(112f) An Exact Reformulation of the Superstructure Optimization Problem

Authors: 
Gopinath, S., Imperial College London
Galindo, A., Imperial College London
Jackson, G., Imperial College London
Adjiman, C. S., Imperial College London
Superstructure optimization or chemical process synthesis involves the selection of the optimal process topology- which is defined by the choice of process units and interconnection of these process units- and the optimal process variables in this topology. The term process unit here refers to a process equipment (such as a distillation column) or a discrete section of the equipment (such as a tray in a distillation column). The superstructure optimization problem is a mixed integer nonlinear programming problem (MINLP).

While MINLPs are NP-hard, the superstructure problem presents some additional unique challenges. The representation and optimization of a process superstructure is not a trivial task. The superstructure proposed by Smith and Pantelides [1] considers possibly full connectivity between all units using mixers and splitters at the inputs and outputs, respectively, of each process equipment in the superstructure. Each process unit is represented by rigorous process models. One of the problems when rigorous process models are used within superstructure optimization, is the fate of a process unit that is not selected during the course of optimization. It is only logical that there is no flow of material in and out of an absent process equipment. However, values of constraint functions of the process unit or their derivatives may become numerically singular or undefined (not lie within the set of real numbers) when the flows are zero. Further, the Jacobian of these constraints may also become rank-deficient [2].

The problem of numerical singularities has been addressed exactly using generalized disjunctive programing (GDP) and the logic based outer approximation algorithm (LBOA) [3]. However, simulation-based superstructure optimization using GDP such as in [4], can lead to an expensive LBOA master problem initialization step. Inexact methods to circumvent singularities include introducing small non-zero flows or the use of short-cut models.

Recently, Gopinath (2018) [5], proposed an exact and simple reformulation of the mixers and splitters that appear in the superstructure, that is amenable to be used both within simulation-based and equation-oriented optimization paradigms. In the proposed approach a pseudo flow-rate is added to a mixer at the inlet of each absent unit so that the flow-rate into all units is strictly positive. Splitters as well are reformulated. The output from an equipment that is present goes to the corresponding splitter without any modification. However, if the equipment is not selected the mass flow at the inlet of the corresponding splitter is driven to zero. Such logical “if-else” relations are implemented using “Big-M” type constraints. In the new formulation zero-flows in process units do not occur, thus the associated issue of singularities is absent.

As a proof of concept, we demonstrate the use of this reformulation for the synthesis of a distillation column. The number of stages in the column and the optimal column degrees of freedom are to be determined for the separation of a binary mixture of n-pentane and n-heptane. The SAFT-γ Mie [6] equation of state is used to predict thermodynamic properties of the various substances. A simulation-based optimization framework is employed. The superstructure MINLP is solved using LBOA. The LBOA master problem may be initialized in a single step by considering all stages in the column to be present (just as in other equation-oriented GDP studies) unlike in the simulation-based GDP approach in [4].

Thus, we demonstrate an exact reformulation of the design problem. The approach allows for the use of rigorous process unit and thermodynamic models in process synthesis, without requiring their reformulation to circumvent singularities.

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  2. Dowling, A. W. (2015). An Equation-based Framework for Large-Scale Flowsheet Optimization and Applications for Oxycombustion Power System Design. Carnegie Mellon University.
  3. Turkay, M, Grossmann, IE (1996). Computers & Chemical Engineering, 20, 959.
  4. Caballero, J. (2015). Logic hybrid simulation-optimization algorithm for distillation design. Computers & Chemical Engineering, 72, 284.
  5. Gopinath, S. (2018). Molecular design, process design and process synthesis of separation systems. Imperial College London.
  6. Papaioannou, V., Lafitte, T., Aveñdano, C., Adjiman, C. S., Jackson, G., Müller, E. A., and Galindo, A. (2014). The Journal of Chemical Physics, 140, 054107.