A simulated annealing-based approach to the optimal synthesis of heat-integrated distillation sequences

Abstract

A simulated annealing-based approach to synthesis of multi-component distillation systems is developed. An encoding procedure that makes use of an integer number series is developed to represent and manipulate the flow sheet structure of the system. With the representation procedure, the overall synthesis problem is formulated as an implicit mixed-integer nonlinear programming (MINLP) problem. A simulated annealing approach suitable for MINLP optimization is adopted and improved to solve the problem. Four example problems, including large-scale ones are solved to illustrate the approach.

Introduction

The objective of synthesis of multi-component distillation systems is to find the separation sequence and the heat integration structure that give the best behaviors in terms of investment and operating costs of the system.

Generally, the synthesis of a heat-integrated distillation system includes two different subtasks. One is to find an optimum flow sheet structure including a separation sequence and a heat integration map, while the other is to determine the levels or values of design/operating parameters (pressure, flowrate, etc.) of the individual columns in the flow sheet so as to arrive at the overall minimum cost for the process. The problems arising from the both aspects, especially when heat integration is considered, are highly coupled and hence should be solved simultaneously. A main challenge for solving such a problem comes from the fact that there are a large number of possible flow sheet structure alternatives for separation of a multi-component mixture, and the key to systematic solutions to the problem is the ability of dealing with the combinatorial problem.

In the last couple of decades, a number of approaches have been proposed for systematic solutions to the problems of synthesis of distillation sequences, including heuristic methods (Seader & Westerberg, 1977), evolutionary techniques (Stephanopoulos & Westerberg, 1976), hierarchical decomposition (Douglas, 1988), superstructure optimization (Andrecovich & Westerberg, 1985; Floudas & Paules, 1988; Yeomans & Grossmann, 1999), and stochastic methods (Chen, Yuan, & Zhong, 1997; Floquet, Pibouleau, & Domenech, 1994; Marcoulaki, Linke, & Kokossis, 2001; Wang, Qian, Yuan, & Yao, 1998), etc. Reviews on distillation system synthesis can be found in Westerberg (1985), Wang et al. (1998), and Yeomans and Grossmann (1999).

Based on a superstructure representation and a mixed-integer nonlinear programming (MINLP) formulation, the mathematical programming approaches to the synthesis problem offer a clear scheme of simultaneous optimization of the configuration and operating parameters. However, the applications for the available methods are often embarrassed by two difficulties. The first comes from the non-convexity of the nonlinear formulation. Even though methods have been developed for some specific class of non-convex MINLP problems (Floudas & Visweswaran, 1990; Yuan & Chen, 1997), it is still difficult to solve general form non-convex problems, which should be assumed for the synthesis problems. The second comes from the combinatorial feature of the problem. In almost all available MINLP methods, the combinatorial problem that is inevitable for the synthesis is handled via branch-and-bound principle, which, in the worst case tends to explore all the combinations of the values of the discrete variables before the optimum is ultimately found. As a result, the mathematical programming approaches available in the literatures were more suitable for solving small or moderate size problems.

Stochastic algorithms based on adaptive random search such as simulated annealing (SA) have been proven to be effective with the ability of global optimization. Differing from traditional methodologies, they deal with the flow sheet structure and the levels of the continuous variables by means of “state”, and explore the states with random search. Apart from its robustness coming from its stochastic nature, this approach offers an extra benefit in that the search can proceed with no direct dependence on the features of the nonlinear functions (for example the type of non-convexity, the gradient, etc.). This is significant for solving process synthesis problems because the difficulties of explicit modeling for the cost function and constraint equations with feasible search space definition can be avoided.

The applications of simulated annealing approach for the synthesis of distillation system have been reported by a number of authors. Floquet et al. (1994) were the first who applied SA technique to the synthesis of distillation system, but they did not address the general problem of continuous variable optimization and heat integration. Chen, Yuan, and Zhong (1997) developed a hybrid SA/LP algorithm based on a decomposition strategy for the synthesis of conventional column separation sequence allowing the use of non-sharp splits and heat integration, but they restricted the continuous sub-problems to LP cases. Recently Marcoulaki et al. (2001) proposed a vector presentation for separation sequencing with the use of more rigorous estimating models.

In fact, a notable characteristic of a distillation flow sheet is its tree structure, which embodies the combinatorial nature of the problem and, at the same time can be used for developing effective representation methods. Stephanopoulos and Westerberg (1976) proposed an evolutionary strategy based on a branching procedure for distillation system sequencing, which pioneered the application of tree sorting principal to distillation system synthesis. Chen, Yuan, and Zhong (1997) developed a SA approach to distillation system synthesis. They applied binary sort tree strategy and generated an effective procedure for manipulating distillation sequence structures.

The present work is to develop a robust SA-based approach to the synthesis of large-scale distillation systems with heat integration. To achieve this, an encoding procedure for representing and manipulating separation sequence as well as heat integration configuration structures is developed based on binary sort tree principle. Based on the coding representation, an SA based method for solving the overall MINLP optimization problem is developed. Example problems of moderate and large scale (problem with a mixture of more than 10 components) are solved effectively to illustrate the proposed approach.