## 6 Case Study Analysis
### Evolution of System Failures Considering the Coupling Effects of Power Flow and Weather  
To explore the evolutionary characteristics of the system during failures, this study provides a detailed analysis of the flow and meteorological coupling effects for various branch disconnections. Observing the changes in load rates and coupling coefficients following branch disconnections allows for a better understanding of flow distribution characteristics under fault conditions and the interactions between branches. The following analysis details the system's response to disconnections of various branches, revealing key features in the fault evolution process. The high-risk fault branches under typhoon conditions are disconnected in sequence: branch 14 at period 3, branch 15 at period 4, and branch 8 at period 5, leading to the changes in load rates and coupling coefficients after disconnection as illustrated in Figure 9. Observing the changes in load rate indicators and coupling coefficients of other branches facilitates the preliminary assessment of the fault chain's evolution trend. After disconnecting branch 8, the load rate of branch 10 reached 0.6145, the highest among all branches, indicating that this disconnection significantly impacted branch 10. Additionally, the coupling coefficient of branch 18 reached 2.9550, indicating a significant impact on its flow. Following the disconnection of branch 14, branch 15 exhibited the highest load rate of 0.6532, while branch 2 had the largest coupling coefficient at 0.2117, indicating significant impacts on these branches. Finally, after the disconnection of branch 15, the load rate of branch 10 again reached a high value of 0.6180, while the coupling coefficient of branch 18 also increased to 2.8120. These data clearly illustrate that at each  
Figure 8: Branches’ fault probability of the IEEE 14 system. stage of fault development, certain branches experienced greater flow changes, highlighting the evolution of system faults. Figure 9: Chart of branch load rates and coupling coefficients. Figures 5 and 9 indicate that disconnecting branch 14 creates a small island in the system, resulting in the failure of one generator and a significant shift in system flow. In contrast to the disconnections of branches 8 and 15, the system's load rate and coupling coefficient indicators exhibited significant changes. Thus, in light of the impacts of extreme weather events on the power grid, particular attention should be given to critical branches that are susceptible to island effects upon disconnection, and these should be reinforced to mitigate failure risks and enhance system stability. As the fault evolves, the fault probabilities of the lines caused by extreme weather and power flow redistribution are shown in Figure 10. The color bar and numbers on the right side of the figure 10 indicate the total fault probability associated with the fault evolution stages caused by extreme weather and cascading failures. The red 'B' followed by a number represents the branches in the system. Beginning at the 6 th period, meteorological factors led to a gradual increase in fault probabilities for several branches. By the 12 th period, the affected area within the system had significantly expanded, particularly for branches 9, 12, and 14, which exhibited fault probabilities exceeding 0.7. This indicates an intensifying impact of the typhoon on critical regions of the power grid, resulting in substantial flow transfers along these branches. During the 18 th and 24 th periods, the distribution of faults within the system intensified further, highlighting the increasing exposure of the power grid's vulnerabilities due to the typhoon's continued impact. To tackle the problem of islands formed during typhoon passage, the maximum island handling method was implemented in the simulation.