A Comparison of Frequency-Dependent Soil Models: Application to the Analysis of Grounding Systems

We present a review and comparison of different models representing the frequency dependence of the soil electrical parameters (conductivity and permittivity). These models are expressed in terms of curve-fit expressions for the soil conductivity and relative permittivity, which are based on experimental data. Six available models/expressions are discussed and compared making reference to two sets of experimental data. It is shown that the soil models by Scott, Smith-Longmire, Messier, and Visacro-Alipio predict overall similar results, which are in reasonable agreement with both sets of experimental data. Differences between the soil models are found to be more significant at high frequencies and for low-resistivity soils. The causality of the considered models is tested using the Kramers-Kronig relationships. It is shown that the models/expressions of Smith-Longmire, Messier, and Portela satisfy the Kramers-Kronig relationships and thus provide causal results. The soil models are applied to the analysis of grounding systems subject to a lightning current. A full-wave computational model is adopted for the analysis. The analysis is performed considering two cases: 1) a simple horizontal grounding electrode, and 2) a realistic grounding system of a wind turbine. Two current waveforms associated with typical first and subsequent return strokes are adopted for the representation of the incident lightning current. In agreement with recent studies, simulations show that the frequency dependence of the soil parameters results in a decrease of the potential of the grounding electrode, with respect to the case where the parameters are assumed to be constant. It is found that the models/expressions by Scott, Smith-Longmire, Messier, and Visacro-Alipio predict similar levels of decrease, which vary from about 2% (ρ
_LV
= 20 Ω·m and first stroke) up to 45% (ρ
_LV
= 10 000 Ω·m and subsequent stroke). On the other hand, the models of Portela and Visacro-Portela predict significantly larger levels of the decrease, especially for very high resistivity soils. Furthermore, in the case of a high resistivity soil (10 000 Ω·m), the Visacro-Alipio expression predicts a longer risetime for the grounding potential rise, compared to the predictions of Scott, Smith-Longmire, and Messier models.