# A Review about the Modeling and Simulation of Electro-Quasistatic Fields in HVDC Cable Systems

^{*}

## Abstract

**:**

## 1. Introduction

_{0}= 8.854 × 10

^{−12}As/(Vm) is the dielectric constant, ε

_{r}is the relative permittivity and $\kappa (T,|\overrightarrow{E}|)$ is the electric field $\overrightarrow{E}$ and temperature T dependent electric conductivity, a stationary electric field is obtained at approximately 10∙τ. Due to the non-linear electric conductivity, a stationary field may be obtained after hours or weeks [1,3,13,14]. For example, Figure 1a shows the electric field within a MI insulation with conductor temperature T(r

_{i}) = T

_{i}= 55 °C, sheath temperature T(r

_{a}) = T

_{a}= 35 °C and applied voltage U = 450 kV. A stationary electric field is seen above the black dashed line. The cylindrical geometry is depicted in Figure 1b, where the electric field is computed along the red evaluation line due to its radial dependency only [11,13].

## 2. High-Voltage Direct Current (HVDC) Cable Insulation Materials and the Non-Linear Electric Conductivity

^{−2}–1 eV, while the depth of deep traps is >1 eV [17,20,21]. The density of the traps is high near the conduction and valence band and decreases with increasing distance, as illustrated in Figure 2b.

_{Trap}, charges remain in traps that may range from a few seconds to years. The time t

_{Trap}is given by:

_{B}= 1.38 × 10

^{−23}J/K is the Boltzmann constant and 1/t

_{Trap,0}≈ (k

_{B}∙T)/h

_{P}is the “escape frequency”, with h

_{P}= 6.626 × 10

^{−34}Js as the Plank constant [17,20,22,23,24].

_{A,1}is the activation energy and K

_{1}and γ

_{1}are constants. All three constants E

_{A,1}, K

_{1}and γ

_{1}are determined by measurements [17,22,23,27]. In [3,14,28,29], the temperature dependency within the hyperbolic sine is neglected and the conductivity is described by:

_{A,2}, K

_{2}and γ

_{2}are different constants. The electric conductivity in (3) and (4) is based on the hopping theory. Using the Poole–Frenkel effect, the conductivity is described by:

_{A,3}, K

_{3}and γ

_{3}that have to be determined by measurements as well. The conductivity models in (3)–(5) are described by analogies to the semiconductor technology. Based on worldwide experience and a narrow range of the operation temperature, the electric conductivity is commonly given in a double exponential form with:

_{0}, α and β [22,30,31,32]. For MI, α ≈ 0.1 °C

^{−1}and β ≈ 0.03 mm/kV [30]. In the case of XLPE, α ≈ 0.1 °C

^{−1}and β ≈ 0.1 mm/kV [31]. Thus, the temperature dependency of both materials is approximately equal, but the electric field dependency is 3.5 times higher in XLPE [22,30,31]. A different formulation of (6) is:

_{0}and α are equal to (6), but the electric field dependency is determined by the constants E

_{Ref}and v. Considering a cable geometry, as depicted in Figure 1b, the approximations E

_{Ref}= U∙exp(−1)/(r

_{a}− r

_{i}) and v = U∙β/(r

_{a}− r

_{i}), with r

_{i}as the radius of the conductor and r

_{a}as the radius of the sheath, hold true [25,26,30].

## 3. Space Charges, Surface Charges and Charge Packets

^{−10}m

^{2}/(Vs) and a density of 0.07–0.1 C/m

^{3}. By contrast, slow charge packets have a mobility of 10

^{−16}–10

^{−14}m

^{2}/(Vs) with a density >0.1 C/m

^{3}[45]. The semicon interface between the polymeric insulation and the electrode shows a blocking behavior for charge packets, resulting in the accumulation of heterocharges and in a decrease of the lifetime [42,46].

## 4. Numerical Simulation of Charge Transport and the Electric Field in Direct Current (DC) Cable Insulations

#### 4.1. Equations to Simulate the Time-Varying Charge and Electric Field Distribution

_{μ}is the density of mobile charge carriers, n

_{t}is the density of trapped charge carriers and κ = n

_{μ}∙μ is the definition of the electric conductivity. Charge generation results from injection processes that is described by the Schottky law [47,52,53,54,55]. Equal to the electric conductivity, the mobility is also a function of the temperature and the electric field [53,56]. Positive and negative trapped and mobile charge carriers have to be considered separately, resulting in 4 equations of the source-term:

_{e,μ}is the density of negative mobile charges, n

_{h,μ}is the density of positive mobile charges (n

_{μ}= n

_{h,μ}+ n

_{e,μ}), n

_{e,t}is the density of negative trapped charges and n

_{h,t}is the density of positive trapped charges (n

_{t}= n

_{h,t}+ n

_{e,t}). The recombination coefficients are S

_{0,1,2,3}and the trapping (detrapping) coefficients for positive and negative charges are B

_{h}(D

_{h}) and B

_{e}(D

_{e}), respectively. The trap density for positive charges is n

_{h,t,0}and for negative charges n

_{e,t,0}[54]. To obtain the density of mobile charges, (12) is solved using the splitting method, which consists of first solving (12) with s = 0 and then solving s

_{1}for mobile electrons and s

_{2}for mobile holes in (15). For the density of trapped charges, only s

_{3}for electrons and s

_{4}for holes are evaluated [47].

_{p}and thermal conductivity λ. The heat source is represented by $\kappa \left(T,\left|\overrightarrow{E}\right|\right){\left|\overrightarrow{E}\right|}^{2}$ [14].

#### 4.2. Discretisation and Numerical Calculation Scheme

**G**

^{T}, the vector of current densities

**j**, the vector of electric dual cell charges

**q**, the permittivity matrix

**M**

_{ε}, the gradient matrix

**G**, the vector of nodal scalar potentials

**Φ**, the electric conductivity matrix

**M**

_{κ}, the vector of nodal temperatures

**u**

_{T}and the boundary conditions for the electric problem

**b**[58]. The discretization of (11) is:

**Φ**(t

^{m+1}) =

**Φ**

^{m+1}is:

_{CFL}, determined by the Courant–Friedrich–Levy (CFL) criterion, for stability reasons [52]. Similarly as with (11), the time discretization of the heat conduction Equation (16), using an explicit Euler method, yields the update scheme:

**M**

_{δ}is the density matrix,

**M**

_{c}is the matrix of the specific heat capacity,

**M**

_{λ}is the thermal conductivity matrix,

**q**

_{T}=

**M**

_{κ}(

**u**

_{T},

**Φ**)

**GΦGΦ**is the vector of heat sources and

**b**

_{T}contains the boundary conditions for the thermal problem [48,49,60,61]. A possible calculation scheme to obtain the electric field and the space charge density, using (8)–(10), is depicted in Figure 4 and using (11) in Figure 5.

^{3}(

**q**= 0) and a constant temperature equal to the environment temperature T

_{∞}. The vanishing space charge density results in a purely capacitive field that is determined by the permittivity and the geometry only. The time integration stops if a predefined time t = t

_{END}or $\Vert {\mathsf{\Phi}}^{m+1}-{\mathsf{\Phi}}^{m}\Vert /\Vert {\mathsf{\Phi}}^{m}\Vert <\eta $ is obtained, where η $\ll $ 1 is the stop threshold and $\Vert {\mathsf{\Phi}}^{m}\Vert $ is the absolute value of the vector

**Φ**at the discrete time step t

^{m}[11,58,59].

## 5. Simulation of the Electric Field and the Space Charge Density within a Cable Insulation

#### 5.1. Transient and Stationary Electric Field under Various Thermal Stresses

_{out}is the heat transmission coefficient [65,66,67,68]. Using Figure 6, the maximum space charge density (Figure 6c) and thus, the maximum field inversion (Figure 6b) are obtained for a stationary temperature distribution, showing the maximum temperature gradient (Figure 6a).

_{r}= const.), the stationary electric field is the solution of:

_{p}and λ, the stationary temperature is the solution of (16) for ∂/∂t = 0 [69] and is given by:

^{b}= exp(b∙ln(a)), (29) is reduced to:

_{V}and is given by:

_{0}(r) = U∙r

^{−1}∙ln(r

_{a}/r

_{i})

^{−1}), a rough estimation of the transient electric field is obtained. With (11) and Figure 6b, a reasonable assumption for the time dependence of the transient electric field is an exponential function. The stationary electric field E(r) is equivalently given by (30) or (33), respectively. With an exponential approach and the time constant τ in (1), the transient electric field is:

_{a}− r

_{i}). Using (7), the constants in Table 1 and the temperature distribution (28), with T

_{i}= 50 °C and T

_{a}= 35 °C, the numerically computed solution and the relative error of the approximation (35) are depicted in Figure 7. The relative error of (35) has the highest values in the vicinity of the conductor and the sheath, during the inversion of the field. The maximum error is about +13.58%. If the electric field approaches the stationary solution, the error decreases and is close to zero.

_{i}= 50 °C and a varying sheath temperature T

_{a}are depicted in Figure 8 [11,22]. As depicted in Figure 8a, the inversion of the electric field is seen at temperature gradients higher than 5 °C. Temperature gradients higher than 20 °C result in a sheath electric field that is higher than the maximum electric field at t = 0, which is about 32 kV/mm. The space charge distribution in Figure 8b for a temperature gradient of ΔT = 0 °C results from the dependency of the conductivity on the electric field. At temperature gradients higher than 15 °C the charge density increases towards the sheath, resulting from the constant δ

_{E}that is <−1 and the negative exponent of the radius r in (34).

_{E}needs to vanish, resulting in E(r) = U/(r

_{a}− r

_{i}). With (30), δ

_{E}vanishes at a temperature gradient T

_{i}− T

_{a}= ln(r

_{a}/r

_{i})/α ≈ 6 °C. Utilizing (33) this temperature gradient corresponds to losses of P

_{V}= 2πλ/α ≈ 10.5 W/m, with λ

_{MI}= 0.167 W/(Km) [30]. With losses higher than 10.5 W/m, the field inversion increases and with lower losses the electric field is higher at the conductor in comparison to the field at the sheath [70]. The temperature coefficient α is approximately equal for XLPE and MI, but the thermal conductivity of XLPE is higher (λ

_{XLPE}= 0.27–0.32 W/(Km)) in comparison to MI. Using an equal XLPE cable insulation, the losses are P

_{V}≈ 17–20 W/m to obtain a homogeneous electric field. Thus, a better heat dissipation of XLPE results in higher transmitted power until the risk of high field values at the outer sheath occurs [5]. According to [25,26] the charge density is separated into two parts. One corresponds to the temperature dependency of the electric conductivity (ρ

_{T}) and one corresponds to the electric field dependency (ρ

_{E}). With a given temperature gradient both charge parts are of the opposite sign. With increasing field dependency (v) the effect of field inversion decreases, but the net charge density (|ρ

_{T}| + |ρ

_{E}|) increases. In comparison to MI, the electric field dependency of XLPE is 3.5 times higher [30,31]. Thus, the net charge density (|ρ

_{T}| + |ρ

_{E}|) is higher in polymeric cable insulations, whereby XLPE usually operate under lower applied electric fields [2,3].

#### 5.2. Fast Calculation of the Steady State Charge Distribution

_{i}= 50 °C and the sheath temperature T

_{a}= 35 °C. Using an applied voltage of 470 kV, oscillations at the sheath start to increase after 5 iterations. High electric field values result in increased conductivity values, having an effect on the computation of the electric potential φ in (36). Decreasing the voltage from 470 kV to 400 kV yields lower and converging field values (Figure 9b). Further possibilities to reduce the charge density and thus, the field strength are a lower temperature gradient and lower constants α and β [71].

#### 5.3. Electric Fields in Power Cables, Considering the Environment

_{p}of the insulation material or the soil are constant, but the thermal conductivity λ of soil or air depends on the humidity and on the temperature. A coupling between the thermal and the electric field results from the thermal heat source $\kappa \left(T,\left|\overrightarrow{E}\right|\right){\left|\overrightarrow{E}\right|}^{2}$. As insulation losses have only a negligible influence on the resulting temperature distribution, the non-linear coupling can be reduced, resulting in a simplified temperature calculation [51,58,69].

_{out}= 52.4 mm and T

_{i}= 55 °C.

_{MI}= 0.167 W/(K∙m) and λ

_{PE}= 0.3–0.4 W/(K∙m) [30,67,80]. Within the range of −30 °C and +90 °C, the thermal conductivity of air λ

_{Air}depends on the temperature and is given by:

_{out}+ 1.5 m). For example, with a soil thermal conductivity of 0.47 W/(K∙m), the temperature at the earth–air interface is T = 32.4 °C and with 2.1 W/(K∙m), the temperature is reduced to T = 26.7 °C.

_{∞}), the effect on the insulation temperature also varies. With increasing T

_{∞}, the temperature gradient and the field stress at the outer sheath decreases. By contrast, the soil around the cable heats up and the temperature at the earth–air interface assumes its maximum value for dry soil (λ = 0.47 W/(K∙m)). With decreasing T

_{∞}, the effect on the environment is reduced, but the insulation is facing higher stress levels [51]. Considering a buried cable pair, a higher influence on the environment is seen, due to the additional heat losses of the second cable (see Figure 13).

_{a}. The metallic sheath has a negligible influence on the environment and the temperature at the earth-air-interface remains unaffected. Thus, one-dimensional electric field simulations are also applicable for cable pairs, if a metallic sheath is considered.

## 6. Simulation of Space Charge Effects at Interfaces and Surfaces and of Moving Charge Packets

#### 6.1. The Stationary Field Distribution, Considering Charges in the Vicinity of Electrodes and Dual-Dielectric Interfaces

#### 6.1.1. Modeling of Charges Close to Electrodes

_{con}(r) yields spatial variations at the conductor (“con”) and f

_{sh}(r) at the sheath (“sh”). The conductivity increases or decreases by a factor of n towards the electrodes and ζ is a constant that defines the gradient of f(r). With (7), the total conductivity is:

_{con}(r = r

_{a}) = 1 and f

_{sh}(r = r

_{i}) = 1, results in f (r) = f

_{con}(r) at the conductor and f (r) = f

_{sh}(r) at the sheath. With the temperature distribution (28), the solution of (41) is:

_{con}= n

_{sh}= 1.5 (n

_{con}= n

_{sh}= 0.75), T

_{i}= 50 °C, T

_{a}= 35 °C, and ζ

_{con}= ζ

_{sh}= (r

_{a}− r

_{i})/10 are depicted in Figure 14 [11,22]. The conductivity of MI is described by (7) and the necessary constants are given in Table 1. Resulting from the negative exponent of f(r) in (42), n > 1 corresponds to a decreasing electric field and, thus, in additional accumulated homocharges and n < 1 describes additional heterocharges. With n = 1 additional homo- or heterocharge distributions vanish and bulk effects are only considered (see Figure 14b).

#### 6.1.2. Modeling of Charges Close to Interfaces of Different Dielectrics

_{r,XLPE}= 2.3) and ethylene propylene rubber (EPR, ε

_{r,EPR}= 2.9) are found in [84]. The applied voltage is U = 30 kV, the conductor radius is r

_{i}= 1.8 mm, with a temperature of T

_{i}= 64 °C and the sheath radius is r

_{a}= 3.9 mm, with T

_{a}= 42 °C. The interface radius is r

_{Int}= 3.25 mm. For the stationary temperature distribution, the thermal conductivity is λ

_{XLPE}= 0.27 W/(K∙m) and λ

_{EPR}= 0.3 W/(K∙m). The space charge distribution is obtained after t = 20,000 s [84,87]. For the electric conductivity, (7) is used, with the constants given in [14]. A sketch of the dual-dielectric-interface is seen in Figure 15a.

_{XLPE}is only valid for r

_{i}≤ r ≤ r

_{Int}and f

_{EPR}is only valid for r

_{Int}≤ r ≤ r

_{a}. The interfacial charges δ

_{Int}are computed with:

_{EPR}(r

_{Int}) being the electric field within the EPR material at the interface (r = r

_{Int}) and E

_{XLPE}(r

_{Int}) is the electric field in the XLPE material at the interface [84,86].

_{Int}= 3.25 mm) increase the electric field of 20%, highlighting the importance of accurate modelling of HVDC components with multiple subcomponents.

#### 6.2. Empirical Conductivity Equation for the Simulation of Heterocharges in Polymeric Cable Insulations

_{con}and K

_{sh}yield conductivity variations at the conductor and the sheath. The constant r

_{x}defines the distance between the conductor (r

_{i}) and the position of the highest gradient (K

_{con}− K

_{sh}= 0.5 at r = r

_{x}) and χ defines the gradient of K

_{con}and K

_{sh}in the vicinity of both electrodes, which has an effect on the magnitude and the shape of the resulting heterocharge distribution [88,89]. Considering a planar insulation, r

_{i}= 0 and r

_{a}= D. To define the constants r

_{x}and χ, different measurements are taken from literature.

_{con}= χ

_{sh}= χ is assumed. To obtain the constants r

_{x}and χ, a “conductivity gradient region” Δ is defined and yields r

_{x}= Δ/2 and 10χ ≈ Δ [88,89].

_{+}and the sheath δ

_{−}are derived from [40] and approximately described by:

_{+}and δ

_{−}are divided by the spatial discretization Δh [96]. Finally, to compare simulations and measurements, the simulation results need to be converted to an equivalent signal, using a Gaussian filter [40].

_{+}= ε

_{0}ε

_{r}E(r

_{i}) and δ

_{−}= ε

_{0}ε

_{r}E(r

_{a}) [97]. In Figure 17, the applied voltage is U = 90 kV, r

_{i}= 4.5 mm, r

_{a}= 9 mm and ε

_{r}= 2.3 [14].

_{x}= Δ/2, 10χ ≈ Δ and Δ are summarized in Table 2. The measurements in [91] and the corresponding simulations are for example seen in Figure 18. To describe the electric conductivity of XLPE and LDPE, (4) is used, where the constants are given in [14,56,89,98,99].

_{x}in (47) are approximated with χ = (0.25∙D)/10 and r

_{x}= (0.25∙D)/2 for a planar insulation and with χ = (0.25∙(r

_{a}− r

_{i}))/10 and r

_{x}= (0.25∙(r

_{a}− r

_{i}))/2 for a cylindrical insulation [88,89]. Utilizing a constant applied voltage with positive and negative polarity, the polarity of the charge density also changes, but the magnitude and the shape remains unaffected. This is called the “mirror image effect” and seen in Figure 18a–f. The effect is discussed in [97], where possible explanations are a spatially varying polarization of the dielectric and the injection of charges at the electrodes [100,101].

#### 6.3. Transient Simulation of Charge Packets within a Cross-Linked Polyethylene (XLPE) Cable Insulation

_{con,ε}and n

_{sh,ε}are the increasing factors at the conductor (“con”) and the sheath (“sh”) and ζ

_{con,ε}and ζ

_{sh,ε}are constants defining the gradient of ε

_{r}(r) at both electrodes. The constant bulk permittivity is ε

_{r,Bulk}. A reasonable modelling approach for moving charges is an electric conductivity, having the shape of a Gaussian pulse and moving through the insulation [45,95]. With the non-linear bulk electric conductivity $\kappa \left(T,\left|\overrightarrow{E}\right|\right)$ the total electric conductivity is:

_{con,κ}and n

_{sh,κ}are the increasing factors, ζ

_{con,κ}and ζ

_{sh,κ}constants for the shape of the pulse and v

_{κ}is the velocity of the Gauss pulse (velocity of the moving charge pulse). The constants for the simulation results are summarized in Table 3. The cylindrical geometry has a conductor radius of r

_{i}= 4.5 mm, a sheath radius of r

_{a}= 9 mm and a bulk permittivity of ε

_{r,Bulk}= 2.25. A time-independent temperature distribution is used and described by (28). For the bulk electric conductivity, the hopping model (4) is used, with the constants E

_{A,2}= 1.48 eV, K

_{2}= 1∙10

^{14}A/m

^{2}and γ

_{2}= 2 × 10

^{−7}m/V [14,81]. A comparison between the measurements and the simulation results, at the time t = 0 s, t = 10.000 s and t = 20.000 s, is seen in Figure 19. The results are obtained along the red evaluation line in Figure 1b.

_{sh,ε}= n

_{sh,κ}= 1 indicate no increase/decrease of the permittivity and conductivity, which makes the corresponding constants ζ

_{sh,ε}and ζ

_{sh,κ}unnecessary. The conductivity factor n

_{con,κ}increases with voltage and temperature, where possible reasons are detrapped charges and thermally activated and electric field assisted injection [20].

- the formation of space charge regions,
- moving regions,
- a shape that is approximately maintained during motion,
- a periodic process (repetitive injection).

_{κ}= 3 × 10

^{−8}–7.5 × 10

^{−8}m/s, depending on the temperature and the electric field. With a mean electric field of E = U/(r

_{a}− r

_{i}) these velocity values correspond to mobility values (μ = v

_{κ}/E) of μ = 6 × 10

^{−15}m

^{2}/(Vs) (a), 3 × 10

^{−15}m

^{2}/(Vs) (b), 2.5 × 10

^{−15}m

^{2}/(Vs) (c) and 3.75 × 10

^{−15}m

^{2}/(Vs) (d). The mobility values are about 10

^{−15}m

^{2}/(Vs), which characterize them as “slow” charge packets [45].

_{Void}= 0), in the absence of partial discharges (PD), is assumed [104,105,106].

## 7. Accuracy of the Electric Field Computation within Cables and Cable Joints

_{i}= 50 °C and T

_{a}= 35 °C [11,59]. The analytic solution, together with the simulation results, using Figure 4 and Figure 5 are depicted in Figure 21a. Both formulations show a good accuracy in the ocular norm. Utilizing the space charge-oriented field formulation (8)–(10), inaccurate results are especially seen at the conductor, while using (11) the error has its maximum value at the sheath (see Figure 21b). With (8)–(10), the maximum error is 0.723%, with an average error of 0.3%. With (11), the maximum error (0.677%) and the average error (0.25%) are slightly lower. Equal to the results in Section 5.2, high electric field values are obtained during the time integration, resulting in inaccurate field values. Increasing the stationary electric field by decreasing the field dependency v (see (30)), the maximum error at the conductor is >1.5% using (8)–(10), but utilizing (11), the maximum error at the sheath has only slightly increased (0.74%) [59].

_{0,FGM}= 7 × 10

^{−11}S/m, E

_{1}= 1.23 kV/mm, E

_{2}= 2.23 kV/mm, N

_{1}= 1, N

_{2}= 0.35, m

_{FGM}= 5.7 × 10

^{−6}m/V and m

_{0}= 2.15 × 10

^{−6}m/V. In (51), the temperature dependency is neglected for simplicity. With field values <1 kV/mm, the material is nearly an insulator with an electric conductivity of 10

^{−10}S/m–10

^{−8}S/m. Between 1 kV/mm and 2.5 kV/mm, the electric conductivity increases of about six orders of magnitude. At fields >2.5 kV/mm, the electric conductivity is nearly constant and approximately 10

^{−2}S/m. Due to the large variety of the conductivity values, the time constant (1) lies between 0.89 s and 8.9 × 10

^{−9}s, with a relative permittivity of the FGM ε

_{r,FGM}= 10 [59,109].

_{END}= 2 μs, with a time step of Δt = 0.0133 μs, yields a relative difference of $\left(\Vert {\mathsf{\Phi}}^{\mathrm{m}+1}-{\mathsf{\Phi}}^{\mathrm{m}}\Vert /\Vert {\mathsf{\Phi}}^{\mathrm{m}}\Vert \right)\cdot 100\text{\hspace{0.17em}}\%<1\text{\hspace{0.17em}}\%$ between the discrete time steps m and m + 1. For simplicity and due to the short time t

_{END}, the electric conductivity of XLPE and insulating silicone rubber (LSR) are assumed to be constant and given by κ

_{XLPE}= 10

^{−15}S/m and κ

_{LSR}= 5 × 10

^{−13}S/m. The relative permittivity of both materials is ε

_{r,XLPE}= 2.3 and ε

_{r,LSR}= 3.5 [59,109].

^{m=13}= 13Δt, as seen in Figure 22d,e.

^{m=12}= 12Δt and the electric conductivity has values of 10

^{−2}S/m (Figure 22e). Consequently, the chosen time step size yields unstable field values. Stable field values are computed with (8)–(10), if the time step size is reduced to Δt = 0.0066 μs. With (11), an averaging process is needed to compute the electric field $\overrightarrow{E}$ = −grad(φ). With (8)–(10), an additional averaging process is needed to compute the charge density (see (8)). Due to the additional averaging process and the non-linear electric conductivity, unstable electric field values are delivered [59].

## 8. The Electric Field of Ground Electrodes, Considering the Effect of Electro-Osmosis

_{0,θ}= 2.5 × 10

^{−1}S/m, b

_{θ}= 1.5 and κ

_{s}= 2 × 10

^{−5}S/m [117]. To consider the temperature dependency, (54) is extended by a correction factor g(T) and finally given by:

_{0,θ}= 8.5 × 10

^{−9}m

^{2}/(Vs) and a

_{θ}= 1.4. Heat losses within the conductor of the ground electrode and within the soil result in a temperature increase where the thermal conductivity decreases with decreasing humidity [89]. Additionally, the thermal conductivity shows a dependency on the temperature over a temperature range of 2 °C–92 °C in [126]. Measurements of the thermal conductivity λ in [77] and [126] are approximated by:

_{θ}= −0.024∙(T-273.15) + 3.049, β

_{θ}= 0.067∙(T-273.15) + 13.867 and γ

_{θ}= 6.67 × 10

^{−4}∙(T-273.15) + 0.299. The measurements of the soil electric conductivity, electro- osmotic conductivity and thermal conductivity, together with their corresponding models (55)–(57) are seen in Figure 23 [127].

^{−9}m

^{2}/(Vs). The effect of thermo-osmosis, due to a temperature gradient, is equally described as (53). By contrast, the thermo-osmotic conductivity is about 3 orders of magnitude lower than k and can be neglected in HVDC applications [128].

_{END}or $\Vert {\theta}^{\mathrm{m}+1}-{\theta}^{\mathrm{m}}\Vert /\Vert {\theta}^{\mathrm{m}}\Vert <\eta $, where η $\ll $ 1 is the stop threshold, are obtained. Compared to the thermal problem, the time constant of the transient electro-osmosis process is higher [129]. Thus, humidity variations immediately affect the thermal conductivity and the temperature, resulting in a weak coupling between the quantities θ and T in Figure 24 [127].

## 9. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Capital letters | |

B_{h} | Trapping coefficient for positive charges [1/s] |

B_{e} | Trapping coefficient for negative charges [1/s] |

D | Thickness of a planar insulation [m] |

D_{h} | Detrapping coefficient for positive charges [1/s] |

D_{e} | Detrapping coefficient for positive charges [1/s] |

$\overrightarrow{E}$ | Electric field [V/m] |

E_{0}(r) | Electric field within a cable insulation at t = 0 [V/m] |

E_{1} | Constant for the electric conductivity of FGM in (51) [V/m] |

E_{2} | Constant for the electric conductivity of FGM in (51) [V/m] |

E_{A,1} | Activation energy in (3) [eV] |

E_{A,2} | Activation energy in (4) [eV] |

E_{A,3} | Activation energy in (5) [eV] |

E_{e} | Energy of charge carriers within the band diagram [eV] |

E_{Ref} | Reference electric field in (7) [V/m] |

E_{Trap} | Trap depth [eV] |

G | Discrete gradient matrix |

G^{T} | Discrete divergence matrix |

I | Current [A] |

$\overrightarrow{J}$ | Current density [A/m^{2}] |

K_{1} | Conductivity constant in (3) [A/m^{2}] |

K_{2} | Conductivity constant in (4) [A/m^{2}] |

K_{3} | Conductivity constant in (5) [S/m] |

K_{con} | Conductivity variations in the vicinity of the conductor in (47) |

K_{sh} | Conductivity variations in the vicinity of the sheath in (47) |

M_{c} | Discrete matrix of the specific heat capacity |

M_{δ} | Discrete density matrix |

M_{ε} | Discrete permittivity matrix |

M_{κ} | Discrete electric conductivity matrix |

M_{λ} | Discrete thermal conductivity matrix |

N_{1} | Constant for the electric conductivity of FGM in (51) |

N_{2} | Constant for the electric conductivity of FGM in (51) |

P_{V} | Losses per length within the conductor [W/m] |

S_{0,1,2,3} | The recombination coefficients for different charge types [m^{3}/(As∙s)] |

T | Temperature [°C] |

T_{a} | Sheath temperature [°C] |

T_{i} | Conductor temperature [°C] |

T_{∞} | Environment temperature [°C] |

U | Voltage [V] |

W | Mean band gab energy [eV] |

Small letters | |

a_{θ} | Constant for the electro-osmotic hydraulic conductivity of soil in (56) |

b_{θ} | Constant for the electric conductivity of soil in (54) |

b | Boundary conditions for the electric problem |

b_{T} | Boundary conditions for the thermal problem |

c_{p} | Specific heat capacity [W∙s/(kg∙K)] |

h_{P} = 6.626∙10^{−34} | Plank constant [J∙s] |

j | Vector of current densities |

k | Electro-osmotic hydraulic conductivity [m^{2}/(Vs)] |

k_{0,θ} | Constant for the electro-osmotic hydraulic conductivity of soil in (56) |

k_{B} = 1.38∙10^{−23} | Boltzmann constant [J/K] |

m | Discrete time index |

m_{0} | Constant for the electric conductivity of FGM in (51) [m/V] |

m_{FGM} | Constant for the electric conductivity of FGM in (51) [m/V] |

n_{con} | Conductivity increasing/decreasing factor at the conductor in (39) |

n_{con,ε} | Conductivity increasing/decreasing factor at the conductor in (49) |

n_{con,κ} | Conductivity increasing/decreasing factor at the conductor in (50) |

n_{EPR} | Conductivity increasing/decreasing factor in (45) |

n_{e,t} | Density of negative trapped charges [As/m^{3}] |

n_{h,t} | Density of positive trapped charges [As/m^{3}] |

n_{e,μ} | Density of negative mobile charges [As/m^{3}] |

n_{h,μ} | Density of positive mobile charges [As/m^{3}] |

n_{e,t,0} | Trap density for negative charges [As/m^{3}] |

n_{h,t,0} | Trap density for positive charges [As/m^{3}] |

n_{sh} | Conductivity increasing/decreasing factor at the sheath in (39) |

n_{sh,ε} | Conductivity increasing/decreasing factor at the sheath in (49) |

n_{sh,κ} | Conductivity increasing/decreasing factor at the sheath in (50) |

n_{XLPE} | Conductivity increasing/decreasing factor in (45) |

q | Vector of electric dual cell charges |

q_{T} | Vector of heat sources |

$\overrightarrow{q}$ | Heat flux [W/m^{2}] |

${\overrightarrow{q}}_{\mathrm{W}}$ | Water flow due to electro-osmosis [m/s] |

r | Radius [m] |

r_{a} | Sheath radius [m] |

r_{i} | Conductor radius [m] |

r_{Int} | Interface radius of two different dielectrics [m] |

r_{out} | Radius of the outer sheath [m] |

r_{x} | Distance between the position of the highest gradient of K_{con} (K_{sh}) and the conductor (sheath) [m] |

s_{1} | Source term for negative mobile charges [A/m^{3}] |

s_{2} | Source term for positive mobile charges [A/m^{3}] |

s_{3} | Source term for negative trapped charges [A/m^{3}] |

s_{4} | Source term for positive trapped charges [A/m^{3}] |

t | Time [s] |

t_{END} | Predefined end time [s] |

t_{Trap} | Time, charges remain in traps [s] |

1/t_{Trap,0} | Escape frequency [1/s] |

u_{T} | Vector of nodal temperatures |

v | Constant for electric field dependency in (7) [-] |

v_{κ} | Velocity of the Gauss pulse (charge packet pulse) [m/s] |

x | Coordinate for the planar insulation [m] |

Greek capital letters | |

∆ | Region of additional spatial variations at the conductor and the sheath [m] |

∆h | Spatial discretization [m] |

Δt | Discrete time step [s] |

Δt_{CFL} | Time step, determined by the Courant-Friedrich-Levy (CFL) criterion [s] |

Φ | Vector of nodal scalar potentials |

Greek small letters | |

α | Constant for temperature dependency in (6) [°C^{−1}] |

α_{out} | Heat transmission coefficient [W/(K∙m^{2})] |

α_{θ} | Constant for the thermal conductivity of soil in (57) |

β | Constant for electric field dependency in (6) [m/V] |

β_{θ} | Constant for the thermal conductivity of soil in (57) [(K∙m)/W] |

𝛾_{1} | Constant for electric field dependency in (3) [K∙m/V] |

𝛾_{2} | Constant for electric field dependency in (4) [m/V] |

𝛾_{3} | Constant for electric field dependency in (5) [m/V] |

γ_{θ} | Constant for the thermal conductivity of soil in (57) [(K∙m)/W] |

δ | Density [kg/m^{3}] |

δ_{Int} | Interface charges between two dielectrics [C/m^{2}] |

δ_{+} | Positive surface charges at the conductor [C/m^{2}] |

δ_{−} | Negative surface charges at the sheath [C/m^{2}] |

𝜀_{0} = 8.854∙10^{−12} | Dielectric constant [As/(Vm)] |

𝜀_{r} | Relative permittivity |

ε_{r,Bulk} | Constant bulk permittivity in (49) |

ζ_{con} | Constant that defines the gradient at the conductor in (39) [m] |

ζ_{con,ε} | Constant that defines the gradient at the conductor in (49) [m] |

ζ_{con,κ} | Constant that defines the gradient at the conductor in (50) [m] |

ζ_{EPR} | Constant that defines the gradient in (45) [m] |

ζ_{sh} | Constant that defines the gradient at the sheath in (39) [m] |

ζ_{sh,ε} | Constant that defines the gradient at the sheath in (49) [m] |

ζ_{sh,ε} | Constant that defines the gradient at the sheath in (50) [m] |

ζ_{XLPE} | Constant that defines the gradient in (45) [m] |

η$\ll $ 1 | Stop threshold for time integration |

θ | Humidity (volumetric water content) [m^{3}/m^{3}] |

κ | Electric conductivity [S/m] |

κ_{0} | Conductivity constant in (6) [S/m] |

κ_{0,FGM} | Constant for the electric conductivity of FGM in (51) [S/m] |

κ_{0,θ} | Constant for the electric conductivity of soil in (54) [S/m] |

κ_{s} | Constant for the electric conductivity of soil in (54) [S/m] |

𝜆 | Thermal conductivity [W/(Km)] |

μ | Mobility of charge carriers [m^{2}/(Vs)] |

ρ | Space charge density [C/m^{3}] |

ρ_{E} | Space charge density that corresponds to the electric field dependency [C/m^{3}] |

ρ_{T} | Space charge density that corresponds to the temperature dependency [C/m^{3}] |

τ | Time constant [s] |

φ | Electric potential [V] |

χ_{con} | Constant to define the gradient of (47) in the vicinity of the conductor [m] |

χ_{sh} | Constant to define the gradient of (47) in the vicinity of the sheath [m] |

ω | Under-relaxation parameter |

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**Figure 2.**(

**a**) Band structure for an insulation material like cross-linked polyethylene (XLPE), where states with different energy levels are formed due to the semicrystalline structure of polyethylene (PE) and impurities within the material. Because of these states, only a mean band gap energy W is defined; (

**b**) density of traps as a function of the trap depth. A high density of shallow traps (trap depth 10

^{−2}–1 eV) is seen near the conduction band and the valence band. With increasing trap depth and thus distance to the conduction band and the valence band, the trap density is decreasing [17,20,21].

**Figure 6.**(

**a**) Transient temperature distribution within a MI insulation; (

**b**) corresponding transient electric field stress; (

**c**) corresponding transient space charge density.

**Figure 7.**(

**a**) Numerically computed electric field distribution; (

**b**) relative error between the numerically computed transient electric field and the approximation (35).

**Figure 9.**Electric field at the conductor and the sheath in a 20 mm-thick MI insulation. (

**a**) with an applied voltage of U = 470 kV, the electric field diverges after 5 iterations; (

**b**) with a reduced voltage of U = 400 kV, the electric field converges [71].

**Figure 10.**Electric field at the conductor and the sheath in a 20 mm thick MI insulation. (

**a**) with an applied voltage of U = 470 kV and ω = 1, the electric field diverges after 5 iterations; (

**b**) with a voltage of U = 470 kV and ω = 0.95, the electric field converges [71].

**Figure 11.**(

**a**) Two-dimensional geometry of a single cable within the earth; (

**b**) the cable consists of conductor, insulating material and outer sheath [51].

**Figure 12.**(

**a**) Simulated temperature distribution inside a MI insulation. The vertical block dotted line is the interface between the insulation and the outer sheath; (

**b**) stationary electric fields stress, resulting from the temperature distribution [51].

**Figure 13.**(

**a**) Temperature distribution around a MI cable pair, with a soil thermal conductivity of 2.1 W/(K∙m) and T

_{∞}= 20 °C; (

**b**) temperature distribution within the insulation [51].

**Figure 14.**(

**a**) Example of the stationary electric field within a MI insulation, considering additional charge accumulation at both electrodes; (

**b**) corresponding space charge distribution [70].

**Figure 16.**Example of a space charge measurement within an XLPE cable insulation. “Filtered” surface charges appear as a Gaussian curve [73].

**Figure 18.**Measured and simulated charge distribution in a XLPE and a LDPE insulation [91]. (

**a**) XLPE, planar, +U; (

**b**) XLPE, planar, −U; (

**c**) XLPE, cylindrical, +U; (

**d**) XLPE, cylindrical, −U; (

**e**) LDPE, planar, +U; (

**f**) LDPE, planar, −U. The absolute value of the applied voltage is |U| = 20 kV.

**Figure 19.**Comparison between a measured (dashed line) and simulated (solid line) space charge distribution within a medium voltage (MV) XLPE cable insulation. (

**a**) t = 0 s, U = 22.5 kV; (

**b**) t = 10,000 s, U = 22.5 kV; (

**c**) t = 20,000 s, U = 22.5 kV; (

**d**) t = 0 s, U = 45 kV; (

**e**) t = 10,000 s, U = 45 kV; (

**f**) t = 20,000 s, U = 45 kV; (

**g**) t = 0 s, U = 90 kV, T

_{i}= 40 °C, T

_{a}= 30 °C; (

**h**) t = 10,000 s, U = 90 kV, T

_{i}= 40 °C, T

_{a}= 30 °C; (

**i**) t = 20,000 s, U = 90 kV, T

_{i}= 40 °C, T

_{a}= 30 °C; (

**j**) t = 0 s, U = 90 kV, T

_{i}= 65 °C, T

_{a}= 45 °C; (

**k**) t = 10,000 s, U = 90 kV, T

_{i}= 65 °C, T

_{a}= 45 °C; (

**l**) t = 20,000 s, U = 90 kV, T

_{i}= 65 °C, T

_{a}= 45 °C [14,95].

**Figure 21.**(

**a**) Stationary electric field, computed with (8)–(11), together with the analytic solution; (

**b**) relative error between both formulations and the analytic solution [59].

**Figure 22.**(

**a**) Geometry of cable joint, using field grading materials; (

**b**) initial electric field; (

**c**) stationary electric field, computed with (11); (

**d**) electric field, computed with (11) at the time t = t

^{n}, with n = 7, 9, 12 and 13; (

**e**) electric field, computed with (8)–(10) at the time t = t

^{n}, with n = 7, 9, 12 and 13. Inaccurate electric field values are seen after n = 12 [59,109].

**Figure 23.**(

**a**) Measurements of the electric conductivity of soil and fit by (55) at a temperature of 25 °C; (

**b**) measurements of the electro-osmotic conductivity of soil and fit by (56); (

**c**) measurements of the thermal conductivity of soil and fit by (57) [77,113,114,120,121,122,123,124,126,127,128].

**Figure 24.**Pseudo code to obtain the time-varying humidity concentration (θ), the temperature (T) and the electric field ($\overrightarrow{E}$) [128].

Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|

U | 450 kV | κ_{0} | 1 × 10^{−16} S/m | T_{∞} | 20 °C |

I | 1500 A | α | 0.1 °C^{−1} | λ | 0.167 W/(K∙m) |

r_{i} | 23.2 mm | β (for (6)) | 0.03 mm/kV | δ∙c_{p} | 2.5 × 10^{6} J/(m^{3} K) |

r_{a} | 42.4 mm | v (for (7)) | 0.7031 | α_{out} | 5 W/(K∙m^{2}) |

ε_{r} | 3.5 | E_{Ref} (for (7)) | 8.622 kV/mm | - | - |

Ref. | χ | r_{x} | Insulation Thickness | Width of Charge Region Δ |
---|---|---|---|---|

[72], U = 15 kV | 8 μm | 40 μm | 300 μm | 0.267∙D |

[73], U = 90 kV | 0.12 mm | 0.60 mm | 4.5 mm | 0.22∙(r_{a} − r_{i}) |

[91], XLPE, planar, +U | 0.052 mm | 0.25 mm | 2 mm | 0.26∙D |

[91], XLPE, planar, −U | 0.052 mm | 0.25 mm | 2 mm | 0.26∙D |

[91], XLPE, cylindrical, +U | 0.0875 mm | 0.44 mm | 3.5 mm | 0.28∙(r_{a} − r_{i}) |

[91], XLPE, cylindrical, −U | 0.0875 mm | 0.44 mm | 3.5 mm | 0.28∙(r_{a} − r_{i}) |

[91], LDPE, planar, +U | 0.052 mm | 0.25 mm | 2 mm | 0.26∙D |

[91], LDPE, planar, −U | 0.052 mm | 0.25 mm | 2 mm | 0.26∙D |

[93], U = 40 kV | 0.052 mm | 0.25 mm | 2 mm | 0.26∙D |

(a) | (b) | (c) | (d) |
---|---|---|---|

U = 22.5 kV | U = 45 kV | U = 90 kV | U = 90 kV |

T_{i} = 65 °C | T_{i} = 65 °C | T_{i} = 40 °C | T_{i} = 65 °C |

T_{a} = 45 °C | T_{a} = 45 °C | T_{a} = 30 °C | T_{a} = 45 °C |

n_{con,ε} = 1.5 | n_{con,ε} = 1.5 | n_{con,ε} = 1.5 | n_{con,ε} = 1.1 |

n_{sh,ε} = 1.0 | n_{sh,ε} = 1.0 | n_{sh,ε} = 1.0 | n_{sh,ε} = 1.0 |

ζ_{con,ε} = 4.5 × 10^{−4} m | ζ_{con,ε} = 4.5 × 10^{−4} m | ζ_{con,ε} = 4.5 × 10^{−4} m | ζ_{con,ε} = 4.5 × 10^{−4} m |

v_{κ} = 3 × 10^{−8} m/s | v_{κ} = 3∙10^{−8} m/s | v_{κ} = 5 × 10^{−8} m/s | v_{κ} = 7.5 × 10^{−8} m/s |

n_{con,κ} = 1.2 | n_{con,κ} = 1.5 | n_{con,κ} = 4.5 | n_{con,κ} = 13 |

n_{sh,κ} = 1.0 | n_{sh,κ} = 1.0 | n_{sh,κ} = 1.0 | n_{sh,κ} = 1.0 |

ζ_{con,κ} = 9 × 10^{−7} m | ζ_{con,κ} = 9 × 10^{−7} m | ζ_{con,κ} = 9 × 10^{−7} m | ζ_{con,κ} = 9 × 10^{−7} m |

t = 0 s | t = 10.000 s | t = 20.000 s | |
---|---|---|---|

(a) | |||

Equations (49) and (50) | 5.3% | 6.1% | 10.2% |

[14] | 5.3% | 6.8% | 8.6% |

(b) | |||

Equations (49) and (50) | 5.5% | 15.0% | 22.1% |

[14] | 5.5% | 12.3% | 17.0% |

(c) | |||

Equations (49) and (50) | 5.4% | 6.0% | 6.3% |

[14] | 5.4% | 9.2% | 14.7% |

(d) | |||

Equations (49) and (50) | 1.3% | 5.8% | 12.5% |

[14] | 1.2% | 32.9% | 34.6% |

**Table 5.**Comparison of different conductivity models and their description of space charge accumulation within the insulation.

Model | Description of Charge Dynamics | Limitations |
---|---|---|

Equations (3)–(7) | Charge accumulation due to a temperature gradient. | Computation of an average charge density of one sign only. No effects at interfaces and surfaces. |

Equations (39) or (45) and (40) | Charge distribution within the insulation and at interfaces and surfaces. | Limited to the stationary case and the additional constants are difficult to determine |

Equation (47) | Description of the stationary bulk and heterocharge distribution in polymeric insulations. | Limited to the stationary case and heterocharges only. |

Equations (49) and (50) | Simulation of transient processes, homo and heterocharges at interfaces and charges within the bulk. | Many additional constants are used that change over time and are determined by space charge measurements. |

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**MDPI and ACS Style**

Jörgens, C.; Clemens, M.
A Review about the Modeling and Simulation of Electro-Quasistatic Fields in HVDC Cable Systems. *Energies* **2020**, *13*, 5189.
https://doi.org/10.3390/en13195189

**AMA Style**

Jörgens C, Clemens M.
A Review about the Modeling and Simulation of Electro-Quasistatic Fields in HVDC Cable Systems. *Energies*. 2020; 13(19):5189.
https://doi.org/10.3390/en13195189

**Chicago/Turabian Style**

Jörgens, Christoph, and Markus Clemens.
2020. "A Review about the Modeling and Simulation of Electro-Quasistatic Fields in HVDC Cable Systems" *Energies* 13, no. 19: 5189.
https://doi.org/10.3390/en13195189