(129e) A Method for the Computation of Many Local Optima in Process Design Problems

Previous work on the optimization of the design parameters of complex processes, especially of reactive distillation processes, has shown that in these problems, a large number of local optima exist [1], [2]. While some of these local optima are close to each other and are most likely caused by early termination of the nonlinear optimization algorithm, many of them are structurally different, i.e. they have different catalyst distributions on the trays and different feed locations. It is therefore of practical interest to identify most, if not all of these local optima in order to make design choices, e.g. by taking additional information into account, like information on the uncertainty of the assumptions made in the cost function, or heuristic knowledge on favorable designs.

Such design optimization problems are usually tackled by mixed-integer nonlinear programming techniques, e.g. DICOPT [3], where the optimization of the structural decisions, of the design parameters and of the operational degrees of freedom is performed simultaneously with the determination of the corresponding state variables. This leads to large optimization problems with several thousand variables and as many complex constraints that are implicitly given by chemical and thermodynamic equilibria and the reaction kinetics. Even without considering integer variables as the number of trays, the resulting nonlinear optimization problems can only be solved reliably after careful reformulation of the equations and scaling of the variables [4] and provide only one local optimum. The application of stochastic multi-start heuristics improves the probability of finding the global optimum, but then the numerical effort increases by typically two orders of magnitude [1], [2].

In this work, an evolutionary algorithm is used to generate initializations for a local search method, such that a large number of local optima is identified. The efficiency of the search is increased by the introduction of so-called tabu zones which exclude regions from the future search. These regions are defined by the initial points of the local search and the corresponding local optima [5]. As it is too complex to handle the full search space that consists of the design variables and the state variables of the resulting process in the EA, the evolutionary search is confined to the space of design variables. The state variables that are needed for the evaluation of the cost function and of the constraints are computed by the same nonlinear solver that performs the local optimization.

The considered reactive distillation process is the heterogeneously catalyzed and kinetically controlled synthesis of methyl-tertiary-butyl-ether (MTBE) from isobutene and methanol in the presence of n-butane at a pressure of 8 bar. The mixture exhibits three binary azeotropes. The aim is to produce MTBE with a purity of 99 % from feed streams of methanol (6.375 mol/s) and isobutene/n-butane (8.625 mol/s, 65.2 % isobutene according to [6]). In the reactive distillation column, the chemical reaction is coupled with the distillation of the reaction mixture. This offers the opportunity to overcome the chemical and thermodynamic limitations. In the conceptual design of reactive distillation columns, the operating parameters, the dimensions and the structure of the column are determined. The design is evaluated by its total annual profit that is composed of product revenues reduced by raw material costs, operating costs and investment costs that depend nonlinearly on the design parameters. The main design variables are the locations of the feeds (initially the number of feed locations is not restricted), the distribution of the catalyst, the diameter of the column and the boil-up and reflux ratios. A design problem hence has 149N+14 continuous degrees of freedom. Here, the number of trays (N) is assumed to be fixed. By the formulation described in [4] empty trays can be included, hence the number of trays is only an upper bound to the effective number of trays.

The design optimization is tackled by a memetic algorithm (MA) that consists of an evolutionary strategy (ES) and a mathematical programming (MP) method. The ES generates new starting points for the MP solver (CONOPT [7]). The local problems are solved in a sequential manner. First, the state variables are computed for the initial point in the space of the design variables. Then the local optimization is performed simultaneously in the space of the design variables and the state variables. This leads to a significant reduction (>75 %) of the computation time needed for the local search.

For problem instances with a fixed number of trays, up to 97 local optima could be located by the application of the new algorithm. In the case of N=40 trays, for instance, 80 local optima have been found. The memetic algorithm generates on average one new local minimum for every 6 calls of the nonlinear solver.

References:

[1] S. Barkmann, G. Sand and S. Engell: Modellierungsansätze für die Design-Optimierung von reaktiven Rektifikationskolonnen, Chemie Ingenieur Technik 80, pp. 107?117, 2008.

[2] S. Barkmann, G. Sand and S. Engell: Optimisation-Based Design of Reactive Distillation Columns, European Congress of Chemical Engineering 6, pp. 435-436, 2007.

[3] J. R. Jackson, I. E. Grossmann: A disjunctive programming approach for the optimal design of reactive distillation columns, Computers and Chemical Engineering 25, pp. 1661-1673, 2001.

[4] G. Sand, S. Barkmann and S. Engell: Structuring of Reactive Distillation Columns for Non-Ideal Mixtures using MINLP-Techniques, European Symposium on Computer-Aided Chemical Engineering 14 (18), pp. 493-498, 2004.

[5] M. Urselmann, G. Sand, S. Engell: A memetic algorithm for global optimization in chemical process synthesis, Proc. Congress of Evolutionary Computation 2009

[6] J. Stichlmair and Th. Frey: Mixed-Integer Nonlinear Programming Optimization of Reactive Distillation Processes, Ind. Eng. Chem. Res. 40, pp. 5987, 2001.

[7] A. S. Drud: CONOPT: A large-scale GRG code, ORSA Journal on Computing 6, pp. 207?216, 1992.