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88 Integrated system design The shadowed area represents all the possible solutions to function F in equation (4-66). The points that fulfill the requirement of Pareto efficiency are in the “Pareto frontier”. Since this is a maximization problem, point C is not Pareto efficient since there are solutions that have both higher values in the functions ηTH (higher efficiency) and ς (lower cost). On the other hand, points A and B have at least one function value that is always a maximum, which makes them Pareto efficient and part of the Pareto frontier. Although the targets of this case are the efficiency and the cost of the system, it is possible to have different targets, with different constrains and parameters depending on the requirement of the optimization. 4-3-1 Solution method Genetic algorithm techniques are attractive options to solve multi-objective optimizations since they have several advantages over other algorithms. They do not require gradient information, making them effective regardless of the shape of the objective functions and constrains. This characteristic can be useful since the optimization process of the cooler presents non linear and non continuous solutions due to the discrete geometries considered in it. Additionally, genetic algorithms are global optimization techniques, which means that they find a global set of solutions rather than a local one. This is the main reason to choose this method for multi-objective optimization since in these kind of problems there may be several solutions which comply with the Pareto optimality criteria [81]. Genetic algorithms belong to the evolutionary algorithms class since they are based on the Darwin’s theory of natural selection. The reproduction cycle of this algorithm is presented in Figure 4-18. START DISCARD NON FIT SOLUTIONS INITIAL POPULATION (x) FITNESS SELECTION OF [ηTH(x),ς(x)] N END CONDITION SATISFIED Y OPTIMAL SOLUTION SET Figure 4-18: General genetic algorithm flow diagram. The initial population represents a randomly generated set of design variables or chromo- somes x. A design vector xi could be represented by a binary code or a set of real values, which is applicable to this case. For a given population there are three operations applied: selection, crossover, and mutation. The selection is based on a criteria called fitness, which is applied on the functions ηTH and ς and it is proportional to the probability of surviving J.S. Bahamonde Noriega Master of Science Thesis NEW GENERATION END MODIFICATION (CROSSOVER, MUTATION)PDF Image | Design method for s-CO2 gas turbine power plants
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