Uncertainty estimation of core safety parameters using cross-correlations of covariance matrix

Figures & data

Figure 1 Correlation of errors between safety parameters “X” and “Y”

Figure 2 Correlation coefficients and frequency distributions of core parameters X and Y

Table 1 Two-group cross sections used in the benchmark problem (unit:1/cm)

	Energy
Material	group	Dg	Σ a,g	νΣ f,g	Σ S,1 → 2
Fuel 1	1	1.500	0.010	0.000	0.020
	2	0.400	0.085	0.135
Fuel 1+Rod	1	1.500	0.010	0.000	0.020
	2	0.400	0.130	0.135
Fuel 2	1	1.500	0.010	0.000	0.020
	2	0.400	0.080	0.135
Fuel 2 + Rod	1	1.500	0.010	0.000	0.020
	2	0.400	0.125	0.135
Reflector	1	2.000	0.000	0.000	0.040
	2	0.300	0.010	0.000

Figure 3 Calculation geometry of simplified small LWR

Figure 4 Variation of relative power and control rod worth (unit:%)

Table 2 Correlations of errors obtained by the direct Monte Carlo sampling and the present method based on the GPT

			RP			CR
		k	Reg. A	Reg. B	Reg. C	Reg. A	Reg. B	Reg. C
k			−0.974	0.801	0.993	−0.992	0.033	0.853
RP	Reg. A	−0.974		−0.916	−0.994	0.995	0.195	−0.713
	Reg. B	0.803	−0.917		0.865	−0.870	−0.572	0.371
	Reg. C	0.993	−0.994	0.866		−1.000	−0.083	0.787
CR	Reg. A	−0.992	0.995	−0.871	−1.000		0.092	−0.781
	Reg. B	0.028	0.199	−0.574	−0.087	0.097		0.549
	Reg. C	0.852	−0.712	0.372	0.787	−0.781	0.547
Uncertainty	Monte	0.726	1.134	0.074	0.330	2.052	0.541	1.115
(%)	Carlo
	Sensitivity	0.726	1.137	0.074	0.330	2.056	0.542	1.113
	analysis

Table 3 Variance of core parameters (×10⁻⁴)

		Monte Carlo	GPT
k		5.27E−1	5.27E−1
RP	Reg. A	1.29E+00	1.29E+00
	Reg. B	5.50E−3	5.54E−03
	Reg. C	1.09E−1	1.09E−1
CR	Reg. A	4.21E+00	4.23E+00
	Reg. B	2.93E−1	2.94E−1
	Reg. C	1.24E+00	1.24E+00

Table 4 Covariance of core parameters (×10⁻⁴)

			RP			CR
		k	Reg. A	Reg. B	Reg. C	Reg. A	Reg. B	Reg. C
k			−8.02E−1	4.31E−2	2.38E−1	−1.48E+00	1.28E−2	6.90E−1
RP	Reg. A	−8.04E−1		−7.71E−2	−3.72E−1	2.31E+00	1.19E−1	−9.01E−1
	Reg. B	4.34E−2	−7.76E−2		2.12E−2	−1.32E−1	−2.30E−2	3.07E−2
	Reg. C	2.38E−1	−3.73E−1	2.13E−2		−6.77E−1	−1.47E−2	2.89E−1
CR	Reg. A	−1.48E+00	2.33E+00	−1.33E−1	−6.79E−1		1.03E−1	−1.79E+00
	Reg. B	1.10E−2	1.22E−1	−2.32E−2	−1.56E−2	1.08E−1		3.31E−1
	Reg. C	6.89E−1	−9.01E−1	3.09E−2	2.89E−1	−1.79E+00	3.30E−1

Figure 5 Neutron flux distribution in the present benchmark configuration

Figure 6 Perturbation of neutron flux distribution due to cross section perturbation in Fuel 2

Figure 7 Typical calculation scheme for commercial PWR geometry

Figure 8 Calculation scheme for estimation of correlation in commercial PWR geometry

Figure 9 Calculation geometry of commercial PWR

Figure 10 Correlation among the errors of relative assembly power and control rod worth in octant core

Figure 11 Uncertainty reduction of control rod worth with or without the prior knowledge of the error of assembly relative power