determination of the computational bias in criticality ... · determination of the computational...

322.1

Determination of the Computational Bias in Criticality Safety Validation of VVER-440/V213

Branislav Vrban1

B&J NUCLEAR ltd.

Alžbetin Dvor 145

900 42 Miloslavov, Slovakia

[email protected]

Jakub Lűley1, Štefan Čerba1, Filip Osuský2 2Institute of Nuclear and Physical Engineering

Slovak University of Technology in Bratislava, Ilkovičova 3

812 19, Bratislava, Slovakia

[email protected], [email protected], [email protected]

ABSTRACT

The key issue in any criticality safety problem is to estimate and to predict the deviation

of calculation from reality. If the calculated value is not equal to its true value bias occurs. In

criticality calculations the computational bias is the difference between the computed and the

actual value of keff. The fundamental assumption is that the computational bias is mostly

caused by errors in the cross-section data. In addition the use of random variables in the

calculation introduces a non-random bias in the computed result as well. The American

National Standards are utilized to predict and bound the computational bias of criticality

calculations. These standards require the validation of the analytical methods and data used in

nuclear criticality safety calculations to quantify the computational bias and its uncertainty.

This paper presents a method for determining the computation bias and bias uncertainty for

VVER-440/V213 reactor. For this analysis a SCALE KENO 3D core model was developed

by B&J NUCLEAR ltd. company. This model is based on technical data and operational

history of NPP Jaslovské Bohunice provided by the Slovenské elektrárne a.s. Several

calculation steps are used to address bias estimation method including sensitivity analysis,

uncertainty analyses and cross section adjustment method. In addition the neutronic similarity

of VVER-440/V213 core to several hundred critical benchmark experiments is evaluated by

the use of three integral indices. The database of the benchmark experiment is based on the

selection and processing procedure VALID provided by the Oak Ridge National Laboratory

and specified in the IHECSBE. The results of all analyses performed are given and discussed

in the paper.

1 INTRODUCTION

The computational bias of criticality safety calculations, defined as difference between

the computed and the actual values of keff, must be established through the validation of the

applied methods to critical experiments. To meet the needs, one may utilize American

National Standard ANSI/ANS-8.1-2014 [1] which allows the use of calculations in the

determination of subcritical limits for the design of fissionable nuclear systems. The

aforementioned validation procedure should be performed with experimental data sufficiently

similar to the system under consideration. The area of applicability of the experiments chosen

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Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, September 5-8, 2016

for validation can be determined via comparison of relevant parameters between two systems

by the use of similarity assessment method. This method is based on sensitivity and

uncertainty analysis, including global integral indices ck, E and G. A computational bias can

be found as a function of these parameters and statistical analysis with extrapolation to target

system can be used to determine an upper subcritical limit (USL). Alternatively the

computational biases may be predicted by the nuclear data adjustment tool TSURFER, which

is based on a GLLS approach [2]. The approaches of bias estimation mentioned above are

applied to SCALE [3] criticality calculations of VVER-440/213 reactor and real operational

conditions. VVER reactors are a special design of Pressurized Water Reactors with a

hexagonal geometry of fuel assemblies, Zirconium-Niobium alloy as fuel rod claddings

material and with the steam generators with a tube sheet in the form of two cylindrical heads.

VVER reactors are the most frequently built reactor types in the world. The first units with

predecessors of VVER-440 type reactors were erected at the Novovoronesh NPP site in 1972

and 1973 [1]. The second step in the development of VVER-440 type reactors was the V-230

design which was mainly constructed in the period from 1973 to 1982. The third step in

VVER-440 development was the V-213 reactor design referred as the second generation of

the standard VVER-440 reactors equipped by many upgrades and enhancements. Slovakia has

four nuclear reactors generating half of its electricity and two more under construction. In

1972, construction of the Bohunice V1 plant commenced, with two VVER-440 V-230

reactors. The first was grid connected in 1978, the second two years later. The V2 units

commenced operation in 1984 and 1985. The Slovak NPP Mochovce with VVER-440/V213

units 1 and 2 were put in operation in the summer of 1998 and the end of year 1999 due to the

construction delay caused by political changes in the early 1990s. As a result of continuous

upgrades implemented in NPP Bohunice V2 the reactors power level was increased from

original thermal power level of 1375 MWth to current 1471.25 MWth in 2010. In order to

extend the fuel residence time in active core accompanied by the improvement of fuel cycle

efficiency the new second generation fuel with an average enrichment of 4.87 wt % 235U and

Gadolinium burnable absorbers (3.35 wt % Gd2O3) was incorporated in fuel loading patterns

of NPP Bohunice and Mochovce unit in the last years. Two another units 3&4 of VVER-

440/213 are currently under construction in Mochovce locality and are planned to be put in

operation in 2017 and 2018 respectively.

2 VVER-440 GEOMETRY MODEL DESRIPTION

Detailed and precise KENO 3D model of the VVER-440/V213 reactor has been

developed for criticality and shielding calculations. The whole-core 3D model shown in Fig.

1-a consists of the reactor in-vessel components such as fuel assemblies (including fuel rods,

upper spacer grid, intermediate spacer grids, supporting grid, mixing grid, central tube and

fuel endings), emergency reactor control assemblies (ERC - absorber and fuel part), core

basket, barrel and the reactor pressure vessel. The boundaries of the created VVER-440

whole-core model are given by the outer surface of the dry shielding, the level of hot-leg

piping and the basement of filtration mechanism.

All the VVER-440 fuel assemblies (FAs) are hexagonal and the fuel rods are placed in

the assembly in a triangular grid pattern. The fuel rod bundle of the assembly is enclosed in a

hexagonal wrapper with the width across the flats equal to 145 mm (the 2nd gen. FA). The FA

and emergency reactor control assemblies are positioned in a hexagonal grid with a spacing of

147 mm. The fuel rods are located in the bundle in a triangular grid pattern with a pitch of

12.3 mm. The fuel rod claddings are made of the E110 zirconium alloy, while the wrapper

tubes of FA and ERC are made of the E125 zirconium alloy.

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a) 3D model geometry b) 30th fuel loading pattern of NPP EBO unit 4

Figure 1: KENO 3D model of the VVER-440/V213 and fuel loading pattern

The outside diameter of fuel rod cladding is 9.1 mm, the inside diameter is 7.75 mm.

The cladding houses a fuel column assembled of UO2 pellets. The rods are filled with helium

and the fuel pellet density is 10.55 g/cm3. Several types of profiled fuel assemblies are used to

maintain power peaking factors under the design limits. A Gd2O3 absorber is integrated with a

mass content of 3.35% into FAs to aid fuel profiling. The fuel loading pattern in the

representative sixth of NPP Bohunice unit 4 during cycle 30 is shown in the Fig. 1-b. The

profiling diagrams with various initial enrichments can be found elsewhere [4]. The FA with

the average enrichment of 4.25 wt % 235U exists in two modifications differing in rod

claddings outer diameters. The fuel rod cladding outer diameter of the first modification is

equal to 9.07 mm and the second modification diameter is identical to other FAs. In addition,

another type of fuel rod bundle with the uniform enrichment of 1.6 wt % 235U is used as a fuel

part of central ERC.

3 CALCULATION METHODOLOGY

3.1 Burnup calculation

An accurate treatment of neutron transport and depletion in modern fuel assemblies

characterized by heterogeneous, complex designs, such as the VVER assembly configuration,

requires the use of advanced computational tools capable of simulating multi-dimensional

geometries. The depletion module TRITON [5], which is part of the SCALE system, was

used to perform depletion simulations for 2D FA and ERC models. The isotopic compositions

of the FAs and ERCs following burnup values shown in Fig. 1-b were calculated by SCALE

6.1.3 system - TRITON depletion sequence and NEWT flux solver. Our modelling approach

is based on parametric study of burnup modelling issues associated with VVER-440 fuel and

on best modelling approach (BMA) extensively described in [4]. In this work the effects of

variations in the depletion parameters, operation history, assembly type, Gadolinium

presence, used energy group structure and time steps are investigated and graphically

illustrated. For the sake of brevity, just the most important options used in burnup

calculations are listed here.

The developed models for the criticality calculations are 2D assembly models with

reflective boundary conditions on all sides which represent infinite radial arrays of

infinite length fuel assemblies.

An unstructured coarse-mesh finite-difference acceleration approach (CMFD) is used

with “partial-current” acceleration scheme. The 25 fine-mesh cells are used in the

322.4


global NEWT grid per coarse-mesh cell. The 40 x 40 global unit coarse-mesh and 4 x

4 individual mesh for each unit is used in the all prepared models.

All models were calculated with the standard SCALE V7-238 multigroup neutron

library based on ENDF/B-VII.0 evaluated data [6].

Fuel pins with burnable absorbers were depleted by constant flux option instead of

constant power approach.

To allow more realistic depletion in outer edge of fuel pin driven by the high thermal

flux in the moderator, the Gadolinium bearing pin is divided into the five regions so

that they are equal-area rings.

The average specific power of each model is derived from the average reactor power

of NPP Bohunice unit 4 during cycle 30 and equals 33.05421 kWth/kgHM. The

average concentration of boron acid (H3BO3) is obtained using the same approach and

reaches cb=2.56 g/kg. The fuel is modelled with temperature of 933 K and the

temperature of structural materials and water coolant is 555 K.

Very fine depletion steps (<0.5 MWd/kgHM) are used before Gadolinium peak

reactivity to tract the fast poison concertation changes. After peak reactivity longer

steps are used but are kept smaller than 1 MWd/kgHM.

The isotopic vectors calculated for the each FA and ERC and associated burnup value

are used in the following criticality calculation.

3.2 Criticality calculation

The core material composition was based on the representative sixth described in model

definition. Following burnup calculation of 59 isotopic vectors of fuel material were defined

and homogeneously distributed in the individual FA’s and ERC’s definitions. All of these

FAs and ERCs were placed to the model of core based on the real loading pattern shown in

the Fig. 1-b. The used temperature specification was the same as in burnup calculation; 933

K for fuel, 555 K for incore structural materials and coolant, 541 K respectively 571 K for

coolant and structural materials depending on the inlet or outlet coolant site of model. The

boric acid concentration (0.09 g/kg) and the position of sixth group of ERCs (232 cm)

correspond to the values of operational day for which the fuel depletion was determined.

Criticality calculation was carried out with four cross section (XS) libraries; Continuous

Energy, 27 group and 238 group ENDF/B-VII, and 44 group ENDF/B-V. In case of

multigroup libraries the self-shielding calculation was necessary to perform, which means that

59 cells calculation using BONAMI/CENTRM/PMC codes were invoked within each

multigoup criticality calculation. For 44 and 238 group libraries a default parameters were

retained during cell calculation but for 27 group library the re-evaluation by CENTRM code

was extended to area of U238 resonances due to course energy structure and library

optimization for shielding transport applications [7]. To achieve acceptable statistical

uncertainty of investigated parameters in peripheral regions of the core from 80 to 900 mill.

neutron histories were calculated. The 44 group ENDF/B-V XS library is used for other

analyses in paper if not stated otherwise.

3.3 Sensitivity analysis

The sensitivity and uncertainty analysis of VVER-440/213 core was performed by the

TSUNAMI-3D code using the SCALE 44-group ENDF/B-V library [3] recommended for

LWR and mixed-oxide lattices. Forward and adjoint transport calculations were carried out

with KENO6 and the sensitivity coefficients were computed by the SAMS module. For the

neutron flux calculations square mesh was placed through the core with a uniform step of 2

322.5


cm in the inner core region. In outer part, the size of the mesh varies from 3 to 4 cm in order

to achieve the best statistical uncertainty. In case of reactor internals and direct core vicinity

the size of mesh was directly proportional to the distance from the core centre in a range from

6 to 50 cm. Based on Standard Perturbation Theory implemented in the TSUNAMI the

sensitivity coefficients can be written in a simple form as follows:

where 𝑆𝑘,𝜎 is the sensitivity coefficient of 𝑘𝑒𝑓𝑓 with respect to 𝜎, which represents nuclear

data like cross sections, fission spectrum or nubar. Symbols 𝐿 and 𝑃 in Eq. (1) are net loss and

production Boltzman operators; 𝛷∗ and 𝛷 are adjoint and forward fluxes respectively. All

information necessary to determine the sensitivity coefficients by Eq. (1) can completely

characterize the investigated system, therefore the sensitivity coefficients serve as the basis

for evaluation of the keff uncertainty induced by cross section data, for the similarity

assessment and for the application of cross section adjustment methods. For validation

purposes the sensitivity coefficients for VVER-440/213 core were also calculated using

SCALE 238 group ENDF/B-VII library. Afterwards the 238 group data were collapsed to 44

group structure for better visual comparison and evaluation of energy profiles.

3.4 Uncertainty and similarity analysis

TSUNAMI-IP utility uses sensitivity data generated by TSUNAMI-3D sequence and

cross section-covariance data stored in the 44GRPCOV library for estimation of the response

uncertainty. The SCALE covariance library includes evaluated covariances obtained from

ENDF/B-VII, ENDF/B-VI [8], and JENDL3.3 [9] for more than 50 materials. ORNL has a

database of pre-calculated sensitivity profiles for several hundred critical benchmark

experiments specified in the International Handbook of Evaluated Criticality Safety

Benchmark Experiments (IHECSBE) [10]. These sensitivities may be input to TSUNAMI-IP

utility, along with calculated sensitivity profile of application system. In our approach 543

benchmark experiments and cases with various energy group structures were used. Three

global integral indices [3] are used in the analysis to assess the similarity of VVER 440/V213

neutronic core design (hereinafter application – index a) and a single experiment (e) on a

system-wide basis for all nuclides and reactions. Each integral index is normalized such that a

value of 1.0 represents complete similarity between application core design and specific

benchmark experiment and the value of 0.0 indicates no similarity. The uncertainty of the

integral response ∆R (for instance keff) on the target integral parameter by the use of XS

sensitivity coefficients denoted by symbol S and XS covariance matrix M is evaluated by the

well-known sandwich formula:

where the impact of the individual reactions and energy groups can be evaluated separately.

The diagonal elements of the resulting matrix, defined as the solution of Eq. (2), represent the

relative variance values for each of the system under consideration. The off-diagonal elements

are the relative covariances between given experiments. These covariances transformed to

correlation coefficients (ck) describe the level of similarity in the predicted response biases

between various systems in the frame of XS induced uncertainties. The E parameter given by

Eq. (3) assesses similarity between two systems based on the magnitude and shape of all

sensitivity profiles.

𝑆𝑘,𝜎 =𝜎

𝑘

∆𝑘

∆𝜎≅

∆𝜎

𝜎

⟨Φ∗(1

𝑘

𝜕𝑃

𝜕𝜎−

𝜕𝐿

𝜕𝜎)Φ⟩

1

𝑘⟨Φ∗𝑃Φ⟩

, (1)

T

RRMSSR

2

, (2)

322.6


The G index assesses the similarity of two systems based on normalized differences in

the energy dependent sensitivity data for fission, capture and scatter. A physical interpretation

of the G index is the ratio of the sum of the sensitivity coefficients of the application that are

covered by the experiment to the sum of the sensitivity coefficients of a given application.

The G index is defined as follows:

where the symbol n stands for the number of application system nuclides, x represents the

reaction and j the summation which is performed over all energy groups. The nuclide-

reaction specific partial integral index based on the same coverage criteria as G is denoted g.

Undercoverage or noncoverage can be penalized with a penalty defined as

where �̃� is a vector of all groupwise sensitivity differences 𝑍�̃� for all nuclides defined as 𝑍�̃� =

𝑆𝑗𝑎 − 𝑆𝑗

𝑒 if |𝑆𝑗𝑎| > 𝑆𝑗

𝑒 otherwise 𝑍�̃� = 0.

3.5 Use of cross section adjustment method

As stated in SCALE manual [3], the TSURFER code uses the generalized linear least-

squares method (GLLS) to consolidate a prior set of measured integral responses (such as keff)

and corresponding calculated values obtained using the SCALE nuclear analysis code system.

The initial estimates for the computed and measured responses are improved by adjusting the

experimental values and the nuclear data used in the transport calculations - taking into

account their correlated uncertainties—so that the most self-consistent set of data is obtained.

By the application of cross section adjustment method, TSURFER provides an estimate for

the computational bias and application uncertainty.

3.6 Determination of Upper Subcritical Limit (USL)

Based on the criteria for subricitality listed in [1], a USL may be determined based on

the analysis of a number of critical systems. Basically two methods of USL estimation are

available. The first method called Confidence Band with Administrative Margin applies a

statistical calculation of the bias and its uncertainty plus and administrative margin to a linear

fit of critical experiment benchmark data. Here calculational bias 𝛽 is defined is given as 𝑘𝑐-1,

where 𝑘𝑐 is the mean value of keff resulting from the calculation of benchmark criticality

experiments. The USL Method 1 is defined as

where Δ𝑘𝑚 is administrative limit, W is the confidence band which accounts for

uncertainties in the experiments, the calculational approach and XS data and ∆𝛽 stands for the

bias uncertainty. Usually adjustments are applied to prevent taking credit for a positive bias

by assuming 𝑘𝑐(𝑥) = 1 everywhere that 𝑘𝑐(𝑥) > 1. In the second method referred as a lower

tolerance band approach, statistical techniques are applied in order to determine a combined

lower confidence band plus subcritical margin. The USL Method 2 is defined as

ea

T

eaSSSSE . (3)

n x j

na

jx

n x j

ne

jx

na

jxSSSG

,

,

',

,

,

,1 , (4)

𝑝𝑔 = √�̃�𝑀�̃�𝑇, (5)

USL1(x) = 1 − Δ𝑘𝑚 − 𝑊 + ∆𝛽, (6)

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where 𝑠𝑃 is the pooled standard deviation for the set of criticality calculations and term

𝐶 𝛼 𝑃⁄ . 𝑠𝑃 provides a band for which there is a probability P with a confidence 𝛼 that an

additional calculation of keff for critical system will lie within the band. The recommended

purpose of Method 2 is to apply it in parallel with Method 1 to verify that the chosen

administrative margin is conservative relative to a purely statistical basic. More details of

USL calculation methods can be found in [12]. A computer program USLSTATS available in

SCALE package is used to evaluate USLs based on Methods 1 and 2.

4 RESULTS

4.1 Criticality calculation

First verification of the prepared material and geometrical model was aimed to

comparison of integral parameter keff. Calculated results based on different energy structure

and evaluated data were subjected to comparison with real operational criticality of NPP

Bohunice V2. All calculated values are presented in Tab. 1. Lowest computational bias was

achieved by calculation with CE library. Best result using multigroup approach was

calculated with 27 group library which can be explained by applied re-evaluation during cell

calculation compared to finer group structures. Positive finding, from the point of

computational time, was also almost identical keff calculated with 44 and 238 group library.

Each multigroup calculation is able to determine local parameters like fission source

distribution, whole core neutron flux and quantities connected to reaction rates like spatial

power distribution.

Table 1: Comparison of keff for different energy structures. Energy

structure CE 27 group 44 group 238 group

keff 1.00465±0.00006 1.00681±0.00003 1.00769±0.00006 1.00751±0.00007

4.2 Sensitivity, uncertainty and similarity analysis

As it was mentioned above, the TSUNAMI sequence computes the contributors to the

application response uncertainty due to the XS covariance data.

Table 2: Uncertainty contribution in keff of VVER-440/213 core

No.

Covariance Matrix

Contributions

to Uncertainty

in keff (% Δk/k) No.

Covariance Matrix

Contributions

to Uncertainty

in keff (% Δk/k)

Nuclide-

Reaction

Nuclide-

Reaction

Due to the

Matrix

Nuclide-

Reaction

Nuclide-

Reaction

Due to the

Matrix

1 239Pu nubar 239Pu nubar 3.99E-01 9 238U nubar 238U nubar 9.39E-02

2 238U n,gamma 238U n,gamma 3.26E-01 10 235U n,gamma 235U n,gamma 8.23E-02

3 239Pu fission 239Pu fission 1.43E-01 11 240Pu n,gamma 240Pu n,gamma 6.99E-02

4 239Pu fission 239Pu n,gamma 1.32E-01 12 239Pu chi 239Pu chi 6.81E-02

5 238U n,n' 238U n,n' 1.31E-01 13 235U fission 235U fission 5.39E-02

6 239Pu n,gamma 239Pu n,gamma 1.26E-01 14 235U fission 235U n,gamma 5.37E-02

7 235U nubar 235U nubar 1.22E-01 15 235U chi 235U chi 4.66E-02

8 135Xe n,gamma 135Xe n,gamma 1.04E-01 16 143Nd n,gamma 143Nd n,gamma 4.26E-02

USL2(x) = 1 − (𝐶 𝛼 𝑃⁄ . 𝑠𝑃) + 𝛽(𝑥), (7)

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The relative standard deviation of VVER-440/213 keff due to XS covariance data is

0.64%. Tab. 2 lists the top 16 covariance matrices that contribute to the keff uncertainty. These

contributors represent more than 98.15% of the total uncertainty induced by XS data. The top

contributor to keff uncertainty is the 239Pu nubar reaction, see Fig. 2. This is due to high values

of burnup reached in the core and, as can be seen in Fig. 1-a, also due to the very high

sensitivities below 1 eV threshold. In case of 239Pu fission large positive sensitivities exist

bellow 1 eV and 100 keV thresholds. The uncertainty associated to the XS data in the energy

range bellow 100 keV is quite large and reaches almost 2.5%. The contribution of 239Pu n,

gamma reaction to the total keff uncertainty is mainly driven by large XS uncertainty above 1

keV. Despite the low sensitivities in this energy range, this contribution is the sixth largest as

it is shown in Tab. 1.

a) b)

Figure 2: Sensitivity and covariance profiles to 239Pu nubar; fission and n,γ reactions

The next presented profiles belong to 135Xe and 143Nd fission products. Here the most

important contributor to keff uncertainty is reaction n, gamma, which is in 135Xe case important

also from the reactivity management point of view. We assume that bellow 1 eV their impact

is driven by quite high sensitivities and above this threshold, by the very large uncertainties as

shown in Fig.3–b.

a) b)

Figure 3: Sensitivity and covariance profiles to 135Xe and 143Nd n,γ reactions

The similarity assessment procedure identified three groups of potential experiments,

where the values of the ck coefficients got over 0.6. However, as it can be seen in Fig. 4, only

cases of experiment MIX-COMP-THERM-002 reached ck values greater than 0.7.

322.9


Figure 4: Graphical results of similarity assessment procedure

The E coefficients for these cases reach unsatisfactory values above 0.6, just one case

got score above 0.73. Nevertheless the big portion (around 40%) of VVER-440 sensitivities is

uncovered (G) by above mentioned cases. More details with description of the first 20

experiments with the highest ck values can be found in Tab. 4. The values of g indices for

nuclide – reaction pairs, having a great impact on the neutron balance of the active core, are

given in Tab. 3. The presented data highlight those nuclide – reaction pairs which are not

sufficiently covered by the MIX-COMP-THERM-002 experiment.

Table 3: Results of nuclide-reaction specific partial integral index g

ID 1H

scatter

1H

total

10B

total

16O

capture

235U

fission

238U

n,n'

238U

scatter

238U

capture

238U

total

239Pu

fission

239Pu

nubar

219 0.87 0.88 1.00 0.66 0.28 0.04 0.16 0.34 0.62 0.99 1.00

As it is stated in [11], Eq. 5 does not provide a proper penalty for dissimilarities

between two sensitivity profiles. However, our results are based on the approach which is

already implemented in SCALE system. As it is recommended, just experiments that exhibit a

certain degree of similarity to the application are used in the penalty assessment calculation.

Data in Tab. 4 show that just 4 cases reach ck values higher than 0.7, therefore our penalty

assessment based on the small amount of experiments cannot be considered as decisive.

According to calculation the standard deviation in the keff of VVER-440/213 due to uncovered

sensitivity coefficients reach 0.26%. Results prove that the most important contributor to

additional penalty, bringing almost 55% of the total value, is 238U n,n’ reaction. The rest

portion mainly consists from contribution of fission products (135Xe, 143Nd and 103Rh).

4.3 Use of cross section adjustment method

Based on the similarity assessment results, 20 experiments listed in Tab. 4 were selected

for inclusion in the adjustment procedure performed by the TSURFER code. In our analyses,

all within-series experiments were assumed to have a correlation of 0.9, and following the

data published in current version of DICE database [13], all experiments not in the same

series were assumed to be uncorrelated. The achieved goodness of fit represented by 𝜒2

parameter reached acceptable value of 1.562E-01. According to calculation the new adjusted

value of keff equals to 1.0075 with XS induced uncertainty of 0.45% what approximately gives

the relative bias of -2.26%. We can conclude that GLLS calculation identified the small

tendency of our calculation approach to overestimate the keff where the 238U n,gamma and 238U

n,n’ nuclide reaction pairs were found to be the most important contributors to the calculated

bias value. As it can be seen in Fig. 5-a), the case of capture reaction adjustments were

applied almost in all reactions above 1 eV. We conclude that these XS adjustments mainly

322.10


influence the effective resonance integral of the system. The adjustments of inelastic reaction

were applied mainly above 1 MeV, but almost constant relative change was applied almost in

all other energies.

Table 4: Uncertainty contribution in keff of VVER-440/213 core

ID IHECSBE ID

Cas

e N

o.

Fis

sile

mat

eria

l

Av

erag

e F

issi

on

Gro

up

En

erg

y

Per

cen

t o

f fi

ssio

n

bel

ow

0.6

25 e

V

Per

cen

t o

f ca

ptu

re

bel

ow

0.6

25 e

V

Neu

tron

gas

tem

per

atu

re

clad

din

g

ck E G

219 MIX-COMP-THERM-002 2 MOX 1.230 eV 0.788 0.489 562 K Zircaloy 0.75 0.74 0.62

















237 MIX-SOL-THERM-002 3 UO2(NO3)2 0.056 eV 0.983 0.92 342 K - 0.63 0.35 0.27

236 MIX-SOL-THERM-002 2 N3O9Pu 0.054 eV 0.985 0.94 340 K - 0.62 0.34 0.25

235 MIX-SOL-THERM-002 1 UO2(NO3)2 0.054 eV 0.985 0.939 340 K - 0.62 0.34 0.26

*In every case the light water is used as the moderator and reflector material.

4.4 Determination of Upper Subcritical Limit

Due to having the small amount of experiments with acceptable level of similarity, the

USLSTATS calculation was performed with all experiments previously used in similarity

procedure. The keff values for these experiments were successfully retrieved from DICE

database. No option to input unique experimental uncertainty for each experiment currently

exists in SCALE environment, therefore uniform experimental uncertainty of 0.3 was used in

calculation. Previously calculated uncertainty due to cross-section covariance data were

automatically added for the all used experiments. In our calculation the W band was

determined at a 95% confidence level. Neither administrative margin nor penalty was applied.

The parameter P was set to 0.99 and 𝛼 to 0.95. The results of USL calculation are illustrated

in Fig. 5-b). In this figure, the upper blue line represents a linear fit to a set of calculation

based on critical experiments. The second green line represents the lower confidence band for

a single additional calculation. This confidence band accounts for uncertainties in the

experiments, the calculational approach and used nuclear data. The third line (short dashes)

represents the USL1 preventing taking credit for a positive bias above the point where the blue

line exceeds criticality. The USL2 function is shown by the yellow colour. It should be noted

that due the small amount of similar experiments the data used in analysis did not pass the test

for normality.

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a) TSURFER relative cross section adjustments b) USLSTATS results

Figure 5: Relative changes in cross sections and ULS results

The calculated values of USL1 are 0.9768 and 0.9510 respectively, which suggests the

relevancy of 5% administrative margin requested by Slovak Regulatory Authority for

subriticality calculations.

5 CONCLUSION

The detailed model of VVER-440/213 reactor was developed by B&J NUCLEAR ltd.

for criticality and shielding analyses including reactor core, core basket, core barrel, pressure

vessel with all internals in an appropriate level of accuracy. Applied simplifications result in

the low computational bias of keff which did not exceed 0.8% in all computational cases.

Special attention was given to the methodology applied to determination of fuel isotopic

vectors modelled in one sixth symmetry core configuration what is one of the reasons of small

calculation bias. Subsequently the defined model was introduced to sensitivity, uncertainty

and similarity analysis. The relative standard deviation of keff induced by XS data was

determined to 0.64%, which is comparable to bias identified in criticality calculations. The

similarity assessment was not so successful. Only 20 benchmarks from more than 500 were

identified with ck value greater than 0.6 and only 4 with ck value greater than 0.7. Low

correlation is accompanied by not sufficient coverage of sensitivity profiles between selected

benchmarks and target VVER-440/213 core. Significant portion of uncovered sensitivity

profiles belong to fission products. The new adjusted of calculated keff value was found by use

of GLLS method. The relative bias between original and adjusted response reached -2.26%.

This result shows the small overestimation of used calculation methodology, adopted cross

section data and applied simplifications. However, the identified overestimation is in principle

consistent with previous results. It should be noted, that the results of TSURFER analyses rely

on the availability of quality uncertainty and correlation data. The determination of Upper

Subcriticality Limit pointed out the necessity of having the administrative margin about the

level of 5% for subcriticality calculations. Moreover, the results of USL method 1 shown that

if more benchmarks suitable for VVER-440 are available, the administrative margin can be

lowered almost to the half of the original value.

Even though our core belongs to PWR’s family the ORNL pre-calculated sensitivity

profiles and benchmarks in DICE database do not cover the Russian technology in a frame of

similarity assessment and biasing methods. Especially the data for fission products will be

welcomed. Finally more effort is needed to extend the DICE database by benchmarks based

on experiments denoted to VVER reactors.

322.12


ACKNOWLEDGMENTS

The authors would like to thank Martin Gajdoš, head of Nuclear safety department at

Slovenské elektrárne, a.s. for valuable comments and support.

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