#
Empirical Conductivity Equation for the Simulation of the Stationary Space Charge Distribution in Polymeric HVDC Cable Insulations^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Empirical Conductivity Model Equation for the Simulation of the Stationary Charge Distribution

_{0}ε

_{r}, where ε

_{0}= 8.854 × 10

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

_{r}is the relative permittivity [9]. The Equations (1)–(3) are solved using the finite-difference method in one dimension. Depending on the geometry, depicted in Figure 1, a uniform grid of spacing Δh in either radial direction or x-direction is utilized, respectively [10]. Simulation results of a 150 μm thick insulating material are given in [11], where a non-uniform grid of minimum nodal distance 0.1 μm is used. To provide a sufficiently accurate spatial discretization for both geometries in Figure 1, the distance between the anode and the cathode is discretized with N = 1500 equidistant grid points in this work.

_{i}, the radius of the outer sheath is r

_{a}, the conductor temperature is T

_{i}, the sheath temperature is T

_{a}, and the applied voltage is U.

_{B}of polymeric insulations is given by

^{−23}J/K is the Boltzmann constant, T is the temperature, $|{J}_{0}|$ is a current density constant, E

_{a}is the activation energy, and γ is a constant to describe the dependency on the electric field [13]. These constants are determined by a fit of Equation (4) to measured data. The constants of XLPE insulations are approximately $|{J}_{0}|$ = 1 × 10

^{14}A/m

^{2}, E

_{a}= 1.48 eV and γ = 2 × 10

^{−7}m/V (see [13]). For an LDPE insulation, the constants are $|{J}_{0}|$ = 0.04224 A/m

^{2}, E

_{a}= 0.84 eV and γ = 4.251 × 10

^{−7}m/V (see [14,15,16]).

_{1}describes the conductivity variations at the inner conductor (anode) and K

_{2}the conductivity variations at the outer sheath (cathode); in (6) and (7), r

_{x}is the distance between r

_{i}and the position of the highest gradient (σ/σ

_{B}(r = r

_{x}) = 0.5). If a planplanar insulation is considered, r → x, r

_{i}= 0 and r

_{a}= L holds.

_{x}= 50 μm are depicted. Increasing the distance constant χ results in a decreasing conductivity gradient and hetero charges distribution, with a slightly spread out charge shape. In Figure 2, a constant bulk conductivity (σ

_{B}= const.) is used for the computation of the stationary space charge density ρ [10].

_{1}, resulting in negative charges at the anode and K

_{2}is describing a conductivity decrease, resulting in positive charges at the cathode. This relationship is obtained from the analytic solution of the stationary charge and electric field distribution, using a conductivity gradient.

_{0}× K

_{1}, where σ

_{0}is a constant conductivity, and a planplanar insulation (see Figure 1a). Using (1), (3), and Gauss law, the stationary charge density is computed by

_{r}= constant) insulation is computed by

_{x}. With L $\gg $ r

_{x}, the constant C is positive. With (11) and Gauss law, the stationary charge density is given by

_{0}× (1 − K

_{2}) positive charges are modeled at the sheath, where the computation is analog to (8)–(10). The electric field is

_{x}and χ, the region of the conductivity gradient is defined by Δ. In Figure 3, (6) and (7) are depicted for a planplanar geometry [10]. To determine the dependency of r

_{x}and χ on Δ, a straight line f(x) = a(x − r

_{x}) + b = (1/Δ)(x − r

_{x}) + 0.5, depicted as the black line in Figure 3b, is used. With f(x = r

_{x}) = 0.5, we define r

_{x}= Δ/2. As a first assumption to describe χ by Δ, we use the gradient of σ/σ

_{B}at x = r

_{x}(equal to the gradient of σ/σ

_{B}at x = L − r

_{x}). For a planplanar geometry, the gradient of σ/σ

_{B}is

_{x}, the gradient at x = r

_{x}or x = L − r

_{x}is equal to 1/(4χ). Using 1/(4χ) as the gradient a for a straight line f(x) = a(x − r

_{x}) + 0.5, the green line in Figure 3b is obtained. Decreasing the gradient from 1/(4χ) to approximately 1/(10χ), depicted as the red dotted curve in Figure 3b, results in the best fit to describe the black line f(x) = (1/Δ)(x − r

_{x}) + 0.5. Thus, the constants r

_{x}and χ are described by r

_{x}= Δ/2 and 10χ ≈ Δ, where now only Δ has to be determined by measurements.

_{x}is one third of the insulation and for the value of Δ = 10χ is half of the insulation thickness. On the contrary, the analytic solution of the charge density (13) and (15) and Figure 2b show a maximum hetero charge density at the conductor, decreasing to zero within the range of Δ. The difference between measurements in [17] and (13) or (15) comes from filtered surface charges, which are considered in the measurements in [17]. The finite resolution of the measurement technique itself filters the surface charges, whereby they spread out and look like a Gaussian curve [8]. Consequently, a measured space charge distribution has its maximum hetero charge values in the vicinity of both electrodes, instead of a position immediately at the electrodes.

_{+}and δ

_{−}) accumulate at both electrodes. These charges are derived from [18] and are approximately described by

## 3. Comparison between Simulated and Measured Space Charge Distribution

_{S}and the measurements are ρ

_{M}.

#### 3.1. Measurements of XLPE Insulation

_{i}and the outer temperature is T

_{a}.

_{x}= 0.25 mm = Δ/2 ≈ (0.26∙L)/2.

_{a}− r

_{i}). The used constants are χ = 0.12 mm = Δ/10 ≈ 0.22 × (r

_{a}− r

_{i})/10 and r

_{x}= 0.6 mm = Δ/2 ≈ 0.22 × (r

_{a}− r

_{i})/2.

#### 3.2. Measurements of LDPE Insulation

_{x}= 40 μm = Δ/2 ≈ 0.267·L/2.

_{x}and the width of filtered surface and hetero charges Δ are seen in Table 2. A comparison between the simulation results and the measurements are seen in Figure 6 [10]. Accumulated hetero charges (including surface charges) are seen in approximately one quarter of the insulation thickness, whereby the width Δ is approximately independent of the geometry or the insulation material (see Table 2). Due to the “mirror image effect” of the charge distribution, the values for χ and r

_{x}are constant, while changing the polarity of the voltage. It is not clear, why the hetero charges accumulate in one quarter of the insulation thickness. The resolution of the PEA method is 1.6 μm for a one dimensional planplanar insulation with a thickness of 25–27,000 μm and 0.1–1 mm for a cable insulation with a thickness of 3.5–20 mm [26]. The resolution is accurate enough to separate between hetero charges and bulk space charges.

_{x}in (6) and (7) with the width Δ (region of filtered surface and hetero charges), the approximation χ = (0.25·L)/10 and r

_{x}= (0.25·L)/2 for a planplanar insulation and χ = (0.25 × (r

_{a}− r

_{i}))/10 and r

_{x}= (0.25 × (r

_{a}− r

_{i}))/2 for a cylindrical insulation is defined.

^{3}(see (8)). The surface charges (17) and (18) are reduced to ${\delta}_{+}={\epsilon}_{0}{\epsilon}_{\mathrm{r}}({r}_{\mathrm{i}})|\overrightarrow{E}({r}_{\mathrm{i}})|$ and ${\delta}_{-}={\epsilon}_{0}{\epsilon}_{\mathrm{r}}({r}_{\mathrm{a}})|\overrightarrow{E}({r}_{\mathrm{a}})|$ (see Figure 7a). A temperature gradient along the insulation results in accumulated bulk space charges, which are lower in the vicinity of the electrodes, compared to the measurements (see Figure 7b).

_{a}, and γ in (4). The resulting space charge distribution depends on many factors e.g., the conductivity, the local electric field, or the electrode material. As a result, it is very difficult to simulate the charge distribution of different references, even if it is the same material, like XLPE [27,28]. Differences between the measurements and the simulations in Figure 4, Figure 5 and Figure 6 are small and the developed model yield results with good agreement to the measurements. The developed conductivity equation shows less differences to measurements compared to a commonly used conductivity equation and thus, the applicability of the formulation.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

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

E_{a} | Constant for the temperature dependency of the bulk electric conductivity [eV] |

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

$|\overrightarrow{{J}_{0}}|$ | Constant for the bulk electric conductivity [A/m^{2}] |

k = 1.38 × 10^{−23} | Boltzmann constant [J/K] |

K_{1} | Conductivity variations at the conductor |

K_{2} | Conductivity variations at the sheath |

L | Thickness of planplanar insulation [m] |

N | Number of grid points |

r | Coordinate for the cylindrical insulation [m] |

r_{a} | Radius of the conductor in cylindrical coordinates [m] |

r_{i} | Radius of the conductor in cylindrical coordinates [m] |

r_{x} | Distance between the conductor (sheath) and the position of the highest gradient of K_{1} (K_{2}) [m] |

T | Temperature [°C] |

T_{a} | Sheath temperature [°C] |

T_{i} | Conductor temperature [°C] |

U | Applied voltage [V] |

x | Coordinate for the planplanar insulation [m] |

γ | Constant for electric field dependency of the bulk electric conductivity [m/V] |

Δ | Gradient region at the conductor and the sheath [m] |

Δh | Distance between two grid points [m] |

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 |

η | Sum of the difference between ρ_{s} and ρ_{M} |

ρ | Space charge density [C/m^{3}] |

ρ_{M} | Measured space charge density [C/m^{3}] |

ρ_{s} | Simulated and filtered space charge density [C/m^{3}] |

σ | Total electric conductivity with hetero charges [S/m] |

σ_{B} | Bulk electric conductivity without hetero charges [S/m] |

φ | Electric potential [V] |

χ | Distance constant to define the conductivity gradient in the vicinity of both electrodes [m] |

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**Figure 2.**Influence of distance constant χ on conductivity gradient and resulting space charge distribution ρ. (

**a**) normalized conductivity σ/σ

_{B}; (

**b**) normalized stationary space charge distribution [10].

**Figure 3.**(

**a**) Gradient region Δ, where the conductivity gradient is present. The position r = r

_{x}= Δ/2 has the highest gradient; (

**b**) to determine χ by Δ, a straight line f(x) = a(x − r

_{x}) + 0.5 = (1/Δ)(x − r

_{x}) + 0.5 is used, where the best approximation is shown by Δ = 10χ [10].

**Figure 6.**Measured and simulated charge distribution in a XLPE and LDPE insulation [6]. (

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

**b**) XLPE, planplanar, −U; (

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

**d**) XLPE, cylindrical, −U; (

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

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

Figure 4a | Figure 4b | Figure 5 |
---|---|---|

U = 40 kV | U = 90 kV | U = 15 kV |

L = 2 mm | r_{i} = 5 mmr _{a} = 9.5 mm | L = 0.3 mm |

T = 27 °C = const. | T_{i} = 65 °CT _{a} = 50 °C | T = 27 °C = const. |

$|\overrightarrow{{J}_{0}}|$ = 1 × 10^{14} A/m^{2} | $|\overrightarrow{{J}_{0}}|$ = 1 × 10^{14} A/m^{2} | $|\overrightarrow{{J}_{0}}|$ = 0.04224 A/m^{2} |

E_{a} = 1.40 eV | E_{a} = 1.48 eV | E_{a} = 0.84 eV |

γ = 2 × 10^{−7} m/V | γ = 2 × 10^{−7} m/V | γ = 4.251 × 10^{−7} m/V |

χ = 65.3 μm | χ = 0.15 mm | χ = 10 μm |

r_{x} = 0.3 mm | r_{x} = 0.68 mm | r_{x} = 45 μm |

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

[20], Figure 4a | 0.052 mm | 0.25 mm | 2 mm | 0.26·L |

[19], Figure 4b | 0.12 mm | 0.60 mm | 4.5 mm | 0.22 × (r_{a} − r_{i}) |

[6], XLPE, planplanar, +U | 0.052 mm | 0.25 mm | 2 mm | 0.26·L |

[6], XLPE, planplanar, −U | 0.052 mm | 0.25 mm | 2 mm | 0.26·L |

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

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

[6], LDPE, planplanar, +U | 0.052 mm | 0.25 mm | 2 mm | 0.26·L |

[6], LDPE, planplanar, −U | 0.052 mm | 0.25 mm | 2 mm | 0.26·L |

[21], Figure 5 | 8 μm | 40 μm | 300 μm | 0.267·L |

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**MDPI and ACS Style**

Jörgens, C.; Clemens, M.
Empirical Conductivity Equation for the Simulation of the Stationary Space Charge Distribution in Polymeric HVDC Cable Insulations. *Energies* **2019**, *12*, 3018.
https://doi.org/10.3390/en12153018

**AMA Style**

Jörgens C, Clemens M.
Empirical Conductivity Equation for the Simulation of the Stationary Space Charge Distribution in Polymeric HVDC Cable Insulations. *Energies*. 2019; 12(15):3018.
https://doi.org/10.3390/en12153018

**Chicago/Turabian Style**

Jörgens, Christoph, and Markus Clemens.
2019. "Empirical Conductivity Equation for the Simulation of the Stationary Space Charge Distribution in Polymeric HVDC Cable Insulations" *Energies* 12, no. 15: 3018.
https://doi.org/10.3390/en12153018