# Effect of Fullerene and Graphene Nanoparticles on the AC Dielectric Strength of Natural Ester

Hocine Khelifa, Eric Vagnon, Abderrahmane Beroual

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Hocine Khelifa, Eric Vagnon, Abderrahmane Beroual. Effect of Fullerene and Graphene Nanoparticles on the AC Dielectric Strength of Natural Ester. Energies, 2023, 16 (4), pp.1995. 10.3390/en16041995. hal-04047619

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Article

# Effect of Fullerene and Graphene Nanoparticles on the AC Dielectric Strength of Natural Ester

Hocine Khelifa* and Eric Vagnon and Abderrahmane Beroual

Univ Lyon, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Ampère, UMR 5005, 69130 Ecully, France

* Correspondence: hocine.khelifa@ec-lyon.fr

**Abstract:** The current study addresses the impact of the addition of fullerene and graphene nanoparticles on the AC breakdown voltage (AC BdV) of natural ester (FR3). The nanofluids (NFs) were prepared following the two-step process, and the AC BdV was performed in compliance with IEC 60156 standards. Five different concentrations of NPs were considered (0.1 g/L, 0.2 g/L, 0.3 g/L, 0.4 g/L, and 0.5 g/L). A Student's t-test was performed to compare the base liquid's AC BdV data with different nanofluids. The experimental data were checked to see if they obeyed the Weibull distribution fitting curve, and the AC BdV at 1%, 10%, and 50% risk levels were then calculated. The performed t-test provides evidence that AC BdV data from the base liquid were different from those of different NFs (except 0.2 g/L fullerene, and 0.1 g/L and 0.4 g/L graphene NFs). It is also shown that the Weibull distribution fit the BdV data of all liquids (except 0.5 g/L fullerene NF), and remarkable improvements of AC BdVs at 1%, 10%, and 50% were observed. The best improvement was obtained with 0.4 g/L fullerene and 0.3 g/L graphene NFs. These results show the importance of using nanofluids as substitutes for the existing insulating liquids for current oil-filled power transformers.

**Keywords:** nanofluids; naturel ester; fullerene nanoparticles; graphene nanoparticles; AC breakdown voltage; statistical analysis; t-test

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**Citation:** Khelifa, H.; Vagnon, E.; Beroual, A. Effect of Fullerene and Graphene Nanoparticles on the AC Dielectric Strength of Natural Ester. *Energies* **2023**, *16*, 1995. https://doi.org/10.3390/en16041995

Academic Editor: Jianchao Cai

Received: 9 January 2023

Revised: 7 February 2023

Accepted: 14 February 2023

Published: 17 February 2023

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**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

Electrical power transformers (PT) often use dielectric fluids for heat dissipation and electrical insulation [1]. The role of these fluids is crucial for adequately operating this type of equipment, which depends, among other things, on the health of the insulating fluids they contain [1]. Therefore, it is essential to develop new fluids to improve the performance of the devices (higher voltage rate and/or reduced volume).

For a time, mineral oil was the preferred solution for oil-filled transformers. This type of oil has been used since the 1930s. It is inexpensive, has remarkable dielectric and heat transfer properties, and is well-compatible with cellulose-based insulating materials [2,3]. Despite these significant benefits, mineral oils have some downsides, such as their flammability, substantial environmental impact, low moisture tolerance, and low flash and fire temperatures [2–5]. Given their characteristics, those liquids are not the best choice for the future [1]. Thus, many academic and industrial researchers have focused on developing safer, non-flammable, and tolerable fluids for the environment. Therefore, new fluids, particularly esters, have appeared on the market [6–8].

For decades, natural esters have been used as insulating liquid in power capacitors. Recently, their utilizations have been extended to large power transformers [9]. Besides being readily biodegradable, natural ester oils show a high flash and poor points compared with mineral oils. Rather than suggesting alternative types of insulating liquids, modifying the existing fluids by adding nanoparticles (called nanofluids) is considered as a possible solution [10–13]. Nanofluids (NFs) are retained for their excellent heat transfer capability [14]
and insulating properties [13,15,16]. As an indication, Segal et al. [17] investigated the impact of iron oxide (Fe₃O₄) NPs on the AC (60 Hz) breakdown voltage (BdV) of mineral oil. These authors showed that, even after thermal ageing, the prepared NFs remain stable and retain their initial magnetic susceptibility. Better AC BdV of NFs was also shown compared to base liquid. Duzkaya and Beroual [18] revealed that adding zinc-oxide (ZnO) NPS to natural ester (Midel 1204) increased the AC BdV by about 5%. This improvement was achieved with 0.1 g/L ZnO NPs. In similar work, Beroual and Duzkaya [19] investigated the impact of adding fullerene NPs on the BdV of Midel 1204. They showed that 0.4 g/L fullerene gives the best improvement (improved by 7.8% than pure natural ester). The concentrations of 0.05 g/L, 0.1 g/L, and 0.2 g/L showed a decreased AC BdV compared to Midel 1204 [19].

In our previous work [20], the impact of fullerene (C₆₀) and graphene (Gr) NPs on AC BdV and partial discharge activity of the synthetic ester Midel 7131 was analyzed. Five concentrations were considered (0.1 g/L, 0.2 g/L, 0.3 g/L, 0.4 g/L, and 0.5 g/L), with different electrode gap distances (0.1 mm, 0.3 mm, 0.5 mm, 0.7 mm, 0.8 mm, 1 mm, 1.5 mm, and 2 mm). The best improvement in AC BdV, i.e., 28.7%, was obtained with 0.4 g/L C₆₀ NPs (0.5 mm electrode gap distance), while the same concentration with Gr NPs and a 0.7 mm electrode gap distance gives the best improvement, that of about 24.4% compared to pure synthetic ester Midel 7131. In addition, we reported that C₆₀ has a positive impact on the partial discharge activity in SE, while the addition of Gr increases this activity.

This work is part of the search for alternative insulating liquids for the existing ones (mineral oils), with less environmental impact and enhanced dielectric performance for their eventual use in high voltage equipment, and especially in power transformers.

This study aims to study the AC BdV of natural ester and natural ester—C₆₀ and Gr NFs, for five concentrations (0.1 g/L, 0.2 g/L, 0.3 g/L, 0.4 g/L, and 0.5 g/L); AC BdV measurements were carried out in compliance with IEC 60156 standards. The Student's t-test was performed to provide statistical evidence about the difference in mean values of different NFs compared to the natural ester FR3, and possible mechanisms behind the improvements are discussed. Then, the AC BdV data were statistically analyzed using the Weibull distribution law; BdVs of 1%, 10%, and 50% risk levels were deduced from the fitting plots.

## 2. Materials and Methods

### 2.1. Materials

In this study, the host liquid (dispersant medium) used to prepare the NFs was the natural ester dielectric fluid FR3, provided by Cargill. FR3 is readily biodegradable and has high/low fire/pour points. The main characteristics of this liquid are summarized in Table 1 [9]. The nano-additives consisted of C₆₀ NPs and Gr nanosheets. C₆₀ NPs and Gr nanosheets were supplied by Sigma-Aldrich (Massachusetts, USA) and SkySpring Nanomaterials (Houston, TX, USA), respectively; their main characteristics are shown in Table 2. Finally, as a dispersant agent, we used oleic acid (cis-9-Octadecenoic acid, purity of >99%), also supplied by Sigma-Aldrich, USA. Oleic acid was used to avoid aggregation/cluster formation and to enhance the stability of NFs.
<table><caption>**Table 1.** Physicochemical properties of natural ester, FR3.</caption><thead><tr><th>Property</th><th>Unit</th><th>FR3</th></tr></thead><tbody><tr><td>Density at 20 °C</td><td>kg/L</td><td>0.92</td></tr><tr><td>Kinematic Viscosity</td><td>mm²/s</td><td></td></tr><tr><td>at 40 °C</td><td></td><td>32–34</td></tr><tr><td>at 0 °C</td><td></td><td>190</td></tr><tr><td>Pour point</td><td>°C</td><td>(–23)–(–18)</td></tr><tr><td>Flash point</td><td>°C</td><td>255</td></tr><tr><td>Fire point</td><td>°C</td><td>350–360</td></tr><tr><td>Water content</td><td>ppm</td><td>4–50</td></tr><tr><td>AC BdV</td><td>kV</td><td>73</td></tr><tr><td>Power Factor at 90 °C</td><td>–</td><td>&lt;0.02</td></tr></tbody></table>

<table><caption>**Table 2.** Nanoparticles' characteristics.</caption><thead><tr><th>NPs</th><th>Fullerene (C<sub>60</sub>)</th><th>Graphene (Gr)</th></tr></thead><tbody><tr><td>Diameter (nm)</td><td>4–8</td><td>20</td></tr><tr><td>Specific surface area (m²/g)</td><td>–</td><td>40</td></tr><tr><td>Density (g/cm³)</td><td>3.4</td><td>5.1</td></tr><tr><td>Purity (%)</td><td>99.5</td><td>99.5</td></tr></tbody></table>

## 2.2. Preparation of Nanofluids

The preparation steps were based on the two-step method [20,21]—the word two-step signifies the isolation of the nanoparticle synthesis from the NFs preparation. The following steps were respected for each NF: (a) purification of the base liquid using a micro membrane filter and a vacuum pump to remove impurities (0.1 mbar), (b) addition of 0.75 wt.% of oleic acid, and mixing the mixture (5 min) using a high shear mixer (HS) at 13,000 rpm, (c) addition of powder NPs (desired amount) to base liquid, and agitating the resulting liquid for 20 min using the HS mixer, (d), and finally, subjecting the NF samples to ultrasonication for 120 min to better disperse the NPs. Five NFs of 400 mL volume, each with NPs concentrations of 0.1 g/L, 0.2 g/L, 0.3 g/L, 0.4 g/L, and 0.5 g/L, were prepared. The 500 W ultrasonication device (operating frequency of 20 kHz) was equipped with a 25 mm solid probe, operating in pulsed mode (67% duty cycle and 15 s of period) with a 60% amplitude setting. The latter was rested for 5 min after every 30 min to prevent the NFs from overheating and damaging the solid probe.

## 2.3. AC Breakdown Measurement

The AC BdV tests were conducted in compliance with IEC 60156 standards by means of a Baur DTA 100C oil tester. The tester consisted of a generator, reaching 100 kV with a 2 kV/s speed rate. The cell consisted of a sphere-to-sphere electrodes configuration (12.5 mm diameter), spaced by an electrode gap of 2.5 mm. Each liquid was stressed 18 times (three series of six measurements), and the average and standard deviation (St.Dev) were calculated. First, the Anderson–Darling fitting test was executed on the BdV outcomes, and then the statistical analysis was performed using the two-parameter Weibull distribution. The Weibull fit curves were used to evaluate the AC BdV at three important probabilities (1%, 10%, and 50%).

# 3. Results and Discussion

## 3.1. Statistical *t*-Test

However, statistically speaking, the improvements could not be observed from the mean AC BdV point of view since there is an evident overlapping. Hence, a statistical comparison of the mean BdV of base liquid and different NFs was performed to conclude the effect of NPs on the BdV performances. Among the statistical tests that allow such analysis is the two-sample *t*-test [22–24]. Therefore, a two-sample *t*-test (also known as an
independent *t*-test) was performed on the data sets (pure NE—NF) to show whether they differed significantly. The *t*-value, the degrees of freedom parameters, the corresponding critical value, and the decision are presented in Table 3 (see Appendix A for the *t*-test explanation).

<table><caption>**Table 3.** *t*-test results summary.</caption><thead><tr><th colspan="6">FR3/NFs with Fullerene</th></tr><tr><th></th><th>0.1 g/L</th><th>0.2 g/L</th><th>0.3 g/L</th><th>0.4 g/L</th><th>0.5 g/L</th></tr></thead><tbody><tr><th scope="row">t-value</th><td>-1.7811</td><td>1.243</td><td>3.9044</td><td>4.7093</td><td>3.1653</td></tr><tr><th scope="row">Degree of freedom (df)</th><td>92.7</td><td>94.4</td><td>94.2</td><td>77.2</td><td>87.5</td></tr><tr><th scope="row">Critical value</th><td>&gt;1.6619</td><td>&lt;1.6617</td><td>&gt;1.6617</td><td>&gt;1.6654</td><td>&gt;1.6628</td></tr><tr><th scope="row">Decision</th><td>different</td><td>indifferent</td><td>different</td><td>different</td><td>different</td></tr></tbody><thead><tr><th colspan="6">FR3/NFs with Graphene</th></tr><tr><th></th><th>0.1 g/L</th><th>0.2 g/L</th><th>0.3 g/L</th><th>0.4 g/L</th><th>0.5 g/L</th></tr></thead><tbody><tr><th scope="row">t-value</th><td>0.4033</td><td>2.9940</td><td>3.4452</td><td>1.5726</td><td>-3.3938</td></tr><tr><th scope="row">Degree of freedom (df)</th><td>86.7</td><td>91.0821</td><td>91.6358</td><td>75.6653</td><td>89.7669</td></tr><tr><th scope="row">Critical value</th><td>&lt;1.6631</td><td>&gt;1.6623</td><td>&gt;1.6621</td><td>&lt;1.6657</td><td>&gt;1.6625</td></tr><tr><th scope="row">Decision</th><td>indifferent</td><td>different</td><td>different</td><td>indifferent</td><td>different</td></tr></tbody></table>

The calculated *t*-values, exceeding the critical value, clearly indicate that the AC BdV data of pure NE differ from those of the various NFs (i.e., all investigated NFs expect 0.2 g/L C₆₀ and 0.1 g/L and 0.4 g/L Gr NFs). Otherwise, the null hypothesis cannot be rejected, and in this case, the AC BdV data of pure NE does not differ from those of the various NFs. Therefore, the negative sign does not affect the comparison, and this sign is considered a reduction of the BdV.

## 3.2. AC Breakdown Voltage

Figures 1 and 2 illustrate the mean AC BdV and St.Dev of base liquid and different NFs with C₆₀ and Gr for different concentrations, respectively. The data and the improvements are numerically depicted in Table 4. As shown in Table 4, the mean AC BdV of 0.1 g/L C₆₀ NF decreased compared to the base liquid. Surpassing this concentration, the AC BdV increased compared to the base liquid. The highest mean AC BdV was observed with 0.4 g/L C₆₀ NF, which gave a 13.08% improvement compared to the base liquid (FR3). Accordingly, the mean AC BdV of Gr-added NF decreased for 0.5 g/L doping concentration, while it showed increased breakdown performance for concentrations of 0.1 g/L, 0.2 g/L, 0.3 g/L, and 0.4 g/L compared to the base liquid. The highest mean AC BdV was observed with 0.3 g/L Gr N, which gives a 12.79% improvement compared to the base liquid.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Mean breakdown voltages versus fullerene (C₆₀) concentrations.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Mean breakdown voltages versus graphene (Gr) concentrations.</figcaption></figure>

<table><caption>**Table 4.** Mean breakdown voltages and standard deviation values of different nanofluids.</caption><thead><tr><th>Concentration</th><th></th><th>FR3</th><th>Fullerene (C<sub>60</sub>)</th><th>Graphene (Gr)</th></tr></thead><tbody><tr><td rowspan="3">0.1 (g/L)</td><td>BdV (kV)</td><td>72.58</td><td>68.49</td><td>73.4</td></tr><tr><td>St. Dev (kV)</td><td>7.34</td><td>6.40</td><td>5.40</td></tr><tr><td>Increment %</td><td>–</td><td>-5.63</td><td>+1.17</td></tr><tr><td rowspan="3">0.2 (g/L)</td><td>BdV (kV)</td><td>72.58</td><td>75.59</td><td>79.3</td></tr><tr><td>St. Dev (kV)</td><td>7.34</td><td>7.21</td><td>6.03</td></tr><tr><td>Increment %</td><td>–</td><td>+4.15</td><td>+9.25</td></tr><tr><td rowspan="3">0.3 (g/L)</td><td>BdV (kV)</td><td>72.58</td><td>81.94</td><td>81.9</td></tr><tr><td>St. Dev (kV)</td><td>7.34</td><td>7.04</td><td>9.09</td></tr><tr><td>Increment %</td><td>–</td><td>+12.89</td><td>+12.79</td></tr><tr><td rowspan="3">0.4 (g/L)</td><td>BdV (kV)</td><td>72.58</td><td>82.08</td><td>75.8</td></tr><tr><td>St. Dev (kV)</td><td>7.34</td><td>4.39</td><td>4.24</td></tr><tr><td>Increment %</td><td>–</td><td>+13.08</td><td>+4.39</td></tr><tr><td rowspan="3">0.5 (g/L)</td><td>BdV (kV)</td><td>72.58</td><td>79.42</td><td>65.10</td></tr><tr><td>St. Dev (kV)</td><td>7.34</td><td>5.50</td><td>5.82</td></tr><tr><td>Increment %</td><td>–</td><td>+9.43</td><td>-10.35</td></tr></tbody></table>

Although most studies suggest that NPs' trapping behavior is responsible for the improvements in the AC BdV of NFs, this mechanism remains a matter of debate. The key to understanding this mechanism, the effect of NPs (conducting, semiconducting, or insulating) on the AC dialectic strength of the base liquid, lies in the appearance of surface charge distribution at the interface between NPs and the base fluid with different conductivities and/or relative permittivities, subjected to a uniform applied electric field (E₀). According to Sima et al. [25] and Morgan and Green [26], the surface of NPs contains induced and polarized charges; induced charges in the case of conducting NPs and polarized charges in the case of insulating NPs (dielectric polarization) [25,26]. Contrary to the idea that a material has only one of the two properties (induced or polarized charges), some materials have both (such as conducting Fe₃O₄ with high relative permittivity, i.e., ε = 80). In addition, there is a difference in charge density on both sides of the particle, which results in an effective or induced dipole across the particle (two hemispheres with opposite signs) aligned with the applied field [26]. This concept is known as polarizability.

According to [27], the mismatch between the base fluid's conductivities and/or relative permittivities and NPs creates a discrete interface region due to the different valence and conduction band positions of the host liquid and the nano-additive. As a result, the massive electrons in the conductive NPs move in the opposite direction of the local electric field, which polarizes them (positive and negative poles).

In the present work, the C₆₀ and Gr NPs are conductive materials (with low relative permittivity, 2.2 for C₆₀ and 6.6 for graphene), so the process of charge appearance was exclusively due to charge induction [25]. Then, the improvement in the AC BdV in a
sphere-to-sphere electrode configuration (quasi-uniform electric field) when adding con-
ducting nanomaterials to natural ester stressed with AC high-voltage source is possibly
due to the electron trapping caused by the formation of the surface charge density. The
electrons that originate from either ionization or injection move quickly and are trapped
by the positively charged pole until saturation is reached. Upon saturation, the negatively
charged NPs are considered motionless compared to the mobility of electrons, slowing the
streamer's propagation by reducing its velocity and enhancing the BdV of NFs compared
to natural ester.

The explanation mentioned above is valid for a uniform applied electric field; that is,
where the relaxation time comes in, where conducting NPs in transformer oil are induced
under the applied electric field quickly, faster than the microseconds or nanoseconds needed
for the streamer development in transformer oils. As a result, the redistribution of the
charge density immediately evens out each time the polarity of the electrodes is reversed.

In the drive to explain the negative impact of adding C₆₀ and Gr on the AC BdV of natural ester, the mechanisms are different due to differences in the concentration range: the lower limit in the case of C₆₀ NF ‘0.1 g/L’ and the upper limit in the case of Gr NF ‘0.5 g/L’.

On the one hand, the negative impact on AC BdV in the case of 0.5 g/L Gr NF could
be attributed to the formation of a conductive path between the electrode spacing due to
the reduced distance between NPs (a change in concentration from 0.4 g/L to 0.5 g/L leads
to the reduced distance between NPs). Another possible reason is the reduction of trapping
capability due to cluster formation (due to Van der Waals attraction forces), which reduces
the effective surface needed for successful trapping events.

On the other hand, the decrease of AC BdV in the case of 0.1 g/L C60 NF is not
negligible and the difference is statistically confirmed (see Table 3). To date it is difficult to
explain this decrease for this low concentration.

The reduction in distance and the formation of clusters justify the existence of an
optimal concentration, ensuring a good compromise between trapping/space among NPs.

3.3. *Weibull Statistical Analysis*

In the following, the experimental results are statistically analyzed using the Weibull
distribution law [28,29]. This statistical law is among the most frequently used probabilistic
distributions to analyze the BdV data of dielectric liquids. Therefore, it constitutes a
valuable tool in predicting breakdown voltages at low-risk levels. Hence, it provides
hopeful information about the health state of the fluid under test. Furthermore, apart
from being beneficial for the probability estimation of a BdV at different risk levels, the
Weibull distribution constitutes a means lifetime estimator of insulation liquid; hence the
equipment. A goodness-of-fit test is required to use the Weibull fit curves safely. The
Anderson–Darling fitting test was performed to check if this distribution law obeys the
experimental data (see Appendix B for the explanation of the Anderson–Darling fitting
test). Table 5 summarizes the results of the AD fitting test. Except for 0.5 g/L C60 NF, the
AC BdV data of all liquids obey the Weibull distribution.

From the above analysis, Weibull fit curves can be used for almost all cases to estimate
AC breakdown stress at different risk levels. For this purpose, the Weibull fit curves should
be plotted. First, the scale (α) and shape (β) parameters were calculated for different liquids,
and the results are depicted in Table 6. Those parameters permit the plot of the Weibull
probability fit curves versus AC BdV (Figures 3 and 4).
<table><caption>**Table 5.** AD goodness-of-fit test results.</caption><thead><tr><th>Concentration (g/L)</th><th>p-Value</th><th>Decision</th></tr></thead><tbody><tr><td>FR3</td><td>0.63</td><td>Accepted</td></tr><tr><td colspan="3">FR3/Fullerene (C<sub>60</sub>) NFs</td></tr><tr><td>0.1</td><td>0.66</td><td>Accepted</td></tr><tr><td>0.2</td><td>0.81</td><td>Accepted</td></tr><tr><td>0.3</td><td>0.60</td><td>Accepted</td></tr><tr><td>0.4</td><td>0.21</td><td>Accepted</td></tr><tr><td>0.5</td><td>0.01&lt;</td><td>Rejected</td></tr><tr><td colspan="3">FR3/Graphene (Gr) NFs</td></tr><tr><td>0.1</td><td>0.15</td><td>Accepted</td></tr><tr><td>0.2</td><td>0.61</td><td>Accepted</td></tr><tr><td>0.3</td><td>0.20</td><td>Accepted</td></tr><tr><td>0.4</td><td>0.45</td><td>Accepted</td></tr><tr><td>0.5</td><td>0.75</td><td>Accepted</td></tr></tbody></table>

<table><caption>**Table 6.** Scale and shape parameters.</caption><thead><tr><th>Concentration (g/L)</th><th>α (Scale)</th><th>β (Shape)</th></tr></thead><tbody><tr><td>FR3</td><td>75.7</td><td>21.50</td></tr><tr><td colspan="3">FR3/Fullerene (C<sub>60</sub>) NFs</td></tr><tr><td>0.1</td><td>71.19</td><td>13.98</td></tr><tr><td>0.2</td><td>75.62</td><td>12.73</td></tr><tr><td>0.3</td><td>85.07</td><td>13.30</td></tr><tr><td>0.4</td><td>84.15</td><td>19.80</td></tr><tr><td>0.5</td><td>82.05</td><td>13.81</td></tr><tr><td colspan="3">FR3/Fullerene (Gr) NFs</td></tr><tr><td>0.1</td><td>73.81</td><td>17.43</td></tr><tr><td>0.2</td><td>82.01</td><td>14.97</td></tr><tr><td>0.3</td><td>85.74</td><td>10.29</td></tr><tr><td>0.4</td><td>77.72</td><td>18.88</td></tr><tr><td>0.5</td><td>67.68</td><td>12.71</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Fullerene NFs Weibull probability plots.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Graphene NFs Weibull probability plots.</figcaption></figure>

Hence, the above curves (except C₆₀ NF at 0.5 g/L) could be used to assess the AC BdV at the most relevant probabilities (i.e., 1%, 10%, and 50%); the results are depicted in Table 7. For 1% probability, the AC BdVs of NFs with C₆₀ and Gr were optimal with concentrations of 0.4 g/L, which give improvements of 30.53% and 19.37% higher than natural ester, respectively (see Table 7). For 10% probability, the best improvements were with concentrations of 0.4 g/L C₆₀ of NPs, with a 20.32% improvement compared to natural ester and 0.3 g/L Gr of NPs, with a 12.81% improvement compared to natural ester. Finally, for 50% probability, the evaluated AC BdVs have the same tendency as the experimental results.

<table><caption>**Table 7.** The AC breakdown voltage for different probability levels, evaluated using Weibull fit curves.</caption><thead><tr><th rowspan="2"></th><th colspan="2">1%</th><th colspan="2">10%</th><th colspan="2">50%</th></tr><tr><th>AC BdV (kV)</th><th>Increment (%)</th><th>AC BdV (kV)</th><th>Increment (%)</th><th>AC BdV (kV)</th><th>Increment (%)</th></tr></thead><tbody><tr><td>FR3</td><td>51.1</td><td>-</td><td>62.5</td><td>-</td><td>73.4</td><td>-</td></tr><tr><td colspan="7">FR3/Fullerene (C<sub>60</sub>) NFs</td></tr><tr><td>(0.1 g/L)</td><td>51.3</td><td>0.39</td><td>60.7</td><td>-2.88</td><td>69.4</td><td>-5.45</td></tr><tr><td>(0.2 g/L)</td><td>54.9</td><td>7.44</td><td>66.0</td><td>5.60</td><td>76.6</td><td>4.36</td></tr><tr><td>(0.3 g/L)</td><td>60.2</td><td>17.81</td><td>71.9</td><td>15.04</td><td>82.8</td><td>12.81</td></tr><tr><td>(0.4 g/L)</td><td>66.7</td><td>30.53</td><td>75.2</td><td>20.32</td><td>82.7</td><td>12.67</td></tr><tr><td>(0.5 g/L)</td><td>58.8</td><td>15.07</td><td>69.8</td><td>11.68</td><td>80.0</td><td>8.99</td></tr><tr><td colspan="7">FR3/Fullerene (Gr) NFs</td></tr><tr><td>(0.1 g/L)</td><td>58.3</td><td>14.09</td><td>66.7</td><td>6.72</td><td>74.3</td><td>1.23</td></tr><tr><td>(0.2 g/L)</td><td>60.3</td><td>18.00</td><td>70.6</td><td>12.96</td><td>80.1</td><td>9.13</td></tr><tr><td>(0.3 g/L)</td><td>54.9</td><td>7.44</td><td>69.0</td><td>10.4</td><td>82.8</td><td>12.81</td></tr><tr><td>(0.4 g/L)</td><td>61.0</td><td>19.37</td><td>69.0</td><td>10.4</td><td>76.3</td><td>3.95</td></tr><tr><td>(0.5 g/L)</td><td>47.2</td><td>-7.63</td><td>56.8</td><td>-9.12</td><td>65.9</td><td>-10.22</td></tr></tbody></table>

## 4. Conclusions

This work aims to study the effect of conducting nanoparticles (C₆₀, and Gr) on the AC BdV of natural ester, FR3, for concentrations of 0.1 g/L, 0.2 g/L, 0.3 g/L, 0.4 g/L, and 0.5 g/L. The following conclusions can be drawn

* The *t*-test showed a significant evident difference in mean AC BdV in the case of natural ester and the different investigated NFs.
* The AC BdV of FR3 increased when adding the C60 and Gr NPs with specified doping concentrations. However, the C60 NFs showed the best improvement of the AC BdV with a 13.08% improvement with 0.4 g/L compared to FR3, while Gr NFs showed the best improvement, i.e., 12.79%, with 0.3 g/L.

* 0.1 g/L C60 NF and 0.5 g/L Gr NF harm natural ester’s mean AC BdV.

* The statistical analysis showed that all the AC BdV outcomes obey the Weibull distribution (except 0.5 g/L C60 NF). So, the fit curves were then used to evaluate the BdVs at different risk levels (1%, 10%, and 50%).

According to the results presented in this work, adding C60 and Gr NPs with well-chosen concentrations to the natural ester FR3 improves the AC BdV.

**Author Contributions:** Conceptualization, H.K., A.B. and E.V.; methodology, H.K., A.B. and E.V.; software, H.K.; validation, A.B. and E.V.; formal analysis, H.K., A.B. and E.V.; investigation, H.K., A.B. and E.V.; data curation, H.K., A.B. and E.V.; writing—original draft preparation, H.K.; writing—review and editing, H.K., A.B. and E.V.; visualization, A.B.; supervision, A.B. and E.V.; project administration, A.B. and E.V.; funding acquisition, A.B. and E.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data reported in the present study are available upon request; please refer to the corresponding author when asking.

**Conflicts of Interest:** The authors declare no conflict of interest.

## Appendix A. Two-Sample t-Test

Given two sets of data (AC BdV of base liquid and different NFs), the t-test is conducted according to the following steps:

1. The calculation of the t-value (two-sample t-test, called the Welch t-test) is given by the formula below [28]:

$$t = \frac{\bar{x}_1 - \bar{x}_2 - \Delta}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \quad (A1)$$

where $\bar{x}_1$ and $\bar{x}_2$ are the mean values of set 1 and 2, respectively, $\Delta$ is the hypothesized difference (0 for equal and 1 for unequal means), $s_1$ and $s_2$ are the standard deviations of set 1 and 2, respectively, and $n_1$ and $n_2$ are the sizes of set 1 and 2, respectively.

2. The degree of freedom (df) for the Welch t-test is given by the formula below [28]

$$df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \quad (A2)$$

The df value is then used to determine the critical values from the t-value table. Because this is a one-tailed test, the alpha level is 0.05.

3. The following step consists of looking up $t_{1-\alpha, df}$ ($\alpha = 0.05$, $1 - \alpha = 0.95$, df) value in the t-table [28], and looking for the critical value, which corresponds to the limit t-value for a known degree of freedom.

a. If t-value $>= t_{1-alpha, df}$, the null hypothesis cannot be rejected in this case, and the two data sets appear different.

b. If t-value $< t_{1-alpha, df}$, the null hypothesis is rejected in this case, and the two data sets do not appear different.
## Appendix B. Anderson–Darling Fitting Test

The conformity check of the AC BdV outcomes to the Weibull distribution was investigated using the Anderson–Darling fitting test (AD-Test) [22–24]; its formula is given below

$$AD^2(x) = -n - \sum_{i=1}^{n} \frac{2i-1}{n} \times [\ln(CDF(x_i)) + \ln(1 - CDF(x_{n+1-i}))] \quad (A3)$$

where, {$x_1 < ... < x_n$} are the ordered AC BdVs points; i and n are the number of data points in the sample and ith sample, respectively. The cumulative distribution function (CDF) of the Weibull distribution is given below:

$$CDF(x) = 1 - e^{-(\frac{x}{\beta})^{\alpha}} \quad (A4)$$

The scale (α) and the shape (β) parameters represent the AC BdV at a 63% risk level and the slope of the Weibull line curve, respectively.

The next step consists of calculating the corresponding p-value using AD statistic [23]

$$p\text{-value} = \begin{cases} e^{(1.294-5.709AD+0.019AD^2)} & AD \ge 0.60 \\ e^{(0.918-4.279AD-1.380AD^2)} & 0.34 < AD < 0.60 \\ 1 - e^{(-8.318+42.796AD-59.938AD^2)} & 0.20 < AD \le 0.34 \\ 1 - e^{(-13.436+101.140AD-223.730AD^2)} & AD \le 0.20 \end{cases} \quad (A5)$$

Finally, the p-value is compared to the significance level (0.05). Hence, if the p-value is less than or equal to 0.05, reject the null hypothesis that the Weibull distribution fits the data; else, accept the null hypothesis that the Weibull distribution fits the data.

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