Temperature Evaluation Considering Gradient Distribution for MV Cable XLPE Insulation Based on Wave Velocity

Abstract

Temperature is an important factor for the service life of cable insulation. To ensure safety, the operating temperature of cables must be monitored. Since optical fiber temperature measurement technology is difficult to be used widely in medium voltage (MV) cables due to cost, this paper proposes a temperature evaluation method based on wave velocity. Firstly, the dielectric constant of cross-linked polyethylene (XLPE) cable insulation under different temperature is obtained through experiment. Based on the result, the relationship curve between wave velocity and temperature is established. The asymmetry effect due to temperature gradient in the cable insulation is discussed via finite element simulation. The effectiveness of obtaining the average insulation temperature of the cable based on wave velocity is validated. In addition, the mechanism of the temperature influence on the cable insulation material’s dielectric constant is analyzed by molecular dynamics simulation, which further deepens understanding of the characteristics of cable insulation materials.

1. Introduction

The power cable is an important piece of equipment in the modern power system, and temperature is the key factor that determines the insulation state and cable service life [1,2]. Cross-linked polyethylene (XLPE) is a common insulation material for power cables. In order to prevent cable insulation failures, the temperature of cable conductors is limited to below 90 °C [3]. For this reason, it is necessary to monitor the XLPE cable insulation temperature. With the continuous integration of renewable energy, the load of distribution of the MV cable is becoming increasingly dynamic. Flexible operation is demanded for the distribution network to ensure safe operation [4]. The power transmission capacity of cables has to be fully utilized, which means more current is loaded into the cable, and the cables often work in high temperatures [5]. To safeguard the MV system, an effective XLPE insulation temperature evaluation method is demanded for MV cables.

Optical fiber-based temperature measurement is commonly used for cable temperature monitoring [6], and has been widely used for high-voltage cables. The optical fiber can be placed in the cable or attached to the surface of the cable. Both methods require the optical fiber to be laid all along the cable. At the same time, a set of laser equipment is needed to achieve the temperature measurement. The high cost of such a system limits its application in distribution network cables. The thermal coupler is another choice for cable temperature monitoring. However, it can only measure the temperature of the specific points where it is placed, which is not suitable for a long-length cable [7] Thus it is urgent to develop a suitable method to monitor the insulation temperature of medium voltage cables in the distribution network.

The transmission line characteristics of a cable can provide operators with information about its intrinsic status. Qi et al. used electromagnetic wave velocities to diagnose cable aging [8,9]. Li et al. obtained the cable input impedance spectrum from the transmission line parameters, based on which the cable defect is located [10]. Broadband impedance spectroscopy based on transmission line theory has also been used to detect and locate cable aging and concentrated defects [11,12].

Temperature will affect the dielectric properties of cable insulation [13], changing its transmission line parameters. Preliminary tests based on vector network analyzers in [14] show that, between 30 °C and 90 °C, the relationship between cable wave velocity and temperature is obvious and measurable. In [15], the change in the reflected wave from the cable termination between 9 °C and 75 °C was observed using the time-domain reflectometry method. This work proves the feasibility of cable temperature monitoring based on electromagnetic wave velocity, while in [16], the temperature monitoring error when using such a technique is discussed. However, no in-depth analysis of this approach exists in the current literature, which is manifested in two aspects: firstly, there is a lack of quantitative characterization for wave velocity-based cable temperature monitoring considering the gradient distribution across the insulation. Secondly, the mechanism for temperature effect on velocity variation is not clear. Therefore, this paper further studies the method to monitor the temperature of MV cable XLPE insulation via wave velocity. The study reveals the temperature dependency of the cable insulation material’s dielectric properties, based on which the quantitative relationship between temperature and electromagnetic wave velocity is established. The temperature gradient in insulation introduces asymmetry and this issue is analyzed by the finite element method. The experimental results show that cable insulation temperature measurement can be realized based on electromagnetic wave velocity. To further explore the mechanism of the temperature effect on the dielectric constant of XLPE, molecular dynamics simulation is performed, which explains the causes involved in the proposed methodology.

2. Methodology

The time-domain reflectometry method is a common method used to locate cable faults [17], in which the relationship between U_in, the signal injected into the cable, and U_out, the signal reflected back from the end of the cable, is shown in Equation (1):

$$U_{\mathrm{out}}=U_{\mathrm{in}}\cdot H\cdot e^{-\gamma l}$$

(1)

where H is the coefficient accounting for signal reflection, γ is the propagation coefficient of the cable, and l is the length of the signal propagation path. Among the three variables, H and l are independent of temperature, while the propagation coefficient γ is temperature dependent.

The propagation coefficient γ can be obtained from the distribution resistance R, inductance L, admittance G, and capacitance C of the cable [18]:

$$\gamma =\sqrt{(R+j\omega L)(G+j\omega C)}=\alpha +j\beta =\alpha +j\frac{\omega }{v}$$

(2)

where α and β are the real and imaginary part of γ, respectively, ω is angular velocity and v is propagation velocity. For single-core coaxial cables, the distributed capacitance per unit length is calculated as follows:

$$C=\frac{2\pi {\epsilon }_0{\epsilon }_r}{\ln (r_{\mathrm{s}}/r_{\mathrm{c}})}$$

(3)

where ε_r is the relative permittivity and r_s and r_c are the insulation radius and conductor radius of the cable, respectively. The following relations can be obtained from Equations (2) and (3):

$$\frac{v}{v_0}=\sqrt{\frac{{\epsilon }_r(T_0)}{{\epsilon }_r(T)}}$$

(4)

From Equation (4), it can be observed that the electromagnetic velocity travelling in the cable is determined by the insulation permittivity, which is affected by temperature. Thus, it is possible to obtain the temperature from the velocity value by using Equation (4) reversely. The schematic diagram of the temperature effect on the electromagnetic wave velocity in the cable is shown in Figure 1, according to which the cable temperature can be traced and monitored based on the wave speed. It should be noted that the dielectric constant in Equation (4) is the equivalent value of the cable insulation. Since the temperature in cable insulation is not the same, the insulation part closer to the conductor has a higher temperature. Thus, this relation in Equation (4) reflects the overall average temperature of the cable insulation. The asymmetry issue brought up by the temperature gradient across cable insulation is discussed in detail in Section 3.

To explore the temperature influence on the dielectric constant of cable insulation material, a test platform based on network analyzer G5061B was built to measure the dielectric constant of XLPE at different temperatures. The results are shown in Figure 2. The dielectric constant is stable in amplitude over the test frequency range. Overall, the dielectric constant decreases with increasing temperature. The change rate is non-linear. More rapid decrease of permittivity is observed at higher temperatures, as observed in Figure 3, which is consistent with the results obtained in [14,15].

In order to make better use of the test results, the relationship between the relative permittivity and temperature of the insulating material is fitted, and the result shown in Equation (5) is obtained.

$${\epsilon }_{\mathrm{XLPE}}=2.053-0.152\times \mathrm{atan}[0.0707\times (T-84.6\left)\right]$$

(5)

Based on Equation (5), Figure 4 is plotted to show the change in wave velocity with temperature, which is small at low temperatures and significantly larger when the temperature rises.

To this end, the relationship between the wave velocity and the insulation temperature in the cable can be obtained. Thus, insulation temperature monitoring based on wave velocity can be realized.

3. Computational Analysis

Temperature will affect the insulation material properties of the cable, which in turn changes the distribution parameters, leading to variation of the electromagnetic wave velocity. From Equation (2), it can be seen that the electromagnetic wave velocity is affected by the resistance R, the inductance L, the admittance G, and the capacitance C. In order to analyze the influence of temperature on the above parameters, a simulation model based on the finite element method was established.

3.1. Model Establishement

Taking the 12/20 kV single core aluminum XLPE cable with a conductor cross-sectional area of 300 mm² as an example, the specific structure and parameters are shown in Figure 5 and

Table 1. Cable parameters.

Symbol (Unit)	Meaning	Value
ρc (Ωm)	Core resistivity	2.82 × 10−8
ρsi (Ωm)	Inner buffer resistivity	0.01
ρso (Ωm)	Resistivity of the outer buffer layer	0.01
ρs (Ωm)	Metal shield resistivity	1.68 × 10−8
r1 (mm)	Core radius	9.5
r2 (mm)	Radius of the insulation	15.6
r3 (mm)	Inner buffer layer radius	16.0
r4 (mm)	Metal shield radius	17.15
r5 (mm)	Radius of the outer buffer layer	17.6
r6 (mm)	Outer sheath radius	20.1
εXLPE	XLPE relative permittivity	2.23–0.001 j
εcs	The relative permittivity of the inner shield	1000
εls	The relative permittivity of the outer shield	1000
εj	Dielectric constant of the outer sheath	2.3
Ri	Thermal resistance of insulation Km/W	0.27
Rb1	Thermal resistance between insulation and shielding Km/W	0.03
Ri	Thermal resistance of insulation Km/W	0.27
Rb1	Thermal resistance between insulation and shielding Km/W	0.03
Rb2	Thermal resistance between shield and outer sheath Km/W	0.02
Rj	Thermal resistance of the outer sheath Km/W	0.07
Qc	Conductor heat capacity J/m/K	694
Qi	Insulation heat capacity J/m/K	946
Qb1	The heat capacity between insulation and shield is J/m/K	89
Qs	Shielding heat capacity J/m/K	413

.

3.2. Simulation Process

As shown in Figure 6, the conductor loss W_c in the cable is calculated according to Equation (6), where I is the current, ρ₀ is the conductivity of the conductor at temperature T₀, T is the actual temperature, α is the temperature coefficient, and l, S are the length and cross-sectional area of the conductor, respectively. The temperature distribution of the single-core cable through 570 A (rated current) is shown in Figure 7, which is derived from Comsol Multiphysics software. The temperature gradient between the inside and outside of the insulation layer is about 10 °C. Further calculations show that when the core temperature is 90 °C, the temperature gradient between the inside and outside of the insulation layer is about 25 °C.

$$W_{\mathrm{c}}=I^2{\rho }_0[1+\alpha (T-T_0)]l/S$$

(6)

The asymmetry issue brought up by the temperature gradient is analyzed as follows. By setting the relationship of Equation (5) to the XLPE material of the finite element model, the distribution parameters of the cable are calculated, based on which the wave propagation velocity can be derived. The influence of temperature gradient on electromagnetic wave velocity can be obtained.

The cable distribution resistance R is calculated by using Ohm’s law as in Equation (7). The distributed inductance L is calculated according to Equation (8).

$$R=\frac{{\displaystyle \int E\cdot \mathrm{d}l}}{I}$$

(7)

$$\{\begin{array}{c}{W}_{\mathrm{m}}=\frac{1}{2}{\displaystyle \int B\cdot H\mathrm{d}s}\\ L=\frac{2W_{\mathrm{m}}}{I^2}\end{array}$$

(8)

where W_m, B, H represent magnetic energy, magnetic induction intensity and magnetic field strength, respectively.

The distributed conductance G and distributed capacitance C of the cable can be calculated from the electric field. The distributed capacitance C depends on the electric energy and the voltage V between the core and the metal shield, as in Equation (9), where E is the field strength and D is the displacement vector, while the distributed conductance G can be solved according to C, as shown in Equation (9), where ε^′ and ε^″ are the real and imaginary parts of the dielectric constant of the cable insulation, respectively.

$$\{\begin{array}{c}{W}_{\mathrm{e}}=\frac{1}{2}{\displaystyle \int E\cdot D\mathrm{d}s}\\ \begin{array}{l}C=\frac{2W_{\mathrm{e}}}{V^2}\\ G=\frac{\omega {\epsilon }^{″}C}{{\epsilon }^{\prime }}\end{array}\end{array}$$

(9)

3.3. Analysis of Results

Table 2. Distributed Parameters Percentage Change at 20–100 °C.

Parameter	Percent Change
R	8.3%
L	0.1%
G	−0.1%
C	−15%

is a calculation based on the temperature-dependent distribution parameters in Figure 8, showing the percentage change for each distribution parameter over the 20–100 °C range.

The distribution parameters of the cable at different frequencies were calculated based on Equations (7)–(9) with Comsol Multiphysics according to the material and structural parameters of the cable in Table 1 [19] and the results are shown in Figure 8. Taking into account the temperature coefficient of aluminum and the dielectric constant as a function of temperature, the resistance and conductance in the distribution parameters are much smaller than those of inductance and capacitance.

In order to further analyze the results, the change in electromagnetic wave velocity with temperature under the influence of different distributed parameters is obtained, as shown in Figure 9. In the figure, the effects of resistance, inductance, conductance, and capacitance on the speed of electromagnetic waves are considered. The correlation between capacitance and temperature is the largest; under the influence of capacitance, the change rate of electromagnetic wave velocity is around 8%, while the correlation between other parameters and temperature is hardly observable. Under the influence of resistance, inductance and conductance, the change rate of electromagnetic wave velocity is less than 0.01%. The above conclusions verify that the electromagnetic wave velocity is mainly affected by the change in cable capacitance.

4. Analysis of Results

4.1. Cable Insulation Temperature Measurement Experiment

In order to verify the correctness of the proposed method, an experimental platform was built to measure the insulation temperature of the cable based on the method.

A single-core cable experimental platform, as shown in Figure 10, was set up in the laboratory to test its wave velocity at different temperatures. The test cables are the same as in Section 3.1. Each cable is about 12 m, and the six cables form a loop of about 72 m. There are a total of seven connectors in the figure, where A is the adapter between the signal source and the cable, B–G are the intermediate connectors of the cable, and connectors 1–3 indicate that three different connectors are used in the experiment. The DC current source connects points B and G to form the current loop indicated by the red arrow, which heats the cable. Two temperature sensors, T1 and T2, are used to measure the temperature of the cable surface.

Pulses are injected into the cable to record the reflected waveforms of the signal at different temperatures, as shown in Figure 11, where the two temperature values correspond to T1 and T2, respectively. In Figure 11a, A–G indicate the reflected pulse at the corresponding connector/termination in Figure 10. While Figure 11b is the enlarged part of Figure 11a.

As can be seen from Figure 11, when the temperature increases, the reflected wave crest shifts to the left, indicating that the wave velocity increases, which is consistent with the previous analysis. In order to correlate the surface temperature of the cable with the insulation temperature, the thermal model of the single-core cable was established, as shown in Figure 12 [20].

According to the thermal model and related parameters in

Table 3. The parameters for cable temperature calculation.

Symbol	Parameters	Value
Wc	Conductor loss W/m	34.5
Tconductor	Conductor temperature °C	-
Tearthscreen	Metal shielding temperature °C	-
Tjacket	Epidermal temperature °C	-
pi, pb1, pb2, pb	Van Wormer coefficient	0.42, 0.5, 0.5, 0.48

, the temperature measured on the cable surface can be converted into the insulation temperature, and the conversion results are shown in

Table 4. The insulation temperature.

Tj °C	Ti °C	Tiv °C
20.4	33.7	32
28.6	42.0	44
37.3	50.7	49
48.7	62.1	61

. In the table, T_j is the outer surface temperature (T1 is taken here), and T_i is the insulation temperature calculated according to the thermal model. According to the change in wave velocity in Figure 11, the insulation temperature T_iv based on wave velocity measurement is obtained by referring to the relationship between wave velocity and insulation temperature in Figure 4. Comparing the measured temperature T_i with the calculated temperature T_iv based on wave velocity, it is found that the error is no more than 6%.

4.2. Discussion of Temperature Effect on Dielectric Constant of Cable Insulation Material

XLPE is a polymer material with a three-dimensional structure formed by cross-linking polyethylene molecules. At present, the common crosslinking processes are peroxide cross-linking, irradiation cross-linking, ultraviolet light cross-linking and silane cross-linking, among which silane cross-linking is the most widely used chemical cross-linking method at present for MV cables. The principle of silane cross-linking is to graft silane to the molecular chain of polyethylene (PE) under the action of water and catalyst to form a cross-network structure of -Si-O-Si-. The basic principle of the chemical reaction is as follows:

(10)

(11)

In this paper, the molecular structure model is established for silane cross-linked polyethylene to study the temperature effect on its permittivity. Firstly, two PE chains with a chain length of 18 are established. The Si atoms and the C atoms are connected on the PE backbone through two C atoms. The cross-linked polyethylene model is established by using the amorphous polymer construction method proposed by Theodorou [21], as shown in Figure 13.

To derive dielectric constant from the model, the simulation process includes geometry optimization, annealing and molecular dynamics simulation. Geometric optimization and annealing can stabilize the energy of the model and make the physical and chemical properties of the model closer to those of the real material. The molecular dynamics simulation is used to obtain the material’s dielectric constant. The three processes are introduced in detail below.

Geometric optimization: The randomly generated initial structure is not a reasonable configuration, and there may be overlapping atoms or unreasonable atomic positions. In order to eliminate this problem, the energy of the system needs to be minimized. The Forcite module was used for geometric optimization to determine the convergence level and reduce the energy of the whole system to avoid possible computational failures in the subsequent molecular dynamics calculations. The truncation radius was selected to be 15.5, and the molecular structure was optimized in 5000 steps to minimize its energy.

Annealing: In molecular dynamics simulation, high temperatures can provide more energy to the system, allowing the system to overcome the energy barrier and reach the lowest energy point in a short period of time. In this paper, the annealing function in the Forcite module is used for eight cycles from 300 K (26.85 °C) to 800 K (526.85 °C) and then back to 300 K (26.85 °C) to produce a stable state for the model.

Molecular dynamics simulation: The annealed model was simulated under the NVT (number, volume and temperature) and NPT (number, pressure and temperature) ensembles, respectively. The initial velocity of the molecules was sampled according to the Maxwell distribution. The temperature was controlled by the Andersen thermostat [22], and the pressure control was carried out by the Berendsen barostat [23]. The entire simulation uses a COMPASS force field. During the simulation, the temperature is set to 25 °C, 35 °C, 45 °C and 82 °C. The total duration of the MD simulation is 3000 ps, and the time step is 1 fs.

Neumann et al. related the crystal dipole moment fluctuations obtained by molecular dynamics simulations to the static permittivity under all boundary conditions, as in Equation (12) [24].

$$\frac{\langle {\left|M\right|}^2\rangle -{\left|\langle M\rangle \right|}^2}{3{\epsilon }_{0}V{k}_{B}T}=\frac{\left(2{\epsilon }_{RF}+1\right)\left({\epsilon }_s-1\right)}{\left(2{\epsilon }_{RF}+{\epsilon }_s\right)}$$

(12)

where M is the total dipole moment of the model, ⟨⟩ is the average calculation, ε₀ is the dielectric constant of vacuum, V is the volume of the sphere, k_B is Boltzmann’s constant, T is the temperature, and ${\epsilon }_{RF}$ is the dielectric continuum.

$${\epsilon }_s=1+\frac{\langle {\left|M\right|}^2\rangle -{\langle \left|M\right|\rangle }^2}{3V{k}_{B}T{\epsilon }_0}$$

(13)

The permittivity can then be calculated from the electric dipole, as in Equation (13). In Equation (13), the dielectric constant at different temperatures is calculated by using the molecular conformational trajectory under NPT relaxation through the perl language. The results shown in Figure 14 are obtained.

Figure 14 shows the results of the MD simulation up to 3000 ps, where the first 2000 ps are used for the equilibrium of the architecture. It can be seen that, after 2000 ps, the model tends to be stable, and the dielectric constant curve of XLPE decreases with the increase in temperature, which is consistent with the experimental results. The dielectric constant represents the polarization capacity of an insulating material, and its variation can be explained by the change in the molecular chain end distance of the polar material with temperature. For XLPE, when the temperature increases, the distance between the ends of the XLPE molecular chain gradually increases, which indicates the irregular movement of the molecule intensifies, hindering the polarization process and slowing down the molecular polarizability [25]. This is shown in Figure 15.

The molecular dynamics simulation reveals that the dielectric constant change with temperature can be related to the temperature effect on the distance of the molecular chain ends.

5. Conclusions

In this paper, a method for evaluating the average insulation temperature of cables based on wave velocity is proposed. The correlation between the dielectric spectrum and temperature of XLPE is measured, and it is found that the dielectric constant decreases with temperature, while its decreasing gradient increases with increasing temperature.

The temperature affects the insulation’s permittivity, leading to changes in the cable’s capacitance, which determines the wave velocity. The temperature gradient in the insulation introduces an asymmetry issue. However, it is feasible to monitor the average cable insulation temperature. The test results show that the error is less than 6%.

To analyze the root cause of the temperature influence on permittivity, molecular dynamics simulation is performed. It is shown that, with temperature increase, the distance between the ends of the XLPE molecular chain increases, hindering the polarization process, which explains the mechanism of the temperature effect on XLPE’s permittivity.