Interaction of Coupled Thermal Effect and Space Chargein HVDC Cables

Abstract

Currently, zero-emission targets require future global energy concepts to be based on renewable energy sources; therefore, huge investments are being made in bulky offshore wind parks worldwide. In this context, there is ongoing and enormous development and a need for HVDC submarine cables (both static and dynamic) to connect offshore wind farms. One of the basic problems when analyzing the operating conditions of HVDC cables is assessing the effects of the load current, which generates thermal and electric fields on the insulation systems in these cables. This article considers the problem of the influence of the thermal effect and space charges—the field effect—on the electrical conductivity of polymeric insulating materials and, thus, on the distribution of the electric field intensity in the cable insulation. An analytical methodology for joint analysis of the thermal-effect- and space-charge-related influence is presented. The critical value of the electric field intensity at which the electrical conductivity is significantly modified under coupled thermal–electric exposure is determined. Special focus is placed on the analysis of the coefficient representing the dependence of the electrical conductivity on the temperature in a much broader range than typically assumed. Hence, the intention of this paper is to highlight the limit values of the electric field strength under the simultaneous action of the space charge and temperature gradient. Recognizing the changes in the electric field intensity value in the insulation is of fundamental importance from the point of view of HVDC cable technology and construction.

1. Introduction

The development of the production of high-voltage direct current (HVDC) cables in the mid-20th century was the result of the growing demand for this type of cable to be installed in submarine cable lines that connected the elements of power systems (e.g., NorNed Norway to Netherlands, Denmark to Germany, and others). The use of high-voltage alternating current (HVAC) cables for this purpose was disadvantageous due to the significant values of the capacitive current and the consequent limited lengths of the installed cable lines [1,2]. The progress of power electronics techniques (direct current rectifiers and inverters) expanded the possibilities of using direct-voltage connections (e.g., “back-to-back”). The effect of significant changes in the technology of high-voltage direct current cables was the dynamic increase in their applications in the emerging HVDC transmission systems during the years 1970–1990.

Currently, zero-emission targets require future global energy concepts to be based on renewable energy sources. Specifically, huge investments are being made in bulky offshore wind parks worldwide (photovoltaic platforms are also targeted in the future). The Global Wind Energy Council reported a total of 64 GW of global offshore wind capacity being in operation at the end of 2022. According to plans, more than 380 GW of offshore wind capacity will be connected by 2032 [3,4]. In this context, there is ongoing and enormous development and a need for HVDC submarine cables (both static and dynamic) to connect offshore wind farms. The basic criterion for the design of high-voltage power cables is the working intensity of the electric field in the insulation system, the permissible value of which is established in relation to the electrical strength of the applied insulating material. The electric field in the insulation of HVAC cables can be practically treated as a field that is described by the equations of the electrostatic field—its distribution is not influenced by the basic operating factor, which is the thermal energy as a result of the load current of the cable core. In the case of HVDC cables, the electric field distribution strongly depends on the insulating material’s electric conductivity, which has both a strong thermal effect and a mutual interaction with an electric field [5,6,7,8,9,10,11,12]. Initially, the insulation in HVDC cables was made of paper that was impregnated with a suitable mineral saturant; since the 1990s, synthetic polymers have been used in them—mainly cross-linked polyethylene (XLPE). With the increasing use of high-voltage DC power cables (especially in offshore transmission systems), the amount of information on the possible causes of changes in their service lives is increasing [9,11,13,14,15,16,17]. Apart from the effect of the heat flux from the cable core (i.e., increased temperature), an additional cause of these changes is assumed to be the formation of space charges in the insulation. This means an increase in the electrical conductivity σ of the insulating materials in this type of cable. The main reasons for the space charge formation in polymeric high-voltage cable insulation include the following [18,19,20,21,22,23,24]:

▪

Charge injection from electrodes—under high electric fields, charge carriers (electrons or holes) can be injected from the conductor or outer semiconducting layers into the insulation. This can occur due to high-voltage stress, temperature, or imperfections at the interface.

▪

Charge trapping in defects and impurities—polymeric insulation materials may contain microscopic defects, impurities, and traps (such as chemical by-products or voids). Charges become trapped in these energy states, leading to localized charge accumulation.

▪

Ionic and electronic conduction—in the case of some polymeric materials, limited conduction of the charge carriers may occur, especially at high temperatures or under prolonged electric fields. Ionic movement can also contribute to the space charge, particularly due to moisture contamination.

▪

Electric field distortion and non-uniformity—space charge accumulation alters the internal electric field distribution, leading to local field enhancement. This effect can accelerate insulation degradation and potentially lead to dielectric breakdown.

▪

Temperature gradient effects—high-voltage cables experience temperature gradients, which can cause thermally activated charge movement. This influences the charge mobility and creates space charge accumulation.

▪

Material aging and degradation—over time, polymeric insulation undergoes aging, oxidation, and structural changes, leading to increased charge trapping sites.

The basis for calculating the electric field intensity distribution is, then, to take the field dependence σ(E) of the electrical conductivity of the insulating materials into account (in addition to the temperature dependence σ(T)). The factor σ(E) that influences the change in the electric field intensity distribution means the action of the electric field (the so-called field effect) strengthening the accumulation of the spatial charges and increasing the electrical conductivity of the material. This factor was not significant in old cable designs with impregnated paper insulation (given the relatively high value of the natural electrical conductivity of the material); however, this can be revealed in modern cables with cross-linked polyethylene insulation. The distribution of the electric field intensity in cable insulation (including its maximum value) depends on the properties of the insulating materials and the influence of the operating factors. There are several techniques for DC electro-thermal insulation stress modeling that have been developed over the years; the most popular are the macroscopic and fluid models [15,16,22,25,26,27,28]. In the case of the macroscopic model, the electrical conductivity is represented as a nonlinear function of the electric field and temperature. For the fluid models, the local charge generation and transport is considered to be a non-homogenous phenomenon. In this model, a charge injection from the electrodes can also be modeled. Increasingly precise experimental measurements of the space charges are also being carried out—both on samples and on mini-cable models; these are mainly based on the pulsed electro-acoustic (PEA) method [21,22,29,30,31,32].

The effects of the space charge accumulation and temperature gradients are highly important in HVDC submarine cable systems. Submarine cables operate under unique conditions, such as deep sea pressure, high thermal dissipation challenges, and long transmission distances, making these effects even more critical [21,24,32]. Additionally, these cables operate in a water environment, where the heat dissipation is different from that of land cables.

The serious effect of the space charge in HV cables results in enhanced local electric stress—hence, the increasing breakdown risk, accelerated material aging and degradation can be mitigated by various strategies. The counteractions refer to the application of optimized material formulations with fewer trap sites, the addition of nanofillers to reduce the charge accumulation, the improvement of the manufacturing processes to minimize defects, and the use of DC voltage conditioning techniques to reduce the charge buildup.

This article presents an analysis of the influence of the thermal effect and space charges on the electrical conductivity of the insulating materials in HVDC cables. The presence of space charges in the insulation (i.e., the field effect) affects the nonlinear dependence of the electrical conductivity of the material on the electric field intensity. An analytical methodology for joint analysis of the thermal-effect- and space-charge-related influence is presented. Hence, the intention of this paper is to highlight the limit values of the electric field strength under the simultaneous action of the space charge and temperature gradient. The critical value of the electric field intensity at which the electrical conductivity is significantly modified under coupled thermal–electrical exposures is determined.

2. Factors Influencing Electrical Conductivity in Insulation of HVDC Cables

The insulation in power cables that operate at high DC voltages is an area of action of an electric field with a spatial distribution that is dependent on the leakage current in the insulation as well as on the cable load current (thus, generating a thermal effect in the insulation). Changes in the electrical conductivity of the insulating material in cables that are under the influence of temperatures under operating conditions are of significant importance in the case of high-DC cables. A change in the nature of the distribution of the electric field intensity in the insulation at a certain value of the load current (i.e., thermal gradient) is referred to as the inversion or reversal of the electric field; this occurs in insulation that is made of materials with a temperature-dependent electrical resistivity and results in a decrease in the value of the field intensity at the surface of the cable core and an increase at the screen on the insulation. This phenomenon is of significant importance when designing cable structures that take different operating conditions into account. The second factor influencing the changes in the electrical conductivity of the insulation (i.e., the influence of spatial charges) is one of the phenomena with high research priority. This results from the complex mechanism of the dependence of electrical conductivity on the electric field intensity.

(a)

Thermal effect in insulation

The power losses in a cable core at an operating current are sources of heat flux, thus creating a thermal effect in the insulation. This effect in the insulation of a power cable is, therefore, related to the temperature difference between the cable core (in which Joule heat is released under the influence of the load current) and the external screen on the insulation. The heat flux W_z from the cable core causes a temperature drop in the thermal resistance R_c of the insulation layer of thickness dx:

$$dT=W_z{R}_c\left(x\right)=W_z{\rho }_c{\displaystyle \frac{dx}{2\pi x}}$$

(1)

where ρ_c is the thermal resistivity of the insulating material, T is the temperature, and x is the distance along the cable radius.

Then, the temperature difference ΔT(x) between the layers of the insulating material at distance x from the cable axis and the outer screen on the insulation (radius r_i) is as follows:

$$\Delta T(x)=T\left(x\right)-T\left(r_i\right)={\int }_x^{r_i}{\displaystyle \frac{W_z{\rho }_{\mathrm{c}}}{2\pi }}{\displaystyle \frac{dx}{x}}={\displaystyle \frac{W_z{\rho }_c}{2\pi }}\ln {\displaystyle \frac{r_i}{x}}$$

(2)

where r_i is the outer radius of the insulation, W_z is the heat flux power from the losses in the cable core, and T(r_i) is the outer temperature on the insulation surface (approximately equal to the ambient temperature). The kinematic temperature distribution in the insulation in the steady state is as follows:

$$T\left(x\right)=T\left(r_i\right)+{\displaystyle \frac{W_z{\rho }_c}{2\pi }}\mathrm{l}\mathrm{n}{\displaystyle \frac{r_i}{x}}$$

(3)

This is the basis for describing the changes in the electrical conductivity of the insulation and, therefore, the thermal effect in the cable:

$$\sigma \left(T\right)={\sigma }_0·\mathrm{e}\mathrm{x}\mathrm{p}\alpha \left[T\left(x\right)-T_0\right]$$

(4)

where σ₀ is the electrical conductivity of the insulating material at reference temperature T₀ (usually, T₀ = 293 [K]), and α is the thermal conductivity coefficient. Its value for XLPE material is usually assumed to be α = 0.1 × [1/K] [21,33].

(b)

Field effect—space charges

In order to take into account the influence of the space charges in the insulation that are generated in an electric field on the electrical conductivity of the insulating material, an additional component with a coefficient β was introduced in the exponent of the expression that describes the macroscopic dependence of the electrical conductivity on the temperature:

$$\sigma \left(x\right)={\sigma }_0·\mathrm{e}\mathrm{x}\mathrm{p}\left\{\alpha \left[T\left(x\right)-T_0\right]+\beta E\left(x\right)\right\}$$

(5)

where β is the coefficient of the influence of the space charges on the electrical conductivity of the insulating material, expressed in mm/kV unit.

Using Formula (2), the dependence of the electrical conductivity on the temperature T and electric field intensity E in model σ(T, E) takes the following form:

$$\sigma \left(x\right)={\sigma }_0·\mathrm{e}\mathrm{x}\mathrm{p}\left\{\alpha \left[T\left(r_i\right)+{\displaystyle \frac{W_z{\rho }_c}{2\pi }}\mathrm{l}\mathrm{n}{\displaystyle \frac{r_i}{x}}-T_0\right]+\beta ·E\left(x\right)\right\}$$

(6)

The value of the β coefficient is determined empirically based on an approximation of the results from a series of measurements. It is assumed that this coefficient is independent of the temperature. Reference [34] presents a summary of the β coefficient values that were obtained in various tests on samples and fragments of XLPE cables. The experimentally obtained value of this coefficient was within a range of 0.018 to 0.347 mm/kV. Currently, the value of β = 0.03 mm/kV is assumed in most research works [21].

The influence of the field effect on the conductivity of an insulating material can be caused by the following:

▪

The field emission of the electric charges from the electrode (Fowler and Nordheim emission), which can occur at electric field strengths that are greater than approximately 10⁸ V/m, limits its practical significance to exceptional cases.

▪

The phenomenon of electric treeing, which means the generation of local electric charges in the micro-discharges in the structure of the material as a result of the content of the conductive impurities (floating conducting particles) in it.

▪

Releasing local trap charges (de-trapping) in the material if the material structure contains such traps (mainly, hole traps).

The volumetric energy density of an electric field with an intensity of 10 kV/mm is about 0.6 × 10⁻⁵ eV/nm³. The value of the electric field energy that is concentrated in an area with molecular dimensions in the order of several dozen nm³ can be estimated at about 0.6 × 10⁻² eV in polyethylene. This value is lower than the activation energy of the trapped charges (an order of 0.5 eV) and may have only stochastic significance under operating conditions. On the other hand, the field energy that is concentrated in the area as before will be about 0.6 eV at an electric field intensity in the order of 100 kV/mm; this is already comparable to the energy of the traps. In laboratory tests on the generation of spatial charges [35], it was found that the density of these charges was 75 C/m³ in a 0.35 mm thick XLPE sample at an electric field strength of 50 MV/m, a temperature of 60 °C, and a gradient of about 114 °C/mm.

(c)

Model of HVDC cable insulation system with a coupled thermal–electric effect

This article analyzes the conditions of the interaction between the electric field and the thermal effect in an HVDC cable when using the model system of coaxial cylinders that is shown in Figure 1a; this represents a cable core of radius r_z and a screen on insulation of radius r_i.

Including the joint effect of the temperature T and electric field E on the electrical conductivity of the material σ(T, E), the above model is an extension of the basic model of insulation conductivity σ(T). The equivalent diagram of the insulating system (containing the locally temperature- and field-variable electrical conductivity σ_i(T, E)) is shown in Figure 1b.

3. Distribution of Electric Field in HVDC Cables in Presence of Thermal Effect and Space Charges

In HVDC cables, the electric field distribution is influenced by both thermal effects (temperature gradients) and space charge accumulation. Unlike AC cables, where the electric field follows a predictable radial distribution based on permittivity, HVDC cables experience insulating material electric-conductivity-controlled field changes. An electric field in the insulation of HVDC cables is the result of leakage current flow under the action of the operating voltage, and its distribution is determined as a voltage drop in the equivalent circuit of the cable insulation resistance. The electric field intensity in the insulation system as a function of the radius x of layer dx in the cable insulation is determined by the following formula:

$$E\left(x\right)={\displaystyle \frac{j\left(x\right)}{\sigma \left(x\right)}}$$

(7)

where x is the radial coordinate, j(x) is the leakage current density in the insulation, and σ is the electrical conductivity of the insulating material.

The dependence of the electrical conductivity of the insulating material on the temperature σ(T) and on the electric field intensity σ(E) (which are elements of the analysis of the influence of the thermal effect and spatial charges on the distribution of the electric field intensity) create the so-called extended model σ(T, E) of the electrical conductivity of a high-voltage direct current cable. For design purposes (especially in the cases of cable accessories—joints, terminations), it is essential to know the value of the electric field intensity at the surface of the cable core and at the external surface of the insulation, along with the changes in the load current of the cable core. The dependence of the electric field intensity on the electrical conductivity of the insulating material in the steady state is expressed by the following formula:

$$E\left(x\right)={\displaystyle \frac{I}{2\pi x}}·{\displaystyle \frac{1}{{\sigma }_0}}·\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\alpha \left[T\left(r_i\right)+{\displaystyle \frac{w_z{\rho }_c}{2\pi }}\mathrm{l}\mathrm{n}{\displaystyle \frac{r_i}{x}}-T_0\right]\right\}·\mathrm{e}\mathrm{x}\mathrm{p}\left[-\beta ·E\left(x\right)\right]$$

(8)

For the purpose of simplification, the following approximation of the function that expresses the effect of the electric field on the electrical conductivity of the insulating material in the above formula is introduced (e—Euler’s number) [21,36]:

$$\exp \left[-\beta ·E\left(x\right)\right]\cong {\left[{\displaystyle \frac{e·E\left(x\right)}{E_z}}\right]}^{-\beta ·E_z}$$

(9)

and introduces the notation E_z of the equivalent uniform electric field intensity in the cable:

$$E_z={\displaystyle \frac{U}{r_i-r_z}}$$

(10)

where U is the operating voltage in the cable, r_i is the insulation radius, and r_z is the cable core radius. An example of the approximating function for coefficient β = 0.03 mm/kV and E_z = 15 kV/mm is shown in Figure 2. Both functions overlap roughly above a field strength value of 8 kV/mm. Then, Formula (8) takes on the following form:

$$E\left(x\right)·{\left[{\displaystyle \frac{e·E\left(x\right)}{E_z}}\right]}^{\beta ·E_z}={\displaystyle \frac{I}{2\pi x{\sigma }_0}}·\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\alpha \left[T\left(r_i\right)-T_0\right]\right\}·\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\alpha \left[{\displaystyle \frac{w_z{\rho }_c}{2\pi }}\mathrm{l}\mathrm{n}{\displaystyle \frac{r_i}{x}}\right]\right\}$$

(11)

To denote the following:

$$A=-\alpha \left[T\left(r_i\right)-T_0\right]; B=-\alpha {\displaystyle \frac{w_z{\rho }_c}{2\pi }}; C={\displaystyle \frac{{\left(E_z/e\right)}^{\beta ·E_z} }{2\pi {\sigma }_0}}$$

(12)

Formula (11) takes on the following form:

$${\left[E\left(x\right)\right]}^{1+\beta ·E_z}=C·{\displaystyle \frac{I}{x}}·\mathrm{e}\mathrm{x}\mathrm{p}\left(A\right)·{\left({\displaystyle \frac{r_i}{x}}\right)}^B$$

(13)

After the following subsequent markings:

$$D=C·\mathrm{e}\mathrm{x}\mathrm{p}\left(A\right); F={\left(D·r_i^B\right)}^{{\displaystyle \frac{1}{1+\beta ·E_z}}}; G={\displaystyle \frac{B+1}{1+\beta ·E_z}}$$

(14)

Formula (13) takes on the following form:

$$E\left(x\right)=F·I^{{\displaystyle \frac{1}{1+\beta ·E_z}}}·x^{-G}$$

(15)

In further calculations, the dependence of the operating voltage U and leakage current I on the electric field strength is taken into account:

$$U={\int }_{r_z}^{r_i}E\left(x\right)·dx=F·I^{{\displaystyle \frac{1}{1+\beta ·E_z}}}·{\int }_{r_z}^{r_i}{x}^{-G}·dx=F·I^{{\displaystyle \frac{1}{1+\beta ·E_z}}}·{\displaystyle \frac{1}{-G+1}}·\left[r_i^{-G+1}-r_{ż}^{-G+1}\right]$$

(16)

$$I={\left\{{\displaystyle \frac{U\left(-G+1\right)}{F\left[r_i^{-G+1}-r_z^{-G+1}\right]}}\right\}}^{1+\beta ·E_z}$$

(17)

By substituting (15) with (17) and performing the transformations, we obtain the distribution of the electric field intensity by taking the effect of the temperature and electric field on the electrical conductivity of the insulating material into account:

$$E\left(x\right)=U·{\displaystyle \frac{1-G}{r_i^{1-G}-r_z^{1-G}}}·x^{-{\displaystyle \frac{B+1}{1+\beta ·E_z}}}$$

(18)

By introducing the description of the B and G symbols in Formula (18), the following final expression for the distribution of the electric field intensity in the cable insulation system is obtained:

$$E\left(x\right)=U·{\displaystyle \frac{1-\frac{1-\alpha \frac{w_z{\rho }_c}{2\pi }}{1+{\beta E}_z}}{r_i^{1-\frac{1-\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}-r_z^{1-\frac{1-\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}}}·x^{-{\displaystyle \frac{1-\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}}$$

(19)

Formula (19) parametrically presents the influence of the following on the distribution of the electric field intensity in the HVDC power cable:

▪

temperature (α coefficient);

▪

field effect (β coefficient);

▪

cable load (causing energy losses of power W_z in the cable core);

▪

type of insulating material (thermal resistance ρ_c).

4. Influence of HVDC Cable Load Current on Electric Field Distribution in Insulation in Presence of Space Charges

In terms of the operating conditions of power cables, the existing thermal effect in the insulation is the main factor that influences the value of the electrical conductivity σ of the insulating materials and, therefore, the distribution of the electric field intensity in the cable. With a variable cable load, the interaction between the thermal effect and the electric field is determined by the size of the operational exposures of the cable-insulating system [37,38]. In a special case of the current load on the cable, the effect of the changes in the electrical conductivity of the insulating materials is the so-called electric field inversion or reversal, meaning a shift of the maximum field intensity from the area near the screen on the cable core to the area near the external screen on the insulation. Presented in this article as the example of a cable with cross-linked polyethylene insulation, this case was analyzed by examining the effect of the cable load current value and, thus, the increased temperature in the insulation as well as the spatial charges on the distribution of the electric field intensity. The inversion phenomenon will occur when the electric field intensity distribution exhibits the following property (for a positive derivative of E(x), the field intensity increases from the cable core to the screen/shell):

$${\displaystyle \frac{dE\left(x\right)}{dx}}=-U·{\displaystyle \frac{\beta E_z+\alpha \frac{w_z{\rho }_c}{2\pi }}{r_i^{\frac{\beta E_z+\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}-r_{ż}^{\frac{\beta E_z+\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}}}·{\displaystyle \frac{1-\alpha \frac{w_z{\rho }_c}{2\pi }}{{\left(1+\beta E_z\right)}^2}}{x}^{-{\displaystyle \frac{2+\beta E_z-\alpha \frac{w_z{\rho }_c}{2\pi }}{1+\beta E_z}}}\ge 0$$

(20)

From this, the operational condition of the field inversion results in the following:

$$w_z\ge {\displaystyle \frac{2\pi }{\alpha ·{\rho }_c}}$$

(21)

This condition means that an inversion will occur when the cable core load exceeds the critical current value I_cr, thus determining the critical power density of the loss flux:

$$I_{cr}=\sqrt{{\displaystyle \frac{2\pi }{\alpha {\rho }_c}}\gamma S}$$

(22)

where S is the cross-section of the cable core, and γ is the conductivity of the core material.

The electric field inversion in the cable insulation system is illustrated in the graph in Figure 3; this was calculated using data on the construction of a 300 kV XLPE-insulated cable [39]:

▪

copper conductor with cross-section S = 1000 mm²;

▪

conductor radius r_z = 17.8 mm, insulation radius r_i = 37.8 mm;

▪

insulation: cross-linked polyethylene with the following properties:

-

electrical conductivity σ = 10⁻¹² S·m⁻¹;

-

thermal conductivity coefficient α = 0.1 1/K;

-

field conductivity coefficient β = 0.03 mm/kV;

-

thermal resistivity—3.5 K·m/W.

When the load current value changed in a range of up to 1500 A, the overturning phenomenon occurred. Figure 3 shows the distributions of the electric field intensity E(x) in the cable insulation at two load current values of 500 A and 1500 A without the participation of the field effect (β = 0 mm/kV) and with its participation (β = 0.03 mm/kV). At the higher value of the β coefficient (0.03 mm/kV), the electric field intensity at the surface of the cable core decreased, while its value increased at the external screen on the insulation. In the case of field reversal, at a critical value of the load current, the above changes in the electric field intensity value apply to the external insulation and the surface of the cable core, respectively.

The graphs in Figure 4 illustrate the effect of the cable load current on the electric field strength values at the conductor surface E(r_z) and at the external insulation surface E(r_i) without the space charge interaction (β = 0 mm/kV) and with the space charge interaction (β = 0.03 mm/kV).

The inversion point occurred at critical current I_cr = 984 A. With the increase of the load current, the thermal loss flux caused the electric field to be pushed out from the area near the cable core to the area near the external screen on the insulation. In a current range from 250 to 1500 A, this means a reduction in the field intensity from 20 to 8 kV/mm at the core surface and an increase from 11 to 22 kV/mm in the remaining insulation area.

5. Characteristic Cases of Electric Field Distribution

A factor that is mainly important in the case of polymeric insulating materials that are susceptible to field or thermal emissions (cross-linked polyethylene) is the influence of the field effect on the electrical conductivity of these insulating materials. Using Formula (19) and assuming the following two values of the β coefficient (β ≠ 0—space charges in insulation; β = 0—no space charges), the distributions of the electric field intensity that characterize these cases were compared: E(x; β ≠ 0) and E(x; β = 0). In both cases, the value of the critical current I_cr is the same, because the electric field does not affect the critical heat flux power on which the field reversal phenomenon depends. Analyzing the influence of the field factor (expressed by the value of the β coefficient) on the distribution of the electric field intensity, the following relationship was found:

$${\displaystyle \frac{dE(x; \beta \ne 0)}{dx}} \ne {\displaystyle \frac{dE(x; \beta =0)}{dx}}$$

(23)

The changes in the slopes of the E(x; β ≠ 0) and E(x; β = 0) functions depend on the value of the cable load current; this means that a field reversal may occur under certain conditions. Meanwhile, the different courses of these functions are the effect of increasing the electrical conductivity of the material by the space charge, which is the cause of the changes in the distribution of the E(x; β ≠ 0) field intensity. If there are no space charges in the insulating material, then the following applies:

(a)

There is no influence of the electric field on the electrical conductivity of the insulating material (β = 0 mm/kV) and the distribution of the electric field intensity is defined in this case by the following formula:

$$E\left(x\right)=U·{\displaystyle \frac{\alpha ·w_z·{\rho }_c}{2\pi \left[r_i^{\alpha \frac{w_z{\rho }_c}{2\mu }}-r_z^{\alpha \frac{w_{z{\rho }_c}}{2\pi }}\right]}}·x^{\alpha {\displaystyle \frac{w_z{\rho }_c}{2\pi }}-1}$$

(24)

and when the cable is unloaded:

(b)

There are no power losses in the cable core, so the heat power flux W_z = 0. The distribution of the electric field intensity is given by the following formula:

$$E\left(x\right)={\displaystyle \frac{U}{x\mathrm{l}\mathrm{n}\frac{r_i}{r_z}}}$$

(25)

The distribution of the electric field intensity that is caused by the flow of the leakage current in the insulation is identical to the distribution of the electrostatic field in the model system of the coaxial cylindrical electrodes in this case.

6. Influence of Voltage Range on Insulation Conductivity in HVDC Cables

Due to the complex and nonlinear dependence of the electrical conductivity of the insulating material on the electric field intensity in the cable, analyzing the influence of the field effect (space charges) on the electric field intensity distribution is a complex problem when taking the insulation temperature into account. In the considerations when explaining these dependencies, Formula (8) was used. By taking its logarithm on both sides, we obtained the following formula:

$$\mathrm{l}\mathrm{n}E\left(x\right)+\beta ·E\left(x\right)=\mathrm{l}\mathrm{n}\left({\displaystyle \frac{I}{2\pi x}}·{\displaystyle \frac{1}{{\sigma }_0}}\right)-\alpha \left[T\left(r_i\right)+{\displaystyle \frac{w_z{\rho }_c}{2\pi }}\mathrm{l}\mathrm{n}{\displaystyle \frac{r_i}{x}}-T_0\right]$$

(26)

where the left side of the formula expresses the electric field as the cause, and the right side shows the effect of its action. In order to perform the analysis over a wide range of electric field strength values (from 1 to 100 kV/mm), the above equation can be represented in the form of two functions—f₁(E) and f₂(E):

$$f_1\left(E\right)=\mathrm{l}\mathrm{n}E\left(x\right)+\beta ·E\left(x\right)$$

(27)

$$f_2\left(E\right)=\mathrm{l}\mathrm{n}E\left(x\right)$$

(28)

Using the following values of the β coefficient (0.003, 0.03, and 0.3 mm/kV), which represent the share of the space charges, the curves of the above functions are presented in Figure 5.

The graph shows that, within the range of the working electric field strength in the cables (i.e., from 1 to 10 kV/mm, and with the β coefficient value being less than about 0.03 mm/kV), the influence of the space charges on the electrical conductivity of the insulating material (and, consequently, on the distribution of the electric field strength) can be neglected. The influence of the space charge can be of significant importance when the working electric field strength exceeds 10 kV/mm. It is worth noting, however, that the effects of the field effect can occur at an electric field strength of even 2 kV/mm (with β = 0.3 mm/kV). The literature sources provide different values of the β coefficient depending on both the type of insulating material and the test method [21,28,34]. The above conclusions are illustrated by the results of the calculations of the electric field strength distributions in the HVDC cable insulation at three voltage values (100, 300, and 500 kV), load currents of 500 and 1500 A, and coefficients β = 0 mm/kV and β = 0.03 mm/kV. The practical ranges of the operating electric field strength at the above cable voltages are as follows: 100 kV: 4–6.5 kV/mm; 300 kV: 11–22 kV/mm, and 500 kV: 18–33 kV/mm. These results are presented in Figure 6. Graph Group (a) in Figure 6 results from the calculations at current I = 500 A, and Group (b) from at 1500 A (i.e., after the field reversal). At a voltage of 100 kV (within the low range of the working electric field intensity), the formation of space charges has no effect on the field intensity distribution (the slopes of the curves are the same in both cases, and the radial distribution of the field is the same); however, this effect is increasingly visible in the remaining cases. The above statement confirms the earlier conclusions that resulted from the theoretical analysis and the courses of functions f₁ and f₂ in Figure 5. In the case of the 300 kV voltage (within the high range of the working electric field intensity), the influence of the field effect on its distribution in the insulation system is visible.

When the slope of the electric field strength distribution curve was greater (Figure 6), the field distribution was steeper. These effects were even more pronounced at the 500 kV voltage. The above results were consistent with the conclusions that resulted from the influence of the core load on the electric field distribution (Figure 4). Both of the energy sources (the thermal effect and electric field) acted on the increase in the conductivity and then on the values of the electric field intensity in the insulating insert. With the numerical results for the above cases of supply voltages being within a range of 100–500 kV, the inversion effect that results from the load currents of 500 and 1500 A, for field effect coefficient β = 0.03 mm/kV, and with its omission β = 0 mm/kV, are listed in

Table 1. β-dependent electric field strength in HVDC cable.

LOAD CURRENT I = 500 A					LOAD CURRENT I = 1500 A “INVERSION”
	E(rz) [kV/mm]		Cable core			E(rz) [kV/mm]		Cable core
β [mm/kV]	100 kV	300 kV	500 kV		β [mm/kV]	100 kV	300 kV	500 kV
β = 0	6.75	20.24	33.74		β = 0	3.00	9.02	15.03
β = 0.03	6.50	18.50	29.77		β = 0.03	3.22	10.63	18.83
delta E	−0.25	−1.74	−3.97		delta E	0.22	1.61	3.80
	E(ri) [kV/mm] Cable outer insul.					E(ri) [kV/mm] Cable outer insul.
β = 0	3.82	11.47	19.12		β = 0	7.15	21.47	35.79
β = 0.03	3.97	12.50	21.52		β = 0.03	6.85	19.33	30.91
delta E	0.15	1.03	2.40		delta E	−0.30	−2.14	−4.88

. The influence of the β coefficient in a wide range of its values (from 0.003 to 0.5 mm/kV) on the electric field intensity at the surface of the cable core E(r_z) and at the external screen on the insulation E(r_i) is interesting. This case is illustrated by the graphs of the E(r_z) = f(β) and E(r_i) = f(β) dependencies that are presented in Figure 7; these assume a voltage of 300 kV and load currents of 500 and 1500 A.

The continuous line indicates the electric field distribution at a load current of 500 A, and the dotted line indicates the course after the inversion for a current of 1500 A. It can be seen that, for the field coefficient β that is above the value of 0.4 mm/kV, the course smoothens, and the influence of its further increase is negligible. For current I = 500 A, a decrease in the field intensity at the surface of the cable core E(r_z) is observed, as is an increase in insulation E(r_i) to the value of the β coefficient that is equal to about 0.3 mm/kV. Above this value of coefficient β, the E(r_z) and E(r_i) field intensities reach steady and equal values.

Theoretically, this means a change in the distribution of the electric field intensity in the insulating system from non-uniform to uniform, and physically, an increase in the electrical conductivity of the insulating material.

In the modeling of the electric field in the cables under the conditions of coupled electrical–thermal effects, values from 0.01 to 0.1 mm/kV are assumed (typically, 0.03 mm/kV). The action of the spatial charges that results from the field effect complements the thermal effect on the conductivity and can affect the change in the distribution of the electric field intensity. Using this fact, there have been proposals for modifying this using special nanoadditives (especially in the application to cable terminations, (e.g., [40,41]) to change its thermal and electrical characteristics in order to improve the distribution of the electric field in the insulating system of the termination. It would be advisable to obtain materials with a significantly reduced effect of temperature on the electrical conductivity (reducing the value of the α coefficient); thanks to this, the effect of the current load of the cable on the radial distribution of the electric field intensity will be mitigated. Such materials (for example, those that are based on silicone rubber) would be advantageous in the insulating structures of cable accessories (joints, terminations).

7. Conclusions

The research problem of understanding the effects of the formations of spatial charges in insulating materials under specific conditions is of particular importance in connection with the development of high-voltage DC cable designs and the use of synthetic polymers in their insulation systems. The interaction of the thermal effect and the electric field and the release of space charges in such insulating materials results in changes in their electrical conductivity, and in consequence, changes in the distributions of the electric field strength in the cable insulation. The electric field intensity at the surface of the cable core and at the external screen on the insulation are of significant importance in such cases. It would also be advisable to develop materials that feature a significantly reduced effect of temperature on their electrical conductivity (reducing the α coefficient value); this would mitigate the effect of the cable current load on the distributions of the electric field intensity.

This article presents a practical range of working electric field intensity values in HVDC cables; these have significant impacts on the electrical conductivity of the insulation.

Hence, intention of this paper is to highlight the limit values of the electric field strength under the simultaneous action of the space charge and temperature gradient. For a space charge that is characterized by a β coefficient value of less than about 0.03 mm/kV, the limit value of the electric field intensity can be assumed to be 10 kV/mm. The influence of the β coefficient within a wide range of values from 0.003 to 0.5 mm/kV on the electric field intensity is shown in two cases: at the surface of the cable core and at the external screen on the insulation; this was in the presence of a thermal effect before and after the electric field inversion effect occurred.

The intention is to present the solution of the electric field distribution inside the structure of power cable, which is influenced by the temperature and in an entangled form by electric field in the analytical form, which will allow researchers and designers to obtain fast results and allow manipulation of the individual coefficients. The presented approach and modeling can then be easily extended for more multilayer structures, as well as assume other forms of parametric descriptions. In this way, the limiting values of the electric field on the core conductor, inside the insulation and on the shield can be calculated, taking into account also the field inversion effect.

Space charges and electrical conductivity are interdependent in dielectric materials. Space charges can alter local electric fields, which influence the movement of the charge carriers and, thus, the material’s conductivity. Conversely, the process of conduction itself can lead to the generation of space charges—particularly in the presence of trapping sites or ionic conduction mechanisms. Understanding this relationship is crucial for designing materials and systems (such as HVDC cables) that must maintain stable electrical properties when exposed to high voltages. Taking not only the temperature but also the effect of other possible factors in an electric field into account, a physical model of the electrical conductivity of synthetic polymers has not yet been fully developed theoretically due to the complex physical and chemical structures of this group of insulating materials.