Statistical Analysis of AC Breakdown Performance of Epoxy/Al₂O₃ Micro-Composites for High-Voltage Applications

Abstract

Thanks to the performance improvement introduced by micro sized functional fillers, application of epoxy composites for electrical insulation purposes has become popular. This paper investigates the dielectric properties of epoxy micro-composites filled with alumina (Al₂O₃). In particular, measurements of relative permittivity, dissipation factor, and electrical breakdown are performed, and a comprehensive statistical analysis on dielectric properties was conducted. AC breakdown strength (AC-BDS) was analyzed for normal distribution using four methods (Anderson–Darling, Shapiro–Wilk, Ryan–Joiner, and Kolmogorov–Smirnov). In addition, the AC-BDS was analyzed at risk probabilities of 1%, 5%, 10%, and 50% using Weibull distribution functions. Both normal and Weibull distributions were evaluated using the Anderson–Darling (A-D) statistic and p-value. Additionally, the log-normal, gamma, and exponential distributions of AC-BDS were examined by A-D goodness-of-fit test. The hypothesis test results of AC-BDS were fit by normal and Weibull distributions, and the compliance was evaluated by p-value and each method statistics. In addition, the experimental results of AC-BDS were fit by log-normal and gamma distributions, and the goodness-of-fit was evaluated by p-value and A-D testing. On the other hand, exponential distribution was not suitable for p-value and A-D testing. The results showed that the distributions of AC-BDS were the best using log-normal distribution. Meanwhile, statistical analysis results verified the apparent effect of temperature on dielectric properties using a paired t-test. The analysis results of this paper not only contribute to better characterization of epoxy/Al₂O₃ micro-composites but also introduce a comprehensive approach for performing statistical analysis for electrical insulation materials.

1. Introduction

The increasing power demand in conventional power systems due to modernization has led to the growing complexity of power grids with the integration of distributed power sources such as renewable energy and energy storage systems. To accommodate various power sources and loads connecting to the grid, installation of various type of power assets and equipment would be required [1]. For stable and safe operation of the power grid, it is essential that such power assets and equipment are designed to show high level of reliability. Typically, the reliability and safety of power equipment are determined by electrical insulation, thus necessitating research on the insulation performance of various electrical materials [2]. Epoxy resin, known for its excellent electrical and thermal properties, has been widely used as an insulating material in power equipment. In order to improve performance of electrical insulation, the approach of developing epoxy composites that include functional fillers have been considered.

The effectiveness of both epoxy micro-composites and epoxy nanocomposites have been reported. Examples of fillers used in epoxy composites include silica (SiO₂), alumina (Al₂O₃), and titania (TiO₂) [3]. As a more significant improvement in performance could be obtained by adding a smaller amount of filler compared to epoxy micro-composites, epoxy nano-composites have received attention. According to [3], nano-sized silica particles contribute to increased performance of thermal conductivity, insulation strength, and PD resistance of epoxy-based insulation materials. Thanks to such improvement, the effectiveness of SiO₂/epoxy nano-composites for applications that involve high level of electrical stress (e.g., HVAC power cables and insulation of electric machines) were studied. In addition, nano-sized alumina fillers improve PD resistance and are mainly applied for miniaturization in rotating machines and winding insulation. Alumina fillers are also known for enhancing mechanical properties. As a result, alumina fillers are applied in gas-insulated switchgear (GIS) spacers, playing a critical role in high-voltage system insulation and providing mechanical support for power equipment [4]. Further research on epoxy-alumina nano-composites has been performed as follows. Asokan et al. [5] investigated the influence of interfacial polarization between the polymer and nano particles on the dielectric properties, highlighting its critical role in dielectric relaxation mechanisms. Preetha and Thomas [6] evaluated the AC breakdown properties of epoxy nano-composites by assessing the insulation durability with varying concentrations of alumina nano-fillers. Furthermore, Pandey and Gupta [7] emphasized the impact of field treeing degradation mechanisms on the electrical performance of epoxy-alumina nano-composites using electroluminescence techniques.

Despite the advantages of nano sized fillers in improving electrical, mechanical, and thermal properties, micro sized fillers are preferred in actual power equipment insulation design due to economic considerations, ease of processing, and reliability. Metro and Selvaraj [8] reported that micro-fillers are significantly cheaper than nano-fillers and more suitable for large-scale production, resulting in substantial cost savings. For example, while 3–5 wt% of nano-fillers can achieve similar property improvements, 20–30 wt% of micro-fillers are required; however, the cost-effectiveness of micro-fillers still prevails [8]. Moreover, micro-fillers have long-established reliability in power equipment [8], while the long-term performance of nano-fillers is yet to be fully verified. Additionally, Bommegowda et al. [9] pointed out that nano-fillers have a tendency to agglomerate, complicating the manufacturing process, whereas micro-fillers disperse more easily, simplifying production and improving quality control. Thus, micro-fillers are preferred for power equipment insulation design. Considering such popularity in practical cases and the industry, therefore, this research selected epoxy/Al₂O₃ micro-composites as the test specimens.

Based on its improved mechanical and thermal performance compared to pure epoxy for electrical insulation applications, research on epoxy/Al₂O₃ micro-composites has been performed [10,11,12,13]. Zhenlian An et al. [10] compared the effects of fluorination treatment on the surface properties of micro-alumina-filled and unfilled epoxy insulators used in SF₆ gas insulation systems. It was reported that fluorination treatment increased surface hydrophilicity and that the presence of Al₂O₃ particles affected the surface structure and agglomeration state of the composite, thereby changing the surface modification characteristics. In addition, Jin Li et al. [11] studied the effect of particle agglomeration state on the dielectric breakdown characteristics of epoxy/micro-alumina (Al₂O₃) composites under temperature gradients. It was demonstrated that micro-alumina particles can alter the electrical tree growth path and dielectric breakdown time under high-voltage and high-temperature gradient conditions. Moreover, Wen Cao et al. [12] analyzed the effects of nano- and micro-alumina (Al₂O₃) and Al(OH)₃ fillers on the flame retardancy properties of epoxy composites in vacuum insulators. In the referenced study [12], micro-fillers demonstrated better combustion resistance than nano-fillers for Al₂O₃. The prior study [13] also compared the electrical performance of epoxy resins filled with micro and nano alumina particles. Ref. [13] emphasized that although nanoparticles more effectively improved the dielectric breakdown resistance, micro alumina is still a favorable choice in terms of large-scale production and cost. Thus, it can be seen that micro-sized Al₂O₃ fillers exhibit superior properties and offer economic advantages when selected.

Meanwhile, high-voltage experiments generally involve statistical processing of experimental data. This is because high-voltage test results can vary due to experimental conditions, environmental factors, and equipment sensitivity. Hence, statistical methods are used to ensure the reliability of the results. In the study by M. Peratchiammal et al. [14], for instance, statistical methods were employed to evaluate a breakdown voltage (BDV) distribution of insulators when analyzing their flashover characteristics. The BDV data were statistically analyzed to assess various distribution models, and comparisons between experimental data were made to identify better-performing models. Other studies have also applied statistical techniques to high-voltage experimental data [15,16,17]. In the work of U. Khaled and A. Beroual [15], the goal was to evaluate how the breakdown strength (BDS) of nanofluids changes when various nanoparticles (Fe₃O₄, Al₂O₃, SiO₂) are added to synthetic ester. In that study, the AC breakdown strength (AC-BDS) was measured using standard electrode configurations, and the collected data were analyzed using both normal and Weibull distributions. Their statistical analysis revealed that in most cases, the Weibull distribution fit the data better. Similarly, H. Khelifa et al. [16] analyzed the probabilities of Weibull and normal distributions for the results of AC breakdown experiments performed on ester synthesized with fullerene and graphene nanofluids. In the research of J. Zhou et al. [17], the study focused on investigating the effect of moisture accumulation on surface flashover performance in high-voltage insulation systems. They measured the flashover voltage under conditions where moisture was evenly or unevenly distributed on the insulator surface and used statistical techniques to analyze the flashover data under different conditions. In that study, log-normal and gamma distributions were also employed for statistical analysis. In previous studies [15,16,17], Weibull distribution analysis was conducted using the Anderson–Darling (A-D) statistic and p-values [18]. The normal distribution analysis, on the other hand, was verified using the Shapiro–Wilk method [19]. Although several distribution models (e.g., Weibull and normal distribution) have been generally considered as a representative distribution model for statistical processing of electrical breakdown data, it is necessary to validate such an approach based on statistical analysis tools, as to be shown later in this paper.

This paper studied the electrical insulation performance of epoxy/Al₂O₃ micro-composites. In order to characterize the specimen, measurement of dielectric properties (i.e., permittivity and loss tangent) and electrical breakdown experiments were performed. From the experiment data, it was possible to characterize the dielectric properties and electrical BDS of epoxy/Al₂O₃ micro-composites. Furthermore, statistical analysis was performed on the data to gain insights on the measurement results and material performance. From such analysis, in particular, the dependence of the specimen permittivity on temperature was observed, and the conformity of electrical breakdown data to several parametric distribution models was observed.

In this study, the conformity of the AC-BDS measurement results of epoxy/Al₂O₃ micro-composites to the laws of normal and Weibull distributions was verified. The Weibull distribution analysis was conducted using the method proposed in [18]. For the normal distribution analysis, four different methods were employed: Shapiro–Wilk (statistic: W) [19], Anderson–Darling (statistic: A-D) [20], Ryan–Joiner (statistic: R-J) [21], and Kolmogorov–Smirnov (statistic: K-S) [22]. The p-values obtained from these methods were used to confirm whether the AC-BDS of epoxy/Al₂O₃ micro-composites satisfied normality. Furthermore, the skewness and kurtosis of the AC-BDS data were analyzed to verify if the shape of the distribution resembled that of a normal distribution. Meanwhile, in the statistical analysis of failure data for insulating materials, distributions such as Weibull, Gumbel, and log-normal are commonly used [23]. Moreover, other distributions, such as Gaussian normal, double-exponential, Wohlmuth’s two-limit, and gamma, have also been considered for processing high-voltage experiment data [24]. Therefore, the effectiveness of the distribution models of log-normal, exponential, and gamma for modeling AC-BDS experiment data of epoxy/Al₂O₃ micro-composites were also studied in this paper. The best-fitting distribution for analyzing the AC-BDS of the epoxy/Al₂O₃ micro-composites was selected using the A-D statistics obtained from the distribution analyses.

The remainder of this paper is organized as follows. In Section 2, the AC-BDS measurement method is presented. Details on the test sample configuration and measurements of the permittivity and the loss tangent are also introduced. Section 3 introduces the normal distribution and Weibull distribution methods, along with distribution fitting techniques, for statistically analyzing BDS. Experiment results of the dielectric properties and electrical breakdown tests are provided in Section 4. Section 5 presents the analysis of the dielectric properties measured under temperature variation in Section 2 through the use of the t-test method. Additionally, the goodness-of-fit for log-normal distribution and the inadequacy of the exponential distribution using the A-D test are discussed in Section 5. Conclusions of the manuscript are provided in the Section 6.

2. Experimental Setup

2.1. Test Samples

The test specimens were epoxy/Al₂O₃ micro-composites with approximately 20 vol% Al₂O₃ micro-particles embedded in an epoxy matrix based on the manufacturing procedure of Figure 1. In case of epoxy composite, the properties of the composites would vary depending on the content ratio of the filler added [25,26]. For example, a decreasing trend in the insulation BDS was observed in epoxy resins containing micro-sized Al₂O₃ fillers beyond a loading of 40 wt% [26]. In addition, Ref. [27] reported that the mechanical properties, such as tensile strength and bending strength, decreased when Al₂O₃ micro-fillers were added at amounts exceeding 40 wt%. In this research, a 20 vol% content ratio of Al₂O₃ filler was considered based on the fact that optimized performance improvement was obtained at 40 wt% in such studies. It is worth noting that a filler loading of 40 wt% approximately corresponds to 20 vol% considering the densities and weight ratios of epoxy resin, curing agent, and micro-sized Al₂O₃ fillers.

As shown in Figure 2, the samples were manufactured in a disc shape with a diameter of 120 mm and an average thickness of 1.05 mm, making them suitable for measuring dielectric BDS through the insulation material. To comprehensively characterize the properties of the epoxy/Al₂O₃ micro-composites, the relative permittivity and dissipation factor, which are related to insulation performance, were measured prior to the AC-BDS test. The Fully Automatic Capacitance and Loss Factor tan$\delta$ Precision Measuring Bridge from Tettex, shown in Figure 3, was used as the measurement equipment. During this process, a test voltage of 1 kV and a frequency of 60 Hz were applied to the upper electrode of the Type 2914 using AC dielectric tester. The capacitance ($C_{DUT}$, unit: pF) and dissipation factor of the epoxy/Al₂O₃ micro-composites specimens were recorded through the output of the Type 2818A.

To investigate the changes in the relative permittivity and dissipation factor with temperature, the temperature of the epoxy/Al₂O₃ micro-composites was varied using Type 2966A equipment. The temperature points were set at 30 °C, 60 °C, 90 °C, 120 °C, and 150 °C, and the temperature was applied through the electrodes of the Type 2914. The entire procedure was conducted in accordance with the IEC 62631-2-1 standard [28], and the measured values for 10 specimens were averaged at each temperature point. The relative permittivity was calculated from the capacitance of the epoxy/Al₂O₃ micro-composites measured with Type 2818A as

$${ϵ}_{DUT}={\displaystyle \frac{C_{DUT}}{{ϵ}_0}}\times {\displaystyle \frac{t_{DUT}}{S_{Electrode}}}$$

(1)

where, $C_{DUT}$ is the capacitance of the specimens, $t_{DUT}$ is the thickness of the epoxy/Al₂O₃ micro-composites specimen, $S_{Electrode}$ is the surface area of the measuring electrode, and ${ϵ}_0$ is the permittivity of vacuum ($8.854\times {10}^{-12}$ F/m).

2.2. AC Breakdown Test

The insulation strength and short-term breakdown time of epoxy/Al₂O₃ micro-composites were obtained according to IEC 60243-1 [29]. The overall test setup is shown in Figure 4. To evaluate the AC insulation characteristics, the setup consisted of a high-voltage source and test electrodes. As shown in Figure 4, the electrode arrangement with embedding must be made for films and boards of up to 3 mm [29]. For the short-term breakdown test, both the ramp test (RT) and step-up test (ST) were considered.

In this study, the RT method was performed such that the electrical stress level is increased in a continuous manner as shown in Figure 5a. In our case, the voltage was ramped up at a rate of 1 kV/s so that the insulation breakdown would occur within 10 to 20 s [29]. The insulation BDS values measured during this test were statistically processed using the Weibull distribution, and the BDS at the 40th percentile was set as the initial voltage for the ST. The ST method incrementally increases the stress level and is classified into 60 s, 10 min, and 30 min STs based on the duration of voltage increase cycles. In this study, the voltage was increased every 60 s. According to the table provided by IEC 60243-1, the voltage was increased by 0.5 kV at 60 s intervals during the breakdown test, as shown in Figure 5b. Since both RT and ST are tests that evaluate the dielectric strength of epoxy/Al₂O₃ micro-composites by penetration, transformer oil was used as the surrounding medium to prevent creepage discharge, as shown in Figure 4.

3. Statistical Methods

3.1. Testing Compliance with Statistical Distribution Laws

This section provides a brief introduction to the four statistical tests that were used to perform statistical analysis of the AC breakdown characteristics. In particular, the statistical test methods that were used include the Shapiro–Wilk test [19], Anderson–Darling test [20], Ryan–Joiner test [21], and Kolmogorov–Smirnov test [22]. These statistical tests were used to verify whether the AC-BDS values of epoxy/Al₂O₃ micro-composites, presented in the following section, follow normal and Weibull distribution laws. For a more detailed understanding of each test method, readers are encouraged to refer to the respective literature [19,20,21,22].

3.1.1. Shapiro–Wilk Test

The Shapiro–Wilk test is one of the most common methods to assess whether a dataset follows the normal distribution, especially effective when the sample size is small. The test statistic W is calculated as

$$W={\displaystyle \frac{{\left({\sum }_{i=1}^n{a}_i{x}_{\left(i\right)}\right)}^2}{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}}$$

(2)

where, $x_{\left(i\right)}$ are the ordered sample data (which is the AC-BDS in this study), $a_i$ are pre-defined constants, n is the number of samples, and $\overline{x}$ is the sample mean [19]. In this research, $\overline{x}$ is the average AC-BDS of epoxy/Al₂O₃ micro-composites. The statistical significance of the test statistic W is determined by comparing the p-value to a significance level of 0.05. In case p-value ≤ 0.05, the null hypothesis (i.e., the data follows the normal distribution) is rejected. That is, the given data set does not conform to the normal distribution. For the case of p-value > 0.05, conversely, the null hypothesis is accepted, meaning the data are normally distributed.

3.1.2. Anderson–Darling Test

The Anderson–Darling test is a versatile method that can be applied to any distribution and is used to determine whether data follow a specific distribution. This test is particularly sensitive to extreme values, making it effective in assessing the goodness-of-fit at the tails of the distribution [20]. The A-D test statistic is calculated as

$$A-D=-n-{\displaystyle \frac{1}{n}}\sum _{i=1}^n\left((2i-1)\left[lnF\left(x_{\left(i\right)}\right)+\mathrm{l}\mathrm{n}\left\{1-F\left(x_{\left(n+1-i\right)}\right)\right\}\right]\right)$$

(3)

where, $F\left(x\right)$ is the cumulative distribution function. A larger Anderson–Darling statistic value (denoted as A-D) indicates that the data deviate more from the specified distribution. Similar to other methods, the null hypothesis is rejected if the p-value is less than the significance level of 0.05. Such rejection implies that the data do not follow the specified distribution.

3.1.3. Ryan–Joiner Test

The Ryan–Joiner test is a method similar to the Shapiro–Wilk test for assessing whether data follow a normal distribution [21]. The Ryan–Joiner test is primarily used in Minitab 19 software. The Ryan–Joiner test evaluates normality by calculating the correlation between the data and the normal scores of the data. A correlation coefficient close to 1 indicates that the population is likely to follow the normal distribution [30]. The Ryan–Joiner statistic (R-J) measures the strength of this correlation (i.e., Pearson correlation coefficient). If the statistic is less than the appropriate critical value, the null hypothesis that the population follows the normal distribution is rejected. The criteria for statistical significance follow the same approach as described in Section 3.1.1 for the Shapiro–Wilk test.

3.1.4. Kolmogorov–Smirnov Test

The Kolmogorov–Smirnov test compares the empirical cumulative distribution function (ECDF) to the theoretical distribution to determine if the data follow the specified distribution [22]. It evaluates the goodness of fit across the entire range of the data. The test statistic K-S is calculated as

$$K-S=\mathrm{m}\mathrm{a}\mathrm{x}\left|F_n\left(x\right)-F\left(x\right)\right|$$

(4)

where $F_n\left(x\right)$ is the empirical cumulative distribution function, and $F\left(x\right)$ is the theoretical cumulative distribution function. The statistic can be computed using software like MATLAB R2024a and Minitab 19. A large K-S value indicates that the data do not follow the tested distribution. Like the other tests, if the p-value is less than the significance level of 0.05, the null hypothesis is rejected.

When a given dataset is provided to statistical software (e.g., Minitab, SPSS, R), the corresponding test statistic is calculated based on the statistical distribution model formulas within each software. In this study, the Shapiro–Wilk test approach (Section 3.1.1) was implemented using statistical software R 4.4.1 [31], and the other approaches were performed using Minitab 19 software [32]. Detailed information on this can be found by referring to each statistical software’s manual.

Based on a short survey on previous studies that performed detailed statistical analysis for high-voltage experiments, it seems that the application of the Kolmogorov–Smirnov [17] approach and the Ryan–Joiner method is rather limited to the Shapiro–Wilk [15,16,33,34] method and the Anderson–Darling [15,33,34] method in high-voltage application studies.

3.2. Goodness-of-Fit Test

In this study, the Anderson–Darling method, which was introduced in Section 3.1.2, was used to study the effectives of different statistical distribution models for modeling the AC breakdown performance. The selection of the statistical distribution for analyzing the AC-BDS of the epoxy/Al₂O₃ micro-composites was based on References [23,24]. According to [23], distributions such as Weibull, Gumbel, and log-normal are used in the statistical analysis of failure data for insulating materials. Furthermore, [24] stated that log-normal, exponential, and gamma distributions are used in high-voltage engineering. Accordingly, this paper analyzes the AC-BDS of the epoxy/Al₂O₃ micro-composites using normal, Weibull, log-normal, exponential, and gamma distributions. At this stage, the best suitable distribution was selected by comparing the Anderson–Darling test statistic (A-D value) with the p-value. Meanwhile, the characteristics of the normal distribution and log-normal distribution are similar. Hence, please note that the normal distribution was excluded from the goodness-of-fit test. The statistical distributions used for the goodness-of-fit test are briefly introduced as follows.

3.2.1. Normal Distribution

The normal distribution is a probability function that describes how the measured experimental values are distributed; it is characterized by a symmetric distribution where most observations cluster around the central peak, with values tapering off sharply as they move away from the mean [35]. Accordingly, the important parameters in this distribution are the mean and standard deviation. The mean and standard deviation for the normal distribution are calculated as follows [24]:

$$\mu \approx x_m=\overline{x}=\sum _{i=1}^n{h}_i{x}_i={\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i$$

(5)

$${\sigma }^2\approx s^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\left(x_i-x_m\right)}^2$$

(6)

where $x_i$ is discrete measured values, $n$ is a limited number, $h_i$ is the rate of occurrence defined as $1/n$, $x_m$ is the arithmetic mean value (or expectation value), and $\mu$ is the empirical estimate of the expectation value. Additionally, $s$ denotes the empirical standard deviation (root mean square deviation, r.m.s.d.), and $\sigma$ is the standard deviation. These parameters are used when analyzing the experimental data (e.g., AC-BDS). In addition, the probability density function (pdf) and cumulative distribution function (cdf) can be obtained using these values. The two probability distribution functions are expressed as follows [24]:

$$D_N(x)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}\exp \left(-{\displaystyle \frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}\right)\mathrm{f}\mathrm{o}\mathrm{r}-\infty <x<\infty$$

(7)

$$F_N(x)={\int }_{-\infty }^x{D}_N\left(x\right)dx$$

(8)

where $D_N\left(x\right)$ is the probability density function of normal distribution and $F_N\left(x\right)$ is its cumulative distribution function. For a detailed explanation of the normal distribution, please refer to [24].

Meanwhile, there are important indicators to consider in a normal distribution. These include skewness, which measures the asymmetry of the data distribution relative to the center, and kurtosis, which characterizes how peaked the shape of the data set is. A skewness value closer to 0 indicates that the distribution is more symmetric. For a data point that shows a perfectly symmetric bell-shaped normal distribution, the skewness would be 0, whereas it may have positive skewness or negative skewness otherwise. The skewness, $S_k$, could be calculated as

$$S_k={\displaystyle \frac{\sum {\left(x_i-\mu \right)}^3·p_i}{{\sigma }^3}}$$

(9)

where $x_i$ represents the measurement value (here, AC-BDS), $\mu$ is the mean value, $\sigma$ is the variance, and $p_i$ is the probability of occurrence of the measured value. In addition, the kurtosis provides information on how concentrated the data are around the center. Kurtosis, $K_u$, can be expressed as

$$K_u={\displaystyle \frac{\sum {\left(x_i-\mu \right)}^4·p_i}{{\sigma }^4}}$$

(10)

and $K_u=3$ implies that the distribution has the same kurtosis as the normal distribution, representing a mesokurtic distribution. While a $K_u$ value being greater to 3 means that the distribution has heavier tails and more outliers, representing a leptokurtic distribution, and K_u being smaller than 3 indicates that the distribution has thinner tails and fewer outliers, representing a platykurtic distribution. It is worth noting that both $S_k$ and $K_u$ values are commonly used during partial discharge (PD) pattern analysis for the classification and identification of defects in electrical insulation [36,37].

3.2.2. Weibull Distribution

The Weibull distribution is useful for determining the probability of breakdown occurring at different rates, as well as estimating the lifespan of insulating materials and power equipment [35]. The Weibull distribution can be described by three parameters: the location parameter $x_0=\gamma$ (initial value, lower extreme value), the 63 % quantile $x_{63}$, and the Weibull index $\beta$ (shape parameter or slope). Sometimes, the scale parameter is referred to as the characteristic life [24]. Sometimes the difference $\alpha =x_{63}-x_0$ is called the scale parameter. According to [24], the three-parameter Weibull distribution, defined by the location, shape, and scale parameters, typically provides a good approximation for the cumulative frequency polygon of a series of measurements. Moreover, the Weibull approximation is generally superior to the approximation of the infinite Gaussian normal distribution with two parameters. However, the two-parameter Weibull distribution without the location parameter is widely applied. Therefore, this study used the two-parameter Weibull distribution without the location parameter. This implies that the starting point of the data distribution in the Weibull distribution is 0, meaning that the location parameter is omitted (i.e., $\gamma =0$). The pdf and cdf of the Weibull distribution are expressed as follows [24]:

$$D_W(x)={\displaystyle \frac{\beta }{\alpha }}{\left({\displaystyle \frac{x}{\alpha }}\right)}^{\beta -1}\exp \left(-{\left({\displaystyle \frac{x}{\alpha }}\right)}^{\beta }\right)$$

(11)

$$F_W(x)=1-\exp \left(-{\left({\displaystyle \frac{x}{\alpha }}\right)}^{\beta }\right)$$

(12)

where $\beta$ represents the shape parameter and $\alpha$ represents the scale parameter; these are the parameters considered in the two-parameter Weibull distribution. For a more detailed understanding of Weibull distribution, readers are encouraged to refer to the respective literature [24,38].

3.2.3. Log-Normal Distribution

The log-normal distribution is used for performing statistical analysis on high-voltage engineering data to analyze the characteristics of insulating materials [23,24]. This distribution is a probability distribution in which the natural logarithm of the data follows a normal distribution. In other words, if a variable $x$ follows a log-normal distribution, the logarithm of that variable, $ln\left(x\right)$, follows a normal distribution. The log-normal distribution is mainly suitable for data that only have positive values and is used when the data are asymmetric with a right-skewed tail. The pdf and cdf of the log-normal distribution are expressed as follows [39]:

$$D_{LN}(x)={\displaystyle \frac{1}{\sigma x\sqrt{2\pi }}}\exp \left(-{\displaystyle \frac{{\left(\mathrm{l}\mathrm{n}\left(x\right)-\mu \right)}^2}{2{\sigma }^2}}\right)\mathrm{f}\mathrm{o}\mathrm{r}x>0$$

(13)

$$F_{LN}(x)={\int }_0^{\infty }{D}_{LN}\left(x\right)dx$$

(14)

where $\mu$ represents the mean of the logarithmic values and $\sigma$ represents the standard deviation of the logarithmic values. This distribution, similar to the normal distribution, allows for the calculation of statistical metrics such as mean and variance. Unlike the normal distribution, however, it provides a more suitable model when the data distribution is skewed to one side. For a more detailed understanding of the log-normal distribution, readers are encouraged to refer to the relevant literature [23,39].

3.2.4. Exponential Distribution

Exponential distribution is one of the continuous probability distributions and is primarily used to model the time intervals between events [40]. In other words, when events occur at a constant rate, the time intervals between those events follow an exponential distribution. Additionally, the exponential distribution possesses the memoryless property, which makes it widely used in reliability analysis due to its mathematical simplicity and well-established methods for statistical estimation and hypothesis testing compared to other distributions. The exponential distribution has only a single parameter, which makes the calculation of the pdf and cdf relatively simple compared to other distributions. The single parameter is the scale parameter $\lambda$, which represents the average rate of event occurrence. The pdf and cdf of the exponential distribution are expressed as follows [40]:

$$D_E(x)=\lambda ·\mathrm{e}\mathrm{x}\mathrm{p}(-\lambda x)\mathrm{f}\mathrm{o}\mathrm{r}x\ge 0$$

(15)

$$F_E(x)=1-\mathrm{e}\mathrm{x}\mathrm{p}(-\lambda x)\mathrm{f}\mathrm{o}\mathrm{r}x\ge 0$$

(16)

For a more detailed understanding of the exponential distribution, readers are encouraged to refer to the relevant literature [24,40].

3.2.5. Gamma Distribution

The gamma distribution is primarily used to model data such as waiting times until an event occurs [41]. It is also used in high-voltage fields to analyze data, such as the Weibull distribution [24]. However, it has the disadvantage of being less useful as a life distribution due to the fact that the shape of the failure rate function is more complex than the Weibull distribution. Nonetheless, there are often cases where it fits well with various lifetime data and is applied. The gamma distribution has shape parameter and scale parameter as its main parameters. The shape parameter indicates how many times an event occurs, and a larger value models the waiting time after more events. The scale parameter represents the average interval between events and is inversely related to the average speed at which events occur. Since this paper used gamma distribution to analyze the AC-BDS, the event was considered the BDS and the time was considered the size of the AC-BDS. The pdf of a gamma distribution with two parameters represents the probability that an event will occur at a specific time, while the cdf represents the total probability that an event will occur up to a given time. The pdf and cdf of the gamma distribution are expressed as follows [41]:

$$D_G(x)={\displaystyle \frac{x^{\beta -1}}{{\alpha }^{\beta }\mathsf{\Gamma }\left(\beta \right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{x}{\alpha }}\right)\mathrm{f}\mathrm{o}\mathrm{r}x>0$$

(17)

$$F_G(x)={\displaystyle \frac{\gamma \left(\beta ,\frac{x}{\alpha }\right)}{\mathsf{\Gamma }\left(\beta \right)}}$$

(18)

where $x$ is a variable representing the time at which an event will occur (used here as the AC-BDS), $\beta$ is the shape parameter (the number of times AC breakdown will occur), $\alpha$ is the scale parameter (the average interval between AC breakdowns), $\gamma (\beta ,{\displaystyle \frac{x}{\alpha }})$ is the incomplete gamma function (the cumulative probability of BDS occurring until a specific AC breakdown), and $\mathsf{\Gamma }\left(\beta \right)$ is the gamma function (the overall probability of AC breakdown occurrence). For a more detailed understanding of the gamma distribution, readers are encouraged to refer to the relevant literature [41].

Additionally, if a deeper understanding of the five distributions introduced in this section (normal, Weibull, log-normal, exponential, and gamma) is needed, readers are referenced to the relevant literature [42,43].

4. Experiment Results

4.1. Dielectric Properties

The average values of the relative permittivity and dissipation factor are summarized in

Table 1. Dielectric properties of epoxy/Al₂O₃ micro-composites at different temperatures.

Temperature (°C)	Relative Permittivity	Dissipation Factor
30	4.94	0.0041
60	5.04	0.0045
90	5.13	0.0051
120	5.19	0.0069
150	5.28	0.0086

. Additionally, the graphs of the relative permittivity and dissipation factor at each temperature are presented in Figure 6. The results of the relative permittivity and dissipation factor measurements exhibited temperature-dependent characteristics. As the temperature increased from 30 °C to 150 °C, both the relative permittivity and dissipation factor showed increases. The relative permittivity showed a limited increase (less than 5%) when the temperature increased from 30 °C to 60 °C and 90 °C. However, the values measured at 120 °C and 150 °C showed increases of approximately 5% and 6.8%, respectively, compared to value measured at 30 °C. Regarding the dissipation factor, measurements at different temperatures were found as follows. Compared to the dissipation factor value of 0.0041 measured at 30 °C, the value measured at 60 °C and 90 °C showed increases of approximately 9.8% and 24.4%, respectively. The values measured at 120 °C and 150 °C showed a noticeable increase compared to the value measured at 30 °C (approximately 68.3% and 109.8%, respectively).

Such results seem to be well aligned with previous studies. For example, F. R. La Mantia et al. reported that the relative permittivity and the dissipation factor showed limited change in the value when the measurement temperature was increased to 90 °C [44]. When the temperature was further increased up to 150 °C, however, the results of [44] reported that both the relative permittivity and the dissipation factor dielectric properties showed more apparent changes in the value as a result of structural changes in the epoxy resin after being exposed to increased thermal stress. Meanwhile, other previous studies have reported on the temperature-dependent dielectric properties of composites filled with Al₂O₃ micro-fillers. Kochetov et al. [45] investigated the thermal and electrical behavior of epoxy-based micro-composites filled with micro-sized silica (SiO₂) and alumina (Al₂O₃) particles. The study demonstrated that as the filler content increased and the measurement temperature rose, the relative permittivity of the composites also increased. These results suggest that the concentration of filler particles and temperature can significantly impact the electrical properties of the composites, supporting the findings of this study.

4.2. Breakdown Strength Analysis

This section presents the results of the RT-BDS and ST-BDS measurements and determine whether the breakdown data set complies with the normal and Weibull distributions [18,19]. These probability distributions are known as the most commonly used models for analyzing the BDS of dielectrics. They serve as useful models in designing and maintaining power equipment by improving the predictability of insulation performance [35].

To verify which probability law the experimental data followed, the hypothesis tests that were introduced in Section 3.1 were applied. The Shapiro–Wilk test [19], Anderson–Darling test [20], Ryan–Joiner test [21], and Kolmogorov–Smirnov test [22] were used to analyze whether the data followed the normal distribution law. Additionally, to determine if the data followed the Weibull distribution law, the Anderson–Darling test [20] was utilized. The statistical analysis was performed using R software [31] and Minitab19 software [32]. The p-value represents the probability of evidence against the null hypothesis. After calculating the p-value using the software, the same statistical procedure was followed by comparing the p-value to a significance level of 5% to determine if the experimental results conformed to the statistical distribution. If the p-value is smaller than the significance level, the null hypothesis is rejected, indicating that the sample data do not conform to the considered statistical distribution law [35]. A smaller p-value provides stronger evidence for rejecting the null hypothesis, and a larger Anderson–Darling statistic (A-D) indicates that the data do not follow the analyzed distribution.

4.2.1. Normal Distribution

Figure 7 presents the normal distribution graph of the AC-BDS values of epoxy/Al₂O₃ micro-composites for the two voltage application methods. Additionally,

Table 2. Hypothesis test of compliance with normal distribution of RT-BDS.

Normality Test Method	Ramp Test
	Statistics	p-Value	Conformity
Shapiro–Wilk	0.939	0.228	Accepted
Anderson–Darling	0.226	0.789	Accepted
Ryan–Joiner	0.993	>0.100	Accepted
Kolmogorov–Smirnov	0.137	>0.150	Accepted

and

Table 3. Hypothesis test of compliance with normal distribution of ST-BDS.

Normality Test Method	60 s Step-Up Test
	Statistics	p-Value	Conformity
Shapiro–Wilk	0.969	0.741	Accepted
Anderson–Darling	0.524	0.160	Accepted
Ryan–Joiner	0.998	>0.100	Accepted
Kolmogorov–Smirnov	0.140	>0.150	Accepted

summarize the results of the normality test performed on the AC-BDS values for each experimental method. While Table 2 shows the result for the RT experiment, Table 3 shows the result for the ST experiment. In each table, the statistics from Shapiro-Wilk [19], Anderson–Darling [20], Ryan–Joiner [21], and Kolmogorov–Smirnov [22] tests are shown along with the p-values for each method. In addition, compliance with the normal distribution is also indicated. At all test statistic levels (Shapiro–Wilk: W, Anderson–Darling: A-D, Ryan–Joiner: R-J, Kolmogorov–Smirnov: K-S), the p-values were all higher than the significance level of 0.05, indicating that the experimental data follow the normal distribution law. Figure 8 shows the histogram of the average distribution of the BDS obtained from both breakdown tests (i.e., RT and ST), and the corresponding skewness and kurtosis values are provided in

Table 4. Skewness and kurtosis values with normal distribution of AC-BDS.

Test Method	Skewness	Kurtosis
Ramp Test	+0.75	+0.80
60 s Step-up Test	−0.19	−0.79

. According to A. Beroual et al. [33], this representation allows for the detection of potential anomalies in the distribution of the BDV across various voltage ranges. In this context, potential anomalies refer to outliers (or abnormal patterns) that may occur in the distribution of RT-BDS or ST-BDS. However, when considering the skewness and kurtosis values, it was observed that the epoxy/Al₂O₃ micro-composites slightly deviates from a perfect normal distribution, where skewness and kurtosis should be 0 and 3, respectively. The experimental results show that the skewness values for RT-BDS and ST-BDS are close to 0, with values of 0.75 and −0.19, respectively. On the other hand, the skewness values deviate somewhat from the normal distribution, with RT-BDS at +0.80 and ST-BDS at −0.79. Additionally, since the peak kurtosis value is less than 1.4, this distribution can be considered slightly platykurtic, as noted in [33,46]. There are various software options available for obtaining skewness and kurtosis. For practical reasons, most statistical software packages, such as R [31] or SPSS (Statistical Package for the Social Sciences) [47], are commonly used. In this study, skewness and kurtosis were calculated using the Python 3.9 software package [48] and its statistical libraries.

4.2.2. Weibull Distribution

Figure 9 presents the Weibull distribution graph for the AC-BDS of epoxy/Al₂O₃ micro-composites, based on different voltage application methods (RT and ST). In the case of Figure 9a, statistical processing was performed on the electric field (AC-BDS), while in Figure 9b, statistical processing was conducted on the voltage (AC-BDV). Both graphs are accompanied by key parameters characterizing the Weibull distribution, including the shape parameter (which indicates the slope of the graph line), the scale parameter (which represents the dispersion and failure level of the data), and the A-D statistic and p-value. The p-value of the experimental data and the compliance with the Weibull distribution are summarized in

Table 5. Hypothesis test of compliance with Weibull distribution of AC-BDS.

Test Method	p-Value	Conformity to Weibull Distribution
Ramp Test	<0.010	Rejected
60 s Step-up Test	>0.250	Accepted

for AC-BDS and

Table 6. Hypothesis test of compliance with Weibull distribution of AC-BDV.

Test Method	p-Value	Conformity to Weibull Distribution
Ramp Test	>0.250	Accepted
60 s Step-up Test	0.091	Accepted

for AC-BDV. In the case of Table 5, ST-BDS indicates that the data follow this probability distribution law because the p-values is greater than 0.05. On the other hand, RT-BDS shows the p-value less than 0.01, indicating that is does not follow the Weibull distribution law. To analyze the cause of this result, additional statistical analysis was performed on AC-BDV, unlike the normal distribution. The results of both experiments (ST and RT) indicate that the data follow this probability distribution law, as the p-value is greater than 0.05 in Table 6. Given that RT-BDV follows the Weibull distribution well, the reason why RT-BDS does not follow the Weibull distribution can be attributed to a thickness of the epoxy/Al₂O₃ micro-composites. This is due to RT-BDS being calculated as the RT-BDV value divided by the specimen thickness. Although the average thickness of epoxy/Al₂O₃ micro-composites was introduced as 1.05 mm in Section 2.1, the actual thickness of each specimen showed a variation between 0.99 and 1.11 mm. Thus, the influence of specimen thickness cannot be ignored as a factor that prevents RT-BDS from following the Weibull distribution.

Table 7. AC-BDS at 1%, 5%, 10%, and 50% cumulative probability from Weibull distribution.

Breakdown Strength Probability (%)	Breakdown Strength (kV/mm)
	Ramp Test	60 s Step-Up Test
1	18.15	17.84
5	19.75	18.74
10	20.50	19.15
50	22.60	20.27

summarizes the AC-BDS at 1% cumulative probability, which is one of the critical parameters from a practical perspective, particularly when designing insulating components [33]. It also includes the AC-BDS at 5%, 10%, and 50% probabilities. This allows for a comparison of the AC-BDS for the micro-composites at each probability level. The RT-BDS and ST-BDS showed differences in growth rates according to the failure probability of the Weibull distribution. Based on a 1% cumulative failure probability, as the cumulative probabilities increased to 5%, 10%, and 50%, RT-BDS increased by approximately 8.8%, 12.9%, and 24.5%, respectively. Additionally, as ST-BDS increased with each cumulative probability, it rose by about 5.0%, 7.3%, and 13.62%, respectively. Overall, as the cumulative probability increased, the growth rate of RT-BDS was greater than that of ST-BDS. This is presumed to be the result of a difference of approximately 0.31 at the lowest 1% cumulative probability for RT-BDS and ST-BDS, respectively.

5. Discussion on Statistical Analysis

5.1. Temperature Dependence of Dielectric Properties

This section explores the temperature dependence of the dielectric properties of epoxy/Al₂O₃ micro-composites using the t-test. According to [49], the t-test is used to assess whether a factor significantly affects experimental results. Therefore, a paired t-test was conducted to determine the impact of temperature changes on the measured dielectric properties. The probability p, which measures the evidence against the null hypothesis, was calculated using R software [31]. In this case, the null hypothesis (H₀) states that there is no significant change in the dielectric properties with temperature, while the alternative hypothesis (H₁) suggests that the dielectric properties change significantly with temperature. The p-value was then compared to the significance level to determine if the data showed a significant difference. Typically, a significance level of 5% is considered reasonable in statistical analysis. If the p-value is less than or equal to the significance level of 0.05, the null hypothesis is rejected, indicating that the dielectric properties do not show significant differences with temperature.

Table 8. Application of paired t-test regarding dielectric properties with different temperatures.

Group	Relative Permittivity		Dissipation Factor
	Statistics t	p-Value	Statistics t	p-Value
A	5.7674	2.7 × 10−4	4.3244	1.9 × 10−3
B	10.661	2.1 × 10−5	7.3442	4.4 × 10−5
C	10.556	2.3 × 10−6	8.8547	9.8 × 10−6
D	8.7395	1.1 × 10−5	11.274	1.3 × 10−6
E	7.0683	5.9 × 10−5	8.6363	1.2 × 10−5
F	7.8335	2.6 × 10−5	7.8994	2.5 × 10−5
G	8.3325	1.6 × 10−5	10.729	2.0 × 10−6
H	5.1203	6.3 × 10−4	6.6813	9.0 × 10−5
I	5.9924	2.0 × 10−4	10.257	2.9 × 10−6
J	4.4111	1.7 × 10−3	6.122	1.8 × 10−4

presents the paired t-test results, with ten groups: (A) 30 °C and 60 °C; (B) 30 °C and 90 °C; (C) 30 °C and 120 °C; (D) 30 °C and 150 °C; (E) 60 °C and 90 °C; (F) 60 °C and 120 °C; (G) 60 °C and 150 °C; (H) 90 °C and 120 °C; (I) 90 °C and 150 °C; (J) 120 °C and 150 °C; comparing the relative permittivity and dissipation factor values to see if there were substantial changes due to temperature. The p-values for each group (from A to J) were much smaller than 0.05 for both the relative permittivity and dissipation factor. These results reject the null hypothesis, confirming that the dielectric properties depend on temperature.

Through these results, even though the changes in relative permittivity and dissipation factor according to the temperature presented in Table 1 seem to be limited, statistical analysis on experiment data highlights the significance of temperature to both the relative permittivity and the dissipation factor. In other words, it can be concluded that the dielectric properties show a dependence on temperature. One notable point is that, unlike previous studies that explained experimental results solely through theoretical analysis, this study demonstrated the changes in relative permittivity and dissipation factor with temperature variation using statistical analysis through the paired t-test.

5.2. Goodness-of-Fit Test

It is well-known that normal distribution, log-normal distribution, exponential distribution, Weibull distribution, and gamma distribution are commonly used for data analysis of insulating materials. In this paper, the AC-BDS values of epoxy/Al₂O₃ micro-composites were analyzed using four distributions (Weibull, log-normal, exponential, and gamma distributions), and the goodness-of-fit was assessed for each distribution based on p-value and test statistics. The most suitable distribution for analyzing the AC-BDS values of epoxy/Al₂O₃ micro-composites was also selected.

The results of the analysis for the three distributions, excluding the Weibull distribution (Figure 9a), are presented in Figure 10, Figure 11 and Figure 12. Additionally, the statistical analysis results of the RT-BDS values obtained for each distribution are summarized in

Table 9. Hypothesis test of compliance with statistical distribution of RT-BDS.

Distribution	Ramp Test
	A-D Statics	p-Value	Conformity
Log-normal	0.533	0.151	Accepted
Exponential	8.403	<0.003	Rejected
Weibull	1.124	<0.010	Rejected
Gamma	0.547	0.178	Accepted

, and those for the ST-BDS values are summarized in

Table 10. Hypothesis test of compliance with statistical distribution of ST-BDS.

Distribution	60 s Step-Up Test
	A-D Statics	p-Value	Conformity
Log-normal	0.181	0.901	Accepted
Exponential	8.566	<0.003	Rejected
Weibull	0.218	>0.250	Accepted
Gamma	0.194	>0.250	Accepted

. The statistical processing results for each distribution were analyzed at a 95% confidence level.

The A-D statistic is an indicator that allows the analyst to determine how well the observed data (in this case, AC-BDS) fit a given distribution. In the analysis of the RT-BDS values shown in Table 9, the log-normal distribution had the smallest value of 0.533, while the exponential distribution had the largest value of 8.403. The A-D values for the Weibull and gamma distribution were found to be 1.124 and 0.547, respectively. Similarly, in the analysis of the ST-BDS values presented in Table 10, the log-normal distribution again had the smallest A-D value of 0.181, while the exponential distribution had the largest value of 8.566. The Weibull distribution was 0.037 and the gamma distribution was 0.013 larger than the log-normal distribution. From these results, it can be concluded that the log-normal distribution is the best distribution for analyzing the AC-BDS values of epoxy/Al₂O₃ micro-composites specimens. In this research, the A-D goodness-of-fit test results showed that the log-normal distribution is the best fit for the experiment data of this study. While normal and Weibull distributions have been widely applied to data analysis in high-voltage experiments [24], the log-normal distribution has been considered as a suitable choice for characterizing -voltage experiment data in statistical standard [23] and relevant studies [17,50].

Meanwhile, the exponential distribution showed considerably larger A-D statistics compared to other statistical distributions. In fact, unlike other distributions, the exponential distribution corresponds to the constant failure rate (CFR) in the bathtub curve, while the other distributions correspond to the increasing failure rate (IFR). The exponential distribution corresponds to the CFR because it reflects the characteristic that the failure rate does not change over time during the useful life of the product [51]. In other words, the exponential distribution is suitable when the failure rate remains constant over time, regardless of the duration. Failures due to deterioration are gradual over time, with components exhibiting an increasing failure rate (i.e., IFR) as they are affected by fatigue, wear, stress, etc. Consequently, the exponential distribution model, which assumes a constant failure rate, is not suitable for explaining failures caused by degradation. This difference in characteristics leads to a significant difference in A-D statistics. Specifically, in Table 10, the p-values for the other distributions are greater than 0.05, indicating statistical significance (Conformity: Accepted). However, for the exponential distribution, the p-value is much smaller than 0.05, leading to the rejection of the null hypothesis that the data follows the exponential distribution (Conformity: Rejected). Therefore, these results are considered to be statistically significant to some extent.

6. Conclusions

This study analyzed the AC-BDS and dielectric properties of epoxy/Al₂O₃ micro-composites and verified the experimental data using various statistical methods. The analysis of dielectric properties showed a trend of increasing relative permittivity and dissipation factor with rising temperatures, and the t-test confirmed that these changes were statistically significant. This emphasizes that the insulation performance is sensitive to temperature changes and provides important insights for predicting the behavior of insulating materials in high-temperature environments.

The AC-BDS of the micro-composites was measured using the ramp test (RT) and 60 s step-up test (ST), and the data were analyzed using various statistical distribution models, including normal and Weibull distributions. ST-BDS data fit well with the Weibull distribution, confirmed by p-values greater than 0.05. On the other hand, RT-BDS showed results that were inconsistent with the Weibull distribution, with a p-value of less than 0.05. However, the breakdown voltage values obtained from RT showed good agreement with the Weibull distribution. This is thought to be due to the standard deviation of the thickness of the epoxy/Al₂O₃ micro-composites. The normal distribution also provided a good fit for the experimental data, with all p-values exceeding the 0.05 threshold. Additionally, the results of the goodness-of-fit test showed that both the log-normal and gamma distributions had p-values greater than the significance level (5%) and exhibited a high degree of fit with low A-D statistics. However, the exponential distribution showed poor conformity, as indicated by higher A-D statistics and p-values lower than 0.05.

In conclusion, the study has provided valuable insights into the breakdown characteristics and dielectric behavior of epoxy/Al₂O₃ micro-composites. Regarding the dielectric properties, the temperature dependence of both the relative permittivity and the dissipation factor were verified using a statistical analysis approach. The statistical analysis confirmed that the Weibull and normal distributions are appropriate for analyzing the AC-BDS of these materials, while the exponential distribution was found to be less suitable. In addition, the goodness-of-fit test results comparing A-D values revealed that the experimental data (AC-BDS of epoxy/Al₂O₃ micro-composites) of this research best fits the log-normal distribution. Given that the log-normal distribution is one of the distributions used in the high-voltage field, our research suggests that it may, as in this study’s findings, be more suitable than the normal or Weibull distributions. Further research is planned to explore the behavior of these materials under higher temperatures to ensure reliability in high-voltage insulation applications. Additionally, various statistical analyses will be conducted to assess the lifetime of these materials over extended testing durations.