In this study, the cable of model YJLW03-64/110 1 × 800 mm² and its intermediate joint were chosen as the research subjects. The axial dimensions of the joint, shown in Table 1, were used as the simulation parameters to establish a three-dimensional electromagnetic-thermal multi-physics coupling model of the cable joint.

The structure and radial thickness of the YJLW03-64/110 1 × 800 mm² intermediate joint are shown in Table 2. In the table, XLPE and PVC refer to crosslinked polyethylene and polyvinyl chloride.

Figure 1 shows the two-dimensional axial cross-sectional view of the cable joint. Tables 3, 4 list the material parameters of electromagnetic field and temperature field respectively for the cable joint model (Xu et al., 2024).

To simplify the calculation of the electromagnetic field, the model and analysis in this paper are based on the following assumptions: First, there is no movement of free charges in the magnetic field. Second, except for the copper conductor and the metal shielding, other materials are isotropic and homogeneous. Finally, when the cable operates at the power frequency (50 Hz), the displacement current density is much smaller than the conduction current and can be neglected. Based on these assumptions and Maxwell’s Equations, the magnetic vector potential A ˙ is introduced and can be expressed as Equation 1 (Jang and Chiu, 2007): ∇ · 1 μ ∇ A ˙ = − J ˙ + j ω σ A ˙ (1) where μ is the magnetic permeability, H/m; A ˙ is the magnetic vector potential, Wb/m; J ˙ is the current density, A/m²; ω is the angular frequency, rad/s; and σ is the electrical conductivity, S/m.

According to heat transfer theory, the steady-state heat conduction equation can be expressed as Equation 2: ∇ · λ ∇ T + Q v = 0 (2) where λ is the thermal conductivity, W/(m·K); T is the temperature, K; and Q v is the heat source per unit volume, W/m³.

For the boundary conditions, the temperature field boundaries must satisfy the effective range of heat transfer, ensuring that the temperature or temperature gradient remains stable. The boundary conditions for the magnetic field must meet the rapid attenuation of the magnetic vector potential. The specific boundary conditions are shown in Figure 2. A cylinder surface forms the boundary of the cable joint. S0 represents the side surface of the cylinder boundary, S1 and S2 represent the top and bottom surface of the cylinder boundary, and S3 represents the outer surface of the cable.

1) Boundary conditions for electromagnetic field

In the air domain, the magnetic vector potential A ˙ rapidly attenuates to 0 T m. Therefore, the magnetic vector potential A ˙ at the boundary of the air domain, located 500 mm radially away from the cable joint, is set to 0 T m, as shown in Equation 3: A ˙ S 0 = 0 (3)

In the axial direction, the cross-section of the cable body at a distance of 3000 mm from the joint is the magnetic insulation boundary, as shown in Equation 4: n × A ˙ S 1 、 S 2 = 0 (4) where n is the boundary normal vector.

2) Boundary conditions for temperature field

Convective heat dissipation consists of two parts: natural convection and thermal radiation. According to the Stefan-Boltzmann law, Equation 5 describes the the thermal radiation for the cable surface: − λ ∂ T ∂ n S 3 = σ 0 ε T f 4 − T a m b 4 (5) where σ 0 = 5.67 × 10 − 8 is the Stefan-Boltzmann constant, W/(m²·K⁴); ε is the surface emissivity; T a m b is the initial ambient temperature of the joint, K; and T f is the surface temperature of the joint, K. Based on the actual operation conditions of the cable, the surface emissivity ε of the joint is taken as 0.6.

In this article, the calculation of convective heat dissipation is based on Equation 6: − λ ∂ T ∂ n S 3 = h T f − T a m b (6) where h is the convective heat transfer coefficient, W/(m²·K), taken as 5.6.

It is generally considered that the temperature gradient of the cable is 0°C/mm at a distance of 3000 mm from the joint center, indicating that the cable temperature no longer changes, as shown in Equation 7: − λ ∂ T ∂ n S 1 、 S 2 = 0 (7)

Contact resistance is caused by insufficient mechanical strength during the crimping of the conductor and the connection tube. The specific resistance value cannot be precisely calculated. Therefore, to simplify the calculation, an equivalent conductivity model is used to simulate the actual conductivity between the connection tube and the joint conductor. Furthermore, the equivalent thermal loss of the contact resistance is calculated. The equivalent model of the contact resistance is shown in Figure 3.

S1 to S5 are the contact surfaces between the connection tube and the joint conductor. σ 1 and R 1 are the electrical conductivity and radius of the cable conductor, respectively, while σ 2 and R 2 are the equivalent electrical conductivity and radius of the connection part, respectively. L is the length of the connection tube.

According to the equivalent conductivity model, the contact coefficient k is employed to represent the magnitude of the contact resistance. The specific expression for the contact coefficient k is: k = 1 σ 2 · l π · R 2 2 / 1 σ 1 · l π · R 1 2 = σ 1 σ 2 · R 1 R 2 2 (8) k = 1 represents the ideal state of the intermediate joint, where there is no contact resistance between the conductor and the connection tube. However, in practical engineering, the contact coefficient k of cable joints is greater than 1, which means there is a large contact resistance between the joint conductor and the connection tube. To ensure that the simulation model reflects the real operation state, only cases where k ≥ 1 are considered in this paper. According to Equation 8, the equivalent conductivity σ 2 at the conductor connection of the intermediate joint is obtained by Equation 9: σ 2 = σ 1 k · R 1 R 2 2 (9)

The electrical conductivity of the conductor is a function of temperature. Therefore, the expression for the equivalent conductivity σ 2 in a temperature-varying environment can be obtained by: σ 2 = σ 20 k · 1 + α T − 20 · R 1 R 2 2 (10) were σ 20 is the conductor conductivity at 293.15 K, S/m; α is the temperature coefficient, 1/K; T is the conductor temperature, K.

The formulas for calculating the current density J ˙ and the thermal loss density Q v inside the intermediate joint are Equations 11, 12: J ˙ = ∇ × 1 μ ∇ × A ˙ (11) Q v = 1 σ J ˙ 2 (12)

By substituting Equation 10 into Equation 12, the expression for the equivalent thermal loss at the conductor connection of the intermediate joint is: Q v = J ˙ 2 σ 2 = J ˙ 2 σ 20 · R 1 R 2 2 · k · 1 + α T − 20 (13)

The equivalent conductivity curves corresponding to different contact coefficients k are plotted in Figure 4.

The equivalent conductivity σ ₂ is related not only to the contact coefficient k but also to the conductor temperature T. The conductivity at the conductor connection is set according to Equation 10. Based on the electromagnetic-thermal coupling model, the equivalent thermal loss for different contact coefficients is calculated. By setting the operation current I = 824 A , the ambient temperature T a m b = 298.15 K , the contact coefficient k = 1 , 3 , 5 , 7 , the simulation results of the equivalent thermal loss distribution at the joint cross-section are shown in Figure 5.

According to the simulation results and Equation 13, the linear conductivity and the operation temperature jointly determine the equivalent thermal loss of the joint conductor. As the contact coefficient k increases, the loss density of the joint cross-section also gradually increases. When the contact coefficient k = 1 , the maximum thermal loss density of the cable joint conductor is only 1.36 × 10⁴ W/m³. When the contact coefficient k = 7 , the maximum thermal loss density of the cable joint conductor is 5.46 × 10⁴ W/m³, an increase of 4.1 × 10⁴ W/m³ compared to k = 1 .

The temperature simulation results of the cable joint operating without contact resistance are shown in Figure 6. The axial direction follows the conductor center, from the left side to the right side. 0 mm and 6,000 mm correspond to the two ends of the model, and the center of the cable joint is located at 3,000 mm. The radial direction is along the cross-section at the middle of the joint, from the conductor center to the outermost layer. The temperature and temperature gradient distributions without contact resistance exhibit the following characteristics:

In the radial direction, the conductor temperature of the cable joint is the highest, and it decreases toward the outer layers, which is consistent with the normal operation condition of the cable. The axial temperature distribution of the cable intermediate joint exhibits a “U” shape, with lower temperature at the joint and higher temperature on both sides of the cable body. The axial temperature difference between the joint and the body conductor is 2°C. This indicates that in an ideal joint model, when both the joint and the cable body are made of copper, the temperature rise of the conductor is primarily determined by the cross-sectional area of the conductor. The larger cross-sectional area of the joint conductor results in lower resistance and thus a slightly lower temperature compared to the cable body.

The largest temperature gradient occurs at both ends of the cable joint, reaching 1.5 × 10⁻³ °C/mm, while the temperature gradient of the cable body at 3,000 mm is nearly reduced to zero. The variation in radial temperature gradient of the cable joint is significantly greater than the variation in radial temperature, so the temperature gradient can better reflect the temperature characteristics of the cable joint. Under the same operation current and ambient temperature, the radial temperature variation of the cable joint is at most 4°C, while the temperature gradient increases from 0°C to 0.1°C/mm. This indicates that the sensitivity of the temperature gradient is much higher than that of the temperature. In the joint model, the maximum temperature gradient is located in the insulation layer, followed by the semi-conductive layer, with the minimum value in the metallic part. Moreover, the temperature gradient distribution in the insulation layer is uneven. The radial temperature gradient variation of the intermediate joint is much greater than the axial variation, indicating that the material properties significantly affect the temperature gradient.

In actual operation, there is significant contact resistance at the conductor connection of the cable joint, resulting in a noticeable temperature rise at the cable joint compared to the body. The high temperature, when acting on the cable joint over a long period, can easily lead to overheating and damage, thus limiting the overall operation current of the cable. By setting the contact coefficient k = 7 and the ambient temperature T a m b = 25 ℃ , an electromagnetic-thermal multi-physics simulation was conducted to calculate the temperature field distributions of the cable joint with contact resistance, as shown in Figure 7.

Because of the contact resistance, the joint temperature increases dramatically by 44°C and is much higher than the body temperature. At the same time, the cable body temperature also rises compared to the temperature without contact resistance. The temperature difference between the joint and the body is 20°C, while the radial temperature difference at the joint is 15°C. Both radial and axial temperature differences are much larger than in the case without contact resistance.

The temperature gradient difference between the joint and the cable body is 1.5 × 10⁻²°C/mm, and the radial temperature gradient difference at the joint is 0.3°C/mm. The change of joint temperature gradient in the radial direction is much greater than in the axial direction. As a good thermal conductor, the copper core has a relatively uniform internal temperature gradient, close to 0°C/mm, while the temperature gradient distribution within the insulation layer is uneven. Regardless of the presence of contact resistance, the radial temperature field distribution of the intermediate joint remains essentially the same. This indicates that the contact resistance does not significantly affect the distribution rules of temperature and temperature gradient in the radial direction.

To further analyze the impact of contact resistance at the cable joint on the temperature field of the cable body, radial temperature and temperature gradient distribution graphs for the cable body are plotted. Figures 8, 9 correspond to the distributions without and with contact resistance ( k = 7 ), respectively. By comparing Figures 6–9, it can be found that:

Without the contact resistance, the conductor temperature in the cable body is 2°C higher than that of the conductor in cable joint due to the difference in cross-sectional area. Additionally, the radial temperature gradient in the cable body is 0.1°C/mm higher than that in the joint. The larger temperature difference between the cable body and the environment leads to this phenomenon. Due to thermal conduction, the contact resistance not only affects the temperature distribution in the joint but also causes an increase in the temperature of the cable body near the joint. The conductor temperature gradient in both the cable body and the joint is 0°C/mm, indicating that there is no significant temperature exchange activity within the conductor.

The air-gap defects in cable joint primarily arise from imperfect manufacturing and the prolonged operation. The accumulation of space charges in cable joint continuously enhances the electric field at the air-gap, leading to increased temperature and partial discharge (Zhu et al., 2014). These discharges accelerate chemical reaction rates, causing further degradation of the cable joint.

Focusing on a single micro air-gap within the insulation layer, this study investigates the impact of air-gap defect on the temperature of the cable joint. Based on the previous simulation model of cable joint, an air-gap defect is set up as shown in Figure 10. The air-gap is situated in the joint insulation and is modeled as a cylindrical shape, with a radius of 0.5 mm and a height of 5 mm, and the material is air.

The loss due to the air-gap discharge varies with the air-gap length. As the length increases, the pulse width of the discharge waveform also increases, while the number of discharge pulses in a cycle shows an upward trend. Additionally, the amplitude of the power loss due to the air-gap discharge increases correspondingly.

The thermal loss of the air-gap defect in cable joint mainly originates from plasma chemical reactions, electron discharges, and charge accumulation. Based on the discharge characteristics of the air-gap (Yang et al., 2017), the power loss density is calculated using Equation 14: q v t = u t × i t V (14) where u t is the discharge voltage of the air-gap, V; i t is the discharge current of the air-gap, A; V is the volume of the air-gap, m³.

The average power loss density of the air-gap discharge in a cycle is obtained by: q v a v g = 1 T ∫ t 0 t 0 + T q v t d t (15)

According to Equation 15, when the air-gap length l is 1 mm, the calculated q v a v g = 197.23 W / m 3 . For further study of the loss of air-gap defect with different lengths, l at 2 mm, 3 mm, 4 mm, and 5 mm are investigated. The thermal loss of the air-gap Q can be described as Equation 16: Q = q v a v g × V (16) where V = π × 0.5 2 × l , m³.

The average power loss density q v a v g and the thermal loss Q of air-gap defect are shown in Table 5.

By incorporating the thermal loss from Table 5 into the simulation model, the distributions of the electric field, volume loss density, temperature, and temperature gradient in the cable joint are shown in Figure 11, and the following conclusions can be drawn:

When there is an air-gap defect in the intermediate joint, the electric field strength at the defect is distorted to 8 kV/mm. The volume loss density caused by the joint air-gap increases by 4 × 10⁻⁴W/m³, indicating that the loss density of the air-gap is extremely low and cannot be reflected in the temperature simulation.

The temperature change at the air-gap is only 0.01°C, demonstrating that a single micro air-gap has a negligible effect on the insulation temperature. The temperature gradient change is only 0.1°C/mm, almost unaffected by the air-gap defect.

However, during the actual operation of the cable, the high electric field strength at the air-gap can further induce partial discharge. Over time, this can lead to the aging of the insulation material, resulting in a more significant temperature rise at the defect area.

When moisture, impurities, and space charges are present in the insulation material, they can form dendritic microchannels under the influence of electric field, referred to as water trees. During cable operation, water molecules gradually accumulate at the defect area under the electric field, causing mechanical damage to the insulation and potentially leading to cable joint failure.

The growth of water trees can occur in two forms: outward growth along the electric field lines and densification growth within the original water tree region. The two growth forms coexist in cables, and the growth rate of water trees increases with the cable aging.

The growth stages of water trees and the constructed defect simulation model are shown in Figure 12. The growth process of water trees can be divided into three stages: the initial stage, the stagnation stage, and the subsequent stage (Zhou et al., 2019; Chen et al., 2016). The stagnation stage differs from the initial and subsequent stages in that the water tree stops extending outward and primarily increases the internal density. This stage is characterized by a significant increase in moisture content within the original water tree region. Due to its stable state and long duration, the stagnation stage is the optimal period for studying water tree defects. Therefore, this paper conducts a simulation study on the water tree model during the stagnation stage. The water tree defect is set in the joint insulation and the material is set as water, with a total height of 5 mm and a spherical densified region radius of 1 mm.

Based on the simulation model of the water tree defect, the results are obtained as shown in Figure 13 and the following conclusions can be drawn:

When a water tree defect is present in the cable joint, the electric field distortion at the defect is the highest, reaching 30 kV/mm, while the electric field distortion in other areas of the joint insulation does not exceed 7 kV/mm. The extremely high electric field strength can easily trigger partial discharge, leading to more severe insulation damage than that caused by air-gap defects. Similarly, the loss density at the water tree defect is the highest, reaching 1.5 × 10⁸W/m³, which is much higher than the loss density in the rest of the joint insulation. The irregular shape of the water tree defect and its high permittivity are the main reasons for the high electric field strength and loss density.

Due to the substantial differences in electric field strength and loss density between the defect area and the surrounding medium, the temperature rise at the defect is more significant. In actual operation, the defect area is more prone to chemical reactions with the surrounding medium, further increasing the temperature at the defect. In the water tree defect model, the highest temperature of the joint is at the water tree defect, reaching 42°C, which is about 20°C higher than in other regions. The maximum temperature gradient of the cable joint is also located at the water tree defect, at 17°C/mm. The impact of the water tree defect on the overall temperature of the cable joint is severe and cannot be ignored.