# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 2 Background on Multi-Objective Optimization  
In this section, some elements of the control methodology used in this work are summarized.  
In multi-objective optimization problems, a number of optimal solutions can be found given a set of conflicting objectives. A Pareto front solution is a methodology that provides data for decision-making considering a trade-off between the divergent objective functions [40].  
For these multi-objective problems, diverse strategies first define a cost function with weights for different goals, and later the optimization problem is solved. An optimization strategy based on Pareto efficiency to optimize two objective optimization problem was developed in [26; 40]. The essential concepts of multi-objective optimization, definitions of Pareto optimal solutions, and Pareto efficiency are described in [29; 40; 41].  
Multi-objective optimization problems are often solved by means of evolutionary algorithms, such as genetic algorithms, due to their practical benefits over traditional optimization techniques [42]. Genetic algorithms can be combined with local searching methods, achieving better computational efficiency [43; 44]. In 2002, a fast elitist non-dominated sorting genetic algorithm (NSGA-II) was designed [44]; it is an enhanced version of the traditional nondominated sorting genetic algorithm (NSGA) [43]. In the current work, the algorithm NSGA-II for tuning the parameters of the proposed control scheme is employed.  
A representative example of a Pareto frontier is shown in Figure 2. The boxed points represent viable solutions to an optimization problem with two objectives to be optimized. Smaller values are preferred to larger ones; that is, the smaller the objective functions F\({}_{1}\) and F\({}_{2}\) for each alternative are, the better the choice is. Point C is dominated by both point A and point B for both criterion (F\({}_{1}\)(A) \(<\) F\({}_{1}\)(C), F\({}_{1}\)(B) \(<\) F\({}_{1}\)(C), F\({}_{2}\)(A) \(<\) F\({}_{2}\)(C), and F\({}_{2}\)(B) \(<\) F\({}_{2}\)(C)); hence, point C cannot be part of the Pareto frontier. Points A and B are not exclusively dominated by any other, and therefore they lie on the Pareto frontier.  
In cases where the different objectives are complex and cannot converge at the same point, an optimal solution can possibly be different from a satisfactory solution. In general, some objectives reach a Pareto frontier and every solution on the frontier are optimal. However, a high-level optimization objective is necessary to analyze the optimal solutions and chose the satisfactory one. MCDM methods are necessary to evaluate the set of different optimal solutions and then select one accordingly to specific preferences. MCDM methods are becoming an increasingly trendy in resolving renewable energy problems because these problems imply multiple and frequent conflicting criteria. As an example, in [45] a comparative analysis of ranking renewable energy sources for electricity generation in Taiwan was carried out using different MCDM methods. Then, they were utilized to quantitatively evaluate and rank all available alternatives. Next, three common MCDM methods are described.