IV. Inverse Uncertainty Quantification of Spent Fuel
4.5. Inverse UQ Workflow and Surrogate Model Validation
This section summarizes the outcomes of the PCE surrogate during the forward evaluation of the model and GSA. Figure 4.1 depicts the SNF decay heat IUQ calculation flow, which includes the following steps:
1) Identify the uncertain assembly parameters to be considered and associate uncertainties as listed in Table 3.1. The surrogate model considers only the uncertain assembly parameters.
2) Create 100 samples using LHS and perturb all of the assembly parameters at the same time.
3) Use the perturbed assembly model parameters as inputs for STREAM decay heat calculations.
4) Build the PCE-based surrogate model using the outputs from the 100 simulations. For each of the assemblies in Table 4.1, a PCE surrogate model is created.
5) For the evaluations of the forward model during the MCMC sampling in the model calibration and IUQ, use the surrogate model instead of STREAM.
6) The statistics of the posterior samples of the assembly model parameters generated via MCMC sampling are postprocessed and analyzed. Analyze the assembly model parameter PDFs, best estimates, and uncertainties which are consistent with the measured decay heat data. The postprocessing and analysis results are reported in Section 4.6.
Since the assemblies listed in Table 4.1 are depleted in various number of cycles, the dimensions of the assembly model parameters vary. As a result, the PCE order and the number of model evaluations needed to compute the PCE coefficients vary. The assembly surrogate model Leave-One-Out Cross-Validation (LOOCV) error is shown in Table 4.2. For the assemblies, the LOOCV error is in multiples of 10-4, indicating that the surrogates can correctly estimate the decay heat given the input parameters. Considering assembly C01, the LOOCV error convergence against the number of samples is shown in Figure 4.2. The use of 100 input/output samples is shown to be enough in Figure 4.2. Eq. (4.21) is used to determine the LOOCV error, where ππ is the mean of the output of the training samples.
ππΏπππΆπ =βππ=1(π(π(π))βπππΆπΈ(π(π)))2
βππ=1(π(π(π))βππ)2 (4.21) The building of 100 PCE models is used to assess the LOOCV error. A total of 99 input/output data are used to train each model. The PCE is then used to estimate the output of the one input
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point that was removed, and the result is compared to the reference solution. The time it takes to develop the PCE surrogates is only a few seconds. Table 4.2 shows the STREAM simulation times on a personal computer for each assembly in Table 4.1. The depletion steps used in the burnup calculations for these assemblies range from 38 to 97. As a result, surrogate models are built for STREAM in order to reduce simulation time in the model calibration and IUQ. The PCE surrogate model is used in Chapter III on the study of forward UQ of modeling parameter induced uncertainties in SNF. In Section 3.6, the accuracy (performance) of the PCE compared against STREAM is demonstrated extensively.
Table 4.2. PCE detail and assembly computation time.
Assembly ID
STREAM run time (minutes)
PCE order
PCE LOOCVa error
Calibration run time (minutes)
E38 4.22 2 1.46E-04 12.01
C20 8.79 3 1.85E-04 13.00
F21 5.03 2 1.39E-04 13.01
E40 3.52 2 2.31E-04 12.01
G11 4.20 2 1.87E-04 13.00
G23 3.65 2 1.81E-04 12.00
I20 3.39 2 1.69E-04 12.00
D27 5.08 3 1.32E-04 12.00
C01 5.79 2 1.75E-04 13.00
0E2 4.22 2 1.96E-04 13.00
3C5 8.60 3 1.19E-04 14.00
4C4 7.47 2 1.46E-04 12.00
0C9 8.50 2 2.45E-05 12.01
5A3 4.08 2 2.44E-04 14.00
a LOOCV: Leave-One-Out Cross-Validation.
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Figure 4.2. LOOCV error convergence against sample number for assembly C01.
The boron concentration, fuel temperature and fuel enrichment have zero effect on the variance of the assembly decay heat, according to the Sobol' sensitivity indices. The moderator temperature, specific power and fuel radius have the greatest impact on the uncertainty of the decay heat, followed by the clad radius and fuel density. This pattern was observed in assembly C01, which was studied in Chapter 3 (Section 3.6, Figure 3.8). As a result, the Sobol' sensitivity indexes are not shown here. Regardless of the cooling times, the same tendency can be seen in all of the assemblies in Table 4.1. It is possible to eliminate the boron concentration, fuel temperature and fuel enrichment from the parameter calibration because they have zero effect on the decay heat variance. The IUQ, on the other hand, includes all eight model parameters of the assemblies. This is due to the fact that this chapter is part of a larger investigation into SNF assembly source terms such as decay heat, gamma source, neutron source, and activity and the unimportant parameters for decay heat are dominant parameters for other source terms (See Figure 3.8). Furthermore, in Section 4.6, the IUQ and calibration results of parameters with zero effect on the decay heat variance are not described or studied in extensively.
Consider the following to put the GSA result in context with the premise of independent assembly model parameters. In neutronics calculations with TH feedback, the moderator temperature, power, boron concentration, power, and fuel temperature are typically impacted. On the output uncertainty, the GSA evaluates the effects of individual inputs (first order effects), interactions between two or more parameters (high order effects), and sums these effects into the
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total effect. Figure 3.8 shows that the total and first order effects are nearly identical, implying that the influence of interactions between, say, fuel temperature and power, on the decay heat variance is minor. Despite the fact that fuel temperature and power should be correlated, Figure 3.8 shows that power impacts decay heat, but fuel temperature has zero effect. This is likely caused by two things: (i) due to the assumption of independence, samples of these two parameters are not drawn from a joint distribution; and (ii) the STREAM assembly calculations in this thesis are based solely on neutronics and do not include TH feedback.