A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) that uses implicit functions to remove algebraic variables and equations from the optimization problem is described. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing it to be solved with second-order nonlinear optimization algorithms. The formulation is shown to lead to more reliable solver convergence when comparing with a full-space simultaneous formulation in challenging case studies involving a chemical looping combustion reactor. In a steady state simulation problem, the implicit function formulation solves 82 out of 100 instances, while the full space formulation solves 42 out of 100 instances. In a dynamic optimization problem, the implicit function formulation solves 189 out of 200 instances, while the full space formulation solves 129 out of 200 instances. In these examples, the nonlinear system of algebraic equations is maintained feasible throughout the optimization, reducing the likelihood that poor conditioning of these equations will cause non-convergence. In bound-constrained instances of the dynamic optimization problem, the full space formulation converges in an average of 83 CPU s while the implicit function formulation converges in an average of 64 CPU s. For the simulation problem, however, the implicit function formulation will require a parallel implementation to be competitive with the full space formulation. On the other hand, our results indicate the potential for the reduced space formulation to be more reliable than the full space formulation for challenging nonlinear DAE optimization problems, without appreciable performance degradation for large problems.
Citation
April 2022