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Voltage instability is one of the main causes of power system blackouts.Emerging technologies such as renewable energy integration, distributed energyresources and demand responses may introduce significant uncertainties inanalyzing of system-wide voltage stability. This paper starts with summarizingdifferent known voltage instability mechanisms, and then focuses on a class ofvoltage instability which is induced by the singular surface of the algebraicmanifold. We argue and demonstrate that this class can include both dynamic andstatic voltage instabilities. To determine the minimum distance to the point ofvoltage collapse, a new formulation is proposed on the algebraic manifold. Thisformulation is further converted into an optimal control framework foridentifying the path with minimum distance on the manifold. Comprehensivenumerical studies are conducted on some manifolds of different power systemtest cases and demonstrate that the proposed method yields candidates for thelocal shortest paths to the singular surface on the manifold for both thedynamic model and the static model. Simulations show that the proposed methodcan identify shorter paths on the manifold than the paths associated with theminimum Euclidean distances. Furthermore, the proposed method always locatesthe right path ending at the correct singular surface which is responsible forthe voltage instability; while the Euclidean distance formulation canmistakenly find solutions on the wrong singular surface. A broad range ofpotential applications using the proposed method are also discussed.

Searching for the Shortest Path to the Point of Voltage Collapse on the Algebraic Manifold