Mutually orthogonal LMP decompositions: Congestion decomposes, losses do not

A useful perspective on Locational Marginal Prices (LMPs) recognizes that, as Lagrange multipliers, they must lie in the null space of a network-structured Jacobian matrix evaluated at the optimal power flow solution. The authors have previously termed vectors in this null space satisfying necessary conditions for optimality ldquoadmissible LMPs.rdquo The work here seeks to use this geometric perspective to clarify some of the concerns that have persisted in widely used LMP formulations: the appropriateness of decomposition approaches, and impact of (uniform cost) reference bus selection. A standard approach decomposes LMPs as a sum of uniform marginal energy, marginal loss, and marginal congestion cost contributions. Employing the null-space perspective, one might seek admissible LMP basis vectors that are likewise partitioned into these three contributions. However, work here will show that such a basis cannot in general be selected to separate the three contributions into mutually orthogonal subspaces. To this end, this paper will first review recent results in the literature that examined coupling of conduction losses to the uniform energy term, using a Taylor expansion perspective. Here, as an alternative, geometric perspective, losses will be shown to rotate a single basis vector within the admissible LMPs; specifically, that vector which appears as the uniform cost basis in the losses case. This is inherently different from the impact of congestion. Simply put, losses are a perturbation to existing equality constraints, while congestion on a line introduces a new active constraint. With these insights established, this paper develops a formulation that represents uniform marginal energy and marginal losses in a single, combined term. It then associates marginal congestion cost (MCC) contributions with mutually orthogonal admissible basis vectors, with these, in turn, orthogonal to the marginal energy/loss term. Our overall result produces a unique LMP decompo- sition, which is independent of choice of reference bus.