Mixed-integer Dynamic Optimization Problem Research Articles

Mixed‐logic dynamic optimization applied to batch distillation process design

AbstractAn approach for the optimal configuration and sequencing of complex batch‐distillation processes is presented. The proposed method is based on formulating a mixed‐logic dynamic optimization problem in which a performance index is minimized subject to a disjunctive dynamic model and constraints. The disjunctive model comprises a given set of equipment modules, such as stacks of trays of different size or feed and product vessels, available for the distillation process under consideration along with logical relationships provided by the design context. The optimization based on this model may yield unconventional design and operational strategies, and thus improve the economic efficiency to a significant extent. The logic‐based solution technique proposed in this work offers some favorable properties when compared to established mixed‐integer dynamic optimization problem solution techniques reported in the literature. The design of a multistage batch‐distillation process serves as an example to illustrate the proposed modeling and solution method.

Determining the Optimal Mode Sequence

The global optimization of linear time varying hybrid systems in the continuous time domain is examined. A method is developed for determining the optimal mode sequence when the transition times are fixed. A reformulation of the problem by introducing binary decision variables while retaining the linearity of the underlying dynamic system is proposed. The linear structure of the resulting mixed-integer dynamic optimization problem is utilized to construct a convex relaxation, which enables the global solution to be found in a finite number of iterations using nonconvex outer approximation

New algorithms for mixed-integer dynamic optimization

Mixed-integer dynamic optimization (MIDO) problems arise in chemical engineering whenever discrete and continuous decisions are to be made for a system described by a transient model. Areas of application include integrated design and control, synthesis of reactor networks, reduction of kinetic mechanisms and optimization of hybrid systems. This article presents new formulations and algorithms for solving MIDO problems. The algorithms are based on decomposition into primal, dynamic optimization and master, mixed-integer linear programming sub-problems. They do not depend on the use of a particular primal dynamic optimization method and they do not require the solution of an intermediate adjoint problem for constructing the master problem, even when the integer variables appear explicitly in the differential–algebraic equation system. The practical potential of the algorithms is demonstrated with two distillation design and control optimization examples.

A Case Study in Simultaneous Design and Control Using Rigorous, Mixed-Integer Dynamic Optimization Models

This article presents an in-depth case study that demonstrates (i) how the process and control design, involving both discrete and continuous decisions, can be simultaneously optimized for systems described by realistic dynamic models and that (ii) mixed-integer dynamic optimization problems, involving thousands of differential−algebraic equations, can now be solved using state-of-the-art algorithms and technology. The study in question involves a distillation column that is subject to time-varying disturbances and time-invariant uncertainty. A new, multicomponent, mixed-integer dynamic distillation model is developed and used to simultaneously determine the number of trays, feed tray location, column diameter, reboiler and condenser surface areas, 5 × 5 control structure, PI-gains, reset times, and set-points, to obtain an economically optimal system that is also guaranteed to give feasible dynamic operation over all uncertainty scenarios.

Design of integrated batch processes with discrete and continuous equipment sizes

This paper presents a mathematical programming approach for the integrated design of batch processes involving multiple processing steps. Detailed dynamic models are used to describe the behaviour of individual batch operations. The mathematical formulation regards both the design ( e.g. equipment items with discrete sizes) and operational characteristics ( i.e. control variable profiles such as reflux ratio, cooling water flowrate) of each task as degrees-of-freedom, the values of which are selected to optimise a particular objective function ( e.g. profit). The solution may be restricted by inequality constraints which hold throughout the task ( e.g. reactor temperature violation) or at the end of the task ( e.g. end-product purity specification). The sizes of the equipment items are treated as discrete decisions and are chosen from a selection of standard sizes. The mathematical formulation leads to a mixed integer dynamic optimisation problem and is solved using the outer approximation/augmented penalty (OA/AP) algorithm.

Mixed-integer dynamic optimization I: problem formulation

A rigorous decomposition approach is presented for mixed-integer dynamic optimization problems. The approach combines dynamic optimization with insight based targeting techniques to decompose the optimization into subproblems providing rigorous upper and lower bounds on the objective. This approach has the potential to eliminate total enumeration of the discrete space, assures termination in a finite number of iterations and yields a rigorous bound on the distance between the solution found and the global solution.

On the design of multicomponent batch distillation columns

This paper presents a mathematical programming approach to the optimal design of batch distillation columns. Both capital and operating cost contributions are taken into account. The column design parameters and operating policy are determined simultaneously. Treating the number of trays in the column as a decision parameter necessitates the introduction of a superstructure that encompasses all possible tray numbers between a lower and an upper bound. The mathematical formulation leads to a mixed integer dynamic optimisation problem which is solved using an outer approximation/augmented penalty algorithm comprising a continuous subproblem involving the optimisation of a column with a fixed number of trays and a master problem involving the solution of a small mixed integer linear programming (MILP) problem that determines the number of trays to be considered at the next iteration of the algorithm. The methodology presented is illustrated by its application to the design of a plant involving a sequence of two batch distillation columns.

Dynamic Optimization in a Discontinuous World

Many engineering tasks can be formulated as dynamic optimization or open-loop optimal control problems, where we search a priori for the input profiles to a dynamic system that optimize a given performance measure over a certain time period. Further, many systems of interest in the chemical processing industries experience significant discontinuities during transients of interest in process design and operation. This paper discusses three classes of dynamic optimization problems with discontinuities: path-constrained problems, hybrid discrete/continuous problems, and mixed-integer dynamic optimization problems. In particular, progress toward a general numerical technology for the solution of large-scale discontinuous dynamic optimization problems is discussed.

Mixed-integer dynamic optimization

A rigorous decomposition approach is presented for mixed-integer dynamic optimization problems. The approach combines dynamic optimization with insight based targeting techniques to decompose the optimization into subproblems providing rigorous upper and lower bounds on the objective. This approach has the potential to eliminate total enumeration of the discrete space, assures termination in a finite number of iterations, and yields a rigorous bound on the distance between the solution found and the global solution.