Alexey S. Matveev, Juan E. Machado, Romeo Ortega et al.
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.