Short Circuit Fault Detection in DAR Based on V-I Characteristic Graph and Machine Learning

Abstract

Dry-type air-core reactors (DAR) are critical components in power systems but are prone to inter-turn short circuit faults which interrupt the symmetry of the winding structure. Inspired by the online detection of transformer winding deformation, the V-I method has been adapted to diagnose short circuit faults in reactors. However, the diagnostic criteria and thresholds of V-I method remain unclear. This paper presents a novel method for determining the threshold for detecting inter-turn short circuit faults in DAR, integrating V-I analysis with machine learning techniques. Specifically, Gradient Boosting Regression (GBR) is used to compute a standard diagnostic criterion value, and curve fitting is also used to define the threshold for identifying inter-turn short circuit faults. The experimental results demonstrate that this method effectively identifies fault conditions in DAR.

1. Introduction

DAR are essential components in power systems. They not only help regulate and stabilize voltage and current within the grid but also improve power transmission efficiency and reduce system losses [1,2,3]. Ensuring the optimal performance of DAR is essential for maintaining the reliability and safety of power systems. However, these reactors are highly susceptible to inter-turn short circuit faults (ISCFs) during operation [4]. Once such a fault occurs, it can lead to the damage of the reactor, disrupt system operation, and even trigger widespread power system failures, causing severe consequences [5]. To ensure the safe and efficient performance of these reactors, accurate and effective fault detection methods are essential. Over the years, researchers have developed various techniques to enhance the accuracy and reliability of ISCF detection. These methods include conventional electrical measurements, frequency response analysis (FRA), vibration analysis, and temperature detection.

Huang proposed an online monitoring approach based on tracking changes in the power angle to detect inter-turn insulation faults in reactors [6]. However, this method is susceptible to frequency fluctuations and harmonic interference, which may introduce errors in power angle measurements. Another approach, known as the fault detection factor (FDF), was introduced based on equivalent resistance to detect slight ISCFs in DAR by considering insulation resistance [7]. While this technique shows promise, it may be limited by its sensitivity to noise interference during the monitoring process. Jin applied an FRA-based approach using eigenvalues derived from the influence law and curve data analysis theory, employing a support vector machine (SVM) model to classify ISCF severity [8]. However, the reliance on the SVM model makes the method vulnerable to insufficient or noisy training data, which could affect its accuracy

Other researchers have explored the use of vibration characteristics for ISCF detection in DAR [9]. Zhu developed an experimental setup using a Laser Doppler Vibrometer to analyze insulation short circuit faults in reactors, demonstrating its effectiveness [10,11]. However, this approach is still in the experimental stage and has not been widely applied in practical engineering scenarios. Temperature-based detection methods have also been investigated, but current techniques mainly measure the average temperature rise, failing to effectively monitor localized temperature variations that are critical for detecting early-stage ISCFs [12,13].

With the advancement of machine learning, researchers have begun applying these techniques to fault detection in various engineering fields. An online sequential extreme learning machine (OS-ELM) was proposed for diagnosing faults in axle box bearings of high-speed EMUs [14]. Similarly, Chao introduced an improved feature extraction method using a grid search-optimized SVM for winding fault diagnosis [15]. And Yu proposed a bearing fault diagnosis method based on a symmetric two-stream convolutional neural network (CNN) and hybrid signal processing techniques, integrating a symmetric parallel CNN-support vector machine model to enhance fault classification under small-sample conditions [16]. However, existing machine learning-based methods often rely on predefined feature extraction and classification models, which may struggle to establish precise diagnostic criteria for inter-turn short circuit faults in DAR, limiting their applicability in complex and nonlinear fault scenarios.

One promising method that has gained significant attention is the V-I analysis, which has been widely used to detect winding mechanical deformation in power transformers. This method examines the relationship between voltage (V) and current (I) waveforms to assess equipment conditions [17]. Due to its simplicity and high sensitivity, the V-I method presents a potential solution for ISCF detection in DAR. However, a major drawback of traditional V-I analysis is its inability to establish clear diagnostic criteria and threshold values, which restricts its accuracy and real-time applicability.

Given the limitations of existing ISCF detection methods in terms of accuracy, sensitivity, and practical implementation, further research is urgently needed. To address these challenges, this study proposes a novel method that integrates GBR with V-I analysis to enhance fault detection in DAR. Unlike conventional V-I methods that lack explicit diagnostic criteria, the proposed approach utilizes GBR to compute optimal weight values for determining fault diagnosis thresholds. By incorporating ensemble learning, GBR improves the robustness of ISCF detection, even in the presence of non-linear fault behaviors and subtle anomalies. This method not only enhances fault detection accuracy but also provides a more reliable and practical framework for real-world ISCF monitoring. Through this research, we aim to offer an effective solution for improving the reliability and accuracy of ISCF detection in DAR. The main contributions are as follows:

• The proposed method integrates the V-I characteristic method with machine learning techniques to enhance fault detection accuracy and reliability. By leveraging data-driven learning, this approach effectively captures complex fault patterns, making it more robust against non-linear behaviors and subtle anomalies in ISCF detection.
• The proposed method utilizes GBR to automatically determine specific fault diagnosis thresholds, eliminating the need for manually set criteria. By learning from data, GBR assigns optimal weight values to fault indicators, ensuring an objective and data-driven approach to ISCF detection, thereby improving reliability and adaptability across different fault scenarios.

The remainder of this study is structured as follows. The fundamental principles of the V-I method, GBR, and the principle of the detection method are introduced in Section 2. The dataset obtained from the artificially simulated inter-turn short circuit faults in these reactor experiments is introduced in Section 3. The experimental results verifying the proposed approaches are analyzed in Section 4. The conclusions and prospects are given in Section 5.

2. Theoretical Analysis

2.1. V-I Method

The V-I method is a well-established technique for analyzing electrical characteristics and detecting faults in power systems, it is used for identifying inter-turn short circuits in DAR. This method involves plotting characteristic figures, which are graphical representations of the relationship between two sinusoidal signals, typically the voltage and current in the reactor coils.

When the reactor receives an AC voltage, the terminal voltage ($V$) and current ($I$) can be expressed as sinusoidal functions of time ($t$). Here, $V_m$ and $I_m$ represent the peak values of voltage and current, $\omega$ is the angular frequency, ${\varphi }_V$ denote the phase angles of the voltage and ${\varphi }_I$ is the phase angles of the current.

$$V(t)=V_m\sin (\omega t+{\varphi }_V)$$

(1)

$$I(t)=I_m\sin (\omega t+{\varphi }_I)$$

(2)

A characteristic figure is obtained by plotting $V(t)$ against $I(t)$. Under normal operating conditions of DAR, characteristic figure typically forms an ellipse, indicating a healthy status. If an inter-turn short circuit fault arises, the shape of the characteristic figure distorts, reflecting changes in the reactor’s electrical characteristics.

The characteristic parameters of the characteristic figure include the bevel angle ($\theta$) of the ellipse, the lengths of the semi-major axis ($a$) and the semi-minor axis ($b$) [18]. These parameters can be calculated as follows:

$$a={\displaystyle \frac{V_m}{\sqrt{1-{\left(\frac{I_m}{V_m}\cos ({\varphi }_V-{\varphi }_I)\right)}^2}}}$$

(3)

$$b={\displaystyle \frac{I_m}{\sqrt{1-{\left(\frac{I_m}{V_m}\cos ({\varphi }_V-{\varphi }_I)\right)}^2}}}$$

(4)

$$\theta ={\tan }^{-1}\left({\displaystyle \frac{I_m\sin ({\varphi }_V-{\varphi }_I)}{V_m-I_m\cos ({\varphi }_V-{\varphi }_I)}}\right)$$

(5)

These parameters provide quantitative measures of the ellipse’s geometry and orientation. Changes in these parameters indicate deviations from normal operating conditions and can help detect inter-turn short circuit faults.

2.2. Gradient Boosting Regression

GBR offers significant advantages in detecting inter-turn short circuit faults in dry-type air-core reactors. Unlike traditional methods relying on single models or linear regression, GBR uses an ensemble of decision trees, iteratively improving accuracy and capturing complex non-linear relationships. Its adaptive error minimization enhances fault detection, even for subtle anomalies. To validate its effectiveness, Section 4.2 compares GBR with other methods, proving its superior accuracy. By leveraging ensemble learning, GBR provides a more robust and reliable approach, making it an optimal choice for fault detection in complex reactor systems.

GBR operates by iteratively training decision trees, with each tree correcting the errors of the previous ones. The final model combines these weak learners through a weighted sum, providing a strong predictive model. The basic idea of GBR can be summarized as follows:

Start with an initial model, assume the goal is to fit a dataset ${\{(x_i,y_i)\}}_{i=1}^n$, where $x_i$ represents the input features and $y_i$ represents the target values. First, the model is initialized with a constant prediction $F_0(x)$, typically the mean of the target values:

$$F_0(x)=\arg {\min }_c{\displaystyle \sum _{i=1}^{n}L}(y_i,c)$$

(6)

Here, $F_0(x)$ represents the initial model prediction, which is typically set as a constant value. The goal is to find the optimal constant prediction $c$ that minimizes the sum of the loss function $L(y_i,c)$. The loss function $L(y_i,c)$ measures the error between the true target values $y_i$ and the constant prediction $c$, with squared error loss $L(y_i,c)={(y_i-c)}^2$ being the most common choice. The notation $\arg {\min }_c$ indicates that we seek the value of $c$ that minimizes this sum over all $n$ training samples.

At each iteration $m=1,2,\dots ,M$, the current model $F_m(x)$ is updated by adding a new model $h_m(x)$:

$$F_m(x)=F_{m-1}(x)+\alpha h_m(x)$$

(7)

This equation updates the model in each iteration. $F_m(x)$ represents the model prediction at iteration $m$, while $F_{m-1}(x)$ is the prediction from the previous iteration. The term $h_m(x)$ is a new weak learner trained to correct the residual errors of the previous model, and $\alpha$ is the learning rate, which controls the contribution of $h_m(x)$ to the updated model.

The new model $h_m(x)$ is trained on the residuals from the previous model. The residuals for the $i\mathrm{th}$ sample at $m\mathrm{th}$ the iteration, which is $r_{m,i}$, are calculated as follows:

$$r_{m,i}=-\left[{\displaystyle \frac{\partial L(y_i,F_{m-1}(x_i))}{\partial F_{m-1}(x_i)}}\right]$$

(8)

Here, $r_{m,i}$ represents the residual for the $i\mathrm{th}$ sample at iteration $m$. It is calculated as the negative gradient of the loss function $L(y_i,F_{m-1}(x_i))$ with respect to the current model prediction $F_{m-1}(x_i)$. The residual indicates how much the current model’s prediction should be adjusted to reduce the loss.

That is, the residual represents the loss function’s gradient concerning the current model’s predictions. A new weak learner $h_m(x)$ is fitted by minimizing the loss function:

$$h_m(x)=\arg {\min }_h{\displaystyle \sum _{i=1}^{n}L}(r_{m,i},h(x_i))$$

(9)

This equation defines the objective of training the weak learner $h_m(x)$, which aims to minimize the sum of the loss function L between the residuals $r_{m,i}$ and the predictions of $h_m(x)$, thereby improving the model’s accuracy. The model’s predictions are updated as [19]:

$$F_m(x)=F_{m-1}(x)+\alpha h_m(x)$$

(10)

Repeat until reaching the predefined $M$ iterations or when the loss function no longer decreases significantly.

In the context of detecting inter-turn short circuit faults in DAR, GBR is used to determine the weights for characteristic parameters derived from characteristic figures. These weights help in calculating a standard diagnostic criteria value that correlates with the fault severity.

Due to the limited size of the fault sample database obtained through simulation, directly attempting to establish a function $f(x)$ that maps the characteristic parameters of characteristic figures to the corresponding fault severity would result in poor performance and low fault identification accuracy. To address this issue, this study proposes an innovative use of GBR. GBR is employed to determine the weights for the characteristic parameters, which are then used to calculate a standard diagnostic criteria value that correlates with fault severity. This approach has led to improved fault identification accuracy.

2.3. Principle of Detection Method

This paper outlines the approach for detecting inter-turn short circuits in DAR as follows:

• Acquire a large volume of V-I data from DAR so that we can plot characteristic figures, obtaining characteristic parameters corresponding to various fault severities.
• Calculate regression coefficients, i.e., weights, using a GBR model.
• Utilize these weights to compute a standard diagnostic criteria value w for assessing fault conditions as follows. Plot two-dimensional scatter plot with standard diagnostic criteria value w on the horizontal axis and fault severity on the vertical axis, and find the fitting curves against two variables.

$$w=a^{\prime }*w_a+b^{\prime }*w_b+{\theta }^{\prime }*w_{\theta }$$

(11)

where $a^{\prime }$, $b^{\prime }$ and ${\theta }^{\prime }$ represent the rate of change of the characteristic parameters of the characteristic figure, compared to its healthy state. And $w_a$, $w_b$ and $w_{\theta }$ represent their respective weights.

4.

Determine the fault threshold $w_0$ by identifying the standard diagnostic criteria value corresponding to the 5% fault severity through curve fitting, with this threshold selection based on [20].

5.

In the next measurement of $w$, if it is greater than $w_0$, the dry-type air-core reactor is determined to have experienced an inter-turn short circuit fault.

The proposed method could be implemented in the following steps as shown in flowchart of Figure 1. By applying a GBR model to large volumes of V-I data from DAR, this method enables the calculation of characteristic parameters that correspond to different fault severities. It enhances fault detection accuracy by establishing a standard diagnostic criteria for fault severity, thereby improving the precision of fault threshold identification.

3. Modeling and Simulation

3.1. Model Parameters and Configuration

This section verified the proposed method based on the modeling and simulation of reactors. To obtain a large amount of V-I data from DAR, this study employed ANSYS Maxwell (2022 R1) electromagnetic simulation software to create models and perform simulations of inter-turn short circuits in DAR. Various fault scenarios were simulated to analyze variations of characteristic parameters. The modeling and simulation focused on a 35 kV single-phase dry-type air-core reactor, with its structural parameters detailed described in

Table 1. Structural parameters of reactor.

Envelopment	Layer	Radius/mm	Height/mm	Number of Turns	Wire Diameter/mm
1	1	2006.5	1846.9	727	2.54
	2	2011.7	1832.0	721
	3	2016.9	1818.4	716
	4	2022.1	1805.9	711
2	5	2085.3	1817.5	675	2.69
	6	2090.8	1806.3	671
	7	2096.3	1796.2	667
	8	2101.8	1787.2	664
3	9	2165.2	1834.9	645	2.85
	10	2171.0	1826.9	642
	11	2176.8	1820.1	640
	12	2182.6	1814.5	638
4	13	2246.2	1810.7	625	2.90
	14	2252.1	1807.0	624
	15	2258.0	1804.6	623
	16	2263.9	1803.4	623
5	17	2327.6	1801.0	622	2.90
	18	2333.5	1801.8	622
	19	2339.4	1803.9	623
	20	2345.3	1807.2	624

. The reactor comprises five layers of packaging, each containing four layers of coils.

In the dry-type air-core reactor shown in Figure 2, which is symmetrical, H represents the coil height, and H1 indicates the location where the inter-turn short circuit fault occurs. When that occurs, it will destroy the symmetry of the structure in a real-world reactor. Based on this, the following model was established in the simulation software: A cylindrical coordinate system in the 2D module was selected based on the reactor’s structural parameters, considering its axi-symmetric properties. The model is shown in Figure 3. This study utilizes unbounded domain analysis, defines the number of turns and coil terminals, and applies excitation through external circuits.

In this study, faults are introduced at both the end and middle of the coils in the 1st, 10th, and 20th layers, with fault severities of 10, 60, and 100 turns, respectively. The fault magnitude is determined by the height of the coil. For instance, considering a 10-turn fault at the end of the first-layer coil, the fault height is calculated by multiplying the wire diameter by the number of faulty turns, as illustrated in Figure 4. Upon short-circuiting the faulty coil in the external circuit, it is divided into two segments: the healthy coil and the faulty coil. This short-circuit configuration enables the faulty coil to be bypassed, as demonstrated in Figure 5.

3.2. Simulation Results

When an inter-turn fault arises in DAR, the simulation result in Layer1 can be seen in Figure 6. The changes in the healthy and fault states are very obvious. The specific changes are as follows:

• When a short circuit fault develops in any position of the inner, middle, or outer coil, whether it is a inter-turn fault at the end or middle, the characteristic figure will change significantly with the count of fault turns. When the count of short circuit turns is small, the health and fault graphs mainly show a change in area. As the count of fault turns increases, fault characteristic figure will rotate significantly clockwise and the area will significantly increase;
• When the same number of turns fault arises at different positions of the same layer coil, as the fault position approaches the middle, the area of the characteristic figure will increase, and the degree of inclination will also slightly increase; When different turns of faults occur at the same position of the same layer of coil, as the count of turns increases, the area of the characteristic figure gradually increases, and the change in inclination degree becomes more obvious. The changes in the characteristic figure are shown in Figure 6;
• When the same fault occurs in coils of different layers, as shown in Figure 7. When the count of fault turns is the same, a short-circuit fault closer to the reactor’s middle causes a more significant change in the characteristic figure. This is because a fault in the middle enhances the mutual inductance effect on the other healthy coils more than a fault at the ends;
• Whether it is a fault in the same layer or a short circuit fault in different layers, the characteristic parameters will be significantly affected by the number of fault turns. The short axis b and bevel angle $\theta$ will increase with the increase of fault turns, with the relative change rate and amplitude of bevel angle $\theta$ being the largest.

The text continues here.

3.3. Simulation with Contact Resistance

In the simulation, the fault representation was modified to account for the contact resistance between the faulty turns. The original model used a solid short-circuit to represent the fault, bypassing the affected turns, which resulted in no current flowing through the shorted turns and all the current passing through the solid short. However, in reality, contact resistance exists between the faulty turns, which significantly affects the current distribution.

To more accurately simulate this fault condition, resistance was introduced into the simulation. The modeling approach is shown in Figure 8, where resistance is added to simulate the actual fault conditions. The current waveform resulting from the introduction of resistance and the corresponding characteristic figure analysis are shown in Figure 9 and Figure 10.

In Figure 9 and Figure 10, 0 ohms represents the case with no contact resistance, simulating a complete short circuit, while the other cases correspond to different resistance values. The simulation investigates the variations in the characteristic figures and current under different insulation failure conditions. It is observed that the characteristic figures and current differ under non-insulation failure conditions. Since studying all possible insulation failure scenarios is complex and numerous, inter-turn short circuit faults are more common and cause more severe damage among the various types of faults in dry-type air-core reactor windings [21]. Therefore, this paper focuses on the study of the completely damaged condition (i.e., inter-turn short circuit fault).

4. Diagnostic Method

4.1. Analysis of Results

Table 2. Characteristic parameters of characteristic figures for partial typical faults of simulated reactors.

Major Axis a	Minor Axis b	Bevel Angle $\mathbf{\theta }$	Fault Severity
1100 (+0.00%)	1.840 (+0.00%)	$0.0082°$ (+0.00%)	0%
1100 (+0.00%)	2.020 (+9.78%)	$0.0180°$ (+119.51%)	1.38%
1100 (+0.00%)	2.429 (+32.01%)	$0.0218°$ (+165.85%)	8.25%
1100 (+0.00%)	2.672 (+45.22%)	$0.0242°$ (+195.12%)	13.76%
1100 (+0.00%)	2.191 (+19.08%)	$0.0315°$ (+284.15%)	1.38%
1100 (+0.00%)	3.188 (+73.26%)	$0.0458°$ (+458.54%)	8.25%
1100 (+0.00%)	3.706 (+101.41%)	$0.0525°$ (+540.24%)	13.76%

presents a portion of the healthy and faulty data for DAR obtained from the simulation and modeling. As part of the data preprocessing and feature engineering process, the a-axis was excluded due to its invariance, ensuring that only the most relevant features were retained for analysis. Specifically, the rate of change of the b-axis and the bevel angle were extracted as key fault indicators, as they exhibited meaningful variations correlated with fault severity.

70% of the fault data was utilized to train GBR model based on the previously described fault simulation. The GBR model was configured with a squared error loss function, a learning rate of 0.1, and 100 estimators. It employed the Friedman mean squared error criterion for measuring node quality, with a maximum tree depth of 3, a minimum of 2 samples required to split an internal node, and at least 1 sample per leaf node. Additionally, all features were considered when searching for the best split. To assess the contribution of each feature to the prediction, feature importance weights were extracted.

To ensure the practical applicability of GBR model in fault diagnosis, SHapley Additive exPlanations (SHAP) was employed to analyze the feature contributions and interactions within the model. Figure 11a presents the SHAP summary plot, which illustrates the overall impact of each feature on the model’s predictions. The b-axis rate of change and the bevel angle are identified as the most influential features, confirming their relevance in fault classification. The spread of SHAP values along the x-axis indicates that both features exhibit non-linear relationships with the predicted fault severity. The color gradient from blue (low feature values) to red (high feature values) further demonstrates how different feature magnitudes contribute to prediction variations. To further explore these relationships, Figure 11b,c provide SHAP dependence plots for b and $\theta$, respectively. Figure 11b demonstrates how variations in b affect model predictions, with the color gradient representing the corresponding values of $\theta$. The plot reveals that higher values of b generally increase the predicted fault severity, but the effect is influenced by $\theta$, suggesting a potential interaction between the two features. Figure 11c, on the other hand, shows that $\theta$ primarily has a negative impact on the predicted fault severity, particularly for lower values, though the interaction with b leads to deviations in this trend. This interpretability analysis using SHAP confirms that the GBR model effectively captures meaningful relationships between the selected features and fault severity. The transparency provided by SHAP enhances the credibility of the proposed method, making it more suitable for real-world industrial fault diagnosis applications.

Furthermore, a 10-fold cross-validation was conducted using a scoring approach where the model was evaluated across different data partitions. The resulting average score of 0.863 confirms the model’s strong stability and generalization ability. The obtained weights are 0.8893 for the minor axis $b$ and 0.1107 for the bevel angle $\theta$. The standard diagnostic criteria value $w$ for determining the inter-turn short circuit in reactors can be then calculated as follows.

$$w=0.8893*b^{\prime }+0.1107*{\theta }^{\prime }$$

(12)

Using the standard diagnostic criteria value w and fault severity, a scatter plot can be generated as shown in Figure 12.

After reprocessing the scatter plot and performing curve fitting, the resulting curve is shown in Figure 13. Prior to performing curve fitting, a one-dimensional interpolation using a linear approach was applied. This interpolation step helped to smooth the data points, providing a continuous representation of the relationship between fault severity and the diagnostic criteria value. Subsequently, the fitting was carried out using a polynomial method based on the least squares approach, with a polynomial degree of 3. This method was chosen to effectively capture the non-linear relationship between the fault severity and the diagnostic criteria value, ensuring a smooth and accurate representation of the data. It can be observed from Figure 13 that the standard diagnostic criteria value $w_0$ at a fault severity of 5% is 37.83. This value is selected as the threshold value for determining the existence of short circuit fault in reactor. Therefore, during the next test of dry-type air-core reactor, if minor axis b and bevel angle $\theta$ are substituted into Equation (12) and the calculated w value exceeds 37.83, it is determined that the reactor has an inter-turn short circuit fault.

4.2. Analysis of Effectiveness

To validate the effectiveness of the threshold for detecting inter-turn short circuit faults in DAR, 30% of the data obtained from simulation models were used. The data comparison is presented in

Table 3. Calculated data to be compared.

W	Fault Severity	W	Fault Severity	W	Fault Severity	W	Fault Severity	W	Fault Severity
2.266771124	0.14	13.70398874	0.15	14.33594437	0.16	14.09811206	0.16	13.73542161	0.16
4.597157342	0.28	16.71633876	0.3	17.37907819	0.31	17.97657983	0.32	17.39463235	0.32
6.852044507	0.41	19.49355305	0.44	20.50937075	4.65	21.4602752	0.48	21.06777457	0.48
8.907223559	0.55	22.26050608	0.59	23.50983681	0.62	24.51010034	0.64	24.11263135	0.64
11.09687294	0.69	24.53396854	0.74	26.07178882	0.78	27.39067663	0.8	26.998176	0.8
12.76818993	0.83	26.64811886	0.89	28.22536151	0.93	29.75324507	0.96	29.25111593	0.96
18.4233842	1.38	35.74876031	1.48	34.61522433	1.55	36.31667603	1.6	35.6900133	1.61
38.01840487	6.88	54.54549733	7.41	57.02600831	7.75	59.75591103	8	58.92586997	8.04
55.40682356	13.76	75.63761777	14.81	80.34028489	15.5	84.30234148	16	78.97029132	16.01
100.885313	27.51	132.7637955	29.63	145.1110494	31.01	154.6869307	32	151.3604999	32.15
255.8463295	50	293.6897678	50	305.9089181	50	312.322	50	304.5767479	50
2.281676211	0.14	13.44563389	0.15	13.95867337	0.16	14.20277221	0.16	13.63076146	0.16
4.641872603	0.28	16.70208275	0.3	17.41385672	0.31	17.98651655	0.32	17.3747589	0.32
7.15576369	0.42	19.5240123	0.45	20.79818484	0.47	21.47518028	0.48	20.79883392	0.48
9.469946664	0.56	22.35058568	0.6	23.94802631	0.63	24.65450739	0.64	24.03313755	0.64
11.50989609	0.7	24.88305206	0.75	26.43015999	0.78	27.45029698	0.8	26.69445682	0.8
13.43524864	0.84	26.93725749	0.9	28.61851121	0.94	29.81783378	0.96	28.93249166	0.96
19.25936723	1.4	33.40229483	1.5	35.22730652	1.56	36.41604328	1.61	35.4064921	1.6
39.9533696	6.98	56.07305635	7.5	58.145188	7.81	60.08414749	8.03	57.96070952	8.01
58.08239883	13.97	77.72163338	14.99	81.64761328	15.63	84.73491355	16.05	81.47307151	16.03
105.8479576	27.93	137.0851969	29.99	148.2986905	31.25	155.5421381	32.1	148.8349491	32.05
262.7926489	50	298.6629982	50	309.976933	50	313.14902	50	300.5597148	50

. It shows that when the standard diagnostic criteria value w exceeds 37.83, the fault severity consistently exceeds 5%, indicating the occurrence of an inter-turn fault in the dry-type air-core reactor and confirming the reliability of the proposed method. Comparing all the data from simulations with this result, out of 110 data points, 7 faults were incorrectly detected, which results in the accuracy rate of 93.64%. Under the same conditions, other machine learning methods, such as support vector machine (SVM) and random forest (RF), were also tested for comparison, achieving accuracy rates of 78.6% and 82.3%, respectively, further demonstrating the effectiveness of the proposed method.

To further assess the robustness and reliability of the proposed model, we conducted additional statistical analysis on the prediction results. Using 10-fold cross-validation, we calculated key performance metrics. The standard deviation of the predictions was 0.4955, indicating a stable distribution of predicted values. The mean squared error (MSE) was 0.0444, reflecting a low prediction error. Additionally, the 95% confidence interval was computed as (0.3609, 0.5057), providing an estimate of where the true predictions are expected to fall. The mean absolute error (MAE) was determined to be 0.0445, further demonstrating the accuracy of the model. These results confirm the model’s strong generalization ability, ensuring its reliability for real-world fault detection applications.

4.3. Experiment on an Actual Dry-Type Air-Core Reactor

A single-phase dry-type air-core reactor was customized for the experimental study, featuring a rated voltage of 350 V, a rated current of 20 A, an inductance of 15 mH, voltage drop of 94.2 V, and operating frequency of 50 Hz. The experimental setup, including all measuring instruments, is shown in Figure 14. The steps for the measurement process are as follows:

• Use a three-phase power supply to output a phase voltage of 15V to power the dry-type air-core reactor;
• A wire was used to simulate inter-turn short circuit faults between different turns of the dry-type air-core reactor, as illustrated in Figure 15;

3.

Measure data through V-I sensors:

4.

An 8-bit digital storage oscilloscope (DSO) is used to capture the V-I signals;

5.

Draw characteristic figures using V-I data and obtain characteristic parameters.

The measured V-I waveform is shown in Figure 16 and the characteristic figures drawn are plotted in Figure 17. And the characteristic parameters gained from the characteristic figures are given in

Table 4. Characteristic parameters of characteristic figures for partial typical faults of actual reactors.

Minor Axis b	Bevel Angle $\mathbf{\theta }$	Fault Severity
2.6435 (+0.00%)	$1.2357°$ (+0.00%)	Healthy
6.1724 (+133.49%)	$3.5065°$ (+183.77%)	Moderate
14.0128(+430.09%)	$6.7321°$ (+444.80%)	Severe

.

Additionally, the current under severe fault conditions shows distortion because the current after the short circuit slightly exceeds the maximum current that the signal source can handle. However, the amplitude and phase remain unchanged. Based on the amplitude and phase, we have redrawn the original distorted figure to better represent the fault characteristics, as shown in Figure 16b.

Using Equation (12), the standard diagnostic criteria value $w$ of moderate and severe short circuit fault can be calculated respectively to be 157.15 and 431.72. They are both greater than 37.83 which means this dry-type air-core reactor has experienced an short circuit fault, confirming the effectiveness of the proposed detection method.

5. Conclusions and Prospects

This study presents a novel approach for detecting inter-turn short circuit faults in DAR, combining V-I analysis with GBR. By leveraging GBR, we effectively determined the weights of characteristic parameters from characteristic figures, which allowed us to compute a standard diagnostic criteria value. Through the analysis of the relationship between this value and fault severity, we established a reliable threshold for fault detection. This approach represents a significant technical advancement by introducing a more adaptive and data-driven method for fault severity analysis. Experimental results confirmed the accuracy and effectiveness of the proposed approach, demonstrating its accuracy and achieving a fault diagnosis rate of 93.64%, thereby validating its potential in improving fault detection in DAR.

This model enables real-time fault detection by continuously acquiring V-I data from sensors in the DAR system. A sampling frequency of 10 kHz ensures the capture of transient fault characteristics, with fault assessments performed at 1-s intervals based on the most recent 100 ms of data. This results in approximately 86,400 assessments per day, allowing for timely detection and response. The Gradient Boosting Regressor (GBR) model operates efficiently on embedded processors such as an ARM Cortex-A72 (1.5 GHz, 4-core), requiring 20 MB of RAM for inference. Each fault diagnosis, including feature extraction and computation, is completed within 50 ms, ensuring minimal delay. If deployed on an industrial PC (e.g., Intel i5-8500, 3.0 GHz), inference time is further reduced to 5 ms, supporting near-instantaneous fault identification. Detected faults are transmitted via Modbus TCP/IP or IEC 61850, enabling seamless integration with SCADA systems. Besides, in practical applications, electrical measurements are susceptible to harmonic interference and environmental noise, which may affect fault detection accuracy. To mitigate these effects, a low-pass filter can be applied to suppress high-frequency noise and harmonics, ensuring signal stability. Additionally, wavelet transform-based denoising can be employed to separate noise from fault-related features while preserving critical signal information. These techniques enhance the robustness of the proposed method, enabling reliable fault detection even under noisy industrial conditions. This real-time detection method enhances grid stability by ensuring rapid fault identification, reducing false alarms, and minimizing downtime in industrial operations.

Despite the promising potential of the proposed model, there are several potential risks associated with its deployment in industrial environments. One of the key concerns is the model’s robustness against adversarial attacks, where biases in training data or sensor measurements could impact the model’s accuracy and reliability. Addressing these risks is crucial to ensure the model’s effectiveness in real-time applications. Additionally, the implementation of this fault detection method must comply with electrical equipment and safety standards, including IEC 60076-6, IEC 60270 and so on. To meet these standards, this method must undergo rigorous validation and testing against real-world DAR fault scenarios to ensure its predictions are both precise and trustworthy. Implementing adequate fail-safe mechanisms is also essential to prevent incorrect fault detection from affecting reactor performance. By adhering to these standards and ensuring reliability, transparency, and security, this method can be effectively integrated into industrial applications for enhancing fault detection in DAR systems. Future work will focus on enhancing the model’s resilience to adversarial attacks and minimizing biases, with the goal of further improving its robustness and performance in practical settings.