Relaxation strategy for the structural optimization of process flowsheets

This paper presents an Equality Relaxation variant to the Outer-Approximation algorithm for solving mixed-integer nonlinear programming (MINLP) problems that arise in structural optimization of process flowsheets. The proposed algorithm has the important capability of being able to explicitly handle nonlinear equations within MINLP formulations that have linear integer variables and linear/nonlinear continuous variables. It is shown that through the explicit treatment of nonlinear equations, the proposed algorithm avoids computational difficulties (e.g. singularities, destruction of sparsity) that are experienced with algebraic or numerical elimination schemes. Also, theoretical properties of the Equality-Relaxation algorithm are discussed, and its performance is demonstrated with a planning problem and a flowsheet synthesis problem. Finally, a simple procedure for structural sensitivity analysis is presented. University Libraries Carnegie Mgiicn University Pittsburgh, ?eria:y;variia 15213 Introduction Process synthesis can be defined as the selection, arrangement, and operation of processing units so as to create an optimal scheme. This task is combinatorial and open-ended in nature and has received a great deal of attention over the past twenty years. An excellent review of research activities in this area can be found in Nishida, Stephanopoulos, and Westerberg (1981). Because the synthesis problem is open-ended, it has motivated the development of quite different approaches. Synthesis methods currently include thermodynamic targets (Linnhoff,1981), heuristic (Douglas, 1985) and evolutionary methods (Stephanopoulos and Westerberg, 1976), and optimization techniques (Grossmann, 1985). This paper will address the structural flowsheet optimization problem that arises in the latter approach. In order to formulate the synthesis problem as a mathematical programming problem, a superstructure must be postulated which includes many alternative designs from which the optimal process will be selected. Superstructures can be developed systematically for homogeneous processes (e.g. heat exchanger networks, separation sequences) while for heterogeneous processes the superstructure is specified by the user based on a preliminary screening of alternatives, possibly through the application of heuristic rules and/or thermodynamic targets. In order to determine the optimal process flowsheet, simultaneous structural and parameter optimization of the superstructure is required. In general, this leads to a mixed-integer optimization problem (see Grossmann, 1985). Most of the previous work on process synthesis that is based on the mixed-integer optimization approach has relied on the use of mixed-integer linear programming (MILP) formulations (e.g. see Papoulias and Grossmann, 1983; Andrecovich and Westerberg, 1985; Hillenbrand, 1984; Floudas et al, 1986; Shelton and Grossmann. 1986). Although these formulations have proved to be quite powerful, they have the limitation that nonlinearities cannot be treated explicitly, and hence they must be approximated through the discretization of operating conditions (see Papoulias and Grossmann, 1983). The need for the explicit handling of the nonlinearities in the synthesis problem motivates the use of mixed-integer nonlinear programming (MINLP). MINLP problems, however, are much more difficult to solve than MILP problems for which Branch and Bound methods perform reasonably well. It should be noted that for process synthesis applications, the MINLP problems have a special structure in which the 0-1 variables appear linearly and the continuous variables appear linearly and nonlinearly (see Duran and Grossmann, 1986a). Current alternatives for solving these MINLP problems include Branch and Bound (Beale, 1977; Garfinkel and Nemhauser, 1972; Little, 1963) , Generalized Benders Decomposition (GBD) (Benders, 1962; Geoffrion, 1972), and Outer-Approximation (OA) method (Duran and Grossmann 1986a; 1986b). Since in structural flowsheet optimization problems the majority of the computational effort is consumed by the solution of a sequence of nonlinear programming problems (NLP), a reasonable measure of the efficiency of the above cited algorithms is the number of NLP subproblems that they must solve. Duran and Grossmann proposed the Outer-Approximation (OA) algorithm with the objective of reducing the number of NLP subproblems that must be solved. Although this algorithm is an efficient method for solving MINLP problems, its main limitation is that it can handle only linear equality and nonlinear / linear inequality constraints. However, the formulation of the process synthesis problem will typically contain many nonlinear equations which describe the performance of process units. In the OA algorithm, nonlinear equations must be eliminated algebraically or numerically. This is usually a nontrivial task which can lead to theoretical and numerical difficulties as will be discussed later in the paper. The main objective of this paper is to present a new variant to the OA algorithm that can explicitly handle nonlinear equality constraints. The proposed algorithm relies on an equality relaxation strategy which has the advantage of not requiring the selection of decision variables, nor the algebraic or numerical elimination of the equations. As will be shown, this algorithm is well suited to solve MINLP problems that arise in structural flowsheet optimization. Basic properties of this algorithm are discussed and its application is illustrated with three example problems. Also, it will be shown how to perform structural sensitivity analysis with the proposed method. Problem Formulation The structural flowsheet optimization problem for process synthesis can be formulated as a MINLP of the following form: z = min c y + fix) s.f. h(x) = 0 g(x) < 0 Ax=a (P) By + Cx < d