DC Power Grids With Constant-Power Loads—Part I: A Full Characterization of Power Flow Feasibility, Long-Term Voltage Stability, and Their Correspondence

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. In Part I of this article, we present a detailed introduction to the problem of power flow feasibility of such power grids, and the associated problem of selecting a desirable operating point which satisfies the power flow equations. We introduce and identify all long-term voltage semistable operating points, and show that there exists a one-to-one correspondence between such operating points and the constant power demands for which the power flow equations are feasible. Such operating points can be found by solving an initial value problem, and a parametrization of these operating points is obtained. In addition, we give a full characterization of the set of all feasible power demands, and give a novel proof for the convexity of this set.

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.…”

“…The matrix $$$ {Y}_{LL} $$$ is assumed to be irreducible (Assumption 2.7, [7]). For $$$ {\mathbf{P}}_c\in {\left({\mathrm{\mathbb{R}}}^{\ast}\right)}^n $$$, using Lemma A.1, we observe that $$$ M:= {Y}_{LL}^{-1}\operatorname{diag}\left({\mathbf{P}}_c\right) $$$ is an irreducible matrix.…”

The Steady States of Isotone Electric Systems

“…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.…”