# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 2 Background on Multi-Objective Optimization
### VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) Method  
The VlseKriterijumska Optimizacija i Kompromisno Resenje (VIKOR) method [54] was created for multicriteria optimization of complex systems in order to find the solution in decision problems with conflicting and noncommensurable criteria [55]. In the VIKOR model, a compromise ranking can be obtained by comparing the measure of closeness to the ideal solution [53]. Each alternative is evaluated according to each criterion function. Thus, a compromise ranking is estimated by comparing the measure of closeness to the ideal alternative. This method has been used for the assessment of sustainable and renewable energy system problems, considering technical, economic, and environmental aspects [56; 57]. The following steps are used for the algorithm VIKOR:  
First, the best \(f_{j}^{+}\) and worst \(f_{j}^{-}\) values of all attribute functions are determined according to (9). \(f_{j}^{+}\) is the positive ideal solution for the j criteria, and \(f_{j}^{-}\) is the negative ideal solution for the j criteria.  
\[\begin{split} f_{j}^{+}=\min\bigl{(}x_{ij}\bigr{)}\,\ i=1,2,\ldots,m\\ \text{for}\ \cos\text{t}\ \text{attribute}\\ f_{j}^{-}=\max\bigl{(}x_{ij}\bigr{)}\,\ i=1,2,\ldots,m\end{split} \tag{9}\]  
Next, the distances from each alternative to the positive ideal solution are calculated and added according to (10) and (11) to obtain \(S_{i}\) and \(R_{i}\). \(S_{i}\) represents the distance rate of the i alternative to the positive ideal solution (best combination), and \(R_{i}\) represents the distance rate of the i alternative to the negative ideal solution (worst combination). The best ranking will be based on \(S_{i}\) values, and the worst ranking will be based on \(R_{i}\) values.  
\[\mathrm{S_{i}}=\sum_{j=1}^{n}w_{j}\ \frac{\left(\mathrm{f_{j}^{+}-x_{ij}}\right)}{ \left(\mathrm{f_{j}^{+}-f_{j}^{-}}\right)} \tag{10}\]  
\[\mathrm{R_{i}}=\max_{i}\left[w_{j}\ \frac{\left(\mathrm{f_{j}^{+}-x_{ij}} \right)}{\left(\mathrm{f_{j}^{+}-f_{j}^{-}}\right)}\right] \tag{11}\]  
Finally, the VIKOR values \(\mathrm{Q_{i}}\) are calculated as (12).  
\[\mathrm{Q_{i}}=\mathrm{v}\left[\frac{\mathrm{S_{i}-S^{+}}}{\mathrm{S^{-}-S^{+ }}}\right]\ +(1-\mathrm{v})\left[\frac{\mathrm{R_{i}-R^{+}}}{\mathrm{R^{-}-R^{+}}}\right] \tag{12}\]  
where \(\mathrm{S^{-}}=\max_{i}(\mathrm{S_{i}})\), \(\mathrm{S^{+}}=\min_{i}(\mathrm{S_{i}})\), \(\mathrm{R^{-}}=\max_{i}(\mathrm{R_{i}})\)\(\mathrm{R^{+}}=\min_{i}(\mathrm{R_{i}})\), and \(\mathrm{v}\) is the weight of the strategy of \"the majority of criteria\" (or \"the maximum group utility\"). \(\left[\left(\mathrm{S^{-}S^{+}}\right)/\left(\mathrm{S^{-}}-\mathrm{S^{+}} \right)\right]\) represents the distance rate from the positive ideal solution of the i alternative's achievements. \(\left[\left(\mathrm{R-R^{+}}\right)/\left(\mathrm{R^{-}-R^{+}}\right]\right.\) represents the distance rate from the negative ideal solution of the i alternative. In general, a \(\mathrm{v}\) value of 0.5 is a compromise attitude of evaluation experts.  
The diverse alternatives are sorted in decreasing order according to \(\mathrm{Q_{i}}\). The best alternative is the one with the minimum value of \(\mathrm{Q}\).