*4.2. Results of Analytical and Numerical Simulations*

The transversal profile for the norm of the magnetic flux density

$$\left|\bar{\underline{\mathbf{B}}}\right| = \sqrt{\left|\underline{\mathbf{B}}\_{\mathbf{x}}\right|^2 + \left|\underline{\mathbf{B}}\_{\mathbf{y}}\right|^2} \tag{10}$$

calculated with relations (8)–(10), is presented in Figure 14 for the cases in which the experimental data obtained with MILTS were available and the values of the rms line currents are known. Conductors (1)–(3) (Figure 1a) carry a symmetric three-phase current, with values specified in Figure 14 (three cases), while the current in the conductors (1 ), (2 ) and (3 ) are zero (line not in service). The geometric coordinates considered in simulations, corresponding to the position of the three conductors at the pillar, are x1 = 5 m, y1 = 35 m, x2 = 8 m, y2 = 28.5 m, x3 = 5 m and y3 = 22 m.

**Figure 14.** Transversal profile of |B| at the pillar—simulations based on analytical results.

A numerical simulation of the line was also performed using COMSOL Multiphysics 6.1. In this case, the electrical constants for the soil, ε<sup>r</sup> and σ can be also taken into account, but these constants depend on the content of water in the soil [17]. In the numerical simulations, the considered values were σsoil = 0.5 S/m and εr soil = 10.

The transversal profile for the magnetic flux density at the height of 1 m and 1.8 m, respectively, above the ground, was computed using the same geometric coordinates for the conductors as before (conductors near the pillar), and the results are presented in Figure 15. Simulations for other numerical values of σsoil and εr soil were also carried out since their value in the onsite experiment is uncertain, but it was found that they have a small influence over B.

**Figure 15.** Transversal profile of |B| at the pillar—simulations in COMSOL: (**a**) at a height of 1 m above the ground, (**b**) at a height of 1.8 m above the ground.

In Figures 14 and 15 the geometric plane of symmetry of the OPL is the plane x = 100 m. In the COMSOL simulations, the physics module magnetic and electric fields (mef) was used, performing a frequency analysis [18]. The plots in Figure 15 correspond to 50 Hz. This physics module takes into account the displacement currents, besides the conduction currents, making the study more accurate. The dimensions of the entire analyzed domain were 400 m in the direction of Ox and 300 m in the direction of Oy, with a 150 m region under the ground.

The results obtained in the COMSOL simulations are in very good agreement with the analytical results. For example, the largest value of B, obtained for I = 193.82 A, is Bmax = 646 nT in COMSOL and 638 nT in MATLAB (analytical results), giving a maximum relative difference between analytical and numerical simulations of 1.2%.

Comparing the results obtained in simulations with the estimated values of B obtained using the hybrid method (Section 4.1), the similarity of the plots in Figures 8 and 14 (or Figure 15) can be observed with inherent irregularities in the curve based on the experiment, due to the particular onsite conditions (as mentioned in 4.1). At the same time, the maximum value of B is 638 nT in the analytical simulations (for I = 193.82 A) and from the estimations based on the measurements Bmax = 605 nT, leading to a maximum relative difference between the simulated and estimated B of 5.4%.

## **5. Discussions**

In order to obtain a complete map of the magnetic flux density produced by an overhead power line (OPL), a large number of simultaneous measurements taken over a significant time interval would be necessary. This approach that takes into account the spatial and temporal variation of B (rms) is not feasible due to the large number of instruments necessary in the survey and the complexity of the entire operation.

That is why characterization of the magnetic fields produced by the OPL is usually performed using a transversal and a longitudinal profile obtained in successive measurements in a number of observation points. This approach is easier to implement and needs only one measurement instrument, but it is not complete because it does not distinguish between the spatial and temporal variations of B and becomes incorrect when the magnetic field has large variations in time, which happens in most cases.

The transversal profile PT obtained through successive measurements and presented in Figure 3 is able to characterize the magnetic field in the direction perpendicular to the line, since B has small variations at the time these data were collected. The small deviations from the real values can be caused by terrain oscillations of levels and small errors in positioning the instrument.

In contrast, the longitudinal profile PL, obtained experimentally through successive measurements and presented in Figure 4, cannot characterize correctly the spatial variation of B along the line in direction Oz, due to the large variations of B during the experiment.

That is why, in this paper, a new method for estimating the longitudinal profile and the transversal profiles for B at different positions along the line (z ∈ [0,d]) is proposed, thus allowing for spatial maps of B to be represented at different moments of time. In the case considered in this study, the map for B covers an area of 60 m × 282 m. In order to obtain the estimated profiles for B, two instruments are used, one for spot measurements in different points (MISM) and one for a long-term survey (MILTS), which record automatically and continuously the values of B in a fixed point. Thus, using only two instruments, 156 values for B are collected using MISM (61 for PT and 95 for PL) and, using MILTS, the values of B can be estimated in 5795 points at any moment of time (17,530 instances of time, according to Table 1). Some of the estimated results are presented in Figures 8–11. Two maps for the magnetic flux density are also presented in Figures 12 and 13, at moments corresponding to the maximum and minimum recorded values of B, respectively.

The first validation of the proposed method is conducted within the method by obtaining two estimated transversal profiles at the mid-distance between two consecutive pillars PTM (Figure 5). The plots in Figure 11 show that the two estimations for the same profile at the same moment of time, obtained using two separate sets of data, are almost identical, with relative differences between the two profiles of under 9%.

In order to verify or validate the proposed method, calculations for B using analytical and numerical methods were also used. The maximum relative difference between the hybrid method and data from analytical simulations, using estimations of the transversal profile at the pillar, was 5.4%.

The two instruments MISM and MILTS were designed, realized and calibrated by the authors, with an uncertainty in the measured value of B of under 5%. Supplementary measurement uncertainty may also occur due to errors in positioning the instrument both in horizontal and in vertical directions and also due to possible errors in the time synchronization of the two instruments.

The proposed hybrid method of estimation may be used to map other sources of electromagnetic field, and, by using an automatic recording of the magnetic flux density and of the electric field, maps for both B and E can be obtained [14,15].
