Multi-Objective Robust Optimization Using a Postoptimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

Abstract

Toward a multi-objective optimization robust problem, the variations in design variables and design environment parameters include the small variations and the large variations. The former have small effect on the performance functions and/or the constraints, and the latter refer to the ones that have large effect on the performance functions and/or the constraints. The robustness of performance functions is discussed in this paper. A post-optimality sensitivity analysis technique for multi-objective robust optimization problems is discussed and two robustness indices are introduced. The first one considers the robustness of the performance functions to small variations in the design variables and the design environment parameters. The second robustness index characterizes the robustness of the performance functions to large variations in the design environment parameters. It is based on the ability of a solution to maintain a good Pareto ranking for different design environment parameters due to large variations. The robustness of the solutions is treated as vectors in the robustness function space, which is defined by the two proposed robustness indices. As a result, the designer can compare the robustness of all Pareto optimal solutions and make a decision. Finally, two illustrative examples are given to highlight the contributions of this paper. The first example is about a numerical problem, whereas the second problem deals with the multi-objective robust optimization design of a floating wind turbine. * Corresponding author: Email: stephane.caro@irccyn.ec-nantes.fr, Tel: +33 (0)2 40 37 69 68, Fax: +33 (0)2 40 37 69 30 Nomenclature x design variable vector p vector of design environment parameters g k the kth constraint f performance function vector F feasible set P Pareto optimal set σ standard deviation µ expected value I RS robustness index against small variations I F feasibility index of a solution I P Pareto optimality index of a solution h(p) probability density function of p I rank individual's ranking N number of discrete values of design environment parameters I RL robustness index against large variations P R (P) the most robust solutions amongst the Pareto optimal solutions P produced power of wind turbine rotor F a the thrust force in the partial load region of a wind turbine rotor γ r the root twist angle γ t the tip twist angle c r the chord length at the root c t the chord length at the tip ω rotor rotational speed r r root radius of the wind turbine IntroductionMany design optimization problems are Multi-Objective Optimization Problems (MOOP) and are subject to uncertainties or variations in their parameters. Robustness is a product's ability to maintain its performance under the variations in its parameters. Robust design aims at maximizing product's robustness. In other words, it aims at minimizing the sensitivity of performance to variations without controlling the source of these variations [1]. Sometimes, the robustness of a product is as important as or even more important than its p...

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