Admissible Locational Marginal Prices via Laplacian Structure in Network Constraints

Standard computations reveal that locational marginal prices (LMPs), being Lagrange multipliers in an optimization problem, must lie in the null space of a Jacobian matrix evaluated at the optimal power flow solution, augmented by columns associated with active line flow limits. The impact of network power flow and active line limit constraints is to confine the LMPs in a subspace that satisfies necessary conditions for optimality. Optimal market clearing can then proceed as a minimization over offer curves confined to admissible LMPs in this subspace. When no line limits are active and losses neglected (e.g., in a dc power flow representation), the matrix in question has a generalized Laplacian structure, and admits only a vector of all equal elements in its null space (verifying the well-known equal incremental cost condition). As line flow limits become active, the null space grows in dimension. Among the phenomena of interest as the dimension of admissible LMPs grows is that of "load pockets"; that is, admissible LMP vectors that can show patterns in which buses partition into zones of approximately equal LMPs, with significant differences between zones. This paper explores an approach to admissible LMP calculation that isolates the topological role of active line flow constraints, independent of offer curve prices. Identification of this admissible subspace can then greatly facilitate computations such as clustering to identify potential zones of differentiated LMPs.