Simultaneous strategies for optimization of differential-algebraic systems with enforcement of error criteria

Abstract

Differential-algebraic systems frequently arise in process engineering. However, optimization of these systems is rarely performed because this task is often difficult and time-consuming. In this paper we explore a simultaneous formulation for solving differential algebraic optimization problems (DAOPs) with enforcement of error criteria for accurate solutions. Here decision variables can be identified as continuous variables but not as functions of time. Building on previous studies for this problem (e.g. Cuthrell and Biegler, AIChE Jl 33, 1257, 1987), we apply orthogonal collocation on finite elements to discretize the ordinary differential equations (ODEs) and solve the resulting nonlinear programming problem with a reduced successive quadratic programming (SQP) method. To control the approximation error we develop two strategies (equidistribution and direct error enforcement) that are embedded within the nonlinear program and adjust the finite element lengths adaptively over the course of the optimization. These approaches are more efficient and reliable, and do not require careful initialization schemes that were required in earlier studies. Moreover, determination of a sufficient number of elements is embedded automatically within the simultaneous strategy and the model is solved only once. These approaches are demonstrated on two hot spot reactor optimization problems. In particular, our results show that accurate solutions are obtained efficiently for the ODEs as part of the optimization, and constraints on stage profiles are very easy to enforce.