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Voltage instability is a major threat in power system operation. The growingpresence of constant power loads significantly aggravates this issue, hencemotivating the development of new analysis methods for both existence andstability of voltage equilibria. Formally, this problem can be cast as theanalysis 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 associateddifferential equation $\dot x=f(x)$. By invoking advanced concepts of dynamicalsystems theory and effectively exploiting its monotonicity, we exhibit allpossible scenarios for existence, uniqueness and stability, of its equilibria.We prove that, if there are equilibria, there is a distinguished one that islocally stable and attractive, and we give some physically-interpretableconditions such that it is unique. Moreover, a simple on-line procedure todecide whether equilibria exist of not, and to compute the distinguished one isproposed. In addition, we show how the proposed framework can be applied tolong-term voltage stability analysis in AC power systems, multi-terminalhigh-voltage DC systems and DC microgrids.

On the Existence and Long-Term Stability of Voltage Equilibria in Power Systems with Constant Power Loads