High Dimensional Three-Periods Locally Ideal MIP Formulations for the UC Problem
The thermal unit commitment (UC) problem often can be formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. The tighter characteristic reduces the search space, therefore, as a natural conse-quence, significantly reduces the computational burden. In the literature, many tightened formulations for single units with parts of constraints were reported without presenting how they were derived. In this paper, a sys-tematic approach is developed to formulate the tight formulations. The idea is using more new variables in high dimension space to capture all the states for single units within three periods, and then, using these state variables systematic derive three-periods locally ideal expressions for a subset of the constraints in UC. Meanwhile, the linear dependence relations of those new state variables are leveraged to keep the compactness of the obtained formulations. Based on this approach, we propose two tighter models, namely 3P-HD and 3P-HD-Pr. The proposed models and other four state-of-the-art models were tested on 51 instances, including 42 realistic instances and 9 8-unit-based instances, over a scheduling period of 24 h for systems ranging from 10 to 1080 generating units. The simulation results show that our proposed MIQP UC formulations are the tightest and can be solved most efficiently. After using piecewise technique to approxi-mate the quadratic operational cost function, the six UC MIQP formulations can be approximated by six corre-sponding mixed-integer linear programming (MILP) formulations. Our experiments show that the proposed 3P-HD and 3P-HD-Pr MILP formulations also perform the best in terms of tightness and solution times.