Corresponding author: Andrey A. Andrianov (

Academic editor: Georgy Tikhomirov

The paper describes in brief the functional capabilities of a computer code for optimizing the neutronics model parameters (neutron data, technological parameters, and their covariance matrices) based on results of reactor physics experiments using conditional nonlinear multi-parameter optimization algorithms. The code’s application scope includes adjustment of neutron constants, technological parameters and their covariance matrices based on integral measurement results, formulation of requiremen117198ts with respect to the neutron data uncertainties for achieving the target accuracies in calculation of the reactor functionals, and estimation of the reactor performance prediction accuracy, as well as the informativity and similarity metrics of reactor physics experiments with respect to each other and in relation to the target reactor system. The paper also considers some examples of using the code to refine the neutronics models of nuclear reactor and fuel cycle systems based on results of reactor physics experiments.

Designs of innovative nuclear reactor and related fuel cycle components are optimized primarily for the purpose of improving the economic performance of systems and their competitive edge with regard for nuclear and radiation safety requirements. A realistic way to reduce the design tolerances and margins that define in the long run the economic efficiency and competitiveness of plants while ensuring, with no constraints, nuclear and radiation safety requirements is to minimize the variety of uncertainties in reactor characteristics associated with nuclear data and technological parameters which are mandatorily estimated in project design. The only way to reduce the uncertainties in prediction of the reactor physics performance is to take into account experimental data via using dedicated algorithms for transfer of these data into calculations known as

However, as a rule, there is a problem of insufficient experimental data from measurements of neutronics performance for systems structurally similar to target designs (due to high cost of respective reactor physics experiments), this affecting the limitation of capabilities for justifying, in a comprehensive manner, the efficiency and safety of facilities under design. It will be timely in this connection to consider and transfer all currently available experimental information, obtained as part of alternative experimental programs, to the target facility which makes it possible to improve the accuracy of predicting the performance of innovative nuclear reactors and related fuel cycle systems while eliminating the need for additional expensive experiments. Improving the procedural and instrumental framework used to assimilate reactor physics experimental data for increasing the accuracy of predicting the reactor performance in conditions of insufficient experimental data is a timely scientific and practical task solving which allows reducing the design tolerances and margins and improving so the economic efficiency and competitive edge of plants under design.

The practically important example described above does not limit the application scope for the procedure to assimilate reactor physics experimental data in nuclear engineering: the procedure is also used to develop benchmark models for reactor physics experiments (a benchmark experiment is an experiment of a reference class with the minimal estimated uncertainties), plan new informative measurements, adjust neutron constants based on integral measurement results, update stationary and time-dependent neutronics models, estimate the uncertainties in reactor characteristics due to neutron data and technological parameters uncertainties, and determine the target accuracies of neutron data and the prediction accuracies for the neutronics performance of nuclear reactor and fuel cycle facilities.

There are multiple possible ways to implement the procedures for assimilating reactor physics experimental data, each with advantages and drawbacks of its own, which can be divided into two groups (see Table

Approaches to neutronics data assimilation

Methods | ||
---|---|---|

Classical approaches (*) | Modern approaches (**) | |

Unconstrained optimization (ANL, CEA, JAEA, JSI, ORNL, IPPE, NIKIET) (***) | Stochastic methods (NRG) (****) | Constrained nonlinear multi-parameter optimization |

Implementation peculiarities, key assumptions | ||

Ease of implementation (reduction to solution of a system of linear algebraic equations) | No need for linearizing the neutronics model | Possibility for taking directly into account the requirements to updated data |

Assumption of normally distributed initial parameters | Applicability for any types of initial parameters distributions | Possibility for ‘contradictory’ experiments to be used in the analysis |

Linearization of the neutronics model via calculation of sensitivity coefficients | Possibility for contradictory experiments to be used in the analysis | Possibility for taking into account nonlinear effects |

Possibility for solving an ‘inverse problem’ to determine required accuracies of initial data | ||

Limitations of the approach, requirements for computational algorithms | ||

No possibility for taking into account requirements to data to be adjusted | Higher requirements to the quality of initial covariance matrices | Higher requirements to the quality of initial covariance matrices |

Requirement for excluding ‘contradictory’ experiments from the analysis | Requirement for using time-consuming algorithms for stochastic optimization and dispersion reduction methods | Requirement for using current algorithms for constrained nonlinear multi-parameter optimization |

No built-in mechanisms for diagnosis of nonphysical solutions | ||

Requirement for undertaking an additional statistical analysis of initial and adjusted data |

(*)

classical approaches dating back to the 1970s – unconstrained optimization (problem solving is reduced to a system of linear algebraic equations);

modern approaches – transition from unconstrained to constrained optimization problem statement (consideration of physical constraints and contradictory experimental data, possibility for implementing alternative adjustment strategies, availability of built-in mechanisms for checking the quality and how ‘physical’ are the solutions, etc.).

Despite the diversity of approaches and software tools for assimilation of reactor physics experimental data, work has been continuously under way to develop and improve these further for the purpose of providing, on their basis, the possibility for reducing uncertainties in the design performance of innovative reactor plants, improve the accuracies of predicting the nuclear safety parameters, identify the most required areas for the neutron data updating, etc.

One of such developed tools is a computer code for optimizing the parameters of neutronics models with regard for reactor physics experiment results – ONIX (

Functionally, the ONIX code allows calculating optimal corrections for the initial neutronics model parameters (neutron data, technological parameters, and their covariance matrices), which minimize the calculation and experimental discrepancies (the objective function – a chi square – is discussed below), with regard for the user-defined requirements to data (e.g., bounds of model parameters adjustments) and the calculation accuracies of the reactor functionals (a set of constraints), determining the required data uncertainties for ensuring the target accuracy of the reactor functional calculation, estimating the bias in the calculated performance of target reactor system and their respective uncertainties when using adjusted initial data, calculating the set of informativeness indicators and similarity of reactor physics experiments to each other and with respect to the target system, and estimating the neutron data and technological uncertainty in the reactor performance.

Specific features of the ONIX code: direct implementation of algorithms for constrained nonlinear multi-parameter optimization (trust-region (

The optimization problem for reactor physics experimental data assimilation solved using the ONIX code can be formulated in general as follows. Let

where _{0}_{i}_{i}_{k}_{k}_{k}_{x}_{E}_{ii}_{i}_{ij}_{i}_{j}

One of the major difficulties encountered in the practical application of the procedure to assimilate reactor physics experimental data consists in the need for analyzing diversified measurements which are often found to be mutually contradictory. Considering such experimental data individually leads to opposite trends in the biases of

The ONIX code implements as well traditional approaches to the neutron constants adjustment with regard for integral experiment results based on unconstrained optimization using the maximum likelihood and the generalized least square methods (

The ONIX code allows estimating different metrics of the informativeness indicators and the mutual similarity of reactor physics experiments and with respect to the target reactor system, as well as of the solution quality diagnosis methods (statistical tests, similarity coefficients, Cook’s distance, calculation of the adjustment potential and motive force indicators, verification of biases obtained for being physical, etc.) (

Since the ONIX supports algorithms for constrained nonlinear multi-parameter optimization, this makes it possible to solve the problem of determining the required accuracies of neutron data which provide for the target accuracies of calculating the neutronics characteristics of reactor and fuel cycle systems. The standard mathematical formulation for the given problem (_{tr}_{R}_{i}_{tr}_{eff}, the recent requirements (_{i}_{i}_{i}

_{R}^{T}_{σ}

where _{i}_{i}^{th} nuclear constant, and _{σ} is the covariance matrix of nuclear data. The optimization problem for defining the requirements to the accuracies of neutron constants can be formulated as follows:

where _{i}^{1/2}} is the vector of nuclear constants uncertainties (standard deviation), _{i,0}_{D} is the diagonal matrix of sensitivity coefficients (_{D}_{ii}_{i}_{D}_{ij}_{σ} is the correlation matrix of neutron constants.

The ONIX calculations of the tests from the OECD Nuclear Energy Agency (

The ONIX computation module has been developed using the Python programming language. To make it easier for users to operate the ONIX code, the module for the initial data preparation and post-processing of calculations results, has been developed with the use of the MS Excel Visual Basic for Applications (

The first example of the ONIX application is problem solving for defining requirements to the accuracy of neutron data that support the target accuracy of prediction for criticality of a lead-cooled fast reactor. The essence of the problem is that in conditions of high requirements to the accuracy of predicting the performance of innovative reactor plants (e.g., the prediction accuracy for _{eff} needs to be δ_{eff} ≤ 0.3 – 0.2% with the initial nuclear data uncertainty being δ_{eff} ≥ 1%), one shall formulate requirements to the accuracy of neutron data that will make it possible to achieve the target accuracy of calculation for parameters of the facility under design. The results of such analysis are used at the OECD

Fig. _{eff} calculation for a lead-cooled fast reactor model at a level of not more than 0.2%. The figure also presents the results of similar estimates from OECD

Relative uncertainties of important neutron reactions defining the _{eff} uncertainty for a lead-cooled fast-reactor model due to nuclear data uncertainties (solid lines – uncertainties achieved, dashed and dotted lines – uncertainties required).

The results obtained demonstrate that achieving the target accuracy of _{eff} at a level of 0.2% without taking into account the integral experiments requires the uncertainties of neutron cross-sections to be reduced substantially (by 7 to 10 times), which is not always feasible through a multitude of differential measurements, primarily because the current experimental techniques lack the required precision.

The second examples of the ONIX application is prediction of performance for neutron multiplying systems based on information on criticality measurements undertaken in critical assembly experiments. As the example for demonstration, the OECD

The results of the ONIX application for solving the problem at hand using different evaluated neutron data sets are presented in Fig.

Calculation and experimental discrepancies in _{eff}: prior to (a) and after (b) the neutron data adjustment. Letter designations for BFS assemblies: А – BFS-35-1; B – BFS-38-2; C – BFS-42; D – BFS-97-1; E – BFS-97-2; F – BFS-97-3; G – BFS-97-4; H – BFS-99-1; I – BFS-99-2; J – BFS-101-1; K – BFS-101-1; L – BFS-101-3.

A priory and a posteriori _{eff} uncertainties for the target system (for the case with a 30% content of plutonium in MOX fuel and a plutonium vector of 96, 4, 0, 0% for ^{239,240,241,242}Pu respectively): crosses – mean value, line inside rectangular boxes – median, rectangular box boundaries – 25 and 75% percentile, whiskers – minimum and maximum values.

The third example of the ONIX code application demonstrates joint adjustment of neutron constants and technological parameters as applied to nuclide kinetics modeling problems. Cases may occur in developing benchmark models of post-irradiation experiments when the discrepancies between calculated and experimental values cannot be eliminated by adjusting only neutron data within the limits of their measurement uncertainties. In such cases, calculation and experimental discrepancies may be caused by other factors (e.g., by an inadequate calculation model, high uncertainties in technological parameters, etc.), so minimizing these discrepancies exclusively by adjusting neutron data may lead to nonphysical results. In cases when the reliability of experimental data is undisputable, it is possible to adjust the calculation model of experiments by adjusting as well the technological parameters in the limits of their respective uncertainties.

Technological parameters (sizes of structural elements, nuclear concentrations, temperatures of materials, etc.) define the conditions for a particular measurement. Apart from the above parameters, essential for nuclide kinetics problems are fuel irradiation modes (irradiation and cooling times, thermal power, etc.). As a rule, specific to this class of problems is major effect of the technological parameter uncertainties (primarily for the initial composition, Table

Uncertainties in one-group cross-sections and initial composition of irradiated sample

Nuclide | Uncertainties in one-group cross-sections (BROND 3.1), % | Uncertainties in initial composition, % | |
---|---|---|---|

(n, fis) | (n, γ) | ||

^{234}U |
1.8 | 25.1 | 33 |

^{235}U |
0.6 | 5.5 | 3.3 |

^{236}U |
0.6 | 3.2 | - |

^{238}U |
0.7 | 4.5 | 0.04 |

^{238}Pu |
2.2 | 26.5 | 50 |

^{239}Pu |
0.7 | 4.3 | 0.2 |

^{240}Pu |
1.4 | 5.1 | 4.5 |

^{241}Pu |
3.2 | 6.6 | 33 |

^{242}Pu |
2.5 | 7.6 | 100 |

^{241}Am |
1.0 | 7.4 | 25 |

^{242m}Am |
2.8 | 17.4 | - |

^{243}Am |
3.0 | 3.7 | - |

^{242}Cm |
12.4 | 19.2 | - |

^{243}Cm |
9.6 | 17.7 | - |

^{244}Cm |
4.2 | 17.8 | - |

The irradiated fuel composition calculation uncertainties due to technological parameters (T) and neutron data (N) uncertainties

Nuclide | Irradiated fuel composition uncertainties, % | |||
---|---|---|---|---|

A priory | A posteriori | |||

T | N | T | N | |

^{234}U |
30 | 2.2 | 2.8 | 1.8 |

^{235}U |
3.0 | 0.5 | 1.0 | 0.4 |

^{236}U |
2.9 | 5.2 | 0.9 | 3.7 |

^{238}U |
0.04 | 0.2 | 0.04 | 0.04 |

^{238}Pu |
14 | 3.1 | 2.7 | 2.0 |

^{239}Pu |
0.13 | 1.2 | 0.13 | 0.2 |

^{240}Pu |
1.8 | 2.7 | 1.5 | 1.5 |

^{241}Pu |
3.9 | 4.8 | 2.9 | 2.5 |

^{242}Pu |
29 | 4.8 | 3.7 | 2.9 |

^{241}Am |
13 | 2.6 | 1.5 | 1.3 |

^{242m}Am |
23 | 6.4 | 0.5 | 1.1 |

^{243}Am |
47 | 7.6 | 5 | 3.7 |

^{242}Cm |
22 | 6.2 | 0.5 | 0.9 |

^{243}Cm |
24 | 20.0 | 0.6 | 2.7 |

^{244}Cm |
57 | 7.9 | 6 | 4.3 |

In this case, combined adjustment of all input data (both technological parameters and neutron data) leads to these being updated (the isotopic composition is updated within the limits of the declared uncertainties), both uncertainty components reduced, and the calculated and experimental values converging.

Table _{0}| ≤ _{0}^{1/2}.

The application scope of the ONIX code is to adjust neutron constants and technological parameters for neutronics models based on the results of reactor physics experiments. The code can be used to update stationary and time-dependent neutronics models of nuclear reactors and fuel cycle facilities, benchmark models of reactor physics experiments, and estimate reactor characteristic uncertainties associated with nuclear data and technological parameters. It can also be used to determine the accuracy of predicting the neutronics performance of reactors and related fuel cycles. Additionally, the code allows for determining the required neutron data uncertainties needed to achieve the target accuracy of neutronics performance prediction. The implemented method for constrained multi-parameter nonlinear optimization significantly expands the application scope of classical approaches to neutronics data assimilation, such as maximum likelihood and generalized least squares methods. This implementation is reduced to the latter when defining the problem for searching for the unconstrained minimum. The proposed procedure allows for solving inverse problems on specifying requirements for neutron data accuracies and determining the optimal set of additional differential and reactor physics experiments necessary to achieve the target accuracy in predicting the neutronics performance of reactor and fuel cycle systems under design.

The research was supported by a grant from the Russian Science Foundation (project number: 23-29-10154).

Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2023, n. 2, pp. 148–161.