Approximating Voltage Stability Boundary Under High Variability of Renewables Using Differential Geometry

TL;DR

The paper tackles voltage stability boundary estimation in renewable-rich grids where operating points vary in high dimensionality. It introduces a differential-geometric framework that uses the power-flow manifold and constructs the Levi-Civita connection to generate geodesics from a given operating point; the boundary is then approximated via a second-order geodesic Taylor expansion, yielding a univariate quadratic for each bus. A conservativeness-enhancing scaling factor $\alpha^k$ is proposed and CPF is used to calibrate it, achieving about a $1000\times$ speedup over continuation power flow on IEEE 14-Bus and 39-Bus systems. The approach provides a global geometric view from local quantities, enabling fast online voltage stability analysis in grids with large renewable variability.

Abstract

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.

Figures (9)

• Figure 1: Illustration of the Levi-Civita Connection.
• Figure 2: IEEE 14-bus voltage stability boundary at different buses.
• Figure 3: IEEE 14-bus voltage stability boundary in different conditions.
• Figure 4: IEEE 14-bus voltage stability boundary in 3D.
• Figure 5: IEEE 39-bus voltage stability boundary at different buses.