Towards Data-Driven Fault Diagnostics Framework for SMPS-AEC Using Supervised Learning Algorithms

Abstract

The service life of aluminium electrolytic capacitors is becoming a critical design factor in power supplies. Despite rising power density demands, electrolytic capacitors and switching devices are the two most common parts of the power supply that age (deteriorate) under normal and diverse working conditions. This study presents a fault diagnostics framework integrated with long-term frequency for a switched-mode power supply aluminium electrolytic capacitor (SMPS-AEC). Long-term frequency condition monitoring (CM) was achieved using the advanced HIOKI LCR meter at 8 MHz. The data acquired during the experimental study can help to achieve the needed paradigm from various measured characteristics of the SMPS/power converter component to detect anomalies between the capacitors selected for analysis. The CM procedure in this study was bound by the electrical parameters—capacitance (Cs), equivalent series resistance (ESR), dissipation factor (DF), and impedance (Z)—-acting as degradation techniques during physical and chemical changes of the capacitors. Furthermore, the proposed methodology was carried out using statistical feature extraction and filter-based correlation for feature selection, followed by training, testing and validation using the selected supervised learning algorithms. The resulting assessment revealed that with increased data capacity, an improved performance was achieved across the chosen algorithms out of which the k-nearest neighbors (KNN) had the best average accuracy (98.40%) and lowest computational cost (0.31 s) across all the electrical parameters. Further assessment was carried out using the fault visualization aided by principal component analysis (PCA) to validate and decide on the best electrical parameters for the CM technique.

1. Introduction

Artificial intelligence (AI) has fundamentally changed the various research fields cutting across prognostics and health management (PHM), subdivided into physics-based, data-driven, and hybrid categories. Power electronic systems are also anticipated to benefit significantly from the integration of power electronics and data analysis as they move toward becoming data-rich systems. A data-driven approach ranges from the data collection stage to making the right decision. This would require vast background knowledge of the power electronic devices and the data collection process has been found to be a difficult task [1,2,3,4].

Power electronics are increasingly common in electrical energy systems nowadays—power electronics process over 70% of the electricity that is utilized by humans. Power electronics are responsible for sending voltages and currents to the load in the most suitable form and use a switching circuits to process and regulate the flow of electrical energy. Power electronics circuits are categorized depending on the input and output type. High power density, high efficiency, low cost, and small size are used to evaluate commercial power electronics converters. Maintaining reliability in these devices can prove difficult, however, when these the above mentioned characteristics are attained [5,6,7]. High junction temperatures in switching devices result from an improved converter topology created to satisfy high-power and high-efficiency systems. Designers can attempt to reduce reliability difficulties by studying the relationship between the electrical and thermal properties of the semiconductor devices used in a power electronic circuit. Common factors that can reduce the reliability of power electronics are temperature (which could result in loss of electrolyte and leakage current), over current (which could result in dry out/electrolyte loss, over voltage (could result in oxide degradation) and environmental conditions [8,9,10,11]. The combined effect of the listed factors could result in a decrease in capacitance, an increase in ESR, and changes in the dissipation factor. Many factors, including capacitance, ESR, and the dissipation factor, were identified as a degradation factors for electrolytic capacitors.

Capacitors, particularly those of the electrolytic variety, have one of the most significant failure rates in electronic circuits, so extra attention should be paid to how they operate. If they are used close to or over their working temperature or voltage ratings, unacceptably high failure rates will occur. “Operating temperature” is the sum of the ambient temperature of the capacitor’s surroundings and the increase brought on by the AC ripple current passing through the capacitor’s equivalent series resistance (ESR). If necessary, the thermal resistance between the capacitor’s core and the surrounding air, the ESR, and the ripple current must be calculated. The ripple current squared times the ESR equals the power dissipated in the capacitor. In order to apply the best de-rating for reliability, availability, and cost concerns, the actual operating voltage and the rated maximum voltage allowed must be compared and studied to assure a reliable design. An alarming “lifetime” or “life validation test” such as 2000 or 4000 h may be stipulated in the manufacturer’s specifications for an aluminium electrolytic capacitor. This is the time until the capacitance, ESR, or leakage current will drift by a predetermined amount when the capacitor is utilized at its maximum rated temperature and voltage [12,13].

Due to their capacity to sustain high voltages for PFC (power factor correction) applications, electrolytic capacitors play a significant role in contemporary power electronics. Solid capacitors, such as ceramic chip capacitors, cannot perform this function. Dielectric breakdown is one of the most frequent electrolytic capacitor failure mechanisms, and it can result in internal short-circuits when the capacitor is charged at a high voltage. Although the control circuit may have protection mechanisms, voltage instabilities during the short may still cause other delicate components to fail. Since the functional properties of the capacitor return to normal after a breakdown, diagnosing such failures is challenging [14,15,16,17].

Devices or components malfunction relatively early on due to flaws in the design, manufacturing processes, or an incompatibility with the operating environment. These defects, known as the early failure period, are corrected by debugging during one of the production steps before shipments for aluminium electrolytic capacitors. The random failure period is consistent, rare, and does not seem connected to the term they have served. Aluminium electrolytic capacitors have experienced fewer catastrophic failures than semiconductors and solid tantalum capacitors. The failure rate during the wear-out period rises with operation time. The electrolyte impregnated in aluminium electrolytic capacitors steadily evaporates and diffuses out of the capacitors through the rubber seal materials since their production is finished, causing a decrease in capacitance and an increase in tan. The capacitors are defined as having “fallen into wear-out failure” when any of these values vary outside the acceptable range of specifications. Their useful life is the capacitors’ serviced period until they enter the wear-out failure period. The failure rate of aluminium electrolytic capacitors can be visualized using the bathtub curve as shown in Figure 1.

The contributions to this study were borne out of a continuous study using the advanced LCR meter in a bid to achieve better performance in terms of data capacity in the capacitors’ diagnosis and they are as follows:

• A framework that prioritizes selecting key electrical parameters (Cs, Rs, D and Z) for diagnostic assessment. This was aided by using the advanced HIOKI 3536 industry standard for the wide application of a seamless data acquisition process involving the three aluminum electrolytic capacitors found in switched-mode power supplies.
• Deploying a statistical feature extraction (time-domain) method that helps to extract the correct information needed for the diagnosis of the capacitors. In addition, filter-based feature selection was adopted that helped to select the best features from their correlation score.
• Adopting a set of supervised learning algorithms for training, testing and validation for the dataset. Providing a decision framework that helps to select the right model and the right electrical parameters.

The remaining section of the paper is arranged as follows: Section 2 provides insight to the motivation behind the study and some related works. Section 3 describes the materials of the electrolytic capacitor, its working principle while Section 4 provides a breakdown of the process in this study for a seamless methodology that can be replicated for industrial application. The data acquisition process, preprocessing and feature engineering are discussed further in Section 5 and Section 6 while the experimental results and diagnostic assessment are detailed in Section 7. The summary of the study is provided in Section 8.

2. Motivation and Related Works

An increasingly important design factor in power supplies is the service life of these electrolytic capacitors. Electrolytic capacitors are the only parts of the power supply that deteriorate despite rising power density needs. The life of the power supply is thus dependent on the type of electrolytic capacitor employed in the design. Additionally, it specifies the end application’s service life or interval in maintaining equipment. The topology, applied ripple current, design layout, lifetime, temperature rating, and local heating effect will differ from one product to another. Under low and high-line input circumstances, they could potentially alter. In increasingly compact designs, external heating impacts sometimes outweigh inside heating effects. The temperature increases that may occur when the power supply is mounted in the application also affect the actual service life.

A review of condition monitoring methods was carried in [18] and classified the availability of CM approaches as online or offline. The methodologies were classified as capacitor ripple current sensor based methods, circuit model based methods, and data and advanced algorithm based methods. Less research on the data-driven/offline/machine learning methods has been conducted, which is one of the motivations behind this study. The first motivation is the established fact of capacitors being a major faulty component in SMPS under normal and diverse conditions. This study suggested an offline parameter estimation technique for a three-order, five-circuit DC-link LC filter. It is based on injecting a voltage signal with the appropriate excitation frequency and amplitude. It uses the BAT algorithm and a global optimization technique to determine the parameters of each circuit element. The suggested approach can be used for railway and metro inverters with an easy-to-use interface for signal injection and an offline maintenance schedule. The calculated values for capacitance and inductance can also be used as inputs for the inverters’ condition monitoring or operation optimization [19]. It creates a parameter observer (PO) and can be used to calculate the equivalent serial resistance and capacity of electrolytic capacitors. The observer is a second-order integral-open-loop system, and its input is the voltage recorded at the capacitor terminals during the process of a two-stage capacitor being discharged through a variable resistor. In order to determine the characteristics of the capacitor, the PO estimates the so-called time constant of the discharging circuit for each of the two stages. A buck DC-DC converter is determined using PO. Two electrolytic capacitors with nominal ratings of 100 F and 470 F were used for the experiment [20]. An ideal frequency range was determined to reduce the impact of angle measurement error on the DF calculation, and a method for the accurate measurement of DF based on the capacitor impedance angle was suggested. The findings from this research show DF as a reliable indicator owing to the fact that it comprises of the capacitance and ESR value [21].

Power electronics are used in practically every industrial automation application, whether it is for generation or transmission. The numerous applications made possible by semiconductor-based power devices include high-voltage direct-current power transmission, flexible AC transmission systems, electric cars, and micro grids. However, as the industry develops, there is a greater requirement for these components to be produced, controlled, and maintained in an efficient manner. The regulation of semiconductor-based power devices is difficult because they are high-frequency, high-switching frequency devices. AI has been shown to be a highly effective method for creating, managing, and sustaining power electronics equipment. The control phase, followed by the maintenance phase, and the design phase, are when AI is most effectively applied in power electronics. The most common categories for the AI techniques employed include regression, classification, optimization, and data structure exploration. However, regression and optimization are the two main objectives. Expert systems, fuzzy logic, metaheuristic techniques, and machine learning are some of the main categories of the extensively used AI techniques in power electronics. When it comes to power-electronic applications, machine learning (ML) performs best. ML is further divided into supervised learning, unsupervised learning, and reinforced learning for use in power electronics. Some of the research trends and an overview of artificial intelligence in power electronics can be found in [22,23,24,25,26,27,28,29,30].

3. Theoretical Backgrounds

3.1. Overview of Capacitor Types, Materials and Usage

Table 1. Selection Criteria and Comparison among Capacitor Types.

Capacitor Types	Material Samples	Merit	Demerit
Electrolytic	Aluminum, Tantanium, Niobium.	It is relatively cheap and with a high capacitance to volume ratio	High leakage current, wide value tolerances, poor equivalent series resistance, limited lifetime.
Polymer	Polystrene, Polycarbonate, Polyamide, PP, Polyester.	Good high frequency performance, suitable for a variety of applications, stable operations for bias voltage conditions	Relatively low temperature resistance, Pulses and transience can cause damages
Film	Polyester Film, Mylar	Small size, higher moisture resistance	Low stability for temperature, hazardous heating for RF applications
Mica	Mica	Durable, stable temperature ability	High cost and prone to moisture
Glass	Glass	Similar Characteristics to Mica, Good radiation resistance.	Expensive
Paper	Paper/Oil-Impregnated, Waxed	Now obsolete but still in use in high voltage applications	low moisture resistance, quick degradation due to moisture
Ceramic Class 1 and 2	Paraelectric (Titanium Dioxide), Ferroelectric (Barium Titanate)	Good frequency response characteristics, Can withstand higher voltages up to 100 volts, lighter in weight, Inexpensive	Capacitance value are limited to below 150 uf, large temperature coefficient, Not stable for use in power supply circuit as output capacitors

shows the criteria for selecting the right capacitor during the design stage and their characteristics. Capacitors are an essential part of any power electronic design. Over time, various device types with diverse characteristics have been produced, making some capacitor technologies particularly suitable for specific applications. To guarantee that the best option is chosen for a given application, designers should thoroughly understand the various types, configurations, and standards.

• Ceramic Capacitors—one of the most widespread capacitor types used to ensure low capacitance. However, this is no longer the case, as the capacitance rating of multilayer ceramic capacitors (MLCC) can reach hundreds of microfarads (F) and they are widely employed in circuits. For obsolete hardware and designs, modern ceramic capacitors can be utilized in place of electrolytic or tantalum capacitors. The difference between an electrolytic and a ceramic capacitor is that the latter provides better performance for less money. Ceramic capacitors are divided into two categories: Class 1 and Class 2. Para-electric ceramics, such as titanium dioxide, are used in Class 1. Ceramic capacitors in this class have high stability, low loss, and a good temperature coefficient of capacitance. Their intrinsic precision is employed in oscillators, filters, and other RF applications. Ceramic dielectrics based on ferroelectric minerals such as barium titanite are used in Class 2 ceramic capacitors. Class 2 ceramic capacitors have a larger capacitance per unit volume than Class 1 capacitors due to their high dielectric constant. However, they have lower precision and stability. They are employed in bypass and coupling situations when the capacitance’s absolute value is not essential [31].
• Film Capacitors—In film capacitors, a thin plastic film serves as the dielectric. The conducting plates on each side of the plastic sheet may have a pair of thin metallization layers. The type of dielectric polymer employed affects the characteristics of capacitors. The various varieties of film capacitors include high voltage breakdown ratings, low equivalent series resistance (ESR) and equivalent series inductance (ESL), and exceptional tolerance and stability, which are all characteristics of polypropylene (PP). They can only be found as leaded devices because of the dielectric’s temperature limitations. PP capacitors are used for circuits that require high power or voltage, such as high-frequency discharge circuits, switch-mode power supplies, and audio systems. For the signal’s integrity, their low ESR and ESL are crucial [32,33].
• Electrolytic Capacitors—Capacitors made of aluminum electrolytically deteriorate over time. Excess gases can be released through a vent on many electrolytes. The electrolyte may dry out as a result of this leak, and the capacitor’s performance may suffer. The oxide layer on the anode of aluminum electrolytic capacitors can also evaporate after a few years. The capacitor must then be repolarized when this occurs. To accomplish this, a capacitor can be exposed to a current-limited voltage. The initial leakage current across the capacitor will be substantial, but as the oxide layer thickens, it will become less relevant. Power electronics uses the surface mount technology (SMT) for electrolytic type of capacitors frequently. They are particularly useful in a range of applications due to their high capacitance and low cost. Initially, they were not employed in large quantities due to their inability to endure certain soldering techniques. Due to improved capacitor design and the use of reflow processes rather than wave soldering, electrolytic capacitors can now be utilized more commonly in surface mount technology. There is usually space on the leaded variants of electrolytic capacitors for the various parameters to be placed on the container. The markings typically include information such as the capacitance value, working voltage, temperature range, and potentially other data. Some big capacitors used in power supplies for smoothing purposes may also contain additional information. The ripple current is a very important element. If the capacitor handles too high a current, it may overheat and fail [34].

All capacitor types have a position in the market, even if that place shifts over time as new technologies and advancements in other capacitor types affect the market. Some capacitors are more effective than others. However, as we have seen, several instances still exist in terms of which one type of capacitor cannot be replaced. Like every other form of electrical components, capacitors are still improving and progressing, propelled forward by the demands of ever-improving technology. Capacitors are frequently seen as a solved technology. However, many current capacitors are far superior to those previously available.

3.2. Output Filter and SMPS Working Principle

The input power is an alternating current (AC), and the output power is a direct current (DC) in this type of SMPS. This AC power is converted to DC using rectifiers and filters. The impacted power factor correction circuits are provided with this unmanageable DC voltage. This is because, near the peak of the voltage, there is a low current pulse inside the rectifier. This includes high-frequency energy, which has an effect on the power factor by lowering it. This is due to power conversion, although we used an AC input source rather than a DC supply. As a result, this block diagram uses a combination of a rectifier and filter to convert AC to DC, and a power amplifier is used to switch an operation [35].

MOSFET transistors have low resistance and can withstand large currents. The switching frequency was designed to keep normal human hearing (above 20 KHz) at a minimum, and the switch’s functioning is regulated by a PWM oscillator. This AC voltage is applied to the transformer’s output once more, as illustrated in the diagram. Otherwise, the voltage level drops. After that, the output filter and corrector are used to fix and smooth the transformer’s output. In comparison to the reference voltage, the reaction circuit controls the output voltage. Figure 2 shows the flow chart of the working principle of a typical AC-DC switched mode power supply.

3.3. Overview of Filter-Based Feature Selection

In some cases, a model’s efficiency might be improved while its computational cost is decreased by reducing the number of input variables. The relationship between each input variable and the target variable is evaluated using statistics-based feature selection approaches, and the input variables with the most robust relationships are selected. These methods can be quick and effective, even if the choice of statistical measures depends on the data type of both the input and output variables. The main divisions of feature selection techniques are supervised and unsupervised, and the subcategories of supervised techniques are intrinsic, wrapper, and filter. Using filter-based feature selection procedures, which rate the correlation or dependence between input variables using statistical measurements, the most important qualities are chosen. The data type of the input variable and the output or response variable must be carefully considered when choosing statistical measures for feature selection. In filter feature selection methods, which apply statistical techniques to evaluate the relationship between each input variable and the target variable, these scores serve as the basis for choosing (filtering) the input variables included in the model. The selection of filter features usually involves correlation-type statistical measurements between input and output variables. As a result, the chosen statistical measures are significantly influenced by the types of variable data. Some of the related work of the filter-based feature selection approach can be seen in [36,37,38,39,40].

4. Proposed Integrated Statistical and Supervised Learning Model

Figure 3 depicts the condition monitoring framework’s suggested architecture. In order to strengthen and improve the distinguishing characteristics amongst the chosen capacitors, filter-based feature selection was introduced. It is interesting to note that the model includes a stage for pre-processing data, three-way feature extraction, correlation-based feature selection, a different set of input variables, ML-based diagnosis, and performance testing of the chosen ML technique. The proposed integrated statistical and supervised learning model framework/approach involves a long-term frequency CM set at 8 MHz. To enhance and improve the performance of the selected models, a statistical approach involving time-domain features’ extraction from each dataset and filter-based feature selection was used to select the right features for training and testing the dataset. Furthermore, the stages of the framework can be categorize as follows:

• Stage I—condition monitoring of stand-alone aluminum electrolytic capacitors.
• Stage II—involves a preprocessing stage, statistical feature extraction and filter-based feature selection.
• Stage III— involves the data normalization process, data cleaning, and concatenation of the dataset before being fed to the supervised machine learning algorithms.
• Stage IV— involves splitting the data into a 70/30 ratio, training of the dataset, testing of the dataset, k-fold cross validation, and classification evaluation metrics.
• Stage V— involves the fault visualization process assisted with the aid of principal component analysis (PCA) that involves the reduction of the features from five to six to two.

5. Data Acquisition Process

The data acquisition process in Figure 4 consists of an offline condition monitoring for SMPS-AEC using the advanced HIOKI 3536 LCR meter. By sensing the current flowing to the measurement target and the voltage across the measurement target’s terminals, LCR meters can determine the impedance (Z) and phase angles. They then calculate measurement variables, including inductance (L), capacitance (C), and resistance (Rs), using the Z and phase angle data. Since the equipment cannot determine which model is most appropriate for a given measurement objective, the equations utilized to calculate these measurement parameters vary depending on whether the instrument is in a series equivalent circuit mode or a parallel equivalent circuit mode. The user must select the suitable equivalent circuit mode to minimise measurement error. The impedance (Z) has a real section (Rs) and an imaginary section (X), and its parameters can be calculated by expanding it on a complex plane [41]. Equations (1)–(5) represent the impedance relationships.

$$Z=R_s+jX=\left|Z\right|\angle \theta$$

(1)

$$\theta =ta{n}^{-1}\left(\frac{X}{R_s}\right)$$

(2)

$$R_s=\left|Z\right|cos\theta$$

(3)

$$X=\left|Z\right|sin\theta$$

(4)

$$\left|Z\right|=\sqrt{{R_s}^2}+X^2$$

(5)

Table 2. Electrolytic Capacitor Experimental Test Conditions.

Functions	Description
Electrical Parameters	Cs–Rs–D–Z
Frequency/Freq-Step	8 MHz/1000 Hz/10 Hz
DC Bias	ON 1.0 volts
Signal Level	0.5 Vrms
Measurement Range	Auto
Speed	SLOW2
LowZ mode	ON

shows the details of the experimental condition used in the data acquisition process. Specific settings are meant to be set by each operator as the measurement target could vary based on conditions. The conditions adopted include a voltage varying concept to replicate a charging and discharging cycle for the capacitors, i.e., the current flows through the capacitor and begins to store charge and discharge when the circuit is stopped. The output resistance of the instrument is decreased and the measuring current is repeatedly supplied in a low impedance high accuracy mode to increase the measurement accuracy. The low-impedance high-precision mode produces more reliable results when measuring a capacitor with a high capacitance of larger than 100 F (and thus low impedance). Likewise,

Table 3. Electrical Signal and its Function.

Parameters	Definition	Functions
Z	Impedance	$Z=\frac{\left|X_c\right|}{Sin\theta }$
D	Loss coefficient/Dissipation Factor	$tan\delta =\frac{ESR}{X_c}$
$Rs$	Equivalent Series Resistance	$ESR=\left|Z\right|Cos\theta$
$Cs$	Capacitance	$Cs=\frac{D}{\omega R}$

shows the description of each selected measurement type from the software along with their functions.

The acquisition process consists of the following stages: Stage A (Calibration)—involves the calibration of the clip to avoid parallax error and improve the accuracy. This was recommended by the manufacturer before proceeding with the experiment. This involves performing open and close connection calibration on the clip with the process shown below in Figure 5.

Stage B (Software)—the option to select at least four electrical signals at once for the monitoring of the SMPS-AEC. This involves selecting the capacitance (Cs), equivalent series resistance (Rs), dissipation factor (D), and impedance (Z). Other parameters selected in the data acquisition process are frequency/frequency step, DC Bias, signal level, measurement range, etc. Figure 6 shows an overview of the software program used in the data acquisition process. First, the USB connection is programmed with the LCR meter, then an electrical signal is selected for each measurement type. Afterwards, the voltage mode to be applied across the test sample is selected which could either be a constant voltage (CV) across the test sample or a varying voltage mode (V). Other important parameters include the acquisition speed of the test sample (SLOW2, FAST) and low impedance is set to ON due the capacitance value of the selected capacitors.

Stage C (File Storage)—during the data acquisition setup, selection of the right file location is required to obtain the stored data. While there is a visualization section on the software that shows the timely capturing of the dataset in columns and rows, there is also a need to have an excel format type dataset for further analysis.

Figure 7 shows the visualization of the four electrical signals captured using the software program with the advanced HIOKI 3536 LCR meter. The capacitance (Cs) against the equivalent series resistance (ESR) is on the left while the dissipation factor (D) against the impedance (Z) is shown on the right.

6. Statistical Feature Extraction and Filter-Based Feature Selection

It is a feature engineering method that gauges how well two sets of values can be distinguished from one another. Correlation matrices are meant to show positive and negative values of the statistical features if available. The measurement parameter is termed a Pearson correlation coefficient that ranges between −1 to +1. Once an F-score value is provided, features greater than the set score will be dropped during feature selection. A specific F-score will be used to determine each feature. The likelihood that this trait will be more discriminative increases with the size of the F-score. The features will then be arranged in ascending order, and the scores will be used to choose which features to employ. To classify types of data, the time domain statistical features were extracted from the capacitor datasets and are shown in

Table 4. The Definitions of the Statistical Features Extracted.

Feature Description	Definitions
Root Mean Square	$X_{rms}=\sqrt{\frac{{\sum }_{i=1}^n{\left(x_i\right)}^2}{n}}$
Mean	$\overline{x}=\frac{1}{n}\left({\sum }_{i=1}^n{x}_i\right)$
Kurtosis	$X_{kurt}=\frac{1}{N}\mathrm{\Sigma }\left(\frac{{\left(x_i-\mu \right)}^3}{\sigma }\right)$
Interquartile range	$upperquarter{Q}_3-lowerquarter{Q}_1$
Median abs deviation	$X_{mad}=\frac{1}{n}{\sum }_{i=1}^n\left|x_i-m\right|$
Skewness	$X_{\mathrm{skew}}=E\left[\left(\frac{{\left(x_i-\mu \right)}^3}{\sigma }\right)\right]$
Max	$X_{max}=max\left(x_i\right)$
Min	$X_{max}=min\left(x_i\right)$
Crest Factor	$X_{CF}=\frac{x_{max}}{x_{rms}}$
Peak factor	$x_{PF}=\frac{x_{max}}{\sqrt{x_s}}$
Wave Factor	$x_{WF}=\frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\left|x_i\right|}^2}}{\frac{1}{n}{\sum }_{i=1}^n\left|x_i\right|}$
Standard error mean	$X_{sem}=\frac{standarddeviation}{\sqrt{n}}$
Standard deviation	$SD=\sqrt{\frac{1}{N-1}{\sum }_{i=1}^N{(x_i-\overline{x})}^2}$
Variance	$VAR=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N}{\left(x_i-\overline{x}\right)}^2}$

. Equation (6) was used to carry out the selection of the best features after their linear correlation. The positive linear matrix shown in the correlation means there is a strong relationship between the variables extracted. A similar correlation matrix can be seen in [11,13] and in statistical feature extraction application [42,43,44,45].

$${\rho }_{X,Y}=\frac{cov(X,Y)}{{\sigma }_X{\sigma }_Y}$$

(6)

where ${\sigma }_X$ and ${\sigma }_Y$ are the standard deviations of X and Y, respectively, while $cov(X,Y)$ is the covariance.

The fundamental idea behind feature selection is to remove non-informative or redundant predictors from the model. Surprisingly, if input parameters unrelated to the target variable are provided, the performance of the machine learning model may suffer. To determine which feature selection methods to use, whether the outcome is supervised or unsupervised should be first determined. The correlation as shown in Figures 8, 10, 12, 14 and 16 was aided by a seaborn library in python. Each dataset (Cs, ESR, Z, DF) underwent a feature extraction process using the time-domain features in Table 4. Afterwards, all features extracted were normalized, concatenated, labelled (as it is a supervised learning approach) and plotted using the seaborn heatmap with the support of the pandas and matplotlib library. To select the best features from the correlation matrix, a correlation score was pegged at 0.9 for the whole dataset.

Overall, a total of 13 features were extracted from the dataset, namely root mean square, mean, kurtosis, interquartile range, median abs deviation, skewness, max, min, crest factor, peak factor, wave factor, standard error mean, standard deviation and variance. As shown in Figure 8, the numerical variables of root mean square, variance and standard deviation had a stronger relationship. Likewise, there are features of multicollinearity with a positive numerical variables of 1. The 10 features selected as shown in Figure 9 included mean, root mean square, interquartile range, maximum, minimum, median abs deviation, kurtosis, skewness, peak factor and wave factor.

Figure 10 shows the correlation of the thirteen (13) time domain features extracted from the ESR dataset. The features selected based on the set correlation are shown in Figure 11, with a total number of 10 features namely, mean, root mean square, variance, interquartile range, maximum, minimum, median absolute deviation, kurtosis, skewness and wave factor. The correlation coefficient in red has a positive value of 1, with the less negative coefficient of −1 represented in blue. In addition, the color coding of the heatmap shows the necessary relationship among the variables and reveals linearity and non-linearity.

Figure 12 shows the correlation relationship among the time-domain statistical features extracted from the impedance dataset. From the heatmap, it can be deduced that there were a greater number of higher numerical values in the range of 0 to 1 and a lower number of numerical values in the range of 0 to −1. Notwithstanding, using the correlation coefficient set using the filter-based approach, the best features were selected as shown in Figure 13. The 7 features selected were mean, variance, interquartile range, maximum, minimum, mean absolute deviation, and wave factor. Overall, the color coding was able to show the linear relationship among the numerical variables.

Figure 14 shows the correlation of the time-domain statistical feature extraction heatmap. From the heatmap, it can be seen that there was a high representation of the higher values of 1 among the first four features which were mean, root mean square, variance, and standard deviation. Likewise, there was a strong relationship between standard error mean and maximum which also converged with the first four features named earlier. However, with the color coding representation, there was a need to remove unnecessary features and to pick the best features to aid the performance of the machine learning models. Figure 15 shows the features selected based on the correlation set. The following seven features were selected: mean, interquartile range, minimum, median absolute deviation, kurtosis, crest factor and wave factor.

Figure 16 shows the correlation heatmap of the time-domain statistical features extracted from the dataset consisting of the four electrical signals acquired from the experimental testbed: capacitance, equivalent series resistance, dissipation factor, and impedance. The color coding shows an even distribution of the positive numerical values in red and distribution of the negative numerical values in blue. However, there is a need for feature selection to be carried out due to multicollinearity. Figure 17 shows the features selected based on the correlation coefficient set and they are as follows: mean, root mean square, minimum, kurtosis, skewness, crest factor, peak factor, wave factor which account for eight in total.

7. Supervised Learning Algorithm Assessment and Discussion

As most machine learning classifiers need discriminant qualities as inputs to attain acceptable diagnostic accuracy, there are significant correlations between derived condition health indicators/features. The feature selection findings are shown in Figure 9, Figure 11, Figure 13 and Figure 15, which should provide adequate and satisfactory diagnostic results and serve as the right condition indicators for the aluminium electrolytic capacitors used in SMPS. For ML-based diagnosis training and assessments, the variation-dimensional selected features are taken from the training and test datasets, respectively. When the concept of combining and removing various input features to be extracted is employed, there is an increase or a reduction based on the input variables chosen. The dataset collected in an excel format had 800,000 rows for each column of capacitance (Cs), equivalent series resistance (ESR), dissipation factor (DF) and impedance (Z). The data splitting process was performed at a ratio of 70–30% using the sklearn model selection function. To avoid over-fitting of the dataset, we deployed 5-fold cross validation for the dataset after training, testing and validation.

Table 5. Classification Models and their Parameters.

ML Classifier	Major Functional Parameters	Parameter Values
DT	max–depth	3
KNN	k	3
RF	n estimators	70
SGD	random–state	101
	loss function = modified Huber
NB	Gaussian, var–smoothing = $1{exp}^{-9}$	–
LR	Regularization	L1, L2
GBC	n estimators	100
SVC	Regularization (C), gamma ($\gamma$)	C = 100, $\gamma$ = auto
Adaboost	n estimators	50

shows the functional parameters of the selected supervised learning algorithms. Due to past knowledge, robustness, and processing costs, we considered the chosen ML-based classifiers. Each selected algorithm has unique parameters and designs, making domain expertise necessary to obtain the best results. The following system specifications were used to implement the machine learning algorithm: processor (AMD Ryzen 7 2700 Eight-Core Processor, 3.20 GHz), installed memory (32 GB RAM), system type (64-bit operating system, x64-based processor).

7.1. Supervised Learning Algorithm Evaluation Metrics

We used the classification result table, namely the confusion matrix, to determine the model performance in classification modelling. The accuracy value generated by the model is typically all that most people look at to gauge how well the model is performing. However, in some situations, we cannot simply gauge the model’s performance by its accuracy (e.g., when the data are imbalanced). We must first comprehend the evaluation metrics. In machine learning, evaluation metrics are crucial for understanding model performance and deciding what advice we may offer based on the research. True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) are the four inputs used to create assessment metrics in classification modelling (FN).

$$\mathrm{Accuracy}=\frac{TP}{TP+FP+TN+FN}$$

(7)

$$\mathrm{Recall}/\mathrm{Sensitivity}=\frac{TP}{TP+FN}$$

(8)

$$\mathrm{Precision}=\frac{TP}{TP+FP}$$

(9)

$$\mathrm{F}1-\mathrm{Score}=\frac{2*\mathrm{Sensitivity}*\mathrm{Precision}}{\mathrm{Precision}+\mathrm{sensitivity}}$$

(10)

where (TP) is the total number of correctly classified predictions, and (TN) is the total number of correctly categorized predictions that are both negative and negative. The (FP) value is the proportion of wrongly classed predictions, while the (FN) value represents the proportion of improperly labelled inputs belonging to a falsely classified class. Since it is the best criterion for implementing and selecting the appropriate model, the model with the lowest (FN) is the best choice [46,47,48,49,50].

7.2. Fault Visualization and Feature Reduction

Interestingly, most of the selected model’s prediction had an accurate score across the metrics. There was a need for a further avenue to select the right electrical parameter for the aluminum electrolytic capacitors, hence the feature reduction and fault visualization process using the principal component analysis (PCA). PCA is a technique that helps in reduction from a high dimension to a low dimension in terms of the number of features. The feature selection plot as shown in Figure 9, Figure 11, Figure 13 and Figure 15 had 10, 10, 7 and 7 features selected, respectively. With the aid of the PCA, these features were reduced to two for the fault visualization and feature reduction. PCA 1 and PCA 2 show the label reduction plot for feature 1 and 2, respectively, with references to their Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22. Initially, the training data were deployed on multi layer perceptron (MLP) but demonstrated poor prediction across each electrical signal. However, we deployed its various hidden layers to plot the fault for each electrical signal. The MLP classifier had hidden layer sizes in the range of 50, 1, and 1,2,3 as shown in the Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22.

Table 6. Global performance comparison of ML models using capacitance (Cs) features.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1-Score (%)	Cost (s)
DT	100.00	100.00	100.00	100.00	0.3267
KNN	99.8611	99.59	99.58	99.58	0.2933
RDF	100.00	100.00	100.00	100.00	15.2800
SGD	99.5833	99.59	99.58	99.58	0.6133
NB	100.00	100.00	100.00	100.00	0.2200
LR	78.7500	81.26	80.83	80.02	0.2933
GB	100.00	100.00	100.00	100.00	43.8567
SVC	100.00	90.53	89.78	89.28	0.5133
Adaboost	99.1667	100.00	100.00	100.00	9.1000

shows the classification report for the capacitance dataset which includes the accuracy, precision, recall and f1 score. Interestingly, observations from the classification report show the decision tree (DT) with 100% accuracy across all the metrics with a computational cost (0.3267 s). Next, with the same performance was the random forest (RDF) with 100% accuracy across the metrics but with a higher computational cost (15.2800). The random forest model is an extension of the decision tree as it consists of many decision trees combined together. This and many other factors could have prompted the high computational cost compared to the decision tree. The k-nearest neighbors and stochastic gradient both had close metrics, with scores of 99.8611%, and 99.5833% and 0.2933 s, and 0.6133 s, respectively. Another outstanding model is the naive Bayes (NB) model with 100% prediction across the metrics and with a computational cost of 0.2200 s. The model with the lowest accuracy represented logistic regression (78.7500%) and a computational cost of 0.2933 s.

Gradient boost (GB) is a better model than Adaboost as it is more robust to an outlier. Hence, the GB presented a score of 100.00% across its metrics but with a higher computational cost (43.8567 s), while Adaboost had 99.1667% accuracy and 100% performance across other metrics but with a lower computational cost (9.1000 s). Overall, with an exceptional prediction across most of the models, it is not enough to decide if the capacitance has the right paradigm to be selected as a diagnostic tool for SMPS-AEC. Hence, there is a need for fault visualization using principal component analysis to reduce the input to two variables. Figure 18 shows the fault visualization plot among the three electrolytic capacitors found in SMPS using the capacitance as an input parameter. It can be seen from the plot without doubt that it is not a good variable for the condition monitoring of the SMPS-AEC.

Table 7. Global performance comparison of ML models using equivalent series resistance (ESR) features.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1-Score (%)	Cost (s)
DT	99.86	100.00	100.00	100.00	0.3200
KNN	100.00	99.21	99.17	99.17	0.3133
RDF	100.00	100.00	100.00	100.00	15.2933
SGD	99.44	100.00	100.00	100.00	0.4333
NB	99.58	100.00	100.00	100.00	0.2200
LR	94.02	95.29	95.14	95.13	0.3067
GB	99.44	100.00	100.00	100.00	55.8500
SVC	100.00	100.00	100.00	100.00	0.4000
Adaboost	99.44	100.00	100.00	100.00	9.3600

shows the classification report for the supervised learning models using the equivalent series resistance (ESR) dataset. Interestingly, ESR was identified as a good diagnostic method that showcases the deteriorating behaviour of an electrolytic capacitor. However, there is a need for a new approach to improve the condition monitoring performance by introducing new ways of data acquisition and new methodological ways that would involve good interpretability for end users. Outstanding performance was achieved across the models selected, with most of the models exceeding 95% accuracy. The decision tree (DT), k nearest neighbors (KNN), stochastic gradient (SGD), naive Bayes (NB), and support vector classifier (SVC) had a computational cost of less than 0.5 s while random forest (RDF), gradient boost (GB), and Adaboost had a higher computational cost (15.2933, 55.8500, and 9.3600), respectively. Overall, the decision of picking a suitable model for use could be made based on computational cost, accuracy and users discretion as all models had good prediction. Furthermore, the fault visualization as shown in Figure 19 indicated ESR as the best diagnostic tool for SMPS-AEC as it was able to discriminate between the three capacitors with capacitor 1 (2200 uf-blue), capacitor 2 (1000 uf-green) and capacitor 3 (470 uf-red).

Table 8. Global performance comparison of ML models using dissipation factor (D) features.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1-Score (%)	Cost (s)
DT	98.19	98.41	98.33	98.34	0.3200
KNN	94.30	92.84	92.50	92.53	0.2400
RDF	98.33	98.41	98.33	98.34	15.1467
SGD	90.97	84.23	75.14	69.50	0.8067
NB	73.61	73.75	73.69	73.68	0.2267
LR	72.22	65.99	60.97	54.89	0.2533
GB	99.72	99.19	99.17	99.17	46.2767
SVC	89.58	84.81	84.44	84.49	0.7333
Adaboost	96.94	96.92	96.19	96.34	8.3533

shows the classification report for the supervised learning algorithms selected to act as a diagnostic assessment for the dissipation factor (D) dataset. It is a common practice during data exploration and analysis to check the dataset for outliers and values. However, there are algorithms with this special characteristics while others require the knowledge of the handler to ensure they are fed with the right dataset to ensure good prediction accuracy. Additionally, some models require the dataset to be normalized or standardized, i.e, rescaling into range of (0, 1) or rescaling to have a mean and standard deviation of 0 and 1, respectively. Overall, the best prediction was achieved with the decision tree (DT) and k nearest neigbors (KNN) model with a 98.19% accuracy and computational cost of 0.3200 s, and 94.30% accuracy and computational cost of 0.2400 s, respectively. The poorest prediction was found with naive Bayes (NB) which presented 73.61% accuracy and logistic regression (LR) which achieved 72.22% accuracy, even though they required a lower computational cost of 0.2267 and 0.2533, respectively. Figure 20 shows the fault visualization using the principal component analysis (PCA) with the dissipation factor dataset. It shows that capacitor 1 (2200 uf) duly separated from the other plane with both capacitor 2 and 3 (1000 uf and 470 uf, respectively).

Table 9. Global performance comparison of ML models using impedance (Z) features.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1-Score (%)	Cost (s)
DT	98.88	99.05	98.89	99.03	0.2933
KNN	99.02	99.18	99.17	99.17	0.2733
RDF	99.44	99.17	99.17	99.17	15.2200
SGD	98.88	98.29	98.19	98.20	0.5533
NB	98.88	99.18	99.17	99.17	0.2467
LR	92.92	91.64	89.03	88.69	0.2800
GB	99.03	98.77	98.75	98.75	39.1967
SVC	99.17	98.91	98.89	98.89	0.3667
Adaboost	98.19	98.20	98.18	98.19	7.9267

shows the classification report of the selected supervised algorithms for the impedance dataset. All algorithms had outstanding prediction in terms of accuracy, precision, recall and f1-score. In terms of the computational cost, the KNN with an accuracy of 99.02% and computational cost of 0.2733 compared to the random forest model with an accuracy of 99.44% and computation cost of 15.2200 s. The NB, SVC and SGD model also had lower computational costs (0.2467, 0.3667, 0.5533 s, respectively) while Adaboost and gradient boost (GB) had higher computational costs (7.9267, 39.1967, respectively). Figure 21 shows the fault visualization plot for the impedance dataset using the principal component analysis (PCA) to reduce the features to two. The electrolytic capacitors were well grouped in their various planes with blue representing 2200 uf, green representing 1000 uf and red indicating 470 uf.

Table 10. Global performance comparison of ML models using all feature sets.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1-Score (%)	Cost (Secs)
DT	82.27	82.89	81.67	81.30	0.6400
KNN	98.83	95.96	95.94	95.94	0.4133
RDF	99.11	98.47	98.44	98.44	16.0533
SGD	72.67	78.88	73.22	70.93	2.0867
NB	66.72	60.97	52.67	50.36	0.2733
LR	49.89	46.52	43.22	43.22	0.3933
GB	99.89	99.89	99.89	99.89	103.2333
SVC	82.83	83.62	81.17	81.51	4.5533
Adaboost	77.39	74.65	71.22	67.75	12.8000

shows the classification report for the selected supervised learning algorithm using the combination of the overall dataset with input parameters of capacitance, ESR, impedance and dissipation factor. Interestingly, the KNN, RDF and GB models had the best prediction accuracy (98.83%, 99.11% and 99.89%) and computational cost (0.4133, 16.0533, and 103.2333 s), respectively. Figure 22 shows the fault visualization plot for the whole dataset and significantly could not discriminate within the dataset. Looking closely at the visuals, it can be seen that there are less clusters in red and green and more blue clusters. This can can be attributed to the statistical feature extraction and selection process as most discriminating features were dropped during the feature selection process. Hence, the approach cannot be recommended for use even though some of the algorithms had good accuracy values. Figure 23 shows the accuracy bar plot of the single features approach with the capacitance in blue, equivalent series resistance in yellow, dissipation factor in green, impedance in red, and all parameters in purple. Overall, it can be seen from the plot that the three supervised algorithms, namely KNN, RDF, and GB, had consistent accuracy in comparison.

Additionally, in terms of computational cost, the KNN model along with its consistency had the lowest average value of 0.32 s while the GB model had an average of 57.68 s but with a more consistent accuracy among the five parameters. Although it is a reliable place to start, classification accuracy frequently faces issues in practice. The fundamental issue with classification accuracy is that it obscures the information required to fully comprehend how well a classification model is performing. With three or more classes, an accuracy of classification of 80% may be achieved, but it is difficult to determine if all classes are predicted equally well or if the model is neglecting one or more classes when the number of classes in the data is not evenly distributed. If 90 out of every 100 records belong to one class, an accuracy of 90% or more may be reached, but this is not considered a good score until the value of the most prevalent class is consistently anticipated. Classification accuracy may obscure information required to assess a model’s effectiveness. Fortunately, a confusion matrix can help in such circumstances. A method for summarizing a supervised learning classification algorithm’s performance is called the confusion matrix. If a dataset has more than two classes or if each class has an unequal amount of observations, classification accuracy alone may be deceiving. A better understanding of the categorization model’s successes and failures can be obtained by calculating a confusion matrix. Figure 24 shows the confusion matrix for the equivalent series resistance (ESR) dataset due to its ability to provide good prediction across all the supervised learning algorithms, including the decision tree model (DT), k nearest neighbors (KNN), random forest (RDF), stochastic gradient (SGD), naive Bayes (NB), logistic regression (LR), gradient boost classifier (GBC), support vector classifier (SVC), and Adaboost classifier.

8. Conclusions and Future Works

This paper presented the integration of statistical feature selection and filter-based feature selection in the fault diagnosis of switched-mode power supplies of aluminum electrolytic capacitors (SMPS-AEC). In our study and data acquisition process, we considered capacitance, ESR, dissipation factor and impedance as a condition monitoring approach, thereby providing a platform to choose which of the electrical signals provide the right paradigm for an aluminium electrolytic capacitor. Indeed, this CM method consumes less time in terms of data acquisition and data analysis than other methodologies used in the past. Furthermore, the devices tested during the experiment were regarded as non-destructive or offline as there are no proven ways yet to assess the reliability of electrolytic capacitors in circuit. Additionally, this study was a continuation of a previous study on condition monitoring for electrolytic capacitors with a range 1 MHz while this new study is with a long-term frequency range of 8 MHz. Overall, improved performance was achieved across the models for each parameter during the increase in data capacity for training, testing and validation compared to using the 1 MHz frequency testing.

The concept of supervised learning algorithm has shown better performance across the electrical signals acquired during the experimental setup. In terms of consistency, the KNN algorithm outperform other algorithms with the least computational cost (0.32 s), accuracy (98.40%) across the electrical signal. The RDF algorithm had good prediction across the electrical signals with an average accuracy and computational cost of (99.376%), (15.40 s) respectively. The GB algorithm had good prediction across the algorithm and electrical signal but with a higher computational cost (57.68 s) and a average accuracy of (99.616%).

The ESR parameter had better prediction across the supervised learning algorithms selected and also from the fault visualization plot standpoint, it was able to discriminate among the capacitors. It can be regarded as the right paradigm for diagnostics of SMPS-AEC. From this perspective and due to the long-term capability of the electrolytic capacitors, the 8 MHz long-term frequency-based approach can be used for condition monitoring/diagnostics approaches in SMPS-AEC. For future studies, we will explore a temperature-dependent approach for the condition monitoring of SMPS-AEC as this was performed at room temperature.