IV. Inverse Uncertainty Quantification of Spent Fuel
4.6. Spent Fuel Model Calibration and Inverse UQ
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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.
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reduce the sample auto-correlation for each parameter. The trace plot can be used to study the chain's evolution, as well as each assembly model parameter posterior distribution. The trace plots of the Markov chain for assembly 5A3 are shown in Figure 4.3. The chains shows good mixing, indicating convergence. Figure 4.3 further shows some assembly 5A3 model parameters posterior distribution after 100,000 iterations. In two cycles of Ringhals-3 PWR operation, assembly 5A3 was irradiated. Kernel density estimation yielded the posterior distributions in Figure 4.3. Figures 4.4, 4.5, and 4.6 show the mean value convergence, the sample auto-correlation, and the convergence of standard deviation values, respectively, for assembly 5A3 model parameters. The MCMC samples converge to the mean and standard deviation of the posterior, as shown in Figures 4.4 and 4.6. The assembly 5A3 MCMC run for Figures 4.4, 4.5, and 4.6 uses five hundred thousand iterations, discarding the first half (burn-in), and keeping every 250th sample (thinning). This was done because before 100,000 iterations, some parameters of assembly 5A3 such as the cycle 1 specific power, moderator temperature, fuel radius, and clad outer radius, and still showed statistical fluctuations in the vicinity of the posterior mean.
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Figure 4.3. Posterior distribution (right) and trace plot of Markov chain (left) for 5A3 assembly.
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Figure 4.4. Mean value convergence for 5A3 assembly parameters.
The convergence of the parameters for other fuel assemblies was verified, and no identical fluctuations were found in the other fuel assemblies. The fluctuations seen in some parameters of assembly 5A3 are statistical, have minimal impact on the mean of the posterior, stays close to the mean, and has minimal impact on the form of the posterior distribution. It requires roughly 1 hour to perform the assembly 5A3 MCMC with 500,000 iterations. Figure 4.5 shows that the sample auto-correlation of the parameters reduces to zero to achieve MCMC sample independence, despite the fact that this requires a large number of samples and iterations. It should be noted that the posterior distribution shape for 5A3 assembly at 500,000 and 100,000 iterations are the same during the MCMC simulation.
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Figure 4.5. Autocorrelation of samples of 5A3 assembly parameters.
Figure 4.6. Standard deviation value convergence for 5A3 assembly parameters.
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The IUQ results are represented by the posterior distributions, which should be employed in future forward UQ. The samples of the posterior should be associated with well established distributions to make subsequent sampling possible and easy. Figure 4.3 highlight a number of things concerning the posterior distributions. The first is that some model parameters of the assembly are not distributed uniformly as we had assumed previously. Second, the cycle 1 specific power and fuel radius distributions appear to be skewed to the left. Despite the fact that this appears to be the case in Figure 4.3 for cycle 2 specific power, it may be stated that the fuel radius obeys the Gumbel-min distribution, whereas cycle 2 and cycle 1 specific powers follow the Weibull distribution. The boron concentration, fuel enrichment, and fuel temperature are all uniformly distributed. These comments apply to only assembly 5A3. The posterior samples of assembly C20 fuel radius, for example, obey a right-skewed distribution known as the Gumbel distribution, otherwise known as the Extreme Value distribution of type I (EV I), whereas the posterior samples of the moderator temperature obey a lognormal distribution. The posterior samples of assembly C01 fuel radius obey a lognormal distribution. As a result, the distribution of the posterior depends on the assembly. Figures 4.7 and 4.8 show the posterior distributions of the moderator temperature and fuel radius samples from assembly C20 and the samples of fuel radius from assembly C01, respectively. The reference [161] contains more information on the Gumbel and Weibull distributions.
Figure 4.7. Posterior distribution of moderator temperature and fuel radius of C20 assembly.
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Figure 4.8. Posterior distribution of fuel radius samples from C01 assembly.
The correlations between several model parameters of the assembly, such as cycle 1 specific power and fuel radius, are not insignificant. This suggests that these input parameters should be sampled jointly during subsequent forward UQ to account for their correlation. Table 4.3 shows the correlations between the model parameters of 5A3 assembly. The correlations are calculated using the samples from the posterior distribution.
Table 4.3. Correlation matrix of posterior samples from assembly 5A3 parametersa.
D E RF TF RC P1 P2 TM BO
D 1.00
E -0.01 1.00
RF -0.17 -0.01 1.00
TF 0.02 0.00 -0.01 1.00
RC -0.03 0.02 -0.13 0.00 1.00
P1 -0.14 0.02 -0.47 -0.02 -0.10 1.00
P2 -0.08 0.00 -0.32 -0.03 -0.04 -0.21 1.00
TM -0.05 0.01 -0.19 0.00 -0.04 -0.14 -0.08 1.00 BO -0.01 -0.02 -0.01 -0.01 0.00 -0.02 0.01 0.01 1.00
a E = fuel enrichment, TF = fuel temperature, BO = boron concentration, D = fuel density, RC = clad outer radius, RF = fuel radius, TM = moderator
temperature, P1 & P2 = specific power cycles 1 & 2.
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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 (𝜎) and mean value (𝜇) 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 𝜎. 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 – 1).
Table 4.4. Parameters of assembly 5A3 model.
Nominal Calibrated Deviation (%)
Parameter 𝜇 𝜎 𝜇 𝜎 𝜇 𝜎
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