# Assessing Voltage Stability in Distribution Networks:
## 5 Case Analysis
### Probabilistic Voltage Stability Assessment under Different Scenarios  
The method proposed by this paper was employed for carrying out probability assessment of LMI under different scenarios, and the PDF and change trend of the indexes were obtained, as shown in Figure 9. Since the input variables contained non-normal variables, the probability density curves of the stability margin exhibited non-normal characteristics. Additionally, as the capacity of photovoltaic integration increases, the probability curves of the margin index generally shift to the right, and their values increase. It is evident that, within a reasonable range of photovoltaic grid-connected capacities, the larger the capacity, the more significant its role in enhancing the system's operational stability. Furthermore, with relatively lower connection capacities, the dispersed integration of PV generators into the system displayed a broader load margin. The wider distribution indicates that, under various operating conditions, the network exhibited higher variability but typically maintained higher stability, and the power grid had a greater ability to adapt to load changes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Scenario**} & \multicolumn{2}{c}{**3 PEM**} & \multicolumn{2}{c}{**3 PEM**} \\ \cline{2-5}  & \multicolumn{2}{c}{**Considering Correlations**} & \multicolumn{2}{c}{**Without Considering Correlations**} \\ \cline{2-5}  & **Relative Error of the Expected Value\%** & **Relative Error of the Deviation\%** & **Relative Error of the Deviation\%** & **Relative Error of the Standard Deviation\%** \\ \hline Scenario 1 & 0.09 & 0.76 & 0.32 & 3.12 \\ \hline Scenario 2 & 0.136 & 1.09 & 0.63 & 4.473 \\ \hline Scenario 3 & 0.153 & 1.18 & 0.78 & 4.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative errors in the expected value and standard deviation of \(I_{\text{LM}}\) derived from two kinds of point-estimation methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**MCS (5000)**} & \multicolumn{2}{c}{**3 PEM**} & \multicolumn{2}{c}{**3 PEM**} \\  & & **Considering Correlations** & **Without Considering Correlations** \\ \hline
**Time (s)** & 7086.5 & 56.62 & 50.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of simulation time among three methods. Figure 9: Probability density curves of \(I_{\text{LM}}\) under different scenarios. In the scenarios with distributed generators integrated into the distribution network, the LMI also displayed a heavier tail distribution, which means that the grid had a higher probability of maintaining a high load-margin under extreme conditions. This was due to multi-point power generation, which could be supplemented by other nodes when some nodes were affected by load or failure, thereby enhancing the overall resilience of the grid and making it more stable in the face of a wide range of operating conditions and potentially high-load events. In contrast, scenarios with centralized access to the distribution network showed a lighter tail, which meant that, under extreme load conditions, the network had difficulty maintaining a high load-margin, increasing the risk of grid collapse in the face of high loads or failures. Particularly in Scenario 3, the higher overall capacity and distributed configuration significantly enhanced the load margin, enabling the grid to better withstand various load fluctuations, and thus enhancing overall stability. In the aforementioned distribution network system scenarios, a warning value for the \(I_{\text{LM}}\) was set at 0.8. Therefore, the severity function of the LMI and the PDF of \(I_{\text{LM}}\) were used to calculate the static voltage instability risk, and the outcomes are presented in Table 3.  
Analysis of the data shows that the instability risk calculated by the 3 PEM considering correlations is essentially consistent with the MCS (5000) method.