Ground Fault in Medium-Voltage Power Networks with an Isolated Neutral Point: Spectral and Wavelet Analysis of Selected Cases in an Example Industrial Network Modeled in the ATP-EMTP Package

Abstract

The paper presents some case spectral analysis of zero-sequence voltages and currents in an example industrial power distribution network. The network layout is based on typical power delivery networks in underground coal mines. Ground fault simulations have been made using an ATP/EMTP program. Due to the high environmental risks, the reliability of the protection relay operation related to their selectivity plays an important role. This paper tries to find the reasons for nonselective operation and unnecessary tripping in extensive mine cable networks, particularly with large power sources of higher-order harmonics. It was found that in transient states—due to the decaying oscillations occurring in complex RLC circuits—the results of short time measurements of the criterion values for ground fault protective relays can be overestimated (particularly for small values of ground resistance) and lead to nonselective tripping of a healthy cable line. Therefore, it might be advisable to increase the integration time used for measuring rms values. Also, if there is a significant level of higher harmonics in the industrial network generated by high-power converters, it should be noted that the higher harmonics of the ground fault current and currents measured by ground fault protection relays assume much higher values, which may also cause nonselective tripping. In this case, it may be advisable to use higher harmonic filters in the measuring circuits and to select a sufficiently high sampling frequency in the digital protective relays.

1. Introduction

The method of grounding the neutral point of the transformer significantly affects the operational properties of the power distribution network. This applies to both the effects of possible ground fault phenomena and to the operation of ground fault protection relays. The most commonly used networks are those with a neutral point grounded directly or indirectly (through resistance or impedance) and with an insulated neutral point. The grounding method used in a particular network often results from historical events and requirements regarding the values of ground fault currents and ground fault overvoltages; therefore, different solutions can be found in individual countries and in different application areas (industry branches). Mining electrical equipment usually operates in very harsh environmental conditions (dust, humidity, methane). This leads to increased fire, shock, and explosion hazards in underground mines [1,2,3]. Therefore, power distribution networks in underground mines often operate with an isolated neutral point, as this configuration significantly reduces the current and energy values at the fault location compared to those occurring in networks with a directly or indirectly grounded neutral point. The reduction in the current value at the fault location occurs due to the fact that this current is determined not so much by the small series impedances of the network elements, as in the case of two-phase and three-phase short circuits, but by the much greater shunt impedances resulting from the earth capacitance of all galvanically connected cable lines. However, a negative aspect of this reduction in ground fault current is the difficulty of detecting these faults quickly and reliably and of selectively (and possibly quickly) tripping only the faulty cable line. This enforces the need for more sophisticated zero-sequence voltage and current measurement systems (like Ferranti transducers for zero-sequence current component measurement, and open delta connected voltage transformers for zero-sequence voltage component measurement) and protective relays. Due to the relatively low values of the zero-sequence currents and voltages in an isolated neutral point network, these measuring systems and protection devices are much more sensitive to disturbances and require more complex calculations and simulations taking into account transients and deformations of the zero-sequence voltages and currents caused by power electronic converters. The complexity of ground fault phenomena (especially transients and nonlinear arc resistances) drives the use of complex multiresolution digital signal processing methods based on wavelet transforms [4,5,6] and empirical mode decomposition (EMD) [7]. It should be noted, however, that despite obvious advances in the use of microprocessor technology and digital signal processing in protective relaying, practically applied protection in hazardous areas is still based on quite simple, strict, and conservative standards and regulations (like directional or nondirectional overcurrent relays or admittance relays). The discrete wavelet transform (DWT) in the applications mentioned above can be considered a very efficient method for digital signal processing (e.g., in the implementation of protection algorithms); however, in this paper, we use the continuous wavelet transform (CWT) as a method to analyze and visualize transient ground fault current waveforms jointly in the time and frequency domain in the first period after ground fault initiation. A comprehensive literature review on ground fault phenomena and their detection and classification is available in [8]. The analytical description of transient states in ground faults is usually focused on the issue of ground fault overvoltages and is based on simple second-order LC models described by a single transient frequency. In our paper, we present a novel two-frequency model described by the fourth-order differential equation. Also, the results of numerical analyses carried out with the use of simulation software were innovatively visualized as scalograms with the use of the continuous wavelet transform—which enabled the presentation of frequency fluctuations and waveform in the first period after the ground fault initiation.

Many research works are devoted to the analysis of ground fault phenomena in MV networks with resonant-grounded [9,10,11] or resistor-grounded [12] neutral and containing overhead lines [4,13,14,15]. However, the operating conditions of these networks differ significantly from those of distribution networks installed in underground mines [1]. In this paper, the focus is on networks with an isolated neutral point containing only cable lines. It is noteworthy that, in order to mitigate the risks associated with high-current value phase-to-phase short circuits, the MV cables used in underground mines are armored and shielded cables. This significantly increases their line-to-ground capacitance [16] and thus their specific ground current value per unit length [1].

2. Materials and Methods

2.1. Theoretical Analysis and Simulation Software

A simplified representation of an unbranched network with an isolated neutral point is depicted in Figure 1. Given the significantly smaller values of the series line impedances compared to the capacitive line-to-ground impedances, the former can be neglected. Similarly, with the network insulation resistance assuming considerably larger values in comparison to the (connected in parallel) capacitive line-to-ground impedances, the latter can also be omitted, as shown in Figure 1b. This simplification facilitates the derivation of the impedance of a Thevenin generator (Figure 1c) and a straightforward equivalent circuit (Figure 1d).

Consequently, this enables a simplified evaluation of ground fault current values across various frequencies, encompassing higher-order harmonics of the fundamental frequency waveform.

Given that the ground current value (I_F) can be expressed according to the simplified Equation (1) derived from Figure 1d, it is evident that the resulting circuit impedance diminishes with increasing frequencies, including higher-order harmonics.

$$I_F(\omega )=\frac{V}{\left|\underset{¯}{Z(\omega )}\right|}=\frac{V}{\sqrt{R_F^2+{\left(\frac{1}{3\omega C_G}\right)}^2}}=\frac{3V\omega C_G}{\sqrt{1+9{\omega }^2{R}_F^2{C}_G^2}},$$

(1)

where I_F—line-to-ground fault current, V—line-to-ground source voltage, R_F—line-to-ground fault resistance, C_G—line-to-ground capacitance, R_G—line-to-ground insulation resistance, and ω—angular frequency of the source voltage.

This phenomenon is especially noticeable for zero or negligible values of the ground resistance (R_F) when Equation (1) can be simplified to Equation (2).

$$I_F(\omega )=\omega \cdot 3V{C}_G$$

(2)

Under such conditions, even modest magnitudes of source voltage harmonics can induce relatively significant values of ground fault current harmonics.

In real networks with a more complex (branched) structure, the flow of currents is much more complicated, as shown in Figure 2. The line-to-ground voltage in the faulty phase is reduced, and those in the other undamaged phases are elevated. Therefore, the damaged phase capacity is discharged (red arrows in Figure 2), and the remaining (healthy) phases’ capacities are additionally charged (blue arrows in Figure 2). The magnitudes of the zero-sequence currents measured by the protection relays are determined by the capacitances of the entire galvanically connected network (Figure 3).

Due to the large number of energy storage elements (including capacitances and inductances), transient dynamics is described by intricate systems of high-order differential equations posing challenges for analytical solutions. Therefore, the adoption of specialized numerical simulation software emerges as a viable methodological approach to dissect and comprehend the intricate phenomena within such a network. We used an ATP/EMTP software package [17]. This package consists of three tools: the interactive graphical user interface (ATPDraw v. 6.3), the simulation tool (TPBIG), and a postprocessor for further analysis and graphical output (PlotXY or Matlab v. R2020a in our case). The whole toolchain is shown in Figure 4.

The Π cable model [17] was used to represent the cable lines. The network design was carried out in the ATP/EMTP package, allowing precise simulation of system behavior. The ATP/EMTP package is widely used for the numerical modeling of transients in power networks [17], including the modeling and testing of the operation of protection relay systems [18], harmonics propagation [19], and power quality problems [20,21]. The ATP/EMTP software is often used to generate data used in the development of new algorithms for the detection and location of ground faults [22,23,24,25]. Other software such as PSCAD [6,26] or Matlab [6,11] can be also used for the modeling and analysis of ground fault phenomena.

2.2. Transient Components of the Ground Fault Current and Their Frequencies and Decay

When a ground fault is initiated, a transient occurs. As the network has a complex layout with a large number of energy-accumulating (capacitances and inductances) and energy-dissipating (resistances) elements, the transient waveforms are described by very high-order differential equations, which can only be solved numerically. An approximate analytical solution is only possible if certain simplifying assumptions are made. In this study, the network model shown in Figure 5a is adopted, with the inductance of the power source (transformer), the ground capacitance of the faulted line, the ground capacitance of the rest of the network, and the resistances and inductances of the faulted line as the most important elements. This model structure allows the earth capacitances of the faulty line and the rest of the network to be connected in parallel, as shown in Figure 5b.

The transient analysis of this model using the Laplace transform is possible based on a superposition method allowing the calculation of the steady-state (easily determined) component waveform to be separated from the calculation of the transient component waveform. The equivalent operator impedance circuit for the transient component and its subsequent simplifications are shown in Figure 6a–d.

The equivalent operator reactance X_z(s) of the parallel LC branch is determined as

$$X_Z\left(s\right)=\frac{\frac{1}{s{C}_{0\Sigma }}\cdot \left(\frac{1}{2s{C}_{0\Sigma }}+\frac{3}{2}s{L}_T\right)}{\frac{1}{2s{C}_{0\Sigma }}+\frac{1}{s{C}_{0\Sigma }}+\frac{3}{2}s{L}_T}=\frac{\frac{1}{2s{C}_{0\Sigma }}+\frac{3}{2}s{L}_T}{s{C}_{0\Sigma }\cdot \left(\frac{3}{2s{C}_{0\Sigma }}+\frac{3}{2}s{L}_T\right)}=\frac{\frac{1}{s{C}_{0\Sigma }}+3s{L}_T}{s{C}_{0\Sigma }\cdot \left(\frac{3}{s{C}_{0\Sigma }}+3s{L}_T\right)},$$

(3)

where L_T—inductance of power transformer windings, and C_0Σ—total capacitance of the entire galvanically connected network.

Therefore,

$$X_Z\left(s\right)=\frac{1+3s^2{C}_{0\Sigma }{L}_T}{3s{C}_{0\Sigma }+3s^3{C_{0\Sigma }}^2{L}_T},$$

(4)

and to simplify the notation, the following R_Σ symbol will be adopted:

$$R_{\Sigma }=R_L+R_F.$$

(5)

Then, the total operator impedance of the entire circuit for the transient component Z_Σ(s) can be determined as

$$Z_{\Sigma }\left(s\right)=R_{\Sigma }+s{L}_L+X_Z\left(s\right)=R_{\Sigma }+s{L}_L+\frac{1+3s^2{C}_{0\Sigma }{L}_T}{3s{C}_{0\Sigma }+3s^3{C_{0\Sigma }}^2{L}_T}.$$

(6)

So

$$Z_{\Sigma }\left(s\right)=\frac{3{C_{0\Sigma }}^2{L}_T{L}_L\cdot s^4+3{C_{0\Sigma }}^2{L}_T{R}_{\Sigma }\cdot s^3+3C_{0\Sigma }\left(L_T+L_L\right)\cdot s^2+3C_{0\Sigma }{R}_{\Sigma }\cdot s+1}{3{C_{0\Sigma }}^2{L}_T\cdot s^3+3C_{0\Sigma }\cdot s},$$

(7)

where L_L—faulted line inductance, and R_Σ—total resistance of the whole circuit (including both cable line cores and line-to-ground fault resistance).

The waveform of the transient component of the earth current I₀(s) can therefore be determined from the following equation:

$$I_{0T}\left(s\right)=V_0\left(s\right)\frac{3{C_{0\Sigma }}^2{L}_T\cdot s^3+3C_{0\Sigma }\cdot s}{3{C_{0\Sigma }}^2{L}_T{L}_L\cdot s^4+3{C_{0\Sigma }}^2{L}_T{R}_{\Sigma }\cdot s^3+3C_{0\Sigma }\left(L_T+L_L\right)\cdot s^2+3C_{0\Sigma }{R}_{\Sigma }\cdot s+1}.$$

(8)

The nature of the waveforms contained in the transient component will be determined by the values of the roots of the characteristic polynomial, i.e., the solutions of the equation

$$Q\left(s\right)=0,$$

(9)

where the characteristic polynomial Q(s) is

$$Q\left(s\right)=3{C_{0\Sigma }}^2{L}_T{L}_L\cdot s^4+3{C_{0\Sigma }}^2{L}_T{R}_{\Sigma }\cdot s^3+3C_{0\Sigma }\left(L_T+L_L\right)\cdot s^2+3C_{0\Sigma }{R}_{\Sigma }\cdot s+1.$$

(10)

Equation (10) describes a polynomial of order 4 that is difficult to solve analytically. However, it can be noticed that for negligibly small values of R_Σ, Equation (10) takes the form of a biquadratic Equation (11), which is much simpler to solve:

$$3{C_{0\Sigma }}^2{L}_T{L}_L\cdot s^4+3C_{0\Sigma }\left(L_T+L_L\right)\cdot s^2+1=0.$$

(11)

Assuming an auxiliary variable

$$q=s^2,$$

(12)

Equation (11) can be written as

$$3{C_{0\Sigma }}^2{L}_T{L}_L\cdot q^2+3C_{0\Sigma }\left(L_T+L_L\right)\cdot q+1=0.$$

(13)

The solution of Equation (13) are two negative-valued roots q₁ and q₂

$$q_1=\frac{-3C_{0\Sigma }\left(L_T+L_L\right)+\sqrt{9{C_{0\Sigma }}^2{\left(L_T+L_L\right)}^2-12{C_{0\Sigma }}^2{L}_T{L}_L}}{6{C_{0\Sigma }}^2{L}_T{L}_L},$$

(14)

$$q_2=\frac{-3C_{0\Sigma }\left(L_T+L_L\right)-\sqrt{9{C_{0\Sigma }}^2{\left(L_T+L_L\right)}^2-12{C_{0\Sigma }}^2{L}_T{L}_L}}{6{C_{0\Sigma }}^2{L}_T{L}_L},$$

(15)

and after some simplification,

$$q_1=-\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)-\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}},$$

(16)

$$q_2=-\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)+\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}.$$

(17)

Equation (11) will therefore have two pairs of conjugate imaginary roots corresponding to angular frequencies:

$${\omega }_1=\sqrt{\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)-\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}},$$

(18)

$${\omega }_2=\sqrt{\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)+\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}}.$$

(19)

This corresponds to the frequencies of

$$f_1=\frac{1}{2\pi }\sqrt{\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)-\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}}=\frac{1}{\sqrt{2}\pi \sqrt{3C_{0\Sigma }\left(L_T+L_L\right)-\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}},$$

(20)

$$f_2=\frac{1}{2\pi }\sqrt{\frac{2}{3C_{0\Sigma }\left(L_T+L_L\right)+\sqrt{3}{C}_{0\Sigma }\sqrt{3\left({L_T}^2+{L_L}^2\right)-4L_T{L}_L}}}=\frac{1}{\sqrt{2}\pi \sqrt{3C_{0\Sigma }\left(L_T+L_L\right)+\sqrt{3}{C}_{0\Sigma }\sqrt{3{\left(L_T+L_L\right)}^2-4L_T{L}_L}}}.$$

(21)

If an additional simplification is adopted to neglect the faulted feeder line inductance L_L, then Equation (11) takes the very simple form:

$$3C_{0\Sigma }{L}_T\cdot s^2+1=0,$$

(22)

whose solution is a single-frequency waveform with a frequency equal to (corresponding in this case to Equation (21))

$$f=\frac{1}{2\pi \sqrt{3C_{0\Sigma }{L}_T}}.$$

(23)

Neglecting the value of the R_Σ resistance significantly facilitates the solution of Equation (10) but makes it impossible to determine directly the coefficients describing the decay of the oscillatory transient waveform components. However, these decay coefficients can be taken into account as a small perturbation [27,28] of Equation (11).

Once the angular frequencies ω₁ and ω₂ have been determined in accordance with Equations (18) and (19), Equation (11) can be written as a product of linear factors:

$$\alpha \cdot \left(s-j{\omega }_1\right)\cdot \left(s+j{\omega }_1\right)\cdot \left(s-j{\omega }_2\right)\cdot \left(s+j{\omega }_2\right)=0,$$

(24)

where the value of the α factor (introduced to shorten the notation) will be equal to

$$\alpha =3{C_{0\Sigma }}^2{L}_T{L}_L,$$

(25)

expressed in s⁴. Equation (24) can be there simplified to the form

$$\alpha \cdot \left(s^2+{{\omega }_1}^2\right)\cdot \left(s^2+{{\omega }_2}^2\right)=0.$$

(26)

If the R_Σ resistance was included in Equation (10), the transient components of the oscillatory waveforms with frequencies f₁ and f₂, being its solution, would be damped sinusoidal waveforms, and the pairs of conjugated complex roots of Equation (10) would take the general form of

$$\left\{\begin{array}{c}{s}_{1,2}=-{\sigma }_1\pm j{\omega }_1\\ s_{3,4}=-{\sigma }_2\pm j{\omega }_2\end{array}\right..$$

(27)

The polynomial Equation (24) can therefore be written as

$$\alpha \cdot \left(s-\left(-{\sigma }_1+j{\omega }_1\right)\right)\cdot \left(s-\left(-{\sigma }_1-j{\omega }_1\right)\right)\cdot \left(s-\left(-{\sigma }_2+j{\omega }_2\right)\right)\cdot \left(s-\left(-{\sigma }_2-j{\omega }_2\right)\right)=0,$$

(28)

so

$$\alpha \cdot \left(\left(s+{\sigma }_1\right)-j{\omega }_1\right)\cdot \left(\left(s+{\sigma }_1\right)+j{\omega }_1\right)\cdot \left(\left(s+{\sigma }_2\right)-j{\omega }_2\right)\cdot \left(\left(s+{\sigma }_2\right)+j{\omega }_2\right)=0,$$

(29)

and

$$\alpha \cdot \left({\left(s+{\sigma }_1\right)}^2+{{\omega }_1}^2\right)\cdot \left({\left(s+{\sigma }_2\right)}^2+{{\omega }_2}^2\right)=0.$$

(30)

Therefore,

$$\alpha \cdot \left(s^2+2{\sigma }_{1}s+{{\sigma }_1}^2+{{\omega }_1}^2\right)\cdot \left(s^2+2{\sigma }_{2}s+{{\sigma }_2}^2+{{\omega }_2}^2\right)=0,$$

(31)

and, finally,

$$\alpha \cdot \left[\begin{array}{l}{s}^4+2s^3\left({\sigma }_1+{\sigma }_2\right)+s^2\left({{\sigma }_1}^2+{{\sigma }_2}^2+4{\sigma }_1{\sigma }_2+{{\omega }_1}^2+{{\omega }_2}^2\right)+\\ +2s\left({\sigma }_1{{\sigma }_2}^2+{\sigma }_1{{\omega }_2}^2+{\sigma }_2{{\sigma }_1}^2+{\sigma }_2{{\omega }_1}^2\right)+\left({{\sigma }_1}^2{{\sigma }_2}^2+{{\sigma }_1}^2{{\omega }_2}^2+{{\sigma }_2}^2{{\omega }_1}^2+{{\omega }_1}^2{{\omega }_2}^2\right)\end{array}\right]=0.$$

(32)

At the same time, the polynomial Equation (26) can be reduced to the form

$$\alpha \cdot \left(s^4+s^2\left({{\omega }_1}^2+{{\omega }_2}^2\right)+{{\omega }_1}^2{{\omega }_2}^2\right)=0.$$

(33)

Subtracting Equation (33) from Equation (32) yields

$$\alpha \cdot \left[\begin{array}{l}2s^3\left({\sigma }_1+{\sigma }_2\right)+s^2\left({{\sigma }_1}^2+{{\sigma }_2}^2+4{\sigma }_1{\sigma }_2\right)+\\ +2s\left({\sigma }_1{{\sigma }_2}^2+{\sigma }_1{{\omega }_2}^2+{\sigma }_2{{\sigma }_1}^2+{\sigma }_2{{\omega }_1}^2\right)+\left({{\sigma }_1}^2{{\sigma }_2}^2+{{\sigma }_1}^2{{\omega }_2}^2+{{\sigma }_2}^2{{\omega }_1}^2\right)\end{array}\right]=0.$$

(34)

By expanding the polynomial Q(s) into a first-order Taylor series around the points ω₁, ω₂ (i.e., omitting the second- and higher-order terms involving σ₁², σ₂², σ₁, σ₂), we obtain

$$\alpha \cdot \left(2s^3\left({\sigma }_1+{\sigma }_2\right)+2s\left({\sigma }_1{{\omega }_2}^2+{\sigma }_2{{\omega }_1}^2\right)\right)=0.$$

(35)

By comparing the polynomial coefficients appearing in Equations (35) and (10) at the respective powers (s and s³), the following system of equations is obtained:

$$\left\{\begin{array}{c}2\alpha {\sigma }_1+2\alpha {\sigma }_2=3{C_{0\Sigma }}^2{L}_T{R}_{\Sigma }\\ 2\alpha {\sigma }_2{{\omega }_1}^2+2\alpha {\sigma }_1{{\omega }_2}^2=3C_{0\Sigma }{R}_{\Sigma }\end{array}\right..$$

(36)

Therefore,

$$\left\{\begin{array}{c}{\sigma }_1+{\sigma }_2=\frac{3}{2\alpha }{C_{0\Sigma }}^2{L}_T{R}_{\Sigma }\\ {\sigma }_2{{\omega }_1}^2+{\sigma }_1{{\omega }_2}^2=\frac{3}{2\alpha }{C}_{0\Sigma }{R}_{\Sigma }\end{array}\right..$$

(37)

The solution to the system of Equation (37) will be of the form

$${\sigma }_1=\frac{3C_{0\Sigma }{R}_{\Sigma }\left(C_{0\Sigma }{L}_T{{\omega }_1}^2-1\right)}{2\alpha \left({{\omega }_1}^2-{{\omega }_2}^2\right)},$$

(38)

$${\sigma }_2=\frac{3C_{0\Sigma }{R}_{\Sigma }\left(1-C_{0\Sigma }{L}_T{{\omega }_2}^2\right)}{2\alpha \left({{\omega }_1}^2-{{\omega }_2}^2\right)}.$$

(39)

By inserting the α value described by Formula (25) and the ω₁ and ω₂ values described by Formulas (18) and (19) into the solution (38) and (39), we obtain, after some transformations,

$${\sigma }_1=R_{\Sigma }\frac{3L_T-3L_L+\sqrt{9{L_T}^2+6L_T{L}_L+9{L_L}^2}}{4L_L\sqrt{9{L_T}^2+6L_T{L}_L+9{L_L}^2}},$$

(40)

$${\sigma }_2=R_{\Sigma }\frac{3L_L-3L_T+\sqrt{9{L_T}^2+6L_T{L}_L+9{L_L}^2}}{4L_L\sqrt{9{L_T}^2+6L_T{L}_L+9{L_L}^2}}.$$

(41)

For the industrial network considered in the further simulation (ATP/EMTP) studies with the aggregated parameters C_0Σ = 6.048 μF, L_L = 2.9808 mH, L_T = 1.109 mH, and R_Σ = 1.07028 Ω, the above calculations were performed to show the following results:

• f₁ = 2194 Hz and σ₁ = 39∙R_Σ = 41.74 s⁻¹ for the 1st transient component.
• f₂ = 606 Hz and σ₂ = 128∙R_Σ = 137.0 s⁻¹ for the 2nd transient component.

The lower-frequency (f₂) transient component therefore decays more than three times faster than the higher-frequency (f₁) transient component. This phenomenon is shown in Figure 7. The decay of both transient components is, of course, even faster if a higher line-to-ground (R_F) resistance value is present at the ground fault location.

The model considered above is a simplified model, which does not take into account the cable line frequency-dependent parameters (skin-effect phenomenon), and the obtained results are conditioned by the specific nature of the analyzed network—i.e., power supply from an HV/MV transformer with large rated power (and therefore the associated low impedance) and high inductance of the considered feeder cable line (due to its long length and the use of shielded cables). Due to the use of individually shielded cable cores, line-to-line capacitances were neglected in the model under consideration. High values of transient waveform frequencies may affect the operation of modern protection devices using digital signal processing, and knowledge of the decay rate of transient waveforms can facilitate the correct selection of the time delay.

2.3. Illustrative Case Study—An Example Network

A simplified MV network model was used to analyze ground fault disturbances in medium-voltage industrial distribution power grids with an isolated neutral point. The network layout was designed on the basis of several typical MV network layouts of Polish coal mines where ground faults have been recorded in the period 2019–2021. The key aspects of this model and the simulation procedures that were used in the study are presented here.

In the examined model, all the network feeders were assumed to be cable lines, consisting of shielded mining cable type YHKGXSFoyn with a cross-section of 3 × 185 mm². The total length of all cables in the galvanically connected network was 10.8 km, and the ground fault current value was 19.76 A. Fault current simulations were carried out considering a fault in a cable leaving from panel 1 of the DS-1 departmental switchgear.

During the experiments, zero-sequence current i₀(t) and zero-sequence voltage u₀(t) waveform data have been recorded in different switchgear panels in the analyzed network model:

• Main switchgear MS-1, panel 1;
• Main switchgear MS-1, panel 2;
• Departmental switchgear DS-1, panel 1;
• Departmental switchgear DS-1, panel 2;
• Departmental switchgear DS-1, panel 3.

The following assumptions have been made for all the simulations performed in the experiment:

• The single line-to-ground fault was forced for 30 ms after the start of the simulation;
• The phase voltage of the faulty line reached the maximum possible value when the fault occurred;
• The switchgear panel of the line in which the single line-to-ground fault occurred was not disconnected until the end of the simulation;
• In order to analyze the steady-state values of zero-sequence current i₀(t) and zero-sequence voltage u₀(t) waveforms, the simulation lasted 500 ms;
• Regardless of the initial value of the line-to-ground fault resistance, this resistance value was assumed to remain constant throughout the whole duration of the fault simulation;
• A sampling rate of 10⁹ samples per second was assumed in the program; this enabled accurate analysis of higher harmonics in the analyzed network;
• The simulation starts from a pre-calculated steady state, and the 30 ms interval is used only to make the signal waveform plots exhibit a more clear difference between the pre-fault state and the after-fault waveforms. As for the moment of initiation of a short circuit, we assumed the most likely case: the insulation is damaged at the moment when the line-to-ground voltage reaches its maximum value.

For the network layout shown in Figure 8, a set of zero-sequence network voltage V₀ and current I₀ time waveforms have been recorded for faults occurring at selected points in panel 1 of DS-1 switchgear (0, 300, 600, 900, 1200, and 1500 m counting from the beginning of the feeder cable) with fault resistance values equal to 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, and 10,000 Ω for each point, respectively.

The network diagram utilized in the simulation, generated using the ATPDraw graphical preprocessor, is depicted in Figure 9. The subsequent section of this paper provides illustrative results corresponding to the selected length values. These results are representative of typical underground MV distribution networks.

Next, the authors directed their attention toward analyzing the rms values of the selected current and voltage waveforms. An important parameter in this analysis was the duration of the measurement window (integration time). The following 3 specific values of the measurement time have been selected:

• 20 ms—This is the duration of one period from the start of the line-to-ground fault, calculated from the moment the line-to-ground fault occurs. The timing duration is not selected by chance. In the context of a ground fault, most of the transient waveform peaks disappear after the initial half-period of the fundamental frequency waveform. Overcurrent protection relays measure the rms value of the current/voltage waveform after a certain duration. To ensure the correct measurement of the rms value, it must be recalculated after a full period, which amounts to 20 ms for a 50 Hz fundamental frequency.
• 80 ms—This is the most common in the Polish industry; it is the duration of a ground fault before the field is disconnected, counted from the moment of line-to-ground fault occurrence. This stems from the safety imperative in the Polish mining industry, driven by factors such as methane hazards; the safety priority is to disconnect all faults, not only line-to-line but also ground faults, without any delay. It is accepted that the fault should be disconnected in less than 100 ms. Since the fault has to be disconnected faster than 100 ms and the protection algorithm has to assess an integral multiple of the fundamental frequency period, the maximum time window duration under these conditions is 80 ms.
• 100 ms—The time required to determine the rms values of steady-state voltages and currents, taken at 400 ms after the line-to-ground fault initiation. The steady state denotes a voltage and current waveform sufficiently temporally distant from the ground fault initiation, allowing transient waveforms to decay by that time. The selected temporal reference is a trade-off, considering factors such as simulation time and the mitigation of fault-induced noise during transients.
• In summary, for the computation of root mean square (rms) values of steady-state voltages and currents, it was decided to select a time instant 20 periods after the ground fault initiation. The choice of a measurement window duration of 100 milliseconds (equivalent to 5 periods of the fundamental frequency) aims to mitigate the potential effects of an unplanned spike resulting from the numerical simulation method. Integrating data over 5 periods effectively “smooths out” such possible signal disturbances, rendering them negligible for the subsequent analysis.

Making the above assumptions, further studies (including spectral analysis) were carried out for two options:

• Transient state in the network without higher harmonics;
• Steady state in a network with higher harmonics.

In the contemporary era of industrial networks using diverse types of power electronic equipment generating higher harmonics, it may seem useless to analyze the phenomena occurring in the network without taking into account higher harmonics. However, in the case of MV distribution networks, in which only the first harmonic is often present, such an approach simplifies further consideration of ground fault phenomena in this type of network and therefore can be useful as a benchmark for comparative analysis.

In the analysis of the case of an industrial network with sources of higher harmonics, the international standard [29] defining the maximum permissible level of higher harmonics in an industrial network was used (

Table 1. Percentage of higher harmonic components adopted for the simulation analysis of ground faults in an example mining network.

Harmonics Order h	Harmonics Value Relative to VPh1h, %
3	5
5	6
7	5

).

The magnitudes of the above harmonics have been set in accordance with the upper limits specified by the standard. In the aforementioned standard, an additional restriction appears, stating that the average value of higher harmonics of voltages measured during any 10 min interval should not exceed 8%. For the data adopted in the above table, this value is 9.27%; however, it is assumed that in the perspective of the 10 min measurement, the contribution of higher harmonics will not exceed the aforementioned 8%.

3. Results

3.1. Analysis of a Ground Fault Transient in the Model Network without Higher Harmonics

In order to obtain data for the analysis and classification of the types of line-to-ground faults, simulations were conducted using the ATP-EMTP software. The simulations involved inducing a single line-to-ground fault at selected locations in panel 1 of the DS-1 departmental switchgear, according to the network configuration shown in Figure 4. Several sets of network voltage and current waveforms for different fault types (with different resistance values) have been recorded.

The obtained results of measurements of the rms values of currents and voltages in the first 20 ms after the fault occurrence were compared with the rms values for the steady state, which made it possible to compare the phenomena occurring in the network during a ground fault transient and in the steady state. Two single line-to-ground fault locations were selected for analysis: at the beginning and at the end of the outgoing cable line in panel 1 of departmental switchgear DS-1 (for x = 0 m—blue graph and for x = 1500 m—red graph). It is noteworthy that all the other characteristics fall in between these two cases.

Analyzing the results obtained, we focused on the characteristic that determines the ratio of the rms value of the zero-sequence component of the V₀ network voltage after the fault occurrence to the rms value of the voltage of any of the phases of V_Ph before the fault occurrence. This ratio is described by Equation (42).

$$V_0=\beta V_{ph}$$

(42)

As is the case in reference [28], it is a common practice to depict the dependence of the β coefficient value on fault resistance value in the form of linear characteristics, but here, we depict our results on a semi-logarithmic scale. The authors hope that this enhances the clarity of these plots and makes it more convenient to analyze and understand the presented phenomena (such as the classification of ground fault types based on the inflection points of the curves visible on the graphs). Within the framework of this study—covering a wide range of fault resistance values 1 Ω … 10 kΩ—these relationships were presented on a semi-logarithmic scale. On the basis of Equation (43),

$$\frac{V_0}{V_{Ph}}=f\left(R_F\right)$$

(43)

there was introduced a classification of ground faults into three categories:

• Low-resistance faults: Those characterized by a relatively low parameter β, ranging from 0.9 to 1 (for steady state). In the analyzed network, these are assumed to be faults whose line-to-ground resistance is between 0 and 80 Ω.
• Medium-resistance faults: Those characterized by a β parameter in the range of 0.2 to 0.9 (for steady state). In the network, they are assumed to include faults whose line-to-ground resistance is between 80 and 800 Ω.
• High-resistance faults: These are faults that are more difficult to detect and are characterized by a β parameter in the range of 0 to 0.2. For the analyzed network, it is assumed that these are faults whose line-to-ground resistance exceeds 800 Ω.

It is important that the above classification, although found in the reference [30], was modified and extended in this work to suit the needs of the analysis. In broad terms, ground faults in an IT network in a hazardous environment can be categorized into three distinct groups:

• Low-resistive (easily detectable but with a high ground fault current magnitude and associated hazard level);
• High resistive (with a low earth fault current magnitude and a reduced danger level but much more difficult to detect);
• An intermediate group between the two defined above (moderate fault current magnitude, moderate detection difficulty, and moderate hazard level).

The scope of the analysis extends to high resistance values (comparable even to the resistance value of the human body in direct or indirect contact scenarios). Line-to-ground currents then assume fairly small magnitudes, theoretically comparable to leakage currents. However, it should be noted that owing to the specific environmental conditions described in the introduction, the regulations mandate stringent high insulation resistance value requirements (at least 1 kΩ/1 V, depending on the rated network voltage), so earth leakage currents tend to be significantly smaller and, above all, (like balanced load currents) are symmetrical, i.e., there is no resulting zero-sequence component. In contrast, ground fault currents exhibit pronounced asymmetry and are therefore more easily detectable by suitable filters (core balance current transformers) [31].

As evident from the characteristics shown in Figure 10, it can sometimes be problematic to select the settings for zero-sequence component current relay protection efficiently in some special cases. Setting the integration time to 20 ms may result in the healthy line being unnecessarily disconnected for low values of fault resistances in addition to the line in which the ground fault occurred. For instance, in the above example, setting the current–overcurrent protection setting to 4 A in bay 3 of the DS-1 switchgear may result in disconnection of this bay even though the short circuit occurred at the beginning of bay 1 of the same switchgear. In the discussed example, the fault resistance would be only a few ohms.

A partial remedy to this issue involves extending the integration time for the calculation of the root mean square (rms) value. Figure 11 shows the same results as Figure 10, however, with the difference that, in this case, the integration time is not 20 ms but 80 ms. The slight modification reduces the significance of the initial distortions for the individual measurements. With this change in integration time, the possibility of selectively disconnecting only the faulty line is increased. A consequence of this adjustment is a prolonged relay response time. However, as demonstrated in the preceding sections, the protective relays in the vast majority of cases successfully disconnect the faulty line within this extended time frame.

Through numerical simulation experiments and a review of the existing literature, it becomes evident that the root mean square (rms) value of individual parameters within the initial 20 ms of a short-circuit duration is notably affected by the distortion in currents and voltages. Example transient waveforms of zero-sequence currents and voltages are presented in Figure 12. Despite the short time duration, high-magnitude pulses and slowly decaying high-frequency zero-sequence current oscillations are distinctly observable within the first period after the occurrence of a ground fault.

According to the references [25,32,33], a network with an isolated neutral point can be treated as a more complex RLC circuit. In RLC circuits, the occurrence of decaying oscillations is possible after a sudden commutation (stepwise change in structure). These decaying oscillations should be regarded as nonperiodic time functions with finite energy.

Thus, it is established that the transient magnitude spectrum is continuous. To enhance clarity, a decision was made to depict (Figure 13) the amplitude spectrum in discrete form for harmonics from 2 to 41. For ease of comparison, the percentage of the higher harmonics in relation to the first harmonic is presented. To streamline the visual analysis of higher harmonics against the first harmonic, the latter has been intentionally omitted from the individual plots.

Transient ground fault waveforms contain, in addition to the step-changing fundamental frequency component, slowly decaying oscillating waveforms with a much higher frequency—clearly visible in Figure 12 (their frequencies are determined not by the source frequency but by the inductances and capacitances occurring in the network—as described in Section 2.2). Therefore, the continuous wavelet transform (CWT) may become a more convenient method for their analysis than the Fourier transform [34,35,36]. This is related to the fact that the Fourier transform is based on sine and cosine functions characterized by infinite (temporally unlimited) support; hence, it cannot provide sufficient localization of fast-changing components. The wavelet transform, on the other hand, is based on functions with a rapidly decaying (time-constrained) support and therefore allows a much more accurate analysis of the variation in the individual components of the relay protection criterion signal. Hence, in order to delve deeper into the electromagnetic transient processes described here, a continuous wavelet transform based on analytical Morse wavelets [37,38] was used in our case study. The continuous wavelet transform makes it possible to watch the fluctuation of the individual components over shorter periods of time.

Instead of using a single (but complex) function of one variable (frequency) ω based on complex exponential support, like in a classical Fourier analysis—or alternatively two orthogonal real-valued functions based on sine and cosine support functions of one variable (frequency) defined over the whole analyzed signal range—the continuous wavelet transform (CWT) allows us to describe a signal with a function of two variables t, a:

$$W(t,a)=\frac{1}{\sqrt{a}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}S(t)\cdot \psi \left(\frac{\tau -t}{a}\right)} d\tau ,$$

(44)

where ψ(t) denotes the so-called mother wavelet, i.e., basis function fulfilling certain mathematical conditions [39] (e.g., finite energy, limited both in time and in frequency space), a is a scale (basis function dilation or constriction—corresponding to frequency change) parameter, and t is a translation (time shift of the analysis window) parameter. This signal description has a local character (valid within the area defined by the time-constrained support of the wavelet applied), thus enabling more accurate analysis of the variability of individual frequency components over shorter time segments. Local matching of wavelet and signal leads to a large transform value.

A wavelet analysis of the transient and steady state was performed using the Matlab Wavelet Toolbox [40]. The outcome of this analysis (scalogram) is depicted in Figure 14.

It can be seen that in the initial (transient) time interval, the contribution from higher harmonics becomes noticeable, and they decay to a negligible magnitude after the period of approx. 40 ms. On the other hand, the 50 Hz fundamental component reaches its final value after approx. 50 ms, which, taking into account the additional time required to measure its rms value, gives a tripping time of 70–90 ms.

It is obvious that at relatively high fault resistance values (R_F > 100 Ω), the percentage of higher harmonics in the current waveform is small; hence, it was consciously decided to omit spectral analysis for high-resistance faults. The first noticeable fact that can be observed is the correspondence of the percentage spectral characteristics of the currents both qualitatively and quantitatively. This makes it possible to determine the spectral character of the fault current on the basis of the selected recorded zero-sequence current I₀ waveform. It was known that for small fault resistances, the higher harmonics would have a significant effect on the I_F current waveform during the initial stage of the fault transient. However, for low R_F values, it is possible (in some cases) that the magnitude of higher harmonics may reach values that are even twice as high as the magnitude of the first harmonic of the zero-sequence current.

What is also important is the fact that the maximum of this value is achieved for the 21st harmonic. Typically, modern digital protection relays are used in mining switchgear samples and record voltage and current waveforms with a sampling frequency of between 1000 and 1600 samples/s. At a fundamental network frequency of 50 Hz, the 21st harmonic frequency is 1050 Hz. This means that it is impossible to efficiently take into account the 21st harmonic value when measuring I₀ and V₀ in the network, which in some extreme situations may lead to disconnection of the healthy cable line in addition to the damaged cable line. As the fault resistance value increases, the share of higher harmonics decreases, and already, when the fault resistance value R_F = 100 Ω, this share is much smaller than 50% of I_1h.

It is noteworthy that the optimal sampling frequency for transient-based digital ground fault protection relays is recommended to be 100 kHz (or at least 20 kHz), as suggested in reference [25].

In the case of zero-sequence voltage V₀ for a ground fault resistance value in the range from 1 to 80 Ω, the content of higher harmonics only once exceeds 10% of V₀—namely for the 21st harmonic when the line-to-ground resistance value R_F = 1 Ω. In the ground fault resistance range between 100 and 500 Ω, the proportion of higher harmonics increases sharply (between the second and tenth harmonics). The voltage magnitude spectrum envelope appears as a homographic function. For higher fault resistances, it can be assumed that the envelope of the zero-sequence voltage V₀ spectrum has the shape of a hyperbola, but the inflection point of this hyperbolic envelope function occurs between the first and second harmonics, so the contribution of higher harmonics in this situation is not significant.

3.2. Steady-State Analysis of a Ground Fault in the Model Network Containing Higher Harmonics

This section of the paper is dedicated to the possible presence of higher harmonics in the network. It was assumed that, according to the standard [29], higher harmonics of voltages can occur in the industrial network. The maximum permissible contribution of higher harmonic voltage values was assumed in accordance with Table 1. The relevant standard imposes two conditions: the first states that none of the harmonics presented may, at any moment in time, exceed 5, 6, and 5% of the contribution of the first harmonic, respectively. The second condition implies that, in any 10 min interval, the total contribution of the higher harmonics must not exceed 8%. The second condition is put this way because in some cases (e.g., the start-up of a hoisting machine fed from a thyristor converter), the 8% condition in a 1 min interval is impossible to meet, but extending it to a 10 min interval is already within the norm. Therefore, in order to consider a worst-case scenario, it was considered that a ground fault occurred at a moment in a short interval when the contribution of higher harmonics was more than 9%, but in a 10 min interval, their contribution would be less than the required 8%. In mining facilities, thyristor-converter-powered hoisting machines are the dominant source of harmonic interference in the mine’s medium-voltage network [41,42]. They are equipped with high-power DC motors powered by controlled (usually 6-pulse or 12-pulse) thyristor rectifiers [43]. Machine speed control is achieved by varying the firing angle of the thyristors, according to a so-called travel diagram, which takes into account the allowable limits of acceleration and deceleration and the speed stabilization of the steady-state travel section. Hence, the greatest disturbances are generated during the machine’s starting and braking [44]. It is the relatively high motor power and the short, but frequently repeated, high-intensity load cycles that cause the hoisting machinery converter drives to have such a high impact on the operating conditions of the mine’s power distribution network [45,46]. Similar problems (however, without such an impact on the environmental safety of the network operation, as described in the introduction) exist in rectifier substations, supplying electrical traction with diverse and rapidly varying loads [47,48,49].

Ground fault simulations have been conducted for the same cases as outlined in Section 3.1, considering the influence of higher-order harmonics in the supply voltage. Example transient waveforms of zero-sequence currents and voltages are depicted (for the case of R_F = 20 Ω) in Figure 15. Heavy waveform distortion is evident not only during the initial period (like in Figure 12) but even in the steady state. This phenomenon may disturb the correct operation of ground fault protection relays and should therefore be taken into account when selecting their settings. An overshoot of the criterion value in the first 20 ms may lead to nonselective tripping of the ground fault overcurrent protective devices, especially when their settings are selected on the basis of the analysis of the earth fault current determined in the steady state.

Comparisons of the rms values of selected parameters for the transient state (assumed as the first 20 ms of the fault duration) and the steady state are presented in Figure 16 and Figure 17. In order to ensure greater clarity, the relationship I₀ = f(R_F) is included in the charts below only for two cable lines outgoing from MS-1 panel 1 and DS-1 panel 1. These lines were chosen because they constitute a direct connection between the transformer supplying the network and the fault site. In addition, the relationship describing zero-sequence current and voltage components I_F = f(R_F) and V₀ = f(R_F) is provided. It can be seen that the rms value of the zero-sequence voltage V₀ depends on the higher harmonics to an almost negligible extent. Therefore, it can be clearly stated that higher harmonics should not affect the operation of zero-sequence voltage protection. However, in the case of zero-sequence currents, it is slightly more important.

At low ground fault resistance values, individual currents in the network with a high content of higher harmonics can reach values up to 15% greater than in the situation without higher harmonics. When ground fault protection is set according to the I₀ > criterion and a small value integration time is used, situations may arise in which the selectivity of ground fault clearing is not ensured—an undamaged cable can be switched off unnecessarily. In order to avoid such a situation, it is recommended in the literature to extend the measurement time over which the waveform will be integrated to values longer than 1 period (e.g., 40 or 60 ms) and to use a higher harmonics filter.

Both methods have their drawbacks. Increasing the integration time increases the tripping time of the protective relay. On the other hand, the higher harmonic filter does not take into account the fact that, in transient time, the distortions are nonperiodic waveforms; hence, the frequency spectrum is not discrete but continuous. In some extreme situations, even the application of the above solutions may still not be sufficient.

Spectral analysis of the same signals as in the previous graphs (Figure 18) for steady state confirms the observations made in those plots. Nevertheless, it is noteworthy to examine the influence of fault resistance on the higher harmonics’ magnitudes. For low-resistance fault values, a certain relationship can be observed, which is best described by Equation (45):

$$\frac{I_0^{kh}}{I_0^{1h}}\approx \frac{V_0^{kh}}{V_0^{1h}}\cdot k$$

(45)

where k—harmonic order, I₀^kh—kth-order harmonic of the zero-sequence current at the given location, I₀^1h—first-order harmonic of the zero-sequence current at the given location, V₀^kh—kth-order harmonic of the zero-sequence voltage in the galvanically connected network, V₀^1h—first-order harmonic of the zero-sequence voltage in the galvanically connected network.

For example, it was assumed that the third harmonic of the phase voltage V_ph^3h before the occurrence of a ground fault was 5%. It turns out that for small ground fault resistance values, the third harmonic of the zero-sequence current (I₀^3h) constituted about 15% of the first harmonic’s value regardless of the measurement point. As the value of R_F increases, the percentage of higher harmonics in the fault current waveform decreases.

In order to highlight steady-state and transient spectral differences between the line-to-ground fault in the network with and without higher-order harmonics in the supply voltage, a differential scalogram has been calculated (difference between scalograms from Figure 14 and Figure 19), as depicted in Figure 20. It can be seen that the difference in the transient period is quite small and quickly decaying, but it still exists in the steady-state period and covers a fairly wide frequency range (several hundred Hz); however, the maximum value of this difference is several percent of the fundamental harmonic value. Thus, analysis of a model MV industrial network (designed to emulate a real coal mine network) has revealed that transients and the contribution of higher harmonics can be important factors influencing the operating conditions of ground fault protection relays in this network. A comparison of the zero-sequence current and voltage waveforms of the criterion values for the ground fault protection relays for the case of a network without higher-order harmonics and for a network with a significant proportion of harmonics in the supply voltage is shown in Figure 21.

4. Conclusions

The spectral (for steady state) and continuous wavelet (for transients) analyses indicate that increased levels of higher harmonics in the zero-sequence current waveform can cause nonselective operation of the ground fault protections, leading to unnecessary tripping of undamaged cable feeder lines and related severe disturbances in the plant operation. Due to the fact (described in Section 2) that, for zero-order sequences, the impedance of the insulated neutral point MV network is capacitive in nature (defined mainly by the values of the line-to-ground capacitances, with a relatively minor contribution from series resistances and inductances), the value of this impedance decreases significantly for higher harmonics of the network voltage. Owing to this phenomenon, even a minor contribution of higher harmonics to the line voltage can result in a large proportion of higher harmonics in the zero-sequence current measured and analyzed by the ground fault protection relays. This problem can become even more significant when there is a temporarily increased contribution of higher harmonics, e.g., in the frequently occurring dynamic states of high power drives supplied by thyristor converters. There has been also presented an innovative two-frequency analytical model of the transient component for a ground fault in an insulated neutral network. A novel derivation of the frequencies of the transient components and their decay coefficients has been described. This model is better suited to the specificities of mining industrial networks with long cable lines, shielded cables, and high-power and low-impedance power transformers. The problem of current waveform distortion poses a challenge to the functionality of ground fault protective relays based on both overcurrent and admittance criteria, so it would be advisable to increase the integration period for calculating the rms values or to incorporate harmonic filters into measurement circuits. The wavelet methods (particularly CWT used in our paper) are not applied directly to the design or setting of the relay protection, but they can help in understanding and analyzing the phenomena occurring during a ground fault and in the description and visualization of criterion signal waveforms.