## 3 Case-Study Methodology and Development
### Methodology and Analysis  
The data were taken from the past six-year history of the transformer-category failure data from the case-study region of the case-study organization. The data were in manual format, and their digitization and digitalization were carried out for the analysis. Initially, the DT-category failure data were recorded in an Excel file. Then, using the Python program, the data from Excel were extracted and plotted in Weibull CDFs and in survival functions, using the reliability library of the Python program. The Python program calculated the  
Figure 14: Weibull survival functions for 33 kV/0.4 kV-400 kVA, 630 kVA DT failures. Figure 13: Weibull survival functions for 33 kV/0.4 kV-160 kVA, 250 kVA DT failures. Figure 15: Weibull survival functions for 33 kV/0.4 kV-800 kVA, 1000 kVA DT failures. alpha and beta values for the Weibull-plot failure data. In addition, the goodness-of-fit values were generated by the same Python program. Figures 4-9 above represent the Weibull CDFs of six years of failure data for 12 different categories of DTs of the selected region of the case-study organization. The data were fitted with complete data and not with incomplete (right censored) data. In the above Weibull CDF plots, the horizontal scale provides measures of failure years for each category of DT. The vertical scale provides the cumulative percentage of the failed category of DTs. The shape factor (beta) provides an indication of the physics of the failure, and the scale (alpha) factor estimates the lifespan of DTs' characteristics. It can be either infant mortality, early wear-out, random failures, or rapid wear-out. In addition, in the above Weibull CDF plots, the dots represent the data points that were collected as failure data for each category of DT. The line represents the theoretical CDF that was fitted to the data points based on the plotted data. The shaded area represents the uncertainty or confidence interval of the fitted Weibull distribution. It shows the range of possible values for the CDF given the uncertainty in the data and the estimation of the parameters. The thickness of the shaded area varies, depending on the amount of DT-failure data that were collected and the degree of uncertainty in the data. Figures 4-9 above also show that, when a large amount of failure data were recorded, the uncertainty in the data were smaller and the shaded area ended up thinner in the Weibull CDFs. This is because, with more data points, the fitted Weibull distribution was able to better approximate the true underlying distribution of the DT-failure data, resulting in a more precise estimation of the parameters of the Weibull distribution. On the other hand, a small amount of DT-failure data were observed to be highly variable, and the uncertainty in the data were larger, resulting in a thicker shaded area. Figure 16: Weibull CDF for all categories of transformer failures. Additionally, the level of confidence chosen to construct the shaded area affected how thick or thin it was. A higher level of confidence resulted in a thinner shaded area and vice versa. The above survival functions from Figures 10-15 provide estimates of the lifespan of DTs. The survival function is plotted on the \(y\)-axis as a function of time on the \(x\)-axis. It is a decreasing function that starts at 1 and declines to 0 as time increases. The shape of the survival function is determined by the shape parameter of the Weibull distribution, and the slope of the survival function at any point is determined by the scale parameter [17; 22]. These survival functions represent the probability that DTs will remain functional beyond a certain time. The survival functions are the complement of the Weibull CDFs of DTs and demonstrate the probability that the time to failure of a DT is greater than a given time t, given that the DT has not failed before time t.  
In reliability and failure analysis, the survival function is more informative than the CDF, since it yields the probability of DTs still working at a certain time [21].