# On Extension of Effective Resistance with Application to Graph Laplacian Definiteness and Power Network Stability

## TL;DR

This paper extends effective resistance concepts to signed graphs, providing new insights into graph Laplacian properties and power network stability, and introduces an OPF model with enhanced stability guarantees.

## Contribution

It generalizes effective resistance to signed graphs, linking it to Laplacian definiteness and power system stability, and proposes a convex OPF model with stability improvements.

## Key findings

- Laplacian positive semi-definiteness linked to positive effective conductance between node sets.
- Number of negative Laplacian eigenvalues bounded by negative edges.
- Convex relaxation of the OPF model achieves global optimality.

## Abstract

This paper extends the definitions of effective resistance and effective conductance to characterize the overall relation (positive coupling or antagonism) between any two disjoint sets of nodes in a signed graph. It generalizes the traditional definitions that only apply to a pair of nodes. The monotonicity and convexity properties are preserved by the extended definitions. The extended definitions provide new insights into graph Laplacian definiteness and power network stability. It is proved that the Laplacian matrix of a signed graph is positive semi-definite with only one zero eigenvalue if and only if the effective conductances between some specific pairs of node sets are positive. Also the number of Laplacian negative eigenvalues is upper bounded by the number of negative weighted edges. In addition, new conditions for the small-disturbance angle stability, hyperbolicity and type of power system equilibria are established, which intuitively interpret angle instability as the electrical antagonism between certain two sets of nodes in the defined active power flow graph. Moreover, a novel optimal power flow (OPF) model with effective conductance constraints is formulated, which significantly enhances power system transient stability. By the properties of extended effective conductance, the proposed OPF model admits a convex relaxation representation that achieves global optimality.