Small-current grounding fault location method based on transient main resonance frequency analysis

1 Introduction

Single-phase-to-ground faults in arc suppression coil grounding distribution networks are typical weak-feature faults.When the fault occurs, the fault current is only several amperes because of the compensation of the arc-suppression coil.The small-current grounding faults has small steadystate zero-sequence current and indistinguishable features.Accurate small-current grounding fault location methods are still a serious problem.

Compared with the transmission network, the distribution network has the characteristics of complex network structure, short lines, and numerous branches.In addition, traditional fault location methods are limited in line-cable hybrid lines and multi-branch lines.

At present, the research on fault location of transmission grids is relatively mature, and the fault location of smallcurrent grounding distribution networks is mostly extended by the transmission grid location methods.According to the adopted data, existing location methods can be divided into two categories:steady-state-data-based methods and transient-data-based methods.

The impedance-based method is one of the most typical steady-state-data-based methods [1-6].Its main principle is to determine the fault location by assuming that the parameters of transmission line are uniform and the circuit impedance of fault line is proportional to the distance length of fault line.But the performance of impedance-based method is greatly affected by the line parameters, fault resistance, and pre-fault power flow [7, 8].Another type of steady-state-data-based methods is voltage-distribution based method.According to the different principles, it can be divided into two types.The first principle is that the fault point has the lowest voltage of the whole line.Through finding the point of lowest voltage, the fault can be located [9, 10].Another principle is the voltages calculated from both sides are the same.So the fault location can be determined by calculating the voltage from both ends of line[11-13].

For the transient-data-based fault location methods, the traveling-wave-based methods are the most typical type.The traveling-wave-based methods achieve fault location by adopting the features of traveling wave refraction with combining the wave velocity and transmission time.Specifically, it can be divided into single-end travelingwave-based methods and double-end traveling-wave-based methods [14-15].Reference [16]proposed a double-end traveling-wave-based method to locate the fault by selecting the first three differences according to the sequence of time of traveling wave head reaching the line end.To solve the problem of different wave velocities of hybrid lines,a normalization algorithm of wave velocities is proposed[17].Some methods achieve fault location based on the relationship by combining the characteristic frequencies of fault transient traveling waves, wave velocities, and the location of fault points [18-20].The traveling-wavebased methods have accurate location results when the measurements are accurate.But its application is limited by the short length of lines in distribution networks and expensive cost.

The medium and low frequency transient information of small current grounded distribution system is abundant.The transient magnitude can reach several times of the steady state magnitude, and the transient is not affected by arc suppression coil compensation, which contains abundant fault information.Another type of transient-data-based fault location method is based on transient feature extraction of corresponding frequency.The relationship between frequency of transient feature and fault distances is firstly proposed in [21].According to the fault waveform, a selfadaptive modified mother wavelet function is established to improve the accuracy of fault location method [22].In these years, the development of distribution-level phasor measurement units (D-PMUs) brings high sampling time-stamped waveform data, which makes transient synchronized information available in distribution networks[23].Some D-PMU-based fault location methods have been proposed [24].For the better application of features,machine learning algorithms have also been adopted in recent researches [25-27].

In this paper, we proposed a novel transient-databased fault location method.Through the main resonance frequency analysis, the frequency bands including the most transient information are determined with considering relative influences.Then, the double-end wavelet transient energy ratio under the corresponding frequency bands is selected as the feature.To fit the non-linear relationship between the extracted features and fault distances, a back propagation neural network (BPNN) is introduced here.Meanwhile, the application of the proposed method in hybrid lines and multi-branch lines is considered and analyzed.The performances are verified by a 4-feeder arc suppression coil grounding distribution network is simulated in PSCAD/EMTDC.

2 Transient main resonance frequency analysis

In this section, we analyze the transient main resonance frequency and its influences.For an RLC series circuit, as shown in Fig.1, the frequency of the sinusoidal voltage source US is variable.R, L, C are corresponding to resistance, inductance, and capacitance, respectively.The equivalent impedance of the port in the above loop can be expressed as follows:

where ω is the angular frequency, and X is the equivalent reactance.Because the frequency effects of the inductor and capacitor are opposite, there must be a frequency ω0 at which X =0 in the above equation.The inductive reactance value in the loop is equal to the capacitive reactance value.In other words, the inductance is equal to the capacitance in this circuit.At this moment, the entire circuit exhibits a purely resistive state, which is defined as the resonance state of the circuit.

Fig.1 RLC series circuit

The equivalent circuit of transient analysis after singlephase small-current grounding fault is shown in Fig.2.R=2(Rs1+RL1)+RL0+3Rk, where Rs1 and RL1 represent the linemode loop resistances,RL0 is the zero-mode loop resistance,and Rk is the fault resistance.L=2(Ls1+LL1)+LL0 represents the sum of the line circuit inductance and the zero-mode loop inductance.Where,Ls1 and Rs1 are the inductance and resistance of the source.LL1 and RL1 are the inductance and resistance of line between fault point to the source.Lp represents the inductance of the arc suppression coil.

Fig.2 Equivalent circuit for the single-phase ground fault transient analysis of a low-current grounded system

When the fault ground resistance is relatively low, and the influence of the arc suppression coil is not considered,the transient process is composed of the energy conversion between L and C.During this period, several resonance states are established, and the transient current can be expressed as follows.

where ICm is the amplitude of capacitance current and φ is the angle of fault current and supply voltage.δ is the decay factor, which is equal to R/2L.ωf is the free oscillation angular frequency of the transient current, which represents a series of frequency values corresponding to the transient current harmonic values.Among the many frequencies, there is a resonance frequency with the highest current energy in the low-frequency band, namely the main resonance frequency.Its calculation formula is as shown in (3).

For a metallic fault,ωf can be simplified as follows.

where Lu0,Lu1, and LT1 are the pre-unit line zero-mode inductance, metallic-mode inductance, and transformer's metallic-mode inductance, respectively.lf is the distance between the fault and the bus line, and C0Σ is the ground capacitance of all the lines.

From the above equation, we find that the main resonant frequency is related to the resistance, capacitance,and inductance of the equivalent loop.In the equivalent loop, the inductance is related to the fault distance, the capacitance is related to the fault distance and the systemto-ground capacitance, and the resistance and fault distance are transition resistance is related.Therefore, the main resonant frequency is determined by the fault distance, the system-to-ground capacitance, and the transition resistance.The farther the fault distance, the lower the main resonance frequency.When the transition resistance is relatively low, the influence can be neglected.When the transition resistance is high enough, the main resonance frequency will decrease accordingly.

The amplitude corresponding to the main resonant frequency component can be expressed as follows:

Clearly, the main resonant frequency component is affected by the fault distance, system-to-ground capacitance,transition resistance, and fault initial phase angle.

3 Double-end wavelet energy ratio

3.1 Wavelet transform and wavelet transient energy spectrum

Wavelet transform is an effective transient information extraction algorithm that can effectively solve the problem of non-stationary signal analysis.At different resolutions,the details of a signal can help characterize different physical structures.The low-resolution details can be used to characterize large structures, and with the increase in the resolution, finer details can be obtained.Through the wavelet transform, the feature information in the full frequency domain can be obtained [28-30].However, a continuous wavelet transform requires a large amount of computation and is unsuitable for practical applications.Although discrete wavelet transform effectively solves this problem, spectral aliasing is observed between the components, which limits the accuracy of the signal analysis.In 1988, Mallat combined existing research and explained the orthogonal wavelet through a mathematical function analysis.A multi-resolution analysis was proposed for the first time.The multi-resolution analysis is based on the function space concept, and the signal is in different levels of space.The basic idea behind decomposition, i.e.,showing a signal with different resolutions, is the use of functions of different scales to continuously approximate the original signal, so this method is also called the multiscale analysis [31].Moreover, Mallat proposed a general construction method for orthogonal wavelets and a fast algorithm for orthogonal wavelet transform, namely the Mallat decomposition algorithm.The Mallat decomposition algorithm, also known as the tower algorithm, involves decomposing a signal through a high-pass filter and lowpass filter layer-by-layer to obtain a series of high-frequency and low-frequency components.This process is equivalent to dividing time-domain signals into frequency bands in the frequency domain.The specific formulae are as follows.

where f(t) is the original signal,j is the number of decomposition layers, and H and G are the wavelet lowpass filter decomposition coefficient and high-pass filter decomposition coefficient in the time-frequency.Aj is the approximation coefficient, which represents the corresponding low-frequency components, and Dj is the detail coefficient corresponding to the high-frequency components.The signal is subdivided layer-by-layer to obtain the characteristic information of the signal in the different frequency bands.

The energy of a signal is expressed as the sum of the squares of the amplitudes of the signals at each point.According to the law of conservation of energy, the energy of a signal in the time domain is equal to its energy in the frequency domain.Therefore, the frequency-domain energy after wavelet transform can also fully represent the signal characteristics.Let the wavelet transform coefficient of the jth layer of the original signal be Wj(k).Then, the wavelet energy of the signal at this scale is defined as the integral of the square of Wj(k) along the time axis:

where N is the sampling window width of the signal.The wavelet energy spectrum of the discrete signal is E =[ E1,E2,…Ej…].After wavelet decomposition, the wavelet energy of j different frequency bands can be obtained.The wavelet energy values of the different frequency bands indicate that the signal is proportional to the total energy in the frequency band.

3.2 Analysis of double-end wavelet energy ratio

The main resonance frequency and its components are affected not only by the fault distance, but also by the fault resistance and fault initial phase angle.Under different fault resistances and fault initial phase angles, the difference in the wavelet energy is relatively large.To avoid the influence of fault resistance and fault initial phase angle on the wavelet energy, the ratio of the two-terminal energy of the fault line is used as the characteristic quantity.

Fig.3 shows the variation in the energy ratio of a singleend wavelet to a double-end wavelet with the fault distance at a certain scale under different transition resistances.Figs.3(a) and (b) show that the zero-mode voltage wavelet energy at the head and at the end of the line varies with the fault distance and fault resistance, respectively.The singleend voltage wavelet energy varies with the fault distance.Moreover, the wavelet energy increases at a higher rate with the decrease in the fault resistance.Fig.3(c) shows the variations in the energy ratio of the wavelets at the head and at the end of the line with the fault distance and resistance.After the two-terminal ratio calculation, the wavelet energy ratio varies with the fault distance;however, it varies only slightly under different fault resistances.Therefore, the ratio of the two-terminal wavelet energy can help eliminate the influence of fault resistance on the wavelet energy.

Fig.3 Variations in the ratio of single-end wavelet energy to double-end wavelet energy with the fault distance under different fault resistances

Similarly, we obtain the variations in the ratio of singleend wavelet energy to double-end wavelet energy of the line at different initial fault angles, as shown in Fig.4.

In summary, when a fault occurs at different locations in a low-current grounded system, the influences of fault resistance and fault initial phase angle can be eliminated by the energy ratio of the two-terminal wavelet.Therefore,we selected the double-end wavelet energy ratio at a scale corresponding to the main resonance frequency as the fault feature.Moreover, we adopt the neural network to fit the nonlinear relationship between the wavelet energy ratio and the fault distance.Thus, the low-current grounded faults can be located.

Fig.4 Variation in the ratio of single-end wavelet energy to double-end wavelet energy under different fault angles

4 Low-current grounding fault location method

4.1 Instruction and parameter settings

To fit the nonlinear relationship between the wavelet energy ratio and the fault distance, the BPNN is selected in this study.The basic idea of the BPNN is to learn certain rules through the training of its own network when there is no definite mathematical mapping equation relationship between the input and output, and obtain the result closest to the expected output value at a specific input value.The basic algorithm is the gradient descent method, which uses the gradient search technique to minimize the mean square error of the actual output value and the expected output value of the network [32].

There are two main factors when training the BPNN:the quality of the extracted features and the setting of the training scenarios.In this study, the extracted features are double-end wavelet energy ratios.To extract the key transient features, the wavelet parameters should be carefully considered.

In this study, “db3” is selected as the mother wavelet based on the experimental test.The number of layers of the wavelet decomposition can be determined from the equation below.

where f is the sampling rate of the waveform.The sampling rate of the waveform data provided by D-PMUs can reach 10 kHz.Therefore, the wavelet decomposition layer is set as 5 under this sampling rate.Fig.5 shows a schematic of the corresponding frequency bands at each scale obtained by the decomposition of the five-layer wavelet transform.Evidently,after the five-layer decomposition, the range of the main resonance frequency is included, and the fault information in the main resonance frequency band is extracted.

Fig.5 Schematic of five-layer wavelet transform decomposition

To analyze the effect of the time window, the spectrum diagrams corresponding to zero-mode currents in the 1 cycle,3/4 cycle, 1/2 cycle, and 1/4 cycle are obtained, as shown in Fig.6.The setting of the time window can only influence the amplitude, not the transient main resonant frequency.In this study, to ensure that the transient information is included, we selected 1 cycle as the time window.

Fig.6 Zero-sequence current spectrum under different time windows

To determine the chosen layers for fault location, the spectral range corresponding to the transient zero-mode current of the entire line should be calculated.Subsequently,the energy ratios under the decomposition layers that cover the spectral range are selected as features.

With an overhead line as an example, Fig.7 shows the transient zero-sequence current spectrum diagram corresponding to different fault locations in the entire line.The main resonance frequency range is between 500 Hz and 1500 Hz, corresponding to the cD2 and cD3 decomposition layers.

Fig.7 Zero-sequence current spectrum of overhead line faults

To verify the effect of the number of layers, the location errors when using different decomposition layers are compared, as shown in Fig.8.As illustrated, the more the number of layers selected, the better the positioning effect.However, an excessive number of layers does not make the fault positioning effect any better.This is because the transient main resonant frequency includes most of the temporary information in the state component, which can represent the basic characteristics of the transient process.When the number of layers is too low, the fault information cannot be fully represented, and the positioning error is very high.When the two-layer wavelet energy ratio is selected as the input sample, the location error is within 500 m.Therefore, the cD2 and cD3 two-layer wavelet energy ratio is selected as the input sample for the neural network.

Fig.8 Location error when using different layers of wavelet energy ratio

4.2 Application of complex line structures

A distribution network contains many hybrid lines(combination of overhead lines and cables) and multibranch lines.In this section, the applicability of the proposed method is analyzed.

Fig.9 Hybrid line fault

Fig.9 shows the fault diagram of the overhead-cable hybrid line.The lengths of the overhead lines and cables are l1 and l2, respectively, and the fault distance is x.After fault occurrence, the spectrum of the zero-sequence current overlaps because of the two different parameters of the overhead line and cable, and the spectrum of the zero-mode current contains two frequencies with high amplitudes.By combining the two frequencies, we can obtain the comprehensive frequency band, and calculate the wavelet energy ratio at the corresponding scales.At this time, the nonlinear relationship between the double-end wavelet energy ratio and the fault distance is a segmentation function.

In the above equation, the piecewise function F is continuous at the overhead line-cable mixing point x =l1.Therefore, if the training samples are sufficient, the neural network can be used to fit the piecewise function.In other words, the proposed method can also be applied to the hybrid lines.

Fig.10 Multi-branch line fault

Fig.10 shows a schematic of a fault occurring in a distribution network with branch lines.The lengths of the three lines are l1,l2, and l3, and the fault may occur at any point on the three lines.The measuring device is installed at three points M, N, and P, and the wavelet energy ratio of MN, NP, and PM is constructed as the location criterion of the line.The corresponding functional relationships are set as F1,F2, and F3 respectively.The corresponding neural networks are trained, and the fault distances x1,x2, and x3 are located using the three networks, respectively.x1,x2,and x3 represent the distances from the fault location to the corresponding ends.Based on the relationship between x1,x2,x3 and l1,l2,l3, the fault branch and the positioning result can be determined.

For the l1 line fault, the network associated with the branch is F1 and F3.The fault distance obtained by the F1 should be less than the length of the l1 branch.The fault distance obtained by F3 minus the length of the l3 branch should be less than the length of the l1 branch and greater than zero.In other words, the network F1 and F3 positioning results satisfy the relationship:

When determining whether a fault occurs on line l1, the location distance from the M terminal is taken as the average value of the location results of F1 and F2 on line l1.

Similarly, any low-current grounded faults occurring at the other lines can also be located.Table 1 presents the location result determination table.

Table 1 Location result decision table

Fault line l1l2l3 Branch decision x l0＜-＜1 1 x l＜2 2 x l＜-＜＜3 3 x l l 3 3 1 x l l 1 1 0＜-＜x l l 2 2 3 Location average x x l x l+--2( )1 1 3 3 x l x l( )x l x l( )+--2 2 2 1 1+--2 3 3 3 2 Distance from M-terminal d=x d l l x=+-1(2 ) d l l x=+-1 (3 )

5 Simulation and verification

To verify the proposed method, a 10 kV four-feeder arc suppression coil grounded distribution network is simulated in PSCAD/EMTDC, as shown in Fig.11.L1 is an 8 km overhead line,L2 is a 6 km cable line, and L3 is a hybrid line with a 5 km overhead line and a 3 km cable line.L4 is a 15 km overhead multi-branch line, where l1,l2, and l3 are 6, 5, and 4 km, respectively.The parameters of the lines are shown below.For the overhead lines, the positive-sequence impedance is Z=0.034+j0.286 Ω/km, and the positive-sequence admittance is Y=j0.406×10-5 S/km.The zero-sequence impedance is Z=0.286+j1.421 Ω/km, and the zero-sequence admittance is Y=j0.155×10-5 S/km.For the cable line, the impedance matrix and admittance matrix are shown below.

The setting of the training scenarios for the BPNN is shown below.

(1) Fault resistance:0.01, 10, 25, 50, 75, 100, 200, 400,and 600 Ω.

(2) Fault angle:0, 10, 20, 30, 45, 60, 75, and 90°.

(3) Fault distance:Starting from the beginning of the line, we set a fault point every 0.5 km.

Fig.11 Simulation topology of a low-current grounded system

5.1 Fault location results for the overhead line

Fig.12 shows the verification result of the fault locations in the overhead line.The location results are represented by “+,” and the errors are shown in the corresponding bar chart.The location errors of the proposed method in the different positions of the overhead line are within 500 m.

Fig.12 Location results of over-head line under different fault distances

To verify the effect of fault resistances, we set the fault distance 3 km away from the bus, and the fault initial angle is set to 90°.The fault resistances are set as 0, 20, 40, 60, 80,150, 300, and 500 Ω.Fig.13 shows the verification results.The errors under the different fault resistances are all within 500 m.The proposed method is less affected by the fault resistances.

Subsequently, the location results under different fault initial angles are verified.The fault initial angles are set to 0, 5, 10, 15, 50, 70, and 90°.The fault location is 6.2 km,and the fault resistance is 10 Ω.Fig.14 shows the location results.Clearly, when the initial phase angle of the fault is small, the location error is relatively greater than the large fault initial angle.The errors remain within 500 m.Thus,the proposed method can accurately locate faults without getting affected by the fault initial angle.

Fig.13 Location results of over-head line under different fault resistances

Fig.14 Location results of over-head line under different fault angles

5.2 Fault location results for the cable line

Similarly, we verified the proposed method for the faults in the cable line.Fig.15 shows the fault location results.As shown, the location errors are within 300 m, thus demonstrating the performance of the proposed method for cable lines.

5.3 Fault location results for the hybrid line

Fig.16 shows the fault location results for the hybrid line.The location errors are all within 300 m, and the proposed method has good performance at positions near the hybrid points.

Fig.15 Location results of cable line under different fault distances

Fig.16 Location results of hybrid line under different fault distances

5.4 Fault location results for a multi-branch line

Finally, the location results for the multi-branch line are verified, as listed in Table 2.The error in the location results under the different positions is less than 600 m.In conclusion, based on the method introduced in Section 4.2,the proposed method can accurately determine the branch line and locate the faults.

Table 2 Location results when faults occur in different positions with branch lines

Fault branch and distance from the fault point to the M end/km Location error/km L1, 0.3 L1 0.089 0.211 Branch result Location result/km

continue

Fault branch and distance from the fault point to the M end/km Branch result Location result/km Location error/km L1, 1.5 L1 1.673 0.173 L1, 4.5 L1 4.904 0.404 L1, 5.7 L1 5.307 0.393 branch point, 6.0 L2 5.841 0.103 L2, 6.3 L2 6.390 0.090 L2, 7.5 L2 8.027 0.527 L2, 10.7 L2 10.397 0.303 L3, 6.2 L3 6.202 0.002 L3, 8.5 L3 8.021 0.479 L3, 9.7 L3 9.100 0.600

6 Conclusion

In this study, we developed a low-current grounded fault location method.The analysis of the transient main resonance frequency showed a correlation between the transient main resonance frequency and the fault distance.With this information, the faults in the system could be located accurately through the double-end wavelet energy ratio corresponding to the main resonance frequency and the BPNN.The verification results showed that the proposed method is less affected by the fault distances, fault resistances, fault initial angle, and line structures.

The proposed method can locate the fault accurately,but it needs sufficient historical data, which may restrict its application.In practice, a practical alternative is to establish an online training mechanism.The initial training data of the fitting model are obtained from simulation or laboratory platforms.Then, during the operation of the distribution network, the training samples are continuously revised through data of actual cases.Another feasible method is training the fitting model by integrating data of multi distribution networks.But this method needs to consider the feature scales and communication.Further research will focus on solving this problem.The proposed method will be relatively complex when the line has a number of branches.The fast branch determination method is worth more attention.

Acknowledgements

This work was supported by National Key R&D Program of China (2017YFB0902800) and Science and Technology Project of State Grid Corporation of China(52094017003D).