# Wind Turbines Optimal Operation at Time Variable Wind Speeds
## 6 Case Study
### PID-Based Control of the PMSG's Optimal Power
\begin{array}{l}1.15\times 10^{5}\frac{dc(t)}{dt}\omega(t)+1.15\times 1 0^{5}\varepsilon(t)^{2}\\ \qquad\qquad=3040.7\Big{(}-3.84\times 10^{-28}\Big{(}3931t^{2}-1.8022\times 1 0^{6}t-1.56\times 10^{10}\Big{)}^{2}\Big{(}3931t-9.0108\times 1 0^{5}\Big{)}\Big{)}\\ \qquad\qquad-\frac{dP_{\text{PMSG\_PID}}(t)}{dt}\\ \frac{dP_{\text{PMSG\_PID}}(t)}{dt}=10\frac{d\omega(t)}{dt}-44.942t-3.8904 \times 10^{5}+2700\omega+9.8032\times 10^{-2}t^{2}+1000\Big{(}\frac{dc(t)}{dt}-1.6 645\times 10^{-2}\Big{)}\\ \qquad\omega(0)=144.09\\ \qquad\varepsilon(0)=1.6645\times 10^{-2}\\ \qquad P_{\text{PMSG\_PID}}(0)=4.6297\times 10^{5}\end{array} \right. \tag{67}\]  
the solutions of which are the \(P_{\text{PMSG\_PID}}(t)\) and \(\omega_{optim}(t)\) curves. In order to highlight the influence of the derivative constant value, \(k_{d}\), on the time variation of the PMSG's optimal power, when using a PID-type regulator, \(P_{\text{PMSG\_PID}}(t)\), several simulations, the results of which are given in Figure 4, were performed for different values of \(k_{d}\). From the PMSG's power time variation presented in Figure 4, it can be noticed that the power oscillations are not diminished when using a PID-type regulator instead of a PI-type regulator, regardless of the derivative constant value, \(k_{d}\). For \(k_{d}=1000\), the PMSG's optimal power curve, \(P_{\text{PMSG\_PID}}(t)\), obtained using a PID-based control, has the smallest oscillations around the PMSG's optimal power curve \(P_{\text{PMSG\_PID}}(t)\), which result from the kinetic motion equation. Figure 4: _Cont._  
Figure 5 shows, by comparison, the time variations of the three powers described below:  
* the maximum wind turbine's power when operating at MPP, defined in Equation (47) (\(P_{WTmax}\))
* the PMSG's optimal power calculated from the kinetic motion equation, defined in Equation (49) (\(P_{PMSG\mathit{optim}}(t)\))
* the PMSG's power obtained by using either the PI- or PID-type regulators, \(P_{PMSG\_PI\_PID}(t)\), resulting from either Equation (62) or Equation (67). Remark 5: The derivative response is proportional to the rate of change of the process variable. Increasing the derivative component will cause the control system to react more rapidly to error. Thus, the derivative term allows one to have bigger P and I coefficients and still keep the loop stable, providing a faster response and a better loop performance. On the other hand, it can cause the output to decrease if the process variable is increasingly rapidly. Most practical control systems use very small derivative times, because the derivative response is very sensitive to noise, and it can make the control system unstable. So, the PI regulator is recommended, unless the process variable is really slow (as in this paper's presented case, due to the large inertia). Figure 6 shows the time variation of the PMSG's angular speed, \(\omega(t)\), and the derivative of the wind speed, \(\frac{dS_{\omega}(t)}{dt}\). Remark 6: At the moment, \(t^{*}\) the derivative of the wind speed is canceled (see Figure 6), and the PMSG's angular speed, \(\omega\), reaches its maximum, \(\omega_{max}\). Thus, the wind speed and the PMSG's angular speed reach their maximum values in the same time instant. Furthermore, the inertial power \(P_{inertial}\), yielding to the equality between the PMSG's power and the wind turbine's power, \(P_{WT}=P_{PMSG}\). Therefore, by using the PI- or the PID-type regulators, the PMSG's power oscillations increase, and they are dependent on the values of the \(k_{p},\ k_{i}\) and \(k_{d}\) constants. For \(k_{p}=10,\ k_{i}=2700\) and \(k_{d}=1000\), the PMSG's power oscillations have the smallest values, for wind speeds varying between 6.24 (m/s) and 6.32 (m/s). As a general conclusion, it can be said that the two proposed controls bring the wind turbine within the MPP operation area, while \(P_{PMSG\_PI\_PID}(t)\) is not identical to \(P_{PMSG\_optim}(t)\), but close to it.