## 2 Proposed Framework for Probabilistic Stability and Performance Assessment
### Definitions, Stability and Performance Indicators  
In a power electronic-based power system, the unstable or lightly damped modes caused significant oscillations or even system instability; therefore, identifying these modes, as well as the most influential parameters on them, was very important. The critical mode was an unstable eigenvalue or weakest damped eigenvalue among all the eigenvalues. Placing the critical mode to the left side of the s-plane can guarantee system stability and improve system performance.  
The following complex equation can represent the eigenvalue of a system:  
\[\lambda=\sigma\pm j\omega_{d} \tag{1}\]  
The real part is the damping factor (\(\sigma\)) and shows the relative system stability. The imaginary part is called the damped frequency (\(\omega_{d}\)) or frequency of oscillations. Another stability and performance indictor is the damping ratio (\(\xi\)), which measures the system performance in terms of overshoot:  
\[\xi=\frac{-\sigma}{\sqrt{\sigma^{2}+{\omega_{d}}^{2}}} \tag{2}\]  
To have some stability margin, the damping factor of the critical mode should be less than a maximum value (\(\sigma\leq\sigma_{\text{max}}\)). This places all closed-loop eigenvalues in a sector, as shown in Figure 2a. Moreover, to limit a system's overshoot, it is desirable to keep the worst damping ratio greater than a minimum value (\(\xi\geq\xi_{\text{min}}\)), as shown in the wedge-shaped sector in Figure 2b. When both damping factor and damping ratio meet the required requirements simultaneously, the system closed-loop eigenvalues are placed in the D-shaped sector in which \(\sigma\leq\sigma_{\text{max}}\) and \(\xi\geq\xi_{\text{min}}\), as shown in Figure 2c.  
Figure 1: Proposed flow diagram for probabilistic stability and performance assessment of a grid-connected VSC.  
It is worth mentioning that the system was exponentially stable for \(\sigma_{\max}<0\) and \(\xi_{\min}>0\), and oscillating or unstable for \(\sigma_{\max}\geq 0\) and \(\xi_{\min}\leq 0\). Therefore, based on the imposed conditions on \(\sigma\) and \(\xi\), and whether the system uncertainty was considered or not, four different definitions for the nominal and robust stability and performance could be defined as follows:  
Nominal stability (NS): the closed-loop system was exponentially stable when the plant was known precisely, i.e.,  
\[\left\{\begin{array}{ll}\sigma<0,\;\xi>0\\ \text{Uncertainties}=0\end{array}\right.\rightarrow\text{Nominal Stability analysis}\]  
Robust stability (RS): the closed-loop system was exponentially stable when there were one or more uncertain parameters in the plant, i.e.,  
\[\left\{\begin{array}{ll}\sigma<0,\;\xi>0\\ \text{Uncertainties}\
eq 0\end{array}\right.\rightarrow\text{Robust Stability analysis}\]  
Nominal performance (NP): the closed-loop system met the performance specifications when the plant was known precisely, i.e.,  
\[\left\{\begin{array}{ll}\sigma_{\min}\leq\sigma\leq\sigma_{\max},\;\xi_{ \min}\leq\xi\leq\xi_{\max}&\rightarrow\text{ Nominal Performance analysis}\\ \text{Uncertainties}=0\end{array}\right.\rightarrow\text{Nominal Performance analysis}\]  
Robust Performance (RP): the closed-loop system met the performance specifications under system uncertainties, i.e.,  
\[\left\{\begin{array}{ll}\sigma_{\min}\leq\sigma\leq\sigma_{\max},\;\xi_{ \min}\leq\xi\leq\xi_{\max}&\rightarrow\text{ Robust Performance analysis}\\ \text{Uncertainties}\
eq 0\end{array}\right.\rightarrow\text{Robust Performance analysis}\]