# DC Model Cable under Polarity Inversion and Thermal Gradient: Build-Up of Design-Related Space Charge

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## Abstract

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## 1. Introduction

- The development of materials with improved performances regarding targeted applications, notably decrease the propensity to store space charge and manage field grading properties;
- The development of physical models for the material behavior: how charges are generated, stored and transported;
- The development of techniques, particularly charge distribution measurement techniques, relevant to the geometry and to the thermal and electrical stresses that are encountered;
- The proposal for materials assessment methodologies for the application: This means that relevant quantities have to be measured and figures of merit for materials provided in order to make systems more safe;
- The implementation of engineering models for stress distribution estimation.

## 2. Challenges and Opportunities for Developing HVDC Cables

#### 2.1. Techniques and Methodologies for Space Charge Assessment

#### 2.2. Challenges Regarding Materials for HVDC Cables

_{2}) [24] and magnesium oxide (MgO) [25] incorporated into low-density polyethylene (LDPE) have been shown to be effective in suppressing space charge. The mechanisms behind those improvements are not completely clear at present [26], neither is the fate of the material integrity in time.

#### 2.3. Conductivity Models

_{a}is the activation energy for conduction at low field, k

_{B}= 8.62 × 10

^{−5}eV/K is the Boltzmann’s constant and other quantities are parameters. The second term with hyperbolic sine function finds physical justification in hopping conduction process as well as ionic conduction for example [32]. The equations used in this kind of approach can fit experimental data. The difficulty, however, is that they do not account for the effect of the nature of electrodes on conductivity, for example, or for the transient processes of conduction (charging currents), and more generally they do not reflect what is called space charge limited conduction. An alternative to macroscopic modelling is to follow the approaches used in semiconductor physics, with bringing details into the charge generation mechanisms, the electronic properties of materials at the interface, etc. These approaches have been developed in the last decades associating the physical concepts issued from semiconductor physics [33,34] and the numerical techniques capable of solving the problems, issued notably from gas physics [35]: The processes of charge injection, charge trapping, detrapping, mobility, etc. are incorporated in the model, with their accompanying set of parameters. One of the main difficulties here is to identify values of model parameters: they are many and cannot be extracted in a straightforward way by independent experiments. The conductivity does not appear explicitly, instead, it is the space- and time-dependent carrier density, the mobility and the local field that describe the transport. Such models have been set up mostly for flat specimen, but extension and resolution in cable geometry is appearing [36]. Also, it was used as a route to model breakdown under DC stress [37]. Although this is a promising route to develop accurate modelling of insulations, at present the full parameterization is still demanding; also the treatment of physical processes like ionization and heterocharge build-up is still in the infancy stage [38]. Owing to their complexity and to the strong dependence on the nature of polymers, such models cannot be easily handled as engineering tools aiming at dimensioning devices. One of the purposes of the present paper is to show how far the macroscopic models can account for the experimental behavior and also to touch the degree of approximation such macroscopic models bring in respect to physical mechanisms at play in insulating polymers.

## 3. Design-Related Space Charge in Model Cables

#### 3.1. Constitutive Equations for the Space Charge

_{i}and r

_{o}are the inner and outer radii of the insulation in the cable.

_{c}and σ

_{c}are the electric field and the conductivity at the reference position r

_{c}. Equations (3) and (4) are equivalent only if σ is homogeneous, i.e., independent of r.

_{c}is the heat power dissipated per unit length of the conductor and λ is the thermal conductivity of the insulation in W/(m K).

_{a}, the charge density can be written as:

#### 3.2. Test Conditions for Conductivity Measurements

#### 3.3. Results for Conductivity

_{app}) and the conduction current (quasi-steady-state current) as follows [43]:

_{app}(I) to Equation (12) was achieved by non-linear curve fitting independently for each temperature level, so providing C(T) and B(T). We then deduced the activation energy E

_{a}by linear regression to B × C vs. T

^{−1}plot (Arrhenius plot) considering that at sufficiently low field the conductivity is approximated by: σ(T) = B(T) × C(T). Finally, as C did not appear significantly temperature-dependent, we considered a constant value. The following final quantities were obtained for conductivity equation: C = 2.15 × 10

^{−7}m/V, A = 3.83 × 10

^{4}S.V/m² and E

_{a}= 0.83 eV.

_{t}≈ 12 kV/mm. The relation to the value of C, which actually controls the threshold, can be deduced as E

_{t}≈ 2.5 /C. Below the threshold, the response tends to follow Ohm’s law, which means that the conductivity is independent from field, but as the field is increasing, it begins to be a non-linear relation. The reason can be space charge effects that accumulate at high fields through, for example, space charge limited current. An alternative would be to have a hopping conduction with charge mobility dependent on the field. The two processes may have different temperature dependencies for the threshold. With the assumption of using a constant C value in the fitting process, supported by the data, we cannot reveal this feature in the modelled temperature-dependent conductivity characteristic.

#### 3.4. Electric Field Simulation Based on Conductivity Data

^{−15}S/m). Therefore, in a 2 h charging time the steady state is reached.

#### 3.5. Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DSO | Digital Signal Oscilloscope |

HVAC | High-Voltage Alternate Current |

HVDC | High-Voltage Direct Current |

LCC | Line Commutated Converter |

LDPE | Low Density Polyethylene |

PEA | Pulsed Electroacoustic |

PPLP | Polypropylene Laminated Paper |

VSC | Voltage Source Converter |

XLPE | Crosslinked Polyethylene |

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**Figure 1.**(

**a**) Schematic of the Pulsed Electroacoustic (PEA) test bench for cable geometry. PVDF—Poly(vinylidene fluoride) is the material for the piezoelectric sensor. The cable is arranged as a loop so that an AC current can be created in it using a current transformer to reach the desired thermal conditions (thermal gradient); (

**b**) Rough signal measured for the calibration step by applying 20 kV on a 4.5 mm thick insulation Medium Voltage (MV) cable [14].

**Figure 2.**Field distribution obtained in the insulation of miniature cable samples with 1.5 mm thick insulation. Measurements were achieved under a thermal gradient of 10 °C, (T

_{in}= 70 °C; T

_{out}= 60 °C), under a voltage of −55 kV applied to the conductor. (

**a**) degassed sample; (

**b**) partly degassed sample. The dashed curve is the geometric field distribution.

**Figure 3.**Space charge density patterns corresponding to field profiles shown in Figure 1. The color scale represents charge density in C/m

^{3}. (

**a**) degassed sample; (

**b**) partly degassed sample.

**Figure 4.**Schematic representation of DC miniature cable used for testing: Conductor radius r

_{1}= 0.65 mm; Inner semicon radius r

_{2}= r

_{i}=1.45 mm; Insulation radius r

_{3}= r

_{o}= 2.95 mm; Outer semicon radius r

_{4}= 3.65 mm.

**Figure 6.**Charging current transients under different fields at a temperature of 50 °C. Currents were measured in 2 s intervals all along the cycle.

**Figure 7.**Conduction current results as a function of field at various temperatures. The solid lines are the results from the fit functions to the conductivity equation.

**Figure 9.**Modeled field distribution profiles in a cable sample with a temperature gradient of 10 ˚C at an applied voltage of 15 kV.

**Figure 10.**Modeled field distribution profiles in a cable sample under voltage inversion applied and at isotherm condition at T = 65 °C.

**Figure 11.**Field distribution profiles in a cable sample with voltage inversion applied from +15 kV to −15 kV and at T gradient = 10 ˚C. The curve indexed 0 V-300 s is the field distribution after 300 s in the short-circuit applied after the negative voltage step.

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**MDPI and ACS Style**

Adi, N.; Vu, T.T.N.; Teyssèdre, G.; Baudoin, F.; Sinisuka, N.
DC Model Cable under Polarity Inversion and Thermal Gradient: Build-Up of Design-Related Space Charge. *Technologies* **2017**, *5*, 46.
https://doi.org/10.3390/technologies5030046

**AMA Style**

Adi N, Vu TTN, Teyssèdre G, Baudoin F, Sinisuka N.
DC Model Cable under Polarity Inversion and Thermal Gradient: Build-Up of Design-Related Space Charge. *Technologies*. 2017; 5(3):46.
https://doi.org/10.3390/technologies5030046

**Chicago/Turabian Style**

Adi, Nugroho, Thi Thu Nga Vu, Gilbert Teyssèdre, Fulbert Baudoin, and Ngapuli Sinisuka.
2017. "DC Model Cable under Polarity Inversion and Thermal Gradient: Build-Up of Design-Related Space Charge" *Technologies* 5, no. 3: 46.
https://doi.org/10.3390/technologies5030046