A 13-State-Space Mathematical Framework for Grid-forming Inverter Stability in Low-inertia Hydro-dependent Power Systems

Abstract

Climate-induced hydrological deficits disrupt traditional electricity architectures globally, forcing hydro-dependent national power grids to accelerate structural transitions toward utility-scale solar photovoltaic generation. While this shift mitigates drought-induced energy deficits, displacing massive synchronous hydro-turbines with power-electronic-interfaced generation introduces severe technical vulnerabilities. Standard solar plants use grid-following (GFL) inverters that possess zero inherent rotating mass. Consequently, their rapid integration drastically erodes total synchronous inertia, leaving networks highly susceptible to critical high-rate-of-change frequency and voltage swings that trigger catastrophic cascading blackouts during short-circuit events. To counteract this low-inertia vulnerability, grid-forming (GFM) inverters equipped with virtual synchronous machine control loops are proposed to mimic stabilising rotational dynamics. However, conventional commercial tuning methodologies treat smart inverters as isolated units, completely neglecting empirical electrical impedance characteristics and structural limits of the surrounding transmission infrastructure. Under severe fault conditions, even advanced GFM devices risk nuisance tripping or unexpected control failure. To address this critical engineering gap without relying on restricted utility data, this paper formulates a generalised, high-fidelity 13-state-space small-signal mathematical model of a GFM-integrated power network. The analytical model seamlessly maps multi-variable dynamic interactions between internal cascaded control loops, virtual power synchronisation mechanisms, physical LC filters, and complex coupling impedances of surrounding transmission lines. Evaluated comprehensively on a standardised IEEE test system, the model undergoes rigorous small-signal eigenvalue trajectory analysis and root-locus mapping. Simulation results isolate precise boundaries where control loop interactions cross into the unstable right-half plane, revealing that standard controller settings fail explicitly under weak grid conditions characterised by high line impedance. Ultimately, this verified mathematical framework establishes a robust, open-access diagnostic blueprint for grid operators to quantify stability margins, identify hidden control interactions, and evaluate low-inertia grid vulnerabilities ahead of full-scale real-world implementation.