Supply Function Equilibrium in a Constrained Transmission System

This paper characterizes equilibrium in an auction market constrained by limited capacities of links in a transportation network. The formulation is adapted to a wholesale spot market for electricity managed by the operator of the transmission system. This paper derives conditions that characterize equilibrium in an auction market of the kind conducted by system operators in the electricity industry. These are wholesale spot markets in which the participants are suppliers (generators) and demanders (utilities and other load-serving entities). Because these participants are spatially distributed, the operator’s allocation of production and consumption of electrical energy is constrained by the capacities of links in the transmission system. Moreover, storage is infeasible, supply must continually match demand, and both net demand and transmission capacities are affected by random shocks. Therefore, participants submit notional supply or demand functions in advance and then in each contingency the operator uses these functions to determine an optimal allocation. Financial settlements in such markets use locational marginal pricing, also called nodal pricing. The operator chooses the allocation in each contingency to maximize the apparent gain from trade subject to the feasibility constraints imposed by limited transmission capacities. (This is the gain from trade “as bid” since the operator treats each supply function as though it reflects the actual marginal cost of production.) This optimization results in a vector λ = (p, μ) of Lagrange multipliers on the energy and capacity constraints. Then a supplier j for which each unit of energy output uses uij units of capacity on transmission link i is paid its “nodal” price pj = p − ∑ i μiuij . Thus, if j submitted the supply function sj(·) then it is assigned to produce sj(pj) units of energy and for this output it is paid pjsj(pj). The formulation is established in Section 1. Sections 2 and 3 characterize a firm’s optimal bidding strategy. In Section 2 this is done for a firm located at a single node in the network. Section 3 addresses the case that a firm controls supply resources located at multiple nodes in the network. Section 4 then applies these results to characterize an equilibrium among the firms. This is an ordinary Nash equilibrium and thus includes “rational expectations” in the sense that each firm anticipates correctly the bidding strategies of all other firms. This is a plausible approximation in wholesale electricity markets because the participants and their costs are known to all, as is the probability distribution of exogenous factors (random variations of demand and of transmission and generator capacities), and the market repeats every hour of every day. Since a few Date: 15 April 2005; corrected April 30. This is a draft — comments welcome! I am indebted to Pär Holmberg for comments on previous drafts. Similar markets are used in other industries, such as gas transmission, but in these industries storage is an important factor that is ignored here. In practice, there are additional constraints that are not addressed by our formulation, such as requirements for reserves (to sustain voltage and to protect against cascading failures of equipment) and dynamic constraints (e.g., “ramp rate” limits on the rate-of-change of generators’ outputs). There are also additional financial aspects (e.g., the operator charges a network management fee, typically in the range of 1%-3% of the energy price, including costs of reserves) and fixed costs of starting-up and operating a generator that we ignore. 1