In this work, we propose a systematic bi-level approach for process optimization under uncertainty that aims to significantly reduce the number of simulations required by the stochastic optimization algorithm after the sampling procedure. The first level of the proposed method implements a data mining technique in the form of clustering on the samples obtained from the probability distribution of the uncertain parameter. The implementation of clustering in the obtained samples leads to the generation of distinct and coherent groups of similar points representing the uncertain parameter values. A cluster center is then defined for each cluster based on statistical criteria and this central point approximately represents the contents of the cluster. At this stage the central point of each cluster is introduced to process simulation and an objective function value is calculated for each cluster center. The second level of the proposed approach involves using the available cluster center points in conjunction with their corresponding objective function values to fit a continuous model that represents the objective function value as a function of the cluster centers. At this point the objective function values of the remaining sample points are calculated based on the developed model, hence avoiding the time consuming simulations. The fitted model provides objective function value predictions that are either identical or lie within very close proximity to the values calculated through simulations. The fitting of the developed model is facilitated by the beneficial cluster-inherent statistical properties as well as from efficient sampling techniques, such as Hammersley sampling, that allow uniform multi-dimensional representation of the uncertain parameter distribution. The proposed approach enables the use of constantly large uncertain parameter samples regardless of the size of the optimization problem addressed or the stage of the performed optimization search. The number of generated clusters is decided by the utilized clustering algorithm based on statistical criteria, is maintained at low, yet sufficient, sample representation levels and presents very limited variations throughout the optimization search. The implementation of clustering and model fitting are fast and computationally insignificant compared to the numerous, time consuming simulations required.
The proposed approach is illustrated through case studies of industrial interest on the design of process flowsheets involving crystallization reactors followed by separation of the produced crystals. A number of uncertainties in raw material and model related data are accounted for in order to maintain a realistic representation of the industrial reality in the utilized process models. Such uncertainties are simultaneously addressed in the synthesis of simple as well as complex process schemes accounting for multiple recycle and bypass streams, varying reaction and separation topologies as well as varying processing conditions. The employed process synthesis framework under uncertainty allows systematic process flowsheet representation, using generic process modules regardless of the processing task addressed, while facilitating comprehensive interconnectivity options within the flowsheet, based on a synthesis framework presented by Papadopoulos and Linke (2004) and Linke and Kokossis (2003). Results indicate the development of robust solutions and significant time savings in computations as compared to solving the same problems without the use of the proposed approach, while merits and shortcomings of the proposed approach are also discussed.
Kim, K. J., Diwekar, U (2002). Hammersley stochastic annealing: Efficiency improvement for combinatorial optimization under uncertainty. IIE Transactions, 34, 761-777.
Painton, L., Diwekar, U. (1995). Stochastic annealing for synthesis under uncertainty. European J. of Operational Research, 83, 489-502.
Papadopoulos,A.I, Linke,P.(2004). On the synthesis and optimization of liquid-liquid extraction processes using stochastic search methods, Comp. Chem. Eng., 28, 2391-2406.
Linke,P.,Kokossis,A.C.(2003) Attainable reaction and separation processes from a superstructure-based method, AIChE J., 49, 1451-1470.