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

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.

Efficient Estimation of Pairwise Effective Resistance

“…The definition of ER 𝑟 (𝑠, 𝑡) is basically the voltage developed between nodes 𝑠 and 𝑡 when a unit current is injected at one and extracted at the other [65]. Intuitively, this physical interpretation benefits analyzing cascading failures [26,59] and power network stability [22,60,61] in electric power networks. Moreover, Spielman and Srivastava [62] theoretically proved that sampling edges in a graph 𝐺 according to their ER values yields a high-quality sparsified graph with a rigorous accuracy guarantee.…”

Observability of leader‐based discrete‐time multi‐agent systems with switching topology

“…Controllability refers to the transferring of the system in a limited time to the desired configuration from any initial configuration when a certain control action is applied to the control system, while the observability refers to whether the initial configuration of such system may be judged by the observations of the system outputs after a period of observation. Recently, the coordination control, especially including controllability and observability, of multi‐agent systems (MASs) has attracted the attention of many scholars [1–13]. For a given MAS with switching topology, such MAS network is observable if the initial states of the network only depends on the evolution topologies of the observed agent(s), which means the observability of MASs focuses on whether the whole network state can be reconstructed by only observing a small number of agents.…”