Paper Title (use style: paper title) [311372]

Numerical determination of magnetic field around a [anonimizat] a [anonimizat]. This paper proposes a mathematical model of magnetic field caused by high voltage conductors of electric power transmission systems using the finite elements method. The numerical computation of magnetic field around of a high voltage 110 kV electrical overhead line is analyzed. The spectrum of magnetic field intensity close to the overhead electric power line and the studied body are analyzed using the ANSYS Multiphysics software package.

Keywords – magnetic field; overhead transmission lines; numerical simulation; [anonimizat]. [anonimizat], which flows in the phase conductors (these, [anonimizat] a [anonimizat]). The intensity of this field depends also on the height and spatial distribution of the conductors. There are other sources (on a small scale) which contribute to the magnetic field intensity: [anonimizat], [anonimizat], [anonimizat]. [anonimizat]. In the literature there are some studies containing an analytical calculation of the magnetic field produced by high voltage power lines. Examples are provided in [1]-[3], [anonimizat], [anonimizat]. Also, the estimation of the magnetic field density at locations under and far from the two parallel transmission lines with different design arrangements is presented in [4]. The effects of conductors’ curvature on the spatial distribution of the magnetic field are presented in [5] [anonimizat], and the power transmission lines’ spans being always parallel to each other. [anonimizat] (also called the First Laplace Formula), carried along the path of every conductor of every spread [6]. The magnetic field produced by electric power lines is usually calculated numerically with the use of a computer. In [7] specific calculations of the magnetic field for a 110 kV overhead line are presented. In [8] the effect of harmonic components at different electromagnetic frequencies is also taken into account. In [9] historical load databases are used to take into account the relations between magnetic field and electrical load patterns. In [10] the magnetic field distribution is calculated in high voltage substations.

[anonimizat] (FEM). The influence of the geometrical configuration of conductors is studied in order to minimize the magnetic field near double circuit high voltage overhead power transmission lines. The six conductors of a 110 kV transposed high voltage transmission line with double circuit are considered.

FEM application for two-dimensional problems of stationary magnetic field

Generally, the finite element approximation is based on partial differential equations as expressions of the solutions defined by a partition of the field study in disjoint elements, called "finite elements", giving the name of the method [11]. One of the main variants of Finite Element Method is the Galerkin Method that based on the use of so-called "weak form" of the field equations, and convenient discretization of the field. For simplicity, consider the plane-parallel magnetic fields. In this case, the magnetic vector potential has the Oz axis, and in the transverse planes z = constant it is presented as a scalar field A (x, y). Conduction currents have also only components after the Oz axis. Therefore, in the cross section planes, the current density J has the character of a scalar field (x, y). As unknowns, magnetic vector potential values are considered in the nodes of a mesh network. To begin with, we shall consider the case of plane-parallel fields, in which the vector potential is oriented in the Oz direction. In the case of plane-parallel fields, these conditions turn into:

 

Consider a stationary magnetic field problem in a domain where there may be current-carrying conductors. The field equations are:

The magnetic circuit law:

The magnetic flux law:

The connecting law:

From these relationships it results:



The calculus domain D is bordered by the closed surface and it is shown in Figure 1. On this surface, border conditions are imposed, by one of the two main types:

Homogeneous Dirichlet conditions:

coming from the boundary condition expressed by the normal component of magnetic induction .

Neumann conditions:

coming from the boundary condition expressed by the tangential component of magnetic field intensity .

We consider a function vector defined on D. The equation is multiplied with this function and then integrates on the domain [11]:

Using the identity:

it results:

Calculus domain

(11)

or, after the application of Gauss-Ostrogradski relationship:

(12)

Using the identities:

(13)

The surface integral can be decomposed into the sum of integrals on surfaces SB și SH:

Let’s consider the set of vector functions Y={U | :

With integrable square value, together with its derivatives,

Satisfying on SD the condition:

(15)

Therefore, in the above equation, if one takes into account the condition on the surface of SH, it results:

This equation represents „the weak form” of the magnetic field equation in scalar potential, and the solution is called the weak solution of the problem. In the case of plane-parallel magnetic fields [11]:

From these relations, we obtain:

(19)

By these, the weak form of the equations becomes:

It is found that the expression which results is similar to that used in the solution through the scalar potential. The vector potential formulation is applicable to both static and dynamic fields with partial orthotropic nonlinear permeability.

Numerical analysis using ANSYS

Numerical simulation is realized using the finite element package ANSYS. The basic magnetic analysis results include magnetic field intensity, magnetic flux density and current densities. These types of evaluations are somewhat different for magnetic scalar and vector formulations.

Consider a system consisting of 6 conductors (double circuit) with 3 conductors on each circuit. The conductors are located at a height of 20 m from earth. Each conductor has a diameter of 30 mm and is located at a distance of 3.6 meters and 5 meters (central conductors) of the tower of power line. The human body has a height of 1.8 m. The following regions with the properties are defined [12]:

human body, with relative permeability εr=700.

air, with relative permeability εr=1.

The power lines are the bare conductors of Aluminum Conductor Steel Reinforced (ACSR) type, having the conductivity σ= 0.8·10-7 S/m and the relative permittivity εr= 3.5 [12], [13]. Phases 1, 2 and 3 are at the electric potential of 110 kV and earth is at the reference electric potential 0 V. The current in the conductors at the moment of the measurements was 48 A for the left-hand side circuit and 60 A for the right-hand side circuit (approximately 35% of its full load value) with a light unbalance between phases.

The next step in the preprocessor phase is mesh generation and load applying upon the elements. We used a mesh with 27499 nodes and 54690 triangular elements. The finite element mesh of the system with six conductors is shown in Figure 2. The boundary conditions and loads to a 2D stationary magnetic field analysis are applied, both on the plane model (key points, lines, and areas) and on the finite element model (nodes and elements) [13]. The solution of magnetic field problems is commonly obtained using potential functions. Two kinds of potential functions, the magnetic vector potential and the magnetic scalar potential are used depending on the problem to be solved.

Finite element mesh

Factors affecting the choice of potential include: field dynamics, field dimensionality, source current configuration, domain size and discretization. In Figure 3 shows the distribution of Oz-axis magnetic vector potential. In ANSYS there is a graphical program that displays the resulting fields in the form of contour and density plots.

Magnetic vector potential values

The magnetic flux density is the first derived result. It is defined as the curl of the magnetic vector potential. The distribution of magnetic flux density is shown in Figure 4. This evaluation is performed at the integration points using the element shape functions [13]:

where: {B} is magnetic flux density, [NA] – shape functions and {Ae} is nodal magnetic vector potential.

The distribution of magnetic flux density around the conductor

The maximum value of the magnetic flux density is obtained around the conductors. In this region, the magnetic flux density has a maximum of 0.237 T for the conductors traversed by a current of 60 A. This current value is much lower than the one used in other references (e.g., [7]), leading to lower magnetic flux density with respect to the one indicated in [7]. In the system analyzed, the current values are relatively low, as the system was designed to serve a local area with consumption of industrial type much higher than the one existing today.

Distribution of the magnetic field around the conductors

Then the magnetic field intensity is computed from the flux density:

where: {H} is magnetic field intensity, [ν] – reluctivity matrix [13].

Nodal values of field intensity and flux density are computed from the integration point value. In Figure 5 shows the distribution of the magnetic field around the 6 conductors and in Figure 6 is shown a detail around a single conductor. The maximum value of the magnetic field intensity is obtained around the conductors. In this region, the magnetic field intensity has a maximum of 628.66 A/m.

Distribution of magnetic field around a single conductor

The magnetic flux density at head level has a value higher than the value obtained from the ankles. In the arm region at a distance of 1.5 m above earth, the magnetic flux density value is 0.28 μT and in the head region at a distance of 1.8 m above earth, the magnetic flux density value is 0.34 μT. The distribution of magnetic flux density around human body is shown in Figure 7.

Distribution of magnetic flux density around human body

The path for the displayed charts is chosen between two points placed symmetrical one from another on the arms. It is also possible to save the plotted results in EMF format (Extended Metafile). Figures 8 and Figure 9 show the chart of the magnetic flux density versus the distance for the conductors region and for ground. In this paper, a very simple application example to point out some important aspects associated with the computation of the magnetic field by transmission lines using 2-D methodologies was intended.

The study focused on magnetic field near the earth's surface and near the line. Therefore, for studies only need the computation of the magnetic field near the midspan or at points sufficiently away from the towers, the adopted modified 2-D modeling is attractive, mainly because it requires a minimum data entrance and a very reduced computational charge. For studies in general, and, mainly for points of computation near the towers, one must employ 3-D methods, in defiance of those request more input data and a bigger computational effort [15].

Chart of magnetic flux density in conductor area

Chart of magnetic flux density in ground

The continuous interest on the effects of the magnetic field is motivated by the presence of different international standards containing the limits imposed to the magnetic field close to the overhead lines [3]. According to admissible limits from the IRPA – International Radiation Protection Association the maximum admissible values for human body exposure into a static magnetic field are set to the following values:

0.5 mT for the whole human body continuing exposure for a work day;

5 mT for the whole human body exposure for 2 hours;

25 mT for the human body extremities exposure.

Method of calculation presented can be useful in the design of a transmission power lines for insulation distance estimation. This could mean that this method of numerical simulation approach can be used to predict the magnetic field generated by high voltage overhead power lines.

Conclusions

This paper has presented numerical computations of magnetic field around of a high voltage 110 kV electrical overhead line. After recalling the mathematical formulations, specific results obtained from the solver ANSYS have been shown. Particular relevance has been given to the presence of the human body in the region in which the magnetic flux density is more significant. The magnetic field intensity of about 629 A/m is below the limit of 800 A/m established by ICNIRP in 2010 for safe occupational exposure [16]. This is relevant to workers operating close to the overhead line without deactivating the line.

The results obtained by simulation confirm that the magnetic field computation for the conductor system and a human body lead to an accurate result. The method addressed in this paper can be useful in the design of the transmission power lines for estimating the insulation distance.

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