## 3 Controller Design Approach
### Proposed GA-Fuzzy-Based Control Schemes
#### 3.3.2 Pitch Angle Control System  
The basic configuration of the WTG system is presented in Figure 4. Figure 5 shows the structure of the proposed pitch angle control scheme that determines the \(\beta\) of each wind turbine. The fraction of extracted wind power converted to mechanical power of the rotor, \(P_{w}\), can be calculated as follows:  
\[P_{w}=\frac{1}{2}\times C_{p}\times V_{w}^{3}\times\rho\times A \tag{2}\]  
where A = \(\pi r^{2}\). Values of \(r\) and \(\rho\) are 23.5 m and 1.25 Kg/m\({}^{3}\), respectively. \(C_{p}\) is a function of both \(\lambda\) and \(\beta\). \(\lambda\) is defined as the ratio of the speed at the blade tip to the wind speed and can be expressed as:  
\[\lambda=\frac{r\times\omega_{w}}{V_{w}} \tag{3}\]  
Then, an approximate expression for \(C_{p}\) as function of \(\lambda\) and \(\beta\) can be given as [51]:  
\[C_{p}=(0.44-0.0167\beta)\text{sin}(\frac{\pi(\lambda-3)}{15-0.3\beta})-0.0184( \lambda-3)\beta \tag{4}\] \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Variable** & **A1** & **A2** & **A3** & **A4** & **A5** & **A6** & **A7** & **A8** & **A9** & **A10** & **A11** \\ \hline Value & 0.078 & 0.058 & 0.066 & 0.05 & 0.031 & 0.017 & 0.013 & 0.082 & 0.072 & 0.076 & 0.039 \\ Variable & A12 & A13 & A14 & A15 & A16 & A17 & A18 & A19 & A20 & A21 & \\ Value & 0.061 & 0.017 & 0.035 & 0.541 & 0.331 & 0.482 & 0.253 & 0.291 & 0.157 & 0.223 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimal membership function parameter values. Figure 4: System configuration of the wind turbine generator (WTG). \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**\(\mathbf{\Delta e_{A}}\)****\(\mathbf{\Delta e_{A}}\)****NL** & **NM2** & **NM1** & **NS** & **Z** & **PS** & **PM1** & **PM2** & **PL** \\ \hline NL & PL & PL & PM2 & PM2 & PM1 & PM1 & PS & PS & Z \\ NM2 & PL & PM2 & PM2 & PM1 & PM1 & PS & Z & NS & NS \\ NM1 & PL & PM2 & PM1 & PM1 & PS & Z & Z & NS & NM1 \\ NS & PL & PM2 & PM1 & PS & PS & Z & NS & NS & NM1 \\ Z & PM2 & PM1 & PS & PS & Z & NS & NM1 & NM1 & NM2 \\ PS & PM1 & PS & PS & Z & NS & NM1 & NM2 & NM2 & NL \\ PM1 & PS & PS & Z & NS & NM1 & NM2 & NM2 & NL \\ PM2 & PS & Z & NS & NM1 & NM2 & NM2 & NL & NL & NL \\ PL & Z & NS & NM1 & NM2 & NM2 & NL & NL & NL & NL \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy control rules. The hydraulic servo system has nonlinear characteristics. However, it can be modeled as first-order lag system. Therefore, in this control scheme, the first-order lag system is considered where the time constant is 1 s. Fuzzy sets and their corresponding membership functions for the proposed controller are presented in Figure 6, while their control rules are discussed in Table 3. The deviation of output power for each WTG and its change with time is then used as input signals for the fuzzy controller, while output is the control signal to the hydraulic servo system that drives the blades according to \(\beta_{CMD}\). The membership functions of the first input are set as Negative (N), Zero (Z), Small (S), Medium1 (M1), Medium2 (M2), and Large (L). Also, the membership functions corresponding to the second input are held as Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), and Positive Large (PL). Meanwhile, the output membership functions are arranged as Zero (Z), xSmall (XS), Small (S), xMedium (xM), xxMedium (xxM), xxMedium (xxxM), Large (L), xLarge (xL), and xxLarge (oxL). The triangular membership function which is the most popular is also used in this control approach. The output of each rule is composed by using the And function with interpretation of the minimum. In the same manner as the first control scheme, the Mamdani interference system is used here.