(372b) Model-Based Identification of Process Operational Design Space Under Uncertainty

Mathematical modelling plays a crucial role in the design and operation of process systems with ensuing models encompassing numerous physical, chemical, biological, economic, etc. phenomena in multi-scale settings and involve making optimal decisions regarding system configuration and its optimal model-predictive control. Importantly, most process models are subject to a number of constraints originating from physical limitations, quality requirements, economic feasibility or regulatory compliance to ensure that both the product and, in many cases, the process itself, satisfy certain Critical Quality Attributes and Key Performance Indicators. The set of process operating parameters that leads to satisfactory performance and product quality is called process Design Space, and its explicit description is an important requirement in many industries, e.g. in (bio)pharma.

Most process models are complex and large in scale or exist only as computer codes which precludes a facile delineation of relationships between individual inputs and outputs. Such relationships are thus studied and visualised through repetitive model evaluations resulting in simplified representations such as scatter plots or response surfaces. However, the exploratory power of such approaches is limited, and they provide only snapshots of the model’s behaviour drawn out of a multidimensional space of parameters. In addition to the problem of scale and structural complexity, all models are subject to uncertainty. While epistemic model uncertainty may be difficult to identify and even more difficult to quantify, aleatory uncertainty (or variability attributed to aleatory uncertainty) in model inputs can in many cases be estimated. These two factors make the identification and explicit description of a process DS a very challenging task.

In this work we propose a systematic methodology for process model analysis under uncertainty which leads to model reduction and explicit identification of process operational DS as a system of linear inequalities involving process parameters.

One of the most widely adopted methods of model analysis under uncertainty is Global Sensitivity Analysis (GSA), which determines how much the variability (including uncertainties) in individual model inputs contributes to the variability in model outputs (such as KPIs and CQAs) thus revealing the uncertainty propagation properties of the model [1,2]. We recently proposed an extension of the GSA methodology to cover a wide range of problems involving inequality constraints (hence named ‘constrained GSA’ or cGSA) imposed on model inputs [3,4]. cGSA is further developed in this work using adaptive sampling of the parameter space and metamodeling. The sensitivity indices obtained through cGSA can be used to identify key parameters whose uncertainty affects outputs to the largest extent and those having negligible effect on the uncertainty of the outputs. Typically, a small fraction of process parameters account for the majority of output variance, which allows the dimensionality of model parameter space to be reduced to a subspace of the most influential inputs while fixing the values of nonessential inputs [5].

An explicit description of the operational DS of the process is then constructed in the form of polytopes (either convex hulls or an alpha-shapes) over feasible sampled points, so that the DS is represented by systems of linear inequalities.

A reduction of parameter space based on the results of cGSA has several advantages for obtaining an explicit description of the DS. First, it allows a significant reduction of the associated computational burden since a metamodel constructed as part of cGSA can be used in place of full process model. Second, since convex hull and alpha-shape descriptions suffer from the curse of dimensionality, a reduction of space dimensionality reduces the corresponding representations. To further reduce the number of linear inequalities in an explicit description of the DS we present an optimisation-based constructive approach in which linear constraints are sequentially added to the system to maximise the volume of the resulting polytope representing the DS.

Since the methodology presented in this work treats the model as a black box, it is applicable to processes encountered in any branch of chemical and process engineering.

References:

[1] Sobol, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 2001. 55(1-3): p. 271-280.

[2] Kucherenko, S., Tarantola, S., Annoni, P. Estimation of global sensitivity indices for models with dependent variables. Computer Physics Communications, 2012. 183(4): p. 937-946.

[3] Kucherenko, S., Klymenko, O.V., Shah, N. Sobol' indices for problems defined in non-rectangular domains. Reliability Engineering & System Safety, 2017. 167: p. 218-231.

[4] Klymenko, O.V., S. Kucherenko, and N. Shah, Constrained Global Sensitivity Analysis: Sobol' indices for problems in non-rectangular domains. 27th European Symposium on Computer Aided Process Engineering, Pt A, 2017. 40a: p. 151- 156.

[5] Kotidis, P., Demis, P., Goey, C.H., Correa, E., McIntosh, C., Trepekli, S., Shah, N., Klymenko, O.V., Kontoravdi, C. Constrained global sensitivity analysis for bioprocess design space identification, Comput. Chem. Eng., 2019, in press.