## 2 Transformer Health Index Estimation Model  
SDM was employed in order to predict the HI of the transformer population, given the limited historical condition parameters data. Figure 1 shows the overall framework for estimating the impending transformer HI using individual condition parameters data. First, the condition parameters data from the transformer population were grouped according to age band from year 1 to 15. Next, the individual condition parameters data of transformers data for every age were fitted into the probability plot to determine the representative distribution. The distribution parameters from year 1 to 15 were then cal culated. The distribution parameters were then plotted and extrapolated to determine the distribution parameters for the next 10 years. Next, the individual condition parameters data were calculated from the extrapolated distribution parameters that were based on the inverse cumulative distribution function (ICDF) of the identified distribution model. The predicted HI was then determined based on the conventional scoring approach. Finally, the predicted HI was compared with computed HI using the Chi-square test and absolute error percentage. The HI of transformer population was modelled based on the statistical approach. The method was used to estimate the predicted HI by calibrating the prediction model based on the condition parameters data. The first condition parameters data that were considered for the input parameters were from dissolved gas analysis (DGA), which included hydrogen (H\({}_{2}\)), methane (CH\({}_{4}\)), acetylene (C\({}_{2}\)H\({}_{2}\)), ethylene (C\({}_{2}\)H\({}_{4}\)), ethane (C\({}_{2}\)H\({}_{6}\)), carbon monoxide (CO), and carbon dioxide (CO\({}_{2}\)). The oil quality analysis (QQA) included dielectric breakdown voltage, interfacial tension, color, acidity, and water content. The furanic compound analysis (FCA) that consisted of 2-furfurraldehyde (2-FAL) was also used in the study to consider the in-service ageing of the solid insulation. It is important to note that  
Figure 1: Modelling framework for future health index of the transformer population. this study does not consider any abnormal data due to the unusual operating environment, internal electrical faults, and electromagnetic interference during in-service. In this study, two-parametric Weibull and normal distribution models were considered. Weibull distribution has been widely used for industrial applications, especially in the field of reliability engineering. It is normally used in the reliability assessment, such as the derivation of reliability indices mean time to failure (MTTF) [25], equipment's failure rate [26; 27; 28], remaining useful life prediction [29; 30], spare parts replacement, and maintenance/replacement strategies [31; 32]. It is a versatile distribution with two critical parameters, which can also describe the characteristics of other types of distributions. The probability density function (PDF) of the Weibull distribution model was computed based on the condition data parameters and it is expressed in Equation (1) [33]. \[f(t)=\frac{\beta}{\alpha}\bigg{(}\frac{t}{\alpha}\bigg{)}^{\beta-1}exp^{-( \frac{t}{\alpha})^{\beta}} \tag{1}\]  
where \(t\) is the lifetime of the transformer, \(\beta\) is the shape parameter, and \(\alpha\) is the scale parameter. \(\alpha\) determines the spread of the data, while \(\beta\) implies the shape of the distribution, whereby it is dimensionless. Different values of the \(\beta\) can affect the behavior of the distribution. For \(3\leq\beta\leq 4\), the shape of the Weibull distribution is similar to that of a normal model. In order too plot the probability of Weibull distribution, the condition parameters data were converted into probabilities using CDF, as shown in Equation (2) [34]. \[F(t)=\int\limits_{0}^{T}f(t)=1-exp^{-(\frac{t}{\alpha})^{\beta}} \tag{2}\]  
Normal or Gaussian distribution has wide application in describing product lifetime data [35], and it can be expressed by Equation (3) [36].