiency of the pumping plant are calculated according to the discharge (Qc) and the head (Hc) required at upstream for the different K values. 3. After that, the absorbed power curves as a function of K are determined, by following the demand curves (Qc Hc) and taking into account the resulting combination of fixed or variable speed pumps. Depending on the volume of water distributed in the network in each period (daily, weekly, monthly, etc.) and the hypothesis of the demanded discharge evolution within each period, the operating time of the installation and the energy consumed (absorbed power multiplied by operation time) can be established, thus deducing the annual energy cost.
2.3.
Third stage
Following the preceding steps, for each K value, the investment annuity and the annual energy cost are obtained. Then, the total annual cost curve is determined for the different K values, obtaining the K value that leads to the minimum total cost. This method can also be used for the network layout of main ring networks, establishing some minimum diameters so that no pipeline disappears. An interesting case could be the analysis of the network supply from several points. Not only this matter but also the analysis of ring networks in general, can be tackled with this method.
3.
Results
In order to apply the methodology, the same network as in Planells et al. (2001, 2005) on design and regulation of the pumping plants was adopted. It should be mentioned that an on-demand irrigation network that irrigates 127.7 ha of citrus fruits in Valencia (Spain) has been used, with a plantation spacing of 5 m 4 m. In the agronomic design, six drips per plant with a nominal discharge of 4 L h1 and a pressure of 10 m are necessary. These conditions originate a minimum pressure of 25 m in the hydrants. In Fig. 1, the location of the hydrants and the superposition of the possible layouts that configure the initial main ring network are shown. The basic characteristics of the irrigation network are included in Table 1.
agricultural water management 88 (2007) 110–118
115
Fig. 1 – Nodes and pipes at the initial main ring network.
The cost equation adopted for the pipes was (Eq. (20)): Ct ¼ 163:69D1:53
(20)
where Ct is the cost of pipelines (s m1); D is the diameter (m). For the peak period (July) the following data were assumed: a daily operation time (DOT) = 18 h, an irrigation time tr = 3.5 h, a number of irrigation subunits per plot Ns = 1 (Planells et al., 2001, 2005) (then, the irrigation time of the plot tp = tr), an irrigation interval Ir = 1 day, irrigation requirements Rr = 4.2 mm day1, and an average application rate of the
system Aar = 1.2 mm h1. With this data it is possible to obtain a first approximation to the problem. The hydrant operation probability will be: p¼
td tr Ns 3:5 1 ¼ 0:194 ¼ ¼ DOT Ir DOT 1 18
(21)
where td is the daily operation time of the hydrant (h). The average discharge of the hydrant will be: PN qa ¼
j¼1
N
qd j
¼
425:7 ¼ 17:03 L s1 25
Fig. 2 – Nodes and pipes at the most inexpensive branched irrigation network.
(22)
116
7.02 23.40 5.76 19.20 5.70 19.00 0.00 0.00 0.33 1.10 1.89 6.30 0.00 0.00 4.47 14.90
33 32 31 30 29 28 27
5.40 18.00
In addition, and in order to simplify, an operating quality of OQ = 99% was used for Cle´ment’s method in all cases, so that U = 2.33. The results obtained lead to the optimum branched irrigation network of Fig. 2. Once the optimum network layout is established, a more accurate procedure could be applied to carry out the optimum size diameters of pipes using, Diopram for instance (Pe´rez et al., 1996), model which is based on the lineal programming technique. Considering different K values (different water speed (v) in the upstream pipe), the respective diameters and the cost of the branched irrigation network are obtained. The solution obtained with different K values is always the same branched irrigation network, although with different pipe diameters and costs. For each sized of the network according to the different K values, a maximum demand curve (Qc Hmax) is determined. This curve corresponds to the discharge concentration in the pipelines which requires the highest upstream head (Hmax), in order to guarantee that all the hydrants have a pressure higher than 25 m, which is the established minimum. The demand curves H = f(Q) resulting for each K value, adjusted to a second degree equation, with Q (m3 s1) and H (m) are shown in Fig. 3 The pumping station is dimensioned for each K value for the upstream discharge Qc (m3 s1) and Hc (m) in the peak period, establishing the type and number of fixed and variable speed pumps to be adjusted to the demand curve. In order to calculate the energy cost, the pumping plant is assumed to be composed of four parallel pumps, their characteristic curves being: H ¼ 54:2 7588Q 2
(23)
h ¼ 47:1Q 690:8Q 2
(24)
where H is the pumping pressure (m); Q the flow (m3 s1); h is the pumping efficiency (decimal). In a first approximation, a constant upstream discharge was supposed, equal to the continuous discharge referred to the DOT for the peak demand month (July) (Planells et al.,
Area (ha) Discharge (qdj) (L s1)
5.49 18.30
8.07 26.90
5.22 17.40
0.00 0.00
7.02 23.40
5.82 19.40
4.98 16.60
26 25 19 18
20
21
22
23
24
Node
5.22 17.40
34
2.58 8.60 4.92 16.40 0.00 0.00 0.00 0.00 2.55 8.50 3.93 13.10 3.84 12.80 5.16 17.20 8.37 27.90 0.00 0.00 5.04 16.80 Area (ha) Discharge (qdj) (L s1)
0.00 0.00
0.00 0.00
8.64 28.80
5.91 19.70
0.00 0.00
4.38 14.60
9 8 7 4 3 2 1
Table 1 – Basic characteristics of the irrigation network
5
6
Node
10
11
12
13
14
15
16
17
agricultural water management 88 (2007) 110–118
Fig. 3 – Maximum demand curves for different K values.
K = 0.6 ðv ¼ 1:81 m s1 Þ
320.6 168.3 369.1 589.5 649.2 1050.5 2202.1 831.1 719.2 404.9 328.1 332.8
7965.3
75691.4
K = 0.7 ðv ¼ 1:33 m s1 Þ
312.5 164.0 359.9 574.7 632.8 1024.0 2146.5 810.1 701.1 394.7 319.8 324.5
7764.6
95824.1
117
117544.6 156376.9 Annual investment cost (s)
140755.7
7532.7 7109.5 7038.9 Annual energy cost (s)
2001). This was calculated by dividing the irrigation water volume to be applied per day (Vd) (Table 2) by DOT. The same discharge was maintained in the remaining months, changing the DOT for each month according to the irrigation volume to be applied (Vd). The average probability of open hydrant ( p = td/DOT) must be constant so that the upstream design flows obtained according to Cle´ment’s methodology remain constant. Since DOT changes each month, td also needs to be changed each month to obtain the same p. To obtain the monthly consumed energy, the power is multiplied by the operation time of the pumping station in that month, and divided by the global efficiency. Assuming that there is one variable speed pump and the remaining three are fixed speed pumps, the electric power (P, in kW) absorbed for each K value is shown in Fig. 4. The monthly and annual energy cost, as well as the annual investment cost resulting, is shown in Table 2. Total costs according to the K value have been represented in Fig. 5, considering the capital recuperation factor CRF = 0.1. From this, it can be deduced that the minimum annual cost is 156,92 s, when K = 0.66.
664.1 383.3 919.5 1519.8 2464.8 4125.0 5360.0 4853.0 2822.4 1008.9 702.4 689.6
2.2 1.3 3.1 5.1 8.3 13.9 18.0 16.3 9.5 3.4 2.4 2.3
0.4 0.3 0.6 1.0 1.6 2.7 3.5 3.2 1.8 0.7 0.5 0.5
283.5 148.9 326.3 520.9 573.6 928.1 1945.4 734.3 635.5 357.8 290.1 294.3
286.3 150.4 329.5 526.2 579.4 937.4 1965.0 741.7 641.9 361.4 293.0 297.2
303.3 159.2 349.1 557.5 613.9 993.3 2082.2 785.9 680.1 382.9 310.3 314.8
Fig. 4 – Electric power absorbed by the network for different K values.
January February March April May June July August September October November December
K = 0.8 ðv ¼ 1:02 m s1 Þ K = 0.9 ðv ¼ 0:80 m s1 Þ K = 1.0 ðv ¼ 0:65 m s1 Þ
td (h) DOT (h) Vd (m3 day1) Month
Table 2 – Monthly and annual energy costs, and annual investment cost in the network
Monthly energy cost (s)
agricultural water management 88 (2007) 110–118
Fig. 5 – Investment, annual energy cost and total cost curves, according to K values.
118 4.
agricultural water management 88 (2007) 110–118
Conclusions
With this methodology, the network layout and the most economic pipe sizing can be optimized at the same time. The procedure allows us to take into account all the possible network layouts, performing the optimization in a direct way (minimizing the total investment cost), obtaining the optimum layout of the branched irrigation network, and the necessary upstream head. The sizing of pipes using the solutions that lead to the optimum network layout is only a first approximation. Therefore, once the network with the optimum layout is determined, a new procedure for the pipe size optimization should be used. The same procedure can be used by simply changing the location of the network supply, thus obtaining the supply location that leads to a lower total cost (investment and energy). An interesting case could be the analysis of the network supply from several points using this methodology. This method could also be used in the network layout optimization of main ring networks, establishing some minimum diameters so that no pipeline disappears.
Acknowledgement The authors wish to thank the Spanish National Research Plan which funded this research through Project AGL2001-1180C02.
references
Aliod, R., Eizaguerri, A., Estrada, C., Perna, E., 1997. Dimensionado y ana´lisis hidra´ulico de redes de distribucio´n a presio´n en riego a la demanda: Aplicacio´n del programa GESTAR. Riegos y Drenajes XXI 92, 22–38. Awumah, K., Bhatt, S.K., Goulter, I.C., 1989. An integer programming model for layout design of water distribution networks. Eng. Optimiz. 15, 57–70.
Bhave, P.R., Lam, C.F., 1983. Optimal layout for branching distribution networks. J. Transp. Eng. 109, 534–547. Cle´ment, R., 1966. Calcul des de´bits dans les re´seaux d’irrigation fonctionnant a` la demande. La Houille Blanche 20 (5), 553–575. Cle´ment, R., Galand, A., 1979. Irrigation par aspersion et re´seaux collectifs de distribution sous pression. Eyrolles, Paris. ´ ptimo de Redes Ramificadas 3.0. Diopram, 2003. Disen˜o O Manual de Usuario. Grupo Multidisciplinar de Modelacio´n de Fluidos. Universidad Polite´cnica de Valencia, Valencia. Granados, A., 1990. Infraestructura de regadı´os. Redes colectivas de riego a presio´n. (2 edicio´n). Escuela Te´cnica Superior de Ingenieros de Caminos Canales y Puertos. Universidad Polite´cnica, Madrid. Labye, Y., Olson, M.A., Galand, A., Tsourtis, N., 1988. Design and Optimisation of Irrigation Distribution Network. Irrigation and Drainage Paper 44. FAO, Rome. Lamaddalena, N., Sagardoy, J.A., 2000. Performance Analysis of On-demand Pressurized Irrigation Systems. Irrigation and Drainage Paper 59. FAO, Rome. Lansey, K.E., Mays, L.W., 1989. Optimization model for water distribution system design. J. Hydraul. Eng. 115, 1401–1418. Martı´nez, J., Gil, A., Rolda´n, J., 1993. Estudio del trazado de redes ramificadas de riego. XI Jornadas Te´cnicas sobre riegos, 2–4th June, Valladolid. Pe´rez, R., Andreu, M., Izquierdo, J., 1996. Disen˜o de redes de distribucio´n. In: Cabrera, E., Espert, V., Garcı´a-Serra, J., Martinez, F., Andris, M., Garcı´a, M. (Eds.), Curso de ingenierı´a hidra´ulica aplicada a los sistemas de distribucio´n de agua. Universidad Polite´cnica de Valencia, Valencia, Spain, pp. 653–727. Planells, P., Tarjuelo, J.M., Ortega, J.F., Casanova, M.I., 2001. Design of water distribution networks for on-demand irrigation. Irrig. Sci. 20, 189–201. Planells, P., Carrio´n, P., Ortega, J.F., Moreno, M.A., Tarjuelo, J.M., 2005. Pumping selection and regulation for water distribution networks. J. Irrig. Drain. Eng. 131 (3), 273–281. Pulido-Calvo, I., Roldan, J., Lo´pez-Luque, R., Gutie´rrez-Estrada, J.C., 2003. Water delivery system planning considering irrigation simultaneity. J. Irrig. Drain. Eng. 129, 247–255. Rios, S., 1988. Investigacio´n operativa. Optimizacio´n. Centro de Estudios Ramo´n Areces S.A., Madrid. Rossman, L.A., 1997. EPANET: User Manual. Risk Reduction Engineering Laboratory Office of Research and Development. United States Environmental Protection Agency, Cincinnati, OH.