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

Jaeckle and MacGregor (AIChE J. 1998, 44, 1105−1118) introduced a data-driven technique to estimate conditions at which a process should operate (i.e., temperature, pressure, and reactant amountsrecipe) 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.