Energy Function Maps Safe Operating Limits for Grid-Forming Virtual Synchronous Generators

As wind turbines, solar farms and battery systems increasingly replace conventional power plants, the converters that connect them to the grid must do more than simply feed electricity into the network. Many are now controlled to behave like synchronous generators, mimicking the inertia, damping and voltage regulation that large rotating machines once provided almost for free. A new study published in Frontiers in Energy Research proposes a way to map exactly how far such a converter can be pushed before the grid connection becomes unsafe, using a transient energy function that treats the converter’s internal voltage as a genuine dynamic variable rather than a fixed quantity.

The research, led by Nan Feng and colleagues at the State Grid Shanghai Electric Power Research Institute, with Tao Niu of Chongqing University, addresses a long-standing blind spot in the analysis of virtual synchronous generators, or VSGs. In a voltage-source VSG, the internal voltage phase angle is set by an active power–frequency control loop, while the internal voltage magnitude is set by a reactive power–voltage control loop. Because the active and reactive power flowing into the grid both depend on the voltage magnitude and the power angle, these two control loops are inherently coupled. Yet many existing stability assessments freeze the internal voltage at a constant value and evaluate stability only at isolated operating points, which cannot describe how the secure operating region changes as power commands vary continuously with renewable output and dispatch.

To capture this coupling, the team modeled the bulk AC grid at the point of common coupling as a Thévenin equivalent voltage source behind an equivalent impedance, and built a third-order nonlinear model of the VSG with three state variables: the power angle, the frequency deviation, and the internal voltage magnitude. On top of this model they constructed a transient energy function that includes, alongside the familiar kinetic energy of frequency deviation and the potential energy of power-angle dynamics, a voltage-related potential energy term arising from the reactive power–voltage control loop. The function contains a logarithmic term in the internal voltage magnitude, so it is mathematically defined only when that voltage remains positive.

The authors show that, under an inductive-network approximation and within the voltage-source operating mode, the energy function is non-increasing along system trajectories when the virtual damping coefficient, the reactive power–voltage droop coefficient, and the voltage magnitude are all positive. This Lyapunov-like property allows the transient stability region to be defined in state space using a critical energy, taken as the minimum energy over the relevant unstable equilibrium points. A positive energy margin means a post-disturbance state remains inside the attraction region of the stable equilibrium; a zero or negative margin means it does not.

Stability alone, however, is not the whole story. The researchers combined the transient energy criterion with internal voltage limits, voltage limits at the point of common coupling, and, in the final security assessment, output current and apparent-power capacity limits. The resulting composite security and stability region in state space is then mapped onto the active–reactive power plane through the power transfer equations, yielding a voltage–power security boundary with three distinct components: an energy boundary from transient stability, a voltage boundary from constraint violations, and a power-feasibility boundary where no equilibrium point exists or the power-transfer limit is reached. An energy margin surface, denoted ΔV, extends this picture into three dimensions, with ΔV greater than zero marking secure operation, zero marking the critical boundary, and negative values marking insecure states.

In a case study of a single VSG connected to a bulk AC grid, scanned over active power from 0.4 to 1.8 per unit and reactive power reference from −0.4 to 0.8 per unit, the method revealed a distinctly non-monotonic boundary. At low active power the energy margins were large and positive everywhere. As active power rose toward 1.5 per unit, instability first appeared in the negative reactive-power region, where the converter absorbs reactive power and weakens its own voltage support: at 1.5 per unit active power and −0.4 per unit reactive power the margin was −0.08, while at 0.4 per unit reactive power it rose to 0.23. At 1.75 per unit active power, only a narrow band near 0.4 to 0.6 per unit reactive power remained close to critical stability.

Extracting the zero-margin contour showed that the critical active power first rises and then falls slightly with increasing reactive power: from 1.41 per unit at −0.4 per unit reactive power, to a peak of 1.76 per unit at 0.4 per unit, then back down to 1.74 per unit at 0.8 per unit, where the point-of-common-coupling voltage approaches its upper limit. Moderate positive reactive support therefore enlarges the secure region, but excessive reactive output contracts it again through voltage constraints. Along the active-power axis at zero reactive power, the margin fell from 1.95 at 0.4 per unit to 0.49 at 1.2 per unit and −0.03 at 1.7 per unit, reflecting how higher output demands a larger steady-state power angle and shrinks the energy distance to the nearest unstable equilibrium.

Time-domain simulations at three representative operating points—one inside the secure region, one near the boundary, and one outside—reproduced the predicted behavior. The secure point damped the disturbance cleanly; the near-boundary point recovered but with visibly weaker damping and slower recovery, consistent with its small positive margin; and the insecure point showed a growing power-angle deviation with voltage responses that violated allowable limits or failed to settle. A quantitative comparison against a conventional method that holds the internal voltage constant found differences of up to 0.14 per unit in the critical active power in the negative reactive-power region, where the constant-voltage benchmark overestimated the secure limit, while the two methods agreed exactly at zero reactive power.

The analysis also quantified how strongly grid strength shapes the boundary. Holding VSG parameters fixed and raising the equivalent grid reactance from 0.1 to 0.2 and then 0.3 per unit reduced the maximum critical active power from roughly 1.95 to 1.76 and then 1.42 per unit. Weaker grid support demands larger power angles for the same output, lowers the critical energy, and makes the point-of-common-coupling voltage more sensitive to power swings, so voltage constraints bind earlier. Varying the reactive power–voltage droop coefficient produced a similarly coupled picture: increasing it from 0.10 to 0.20 per unit raised the critical active power at zero reactive power from about 1.55 to 1.67 per unit, and further increases helped in moderate reactive ranges, but at high reactive output a coefficient of 0.30 per unit actually contracted the boundary because the voltage upper limit became active sooner.

The authors are careful to state the method’s limits. The model applies to the voltage-source operating mode before current limitation or controller saturation is triggered; once the converter enters current-limited operation it can no longer be treated as an ideal controlled voltage source, and current and capacity limits are instead applied as additional feasibility boundaries. Line resistance, switching-level dynamics, control delays, protection logic and interactions among parallel VSGs are left for future work involving detailed electromagnetic transient simulation and hardware-in-the-loop testing. Within those bounds, the framework offers grid operators something previous single-point analyses could not: a continuous map of the secure operating region in the power plane, complete with a quantified margin showing how far any given operating point sits from the edge of instability.

Subject of Research: A transient energy function method for determining the voltage–power security and stability boundary of voltage-source virtual synchronous generators connected to bulk AC grids.

Article Title: Analysis of voltage-power security and stability boundaries for virtual synchronous control based on a transient energy function

Article References: Feng, N., Guo, Q., Pan, A., Feng, Y., Zhang, Y., & Niu, T. (2026). Analysis of voltage-power security and stability boundaries for virtual synchronous control based on a transient energy function. Frontiers in Energy Research, 14, Article 1900814. https://doi.org/10.3389/fenrg.2026.1900814

Image Credits: AI Generated

DOI: 10.3389/fenrg.2026.1900814

Keywords: virtual synchronous generator, grid-forming converter, transient energy function, stability boundary, voltage-power coupling, reactive power control, power system stability, renewable energy integration, point of common coupling, grid strength, energy margin, security region