  In this paper, we consider the polyhedral structure of the unit commitment
polytope. In particular, we provide the convex hull results for the problem
under the following different settings: 1) the convex hulls for the integrated
minimum-up/-down time and ramping polytope under the general $T$ time period
setting in which the ramping rate equals to the gap between the generation
upper and lower bounds and equals to half of the gap between the generation
upper and lower bounds, respectively, 2) the convex hull for the integrated
minimum-up/-down time and ramping-up polytope for the problem under the general
$T$ time period setting, and 3) the convex hull for the integrated
minimum-up/-down time and ramping-down polytope for the problem under the
general $T$ time period setting.
