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Unidirectional Radial-Air-Turbine OWC Wave Energy Converters

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Unidirectional Radial-Air-Turbine OWC Wave Energy Converters ( unidirectional-radial-air-turbine-owc-wave-energy-converters )

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Energies 2019, 12, x FOR PEER REVIEW 9 of 22 Energies 2019, 12, 2791 9 of 22 where, 􏵶􏵂 and 􏵶􏵷 are the design variables, 􏵸 the tuning parameter and 􏵹 the number of parameters. where, x and x are the design variables, a the tuning parameter and n the number of parameters. In the CCiD, an ojptimal design space is considered with two criteria: the degree of non-orthogonality In the CCD, an optimal design space is considered with two criteria: the degree of non-orthogonality of regression terms (or variation inflation factor (VIF)), and the position of sample points (Leverages of regression terms (or variation inflation factor (VIF)), and the position of sample points (Leverages or the diagonal elements of the design matrix) [46]. Using this method, the design space contains a or the diagonal elements of the design matrix) [46]. Using this method, the design space contains a centre point, 2􏵹 design points located at the −α and +α position on each axis of the selected input centre point, 2n d􏵜e􏴬s􏵺ign points located at the −α and +α position on each axis of the selected input parameters and 2 factorial points located at −1 and +1 positions along the diagonals of the input parameters and 2n− f factorial points located at −1 and +1 positions along the diagonals of the input parameters space. Where α is selected such that both the maximum VIF and the maximum leverage parameters space. Where α is selected such that both the maximum VIF and the maximum leverage are are the minimum possible and 􏵻 is the fraction of the factorial design and is a function of 􏵹. As an the minimum possible and f is the fraction of the factorial design and is a function of n. As an example, example, CCD for two design variables consists of four factorial points, four axial points, and one CCD for two design variables consists of four factorial points, four axial points, and one central point central point as schematically shown in Figure 7. In this study, seven input parameters were 􏵵 6 as schematically shown in Figure 7. In this study, seven input parameters were considered and 2 considered and 2 fractional factorial designs used, which halved the number of experiments from 7 fractionalfactorialdesignsused,whichhalvedthenumberofexperimentsfromthe2 factorialdesigns. the 2􏵼 factorial designs. Figure 7. Central composite design for two design variables at two levels (Jung et al., 2016). Figure 7. Central composite design for two design variables at two levels (Jung et al., 2016). The response surface function is used in the next step to fit the actual analysis data characterized The response surface function is used in the next step to fit the actual analysis data characterized by the DOE and sample a surrogate model. The response surface optimisation is used to perform by the DOE and sample a surrogate model. The response surface optimisation is used to perform an an indirect optimisation analysis and evaluate the optimum candidate design predicted by various indirect optimisation analysis and evaluate the optimum candidate design predicted by various methods [47]. It provides a smooth and continuous mathematical formulation by interpolating between methods [47]. It provides a smooth and continuous mathematical formulation by interpolating discrete design points of the DOE. The response surface optimisation method allows the design points between discrete design points of the DOE. The response surface optimisation method allows the to be predetermined by the DOE and permits simultaneous solving of the response-surface design design points to be predetermined by the DOE and permits simultaneous solving of the response- points and multiple optimisations. In the current study, the Genetic Aggregation (GA) response surface design points and multiple optimisations. In the current study, the Genetic Aggregation (GA) surface algorithm was used to predict the optimum design point. GA is a meta model that selects response surface algorithm was used to predict the optimum design point. GA is a meta model that the most appropriate response surface for each output parameter based on the genetic algorithm. selects the most appropriate response surface for each output parameter based on the genetic It solves different response surfaces in parallel, analyses them regarding their accuracy and the stability algorithm. It solves different response surfaces in parallel, analyses them regarding their accuracy in the cross-validation and can be a single response surface or a combination of several different and the stability in the cross-validation and can be a single response surface or a combination of response surfaces [46]. In the optimisation step, the genetic algorithm was used as the optimiser several different response surfaces [46]. In the optimisation step, the genetic algorithm was used as which is a well-known approach in turbomachinery design optimisation) [44,45,47]. In the genetic the optimiser which is a well-known approach in turbomachinery design optimisation) [44,45,47]. In algorithm, feasible solutions are specified according to the bounds of the optimisation problem and the genetic algorithm, feasible solutions are specified according to the bounds of the optimisation the optimal solution is explored by analysing the maximum allowable Pareto front [48]. In this study, problem and the optimal solution is explored by analysing the maximum allowable Pareto front [48]. maximization of the total to static efficiency (defined in Equation (3)) was specified as the optimisation In this study, maximization of the total to static efficiency (defined in Equation (3)) was specified as objective. In addition, other turbine characteristics such as torque coefficient (Equation (1)), input the optimisation objective. In addition, other turbine characteristics such as torque coefficient power coefficients (Equation (2)) and flow coefficient (Equation (4)) were set as the secondary output (Equation (1)), input power coefficients (Equation (2)) and flow coefficient (Equation (4)) were set as parameters. Figure 8 shows the iterative process of the optimisation study. the secondary output parameters. Figure 8 shows the iterative process of the optimisation study.

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