(216g) Generalization of a Tailored Approach for Dynamic Simulation and Optimization

There exist in academia several packages for the efficient solution of dynamic optimization problems. The major problem of these packages lies on the fact that they are somehow restricted either to a special type of dynamic model equations or to a specific application, e.g. dynamic control problem, (state &) parameter estimation, optimal design of experiments. Moreover, dynamic problems in chemical engineering are generally defined as differential algebraic equation system (DAE), namely, a full implicit DAE. For numeric purposes, special restrictions are added so as to reduce the problem to a quasi-linear semi-explicit or implicit system of index one. Consequently, the implementation of a specific dynamic model implies firstly a fault prone reformulation of the original equation system in order to reduce the index of the DAE-system. An additional problem represents the definition of the quasi linear mass matrix. Using component property equations (e.g. from ChemCad® data) or even having a model of more than one component (e.g. models in separation processes) involves a complex transformation of the original problem.

Furthermore, in contrast to the simultaneous approach for the optimization problem solution, the sequential approach (feasible path method) corresponds to the intuitive way of defining an optimization problem by a process engineer. This is basically because of the fact that prior to the definition of the optimization problem simulation studies are often carried out e.g. in order to make sensitivity analysis or to find a good starting guess for the free decisions. By this means, the optimization problem size decreases significantly depending only on the number of discretized controls. The challenge lies then in reducing the computational effort, which is mainly determined by the integration of the model equations together with the sensitivities.

Based on the aforementioned considerations, in this work, we propose a new solver for the integration of DAE systems. The main advantage is its general interface for the definition of dynamic process models, which corresponds to the fully implicit type and with an index of arbitrary order. In addition, it can be used in a framework together with any NLP solver following the sequential optimization approach. The proposed solver consists first in a robust integration scheme based on orthogonal collocation on finite elements and a subsequent solution of the nonlinear equations system. An internal numerical differentiation scheme enables the efficient calculation of dynamic sensitivities, which are then used by the optimizer in order to solve the posed optimization problem.

The new solver has also an interface to Matlab®, and the corresponding sparse version to Fortran®. Whereas both solvers have shown to be very robust, compared to other commercially available solver in both programming languages, the sparse version has shown to be especially efficient in solving large scale problems such as those resulting from PDE systems. In order to demonstrate the performance of the proposed solver, some applications such as:

? finding the minimum time and optimal control trajectory for a multi-component distillation column model subject to product changeover

? estimation of adsorption parameters for a chromatographic column model

? optimal experimental design for a fed-batch reactor model

? Sensitivity studies of a fixed bed tubular reactor model

Acknowledgment: The authors acknowledge the support from the Collaborative Research Center SFB/TR 63 InPROMPT ?Integrated Chemical Processes in Liquid Multiphase Systems? coordinated by the Berlin Institute of Technology and funded by the German Research Foundation.