Laplacian Structure in Power Network Constraints and Inherent Zonal Price Regions

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 associated with power flow and line flow limit constraints in a power network. When no line limits are active, the matrix in question has a nearly Laplacian structure, and must admit a vector of all equal elements in its null space (verifying the well- known equal incremental cost condition - that all LMPs must be equal in a lossless, unconstrained system). When line flow limits are active, the null space grows in dimension, and admissible LMP vectors can show patterns in which buses partition into regions of approximately equal LMPs. We claim that this phenomena arises from the same near-Laplacian structure in the power flow Jacobian that gives rise to coherency in electromechanical dynamics. For coherency problems, Fiedler vector computations have been previously exploited for graph partitioning to identify coherent buses. Using similar concepts, this paper will explore a new computational approach to identifying network partitions in LMP computations, giving rise to "inherent" zonal price regions.