(659f) A MINLP Approach to Model-Based Data Mining for the Quick Development of Nonlinear Dynamic Models

Simple and reliable
phenomenological models always represent an attractive and powerful instrument in
several chemical and biological industrial processes. A trustworthy model can
potentially predict the response of a system outside the investigated range of experimental
conditions and can be fruitfully exploited for the purposes of process design
and non-empirical process optimization. Following the seminal work by Box and
Lucas [1] a number of works appeared in the scientific literature regarding the
discrimination among candidate models and the precise identification of the
model parameters through model-based design of experiments (MBDoE) techniques
for model discrimination and parameter precision (PP) in nonlinear dynamic
systems [2,3]. However, the application of these techniques always starts from
the availability of an existing set of candidate models while direct guidelines
for subsequent model improvement are not obvious. A well-established systematic
technique for quick model development and enhancement has not been proposed yet.

A model may be affected by two types of weaknesses: i)
a structural weakness, intrinsically associated to the mathematical structure
of the equations and to model identifiability [4]; ii) a descriptive
weakness (i.e. the model can be weak on representing the system under certain
experimental conditions because of an incorrect or incomplete set of
equations). In this work a new approach is presented to guide the improvement
of the model through the introduction of a method for the automated detection
of the second-type weaknesses. The proposed
method is based on the solution of a MINLP problem whose aim is the
maximization of the likelihood function [5] acting on the following variables: i)
the set of parameters of the candidate model (parameters are treated as
continuous variables); ii) a set of user-defined binary variables. The
user-defined binary variables act like switchers, including or removing
experimental data from the parameter estimation problem, on single or groups of
measurements (e.g. one could be interested in evaluating the model’s capability
in the prediction of certain experimental conditions grouping the measurements
gathered in the same experiment under the same switcher). The method works as a
model-based data mining (MBDM) filter for parameter estimation (PE)
simultaneously removing the experimental results that the model is not able to
fit taking into account the measurement uncertainty. It allows for a quick,
automated mapping in the space of experimental conditions in terms of good and bad
model predictive capabilities, preventing the thoughtless use of fake optimal
process points located in regions where the model is not reliable. The
presented MBDM-PE technique is proposed as part of a wider framework for a systematic
approach to model identification which is synthetically shown in Figure 1a. The
procedure starts with a candidate model and preliminary statistical information
on measurement errors. The distribution of the residuals associated to the
measurements removed by the MBDM-PE filter identifies the main weaknesses of
the proposed model, highlighting both critical experimental conditions and
specific measured variables whose associated prediction is very poor. This
provides a feedback to the gradual improvement of the model itself. Once the
enhanced model is known to give a satisfactory description of the phenomenon in
the investigated experimental conditions, statistically unsatisfactory
parameter estimates are amended performing additional experiments designed
through known MBDoE methodologies for improving the parameter precision [1,2,3].

The MBDM-PE filtering
technique has been successfully applied in a case study on the identification
of a simplified reaction mechanism for the partial oxidation of methanol to
formaldehyde over a silver catalyst in micro-reactor devices [6]. The
investigation has been carried out adopting the optimization tool implemented
in gPROMS Model Builder. The filter has automatically identified the poor predictive
capabilities offered by a candidate kinetic model in the low temperature region
of the experimental design space as shown in Figure 1b where the filtered (removed)
experimental data have been highlighted with black circles. The underestimation
of the conversion of methanol and the overestimation of selectivity for the
formaldehyde indicate a parallel reaction of complete oxidation for methanol,
non-negligible at low temperatures.  

Future works will focus on
the development of meaningful methods for assessing the model reliability in
non-investigated intermediate experimental conditions and systematic approaches
for model building, given the model structure, for the quick, automated
identification of phenomenological models.

Figure 1. (a) Systematic approach for model identification
implementing the MBDM-PE filtering technique; (b) Results given by the MBDM
filter applied to a simplified kinetic model for partial methanol oxidation.
Solid lines represent model predictions; scattered triangles and squares
represent measurements collected in four experiments at different temperature. The
experiment switchers in the upper part of the graph represent the final values evaluated
by the solver for the user-defined binary variables indicating which
experiments have been included in the PE problem (1) and which have been removed
(0). Removed experimental data are also highlighted with black circles. The
grey-coloured side in the plot indicates the region of low model reliability.

References

[1]           G.E.P.
Box, H.L. Lucas (1959). Design of experiments in non-linear situations, Biometrika,
46, 77-90.  

[2]           D. Espie,
S. Macchietto (1989). The optimal design of dynamic experiments. AIChE
Journal , 35(2), 223-229.

[3]           F. Galvanin, S. Macchietto and F. Bezzo (2007). Model-based design of parallel experiments.
Ind. Eng. Chem. Res., 46, 871-882.

[4]           F. Galvanin, C.C. Ballan, M.
Barolo and F. Bezzo (2013). A general model-based design of experiments
approach to achieve practical identifiability of pharmacokinetic and
pharmacodynamic models. J. Pharmacokinet. Pharmacodyn., 40, 451-467.

[5]           Yonathan Bard (1974), Nonlinear parameter
estimation, ACADEMIC PRESS, New York and London.

[6]           F. Galvanin, E. Cao, N. Al-Rifai, V. Dua and A. Gavriilidis
(2015). Optimal
design of experiments for the identification of kinetic models of methanol
oxidation over silver catalyst. Chemistry Today, 33(3), 51-56.