## 6 Discussion and Future Works  
This work aimed to develop a comprehensive probabilistic framework for the robustness analysis of VSCs for power system applications. From the provided perspective, it was possible to measure system stability and performance using statistical properties of operational indices, which are of utmost importance. Moreover, based on the provided statistical information, the likelihood of a specific condition could be easily calculated, and relationships between the system stability and the risk and reliability assessment were identified. Therefore the proposed framework and defined indices could broaden the existing knowledge on robust stability and performance analysis of VSCs and provide new insights for future works. To evaluate the applicability of the proposed framework, the impact of uncertainties in the grid SCR, operating point conditions, and control-loop interactions on system stability and the performance of a grid-connected VSC were thoroughly investigated. It is worth remarking that the most crucial stability and performance indicators were the maximum damping factor and minimum damping ratio, which were evaluated for the specified level of plant uncertainty. It was found that the grid SCR was the most influential parameter on the system stability and performance among the different parameters and could seriously limit the VSC's performance. Lower Grid SCR results in higher sensitivity of the critical mode to grid inductance variations, which appear in wider PDFs or lower slopes of CDFs. CDFs could also determine the risk of system instability or the likelihood of a particular condition. In contrast, the deterministic methods were only treat as binary. Figure 14: Experimental results showing the system response under strong grid conditions (SCR = 5.3) and different power levels (\(P\) = 5 kW, 2.5 kW) and PLL bandwidths (\(f_{PLL}\) = 8 Hz, 24 Hz). Figure 13: Experimental results showing the system response under weak grid conditions (SCR = 1.63) and different power levels (\(P\) = 5 kW, 2.5 kW) and PLL bandwidths (\(f_{PLL}\) = 8 Hz, 19 Hz). It was also concluded that the stability and performance of VSCs were not affected by the inverter power level for strong grids. In contrast, the opposite observation was observed for weak grids. Increasing the inverter loading increased the stability margin at first. However, at higher levels, stability margins were again reduced due to higher coupling and interactions between different control parts. The impact of PLL bandwidth was also investigated with respect to the system stability and performance. Under very low PLL bandwidths, VSCs and power grids could be assumed decoupled due to very narrow PDFs in the outputs. However, the stability margin was also insufficient, which was not desirable. It could be expected that increasing the PLL bandwidth increases the stability margin; however, PDFs could become broader than before, which could increase coupling between the grid and the VSC and increase sensitivity to grid inductance variations. Additionally, system instability might occur for higher PLL bandwidths under weak grid conditions; therefore, low and high PLL bandwidths might reduce the stability margin and drive the system to instability. Consequently, this parameter should be carefully selected. It is worth noting that for the usual selection of control parameters, fast dynamic parts of the VSC, such as the current controller, highpass filter, control delay, etc., had a significant impact on the minimum damping ratio. Moreover, outer and slow control parts such as AVC, DVC, PLL, etc., influenced the maximum damping factor. For strong grids, AVC usually had a lower bandwidth among the different control loops, and it had a significant impact on the maximum damping factor. For weak grids, the damping factor of the critical mode was more sensitive to PLL bandwidths, and different control-loop interactions influenced it remarkably.