Collision Probability Calculation Research Articles

Probabilistic model for vessel-bridge collisions in the Three Gorges Reservoir

Based on a field observation on vessel transit path of three bridges over the Yangtze River in the Three Gorges Reservoir, and an analysis of the geometric probabilistic model of transiting vessels in collision probability calculation, the aberrancy angle and vessel velocity probabilistic model related with impact force, a probabilistic model is established and also verified by goodness-of-fit test. The vessel transit path distribution can be expressed by the normal distribution model. For the Three Gorges Reservoir, the mean and standard deviation adopt 0.2w and 0.1w, respectively (w is the channel width). The aberrancy angle distribution of vessels accepts maximum I distribution model, and its distribution parameters can be taken as 0.314 and 4.354. The velocity distribution of up-bound and down-bound vessels can also be expressed by the normal distribution model.

Light Water Reactor Equilibrium Cycle Search Methodology for Assembly-Level Fuel Cycle Analysis

We have developed a light water reactor (LWR) equilibrium cycle search algorithm that is similar to the REBUS-3 fast reactor methodology but with depletion capabilities typically employed for LWR analysis. Our LWR methodology projects the original coupled nonlinear isotopic balance equations to a series of equations that are piecewise linear in time. Iterations are performed on microscopic reaction rates until the linearized isotopic balance equations yield an ultimate equilibrium state. We further reduce the computational burden associated with LWR analysis by approximating global depletion calculations with assembly-level, collision probability calculations performed by the CASMO-3 code. We demonstrate the benefits of our equilibrium cycle methodology by calculating the true equilibrium Pu inventory of two configurations: a heterogeneous assembly configuration that contains both low enriched UO2 and mixed oxide (MOX) fuel pins and a homogeneous configuration comprising a 2 × 2 colorset arrangement of MOX and low enriched UO2 assemblies. For each configuration our methodology yields a true equilibrium Pu inventory with only 12 CASMO-3 lattice physics calculations. As a validation, an inventory extrapolation technique is used to arrive at a quasi-equilibrium cycle for both LWR configurations. The extrapolated technique yields a similar Pu inventory and isotopic composition but requires 65 lattice physics calculations.

Power Reconstruction of Fuel Rods by Support Vector Regression for CANDU Reactors

A support vector regression (SVR) model has been presented for reconstructing fuel rod powers from Canada deuterium uranium core calculations performed with a coarse-mesh finite difference diffusion approximation and single-assembly lattice calculations. The SVR is to nonlinearly map the original data into a higher dimensional feature space. Parameters related to the SVR are optimized by a genetic algorithm using the partial core calculation results of two 6 times 6 fuel bundle models (for training data). Verification has been conducted for two other partial core benchmark problems composed of 6 times 6 and 3 times 3 fuel bundles (for test data). The reconstructed fuel rod powers are compared with the reference solutions obtained with the detailed collision probability calculations using the HELIOS lattice analysis code. It is known from simulation results that the proposed rod power reconstruction algorithm is accurate, yielding the error due to the reconstruction scheme of less than 0.35%.

Satellite Collision Probability Enhancements

This work refines collision probability calculations for rectangular object shapes of unknown orientation and compares those results to their representations as spheres. Conjunction probability analysis for spherical objects exhibiting near-linear relative motion is accomplished by combining covariances and physical object dimensions at the point of closest approach. The resulting covariance ellipsoid and hardbody can then be projected onto the plane perpendicular to relative velocity. Collision potential is determined from the object’s physical footprint on the projected, two-dimensional, probability density space. In the absence of object attitude information, a footprint that completely defines the region where the two objects might touch must be created. This footprint can then be rotated to determine the orientation that produces the largest probability making it the most conservative estimate for the given conjunction conditions. A further enhancement is presented to address the maximum possible probability for a given miss distance while also assessing the sufficiency of the orbital data to meaningfully support the calculations.

Optimized Algorithm for Collision Probability Calculations in Cubic Geometry

An optimized algorithm for implementing a recently developed method of computing collision probabilities (CPs) in three dimensions is reported in this work for the case of a homogeneous cube. Use is made of the geometrical regularity of the domain to rewrite, in a very compact way, the approximate formulas for calculating CPs in general three-dimensional geometry that were derived in a previous work by the author. The ensuing gain in computation time is found to be substantial: While the computation time associated with the general formulas increases as K2, where K is the number of elements used in the calculation, that of the specific formulas increases only linearly with K. Accurate numerical results are given for several test cases, and an extension of the algorithm for computing the self-collision probability for a hexahedron is reported at the end of the work.

A Combined Method to Evaluate the Resonance Self Shielding Effect in Power Fast Reactor Fuel Assembly Calculation

A combined method for evaluating the resonance self shielding effect in a two-dimensional power fast reactor fuel assembly is proposed. This method uses the sub-group method to evaluate the self shielding effect of heterogeneous cell and collision probability method in the ultra-fine energy groups to deal with the resonance interference effect between different resonant nuclides. In the present paper, a comparison between the table look-up method and the sub-group method is carried out and it is shown that the latter is superior to the former in a view of evaluating the resonance self shielding effect. These methods have a common defect that it is impossible to treat the resonance interference effect between different resonant nuclides. It can be overcome by using “a correction factor of the resonance interference effect” obtained by the collision probability calculation in the ultra-fine energy groups in a single fuel pin cell model. The microscopic effective cross section obtained by this proposed method agrees well with that by continuous energy Monte Carlo code within 5% relative difference above 100 eV. The k ∞ value is affected about 0.1%Δk/k by the use of the correction factor.

Space Vehicle Maneuver Method to Lower Collision Risk to an Acceptable Level

A method for determining the maneuver required by a space vehicle to reduce its collision probability with another tracked object below a predefined threshold was developed. The methodology employs efficient computationalschemes for computing collision probability, propagating space vehicles, and three-dimensional root finding. The collision probability calculation was improved by reducing a two-dimensional integral to a one-dimensional integral, thereby realizing a significant reduction in computational effort. In addition, a method to circumvent high-fidelity propagation of the space objects from the time of the maneuver to the time of closest approach was found. The method essentially replaces high-fidelity propagation with simple two-body Keplerian propagation. A gradient technique is used to find the maneuver direction. The maneuver magnitude is found using a one-dimensional root finding scheme that takes advantage of the linear relationship between the maneuver velocity and resulting displacement. The method was applied to an actual case involving two geostationary satellites whose collision probability was unacceptably high. For various times before closest approach, the maneuver required to reduce the collision probability below a required threshold was found and provided to satellite operators. A maneuver time was selected and the associated maneuver was executed by satellite operators. Postmaneuver tracking of the satellites indicated that the maneuver worked as planned and reduced the collision probability to the desired level.

A Combined Method to Evaluate the Resonance Self Shielding Effect in Power Fast Reactor Fuel Assembly Calculation

A combined method for evaluating the resonance self shielding effect in a two-dimensional power fast reactor fuel assembly is proposed. This method uses the sub-group method to evaluate the self shielding effect of heterogeneous cell and collision probability method in the ultra-fine energy groups to deal with the resonance interference effect between different resonant nuclides. In the present paper, a comparison between the table look-up method and the sub-group method is carried out and it is shown that the latter is superior to the former in a view of evaluating the resonance self shielding effect. These methods have a common defect that it is impossible to treat the resonance interference effect between different resonant nuclides. It can be overcome by using “a correction factor of the resonance interference effect” obtained by the collision probability calculation in the ultra-fine energy groups in a single fuel pin cell model. The microscopic effective cross section obtained by this proposed method agrees well with that by continuous energy Monte Carlo code within 5% relative difference above 100 eV. The k ∞ value is affected about 0.1%Δk/k by the use of the correction factor.

Pin power reconstruction for CANDU reactors using a neuro-fuzzy inference system

A neuro-fuzzy inference system has been developed for reconstructing fuel pin powers from Canada deuterium uranium (CANDU) core calculations performed with a coarse-mesh finite difference diffusion approximation and single-assembly lattice calculations. The neuro-fuzzy inference system is trained by a genetic algorithm and a least-squares method using the partial core calculation results of two 6/spl times/6 fuel bundle models. Verification tests have been performed for two partial core benchmark problems composed of other 6/spl times/6 and 3/spl times/3 fuel bundles. The reconstructed pin powers are compared with the reference solutions obtained with the detailed collision probability calculations using the HELIOS lattice analysis code. The results indicate that the proposed reconstruction algorithm is accurate, yielding the error due to the reconstruction scheme of less than 0.5%.

Application of theBnTheory to Unit Cell Calculations

The integral transport equation is solved in square unit cells by assuming the existence of a fundamental mode. The equations governing the B{sub n} method are given without making the small buckling approximation. First, the angular flux is factorized into two parts: a periodic microscopic fine-structure flux and a macroscopic form with no angular dependence. The macroscopic form only depends on a buckling vector with a given orientation. The critical buckling norm, along with the corresponding fine-structure flux, is obtained using collision probability calculations that are repeated until criticality is achieved. The procedure allows the periodic or reflective boundary conditions of the unit cell to be taken into account using closed-form contributions obtained from the cyclic tracking technique. Numerical results are presented for one-group heterogeneous cell problems with isotropic and linearly anisotropic scattering kernels, some of which include void regions.

Collision Probability Calculation and Multigroup Flux Solvers Using PVM

Collision probability evaluation and flux computation are the most time-consuming aspects of applications based on the linearized time-independent transport equation. Parallelization for collision probability calculation and multigroup flux computation are investigated. Particular techniques pertinent to the two-step energy/space iterative process of solving a multigroup transport equation are described. The parallel performance is studied in cases where the cyclic tracking technique is used to integrate collision probability. Parallelization is achieved by distributing either different energy groups or different regions on a set of processors. These algorithms were tested on a four-processor IBM SP2 and an eight-processor SPARC 1000 as well as on networks of workstations using the public domain PVM library. Typical run times are provided for unit cell calculations.

An Accurate Method to Interpolate First-Flight Collision Probabilities

A method is proposed to approximately calculate first-flight collision probabilities for use in resonance integral calculations with good accuracy and within a short computing time. In the proposed method, collision probabilities at three different energy points are rigorously calculated and stored beforehand. They are used to obtain approximate collision probabilities at an arbitrary energy point by Takahashi's zero-th approximation. Then, these approximate values are used to obtain more accurate values by quadratic interpolation with the neutron total cross section change as an interpolation parameter. The proposed method was applied to the calculation of collision probabilities in a two-region LWR fuel pin cell, and it was found that the collision probabilities can be calculated with an error of less than 1%, a great improvement in accuracy compared to the zero-th approximation, when the total cross section of the fuel region changes 30 times. The proposed method shortens computing time of lattice cell neutronic calculation codes based on the first-flight collision probability method.

An Accurate Method to Interpolate First-Flight Collision Probabilities.

A method is proposed to approximately calculate first-flight collision probabilities for use in resonance integral calculations with good accuracy and within a short computing time. In the proposed method, collision probabilities at three different energy points are rigorously calculated and stored beforehand. They are used to obtain approximate collision probabilities at an arbitrary energy point by Takahashi's zero-th approximation. Then, these approximate values are used to obtain more accurate values by quadratic interpolation with the neutron total cross section change as an interpolation parameter. The proposed method was applied to the calculation of collision probabilities in a two-region LWR fuel pin cell, and it was found that the collision probabilities can be calculated with an error of less than 1%, a great improvement in accuracy compared to the zero-th approximation, when the total cross section of the fuel region changes 30 times. The proposed method shortens computing time of lattice cell neutronic calculation codes based on the first-flight collision probability method.

Equilibrium Cycle Calculations for Transuranics Recycling in Pressurized Water Reactors

Assembly-level collision probability calculations are performed with the CASMO code to study the feasibility and efficiency of recycling transuranics (TRUs) in pressurized water reactors (PWRs). A zero-dimensional equilibrium cycle approach is combined with an extrapolation algorithm to estimate equilibrium cycle configurations in a three-zone PWR core. A whole-core implementation of mixed-oxide assemblies containing TRUs may require an unusually large loading of TRUs, but a (2 [times] 2) colorset arrangement of TRU and uranium assemblies, with a sufficient [sup 235]U enrichment, may yield a reasonable PWR configuration. Colorset arrangements will not result in net depletion of TRUs but will slow down the accumulation of TRUs in a continuing PWR economy.

A closed form solution of the Neutron Evacuation Problem from an expanding, purely absorbing sphere

A rigorous solution of an initial value problem for the neutron transport equation in a purely absorbing sphere, the material of which is radial motion, illustrates the application of the Vladimirov transformation to the solution of the Cauchy problem in spherical geometry. The resulting method is based on the solution of integral transport equations along characteristics drawn in the space-angle-time domain obtained via the application of ray-tracing techniques known from the algorithms applied for the calculation of collision probabilities.

Anisotropic scattering for integral transport codes. Part 2. Cyclic tracking and its application to xy lattices

The integral transport equation is solved in Cartesian 2-D geometries in the case where anisotropic effects are taken into account. A model for anisotropic collision probability (CP) calculations based on an angle-dependent representation of the collisions has been developed. First, a general cyclic tracking procedure is set up such that specular boundary conditions are accurately treated. The computation of angle-dependent CP factors is then performed by adding closed-form contributions resulting from infinite line integration on all cyclic tracks corresponding to the same angle. The CPs up to any anisotropic order are simply obtained from a linear combination of these angle-dependent factors. Comparisons with other authors' solutions are presented in order to show that this model preserves overall accuracy. Extensions to 3-D rectangular assemblies are also discussed.

Anisotropic scattering for integral transport codes. Part 1. Slab assemblies

The integral transport equation is solved in slab geometries in the case where anisotropic effects are taken into account. A model for anisotropic collision probability (CP) calculations based on an angle-dependent representation of the collisions has been developed. This model allows exact boundary conditions to be taken into account without resorting to an infinite series; it leads to simple integral formulæ for the CP, and the only choice remaining is an angular quadrature set. In this integral formulation of the transport equation discretized by a Galerkin scheme, no ray effect has been observed in the results although very few angles are chosen. Numerical results are compared to analytic solutions and to those obtained with ANISN. The comparisons show that this model preserves overall accuracy. Extensions to the 2- and 3-D rectangular assemblies are discussed.

A Transport Method for Treating Three-Dimensional Lattices of Heterogeneous Cells

A new ray-tracing method for the calculation of collision probabilities within arbitrary three-dimensional geometries has been developed. This method is used to discretize the neutron transport equation for heterogeneous rectangular cells containing zones of mixed cylindrical and rectangular geometry. For multicell applications, the interface current (IC) method provides the coupling between cells. The solution to the IC equations over multicell domains consisting of rectangular three-dimensional cells is improved by using an alternate direction implicit iteration scheme with variational acceleration. Results include comparisons of this technique with SHETAN for simple geometries and the analysis of a three-dimensional extension of a two-dimensional 15 × 15 pressurized water reactor benchmark problem.

The Evaluation of Aircraft Collision Probabilities at Intersecting Air Routes

In this article formulas useful for the calculation of aircraft mid-air collision probabilities at intersecting air routes are developed. The aircraft overlap density, a key parameter in the calculation, is expressed as a function of the intersection angle of air routes, aircraft speeds, the nominal separation, and parameters in the position-error distribution. Two representative probability models for position errors are used to illustrate the computational procedures suggested. A method of approximation over large navigation systems is also proposed.The calculation of aircraft mid-air collision probabilities is an important step in the design of air-route structure and is a key element in the specification of separation standards and minimum requirements of facilities reliability and accuracy. A thorough analysis of collision probabilities for long-range parallel air routes over oceanic areas was reported by P. G. Reich in three research documents and in a series of papers published in this Journal in 1966. Analysis of collision probabilities at intersecting air routes typical in off-shore and continental areas, however, has not been fully explored to date. The only direct reference available on this subject appears to be the work by Bellantoni,7 in which a collision problem was formulated in the three-dimensional space and was apt to deal with airport terminal landing or take-off situations. For circumstances where horizontal linear air routes intersect one another at preassigned positions, the analysis provided by Bellantoni needs to be simplified and refashioned for use in extensive applications.

An Evaluation of Different Methods for Space-Dependent Resonance Shielding of Cross Sections in Fast Reactors

Within the framework of the Abajan, Bazazjantz, Bondarenko, and Nikolaev (ABBN) shielding factor approach, several methods have been proposed in the past for the treatment of resonance shielding of multigroup cross sections in heterogeneous cells of fast reactor lattices. First, the approximations made in the different methods and their interrelationship is briefly reviewed. Then, three recent efficient methods proposed by Bitelli et al., by Tone, and by Kujawski and Protsik are numerically compared by checking against exact fine-group collision probability calculations. It is found that the method of Bitelli et al. may give erroneous results, while the methods of Tone and of Kujawski and Protsik show satisfactory agreement with the exact calculations.