New MI-SOCP formulations for the Unit Commitment problems with ramping constraints

The Unit Commitment (UC) problem in electrical power production requires to optimally operate a set of power generation units over a short time horizon (one day to a week). Operational constraints of each unit depend on its type (e.g., thermal, hydro, nuclear, \ldots), and can be rather complex; typical ones concern minimum and maximum power output, minimum up- and down-time, start-up and shut-down limits, ramp-up and ramp-down limits. Also, the objective function is often nonlinear. Thus, even the Single-Unit Commitment (1UC) problem, in which only one unit is present, has a rich combinatorial structure. In this work we present the first MINLP formulation that describes the convex hull of the feasible solutions of (1UC) comprising all the above constraints, and Second-Order Cone representable power generation costs. The new formulation has a polynomial number of both variables and constraints, and it is based on the efficient Dynamic Programming algorithm proposed in~\cite{FG06} together with the \emph{perspective reformulation} technique proposed in~\cite{FG06a}. We then analyze the effect of using it to develop tight formulations for the more general (UC). Since the formulation, despite being polynomial-size, is rather large, we also propose two new formulations, based on partial aggregations of variables, with different trade-offs between quality of the obtained bound and cost of the solving the corresponding continuous relaxation. Our results show that navigating these trade-offs may lead to improved performances for the partial enumeration approach used to solve the problem.