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457352 Parallel Solution of Parameter Estimation Problem for Polymer Models Using Multiple Grade Transition Curves

**Parallel solution of parameter estimation problem for polymer models using multiple grade transition curves**

Francisco Trespalacios and Thomas Badgwell

ExxonMobil Research & Engineering Company

The dynamic modelling of grade transitions in polymer production typically requires the solution of a large scale differential-algebraic equation (DAE) system. These models involve many parameters of different types, including kinetic constants, material properties, diffusion coefficients, etc. Not only can these parameters differ from laboratory estimates, they can also change with time. For this reason, efficient parameter estimation is essential if such models are to be used in industrial polymerization applications.

One of the approaches for the parameter estimation problem is to solve a least-squares problem. In this problem, the objective is to find the value of the desired parameters that minimizes the difference between the predicted output and real data (or its variant errors-in-variables (EVM) problem, in which both the input and the output are considered as variables). With this approach, solving the parameter estimation problem requires the solution of an optimization problem with DAE constraints. For example, if one data set is available for a transition between two products, the parameter estimation problem solves a least-squares problem subject to the DAE system that defines that grade transition. If there are multiple data sets of multiple transitions, and even multiple models, the DAE system that constraints the optimization problem quickly grows in size.

In the parameter estimation problem, each scenario (which represents a data set and its corresponding DAE system) has a set of local variables and parameters. The only links between different scenarios are the parameters being estimated. Therefore, while the parameter estimation problem can increase to intractable sizes with many scenarios, the problem is sparse. In order to solve such problems, it is necessary to exploit the problem structure.

Two main approaches are typically used to solve this type of problem: the sequential approach and the simultaneous approach. In the sequential approach an optimizer updates the parameters and passes them to a DAE solver, solving each scenario independently. In the simultaneous approach, the parameter optimization and solution of DAE systems is solved together. This is achieved by a full discretization of the differential equations, yielding a large scale NLP problem. Developing solution frameworks for the sequential approach is relatively simple, but it requires solving each DAE system multiple times. The second approach requires the solution of a much larger but sparse problem. The work by Zavala et al. [1] shows the advantages for the simultaneous approach, especially for large scale systems.

In this work, motivated by a real application for Unipol polymerization reactors, we extend the approach suggested by Zavala et al. [1] in two ways. First, we consider a heterogeneous structure in which we allow different reactor models for each sub-problem. Second, we consider much larger problems with several data sets and multiple polymer grade transitions. We solve the problem with multiple cores (one for each scenario), using PIPS-NLP [2]. PIPS-NLP allows the modeler to specify first stage (global parameters) and second stage (local) variables. The solver parallelizes the linear algebra, allowing considerable speed-ups in the large scale dynamic optimization problem.

The application we present in this work is copolymerization of propylene and ethylene, using a bubbling fluidize bed reactor [3,4]. We present the performance of the parallelized solution method for the parameter estimation of this problem. The results show that the problem scales well when using multiple processors. The results also show that it is possible to solve the problem in reasonable time, even with multiple data sets of multiple transitions.

[1]. Zavala, Victor M., Carl D. Laird, and Lorenz T. Biegler. "Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems." Chemical Engineering Science 63.19 (2008): 4834-4845.

[2]. Chiang, Naiyuan, Cosmin G. Petra, and Victor M. Zavala. "Structured nonconvex optimization of large-scale energy systems using PIPS-NLP." Power Systems Computation Conference (PSCC), 2014. IEEE, 2014.

[3]. Choi, Kyu-Yong, and W. Harmon Ray. "The dynamic behaviour of fluidized bed reactors for solid catalysed gas phase olefin polymerization." Chemical Engineering Science 40.12 (1985): 2261-2279.

[4] McAuley, K. B., and J. F. MacGregor. "Optimal grade transitions in a gas phase polyethylene reactor." AIChE Journal 38.10 (1992): 1564-1576.

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