: Documentation is given for some subroutines which compute potentials and other functions. A set of subroutines uses rational approximations to compute Bessel functions of integral order. One subroutine uses the Debye approximation for the efficient computation of Bessel functions of complex argument and complex order. Empirical formulae have been developed to express the limiting boundaries of the modes of computation. (Author)

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Computation of Special Functions

The aim of this document is to provide an extended description and application guide of methods belonging to the so-called Numerical Electromagnetic Analysis (NEA) applied to the calculation of electromagnetic transients in power systems. As known, the accurate computation of electromagnetic transients is a fundamental requirement of several studies in the area of power systems. Lightning and switching studies are, for instance, typical subjects where the accuracy of transient’s computation has a direct influence to the proper sizing of components like insulators and breakers. Traditional approaches adopted since now were based on the combination of circuit and transmission lines theories. These approaches, analytically and numerically validated by numerous contributions to the literature, rely on specific assumptions that are inherently relaxed by NEA methods. Indeed, NEA methods mostly rely on the numerical solution of the full-wave Maxwell’s equations and, in this respect, the assessment of their accuracy, as well as the description of the various numerical methodologies, have motivated the preparation of this Technical Brochure. In this context, this guide will first discuss the general aspects and limitations associated to classical circuit and transmission lines theories. In particular, the guide will make reference to the modelling approaches used to represent the most typical power system components, like transmission lines, grounding systems, towers etc., within EMTP-like simulation tools (Electromagnetic Transient Program). A first comparison with the most typical NEA methods is presented in order to discuss the main differences and better support the contents of this guide. Then, the guide focuses on the analytical formulation of the most used NEA methods like, the Finite-Difference Time-Domain (FDTD), the Transmission Line Matrix ‘TLM’, the Finite-Element Method in Time Domain (FEMTD), the Method of Moment (MoM) and the Partial Element Equivalent Circuit (PEEC) method. A further remark refers to the comparative analysis of NEA vs EMTP-like simulation approaches. Such an assessment, addressed in this document in various sections, aims at stressing the advantages and drawbacks of both methods. Indeed, NEA methods, although characterized, in general, by better accuracies, result into non-negligible computation times that require the availability of specific computation environments. Such a characteristic is due to the inherent numerical complexity of NEA solvers that require the treatment of large amount of data that, additionally, have an influence on the results accuracy. To this end, the last part of the guide refers to the benchmarking of the various NEA methods by means of typical test cases. In this respect, the members of the Cigre WG C4.501 agreed to include a specific section of the brochure aimed at providing the NEA-computed electromagnetic transients with reference to the most typical test cases like: (i) lightning surge calculation in substations, (ii) influence of grounding on lightning surge calculation in substations, (iii) surge voltages on overhead lines, (iv) lightning-induced surges on distribution lines, (v) LEMP and induced surges calculation in overhead lines above a lossy ground and (vi) simulations of very fast transients (VTFs) in GIS. Additionally, as NEA methods represent power tools for the computation of parameters of power systems components, the guide has also provided benchmarking examples for the following assessments: (a.) surge characteristics of transmission towers, (b.) surge characteristics of grounding electrodes, vertical grounding rods, horizontal grounding electrode and complex grounding configurations, (c.) influence of grounding on surge propagation in overhead transmission lines, (d.) propagation characteristics of PLC signals along power coaxial cables, (e.) lightning surge characteristics of wind-turbine towers.

Guideline for Numerical Electromagnetic Analysis Method

When evaluating the risks resulting from the effects of electromagnetic interference (EMI) on buried pipelines running near ac electrified traction systems, the harmonic distortion in railway systems should be considered. This is because the tractive units enforce current harmonics on the interference source, in addition to the fundamental frequency. The presence of harmonics may exacerbate voltages accumulating on buried pipelines running in close vicinity to traction systems. Such voltages may cause danger to persons or may cause damage to the pipelines, in the long term. However, the technical standards on pipelines-traction systems’ separation disregard the impact of harmonics. They are based on the fundamental frequency induction. This paper uses state-of-the-art modeling techniques to holistically assess EMI on underground pipelines near ac traction systems. This is achieved through a realistic modeling of the infrastructure associated with a traction system and by accounting for the presence of harmonics in the contact line system (i.e., interference source).

Effects of Electromagnetic Interference on Underground Pipelines Caused by the Operation of High Voltage AC Traction Systems: The Impact of Harmonics

The goal of this article is to investigate the effect of frequency-dependent soil models on the performance of grounding electrodes subjected to lightning strikes. Several soil models are examined while accounting for the variation of soil resistivity and permittivity as a function of the lightning current frequency spectrum. The analysis is performed for a homogeneous soil and a two-layer horizontally stratified soil. The impact of the frequency-dependent soil parameters on the ground potential rise (GPR) of simple grounding electrodes subjected to lightning is analyzed and discussed. The analysis is performed in the frequency domain and in the time domain. A wind turbine and its grounding system are also considered in this article. Special attention is given to the case of indirect lightning, rarely mentioned in the literature. The GPR of the grounding electrodes is examined when the frequency dependence of the soil is taken into account and the lightning channel is located at close distances to the electrodes. Indeed, the level of induced electromagnetic fields caused by a nearby lightning channel can still be too high and potentially dangerous. The computations are performed using an efficient Method of Moments (MoM) numerical tool based on surface-wire integral equations for a stratified medium in the frequency range from dc to several MHz. Numerical results demonstrate that the frequency dependency of the soil parameters results in a decrease of the potential rise of the grounding electrodes, with respect to the case where the parameters are assumed constant.

Impact of Frequency-Dependent Soil Models on Grounding System Performance for Direct and Indirect Lightning Strikes

One of the most accurate approaches to the lightning-related transient analysis of grounding systems in layered soil is based on the method of moments solution of the integral form of the Maxwell equations. The practical application of this rigorous model to large systems is hindered by a great amount of computer time required for the numerical integration of oscillatory and slowly convergent Sommerfeld integrals (SI). As a result, approximate methods that avoid the solution of SI, e.g., image theory-based methods, are often used for computation of the response of large grounding systems. In this article, we extend the application of the rigorous model for large grounding systems in multilayered soil by applying a procedure that minimizes the number of direct computations of the SI using interpolation over a grid of a small number of sample points. We implement a three-dimensional interpolation scheme that adapts to the problem to minimize the interpolation grid, maintaining the accuracy of the rigorous approach. The developed model is applied for the analysis of transient voltages that might couple to secondary cables and disrupt the operation of connected equipment in case of lightning. The analysis shows that there are cases when the image theory-based models might underestimate the possibly harmful transient voltages.

Fast and Accurate Transient Analysis of Large Grounding Systems in Multilayer Soil

This paper presents a theoretical model for the analysis of grounding systems located in multilayer soils in which arbitrary heterogeneities (finite volumes) are embedded. The presence of the heterogeneities is handled through a boundary element method, whereas that of the horizontal soil layers is analytically handled using image theory (or any other equivalent technique). Numerical results are presented for several cases and are shown to agree with limiting cases. The results also clearly show that the presence of finite heterogeneities in multilayer soils can have a very strong impact on the safety performance of a grounding system.

Analysis of Grounding Systems in Horizontal Multilayer Soils Containing Finite Heterogeneities

When measuring the ground impedance of an electrically isolated grounding system, it sometimes happens that the test electrodes are placed at considerably greater distances from the installation under test than the maximum electrode spacing used during the soil resistivity measurements carried out during the predesign phase of the grounding system. As a result, ground impedance measurements carried out with the fall-of-potential method may contain valuable supplemental information about deep soil strata that can improve grounding system performance predictions made during the design phase with computer modeling software and explain discrepancies encountered between predicted and measured ground impedance values. These data can be also used during subsequent grounding design work associated with facility expansion. It is shown in this paper how, in the absence of interfering metallic infrastructure, such ground impedance measurements can be converted into apparent soil resistivity values corresponding to deeper soil layers, with an example showing how dramatic improvements in grounding system performance predictions can be obtained.

Using Fall-of-Potential Measurements to Improve Deep Soil Resistivity Estimates

A full-wave solution method of Maxwell's equations is adopted for the analysis of the performance of grounding, bonding, and shielding systems of a metallic enclosure subjected to direct and indirect lightning strikes. The method is based on a numerical solution of the electric field integral equation by the method of moments. An array of conducting wires and metallic surfaces is used to represent the lightning channel and structures in its proximity. An improved antenna theory model approach based on the concept of the distributed source is adopted for the electromagnetic analysis of indirect lightning. The methodology enables an efficient electromagnetic compatibility analysis of a combined network of metallic plates, wires, and cables exposed to direct and indirect lightning strikes. The effectiveness of most common protective schemes—namely grounding, shielding, and bonding—is examined for cables located inside a metallic enclosure with apertures. A safety assessment analysis is performed for the case of indirect lightning and is compared to the case where the lightning current is injected directly into the top of the enclosure.

On the Performance of Grounding, Bonding, and Shielding Systems for Direct and Indirect Lightning Strikes

The shielding performance of new spherical polyhedral structures illuminated by an electromagnetic plane wave is investigated. The spherical shield screens consist of a juxtaposition of wire conductors and metallic screens distributed according to the regular pattern of a spherical polyhedron. It is shown that at low frequency, the wire frame screen provides high levels of shielding effectiveness, while its counterpart of metallic surfaces has a similar behavior for higher frequency bands. Moreover, the shielding characteristics of a combined wire-surface spherical surface are reported and compared with the cases of a single screen of wire conductors or metallic surfaces. The reported numerical results illustrate the advantage of using the combined configuration. It is observed that the combined wire-surface spherical screen can be designed in order to be effective simultaneously against interference at the low and high frequency limits for any polarization of the incident field.

Simon Fortin

Papers

Impact of Frequency-Dependent Soil Models on Grounding System Performance for Direct and Indirect Lightning Strikes

Analysis of Grounding Systems in Horizontal Multilayer Soils Containing Finite Heterogeneities

Using Fall-of-Potential Measurements to Improve Deep Soil Resistivity Estimates

On the Performance of Grounding, Bonding, and Shielding Systems for Direct and Indirect Lightning Strikes

Electromagnetic shielding properties of spherical polyhedral structures generated by conducting wires and metallic surfaces