Improved probabilistic neural networks with self-adaptive strategies for transformer fault diagnosis problem

Abstract

Probabilistic neural network has successfully solved all kinds of engineering problems in various fields since it is proposed. In probabilistic neural network, Spread has great influence on its performance, and probabilistic neural network will generate bad prediction results if it is improperly selected. It is difficult to select the optimal manually. In this article, a variant of probabilistic neural network with self-adaptive strategy, called self-adaptive probabilistic neural network, is proposed. In self-adaptive probabilistic neural network, Spread can be self-adaptively adjusted and selected and then the best selected Spread is used to guide the self-adaptive probabilistic neural network train and test. In addition, two simplified strategies are incorporated into the proposed self-adaptive probabilistic neural network with the aim of further improving its performance and then two versions of simplified self-adaptive probabilistic neural network (simplified self-adaptive probabilistic neural networks 1 and 2) are proposed. The variants of self-adaptive probabilistic neural networks are further applied to solve the transformer fault diagnosis problem. By comparing them with basic probabilistic neural network, and the traditional back propagation, extreme learning machine, general regression neural network, and self-adaptive extreme learning machine, the results have experimentally proven that self-adaptive probabilistic neural networks have a more accurate prediction and better generalization performance when addressing the transformer fault diagnosis problem.

Keywords

Classification self-adaptation probabilistic neural network back propagation general regression neural network extreme learning machine fault diagnosis

Introduction

Fault diagnosis (FD)^1,2 begins with mechanical equipment FD. With the increment of the technical level of modern equipment and complexity, the effects of equipment failure on the production are also significantly increased. Therefore, in order to ensure that the equipment is reliable, operates effectively, and gives full importance to its effectiveness, it is necessary to develop FD technology. In FD, the state information of facilities in operation or relatively static conditions is fully investigated through the technology of modern testing, monitoring and computer analysis, and other methods. Then, the technical status of the equipment is analyzed; the nature and the cause of the faults are determined with the aim of forecasting the trend of faults and preparing for the necessary countermeasures. FD technology can be used to diagnose the fault symptoms and causes. It is advantageous to eliminate the fault and safety hazards as soon as possible, to avoid unnecessary loss, so it has high economic and social benefits.

Probabilistic neural network (PNN), first proposed by Specht³ in 1990, is a kind of parallel algorithm. It is developed based on Bayes classification rules and Parzen window probability density function (PDF) estimation method. PNN is a kind of widely used artificial neural network (NN), which has a simple structure. In practical application, especially in solving classification problems, its advantage is that linear learning algorithm in PNN is well capable of achieving results of the nonlinear learning algorithms, while preserving the high accuracy of nonlinear algorithm. In PNN, the network weights are the model of the sample distribution, so the network need not train. Therefore, it can meet the real-time processing on the training requirements.

Although PNN has various advantages mentioned above, it has an inherent shortcoming. In PNN, one of the most parameters is Spread that has great influence on PNN’s performance. The Spread that is also so-called smoothing parameter is the standard deviation (SD) of the underlying Gaussian distribution, and it is a determinant of the receptive width of the Gaussian window for the PDF of the training set.⁴ If the Spread is improperly selected, it cannot guide PNN’s training; hence, PNN will generate bad prediction results.⁴ In this article, a self-adaptive strategy is incorporated into PNN and then a self-adaptive probabilistic neural network (SaPNN) is proposed. In SaPNN, the best Spread can be self-adaptively selected all the time; thus, SaPNN can always get the best prediction accuracy. In addition, in order to reduce the computational requirements of SaPNN, two simplified strategies are combined with the proposed SaPNN and then two variant of SaPNNs (SSaPNN1 and SSaPNN2) are proposed. These two simplified SaPNNs are able to reach the same prediction accuracy with SaPNN using less computational requirements. In variants of SaPNNs, Spread can be self-adaptively selected; therefore, there is no parameters to be adjusted in the training process. The transformer FD problem is addressed by the variants of SaPNNs. By comparing them with basic PNN, and the traditional back propagation (BP),^5,6 extreme learning machine (ELM),^7–10 general regression neural network (GRNN),^11,12 and self-adaptive extreme learning machine (SaELM),¹³ the results have experimentally shown that SaPNNs have a more accurate prediction rate and better generalization performance when addressing the transformer FD problem.

The remainder of this article is organized as follows: section “Preliminaries” reviews the related preliminaries, including transformer FD problem and PNN. Sections “SaPNN” and “SaPNN model for transformer FD problem” represent the framework of variants of SaPNNs and the detailed classification using three SaPNNs. In section “Simulation results,” a series of comparison experiments on FD problem are conducted. Section “Conclusion” provides our concluding remarks and points out our future work orientation.

Preliminaries

Transformer FD

Abnormal condition or information will be generated when the transformer is getting out of order to a certain degree. Fault analysis is to collect the abnormal phenomenon or information of the transformer, which is then analyzed. According to the analyses of these phenomena or information, the type of fault, the severity, and fault location are determined. Therefore, the purpose of transformer FD is first to correctly determine whether the current state of the normal operation of the equipment is in normal or abnormal state. If the transformer is in abnormal state, the nature, type, and cause of the fault are judged. The fault may be insulation fault, overheating fault, or mechanical failure. If it is insulation fault, it may be insulation aging, moisture or discharge fault; if it is discharge fault, which type of discharge should be further determined. Transformer FD is also to predict possible development of fault according to the fault information or its processed results. That is to say, the fault severity and development trend should also be diagnosed. Subsequently, some measures are put forward to control, prevent, and eliminate the fault. The reasonable method of equipment maintenance and the corresponding anti-accident measures are also put forward. Some improved suggestions are proposed with respect to the equipment design, manufacture, and assembly, which can provide scientific basis and suggestions for the modernization of equipment.

Analysis of dissolved gas in transformer oil is an important method for the diagnosis of transformer internal faults. In China, the improved three-ratio method¹⁴ is widely used in a large number of applications at present. However, the three-ratio method as the discrimination criteria of the transformer FD has two problems, which are coding defect and critical value criterion defect.¹⁴

Various methods have been proposed to solve FD problems. Wang et al.¹⁵ put forward a novel nearest prototype classifier to diagnose faults in a power plant and then an improved particle swarm optimization (PSO) was used to optimize the position of the prototypes. Sun et al.¹⁶ put forward an enhanced accuracy of FD method in Smart Grid via rough sets together with genetic algorithm (GA) and Tabu search (TS). After the simulated fault and system data were generated, Zhao et al.¹⁷ formulated the FD problem as an optimization problem, which was then solved by an improved differential evolution (DE). Tang et al.¹⁸ established a multi-fault classification model via the support vector machine (SVM) trained by chaotic PSO that was applied to the FD of rotating machines. Fault diagnosis and isolation (FDI) on industrial systems was formulated as an optimization problem, and it was then addressed by PSO and ant colony optimization (ACO) algorithms.¹⁹ By integrating the theory of fuzzy sets, pairwise probabilistic multi-label classification, and decision-by-threshold, a new framework of simultaneous FD, called fuzzy and probabilistic simultaneous fault diagnosis (FPSD), was proposed by Vong et al.²⁰ Xia et al.²¹ put forward a multi-objective unsupervised feature selection algorithm (MOUFSA) that is verified by nine UCI datasets and five fault recognition datasets. Fathabadi²² proposed a soft computing method via discrete wavelet transform and a hardware via two-stage finite impulse response to detect short-circuit faults in power transmission lines. Qin et al.²³ proposed a method to recognize power cable fault types via an annealed chaotic competitive learning network. Kang et al.²⁴ put forward a highly reliable FD scheme for incipient low-speed rolling element bearing failures. In this method, a binary bat algorithm (BBA)²⁵ is used to filter discriminative fault features.

Another one of the most representative paradigms for solving FD problem is NNs. NN is an effective and efficient problem-solving algorithm, which has been successfully used in several practical problems. In order to improve the performance of NNs, some state-of-the-art developed intelligent algorithms have been incorporated into NNs, such as GA,²⁶ shuffled frog leaping algorithm (SFLA),²⁷ and PSO.²⁸

Artificial NN has the advantages of distributed parallel processing, self-adaption, self-learning, associative memory, and nonlinear mapping, which has opened up a new way to solve this problem. In our current work, three improved versions of PNNs in combination with three-ratio method are used to deal with transformer FD problem, which can be found in the following sections in detail.

PNN

PNN is a kind of feedforward network with the development of radial basis function network, and its theoretical basis is the minimum Bayesian risk criterion (i.e. Bayesian decision theory). PNN, as a kind of radial basis function NN, is suitable for pattern classification. When the value of Spread is close to 0, it constitutes the nearest neighbor classifier. When the value of Spread is large, it constitutes the nearest neighbor classifier of several training samples. The model of PNN is composed of four layers, which are input layer, pattern layer, summation layer, and output layer, and its basic structure can be shown in Figure 1.

Figure 1.

Structure of PNN.

The input layer receives the value of the training sample and transfers the feature vector to the network. The number of neurons in input layer is equal to the dimension of the sample vector. The matching relation between feature vector and each pattern in training set is calculated in the pattern layer, and the number of neurons in pattern layer is equal to the sum of training samples for all categories. Thus, the output of each pattern neuron in this layer can be given as

f (X, W_{i}) = \exp [- \frac{{(X - W_{i})}^{T} (X - W_{i})}{2 δ^{2}}]

(1)

where W_i is the weight between input layer and pattern layer and δ is smoothing factor that plays a vital role in classification problem.

The third layer is summation layer, which is the accumulation of the probability of a certain class, according to equation (1) and then the PDF of the fault mode is obtained. Each class that has only one layer unit is only connected with the pattern layer unit of its own class, while it is not connected with the other units in the pattern layer. Therefore, the output of the neurons of its own class in the summation layer will simply be added, while the output is independent of the output of the pattern layer which belongs to other classes. The output of the summation layer is proportional to the probability density of the kernel. The probability statistics of all kinds of the output layer can be obtained by normalizing the output layer. The output decision-making layer is composed of a simple threshold discriminator, whose role is to choose a maximum inspection probability density of neurons as the output of the whole system in various fault modes of the probability density estimation. The neuron in output layer is a kind of competitive neuron, and each neuron corresponds to a data type (i.e. fault mode). The number of neurons in output layer is equal to the number of classes of training sample data. Output layer receives all PDF from summation layer, and the output of neuron with the biggest PDF is 1; that is to say, the corresponding class is the exact pattern recognition class to be determined, while the output of the other neurons is 0.

FD method based on PNN is a widely accepted decision-making method in probability statistics. It can be described as follows: Suppose there are two known fault modes $θ_{A}$ and $θ_{B}$ with regard to a fault sample X = (x₁, x₂, …, x_n) to be judged

\begin{matrix} If h_{A} l_{A} f_{A} (X) > h_{B} l_{B} f_{B} (X) holds, then X \in θ_{A} \\ If h_{A} l_{A} f_{A} (X) < h_{B} l_{B} f_{B} (X) holds, then X \in θ_{B} \end{matrix}

where h_A and h_B are, respectively, the prior probability of the fault modes $θ_{A}$ and $θ_{B}$ (h_A = N_A/N, h_B = N_B/N); N_A and N_B are, respectively, the number of training samples for failure modes $θ_{A}$ and $θ_{B}$ , and N is the total number of training samples. l_A is the cost factor when X is classified into $θ_{A}$ while it belongs to $θ_{B}$ in reality. Similarly, l_B is the cost factor when X is classified into $θ_{B}$ while it belongs to $θ_{A}$ in reality. f_A(X) and f_B(X) are the PDF of the fault modes $θ_{A}$ and $θ_{B}$ , respectively, and they cannot usually be obtained accurately, and their statistical values are only achieved according to the existing fault samples.

In 1962, Parzen²⁹ put forward a PDF estimation method from a known random sample. In this method, if the sample size is large enough, it can continuously and smoothly approach the original PDF. The PDF estimation obtained by the Parzen method is as follows

f_{A} (X) = \frac{1}{{(2 π)}^{P / 2} δ^{P}} \frac{1}{m} \sum \exp [- \frac{{(X - W_{ai})}^{T} (X - W_{ai})}{2 δ^{2}}]

(2)

where m is the number of training samples of the fault mode $θ_{A}$ ; δ is the smoothing parameter, which can determine the width of the bell-shaped curve in which the sample point is considered as the center.

SaPNN

As discussed before, for the basic PNN, the selection of Spread is a difficult problem. In the present work, a self-adaptive strategy and two simplified technologies are originally proposed in order to select the best Spread. The framework of three improved PNNs can be described in the following.

SaPNN

First, a self-adaptive strategy is incorporated into the basic PNN, and self-adaptive PNN (SaPNN) is then proposed. In SaPNN, the best Spread is used to form the NNs and to guide the prediction all the time. The mainframe of SaPNN can be shown in Algorithm 1.

Algorithm 1:

SaPNN algorithm

Begin

Step 1:Initialization. Set the minimum and maximum of Spread Spread_min and Spread_max; set the maximal generation MaxIter and current generation counter G = 1; initialize the interval of Spread Spread_Interval.

Step 2:While Spread_Interval is not less than 0.1 do

Calculate all the possible Spread (say Spread₁, …, Spread_M) according to Spread_min, Spread_max, and Spread_Interval.

for j = 1: M (all the possible Spread between Spread_min and Spread_max) do

While G < MaxIter do

Construct the basic PNN with Spread_j.

Train and predict as shown in section “PNN.”

Evaluate the prediction results.

G = G + 1.

end while

end for j

Select the best Spread Spread_best with minimum error.

Update Spread_min, Spread_max and Spread_Interval according to equations (4)–(6).

Step 3: end while

Step 4: Output the best Spread and prediction accuracy.

End.

In Algorithm 1, Spread_min and Spread_max indicate the minimum and maximum of Spread. Spread_Interval is the interval of Spread. That is to say, Spread₁ =Spread_min, Spread₂ = Spread_min + Spread_Interval, Spread₃ = Spread_min + 2 × Spread_Interval, …, Spread_j = Spread_min + (j − 1) × Spread_Interval, …, Spread_M =Spread_max (j = 1, 2, …, M). Here, M is the number of spread that can be given as

M = round (\frac{Sprea d_{\max} - Sprea d_{\min}}{Sprea d_{Interval}}) + 1

(3)

where round(x) rounds each element of x to the nearest integer.

For each Spread (Spread_j, j = 1, 2, …, M), SaPNN is formed and performs the training and testing process. In order to reduce the influence of randomness, MaxIter independent runs are implemented for each Spread_j. After that, the best Spread Spread_best is selected that has the minimum error. Spread_min, Spread_max, and Spread_Interval are then adjusted according to Spread_best, as shown in equations (4)–(6)

Sprea d_{\min} = max (Sprea d_{best} - Sprea d_{Interval}, Sprea d_{\min})

(4)

Sprea d_{\max} = min (Sprea d_{best} + Sprea d_{Interval}, Sprea d_{\max})

(5)

Sprea d_{Interval} = 0.1 \times round (\frac{Sprea d_{Interval}}{2 \times 10})

(6)

In general, the minimal Spread is 0.1, so the Spread_Interval is updated as equation (6). When Spread_Interval is less than 0.1, SaPNN stops and outputs the best Spread Spread_best and final best prediction accuracy for this case.

SSaPNN1

As mentioned above, in SaPNN, the Spread_min, Spread_max, and Spread_Interval are updated many times. In order to reduce updating times and accelerate convergent speed, a simplified version of SaPNN, called SSaPNN1, is proposed. In SSaPNN1, Spread_min, Spread_max, and Spread_Interval are updated only once. At iteration 2, the Spread_Interval is set to 0.1 directly after updating Spread_min and Spread_max according to equations (4) and (5). After that, the SSaPNN1 stops and outputs the best Spread Spread_best and final best prediction accuracy for this case. Its framework can be given in Algorithm 2.

Algorithm 2:

SSaPNN1 algorithm

Begin

Step 1:Initialization. Set the minimum and maximum of Spread Spread_min and Spread_max; set the maximal generation MaxIter, current iteration counter t = 1 and current generation counter G = 1; initialize the interval of Spread Spread_Interval.

Step 2: While t≤2 do

Calculate all the possible Spread (say Spread₁, …, Spread_M) according to Spread_min, Spread_max, and Spread_Interval.

for j = 1: M (all the possible Spread between Spread_min and Spread_max) do

While G < MaxIterdo

Construct the basic PNN with Spread_j.

Train and predict as shown in section “PNN.”

Evaluate the prediction results.

G = G + 1.

end while

end for j

Select the best Spread Spread_best with minimum error.

Update Spread_min and Spread_max according to equations (4) and (5).

Spread_Interval = 0.1.

t = t + 1.

Step 3: end while

Step 4: Output the best Spread and prediction accuracy.

End.

SSaPNN2

As mentioned in section “SaPNN,” the number of PNN evaluations at each iteration is equal to the number of spread M. At iteration 2, the PNN with certain Spread has been implemented at iteration 1, therefore, there is no need to be implemented again. On contrast, the results of prediction accuracy at iteration 1 can be used directly at iteration 2. The computational requirements of SaPNN can be further reduced, and another simplified version of SaPNN, called SSaPNN2, is proposed. Its framework can be given in Algorithm 3.

Algorithm 3:

SSaPNN2 algorithm

Begin

Step 1: Initialization. Set the minimum and maximum of Spread Spread_min and Spread_max; set the maximal generation MaxIter, current iteration counter t = 1 and current generation counter G = 1; initialize the interval of Spread Spread_Interval.

Step 2: While Spread_Interval is not less than 0.1 do

Calculate all the possible Spread according to Spread_min, Spread_max, and Spread_Interval.

Remove the repeated Spread at iteration t (t≥2), and the remained Spread is Spread₁, …, Spread_M.

for j = 1:M (all the possible Spread) do

While G < MaxIter do

Construct the basic PNN with Spread_j.

Train and predict as shown in section “PNN.”

Evaluate the prediction results.

G = G + 1.

end while

end for j

Select the best Spread Spread_best with minimum error.

Update Spread_min, Spread_max and Spread_Interval according to equations (4)–(6).

t = t + 1.

Step 3: end while

Step 4: Output the best Spread and prediction accuracy.

End.

In Algorithm 3, at iteration 1, SSaPNN2 has the same implementation process with SaPNN. While the repeated Spread at iteration 1 will be removed. This operation will significantly reduce its computational requirements without losing the feature of SaPNN. This process will be repeated until the Spread_Interval is less than 0.1. Finally, SSaPNN2 outputs the best Spread Spread_best and the best prediction accuracy for a certain problem.

SaPNN model for transformer FD problem

In any network model, the selection of the input feature vector must be able to correctly reflect the characteristics of the problem. If the fault feature does not include enough identification information or the information that cannot be extracted to reflect the fault characteristics, the diagnosis results will be greatly affected. Dissolved gas analysis in oil can well reflect the transformer latent fault, and the improved three-ratio method has the highest accuracy in a variety of diagnostic methods. Herein, three pairs of the ratio of dissolved gas in oil content are considered as the input feature vectors of the NN, and the output feature vector is the type of transformer fault.

PNN has the advantages of simple structure and simple training, and it has strong nonlinear classification ability. Herein, the fault sample space is mapped to the fault pattern space, forming a diagnosis system that has stronger fault-tolerant ability and the ability of the self-adaptive structure, so as to improve the accuracy of FD.

In our current work, the dissolved gas in the oil is first analyzed, and the FD model based on SaPNNs is then established, which is based on the improved three-ratio method. Based on the analyses above, the FD model using SaPNNs is designed, and its flowchart can be shown in Figure 2.

Figure 2.

Flowchart of SaPNNs for the transformer fault diagnosis problem.

Simulation results

In this section, the performance of variants of SaPNNs when addressing transformer FD problem is fully investigated from various respects. The influence of Spread on PNN is studied, followed by the comparisons between PNN and variants of SaPNNs. Variants of SaPNNs are compared with PNN and other four NNs, including BP, ELM, GRNN, and SaELM.

To allow a fair comparison, all the experiments were conducted on a PC with a Pentium IV processor running at 2.0 GHz, 512 MB of RAM and a hard drive of 160 GB. Our implementation was compiled using MATLAB R2012a (7.14) running under Windows XP3.

Here, the Spread used in this article is between 0.1 and 4.9, that is, Spread_min = 0.1 and Spread_max = 4.9. In order to remove the randomness of NNs, 1000 independent runs are performed in the following experiments.

The data studied in this article are the matrix of the 33 × 4 dimension. The first three columns are the data of the improved three-ratio method, and the fourth column is the classification output, that is, the type of transformer fault. The first 23 samples and the last 10 samples are considered as training samples and test samples of SaPNNs, respectively.

Influence of Spread on PNN

In this section, the influence of Spread on PNN is studied for transformer FD problem. In order to fully investigate this influence, the interval is set to 0.1, that is, Spread_Interval = 0.1, Spread = 0.1, 0.2, 0.3, …, 4.8, 4.9. The results are recorded in Table 1 and Figure 3. The best value obtained by each method is bold.

Table 1.

Accuracy of PNN with Spread_min = 0.1 and Spread_max = 4.9 for the transformer fault diagnosis problem.

Spread	Training set	Test set	Spread	Training set	Test set
0.1	23	7	2.6	21	6
0.2	23	7	2.7	21	6
0.3	23	8	2.8	21	6
0.4	23	8	2.9	21	6
0.5	23	8	3.0	21	6
0.6	23	8	3.1	21	4
0.7	23	8	3.2	21	4
0.8	23	8	3.3	20	4
0.9	22	9	3.4	19	4
1.0	22	9	3.5	18	4
1.1	22	9	3.6	17	4
1.2	22	8	3.7	17	4
1.3	22	8	3.8	16	4
1.4	22	8	3.9	16	4
1.5	21	8	4.0	16	4
1.6	21	8	4.1	15	4
1.7	21	8	4.2	15	4
1.8	21	8	4.3	15	4
1.9	21	8	4.4	15	4
2.0	21	7	4.5	15	4
2.1	21	7	4.6	15	4
2.2	21	6	4.7	15	4
2.3	21	6	4.8	15	4
2.4	21	6	4.9	15	4
2.5	21	6

Figure 3.

Prediction of test samples (PNN) with Spread_min = 0.1 and Spread_max = 4.9 for the transformer fault diagnosis problem.

From Table 1, for training set, PNN has the best prediction accuracy when Spread is in [0.1, 0.8]. With the increment of Spread, the prediction accuracy becomes less and less, and this declining trend reaches the minimal accuracy 65.22% (15/23) when Spread = 4.9, which can also be shown in Figure 3. In addition, for test set, PNN has the best prediction accuracy when Spread is in [0.9, 1.1] with the maximal accuracy 90.00% (9/10). With the increment of Spread, the prediction accuracy first increases from 0.1 to 0.9 and then decreases from 1.1 to 4.9. Finally, PNN has the worst prediction accuracy when Spread is in [3.1, 4.9]. This trend can be clearly shown in Figure 3. The prediction accuracy of PNN for training set and test set has the similar trend although they do not reach the optimal at the same time.

From the above analyses, Spread has great influence on the performance of PNN, so the selection of Spread is of vital importance in PNN. It has experimentally shown that PNN performs the best when Spread is in [0.9, 1.1]. Spread is set to 0.9 in other experiments.

In addition, in order to get the best Spread, 49 implementations of PNN are performed in this experiment. In the next section, three self-adaptive strategies are studied in order to decrease the computational requirements.

Influence of variants of SaPNNs

In this section, three variants of improved PNNs (SaPNN, SSaPNN1, and SSaPNN2) are fully investigated for transformer FD problem.

SaPNN

Here, SaPNN is studied with initial Spread_Interval = 0.8. The results are recorded in Table 2. The best value obtained by each method is bold.

Table 2.

Prediction of test samples (SaPNN) with Spread_min = 0.1 and Spread_max = 4.9 for the transformer fault diagnosis problem.

Iteration	Interval	Range	Evaluations	Spread	Training set	Test set
1	0.8	[0.1, 4.9]	7	0.1	23	7
				0.9	22	9
				1.7	21	8
				2.5	21	8
				3.3	20	6
				4.1	15	4
				4.9	15	4
2	0.4	[0.1, 1.7]	5	0.1	23	7
				0.5	23	8
				0.9	22	9
				1.3	22	8
				1.7	21	8
3	0.2	[0.5, 1.3]	5	0.5	23	8
				0.7	23	8
				0.9	22	9
				1.1	22	9
				1.3	22	8
4	0.1	[0.7, 1.1]	5	0.7	23	8
				0.8	23	8
				0.9	22	9
				1.0	22	9
				1.1	22	9

First, we have Spread_min = 0.1, Spread_max = 4.9, and Spread_Interval = 0.8, that is, Spread can be equal to 0.1, 0.9, 1.7, 2.5, 3.3, 4.1, and 4.9. Accordingly, the number of evaluations for PNN is 7 (nEval₁ = 7). SaPNN reaches the maximal prediction accuracy 100.00% (23/23) when Spread = 0.1 for training set, while for test set, SaPNN reaches the maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9. Therefore, the best Spread Spread_best is selected and set to 0.9 at this iteration. For this case, it can also be shown in Figure 4.

Figure 4.

Prediction of test samples (SaPNN) with Spread_min = 0.1, Spread_max = 4.9, and Spread_Interval = 0.8 for the transformer fault diagnosis problem.

Second, Spread_min, Spread_max, and Spread_Interval are updated according to equations (4)–(6), as shown in equations (7)–(9)

\begin{matrix} Sprea d_{\min} & = max (Sprea d_{best} - Sprea d_{Interval}, Sprea d_{\min}) \\ = max (0.9 - 0.8, 0.1) \\ = 0.1 \end{matrix}

(7)

\begin{matrix} Sprea d_{\max} & = min (Sprea d_{best} + Sprea d_{Interval}, Sprea d_{\max}) \\ = min (0.9 + 0.8, 4.9) \\ = 1.7 \end{matrix}

(8)

\begin{matrix} Sprea d_{Interval} & = 0.1 \times round (\frac{Sprea d_{Interval}}{2 \times 10}) \\ = 0.1 \times round (\frac{0.8}{2 \times 10}) \\ = 0.4 \end{matrix}

(9)

Herein, we have Spread_min = 0.1, Spread_max = 1.7, and Spread_Interval = 0.4, that is, Spread can be equal to 0.1, 0.5, 0.9, 1.3, and 1.7. Accordingly, the number of evaluations for PNN is 5 (nEval₂ = 5). SaPNN has the best prediction accuracy of 100.00% (23/23) when Spread = 0.1 and 0.5 for training set, while for test set, SaPNN has the best maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9. Therefore, the best Spread Spread_best is set to 0.9 at this iteration. For this case, it can also be shown in Figure 5.

Figure 5.

Prediction of test samples (SaPNN) with Spread_min = 0.1, Spread_max = 1.7, and Spread_Interval = 0.4 for the transformer fault diagnosis problem.

Third, similar to equations (7)–(9), Spread_min, Spread_max, and Spread_Interval are updated. Herein, we have Spread_min = 0.5, Spread_max = 1.3, and Spread_Interval = 0.2, that is, Spread can be equal to 0.5, 0.7, 0.9, 1.1, and 1.3. Accordingly, the number of evaluations for PNN is 5 (nEval₃ = 5). For this case, SaPNN has the best prediction accuracy of 100.00% (23/23) when Spread = 0.5 and 0.7 for training set, while for test set, SaPNN has the best maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9 and 1.1. Therefore, the best Spread Spread_best is set to 0.9 at this iteration (of course, 1.1 can also be selected). For this case, it can also be shown in Figure 6.

Figure 6.

Prediction of test samples (SaPNN) with Spread_min = 0.5, Spread_max = 1.3, and Spread_Interval = 0.2 for the transformer fault diagnosis problem.

Finally, similarly, we have Spread_min = 0.7, Spread_max = 1.1, and Spread_Interval = 0.1, that is, Spread can be equal to 0.7, 0.8, 0.9, 1.0, and 1.1. Accordingly, the number of evaluations for PNN is 5 (nEval₄ = 5). For this case, SaPNN has the best prediction accuracy of 100.00% (23/23) when Spread = 0.7 and 0.8 for training set, while for test set, SaPNN has the best maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9, 1.0, and 1.1. Therefore, the best Spread Spread_best is set to 0.9 at this iteration (of course, 1.0 and 1.1 can also be selected). For this case, it can also be shown in Figure 7.

Figure 7.

Prediction of test samples (SaPNN) with Spread_min = 0.7, Spread_max = 1.1, and Spread_Interval = 0.1 for the transformer fault diagnosis problem.

Because the Spread_Interval reaches minimum (0.1), the SaPNN stops, and the best Spread Spread_best = 0.9 with the best prediction accuracy of 90.00% (9/10).

From the above analyses, the total implementations are 22 (7 + 5 + 5 + 5), that is, only 44.90% (22/49) of PNN. That is to say, the computation requirements decrease to the half without losing the advancement of PNN. The prediction accuracy of SaPNN with best Spread Spread_best = 0.9 for training set and test set can be shown in Figures 8 and 9.

Figure 8.

Prediction of training samples (SaPNN) with Spread = 0.9 for the transformer fault diagnosis problem.

Figure 9.

Prediction of test samples (SaPNN) with Spread = 0.9 for the transformer fault diagnosis problem.

SSaPNN1

In this section, SaPNN is further simplified and then the simplified SSaPNN1 is studied with initial Spread_Interval = 0.8. The results are recorded in Table 3. The best value obtained by each method is bold.

Table 3.

Prediction of test samples (SSaPNN1) with Spread_min = 0.1 and Spread_max = 4.9 for the transformer fault diagnosis problem.

Iteration	Interval	Range	Evaluations	Spread	Training set	Test set
1	0.8	[0.1, 4.9]	7	0.1	23	7
				0.9	22	9
				1.7	21	8
				2.5	21	6
				3.3	20	4
				4.1	15	4
				4.9	15	4
2	0.1	[0.1, 1.7]	17	0.1	23	7
				0.2	23	7
				0.3	23	8
				0.4	23	8
				0.5	23	8
				0.6	23	8
				0.7	23	8
				0.8	23	8
				0.9	22	9
				1.0	22	9
				1.1	22	9
				1.2	22	8
				1.3	22	8
				1.4	22	8
				1.5	21	8
				1.6	21	8
				1.7	21	8

First, we have Spread_min = 0.1, Spread_max = 4.9, and Spread_Interval = 0.8, that is, Spread can be equal to 0.1, 0.9, 1.7, 2.5, 3.3, 4.1, and 4.9. For this case, it is the same as in section “SaPNN.” Accordingly, the number of evaluations is 7 (nEval₁ = 7). SaPNN reaches the maximal prediction accuracy of 100.00% (23/23) when Spread = 0.1 for training set, while for test set, SaPNN reaches the maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9. Therefore, the best Spread Spread_best is selected and set to 0.9 at this iteration.

Subsequently, Spread_min and Spread_max are updated according to equations (4)–(6), as shown in equations (7) and (8). After that, the Spread_Interval is set to 0.1 directly.

Herein, we have Spread_min = 0.1, Spread_max = 1.7, and Spread_Interval = 0.1, that is, Spread can be equal to 0.1, 0.2, 0.3, …, 1.5, 1.6, and 1.7. Accordingly, the number of evaluations for PNN is 17 (nEval₂ = 17). SSaPNN1 has the best prediction accuracy of 100.00% (23/23) when Spread is in [0.1, 0.8] for training set, while for test set, SSaPNN1 has the best maximal prediction accuracy of 90.00% (9/10) when Spread is in [0.9, 1.1]. Therefore, the best Spread Spread_best is set to 0.9 at this iteration.

Because the Spread_Interval reaches minimum (0.1), the SSaPNN1 stops, and the best Spread Spread_best = 0.9 with the best prediction accuracy of 90.00% (9/10).

From the above analyses, the total implementations are 24 (7 + 17), that is, only 48.98% (24/49) of PNN. The computation requirements of SSaPNN1 are a little more than SaPNN (44.90%, 24/49), but Spread_min, Spread_max, and Spread_Interval are only updated once. Therefore, it is much simpler than SaPNN.

SSaPNN2

In this section, another simplified version, SSaPNN2, is used to solve transformer FD problem. Similarly, initial Spread_Interval is set to 0.8. The results are recorded in Table 4. The best value obtained by each method is bold.

Table 4.

Prediction of test samples (SSaPNN2) with Spread_min = 0.1 and Spread_max = 4.9 for the transformer fault diagnosis problem.

Iteration	Interval	Range	Evaluations	Best Spread	Local best	Global best	Spread	Training set	Test set
1	0.8	[0.1, 4.9]	7	0.9	9	9	0.1	23	7
							0.9	22	9
							1.7	21	8
							2.5	21	8
							3.3	20	6
							4.1	15	4
							4.9	15	4
2	0.4	[0.1, 1.7]	2	0.9	8	9	0.5	23	8
2	0.4	[0.1, 1.7]	2	0.9	8	9	1.3	22	8
3	0.2	[0.5, 1.3]	2	0.9	9	9	0.7	23	8
3	0.2	[0.5, 1.3]	2	0.9	9	9	1.1	22	9
4	0.1	[0.7, 1.1]	2	0.9	9	9	0.8	23	8
4	0.1	[0.7, 1.1]	2	0.9	9	9	1.0	22	9

First, we have Spread_min = 0.1, Spread_max = 4.9, and Spread_Interval = 0.8, that is, Spread can be equal to 0.1, 0.9, 1.7, 2.5, 3.3, 4.1, and 4.9. For this case, it is the same with sections “SaPNN” and “SSaPNN1.” Accordingly, the number of evaluations is 7 (nEval₁ = 7). SSaPNN2 reaches the maximal prediction accuracy of 100.00% (23/23) when Spread = 0.1 for training set, while for test set, SaPNN reaches the maximal prediction accuracy of 90.00% (9/10) when Spread = 0.9. Therefore, the best Spread Spread_best is selected and set to 0.9 at this iteration.

Second, Spread_min, Spread_max, and Spread_Interval are updated according to equations (4)–(6), as shown in equations (7)–(9).

Herein, we have Spread_min = 0.1, Spread_max = 1.7, and Spread_Interval = 0.4, that is, Spread can be equal to 0.1, 0.5, 0.9, 1.3, and 1.7. Looking carefully at Spread, the prediction accuracy for training set and test set when Spread = 0.1, 0.9, and 1.7 has been found at previous iteration. Therefore, Spread can only be equal to 0.5 and 1.3. Accordingly, the number of evaluations for PNN is 2 (nEval₂ = 2). SSaPNN2 has the best prediction accuracy of 100.00% (23/23) when Spread = 0.5 for training set, while for test set, SSaPNN2 has the best maximal prediction accuracy of 80.00% (8/10) when Spread = 0.5 and 1.3. Therefore, the best Spread Spread_best at this iteration is 0.5, and the global best Spread Spread_best is set to 0.9 up to now.

Third, similar to equations (7)–(9), Spread_min, Spread_max, and Spread_Interval are updated. Herein, we have Spread_min = 0.5, Spread_max = 1.3, and Spread_Interval = 0.2, that is Spread can be equal to 0.5, 0.7, 0.9, 1.1, and 1.3. Similarly, the prediction accuracy for training set and test set when Spread = 0.5, 0.9, and 1.3 has been found before. Therefore, Spread can only be equal to 0.7 and 1.1. Accordingly, the number of evaluations for PNN is 2 (nEval₃ = 2). For this case, SSaPNN2 has the best prediction accuracy of 100.00% (23/23) when Spread = 0.7 for training set, while for test set, SSaPNN2 has the best maximal prediction accuracy of 90.00% (9/10) when Spread = 1.1. Therefore, the best Spread Spread_best at this iteration is 1.1, and the global best Spread Spread_best is set to 0.9 up to now.

Finally, similarly, we have Spread_min = 0.7, Spread_max = 1.1, and Spread_Interval = 0.1, that is, Spread can be equal to 0.7, 0.8, 0.9, 1.0, and 1.1. After removing the repeated Spread, Spread can be equal to 0.8 and 1.0. Accordingly, the number of evaluations for PNN is 2 (nEval₄ = 2). For this case, SSaPNN2 has the best prediction accuracy of 100.00% (23/23) when Spread = 0.8 for training set, while for test set, SSaPNN2 has the best maximal prediction accuracy of 90.00% (9/10) when Spread = 1.0. Therefore, the best Spread Spread_best at this iteration is 1.0, and the global best Spread Spread_best is set to 0.9 up to now.

Because the Spread_Interval reaches minimum (0.1), the SSaPNN2 stops, and the best Spread Spread_best = 0.9 with the best prediction accuracy of 90.00% (9/10).

From the above analyses, the total implementations are 13 (7 + 2 + 2 + 2), that is, only 26.53% (13/49) of PNN. The computational requirements of SSaPNN2 are much less than SaPNN (44.90%, 24/49), although SSaPNN2 has the same updating number with SaPNN for Spread_min, Spread_max, and Spread_Interval.

Comparisons of SaPNNs with BP, ELM, GRNN, and SaELM

In order to further prove the superiority of SaPNNs when addressing transformer FD problem, three versions of SaPNNs are further compared with BP, ELM, GRNN, and SaELM.

For BP NN, epochs = 200, learning rate = 0.1, and objective = 0.00004. For GRNN, cyclic training method is used in order to select the best Spread value, making GRNN achieve the best prediction. For ELM, the number of neurons in the hidden layer is 20, and the activation function is S function. The parameters used in SaELM are set as follows: width factor Q = 2, scale factor L = 4, N_min = 5, N_max = 120, N_Interval = 5, and the maximum generation MaxIter = 2000. For PNN, Spread is set to 1.5. The results are recorded in Table 5. The best value obtained by each method is bold.

Table 5.

Accuracy of BP, ELM, GRNN, PNN, SaELM, SaPNN, SSaPNN1, and SSaPNN2 for the transformer fault diagnosis problem.

	Training set				Test set
	Mean	Best	Worst	Standard deviation	Mean	Best	Worst	Standard deviation
BP	16.62	23.00	2.00	4.24	6.24	10.00	0	2.08
ELM	23.00	23.00	23.00	0	3.91	9.00	1.00	1.21
GRNN	8.92	12.00	5.00	1.48	6.09	7.00	4.00	0.48
PNN	21.00	21.00	21.00	0	8.00	8.00	8.00	0
SaELM	23.00	23.00	23.00	0	6.20	6.62	5.66	0.13
SaPNN	23.00	23.00	23.00	0	9.00	9.00	9.00	0
SSaPNN1	23.00	23.00	23.00	0	9.00	9.00	9.00	0
SSaPNN2	23.00	23.00	23.00	0	9.00	9.00	9.00	0

BP: back propagation; ELM: extreme learning machine; GRNN: general regression neural network; PNN: probabilistic neural network; SaELM: self-adaptive extreme learning machine; SaPNN: self-adaptive probabilistic neural network; SSaPNN1: simplified self-adaptive probabilistic neural network 1; SSaPNN2: simplified self-adaptive probabilistic neural network 2.

From Table 5, for training set, the comprehensive performance of ELM, SaELM, and variants of SaPNNs is identical and is better than BP, GRNN, and PNN. Except BP and GRNN, all the other NNs have the same SD that is 0. For test set, the average prediction accuracy of variants of SaPNNs is better than BP, ELM, GRNN, PNN, and SaELM (6.24/10 = 62.40%, 3.91/10 = 39.10%, 6.09/10 = 60.90%, 8/10 = 80.00%, 6.20/10 = 62.00%). Although BP has the best prediction accuracy (10/10 = 100.00%), the worst prediction accuracy is (0/10 = 0.00%). The results have proven that variants of SaPNNs are suitable for transformer FD problem.

Conclusion

In order to remove the influence of Spread in PNN, a self-adaptive strategy is incorporated into the basic PNN and then SaPNN is proposed. In SaPNN, the best Spread can be self-adaptively selected; therefore, SaPNN can always reach the best prediction accuracy. Moreover, in order to further reduce the computational requirements, two simplified strategies are added to the proposed SaPNN and then two versions of simplified SaPNN (SSaPNN1 and SSaPNN2) are proposed. The simulation results indicate that SSaPNN1 and SSaPNN2 perform more effectively and efficiently than SaPNN while preserving the feature of SaPNN. In variants of SaPNNs, Spread can be well selected; therefore, there is no parameters to be adjusted in the training process. The transformer FD problem is addressed by variants of SaPNNs, and their performance is fully investigated from various respects. In addition, by comparing them with the basic PNN, and BP, ELM, GRNN, and SaELM, the results have experimentally proven that SaPNNs have a more accurate prediction rate and better generalization performance when addressing the FD problem.

Although variants of SaPNNs have shown their advantages over the basic PNN, and BP, ELM, GRNN, and SaELM when addressing transformer FD problem, the following points should be clearly provided in our future research. First, the results will be analyzed by other methods in our future research, such as t-test. Second, the proposed SaPNNs will be used to address other classification and prediction problems. Finally, some other new metaheuristic algorithm, such as monarch butterfly optimization (MBO),³⁰ will be combined with SaPNNs, and this combination will surely further significantly improve the performance of SaPNNs.

Footnotes

Academic Editor: Siamak Talatahari

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the Fundamental Research Funds for the Central Universities under grant number 2015XKMS051, a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Cooperative Innovation Center of Jiangsu Province.

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Spread	Training set	Test set	Spread	Training set	Test set
0.1	23	7	2.6	21	6
0.2	23	7	2.7	21	6
0.3	23	8	2.8	21	6
0.4	23	8	2.9	21	6
0.5	23	8	3.0	21	6
0.6	23	8	3.1	21	4
0.7	23	8	3.2	21	4
0.8	23	8	3.3	20	4
0.9	22	9	3.4	19	4
1.0	22	9	3.5	18	4
1.1	22	9	3.6	17	4
1.2	22	8	3.7	17	4
1.3	22	8	3.8	16	4
1.4	22	8	3.9	16	4
1.5	21	8	4.0	16	4
1.6	21	8	4.1	15	4
1.7	21	8	4.2	15	4
1.8	21	8	4.3	15	4
1.9	21	8	4.4	15	4
2.0	21	7	4.5	15	4
2.1	21	7	4.6	15	4
2.2	21	6	4.7	15	4
2.3	21	6	4.8	15	4
2.4	21	6	4.9	15	4
2.5	21	6

Spread	Training set	Test set	Spread	Training set	Test set
0.1	23	7	2.6	21	6
0.2	23	7	2.7	21	6
0.3	23	8	2.8	21	6
0.4	23	8	2.9	21	6
0.5	23	8	3.0	21	6
0.6	23	8	3.1	21	4
0.7	23	8	3.2	21	4
0.8	23	8	3.3	20	4
0.9	22	9	3.4	19	4
1.0	22	9	3.5	18	4
1.1	22	9	3.6	17	4
1.2	22	8	3.7	17	4
1.3	22	8	3.8	16	4
1.4	22	8	3.9	16	4
1.5	21	8	4.0	16	4
1.6	21	8	4.1	15	4
1.7	21	8	4.2	15	4
1.8	21	8	4.3	15	4
1.9	21	8	4.4	15	4
2.0	21	7	4.5	15	4
2.1	21	7	4.6	15	4
2.2	21	6	4.7	15	4
2.3	21	6	4.8	15	4
2.4	21	6	4.9	15	4
2.5	21	6

Spread	Training set	Test set	Spread	Training set	Test set
0.1	23	7	2.6	21	6
0.2	23	7	2.7	21	6
0.3	23	8	2.8	21	6
0.4	23	8	2.9	21	6
0.5	23	8	3.0	21	6
0.6	23	8	3.1	21	4
0.7	23	8	3.2	21	4
0.8	23	8	3.3	20	4
0.9	22	9	3.4	19	4
1.0	22	9	3.5	18	4
1.1	22	9	3.6	17	4
1.2	22	8	3.7	17	4
1.3	22	8	3.8	16	4
1.4	22	8	3.9	16	4
1.5	21	8	4.0	16	4
1.6	21	8	4.1	15	4
1.7	21	8	4.2	15	4
1.8	21	8	4.3	15	4
1.9	21	8	4.4	15	4
2.0	21	7	4.5	15	4
2.1	21	7	4.6	15	4
2.2	21	6	4.7	15	4
2.3	21	6	4.8	15	4
2.4	21	6	4.9	15	4
2.5	21	6