An Approach for Detecting Faulty Lines in a Small-Current, Grounded System Using Learning Spiking Neural P Systems with NLMS

Abstract

Detecting faulty lines in small-current, grounded systems is a crucial yet challenging task in power system protection. Existing methods often struggle with the accurate identification of faults due to the complex and dynamic nature of current and voltage signals in these systems. This gap in reliable fault detection necessitates more advanced methodologies to improve system stability and safety. Here, a novel approach, using learning spiking neural P systems combined with a normalized least mean squares (NLMS) algorithm to enhance faulty line detection in small-current, grounded systems, is proposed. The proposed method analyzes the features of current and voltage signals, as well as active and reactive power, by separately considering their transient and steady-state components. To improve fault detection accuracy, we quantified the likelihood of a fault occurrence based on feature changes and expanded the feature space to higher dimensions using an ascending dimension structure. An adaptive learning mechanism was introduced to optimize the convergence and precision of the detection model. Simulation scheduling datasets and real-world data were used to validate the effectiveness of the proposed approach, demonstrating significant improvements over traditional methods. These findings provide a robust framework for faulty-line detection in small-current, grounded systems, contributing to enhanced reliability and safety in power system operations. This approach has the potential to be widely applied in power system protection and maintenance, advancing the broader field of intelligent fault diagnosis.

1. Introduction

In China, the distribution network that operates below 35 kV is referred to as a small-current, grounded system [1,2]. A single-phase grounding fault in such a system, if not addressed promptly, can lead to significant disruptions in the network’s operation and, in severe cases, can pose serious safety risks. Therefore, it is crucial to quickly identify and isolate the faulty line. Research on small-current, single-phase grounding fault detection focuses on developing methods for fast and accurate faulty line identification. Advancing faulty line selection technology for small-current, grounded systems is essential to ensure the reliable operation of distribution networks and to enable the full automation of transformer substations. However, due to the subtle nature of fault characteristics, compounded by communication interference and the inherent variability within the distribution network, it is challenging for operators to quickly pinpoint the faulty line. Thus, improving fault detection and line selection technology is vital for the efficient and accurate resolution of single-phase ground faults in small-current, grounded systems.

First of all, the line selection method based on a steady-state signal [3,4,5,6,7,8,9] is a more mature method in China, which has high accuracy rate and high sensitivity, but it is not suitable for intermittent transient ground faults. The transient line selection method [10,11,12,13] can make up for the shortcomings of the steady-state line selection method to a certain extent but, in the event of a high impedance ground fault, the sensitivity of the transient line selection method is reduced or even failed because there are few transient signals contained in the fault signal. Therefore, both the steady-state line selection method and the transient line selection method have certain limitations. The line selection method based on information fusion technology [14,15,16,17] has been favored by an increasing number of researchers in view of the shortcomings of the low reliability of the single line selection method. The line selection method based on neural networks [18,19,20,21,22,23,24,25] is another one of these type of methods, and it mainly operates through sample training to change the synaptic weight value of the neural network so as to dynamically adjust the weight of each line selection fault feature input to the neural network and to improve the accuracy of line selection.

Spiking neural networks (SNNs), considered a third-generation artificial neural network (ANN) model, have garnered significant attention due to their efficiency [26,27,28,29]. These networks have been utilized in various fields, such as signal and image processing, classification, epilepsy and seizure detection, robotic control, and time-series forecasting. In parallel, over the past two decades, other computational paradigms have emerged alongside ANN developments. Membrane computing (MC), a branch of natural computing, was proposed by Gh. Păun in 1998 [30,31]. MC, also called P systems, is inspired by the structure and the function of living cells [32]. Currently, P systems can be divided into (1) cell-like P systems [33]. (2) Tissue-like P systems [34,35]. (3) Neural-like P systems [36,37,38]. In recent years, spiking neural P systems (SN P systems) [39,40,41,42], which are one of the neural-like P systems, have been studied as both theoretical models and as real applications, like in optimization problems [29,43,44,45,46,47,48], fault diagnosis [49,50,51], arithmetic calculators [45,52], robot controllers [53,54], image processing [55,56,57,58], and medical image analysis [29,59,60,61,62,63].

Therefore, this paper is dedicated to accomplishing the following objectives. Initially, the SNP system is extended to encompass applications in the selection of lines affected by small-current ground faults. Secondly, this paper aimed to enhance the precision of faulty line detection within small-current, grounded systems. In China, the majority of medium-voltage distribution networks operate with a small-current, grounded system. Furthermore, it is estimated that 70–80% of the faults occurring in these distribution networks are due to single-phase grounding [64,65,66,67]. Consequently, identifying single-phase-to-ground faults in small-current, grounded systems is a prevalent diagnostic challenge in power systems [68,69,70,71]. To address this, a novel approach for detecting faulty lines in small-current, grounded systems using the LSNPSn was introduced to boost detection accuracy. The process begins with an analysis of eight unique characteristics of three-phase current signals, three-phase voltage signals, active power, and reactive power within the small-current, grounded system. Following this, a faulty line detection model was constructed using the LSNPSn.

The structure of this paper is as follows. Section 2 describes the LSN P systems, detailing their structure. Section 3 analyzes, in detail, the changes in multiple characteristic quantities when a small-current ground fault occurs through simulation. Section 4 proposes a faulty line selection model based on LSNPSn. Section 5 presents and analyzes the experimental results. Finally, a discussion and conclusions are drawn in Section 6.

2. Preliminaries

LSN P systems are forward neural networks composed of multiple propositional neurons and regular neurons, and its structure is shown in reference [72,73]. LSN P systems combine high-dimensional coding with a unique weight adjustment mechanism, and its structural expression is as follows:

$$\Pi =\left(\begin{array}{c}a,\left\{\begin{array}{c}{\varphi }_{p1}^1,...,{\varphi }_{pk}^1,{\varphi }_{p1}^3,{\varphi }_{p1}^5\end{array}\right\},\{{\varphi }_{r1}^2,...,{\varphi }_{rk}^2,{\varphi }_{r1}^4,...,{\varphi }_{rk}^4\},syn,lN,OUT\end{array}\right),$$

where

1.

Let b represent a spike;

2.

The configuration of the proposition neuron is defined as $\begin{array}{ccc}\hfill {\varphi }_{qi}^m& =& \{0,w_{ij}^m,{\zeta }_i^m,{\gamma }_i^m\},\hfill \end{array}$ where m denotes the layer index.

(a)

0 indicates the absence of an initial potential in the proposition neurons.

(b)

The weights follow the forms $w_{ij}^1=\mathrm{rand}(0,1)$ and $w_{ij}^3=1$, where $1\le i\le k$, $1\le j\le n$.

(c)

The firing rules are included in the set ${\gamma }_i^m$, where each rule follows the pattern $q_i^1:q_i^1:{\mathcal{F}}^1/b^{{\delta }_i}\to b^{{\delta }_i}$, with ${\mathcal{F}}^1=\{{\delta }_i\ge 0\}$ and $1\le i\le k$. The final neuron is termed the bias neuron, and $\begin{array}{c}\hfill {\delta }_k=1.q_1^3:{\mathcal{F}}^3/b^p\to b^p\end{array}$, where ${\mathcal{F}}^3=\{p\ge 0\}$, and $\begin{array}{c}\hfill p={\varphi }_1\vee {\varphi }_2\vee \cdots \vee {\varphi }_n\end{array}$. Here, ${\varphi }_j$ will be defined in the following paragraph. The rule $q^5:b\to b$ lacks a firing condition, implying it is always active when the incoming potential reaches 1.

(3)

A rule neuron is characterized by ${\varphi }_{tj}^l=\left(0,1,1,{\eta }_j^l,{\gamma }_j^l\right)$, where l specifies the layer.

(a)

The initial potential for rule neurons is set to zero.

(b)

All rule neurons have a fixed factor of 1.

(c)

The synaptic weight from rule neurons is designated as 1, implying that the output weights of standard neurons do not influence the weight adjustment in the LSN P system.

(d)

Firing rules for ${\varphi }_{t1}^2,\dots ,{\varphi }_{tn}^2$ are structured as $q_j^2:{\mathcal{F}}^2/b^{{\varphi }_j}\to b^{{\varphi }_j}$, where ${\mathcal{F}}^2=\{{\varphi }_j\ge {\eta }_j^2\}$, ${\varphi }_j=({\xi }_{1j}\otimes {\delta }_1)\oplus ({\xi }_{1j}\otimes {\delta }_1)\oplus ...\oplus ({\xi }_{1j}\otimes {\delta }_1)$, with $\begin{array}{c}\hfill 1\le j\le n\end{array}$. Neurons ${\varphi }_{t1}^4,{\varphi }_{t2}^4,\dots ,{\varphi }_{tn}^4$ have the spiking rules $q_1^4:{\mathcal{F}}_j^4/b^{{\varphi }_j}\to b;f_j$, where ${\mathcal{F}}_j^4=\{ph{i}_j\ge p\}$. This ignition rule signifies that the output spiking value from the previous layer’s proposition neurons functions as the ignition threshold for the current layer’s neurons. Only one neuron in this layer activates, and, after the specified delay steps, a spike is generated by the following layer’s proposition neurons.

(4)

$$\begin{gathered} \mathrm{syn}=\{({\varphi }_{q1}^1,{\varphi }_{t1}^2),({\varphi }_{q1}^1,{\varphi }_{t2}^2),\dots ,({\varphi }_{qk}^1,{\varphi }_{tn}^2),({\varphi }_{t1}^2,{\varphi }_{q1}^3),({\varphi }_{t2}^2,{\varphi }_{q1}^3),\dots ,({\varphi }_{tn}^2, \\ {\varphi }_{q1}^3),({\varphi }_{q1}^3,{\varphi }_{t1}^4),({\varphi }_{q1}^3,{\varphi }_{t2}^4),\dots ,({\varphi }_{q1}^3,{\varphi }_{tn}^4),({\varphi }_{t1}^4,{\varphi }_{q1}^5),({\varphi }_{t2}^4,{\varphi }_{q1}^5),\dots ,({\varphi }_{tn}^4,{\varphi }_{q1}^5)\} \end{gathered}$$

The LSN P system can be approximated as a high-dimensional analysis method with low-dimensional input information. The initial layer acts as the data preprocessing stage to yield standardized input data, while the subsequent layer functions as the input layer for the learning spiking neural membrane system, and it is primarily tasked with deriving high-dimensional representations from low-dimensional data. The second and third layers are connected by a set of synaptic weights on the synapse. The maximum input spiking value will be output in the fourth layer and, after comparing it with the output spiking value of the second layer neuron, only one neuron will be activated and emit spiking to the next layer of propositional neurons after a certain delay step. The step number of delayed emission spiking will be used as a basis for determining different types of input data. In the event that the output category fails to correspond with the actual category, the synaptic weights proceed to be repeatedly modified between the second and third layers until a match with the real category is achieved.

3. Simulation and Analysis

Due to the relatively limited amount of data in the actual dispatch data, and due to the fact that the fault type and distribution were not complete enough, a simulation model of small-current grounding faulty line selection was built based on Simulink tools so as to analyze the changes in voltage, current, and other characteristic quantities of the line in the occurrence of single-phase grounding faults in the small-current, grounded system in a more in-depth manner. Through the construction and simulation of the model, a more comprehensive understanding of the system response when the fault occurs could be achieved, which, in turn, provides a more accurate basis for faulty line selection.

In low- and medium-voltage distribution networks, common grounded systems mainly include a neutral-point, ungrounded system and neutral-point, grounded system with the arc suppression coil. In order to comprehensively analyze the characteristics of these two systems in the occurrence of small-current, single-phase grounding faults, a thesis was constructed involving the building of the corresponding models in Section 3.1 and Section 3.2, respectively. As shown in Section 3.1, a simulation model of a neutral-point, ungrounded system was constructed, and the changes in three-phase currents, voltages, and other characteristic quantities when a small-current single-phase grounding fault occurred were verified by a simulation analysis of the model. As shown in Section 3.2, a neutral-point, grounded system with the arc suppression coil was further constructed. This model was constructed to investigate the effect of the arcing coil on the fault characteristics of the system. Similarly, through the simulation analysis of this model, this thesis verified the changes in the three-phase currents, voltages, and other characteristic quantities when a small-current single-phase grounding fault occurs, and it was able to summarize the rule of change in these characteristic quantities.

3.1. Neutral-Point, Ungrounded System

The dispatching data of a small-current, grounded system in medium- and low-voltage distribution networks mainly cover the phase current; zero-sequence current; the active and reactive power of a transmission line and the line voltage; and the phase voltage and zero-sequence voltage of a bus. These data are valid values and provide critical information about the operating status of the power grid. However, it is necessary to generate the simulation scheduling datasets for verification and analysis in order to better understand the data characteristics of small-current, single-phase ground faults.

To this end, this paper used the Simulink tool to build a simulation model of medium- and low-voltage distribution networks, as shown in Figure 1. The simulation model included a 110 kV power supply, a 110 kV-to-10.5 kV transformer, a switch-on switch, a transmission line, a load, a signal acquisition module, and a ground fault generator. In the simulation model, the ideal power supply with infinite internal impedance was used to simulate the actual situation to ensure the accuracy of the simulation. At the same time, the data acquisition module was responsible for collecting the scheduling data generated during the operation, which provided convenience for the subsequent analysis. In addition, the model had a total of six load lines, and their lengths were appropriately extended to more closely simulate the actual operation. The length of these load lines were 18 km, 17 km, 11 km, 16 km, 16 km, and 16 km, which together formed the main part of the system model. In the transmission line parameter setting, the actual situation was also fully considered. The zero-sequence parameter $r_0$ of the transmission line was 0.23 $\mathrm{\Omega }$/km, $l_0$ was $5.478\times {10}^{-3}$ H/km, $c_0$ was $1.369\times {10}^{-8}$ F/km, the positive-sequence parameter $r_1$ was $0.17\mathrm{\Omega }$/km, $l_1$ was $1.21\times {10}^{-3}$ H/km, and $c_1$ was $1.969\times {10}^{-8}$ F/km. The accurate setting of these parameters made the model closer to the actual operating state of the power grid, thus improving the reliability and accuracy of the results. It can be seen from Equation (1) that the selection of the arc suppression coil was only related to the ground capacitance of the distribution network.

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\omega L}}& =1.1\ast 3\omega C_{\varphi }\hfill \\ \hfill L& ={\displaystyle \frac{1}{3.3w^2{C}_{\varphi }}}\hfill \end{array}.$$

(1)

As shown in Figure 1, when the neutral point n2 node of the transformer was operated, the system switched to a neutral ungrounded configuration. Then, the simulation time was set to 0.2 s, and the A ground fault was simulated at 10 km of Phase A of Line 1, the specific time of the fault was 0.1 s, and the fault ended at 0.15 s.

In the neutral ungrounded system, there was no real electrical circuit between the capacitors of the distribution network and the ground, which led to the specific change mode of the bus phase voltage when the single-phase ground fault occurred. As shown in Figure 2, three-phase voltage curves can be clearly seen. Among them, the voltage amplitude of the fault Phase A decreased significantly from the original 8.45 kV to 0.25 kV. At the same time, the voltage amplitude of the non-fault B and C phases increased, reaching 14.82 kV.

Further analysis of these voltage changes showed that the amplitude of the sinusoidal signal was equal to $\sqrt{2}$ times the RMS value. Therefore, it could be calculated that the RMS phase voltage of Phase A decreased from the original 5.94 kV to 0.18 kV, while the RMS voltage of Phase B and Phase C increased from 5.94 kV to 10.48 kV, which was exactly equal to the bus voltage.

When Phase A of Line 1 was grounded, the bus bar voltage changed, as shown in Figure 3. The three interval lines in the figure represent the three-phase bus voltage. It can be observed that the amplitude of the line voltage was maintained at 15.00 kV, and its RMS value was 10.60 kV. This showed that the single-phase grounding fault did not affect the magnitude and symmetry of the three-phase line voltage in an ungrounded neutral system. Therefore, the medium- and low-voltage distribution network system was still able to maintain its operating state when such faults occurred, although the fault needed to be handled in time to ensure the safety and stability of the grid.

When the A phase of Line 1 was grounded, the zero-sequence current recording waveform and RMS waveform changed in each load line, becoming the target of attention. Figure 4 shows the zero-sequence current recording waveform, from which the changing trend of the zero-sequence current in different lines could be clearly observed. When single-phase grounding fault occurred in Line 1, its zero-sequence current waveform was opposite to that of other normal lines. In order to further understand the characteristics of the zero-sequence current shown in Figure 5, the zero-sequence RMS current waveform was further analyzed. Figure 5 provides an intuitive display of the zero-sequence current RMS value so that the relationship between the zero-sequence current of each line could be more accurately grasped.

Next, the relationship between the zero-sequence current in faulty Line 1 and the zero-sequence currents in the normal lines were compared, as shown in Figure 6. Figure 6 shows the waveform of the sum of zero-sequence current for normal lines, the zero-sequence current in faulty Line 1, and the sum of the zero-sequence current for all lines. It can be seen that the sum of the zero-sequence current for normal lines was in the opposite direction of the zero-sequence current in the faulty line, and the sum of the zero-sequence current for normal lines was exactly equal to the zero-sequence current of the faulty line. This means that, in the event of a single-phase ground fault, there was a balanced relationship between the zero-sequence current of the faulty line and the normal lines.

Figure 7 further illustrates the total zero-sequence current for standard lines, the zero-sequence current in faulty Line 1, and the RMS waveform of the zero-sequence current for all lines. From an RMS value perspective, it is evident that the direction of the zero-sequence current RMS values for standard lines is contrary to that of the faulty line, and the total RMS value of the zero-sequence current for standard lines equals the RMS value of the zero-sequence current in the faulty line. Moreover, the aggregate RMS value of the zero-sequence current for all lines was zero, which thoroughly demonstrates the characteristics of the zero-sequence current during a single-phase grounding fault in a neutral ungrounded system.

In summary, according to the simulation results of the single-phase grounding fault of the neutral ungrounded system, the following conclusions can be drawn: when the single-phase grounding fault occurs in the system, the zero-sequence current RMS value in the faulty line will become the maximum value among all the lines under the same bus. The zero-sequence current RMS value of the faulty line is equal to the sum of the zero-sequence current RMS values for the normal lines under the same bus. At the same time, the direction of the zero-sequence current of the faulty line is opposite to that of the non-faulty line. These characteristics provide an important theoretical basis for fault analysis and handling.

3.2. The Neutral-Point Grounded System with the Arc Suppression Coil

In a neutral ungrounded system, the fault-point current can be calculated by Equation (2):

$$I_f=3U_{\phi }\omega C_{\varphi }.$$

(2)

When the ground current exceeds the critical value, the arc phenomenon will occur. This phenomenon means a greater, more serious failure risk for the distribution network. Therefore, when the ground capacitance of the distribution network accumulates to a certain extent, the RMS values must be taken to reduce this risk. One of the common practices is to ground the neutral point through the arc suppression coil.

In specific operations, the transformer neutral Point n2 in Figure 1 is connected to the arc suppression coil so that a system with a neutral point grounded through the arc suppression coil is obtained. In the simulation experiment, the simulation time was set to 0.2 s, and the ground fault of Phase A at 3 km of Line 1 was simulated. The time when the fault occurred was 0.1 s, and the time when the fault ended was 0.15 s. Through comparative analysis, it was found that the phase voltage and line voltage change law of the neutral grounded system was the same as that of the ungrounded system, which indicates that the access of the arc suppression coil did not change the basic voltage change law of the system.

However, the zero-sequence current and reactive power changes in the lines were different from those in the ungrounded system. This was because the access of the arc suppression coil changeed the grounding mode of the system, which affected the distribution of the current and the transmission of power. These changes provide valuable clues for understanding the operating characteristics of the neutral grounded system with an arc suppression coil.

When the neutral point passes through the arc suppression coil grounded system, once Phase A of Line 1 is grounded, the zero-sequence current waveform and RMS waveform in the line will show different changing trends. As shown in Figure 8, the amplitude of the zero-sequence current waveform of the six loaded lines was different. The waveform of the faulty line was different from that of the other lines. Figure 9 further shows the RMS values waveforms of the zero-sequence currents of these lines, which the changes in the strength of the zero-sequence currents of each line can be more intuitively understood.

When the neutral point is grounded through the arc suppression coil, the effect of the Phase A ground fault of Line 1 on the zero-sequence current becomes complicated. As shown in Figure 10 and Figure 11, the waveform and RMS waveform of the sum of the zero-sequence current for normal lines, the zero-sequence current in faulty Line 1, and the sum of the zero-sequence currents for all lines showed specific rules. From these waveforms, it can be observed that the sum of the zero-sequence currents for the normal lines was opposite to the zero-sequence current in the faulty line. This was because the inductive current generated by the arc suppression coil was in the opposite direction of the capacitive current generated by the normal lines, and it was also in the same direction as the zero-sequence current in the faulty line. This characteristic makes it such that the arc suppression coil can reduce the fault current, but, at the same time, it may also cause the zero-sequence current RMS value of the faulty line to no longer be the maximum value in all of the lines. In addition, it may even become the minimum value, which greatly increases the difficulty of line selection.

In addition, when a single-phase ground fault occurs in Line 1, the reactive power of each line will also significantly change. This feature can be used as one of the important bases for the selection of single-phase grounding faulty lines. However, it should be noted that other faults in the distribution network may also cause fluctuations in reactive power, so the single-phase ground faulty line cannot be determined solely by the change in reactive power.

To summarize, when a single-phase grounding fault occurs in a neutral grounded system via an arc suppression coil, the zero-sequence current RMS value of the faulty line connected under the same bus may no longer be the maximum value. The sum of the zero-sequence currents of all lines is equal to the arc suppression coil current. When the fault occurs and ends, the reactive power of the faulty line changes the most. These characteristics provide important factors for identifying and dealing with single-phase grounding faulty line selection. Generally speaking, the occurrence of a small-current, single-phase grounding fault is accompanied by a change in the three-phase current, three-phase voltage, active power, and reactive power of the load electrical line; as such, the small-current single-phase grounding faulty line can be selected according to the change characteristics of these scheduling data.

Through the feature analysis in this section, you can clearly see the difference between the normal lines and faulty lines. These differences provide RMS values to detect the faulty lines in the small-current, grounded system. These characteristics can be further explored and utilized to improve the accuracy and efficiency of fault detection and to provide a strong guarantee for the safe and stable operation of a distribution network.

4. Faulty Line Selection Models with LSNPSn

In this section, a learned spiking neural P system with NLMS (LSNPSn) will be used for faulty line selection. Section 2 introduced a definition of the LSN P system, and this section describes the characteristics of LSNPSn and the faulty line selection model [28]. Section 4.1 describes the network structure and adaptive learning mechanism based on LSNPSn. Section 4.2 describes the small-current grounding faulty line selection methods. Section 4.3 describes the pseudocode of the algorithm.

4.1. Learning Spiking Neural P Systems with NLMS

The structure of the LSNPSn consists of an input layer, a preprocessing layer, a categorization layer, and an output layer (e.g., reference [28]. Data were imported from the environment into the input layer (the data format can be one-dimensional or multi-dimensional). The preprocessing layer accepts the input layer’s dimensionally enhanced data. The learning weights between the preprocessing and classification layers were updated and the classification results were output from the output layer.

An LSNPSn was constructed [28] as follows:

$$\mathrm{\Pi }=({\varphi }_{1,1},{\varphi }_{2,1},\dots ,{\varphi }_{n,1},{\varphi }_{1,2},{\varphi }_{2,2},\dots ,{\varphi }_{k,2},{\varphi }_{1,3},{\varphi }_{2,3},\dots ,{\varphi }_{n,3},{\varphi }_{1,4},syn,in,out),$$

(3)

where

(1) ${\varphi }_{1,1},{\varphi }_{2,1},\dots ,{\varphi }_{n,1}$ denotes n neurons in the initial layer of the LSNPSn, and the expressions for ${\varphi }_{1,1},{\varphi }_{2,1},\dots ,{\varphi }_{n,1}$ are given as below:

$${\varphi }_{i,1}=(z_{i,1},Q_{i,1},z_{i,1}\left(0\right)),1\le i\le n,$$

(4)

where:

  (a) $z_{i,1}$, for $1\le i\le n$, signifies a quantitative variable within the i-th neuron ${\varphi }_{i,1}$ of the starting layer in LSNPSn. Here, $m\in \mathbb{R}$ denotes the total number of neurons present in the first layer.

  (b) $Q_{i,1}$, for $1\le i\le n$, characterizes the creation function of the i-th neuron ${\varphi }_{i,1}$ in this layer. The structure of $Q_{i,1}$ is delineated as follows:

$$Q_{i,1}:f\left(z_{i,1}\right)=z_{i,1},1\le i\le n,$$

(5)

  (c) $z_{i,1}\left(0\right)$, for $1\le i\le n$, is the starting value of the variable resides within the neuron, and ${\varphi }_{i,1}$ is within the first layer.

(2) ${\varphi }_{1,2},{\varphi }_{2,2},...,{\varphi }_{k,2}$ represents the l neurons located in the second tier of LSNPSn, which are characterized by the following definitions:

$${\varphi }_{i,2}=(z_{i,2},Q_{i,2},z_{i,2}\left(0\right)),1\le i\le l,$$

(6)

where:

  (a) $z_{i,2}$, for $1\le i\le l$, is a variable in the i-th neuron ${\varphi }_{i,2}$ in the second layer of LSNPSn, with $l\in \mathbb{R}$ representing the count of neurons in this layer.

  (b) $Q_{i,2}$, for $1\le i\le l$, is the generation function for each neuron, and it is structured as follows:

$$Q_{i,2}:f\left(z_{i,2}\right)=z_{i,2},1\le i\le l,$$

(7)

  (c) $z_{i,2}\left(0\right)$, for $1\le i\le l$, represents the initial value of the variable in neuron ${\varphi }_{i,2}$ of this layer.

(3) ${\varphi }_{1,3},{\varphi }_{2,3},\dots ,{\varphi }_{n,3}$ represents the n neurons in the third layer, with its forms described as follows:

$${\varphi }_{i,3}=(z_{i,3},Q_{i,3},z_{i,3}\left(0\right)),1\le i\le n,$$

(8)

where:

  (a) $z_{i,3}$, for $1\le i\le n$, is the variable in the i-th neuron ${\varphi }_{i,3}$ in the third layer, with $n\in \mathbb{R}$ as the neuron count.

  (b) $Q_{i,3}=\{Q_{i,3}^1,Q_{i,3}^2\}$, for $1\le i\le n$, describes a limited collection of creation functions for the neuron ${\varphi }_{i,3}$, which is defined by the following:

$$Q_{i,3}^1:f\left(z_{i,3}\right){=1|}_U,1\le i\le n,Q_{i,3}^2:f\left(z_{i,3}\right)=0,1\le i\le n,$$

(9)

where $U=max(z_{1,3},z_{2,3},\dots ,z_{n,3})$.

  (c) $z_{i,3}\left(0\right)$, for $1\le i\le n$, represents the initial setting of the variable within the neuron ${\varphi }_{i,3}$.

(4) $syn=\left\{({\varphi }_i,{\varphi }_j)\right|1\le i\le o\mathrm{and}1\le j\le o\mathrm{with}i\ne j\}$ represent neuron synapses. Every synapse is denoted as $S_{ij}=((i,j),v_{ij})$, where $v_{ij}$ is the weight for the connection $\left(i,j\right)$. The initial weights between the layers vary, and the weights $\left(v_{i,j}^1,1\le i\le o\mathrm{and}1\le j\le l\right)$ between the initial and second layers follow an ascending order. Weights $\left(v_{i,j}^2,1\le i\le l\mathrm{and}1\le j\le o\right)$ are adapted via a learning mechanism. The final layer’s weights are set to $v_i^3=i$ for $1\le i\le o$.

(5)$input=\{{\varphi }_{1,1},{\varphi }_{2,1},\cdots ,{\varphi }_{m,1}\}$ is the set of input neurons;

(6)$output={\varphi }_{1,4}$ represents the output neuron.

Notably, the core targets of the LSNPSn were expressed as numerical parameters. Unlike the spike count method, LSNPSn with numerical parameters can directly tackle classification tasks involving real values. Additionally, in the third layer, the processing function is divided into two types: ${\varphi }_{i,3}^1:f\left(z_{i,3}\right){=1|}_U$ and $\varphi {i,3}^2:f\left(z_{i,3}\right)=0$. Here, _U is a threshold condition. If $z_{i,3}$ exceeds _U, the processing function ${\varphi }_{i,3}^1$ is triggered; otherwise, ${\varphi }_{i,3}^2$ is executed.

In the LSNPSn framework, $z_{ij}$ denotes the numerical value in the ith node within the jth layer. If $z_{ij}$ participates in the processing function $f\left(z_{ij}\right)$, it will reset to a zero post-calculation; otherwise, $z_{ij}$ retains its previous state. If the value $a_0$ is produced by the presynaptic node and the synapse weight is denoted by ${\omega }_0$, then $z_{ij}$ will update to $a_0{\omega }_0$, which is represented as $(z_{ij}=a_0{\omega }_0)$.

When the LSNPSn system progresses from one configuration state to the next, the transformation is noted as $P_t\to P_{t+1}$. Here, $P_t$ represents the entire set of numerical values at time t, and it is denoted as $P_t=(z_{1,1}\left(t\right),z_{2,1}\left(t\right),\dots ,z_{m,1}\left(t\right),z_{1,2}\left(t\right),\cdots ,z_{k,2}\left(t\right),z_{1,3}\left(t\right),z_{2,3}\left(t\right),$$\cdots ,z_{n,3}\left(t\right),z_{1,4}\left(t\right))$. If the LSNPSn has finite execution cycles, they are represented by $(P_0\to P_1\to \cdots \to P_t\to \cdots \to P_h)$, where h denotes the maximum cycle count.

Within LSNPSn, the Widrow–Hoff update rule (least mean square, LMS [74,75,76,77]) updates the synapse weights between the second and third layers. This thesis employs an adaptive normalization technique, i.e., the normalized least mean square (NLMS) [78,79,80,81], to dynamically adjust the weights. The NLMS formula is shown in Equation (10):

$$\begin{array}{c}\hfill {\omega }_{ij}(t+1)={\omega }_{ij}\left(t\right)+{\displaystyle \frac{\beta }{\kappa +|z_i{\left(t\right)|}^2}}(e_j-o_j)z_i\left(t\right),\end{array}$$

(10)

where ${\omega }_{ij}\left(t\right)$ and ${\omega }_{ij}(t+1)$ are the weights before and after the update, respectively. Here, $e_j$ represents the target output for node j, $o_j$ is the observed output of node j, and $z_i\left(t\right)$ is the state of node i at time t. The term ${\displaystyle \frac{\beta }{\kappa +|z_i{\left(t\right)|}^2}}$ defines the adaptive step size, with $\kappa$ being a regularization factor to avoid zero division.

The NLMS algorithm introduces normalization into LMS, adjusting the step size based on the current input state. It provides an initially larger step size, enhancing the convergence speed; as it approaches convergence, the step size is reduced to maintain high precision. This makes the algorithm resilient to variations in input amplitude.

4.2. Small-Current Grounding Faulty Line Selection Methods

The LSNPSn-based, small-current ground faulty line selection method is shown in Figure 12. First, in the input layer, eight neurons are set up, corresponding to the eight feature quantities of the scheduling data. In this way, when the scheduling data are input into the model, each feature quantity is transmitted and processed through the corresponding neuron.

Next, the preprocessing layer is entered. In this layer, the ascending weights are generated based on the spiking values entering the output layer. These ascending weights play a role in calculating the spiking values of the neurons in the preprocessing layer. Through the ascending weights calculation, the spiking values of the preprocessing layer neurons can be obtained, which not only contain the feature information of the original data, but also incorporate the model’s learning capability.

Based on the initialized classification weights, the input values of the classification layer are calculated. Each neuron in the categorization layer contains two production functions, and one of the important parameters is Threshold T. This Threshold T is determined by the spiking value of the neurons in the categorization layer at this point in time, which determines whether or not the neuron will be activated and pass information to the next layer.

In the categorization layer, those neurons whose spiking values can reach Threshold T will pass a 1 to the output layer neuron through the output category weights, indicating that the category is selected, while the rest of the neurons will pass a 0 to the output layer neuron, indicating that it is not selected. The output category weights are initialized based on the number of categories in the dataset, and they reflect the model’s ability to recognize and distinguish between different categories.

When the output layer receives information from the categorization layer, it makes a judgment based on the output category weights and determines the final output category. This output category is the result of the model’s categorization of the dispatch data and the prediction of the small-current grounding faulty line selection.

The next step determines whether all of the lines have been detected. If not, the model returns to the preprocessing layer and continues the routing for the next line. When all lines are detected, the wire routing results of all lines are output.

In this way, the small-current grounding faulty line selection based on the learned numerical impulse neuromembrane system is accomplished. This method can automatically process and analyze the dispatch data to achieve fast and accurate small-current ground faulty line selection.

4.3. The Pseudocode of the Algorithm

The learning process of the LSNPSn consists of three sub-processes: The first layer is the preprocessing layer, and its number of neurons is the same as the number of features of the power system dispatching data. The second layer is the input layer, which is used to obtain the high-dimensional representation of the input data. The third layer is the hidden layer, and the synaptic weights of neurons between the second and third layers are changed by learning. The next layer is the comparison layer, which outputs the maximum pulse value of the neurons in this layer. The fifth layer is the selection layer, which activates the corresponding regular neuron and delays the firing of pulses to the next layer of neurons. The last layer is the output layer, where the firing rules of the neurons will be excited and a pulse will be fired into the environment to obtain the sequence number of the faulty line. If the output ordinal number is inconsistent with the real ordinal number, NLMS is used to adjust the weight between the output layer and the hidden layer to achieve higher line selection accuracy on the training set. Finally, the trained model is saved and the small-current ground faults line selection accuracy rate $Acc$ is output. The specific pseudocode of the LSNPSn is described in Algorithm 1.

Algorithm 1 Faulty line detection models with LSNPSn

Require:  Simulated power system dispatch dataset; the synapse weights $w_{i,j}^1,1\le i\le k,AND 1\le j\le n$; the learning step size $\alpha$; the regularization factor c; and the initial values of each neurons. 1: Divide the datasets into three quarters for training and one quarter for testing. In addition, the features of the sample data are inputted from the environment to the neurons in the first layer ($i{n}_1=x_1,i{n}_2=x_2,\cdots ,i{n}_m=x_m$). 2: Determine synaptic weights $w_{i,j}^1,1\le i\le mAND1\le j\le k$. 3: $x_{i,2},1\le i\le k,$ is computed by the output data of the first layer ($x_{1,2},x_{2,2},...,x_{k,2}={\mathbf{X}}_{\mathbf{1}}{\mathbf{W}}_{\mathbf{1}}$), where ${\mathbf{X}}_{\mathbf{1}}$ is the input data of the first layer (${\mathbf{X}}_{\mathbf{1}}=(i{n}_1,i{n}_2,\cdots ,i{n}_m$), and ${\mathbf{W}}_{\mathbf{1}}$ represents the ascending matrix. 4: Initialization parameters: K, $\alpha$, EPOCH, W. 5: Data training:. 6: According to the value of K, the order of the dimension is raised. The dimension weights are generated according to the spike value entering the input layer, and the spike value of neuron ${\varphi }_1^2$, ${\varphi }_2^2$,..., ${\varphi }_k^2$ in the pre-processing layer is calculated by the dimension weight. 7: while ($t\le Maxt$) do 8:  Based on the initialized learning weights $w_{i,j}^2,1\le i\le kAND1\le j\le n$, the input values ${\mathbf{X}}_{\mathbf{3}}=x_{1,3},x_{2,3},...,x_{n,3}$ of neurons ${\varphi }_1^3$, ${\varphi }_2^3$,⋯, ${\varphi }_n^3$ in the third layer are calculated. 9:  Each neuron in neurons ${\varphi }_1^3$, ${\varphi }_2^3$,⋯, ${\varphi }_n^3$ contain two production functions, and the threshold is T, and Threshold T is determined by the spike value of neurons ${\varphi }_1^3$, ${\varphi }_2^3$,⋯, ${\varphi }_k^3$ at this time. Neurons whose spike values can reach Threshold T will pass 1 through the output class weight to output layer neuron ${\varphi }_1^4$, while the remaining neurons will pass 0 to output layer neuron ${\varphi }_1^4$. 10:   The data class is obtained from the output value of neuron ${\varphi }_1^4$. 11:   Calculate the difference e between the output class and the label class and use it to update the classification weight $W_2$. $w_{ij}^2(t+1)=w_{ij}^2\left(t\right)+{\displaystyle \frac{\alpha }{c+{\mid x_i\left(t\right)\mid }^2}}(d_j-y_j)x_i\left(t\right)$. 12: end while 13: Save the small-current ground faulty line selection model $w_{i,j}^1,1\le i\le k,AND1\le j\le n$ and verify the test sets. Ensure:  Accuracy of the line selection for small-current ground faults: $Acc$.

5. Experimental Results and Analysis

To assess the viability of the suggested method, the distribution network system depicted in Figure 1 was emulated using MATLAB/Simulink (2022b), generating a dataset for training and testing purposes to validate the faulty line detection capability. The collected data were then processed through the LSNPSn-based model for small-current, grounded faulty line identification to derive the selection outcomes. The entire experimental procedure was conducted on a workstation equipped with MATLAB, Python 3.10, a GPU 3080, 32 GB RAM, and the Windows 10 operating system.

5.1. The Description of the Datasets

During the simulation experiment, we gathered a set of 8 characteristics for each of Lines 1 through 6, amounting to a comprehensive dataset of 48 features. These included the three-phase currents, three-phase voltages, active power, and reactive power. Through simulation, we acquired 6000 datasets for both the ungrounded neutral point and the neutral point grounded with an arc suppression coil. An additional 6000 datasets were sourced from real-world measurements. These data were then organized to facilitate the selection model for identifying small-current ground faults in the lines.

5.2. Selecting Parameters

To address the challenge of selecting small-current ground faults in the six lines, it was necessary to specifically configure and fine tune the model’s parameters.

The decision to use an ascending order was primarily based on the intricacy of the issue at hand; for this experiment, the ascending order of dimensions was fixed at three. Additionally, the regularization parameter c was commonly set to 0.1, the learning rate for LSNPSn was established at 0.05, and the maximum iteration count was capped at 300. The synaptic weights connecting the output layer to the hidden layer were initialized within the range of 0 to 1. The two simulated dataset groups consisted of 4200 training examples and 1800 test examples, while the actual datasets comprised 14,400 training examples and 4800 test examples. Once all of the training set data were processed by LSNPSn, the error function was computed by comparing the actual line selection output with the desired line selection outcome. During the iterative training phase, the synaptic weights of LSNPSn were modified using the error and the NLMS algorithm, ultimately yielding a trained model for small-current ground faulty line selection.

5.3. Experimental Results

In this paper, the simulation data and the actual data of the distribution network are used to select faulty lines to verify the application effect of LSNPSn. In this paper, the simulation data and the actual data of the distribution network were used to select faulty lines to verify the application effect of LSNPSn.

5.3.1. The Datasets of the Neutral-Point Ungrounded System

The dataset for systems with an ungrounded neutral point comprised 6000 samples, each featuring 48 attributes and were classified with numbers ranging from 0 to 5, which corresponded to fault scenarios across the six lines. The distribution of the dataset for training and testing purposes was 4200 and 1800 samples, respectively.

In this context, the performance metrics ($Accuracy$ and $MSE$) were utilized to assess the efficacy of the LSNPSn model. The $MSE$ metric was applied to gauge the training convergence rate, while the $Acc$ metric was used to determine the correctness of predictions on the test dataset. The $MSE$ and $Acc$ were computed using Equation (11) and Equation (12), respectively.

$$MSE={\displaystyle \frac{1}{T_N}}\sum _{k=1}^{T_N}{(d_i-y_i)}^2,$$

(11)

$$Acc={\displaystyle \frac{R_N}{T_N}}\times 100\%,$$

(12)

where $R_N$ denotes the count of samples that were accurately classified, while $T_N$ signifies the overall number of samples. Additionally, $d_i$ refers to the actual output class, and $y_i$ is the predicted output class.

To test the convergence of LSNPSn, 20 independent experiments were implemented. The mean $MSE$ and confidence interval based on LSNPSn in the datasets of the neutral-point ungrounded system faulty line selection problem are displayed in Figure 13. Figure 13a,b are the mean $MSE$ and confidence interval based on LSNPS and LSNPSn, respectively. As can be seen from Figure 13, the convergence of LSNPSn was better than LSNPS during the training. Meanwhile, in the testing sets, the accuracy rates of the LSN P systems and LSNPSn were able to achieve 98.9% and 98.94%, respectively. Compared to the LSN P systems, the adaptive adjustment mechanism of the LSNPSn enabled the system to adapt and adjust to changes in the environment, leading to faster convergence, which resulted in LSNPSn having better convergence compared to the LSN P systems.

5.3.2. The Datasets of the Neutral-Point Grounded System with the Arc Suppression Coil

The datasets of the neutral-point grounded system with the arc suppression coil also consisted of 6000 samples. Each sample contained 48 features. The size of the training and testing sets was 4200 and 1800, respectively.

The 20 independent experiments were implemented to test the performance of the LSN P systems and LSNPSn. The mean $MSE$ and confidence interval based on the LSN P systems and LSNPSn in the datasets of the neutral-point grounded system with the arc suppression coil faulty line selection problem are displayed in Figure 14. Figure 14a,b show the mean $MSE$ and confidence interval based on LSNPS and LSNPSn, respectively. As can be seen from Figure 14, the convergence of LSNPSn was better than LSNPS during the training.

Ultimately, the results for various scenarios within the power system were obtained, as presented in

Table 1. Line selection results.

Methods	Systems	Results
LSNPS	Neutral-point ungrounded system	98.90%
	Neutral-point grounded system with the arc suppression coil	99.00%
SVM	Neutral-point ungrounded system	97.33%
	Neutral-point grounded system with the arc suppression coil	98.17%
kNN	Neutral-point ungrounded system	98.62%
	Neutral-point grounded system with the arc suppression coil	98.91%
LSNPSn	Neutral-point ungrounded system	98.94%
	Neutral-point grounded system with the arc suppression coil	99.17%

. The data for the neutral-point ungrounded system indicated that the line selection accuracy rate of LSNPS, SVM, and kNN were 98.9%, 97.33%, and 98.91%, respectively, while that of LSNPSn was 98.94%. For the data of the neutral-point grounded system with the arc suppression coil, the line selection accuracy rate of LSNPS, SVM, and kNN were 99%, 98.17%, and 98.91%, respectively, whereas LSNPSn exhibited an accuracy rate of 99.17%. These results demonstrate the performance superiority of LSNPSn.

6. Conclusions

This study presents an innovative technique that employs spiking neural P systems trained with NLMS for identifying faulty lines within a small-current grounded system. By integrating the steady-state attributes obtained from the real-world distribution network data analysis, a distinction can be made between the operational and defective lines, which enables the detection of faults in the grounded system. The proposed method’s viability and accuracy are confirmed through experiments conducted on both synthetic and actual datasets.

However, in response to the measurement uncertainties and noise in real-world scenarios, as well as the parasitic effects or measurement errors, we did not conduct separate treatments in this study. Although the model introduced NLMS to improve convergence speed, and the experimental results verified the stability of the convergence of the proposed method, these interferences still affected the accuracy of the model. Therefore, in future work, we will investigate reducing the impact of these interferences during the data pre-processing stage and aim to improve the accuracy of the faulty line selection and the reliability of the results.