Short-Circuit Damage Diagnosis in Transformer Windings Using Quaternions: Severity Assessment through Current and Vibration Signals

Abstract

Short circuits occurring between turns within the windings are widely known as one of the primary causes of damage in electrical transformers; as a result, early detection plays a fundamental role in preventing further and more serious damage. This study introduces a novel approach that relies on the analysis of current and vibration signals, specifically employing the analysis of quaternion signals, to effectively detect short circuits within electrical transformers., offering an identification of conditions ranging from a healthy state to six levels of short circuit turns. in a no-load transformer, i.e., 0, 5, 10, 15, 20, 25 and 30 SCT. This proposed method employs quaternion rotation to extract statistical features that can be used to classify the condition of the transformer. To evaluate the effectiveness of the proposed methodology, an experimental validation is carried out using a 1.5 kVA transformer, comparing its performance against other existing methods. The results demonstrate the feasibility of the proposal, accurately identifying various levels of SCT, achieving an accuracy of 97.5%, using only 100 samples with the k nearest neighbors method.

1. Introduction

Electric transformers play a crucial role in efficiently transmitting and distributing electricity within a power system. They are indispensable for minimizing energy losses during long-distance transmission. However, like other electrical devices, transformers are vulnerable to faults that can result in damage, reduced efficiency, and even catastrophic failures. Among the most prevalent faults is the short-circuit turn (SCT), which is a fault that occurs when the insulation between the turns of a coil breaks down and allows the turns to encounter each other, creating a short circuit path. SCTs in electrical transformers are faults that can cause several problems, including overheating, electrical arcing, and the risk of fire. Therefore, it is essential to detect and repair SCTs as soon as possible to prevent further damage to the transformer and ensure safe and reliable operation.

In the recent literature, numerous approaches have been proposed to diagnose transformer faults, covering a wide spectrum of techniques ranging from timing analysis to advanced time-frequency techniques. Among these approaches, the Fourier transform remains a classic and widely employed technique for assessing the condition of transformers. Some applications of Fourier Transform can be found in works such as [1], where vibroacoustic signals are utilized for diagnosing power medium-voltage/low-voltage issues in transformers. In [2] a modified version of the Discrete Fourier Transform known as the empirical Fourier Transform is introduced, which is applied to measure specific segments of a current waveform for detecting internal faults, inrush current, and current transformer saturation. Despite the great use of the fast Fourier transform, this method has several problems, such as limited time resolution and the assumption that the signal is stationary and periodic. Other frequency techniques, including the wavelet transform, have been employed for fault detection in transformers. The wavelet transform possesses advantageous properties such as multiresolution analysis and the capability to handle non-stationary signals, making it an invaluable tool for distinguishing between transformer inrush currents and power system fault currents [3,4,5]. The works use wavelet transform to assess the condition of the winding in a transformer when a short circuit occurs through the vibroacoustic signal analysis, as is presented in [6] and in [7], which employ a probabilistic WT for assessing partial discharges in transformers. Furthermore, various novel methods have emerged to enhance the condition monitoring of electrical transformers, Among them are those based on decomposing the signal into sub-frequency bands to improve its analysis. One notable technique is the empirical mode decomposition (EMD), which has gained widespread usage in diagnosing electric machines; this method is used in [8] for discrimination of inrush current from internal fault of the power transformer. Shang et al. [9] employed the Complete Ensemble EMD (CEEMD) technique, entropy computation, and multiple neural networks to diagnose the transformers, while Hong et al. [10] utilized the variational mode decomposition (VMD) method to identify the degradation of transformer windings. Different ways have also been developed, such as using fractal dimension algorithms and classifiers to diagnose potential faults in transformers. In other examples, Ref. [11] uses the fusion of wavelet transform and the EMD method to diagnose internal faults from magnetization conditions in power transformers and [12] uses a combination of fractal dimension algorithms and classifier transformers to diagnose faults in transformers automatically.

Although the previous studies have obtained promising results, there is still a need to develop and apply more efficient methods in terms of accuracy and computational burden, mainly considering that detecting faulty transformers using vibration analysis is not accessible due to various factors such as the burden of high-level noise, nonstationary properties of the vibration, and the current signals of the transformer. Recently, the use of quaternions has been presented as an effective tool for fault diagnosis, as shown by their growing implementation in various studies. For instance, Contreras et al. [13] utilized current and vibration signals to detect faults in electric motors, while Zhou et al. [14] introduced a novel signal decomposition technique based on quaternions used to identify gear faults Zou et al. [15] proposed the quaternion block sparse representation for signal recovery and classification which is applied in color face recognition. They also proposed the quaternion block sparse representation for signal recovery and classification applied in color face recognition. Moreover, quaternions have proven effective in spectral analysis for diagnosing bearings and bevel gear faults in [16,17] respectively. These notable applications highlight the versatility and potential of quaternions in fault detection across the different components of electrical machines.

This study introduces quaternion signal analysis QSA to diagnose SCTs in electrical transformers. The method involves the creation of a model that captures the transformer’s concurrent behavior for four physical variables in the time domain. This method considers each sample as a point, forming shape trajectories over time and providing high levels of accuracy from a small number of samples. The obtained QSA-based model and its statistical features, i.e., variance and standard deviation, are used to classify various severities of SCTs (i.e., 0, 5, 10, 15, 20, 25, and 30 SCTs) in a real electrical transformer using as physical variables the current signal and the vibration signals acquired from tri-axial accelerometer. As a classifier, the k-Nearest-Neighbors (KNN) method is used. As a final observation, this study contributes by proposing a new methodology based on the QSA technique that allows fast processing using a small number of samples while achieving good accuracy results above 97.5% during classification.

This paper is organized into six sections. In Section 1, an introduction and the related works concerning SCT damage diagnosis are presented. Section 2 describes the background of QSA, statistical features, and the KNN method. In Section 3, the experimental setup is presented. The results are shown in Section 4, and discussions and the future works of the proposed methodology are presented in Section 5. Finally, conclusions are drawn in Section 6.

2. Methodology

The quaternions theory is the basis of this development. The definitions of quaternion displacement and the rotation of quaternions are presented below, as well as the flowchart of the methodology applied in this work.

2.1. Quaternions and Rotation

The quaternion is a structure with four values, one in the real part and three in the imaginary part, as in Equation (1). In this equation, the values $q_0$, $q_1$, $q_2$, and $q_3$ are coefficients changing through time [18].

$$q\left(t\right)=q_0\left(t\right)+q_1\left(t\right)i+q_2\left(t\right)j+q_3\left(t\right)k$$

(1)

The displacement of $q\left(t\right)$ is obtained from the sample in $\Delta$t time, as is presented in Equation (2), where $q_d\left(t\right)$ is the delayed quaternion with respect to $q\left(t\right)$. The values $q_{d0}$, $q_{d1}$, $q_{d2}$, and $q_{d3}$ are coefficients changing through time.

$$q_d\left(t\right)=q(t+\Delta t)=q_{d0}\left(t\right)+q_{d1}\left(t\right)i+q_{d2}\left(t\right)j+q_{d3}\left(t\right)k$$

(2)

The quaternion signals $q\left(t\right)$ and $q_d\left(t\right)$ are related through the rotation quaternion signal $q_r\left(t\right)$, i.e., $q_r\left(t\right)$ is the quaternion signal that describes how to move from $q\left(t\right)$ to $q_d\left(t\right)$ as shown in Equation (3) [19]:

$$q_d\left(t\right)=q_{rot}\left(t\right)·q\left(t\right).$$

(3)

The aforementioned equation implies that one sample of the rotated quaternion $q_r$ can be determined by q and $q_d$, as is shown in Equation (4) [18]. Each sample of the rotation is a model that describes the behavior among samples of the signal.

$$q_r=\left[\begin{array}{ccc}1-2q_{d2}^2-2q_{d3}^2& 2(q_{d1}{q}_{d2}+q_{d0}{q}_{d3})& 2(q_{d1}{q}_{d3}-q_{d0}{q}_{d2})\\ 2(q_{d1}{q}_{d2}-q_{d0}{q}_{d3})& 1-2q_{d1}^2-2q_{d3}^2& 2(q_{d2}{q}_{d3}+q_{d0}{q}_{d1})\\ 2(q_{d1}{q}_{d3}+d_{d0}{q}_{d2})& 2(q_{d2}{q}_{d3}-q_{d0}{q}_{d1})& 1-2q_{d1}^2-2q_{d2}^2\end{array}\right]\left[\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right]$$

(4)

The rotation signal $q_r\left(t\right)$ is presented as natural quaternions with coefficients $q_{r0}$ equal to zero because the rotation is developed in three dimensions as shown in Equation (5).

$$q_r\left(t\right)=q_{r0}\left(t\right)+q_{r1}\left(t\right)i+q_{r2}\left(t\right)j+q_{r3}\left(t\right)k$$

(5)

2.2. QSA Method

This paper proposes a methodology that combines the QSA method and a classifier to classify SCT levels in transformers automatically. The algorithm presents three stages as shown in Figure 1: (1) QSA, (2) Statistical calculus, and (3) Training and evaluation.

In the QSA stage, the quaternion signals are formed, rotated and the modulus of each rotation is obtained. After that, some samples are selected by a data window to calculate the statistical values.

The quaternion signal $q\left(t\right)$ is formed by coefficients of the transformer current signal $I\left(t\right)$ and three accelerations, which represent vibrations $V_x\left(t\right)$, $V_y\left(t\right)$, and $V_z\left(t\right)$, as is shown in Equation (6). The current and vibration signals have the advantage of being able to simultaneously be obtained while the transformer is in operation, allowing for real-time online monitoring. The QSA method requires four values, and since the magnetostriction phenomenon correlates the current and vibration signals, the application of this method using these signals becomes feasible. The current signal is measured in amperes and the vibration signals in g force. All signals are discrete with N samples in time.

$$q\left(t\right)=I\left(t\right)+V_x\left(t\right)i+V_y\left(t\right)j+V_z\left(t\right)k$$

(6)

Similarly, the delayed quaternion signal $q_d\left(t\right)$ is formed by the corresponding current signal and vibration signals coefficients in the displaced time $\Delta t$, called $I_d$, $V_{xd}$, $V_{yd}$, and $V_{zd}$ as is shown in Equation (7).

$$q_d\left(t\right)=I_d\left(t\right)+V_{xd}\left(t\right)i+V_{yd}\left(t\right)j+V_{zd}\left(t\right)k$$

(7)

One sample of the signals $q\left(t\right)$ and $q_d\left(t\right)$ are selected from the same time. As is presented in Figure 2, $q_r$ is calculated through the present and delayed sample to obtain one quaternion model.

A set of models is obtained through the quaternion rotation of the signals $q\left(t\right)$ and $q_d\left(t\right)$. The natural quaternion of rotation is obtained from (8) which is originally (4). $N-1$ behavior models are obtained in the full signal.

$$q_r=\left[\begin{array}{ccc}1-2V_{yd}^2-2V_{zd}^2& 2(V_{xd}{V}_{yd}+I_d{V}_{zd})& 2(V_{xd}{V}_{zd}-I_d{V}_{yd})\\ 2(V_{xd}{V}_{yd}-I_d{V}_{zd})& 1-2V_{xd}^2-2V_{zd}^2& 2(V_{yd}{V}_{zd}+I_d{V}_{xd})\\ 2(V_{xd}{V}_{zd}+I_d{V}_{yd})& 2(V_{yd}{V}_{zd}-I_d{V}_{xd})& 1-2V_{xd}^2-2V_{yd}^2\end{array}\right]\left[\begin{array}{c}{V}_x\\ V_y\\ V_z\end{array}\right]$$

(8)

The modulus of each quaternion in all the signal $q_r\left(t\right)$ is calculated. After that, M consecutive modulus are selected to create the window vector $w\left[m\right]$, as shown in (9).

$$w\left[m\right]=\left[|q_r\left(t\right)|,|q_r(t+\Delta t)|,|q_r(t+2\Delta t)|\dots |q_r(t+(M-1)\Delta t)|\right]$$

(9)

2.3. Statistical Calculus

After $w\left[m\right]$ is obtained, the statistical values are calculated to determine its evolution in time domain. Multiple statistics were analyzed to select which present isolated values belong to each SCT, resulting in the Variance ($VA$) and Standard deviation ($SD$), as shown in (11) and (12). The Mean ($\mu$) is obtained as in (10) to calculate the selected statistical indicators.

$$\begin{array}{cc}& \mu =\frac{1}{L}\sum w\left[l\right]\end{array}$$

(10)

$$\begin{array}{cc}& VA=\frac{1}{L}\sum {\left(w\left[l\right]-\mu \right)}^2\end{array}$$

(11)

$$\begin{array}{cc}& SD=\sqrt{VA}\end{array}$$

(12)

The $VA$ and $SD$ are used to train the KNN method and to classify SCTs. In Figure 3 the SD-VA combination of each SCT condition is presented. As can be observed, these two features enable the construction of clear patterns to differentiate between SCT conditions.

2.4. Training and Classification

Training and classification are important aspects to create an automatic fault detection system. During training, the models learn from the labeled data, gaining insights and patterns to make accurate predictions or classifications, while the classification enables the evaluation of our system performance. KNN is a widely used classification algorithm known for its simplicity and versatility. It offers an intuitive and straightforward implementation without imposing any assumptions about the underlying data distribution. The KNN method is a non-parametric, supervised machine learning tool [20]. Some samples from each class are used to train the KNN, and the training pattern is obtained. After that, the KNN is ready to classify other sample data. This algorithm compares the proximity among one sample’s data, and each group of samples according to the obtained pattern. It selects the nearest group to the data and it is classified and labeled as shown in Figure 4 [21].

In this work, the KNN pattern of each group is formed using sets of statistical values $VA$, $SD$, and the real SCT results. The KNN patterns are created in MATLAB selecting one statistical set of each characteristic to extract multiple $w\left[m\right]$ and sweeping through all samples of $q_r$.

The QSA and statistical methods are applied to the rest of the archives to evaluate them using the calculated KNN pattern. The statistical values are compared to obtain a label with the classification of SCTs.

3. Experimental Setup

To test the proposed approach, a real transformer under different SCT conditions is employed. The experimental setup is presented in Figure 5, where the transformer, sensors, and data acquisition system ($DAS$) are shown. A 1.5 kVA single-phase transformer is operated at 120 V with 135 turns in its primary winding. To produce SCT damage in a controlled way, the transformer was intentionally modified. It was unwinding and rewinding; during the rewinding, various taps were obtained every five turns. Each tap generally represents a reduction in the winding, simulating a short-circuit condition. In a short-circuit condition, the length of the wire that makes up the winding is reduced. In this way, when the connection is made at the ends of the transformer primary, a healthy connection is represented, meaning 0 SCTs. On the other hand, when the connection is made at the first tap, the first faulty condition is produced, indicating a reduction of 5 turns and, consequently, simulating a faulty condition of 5 SCTs. These connections are also made for the second, third, fourth, fifth, and sixth taps, representing 10, 15, 20, 25, and 30 SCTs, respectively. It is worth noting that the higher the number of SCTs, the greater the severity of the damage. Additionally, an autotransformer is used to de-energize the transformer.

A Fluke measures the current i200s current clamp, and the vibrations are measured by a tri-axial accelerometer KISTLER, model 8395A with a resolution of 400 mV/g until ±10 g. The $DAS$ is a National Instrument NI-USB 6211 board that allows for converting the sensor data into 16 bits for storage on a PC with a frequency sample of 6000 samples/s during 2.3 s in a stable state. Twenty files that contain one current signal and three vibration signals were obtained to test the QSA method with 0, 5, 10, 15, 20, 25, and 30 SCT. The 0 SCT case represents the healthy transformer.

Table 1. Specifications of the equipment used.

Equipment	Specifications
Transformer	1.5 kVA, 135 turns, voltage input 120 V
Autotransformer	Powerstat, type 116CU-3, Input 240 V, Output 0–280 V
Current clamp	Fluke i200s, 10 mV/A
Triaxial accelerometer	KISTLER, model 8395ª, resolution = 400 mV/g, measurement range ±10 g
Data acquisition system	NI-USB 6211, ADC of 16 bits with an input range of ±10 V, sampling frequency used = 6000 samples/s

presents a summary of the equipment used Examples of the current and vibration signals for the healthy transformer and 30 SCT are shown in Figure 6.

Figure 7 shows the graphical interface to apply the proposed methodology, it is divided into three sections, signals, analyzed signals, and diagnosis. In the signals section, the captured vibration and current signals are displayed. The user can choose between reading a previously stored file or performing a new signal acquisition. This section also requests the sampling frequency as well as the acquisition time. In the analyzed signals section, the window of the signal to be analyzed is shown, the user can choose between 100, 500, 1000, or 8000 samples. They can also select which sample to start analyzing from the original signals. Finally, in the diagnosis section, the user selects which classifier to use, and the result is shown through an LED light that indicates any of the SCT severities.

4. Results

This section presents the results of the proposed method. At first, the mean, minimum, and maximum accuracies of each SCT are presented by changing the number of window samples to check the method behavior. After, the general accuracy of the proposed method is compared changing the classification method by a Neural Network (NN) and a decision tree (DT). This additional step is carried out to check the QSA discrimination obtained and its adaptation to other classification methods. Finally, the proposed method is compared with the literature.

Initially, the results of the accuracy analysis are presented. The classification accuracy results are presented in confusion matrices to obtain the system behavior. An example of a confusion matrix is shown in

Table 2. Confusion matrix.

Classified	Real SCT
SCT	0	5	10	15	20	25	30
0	1.0	0.0317	0	0	0	0	0
5	0	0.9384	0.0013	0	0	0	0
10	0	0.0300	0.9987	0	0	0	0
15	0	0	0	1.0	0	0	0
20	0	0	0	0	1.0	0.0034	0
25	0	0	0	0	0	0.9966	0.0044
30	0	0	0	0	0	0	0.9956

. The real SCT condition is presented in each column of the confusion matrix, and the classified/predicted SCT condition is presented in each row. The normalized classification accuracies are shown from 0 to 1.0, 0 being equal to 0% and 1.0 equal to 100%. The values of the diagonal cells correspond to the normalized percentage of the tests for each SCT condition that was correctly classified. In this regard, the accuracy values of 1 for the conditions of 0, 15, and 20 SCTs indicate that 100% of tests were correctly classified.

The results of Table 2 belong to the first KNN training for a data window size of 100. To have a general statistical performance with different data window sizes. The classification process is repeated forty times using different file combinations and various data window sizes, i.e., 100, 500, 1000, 5000, and 8000 samples to test. From the obtained accuracy values, their mean, minimum, and maximum values are computed and presented in

Table 3. Mean, minimum, and maximum accuracy values of SCT classification with various data window sizes.

	Window Size
	100	500	1000	5000	8000
SCTs	Mean (min,max)	Mean (min,max)	Mean (min,max)	Mean (min,max)	Mean (min,max)
0	0.9889 (0.94, 1.0)	0.9998 (0.99, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)
5	0.8758 (0.78, 0.93)	0.9310 (0.75, 1.0)	0.9460 (0.77, 1.0)	0.9439 (0.65, 1.0)	0.9632 (0.82, 1.0)
10	0.9811 (0.91, 1.0)	0.9987 (0.99, 1.0)	0.9990 (0.99, 1.0)	0.9996 (0.99, 1.0)	0.9999 (0.99, 1.0)
15	0.9997 (0.99, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)
20	0.9980 (0.99, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)	1.0 (1.0, 1.0)
25	0.9918 (0.97, 1.0)	0.9947 (0.98, 1.0)	0.9958 (0.98, 1.0)	0.9974 (0.97, 1.0)	0.9983 (0.98, 1.0)
30	0.9929 (0.96, 1.0)	0.9969 (0.98, 1.0)	0.9977 (0.98, 1.0)	0.9988 (0.99, 1.0)	0.9995 (0.99, 1.0)

. For example, the cell for 0 SCTs and 100 samples as window size have the values 0.9889 (0.94, 1.0), indicating that the mean for the forty accuracy values is 0.9889, the minimum obtained value is 0.94, and the maximum obtained value is 1.0. The best classification results are obtained using 5000 and 8000 samples, where three classes present 100% accuracy and three classes are near 99%. The 8000 samples are obtained in 1.33 s, which represents 58% of the total acquisition time, and 5000 samples are obtained in 0.83 s, which represents 36% of the total acquisition time. Additionally, the windows with smaller sample number, such as 500 and 1000 present similar accuracies in these classes, except in 5 SCTs which is understandable since it is a low severity damage. They represent 3.62% and 7.25% of the total acquired samples, which means that high accuracies are obtained using a very low number of samples. The accuracy in using a window of 100 samples remains near 99%, being a good option in classification despite the reduction in the accuracy of some classes. Therefore, it is concluded that a reliable diagnosis can be obtained with a small number of samples, which is desirable for applications that require fast responses, as in the case of protections; in the same regard, a slight improvement in the accuracy values can be obtained, but an increase in the number of samples can be taken into account. The best option depends on the application.

The box and whisker plots shown in Figure 8 show the comparison among the mean accuracies in the classification of SCTs using data window sizes of 100, 500, 1000 and 5000 samples. These plots show the distribution of the obtained accuracies. For instance, the higher accuracy distribution observed for the condition of 5 SCTs indicates that the obtained values are more dispersed and, consequently, more difficult to classify in the system since their range of possibilities is bigger. This issue can be seen in the plot for the case of 100 samples, since some outliers (red marks) for the 10 SCT condition overlap with the range of the 5 SCTs. The 8000 and 5000 window sample plots are similar.

The QSA method is implemented using a feedforward and fully connected NN classification to compare it with the results obtained using KNN. The NN is applied with Bayesian optimization, 30 objective function evaluations, an activation function for classification by sigmoid function, and 10 values in each dimension. The results are presented in

Table 4. General accuracy comparison among KNN and NN classification.

	Window Size
	100	500	1000	5000	8000
SCTs	KNN,NN	KNN,NN	KNN,NN	KNN,NN	KNN,NN
0	0.9889, 0.9835	0.9998, 1.0	1.0, 1.0	1.0, 0.9877	1.0, 1.0
5	0.8758, 0.8690	0.9310, 0.7470	0.9460, 0.6767	0.9439, 0.5839	0.9632, 0.6922
10	0.9811, 0.9859	0.9987, 0.9959	0.9990, 0.9968	0.9996, 0.9658	0.9999, 0.9605
15	0.9997, 0.9981	1.0, 0.9721	1.0, 0.9720	1.0, 0.8911	1.0, 0.9737
20	0.9980, 0.9948	1.0, 0.9676	1.0, 0.9627	1.0, 0.9285	1.0, 0.9333
25	0.9918, 0.9876	0.9947, 0.9079	0.9958, 0.9031	0.9974, 0.8640	0.9983, 0.8601
30	0.9929, 0.9790	0.9969, 0.9958	0.9977, 0.9817	0.9988, 0.9767	0.9995, 0.9652

, where the KNN method shows higher accuracy than the NN method for all data window sizes and classifications. KNN uses fewer resources than NN; hence, the QSA method using KNN is the best option with high accuracy, low window samples, and low processing.

In the same way, the DT classification is applied to the QSA method. It presents similar accuracies with a low classification process as is shown in

Table 5. General accuracy comparison among KNN and DT classification.

	Window Size
	100	500	1000	5000	8000
SCTs	KNN,DT	KNN,DT	KNN,DT	KNN,DT	KNN,DT
0	0.9889, 0.9831	0.9998, 0.9997	1.0, 1.0	1.0, 1.0	1.0, 1.0
5	0.8758, 0.8615	0.9310, 0.9266	0.9460, 0.9332	0.9439, 0.9653	0.9632, 0.9568
10	0.9811, 0.9786	0.9987, 0.9988	0.9990, 0.9994	0.9996, 0.9996	0.9999, 0.9996
15	0.9997, 0.9983	1.0, 1.0	1.0, 1.0	1.0, 1.0	1.0, 1.0
20	0.9980, 0.9923	1.0, 0.9996	1.0, 0.9999	1.0, 1.0	1.0, 1.0
25	0.9918, 0.9940	0.9947, 0.9961	0.9958, 0.9923	0.9974, 0.9974	0.9983, 0.9988
30	0.9929, 0.9935	0.9969, 0.9944	0.9977, 0.9949	0.9988, 0.9991	0.9995, 0.9997

, however, there are cases where the DT method presents the best results as in the case of 5000 samples. The KNN classification method presents the best accuracy using 100, 500, and 1000 window samples. The condition with 5 SCTs yielded the lowest classification accuracy across all three classifiers. Particularly noteworthy are the results obtained by the NN classifier, which exhibited the poorest performance with an average accuracy of 71.7%. The accuracies in 8000 samples are close to the same value. Therefore, the KNN method has general advantages over the DT method in accuracy with a small window size.

The comparison among the accuracies of the classification methods is presented in Figure 9. As can be observed, the three proposed classifiers exhibit similar and high values of accuracy for all the window sizes and all the SCT conditions; however, KNN presents a slight advantage. The worst values occur in 5 SCTs applying the three classification methods; they are squared with dotted lines, regardless of the window size.

5. Discussion

Table 6. Comparison between the proposed method and others reported in the literature.

Method	Samples	Signal Type	Statistical Indices	Classes	Classification	Accuracy
Frequency-response	40,000,000	Voltage and	SD	Multiple short	ANN	0.9
analysis [23]		current		cuts
Wavelet-based [24]	15,360	Current	Energy	1–5% short-circuited	—	1.0
				winding
Statistical time	30,000	Vibrations	Entropy, Kurtosis	0–35 SCTs	SVM	0.9682
features [22]			SMR, RMS, SD, VA
Fractal dimension	18,000	Vibrations	—	0–35 SCTs	KNN	0.9962
algorithms [12]
	100					0.9755
Quaternion signal analysis (QSA)	5000	Vibrations	SD, VA	0–35 SCTs	KNN	0.9914
	8000	and current				0.9952

compares the presented method against other works reported in the literature to highlight its advantages. First, it is worth noting that the proposed method uses only 100 samples to obtain a 97.55% general accuracy, which is a near value compared with the presented works from 90% to 100% of accuracy. The proposed method increases its general accuracy when the sample number is increased, but it stays above 99% from 8000 samples. Similar accuracies are obtained in [12,22]. The previous studies utilized vibration signals [12,22] and electrical signals [23,24] as input parameters. In contrast, our proposed approach combines both vibration and current signals to extract two statistical features. Unlike the previous research that relied on a single statistical index [23,24], this method achieves high accuracy without the need for spatial transformations The maximum number of samples processed in our work is less than the number of samples used in the presented works, so our work is efficient in processing time, since it only requires 100 samples to obtain a comparable percentage of precision. Additionally, the presented method is developed in the time domain, which avoids spatial transforms (e.g., wavelet transform), proposing a more efficient processing tool.

In summary, the QSA method is efficient in SCT classification using statistical values such as variance and standard deviation, which are low-processing calculations. The general accuracies are from 99% to 100% among different classes and window samples, except in 5 SCTs, which vary from 87% to 96%. The method obtains good classification in a short time having a low number of samples and amount of processing in the time domain. In addition, low resources in the KNN algorithm allow the implementation of the complete method in a portable processor for real analysis in future works adding other fault classes.

6. Conclusions

A power supply is indispensable in daily life, and a crucial element in ensuring its reliability is the transformer. Therefore, its maintenance and prevention of failures through the development and application of accurate methods that timely identify SCTs are imperative. In this regard, this work presents a novel method for SCT fault detection. The proposed method, in summary, utilizes the QSA method within a time-space framework, two statistical indices, and a machine-learning classifier.

In this work, it was found that the processing of the current signal and the vibration signals with the QSA method allows for a fast and effective diagnosis, since only 100 samples of the signals are required two achieve 97.55% of accuracy. Thus, the proposed method contributes to minimizing both sample acquisition and processing, using simple operations. These features help to evaluate the transformers in short operating times and prevent the system from being exposed to major faults. It is important to highlight that a larger number of samples increases the accuracy values, 8000 samples being the window size that allows for 99.52% accuracy. Moreover, after testing multiple statistical indices, it was found that VA and SD provide suitable patterns to differentiate between SCT conditions, having the advantage of requiring simple mathematical operations for their computation. Finally, although three different classifiers were tested (i.e., KNN, NN, and DT), the best accuracy results were obtained using the KK classifier for all the analyzed window sizes.

Despite the obtained promising results, it is important to mention that there are still several opportunities for the future research to overcome some detected shortcomings. For instance, classification results of 87.58% were obtained for the condition of 5 SCTs, which is not a good enough value compared with the other ones obtained, which were higher than 98%. Although this low value is somehow expected since the fault can be considered of low severity and, consequently, the generated signals (current and vibrations) are slightly affected, making difficult the extraction of more useful information, it is fundamental that we continue the research into other methods that can be more effective for incipient damages.

The future work will explore the detection of SCTs and other faults in transformers operating under both linear and non-linear loads in order to provide more robust methods. Furthermore, we will explore the application of deep learning techniques in order to enhance diagnostic accuracy, mainly considering incipient damages.