An Entropy–Envelope Approach for the Detection and Quantification of Power Quality Disturbances

Abstract

The importance of power quality has increased these days due to the growth in the use of renewable energies and nonlinear loads. Although the use of renewable energies provides power generation sources that help in reducing greenhouse gas emissions, they might have a detrimental effect on the power quality due to their intermittency and dependence on weather conditions. Due to the importance of keeping an optimal power quality, in this work, a novel methodology is developed whose main contribution relies on the use of entropy features and envelope analysis for the detection and quantification of power quality disturbances. The proposed method is implemented within a machine learning framework, where linear discriminant analysis (LDA) is employed to optimize entropy-based features. Subsequently, a neural network classifier performs an automatic classification and quantifies the magnitude of affectation associated with grid disturbances. The training is performed using synthetic signals, and validation is conducted with real signals from a photovoltaic park and from an IEEE working group. The results obtained are compared with those provided by other methodologies proving the accuracy and the viability of the proposed approach.

1. Introduction

Conventional power generation has raised concerns due to its contribution to climate change because of the need to burn fossil fuels. This type of generation represents up to 80% of the total energy generated on the planet. To address this situation, the use of renewable energy power generation has increased, and it is expected that, by 2050, 30.9% of the power consumption worldwide will come from renewable energy systems [1]. Photovoltaic (PV) generation is one of the most widely implemented technologies; however, it presents an intermittency in power generation due to weather conditions that can cause disturbances that compromise power quality (PQ) [2], potentially damaging grid-connected loads. The PQ term refers to shape deviations in power signals compared to a pure sinusoidal wave, as defined by IEEE 1159-2019 [3]. In terms of distributed generation grids, some of the most common power quality disturbances (PQDs) are sag, swell, fluctuations, harmonics, and impulsive or oscillatory transient events. In this regard, several techniques have been developed for the identification and classification of PQDs.

Classical techniques use a time, frequency, or time–frequency transform to extract relevant information regarding the existence of a particular disturbance. For frequency analysis, the most used technique is the Fourier transform (FT). Although this technique provides satisfactory results, it is limited because it only provides amplitude and frequency information, and it lacks data about the behavior of the signal in time. Moreover, the FT is only effective for the detection of stationary disturbances, and it fails when transient events appear, as seen in [4]. To overcome these limitations, other specialized techniques are being used that provide time–frequency analysis, such as Empirical Mode Decomposition (EMD) [5], the Wavelet Transform (CWT) [6], the Hilbert–Huang transform (HHT) [7], and Improved S-transform (IST) [8], among others. Due to their ability to perform an adaptive time and frequency trace, these methods are suitable for analyzing not only stationary issues but also transient, nonlinear, and non-stationary signals. While these techniques can offer a comprehensive analysis of signals in a time–frequency manner, they can present some drawbacks like the need for a complex and problem-specific configuration, as well as the existence of mode mixing. Nevertheless, the HHT has some characteristics that make it desirable for the identification of PQDs, as shown in [7]; for instance, it can detect variations in the instantaneous power as well as performing a signal envelope tracking that makes the detection of amplitude-related disturbances easier.

All the aforementioned techniques are able to extract features and patterns helpful for detecting the appearance of a disturbance; nevertheless, they cannot perform a classification task by themselves. In this sense, machine learning (ML) is one of the preferred solutions. One work that focuses on ML is described in [9], where it applies a complex image processing derived from the spectrograms of signals generated using CWT, yielding good results. In this regard, the use of ML-based methodologies has several advantages in automating the entire process, minimizing the human factor. Yet it requires great computational resources to operate. To reduce the computational burden, the work developed in [10] proposes an approach based on a three-step classification process using the Intrinsic Mode Functions (IMF) obtained by means of the EMD and reducing the data amount through a linear discriminant analysis (LDA) to simplify the classification of the PQD. Additionally, the work in [11] implements a four-step methodology involving the generation of PQD classes, feature extraction, the selection of highly discriminating features, and then the classification of PQDs using the selected features, which are mostly the necessary steps for carrying out any ML-based methodology. These approaches have an advantage in terms of the amount of data needed to develop a model, as well as the computational burden for its implementation. Notwithstanding, it is important to notice that, as with any ML approach, they rely on the extraction of features to characterize the disturbance behavior. In this sense, most of the work reported so far use statistical features due to their great ability to model the behavior of a system. However, they are prone to fail in noisy environments. Thus, the use of different features to track the PQD behavior may be useful. In this sense, entropy features have shown promising results in various fields, including biomedicine [12], economics [13], electronics [14], and the detection of failures in mechanical systems [15]. Due to their structure, these features specialize in quantifying the degree of uncertainty in a system. Consequently, they are good at predicting system behaviors with minimal information. This has sparked the interest of several researchers for their application in different fields, making them potential candidates for improving the results delivered by statistical features in the characterization of PQDs.

The main contribution of this work relies on the use of entropy features to characterize both the raw electric signal and its envelope. This allows us to extract information regarding instantaneous power and amplitude variations for the steady and transient state in a single signal. The use of entropy features improves the tracking of PQDs because it can detect irregular behavior even in noisy environments. Additionally, an LDA is performed to reduce the computational effort by discarding non-relevant and redundant information that could be integrated into the database [16]. Also, an artificial neuronal network (ANN) is implemented to perform the automatic classification task. Moreover, the proposed methodology can also quantify the amplitude and duration of the PQD by specifying thresholds in amplitude and time over the signal envelope. The contribution of this work can be summarized as follows:

• We use entropy features to characterize a PQD, allowing us to obtain a good description of the disturbances, even in noisy environments.
• We perform an analysis of the raw electrical signal but also of the signal envelope, enhancing the performance in the detection of amplitude-related disturbances like sag and swell.
• We reduce the computation burden by using a reduced number of features and LDA.
• The proposed methodology can automatically detect, classify, and quantify the disturbance present in the power grid.

The proposed methodology is intended to be a tool for working in grids with constant power fluctuations caused by the nonlinearity of the loads and the use of renewable energies. This way, it is possible to detect the occurrence of a PQD. The proposed approach is validated using synthetic signals for the training and real-world signals provided by a work group from the IEEE and from a real PV plant located in central Spain for the validation. The results prove that the proposed methodology can accurately detect and classify transient and stationary disturbances in real environments and provide the magnitude of the phenomena that marks a difference over other existing similar methodologies, proving its suitability for being implemented in condition monitoring systems. The rest of this paper is structured as follows: Section 2 presents the theoretical background, including the foundational concepts necessary to understand the subject at a broad level. Section 3 details the methodology developed in this study, explaining the implemented process step by step. Section 4 discusses the results obtained through the proposed methodology, along with comparisons to similar works. Finally, Section 5 presents the conclusions of the study.

2. Theoretical Background

This section introduces the main topics that must be known to implement the proposed methodology.

2.1. Power Quality

PQ is a term that refers to any disturbance that deforms the voltage or current signal of any supply grid. This distortion can affect four different aspects: amplitude, continuity, waveform, and frequency. Due to their importance, some of the most studied disturbances are sag (SAG), swell (SWL), flicker (FLC), harmonics (HAR), impulsive transient (IMP), and oscillatory transient (OSC). All these PQDs are defined by the standard IEEE 1159-2019 [3]. This work focuses on these six disturbances, along with the healthy (HLT) condition. Three of these disturbances are directly related to the amplitude of the signal; the SAG represents a loss in the rms amplitude within a range from 0.1 to 0.9 p.u. (per unit) and with a duration greater than 5 cycles in 60 s. The SWL is the increase in the magnitude signal around 1.1 p.u. in terms of the rms value and durations of 5 cycles in 60 s. The FLC is defined as short-term periodic voltage fluctuations in the power supply system. An IMP can modify the magnitude of a signal; it produces changes with unidirectional polarity and occurs in short periods of time. Regarding the disturbance that can modify the frequency of the waveform, the HAR are spurious frequencies that are integer multiples of the fundamental frequency component, and they cause a waveform distortion. Finaly the OSC is a disturbance that modifies the amplitude of the signal with polarity changes in short time periods.

2.2. Hilbert–Huang Transform

HHT is a nonlinear and non-stationary data analysis technique based on EMD. It assumes that the decompositions generated by the EMD signal can be decomposed into IMF, which represents the frequency components of the signal under analysis, allowing for the application of the HHT for both stationary and transient signals, obtaining the instantaneous frequency as a time function with an amplitude distribution of the signal. Thus, by applying the HHT technique, it is possible to calculate the instantaneous frequency and the amplitude (1) that allow us to obtain the characteristics of the signal fluctuations in different time scales that are contained in the signal.

$$x_a\left(\mathrm{t}\right)=F^{-1}\left(F\left(x\right)2U\right),$$

(1)

where $F$ is the Fourier transform, $U$ is the unit step function, and $x$ is the signal data. The amplitude envelope is given by the absolute magnitude of the analytic signal $x_a$.

2.3. Entropy Features

The extraction of features to characterize a phenomenon has been widely used by a variety of disciplines because they describe the behavior of a system in an objective manner, obtaining both quantitative and qualitative information. Entropy features are associated with the measure of the uncertainty of the information; this makes them suitable for describing systems where their characteristics are unknown. Among all the entropy features there are some that have proven to be effective in signal analysis. For instance, the fuzzy entropy (FuzzEn) is a concept used to quantify the ambiguity existing in a system in the face of any change that occurs throughout the signal to be analyzed, since it compares the symmetry between points, finding the difference created by changes in the signal behavior [17], In this context, FuzzEn demonstrates the capability to effectively capture the increase in signal complexity induced by transients or harmonics. On the other hand, distribution entropy (DistEn) takes full advantage of information inference, calculating the vector-to-vector distance created between spaces, thus calculating the probability of the population density, improving the ability to capture the characteristics of nonlinear systems [18]. This makes it ideal for detecting oscillatory transients and voltage flicker, as it can easily distinguish them from a pure sine wave. Furthermore, slope entropy (SlopEn) is an estimator based on symbolic patterns and magnitude information, proving to be an option to obtain patterns regarding the magnitude of a signal [19]; given its ability to rapidly detect abrupt variations in the signal trend, it can effectively identify the occurrence of voltage sags or swells, as well as the appearance of impulsive transients. Additionally, increment entropy (IncrEn) focuses on the detection of increases in magnitude in the signal; being highly sensitive to hidden dynamic changes in the signals, it is particularly effective at detecting discontinuities of the waveform, such as notches, spikes, and high-frequency harmonics. Finally, Renyi entropy (RenyiEn) has proven to be a versatile feature focused on obtaining distribution probabilities, making it possible to evaluate small values against large values in large databases [20]; this property makes it especially suitable for the detection of transient events, enabling a reliable characterization of short-duration disturbances. By employing this diverse set of entropy measures, a powerful and multidimensional feature vector can be constructed. FuzzyEn provides noise resilience, DistEn captures dynamic richness, SlopeEn enables rapid trend analysis, IncrEn focuses on signal smoothness, and RenyiEn offers a tunable sensitivity to both common and rare disturbances. Every one of the features described is computed according to a mathematical model. Thus,

Table 1. Description of the mainly used entropy features.

No.	Entropy Feature	Equation	Description
(2–3)	Fuzzy Entropy	$Ps1={\displaystyle \frac{1}{N-m+1}}{\displaystyle \sum _{i=1}^{N-m+1}}{\varphi }_m^r\left(i\right)$	Ps1 is the average value of the fuzzy similarities for length m, and Ps2 helps to capture the regularity on an extended scale. N is the length of the time series, m is the length of the pattern, and r is the tolerance threshold which determines the scale of similarity of the patterns. ${\varphi }_m^r\left(i\right)$ is the fuzzy similarity function for patterns of length m [21].
		$Ps2={\displaystyle \frac{1}{N-m}}{\displaystyle \sum _{i=1}^{N-m}}{\varphi }_{m+1}^r\left(i\right)$
(4)	Distribution Entropy	$DistEn\left(m,M\right)=-{\displaystyle \sum _{t=1}^M}{P}_t{\log }_2{P}_t$	$m$ indicates the number of dimensions of the matrix and $M$ is the number of histograms, while $P_t$ is estimated as the probability of each $M$ element.
(5)	Slope Entropy	$SlopEn\left(X,m,\gamma ,\delta \right)={\displaystyle \sum _{i=1}^r}{P}_r\ln P_r$	$X$ is the time series to be developed, according to size $m$; $\gamma$ and $\delta$ represent the limits of the partitions for each dimension conformed by $m;$ and $P_r$ represents the total number of patterns vs. times of the number of occurrences.
(6)	Increment Entropy	$IncrEn\left(m\right)={\displaystyle \sum _{i=1}^{{(2R+1)}^m}}p\left(w_n\right)\log p\left(w_n\right)$	$m$ is the order of the function, $R$ quantifies the resolution of the function, and $p\left(w_n\right)$ is the relative frequency for each element evaluated.
(7)	Renyi Entropy	$R_{\propto }={\displaystyle \frac{1}{1-\propto }}{\log }_2\left({\displaystyle \sum _n}{\displaystyle \sum _k}{P}_x^{\propto }\left(n,k\right)\right)$	$\propto$ is the distribution measure, $K$ represents the frequency discrete variable and $n$ the temporal discrete variable, and $P_x$ is the probability of the distribution along the discrete signal.

summarizes all the equations required for the calculation of the entropy features.

2.4. Dimensionality Reduction

Although the use of different features to characterize signals has become very popular in strategies for classification tasks in condition monitoring strategies, the proper selection of the relevant features is complicated. Moreover, it is common to obtain a large feature set that must be reduced to optimize the characterization task. The best-known techniques for performing dimensionality reduction are LDA and principal component analysis (PCA). The performance of each technique depends on the characteristics of the system, as well as the application area. PCA, being an unsupervised technique, focuses on maximizing the variance of the information by generating a new projection from the original information, so it is highly recommended for databases whose behavior is unknown to the user. In contrast, LDA is a supervised technique that focuses on finding the maximum linear separation between classes. This makes it ideal for use in systems with multiple classes and, therefore, recommended to be applied in monitoring methodologies where classes are well known to the user [10]. Therefore, in this work, LDA is used to perform the dimensionality reduction.

2.5. Magnitude of the PQD

Besides detecting the existence of a PQD in a grid, it is also important to quantify it. Each disturbance requires different parameters to be quantified depending on its nature. For example, transient disturbances are measured in terms of the amplitude and duration of the disturbance, whereas steady-state disturbances only require an amplitude value. The amplitude of each disturbance is obtained from the difference between the ideal value and the real one. These parameters can be hard to find over the electrical signal, but, when the envelope is used, the moment where the disturbance starts and finishes as well as its amplitude can be easily identified by establishing thresholds. Regarding the amplitude, the thresholds are set following the definition of the disturbances according to international standards like IEEE 1159 and IEC 61000. That is, in the case of a SAG, there is a temporary reduction in the amplitude below 0.9 p.u. with respect to the nominal value; for a SWL, a temporary increase beyond 1.1 p.u. occurs. On their part, fluctuations correspond to small, random, or periodic variations in amplitude, while in an IMP there is an almost total voltage drop over a short time interval. HAR introduced distortion due to frequency components that are multiples of the fundamental. Finally, the OSC is characterized by a high-frequency transient superimposed on the base signal, whose magnitude is defined by the maximum impulse amplitude relative to the nominal value. The mathematical formulation and the description of the disturbances can be seen in

Table 2. Magnitude definition for every PQD.

No.	PQ	Equation	Description
(8)	SAG	$M_{Sag}=\left(1-{\displaystyle \frac{{\overline{A}}_{sag}}{A_{nom}}}\right)\times 100\%$	The value of ${\overline{A}}_{Sag}$ is the value of the upper envelope, and the $A_{nom}$ is the mean of the upper envelope of the HLT signal.
(9)	SWELL	$M_{SwL}=\left(1-{\displaystyle \frac{{\overline{A}}_{swell}}{A_{nom}}}\right)\times 100\%$	The value of ${\overline{A}}_{Swell}$ is the value of the upper envelope, and the $A_{nom}$ is the mean upper envelope of the HLT signal.
(10)	FLC	$M_{FCL}= {\displaystyle \frac{{\overline{\sigma }}_{FLC}}{A_{nom}}}\times 100$%	The value of ${\overline{\sigma }}_{FLC}$ represents the standard deviation of the upper envelope, and $A_{nom}$ is the mean upper envelope of the healthy signal.
(11)	HAR	$TDH={\displaystyle \frac{\sqrt{{\sum }_{h=2}^H{v}_h^2}}{V_1}}\times 100\%$	Here, $V_1$ represents the RMS value of the fundamental frequency, while $V_h$ denotes the RMS value of the h-th harmonic component obtained through spectral decomposition (e.g., FFT). $H$ indicates the highest harmonic order. The THD expresses the degree of waveform distortion.
(12)	IMP	$M_{imp}=\left(1-{\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}\left({\overline{A}\upsilon }_{imp}\right)}{\mathrm{m}\mathrm{a}\mathrm{x}({A\upsilon }_{nom)}}}\right)\times 100\%$	$M_{1,\mathrm{imp}}$ calculates the percentage reduction in peak voltage by comparing the maximum value of the IMP signal $\mathsf{\Delta }{v}_{\mathrm{imp}}$ to that of the HLT signal $\mathsf{\Delta }{v}_{\mathrm{nom}}$.
(13–14)	OSC	$M_{peak}={\displaystyle \frac{max\left(A_{osc}\right)-max\left(A_{nom}\right)}{max\left(A_{nom}\right)}}\times 100\%$ $M_{RMS}={\displaystyle \frac{{RMS}_{OSC}-{RMS}_{nom}}{{RMS}_{nom}}}\times 100\%$	Here, ${maxA}_{osc}$ calculates the max value in the upper envelope, and $max\left(A_{nom}\right)$ calculates the maximum value of the healthy upper envelope; this gives the $M_{peak}$ peak magnitude affectation. In the case of $M_{RMS}$, it measures the relative change in energy between the OSC and the HLT signal.

.

3. Methodology

This section, presents and describes the methodology proposed for the detection and classification of PQD using the Hilbert–Entropy approach. The analysis conducted in this study involves the fusion of different signal-processing techniques, and it is divided into four different stages, as shown in Figure 1: HHT estimation, entropy feature extraction, data fusion with dimensionality reduction, and ANN-based disturbance classification.

3.1. Envelope Calculation

As the first step in the methodology, the HHT is computed over the voltage signal of the power grid to obtain information regarding the instant power and the envelope of such signal. The amplitude envelope is obtained from the magnitude of the analytic signal. This process emphasizes the amplitude variations in the signal over time, allowing the generation of a data set that highlights the changes in signal magnitude while preserving its original characteristics in both frequency and time. The HHT is applied considering the whole length of the voltage signal; thus, the result is two new signals of the same length as the original one. The data set resulting from the computing of the upper and lower envelope HHT is used along with the raw signal to increase the amount of information regarding a specific disturbance. Therefore, it is expected that the following stages can perform a proper detection in an accurate and easy way.

3.2. Entropy Feature Extraction

In this step, the most representative characteristics of the signal are extracted using entropy features. In this case, six features are calculated: FuzzEn, DistEn, SlopEn, IncrEn, and RenyiEn. It is worth mentioning that the FuzzEn feature delivers two values: the first is the average value of the fuzzy similarities (Ps1), and the second is the regularity on an extended scale of the time series (Ps2). Thus, the six features computed for every signal and envelope are FuzzEn (Ps1, Ps2), DistEn, SlopEn, IncrEn, and RenyiEn. The number of features to be used is important because it determines the length of the feature matrix that is used in the next stage. This feature calculation is applied to both the original voltage signals and the HHT envelope. Also, it is worth mentioning that a large set of signals is required to properly train the proposed method; therefore, in this step, two different matrices with a size of 7n × 6 are obtained, one for the raw voltage signal and another for the envelope signals. The n rows of the feature matrix represent the total number of signals for the seven different conditions that are considered in this work (HLT, SAG, SWL, FLC, HAR, IMP, and OSC). In addition, the six columns of the feature matrix are the six features calculated according to Table 1.

3.3. Data Fusion and Dimensionality Reduction

In the third step, a data fusion and dimensionality reduction process for the obtained data are applied through the LDA technique. Here, the two feature matrices calculated in the previous step are joined together by placing them side by side. As a result, a single $7n\times 12$ matrix is obtained. This matrix is used as input for the LDA technique, which treats the features from the raw signal and the features from the HHT as a single data set and delivers a $7n\times 3$ matrix as an output containing the most relevant features. Since the LDA technique allows us to reduce the amount of information preserving the main characteristics of the original data, the three features delivered by this technique describe the behavior of each one of the disturbances treated in this work. This step is important to reduce the computational burden as well as to improve the results in the classification process, even with a simple classifier.

3.4. ANN-Based Disturbance Classification

The classification process is implemented using a classical ANN-based classifier. The training is performed through a cross-validation methodology, ensuring that the model achieves high accuracy without overfitting. The structure of the classifier is composed of an input layer, two hidden layers, and an output layer. In this neural network, the input layer consists of three input neurons, where every neuron receives one of the three features delivered by the LDA in the previous step. Additionally, the hidden layer is composed of two layers, the first one with ten neurons and the second with twenty neurons, with results sufficient for achieving accurate results. Finally, seven neurons are considered in the output layer, representing each of the PQDs evaluated. The activation functions use the ReLU (Rectified Linear Unit), which introduces nonlinearity and helps avoid vanishing gradients. The optimizer uses L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno), a Quasi-Newton optimization algorithm ideal for small- to medium-sized data sets. One advantage is that it does not require a manual assignment of the learning rate; this parameter is adjusted internally during the training process. Finally, the loss function is cross-entropy loss, which measures how well the predicted probabilities match the true class labels and is well suited for multi-class classification.

3.5. Disturbance Quantification

In cases where the classifier detects the existence of a PQD, the magnitude of the disturbance is calculated. This calculation is performed using a specific formulation configured to each type of disturbance. The amplitude of the disturbance is calculated using the definitions presented in the standard IEEE std. 1159-2019. The values presented in the standard are set as thresholds, and the amplitude of the disturbance is obtained by comparing the amplitude of the envelope in the presence of the disturbance against the amplitude of the envelope in healthy conditions. Moreover, the time when the disturbance starts can be identified by looking at the first sample where the amplitude surpasses the established threshold, and the end of the disturbance is identified by the point where the envelope recovers the normal values.

4. Results and Discussion

The proposed methodology is validated and tested using two different study cases. The first case is performed with the purpose of training the model and performing a first validation using synthetic signals generated by the software. The second case considers real signals from two different sources: a well-known and previously studied set of real signals coming from the IEEE 1159.2 workgroup taken from [22], and signals from a PV generation plant located in Spain.

4.1. Study Case I: Synthetic Signals

Since the training process of every machine learning strategy requires a large data set to present different scenarios of the same condition, a set of synthetic signals are generated by the software. All of these signals are constructed by following the considerations marked by standard IEEE 1159-2019 [3] and the mathematical formulation presented in [22]. The resulting test bench considers seven different conditions: the healthy signal and the six PQDs under study (SAG, SWL, FLC, HAR, IMP, and OSC). These signals are generated with an amplitude of 1 p.u., a duration of 0.3 s, a fundamental frequency of 60 Hz, and a sampling frequency of 8 kHz. Figure 2 shows some examples of the resulting signals for the six PQDs. In this stage, 500 different signals are generated for each one of the seven conditions studied, i.e., the training database is formed by a total of 3500 signals. It is important to mention that, in all cases, the duration, severity, and time of the first appearance of the disturbance are randomly generated; this way, it achieves a set that comprises many different cases of the same disturbance.

Once this database has been generated, the envelope of all the signals is calculated using the HHT to obtain a second set of signals with the same length as the original voltage signal. Since this second database is obtained from the first one, it is expected that both provide meaningful information regarding the condition of the signal. Subsequently, both signals are characterized by means of entropy features; in this case, they are evaluated by six features: the two values delivered by fuzzy entropy (see Table 1), distribution entropy, slope entropy, increment entropy, and Renyi entropy. Therefore, at the end of the feature extraction stage, two feature matrices are obtained, the first one with the features of the original voltage signal and the second with the features of the HHT envelope. Both matrices have a 3500 × 6 length, where the 3500 rows represent the total synthetic signals generated for the seven conditions under study, and the 6 rows are the total entropy features computed for every signal. After these matrices are obtained, it is necessary to perform the data fusion and dimensionality reduction process using the LDA technique. In this aspect, the final data set to be used is formed by concatenating the two feature matrices (original + envelope) in a single 3500 × 12 matrix. The resultant matrix is used as input for the LDA, which is responsible for reducing the number of features preserving the relevant information to describe the behavior of the PQD.

In this work, after applying the LDA technique, the three most representative discriminant features were selected. These components retain 98.29% of the total discriminative variance, indicating that most of the separability information among the seven evaluated classes is preserved within the three-dimensional feature space. This ensures that the dimensionality reduction process maintains the discriminative power of the original 12 features while reducing computational load. The result of applying the LDA to the feature set is shown in Figure 3. It is observed that the technique performs a data clustering where each identified group corresponds with one of the PQD under analysis. A good separation of the disturbances is observed in Figure 3, with a light overlap among the HLT, SAG, and SWL conditions. This situation is explained by the fact that these three conditions are very similar to each other. Although the results shown in Figure 3 depict separated groups, it is important to mention that the purpose of the LDA is not to be a classifier.

As previously mentioned, 500 synthetic signals are generated for every condition under test and 400 (80%) are used for training, whereas 100 (20%) are left for testing.

Table 3. Confusion matrix achieved by NN structure during the test for all assessed conditions, and a classification report.

		True Class							Classification Report
		HLT	SAG	SWL	FLC	HAR	IMP	OSC	Precision	Recall	F1-Score
Assigned Class	HLT	98.2	0.4	1.4	0	0	0	0	0.9592	0.9400	0.9600
	SAG	0.8	99.0	0.2	0	0	0	0	0.9800	0.9800	0.9851
	SWL	2.6	0	97.0	.04	0	0	0	0.9505	0.9600	0.9697
	FLC	0	0	0.2	99.4	0.4	0	0	0.9900	0.9900	0.9849
	HAR	0	0	0	0	99.6	0.4	0	0.9901	1.0	0.9901
	IMP	0	0	0	0	0	99.6	0.4	1.0	1.0	1.0
	OSC	0	0	0	0	0	0	100	1.0	1.0	1.0

shows the confusion matrix for the test process. Some of the signals that exhibit the highest degree of confusion in the classification are HLT, SAG, and SWL. This phenomenon can be attributed to several factors. One of these factors is that the maximum value and the RMS value of the SAG and SWELL conditions can be very similar to those presented in the HLT condition depending on the duration and magnitude of the event. Also, in terms of their amplitude profile, a SAG may exhibit a sharp but brief drop in amplitude, while a mild SWELL presents a sustained elevation, but, in both cases, after the disturbance, the signal becomes HLT. Moreover, if the amplitude of a SAG or SWELL remains close in nominal value, both can mimic the behavior of a healthy signal, especially in systems with dynamic loads or a limited temporal resolution. As a result, although these disturbances are among the most characteristic when visually inspected, they often pose greater challenges for automatic classification algorithms due to the similarities in their profiles. This phenomenon is clearly observed in the projection shown in Figure 3, where, despite an exhaustive feature extraction process, the classes remain closely clustered, highlighting the difficulty in achieving an accurate classification.

For the case of the FLC, HAR, IMP, and OSC conditions, it achieved a 100% accuracy. This is an expected situation because, from Figure 3, it is observed that there is a higher separation among them, i.e., they are easier to identify. Thus, the accurate performance demonstrates that the proposed methodology can detect and classifying various PQDs, making it a valuable diagnostic tool for evaluating power grid quality since, in general, the classification accuracy rounds to 98.8%, as shown in Table 3.

In order to provide a better indicator of the overall performance of the classification task, in Table 3, the precision, recall, and F1-score are summarized for each class. It can be observed that the model generalizes very well, achieving an overall accuracy of 98.43% without showing any strong bias toward any class. All classes obtained F1-score values above 94%, demonstrating a balanced performance between false positives and false negatives throughout the training process. By using the tic-toc function in Python 3.12.11, it is possible to accurately measure the ANN training time, which averaged 28.03 s. However, once the network has been trained, the inference process is carried out in 1 millisecond; since the training is carried out only once, the time that really interests us is the last one, showing that this approach allows us to reduce the computational burden compared to other methods.

This can be compared with the work developed in [11]. Although it shows excellent results, it requires nearly twice as much data for training compared to the proposed methodology. The auto-encoder process presented in [11] is responsible for reducing around 80 different features, generating a considerable computational load during the training phase. Moreover, the authors in [11] report an inference time of 16.5 milliseconds, a higher time than the one reached for the entropy–envelope approach developed in this work. In the case of study [10], which also achieved outstanding results, a complex mathematical process was employed to extract values through the computation of eigenvalues applied to a series of complex IMFs using ICEEMDAN (Complete Ensemble Empirical Mode Decomposition with Adaptive Noise) combined with the application of the LDA technique through adaptive KNN with outlier exclusion. However, as can be inferred, the mathematical and operational procedures required for this approach involved the use of specialized software, such as MATLAB R2025a, which increased the overall computational load needed to carry out the process. Therefore, in the proposed methodology, although some similar procedures were used, the process was optimized in such a way that the tools employed can be implemented using different software platforms, such as Python, without imposing a significant computational load and with relatively fast execution. Consequently, the concept of efficiency can be applied to the proposed methodology when compared to other similar approaches.

To guarantee the correct calculation of the entropy features, some parameters must be properly set. These parameters are different for every feature, and they are described next. The parameters used for the implementation of the FuzzEn were as follows: the embedding dimension $m=2$, which defines the number of consecutive data points used to form vectors for pattern comparison; the time delay $\tau =1$, which determines the spacing between elements in each embedded vector—this time delay indicates that the points are adjacent in time. The tolerance parameter $r\left[0\right]=0.2$ represents the similarity threshold, typically set as a fraction of the signal’s standard deviation, while the shape parameter $r\left[1\right]=2$ of the fuzzy function controls how sharply the similarity decays with distance. Similarly, for DistEn, the selected parameters were $m=2$, $\tau =1$, and the comparison of the number of interval distributions $B=6$, which determines how the signal amplitude is discretized in intervals; the entire signal was used as the window length. The values required for SlopeEN are the slope thresholds, which are set as $\alpha =[0.7,1.4]$. These thresholds define the boundaries for categorizing the slope between consecutive signal points into symbolic patterns; the entire signal may be used as the window length. Similarly, for IncrEn, the values of $m=2$ and $\tau =1$ are selected; the new parameter for this function is the quantization level $q=6$, which defines how the increment values are discretized into symbolic categories; the entire signal is used as the window length. Finally, RenyiEn requires the following parameters: an embedding dimension $m=1$, meaning that the entropy is computed directly on the raw signal without embedding; and the Rényi entropy order $\alpha =4$, which emphasizes dominant patterns and reduces the influence of rare events; the entire signal is used as the window length.

For a better understanding of how the use of several features is preferred over the use of a single feature, a comparative analysis was carried out. This analysis consists of using each entropy feature individually as an input for the neural-based classifier and comparing the results achieved by each feature itself against the results delivered by the fusion of all the features and the LDA for dimensionality reduction. The results from this analysis can be observed in

Table 4. A comparative analysis showing the individual performance of different entropy features.

		F1-SCORE
		FuzzEn (Ps1)	FuzzEn (Ps2)	DistEn	SlopeEn	IncreEn	RenyiEn	Proposed
Assigned Class	HLT	0.3636	0.6552	0.0000	0.7280	0.2895	0.7901	0.9600
	SAG	0.8776	0.8586	0.3030	0.7861	0.7162	0.6502	0.9851
	SWL	0.9314	0.9366	0.2196	0.3478	0.6911	0.2336	0.9697
	FLC	0.7042	0.7909	0.0198	0.5381	0.4593	0.5283	0.9849
	HAR	0.9694	0.9694	0.0571	0.5899	0.6173	0.9950	0.9901
	IMP	0.8475	0.9014	0.5929	0.8070	0.9901	0.6111	1.0
	OSC	0.9851	0.9851	0.6755	0.7570	0.9800	0.9851	1.0
	ACCURACY	0.8257	0.8743	0.3143	0.6586	0.6929	0.7114	0.9843

. It is noticed that each entropy feature achieves different levels of performance when they are individually used for identifying different PQDs. FuzzEN reaches a value of 0.8743 for the F1-Score. This is the highest value reached by a single feature. However, when all the features are fused by the LDA, the overall performance shows a 0.9843 accuracy, demonstrating that it is better to use all the features instead of only one. Moreover, some features, like DistEn, present a low performance when used individually; however, when they work along with the other features, it is possible to obtain a better overall performance. In this sense, it can be inferred that the high performance of the proposed methodology is only possible through the combination of the distinct contributions of the various entropy features.

4.2. Study Case II: Real Signals from the IEEE Working Group and the PV Generation Plant

To demonstrate the effectiveness of the proposed method under real situations, some signals directly taken from the IEEE 1159.2 working group database have been studied [22]. The signals selected from this set have a 0.3 s duration, fundamental frequency of 60 Hz, and sampling frequency of 8 kHz. IEEE 1159 emphasizes that the sampling frequency must be selected based on the specific phenomenon under investigation. A rate of 8 kHz is suitable for general power quality analysis [3]. Regarding the signals obtained directly from the IEEE study group, it is not possible to know how the instrumentation and measurement procedure was performed. However, this data set has been widely used in previously reported works, where it has been demonstrated that they provide relevant and useful information, and no signs have been found of measurement errors. Moreover, they belong to a prestigious international organization, so it suggests that all the signals were acquired following proper procedures to avoid measurement errors or external noise interference. Regarding the signals from the photovoltaic facility, it is important to recall that they were acquired using a proprietary DAS. In this system all the components were carefully selected to accomplish the requirements established by the IEC and IEEE standards for the measurement of electric signals in power quality applications (sampling frequency, ADC resolution, electromagnetic protection). Moreover, the system was calibrated using the FLUKE 435 Series II Power Quality Energy Analyzer, with a tolerance below 0.1%. This way, it is possible to guarantee that the measurement error always remains low. Moreover, it must be mentioned that, although the noise levels present in the analyzed signals were always relatively low, one of the benefits of using entropy features is the high performance, even in noise conditions. This is one of the reasons why entropy features were selected, and they proved to be effective for the characterization of PQD. To complete this second study case, we considered signals from a PV generation plant located in south-central Spain. The PV system generates a total of 30 MW (Watts). Nevertheless, the measurements are made for a three-phase PV inverter with a rated power of 100 kW and a nominal voltage of 230 $V_{rms}$ at 50 Hz. The measurements are taken on the AC side of the PV inverter; that is the device responsible for supplying power to the loads connected to the grid. The signals are collected and stored using a proprietary data acquisition system (DAS) built using field-programmable gate array (FPGA) technology. This DAS is capable of simultaneously capturing the voltages and currents from the three phases at a sampling frequency of 8 kHz, with a 12-bit resolution. All the signals are normalized to obtain a p.u. standard representation before being analyzed by the proposed methodology, as shown in Figure 4.

Then, by using the HHT–entropy approach, it was possible to detect the six PQDs previously discussed under real scenarios. In Figure 4a, an impulsive transient appears during the second cycle of the signal. It is worth mentioning that this disturbance was detected using the proposed approach, and it was able to detect it despite the short duration and severity of the disturbance. In addition, Figure 4b shows a swell condition.

It is observed that, from the beginning of the signal, the threshold of 1.1 p.u. (red dotted line) established by the standard IEEE 1159-2019 is surpassed. After three cycles, the amplitude of the signal decreases and the swell condition disappears, but the disturbance returns during the last two cycles of the signal. Figure 4c,d show voltage fluctuations and harmonic distortion, which are two stationary conditions. The fluctuation of the signal amplitude is notorious in Figure 4c, whereas the signal in Figure 4d is clearly distorted due to the presence of harmonics.

On the other hand, Figure 4e shows a severe SAG condition. In this figure the red line indicates the 0.9 p.u. threshold that determines the existence of a SAG. It is observed that the signal remains below this threshold almost all the time, and it only recovers the healthy condition during the last three cycles. In fact, two different severities can be observed: from 0 to 0.12 s the 50% attenuation is observed, and from 0.12 to 0.23 s the severity reaches a value around 60%. In both cases, the sag severities are high, but the merit of the proposed methodology relies on the fact that it allows the automatic detection of the disturbance without a visual inspection. Finally, Figure 4f depicts an oscillatory transient with a duration of almost half a cycle.

Transient disturbances are usually hard to detect because of their short duration, and the proposed methodology can detect them properly, proving that it can be considered as a tool to cope with renewable generation systems to increase their reliability by allowing them to detect disturbances when they appear. It must be mentioned that the conditions presented in Figure 4a–d are conditions taken from the database of the IEEE 1159.2 workgroup, and the images from Figure 4e,f represent signals from the PV plant.

Here, it is important to say that the IEEE signals were measured in a 60 Hz grid, whereas the signals from the PV plant were measured in a 50 Hz grid. In this sense, one of the advantages of the proposed approach lies in the fact that it can be used for the monitoring of PQD in any power grid, regardless of its fundamental frequency and nominal voltage.

To validate the effectiveness of the proposed approach, a comparative analysis is performed with works previously published in the related literature.

Table 5. Comparison of the performance of the proposed methodology against other approaches.

Ref.	PQD	Data Type	Technique	Advantage	Dis-Advantage	Total Performance %
[7]	Transient, sag, swell, harmonics, notch, flicker, DC offset.	Synthetic/Emulated	EMD/HHT/ Random Forest Technique	Higher learning speed; superior classification accuracy.	It is necessary to process the magnitude response, frequency response, phase response, and Hilbert energy spectrum for a good response of the methodology.	97.40%
[9]	Normal, impulsive, interruption, oscillatory, sag, swell.	Synthetic	CWT/2D-CNN	High precision, time, and frequency information that characterizes several PQ issues.	Network training time is extensive and requires lots of computational power.	96.67%
[10]	Normal, interruption, swell, sag, flicker, harmonics, oscillatory transient.	Synthetic/Emulated	EMD/Feature extraction/LDA/K-nearest neighbor	Robustness and high accuracy even under different conditions of noise.	IMFs will affect the classifier’s accuracy, such as noise and pseudo. At the same time, the overall computation time will increase.	97.40%
[11]	Normal, sag, swell, interruption. flicker, harmonics, oscillatory transient.	Synthetic/Real	FFT/EMD/NN	Robustness and high accuracy even under different conditions of noise.	It is necessary to tune of the corresponding hyperparameters.	99.47%
Proposed	Normal, sag, swell, flicker, harmonics, impulsive, oscillatory.	Synthetic/Real	HHT/Entropy/LDA/NN	Higher learning speed; superior classification accuracy.	Some of the techniques applied may be difficult to apply in systems with low processing power, making them difficult to apply in microprocessors.	98.8%

summarizes the findings from previous research, specifying the conditions assessed and the classification techniques applied. The results presented in this comparative analysis highlight the different approaches for the detection and classification of PQDs; each one demonstrates excellent outcomes. A common feature among these methods is the use of advanced signal analysis techniques, such as EMD or CWT, as an initial or secondary step in signal decomposition to ensure accurate processing. Although these techniques yield accurate results, a significant disadvantage is the computational load they impose. This is due to the need to decompose the data into IMF, which substantially increases the volume of data to be analyzed. While this additional data may initially seem trivial, it is important to note that, in power quality analysis, an 8 kHz sampling rate is required, resulting in a considerable data volume. Consequently, the approach proposed in this study demonstrates comparable results while addressing these computational challenges. In this sense, the envelope obtained by means of the entropy features and the envelope calculation is very helpful for isolating the disturbance facilitating the detection and quantification of the affectation. Moreover, entropy features are designed to measure the level of uncertainty in a signal; since PQDs cause unexpected sudden changes in the signal behavior, these features provide an effective characterization of the disturbances. The proposed methodology allows for detecting the existence of a specific PQD and its magnitude, and therefore it is possible to determine the cause of the disturbance and then to perform a cause–effect correlation; this will allow future works to mitigate the occurrences of specific PQDs. Finally, it is important to mention that the proposed methodology can detect the disturbance, but it cannot provide the time when it occurs. Notwithstanding, by using short windows for analysis, it is easy to visualize the moment when disturbance arises.

Finally, Figure 5a presents the affectation magnitude computed through the proposed methodology, applied to the scenario depicted in Figure 4f. In this case, both the peak amplitude and the distortion energy associated with the root mean squared value of the oscillatory transient are quantitatively assessed. The results demonstrate the method’s capability to reliably identify the disturbance and accurately estimate its impact. Moreover, it is identified that the disturbance starts at a time of 0.07 s, and it finishes at a time of 0.09 s, i.e., the disturbance presents a duration of 20 ms. The analysis focused on power quality evaluates the amount of distortion present in a signal; therefore, it is essential to know the quantitative parameters that describe how much a signal diverges from the ideal one. Due to the nature of the disturbance presented in Figure 5a, $M_{PEAK}$ and $M_{RMS}$ are used to quantify the intensity of the event. These indices allow us to understand the effects that the disturbance can cause on the grid or on the loads attached to the grid. Moreover, they can be used for other methodologies, different than this one, whose purpose is to mitigate the disturbances that appear in the power grid.

To address the reliability of the quantification indices, Figure 5b shows the envelope of the oscillatory transient signal and the envelope of a healthy signal. In both cases, the maximum amplitudes reached by the envelopes are shown. The difference between both peaks is the parameter described in Table 2 as $M_{peak}$, and it represents the amplitude deviation caused by the oscillatory transient. This deviation corresponds to approximately 0.0267 p.u., which matches the 2.67% reported. This value indicates a noticeable instantaneous increase in the signal magnitude due to the disturbance. Additionally, in Figure 5c, the same envelopes from Figure 5b are presented, accompanied by their RMS values. The difference between both RMS values represents the energy reduction caused by the oscillatory transient. This difference corresponds to the 5.38% reported for the $M_{rms}$ index, showing a noticeable decrease in the system’s effective power during the disturbance. This type of analysis highlights the importance of not only classifying the PQD phenomenon, but also providing quantitative statistics, which can make a significant difference. Timely attention is essential when disturbances are recurrent or exceed established thresholds, as a delayed mitigation can lead to equipment degradation, data loss, or operational instability.

5. Conclusions

In the present work, a strategy is presented for the detection and classification of PQDs specifically applicable to power grids with PV penetration. The strategy considers the development of a monitoring system based on machine learning, taking advantage of the benefits of feature extraction through the application of entropy features, which have proven to be an excellent option for the characterization of system behavior in noisy environments. The most representative contribution of this work is the fusion of HHT with entropy features, which are effective in characterizing and isolating PQDs, allowing for an accurate detection and classification even of the magnitude of the phenomena. Additionally, the use of the LDA allows the unification of two different feature vectors, one coming from the original voltage signals and the other being generated by the application of the envelope, while reducing the computational load required for the identification and classification of PQDs by eliminating the redundancy of information. Although the system demonstrates robustness under certain conditions by performing a normalization and dimensionality reduction process, it faces certain limitations when dealing with signals with low sampling frequencies and sample consistency, since it is designed to process a minimum of 13 cycles of the waveform to achieve an optimal performance. Since the penetration of renewable energies is becoming more popular every day, it is important to have monitoring strategies that allow us to properly detect power fluctuations and disturbances related to the variation in environmental conditions. In this regard, the proposed methodology is intended to cope with the improvement in the reliability of renewable generation systems by providing tools for the detection of disturbances to determine the best preventive and corrective actions. Regarding its implementation in embedded systems, this perspective is somewhat limited due to the complexity of the techniques used, such as LDA. However, ongoing developments in hybrid systems are opening new possibilities, enabling independent processing between a microchip and an FPGA. These advances could, from a future perspective, allow the full implementation of the proposed methodology in a real-time monitoring system. As a perspective in future research, the capabilities of the proposed methodology are expected to increase to identify not only the existence of a disturbance but also the time where it starts and ends to facilitate the determination of causes of the disturbance.