Calculation Of Sensitivity Coefficients Research Articles

A random walk based methodology to calculate the sensitivity coefficients in inverse radiant boundary design problems

Radiative heating furnaces are widely used for heat treatment applications in manufacturing industries. Strict requirements on heat treatment characteristics of the workpieces require precise control of furnace heater temperatures. Numerical techniques for the solution of inverse heat transfer problems can be used for the optimization of the radiant heater temperatures. In gradient-based optimization techniques, the optimization is carried out using the sensitivity coefficients between the heater and workpiece temperatures. In this study, we present a methodology for calculating the sensitivity coefficients for the solution of inverse boundary design problems involving transient heating of solid workpieces in a three-dimensional radiant furnace. In our methodology, the sensitivity coefficients on the selected design points within the workpieces are calculated by analytical differentiation of design point temperature formulations derived utilizing the floating random walk method and radiative heat flux formulation with exchange factors. The methodology for calculating sensitivity coefficients has been verified through applications to both a one-dimensional scenario and the three-dimensional radiant furnace model. The developed methodology was integrated into a gradient-based optimization algorithm in two test cases, to determine the furnace heater temperatures required to achieve the desired temperature histories at design points within the workpieces. First test case focuses reconstruction of an assumed heater temperature and examines the impact of design point configuration and input data noise on convergence behavior and calculated estimation. The findings reveal that increasing the number of design points in the workpieces increases solution accuracy, with transverse positioning of the design points in the workpieces resulting in heater temperatures closest to the assumed values. Second test case focused on estimating required heater temperatures for spatially uniform heating pattern on the workpieces. With 72 transversely positioned design points, we achieved uniform heating at these points with a maximum relative temperature deviation of 0.26%.

Computation of sensitivity coefficients in fixed source simulations with SERPENT2

Within the scope of the implementation of a nuclear data pipeline aiming at producing the best possible evaluated nuclear data files, a major point is the production of relevant sensitivity coefficients when including integral benchmark information. Thanks to recent code modifications in the Monte Carlo code Serpent2, it is now possible to produce these coefficients in fixed source simulations. The manuscript describes the verification of this implementation against the deterministic transport code susd3d. The study is completed by an analysis of the computational cost (running time and memory allocation) associated with such calculations with Serpent2. The relative difference between the sensitivity coefficients produced by Serpent2 and susd3d is of the order of the percent at most, except for the low energy range where the lack of neutrons prevents from reducing the Monte Carlo uncertainties. The computational cost of such calculation is similar to the one of criticality calculation mode, although the OpenMP scalability should be further improved.

Research on the calculation method of sensitivity coefficients of reactor power to material density based on Monte Carlo perturbation theory

The ability to calculate the material density sensitivity coefficients of power with respect to the material density has broad application prospects for accelerating Monte Carlo-Thermal Hydraulics iterations. The second-order material density sensitivity coefficients for the general Monte Carlo score have been derived based on the differential operator sampling method in this paper, and the calculation of the sensitivity coefficients of cell power scores with respect to the material density has been realized in continuous-energy Monte Carlo code RMC. Based on the power-density sensitivity coefficients, the sensitivity coefficients of power scores to some other physical quantities, such as power-boron concentration coefficients and power-temperature coefficients considering only the thermal expansion, were subsequently calculated. The effectiveness of the proposed method is demonstrated in the power-density coefficients problems of the pressurized water reactor (PWR) moderator and the heat pipe reactor (HPR) reflectors. The calculations were carried out using RMC and the ENDF/B-VII.1 neutron nuclear data. It is shown that the calculated sensitivity coefficients can be used to predict the power scores accurately over a wide range of boron concentration of the PWR moderator and a wide range of temperature of HPR reflectors.

Sensitivity analysis of volatile organic compounds to PM2.5 concentrations in a representative industrial city of Korea

This study aims to analyze the sensitivity of volatile organic compounds (VOCs) to ambient concentrations of fine particles (PM2.5) in the representative industrial city of Ulsan, Korea. For the calculation of sensitivity coefficients between VOCs and PM2.5 (SVOCs-PM2.5), PM2.5 data were obtained from an air quality monitoring station, and their corresponding 6-h average concentrations of VOCs (alkanes, alkenes, aromatics, and total VOCs) were measured at the Yeongnam intensive air monitoring station. The air monitoring period was divided into the warm-hot season (May–October 2020) and the cold season (November 2020–January 2021). The sensitivity coefficients in the low pollution period of PM2.5 (5 < PM2.5 ≤ 15 μg/m3) were higher and much higher than those in the medium pollution period (15 < PM2.5 ≤ 35 μg/m3) and high pollution period (35 < PM2.5 ≤ 50 μg/m3), respectively. This result indicates that the change ratios of PM2.5 concentrations to the background (PM2.5 ≤ 5 μg/m3) per unit concentration change of VOCs (particularly alkenes) in the high PM2.5 pollution period were much higher than those in the low pollution period. This also indicates that PM2.5 concentrations above 35 μg/m3 were more easily affected by the unit concentration change of VOCs (particularly alkenes) than those below 15 μg/m3. The average sensitivity coefficients during the cold season increased in a range of 23–125% as compared to those during the warm-hot season, except the alkenes-PM2.5 sensitivity with a decrease of 7%. It means that the impact of VOCs (except alkenes) on PM2.5 concentrations was relatively low in the cold season. However, in the cold season, the alkenes might contribute more to PM2.5 formation, particularly over the high pollution period, having PM2.5 > 35 μg/m3, than other VOC groups. The result of this study can be a basis for establishing PM2.5 management plans in industrial cities with large VOC emissions.

Depletion Perturbation Theory in decay heat calculation context

This article describes an implementation of Depletion Perturbation Theory (DPT) using deterministic neutronics code APOLLO3® alongside depletion code MENDEL. We use DPT to calculate the sensitivity coefficients of decay heat (DH) to various nuclear data such as reaction cross-sections, fissions yields, and reaction energy. DPT allows for the coupling of the Bateman, Boltzmann, and normalization equations, and has the advantage of needing only one calculation per cooling time to compute the sensitivity coefficient to all input parameters, as opposed to usual finite differences schemes which need as many calculation as there are input parameters. The calculation of sensitivity coefficients is done using both the DPT methodology and a finite differences scheme as a mean of numerical verification. Both methods showed good agreement, although we found that DPT should be used with care when studying short cooling times. Sensitivity coefficients were then used to perform deterministic uncertainty propagation, the resulting uncertainty calculation were used to compare the effect of various coupling configurations.

Uncertainty quantification calculations of nuclide number densities of Gd-bearing fuel rods in light water reactors

The Post-Irradiation Examination (PIE) data are quite useful to validate the evaluated nuclear data. Recently, a new experimental program, REGAL (the Rod-Extremity and Gadolinia AnaLysis), has been proposed and launched. During this program, the PIE data are acquired for the Gd-bearing fuel rods during (or after) the initial Gd burnout. These PIE data would be different from those for the normal UOX or MOX fuel rods from a viewpoint of nuclide transmutation process. In the present work, we attempt to quantify this difference through calculating sensitivity coefficients of nuclide number densities of the Gd-bearing rods using the depletion perturbation theory and performing uncertainty quantification calculations with the sensitivity coefficients and the nuclear data covariance data. Through the numerical analyses, we found that the impact of the uncertainties of the nuclear data of Gd isotopes on the number densities of actinoids is negligible. The nuclear data-induced uncertainties of Gd isotopes number densities are relatively large during the Gd depletion, and those become small after the Gd burnout. After the Gd burnout, the relative standard deviations of Gd-155 and -157 number densities are approximately 6% and 13%, respectively, and the nuclear data of the parent nuclides of these Gd isotopes, Gd-154 and -156, are also important as well as those of Gd-155 and -157.

Evaluation of the measurement uncertainty during the tensile tests of high-strength bolts

A developed methodology for evaluating measurement uncertainty during the tensile tests of metals and alloys is presented. In this connection, the various sources of uncertainty are considered. The object of studies included high-strength bolts made of 40Kh “Selekt” steel, which were subjected to tensile tests according to the GOST 1497-85 State Standard using a UMM-100 universal tensile testing machine. The basic sources of uncertainty in the measurement of relative elongation and reduction were shown to include repeated measurements of relative elongation; errors of measuring the finite length by a vernier calliper and marking the initial length of a calculated section, as well as the measurement error of the tensile testing machine. These also include operator reading error, repeated relative reduction measurements, as well as the error of measuring the sample diameter by a micrometer before and after the tests. During the measurements, temperature deviation was demonstrated to constitute an additional source of uncertainty when the ambient temperature is different from the standard temperature value ((20±5)°C). Assumptions underlying laws describing the distribution of input values were assigned. Tensile tests are shown to be characterised by two components of uncertainty evaluated as types A and B. A mathematical model constructed for measuring relative elongation and relative reduction during tensile tests is presented. The standard uncertainties of input values are evaluated based on the assumed laws of their distribution. A correlation between the final length of the calculated section, the diameter of the sample following the test, and the applied force, is revealed. Expressions for the calculation of sensitivity coefficients, which characterize variations in the output value (relative elongation) depending on variations of input values, were obtained. The total and extended measurement uncertainties were evaluated. Based on the performed studies, a procedure for evaluating measurement uncertainty when carrying out tensile tests of high-strength bolts was described. The evaluation of measurement uncertainty during the product testing was shown to be a rather labourconsuming work. In this regard, the authors propose the development of procedures for evaluating the measurement uncertainty during the tests with their inclusion into regulatory documents for control methods.

GROWTH AND PHYSIOLOGICAL RESILIENCE OF PINE FORESTS IN UKRAINIAN POLISSIA

The main prerequisite for this study is the use of forest resources corresponding to the principles of sustainable forest management. The purpose of this study is to figure out the growth characteristics of pine stands and their physiological response to adverse factors. The experimental material (cores) was selected from pine forests of Ukrainian Polissia using Haglöf increment borer at breast height of 1.3 m. The number of annual rings and the parameters of radial increment were found using the ImageJ software. The result was a tree-ring chronology of sample trees. Statistical analysis of the experimental data proved that the radial increment variability decreases with age, and it ranges within 0.99-2.78 mm. The average radial increment value in the data set under study is 1.79 mm. The average number of annual rings of Scots pine (Pinus sylvestris L.) trees is 80: the minimum is 61, the maximum is 92. The correlation analysis of experimental data proved that the pairwise correlation coefficients of radial increment (-0.54) and current increment by diameter (-0.53) have an inverse relationship with the age of trees, and diameter at breast height with age – a direct relationship (0.87). The developed mathematical models of the dynamics of the width of the annual ring, the diameter at breast height and the current increment by diameter allow estimating the growth characteristics of Scots pine trees throughout their life. The obtained results were compared with the growth tables of fully stocked (at a relative stocking of 1.0) stands. The adequacy test of the developed mathematical models proved the accuracy of the given patterns and is as follows: for the width dynamics of the annual ring – 0.46; the diameter at breast height – 0.78, and the percentage of current increment by diameter – 0.51. Based on standardisation of individual chronologies by calculating sensitivity coefficients, no significant physiological response was established. Accordingly, the impact of short-term stress reactions is insignificant. The maximum resistance of pine stands to adverse environmental factors is achieved at the age of 50-60 years. This study is important to evaluate the impact of climate change and other adverse factors on the growth of pine stands and forecasting the dynamics of biometric indices. The obtained results can be used by the specialists at IA “Ukrderzhlisproekt” to update biometric indices and substantiate the use of forest resources

On the use of criticality and depletion benchmarks for verification of nuclear data

The problem of calculating sensitivity coefficients to nuclear data in burnup problems is addressed. A non-intrusive method based on the random sampling of nuclear data and on linear regression is proposed to overcome the limitations of dealing with a complex coupled problem. Sensitivity plots are reported for the actinide concentrations that build up in irradiated fuel and that have the largest impact for the safe and economical management of spent nuclear fuel. Despite using a simplified pincell model, the range of applicability of the calculated sensitivities is further discussed. The sensitivity study is corroborated by an analysis of how different nuclear data libraries perform in the prediction of the neutron multiplication factor in a burnup problem. The origin of the discrepancies between the results obtained using the JEFF-3.3 and ENDF/B-VIII.0 nuclear data libraries are investigated.

Sensitivity analysis of control parameters errors and current parameters to motion accuracy of underwater glider using Sobol’ method

Control parameters errors and current can weaken the motion accuracy of underwater gliders, which will obstruct the achievement of high accuracy exploration missions. To improve the glider motion accuracy more pertinently, it is necessary to identify the key factors that affect the glider motion accuracy. To resolve the above problem, we employ Sobol’ method to analyze the sensitivity of control parameters errors and current parameters to the glider motion accuracy. The control parameters errors include the net buoyancy adjustment amount error, the movable mass block translation amount error and the movable mass block rotation amount error. The current parameters include the current intensity, action depth and direction. First, the dynamic model of an underwater glider is established, and this model considers the effects of hull deformation, water density variation and current simultaneously. Then the dynamic model is validated by experimental data. After the evaluation parameter of motion accuracy is given, the glide motion accuracy is preliminarily analyzed, and then the sensitivity analysis is carried out. In this paper, we calculate the sensitivity coefficients by Monte Carlo method. To reconcile calculation efficiency with calculation accuracy, a convergence rule is proposed to reduce the number of sample points that are used for calculating sensitivity coefficients as much as possible. Besides, based on simulation results of dynamic model, the surrogate models are established to participate in the calculation of sensitivity coefficients, which can improve analysis efficiency further. For two typical motion modes, analysis results identify the key factors affecting the glider motion accuracy and illustrate the form of their effect on the glider motion accuracy. This research may be valuable for the achievement of the glider precision operation.

UNCERTAINTY ANALYSIS OF SUB-EXERCISES IN UAM-SFR BENCHMARK WITH THE MCS CODE

Uncertainty analysis in Modelling (UAM) for Design, Operation and Safety Analysis of Sodium-cooled Fast Reactors (SFRs) has been formed by OECD/NEA to assess the effect of nuclear data uncertainties on parameters of interest in SFR analysis. In this paper, sub-exercises of a medium 1000 MWth metallic core (MET-1000) and a large 3600 MWth oxide core (MOX-3600) are tested by a Monte Carlo code MCS to perform uncertainty analysis. Classical perturbation theory and generalized perturbation theory are used to calculate sensitivity coefficients. Uncertainty is calculated by multiplying the sensitivity coefficients and relative covariance matrix from ENDF/B-VII.1 library.

Geometric sensitivity analysis of generalized response function with RMC code

GEneralized Adjoint Responses in Monte Carlo (GEAR-MC) method has been developed to perform sensitivity analysis of reaction rate ratios, which was to compute sensitivity coefficients of reaction rate ratios to nuclear data e.g. nuclear cross sections. In this work, the GEAR-MC method is extended to calculate sensitivity coefficients of reaction rate ratios to geometric parameters for the first time and is implemented in the Reactor Monte Carlo (RMC) code. This newly developed geometric sensitivity analysis capability is tested by four benchmark problems. The results are shown to agree within 4% and 3 σ with reference results obtained by the direct perturbation method for all cases. Moreover, the contributions from different terms to the sensitivity results are compared. It is shown that, for Godiva, Hemisphere Godiva and Jezebel benchmarks, the intergenerational term contributes much to sensitivity coefficients, and this term is necessarily considered correctly in the practical application.

Coordinated voltage control of renewable energy power plants in weak sending-end power grid

The utilization of renewable energy in sending-end power grids is increasing rapidly, which brings difficulties to voltage control. This paper proposes a coordinated voltage control strategy based on model predictive control (MPC) for the renewable energy power plants of wind and solar power connected to a weak sending-end power grid (WSPG). Wind turbine generators (WTGs), photovoltaic arrays (PVAs), and a static synchronous compensator are coordinated to maintain voltage within a feasible range during operation. This results in the full use of the reactive power capability of WTGs and PVAs. In addition, the impact of the active power outputs of WTGs and PVAs on voltage control are considered because of the high R/X ratio of a collector system. An analytical method is used for calculating sensitivity coefficients to improve computation efficiency. A renewable energy power plant with 80 WTGs and 20 PVAs connected to a WSPG is used to verify the proposed voltage control strategy. Case studies show that the coordinated voltage control strategy can achieve good voltage control performance, which improves the voltage quality of the entire power plant.

Sensitivity analysis of the model of tuberculosis

In this contribution, a sufficiently simple, yet realistic model of TB propagation and control is studied in detail. We compute the controlled basic reproduction number R0(uB, uD) for the extended model including the action of two types of controls and analyze the sensitivity of R0(uB, uD) with respect to both controls following the recently devised approach. It is shown that the computed sensitivity coefficients admit a clear epidemiological interpretation and can be used in assessing the efficacy of the respective types of intervention.

Computational aspects of Simplification of Mathematical Models of Chemical Reaction Systems

The primary objective of this study is to develop a procedure for simplifying the mathematical model of a chemical reaction by reducing the number of steps and species in the reaction scheme. The procedure is based on the sensitivity analysis of the functional model to the change in the model parameters. The application of this procedure makes it possible to accelerate and improve the construction of mathematical models for chemical reactions, as this does not require the calculation of sensitivity coefficients at various time points or sensitivity matrix analysis. The efficiency of using this procedure was demonstrated by comparative results of modeling of formaldehyde oxidation in terms of detailed and reduced schemes. Computational aspects of the solution to the problems of chemical kinetics by the global sensitivity method using the example of the formaldehyde oxidation reaction are presented.

Comparison of MCM and GUM Method for Evaluating Measurement Uncertainty of Wind Speed by Pitot Tube

“The Guide to the Expression of Uncertainty in Measurement” (GUM) method has many disadvantages when evaluating the uncertainty of wind speed measured by pitot tube, and the “Monte Carlo method” (MCM) is put forward to evaluate the uncertainty of wind speed. The measured quantity value of wind speed by the S-shaped pitot tube is the research object, and the uncertainty is evaluated by GUM and MCM, respectively, and the simulation test is carried out. GUM has errors due to its adoption of approximate linear model. MCM adopts real simulation strategy and has higher credibility than the GUM method, so the GUM method can be validated by the MCM method. When the partial input quantity is changed into triangular distribution, uniform distribution and arcsine distribution, the GUM method is found to be inapplicable. In the evaluation of MCM method, the complex calculation of sensitivity coefficient is avoided, and some uncertainty with small influencing quantity is not needed to be discarded. The evaluation result is more complete. Through comparing the uncertainty evaluation results of MCM before and after the water vapor correction term is discarded, the influence of the water vapor correction term is quantified. Therefore, compared with the GUM method, the MCM method has more advantages when wind speed is measured by pitot tube.

Computing sensitivity coefficients by using complex differentiation: Application to heat conduction problem

A simple and efficient computational method for Sensitivity Coefficient calculation is presented. Sensitivity analysis are carried out for a simple one-dimensional heat conduction problem with respect to two parameters: thermal conductivity (k) and volumetric heat capacity (c). Results obtained by finite difference and complex step differentiation are compared with analytical solution in non-dimensional form. With an arbitrary small step-size, the complex differentiation method can achieve near analytical accuracy in languages that support complex arithmetic. Further implementation of complex step differentiation in Matlab environment is explained with some precautions.

Coordinated voltage control scheme for VSC‐HVDC connected wind power plants

This study proposes a coordinated voltage control scheme based on model predictive control for voltage source converter (VSC)-based high voltage direct current connected wind power plants. In the proposed scheme, voltage regulation capabilities of VSC and wind turbine generators (WTGs) are fully utilised and optimally coordinated. Two control modes, namely the operation optimisation mode and corrective mode, are designed to coordinate voltage control and economic operation of the system. In the first mode, the control objective includes the bus voltages, power losses and dynamic Var reserves of WTGs. Only the terminal voltages of WTGs are considered in the second mode. The predictive model of the system including VSC and WTGs is developed firstly. The calculation of sensitivity coefficients is done by an analytical method to improve the computational efficiency. Simulation results are presented to demonstrate the effectiveness of the proposed controller and the control performance is compared with conventional optimal control and loss minimisation control. Besides, the robustness of the proposed controller to communication time delay and measurement errors is investigated in the last.

Anomalous phosphine sensitivity coefficients as probes for a possible variation of the proton-to-electron mass ratio

A robust variational approach is used to investigate the sensitivity of the rotation-vibration spectrum of phosphine (PH$_3$) to a possible cosmological variation of the proton-to-electron mass ratio, $\mu$. Whilst the majority of computed sensitivity coefficients, $T$, involving the low-lying vibrational states acquire the expected values of $T\approx-1$ and $T\approx-1/2$ for rotational and ro-vibrational transitions, respectively, anomalous sensitivities are uncovered for the $A_1\!-\!A_2$ splittings in the $\nu_2/\nu_4$, $\nu_1/\nu_3$ and $2\nu_4^{\ell=0}/2\nu_4^{\ell=2}$ manifolds of PH$_3$. A pronounced Coriolis interaction between these states in conjunction with accidentally degenerate $A_1$ and $A_2$ energy levels produces a series of enhanced sensitivity coefficients. Phosphine is expected to occur in a number of different astrophysical environments and has potential for investigating a drifting constant. Furthermore, the displayed behaviour hints at a wider trend in molecules of ${\bf C}_{3\mathrm{v}}\mathrm{(M)}$ symmetry, thus demonstrating that the splittings induced by higher-order ro-vibrational interactions are well suited for probing $\mu$ in other symmetric top molecules in space, since these low-frequency transitions can be straightforwardly detected by radio telescopes.

Mechanism reduction of chemical reaction based on sensitivity analysis: development and testing of some new procedure

This paper proposes a procedure for simplifying the mathematical model of a chemical reaction by reducing the number of steps and species in the reaction scheme. This procedure is based on the sensitivity analysis of the model functional to the change in the model parameters. The application of this procedure makes it possible to accelerate and improve the construction of mathematical models for chemical reactions, as this does not require the calculation of sensitivity coefficients at various time points or sensitivity matrix analysis. This scheme was predicted the same behavior of the main reaction components as the detailed reaction scheme but includes much fewer steps. The efficiency of using this procedure was demonstrated by comparative results of modeling of formaldehyde oxidation and hydrogen oxidation in terms of detailed and reduced schemes. The sensitivity analysis was performed by Sobol’s method.