*3.2. AC Breakdown Voltage*

According to the mean AC BDVs displayed in Figure 5 and Table 3, iNF and sNF samples present enhanced dielectric strength by 4.1% and 16.3%, respectively. The standard deviation of the collected AC BDV events is approximately the same for the iNF with respect to that of the base oil. On the contrary, the standard deviation of the results regarding sNF is increased.

**Figure 5.** Mean AC BDVs and standard deviation of the NF and base oil samples.


**Table 3.** Descriptive statistics of the compared samples.

The estimation of the BDV in low probability levels is of major importance as well. The BDV can be considered as a random variable that follows normal or/and Weibull distribution [7,33–36]. The cumulated distribution functions (CDF) of the normal and Weibull random variable *x*, expressing the breakdown voltage, are given by (1) and (2), respectively:

$$F\_{(\mathbf{x})} = \frac{1}{2} \left| 1 + \text{erf} \left( \frac{\mathbf{x} - \mu}{\sigma \sqrt{2}} \right) \right| \tag{1}$$

$$F\_{(x)} = 1 - e^{-\frac{x}{\tilde{\rho}}a} \tag{2}$$

where, μ is the mean value; σ is the standard deviation; α is the shape parameter and β is the scale parameter.

Firstly, Anderson–Darling goodness of fit test is performed to determine whether the sample of the BD events are normally distributed. The same procedure is followed for Weibull distribution too, because it is expected to give more precise analysis as it does not make assumptions of the skewness and kurtosis. At 5% significance level, the p-value is higher for all the samples under investigation both for normal and Weibull distribution, as shown in Table 4.


**Table 4.** Goodness of fit test (Anderson-Darling).

Figure 4 depicts the frequency density plot of the BD events along with adjustment to normal distribution per each sample under investigation. Such a display is necessary for the detection of possible anomalies in the distribution of the AC BDV [7,23,34]. From the plots of Figure 6, along with the goodness of fit test results; it is concluded that none of them could be rejected for the following statistical analysis.

The breakdown voltages at 1, 10, and 50% probability levels of base, iNF and sNF (Table 5) were calculated from the probability density plots for both distributions, which are demonstrated in Figure 7. The choice of these probability levels corresponds to their importance for the operation of the transformer. On the one hand, U50% gives an assessment for the expected breakdown voltage, while on the other hand, U10% represents an indication on the initiation threshold of ionization, and therefore assists in concluding about the reliability of the transformer oil sample [7,23,32–35]. Last but not least, the BDV at 1% cumulative probability is necessary information for the designing of electric equipment. U1% corresponds to the limit of voltage for safety and continuous operation of the appliance [37].

**Figure 6.** Frequency density plots of the breakdown events for each sample in question with conformity to Normal distribution.

**Table 5.** Breakdown Voltages at 1, 10 and 50% probability levels for normal and Weibull distributions.


**Figure 7.** Probability density plots of the breakdown events for each sample in question according to Normal and Weibull distributions.

Based on the results of Table 5, both iNF and sNF demonstrate enhanced BDVs in low cumulative probability levels, with sNF showing the highest improvement with respect to the natural ester oil sample by around 16.2, 15.6, and 13.7% for U50%, U10%, and U1%, respectively.

From the U10% and U1% BD probability results, it is evident that the addition of SiC NPs could assure more reliable nominal operation of the transformer and delay the initiation of streamer propagation [23,35,37].
