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This paper extends the definitions of effective resistance and effectiveconductance to characterize the overall relation (positive coupling orantagonism) between any two disjoint sets of nodes in a signed graph. Itgeneralizes the traditional definitions that only apply to a pair of nodes. Themonotonicity and convexity properties are preserved by the extendeddefinitions. The extended definitions provide new insights into graph Laplaciandefiniteness and power network stability. It is proved that the Laplacianmatrix of a signed graph is positive semi-definite with only one zeroeigenvalue if and only if the effective conductances between some specificpairs of node sets are positive. Also the number of Laplacian negativeeigenvalues is upper bounded by the number of negative weighted edges. Inaddition, new conditions for the small-disturbance angle stability,hyperbolicity and type of power system equilibria are established, whichintuitively interpret angle instability as the electrical antagonism betweencertain two sets of nodes in the defined active power flow graph. Moreover, anovel optimal power flow (OPF) model with effective conductance constraints isformulated, which significantly enhances power system transient stability. Bythe properties of extended effective conductance, the proposed OPF model admitsa convex relaxation representation that achieves global optimality.

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