# Wind Turbines Optimal Operation at Time Variable Wind Speeds
## 6 Case Study
### PMSG's Optimal Power Determination
\begin{array}{ll}P_{PMSGoptim}(t)=&3040.7\big{(}-1.5724\times 10^{-6}t^{2}+7.2086 \times 10^{-4}t+6.24\big{)}^{3}\\ &-1.15\times 10^{5}(23.091)^{2}\big{(}7.2086\times 10^{-4}-3.1448\times 10^{-6}t \big{)}\\ &\times\big{(}-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4}t+6.24\big{)} \end{array}  \tag{49}\]  
Figure 2 below presents the time variations of the maximum wind turbine's power and the PMSG's optimal power. Integrating the PMSG's optimal power described by Equation (49), for a time interval from 0 (s) to 570 (s), the value of the energy delivered by the PMSG to the main grid, \(E_{E}\), is the below:  
\[E_{E}(570)=\int_{0}^{570}\big{(}P_{PMSGoptim}(t)\big{)}dt=4.663\times 10^{8} \tag{50}\]The maximum value of the captured wind energy,\(E_{0}\), during the time interval between 0 (s) and 570 (s), is obtained by integrating the wind turbine's maximum power into (47):  
\[E_{0}(570)=\int_{0}^{570} \left(3040.7\Big{(}-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4}t+6. 24\Big{)}^{3}\right)dt \tag{51}\] \[=4.2835\times 10^{8}\]  
Remark 3: The value of the captured wind energy is less than the value of the delivered energy, the difference between the two values being found in the kinetic energy:  
\[E_{kinetic}=J\frac{\omega\left(570\right)^{2}-\omega\left(0\right)^{2}}{2}=1. 15\times 10^{5}\frac{141.78^{2}-144.09^{2}}{2}=-3.7971\times 10^{7} \tag{52}\]  
Then, using Equations (50) and (52), Equation (15) becomes  
\[E_{0}=E_{E}+E_{kinetic}=4.663\times 10^{8}-3.7971\times 10^{7}=4.2833\times 10^{8} \tag{53}\]  
which is equal to \(E_{0}(570)=4.2835\times 10^{8}\) (\(J\)) from (51). Figure 1: Time variations of the approximated wind speed, \(S_{w}(t)\), of its derivative, \(\frac{dS_{w}(t)}{dt}\), and of the measured the wind speed, \(S_{wmass}(t)\). Figure 2: Powers variations in time: \(P_{PMSGoptim}(t)\) and \(P_{WTmax}\). Thus, the energy balance in Equation (15) is checked. Remark 4: For high values of the moment of total inertia, \(J\), and of the wind speed derivative, \(\frac{dS_{w}(t)}{dt}\), the value of the inertial power, \(P_{initial}\), is comparable with the wind turbine's power values, \(P_{WIT}\), and the PMSG's optimal power values, \(P_{PMSGoptim}\). For example, for the time instant, \(t=570\) (s) the wind speed derivative in Equation (50) has the below value:  
\[\frac{dS_{w}}{dt}(570)=-3.1448\cdot 10^{-6}570+7.2086\cdot 10^{-4}=-1.071\cdot 10^{ -3} \tag{54}\]  
while the power's values are:  
* the inertial power from Equation (22), \[P_{inertial}=-4.0347\times 10^{5}\] (55)
* the optimal PMSG's power from Equation (49), \[P_{PMSGoptim}=11.073\times 10^{5}\] (56)
* the maximum wind turbine's power from Equation (47), \[E_{0}(570)=\int_{0}^{570} \left(3040.7\Big{(}-1.5724\times 10^{-6}t^{2}+7.2086\times 10^{-4}t+6. 24\Big{)}^{3}\right)dt \tag{51}\]0 (57)  
It can be observed that the value of the inertial power is negative, while the wind speed derivative is negative and the PMSG's angular speed decreases. Thus, in conclusion, the moment of total inertia, \(J\), has a huge impact on the wind turbine's operation mode. It also has to be noted that, within the time interval between 0 (s) and 230 (s), the wind speed rises and the PMSG's optimal power increases, due to the augmented value of the maximum wind turbine's power, which depends on the wind speed cube. The wind turbine's power is the sum of two powers: the PMSG's optimal power and the inertial power, as can be deduced from Equation (17). This is greater than the PMSG' power, because the value of the inertial power is positive, while the PMSG's angular speed increases. During this time period, the captured wind energy, \(E_{0}\), is included in the delivered energy, \(E_{E}\), and in the kinetic energy, \(E_{kinetic}\). For the time instant \(t^{*}=230\) (s), the PMSG's output power value is equal to the wind turbine's power value, the PMSG's angular speed value being at its maximum.