# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 3 Control Methodology
### Description of the Control Blocks
#### 3.3 Multi-Objective Optimization by Genetic Algorithms  
In previous sections, the blocks contained in the proposed control scheme have been explained. In this section, the methodology for tuning the parameters of these control blocks is described. It is addressed as an optimization problem that combines two objective functions \(f_{1}\) and \(f_{2}\) to be minimized. These indices correspond to the integral of absolute error (IAE) of the generator speed \(\omega_{\text{g}}\) and the cumulative variation rate (CVR) of the tower f-a displacement \(\text{x}_{\text{t}}\), and they are given in (24):  
\[ \begin{split}\text{f}_{1}&=\text{IAE}_{\text{avg}}= \int_{\text{f}_{0}}^{\text{t}}\abs{\omega_{\text{g}_{\text{rated}}}-\omega_{ \text{g}}(\text{t})}\dt\\ \text{f}_{2}&=\text{CVR}_{\text{x}_{\text{t}}}= \int_{0}^{\text{t}}\abs{\text{x}_{\text{t}}(\text{t}_{\text{k}})-\text{x}_{ \text{t}}(\text{t}_{\text{k}-1})}\dt\end{split}  \tag{24}\]  
The two conflicting objectives in (24) are combined to be optimized using multi-objective genetic algorithms. The proposed cost function seeks simultaneously to reduce the tower fatigue by minimizing fore-aft displacements while tries to maintain the generator speed close to its rated value. Due to the nonlinearity of these performance indices, the PI tuning that minimizes them must be formulated as a nonlinear optimization problem [66; 67]. The optimization procedure is performed by means of simulations of each identified linear model in its operation point where the controller must maintain the generator speed at its rated value of 1173.7 rpm for wind step changes of \(\pm 0.5\) m/s from its nominal value at that point. The six operation points are related to the wind speed values: 12, 14, 16, 18, 20, and 22 m/s. The simulations also consider the noise presented by the nonlinear model on the variables \(\omega_{\text{g}}\) and \(\text{x}_{\text{t}}\) to obtain more realistic results. Each controller has four parameters to be tuned: \(\text{K}_{\text{p}}\) and \(\text{T}_{\text{i}}\) (PI controller), \(\text{K}_{\text{f-a}}\) (ATDC), and \(\text{K}_{\text{ff}}\) feedforward gain. The search range of the \(\text{K}_{\text{p}}\), \(\text{T}_{\text{i}}\), and \(\text{K}_{\text{f-a}}\) is limited from \(10^{-4}\) to 30, being negative for \(\text{K}_{\text{p}}\). The \(\text{K}_{\text{ff}}\) gain ranges from 0 to 1. The main options configured in the genetic algorithm are as follows: a population size of 1000, an elite count of 0.05 times the population size for reproduction with a crossover fraction of 0.8. Due to the presence of two objective functions, a set of Pareto-optimal solutions with 50 possible solutions is derived in each operation point. In each Pareto front of solutions, there is one optimal solution based on each objective function. Figure 6 shows the resultant Pareto front for the operating point with wind speed of 12 m/s. The optimal solutions based on each single objective function \(\text{f}_{1}=\text{IAE}_{\text{avg}}\) and \(\text{f}_{2}=\text{CVR}_{\text{xt}}\) are also pointed out. MCDM methods help to evaluate the different alternatives according to various criteria to obtain the most suitable solutions. Each approach has its advantages and disadvantages. There is no method which is better than another; however, the selection of an approach depends mainly on the preferences of the selected criteria. Therefore, decision makers do not restrict themselves to one method and may achieve nonidentical results using different methods. In this case, decision-makers require using integrated methods for the final decision. The average ranking of alternatives obtained by different methods can be a good choice [68]. Figure 6: Pareto front points obtained for a wind speed of 12 m/s. For each Pareto front, the following procedure is performed: first, the SAW, TOPSIS, and VIKOR method are applied to each one of the 50 possible solutions of the front. Next, the solutions are sorted from the best to the worst for each method.