Detection and prediction of thimble tube defects using artificial neural networks

Abstract

The reliability of thimble tubes plays a critical role for maintaining the safety of a nuclear power plant. The defect depth needs to be quantified and predicted to support the operational decision-making. This paper presents a method to quantify the defects on thimble tube wall based on the analyzation of eddy current testing (ECT) data. Then, a method using artificial neural network (ANN) to predict the detect depth is studied. The tubes are divided into 2 shapes and four regions according to their positions and the data of each region and each shape is expanded by mean interpolation. A prediction model based on ANN is constructed for each shape in each region. The experimental results show that the model can predict the signal of the next year according to the signal of the previous three years with mean absolute percentage error less than 16%.

Keywords

Thimble tube eddy current testing artificial neural network quantification prediction

1. Introduction

An in-core flux thimble tube (hereinafter referred to as thimble tube) is the channel for the neutron flux detector to enter and exit the reactor fuel assembly, as shown in Fig. 1(a). During the service, thimble tubes are worn by the fluid in the reactor. Due to the relatively stable and enclosed environment, the shapes of the defects depend on the wear positions of the thimble tube. There are mainly two kinds of defects, namely: ring-shaped and wedge-shaped defects, as shown in Fig. 1(b) and (c).

Fig. 1.

(a) The structure of thimble tube (b) ring-shaped defect and (c) wedge-shaped defect.

The evaluation of the defect size is significant to the safety of a nuclear power plant [1]. Since the Office of Nuclear Reactor Regulation, Nuclear Regulatory Commission (NRC) released the reports on the defects of thimble tube in some countries’ nuclear power plants [2], the detection and evaluation of thimble tube defects has gradually become an important research topic. The most common inspection method for thimble tube is eddy current testing (ECT), which is an efficient and economical technology for conductive structures inspection [3]. After obtaining the inspection data, it is important to adopt proper algorithms to analyze the data, so as to evaluate the structure quantitatively [4]. In recent years, neutral network is investigated for defect classification and quantification [5,6]. In addition, the decision makers of a nuclear power plant need more than just current deficiencies information, but also the most likely defect size in the near future to guide the operation strategy. The site survey shows that the tube position in the reactor affects the development of the defect significantly. Therefore, the position and the part of the tubes should be taken into consideration to predict the defect size. In this paper, the relationship of the signal and defect dimensions are established based on simulation and experimental data. Then the data of thimble tubes in the recent 10 years are analyzed to construct the prediction model.

2. Defect qualification

A finite element method (FEM) model is developed to calculate the multi-frequency signals of defects with different dimensions. The governing equations of the model are the reduced magnetic vector potential (RMVP) formulations [7]. The mesh of the model is generated using Python and the large sparse matrices are solved using Fortran. This study simulates ring-shaped and wedge-shaped defects with different depths and wear angles. The specific defect parameters are listed in Table 1. The outer diameter, inner diameter and length of the thimble tubes are 8.6 mm, 5.2 mm and 80 mm, respectively. The conductivity, relative magnetic permeability and relative permittivity of the tube are 1010000 S/m, 1 and 1, respectively. A bobbin probe scans inside the tube to inspect defects on the tube wall. The geometry dimensions of the probe are as follows: outer diameter 4.32 mm, inner diameter 4.01 mm, height 2.2 mm. The number of turns of the wires in the probe is 500. Figure 2 shows an example model of a wedge defect and the amplitude of the defect signal versus the axial position.

Table 1
The defect parameters of the simulation model.

Frequencies (absolute channel): 20 kHz, 40 kHz, 80 kHz, 160 kHz

Defect shape Wedge Ring

Parameters Depth (of total) Wear angle Depth (of total) Wear angle

Value 10% to 80% Step distance 10% 60° to 360° Step distance 30° 10% to 80% Step distance 10% 75° to 345° Step distance 30°

Frequencies (absolute channel): 20 kHz, 40 kHz, 80 kHz, 160 kHz
Value	10% to 80% Step distance 10%	60° to 360° Step distance 30°	10% to 80% Step distance 10%	75° to 345° Step distance 30°

Fig. 2.

(a) An example model of a wedge defect and (b) the signal amplitude of the defect versus the axial position.

Fig. 3.

The relationship of phase-amplitude-wear angle-wear depth in 160 kHz simulation of wedge defects.

It seems that the signal has a peak when the probe passes the defect. The amplitude and phase of the signal is affected by the defect dimensions. To analyze the signals of the defects, the amplitude and phase of the signal peaks are summarized in Fig. 3, where the x-axis is the phase and the y-axis is the amplitude of the signal peak. Each solid line in Fig. 3 presents a curve of the defects with the same circumferential wear angle and different depths (varying from 10% to 80% of the tube wall). Meanwhile, a dashed line in the figure indicates the curve of the defects with the same depth but different circumferential wear angles. It seems that if there is only one variable and all the other parameters are kept constant, the signal is monotonicity correlated with the defect size.

The simulation model is validated and calibrated by comparing its output with experimental results. To this aim, some experimental samples with machined defects are fabricated. The material of the tube is 316L cold-worked stainless steel, the geometry dimensions of which are the samples as that shown in the simulation model. Photographs of the two kinds of defects are shown in Fig. 4. The signal of a through-hole with 1.7 mm diameter is utilized as the reference for calibration. The calibration coefficients are obtained by calibrating its amplitude to 7 V and phase to 20°.

Fig. 4.

Ring-shaped defect (a) and wedge-shaped defect (b) samples.

Table 2

Defect depth and signal of the experimental wedge samples

Defect number	1	2	3	4	5	6
Signal amplitude (V)	13.03	10.06	14.56	11.55	19.15	7
Signal phase (°)	58	47	73	44	92	31
Wear angle (°)	165	165	105	225	105	195
Defect depth (mm)	0.85	0.68	1.02	0.68	1.19	0.51
Quantification result (mm)	0.782	0.697	0.978	0.645	1.134	0.522
Relative error	8%	2.5%	4.2%	5.1%	4.7%	2.3%

To quantify the defect depth, the amplitude and phase of the defect signal measured experimentally are mapped to the figure shown in Fig. 3. Then the most likely defect depth and circumferential angle are calculated. The defect depth is the most important evaluation index for maintaining nuclear power safety. Therefore, the accuracy of defect depth quantification is studied in this paper. Six signals of wedge defects are selected as examples to verify the quantification accuracy. The depth and wear angle of the six defects are shown in Table 2, as well as the signal amplitudes and phases of the signals obtained with an absolute bobbin probe at frequency 160 kHz. The qualification results are also presented in Table 2. It is seen that the relative error range of the defect depth calculated from the experimental data is 2.3% to 8%, and the average error is 4.47%. The error of the quantification is mainly due to the fact that the parameters used in the simulation model, such as the conductivity of the sample, the lifting distance of the probe and the excitation current density, cannot be exactly the same as those in the experiments, and the noise in the experiments is unavoidable.

3. Defect prediction

3.1. Data processing

Thimble tubes in nuclear power plants are inspected periodically. Defect prediction involves using historical data and statistical analysis algorithms to identify areas with high risk of defects. The goal is to prioritize inspections and maintenance efforts, allowing for early detection and prevention of potential issues. Defect prediction allows the decision makers to identify potential issues early, enabling timely maintenance and repair actions before defects lead to catastrophic failures, which plays a crucial role in ensuring the safety and reliability. The data of thimble tubes in nuclear power plants over the last 10 years was collected, which was measured by absolute bobbin probes at four frequencies, namely 20 kHz, 40 kHz, 80 kHz, and 160 kHz. To reduce the experimental noise and the effect of the support structure, the data of two frequencies is mixed during the pre-processing. In addition, for the anomalous data where the signal amplitude does not change monotonically with each passing year, a correction factor is determined based on the average rate of change of the overall signal over the years, and then multiplied by the value of the previous year of the anomalous data to obtain the corrected value. For missing data in some years, additional information is provided by averaging the signals from previous and subsequent years.

Considering the effect of fluid flow velocity and flow direction on the defect, the development trend of defects in different regions is different. Therefore, the thimble tubes in a nuclear power plant are divided into four regions, as shown in Fig. 5. In addition, the defects of different shapes need to be distinguished. Therefore, different shapes of defects in different locations are predicted separately. The total amount of data collected in field is about 10,000. However, considering the partitioning of the nuclear reactor, the division of the thimble tubes into different shapes and the combination of data at the time of conversion to input, the size of data set for each model training is not sufficient, so the data should be expanded. Therefore, mean interpolation is used in data processing, which inserts two additional new data in the middle of every two-neighboring data. Figure 6 shows the distribution of data before and after the interpolation.

Fig. 5.

Schematic diagram of nuclear reactor thimble tube partitioning.

Fig. 6.

The data distribution before (a) and after (b) the interpolation.

3.2. Prediction with artificial neural network

To predict the defect signals according to the amplitude and phase data in previous years, a prediction model based on ANN is built for each shape in each region, resulting in eight prediction models in total. Each model consists of a three-layer network, which contain 12, 14, and 8 neurons, respectively. In this study, the data of four consecutive years is considered as a data set, with the data of the first three years as input and the last year’s data as output. The dataset is divided into two parts: a training set and a testing set. The ANN model is trained using the training dataset.

The performance of the trained model is assessed using the testing dataset. The Mean Absolute Percentage Error (MAPE) is chosen as the evaluation criterion, as shown in Eq. (1). $\begin{eqnarray}\text{MAPE}=\mathop{\sum }_{i=1}^{n}\frac{1}{n}\left|\frac{{\hat{y}}_{i}-y_{i}}{y_{i}}\right|\end{eqnarray}$ (1) where ${\hat{y}}_{i}$ is predicted value and y_i is true value.

Taking the data of ring defects in region 1 as an example, the Epoch-MAPE curves of the ANN model are presented in Fig. 7(a). It seems that the loss decreases smoothly over epochs, which approach 7% after about 150 epochs. Meanwhile, the effectiveness of the partitioning of the thimble tubes in nuclear power plant is verified. A similar ANN is constructed to predict the signal using all the data without the region division and shape division, the epoch-MAPE curves of which are shown in Fig. 7(b). It is found that the division of the thimble tubes into four regions and two shapes is helpful for defect prediction.

Fig. 7.

(a) The epoch-MAPE curves of the ANN model for the data in region 1, ring-shaped defects, (b) The epoch-MAPE curves for all the data without the region division and shape division.

As a comparison, this paper also builds a convolutional neural network (CNN) prediction model. It consists of a convolutional layer, a planar layer and a fully connected layer with 10 neurons. The model also converges after about 150 epochs. The accuracies of the two models are compared in Table 3. For the ANN model, the MAPE varies from 3.58% to 15.82% for different shapes and regions. The MAPE of the CNN model varies from 9.64% to 20.13% for different shapes and regions. It seems that the ANN model performs slightly better than the CNN model. However, it is worth noting that the choice between ANN and CNN models depends on the specific characteristics of the prediction problem, the nature of the data, and the domain expertise of the practitioner. Proper experimentation and evaluation of both model types are necessary in the future study to determine which one best suits the task.

Table 3

The MAPE of test data based on ANN and CNN model

Location	Region1, R	Region 1, W	Region 2, R	Region 2, W
MAPE(ANN)	7.31%	6.62%	9.85%	7.44%
MAPE(CNN)	11.43%	10.55%	13.23%	15.63%
Location	Region 3, R	Region 3, W	Region 4, R	Region 4, W
MAPE(ANN)	3.58%	5.01%	15.82%	7.13%
MAPE(CNN)	9.64%	12.55%	20.13%	9.68%
Where R means ring-shaped defects and W means wedge-shaped defects.

4. Conclusion and future work

This paper studies the method of quantifying the defect depth on a thimble tube from the features of ECT signals and the method of predicting the defect signal from the data for previous three years using an ANN model. A FEM model is utilized to generate the multi-frequency signals of defects with different shapes and dimensions. The simulation results show that the signal has a peak when the probe passes the defect, whose amplitude and phase is a function of the defect dimensions. The simulation model is validated and calibrated by comparing its output with experimental results. Then a defect quantification method is proposed, which maps the amplitude and phase of the signal measured in the experiment into the characteristic curves that are generated by the simulation model. Then the most likely defect depth and circumferential angle are calculated. The average relative error of the defect depth calculated is 4.47%. Data of nuclear power plant collected in the last 10 years are utilized to study the possibility of predicting the defect from the data. The thimble tubes in a nuclear power plant are divided into four regions considering the effect of fluid flow velocity and flow direction on the defect. In addition, as the defect shapes at different positions are different, the defects in different locations and regions are predicted separately. A prediction model based on ANN is built for each position in each region to predict the amplitude and phase based on the previous three year’s signals. The MAPE of the ANN models varies from 3.58% to 15.82% for different positions and regions. In the future, more simulation and experimental should be collected to further test and optimize the algorithm. This method can also be used to predict similar defects on other structures, such as wear on steam generator heat transfer tubes.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 52277014.

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Frequencies (absolute channel): 20 kHz, 40 kHz, 80 kHz, 160 kHz
Defect shape	Wedge		Ring
Parameters	Depth (of total)	Wear angle	Depth (of total)	Wear angle
Value	10% to 80% Step distance 10%	60° to 360° Step distance 30°	10% to 80% Step distance 10%	75° to 345° Step distance 30°