We derive a new upper bound on the minimum rank of matrices belonging to an affine slice of the positive semidefinite cone, when the affine slice is defined according to a system of sparse linear matrix equations. It is shown that a feasible matrix whose rank is no greater than said bound can be computed in polynomial time. The bound depends on both the number of linear matrix equations and their underlying sparsity pattern. For certain problem families, this bound is shown to improve upon well known bounds in the literature. Several examples are provided to illustrate the efficacy of this bound.

10aRM14-0021 aLouca, Raphael1 aBose, Subhonmesh1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/bound-minimum-rank-solutions-sparse01755nas a2200193 4500008003900000020002200039245006800061210006800129260004000197300001600237520110400253653001901357653003001376653002801406653001301434100002101447700001801468856007501486 2014 d a978-1-4799-7746-800aVariability and the Locational Marginal Value of Energy Storage0 aVariability and the Locational Marginal Value of Energy Storage aLos Angeles, CA, USAbIEEEc12/2014 a3259 - 32653 aGiven a stochastic net demand process evolving over a transmission-constrained power network, we consider the system operator's problem of minimizing the expected cost of generator dispatch, when it has access to spatially distributed energy storage resources. We show that the expected benefit of storage derived under the optimal dispatch policy is concave and non-decreasing in the vector of energy storage capacities. Thus, the greatest marginal value of storage is derived at small installed capacities. For such capacities, we provide an upper bound on the locational (nodal) marginal value of storage in terms of the variation of the shadow prices of electricity at each node. In addition, we prove that this upper bound is tight, when the cost of generation is spatially uniform and the network topology is acyclic. These formulae not only shed light on the correct measure of statistical variation in quantifying the value of storage, but also provide computationally tractable tools to empirically calculate the locational marginal value of storage from net demand time series data.

10aenergy storage10aLocational marginal value10areliability and markets10aRM11-0061 aBose, Subhonmesh1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/variability-and-locational-marginal