Tighter Approximated MILP Formulations for Unit Commitment Problems

Abstract

The short-term Unit Commitment (UC) problem in hydro-thermal power generation is a largescale, Mixed-Integer NonLinear Program (MINLP), which is difficult to solve efficiently, especially for large-scale instances. It is possible to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by a Mixed-Integer Linear Program (MILP); applying the available efficient general-purpose MILP solvers to the resulting formulations, good quality solutions can be obtained in a relatively short amount of time. We build on this approach, presenting a novel way to approximating the nonlinear objective function based on a recently developed class of valid inequalities for the problem, called "Perspective Cuts". At least for many realistic instances of a general basic formulation of UC, a MILP-based heuristic obtains comparable or slightly better solutions in less time when employing the new approach rather than the standard piecewise linearizations, while being not more difficult to implement and use. Furthermore, "dynamic" formulations, whereby the approximation is iteratively improved, provide even better results if the approximation is appropriately controlled.Key words: Hydro-Thermal Unit Commitment, Mixed-Integer Linear Program Formulations, Valid Inequalities. 3. NomenclatureThe notation used throughout the paper is stated below. For unit consistency, note that hourly intervals are considered. Constants: n number of time intervals (hours) T set of all time periods P set of thermal units H set of hydro cascades, each comprising one or more basin units H(h) set of individual hydro units cascade h ∈ H is composed of B(j) set of the immediate predecessors of hydro unit j t kj water time delay from plant k ∈ B(j) to the basin feeding hydro unit j

A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes

“…This results in (5) and (1) coincident for u ∈ {0, 1} n , hence (PRef) is a "good" reformulation of (MINLP) since its continuous relaxation, called the Perspective Relaxation (PRel), provides significantly stronger bounds than the continuous relaxation of (MINLP) [5,6,1,8,9]. We remark that u i f i (p i /u i ) for u i ≥ 0 is the perspective function of f i (p i ), a well-known tool in convex analysis, hence the name.…”

“…This procedure can easily be implemented by using the standard tools made available by off-the-shelf solvers such as Cplex. Again, this is usually more efficient than approaching (MINLP) directly [5,6,8].…”

“…We will not describe the complete formulation here for space reasons; the interested reader is referred e.g. to [7,8]. It is however worth mentioning that thermal units are subject to minimum up-and down-time and ramp rate constraints, hydro units are subject to mass balance and reservoir volume constraints, while the interconnecting electrical network adds spinning reserve and capacity constraints.…”

“…We have compared P/C and CP on a test bed of randomly generated realistic instances already employed in [5,7,8], and freely available at the previously mentioned web address. In the tables, "p" is number of thermal units (hence n = 24p, as the time discretization is hourly on daily instances) and "h" is the number of hydro units.…”

“…Semi-continuous variables are very often found in models of real-world problems such as production planning problems [17,5,7,8], financial trading and planning problems [10,6], and many others [4,1,9]. These are variables that are constrained to either assume the value 0, or to lie in some given convex compact set P ⊆ R m ; in our applications P will always be a polyhedron.…”