Carl A. Schweiger

Interaction of Design and Control: Optimization with Dynamic Models

Process design is usually approached by considering the steady-state performance of the process based on an economic objective. Only after the process design is determined are the operability aspects of the process considered. This sequential treatment of the process design problem neglects the fact that the dynamic controllability of the process is an inherent property of its design. This work considers a systematic approach where the interaction between the steady-state design and the dynamic controllability is analyzed by simultaneously considering both economic and controllability criteria. This method follows a process synthesis approach where a process superstructure is used to represent the set of structural alternatives. This superstructure is modeled mathematically by a set of differential and algebraic equations which contains both continuous and integer variables. Two objectives representing the steady-state design and dynamic controllability of the process are considered. The problem formulation thus is a multiobjective Mixed Integer Optimal Control Problem (MIOCP). The multiobjective problem is solved using an ∈-constraint method to determine the noninferior solution set which indicates the trade-offs between the design and controllability of the process. The (MIOCP) is transformed to a Mixed Integer Nonlinear Program with Differential and Algebraic Constraints (MINLP/DAE) by applying a control parameterization technique. An algorithm which extends the concepts of MINLP algorithms to handle dynamic systems is presented for the solution of the MINLP/DAE problem. The MINLP/DAE solution algorithm decomposes the problem into a NLP/DAE primal and MILP master problems which provide upper and lower bounds on the solution of the problem. The MINLP/DAE algorithm is implemented in the framework MINOPT which is used as the computational tool for the analysis of the interaction of design and control. The solution of the MINLP/DAE problems is repeated with varying values of ∈ to generated the noninferior solution set. The proposed approach is applied to three design/control examples: a reactor network involving two CSTRs, an ideal binary distillation column, and a reactor/separator/recycle system. The results of these design examples quantitatively illustrate the trade-offs between the steady-state economic and dynamic controllability objectives.

Mixed-Integer Nonlinear Optimization in Process Synthesis

The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a Mixed-Integer optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as Mixed-Integer Nonlinear Programming (MINLP) problems.

Process Synthesis, Design, and Control: A Mixed-Integer Optimal Control Framework

Abstract A mixed-integer optimal control framework for analyzing the interaction of process synthesis, design, and control is presented in this paper. The approach integrates the economic design and dynamic controllability into a multiobjective Mixed-Integer Optimal Control Problem (MIOCP). The problem formulation includes dynamic models and incorporates both discrete and continuous decisions. An algorithm for the solution of the MIOCP is developed based on the principles of Generalized Benders Decomposition for mixed-integer nonlinear optimization. The algorithm is used to determine the trade-offs between the economic design and dynamic controllability of a reactor-separator-recycle system.

Synthesis of optimal chemical reactor networks

Abstract The synthesis of optimal reactor networks using a superstructure based approach is considered. The fundamental units in the superstructure are the continuous stirred tank reactor (CSTR) and a cross flow reactor (CFR). The mathematical modeling leads to an optimal control formulation which is solved using a control parameterization technique. The approach is applicable to general reaction mechanisms and is applied to a complex nonisothermal reaction problem.

Generalized Geometric Programming Problems

In this chapter, we will discuss test problems that arise from generalized geometric programming applications. For a thorough theoretical and algorithmic exposition of global optimization approaches for generalized geometric programming problems, the reader is directed to the article of Maranas and Floudas (1997), and the book by Floudas (2000).

The MINOPT Modeling Language

MINOPT is a modeling language for solving a broad range of optimization problems formulated as mathematical programs. The motivation for developing MINOPT was the need for a framework capable of addressing mixed-integer nonlinear programming problems as well as problems involving dynamic problems described by differential and algebraic equations. The result is a comprehensive system that can address linear programs, mixed-integer linear programs, nonlinear programs, nonlinear programs with differential and algebraic constraints, mixed-integer nonlinear programs with differential and algebraic constraints, optimal control problems, and mixed-integer optimal control problems. MINOPT features an expressive modeling language, implementations of Generalized Benders Decomposition and Outer Approximation algorithms, and interfaces to commercial optimization software. Examples are presented to illustrate the features of the modeling language.

Quadratic Programming Problems

In this chapter nonconvex quadratic programming test problems are considered. These test problems have a quadratic objective function and linear constraints. Quadratic programming has numerous applications (Pardalos and Rosen (1987), Floudas and Visweswaran (1995)) and plays an important role in many nonlinear programming methods. Recent methods of generating challenging quadratic programming test problems and disjointly constrained bilinear programming test problems can be found in the work of Vicente et al. (1992) and Calamai et al. (1993). Furthermore, a very broad class of difficult combinatorial optimization problems such as integer programming, quadratic assignment, and the maximum clique problem can be formulated as nonconvex quadratic programming problems.

Twice Continuously Differentiable NLP Problems

Twice continuously differentiable NLPs represent a very broad class of problems with diverse applications in the fields of engineering, science, finance and economics. Specific problems include phase equilibrium characterization, minimum potential energy conformation of clusters and molecules, distillation sequencing, reactor network design, batch process design, VLSI chip design, protein folding, and portfolio optimization.

Nonlinear Systems of Equations

The solution of nonlinear systems of equations is an important problem in engineering. Applications include identifying the multiple steady states of a reactor network (Folger, 1986; Schlosser and Feinberg, 1994), predicting the azeotropes formed in a nonideal mixture (Fidkowski et al., 1993; Maranas et al., 1996), finding the steady-states of a flowsheet or part of a flowsheet (Zhang, 1987; Bullard and Biegler, 1991; Wilhelm and Swaney, 1994; Bekiaris et al., 1993), identifying equilibrium points in multiphase systems (Heidemann and Mandhane, 1973; Seader et al., 1990) .

Semidefinite Programming Problems

Semidefinite programming involves the minimization of a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Several types of problems can be transformed to this form. This constraint is in general nonlinear and nonsmooth yet convex. Semidefinite programming can be viewed as an extension of linear programming and reduces to the linear programming case when the symmetric matrices are diagonal.