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.

%B 2016 American Control Conference (ACC) %I IEEE %C Boston, MA, USA %P 6501 - 6506 %8 08/2016 %R 10.1109/ACC.2016.7526693 %0 Conference Paper %B 2014 IEEE 53rd Annual Conference on Decision and Control (CDC) %D 2014 %T Variability and the Locational Marginal Value of Energy Storage %A Subhonmesh Bose %A Eilyan Bitar %K energy storage %K Locational marginal value %K reliability and markets %K RM11-006 %XGiven 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.

%B 2014 IEEE 53rd Annual Conference on Decision and Control (CDC) %I IEEE %C Los Angeles, CA, USA %P 3259 - 3265 %8 12/2014 %@ 978-1-4799-7746-8 %R 10.1109/CDC.2014.7039893