Investigation of Survival/Hazard Rate of Natural Ester Treated with Al 2 O 3 Nanoparticle for Power Transformer Liquid Dielectric

: Increasing usage of petroleum-based insulating oils in electrical apparatus has led to increase in pollution and, at the same time, the oils adversely affect the life of electrical apparatus. This increases the demand of Mineral Oil (MO), which is on the verge of extinction and leads to conducting tests on natural esters. This work discusses dielectric endurance of Marula Oil (MRO), a natural ester modiﬁed using Conductive Nano Particle (CNP) to replace petroleum-based dielectric oils for power transformer applications. The Al 2 O 3 is a CNP that has a melting point of 2072 ◦ C and a low charge relaxation time that allows time to quench free electrons during electrical discharge. Al 2 O 3 is blended with the MRO and Mineral Oil (MO) in different concentrations. The measured dielectric properties are transformed into mathematical equations using the Lagrange interpolation polynomial functions and compared with the predicted values either using Gaussian or Fourier distribution functions. Addition of Al 2 O 3 indicates that 0.75 g/L in MRO has an 80% survival rate and 20% hazard rate compared to MO which has 50% survival rate and 50% hazard rate. Considering the measured or interpolated values and the predicted values, they are used to identify the MRO and MO’s optimum concentration produces better results. The test result conﬁrms the enhancement of the breakdown voltage up to 64%, kinematic viscosity is lowered by up to 40% at 110 ◦ C, and ﬂash/ﬁre points of MRO after Al 2 O 3 treatment enhanced to 14% and 23%. Hence the endurance of Al 2 O 3 in MRO proves to be effective against electrical, physical and thermal stress. The relationship between hottest spot temperature and long-term operations for both mineral oil and vegetable oil-based transformer are determined under medium voltages.


Introduction
The power transformer (PT) is vulnerable and is the most expensive power system network equipment. It provides functions such as stepping up and stepping down the voltage between electrical power generating stations. The PT works at 100% load for 24 h throughout the day whereas the distribution transformer works at 50% or 70% of the full load [1]. Increase in the power demand across the world is persistent and it is expected to be double in the forthcoming years as per the predicted data of International Energy Associations [2]. This demand has caused the rise in PT numbers in the power system network. The report released by allied market research indicates that countries in the Asia-Pacific (APAC) are the biggest market for the future transformer industries [3]. Here, a projected growth of USD 3 million is expected by 2025, which almost strikes a peak of 6.9% Compound Annual Growth Rate (CAGR) from the year 2020 [4]. This growth is brought about by the huge demand for power, rapid urbanization, and PT's replacement. The report published by the Allied Market Research estimated that the consumption of MO in 2014 was nearly 1437.8 million litres [5]. It further predicted that MO's sale was to reach 3.4 billion United States dollar by 2020 which represents a 6.3% increase from 2015. Upon extrapolating the data with the same growth rate up to 2030, the consumption of MO conditions. The functions are used to derive the bath-tub curves used in the reliability analysis, but in our case, we focus on the response of oils to nanoparticle and their stability to accelerated ageing.

Marula Oil (MRO)
Marula or Sclerocarya birrea is a drought-resistant plant species, found in Southern Africa, Botswana. The tree can grow a diameter of 2 m and produces fruits [26]. The nuts are crushed to produce MRO, which has alcohol content range from 1% to 7% [27]. The oil contains amino acids, fatty acids, oleic and linoleic, flavonoids, catechins, and procyanidin and high amounts of antioxidants [27]. Oleic acid in MRO has a melting point of 360 • C, which helps the antioxidants to resist high-temperature stress. MRO naturally contains oleic acid, which shows high temperament against temperature stress. In addition, the oil contains vitamin-E (tocopherol), which naturally inhibits free electrons during the discharge phenomena [24]. Natural esters containing antioxidants helps to prevent formation of valence electrons during electron avalanche. Thus, natural ester retards ionization delays breakdown in oil. On the other hand, the procyanidin in the oil acts as an excellent antioxidant and inhibits reactive oxygen, thereby preventing the oil from ionization.

Conductive Nanoparticle (CNP)
The theory of nanoscience has its application as a drug carrier in various fields of engineering and health. These nano-sized particles (10 nm) are highly stable, and it is complicated to predict their physio-chemical characteristics. The application of CNP in the transformer liquid dielectric starts with the nanoparticle's ability to capture free electrons during the discharge or streamer application [28]. Discharge or streamer is initiated by the ionization; a process by which valence electrons are separated from shells and start bridging a conductive path between electrodes. This establishes a short-circuited path in liquid dielectric and results in dielectric breakdown [21,24]. The CNP traps the fast-moving electrons during the discharge mechanism and slows the process either by neutralizing or slowing down the ionization. This increases the breakdown strength of the liquid dielectric and reduces ageing. Unlike magnetic and semi-conductive nanoparticles, CNP is influenced by the direction of the electric field. This leads to CNP coming into action whenever there exists a strong field between the two electrodes [29]. Al 2 O 3 shows very low charge relaxation time, metal chelation, and electron scavenging properties. This retards free electrons that are responsible for electron avalanche, reducing energy density of avalanche chain. It is also known that CNP improves the breakdown voltage at a concentration of 0.5 g/L in the host fluid as mentioned by Raymon et al. [28]. Al 2 O 3 possesses amphoteric structure and it does not react with water in the presence of Natural Esters. Since Al 2 O 3 is slower to saturate in water, it is used for adsorption purification of oils as adsorbent to capture hydrocarbon impurities from the air. The effect of Al 2 O 3 has been well utilized in this study to understand the behavior in MRO and MO, and its suitability as an additive in liquid dielectric coolant for the transformer.

Dielectric Behavior of Host Fluids
MRO and inhibited MO are collected from Botswana and India, respectively. Both MRO and MO are considered as host fluids, and their dielectric characteristics are measured as per IEC (International Electrotechnical Commission) [30] and ASTM (American Society for Testing and Materials) International [31][32][33][34] standards and presented in Table 1.
The dielectric characteristics like breakdown voltage, kinematic viscosity and flash/fire point are considered in the statistical study. These parameters are much likely to influence other parameters of the liquid dielectric; hence, it is necessary to investigate the behavior analytically. The analysis involves the estimation of survival rate and hazard rate by populating the data points of the dielectric characteristics of base fluids as density and cumulative probability distributions. The study makes use of the dielectric parameters such as breakdown voltage, kinematic viscosity, flash point, fire point to validate them with hazard function and survival function. Breakdown voltage of the liquid dielectric is defined as the maximum withstand voltage capacity of the liquid dielectric under standard room temperature and pressure. Usually, the breakdown voltage is measured by filling the measuring cup (test cup) with 500 mL of the liquid, spherical electrodes fully submerged in the liquid. Gap spacing between electrodes is 2.5 mm and 50 Hz ac test voltage is increased using the control knob at a rate of 2 kV/s until breakdown. The voltage is recorded as breakdown voltage of the sample. Between the successive measurements, a 2 min relaxation time is given, and the test cup is inspected for the carbon formation, then the oil is slowly stirred to avoid formation of bubbles. The test is repeated according to IEC 60156 standards and measurements are recorded [30]. Similarly, the kinematic viscosity is measured using a Redwood viscometer. Kinematic viscosity is defined as the shear stress of the liquid against its flow. The test cup of the Redwood viscometer is filled with the liquid and placed on the water bath, which is heated using the electric heater. The liquid temperature is measured using the thermometer setup and the orifice is controlled manually to drain the oil to a vessel placed under the viscometer. The time of flow is measured using the stopwatch and kinematic viscosity is calculated by considering the pipette and burette constants according to ASTM D445 standards [32]. On the other hand, the fire point and flash point are measured using the Pensky-Martens closed cup apparatus according to ASTM D93 standards [33]. The test cup is filled with the oil and placed in a water bath then heated using an electric heater. The temperature of the liquid is measured using a thermometer and flame is introduced to a small opening in the top surface of the test cup. At some temperature, the liquid begins producing vapour that can be a combustible source when oxygen is introduced to the source in the presence of a fire source. This phenomenon is termed as flash point (smoke point). Likewise, when the combustion becomes continuous for at least 5 s, it is called as fire point (ignition point). Although there are no direct connections between the dielectric properties, they can be individually affected by factors like concentration of the nanoparticles or temperature, which is used to describe the changes in the dielectric properties.

Assessment of Breakdown Voltage of Host Fluids
The dielectric breakdown voltage of MRO and MO is distributed as a density plot shown in Figure 1. Readings were taken two times a day for 30 successive days. During this period, the samples are stored in a dark container to protect the fluid from photo oxidation and dust particles. Density plot is a continuous plot of variables that are random in space. The breakdown strength of MRO is higher than that of MO, with MRO showing a wider range of breakdown voltage as compared to MO. MRO's breakdown voltage limits are from 36 kV to 47 kV while MO falls between 22 kV and 33 kV. The maximum strength of MRO breakdown voltage lies between 45 kV to 47 kV. On the other hand, MO shows endurance between 22 kV to 25 kV and 30 kV to 33 kV. This indicates that most of the breakdown strength's measured values lie within these indicated regions for both MRO and MO. Breakdown endurance is due to the presence of CNP that retards free electrons detached from valence shell and then reduces the energy density of the electron chain between electrodes. Figure 2 illustrates the cumulative probability of random breakdown voltage of MRO and MO that occur at different voltage instances (sequential events). The sequential events are independent of each other, and it is critical not to have two events occur simultaneously [35].  A wider range of breakdown voltage of the MRO indicates that it experiences a lesser effect on the successive breakdown events. The density plot also shows a quick recovery between breakdown events. Such recovery is due to the presence of natural antioxidants in MRO. Hence, it is hypothetical and practical to conclude that MRO has a lesser propensity to electrical stress. Reliability analysis such as survival function and hazard function are used in this study to understand and compare the endurance of the liquid dielectrics. Survival function determines the number of data points that survive over time or in other words the data points that fail in the expected duration of time. Survival function is determined using the Kaplan-Meier function. A simple way to calculate the distribution of the survival function is through a hypothesis which evaluates the risk over a constant time i.e., λ(t) = λ. From the expression of distribution function; F(t) = 1 − exp(−λt), the survival function is; S(t) = exp(−λt), can be expressed as S(t) = 1 − F(t). Likewise, hazard function indicates the rate at which the data points experience hazard over time or in other words it is the likelihood or frequency of failure per unit time. The above survival and hazard functions are used to determine the reliability of the liquid dielectric. Such mathematical analysis is therefore essential to realize the bathtub function of the liquid dielectric. When the distribution becomes exponential, the hazard function becomes; λ(t) = F(t)/S(t) [36]. According to Figure 3, the proportion of MRO surviving 40 kV is 70%. On the other hand, 80% of the time MO survives only 26 kV and 20% of the time it survives 30 kV. MRO shows 50% survivalat 44 kVwhile MO is at 28 kV. This indicates that MRO has a superior survival rate than MO. From Figure 4, at 50% cumulative hazard, MRO is less likely to fail until it reaches a breakdown voltage of 45 kV. It is also evident that MO experiences a 50% hazard at 32 kV and above voltages. Both λ(t) and H(x) from Figures 3 and 4 shows some data exceeding the confidence bounds of MO. This clearly indicates that MRO is likely to follow the upper survival bounds. Most of the time, it is less likely to fail above 40 kV. Here, the Weibull parameters are obtained from the generated Figures 3 and 4 with a shape parameter "K > 1". This shows the rate of failure increases with time as ageing goes on with time. Similarly, the scale parameter "λ" is assumed to be greater than "0" to spread the distribution evenly. Figure 5 shows the failure probability of MRO and MO for 30 days from the breakdown voltage tests as per IEC standard [30]. The probability of failure of the dielectric fluid is the ability of the fluid to show a constant breakdown strength during successive testing. The probability of failure is measured at the breakdown voltages of MRO (47 kV) and MO (32 kV). Typically, MRO shows no changes upto 5 days, while MO shows 40% probability to breakdown from day 1 of testing. By day 15, MO has 87% breakdown probability which is a 47% rise from day 1 while MRO has a 60% rise for the same period. There is an inflexion point on day 23, where MRO shows the same rate of failure as MO. MO has shown slower growth in failure rate even though its probability of failure was initially 40% higher than that of MRO. This indicates that electrical stress has a greater impact on MRO than MO from the inflexion point on day 23. In this case, MRO needs enhancement by adding a CNP such as Al 2 O 3 .

Preparation of the Nanofluids
The nanofluids are prepared by amalgamation of MRO and MO with nano sized Al 2 O 3 in different concentrations from 0.1 g/L to 2 g/L with concentration increase interval of 0.25 g/L. They are heated up to 100 • C and treated in an ultrasonication bath with 30 kHz as mixing frequency. The oil moisture is removed during the heating process. The nanofluid is collected in a closed container after the procedure and kept at room temperature out of reach of sunlight to avoid ultraviolet and photo oxidation. The nanofluid prepared using MRO shows a fine dispersion of Al 2 O 3 since it contains oleic acid which naturally allows the additive to float in the host fluid. Comparatively, the MO-based nanofluid shows agglomeration after 5 h and needs repeated dispersion using the ultrasonication bath.

Nanoparticle Impact on the Breakdown Voltage
The impact of nanoparticle on the breakdown voltage of MRO and MO is presented for varying concentrations starting from 0.1 g/L, and 0.25 g/L increment upto 2 g/L. The addition of Al 2 O 3 has resulted in improved breakdown voltage of the host fluids as seen from Table 2. There is a constant rise in the breakdown voltage when Al 2 O 3 is added from the concentration of 0.1 g/L to 0.75 g/L and it begins to decline after further additions of Al 2 O 3 . This variation is seen from Table 2 where the highest enhancement of breakdown voltage is measured at 0.75 g/L with 26.6% for MRO and with 31.2% for MO. A composite cross-section of liquid-nanoparticle helps to reduce the random nature of breakdown of fluid by energy reduction achieved by field grading. This implies that breakdown is an extreme-value process. For further additions of Al 2 O 3 at 1.75 g/L, enhancement is lost for MRO, showing a decline of 4.4% from the host's original value. A similar behaviour is measured for MO at 2 g/L with a 6.7% decline. Both oils do not experience a major stochastic change in the breakdown voltage with the addition of CNP. % enhancement from Table 2 indicates that addition of nanoparticles increases the electron trapping density of the liquid when an ac breakdown voltage is applied between the electrodes. The increase in the kinematic viscosity by the addition of CNP above 0.75 g/L distracts the capture crosssection of the liquid molecules. As a result, the mean free path is very short for breakdown to occur, which gradually decreases breakdown voltage after 0.75 g/L concentration of CNP. A polynomial interpolation function is developed by considering the breakdown voltage of MRO and MO with the concentration of the Al 2 O 3 . The Lagrange interpolation polynomial is found for a given set of different breakdown voltage readings. The polynomial function P(x) is calculated for points 'x j ' which is given in Equations (1) and (2).
The Lagrange Interpolation helps to find the polynomial which takes certain values at arbitrary points. When more data points are used for developing a polynomial function, greater data turbulence is observed between data points. The breakdown voltage data is used to develop a mathematical function by concentrating on four limits for reducing the polynomial and data turbulence order. Equations (3) and (4) represent the Lagrange polynomial interpolation functions of MRO and MO, respectively, as calculated for the breakdown voltage variations for the addition of Al 2 O 3 . Here, U(x) is breakdown voltage at the instant and U is the maximum withstand capacity of oil. The generated generalized form of polynomial function for MRO and MO is given in Equation (5). Equation (5) is then used to estimate arbitrary values of the breakdown voltage of the host fluids transformed using the nanoparticle.
A Gaussian function shown in Equation (6) is a function for predicting arbitrary real constants. It is a probability density function that calculates the value of the distribution function for the specified concentrations between a range. Here, the concentration is a random value "x", which is usually distributed as seen from Figure 6a,b. Though the data points are typically distributed, from Table 3, the measured maximum breakdown voltage is higher than the predicted voltages. The predicted values face deviation of 0.82 kV and 0.24 kV for MRO and MO, respectively, with low root mean square error (RMSE). This shows that there is no data disorder for the confidence interval of 95%. Lagrange interpolation is useful in finding arbitrary points and the development of the mathematical functions. At 95% confidence, a maximum conformity (R 2 ) is achieved in Lagrange interpolation polynomial function. Another reason for using the Lagrange interpolation polynomial is that for a given set of points (with no two values equal), the function assumes the lowest degree at each value the corresponding value. This helps the functions to coincide with each other at each point. Other interpolation functions seem to be not fitting well with the arbitrary points.

Assessment of Survival and Hazard Functions of Breakdown Voltage
The density plot of the breakdown voltage for the fluids with varying concentrations of Al 2 O 3 is presented in Figure 7a,b. From Figure 7a, the frequency of breakdown voltage for MRO is highest when 0.75 g/L of the nanoparticle is added, and density (%) begins to rise for the breakdown voltage from 59 to 67 kV. Similarly, without the addition of nanoparticles MRO has the lowest density of 10% with a range of 33 to 47 kV while a maximum of 6% density is observed for the 0.  The cumulative hazard function for the breakdown voltage of MRO and MO under varying concentrations of Al 2 O 3 is presented in Figure 9a,b. There are two to three outbounds of the cumulative hazard as seen in Figure 9a for Al 2 O 3 concentrations of 0 g/L, 0.5 g/L, and 0.75 g/L. For the 2 g/L concentration as seen from Figure 9a,b, with both overreaching upper and lower limits of Weibull distribution, the hazard rate is 70%. The data for MRO at 0.5 and 0.75 g/L concentrations has closely followed the Weibull distribution as shown in Figure 9a. Moreover, 0.75 g/L addition shows a narrow distribution which is less likely to survive than the other concentrations. In the event of cumulative breakdown voltage, the hazard rate for MO is higher as seen in Figure 9b for the 2 g/L Al 2 O 3 concentration. Only in the 0.5 g/L and 0.75 g/L concentrations of Al 2 O 3 the hazard functions follow the Weibull distribution with wider extremities out of the 95% confidence limit. It is also observed that the survivors of MRO are higher than those of MO from Figure 10a,b. The data of MO is completely out of the 95% confidence limits for 0 g/L and 2 g/L concentrations of Al 2 O 3 . From observation, it can be concluded that the wider the range of the survivor breakdown voltage, the lesser the probability of survival as can be seen from Figures 9b  and 10b. However, the trend is narrow for MRO, especially for the 0.75 g concentration, which shows the highest data survival from Figures 9a and 10a.

Nanoparticle Impact on Kinematic Viscosity
The kinematic viscosity of MRO and MO for the various concentrations of Al 2 O 3 and temperature is studied and presented in Table 4. The kinematic viscosity of MRO and MO is measured at temperatures of 0 • C, 30 • C, 60 • C, 90 • C, and 110 • C while increasing the concentration of Al 2 O 3 to assess the molecular relaxation of the fluid at the same time. The results show a promising reduction in kinematic viscosity at 0.75 g/L for all temperatures with a reduction of 40% at 0 • C, 66% at 30 • C, 54% at 60 • C, 66% 90 • C, and 62% at 110 • C for MRO. The interpolation polynomial function for the kinematic viscosity under varying temperatures is shown in Equations (7)- (11). Here, V(x) is kinematic viscosity at the instant and V is the actual measured kinematic viscosity. The generalized interpolation function for the kinematic viscosity under various temperatures is presented in Equation (12).
V(x) V = −9.8904x 5 + 30.303x 4 − 31.3381x 3 + 12.7129x 2 − 2.0508x 1 + 1 (10) f (x) = a 0 + a 1 cos(xw) + b 1 sin(xw) (13)  The concentration at which the kinematic viscosity changes with temperature is estimated using Fourier distribution with Equation (13) and its parameters are presented in Table 5. The interpolation function is used to determine the arbitrary points between the concentrations of Al 2 O 3 . The data is used to determine the Fourier distribution through which the predicted curve is generated. The measured and predicted distribution of MRO's kinematic viscosity is presented in Figure 11a-e. The dispersion of the nanoparticle is well examined up to a concentration of 0.75 g/L Al 2 O 3 . Above 0.75 g/L, MRO begins to show viscosity rise. The dispersion of nanoparticles in the host fluid never agglomerate due to the presence of oleic acid [37,38] in MRO. The presence of oleic acid acts as surface coating for the CNP and reduces the density in liquid. Moreover, the surface coating protects the CNP from thermal stresses and releases CNP at the time of ionization process. Both low and high temperatures affect the viscosity of MRO. The negative sign (from Table 5) in the decrement clearly shows that viscosity reduction is significant when compared to the host fluid at 0 or 0.1 g/L concentration of A similar effect from Table 6 can be seen in MO with an impressive reduction at 0.75 g/L at all temperatures with the drop of 46.67% at 0 • C, 53.85% at 30 • C, 50% at 60 • C, 42.86% at 90 • C, and 75% at 110 • C. Unlike MRO, MO shows saturation at all temperatures above 0.75 g/L concentration of Al 2 O 3 . The interpolation polynomial function for various temperatures is presented in Equations (14)- (18). Here the trend shows better conformity (R 2 ) than in MRO, with a low RMSE. The generalized interpolation function for the kinematic viscosity under various temperatures is presented in Equation (19). The predicted curve is generated using the Fourier distribution given in Equation (20), and its parame-ters are presented in Table 7. Comparison of measured and predicted curves for MO is presented in Figure 12a-e.

Assessment of Survival and Hazard Functions of Kinematic Viscosity
The density plots from Figures 13 and 14 show the temperature effect on MRO and MO's kinematic viscosity at various concentrations of Al 2 O 3 . The concentration of data points of viscosity at high temperatures for MRO is populated within the range of less than 25 cSt while MO has a wider range up to 32 cSt. It can be seen that at concentration of 1 g/L of Al 2 O 3 , the viscosity of MO increases by 22% from the host fluid viscosity value. From Figures 15 and 16, the cumulative probability of MRO and MO at various concentrations of Al 2 O 3 indicates that above 50% probability the temperature has reduced the viscosity of the fluids. There exists inverse relationship between temperature and oil's viscosity according to the above results and results of [21,24]. According to that, when temperature increases, the viscosity begins to reduce. Moreover, temperature rise increases the volume of the oil and reduces consumption of oil in transformers. It can also be observed that at 50% probability, MRO and MO at 110 • C have a viscosity less than 11 cSt and 7 cSt, respectively.
It is evident from Figures 15 and 16 that molecular relaxation is highly effective in MRO than MO. At a temperature of 90 • C from Figure 15, the kinematic viscosity begins to shift from 23 cSt at 60 • C to 12 cSt, which is not realized in MO. Moreover, both Figures 15  and 16 show saturation of viscosity above the 70% probability with a much steeper increase than the previous curve or inflexion points. The MO's saturation in Figure 16 supports the argument that irrespective of temperature, the concentration of Al 2 O 3 begins to increase the viscosity followed by agglomeration.       The nanoparticle concentration effect on kinematic viscosity reverses after 0.75 g/L in MO as seen from Figure 20 with a 20% probability. However, the impact on MRO from Figure 18 is slower and shows that at 30 • C MRO experiences saturation at 0 • C with 20% probability. Whenever a steeper rate is observed from Figures 19 and 20, there is less survival within the 95% confidence bounds, which is a risk for both MRO and MO. There is more significant data turbulence in the steeper curve with insufficient proximity to the Weibull curve, which is higher for MO as seen in Figure 22 than MRO in Figure 23. The cumulative hazard, on the other hand, supports the conclusion devised from the survival rate of MRO and MO. From Figures 21 and 22, 20% of the data points experience hazards whenever the saturation starts. It is also desirable to maintain a minimum concentration of Al 2 O 3 to avoid agglomeration and thermal conductivity alteration that results in local hot spots.

Nanoparticle Impact on Flash Point and Fire Point
MRO and MO's thermal characteristics are well discussed by calculating the absolute difference and relative difference between the flash and fire points. At various concentrations of Al 2 O 3 in MRO and MO, the flash and fire points are modified as seen from Table 8. Al 2 O 3 is an excellent heat conduction additive that significantly alters the fluids thermal characteristics. Although Al 2 O 3 increases the viscosity of the fluids above 0.75 g/L concentration, it also affects the flash and fire points. The addition of nanoparticles increases the viscosity and reduces thermal conduction. Even though viscosity decreases with an increase in temperature, it increases above 0.75 g/L addition, resulting in a rise in thermal conduction. Effectively, this leads to a drop-in flash and fire points above 0.75 g/L concentration. Comparing the absolute and relative difference of MRO and MO's flash and fire points from Table 8, there is a significant absolute and relative difference of 34 • C and 12.41% observed for MRO. MRO absolute difference is almost four times higher than MO. On the other hand, MO has shown 8 • C and 4.79%, which need rapid extinction whenever smoke is identified. The drop is well predictable in MO as it shows clear evidence of the temperature drop below the base values of flash point and fire point. It is also evident to have a minimum of 7 to 10% relative difference between the flash and fire points reported by Raymon et al. [5]. The Lagrange interpolation polynomial function for flash point and fire point for MRO and MO is presented in Equations (21)- (24). Here, FLP(x) and FRP(x) is fire point and flash point measured at the instant. The generalized Lagrange function and Fourier function (predicted) for the varying concentration of Al 2 O 3 is given in Equations (25) and (26). The parameter for Fourier function is presented in Table 9. The graphs obtained using Lagrange and Fourier functions shown in Figure 23a-d show lower data deviation for MRO and higher data deviation for MO, which is a clear indication that the actual parameter is closely predicted with a deviation of ±3 • C and ±0.2 g Al 2 O 3 concentration.
f (x) = a 0 + a 1 cos(xw) + b 1 sin(xw) The density (%) and cumulative curves for the flash point and fire point of MRO and MO are presented in Figure 24a,b and Figure 25a,b. From Figure 25a, MRO has the highest data traces covering 230 to 291 • C, which is 58% higher than MO as seen in Figure 25a. The density of MO shows the highest residues in the range 150 to 163 • C from Figure 24a. Figure 25b shows that its probability to maintain 160 • C flash points is only 40%. MRO is likely to keep 240 • C with a probability of 40%, which is almost an 80 • C difference to ensure the safety of solid transformer insulation and core components. Figure 25b shows that MRO's fire point within the 40 to 60% probability range is nearly 258 to 268 • C. Similarly, for MO at 40 to 60% probability, the range is between 158 to 160 • C. This in particular shows that the temperature difference between the flash point and fire point of MRO and MO is 10 • C and 2 • C, respectively, which indicates that MRO is comparatively safer than MO.
Therefore, the determination of survival and hazard for the flash point is adequate to discuss because the relative percentage between the flash point and fire point is already discussed. The survival rate and cumulative hazard rate of MRO and MO's flash point characteristics are presented in Figure 26a,b. At 50% probability, the data points of fire point of MO leading to survival is 159 • C and MRO is 242 • C. At 90% probability, MO begins to show a poor survival rate and increased hazard, while MRO shows less hazard rate. Moving from the 50% to 90% probability, MO can survive at a maximum of 140 • C. On the other hand, MRO shows survival of up to 222 • C. There is a difference in the temperature drop from 50% probability to 90% probability observed for MO and MRO which is 19 • C and 20 • C, respectively. The decline is mainly due to the concentration rise of Al 2 O 3 above 0.75 g/L, thereby affecting saturation in the fluid's dielectric properties.

Conclusions
The results of statistical analysis shows the treatment of natural esters like Marula oil with Al 2 O 3 enhances the oil's electrical, physical, and thermal characteristics. The survival function and hazard function enable analysis of the host fluids' dielectric properties treated with Al 2 O 3 , as well as the electron scavenging properties of Al 2 O 3 . An oil's stability is accessed by subjecting the oil to accelerated heating, it also shows the temperament of the oil after accelerated ageing. At the same time, it helps to derive mathematical function that actually fits the linear relationship of the parameter, which are then used for prediction of the future events using the supervised and unsupervised methods. The rate of increase in breakdown voltage at 0.75 g/L concentration of Al 2 O 3 shows the Al 2 O 3 nature helps to capture the free electrons generated during the inception of ionization. The addition of CNP is effective at electron trapping, reducing the mean free path between the two electrons. The increase in the valence electron trap density particularly with the CNP is moreover reduced with a small concentration (0.75 g/L) of Al 2 O 3 than the higher concentration. Thus, the addition of Al 2 O 3 delays ionization and improves the breakdown voltage of the MRO. The presence of oleic acid in MRO ensures a fine dispersion without the need to coat the CNP with oleic acid and at the same time improves the dielectric breakdown voltage of the insulating fluid. MRO treated with CNP is thermally stable and show endurance for an extended time. A higher absolute and relative difference is seen in MRO than MO. This indicates that the fluid possesses high heat resistance and immunity to temperature stress. The statistical reliability analysis helps to do reliability analysis (time to failure) and enables one to evaluate the lifetime behavior of the insulating oil when subjected to ageing. According to the observations recorded for the different samples, MO become no longer useful and loses its endurance. Such modeling is normally required when actual testing is complicated or impractical. The addition of Al 2 O 3 to MRO acts as a superior nano-ester insulating fluid and it will perform well in actual transformer conditions. The superior nano-ester insulating fluid can serve various power apparatus like circuit breakers, reactors, capacitors, and cables. Thus, maintaining the life of the equipment for an extended period without the risk of failure.

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