Risk Probabilistic Characteristics for Contaminated Porcelain Insulator in the Egyptian Sinai Desert

. Transmission lines in the desert are exposed to the desert environment, which includes sandstorms as one of its hallmarks. A conductive layer develops with prolonged sand deposition and the presence of moisture, ambient humidity, and dew. Te ensuing leakage current causes surface discharge, which limits the life of the insulator and interrupting the power supply. Te locations of power lines in the Egyptian Sinai desert, where sandstorms are known to occur frequently, are exposed to such a risk. In order to estimate the danger of insulator failure, this paper studies the fow of leakage current on porcelain insulators that have been contaminated with sand. Tis work relies on accurate data collected and published in a prior study regarding Sinai, which mainly focused on contaminating sand’s grain sizes. Porcelain insulator is simulated using fnite element method to determine the leakage current that results on its contaminated surface. Te probabilistic characteristics of the insulator’s leakage current are derived using Monte Carlo technique, allowing for the risk assessment of insulator failure. Tis assessment can be used to justify the suitability of using this kind of insulator in Sinai.


Introduction
Transmission lines in the desert area are subject to harsh environmental conditions, including sandstorms. During sandstorms, the sand particles can also accumulate on the insulators and other components of the transmission lines, leading to reduced performance and increased risk of failure [1][2][3][4]. When an insulator's surface becomes contaminated with pollutants such as dust, salt, or other conductive materials, it can create a conductive path for electrical current to fow through. Tis can lead to a leakage current, which can cause heating and deterioration of the insulator's surface [5][6][7]. Over time, this can weaken the insulation and increase the risk of a power outage or even a catastrophic failure of the power line. To prevent leakage current on insulators surfaces, regular maintenance and cleaning are necessary such as water washing, mechanical brushing, or abrasive blasting. Ultimately, the cost of insulator cleaning must be weighed against the potential costs of a power outage or insulation failure, which can be much more signifcant. Te level of contamination and salinity, which in turn infuences the conductivity of the contamination layer, determines the amount of the leakage current on a polluted insulator surface [8][9][10][11][12]. Porcelain insulators are commonly used in high voltage power lines due to their excellent electrical and mechanical properties. However, the surface of porcelain insulators becomes contaminated with pollutants and afects their performance [13,14]. In desert atmospheric weather, the thickness of the pollution layer that built up on the insulator surface is dependent on the type of soil present in the area as well as the size of the contaminating sand grains [15]. Te leakage current would fow on the insulator's surface when sand is deposited and there is a signifcant source of wetting, such as dew in the morning. Tis current can cause heating of the conductive sand layer, which in turn can lead to the formation of dry bands. Tese dry bands can then result in surface fashover, which can damage the insulator and cause power outages. It is therefore important to monitor the contamination level on insulators to prevent such occurrences [16][17][18][19][20].
It is clear that there have been several previous studies conducted to investigate the efects of pollution on insulators. Tese studies have employed various analytical and experimental methods to estimate parameters such as current density distribution, fashover, and leakage current under diferent conditions. For example, a study estimated the risk and performance of composite insulator in Sinai desert [15,21], while another study computed the current density distribution and fashover along the polluted insulator surface in desert pollution [1,22]. Another one considered the amount of salt in the contaminated layer when simulating the leakage current [4,23]. In addition, numerical and experimental investigations were used in some studies to calculate the fashover on porcelain polluted insulators and estimate the dielectric strength [2,24]. Other investigations have used statistical methods to assess the risk of leakage current caused by various contamination layers along insulators such as glass or silicone rubber [3,10]. Tese studies are important in understanding the behavior of insulators under diferent pollution levels and designing appropriate measures to maintain their performance [25].
Tis research focuses on the primary cause of insulator failure under power-frequency voltage, which is leakage current on surface pollutants. Te 3D COMSOL software program, which is based on the fnite element approach, is used to simulate insulators. A conventional two-shed porcelain insulator that could be utilized on 220 kV power lines is used as a simulated case study. Tese leakage current distributions are computed with various sand conductivities and sand grain thicknesses using actual data from a previous study based on sand samples gathered from the Sinai desert near the transmission line [9]. In that investigation, the salinity and subsequent conductivity of random samples of soil from the desert are used to determine the statistical distributions of the size of the sand grains. Te overall probability density distribution of leakage current is derived using statistical distribution mapping based on the estimated efects of sand grain size and salinity on the resulting leakage current. Finally, the risk of insulator failure is calculated using the cumulative statistical distribution of leakage current. Tis research provides valuable insights into the factors that contribute to insulator failure under powerfrequency voltage and can help in developing strategies to prevent such failures.

Methods
Tis research uses a 220 kV porcelain long rod insulator with technical specifcations data according to IEC 60274 and IEC 60383. Tese specifcations defne the requirements for insulators used in high voltage power systems. Te dimensions of this insulator are provided in Table 1, and the insulator is shown in Figure 1(a). Te dimensions of the insulator are important factors in determining its electrical performance and its ability to withstand the environmental factors. Te use of standardized specifcations and testing procedures helps ensure that insulators used in power systems meet minimum performance requirements and are safe and reliable.
Te insulator model was built using the fnite element method software by defning the material of porcelain. Since analyzing the distribution of leakage current along the entire insulator is a time-consuming procedure that is unnecessary, a sector consisting of two sheds with a creepage distance of 355.3 mm is selected to simulate the leakage current. Te boundary conditions, including local potential and electric feld, resulting from those conditions were set around this sample sector of the insulator. A suitable fnite element meshes were employed to the sector of insulator as shown in Figure 1(b). Te tangential (surface) current is then produced from the directional components of the leakage current density as a result from the software. Tis step analyzes the distribution of leakage current in the selected sector of the insulator.

Results and Discussion
Te boundary conditions are applied to the chosen sample insulator simulation section, where the potentials at the sample sector's two ends, as determined by the global analysis. Te values of the potentials at the ends of the sample are 53.142 kV and 48.952 kV. Te potential distribution is infuenced by several factors, including the insulator's shape, surface condition, and the electric feld in the surrounding environment. Te distribution may be nonlinear due to many factors such as the surface leakage current, corona discharge, tracking, and erosion [26]. Tis work assumes a linear potential distribution in clean and contaminated cases. Sand grains with diameters between 1 and 2 mm, which predominate in Sinai according to statistical distributions of sand particle sizes published in a previous study [15], are selected. Tis range of grain sizes is used in the research, and it is assumed that sufcient accumulation will result in an equal-thickness contaminated layer. Tus, the contaminated layer along the entire insulator has a thickness ranging from 1 mm to 2 mm. In addition, a chemical analysis of the collected samples indicated the pollution layer's equivalent salt content (ESC, measured in mg of salt per g of sand). It was found that the most likely salinity range to be found in Sinai was between 0.5 and 1.5 mg salt/g sand [15].
Te solution salinity is frst derived from the below expression in order to transform the salt content expressed , as provided by the chemical analysis, into pollution layer electrical conductivity (μS/m) [9,17].
where S a is the salinity of the solution and W is the quantity of sand (mg) that is placed on insulator's surface with a specifc volume of water. Salinity of the layer is then linked to the solution's electrical conductivity as follows [4]: where σ 20 is the conductivity (μS/m) at a temperature of 20°C. Te quantity W can be written out using the lattice geometry theories as follows: where λ is the lattice arrangement density, which is the measure of how much sand actually occupy a certain region, and ρ is the specifc gravity of wet sand (1.92 g/ml) [15]. According to calculations, the parameter λ can be within the range from 0.523 to 0.740 depending on how compact the system [17]. Tis work uses lattice arrangement density of 0.523 as a worst-case study to get higher quantity of sand on insulators.
To get the necessary electrical conductivity, the aforementioned relations are applied over the ESC's specifed operating range. With the value W � 2.1 g/ml, Table 2 displays the various conductivities of sand grains taken from the Sinai desert according to their ESC range. In order to fnd the statistics of the leakage current, the values shown in Table 2 are considered in polluted insulator simulation. Te impact of those conductivities in each contaminated layer of the leakage current on the insulator surface is simulated.
Te distribution of leakage current along the sector of insulator creepage distance is shown in Figure 2 for 1 mm thickness of contamination layer with conductivity of 284.9 μS/cm. Te surface leakage current is calculated by the surface integration of the current densities, and it is 65.5 mA. Figure 3 shows the impacts of various conductivities on the current density distribution on insulator creepage distance at 2 mm thickness of contaminated layer. Also, Figure 4 shows the impact of diferent contamination layer thickness on the current density distribution at conductivity 284.9 μS/cm. All the results for surface leakage current are summarized in Table 3 for diferent cases of conductivities and contamination layer thicknesses.

Dependence of Leakage Current on Sand Grain Size and
Conductivity. Numerical derivation is involved to fnd the relationship between leakage current with contamination layer conductivity and sand grain sizes. Te relationship between leakage current and conductivity with diferent grain sizes is shown in Figure 5. In addition, the relationship between leakage current and grain size at diferent conductivities is shown in Figure 6. Te interdependence of the leakage current on contamination conductivity and the thickness of grain size on insulator surface are used to derive the statistics of leakage current, which become the basis for the insulator failure risk assessment. Tis joint dependence of leakage current is derived numerically and the characteristics are depicted in Figure 7. According to these   analyses, the results indicate that changing the conductivity has a signifcant efect on the leakage current, while the efect of grain size is relatively small.

Probability Density Function of Leakage Current.
Te statistical analysis of the leakage current in terms of two variables: contamination layer thickness and its conductivity is derived depending on the abovementioned data. Tese variables are treated as random variables, which mean they can be described statistically. Te leakage current can also be considered a random variable, with its probability density distribution being a combination of the conductivity and thickness of the contamination layer. Te paper also assumes that the random variables x (conductivity) and y (grain size) are independent. Tis means that the probability density function of the random variable z (leakage current) can be calculated as the product of the probability density functions of x and y, i.e., P(z) � P(x) * P(y).    Depending on the samples collected of sands in Sinai [9], the frequency distribution of ESC was calculated and then the probability density function P(x) was derived. It was found that P(x) is represented as Beta distribution; its average is 298.7 μS/cm, and standard deviation is 557.4 μS/cm [9]. Also, the frequency distribution of grain size was calculated and then the probability density function P(y) was derived. It was found that P(y) is represented as lognormal distribution; its average is 0.401 mm, and standard deviation is 0.346 mm [15].
Te Monte Carlo technique is used to derive the probability density function of leakage current, P(z), based on the data mentioned earlier. Te Monte Carlo technique is a mathematical simulation technique that is often used to incorporate risk into quantitative analysis [15]. Te Monte Carlo simulation involves creating models of potential outcomes and replacing each factor that has intrinsic uncertainty with a range of values, represented by a probability distribution. Ten, the simulation repeatedly calculates the outcomes using a new set of random values drawn from the probability distributions. Depending on the number of uncertainties and the ranges assigned to them, the Monte Carlo simulation may require thousands or tens of thousands of recalculations. Te probability function of leakage current is calculated by repeatedly generating new sets of random values of the variables (i.e., conductivity and the grain size). Te procedure for carrying out these calculations is described in detail in reference [15]. Te overall probability of leakage current is formed using a sufciently large sample of calculated values, which are displayed in Figure 8. It is found that the best distribution to represent the leakage current is Weibull distribution. Te probability density function is expressed as follows: Te statistical parameters in (4) are λ � 41.4 and k � 0.52. Te mean value of leakage current is 81 mA, and the standard deviation is 22 mA with a square error of 0.02348. Te average value represents the expected value of the leakage current based on the given statistical parameters. It is worth noting that these statistics data will vary for diferent cases, as they depend on the specifc conditions and variables that are involved in each case. Terefore, the statistical analysis should be carried out for each specifc case separately, in order to obtain accurate and relevant results.

Risk Failure Assessment.
Te probability distribution of leakage current can be used to evaluate the chance of insulator failure. Te cumulative probability of the leakage current can be generated using the probability density distribution. Tis cumulative probability function shows the maximum limits of the leakage current that should not be exceeded for certain risk value and thus indicates the probability of insulator failure. Te resulting cumulative probability function can be displayed graphically, as shown    Journal of Electrical and Computer Engineering 5 in Figure 9. Te fgure shows the danger of insulator failure in conditions of desert pollution for a specifc insulator located in the currently stated environment and operating at a 220 kV power line. Te critical leakage current value of 200 mA is used as an example to illustrate the risk of insulator failure. According to Figure 9, this critical leakage current value indicates a 68% probability of insulator failure. In summary, the probability distribution of leakage current provides a useful tool for evaluating the risk of insulator failure under diferent conditions. Te resulting cumulative probability function shows the probability of exceeding a given leakage current value and can be used to inform decisions and actions related to insulator maintenance and replacement.

Conclusion
Tis work describes the process of simulating the leakage current density on porcelain insulator under various conditions of desert pollution using the fnite element method. Te simulation considers diferent contaminated layer thickness and salinity (conductivity) levels. Te result of the simulation is the total magnitude of leakage current on a porcelain insulator sector under the given conditions. Based on the simulation results, the relation between leakage current, conductivity, and grain size is estimated numerically. Te leakage current is expressed statistically as a bivariate random variable with conductivity and grain size as independent variables. Te Monte Carlo technique is used to derive the probability density function of leakage current from this bivariate distribution.
Te resulting probability density function is ftted to a lognormal distribution with a mean leakage current value of 81 mA. Tis statistical distribution provides an accurate representation of the leakage current distribution under the given conditions. Te cumulative probability function of the leakage current is used in this study as a direct tool for assessing the risk of insulation failure. Tis function shows the probability that a given leakage current value will be exceeded, and thus, it indicates the risk of insulator failure.
In the future work, it is required to study the performance of diferent types of electrical insulator (silicone rubber and glass) at higher voltage system of 500 kV. Tis is also existed in Sinai desert and is afected by the same contamination conditions.

Data Availability
Te data used to support the study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.