A protocol for the estimation of parameters in process models

Abstract

As our understanding of chemical processes increases the models created to describe them also increase in complexity. These models usually consist of sets of differential equations, containing multiple response variables which are a function of multiple input or design variables and a potentially large number of parameters. In most cases the equations are nonlinear in the inputs and parameters, and must be solved by numerical integration. An example of such a model is the Wat poly polymerization model (Gao and Penlidis, 1996 and 1998) developed in the polymer research group at the University of Waterloo. To be able to use these models effectively the parameters that they contain have to be known.

The bulk of the literature dealing with parameter estimation has only considered small models. At present in estimating parameters for large process models, there are two shortcomings in the existing knowledge about parameter estimation. The first is, how effective is the present parameter estimation methodology when applied to large models, and the second is, can any advantage be gained from considering the parameter estimation problem as a whole. This work will try to address this limitation, by revisiting the parameter estimation process and developing a protocol for the estimation or updating of the parameters within process models.

The projected use of the parameter estimation protocol is as part of a model based experimentation program. Therefore it considers actual experimental conditions, where the number of experiments that can be carried out is limited due to the expense of performing experiments and analysis.

In the development of a parameter estimation protocol all of the steps of the parameter estimation will be revisited. The parameter estimation steps are: parameter sensitivity analysis, statistical design of experiments, estimation of parameters and confidence regions. Where these four steps correspond to answering the following questions; 1. Is it possible to estimate the parameters with the chosen responses and which response or responses will provide the most information? 2. At what conditions (i.e. temperature, conversion, initial feed composition) should the data in the experiment be collected? 3. What is the best method to estimate the parameters with the data that was collected? 4. How good are the parameters that were estimated?

Of the parameter sensitivity analysis methods available it was found that the best method to present sensitivity information is a plot of the gradient values of the responses with respect to the parameters. The gradients are normalized and plotted as a function of independent variables such as initial feed composition, time or conversion. This is performed so that responses of different magnitudes and different measurement errors can be compared to each other.

In designing experiments the D-optimality criterion is generally used. One of the implementation challenges in designing experiments is local optima. One possible method found to deal with this difficulty, is to provide the optimization algorithm with a good initial guess that is based on the information provided by the gradient plots.

To estimate the parameters with multiple responses the Determinant criterion is used. When estimating multiple parameters in a large model a large number of local optima can exist. To overcome this difficulty, different approaches are available, such as a robust optimization algorithm ( e.g. simulated annealing) or the use of multiple starting points.

Confidence regions of the parameter estimates will provide a measure of the quality of the parameter estimates. The true shape approximate level confidence regions were found to be an adequate compromise between information provided and computation required. It was found that the true shape joint confidence regions can be incorrect if multiple responses are used and the sample size is small.

The parameter estimation protocol is a series of actions or steps that can be followed in the course of obtaining parameter estimates. By following these actions the overall parameter estimation procedure can be more efficient and some pitfalls such as local optima and incorrect confidence regions: may be dealt with in an appropriate manner. To illustrate the application of the protocol, three case studies are presented. These case studies illustrate some of the problems that may be encountered in the parameter estimation process and how the proposed protocol can aid in overcoming them.