The min-up/min-down unit commitment polytope

Abstract

The min-up/min-down unit commitment problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. A full polyhedral characterization of the MUCP with only one production unit is provided by Rajan and Takriti (Minimum up/down polytopes of the unit commitment problem with start-up costs. IBM Research Report, 2005). In this article, we analyze polyhedral aspects of the MUCP with n production units. We first translate the classical extended cover inequalities of the knapsack polytope to obtain the so-called up-set inequalities for the MUCP polytope. We introduce the interval up-set inequalities as a new class of valid inequalities, which generalizes both up-set inequalities and minimum up-time inequalities. We provide a characterization of the cases when interval up-set inequalities are valid and not dominated by other inequalities. We devise an efficient Branch and Cut algorithm, using up-set and interval up-set inequalities.