## 4 Case Study
### Vulnerable Node Identification  
In order to prove the validity of the proposed method for vulnerable node identification and obtain the system state when one node operation state changes, the three-phase fault was set to trigger a sudden operation state change of each node each time, and obtain the voltage fluctuation of the power grid. In each simulation, we just set the fault on one node for one time. Therefore, the test in PSCAD was simulated sixty-nine times, since there are sixty-nine nodes in a PG and E 69-node test system. In Appendix A, we also did the same test thirty-nine times, since there are thirty-nine nodes in an IEEE 39-node test system. Regarding the node which impacts the system operation state most as the vulnerable node, the vulnerability index of each node was added into the objective function Equation (9) to ensure the vulnerable nodes will be protected as much as possible when branch 2-3  
\begin{table}
\begin{tabular}{l c c c} \hline
**DG** & **Node** & **DG Type** & **Max Power/kW** & **Effective Power/kW** \\ \hline DG1 & 36 & Uncontrollable & 50 & 47.5 \\ DG2 & 5 & Controllable & 200 & 200 \\ DG3 & 19 & Uncontrollable & 380 & 361 \\ DG4 & 52 & Controllable & 1700 & 1700 \\ DG5 & 32 & Uncontrollable & 40 & 38 \\ DG6 & 65 & Controllable & 100 & 100 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of DGs.  
\begin{table}
\begin{tabular}{c c} \hline
**Controllable Type** & **Load Node No.** \\ \hline Completely controllable & 13, 26, 27, 34, 35, 39–41, 43, 44, 48, 53–58, 66–69 \\
40\% Controllable & 11, 21, 38 \\ Uncontrollable & Remaining load nodes \\ \hline \end{tabular}
\end{table}
Table 3: Types of load controllability.  
Figure 3: Structure of a PG and E 69-node system.  
is broken. A PG and E 69-node system, as shown in Figure 4, was taken as the experimental subject to build a simulation model in PSCAD, and the experiments of the three-phase ground fault were conducted for each node. Then, the voltage fluctuation levels of the nodes in the case of a fault are obtained, as shown in Figure 4.  
From Figure 4, it can be seen that the voltage fluctuation level caused by three-phase node fault at different nodes varied, and there was a major difference in the influence of neighboring nodes upon transient operation stability of the power grid. Therefore, to verify the correctness of the proposed vulnerable node identification method, the vulnerability index and voltage fluctuation amplitude were correlated, hence obtaining the results shown in Figure 5.  
Figure 5 is the scatter diagram of the correlation between the node vulnerability index and the voltage fluctuation amplitude caused by the node three-phase grounding fault, which was obtained using the proposed node vulnerability identification method. From Figure 5, it can be seen that the obtained node vulnerability index was basically positively correlated with the voltage fluctuation amplitude. Meanwhile, due to the differences in voltage fluctuation diffusion and maximal voltage fluctuation amplitude caused by node fault, although the average value of voltage fluctuation amplitude was close, there was still a difference in node vulnerability. For example, the vulnerability of the most vulnerable nodes 8 and 9 were, respectively, 0.3757 and 0.387, and the average voltage fluctuation in the case of a fault were, respectively, 0.0670 p.u. and 0.0676 p.u., with a maximal voltage fluctuation of 0.2554 p.u. and 0.2564 p.u. Hence, according to the assessment, node 9 was more vulnerable than node 8.  
Appendix A is the simulation calculation of node vulnerability in an IEEE 39-node distribution system, hence, further verifying the validity of the proposed node vulnerability identification method.  
Figure 4: Voltage fluctuation level of nodes in the distribution system during a fault.  
Figure 5: Correlation scatter diagram between voltage fluctuation and node vulnerability.