(340l) Simultaneous Design, Control and Selection of Finite Elements: A Hamiltonian Function-Profile-Based Approach.

Multiple studies have addressed the solution for integrated design and control problems where multiple aspects such as process design, process economics and the dynamic operability of the process have been considered simultaneously [1-3]. The formulation of optimal process design and control problems often involve the solution of differential-algebraic equation models (DAEs), which are commonly discretized and represented as a set of nonlinear algebraic equations. Orthogonal collocation on finite elements (OCFE) is one of the most attractive numerical techniques used for discretization of DAEs since they offer adequate accuracy and numerical stability. One key aspect in OCFE is the selection of the number of finite elements, which is often decided based on a priori simulations or process heuristics. Thus, the effect of the discretization scheme on the optimal solution is not often assessed or even considered during the solution of integrated design and control problems. Studies involving the specification of the number of finite elements in the design of optimal control trajectories for chemical engineering applications have been reported, e.g. separation and reactive processes [4], crystallization [5], power generation [6]. Consequently, the optimal selection of the number of finite elements is an important aspect to consider for integrated design and control problems. Considering the complexity of integrated design and control problems, the optimal selection of the discretization mesh is still an open issue that have not been studied in the literature.

Studies in singular control and dynamic optimization have presented new methodologies for the optimal selection of finite elements in the context of OCFE [7, 8]. These methodologies have been tested on a few applications, e.g. optimal control of complex chemical reacting systems [7] and optimal trajectories for landing of spatial rockets [9]. For autonomous systems, the Hamiltonian function is known to be continuous and constant over time even if the control profile is not continuous. This attractive property of the Hamiltonian function has been used in [7, 8] as the main criterion for the selection of the number of finite elements. Nevertheless, these studies have focused on the solution of nonlinear programing (NLP) models, i.e. models involving only continuous variables. In formulations involving discrete (integer) variables representing structural decisions in optimal process design and control problems, the selection of the number of finite elements may lead to suboptimal solutions if the control profiles are not estimated with sufficient accuracy. Therefore, the implementation of methodologies for the optimal selection of the number of finite elements for optimal process design and control problems becomes critical to guarantee accurate solutions.

In this work, we extend the application of an existing methodology proposed for optimal selection of the number of finite elements based on the Hamiltonian function to consider the simultaneous design and control of chemical manufacturing processes. Our implementation addresses the solution of integrated design and control problems involving structural decisions and product grade transitions. We use a set of nested optimization problems for the selection of the optimal discretization mesh and the solution of optimal process design and control formulations. Moreover, we implement a moving finite element approach to adjust element sizes; this allows a reduction on the required number finite elements by reducing the element length in those regions where the process model is steep. Nested programming problems are solved iteratively in a sequential fashion. Integer variables are sequentially selected using an adapted version of a branch and bound strategy; this allows the systematic solution of NLP models with fixed values on the integer variables. Our implementation is illustrated with a case study that aims to optimize a reactor network superstructure that requires to manufacture a set of products with different quality grades during operation in closed-loop. We explore the effect of the selection of the number of finite elements on the solution where consideration of discrete variables introduces discontinuities to the optimization model. The results show that selecting an adequate number of elements for problems involving process dynamics and integer decisions is not trivial, i.e. an improper choice of the number of elements may result in suboptimal solutions.

References

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