# Impedance-Based Stability Analysis of Grid-Connected Inverters under the Unbalanced Grid Condition
## 3 The Simplified Stability Analysis Method Based on the GNC and Verification
### Simulation Verification of the Stability Analysis Results  
To illustrate the importance of considering three-phase unbalance and the frequency-coupling effect for stability analysis of the grid-connected inverter system, the Nyquist curves of the grid-connected inverter system with and without consideration of three-phase unbalance and the frequency-coupling effect are drawn in Figure 7. The main system parameters are listed in Table 2. To simulate the three-phase unbalanced grid condition, 2.13 mH extra inductance is added to the \(a\)-phase grid impedance, which makes the \(a\)-phase grid inductance 7.13 mH.  
It can be concluded that with the parameters in Table 2, the Nyquist curves without considering the three-phase unbalance or frequency-coupling effect do not encircle the \((-1,\,j0)\) point. This indicates that the perturbation cannot trigger any oscillations, and the grid-connected inverter system is stable. But the Nyquist curves considering negative sequence impedance and the frequency-coupling effect encircle the \((-1,\,j0)\) point. This means that the perturbation will trigger system oscillations. Furthermore, the predicted oscillation frequencies should be 42 Hz and 58 Hz, which correspond to the intersection frequencies of the Nyquist curves and the unit circle, respectively.  
Different stability analysis results are given by the Nyquist curves with and without considering the three-phase unbalance and frequency-coupling effect. To verify the above stability analysis results, the time-domain simulation results are shown in Figure 8. At 1.0 s, the \(a\)-phase grid inductance is changed from 5 mH to 7.13 mH to simulate the perturbation and unbalanced grid impedance. Figure 8a shows that the inverter can work stably before 1.0 s. Figure 8b illustrates the current waveforms from 2.0 s to 2.5 s, which show when the grid inductance changes suddenly and, with the parameters listed in Table 2, the current starts to oscillate. According to the FFT analysis results of the oscillated current at the PCC shown in Figure 8c, there are several components at different frequencies, with positive sequence components at 42 Hz and 58 Hz and negative sequence components at 42 Hz and 58 Hz.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Grid voltage & 380 V & Grid frequency & 50 Hz \\ DC-link voltage & 800 V & Switching frequency & 10 kHz \\ Filter inductance & 1.5 mH & Filter resistance & 0.05 \(\Omega\) \\ Damping coefficient & 1.414 & Grid inductance & 5 mH \\ Proportional gain of PLL controller & 0.278 & Integral gain of PLL controller & 11.99 \\ Proportional gain of current controller & 1.2 & Integral gain of current controller & 72.5 \\ \(d\)-axis current reference & 50 A & \(q\)-axis current reference & 0 A \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters and values of the DSOGI-PLL-based grid-connected inverter system for stability analysis.  
Figure 7: Comparison of different Nyquist curves of the DSOGI-PLL-based inverter.  
The time-domain simulation results are consistent with the stability analysis results predicted by the Nyquist curves based on the impedance model considering the three-phase unbalance and frequency-coupling effect. Hence, it can be concluded that the Nyquist curves based on the derived frequency-coupled impedance model in this paper can not only accurately predict the system instability but also the system oscillation frequency. However, the stability analysis given by the Nyquist curves based on the impedance model without considering the three-phase unbalance or frequency-coupling effect cannot predict the instability or oscillation frequency of the system.