DC Power Grids With Constant-Power Loads—Part II: Nonnegative Power Demands, Conditions for Feasibility, and High-Voltage Solutions

Abstract

In this two-part article, we develop a unifying framework for the analysis of the feasibility of the power flow equations for dc power grids with constant-power loads. Part II of this article, explores further implications of the results in Part I. We present a necessary and sufficient linear matrix inequality (LMI) condition for the feasibility of a vector of power demands (under small perturbation), which extends a necessary condition in the literature. The alternatives of these LMI conditions are also included. In addition, we refine these LMI conditions to obtain a necessary and sufficient condition for the feasibility of nonnegative power demands, which allows for an alternative approach to determine power flow feasibility. Moreover, we prove two novel sufficient conditions, which generalize known sufficient conditions for power flow feasibility in the literature. Finally, we prove that the unique long-term voltage semistable operating point associated to a feasible vector of power demands is a strict high-voltage solution. A parametrization of such operating points, which is dual to the parametrization in Part I, is obtained, as well as a parametrization of the boundary of the set of feasible power demands.

The Steady States of Isotone Electric Systems

“…Because a ≤ b are arbitrary positive vectors we obtain that Ψ is an injective function. 9 The vector y ∈ R * + n is an element of order two for S if y ̸ = S(y) and S • S(y) = y. 10 A subset of a partially ordered set is a chain if it is totally ordered with respect to the induced order.…”

“…A second example comes from the papers [8] and [9] where a DC power grid model with constant-power loads at steady state is considered. For a given power grid with n loads and m sources the voltage potentials V, the power P, and the Kirchhoff matrix Y are partitioned according to whether nodes are loads (L) and sources (S)…”

“…The equation (1.3) is feasible for P c if it has a positive solution V L (operating point associated to P c ). In the papers [8] and [9] is presented a study of the solutions of (1.3) and are obtained necessary and sufficient conditions for feasibility.…”

The steady states of isotone electric systems

“…In papers [7] and [8] is considered a DC power grid model with constant‐power loads at steady state. For a given power grid with $$$ n $$$ loads and $$$ m $$$ sources, the voltage potentials $$$ \mathbf{V} $$$, the power $$$ \mathbf{P} $$$, and the Kirchhoff matrix $$$ Y $$$ are partitioned according to whether nodes are loads ( $$$ L $$$) and sources ( $$$ S $$$); more precisely, we have $$$$ \mathbf{V}=\left(\begin{array}{c}{\mathbf{V}}_L\\ {}{\mathbf{V}}_S\end{array}\right)\in {\mathrm{\mathbb{R}}}^{n+m},\mathbf{P}=\left(\begin{array}{c}{\mathbf{P}}_L\\ {}{\mathbf{P}}_S\end{array}\right)\in {\mathrm{\mathbb{R}}}^{n+m},\kern0.3em \mathrm{and}\kern0.40em Y=\left(\begin{array}{ll}{Y}_{LL}& {Y}_{LS}\\ {}{Y}_{SL}& {Y}_{SS}\end{array}\right)\in {\mathcal{M}}_{n+m}\left(\mathrm{\mathbb{R}}\right).$$…”

“…Equation () is feasible for $$$ {\mathbf{P}}_c $$$ if it has a positive solution $$$ {\mathbf{V}}_L $$$ (operating point associated to $$$ {\mathbf{P}}_c $$$). In the papers [7] and [8] is presented a study of the solutions of () and are obtained necessary and sufficient conditions for feasibility.…”