## 4 Results and Discussions  
Using the previously presented dynamic model, a numerical response is targeted for a 100 kW wind system with parameter values provided in Table 3.  
The wind turbine has a nominal power of 100 kW, generated by a wind rotor with a diameter of 22.5 m at a nominal wind speed of 10.5 m/s. The start-up of the wind  
Figure 3: The mechanical characteristic of a wind rotor modeled by four linearized zones. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(z_{i}\) & \(i_{0}\) & \(i_{a}\) & \(a\) [kNms] & \(\eta\) & \(J\) [kgm\({}^{2}\)] \\ \hline \(z_{1}=19\) & & & & \(a_{R1}=-0.328\), \(b_{R1}=1.866\) & \\ \(z_{2}=135\) & \(i_{01}=-14.8052\) & \(i_{a1}=15.8052\) & \(a_{R2}=-11.536\), \(b_{R2}=-20.533\) & \(\eta_{g}=0.9560\) & \(J_{1}=0.1\cdot 10^{3}\) \\ \(z_{3}=58\) & \(i_{02}=2.0837\) & \(i_{a2}=-1.0837\) & \(a_{R3}=2.298\), \(b_{R3}=34.014\) & \(\eta_{1}=0.9195\) & \(J_{2}=1\cdot 10^{3}\) \\ \(z_{4}=20\) & \(i_{aG}=16.8889\) & \(i_{aG}=16.8889\) & \(a_{R4}=20.259\), \(b_{R4}=121.065\) & \(\eta_{2}=0.8468\) & \(J_{H}=200\cdot 10^{3}\) \\ \(z_{5}=97\) & & & \(a_{G}=0.368\), \(b_{G}=27.975\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constant intrinsic parameters of the wind system. system is performed from rest, and the electric generator load is applied at the time when \(\omega_{G}=b_{G}/a_{G}=76\) s\({}^{-1}\). Thus, the wind system goes through three phases:  
1. In the first phase, the mechanical energy generated by the wind rotor is used exclusively to overcome inertial resistance (implicitly, to accelerate the system);
2. In the second phase, when the generator is coupled to the grid, the generator-resistant torque is added to the inertial load;
3. In the third phase, the wind turbine enters into a steady state (i.e., zero accelerations), obtaining the operating point described by the values of the angular velocities and torques, as well as the powers of all the shafts of the wind system, Table 4.  
The results of the numerical simulation in MATLAB-Simulink R2014a, based on the equation of motion (9) and all the other equations of the analytical model, are depicted in Figures 4-6. The diagrams in these figures highlight the two moments of time delimiting the three phases of the WT transition from rest to steady-state: the electric generator enters the load at \(\approx\)260 s, and the stabilization of the system takes place at \(\approx\)280 s (i.e., the starting time). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shaft** & **Torque** & **Angular Speed** & **Power** \\  & **[kNm]** & **[s\({}^{-1}\)]** & **[kW]** \\ \hline R \(\equiv\) H & 23.208 & 4.704 & 109.17 \\ H1 & 21.600 & 4.704 & 101.61 \\ H2 & 1.608 & 4.704 & 7.56 \\ \(1\equiv\) GR & \(-1.256\) & 74.317 & \(-93.34\) \\ \(2\equiv\) GS & 1.256 & \(-5.097\) & \(-6.40\) \\ G & \(-1.256\) & 79.414 & \(-99.75\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Operating point of the wind system in steady-state. Figure 4: The wind system dynamic response: (**a**) input vs. output angular speeds; (**b**) input vs. output angular accelerations; (**c**) wind rotor vs. generator torques; (**d**) wind rotor vs. generator powers. Figure 4 shows comparatively the time variations of the kinematic parameters, torques, and powers from the system input vs. output. Worth noting the quasi-linear variation of all input parameters and output motions in the first part of the start-up phase 1, characterized by \(T_{G}\) = 0 (i.e., the generator runs idle) and wind rotor operation on the zone I (see Figure 3). Since the torque generated by the wind rotor has low values at start-up (\(T_{R}\approx\) 2 kNm, Figure 4c), the system requires a long time (\(\approx\)220 s) of slow increase in speed (Figure 4a) and implicitly in angular acceleration (Figure 4b) as a result of inertial resistance. Once the wind rotor enters zone II (time \(t_{1}\approx\) 220 s), the system is rapidly accelerated as a result of the greater torque extracted by the wind rotor from the wind.