# Reduction of Prediction Uncertainty

To accurately calculate neutronics parameters, we have to use reliable calculation methods and nuclear data. For this purpose, we can use valuable measured data obtained from fast critical assemblies and fast reactors by applying the bias factor method [13] and the cross-section adjustment method [14]. In these two methods, it is necessary to consider that there are two kinds of errors, systematic and statistical errors, in measured and calculation errors. Here we propose a method to remove the systematic errors to improve prediction accuracy. Measured data *Re* have a systematic error *R*eb and a statistical error *R*es and are expressed by

*Re* ¼ *Re*0ð1 þ *R*eb þ *R*esÞ ð17:35Þ

where *Re*0 is the true value. Also, calculated neutronics parameters *Rc* are expressed by

*Rc* ¼ *Rc*0ð1 þ *R*cb þ *R*cs þ *S*ΔσÞ ð17:36Þ

where *Rc*0 is the true value, *R*cb is systematic error, *R*cs is statistical error from calculation methods, and *S*Δσ is the error from cross-section error. To eliminate the systematic errors in measurements and calculations, we consider the ratio of measurement to calculation, called bias factors:

__ ____R____e____ __ __ ____ ____1____ ____þ____ ____R____ ____eb____ ____ ____þ____ ____R____ ____es____ ____ __ 17:37

*f* ¼

*c* ¼ 1 þ *R*cb þ *R*cs þ *S*Δσ ð Þ

Because the average of statistical errors becomes zero, the variance of *f* becomes

*V*ð*f* Þ ¼ *V*ð*R*esÞ þ *V*ð*R*csÞ þ SWS*T* ð17:38Þ

where *W* is the variance of nuclear data used. In deriving Eq. (17.38) it was assumed that all the systematic and statistical errors are smaller than unity and that there is no correlation between statistical errors of measurements and calculations. From Eq. (17.38) we can say that if there is no statistical error, the bias factor *f* is within the range of

p ð Þ ð17:39Þ

with the confidence level of 65 %(*c* ¼ 1), 95 % (*c* ¼ 2), or 99 %(*c* ¼ 3). Therefore, if *f* is outside the range, we can say in the foregoing confidence level, there is a systematic error of

*R*eb *R*cb ¼ j1 *f* j *c*σ ð17:40Þ

For sodium void calculations, calculated values are the sum of positive nonleakage components and negative leakage components. The negative leakage components are difficult to estimate because the transport effect has to be considered in calculating the neutron steaming. Therefore, there may be a nonnegligible

systematic error in the leakage term *RL* when the void pattern is leaky. By

considering such a void pattern, we can discard the leakage term in systematic errors. Thus, we can determine the systematic errors. After the removal of the systematic errors, we can apply the cross-section adjustment method or the bias factor method to improve the calculation accuracy. In the cross-section adjustment method [13], the adjusted cross section is determined so as to minimize the functional *J*

194 T. Takeda et al.

*J* ¼ ð*T T*0Þ*W*-1ð*T T*0Þ*t* þ ½*Re Rc*ð*T*Þ]½*Ve* þ *Vm*]-1½*Re Rc*ð*T*Þ]*t* ð17:41Þ

where *W* is the cross-section covariance data, and *Ve* and *Vm* are the variance of measured data and calculation method, respectively. In Eq. (17.41), we replace *Re* and *Rc*(*T* ) by

*Re* ! *Re Re*0*R*eb

*Rc*ð*T*Þ ! *Rc*ð*T*Þ *Rc*0*R*cb ð 17:42Þ

because the true values are unknown, they are approximated by *Re* and *Rc*. The adjusted cross section is given by

h 0 i

*T* ¼ *T*0 þ WG*t*½GWG*t* þ *Ve* þ *Vm*] *Re*ð1 *R*ebÞ *Rc*ð*T*0Þð1 *R*cbÞ ð17:43Þ

The covariance of the adjusted cross section is expressed by

*W*0 ¼ *W* WG½GWG*t* þ *V* þ *Vm* ]GW ð17:44Þ

This expression is the same as in [13], but we have to use the adjusted cross section shown in Eq. (17.43).

Using the adjusted cross section, the MA transmutation rate can be estimated by

*R*ð2Þ ð2Þ ð2Þ

*c* ð*T*Þ ¼ *Rc* ð*T*0Þ þ *G* ð*T T*0Þ ð17:45Þ

where the superscript (2) indicates the MA transmutation rate of the target reactor.

The variance, the uncertainty, of the MA transmutation rate is given by

*V*h*R*ð2Þ i ð2Þ 0 ð2Þ*t* ð2Þ ð12Þ ð12Þ*t t*

*c* ð*T*Þ ¼ *G* W G þ *Vm* NV*m Vm N* ð17:46Þ

where *V*ð2Þ is the covariance of the calculation method used for the MA transmutation rate in the target reactor core, *V*ð12Þ is the correlation between the calculation method errors for the critical assemblies and the target core, and *N* is defined by

*N* ¼ *G*ð2ÞWGð1Þ*t*n*G*ð1ÞWGð1Þ*t* þ *V*ð1Þ ð1Þo-1

*e* þ *Vm* ð17:47Þ

We will estimate the MA transmutation amount and the uncertainties by using these methods.

17 Method Development for Calculating Minor Actinide Transmutation in a Fast.. . 195

# Conclusion

To realize the harmonization of MA transmutation and sodium void reactivity, the MA transmutation fast reactor core concept, with an internal blanket between the MA-loaded core fuel region and the sodium plenum above the core fuel, was proposed. The feature of this core concept is that sodium void reactivity can be greatly reduced without spoiling core performance for normal operation.

To accurately evaluate neutronics parameters in a MA transmutation fast reactor, we improved the calculation methods for estimating MA transmutation rates and safety-related parameters such as sodium void reactivity. For the MA transmutation rate, we introduced a definition of MA transmutation for individual MA nuclides and a method for calculating the MA transmutation rates. To evaluate the prediction accuracy of neutronics parameters, we proposed a new method that can eliminate systematic errors of measurements and calculations, and introduced a method to reduce the prediction uncertainty based on the cross-section adjustment method or the bias factor method. Furthermore, we improved the sensitivity, which is necessary to evaluate the uncertainty, by considering the effect of self-shielding.

Acknowledgments A part of the present study is the result of “Study on minor actinide transmutation using Monju data” entrusted to University of Fukui by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT).