(59d) Hybrid Bayesian-Based Surrogate Optimization for Sustainable Process Design within Planetary Boundaries

The design of sustainable processes is attracting increasing interest due to the need to meet the Sustainable Development Goals and climate targets on time. Given a set of unit operations to transform raw materials into the desired products, the goal is to find their optimal combination and operating conditions in order to maximize sustainability performance while meeting the product demand and a set of technical constraints.

Among the different approaches, the preferred one is to use mathematical modeling where the design task is formulated in mathematical terms as an optimization problem with a multi-objective objective function and a set of constraints. This model can then be implemented following a monolithic approach or using a simulation-optimization framework relying on process simulators. The advantage of the latter is that it exploits the modeling capabilities of standard process simulation packages, which include thermodynamic packages and detailed models of unit operations. However, the presence of numerical noise makes it challenging to estimate derivatives accurately, resulting in a prohibitive number of function evaluations or non-convergence to feasible points during process optimization or sensitivity analyses [1].

Alternatively, surrogate formulations of first principles models are used to improve their numerical robustness. One approach to generating surrogate models is to create a direct mapping between the degrees of freedom of the process and the objective function to be optimized, commonly referred to as a black-box surrogate model. However, this direct input-output mapping in black-box surrogate models results in a lack of interpretability, and such models are prone to overfitting and poor generalization [2].

Another approach for surrogate model formulation is to combine first principles knowledge via mechanistic equations with black-box models, leading to a hybrid surrogate model. Mainly artificial neural networks (ANNs) and kriging have been used to create such hybrid models, where individual process units are replaced with surrogate models and are complemented with first principles knowledge. For example, ANNs have been used in superstructure-based optimization by Henao and Maravelias [3] for a maleic anhydride production case study and by Fahmi and Cremaschi [4] for a biodiesel production plant. Kriging has been employed by Quirante and Caballero [5] and Quirante et al. [6] for the optimization of a sour water stripping plant and the vinyl chloride monomer production process, respectively. Furthermore, if global optimization using ANNs or kriging is sought, tailored optimization approaches can be applied, such as in Bongartz et al. [7], where they introduced a global optimization algorithm based on the propagation of McCormick relaxations in a reduced space.

As an alternative to ANNs and kriging, the field of symbolic regression, which generates closed-form analytical equations based on expression trees [8], has been gaining traction in recent years. These closed-form analytical equations can then be implemented relatively easily in an algebraic modeling system interfacing with state-of-the-art global solvers. In this work, we explore the use of symbolic regression based on Bayesian learning [9] for the global optimization of a hybrid surrogate model of the green methanol production process. To create the hybrid model, the closed-form analytical equations of individual process units are complemented by a mechanistic backbone of mass and energy balances. The hybrid model is formulated as a mixed integer nonlinear programming (MINLP) problem and is optimized using the global solver BARON [10].

To quantify the environmental dimension of sustainability, we incorporate into the MINLP explicit constraints based on life cycle assessment principles and the planetary boundaries concept [11,12]. The latter represent ecological limits of the Earth system that should never be surpassed, including climate change, biodiversity loss, fresh water use and biochemical flows, among others. The impact on these boundaries is quantified via tailored damage models recently developed [13]. The main advantage of the method is that it allows to globally optimize the surrogate model built via symbolic regression in a standard modeling system, in contrast to other global optimization approaches that would apply tailored computationally intensive algorithms iteratively to the surrogate formulation.

Our results show that our approach is able to efficiently identify Pareto sets, while the hybrid surrogate modeling approach outperforms other black-box surrogate modeling approaches in the single and multi-objective global optimization of the green methanol production process.

References

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