Fourth, a novel application of decay heat metrics based on inverse UQ (IUQ) and model calibration is proposed. Furthermore, this improves the agreement between the decay heat calculation and experiment and reduces the uncertainty about the decay heat.
Introduction
- Spent Nuclear Fuel
- Elements of Nuclear Reactor Computer Simulation
- Literature Review of Spent Fuel Radiation Source Terms
- Thesis Motivations and Contributions
- Thesis Outline
The amount of landfill space required for disposal depends on the decay heat and reactivity of the SNF. Applications can be found in spent fuel research [29], nuclear safeguards and nuclear criticality non-proliferation studies [32], prediction of reactor core behavior during different core power levels and loss of flux accidents [33], prediction of reactor core optimization of sample neutron characteristics core fueling [38,39], sensitivity and uncertainty analysis [40] and core fuel management [41].
Spent Nuclear Fuel Radiation Source Terms
- Implementation in STREAM
- Verification
- Methods of Decay Heat Measurements
- Calorimetric and Gamma Scanning Method
- Decay Heat Measurements Benchmark
- LWR Fuel Assembly Decay Heat Measurements Benchmark
- Experimental Validation against Decay Heat Measurements
Another report [98] shows that the uncertainty of the US PWR fuel composition decay heat measurement (i.e. GE-Morris and HEDL) is 2%. The decay heat of activation of structural elements in the cladding and spacers, such as stainless steel, is estimated and included in the fuel composition's decay heat.
Forward Uncertainty Quantification of Spent Fuel
Stochastic Sampling of Nuclear Reaction Microscopic Cross section Covariance
In the above equations, the nominal and perturbed multigroup microscopic cross sections are denoted by 𝝁 and 𝒙, respectively. The vector z is obtained by sampling from a standard normal distribution, and C is the multigroup microscopic cross-section covariance matrix calculated by the NYJOY code.
Generation of Perturbed Fission Yield Data
Details of the stochastic sampling of the ENDF/B-VII.1 covariance data implemented in STREAM are described here to make the thesis self-contained, but can be found in reference [105]. One of the capabilities of this code is to produce fission yield covariance matrices using the Bayesian/GLS method described above [114].
PCE Surrogate Model Formulation and Global Sensitivity Analysis
Uncertain input parameter vector variables, expansion coefficients, and multivariate basis rank are in bold italics. The Sobol indices can be obtained by post-processing the PCE coefficients as described in terms of the basis functions depending on input parameters other than (𝜉𝑖, …, 𝜉𝑠).
Description of Fuel Assembly Test Case
A script interface and system call can be used to connect STREAM to the UQ instrument for single parameter perturbation. Although there are correlations between the input parameters in Table 3.1, the uncertainty analysis ignores these correlations.
Uncertainty Quantification Results
- UQ due to Modeling Parameters using Surrogate Models
- Uncertainty Quantification due to Nuclear Data
- Discussion of Nuclear Data and Modeling Parameter Induced Uncertainties
The relative validation error for the gamma source after 23 years is smaller than at discharge, as shown in Table 3.6. Statistical fluctuations and errors are present in the results due to the limited size of the analyzed sample. The uncertainties caused by the finite size of the input data are estimated using the bootstrap method [121] for the pointwise predictions of the surrogate model.
Figures 3.10b – 3.21b show the evolution of uncertainties caused by modeling parameters during combustion and cooling.
Propagation of Radiation Source Term Uncertainties in Cask Shielding
- Description of TN-32 SNF Cask
- The Monte Carlo Codes
- Shielding Calculation Conditions
- Radiation Source Spectra Results and Uncertainties
- Cask Dose Uncertainties
The fitting neutron and photon source uncertainties calculated from the nuclear perturbed data and the perturbed model parameters are shown in Figure 3.24 as a function of energy groups. Comparing Figure 3.24 and Table 3.11, it can be seen that although the contribution of the fission yield to the neutron source uncertainty is low at about 0.3%, the contribution to the neutron dose uncertainty is about 4% after propagation. Comparing Figure 3.21 and Table 3.11, we note that the total uncertainty of the neutron source in 23 years of cooling is about 11%, while the neutron dose rate at the boiler surface has a total of about 13% uncertainty from all perturbed data .
Most importantly, based on the results shown in the figures and Table 3.11, it is unnecessary to perturb the fission yield to determine the total neutron dose uncertainty.
Inverse Uncertainty Quantification of Spent Fuel
Bayesian Method
If the data set 𝚵 is measured and the sample points Ξ = (𝜉1, … 𝜉𝑛)𝑇 are considered to be generated independently of Eq. 4.1), the probability function, which depends on the hyperparameters, is as follows: 4.4) can be understood as the probability of observing the current data set assuming that 𝚵~𝜋(𝜉|𝜽), i.e. the probability that 𝚵 is actually measured if 𝜽 are true hyperparameters. The posterior distribution of the hyperparameters can now be an update of the prior distribution in Eq. where 𝑍 is the evidence or marginal probability and a normalization constant that ensures uniform integration of the distribution:.
The Bayesian technique determines the posterior distribution in Eq. 4.5) in the IUQ context and then identifies the ideal distribution of the parameters 𝚵 given the current and available dataset.
Inverse Uncertainty Analysis under Bayesian Framework
Markov Chain Monte Carlo
Quantities that can be derived from the posterior distribution can then be approximated by the Markov chain. The extent to which a proposal is accepted or rejected is also determined by how closely the proposal distribution matches the posterior distribution. The MCMC techniques, due to their iterative tendency, are affected by convergence challenges that come from correlations between the parameters of the posterior distribution.
Consequently, the higher the probability of accepting the sampled input parameters as candidates from the posterior distribution.
Description of Fuel Assembly Test Cases
For the details described in this chapter, sample correlations of the composition model parameters are not considered. Thermal hydraulics (TH) feedback is not used in the neutronics calculations in this chapter. The Bayesian method and MCMC algorithms in this chapter use the built-in libraries of the UQLab tool [156].
The purpose of GSA in this chapter is to find input parameters that do not need calibration.
Inverse UQ Workflow and Surrogate Model Validation
The boron concentration, fuel temperature, and fuel enrichment have no effect on the variance of the assembly decay heat according to Sobol's sensitivity index. The moderator temperature, specific power, and fuel radius have the greatest influence on the uncertainty of the decay heat, followed by cladding radius and fuel density. It is possible to eliminate the boron concentration, fuel temperature, and fuel enrichment from the parameter calibration because they have no effect on the decay heat variance.
Furthermore, in Section 4.6, the IUQ and calibration results of parameters with no effect on the decay heat variance are not described or studied in detail.
Spent Fuel Model Calibration and Inverse UQ
The samples from the posterior should be associated with well-established distributions to make subsequent sampling possible and easy. Figures 4.7 and 4.8 show the posterior distributions of the moderator temperature and fuel radius samples of composition C20 and the samples of fuel radius of composition C01, respectively. The correlations between various model parameters of the assembly, such as cycle 1 specific power and fuel radius, are not insignificant.
The nominal values in Table 4.4 are from the prior distribution, while the calibrated values are derived from samples from the posterior distribution.
Impact of Calibrated Model Parameters
The Kolmogorov-Smirnov criteria [162] are used to select the distribution that best matches the posterior samples. The posterior samples for the outer coated beam and the fuel beam were found to obey the beta distribution. This second UQ now uses samples of the modeling parameters drawn from the posterior distribution after calibration of the IUQ and the model.
The 'posterior' column indicates the uncertainty when the modeling parameters are samples from the posterior distribution.
Machine Learning System for Spent Fuel Analysis
Dataset of Measured Spent Fuel Assembly Decay Heat
Chapter II and section 2.6 present the measured decay heat data of the SNF LWR assembly. Although these measurement data were obtained to support the validation of computational tools used in SNF analysis, it is decided to use these data for more applications such as machine learning and inverse uncertainty quantification (see Section IV). This becomes more reasonable when we consider that obtaining fuel cell decay heat measurement data is expensive.
Thus it would be desirable to have a tool that could learn relationships between measurements and associated fuel assemblies so that the knowledge learned could be used to make predictions about other assemblies.
Exploratory Decay Heat Measurement Data Analysis
For example, it can be observed that the measured decay heat has a similar relationship to the burnout and enrichment. An input function space that includes both combustion and enrichment may not improve the accuracy of the decay heat estimate, as they have similar correlation coefficients of 0.20 and 0.16, respectively, with the decay heat. It quantifies the amount of increase in the variance of a regression coefficient estimate due to collinearity.
For the BWR dataset, the same VIF trends are noted and similar correlations (between measured decay heat and cooling time; between combustion and enrichment; between enrichment and uranium mass) are observed as in the PWR dataset.
Selection of Input Features
From the physics of the problem, a lot of information about the assembly design and irradiation history is related, and there could be redundancies in the input space. Another reason why these features were chosen is that they could be obtained for any other set besides those in the decay heat data set. This is an advantage as the models trained in this work could then be used to predict the integral decay heat of assemblies released from any LWR provided the input data are within the ranges listed in Table 5.3.
And that enrichment and uranium mass account for high multicollinearity in the input space.
Synthetic Data Generation
Regarding the generation of synthetic data, the uncertainties determined in step ii for the input features can be obtained from the literature [10,169]. It is possible to determine the actual probability distribution of the input features which will then be used in the generation of synthetic data. An important feature of synthetic data is that it has the same statistical characteristics as the original data.
Synthetic data should preserve the statistical properties (mean, variance, and correlations) of the original data set.
Methodology of Training and Testing
After training with the original data set to determine baseline performance, we then used the synthetic data in a second training set. Consequently, the performances of models developed with synthetic data are evaluated on a 10% test set of the original data set. Since there is relatively more synthetic data, training with synthetic data was performed once per model.
At a minimum, the models developed with synthetic data should have similar error properties to the models based on the original data set if the synthetic data reflect the complex relationships between the features of the original data set.
Machine Learning Methods
- Gaussian Process
- Support Vector Machines
- Neural Network
The doctor's statistics, i.e. the mean and variance, are then determined conditional on the actual inputs and outputs [176]. The sum of the bias and weighted inputs is the input to the transfer function. A single layer of NN consists of neurons and transfer functions, and many layers can be connected in a NN.
The scheme of the NN used in this work is shown in Figure 5.5, where W represents the weight vector, b is the bias vector, 𝑓(∙) de.
Model Evaluation
- Model Performance
Comparing Tables 5.5 and 5.7, the RMSE and MAE of the GPR and NN models on the synthetic data set are comparable to those of the original data set, considering the test set. Nevertheless, the accuracy of the SVM model used to generate the synthetic data is guaranteed with the performances shown in Table 5.7. Moreover, this guarantees the accuracy of the output of the models built with the synthetic data.
The result is shown in Table 5.9, where it can be seen that the synthetic data model of the SVM outperforms that of the bootstrap.
Uncertainty Quantification of Machine Learning Models
- Uncertainty Quantification due to Perturbation of Input Features
- Uncertainty Quantification due to Training Data Selection and Size
The results of the UA due to uncertainties in the input data are shown in figures. The second UA is due to the selection of training data and the size of the input data used in the training. These are then used to calculate the standard deviation of the ML model's point predictions.
Furthermore, the amount of uncertainty present in the point estimates will be significant if the models are used for extrapolation, i.e., the input features are outside the range shown in Table 5.3.
Conclusions
Future Work
Shin, Validation of Lattice Physics Code STREAM for Predicting Spent Water Reactor Isotopic Inventory of Nuclear Fuels, Ann. Bamidele Ebiwonjumi, Deokjung Lee, “Bayesian Method and Polynomial Chaos Expansion Based on Inverse Uncertainty Quantification of Consumed Fuel Using Decay Heat Measurements,”. Bamidele Ebiwonjumi, Alexey Cherezov, Siarhei Dzianisau, Deokjung Lee, “Machine Learning of Light Water Reactor Spent Nuclear Fuel Assembly Decay Heat Measurements,”.
34; Validation of Lattice Physics Code STREAM for Predicting the Isotopic Inventory of Spent Nuclear Fuel Under Pressurized Water Reactor,” Ann.
