# Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations

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## Abstract

Deterministic global methods for flowsheet optimization have almost exclusively relied on an equation-oriented formulation where all model variables are controlled by the optimizer and all model equations are considered as equality constraints, which results in very large optimization problems. A possible alternative is a reduced-space formulation similar to the sequential modular infeasible path method employed in local flowsheet optimization. This approach exploits the structure of the model equations to achieve a reduction in problem size. The optimizer only operates on a small subset of the model variables and handles only few equality constraints, while the majority is hidden in externally defined functions from which function values and relaxations for the objective function and constraints can be queried. Tight relaxations and their subgradients for these external functions can be provided through the automatic propagation of McCormick relaxations. Three steam power cycles of increasing complexity are used as case studies to evaluate the different formulations. Unlike in local optimization or in previous sequential approaches relying on interval methods, the solution of the reduced-space formulation using McCormick relaxations enables dramatic reductions in computational time compared to the conventional equation-oriented formulation. Despite the simplicity of the implemented branch-and-bound solver that does not fully exploit the tight relaxations returned by the external functions but relies on further affine relaxation at a single point using the subgradients, in some cases it can solve the reduced-space formulation significantly faster without any range reduction than the state-of-the-art solver BARON can solve the equation-oriented formulation.

## Keywords

Global optimization Process design Sequential modular Branch-and-bound Relaxation of algorithms## Notes

### Acknowledgements

This work received funding through the “Competence Center Power to Fuel” of RWTH Aachen University and project “Power to Fuel” of JARA Energy, both of which are funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities, as well as from the German Federal Ministry of Education and Research (BMBF) under grant number 03SFK2A. The responsibility for the content lies with the authors. The authors would also like to thank Jaromił Najman, Hatim Djelassi, and Wolfgang Huster for helpful discussions.

## Supplementary material

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