Research on Typical Decay-like Fracture Defects of Composite Insulators Based on Electro-Thermal Coupling

Abstract

In response to the typical decay and fracture defects of composite insulators, a three-dimensional electrically and thermally coupled simulation physical model was constructed based on the finite element method, and the local electric field distortion and temperature rise were analyzed. The study confirms that the insulator interface’s axial electric and thermal fields show a U-shaped curve; the interface field strength is the largest. There is an electric field gradient difference between the mandrel and the sheath, and the thermal field is concentrated at the mandrel and the interface. The field strength at the edge of the defect is the largest, the aberrant electric field at the defect shows a sawtooth shape, and the temperature rise is concentrated in the defect area. The degradation is fast in the air gap, the etching hole diameter direction, and the carbonation channel axial direction. The larger the defect volume, the larger the aberration in the electric field and temperature rise. Water vapor air gaps, breakdown holes, and carbonized channels have the most pronounced electric field and temperature changes. The functional relationship between electric field aberration, temperature rise, and defect volume is established. The results provide a basis for the protection of insulator decay-like fracture.

1. Introduction

A total of 9 million composite insulators, because of their high strength, anti-fouling flash, low cost, ease of installation, and maintenance advantages, according to statistics, are used at all levels of power grids [1,2,3]. However, with the increase in operating years, composite insulators gradually exhibit problems, such as crispy fracture, decay-like fracture, unexplained flashover, bird-pecking outer insulation damage, etc. One of them is a kind of lousy insulator with abnormal fracture accidents in recent years. The decay-like fracture form is entirely different from the known fracture form of composite insulators; various accidents that affect the stability of the power system may occur, such as short circuit tripping, dropped wires, and power lines, resulting in substantial economic losses [4,5,6]. Therefore, it is necessary to study composite insulator cracking to actively respond to the dual-carbon strategy and build a new low–loss power system.

At present, scholars at home and abroad have carried out much research on composite insulators with decay-like fracture. They are committed to exploring the characteristics and mechanisms of decay-like fracture. The literature [7], through the testing and analysis of decay-like fracture insulators, shows that decay-like fracture belongs to the category of abnormal fracture, which is the result of the synchronous actions of mechanical stress, acidic medium, discharge, moisture, and so on. The literature [8,9] has studied and analyzed the formation, characteristics, and heating mechanism of the fracture of composite insulators and pointed out that fractures are formed by interface defects (between the sheath and the core bar), internal defects, and heating-induced local discharge. The literature [10] has tested composite insulator interfaces with different connecting materials under humid conditions for discharge aging, and the results showed that the hydrolysis reaction led to the degradation of the epoxy substrate at the interface of composite insulators. The literature [11] has completed surface discharge test designs and analyzed the compositional fluctuation of cracked composite insulator mandrel materials. Meanwhile, it was concluded that the hydrolysis and carbonization of epoxy resin under partial discharge conditions leads to different cracking and decay-like fracture characteristics. The literature [12] has simulated the silvering degradation process of hot and humid composite insulators and pointed out that moisture at the interface increases the dielectric constant and dielectric loss factor (at the aging point of the mandrel) and increases the abnormal temperature. The literature [13] has explored the cracking problem of 500 kV composite insulators based on theoretical thermodynamics, chemistry, and optics methods. The results show that the degradation of epoxy-based materials at the interface of composite insulators leads to glass corrosion, ion exchange, and hydrolysis of mandrel materials, which are caused by hydrolysis reactions. Coronation, ion exchange, and hydrolysis are the main degradation mechanisms of mandrel materials. The literature [14] has analyzed the mechanism of various aging factors during aging. The authors found that mandrels are significantly warmed under multiple stresses and high electric fields. The literature [15] has investigated two influencing factors, the air gap and 3D modeling of 110 kV composite insulators (water vapor-based). The authors completed the analysis of the effect of interface defect thickness, length, and span on the overall and local electric field distribution based on the finite element method. The literature [16] has calculated and analyzed the electric field around the carbonized mandrels of composite insulators. The analysis confirmed that the reason for this is that the carbonized channel distorts the spatial electric field distribution of the insulator string, and the smaller the distance between the high-voltage end and the channel, the more likely it is to cause distortion. The literature [17] has constructed a cubic function relationship between the surface temperature rise of composite insulators and the length of the carbonized channel of the core rod based on the influence of the defect location and the ambient wind speed on the thermal field distribution on the insulator surface under the carbonization defect of the core rod of 110 kV composite insulators. The literature [18] has simulated and analyzed the electric field distribution characteristics under different damage locations, lengths, widths, depths, and surface fouling conditions (composite insulator sheath damage defects). The literature [19] has established a simulation model of electro-thermal coupling of decay-like insulators, completed the statistical analysis of the infrared library of composite insulators (anomalous heating) through laboratory tests and field inspections, and obtained the temperature gradient coefficients simultaneously. The literature [20] has analyzed and studied the observations, electric field distribution, temperature rise, and discharge results of 500 kV decay-like insulators with silicone rubber sleeves and mandrels. It was found that the temperature rise and field distribution were more pronounced in decay-like insulators compared to other insulators. The literature [21] has analyzed the improvement in temperature rise due to polarization loss and leakage conduction loss of decay-like samples under low and high humidity conditions. The authors confirmed through a comparative study that other insulators contributed less to the temperature rise of decay-like rotten samples than the polarization loss. The literature [22] has completed the modeling and temperature rise calculations of corroded specimens under high and low humidity conditions and the analysis of heat sources for sheath aging based on the finite element method. From the simulation conclusions, it can be seen that the corrosion of the mandrel, the humidity at the core–sheath interface, and the aging layer of the sheath lead to the rise in temperature of the corroded specimen. The literature [23] has simulated the electro-thermal connection of conventionally operated, moisture-heat aged, partially coated, and integrally coated composite insulators and found that the electric field was distorted (defective insulators) along with a significant temperature rise. The literature [24] has completed the design of a heat transfer model (mandrel heating). The authors carried out simulation and experimental studies, and the results showed that when the composite insulator mandrel was heated, the radial temperature in the silicone rubber jacket was logarithmically distributed. The surface temperature of the mandrel was a first-order function of the jacket’s surface temperature. The literature [25] has attributed the increase in temperature of composite insulator samples to surface currents, and the simulation results confirmed the scholars’ view. The literature [26] has used infrared and ultraviolet imaging techniques to monitor polymer insulators; also, the finite element method was used to simulate the electrical behavior of corona rings and insulators. From the simulation results, it can be seen that high insulation temperature leads to polymer degradation. The literature [27] has analyzed a composite insulator’s internal heat transfer process using finite element simulations. It was modeled using pulsed and continuous thermography, and variations in the infrared heat field on its surface were recorded. Temperature and time difference curves were plotted. The literature [28] has investigated the measured results of outdoor polymer insulator shell materials, including their thermal, electrical, and mechanical properties. The results showed that the breakdown voltage strength of silicone rubber EPDM blends exceeded that of silicone rubber.

The results of the above studies were only studied experimentally for the fracture mechanism or simulated and analyzed only for the defects inside and outside of a single medium. Still, they did not consider the dynamics and correlation with the development of insulator decay-like fracture. There is little consideration of the defective parts of the current’s thermal effect, so it is not easy to get close to the actual situation. The topic was based on the principle of electric thermal coupling to complete the design of a three-dimensional simulation model (500 kV composite insulator). The thermal and local electric field distribution characteristics of composite insulators with transverse corrosion holes, carbonized conductive channels, and interface air gaps under the background of three decay-like defects were analyzed and discussed. The disturbance of local electric field aberration and temperature rise due to fluctuations in the decay-like defect volume was analyzed, and the relationship between the electric field aberration rate, temperature rise, and the change in the decay-like defect volume was constructed based on the fitting analysis. The fitting analysis establishes the quantitative calculation expression of the volume change of decay-like defects, the electric field distortion rate, and the temperature rise. From a theoretical perspective, this lays the foundation for implementing a decay-like fracture prevention plan for composite insulators.

2. Simulation Principle and Model Building

2.1. Simulation Principle

Composite insulator class decay aging is often due to sheath and mandrel interface bonding failure. Water and other dielectrics invade the interface air gap, resulting in an air gap at the electric field enhancement of the formation of partial discharge and current thermal effects; water in the electric field polarization effect will occur, resulting in insulator polarization loss caused by the insulator’s abnormal heating, and will be with the mandrel glass fibers during the ion exchange and slow hydrolysis of epoxy resin. At the same time, under discharge conditions in the air, nitrogen (N₂) has a more robust penetration capacity as nitrogen dioxide (NO₂), and the nitrogen dioxide and water (H₂O) reaction generates nitric acid (HNO₃), which erodes the epoxy resin in the mandrel further, so electric corrosion, acid corrosion, heat, and other factors together will cause carbonization of the mandrel and sheath breakdown, resulting in the reduction of the mechanical strength of the insulator in the class of decay and fracture. The following Figure 1 below shows the principle of degradation development of a decay fracture-like composite insulator.

According to the above composite insulator decay-like fracture principle, construction of the composite insulator electric-thermal coupling model mainly studies the composite insulator in different parts of the occurrence of decay-like fracture, the occurrence of electric and thermal field distribution characteristics, and the heating characteristics of the degraded parts of the insulator itself, which will be affected by the current distribution and electric field strength and are the focus of this paper’s research. Hence, the simulation model adopts the electro-thermal coupling principle to solve the problem.

When considering that the heat is generated by Joule heat, the electric field interface module in the simulation is adopted in the current model, and its mathematical model can be described as follows:

$$\nabla ·J=Q_{j,v}$$

(1)

$$J=\sigma E+J_e$$

(2)

$$E=-\nabla V$$

(3)

where ∇ is the gradient operator; J is the current density, A/m²; σ is the material conductivity, S/m; Q_j,v is the free charge, C; E is the electric field strength, V/m; J_e is the externally generated current density, A/m²; and V is the voltage, V, at any point in the electric field.

The simulation of dielectric heating is usually accomplished at a Joule heating multiphysics field interface; this multiphysics interface corresponds to the coupling of a solid heat transfer interface and a current interface. The defect site is represented by the heat source [Q_e], for which the dielectric loss density forms the heat. The following mathematical model describes the thermal field module:

$$\rho C_p{\displaystyle \frac{d{T}_2}{dt}}+\rho C_{p}u·{\nabla }_{T_2}=\nabla ·\left(k{\nabla }_{T_2}\right)+Q_e$$

(4)

$$Q_e=\nabla J·E=\omega C{U}^{2}tg\delta$$

(5)

$$q=\nabla ·-k{\nabla }_{T_2}=h\left(T_{ext}-T_2\right)$$

(6)

where ∇ is the gradient operator; C_p is the specific heat capacity, J/(kg·k); k is the thermal conductivity, W/(m·k); h is the heat transfer coefficient, W/(m²·k); u is the velocity matrix of the surrounding fluid, m/s, computed from the fluid field; Q_e is the heat source, calculated from the electric field; E is the strength of the electric field, V/m; ω is the angular frequency, (rad/s); C is the equivalent capacitance, µF; tgδ is the dielectric loss factor, q is the heat flow density, J/(m²·s); T₂ is the conductor temperature; and T_ext is the ambient temperature, °C.

2.2. Simulation Model

The research object model is the FXBW4-500/210-rod suspension composite insulator: umbrella skirt type for one large and two small (a large umbrella and two tiny umbrellas for a unit); a total of 90 large and tiny umbrellas; large umbrella spacing of 130 mm; the sizes of the umbrella skirt diameter are 174 mm and 88 mm; 30 mm is the core rod’s diameter value; and the sheath’s thickness is 5 mm. The following

Table 1. Structural parameters of composite insulator.

Structural Parameters	Numerical Value
Rated voltage/kV	500
Rated mechanical load/KN	210
Structure height/mm	4450
Insulation distance/mm	4000
Nominal creepage distance/mm	13,750
Wet frequency 1 min withstand voltage/kV	925
Lightning full-wave impulse withstand voltage/kV	2050

shows the structural parameters of the composite insulator.

The local thermal and electric field distribution of decay-like fracture defects in composite insulators is the focus of the analysis and exposition of this topic. This paper ignores the model construction of external factors (tower, conductor, etc.). Only in this way can the trend of the electric and thermal fields of the composite insulator itself (composite insulator decay fracture) be more accurately reflected. With the support of finite element simulation software, the construction of the three-dimensional model of the composite insulator shown in Figure 2a,c can be carried out. This is done to close the composite insulator’s actual size and ensure the simulation calculation’s accuracy. A 5 m × 1.5 m × 1 m rectangular air domain is completed, as shown in Figure 2b. In order to simulate the electric field distribution of a 500 kV composite insulator under actual conditions, a zero potential is applied to the low-voltage end, and a voltage of 500 kV × 1.1 × √2/√3 = 449 kV is applied to the terminal connector (high-voltage end). The following table lists some of the material parameters used in the simulation model. The settings of the material parameters in the simulation model are shown in

Table 2. Simulation material parameters.

	Air	Sheath	Umbrella Skirt	Mandrels	Fixture
Material	Air	Silicone Rubber	Silicone Rubber	GlassFiber	Steel
Relative dielectric constant	1	3.5	3.5	5	1 × 108
Electrical conductivity (S/m)	5 × 10−10	1 × 10−10	1 × 10−7	1 × 10−11	6 × 107
Thermal conductivity (W/m·K)	0.0011	0.27	0.27	0.3	33

.

In order to improve the accuracy of the analysis of composite insulator decay defects on insulator thermal and electric field interference, the convective heat transfer coefficient of air (near the insulator) is set to 0.23 W/(m²·K). According to previous cases of composite insulator decay, the actual operation of composite insulator decay shows that decay and fracture tend to develop simultaneously along the axial and radial directions. Three major decay features mainly characterize the decay defects: (1) the sheath and mandrel interface failure area between the high-pressure end and the high-pressure end connected through the carbonization channel, (2) mandrel fracture near the interface between the sheath and the glass mandrel failure area, and (3) the appearance of several sheaths from the inside to the outside of the development of transverse corrosion holes. The composite insulator decay-like fracture degradation process and development direction are shown in Figure 3.

Therefore, the geometrical characteristics of the three decay-like fracture defects must be assumed for the simulation.

Assumption 1.

According to the material properties and shape characteristics of the carbonization channel of the fractured insulator mandrel, the carbonization channel is simulated by the cylindrical conductive channel formed by the carbonization of the mandrel, and the location of the conductive channel is located at the high-voltage end.

Assumption 2.

According to the shape characteristics of the interface failure air gap, when the actual insulator is fractured, a curved column can be used to simulate the interface air gap. The interface failure is located between the first piece of the umbrella skirt and the fixture on the high-voltage side.

Assumption 3.

Due to the significant difference in the shape of the transverse erosion hole of the fractured insulator, in order to be closer to the actual situation, this paper sets the shape of the transverse erosion hole as a rectangle, and the location of the transverse erosion hole is also located in the high-voltage side of the first piece of parachute skirt between the gold fixture.

Figure 4 shows the three defects mentioned above. In order to better reflect the dynamic development process of the decay-like defects, this paper changes the volume of the three defects to indicate the change in their size and no longer conducts a targeted study of the relevant volume components.

3. Simulation Analysis

3.1. Simulation Analysis of Intact Composite Insulators

The cross-section electric field and heat field distribution of the intact composite insulator, as well as the axial electric field and heat field distribution of the interface, are shown in Figure 5. The radial cross-sectional electric field appears in a more uniform distribution state, and compared with the electric field strength of the core rod, the interface and sheath layer are significantly higher (refer to Figure 5a,b). The interface between the core rod and the sheath showed the highest cross-sectional electric field strength (E_max), 1.02 × 10⁵ v/m. The core rod and interface are the critical areas of thermal field distribution. Compared with the core rod and interface temperature, the temperature of the sheath is lower, and the temperature rise of the cross-section is smaller, with an overall temperature of 20.2 °C.

The axial electric field U-curve distribution at the interface is detailed in Figure 5c,d. The electric field strength j is concentrated around the high- and low-voltage ends. The electric field strength around the high-voltage end, especially in the gold fixture, is more likely to fluctuate. Most of the field strength is distributed at the front end of the shaft and the umbrella skirt (near the high- and low-voltage ends). The surface field strength of the insulator is negatively correlated with the distance from the high-voltage end until it rises near the low-voltage end. The maximum field strengths at the high-voltage and low-voltage ends are 8.12 × 10⁶ v/m and 3.55 × 10⁶ v/m, respectively, but the latter is approximately equal to 43.72% of the former. The overall shape of the insulator thermal field distribution is also close to the U-shape; the high-voltage end and the low-voltage end are the main high-temperature parts. There is also a high-temperature zone near the front end of the umbrella skirt and mandrel near the HV and LV ends. The farther away from the HV and LV ends, the lower the temperature in this area. The maximum temperatures at the low and high-pressure ends are 21.65 °C and 27.83 °C, respectively.

From the above analysis, it can be concluded that high temperatures and strong electric fields are usually concentrated around composite insulator’s high- and low-voltage ends. However, the high-voltage end has the highest electric field strength and temperature compared with the medium- and low-voltage end areas. Under the same circumstances, cracking and rupture are more harmful. Therefore, the occurrence of cracking defects near the high-voltage end will be the focus of the analysis and discussion of this topic.

3.2. Simulation Analysis of Composite Insulator Decay-like Fracture Defects

3.2.1. Interface Air Gap Defects

The leading cause of interface defects is bonding failure between the mandrel and the sheath interface, resulting in air gaps and water infiltration. The air gap will lead to a partial discharge at the interface, converting nitrogen (N₂) in the air into nitrogen dioxide (NO₂), which then reacts with the infiltrated water to generate nitric acid, further destroying the bonding properties at the interface and reducing the insulating properties of the insulator. In order to study in detail the effects of infiltrated air and water on the electric and thermal fields near the interface of the mandrel and sheath, the defective medium was set to be air and water. The water relative dielectric constant and conductivity were set to 81 and 5.5e⁻⁶ S/m, respectively.

Firstly, the composite insulator interface air gap model was established. Figure 6 shows the simulation results. This indicates that the cross-sectional electric field of insulators will be comprehensively enhanced under air gap defects. A maximum electric field strength E_max of 1.55 × 10⁵ v/m will be formed at the edge of the defect, which is significantly higher than the average electric field strength nearby—exceeding the interface field strength of a complete insulator by 51%. The interface and core rod are still the thermal field distribution gathering places. The temperature rise at the defect site reaches 2.1 °C, forming a maximum of 22.3 °C in the cross-section.

As shown in Figure 7, the trend of the electric field is similar to that of the thermal field (along the arc length and longitudinal direction) as long as there is an air gap at the interface. The trends of the electric and thermal fields along the direction of the air gap thickness are opposite. The electric field formed along the air gap thickness direction and arc length direction is stronger than the electric field formed along the air gap length direction. The maximum values of the field strength along the arc length, thickness, and length of the air gap are 1.451 × 10⁵ v/m, 1.326 × 10⁵ v/m, and 1.063 × 10⁵ v/m, respectively. The localized electric field along the arc length and length of the air gap undergoes aberrations, and the peaks of the aberrated electric field are 1.381 × 10⁵ v/m and 1.011 × 10⁵ v/m, respectively. The temperatures of the three cases are centered around 22.3 °C. It can be seen that when interfacial defects occur, the temperature of the interface is not significantly reduced. It can be seen that when the interfacial defect occurs, there are many electric field distortions along the arc length of the air gap and one electric field distortion along the air gap length. Therefore, radial and axial degradation at the interfacial defects co-occur, but the radial degradation rate is faster than the axial degradation rate.

For the water and air medium parameters set above, the interface air gap defects are the air gap and water vapor air gap with volumes of 0.03 mm³, respectively. Figure 8 shows the changes in the electric field distribution along the entire interface axis (air gap or water vapor gap) of the composite insulator when the interface air gap is generated. The insulator’s overall interface axis electric field distribution shows no apparent changes; interface defects appear in the pronounced local electric field waveform changes. The local electric field intensity at the air gap is distorted from 1.5 × 10⁵ v/m to 1.63 × 10⁵ v/m, and the local electric field intensity at the water vapor air gap is distorted from 1.5 × 10⁵ v/m to 2.04 × 10⁵ v/m. The distorted electric field intensity at the two defects is higher than the surrounding average electric field intensity.

In order to study the electric field distortion and thermal field changes at the defects of composite insulators before and after the generation of the air gap while keeping the position and shape of the defects unchanged, the size of the air gap volume was changed to 0.25 mm³, 0.84 mm³, 1.99 mm³, and 3.87 mm³, respectively. The medium parameters were set to air and water vapor for the simulation calculations, and data collection was performed for every change in volume. These are shown in Figure 9 as well as in Figure 10. Figure 9 shows that, after the appearance of the interface air gap defects, increasing the air gap volume synchronously increases the peak intensity of the distorted electric field at the defect (at the defect location) to 1.78 × 10⁵ v/m, 1.98 × 10⁵ v/m, 2.36 × 10⁵ v/m, and 2.8 × 10⁵ v/m, respectively. Increasing the air gap volume further accentuates the distortion of the electric field intensity profile (at the defect location). The upward and downward variations of the aberration curve show a jagged shape. Once water vapor penetrates the air gap, it continuously intensifies the waveform distortion of the air gap electric field (in raw air), as shown in Figure 10. The distorted peak electric field strength of the water vapor gap increases to 2.617 × 10⁵ v/m, 3.1217 × 10⁵ v/r, 3.43 × 10⁵ v/m, and 4.84 × 10⁵ v/m, in that order.

Figure 11 shows another thermal field calculation process and results diagram. Before and after water vapor infiltration, the temperature near the defects along the interface decreases with increasing distance when the volume ranges from 0.03 mm³ to 1.99 mm³. However, there is no noticeable change in the overall situation, and it tends to range between 22 °C and 28 °C; the temperature increases by about 2 °C compared to the intact state. When the volume of the defects is 3.87 mm³, the thermal field near the interface defects is further warmed up between 23 °C and 32.5 °C, and the temperature increases by about 3 °C compared to the intact state. Before and after water vapor infiltration, as shown in Figure 12, increasing the distortion field strength will simultaneously increase the temperature of the interface air gap; the temperature rise after water vapor infiltration is slightly higher than that of the air gap, and the highest temperatures occur in the water vapor air gap and the air gap area with 22.3 °C, 22.3 °C, 22.3 °C, 22.3 °C, and 23.7 °C and 22.3 °C, 22.4 °C, 22.4 °C, 22.4 °C, and 23.8 °C, respectively.

It can be seen that the thermal field and electric field distribution of composite insulators are affected before and after water vapor infiltration. If water vapor appears on the outer surface of the insulator’s umbrella skirt, it is easy for a water film to accumulate and may cause a dirty flash. At the same time, water vapor will also gradually penetrate the sheath to reach the failure of the adhesive interface; the formation of the water vapor gap and the water vapor aberration of the electric field are mainly due to the change in dielectric constant and conductivity before and after the infiltration of water vapor and the insulator’s temperature rise effect. The main current flow through the dielectric loss is caused by dielectric heating, so the water vapor infiltration caused by the temperature rise and electric field distortion is slightly larger.

From the above analysis, it can be concluded that composite insulator interface bonding failure results in air and water vapor infiltration. Compared with the axial electric field distortion intensity, the failure gap is more likely to appear as noticeable radial electric field distortion. Compared with the electric field intensity around the interface, the local electric field distortion intensity is higher. The core rod and sheath material will heat and crack under long-term partial discharge, resulting in sheath breakdown, and part of the region of the core rod may appear as a carbonized layer, causing corrosion of the core rod and sheath, and ultimately leading to fracture of the composite insulator.

3.2.2. Mandrel Carbonization Defects

From Section 3.2.1, it can be seen that two different media air gap defects will cause local distortion of the electric and thermal fields, resulting in partial discharges and temperature rise. The generation of nitric acid will further reduce the degree of interfacial adhesion and corrosion of the glass mandrel, resulting in carbonization of the mandrel conductive carbonized channel. Based on the location of carbonized conductive channel defects in the inner mandrel of actual decay samples, broken composite insulators, and related research on the materials, it is concluded that the material is a wholly carbonized glass fiber with high thermal conductivity and low resistivity characteristics, so the material of the carbonized conductive channel of the mandrel in the simulation model was set as carbon. The relative dielectric constant and conductivity were set to 20 and 1e3 S/m, respectively.

According to the above parameters to establish the composite insulator carbonized conductive channel model, the simulation results in Figure 13 show that, under the condition of forming a carbonized conductive channel, the overall electric field of the composite insulator cross-section increases significantly, forming a maximum electric field strength of 1.03 × 10⁸ v/m, with the edge of the defect as its key aggregation area, which is 1000 times the field strength at the interface of the intact insulator. The interface between the sheath layer and the defect becomes the critical area of its thermal field distribution. The defect area and cross-sectional temperatures are 28.7 °C and 29 °C, respectively, and the temperature rises at the whole, and the defect is more significant, respectively, 8.9 °C and 8.2 °C.

Meanwhile, from Figure 14, it can be seen that the temperature and electric field strength of the conducting channel shows an aberration increase in the length direction and a decrease in the diameter direction; the electric field strength is more significant in the length direction, and the temperature is slightly lower in the diameter direction compared to that in the diameter direction. The maximum field strength in the length direction is 1.284 × 10⁸ v/m, and the maximum field strength in the radial direction is 1.67 × 10⁷ v/m. Localized electric field distortion is formed along the channel’s length direction, and the distortion’s peak field strength is 8.114 × 10⁷ v/m. The electric field has no radial distortion, and the two temperatures are centrally distributed around 28 °C. It can be seen that when the mandrel has a conductive channel, the mandrel will have defects in both the radial and axial directions, and deterioration will co-occur. However, the axial degradation rate is faster than the radial degradation rate.

To analyze the thermal field fluctuation defects and electric field aberrations caused by composite insulators based on carbonized conductive channels, it was necessary to change the size of the conductive channel volume to 60 mm³, 480 mm³, 1620 mm³, 3840 mm³, and 7500 mm³ for the simulation calculations, respectively, under the condition of keeping the location and shape of the defects unchanged, and the data were collected once for every change in volume. The data in Figure 15a show the defect conditions of the electric field solution results. Increasing the volume of the conductive channel synchronously increases the electric field intensity in the defect region (along the sheath surface), leading to the loss of smoothness, up-and-down fluctuations (jagged) of the electric field curve, and further increasing the peak electric field intensity of the local aberrations, with peaks at 4.54 × 10⁵ v/m, 3.28 × 10⁶ v/m, 7.2 × 10⁶ v/m, 8.14 × 10⁶ v/m, and 8.26 × 10⁶ v/m. Figure 15b shows the calculation results of the thermal field; the original channel volume of 60 mm³ increases to 7500 mm³ under the condition of establishing a conductive channel. As the defect volume increases, the temperature at the defect location (along the sheath surface) increases further, with maximum temperatures of 27.78 °C, 32.03 °C, 36.17 °C, 40.6 °C, and 43.39 °C. The higher the temperature and the electric field strength along the surface of the sheath, the closer to the high-voltage end. As shown in Figure 16, under the condition that the defect volume increases from 60 mm ³ to 7500 mm ³, the distortion field strength leads to a simultaneous increase in the maximum temperature of the defect area, with the highest values of 28.8 °C, 31.6 °C, 35.4 °C, 39.4 °C, and 42.9 °C.

Therefore, it can be concluded that the electric field distortion intensity (axial electric field defects) formed by composite insulators after the carbonization of conductive channels significantly exceeds the radial electric field distortion intensity, further highlighting the local distortion of electric field intensity and significantly increasing the defect temperature. This is due to the mandrel at the defective fiberglass material carbonization, where the permittivity and conductivity increase, making the dielectric loss higher, and resulting in higher heat.

3.2.3. Transverse Galvanic Corrosion Hole Defects

From Section 3.2.1 and Section 3.2.2, it can be seen that composite insulators at the beginning of the decay-like aging stage usually exhibit a single point of failure, and then due to local discharge, acidic media, and other factors, it gradually expands in the decay-like aging development process, and radial and axial degradation co-exist during erosion. The insulator’s electrical and insulation properties will decrease due to axial carbonization (along the surface of the axis) of the conductive channel, high amplitude overvoltage will occur along the radial internal breakdown, and the breakdown aperture is also the transverse corrosion holes.

The simulation results are shown in Figure 17 with the above parameters to establish the composite insulator transverse electrical etching hole model. After the etching hole is generated, the composite insulator cross-section electric field shows an overall elevation, forming the highest electric field intensity E_max of 1.45 × 10⁵ v/m. The same is concentrated in the edge of the defects, and compared with complete composite insulators, the interface field strength of composite insulators increases by 42%. The distribution of its thermal field is mainly concentrated in the mandrel, interface, and interface defects; the temperature peaks at the cross-sectional area, the temperature in the defect zone is 22.3 °C, and the temperature rise is 2.1 °C.

When transverse electrode holes are present in the sheath, as shown in Figure 18, the trends of the electric and thermal fields along the width and length directions of the etched holes are close to each other. The development of thermal and electric fields along the depth direction of the etch hole is opposite to the former. The electric field intensity along the etch hole width direction and depth direction is more significant than that along the etch hole length direction, and the maximum field intensity along the etch hole length direction is 1.685 × 10⁵ v/m, that along the etch hole width direction is 1.26 × 10⁵ v/m, and that along the depth direction is 1.483 × 10⁵ v/m, respectively. The localized electric field along the etch hole width and length directions is distorted, and 1.667 × 10⁵ v/m and 1.26 × 10⁵ v/m are the peaks of the distorted electric field. These localized distorted fields increase the electric field in the direction of the depth of the etch hole, gradually deepening the depth of the etch hole.

Meanwhile, it can be seen from the axial and radial electric fields that the localized aberrant electric fields are mainly concentrated in the middle of the width direction and length direction of the etch holes, which leads to the inconsistent development of etch holes and the formation of etch holes that are small on the outside and large on the inside. Therefore, radial and axial degradation will occur at the sheath when transverse electrical etch holes appear in the sheath defects. However, the radial degradation rate is faster than the axial degradation rate.

The position and shape of the defects remained unchanged to analyze and explore the thermal field changes and local electric field distortions of composite insulators before and after the occurrence of transverse galvanic corrosion holes. The volume size of the transverse galvanic corrosion holes was changed to 40 mm³, 240 mm³, 720 mm³, 1600 mm³, and 3000 mm³, respectively. These were simulated sequentially and separately; every time the volume changed, the data were collected. The electric field results are shown in Figure 19a and Figure 20, and the electric field curve becomes rougher and rougher. Figure 19a shows that when the defect volume increases from 40 to 3000 mm³, the electric field along the surface of the sheath at the defect is aberrant, the aberrant electric field strength increases sequentially, and the electric field curve becomes rougher and rougher, showing up and down fluctuations. The peak values of the distorted electric field strength are 1.46 × 10⁵ v/m, 1.64 × 10⁵ v/m, 1.9 × 10⁵ v/m, 2 × 10⁵ v/m, and 3.94 × 10⁵ v/m, respectively.

Figure 19b and Figure 20 are graphical representations of the thermal field solution data, where the temperature along the surface of the sheath at the defects does not change significantly when the transverse galvanic corrosion holes become larger in volume, and the temperature change is relatively concentrated between 20 °C and 27 °C. With increasing volume of the defect, the maximum value of the aberration field strength at the defect increases. The maximum value of the temperature at the defect increases as well, and the maximum values of the temperature are 22.4 °C, 22.6 °C, 22.9 °C, 23 °C, and 23.1 °C, respectively.

From the above analysis, it can be seen that the radial electric field distortion intensity is greater than the axial electric field distortion intensity after the emergence of transverse galvanic corrosion hole defects. In addition, for the volume of 3000 mm³, the distortion intensity and temperature rise are the largest, which is due to the volume of the transverse galvanic corrosion holes at this depth being equal to the thickness of the sheath, that is, the sheath within the breakdown, at this time, the defects at the junction of the phases of the sheath, air, and the mandrel and the relative permittivity and conductivity increase caused by the electric field aberration and temperature rise.

4. Establishment of Electric Field Distortion Rate and Temperature Rise Formulas for Decay-like Defects

4.1. Establishment of Formulas for Calculating the Electric Field Distortion Rate and Temperature Rise of Shortening Defects

Based on the above simulations and analyses, it can be seen that the changes in the volume of decay-like defects of the three composite insulators mentioned above are related to the electric field distortion and temperature rise in the vicinity of insulator defects. In order better to describe the change of the local distortion field and temperature rise with the volume of decay-like defects, this paper analyzes and explores in depth the effect of the fluctuating volume of the decay-like fracture defects on the distribution of the local thermal field and electric field by taking the rate of the electric field distortion and the magnitude of the temperature rise as the variables and quantitative mathematical equations are constructed. The following equations describe the electric field distortion rate and temperature rise:

$$\partial ={\displaystyle \frac{E_{max}-E_0}{E_0}}$$

(7)

where $\partial$ is the electric field distortion rate; $E_{max}$ is the maximum value of electric field distortion at the defect, v/m; and $E_0$ is the maximum electric field strength value at the defect when the insulator is intact, v/m.

$$∆T=T_{max}-T_0$$

(8)

where $∆T$ is the temperature rise; $T_{max}$ is the maximum temperature value at the defect, °C; and $T_0$ is the maximum temperature value at the defect when the insulator is intact, °C.

Through the simulation analysis above, the maximum value of the electric field strength and the maximum value of the temperature at each defect when the insulator is intact, the maximum value of the electric field distortion at the defect, and the maximum value of the temperature at the defect are substituted into Equations (7) and (8), and the results are shown in

Table 3. Distortion rate and temperature rise at the interfacial air gap.

Interfacial Air Gap
Volume (mm3)	Air Gap Electric Field Distortion Rate	Water Vapor Gap Electric Field Distortion Rate	Air gap Temperature Rise (°C)	Water Vapor Gap Temperature Rise (°C)
0.03	0.09	0.36	0.8	0.8
0.25	0.19	0.75	0.8	0.9
0.84	0.32	1.08	0.8	0.9
1.99	0.58	1.29	0.8	0.9
3.87	0.87	2.23	2.2	2.3

,

Table 4. Distortion rate and temperature rise at the conducting channel.

Conductive Channel
Volume (mm3)	Electric Field Distortion Rate	Temperature Rise (°C)
60	0.05	0.9
480	6.59	3.7
1620	15.67	7.5
3840	17.84	12
7500	18.12	15

and

Table 5. Distortion rate and temperature rise at transverse galvanic corrosion holes.

Transverse Corrosion Holes
Volume (mm3)	Electric Field Distortion Rate	Temperature Rise (°C)
40	0.19	0.9
240	0.36	1.1
720	0.58	1.4
1600	0.66	1.5
3000	2.27	1.6

.

From Table 3, Table 4 and Table 5, with the increase in the defect volume, the decay-like fracture defects at the electric field distortion rate and temperature rise show an upward trend. Figure 21 shows the average electric field distortion rate and average temperature rise size, according to the size of average electric field distortion rate row for the conductive channel > water vapor gap > transverse corrosion holes > air gap, according to the size of the temperature rise row for the conductive channel > transverse corrosion holes > water vapor gap > air gap. From the above analysis, it can be seen that interface bonding failure and water vapor intrusion are the causes of insulator cracking, the core rod carbonized conductive channel accelerates the main factors of cracking, and transverse corrosion holes are caused by the core rod carbonized conductive channel caused by the local electric field aberration caused by the breakdown. The results are basically in line with the principle of cracking and the development process of composite insulator cracking.

4.2. Model Validation of Electric Field Distortion Rate and Temperature Rise for Decay-like Defective Insulators

In order to verify the accuracy of the established simulation model, the measured average electric field distortion rate of 0.39 near the high-voltage end of the 500 kV early decayed and fractured insulator in the literature [8] and the measured minimum abnormal temperature rise of 30 K of the decayed and fractured insulator identified in the literature [20] were selected for the control group. The two datasets were used as the control group to compare the average electric field distortion and average temperature rise of the interfacial air gaps, the conductive channels, and the transverse galvanic erosion holes, respectively. The simulation calculation results were compared to verify the correctness of the simulation model.

Through the comparative analysis in Figure 22, it can be found that the average electric field distortion rate and temperature rise of the three defective insulators calculated in the simulation were more significant than the average electric field distortion rate and abnormal temperature rise of early decay-like insulators, which satisfied the primary conditions of the insulators’ decay-like fracture, and therefore fully proved the accuracy of the constructed simulation model of the insulators’ electro-thermal coupling, which could be used for the subsequent study of the electric field distortion rate of the decay-like insulators and the law of the abnormal temperature rise.

4.3. Establishment of Equations for the Electric Field Distortion Rate and Temperature Rise Versus Volume Fitting for Decay-like Defects

Now, the simulation results of electric field distortion rate and temperature rise are fitted to analyze the effect of defect volume change on the local electric field distortion rate and temperature rise. Because the electric field distortion and temperature rise laws of the water vapor gap and the air gap are close to each other, only the air gap is fitted. The water vapor air gap is no longer fitted. The fitting equations of the electric field distortion rate and temperature rise and the fitting diagrams are shown as follows, where the vertical coordinates are the electric field distortion rate ${\partial }_1$ and temperature rise ${∆T}_1$ at the interfacial air gap and the electric field distortion rate ${\partial }_2$ and ${∆T}_2$ at the conductive channel, and the vertical coordinates are the electric field distortion rate ${\partial }_3$ and ${∆T}_3$ at the transversal erosion holes. The horizontal coordinates are the change in the volume of the interfacial air gap ${∆V}_1$ and the volume change of the conductive channel ${∆V}_2$, and the horizontal coordinate is the volume change of ${∆V}_3$. According to the trend of the data points, the exponential function and polynomial function are used to fit the data points, respectively, and the fitted equation and the fitted Figure 23 can be obtained as follows:

$${\partial }_1=1.38679-1.29204\ast {0.78932}^{{∆V}_1}$$

(9)

$${∆T}_1=0.79142+0.11066{∆V}_1-0.17878{{∆V}_1}^2+0.06311{{∆V}_1}^3$$

(10)

$${\partial }_2=18.19561-19.65211\ast {0.99883}^{{∆V}_2}$$

(11)

$${∆T}_2=15.91035-15.02996\ast {0.99964}^{{∆V}_2}$$

(12)

$${\partial }_3=0.14327+0.00113{∆V}_1-9.20326e^{-7}{{∆V}_1}^2+2.59929e^{-10}{{∆V}_3}^3$$

(13)

$${∆T}_3=1.57957+0.73037\ast {0.9982}^{{∆V}_3}$$

(14)

The calculations show that the fitting effect indicators, R squared (COD), of the above fitting functions are 0.99826, 0.99967, 0.99846, 0.9977, 0.99994, and 0.99267, respectively, which are very close to 1, thus indicating that the fitting effect is good. The fitting results confirmed that synchronizing the defect volume increases the two parameter values of the rise in electric field temperature and the distortion rate at the interface air gap. As shown in Figure 23, in the volume range of 0.03 to 1.99 mm³, the growth rate is not particularly fast, and the rate of increase increases after the volume is more significant than 3.87 mm³. Increasing the defect volume usually synchronously increases the electric field temperature rise and distortion rate at the conductive channel site; the rate of increase of the electric field distortion rate is greater than the rate of temperature rise between volumes of 60 and 1620 mm³. The electric field distortion rate increase is lower than the temperature rise rate when the volume exceeds 1620 mm³. Increasing the defect volume will simultaneously increase the two parameter values of the rise in electric field temperature and the distortion rate in the lateral erosion hole area. The volume change range is 40–1600 mm³, and compared with the rising speed under the condition of the volume exceeding 1600 mm³, the growth of the electric field distortion rate is slightly lower; the temperature rise rate will decrease when the volume exceeds 720 mm³. By analyzing the fitting results, it can be observed that if we continue to increase the defect volume, the temperature and field aberration rate parameters will inevitably increase further.

4.4. Fitting Error Analysis and Test

The smaller the fitting error of the curve, the better the fitting effect because the more ideal the fitting effect, the more advantageous the accuracy of the equation for solving the electric field temperature rise and distortion rate at the decay-like fracture defect site of the composite insulator and the verification process of the fitting effect. The equations for the rise in the electric field temperature and the distortion rate in the region of decay defects are solved with high accuracy. The rise in the electric field temperature and the aberration rate (for each decay defect) obtained from the simulation and solved equations are substituted into Equation (15). Figure 24 shows the specific detection results. The fitting errors of the electric field distortion rate of the 15 simulation and formula calculation groups are all within ±5%. The fitting errors of the temperature rise of the 15 simulation and formula calculation groups are all within ±5%; the overall error has not exceeded the acceptable range. This also confirmed that the fitting accuracy of the two parameters of electric field temperature rise and distortion rate at the decay-like fracture defect site was consistent with standard expectations.

$$\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}={\displaystyle \frac{\left|Simulation value-Formula value\right|}{Simulation value}}\times 100\%$$

(15)

5. Conclusions

Three decay-like defects of interfacial air gaps, carbonized conductive channels, and transverse corrosion holes occur in decay-like fractured composite insulators. According to the characteristics of the decay-like fracture defects, a three-dimensional electro-thermal coupling (500 kV composite insulator) simulation model was constructed to analyze and discuss the interference state of similar decay-like defects on the local thermal and electric field distribution of composite insulators, and the following conclusions were drawn:

• The distribution of the radial cross-sectional electric field (intact composite insulator) is relatively balanced. An objective difference in the electric field gradient exists between the core rod and the sheath. The field strength of the core rod is slightly smaller than that of the sheath, and there is an extreme field strength at the interface. The interface and core rod become the thermal field’s main distribution area, and the core rod’s temperature is higher than that of the sheath layer. There are differences in the axial thermal field and electric field distribution within the interface area. The electric field at the low- and medium-voltage ends is not high, and the temperature rise and voltage drop are mainly distributed around the high- and low-voltage ends.
• When a composite insulator has a decay-like fracture defect, the increase in the overall electric field across the insulator cross-section leads to a simultaneous increase in temperature. The average field strength in the vicinity is more vital than that in the center of the defect, and the mandrel, the interface, and the defect become the concentrated areas of thermal field distribution. By contrast, the edge of the defect becomes the area of maximum field strength distribution. The radial degradation of the interfacial air gaps and transverse galvanic holes is rapid, as is the axial degradation of the conductive channels.
• When the volume of a decay-like defect increases, the waveform of the electric field located at the defect is distorted. The distorted electric field profile varies up and down (sawtooth) and increases the temperature. When water vapor intrudes into the interface, the electric field distortion and temperature rise further. The presence of conductive channels and breakdowns within the sheath distort the electric field strength and make the temperature rise pronounced.
• Through the fitting analysis, the simulation results aligned with the principle and development process of composite insulator fracture. With the change in defect volume, the conductive channel electric field distortion rate, conductive channel temperature rise, interface air gap electric field distortion rate, and transverse corrosion hole temperature rise are approximately an exponential function of the growth trend, and the transverse corrosion hole electric field distortion rate and interface air gap temperature rise are approximately a cubic function of the growth trend. From the fitting results, it can be further hypothesized that if the defect volume is further increased in actual operation, the field aberration and temperature rise enhancement will be more pronounced, accelerating the insulator cracking process.