Optimization of Batch Operating Policies. Part I. Handling Multiple Solutions#

Abstract

introduced a data-driven technique to estimate conditions at which a process should operate (i.e., temperature, pressure, and reactant amountssrecipe) in order to yield a final product with a desired set of quality characteristics. Their proposed technique utilizes empirical latent variable models that are fitted to historical process data from existing process grades. This paper extends the methodology to include estimation of the entire set of time-Varying profiles for the manipulated Variables for batch processes. The problem is formulated in an optimization framework to include both equality and inequality constraints in the objective function. Since often the solution is not unique, the locus of the multiple solutions (defined as the null space) is studied and approaches to selecting the best solution for the final variable settings and trajectories are discussed. Finally, a parallel approach based on a deriVatiVe-augmented model is suggested that avoids considering null spaces to select the final design. An industrial batch digester from the pulp and paper industry is considered throughout the paper to explain and illustrate the key concepts.

Mathematical Tools for the Quantitative Definition of a Design Space

“…The author suggests estimating the hypothetical operating conditions at the manufacturing scale that are necessary to mimic the effect of scaling on quality. This can be accomplished using an optimization approach as discussed in García-Muñoz et al [117,118].…”

Advanced Process Decision Making Using Multivariate Latent Variable Methods

“…Yacoub and MacGregor [45] and García-Muñoz et al [48,49] formulated the model inversion problem defined in the previous section as an optimization problem. With respect to the direct inversion solution shown in Eq.…”

“…Build the LVM model between (properly preprocessed) X and Y, after checking their statistical rank to confirm the presence of any null space [48];…”

“…Γ is a weight diagonal matrix: variables that are more important for the specific applications should be weighted more heavily. García-Muñoz et al [48] suggested the use of the fraction of the total variance explained by each variable as a weighting factor, with a value of 0 to be used for those variables for which equality constraints are not assigned. The second term of Eq.…”

“…For a more thorough discussion on the terms in Eqs. (22)(23)(24), the reader is referred to Yacoub and MacGregor [45], García-Muñoz et al [48,49,51] and Tomba et al [46,52].…”