# Wind Turbines Around Cut-In Speed: Startup Optimization and Behavior Analysis Reported to MPP
## 3 Mathematical Model of the Wind Turbine
The determination of _MM-WT_ is based on experimental data from the 1.5 MW _WT_ of type Fuhrhlander FL MD 70, within the Oravita wind farm. The mathematical model of _WT_ defined by the experimental power characteristics, \(P_{WT}(\omega,V)\), [4; 16; 19; 30] was initially deduced, at a fixed blade tilt angle, \(\beta=0\). The value of the maximum power is obtained at \(\omega_{OPTIM}\), by canceling the power derivative:  
\[\frac{dP_{WT}}{d\omega}=0 \tag{21}\]  
results in:  
\[\omega-V\cdot c+b\cdot c\cdot\omega=0 \tag{22}\]  
and from here [29; 37; 38]:  
\[\omega_{OPTIM}=\frac{c}{1+b\cdot c}\cdot V=k\cdot V \tag{23}\]  
The period was defined as an interval, during which the wind speed remains steady, allowing the wind turbine to function at its maximum efficiency, the maximum power point. The measured values for wind speed, output power, \(P_{EG}\) and _MAS_ at the generator, \(\omega\), the experimental data were collected [30]: \(\omega_{OPTIM}=204.39\) rad/s, \(V=8.5\) m/s, \(P_{WT-MAX}=1500\) kW. From the experimental data, we obtained \(\omega_{OPTIM}/V=24.046\), or \(c/(1+b\cdot c)=24.046\). This is the first equation of the system in the unknowns \(a\), \(b\), and \(c\). The value of the maximum power of the _WT_ corresponds to the optimal _MAS_, \(\omega_{OPTIM}\):  
\[\frac{V}{\omega_{OPTIM}}=\frac{1+b\cdot c}{c} \tag{24}\]  
and it is:  
\[P_{WT-MAX}(V)=a\cdot\left(\frac{1+b\cdot c}{c}-b\right)\cdot e^{-c\left(\frac {1+b\cdot c}{c}\right)}\cdot V^{3}=\frac{a}{c}\cdot e^{-1-c\cdot b}\cdot V^{3} \tag{25}\]  
Results:  
\[k_{p}=\frac{a}{c}\cdot e^{-1-c\cdot b} \tag{26}\]Analogous to \(\omega_{OPTIM}\), from the experimental data, the proportionality factor \(k_{p}\) has the value:  
\[k_{p}=\frac{P_{WT-MAX}}{V^{3}}=\frac{1500}{8.5^{3}}=2.4425 \tag{27}\]  
Thus, the second equation of the system results in the unknowns a, b and c:  
\[\frac{a}{c}\cdot e^{-1-c\cdot b}=2.4425 \tag{28}\]  
These two equations are essential in determining _MM-WT_ and in establishing _MPP_, the \(\omega_{OPTIM}\) and \(P_{WT-MAX}\) coordinates. To determine the unknown variables \(a\), \(b\), and \(c\), an extra equation is necessary. However, this equation is not critical for establishing the coordinates of the Maximum Power Point (_MPP_), which can instead be derived from the ratio: \(\omega_{OPTIMAL}/\omega_{MAXIM}\), where \(\omega_{MAXIM}\) is the maximum mechanical angular speed, at the idle operation of the WT. The value of the \(\omega_{OPTIM}/\omega_{MAX}\) ratio generates the third equation. Since the _WT_ power is zero at idle:  
\[P_{WT}=a\cdot\left(\frac{V}{\omega}-b\right)\cdot e^{-c\cdot\frac{V}{2}}\cdot V ^{3}=0 \tag{29}\]  
at _MAS_\(\omega=\omega_{MAXIM}\) results:  
\[\omega_{MAXIM}=\frac{V}{b} \tag{30}\]  
Substituting in the value of the \(\omega_{OPTIMAL}/\omega_{MAXIM}\) ratio, we obtain:  
\[\omega-V\cdot c+b\cdot c\cdot\omega=0 \tag{22}\]0  
The value of the ratio \(\omega_{OPTIM}/\omega_{MAXIM}\) generates the 3. equation. Since the _WT_ power is zero at idle, from the power conversion coefficient, \(C_{p}(\lambda)\) for a three-bladed _WT_[16; 19; 20], we obtain e\({}^{-1-0.725\;66}\) = 0.17806, \(\omega_{OPTIM}/\omega_{MAXIM}\) = 0.42051. The following illustrates how the system of equations led to the determination of the unknown variables \(a\), \(b\), and \(c\):  
\[\left\{ \begin{array}{l}\frac{c}{1+(b\cdot c)}=24.046\\ \frac{d}{c}\cdot e^{-1-c\cdot b}=2.4425\\ \frac{b\cdot c}{(b\cdot c)+1}=0.42051\end{array} \right. \tag{32}\]  
with the solution: \(a\) = 569.2, \(b\) = 1.7488\(\cdot\)10\({}^{-2}\), \(c\) = 41.495.