Overcoming the modeling bottleneck: A methodology for dynamic gray-box modeling with optimized training data

Abstract

For model-based control and real-time optimization, the existence of sufficiently reliable process models is a prerequisite. The development of such models often is the bottleneck in developing advanced solutions and particularly challenging in cases where the underlying physico-chemical phenomena are not fully understood. When process data is available, machine learning (ML) approaches can be used to build data-based models but the validity of such models is limited to the conditions for which sufficiently rich training data was available. To combine the flexibility of ML-models and the reliability of purely mechanistic models, gray-box models can be applied. In this work we consider gray-box models in which ML-submodels are embedded into differential–algebraic equations that have been derived from first principles. To identify the ML-submodels is challenging as no explicit input/output data for the submodels is available. Therefore, suitable training data must first be estimated from the available data. We propose a systematic methodology for developing gray-box models in a sequence of steps, employing specific regularizations and numerical techniques. We apply the methodology to two case studies: a distillation column with a reaction occurring in the rectifying section, and a fermentation process of a sporulating bacterium. The value of the fermentation model is demonstrated by its use in dynamic process optimization.