Characterization of Feasible LMPs: Inclusion of Losses and Reactive Power

When formulated within a security constrained optimal power flow, the vector of locational marginal prices (LMPs) are a subset of Lagrange multipliers, and must lie in the null space of a Jacobian matrix associated with power flow and line flow limit (and possibly other) constraints in a power network. This paper builds on previous work by the authors, demonstrating the close relation of this matrix null space problem to the underlying graph of the network, and showing that structural information regarding admissible patterns of LMPs can thereby be obtained independent of any consideration of the nature of generator offers. We showed in previous work that the load pocket phenomena in LMPs can be interpreted in terms of generalized Laplacian structure that is inherent in the lossless, active power flow Jacobian. This paper seeks to explore the degree to which these types of structural insights persist when losses and reactive power are included in the LMP calculation, when the Jacobian matrix of interest deviates from generalized Laplacian structure. To this end, we propose a new computational approach to identify feasible LMPs that exploits algebraic features of the null space calculation that remains valid even when the greater complexity of loss and reactive power balance is exactly represented. A numerical example is presented to compare the structural features of feasible LMPs in the lossless, active-power-only case, versus those obtained with full treatment of reactive power balance and losses.