**1. Introduction**

Electrical energy transport and distribution is done through overhead power lines (OPLs) as well as, where necessary, through underwater sea cables buried in the sea bed. The development of industrial and household applications both in urban and in rural areas lead to an expansion of the distribution networks so as to satisfy consumer demands. This means a network of high voltage power lines, as well as medium and low voltage lines is geared towards the end consumer. Overhead power lines, which host currents in the order of hundreds of amps, produce magnetic fields whose levels must be monitored due to the possible adverse effects on people and other living beings.

Overhead power lines work in the extra low frequency domain of the electromagnetic spectrum (ELF), i.e., 30–300 Hz. The current view, resulting from numerous researches and measurements, is to achieve a prudent avoidance of the exposure to electric and magnetic fields, both in occasional and in occupational situations.

Even if the maximum admissible values recommended by several international regulating authorities are rather large (of the order of 100 μT for the magnetic flux density), in cases of exposure for long periods of time, the threshold for B must be significantly smaller, especially for children, hence the necessity of a long-term survey and that of a complete characterization of the magnetic field exposure for the general population [1–4].

**Citation:** Pavel, I.; Petrescu, C.; David, V.; Lunca, E. Estimation of the Spatial and Temporal Distribution of Magnetic Fields around Overhead Power Lines—A Case Study. *Mathematics* **2023**, *11*, 2292. https:// doi.org/10.3390/math11102292

Academic Editor: Jacques Lobry

Received: 23 April 2023 Revised: 10 May 2023 Accepted: 13 May 2023 Published: 15 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Numerous researches addressed the problem of determining, both analytically and experimentally, the electric and magnetic field produced by OPLs, trying to include various aspects such as ground proximity or the proximity of other conductors, the deformation of line conductors due to gravity, etc. In the following, a brief overview of some recent papers that address this subject is given.

In [5], the formulae for computing the electric and magnetic field produced by a line current at high altitudes (110 km above the Earth's surface) are developed. The model considers a flat Earth surface and a multilayer soil structure, with different electric properties, and uses a series expansion technique and the complex image method, as well

as exact integral expressions for <sup>−</sup> *<sup>E</sup>* and <sup>−</sup> *B* components based on Maxwell's equations and boundary conditions at the Earth's surface.

In [6], the authors use double complex numbers to incorporate in one expression the

components of <sup>−</sup> *B*, *B*<sup>x</sup> and *B*<sup>y</sup> (and the fact that these are complex numbers). Closed form

expressions are derived for <sup>−</sup> *B*, based on the Biot Savart Laplace (BSL) formula.

In [7], the relations for the computation of the electric and magnetic fields produced by finite length OPLs, using the classical expression for the magnetic vector potential, are established.

Paper [8] considers a 3D model of OPLs, with deformation of the lines due to gravity. The study combines the classical simulation current method (SCM) which replaces the actual unknown distribution of currents with simulated currents (in order to determine the magnetic fields) with two stochastic optimization methods, Particle Swarm (PS) and Differential Evolution (DE), used to optimize (determine) the position and number of simulated currents.

Another optimization method, Ant Lion Optimization (ALO), is used in [9] to determine the position of the overhead power line conductors that minimize both the electric and magnetic fields at a height of 1 m above the ground. This method of field reduction implies, however, additional costs, due to the necessity to re-design OPLs.

In [10], the authors propose a method for the analysis of multi-circuit overhead transmission lines using artificial neural networks (ANN) trained to estimate the real and imaginary parts of *B*x and *B*y at arbitrary points (x, y). The results are compared with BSL simulations and experimental measurements.

The magnetic field produced by submarine power cables is studied in papers such as [11], where closed form solutions for thin wires with a helical shape are established, as well as in [12], where both modelled and measured results for B are presented and the geomagnetic field is also taken into account.

The main objective of the present paper is to obtain a complete characterization of the magnetic field generated by overhead power lines in cases of long-term exposure, taking into account that B (rms) has large and unpredictable variations in time. The originality and novelty of the paper resides in the proposed hybrid method for estimating B using a correlation between spot measurements and long-term survey data. Both these measurements are made simultaneously using two measuring devices realized by the authors. Thus, maps with the spatial distribution of B in a large area near the OPL can be obtained for any moment of time, without the need of using a large network of sensors for measuring the magnetic flux density.

The results obtained using the proposed hybrid method and our instruments are in good agreement with the results obtained in simulations.

The organization of the paper is as follows: Section 2 presents the materials and methods used in the study, Section 3 presents the performed measurements, Section 4 presents the results of simulations, Section 5 discusses the obtained results and Section 6 is reserved for the conclusions of the paper.
