(364f) Finite Element Refinement and Selection on the Integration of Design and Control: A Hamiltonian Function-Profile-Based Approach.

Nowadays, integration of design and control is considered as an attractive and widely accepted approach for optimal process design [1]. Multiple studies have addressed the solution for integrated design and control problems considering aspects such as process design, process economics, and the dynamic operability of the process among others [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. In chemical engineering, orthogonal collocation on finite elements (OCFE) is one of the most widely used techniques for the discretization of DAEs. OCFE offers adequate accuracy and numerical stability if the number of collocation points and the number of finite elements is selected properly [4]. Thus, one key aspect in the implementation of OCFE is the selection of the number of finite elements (i.e., the mesh size for the discretization), which is often decided based on a priori simulations or process heuristics [5]. Also, the selection on the number of finite elements is related to the size of the optimization model. Traditionally, finite elements in an OCFE strategy are selected to be uniformly distributed (equidistribution), i.e., finite elements have the same length. 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[6]-[8]. In a study presented by Wright K. [9], they found that in cases where the accuracy of the approximation by the number of discretization points is low, in special for nonsmooth solutions to the differential equations, equidistribution of finite elements may not return a unique solution. Hence, the selection of the number and length 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 addressed in the open literature.

Studies in optimal control and dynamic optimization have presented new methodologies for the optimal selection of finite elements in the context of OCFE [9]. Those methodologies have been tested on a few applications, e.g., optimal control of complex chemical reacting systems [8]. For autonomous systems, the Hamiltonian function is known to be continuous and constant over time even if the control profile is not continuous [10]. This attractive property of the Hamiltonian function has been used in [11] as the main criterion for the selection of the number of finite elements. 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 by Chen et al. [11] for the selection of the number of finite elements for optimal control. We present an algorithmic methodology for the refinement and selection of the number of finite elements for integration of design and control. proposed methodology is based on the profile of the Hamiltonian function associated to the optimization problem for simultaneous design and control. Similarly, the accuracy of the approximation at noncollocation points is taken into account as a mathematical criterion on the length of the finite elements. Our implementation addresses the solution of integrated design and control problems involving structural decisions and product grade transitions. The proposed decomposition algorithm involves 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 finite element sizes. This allows a reduction on the required number of finite elements by reducing the element length in those regions where the process model is steep. Similarly, the algorithm can add finite element in those regions with nonsmooth profiles in the controlled variables, where higher accuracy in the approximation is required. A sequence of programming problems is solved iteratively for the refinement of the length and number of finite elements. Integer variables are sequentially selected using an adapted version of a branch and bound strategy. This allows for the systematic solution of NLP models with fixed values on the integer variables. Our algorithmic framework is illustrated with a case study that aims to optimize a reactor network superstructure that need to meet specific process design goals, e.g., meet product quality and dynamic feasibility constraints during operation. We explore the effect of the selection of the number of finite elements on the solution where discrete variables are considered. The results show that selecting an adequate number of finite 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 . The proposed methodology allows to make a better decision on the selection of the size of the discretization for optimization purposes thereby avoiding the computation of suboptimal solutions for integration of design and control problems.

References

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[11] Chen, W., Ren, Y., Zhang, G., & Biegler, L. T. (2019). A simultaneous approach for singular optimal control based on partial moving grid. AIChE Journal, 65(6), e16584.