Generalized regression neural network

To deal with sample data comprising complicated multi-dimensional variables, prior research concentrated on PCA, clustering, and Kalman filter methods to perform signal analysis of the SPND unit. These techniques can be used to rapidly establish models for axial SPNDs in the same detector assembly. In this study, we aimed to analyze the spatial correlation between the radially distributed SPND variables simultaneously in the plane of the reactor core. The aforementioned technique is not appropriate for large-scale and nonlinear data feature mining; however, the latest neural network algorithms that an distribute information among neurons have significant robustness and can quickly analyze complex nonlinear relationships. As a typical radial basis neural network model based on nonlinear regression theory, GRNN can process high-dimensional data and mine the mutual effects among multiple SPND current features because of its capabilities of nonlinear mapping and high learning speed. Unlike other types of artificial neural networks, GRNN have straightforward structures and training processes. Therefore, we used GRNN as the constituent architecture of the SPND twin model. As shown in Fig. 4, the GRNN is composed of the input, pattern, summation, and output layers. The input and output vectors of the corresponding network are $X=[x_1\mathrm{,}{x}_2\mathrm{,...,}{x}_{n-1}{]}^{\text{T}}$ and $Y=[y_1\mathrm{,}{y}_2\mathrm{,...,}{y}_k{]}^{\text{T}}$, $n-1$ and $k$ are the dimensions of $X$ and $Y$ respectively. The number of neurons in the input layer is equal to the dimensions of the input vector, and each neuron is a simple distribution unit that directly transmits the input to the pattern layer. The pattern layer is a radial base layer, whose number of neurons is equal to the number of learning samples, $n-1$. The neuron transfer function $P_i$ is typically a Gaussian function and can be written as $\begin{array}{c}{P}_i=\exp \left[-{\left(X-X_i\right)}^{\text{T}}\left(X-X_i\right)/2{\sigma }^2\right]\mathrm{,}\\ i=\mathrm{1,2,...,}n-1\end{array}$ (1) where $X_i$ is the training sample corresponding to the $i$-th neuron, $T$ is the transpose of the matrix, and $\sigma$ is a smoothing factor. The smaller $\sigma$ is, the stronger is the approximation ability of $P_i$ for the samples.

Fig. 4Full-screen View

Structure of the GRNN

There are two types of neuronal calculation formulas in the summation layer. One is to sum the outputs of all the neurons in the pattern layer. The connection weight between each neuron and the pattern layer is set to one. The transfer function can be expressed as $S_{\text{R}}={\displaystyle \sum _{i=1}^m{P}_i}\mathrm{.}$ (2) The other type of calculation formula is to sum the outputs of all neurons in the pattern layer by weight. The transfer function can be described as $S_{\text{M}j}={\displaystyle \sum _{t=1}^m{w}_{ij}}{P}_i\mathrm{,0.3}pcj=\mathrm{1,2,...,}k$ (3) where $w_{ij}$ represents the weight of the $i$-th neuron in the pattern layer connected to the $j$-th neuron element in the summation layer, $S_{\text{M}j}$ represents the summation value of the $j$-th neuron in the summation layer.

The output layer is calculated by dividing the values obtained using the two formulas in the summation layer baserd on the following equation: $y_j=\frac{S_{\text{M}j}}{S_{\text{R}}},$ (4) where $y_j$ is the output of the $j$-th output-layer neuron.

GRNN can be applied to construct different SPND regression submodels that are integrated to form a twin model that outputs twin data.

Construction of the twin model

The state analysis of SPND signals with the application of DT can provide a more efficient and intelligent service for monitoring the neutron flux distribution and its rate. The premise of constructing a twin model is to extract sufficient and effective feature information from physical entities. The reactor core contains $n$ sets of symmetrically distributed neutron flux measurement detector assemblies. Although SPNDs operate independently of each other, the multidimensional currents they produce reveal a strong cooperative link in a shared environment. The features of the nearby signals reflect the characteristics of a specific SPND signal. The relationship between SPND signals can be explained by the following equations: $｛\begin{array}{l}S=｛s_1\mathrm{,}{s}_2\mathrm{,...,}{s}_n｝\hfill \\ D=｛d_1\mathrm{,}{d}_2\mathrm{,...,}{d}_n｝\hfill \\ U=｛U_1\left(S\right)\mathrm{,}{U}_2\left(S\right)\mathrm{,...,}{U}_n\left(S\right)｝\hfill \end{array},$ (5) where $S$ is the set of all SPND individuals at the same height in the reactor core, $D$ is the set of sample values for each variable in $S$, and $U$ is the set used to denote the correlations between the variables. $U_i(S)$ describes how the signal characteristics of any variable in $S$ are collectively represented by the remanent surrounding variables. As seen in Fig. 5, the reference network $U_i(S)$ is expressed as the connectivity between the label variable and other variables. The data of the label variable can be twinned by analyzing the network of relationships among the variables. By studying the characteristics of the associated $｛s_1\mathrm{,}{s}_2\mathrm{,...,}{s}_{n-1}｝$, the inherent information of the label variables can be obtained. Thus, each variable in $S$ can also be collectively described by the others. The twin model is built based on relationship networks, and GRNN is used to construct specific submodels to form a twin body. By dividing the set $D$, the $n-1$-dimensional variables are the features $I_i$ and the remaining single variable is the training label $T_i$. After an alternate division of $n$ groups, the data patterns between $I_i$ and $T_i$ are expressed as follows. $\left(I_i|T_i\right)=\left[\begin{array}{ccccccc}{d}_1& d_2& d_3& \cdots & d_{n-2}& d_{n-1}& d_n\\ d_1& d_2& d_3& \cdots & d_{n-2}& d_n& d_{n-1}\\ d_1& d_2& d_3& \cdots & d_{n-1}& d_n& d_{n-2}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ d_1& d_3& d_4& \cdots & d_{n-1}& d_n& d_2\\ d_2& d_3& d_4& \cdots & d_{n-1}& d_n& d_1\end{array}\right]$ (6) The division result contains $n$ rows of data patterns, and each dimensional variable of SPND signals in set $D$ is designated as a label for its corresponding data pattern. For instance, SPND $s_n$ is identified as the label of the first data pattern. Similarly, SPND $s_i$ is considered as the label of the $(n$+1-$i)$-th data pattern. To thoroughly explore the correlation between any target variable and the remaining variables in each pattern, GRNN was utilized to train the submodels to learn the state relationships between SPND features and labels. Subsequently, $n$ groups of submodels were built after repeated training. An integrated organism $M$ is combined as follows: $M=｛M_1\mathrm{,}{M}_2\mathrm{,...,}{M}_n｝,$ (7) where $M_i$ represents the trained GRNN submodels for different data patterns. After learning the algorithm, each GRNN submodel outputs the predicted value of the target SPND when its input are $(n-1)$-dimensional feature vectors. All submodels are combined into $M$, which produces $n$ groups of estimated data and explains the commonality of signal changes in all SPND components. The real SPND signals can be twinned using these virtual values. The residual between the output results of the twin model and the real values can be calculated in the given operating scenario to determine the faulty SPNDs, and the following status assessment and potential fault tolerance can then be performed.

Fig. 5Full-screen View

The network of relationship $U_i(S)$

Fault detection and tolerance

Measuring the residual between the outputs of the twin model and the actual values allows us to ascertain whether faults exist in the detector assemblies. Statistical tests can generally be used to determine the deviation points in a residual sequence. Traditional statistical techniques are sensitive to the presence of outliers because inaccurate data points can distort the mean and standard deviation of a data sequence. In this section, ESD was employed to detect possible outliers in the residual sequence and discriminate the fault from the original signals. ESD can maintain a good elimination effect when several anomalies exist concurrently in the data. We assumed that outliers exist in the residual sequence. The maximum number of outliers was predetermined as $r$, and $r$ rounds of separate tests were performed by calculating the variable of the statistical test, the formula of which is as follows: $R_i=\max \frac{|E_i-\widehat{E}|}{z}\mathrm{,}$ (8) where $E_i$ is an observed point and $\widehat{E}$ and $z$ denote the sample mean and standard deviation, respectively. The observed point that maximizes $|E_i-\widehat{E}|$ and deviates the most from the mean should be located and eliminated during the calculation. The aforementioned statistic is then recalculated using the remaining $n$-1 observed points. The process is repeated until $r$ observed points have been removed, resulting in the obtained $r$ test statistics $R_1$, $R_2$, ..., $R_r$. Corresponding to the $r$ test statistics, $r$ critical values are calculated as follows: ${\lambda }_i=\frac{(n-i)t_{p\mathrm{,}n-i-1}}{\sqrt{\left(n-i+1+t_{p\mathrm{,}n-i}^2\right)\left(n-i+1\right)}}$ (9) where $t_{p\mathrm{,}v}$ represents the 100p percentage points from the $t$ distribution with $v$ degrees of freedom and $p=1-\frac{\alpha }{2\left(n-i+1\right)}$. $\alpha$ denotes the test confidence level. The number of outliers is determined by finding the largest $i$ such that $R_i$ > ${\lambda }_i$, and thus, the initial hypothesis can be true.

As mentioned above, the simultaneous failures of excessive SPNDs in detector assemblies can lead to anomalously measured data, which cannot accurately represent the actual operational status of the system. A judgment based on fault signals will cause the system to fail to reach its normal operating condition and even cause personnel to misoperate. The capacity of the twin model to perform an effective analysis suffers because of the disordered input, and the reliability and validity of the detection results cannot be demonstrated by the residual information calculated from a broken model. For the CNFMS to be dependable and secure, the fault detection function must be rehabilitated. In this case, we introduce an efficient fault-tolerant strategy, which is implemented before multi-fault detection. The most important factor to consider when designing a fault tolerance solution is that the information that should be available to the entire system after an unexpected failure must be identical to the information that would have been obtained if the SPNDs were not faulty. The main objective was to achieve data recovery, including troubleshooting, data substitution, and data verification.

In the presence of multiple unknown fault sources, fault troubleshooting must be performed using multidimensional data. A possible selection approach involves selecting the vector with the largest calculated residual and performing data recovery first. When executing data substitution, the values of the faulty variables should be replaced with normal signals. To improve the accuracy of the replaced data, the KNN classifier was first used to select $K$ instance points from the historical database in the nearest neighbor of the feature variable of the fault points to form a new signal sample. The operating condition of $K$ nearest-neighbor points was the closest to that of the fault signals, and their variable characteristics were in a division range similar to the original signal characteristics. The weighted average operation was utilized for the values of these $K$ points, which produced the corrected signal values. Finally, the fault signal values were entirely replaced by the revised values.

After data substitution, the Jensen–Shannon (JS) divergence [43] was employed to confirm the modified data and guarantee the consistency of the replacement. It can describe the differences between the probability distributions of variables, which perform well in the similarity measurement. For the two probability distributions, the interval value of the JS divergence is $\left[\mathrm{0,1}\right]$. The two distributions are more similar and vice versa as the value decreases. The degree of JS divergence can be used to fully evaluate the suitability of the modified data. Repeating the preceding steps for a multi-fault sample can reduce the number of faulty SPNDs after recovery until the detection results are satisfactory. When the satisfied sample is input into the twin model, the output is stable and controlled within a small error. In this case, the previous fault-detection approach is adequate and fault tolerance is no longer required.

Performance of the twin model

With the continuous reduction in neutron flux during shutdown, the number of neutrons captured by the emitter decreased, resulting in a proportional decrease in the currents generated by the 44 SPNDs. To extract valuable information from physical SPNDs under different power levels and build a twin model based on it, submodels for 44 dimensional signal variables were executed. First, the training set was segmented round after round, and 44 groups of data patterns with several feature variables $I_i$ and label variable $T_i$ were obtained as follows: $\left(I_i|T_i\right)=\left[\begin{array}{ccccccc}{d}_1& d_2& d_3& \cdots & d_{42}& d_{43}& d_{44}\\ d_1& d_2& d_3& \cdots & d_{42}& d_{44}& d_{43}\\ d_1& d_2& d_3& \cdots & d_{43}& d_{44}& d_{42}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ d_1& d_3& d_4& \cdots & d_{43}& d_{44}& d_2\\ d_2& d_3& d_4& \cdots & d_{43}& d_{44}& d_1\end{array}\right]$ (10) Each data pattern in $(I_i|T_i)$ was input into GRNN for the model to learn the correlation between signal variables, and the 44 submodels were well-trained. The superiority of GRNN over the other models is ascribed to the convenience of its parameter setting. The performance of GRNN can only be changed by setting the smoothness factor $\sigma$ in the kernel function. To improve the accuracy of the model, the root mean square error (RMSE) was employed as the loss function to optimize the value of $\sigma$ for the submodels. Twenty percent of the training set was selected as the verification set and a cross-verification approach was used to obtain accurate parameter optimization results. As shown in Table 1, each submodel obtained the minimum RMSE with the corresponding setting of the optimal $\sigma$. The optimized parameters helped improve the performance of the twin model in the verification set.

To demonstrate the superiority of GRNN in modeling more clearly, we compared the proposed method with several typical approaches, including convolutional neural networks (CNN), multilayer perceptron (MLP), extreme gradient boosting (XGB), decision tree (DT), SVM, and Bayesian ridge (BR). All methods were implemented using the same data samples from the test set to compare the output accuracies. As shown in Fig. 7, the twin model formed by GRNN had the lowest RMSE between the model outputs and actual signal values. DT and XGBoost-based models, as common tree regression algorithms, had slightly larger errors than the GRNN-based model, which can easily overfit SPND signals and reduce precision. The learning abilities of the CNN and MLP models for SPND signal features were poorer than that of the GRNN model. These models are suitable for dominant feature extraction but tend to ignore the correlation between the part and whole. SVR- and BR-based models are appropriate for handling small samples instead of analyzing larger multi-dimensional signals. Compared to typical machine learning algorithms, the twin model established based on GRNN can extract rich feature information from isolated time-varying data with lower errors and higher prediction ability. In the subsequent fault detection phase, the errors between the model outputs and actual signals can be used to determine the fault location. The estimated values of the twin model can help improve the in-core power measurement by avoiding interference from fault information.

Fig. 7Full-screen View

(Color online) Prediction errors of different algorithm-based models

Fault detection results under different conditions

The novel contribution of this study is the accurate identification of faulty SPNDs in numerous assemblies using a twin model. The signal characteristics analyzed by the twin model can help examine the fault characteristics. In this section, single- and multi-point SPND fault signals were simulated through experiments to determine the effectiveness of fault detection under different conditions. The faulty SPND signals of diverse measurement channels were simulated by adding different percentages of bias to the normal signals based on the original operating conditions.

For single-point fault detection, as shown in Fig. 8, the deviation between the output of the twin model and the actual normal value is very small before introducing faults into the test set, and the data fit well. In the simulation, a bias of 50% of the normal signal value was introduced into SPND#12 at a time point after the normal signal. In this case, the twin model could maintain a normal output, but the failure of SPND#12 caused the calculated RMSE to fluctuate significantly at this point. The results of the ESD test show that the RMSE of the 12th point deviates the most, which confirms the existence of a faulty SPND#12.

Fig. 8Full-screen View

Comparison of the output values of the twin model with the actual values and the RMSE of the models in the two cases

In reality, complex and changeable interference factors usually cause fault signals to vary. Different types of common fault behaviors mainly include bias, precision degradation, and complete failure, the states of which are abnormal owing to the influence of external environmental factors. The deviation in the signals also varies at the same power level that is used to measure the extent of the magnitude of failures. Experiments (15 groups × 130 rounds) were conducted to compare the fault-detection efficiency under different fault types and deviation rates. The mean accuracy, an evaluation index, is the proportion of the number of successful detections in each group of experiments to the total number of experiments. Table 2 provides the calculation results for fault detection accuracy. Except for the detection accuracy of 99.23% in "5% drifting", the mean accuracies of other cases were 100%, indating that the fault variables identified by the ESD test from errors are consistent with the actual faulty SPNDs. Therefore, the proposed method can maintain an extremely high correct detection rate for single-point SPND faults under various conditions.

Similarly, it is possible for multiple SPNDs to fail simultaneously. The method for detecting multi-point faults is the same as that for detecting single-point faults. However, this is still a challenge because multiple faulty signals can contaminate the input of the twin model, leading to rising nondeterminacy of the outputs. In this study, we analyzed the impact of the number of faulty SPNDs on the output of a twin model and the fault detection accuracy. Because the type and deviation rate of the failure size had little influence on the detection accuracy, the added simulated fault signals were random in the multi-point fault experiment (44 groups × 150 rounds). The experimental results and comparisons are shown in Fig. 9.

Fig. 9Full-screen View

(Color online) Influence of the number of faults on the detection accuracy of different algorithm-based models

It was observed that the accuracy of multi-point fault detection remained at 100% when the number of faulty SPNDs was less than 26. In other words, the proposed method can perform a relatively efficient detection when approximately half of the 44 SPNDs fail. As the number of failures continues to increase, the accuracy begins to gradually decrease and approaches 50%, which means that approximately half of the faulty SPNDs detected by ESD are true. Unacceptable data deviating from the normal range is the reason for the occurrence of a sharp fall. Excessive abnormal inputs strongly affect the original function of the twin model and cause it to collapse, hindering the realization of subsequent detection. With ESD test calculation, the multi-point fault detection accuracy of the models based on other algorithms declined earlier than that of GRNN-based models under identical conditions. Their accuracies decreased and slowed to different degrees with the further expansion of failure. Finally, they became more reposeful along their tails. The method proposed in this study performs well for both single- and multi-point SPND fault detection.

Evaluation of fault tolerance

The previous analysis proved the superiority of using the twin model to detect multiple faults. However, there is still a deficiency in dealing with excessive failures, that is, the detection accuracy decreases significantly when the number of faulty signals exceeds the upper limit. Therefore,efficient fault-tolerant solution is required to restore abnormal data. For multi-dimensional fault data points, the KNN algorithm can help search for $K$ operating points that are closest to the faulty signal features in the historical database. On this basis, WKNN regression was employed to obtain more accurate values by assigning a greater weight to the nearest neighbors. The selection of the value of $K$ has a significant impact on the results of the regression analysis. This is because an approximation error with a smaller $K$ can make the model more complicated and can likely lead to overfitting, resulting in a larger estimation error. According to Fig. 10, WKNN has a lower error than KNN without weighting. When $K$=8, WKNN has the smallest RMSE (2.2217). Hence, the hyperparameter $K$ in WKNN used in this study was set to eight.

Fig. 10Full-screen View

(Color online) Optimization of K in KNN and WKNN

To verify the feasibility of the fault tolerance scheme mentioned above, we selected 100 points of a continuous time sequence under the full-power platform from the test set, and random faults were introduced into SPNDs from the 25th point to the 100th point in the example. In fault-tolerant processing, data recovery is first performed for the SPND variable whose RMSE is the largest. Taking SPND#27 as an example, Fig. 11 shows that the deviation between the fault and original values is distinct ($RMSE=0.0387$), which significantly affects normal system monitoring. From the perspective of the twin model, the deviation between the predicted and original values was high ($RMSE=0.0488$), making it incapable of meeting the requirements of high-precision control. In the implementation of WKNN, the deviation distance between the regression value and the original values was the shortest ($RMSE=0.0044$), and the data recovery accuracy was enhanced compared with that of the twin model. Through data recovery, WKNN can perform more reliable correction and can maintain stable operation of the system.

Fig. 11Full-screen View

(Color online) Signal value of SPND#27 after data recovery

For data verification, the JS divergence was employed to prove the similarity of the probability distribution of the values. After the calculation, the divergence values of the fault values, predicted values of the twin model, and regression values of WKNN with the original values were 4.3409×10^-5, 7.5058×10^-7, and 5.6829×10^-7, respectively. The regression values of WKNN have advantages in terms of similarity measurement which verifies the remarkably similar probability distribution and high recovery performance.

We further evaluated the performance of the fault-tolerant method on a sample filled with faulty signals. The impact of the number of recovered variables on the average RMSE of the model outputs and fault detection accuracy is illustrated in Fig. 12. In multi-point fault detection, the obtained detection accuracy with no measures for recovering data was 52.18%, which is the same as before. This is mainly attributed to the fact that the output error of the twin model is much larger than the normal range, which provides completely false knowledge and leads to the inability of the ESD test to determine that all SPND variables have failed. As the data recovery progresses, the faulty variables are substituted by the recovered data. Subsequently, the average RMSE decreases, and the detection accuracy increases gradually. When the number of recovered variables reaches approximately 18, the accuracy increases to approximately 100%. In addition, there is no additional need to recover from other failures in this case.

Fig. 12Full-screen View

Performance of fault tolerance

Finally, the effects of the fault tolerance strategy on the improvement of twin-model-based fault detection for SPNDs are discussed. As shown in Table 3, the number of faulty SPNDs was divided into five intervals in the multi-point fault detection experiment (150 groups): less than 25, 25–30, 30–35, 35–40, and more than 40. Without the introduction of fault tolerance, the mean fault detection accuracies in the five intervals were 100%, 72.23%, 53.29%, 52.91% and 53.88%. After recovering the faulty data, the number of successfully detected faulty SPNDs increased, resulting in accuracies of 100%. This change confirms that the provision of subsequent fault-tolerant measures to the twin model can efficiently enhance the detection accuracy and improve the capacity to parse the signal correlation for the model, which is sufficient for maintaining healthy monitoring of the system.