Start-Up/Shut-Down MINLP Formulations for the Unit Commitment with ramp constraints

In [1] the first MIP exact formulation was provided that describes the convex hull of the solutions satisfying all the standard operational constraints for the thermal units: minimum up- and down-time, minimum and maximum power output, ramp (including start-up and shut-down) limits, general history-dependent start-up costs, and nonlinear convex power production costs. That formulation contains a polynomial, but large, number of variables and constraints. We present two new formulations with fewer variables defined on the shut-down period and computationally test the trade-off between reduced size and possibly weaker bounds.