Yield Estimation for a Single Purpose Multi-Reservoir System Using LP Based Yield Model

Application of optimization techniques for determining the optimal operation policy for reservoir is a major area in water 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 reevaluate the annual yield available from the reservoirs for irrigation. This paper extends the basic yield model and presents 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.


Introduction
Linear Programming (LP) is a commonly used optimization approach in water resources management.It is concerned 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 optimization was achieved by decomposing reservoir systems in time and space.These early models were predominantly deterministic, that is, they did not take into account the stochastic nature of inflows but rather were based on long-term average seasonal or monthly flows.However, they have gradually been improved.For example, Loucks [1] developed a stochastic LP technique for a single reservoir subject to random, serially correlated, flows.Subsequently, much more complicated stochastic models have been developed to reflect more realistically 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 approaches that has been developed specifically for modeling 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, hydropower production which involved constrained linear optimization.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 tandem using linear programming (LP), dynamic programming (DP), artificial neural networks (ANN), hedging rules (HRs), and simulation.An LP-based yield model (YM) has been used to reevaluate the annual yields available from the Mula reservoir for water supply and irrigation.
This study presents a methodology to optimize the design 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 model for reservoir operation for a monthly time step.
2) Comparison of yield model and actual irrigation releases for single purpose irrigation reservoirs in Manar River.
3) To draw the conclusions from the interpretation of results obtained.

Reservoir Yield Model
The conceptualisation and details of the yield model on which the present model development is based are presented 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 allowable deficit in yield) during failure years can be specified.This is achieved by specifying a failure fraction for the yield during the failure years.The factor θ p,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 , , 1 2 Constraint 1) Over-year storage continuity The over-the-year capacity is governed by the distribution 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 possible 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, 2) Over-year active storage volume capacity The active over-year reservoir capacity (Y 1 ) required for delivering a safe or firm annual yield in Upper Manar reservoir and active over-year reservoir capacity (Y 2 ) required for delivering a safe or firm annual yield in Lower Manar reservoir 3) Within-year storage continuity , Any distribution of the within-the-year yields differing from that of the within-the-year inflows may require additional active reservoir capacity.The maximum of all the within-the year storage volumes is the within-thes s year storage capacity.In the above equation, , and s s are the initial and the final within-theyear 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 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 (Overyear) Copyright © 2013 SciRes.JWARP 6) Total reservoir capacity    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 γ t (the fraction of the annual evaporation volume loss that occurs in within-year period t) is computed by taking the ratio of the average monthly evaporation depth to the average annual evaporation depth at respective reservoirs.Th he γ t ar Table 2.

Application of the Yield Model in Assessment of Manar River Yield
The approximate model which includes within year periods 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 β t ' 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 θ p,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 θ p,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 θ p,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 storage 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 θ p,j = 1 and θ p,j = 0.00 respectively.Calculated annual yield of Upper Manar-Limboti reservoir by yield model is 52.44 and 107.24MCM respectively and for Lower Manar-Limboti reservoir is 42.76 and 107.27MCM respectively in Multi reservoir yield model analysis.Within-period water releases are shown in Table 3.

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 demands of the reservoir operation system is considered for  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 reservoir.The data available on actual releases of only 6 years is used for comparison.As per the Table 4   Yield model provides a better alternative to the determin size and provides sufficiently accurate results.It also allows determination of annual yield with a given reliability less than the maximum reliability.There is also a provision of determining the percentage of annual yield to be supplied during failure years.

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 during 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 systems 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 candidate solutions to be evaluated and their proximity to the optima solution.

Appendix: Notation
The following symbols are used in this paper:

Figure 4 .
Figure 4. Comparison of ac releases and releases tai fr eld model.ont ater r , m de and ly ield by yield model.From the figure it is very clear that in the month of June, December and January the reservoir 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 Annual firm Lower Manar reservoir yield.
Initial storage of Upper Manar reservoir at the beginning of year j.Initial storage of Lower Manar reservoir at the g of year j. 10

2.1. System Description: Manar River The
 of the over-the-y ithin-the-year stor-7) Proportioning of yield in within-year periods   K and 2,t K defines a predete tio l reservoir yield to be supplied in the with yield in period t.The tions , a tributary of Manjara River in Godaanar River pper Manar rmined frac n of annua in-year Equa (1) to (15) present the Multi-reservoir yield model for Upper Manar and Lower Manar reservoir in Manar River.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-

Upper Manar for Irrigation Purpose 2-Lo Irrigat wer Manar for ion Purpose Manar river Inflow Evaporation Evaporation Spill Spill 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 Figures2 and 3. 2.1.1.

Irrigation Parameters (K t ) for Upper Manar Limboti Reservoir and Lower Manar Barul Reservoir The
monthly proportions of the annual irrigation targets (K t values) are worked out by considering the cropping patterns and irrigation intensities recommended by the agricultural officer.K t defines a predetermined fraction of reservoir yield the within-year period t.The K value t are given in

Table 1 . Silent features of Upper Manar and Lower Manar project in Manar River.
Month

Table 3 . Representing the monthly water releases for irriga
Safe Reservoir Yield (MCM) θ p,j = 1.00 the comparison.The Upper Manar-Limboti reservoir is recently constructed and has started operating from October 2010.Water releases data is not available for it hence only Lower Manar-Barul reservoir is taken for the comparison.Table

tion by approximate YM (Multi-reservoir) in Manar River.
the actual release from the reservoir is maximum 98.79 MCM in the year 2000-2001 and minimum is 68.49MCM in year 2003-2004.Figure 4 shows comparison between

Table 4 . Values of actual demand, actua ses and yield model (YM with 7 able θ p,j = 0.0
T e yield m del is stud d for the tw cases, mode