## 3 Generator Side Converter Control  
The mechanical power captured from wind turbine is governed by the following equation:  
\[P_{m} = 0.5\,\rho\,A\,C_{p}\,\upsilon_{w}^{\ 3} \tag{1}\]  
Where \(P_{m}\) is the mechanical output power of the wind turbine (Watt), \(\rho\) is the Air density (Kg/m\({}^{3}\)), \(A\) is the swept area (m\({}^{2}\)), \(C_{p}\) is the power coefficient of the wind turbine and \(\upsilon_{w}\) is the wind speed (m/second). Consequently, the output energy is determined by the power coefficient (\(C_{p}\)) of wind turbine if the swept area, air density, and wind speed are assumed to be constant. \(C_{p}\) is function in tip speed ratio (\(\lambda\)) and pitch angle (\(\beta\)) in degree. If \(\beta\) is equal zero, in this case \(C_{p}\) is only function in \(\lambda\) as shown in (2), and \(\lambda\) is function of rotor mechanical speed, rotor radius of blade and wind speed as indicated in (3). \[C_{p}(\lambda) = \frac{60.04-4.69\lambda}{\lambda}e^{\left(\frac{-21+0.735\lambda} {\lambda}\right)}+\frac{0.0068\lambda}{1-0.035\lambda} \tag{2}\] \[\lambda = \frac{\omega_{r}\,R}{\upsilon_{w}} \tag{3}\]  
Where \(\omega_{r}\) is the rotational speed (rad/second) and \(R\) is the radius of blade (m). The relation between \(C_{p}\) and \(\lambda\) when \(\beta\) equal zero degree is shown in Figure 2. It can be noticed that the optimum value of \(C_{p}\) is about 0.48 for \(\lambda\) equal 8.1. Maximum power extraction from wind turbine can be achieved when the turbine operate at maximum \(C_{p}\) (\(C_{p\text{-}opt}\)). Therefore, it is necessary to adjust the rotor speed at optimum value of tip speed ratio (\(\lambda_{opt}\)) with wind speed variation. Figure 1: Power circuit topology of a variable speed stand-alone wind energy supply system. Using (1) and (3), the optimum output power from wind turbine can be written as:  
\[P_{m\_opt}=0.5\ \rho\ A\ C_{p\_opt}\left(\frac{\omega_{r\_opt}\ R}{\lambda_{opt}} \right)^{3}=K_{opt}\left(\omega_{r\_opt}\ \right)^{3} \tag{4}\]  
Where:  
\[K_{opt}=0.5\ \rho\ A\ C_{p\_opt}\left(\frac{R}{\lambda_{opt}}\right)^{3} \tag{5}\]  
\[\omega_{r\_opt}=\frac{\lambda_{opt}}{R}\upsilon_{w}=K_{\omega}\upsilon_{w} \tag{6}\]  
Figure 3 indicates that the mechanical powers generated by the turbine as a function of rotor speeds for different wind speeds. The maximum power extraction within the allowable range can be achieved if the controller can properly follow the optimum curve with variation of wind speed. The generator side converter (a single switch three phase mode rectifier) is controlled to extract maximum power from available wind turbine power. Hence, the wind turbine can produce maximum power when the turbine operates at optimum value of \(C_{p}\) (\(C_{p\_opt}\)). So it is necessary to adjust the rotor speed at optimum value of the tip speed ratio (\(\lambda_{opt}\)). Figure 2: The relation between power coefficient (\(C_{p}\)) and tip speed ratio (\(\lambda\)). The structure of the proposed control strategy of the switch mode rectifier is shown in Figure 4. The objective of this control is to control the duty cycle of DC-DC boost switch \(S_{I}\) which is shown in Figure 1 to extract maximum power from available wind power. The control algorithm includes the following steps:  
* Estimate generator speed, \(\omega_{g}\)
* Generator speed of PMSG can be estimated by measuring the average output voltage \(V_{d}\) and current \(I_{d}\) of the uncontrolled bridge rectifier and by knowing the parameters of used PMSG in this paper which are given in \[C_{p}(\lambda) = \frac{60.04-4.69\lambda}{\lambda}e^{\left(\frac{-21+0.735\lambda} {\lambda}\right)}+\frac{0.0068\lambda}{1-0.035\lambda} \tag{2}\]0 [20]. * The relation among generator speed, PMSG parameters, \(V_{d}\) and \(I_{d}\) is governed by (7). \[C_{p}(\lambda) = \frac{60.04-4.69\lambda}{\lambda}e^{\left(\frac{-21+0.735\lambda} {\lambda}\right)}+\frac{0.0068\lambda}{1-0.035\lambda} \tag{2}\]1 (7) where \(\omega_{g}\) is the generator speed in (rad/second).