# Wind Turbines Optimal Operation at Time Variable Wind Speeds
## 6 Case Study  
### PMSG's Optimal Power Determination  
In what follows, the objective is to determine the optimal values of the PMSG's power \(P_{PMSG\mathit{optim}}(t)\), delivered by one of the wind turbines within [21]. Since the \(P_{PMSG\mathit{optim}}(t)\) depends on \(S_{w}(t)\), the main input parameter that could be measured is the time-varying wind speed. The wind farm in [21] provided the wind speed measurements, sampled with a step of 30 (s), as shown in the below presented Table 1.  
Based on the data from [21], [PERSON] et al. [3] demonstrates that  
\[\frac{\omega_{qptm}}{\omega_{\max}}=0.60606 \tag{34}\]  
and  
\[k_{1}=23.091 \tag{35}\]  
and  
\[k_{2}=3040.7 \tag{36}\]  
Thus, employing Equation (34) in Equation (7), it results to  
\[bc=1.5385 \tag{37}\]  
Using Equations (35) and (37) in Equation (4), it yields to  
\[b=2.6247\times 10^{-2}\text{ and }c=58.617 \tag{38}\]  
Using Equations (36) and (38) in Equation (9), the value of \(a\) parameter is obtained as  
\[a=2.2566\times 10^{6} \tag{39}\]  
Finally, employing Equations (38) and (39), Equation (1) becomes  
\[P_{WT}=2.2566\times 10^{6}\bigg{(}\frac{S_{w}}{\omega}-2.6247\times 10^{-2} \bigg{)}e^{-58.617\frac{5w}{\omega}}S_{w}^{3} \tag{40}\]  
Accuracy in finding the PMSG's optimal power curve can be achieved via the accuracy of the modelled wind speed. Thus, considering a parabolic shape for the approximated wind speed time variation, described as  
\[S_{w}(t)=mt^{2}+nt+6.24 \tag{41}\] \begin{table}
\begin{tabular}{c c} \hline t (s) & \(\mathbf{S_{wmeas}(t)}\) (m/s) \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Measured wind speed values over time. and using the measurements data from Table 1, the below equations system results:  
\[\left\{ \begin{array}{l}6.32=m270^{2}+n270+6.24,\text{ for }t=270\text{ [s] and }S_{wmax}=6.32\text{ [m/s]}\\ 6.14=m570^{2}+n570+6.24,\text{ for }t=570\text{ [s] and }S_{wmin}=6.14\text{ [m/s]}\end{array} \right. \tag{42}\]  
the solutions of which are  
\[m=-1.5724\times 10^{-6},\text{ }n=7.2086\times 10^{-4} \tag{43}\]  
Finally, an approximation for the wind speed time variation is obtained, as depicted below and shown in Figure 1.  
\[S_{w}(t)=-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4}t+6.24 \tag{44}\]  
The first-order derivative of the wind speed time variation from Equation (44), which is also depicted in Figure 1, can be obtained by using the below equation:  
\[\frac{dS_{w}(t)}{dt}=\frac{d\big{(}-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4} t+6.24\big{)}}{dt}=-3.1448\times 10^{-6}t+7.2086\times 10^{-4} \tag{45}\]  
The approximated wind speed reaches a maximum value for the time instant \(t^{*}\) (see Figure 1), obtained by canceling its derivative in Equation (45), \(\frac{dS_{w}(t)}{dt}=0\), whose solution is \(t^{*}=229.22\) (s). Remark 2: In Figure 1, it can be seen that the approximated wind speed time variation depicted in (44) is very close to the measured wind speed curve, \(S_{wmeas}(t)\), obtained as depicted in Table 1.  
The maximum value of the wind turbine's power can be determined from Equations (18) and (36), using the below relationship:  
\[P_{WTmax}(t)=3040.7S_{w}^{3}(t) \tag{46}\]  
which, by replacing \(S_{w}(t)\) with Equation (44), becomes  
\[P_{WTmax}(t)=3040.7\big{(}-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4}t+6.24 \big{)}^{3} \tag{47}\]  
Following Equations (35) and (36), and replacing the moment of total inertia with the value of \(J=1.15\times 10^{5}\big{(}\text{kg}/\text{m}^{2}\big{)}\), which is known from [21], the PMSG's optimal power curve, \(P_{PMSGoptim}(t)\), defined by Equation (21), for a certain wind speed time variation, \(S_{w}(t)\), can be written as below:  
\[P_{PMSGoptim}(t)=3040.7S_{w}^{3}(t)-1.15\times 10^{5}(23.091)^{2}\frac{dS_{w}(t)} {dt}S_{w}(t) \tag{48}\]  
Based on the wind speed time variation, given in Figure 1, it becomes  
\[