  The thermal unit commitment (UC) problem has historically been formulated as
a mixed integer quadratic programming (MIQP), which is difficult to solve
efficiently, especially for large-scale systems. The tighter characteristic
reduces the search space, therefore, as a natural consequence, significantly
reduces the computational burden. In literatures, many tightened formulations
for a single unit with parts of constraints were reported without presenting
explicitly how they were derived. In this paper, a systematic approach is
developed to formulate tight formulations. The idea is to use more binary
variables to represent the state of the unit so as to obtain the tightest upper
bound of power generation limits and ramping constraints for a single unit. In
this way, we propose a multi-period formulation based on sliding windows which
may have different sizes for each unit in the system. Furthermore, a
multi-period model taking historical status into consideration is obtained.
Besides, sufficient and necessary conditions for the facets of single-unit
constraints polytope are provided and redundant inequalities are eliminated.
The proposed models and three other state-of-the-art models are tested on 73
instances with a scheduling time of 24 hours. The number of generators in the
test systems ranges from 10 to 1080. The simulation results show that our
proposed multi-period formulations are tighter than the other three
state-of-the-art models when the window size of the multi-period formulation is
greater than 2.
