Abstract
A technique for optimizing large-scale differential-algebraic process models under uncertainty using a parallel embedded model approach is developed in this article. A combined multi-period multiple-shooting discretization scheme is proposed, which creates a significant number of independent numerical integration tasks for each shooting interval over all scenario/period realizations. Each independent integration task is able to be solved in parallel as part of the function evaluations within a gradient-based non-linear programming solver. The focus of this paper is on demonstrating potential computation performance improvement when the embedded differential-algebraic equation model solution of the multi-period discretization is implemented in parallel. We assess our parallel dynamic optimization approach on two case studies; the first is a benchmark literature problem, while the second is a large-scale air separation problem that considers a robust set-point transition under parametric uncertainty. Results indicate that focusing on the speed-up of the embedded model evaluation can significantly decrease the overall computation time; however, as the multi-period formulation grows with increased realizations, the computational burden quickly shifts to the internal computation performed within the non-linear programming algorithm. This highlights the need for further decomposition, structure exploitation and parallelization within the non-linear programming algorithm and is the subject for further investigation.
Highlights
The optimization of process systems under uncertainty is important and, in many cases, necessary for capturing realistic solutions to the optimal operation and design of physical systems
A popular stochastic optimization approach that has emerged over the last several decades is chance constraint programming (CCP), in which constraints are relaxed according to a particular probability distribution
We have presented a parallel computing approach for large-scale dynamic optimization under uncertainty that targets the decomposition of the embedded differential-algebraic equation model
Summary
The optimization of process systems under uncertainty is important and, in many cases, necessary for capturing realistic solutions to the optimal operation and design of physical systems. Robust optimization formulations are conveniently posed as min-max problems, where the idea is to minimize the maximum impact of uncertainty on the performance index subject to the largest possible constraint violation (i.e., worst-case analysis) Recent work in this direction is discussed by Diehl et al [8] and Houska et al [9], who provide a framework for robust optimal control of dynamic systems. We are primarily concerned with the use of dynamic process models described by a system of differential-algebraic equations (DAE) and the efficient incorporation of uncertainty using multi-period optimization This approach can be used to fully or partially address stochastic or robust optimization formulations, depending on how one characterizes uncertainty. Some concluding remarks are provided and future work noted
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