During normal reactor operation, the major source of specific activity in the coolant is tramp uranium, which has a relatively low level of specific activity. When the fuel cladding is defective, the FPs released from the gap in the fuel cladding dominate the specific activity, that is, in a rather high level. It is obvious that a large defect size will result in a high level of specific activity. As a result, the coolant’s specific activity serves as a fingerprint for predicting the status of fuel failure. While it should be noted that if the defect size is very small and the tramp uranium mass is considerable, the tramp uranium contribution may overwhelm the defective fuel rod contribution.

The framework for the detection approach is depicted in Figure 1, which is divided into three parts. The blue parts illustrate how the training, validation, and test datasets are generated. Due to the extremely low probability of fuel failure and pursuit of zero fuel rod failure, there is no appropriate fuel failure dataset for training the network. Thus, the dataset is generated using the FP release model, which includes the Booth-type diffusion model and first-order kinetic model, as detailed in the previous study (Dong et al., 2019). The Booth-type diffusion and first-order kinetic models are proven to be appropriate for calculating fission gas release fraction (Lewis et al., 2017). The green parts denote the establishment of a system for detecting fuel failures. The system is trained and validated on the dataset generated by the blue-outlined technique. The red parts demonstrate how the system is being used to predict the status of a fuel failure.

In this study, an improved method for fuel failure detection is proposed based on the cascade-forward neural network (CFNN). The CFNN is a type of feedforward neural network, in which there is a direct connection between every two layers of the network. It means that there is an additional connection between the input layer and output layer for a regular three-layer neural network. The advantage of the CFNN is that it accommodates the nonlinear relationship between the input and output by not eliminating the linear relationship between the two (Warsito et al., 2018). Thus, it is suitable for establishing a relationship between the specific activity in the coolant and degree of fuel failure.

Figure 2 illustrates the structure of the CFNN utilized to identify fuel failures. The network’s input is a vector containing the normalized specific activity of FPs in the coolant, and the network’s output is the degree of fuel failure, which is encoded in one-hot form. The degree is divided into six categories: degree 1 denotes an unbroken fuel cladding, whereas degrees 2–6 denote a defective fuel cladding. Correspondingly, a high degree indicates a big defect size. After going through a Softmax layer, the value of the neuron represents the probability of that degree of failure occurring.

Perez and Hamawi (2017) show that typical elements in design basis source term determinations can be represented by Kr, Xe, Br, I, Rb, Cs, Sr, Ba, Mo, and Te. Among these elements, Kr, Xe, I, and Cs are most important FPs in fuel failure detection. The nuclides mainly considered in the study are Kr, Xe, I, and Cs, which have a relatively large fission yield and a relatively long half-life, including Kr-85, I-131, I-133, Xe-133, Xe-135, and Cs-137. Simultaneously, considering the decay chain and the representativeness of nuclides in each group, the generated dataset has specific activities of 23 nuclides: Br-83, Kr-85, Kr-85m, Kr-87, Kr-88, Sr-90, Te-131, Te-131m, I-129, I-131, I-133, I-134, I-135, Xe-133, Xe-133m, Xe-135, Xe-135m, Xe-138, Cs-134, Cs-134m, Cs-137, Cs-139, and Pr-143, which are used as the input of the neural network. Then, the neural networks using 23 nuclides as input are trained in this study. Table 1 shows typical samples of the input and output of the neural network.

Prior to training the neural network, various parameters must be adjusted to make the neural network work well. The number of neurons in the hidden layer is obtained empirically using the following formula (Dong et al., 2019): l = 0.43 n m + 0.12 m 2 + 2.54 n + 0.77 m + 0.35 + 0.51 , (4) where m, n, and l are the corresponding number of neurons in the output, input, and hidden layer, respectively.

Eq. 4 is proven to be an efficient formula to determine the number of neurons in the hidden layer. Since it is shown that the neural networks with the number of hidden layer neurons around the formula all have good performance, the number of hidden layer neurons is determined directly by the formula. The number of hidden layer neurons is 11 for an input of 23 nuclides.

As Figure 2 shows, neurons pass their value to the next layers. Besides the final layer, an activation function, rectified linear unit (ReLu) function, is applied between the two layers to make the neural network present the nonlinear relationship. The ReLu function helps solve the gradient vanishing problem and increases training efficiency (Choi et al., 2021). The Softmax function is applied as the activation function in the final layer: y i = e x i ∑ j e x j , (5) where x _i and x _j indicate neurons in the previous layer.

After applying a Softmax function, the values of the neurons are normalized to [0,1], and the summation of each value of neurons in the final layer is 1. Then, the value of neurons can represent the probability of the corresponding degree of failure. The advantage of Softmax function is that it can improve the discrimination between the output neurons and hence increase learning efficiency.

Accompanying the Softmax function, cross-entropy is selected as the loss function (de Boer et al., 2005): L = − 1 N ∑ i = 1 N ∑ j y j i ⁡ log y ̂ j i , (6) where N is the total number of samples, i denotes the sample number, j denotes the neurons in the final layer, y j ( i ) is the real value in the training sample, and y ̂ j ( i ) is the output value of the neural network. Using Softmax function and cross-entropy together will make the training process faster and more stable.

Adam is selected as the optimizer (Kingma and Ba, 2017), and the neural network is trained with a learning rate of 0.01 balancing the convergence and training speed. The loss of training and validation is shown in Figure 3. It can be seen that the loss of training and validation almost does not decrease after 1,000 epochs. In addition, aiming to avoid the discrepancy of single training, the neural network is trained 10 times individually and averaged to inference the degree of fuel failure.

The dataset utilized to train the ANN is imbalanced, with the intact condition prevailing. As a consequence, the trained neural networks have a tendency to categorize degree 2 or 3 (small defects) as degree 1 (intact), which significantly underestimates the degree of fuel failure. To overcome this issue, a decision tree (DT) such as the classifier is developed, as shown in Figure 4.

Two distinct forms of the CFNN are developed in the improved method. The CFNN1 is used for predicting whether the fuel cladding is defective. Its structure is the same as shown in Figure 2, except that the output layer contains just two neurons. If the output of the CFNN1 indicates that the fuel cladding is intact, the degree of fuel failure can be assessed to 1. Otherwise, the CFNN2 is used to predict the degree of fuel failure when the fuel cladding is defective. The output of the CFNN2 is the probability of the corresponding degree under defective condition.

Due to the fact that the neural network is a black-box used to make a connection between the input and output, it is incapable of explaining the mechanisms behind physical phenomena. While the output of the neural network depends on the magnitude of the weights between connections, the contribution of input variables can be determined by interpreting the weights of the trained neural network. This is meaningful information for reactor operators to select the appropriate FPs predicting the status of fuel failure.

A method to determine the relative importance of the input of the neural network is proposed first by Garson, (1991) and simplified by Gevrey et al., (2003). The method essentially involves partitioning the hidden-output connection weights of each hidden neuron into components associated with each input neuron.

For each hidden neuron, the absolute value of the input hidden layer connection weight is divided by the sum of the absolute value of the input hidden layer connection weight of all input neurons: Q n l = W n l ∑ n W n l , (7) where W denotes the weight between neuron connections, n denotes the input layer, and l denotes the hidden layer.

Once the parameter Q _nl is determined, the sum of Q _nl for each hidden neuron is divided by the sum of each hidden neuron of the sum for each input neuron of Q _nl . The relative importance of all output weights attributable to the given input variable is then obtained. Then, the relative importance of all output weights for a given input variable can be obtained: R I n = ∑ l Q n l ∑ l ∑ n Q n l . (8)

Apart from the connection between the input and hidden layer, there are also connections between the input and other layers. Then, the relative importance should also include other connections and be averaged: R I n a v e = 1 3 R I n 1 → 2 + R I n 1 → 3 + R I n 1 → 4 . (9)

It is known that hundreds of nuclides are produced during reactor operation, and their fission yields and decay constants are significantly different, which may vary several orders of magnitude. Because using all nuclides for fuel failure detection is uneconomical and impracticable, it is vital to identify the key nuclides for predicting the status of the fuel cladding.

The neural networks in this study use 23 nuclides as the input, mainly including noble gases, iodine, and cesium. Due to their relatively large fission yield and long half-life, these types of nuclides are already frequently utilized in the isotopic ratio method for fuel failure detection. In order to evaluate the contribution of these nuclides to the determination of the degree of fuel failure and support the further selection of FPs, a sensitivity analysis is performed using the method provided in Section 2.6.

After training the neural network, the connection weights are acquired. Then, Q _nl in Eq. 7 can be easily calculated. After obtaining Q _nl , the contribution of the input variable can be calculated using Eq. 8. Since there are connections between the input and other layers, it should be noted that the contribution derived by Eq. 8 needs to be averaged.

The contribution of FPs derived by sensitivity analysis of 10 trained neural networks is shown in Figure 8. It can be obviously seen that the most important nuclides are I-131, Xe-133, Xe-135, and Kr-85, which is consistent with the prevalent knowledge. These nuclides are already usually used in the isotopic ratio method for fuel failure detection (Menéndez, 2009; Li et al., 2017; Lyu and Xiong, 2019). These four nuclides are critical in the fuel failure detection process. Besides these four nuclides, it can be seen that Xe-133m, Sr-90, and I-133 also have a relatively high contribution.

The previous study has shown that increasing the quantity of input nuclides can improve detection accuracy. While considering the efficiency and limitation of the detection apparatus, some nuclides may be ignored. Although the neural network can be simplified by ignoring some FPs, it is better to not ignore the aforementioned seven nuclides.

The ideal neural network should make extensive use of FP-specific activity. While due to the detection limitation of the detector, only a few nuclides can be used as the input of the neural network. Since the specific activity of FPs in the coolant is supposed to be detected by a cadmium zinc telluride (CZT) detector, the nuclides used as the input are selected based on matching the energy spectrum of the CZT detector and generated dataset. A typical spectrum of FPs derived by the CZT detector is shown in Figure 9, and the generated dataset has specific activities of 23 nuclides as stated in Section 2.3; the nuclides used as the input are selected as I-131, I-133, Xe-133, and Xe-135 combining the sensitivity analysis in Section 3.2. Then, the neural networks with an input of four nuclides are trained based on the improved method and used for practical application, which is consistent with the previous study.