# 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
### Experimental Verification of the Stability Analysis Results  
In this section, the hardware experimental setup of the grid-connected inverter system is built to further verify the theoretical stability analysis results based on the proposed frequency-coupled impedance model. The AC side is directly connected to the grid through a transformer, and the DC source IT6516C is used to maintain the DC-side voltage. The power hardware module is YXPHM-TP210b-I, which contains the digital signal processor (DSP), IGBTs, and the signal processing board. The digital signal processor chip TMS320F28335 is adopted to control the whole grid-connected system. The experimental setup is illustrated in Figure 9, and the parameters of the experimental setup are listed in Table 3. And to simulate the unbalanced grid condition, an extra 3.0 mH inductance is added to the _a_-phase grid impedance. Therefore, the grid impedance is unbalanced throughout the hardware experiment. Figure 8: Current waveform at the PCC with different _a_-phase grid inductances: **(a)** Waveform from 0.5 to 1.0 s with 5 mH _a_-phase grid inductance **(b)** Waveform from 2.0 to 2.5 s with 7.13 mH _a_-phase grid inductance. **(c)** FFT analysis results of the current from 2.0 to 3.0 s.  
Figure 9: The configuration of the experimental setup. Based on the proposed stability analysis method in Section 3, we can draw the Nyquist diagram of the grid-connected inverter system with the experimental parameters as shown in Figure 10. In Figure 10, the Nyquist curves of the inverter system with different current loop bandwidths (100 Hz and 500 Hz) are illustrated. When the current loop bandwidth is reduced from 500 Hz to 100 Hz, the grid-connected inverter system becomes unstable. According to the intersection of the Nyquist curves and the unit circle, the predicted oscillation frequencies are 63.5 Hz and 36.5 Hz. Based on the hardware experimental setup shown in Figure 9 and the experiment parameters listed in Table 3, the hardware experiments of the grid-connected inverter system are carried out to verify the instability phenomenon as the reduction in the current loop bandwidth as well as the corresponding theoretical analysis results. Figures 11 and 12 show the three-phase waveforms and FFT analysis results of the grid-connected current waveforms at the PCC with current loop bandwidths of 500 Hz and 100 Hz, respectively. The conversion ratio of the current probe to measure the grid-connected current is 1 V/10 A. It can be observed from Figure 11 that the grid-connected inverter system can operate stably when the current loop bandwidth is 500 Hz. Due to the unbalanced grid impedance, an obvious negative sequence harmonic component at 50 Hz exists. It is obvious from Figure 12 that the current waveforms begin to oscillate when the current loop bandwidth is reduced to 100 Hz. It can be seen from the FFT analysis results that, in addition to the fundamental frequency component, positive sequence components  
Figure 10: Nyquist curves of the inverter system with different current loop bandwidths for the experimental parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Grid voltage & 108 V & Grid frequency & 50 Hz \\ DC-link voltage & 300 V & Switching frequency & 10 kHz \\ Filter inductance & 2 mH & Filter resistance & 0.035 \\ Damping coefficient & 1.414 & Grid inductance & 6 mH \\ Proportional gain of PLL controller & 1.1246 & Integral gain of PLL controller & 112.58 \\ Proportional gain of current controller & 4.317 & Integral gain of current controller & 4660 \\ \(d\)-axis current reference & 5 A & \(q\)-axis current reference & 0 A \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters and values of the DSOGI-PLL-based grid-connected inverter system for experimental verification. at 34 Hz and 66 Hz and negative sequence components at 34 Hz, 50 Hz, and 66 Hz exist.