**Abstract.** Efficient and robust techniques for the optimization
of dynamic chemical processes are presented. In particular, we address
the solution of large, multistage optimal control and design optimization
problems for processes described by DAE models of index one. Our boundary
value problem approach (a simultaneous solution strategy) is based on
a piecewise parametrization of the control functions and a multiple
shooting discretization of the DAEs, combined with specifically tailored
SQP strategies. Unlike other simultaneous strategies based on collocation,
direct use is made of existing advanced, fully adaptive DAE solvers. The
present work provides a considerable extension and generalization of
the classical direct multiple shooting approach for ODE models (Bock
and Plitt, 1984).

In order to exploit the structure of large, sparse DAE process models with many algebraic variables, a new partially reduced SQP strategy has been developed. The structured NLP problem resulting from the discretization of the optimization boundary value problem is projected onto the reduced space of differential variables plus control parameters, utilizing the natural decomposition of the discretized states into differential and algebraic variables. It is demonstrated that this projection can be obtained very efficiently through direct computation of the reduced linearized constraint system via directional derivatives. In addition, the partially reduced SQP framework allows to conserve certain structural features of the discretized problem and to handle inequality constraints in a straightforward manner. Like the original full-space boundary value problem approach, the reduced-space formulation lends itself well to parallel computation since only local projections are used.

A practical implementation of the new reduced-space strategy is provided within the modular optimal control package MUSCOD-II (Leineweber, 1995). The recently developed BDF code DAESOL (Bauer et al., 1997) is used for the efficient calculation of the required directional sensitivities. Apart from a number of multistage optimal control problems from the literature, three chemical process applications of MUSCOD-II are discussed:

- Optimal control of a fed-batch fermentation process.
- Optimization of combustion processes in 1D steady-state tube reactors.
- Optimal control of batch distillation processes.

In the context of the first application, some extensions for real-time optimization are addressed, e.g., the use of precalculated exact Hessians for the fast reoptimization after disturbances. The second problem demonstrates the suitability of our approach for the large, stiff differential equation models that are common in chemical kinetics applications. For this problem, we present solutions obtained on a parallel computer. The third application involves sparse nonlinear DAE models with up to several hundred states, most of which are algebraic. This serves as a typical example of the class of large-scale dynamic optimization problems which can be efficiently solved by the reduced-space boundary value problem approach presented.

**Keywords.** Large-Scale Optimal Control, Index One DAEs, Multiple
Shooting, Structured Full-Space and Reduced SQP Methods, Directional
Derivatives, Sparse Equation Systems, Parallel Computation, Real-Time
Optimization, Dynamic Chemical Processes.

Correspondence should be addressed to daniel.leineweber.dl@bayertechnology.com |