Nuclear Science and Engineering ISSN: 0029-5639 (Print) 1943-748X (Online) Journal homepage: http://www.tandfonline.com/loi/unse20 A Data Identification Method for the Improvement of a Cross-Section Model Aldo Dall’Osso To cite this article: Aldo Dall’Osso (2006) A Data Identification Method for the Improvement of a Cross-Section Model, Nuclear Science and Engineering, 154:2, 241-246, DOI: 10.13182/NSE06A2630 To link to this article: http://dx.doi.org/10.13182/NSE06-A2630 Published online: 10 Apr 2017. Submit your article to this journal Article views: 1 View related articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=unse20 Download by: [Australian Catholic University] Date: 25 October 2017, At: 11:44 NUCLEAR SCIENCE AND ENGINEERING: 154, 241–246 ~2006! A Data Identification Method for the Improvement of a Cross-Section Model Aldo Dall’Osso* Downloaded by [Australian Catholic University] at 11:44 25 October 2017 Framatome ANP Tour Areva, 92084 Paris La Défense Cedex, France Received April 21, 2005 Accepted November 3, 2005 Abstract – The accuracy of a neutronics model depends not only on the validity of the equations that are solved but also on the quality of the cross-section model. This last is currently constituted by a set of correlations, the parameterized tables, relating the data of the neutronics problem to the local conditions. The more the correlations represent the local conditions, the more the results will be accurate. For a simulation model, this means that the results will be closer to the measurements. The goal of the data identification method presented is to solve a constrained inverse problem and to obtain the parameters of some further correlations that will enhance the accuracy of the results. The constraint imposed minimizes the error committed in solving the diffusion equation, using as reference the results of a more accurate computer code or the measurements performed for in-core flux maps. Some purely numerical examples and an application in conjunction with in-core measurements illustrate the method. I. INTRODUCTION 2. moderator temperature, which modifies the Maxwellian distribution of the thermal neutrons The accuracy of a neutronics model depends not only on the validity of the equations that are solved but also on the quality of the cross-section model. A crosssection model provides correlations relating the cross sections to the local conditions of the reactor. As the physical conditions vary inside the reactor, the physical properties, the cross sections, are space dependent. Cross sections depend on local conditions in two ways: 3. moderator density, which acts directly as H 2O and B ~soluble boron! number density change and indirectly in influencing the neutron slowing down 4. fuel temperature, which acts mainly in the resonance capture of the heavy nuclides. The influence of these parameters is instantaneous and delayed via the nuclides buildup. Some historical parameter is used to represent this effect. Multiparameterized tables technique enables relating the cross sections to the local physical parameters. They are computed by detailed transport codes that model all the aspects of neutron slowing down and spatial propagation. Nevertheless, despite the good accuracy of these codes, cross-section tables contain some approximations. When used by a code representing the whole reactor, these approximations can cause a deviation of the result with respect to a more accurate reference calculation or to the result of in-core measurements. The reference results can be taken as a target, and some functions of core physical parameters can be taken as the means to reach it. The goal of the method proposed in this paper is 1. directly, via the nuclides number density 2. indirectly, via the neutron flux shape versus space and energy, on which is carried out the homogenization to obtain few-group constants. The main local physical parameters representing the local conditions are the following: 1. nuclide density, which acts directly via microscopic cross sections *E-mail: aldo.dallosso@framatome-anp.com 241 Downloaded by [Australian Catholic University] at 11:44 25 October 2017 242 DALL’OSSO to determine this kind of function. The principle at the basis of the method is to determine correlation parameters that minimize the error on the neutron balance equations in the least-squares sense. This method is an application of inverse problem techniques of which the goal is to identify some input parameters of the model. More precisely, since the operator ~or some of its coefficients! is not computed with the objective of reproducing the measurements but is only adjusted to minimize the error, we shall call it a constrained inverse problem. This kind of problem was already approached in neutronics in contexts rather close to ours,1 to adjust the matrix of the system corresponding to a core calculation or to determine cross sections using measurements by solving the diffusion equations, where flux are the data and the cross sections are the unknown parameters.2 Other works on parameter estimation approach the neutron kinetics problem in order to tune a simplified model on a more detailed one.3 Adjustment techniques based on generalized perturbation theory have also been proposed.4 II. METHOD To describe the principle of our method, we write the neutron balance equation in diffusion theory on its discretized form, in which we express the cross sections, such as they are produced by the cross-section model. The equation in node n for the fast group is (f ~an, n, g Fn,1 ⫺ af, n, g Ff,1 ! ⫹ ~S a, n,1 ⫹ S r, n !Fn,1 ⫺ 1 k eff 2 ( nS f, n, g Fn, g ⫽ 0 , ~1! g⫽1 and for the thermal, ⫺S r, n Fn,1 ⫹ ( ~an, n, g Fn,2 ⫺ af, n, g Ff,2 ! f ⫹ S a, n,2 Fn,2 ⫽ 0 , ~2! discretization used. In our case, they are calculated with the nodal expansion method. The solution of these equations provides the calculated flux distribution. It differs from the estimated flux distribution coming from in-core measurements, which we indicate by FM, n, g and will call pseudomeasured flux in the remainder of the paper. If one introduces FM, n, g in Eqs. ~1! and ~2!, these equations will not be satisfied anymore. Even though it is not possible to satisfy the equations in changing at the same time the cross-section distribution, we can determine, by a least-squares approach, some corrective correlations to the cross sections to minimize the error. II.A. The Least-Squares Formulation In its basic formulation, the least-squares method allows us to determine the function that best fits a set of reference points, when the relationship between the dependent variables and the independent variables is known. In our problem, this dependence is not known in a direct way, and one cannot thus directly minimize the difference between calculated flux and pseudomeasured flux. We regard it as an error to minimize the one that is made on the application of the neutron balance operator. Indeed, one can consider the difference between the two sides of Eqs. ~1! and ~2! like an estimator of the error made by solving the neutron balance equation. This difference represents the excess or the defect of neutrons produced in the unit of time and the unit of volume. This deviation can be linked with inaccuracies in the calculation of the cross sections used in the calculation. These inaccuracies could result in particular from the approximations in the relationships between the cross sections and the core state parameters, relationships defined in the multiparameterized tables. They can also be related to the lack of modeling of some physical phenomena. Let us indicate by dS the correction that one should make to the cross sections to reduce these deviations and introduce it, as well as the pseudomeasured flux, in Eqs. ~1! and ~2! after having made the following definition: f where S a, n, g ⫽ absorption cross section ~g ⫽ 1,2! S r, g ⫽ removal from fast to thermal cross section nS f, n, g ⫽ fission neutron production cross section af, n, g ⫽ coupling coefficients between the nodes resulting from the discretization of the divergence of the neutron current. Sums appearing in Eqs. ~1! and ~2! are applied to the nodes f that have a common face with node n. In general, the coupling coefficients af, n, g depend on the method of 冉 L n, g ⫽ ( an, n, g ⫺ af, n, g Ff, g Fn, g 冊 . ~3! We thus obtain the expression of the error «n, g on the application of the neutron balance operator: «n,1 ⫽ L n,1 FM, n,1 ⫹ ~S a, n,1 ⫹ dS a, n,1 ⫹ S r, n ⫹ dS r, n !FM, n,1 ⫺ 1 k eff 2 ( ~nS f, n, g ⫹ dnS f, n, g !FM, n, g , ~4! g⫽1 «n,2 ⫽ ⫺~S r, n ⫹ dS r, n !FM, n,1 ⫹ L n,2 FM, n,2 ⫹ ~S a, n,2 ⫹ dS a, n,2 !FM, n,2 . NUCLEAR SCIENCE AND ENGINEERING VOL. 154 ~5! OCT. 2006 IMPROVEMENT OF CROSS-SECTION MODEL Although it is not possible to determine the dS’s for each one of the nodes in the core with the information provided by a reference solution of the diffusion equation, it is possible to determine correlations giving the corrections dS for the whole core. The relationships that express the dependence of the corrections to the cross sections with respect to the physical parameters of the core are expressed by the following order 2 polynomials ~but the choice could be extended to other kinds of functions!: 3 2 j a a, g, i, j X i, n ( i⫽1 j⫽1 dS a, n, g ⫽ a a, g,0,0 ⫹ ( Downloaded by [Australian Catholic University] at 11:44 25 October 2017 3 dS r, n ⫽ a r,0,0,0 ⫹ ( , and 3 2 j a f,2, i, j X i, n ( i⫽1 j⫽1 , ~6! where X i, n indicates the values of the physical parameters ~burnup, water density, fuel temperature, etc.! in node n and a k, n, g represents the required parameters of the correlations. We expressly avoided introducing a correction for nS f,1 . Indeed, Eq. ~1! contains two terms multiplying F1 , and one cannot find independent equations for the parameters of the correlations relating to S a,1 and nS f,1 . Although S r is also a term multiplying F1 , it appears at the same time in the second equation and this removes any ambiguity. In order to not leave nS f,1 uncorrected, we introduce a relationship defining its correction. We use the following relationship of proportionality between the cross sections and the corrections: dnS f, n,1 ⫽ dnS f, n,2 nS f, n,1 nS f, n,2 . ~7! To simplify the method, and in the current state of the tests carried out, no correction is applied to the diffusion coefficients. A correction of the diffusion coefficients would affect the coupling coefficients appearing in Eq. ~3!. If we substitute Eq. ~6! in Eqs. ~4! and ~5!, we raise square and sum over all nodes n, we obtain an expression of the mean-square error as a function of the parameters a k, n, g appearing in Eq. ~6!: N «sq, g ⫽ «n,2 g ( n⫽1 . ~8! From Eq. ~8!, two systems of equations for the fast group and for the thermal are obtained while imposing that the derivative of esq, g with respect to each one of the paramNUCLEAR SCIENCE AND ENGINEERING ]«sq, g ]a k, g ', i, j VOL. 154 ⫽0 , ~9! where subscript k takes the values a ~absorption!, r ~removal from group 1 to 2!, and f ~fission!. The solution of this system provides the parameters of the correlations that will approach the model to the measure. II.B. Adjustment of the Cross-Section Model by Residual Correlations i⫽1 j⫽1 dnS f, n,2 ⫽ a f, g,0,0 ⫹ ( eters a k, n, g appearing in Eq. ~6! be null. This is expressed by the relationship , 2 ( a r,0, i, j Xi,j n 243 OCT. 2006 In the choice of the correlations expressed by Eq. ~6!, we can follow the same formulation established in the multiparameterized tables ~order 2 polynomials!, a function of the physical parameters representing the core under the current operating conditions ~burnup, water density, fuel temperature, etc.!, or add new relationships relating the cross sections to new parameters. In the first case, we have corrective correlations, while in the second case, we have improving model correlations. When using our method to determine corrective functions, physical parameter X i, n indicates for i ⫽ 1 the burnup, for i ⫽ 2 the water density, for i ⫽ 3 the square root of the fuel temperature: X 1, n ⫽ Bu n , X 2, n ⫽ rH2O, n , and X 3, n ⫽ M Tfuel, n . ~10! When using our method to determine model-improving functions, X i, n indicates physical parameters not used in the cross-section tables. In our analysis of the method, we used the spectral index variation with respect to infinite medium conditions: X 1, n ⫽ Fn,1 Fn,2 ⫺ S a, n,2 S r, n . ~11! This physical parameter takes into account dependence of the cross sections on the environment. If we have three independent physical parameters, the system relative to the fast group consists of 14 equations with 14 unknown parameters of the correlations. The system relative to the thermal group also consists of 14 equations with 14 unknown parameters of the correlations. We initially solve the thermal system whose unknown parameters of the correlations are a r,0, i, j and a a,2, i, j .The parameters a r,0, i, j thus calculated are then introduced into the other system, which is solved in its turn to determine the parameters a a,1, i, j and a f,1, i, j . 244 DALL’OSSO Downloaded by [Australian Catholic University] at 11:44 25 October 2017 III. METHODOLOGICAL ASPECTS The method of adjustment presented in this paper is based on the assumption of existence of a relationship between the sought data and taken measurements. Good results could be obtained only if the relationship indeed exists. This method is to be used only when the deviations observed on the flux distribution are allotted to approximations in the formalism of the multiparameterized tables. It asks many precautions in the implementation, because, like any best-fit method by least-squares, it cannot be applied in a black box way. This remark is even more important in our method because one does not directly control the difference between the calculated flux and the reference. Indeed, control is carried out indirectly while acting on the error made in the application of the neutron balance operator. We will therefore discuss some aspects of this method. Concerning the robustness, as for any application of the least-squares method, for the algorithm to work correctly, the set of points must be organized in a regular way. It is necessary thus that the range of values of water density and fuel temperature be rather broad and that these variables not be correlated to each other. This requires that the calculation be made in a condition of full power or that several sets of measurements be used at the same time. Concerning the burnup, it is necessary that the core be sufficiently depleted. In a contrary case, the correlation on this physical parameter must be eliminated. Therefore, in this situation, the number of equations to be solved is reduced. The correlations described in the previous sections depend on the composition of the nodes. The sums on the nodes have thus to be extended to the nodes of the same composition. The parameters of the correlations are, in general, a function of the composition. In Sec. II.B, we showed the possibility of deducing 28 parameters of the correlations. If this is true from a mathematical point of view, difficulties can arise from a numerical point of view. Indeed, the greater the number of parameters, the more the matrix of the systems approaches singularity. This is a known behavior in least-squares problems. The reason is that very often data to be fitted do not clearly distinguish between two or more of the fit functions provided. The more functions that are provided, the greater are the chances that two functions fit the data equally well ~see Ref. 5 for a discussion on this subject!. In our method, this aspect is enhanced by the fact that the fit functions do not enter directly in the least-squares process, but they are filtered by the diffusion equations in which they are inserted, and regularization techniques such as singular value decomposition do not significantly improve the method. To prevent the matrix of the systems from being too close to singularity, it is necessary to use the smallest number of parameters in the correlations ~which reduces the number of fit functions!. This recommendation is reinforced by the fact that the number of points of measure in the core is much lower than the number of calculation points. A big part of the pseudomeasured flux distribution is an extension of measurement ~see Sec. IV! and thus does not bring information in the best-fit process. The calculated parameters satisfy a total minimization of the error in the core, and some local deviations can continue to remain after the correction of the cross sections. It is obvious that reliable parameters of the correlations could be provided only if the error on the flux really depends on approximations on the cross-section model. This can be evaluated by analyzing the diagonal of the inverse of the matrix of the systems appearing in Eq. ~9!. Indeed, according to the theory of the least-squares method, each element of the diagonal represents the variance associated with the corresponding parameter. The error analysis by applying Eqs. ~4! and ~5! also allows evaluating the relevance of the correlations. Other techniques can be used to check the quality of the parameters of the correlations. An immediate test is the reexecution of the flux calculation with application of the parameters of the correlations, which must approach the reference. Note that even though the system composed by Eqs. ~1! and ~2! is nonlinear, because the cross sections depend on flux via the thermal feedback, the system composed by Eq. ~9! is in general linear. Therefore, the calculation of the parameters of the correlations is made in a direct way. We could obtain a nonlinear system only if the parameters of the correlations appear as an argument of nonlinear functions. A question about the use of the flux as a physical parameter on which the identification process is based could arise. Since the parameterized cross sections are obtained from transport calculations by forcing the twogroup diffusion equation to preserve infinite-medium assembly reaction rates, the two-group diffusion fluxes are not directly comparable with measured fluxes and it would be more consistent to base the identification on reaction rates. In our method, the consistence is granted by the fact that pseudomeasured fluxes are inferred from measured reaction rates ~detector readings!, as described in Sec. IV, using cross sections that have the property to preserve infinite-medium detector reaction rates. Inserting the pseudomeasured fluxes into the node-by-node diffusion equation enables preservation of the neutron balance, a kind of global reaction rate, instead of preserving the detector reaction rate. Preserving the detector reaction rate could not provide easy-to-use mathematical conditions allowing correcting cross sections appearing in the diffusion equation. IV. ELEMENTS OF IMPLEMENTATION The data input of the equations presented in Sec. II is the pseudomeasured flux distribution deduced from NUCLEAR SCIENCE AND ENGINEERING VOL. 154 OCT. 2006 IMPROVEMENT OF CROSS-SECTION MODEL theoretical flux by an interpretation of in-core measurements. Calculation of the parameters of the correlations for the correction of the cross sections is as follows: Downloaded by [Australian Catholic University] at 11:44 25 October 2017 1. calculation of the distribution of pseudomeasured power by extension of the deviations between calculation and measure 2. conversion of the distribution of pseudomeasured power into pseudomeasured flux distribution 3. solution of the equations described in Sec. II 4. calculation of flux with thermal feedback by taking into account the residual correlations. Step 1 is carried out by computing the deviation between predicted and measured reaction rates in the detector position, converting it into relative power deviation and extending it to the whole core by polynomial interpolation. The extension is performed by determining the coefficients of an order 2 polynomial of the coordinates x and y of each horizontal plane of nodes, by least-squares fit of the deviations in the detector positions, and interpolating it in all the nodes of the core. Step 2 is based on the assumption that the spectrum of numerical neutron flux Fn, g is correct and is thus preserved in the distribution of pseudomeasured flux FM, n, g . By indicating by Pn the pseudomeasured power in node n, the pseudomeasured flux FM, n, g is thus given by solving the following system of equations: and FM, n,2 ⫽ Fn,1 Fn,2 . ~12! Step 4 is performed to verify the quality of the parameters of the correlations. The procedure to obtain the pseudomeasured flux ~steps 1 and 2! is the same used by Framatome-ANP to reset the three-dimensional on-line code for in-core measurements.6 V. RESULTS The method has been applied in two ways: 1. purely numerical experiments to verify the capability of the method to identify the data 2. application of the method to self-powered neutron detector ~SPND! measurements from a 1300-MW reactor. We have tested the capability of the method to identify inaccuracies in the cross-section tables only by numerical experiments. Applications of the method using incore measurements did not identify any significant NUCLEAR SCIENCE AND ENGINEERING VOL. 154 discrepancy in cross-section tables. This shows that the dependence of the cross sections on the classical physical parameters such as burnup, water density, fuel temperature, etc., is correctly treated by the tables. Nevertheless, the application of the method to real measurements has revealed some physical effect that was not taken into account by the cross-section tables ~see later text!. The method applied to the determination of corrective functions of the cross-section model was tested in various situations, among them a very simple onedimensional subcritical core model and a threedimensional core model of the 1300-MW type. The so-called pseudomeasured flux distribution is the result of a reference calculation. The tests were carried out according to the following procedure: 1. execution of a calculation with the initial distribution of cross sections simulating a distribution affected by uncertainty 2. choice of a correlation and a set of values to be applied to the parameters 3. calculation of the distribution of cross sections with these correlations and determination of the corresponding flux distribution ~reference distribution!, thus simulating a pseudomeasured flux distribution 4. determination of the parameters of the correlations by solving Eq. ~9!; these parameters must be very close to those imposed in step 2 kS f, n,1 FM, n,1 ⫹ kS f, n,2 FM, n,2 ⫽ Pn FM, n,1 245 OCT. 2006 5. calculation ~for checking! of flux with the distribution of cross sections calculated with the parameters of the correlations determined in step 4; this flux distribution must be very close to those calculated with step 3. These calculations ~numerical experiments! showed a good behavior of the identification method. The parameters of the correlations obtained by solving system ~9! were equal to the expected values, and the flux distribution was identical to the reference distribution. The method applied to the determination of improving functions of the cross-section model was tested on a three-dimensional core model of the 1300-MW type at 1260 MWd0tonne from the beginning of its 13th fuel cycle. The pseudomeasured flux distribution comes from in-core measurements. The procedure described in Sec. IV was applied to obtain the pseudomeasured flux distribution from SPND. The core is at full power with regulation group ~group R! inserted at 9%. A model improving correlation was sought on the basis of the following eight parameters of the correlations: a a,1,1,1 , a a,1,1,2 , a a,2,1,1 , a a,2,1,2 , a r,0,1,1 , a r,0,1,2 , a f,1,1,1 , a f,1,1,2 , 246 DALL’OSSO TABLE I Core Parameters Relative to the Reference, Initial, and Adjusted Power Distribution for a 1300-MW Three-Dimensional Core Model Core Parameter Power axial offset ~%! Power radial offset ~%! Core power pic ~Fq ! Axial power pic ~Fz ! Enthalpy rise factor ~FDH ! Maximum radial power pic ~Fxy, max ! Reference a Initial b Adjusted c ⫺1.77 12.6 1.56 1.15 1.33 ⫺1.76 9.4 1.60 1.16 1.33 ⫺1.75 13.2 1.55 1.15 1.32 1.54 1.49 1.50 a From in-core measurements. Predicted without application of residual correlations. c Predicted with application of residual correlations. Downloaded by [Australian Catholic University] at 11:44 25 October 2017 b model. But the role of the method can be limited to determining which new correlation fits the observed behavior. The parameters of the new correlations can be determined by adding new calculations in the building of cross-section tables. Following this approach, Framatome ANP has recently improved the physical model in its core simulator code: New correlations based on the spectral index have been added in the multiparameterized tables. If introduced in a three-dimensional on-line coremonitoring system, this method can be used as a tool to gain experience from measurements and refine the model on-line. Framatome ANP has successfully tested its own core-monitoring system 7,8 and plans to use this method to improve its cross-section model, namely, fission product cross sections as a function of burnup. REFERENCES corresponding to second-order correlations in the spectral index variation according to Eq. ~11! for all the cross sections, with order 0 terms set to 0. The correlation on dnS f,2 was dropped ~dnS f,2 was considered 0! and replaced by a correlation on dnS f,1 . A set of parameters for each fuel type was calculated. The results are presented in Table I, which shows that with the application of the residual correlations on the cross-section distribution, the power distribution comes closer to the reference. The mean-square error changed from 2.0 to 1.7%, and the maximum error passed from 4.4 to 2.6%. The fact that the error is not reduced to zero means that only a part of the error can be attributed to the lack of modeling of the environmental spectrum variation. VI. CONCLUSION We have presented a method of determining correlations relating the cross sections to some local core physical parameters ~burnup, water density, fuel temperature, and spectral index! in order to approach a neutronics model to a reference. This reference can result from measurements or calculations. Numerical benchmarks have shown that the method has good identification properties and can positively identify a correlation when this correlation exists. When used to approach a core model to measurements, this method can be considered as a way to measure some physical effect that is not seen by our modeling. In this context it can be used to explore new correlations that could improve the cross-section model. The parameters of the correlations can be used as they are calculated, as an experimental adjustment of the cross-section 1. Y. RONEN, D. G. CACUCI, and J. J. WAGSCHAL, “Determination and Application of Generalized Bias Operators Using an Inverse Perturbation Approach,” Nucl. Sci. Eng., 77, 426 ~1981!. 2. R. SANCHEZ and N. J. McCORMICK, “Inverse Problem Calculations for Multigroup Diffusion Theory,” Nucl. Sci. Eng., 83, 63 ~1983!. 3. R. ACCORSI, M. MARSEGUERRA, E. PADOVANI, and E. ZIO, “Neural Estimation of First-Order Sensitivity Coefficients: Application to the Control of a Simulated Pressurized Water Reactor,” Nucl. Sci. Eng., 132, 326 ~1999!. 4. A. GANDINI, “Uncertainty Analysis and Experimental Data Transposition Methods Based on Perturbation Theory,” Handbook of Uncertainty Analysis, Y. RONEN Ed., CRC Press, Boca Raton, Florida ~1988!. 5. W. H. PRESS, S. A. TEUKOLSKY, W. T. VETTERLING, and B. P. FLANNERY, Numerical Recipes in Fortran 77—The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge ~1992!. 6. A. DALL’OSSO, “A Method to Reset a 3D On Line Core Model on Incore Measurements,” Proc. Int. Conf. New Frontiers of Nuclear Technology: Reactor Physics, Safety and HighPerformance Computing (PHYSOR 2002), Seoul, Korea, October 7–10, 2002, 1C-04 ~2002! ~CD-ROM!. 7. J. L. MOURLEVAT, “Advantages of a Fixed In-Core Based System for an On-Line Margins Monitoring,” Book of Abstracts of ICONE 9, Nice, France, April 8–12, 2001 ~2001!. 8. A. DALL’OSSO, G. MARTINEZ, and G. RIO, “Core Monitoring and Prediction with a 3D On Line Model,” Proc. Int. Topl. Mtg. Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications (M&C 2005), Avignon, France, September 12–15, 2005 ~2005! ~CD-ROM!. NUCLEAR SCIENCE AND ENGINEERING VOL. 154 OCT. 2006

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