# A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints

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## Abstract

This paper provides the convex hull description of the single thermal Unit Commitment (UC) problem with the following basic operating constraints: (1) generation limits, (2) start-up and shut-down capabilities, and (3) minimum up and down times. The proposed constraints can be used as the core of any unit commitment formulation to strengthen the lower bound in enumerative approaches. We provide evidence that dramatic improvements in computational time are obtained by solving the self-UC problem and the network-constrained UC problem with the new inequalities for different case studies.

## Keywords

Unit commitment (UC) Mixed-integer programming (MIP) Facet/convex hull description## Mathematics Subject Classification

90C11 90C57 90C90## Notes

### Acknowledgments

The authors thank Laurence Wolsey, Santanu Dey, Antonio Frangioni, and Paolo Ventura for useful discussions on the paper.

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