INTRODUCTION

The problem of geomagnetically induced currents (GICs) is acute because of electric power line failures occurring as a result of extreme geomagnetic disturbances [1, 2] and consequent economic losses; the load on the power system increases, though without large outages [3]. It is especially important to study the negative impact GICs have on the operation of electrical conductor systems in Russia because of the long length of Russian high-voltage transmission lines, including those located at high latitudes [4]. Electrical equipment malfunctions can be caused by premature aging of components of high-voltage transformers due to the cumulative effect of even moderate GICs [5]. Because of hystereses in transformers, GICs of only a few amperes could potentially endanger protective relays [6].

There is no clear correlation between the amplitude of geomagnetic disturbances and GICs. This is due to the integral nature of the connection between geomagnetic disturbances and the voltage induced in a circuit formed by a power transmission line, ground connection, and conductive layers of Earth’s crust. The size of this effective circuit depends on the frequency and spatial scale of a geomagnetic disturbance. Even moderate geomagnetic disturbances can cause intense GICs [7]. Works on GIC-caused damage at middle and low latitudes are growing in number [8], though geomagnetic disturbances are usually of less magnitude at these latitudes than at high latitudes.

Research on GICs mostly involves analyzing magnetic storms and substorms, (i.e., aperiodic bay disturbances with abrupt onsets). Some works describe GIC bursts of extreme amplitudes, which result from Pi3 pulsations (i.e., quasiperiodic series of magnetic pulses [9, 10]). However, there are there is virtually no literature on GICs caused by Pc5 geomagnetic pulsations, although these GICs of moderate intensity that last several hours can be more dangerous for the long-term normal operation of electric power networks than short-term bursts of GICs at the onset of substorms and storms.

The aim of this work was to study GICs detected at the Vyhodnoy station (VKH) and geomagnetic pulsations in the range 1.4 to 5.6 mHz recorded by the IMAGE magnetometric stations closest to the VKH [11]. The range of frequencies we consider includes quasi-monochromatic Pc5 pulsations (with dominant frequencies f > 2 mHz) and broadband Pi3 pulsations (with dominant frequencies f < 2 mHz).

RECORDED DATA AND DATA PROCESSING

There is a unique GIC monitoring network on the Kola Peninsula and in Karelia. It is based on the substations of a 330 kV transmission line [12]. The transformer neutral current can be measured in the range of 1 to 120 A [13] with a sampling rate of 1 min.

We analyzed geomagnetic pulsations recorded at the KEV station of the IMAGE magnetometric network (the one closest to the VKH station), and geomagnetic observation data of the KIL and SOD stations located to the west and south of the KEV station, respectively. The coordinates of the stations are given in Table 1. The initial sampling rate of records made at the IMAGE stations was 10 s. In this work, these data are reduced to a 1 min sampling interval like that of the data from measuring GIC.

Table 1.   Coordinates and other characteristics of observing stations
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Geomagnetic pulsations were selected automatically according to [14], followed by selective visual control. For the selected intervals of geomagnetic and GIC pulsations, the power spectral density (PSD) was estimated according to Blackman and Tukey [15] using a running window 64 points long (3840 s) with a step of 300 s. The spatial scale of pulsations was estimated from the cross spectrum of pulsations measured at two stations using the parameters spectral coherence γ2, phase difference Δφ, and ratio of spectral power densities R = S11/S22, where S11 and S22 are the power autospectra of the pulsations measured at points 1 and 2. The spectral coherence is the normalized cross power spectrum: γ2 = \({{S}_{{12}}}S_{{11}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2}}}S_{{22}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2}}},\) where S12 is the cross power spectrum. Spectral coherence γ2 varies between 0 and 1, and γ2 ≈ 1 in a wide range of frequencies. This means the spectral composition is virtually the same, and the phase difference is constant for every frequency in the considered time interval.

We analyzed two horizontal components of the geomagnetic field, BX and BY, oriented along the geographic meridian and parallel, respectively. We did not transform the geographic coordinates to the geomagnetic coordinates because the angle between the geographic and geomagnetic meridians was small for the relevant stations.

Typical Pc5 pulsations exhibit horizontal components with similar amplitudes and high coherence [16], which made it possible to introduce parameters that describe the total amplitude [17]. Here we analyze the effect of these geomagnetic components separately to determine the influence of pulsation polarization could have on the generation of GICs.

The calculated spectra were used to obtain the coefficients of correlation between temporal variations in logarithms Pj(f) of the PSDs of GICs and components Pb(f) of the magnetic field for every frequency, along with the coefficients of linear regression of the relationship between Pj and Pb. Pulsations were divided into large-scale and small-scale for a specified direction in order to consider the spatial scale.

RESULTS

Figure 1 shows data from observing pulsations of geomagnetic components and GICs. Pulsations with an average peak-to-peak amplitude of around 20 nT are observed in both horizontal components at the KEV station for at least 3 h in the local morning sector (MLT). The visible period of oscillation is around 4 min. Similar oscillations are visible in measurements of the current with an average peak-to-peak amplitude of around 10 A. Oscillations developed outside the magnetic storm. Index Dst was above −25 nT for 7 days before the given time interval. The auroral activity determined using the AE index remained weak (AE < 100 nT) until 05:00 UT (i.e., pulsations started developing with no appreciable auroral disturbances).

Fig. 1.
figure 1

Pc5 pulsations registered simultaneously on March 11, 2015 (day 70), in both the current of the transformer neutral at the VKH station and the geomagnetic field at the KEV station.

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The Severnyi Tranzit Power Transmission Line runs north–south, so GICs are expected to be caused mainly by pulsations of the BY component of the magnetic field. This effect is statistically apparent from the spectra of pulsations recorded over a long period. Figure 2a shows coefficient C of linear correlation (averaged over all Pc5/Pi3 intervals of 2015) between logarithms of the PSDs of current and two magnetic field components versus frequency f. The two components exhibit qualitatively similar C(f): there is a weak maximum in the low-frequency region (f = 1.5 mHz), a minimum at 2.7 mHz, and growth in the high-frequency region of the spectrum with two broad maxima at 3.3 and 4.8 mHz. At the same time, the coefficient of correlation between GIC and BY is higher than between GIC and BX throughout the considered range of frequencies.

Fig. 2.
figure 2

Frequency dependences of the coefficients of (a) correlation and (b) linear regression between logarithms of the spectral power of GIC variations and horizontal components of magnetic field, averaged over all selected Pc5 intervals (2015).

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Figure 2b shows the spectrum of coefficient K of linear regression between logarithms of the PSDs of GIC and the horizontal components of the magnetic field: Pj(f) = K(f)Pb(f) + const. The resulting K = 0.65–0.8 means that Pj grows with Pb slower than linear. The dependence of GIC on BY is stronger than on the BX component at all frequencies, so we used data on BY to analyze the relationship between Pc5/Pi3 pulsations and GICs.

The role of the spatial scale of pulsations in the generation of GICs can depend on the angle between the direction of the electric transmission line and the direction of changes in the field of pulsations. The ratio of pulsation amplitudes and the coherence are important if the direction is perpendicular to the electric power line. We divided geomagnetic pulsations into east–west large- and small-scale. Pulsations are considered east–west large scale if (1) the spectral coherence between the records of the KEV and KIL stations (which lie in a direction parallel to the lines of latitude) is greater than the threshold value γ2 > \(\gamma _{{\text{b}}}^{{\text{2}}};\) and (2) ratio R of the spectral powers is close to 1 (|R − 1| < δ). A similar criterium can be introduced for the north–south direction according to data from the KEV and SOD stations. We must consider the phase difference when it comes to direction parallel to the power transmission line, since it determines the time within which pulsation-related changes in the magnetic flux have the same sign on a segment of the circuit. Conditions (1) and (2) are therefore supplemented with the condition of a small phase difference, |Δϕ| < ϕb.

Figure 3 shows the results from a correlation analysis of variations of the PSDs of GIC and geomagnetic pulsations, calculated for all selected Pc5/Pi3 intervals. The pulsations are divided into large- and small-scale using threshold values \(\gamma _{{\text{b}}}^{{\text{2}}}\) = 0.7, δ = 0.33, and ϕb = 30°. In the Pi3 range of frequencies (f < 2 mHz), coefficients of correlation ~0.75 are virtually independent of the scale of pulsations in east–west and north–south directions. In the high-frequency region of the spectrum (f > 2 mHz for large east–west pulsations and f > 3 mHz for large north–south pulsations), the correlation between the PSDs of GICs and large geomagnetic pulsations grows along with frequency, reaching 0.85.

Fig. 3.
figure 3

Frequency dependence of the coefficient of correlation between logarithms of spectral power of variations in the GIC and BY component of magnetic field, averaged over all selected Pc5/Pi3 intervals (2015) for groups of pulsations classified with their spacial scale along E–W (a) and N–S (b) directions. Large-scale pulsations are denoted by L; small-scale ones, by S; E–W and N–S denote east–west and north–south directions, respectively.

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The integral frequency-averaged parameters describing the relationship between geomagnetic pulsations and GICs are easy to use, though they lack the detail of frequency dependences similar to those shown in Figs. 2 and 3, which are used in GIC intensity forecasting applications. In a layered inhomogeneous medium, the relationship between the magnitudes of GICs and geomagnetic disturbances is generally determined by the impedance of Earth’s surface [17]. There is usually no detailed information on the conductivity depth distribution, so simplified models are used that have simple analytical expressions. Let us see how the choice of a frequency-averaged criterion depends on the presence of frequency f found in the spectrum of pulsations and possible variants of the conductivity depth distribution:

(1) An electrotechnical model in which the size of the circuit does not depend on frequency and expression J ~ dB/dt (J ~ fB, where J and B are the components of the Fourier spectrum) can be used if there is a narrow conducting layer at a finite depth;

(2) If the conductivity of Earth’s crust is independent of depth, a frequency dependence of the J ~ f 1/2B type is used [18];

(3) Individual quasi-monochromatic pulsations can be studied using a dependence of the J ~ B type.

To choose an integral (frequency-averaged) parameter to describe the relationship between the spectral power of GICs and geomagnetic pulsations, we divided all selected Pc5/Pi3 intervals (belonging to the same year) into subsequences of equal length and then analyzed the distribution with respect to partial coefficients of correlation for all the models. Figure 4 shows the results for a subsequence of length N = 240 as a cumulative probability function; i.e., probability P(C1) = P(C > C1). When coefficients of correlation are high (C > 0.6), there is a considerable difference between models 1, 2, and 3, while models 1 and 2 are quite similar to each other. The relationship between the amplitudes of geomagnetic pulsations in the Pc5/Pi3 range (1.4–5.6 mHz) and GICs generated in the quasi-meridional transmission line on the Kola Peninsula can thus be described using a dependence of the J = cWBY type, where c = const and WBY = f αBY. Parameter WBY describes the relationship (integral in respect of the whole range under study) between the amplitude of the pulsations of GICs and the BY component of the magnetic field; exponent WBY is mainly determined by the spatial distribution of conductivity and lies in the range from 0.5 to 1 for the given region.

Fig. 4.
figure 4

Cumulative probability function for partial coefficients of correlation (N = 240) between logarithms (integral over the 1.4–5.6 mHz band) of the spectral power of variations of GIC and BY component of magnetic field; the probability function is averaged over all selected Pc5/Pi3 intervals (2015) for weight functions corresponding to different models of conductivity.

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DISCUSSION AND CONCLUSIONS

Analysis of intervals of Pc5/Pi3 pulsations of the geomagnetic field and GICs shows that the relationship between the spectral power of geomagnetic pulsations and geomagnetically induced currents (GICs) in a north–south electric power transmission line is more pronounced with respect to the west–east component (BY) than the north–south component (BX) of the geomagnetic field. This is apparent from the greater coefficients of correlation and linear regression.

The coefficient of correlation in the high-frequency region of the spectrum is affected by the spatial scale of pulsations in both the north–south and west–east directions. In the low-frequency region (f < 2 mHz) of the considered range, the coefficients of correlation depend weakly on the scale of pulsations, while in the high-frequency spectral region (f > 2 mHz for large west–east pulsations and f > 3 mHz for large north–south pulsations), the correlation between the amplitude of GICs and large-scale geomagnetic pulsations grows along with frequency and exceeds the one between GIC amplitudes and small-scale geomagnetic pulsations. Large-scale Pc5 geomagnetic pulsations thus exhibit stronger correlations with GICs than small-scale ones.

Many authors (e.g. [19]) assume that the intensity of GICs is proportional to the derivative of geomagnetic field variations with respect to time (dB/dt). This approximation is logical only for a special type of conductivity depth distribution; the frequency-dependent relationship between the amplitudes of GICs and magnetic field variations generally requires numerical investigation. In many cases, the relationship between GIC amplitudes J(f) and geomagnetic pulsations B(f) at a given frequency f can be approximated by power dependence J(f) ~ f αB(f).

Given typical values of conductivity of Earth’s crust and frequencies of several millihertz, GICs penetrate approximately to skin depth. At Earth’s surface, the impedance relationship \(\vec {E}\)(f) = Z(f)\(\vec {H}\)(f) describes the relationship between the amplitudes of the vectors of the horizontal electric and magnetic components, \(\vec {E}\) = {EX, EY} and \(\vec {B}\) = {BX, BY}, respectively (a plane-wave approximation), where Z(f) is the impedance of Earth’s crust, determined by the resistivity-depth relation ρ(z). There is no need to synthesize telluric field E(t) if we introduce proxy-telluric field \({{E}_{{\text{p}}}}(t)\) = \(\left| {{{F}^{{ - 1}}}\left\{ {Z(f)B(f)} \right\}} \right|,\) where F −1 is the inverse Fourier transform [20], and assume the conductivity of Earth’s crust is uniform: Z(f) ~ f 1/2 [21].

Analysis of the partial coefficients of correlation between the spectral power density of pulsations of the BY component of the geomagnetic field and GICs generated in the quasi-north–south electric power transmission line on the Kola Peninsula shows that the greatest correlation is at 0.5 < α < 1 for power dependence J ~ f αBY. Value α = 0.5 corresponds to constant conductivity, while α = 1 corresponds to the constant effective height of the circuit formed by the electric power transmission line and ground currents. The weak dependence of the correlation coefficient on parameter α in the specified range allows us to use one of these simple models to make estimates. The difference between the coefficients of correlation, which is due to conductivity depth distribution, turns out to be smaller than the difference between the large- and small-scale pulsations.