# Nonlinear Model Predictive Control for Doubly Fed Induction Generator with Uncertainties
## 3 Simulation Results  
The numerical values of the parameters for the DFIG system can be found in Table 1.  
To accurately model the induction generators within the wind turbine structure, it's essential to incorporate a wind model into the system as the primary input applied to the wind turbine. The wind model significantly influences the system's output characteristics and behavior, and a more realistic representation leads to a more reliable output response. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline \(R_{s}\) & 0.0 & p.u. \\ \hline \(X_{s}\) & 0.1 & p.u. \\ \hline \(R_{r}\) & 0.01 & p.u. \\ \hline \(X_{r}\) & 0.08 & p.u. \\ \hline \(X_{m}\) & 3 & p.u. \\ \hline \(H\) & 3 & p.u. \\ \hline \(T_{\epsilon}\) & 0.01 & s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical values of DFIG system parameters. In a suitable nonlinear approximation, the wind behavior can be represented by a white noise signal passed through a Kalman filter. The wind speed curve is illustrated in Figure 2.  
The initial step to assess the functional accuracy of the designed nonlinear MPC based on the Lyapunov function is to examine the curves obtained for the system outputs resulting from applying the reference inputs. After applying the input of Figure 2 to the system, its outputs, which include rotor speed (\(\omega_{m}\)), pitch angle (\(\beta\)), active power (\(P\)), and reactive power (\(Q\)), are shown in Figures 3-6. Figure 3: Stabilization of the generator rotor angular speed by applying a fixed input. Figure 2: Variable wind speed for DFIG system. In the performed simulations, the steady state values for the system outputs are considered as follows:  
\[\left[P\quad Q\quad\omega_{m}\quad\beta\right]_{ref}=\left[2\ MW\quad 0\ MVar\quad 1\ p.u \quad 2\ deg\ \right]\]  
As depicted in Figure 3, the rotor's angular speed experiences minor fluctuations around the 1 pu speed after surpassing its overshoot. Eventually, it stabilizes at the value of 1 pu. As the oscillations' amplitude is extremely small, they can be disregarded, and the rotor's angular velocity response can be considered stable. In Figure 4, it is apparent that the pitch angle curve of the generator exhibits a response closely mirroring the angular speed of the rotor. Following the overshoot point, the pitch angle curve undergoes oscillations around a consistent value. Upon the completion of the transient period, the pitch angle curve stabilizes, fluctuating around \(2^{\circ}\), with negligible amplitude in these fluctuations. Consequently, it can be asserted that this response is entirely stable. Figure 4: Stabilization of the generator pitch angle by applying a fixed input. Figure 5: Stabilization of the generator’s active power by applying a fixed input. In Figure 5, it can be observed that the reactive power response curve has reached 3 MW during the transient period and then oscillates around 2 MW with a small range of fluctuations. Similar to the previous curves, the response can be considered stable due to the minor fluctuation range. Figure 5 depicts the stabilization of the generator's active power through the application of a fixed input. As shown in Figure 6, the reactive power curve does not have an overshoot point. After reaching zero reactive power, it stabilizes within a numerical range near zero. Upon careful examination of Figures 3-6, it is evident that the system outputs for the nonlinear MPC law based on the Lyapunov function are significantly more favorable compared to the results obtained with conventional control strategies. The comparison of the obtained results indicates that the curves have exhibited significantly better performance in terms of convergence speed, amplitude, and the number of oscillations. This serves as evidence of the superiority of Lyapunov-based nonlinear MPC over conventional control methods. Additionally, it is evident that the noise in the output curves is significantly reduced.