Approximate Programming Method Research Articles

Optimal dynamic trajectory planning for linearly actuated platform type parallel manipulators having task space redundant degree of freedom

Two types of dynamic trajectory planning problems associated with task space redundant degree of freedom of parallel manipulators are investigated. The first problem involves synthesizing the point-to-point trajectory of the moving platform such that the distribution of the cutting force of the tool bit along a specified machining contour is optimized. The second problem is to determine the maximum constant cutting force along the contour while maintaining an optimal distribution of the actuator forces. The numerical algorithms developed for solving these problems are based on the method of approximate programming and take into account all of the dynamic and kinematic constraints imposed on the system.

Optimization of Fuel Rod Enrichment Distribution to Minimize Rod Power Peaking throughout Life within BWR Fuel Assembly.

A practical method was developed for determining the optimum fuel enrichment distribution within a boiling water reactor fuel assembly. The method deals with two different optimization problems, i.e. a combinatorial optimization problem grouping fuel rods into a given number of rod groups with the same enrichment, and a problem determining an optimal enrichment for each fuel rod under the resultant rod-grouping pattern. In solving these problems, the primary goal is to minimize a predefined objective function over a given exposure period. The objective function used here is defined by a linear combination: C1X2XG , where X and XG stand for a control variable to give the constraint respectively for a local power peaking factor and a gadolinium rod power, and C 1 and C 2 are a user-definable weighting factor to accommodate the design preference.The algorithm of solving the combinatorial optimization problem starts with finding the optimal enrichment vector without any rod-grouping, and promising candidates of rod-grouping patterns are found by exhaustive enumeration based on the resulting fuel enrichment ordering, and then the latter problem is solved by using the method of approximation programming. The practical application of the present method is shown for a contemporary 8x8 Pu mixed-oxide fuel assembly with 10 gadolinium-poisoned rods.

A Two-Step Method for Developing a Control Rod Program for Boiling Water Reactors

This paper reports on a two-step method that is established for the generation of a long-term control rod program for boiling water reactors (BWRs). The new method assumes a time-variant target power distribution in core depletion. In the new method, the BWR control rod programming is divided into two steps. In step 1, a sequence of optimal, exposure-dependent Haling power distribution profiles is generated, utilizing the spectral shift concept. In step 2, a set of exposure-dependent control rod patterns is developed by using the Haling profiles generated at step 1 as a target. The new method is implemented in a computer program named OCTOPUS. The optimization procedure of OCTOPUS is based on the method of approximation programming, in which the SIMULATE-E code is used to determine the nucleonics characteristics of the reactor core state. In a test in cycle length over a time-invariant, target Haling power distribution case because of a moderate application of spectral shift. No thermal limits of the core were violated. The gain in cycle length could be increased further by broadening the extent of the spetral shift.

Optimization of axial enrichment and gadolinia distributions for BWR fuel under control rod programming.

The axial enrichment and gadolinia distributions of BWR (boiling water reactor) fuel are optimized under control rod programming. The objective of the problem is to minimize the average enrichment required to reach a planned EOC (end-of-cycle) with criticality condition and axial power peaking constraint. A method of approximation programming is employed as the basis for the solution method. Resulting linear programming problem at each iteration step is solved by means of goal programming algorithm. The method is applied to the initial fuel for a typical BWR/5 represented by an axial one-dimensional core model Two-region analysis leads to the conclusion that the core bottom should be depleted during the cycle so that the power shifts to the core top at EOC. The enrichment and gadolinia distributions are determined to maximize EOC power peaking within a limit. The optimal solution of a 24-region fuel with a power peaking limit of 1.4 saves 10.6% in uranium ore compared with a uniform fuel depleted with a H...

A Mathematical Method for Boiling Water Reactor Control Rod Programming

A new mathematical programming method has been developed and utilized in OPROD, an existing computer code for automatic generation of control rod programs as an alternative inner-loop routine for the method of approximate programming. The new routine is constructed of a dual feasible direction algorithm, and consists essentially of two stages of iterative optimization procedures Optimization Procedures I and II. Both follow almost the same algorithm; Optimization Procedure I searches for feasible solutions and Optimization Procedure II optimizes the objective function. Optimization theory and computer simulations have demonstrated that the new routine could find optimum solutions, even if deteriorated initial control rod patterns were given.

On the Use of Nonlinear Programming in Real Time Control of Process Industries

In process industries there is an increasing trend in the -use of computers anchor distributed digital control. The invention of powerful 16 hit and 32 bit microprocessors is opening avenues for realising advanced control algorithms in practice and control of plant with distributed intelligence. Over the years, considerable work has been done towards evolving an appropriate configuration 1 of distributed digital control system for plant control. In many system configurations a computer has been chosen at the upper tier for carrying out plant optimization and management information report generation, leaving the regulatory control and monitoring functions to the processors in the bottom 2 , Optimization of an operating process unit performed in real time with an on line process control computer has a number of significant advantages over an optimization executed by an off-line computer. While the concept of Dynamic Matrix Control 3 to address the control of a plant at its economic optimum is new, the concept of process control computer assisting optimization by moving one set of operating conditions to another is well accepted in industry. A good account of this use of static optimization in computer control of process industries is available in reference 4. If. Two popular techniques e. g linear and nonlinear programming have found place in the on line optimization task 4 . Since the mathematical model of the plant and the constaint functions are nonlinear 5 , it is appropriate to use nonlinear programming technique to carry out the on line optimization. Various algorithms are available for solving nonlinear programming (NLP) problem 7 . An account of user experience on some of them is available in reference 8. Desirable features of NLP software has been discussed by Lasdon 9 . A brief description of the performance of M.J. Box's complex method for on line optimization is available in reference 10. This paper is conceited with making a brief survey of current industrial practice of real time plant optimization using nonlinear programming techniques. An account of desirable features of a good nonlinear programming software and relationship of software packages for on line execution from various data sources e.g, historical data base of the plant, operator data entry, input from Research and development activities will be included. The performance of various techniques e.g. successive linear programming (SLP) or Method of approximation programming (MAP), generalised reduced gradient (GRG) and complex method of M.J. Box will he evaluated in the context of on line application. Case studies from petroleum industries and thermal power plant will be chosen for this purpose.

Nonlinear Optimization by Successive Linear Programming

Successive Linear Programming (SLP), which is also known as the Method of Approximation Programming, solves nonlinear optimization problems via a sequence of linear programs. This paper reports on promising computational results with SLP that contrast with the poor performance indicated by previously published comparative tests. The paper provides a detailed description of an efficient, reliable SLP algorithm along with a convergence theorem for linearly constrained problems and extensive computational results. It also discusses several alternative strategies for implementing SLP. The computational results show that SLP compares favorably with the Generalized Reduced Gradient Code GRG2 and with MINOS/GRG. It appears that SLP will be most successful when applied to large problems with low degrees of freedom.

Optimum Path Planning for Mechanical Manipulators

To assure a successful completion of an assigned task without interruption, such as the collision with fixtures, the hand of a mechanical manipulator often travels along a preplanned path. An advantage of requiring the path to be composed of straight-line segments in Cartesian coordinates is to provide a capability for controlled interaction with objects on a moving conveyor. This paper presents a method of obtaining a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain a minimum traveling time, under the constraints of composite Cartesian limit on linear and angular velocities and accelerations. Because of the involvement of a linear performance index and a large number of nonlinear inequality constraints, which are generated from physical limitations, the “method of approximate programming (MAP)” is applied. Depending on the initial choice of a feasible solution, the iterated feasible solution, however, does not converge to the optimum feasible point, but is often entrapped at some other point of the boundary of the constraint set. To overcome the obstacle, MAP is modified so that the feasible solution of each of the iterated linear programming problems is shifted to the boundaries corresponding to the original, linear inequality constraints. To reduce the computing time, a “direct approximate programming algorithm (DAPA)” is developed, implemented and shown to converge to optimum feasible solution for the path planning problem. Programs in FORTRAN language have been written for both the modified MAP and DAPA, and are illustrated by a numerical example for the purpose of comparison.

A pattern synthesis of circular arrays by phase adjustment

A problem of sidelobe suppression of a uniformly excited circular array of monopoles through the adjustment of the excitation phases is dealt with. For this purpose, the method of approximation programming (MAP) is used to determine the optimum excitation phases which minimize the sidelobe level. Galerkin's method is used to analyze the mutual coupling between array elements. The resulting radiation pattern has equal amplitude sidelobes that are lower than those obtained with a cophasal array. The resulting radiation pattern of a 32-element double-ring array, operating at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> -band, was measured and reasonable agreement with theory was obtained.

Boiling Water Reactor Control Rod Programming Using Heuristic and Mathematical Methods

OPROD, a computer code for automatic generation of control rod programming that has successfully been applied to an older boiling water reactor (BWR), has experienced some difficulties when applied to a BWR of larger power density and stronger heterogeneity. To improve the performance, a heuristic algorithm that is derived from accumulated experience has been introduced to search for a feasible rod pattern that satisfies all constraints. Application of this algorithm to an initial cycle of an 800-MW(e) BWR of high heterogeneity has been very successful. It has been demonstrated that the proposed algorithm is capable of finding a feasible rod pattern, even starting from an all-rods-out pattern. Some improvement was also made in the method of approximation programming (MAP) algorithm. The temporal constraint relaxation method is shown to be effective in finding an optimal control rod pattern in MAP starting from a guess pattern that is not feasible.

A Mathematical Programming Model for Allocation of Natural Gas

This paper presents a methodology for the allocation of natural gas. The model consists of several objective functions, a set of linear constraints, and a set of nonlinear constraints. The objective functions represent allocation in various categories and can be optimized sequentially. The linear constraints represent the conservation of flow equations for the pipeline network and various accounting relationships. The nonlinear constraints represent the momentum balance necessary for each pipe segment, compressor, or valve. The nonlinear constraints are linearized in a method similar to the method of approximate programming (MAP). A matrix generator is used to create the necessary files for the program execution. We have solved example problems with over 250 linear constraints, 240 nonlinear constraints, and 800 structural columns.

A Method for Generating a Control Rod Program for Boiling Water Reactors

The OPROD computer code has been developed to generate a long-term control rod program, a series of control rod patterns that optimize a cycle length within various operational constraints. In the algorithm, the optimization problem is decomposed into two hierarchies. In the inner loop, a time-invariant target power distribution is assumed, and a control rod pattern is determined so as to best fit the power distribution to the target within the constraints at each burnup step. The target is then improved in the outer loop to achieve a longer cycle length. The code consists of two major parts: a three- dimensional boiling water reactor (BWR) core simulator and MAP, the method of approximate programming. It readily generates a long-term control rod program of BWRs without trial search by core-management engineers. The OPROD has therefore facilitated prompt response to varying operating conditions and the investigation of a conflicting relationship between the thermal limitation and the cycle length. (auth)

Zur methode der approximierenden optimiernng

To meet difficulties of convergence and “stepsize-choosing” within the Method of Approximation Programming (MAP) of GRIFFITH and STEWART a modified approximation method for nonlinear programming problems is given. It is proved to be convergent for convex problems. The running times and gained numerical experience is discussed for numerous test problems.

Optimization of Control Rod Programming and Loading Pattern in Multiregion Nuclear Reactor by the Method of Approximation Programming

A nonlinear programming technique was applied to a one-dimensional multiregion slab reactor to optimize control rod programming and fuel loading pattern simultaneously.Original equations and constraints which were continuous in space and time were discretized and further linearized to use linear programming repeatedly.Numerical results were confirmed by the previously developed two-region burnup space theory. Furthermore, the more quantitative evaluation of burnup optimization and the determination of more realistic control rod programming became possible by the increased degree of control freedom.