# Wind Turbines Around Cut-In Speed: Startup Optimization and Behavior Analysis Reported to MPP
## 3 Mathematical Model of the Wind Turbine  
The mathematical models of \(WT\) are relatively well presented in the literature [4,8,19]. In order to achieve, at a given wind speed, a maximum capture of wind energy, the \(WT\) must operate at the optimal \(MAS\), \(\omega_{OPTIM}\), in the maximum power point, \(MPP\)[22,27,34]. The mathematical model of \(WT\), in the authors' conception, is the characteristic of \(WT\) power, the function \(P_{WT}(\omega,\,V,\,\beta)\), with \(V\)--wind speed [m/s] and \(\beta\)--the angle of inclination of the blades [28,35,36]:  
\[P_{WT}(\omega,\,V,\beta)=\rho\cdot\pi\cdot R_{p}^{2}\cdot C_{p}(\lambda,\beta )\cdot V^{3} \tag{15}\]  
where \(\rho\) is the density of the air in the operating location of the \(WT\), \(R_{p}\) blades radius, \(C_{p}(\lambda)\) is the power conversion coefficient, and \(\lambda=\omega\cdot R_{p}/V\). The power conversion coefficient, \(C_{p}(\lambda)\), at \(WT\) with three blades and the inclination angle of the blades \(\beta\), fixed, is determined with the relationship [19,20,22,23]:  
\[C_{p}(\lambda)=c_{1}\cdot\left(\frac{c_{2}}{\Lambda}-c_{3}\right)\cdot e^{- \left(\frac{c_{4}}{\Lambda}\right)} \tag{16}\]  
where \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\) are constructive constants, given in the catalog. \[\frac{1}{\Lambda}=\frac{1}{\lambda}-0.035=\frac{V}{R_{p}\cdot\omega}-0.035= \frac{V}{1.5\cdot\omega}-0.035 \tag{17}\]  
In wind turbines with blades having an adjustable angle of inclination, the power conversion coefficient, \(C_{p}(\lambda,\,\beta)\) is determined by the relation [23,28]:  
\[C_{p}(\lambda,\beta)=c_{1}\cdot\left(\frac{c_{2}}{\Lambda}-c_{3}\right)\cdot e ^{-\left(\frac{c_{4}}{\Lambda}\right)} \tag{18}\]  
where the parameter \(\Lambda\) has the form:  
\[\frac{1}{\Lambda}=\frac{1}{\lambda}\exp(-d\cdot\beta)-0.035=\frac{V}{R_{p} \cdot\omega}\exp(-d\cdot\beta)-0.035=\frac{V}{1.5\cdot\omega}\exp(-d\cdot \beta)-0.035 \tag{19}\]  
By substituting \(\Lambda\), the power conversion coefficient, \(C_{p}(\lambda,\,\beta)\), is obtained in the form:  
\[ \begin{array}{l}C_{p}(\lambda,\beta)=c_{1}\cdot\left(\frac{c_{2}}{\Lambda}- c_{3}\right)\cdot e^{-\left(\frac{c_{4}}{\Lambda}\right)}=c_{1}\Big{(}c_{2} \cdot\frac{V}{1.5\cdot\omega}\exp(-d,\beta)-0.035-c_{3}\Big{)}\cdot e^{-c_{4} \left(\frac{V}{1.5\cdot\omega}-0.035\right)}=\\ a_{1}\cdot\left[\frac{V}{\omega}exp(-d,\beta)-b\right]\cdot e^{-c\cdot\frac{V} {\omega}}\end{array}  \tag{20}\]  
where the values of the parameters \(a\), \(b\), \(c\) and \(d\) are determined from the experimental data. In this paper, we present a mathematical model of _WT_, _MM-WT_, deduced by measuring the wind speed, \(V\), the power output by the generator, \(P_{EG}\) and the _MAS_ at the generator, or _WT_, \(n/\omega\). As atmospheric conditions, mechanical stresses and even turbine geometry change over time, and the initial _MM-WT_ changes over time. Therefore, a periodic recalculation of it is required based on measurements of the fundamental quantities: wind speed, power and _MAS_, at the electric generator. In most of the works [11; 37; 38] the existing wind systems are treated on the basis of inadequate _MM-WT_ and without efficient processing of experimental data. The experimental data can be analyzed through various algorithms, methodologies, and specified conditions [39; 40]. On the basis of _MM-WT_, the dependence of the optimal mechanical angular speed, \(\omega_{OPTIM}(V)\), on the wind speed, \(V\)--reference quantity in the r control algorithms, is determined for wind systems operating at time-varying wind speeds. A maximum capture of wind energy, at wind speeds that vary significantly over time, can only be achieved on the basis of a valid, constantly updated _MM-WT_. It is preferable that the determination of the _MM-WT_ is carried out during the operation of the wind system. To ensure the operation of the wind turbine at maximum power, it is essential to know the mathematical model of _WT_ as accurately as possible.