The Ramping Polytope and Cut Generation for the Unit Commitment Problem

Abstract

We present a perfect formulation for a single generator in the unit commitment problem, inspired by the dynamic programming approach taken by Frangioni and Gentile. This generator can have characteristics such as ramp up/down constraints, time-dependent start-up costs, and start-up/shut-down limits. To develop this perfect formulation we extend the result of Balas on unions of polyhedra to present a framework allowing for flexible combinations of polyhedra using indicator variables. We use this perfect formulation to create a cut-generating linear program, similar in spirit to lift-and-project cuts, and demonstrate computational efficacy of these cuts in a utility-scale unit commitment problem.

Start-up/Shut-Down MINLP Formulations for the Unit Commitment with Ramp Constraints

“…This is why we also presented two additional MIP formulations which trade a weaker bound for fewer variables. We mention that three independent groups obtained a similar result restricted to linear objective function: the first was [6,7] and then [8,9] also appeared with very similar structure but with different proof techniques.…”

Electricity Market Design and Zero-Marginal Cost Generation

“…An expanded unit commitment characterization can provide convex hull prices with most of the existing unit commitment constraints including ramping constraints [35]. Reformulating the original problem by adding constraints and variables, to construct an equivalent master problem that characterizes the convex hull provides exact pricing solutions and reports good computational performance [36][37][38]. However, the reformulated problem may be hard to connect to the native formulation used by the system operator.…”

A novel matching formulation for startup costs in unit commitment

“…The objective function (1a) minimizes the cost to operate the system, with respect to generators' piecewise production costs, minimum running costs, and startup costs. Constraints (1b-1r) are standard in UC formulations without time-varying startup costs [13,17]. Constraint (1b) balances supply and demand, while Constraint (1c) ensure sufficient (spinning) reserves are available.…”

“…This result is extended in [10] to generators with startup and shutdown power constraints. Inequalities to tighten the formulation of the ramping process are considered in [5,13,21,23].…”