Managing Large-Scale Uncertainty In Process Flow Sheet Synthesis and Design

Introduction

Design and synthesis of processes in chemical, biochemical and pharmaceutical industry involve a great deal of uncertainty in the external, measured and model data. Including the uncertainty into design approaches brings the flexibility to process flow sheets, however, it requires transformation of deterministic mathematical formulation into the stochastic problem, and further, into its deterministic equivalent. The latter represents much larger mathematical model than the deterministic, therefore, the computational effort to find its optimal solution could be extremely high or even impossible. This is particularly problematic in large-scale process systems with several tens or even hundreds of uncertain parameters. Various approaches have been proposed for reducing the number of scenarios in stochastic programming, e.g. Karuppiah et al. 2010. Authors of this contribution have also proposed a few approaches for MINLP synthesis of flexible processes flow sheets (Novak Pintarič and Kravanja, 2004). The emphasize of this contribution is on the approaches for solving flow sheet problems with large number of uncertain parameters (up to 100) that have not yet been managed satisfactorily.

Outline of the approach for managing large-scale uncertainty

The proposed approach is based on a two-stage stochastic programming formulation with fixed recourse. Its main feature is the approximation of stochastic result by using a small set of discrete points while ensuring flexibility of flow sheet design within the pre-specified variations of uncertain parameters. The main effort is on the transformation of the flow sheet’s stochastic formulation into its minimum deterministic equivalent. The latter is defined over a considerably reduced set of scenarios (points) which are determined in advance either for fixed process topology in the case of process design, or for topological alternatives in process synthesis by using a mixed integer nonlinear programming (MINLP) approach. The reduced set of scenarios is composed of the nominal point and the minimum set of vertex points, called critical vertices. The nominal point could be a mean or a mode value for skewed distributions. The critical vertices define the values of the first-stage (design) variables in the flow sheet model, while the expected cost of the second-stage (operating) variables is approximated at the nominal point.

Identification of critical vertices

Identification of critical vertices is the most difficult step of the proposed approach. First, a sensitivity analysis of the influence of uncertain parameters on design variables is carried out. The parameters are classified into three groups based on the results of the sensitivity analysis: a) parameters with no influence, b) parameters with monotonic influence, and c) parameters with non-monotonic influence on design variables. The non-influential parameters could be fixed at any arbitrary value in the deterministic equivalent model. Those parameters with monotonic influence are fixed to the worst-case values, i.e. either the lower or upper bound.

Determination of critical vertices is performed only for those parameters that show non-monotonic influence during the sensitivity analysis. In previous work, authors have proposed several approaches for identification of critical points ranging from the exact formulation based on the Karush-Kuhn-Tucker conditions to the heuristic ones (Novak Pintaric and Kravanja, 2008). The common idea of all these methods was to identify those critical values of uncertain parameters that determine the largest values of first stage-variables at minimum cost. This requires the solution of min-max problem which could not be solved straightforwardly.

While not all of the proposed methods are suitable to manage large-scale processes with large number of uncertain parameters, in recent research, two methods for finding the critical vertices have been further investigating: approximate one-level method (Novak Pintaric and Kravanja, 2008), and a novel one, a two-stage method using the external functions in GAMS.

a) Approximate one-level method

In this method, a simple deterministic model is solved for each first-stage variable. Both objectives, minimization of the cost function and maximization of the design variable, are accomplished simultaneously in this model, while uncertain parameters are treated as optimization variables. Approximate min-max solution is obtained by a composite objective function, which consists of the minimum cost function from which the design variable is subtracted multiplied by a large positive constant (big-M). The results of the model are the critical values of uncertain parameters that maximize design variable under study. The maximum number of critical points would be as high as the number of the first-stage variables, but is usually considerably lower. The identified vertices can be further reduced by using a set-covering procedure. A drawback of one-level method is its sensitiveness to the value of big-M constant which could influence the selection of critical vertices. This challenge is studied further in this research.

b) Two-level method with external functions

This method starts with the solution of the deterministic problem at a single, i.e. the nominal point. As the solution obtained is inflexible, the objective value represents a lower bound for minimization problem. In the next iterations, critical vertices are added to the model one by one, and dimensionality of the problem gradually grows from iteration to iteration. The additional critical point at every major iteration is calculated simultaneously as the optimization variable by using the external functions in GAMS program, which enable to solve min-max problem iteratively. Adding the critical points into the model forces the objective value to rise. The complete minimum set of critical vertices is found when the objective value stops rising. The case-studied examples indicate fast convergence of this method and precise identification of critical vertices, however, a termination criterion still needs to be improved. The advantage of this method is also that the iteration at which all critical vertices are identified represents a solution of deterministic equivalent problem whose result is the approximate flexible flow sheet design.

 

Conclusions

This contribution presents an approach for approximating a stochastic model of process flow sheet under uncertainty by a minimum deterministic equivalent. The main goal was to improve and upgrade those approaches developed by authors in the past in order to be able of managing industrial flow sheet problems with high level of uncertainty.

Minimum deterministic equivalent of stochastic flow sheet model is defined over significantly reduced set of scenarios composed of the nominal point and critical vertices. Prescreening approach was introduced in order to identify those uncertain parameters that influence design variables. Two methods are proposed for efficient and quick identification of critical vertices in large-scale problems with large number of uncertain parameters. Some illustrating examples will be presented in order to illustrate the main characteristics of the proposed approach, as well as a synthesis of large-scale bio-ethanol process via MINLP. By managing large-scale uncertainty in process flow sheet synthesis and design, mathematical programming evolves towards the efficient supporting tool for investment decision-making in process industries.

References

Karuppiah R., Martín M., Grossmann I. E.  A simple heuristic for reducing the number of scenarios in two-stage stochastic programming. Comput. Chem. Eng. 2010, 34, 1246–1255.

Novak Pintarič Z., Kravanja Z. A strategy for MINLP synthesis of flexible and operable processes.

Comput. Chem. Eng.  2004, 28, 1105–1119.

Novak Pintarič Z., Kravanja Z. Identification of critical points for the design and synthesis of flexible processes. Comput. Chem. Eng. 2008, 32, 1603–1624.