A Unified VSG–VAM Energy Function for Stability Assessment of Coupled Electric–Thermal Microgrids

This paper develops a measurable unified electric–thermal energy indicator for converter-dominated microgrids, containing virtual synchronous generator (VSG) and virtual asynchronous machine (VAM) dynamics. The principal contribution comprises the scalar energy indicator HΣ and its normalized margin ηstab; together, they combine active-power–frequency, reactive-power–voltage, thermal-flow, current-boundary, and electric–thermal coupling information obtained from point-of-common-coupling measurements and thermal states. A bounded event-triggered thermal–electric action is used only as a secondary demonstration of how the indicator can support current-constrained coordination; it is not proposed as a replacement for the inner VSG controller. Local positive-definiteness, dissipation, invariant-sublevel-set, and two-time-scale conditions are derived in an explicitly defined operating domain. The framework was evaluated in a three-phase Simulink/Simscape model using a 26-run matrix of disturbance and parameter cases, together with three severe-case baselines and four reduced-order channel-isolation cases. For voltage-sag depths of 5–20%, the event increase in HΣ/Hcrit rose from 0.137 to 0.385. In the severe 20% combined case, bounded coordination increased the minimum margin from 0.048 to 0.138, with 0.0897 kWh of heat-load reduction; however, its trajectory was identical to that of the matched current-threshold trigger, so no controller-performance superiority is claimed. Omitting Hqv, Hth, Hlim and Hcpl eliminated 100.0%, 99.7%, 90.8% and 87.3%, respectively, of the corresponding event-window sensitivity. Under deep-current saturation, the limiter gain fell to 0.466 and ηstab reached zero, numerically identifying a boundary beyond which the local stability guarantee must not be extrapolated.