Degeneracy and duality gap on optimization-based electricity auctions

Abstract

Degeneracy and duality gap can exist even in very simple optimization-based electricity auctions, and they have important effects on the meaning and usefulness of dual variables to set market price. An optimization model that describes a simple-bids auction, in which participants only submit the offered power quantity and its unitary price, is formulated in the paper. Using duality theory, the authors prove that Lagrange multipliers can be used to set the market price and that, in the absence of degeneracy, they reflect marginal pricing and are equivalent to a first-price auction. The conditions under which multiple primal solutions exist for this model are described. For a multi-part bids auction in which bidders submit both a startup and variable cost coefficient, they find that, in the absence of duality gap, Lagrange multipliers used as market prices lead to the recovery of all the costs submitted by the scheduled bidders. When a duality gap exists, the dual variables do not recover all the costs; even more, the cost not recovered is equal to the magnitude of the duality gap. The authors describe the conditions under which certain type of multiple primal solutions can be identified. Throughout the paper, several illustrative auction examples are presented.