# Wind Turbines Optimal Operation at Time Variable Wind Speeds
## 3 PMSG's Optimal Power Determination based on the Kinetic Motion Equation  
The kinetic motion equation, as depicted in [2], is as shown below  
\[J\frac{d\omega}{dt}=T_{WT}-T_{PMSG} \tag{10}\]  
where \(J\) is the moment of total inertia, \(\omega\) is the PMSG's angular speed, \(T_{WT}\) is the wind turbine's developed torque, \(T_{PMSG}\) is the PMSG's developed torque, and all of them are related to the PMSG's shaft.  
Multiplying (10) by \(\omega\), the following expression results  
\[J\frac{d\omega}{dt}\omega=P_{WT}-P_{PMSG} \tag{11}\]  
where:  
* the first term, which is dependent on the moment of total inertia, \(P_{inertial}\) is called the \"inertial power\", \[P_{inertial}=J\frac{d\omega}{dt}\omega\] (12)* the second term, \(P_{WT}\), which is dependent on the PMSG's angular speed, \(\omega\), and on the wind speed, \(S_{w}\), as depicted by Equation (1), represents the \"wind turbine's power\".
* the third term, which is dependent on the PMSG's angular speed, \(\omega\), and on the load resistance, represents the PMSG's power, \(P_{PMSG}\). Therefore, Equation (11) can also be written as \[P_{inertial}=P_{WT}-P_{PMSG}\] (13) Then, integrating Equation (11), the below relationship can be obtained as \[J\frac{\omega_{k}^{2}-\omega_{k-1}^{2}}{2}=\int_{t_{k-1}}^{t_{k}}P_{WT}dt-\int _{t_{k-1}}^{t_{k}}P_{PMSC}dt=\frac{P_{WT\_avg}}{\Delta t}-\frac{P_{PMSG\_avg}}{ \Delta t}\] (14) where \(\omega_{k}\) is the PMSG's angular speed at time instant \(k\), \(\omega_{k-1}\) is the PMSG's angular speed at time instant \(k-1\), \(P_{WT\_avg}\) is the average power delivered by the wind turbine, and \(P_{PMSG\_avg}\) is the PMSG's average power.  
Thus, from Equation (14), the energy balance of the wind turbine was obtained as  
\[E_{0}=E_{E}+E_{kinetic} \tag{15}\]  
where:  
* the first term, \(E_{0}\), represents the wind energy captured by the wind turbine during the time interval, \(\Delta t\), \[E_{0}=\int_{t_{k-1}}^{t_{k}}P_{WT}dt=\frac{P_{WT\_avg}}{\Delta t}\] (16)
* the second term, \(E_{E}\), represents the energy delivered by the PMSG to the main grid, \[E_{E}=\int_{t_{k-1}}^{t_{k}}P_{PMSG}dt=\frac{P_{PMSG\_avg}}{\Delta t}\] (17)
* the third term, \(E_{kinetic}\), represents the kinetic energy, \[E_{kinetic}=J\frac{\omega_{k}^{2}-\omega_{k-1}^{2}}{2}\] (18) where the variation of the kinetic energies, \(J\frac{\omega_{k}^{2}-\omega_{k-1}^{2}}{2}\), can be determined very precisely by measuring the PMSG's angular speed, \(\omega\).  
Remark 1: The kinetic energy is the useful energy that also includes the PMSG's efficiency, thus generating the additional electrical power from the PMSG's terminals.  
The average power delivered by the PMSG can be determined from Equation (14) by using the below relationship:  
\[P_{PMSG\_avg}=P_{WT\_avg}-\left(J\frac{\omega_{k}^{2}-\omega_{k-1}^{2}}{2} \right)/\Delta t \tag{19}\]  
Similar to Equation (19), the optimal values of the PMSG's power are obtained for the MPP operation when its angular speed, \(\omega(t)\), becomes equal to the PMSG's optimal angular speed, \(\omega_{optim}(t)\), which depends linearly on the time-varying wind speed, \(S_{w}(t)\), as described by Equation (3). Thus, \(\omega(t)=\omega_{optim}(t)\) and by employing Equation (3) in Equation (11), the result is that  
\[P_{PMSGoptim}(t)=P_{WT\max}(t)-Jk_{1}\frac{dS_{w}(t)}{dt}k_{1}S_{w}(t) \tag{20}\]which, replacing Equation (9) in Equation (20), becomes  
\[P_{PMSG\_PI}(t)=k_{2}S_{w}^{3}(t)-Jk_{1}^{2}\frac{dS_{w}(t)}{dt}S_{w}(t) \tag{21}\]  
where  
\[P_{inertial}(t)=Jk_{1}^{2}\frac{dS_{w}(t)}{dt}S_{w}(t) \tag{22}\]  
In conclusion, based on the measured time-varying wind speed values, \(S_{w}(t)\), the optimal values of the PMSG's power, \(P_{PMSG\_PI}(t)\) can be determined using Equation (21). In essence, this means that the wind turbine operates at the MPP, and thus, the PMSG's angular speed, \(\omega(t)\), tends towards the optimal value, \(\omega_{optim}(t)\).