Diagnostics of Early Faults in Wind Generator Bearings Using Hjorth Parameters
Abstract
:1. Introduction
2. Methodology
2.1. Hjorth’s Parameters
- Activity is defined as the zeroth-order spectral moment (), given by Equation (1), and is expressed by the variance () of the signal amplitude (y), representing the surface envelope of the power spectrum in the time domain.
- Mobility represents the second-order spectral moment (), expressed by Equation (2), as the square root of the ratio between the variance () of the first-order derivative of the signal () and the variance of the signal. A measure of the standard deviation of the slope compared to the standard deviation of the amplitude is established, often known as the mean frequency.The fact that mobility is a slope measure relative to the mean makes it dependent solely on the waveform shape.
- Complexity is given by the fourth-order spectral moment (), defined by Equation (3), as the square root of the ratio between the variance () of the second-order derivative of the signal amplitude () and the variance of the first-order derivative of the signal. A measure of the similarity of the waveform under study to a sinusoidal wave is established, expressing a change in the frequency of the analyzed signal.
2.2. Feature Engineering and Machine Learning
2.2.1. Feature Extraction
- The standard deviation (STD) is given by Equation (4), where N represents the number of points composing the signal, represents the mean value of the signal amplitude, and is the amplitude of the signal at point i, with the SDT being a measure of data dispersion around the mean value.
- The root mean square (RMS) is expressed by Equation (5), quantifying the average power contained in the signal, serving as a metric for detecting vibration levels yet not being sensitive to early-stage faults.
- The skewness (SKW) assesses how far the signal distribution deviates from a normal distribution, and faults can lead to an increase in signal skewness, as expressed by Equation (6).
- Kurtosis is a measure of the data concentration around the central tendency measures of a normal distribution, given by Equation (7).
- The peak value () checks for the highest absolute value of the signal, given by Equation (8), where X represents the signal amplitude, and an increase in its value may indicate the occurrence of faults.
- The waveform length (WL) provides information about the signal frequency, calculated by Equation (9), where P represents the number of signal points and represents the difference between the amplitude of the current sample i and that of the next sample.
- The crest factor (CF) aims to overcome the limitation encountered by the RMS value for sensitivity to early-stage faults, expressed by Equation (10), which is the division of the peak value by the RMS value.The peak value has a greater sensitivity to early-stage faults, but as the fault progresses, the RMS value increases faster than the peak value, causing the CF value to decrease in the later stages.
- The factor K (FK) aims to combine the sensitivity of the peak value for early-stage faults and the sensitivity of the RMS value for later-stage fault detection. It is expressed as the product of the two metrics, as in Equation (11).
- The impulse factor (IF) compares the maximum value of the signal to the signal’s mean and is expressed by Equation (12), where represents the signal’s mean value.
- The form factor (FF), given by Equation (13), is defined as the ratio between the RMS value and the mean value of the signal, becoming dependent on the signal’s shape and independent of the signal’s dimensions.
2.2.2. Machine Learning
- Logistic regression (LR) is a statistical method for binary classification that employs input variables to calculate the probability of an event occurring. Utilizing the logistic function to convert values to probabilities ranging from 0 to 1, LR is useful for categorical and binary classification problems [25].
- Decision tree (DT) is a method that predicts outcomes by generating a tree-like structure of decisions based on input features. It divides data into subsets recursively, beginning with the root node, using features that best separate between classes. Leaf nodes represent the result of the predictions. DTs are interpretable, applicable to different fields, and facilitate the hierarchical visualization of decision-making processes [26].
- Random forest (RF) is a DT ensemble-based classification method. Since the decision trees in the RF are generated independently from random samples, there is a low association between the trees. Afterward, voting takes place using the classifications generated by each tree, and the class with the most votes is used to predict the presented sample [27].
- The support vector machine (SVM) algorithm searches for the optimal hyperplane for class separation, and various hyperplanes can be used to divide classes [28]. Nonetheless, the optimal hyperplane is determined by utilizing the most similar samples between the classes, which are the coordinates from which the support vectors are derived. The objective is to maximize distances in both directions to identify the hyperplane with the most significant separation, providing superior generalization [29].SVM can classify datasets that are not linearly separable by utilizing a kernel that determines the relationship between higher-dimensional data to identify the separability plane [30].
- The k-nearest neighbors (k-NN) classifier is based on the distance between the new sample to be classified and the other samples. The class of the new sample is determined by the majority class among the nearest neighbors. The parameter k specifies the number of closest points (neighbors) observed during classification, where small k values can lead to less stable results. In contrast, larger k values produce more stable results with increased errors. The Euclidean, Manhattan, or Minkowski functions can compute the distance between points [31].
3. Experimental Setup and Dataset
4. Results and Discussion
4.1. Signal Separation
4.2. Classification
4.3. Comparative Analysis
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
CF | Crest factor |
CWRU | Case Western Reserve University |
DT | Decision tree |
EEG | Electroencephalogram |
EFS | Exhaustive feature selection |
FF | Form factor |
ICP | Integrated circuit piezoelectric |
IF | Impulse factor |
IMS | Intelligent maintenance systems |
k-NN | k-nearest neighbors |
LR | Logistic regression |
RF | Random forest |
RMS | Root mean square |
SKW | Skewness |
STD | Standard deviation |
SVM | Support vector machine |
WL | Waveform length |
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Quantity of Samples | Bearing Fault | Fault Location | |
---|---|---|---|
Experiment 1 | 2156 | Bearing 3 Bearing 4 | Inner race Roll |
Experiment 2 | 984 | Bearing 1 | Outer race |
Experiment 3 | 4448 | Bearing 3 | Outer race |
Day of Fault | Number of Healthy Samples | Number of Faulty Samples | |
---|---|---|---|
Bearing 3—Experiment 1 | 33 | 1910 | 246 |
Bearing 4—Experiment 1 | 25 | 1540 | 616 |
Bearing 1—Experiment 2 | 4.8 | 704 | 280 |
Classifier | Hyperparameter | Tested Values | Selected |
---|---|---|---|
LR | C | 0.2, 2, 20, 80 | 0.2 |
Penalty | L2, Elasticnet | L2 | |
Solver | lbfgs, liblinear, sag, saga | lbfgs | |
DT | Criterion | Gini, Entropy | Entropy |
Tree depth | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 10 | |
Data points to split a node | 8, 10, 12 | 10 | |
Tree depth | 10, 20, 30 | 10 | |
Attributes to split a node | 2, 3 | 3 | |
RF | Data points to split a node | 8, 10, 12 | 10 |
Minimum allowed data in a leaf | 3, 4, 5 | 3 | |
Number of trees | 100, 150, 200, 250 | 100 | |
C | 0.1, 5, 10, 20, 50 | 50 | |
SVM | Kernel coefficient | 0.001, 0.01, 0.1, 1 | 1 |
Kernel | RBF, Linear | RBF | |
k-NN | Number of neighbors (k) | 3, 5, 7, 9, 11 | 11 |
Weights | Uniform, Distance | Distance | |
Metric | Minkowski, Euclidian, Manhattan | Minkowski |
Classifier | |||||
---|---|---|---|---|---|
Metric | LR | DT | RF | SVM | -NN |
Accuracy | 0.82 | 0.98 | 0.99 | 0.97 | 0.98 |
Precision | 0.85 | 0.98 | 0.98 | 0.97 | 0.98 |
Recall | 0.58 | 0.96 | 0.98 | 0.94 | 0.96 |
F1 score | 0.59 | 0.97 | 0.98 | 0.95 | 0.97 |
2.54 | 3.98 | 122.64 | 25.62 | 0.73 | |
0.0007 | 0.001 | 0.015 | 0.072 | 0.011 |
Classifier | |||||
---|---|---|---|---|---|
Metric | LR | DT | RF | SVM | -NN |
Accuracy | 0.84 | 0.99 | 0.99 | 0.97 | 0.99 |
Precision | 0.91 | 0.98 | 0.99 | 0.97 | 0.98 |
Recall | 0.61 | 0.98 | 0.99 | 0.94 | 0.98 |
F1 score | 0.63 | 0.98 | 0.99 | 0.96 | 0.98 |
2.49 | 3.38 | 105.05 | 22.41 | 0.73 | |
0.0019 | 0.0022 | 0.023 | 0.051 | 0.012 |
Classifier | Features |
---|---|
LR | STD and |
DT | WL and FF |
RF | RMS and WL |
SVM | STD and WL |
k-NN | RMS and WL |
Classifier | |||||
---|---|---|---|---|---|
Metric | LR | DT | RF | SVM | k-NN |
Accuracy | 0.82 () | 0.98 () | 0.98 () | 0.96 () | 0.98 () |
Precision | 0.86 () | 0.97 () | 0.98 () | 0.96 () | 0.97 () |
Recall | 0.56 () | 0.96 () | 0.97 () | 0.92 () | 0.97 () |
F1 score | 0.56 () | 0.97 () | 0.97 () | 0.94 () | 0.97 () |
0.77 () | 1.21 () | 89.52 () | 24.33 () | 0.51 () | |
0.001 () | 0.002 () | 0.018 () | 0.065 () | 0.004 () |
Classifier | |||||
---|---|---|---|---|---|
Metric | LR | DT | RF | SVM | k-NN |
Accuracy | 0.82 () | 0.98 () | 0.99 () | 0.96 () | 0.98 () |
Precision | 0.91 () | 0.98 () | 0.98 () | 0.96 () | 0.98 () |
Recall | 0.57 () | 0.97 () | 0.98 () | 0.93 () | 0.97 () |
F1 score | 0.58 () | 0.97 () | 0.98 () | 0.94 () | 0.98 () |
0.81 () | 1.24 () | 86.14 () | 24.39 (⇑ 4.8%) | 0.53 (⇓ 27.4%) | |
0.001 () | 0.002 () | 0.025 () | 0.060 () | 0.004 (⇓ 55.5%) |
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Santos, A.C.; Souza, W.A.; Barbara, G.V.; Castoldi, M.F.; Goedtel, A. Diagnostics of Early Faults in Wind Generator Bearings Using Hjorth Parameters. Sustainability 2023, 15, 14673. https://doi.org/10.3390/su152014673
Santos AC, Souza WA, Barbara GV, Castoldi MF, Goedtel A. Diagnostics of Early Faults in Wind Generator Bearings Using Hjorth Parameters. Sustainability. 2023; 15(20):14673. https://doi.org/10.3390/su152014673
Chicago/Turabian StyleSantos, Arthur C., Wesley A. Souza, Gustavo V. Barbara, Marcelo F. Castoldi, and Alessandro Goedtel. 2023. "Diagnostics of Early Faults in Wind Generator Bearings Using Hjorth Parameters" Sustainability 15, no. 20: 14673. https://doi.org/10.3390/su152014673
APA StyleSantos, A. C., Souza, W. A., Barbara, G. V., Castoldi, M. F., & Goedtel, A. (2023). Diagnostics of Early Faults in Wind Generator Bearings Using Hjorth Parameters. Sustainability, 15(20), 14673. https://doi.org/10.3390/su152014673