## 6 Model Validation
### Test Case 1: A 3 kW DFIG-Based Wind Turbine Connected to the Electric Grid  
For the harmonic analysis of the DFIG, the results obtained are directly usable as initial values in a steady-state model of the machine. A wind speed of 10 m/s is considered input data. For this case, the induction generator is excited with a sinusoidal three-phase balanced voltage of 436 V at 60 Hz in the stator winding from the electrical grid and a non-sinusoidal three-phase balanced voltage of 22.5 V at 45 Hz applied to the rotor winding generated by the B2B power converter, which produced the harmonic components of order: 5 th,7 th, 11 th, 13 th, 17 th, 19 th, 23 th, 25 th, and 29 th, known as the characteristic harmonics of the B2B power converter. The magnitude and angle of the voltage harmonic components are presented in Table 4. The waveform of the rotor voltage is shown in Figure 10; furthermore, current waveforms from measurements and simulations are displayed in Figures 11 and 12, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multicolumn{4}{c}{**Wind Speed**} \\ \cline{2-5}  & 5 **m/s** & 7 **m/s** & 10 **m/s** & 14 **m/s** \\ \hline \(P\) & 2070 W & 2250 W & 3450 W & 4140 W \\ \hline \(Q\) & 689 Vars & 674 Vars & 1149 Var & 1378 Var \\ \hline \(P_{s}\) & 2484 W & 2577 W & 2906 W & 3116 W \\ \hline \(P_{r}\) & \(-\)414 W & \(-\)328 W & 564 W & 1048 W \\ \hline \(U_{s}\) & 427 V & 432 V & 436 V & 438 V \\ \hline \(\delta_{s}\) & 88.7\({}^{\circ}\) & 126.4\({}^{\circ}\) & 36.6\({}^{\circ}\) & 274.1\({}^{\circ}\) \\ \hline \(U_{r}\) & 212.3 V & 87.1 V & 22.5 V & 146.6 V \\ \hline \(\delta_{r}\) & \(-\)63.4\({}^{\circ}\) & 25.1\({}^{\circ}\) & \(-\)123.8 & 40.9\({}^{\circ}\) \\ \hline \(I_{s}\) & 1.45 A & 3.833 A & 5.45 A & 10.05 A \\ \hline \(\varphi_{s}\) & 27.6\({}^{\circ}\) & \(-\)96.1\({}^{\circ}\) & 42.2\({}^{\circ}\) & \(-\)103.2\({}^{\circ}\) \\ \hline \(I_{r}\) & 4.35 A & 5.74 A & 9.46 A & 15.07 A \\ \hline \(\varphi_{r}\) & \(-\)2.21\({}^{\circ}\) & \(-\)155\({}^{\circ}\) & 49.2\({}^{\circ}\) & 127.4\({}^{\circ}\) \\ \hline \(s\) & \(-\)0.188 & \(-\)0.06 & \(-\)0.756 & \(-\)0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Steady-state operating condition of the DFIG with P.F. = 1. The rotor current harmonic caused by the switching of the rotor-side converter of the B2B power converter appears at frequencies of \(f_{rh}~{}=~{}(6h\pm 1)sf_{es},~{}h~{}=~{}0,1,2,3,\)... i.e., at the 5 th, 7 th, 11 th, 13 th\(\ldots\) of the fundamental frequency in the currents of the induction generator as summarized in Table 5, where the current waveforms presented in this study are attained by applying the solution of Equations (38) and (39) in the previous section. Table 5 also shows the current THD index of the current waveforms, both stator and rotor are presented in Figures 11 and 12. The rotor harmonic currents establish rotating magnetic fields in the airgap, inducing currents of corresponding frequencies to the stator winding. The DC current distortion introduced by the grid-side converter is also reflected in the rotor currents, which causes the appearance of additional harmonics. The full harmonic content of the \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Harmonic Components**} \\ \cline{2-9}  & **5 th** & **7 th** & **11 th** & **13 th** & 17 th & **19 th** & **23 rd** & **25 th** \\ \hline Voltage & 4.3 & 3.01 & 1.98 & 1.63 & 1.03 & 0.88 & 0.73 & 0.67 \\ \hline Degrees & \(-\)84.3 & 63 & 161.1 & 100.7 & \(-\)7.9 & 26.3 & 40.1 & 251.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Harmonic components of the rotor voltage for test case 1. Figure 11: Stator current. Figure 12: Rotor current. Figure 10: Rotor voltage. rotor current is then given by \(f_{th}\ =\ \left|(6h\pm 1)s\pm 6m\right|f_{cs},\ h,m\ =\ 0,1,2,3,\ldots\) setting \(m=0\) yields the \"conventional\" harmonic frequencies caused by the rotor-side converter.