It is difficult to obtain the current waveform of series arc faults from actual distribution wires because of the uncertainty of the occurrence time and location of series arc faults. This article sets up a series of arc fault simulation experiment environments, which are composed of a power supply, series arc generator, signal acquisition module, and loads inspection (General Administration of Quality Supervision, 2014). The schematic diagram of the series arc fault simulation experiment is shown in Figure 1.

In this article, an arc generator is chosen to simulate the generation of arc faults. The series of arc generator is mainly composed of two electrodes. One electrode which is regarded as a mobile electrode is a carbon-graphite rod with a diameter of 6 ± 0.5 mm. The arc burning end of the electrode is made into a tip and equipped with a sliding block. The clearance between the two electrodes can be controlled by adjusting the horizontal adjusting knob. The other can be a 6 ± 0.5-mm-diameter copper rod set as a fixed electrode. The arc ends of both electrodes should be kept clean to allow for repeatability of arcing. The two electrodes are connected in series by wire, with one end connected to a load and the other to the power supply. A stable arc can be formed by adjusting the horizontal adjustment knob so that the two electrodes are separated at proper distances. The schematic diagram of the device is shown in Figure 2. The physical map is shown in Figure 3.

The signal acquisition module is composed of a current transformer and a filter amplifying circuit and is responsible for collecting arc current signals. The current is converted into a voltage signal through a current transformer and a sampling resistor, then filtered and amplified by the circuit, and finally, the current signal is sampled using an oscilloscope. For the acquisition of the current signal, the low-frequency and high-frequency mutual inductors are used to collect the low-frequency and high-frequency current waveforms, respectively. The low-frequency mutual inductor collects the low-frequency current and outputs the low-frequency current component signal through the low-pass filtering and amplifying circuit. The low-pass filtering circuit consists of an RC low-pass filter. The cut-off frequency is configured according to 1/2πRC to about 1 kHz. The high-frequency mutual inductor collects the high-frequency current and outputs the high-frequency current component signal through the high-pass filtering and amplifying circuit. The high-pass filtering circuit consists of an RC high-pass filter, and the cut-off frequency is configured to be about 1 kHz.

According to GB/T31143-2014 “General Requirements for Series Arc Fault Detection Device (AFDD)" issued by the General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China in 2014, it is stipulated that AFDD must meet the inhibitory load shielding test. Seven shielded loads are specified in the standard; they are vacuum cleaners, switching power supplies, motor loads with capacitive start (such as vacuum cleaners and compressors), electronic light regulators, resistive loads, electric drill loads, and halogen lamps. Therefore, resistance, electric kettle, electric drill, and vacuum cleaner are taken as the loads. The main hardware configuration required for the experiment is shown in Table 1.

Experiments were carried out at room temperature, the power supply is connected to the arc generator through the isolated power supply, the other end of the arc generator is connected to the load, and the wire of the load end passes through the mutual inductor. The current signal enters the signal acquisition module through the sampling resistor, and the output end of the signal acquisition module is connected to the oscilloscope. The waveform displayed by the oscilloscope is the voltage value, which actually reflects the current waveform in the line. The field diagram of the series arc fault simulation experiment is shown in Figure 4.

The horizontal adjustment knob of the arc generator is adjusted to control the generation of the arc. The sampling frequency of the oscilloscope is set at 62.5 kHz, and the sampling time of each group of waveforms is 320 ms, with a total of 16 cycles. The experiment obtains the low-frequency and high-frequency current waveforms of resistors, electric kettles, electric drills, and vacuum cleaners during normal operation and arc faults, as well as the current waveforms of switching power supplies and electric drills at the moment of startup. For the convenience of subsequent data analysis, the collected data are normalized in MATLAB. Figures 5–8 show the low-frequency and high-frequency arc current waveform of four cycles under different loads.

At 0.04 s, the series arc generator simulates the occurrence of series arc faults, that is, the waveform of the first two cycles is in a normal working state, and the series arc faults occur in the last two cycles. It can be seen from the waveform figure that when the electric kettle and resistance work normally, the low-frequency current waveform is a sine wave of 50 Hz, and the high-frequency current signal waveform has a small number of high-frequency pulses. When a series of arc faults occurs, the low-frequency waveform appears and has burrs at the peak, while the high-frequency waveform changes obviously and there are a large number of high-frequency pulses. When the electric drill and vacuum cleaner work normally, the low-frequency current waveform has the “flat shoulder,” which is similar to the low-frequency waveform when the series arc fault occurs between the electric kettle and the resistance. At the same time, the high-frequency current signal waveform also has a small number of high-frequency pulses. When a series of arc faults occurs, the low-frequency waveform changes dramatically, burrs increase, waveform amplitude decreases, waveform distortion is serious, the high-frequency waveform amplitude increases, and there are a large number of high-frequency pulses.

Time domain characteristics refer to the description of signal waveform with time as a variable, which is an important indicator to measure signal characteristics. Characteristic quantities in the time domain are usually divided into dimensionless and dimensional characteristic quantities. Dimensionless characteristics are not sensitive to the change of load and can more intuitively represent the status information of normal operation and fault of load. Kurtosis is often used in the field of bearing fault diagnosis. It has nothing to do with bearing speed and size, etc. It is sensitive to impact signals and is suitable for the description of surface damage faults. It can be seen from the arc fault current waveform diagram in Figures 3–6 that the current waveform will be distorted and high-frequency pulses will appear when an arc fault occurs. These signals are similar to impulse signals. Therefore, this article uses kurtosis as a waveform time domain feature to calculate. The waveform factor is the ratio of the effective value to the rectified average value. When an arc fault occurs, the waveform of the low-frequency current component will be distorted, the periodicity will be destroyed, and both the effective value and the rectified average value will change, so its shape factor can be calculated. The crest factor is defined as the ratio of the peak-to-peak value to the effective value of a signal. When an arc fault occurs, the low-frequency current component will appear “burr,” and its peak-to-peak value will become larger, so the arc fault can be described by calculating the change in the value of the crest factor. The impulse factor refers to the ratio of the peak value of the signal to the rectified average value. Similar to the crest factor, arc faults can also be described by the pulse factor. The margin factor is the ratio of the peak value of the signal to the rms amplitude. Crest factor, impulse factor, and margin factor, like kurtosis, are all indicators used to detect whether there is a shock in a signal. In this article, kurtosis, waveform factor, crest factor, pulse factor, and margin factor are selected as five dimensionless indexes for time domain characteristic extraction. Low-frequency current waveform of two cycles, i.e., 20 ms, and 2,500 points of sampling points N were selected as an analysis sample. Kurtosis, waveform factor, crest factor, pulse factor, and margin factor of low-frequency current waveform were calculated in the time domain, and the five characteristic quantities were marked as X₁, X₂, X₃, X₄, and X₅ in turn. The expression of each time domain characteristic quantity is shown in Table 2.

In Table 2, x i represents the current sample at the ith sampling point, i = 1,2,3..., N; μ is the mean of x i , σ is the standard deviation of x i , and E represents the mathematical expectation. The time domain characteristic quantities of 100 samples were calculated for each load. Table 3 shows the average time domain characteristics values of different loads.

It can be seen from Table 2 that the crest factor, pulse factor, and margin factor of each load increase when a series of arc faults occurs compared with normal operation. The value of the waveform factor decreases when a series of arc faults occurs. For kurtosis, the values of electric kettles and electric drills will decrease in the event of a series of arc faults, and the values of resistance and vacuum cleaners will increase. Under the condition of a single load, the threshold value can be set to determine whether the series of arc faults occurs. But in the actual line, load condition cannot be determined in advance, and threshold setting will be difficult. It can be seen that the high-frequency current waveform changes dramatically when series arc faults occur, and more series arc fault characteristics can be obtained in the high-frequency waveform, so it is necessary to analyze the high-frequency current waveform.

It is impossible to calculate the characteristic values of the high-frequency current waveform in the time domain because the waveform of the high-frequency current waveform is very drastic. Therefore, characteristic extraction is carried out in the frequency domain.

VMD is a novel adaptive and completely non-recursive signal analysis method provided by Dragomiretskiy and Zosso. (2014) for EMD’s sensitivity to noise and signal sampling. To establish and solve the variational problem as the core, based on the classical Wiener filter, Hilbert transform and mixes as the basis of expansion solution, intrinsic mode function, and their respective central frequencies are obtained through each intrinsic mode function to reconstruct the signal. The reconstructed signal can smoothly reproduce the input signal. VMD is the sum of the input signal f(t) decomposed into K sub-signals (i.e., IMF components) and the remainder: f ( t ) = u r ( t ) + ∑ k = 1 K u k ( t ) , (1) where u _k (t) is the kth IMF component, and u _r (t) is the remainder.

The IMF component is a function of amplitude and frequency modulation: u k ( t ) = A k ( t ) cos ( φ k ( t ) ) , (2) where φ _k (t) is a non-decreasing function, that is, φ ^′ _k (t)≥0,k ≤ K; A _k (t) represents the envelope A _k (t) ≥ 0; k ≤ K.

The VMD algorithm requires the bandwidth and minimum of all IMF components. The solution of the constrained variational problem is constructed as follows: m i n { u k ( t ) } , { ω k } { ∑ k = 1 K ‖ ∂ t [ ( δ ( t ) + j π t )   ∗   u k ( t ) ] e − j ω k t ‖ 2 2 } , (3) where ω _k is the central frequency of the kth IMF component, ω _k = φ ^′ _k (t); δ(t) is the Dirac function.

In Formula (3), quadratic penalty term and Lagrange multiplier are introduced to solve the variational problem, making it unconstrained. The augmented Lagrange function is obtained as follows: L { [ u k ( t ) ] , [ ω k ] , λ ( t ) } = α ∑ k = 1 K ‖ ∂ t [ ( δ ( t ) + j π t )   ∗   u k ( t ) ] e − j ω k t ‖ 2 2 + ‖ f ( t ) − ∑ k = 1 K u k ( t ) ‖ 2 2 + 〈 λ ( t ) , f ( t ) − ∑ k = 1 K u k ( t ) 〉 , (4) where λ(t) is the Lagrange multiplier and α is the penalty factor.

The detailed iterative solution steps of modal components u _k (t), central frequency ω _k , and λ _k (t) in Formula (4) can be referred to as the solution steps in the article by Dragomiretskiy K and Zosso D (2014). According to the aforementioned principle, the VMD algorithm is used in MATLAB for waveform decomposition. According to the study of K and α in an article by Ma et al. (2020), the number of decomposition and the penalty factors were set at K = 4 and α = 2000, respectively. Other parameters in the VMD algorithm are set as the default values of the algorithm in an article by Liu et al. (2021). The high-frequency current waveform is taken as an example when an electric kettle works normally and a series of arc faults occurs. For the convenience of analysis, the waveform data were normalized, and then VMD decomposition was carried out to obtain four IMF components, and the corresponding spectrum of each component was obtained by Fast Fourier Transformation in MATLAB. The decomposition results are shown in Figures 9–11. At the same time, the EMD algorithm was used for the same series arc fault waveform to obtain each IMF component and its spectrum after decomposition, as shown in Figure 12.

Figure 9 shows the original current waveform when the electric kettle is in normal operation and series arc fault occurs; Figure 10 shows the decomposition result of VMD algorithm when the electric kettle is in normal operation; Figures 11, 12, respectively, show the decomposition result of VMD and EMD algorithms of the same high-frequency component of the series arc fault. As can be seen from Figure 12, the EMD algorithm decomposes the high-frequency signal into nine components, and the IMF1–9 components are arranged according to the central frequency from large to small. Among them, both IMF1 and IMF2 appear in the frequency band around 5 kHz, with an over-decomposition phenomenon. The center frequency distribution of the IMF2 component is not obvious, including the frequency band [5000 Hz and 10000 Hz], and there is the phenomenon of mode aliasing. In addition, it can be seen from the amplitude–frequency diagram of IMF5–9 components that the component is lower than 1kHz, which is due to the frequency band attenuation of the RC high-pass filter, but it is not needed for the high-frequency component analysis in this article. It can be seen from Figure 11 that the high-frequency current component is decomposed into four IMF components by the VMD algorithm, which are independent of each other without modal aliasing, and the decomposition effect is significantly better than that of the EMD algorithm. The center frequency of each IMF based on the VMD algorithm is shown in Table 4.

Energy entropy can measure the regularity of time series and the energy characteristics of signals in different frequency bands (Jin et al., 2021). When the series arc fault occurs, the current will change and the energy will also change. The energy entropy of mth IMF component is calculated as: H E m = ∑ i = 1 N ( x m ( i ) ) 2 ∑ m = 1 K ∑ i = 1 N ( x m ( i ) ) 2 lg ∑ i = 1 N ( x m ( i ) ) 2 ∑ m = 1 K ∑ i = 1 N ( x m ( i ) ) 2 , (5) where x _m (i) is the value of the ith point of the mth IMF component, m = 1,2,3, … ,K. s m = v a r m var r + ∑ k = 1 K v a r k , (6) where var _m is the variance of mth IMF component, m = 1,2,3, … , K and var _r is the variance of the remainder.

The energy entropy and variance contribution ratio of each IMF in Figures 8, 9 were calculated according to the aforementioned formula. Table 5 shows the calculation results.

It can be seen from Tables 4, 5 that the IMF1 component has a center frequency of less than 1 kHz, which is due to the frequency band attenuation of the high-pass filter, but it is not needed for the high-frequency waveform analysis in this article. The frequency of the high-frequency current waveform in this article is set above 1 kHz, so only the IMF component larger than 1 kHz needs to be studied. When the center frequency of the IMF component is greater than 1KHZ, the IMF2 variance contribution rate of the normal operating current is the largest, and the energy entropy is 0.122. When a series of arc faults occur, the variance contribution ratio of IMF1 is the largest, and the energy entropy is 0.155, which increases obviously. The corresponding energy entropy of IMF with the largest variance contribution ratio was calculated for 100 groups of normal working and 100 groups of series arc fault samples, as shown in Figure 13.

As can be seen from Figure 11, the corresponding energy entropy of IMF with the largest variance contribution ratio in normal operation is less than 0.15, and the corresponding energy entropy of IMF with the largest variance contribution ratio in series arc fault is greater than 0.15. Therefore, the energy entropy corresponding to IMF with the largest variance contribution ratio can be taken as a characteristic value and denoted as X _6.

In order to improve the diagnosis ratio of series arc fault and realize the diagnosis under different load conditions, the load working state is marked as X₇, “0” means normal operation, “1” means series arc fault, and the series arc fault characteristic vector is constructed with the six time–frequency characteristic quantities in this article. The characteristic vectors of some experimental samples are shown in Table 6.

Random forest algorithm is an algorithm that integrates multiple decision trees through the idea of ensemble learning (Li et al., 2020). Its basic unit is the decision tree. In this article, the decision tree algorithm selects CART [Jiang et al. (2021), Ali et al. (2012)], and the Gini coefficient minimization criterion is used for characteristic selection in CART. The series arc fault diagnosis flow chart based on random forest algorithm is shown in Figure 14:

In this article, 1000 training samples were selected with 250 for each load, including 100 normal samples and 150 series of arc fault samples. Characteristic quantities n = 6. The number of decision trees is T = 100. The diagnosis model was trained, and the untrained load samples were tested. Figures 15A–F shows the diagnostic results of different types of loads based on the random forest algorithm. From Figures 15A–F, the information shown in Table 7 can be obtained. The random forest algorithm has ideal fault diagnosis effects and high diagnosis accuracy for electric kettles, hair dryers, electric drills, switching power supplies, and vacuum cleaners.

It can be seen from Table 7 that in the series arc fault detection model based on random forest, the accuracy ratio of load detection under a normal working state is higher than 96%. The fault detection accuracy of load in a series arc fault state is higher than 96%. The comprehensive detection ratio was above 97%. The detection effect is very good.

In the actual distribution lines, the loads are varied and mixed. In order to verify the validity of the aforementioned diagnostic model, series arc fault simulation experiments of switching power supply, hair dryer, and mixed load are added in this article. Switching power supply parameters: BSD-36 P-60 W, input 220 VAC 50 Hz, and output 36 VDC 60 W. Hair dryer parameters: 220 VAC 1600 W. According to the time domain and frequency domain characteristic extraction methods proposed in this article, the time–frequency characteristic values are extracted, the characteristic vector is constructed, and a new load training sample diagnosis model is added based on the random forest algorithm training, and then the fault diagnosis is carried out. The diagnosis results are shown in Table 8.

As can be seen from Table 8, in the diagnosis of the new loads and mixed load types, the accuracy of the original diagnosis model decreases to 94.91% and 91.13%, respectively, and the detection effect is lower than that of the original four loads. Therefore, new loads and mixed loads were added to the original training samples to optimize the diagnostic model. The results in Table 8 show that the recognition efficiency of the new diagnostic model has reached more than 97%, and the recognition effect is significant. For more load cases, new training samples can be added to improve the diagnosis model for diagnosis.