# Assessing Voltage Stability in Distribution Networks:
## 5 Case Analysis
### Method Verification  
To validate the calculation effect of the method presented in various scenarios of PV integration into the distribution network, the following three scenarios were defined:  
* Scenario 1: PV generators are centrally connected to node 33, and a total connection capacity of 1.5 MW;
* Scenario 2: PV generators are dispersedly connected to nodes 33, 30, 25, 18, and 12, with each node having a rated output of 0.2 MW and a total network connection capacity of 1 MW;
* Scenario 3: PV generators are dispersedly connected to nodes 33, 30, 25, 18, and 12, with each node having a rated output of 0.3 MW, and a total network connection capacity of 1.5 MW. A considerable amount of Monte Carlo simulations were carried out, and it was found that the results tended to be stable after 5000 sampling instances. With the aim of validating the precision and efficiency of the method, results from the MCS using 5000 samples were used as a comparison. The proposed method, the traditional 3 PEM method, and the MCS method, were then used to perform probabilistic analyses of the distribution network system under the three scenarios described. To calculate the voltage stability critical points, it was assumed that the entire system load grew at a constant power factor. Figures 6-8 present the voltage stability critical values at various nodes, and the cumulative distribution functions the system LMI for the three scenarios using different methods. In these figures, Case 1 employs the MCS method, taking into account variable correlations (5000 samples); Case 2 uses the method proposed by this paper, considering variable correlations; Case 3 uses the traditional 3 PEM, without considering variable correlations. The simulations demonstrate that the presented method can precisely calculate the voltage stability critical values at each node and the load margin of the entire network when the loads at all nodes uniformly increase according to a constant power factor. From (a) in Figures 6-8, it is evident that, as distributed photovoltaics are integrated into the distribution network, the overall voltage at network nodes increases, and the increase is more pronounced the closer to the connection node they draw. This integration improves the SVS in the distribution network. However, centralized single-point connections can lead to voltage exceedances as capacity increases, posing a threat to the stable operation of the distribution network. Dispersed multi-point connections more effectively limit voltage fluctuations, maintaining the reliable and stable operation of the system. Comparing the three methods in each scenario, the curve for Case 2 essentially coincides with that of Case 1, while the curve for Case 3 shows a slight deviation. Moreover, as the  
Figure 5: \(\lambda\)-\(V\) curve of Node 16. To further verify the computational accuracy of the method presented, the relative error indicators \(\varepsilon_{\mu}\) for the expected value, and \(\varepsilon_{\sigma}\) for the standard deviation of the \(I_{\text{LM}}\) index, as shown in Equations (31) and (32) [37], are used as accuracy-assessment indicators for the method proposed in this paper. \[\varepsilon_{\mu}=\left|\frac{\mu_{\text{MCS}}-\mu_{\text{PEM}}}{\mu_{\text{ MCS}}}\right|\times 100\% \tag{31}\]  
\[\varepsilon_{\sigma}=\left|\frac{\sigma_{\text{MCS}}-\sigma_{\text{PEM}}}{ \sigma_{\text{MCS}}}\right|\times 100\% \tag{32}\]  
where \(\mu_{\text{MCS}}\) and \(\mu_{\text{PEM}}\) are the expected values of the \(I_{\text{LM}}\) index obtained by the MCS and 3 PEM methods, respectively. \(\sigma_{\text{MCS}}\) and \(\sigma_{\text{PEM}}\) are the standard deviations of the \(I_{\text{LM}}\) index as calculated by the MCS and 3 PEM methods, respectively. From the statistical data in Table 1, it is observed that the relative error in the expected values of the algorithm proposed in this paper do not exceed 0.2%, and the relative error in the standard deviation is less than 1.18%.