**2. Materials and Methods**

Magnetic flux density can be expressed in a three-orthogonal system as:

$$\mathbf{B} = \sqrt{\mathbf{B}\_1^2 + \mathbf{B}\_2^2 + \mathbf{B}\_3^2} \tag{1}$$

where B1, B2 and B3 are the rms values of the components along the orthogonal directions. If the orthogonal system is Oxyz, then the three components are Bx, By and Bz. This simple observation ensures the fact that if the measurements are done with a sensor that determines the rms value of B, then the result is the same, irrespective of the sensor orientation.

The usual approach in monitoring the magnetic field produced by OPL is to trace a longitudinal and a transversal profile of B. If a mesh with a step size of 1 m is considered and measurements are conducted using only one instrument, the map for B is correct, if the current in the OPL is constant (rms) during the entire survey. Due to the fact that line currents are unpredictable, the magnetic field may vary significantly in the observation points, so a time variation for B (rms) must be also determined.

The method proposed in this paper takes into account both the spatial and the temporal variation of B (rms). The results are based on experimental determinations onsite and also on measurements and simulations (being thus a hybrid method). The experimental results are compared with the results obtained through numerical simulations that use the OPL configuration and the values of the line currents.

As a result of applying the proposed method, the spatial distribution (map) for B is obtained in a rectangular region with the dimensions (in this case study) of 60 m in transversal direction and 282 m in longitudinal direction, for any observation time during the automatic monitoring interval.

The geometric configuration of the OPL (2D transversal view) is outlined in Figure 1a and in Figure 1b. The real view of the line, with the measuring instruments, is presented.

**Figure 1.** (**a**) 2D cross section of the overhead power line, (**b**) real view of the OPL with the two measuring instruments (MILTS and MISM).

#### *2.1. Measurement Procedure*

The measurements for B are carried out using two instruments for automatic survey of the magnetic field, which were designed by the authors and are presented in Section 3.1 (Instrumentation). One of the instruments, MILTS, used for long-term survey, is placed in a fixed point under the geometric center of the OPL, at the mid-distance between two consecutive pillars. This instrument performs a survey for several hours, but this time can be increased to days or even months if proper housing and power supply is available. The long-term survey performed by MILTS records the rms values of B for almost 6 h (5 h and 31 min) with a time step of about 1 s.

The second measurement instrument, MISM (Figure 1b), realizes spot measurements for B in the specified points of the longitudinal (LD) and transversal (TD) directions (with respect to the OPL), at points 1 m apart for TD and 3 m apart for LD. Unlike the procedure presented by the authors in [13], which used a simple commercial instrument for spot measurements, the procedure used in this paper is based on two automatic monitoring systems.

In this way, besides a very good synchronization between the two instruments (MISM and MILTS), i.e., the spot measurements data and the long-term survey data necessary for the proposed hybrid method, supplementary information regarding B are obtained in all the observation points.

The time for completing the spot measurements with MISM is approximately 20 min for the transversal profile and 40 min for the longitudinal profile, in the case studied experimentally in this paper. Two transversal profiles were traced, one at the pillar (z = 0) and one at the mid-distance between consecutive pillars (z = d/2), where d is the distance between the two pillars. The second transversal profile was traced in order to verify and compare the results obtained with the proposed method, as will be later detailed.

The number of spot measurements in longitudinal direction can be reduced to one half due to the symmetry of the magnetic field with respect to the transversal middle plane z = d/2. A total of 61 spot measurements were taken in transversal direction at times tT1, tT2,..., tT61 and 95 measurements in longitudinal direction at times tL1, tL2,..., tL95.

## *2.2. Proposed Hybrid Method*

Due to the fact that the current rms values have a permanent variation, in this paper, a method is proposed for the estimation of the magnetic field produced by the OPL both in transversal and in longitudinal directions for a specified time, e.g., tT1, for the entire transversal profile (EPT(tT1)) and similarly at tL1 for the entire longitudinal profile (EPL(tL1)).

The estimated magnetic flux density, Be, is determined using the measured values Bm and, also, taking into account the time variation of B (rms) recorded in one single fixed point (z = d/2) during the long-term survey, BLTS.

The transversal profile is traced sequentially, resulting in a set of data for Bm in 61 evenly spaced locations, namely Bm measured at point P1 at time tT1, and so on, Bm(P1, tT1), Bm(P2, tT2),..., Bm(P61, tT61). In the transversal profile estimated at time tT1 (EPT(tT1)), the estimated value coincides with the measured value only at the first point P1:

$$\mathbf{B\_{c}(P\_{1}, \mathsf{tr}\_{\mathsf{T}1}) = B\_{\mathsf{m}}(P\_{1}, \mathsf{tr}\_{\mathsf{T}1})} \tag{2}$$

The next values will be estimated based on the values measured with MISM in subsequent points P2, P3, ... , P61, taking into account the time variation of B recorded with MILTS at the fixed point. Thus, at point P2 the estimated value Be is the following:

$$\mathbf{B\_{c}(P\_{2},t\_{\Gamma1}) = B\_{\rm m}(P\_{2},t\_{\Gamma2}) \* B\_{\rm LTS}(P,t\_{\Gamma1}) / B\_{\rm LTS}(P,t\_{\Gamma2})}\tag{3}$$

Relation (3) is based on system linearity and can be obviously rewritten in the form:

$$\frac{\mathbf{B\_{c}(P\_{2}, \mathsf{tr}\_{\mathsf{T}1})}}{\mathbf{B\_{m}(P\_{2}, \mathsf{tr}\_{\mathsf{T}2})}} = \frac{\mathbf{B\_{\mathrm{LTS}}(P\_{\mathsf{L}}, \mathsf{tr}\_{\mathsf{T}1})}}{\mathbf{B\_{\mathrm{LTS}}(P\_{\mathsf{L}}, \mathsf{tr}\_{\mathsf{T}2})}} \tag{4}$$

A similar procedure is employed for estimating the longitudinal profile, EPL(t), at a given time t, using the data measured with MISM for B along the line and the values recorded by MILTS in a fixed observation point P.

According to this procedure, the estimated values should be the same as those measured simultaneously in all the 61 discretization points for the transversal profile and in all the 95 points for the longitudinal profile (if such an experiment were realized), but this would require a very large number of instruments (MISM), thus being impractical.

Using the consecutive measurements for one transversal profile (PT) and one longitudinal profile (PL), measured with MISM, and the values for B obtained during the long-term survey in a fixed point (measured with MILTS), the transversal profiles for any z = k·d/N, k =0, N, where d/N is the step size in longitudinal direction, can be estimated for any moment of time. At the same time, the longitudinal profiles for any time during the observation procedure with MILTS can be also estimated.

Using this method, a matrix of 61 × 95 points (for the considered step size), containing the estimated values of B in a rectangular region under the OPL covering an area of 60 m × 282 m, is obtained for any moment of time during the survey with MILTS. Maps with the estimated values of B can be generated in MATLAB using these matrices.
