Advanced hybrid optimization methods for the design of complex separation processes

Highlights

• Evaluation of two advanced hybrid optimization methods for MINLP problems.

• Global optimization for simple column design verified by comparison with BARON.

• EA-based hybrid algorithm more efficient than MSGS-based hybrid algorithm.

• Optimization-based design of energy-integrated process and solvent selection.

• Illustration for extractive distillation process for acetone-methanol separation.

Abstract

While the need for more efficient and integrated processes for the separation of non-ideal and azeotropic mixtures mandates model-based design methods, the resulting mixed-integer problems are highly nonlinear and particularly hard to solve without proper initialization. In order to handle such complex design problems, two hybrid optimization methods are presented in the current contribution. Both methods rely on a nested approach, which integrates a polylithic modeling and solution strategy, based on the solution of successively refined nonlinear programming problems, with an upper level metaheuristic for the initialization and optimization of discrete design decisions. Both methods are evaluated for two case studies. The optimization of a simple column design problem allows for the direct comparison with an available global deterministic optimization solver for identifying the global optimal solution. The optimization of an extractive distillation process with simultaneous solvent selection and energy integration illustrates the potential complexity that can be handled effectively.

Introduction

Economic and environmental improvements, especially in mitigating global warming, mandate the development of more efficient and sustainable processes in the chemical industry. This requires the consideration of different means for energy and mass integration as well as the processing of more complex bio-based feedstocks. While downstream separation processes account for approximately 50% of the overall energy demand of a chemical plant (Kiss and Smith, 2020), especially the processing of bio-based feedstocks requires the separation of complex azeotropic mixtures (Skiborowski et al., 2014). Yet, the design and optimization of the mostly hybrid process concepts, which oftentimes require the selection of suitable mass separating agents (MSAs), remains a challenging and tedious task, often resulting in sub-optimal local solutions.

Even in the case that all necessary information to accurately model the competing process options is available or can be estimated by means of predictive models, the resulting design problems are usually represented by non-convex mixed-integer nonlinear programming (MINLP) problems with a considerable number of design degrees of freedoms (DDoFs). Given a sufficient initialization, e.g., derived from initial simulation studies, local deterministic solvers are very efficient, but bare the risk to get stuck in sub-optimal local solutions (Skiborowski et al., 2015b). In contrast, global deterministic solvers, such as BARON (Tawarmalani and Sahinidis, 2005), ANTIGONE (Misener and Floudas, 2014), SCIP (Achterberg, 2009) or COUENNE (Belotti et al., 2009), compute the global optimal solution, while a direct application to complex nonlinear models of strongly integrated processes is still not feasible (Kallrath, 2013).

Recent progress in deterministic global optimization is impressive. In order to overcome the problem of poor relaxations, different publications propose improved convex or concave relaxations. Najman et al. (2019) presented a profound study on possible relaxations and envelopes for various functions that are frequently represented in chemical engineering problems, as e.g., in modeling saturation pressure, enthalpy of vaporization, and Gibbs free energy, providing mathematical proof and computational validation for a broad range of parameter sets in the DIPPR 801 database (Design Institute for Physical Properties, 2016). Ballerstein et al. (2015) presented a tailored bounding strategy that exploits the monotonicity of composition profiles in distillation for ideal mixtures, showing impressive performance improvements for the optimization of stand-alone distillation columns as well as hybrid distillation/melt-crystallization processes. Further extension of the approach by a linear transformation of the composition variables was presented by Mertens et al. (2016), who also presented a proof of its monotonicity (Mertens et al., 2018). While limited to ideal mixtures, the results indicate a significant improvement with respect to the computational effort, even without further bound tightening.

Since the monotonicity reformulation cannot be applied for non-ideal mixtures, Keßler et al. (2019) suggested a surrogate-based optimization that builds on Kriging models and an iterative sampling algorithm. In a similar approach, Carpio et al. (2018) addressed different problems including the optimization of a reactor network, a heat exchanger network, and an extractive distillation (ED) process. Although only continuous operational DDoFs were optimized, the so-called modified probability of improvement method showed great potential compared to published results for those chemical engineering case studies. The huge potential of exploiting surrogate models in the context of process optimization was further demonstrated for various sub-models, including the flash calculations by means of artificial neural networks (ANNs) (Schweidtmann et al., 2019), the computations of liquid-liquid equilibrias (LLEs) (Kunde, Keßler, Linke, McBride, Sundmacher, Kienle, 2019, Nentwich, Engell, 2019), or other highly nonlinear functions (Chen, 2019). The derived surrogate models enable a computationally robust and efficient optimization due to simpler and smooth mathematical models. However, the derivation of these models requires a considerable number of evaluations of the original model and an accurate training of the surrogate model to derive the optimum solution of the original problem. Consequently, the effort and complexity of deriving an accurate surrogate model need to be accounted for in a comparison with a direct optimization of the original model.

In contrast to the use of surrogate models, optimization problems can also be solved more reliably and efficiently by means of reduced-space formulations that integrate implicit functions to solve particularly complex sets of equations. As recently summarized by Seidel et al. (2020) such decomposition has been first proposed by Kravanja and Grossmann (1996) and since then been exploited by several authors for various process design problems (Caballero, 2015, Tolsma, Clabaugh, Barton, 2002, Skiborowski, Wessel, Marquardt, 2014, Skiborowski, Harwardt, Marquardt, 2015, Recker, Skiborowski, Redepenning, Marquardt, 2015, Manassaldi, Mussati, Scenna, Mussati, 2019). While most available deterministic global optimization solvers do not support implicit functions and require a symbolic implementation of the complete model, Bongartz and Mitsos (2017) have presented the application of their in-house solver MAiNGO (Bongartz et al., 2018) for exploiting such implicit functions in reduced-space formulations. The impressive potential of this approach has furthermore been highlighted for flowsheet optimization problems by Bongartz and Mitsos (2019). As pointed out by the authors, this potential depends largely on the identification of suitable model formulations on the unit operation and flowsheet level. This statement holds for all of the referenced studies, as the success in process optimization depends heavily on the combination of a suitable algorithmic approach and model formulation.

Besides deterministic global optimization methods there have also been a wide variety of studies that promote the use of metaheuristics, which are either local search strategies, like simulated annealing or global search methods, which are mostly population-based methods, such as evolutionary and genetic algorithms or algorithms based on the concept of swarm intelligence. A brief review of these methods in process optimization is provided by Skiborowski et al. (2015b), who also point out the benefits of hybrid optimization algorithms, which combine efficient gradient-based local optimization with a global search method. As indicated by Molina et al. (2010), especially the combination of an evolutionary algorithm (EA) and an efficient local optimization algorithm can greatly improve the effectiveness of the optimization. Such a combination has been termed memetic algorithm (MA) by Moscato (1989) and can be interpreted according to the evolutionary theory of Lamarck (Weicker, 2015). MAs have been demonstrated to show superior performance for different process design problems. Especially the combination of a problem-specific EA for optimization and a mathematical programming (MP) method that solves challenging continuous sub-problems, developed in the research group of Prof. Engell at TU Dortmund, showed large prospect and was demonstrated for the optimization of reactive distillation columns (Urselmann et al., 2011) with optional external reactors (Urselmann and Engell, 2015), two-stage stochastic programming for early-stage conceptual design (Steimel and Engell, 2015), and the optimization of models in commercial flowsheet simulators (Janus et al., 2017). In a similar fashion, Skiborowski et al. (2015b) proposed a hybrid optimization approach, which also builds on a combination of an EA and a MP method, while the latter solves full MINLP problems through a polylithic modeling and solution approach. This modification results in a considerable reduction of the search space covered by the EA, allowing for a reduction and simplification of the optimization problems addressed by the MP method since decisions with a decisive impact on the model structure, such as the choice of a MSA, are fixed by the EA. While the results of several process design studies indicated a huge potential of the proposed combination, no comparison with a deterministic global optimization was performed and the possible extension in terms of problem complexity was only indicated.

Both of these points are addressed in the current contribution, which presents a further extension of the previously developed hybrid optimization approach as well as an alternative approach that substitutes the stochastic EA with a deterministic multi-start grid search (MSGS) with consecutive grid refinement. In order to allow for a comparison with state-of-the-art deterministic global optimization methods, a first indicative case study investigates the optimization of a single distillation column, showcasing the efficiency and effectiveness of the hybrid methods in identifying the global optimal solution. The second case study illustrates the level of complexity that can be handled by the hybrid approach and which is well beyond the current range of applicability of a direct deterministic global optimization. For this purpose, an extractive distillation process is optimized considering the choice of mass separating agents as well as potential flowsheet modification for energy integration. Both methods are described in Section 2, while the results for the two case studies are further presented in Section 3. Some conclusions and outlook on future work are finally provided in Section 4.