New Mixed-Integer Nonlinear Programming 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. Operational constraints of each unit depend on its type and can be rather complex. For thermal units, 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, and nonlinear objective function. In this work, we present the first mixed-integer nonlinear program formulation that describes the convex hull of the feasible solutions of the single-unit commitment problem comprising all the above constraints and convex 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 by Frangioni and Gentile together with the perspective reformulation. The proof that the formulation is exact is based on a new extension of a result previously only available in the polyhedral case, which is potentially of interest in itself. We then analyze the effect of using it to develop tight formulations for the more general UC problem. Because the formulation is rather large, we also propose two new formulations, based on partial aggregations of variables, with different trade-offs between quality of the bound and cost of solving the continuous relaxation. Our results show that navigating these trade-offs may lead to improved performances.