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Tailored Indirect Algorithms for Efficient On-line Optimization of Batch and Semi-Batch Processes
Abstract and Figures
The trend towards high-quality, low-volume and high-added value production has put more emphasis on semi-batch processing due to its increased flexibility of operations. Dynamic optimization plays an important role toward improving the operation of batch and semi-batch. In addition, nonlinear model predictive control (NMPC) is also an important tool for the real-time optimization of batch and semi-batch processes under uncertainty. However, the transient behaviour as well as the flexibility decrease with respect to time make the optimization of such processes very challenging. The preferred strategy to solve constrained nonlinear dynamic optimization problems is usually to use a so-called direct method. Nevertheless, based on the problem type at hand and the solution algorithm used, direct methods may lead to computational complexity. In particular, the large prediction horizons required in shrinking-horizon NMPC increase the real-time computational effort because of expensive matrix factorizations in the solution steps, especially at the beginning of the batch. The computational delay associated with advanced control methods is usually underestimated in theoretical studies. However, this delay may contribute to suboptimal or, worse, infeasible operation in real-life applications. Alternatively, indirect methods based on Pontryagin’s Minimum Principle (PMP) could efficiently deal with the optimization of batch and semi-batch processes. In fact, the interplay between states and co-states in the context of PMP might turn out to be computationally quite efficient. The main indirect solution technique is the shooting method, which however often leads to convergence problems and instabilities caused by the integration of the co-state equations forward in time. Generally, it has been extensively argued that indirect methods are non-convergent and inefficient for constrained problems. However, this study proposes an alternative, convergent and effective indirect solution technique. Instead of integrating the states and co-states simultaneously forward in time, the proposed algorithm parameterizes the inputs, and integrates the state equations forward in time and the co-state equations backward in time, thereby leading to a gradient-based optimization approach. Constraints are handled by indirect adjoining to the Hamiltonian function, which allows meeting the active constraints explicitly at every iteration step. The performance of the solution strategy is compared to direct methods through three different case studies. The results show that the proposed PMP-based quasi-Newton strategy is effective in dealing with complicated constraints and is quite competitive computationally. In addition, this work suggests using the proposed indirect solution technique in the context of shrinking-horizon NMPC under uncertainty. Uncertainties can be handled by the introduction of time-varying backoff terms for the path constraints. The resulting NMPC algorithm is applied to a two-phase semi-batch reactor for the hydroformylation of 1-dodecene in the presence of uncertainty, and its performance is compared to that of NMPC that uses a direct simultaneous optimization method. The results show that the proposed algorithm (i) can enforce feasible operation for different uncertainty realizations both within batch or from batch to batch, and (ii) is significantly faster than direct simultaneous NMPC, especially at the beginning of the batch. In addition, a modification of the PMP-based NMPC scheme is proposed to enforce active constraints via tracking and reduce the real-time computational load further. This thesis also details the combination of an indirect solution scheme together with a parsimonious input parameterization. The idea is to parameterize the sensitivity-seeking inputs in a parsimonious way so as to decrease the computational load of constrained nonlinear dynamic optimization problems. The proposed method is tested on the simulated examples of a batch binary distillation column with terminal purity constraints and a two-phase semi-batch hydroformylation reactor with a complex path constraint. The performance of the proposed indirect parsimonious solution scheme is compared with those of a fully parameterized PMP-based and a direct simultaneous solution approaches. It is observed that the combination of the indirect approach with parsimonious input parameterization can result in significant reduction in computational time. Finally, in this work, the combination of simple solution models with parsimonious input parameterization in the context of shrinking-horizon NMPC is suggested in order to minimize the computational delay in feedback. Solution models exploit the nominal optimal solution to suggest parsimonious parameterizations (especially for sensitivity-seeking arcs) that lead to fast optimization. The proposed approach is illustrated in simulation on two case studies in the presence of uncertainty, namely a binary batch distillation column and a semi-batch reactor. The results show that the suggested parsimonious shrinking-horizon NMPC (i) performs very similarly to the standard (fully parameterized) shrinking-horizon NMPC in terms of cost, (ii) is computationally much faster than the standard shrinking-horizon NMPC especially at the beginning of the batch, (iii) is robust to plant-model mismatch
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