Slavko Vujević

NUMERICAL ANALYSIS OF EARTHING GRIDS BURIED IN HORIZONTALLY STRATIFIED MULTILAYER EARTH

This paper describes a new numerical procedure for analysing earthing grids buried in horizontally stratified multilayer earth. The procedure is very efficient and general. The total number of layers and the total number of metallically disconnected earthing grids are completely arbitrary. A single earthing grid can be positioned in several layers. The procedure is based on an integral equation formulation. Earthing grid conductors are subdivided into segments and the average potential method is used. Efficiency and generality of the computation procedure are based on the successful application of numerical approximations of two kernel functions of the integral expression for the potential distribution within a single layer which is caused by a point current source. Each kernel function of the observed layer is approximated using a linear combination of 15 exponential functions. Extension from the point source to a segment of the earthing grid conductor is done by integrating the potential contribution due to a line of point current sources along the segment axis. This computational procedure gives highly accurate results in a short execution time.

Potential distribution for a harmonic current point source in horizontally stratified multilayer medium

Purpose – This paper aims to describe a numerical procedure for approximating the potential distribution for a harmonic current point source, which is either buried in horizontally stratified multilayer earth, or positioned in the air. The procedure is very efficient and general. The total number of layers and the source position in relation to the medium model layers are completely arbitrary.Design/methodology/approach – The efficiency of the computation procedure is based on the successful application of the numerical approximation of two kernel functions of the integral expression for the potential distribution within an arbitrarily chosen layer of the medium model. Each kernel function of the observed layer is approximated using a linear combination of 15 real exponential functions. Using these approximations and the analytical integration based on the Weber integral, a simple expression for numerical approximation of potential distribution within boundaries of the observed medium layer is given. Pote...

ACCURACY OF APPROXIMATE FORMULAS FOR INTERNAL IMPEDANCE OF TUBULAR CYLINDRICAL CONDUCTORS FOR LARGE PARAMETERS

Exact formulas for internal impedance per unit length of tubular cylindrical conductors energized by time-harmonic current involve Bessel functions. These functions are deflned by inflnite series, which yield unstable and often erroneous results for complex arguments of large magnitudes. Although it is well known how to evaluate Bessel functions numerically and many routines are now available to perform the actual computation, the available software routines often fail when computing equations that consist of a product and a quotient of Bessel functions under large complex or real arguments. For such cases, difierent approximate formulas can be used. In this paper, three types of approximate formulas for internal impedance of tubular cylindrical conductors are compared with respect to numerical stability and accuracy.

Time-harmonic current distribution on conductor grid in horizontally stratified multilayer medium

This paper presents a novel time-harmonic electromag- netic model for determining the current distribution on conductor grids in horizontally stratifled multilayer medium. This model could be seen as a basis of the wider electromagnetic model for the frequency-domain transient analysis of conductor grids in multilayer medium. The total number of layers and the total number of conductors are completely arbitrary. The model is based on applying the flnite element technique to an integral equation formulation. Each conductor is subdivided into segments with satisfying the thin-wire approximation. Complete elec- tromagnetic coupling between segments is taken into account. The computation of Sommerfeld integrals is avoided through an efiective approximation of the attenuation and phase shift efiects. Computation procedure for the horizontally stratifled multilayer medium is based on the successful application of numerical approximations of two kernel functions of the integral expression for the potential distribution within a single layer, which is caused by a point source of time-harmonic cur- rent. Extension from the point source to a segment of the earthing grid conductors is accomplished through integrating the potential con- tribution due to the line of time-harmonic current source along the segments axis.

3D Computation of the Power Lines Magnetic Field

In this paper, a 3D quasi-static numerical algorithm for computation of the magnetic field produced by power lines is presented. These power lines can be overhead power line phase conductors and shield wires or buried cable line phase conductors. The basis of the presented algorithm is the application of Biot-Savart law and the thin-wire approximation of cylindrical conductors. The catenary form of the power line conductors is approximated by a set of straight cylindrical segments. By summing up contributions of all conductor segments, magnetic field distribution is computed. On the basis of the presented theory, a FORTRAN program PFEMF for computation of the magnetic flux density distribution was developed. For each conductor catenary, it is necessary to define only global coordinates of the beginning and ending points and also the value of the longitudinal phase conductor current. Global coordinates of beginning and ending points of each catenary segment are generated automatically in PFEMF. Numerical results obtained by program PFEMF are compared with results obtained by simple 2D model and results obtained using software package CDEGS.

High-Accurate Numerical Computation of Internal Impedance of Cylindrical Conductors for Complex Arguments of Arbitrary Magnitude

In this paper, a numerical model is proposed for computing the per-unit-length internal impedance of tubular cylindrical conductors for complex arguments of arbitrary magnitudes. The proposed model either numerically solves the exact formula for internal impedance consisting of modified Bessel functions or utilizes asymptotic approximations of modified Bessel functions when applicable. It is shown that the results obtained by the proposed model are highly accurate and numerically stable. The proposed model is also applicable for solid cylindrical conductors.

Mathematical model of lightning stroke development

This paper presents a mathematical model of the lightning stroke path development. Any lightning model is a mathematical construct design to reproduce certain aspects of the physical process involved in the lightning discharge. This mathematical model resides on the underlying observed physical properties of the lightning stroke development. It uses Monte Carlo based algorithm in order to simulate stochastic and dynamic movement of the lightning stroke stepped leader descent. Influence of the lightning stroke current amplitude on the path development is also taken into account. Two examples of the lightning stroke stepped leader propagation paths are presented.

A surface charge simulation method based on advanced numerical integration

An advanced method for computation of electric field intensity is presented.Subparametric spatial 2D finite elements are developed.Expressions for self and mutual coefficients of 2D finite element nodes are derived.The problem of singularity is solved using advanced double 2D numerical integration. Numerical models for computing low-frequency electromagnetic fields can contain spatial 2D finite elements, which are numerically most demanding due to problem of singularity. In this paper, an advanced time-harmonic quasistatic surface charge simulation method for computation of scalar electric potential and electric field intensity distribution is presented. Subparametric spatial 2D finite elements with an arbitrary number of nodes for description of surface charge density distribution are developed. The problem of singularity that occurs in the double 2D integration over these elements is solved using an originally developed advanced numerical integration based on 2D Gaussian quadrature. Self and mutual coefficients of spatial 2D finite element nodes are numerically computed and included in the system of linear equations for surface charge density distribution computation. The accuracy of the computer program, based on the presented model, is shown in the chosen numerical example with known analytical solution. Numerical model and advanced integration presented herein could be easily extended to non-homogeneous regions and multilayer problems using the image method.

Computation of the overhead power line electromagnetic field

This paper presents a novel computational algorithm for the determination of the overhead power line electromagnetic fields. Algorithm is based on the separation of the phase conductor currents into its longitudinal and transversal components and application of the Biot-Savart law for the computing of the magnetic field. Longitudinal component of the electric field is computed from vector magnetic potential, while two other components of the electric field are computed from the transversal currents, which are obtained by the average potential method. Typical example of the 110 kV overhead power line is also considered.

2D computation and measurement of electric and magnetic fields of overhead electric power lines

In this paper a 2D numerical algorithm is presented for the computation of electric and magnetic fields of power lines. The numerical algorithm for the electric field intensity computation takes into account a short power line and approximates the conductor charge density by a constant. The numerical algorithm for magnetic flux density computation is based on the application of the Biot-Savart law. The computed results are compared to measurements taken underneath a 400 kV power line. The computed results and measurements prove to be in good agreements keeping in mind that the sag of the power line section is approximated with a horizontal straight line. Both the computed results as well as measurements confirm the fact that the electric and magnetic fields of high voltage power lines are well within the prescribed limits.