(393d) Optimization of Chemical Process Flowsheets with Ordered Discrete Structures: A Hybrid Deterministic-Stochastic Algorithm

The optimization of Mixed-Integer Nonlinear Programming (MINLP) chemical process flowsheets is often perceived as a challenging task due to the nonlinearity of mass and energy balances in chemical processes, and the nonlinear interactions between discrete and continuous variables. Deterministic and stochastic methods have been proposed to address MINLP flowsheet optimization problems. Deterministic strategies offer benefits such as local or global optimality guarantees for problems involving well-behaved (e.g., differentiable) functions, while stochastic techniques are preferred when handling highly non-convex problems involving conflicting objectives or a large number of integer variables. Despite their benefits, both methods have been criticized due to their inherent limitations. A practical limitation of deterministic techniques is their lack of integration with process flowsheet simulators [1]. Although this limitation is overcome by stochastic algorithms, these methods often exhibit slow convergence rates. Also, different solutions that may not necessarily satisfy local or global optimality can be obtained from multiple runs due to the stochastic nature of these strategies [2].

Hybrid stochastic-deterministic optimization methods have been proposed to counterbalance the benefits and drawbacks of stochastic and deterministic methods for flowsheet optimization. Previous studies have presented sequential hybrid strategies where a stochastic solution is improved using deterministic optimization; however, this strategy may not return local optimality guarantees when using a chemical process simulator without MINLP capabilities [3]. Other works have developed nested hybrid methods such as memetic algorithms that optimize discrete variables stochastically in an outer loop, while NLP optimization with discrete variables fixed is used in an inner loop [4]. Although memetic algorithms have demonstrated improvements in solution quality and computational performance, they may not guarantee local optimality for discrete variables. These limitations can be addressed by developing hybrid strategies able to return solutions with local optimality guarantees for both discrete and continuous variables. For this purpose, deterministic MINLP optimizers that can be integrated with flowsheet simulators must also be investigated; otherwise, local optimality for both discrete and continuous variables could not be guaranteed.

This study brings novelty to the field of chemical process flowsheet optimization by presenting a new hybrid deterministic-stochastic architecture, where the stochastic and the deterministic strategies are executed in parallel, i.e., the stochastic algorithm runs in one processor whereas the deterministic strategy runs in an alternate processor. This framework enables the exchange of information between the two optimization algorithms thus allowing faster convergence when compared to the sequential execution of the deterministic and the stochastic algorithms. The proposed parallelization also helps the hybrid algorithm to find multiple locally optimal solutions, which may provide a basis for the comparison of different competitive designs. Another novelty of this work is the development of a deterministic MINLP algorithm that is integrated within the proposed hybrid method that is capable of 1) interacting with chemical process flowsheet simulators, which are often limited to perform NLP optimization; 2) returning locally optimal solutions for both discrete and continuous variables, which increases the reliability of the optimization results, and 3) updating the search bounds of continuous and discrete variables, which is a feature that has not been incorporated within hybrid optimization strategies. In contrast to traditional sequential and nested hybrid arrangements, discrete and continuous variables are optimized by both the deterministic and the stochastic techniques simultaneously. A salient feature is that the exploration of variables performed by the stochastic technique is not constrained to a single variable type as in traditional memetic algorithms, e.g., discrete variables only. To the authors’ knowledge, a hybrid strategy with a parallel structure between the stochastic and the deterministic methods for flowsheet optimization is not available in the literature. The proposed hybrid strategy is explained in detail next.

The hybrid algorithm proposed in this work was designed to optimize chemical process flowsheets involving continuous variables (e.g., flowrates and heat duties) and ordered discrete decisions (e.g., number of units and location of interconnecting flows). The proposed hybrid algorithm combines a stochastic method (SM) with a deterministic optimization algorithm available in the literature: the deterministic Discrete-Steepest Descent Algorithm (DSDA) [5]. The latter method solves a problem by iteratively fixing discrete decisions (master problem) and solving for the remaining continuous variables using NLP optimization (subproblems). To verify the local optimality of discrete decisions, the DSDA evaluates a neighborhood of a candidate solution (e.g., evaluating the effect of adding or removing a unit operation from the current flowsheet), whereas the local optimality of continuous variables is assessed by an NLP solver. In this study, we integrate the DSDA within a variable bounding (VB) algorithm, i.e., a method that implements a bisection algorithm that relies on the convergence/failure of the NLP subproblems to update the search bounds of continuous variables. These updated search bounds avoid convergence failure of the NLP subproblems while guaranteeing local optimality for the solution returned. The proposed parallel method begins by executing the SM, using the variable bounds initially specified by the user. Simultaneously, the DSDA-VB iteratively retrieves the best solution found by the SM and uses this information to initialize a local search. If the most recent locally optimal solution identified DSDA-VB shows an improvement in the objective function, then the bounds and solution reported by DSDA-VB are used to update the bounds of the SM, i.e., bounds of continuous variables are updated in the SM based on the VB strategy solution, while bounds for discrete decisions of the SM are constrained within a neighborhood of the solution found by DSDA-VB. This parallel execution continues until the gap between the best solution from both algorithms satisfies a user-defined tolerance. This interdependence between the DSDA-VB and the SM results in an enhanced convergence performance of both algorithms towards improved local solutions.

The proposed method is tested by performing the optimal design of a sequence of reactive, extractive, and traditional distillation columns, using the total annual cost as the objective function and Aspen Plus as the process simulator. This case study involves 15 continuous degrees of freedom (i.e., reflux ratios, output flows, and column diameters) and 13 discrete degrees of freedom (i.e., number of stages, feed locations, and number of trays in the reactive section). Rigorous stage-by-stage nonlinear models combining mass/energy balances and non-ideal equilibrium relationships for each unit are considered. Differential Evolution with Tabu List (DETL) is used as the SM, which is also used as a benchmark to compare the results obtained by the present method. DETL was selected due to its effectiveness in solving highly non-linear and potentially non-convex flowsheet optimization problems. A set of standard tuning parameters was adopted, and a time limit of minutes was established for both the traditional DETL and the proposed hybrid algorithm. Under these conditions, the results show improvements in convergence rate when comparing the proposed method with the traditional DETL, e.g., at the time limit, the relative gap between the deterministic and stochastic objectives within the proposed hybrid method was below 1%, while the gap between the best objective function from the hybrid method and the traditional DETL remained at 48%. In contrast to traditional hybrid approaches (e.g., memetic algorithms), the optimal discrete and continuous variables obtained from our approach satisfy local optimality, which is possible thanks to the parallel hybridization of the SM with the DSDA-VB method.

References

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