Linear Programming Approach for Irrigation Scheduling A case Study

Transcription

1 Lnear Programmng Approach for Irrgaton Schedulng A case Study H. MD. AZAMATHULLA, Senor Lecturer, Rver Engneerng and Urban Dranage Research Centre (RAC), Unverst Sans Malaysa, Engneerng Campus, Nbong Tebal, Pulau Pnang, Malaysa; emal: redacazamath@eng.usm.my, mdazmath@gmal.com (author for correspondence) AMINUDDIN AB GHANI, Professor, RAC, Unverst Sans Malaysa, emal: redac02@eng.usm.my NOR AZAZI ZAKARIA, Professor, RAC, Unverst Sans Malaysa, emal: redac01@eng.usm.my CHANG CHUN KIAT, Scence Offcer, RAC, Unverst Sans Malaysa, emal: redac10@eng.usm.my Abstract There s an ncreasng awareness among rrgaton planners and engneers to desgn and operate reservor systems for maxmum effcency to maxmze ther benefts. Accordngly, sgnfcant wor has been done on reservor operaton for nown total rrgaton demand and on the optmal allocaton of water avalable to crops at the farm level.very few studes have been conducted to derve optmal reservor operaton polces ntegratng the reservor operaton wth the on-farm utlsaton of water by the varous crops. Ths present paper deals wth the development of model -- Lnear Programmng (LP) -- to be appled to real-tme reservor operaton n an exstng Chller reservor system n Madhya Pradesh, Inda. Keywords: Croppng pattern, Water resource management, Irrgaton management, Optmzaton 1. Introducton In most developng countres, a huge share of the lmted budget goes to creatng facltes for rrgaton. Constructon of reservors requres very hgh nvestment and also causes socoeconomc and envronmental ssues. Water n the reservor has multple clamants and needs to be optmally utlzed to generate maxmum benefts through proper operaton, whch must reman consstent despte uncertan future nflows and demands. Accordng to the World Commsson on Dams, many large storage projects worldwde are falng to produce the antcpated benefts (Labade, 2004). Smlarly, small storage projects made for local areas n developng countres, le Inda, are also falng to meet expectatons. The man cause dentfed at varous levels of dscusson, as reported by Labade (2004), s nadequate consderaton of the more mundane operaton and mantenance ssues once the project s completed. For exstng reservors, optmum operaton s crtcal, snce all the expected benefts are based on tmely water releases to meet the stpulated demand. Real-tme operaton of a reservor requres mang relatvely quc decsons regardng releases based on short-term nformaton. Decsons are dependant on the storage n the reservor and nformaton avalable n the form of forecast hydrologc and meteorologcal parameters. Ths s especally mportant durng floods and power generaton, where the system has to respond to changes very qucly and may need to adapt rapdly (Mohan et al. 1991). For reservor systems operated for rrgaton schedulng, real-tme operaton s not very common because of longer decson steps. Tradtonally, the reservors meant for rrgaton purposes are operated on heurstcs and certan rules derved from prevous experences. Ths defes the concept of water-management; much of the water s lost, whch n turn leads to loss of revenue. In the early 1960s, mathematcal programmng technques became popular for reservor plannng and operaton; pertnent lterature s avalable. An excellent revew of the topc s gven by Yeh (1985), followed by Labade (2004) and Wurbs (1993). Along wth smulaton studes, Lnear Programmng (LP), Dynamc Programmng (DP) and Non Lnear Programmng (NLP) are the most popular modellng technques. A comparatve study on the applcablty and computatonal dffcultes of these models s presented by Mujumdar and Narular (1993). Many of the aforementoned technques have been mplemented n realstc scenaros, and many reservor systems worldwde are operated based on the decson rules generated from these technques. However, there exsts a gap between theory and practce, and full mplementaton has not been acheved yet (Labade, 2004). 1

2 14 & 15 February 2009 Kuchng, Sarawa The basc dffculty a reservor manager faces s to tae a real-tme optmum decson regardng releases accordng to the future demand and nflow. Ths leads to the problem of optmzaton of the stochastc doman. Two approaches of stochastc optmzaton are practsed: ) Explct Stochastc Optmzaton (ESO), whch wors on probablstc descrptors of random nputs drectly and ) Implct Stochastc Optmzaton (ISO), whch s based on hstorcal, generated or forecasted values of the nputs through the use of Tme Seres Analyss or other Probablstc approaches. The ESO approach has computatonal dffcultes; ISO methods are smple, but requre an addtonal forecastng model for real tme operaton. In the case of rrgaton reservors, decson mang at the reservor level depends upon the water demand arsng at the feld level. In order to operate the reservor n the best possble way, t becomes mperatve to understand the processes occurrng n the crop-sol-water-atmosphere system. Ths helps not only n the estmaton of accurate demands, but also ensures optmum utlsaton of water. If the processes at the feld level are also modelled properly and ntegrated wth the reservor level model, the goal of water management can be acheved n the best possble way. Dudley et al. (1971) poneered the ntegraton of the systems n the determnaton of optmal rrgaton tmng under lmted water supply usng a Stochastc DP model. Dudley and hs assocates then mproved the model (Dudley and Burt, 1973; Dudley, 1988; Dudley and Musgrave, 1993). Vedula and Mujumdar (1992, 1993) and Vedula and Nagesh Kumar (1996) have also contrbuted to ths area. Ther approach was to derve a steady state reservor operaton polcy whle maxmzng the annual crop yeld. DP-SDP and LP-SDP were used n the modellng. However, for real-tme reservor operaton, Vedula and Nagesh Kumar (1996) stressed the need to forecast nflows and ranfall n the current season to mplement the steady state operaton polcy. As a result, the ESO model has to be supplemented wth an ISO model to get a polcy for the current perod. As an extenson to the wor of Vedula and Mujumdar (1992), a sgnfcant contrbuton to the real-tme reservor approach was presented by Mujumdar and Ramesh (1997). They addressed the ssue of short term real-tme reservor operaton by forecastng the nflow for the current perod, a crop producton state varable and a sol mosture state varable. Ther wor was based on SDP, but had all the lmtatons of SDP regardng the curse of dmensonalty. Aganst ths bacground, a model for the dervaton of real-tme optmal operatng polcy for a reservor under a multple crop scenaro s proposed n the present study. The prmary ssue s that the reservor gets nflows durng the wet season (monsoon season) and s operated for rrgaton n the dry season (non-monsoon season). The reservor storage and the sol mosture level are consdered to be the prncpal state varables, and the rrgaton depths are the decson varables. An optmal allocaton model s embedded n the ntegrated model to evaluate the rrgaton water depth suppled to dfferent crops whenever a competton for water exsts amongst varous crops. The model also serves as an rrgaton-schedulng model because t specfes the amount of rrgaton for any gven fortnght. The mpact on crop yeld due to water defcts and the effect of sol mosture dynamcs on crop water requrements are taen nto account. Moreover, a root growth model s adopted to consder the effects of varyng root depths on mosture transfer. The only stochastc element n the season s the evapotranspraton. The handlng of stochastcty has been accomplshed through dependablty based forecastng n an ISO model. The rest of the varables, such as sol mosture status and the reservor storage status, at the begnnng of any perod are consdered to be state varables. The basc formulaton s based on a LP model and s later transformed nto a GA framewor. 2. The Model Formulaton and Concept The real-tme operaton model proposed n the present study ntegrates the reservor level and a feld level decson (Fgure 3). It consders the sol-mosture status and the reservor storage as the state varables and the appled rrgaton depths as decson varables. The formulaton s based on the conceptual model for sol mosture accountng and the reservor storage contnuty relatonshps. A major emphass s lad on mantanng sol mosture n a state such that the evapotranspraton from the crops taes place at a rate that acheves better results n the form of ncreased yelds from the crops. To assess the tmng of rrgaton water applcaton, the sol mosture status of the crop s an mportant parameter. Whenever the sol mosture status approaches a crtcal lmt, rrgaton s appled. Thus, the sol mosture status s montored ether by physcal measurement or through sol mosture models. Sol mosture models are more popular snce they do not requre a lot of nstrumentaton to be nstalled n the feld. Sol mosture models can be formulated ether by a physcal approach (Fedders et al., 1978) or a conceptual approach (Rao, 1987). The conceptual approach has been used by Rao et al. (1988), Rao et al. (1990) and 2

3 Hajlal et al. (1998) for the problem of rrgaton schedulng. Vedula and Mujumdar (1992) utlsed the conceptual model n ther study. The same concept s adopted n the present study. Fgure 3 Flow chart of real-tme operaton of reservor 3

4 14 & 15 February 2009 Kuchng, Sarawa 3. The Conceptual Model In the conceptual model for the Crop-Sol-Water-Atmosphere (CSWA) system, the basc assumpton s that the sol acts as a reservor, the man nputs to the reservor are ranfall rrgaton, and the man outputs are evapotranspraton, percolaton and dranage. The extent of the reservor s consdered to be up to the effectve root zone at the partcular tme. The sol water reservor s governed by a contnuty equaton: IRR + AET = RF (1) The conceptual model stated by Eq. 1 s used to compute the rrgaton to be appled for the LP model wth area as a decson varable. The followng parameters are mportant for the conceptual model. Fgure 1 shows the setch for the conceptual reservor. In the context of the conceptual model two parameters are mportant: RF IRR AET Fgure 1 Conceptual model Varaton of Evapotranspraton wth the Avalable Sol Mosture Evapotranspraton as a functon of the avalable sol mosture s expressed as: AET = PET f a a Zww (2) or aa AET = PET (3) Zww where AET s the actual evapotranspraton that has occurred from crop n fortnght (mm), PET s the potental evapotranspraton n a partcular geographcal locaton (mm), Zww s the crtcal avalable mosture lmt (mm/cm) = (Zf Zw) d, Zf s the feld capacty for the sol (mm/cm), Zw s the permanent wltng pont for the sol (mm/cm), d s the depleton factor and assumed to be 0.5 n the present study, and aa s the average avalable sol mosture over a fortnght (mm/cm). The average avalable sol mosture over a fortnght s gven by +1 a + a a a = 2.0 where a = Zw f otherwse a = Zww A smlar expresson can be used for a < Zww +1 a. 4

5 Root Zone Depth Growth The root depth data n relaton to the tme stages are prepared accordng to the Lnear Root Growth Model (adopted by Narular, 1995). The model assumes that maxmum root depth s acheved at the start of the yeld formaton stage. It remans at the maxmum depth untl the maturty stage. A mnmum depth of 15 cm s consdered n the frst fortnght to account for the condtons of bare sol and an area wth sparse crops. The root depth model s shown n Fgure 2. Lfe span of group Growth stages of group V F G Root Depth Max. Depth Fgure 2 Root Depth growth model Relatve Yeld Rato The yeld of a crop s affected by water defcts and the rate of evapotranspraton. The rate of evapotranspraton tends to decrease dependng on the avalable mosture content. There are many methods to model the phenomenon. However, the model used n the present study s the most commonly-adopted model. The relatve yelds are computed on the bass of the expresson gven by Doorenbos and Kassam (1979) Y Y a m AET 1 Ky 1 (4) PET = Equaton (4) gves a yeld rato for a sngle perod only. However, the aggregate effect of mosture defcts over all fortnghts of crop growth s also evaluated. The fnal yeld ratos computed for the crop durng varous tme perods of a season s computed by a multplcatve model (Rao et al., 1990). The determnaton of the yeld rato s very mportant snce they reflect the operaton polcy for an rrgaton system. The expresson s gven by Y Y a m ncr = AET 1 Ky 1 = 1 PET (5) Water Requrements of the Crops The model derved for an optmal crop pattern uses predetermned rrgaton demands. On the bass of ths, the optmsaton model selects an approprate area for an ndvdual crop. The rrgaton demands are determned usng the conceptual model stated n Eq. 1. The rrgaton requrements may be calculated by substtutng a value of crtcal sol mosture content nstead of sol mosture n ether of the fortnghts and +1 and replacng the values of actual evapotranspraton by potental evapotranspraton and rearrangng the terms of Eq. 1: cr +1 ( ) PET IRR = + (6) 5

6 14 & 15 February 2009 Kuchng, Sarawa where cr s the crtcal sol mosture content below whch the actual evapotranspraton may fall below the potental rate. 4. Integrated LP Formulaton In the objectve functon, the weghted sum of all the actual evapotranspraton values s maxmsed. The weghts are assgned accordng to the yeld response factors for ndvdual crops n ndvdual perods. The objectve s to maxmse the actual evpotranspraton rate to mnmse the defcts n the yelds. The avalable sol mosture n any tme perod n the objectve functon s ndrectly maxmsed: ncr np a + a MaxZ = = 1 = Ky Zww (7) subject to the followng constrants: 1. Sol mosture contnuty a + a 2.0 PET Zww IRR + = RF (8) where a b = ZW wth physcal bounds (9) a (11) 2. Reservor contnuty A S ncr + 1 IRR * AREA B S + = ID Ao RE (12) = 1 Eff S (Maxmum Reservor Capacty M m 3 ) (13) (10) 5. Crop Smulaton Model The optmsaton model presented above yelds some rrgaton depth values that are based on forecasted values for the reference evapotranspraton. Ths reference evapotranspraton, n turn, s based on a dependablty model. However, the actual evapotranspraton value dffers from these values, and thus, before gong nto the next fortnght, the sol mosture status must be updated wth the appled rrgaton and actual clmatc factors. The formulaton for crop smulaton s as follows: Frst compute the fnal sol mosture wth the followng relaton ( + IRR Fc + ARF )/ = (14) or If +1 < 3. 1 Fc IRR = Fc ( ) Fc ZW + ARF + 1 (15) 6

7 1 1 Fc Fc Fc = + Zw + ARF + IRR (16) or 1 1 Fc Fc Fc = + IRR + Zw (17) The computed sol mosture status of the crops s used n the next fortnght to compute the demand. 6. Stochastc Analyss of Evapotranspraton It was prevously stated that the data regardng the clmatc factors s uncertan n nature and the determnaton of these factors beforehand s mpossble. However, there s a general trend to assume the expected values for these factors and carry out the operaton. The concept does not gve a clear pcture of the actual scenaro and the approprate weghts for the ndvdual growth stage of the crops are not assgned. The present study proposes a dfferent method of forecastng the expected values for the clmatc factors. The method of analyss starts wth the computatons of dependablty values of reference evapotranspraton factors from the avalable data. The dependablty of realsaton of any stochastc varable s defned as the probablty of equallng or exceedng that varable wth a partcular value. Mathematcally, ( x X ) P (18) where P (.) s the probablty and x s the varable under consderaton and X s a stpulated value of the varable. A tradtonal method of estmaton of the dependablty value s the use of standard frequency formulae (e.g. Webull s formula or Hazen s formula). In the present study, a detaled probablty analyss for the data s performed. The data s ftted to a standard probablty dstrbuton and the best fttng dstrbuton s tested through the Kolmogorov Smrnov Test (Haan, 1977). Once the values correspondng to dfferent dependabltes are evaluated, dependablty values for reference evapotranspraton are assumed to be dfferent n dfferent growth stages. The analyss s performed on the bass of the yeld response factor. A hgh yeld response factor sgnfes greater senstvty towards the defcts, and thus, a hgher level of dependablty s assumed for the evapotranspraton data and a lower level of dependablty s assumed for the ranfall data. Ths wll ensure a hgher value of rrgaton requred for the crop n the senstve perod. As a result, the crop wll be safeguarded aganst any poor mosture content condtons. 7. LP Model Formulaton for Optmal Croppng Pattern At the start of each dry season, dependng on the storage volume n the reservor, the crop pattern must be determned. To evaluate the crop pattern, another LP model s used. In ths model, rrgaton depths are calculated from Eq. (6). The formulaton s as follows: The objectve functon s MaxZ = C 1 X 1 + C 2 X 2 + C 3 X 3 (19) whch s subject to the followng constrants: 1. Total avalable area X 1 +X 2 +X 3 A (20) where X 1, X 2, and X 3 are the decson varables related to the area of ndvdual crops;c 1, C 2, and C 3 are the cost coeffcent for each crop n Indan Rupees (1 US $ = 50 INR); and A s the maxmum area avalable for rrgaton. 2. Area of each ndvdual crop: 7

8 14 & 15 February 2009 Kuchng, Sarawa The area under each crop s requred to be constraned; thus, there are lower and upper bounds on the area under each crop. The lower bounds ndcate the mnmum area that can be allocated to a crop, whle the upper bound ndcates the maxmum. In the present study, the lower bounds were defned for all the crops except cash crops, whle the upper bounds were defned consderng the present croppng pattern. The constrants can be expressed as L X M where L corresponds to the lower bound of the area for the th crop and M corresponds to the upper bound on the area of the th crop. (21) 8. Model Applcaton The developed models were appled to the Chller reservor system n Madhya Pradesh, Inda (Lattude 23 o 23 N and Longtude 76 o 18 E). In the central part of Inda, many reservor projects have been constructed for rrgaton, but no rrgaton s avalable from these reservors durng the monsoon perod (from June to September). The area receves about 90 to 95 % of ts ranfall durng the Monsoon season. The ranfall then becomes runoff to the reservors. These reservors are desgned to contan the runoff n the monsoon season, but there s no runoff durng non-monsoon months. The present formulatons are specally suted for these types of reservors. Non-monsoon ranfall s rare and provdes lttle runoff. A systematc data base was prepared for the varous physcal features of the reservors, ncludng the meteorologcal and hydrologcal data such as evapotransraton, detals of crops n the command area, detals of net returns from ndvdual crops and sol propertes collected from the College of Agrculture, Indore, Inda. 9. Results and Dscusson Optmum Crop Pattern A separate computer program was run before the real tme operaton program to determne the optmum crop pattern for all possble storage values. The results of the optmum crop pattern are stated n Table 1. The results ndcate that from a storage level of M m 3 to a storage level of M m 3, the croppng pattern s same as the one that has been adopted n the project formulaton. However, below a storage level of M m 3, the crop pattern changes suddenly, and wheat (ordnary) s not recommended by the model. The area of wheat (hybrd) also gets reduced when the ranfall storage s below ths level. However, the area for Gram s full, up to a storage level of M m 3. The change n croppng pattern ndcates that effcent water usage s mantaned. Table 1 Optmum Croppng Pattern for Dfferent Lve Storage Values Lve storage (M m 3 ) Area (ha) for dfferent crops Wheat (ordnary) Gram Wheat (hybrd) Results from Real-Tme Operaton Model The real-tme operaton model gves an optmal operatng polcy for the avalable storage n the present fortnght consderng the future. The model also yelds the values of rrgaton to be appled to ndvdual crops n the felds. In the wae of defcent water supples, the model dstrbutes the avalable water over the tme for dfferent crops optmally. The sample results of the present model are stated n Table 2. The avalable mosture to the crops s not affected, and generally the sol remans at the upper lmt of the avalable sol-mosture. Ths 8

9 s because the crop pattern s predcted accordng to the avalablty of the storage n the reservor. The results are ndcatve of successful applcaton of the real-tme operaton strategy proposed n the present wor. Table 2 Sample Results Showng the Sol Mosture, Avalable Sol Mosture, Storage, and Irrgaton to be appled for Dfferent Crops for a Real-Tme Reservor Operaton Model (LP) Lve Storage n the Reservor 31.1 M m 3 PARAMETER FORTNIGHT Reservor Storage (M m 3 ) Crop Wheat (ordnary) 1) Sol Mosture (mm/cm) ) Avalable sol Mosture (mm/cm) ) Appled Irrgaton (mm) Crop GRAM 1) Sol Mosture (mm/cm ) Avalable sol Mosture (mm/cm) ) Appled Irrgaton (mm) Crop Wheat (hybrd) 1) Sol Mosture (mm/cm ) Avalable sol Mosture (mm/cm) 3) Appled Irrgaton (mm) Relatve Yeld Ratos Relatve yeld ratos computed for dfferent crops at dfferent lve storage values are shown n Table 3. The relatve yeld ratos for all the crops become one f lve storage n the reservor s equal to or greater than M m 3. The GA model s found to be better for applcaton n real world operaton of the reservor. Table 3 Relatve Yeld Rato for Dfferent Lve Storage Values Computed Wth a Real-Tme Reservor Operaton Model Lve storage (M m 3 ) Wheat (ordnary) Relatve yeld rato for dfferent crops LP Gram Wheat (hybrd) Concluson A real-tme model usng an ntegrated Lnear Programmng Model for a reservor system meant for rrgaton has been developed n the present study to obtan an optmal reservor operatng polcy that ncorporates feld level decsons, whle also decdng the approprate tme and amount of water to release from the reservor. 9

10 14 & 15 February 2009 Kuchng, Sarawa From the analyss, the followng conclusons can be drawn: The developed model can be successfully appled to rrgaton supportng reservor systems. Furthermore, the models ensure an optmum reservor release over dfferent tme perods. In addton, they also ensure optmum allocaton of the avalable water over the dfferent crops n the felds. Whle allocatng the water to dfferent crops n the felds, the model taes nto account the crtcal growth stages of the crops and allocates suffcent water to each crop to safeguard t aganst any ll effects of water defcts. The optmum crop pattern model used n the study wll only allow productve rrgaton, so the amount of wasted water s reduced. Acnowledgements The authors would le to express sncere thans to Unverst Sans Malaysa for the fnancal support of ths wor. Nomenclature AET Actual evapotranspraton n perod from crop (mm) Actually occurrng potental evapotranspraton n perod (mm) ARF Actual ranfall value n the fortnght A and B K Constants relatng the storage to reservor evaporaton A o Area of spread at dead storage level d Depleton factor Effectve root zone depth of a crop n perod (cm) +1 Effectve root zone depth of a crop n perod +1 (cm) Eff Overall effcency Fc Crop evapotranspraton coeffcent ID Industral supply from the reservor (mandatory release) IRR Irrgaton appled to crop n stage (mm) Ky Yeld response factors for a crop n perod PET Potental evapotranspraton n a partcular geographcal locaton (mm) RE Rate of evaporaton n fortnght RF Ranfall n perod (mm) S Reservor storage at the begnnng of perod S +1 Reservor storage at the end of perod Zf Feld capacty for the sol (mm/cm) Zw Permanent wltng pont for the sol (mm/cm) Zww Crtcal avalable mosture lmt (mm/cm) +1 Y a Y m Intal sol mosture n the tme stage n for a crop (mm/cm) Fnal sol mosture n a partcular tme stage for a partcular crop (mm/cm) Actual crop yeld Maxmum crop yeld 10

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