Efficient design space identification for superstructure optimization subjected to parameter uncertainty

In superstructure optimization of processes and energy systems, the design space is defined as the combination of unit considerations, process conditions and model parameters. Many of the parameters are subjected to uncertainty, such as resource and product prices or impact factors. Investment decisions on industrial energy systems for example are normally based on short-term profitability criteria and assumptions on current and future investment, operating and resource costs (Turton et al., 2008). However, those assumptions are often proven to be wrong retro-perceptively (Moret, 2017). When designing systems and providing assistance to informed decision-making on investments, it is crucial to ensure the robustness of the configurations against this uncertainty, thus to make sure a selected design is attractive despite variations in design space parameters. This might be challenging, since modeling complex systems under uncertainty can be computationally expensive (Beland and Nair, 2016). The aim of this research is to support the efficient generation of a set of meaningful solutions for a decision-maker by applying a Bayesian methodology to predict Pareto-optimal solutions that are robust under parameter uncertainty. Firstly, an initial dataset is generated by running the original optimization model for a set of design space configurations. Design space configurations in this regard contain sets of parameters subjected to uncertainty. The concept of robustness is addressed and quantified by recalculating -a posteriori to optimization- the 95% of the desired objectives for each solution under the assumed parameter distribution. These newly calculated objectives are used to define the robust Pareto-front. A set of design space parameters and the corresponding solution obtained from optimization is evaluated on their performance of the robust objectives regarding Pareto-optimality. For this, the 𝜀-Pal algorithm for Pareto-front identification by means of machine learning presented by Zuluaga et al., 2016 and implemented by Jablonka et al., 2021 is applied. The algorithm is based on an adaptive learning concept which systematically identifies the next best function evaluation to improve the confidence of the Pareto-frontier identification. For making predictions, Gaussian Process Regression models are applied. The algorithm identifies design points with a high probability of being Pareto-optimal regarding the defined objectives, and evaluates them by calling the original model. The confidence of the Pareto-front prediction is increasing, while simultaneously, the relevant design space is reduced, as in each iteration points are either identified as Pareto-optimal or discarded. The described methodology is applied to the efficient design of an integrated industrial biorefinery, where a Kraft pulp mill is enriched with biomass-based process units that convert excess electricity and biogenic residual streams such as bark and black liquor to storable en