## 4 System Operation in the Optimal Zone: Analysis of Various Dependencies
### The Dependencies of Frequency, Steady-State Current, and Power Factor in Relation to Wind Speed  
The static frequency, f, is adjusted to enable the operation of the wind turbine at maximum power points (_MPPs_) under optimal conditions, where \(\omega_{OPTIM}\) = 32.026\(\cdot\)_V_, which is contingent upon the wind velocity, \(V\). We consider the stationary regime, where the powers are equal, \(P_{WT}(\omega,\)\(V)=P_{EG}(\omega,f)\). The wind speed varies from 3 m/s to 10 m/s, yielding the following results. For \(V\) = 3 m/s:  
\[ \begin{split} 55\cdot V^{3}=-\frac{\frac{3\cdot 4\cdot 6^{2}\left( \frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}{\left(38179\cdot 10^{-2} +\frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}+\left(\frac{131}{80} \right)^{2}}{\omega=32.026\cdot V}\\ f=15.233\ \mathrm{[Hz]}\end{split}  \tag{46}\]  
For \(V\) = 4 m/s:  
\[ \begin{split} 55\cdot V^{3}=-\frac{\frac{3\cdot 4\cdot 6^{2}\left( \frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}{\left(38179\cdot 10^{-2} +\frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}+\left(\frac{131}{80} \right)^{2}}{\omega=32.026\cdot V}\\ f=20.284\ \mathrm{[Hz]}\end{split}  \tag{47}\]  
For \(V\) = 5 m/s:  
\[ \begin{split} 55\cdot V^{3}=-\frac{\frac{3\cdot 4\cdot 6^{2}\left( \frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}{\left(38179\cdot 10^{-2} +\frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}}{\omega=32.026\cdot V} \\ f=25.231\ \mathrm{[Hz]}\end{split}  \tag{48}\]  
For \(V\) = 6 m/s:  
\[ \begin{split} 55\cdot V^{3}=-\frac{\frac{3\cdot 4\cdot 6^{2}\left( \frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}{\left(38179\cdot 10^{-2} +\frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}+\left(\frac{131}{80} \right)^{2}}{\omega=32.026\cdot V}\\ f=30.44\ \mathrm{[Hz]}\end{split}  \tag{49}\]  
For \(V\) = 7 m/s:  
\[ \begin{split} 55\cdot V^{3}=-\frac{\frac{3\cdot 4\cdot 6^{2}\left( \frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}{\left(38179\cdot 10^{-2} +\frac{38179\cdot 10^{-2}}{1-\frac{101915}{7}\omega}\right)}+\left(\frac{131}{80} \right)^{2}}{\omega=32.026\cdot V}\\ f=35.347\ \mathrm{[Hz]}\end{split}  \tag{50}\]  
The variation of the static current with wind speed is illustrated in Figure 8.  
At a wind speed of 10 m/s, the nominal current \(I_{N}\) is determined to be 98 A. A linear variation of the current, expressed as \(I=9.8\cdot V\), is validated by the relationship between the steady-state current and wind speed illustrated in Figure 9. Consequently, the electrical power delivered to the grid is  
\[P_{EI}=3\cdot U\cdot I\cdot\cos\varphi=3\cdot 4.6\cdot 5\cdot V\cdot 9.8\cdot V \cdot I\cdot\cos\varphi=676.2\cdot V^{2}\cdot\cos\varphi \tag{56}\]  
From the equality of the turbine powers, \(P_{WT}(V)=55\cdot V^{3}\), with the generator \(P_{WT}(V)=P_{EG}(V)\), the generator power factor \(cos\varphi\) value is obtained:  
\[\cos\varphi=\frac{55\cdot V^{3}}{676.2\cdot V^{2}}=8.1337\cdot 10^{-2}\cdot V \tag{57}\]  
Figure 9 illustrates that the power factor ranges from 0.24401 to 0.81337, with the minimum value occurring at a wind speed of 3 m/s and the maximum value at a wind speed of 10 m/s. Figure 8: Dependence of the stator current on wind speed. Figure 7: Dependence of the frequency on wind speed. The speed of the wind significantly affects both the quality of power delivered to the grid, as indicated by the power factor, and the quantity of power supplied, which is dependent on the cube of the wind speed. The higher the wind speed value, the higher the quality and value of power delivered to the grid. Wind System Behavior at Time-Varying Wind Speeds by Estimating the Wind Energy Captured by the Turbine  
At constant wind speeds over time, the wind turbine is capable of operating at the maximum power point, _MPP_, by utilizing a suitable power prescription at the _EG_[8; 26].