## 5 The Seismic Vulnerability of an \(\pm\)800 kV Suspended Filter Capacitor
### The Single-Variable-Input AdaBoost Regressor and Vulnerability Curve  
To build the AdaBoost regressors, the data of mean and standard deviation values of the responses and PGA are used as the output and input data, respectively, and then the relationships can be replaced by the regressors after training. From Figure 14, we can see that the relationship between the mean value of the responses and PGA exhibits a nearly linear line. Therefore, linear model-based regressors are more reasonable for such substitutes. The AdaBoost regressor, that has been introduced in Section 4.3, requires a series of individual estimators, such as a logistic regression model, ridge regression model, support vector machines, decision tree regression model, etc. Since the nearly linear line is shown in Figure 14 and there is only one input of PGA, a logistic regression model is determined as the individual estimator in the AdaBoost regressors. Using the data of mean and standard deviation as training samples, two AdaBoost regressors were obtained, where the hyperparameters include the number of individual estimators, the penalty function, the inverse of regularization strength, the maximum number of iterations, etc. In this work, the penalty function was set to the widely used L2 penalty term, and the maximum number of iterations was set to 100. Given the tuning range of 5-30 for the number of individual estimators, and the tuning range 0.001-1 for the inverse of the regularization strength, the final number of individual estimators is 10, and the final inverse of the regularization strength is 0.015. The optimal values are obtained through multiple calculations in terms of grid search. The subsequent prediction results will illustrate the accuracy. Then, a series of discrete PGA points were given from 0 to 1.2 g with a step of 0.01 g. Based on these PGA points, the regressors can output the mean values  
Figure 14: The statistic results (the mean and standard deviation) of responses versus PGA. Figure 13: The mean and standard deviation values of responses under each parameter combination. and standard deviation values of the responses, respectively. The predicted results are shown in Figure 15, in which the data points represent the original results. After getting the regressors mapping the PGA to the mean and standard deviation of the responses, the mean and standard deviation can be directly obtained at any PGA. Then, a Gaussian distribution can be determined based on the mean \(\mu_{i}\) and standard deviation \(\sigma_{i}\), whose cumulative distribution function (CDF) under a given limit state \(MS\) is as shown in Equation (4):  
\[\text{CDF}(MS|\mu_{i},\sigma_{i})=\frac{1}{\sigma_{i}\sqrt{2\pi}}{\int_{- \infty}^{MS}e^{\frac{-{(t-\mu_{i})}^{2}}{2\sigma_{i}^{2}}}}\,dt \tag{4}\]  
where CDF represents the probability of the response prediction value less than the material strength \(MS\). Therefore, the probability that the response prediction value exceeds the material strength \(MS\) can be expressed as Equation (5):  
\[p(\mu_{i},\sigma_{i})=1-\text{CDF}(MS|\mu_{i},\sigma_{i})=1-\frac{1}{\sigma_{ i}\sqrt{2\pi}}{\int_{-\infty}^{MS}e^{\frac{-{(t-\mu_{i})}^{2}}{2\sigma_{i}^{2}}} }\,dt \tag{5}\]  
where \(p\) represents the failure probability, \(\mu_{i}\) and \(\sigma_{i}\) represent the mean and standard deviation of the responses at the \(i\)-th point, and \(MS\) represents the material strength. The failure strength of the suspension insulators is 70 MPa according to the manufacturer. Going through every PGA point, the data points about failure probability and PGA can be obtained, as shown in Figure 16. According to the seismic vulnerability theory for electrical equipment mentioned in Section 4.1, these points can be substituted by a log-normal distribution function. Therefore, a log-normal distribution curve was fitted using the scatters as seen in Figure 16.  
Figure 16: Failure probabilities predicted by single-variable-input AdaBoost regressors, as well as the fitted seismic vulnerability curve.