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Journal of Water Resource and Protection, 2013, 5, 28-34 http://dx.doi.org/10.4236/jwarp.2013.57A005 Published Online July 2013 (http://www.scirp.org/journal/jwarp) Yield Estimation for a Single Purpose Multi-Reservoir System Using LP Based Yield Model Deepak V. Pattewar1, Kalpeshkumar M. Sharma1, P. D. Dahe2 1Civil Engineering Department, M. G. M’s College of Engineering, Nanded, India. 2Civil Water Management, S. G. G. S College of Engineering and Technology, Nanded, India Email: deepak_pattewar@yahoo.co.in, kalpeshkumar_sharma@yahoo.com Received April 17, 2013; revised May 22, 2013; accepted July 1, 2013 Copyright © 2013 Deepak V. Pattewar et al. This is an open access article distributed under the Creative Commons Attribution Li- cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Application of optimization techniques for determining the optimal operation policy for reservoir is a major area in wa- ter resources planning and management. Linear programming, ruled by evolution techniques, has become popular for solving optimization problems in diversified fields of science. An LP-based yield model (YM) has been used to re- evaluate the annual yield available from the reservoirs for irrigation. This paper extends the basic yield model and pre- sents a yield model for a multiple-reservoir system consisting of single-purpose reservoirs. Optimum yield of reservoirs system is calculated by yield model. The objective is to achieve prespecified reliability for irrigation and to incorporate an allowable deficit in the annual irrigation target. The yield model is applied to a system of two reservoirs in the Manar River in India. This model can act as a better screening tool in planning by providing outputs that can be very useful in improving the efficiency and accuracy of detailed analysis methods such as simulation. Keywords: Yield Model; Reservoir Operation; Irrigation Releases; Manar River 1. Introduction Linear Programming (LP) is a commonly used optimiza- tion approach in water resources management. It is con- cerned with solving a special type of problem; one in which all relations among the variables are linear, both in constraints and the objective function to be optimized. An application of LP to reservoir operations has varied from simple straightforward allocation of resources to complex situations of operation and management. In the past, limitations of computing power meant that optimi- zation was achieved by decomposing reservoir systems in time and space. These early models were predomi- nantly deterministic, that is, they did not take into ac- count the stochastic nature of inflows but rather were based on long-term average seasonal or monthly flows. However, they have gradually been improved. For ex- ample, Loucks [1] developed a stochastic LP technique for a single reservoir subject to random, serially corre- lated, flows. Subsequently, much more complicated sto- chastic models have been developed to reflect more real- istically stream flow stochasticity, evaporation losses and more complex systems involving multiple reservoirs (Dandy G.C., Connarty M.C and Locks D.P. [2]; William W. G. Yeh [3]). Under certain assumptions, non-linear problems can be linearized and LP equations solved by iteration or approximation procedures. The program MODSIM is a generic program based around LP ap- proaches that has been developed specifically for model- ing water resources systems and reservoir operation by Labadiee [4]. Sinha A.K., Rao B.V. and Lall U. [5] have studied optimal reservoir operation for irrigation, hydro- power production which involved constrained linear op- timization. Dahe P.D. and Srivastava D.K. [6] developed the basic yield model and presented a multiple yield model for a multiple reservoir system consisting of single purpose and multipurpose reservoirs. The objective was to achieve pre specified reliabilities for irrigation and energy generation and to incorporate an allowable deficit in the annual irrigation target. The results were analyzed for four cases. Srivastava D.K. and Taymoor A. Awachi [7] developed nested models which were applied in tan- dem using linear programming (LP), dynamic program- ming (DP), artificial neural networks (ANN), hedging rules (HRs), and simulation. An LP-based yield model (YM) has been used to reevaluate the annual yields avai- lable from the Mula reservoir for water supply and irri- gation. C opyright © 2013 SciRes. JWARP D. V. PATTEWAR ET AL. 29 This study presents a methodology to optimize the de- sign of the multi-reservoir irrigation system by taking monthly inflow and initial storage and tries to predict the maximum possible releases using Linear programming based Yield model. The specific objectives of the present study can be stated as fallows: 1) To develop a Linear Programming based yield mo- del for reservoir operation for a monthly time step. 2) Comparison of yield model and actual irrigation re- leases for single purpose irrigation reservoirs in Manar River. 3) To draw the conclusions from the interpretation of results obtained. 2. Reservoir Yield Model The conceptualisation and details of the yield model on which the present model development is based are pre- sented in Loucks et al. [8]. When reservoir yield with reliability lower than the maximum reliability is to be determined, the extent of availability of yield (or the al- lowable deficit in yield) during failure years can be specified. This is achieved by specifying a failure frac- tion for the yield during the failure years. The factor θp,j is used in the model to define the extent of available yield during failure years. The objective of this model is to maximize the yield for given capacity of the reservoir. Let p denotes the exceedence probability for the yield. The index j refers to a year and index t refers to a within-year period. In this model only the firm yield is used. The yield model given by Dahe and Srivastava [6] to determine single yield from a reservoir is as follows. The formulation of the yield model is as follows: Objective function Maximize ,, 12 f pfp Oy Oy o 1, 1, (1) Constraint 1) Over-year storage continuity o, 1, 11,1,,1, fp j jpjjj j s IOy SpEls o 1, 2, (2) o, 2, 12,2,,2,2, fp j jpj jjjj s IOySp ElSps ,, 12 (3) The over-the-year capacity is governed by the distri- bution of annual stream flows and the annual yield to be provided. The maximum of all the over-the-year storage volumes is the over-the-year storage capacity. It is possi- ble to specify a failure fraction to define the allowable deficit in annual reservoir yield during the failure years in a single-yield problem. In the above equation, f pfp Oy Oy o 1, 1 is the safe (firm) annual yield from Up- per Manar reservoir and Lower Manar reservoir with o 1, and reliability p. j s j s are the initial and the final over-the-year active storages in year j for the Upper Manar reservoir and similarly for the Lower Manar o 2, 1 o 2, j s and j s respectively; 1, j I2, and j I 1, are the inflows in year j (Upper Manar and Lower Manar in Manar River); θp,j is the failure fraction defining the pro- portion of the annual yield from reservoir to be made available during the failure years to safeguard against the risk of extreme water shortage during the critical dry periods (θp,j lies between 0 and 1, i.e., for a complete failure year θp,j =0, for a partial failure year 0 < θp,j <1, and for a successful year θp,j =1); j El 2, and j El 1, = evaporation loss in year j and j Sp 2, and j Sp o 1, 11j excess- release (spills) in year j; 2) Over-year active storage volume capacity s Y o 2, 12j (4) s Y , 1,11,11,1, ,11, wfp ttw ttt fpt t (5) The active over-year reservoir capacity (Y1) required for delivering a safe or firm annual yield in Upper Manar reservoir and active over-year reservoir capacity (Y2) required for delivering a safe or firm annual yield in Lower Manar reservoir 3) Within-year storage continuity OyElOyEl s , 2, 12,22, 2, ,21, ,2, 0.10 wfp tt t t tt tw s (6) f pfpt sOyEl Oy ElOys 1, 12, 1 , ww tt (7) Any distribution of the within-the-year yields differing from that of the within-the-year inflows may require ad- ditional active reservoir capacity. The maximum of all the within-the year storage volumes is the within-the- s s year storage capacity. In the above equation, 1, 2, , ww tt and s s 1,t are the initial and the final within-the- year active storages at time t; and 2,t are the ratio of the inflow in time t of the modelled critical year of record to the total inflow in that year; and and are the within-the-year evaporation losses during 1,t El 2,t El time t. The inflows and the required releases are just in balance. So, the reservoir neither fills nor empties during the critical year. 4) Definition of estimated evaporation losses (Over- year) 1, 11, 0 1,11, 11,1 10 2 ww tt r jj t t ss EEsEl (8) Copyright © 2013 SciRes. JWARP D. V. PATTEWAR ET AL. 30 2, 1 0 2,22, 1 2, 2, 10 2 ww tt jj t t ss EEs (9) 5) Definition of estimated evaporation losses (Within- year) 2 r El 1, 11, o 11,1 1,1 10 2 tt tr tc r t EE s El (10) ww ss 2, 1 o 22,2 10 2 tc r EE s 2, 2,2 ww tt tr t ss El 6) Total reservoir capacity 11,1 1 w t Ys Ya 2 Ya (13) Sumear and the w age capacities is equal to the active storage capacity of the reservoir. (11) (12) 22,1 w t Ys of the over-the-yithin-the-year stor- 7) Proportioning of yield in within-year periods , 1, ,1,1 p t fp t f OyK Oy (14) , 2, ,2,2 p t fp t f OyK Oy (15) 1,t K and 2,t K defines a predetetio l reservoir yield to be supplied in the with yield in period t. The tions , a tributary of Manjara River in Goda- anar River pper Manar rmined fracn of an- nuain-year Equa (1) to (15) present the Multi-reservoir yield model for Upper Manar and Lower Manar reservoir in Manar River. 2.1. System Description: Manar River The Manar River vari basin, Maharashtra states in INDIA. In M two medium project has constructed i.e. U and Lower Manar reservoir for irrigation preposes Fig- ure 1. Table 1 is the silent features of Upper Manar Pro- 1 2 1- Upper Mana r for Irrigation Purpose 2-Lo Irrigat wer Manar for ion Purpose Manar river Inflow Evaporation Evaporation Sp ill Spil l Figure 1. Line diagram of reservoir system on Manar River. ject Limboti reservoir and Lower Manar Project-Barul reservoir. A 37 years historic inflow data for the system considered is available as shown in Figures 2 and 3. 2.1.1. Irrigation Parameters (Kt) for Upper Manar Limboti Reservoir and Lower Manar Barul Reservoir The monthly proportions of the annual irrigation targets (Kt values) are worked out by considering the cropping patterns and irrigation intensities recommended by the agricultural officer. Kt defines a predetermined fraction of reservoir yield the within-year period t. The K value t are given in Table 2. s Figure 2. Inflow at Upper Manar-Limboti reservoir. Figure 3. Inflow at Lower Manar-Barul reservoir. Table 1. Silent features of Upper Manar and Lower Manar project in Manar River. Particulars Upper Manar Lower Manar Irrigation Purpose Irrigation Purpose Scope of Scheme Location Manar River at Limboti Manar River at Barul Catchment area 987.60 Sq Km 1585.08 Sq Km Gross storage capacity107.98 MCM 146.92 MCM Capacity of Live Storage 75.71 MCM 138.21 MCM Capacity of Dead Storage 32.27 MCM 8.71 MCM 75% dependable yield(Project Report)162.50 MCM 205.76 MCM Copyright © 2013 SciRes. JWARP D. V. PATTEWAR ET AL. Copyright © 2013 SciRes. JWARP 31 w approximation, irrigation and evaporatised iodel fo Mana Month Table 2. Within-year infloon parameters un the yield mr reservoirs on r River. 1,t 2,t γt1 γt2Kt1 Kt2 June 0.0522 0.0560 0.133 0 0.04430.0215 0.1240 July 0.1580 0.1670 0.0686 0.0492 0.0151 0.0000 0589 0.0417 0.0186 0.0000 0.1490 0.1460 0.0575 0.0858 0.1618 0.0446 0.0363 0.0363 0.0540 0.0759 0. M Aug 0.2967 0.2953 0. Sept 0.2289 0.2209 0.0568 0.0880 0.0299 0.0336 Oct Nov 2347 0.1855 Dec 0.0186 0.0186 0.0450 0.0973 0.1794 0.1840 Jan 0.0160 0.0160 0.0547 0.0977 0.0842 0.1860 Feb 0.0103 0.0103 0.0623 0.1041 0.0634 0.0910 arch Apr 0.0112 0.0112 0.1004 0.1173 0.0667 0.0293 il 0.0119 0.0119 0.1302 0.1400 0.0706 0.0230 May 0.0109 0.0109 0.1787 0.0585 0.0541 0.0990 2.1.2.ximationtical Withinnflows Values for Uervoir Lower Man Reservoi βt vae based on monthly floe βt val- ues baseon average hly flows fovoir are given ble 2. .1.3. Evaporation Parameters of Reservoirs γ voirs and an volume at dead storage elevation for respec- tive reservoirs. The storage-area and storage-elevation e stor- e values of te given in the d Modelbserverical inflowanar for 37(1969-2e used in ation e yieldshe reseOut of the of 9 st flow 3rd, 4th, th, 18th, 23, 29th th) 197172, 1985, 1986, 193, and 20% of ths) were asss the mon faiars in bervoir, de the modified method of determining failure years by el. Thus remaining 28 years were successful roject reliability. Thus, elve within year periods s based on for the anal The within Appro of Cri-Year I (βt) pper Manar Limboti Res andar Barul r lues araveragews. Th d in Ta montr reser Yiel. The od histos of M River years 005) arcomput of th from trvoirs. th se a set rd th lowe years (16 , 17, 25 and 36 1997 , 19 04 (≈25 4, 198 e year 91, 199 umed a comlure yeoth restermined by 2t The average monthly evaporation depth at all the reser- is obtained from the Water Resources Department vailable project reports. The evaporatio loss due to dead storage E01 = 8.158 and E02 = 11.30 are obtained by product of the average annual evaporation depth and the area relationship is taken for study. A linear fit for th age-area data for each reservoir above the dead storage is obtained from the storage area relationship. The evapora- tion volume loss rate 10.2880 r El and 20.413 9 r El are obtained by taking the product of the slope of the area elevation curve linearized above dead storage and the average annual evaporation depth at respective reservoirs. The parameter γt (the fraction of the annual evaporation volume loss that occurs in within-year period t) is com- puted by taking the ratio of the average monthly evapo- ration depth to the average annual evaporation depth at respective reservoirs. Thhe γt ar Table 2. 3. Analysis and Results 3.1. Application of the Yield Model in Assessment of Manar River Yield The approximate model which includes within year pe- riods for only one modelled critical year is known as the yield mod years representing 75% annual p thirty seven over year and tw were considered for analysis. The value of βt’ average monthly flows have been considered ysis and are presented in the Table 2. year yields from the reservoir for irrigation in a month are represented as a fraction of its annual yield. With the provision of θp,j , the extent of failure in the annual yield from the reservoir during failure years was monitored as clear guidelines were not established for deciding its value. The value of θp,j for the project was determined using the YM with an objective to minimize its value. In Manar River, irrigation originally being the main project target was considered as a single yield or firm yield from the reservoir. The annual project reliability for irrigation was kept equal to 75%. The value of θp,j was found to increase with the decrease in the annual yield from the reservoir. In Manar River two reservoirs (Upper Manar- Limboti reservoir and Lower Manar-Barul reservoir) are constructed for the irrigation purposes. For Upper Manar-Limboti reservoir with active stor- age capacity of 75.71 MCM and for Lower Manar-Barul reservoir with active storage capacity of 95.71 MCM, the yield is found out for Safe reservoir yield θp,j = 1 and θp,j = 0.00 respectively. Calculated annual yield of Upper Manar-Limboti reservoir by yield model is 52.44 and D. V. PATTEWAR ET AL. 32 107.24 MCM respectively and for Lower Manar-Limboti reservoir is 42.76 and 107.27 MCM respectively in Multi reservoir yield model analysis. Within-period water re- leases are shown in Table 3. 3.2. Comparison of YM and Actual Releases in Lower Manar-Barul Reservoir The main objective is to compute the yield that should be released to fulfill the total demand. Comparison of actual demand, releases and yield which are obtained from the model used is as follows. Multi-reservoir yield model based on the monthly inflow and monthly irrigation de- mands of the reservoir operation system is considered for Table 3. Representing the monthly water releases for irriga Safe Reservoir Yield (MCM) θp,j = 1.00 the comparison. The Upper Manar-Limboti reservoir is recently constructed and has started operating from Oc- tober 2010. Water releases data is not available for it hence only Lower Manar-Barul reservoir is taken for the comparison. Table 4 gives the output of the model used for 75% reliable yield as well as demand and actual releases in the years which are considered in Lower Manar-Barul res- ervoir. The data available on actual releases of only 6 years is used for comparison. As per the Table 4 the ac- tual release from the reservoir is maximum 98.79 MCM in the year 2000-2001 and minimum is 68.49 MCM in year 2003-2004. Figure 4 shows comparison between tion by approximate YM (Multi-reservoir) in Manar River. 75% reliable Yield (MCM) θp,j = 0.00 Month Upper Manar Limboti Reservoir Lower Manar Barul Reservoir Total Yield (MCM) Upper Manar Limboti Reservoir Lower Manar Barul Reservoir Total Yield (MCM) June 1.127 5.302 6.429 2.305 13.301 15.606 July 0.719 0.000 0.719 1.619 0.000 1.619 1.994 0.000 1.994 Sep3.004 3.206 3.604 6.810 17.352 4.784 22.136 25.170 19.898 45.068 19.239 19.737 38.976 9.030 19.952 28.982 6.799 9.761 16.560 March 3.497 1.253 4.750 7.153 3.143 10.296 A May 2.8070 5.801 16.420 Aug 0.975 0.000 0.975 t 1.568 1.436 Oct 8.485 1.907 10.392 Nov 12.308 7.932 20.240 Dec 9.408 7.868 17.276 Jan 4.415 7.954 12.369 Feb 3.324 3.891 7.215 pril 3.702 0.983 4.685 7.571 2.467 10.038 37 4.233 7.10.619 Total 52.437 42.765 95.207 107.239 107.27 214.515 Table 4. Values of actual demand, actuases and yield model (YM with 7able θp,j = 0.0 Actual Water Releases in ys 2000-2005 (MCM l relea5% reli0). ear) SMonth M ) Actual Demand (MCM) 2001 2001-2002-200320042004-2006 release M) r. No Multi Y (MCM 2000-20022003-2005 2005- Average Water (MC 6 June 25.51 25 111.57 8.5 10.1.63 95 13.30 12.1.29 46 110. 7 July 0 0 0 0 0 0 0 00 Aug 0 0 0 0 0 0 0 00 Sept 6.91 32 3.3.14 2.3 2.15 97 Oct 9.18 41 4.4.16 3.06 3.18 94 Nov 38.17 32 17.31 12.71 15.17.4 38 Dec 37.86 18 16.17.17 12.61 15.26 25 Jan 38.27 37 16.17.36 12.74 15.44 42 Feb 18.72 99 8.8.49 6.23 7.53 ar 6.03 9 2.73 2 2.2.75 4 Apr2.03 9.78 9.28 eld 98.79 91.78 0 0. 8 0 0. 9 3.60 3.06 83 3.2. 10 4.78 4.06 76 4.3. 11 19.89 18.16.9 65 16. 12 19.73 18.76 52 17.16. 1 19.95 18.94 69 17.16. 2 3 M 9.76 3.14 8. 2. 29 2.67 66 8. 47 8.03 2.59 il 2.46 4.73 2.27 2.09 2.15 1.56 1.94 2.16 5 Yi May 10.61 107.26 20.37 205.75 01 9.24 6.78 8.35 9. 07 93.32 68.49 84.33 93. 8.74 88.30 Copyright © 2013 SciRes. JWARP D. V. PATTEWAR ET AL. 33 Figure 4. Comparison of acreleases and releases tai freld model. ontater r, m deand ly ield by yield model. From the figure it is very clear that in the month of June, December and January the reser- voir releases are comparable with the yield model, where as the actual demand is very large as compared to the actual releases from the reservoir except in the month February, March and April. It can be seen from the Fig- ure 4 that the releases are negligible in the period of Kharif Crop i.e. June, July, August, September and mid of October. Whereas the releases are more in the period of Rabbi Crop (i.e. from October to February) and in Hot Weather crop period (i.e. from February to May). The Yield model can be used for yield assessment with specified reliabilities and thus assists in the effectiv istic full optimization model by the way of reduction in l hoieo whicfe r yield model and yield l with reliabf flowmplete failure. Ihe case of complete failure, the annual firm pro- ved is zerouring thelure yearhe yieldel is le of consideringreliabif annuad. It in the coof come or partial failure g the yearshe yield model for rvoir m developed in thaddresses the ts oforatiesiiabilitidif- purpas we alldeficit criterion nnualation target in a roir systcon- g of a Mlti reserrrigatistem. It can act as tter scrg tool ing. Being an option el, no inlicy is eeded for the analysis of res- ir syste mis gennough ould ilar reir sys REFERENCES tual ob- nedom yi mhly weleasesonthlymand month y e management and design of irrigation reservoir system. Yield model provides a better alternative to the determi- n size and provides sufficiently accurate results. It also allows determination of annual yield with a given reli- ability less than the maximum reliability. There is also a provision of determining the percentage of annual yield to be supplied during failure years. 4. Conclusion The study of multi-reservoir operation in Manar River is carried out using LP based yield model. Identification and screening of the feasible solution to provide potential candidates for detailed evaluation is a crucial stage dur- ing the search for optimal solution of real life problems. Mathematical optimization models play a vital role in this regard. The overall effort in handling real life sys- tems can be significantly reduced with screening models capable of better representing the system and providing fewer and more accurate candidate solutions for detailed evaluation which is proportional to the number of candi- date solutions to be evaluated and their proximity to the optimasolution. Te yield mdel is studd for the tw cases, mode h are sa 75% reservoi ility o with co n tyield id d fais. T mod capab the lity ol yiel can alsocludencept plet durin syste failure. T is study successfully rese aspec incorpng the dred reles for ferentoses, ll as anowable for a irrigeservems sistinuvoir ion sy a beeeninin planntimiza mod ervo itial po ems. Th n odel eral eand c be applied to simservotems. [1] D. P. Loucks, “Computer Models for Reservoir Regula- tion,” Journal of the Sanitary Engineering Division, Vol. 94, No. 4, 1968, pp. 657-671. [2] G. C. Dandy, M. C. Connarty and D. P. Loucks, “Com- parison of yield assessment of multiple reservoir sys- tems,” Journal of Water Resources Planning and Man- agement, Vol. 123, No. 6, 1997, pp. 350-358. [3] W. W. G. Yeh, “Reservoir Management and Operation s Models: A State-of-the-Art Review,” Water Resources Research, Vol. 21, No.12, 1985, pp. 1797-1818. doi:10.1029/WR021i012p01797 [4] J. W. Labadie, “Optimal Operation of Multireservoir Systems: State-of-the-Art Review,” Journal of Water Resources Planning and Management, Vol. 130, No. 2, 2004, pp. 93-111. doi:10.1061/(ASCE)0733-9496(2004)130:2(93) [5] A. K. Sinha, B. V. Rao and U. Lall, “Yield Model for Screening Multi Purpose Reservoir Systems,” Journal of Water Resources Planning and Management, Vol. 125, No. 6, 1999, pp. 325-332. doi:10.1061/(ASCE)0733-9496(1999)125:6(325) [6] P. D. Dahe and D. K. Srivastava, “Multipurpose Multi- yield Model with Allowable Deficit in Annual Yield,” Journal of WRPM, Vol. 128, No. 6, 2002, pp. 406-414. [7] D. K. Srivastav and T. A. Awchi, “Storage-Yield Evalua- tion and Operation of Mula Reservoir, India,” Journal of Water Resources Planning and Management, Vol. 135, No. 6, 2009, pp. 414-425. doi:10.1061/(ASCE)0733-9496(2009)135:6(414) [8] D. P. Loucks, J. R. Stedinger and D. A. Haith, “Water Resource Systems Planning and Analysis,” Prentice-Hall, Inc., Englewood Cliffs, 1981. Copyright © 2013 SciRes. JWARP D. V. PATTEWAR ET AL. 34 Appendix: Notation The following symbols are used in this paper: , 1 fp Oy Annual firm Upper Manar reservoir yield. , 2 fp Oy Annual firm Lower Manar reservoir yield. o 1, 1j s Initial storage of Upper Manar reservoir at the beginning of year j. Initial storage of Lower Manar reservoir at the g of year j. begin- r at the 1, j year j. 2, j Sp Excess release in Lower Manar reservoir in year j. 1, j I Annual inflow at Upper Manar reservoir site in year j. 2, j I Annual inflow at Upper Manar reservoir site in year j. nin o 1, j s Final storage of Upper Manar reservoir at the beginning of year j. o 2, j s Final storage of Lower Manar reservoi beginning of year j. Sp Excess release in Upper Manar reservoir in 1, 2, , j j ElEl = Annual evaporation volume loss from reservoir in year j. 12 , tt ElEl = Evaporation volume loss from reservoir in period t. 1, 2, , tt = Fraction of total annual yield for assumed critical period inflow in Uppe Manar reservoir and Lower Manar reservoir. 12 ,YY = Over-year storage capacity of Upper Manar reservoir and Lower Manar re ,Ya Ya = Total active storage capacity of servoir. Upper 12 Manar reservoir and Lower Manar reservoir. 1, 2, , tt K K = Percentage fraction of annual irrigation target in period t in Upper Manar and Lower Manar res- ervoir. 1, 12, 1 , ww tt s s = Initial within-year storage volume in pe- riod t in Upper Manar reservoir and Lower Manar reser- voir. 1, w t s = Final within-year storage volume in period t in Upper Manar reservoir. MCM = Million Cubic Meter. Copyright © 2013 SciRes. JWARP |