Small-Signal Stability Condition of Inverter-Integrated Power Systems: Closed-Form Expression by Stationary Power Flow Variables
Taku Nishino, Yoshiyuki Onishi, Takayuki Ishizaki
TL;DR
The paper addresses small-signal stability of inverter-integrated power systems by deriving a necessary-and-sufficient condition that can be written as semidefinite inequalities depending only on the synchronous reactances, network susceptance $B$, and the stationary power-flow distribution. The approach uses equilibrium-independent passivity to obtain a decentralized, closed-form criterion that separates local device contributions from global network effects, relying on the stationary power-flow solution rather than full dynamics. The main contributions include a precise matrix-inequality characterization with explicit definitions of $\gamma_i(\varrho_i^*;X_i)$, $\Gamma_i(\varrho_i^*;X_i)$, and $L(\theta_e^*,V^*;B)$, along with proofs of sufficiency and necessity, and a numerical example showing grid-forming inverters improving synchronization while grid-following inverters can destabilize the system. The results offer a scalable stability test for grids with mixed inverter/generator resources and provide insight into how stationary power-flow distributions govern small-signal behavior in lossless networks.
Abstract
This paper shows that a necessary and sufficient condition for the small-signal stability of an inverter-integrated power system can be expressed in terms of semidefinite matrix inequalities determined only by the synchronous reactance of the components, the susceptance matrix of the transmission network, and the stationary values of the power flow distribution. To derive the stability condition, we consider a class of grid-forming inverters corresponding to a singular perturbation of the synchronous generator. The resulting matrix inequality condition, which has twice as many dimensions as the number of buses and is independent of the dynamics of the connected components, is expressed in terms of each component compensating in a decentralized manner for the loss of frequency synchronization caused by the reactive power consumption in the transmission network. A simple numerical example using a 3-bus power system model shows that a grid-forming inverter load improves power system synchronization, while a grid-following inverter load disrupts it.