## 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
This can be accomplished through the Perturb and Observe Method, which is thoroughly examined in the literature [28] concerning the operation of wind installations in energetically optimal areas under stable wind speeds. Given the variable wind speeds over time [11; 12; 43], the issue becomes complex due to the significant inertia moments. To operate within the optimal energy-efficient range, the mechanical angular speed must correspond with the temporal variations in wind speed. As will be demonstrated subsequently, employing the mathematical model of _WT_, if the wind speed varies over time while the power output of the electric generator stays constant, the maximum wind energy that the turbines can capture will not be achieved. It is known that the optimal power value at the generator is contingent upon the mechanical angular speed, which is in turn is influenced by the wind speed. Owing to the significant inertia moment [12; 42], it is not feasible for the operation to consistently maintain the _MPP_, as the wind turbine's rotational speed is unable to sufficiently adjust to the swift variations in wind speed over time [16; 42], which requires an optimal rotational speed from an energy efficiency standpoint [11]. It is considered that the wind speed changes from \(V_{1}=8\) m/s to \(V_{2}=10\) m/s, having a sinusoidal variation of the form  
\[V(t)=9+\sin 9.43\cdot 10^{-2}\cdot t \tag{58}\]  
At t = 0, the wind speed has the value \(V(0)=9\) m/s and the optimal _MAS_, \(\omega_{OPTIM}\)-corresponding to the speed of 9 m/s is  
\[\omega_{OPTIM}=32.026\cdot V=32.026\cdot 9=288.23\ \text{rad./s} \tag{59}\]  
Figure 9: The dependence of the power factor on wind speed. At \(\omega_{OPTIM}\) = 288.23 rad/s, the turbine operates at its maximum power point (Figure 10), with a constant generator power output. \[P_{EC}(0)=33\cdot 9^{3}=24057\;[W]\;\text{rad./s} \tag{60}\]  
The turbine power characteristics are  
For \(V_{1}\) = 8 m/s:  
Figure 9.0  
For \(V_{2}\) = 10 m/s:  
Figure 9.1  
For \(V_{3}\) = 9 m/s:  
Figure 9.2  
From the power equation  
Figure 9.3  
with the total moment of inertia of the generator and turbine system, \(J\) = 0.7 kg m\({}^{2}\), the obtained equation of motion is  
Figure 9.4  
or  
\[\left\{ \begin{array}{l}0.7\cdot\frac{d\omega}{dt}\cdot\omega=19182\left[ \frac{\left(9+\sin 9.43\cdot 10^{-2}\cdot t\right)}{\omega}-1.3877\cdot 10^{-2} \right]\cdot e^{-57.65\cdot\left(\frac{9+\sin 9.43\cdot 10^{-2}\cdot t }{\omega}\right)}.\\ \left(9+\sin 9.43\cdot 10^{-2}\cdot t\right)^{3}-33\cdot 9^{3}\\ \frac{dE_{W}}{dt}=19182\left[\frac{\left(9+\sin 9.43\cdot 10^{-2}\cdot t \right)}{\omega}-1.3877\cdot 10^{-2}\right]\cdot e^{-57.65\cdot\left(\frac{9+\sin 9.43\cdot 10^{-2} \cdot t}{\omega}\right)}.\\ \left(9+\sin 9.43\cdot 10^{-2}\cdot t\right)^{3}\\ \omega(0)=288.23\\ E_{W}(0)=0\end{array} \right. \tag{66}\]  
Figure 10: Turbine power characteristics. By solving the differential equations system, the _MAS_ variation, \(\omega\), is obtained as illustrated in Figure 11, under the condition that the power at the generator remains constant at 24,057 W.  
The wind energy harnessed by the turbine over a time span of 3500 s, \(E_{\text{W}}(3500)\), is derived from the integration of the differential equation  
\[\frac{dE_{W}}{dt}=P_{\text{WT}}[W] \tag{67}\]  
resulting in \(E_{\text{W}}(3500)=8.4251\times 10^{7}\) J.