Nonlinear Switching Transient Field Simulation of Cable Joint without Residual Charge

The analysis of the impulse voltage on the internal electric field of the cable joint plays a key role in studying the breakdown of the joint. Based on the finite element method, a three-dimensional electromagnetic field simulation model of the cable joint is established in this paper. Simulation results show that the voltage at the head of the cable joint reaches about twice the impulse voltage. The increase of the conductivity of semi-conductive material also leads to the increase of electric field intensity. Then, several points and curves at different positions are selected for further analysis in this paper. Among them, the electric field distortion at the edge of the high voltage shield is the most serious and the electric field in the air gap is the least.


Introduction
With the increasing use of power cables in medium voltage distribution networks, the number of cable joints in use is increasing. The quality of the cable joint manufacturing and installation varies, making the cable joint a weak link in medium voltage cables. Once the insulation performance of the cable joint is deteriorated, it will cause partial discharge and even cause serious accidents such as explosions, which will affect the safe operation of the power grid. During the switching operation, the overvoltage formed during the transient process prop-agates through the cable joints in the form of electromagnetic waves. Due to the existence of refraction and reflection phenomena, the components of each traveling wave are superimposed, which makes the transient overvoltage rises extremely fast, with large amplitude and high frequency, posing a great threat to the insulation of cable joints. When the conductivity of the semi-conductive material in the cable joint changes, it will affect the electric field distribution of the cable joint and seriously threaten the operational reliability of the cable joint [1].
In order to assure reliable operation of cable joint, a lot of research has been conducted on the space charge characteristics, interface breakdown and flashover characteristics, and optimization of the accessory structure, so as to minimize the possibility of problems in the production, installation and use of cable ac- [6]. These studies are based on steady-state fields and cannot reflect transient processes during switching operations. Therefore, in order to more accurately study the overvoltage and transient electromagnetic fields in cable joints, a full-wave simulation analysis of the three-dimensional electromagnetic field model is needed.
Based on the finite element method, a three-dimensional electromagnetic field simulation model of the cable joint is established in this paper. Under the shock wave form of double exponential wave, the electric field distribution and overvoltage in the cable joint are calculated, and the influence of the conductivity of the semi-conductive material on the electric field distribution and overvoltage is analyzed.

Structure and Material of Model
The structure of the 10 kV AC cable connector is shown in Figure 1. The model used in this paper is obtained by rotating the axisymmetric model around the axis of symmetry. The model consists of 9 parts: conductive copper, inner semi-conductive layer, main cable insulation layer, outer semi-conductive layer, stress cone, high-voltage shield, reinforced insulation layer and insulation shield. The material parameters of each part are shown in Table 1.

Control Equation
To calculate the transient field, Maxwell's equations need to be solved. In the region without charge density and current density, the existence of the potential φ can be ignored, and only the magnetic vector potential A is used to represent the electric field strength and magnetic field strength. The control equation is shown in Equations (1) as follow.
where t is time, μ r is relative permeability, μ 0 is the permeability of vacuum, σ is electricity conductivity, ε 0 is the permittivity of vacuum, ε r is relative permittivity.
The head and end of the cable joint are set as lumped port I and lumped port II, respectively, and the relationship between the voltage U port and current I port of the port is expressed as port port port where Z port is the wave impedance of lumped port. The wave impedance of the lumped port at the head is set to 5 Ω. Although the internal impedance of the power supply should actually change with frequency, it is approximated in this paper with a small resistance. The impedance of the lumped port at the end is set to 5 GΩ to simulate an open circuit condition in the laboratory. And the curve of the overvoltage at the end of the cable joint is extracted from this port.
A wave excitation is applied at the head of the joint, and the waveform is shown in Figure 2. The equation of the impulse wave is expressed as Since the influence of residual charge is not considered, the initial condition of the magnetic vector potential A is set as ( )

Results and Discussion
According to the methods and settings above, the overvoltage of lumped ports and the electric field distribution under different conductivity of semi-conductive materials are simulated.

Simulation Analysis of Overvoltage under Different Conductivity
The voltage of lumped port I and impulse voltage under different conductivity is compared in Figure 3. It can be seen from Figure 3(a) that the voltage of lumped port I is also in the form of a double exponential function, reaching twice the applied voltage. Figure 3(b) is enlarged by the voltage peak time of Figure 3(a). As can be seen from Figure 3(b), when the conductivity changes from 5 S/m to 20 S/m, the voltage gradually decreases with a small change.
And the voltage of lumped port II under different conductivity is compared in Figure 4. Obviously, the voltage oscillation amplitude of lumped port II is larger, and the voltage decreases with the increase of conductivity. Figure 5 shows the distribution of the electric field under different conductivity when the voltage reaches its peak. As the conductivity increases, the maximum electric field intensity increases, but the growth rate slows down.    selected, as shown in Figure 6. Figure 7 shows the electric field intensity at point A, B, C and D under different conductivity over time. It can be seen that the conductivity has a greater effect on point A and D, and a smaller effect on point B and C. As the conductivity increases, the electric field intensity at point A increases. But the conductivity has the opposite effect on the electric field intensity at points B, C and D.

Analysis of Electric Field Distribution under Different Conductivity
Among the four points, the electric field distortion is the most serious at point B. Figure 8 shows the distribution of electric field intensity along the curve l 1 , l 2 , l 3 and l 4 when the voltage reaches its peak under different conductivity. As the conductivity increases, the growth rate of electric field intensity gradually saturates. Among the four curves, the electric field intensity on curves l 1 and l 2 is larger, while the electric field intensity on curves l 4 is the smallest. Energy and Power Engineering