  This paper proposes a novel method rooted in differential geometry to
approximate the voltage stability boundary of power systems under high
variability of renewable generation. We extract intrinsic geometric information
of the power flow solution manifold at a given operating point. Specifically,
coefficients of the Levi-Civita connection are constructed to approximate the
geodesics of the manifold starting at an operating point along any interested
directions that represent possible fluctuations in generation and load. Then,
based on the geodesic approximation, we further predict the voltage collapse
point by solving a few univariate quadratic equations. Conventional methods
mostly rely on either expensive numerical continuation at specified directions
or numerical optimization. Instead, the proposed approach constructs the
Christoffel symbols of the second kind from the Riemannian metric tensors to
characterize the complete local geometry which is then extended to the
proximity of the stability boundary with efficient computations. As a result,
this approach is suitable to handle high-dimensional variability in operating
points due to the large-scale integration of renewable resources. Using various
case studies, we demonstrate the advantages of the proposed method and provide
additional insights and discussions on voltage stability in renewable-rich
power systems.
