  Voltage instability is a major threat in power system operation. The growing
presence of constant power loads significantly aggravates this issue, hence
motivating the development of new analysis methods for both existence and
stability of voltage equilibria. Formally, this problem can be cast as the
analysis of solutions of a set of nonlinear algebraic equations of the form
$f(x)=0$, where $f:\mathbb{R}^n \mapsto \mathbb{R}^{n}$, and the associated
differential equation $\dot x=f(x)$. By invoking advanced concepts of dynamical
systems theory and effectively exploiting its monotonicity, we exhibit all
possible scenarios for existence, uniqueness and stability, of its equilibria.
We prove that, if there are equilibria, there is a distinguished one that is
locally stable and attractive, and we give some physically-interpretable
conditions such that it is unique. Moreover, a simple on-line procedure to
decide whether equilibria exist of not, and to compute the distinguished one is
proposed. In addition, we show how the proposed framework can be applied to
long-term voltage stability analysis in AC power systems, multi-terminal
high-voltage DC systems and DC microgrids.

Systems and Control