G. M. Ostrovsky

Flexibility analysis and optimization of chemical plants with uncertain parameters

Abstract In the paper two new feasibility tests are given. The first test gives the sufficient condition of feasibility of a design. The second test gives a sufficient condition of infeasibility of a design. The values of the tests give the upper and lower bounds of the value of the feasibility test introduced by Halemane and Grossmann ( AIChE J. 29 (3), 425–433, 1983). The procedure for determination of the value of the feasibility test of Halemane and Grossmann based on the use of the “branch and bound” method is suggested. The approaches to solving a one-stage programming problem and a two-stage programming problem are considered. Two problems are introduced and that permits to obtain upper and lower bounds of the criterion in the two stage programming problem. Application of the suggested approaches is illustrated by three examples.

An approach to solving a two-stage optimization problem under uncertainty

In this paper, two new algorithms for solving a two-stage optimization problem are given. The problem arises during design of chemical processes when there is some uncertainty in original data. These algorithms are extensions of an algorithm suggested by Ostrovsky et al. (1994) [Computers chem. Engng 18(8), 755–767 (1995)]. A distinctive feature of the algorithms is that during an execution of iterations the upper and the lower estimates of the optimal value of an objective function are calculated and the operation of partitioning the uncertainty region is performed. The application of one of the suggested algorithms is illustrated by three examples.

On the solution of mixed-integer nonlinear programming models for computer aided molecular design

This paper addresses the efficient solution of computer aided molecular design (CAMD) problems, which have been posed as mixed-integer nonlinear programming models. The models of interest are those in which the number of linear constraints far exceeds the number of nonlinear constraints, and with most variables participating in the nonconvex terms. As a result global optimization methods are needed. A branch-and-bound algorithm (BB) is proposed that is specifically tailored to solving such problems. In a conventional BB algorithm, branching is performed on all the search variables that appear in the nonlinear terms. This translates to a large number of node traversals. To overcome this problem, we have proposed a new strategy for branching on a set of linear branchingfunctions, which depend linearly on the search variables. This leads to a significant reduction in the dimensionality of the search space. The construction of linear underestimators for a class of functions is also presented. The CAMD problem that is considered is the design of optimal solvents to be used as cleaning agents in lithographic printing.

Optimization problem of complex system under uncertainty

Abstract Design of chemical processes (CP) is usually performed in a condition of some uncertainty of original physical and chemical information. In connection with this the important problem of the evaluation of flexibility of CP (the ability of CP to preserve its capacity for work) arises. Halemane and Grossmann (1983) (Optimal process design under uncertainty. A.I.Ch.E. J., 29 (3), 425–433.) introduced the flexibility (feasibility) function which permits evaluation of CP flexibility. But direct calculation of the flexibility function is reduced to solving very hard nondifferentiable and multiextremal optimization problems. In connection with this we give an effective method of calculation of the flexibility function. It is reduced to an iteration procedure on each iteration of which usual nonlinear programming problems are solved. On the basis of this method we consider two algorithms for solving the two stage optimization problem.

UNCERTAINTY AT BOTH DESIGN AND OPERATION STAGES

Process uncertainty is almost always an issue during the design of chemical processes (CP). In the open literature it has been shown that consideration of process uncertainties in optimal design necessitates the incorporation of process flexibility. Such an optimal design can presumably operate reliably in the presence of process and modeling uncertainty. Halemane and Grossmann (1983) introduced a feasibility function for evaluating CP flexibility. They also formulated a two-stage optimization problem for estimating the optimal design margins. These formulations, however, are based implicitly on the assumption that during the operation stage, uncertain parameters can be determined with enough precision. This assumption is rather restrictive and is often not met in practice. When available experimental information at the operation stage does not allow a more precise estimate of some of the uncertain parameters, new formulations of the flexibility condition and the optimization problem under uncertainty are needed. In this article, we propose such formulations, followed by some computational experiments.

A unique approach for solving sub-problems in flexibility analysis

In the design of a chemical process (CP), certain design specifications (for example those related to process economics, process performance, safety, and the environment) must be satisfied. During the operation of the plant, since design models have uncertainties associated with them, we need to ensure the flexibility of the CP. This means that within the region of uncertainty, all design specifications must be satisfied. In recent years, research has focused on the development of methods for flexibility analysis of the CP. There are three main sub-problems associated with flexibility analysis, namely evaluation of CP flexibility, evaluation of CP structural flexibility anddetermination of the optimal regime over which the flexibility of the CP is guaranteed. We have developed a general approach to solving the sub-problems based on the split and bound strategy.

Multicriteria Optimization under Uncertainty: Average Criteria Method

This paper discusses chemical process models for which the only uncertainties of interest are model parameters. In an earlier paper the authors addressed multicriteria optimization in the presence of model and process uncertainty at the design stage. Specifically the authors discussed extensions of the average criterion method, the worst-case strategy and the e -constraint method under the following conditions: (a) at the design stage the only information available about the uncertain parameters is that they are enclosed in a known uncertainty region T , and (b) at the operation stage, process data is rich enough to allow the determination of exact values of all the uncertain parameters. The suggested formulation assumed that at the operation stage, certain process variables (called control variables) could be tuned or manipulated in order to offset the effects of uncertainty. This formulation made the conventional assumption that there was only one type of uncertain parameters. In this paper, the authors consider the more realistic case, where the uncertain parameters fall under at least two classes at the operation stage, namely (a) those that can be determined with enough accuracy and (b) those that cannot be determined with such accuracy given the available process data. The case study is an application to a direct methanol fuel cell.

Evaluation of Chemical precesses flexibility

Abstract Design of chemical processes (CP) is usually performed in condition of some uncertainty of original physical and chemical information. In connection with this the important problem of the evaluation of flexibility of CP (the ability of CP to preserve its capacity for work) arises. Grossmann introduced the flexibility function which permits to evaluate CP flexibility. But direct calculation of the flexibility function is reduced to solving very hard nondifferentiable and multiextremal optimization problem. In connection with this we gave an effective method of calculation of flexibility function. It is reduced to an iteration procedure on each iteration at which usual non-linear programming problem is solved.

A Global Optimization Strategy and Its Use in Solvent Design

Solvent design can be modeled as a mixed integer nonlinear programming problem (MINLP) in which discrete variables denote the presence or absence of molecular structural entities and to what extent they occur in the pure component compound or mixture. On the other hand, continuous variables denote process variables such as temperature and flow rates. In the MINLP model the number of discrete variables can range from several tens to several hundreds. Therefore the use of the standard branch-and-bound method for solving the problem can be computationally intensive since all the variables (discrete and or continuous) must be used as branching variables. To overcome this problem, we have proposed a new strategy in which branching is done using branching functions instead of all the search variables. This approach results in a decrease in the number of branching variables. During branch and bound, the bounding operation is performed in the search variables space, while the branching operation is performed in a reduced dimension space defined by the branching (or splitting) functions. The branching functions are determined from the special tree function representation of both the objective function and constraints. The suggested MINLP solution approach is demonstrated on a solvent design application.

Split and bound method for process optimization under parametric uncertainty

We discuss methods for solving steady state process optimization problems under parametric uncertainty. The problem is formulated as a two-stage optimization problem (TSOP) which is inherently multiextremal and nondifferentiable. An indirect approach (split and bound method, SB) has been developed to address the nondifferentiability issue. The SB method iteratively solves for lower and upper bounds of the TSOP objective function, such that in the limit these bounds sandwich the optimal solution to within a given tolerance, thus avoiding the explicit solution of the nondifferentiable TSOP. We have introduced a linearization approach, which can lead to significant computational savings. Heuristics are proposed for partitioning and selection of critical points for the lower bound problem. We illustrate the proposed approach with one computational experiment