ABSTRACT
The review of some applications of evolutionary algorithms to reservoir operations is presented in this study. Reservoir operation is a complex multi-objective optimization problem with many conflicting objectives and constraints. Evolutionary algorithms are stochastic search algorithms which have a lot of applications in water resources management. They have the ability to generate non-dominated solutions which converge to Pareto optimal front to multi-objective problems in one simulation run. The results presented in the literatures reviewed show that evolutionary algorithms are good algorithms for solving complex, non-linear, convex and multidimensional reservoir problems. They produce tradeoffs to reservoir operation problems from which a reservoir operator can choose a solution applicable to him.
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DOI: 10.3923/ajsr.2011.16.27
URL: https://scialert.net/abstract/?doi=ajsr.2011.16.27
INTRODUCTION
Multi-objective optimization techniques are widely used in water resources management. They are especially good optimizers with few control parameters. They are easy to apply once the objective functions and constraints are formulated. Water resources management problems are mostly formulated as multi-objective with many conflicting objectives and constraints. Certainly, many Multi-Objective Evolutionary Algorithms (MOEAs) are developed to solve these water resources multi-objective problems. A number of improvements are made daily in the world of evolutionary research to address the shortcomings of previous algorithms. Nowadays, we have several MOEAs producing Pareto optimal solutions in few minutes. The results produced by these algorithms are useful for decision makers in ensuring equitable distribution of resources among competing users.
Evolutionary Algorithms (EAs) as robust optimization techniques have the ability to find multiple Pareto optimal solutions in one single simulation run because of their population-approach. They are general purpose stochastic search methods simulating natural selection and biological evolution (Salman et al., 2007). They maintain a population of potential (or candidate) solutions to a problem. They are biologically-inspired optimization algorithms, imitating the process of natural evolution and are becoming important optimization tool for several real world applications. They use a set of solutions called population to converge to the optimal solution. They are less susceptible to problem dependent characteristics, such as the shape of the Pareto front (convex, concave or even discontinuous) and the mathematical properties of the search space, whereas these issues are of concerns for mathematical programming techniques for mathematical tractability (Rakesh and Babu, 2005). Many applications of evolutionary algorithm in water resources are widely available in the literatures. The application of Differential Evolution (DE) algorithm to a constrained optimization problem of irrigation water use is presented by Adeyemo and Otieno (2009d). Irrigation water, amounting to 10 Mm3 is optimized to irrigate 2500 ha of land where 16 different crops are planted on different areas of land. Also a comparative study using linear programming (LP) is performed. The results are compared to study the effectiveness of the DE algorithm. The different areas of land where the crops are to be planted to maximize the Total Income (TI) in monetary terms (South African Rand, ZAR) are optimized. Ten strategies of DE are tested with this problem varying the population size (NP), crossover constant (CR) and weighting factor (F). It is found that strategy 1, DE/rand-1-bin, with values of NP, CR and F of 160, 0.95 and 0.5, respectively, obtains the best solution most efficiently. DE is considered comparable to LP in solving this optimization problem. Also, DE techniques are applied to determine the optimum cropping pattern in the Lower Orange catchments of South Africa (Otieno and Adeyemo, 2009, 2010). Three different cropping patterns are studied to determine the best cropping pattern for the study area in terms of maximum total income generated from farming, irrigation water use and total man-days required on the farm. The objective of the study is to explore the ten different strategies of DE to determine the optimum cropping pattern in each of the farmlands studied. Each farmland is studied to determine the best DE strategy for this model. From the analysis of the results, DE strategies with binomial crossover method outperform DE strategies with exponential crossover method. It is suggested that DE/best/1/bin is the best strategy for this model taking the minimum number of iterations before convergence and the maximum total net benefit generated as criteria. It can be concluded that DE can be used for determining optimum cropping pattern in a water scarce environment with necessary modifications to this model. A study by Otieno and Adeyemo (2009a) presents a mathematical model of cropping patterns solved using novel Differential Evolution (DE) algorithm that will derive the maximum total net benefit (TNB) from farming when constrained by irrigation water. Three cropping patterns are studied to find the optimum cropping pattern for each of the farm units that will derive the maximum TNB.
Multiobjective formulation for a problem is primarily due to the fact that it may not be possible to have single solution which optimizes all objectives. Therefore an algorithm that gives a large number of alternative solutions lying on or near the Pareto-optimal front is of great practical value. Evolutionary Algorithms (EAs) are different from conventional algorithms for non-linear optimization since they use only objective function information instead of derivatives or other auxiliary information of the problems (Fan et al., 2006). In addition, they aim at finding the optima from a population of points in parallel rather than from a single point. These features make them attractive for addressing complex engineering problems. Over the past years, a number of multi-objective evolutionary algorithms have been developed. Recently, multiobjective differential evolution algorithm named MDEA was suggested by Adeyemo and Otieno (2009b). The new algorithm adjusts the selection scheme of traditional DE to solve multi-objective problems. The algorithm also modifies the domination criteria for the population. The offspring generated in subsequent generations are improved before domination check is performed on the population in the final generation. Moreover, trial solution replaces the target solution if it is better or equal in all the objectives. The proposed algorithm has been successfully applied to five common test problems and an engineering cantilever design problem. Good spread of quality Pareto optimal solutions are achieved. The algorithm produces more Pareto optimal solutions than the previous algorithms and retains the fast convergence and diversity exhibited by DE in global optimization. The algorithm is a good choice for solving many practical engineering problems with ease.
The procedures of EAs are initialization, mutation, crossover and selection. Populations of individuals which are potential solutions are first randomly generated. Each solution is assessed by using fitness function. A selection process is applied in each iteration to form a new population which will be better than the previous population. The selection is biased towards the solution that has better fitness function. In each iteration, the solutions undergo mutation and crossover to mimic the natural evolution technique. The iteration continues until convergence is reached.
Evolutionary Algorithms (EAs) are global optimization heuristics that search for optima using a process that is analogous to Darwinian natural selection. Since their inception in the 1960s, evolutionary algorithms have been used in a tremendous array of applications. The growing popularity of evolutionary algorithms stems from their ease of implementation and robust performance for difficult engineering and science problems.
A criticism of Evolutionary Algorithms (EAs) might be the lack of efficient and robust generic methods to handle constraints. The most widespread approach for constrained search problems is to use penalty methods. EAs have received increased interest during the last decade due to the ease of handling multiple objectives. A constrained optimization problem or an unconstrained multi-objective problem may, in principle, be two different ways to pose the same underlying problem.
The most popular Evolutionary Algorithm (EA) is genetic algorithm. Although many genetic algorithm versions have been developed, they are still time consuming. In order to overcome this disadvantage, the evolution strategy called DE has been recently proposed by Price and Storn (1997). It has been applied to several engineering problems in different areas.
WATER RESOURCES MANAGEMENT USING EVOLUTIONARY ALGORITHM
Water resources management is a multi-objective optimization problem. It is a difficult task to estimate reservoir operating policies that maximize all the benefits provided by these reservoirs and also minimize their adverse impacts. It is a complex decision making process which will involve a number of variables, risks, uncertainties and also conflicting objectives. Reservoirs serve many purposes. They are used to drive turbines to generate electricity. They are used to supply water for irrigation, city and industrial uses and also for flood protection. Reservoirs may be built to satisfy a single purpose or multipurpose. Some of the purposes are conflicting in nature. For example for power generation, the reservoir should be as full as possible to increase the head, whereas for flood protection, it should be empty to provide for maximum storage of flood waters if flood occurs.
Many optimization techniques have been applied to water resources management in the past. These include Linear Programming (LP); Nonlinear Programming (NLP); Dynamic Programming (DP); Stochastic Dynamic Programming (SDP); and Heuristic Programming such as Genetic Algorithms, Shuffled Complex Evolution, Fuzzy logic and Neural Networks, Differential Evolution etcetera. Sarker and Ray (2009) analyse multi-objective optimization problems and provide useful insights about solutions that are generated using population-based approached. Crop-planning problem as a multi-objective optimization model is formulated. Well-known multi-objective evolutionary algorithm called NSGAII and their proposed Multi-Objective Constrained Slgorithm (MCA) are compared. The study by Saravanan (2008) unravels the complexity of water management institutions by analysing the interactive nature of actors and rules to a particular water-related problem, using a systems approach in a hamlet in the Indian Himalayas. Azamathulla et al. (2008) present a study to deal with the development and comparison of two models; a Genetic Algorithm (GA) and Linear Programming (LP) to be applied to real-time reservoir operation in an existing Chiller reservoir system in India. Their performance is analysed and from the results, the GA model is found to be superior to the LP model. Optimal water allocation and cropping patterns for the Jordan Valley, taking into consideration variations in expected incomes from agricultural production and rising water prices are studied by Doppler et al. (2002). Their calculations were based on information available on water supplies, areas under irrigation and market conditions and used linear programming models for determining solutions that maximize gross margins and minimize potential variations in these gross margins. The results indicated that optimizing cropping patterns and the allocation of irrigation water still has a substantial potential to increase the financial return from agriculture.
Adeyemo and Otieno (2009e) presents a multi-objective differential evolution algorithm (MDEA) technique that is adapted to crop planning in a farmland in the Vaalharts irrigation scheme (VIS) in South Africa. The objectives of the model are to maximize the total net profit (NF) in monetary terms (South African Rand, ZAR) generated on the farm by planting four different crops, maximize total planting area (m2) and minimize the irrigation water use (m3). From the analysis of the results, it is found that the proposed MDEA can be used for solving crop planning problem and generate non-dominated solutions from where a farmer can select a solution that suits his particular situation. Further in their study, they present four strategies Multi-Objective Differential Evolution Algorithm (MDEA) (Adeyemo and Otieno, 2010). The four strategies namely, MDEA1, MDEA2, MDEA3 and MDEA4 are adapted to solve the multi-objective crop planning model with multiple constraints in a farmland in South Africa. They conclude that MDEA1 and MDEA2 strategies with binomial crossover method are better for solving the crop planning problem presented than MDEA3 and MDEA4 strategies with exponential crossover method. Also, MDEA is proposed for irrigation planning by Adeyemo and Otieno (2009c). The model in their study is adapted to Vanderkloof dam along the Orange River to determine the optimal monthly release for irrigation release requirements. It is concluded that the proposed algorithm can also be adapted to many multi-objective optimization of water resources systems in South Africa and similar area to generate optimal trade-offs. Other applications of differential evolution in water resources management are available in the literatures (Adeyemo and Otieno, 2009a; Adeyemo et al., 2008; Janga and Nagesh, 2006, 2007).
In another study, a tolerance based Fuzzy Goal Programming (FGP) and a FGP based Genetic Algorithm (GA) model for nutrient management decision-making for rice crop planning in India are presented. In the proposed model, fuzzy goals such as fertilizer cost and rice yield are included in the decision making process (Sharma and Jana, 2009). Montalvo et al. (2008) applied one of the variants of Particle Swarm Optimization (PSO) as one of these evolutionary algorithms to two case studies: The Hanoi water distribution network and the New York City water supply tunnel system. Both cases occur frequently in the related literature and provide two standard networks for benchmarking studies. This allows them to present a detailed comparison of their new results with those previously obtained by other researchers.
In the field of water resources engineering, particularly reservoir operations, Genetic Algorithm (GA) has been proved to be computationally superior to traditional methods like linear programming, non linear programming and dynamic programming. Two types of genetic algorithms, real-coded and binary-coded were applied to the optimization of a flood control reservoir model (Chang et al., 2005). Wardlaw and Sharif (1999) explored the potential of alternative GA formulations in application to real time reservoir operation. They found that: (a) GA has the potentiality to large-finite horizon multireservoir system problems where objective function is complex; (b) GA needs no initial trial release policy; (c) easily applicable to nonlinear problems and (d) GA can generate several solutions that are close to the optimum. Several other studies have shown the application of GA to water resources management (Akter and Simonovic, 2004; Chang et al., 2005; Kumar et al., 2006; Raju and Nagesh, 2004; Sharma and Jana, 2009; Tospornsampan et al., 2005; Wardlaw and Sharif, 1999).
RESERVOIR OPERATION MODELS
Complex constrained multiobjective optimization problems are solved by evolutionary algorithms. This ability makes EA a good choice for solving reservoir operation problems which present multiple conflicting objectives and constraints. The objective functions are always non-linear, con-convex and multi-modal functions. It is characterized with many decision variables, multiple conflicting objectives as well as considerable risk and uncertainty.
Reservoir can be operated to maximize hydropower generation while satisfying other uses like irrigation, downstream water requirement and flood control. The objective function can be formulated for 12 months period as follows:
(1) |
The constraints are as follows:
(2) |
(3) |
(4) |
(5) |
where, E is the hydropower energy, Ht is the height of water above the turbine at the end of month t which has non-linear correlation with the volume of water in the reservoir. It is also dependent on the shape of the reservoir and the position of the turbine, η is the efficiency of the turbines, Rtt is the volume of water released to the turbine in month t, St is the reservoir storage at the end of month t, S0 is the starting end of period storage, It is the inflow to the reservoir in month t, Spt is the water spill at month t.
When a reservoir is a multipurpose reservoir, the multiple objectives set in. In this case, the reservoir may be required to maximize hydropower generation, maximize the irrigation water releases and minimize the flood risk. The constraints may be to satisfy the minimum downstream water requirement, the minimum height required for hydropower generation and the height should be less than the maximum height to prevent spilling.
Multiobjective Differential Evolution (MODE) algorithm was presented by Reddy and Kumar (2007) with an application to a case study in reservoir system optimization. The algorithm was applied to a multiobjective reservoir operation problem and provides many alternative Pareto optimal solutions with uniform coverage and convergence to true Pareto optimal fronts. The multiple objectives of the reservoir system are minimizing flood risk, maximizing hydropower production and minimizing irrigation deficits in a year. The model is subject to physical and technical constraints.
The model was formulated for ten daily operations with the objectives of maximizing hydropower production (f1) and minimizing the annual sum of square deficits of irrigation release from demands (f2) which are presented in Eq. 6-12.
Objective functions:
(6) |
(7) |
Where:
(8) |
Subject to the following constraints:
(9) |
(10) |
(11) |
(12) |
Where:
Pi,t | = | Hydropower energy produced (x106 kWh) in the ith power house (i = 1, 2) during period t (t = 1, 2,..., 36) |
NT | = | Total number of time periods |
Ki | = | Power coefficient |
RPi,t | = | Amount of water released to ith turbine during period t |
Hi,t | = | Average head available during period t and is expressed as a nonlinear function of the average storage during that period |
IRt | = | Irrigation release in period t |
IDt | = | Maximum irrigation demand in period t |
Rpt,1min | = | Minimum release to be made to meet hydropower requirements |
St | = | Initial storage volume during time period t |
It | = | Inflow into the reservoir |
EVPi | = | Evaporation losses (is a nonlinear function of the average storage) |
OVFt | = | Overflow from the reservoir |
stmin and stmax | = | Minimum and maximum storages allowed in time period t, respectively |
IRtmin and IRtmax | = | Minimum and maximum irrigation releases respectively in time period t |
TCi | = | Turbine capacity of power plant i (i = 1, 2) |
Fig. 1: | Non dominated solutions obtained using MODE and NSGA-II. BCS is the best compromised solution obtained for MODE solution set |
In addition to the previous constraints, it is to be ensured that end storage of the last period of the year is greater than or equal to the initial storage of the first period of the next year. This reservoir operation model contains a total of 108 decision variables. Figure 1 shows the non-dominated solutions obtained from the model results using multi-objective differential evolution (MODE) and non-dominated sorting genetic algorithm version II (NSGA-II).
In real time operation of reservoir for maximum irrigation release, evolutionary algorithms are also used. The model considers the reservoir level and a filed level decision (Azamathulla et al., 2008). Soil moisture status, the reservoir storage as the state variables and the applied irrigation depth as decision variables are considered. The soil moisture should be maintained to achieve maximum yields of crops. Conceptual model (Rao et al., 1988) or physical approach (Feddes et al., 1978) may be used for soil moisture models. In conceptual models, the soil acts as a reservoir, with inputs as rainfall and irrigation with outputs as evapotranspiration, percolation and drainage. The continuity equation for soil water reservoir is given in Eq. 13.
(13) |
The objective function of the model is to maximize the actual evapotranspiration rate to minimize the deficits in the yields.
(14) |
Subject to the following constraints:
• | Soil moisture continuity |
• | Reservoir continuity |
The full details of this are given in Md. Azamathulla et al. (2008).
Fig. 2: | Monthly reservoir storage volume for different probability of inflow |
Genetic algorithm was used to optimize the operation of the multipurpose Jiroft dam reservoir in Hashemi et al. (2008). The probability of inflow for a period of 12 months was considered. In their model, the difference between demand and release while satisfying the continuity was minimized. Their objective function is defined as:
(15) |
Where:
Rt | = | Release of water from reservoir in month i |
Dt | = | Downstream demand for water in month i |
St | = | Storage volume of water at the beginning of each month t |
Et | = | Evaporation in month t |
It | = | Inflow to the reservoir in month t |
Generally, many methods are used to convert constrained problem to unconstrained. The fitness function and constraints are converted to an unconstrained function using penalty function method in their study. In their results, the generated different curves for different inflow probabilities are given in Fig. 2.
SOME APPLICATIONS OF EVOLUTIONARY ALGORITHMS TO RESERVOIR OPERATION
Evolutionary algorithms are very useful in solving multiobjective optimization problem of reservoir operation. Many algorithms are tested using different models to solve non linear, convex and multidimensional reservoir problems. Chang and Chang (2009) applies a multi-objective evolutionary algorithm, the non-dominated sorting genetic algorithm (NSGA-II) to examine the operations of a multi-reservoir system in Taiwan. A daily operational simulation model is developed to guide the releases of the reservoir system and then to calculate the Shortages Indices (SI) of both reservoirs over a long-term simulation period. The objective is to minimize the SI values using NSGA-II through identification of optimal joint operating strategies. There is a better operational strategy that would reduce shortage indices for both reservoirs using a 49-year data set. In another study, Genetic Algorithm (GA) application was demonstrated by Chang (2008). Real-time flood control of a multi-purpose reservoir was proposed. The optimization model with linguistic description of requirements and existing regulations for rational operating decisions was studied. The approach involves formulating reservoir flood operation as an optimization problem using GA as a search engine. GA was used to search a global optimum of a mixture of mathematical and non-mathematical formulations. When some constraints were violated because of a great number of constraints and flood control requirements, a proper penalty strategy for each parameter was proposed to guide the GA searching process.
A new multiobjective optimization algorithm called Macro-evolutionary multi-objective genetic algorithm (MMGA) was developed and applied to reservoir operation by Chen et al. (2007). The algorithm is relatively easy to implement and yielded a better spread of solutions than NSGA-II. MMGA was applied to rule-curve optimization for a multipurpose reservoir system. The problem is non linear with mixed integer variables and a complex non-convex Pareto frontier. The operators can make decisions regarding releases for water supply and hydropower generation from the operating rule curves defined long-term target storage levels and target releases. From their results, MMGA found an acceptable spread of solutions on the Pareto front with a relatively low diversity metric. Another algorithm based on GA is called Self-Learning Genetic Algorithm (SLGA). It is an improved version of the SOM-Based multiobjective genetic algorithm (SBMOGA) presented by Hakimi-Asiabar et al. (2009a, b). SLGA was used to derive optimal operating policies for a three-objective multi-reservoir system Hakimi-Asiabar et al. (2009a, b). The objectives are supplying water demand, generating hydropower energy and controlling water quality in downstream river. The applicability and efficiency of the proposed methodology was evaluated by developing the optimal operating policies for the Karron-Dez multi-reservoir system in Iran. The results show that SLGA can outperform the existing multi-objective optimization models. It has capability of solving large scale multireservoir, multipurpose reservoir operation optimization problems.
In a similar study, Malekmohammadi et al. (2009) presents a Bayesian Network (BN) for developing monthly operating rules for a cascade system of reservoir for irrigation and flood control. The inputs of the BN are monthly flows, reservoir storages at the beginning of the month and downstream water demands. The long-term optimization model in monthly scale is formulated to minimize the expected flood and agricultural water deficit damages. A short term optimization model which provides the optimal hourly releases during floods is utilized and linked to a flood damage estimation model. Also Kerachian and Karamouz (2007) studied a model which includes a GA-based optimization model linked with a reservoir water quality simulation model. The model is based on the objective function of the optimization model on the Nash bargaining theory to maximize the reliability of supplying the downstream demands with acceptable quality, maintaining a high reservoir storage level and preventing quality degradation of the reservoir. The models are applied to the Satarkhan reservoir in Iran and the results of the optimization model are used to generate operating rules for water release considering water quality and quantity objectives. The operating rules are developed for the two reservoir operation optimization model using water quality and quantity objectives which are generated. A study by Chaves and Kojiri (2007) shows the application of Stochastic Fuzzy Neutral Network (SFNN) trained by a GA based model for deriving reservoir operational strategies considering water quality and quantity objectives. It yielded a quasi optimal solution. The SFNN was successfully applied to the optimization of the monthly operational strategies considering maximum water utilization and improvement on water quality.
Artificial Neural Networks (ANN) was used to solve a novel intelligent reservoir operation system presented by Chaves and Chang (2008). The developed intelligent system was applied to the operation of the Shihmen Reservoir in North Taiwan to investigate its applicability and practicability. The reservoir has five decision variables. The results show that ANN improved the operation performance of the reservoir when compared to its current operational strategy. Elferchichi et al. (2009) presents a methodology that analyzes the adequacy of the difference between supply and demand. The operation of reservoirs in an on-demand irrigation system using a stochastic methodology based on real-coded GA. The model determines adequate inflow hydrographs to ensure the optimal regulation of reservoirs during the peak demand period. The model was applied and tested on the Sinista Ofanto irrigation scheme (Foggia, Italy).
Another optimization technique called honey-bee mating was presented by Afshar et al. (2007) in handling the single reservoir operation optimization problems. The honey-bee mating process has been considered as a typical swarm-based approach to optimization in which the search algorithm is inspired by the process of real honey-bee mating. Their results were comparable to the results of the well-developed traditional linear programming (LP) solvers such as LINGO 8.0.
CONCLUSIONS
Reservoir operation models are presented in single and multi-objectives. Multi-objective evolutionary algorithms are shown to be capable of solving these models. Several algorithms based on evolutionary algorithms are presented with applications to reservoir operation models. The reservoir operations are always formulated with nonlinear, convex and multidimensional objective functions and lots of constraints. Several studies reviewed generate the Pareto optimal fronts to the models. The evolutionary algorithms presented outperform other algorithms like linear programming when they are compared. The non-dominated solutions are lying near or on the Pareto front and diverse on the Pareto front thereby fulfilling the two goals of a good multi-objective evolutionary algorithm which are; discover solutions as close to the Pareto-optimal solutions as possible and find solutions as diverse as possible in the obtained non-dominated front (Deb, 2001).
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