When a single-phase grounding fault occurs at the neutral point of an arc suppression coil grounding system, the transient capacitive current, transient inductive current, and transient grounding current at the fault point can be analyzed using the equivalent circuit shown in Figure 1.

As shown in Figure 1, the transient equivalent circuit describes the dynamics of single-phase grounding faults in resonant grounding systems. The transient response of the system can be analyzed by considering the interaction of capacitive and inductive components. The oscillatory behavior at the fault point is primarily driven by the energy exchange between the inductance L 0 and the capacitance C , while the damping effect is determined by the resistive components.

Due to the high natural frequency of the transient capacitive current and the fact that the inductance of the arc suppression coil L is much greater than L 0 , the effects of R L and L shown in Figure 1 can be neglected. Therefore, by using the series circuit composed of L 0 , R 0 and C , and applying the zero-sequence sinusoidal voltage u 0 across it, the transient response of the capacitive current i C can be effectively determined as shown in Equation 1. R 0 i c + L 0 d i c d t + 1 C ∫ 0 1 i c d t = U m ⁡ sin ω t + φ (1) where U m is the amplitude of the phase voltage; φ is the initial phase angle of the fault.

Therefore, the transient capacitive current i C includes the transient free oscillation component i C o s and the steady-state power frequency component i C s t .

At the initial time, the initial condition satisfies the following conditions as described in Equation 2: I C o s + I C s t = 0 I C m = U m ω C (2)

By performing Laplace transform operations, the relevant mathematical expressions can be obtained: i C = i C o s + i C s t = I C m ω f ω sin φ sin φ f t − cos φ cos φ f t e − δ t + cos ω t + φ (3) In Equation 3, i C represents the total transient capacitive current, which is a combination of the transient free oscillation component i C o s and the steady-state power frequency component i C s t . Here, I C m is the amplitude of the steady-state current component, ω is the angular frequency of the system, and ω f is the natural angular frequency of the system. The phase angles φ and φ f correspond to the steady-state and transient components, respectively, while δ is the damping coefficient, which affects the rate at which the transient oscillations decay.

In power systems, the time constant τ C is considered a constant, and its value has a significant impact on the system’s dynamic response. The magnitude of τ C directly determines the damping rate of the system’s free oscillations: when τ C is large, the system’s free oscillations decay more slowly; conversely, when τ C is small, the decay rate of the free oscillations increases. Additionally, the free oscillation component i C o s in the power system is influenced by sin ⁡ φ and cos ⁡ φ and can be generated under ground faults at any phase angle φ . Specifically, when the phase angle φ equals zero, the value of the free oscillation is minimal; when the phase angle φ equals π / 2 , that is, under a ground fault at the voltage peak, the value of the free oscillation is maximal, occurring at a quarter of the free oscillation period T f . In this case, the amplitude of i C o s is at its maximum and is proportional to the ratio of the system’s ω f to ω . The larger this ratio, the higher the maximum amplitude γ C max of the free oscillation component.

When a single-phase ground fault occurs and the voltage of the faulted phase drops to 0, no additional transient capacitive current free oscillation component is generated. Under this condition the voltage difference across the capacitor is 0, preventing a rapid change in the charge within the capacitor and resulting in minimal transient capacitive current. In this case, the amplitude of the transient free oscillation current reaches its minimum value when the time is equal to half of the free oscillation period T f . This minimum amplitude can be described by the following Equation 4: i C o s ⁡ min = I C m ω f ω e T f / 2 T C (4)

Alternatively, it can be calculated using the ratio: γ C min = ω f ω e T f / 2 T C (5) Equation 5 indicate that, when the phase angle φ is zero, the transient capacitive current has a free oscillation component equal in amplitude to the power frequency capacitive current.

Analyzing the free oscillation current is crucial because it is directly related to the stability and safety of the system. The calculation of the natural angular frequency ω f is a core part of this analysis, and the derivation process is shown in Equation 6: ω f = ω 0 2 − δ 2 ω 0 = 1 L 0 C T 0 = 2 π L 0 C (6)

The attenuation ratio of the current amplitude, γ C , reflects the degree of current amplitude attenuation after a time period of T 0 / 2 and can be calculated using the following Equation 7: γ C = e R 0 T 0 4 L 0 (7)

This ratio is generally used to assess the damping characteristics of the power grid. Experimental observations have found that the damping ratio in overhead line networks typically ranges between 1.5 and 2.0, indicating significant current attenuation after half a cycle.

Experimental results indicate that the dynamic response of cable lines is influenced by their physical characteristics. The free oscillation frequency of cable lines is relatively high, typically ranging from 1,500 to 3,000 Hz. This characteristic results in a shorter response time during transients, allowing cable lines to reach a steady state more quickly. In contrast, the free oscillation frequency of overhead lines usually ranges from 300 to 1,500 Hz.

When analyzing the transient process of an arc suppression coil, the first consideration is the expression of the magnetic flux in the core. During the transient process, the magnetic flux in the core remains the same as it is in the unsaturated state. Therefore, the known expression for the magnetic flux under static conditions can be directly used to determine the magnetic flux in the core during the transient process, which in turn allows for the calculation of the neutral point current through the arc suppression coil. As shown in Figure 1, according to Kirchhoff’s second law, we can derive the following expression for the magnetic flux, as described in Equation 8: U m ⁡ sin ω t + φ = R L i L + N d ϕ L d t (8) where N is the number of turns in the arc suppression coil in the system; ϕ L is the magnetic flux passing through the iron core of the arc suppression coil.

The relationship between the current through the arc suppression coil and the magnetic flux linkage can be expressed as shown in Equation 9: L I L = N ϕ L (9)

Since the magnetic flux through the core of the arc suppression coil is zero. Substituting I L into Equation 8, the magnetic flux ϕ L can be obtained as: ϕ L = ϕ s t ω L Z cos φ + ε e − t τ L − cos ω t + φ + ξ ϕ s t = U m N ω ξ = arctan R L ω L Z = R L 2 + ω L 2 (10) where ϕ s t is the steady-state magnetic flux; ξ is the phase angle of the compensating current; Z is the impedance of the arc suppression coil; τ L is the time constant of the inductive circuit.

Since R L ≪ ω L , the impedance of the arc suppression coil can be approximated as Z ≈ ω L , and the phase angle of the compensation current, denoted by ξ , can be considered to be zero. Equation 10 can be simplified to Equation 11: I L = I L m cos ⁡ φ e − t τ L − cos ω t + φ I L m = U m / ω L (11) where I L m is the amplitude of the inductive current.

In a low-current grounding system, the transient process at a single-phase ground fault point is mainly composed of transient capacitive current and transient inductive current. The values are described in Equation 12: i D = i C + i L = I C m − I L m cos ω t + φ + I C m ω f ω sin ⁡ φ ⁡ sin ω f t − cos ⁡ φ ⁡ cos ω f t e − t τ C + I L m ⁡ cos ⁡ φ e − t τ L (12)

In the equation: the first term is the steady-state component of the ground fault current, and the remaining two terms are the transient components of the ground fault current.

In power systems, the amplitude of transient capacitive current is typically much larger than that of transient inductive current, and it decays more rapidly as well. Whether the neutral point of the power grid is ungrounded or grounded through an arc suppression coil, the amplitude and frequency of the transient grounding current are primarily influenced by the transient capacitive current, with its amplitude also being dependent on the initial phase angle.

Based on the above analysis, the transient frequency during a single-phase grounding fault in a low-current grounding system typically ranges from 300 to 1,500 Hz (Abdel-Fattah and Lehtonen, 2010). This frequency determines the duration of the initial half-wave and provides essential guidance for selecting the data window length for the ResNet-50 model.

To superimpose the zero-sequence voltage and zero-sequence current on a single image, it is necessary to normalize the sampled values of the zero-sequence voltage and zero-sequence current, as shown in the following Equation 13: u 0 n o r t = u 0 t max u 0 t , t ∈ t 0 , t 0 + 1 m s i 0 k n o r t = i 0 k t max i 0 k t , t ∈ t 0 , t 0 + 1 m s (13) where u 0 is the sampled value of the bus zero-sequence voltage for the first half-wave, and i 0 k is the sampled value of the zero-sequence current for the kth feeder. u 0 n o r and i 0 k n o r are the normalized zero-sequence voltage and the normalized zero-sequence current of the kth feeder, respectively. t 0 is the fault occurrence time.

As shown in Figure 2, the waveforms of a single-phase ground fault under normal line conditions are presented. During the first half wave after the fault occurs, the zero-sequence current and zero-sequence voltage maintain the same phase relationship. Figure 3 shows the waveforms of a single-phase ground fault in a faulty line. In this scenario, during the first half wave after the fault occurs, the zero-sequence current and voltage are in opposite phases. This difference can be used to distinguish normal lines from faulty lines.

Figures 4, 5 show the preprocessed waveforms for the normal line and the faulty line, respectively. These figures highlight the transient process and normalize the voltage and current waveforms to make the features more apparent. This helps the ResNet-50 model in fault line selection.

ResNet-50 is a type of deep convolutional neural network. It was proposed by Kaiming He et al. (He et al., 2016), where the concept of residual learning was first introduced. This effectively alleviates the problems of gradient vanishing and gradient explosion, significantly improving the algorithm’s reliability. Specifically, the residual module includes two or more convolutional layers, as well as a skip connection that directly adds the input to the output, as shown in Figure 6. The residual block, illustrated in Figure 6, consists of a series of weight layers followed by Rectified Linear Unit (ReLU) activation functions. The input x is passed through these layers to produce f(x), which is then added to the original input x to form the output f(x)+x. This skip connection helps in preserving the gradient flow during backpropagation, thereby mitigating the issues of vanishing and exploding gradients.

The loss function used for training the ResNet-50 model is the Cross-Entropy Loss, which is well-suited for classification tasks. This function measures the difference between the predicted class probabilities and the true class labels, guiding the optimization process to minimize the prediction error.

This model includes an input module, 4 residual modules, and an output module. In this paper, the number of neurons in the final fully connected layer is changed to 2 to adapt it for single-phase ground fault line selection. The network structure is shown in Table 1.

Applying ResNet-50 to small current grounding fault line selection, the fault identification algorithm flowchart is shown in Figures 7, 8. The specific process is as follows.

(1) Collect zero-sequence voltage and zero-sequence current data at the time of the fault, intercept the data within 1 m of the fault occurrence, preprocess it, and superimpose the waveforms to form a dataset.

To verify the accuracy of the proposed method, the PSCAD tool was used to simulate the system, and a 110/10 kV simulation grid was constructed. The small current grounding system established in this study is shown in Figure 9. The model and parameters are as follows: the power source is an infinite power source, the transformer capacity is 1,000 MVA, the transformation ratio is 110/10 kV, and there are four outgoing lines connected to the substation bus. The lengths of the feeders are as follows: 20 km (Line 1), 15 km (Line 2), 10 km (Line 3), and 5 km (Line 4). The line parameters are unit positive-sequence impedance Z 1 = 0.17 + j 0.38 Ω / km , unit zero-sequence impedance Z 0 = 0.23 + j 1.72 Ω / km , and the end of each line is equipped with a transformer and load.

By selecting different fault initial phase angles, grounding transition resistances, fault point distances, and arc suppression coil compensation degrees, different conditions of single-phase grounding faults are simulated. The types and values of the varying parameters are shown in Table 2, in which both full compensation and over-compensation scenarios are included to ensure a comprehensive evaluation of the ResNet-50 model under various conditions.

In a noise-free environment, by setting different compensation methods, single-phase grounding fault simulations are performed for different positions on lines L1 to L4, with various initial fault phases and different grounding resistances. As shown in Table 2, the dataset consists of 1,536 samples, with 80% used for training, 10% for validation, and 10% for testing. This division ensures that the ResNet-50 model is trained, validated, and tested on separate data subsets to accurately assess its performance. These simulated waveforms are input into the ResNet-50 model for training, with a batch size of 32 and 50 iterations. Ten percent of the dataset is randomly selected as the validation set for verification, and the validation accuracy reaches 100%.

The ResNet-50 model training results are shown in Figures 10, 11. Figure 10 illustrates the algorithm loss, which gradually converges, indicating effective model training. Figure 11 displays the validation accuracy, which stabilizes at 100%, indicating that the model has high accuracy in a noise-free environment. These results demonstrate that the ResNet-50 model can accurately identify and classify single-phase grounding faults under ideal conditions.

Noise refers to irregular information that is unrelated to the original signal. The ratio of the original signal to the noise is called the signal-to-noise ratio (SNR). To verify the noise resistance capability of the proposed method, 30 Decibel (dB) Gaussian white noise was added to the fault zero-sequence current and voltage, and tests were conducted. The test results are as follows.

Figures 12, 13 show the waveforms under normal and fault line noise environments. Despite the presence of significant noise, Figure 12 shows that in a normal line noise environment, the zero-sequence current and zero-sequence voltage maintain the same phase relationship. Figure 13 illustrates that in a fault line noise environment, the zero-sequence current and voltage are in opposite phases. These results indicate that even when the data is corrupted by noise, the distinct characteristics of the normal and fault conditions remain evident.

The verification results show that the waveform validation accuracy in a noisy environment is still 99.77%, indicating that even in a 30 dB noise environment, the algorithm can still efficiently perform single-phase grounding fault line selection. This high validation accuracy demonstrates the model’s robustness and its ability to effectively filter out noise while retaining the essential features needed for accurate fault detection.

To verify the effectiveness of the proposed method in a real-world environment, waveform data recorded on-site in Xi’an, Shaanxi, were used. These data were collected using advanced recording equipment provided by NARI Relays, a subsidiary of the State Grid Corporation of China. The waveform data accurately represent the voltage and current conditions during single-phase grounding faults in an actual distribution network. A total of 25 on-site waveforms were input into the trained ResNet-50 model for verification. The verification results demonstrate that the line selection accuracy for all test waveforms is 100%.

Figure 14 shows some of the on-site recorded waveforms, including voltage, normal current, and fault current. The figure clearly illustrates the fault occurrence time and its impact on the system: the single-phase grounding fault occurred at approximately 75 milliseconds, causing a significant disturbance in the voltage waveform within the first millisecond. Both the normal current and fault current waveforms exhibit pronounced spikes, with opposite phases, immediately following the fault occurrence. The verification results confirm the reliability and adaptability of the proposed method under various fault conditions, providing strong support for fault detection in actual distribution networks.