## 4 Simulation Results
The balanced 3-phase condition is distorted by the voltage drop from 1 to 1.2 s. Figure 18c shows the detected positive sequence voltage waveform using SOGI-FLL. Unlike that shown in Figure 18b, the magnitude of voltage is slightly reduced and it maintains a balanced 3-phase condition. The 3-phase input current waveform is illustrated in Figure 18d. Unlike the voltage in Figure 18b, the changes in the amplitudes of the phases differ, and the shape of the waveform is severely distorted. Figure 18e shows the center frequency \(\omega^{\prime}\) extracted from SOGI-FLL. Only a slight ripple is generated, and it does not significantly differ from the frequency value in the steady state. Figure 17: Waveform of voltage when harmonics is included in input side; (**a**) the 3-phase input voltage with harmonic components; and, (**b**) the 3-phase input voltage detected by the SOGI-FLL. Figure 18: \(Cont\). Figure 18: Waveforms of simulation results when \(b\) phase has a voltage drop; (**a**) the dc-link voltage; (**b**) the 3-phase input voltage with voltage unbalance; (**c**) the 3-phase input positive voltage detected by SOGI-FLL; (**d**) the 3-phase input current; and, (**e**) the center frequency extracted from SOGI-FLL. Figure 19 shows the positive and negative sequence voltages and the phase angle generated by SOGI-FLL. While only negligible negative sequence voltage occurs before the unbalance condition, the negative sequence voltage increases immediately after the unbalance condition. In addition, while the phase angle of the positive sequence voltage is stably tracked, the phase angle of the negative sequence component is rapidly changed. Figure 20 shows the waveforms when the grid frequency decreased from 60 Hz to 45 Hz. Grid frequency is reduced to 45 Hz from 1 to 1.2 s, while the power is supplied to the 12 kW load, and the grid frequency is then recovered to 60 Hz. Figure 20a shows that the dc-link output voltage waveform maintains an almost constant voltage, even when the frequency decreases. Figure 20b shows the 3-phase input voltage waveform. When the frequency is decreased from 1 to 1.2 s, the period increases. Figure 20c shows the positive sequence voltage waveform that was detected by SOGI-FLL. As shown in Figure 20b, since only negligible distortion occurs in the voltage, only the period changes without large amplitude change. Figure 20d shows the 3-phase input current waveform. It can be seen that, while the input current also only changes the period, it but does not cause large distortion. Figure 20e shows the center frequency \(\omega^{\prime}\), as extracted from SOGI-FLL. As soon as the grid frequency changes, the FLL follows the grid frequency and synchronizes the output center frequency to the grid frequency. The FLL gain value is designed considering the frequency tracking time of 0.1 s, but the steady state takes slightly more time than the designed time due to the change of the grid conditions. Figure 19: Waveforms of simulation results when voltage drop is occurred; (**a**) the positive and negative sequence voltage; and, (**b**) the phase angle of the positive and negative sequence voltage. Figure 20: Waveforms of simulation results when grid frequency decreased from 60 Hz to 45 Hz drop; (**a**) the dc-link voltage; (**b**) the 3-phase input voltage; (**c**) the 3-phase input positive voltage detected by SOGI-FLL; (**d**) the 3-phase input current; and, (**e**) the center frequency that was extracted from SOGI-FLL. In Figure 21, the conventional synchronous reference frame-phase locked loop (SRF-PLL) method [25] and SOGI-FLL method are compared to each other when the 3-phase grid voltage frequency is decreased from 60 Hz to 45 Hz. During power supply to 50 kW load, the grid frequency decreases to 45 Hz from 1 to 1.2 s, and then recovers to 60 Hz. Figure 21 shows that the \(d\)-axis voltage in a synchronous reference frame oscillates significantly when the grid frequency changes, and the dc-link voltage oscillation ranges from \(\pm\)0.5 to 0.9 V\({}_{\text{dc}}\).