**4. Conclusions**

An empirical conductivity model equation was developed for XLPE and LDPE insulation materials. As various space charge measurements in literature showed accumulated hetero charges in the vicinity of both electrodes, to simulate these charges, a commonly established, but insufficient accurate nonlinear field and temperature dependent conductivity model was used and improved by two sigmoid functions to create a conductivity gradient at both electrodes. The model was validated with its steady state space charge distribution and resulted in realistic accumulation of hetero charges typically positioned in about one quarter of the insulation thickness at each electrode. A comparison of simulation and measurement results validated the presented model, showing a good agreemen<sup>t</sup> and an improvement in comparison to those of previously established conductivity models.

**Author Contributions:** Conceptualization, C.J. and M.C.; Methodology, C.J. and M.C.; Software, C.J.; Validation, C.J.; Formal Analysis, C.J. and M.C.; Investigation, C.J. and M.C.; Resources, C.J. and M.C.; Data Curation, C.J. and M.C.; Writing-Original Draft Preparation, C.J. and M.C.; Writing-Review & Editing, C.J. and M.C.; Visualization, C.J. and M.C.; Supervision, M.C.; Project Administration, M.C.; Funding Acquisition, M.C.

**Funding:** This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under the gran<sup>t</sup> number CL143/17-1. **Acknowledgments:** This work was supported by the Deutsche Forschungsgemeinschaft DFG (grant number CL143/17-1). **Conflicts of Interest:** The authors declare no conflict of interest.
