1. Introduction

In the context of electrical engineering, it is crucial to maintain the reliability and availability of traditional electrical systems while ensuring a secure and stable network. To achieve this, an effective maintenance strategy involves monitoring the health of components using simple and cost-effective analytical methods. Researchers have developed health monitoring schemes for various applications, including electric machines ⁽¹⁾ and power devices ⁽²⁾.

More recently, these techniques have been applied to assess the condition of electric capacitors due to their high failure rate compared to other electrical devices ^(3-6). Capacitor aging, which results in changes in capacitor parameters over time, is primarily caused by thermal, electrical, and chemical factors in metalized polymer-film capacitors. Capacitors have the highest rate of component degradation ^(7,8); most systems' commonly used capacitor types are high-capacitance electrolytic capacitors (Al-Caps), and metalized polypropylene-film capacitors.

The major mechanisms of degradation of different capacitors are different. Increasing thermal pressure and ambient temperature are the main reasons for the degradation of Al-Caps, which cause continuous vaporization of the electrolyte in the capacitor, consequently resulting in a substantial decrease of the capacitance and escalation of the equivalent series resistance (ESR). Therefore, the capacitance and ESR are suitable measures of aging in Al-Caps ⁽⁹⁾. The principal mechanism of degradation for metalized polypropylene-film capacitors, also known as Film-Caps, is high-voltage stress, which typically results in a reduction of capacitance and a simultaneous increase in the equivalent series resistance (ESR). However, due to the minimal changes in ESR, capacitance is often considered as the degraded feature.

All the aforementioned techniques necessitate additional hardware or complex procedures. With the advancement of technology, artificial intelligence (AI) models have gained popularity and can offer potential solutions for fault detection, including the identification of arc faults ^(10-15). Several sophisticated models, such as neural networks and adaptive neuro models, have been utilized to assess the condition of Al-Caps ^(16-20). The adaptive neuro algorithms distinguish the aging burden of Al-Caps based on the elderly links amongst the factors and actual features using curvature fitting methods. This method could study the health state from the response data caused by the capacitor's usual and aging fault conditions. It is important to note that Al-Caps and Film-Caps have different characteristics, and most research has focused on Al-Caps. In contrast, research on parameter estimation for Film-Caps, particularly based on AI algorithms ⁽²⁰⁾, is still limited, and a comprehensive estimation strategy has yet to be developed. Therefore, there is a need for further research on film capacitors. The present study proposes a condition monitoring strategy that employs frequency signal analysis to assess the health of film capacitors in a three-phase AC-DC converter. The study uses the discrete wavelet transform (DWT) to analyze the capacitor current, which is then normalized by indexes and used as input for the learning algorithms. In addition, capacitor voltage, as well as output current, and output voltage are examined utilizing the DWT and fast Fourier transform (FFT) for comparison. In this study, various indexes including root-mean-squared value, variance, average, and median, are utilized as inputs for artificial intelligent models to investigate factors affecting film capacitors. Eight learning algorithms are implemented to monitor the health status of film capacitors. The results show that utilizing the discrete wavelet transform combined with indexes for capacitor current yields a high accuracy of approximately 99.85%. The subsequent sections of this manuscript are organized as follows. Section 2 outlines the characteristics of film capacitors. Section 3 describes the AI models and the configuration of input parameters for the models. Section 4 evaluates the estimated parameters of the AI models using various input configurations. Finally, Section 5 provides recommendations for capacitor diagnosis and suggests avenues for future research.

2. Properties of Film Capacitors

fig. 1illustrates the typical degradation curve of a capacitor, where the capacitance value decreases with time. For Film-Caps, the capacitor is considered to have reached its end-of-life (EOL) when there is a 2% decrease in the capacitance value. A corresponding circuit of a Film-Cap is displayed in fig. 2. The impedance of the capacitor is given as

The main degradation mechanism for Film-Caps is high-voltage pressure, which leads to a reduction in capacitance and typically an increase in equivalent series resistance (ESR). However, since the variation in ESR is usually insignificant, capacitance is typically chosen as the primary degraded factor. The C value is evaluated by the relationship of capacitor voltage and current as follows:

Fig. 1. Degradation curve of the film capacitor

Fig. 2. Simplified circuit of a film capacitor

Fig. 3. Three-phase inverter system

3. Artificial Intelligent Models and Signal Analysis

3.1 TiArtificial Intelligent Modelstle

This study utilized eight distinct learning algorithms for estimating the capacitance values of Film-Caps. These algorithms include Gaussian decision tree (DT), process regression (GPR), linear regression (LR), support vector machine (SVM), ensemble learning regression (ELR), gated unit current (GRU), deep neural network (DNN), and long short-term memory (LSTM). ELR utilizes a set of models that employ learning advancements to an identified challenge. The set of models is integrated to attain the ultimate estimate. Ensemble regression goals to link various models to boost the predictable exactness using an objective mathematical function ⁽²¹⁾. The GPR is a probabilistic supervised machine learning (ML) that is regularly used for category and regression purposes. The GPR algorithm produces predictions by participating the abovementioned statistics ⁽²²⁾. The SVM reveals the greatest borderline, differentiating components into identifiable classes ⁽²³⁾. The SVM function used to predict new values is given by

(4)

$f(x)=\sum_{n=1}^{N}\left(a_{n}-a_{n}^{*}\right)(x_{n}^{'}x)+b$

where $x$ is the observed data, $a_{n}$ and $a^{*}_{n}$ are Lagrange multipliers, N is the number of observations, and b is the bias. LR is a linear methodology used to demonstrate the association between one or more expressive variables and a scalar response. The associations are obtained via linear predictor tasks whose unknown parameters are predicted from the statistics ⁽²⁴⁾. The DT formation involves the tree stream diagram, which depicts the entire probable options and outcomes. In addition, this formation shows the guesses following the altered points. The general form of the DT is expressed as

(5)

$\operatorname{Min}_{D T}=\sum_{n=1}^N\left(y_i-\widehat{y_i}\right)^2+\alpha(T)$

where $y_{i}$ and $\widehat{y_i}$ are the actual and predicted values, respectively. T is the node number of the tree, and $\alpha$ is the optimal coefficient. This coefficient controls the tradeoff between tree complexity and accuracy. The source point stands for initiating the DT, and the decisions produced at the "leaves" will achieve the formation" ⁽²⁵⁾. Deep learning (DL) models enable computers to mimic the conventional computational abilities of the human brain, such as learning. To achieve this, a progression model is initiated in DL to train and categorize components using abundant forms of data, such as pictures, scripts, or signals. DL models possess the capability to attain superior accuracy and surpass the cognitive capacity of humans in performing various tasks. A large database of statistics is commonly consumed to educate DL models ^(11,12). The DL model are exercised in this study to guesstimate the capacitance. fig. 5presents the typical structure of a DL algorithm. The entered layout, hidden middle layouts, and output layout are the standard layouts in the DL formation. The middle part comprises several layers. DNN comprises two middle layers, whereas LSTM and GRU have three middle-layer configurations. The middle layers of the GRU, DNN, and LSTM are called fully connected, GRU, and LSTM layers, respectively. The trial-and-error technique is used to elect the hidden layouts. The elected layouts of the DL approaches afford the highest effectiveness among numerous potential layouts. However, other probable layouts that may correspondingly be applicable also exist.

Fig. 4. Experimental waveforms of the investigated three-phase inverter: (a) Al-Cap and (b) Film-Cap waveforms

3.2 Signal Analysis

In signal analysis, the time and frequency domains are commonly used. However, in the case of Film-Caps, the capacitance dominates the frequency spectrum, making it necessary to focus on frequency domain data exploration. This study utilizes both FFT and DWT to analyze the data, followed by estimating the capacitor's capacitance using various indices such as RMS, variance, average, and median. The low-frequency range of the signals and indices are obtained using low-pass filters (LPFs) after FFT. For DWT analysis, the data is decomposed using a Daubechies-type wavelet (db5), with the DWT modal maxima revealing the impulsive variation characteristics of the signal. In general, the FFT and DWT indicators associated with the capacitor are capable indicators of the capacitor monitoring compared to indicators associated with the load.

Fig. 5. Typical DL layout structure

Fig. 6. Film-Caps estimation scheme

4. Artificial Intelligent in Capacitance Estimation of Film-Cap

fig. 6illustrates the guesstimate strategy for the Film-Caps. Capacitor and load currents and voltages are recorded and saved, and then processed using DWT and FFT analysis. Two frequency ranges, LPF1 (1-5 kHz) and LPF2 (5-10 kHz), are used to filter the FFT signals to investigate their impact on accuracy. Index values are then derived from the analyzed signals, resulting in six combinations of inputs. These include raw signals from FFT analysis (FFT-raw), indexes from raw FFT signals (FFT-raw indexes), indexes from LPF1 (LPF1&indexes) and LPF2 (LPF2&indexes) of the FFT analysis, raw signals from DWT analysis (DWT-raw), and indexes from DWT analysis (DWT-indexes). The training and testing data are split equally in terms of sample size. The best technique is selected based on having the lowest error (highest accuracy) for all input combinations compared to the actual parameter values. The errors are reported as follows:

Fig. 7. Average errors of the estimated capacitances for Film-Caps using DWT analysis with and without indexes

fig. 8displays the mean errors in the estimated capacitances using FFT analysis with and without filters and indexes. LPF1 & indexes exhibits the best overall accuracy among the four input combinations, whereas FFT-raw has the lowest overall accuracy. The other input combinations provide average performances. fig. 9illustrates the comparison between the two LPFs (LPF1 and LPF2). LPF1 offers higher accuracy than LPF2; this could be explained by the fact that LPF1 has a low-spectrum frequency range compared with LPF2. Hence, the unique characteristics of the capacitance are more visible and detectable than using LPF2. However, the differences in the overall errors are insignificant.In fig. 10, the performance of DWT and FFT analyses is compared. It is observed that when ML techniques are used, the FFT analysis yields lower errors compared to DWT analysis, whereas the DWT analysis performs better with DL techniques. However, the difference in error between DWT and FFT analyses with ML techniques is not significant compared to DL techniques. The DL techniques, especially DNN and GRU, show better performance than ML techniques for both DWT and FFT analyses. Additionally, DWT analysis does not require filters, unlike FFT analysis, which reduces the complexity of the monitoring system.In figure 11, it can be observed that input signals related to the capacitor provide higher accuracies compared to load-related inputs for both DWT and FFT techniques. Specifically, the capacitor current input yields excellent performance with about 99.85% accuracy when DWT-indexes are employed, whereas the accuracies are lower when using raw signals. The measurement of capacitor current can be achieved directly or indirectly. The direct method involves introducing additional current sensors, which can be costly and challenging to implement when dealing with a large number of capacitors. On the other hand, the indirect method involves obtaining the capacitor current using converter correlation and switching states, which is a simple and low-cost technique.In fig. 12, a comparison of the performances of the eight AI models is presented. It is shown that DL models,

Fig. 8. Average errors of the estimated capacitances for Film-Caps using FFT analysis with and without filters and indexes

Fig. 9. Average errors of the estimated capacitances for Film-Caps with FFT analysis, indexes and filters

Fig. 10. Average errors of the estimated capacitances for Film-Caps with DWT and FFT analyses

Fig. 11. Average errors of the estimated capacitances for Film-Caps with different input signals

5. Comparison of Conditions Monitoring Results of Film Capacitors and Electrolytic Capacitor in DC to AC Converters

A comparison procedure is implemented for Al-Caps ⁽²⁰⁾ and Film-Caps; however, this study mainly focuses on the health monitoring of Film-Caps. Thus, other comparisons of Al-Caps are omitted herein; the performances regarding the input signals and learning algorithms are illustrated in fig. 13. In addition, the ranges of LPF1 (100–1000 Hz) and LPF2 (1–3 kHz) for Al-caps differ from those of the film-caps. This difference was intended to investigate the effects of the imbalance ranges of the LPFs on the estimated capacitances. For Film-Caps, the capacitor signals and DWT analyses outperform those of the load signals and FFT analyses, respectively. Additionally, the performing of LPF1 is outstanding compared with that of LPF2. In ⁽²⁶⁾, the odd harmonics are injected into a PV system to guesstimate the capacitances. These injected signals were placed in the LPF1 scale, meaning that the vital characteristics of the capacitance were located in the low-frequency range. This is the reason why the precision of LPF1 is greater than that of LPF2. The general errors of DL models are lower than those of ML models. LSTM shows the lowest overall error among the eight learning algorithms. The hidden structures of DL techniques for Film-Caps and Al-Caps are different; thus, LSTM is the best technique for Al-Caps, whereas GRU is the most suitable method for Film-Caps. Additionally, the differences in the waveform shapes (fig. 4), which were reflected as the magnitude differences in the frequency spectrum, maybe another critical factor for the differences in the learning algorithm performances.

Fig. 12. Average errors of the estimated capacitances for Film-Caps with different learning algorithms

Fig. 13. Performance comparisons of the estimated capacitances between Al-Caps ⁽²³⁾ and Film-Caps with different inputs and learning algorithms: (a) DWT+indexes; (b) LPF1+indexes; (c) LPF2+indexes; (d) comparison of learning algorithms.

6. Conclusions

The present study aimed to evaluate the accuracy of capacitance estimation for Film-Caps using a combination of capacitor current signal, DWT analysis algorithms, and indexes. Due to the dominant presence of capacitance in the frequency spectrum, the study focused on exploring data in the frequency domain. The results showed that DWT analysis outperformed FFT analysis in terms of estimated accuracy. The appropriate filter range for FFT analysis was found to be between 100 Hz to several kHz. The comparison between Al-Caps and Film-Caps demonstrated the effectiveness of low-frequency filters. However, the DWT approach did not require filters to extract important features, thereby reducing the complexity and estimation time. Additionally, incorporating capacitor-related signals into the inputs of AI models resulted in lower errors in capacitance estimation. The current flowing through a capacitor can be measured either directly or indirectly. A direct method involves installing additional current sensors, which can be costly and challenging to implement when the number of capacitors increases. An alternative method is to use the converter correlation and switching states to obtain the capacitor current indirectly, which offers a simple and low-cost solution. The DWT technique combined with indexes for capacitor current estimation achieves an excellent accuracy of approximately 99.85%. All AI models exhibit poor accuracy when utilizing load current and voltage data due to various factors, including switching noise, nonlinearity, and signal degradation during conduction in multi-phase networks. DL models outperform ML models due to their increased number of neurons and layers. This results in superior handling capabilities of DL models compared to ML models, resulting in lower general errors, regardless of whether DWT or FFT is used. One challenge when utilizing DL models is selecting the appropriate hidden layer configuration, which may require a trial-and-error approach, resulting in additional time to determine the optimal configuration.