Power quality disturbances classification using autoencoder and radial basis function neural network

Venkataramana Veeramsetty; Aitha Dhanush; Aluri Nagapradyullatha; Gundapu Rama Krishna; Surender Reddy Salkuti

doi:10.1515/ijeeps-2023-0143

Article Open Access

Power quality disturbances classification using autoencoder and radial basis function neural network

Venkataramana Veeramsetty , Aitha Dhanush , Aluri Nagapradyullatha , Gundapu Rama Krishna and Surender Reddy Salkuti

Published/Copyright: September 25, 2023

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From the journal International Journal of Emerging Electric Power Systems Volume 25 Issue 6

Abstract

The classification of power quality (PQ) disturbances is a critical task for both utilities and industry. PQ issues cause power system equipment to fail. PQ disruptions also cause significant disruption in the paper and semiconductor industries, with significant financial implications as well as technological difficulties. Deep learning based approaches are used for automatic PQ disturbance classification, which requires huge amounts of data. A PQ disturbance dataset consisting of 12 PQ disturbances is developed using wavelet transform and MATLAB software. In this paper, an autoencoder is used to reduce the dimensionality of power quality disturbances data from higher dimensionality space, which consists of 72 input features, to lower dimensionality space, which consists of 21 input features. Based on data extracted from the autoencoder, a radial basis function neural network is used to identify the type of PQ disturbances. Based on the simulation results, it is observed that radial basis function neural network is able to distinguish the type of PQ disturbance with 92 % accuracy.

Keywords: power quality; machine learning; radial basis function neural network; autoencoder

Nomenclature

α: scaling factor corresponds to severity of disturbance
α ₃: magnitude corresponds to third order harmonic
α ₅: magnitude corresponds to fifth order harmonic
α ₇: magnitude corresponds to 7th order harmonic
Δb _h: change in bias connected to neurons in latent space
Δb _k: change in bias connected to neurons in output layer
Δ W h k n: change in weights connected to nth neuron in output layer
Δ W i h n: change in weights connected to nth neuron in latent space
η: learning rate
μ _h: centriod mean
σ: standard deviation
A: signal amplitude
b _k: bias parameters at output layer
net _k: net input to output layer
O _h: output at hidden layer
t ₁: starting time for PQ disturbance
t ₂: stop time for PQ disturbance
W _hk: weights between output and hidden layer
W h k n: weights connected to nth neuron in output layer
ADMN: analog–digital mixing network
AE: autoencoder
ANN: artificial neural network
BPNN: back propagation neural network
DR: dimensionality reduction
ED: Euclidean distance
FFNN: feed forward neural network
p: width parameter
PNN: probabilistic neural network
RBFNN: radial basis function neural network
RMS: root mean square
SD: standard deviation
SVM: support vector machine
WMSVM: wavelet multi class SVM

1 Introduction

Power quality (PQ) is defined as a combination of voltage and current quality. The quality of the voltage is affected by variations from the ideal voltage. Whereas current quality is the deviation from ideal current waveform [1]. Power quality identification is necessary to protect sensitive loads from power quality disturbing loads by installing protection equipment [2]. Computer, telecommunications, electronics manufacturing, pharmaceutical, and data processing centres are all concerned about power quality. As a result, it is critical to comprehend the quality of power delivered to enterprises via the process of acquiring, analysing, and interpreting data into valuable information. As a result, effective detection and classification of these disturbances is required so that control action may be undertaken [3]. Now a days deep learning or machine learning approaches are using for effective classification of PQ disturbances.

Huge amount of data is required to classify the power quality disturbance using any machine learning or deep learning models with good accuracy and the data volume increases with increasing in number PQ disturbances. Complexity of the model increases with number of input features in the data and may cause model to become overfit model. To avoid this limitation, dimensions of the data needs to reduced using dimensionality reduction (DR) techniques. DR is key machine learning technique used to convert data from higher dimensional space to lower dimensionality space [4, 5] in order to build a predictive machine learning models with less number of model parameters. This process leads to use less memory space to deploy deep learning models on edge devices and less computation time [6]. DR is a method of extracting enough information from data to determine the prediction by removing noise and redundant information that causes over-fitting and improves discriminating.

Classification of PQ disturbances like sag, swell, harmonics, transients and interruption using artificial neural networks is proposed in [7]. In this paper, very few PQ disturbances are considered but in real time exist many more PQ disturbances. Classification of all 12 PQ disturbances those are discussed in current article using decision tree is implemented in [8]. The proposed approach suffering with over-fit problem as authors used decision tree algorithm and not used any dimensionality reduction techniques. In [9], decision tree based approach has been used to classify PQ disturbances. In this paper, authors considered only few combined multiple PQ disturbances, and due to usage decision tree solution has over-fit problem, and authors not considered any DR techniques. A probabilistic neural network-based technique was utilised to classify PQ disturbances in [10]. The authors evaluated just a few mixed multiple PQ disturbances in their research, and they did not consider any DR approaches.

In [11], support vector machine based approach has been used to classify PQ disturbances. In this paper, authors considered only few combined multiple PQ disturbances, and Gram–Schmidt approach is used as a dimentionality reduction technique. In [12], authors proposes the use of artificial neural networks to classify PQ disturbances such as sag, interruption, flicker, transients, and interruption. In this study, just a few PQ disturbances are studied, however there are many more in real life. Single PQ disturbance classification using feed forward neural network is proposed in [13]. The main limitation in this work is authors have not considered occurrence of multiple PQ disturbances. In [14], a single PQ disturbance categorization utilising a probabilistic neural network is presented. The fundamental shortcoming of this work is that the authors did not account for the occurrence of numerous PQ disruptions. Identification of type of PQ disturbance using ANN is discussed in [15]. In this article, authors have not considered all combined PQ disturbances and also there is no discussion on dimensionality reduction.

Classification of single PQ disturbances like sag, swell, interruption, harmonics and transients are discussed in [16]. The authors did not address mixed PQ disturbances in this study, and there is no mention of dimensionality reduction. Classification of single PQ disturbances like sag, swell, interruption and harmonics using support vector machine is discussed in [17]. The authors did not address mixed PQ disturbances in this work, and dimensionality reduction is not mentioned. In [18], authors discusses utilising ANN to identify the type of PQ disruption. The authors did not address all coupled PQ disturbances in this study, and there is no consideration of dimensionality reduction. In [19], authors implements the classification of the 12 PQ disturbances presented in this paper using probabilistic neural networks. The suggested method did not take into account any dimensionality reduction approaches.

In this paper total 12 PQ disturbances i.e., voltage sag, voltage swell, interruption, flicker, harmonics, transients, swell with harmonics, sag with harmonics, interruption with harmonics, flicker with harmonics, flicker with swell and flicker with sag are considered for classification. All these disturbances are generated in MATLAB and these signal decomposed using wavelets and multi-resolution analysis. Statistical features like mean, RMS, standard deviation, range, energy, entropy, skewness and kurtosis are extracted from each decomposed signals. These features are used as input features to classify the PQ disturbances. Autoencoder (AE) is one of the dimensionality reduction techniques is used to reduce the number of features in the data. Radial basis function neural network (RBFNN) is one of the machine learning models is used to classify the PQ disturbances.

The main contributions of this paper are as follows:

All varieties of PQ disturbances are considered to classify
AE is used for the first time to reduce the dimensions of the PQ disturbances
Wavelet transform with Daubechies mother wavelet is used to decompose the PQ disturbance signals
RBFNN is used for the first time to classify both single and combined PQ disturbances

The remaining part of the paper is organized as follows: Section 2 describes the methodology, Section 3 presents the analysis of the simulation results, and Section 4 provides the conclusions of this work. Appendix A sections demonstrate RBFNN training and prediction with a small sample of data.

2 Methodology

This section presents functioning of autoencoder (AE) and RBFNN, details of PQ disturbances dataset and error metrics that are used to obtain the performance of the AE and RBFNN model.

2.1 PQ disturbances dataset

A PQ disturbances dataset is developed by considering 12 PQ disturbances like voltage sag, voltage swell, interruption, flicker, harmonics, transients, swell with harmonics, sag with harmonics, interruption with harmonics, flicker with harmonics, flicker with swell and flicker with sag, and with one fundamental signal. The mathematical modelings used to generate single disturbance signals in MATLAB are shown in equations (1)–(6) and single disturbance signals are shown in Figure 1 and for multiple disturbance signals like swell with harmonics, sag with harmonics, interruption with harmonics, flicker with harmonics, flicker with swell and flicker with sag, the mathematical modeling are shown in equations (7)–(12) respectively and these combined multiple disturbance signals are shown in Figure 2.

(1) V sag ( t ) = A [ 1 − α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] sin ( ω * t )

(2) V swell ( t ) = A [ 1 + α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] sin ( ω * t )

(3) V interruption ( t ) = A [ 1 − α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] sin ( ω * t )

(4) V flicker ( t ) = A [ 1 + α f sin ( β ω t ) ] sin ( ω t )

(5) V harmonics ( t ) = A [ α 3 sin ( 3 ω t ) + α 5 sin ( 5 ω t ) + α 7 sin ( 7 ω t ) ]

(6) V transients ( t ) = sin ( ω t ) + A [ u ( t − t 1 ) − u ( t − t 2 ) ] * e − t t y * sin ( f n ω t )

(7) y ( t ) = A [ 1 + α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] α 1 sin ( ω t ) + α 3 sin ( 3 ω t ) + α 5 sin ( 5 ω t )

(8) y ( t ) = A [ 1 − α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] α 1 sin ( ω t ) + α 3 sin ( 3 ω t ) + α 5 sin ( 5 ω t )

(9) y ( t ) = A [ 1 − α ( u ( t − t 1 ) − u ( t − t 2 ) ) ] α 1 sin ( ω t ) + α 3 sin ( 3 ω t ) + α 5 sin ( 5 ω t )

(10) y ( t ) = A ( 1 + α f sin ( β ω t ) ) sin ( ω t ) α 1 sin ( ω t ) + α 3 sin ( 3 ω t ) + α 5 sin ( 5 ω t )

(11) y ( t ) = A ( 1 + α f sin ( β ω t ) ) sin ( ω t ) [ 1 + α ( u ( t − t 1 ) − u ( t − t 2 ) ) ]

(12) y ( t ) = A ( 1 + α f sin ( β ω t ) ) sin ( ω t ) [ 1 − α ( u ( t − t 1 ) − u ( t − t 2 ) ) ]

Figure 1:

Single PQ disturbance signals those are considered in this paper. (a) Sag signal with ⍺ = 0.4, t ₁ = 0.02 and t ₂ = 0.06. (b) Swell signal with ⍺ = 0.4, t ₁ = 0.02 and t ₂ = 0.06. (c) Interruption signal with ⍺ = 0.95, t ₁ = 0.02 and t ₂ = 0.06. (d) Flicker signal with ⍺ = 0.35 and β = 6. (e) Harmonics signal with ⍺ ₃ = 0.3, ⍺ ₅ = 0.1 and ⍺ ₇ = 0.15. (f) Transient signal with f _n = 200, A = 5, t ₁ = 0.05 and t ₂ = 0.052.

Figure 2:

Combined PQ disturbances signal those are considered in this paper. (a) Combined Sag-Harmonics signal with ⍺ = 0.5, ⍺ ₃ = 0.15, ⍺ ₅ = 0.2, ⍺ ₇ = 0.25, t ₁ = 0.02 and t ₂ = 0.06. (b) Combined Swell-Harmonics signal with ⍺ = 0.5, ⍺ ₃ = 0.15, ⍺ ₅ = 0.2, ⍺ ₇ = 0.25, t ₁ = 0.02 and t ₂ = 0.06. (c) Combined Interruption-Harmonics signal with ⍺ = 0.95, ⍺ ₃ = 0.15, ⍺ ₅ = 0.2, ⍺ ₇ = 0.25, t5 ₁ = 0.02 and t ₂ = 0.06. (d) Combined Flicker-Harmonics signal with ⍺ = 0.2, ⍺ ₃ = 0.1, ⍺ ₅ = 0.2, ⍺ ₇ = 0.3 and β = 6. (e) Combined Flicker-Sag signal with ⍺ = 0.4, β = 0.5, t ₁ = 0.02 and t ₂ = 0.06. (f) Combined Flicker-Swell signal with ⍺ = 0.4, β = 0.5, t ₁ = 0.02 and t ₂ = 0.06.

The Multi-Resolution Analysis (MRA) technique along with daubechies mother wavelet is used to decompose the PQ signals at level 8. Every PQ signal is decomposed into 8 detailed coefficient signals and 1 approximate coefficient signal. Structure of signal decomposition with level 8 is shown in Figure 3.

Figure 3:

Decomposition of PQ disturbance signal using MRA with eighth level.

From each signal total 8 statistical features i.e., mean, standard deviation (SD), rms value (RMS), energy, entropy, skewness, kurtosis and range are extracted using the mathematical modeling shown in equations (13)–(20) respectively. Total number of features extracted from 9 decomposed signals are 9 × 8 = 72. Total 750 sample signals for 12 PQ disturbances and 1 fundamental signal are used while preparing the dataset, and the shape of the dataset is 750 × 72.

(13) Mean = 1 N ∑ j = 1 N A 8 j or d 1 − − 7 j

(14) SD = 1 N ∑ j = 1 N A 8 j − μ A 8 2 or d 1 − − 7 j − μ d 1 − − 7 2

(15) RMS = 1 N ∑ j = 1 N A 8 j 2 or d 1 − − 7 j 2

(16) Energy = ∑ j = 1 N ( A 8 j 2 or d 1 − − 7 j 2 )

(17) Entropy = − ∑ j = 1 N A 8 j 2 log A 8 j 2 or d 1 − − 7 j 2 × l o d d 1 − − 7 j 2

(18) Skewness = E ( A 8 − μ A 8 ) 3 σ 3 or E ( d 1 − − 7 − μ d 1 − − 7 ) 3 σ 3

(19) Kurtosis = E ( A 8 − μ A 8 ) 4 σ 4 or E ( d 1 − − 7 − μ d 1 − − 7 ) 4 σ 4

(20) Range = Max ( A 8 , d 1 − − 7 ) − Min ( A 8 , d 1 − − 7 )

2.2 Dataset compression using autoencoder

Autoencoder is a neural network that is used to reconstruct its input [20], [21], [22]. It has two parts i.e., encoder and decoder [23]. Encoder is used to get compressed features at latent space [24], [25], [26], [27], whereas decoder is used to reconstruct the original inputs from the compressed latent space. In autoencoders, the learning algorithm performs back propagation by assigning input data as target values [28]. In each hidden layer and in latent space ReLu activation function [29], [30], [31] is considered and that is mathematically modelled as shown in equation (21). The gradient of ReLu activation function is zero for z ≤ 0 and whereas for z > 0 the gradient is equal to 1. Linear activation function is considered for output layer of autoencoder that is mathematically modelled as shown in equation (22). In equation (22), “w” represents weight matrix between latent space and output layer, “f(z)” represents output of latent space and “b” represents bias parameters in output layer.

(21) f ( z ) = max ( 0 , z )

(22) g ( w , f ( z ) ) = w T f ( z ) + b

Architecture of the autoencoder is shown in Figure 4. In Figure 4, Mean, Standard Deviation, RMS, Range, Energy, Kurtosis, Skewness and Entropy represents original input features (F) in the PQ disturbance dataset. Whereas Mean’, Standard Deviation (Std’), RMS’, Range’, Energy’, Kurtosis’, Skewness’ and Entropy’ are the output (F′) at output layer while expecting same input features which are given to input layer. X1, X2, …, X64 are the reduced extracted features from input features in the original PQ disturbances data. Performance of the autoencoder is observed in terms of training and validation loss i.e., mean square error [32], [33], [34] as shown in equation (23).

(23) loss = 1 n ∑ i = 1 n sample ∑ k = 1 n o F k − F k ′ 2

where F ϵ {Mean, Standard Deviation, RMS, Range, Energy, Kurtosis, Skewness, Entropy} and F′ ϵ {Mean’, Standard Deviation(Std’), RMS’, Range’, Energy’, Kurtosis’, Skewness’, Entropy’}.

Figure 4:

Autoencoder architecture.

2.3 PQ disturbance classification using radial basis function neural network

Radial Basis Function Neural Network (RBFNN) consists one input layer, one hidden layer and one output layer [35], [36], [37]. Input layer consists 21 neurons as number of input features extracted from autoencoder is equal to 21. Output layer consists 13 neurons with softmax activation function as this model is used to classify total 12 PQ disturbances and one fundamental signal. The mathematical modeling of softmax activation function is shown in equation (26). Whereas as the number of neurons in hidden layer or also called as centroids is the hyper parameter, going to tune it for best model. Unlike ANN [38, 39], training of RBFNN includes both supervised and unsupervised. The complete algorithm to train RBFNN model based on stochastic gradient descent optimizer [40] is shown in Algorithm 1. As per the Algorithm 1. The training between input layer and hidden layer is based on unsupervised learning using Euclidean distance [41], whereas the training between output layer and hidden layer is based on supervised learning. The weights and bias parameters between hidden and output layers are updated using stochastic gradient descent optimizer. Gaussian activation function [42] is used in hidden layer and its mathematical modeling is shown in equation (24). Whereas standard deviation σ [43] is calculated based on width factor “p” as shown in equation (25). The RBFNN architecture for PQ disturbances classification is shown in Figure 5.

Figure 5:

RBFNN architecture for PQ disturbances classification.

The performance of RBFNN is observed in terms of accuracy as shown in equation (27)

(24) g ( x , μ , σ ) = e − ( x − μ ) 2 2 σ 2

(25) σ ( p , μ i ) = 1 p ∑ ( μ i − μ j ) 2

(26) F ( y k ) = e y k ∑ i = 1 n o e y i

(27) Accuracy = Total number of correctly identified samples Total number of samples

Algorithm 1.

. RBFNN training algorithm

Step 1: Read input and output features of PQ disturbances dataset ⊳ X = {X2, X7, X12, X15, X16, X24, X25, X26, X28, X29, X33, X35, X37, X39, X40, X41, X42, X51, X57, X60, X61} and Label, Where X is the set of extracted features from autoencoder

Step 2: Initialize number of centriods (neurons) in hidden layer, weights W _hk, bias parameter b _k, learning rate η and tol=1

Step 3: Randomly pick few samples from training dataset and assign as mean vector (μ) for each neuron/centriod.

while tol ≥ 0.001 do ⊳ Unsupervised learning between input and hidden layer of RBFNN

Step 4: Calculate the Euclidean distance (ED) between each sample and mean vector using equation (28)

(28) E D = ( X − μ ) 2

Step 5: Assign the sample which has minimum ED to that particular centroid and update mean value (μ) as an average of all assigned samples to that particular centroid.

Step 6: Calculate tolerance (tol) as the maximum difference between old and new mean values among all centriods.

end while

Step 7: Calculate the standard deviation (σ) using equation (29)

(29) σ = 1 P ∑ j ϵ P _ n e i g h b o u r ( C i − C j ) 2

Step 8: Calculate the output of each hidden neuron (h) using equation (30)

(30) O h = exp − ( X − μ h ) 2 2 σ h 2

while del _W ≥ 0.001 do ⊳ supervised learning between output and hidden layer of RBFNN

Step 9: Calculate input of the RBFNN output layer using equation (31)

(31) n e t k = W h k T O h + b 0

Step 10: Calculate output of the RBFNN output layer using equation (32)

(32) O k = e n e t k ∑ k = 1 n k e n e t k

Step 11: Update W _hk using equation (33) and b _k using (34)

(33) W h k = W h k + η ( 1 − O k ) O h

(34) b k = b k + η ( 1 − O k )

Step 12: find del _W as maximum change among W _hk and b _k

end while

Step 13: Store model in terms of model parameters W _hk, μ, σ and b _k, and architecture

3 Results

Various samples of 12 PQ disturbances and one fundamental signal are generated using MATLAB. Each sample is decomposed into 8 detailed coefficient signals and 1 approximate signal. Total 8 features are extracted from each decomposed signal. The final PQ disturbances dataset comprises a 750 samples with 72 features (9 × 8 = 72). Further, this dataset is compressed to a new dimension 750 × 21 using autoencoder. The compressed dataset is used to classify PQ disturbances using RBFNN. PQ disturbances dataset is split in 90 %:10 % for training and testing respectively. Based on this ratio the shape of training and testing data for each model is shown in Table 1. The following procedure is used for the PQ disturbances classification. All these tasks are implemented using python programming in visual studio code.

Generation PQ disturbances in MATLAB
Dimensionality reduction using autoencoder
PQ disturbances classification using RBFNN

Table 1:

Training and testing data size.

Task	Dataset	Model	Data Shape
Task	Dataset	Model	Training	Testing
Data compression	PQ disturbances	Autoencoder	675 × 72	75 × 72
PQ disturbances classification	Compressed PQ data	RBFNN	675 × 21	75 × 21

3.1 Data insights

The original PQ dataset that is prepared based on PQ disturbances generated in MATLAB and the compressed dataset that is extracted using autoencoder are available at public repository at https://data.mendeley.com/datasets/nkdpg8mn4f/3. Table 2 presents statistical information of the original PQ disturbances dataset.

Table 2:

Statistical information of original PQ disturbances dataset.

Parameter	Mean-d1	Mean-d2	Mean-d3	Mean-d4	Mean-d5	Mean-d6	Mean-d7	Mean-d8	Mean-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.00	0.01	−0.01	0.02	−0.16	0.12	0.13	0.06	1.19
Std	0.02	0.05	0.11	0.17	0.62	0.33	0.64	1.07	3.98
Min	−0.07	0.00	−0.15	−0.03	−2.19	−0.14	−0.26	−1.18	−0.25
25 %	0.00	0.00	0.00	0.00	0.00	−0.01	−0.10	−0.32	−0.18
50 %	0.00	0.00	0.00	0.00	0.00	0.01	−0.09	−0.26	−0.15
75 %	0.00	0.00	0.00	0.00	0.01	0.08	−0.04	−0.09	−0.09
Max	0.02	0.92	2.11	3.32	3.81	1.93	2.49	3.97	15.92

Parameter	Std-d1	Std-d2	Std-d3	Std-d4	Std-d5	Std-d6	Std-d7	Std-d8	Std-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.28	0.18	0.24	0.48	0.82	2.02	6.22	4.03	2.44
Std	0.94	0.51	0.43	0.40	0.45	0.23	0.74	0.45	0.40
Min	0.00	0.00	0.00	0.00	0.00	−0.02	−0.11	−0.60	−0.18
25 %	0.00	0.00	0.02	0.09	0.45	1.90	5.85	3.77	2.28
50 %	0.00	0.03	0.12	0.42	0.64	2.03	6.24	4.00	2.41
75 %	0.01	0.04	0.19	0.69	1.11	2.14	6.56	4.17	2.75
Max	3.92	2.05	1.63	1.66	2.24	2.73	8.35	5.83	3.21

Parameter	Rms-d1	Rms-d2	Rms-d3	Rms-d4	Rms-d5	Rms-d6	Rms-d7	Rms-d8	Rms-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.01	0.03	0.12	0.43	0.76	2.02	6.17	3.95	3.23
Std	0.01	0.03	0.12	0.36	0.42	0.25	0.74	0.45	3.38
Min	0.00	0.00	0.01	0.07	0.34	1.54	4.77	3.25	1.30
25 %	0.00	0.00	0.02	0.09	0.45	1.89	5.76	3.68	2.23
50 %	0.00	0.02	0.11	0.41	0.65	2.02	6.16	3.89	2.35
75 %	0.01	0.03	0.16	0.59	0.99	2.12	6.47	4.08	2.68
Max	0.19	0.27	1.07	3.57	7.08	6.61	17.08	11.47	16.21

Parameter	Range-d1	Range-d2	Range-d3	Range-d4	Range-d5	Range-d6	Range-d7	Range-d8	Range-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.11	0.23	0.75	2.17	3.69	6.75	19.37	12.78	10.39
Std	0.13	0.32	0.77	1.59	1.93	1.32	3.44	1.78	2.08
Min	0.01	0.03	0.13	0.54	0.99	1.98	5.94	3.74	2.23
25 %	0.03	0.05	0.19	0.63	1.91	5.87	17.36	11.89	9.78
50 %	0.06	0.16	0.66	2.05	3.39	6.20	18.64	12.42	10.13
75 %	0.13	0.21	0.87	2.97	5.03	7.24	19.29	13.24	11.89
Max	1.07	1.81	5.49	9.94	10.19	13.81	36.96	24.75	16.16

Parameter	Energy-d1	Energy-d2	Energy-d3	Energy-d4	Energy-d5	Energy-d6	Energy-d7	Energy-d8	Energy-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.10	1.23	13.96	72.98	87.89	269.42	1317.73	282.12	282.11
Std	0.20	2.77	29.91	110.31	86.18	48.81	253.03	53.41	53.45
Min	0.00	0.01	0.10	0.59	2.01	7.30	25.36	12.76	9.88
25 %	0.00	0.01	0.13	1.91	25.46	237.25	1171.95	245.94	245.94
50 %	0.04	0.54	6.06	41.95	53.00	265.14	1297.23	271.90	271.90
75 %	0.09	1.16	12.96	87.30	124.42	292.13	1420.98	298.07	298.07
Max	1.87	26.77	271.49	687.52	632.94	476.96	2299.36	577.59	577.59

Parameter	Kurtosis-d1	Kurtosis-d2	Kurtosis-d3	Kurtosis-d4	Kurtosis-d5	Kurtosis-d6	Kurtosis-d7	Kurtosis-d8	Kurtosis-A8
Count	750	750	750	750	750	750	750	750	750
Mean	192.67	55.65	21.08	11.53	3.75	2.20	4.04	2.46	7.54
Std	303.77	163.15	50.49	30.99	8.31	12.95	64.23	14.78	14.63
Min	0.24	1.58	1.55	1.59	1.46	1.41	1.53	1.61	3.59
25 %	14.42	2.24	2.25	2.41	2.00	1.55	1.55	1.75	6.30
50 %	41.77	3.42	3.14	3.23	2.56	1.69	1.64	1.82	6.77
75 %	317.72	29.49	16.48	4.46	3.56	1.86	1.79	2.08	8.03
Max	1956.16	866.43	354.49	273.64	196.74	356.46	1760.75	406.50	406.50

Parameter	Skewness-d1	Skewness-d2	Skewness-d3	Skewness-d4	Skewness-d5	Skewness-d6	Skewness-d7	Skewness-d8	Skewness-A8
Count	750	750	750	750	750	750	750	750	750
Mean	−2.60	0.88	0.60	0.18	0.10	0.34	1.71	0.54	−1.49
Std	7.04	5.53	2.10	2.00	1.01	9.23	48.41	9.63	9.71
Min	−44.00	−26.16	−12.96	−13.49	−4.67	−0.36	−0.35	−0.29	−2.39
25 %	−4.00	0.01	0.00	−0.02	−0.02	−0.03	−0.12	0.15	−2.14
50 %	−1.05	0.10	0.07	0.00	0.04	0.00	−0.06	0.19	−1.81
75 %	−0.36	2.19	1.56	0.55	0.12	0.04	−0.03	0.21	−1.67
Max	42.45	27.78	17.23	14.42	22.12	252.66	1325.72	264.03	264.03

Parameter	Entropy-d1	Entropy-d2	Entropy-d3	Entropy-d4	Entropy-d5	Entropy-d6	Entropy-d7	Entropy-d8	Entropy-A8
Count	750	750	750	750	750	750	750	750	750
Mean	0.89	2.03	3.13	3.53	3.47	1.73	1.25	1.04	1.69
Std	0.77	0.97	0.97	0.49	0.37	0.21	0.17	0.15	0.49
Min	0.01	0.82	1.74	2.38	1.98	1.29	1.00	0.85	0.00
25 %	0.02	0.92	2.03	3.14	3.24	1.59	1.16	0.96	1.72
50 %	1.10	2.33	3.54	3.56	3.57	1.68	1.16	0.96	1.74
75 %	1.48	2.82	3.97	3.90	3.75	1.86	1.32	1.19	1.97
Max	2.83	4.07	4.81	5.02	4.26	2.52	1.78	1.72	2.43

3.2 Optimal autoencoder architecture for data compression

The performance of various autoencoder architectures is observed in terms of training and validation loss as shown in Table 3. From the Table 3, it is observed that validation loss of autoencoder is increasing with decrease in latent space. When autoencoder trying to compress the data more with less number of neurons in latent space then error between original data and reconstructed data at output layer is increasing. In Table 3, data volume corresponding to latent space having neurons 50 and 43 is less in comparison with the latent space having neurons 36 and 28 even though more data compression happened in later case. It is due to more number of columns with all zero values and these columns are dropped later leads less data volume.

Table 3:

Performance of various autoencoder architectures.

Latent space	Training loss	Validation loss	Data volume
68	0.002	0.0017049	14,784
64	0.003	0.0024596	11,655
57	0.007	0.0046854	10,234
50	0.009	0.0055879	7440
43	0.220	0.0184691	7425
36	0.222	0.0186251	10,125
28	0.225	0.0188132	10,125
21	0.224	0.0184358	4050
14	0.221	0.0186929	3375

Optimal autoencoder architecture is identified based on value of objective function. Objective function is computed based on 10 % weighted of normalized data volume (NDV) and 90 % weighted of normalized validation loss (NVL). Table 3 shows objective function values for various autoencoder architectures. From the Table 3, it is observed that autoencoder with 64 neurons in latent space has minimum objective function value i.e., 0.2486. Hence, the autoencoder with 64 neurons in the latent space is considered as optimal autoencoder to compress the PQ disturbances data.

The optimal autoencoder is used to reduce the dimensions of the original PQ dataset from the shape 750 × 72 into a compressed data having the shape 750 × 64. From the Figure 6 it is observed that, in the compressed data the features like X1, X3–X6, X8–X11, X13, X14, X17–X23, X27, X30–X32, X34, X36, X38, X43–X50, X52–X56, X58, X59 and X62–X64 have more than 100 zero values, and these columns not able to help to classify the PQ disturbances. Hence, these columns are removed from the compressed dataset. Now the shape of the compressed dataset becomes 750 × 21. This is final compressed dataset that can be used to classify any PQ disturbance using AI models.

Figure 6:

Number of zero values in each column.

3.3 Optimal RBFNN model to classify the PQ disturbances with original PQ dataset

The RBFNN architecture that have maximum validation accuracy is considered as optimal model for PQ disturbances classification. Various RBFNN architectures are developed by tuning the number of centroids/neurons and width parameter (P) in hidden layer. The training and validation accuracy values for various RBFNN architecture with batch size 32 for PQ disturbances classification based on original PQ dataset consists of 750 samples and 72 features is shown in Tables 4 and 5 respectively. From the Table 5, it is observed that RBFNN architecture with 16 centroids and width parameter 2 is considered as optimal model due highest validation accuracy i.e., 98.7 % and From the Table 4, it is observed that the corresponding training accuracy is 89.33 %.

Table 4:

Training performance of various RBFNN architectures with original PQ dataset.

No. of centriods	Width parameter (P) & batch size 32
No. of centriods	2	4	6	8	10	12	14	16	18	20	22	24	26
11	0.8844	0.8415	0.8089	0.7585	0.6874	–	–	–	–	–	–	–	–
16	0.8933	0.8741	0.8504	0.8504	0.8133	0.7422	0.7259	0.6533	0.0000	0.0000	0.0000	0.0000	–
21	0.9052	0.8904	0.8770	0.8607	0.8385	0.8415	0.8207	0.7837	0.7126	0.6563	0.0000	0.0000	–
26	0.9185	0.9126	0.8933	0.8681	0.8652	0.8474	0.8444	0.8356	0.8089	0.7926	0.7304	0.6726	0.6385
32	0.9156	0.9156	0.8919	0.8889	0.8741	0.8533	0.8519	0.8474	0.8459	0.8415	0.8133	0.7985	0.7467
37	0.9215	0.9126	0.9081	0.8919	0.8815	0.8696	0.8593	0.8607	0.8519	0.8489	0.8504	0.8267	0.8000
42	0.9244	0.9170	0.9111	0.8993	0.8919	0.8800	0.8770	0.8563	0.8622	0.8533	0.8519	0.8356	0.8400
47	0.9274	0.9185	0.9081	0.8978	0.8919	0.8815	0.8770	0.8593	0.8578	0.8593	0.8548	0.8385	0.8459
53	0.9259	0.9259	0.9230	0.9111	0.8978	0.8919	0.8800	0.8756	0.8696	0.8593	0.8607	0.8593	0.8370

	28	30	32	34	36	38	40	42	44	46	48	50	52

11	–	–	–	–	–	–	–	–	–	–	–	–	–
16	–	–	–	–	–	–	–	–	–	–	–	–	–
21	–	–	–	–	–	–	–	–	–	–	–	–	–
26	–	–	–	–	–	–	–	–	–	–	–	–	–
32	0.7052	0.6889	0.6533	–	–	–	–	–	–	–	–	–	–
37	0.7630	0.7244	0.7067	0.6859	0.6578	–	–	–	–	–	–	–	–
42	0.8044	0.7600	0.7600	0.7170	0.7096	0.6889	0.6607	0.6593	–	–	–	–	–
47	0.8237	0.7970	0.7911	0.7689	0.7556	0.7244	0.7244	0.7111	0.6667	0.6667	–	–	–
53	0.8385	0.8207	0.8237	0.7970	0.7970	0.7867	0.7733	0.7422	0.7274	0.7215	0.7096	0.6593	0.6430

Table 5:

Performance of various RBFNN architectures with original testing PQ data.

No. of centriods	Width parameter (P) & batch size 32
No. of centriods	2	4	6	8	10	12	14	16	18	20	22	24	26
11	0.9333	0.9067	0.8400	0.7067	0.5333	–	–	–	–	–	–	–	–
16	0.9867	0.9467	0.8533	0.8667	0.8000	0.6800	0.6267	0.5467	–	–	–	–	–
21	0.9600	0.9600	0.8533	0.8667	0.8400	0.8400	0.8533	0.8000	0.6933	0.6667	–	–	–
26	0.9333	0.9333	0.8667	0.8667	0.8400	0.8267	0.8267	0.8667	0.8667	0.8533	0.7467	0.7600	0.7733
32	0.9333	0.9333	0.9200	0.8667	0.8533	0.8267	0.8267	0.8267	0.8667	0.8800	0.8800	0.8533	0.8000
37	0.9600	0.9333	0.9067	0.8667	0.8667	0.8400	0.8267	0.8267	0.8133	0.8933	0.9067	0.8933	0.8533
42	0.9467	0.9200	0.8933	0.9200	0.8667	0.8400	0.8267	0.8267	0.8400	0.8400	0.8933	0.9067	0.9067
47	0.9333	0.9333	0.9067	0.8800	0.8667	0.8533	0.8267	0.8267	0.8267	0.8533	0.9067	0.9067	0.8933
53	0.9333	0.8933	0.9067	0.8933	0.8400	0.8400	0.8400	0.8267	0.8267	0.8400	0.9067	0.9067	0.9067

	2	4	6	8	10	12	14	16	18	20	22	24	26

11	–	–	–	–	–	–	–	–	–	–	–	–	–
16	–	–	–	–	–	–	–	–	–	–	–	–	–
21	–	–	–	–	–	–	–	–	–	–	–	–	–
26	–	–	–	–	–	–	–	–	–	–	–	–	–
32	0.8000	0.7867	0.8400	–	–	–	–	–	–	–	–	–	–
37	0.8267	0.8133	0.8400	0.8667	0.8000	–	–	–	–	–	–	–	–
42	0.8800	0.8267	0.8400	0.8000	0.8267	0.8267	0.8533	0.8533	–	–	–	–	–
47	0.8933	0.8933	0.8800	0.8533	0.8267	0.8000	0.7867	0.8000	0.8667	0.8667	–	–	–
53	0.9067	0.9200	0.9200	0.8933	0.8800	0.8800	0.7867	0.8000	0.7867	0.7600	0.7600	0.8133	0.8133

The optimal RBFNN architecture for PQ disturbance classification with 16 centroids and width factor 2 is trained with various batch sizes i.e., 8, 16, 32, 64, 128, 256 and 512. The validation accuracy of the model for various batch sizes is presented in Figure 7. From the Figure 7, it is observed that RBFNN model that trained with batch size 32 has maximum validation accuracy i.e., 98.7 % and the corresponding training accuracy is 89.33 %. Hence the RBFNN model with 16 centroids, width factor 2 and trained with batch size 32 is considered as optimal model for PQ disturbance classification based on original data consists of 750 samples with 72 features.

Figure 7:

Performance of various RBFNN architecture with various batch sizes on original PQ dataset.

Converging characteristics of RBFNN architecture with 16 centroids, width factor 2 and with batch size 32 is shown in Figure 8. Figure 8a represents training accuracy of RBFNN model with original PQ dataset. Figure 8b represents training loss of RBFNN model with original PQ dataset. Figure 8c represents validation accuracy of RBFNN model with original PQ dataset. Figure 8d represents validation loss of RBFNN model with original PQ dataset. From the Figure 8, it is observed that the RBFNN model that trained with batch size 32 has maximum validation accuracy i.e., 98.7 %, also the difference between both training and validation accuracy values is less hence the model is well trained without under-fit and over-fit problems.

Figure 8:

Converging characteristics of RBFNN for PQ disturbances classification with original PQ dataset. (a) Training accuracy with original PQ dataset. (b) Training loss with original PQ dataset. (c) Validation accuracy with original PQ dataset. (d) Validation loss with original PQ dataset.

The confusion matrix that represents number of correct and incorrect PQ disturbances classification with training data from original PQ disturbances dataset is shown Figure 9 and with testing data from original PQ disturbances dataset is shown Figure 10.

Figure 9:

Confusion matrix for PQ disturbances classification with training data from original PQ disturbances dataset.

Figure 10:

Confusion matrix for PQ disturbances classification with testing data from original PQ disturbances dataset.

3.4 Optimal RBFNN model to classify the PQ disturbances with compressed PQ dataset

The RBFNN architecture that have maximum validation accuracy is considered as optimal model for PQ disturbances classification. Various RBFNN architectures are developed by tuning the number of centroids/neurons and width parameter (P) in hidden layer. The training and validation accuracy values for various RBFNN architecture with batch size 32 for PQ disturbances classification is shown in Figures 11 and 12 respectively. The RBFNN architecture with 32 centroids and width parameter 6 is considered as optimal model due highest validation accuracy i.e., 90.67 % and the corresponding training accuracy is 86.96 %.

Figure 11:

Training performance of various RBFNN architectures.

Figure 12:

Performance of various RBFNN architectures with testing data.

The optimal RBFNN architecture for PQ disturbance classification with 32 centroids and width parameter 6 is trained with various batch sizes i.e., 8, 16, 32, 64, 128, 256 and 512. The validation accuracy of the model for various batch sizes is presented in Figure 13. From the Figure 13, it is observed that RBFNN model that trained with batch size 64 has maximum validation accuracy i.e., 92 % and the corresponding training accuracy is 90 %. Hence the RBFNN model with 32 centroids, width factor 6 and trained with batch size 64 is considered as optimal model for PQ disturbance classification.

Figure 13:

Performance of various RBFNN architecture with various batch sizes.

Converging characteristics of RBFNN architecture with 32 centroids, width factor 6 and with batch size 64 is shown in Figure 14. From the Figure 14, it is observed that the RBFNN model that trained with batch size 64 has maximum validation accuracy i.e., 92 %, also the difference between both training and validation accuracy values is less hence the model is well trained without under-fit and over-fit problems.

Figure 14:

Converging characteristics of RBFNN for PQ disturbances classification. (a) Training accuracy. (b) Training loss. (c) Validation accuracy. (d) Validation loss.

The confusion matrix that represents number of correct and incorrect PQ disturbances classification with training data from compressed PQ disturbances dataset is shown Figure 15 and with testing data from compressed PQ disturbances dataset is shown Figure 16. From the Figure 15, it is observed that total number of correctly classified disturbances is equal to sum of diagonal elements in the confusion matrix i.e. 609, and the training accuracy is 609/675 = 90 %. Similarly, from the Figure 16, it is observed that total number of correctly classified disturbances is equal to sum of diagonal elements in the confusion matrix i.e. 69, and the training accuracy is 69/75 = 92 %.

Figure 15:

Confusion matrix for PQ disturbances classification with training data from compressed PQ disturbances dataset.

Figure 16:

Confusion matrix for PQ disturbances classification with testing data from compressed PQ disturbances dataset.

3.5 Comparative analysis

The proposed autoencoder plus radial basis function neural network (AE + RBFNN) is compared with RBFNN model for PQ disturbances classification in terms traning and validation accuracy, and model complexity as shown in Table 6. From Table 6, it is observed that performance of the model RBFNN is 5.7 % more accurate than AE + RBFNN but complexity of the model in terms of ED calculations and model parameters is increased by 20 %.

Table 6:

RBFNN vs. AE + RBFNN.

Model	Training accuracy	Validation accuracy	Input neurons	Centroids	Output neurons	Model parameters	ED calculations
RBFNN	89.3 %	98.6 %	72	16	13	221	1152
AE + RBFNN	90.2 %	92 %	21	32	13	429	672

The proposed AE + RBFNN model is validated by comparing with decision tree, random forest and ANN in terms of validation accuracy on classification of PQ disturbances with the compressed PQ dataset and these values are presented in Table 7. From the Table 7, it is observed that the proposed AE+RBFNN model performing better that decision tree, random forest and ANN in terms of validation accuracy. Both decision tree and random forest are suffering with overfit problem.

Table 7:

Comparative analysis.

Model	Training accuracy	Validation accuracy
AE + RBFNN	90.2 %	92 %
Decision tree	100 %	90.7 %
Random forest	100 %	90.7 %
ANN	86.5 %	90.7 %

4 Conclusions

The term power quality (PQ) refers to the combination of voltage and current quality. Variations from the optimum voltage have an effect on voltage quality. Current quality, on the other hand, is the divergence from the ideal current waveform. Power quality identification is necessary to protect sensitive loads from power quality disturbances by installing protection equipment. In this paper, 750 × 72 size PQ dataset is developed with 12 PQ disturbances i.e., voltage sag, voltage swell, interruption, flicker, harmonics, transients, swell with harmonics, sag with harmonics, interruption with harmonics, flicker with harmonics, flicker with swell and flicker with sag and with one fundamental signal. The PQ signals are decomposed at level 8 using the Multi-Resolution Analysis (MRA) approach and the daubechies mother wavelet. Every PQ signal is broken down into eight detailed coefficient signals and one approximation coefficient signal. From each broken signal, 8 features like Mean, Standard Deviation, RMS, Range, Energy, Kurtosis, Skewness, Entropy are extracted. This original PQ dataset with shape 750 × 72 is compressed to 750 × 21 shape with an autoencoder having training and testing losses 0.003 and 0.0025 respectively. Based on the objective function examined in this research, the autoencoder with 64 neurons in the latent space is regarded the optimal autoencoder. A RBFNN model with training accuracy 90 % and validation accuracy 92 % is developed to classify the 12 PQ disturbances and 1 fundamental signal with compressed PQ dataset. The proposed RBFNN model validated by comparing with decision tree, random forest and artificial neural networks. Both decision tree and random forest had over-fit problem. Where as simple ANN not performing will in comparison with RBFNN in both training and testing phases.

Corresponding author: Surender Reddy Salkuti, Department of Railroad and Electrical Engineering, Woosong University, Daejeon 34606, Republic of Korea, E-mail: surender@wsu.ac.kr

Funding source: Woosong University

Award Identifier / Grant number: Woosong University’s Academic Research Funding - 2023

Appendix A

A.1 Radial basis function neural network (RBFNN)

This section presents step by step procedure for training of RBFNN and also explains how trained RBFNN is used to classify the PQ disturbances. The sample architecture of RBFNN that used in this section to explain the training and prediction is shown in Figure 17.

Figure 17:

Sample RBFNN architecture.

The initial weight matrix (W _h k) is shown in the equation (35), the bias parameters at output layer are shown in equation (36), and the learning rate eta = 0.1. The sample data that is used to explain the RBFNN training and prediction is shown in Table 8.

(35) W h k = 0.1 0.2 − 0.1 0.3 0.2 − 0.2

(36) b k = 0.1 0.2

Table 8:

Sample data for temperature forecasting.

F1	F2	F3	F4	F5	PQ dist.
3.7	2.9	2.6	0.4	2.4	2
4.1	2.9	2.6	0.4	2.6	2
4.7	1.9	2	0.8	3.1	1
4.8	1.9	2	0.9	3.2	1
4.9	1.8	1.9	1.0	3.3	1
4.8	1.8	0.7	0.3	2.9	2

A.2 Unsupervised learning between input and hidden layers

Randomly pick three samples from the data as the mean vector for each centriod i.e., mu ₁ = [3.7 2.9 2.6 0.4 2.4], μ ₂ = [4.1 2.9 2.6 0.4 2.6] and μ ₃ = [4.7 1.9 2 0.8 3.1]. Calculate the Euclidean distance between each sample and mean vector using equation (28)

Iteration-1

Sample-1: X = [3.7 2.9 2.6 0.4 2.4]

ED1 = ( 3.7 − 3.7 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.4 ) 2 ) = 0
ED2 = ( 3.7 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.6 ) 2 ) = 0.447
ED3 = ( 3.7 − 4.7 ) 2 + ( 2.9 − 1.9 ) 2 + ( 2.6 − 2.0 ) 2 + ( 0.4 − 0.8 ) 2 + ( 2.4 − 3.1 ) 2 ) = 1.73

Sample-2: X = [4.1 2.9 2.6 0.4 2.6]

ED1 = ( 4.1 − 3.7 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.6 − 2.4 ) 2 ) = 0.44
ED2 = ( 4.1 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.6 − 2.6 ) 2 ) = 0
ED3 = ( 4.1 − 4.7 ) 2 + ( 2.9 − 1.9 ) 2 + ( 2.6 − 2.4 ) 2 + ( 0.4 − 0.8 ) 2 + ( 2.6 − 3.1 ) 2 ) = 5.93

Sample-3: X = [4.7 1.9 2.0 0.8 3.1]

ED1 = ( 4.7 − 3.7 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.8 − 0.4 ) 2 + ( 3.1 − 2.4 ) 2 ) = 1.73
ED2 = ( 4.7 − 4.1 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.8 − 0.4 ) 2 + ( 3.1 − 2.6 ) 2 ) = 1.43
ED3 = ( 4.7 − 4.7 ) 2 + ( 1.9 − 1.9 ) 2 + ( 2.0 − 2.0 ) 2 + ( 0.8 − 0.8 ) 2 + ( 3.1 − 3.1 ) 2 ) = 0

Sample-4: X = [4.8 1.9 2.0 0.9 3.2]

ED1 = ( 4.8 − 3.7 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.9 − 0.4 ) 2 + ( 3.2 − 2.4 ) 2 ) = 1.86
ED2 = ( 4.8 − 4.1 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.9 − 0.4 ) 2 + ( 3.2 − 2.6 ) 2 ) = 1.57
ED3 = ( 4.8 − 4.7 ) 2 + ( 1.9 − 1.9 ) 2 + ( 2.0 − 2.0 ) 2 + ( 0.9 − 0.8 ) 2 + ( 3.2 − 3.1 ) 2 ) = 0.173

Sample-5: X = [4.9 1.8 1.9 1.0 3.3]

ED1 = ( 4.9 − 3.7 ) 2 + ( 1.8 − 2.9 ) 2 + ( 1.9 − 2.6 ) 2 + ( 1.0 − 0.4 ) 2 + ( 3.3 − 2.4 ) 2 ) = 2.07
ED2 = ( 4.9 − 4.1 ) 2 + ( 1.8 − 2.9 ) 2 + ( 1.9 − 2.6 ) 2 + ( 1.0 − 0.4 ) 2 + ( 3.3 − 2.6 ) 2 ) = 1.79
ED3 = ( 4.8 − 4.7 ) 2 + ( 1.9 − 1.9 ) 2 + ( 2.0 − 2.0 ) 2 + ( 0.9 − 0.8 ) 2 + ( 3.2 − 3.1 ) 2 ) = 0.37

Sample-6: X = [4.8 1.8 0.7 0.3 2.9]

ED1 = ( 4.8 − 3.7 ) 2 + ( 1.8 − 2.9 ) 2 + ( 0.7 − 2.6 ) 2 + ( 0.3 − 0.4 ) 2 + ( 2.9 − 2.4 ) 2 ) = 2.51
ED2 = ( 4.8 − 4.1 ) 2 + ( 1.8 − 2.9 ) 2 + ( 0.7 − 2.6 ) 2 + ( 0.3 − 0.4 ) 2 + ( 2.9 − 2.6 ) 2 ) = 2.32
ED3 = ( 4.8 − 4.8 ) 2 + ( 1.8 − 1.9 ) 2 + ( 0.7 − 2.0 ) 2 + ( 0.3 − 0.8 ) 2 + ( 2.9 − 3.1 ) 2 ) = 1.41

From the calculations, observed that sample1has minimum Euclidean distance from centriod1. So mean vector for centriod-1 is updated as sample-1 itself. Similarly, sample2 has minimum Euclidean distance from centriod2. So mean vector for centriod-2 is updated as sample-2 itself. Sample3, sample4, sample5 and sample6 has minimum Euclidean distance from centriod3. So, mean vector for centriod-3 is updated as mean of Sample3, sample4, sample5 and sample6. The updated mean values of centriod1, centriod2 and centriod3 are shown in equation (111) and (112) and equation(113) respectively.

(37) μ 1 = 3.7 2.9 2.6 0.4 2.4

(38) μ 2 = 4.1 2.9 2.6 0.4 2.6

(39) μ 3 = 4.78 1.86 1.72 0.76 3.12

Iteration-2

Sample-1: X = [3.7 2.9 2.6 0.4 2.4]

ED1 = ( 3.7 − 3.7 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.4 ) 2 ) = 0
ED2 = ( 3.7 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.6 ) 2 ) = 0.447
ED3 = ( 3.7 − 4.79 ) 2 + ( 2.9 − 1.85 ) 2 + ( 2.6 − 1.66 ) 2 + ( 0.4 − 0.75 ) 2 + ( 2.4 − 3.12 ) 2 ) = 1.92

Sample-2: X = [4.1 2.9 2.6 0.4 2.6]

ED1 = ( 4.1 − 3.7 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.6 − 2.4 ) 2 ) = 0.44
ED2 = ( 4.1 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.6 − 2.6 ) 2 ) = 0
ED3 = ( 4.1 − 4.79 ) 2 + ( 2.9 − 1.85 ) 2 + ( 2.6 − 1.66 ) 2 + ( 0.4 − 0.75 ) 2 + ( 2.6 − 3.12 ) 2 ) = 1.65

Sample-3: X = [4.7 1.9 2.0 0.8 3.1]

ED1 = ( 4.7 − 3.7 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.8 − 0.4 ) 2 + ( 3.1 − 2.4 ) 2 ) = 1.73
ED2 = ( 4.7 − 4.1 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.8 − 0.4 ) 2 + ( 3.1 − 2.6 ) 2 ) = 1.46
ED3 = ( 4.7 − 4.79 ) 2 + ( 1.9 − 1.85 ) 2 + ( 2.0 − 1.66 ) 2 + ( 0.8 − 0.75 ) 2 + ( 3.1 − 3.12 ) 2 ) = 0.29

Sample-4: X = [4.8 1.9 2.0 0.9 3.2]

ED1 = ( 4.8 − 3.7 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.9 − 0.4 ) 2 + ( 3.2 − 2.4 ) 2 ) = 1.86
ED2 = ( 4.8 − 4.1 ) 2 + ( 1.9 − 2.9 ) 2 + ( 2.0 − 2.6 ) 2 + ( 0.9 − 0.4 ) 2 + ( 3.2 − 2.6 ) 2 ) = 1.57
ED3 = ( 4.8 − 4.79 ) 2 + ( 1.9 − 1.85 ) 2 + ( 2.0 − 1.66 ) 2 + ( 0.9 − 0.75 ) 2 + ( 3.2 − 3.12 ) 2 ) = 0.33

Sample-5: X = [4.9 1.8 1.9 1.0 3.3]

ED1 = ( 4.9 − 3.7 ) 2 + ( 1.8 − 2.9 ) 2 + ( 1.9 − 2.6 ) 2 + ( 1.0 − 0.4 ) 2 + ( 3.3 − 2.4 ) 2 ) = 2.08
ED2 = ( 4.9 − 4.1 ) 2 + ( 1.8 − 2.9 ) 2 + ( 1.9 − 2.6 ) 2 + ( 1.0 − 0.4 ) 2 + ( 3.3 − 2.6 ) 2 ) = 1.79
ED3 = ( 4.9 − 4.79 ) 2 + ( 1.8 − 1.85 ) 2 + ( 1.9 − 1.66 ) 2 + ( 1.0 − 0.75 ) 2 + ( 3.3 − 3.12 ) 2 ) = 0.38

Sample-6: X = [4.8 1.8 0.7 0.3 2.9]

ED1 = ( 4.8 − 3.7 ) 2 + ( 1.8 − 2.9 ) 2 + ( 0.7 − 2.6 ) 2 + ( 0.3 − 0.4 ) 2 + ( 2.9 − 2.4 ) 2 ) = 2.51
ED2 = ( 4.8 − 4.1 ) 2 + ( 1.8 − 2.9 ) 2 + ( 0.7 − 2.6 ) 2 + ( 0.3 − 0.4 ) 2 + ( 2.9 − 2.6 ) 2 ) = 2.32
ED3 = ( 4.8 − 4.79 ) 2 + ( 1.8 − 1.85 ) 2 + ( 0.7 − 1.66 ) 2 + ( 0.3 − 0.75 ) 2 + ( 2.9 − 3.12 ) 2 ) = 1.14

(40) μ 1 = 3.7 2.9 2.6 0.4 2.4

(41) μ 2 = 4.1 2.9 2.6 0.4 2.6

(42) μ 3 = 4.79 1.85 1.66 0.75 3.12

The values of Euclidean distance between centroids are calculated by using,

ED _c1−c2 = ED _c2−c1 = ( 3.7 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.6 ) 2 ) = 0.447
ED _c1−c3 = ED _c3−c1 = ( 3.7 − 4.79 ) 2 + ( 2.9 − 1.85 ) 2 + ( 2.6 − 1.66 ) 2 + ( 0.4 − 0.75 ) 2 + ( 2.4 − 3.12 ) 2 ) = 1.93
ED _c2−c3 = ED _c3−c2 = ( 4.1 − 4.79 ) 2 + ( 2.9 − 1.85 ) 2 + ( 2.6 − 1.66 ) 2 + ( 0.4 − 0.75 ) 2 + ( 2.6 − 3.12 ) 2 ) = 1.69

Euclidean distance between centroids are shown in Table 9. From the Table 9, it is observed that centroid-3 is near centroid-1 and centroid-1.

Table 9:

Euclidean distance between centroids.

	C1	C2	C3
C1	0	0.447	1.93
C2	0.447	0	1.69
C3	1.93	1.69	0

Calculate the standard deviation (Σ) using equation (29) for each centroid by considering width factor (P) as 2.

Standard deviation for centroid-1 i.e., σ ₁

σ 1 = 1 2 [ ( ( 3.7 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.6 ) 2 ) ] = 0.316

Standard deviation for centroid-2 i.e., σ ₂

σ 2 = 1 2 [ ( ( 3.7 − 4.1 ) 2 + ( 2.9 − 2.9 ) 2 + ( 2.6 − 2.6 ) 2 + ( 0.4 − 0.4 ) 2 + ( 2.4 − 2.6 ) 2 ) ] = 0.316

Standard deviation for centroid-3 i.e., σ ₃

σ 3 = 1 2 [ ( ( 4.1 − 4.79 ) 2 + ( 2.9 − 1.85 ) 2 + ( 2.6 − 1.66 ) 2 + ( 0.4 − 0.75 ) 2 + ( 2.6 − 3.12 ) 2 ) ] = 1.199

Calculate the output of first hidden neuron (h = 1) using equation (30) as shown below

sample-1: X = [3.7 2.9 2.6 0.4 2.4]

o 1 1 = e − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 1.00

sample-2: X = [4.1 2.9 2.6 0.4 2.6]

o 2 1 = e − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.37

sample-3: X = [4.7 1.9 2.0 0.8 3.1]

o 3 1 = e − [ 4.7 , 1.9 , 2.0 , 0.8 , 3.1 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.00

sample-4: X = [4.8 1.9 2.0 0.9 3.2]

o 4 1 = e − [ 4.8 , 1.9 , 2.0 , 0.9 , 3.2 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.00

sample-5: X = [4.9 1.8 1.9 1.0 3.3]

o 5 1 = e − [ 4.9 , 1.8 , 1.9 , 1.0 , 3.3 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.00

sample-6: X = [4.8 1.8 0.7 0.3 2.9]

o 6 1 = e − [ 4.8 , 1.8 , 0.7 , 0.3 , 2.9 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.00

Calculate the output of second hidden neuron (h = 2) using equation (30) as shown below

sample-1: X = [3.7 2.9 2.6 0.4 2.4]

o 1 2 = e − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.37

sample-2: X = [4.1 2.9 2.6 0.4 2.6]

o 2 2 = e − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 1.00

sample-3: X = [4.7 1.9 2.0 0.8 3.1]

o 3 2 = e − [ 4.7 , 1.9 , 2.0 , 0.8 , 3.1 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.00

sample-4: X = [4.8 1.9 2.0 0.9 3.2]

o 4 2 = e − [ 4.8 , 1.9 , 2.0 , 0.9 , 3.2 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.00

sample-5: X = [4.9 1.8 1.9 1.0 3.3]

o 5 2 = e − [ 4.9 , 1.8 , 1.9 , 1.0 , 3.3 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.00

sample-6: X = [4.8 1.8 0.7 0.3 2.9]

o 6 2 = e − [ 4.8 , 1.8 , 0.7 , 0.3 , 2.9 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.00

Calculate the output of third hidden neuron (h = 3) using equation (30) as shown below

sample-1: X = [3.7 2.9 2.6 0.4 2.4]

o 1 3 = e − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] − [ 4.79 , 1.85 , 1.66 , 0.75 , 3.12 ] 2 2 * 1.19 9 2 = 0.27

sample-2: X = [4.1 2.9 2.6 0.4 2.6]

o 2 3 = e − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] − [ 4.79 , 1.85 , 1.66 , 0.75 , 3.12 ] 2 2 * 1.19 9 2 = 0.37

sample-3: X = [4.7 1.9 2.0 0.8 3.1]

o 3 3 = e − [ 4.7 , 1.9 , 2.0 , 0.8 , 3.1 ] − [ 4.79 , 1.85 , 1.66 , 0.75 , 3.12 ] 2 2 * 1.19 9 2 = 0.96

sample-4: X = [4.8 1.9 2.0 0.9 3.2]

o 4 3 = e − [ 4.8 , 1.9 , 2.0 , 0.9 , 3.2 ] − [ 4.79 , 1.85 , 1.66 , 0.75 , 3.12 ] 2 2 * 1.19 9 2 = 0.95

sample-5: X = [4.9 1.8 1.9 1.0 3.3]

o 5 3 = e − [ 4.9 , 1.8 , 1.9 , 1.0 , 3.3 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.94

sample-6: X = [4.8 1.8 0.7 0.3 2.9]

o 6 3 = e − [ 4.8 , 1.8 , 0.7 , 0.3 , 2.9 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.66

The final dataset that is used to update the weights (W _hk) and bias (b _k) is shown in Table 10.

Table 10:

Data that used to update weights and bias between hidden layer and output layer.

Sample	O ₁	O ₂	O ₃	O _A
1	1.00	0.37	0.27	2
2	0.37	1.00	0.37	2
3	0.00	0.00	0.96	1
4	0.00	0.00	0.95	1
5	0.00	0.00	0.94	1
6	0.00	0.00	0.66	2

A.3 Supervised learning between hidden and output layer

Iteration-1

Sample-1 = [1.00 0.37 0.27]

Calculate input of the RBFNN output layer using equation (31)
(43) n e t k = 0.1 − 0.1 0.2 0.2 0.3 − 0.2 * 1.00 0.37 0.27 + 0.1 0.2 = 0.217 0.457
Calculate output of the RBFNN output layer using equation (32)
(44) o 1 = e [ 0.217 ] / e [ 0.217 ] + e [ 0.457 ] = 0.44

(45) o 2 = e [ 0.457 ] / e [ 0.217 ] + e [ 0.457 ] = 0.56
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 2 i.e., out of the two neurons in the output layer, second neuron is activated.Therefore, O2 is used to calculate updated weight and bias matrices.
(46) W h k = 0.2 0.3 − 0.2 + 0.1 * ( 1 − 0.56 ) * 1.00 0.37 0.27 = 0.244 0.316 − 0.188

(47) b k = 0.2 + 0.1 * ( 1 − 0.56 ) = 0.244
Therefore, Updated weight and bias matrices after passing sample-1 are,
(48) W h k = 0.1 0.244 − 0.1 0.316 0.2 − 0.188 b k = 0.1 0.244

Sample-2 = [0.37 1.00 0.37]
Calculate input of the RBFNN output layer using equation (31)
(49) n e t k = 0.1 − 0.1 0.2 0.244 0.316 − 0.188 * 0.37 1.00 0.37 + 0.1 0.244 = 0.111 0.581
Calculate output of the RBFNN output layer using equation (32)
(50) o 1 = e [ 0.111 ] / e [ 0.111 ] + e [ 0.581 ] = 0.385

(51) o 2 = e [ 0.581 ] / e [ 0.111 ] + e [ 0.581 ] = 0.615
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 2 i.e., out of the two neurons in the output layer, second layer is activated.Therefore, O2 is used to calculate updated weight and bias matrices.
(52) W h k = 0.244 0.316 − 0.188 + 0.1 * ( 1 − 0.615 ) * 0.37 1.00 0.37 = 0.258 0.355 − 0.174

(53) b k = 0.244 + 0.1 * ( 1 − 0.615 ) = 0.283
Therefore, Updated weight and bias matrices after passing sample-2 are,
(54) W h k = 0.1 0.258 − 0.1 0.355 0.2 − 0.174 b k = 0.1 0.283

Sample-3 = [0.00 0.00 0.96]
Calculate input of the RBFNN output layer using equation (31)
(55) n e t k = 0.1 − 0.1 0.2 0.258 0.355 − 0.174 * 0.00 0.00 0.96 + 0.1 0.283 = 0.292 0.116
Calculate output of the RBFNN output layer using equation (32)
(56) o 1 = e [ 0.292 ] / e [ 0.292 ] + e [ 0.116 ] = 0.544

(57) o 2 = e [ 0.116 ] / e [ 0.292 ] + e [ 0.116 ] = 0.116
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(58) W h k = 0.1 − 0.1 0.2 + 0.1 * ( 1 − 0.554 ) * 0.00 0.00 0.96 = 0.1 − 0.1 0.244

(59) b k = 0.1 + 0.1 * ( 1 − 0.544 ) = 0.146
Therefore, Updated weight and bias matrices after passing sample-3 are,
(60) W h k = 0.1 0.258 − 0.1 0.355 0.244 − 0.174 b k = 0.146 0.283

Sample-4 = [0.00 0.00 0.95]
Calculate input of the RBFNN output layer using equation (31)
(61) n e t k = 0.1 − 0.1 0.244 0.258 0.355 − 0.174 * 0.00 0.00 0.95 + 0.146 0.283 = 0.377 0.117
Calculate output of the RBFNN output layer using equation (32)
(62) o 1 = e [ 0.377 ] / e [ 0.377 ] + e [ 0.117 ] = 0.565

(63) o 2 = e [ 0.117 ] / e [ 0.377 ] + e [ 0.117 ] = 0.434
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(64) W h k = 0.1 − 0.1 0.244 + 0.1 * ( 1 − 0.565 ) * 0.00 0.00 0.95 = 0.1 − 0.1 0.285

(65) b k = 0.146 + 0.1 * ( 1 − 0.565 ) = 0.189
Therefore, Updated weight and bias matrices after passing sample-4 are,
(66) W h k = 0.1 0.258 − 0.1 0.355 0.285 − 0.174 b k = 0.189 0.283

Sample-5 = [0.00 0.00 0.94]
Calculate input of the RBFNN output layer using equation (31)
(67) n e t k = 0.1 − 0.1 0.285 0.258 0.355 − 0.174 * 0.00 0.00 0.94 + 0.189 0.283 = 0.457 0.119
Calculate output of the RBFNN output layer using equation (32)
(68) o 1 = e [ 0.457 ] / e [ 0.457 ] + e [ 0.119 ] = 0.585

(69) o 2 = e [ 0.119 ] / e [ 0.457 ] + e [ 0.119 ] = 0.416
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated. Therefore, O1 is used to calculate updated weight and bias matrices.
(70) W h k = 0.1 − 0.1 0.285 + 0.1 * ( 1 − 0.585 ) * 0.00 0.00 0.94 = 0.1 − 0.1 0.324

(71) b k = 0.189 + 0.1 * ( 1 − 0.585 ) = 0.231
Therefore, Updated weight and bias matrices after passing sample-5 are,
(72) W h k = 0.1 0.258 − 0.1 0.355 0.324 − 0.174 b k = 0.231 0.283

Sample-6 = [0.00 0.00 0.66]
Calculate input of the RBFNN output layer using equation (31)
(73) n e t k = 0.1 − 0.1 0.324 0.258 0.355 − 0.174 * 0.00 0.00 0.66 + 0.231 0.283 = 0.445 0.168
Calculate output of the RBFNN output layer using equation (32)
(74) o 1 = e [ 0.445 ] / e [ 0.445 ] + e [ 0.168 ] = 0.569

(75) o 2 = e [ 0.168 ] / e [ 0.445 ] + e [ 0.168 ] = 0.432
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 2 i.e., out of the two neurons in the output layer, second neuron is activated.Therefore, O2 is used to calculate updated weight and bias matrices.
(76) W h k = 0.258 0.355 − 0.174 + 0.1 * ( 1 − 0.432 ) * 0.00 0.00 0.66 = 0.258 0.355 − 0.136

(77) b k = 0.283 + 0.1 * ( 1 − 0.432 ) = 0.339
Therefore, Updated weight and bias matrices after passing sample-6 are,
(78) W h k = 0.1 0.258 − 0.1 0.355 0.324 − 0.136 b k = 0.231 0.339

Iteration-2

Sample-1 = [1.00 0.37 0.27]

Calculate input of the RBFNN output layer using equation (31)
(79) n e t k = 0.1 − 0.1 0.324 0.258 0.355 − 0.136 * 1.00 0.37 0.27 + 0.231 0.339 = 0.381 0.692
Calculate output of the RBFNN output layer using equation (32)
(80) o 1 = e [ 0.381 ] / e [ 0.381 ] + e [ 0.692 ] = 0.423

(81) o 2 = e [ 0.692 ] / e [ 0.381 ] + e [ 0.692 ] = 0.577
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 2 i.e., out of the two neurons in the output layer, second neuron is activated.Therefore, O2 is used to calculate updated weight and bias matrices.
(82) W h k = 0.258 0.355 − 0.136 + 0.1 * ( 1 − 0.577 ) * 1.00 0.37 0.27 = 0.301 0.370 − 0.125

(83) b k = 0.339 + 0.1 * ( 1 − 0.577 ) = 0.382
Therefore, Updated weight and bias matrices after passing sample-5 are,
(84) W h k = 0.1 0.301 − 0.1 0.370 0.324 − 0.125 b k = 0.231 0.382

Sample-2 = [0.37 1.00 0.37]
Calculate input of the RBFNN output layer using equation (31)
(85) n e t k = 0.1 − 0.1 0.324 0.301 0.370 − 0.125 * 0.37 1.00 0.37 + 0.231 0.382 = 0.288 0.817
Calculate output of the RBFNN output layer using equation (32)
(86) o 1 = e [ 0.288 ] / e [ 0.288 ] + e [ 0.817 ] = 0.371

(87) o 2 = e [ 0.817 ] / e [ 0.288 ] + e [ 0.817 ] = 0.629
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(88) W h k = 0.1 − 0.1 0.324 + 0.1 * ( 1 − 0.371 ) * 0.37 1.00 0.37 = 0.319 0.408 − 0.111

(89) b k = 0.231 + 0.1 * ( 1 − 0.371 ) = 0.419
Therefore, Updated weight and bias matrices after passing sample-3 are,
(90) W h k = 0.1 0.314 − 0.1 0.408 0.324 − 0.111 b k = 0.231 0.419

Sample-4 = [0.00 0.00 0.95]
Calculate input of the RBFNN output layer using equation (31)
(91) n e t k = 0.1 − 0.1 0.367 0.314 0.408 − 0.111 * 0.00 0.00 0.95 + 0.275 0.419 = 0.624 0.313
Calculate output of the RBFNN output layer using equation (32)
(92) o 1 = e [ 0.624 ] / e [ 0.624 ] + e [ 0.313 ] = 0.577

(93) o 2 = e [ 0.313 ] / e [ 0.624 ] + e [ 0.313 ] = 0.423
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(94) W h k = 0.1 − 0.1 0.367 + 0.1 * ( 1 − 0.577 ) * 0.00 0.00 0.95 = 0.1 − 0.1 0.407

(95) b k = 0.275 + 0.1 * ( 1 − 0.577 ) = 0.317
Therefore, Updated weight and bias matrices after passing sample-4 are,
(96) W h k = 0.1 0.314 − 0.1 0.408 0.407 − 0.111 b k = 0.317 0.419

Sample-5 = [0.00 0.00 0.94]
Calculate input of the RBFNN output layer using equation (31)
(97) n e t k = 0.1 − 0.1 0.407 0.314 0.408 − 0.111 * 0.00 0.00 0.94 + 0.317 0.419 = 0.699 0.314
Calculate output of the RBFNN output layer using equation (32)
(98) o 1 = e [ 0.699 ] / e [ 0.699 ] + e [ 0.314 ] = 0.595

(99) o 2 = e [ 0.314 ] / e [ 0.699 ] + e [ 0.314 ] = 0.405
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 1 i.e., out of the two neurons in the output layer, first neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(100) W h k = 0.1 − 0.1 0.407 + 0.1 * ( 1 − 0.595 ) * 0.00 0.00 0.94 = 0.1 − 0.1 0.445

(101) b k = 0.317 + 0.1 * ( 1 − 0.595 ) = 0.358
Therefore, Updated weight and bias matrices after passing sample-5 are,
(102) W h k = 0.1 0.314 − 0.1 0.408 0.445 − 0.111 b k = 0.358 0.419

Sample-6 = [0.00 0.00 0.66]
Calculate input of the RBFNN output layer using equation (31)
(103) n e t k = 0.1 − 0.1 0.445 0.314 0.408 − 0.111 * 0.00 0.00 0.66 + 0.358 0.419 = 0.652 0.345
Calculate output of the RBFNN output layer using equation (32)
(104) o 1 = e [ 0.652 ] / e [ 0.652 ] + e [ 0.345 ] = 0.576

(105) o 2 = e [ 0.345 ] / e [ 0.652 ] + e [ 0.345 ] = 0.345
update W _hk using equation (33) and b _k using (34)
For the above sample, predicted output of RBFNN is 2 i.e., out of the two neurons in the output layer, second neuron is activated.Therefore, O1 is used to calculate updated weight and bias matrices.
(106) W h k = 0.314 0.408 − 0.11 + 0.1 * ( 1 − 0.345 ) * 0.00 0.00 0.66 = 0.314 0.408 − 0.073

(107) b k = 0.317 + 0.1 * ( 1 − 0.345 ) = 0.476
Therefore, Updated weight and bias matrices after passing sample-5 are,
(108) W h k = 0.1 0.314 − 0.1 0.408 0.445 − 0.073 b k = 0.358 0.476

Appendix B: Prediction using trained RBFNN

RBFNN is trained for two iterations with 6 samples. After training updated weights (W _hk), bias (b _k), centroids (μ) and standard deviation (σ) are shown below

(109) W h k = 0.1 0.314 − 0.1 0.408 0.445 − 0.07

(110) b k = 0.358 , 0.476

(111) C e n t r o i d 1 = μ 1 = 3.7 , 2.9 , 2.6 , 0.4 , 2.4

(112) C e n t r o i d 2 = μ 2 = 4.1 , 2.9 , 2.6 , 0.4 , 2.6

(113) Centroid 3 = μ 3 = 4.78 , 1.86 , 1.72 , 0.76 , 3.12

(114) σ 1 = σ 2 = 0.316 σ 3 = 1.199

Now trained RBFNN is predict the output variable based on inputs {4.0, 1.7, 1.7, 1.1, 2.8} as shown below

Calculate output of each hidden neuron using centroids (μ ₁, μ ₂, μ ₃) and standard deviation (σ ₁, σ ₂, σ ₃) as shown below
o 1 = e − [ 4.0 , 1.7 , 1.7 , 1.1 , 2.8 ] − [ 3.7 , 2.9 , 2.6 , 0.4 , 2.4 ] 2 2 * 0.31 6 2 = 0.00

o 2 = e − [ 4.0 , 1.7 , 1.7 , 1.1 , 2.8 ] − [ 4.1 , 2.9 , 2.6 , 0.4 , 2.6 ] 2 2 * 0.31 6 2 = 0.00

o 3 = e − [ 4.0 , 1.7 , 1.7 , 1.1 , 2.8 ] − [ 4.78 , 1.86 , 1.72 , 0.76 , 3.12 ] 2 2 * 0.31 6 2 = 0.085
Calculate output of output neurons using weights (W _hk), bias (b _k) and output of hidden neurons as shown below
(115) n e t k = 0.1 − 0.1 0.445 0.314 0.408 − 0.07 * 0.00 0.00 0.085 + 0.358 0.476 = 0.652 0.423

(116) o 1 = e [ 0.652 ] / ( e [ 0.652 ] + e [ 0.423 ] ) = 0.6

(117) o 2 = e [ 0.423 ] / ( e [ 0.652 ] + e [ 0.423 ] ) = 0.4

As the first output neuron has highest probability, the given sample corresponding to label “1” PQ disturbance.

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no conflict of interest.
Research funding: Woosong University’s Academic Research Funding – 2023.
Data availability: Not applicable.

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Received: 2023-05-01

Accepted: 2023-09-06

Published Online: 2023-09-25

Published in Print: 2024-12-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ijeeps-2023-0143

Keywords for this article

power quality; machine learning; radial basis function neural network; autoencoder

Creative Commons

BY 4.0

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