IV. Inverse Uncertainty Quantification of Spent Fuel
4.7. Impact of Calibrated Model Parameters
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Model calibration is used to update the assembly model parameters based on the inverse analysis. Table 4.4 shows the nominal and calibrated standard deviation (π) and mean value (π) of the model parameters of 5A3 assembly. For those parameters that are important (clad outer radius, moderator temperature, cycle 2 specific power, cycle 1 specific power, fuel density, fuel radius), there is a decrease in the π. The deviation of the calibrated and nominal standard deviation and mean values is very small for the least important parameters (235U enrichment, fuel temperature, and boron concentration), likely because their impact on the variance of the decay heat is negligible, as previously explained in Section 4.6. The shape of the prior and posterior distributions for the least important parameters are also identical, which is why their IUQ, and calibration results are not discussed at length in this section. The nominal values in Table 4.4 originate from the prior distribution, while the calibrated values are derived from samples of the posterior distribution.
Table 4.4 shows the relative difference in percent between the calibrated and nominal values, defined as (calibrated/nominal β 1).
Table 4.4. Parameters of assembly 5A3 model.
Nominal Calibrated Deviation (%)
Parameter π π π π π π
Fuel density, g/cc 10.2700 0.0417 10.2759 0.0410 0.057 -1.679
235U enrichment, wt.% 2.1000 0.0167 2.0995 0.0167 -0.024 0.000 Pellet radius, cm 0.40955 0.0025 0.41094 0.0020 0.339 -20.000 Clad outer radius, cm 0.47500 0.0083 0.47566 0.0082 0.139 -1.205 Fuel temperature, K 900.000 30.0000 900.036 30.0039 0.004 0.013 Cycle 1 specific power, W/g 11.270 0.1882 11.340 0.1676 0.621 -10.946 Cycle 2 specific power, W/g 33.350 0.5569 33.485 0.5329 0.405 -4.310 Mod. Temp., K 577.000 11.5400 578.322 11.4190 0.229 -1.049 Boron Conc., ppm 650.000 13.0000 650.0687 13.0027 0.011 0.021
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decay heat (C β E), where C represents the calculated decay heat by STREAM and E represents the decay heat measurement. In the nominal situation, Table 4.5 contains two cases where the C β E results are outside the measurement uncertainties (at 95% confidence level (CL)). These cases happen at cooling times of 6950 days and 6977 days, respectively, for assemblies C20 and 5A3.
The C β E results now lie within the range of measurement uncertainties, thanks to the improved model parameters obtained through calibration.
Table 4.5. Comparison of calibrated and nominal decay heat results of STREAM.
Nominal result Calibrated result Assembly
ID
Decay time (days)
C β E (W)
C/E β 1 (%)
C β E (W)
C/E β 1 (%)
Exp. Uncer. (W) (95% CL)
5A3 6972 -5.43 -2.28 -1.91 -0.80 9.02
5A3 6975 -4.42 -1.87 -0.90 -0.38 9.00
5A3 6977 -11.20 -4.60 -7.68 -3.16 9.10
5A3 7291 -2.09 -0.90 1.39 0.60 8.92
5A3 7691 -3.18 -1.39 0.24 0.11 8.87
5A3 9467 -2.11 -1.01 1.09 0.52 8.61
0C9 6551 7.39 1.50 1.55 0.31 12.76
3C5 6948 2.40 0.49 0.88 0.18 12.69
3C5 8713 5.65 1.28 4.24 0.96 12.04
4C4 6572 -5.32 -1.26 -1.93 -0.46 11.74
0E2 5823 -9.12 -1.55 -3.25 -0.55 14.19
0E2 6389 -5.84 -1.03 -0.14 -0.03 13.87
0E2 6390 -7.57 -1.33 -1.87 -0.33 13.89
0E2 7826 -3.53 -0.67 1.78 0.34 13.22
0E2 7837 -7.01 -1.33 -1.70 -0.32 13.27
0E2 7970 -4.96 -0.95 0.31 0.06 13.19
C20 6950 13.40 3.22 10.72 2.58 11.65
C20 6951 3.17 0.75 0.50 0.12 11.80
D27 7673 -8.40 -1.84 -2.89 -0.63 12.24
E38 7999 -7.51 -2.00 -2.82 -0.75 11.07
E38 8000 -5.55 -1.48 -0.86 -0.23 11.04
F21 7376 -8.18 -1.94 -4.90 -1.17 11.72
F21 9377 -4.51 -1.19 -1.51 -0.40 11.10
F21 9524 -3.15 -0.84 -0.16 -0.04 11.04
G11 6990 -8.56 -2.06 -3.16 -0.76 11.66
G23 6984 -7.07 -1.68 -2.82 -0.67 11.72
I20 6588 -8.23 -2.04 -3.31 -0.82 11.47
E40 8075 -8.56 -2.25 -3.54 -0.93 11.14
C01 8468 1.33 0.32 0.13 0.03 11.65
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Figure 4.9 shows the fitted distributions for the posterior samples of assembly 5A3 model parameters. The frequency histogram of samples is also included in this figure. The posterior samples are associated with different parametric probabilistic distributions using UQLab statistical inference capability by calculating the cumulative distribution functions, PDFs, parameters, and moments. The Kolmogorov-Smirnov criteria [162] is used to choose the distribution that matches best the samples of the posterior. Table 4.4 contains the moments of the fitted distributions under the column of βCalibratedβ result. Only distributions that deviate from the prior uniform distribution are shown with fitted distributions. The fuel radius posterior samples are fitted to the Gumbel-min distribution, whereas the specific powers in cycles 2 and 1 are matched to the Weibull distribution. The Gumbel-min distribution is otherwise referred to as the SEV Type I distribution or the smallest extreme value (SEV) distribution.
Figure 4.9. Fitting the posterior PDFs of assembly 5A3 model parameters. The prior samples follow a uniformly distribution.
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For the example of assembly C01, a different prior was tested. To examine the impact on the posterior distribution, starting with a prior that is normally distributed. The samples of the posterior for the clad outer radius and fuel radius were found to obey the beta distribution. This indicates that the assembly C01 fuel radius of is not distributed uniformly or normally. Figure 4.10 depicts the posterior and prior distributions of assembly C01 clad outer radius and fuel radius.
Figure 4.10. Posterior and prior distributions of assembly C01 clad outer radius and fuel radius.
The posterior samples obey a beta distribution when the prior samples follow a Gaussian distribution.
To end this chapter, the modeling parameter uncertainties for the assembly C01 are propagated again. This second UQ now employs the modeling parameter samples obtained from the posterior distribution after the IUQ and model calibration. Consequently, the PDF of the model parameters now differ from those assumed in Table 3.1 for the prior. This second UQ has two purposes: to validate the posterior samples in subsequent decay heat calculation and to examine posterior effect on the new decay heat uncertainty. Table 4.6 shows the results of this second UQ in comparison to those of the first UQ which are reported in Sections 3.5 and 3.6. Only the decay heat is investigated. Table 4.6 shows that there is a reduction in the uncertainty when we use the posterior samples of the model parameters in a new decay heat calculation. The βpriorβ column shows the uncertainty when the modeling parameters use the prior information in Table 3.1 which are based on expert judgement and assumed PDFs. The βposteriorβ column indicates the uncertainty when the modeling parameters are samples from the posterior distribution which is
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based on the IUQ, and model calibration described in this chapter. The absolute uncertainty reduces by about 40%. This reduction can be attributed to the following: (i) the modeling parameters uncertainties such as manufacturing tolerances are no longer based on expert judgement, assumptions or personal opinions but based on a mathematical basis of statistical framework (ii) the PDFs of the parameters are no longer assumed but now known to some degree of accuracy and (iii) the correlations between the parameters are now accounted for in the posterior samples. In Chapter III, the modeling parameter induced uncertainty of the decay heat in Table 3.1 is around 2%, whereas it is now 1.2%. By implication, the overall uncertainty in Table 3.1 reduces to 2.2%.
Table 4.6. Absolute uncertainty of assembly C01 decay heat calculation results.
Prior Posterior Deviation (%) 1,000 samples 8.02 W 4.82 W -39.90
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