Optimization of Batch Operating Policies. Part II. Incorporating Process Constraints and Industrial Applications Salvador Garcı ´a-Mun ˜ oz, †,‡ John F. MacGregor,* ,† Debashis Neogi, § Bruce E. Latshaw, § and Sanjay Mehta § Department of Chemical Engineering, McMaster UniVersity, 1280 Main Street West, Hamilton, ON, L8S 4L7 Canada, and Air Products and Chemicals, 7201 Hamilton BouleVard, Allentown, PennsylVania 18195 In the first part of this series [Ind. Eng. Chem. Res. 2006, 45, 7856-7866], data-driven approaches, based on partial least squares (PLS) models built from historical batch data, were used to find optimal batch operating trajectories that would yield a desired vector of final product quality attributes. The method allowed for the inclusion of univariate and multivariate constraints on the set of desired final product quality attributes and presented approaches for handling multiple solutions. In this paper, the technology is further extended to include constraints in the process operating trajectories themselves. The methodology is successfully applied to an industrial batch polymerization process where the batch trajectories are designed to achieve specific properties of the final polymer while consuming the minimal amount of time for the batch run. 1. Introduction Batch processes play a key role in the manufacturing of high value added chemicals such as pharmaceuticals and polymer resins. Process systems engineering (PSE) has played a crucial role in developing techniques to improve the operation for batch units. At the design phase where the geometry and nominal operation is determined, fundamental models are traditionally used to aid the design exercise. Once a class of products is in production, any improvements or optimization of the product quality or process operation can use empirical models built from the accumulated database. Multivariate latent variable models (LVM) have proven to be efficient at modeling batch process operation 2 and the sources of uncertainty that enter the system. 1.1. Background. The redesign of process operating condi- tions to achieve a new set of final product quality attributes, based on partial least squares (PLS) models built from historical data, was first discussed by Jaeckle and MacGregor. 3,4 In those works the objective was to use historical databases on process operation over a range of product grades to find new operating conditions that might achieve a desired set of new product properties. This was referred to as a product design problem. The solution was provided by an inversion of the linear or nonlinear PLS models. The first paper in this series 1 extended the approach to the estimation of complete batch trajectories and considered optimization based approaches to handle multiple solutions and constraints in the final product properties. An alternative approach, proposed to introduce more information about the derivatives of the trajectories into the multiway model, was shown to provide smoother solutions and eliminate the multiplicity of solutions. 1.2. Contributions of this Work. The way that the variable trajectories were estimated in the first paper 1 does not allow for the introduction of constraints in the process variable trajectories into the optimal design criterion. The focus of this second paper is to reformulate the optimization to include additional objectives and constraints on the batch operation into the design, and to present results on an industrial application of the method. The industrial problem involves the design of new operating trajectories and initial conditions (recipes) for a batch polymerization process to achieve a desired set of end product properties while using the shortest possible batch duration. The development of several new polymer grades is considered and some of the solutions are implemented in an industrial scale pilot plant. 2. PLS Modeling of Batch Processes This paper follows the same notation as in past work 1–7 on multivariate modeling of batch processes. A data set from a batch process consists of I batches, and measurements on J variables taken throughout the duration of the batch (variable trajectories). Due to variations in the time duration of the batches, there may be different numbers of observations throughout each batch. However, after alignment of the batch data using a maturity or evolution index variable 6 (or other approach 5 ) one is left with a fixed number of observations at K values of the maturity variable throughout each batch. After unfolding the three-way batch data array X as proposed by Nomikos and MacGregor, 6,7 the two-way matrix of variable * To whom correspondce should be addressed. Tel.: (905) 525-9140 ext 24951. Fax: (905) 521-1350. E-mail: macgreg@mcmaster.ca. † McMaster University. ‡ Current address: Pfizer Global Research and Development, 445 Eastern Point Road, Groton, CT 06340. E-mail: Salvador.Garcia- Munoz@pfizer.com. § Air Products and Chemicals. Figure 1. Structure of unfolded and aligned batch data. Figure 2. Data structure with PLS score matrix T and new vectors corresponding to the desired trajectory optimization solution. Ind. Eng. Chem. Res. 2008, 47, 4202–4208 4202 10.1021/ie071437j CCC: $40.75  2008 American Chemical Society Published on Web 05/15/2008