رویکرد فازی الگوریتم ژنتیک برای بهینه سازی مخازن موج

One objective of a pipeline system is to provide customers with a low cost water supply of acceptable quality. Despite the many studies conducted on the optimization of pipeline systems under steady state conditions, there is still much to be learned about their operation under transient conditions. Negative and positive transient pressure surges travel along the pipeline, which may cause damage to the system. It is, therefore, necessary to study unsteady flows or transient conditions in pipeline systems. Low pressure transient waves have considerable potential to draw contaminants through leaks into a pipeline system. A motivation for considering the transient waves arises from water quality considerations [1] and [2]. Transient pressure waves occur in pipelines due to changes in fluid velocity typically caused by pump power failure or valve movement. When velocities in a pipe system change so rapidly that the elastic properties of the pipe and liquid must be considered in an analysis, there is a hydraulic phenomenon commonly known as water hammer. The governing equations of water hammer include two independent partial differential equations, the conservation of mass and momentum equations. The most general and well-known technique for solving these equations is the method of characteristics. Controlled valve movement, pump inertia control, pressure relief valves, air valves, surge tanks (open-end or one-way surge tanks), and air chambers are some of the transient protection devices and methods [3]. Vítkovský et al. [4] applied a forward transient technique and the Genetic Algorithm (GA) optimization technique for leak detection and calibration of pipe internal roughness in water distribution systems. Stephenson [5] presented design nomographs to simplify the process of sizing air vessels for water hammer protection of pumping pipelines. Jung and Karney [1] optimized the location, size, and number of transient control devices in water distribution networks using GA and Particle Swarm Optimization (PSO). They examined a gravity network with different protection strategies in each case. Transient pressure waves were caused by valve closures. Izquierdo et al. [6] used a neural network to optimize the design of air vessels based on system parameters to achieve permissible heads during a hydraulic transient. The purpose of the present work is to optimize the design of transient control devices to prevent water column separation after pump power failure. The optimization algorithm is combined with a transient simulation program to achieve the optimal solution. The large search space of the problem is maintained using a genetic algorithm. The GA deals with a large number of discrete or continuous variables, does not require a derivable objective function, explores a wide search space simultaneously, provides a population of optimum solutions, and works with numerically generated data, experimental data, or analytical functions in extremely complex problems [7]. In this paper, GA is improved using fuzzy inference systems. A fuzzy decision making is incorporated in the GA approach to improve its fitness evaluation process and its capability for handling constraints. The fitness evaluation in this paper does not incorporate cost directly. Each chromosome is evaluated using a fuzzy decision defined after transient analysis. The basic idea underlying fuzzy logic was suggested by Zadeh [8]. In general, fuzzy logic is concerned with formal principles of approximate reasoning, while classical two-valued logic (true or false) is concerned with formal principles of reasoning. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Two of the main concepts that play an important role in many applications of fuzzy logic are the concepts of linguistic variable and fuzzy if-then rules [9]. For example, height is a linguistic variable when its values are defined to be low, medium or high. Each linguistic value is represented as a fuzzy set that is characterized by a membership function, usually taking values between 0 and 1. In general, a fuzzy rule can be represented as: If x1 is A1 and x2 is A2 and …xn is An then y1 is B1 and y2 is B2 and …ym is Bm,If x1 is A1 and x2 is A2 and …xn is An then y1 is B1 and y2 is B2 and …ym is Bm, where x1,x2,…,xn,y1,y2,…,ymx1,x2,…,xn,y1,y2,…,ym are linguistic variables, and A1,A2,…,An,B1,B2,…,BmA1,A2,…,An,B1,B2,…,Bm are their respective linguistic values. The goal of using fuzzy systems is to put human knowledge into engineering systems in a systematic, efficient, and analyzable order. Fuzzy systems are knowledge-based or rule-based systems and work very well for many engineering problems [10]. Goulter and Bouchart [11] used fuzzy sets combined with linear programming for network cost minimization. Vamvakeridou-Lyroudia [12] used fuzzy sets for pressure and velocity constraint violation in a dynamic programming algorithm, for optimal design of water supply networks. Xu and Goulter [13] presented a fuzzy linear program optimization method in which the capital costs of the network were minimized while maintaining the nodal heads at demand nodes within a satisfactory region, as defined by the customers at those nodes. Revelli and Ridolfi [14] simulated uncertain parameters, like the roughness coefficient of pipes and the demands of the network, using fuzzy theory. Vamvakeridou-Lyroudia et al. [15] used a fuzzy multi-objective optimization model (minimizing cost and maximizing a benefit-quality function) to the “Anytown” water distribution network. They used genetic algorithms, combined with fuzzy reasoning, for benefit-quality evaluation. They showed that their model manages to find a better solution than any other previous approach in terms of cost, despite the multiple criteria applied for the benefit function being more extensive and stricter. Amirabdollahian et al. [16] applied a fuzzy genetic algorithm to obtain the least-cost design of looped water distribution networks. They used a fuzzy decision system to eliminate the traditional use of the penalty function in the genetic algorithm. They concluded that their proposed method yielded solutions with reduced costs. Mamdani and Sugeno are two types of fuzzy inference system that are tested and compared in this paper. In Mamdani-type inference, the output membership functions are fuzzy sets, but Sugeno output membership functions are constant. The Mamdani method is intuitive and has widespread acceptance, while the Sugeno method is computationally efficient and works well with optimization and adaptive techniques. The Sugeno method also has guaranteed continuity of the output surface [17]. To demonstrate the effectiveness of the proposed hybrid approach, three cases are presented.

This paper applies Genetic Algorithms (GA) combined with Fuzzy Inference Systems (FIS) for optimization of the location and size of surge tanks in a water system consisting of a series of pipes. The fitness evaluation in this paper does not incorporate cost directly. Minimizing the size of the surge tanks results in minimized costs. Maintaining the desirable final liquid height in the surge tanks leads to a least cost design (the least volume of material used in the walls, bottom, and roof) for these devices. The results obtained from the present work indicate that determination of the fitness value for each chromosome and satisfaction of the constraints can be simply accomplished using the fuzzy inference system. A generalized method is developed for evaluating the fitness value that does not change with dimensional characteristics, time (implying inflation and discount rates in costs), and location. The stopping criteria of the optimization algorithm become rational, and local optimum solutions are avoided by normalizing the fitness value in the range [0, 1]. The number, size, and location of the surge tanks have no significant effects on the volume of consumed water to overcome water column separation. In other words, the amount of water consumed by one-way surge tanks to effectively handle water column separation is approximately constant and does not change with changes in surge tank geometry or location. Mamdani and Sugeno methods yield similar results for the demonstrated cases. Fulfillment of the minimum pressure constraint and critical submergence shows the effectiveness of the fuzzy-genetic method proposed.