A Hybrid Machine Learning and NGO Algorithm Approach for Fault Classification and Localization in Electrical Distribution Lines

Abstract

Today’s distribution networks are becoming increasingly complex, necessitating highly accurate and robust fault diagnosis methods. Traditional methods based on impedance or traveling waves often lack flexibility and precision in these dynamic environments. This study proposes a hybrid approach based on the synergy between machine learning (ML) techniques and a recent metaheuristic, the Northern Goshawk Optimizer (NGO). Fault location is performed using a cubic spline interpolation model. Classification is handled by a decision tree, while fault resistance—a key parameter that significantly influences diagnostic performance—is optimized using the NGO algorithm. The effectiveness of the proposed method is evaluated through a series of experiments conducted on the IEEE 34-bus test network. These experiments encompass various fault scenarios (single line-to-ground, line-to-line, double line-to-ground, and three-phase faults) as well as voltage and load variation conditions. Fault resistance values considered in the study are 0, 10, 50 and 100 ohms. The results highlight the robustness and efficiency of the hybrid approach, achieving an accuracy rate of up to 99.999% in fault location. This level of performance enables reliable identification of both the fault location and the affected line.

1. Introduction

For modern transmission and distribution networks to be reliable, stable, and resilient, they need to be able to find, classify, and locate faults accurately. The move to active distribution networks (ADNs), which have a lot of distributed generation (DG), renewable energy sources, storage systems, and bidirectional power flows, has made operations much more complicated. These changing features make traditional protection schemes less effective and call for intelligent, adaptable, and computationally efficient fault diagnosis frameworks.

Signal-processing-based methods have been used for a long time to find faults and extract temporary features. Wavelet-based and frequency-domain approaches were investigated by Neri Junior et al. [1], Fault location on radial distribution systems using wavelets and artificial neural networks with a new data processing feature, and in [2], Enhancing fault location accuracy in transmission lines using transient frequency spectrum analysis: an investigation into key factors and improvement strategies. Variational mode decomposition combined with ensemble bagged trees was proposed in [3], Voltage-Based Fault Detection in PV-Integrated Networks Using VMD and Ensemble Learning. Wavelet entropy and neural networks were integrated in [4], Fault Section Identification Using Wavelet Entropy and Neural Networks, while discrete wavelet transform combined with deep neural networks was introduced in [5], DWT–DNN-Based Fault Detection in Hybrid Energy Microgrid Clusters. Frequency spectrum analysis was further explored by Aslan et al. [6], Neural Network-Based Fault Detection Using Frequency Spectra, and traveling-wave frequency characteristics were analyzed in [7], Frequency Analysis of Traveling Waves for Distribution Fault Location.

Traveling-wave-based localization methods continue to be significant owing to their theoretical accuracy. The concept of time-matrix-based traveling-wave localization was introduced in [8], titled Time Matrix of Traveling Waves for Fault Location in DG Networks. The development of single-ended and two-terminal methods is documented in [9], Single-Ended Traveling Wave Fault Location Using Arrival Time and Wave Polarity [10], Two-Terminal Traveling Wave Fault Location for Wind-Connected Compensated Lines, and by Jin et al. [11], Two-Terminal Fault Location Method for HVAC Cable Lines with Frequency-Dependent Parameters. In the study by Wang et al. [12], Traveling-Wave-Based Fault Location Considering Wavefront Distortion, the effects of wavefront distortion were investigated. In [13], Impedance-Based Fault Location in Unbalanced Overhead Radial Networks, impedance-based formulations for unbalanced radial networks were suggested. Shu et al. introduced parameter-independent methods for multi-terminal lines in [14], Fault Location Method for Three-Terminal Lines Without Line Parameters. The study in [15], Machine Learning-Based Fault Localization in Multi-Terminal Transmission Lines, looked into two-terminal methods for multi-terminal transmission lines. Even though these methods work very well when everything is perfect, they are affected by how well they are synchronized, how much noise there is in the measurements, and how uncertain the parameters are.

Researchers investigated PMU-based and state-estimation methods to improve observability. In [16], Optimal PMU Placement Using Enhanced Feature Selection Techniques, the authors discussed the best places to put PMUs. In [17], Robust Three-Phase State Estimation Method for Active Distribution Networks, the authors proposed improved estimation techniques. Dashtdar et al., in [18], Optimization-Based Fault Location Using PMU-Assisted State Estimation, characterized fault location as an optimization problem. Even though these methods are more accurate, they rely a lot on the cost of communication infrastructure and installation, which could make it hard to scale them up.

The machine learning (ML) and deep learning (DL) techniques have shown a lot of promise for modeling nonlinear systems. In [19], Comparative Study of ML Models with Random Search Optimization, comparative ML analyses were done. In Najafzadeh et al. [20], Optimized Machine Learning Algorithms for Transmission Line Fault Diagnosis, optimized ML fault diagnosis methods were suggested. Artificial neural networks were utilized in [21], ANN-Based Fault Indicator Validation for Section Identification, Zhou et al. [22], Backpropagation Neural Network Optimized for Fault Location, and by Alhanaf et al. [23], Deep Neural Network-Based Fault Detection Scheme for Smart Grids. Hybrid classifiers were developed by Jamali et al. [24], Hybrid Classifier for Active Distribution Networks, to address limited data conditions, and El Ghaly et al. [25], Hybrid Multi-Target Ensemble Classifier under Limited Data Conditions, proposed a multi-target ensemble learning strategy to enhance classification robustness. Integration of ML frameworks for identification, classification, and localization was presented in [26], Integrated Machine Learning Methods for Comprehensive Fault Diagnosis. Nguyen et al. [27] proposed deep learning architectures like CNN–LSTM models in their study, Hybrid CNN–LSTM Model for Fault Classification and Location in Transmission Systems. Herrada et al. [28] used LSTM-based approaches in their study, LSTM-Based Fault Detection in Smart City Distribution Systems, and ref. [29] used temporal convolutional networks in their study, Temporal Convolutional Neural Network for Overhead Line Fault Classification. The combination of transfer entropy with an improved AlexNet architecture for fault segment localization was investigated in [30], Fault Segment Localization Using Transfer Entropy and an Improved AlexNet Architecture, where feature extraction and deep convolutional learning were integrated to enhance localization performance. Graph convolutional networks were introduced in [31], Graph Convolutional Network for Fault Localization in High-DG Networks, and domain-adaptive TGATv2 algorithms were proposed in [32], Domain-Adaptive Fault Location Using TGATv2. Khan et al. [33], Enhancing Fault Classification Using VAE-Generated Synthetic Data, looked into making synthetic data with variational autoencoders. The study in [34], Deep Learning and Digital Twin-Based Intelligent Fault Detection, investigated putting digital twins together. Rafique et al. [35], End-to-End Deep Learning Approach for Fault Diagnosis, showed end-to-end deep learning solutions. While these methods perform well for classification, they typically require large labeled datasets and are computationally intensive.

Hybrid intelligent and optimization-based frameworks have interested many researchers. Dai et al. [36] presented an advanced heuristic optimizer in their work, Improved Subtraction-Average-Based Optimizer for Fault Section Location in ADNs. In [37], Voltage Change-Based Fault Location Using Genetic Algorithm, genetic-algorithm-based solutions were suggested. In [38], Dynamic Quantum Genetic Algorithm for Active Network Fault Localization, quantum genetic algorithms were created. In Xu et al. [39], Hybrid PSO–Genetic Algorithm for Successive Fault Location, hybrid PSO–GA strategies were introduced. In [40], Multi-Agent Framework for Fault Location and Cyber-Attack Detection, multi-agent frameworks for finding faults and cyber-attacks were presented. In [41], Distributed Sensing and Fuzzy C-Means-Based Fault Location, fuzzy-clustering-based distributed sensing strategies were created. In [42], Integrated Fault Location and Isolation Using MDP, Markov decision process-based integrated fault location and isolation were suggested. Successive optimization strategies for DG-integrated lines were introduced in [43], Intelligent Fault Location Scheme for Distribution Lines with Distributed Generators. For compensated transmission lines, Swetapadma et al. [44], Hybrid Fault Location Method for Fixed Series-Compensated Lines, developed a specialized hybrid strategy to address impedance variation and compensation-induced nonlinearities.

Additional studies focused on battery sizing for fault tolerance in wind microgrids [45], Battery Sizing in Wind Microgrids from a Fault Tolerance Perspective, islanding detection through reactive power variation [46], Data-Driven Islanding Detection Based on Reactive Power Variation, hierarchical fuzzy Petri nets [47], Time-Sequence-Based Hierarchical Fuzzy Petri Nets for Fault Diagnosis, and high-impedance fault detection taxonomies the study by in Mishra et al. [48], Taxonomy of High-Impedance Fault Detection Algorithms. Case studies like Azeroual et al. [49], Fault Detection in Distributed Generation Networks: Case Study of Kenitra City, and intelligent high-voltage transmission fault detection frameworks [50], Intelligent Fault Detection and Location in High-Voltage Transmission Lines, show more real-world examples of how to use these technologies. The studies [51], Overview of AI-Driven Fault Detection and Localization Methods [52], Review of Machine Learning Tools for Fault Diagnosis in Active Grids, and Cao et al. [53], AI- and Signal-Processing-Based Fault Diagnosis Techniques in Distribution Networks, all give detailed overviews.

Although extensive research has been conducted, several problems remain unresolved. For instance, signal processing techniques are susceptible to noise and synchronization errors, PMU-based approaches rely heavily on infrastructure, deep learning models demand substantial data and computational resources, and conventional metaheuristic algorithms often suffer from premature convergence and parameter sensitivity within the multi-modal optimization landscapes typical of ADNs.

Consequently, a robust optimization framework is essential to maintain population diversity, enhance global search capabilities, and ensure stable convergence under dynamic conditions. To address these limitations, this study proposes a Northern Goshawk Optimization (NGO)-based fault location framework designed to optimize the exploration–exploitation balance while minimizing parameter dependence and retaining computational efficiency for real-time ADN applications.

The existing literature highlights a growing reliance on intelligent, hybrid, and optimization-driven techniques to address modern power system complexities. Building on this, the present work introduces a hybrid methodology for fault classification and localization in electrical distribution networks. The primary contribution integrates a cubic-spline-based fault localization model with a decision tree for fault classification coupled with the NGO algorithm, which is specifically utilized to accurately estimate fault resistance.

While algorithms like PSO, GA, and GWO are widely utilized for fault localization, they frequently demonstrate premature convergence, slow execution, or high parameter sensitivity in highly nonlinear active distribution networks. The NGO algorithm was selected for this study because of its robust convergence, minimal parameter sensitivity, and superior capability in navigating multi-modal, non-convex objective functions, thereby guaranteeing more reliable fault detection and localization.

Table 1. Comparative analysis of NGO vs. classical metaheuristics.

Algorithm	Convergence Behavior	Sensitivity to Parameters	Applicability to Active Distribution Networks (ADNs)	Remarks
PSO (Particle Swarm Optimization)	Moderate; can get trapped in local minima	High; requires careful tuning of inertia weight and acceleration coefficients	Limited in highly nonlinear or multi-modal ADN scenarios	Efficient for simple networks but less reliable under DG penetration
GA (Genetic Algorithm)	Slow; may require many generations	Moderate; crossover and mutation rates need tuning	Can handle nonlinear objectives but computationally intensive for large ADNs	Effective for global search but convergence may be slow
GWO (Grey Wolf Optimizer)	Good exploration, but may stagnate	Moderate; performance sensitive to leader hierarchy parameters	Suitable for medium-complexity networks; may underperform in dynamic ADNs	Balance between exploration/exploitation, but parameter tuning crucial
NGO (Nelder–Goldberg Optimizer)	Fast; robust global and local search	Low; minimal parameter sensitivity	Highly suitable for complex, nonlinear, multi-modal ADNs	Handles multi-modal objective functions efficiently, ensures accurate fault localization and faster convergence

briefly summarizes the advantages of selecting the NGO method over alternative techniques documented in the literature.

The technical contributions of this paper are summarized as follows:

• A fault localization model based on the cubic spline interpolation method is developed.
• The NGO algorithm is applied as an optimization method for fault resistance estimation.
• A machine learning decision tree is used for classification.
• A comprehensive dataset is generated using MATLAB/Simulink (R2020a) based on the modified IEEE 34-bus distribution network, covering various fault scenarios (fault types, load variations, fault resistance levels, and voltage conditions).

The paper is divided into five sections. Section 2 describes the redundancy problem formulation and modeling. The intelligent fault location and classification method is presented in Section 3. Section 4 presents a numerical example and computational results to demonstrate the efficacy of the suggested methodology. The conclusions and future research are presented in Section 5.

Table 2. Summary of previous methods for fault classification and localization in electrical networks.

Ref.	IEEE Test System/Case Study	Objective Function	Optimization/Learning Method	Category	Main Limitation
[1]	Distribution feeder	Minimize classification error	Wavelet + ANN	Signal Processing + ML	Sensitive to noise and parameter selection
[2]	Transmission line	Fault distance accuracy	Transient frequency spectrum analysis	Signal Processing	Sensitive to noise, high sampling required
[3]	PV-integrated network	Voltage variation	VMD + ensemble trees	ML/DL	May not generalize to high-DG penetration
[4]	Distribution feeder	Entropy-based error	Wavelet + neural networks	Signal Processing + ML	Sensitive to noise
[5]	Microgrid cluster	Detection accuracy	DWT + DNN	Signal Processing + ML	May require high sampling
[6]	Distribution feeder	Spectral feature loss	Neural networks on frequency spectra	Signal Processing	Requires spectral accuracy
[7]	Distribution network	Frequency deviation	Traveling wave frequency analysis	Signal Processing	Requires precise timing
[8]	DG network	Traveling wave timing error	Time matrix method	Signal Processing	Needs accurate measurement
[9]	Transmission line	Arrival time error	Single-ended traveling wave	Signal Processing	Sensitive to noise
[10]	Wind-compensated line	Traveling wave mismatch	Two-terminal method	Signal Processing	Sensitive to parameter variations
[11]	HVAC cable	Frequency-dependent impedance error	Two-terminal analytical method	Signal Processing	Needs precise line parameters
[12]	Transmission line	Wavefront distortion error	Traveling-wave-based	Signal Processing	Sensitive to distortion modeling
[13]	Unbalanced radial network	Impedance mismatch	Analytical/impedance method	Signal Processing	Limited to radial networks
[14]	Three-terminal line	Fault distance error	Parameter-independent method	Signal Processing	Accuracy affected by topology changes
[15]	Multi-terminal line	Localization error	ML-based	ML	Scalability to very large networks
[16]	IEEE 34-bus system	PMU placement cost and observability	Feature selection optimization	PMU/State Estimation	Installation cost and scalability
[17]	Active distribution network	State estimation error	Robust three-phase estimation	PMU/State Estimation	Dependent on PMU deployment
[18]	IEEE 34-bus + PMU	State estimation error	Optimization-based	PMU/Optimization	PMU deployment cost
[19]	IEEE test system	Classification accuracy	Random search ML optimization	ML	Limited global search capability
[20]	Transmission line	Fault diagnosis accuracy	Optimized ML algorithms	ML	Dataset-dependent performance
[21]	Distribution feeder	Fault indicator validation error	ANN	ML	Requires training data quality
[22]	Distribution system	Fault distance error	Backpropagation neural network	ML	May converge slowly
[23]	Smart grid	Detection accuracy	Deep neural network	ML/DL	Large dataset requirement
[24]	Active distribution network	Classification accuracy	Hybrid classifier	ML/DL	Needs extensive training
[25]	Distribution system	Multi-target classification error	Multi-target ensemble ML	ML/DL	Limited scalability with very large networks
[26]	Distribution network	Integrated identification and location loss	Combined ML methods	ML/DL	Complex framework, high computation
[27]	Transmission system	Classification and location accuracy	Hybrid CNN–LSTM	ML/DL	High data requirement, computational cost
[28]	Smart city network	Classification accuracy	LSTM	ML/DL	High computational cost
[29]	Overhead line model	Classification error	Temporal CNN	ML/DL	High data requirement
[30]	Distribution network	Fault segment localization loss	Transfer entropy + improved AlexNet	ML/DL	May require feature tuning
[31]	High-DG network	Localization loss	Graph convolutional network	ML/DL	Sensitive to topology changes
[32]	DG network	Fault location loss	Domain-adaptive TGATv2	ML/DL	Complexity and data dependency
[33]	Transmission line	Classification accuracy	VAE + ML	ML/DL	Synthetic data may not capture all real conditions
[34]	Smart grid	Detection error	Deep learning + digital twin	ML/DL	High computational requirement
[35]	Transmission line	End-to-end fault loss	Deep learning	ML/DL	Data-intensive
[36]	Active distribution network	Minimize fault section mismatch	Subtraction-average-based optimizer	Hybrid and Optimization	May converge to local optima
[37]	Distribution line	Voltage change	Genetic algorithm	Metaheuristic	Premature convergence
[38]	Active network	Fault localization error	Dynamic quantum genetic algorithm	Metaheuristic	Complexity, sensitive to parameters
[39]	DG line	Distance error	Hybrid PSO–GA	Metaheuristic	Premature convergence possible
[40]	Distribution system	Fault localization accuracy	Multi-agent system	Hybrid	Communication infrastructure dependent
[41]	Distribution system	Clustering error	Fuzzy C-means	Hybrid	Sensitive to initial cluster selection
[42]	Distribution system	Location and isolation reward	Markov decision process	Hybrid	Requires accurate reward modeling
[43]	DG-integrated lines	Fault localization error	Intelligent scheme	Hybrid	May not handle all fault types
[44]	Fixed series-compensated line	Distance error	Hybrid method	Hybrid	Specific to compensated lines
[45]	Wind microgrid	Reliability index	Battery sizing optimization	Hybrid	Focused only on battery sizing
[46]	DG system	Reactive power variation	Data-driven	ML	May fail under unmodeled dynamics
[47]	Distribution network	Sequence-based fault detection	Hierarchical fuzzy petri nets	Hybrid	Complex rule definition
[48]	Review	—	Taxonomy of high-impedance fault detection	Review	—
[49]	DG network, Kenitra City	Detection accuracy	Data-driven	ML	Case-specific, limited generalization
[50]	High-voltage transmission	Detection accuracy	Intelligent method	Hybrid	Large-scale deployment challenging
[51]	Review	—	Survey	Review	—
[52]	Review	—	Survey	Review	—
[53]	Review	—	Survey	Review	—

categorizes previous studies according to their datasets and methodologies in order to offer a more comprehensive understanding of the current methods.

2. Problem Definition

Figure 1 illustrates a three-phase transmission line. A single line-to-ground fault occurs at an arbitrary point $F$ on the line, located at a distance $x$ from the sending end (bus $S$) [54].

The short-circuit current magnitude depends on several parameters, some of which remain constant while others are variable.

In the above:

$V_{s_p}$: the post-fault sending-end voltage of bus S corresponding to phase $p$.

$I_{s_p}$: the post-fault sending-end current corresponding to phase $p$.

$V_{F_p}$: the post-fault voltages at the fault location corresponding to phase $p$.

$I_{F_p}$: the post-fault current at the fault location corresponding to phase $p$.

$R_F$: the fault resistance in $\left(\mathsf{\Omega }\right).$

$x$: the distance between the sending end and the fault location, expressed in meters or kilometers.

$p$: the phase, a, b or c.

$I_{a,b,c}$: the post-fault load current.

$Z_{a, b, c}$: the load impedance.

$L$: the transmission line length.

The sending-end voltages are given by (1), which describes the steady-state fault conditions:

$$V_s=x Z_{line} I_s+V_F$$

(1)

where:

$$V_F=\left\{\left(L-x\right)Z_{line}+Z_{a, b, c}\right\} \left(I_s-I_F\right)$$

(2)

then, the current at the sending end is given by:

$$I_s={\left[L Z_{line}+Z_{a, b, c}\right]}^{-1}\left\{V_s+I_F \left[\left(L-x\right) Z_{line}+ Z_{a, b, c}\right]\right\}$$

(3)

To estimate the value of the current $I_s$, it is first necessary to evaluate the fault current as well as the load impedance $Z_{a, b, c}$. A numerical method presented in [54] is used to estimate the value of the fault current and the load impedance based on the harmonic components of the transient regime. However, these methods make it difficult to accurately determine the exact value of the current at the sending end. To overcome this limitation, current transformers on each busbar are utilized to directly measure the fault current under steady-state conditions. Moreover, the load impedance variation is replaced by the power measurement by the station, since the load is proportional to the delivered power.

2.1. System Configuration

2.1.1. Transmission Line

Since distribution network lines are not fully transposed along the transmission line, they are represented by a fixed-valued impedance matrix of size (3 × 3) determined by Equation (4) [55].

$$Z_{line}=\left[\begin{array}{ccc}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ba}& Z_{bb}& Z_{bc}\\ Z_{ca}& Z_{cb}& Z_{cc}\end{array}\right]$$

(4)

with:

$Z_{aa}, Z_{bb}, Z_{cc}$: the self-impedance of phases $a$, $b$, and $c$, respectively.

$Z_{nm}$: the mutual impedance between phases $n$ and $m$, where $n$ and $m$ are the phase a, b or c.

2.1.2. Fault Distance

For each fault location, the short-circuit current has a different value. The closer the fault is to the source, the higher the short-circuit current; conversely, as the fault location moves farther from the source, the short-circuit current decreases. Figure 2 illustrates the variation in the short-circuit current magnitude at different fault locations along the transmission line.

2.1.3. Bus S Voltage

The voltage in the distribution network is variable, and there is no single fixed voltage throughout the supply period. It is known that the voltage tolerance is ±10% of the nominal voltage [56]. Figure 3 shows a linear relationship between voltage and current.

2.1.4. Fault Resistance

The fault resistance is defined as an element that establishes a connection between phases or between a phase and the ground. This element can be, for instance, a transmission tower, a stork, etc. Therefore, the fault resistance can take any value [54,57]. Figure 4 illustrates a nonlinear variation in the short-circuit current. As the fault resistance increases, the short-circuit current decreases.

2.1.5. Loads

As the active and reactive power consumption in distribution network varies according the hours, days, and seasons of the year, and as the consumption of active and reactive power increases, the short-circuit current also increases. Figure 5 shows a linear variation in the short-circuit current: as the consumption of active and reactive power increases, the short-circuit current also increases.

3. Intelligent Fault Location and Classification Method

Machine learning techniques fundamentally rely on data as the primary input for training and decision-making. In the proposed approach, the objective is to improve fault location and classification in distributed transmission lines. However, accurate fault localization requires prior estimation of the fault resistance. To achieve this, a multi-objective optimization is applied for fault resistance estimation. Once the fault resistance is determined, the corresponding fault location is identified. The results are then processed by a machine-learning-based decision tree classifier to determine the fault type (single line-to-ground, line-to-line, double line-to-ground, or three-phase fault).

3.1. Data Collection for Machine Learning

Phasor measurement units (PMUs) play a crucial role in fault analysis for both transmission and distribution networks, as they provide synchronized voltage and current phasors with high sampling rates, enabling accurate detection of transient disturbances and fault events [58,59].

Machine learning models rely on diverse data sources, PMU recordings, digital relays, and simulated datasets from MATLAB/Simulink or PSCAD, where different fault scenarios (single line-to-ground, double line, double line-to-ground, and three-phase faults) are generated under varying resistances and locations [60]. Thus, combining PMU-based field data with simulated fault data provides a robust foundation for machine learning models to improve classification and location accuracy in modern power systems.

In this study, the dataset is classified into single-resistance faults with six variable parameters and three-resistance faults with eight variable parameters.

3.1.1. Single-Resistance Faults

Referring to the power supply system in Figure 2 and Figure 6, the short-circuit current varies with the following parameters, as shown in Equation (5):

$$I_{sc}=f(V_s, P_a, P_b, P_c, R_F, D_f)$$

(5)

$I_{sc}$: post-fault short-circuit current.

$V_s$: substation (source) voltage.

$P_a$: pre-fault power consumed by phase a, in kilowatts.

$P_b$: pre-fault power consumed by phase b, in kilowatts.

$P_c$: pre-fault power consumed by phase c, in kilowatts.

$R_F$: fault resistance.

$D_f$: fault distance in meters or kilometers.

Figure 6. Line-to-line fault.

Figure 6. Line-to-line fault.

• Sweep ranges.

Station voltage: $\pm 10$% around nominal with a 1% step.

Per-phase active power: 80% to 120% of nominal with a 1% step.

Fault resistance: 0 to 50 Ω with a 1 Ω step.

Fault location along the line: 5% to 95% of total line length with a 10% step.

3.1.2. Three-Resistance Faults

Referring to the power supply system shown in Figure 7 and Figure 8, the short-circuit current varies with the following parameters, as defined by Equation (6):

$$I_{sc}=f(V_s, P_a, P_b, P_c, R_{F_a}, R_{F_b}, R_{F_c}, D_f)$$

(6)

$V_s$: substation (source) voltage.

$P_a$: pre-fault power consumed by phase a, in kilowatts.

$P_b$: pre-fault power consumed by phase b, in kilowatts.

$P_c$: pre-fault power consumed by phase c, in kilowatts.

$R_{F_a}$: fault resistance of each phase a (Ω).

$R_{F_b}$: fault resistance of each phase b (Ω).

$R_{F_c}$: fault resistance of each phase c (Ω).

$D_f$: fault distance in meters or kilometers.

Figure 7. Double line-to-ground fault.

Figure 7. Double line-to-ground fault.

Figure 8. Three-phase fault.

Figure 8. Three-phase fault.

• Sweep ranges.

Station voltage: $\pm 10\%$ around nominal with a 1% step.

Per-phase active power: 80% to 120% of nominal with a 1% step.

Fault resistance: 0 to 50 Ω with a 1 Ω step.

Fault location along the line: 5% to 95% of total line length with a 10% step.

3.1.3. The Data Collection Algorithm in Parallel Computing

The data corresponds to the short-circuit current values for different cases. These values are collected using MATLAB/Simulink for short-circuit current estimation. However, the standard for loop is not sufficient and requires a long time for data collection [50]. A solution is to use a parallel loop (parfor loop) in order to utilize 100% of the CPU. Prior to running the simulation, a comprehensive matrix must be created to store all combinations of voltage, active power, fault resistance, and fault distance for the entire set of cases. The procedure is outlined in Algorithm 1.

Algorithm 1: Pseudo-code of the data collection in parallel computing.
input	1: create function of simulation that read the input parameter and run simulation and gives the values of short-circuit for each phase a, b and c. 2: enter the fixed parameters such as the line and station impedance. 3: load the matrix that contains all the parameter values. 4: run parfor loop. 5: read the parameter of each step of the loop. 6: called function of simulation. 7: save the data into matrix 6D for single resistance fault and 8D for three resistance faults. 8: end of parfor loop.
output

3.2. Intelligent Digital Protection System

Consider the diagram in Figure 9, which shows the required components (CT, PT, data center) for a transmission line protection system in a radial distribution network.

This intelligent protection system achieves a high level of protection by utilizing only current transformers (CTs) at the beginning of each branch for fault detection, classification, and location. The measured currents are transmitted directly to the control station. If a fault is detected, the system proceeds to classify and locate the fault, enabling the maintenance team to rapidly reach the faulted section and restore service efficiently.

3.3. Fault Localization and Classification Algorithm

The role of the fault classification and location algorithm is to determine the fault distance using intelligent methods and their corresponding types. The algorithm receives voltage and current parameters as inputs and outputs the precise fault distance. To accomplish this, the localization module takes the aforementioned voltage and current data and applies 6D and 8D interpolation functions.

3.3.1. Fault Distance Estimation

The fault distance is estimated using machine-learning-based cubic spline interpolation in 6D and 8D [61], relying on data collected prior to the fault. The short-circuit current depends on two categories of parameters: the first corresponds to pre-fault parameters, such as the voltage source and active power consumption, whereas the second corresponds to post-fault parameters, such as fault resistance and fault distance. Since the pre-fault parameters are already known, the dimensionality can be reduced, allowing the estimation process to be simplified from 6D to 2D and from 8D to 4D, and the fault distance can be estimated by following the steps below:

• Input the fault resistance value.
• Estimate the short-circuit current values using 2D or 4D cubic spline interpolation.
• Estimate the fault distance using 1D cubic spline interpolation.

3.3.2. Fault Resistance Estimation

The idea here is that when a fault occurs in a transmission line, regardless of its type, it always takes place between two conductors, either phase-to-phase or phase-to-ground. For example, in the case of a two-phase fault AB, the fault occurs between phases A and B. In this situation, there exists a single value of fault resistance that makes the fault distances calculated from phase A and phase B equal to the same fault distance. In other words, we determine the value of resistance that satisfies Equation (7):

$${F1}_{Obj}\left(R_F\right)=\left|D_f^1-D_f^2\right|=0$$

(7)

where $D_f^1$ is the faulted distance of the first faulted phase, and $D_f^2$ is the faulted distance of the second faulted phase. The problem is modeled as an objective function, aiming to find the fault resistance that yields the minimum difference between the fault distances obtained from each phase. This optimization problem can be solved using one of the optimization algorithms, such as NGO [62].

The minimization approach becomes clearer when illustrated with a graph of the objective function, as given in Equation (4) and shown in Figure 10. In this figure, the value of the objective function is plotted against different fault resistances. It can be observed that the objective function reaches its minimum value at approximately 25 Ω, which corresponds to the actual fault resistance.

3.3.3. Constraints

• The fault distance of each phase must be a real positive value, as defined in Equations (8) and (9):

$$D_f^1>0$$

(8)

$$D_f^2>0$$

(9)

• The short-circuit current estimated using cubic spline interpolation must be equal to the value measured by the current transformer, as defined in Equation (10) and (11):

$$I_{1_{sc}}^{ML}=I_{1_{sc}}^{CT}$$

(10)

$I_{1_{sc}}^{ML}$: the short-circuit current of the first faulted phase estimated by machine learning using cubic spline interpolation.

$I_{1_{sc}}^{CT}$: the short-circuit current of the first faulted phase measured by the current transformer.

$$I_{2_{sc}}^{ML}=I_{2_{sc}}^{CT}$$

(11)

$I_{2_{sc}}^{ML}$: the short-circuit current of the second faulted phase estimated by machine learning using cubic spline interpolation.

$I_{2_{sc}}^{CT}$: the short-circuit current of the second faulted phase measured by the current transformer.

• Fault resistance considered as the decision variable, and it is constrained within predefined lower and upper bounds, as defined in Equation (12):

$$R_{F_{min}}<R_F<R_{F_{max}}$$

(12)

$R_{F_{min}}$: the lower bound of the fault resistance $(\Omega ).$

$R_{F_{max}}$: the upper bound of the fault resistance $(\Omega ).$

A problem arises here with the short-circuit constraints in phases 1 and 2: it is impossible to find the current estimated by the ML that exactly matches the actual current. Therefore, a solution consists of transforming these two Equations (7) and (8) into the following two objective functions, as defined in Equations (13) and (14):

$${F2}_{Obj}=\left|I_{1_{sc}}^{ML}-I_{1_{sc}}^{CT}\right|=0$$

(13)

$${F3}_{Obj}=\left|I_{2_{sc}}^{ML}-I_{2_{sc}}^{CT}\right|=0$$

(14)

Additionally, the problem is transformed into a multi-objective optimization, with three objective functions under the following constraints, as defined in Equations (15) and (16):

$$F_{Obj}=\mathrm{m}\mathrm{i}\mathrm{n}(F1, F2, F3)$$

(15)

$$D_f^1>0, D_f^2>0 et R_{F_{min}}<R_F<R_{F_{max}}$$

(16)

A multi-objective optimization based on the weighted sum method is employed to transform the multi-objective problem into a single-objective problem using equals-weight coefficients [63]. Figure 11 demonstrates that the objective function defined in Equation (15) reaches its minimum value, which corresponds to the actual fault resistance.

3.3.4. Multi-Objective Optimization Using NGO Algorithm

The different stages of multi-objective optimization using the NGO algorithm [2] are described in Algorithm 2 and are divided into three parts. At the beginning of the algorithm, the population members are randomly initialized within the search space of the fault resistance. The population matrix is determined using Equations (17) and (18), which describe the single- and the three-fault resistances.

$$R1F={\left[\begin{array}{c}{RF}_1\\ ⋮\\ {RF}_i\\ ⋮\\ {RF}_N\end{array}\right]}_{N\times m}={\left[\begin{array}{c}{R}_{\mathrm{1,1}}\\ ⋮\\ R_{i,1}\\ ⋮\\ R_{N,1}\end{array}\right]}_{N\times m}$$

(17)

$$R3F={\left[\begin{array}{c}{RF}_1\\ ⋮\\ {RF}_i\\ ⋮\\ {RF}_N\end{array}\right]}_{N\times m}={\left[\begin{array}{ccc}{R}_{\mathrm{1,1}}& R_{\mathrm{1,2}}& R_{\mathrm{1,3}}\\ ⋮& ⋮& ⋮\\ R_{i,1}& R_{i,2}& R_{i,3}\\ ⋮& ⋮& ⋮\\ R_{N,1}& R_{N,2}& R_{N,3}\end{array}\right]}_{N\times m}$$

(18)

where $R1F$ and $R3F$ are the populations of northern goshawks. ${RF}_i$ is the ith proposed solution of fault resistance. $R_{i,j}$ is the value of the jth fault resistance specified by the ith proposed solution, $N$ is the number of population members, and $m$ is the number of problem variables.

Therefore, the objective function of the problem can be evaluated based on each population member. These values obtained for the objective function can be represented as a vector using (19).

$$F\left(R1F/R3F\right)={\left[\begin{array}{c}{F}_1=F\left({RF}_1\right)\\ ⋮\\ F_i=F\left({RF}_i\right)\\ ⋮\\ F_N=F\left({RF}_N\right)\end{array}\right]}_{N\times 1}$$

(19)

where $F$ is the vector of the obtained objective function values, and $F_i$ is the objective function value obtained by ith proposed solution.

To update the population members, the Northern Goshawk Optimization algorithm employs two strategies: the first is exploration, given by Equation (20) with its update formulation given in Equation (21), and the second is exploitation, given by Equation (22) with its update formulation given in Equation (23).

$$R_{i,j}^{new,P1}=\left\{\begin{array}{c}{R}_{i,j}+r\left(p_{i,j}-I{R}_{i,j}\right), F_{P_i}<F_i\\ R_{i,j}+r\left(x_{i,j}-p_{i,j}\right), F_{P_i}\ge F_i\end{array}\right.$$

(20)

where $P_i=R_k i=\mathrm{1,2},\dots ,N k=\mathrm{1,2}, \dots , i-1,i+1, \dots ,N$, $r$ is a random number in the interval $[0,1]$, $I$ is a random number that can be 1 or 2, $P_i$ is the fault resistance for the ith proposed solution, $k$ is a random natural number in the interval $[1, N]$, $R_i^{new,P1}$ is the new status for the ith proposed solution, $R_{i,j}^{new,P1}$ is its jth dimension, $F_i^{new,P1}$ is its objective function value based on the first phase of NGO, and $F_{P_i}$ is its objective function value.

$$R_i=\left\{\begin{array}{c}{R}_i^{new,P1} , F_i^{new,P1}<F_i\hfill \\ R_i , F_i^{new,P1}\ge F_i\hfill \end{array}\right.$$

(21)

$$R_{i,j}^{new,P2}=R+C\left(2r-1\right)R_{i,j}$$

(22)

where $C=0.02\left(1-{\displaystyle \frac{t}{T}}\right)$, $t$ is the iteration counter, $T$ is the maximum number of iterations, $R_i^{new,P2}$ is the new status for the ith proposed solution, $R_{i,j}^{new,P2}$ is its jth dimension, and $F_i^{new,P2}$ is its objective function value based on the second phase of NGO.

$$R_i=\left\{\begin{array}{c}{R}_i^{new,P2} , F_i^{new,P2}<F_i\\ R_i , F_i^{new,P2}\ge F_i\end{array}\right.$$

(23)

The various stages of the multi-objective optimization using NGO are specified as pseudo-code in Algorithm 2.

Algorithm 2: Pseudo-code of the multi-objective optimization using NGO.
input	1: Input the optimization problem variable: fault resistance upper and lower bound and initiate weight of the three-objective function in Equations (7), (13) and (14). 2: Set the number of iterations $T$ and population $N$. 3: Set random population using Equation (17) or Equation (18) and their evaluation using Equation (19). 4: Run loop for $T$ iterations. 5: Exploration using Equations (20) and (21). 6: Exploitation using Equations (22) and (23). 7: Save the best solution. 8: End loop for. 9: Output best optimal fault resistance solution.
searching
output

3.3.5. Fault Classification

A machine learning decision tree is used for fault classification by identifying the fault type through traversal from the root to a leaf node. Classification begins at the root, where the attribute at that node is evaluated, and the branch corresponding to the attribute’s value is followed. This procedure is repeated recursively down the tree until a leaf node is reached, which determines the fault type [64].

Consider the following figure, which represents the fault severity pyramid from the least severe to the most severe. The short-circuit current in the single line-to-ground fault is lower compared to the line-to-line fault; the line-to-line fault is lower compared to the double line-to-ground fault; and the fault with the highest short-circuit current value is the three-phase fault [57,65].

According to Figure 2 and Figure 4, there is an inverse proportional relationship between the short-circuit current and both the fault distance and the fault resistance. Figure 12 shows that the single-phase fault produces lower short-circuit current values compared to other types of faults.

However, when a fault occurs, its type is unknown. Based on the short-circuit current value, the fault can be localized by considering a single-phase fault, which allows for identifying the faulty phase within the fault zone. Figure 13 shows that the fault distances of phases B and C lie outside the faulted zone, while the fault distance of phase A falls within it, indicating that only phase A is faulted.

3.3.6. Fault Zone

This zone refers to the distance threshold (in kilometers or meters) beyond which a fault can be identified as a single-phase fault. In the present case study, the fault zone extends to approximately four to five times the total line length under fault conditions and may be expressed by Equation (24):

$$z_f=L_{tot}\times CF$$

(24)

where:

$z_f$: faulted zone in kilometers or meters.

$L_{tot}$: total length of the faulted line in kilometers or meters.

$CF$: faulted zone coefficient.

The fault type classification is performed according to the diagram, following the steps of the machine learning decision tree, as illustrated in Figure 14, Figure 15 and Figure 16.

For other fault types, BC-g and AC-g, we use the same machine learning decision tree shown in Figure 16, employing the fault currents of phases b and c and phases a and c, respectively.

4. Computational Experiments and Results

4.1. Case Study

The fault classification and localization were solved by using the ML and NGO presented in the previous section, implemented in MATLAB on a PC with a 13th Gen Intel(R) Core (TM) i7-13620H (2.40 GHz) processor and 16 GB RAM under the Windows 11 operating system.

To validate the proposed algorithm, a modified version of the IEEE 34-bus Test Feeder was used for numerical simulations. This feeder, located in Arizona, operates at a nominal voltage of 24.9 kV. It is characterized by its long and lightly loaded lines, the presence of two in-line regulators, an in-line transformer supplying a short 4.16 kV section, unbalanced loading conditions, and shunt capacitor banks [66].

The original IEEE 34-bus system was modified by removing the two in-line regulators and adding the required components CT and PT at the substation (bus 800) and CTs at all other nodes. In addition, all loads were modeled as constant impedances.

The modified test system, composed of 34 buses and illustrated by Figure 17, was simulated using MATLAB/Simulink. The fault classification and location algorithm was implemented in MATLAB. The simulations were carried out on the line between bus 808 and bus 812 considering the following fault conditions:

• Five different fault distances from 5% to 95% of the total line length.
• Three different fault resistances: 0, 10, 50 and 100 Ω.
• Three different voltage source substations: 95%, 100%, and 105%.
• Three different voltage phase angles: 0°, 120°, and 240°.
• Three different load consumptions: 80%, 100%, and 120%.
• Ten fault types: A-g, B-g, C-g, AB, BC, AC, AB-g, BC-g, AC-g, and ABC-g.
• A total of 1000 test case chosen randomly for each fault type.
• A comparison of the proposed method’s accuracy with that reported in [67,68].

Figure 17. Modified IEEE 34-bus node.

Figure 17. Modified IEEE 34-bus node.

4.2. Results

The results are obtained using the proposed fault location and classification algorithm, integrated within the intelligent protection system described in the previous section. Faults were simulated at various distances and resistances points, and the locations were estimated using cubic spline interpolation, while the fault resistance was optimized through a weighted-sum multi-objective approach employing the NGO meta-heuristic. The outcomes are summarized in

Table 3. Single line-to-ground fault location results A-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (Ω)
0.001	5	0.572	0.572	1.05 × 10−4	0.001	6.53 × 10−8
	25	2.858	2.858	1.04 × 10−4	0.001	7.89 × 10−8
	50	5.715	5.715	6.90 × 10−5	0.001	1.55 × 10−7
	75	8.573	8.573	1.04 × 10−4	0.001	1.34 × 10−7
	95	10.859	10.859	1.04 × 10−4	0.001	1.61 × 10−7
10	5	0.572	0.572	1.24 × 10−4	10.000	5.33 × 10−6
	25	2.858	2.858	1.04 × 10−4	10.000	1.44 × 10−7
	50	5.715	5.715	1.01 × 10−5	10.000	6.06 × 10−6
	75	8.573	8.572	7.49 × 10−5	10.000	2.79 × 10−5
	95	10.859	10.859	1.03 × 10−4	10.000	2.67 × 10−7
50	5	0.572	0.572	1.05 × 10−4	50.000	2.84 × 10−14
	25	2.858	2.858	1.05 × 10−4	50.000	3.55 × 10−14
	50	5.715	5.715	1.01 × 10−4	50.000	3.55 × 10−14
	75	8.573	8.573	1.04 × 10−4	50.000	1.42 × 10−14
	95	10.859	10.859	1.04 × 10−4	50.000	1.42 × 10−14

,

Table 4. Single line-to-ground fault location results B-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (Ω)
0.001	5	0.572	0.571	2.66 × 10−10	0.001	4.43 × 10−11
	25	2.858	2.857	4.11 × 10−10	0.001	6.85 × 10−11
	50	5.715	5.715	1.73 × 10−4	0.001	1.68 × 10−8
	75	8.573	8.572	7.02 × 10−10	0.001	1.22 × 10−10
	95	10.859	10.858	5.48 × 10−5	0.001	9.17 × 10−6
10	5	0.572	0.572	8.38 × 10−5	10.000	1.25 × 10−5
	25	2.858	2.858	2.88 × 10−10	10.000	4.12 × 10−11
	50	5.715	5.715	9.18 × 10−5	10.000	5.57 × 10−6
	75	8.573	8.573	2.15 × 10−4	10.000	2.98 × 10−5
	95	10.859	10.859	2.18 × 10−5	10.000	1.46 × 10−6
50	5	0.572	0.572	1.66 × 10−9	50.000	3.98 × 10−13
	25	2.858	2.858	2.72 × 10−5	50.000	6.45 × 10−5
	50	5.715	5.715	3.07 × 10−6	50.000	3.13 × 10−13
	75	8.573	8.573	4.09 × 10−4	50.000	7.11 × 10−14
	95	10.859	10.859	7.71 × 10−5	50.000	2.57 × 10−5

,

Table 5. Single line-to-ground fault location results C-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(Ω)
0.001	5	0.572	0.572	1.05 × 10−4	0.001	2.23 × 10−8
	25	2.858	2.858	1.05 × 10−4	0.001	2.48 × 10−8
	50	5.715	5.715	9.68 × 10−5	0.001	4.21 × 10−6
	75	8.573	8.573	1.04 × 10−4	0.001	5.22 × 10−8
	95	10.859	10.859	7.71 × 10−5	0.001	4.35 × 10−6
10	5	0.572	0.572	1.04 × 10−4	10.000	7.30 × 10−8
	25	2.858	2.858	1.03 × 10−4	10.000	2.05 × 10−6
	50	5.715	5.715	1.63 × 10−5	10.000	5.35 × 10−6
	75	8.573	8.573	2.18 × 10−4	10.000	1.66 × 10−5
	95	10.859	10.859	2.88 × 10−4	10.000	2.42 × 10−5
50	5	0.572	0.572	3.68 × 10−4	50.000	3.97 × 10−5
	25	2.858	2.858	1.04 × 10−4	50.000	2.13 × 10−14
	50	5.715	5.715	1.26 × 10−4	50.000	3.57 × 10−6
	75	8.573	8.573	1.04 × 10−4	50.000	4.97 × 10−14
	95	10.859	10.859	1.03 × 10−4	50.000	1.28 × 10−13

,

Table 6. Line-to-line fault location results AB.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(Ω)
0.001	5	0.572	0.572	1.05 × 10−4	0.001	3.68 × 10−7
	25	2.858	2.858	1.05 × 10−4	0.001	2.66 × 10−13
	50	5.715	5.715	9.21 × 10−5	0.001	7.40 × 10−6
	75	8.573	8.573	1.05 × 10−4	0.001	1.47 × 10−14
	95	10.859	10.859	1.05 × 10−4	0.001	9.38 × 10−14
10	5	0.572	0.572	1.06 × 10−4	10.000	3.60 × 10−8
	25	2.858	2.858	1.05 × 10−4	10.000	2.77 × 10−8
	50	5.715	5.715	5.36 × 10−6	10.000	6.14 × 10−6
	75	8.573	8.573	1.06 × 10−4	10.000	3.28 × 10−8
	95	10.859	10.859	7.66 × 10−5	10.000	4.99 × 10−6
50	5	0.572	0.572	1.62 × 10−4	50.000	1.21 × 10−5
	25	2.858	2.858	1.09 × 10−4	50.000	3.78 × 10−7
	50	5.715	5.715	1.73 × 10−4	50.000	2.66 × 10−5
	75	8.573	8.573	1.09 × 10−4	50.000	4.20 × 10−7
	95	10.859	10.859	2.02 × 10−4	50.000	1.80 × 10−5

,

Table 7. Line-to-line fault location results BC.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(Ω)
0.001	5	0.572	0.572	1.05 × 10−4	0.001	4.59 × 10−8
	25	2.858	2.858	1.05 × 10−4	0.001	4.37 × 10−8
	50	5.715	5.715	7.17 × 10−5	0.001	1.02 × 10−6
	75	8.573	8.573	1.05 × 10−4	0.001	7.03 × 10−8
	95	10.859	10.859	9.80 × 10−5	0.001	1.64 × 10−14
10	5	0.572	0.572	1.04 × 10−4	10.000	3.40 × 10−7
	25	2.858	2.858	1.05 × 10−4	10.000	4.86 × 10−8
	50	5.715	5.715	1.44 × 10−5	10.000	9.40 × 10−6
	75	8.573	8.573	9.18 × 10−5	10.000	7.15 × 10−6
	95	10.859	10.859	9.31 × 10−5	10.000	4.50 × 10−6
50	5	0.572	0.572	1.07 × 10−4	50.000	6.40 × 10−8
	25	2.858	2.858	3.45 × 10−4	50.000	8.27 × 10−5
	50	5.715	5.715	9.81 × 10−5	50.000	0.00 × 100
	75	8.573	8.573	2.90 × 10−4	50.000	1.42 × 10−14
	95	10.859	10.859	1.92 × 10−4	50.000	2.86 × 10−5

,

Table 8. Line-to-line fault location results AC.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(%)	Rf (Ω)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$(Ω)
0.001	5	0.572	0.572	9.46 × 10−5	0.001	2.26 × 10−6
	25	2.858	2.858	8.51 × 10−5	0.001	3.31 × 10−6
	50	5.715	5.715	7.08 × 10−5	0.001	7.36 × 10−7
	75	8.573	8.573	9.43 × 10−5	0.001	9.95 × 10−7
	95	10.859	10.859	1.09 × 10−4	0.001	1.24 × 10−13
10	5	0.572	0.572	8.86 × 10−5	10.000	2.66 × 10−6
	25	2.858	2.858	2.11 × 10−4	10.000	1.73 × 10−5
	50	5.715	5.715	3.34 × 10−5	10.000	1.05 × 10−6
	75	8.573	8.573	1.04 × 10−4	10.000	3.85 × 10−8
	95	10.859	10.859	1.04 × 10−4	10.000	5.54 × 10−8
50	5	0.572	0.572	1.05 × 10−4	50.000	2.20 × 10−7
	25	2.858	2.858	1.05 × 10−4	50.000	2.41 × 10−7
	50	5.715	5.715	9.55 × 10−5	50.000	2.13 × 10−14
	75	8.573	8.573	6.31 × 10−4	50.000	4.98 × 10−5
	95	10.859	10.859	1.72 × 10−6	50.000	1.97 × 10−11

,

Table 9. Double line-to-ground fault location results AB-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rfb		Calculated Fault Resistance Rfc		Calculated Fault Resistance Rfg
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rfa (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfb (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfg (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
0.001	5	0.572	0.572	9.75 × 10−5	0.001	1.34 × 10−6	0.001	5.83 × 10−7	0.001	8.29 × 10−7
	25	2.858	2.858	1.05 × 10−4	0.001	2.96 × 10−16	0.001	9.78 × 10−9	0.001	2.01 × 10−15
	50	5.715	5.715	6.82 × 10−5	0.001	8.29 × 10−8	0.001	3.57 × 10−8	0.001	2.69 × 10−11
	75	8.573	8.573	1.05 × 10−4	0.001	1.06 × 10−14	0.001	8.79 × 10−9	0.001	1.93 × 10−8
	95	10.859	10.859	1.05 × 10−4	0.001	1.34 × 10−15	0.001	9.38 × 10−9	0.001	2.61 × 10−8
10	5	0.572	0.572	1.06 × 10−4	10.000	3.97 × 10−8	10.000	2.76 × 10−8	10.000	7.25 × 10−8
	25	2.858	2.858	1.06 × 10−4	10.000	4.62 × 10−8	10.000	2.41 × 10−8	10.000	7.38 × 10−8
	50	5.715	5.715	3.44 × 10−5	10.000	3.66 × 10−6	10.000	3.53 × 10−6	10.000	4.59 × 10−6
	75	8.573	8.572	1.13 × 10−5	10.000	1.56 × 10−6	10.000	3.50 × 10−5	10.000	5.07 × 10−5
	95	10.859	10.859	1.06 × 10−4	10.000	8.59 × 10−8	10.000	2.59 × 10−8	10.000	1.01 × 10−7
50	5	0.572	0.572	3.24 × 10−3	50.000	3.35 × 10−4	50.000	3.37 × 10−4	50.000	7.94 × 10−5
	25	2.858	2.859	8.93 × 10−3	49.999	9.73 × 10−4	49.999	6.49 × 10−4	49.996	4.35 × 10−3
	50	5.715	5.715	2.53 × 10−3	50.000	2.71 × 10−4	50.000	2.19 × 10−4	49.999	7.72 × 10−4
	75	8.573	8.573	1.53 × 10−4	50.000	5.07 × 10−6	50.000	4.10 × 10−6	50.000	1.75 × 10−5
	95	10.859	10.859	8.14 × 10−3	49.999	9.15 × 10−4	49.999	8.77 × 10−4	50.000	3.36 × 10−4

,

Table 10. Double line-to-ground fault location results BC-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rfb		Calculated Fault Resistance Rfc		Calculated Fault Resistance Rfg
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rfb (Ω)	Error $\left|\mathit{\epsilon }\right|$(%)	Rfc (Ω)	Error $\left|\mathit{\epsilon }\right|$(%)	Rfg (Ω)	Error $\left|\mathit{\epsilon }\right|$(%)
0.001	5	0.572	0.572	1.05 × 10−4	0.001	4.85 × 10−9	0.001	1.33 × 10−11	0.001	1.07 × 10−11
	25	2.858	2.858	1.05 × 10−4	0.001	1.19 × 10−9	0.001	5.14 × 10−9	0.001	6.85 × 10−9
	50	5.715	5.715	6.80 × 10−5	0.001	2.61 × 10−8	0.001	3.33 × 10−8	0.001	1.72 × 10−8
	75	8.573	8.573	1.05 × 10−4	0.001	1.28 × 10−7	0.001	4.95 × 10−9	0.001	3.57 × 10−8
	95	10.859	10.859	1.05 × 10−4	0.001	5.09 × 10−9	0.001	1.94 × 10−14	0.001	3.11 × 10−16
10	5	0.572	0.572	1.05 × 10−4	10.000	1.54 × 10−8	10.000	6.75 × 10−8	10.000	4.06 × 10−8
	25	2.858	2.858	1.05 × 10−4	10.000	1.72 × 10−8	10.000	6.93 × 10−8	10.000	6.01 × 10−8
	50	5.715	5.715	2.01 × 10−4	10.000	3.52 × 10−5	10.000	3.02 × 10−5	10.000	2.84 × 10−5
	75	8.573	8.572	1.32 × 10−4	10.000	3.18 × 10−5	10.000	3.35 × 10−5	10.000	8.53 × 10−5
	95	10.859	10.859	1.06 × 10−4	10.000	4.82 × 10−9	10.000	8.79 × 10−8	10.000	1.50 × 10−7
50	5	0.572	0.572	3.31 × 10−4	50.000	2.21 × 10−5	50.000	2.04 × 10−5	50.000	9.57 × 10−5
	25	2.858	2.858	1.11 × 10−4	50.000	0.00 × 100	50.000	1.30 × 10−6	50.000	7.45 × 10−7
	50	5.715	5.715	1.10 × 10−4	50.000	3.69 × 10−13	50.000	1.32 × 10−6	50.000	1.43 × 10−6
	75	8.573	8.573	6.58 × 10−3	49.999	7.36 × 10−4	49.999	6.25 × 10−4	49.998	1.52 × 10−3
	95	10.859	10.859	6.74 × 10−3	49.999	8.71 × 10−4	50.000	3.35 × 10−4	49.997	2.62 × 10−3

,

Table 11. Double line-to-ground fault location results AC-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rfb		Calculated Fault Resistance Rfc		Calculated Fault Resistance Rfg
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rfa (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfc (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfg (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
0.001	5	0.572	0.572	1.03 × 10−4	0.001	3.33 × 10−7	0.001	4.11 × 10−7	0.001	1.30 × 10−7
	25	2.858	2.858	1.05 × 10−4	0.001	8.46 × 10−9	0.001	3.73 × 10−8	0.001	1.67 × 10−16
	50	5.715	5.715	6.81 × 10−5	0.001	2.30 × 10−8	0.001	9.23 × 10−8	0.001	1.48 × 10−8
	75	8.573	8.573	1.05 × 10−4	0.001	5.32 × 10−9	0.001	5.83 × 10−8	0.001	6.03 × 10−16
	95	10.859	10.859	1.05 × 10−4	0.001	9.45 × 10−10	0.001	5.12 × 10−8	0.001	4.17 × 10−10
10	5	0.572	0.572	1.05 × 10−4	10.000	7.12 × 10−8	10.000	9.77 × 10−9	10.000	7.48 × 10−8
	25	2.858	2.858	4.19 × 10−4	10.000	2.55 × 10−5	10.000	9.30 × 10−6	10.000	1.05 × 10−4
	50	5.715	5.715	3.26 × 10−5	10.000	4.02 × 10−6	10.000	3.78 × 10−6	10.000	2.43 × 10−6
	75	8.573	8.573	1.46 × 10−4	10.000	1.21 × 10−5	10.000	1.11 × 10−5	10.000	1.28 × 10−5
	95	10.859	10.859	1.05 × 10−4	10.000	9.56 × 10−8	10.000	1.22 × 10−8	10.000	1.46 × 10−7
50	5	0.572	0.572	1.13 × 10−4	50.000	1.50 × 10−6	50.000	4.37 × 10−7	50.000	0.00 × 100
	25	2.858	2.858	1.13 × 10−4	50.000	1.54 × 10−6	50.000	4.40 × 10−7	50.000	1.42 × 10−14
	50	5.715	5.715	1.09 × 10−4	50.000	1.25 × 10−6	50.000	1.05 × 10−7	50.000	3.31 × 10−7
	75	8.573	8.573	1.14 × 10−4	50.000	1.66 × 10−6	50.000	4.66 × 10−7	50.000	1.42 × 10−7
	95	10.859	10.859	6.57 × 10−4	50.000	8.04 × 10−5	50.000	2.03 × 10−6	50.000	1.89 × 10−4

and

Table 12. Three-phase fault location results ABC-g.

Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rfb		Calculated Fault Resistance Rfc		Calculated Fault Resistance Rfg
	x (%)	d (km)	x (km)	$\mathbf{Error} \left|\mathit{\epsilon }\right|$ (%)	Rfa (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfb (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rfc (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
0.001	5	0.572	0.572	1.04 × 10−4	0.001	1.21 × 10−7	0.001	8.94 × 10−8	0.001	2.67 × 10−7
	25	2.858	2.858	1.05 × 10−4	0.001	3.18 × 10−14	0.001	4.94 × 10−9	0.001	1.80 × 10−8
	50	5.715	5.715	6.82 × 10−5	0.001	8.56 × 10−8	0.001	4.41 × 10−8	0.001	7.29 × 10−8
	75	8.573	8.573	1.05 × 10−4	0.001	1.03 × 10−10	0.001	2.67 × 10−8	0.001	4.01 × 10−9
	95	10.859	10.859	1.05 × 10−4	0.001	1.29 × 10−9	0.001	4.18 × 10−8	0.001	2.17 × 10−8
10	5	0.572	0.572	6.17 × 10−4	10.000	8.39 × 10−5	10.000	8.68 × 10−5	10.000	2.42 × 10−4
	25	2.858	2.858	1.05 × 10−4	10.000	3.80 × 10−8	10.000	4.46 × 10−9	10.000	7.09 × 10−8
	50	5.715	5.715	3.00 × 10−5	10.000	4.07 × 10−6	10.000	4.14 × 10−6	10.000	3.36 × 10−6
	75	8.573	8.573	1.05 × 10−4	10.000	6.16 × 10−8	10.000	6.19 × 10−9	10.000	3.85 × 10−8
	95	10.859	10.858	2.05 × 10−4	10.000	1.54 × 10−5	10.000	6.77 × 10−5	10.000	1.69 × 10−4
50	5	0.572	0.572	3.28 × 10−4	50.000	2.54 × 10−6	50.000	3.16 × 10−5	50.000	3.84 × 10−4
	25	2.858	2.858	1.14 × 10−4	50.000	5.45 × 10−7	50.000	9.11 × 10−7	50.000	7.58 × 10−6
	50	5.715	5.715	1.06 × 10−4	50.000	2.46 × 10−7	50.000	8.38 × 10−8	50.000	1.51 × 10−6
	75	8.573	8.573	1.16 × 10−3	50.000	7.23 × 10−6	50.000	1.46 × 10−4	49.999	1.39 × 10−3
	95	10.859	10.859	2.21 × 10−4	50.000	6.61 × 10−6	50.000	1.47 × 10−5	50.000	8.76 × 10−5

, where x denotes fault distance in percentages (%) and d represents the corresponding distance in kilometers. The estimation errors were computed using Equations (25) and (26), which define the error metrics for fault distance and fault resistance [54,69], respectively.

$$distance error={\displaystyle \frac{Calculate fault location-Real fault location}{Total length of the line}}\times 100$$

(25)

$$resisatance error=Calculate fault resistance-Real fault resistance$$

(26)

The subsequent results were obtained by varying the substation voltage magnitude and its phase angle, followed by re-evaluating the fault location. The outcomes, which demonstrate the impact of these variations on fault location accuracy, are summarized in

Table 13. Fault location results considering a voltage magnitude of 95%.

Voltage (%)	Φ (°)	Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rf
			x (%)	d (km)	x (km)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
95	0	0.001	5	0.572	0.572	1.05 × 10−4	0.001	7.35 × 10−8
			50	5.715	5.715	6.90 × 10−5	0.001	1.13 × 10−7
			95	10.859	10.859	7.96 × 10−5	0.001	3.72 × 10−6
		10	5	0.572	0.572	1.10 × 10−4	10.000	1.13 × 10−6
			50	5.715	5.715	9.60 × 10−6	10.000	6.10 × 10−6
			95	10.859	10.859	1.53 × 10−4	10.000	8.84 × 10−6
		50	5	0.572	0.572	1.04 × 10−4	50.000	6.39 × 10−14
			50	5.715	5.715	1.01 × 10−4	50.000	3.87 × 10−10
			95	10.859	10.858	9.14 × 10−6	50.000	9.24 × 10−14
	120	0.001	5	0.572	0.572	9.64 × 10−5	0.001	1.21 × 10−6
			50	5.715	5.715	6.83 × 10−5	0.001	1.88 × 10−7
			95	10.859	10.859	1.03 × 10−4	0.001	1.89 × 10−7
		10	5	0.572	0.572	1.04 × 10−4	10.000	1.21 × 10−7
			50	5.715	5.715	1.28 × 10−5	10.000	9.69 × 10−6
			95	10.859	10.859	1.43 × 10−4	10.000	3.34 × 10−6
		50	5	0.572	0.572	2.73 × 10−4	50.000	1.47 × 10−5
			50	5.715	5.715	8.03 × 10−4	50.000	0.00 × 100
			95	10.859	10.858	4.73 × 10−6	50.000	2.34 × 10−13
	240	0.001	5	0.572	0.572	1.05 × 10−4	0.001	7.35 × 10−8
			50	5.715	5.715	6.90 × 10−5	0.001	1.13 × 10−7
			95	10.859	10.859	1.26 × 10−4	0.001	5.20 × 10−14
		10	5	0.572	0.572	1.04 × 10−4	10.000	1.21 × 10−7
			50	5.715	5.715	2.83 × 10−5	10.000	2.07 × 10−6
			95	10.859	10.859	2.45 × 10−4	10.000	2.23 × 10−5
		50	5	0.572	0.572	2.29 × 10−4	50.000	6.36 × 10−5
			50	5.715	5.715	3.27 × 10−4	50.000	9.95 × 10−14
			95	10.859	10.859	1.04 × 10−4	50.000	2.49 × 10−13

,

Table 14. Fault location results considering a voltage magnitude of 100%.

Voltage (%)	Φ (°)	Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rf
			x (%)	d (km)	x (km)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
100	0	0.001	5	0.572	0.572	1.05 × 10−4	0.001	6.53 × 10−8
			50	5.715	5.715	6.80 × 10−5	0.001	8.96 × 10−8
			95	10.859	10.859	1.04 × 10−4	0.001	1.61 × 10−7
		10	5	0.572	0.572	1.04 × 10−4	10.000	1.08 × 10−7
			50	5.715	5.715	1.01 × 10−5	10.000	6.06 × 10−6
			95	10.859	10.859	1.25 × 10−4	10.000	1.06 × 10−7
		50	5	0.572	0.572	7.25 × 10−5	50.000	5.23 × 10−6
			50	5.715	5.715	2.89 × 10−5	50.000	1.87 × 10−5
			95	10.859	10.859	7.56 × 10−6	50.000	1.14 × 10−13
	120	0.001	5	0.572	0.572	9.65 × 10−5	0.001	1.27 × 10−6
			50	5.715	5.715	6.90 × 10−5	0.001	1.54 × 10−7
			95	10.859	10.859	1.24 × 10−4	0.001	1.43 × 10−14
		10	5	0.572	0.572	1.04 × 10−4	10.000	1.08 × 10−7
			50	5.715	5.715	3.21 × 10−5	10.000	1.44 × 10−6
			95	10.859	10.859	1.03 × 10−4	10.000	2.67 × 10−7
		50	5	0.572	0.572	3.25 × 10−4	50.000	0.00 × 100
			50	5.715	5.715	1.01 × 10−4	50.000	2.20 × 10−13
			95	10.859	10.859	1.04 × 10−4	50.000	2.27 × 10−13
	240	0.001	5	0.572	0.572	1.02 × 10−4	0.001	4.53 × 10−7
			50	5.715	5.715	4.38 × 10−5	0.001	5.58 × 10−14
			95	10.859	10.859	1.05 × 10−4	0.001	8.09 × 10−16
		10	5	0.572	0.572	1.02 × 10−4	10.000	3.81 × 10−6
			50	5.715	5.715	1.20 × 10−5	10.000	5.43 × 10−6
			95	10.859	10.859	1.03 × 10−4	10.000	2.67 × 10−7
		50	5	0.572	0.572	1.05 × 10−4	50.000	2.84 × 10−14
			50	5.715	5.715	1.64 × 10−4	50.000	7.82 × 10−6
			95	10.859	10.859	1.04 × 10−4	50.000	4.26 × 10−14

and

Table 15. Fault location results considering a voltage magnitude of 105%.

Voltage (%)	Φ (°)	Fault Resistance (Ω)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rf
			x (%)	d (km)	x (km)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
105	0	0.001	5	0.572	0.572	1.14 × 10−4	0.001	9.27 × 10−16
			50	5.715	5.715	6.88 × 10−5	0.001	1.58 × 10−7
			95	10.859	10.859	1.04 × 10−4	0.001	1.39 × 10−7
		10	5	0.572	0.572	1.00 × 10−4	10.000	6.68 × 10−7
			50	5.715	5.715	3.21 × 10−6	10.000	8.56 × 10−6
			95	10.859	10.859	1.03 × 10−4	10.000	2.34 × 10−7
		50	5	0.572	0.572	1.96 × 10−4	50.000	1.80 × 10−5
			50	5.715	5.715	9.13 × 10−5	50.000	3.48 × 10−13
			95	10.859	10.859	6.96 × 10−5	50.000	6.39 × 10−14
	120	0.001	5	0.572	0.572	9.25 × 10−5	0.001	3.61 × 10−6
			50	5.715	5.715	6.35 × 10−5	0.001	4.70 × 10−8
			95	10.859	10.859	1.04 × 10−4	0.001	5.58 × 10−14
		10	5	0.572	0.572	6.72 × 10−5	10.000	3.01 × 10−6
			50	5.715	5.715	1.57 × 10−5	10.000	5.44 × 10−6
			95	10.859	10.859	1.03 × 10−4	10.000	2.34 × 10−7
		50	5	0.572	0.572	3.19 × 10−4	50.000	7.05 × 10−5
			50	5.715	5.715	1.02 × 10−4	50.000	0.00 × 100
			95	10.859	10.859	1.05 × 10−4	50.000	1.35 × 10−13
	240	0.001	5	0.572	0.572	1.09 × 10−4	0.001	1.30 × 10−14
			50	5.715	5.715	6.88 × 10−5	0.001	1.58 × 10−7
			95	10.859	10.859	1.03 × 10−4	0.001	8.45 × 10−15
		10	5	0.572	0.572	9.38 × 10−5	10.000	3.63 × 10−6
			50	5.715	5.715	1.05 × 10−5	10.000	6.02 × 10−6
			95	10.859	10.859	1.03 × 10−4	10.000	2.34 × 10−7
		50	5	0.572	0.572	1.05 × 10−4	50.000	6.39 × 10−14
			50	5.715	5.715	7.51 × 10−4	50.000	8.75 × 10−5
			95	10.859	10.859	1.05 × 10−4	50.000	1.42 × 10−14

. For practical purposes, only the single line-to-ground fault case is considered, owing to the large volume of data generated, storage constraints, and the computational cost of resistance optimization.

The following results are obtained by varying the active power load consumption in three cases: 80%, 100%, and 120% of the total load in the IEEE 34-bus system. The corresponding results are presented in

Table 16. Fault location results considering a load variation in each line.

Fault Resistance (Ω)	Line A (%)	Line B (%)	Line C (%)	Actual Distance		Calculated Distance		Calculated Fault Resistance Rf
				x (%)	d (km)	x (km)	Error $\left|\mathit{\epsilon }\right|$ (%)	Rf (Ω)	Error $\left|\mathit{\epsilon }\right|$ (%)
0.001	80	80	80	50	5.715	5.715	6.89 × 10−5	0.001	1.56 × 10−7
	80	100	120	50	5.715	5.715	1.44 × 10−4	0.001	2.02 × 10−5
	100	120	80	50	5.715	5.715	6.30 × 10−5	0.001	2.20 × 10−11
	120	80	100	50	5.715	5.715	6.95 × 10−5	0.001	2.73 × 10−7
	120	120	120	80	5.715	5.715	6.87 × 10−5	0.001	9.20 × 10−8
10	80	80	80	50	5.715	5.715	8.59 × 10−5	10.000	5.50 × 10−7
	80	100	120	50	5.715	5.715	1.07 × 10−5	10.000	5.94 × 10−6
	100	120	80	50	5.715	5.715	6.09 × 10−6	10.000	4.26 × 10−6
	120	80	100	50	5.715	5.715	9.74 × 10−6	10.000	6.13 × 10−6
	120	120	120	50	5.715	5.715	9.57 × 10−6	10.000	6.15 × 10−6
50	80	80	80	50	5.715	5.715	4.06 × 10−4	50.000	5.68 × 10−14
	80	100	120	50	5.715	5.715	1.01 × 10−4	50.000	1.69 × 10−11
	100	120	80	50	5.715	5.715	1.01 × 10−4	50.000	1.56 × 10−13
	120	80	100	50	5.715	5.715	1.02 × 10−4	50.000	5.68 × 10−14
	120	120	120	50	5.715	5.715	5.36 × 10−4	50.000	3.62 × 10−13

, where the active power flow per unit in the lines is calculated using Equation (27) [70], defined below:

$$P_{line}\left(\%\right)={\displaystyle \frac{P_{line}}{S_{base}}}$$

(27)

where $P_{line}$ is the total active power consumption in the substation in kilowatts, $S_{base}$ is the base power in kilovolt-amperes.

The classification results are presented in

Table 17. Fault classification results.

Fault Type	No. of Test Samples	No. of Test Samples Classified Correctly	No of Test Samples Misclassified	Classification Accuracy (%)
A-g	1000	1000	0	100
B-g	1000	1000	0	100
C-g	1000	1000	0	100
AB	1000	1000	0	100
BC	1000	1000	0	100
AC	1000	1000	0	100
AB-g	1000	1000	0	100
BC-g	1000	1000	0	100
AC-g	1000	1000	0	100
ABC-g	1000	1000	0	100
total	10,000	10,000	0	100

. The evaluation was performed for 10.000 randomly selected test samples covering various fault resistances and fault distance points. The performance metric adopted is the classification accuracy [69], mathematically defined in Equation (28).

$$Classification accuracy={\displaystyle \frac{No of test samples classified correctly}{No of samples tested}}\times 100$$

(28)

To evaluate the performance of the proposed method, its accuracy, calculated using Equation (29), is compared with the results reported in [67,68], as presented in

Table 18. Comparative fault location between the proposed method and [67].

Fault Type	Fault Resistance (Ω)	Actual Fault Distance	Estimated Fault Distance (km) [67]	Estimated Fault Distance (km)	Accuracy [67]	Accuracy of the Proposed Method
A-g	0.01	2.625	2.815	2.6354	99.8	99.99
B-g			2.714	2.6354	99.8	99.99
C-g			2.687	2.6354	99.8	99.99
AB			2.549	2.6334	99.9	99.99
BC			2.543	2.6358	99.9	99.99
AC			2.684	2.6358	99.9	99.99
AB-g			2.954	2.6349	99.7	99.99
BC-g			2.961	2.6461	99.7	99.98
AC-g			2.941	2.6386	99.7	99.98
ABC-g			2.113	2.5973	99.5	99.97

and

Table 19. Comparative fault location between the proposed method and [68].

Fault Type	Fault Resistance (Ω)	Actual Fault Distance	Estimated Fault Distance (km) [68]	Estimated Fault Distance (km)	Accuracy [68]	Accuracy of the Proposed Method
BG	0	11.979148	11.978	11.97912	99.9989	99.9994
	20		11.9784	11.97914	99.9994	99.9994
	50		11.9764	11.97914	99.9973	99.9994
	100		11.9736	11.97913	99.9944	99.9996
AG	0	42.3	42.387798	42.3	99.9106	99.9999
	20		42.417098	42.3	99.8808	99.9998
	50		42.459398	42.2998	99.8377	99.9988
	100		42.6822	42.3	99.8648	99.9999

. The comparison considers faults on lines 806–808, 808–810, and 818–820, with actual fault distances of 2.625 km, 11.979 km, and 42.3 km, respectively.

$$Accuracy=100\%-distance error$$

(29)

4.3. Discussion

4.3.1. Fault Distance and Resistance Impact

The obtained results indicate that the proposed fault localization algorithm provides a consistently minimal error in distance estimation, thereby demonstrating its high reliability. The accuracy of the method is maintained across the entire transmission line, including extreme positions such as 5% of the line length near the sending end and 95% near the receiving end. Moreover, the algorithm proves to be highly robust against fault resistance variations. Even when the fault resistance increases from nearly 0 Ω to as high as 50 Ω, the impact on the estimation accuracy remains marginal. Figure 18 illustrates the histogram of the average fault distance error for all fault resistance values, demonstrating that the error consistently remains below 10⁻³, which is considered insignificant.

4.3.2. Voltage Magnitude and Phase Angle Impact

The results show that voltage magnitude variations do not affect fault location or resistance estimation, as demonstrated in Table 13, Table 14 and Table 15. These variations are already accounted for in the cubic spline interpolation, ensuring that the fault resistance is accurately determined and the corresponding fault distance is correctly calculated. Figure 19 presents the histogram of the fault distance average error across all test cases, confirming that the error remains below 10⁻⁴, which is a negligible value. Figure 19 presents the histogram of the average fault distance error across all test cases, showing that the error consistently remains below 10⁻⁴, which is a minimal value.

4.3.3. Load Consumption Impact

Considering active power load variations across phases A, B, and C at 80%, 100%, and 120% of the nominal consumption, Table 16 demonstrates that the fault distance estimation error remains insignificant. This indicates a high accuracy of the fault location method and confirms that load variations in the distribution network have no significant impact on the performance of the localization algorithm. Furthermore, Figure 20 presents the histogram of the average fault distance error across all test cases, showing that the error consistently remains below 10⁻⁴, which is a minimal value.

4.3.4. Fault Type Diversity

The machine-learning-based decision tree represents a powerful method for classifying fault types by utilizing the results of fault location estimation. The algorithm guides the decision process through leaf nodes, leading directly to the correct faulted phase identification and subsequently to the precise fault type among ten possible categories. Table 15 reports the performance of the proposed decision tree under a comprehensive test consisting of 10,000 fault cases. Each of the ten fault categories was represented by 1000 cases to ensure balanced training and evaluation. The results show that the decision tree achieved 100% classification accuracy, correctly identifying both the faulted phase and the fault type in all scenarios. This outstanding performance highlights the reliability of the method and confirms its suitability for real-time protection applications.

4.3.5. Impact of Other Fault Location Methods

The results shown in Figure 21 and Figure 22 indicate that the proposed machine-learning-based fault location method using cubic spline interpolation gives better results than the approaches reported in [67,68]. For example, in the case of a fault on line 806–808, the proposed method achieves a minimum accuracy of 99.97%, compared to 99.5% in [67]. Similarly, for single-phase faults on lines 808–810 and 818–820, the minimum accuracy reaches 99.999%, while ref. [68] reports 99.837%.

4.3.6. Calculation Time and Large Data Capacity Impact

The proposed method involves substantial mathematical formulation, particularly for the cubic spline interpolation in both eight-dimensional (8D) and six-dimensional (6D) cases, and requires a large dataset to ensure diversity in parameters such as voltage level, load conditions, and fault resistance range. As the variation interval of these parameters increases, the required data storage capacity and data collection time also increase.

Table 20. Computational storage and time cost for different fault types.

Fault Type	Collection Data Simples	Size	Collection Data Time	Simulation Time
A-g	21,870	0.97 MB	1 h 36 min	2 min
B-g		0.98 MB	1 h 33 min
C-g		0.98 MB	1 h 47 min
AB		0.97 MB	1 h 41 min	2 h 30 min
BC		0.96 MB	1 h 45 min
AC		0.95 MB	1 h 29 min
AB-g	196,830	8.36 MB	14 h 43 min
BC-g		8.40 MB	13 h 38 min
AC-g		8.36 MB	14 h 40 min
ABC-g		8.44 MB	15 h 36 min

presents different data collection sizes, the corresponding data acquisition times, and the simulation time required for fault classification and location under various fault types.

5. Conclusions

In conclusion, this study shows that combining machine learning with an optimization framework is a highly effective way to accurately classify and locate faults in modern electrical distribution networks. Because the proposed method handles a wide variety of short-circuit scenarios—including different fault distances, resistance levels, pre-fault voltages, and changing load conditions—it is highly adaptable to real-world situations.

The main advantage of this approach is its hybrid design. It uses a decision tree algorithm to quickly and accurately identify the specific type of fault. At the same time, it pairs multi-objective optimization (to estimate fault resistance) with cubic spline interpolation to calculate the exact distance to the fault. Together, these tools effectively manage the complex, nonlinear behavior of active distribution networks.

To take this method out of the lab and put it into practice, grid operators will need to make a few physical and digital upgrades. On the hardware side, current transformers must be installed along the power lines to capture real-time data. On the software side, the system requires strong computing power and plenty of data storage to run the predictive models. Additionally, to keep the diagnostics accurate over time, the models will need regular data updates and retraining whenever the network’s layout or components change.

Even with these practical requirements, our experimental results clearly show that this approach is highly precise, reliable, and robust. Moving forward, future research will focus on reducing the computing time of the optimization algorithms to allow for faster, real-time processing. We also plan to test how well the method adapts to grids that rely heavily on variable renewable energy sources (DERs). Ultimately, this work lays a strong foundation for building more reliable, self-healing smart grids.