Numerical Methods for Dynamic Optimization of (Bio)Chemical Processes under Stochastic Parametric Uncertainty

Sustainable design and operation are key requirements for the current chemical process industry. To achieve these, mathematical process models and computer aided process engineering are key tools. Mathematical models are however only an approximation of the process under study and uncertainty is inherently present due to: (i) structural or parametric model uncertainty, (ii) exogenous disturbances or (iii) process variability. Consequently, this uncertainty is also present when model-based optimization techniques are used. Not accounting for this uncertainty can lead to erroneous model predictions. Using these erroneous model predictions potentially results in an overestimation of the actual process performance or unsafe process operation.

The overall goal of this PhD research is to study how stochastic parametric uncertainty can be accounted for in an efficient manner in model-based optimization approaches to enable a more sustainable process operation. The assumption of stochastic parametric uncertainty entails that the uncertainty is modeled in the model parameters, i.e., parametric uncertainty, and it is considered that the parametric uncertainty can be described by a known probability distribution (i.e., stochastic) which is obtained from a previous identification procedure. To achieve this overall goal, three model-based approaches have been studied: (i) model-based experiment design, (ii) model-based optimization/simulation for achieving a better process understanding and (iii) multi-objective optimization and decision making. Four contributions have been made with respect to these three aspects. 

The first two contributions are related to accounting for uncertainty in model-based experiment design (OED). Firstly a sampling-based stochastic OED formulation based on non-intrusive polynomial chaos expansion has been presented. This approach has been compared with a sigma points approach for a benchmark Williams-Otto reactor case study. As a second contribution, the earlier presented formulation has been compared with an approximate robust sensitivities based OED formulation. Two case studies have been implemented: (i) a Lotka Volterra fishing problem in which the focus was on robustness with respect to information content and (ii) a jacketed tubular plug flow reactor in which the emphasis was on robustness with respect to constraint satisfaction. As a conclusion guidelines have been presented on which approach to use for OED under uncertainty.

The third contribution is related to dynamic optimization of biological networks under parametric uncertainty. It has been investigated how three stochastic uncertainty propagation techniques, the linearization, sigma points and polynomial chaos expansion approaches, can be used in the frame of predicting the regulation of metabolic pathways in biological networks under stochastic parametric uncertainty. A critical comparison of these three techniques has been made and two biological network case studies have been investigated: (i) the minimization of intermediate metabolite accumulation in a basic three-step linear pathway model and (ii) the multiobjective optimization (i.e., the minimization) of the final time and enzymatic cost in a glycolysis inspired network model.  The results are discussed from both a mathematical and a physical/biological point of view, showing that each of the studied uncertainty propagation strategies offered a reduction in constraint violations and consistent performance predictions.

The fourth and final contribution of this PhD dissertation addresses the need for efficient algorithms for multi-objective optimization under parametric uncertainty. The computational cost of multi-objective optimization problems increases with the number of Pareto points to be computed and accounting for uncertainty increases this computational cost even further. An algorithm is presented that constructs Pareto ellipsoids from the variance and expected value approximations on the objective functions computed with state-of-the-art stochastic parametric uncertainty propagation techniques: linearization, sigma points and polynomial chaos expansion. These Pareto ellipsoids comprise the actual values the objective functions can take in practice. The trade-off between Pareto points is now extended to a trade-off between Pareto ellipsoids and is used in a divide and conquer strategy for multiobjective optimization. The novel algorithm has been applied to the bi-objective optimization of the batch fermentation of glucose to gluconic acid by Pseudomonas ovalis and the tri-objective optimization of a plug flow reactor in which a state constraint on the reactor temperature has to be satisfied. As a lower number of Pareto points (or better: Pareto ellipsoids) are computed, the computational cost is also reduced when compared with standard NBI (normal boundary intersection, a standard scalarization based multi-objective optimization method that doesn’t account for uncertainty on the Pareto points)  and a divide and conquer algorithm that does not account for uncertainty on the Pareto points.