## 4 Results of Probabilistic Stability and Performance Assessment
### The Grid SCR Impact on System Stability and Performance  
#### 4.1.1 Critical Mode Locus in the S-Plan  
This subsection investigates the impact of grid SCR variations on the system stability and performance. In this respect, five different levels for grid SCR (10, 5, 2.5, 2, and 1.67) were considered and a proper normal PDF was defined for each of them, or equivalently for grid inductances in Figure 6a. The mean values were equal to each SCR level, and the standard deviation was equal to 6.67% of the mean value. Therefore, the sampled grid SCR and corresponding inductance had a 99.9% confidence of being within \(\pm\)20%. It is worth mentioning that a strong grid was introduced when SCR = 5 and 10, 2 < SCR < 3 represented a weak grid, and SCR < 2 corresponded to a very weak grid [11]; therefore, the following analysis could cover a wide range of probable operating conditions of a power system. Figure 6b depicts variations in the system critical mode under different grid conditions. Before explaining the obtained results, it is worth remarking that the critical mode was computed based on the proposed framework in Figure 1. In such a way, at first, the  
Figure 6: Probabilistic assessment and transient response of a grid-connected VSC. (**a**) Probability density functions (PDFs) of a normal distribution considering different levels of grid SCR; (**b**) critical mode locations in the complex s-plane due to considered uncertainties; (**c**) time-domain simulations showing inverter current under 10% voltage sag at grid voltage and different grid SCRs (\(P\) = \(P_{n}\)); numbers indicate the operation under different grid SCRs, i.e., 1, 2, and 3 represent grid SCR equal to 1.51, 1.67, and 2, respectively. Monte Carlo sampling method drew a random grid inductance value from the PDFs of grid inductances (defined in Figure 6a). The sampled grid inductance, the inverter power level, and other fixed control and power system parameters were then used to calculate the steady-state operating point in (3). This information built the stability model up in (A1). Eigenvalue analysis then estimated the critical mode. This procedure was repeated \(N_{C}\) times, and the results were plotted and are shown in Figure 6b. Additionally, the statical properties of the critical mode and corresponding PDFs and CDFs could also be assessed for this case and other inverter powers and PLL bandwidths, which the following subsections discuss. As shown in Figure 6b, under a weaker grid, the potential locations of the critical modes covered a larger area, meaning there was a higher sensitivity to grid inductance variations. Moreover, when the SCR decreased from 10 to 2.5 and then from 2.5 to 1.67, two opposite behaviors could be seen. In the first part, the system damping increased, while by further SCR reduction, the stability margin decreased. The main reasons behind this phenomenon could be explained as follows. As discussed in [40], a VSC with an inverter-side current control has a smaller stability margin under higher SCR; therefore, by reducing the SCR from 10 to 2.5, the system stability margin increased. However, a further SCR reduction resulted in higher grid coupling and control-loop interactions due to PLL and higher grid inductances, which could degrade the system stability and even cause instability. In addition to this, the AVC could also significantly impact the damping factor in higher SCRs due to its lower bandwidth among different control loops. In this situation, the worst damping factor could be approximated as \(\sigma_{\text{max}}=-\omega_{B,AVC}\). For the studied system, the AVC bandwidth changed from 5.38 rad/s (0.85 Hz) to 32.31 rad/s (5.14 Hz), according to \(\omega_{B,AVC}=k_{ia}.\omega_{1}.L_{g}\)[41]. Therefore, a higher SCR (SCR : 2.5 \(\to\) 10) resulted in a lower AVC bandwidth and, consequently, a smaller damping factor (\(\sigma_{\text{max}}=-\omega_{B,AVC}:-21\to-5.1\)), which can also be seen from Figure 6b.