Comparison between genetic algorithm and linear programming
approach for real time operation
H. Md. Azamathulla
a,*
, Fu-Chun Wu
b, Aminuddin Ab Ghani
a,
Sandeep M Narulkar
c, Nor Azazi Zakaria
a, Chun Kiat Chang
aa
River Engineering and Urban Drainage Research Centre (REDAC), Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Pulau Pinang, Malaysia
b
Department of Bio-Environmental Systems Engineering, National Taiwan University, Taipei, Taiwan
c
Shri Govindram Seksaria Institute of Technology & Science, Indore-452003, M.P., India Received 25 July 2007; revised 22 September 2008; accepted 6 October 2008
Abstract
There is an increasing awareness among irrigation planners and engineers to design and operate reservoir systems for maximum efficiency to maximise their benefits. Accordingly, significant work has been done on reservoir operation for known total irrigation demand and on the optimal allocation of water available to crops at the farm level. This present paper deals with the development and comparison of two models e a Genetic Algorithm (GA) and Linear Programming (LP) e to be applied to real-time reservoir operation in an existing Chiller reservoir system in Madhya Pradesh, India. Their performance is analysed, and from the results, the GA model is found to be superior to the LP model. Ó 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Keywords: Cropping pattern; Water resource management; Irrigation management; Optimization
1. Introduction
In most developing countries, a huge share of the limited budget goes to creating facilities for irrigation. Construction of reservoirs requires very high investment and also causes socioeconomic and environmental issues. Water in the reser-voir has multiple claimants and needs to be optimally utilised to generate maximum benefits through proper operation, which must remain consistent despite uncertain future inflows and demands. According to the World Commission on Dams, many large storage projects worldwide are failing to produce the anticipated benefits (Labadie, 2004). Similarly, small storage projects made for local areas in developing countries,
like India, are also failing to meet expectations. The main cause identified at various levels of discussion, as reported by
Labadie (2004), is inadequate consideration of the more
mundane operation and maintenance issues once the project is completed. For existing reservoirs, optimum operation is critical, since all the expected benefits are based on timely water releases to meet the stipulated demand. Real-time operation of a reservoir requires making relatively quick decisions regarding releases based on short-term information. Decisions are dependant on the storage in the reservoir and information available in the form of forecast hydrologic and meteorological parameters. This is especially important during floods and power generation, where the system has to respond to changes very quickly and may need to adapt rapidly (Mohan et al., 1991). For reservoir systems operated for irri-gation scheduling, real-time operation is not very common because of longer decision steps. Traditionally, the reservoirs meant for irrigation purposes are operated on heuristics and certain rules derived from previous experiences. This defies
* Corresponding author. Tel.:þ604 599 5867; fax: þ604 594 1036. E-mail addresses: mdazmath@gmail.com, redacazamath@eng.usm.my
(H. Md. Azamathulla),fcwu@ntu.edu.tw(F.-C. Wu),redac02@eng.usm.my
(A.Ab. Ghani), snarulkar@gmail.com (S.M. Narulkar), redac01@eng.usm. my(N.A. Zakaria),redac10@eng.usm.my(C.K. Chang).
1570-6443/$ - see front matterÓ 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jher.2008.10.001
Journal of Hydro-environment Research 2 (2008) 172e181
the concept of water-management; much of the water is lost, which in turn leads to loss of revenue.
In the early 1960s, mathematical programming techniques became popular for reservoir planning and operation; pertinent literature is available. An excellent review of the topic is given
byYeh (1985), followed byLabadie (2004)andWurbs (1993).
Along with simulation studies, Linear Programming (LP), Dynamic Programming (DP) and Non-Linear Programming (NLP) are the most popular modelling techniques. A comparative study on the applicability and computational difficulties of these models is presented by Mujumdar and Narulkar (1993).
Recently, the research community has diverted their attention towards soft computing techniques, such as Evolu-tionary Algorithms (EA) and principally Genetic Algorithms (GA). Genetic Algorithms are search algorithms based on the mechanism of natural selection and natural genetics. GA modelling is gaining importance because of its robust random
search capability and near global optimal values. It originated in the mid 1970s (Holland, 1975) and emerged as a powerful optimization approach. An excellent introduction to GA is given byGoldberg (1989), and several recent researchers have summarised the essentials of genetic algorithm modelling
(Deb, 1995; Wang, 1991; Oliveira and Loucks, 1997; Wardlaw
and Sharif, 1999) (Fig. 1). The application of GA to ground water problems (Ritzel et al., 1994; Aly and Peralta, 1999; Reed et al., 2003; Hilton and Culver, 2005; Espinoza et al., 2005), and water distribution network problems (Dandy and
Engelhardt, 2001;Wu and Simpson, 2001) has been studied
extensively.
The application of GA to reservoir operation problems began in the last decade of the 21st century. East and Hall
(1994)applied GA to a four-reservoir problem. The objective
was to maximise the benefits from power generation and irrigation water supply, subject to constraints on storage and releases from the reservoirs.Fahmy et al. (1994)compared the performance of GA with that of dynamic programming and found that GA was better. Oliveira and Loucks (1997)used it to evaluate operating rules for multi-reservoir systems, demonstrating that GA could be used to identify effective operating policies. A brief review of GA applications to water resources problems can be found in the work ofWardlaw and
Sharif (1999). Sharif and Wardlaw (2000) applied a genetic
algorithm for the optimization of a multi-reservoir system in Indonesia (Brantas Basin) by considering the existing devel-opment situation in the basin and two future water resource development scenarios. They proved that GA is able to produce solutions very close to those produced by dynamic programming. Kim and Heo (2004) applied multi-objective Nomenclature
AETk
i actual evapotranspiration in periodk from crop i (mm)
APETk actually occurring potential evapotranspiration in period k (mm)
ARFk actual rainfall value in the fortnight k
Ak and BK constants relating the storage to reservoir evaporation
Ao area of spread at dead storage level
d depletion factor
EDki effective root zone depth of a cropi in period k (cm)
EDkþ1i effective root zone depth of a crop i in period kþ 1 (cm)
Eff overall efficiency
Fkcik crop evapotranspiration coefficient
ID industrial supply from the reservoir (mandatory release)
IRRk
i irrigation applied to cropi in stage k (mm) Kyk yield response factors for a crop i in period k PETk
i potential evapotranspiration in a particular geographical location (mm)
REk rate of evaporation in fortnightk RFk rainfall in period k (mm)
Sk reservoir storage at the beginning of period k Skþ1 reservoir storage at the end of period k Zf field capacity for the soil (mm/cm)
Zw permanent wilting point for the soil (mm/cm)
Zww critical available moisture limit (mm/cm) qki initial soil moisture in the time stage k in for
a cropi (mm/cm)
qkþ1i final soil moisture in a particular time stagek for a particular crop i (mm/cm)
Yai actual crop yield
Ymi maximum crop yield
GAs to optimize the multi-reservoir system of the Han River basin in South Korea. A curve was derived that identified the population points that define optimal solutions. Ahmed and
Sarma (2005) developed a GA model to derive the optimal
operating policy and compared its performance with that of Stochastic Dynamic Programming (SDP) for a multi-purpose reservoir.Nagesh Kumar et al. (2006)presented a GA model to obtain an optimum operating policy and crop water allo-cation from an irrigation reservoir.
A major characteristic of any reservoir is that it is operated in a Multi-Objective framework.Shie-Yui et al. (2004)applied a Multiple Objective GA to the Chaliyar river basin system in India to maximise the Irrigation and Hydropower. Their model generated a Pareto front, which was then translated to operator instructions through a neural network. Jothiprakash and
Ganesan (2006)developed a GA model for operating policies
for Pechiparai reservoir in India. Reddy and Nagesh Kumar
(2006) presented a Multi-Objective Evolutionary Algorithm
(MOEA) to obtain an optimal operation policy for a multi-purpose reservoir system. The Particle Swarm Optimization technique is another computing technique that has been applied to the reservoir operation problem by Reddy and
Nagesh Kumar (2007).
Many of the aforementioned techniques have been imple-mented in realistic scenarios, and many reservoir systems worldwide are operated based on the decision rules generated from these techniques. However, there exists a gap between theory and practice, and full implementation has not been achieved yet (Labadie, 2004).
The basic difficulty a reservoir manager faces is to take a real-time optimum decision regarding releases according to the future demand and inflow. This leads to the problem of optimization of the stochastic domain. Two approaches of stochastic optimization are practised: i) Explicit Stochastic Optimization (ESO), which works on probabilistic descriptors of random inputs directly and ii) Implicit Stochastic Optimi-zation (ISO), which is based on historical, generated or fore-casted values of the inputs through the use of Time Series Analysis or other Probabilistic approaches. The ESO approach has computational difficulties; ISO methods are simple, but
require an additional forecasting model for real-time
operation.
In the case of irrigation reservoirs, decision making at the reservoir level depends upon the water demand arising at the field level. In order to operate the reservoir in the best possible way, it becomes imperative to understand the processes occurring in the cropesoilewatereatmosphere system. This helps not only in the estimation of accurate demands, but also ensures optimum utilisation of water. If the processes at the field level are also modelled properly and integrated with the reservoir level model, the goal of water-management can be achieved in the best possible way. Dudley et al. (1971) pio-neered the integration of the systems in the determination of optimal irrigation timing under limited water supply using a Stochastic DP model. Dudley and his associates then improved the model (Dudley and Burt, 1973; Duldley, 1988;
Dudley and Musgrave, 1993). Vedula and Mujumdar (1992,
1993) and Vedula and Nagesh Kumar (1996) have also
contributed to this area. Their approach was to derive a steady state reservoir operation policy while maximising the annual crop yield. DP-SDP and LP-SDP were used in the modelling. However, for real-time reservoir operation,Vedula and Nagesh Kumar (1996)stressed the need to forecast inflows and rainfall in the current season to implement the steady state operation policy. As a result, the ESO model has to be supplemented with an ISO model to get a policy for the current period. As an
extension to the work of Vedula and Mujumdar (1992),
a significant contribution to the real-time reservoir approach
was presented by Mujumdar and Ramesh (1997). They
addressed the issue of short-term real-time reservoir operation by forecasting the inflow for the current period, a crop production state variable and a soil moisture state variable. Their work was based on SDP, but had all the limitations of SDP regarding the curse of dimensionality.
Against this background, a model for the derivation of real-time optimal operating policy for a reservoir under a multiple crop scenario is proposed in the present study. The primary issue is that the reservoir gets inflows during the wet season (monsoon season) and is operated for irrigation in the dry season (non-monsoon season). The reservoir storage and the soil moisture level are considered to be the principal state variables, and the irrigation depths are the decision variables. An optimal allocation model is embedded in the integrated model to evaluate the irrigation water depth supplied to different crops whenever a competition for water exists amongst various crops. The model also serves as an irrigation scheduling model because it specifies the amount of irrigation for any given fortnight. The impact on crop yield due to water deficits and the effect of soil moisture dynamics on crop water requirements are taken into account. Moreover, a root growth model is adopted to consider the effects of varying root depths on moisture transfer. The only stochastic element in the season is the evapotranspiration. The handling of stochasticity has been accomplished through dependability based forecasting in an ISO model. The rest of the variables, such as soil moisture status and the reservoir storage status, at the beginning of any period are considered to be state variables. The basic formu-lation is based on a LP model and is later transformed into a GA framework.
2. The model formulation and concept
The real-time operation model proposed in the present study integrates the reservoir level and a field level decision (Fig. 2). It considers the soil moisture status and the reservoir storage as the state variables and the applied irrigation depths as decision variables. The formulation is based on the conceptual model for soil moisture accounting and the reser-voir storage continuity relationships. A major emphasis is laid on maintaining soil moisture in a state such that the evapo-transpiration from the crops takes place at a rate that achieves better results in the form of increased yields from the crops.
To assess the timing of irrigation water application, the soil moisture status of the crop is an important parameter.
Whenever the soil moisture status approaches a critical limit, irrigation is applied. Thus, the soil moisture status is moni-tored either by physical measurement or through soil moisture models. Soil moisture models are more popular since they do not require a lot of instrumentation to be installed in the field. Soil moisture models can be formulated either by a physical approach (Fedders et al., 1978) or a conceptual approach (Rao, 1987). The conceptual approach has been used byRao et al.
(1988), Rao et al. (1990) and Hajilala et al. (1998) for the
problem of irrigation scheduling. Vedula and Mujumdar
(1992)utilised the conceptual model in their study. The same
concept is adopted in the present study.
3. The conceptual model
In the conceptual model for the CropeSoileWatere Atmosphere (CSWA) system, the basic assumption is that the
soil acts as a reservoir, the main inputs to the reservoir are rainfall irrigation, and the main outputs are evapotranspiration, percolation and drainage. The extent of the reservoir is considered to be up to the effective root zone at the particular time. The soil water reservoir is governed by a continuity equation: qkþ1i EDkþ1i qk iED k i IRR k iþ AET k i ¼ RF k ð1Þ
The conceptual model stated by Eq.(1)is used to compute the irrigation to be applied for the LP model with area as a deci-sion variable. The following parameters are important for the conceptual model.
3.1. Variation of evapotranspiration with the available soil moisture
Evapotranspiration as a function of the available soil moisture is expressed as:
AETki ¼ PETk i ifa k ai Zww ð2Þ or AETki ¼ a k ai ZwwPET k i ð3Þ whereAETk
i is the actual evapotranspiration that has occurred from crop i in fortnight k (mm), PETk
i is the potential evapotranspiration in a particular geographical location (mm),
Zww is the critical available moisture limit (mm/
cm)¼ (Zf Zw) d, Zf is the field capacity for the soil (mm/ cm),Zw is the permanent wilting point for the soil (mm/cm), d is the depletion factor and assumed to be 0.5 in the present study, and akai is the average available soil moisture over a fortnight (mm/cm). The average available soil moisture over a fortnight is given by akai¼ ak iþ akþ1i 2:0 whereaki ¼ qk i Zw if a k i< Zww otherwise ak i ¼ Zww
A similar expression can be used for akþ1i . 3.2. Root zone depth growth
The root depth data in relation to the time stages are prepared according to the Linear Root Growth Model (adopted
by Narulkar, 1995). The model assumes that maximum root
depth is achieved at the start of the yield formation stage. It remains at the maximum depth until the maturity stage. A minimum depth of 15 cm is considered in the first fortnight to account for the conditions of bare soil and an area with sparse crops.
3.3. Relative yield ratio
The yield of a crop is affected by water deficits and the rate of evapotranspiration. The rate of evapotranspiration tends to decrease depending on the available moisture content. There are many methods to model the phenomenon. However, the model used in the present study is the most commonly-adopted model. The relative yields are computed on the basis of the expression given byDoorenbos and Kassam (1979)
Yai Ymi ¼ 1 Kyk 1AET k i PETk i ð4Þ Eq. (4)gives a yield ratio for a single period only. However, the aggregate effect of moisture deficits over all fortnights of crop growth is also evaluated. The final yield ratios computed for the crop during various time periods of a season is computed by a multiplicative model (Rao et al., 1990). The determination of the yield ratio is very important since they reflect the operation policy for an irrigation system. The expression is given by Yai Ymi ¼Y ncr i¼1 1 Kyk 1AET k i PETk i ð5Þ
3.4. Water requirements of the crops
The model derived for an optimal crop pattern uses pre-determined irrigation demands. On the basis of this, the optimization model selects an appropriate area for an indi-vidual crop. The irrigation demands are determined using the conceptual model stated in Eq. (1). The irrigation require-ments may be calculated by substituting a value of critical soil moisture content instead of soil moisture in either of the fortnights k and kþ 1 and replacing the values of actual evapotranspiration by potential evapotranspiration and rear-ranging the terms of Eq. (1):
IRRki ¼ qcr EDkþ1i ED k i þ PETk i ð6Þ
where qcris the critical soil moisture content below which the actual evapotranspiration may fall below the potential rate. 4. Integrated LP formulation
In the objective function, the weighted sum of all the actual evapotranspiration values is maximised. The weights are
assigned according to the yield response factors for individual crops in individual periods. The objective is to maximise the actual evpotranspiration rate to minimise the deficits in the yields. The available soil moisture in any time period in the objective function is indirectly maximised:
MaxZ¼X ncr i¼1 Xnp k¼1 ak iþ a kþ1 i 2:0 Kyk Zww ð7Þ
subject to the following constraints: 1. Soil moisture continuity qkþ1i EDkþ1i q k iED k i IRR k iþ ak iþ akþ1i 2:0 PET Zww¼ RF k ð8Þ where qkþ1i akþ1i b kþ1 i ¼ ZW ð9Þ
with physical bounds
qkþ1i 4:0 ð10Þ akþ1i 0:9 ð11Þ 2. Reservoir continuity AkSkþ1 BkSkþX ncr i¼1 IRRk iAREA k i Eff ¼ ID AoRE k ð12Þ
Skþ1 31:1Maximum Reservoir Capacity Mm3 ð13Þ
5. Crop simulation model
The optimization model presented above yields some irri-gation depth values that are based on forecasted values for the reference evapotranspiration. This reference evapotranspira-tion, in turn, is based on a dependability model. However, the actual evapotranspiration value differs from these values, and thus, before going into the next fortnight, the soil moisture status must be updated with the applied irrigation and actual climatic factors. The formulation for crop simulation is as follows:
First compute the final soil moisture with the following relation qk i¼ qkþ1i EDkþ1i þ IRR k i Fkc k iAPET kþ ARFk=EDk i ð14Þ If qkþ1i < 3:1 EDkþ1i qkþ1i ¼ qkiEDk iþ IRRkþ1i Fkckþ1 i APETkþ1 2:0 þ Fkckþ1 i APETkþ1 2:0 ZWþ ARF kþ1 EDkþ1i Fkckþ1 i APETkþ1 2:0 ð15Þ
or qki¼ qk1 i EDk1i Fkc k iAPET 2:0 þFkc k iAPET 2:0 Zwþ ARF k þ IRRk i Fkck iAPET 2:0 =ED k i ð16Þ or qk i ¼ qk1i EDk1i Fkck iAPET 2:0 þ IRRk i þFkc k iAPET 2:0 Zw = EDk i Fkck iAPET 2:0 ð17Þ The computed soil moisture status of the crops is used in the next fortnight to compute the demand.
6. Stochastic analysis of evapotranspiration
It was previously stated that the data regarding the climatic factors is uncertain in nature and the determination of these factors beforehand is impossible. However, there is a general trend to assume the expected values for these factors and carry out the operation. The concept does not give a clear picture of the actual scenario and the appropriate weights for the indi-vidual growth stage of the crops are not assigned. The present study proposes a different method of forecasting the expected values for the climatic factors. The method of analysis starts with the computations of dependability values of reference evapotranspiration factors from the available data. The dependability of realisation of any stochastic variable is defined as the probability of equalling or exceeding that variable with a particular value.
Mathematically,
Pðx XÞ ð18Þ
where P (.) is the probability and x is the variable under consideration and X is a stipulated value of the variable. A traditional method of estimation of the dependability value is the use of standard frequency formulae (e.g. Wiebull’s formula or Hazen’s formula). In the present study, a detailed proba-bility analysis for the data is performed. The data is fitted to a standard probability distribution and the best fitting distri-bution is tested through the Kolmogorov Smirnov Test (Haan, 1977).
Once the values corresponding to different dependabilities are evaluated, dependability values for reference evapotrans-piration are assumed to be different in different growth stages. The analysis is performed on the basis of the yield response factor. A high yield response factor signifies greater sensitivity towards the deficits, and thus, a higher level of dependability is assumed for the evapotranspiration data and a lower level of dependability is assumed for the rainfall data. This will ensure a higher value of irrigation required for the crop in the sensitive period. As a result, the crop will be safeguarded against any poor moisture content conditions.
7. LP model formulation for optimal cropping spattern At the start of each dry season, depending on the storage volume in the reservoir, the crop pattern must be determined. To evaluate the crop pattern, another LP model is used. In this model, irrigation depths are calculated from Eq. (6). The formulation is as follows:
The objective function is
MaxZ¼ C1X1þ C2X2þ C3X3 ð19Þ
which is subject to the following constraints: 1. Total available area
X1þ X2þ X3 A ð20Þ
whereX1,X2, and X3are the decision variables related to the
area of individual crops;C1,C2, andC3are the cost coefficient
for each crop in Indian Rupees (1 US$¼ 45 INR); and A is the maximum area available for irrigation.
2. Area of each individual crop:
The area under each crop is required to be constrained; thus, there are lower and upper bounds on the area under each crop. The lower bounds indicate the minimum area that can be allocated to a crop, while the upper bound indicates the maximum. In the present study, the lower bounds were defined for all the crops except cash crops, while the upper bounds were defined considering the present cropping pattern. The constraints can be expressed as
Li Xi Mi ð21Þ
whereLicorresponds to the lower bound of the area for the
ith crop and Micorresponds to the upper bound on the area of
the ith crop.
8. Genetic algorithm model formulation
The genetic algorithm is a search procedure based on the mechanism of natural selection and natural genetics, which combines artificial survival of the fittest with genetic operators abstracted from nature (Holland, 1975). The structure of the genetic algorithm differs from the more traditional optimiza-tion methods in four ways:
1. A GA typically uses a coding of the decision variable set, not the decision variable itself.
2. A GA searches from a population of decision variable sets, not a single decision variable set.
3. A GA uses the objective function itself, not the derivative information.
4. A GA uses probabilistic, not deterministic, search rules (Goldberg, 1989).
The various steps involved in a genetic algorithm are well presented in Goldberg (1989). The following procedure is adopted in the present study:
For a single reservoir system, there are fortnightly releases in the optimization horizon; the chromosome representing
a solution to such a problem consists of 11 genes (sub-string) that represent the decision variables (releases). The binary bits are generated using random numbers. The fitness function is set to the annual sum of the squared deviation from the desired irrigation release and desired storage volume.
After the fitness function evaluation, the crucial mechanism of theSurvival of the Fittest is applied to the strings based on the Roulette Wheel Method (Goldberg, 1989) to form the next generation. The strings with better fitness will have a higher probability of being copied to the next generation. In this way, weak solutions are eliminated and strong solutions survive to form the next generation, which clearly depicts Darwin’s theory of the survival of the fittest. In the present study, some populations will be eliminated and some will be copied based on the Roulette Wheel Method.
Since the string is lengthy, uniform crossover was selected,
which operates on individual genes of the selected
chromosomes and each gene is considered in turn for cross-over or exchange. An important aspect of the crosscross-over application in binary coding is that the crossover should occur only at gene boundaries because each gene consists of bits, and the crossover may split the genes. Mutation is an impor-tant process that permits new genetic material to be introduced to a population. In the present study, a modified uniform mutation operator has been used. Modified uniform mutation permits the modification of a gene by a specified amount, which may be either positive or negative. After mutation, the fitness function is evaluated again to find the optimal values of the decision variables.
9. Model application
The developed models were applied to the Chiller reservoir system in Madhya Pradesh, India (Latitude 2323’N and Longitude 7618’E). In the central part of India, many reser-voir projects have been constructed for irrigation, but no irrigation is available from these reservoirs during the monsoon period (from June to September). The area receives about 90e95% of its rainfall during the Monsoon season. The rainfall then becomes runoff to the reservoirs. These reservoirs are designed to contain the runoff in the monsoon season, but there is no runoff during non-monsoon months. The present formulations are specially suited for these types of reservoirs. Non-monsoon rainfall is rare and provides little runoff. A systematic database was prepared for the various physical features of the reservoirs, including the meteorological and hydrological data such as evapotransiration, details of crops in the command area, details of net returns from individual crops
Table 1
Optimum cropping pattern for different live storage values. Live storage (M m3) Area (ha) for different crops
Wheat (ordinary) Gram Wheat (hybrid)
4.3230 e 342.910 120.00 8.2379 e 427.580 500.00 12.3246 e 1084.015 500.00 15.8632 e 1100.000 855.00 20.7581 e 1100.000 1434.00 26.0986 300.0 1100.000 1700.00 28.8610 300.0 1100.000 1700.00 30.1250 300.0 1100.000 1700.00 31.1000 300.0 1100.000 1700.00 Table 2
Sample results showing the soil moisture, available soil moisture, and storage, irrigation to be applied,for different crops for a real-time reservoir operation model (GA). Live storage in the reservoir 31.1 M m3.
Parameter Fortnight
1 2 3 4 5 6 7 8 9 10 11
Reservoir storage (M m3) 29.25 27.35 26.22 21.98 17.28 15.7 12.52 6.60 4.43 3.15 3.12
Crop Wheat (ordinary) 1) Soil moisture (mm/cm) 4.0 4.0 3.9 3.14 4.0 3.61 3.30 3.18 4.0 e e 2) Available soil moisture (mm/cm) 0.9 0.9 0.85 0.9 0.9 0.9 0.9 0.9 0.9 e e 3) Applied irrigation (mm) 58.85 89.89 92.4 37.68 162.9 0.0 0.0 9.91 102.4 e e Crop Gram 1) Soil moisture (mm/cm) 3.9 4.0 3.05 3.44 4.0 3.53 3.24 3.2 4.0 e e 2) Available soil moisture (mm/cm) 0.9 0.9 0.85 0.9 0.9 0.9 0.9 0.9 0.9 e e 3) Applied irrigation (mm) 68.76 61.69 24.3 54.82 64.52 0.0 12.5 23.0 73.2 e e
Crop Wheat (hybrid) 1) Soil moisture (mm/cm) e e e 4.0 4.0 3.28 3.34 4.0 3.38 3.19 3.46 2) Available soil moisture (mm/cm) e e e 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 3) Applied irrigation (mm) e e e 94.97 72.79 47.0 90.2 170.2 0.0 36.9 0.0
and soil properties collected from the College of Agriculture, Indore, India.
10. Results and discussion 10.1. Optimum crop pattern
A separate computer program was run before the real-time operation program to determine the optimum crop pattern for all possible storage values. The results of the optimum crop pattern are stated in Table 1. The results indicate that from a storage level of 31.10 M m3to a storage level of 26.06 M m3, the cropping pattern is same as the one that has been adopted in the project formulation. However, below a storage level of 26.06 M m3, the crop pattern changes suddenly, and wheat (ordinary) is not recommended by the model. The area of wheat (hybrid) also gets reduced when the rainfall storage is
below this level. However, the area for Gram is full, up to a storage level of 15.83 M m3. The change in cropping pattern indicates that efficient water usage is maintained.
10.2. Results from real-time operation models (GA & LP) The real-time operation model gives an optimal operating policy for the available storage in the present fortnight considering the future. The model also yields the values of irrigation to be applied to individual crops in the fields. In the wake of deficient water supplies, the model distributes the available water over the time for different crops optimally. Two sample results of the present model are stated inTables 2 and 3. The available moisture to the crops is not affected, and generally the soil remains at the upper limit of the available soil moisture. This is because the crop pattern is predicted according to the availability of the storage in the reservoir. The
Table 3
Sample results showing the soil moisture, available soil moisture, storage, and irrigation to be applied for different crops for a real-time reservoir operation model (LP). Live storage in the reservoir 31.1 M m3.
Parameter Fortnight
1 2 3 4 5 6 7 8 9 10 11
Reservoir storage (M m3) 29.28 28.17 26.30 22.22 19.68 14.64 10.87 5.62 4.24 3.63 3.60 Crop Wheat (ordinary)
1) Soil moisture (mm/cm) 3.76 3.89 3.84 3.07 3.54 3.30 3.22 3.17 4.0 e e 2) Available soil moisture (mm/cm) 0.9 0.9 0.9 0.87 0.9 0.9 0.9 0.9 0.9 e e 3) Applied irrigation (mm) 53.62 90.63 92.87 36.04 163.9 8.44 23.02 19.94 102.6 e e Crop Gram 1) Soil moisture (mm/cm) 3.90 3.07 3.28 3.15 3.4 3.28 3.66 3.23 3.47 e e 2) Available soil moisture (mm/cm) 0.9 0.87 0.9 0.9 0.9 0.9 0.9 0.9 0.9 e e 3) Applied irrigation (mm) 68.76 22.27 60.67 41.59 26.96 37.64 53.15 0.00 33.17 e e
Crop Wheat (hybrid) 1) Soil moisture (mm/cm) e e e 4.00 3.06 3.48 3.32 3.28 3.38 3.18 3.19 2) Available soil moisture (mm/cm) e e e 0.9 0.86 0.9 0.9 0.9 0.9 0.9 0.9 3) Applied irrigation (mm) e e e 94.21 37.19 127.9 78.89 162.9 0.00 36.09 0.0 Table 4
Relative yield ratio for different live storage values computed with a real-time reservoir operation model (GA).
Live storage (M m3) Relative yield ratio for different crops GP Relative yield ratio for different crops LP
Wheat (ordinary) Gram Wheat (hybrid) Wheat (ordinary) Gram Wheat (hybrid)
4.3230 e 0.982 0.975 e 0.9677 1.000 8.2362 e 1.000 1.000 e 0.9083 1.000 12.3246 e 0.896 1.000 e 0.9576 1.000 15.8632 e 0.898 0.995 e 0.989 1.000 20.7581 e 0.951 0.944 e 0.987 0.911 26.0986 1.000 1.000 0.960 1.000 0.987 0.952 28.8610 1.000 1.000 1.000 1.000 0.987 1.000 30.1250 1.000 1.000 1.000 1.000 1.000 1.000 31.1000 1.000 1.000 1.000 1.000 1.000 1.000
results are indicative of successful application of the real-time operation strategy proposed in the present work.
10.3. Relative yield ratios
Relative yield ratios computed for different crops at different live storage values are shown inTable 4. The relative yield ratios for all the crops become one if live storage in the reservoir is equal to or greater than 28.89 M m3. The GA model is found to be better for application in real world operation of the reservoir.
11. Conclusion
A real-time model using an integrated Linear Programming & Genetic Algorithm Model for a reservoir system meant for irrigation has been developed in the present study to obtain an optimal reservoir operating policy that incorporates field level decisions, while also deciding the appropriate time and amount of water to release from the reservoir.
From the analysis, the following conclusions can be drawn: The developed models can be successfully applied to irri-gation supporting reservoir systems. Furthermore, the models ensure an optimum reservoir release over different time periods. In addition, they also ensure optimum allocation of the available water over the different crops in the fields. While allocating the water to different crops in the fields, the model takes into account the critical growth stages of the crops and allocates sufficient water to each crop to safeguard it against any ill effects of water deficits. The optimum crop pattern model used in the study will only allow productive irrigation, so the amount of wasted water is reduced. The GA model gives better yields when compared to the LP model. Moreover, GAs are well suited to the solution of irrigation scheduling problems. They are robust and very efficient and can easily be run with a range of objective functions. The developed GA could be applied to more complex problems with little difficulty.
Acknowledgements
The authors would like to express sincere thanks to National Science council (NSC), Taiwan for the financial support of this work. The first author is also thankful to Universiti Sains Malaysia for providing an Incentive Research Grant.
References
Ahmed, J.A., Sarma, A.K., 2005. Genetic algorithm for optimal operating policy of a multipurpose reservoir. Water Resour. Manag. 19 (2), 145e161. Aly, A.H., Peralta, R.C., 1999. Comparison of a genetic algorithm and mathematical programming to the design of groundwater clean up system. Water Resour. Res. 35 (8), 2415e2426.
Dandy, G.C., Engelhardt, 2001. The optimal scheduling of water main replacement using genetic algorithm. J. Water Resour. Plann. Manag. ASCE 127 (4), 214e223.
Deb, Kalyanmoy, 1995. Optimization for Engineering Design. Prentice Hall of India.
Doorenbos, J., Kassam, A.H., 1979. Yield Response to Water. In: Irrigation and Drainage Paper, vol. 33. FAO, Rome.
Dudley, N.J., Howell, D.T., Musgrave, W.F., 1971. Optimal intraseasonal irrigation water allocation. Water Resour. Res. 7 (4), 770e788.
Dudley, N.J., Burt, O.R., 1973. Stochastic reservoir management and system design for irrigation. Water Resour. Res. 9 (3), 507e522.
Duldley, N.J., 1988. A single decision-maker approach to irrigation reservoir and farm management decision making. Water Resour. Res. 24 (5), 633e640. Dudley, N.J., Musgrave, W.F., 1993. Economics of water allocation under
certain conditions. In: Biswas, A.K. (Ed.), Water for Sustainable Devel-opment in the Twenty-First Century. Oxford University Press, Delhi. East, V., Hall, M.J., 1994. Water resources system optimization using
genetic algorithms. In: Proceedings of the first International Confer-ence on Hydroinformatics, Baikema, Rotteram, The Netherlands. pp. 225e231.
Espinoza, F.P., Minska, B.S., Goldberg, D.E., 2005. Adaptive hybrid genetic algorithm for ground water remediation design. J. Water Resour. Plann. Manag. ASCE 131 (1), 14e24.
Fahmy, H.S., King, J.P., Wentzel, M.W., Seton, J.A., 1994. Economic Opti-mization of River Management Using Genetic Algorithms. Am. Soc. of Agricultural Engrs., St. Joseph, Mich. Paper no.943034, ASCE 1994 Int. Summer Meeting.
Fedders, R.A., Kowalic, P.S., Zarandy, H., 1978. Simulation of Field Water Use and Crop Yield. Centre for Agricultural Publishing and Documenta-tion, Wganingen.
Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization AND Machine Learning. Addison eWesley, Reading, Mass.
Hilton, C.A.B., Culver, T.B., 2005. Ground water remediation design under uncertainty using genetic algorithm. J. Water Resour. Plann. Manag. ASCE 131 (1), 25e34.
Haan, C.T., 1977. Statistics Methods in Hydrology. Iowa State Press, Iowa. Hajilala, M.S., Rao, N.H., Sarma, P.B.S., 1998. Real time operation of
reservoir based canal irrigation systems. Agric. Water Manag. 38, 103e122.
Holland, J.H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press, Cambridge Mass.
Jothiprakash, V., Ganesan, S., 2006. Stochastic dynamic programming model for optimal policies of a multi-purpose reservoir. J. Water Resour. Manag. 20, 917e929.
Kim, T., Heo, J.H., 2004. Multi-reservoir system optimization using multi-objective genetic algorithms. In: Proceedings of the World Water and Environmental Resources Congress, 27th Jun. to 1st Jul. WERI, ASCE, Salt Lake City UT, pp. 1e10.
Labadie, J.W., 2004. Optimal operation of multi reservoir systems: state-of-the-art review. J. Water Resour. Plann. Manag. 130 (2), 93e111. Mohan, S., Raman, S., Premganesh, G., 1991. Real-time reservoir operation.
IWRS 11 (1), 35e37.
Mujumdar, P.P., Narulkar., Sandeep, 1993. Optimisation Models for Multi-reservoir Planning and Operation, Hydrology Review, vol. VIII. Indian National Committee on Hydrology, Roorkee, India, No. 1, pp. 29e52. Mujumdar, P.P., Ramesh, T.S.V., 1997. Real time reservoir operation for
irri-gation. J. Water Resour. Res. vol. 33 (No. 5), 1157e1164.
Nagesh Kumar, D., Srinivasa Raju, K., Ashok, B., 2006. Optimal reservoir operation for irrigation of multiple crops using genetic algorithms. J Irrig. Drain. Eng., ASCE 132 (2), 123e129.
Narulkar, S.M., 1995. Optimum real-time operation of multi reservoir systems for Irrigation scheduling. Ph.D thesis submitted at I.I.T., Bombay, India Oliveira, R., Loucks, D.P., 1997. Operating rules for multi reservoir system.
Water Resour. Res. 33 (4), 839e852.
Rao, N.H., 1987. Field test for a simple soil-water balance model for irrigated areas. J. Hydrol. 91, 179e186.
Rao, N.H., Sarma, P.B.S., Chander, S., 1988. Irrigation scheduling under a limited water supply. Agri. Water Manag. 15, 165e175.
Rao, N.H., Sarama, P.B.S., Chander, S., 1990. optimal multicrop allocations of seasonal and interseasonal irrigation water. Water Resour. Res. 26 (4), 551e559. Reddy, J.M., Nagesh Kumar, D., 2006. Optimal reservoir operation using multi-objective evolutionary algorithm. Water Resour. Manag. 20 (6), 861e878. Springer.
Reddy, J.M., Nagesh Kumar, D., 2007. Optimal reservoir operation for irri-gation of multiple crops using elitist-mutated particle swarm optimization. Hydrol. Sci. J. 52 (4), 686e701. IAHS Press, UK.
Reed, P., Minsker, B.S., Goldberg, D.E., 2003. Simplifying multiobjective optimization: an automated design methodology for the nondominated sorted genetic algorithm-II. Water Resour. Res. 39 (7), 1196, doi: 10.1029/2002 WR001483.
Ritzel, B.J., Eheart, J.W., Ranjithan, S., 1994. Using genetic algorithms to solve multiple objective ground water pollution containment. Water Resour. Res. 30 (5), 1589e1604.
Sharif, M., Wardlaw, R., 2000. Multireservoir system optimization using genetic algorithms case study. J. Comp. Civ. Eng. ASCE 14 (4), 255e263. Shie-Yui, L., Al-Fayyaz, T.A., Sai, L.K., 2004. Application of evolutionary algorithms in reservoir operations. J Inst. Eng., Singapore 44 (1), 39e54. Vedula, S., Mujumdar, P.P., 1992. Optimal reservoir operation for irrigation of
multiple crops. Water Resour. Res. 28 (1), 1e9.
Vedula, S., Mujumdar, P.P., 1993. Modelling for Reservoir Operation for Irrigation. Proceedings of International Conference on Environmentally Sound Water Resources Utilisation, Bangkok.
Vedula, S., Nagesh Kumar, D., 1996. An integrated model for optimal reser-voir operation for irrigation of multiple crops. Water Resour. Res. 32 (4), 1101e1108. American Geophysical Union.
Wang, Q.J., 1991. The genetic algorithm and its application to calibrating conceptual rainfall-runoff models. Water Resour. Res. 27 (9), 2467e2471. Wardlaw, R., Sharif, M., 1999. Evaluation of genetic algorithms for optimal reser-voir system operation. J. Water Resour. Plann. Manag. ASCE 125 (1), 25e33. Wu, Z.Y., Simpson, A.R., 2001. Competent genetic evolutionary optimization
of water distribution systems. J. Comput. Civ. Eng. 15 (2), 89e101. Wurbs, R.A., 1993. Reservoir system simulation and optimization models. J.
Water Resour. Manag. ASCE 119 (4), 455e472.
Yeh, W.W.G., 1985. Reservoir management and operation models: a state of the art review. Water Resour. Res. 21 (1), 1797e1818.
