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uncertainty calculations using the 3D neutron transport
code CRONOS2 at an industrial scale
J. Gaillet, T. Bonaccorsi, G. Noguere, G. Truchet
To cite this version:
J. Gaillet, T. Bonaccorsi, G. Noguere, G. Truchet. Development and validation of sensitivity and un-certainty calculations using the 3D neutron transport code CRONOS2 at an industrial scale. PHYSOR 2018: Reactors Physics paving the way towards more efficient systems, Apr 2018, Cancun, Mexico. �hal-02416222�
Development and validation of sensitivity and uncertainty calculations using
the 3D neutron transport code CRONOS2 at an industrial scale
J. Gaillet
1, T. Bonaccorsi
1, G. Noguère² and G. Truchet
11
Commissariat à l’énergie atomique et aux énergies alternatives
CEA/DEN/DER/SPRC/LPN
13108 Saint-Paul-Lez-Durance, France
2
Commissariat à l’énergie atomique et aux énergies alternatives
CEA/DEN/DER/SPRC/LPN
13108 Saint-Paul-Lez-Durance, France
Julien.gaillet@cea.fr, Thomas.bonaccorsi@cea.fr, Gilles.noguere@cea.fr,
Guillaume.truchet@cea.fr
ABSTRACT
Evaluating uncertainties on reactivity is a major issue for conception of nuclear reactors. These uncertainties mainly come from the lack of knowledge on nuclear and technological data. The common method for the propagation of the nuclear data uncertainties consists in using the Standard Perturbation Theory to calculate reactivity sensitivity to the desire nuclear data. In such a method, sensitivities are combined with a priori nuclear data covariance matrices, such as the COMAC set developed by CEA. In this study, COMAC nuclear data uncertainties have been propagated on the BEAVRS benchmark using a classical two-step APOLLO2/CRONOS2 scheme. A procedure implementing Standard Perturbation Theory has been created using the 3D neutron transport code CRONOS2 in order to allow a full-core uncertainty propagation with 4 groups industrial scheme. CRONOS2 sensitivities are compared to sensitivities calculated by the IFP method in Monte Carlo codes. For the purpose of the tests, dedicated covariance matrices have been created by condensation from 49 to 4 groups of the COMAC matrix. In conclusion, CRONOS2 sensitivities are agreed with the sensitivities calculated by the IFP method, which validates the calculation procedure. In addition, reactivity uncertainty calculated by this method is close to values found for this type of reactor. Finally, the calculation method allows computing sensitivities quickly with a good accurate.
KEYWORDS: sensitivity, covariance, nuclear data
1. INTRODUCTION
The studies of conception and safety as well as the exploitation of the reactor require simulation tools which have to be adaptative, reliable and able to predict fission chain reactions. They use nuclear and technological data. Full core determinist calculation is usually made in two steps. The first step consists on resolve Boltzmann’s equation at an assembly scale in two dimensions with a high number of energy groups and a fine spatial mesh. This calculation allows storing condensed self-shielded and homogenized cross sections as a function of different parameters such as burnup. Then, these cross sections are used in a 3D “core code” resolving the diffusion equation with a low number of energy groups. The calculation scheme used in this paper is based on the cell code APOLLO2 [1] and the “core code” CRONOS2 [2]. VVUQ methodology [3] enables to verify that determinist codes reproduce correctly physical models, resolve equations without error and that biases and uncertainties are controlled to guarantee their
prediction. The calculated values contain biases and uncertainties. The quantification of these uncertainties (due to nuclear and technological data) is done by assimilating integral experiments or propagating uncertainties linked to input data.
A way to propagate uncertainties on reactivity in determinist schemes consists on calculating sensitivities to nuclear data using Standard Perturbation Theory [4][5][6]. Then, sensitivities are associated to covariance matrix of nuclear data to obtain uncertainties. Another way to propagate uncertainties consists on sampling data input by random selection in their statistic distribution [7]. More recently, developments in Monte Carlo codes allowing to calculate adjoint flux and sensitivities by Iteration Fission Probability (IFP) method has been realized [8][9].
Although Monte Carlo methods are directly applicable on 3D modeled core, they can be quite long and expensive or have convergence problems. In addition, Monte Carlo codes cannot do depletion calculation. The goal of this paper is to suggest a method able to propagate nuclear data uncertainties with short calculation time compatible with industrial constraints on a 3D modeled core. For this, a CRONOS2 procedure calculating reactivity sensitivities to nuclear data by Standard Perturbation Theory has been developed and tested on BEAVRS benchmark. The obtained sensitivities are compared to IFP calculations and are used in uncertainty calculations on reactivity.
2. UNCERTAINTIES PROPAGATION THEORY 2.1. Uncertainty calculation formula
In the framework of uncertainties propagation, the studied real system is modeled by a numeric model g which input data are uncertain variables such as nuclear data. The numerical model then computes variables of interest which can be in our case reactivity.
(1) The first step of uncertainties propagation consists on identify uncertainties sources which will have an influence on calculated values. Then, these uncertainties sources must be quantified to be modeled in the code’s input by distributions. Consequently, the input data dispersion can be propagated through numerical model g to deduce average and variance on Y variable. One of methods able to estimate these quantities is the quadratic cumulative formula. It is based on two basic results of probability:
, (2) With are random variables and are real. represents covariance between
two random variables and . It is defined as :
(3) In this expression, for i, j [1,..,d]², we have :
If g is a linear model, , these equations give average and variance of Y. The quadratic cumulative method consists on suppose that input parameters variations are small around their average. Consequently, g can be linearized by a Taylor development at order 2 around :
(5)
This expression can be used to calculate average and variance of Y at first order: , (6) In addition, if input parameters ( are independent, in other words, if covariance are equal to
zero, these equations become :
,
(7) This last equation represents the quadratic cumulative formula. In the case of not independent input
parameters, variance of Y can write more easily :
(8)
S is the sensitivities vector and M is the covariance matrix :
, (9)
With . In the following of this paper, Y standard deviation will be calculate with relative values. Consequently, this will write:
(10) , (11)
The advantage of this method is that uncertainty calculation becomes a simple matrix product and uncertainty can be breakdown by reaction and isotope. However, it is based on linear approximation of model response and it main drawback is that it requires sensitivities vector knowledge. Sensitivity calculations can be also very expensive, so disposing a calculation method able to calculate them quickly is interesting.
2.2. Perturbation theory
2.2.1. Definition of adjoint flux
Adjoint flux is used to quantify the notion of neutronic importance. In fact, a neutron can have a more or less significant contribution to maintaining fission chain reaction depending of his energy or location. This importance can be quantified scoring the average number of his descendants. Consequently, adjoint flux notion does not make physical sense for sur-criticality system. For a critical reactor, adjoint flux will be the limit of adjoint flux in sub-criticality system when reactivity moves towards zero. To define it from a mathematical point of view, we define the following scalar product as the integral on phases space:
(12) If M is an operator, it adjoint operator M+ is defined by:
(13) In the form of operators, transport equation is at stationary state:
(14)
F is the neutrons production operator, it represents neutrons emitted at energy E by neutrons of energy E’
that have induced fissions. A is the disappearance operator, it represents neutrons at energy E which are lost by absorption or leakage. k is the effective multiplication coefficient.
(15)
(16) In determinist codes, phases space discretization allows to write transport equation in matrix form. So, adjoint operators A+ and F+ are defined by a switch of transfer senses and a switch of odd derivatives. In
fact, the derivative adjoint is its opposite and the real matrix adjoint is its transposed [10]. (17)
(18) By analogy with forward transport equation, adjoint transport equation is defined by :
(19) The term is the adjoint flux which is the solution of adjoint transport equation. The eigenvalue of this equation is equal to direct equation eigenvalue [11].
(20)
2.2.2 Exact perturbation theory
Exact perturbations enables to calculate a reactivity difference due to perturbation of input parameter without doing hypothesis on perturbation magnitude. We define a core in it undisturbed state, subscripted 1, and the same core in it disturbed state (in which a concentration or a cross section is disturbed), subscripted 2. For these two states, transport equations are:
(21) By subtracting them and by grouping terms, then by introducing scalar product, we have :
(22) (23) As we have :
(24) And by definition of adjoint flux, we can write:
(25) (26) Finally, the reactivity difference between disturbed and undisturbed states is obtained by :
(27) The main drawback of this theory is the need for doing two calculations: one for adjoint flux calculation in undisturbed state and the other for flux calculation in disturbed state. It is therefore necessary to do n+1 calculations to obtain the reactivity difference for n perturbations.
2.2.3 Standard Perturbation Theory (SPT)
This theory is a first order approximation of exact theory. A comparison between them is therefore necessary to check approximation relevance. The SPT is used to calculate reactivity sensitivities. It is based on small perturbations hypothesis, that is . Consequently, we can write :
(28) The variation of effective multiplication coefficient in the undisturbed state is given by :
If perturbation is on σ parameter in the energy group g, the expression of sensitivity in the first order approximation in pcm/% is: (30)
Unlike exact theory, this formula takes advantage requiring only two flux calculations (one for direct flux and the other for adjoint flux) to have access to all reactivity sensitivities. The following equations develop terms of sensitivities formula for each reaction. The denominator will be noted .
(31) Capture reaction sensitivity:
(32) Fission reaction sensitivity:
(33) Neutron multiplicity sensitivity:
(34) Spectrum sensitivity: (35)
2.3 Covariance matrix and condensation
Uncertainties calculations also require knowing covariance matrix of nuclear data. In CRONOS2, uncertainties are computed to four energy groups. So, a condensation of 49 groups COMAC matrix [12][13] to 4 groups has been done preserving total variance. Correlations coefficients between group i of reaction 1 and group j of reaction 2 are defined by :
is absolute uncertainty of reaction 1 cross section in group i. The chosen condensation method preserves the total variance for a given isotope. Variances are calculated by quadratic cumulative formula. The correlations matrix of two cross sections reac1 and reac2 for a same isotope is noted . Matrix whose diagonals are relative uncertainties of cross sections reac1 and reac2 are noted and
. Consequently, covariance matrix is:
(37) The reactivity sensitivities vector to cross section reac1 is noted . They has been calculated by APOLLO2 with 49 energy groups. So, for each energy group of condensed matrix, the variance of reac1 rate for an isotope and covariance between reactions reac1 and reac2 are :
(38) In this case, relative uncertainties of cross sections reac1 and reac2 and correlation coefficient in the
condensed matrix are :
, (39)
Figure 1.
Composition of BEAVRS core and number and positions of pyrex pin in cycle 1
3. CALCULATION SCHEME OF BEAVRS
The calculations scheme in this paper is a two step APOLLO2/CRONOS2 scheme applied on BEAVRS benchmark [14]. It is a PWR reactor with three different enrichments (see Fig. 1). Each assembly is modeled in 2D in APOLLO2. Input data are nuclear data from JEFF-3.1.1 library [15] and technological data from BEAVRS benchmark. Self-shielding is realized at 281 energy groups corresponding to the SHEM mesh [16]. It is the optimized mesh for self-shielding which is performed with Pij method. Then, a condensation to 49 energy groups is done to improve time calculation while maintaining a good accuracy. Flux is calculated with the MOC method. Finally, self-shielding cross sections are condensed to 4 energy groups and are stored. Equivalent coefficients transport/diffusion allowing to preserve reactions rates
between the two codes are also calculated. The second step of this calculation scheme is at core scale using CRONOS2. It uses stored cross sections computed for each assembly by APOLLO2 and it solves flux calculation in diffusion theory with 4 energy groups thanks to MINOS solver. Then, the CRONOS2 procedure developed for sensitivities calculation can be called. Firstly, adjoint flux is calculated. Secondly, CRONOS2 structures which contain cross sections of selected isotopes by reaction are created. They allow calculating production and disappearances operators. Consequently, the scalar product with adjoint flux can be calculated to have production term . In addition,
expression of sensitivity can be
implemented and have to be breakdown by isotope, by reaction and by energy group, that is
equations (32), (33), (34) and (35). Then, the split by energy group is made thanks to unitary sources which are equal to 1 in the wanted energy group and 0 in the other groups. Finally, the numerator scalar product of equations (32), (33), (34) and (35) can be calculated and also sensitivity. Results of this procedure are shown in paragraph 5.3.Table I.
Comparison between exact and direct perturbations on concentration in CRONOS2 code
Isotope
Perturbation on concentration
Direct perturbation
(pcm) Exact perturbation (pcm)
Gap exact perturbation – direct perturbation (pcm) U234 1% -1,39 -1,38 0,01 U235 1% 358,13 358,18 0,05 U238 1% -217,46 -214,83 2,63 B10 1% -163,15 -163,14 0,01 Fe56 1% -1,79 -1,71 0,08 Sn112 10% -0,03 -0,02 0,01 Sn115 10% -0,04 -0,03 0,01 Sn116 10% -0,14 -0,13 0,01 Sn117 10% -0,15 -0,14 0,01 Sn118 10% -0,12 -0,11 0,01 Sn119 1% -0,10 -0,09 0,01 Sn120 10% -0,05 -0,05 0,00 Sn124 10% -0,04 -0,03 0,01 Zr90 1% -1,19 -0,93 0,26 Zr91 1% -6,65 -6,61 0,04 Zr92 1% -2,18 -1,94 0,24 Zr94 1% -0,89 -0,69 0,20 Zr96 1% -0,89 -0,87 0,02
4. RESULTS ON SENSITIVITIES CALCULATION
All calculations in this part are done on BEAVRS benchmark for critical configuration with All control Rods Output (ARO) and 975 ppm of bore at Hot Zero Power conditions.
5.1. Verification of integral terms of perturbation theory
Several integral terms sensitivities expression are computed and has to be verified. For it, developments are adapted to exact perturbation. The introduced perturbation is done on the concentration by isotope. Generally, it is 1% perturbation but for the least sensitive isotopes, it can be a perturbation of 10% to see a
better effect on the reactivity. The adjoint flux and reactivity are calculated with this modification whereas direct flux is obtained before perturbation. For a perturbation of p% on concentration, the exact perturbation formula is :
with
(40) So, exact formulation looks like SPT formula. The only difference is the adjoint flux which is calculated in disturbed state for exact formulation. Exact perturbations have to be compared to direct perturbations:
(41) These two methods are compared in Table I. In all cases, the difference is of the order of pcm. So, exact perturbations are agreed with direct perturbations, that is calculation methods used are correct.
Table II. Sensitivities
calculated by CRONOS2 and comparison with exact perturbations
Isotope Pertubation on concentration Exact perturbation (pcm/%) CRONOS2 Concentration Sensibility (pcm/%)
Gap Exact perturbation - Sensibility (%) U234 1% -1,38 -1,38 0,00 U235 1% 358,18 362,82 1,30 U238 1% -214,83 -215,80 0,45 B10 1% -163,14 -153,00 -6,22 Fe56 1% -1,71 -1,71 0,00 Sn112 10% -0,02 -0,02 0,00 Sn115 10% -0,03 -0,03 0,00 Sn116 10% -0,13 -0,13 0,00 Sn117 10% -0,14 -0,14 0,00 Sn118 10% -0,11 -0,11 0,00 Sn119 1% -0,09 -0,09 0,00 Sn120 10% -0,05 -0,04 -20,00 Sn124 10% -0,03 -0,03 0,00 Zr90 1% -0,93 -0,94 1,08 Zr91 1% -6,61 -6,63 0,30 Zr92 1% -1,94 -1,95 0,52 Zr94 1% -0,69 -0,70 1,45 Zr96 1% -0,87 -0,87 0,00
5.2. Impact of first order approximation on sensitivities
The CRONOS2 procedure calculates concentration and nuclear data sensitivities by isotope, energy group and reaction. To obtain this, there is only one CRONOS2 calculation which time is one hour. Results are shown in Table II for concentration sensitivities. They are a first order approximation. So, they must be compared to exact perturbation to estimate the impact of this approximation. The gap between SPT sensitivities and exact perturbation are also shown in Table II. For each isotope, it is nearly 1% except for
Sn120. However, this isotope has a negligible effect on reactivity and this difference can result from numerical approximation. This confirms that first order approximation is correct for nuclear data.
Table III.
CRONOS2 SPT sensitivities compared to IFP sensitivities from RMC code [9]
Isotope Reaction CRONOS2 Sensibility (pcm/%) IFP(RMC) (pcm/%) [9]
Gap IFP(RMC) - CRONOS2 (%) U234 Capture -1,42 -1,50 -5,33 Fission 0,04 0,04 0,00 Neutron mult. 0,07 0,07 0,00 U235 Capture -97,26 -99,60 -2,35 Fission 460,09 441,00 4,33 Neutron mult. 931,92 927,00 0,53 U238 Capture -259,32 -217,00 19,50 Fission 44,15 46,10 -4,23 Neutron mult. 68,64 72,90 -5,84 B10 Capture -163,91 -153,00 7,13 Zr90 Capture -0,94 -1,78 -47,19 Zr91 Capture -6,63 -5,06 31,03 Zr92 Capture -1,95 -1,82 7,14 Zr94 Capture -0,70 -0,64 9,37 Zr96 Capture -0,87 -0,67 29,85 Sn112 Capture -0,02 -0,02 0,00 Sn115 Capture -0,03 -0,06 -50,00 Sn116 Capture -0,13 -0,13 0,00 Sn117 Capture -0,14 -0,12 16,67 Sn118 Capture -0,11 -0,09 22,22 Sn119 Capture -0,09 -0,09 0,00 Sn120 Capture -0,04 -0,04 0,00 Sn124 Capture -0,03 -0,03 0,00 H2O Capture -65,80 -64,50 2,02
Table IV.
CRONOS2 SPT sensitivities compared to IFP sensitivities from TRIPOLI4 code
Reaction C2 SPT (pcm/%) IFP T4 (pcm/%) Gap C2 - IFP T4 (%) U234 Capt+Fiss -1,38 -1,46 +/- 1.97E-03 -5,48U235 Capt+Fiss 362,82 336,00 +/- 0.158 7,98 U238 Capt+Fiss -215,80 -174,00 +/- 6.12E-02 24,02
5.3. Validation of sensitivities calculation
CRONOS2 sensitivities have been compared to IFP reference calculations coming from RMC code [9] (see Table III) for each isotope and each reaction and coming from TRIPOLI4 code (see Table IV). For the two Monte Carlo codes, differences between them and CRONOS2 are small for the most sensitive isotope (U235, U238, H2O and B10) but can be more important for less sensitive isotopes. Their sensitivities are so weak that these differences will not have consequence on uncertainties calculations. Finally, sensitivities calculations by SPT in CRONOS2 code are validated.
5. UNCERTAINTIES CALCULATION ON REACTIVITY
CRONOS2 sensitivities are combined with four energy groups covariance matrix to estimate uncertainty on reactivity. Results by isotope and reaction are given in Table IV. The total uncertainty is 532 pcm. This result is agreed with uncertainty calculated on other similar PWR [17].
Table V.
Uncertainty on reactivity due to nuclear data
Capture Fission Total (pcm)
U235 133,69 152,47 202,78 U238 264,90 357,30 444,79 B10 76,66 0,00 76,66 Zr90 10,30 0,00 10,30 Zr91 41,75 0,00 41,75 Zr92 35,34 0,00 35,34 Zr94 2,72 0,00 2,72 Zr96 6,95 0,00 6,95 H2O 188,00 0,00 188,00 Total (pcm) 363,89 388,47 532,29 6. CONCLUSIONS
This paper proposed a determinist calculation method for reactivity sensitivities due to nuclear data at industrial scale for a 3D modeled core. Unlike Monte Carlo methods which are expensive and time-consuming, the developed CRONOS2 procedure using SPT takes advantage of using a short calculation time. It has been validated comparing them to IFP sensitivities. Differences between these two methods are small, especially for the most sensitive isotopes. These sensitivities have enabled to estimate reactivity uncertainty du to nuclear data for the BEAVRS reactor. This uncertainty is agreed with that calculated by other authors. To resume this work, this study shows the capacity of CRONOS2 code to compute sensitivities in diffusion with only 4 energy groups. For the user, sensitivities calculation comes down to a simple procedure to obtain all reactivity sensitivities with only one calculation. For the qualification, this development enables to have quickly an estimation of reactivity uncertainties. For the future, this development requires to be tested with other calculation scheme to study the impact of self-shielding choice or equivalent coefficients which would preserve reaction rate coupled with adjoint flux.
ACKNOWLEDGMENTS
This template was adapted from the template for M&C + SNA 2007 posted on the Internet. The author wishes to thank G. Prulhière for the development of BEAVRS scheme and C. Magnaud for her help on the implementation of sensitivities calculation in CRONOS2 code.
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![Table III. CRONOS2 SPT sensitivities compared to IFP sensitivities from RMC code [9]](https://thumb-eu.123doks.com/thumbv2/123doknet/12910667.372410/11.918.105.819.197.828/table-iii-cronos-spt-sensitivities-compared-ifp-sensitivities.webp)
