Pellet-clad interaction (PCI) is one of the major issues in fuel rod design and reactor core operation in water cooled reactors. The prediction of fuel rod failure by PCI is studied in this paper by the method of radial basis function neural network (RBFNN). The neural network is built through the analysis of the existing experimental data. It is concluded that it is a suitable way to reduce the calculation complexity. A self-organized RBFNN is used in our study, which can vary its structure dynamically in order to maintain the prediction accuracy. For the purpose of the appropriate network complexity and overall computational efficiency, the hidden neurons in the RBFNN can be changed online based on the neuron activity and mutual information. The presented method is tested by the experimental data from the reference, and the results demonstrate its effectiveness.
The reactor core of Light Water Reactors (LWRs) holds fuel assemblies of fuel rods, which consist of zirconium alloy tubes containing uranium dioxide pellets. The Zr-alloy cladding is the first containment barrier for fission products. Due to water pressure, the cladding creeps down until contact with the pellet occurs after a few operating cycles. In the case of a power increase, this Pellet-Cladding Interaction (PCI) induces large stresses in the cladding that might lead to fuel rod failure [
It is well known that the development of the crack before clad failure is difficult to detect. The usual methods for calculating PCI mainly are finite elements models [
This paper is structured as follows. Section
Taking into account that the performance of an RBFNN is heavily dependent on its architecture, many researchers have focused on self-organizing methods that can be used to design the architecture of three-layered RBFNN. In order to design the structure of the RBFNN automatically, Han et al. [
Figure
The structure of radial basis function neural network (RBFNN).
The FS-RBFNN is based on the average firing rate of the hidden neurons and the MI intensities between the hidden layer and the output layer. Firstly, the activities of the hidden neuron are evaluated according to the average firing rate. The hidden neurons which have a high firing rate are divided into new neurons. Secondly the MI value is used to adjust the network structure; that is, the value of the MI is used as a measure of the connectivity between the hidden layer and the output layer; connections which have a small MI value will be pruned in order to simplify the structure of the network. Finally the gradient-descent method, which is used to adjust the values of the parameters, ensures the exactitude of the FS-RBFNN.
The main steps of the proposed FS-RBFNN algorithm can be summarized as follows.
Create an initial RBFNN consisting of three layers, an input layer, a hidden layer, and an output layer. The number of neurons in the input and output layers is the same as the number of input and output variables in the problem that is being solved. The number of neurons in the hidden layer is randomly generated. Initialize all the parameters: the centers, radii, and connection weights of the RBFNN are all uniformly distributed with a small range.
For the input sample
The centers and radii are adjusted by the gradient method proposed in [
Compute the active firing rate,
Split the
If
Delete the connection between the hidden neuron
Figure
Flow chart of the FS-RBFNN.
Water cooled reactor fuel is subjected to several operational restrictions in order to secure cladding integrity under various classes of anticipated operational transients. Typically, only a few failures occur during a reactor start-up or Condition 2 transients. They are avoided by controlling the magnitude, and the rate of power increases to maintain the cladding tensile stress below the threshold stress or by design measures that increase the threshold for given ramp conditions. Maximum allowable operating conditions are established on the basis of extensive calculations involving analyses of multiple transients, multiple irradiation histories for every rod in a large portion of a reactor core. Pellet-induced cladding failures occurring during power transients can be classified as follows [
SCC is described as follows: Stress corrosion cracking (SCC) is initiated on the inside surface of the cladding due to the sustained strain from pellet-cladding mechanical interaction (PCMI) and the presence of a caustic agents such as iodine and cesium fission products. The tensile stress for this type of defect is below the yield strength of the cladding.
PCMI is described as follows: cladding ductility is exceeded due to strain from pellet thermal expansion and fission gas-induced swelling. The tensile stress required for this type of defect exceeds the yield strength of the cladding. Since PCMI cracks may initiate either on inside or outside cladding surfaces, this failure mechanism also includes outside-in failures, some of which are presently attributed to the local hydriding weakening of the cladding.
Once PCI failures were recognized as a manifestation of PCMI and SCC, it was evident that four factors would be simultaneously necessary for failure to occur: sufficient stress, sufficient time, a susceptible material, and the right chemical environment. However, the four factors are hard to detect during operation or power transients, and the mechanism of the whole PCI-induced clad failure process is so complex, which consists of nuclear dynamics, thermohydraulics, mechanics of materials, and so on. We solve the problem by a “black box method,” which only focuses on the inputs and outputs of the system but do not consider its mechanism. The four factors of the PCI failures mentioned before are the results of the power transient and history, not the sources. As for the power transient and history, the power increase (
In this section, the main objective is to develop a PCI failure probability prediction model using the proposed FS-RBFNN. The FS-RBFNN was programmed with Matlab version 8.3 and was run on an Intel(R) Core(TM) i7 with a clock speed of 3.4 GHz and 4 GB of RAM, under a Microsoft Windows 8.1 environment.
The FS-RBFNN can be designed to update its input-output performance, resulting in continuous, online, self-correcting models. In the experiment, the most important parameters affecting PCI are selected: power increase (
As the various units of the inputs and output, the samples are normalized as
First, the data Douglas-1 is trained, the inputs here are
The RBFNN trained by Douglas-1.
It can be seen from Figure
The Douglas-1 test results of the RBFNN trained by Douglas-1.
Test number | Predicted value | Real value |
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1 | 0.6571 | 1 |
2 | 0.3245 | 1 |
3 | 0.2431 | 1 |
4 | 0.8250 | 1 |
5 | 0.3045 | 1 |
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Tests numbers 1–5 are the defect points, and tests numbers 6–20 are the nondefect points in the literature. The real value in Table
As to the training of the data Douglas-2, the inputs are
The RBFNN trained by Douglas-2.
It can be seen from Figure
The Douglas-2 test results of the RBFNN trained by Douglas-2.
Test number | Predicted value | Real value |
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1 | 0.1345 | 1 |
2 | 0.9745 | 1 |
3 | 0.0156 | 1 |
4 | 0.6825 | 1 |
5 | 0.4245 | 1 |
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The results in Table
Tables
In this paper, a self-organizing neural network named FS-RBFNN is used to predict the PCI failure probability. Based on its advantages of varying its hidden layer dynamically and reducing the calculation time, this neural network model can maintain its prediction accuracy and efficiency, which are suitable features for online calculation when a reactor is operating. Although other mechanistic models are probably more accurate than the present neural network model, the benefit of the latter is the faster calculation with moderate accuracy. Comparing to a simple criterion, the FS-RBFNN method can be used by the input data processing instead of the PCI mechanism analysis. The calculation results based on the data from existing literature demonstrate the effectiveness of the present method. According to the rapid and accurate prediction results, the PCI failure condition after a power ramp can be obtained. Therefore, it is beneficial for early warning and dealing with the possible accident of clad failure. Finally, it should be stressed that the applicability of FS-RBFNN relies on the availability of large number of experimental data.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Science Foundation of China under Grant 11405125, China Postdoctoral Science Foundation Funded Project under Grant 2014M562420, and the Fundamental Research Funds for the Central Universities.