Optimization of Reservoir Operation Using Combined Genetic Algorithm and Wavelet Transform Method

Dams construction are high cost projects, and therefore, optimal using of reservoir storages behind dams is an important task in water resources management. In last decades, various optimization techniques have been used for reservoir operation. Among the new methods, Genetic Algorithm (GA) and wavelet transform (WT) have been widely used in recent years. In this study, GA as well as combination of GA and four WT (Coif5, Haar, Db10 and Sym4) has been investigated in optimal operation of Reservoir storages. Using these techniques operation of the multipurpose Vanyar dam reservoir in Tabriz, Iran has been optimized. Results were compared using total monthly square deviation (TSD) of release and demand. The best results were obtained for the case of combined GA and WT sym-4 with least TSD equal to 17.453 and reliability of 84%.


INTRODUCTION
Shortage of fresh water resources through the world necessitates conservation of these resources. So dam reservoirs with a huge amount of stored water should operate optimally. To achieve this many optimization techniques such as linear programming, dynamic programming, nonlinear programming and GA have been used to optimize the operation policies of reservoirs.
GA which is based on Darwin , s principle of evolution was first proposed by Holland in 1960 and developed by him and his students and colleagues during 1960 to 1970, Goldberg and Deb. [8] . However complete introduction to GA was given by Goldberg [7] . Fahmy et.al [6] used GA to optimize the operation of a reservoir system; they compared results with the dynamic programming results, and stated that GA is potentially applicable for large river catchments. Esat and Hall [5] used GA to optimize the operation of a fourreservoir system, where, the objective was to maximize the benefits of hydropower generation and irrigation water supply subject to constraints on storage and releases from reservoirs. They concluded that the GAs have significant potential for optimization of water resources systems and clearly demonstrated the advantage of GAs over standard dynamic programming techniques in terms of computational requirements. Oliveira and Louks [11] used GAs to develop operating policies of multi-reservoir systems and concluded that GAs are practical and robust methods of effective operating policies. Wardlaw and Sharif [15] used GA to optimize operation of a deterministic finite horizon multi-reservoir system and concluded that the approach can be easily applied to nonlinear and complex systems. Ahmed and Sarma [2] used GA to determine optimal operation policy of a multi-purpose reservoir. They compared the GA results with the results of dynamic programming (SDP) method and demonstrated that GAs have better performance in comparison with SDP. Wavelet transform was presented for the first time by Grossman and Morlet [9] . Addison et.al [1] used a laboratory model (flume) to investigate turbulent flow around different bodies in different Reynolds numbers in open channels. They used the Mexican Hat Continuous wavelet transform and the Dabchiz discontinuous one and compared the results of these two wavelets. They emphasized on application of wavelet transform in hydraulics. Santos et.al. [12] , used the Morlet wavelet transform to analyze monthly rainfall data of Matsouyama city to determine the periods of low and high variations. They recommended that this wavelet transform also can be used to analyze erosion model of river catchments. Anctil and Coulibaly [3] used wavelet transform to analyze annual stream flow of south Kebek city in Canada. Tantanee et.al. [14] used a wavelet filter and presented a forecasting rainfall model. In this study, GA and the combined models of GA and four wavelet transforms (Coif5, Haar, Db10 and Sym4) have been used to optimize the operation policy of the multipurpose vanyar dam reservoir.

MATERIAL AND METHODS
Genetic Algorithm: Genetic Algorithms are stochastic search algorithms that work based on natural evolution principals and in fact, are taken from Darwin theory. Based on this theory, among a population, individuals with more fitness, accept more adaptation and survive, while, those with less fitness have less adaptation and die out. As a result, with spending time, number of individuals with suitable manner and structure increases, and population goes toward evolution and improvement. Since powerful members have more chance for surviving, affect more in producing new generation. So, more number of new generation individuals will be powerful members that usually succeed in competition. Also new generation individuals will have anoxic property of their parent from their genes. In summary, genetic algorithm may be explained as: 1. Problem is stated with genetic language, that is, variables are encoded and usually introduced by strings of 0 and 1 2. An initial population is produced in a probabilistic manner. 3. Pairs of individuals are selected from the initial population as the parents for producing the next generation. 4. The new generation is produced by mating these parents 5. New generation will substitute the previous generation using special procedures 6. Generation process will continue as far as the stopping criteria are satisfied.
This algorithm has the advantages of operating with the coding of parameters set instead of parameters themselves, searching from a population not from an individual, using objective function information and probabilistic transformation methods.
Wavelet transformation: Wavelet transformation is an analytical tool of regular signals for simulation of time information. Through wavelet transformation, a new function will be produced from fundamental functions, such that the main function is mother wavelet and the new function is denoted as daughter wavelet. Wavelet theory is similar to Fourier analysis, but instead of sine and cosine functions of Fourier series, different wavelet functions can be used [13] . Four Wavelet transforms that have been used in this study are as follows: Haar wavelet: The mathematical relation of Haar mother wavelet is: The mathematical relation of Haar scale function is: The Haar wavelet curve is presented in Fig (1-a), Chang et.al.(2005) [4] .
Symlet wavelet: The symlet wavelets do not have explicit mathematical functions. They are symmetric functions which can be shown by sym N. In this work sym-4 has been combined with G.A. for optimizing the operation of reservoirs. [10] , Fig1-b.
Coiflet wavelet: These wavelets in comparison with Daubechies wavelets are more symmetric. Also, these wavelets do not have an explicit mathematical relation. The Coif5 wavelet has been used in this study, Fig (1 Daubechies wavelet: The daubechies wavelets do not have a explicit mathematical function and usually are not symmetric. They are identified by dbN symbol, which N denotes the order of the wavelet. In this study, Daubechies wavelet with order 10 (Db10) has used, Fig  (1-d). The Vanyar dam is built on Agichay river at 5kilometers north-east of Tabriz city in Iran. The dam site is located at the elevation of 1485-m from sea-level, at 46° 23 / E-longitude and 38° 05 / N-latitude. The storage volume of this dam reservoir at normal pool level is 361.2 million cubic meters. This storage volume can supply irrigation water for 40000 hectares of land. The minimum storage volume of reservoir equals to 84 million cubic meters .The average flow rate of Agichay-river at the dam-site is 14.14m 3 /sec. In this study, the volumes of 50-years monthly stream flow of the Agichay River have been used. Monthly demands of water and average monthly inflows to the Vanyard dam reservoir are given in table-1. Optimization model:

Objective function
Minimize Where x is a m-dimensional vector, and g(x) is objective function. Then, the fitness function may be defined as: Where φ is penalty function and r° is penalty coefficient. In this study, a value of 100 was found for the penalty coefficient by trial & error. Also the sum of the squared differences between monthly release of water from reservoir and downstream demands is considered as objective function and is defined as follows: where r t represents released water from reservoir in month t and d t denotes downstream demand for water in the same month. This type of objective function has been used by Ahmed and Sarma (2005) [2] .
Decision variables: In this study 24-decision variables have been used. These variables include monthly release of water from reservoir and reservoir storage volume through a year.
Operation policy: The operation policy used in this optimization model is a linear function as: R t = a t (S t + I t ) +b t , t= 1,2,…,12 Where R t represents outflow from reservoir during month t, S t is storage volume at the beginning of month t and I t is the monthly inflow to the reservoir at month t. R t , S t and I t are in million cubic meters. Parameters a t and b t are constant coefficients of operation policy at month t. So, release of water from reservoir is a function of reservoir storage volume at the beginning of month t and inflow to the reservoir during that month.

Constraints
Storage volume constraints: Reservoir storage volume varies between maximum and minimum levels. The maximum storage volume of reservoir is at the normal pool level (S max ) and the minimum storage level is limited to the reservoir dead storage level (S min ).
S min ≤ S t ≤ S max ∀ t= 1,2,…,12 Outflow constraints: Release of water from reservoir in a particular month should be a positive value greater than downstream demand in the same month. D t ≤ R t ∀ t= 1,2,…,12 (10) D t denotes water demand during month t in million cubic meters.
Continuity Constraint: S t+1 = S t + I t -R t -E t ∀ t , t= 1,2,…,12 (11) E t is the evaporation volume from the reservoir surface during month t in million cubic meters.

RESULTS AND DISCUSSION
The Optimum results obtained by applying G.A and four combined models of wavelet transform and G.A were compared, as shown in Table 3 The computed TSD values for five models are given in table 5. The monthly percentages of satisfying demands using the combined model of GA-W-Sym4 are shown in table-6.
The optimum operation policy coefficients obtained using the combined model of GA-W-Sym4 is given in table 7.     Table 6: Monthly percentages of satisfying demands using the GA-W-Sym4 model

CONCLUSION
The G.A approach and four combined models of G.A and wavelet transforms were used to optimize the operation of the Vanyard dam reservoir. Results of applying GA showed that downstream reservoir demands can be satisfied with 67% reliability. Among the four combined models, the best results belong to the combined model of GA and W-Sym4 transform with least value of TSD equal to 17.453. So, this model is recommended for the optimization of the Vanyard reservoir operation. Based on the results of this combined model (table-5), the maximum storage volume of 360.72 millions cubic meters is obtained in June, while the reliability of satisfying demand in this month is 100%. It is concluded that, combination of GA and a suitable wavelet transform model can improve the ability of GA to optimize the operation of multipurpose reservoirs.