Research on Time–Frequency Domain Characteristics Analysis of Fault Arc Under Different Connection Methods

Abstract

Arc fault detection is a key technology for preventing electrical fires. However, existing research has primarily focused on series connections, with insufficient attention paid to parallel load conditions, which are prevalent in real-world residential electricity usage. In accordance with the UL 1699 and IEC 62606 standards, this study established an experimental platform for arc faults, incorporating seven single loads (categorized into four types) and nine multi-load combinations. A systematic analysis of the differences in time–frequency characteristics under different connection modes was conducted. Time-domain and frequency-domain analyses revealed that under parallel connection the dispersion of arc fault time-domain characteristics decreases by more than 50% and the fundamental frequency component increases significantly. For parallel multi-load scenarios, the fundamental component of resistive combinations can reach 90%, while the frequency variance of inductive combinations can be as high as 400,000. By elucidating the time–frequency domain characteristics of parallel arc faults, this study proposes an optimized feature parameter analysis scheme for electrical fire monitoring systems. Based on this, this paper proposes an arc fault detection method using the Dual-Channel Convolutional Neural Network (DCNN). The method achieves 97.09% recognition accuracy for arc faults with different connection modes. Comparative experiments with other models and ablation studies show that the model attains 98.52% detection accuracy, verifying the effectiveness of the proposed method. This approach can significantly improve the accuracy of arc fault detection in multi-load environments, thereby enabling early warning of electrical circuit faults and potential fire hazards.

1. Introduction

With the development of modern society, electric energy has become an indispensable part of our daily lives. However, electrical fire accidents occur frequently, with serious consequences affecting people’s lives and property safety. According to statistics, a total of 55,000 electrical fires occurred in 2022, accounting for over 30% of the total number of fires [1]. In 2017, a fire broke out in an apartment in Daxing District, Beijing, due to an electrical wiring fault, resulting in the death of 19 people. In 2018, a hotel in Harbin caused 20 deaths due to a short circuit of the air conditioning power line and aging insulation layer caused by long-term overload operation of electrical lines. These accidents show that the local high temperature generated by the fault arc can quickly ignite the surrounding combustible materials, However, existing detection technologies lack sufficient research on the parallel load conditions commonly encountered in actual electricity usage, making it difficult to effectively predict such fire risks. In the field of arc fault detection, scholars at home and abroad have made some achievements. Zhang Guanying et al. [2] proposed a method for wavelet threshold denoising based on the subtraction of adjacent periods of line current; Di Zhenguo et al. [3] proposed a DC arc fault detection method based on weighted differential current. By extracting multi-frequency band characteristic parameters within 20~65 KHz and setting threshold criteria, the detection accuracy was significantly improved, and its effectiveness in low-voltage DC systems was verified. Gao Haiyang [4] overcame the difficulty in distinguishing traditional current characteristics by analyzing the voltage distortion characteristics of the load side and established an arc fault detection method based on wavelet energy spectrum entropy. Wang et al. [5] proposed an arc fault detection method based on EWT composite entropy and multi-domain feature fusion to achieve high-precision detection of more than 98%. Aires [6] established a wavelet-based limited ground fault protection, which uses high frequency components to quickly detect transformer turn-to-turn grounding faults and is more efficient than traditional REF units. Gong [7] proposed a DNN arc fault detection model based on wavelet decomposition and eigenvalue compression, which has good detection effect on both training set load and non-training set load. However, most of the existing research focuses on the analysis of series arc characteristics, and research on the time–frequency domain characteristics of arc under parallel load is insufficient, which leads to high false alarm rate and missing alarm rate in practical application. In view of the above problems, this paper systematically analyzes the difference of time- and frequency-domain characteristics between series and parallel connection modes. Based on this, this paper proposes an arc fault detection method using FF-DCNN. By integrating time–frequency domain feature fusion of dual channels and optimizing the model, it fully extracts the time-domain and frequency-domain features from current signals, significantly improving the detection accuracy of arc faults. The method can accurately identify arc faults under different connection modes and achieves extremely high detection accuracy on the validation set. Experiments show that the model converges rapidly with small loss fluctuation and high stability, providing new ideas and technical support for the safety monitoring of electrical equipment.

2. Materials and Methods

2.1. Construction and Data Acquisition of Arc Fault Experimental Platform

2.1.1. Construction of Arc Fault Test Platform with Different Connection Modes

According to the UL1699 standard [8], this study uses the electrode separation method to generate electronic arc and build an arc fault test platform. The experimental platform is mainly composed of the following core components: 220 V/50 HZ AC power supply, circuit protection switch, arc fault simulation device, digital oscilloscope, and various types of test loads. It is used to study the fault detection of series arc. The research premise of this experiment is that all tests were conducted under the conditions of 220 V/50 Hz.

Due to the complexity of parallel operation of different loads in real situations, the test platform is built to realize the function of parallel operation of a variety of test loads and fault arc devices. According to the IEC 62606 standard [9], the test platform is designed as shown in Figure 1 and Figure 2 for parallel arc fault detection. In order to better analyze the influence of different loads on arc fault detection, the other branch uses fixed resistance as the load, which will not introduce too complex additional interference.

The arc fault generation device, shown in Figure 3, primarily comprises a moving electrode and a stationary electrode. The moving electrode is a 6.0 mm diameter copper rod with a conical tip, while the stationary electrode is a 6.0 mm diameter carbon rod with a flat end. A stable arc discharge is maintained by retracting the moving electrode via an adjustment knob to create a preset gap. And the copper–carbon electrodes used for arc generation in this experiment are uniformly replaced with new ones after every 200 tests.

2.1.2. Design of Test Conditions

In the load selection, we referred to the provisions of the national standard GB/t31143-2014 [10] and the complex variety of electrical loads in the power supply and distribution lines are considered, and the arc current also presents complex nonlinear characteristics. This study complies with the provisions on test circuits and load conditions specified in the international standard IEC 62606 [9]. This standard explicitly lists the most common load types in residential low-voltage distribution systems that are prone to arc faults, such as resistive loads (electric kettles), inductive loads (electric fans, induction cookers), power electronic loads (energy-saving lamps, laptops), and switching power supply loads (microwave ovens), as shown in

Table 1. Load condition scheme of series arc fault test.

Number	Load Type	Load Name	Core Components
a	Resistive load	Electric kettle	Heating element
b		Hair dryer	Motor
c	Inductive load	Induction cooker	Insulated gate bipolar transistor
d		Electric fan	Motor rotor
e	Power electronic load	Energy-saving lamps	Electronic driver
f		Laptop computer	Switching mode power supply
g	Switching power supply load	Microwave oven	Magnetron

. Therefore, the 7 types of single loads and 9 types of multi-load combinations adopted in this study cover the core load types specified by the standard and simulate the common parallel power consumption scenarios in actual households. This standard-based experimental design ensures that the analyzed time–frequency domain characterization laws of arc faults can effectively represent the conditions in real power consumption environments.

To approximate real-world scenarios and comprehensively account for parallel load conditions across different types of load combinations, seven single-load and nine multi-load combination conditions were designed, as detailed in

Table 2. Load condition scheme of parallel arc fault test.

Number	Load Type	Load Name
a	Resistive load	Electric kettle
b	Inductive load	Hair dryer
c	Electronic load	Induction cooker
d	Switching power supply load	Electric fan
e	Resistive and inductive loads	Energy-saving lamps
f	Inductive and inductive	Laptop computer
g	Resistive and resistive	Microwave oven
h	Resistive and power electronic	Electric kettle + Electric fan
i	Resistive and inductive	Induction cooker + Electric fan
j	Inductive and power electronic	Electric kettle + Electric kettle
k	Resistive and power electronic	Electric kettle + Energy-saving lamp
l	Resistive and power electronic	Hair dryer + Electric fan
m	Power electronic and switching power supply	Induction cooker + Energy-saving lamp
n	Resistive load	Hair dryer + Energy-saving lamp
o	Inductive load	Hair dryer + Laptop computer
p	Electronic load	Energy-Saving lamp + Microwave oven

. Current waveforms under both normal and arc fault conditions were collected for each setup. This data collection facilitates subsequent arc fault detection and enables a comparative analysis with series connection modes. Each operating condition was tested three times.

2.1.3. Test Data Collection and Pretreatment

Due to the randomness of arc generation in the actual situation, the voltage data only has obvious changes at both ends of the arc gap, so the voltage signal is not collected, and only the current signal is collected. During the test, the current signals of normal and arc fault states of different load types are collected and stored. Set the sampling frequency of the oscilloscope as 62.5 khz and the sampling time as 4 s. It should be noted that the 4 s of data collected in each test is intended to capture a complete and continuously stable arc discharge process rather than multiple independent transient ignition events. In this experiment, the electrode separation method combined with a preset fixed gap is adopted to generate and maintain a quasi-steady-state arc. To ensure that subsequent analyses are based on stable arc characteristics, the middle stable region of the 4 s raw data, specifically, the data from 1.0 s to 3.5 s after the arc, is confirmed to be established. Then before it extinguishes, it is manually intercepted for all time-domain and frequency-domain analyses. This data processing method effectively eliminates the transient impact during the initial arc ignition stage and the unstable phase at the final extinction stage, thereby ensuring the accuracy and representativeness of the extracted features. Repeat the test process many times under each working condition and collect multiple groups of test current data.

Normalize the data samples of different scales. The linear standardization is also called the minimum–maximum standardization, which takes the difference between the maximum and minimum values of each feature of the sample as the scaling factor, as shown in Formula (1) [11].

$${\mathrm{x}}^{\prime }={\displaystyle \frac{\mathrm{x}-\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{x})}{\mathrm{m}\mathrm{a}\mathrm{x}(\mathrm{x})-\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{x})}}$$

(1)

Although linear standardization can normalize data points to any cardinality, once the new sample value exceeds the boundary of the maximum and minimum value, the normalized eigenvalue will be unstable. However, the maximum absolute value standardization only uses the absolute value of the maximum characteristic value, with a value range of [−1, 1], which is applicable to cases where the distribution range of characteristic values is unknown or the difference is large, as shown in Formula (2) [9].

$${\mathrm{x}}^{\prime }={\displaystyle \frac{\mathrm{x}}{‖\mathrm{m}\mathrm{a}\mathrm{x}(\mathrm{x})‖}}$$

(2)

By maximum absolute value standardization of the collected data, the difference of samples is reduced and the reliability is increased.

2.2. Theoretical Analysis of Frequency-Domain Characteristics

2.2.1. Time-Domain Feature Theory Analysis

According to the time-domain analysis method in the field of arc fault detection, three time-domain characteristic statistics of absolute mean value, peak–peak value, and variance are selected for analysis [12].

Absolute mean value can reflect the overall energy level of the signal and calculate the power frequency cycle mean as the core characteristic parameter. Its calculation method is shown in Equation (3):

$${\mathrm{I}}_{\mathrm{a}}={\displaystyle \frac{1}{\mathrm{N}}}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left|\mathrm{i}(\mathrm{n})\right|}$$

(3)

where I_a is the average current, i(n) is the instantaneous current sampling value, and N is the number of sampling points of arc current signal per cycle. Peak–peak value is used to calculate the difference between the maximum and minimum values of a sample in a period, which characterizes the dynamic range of the signal and reflects the fluctuation amplitude of the signal. It is suitable for detecting transient changes or abnormal events. Variance is the average value of the square of the signal amplitude deviation from the mean value, which can quantify the degree of dispersion of signal characteristics. The calculation method is shown in Equation (4):

$${\mathrm{I}}_{\mathrm{v}\mathrm{a}\mathrm{r}}={\displaystyle \frac{1}{\mathrm{N}-1}}{{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}({\mathrm{i}}_{\mathrm{n}}-\overline{\mathrm{i}})}}^2$$

(4)

where $\overline{\mathrm{i}}$ is the mean value of the signal. The variance can effectively characterize the fluctuation characteristics of the signal and has a good description ability for the amplitude variation and frequency components of the signal.

2.2.2. Frequency-Domain Characterization Theory Analysis

In the frequency-domain analysis methods, the common ones are fast Fourier transform, power spectral density function calculation center frequency, frequency variance, and so on. Fast Fourier transform (FFT) is a fast algorithm of DFT. It uses the symmetry, periodicity, and redundancy of DFT to reduce the computational complexity of DFT. The expression of FFT is shown in Equation (5) [13]:

$$\mathrm{X}[\mathrm{n}]={\displaystyle {\int }_{-\infty }^{\infty }\mathrm{x}[\mathrm{t}]{\mathrm{e}}^{-\mathrm{j}2\mathrm{\pi }\mathrm{f}\mathrm{t}}}\mathrm{d}\mathrm{t}$$

(5)

Among them, the frequency component distribution of the signal is explained, and the relative relationship between different frequency components can be further calculated from the spectrum distribution. Power spectral density function (PSD) is calculated by calculating the spectrum distribution of the signal at different frequencies through Fourier transform, and further frequency-domain characteristic indexes can be obtained: center frequency, average frequency, root mean square frequency, frequency variance, etc. Among them, the center frequency can reflect the main distribution range of the signal power spectrum. Frequency standard deviation is the radius of inertia center to the center frequency. If the spectrum amplitude near the center of gravity is large, the frequency standard deviation is small; if the spectrum near the center of gravity is small, the frequency standard deviation is large. The frequency variance is the square of the frequency standard deviation. Both the frequency standard deviation and the frequency variance describe the energy dispersion of the power spectrum. The calculation formula of center frequency and frequency variance is shown in Equations (6) and (7) [14]:

$$\mathrm{F}\mathrm{C}={\displaystyle \frac{{\displaystyle {\int }_0^{+\infty }\mathrm{f}\mathrm{P}(\mathrm{f})\mathrm{d}\mathrm{f}}}{{\displaystyle {\int }_0^{+\infty }\mathrm{P}(\mathrm{f})\mathrm{d}\mathrm{f}}}}$$

(6)

$$\mathrm{V}\mathrm{F}={\displaystyle \frac{{\displaystyle {\int }_0^{+\infty }(\mathrm{f}-\mathrm{F}\mathrm{C}{)}^2\mathrm{P}(\mathrm{f})\mathrm{d}\mathrm{f}}}{{\displaystyle {\int }_0^{+\infty }\mathrm{P}(\mathrm{f})\mathrm{d}\mathrm{f}}}}$$

(7)

2.2.3. Wavelet Transform Theory

Wavelet transform is a method that can realize multi-scale decomposition of signal characteristics. It can extract the time–frequency characteristics of the measured signal at the same time. In addition, wavelet transform can effectively analyze non-stationary signals, especially highlighting the specific characteristics of the signal. Since the current sample is typical time series data, in order to effectively characterize the time–frequency characteristics of the arc fault current signal, the signal is transformed into a wavelet time–frequency map, which can highlight the characteristic information of the current sample, as shown in Equations (8) and (9) [15].

Wavelet transform decomposes the original signal into the sum of a series of wavelets by using the wavelet base after translation and scaling, as shown in Equation (8):

$$\mathrm{W}\mathrm{T}(\mathrm{\alpha }, \mathrm{\tau })={\displaystyle \frac{1}{\sqrt{\left|\mathrm{\alpha }\right|}}}{\displaystyle {\int }_{-\infty }^{+\infty }\mathrm{f}(\mathrm{t}){\mathrm{\psi }}^{\ast }}({\displaystyle \frac{\mathrm{t}-\mathrm{\tau }}{\mathrm{\alpha }}})\mathrm{d}\mathrm{t}$$

(8)

where f(t) is the energy-limited signal, and ψ is the wavelet basis function. In addition, the wavelet transform has two variables: scale α and translation τ. The scale α controls the scaling of the wavelet function corresponding to frequency, and the translation τ controls the translation of the wavelet function corresponding to time.

The Continuous Wavelet Transform (CWT) defines ${\mathrm{f}}^{\left(\mathrm{d}\right)}\left(\mathrm{t}\right)$ as the current signal of the $d$-th feature of the original current sample data. For any energy-limited signal, the CWT formula is shown in Equation (9):

$$\mathrm{C}\mathrm{W}\mathrm{T}(\alpha ,\tau )={\displaystyle \frac{1}{\sqrt{\left|\mathrm{\alpha }\right|}}}{\displaystyle {\int }_{-\infty }^{+\infty }{\mathrm{f}}^{(\mathrm{d})}(\mathrm{t})\overline{\mathrm{\psi }}}({\displaystyle \frac{\mathrm{t}-\mathrm{\tau }}{\mathrm{\alpha }}})\mathrm{d}\mathrm{t}$$

(9)

In the formula, $\overline{\mathsf{\psi }\left(\mathrm{t}\right)}$ denotes the complex conjugate of the wavelet basis function, and $\mathrm{C}\mathrm{W}\mathrm{T}(\mathsf{\alpha },\mathsf{\tau })$ represents the time–frequency coefficient obtained after the CWT operation. When performing wavelet transform, the selection of the wavelet basis function has a significant impact on the analysis results. The Morlet wavelet is one of the commonly used wavelet basis functions: it has good resolution in the low-frequency region and relatively gentle attenuation in the high-frequency region, ensuring excellent time–frequency aggregation and resolution capabilities.

3. Results

3.1. Time–Frequency Domain Characteristic Analysis Based on Different Load Types

3.1.1. Time-Domain Characteristic Analysis of Different Loads

The distribution diagram of the absolute mean value of the current, as shown in Figure 4, was plotted. For each load type, the average value of repeated experiments was selected. Blue and orange correspond to the characteristic quantities of samples in the normal state and the arcing state, respectively.

As can be seen from Figure 4, except for the power electronic loads (e) and (f) (where the average current increases in the arc fault state compared to the normal state), the average current of other loads decreases significantly (with varying degrees of reduction). From the data distribution, it can be observed that after arcing occurs in resistive loads (a) and (b), the average value distributions are relatively scattered, with differences of 1.98 and 0.67, respectively. This indicates that the shoulder phenomenon exhibits obvious randomness.

The peak–peak value distribution diagram of samples in the normal state and the arcing state is shown in Figure 5.

The peak–peak value distribution can reflect the intensity of current burrs. As can be observed from Figure 5, after the occurrence of arc faults, the inductive load (d) and power electronic loads (e) and (f) exhibit a large distribution of peak–peak value, indicating that these loads generate intense spikes and burrs when arc faults occur. In contrast, the peak–peak value distributions of resistive loads and switching power supply loads are relatively concentrated, showing a slight decrease after the occurrence of arc faults.

The current variance distribution diagram of samples in the normal state and the arcing state is shown in Figure 6.

Due to the differences in information dimensions contained in various characteristic parameters, it is difficult to achieve accurate identification of arc faults using a single characteristic parameter. Therefore, this study introduces the ratio analysis method for characteristic parameters between the arc fault state and the normal state. This method can effectively reduce the impact of load types on identification results and has broader engineering applicability.

Table 3. Ratios of time-domain characteristic quantities for different load types.

Load Type	Load	Absolute Mean Value Ratio	Peak–Peak Value Ratio	Variance Ratio
Resistive load	Electric kettle	0.869	0.980	0.901
	Hair dryer	0.941	0.972	0.906
Inductive load	Induction cooker	0.980	0.954	0.870
	Electric fan	0.984	1.039	1.032
Power electronic load	Energy-saving lamps	1.648	7.515	10.824
	Laptop computer	2.670	11.888	14.291
Switching power supply load	Microwave oven	0.922	0.914	0.881

presents the experimental ratios of time-domain characteristic quantities under different load types.

Based on the experimental data in Table 3, when an arc fault occurs in the circuit, the absolute average value of the current under all load conditions (except for power electronic loads) decreases by more than 2% compared with the normal operating conditions. Additionally, except for the inductive load (electric fan) and power electronic loads, the ratios of time-domain characteristic quantities of other loads are all less than 1, indicating that the time-domain characteristics in the fault state are lower than those in the normal state.

Notably, among the four different load types, although the variance ratios of resistive loads, inductive loads, and switching power supply loads do not differ significantly from their absolute average value ratios and peak-to-peak value ratios, the variance ratio of power electronic loads is 1.44 to 6.57 times that of the other two time-domain analysis ratios, which is much larger than the latter. This demonstrates that variance analysis can better distinguish between the normal state and the fault state, providing important data support for subsequent arc fault identification.

3.1.2. Frequency-Domain Characteristic Analysis of Different Loads

Due to significant differences in the content rate of harmonic currents among electrical equipment of different load types, the characteristic information obtained via FFT is used to further calculate the frequency component distribution. As shown in Figure 7, the harmonic content distributions of different load types are presented, where the serial numbers of load types correspond to those listed in Table 1.

Based on the analysis of Figure 7, there are significant differences in the frequency spectra of different load types between the normal state and the arc fault state. In the normal state, as can be observed from Figure 7a, the fundamental wave components of resistive and inductive loads reach 90%. For power electronic loads, the contents of the fundamental wave, 3rd, 5th, and 7th harmonics are distributed within the range of 15–30% with relatively uniform distribution. For switching power supply loads, the 3rd and 5th harmonics are more prominent, accompanied by an increase in high-order harmonic components. Under the same state, the frequency spectrum distributions of loads of the same type are almost consistent, while those of different load types vary significantly. Such differences reflect the influence of different load types on frequency-domain characteristics.

After the occurrence of arc faults, a comparison between Figure 7a,b shows that the degrees of change in the frequency spectrum responses of different loads vary. However, for most loads, the fundamental frequency components of samples in the arcing state decrease by 0.08 to 0.23 times compared with those in the corresponding normal state, while the high-order harmonic components increase. The high-order harmonic contents of resistive and inductive loads increase by 0.95 to 2.03 times, but the induction cooker load maintains the highest fundamental wave content of 91.4% with little change in harmonic content. For the power electronic load group, the wide-band components increase in the arcing state, resulting in a more uniform frequency distribution. The fundamental wave content of switching power supply loads increases by 21%, with an overall slight change.

3.1.3. Time–Frequency Domain Characteristic Analysis of Different Loads

Taking the electric kettle as an example, Figure 8 presents the wavelet time–frequency diagram of the electric kettle after Continuous Wavelet Transform (CWT). In the time–frequency diagram, the lighter the color, the greater the energy it represents.

To improve the quality of the time–frequency diagram, the scale magnification method is used to enhance the contrast between the fault state and the normal state. By performing scale transformation on the arithmetic sequence with the starting point as $totalscal$, the ending point as 1, and the step size as −1, the scale sequence $scales$ is obtained, as shown in Equation (10):

$$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{s}=\{\mathrm{\alpha }/\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l},\cdots ,\mathrm{\alpha }/(\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}-1),\mathrm{\alpha }/4,\mathrm{\alpha }/2,\mathrm{\alpha }\}$$

(10)

By magnifying $scales$ by a factor of 2ⁿ, the display area of the arc fault signal in the time–frequency diagram can be effectively expanded. To enhance the contrast between high-frequency and low-frequency signals, a logarithmic scaling method is used to compress the dynamic range of the wavelet coefficients, making the details of the low-amplitude portions more prominent. Figure 9 shows the enhanced time–frequency diagram of the electric kettle. Compared with the original image, both the brightness and contrast of the enhanced image have been improved, which facilitates subsequent feature extraction. The time–frequency diagrams of other loads will also be processed in the same way as described above.

3.2. Time–Frequency Domain Characteristic Analysis Based on Single-Load Series Parallel Mode

3.2.1. Time-Domain Characteristic Analysis of Single Load

The different connection modes of single load can be divided into series and parallel. The left figure of Figure 10 shows the absolute average value of the current of a series load, and the right figure shows the distribution of the absolute mean value of the current of a single load in parallel mode.

As shown in Figure 10, the absolute mean values of different load types are more uniformly distributed under the parallel connection condition than under the series condition. Furthermore, the mean values during fault states are consistently lower than those during normal states for all loads. Specifically, the absolute mean values for resistive, inductive, and switching power supply loads are lower in the parallel mode compared to the series mode. In contrast, power electronics loads exhibit a higher mean value in the parallel connection.

The two-factor analysis of variance was used to compare the variance of normal state and fault state in different connection modes, as well as the ratio of characteristic quantities between fault and normal. Taking electric kettle load as an example, the analysis diagram of average current under different connection modes was drawn, as shown in Figure 11.

In Figure 11, statistical quantity ‘a’ denotes the mean difference of the normal state between different connection modes, while ‘b’ represents that of the fault state. Similarly, ‘c’ is the ratio of the characteristic quantity in the fault state to that in the normal state for the series mode, and ‘d’ is the corresponding ratio for the parallel mode. Furthermore, based on a two-factor analysis of variance (ANOVA), the comparison of the mean statistics for single loads under different connection modes is presented in

Table 4. Comparison of average statistics of single load under different connection modes.

Load type	Mean Difference Under Normal Condition for Different Connection	Mean Difference Under Fault Condition for Different Connection	Ratio of Characteristic Signatures Under Series Connection Mode	Ratio of Characteristic Signatures Under Parallel Connection Mode
Electric kettle	0.765	0.827	0.869	0.940
Hair dryer	0.645	0.629	0.941	0.917
Induction cooker	0.613	0.573	0.980	0.916
Electric fan	0.985	0.968	0.984	0.953
Energy-saving lamps	61.731	35.618	1.648	0.951
Laptop computer	29.613	10.334	2.670	0.932
Microwave oven	0.763	0.768	0.922	0.927

.

As shown in Table 4, there is a significant difference in the mean values of single loads between parallel and series connection modes. Under normal conditions, the mean value for the power electronic energy-saving lamp in parallel mode is 61 times higher than that in series mode. Similarly, under fault conditions, the mean value for the power electronic laptop computer in parallel mode is 10 times higher. In contrast, the mean values for the other resistive, inductive, and switching power supply loads are lower in the parallel connection mode than in the series mode. A comparison of the fault-to-normal characteristic ratio under different connection modes reveals that in parallel mode, the mean value after an arc fault occurs is approximately 90% of the normal state value. This indicates that the time-domain characteristic ratio in parallel mode provides a more distinct criterion for arc fault detection compared to the series mode.

The variance distribution diagram under different connection modes of single load is drawn as shown in Figure 12. Generally speaking, the dispersion degree of normal and fault states of parallel mode is lower than that of series mode, and the variance distribution is more concentrated than that of series mode.

As mentioned above, the comparison of the average statistics of single load under different connection modes is shown in

Table 5. Comparison of difference statistics under different connection modes.

Load Type	Mean Difference Under Normal Condition for Different Connection	Mean Difference Under Fault Condition for Different Connection	Ratio of Characteristic Signatures Under Series Connection Mode	Ratio of Characteristic Signatures Under Parallel Connection Mode
Electric kettle	0.576	0.580	0.901	0.907
Hair dryer	0.347	0.397	0.906	1.036
Induction cooker	0.870	1.000	0.870	1.006
Electric fan	0.928	0.898	1.032	0.999
Energy-saving lamps	142.939	131.977	10.824	0.998
Laptop computer	328.385	22.972	14.291	0.998
Microwave oven	0.449	0.510	0.881	0.999

.

As shown in Table 5, the variance for resistive, inductive, and switching power supply loads in parallel connection is lower than that in series connection. In contrast, the mean value for the power electronic energy-saving lamp in parallel mode is 131 to 142 times higher than that in series mode. A comparison of the fault-to-normal variance ratio under different connection modes reveals a more uniform distribution of variance across load types in the parallel mode. Furthermore, the variance in the fault state is generally lower than or approximately equal to that in the normal state.

3.2.2. Frequency-Domain Characteristic Analysis of Single Load

FFT is used to analyze the spectrum of the sample data with the sampling rate of 65 khz and to further calculate the frequency component distribution. The harmonic order distribution diagram under the single-load parallel connection mode is presented in Figure 13.

Under normal conditions, loads of the same type exhibit similar harmonic spectra in series connection, with notable differences between different load types. In parallel connection, the influence of individual loads on the frequency distribution diminishes, leading to a convergence in the harmonic distribution between power electronic loads and resistive/inductive loads. Furthermore, the parallel mode exhibits a higher fundamental component and fewer high-order harmonics compared to the series mode.

Following an arc fault, the fundamental content of capacitive loads drops from 90% to below 80% in series mode, while the decrease in parallel mode is only to approximately 80%. Conversely, the fundamental content of power electronic loads increases significantly, from 20% in series mode to 80% in parallel mode. A unique behavior is observed in the microwave oven (a switching power supply load): its fundamental content increases in series mode but decreases in parallel mode after a fault. Although the parallel mode still generally exhibits a trend of decreasing fundamental frequency and increasing high-order harmonics during a fault, the magnitude of these changes is less pronounced than in the series mode. This unique phenomenon can be explained by the differential impact of fault locations on the operating mode of the switch-mode power supply. According to the fault physics underlying the UL 1699 [8] and IEC 62606 [9] standards, in the case of series faults, the arc impedance is directly connected in series with the microwave oven, resulting in a severe drop and distortion of the input voltage at its terminals. This condition, which conforms to the standard fault characteristics, is highly likely to cause the internal switch-mode power supply to cease high-frequency switching due to undervoltage or waveform anomalies, degenerating into the periodic charging and discharging of the front-end rectifier filter capacitor. Consequently, the microwave oven exhibits resistive-like characteristics with a relative increase in the fundamental frequency component.

In contrast, under parallel fault conditions, the microwave oven remains connected to a relatively normal bus voltage, and its switch-mode power supply strives to maintain normal operation. However, the parallel arc path, as a high-frequency noise source defined by the standards, significantly shunts the high-frequency harmonic currents, leading to a relative decrease in the fundamental frequency component of the bus current. Therefore, the same load exhibits opposing frequency-domain characteristics under different connection modes due to fundamental changes in its own operating state.

The center frequency and frequency variance of a single load under different connection modes are calculated by power spectrum (PSD), and the average value of repeated tests is shown in Figure 14.

As shown in Figure 14, under the parallel connection mode, the center frequency of the inductive load (induction cooker) is close to 200 Hz, while the frequency distributions of other load types are concentrated around 50 Hz. The results indicate that the center frequency and frequency variance in the arc state are significantly lower in the parallel mode than in the series mode. Its mechanism mainly stems from the interaction between inductive characteristics and the nonlinearity of arc faults. The impedance of inductive loads increases with the rise in frequency, which suppresses the high-frequency harmonic currents generated by arc faults, resulting in the current being rapidly cut off and reignited when crossing zero. This abrupt intermittent process is equivalent to a high-frequency current pulse, which shifts the spectral energy to higher frequency bands, thereby significantly elevating the center frequency. In contrast, the impedance of resistive loads remains unchanged with frequency variations, making it difficult for the high-frequency energy of arc faults to accumulate. Consequently, the changes in their frequency spectra and center frequency are relatively gentle. Specifically, the fault-to-normal-state ratio range for the center frequency is reduced from 0.83–599.75 to 0.76–1.49, and the ratio range for frequency variance is reduced from 1.30–3143.17 to 0.14–1.78. This suggests that the series connection mode contains richer harmonic components, whereas the parallel connection mode is characterized by a higher fundamental frequency content.

3.3. Time–Frequency Domain Characteristic Analysis Based on Multi-Load Series Parallel Mode

3.3.1. Multi-Load Time-Domain Characteristic Analysis

Figure 15 presents the absolute mean current values for multiple loads, comparing their distribution in series mode (left) with that in parallel mode (right).

As shown in Figure 15, the mean values of different types of loads are more evenly distributed in parallel mode, and the mean values of fault states are lower than those of normal states. The absolute mean value of different types of loads in parallel mode is lower than the absolute mean value of resistive, inductive, and switching power supply load in series connection mode, and higher than that of power electronic loads.

The variance distribution diagram under different connection modes of multiple loads is drawn as shown in Figure 16.

Overall, the dispersion of both normal and fault states in the parallel mode is lower than in the series mode, with a more concentrated variance distribution. In parallel mode, the mean value in the fault state is consistently lower than that in the normal state across all load types. The variance for different load types in parallel configuration is generally higher than that of power electronic loads alone. For instance, the combination of a resistive load and an electronic load (as in condition (j)) exhibits a variance significantly higher than other parallel conditions. Furthermore, multi-load combinations involving both power electronic and inductive loads show an even greater degree of variance dispersion. This indicates that the parallel mode, to some extent, preserves the distinct time-domain characteristics inherent to different load types.

3.3.2. Frequency-Domain Characteristic Analysis of Multiple Loads

A spectral analysis of the data was performed using the FFT method to calculate the distribution of frequency components. Figure 17 presents the resulting harmonic current distribution for the multi-load parallel connection mode.

Under normal conditions, parallel combinations involving two different load types show that those containing resistive loads exhibit higher fundamental content than those with inductive loads. For instance, condition (j)—the parallel connection of two resistive loads—demonstrates the highest fundamental content in the fault state. Among parallel configurations, those with inductive loads generally yield lower fundamental content, while those incorporating switching power supply loads show the lowest normal-state fundamental content, at only 80%. Following an arc fault, a reduction in the fundamental component and an increase in higher-order harmonics are still observed in parallel mode; however, this trend is less pronounced in multi-load parallel configurations than in series mode.

The center frequency and frequency variance for the multi-load connection modes were calculated based on the power spectrum. The load types are listed in Table 2, and the average distribution from repeated tests is shown in Figure 18.

As shown in Figure 18, the center frequency and frequency variance for multi-load conditions involving the inductive induction cooker (conditions (i) and (m)) in parallel mode are significantly higher than those of other load combinations, reaching 150 Hz and 400,000, respectively. This trend is consistent with observations from single-load parallel connections, where the inductive cooker also exhibited much higher values compared to other loads clustered around 50 Hz. This consistency indicates that the multi-load parallel configuration preserves the frequency-domain characteristics of individual load types to some extent. Furthermore, the center frequency and frequency variance following an arc fault are markedly lower in the parallel mode than in the series mode. This suggests that the series connection contains richer harmonic components, whereas in the multi-load parallel mode, the difference in frequency variance between the arc fault and normal states is relatively small.

4. Arc Fault Recognition Method Based on FF-DCNN

4.1. Data Prepocessing and DCNN Model Structure

4.1.1. Dataset Establishment

To reasonably evaluate the generalization performance of the machine learning model and verify the reliability of the detection algorithm, each current sample data is labeled with 0 for normal samples (negative samples) and 1 for fault samples (positive samples). According to the 8:1:1 allocation ratio, the original current signal samples collected by the arc fault test platform are divided to construct three independent datasets, namely, the training subset, validation subset, and test subset. This ensures the independence of model training, parameter tuning, and performance evaluation. Specifically, the training subset is used for network parameter learning, the validation subset for hyperparameter optimization, and the test subset for the final evaluation of generalization ability as shown in

Table 6. Division Ratios of Datasets.

Signal Type	Label	Training Set	Validation Set	Test Set
Number of normal samples	0	540,175	67,521	67,521
Number of faulty samples	1	201,619	25,202	25,202
sample size		741,194	92,723	92,723

.

4.1.2. Data Preprocessing

(1)

1D Data Preprocessing

In this experiment, the raw signals of normal and arc fault states are collected as one-dimensional data. To enable more effective feature extraction, an overlapping sampling approach with time windows is applied to this one-dimensional data. Time windows are generally categorized into rolling window and sliding window techniques: the rolling window uses a window width equal to the moving stride, which may lead to insufficient extraction of key features; in contrast, the sliding window sampling method adopts a moving stride smaller than the window width, allowing consecutive data windows to capture more feature information. Choosing an appropriate stride can not only reduce computational overhead and information redundancy but also effectively extract features and enhance model performance. This sampling method enables better capture of the dynamic features of raw data during the transition from normal to fault states, facilitating a more comprehensive analysis of the development trend and variation pattern of faults. Figure 19 illustrates a schematic diagram of raw current data segmentation using the sliding window method.

Based on the typical time-domain and frequency-domain feature quantities selected through the time–frequency domain feature analysis in the previous chapter, feature extraction is performed on the one-dimensional signals, yielding two distinct feature vectors: variance and frequency variance. The schematic diagram of this process is illustrated in Figure 20.

The decomposed feature vectors are concatenated with the original data. Herein, blue represents the raw signal data, yellow denotes the variance, and green indicates the frequency variance. The concatenated feature vector set is used as the one-dimensional data for time–frequency domain feature fusion and serves as the input data for the 1D-CNN.

(2)

2D Data Preprocessing

To enhance the model’s capability in extracting time–frequency domain fusion features, the time–frequency domain features of the input signals from the two-dimensional channel are adopted. As analyzed in the previous chapter, the Continuous Wavelet Transform (CWT) is a key technique in time–frequency domain analysis. In this study, the data sampled via the sliding window are converted into two-dimensional data through CWT for input. This two-dimensional data carries more abundant time–frequency domain fusion features, which not only highlights the characteristics of the original current samples but also further strengthens the feature extraction of fused temporal- and frequency-domain information. Theoretically, this approach contributes to improved stability and accuracy of recognition.

4.1.3. FF-DCNN Model Structure

The convolutional neural network architecture for dual-channel time–frequency domain feature fusion proposed in this chapter is primarily composed of a data input module, the FF-DCNN network model, a feature extraction module, and a fault recognition module, with its architectural diagram illustrated in Figure 21. Specifically, the network model extracts time–frequency domain features corresponding to normal states and arc fault states from both one-dimensional and two-dimensional data. Subsequent to the in-depth fusion and iterative learning of these extracted features, the integrated feature vectors are fed into the fully connected layers and the Softmax layer for fault classification, thereby yielding the final recognition results of arc faults.

The structural parameter configuration of the FF-DCNN model is detailed in

Table 7. Specific parameter settings for the FF-DCNN model.

	1D Feature Extraction				2D feature Extraction
	Network Layer	Number of Convolutional Kernels	Size	Stride	Network Layer	Number of Convolutional Kernels	Size	Stride
Feature extraction layer	Conv1D MaxPool1D Conv1D MaxPool1D Conv1D MaxPool1D	16 ReLU / 32 ReLU / 32 ReLU /	3 × 1 2 × 1 3 × 1 4 × 1 3 × 1 4 × 1	2 2 2 4 2 4	Conv2D	48	11 × 11	1
					ReLU
					MaxPool2D	/	3 × 3	2
					Conv2D	128	5 × 5	2
					ReLU
					MaxPool2D	/	3 × 3	2
					Conv2D	192	3 × 3	1
					ReLU
					Conv2D	192	3 × 3	1
					ReLU
					Conv2D	128	3 × 3	1
					ReLU
					MaxPool2D	/	3 × 3	2
	Flatten	/			Flatten	/
Feature fusion	Feature splicing
full connectivity layer	full connectivity layer
Output	Output

. The feature extraction module of the model consists of multiple network layers, where each layer clearly specifies the number of convolution kernels, kernel size parameters, sliding stride, and the adopted ReLu activation function.

The equations of the convolutional neural network in this paper are shown in Equations (11)–(14) [16]. Among them, the input data of the 1D-CNN is a feature vector set formed by concatenating the feature vectors decomposed from the time domain and frequency domain with the original data, which is fed into the 1D-CNN for training. The 1D-CNN mainly consists of three convolutional layers, three max-pooling layers, and one flatten layer. Specifically, the one-dimensional input data is first fed into the convolutional layer of the 1D-CNN, and its calculation formula is given in Equation (11):

$${\mathrm{y}}_{1\mathrm{D}}^{\mathrm{j},\mathrm{l}}=\mathrm{f}({\displaystyle \sum _{\mathrm{i}\in {\mathrm{M}}_{\mathrm{j}}}{\mathrm{\omega }}_{\mathrm{i}\mathrm{j}}^{\mathrm{l}}}\ast {\mathrm{x}}_{1\mathrm{D}}^{\mathrm{i},\mathrm{l}-1}+{\mathrm{b}}_{\mathrm{j}}^{\mathrm{l}})$$

(11)

Herein, ${\mathrm{y}}_{1\mathrm{D}}^{\mathrm{j},1}$ denotes the j-th output feature matrix in the l-th layer, * represents the convolution operation, ${\mathrm{x}}_{1\mathrm{D}}^{\mathrm{i}.\mathrm{l}-1}$ denotes the i-th input feature matrix in the one-dimensional layer, ${\mathrm{w}}_{\mathrm{i}\mathrm{j}}^{\mathrm{l}}$ denotes the weight parameter matrix of the l-th layer, ${\mathrm{b}}_{\mathrm{j}}^{\mathrm{l}}$ denotes the corresponding bias term, and $\mathrm{f}\left(·\right)$ represents the nonlinear activation function. The subsequent pooling layer adopts max-pooling, and its formula is given in Equation (12):

$${\mathrm{O}}_{1\mathrm{D}}^{\mathrm{j},\mathrm{l}}=\mathrm{m}\mathrm{a}{\mathrm{x}}_{\mathrm{u}=0}^{\mathrm{n}-1}\mathrm{m}\mathrm{a}{\mathrm{x}}_{\mathrm{v}=0}^{\mathrm{n}-1}{\mathrm{I}}_{1\mathrm{D}}^{\mathrm{j}\cdot \mathrm{s}+\mathrm{u},\mathrm{l}\cdot \mathrm{s}+\mathrm{v}}$$

(12)

Herein, ${\mathrm{O}}_{1\mathrm{D}}^{\mathrm{j},\mathrm{l}}$ denotes the j-th output feature matrix in the l-th layer, $\mathrm{m}\mathrm{a}\mathrm{x}$ represents the max operation, the pooling window size is $\mathrm{n}\times \mathrm{n}$,$\mathrm{s}$ is the stride, and ${\mathrm{I}}_{1\mathrm{D}}^{\mathrm{j}+\mathrm{s}·\mathrm{u},\mathrm{j}+\mathrm{s}·\mathrm{v}}$ is the corresponding input feature matrix. Finally, the features are flattened via a flatten layer to output the feature matrix $y_{1D}^{output}$.

For the 2D-CNN, the input data is the time–frequency image obtained by CWT, which is fed into the 2D-CNN for training. The 2D-CNN mainly consists of five convolutional layers and three max-pooling layers. Specifically, the two-dimensional input data ${\mathrm{X}}_{2\mathrm{D}}$ is fed into the convolutional layer of the 2D-CNN, and its calculation formula is given in Equation (13):

$$Y_{2\mathrm{D}}^{\mathrm{j},\mathrm{l}}=\mathrm{f}({\displaystyle \sum _{\mathrm{i}\in {\mathrm{M}}_{\mathrm{j}}}{\mathrm{\omega }}_{\mathrm{i}\mathrm{j}}^{\mathrm{l}}}\ast {\mathrm{X}}_{2\mathrm{D}}^{\mathrm{i},\mathrm{l}-1}+{\mathrm{b}}_{\mathrm{j}}^{\mathrm{l}})$$

(13)

Herein, ${\mathrm{Y}}_{2\mathrm{D}}^{\mathrm{j}.\mathrm{l}}$ denotes the j-th 2D output feature matrix in the l-th layer, and ${\mathrm{X}}_{2\mathrm{D}}^{\mathrm{I},\mathrm{L}-1}$ denotes the i-th 2D output feature matrix in the $\left(\mathrm{l}-1\right)$-th layer. The subsequent pooling layer adopts max-pooling, and the features are flattened via a flatten layer to output the feature matrix $Y_{2D}^{output}$.

The features output by the 1D convolutional channel and 2D convolutional channel are concatenated to form the output of the feature fusion layer: ${\mathrm{Y}}_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}=\left[{\mathrm{y}}_{1\mathrm{D}}^{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}},{\mathrm{Y}}_{2\mathrm{D}}^{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}}\right]$. The fused feature data is then fed into the fully connected layers and the Softmax layer to obtain the final output $\widehat{\mathrm{y}}$, whose calculation formula is given in Equation (14):

$$\widehat{\mathrm{y}}=\mathrm{\sigma }({\mathrm{\omega }}_{\mathrm{R}}{\mathrm{Y}}_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}+{\mathrm{b}}_{\mathrm{R}})$$

(14)

In this equation, ${\mathrm{w}}_{\mathrm{R}}$ and ${\mathrm{b}}_{\mathrm{R}}$ represent the weight matrix and bias vector between the deep fusion feature connection layer and the final linear regression layer, while $\sigma$ denotes the Softmax classifier. The features extracted by the two channels are fed into the fully connected layer for label classification. Each neuron in the fully connected layer is connected to all learnable-weight neurons in the previous layer.

4.2. FF-DCNN Model Optimization

4.2.1. Model Evaluation Metrics

To evaluate the performance of arc fault classification models, objective metrics are generally adopted. As arc fault detection and classification models fall into the category of classification tasks, normal current samples in the test set are labeled as negative samples and fault samples as positive samples. In the model performance evaluation framework, the binary classification results after training can be systematically divided into four scenarios: True Positive (TP), referring to actual positive samples correctly predicted; False Positive (FP), actual negative samples incorrectly identified as positive; True Negative (TN), actual negative samples consistently predicted; and False Negative (FN), actual positive samples not correctly recognized. In the development of deep learning, the confusion matrix is commonly used to compare classification results for its clear reflection of mutual misclassification between categories, and the four scenarios of the confusion matrix for the arc fault binary classification task are shown in

Table 8. Confusion matrix.

Actual Class	Predicted Class
Positive Sample	TP	FN
Negative Sample	FP	TN

.

Based on the confusion matrix, accuracy is another commonly used evaluation metric in model assessment. Defined as the ratio of correctly classified samples to the total number of samples, its calculation formula is given in Equations (15)–(18) [16]. This metric can intuitively reflect the overall classification performance of the model; however, it has limitations in specific scenarios: when the sample distribution is imbalanced, the classifier fails to truly reflect the model’s ability to recognize minority classes.

$$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}={\displaystyle \frac{\mathrm{T}\mathrm{P}+\mathrm{T}\mathrm{N}}{\mathrm{T}\mathrm{P}+\mathrm{T}\mathrm{N}+\mathrm{F}\mathrm{P}+\mathrm{F}\mathrm{N}}}$$

(15)

Precision is used to measure the accuracy of positive class predictions, and its calculation formula is shown in Equation (16):

$$\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}={\displaystyle \frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}}}$$

(16)

Recall is used to evaluate the classifier’s ability to cover positive samples, which is defined as the proportion of samples correctly identified as positive among all actually positive samples, with its calculation formula shown in Equation (17):

$$\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}={\displaystyle \frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N}}}$$

(17)

As a comprehensive evaluation metric integrating precision and recall, the F1-Score enables a more reasonable assessment of model performance, with its calculation formula presented in Equation (18):

$$\mathrm{F}1\mathrm{-score}={\displaystyle \frac{2\cdot \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\cdot \mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}}{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}+\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}}}$$

(18)

All the aforementioned evaluation metrics range between [0, 1], with higher values indicating better performance of the arc fault detection model.

4.2.2. Loss Function

In the process of training the convolutional neural network model, a loss function must first be defined to measure the discrepancy between the model’s predicted values and the true values. By calculating the loss function, the model parameters can be updated to reduce optimization errors until they continuously approach the target values. The commonly used loss functions mainly fall into two categories: the mean squared error (MSE) function and the cross-entropy loss function (Loss) [17]. Among them, MSE measures the average of the squared differences between the predicted values and the true values, with its formula presented in Equation (19):

$$\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}})}}^2$$

(19)

where $\mathrm{n}$ denotes the total number of data samples, ${\mathrm{y}}_{\mathrm{i}}$ represents the true label of the data sample, and ${\widehat{\mathrm{y}}}_{\mathrm{i}}$ denotes the label predicted by the dual-channel neural network. Since MSE represents the discrepancy between predicted values and true values through mean squared error, which amplifies the error of outliers, it is generally applicable to regression tasks.

In contrast, the cross-entropy loss function quantifies the deviation between the predicted probability distribution and the true labels by calculating the information entropy discrepancy between them. When the cross-entropy loss function reaches its minimum value, the predicted probability distribution output by the model achieves the optimal match with the true label distribution—that is, the model’s prediction results have the highest consistency in probability distribution with the actual situation. This function is suitable for classification tasks. Therefore, the binary cross-entropy is selected as the loss function in this study, and its calculation formula is presented in Equation (20):

$$\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}=-{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}\mathrm{l}\mathrm{o}\mathrm{g}{\widehat{\mathrm{y}}}_{\mathrm{i}}+(1-{\mathrm{y}}_{\mathrm{i}})\mathrm{l}\mathrm{o}\mathrm{g}(1-{\widehat{\mathrm{y}}}_{\mathrm{i}}))}$$

(20)

During the model training process, minimizing the cross-entropy loss value can continuously improve classification accuracy and enhance the model’s generalization ability on the validation and test sets. To prevent overfitting, the Dropout method is adopted in this study, with the dropout rate set to 0.5, aiming to improve both the training efficiency and generalization performance of the model.

4.3. Verification Results and Analysis of the FF-DCNN Model

4.3.1. Comparison of Different Model Experiments

To verify the arc fault recognition performance of the proposed FF-DCNN model under different load types and connection modes, this chapter selects the following models for experimental comparison while keeping the dataset and initial training parameters unchanged [18,19,20,21,22,23,24,25]: CNN, Long Short-Term Memory (LSTM), Transformer, DCNN, and FF-DCNN.

The method of repeating training five times and taking the average value was adopted to record the average accuracy and loss curves of different models. Figure 22 presents the comparative visualization results of various models in arc fault detection. As observed from the figure, during the training iteration process, the accuracy of all models showed a trend of rapid increase followed by gradual convergence. For the DCNN-based model, its accuracy rose from 27.23% to 73.07% in the first 10 iterations, then entered a plateau phase, and did not surge to around 95% until after 72 iterations. Although the Transformer-based model exhibited a fast convergence rate in the early stage of iteration, significant fluctuations occurred when its accuracy reached 73%, and it ultimately converged to a relatively low accuracy. In contrast, the traditional CNN-based and LSTM-based models had a slower accuracy growth rate compared to other models, with gradual convergence achieved only at the 75th iteration. Notably, the FF-DCNN model proposed in this study achieved an extremely rapid growth rate: its accuracy soared from 56.33% to 95.43% in the first 10 iterations, while the loss value dropped from 0.982 to 0.077. Afterward, with slight fluctuating growth, the accuracy converged to approximately 98%. The proposed model not only obtained a high average classification accuracy but also maintained a low average loss value. In addition, the model demonstrated excellent stability during the convergence process, fully verifying that compared with other comparative methods, it possesses stronger generalization performance and robustness, and can effectively adapt to diverse test scenarios.

Table 9. Comparison results of different models.

Model	Accuracy	Precision	Recall	F1-score
CNN	97.62%	97.41%	97.54%	97.59%
LSTM	96.19%	95.86%	96.08%	96.08%
Transformer	73.31%	73.75%	73.75%	62.03%
DCNN	96.19%	95.86%	95.74%	96.08%
FF-DCNN	98.52%	98.41%	98.29%	98.29%

details the performance results of different comparative models in arc fault detection. To ensure the reliability of the experimental results, each model was trained five times repeatedly, with the average value taken for final evaluation.

As can be seen from Table 9, the Transformer-based model achieves relatively low recognition accuracy and precision, with the accuracy merely reaching 73.31%. In comparison, the CNN-based and DCNN-based models outperform the LSTM and Transformer models in recognition precision, indicating that the inductive bias of convolutional neural networks offers more appropriate advantages in feature extraction and recognition of arc faults. Notably, the FF-DCNN model proposed in this study exhibits optimal performance. Compared with the other four model methods, it achieves an accuracy as high as 98.52%, a precision of 98.41%, and an F1-score of 98.29%. Its accuracy shows a significant improvement, with a minimum increase of 0.9% and a maximum increase of 25.21% compared to other models. This superior performance is attributed to the model designed in this study, which integrates time-domain and frequency-domain features and adopts a dual-channel parallel approach for feature extraction, enabling the model to more efficiently learn the data features of arc faults.

4.3.2. Ablation Study

To verify the performance gain of the dual-channel time–frequency fusion mechanism over the single-channel architecture, this study designs two groups of controlled experiments for ablation study [26]: the first group retains only the 1D-CNN, taking the one-dimensional vector formed by concatenating the original vibration signals and time–frequency features as the network input; the second group retains only the 2D-CNN independently, directly using the two-dimensional spectrogram generated by time–frequency domain transformation as the feature representation. The specific experimental results are detailed in

Table 10. Results of the model ablation test.

Method	Accuracy	Precision	Recall	F1-Score
Retain only the 1D-CNN	97.50%	97.64%	97.34%	97.36%
Retain only the 2D-CNN	95.29%	94.88%	95.07%	95.07%
FF-DCNN	98.52%	98.41%	98.28%	98.29%

. By adopting the control variable method, the experimental design effectively verifies the improvement effect of dual-channel feature fusion on model accuracy.

As indicated by the results in Table 10, compared with the model retaining only the 1D-CNN, the model with only the 2D-CNN retained exhibits a decrease of 2.21% in accuracy, 2.76% in precision, 2.27% in recall, and 20.29% in F1-score. This may be attributed to a certain degree of information loss that occurs when converting the original current data into wavelet time–frequency diagrams. In contrast, compared with the model retaining only the 1D-CNN, the FF-DCNN model achieves an increase of 0.98% in accuracy, 0.77% in precision, 0.94% in recall, and 0.93% in F1-score. When compared with the model retaining only the 2D-CNN, the FF-DCNN model shows improvements of 3.23% in accuracy, 3.53% in precision, 3.21% in recall, and 3.22% in F1-score.

The experimental results demonstrate that the constructed FF-DCNN network, by introducing the dual-channel time–frequency feature fusion mechanism, exhibits significant advantages in feature representation capability and model generalization performance. Compared with the single-channel architecture, the proposed model can more fully capture the characteristic patterns of time-domain and frequency-domain distributions in arc fault signals, making the model’s detection and recognition more effective.

4.4. Verification Results and Analysis of Different Load Types

To evaluate the arc fault detection performance of the proposed model under different load types, Figure 23 presents the experimental results of the model proposed in this study across various loads.

Table 11. Comparison results of different load type tests.

Number	Load Type	Load	Accuracy	Precision	Recall	F1-Score
1	Resistive load	Electric kettle	99.99%	99.94%	98.62%	98.92%
2		Hair dryer	97.40%	96.30%	97.68%	96.92%
3	Inductive load	Induction cooker	99.99%	99.79%	99.87%	99.98%
4		Electric fan	99.99%	99.18%	99.17%	99.45%
5	Power electronic load	Energy-saving lamps	99.52%	98.09%	98.10%	98.19%
6		Laptop computer	99.89%	99.13%	99.79%	99.56%
7	Switching power supply load	Microwave oven	99.20%	99.17%	99.23%	99.17%

presents the experimental results of the proposed model in this study under different load types.

As can be observed from Figure 23, the proposed FF-DCNN model demonstrates excellent accuracy across different load types. Notably, the accuracy varies with load types: the accuracy reaches over 98% for electric kettles, induction cookers, laptops, and microwave ovens. Among them, the inductive loads, specifically induction cookers and electric fans, achieve the highest recognition accuracy, with all four evaluation metrics approaching 100%. In contrast, the accuracy for hair dryers is significantly lower than that of other loads, hovering around 97%.

4.5. Verification Results and Analysis of Different Connection Modes

To evaluate the arc fault detection performance of the proposed model under different connection modes, Figure 24 presents the experimental results of the model proposed in this study with various connection modes.

Table 12. Comparison results of different connection modes.

Number	Load Type	Load Name	Accuracy	Precision	Recall	F1-Score
1	Resistive load	Electric kettle	98.01%	97.12%	97.71%	97.72%
2	Inductive load	Hair dryer	97.01%	96.25%	96.99%	96.21%
3	Electronic load	Induction cooker	97.34%	97.24%	96.93%	96.92%
4	Switching power supply load	Electric fan	98.12%	98.01%	98.17%	98.00%
5	Resistive and inductive loads	Energy-saving lamps	97.43%	98.27%	98.14%	98.13%
6	Inductive and inductive	Laptop computer	98.04%	98.13%	97.79%	97.96%
7	Resistive and resistive	Microwave oven	98.34%	98.24%	98.13%	98.23%
8	Resistive and power electronic	Electric kettle + Electric fan	95.69%	95.23%	95.67%	95.13%
9	Resistive and inductive	Induction cooker + Electric fan	98.80%	98.67%	99.16%	99.66%
10	Inductive and power electronic	Electric kettle + Electric kettle	99.16%	99.06%	98.98%	99.06%
11	Resistive and power electronic	Electric kettle + Energy-saving lamp	97.73%	94.13%	97.16%	97.17%
12	Resistive and power electronic	Hair dryer + Electric fan	92.87%	92.85%	91.76%	92.48%
13	Power electronic and switching power supply	Induction cooker + Energy-saving lamp	95.37%	94.89%	95.46%	96.01%
14	Resistive load	Hair dryer + Energy-saving lamp	92.76%	92.85%	92.46%	92.73%
15	Inductive load	Hair dryer + Laptop computer	97.76%	98.12%	98.06%	97.95%
16	Electronic load	Energy-Saving lamp + Microwave oven	99.08%	98.37%	97.86%	97.97%

presents the experimental results of the proposed model in this study under different connection modes.

A comparison of Table 10 and Table 11 reveals that the arc recognition accuracy of single loads under the parallel connection mode is generally lower than that under the series connection mode. Among them, the induction cooker load exhibits the maximum accuracy decrease of 2.66% compared with the series connection mode, while the minimum decrease is 0.39%. This indicates that during the detection process, the current signals under the parallel load connection are subject to more interference than those under the single-load series connection, making arc fault recognition more challenging. Under the multi-load parallel connection mode, the accuracy under complex load conditions involving hair dryers is relatively low, hovering around 92%. This may be attributed to the burrs contained in the current waveform when hair dryers are in operation. However, the overall arc recognition accuracy across different connection modes reaches 97.09%, demonstrating that the proposed detection model achieves excellent performance in arc fault recognition under various connection modes.

5. Discussion

Through the development of a series–parallel arc fault test platform, this study systematically reveals how connection modes influence arc characteristics. Comparative analysis of single- and multi-load scenarios clarifies the impact of wiring configuration on detection features. The principal conclusions are as follows:

• Simulations of real-world parallel loads (four single, nine multi-load types) showed that parallel arcs reduce time-domain dispersion by >50% and increase fundamental content. These findings are directly applicable to refining electrical fire monitoring algorithms.
• Under different connection configurations of a single load, a comparison of time–frequency domain characteristics between series and parallel connections reveals that, in the case of parallel connection, although significant differences in load types persist, the waveform characteristics of different loads become submerged, and high-frequency harmonics are attenuated, resulting in notable convergence behavior. The distribution of time-domain features becomes more uniform in the parallel configuration, making it easier to determine the occurrence of arc faults using the fault-to-normal ratio of time-domain characteristic parameters compared to the series configuration. Furthermore, under parallel connection, the influence of load type on the distribution of frequency-domain characteristics is reduced. Although the fundamental frequency component decreases and higher harmonic components increase during arc fault conditions, this trend is less pronounced in the parallel connection than in the series connection.
• Under the parallel multi-load connection mode, a comparison of the time–frequency domain characteristics between series and parallel connections of different load types reveals that the underlying physical mechanisms mainly stem from the current superposition in parallel paths and changes in system impedance characteristics. In a parallel circuit, multiple loads are connected to the same power supply, and the currents of each branch superpose at the nodes, resulting in a smoother total current waveform. First, the current superposition and harmonic cancellation effects cause high-frequency harmonics generated by different loads to partially cancel each other out at the nodes due to phase differences, thereby reducing the overall harmonic content and enhancing the significance of the fundamental wave component. Second, the influence of system impedance cannot be ignored. Parallel paths reduce the equivalent impedance of the system, and high-frequency noise generated by arcs is more easily shunted and attenuated in low-impedance paths, leading to more concentrated frequency-domain characteristics. In addition, the nonlinear characteristics of arcs exhibit more complexity in the parallel structure—the randomness and instability of arcs are “averaged out” by the steady-state characteristics of multiple loads, resulting in a reduction in the dispersion of time-domain characteristics. Therefore, after an arc fault occurs, both the centroid frequency and frequency variance in the parallel connection decrease more significantly than those in the series connection. This indicates that in arc faults with multi-load parallel connections, harmonic components are significantly reduced due to the aforementioned mechanisms, revealing the convergent modulation effect of the parallel structure on the harmonic characteristics of arc faults.
• This study observes that the composite characteristics of multiple parallel loads are not a simple linear superposition of the characteristics of their constituent loads. In accordance with the provisions of the UL 1699 [8] and IEC 62606 [9] standards, the key finding of this experiment is that under multiple parallel loads, the inherent characteristics of each load (such as the inductive spikes of motors and the broadband noise of switching power supplies) undergo mutual modulation and coupling in the circuit. This results in the overall characteristics being not an arithmetic sum of individual components but rather exhibiting the more concealed “emergent” phenomena highlighted by the standards—including the cancellation or enhancement between harmonics of different loads—and the overall frequency response shift caused by impedance matching changes. Consequently, detection algorithms designed based on single-load characteristics are prone to failure under such complex operating conditions, which inversely demonstrates the necessity and superiority of the standard-based test method adopted in this study that covers multiple load combinations.
• To address the challenge of high difficulty in arc fault detection under complex parallel load environments, this study proposes a time–frequency domain feature fusion detection method based on the Dual-Channel Convolutional Neural Network (FF-DCNN). This method extracts the time–frequency features of one-dimensional current signals and the deep features of two-dimensional wavelet time–frequency diagrams, respectively, achieving high-robustness and high-precision recognition of arc faults. It maintains an average accuracy exceeding 97% under different connection modes and load types, which is significantly superior to traditional models. The algorithm not only provides a new technical approach for electrical fire monitoring but also has its engineering application value initially verified: the prototype of a dedicated detection chip configured based on this method has now entered the testing phase. Preliminary results indicate that its detection accuracy can reach over 90%, laying a solid foundation for the official launch of subsequent products.
• This study reveals the variation patterns of arc fault characteristics under different connection configurations. The findings not only provide theoretical support for arc fault detection techniques but, more importantly, offer a foundational basis for predicting abnormal temperature rises in electrical circuits and developing early fire warning systems. In future research, we will combine numerical simulation with the FF-DCNN method proposed in this study, which will further deepen the understanding of the intrinsic nature of arc faults.